Boundary Layers for 2D Stationary Navier-Stokes Flows over a Moving Boundary by Sameer S. Iyer Sc.B, Brown University; Providence, RI, 2012 A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Division of Applied Mathematics at Brown University PROVIDENCE, RHODE ISLAND May 2018 c Copyright 2018 by Sameer S. Iyer This dissertation by Sameer S. Iyer is accepted in its present form by The Division of Applied Mathematics as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date Yan Guo, Ph.D., Advisor Recommended to the Graduate Council Date Govind Menon, Ph.D., Reader Date Benoit Pausader, Ph.D., Reader Approved by the Graduate Council Date Andrew G. Campbell, Dean of the Graduate School iii Vita The author received his Sc. B. in Applied Mathematics from Brown University in May of 2012. In the Fall of 2013, he enrolled as a Ph.D. student in the Division of Applied Mathematics at Brown University. While at Brown, he has worked under the supervision of Professor Yan Guo. His preprints and publications include (Iye17a), (Iye16), (Iye17b), (IZ17), (IS16). iv Preface and Acknowledgments First and foremost, I would like to express my deepest gratitude to my advisor, Yan Guo. During the last four years, Yan has spent countless hours teaching me mathematics during which the subject of partial di↵erential equations came alive and became ripe for investiga- tion. Through his patience, dedication, generosity, and unbounded energy, Yan has taught me to strive for the highest ideals as a scientist and has instilled in me the notions of rigorous and relentless scientific pursuit. I am honored to be his student. I would like to thank my thesis readers Govind Menon and Benoit Pausader. Govind taught my first ODE course (and several more along the way) and has been supportive ever since. Benoit has played a crucial role introducing me to exciting PDE topics which I foresee being a large part of my future career, and has continuously encouraged my pursuit of these areas. I am grateful to both for sharing, on many occasions, their perspectives on our discipline which has played a big role in shaping my own points of view. I would like to thank Bjorn Sandstede for bringing me to the realm of applied analysis when I was an undergraduate, from whom I have learned many things, and with whom I have had a successful and exciting collaboration. I would also like to thank the several other faculty at Brown from whom I have learned partial di↵erential equations: Constantine Dafermos, Hongjie Dong, Justin Holmer, and Walter Strauss. Thanks to Basilis Gidas for always being available to share his unique perspectives on mathematics and for being supportive from the start. Thanks also to the many other teachers from my Brown undergraduate days who initially opened my eyes to v higher mathematics, in particular Alex Kontorovich and Nimish Shah. Thanks to my officemates Ian Alevy, Michael Burkhart, and Michael Snarski for valuable friendship during my time as a graduate student. Thanks also to the following people with whom I became friends while they were postdocs at Brown, and who have helped me along the way: Francesco di Plinio, Klaus Widmayer, and Sona Akopian. Thanks to the many people in the Division of Applied Mathematics at Brown who made the last five years a great time for me to develop as a scientist. Finally, on a personal note, I would like to thank my Mom, Dad, and Briana for all the love and support. My dad was my very first teacher in mathematics who instilled in me a sense of curiosity, wonder, and marvel that has set the tone for my scientific pursuits. My mom has been a steadfast source of love and support from the beginning; a calm and reliable force in ever-changing circumstances and always cheering me on. Briana’s love, companionship, and encouragement has been extraordinary, helping me always take fresh perspectives, to believe in myself, to pursue my passions, and to maintain strong positive energy throughout. Nothing would be possible without all of you! vi Abstract of “Boundary Layers for 2D Stationary Navier-Stokes Flows over a Moving Bound- ary”, by Sameer S. Iyer, Ph.D., Brown University, May 2018 In this thesis, we study Prandtl’s boundary layer theory for 2D, stationary, incompressible Navier-Stokes flows posed on domains with boundaries. The boundary layer hypothesis posed by Prandtl in 1904 asserts that for flows with low viscosities, the e↵ect of friction is prominent near physical boundaries and negligible in the bulk. Verifying this mathematically is, in general, an important open problem. We establish the validity of Prandtl’s hypothesis in three settings, all under the hy- pothesis of a translating boundary. The first setup we consider are flows on the exterior of a rotating disk. Here, the e↵ect of the curved boundary poses the main obstacle to the analysis. Second, we study flows for which the tangential variable extends globally. Here, obtaining strong, integrable decay estimates is the main task. Third, we study boundary 1/2 layers in the presence of a non-shear Euler flow. Here, a singularity of size " is created at leading order, which we control by establishing estimates weighted from the far-field. A fourth contribution of this thesis is the construction of a one-parameter family of stationary, 2D Navier-Stokes solutions which are arbitrarily close to the Couette flow. Contents Vita iv Preface and Acknowledgments v 1 Introduction 1 1.1 Boundary Layers and the Inviscid Limit: . . . . . . . . . . . . . . . . . . . . . 2 1.2 Previous Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 The Asymptotic Expansion: . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.2 The Remainder System . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Thesis Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.1 Steady Prandtl Boundary Layer Expansions over a Rotating Disk . . 14 1.3.2 Global Steady Prandtl Expansion Over a Moving Boundary . . . . . . 18 1.3.3 Steady Prandtl Expansions over a Moving Boundary: Nonshear Euler Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3.4 Stationary Inviscid Limit to Shear Flows . . . . . . . . . . . . . . . . . 25 2 Steady Prandtl Boundary Layer Expansions over a Rotating Disk 27 2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2.1 Boundary Layer Expansion . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2.2 Boundary Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.3 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3 Function Spaces and Embedding Theorems . . . . . . . . . . . . . . . . . . . 47 2.3.1 Basic Properties of ⇤-spaces . . . . . . . . . . . . . . . . . . . . . . . . 47 2.3.2 Properties of Z, I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.3.3 Properties of Z, II: Low Regularity Embeddings . . . . . . . . . . . . 51 2.3.4 Properties of Z, III: High Regularity Embeddings . . . . . . . . . . . . 56 2.4 Construction of Approximate Solutions . . . . . . . . . . . . . . . . . . . . . . 60 2.4.1 Prandtl-0 Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.4.2 Euler-1 Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.4.3 Prandtl-1 Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.4.4 Estimates on Profile Errors . . . . . . . . . . . . . . . . . . . . . . . . 97 2.5 Energy Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 2.6 Positivity Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 2.7 Pressure Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 vii 2.8 Linearized Existence and Uniqueness for Navier-Stokes Remainders . . . . . . 130 2.9 High Regularity Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 2.10 Nonlinear Existence and Uniqueness for Navier-Stokes Remainders . . . . . . 143 3 Global-in-x Prandtl Layers over a Moving Boundary 148 3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 3.3 Overview of Profile Constructions . . . . . . . . . . . . . . . . . . . . . . . . . 166 3.4 Asymptotics of Prandtl Layer, u0p : . . . . . . . . . . . . . . . . . . . . . . . . 169 3.4.1 Existence of Front Profile . . . . . . . . . . . . . . . . . . . . . . . . . 170 3.4.2 Zeroeth Prandtl Layer, u0p . . . . . . . . . . . . . . . . . . . . . . . . . 176 3.5 Euler-1 Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 3.5.1 Derivation of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 188 3.5.2 Uniform Decay Estimates . . . . . . . . . . . . . . . . . . . . . . . . . 190 3.6 Prandtl Layer 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 3.6.1 Derivation of Linearized Prandtl Equations: . . . . . . . . . . . . . . . 201 3.6.2 Global in x Existence and Decay: . . . . . . . . . . . . . . . . . . . . . 207 3.7 Intermediate Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 3.7.1 Construction of Euler Layer, [uie , vei ] . . . . . . . . . . . . . . . . . . . 220 3.7.2 Construction of Prandtl Layer, [uip , vpi ] . . . . . . . . . . . . . . . . . . 224 3.7.3 Final Prandtl Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 3.8 Overview of a-priori Z-Norm Estimates . . . . . . . . . . . . . . . . . . . . . 257 3.9 The Function Space Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 3.9.1 Elliptic Estimates and the Spaces Yi . . . . . . . . . . . . . . . . . . . 264 3.9.2 Embedding Theorems for the Space Z . . . . . . . . . . . . . . . . . . 276 3.9.3 Function Space, Z(⌦N ) . . . . . . . . . . . . . . . . . . . . . . . . . . 284 3.10 Navier-Stokes Remainders: Energy Estimates . . . . . . . . . . . . . . . . . . 286 3.10.1 Energy Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 3.10.2 Positivity Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 3.10.3 Second Order Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 3.10.4 Third Order Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 3.11 Nonlinear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 3.11.1 a-priori Estimate of Nonlinearities . . . . . . . . . . . . . . . . . . . . 343 3.12 Overview of Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . 359 3.13 Step 1: Invertibility of Weighted Stokes Operator, S↵ . . . . . . . . . . . . . 362 3.14 Step 2: Compact Perturbations, S↵ + T [ ] . . . . . . . . . . . . . . . . . . . 369 3.15 Step 3: Nonlinear Existence of Auxiliary Systems . . . . . . . . . . . . . . . . 393 3.16 Step 4: Nonlinear Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 3.17 Step 5: Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 4 Steady Prandtl Layers over a Moving Boundary: Nonshear Flows 426 4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 4.3 Energy Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 4.4 Positivity Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 4.5 Weighted Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 4.5.1 The Korn’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 4.5.2 Summary of L2 Estimates: . . . . . . . . . . . . . . . . . . . . . . . . 464 viii 4.6 Uniform Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 4.7 Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 4.8 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 4.9 Construction of Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 4.9.1 Specification of Ru . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 4.9.2 Specification of Rv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 4.9.3 Construction of Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 4.9.4 Remainder System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 4.10 Existence and Uniqueness of Remainder . . . . . . . . . . . . . . . . . . . . . 497 5 Stationary Inviscid Limit to Shear Flows 507 5.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 5.3 Linear Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 5.3.1 Energy Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 5.3.2 Positivity Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 5.4 Evaluation of Right-Hand Sides . . . . . . . . . . . . . . . . . . . . . . . . . . 519 5.5 Construction of Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 5.5.1 Formal Asymptotic Expansion . . . . . . . . . . . . . . . . . . . . . . 523 5.5.2 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 5.5.3 Boundary Layer Equations . . . . . . . . . . . . . . . . . . . . . . . . 529 ix List of Figures 1.1 Boundary Layer Separation (Reprinted from (OS99), page 9). . . . . . . . . . 8 1.2 Flow over a Moving Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Flow past a rotating disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 x Chapter One Introduction 2 1.1 Boundary Layers and the Inviscid Limit: In this thesis, I will study 2D, incompressible, stationary flows at low viscosities for which the Navier-Stokes equations read: 9 " " " "> > u · ru + rP = " u > > > = " (1.1.1) r·u =0 > > > > > u|@⌦ = 0. ; Here, u" = (u" (x, Y ), v " (x, Y )) : R2 ! R2 is a vector field defined on an open set ⌦ ⇢ R2 . Describing the asymptotic behavior of (1.1.1) as the viscosity vanishes (" # 0) in the presence of boundaries is a fundamental challenge in partial di↵erential equations. This is due to the disparity between the no-slip boundary condition, u" |@⌦ = 0, for viscous flows and the no-penetration condition, u0 · n|@⌦ = 0 for inviscid flows. Prandtl’s boundary layer hypothesis, (Pra04), rectifies this boundary mismatch, and can be expressed mathematically as: Y p u" (x, Y ) = u0e (x, Y ) + u0p (x, p ) + "u for 0 < " << 1, (1.1.2) " where u0e = (u0e , ve0 ) is a prescribed Euler flow, u0p = (u0p , vp0 ) is a constructed boundary layer, and u = (u, v) is the remainder. For the purposes of being concrete, we will think Y of Y = 0 as the no-slip boundary, and thus the scaling y = p " zooms in to a strip near the boundary. The remainder can be thought of as a function of either Y or the scaled Y variable p " . In this thesis, we will work in boundary layer coordinates, meaning we think of u = u(x, y). It is worthwhile to provide historical context regarding (1.1.2) (see (SG00) for more de- tails). (1.1.2) was introduced by Prandtl in 1904, (Pra04). At the time, the Euler equations, 3 which govern u0e , were well known. Over the 1800’s, the Eulerian theory was vigorously in- vestigated by the likes of Kelvin, Helmholz, Rayleigh, Riemann, and many other influential mathematicians. Moreover, the Euler equations saw great agreement with certain physical experiments such as free surface ocean waves. However, the Euler theory saw great disparity with physical experiments which involved solid boundaries (such as pipe-flow). Because of this, experimentalists largely discarded the Euler theory. The Navier-Stokes equations were also well known by the turn of the 1900’s. However, they were also not being used in practice for two chief reasons. First, they were more complicated than the Euler equations from the point of view of special solutions. Second, the fluids that mattered the most were air and water, which have very low viscosity. Due to this, the general perception at the time was that the Navier-Stokes equations could not be very di↵erent from the Euler equations. It was this that Prandtl disagreed with, and how he distinguished himself. He realized that, even for low viscosities, the Navier-Stokes and the Euler equations were very di↵erent in the presence of boundaries. He moreover characterized precisely how the di↵erence should behave using the object u0p in (1.1.2). In so doing, he linked these equations, and also found great agreement to experiments. For this, he is credited as being the father of modern day aerodynamics. An expansion of the form (1.1.2) is very natural from the point of view of singularly perturbed systems. For instance, one could consider the ODE "@Y '(") + '(") = 1 on the domain Y > 0, with boundary condition '(") (0) = u0 6= 1. This has explicit solution: Y '(") = 1 |{z} + (u0 1)e " . | {z } Inviscid Profile Boundary Layer From this basic ODE example, we may gather several heuristics that will carry over the more complicated situation of Navier-Stokes and Euler. First, the boundary layer decays 4 rapidly (exponentially) away from the boundary, Y = 0. Second, it decays in a new, scaled Y Y variable which in this case is " , whereas in the case of Navier-Stokes is y = p " (as shown in (1.1.2)). A major implication of this scaling is that topology matters when speaking of 1 inviscid limits. Indeed, the boundary layer profile, uBL from (1.1.2) is size O(" 2p ) in Lp , and so vanishes in the limit as " ! 0 for all p < 1. For this reason, we view the identity (1.1.2) as most important (and most challenging) in L1 . One of Prandtl’s major contributions in his seminal work, (Pra04), was to obtain the Y proper scaling of the boundary layer variable y = p " . We briefly outline his argument. Suppose one has, to leading order, the expansion: 2 3 2 3 2 3 " 0 Y u 1 6 7 6 7 6 p u (x, ) ! 7 4 5=4 5+4 5 for some ! > 0. " 0 Y v 0 vp (x, ! ) As the profiles u0p , vp0 are expected to decay rapidly in their variable, this expansion implies the thickness of the boundary layer is size !. Thus, at Y = 0, one has the no- slip boundary condition (u" , v " )|Y =0 = (0, 0), whereas at Y = ! one has the “matching condition” of (u" , v " )|Y =! = (1, 0). This implies that the relative orders of the gradients should be: @Y u" ⇠ o( !1 ), whereas @x u" ⇠ @Y v " ⇠ o(1). This then indicates that one should scale out to the following normalized variables and unknowns: Y v " (x, Y ) y= , U " (x, y) = u" (x, Y ), V " (x, y) = , P " (x, y) = p" (x, Y ). ! ! Inserting the new scaled variables into the Navier-Stokes equations gives: " " U " Ux" + V " Uy" + Px" = "Uxx " + 2 Uyy ! ⇣ ⌘ P" " " y ! U " Vx" + V " Vy" + " = "!Vxx + Vyy ! ! Ux" + Vy" = 0. 5 The final ingrediant is Prandtl’s physical insight that within the boundary layer, the convective terms are of the same order as the viscosity in the normal, Y , direction. This is p achieved if ! = ". Subsequently retaining the leading order terms in " yields the Prandtl equations: u ¯u¯x + v¯u ¯y u ¯yy = @x PE on R+ ⇥ R+ u ¯x + v¯y = 0, (1.1.3) [¯ u, v¯]|y=0 = [0, 0], ¯|y!1 ! u0e (0), u ¯|x=0 = u u ¯0 (y). u, v¯] = u0e (0) + u0p are expected to describe the leading order of the The unknowns [¯ Navier-Stokes velocity field. A key observation to start with is that Equation (1.1.3) is a parabolic equation, with x occupying the time-like variable and y occupying the space-like variable. The boundary at {y = 0} corresponds to a physical boundary past which the flow is occurring. On the other hand, the data prescribed at the {x = 0} boundary corresponds to an in-flow profile. Using [¯ u, v¯] in place of the Navier-Stokes velocity field has revolutionized many applied sciences, in particular aerodynamics. System (1.1.3) represents a significant simplification from (1.1.1) as parabolic initial-boundary value problems are in general computationally simpler than elliptic boundary value problems. Moreover, explicit solutions to (1.1.3) can be written down. This makes the computation of important physical quantities, such as drag over an airfoil, feasible and efficient. However, from the point-of-view of PDE, there are at least two important questions: Q1. Establish well-posedness or ill-posedness of the Cauchy problem (1.1.3). Q2. Establish the validity of expansion (1.1.2). These two questions are obviously related. On the one hand, if one does not know the answer to Q2 in the affirmative, then a study of Q1 cannot indicate any useful information 6 regarding the Navier-Stokes velocity field. On the other hand, one would, at the bare minimum, require well-posedness of system (1.1.3) and suitably strong estimates over [¯ u, v¯] if one hopes to answer Q2 in the affirmative. Oleinik’s seminal work, (OS99), lists the major open problems in the boundary layer theory from a mathematical perspective, of which the first such problem is Q2: Is it possible to give a strict mathematical justification of [1.1.2] and find the limits of applicability of Prandtl’s hypothesis. 1.2 Previous Works Let us first mention that Prandtl’s original work, (Pra04), was developed in the context of 2D, stationary flows. Oleinik’s open problem regarding Q2 also concerns the 2D, stationary setting. Of course, one could also consider unsteady flows, for which many works exist regarding both Q1 and Q2. A discussion of these works would lead us astray, so we point the reader to (E00) for a comprehensive set of references regarding the unsteady theory. We will exclusively restrict our point of view to 2D stationary flows. Let us now discuss work regarding Q1, system (1.1.3). Oleinik proved the following results in (OS99): Theorem 1.2.1 (Oleinik, (OS99), Local Existence). Suppose u ¯0 (y) > 0 for y > 0, u ¯0 (0) = ¯00 (0) > 0, u 0, u ¯0 is sufficiently regular, and appropriate compatibility conditions are satisfied u, v¯] to (1.1.3) for x 2 (0, L) for some L > 0. at x = 0, y = 0. Then there exists a solution [¯ ¯|y=0 > 0, u On this interval, the following are satisfied: @y u ¯ > 0, and u, uy , uyy , ux = vy are all continuous and bounded. Theorem 1.2.2 (Oleinik, (OS99), Global Existence). Suppose, in addition to the hypothe- ses in Theorem 1.2.1, that @x PE  0. Then L can be taken to be arbitrary. 7 The methods used to establish Theorems 1.2.1 - 1.2.2 rely on a nonlinear change of coordinates to (x, (x, y)) where is the stream function associated to [¯ u, v¯]. Such a change of coordinates is clearly possible due to the positivity of u ¯0 for y > 0. Under this change, p letting w = u¯2 , the following degenerate parabolic equation is derived: wx = ww 2@x PE . This procedure is known as the “von-Mises” transformation. Subsequently, Oleinik establishes maximum principles that apply to this transformed equation. In the favorable pressure gradient case of @x PE  0, it is natural to investigate the asymptotic in x (or “downstream”) behavior of [¯ u, v¯]. This was investigated by Serrin in (Ser67). First, one seeks a self-similar solution, f 0 ( pyx ), to the Prandtl equations, (1.1.3). Such a solution is known as the Falkner-Skan solution. These similarity solutions are unique once the outer streaming velocity U (x) = limy!1 u ¯ is prescribed. In (1.1.3) we are taking U (x) = 1 for simplicity. Serrin then establishes convergence to these self-similar solutions (we state below a simplified version of his result): Theorem 1.2.3 (Serrin, (Ser67)). Let @x PE  0. Let f 0 denote the unique Falkner-Skan self-similar solution specific to the boundary data provided in (1.1.3). Then the following asymptotics are valid |¯ u f 0 | = o(1) as x ! 1. Again, central to Serrin’s approach is the von-Mises transformation and maximum prin- ciple arguments applied to the resulting quasilinear, degenerate, parabolic equation. From the point of view of phenomena, the favorable pressure gradient condition in The- orems 1.2.2 and 1.2.3 prevents “boundary layer separation”. Separation corresponds to a detachment of the boundary layer from the physical boundary. Figure 1.1 helps visualize such a process. After the point x0 in Figure 1.1, the velocity field points in the negative x-direction ¯|y=0 (x0 ) = 0. This subsequently creates a pinching o↵ of the boundary layer. The and @y u natural notion of singularity for [¯ ¯|y=0 # 0. While Theorem 1.2.2 shows u, v¯] is thus when @y u 8 Figure 1.1: Boundary Layer Separation (Reprinted from (OS99), page 9). that this cannot occur if @x PE  0, the following, very recent, result establishes the converse result: Theorem 1.2.4 (Dalibard-Masmoudi, (DM18)). Let @x PE = 1. Then there exists an ¯0 (y) satisfying the hypothesis of Theorem 1.2.1, and 0 < x0 < 1 such that the initial data u solution [¯ ¯|y=0 (x) > 0 on 0 < x < x0 , and @y u u, v¯] exists on (0, x0 ), satisfies @y u ¯|y=0 (x) # 0 as x ! x0 . Although Theorems 1.2.1 - 1.2.4 paint a relatively well-developed picture of Q1, there have been very few approaches to answering Q2 in the stationary, 2D setting. From a PDE standpoint, the fundamental challenge to prove expansion (1.1.2) is to control the remainder, u. Due to the multiple scales (@y of uBL creates a singularity of O( p1" )), there are no natural coercive quantities that yield estimates on u uniformly in ". For 2D, stationary flows, the work (GN17) overcomes this difficulty, in the presence of a moving boundary. In (GN17), the fluid domain, ⌦, is (0, L)⇥R+ . The crucial no-slip condition is prescribed for the Navier-Stokes vector field at {y = 0}. In addition, this boundary is assumed to be translating to the right at velocity ub = 1 > 0. Physically, one may imagine a fluid flowing over a plate, which in turn is translating. In contrast, the {x = 0} and {x = L} are in-flow and out-flow boundaries respectively. To visualize the parameter and the translating boundary, one should keep in mind Figure 1.2. The flow field of Figure 1.1 has 9 Figure 1.2: Flow over a Moving Boundary thus been modified to include a transition from (1 ) to 1 as opposed to from 0 to 1. This creates a boundary layer of size 2 (0, 1). On this domain, Guo-Nguyen prove: Theorem 1.2.5 (Guo-Nguyen, (GN17)). Let the parameter in the expansion (1.2.1) 2 (0, 14 ). Assume the boundary and initial data for all the profiles are given C 1 functions that decay exponentially in their respective arguments at infinity. Without loss of generality assume ue (0) = 1. Assume the boundary velocity 1 > 0. Then (1.2.1) holds on the domain [0, L] ⇥ R+ , for 0 < L << 1 sufficiently small relative to universal constants. Moreover, the remainder, u, obeys the following estimates: p p kr" {u, v}k2 + " 2 k{u, "v}k1 . 1, r" := ( "@x , @y ). Corollary 1.2.6 (Inviscid Convergence in L1 ). Y p ku" (x, Y ) u0e (Y ) u0p (x, p )k1 . ", " 10 p Y p 1 kv " (x, Y ) "vp0 (x, p ) "ve1 (x, Y )k1 . " 2 + 2 " The methods introduced in (GN17) are distinguished from those introduced in Theorems 1.2.1 - 1.2.2. Indeed, the central object of study in (GN17) is the remainder solution in 1.1.2, u ¯, which solves a linearized Navier-Stokes equation (specifically, see 1.2.6). This system is far more complicated than the Prandtl equation, and is not amenable to transformations such as the von-Mises transformation, nor to maximum principle methods. (GN17) thus introduce viable energy based methods to analyze u ¯. However, at the level of energy there is a competition between the viscosity term @yy u and the convection terms. Thus, delicate analysis is required to properly isolate coercive operators and prove their corresponding boundedness properties. This is manifested through the two-tiered structure that will be discussed in (1.2.13) - (1.2.14). 1.2.1 The Asymptotic Expansion: (GN17) considers the following Ansatz (letting 2 (0, 14 ) be a free parameter): p h 1 i 1 U " = u0e (Y ) + u0p (x, y) + " ue (x, Y ) + u1p (x, y) + " 2 + u(x, y) p 1 V " = vp0 (x, y) + ve1 (x, Y ) + "vp1 (x, y) + " 2 + v(x, y) (1.2.1) p p 1 P" = "Pe1 (x, Y ) + "Pp1 (x, y) + "Pp2 (x, y) + " 2 + P (x, y) Let us discuss each of the terms in the above expansion, as many of the themes will carry over to the expansions we validate in this thesis. The profile [u0e (Y ), 0] is a given solution to the Euler equations, which one should think of heuristically as [1, 0]. Subsequently following the Euler profiles is an alternating sequence of boundary layer profiles, indexed by [uip , vpi ] (“p” for Prandtl) and Eulerian profiles, indexed by uie , (“e” for Euler). One may read-o↵ the boundary conditions at {y = 0} from the above expansions by enforcing the no-slip 11 condition at each order. Thus, we take: [u0e + u0p ]|y=0 = 1 , [u1e + u1p ]|y=0 u|y=0 = 0, [vp0 + ve1 ]|y=0 = 0, vp1 |y=0 = 0, v|y=0 = 0. [u0p , vp0 ] are Prandtl boundary layer profiles, which solve the nonlinear Prandtl equations: (u0e (0) + u0p )@x u0p + (vp0 vp0 (0))@y u0p @yy u0p = 0, @x u0p + @y vp0 = 0, (1.2.2) u0p |x=0 = u0,p (y), u0p |y=0 = ub u0e (0), @yk u0p |y!1 = 0 for all k 0. The profiles [u1p , vp1 ] also satisfy a parabolic equation, the linearized Prandtl equation: (u0e (0) + u0p )@x u1p + u0px u1p + (vp0 + ve1 )@y u1p + (vp1 vp1 (0))@y u0p @yy u1p + Ppx 1 = f1 , (1.2.3) u1px + vpy 1 = 0. (1.2.4) In contrast, the profiles [u1e , ve1 ] satisfy an elliptic boundary value problem in the variables (x, Y ), which upon going to the vorticity formulation reads: u0e ve1 + u0eY Y ve1 = 0, @x u1e + @Y ve1 = 0 (1.2.5) ve1 |x=0 = V0 (Y ), ve1 |x=L = VL (Y ), ve1 |Y =0 = vp0 |y=0 . 12 1.2.2 The Remainder System Let us briefly discuss the central ideas introduced in (GN17). The focus of our discussion will be entirely on the remainder solution [u, P ], which satisfies the following system: "u + @x P + Su (u, v) = N1 (u, v) + f, @y "v + P + Sv (u, v) = N2 (u, v) + g, (1.2.6) " ux + vy = 0, together with boundary conditions: [u, v]|x=0 = 0, [u, v]|y=0 = 0, [u, v]|y!1 = 0, (1.2.7) [uy + "vx ]|x=L = 0, [P 2"ux ]|x=L = 0. The boundary condition at {x = L} is known as the “Stress-Free” boundary condition, which takes the place of Neumann boundary condition for fluid systems and corresponds to the vanishing of the Cauchy stress tensor. Let us now discuss the various quantities appearing in (1.2.6). The terms on the right-hand side, (f, g), are forcing terms that arise from previously constructed layers. The terms N1 , N2 are quadratic interactions of u with itself: N1 (u, v) := uux + vuy , N2 (u, v) := uvx + vvy . (1.2.8) The terms Su (u, v) and Sv (u, v) contain linearizations around previously constructed layers: Su (u, v) := us ux + usx u + vs uy + usy v, (1.2.9) Sv (u, v) := us vx + vsx u + vs vy + vsy v, (1.2.10) 13 p p us := u0e + u0p + "[u1e + u1p ], vs := vp0 + ve1 + "vp1 . (1.2.11) Let us also define the Rayleigh operator, which can be thought of as the “main” com- ponents of (Su , Sv ): 0 1 B us vy + usy v C R(u, v) := @ A (1.2.12) us vx usx v We will close by stating the basic structure of the linear estimates in (GN17): Proposition 1.2.7 (Linear Estimates in (GN17)). kuy k22 . O(L)kr" vk22 + Forcing and Nonlinear Terms (1.2.13) kr" vk22 . kuy k22 + Forcing and Nonlinear Terms. (1.2.14) Estimate (1.2.13) is obtained via an energy estimate, whereas estimate (1.2.14) is ob- tained by identifying a cancellation enjoyed by the Rayleigh operator, (1.2.12). It is clear that by bringing L small, the sequence (1.2.13) and (1.2.14) can be combined to achieve linear control over kr" {u, v}k2 . Let us briefly highlight the main mechanism behind the estimate (1.2.14). A natural way to generate positivity over kr" vk2 is to take the L2 inner product of R(u, v) against r? " v. Doing this produces: Z Z usyy 2 us vy2 + v ◆& S ◆ S vy2 . 2 v R Thus, Guo-Nguyen design a modified multiplier us producing instead the quantity vy2 + usyy 2 us v . This quantity is subsequently shown to be coercive over kvy k22 using a special cancellation identity. 14 1.3 Thesis Results My thesis furthers the Prandtl theory in the following three ways: I proved the validity of (1.1.2) upon introducing curvature e↵ects ((Iye17a)), infinite tangential length scales ((Iye16)), and non-shear e↵ects ((Iye17b)), and characterized quantitatively how these flow features influence the expansion (1.1.2), all in the setting of a moving boundary. It is convenient to state a unifying “meta-theorem”: Meta-Theorem ((Iye17a), (Iye16), (Iye17b)). For each of the three setups in (Iye17a), (Iye16), (Iye17b) (described below), the expansion (1.1.2) holds, uBL is smooth and rapidly decaying, and most importantly the remainder is controlled in a particular norm Z (that changes depending on the particular theorem, but controls L1 in all cases) ||u||Z . 1. This p therefore implies ||u" u0 uBL ||1 . ". The spirit of the above “meta-theorem” is as follows: the main challenge in the following analyses will be to control u. This task is far from generic; rather the particulars of the model (be it the geometry of the domain, spatial length scales present, or the structure of the Euler flow) dictate the norm Z, which must be determined and controlled through careful analysis. The eventual form of || · ||Z tracks refined structural properties of u (e.g. growth/ decay rates in various spatial regimes) which change dramatically depending on the model. We now turn to specifics. For the sake of presentation, we will continue to refer to the model expansion (1.1.2) as opposed to more refined expansions such as (1.2.1). 1.3.1 Steady Prandtl Boundary Layer Expansions over a Rotating Disk The first setup I considered, (Iye17a), was a shear Euler flow on the exterior of a rotating disk. The purpose is to investigate the influence of a curved boundary on (1.1.2). More precisely, the domain I consider is given in polar coordinates via ⌦ = (!, r) 2 (0, ✓0 ) ⇥ 15 Figure 1.3: Flow past a rotating disk (R0 , 1). Moreover, the disk is assumed to be rotating at velocity 1 > 0. This setup is displayed in Figure 1.3. On this domain, the boundary layer scaling is defined in the following way: r R0 p R := R0 + p , r = R0 + "(R R0 ). " These scalings distinguish the curved boundary case from the Euclidean case because: Z Z Z Z Z Z p p u2 r dr d! = ✏ u2 (!, R)r dR d! 6= ✏ u2 (!, R)R dR d!. (1.3.1) Thus, Eulerian L2 norms cannot be compared directly to Prandtl L2 norms as in the Euclidean case, but rather to the measure r(R) dR d!. The Z-norm is designed to adapt to the scaling above by included specially selected weights r , and is defined via: Definition 1.3.1 (The norm Z). Z Z Z Z ||u, v||2Z = u2! r + u2R r1+ d!dR + ✏v!2 r + |@R (rv)|2 r d!dR ✓Z Z ◆1/q ✓Z Z ◆1/q +✏ u2q Rr q+↵ d!dR +✏ 2q q+↵ vR r d!dR ✓Z Z ◆1/q ✓Z Z ◆1/q q +✏ u2q ! d!dR +✏ +1 v!2q r 2p d!dR , (1.3.2) 16 where q = 1 + 0 , 0 arbitrarily small but positive, 2 (0, 14 ). Let p be the Holder q q conjugate of q, and 0 < p  ↵  2 . Most importantly, will be taken in the interval 1 1 2p  < 1. The space Z depends on the weight , but we will refrain from depicting this explicitly. The Z-norm should be thought of as a scaled, weighted H 1 norm, together with some H 1+ quantities. These H 1+ quantities are utilized to control nonlinear quantities: instead of estimated terms of the form kuruk2 using L1 ⇥ L2 estimates, it turns out to be more useful to use Lp ⇥ L2+ estimates for p very large. This in turn is because we are able to control Lp with r-weights in a more refined manner (see for instance Lemma 1.3.3 below). Theorem 1.3.2. Let 2 (0, 1) from Figure 1.2, let " << ✓0 << 1 be sufficiently small relative to universal constants. Suppose appropriate boundary data are provided for the profiles appearing on the right-hand side of (1.1.2). Then there exists a unique solution, [U " , V " , P " ] to the Navier-Stokes equation, (1.1.1), which can be expressed in the expan- sion (1.1.2). Moreover, the following estimates are valid (consult (1.3.2) for the Z-norm specification): ku0p , vp0 , u1e , ve1 , u1p , vp1 k1 + ku, vkZ . 1. The main mathematical difficulties I encounter to control u is that the boundary layer scaling and curvature of the boundary interact in a nontrivial way and that exterior domains are not Poincare domains, which produces weak energy estimates. To handle these difficul- ties, I designed and controlled several radially weighted norms, established their embedding theorems, and used this calculus to control u. Let us now introduce these norms. Let be arbitrary for now; it will be specifically based on the nonlinear iteration. Z Z ||u||2L2⇤,:= u2 (!, R)r dRd!, Z Z ||u||2A := u2R r1+ + ✏u2! r 1+ + ✏u2 r 1+ dRd!, 17 Z Z ||v||2B := r |@R (rv)|2 + ✏v!2 r + ✏v 2 r dRd!. The essential property satisfied by norms A and B is loss of one weight of r when going from A to B. This property is independent of the choice of . The scheme of linear estimates reads for any 2 [ 1, 1]: p ||u||2A . ✓0 ||v||2B + ✓0 ||P ||2L2⇤, + ||f, ✏g||2L2 , (1.3.3) ⇤,2+ p ||v||2B . ||u||2A + (1 )2 ||P ||2L2⇤, + ||f, ✏g||2L2 , (1.3.4) ⇤,2+ p ||P ||2L2⇤, . O(✓0 )||u||2A + O(✓0 )||v||2B + ||f, ✏g||2L2 . (1.3.5) ⇤,2+ Thus, in (2.2.31) there is a loss of one weight r, whereas in (1.3.4) we gain back this weight. This staggered structure is essentially due to the divergence free relation: u! ' rvR . At the culmination of (2.2.31) - (1.3.5), the choice of is still arbitrary. Let us now consider a sample trilinear term with = 1: Z Z ⇣ ⌘ + 12 + 12 1 1 ✏ vuR · @R (rv)  ✏ ||vr 2 ||L1 ||uR ||L2 ||r 2 @R (rv)||L2 (1.3.6) + 12 1 ✏ ||vr 2 ||L1 ||u, v||2A[B . However, kvr1/2 k1 is far out of reach of A [ B with = 1. We must thus push much larger to estimate this trilinear quantity. However, = 1 faces a criticality created from the forcing term: Z Z u2e 2+ 1 kf k2L2 = r = = 1. ⇤,2+ r4 r We must thus aim for selecting = 1 . For such a , we establish weighted H 1 ,! Lp embeddings of the type: 18 Lemma 1.3.3 (Embedding Theorems). ✓Z Z ◆ p1 p p ✏1/2 vp r 2 1+ 2 dRd!  Cp ||v||B for 2  p < 1, ✓Z Z ◆ p1 p 1 up r 2 r2  Cp ||u||A[B for 4  p < 1, ✓Z Z ◆ p1 p p u r 2  Cp ||u||A[B for 2  p < 4, In the end, we find a small region of 2 (1 , 1) which is reflected in the Z-norm above for which all nonlinear quantities can be controlled. 1.3.2 Global Steady Prandtl Expansion Over a Moving Boundary In my second work, (Iye16), I considered flows on the quadrant [0, 1) ⇥ [0, 1). The impor- tant feature here is that the x-coordinate is unbounded. Asymptotic in x behavior has long occupied a central role in the boundary layer theory due to the possibility of boundary layer separation, as has been discussed in Theorems 1.2.1 - 1.2.4. Theorem 1.3.4 is a global-in-x generalization of Theorem 1.2.5. Theorem 1.3.4. Let the outer Euler flow be fixed at [u0e , ve0 ] = [1, 0]. Let 0 <  0 << 1 from Figure 1.2, where 0 is sufficiently small relative to universal constants. Then the expansions (1.1.2) hold in the quadrant [0, 1) ⇥ [0, 1), and the following estimates are valid: kui , vei , uip , vpi k1 + ku, vkZ . 1. Let us compare the above result with Theorem 1.2.5. There exists a scaling L = L( ) present in Theorem 1.2.5 such that as ! 1, L( ) # 0 and as ! 0, L( ) " 1. This is analogous to a “large-data, local” result, where corresponds to the size of the data, L 19 plays the role of the time scale. This is substantially di↵erent than Theorem 1.3.4, which is analogous to a “small-data, global” result. Typically, in order to break scaling from a large-data local theorem, one needs to obtain sufficiently strong decay estimates of various quantities involved. This is encoded by the specification of the Z-norm, which we will now discuss. Definition 1.3.5. The norm Z is defined through: 1 p 1 ||u, v||Z :=||u, v||X1 \X2 \X3 + ✏N2 ||u, v||Y2 + ✏N3 ||u, v||Y3 + ✏N4 ||ux 4 , "vx 2 ||L1 p 3 5 1 + ✏N5 sup || "vx x 2 , ux x 4 ||L1 + "N6 sup ||uy x 2 ||L2y x 20 x 20 hZ 1 p i 12 + ✏ N7 x4 || "vxx ||2L1 y dx . (1.3.7) 20 Here, Ni , are large numbers which will be specified in (3.9.103) - (3.9.105). They depend only on universal constants. The parameter n from (3.2.14) - (3.2.15) will be taken much larger than any of the Ni . The norms || · ||Yi are elliptic norms defined in (3.9.6) - (3.9.7). For the purposes of discussing the main result, we can refrain from being too specific with regards to the definitions of the Yi norms. However, the norms || · ||Xi are important: p ku, vkX1 := kuy k2 + kr" v · xk2 , ku, vkXi :' k(x · @x )i {u, v}kX1 for i = 2, 3. The norm X1 is estimated in a two-step procedure analogous to (1.2.13) - (1.2.14): p kuy k22 . O( )kr" v · xk22 , p kr" v · xk22 . kuy k22 . p The first crucial point here is the loss of a factor of x, which is due to the self- 20 y similarity of the Prandtl layers: u0p ⇠ ⇤ ( px ), where ⇤ is a rapidly decaying profile in its argument. We establish this self-similarity by extracting an ODE governing ⇤ in the self-similar variable, z = py , and proving asymptotic convergence to this profile for general x p initial data. The second crucial point is the gain back of x weight in the second step. This in turn relies crucially on sharp, point-wise estimates on the Poisson kernel that we establish in order to conclude decay for the higher-order Euler layers: |vei | . x 1/2 . The norms X2 , X3 establish X1 control over anisotropic scaling vector-fields x·@x applied to the solution {u, v}. In turn this relies on commutation properties of @x with the linearized equation, (1.2.6). R This linearized strategy forces us to estimate trilinear quantities of the form vuy vy · x, p which in turn demands a sharp decay estimate for kv xk1 . Obtaining the decay for v using the norms Xi is an extremely delicate matter, in which key quantities must overcome the critical Hardy inequality. To see this, we first use the Hy1 (R+ ) ,! L1 y (R+ ) Sobolev embedding (ignoring factors of "): 1 1 1 sup ||vx 2 ||L1 y  sup ||v||L2 2 · sup ||vy x||L2 2 . (1.3.8) y y x 1 x 1 x 1 For the first quantity on the right-hand side above, we write: Z Z @x v 2 dy = 2vvx dy. (1.3.9) p 1 1 Recall now the quantities, ||uy , "vx x 2 , vy x 2 ||2L2 , which are controlled on the left-hand xy sides (3.2.45), (3.2.46) and constitute the energy norm X1 . The right-hand side above fails 1 to be x-integrable, precisely because of criticality of Hardy’s inequality with power x 2 in L2 : Z Z 1 1 1 | vvx dy dx|  ||vx 2  ||vx x 2 ||2L2xy . ||L2xy ||vx x 2 ||L2xy @ (1.3.10) 21 To avert this, we move to higher-order derivatives, which invokes the full strength of the 3 norm Z. Indeed, suppose we knew vx ⇠ x 2 , then coupled with the boundary condition 1 v ! 0 as x ! 1, this would immediately imply v ⇠ x 2 . Establishing the decay rate, 3 ||vx ||L1 y  x 2 , then becomes the goal, which requires us to use the full set of vector- fields X2 , X3 . Our main uniform estimates, given in Lemmas 3.9.15, 3.9.17 are given by the following sequence: 1 3 1 1 ||vx 2 ||L1 xy . ||vx x 2 ||L1 xy . sup ||vx x||L2 2 · sup ||vxy x2 ||L2 2 . ||u, v||X1 \X2 \X3 . (1.3.11) y y x 1 x 1 The key point is that the quantities ||vx x||L2y and ||vxy x2 ||L2y appearing above do not face issues of Hardy-criticality present in (1.3.10). There are several other components to the norm Z, which I now discuss: • Anisotropic Mixed Norms: Due to the anisotropy of the above procedure, some non- linear terms cannot be estimated through L1 L2 L2 type estimates. These must estimated using mixed norms, which we include in the above Z-norm and subsequently show can be controlled. • Elliptic Norms: Due to the presence of boundaries, we must technically work with cut-o↵ vector fields, that is (x 10) · x · @x . In order to supplement the norm with ample control on x ⇠ o(1) region, we must appeal to elliptic estimates available for the Stokes operator, which in turn scale poorly with ". • Null Forms: In order handle the large negative power of " appearing in the elliptic n norms, we must propagate the expansion of (1.1.2) up to high order, " 2 , all the while preserving the baseline decay estimates and self-similar estimates which are attained by propagating control of py . The most dangerous contributions towards x establishing this are Euler-Euler interactions, because weights of py are unfavorable x for such interactions. The crucial observation is that these interactions enjoy a special 22 property, which is that they are of “gradient-type”. That is, there exists a potential function whose gradient is exactly the vector of all Euler-Euler interactions. This is due to the special Cauchy-Riemann structure of the Euler flows. Such a potential can be exploited by adding it into the pressure to cancel out these interactions. • Existence and Uniqueness: Existence in our space Z is achieved via compactness methods which rely on specially selected weights which we append to an approximating sequence of systems. These weights must be compatible with the scheme of estimates designed above. For uniqueness, we reapply the above estimates with weaker weights. It turns out only a small interval of weaker weights can work with the aforementioned scheme of estimates. 1.3.3 Steady Prandtl Expansions over a Moving Boundary: Nons- hear Euler Flows In the work (Iye17b), I considered again the domain ⌦ = (0, L) ⇥ R+ , for 0 < L << 1 small relative to universal constants. Let the boundary have velocity 0 < ub . In this case, suppose the prescribed Euler flow, [u0e , ve0 ] is not shear. This means that u0e = u0e (x, Y ) is allowed to have x-dependence. I prove for “weakly nonshear” Euler flows (as specified by assumption 1.3.13), the following: Theorem 1.3.6. Let the motion of the boundary equal ub > 0. Consider an Euler flow [u0e (x, Y ), ve0 (x, Y )] satisfying the following hypothesis: 0 < c0  u0e  C0 < 1, (1.3.12) ve0 || ||L1 << 1, and (1.3.13) Y ||Y k rm ve0 ||L1 < 1 for sufficiently large k, m 0, (1.3.14) ||Y k rm u0e ||L1 < 1 for sufficiently large k 0, m 1. (1.3.15) 23 Let the interval L be sufficiently small relative to universal constants. Suppose in ad- dition that the boundary data described above are prescribed, assumed to be smooth and rapidly decaying in their arguments, and satisfy suitable compatibility conditions. Then the expansions (1.1.2) is valid, and the remainder solutions [u, v, P ] exist in the space Z and satisfy the estimate: ||u, v||Z . 1. (1.3.16) Let us briefly highlight the physical importance of developing a method to handle non- shear Eulerian flows. A classical setup from fluid mechanics deals with horizontal flows past a rotating disk, see for instance (SG00). Such a flow is non-shear, as in the set-up considered here. In the simpler case when the flows are actually circular (and therefore shear), as opposed to horizontal, in the presence of a rotating disk, Theorem 1.3.2 develops machinery to handle the geometry of the boundary. The present article can be viewed as a first step in studying non-shear flows, without adding the complexities of a curved boundary. Mathematically, the main difficulty of the analysis can be seen by explicitly writing the form of vs , which presents a singularity of size p1 at the leading order that must be " addressed by our analysis: v0 p vs = pe + vp0 + ve1 + "vp1 . (1.3.17) " In this case, we control this singularity by adding far-field controls over u in the Z-norm, which is defined via: p ||u, v||Z :' ||r" u · y||L2 + kr" vkL2 + ||r2" u · y||L2 + " 2 ||u, "v||L1 . Let us briefly demonstrate how this norm will be utilized and estimated. The scheme of 24 linear estimates now read: kuy k22 . O(L)kr" vk22 , (1.3.18) ve0 kr" vk22 . kr" u · yk22 + kuy k22 , (1.3.19) Y 1 kr" u · yk22 + kr2" u · yk22 . kuy k22 + kr" vk22 . (1.3.20) Let us highlight estimate (1.3.19) in which the following term creates a loss of one weight y: Z Z v0 v0 v0 | pe uy vy | | p e uy vy y| . e kvy k2 kuy yk2 " "y Y 1 Estimate (1.3.20) is the main contribution of this work. It is achieved by introducing a vector field: G := y 2 (1 x)@y . The factor of 1 x is sometimes known as a “Ghost Weight”. The reason is because in absolute value, for x small, 1 x ⇠ 1. However, at the level of one derivative, @x {1 x} is not similar at all to @x 1. Thus, the inclusion of (1 x) can amplify terms which have extra integration by parts available in the x direction. This is utilized in the present context to generate the following positive term: Z Z Z G{us @x u} · uy ' us uxy uy y 2 (1 x) ' u2y y 2 . The remaining task in the estimate is to ensure that G interacts favorably with the remaining quantities in the equation (1.2.6). 25 1.3.4 Stationary Inviscid Limit to Shear Flows We will now consider flows in a channel, that is (x, Y ) 2 (0, L) ⇥ (0, 2). In addition, assume the {Y = 2} is translating rightward at velocity ub 0. We will consider Euler shear flows, [u0e , ve0 ] = [µ, 0] that themselves satisfy no-slip: µ(0) = 0, µ(2) = ub , (1.3.21) for which there is no leading order boundary layer. Denote now the asymptotic expan- sion: 0 1 0 1 3 3 3 " 1 1 2 u2 + " 2 u2 + " 2 + u u B C B µ + "u e + "u p + " e p C u" := @ A = @ A. (1.3.22) 3 3 3 v" "ve1 + " 2 vp1 + " 2 ve2 + "2 vp2 + " 2 + v Theorem 1.3.7 (joint with Chunhui Zhou). Let ub 0 in (5.2.3). Let µ(y) 2 C 1 ([0, 2]) be a given function, satisfying the conditions: µ(0) = 0, µ(2) = ub , (1.3.23) @yj µ(0) = @yj µ(2) = 0 for 2  j  N0 , (1.3.24) @y µ(0) > 0, |@y µ(2)| > 0. (1.3.25) where N0 < 1 and large but unspecified. Let also standard compatibility conditions at the corners of ⌦ be prescribed for the layers in us . Then there exists a unique solution, u" satisfying the Navier-Stokes equations, (5.2.3), such that: ||u" µ||1 + ||v " ||1  c0 (µ)". (1.3.26) 26 The constant c0 (µ) satisfies: µ000 c0 (µ) . || ||W 100,1 . (1.3.27) µ Our ultimate interest is motivated by Yudovich’s ninth problem, (Yud03). Classical ex- periments starting with Reynolds have shown that unsteady flows in a 2D channel that start near Couette or Poiseulle flow do not converge to these flows. This indicates the existence of infinitely many stationary solutions to Navier-Stokes “near” Couette or Poiseulle. Establish- ing the existence of these solutions is an open problem. Our second result, Corollary 5.2.2, produces stationary solutions sufficiently close to Couette, assuming x 2 [0, L], L << 1, and a moving boundary at y = 2. Our construction is local in x, and is a first step towards obtaining these global-in-x solutions. Corollary 1.3.8. Let any ↵ > 0 be prescribed, which could depend on ". Let µ ˜ be prescribed to satisfy the vanishing conditions: @yk µ ˜|y=0 = @yk µ ˜|y=2 = 0 for 0  k  N0 . There exists a unique solution, u" to (5.2.3) with ub = 2 such that: ⇣ ⌘ ||u" µ(y) ||1 + ||v " ||1 . ↵". y + ↵˜ (1.3.28) Proof. One can obtain this by applying Theorem 5.2.1 with µ(y) = y + ↵˜ µ(y), where µ ˜ vanishes at high order near y = 0, 2. In this case, the constant c0 (µ) . ↵. Remark. The requirement of ub = 2 is so that the no-slip condition is satisfied by the Couette flow. We do not use this motion of the boundary anywhere in the proof. Chapter Two Steady Prandtl Boundary Layer Expansions over a Rotating Disk 28 2.1 Abstract This chapter concerns the validity of the Prandtl boundary layer theory for steady, incom- pressible Navier-Stokes flows over a rotating disk. We prove that the Navier-Stokes flows can be decomposed into Euler and Prandtl flows in the inviscid limit. In so doing, we de- velop a new set of function spaces and prove several embedding theorems which capture the interaction between the Prandtl scaling and the geometry of our domain. 2.2 Introduction We consider the steady incompressible Navier-Stokes equations on the domain ⌦ = (0, ✓0 ) ⇥ (R0 , 1) in polar coordinates. The boundary @⌦ then consists of three components: {! = ✓0 }, {! = 0}, {r = R0 }. In cartesian coordinates, the equations read: 9 ¯ ¯ ¯ ¯ ¯ U U x + V U y + Px = ✏ U >> > > > = ¯ ¯ ¯ ¯ ¯ U V x + V V y + Py = ✏ V > in ⌦ (2.2.1) > > > > ¯x + V¯y = 0. U ; In polar coordinates, the equations read (KC04, Page 739): ✓ ◆9 U U! UV P! Ur U!! U 2 > + V Ur + + = ✏ Urr + + 2 + V ! > > r r r r r r2 r2 > > ✓ ◆ > = U V! U2 1 V!! V 2 in ⌦ (2.2.2) + V Vr + Pr = ✏ Vrr + Vr + 2 U ! > r r r r r2 r2 > > > > > ; U! + @r (rV ) = 0. 29 ¯ and V¯ represent the horizontal and vertical velocities of the flow, and U, V Here, U represent the angular and radial velocities of the flow. The Navier-Stokes equations are taken together with the no-slip boundary conditions on the boundary {r = R0 }. We suppose that the disk of radius R0 is rotating counter-clockwise with a constant angular velocity of ub > 0. The no slip boundary condition in our case is then U |r=R0 = ub and V |r=R0 = 0. (2.2.3) The boundary conditions at {! = 0} and {! = ✓0 } will be prescribed in the text. We study the limit as ✏ ! 0. Formally, one expects solutions to the above Navier-Stokes equations to converge to solutions of the Euler equations with ✏ = 0, but this does not happen due the mismatch at the boundary between the no slip condition enforced for solutions to Navier Stokes equations and the no normal flow condition enforced for solutions to Euler equations. To account for this mismatch, in 1904 Ludwig Prandtl proposed the formation of a p boundary layer of size ✏ near the boundary, such that the Navier-Stokes flow can be decomposed into the sum of the Euler flow and the boundary layer flow. This is regarded as one of the most important ideas in fluid mechanics in the last century, and the theory has led to astounding developments in the applied sciences. Indeed, many phenomena in fluids such as wake flows and plane jet flows are described by the Prandtl theory (SG00). Despite this, a rigorous mathematical justification of the boundary layer theory remains open in general. For unsteady flows, there are several interesting results, see for instance (SC98a), (SC98b), (Mae14), (Asa91), (MT08). The work of Guo and Nguyen, (GN17), is the first result establishing validity of the boundary layer expansion for steady state flows in a rectangular domain over a moving plate. They do so using a combination of energy es- 30 timates, elliptic estimates, and a new positivity estimate obtained via the vorticity multi- ⇣ ⌘ ⇣ ⌘ plier @y uvs ✏@x uvs . The main goal of this paper is to generalize Guo and Nguyen’s method in the presence of geometric curvature e↵ects in order to establish the validity of the boundary layer theory for steady flows over a rotating disk. 2.2.1 Boundary Layer Expansion We denote by u0e to be an outer Euler shear flow which is radial: u0e = u0e (r). (2.2.4) Such a shear flow describes an Euler fluid which rotates counterclockwise. On the bound- ary {r = R0 }, we denote by ue = u0e (R0 ) and assume that ue > 0. We also suppose that the disk of radius R0 is rotating at an angular velocity ub > 0. We now scale to boundary layer variables in the following way: Boundary Layer Scaling: Euler Scaling: r pR0 p R = R(r) = R0 + ✏ , r = r(R) = R0 + ✏(R R0 ). p p Note that r, R R0 > 0 and that @R r(R) = ✏, @r R(r) = 1/ ✏. We scale to boundary layer velocities and pressure in the following way: 1 U ✏ (!, R) = U (!, r), V ✏ (!, R) = p V (!, r), P ✏ (!, R) = P (!, r). (2.2.5) ✏ The boundary layer velocities and pressure satisfy the following scaled Navier-Stokes equations: p U ✏ U!✏ ✏ ✏ ✏ 1 ✏ p U✏ U✏ U✏ ✏ 3/2 V! ✏ ✏ + V UR + ✏ U V + P! = URR + ✏ R + ✏ !! ✏ + 2✏ , (2.2.6) r r r r r2 r2 r2 31 U ✏ V!✏ 1 (U ✏ )2 1 p V✏ V✏ V✏ p U✏ + V ✏ VR✏ p + PR✏ = VRR ✏ + ✏ R + ✏ !! ✏ 2 ✏ 2! , r ✏ r ✏ r r2 r r U!✏ + @R (rV ✏ ) = 0. We start with the following formal expansion: p p + 12 U ✏ (!, R) = u0e (!, r) + u0p (!, R) + ✏u1e (!, r) + ✏u1p (!, R) + ✏ u✏ (w, R), (2.2.7) p V ✏ (!, R) = vp0 (!, R) + ve1 (w, r) + ✏vp1 (!, R) + ✏ +1/2 ✏ v (!, R), (2.2.8) p p 1 P ✏ (!, R) = Pe0 (r) + Pp0 (!, R) + ✏Pe1 (!, r) + ✏Pp1 (!, R) + ✏Pp2 (!, R) + ✏ 2 + P ✏ (!, R). (2.2.9) According to the expansions (2.2.7) - (2.2.8), the Prandtl decomposition up to leading order is then: U (!, r) = U ✏ (!, R) ⇡ u0e (!, r) + u0p (!, R), (2.2.10) p p p V (!, r) = ✏V ✏ (!, R) ⇡ ✏vp0 (!, R) + ✏ve1 (!, r). (2.2.11) [u0p , u1e , u1p ] and [vp0 , ve1 , vp1 ] are approximate boundary layers to be constructed, after which the remainders u✏ , v ✏ must be constructed and controlled. We insert the expansions into the scaled Navier-Stokes equations, and obtain the di↵erent orders of the errors Ru and Rv , which are detailed in equations (2.4.1) - (2.4.19). The scaled divergence free condition is enforced at each stage of the expansion. So, for example, for the Prandtl-0 layer, we enforce p 0 u0p! = @R (rvp0 ) = ✏vp rvpR0 , and for the Euler-1 layer, we enforce u1e! = @r (rve1 ) = ve1 1 rver . We define the following notation which will be in use throughout the paper: p p Definition 2.2.1. us = u0e + u0p + ✏u1e , vs = vp0 + ✏ve1 , and uapp = us + u1p , vapp = vs + vp1 . Once the velocities of each layer has been constructed, the pressures are defined using the radial error contributions up to and including the ✏0 contributions. The ✏ 1 order equation, 32 1/2 (2.4.7), for instance, dictates that the initial Prandtl pressure is constant in R. The ✏ order error, (2.4.8), is the Euler-0 pressure as given by the Euler equation for the shear radial flow u0e (r). We then estimate the error caused by this definition in the angular equations. 2.2.2 Boundary Data The no-slip boundary conditions at {r = R0 } must be enforced for each order of the expan- sion in (2.2.7, 2.2.8). Since the outer Euler flow u0e is given, we have: Boundary Conditions on {r = R0 }: u0e (R0 ) + u0p (!, R0 ) = ub , u1e (!, R0 ) + u1p (!, R0 ) = 0, u✏ (!, R0 ) = 0, (2.2.12) vp0 (!, R0 ) + ve1 (!, R0 ) = 0, vp1 (!, R0 ) = 0, v ✏ (!, R0 ) = 0. (2.2.13) Boundary Conditions on {! = 0}: u0p (0, R) = u ¯0 (R), u1p (0, R) = u ¯1 (R), u1e (0, r) = u1b (r), (2.2.14) ve1 (0, r) = Vb0 (r), u✏ (0, R) = v ✏ (0, R) = 0. (2.2.15) Boundary Conditions on {! = ✓0 }: ve1 (✓0 , r) = Vb1 (r), (2.2.16) ✏v!✏ + ru✏R = 0 and P ✏ r = 2✏u✏! . (2.2.17) Boundary Conditions as r ! 1: u0p (!, R), u1p (!, R), [uje , vej ](!, r) ! 0 as r, R ! 1. (2.2.18) 33 We impose the following compatibility conditions for the Euler boundary conditions: Vb0 (R0 ) = vp0 (0, R0 ), Vb1 (R0 ) = vp0 (✓0 , R0 ). (2.2.19) The boundary conditions for the remainders (u✏ , v ✏ ) in (2.2.12) - (2.2.13) and (2.2.15) are the no-slip conditions, and the condition in (2.2.17) is the stress-free condition. 2.2.3 Main Result In order to state our main result, we must first define the norm Z, which is the Prandtl layer norm in which we close our nonlinear analysis: Definition 2.2.2. Z Z Z Z ||u, v||2Z = u2 r + u2! r + u2R r1+ d!dR + ✏v!2 r + |@R (rv)|2 r d!dR ✓Z Z ◆1/q ✓Z Z ◆1/q +✏ u2q Rr q+↵ d!dR +✏ 2q q+↵ vR r d!dR ✓Z Z ◆1/q ✓Z Z ◆1/q q +✏ u2q ! d!dR +✏ +1 v!2q r 2p d!dR , (2.2.20) where q = 1 + 0 , 0 arbitrarily small but positive, 2 (0, 14 ). Let p be the Holder q q conjugate of q, and 0 < p  ↵  2 . Most importantly, will be taken in the interval 1 1 2p  < 1. The space Z depends on the weight , but we will refrain from depicting this explicitly. Theorem 2.2.3. Let ub > 0 and u0e (r) be a given Euler shear flow such that the derivatives @rk u0e (r), k 1 decay exponentially. Suppose the boundary data in (2.2.12 - 2.2.18) are prescribed. Suppose that u ¯0 and u ¯1 decay exponentially fast in their arguments, that the compatibility conditions (2.2.19) are satisfied, and that |Vb0 Vb1 | . ✓0 for small ✓0 . Suppose further that min{ub , u0e + u ¯0 } > 0. There exists a positive angle ✓0 which depends on the 34 prescribed data such that for 2 (0, 14 ) and 2 (0, 1) sufficiently close to 1, the asymptotic expansions given in equations (2.2.7 - 2.2.9) are valid. The approximate solutions appearing in the expansion are those constructed in Theorems 2.4.1, 2.4.5, and 2.4.12, and the Navier- Stokes remainder satisfies ||u✏ , v ✏ ||Z  C0 . Corollary 2.2.4 (Inviscid Lp convergence). Under the assumptions of Theorem 2.2.3, we have the following inviscid Lp convergence: ✓Z Z ◆ p1 ✓Z Z ◆ p1 1 |U (!, r) u0e (r)|p r drd! + p |V (!, r)| r drd!  C✏ 2p for 2  p < 4, and for arbitrarily close to 1, and ✓Z Z ◆ p1 ✓Z Z ◆ p1 1 |U (!, r) u0e (r)|p rdrd! + p |V (!, r)| rdrd!  C✏ 2p for 4  p < 1, where U and V are the original Navier-Stokes flows appearing in equation (2.2.2). In L1 we have the following convergence: 1 sup |U (!, r) u0e (r) u0p (!, R)| . ✏ 4 + 2 , (2.2.21) (!,r)2⌦ p p 1 sup |V (!, r) ✏vp0 (!, R) ✏ve1 (!, r)| . ✏ 4 + 2 , (2.2.22) (!,r)2⌦ Function Space Preliminaries We briefly discuss the relevant function spaces in which we develop our analysis. Only basic definitions are given here because they are required to follow the steps of the outline below. The details of our functional analytic setup are presented in Section 2.3. The interaction 35 between the Prandtl scaling and the geometry of our domain manifests itself in the functional framework of our analysis for the following reason: Z Z Consider an L2 function, u ¯, in Euler coordinates. By definition, this means ¯2 (!, r)rdrd! < u 1. The corresponding scaled function in Prandtl coordinates is given by u(!, R) = u ¯(!, r). The Eulerian L2 norm scales down to: Z Z Z Z Z Z p p ||u||2L2 (Euler) = ¯2 rdrd! = u ✏ u2 (!, R)rdRd! 6= ✏ u2 (!, R)RdRd!. Due to the mismatch between the scaled Euler L2 norm and the actual Prandtl L2 norm, we must work in a new set of function spaces (notationally depicted as || · ||⇤ and variants thereof) which are natural to our problem, and build the corresponding analytic machinery we require to do our nonlinear analysis. Motivated by this, we define the following Prandtl- layer version of the L2 norm. Z Z Definition 2.2.5. ||u||2L2⇤ := u2 rdRd!. @! Applying this same analysis to the derivative operator, r = r , @r , motivates the following definition: Definition 2.2.6. ||u||H⇤1 := ||u||L2⇤ + ||r⇤ u||L2⇤ where r⇤ = @r! , @R and similarly for ✓ ◆ @!! @!R H⇤2 where r2⇤ has components , , @ RR . Occasionally we will refer to r⇤,✏ which ⇣ ⌘ r2 r p ✏@! has components r , @R . The corresponding weighted variants of these norms will be Z Z denoted with two subscripts: ||u||2L2 := u2 (!, R)r dRd!. ⇤, Whenever we write Lp without any subscripts, this means the usual Lp in either the Prandtl layer or the Euler layer, which will be clear from context. The following are the norms in which energy estimates will be obtained: Z Z 2 Definition 2.2.7. ||u||A := u2R r1+ + ✏u2! r 1+ + ✏u2 r 1+ dRd!. 36 Z Z Definition 2.2.8. ||v||2B := r |@R (rv)|2 + ✏v!2 r + ✏v 2 r dRd!. Z Z By the Fundamental Theorem of Calculus and Holder’s inequality: u2 r dRd!  Z Z ✓02 u2! r dRd! if u|!=0 = 0. This paired with the divergence-free condition, u! = Z Z @R (rv), yields: u2! + u2 r dRd!  ||v||2B . Motivated by this, we define the following norm: Z Z Definition 2.2.9. ||u||2X := u2 r + u2! r + u2R r1+ . With these definitions in hand, we detail the steps of our analysis. Outline of Proof Inserting the boundary layer expansions (2.2.7 - 2.2.9) into the scaled Navier-Stokes system (2.2.6), we obtain the following system for the Navier-Stokes remainders (for the remainder of this section, we replace u✏ , v ✏ , P ✏ by u, v, P for notational ease): p p 1 1 ✏ ✏ 1 us u! + us! u + usR v + vs uR + vs u + u s v + P! (2.2.23) r r p r r r ✏ ✏ ✏ 2 3/2 uRR ur u!! + 2 u ✏ v! = f, r r2 r r2 1 1 2 1 1 us v! + vs! u + vs vR + vsR v p u s u + PR (2.2.24) r r r ✏ ✏ p p ✏ ✏ 2 ✏ ✏ vRR vR v!! + 2 u! + 2 v = g, r r2 r r 1 p 1 u! + ✏ v + vR = 0. (2.2.25) r r where ✓ p ◆ 1 p + 12 1 ✏ f (!, R) = ✏ 2 Ru ✏Ru,p ✏ uu! + vuR + uv , (2.2.26) r r ✓ ◆ 1 p + 12 1 1 u2 g(!, R) = ✏ 2 Rv ✏Rv,p ✏ uv! + vvR p . (2.2.27) r ✏ r 37 Here, Ru , Rv are the remainders from the approximate solutions uapp , vapp , whose precise definitions are given in (2.4.1) - (2.4.19). Ru,p , Rv,p are the linearizations of the Navier- Stokes remainders around the Prandtl-1 layer, which precisely are given by: p p 1 1 1 ✏ 1 ✏ 1 Ru,p = up u! + u1p! u + u1pR v + vp1 uR + vp u + u v, (2.2.28) r r r r p 1 1 1 2 Rv,p = u1p v! + vp! u + vp1 vR + vpR 1 v p u1 u. (2.2.29) r r r ✏ p The NS remainders u, v satisfy the following boundary conditions: [u, v]|!=0 = [u, v]|R=R0 = 0, ✏v! + ruR = 0 and P r = 2✏u! on {! = ✓0 }. (2.2.30) Step I: Construction of Approximate Solutions We first construct the approximate solutions uapp , vapp such that the resulting remainder terms Ru and Rv are higher order in ✏. This involves three stages: constructing the Prandtl- 0 layers (u0p , vp0 ) using equations (2.4.1, 2.4.7), the Euler-1 layers (u1e , ve1 ) using the equations (2.4.2, 2.4.10), and the Prandtl-1 layers (u1p , vp1 ) using the equation (2.4.3). The divergence free conditions are enforced at each stage, and the boundary conditions are given in (2.2.12 - 2.2.18). The method of constructing the approximate solutions is as follows: the Prandtl-0 layer angular velocity, u0p , is constructed via a von-Mises transformation, for which the assumption min{ub , u0e + u ¯0 } > 0 is crucial. The radial velocity vp0 is then obtained via the divergence R1 free condition: rvp0 = R u0p! . This choice creates rapid decay as R ! 1 for the Prandtl-0 layers, but as a consequence a boundary condition for vp0 |R=R0 cannot be enforced. The second stage of the construction addresses the Euler-1 layer, (u1e , ve1 ) which is de- signed to correct for the normal boundary velocity of vp0 |R=R0 by enforcing vp0 |R=R0 + 38 ve1 |R=R0 = 0. After passing to a vorticity formulation for ve1 , this layer is obtained via standard methods from the second order elliptic theory. The last stage of the construction addresses the Prandtl-1 layer, (u1p , vp1 ). The boundary conditions on {r = R0 } are u1p |R=R0 = u1e |R=R0 and vp1 |R=R0 = 0. These are designed such that uapp |R=R0 = 0 and vapp |R=R0 = 0. This construction relies on the positivity estimate, which will be discussed in Step III. RR After these three stages, we evaluate the remaining error, |Ru |2 + ✏|Rv |2 r2+ . The weight of r2+ must be included because Ru , Rv are contained in f, g in the equations (2.2.26 - 2.2.27), and r2+ accompanies f, g in the linear estimate (2.2.48). Interestingly, the angular RR 02 error term arising from equation (2.4.4), |ue | r 2 , is infinite in the critical case of = 1, which is the reason the convergence in Corollary 2.2.4 cannot include = 1 for p < 4, which in turn would correspond to the usual Lp convergence. Despite this, we make use of the delicate embedding theorems we prove in Section 2.3 in order to recover Lp convergence for p 4. The construction of the approximate solutions and evaluation of the resulting error culminates in the following: Theorem 2.2.10. Under the assumptions of Theorem 2.2.3, there exist approximate solu- ✓Z Z ◆ 12 ✓Z Z ◆ 12 p 3 tions such that 2 2+ Ru r dRd! + ✏ 2 2+ Rv r dRd! . ✏ 4  if 0  < 1 and for  > 0 but arbitrarily small. Moreover, the approximate solutions satisfy the various estimates which appear in Theorems 2.4.1, 2.4.5, and 2.4.12. Step II: Energy Estimate In this step, we obtain the natural energy estimate associated to the linearized system (2.2.23) - (2.2.25). 39 Theorem 2.2.11. ||u||2A . ✓0 ||v||2B + ✓0 ||P ||2L2 + ||f ||2L2 + ✏||g||2L2 for 2 [0, 1] and ✏ << ✓0 . ⇤, ⇤,2+ ⇤,2+ (2.2.31) This estimate is generated by applying the multiplier (r1+ u, ✏r1+ v) to equations (2.2.23 - 2.2.24). Once the weight r1+ is fixed for the angular multiplier, the divergence free condition (2.2.25) forces a loss of one factor of r. We illustrate this by multiplying the right-hand side of (2.2.23) by r1+ u, yielding: Z Z Z Z Z Z Z Z Z Z f r1+ u . f 2 r2+ + u2 r . f 2 r2+ + ✓02 u2! r Z Z Z Z Z Z . f 2 r2+ + ✓02 |@R (rv)|2 r . f 2 r2+ + ✓02 ||v||2B . (2.2.32) Thus ||v||B must appear in our estimate, which features an extra factor of r as com- pared to ||v||A according to Definitions 2.2.7 and 2.2.8. The strongest weight for the radial multiplier which is then consistent with the presence of ||v||B is ✏r1+ v. Step III: Positivity Estimate In this crucial step, we estimate ||v||B in terms of ||u||A . Such an estimate must overcome two difficulties. First and foremost, multiplying equation (2.2.24) by a multiplier which is O(✏v), RR as in Step II, formally results in control over ✏ |r✏ v|2 , which is too weak in the inviscid limit. This lack of a basic order-one estimate of v is the most fundamental difficulty in the boundary layer theory. In the case of a rectangular geometry, Guo and Nguyen overcame ⇣ ⌘ ⇣ ⌘ this difficulty by using the vorticity multiplier @y uvs ✏@x uvs (GN17). Second, since ||v||B , which appears on the right-hand side of estimate (2.2.31), contains an extra factor of r when compared to ||u||A , the positivity estimate must recover this factor. This difficulty 40 is new to our problem due to the geometry of our domain ⌦. The starting point is the following calculation, which we use in Section 2.6 and through- out the construction of the approximate solutions. We temporarily ignore boundary contri- butions as we shall apply this calculation to functions v vanishing on relevant parts of the boundary. Lemma 2.2.12 (Positivity Calculation). Z Z Z Z Z Z ✓ ◆2 rv r |@R (rv)|2 . r |@R (rv)|2 + r us usRR + ✓0 ||v||2B . (2.2.33) us Proof. For the sake of simplicity, we select the = 0 case to showcase initially. The case for general which we shall need involves controlling a few more terms, and is proved rigorously in Section 2.6. Z Z Z Z Z Z v rv v |@R (rv)|2 = us )|2 = |@R (r |@R ( )us + r usR |2 (2.2.34) us us us Z Z Z Z ✓ ◆2 Z Z ✓ ◆ rv 2 v rv rv = |@R ( )| |us |2 + r2 u2sR + 2 @R us usR us us us us Z Z ✓ ◆ Z Z ✓ ◆2 rv 2 2 rv = |@R | us us usRR , us us Z Z Z Z Z Z Z Z v rv 2 2 r2 v 2 2 |@R (rv)|2 = |@R (r us )|2 . |@R ( )| us + u , (2.2.35) us us u2s sR Z Z Z Z Z !2 Z Z Z R R r2 v 2 2 rv rv 2 u = u2sR @R ( )  u2sR (R R0 ) |@R ( )| (2.2.36) u2s sR R0 us R0 us Z Z rv 2 . |@R ( )| . us Recalling that min us > 0, and inserting (2.2.36) in (2.2.35) and then into (2.2.34) yields the desired estimate. 41 The key calculation is that in estimate (2.2.36), in which we’ve used the rapid decay of usR to conclude: Z 1 sup u2sR (R R0 )dR < 1 (2.2.37) !2[0,✓0 ] R0 This will be proven rigorously in equation (2.5.27). Moreover, as in (GN17), our posi- tivity estimate relies on the profile us > 0, which in turn relies upon our assumption that ub > 0. This is the reason our analysis does not treat the case of a non-rotating boundary. This lemma is used to prove the following: Theorem 2.2.13 (Positivity Estimate). For 2 [0, 1], Z p ||v||2B + ✏u2! r 1 .||u||2A + (✓0 + ✏)||v||2B + (1 )2 ||P ||2L2 + ||f ||2L2 + ✏||g||2L2 . ⇤, ⇤,2+ ⇤,2+ !=✓0 (2.2.38) In particular for ✓0 and ✏ small enough, this establishes control of ||v||B in terms of ||u||A : Z ||v||2B + ✏u2! r 1 .||u||2A + (1 )2 ||P ||2L2 + ||f ||2L2 + ✏||g||2L2 . (2.2.39) ⇤, ⇤,2+ ⇤,2+ !=✓0 The essential mechanism behind the positivity estimate is to capitalize on the order 1 ap- us pearance of vR in the positive profile term r u! in equation (2.2.23) through the divergence r2 v r1+ v free condition. We apply the multiplier (r @R ( ), ✏@! ( )) to equations (2.2.23 - us us 2.2.24), which is formally a weighted vorticity multiplier. The weights are designed care- fully to capture the ||v||B norm using the profile terms from (2.2.23 - 2.2.24). We highlight this using the three important profile terms below: Z Z Z Z Z Z us r2 v usR u! r @ R ( )⇡ r u! @R (rv) + r1+ v@R (rv) r us us 42 Z Z Z Z 1 usR ⇡ r |@R (rv)|2 + r @R ((rv)2 ), (2.2.40) 2 us Z Z ✓ ◆ Z Z v ✏ us v ! r @ ! ⇡ ✏ r v!2 , (2.2.41) us Z Z Z Z Z Z r2 v 1 usR u2sR 2 vusR r @R ( )⇡ r @R ((rv)2 ) r2+ v . (2.2.42) us 2 us u2s Summing (2.2.40 - 2.2.42), integrating by parts, and using Lemma (2.2.12) yields: Z Z Z Z Z Z 2 usRR 2 (2.2.40 2.2.42) ⇡ r |@R (rv)| ✏ r v!2 r2+ v (2.2.43) us Z Z Z Z . r |@R (rv)|2 ✏ r v!2 . (2.2.44) Once the important quantities from ||v||B have been extracted, the rest of the proof proceeds by estimating the remaining terms after the multiplier is applied to the equations (2.2.23) - (2.2.24). Step IV: Pressure Estimate The pressure term ||P ||L2⇤, must be included in Theorems 2.2.11 and 2.2.13 due to geometric e↵ects. The choice of = 0 for the energy estimate multiplier in Theorem 2.2.11 forces the pressure term to drop out whereas the choice of = 1 is required by Theorem 2.2.13 in order for this term to vanish. The lack of a consistent choice of which simultaneously forces the pressure to drop out of Theorems 2.2.11 and 2.2.13 requires us to estimate ||P ||L2⇤, in the following: Theorem 2.2.14. For 2 [0, 1] and ✏ << ✓0 , p ||P ||2L2 . C1 (✓0 , ✏)||u||2A + C2 (✓0 , ✏)||v||2B + ||f, ✏g||2L2 . (2.2.45) ⇤, ⇤, Here, Ci (✓0 , ✏) ! 0 as either ✓0 ! 0 or ✏ ! 0. 43 We emphasize that this estimate is new in our analysis due to the presence of geometric e↵ects, and therefore did not appear in (GN17). Moreover, it is surprising that the ||u||A and ||v||B terms appearing on the right-hand-side of Theorem 2.2.14 are accompanied by small parameters. The estimate relies on the existence of a vector field, A1 = (a(!, R), b(!, R)) such that div(A1 ) ⇡ P , and ||A1 ||H⇤1 . ||P ||L2⇤ , which is guaranteed to exist for P 2 L2 by (Orl98, Page 27) and the estimates we establish in Claims 5 - 8. Moreover, A1 can be selected to vanish on the Dirichlet portions of the boundary, {R = R0 } and {! = 0}. Given this vector field, we apply the multiplier (ar , ✏br ) to (2.2.23 - 2.2.24). The weighted vector field is used as our multiplier in order to estimate the correct weight on the Pressure term: Z Z Z Z Z Z Z Z P! a! p b r a + PR br ⇡ P( + ✏ + bR )r ⇡ P div(A1 )r ⇡ P 2r . r r r (2.2.46) Once ||P ||L2⇤, has been extracted, the rest of the proof proceeds by controlling the terms arising from applying the multiplier (ar , ✏br ) to (2.2.23 - 2.2.24). Summary of Linear Analysis in Steps II - IV: Putting estimates (2.2.31), (4.4.5), (2.2.45) together yields the full energy estimate for the linearized system in (2.2.23 - 2.2.25): Z p ||u||2A + ||v||2B + ||P ||2L2 + ✏u2! r 1 . ||f, ✏g||2L2 . (2.2.47) ⇤, ⇤,2+ !=✓0 p When paired with the divergence-free condition, we can upgrade u, u! from order ✏ to order 1: Z p ||u||2X + ||v||2B + ||P ||2L2 + ✏u2! r 1 . ||f, ✏g||2L2 . (2.2.48) ⇤, ⇤,2+ !=✓0 44 The existence of a unique solution to the linear problem (2.2.23 - 2.2.25) is then given by an application of Schaefer’s fixed point theorem: Theorem 2.2.15. Let us and vs be the approximate solutions as defined in equations (2.2.7, 2.2.8). Then there exists a unique solution [u, v, P ] to the system in (2.2.23 - 2.2.25) on the domain ⌦ together with the boundary conditions (2.2.30) which satisfies estimate (2.2.48) uniformly in ✏ and small ✓0 . Step V: High Regularity Estimates In this step, we obtain higher regularity estimates for solutions to the problem (2.2.23 - 2.2.25). To do so, we rewrite the equations (2.2.23 - 2.2.25) by moving the profile-dependent terms to the right-hand-side: p ✏ ✏ ✏ 2 3/2 1 uRR ur u!! + 2 u ✏ v! + P! = f˜, (2.2.49) pr r2 rp r2 r ✏ ✏ 2 ✏ ✏ 1 vRR vR v!! + 2 u! + 2 v + PR = g˜, (2.2.50) r r2 r r ✏ where ✓ p p ◆ 1 1 ✏ ✏ f˜ = f us u! + us! u + usR v + vs uR + vs u + us v , (2.2.51) r r r r ✓ ◆ 1 1 2 1 g˜ = g us v! + vs! u + vs vR + vsR v p us u . (2.2.52) r r r ✏ From this point of view, we formally expect high regularity estimates using the standard p theory of the Stokes equation: ||u, v||H˙ 2  ✏ M ||f˜, ✏˜ g ||L2⇤ for some potentially large value ⇤ M . This is only a formal estimate, however, because the corners of ⌦, (! = 0, R = R0 ) and (! = ✓0 , R = R0 ), obstruct the H 2 regularity of the standard Stokes problem. To account for this, we use the results of (Orl98) to recover H 3/2 regularity for the solutions near the corners. Precisely, the main result of this section is: 45 Lemma 2.2.16. The solutions u and v can be decomposed into u = u1 +u2 and v = v1 +v2 , where u2 , v2 are supported near the corners of the domain ⌦ in a region ⌦2 satisfying (!, R) 2 p ⌦2 implies R R0  1, r R0  ✏. The decomposition obeys the following estimates: p |u1 ||H˙ 2 + ||v1 ||H˙ 2 + ||u2 rm , v2 rm ||H 3/2 . ✏ M ||f˜, g ||L2⇤,2+ ✏˜ (2.2.53) ⇤,2+ ⇤,2+ for some possibly large value of M , where the constant is independent of ✓0 , and for m arbitrarily large. Step VI: Nonlinear Analysis In the final step, we use a contraction mapping to obtain existence and uniqueness of the + 12 nonlinear problem (2.2.23) - (2.2.30). Consider a sample nonlinear term, ✏ vuR , from equation (2.2.26). According to the right-hand-side of estimate (2.2.48), we must estimate the || · ||L2⇤,2+ norm of the nonlinearity: Z Z ⇣ ⌘2 ✓Z Z ◆ p1 ✓Z Z ◆ q1 + 12 + 12 ||✏ vuR ||2L2 = ✏ vuR r 2+ ✏ 2 +1 2p p 1+ p v r u2q Rr q+↵ . ⇤,2+ (2.2.54) We have used Holder’s inequality and we suppose for this discussion that the technical ↵ 1 parameter ↵ satisfies q p in order to make the above inequality valid. We think of p p as being very large and so q = p 1 is very close to 1. This calculation then motivates two features of our norm Z. RR R R p 2p First, Z must control u2p and ( ✏v) for large p, together with the appropriate choice of weights r. Typically, the H 1 (R2 ) ,! L2p (R2 ) embedding yields the desired control, but cannot be applied in our setting for two reasons. First, the standard embeddings apply 46 to integrals taken against the usual measure RdRd! and for the usual gradient operator @! r= R , @R , whereas in our setting the measure is r(R)dRd! and r is replaced by r⇤ . Second, we must precisely determine the weight of r(R) that can be controlled by our weighted energy norms, X and B. As such, we establish the required embeddings from scratch. The second ingredient which is built into Z are high regularity quantities, because as seen RR in estimate (2.2.54), Z must control |ru, r✏ v|2q . In order to close a contraction mapping argument, we must in turn control these high regularity quantities (see Definition 2.2.2). The main result in this direction, which serves as the driving force behind the contraction mapping argument, is: 0 Theorem 2.2.17. For u, v solutions to the system (2.2.23 - 2.2.25), there exists a >0 0 q such that if q = 1 + and p = q 1, we have: 1 p ✏ 4 ||uR ||L2q + ✏ 4 ||vR ||L2q + ✏ 4 ||u! ||L2q + ✏ 4 + 2 ||v! ||L2q . ||f˜, g ||L2⇤,2+ ✏˜ (2.2.55) ⇤,q+↵ ⇤,q+↵ ⇤,0 ⇤, 1 q q for all such that max{1 q , 2}   1, where > 0 and we can take 0 < p ↵ 2 . The essence of the proof of Theorem 2.2.17 is as follows: since 2q is only slightly larger than 2, we interpolate between the H⇤1 estimate in (2.2.48) which is uniform in ✏ and the high regularity estimates in Lemma 2.2.16, which scale poorly in ✏. Again, these interpolations p are highly sensitive to the weights r which can be controlled by ||f˜, ✏˜ g ||L2⇤,2+ , and are also taking place in our ⇤ spaces, and so must be developed from scratch. The required embedding theorems are proven in Section 2.3, and Theorem 2.2.17 is proved in Section 2.9. With Theorem 2.2.17 in hand, we are able to close a contraction mapping argument in the space Z, which we do in Section 2.10: Theorem 2.2.18 (Nonlinear Existence and Uniqueness in Z). For 2 (0, 1) sufficiently close to 1 there exists unique Navier-Stokes remainders (u, v, P ) to the system (2.2.23 - 2.2.29) such that ||u, v||Z < 1. 47 From here, the main result in Theorem 2.2.3 follows immediately. 2.3 Function Spaces and Embedding Theorems 2.3.1 Basic Properties of ⇤-spaces Here we establish a few basic facts which will be in use throughout our analysis: Lemma 2.3.1. Holder’s Inequality for Lp⇤, : For p, q Holder conjugates, ||uv||L1⇤,  ||u||Lp⇤, ||v||Lq⇤, Proof. Z Z Z Z Z Z ✓ ◆1/p ✓ ◆1/q r r r ||uv||L1⇤, = |uv|r = |uv| RdRd! = |uv| RdRd! R R R Z Z ✓ ◆1/p !p ! p1 Z Z ✓ ◆1/q !q ! q1 r r  |u|p RdRd! |v|q Rd!dR R R = ||u||Lp⇤, ||u||Lq⇤, . The third inequality above is the usual Holder inequality against the standard measure RdRd!. Lemma 2.3.2. The space Lp⇤, (⌦) endowed with the norm || · ||Lp⇤ , is a Banach Space for 1  p < 1. Z Z Z Z Z Z p p Proof. It is clear that u r = 0 () u = 0 and that (cu) r = c p up r = Z Z cp up r . We check the triangle inequality by using the corresponding triangle inequality 48 for the usual Lp norm which corresponds to the weight RdRd!: u+v /p u /p v /p ||u + v||Lp⇤, = || r ||Lp  || r ||Lp + || r ||Lp = ||u||Lp⇤, + ||v||Lp⇤, . R1/p R1/p R1/p We must argue that Lp⇤, (⌦) is complete under this norm. Suppose {un } is a Cauchy un sequence () { 1/p r /p } is Cauchy in the usual Lp norm, so there exists a limit function R un /p ¯ such that 1/p r /p ! u u ¯ = u Rr 1/p , so we have: ¯ in Lp . Define u R Z Z Z Z p un r /p |un u|p r dRd! = r /p u RdRd! ! 0. R1/p R1/p Lemma 2.3.3. The space L2⇤, (⌦) is a Hilbert space, endowed with the inner product (u, v) = Z Z uvr dRd!. Proof. By the Holder’s inequality (established above), the inner product is well defined as a mapping L2⇤, ⇥ L2⇤, ! R. Moreover, it is easy to see linearity, symmetry, and non- degeneracy of the inner product. By the previous lemma, this inner product induces a norm, and the space is complete with respect to this norm. In general, many properties of the usual Lp will be inherited by Lp⇤, because the map /p T : Lp⇤, ! Lp given by T (f ) = f Rr 1/p is a linear isometry. For instance, the characterization of the dual space to Lp⇤, follows trivially from this observation: ⇣ ⌘⇤ Lemma 2.3.4. Lp⇤, = Lq⇤, where the superscript ⇤ denotes (as always) the dual space. Proof. Given a bounded linear functional I : Lp⇤, ! R, I T 1 is a bounded linear functional Z Z Lp ! R, and is therefore given by f¯ ! f¯g¯RdRd! for some g¯ 2 Lq , where f¯ = 49 Z Z Z Z f¯g¯RdRd! and that p f Rr1/p . Letting g = g¯R1/q r q , we readily check f gr dRd! = ||g||Lq⇤, = ||¯ g ||Lq . This immediately implies reflexivity for 1 < p < 1, and thus we will be able to obtain weak subsequential limits from sequences bounded uniformly in ⇤, spaces in the usual manner. We’ll need a few more facts: Lp ⇤, Lemma 2.3.5. If un ! u for any 1  p < 1 and any weight  2 R, a subsequence a.e. u nk ! u. un p u  Lp a.e. Proof. Define u ¯n = 1/p ¯n ! u r and u = 1/p r p . Then u ¯, so a subsequence u ¯ nk !u ¯, R R a.e. which immediately implies unk ! u. Lemma 2.3.6 (Density of Cc1 in Lp⇤, ). For 1  p < 1 and any weight  2 R, we have that Cc1 is dense in Lp⇤, . f  Proof. Given an f 2 Lp⇤, , define f¯ = 1/p r p which is now in the usual Lp . By density, R Lp RR p n there exists ¯n ! f¯ () f¯ ¯n RdRd! ! 0. Now define 1/p r/p = ¯n , which R Lp⇤, immediately yields n ! f and moreover n 2 Cc1 (⌦) because R R0 . 2.3.2 Properties of Z, I In this subsection, we prove the first basic property of the space Z: Lemma 2.3.7. The space Z together with the norm ||u, v||Z defined above is a Banach space. 50 Proof. Nondegeneracy and homogeneity of the ||·||Z follows from the definition. The triangle inequality follows from applying it separately to each component, showing || · ||Z is a norm. We must verify completeness. We first show that weak derivatives, when they are elements of the space Lr⇤, for any r 2 and weight , are unique within this space. Suppose we have two weak radial derivatives u1R and u2R in Lr⇤, of u. Then Z Z u1R u2R d!dR = 0 for all 2 Cc1 (⌦). Since the support of is compact, we Z Z also have (u1R u2R ) r d!dR = 0 for all 2 Cc1 (⌦). Let s be the Holder conjugate to r, and select an arbitrary f 2 Ls⇤, = (Lr⇤, )⇤ . By Lemma 2.3.6 approximate f by n in the norm Ls⇤, . Z Z Z Z Z Z (u1R u2R )f r d!dR = (u1R u2R )(Ls⇤, lim) nr  = lim (u1R u2R ) nr  = 0. RR We have exchanged the Ls⇤, lim and by using Holder’s inequality: Z Z ✓Z Z ◆ r1 ✓Z Z ◆ 1s r s  u1R u2R ( n  f ) r d!dR  u1R u2R r  | n f| r Since the right-hand side goes to zero in the above inequality, we are able to switch the limit and integral. Thus, u1R u2R has operator norm 0, and the only such element is the 0 element, showing radial derivatives are unique within the class Lr for any r and . The choice of radial derivative as opposed to angular derivative was without loss of generality, so the above uniqueness result holds for angular derivative as well. Suppose {un } is Cauchy in Z. Then in particular {un } is Cauchy in L2⇤, , so there exists L2⇤, L2⇤, a limit u such that un ! u by completeness of L2⇤, . By the same argument, un! !v L2⇤, 1 and unR ! w. We must verify v = u! and w = uR . Let 2 CC (⌦). Then: Z Z Z Z Z Z Z Z w dRd! = L2⇤, lim unR = lim unR = lim un R 51 Z Z Z Z = L2⇤, lim un R = u R. We can exchange the L2⇤, limit and integral again by Holder’s inequality. Since uR is the unique element in L2⇤, satisfying the above equality, we have uR = w. The identical argument shows u! = v. We now turn to the ||uR ||L2q term. Again by completeness, there exists some limit ⇤,q+↵ L2q ⇤,q+↵ function, w (we will abuse notation), such that unR ! w. By passing to a subsequence, a.e. L2⇤, we can assume unR ! w. But we know from earlier that unR ! uR so a further subsequence must converge almost everywhere to uR . Since every subsequence of an a.e. converging sequence must also converge a.e. to the same limit function, we have w = uR . Since all of the weights above were done in full generality, we can repeat these arguments for all of the terms in the norm. This proves completeness since we have exhibited a single element u which serves as the Z limit of the Cauchy sequence un . Corollary 2.3.8. The spaces A, B, and X are individually Banach spaces. 2.3.3 Properties of Z, II: Low Regularity Embeddings We prove weighted embedding theorems which replace the usual H 1 ,! Lp Sobolev embed- ding in R2 . This style of argument will be applied repeatedly in this paper. For this section, we suppose that u, v satisfy the boundary conditions displayed in (2.2.30) and satisfy the divergence free condition in (2.2.25). ✓Z Z ◆ p1 p Lemma 2.3.9. ✏1/2 v p rp/2 1+ 2 dRd! . ||v||B for 2  p < 1. 52 Proof. First consider the case p = 2. In this case the exponent on the weight r is , and so we have the result by definition of ||v||B . Next, consider the case p = 4. We express: Z ! Z R v 4 r1+2 = v 2 r v 2 r1+ = @! (v 2 )r @y (v 2 r(y)1+ ) 0 R0 Z ! Z R Z ! Z R p ⇡ vv! r vvR r(y)1+ + vv! r v 2 ✏r(y) 0 R0 0 R0 Z ✓0 Z 1 Z ✓0 Z 1 p . |vv! |r |vvR |r1+ + ✏ |vv! |r v2 r . (2.3.1) 0 R0 0 R0 Integrating both sides in d!dR and applying Holder yields: Z Z Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 4 1+2 v r . v r 2 v!2 r 2 2+ vR r ✓Z Z ◆ 32 ✓Z Z ◆ 12 p 2 + ✏ v r v!2 r . (2.3.2) We now have ✏p/2 = ✏2 to distribute among the right hand side of the inequality, which yields the desired result. For p 2 (2, 4), we interpolate: ✓Z Z ◆ p1 ✓Z Z ◆ p1 p p 1 p 1+ 2+2 p 1 v r 2 2 dRd! = |vr | r dRd! ✓Z Z ◆ ✓2 ✓Z Z ◆14✓ 1 1  |vr 2+2 | r 2 1 dRd! |vr 2+2 4 | r 1 dRd!  ||v||B . (2.3.3) Now for p 4 we can proceed inductively via the calculation: Z ! Z R p p p p p p p p p p p p p p vp r 2 1+ 2 = v2r4 1+ 4 v2 r4+ 4 = v2 1 v! r 4 1+ 4 v2 1 vR r(y) 4 + 4 0 R0 53 Z ! Z R p p 1 p 1+ p p p p + ✏ v2 v! r 4 4 v 2 r(y) 4 + 4 1 . (2.3.4) 0 R0 Taking absolute value, integrating, and using Holder yields: Z Z Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 p p p 2 | vp r 2 1+ 2 | vp 2 r 2 1+ 2 (p 2) v!2 r 2 2+ vR r ✓Z Z ◆ ✓Z Z ◆ 12 ✓Z Z ◆ p p 2 p p p + ✏ vp 2 r 2 1+ 2 (p 2) v!2 r v2 r4+ 4 1 . (2.3.5) If p is an even integer, the absolute values on the left-hand side can be removed. We therefore establish the inequality for even integers successively starting at p = 6 (since p = 4 has been computed directly), and then interpolate in between. ✓Z Z ◆ p1 ✓Z Z ◆ p1 p 1 p Lemma 2.3.10. up r 2 r2 . ||u||X for 4  p < 1 and up r 2 . ||u||X for 2  p < 4. RR Proof. For p = 2, we have u2 r  ||u||2X by definition of the norm. We compute the case p = 4: Z Z Z ! ! R p R 4 2 + 12 2 2 1 2+ 1 2+ 2 1 u r =u r u r ⇡ uu! r uuR r(y) + ✏ u r(y) 2 0 R0 R0 Z ✓0 Z 1 Z ✓0 Z 1 1 1  |uu! |r |uuR |r 2 + + |uu! |r u2 r 2 . (2.3.6) 0 R0 0 R0 Integrating both sides over d!dR and applying Holder’s inequality yields: Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 ✓Z Z ◆1/2 ✓Z Z ◆1/2 + 12 u4 r 2 . u2 r u2! r u2 r u2R r1+ 54 ✓Z Z ◆1/2 ✓Z Z ◆1/2 ✓Z Z ◆ 1 2 + u r u2! r u r 2 2  ||u||4X . (2.3.7) We now interpolate for p 2 (2, 4): ✓Z Z ◆ p1 ✓Z Z ◆ ✓2 ✓Z Z ◆14✓ p 2 4 2 |ur 2 | dRd!  u r dRd! u r dRd!  ||u||X . Once the above estimate for p 2 (2, 4) has been established, the desired estimate can be inductively established via: Z ! Z R 1 p p p p p 1 p p p p 1 up r 2 + 2 = u2 r 4 u2 r 4 +2 = u2 1 u! r 4 @R (u 2 r(y) 4 +2 ) 0 R0 Z ! Z R Z ! Z R p 1 p p 1 p 1 p p p p p 1 = u2 u! r 4 u2 uR r(y) 4 +2 + ✏ u2 1 u! r 4 u 2 r(y) 4 2 . 0 R0 0 R0 (2.3.8) Taking absolute values and applying Holder’s inequality yields: Z Z Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 1 p up r 2 + 2 . up 2 r 2 (p 2) u2! r u2R r1+ Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 p p p 1 + ✏ u r 2 4 2 u2! r u p 2 r 2 (p 2) . (2.3.9) For p 4 all of the quantities in the above estimate are inductively controlled by powers of ||u||X . 55 Remark. This argument is reminiscent of the proof of the classical Gagliardo-Nirenberg- Sobolev inequality. This method can be used to yield a direct proof of the standard H 1 ,! Lp embedding in R2 by replacing the weights r by R. The advantages of the direct approach above is the avoidance of defining the fractional Sobolev spaces H s and consequently the avoidance of appealing to the Fourier Transform. Indeed, in the classical case, one must argue H 1 ,! H s ,! Lp for 0  s < 1 and 2  p < 1 because the Sobolev exponent is critical. The drawbacks are that this method must take into account the behavior of u on the boundary @⌦, and that it doesn’t directly apply to more complex domains. We now prove an embedding of the type Z ,! L1 , from which Corollary 2.2.4 follows directly: 0 1 Lemma 2.3.11. Given > 0, let q = 1 + 0 . For 1+ 0   1, 1 1 1 ✏ 2 + 4q ||u||L1 + ✏ 2 + 4q + 2 ||v||L1 . ||u, v||Z . (2.3.10) Proof. Since 2q = 2(1 + 0 ) > 2, by Morrey’s Inequality: ✓Z Z ◆ 2q1 ✓Z Z ◆ 2q1 |u(¯ ¯ . ||ru||L2q + ||u||L2q = ! , R)| 2q |ru| RdRd! + 2q |u| RdRd! . (2.3.11) 1 Multiplying by ✏ 4q + 2 yields: ✓Z Z ◆ 2q1 ✓Z Z ◆ 2q1 1 ✏ 4q + 2 |u(¯ ¯ . ✏2 ! , R)| |ru|2q rdRd! + ✏2 |u|2q rdRd! ✓Z Z ◆ 2q1 ✓Z Z ◆ 2q1 . ✏2 |r⇤ u|2q rdRd! + ✏2 |u|2q rdRd! . ||u||Z . (2.3.12) ⇣u ⌘ ⇣u ⌘ ! ! where we have used |ru| = | , uR | . | , uR | = |r⇤ u|. The condition 1 1+ 0  R r 56 ensures ||u||L2q  ||u||L2q  ||u||Z by Lemma 2.3.10. The proof for v works identically, ⇤,1 ⇤, q 1 1 ⇣R R q ⌘ 2q1 where the extra factor of ✏ 2 is required as the ||v||Z contains ✏ 2 + 2 v!2q r 2p . 1 Remark. Note that the condition in Definition 2.2.2, 1 2p  < 1, implies the condition 1 1+ 0   1. 2.3.4 Properties of Z, III: High Regularity Embeddings In this subsection, we provide careful estimates which will yield control of the high regularity quantities appearing in || · ||Z . Throughout this section, u, v are assumed to satisfy the boundary conditions displayed in (2.2.30) and the divergence free condition in (2.2.25). Lemma 2.3.12. Let 2 [ 12 , 1]. There exists a 0 > 0 such that if q = 1 + 0 , then 1 1 1 1 ||uR ||L4 . ||u||X 2 ||u||H 2 ˙2 ; and ||vR ||L4 . ||v||X 2 ||v||H 2 ˙2 (2.3.13) ⇤,2+ 2↵ ⇤,2+ ⇤,2+ 2↵ ⇤,2+ q q q q 0 q 1+ for all 0  ↵  2 . Moreover, ↵ can be selected such that p = p ↵ 2 , where p is 0 the Holder conjugate of q by taking small enough. 1 p Proof. After noticing that v = vR = 0 on the boundary {! = 0} and vR = r (u! + ✏v) =0 on the boundary {R = R0 }, the u and v estimates follow in an identical manner, so we focus on u. We express: ↵ 0 1 0 3 u4R r2+2 q = u4R r2+↵ = u2R r 2 +↵ u2R r 2 . (2.3.14) 1 3 Since 2, the quantity u2R r 2 is integrable and lies in the Sobolev space W 1,1 (R+ ) for each fixed ! by the definition of H˙ ⇤,2+ 2 , implying that this quantity decays at 1. This 57 enables us to write: Z 1 Z 1 Z 1 p u2R r3/2 = @R (u2R (y)r3/2 )dy  |uR uRR |r3/2 dy + ✏ |u2R |r1/2 dy. (2.3.15) R R0 R0 We also note that u = uR = 0 on the boundary ! = 0. Thus, we are able to write: Z ! Z ✓0 1 0 1 0 0 u2R r 2 +↵ = uR uR! r 2 +↵  |uR uR! |r1/2+↵ . (2.3.16) 0 0 Multiplying the previous two inequalities, integrating and applying Holder’s inequality yields: Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 ✓Z Z ◆1/2 ✓Z Z ◆1/2 0 0 0 u4R r2+↵ . u2R r1+↵ u2RR r2 u2R! r↵ u2R r . ||u||2X ||u||2H˙ 2 , (2.3.17) ⇤,2+ where we use that ↵0 = 2 ↵q  . Taking fourth roots yields the result. 1 1 Lemma 2.3.13. There exists a 0 such that for q = 1 + 0 , ||u! ||L4⇤,0 . ||u||X 2 ||u||H 2 ˙2 for ⇤,2+ 2 [ 12 , 1]. Proof. Writing u4! = u2! u2! = u2! r u2! r and recalling that u = u! = 0 on R = R0 , enables us to write: Z R Z R Z R p u2! r = @R (u2! r )dy = r u! u!R + ✏ u2! r 1 . (2.3.18) R0 R0 R0 Using the divergence free condition, u! = @R (rv), we have u! = 0 on ! = 0. Therefore 58 we can write: Z ! Z ! 3 u2! r = u! u!! r = u! r 2 u!! r 2 . 0 0 Multiplying the two equalities above together, taking absolute values, and applying Holder yields: Z Z ✓Z Z ◆ ✓Z Z ◆1/2 ✓Z Z ◆1/2 u4!  u2! r u2!! r 3 u2!R r  ||u||2X ||u||2H˙ 2 ⇤,2+ (2.3.19) 1 where we have used 2  1) 3  2+ . p Lemma 2.3.14. There exists 0 such that for q = 1 + 0 , we have ✏||v! ||L4 2 . ⇤, q 1 1 ||v||B ||v||H˙ 2 2 2 for any > 0 and 1 q   1. ⇤,2+ 2 0 0 0 Proof. Temporarily writing = q , we proceed to write: v!4 r = v!2 r r v!2 r . Using that v = v! = 0 on the boundary R = R0 , we can write Z R Z R Z R p v!2 r = @y (v!2 r )dy = v! v!R r + ✏ r 1+ v!2 . R0 R0 R0 r Next, we recall the boundary conditions at ! = ✓0 are v! = uR ! v!2 (✓0 , R) = ✏ r2 uR (✓0 , R)2 . As such, we write: ✏2 0 Z ✓0 0 r2 0 v!2 r r = uR (✓0 , R) + 2 v! v!! r . ✏2 ! 59 Taking absolute values and multiplying the previous two equalities together yields: 0 Z 1 Z ✓0 Z 1 0 r2 0 v!4 r . uR (✓0 , R)2 |v! v!R |r dy + |v! v!! |r d! |v! v!R |r dy ✏2 R0 0 R0 0 Z 1 Z ✓0 Z 1 r2 2 p 0 p + uR (✓0 , R) ✏ v!2 r 1+ + |v! v!! |r d! ✏ v!2 r 1+ dy. ✏2 R0 0 R0 Integrating over d! and dR yields: Z Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 0 1 1 2 0 v!4 r . 2 r 2 uR (✓0 , R) dR 2 v! r 2 v!R r ✏ R0 ✓Z Z ◆1/2 ✓Z Z ◆1/2 ✓Z Z ◆1/2 ✓Z Z ◆1/2 2 3 2 0 2 2 + v! r v!! r v! r v!R r Z 1 Z Z 1 0 + 3/2 r2 u2R v!2 r 1+ ✏ R0 ✓Z Z ◆ 12 ✓Z Z ◆ 12 Z Z p 0 + ✏ v!2 r 2 v!! r 3 2 v!2 r 1+ . (2.3.20) Z ✓0 As u = uR on the boundary ✓ = 0, we can write uR (✓0 , R) = uR! ) uR (✓0 , R)2 . Z 0 ✓0 u2R! . Inserting this into (2.3.20): 0 Z Z Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 0 1 1 ✓0 2 0 v!4 r . r u 2 R! v 2 ! r v 2 !R r ✏ 2 R0 0 ✓Z Z ◆1/2 ✓Z Z ◆1/2 ✓Z Z ◆1/2 ✓Z Z ◆1/2 0 + v!2 r v!! r 3 2 v!2 r 2 v!R r Z Z Z Z 1 0 + 3/2 r2 u2R! v!2 r 1+ ✏ ✓Z Z ◆ 12 ✓Z Z ◆ 12 Z Z p 2 2 3 2 0 + ✏ v! r v!! r v!2 r 1+ . (2.3.21) We multiply estimate (2.3.21) by ✏2 , and according to the weights we are able to estimate 60 for the H˙ ⇤,2+ 2 terms, we require: 0 0 0 2  () 2 2 () 1  , (2.3.22) 2 and 0 0 0 1 3 2  2 () 2 2 4 ()  . (2.3.23) 2 2 The lemma has been proved. Remark. The above estimate for v! is the most delicate of the high-order estimates as it relies on the stress-free boundary condition placed at the boundary {! = ✓0 }. For our q purposes in Section 2.10, we will take = 2p in which case the valid interval for is 1 2p   1. 2.4 Construction of Approximate Solutions In this section, we construct the approximate solutions in the expansion (2.2.7, 2.2.8), and estimate the corresponding errors Ru and Rv . First, we record the errors Ru , Rv , which are obtained by inserting the expansion (2.2.7 - 2.2.9) into the scaled NS system (2.2.6): Angular Error, Ru : 1 0 1 0 ✏0 order error: (u + u0p )u0p! + (vp0 + ve1 )u0pR + Pp! u0pRR ; (2.4.1) r e r 1 0 0 1 1 u u u v P ✏1/2 order error, Euler: e! e + ve1 u0er + e e + e! ; (2.4.2) r r r 1 u0e u1p! u0p ✏1/2 order error, BL: (u1e + u1p )u0p! + + (u1e! + u1p! ) + vp0 (u0er + u1pR ) r r r 0 0 1 p u v e p 1 Pp! 1 0 + ve1 u1pR + ✏u0er vp1 + vp1 u0pR + + u0p (vp0 + ve1 ) + u1pRR u ; (2.4.3) r r r r pR 61 1 1 ✏1 order error: (u + u1p )@! (u1e + u1p ) + (vp0 + ve1 )u1er + vp1 u1pR (2.4.4) r e 1 1 1 1 0 + (u0e + u0p )vp1 + (u1e + u1p )(vp0 + ve1 ) + PP2 ! u0err (u + u1pR ) r r r r er 1 0 1 u + (u0 + u0p ); r2 p!! r2 e 1 1 1 1 1 ✏3/2 order error: vp1 u1er + (u1e + u1p )vp1 u1err u (u + u1p!! ) (2.4.5) r r er r e!! 1 2 0 + 2 (u1e + u1p ) (v + ve!1 ); r r2 p! 2 1 ✏2 order error: 2 vp! . (2.4.6) r After splitting into Euler and Boundary Layer variables, we have the following errors for the Radial component: Radial Errors, Rv : 1 ✏ order error, BL: PP0 R ; (2.4.7) 1/2 0 (u0e )2 ✏ order error, Euler: Per ; (2.4.8) r (u0p )2 u0e u0p ✏ 1/2 order error, BL: PP1 R 2 ; (2.4.9) r r u0 1 u1 u0 ✏0 order error, Euler: Per 1 + e ve! 2 e e; (2.4.10) r r 1 ✏0 order error, BL: PP2 R + u0 v 0 + u0p @! vp0 + ve1 + (vp0 + ve1 )vpR 0 (2.4.11) r e p! 2 (u1e + u1p )u0p + u1p u0e 0 vpRR ; (2.4.12) r 1 1 1 2 ✏1/2 order error, Euler: u1e ve! 1 + ve1 ver 1 (u ) ; (2.4.13) r r e 1 1 0 1 ✏1/2 order error, BL: ue vp! + u1p @! (vp0 + ve1 ) + ((u0e + u0p )vp! 1 ) + vp1 vpR 0 (2.4.14) r r 1 1 0 2 + vp0 (ver 1 1 + vpR ) + ve1 (vpR 1 ) (2u1e u1p + (u1p )2 ) v + (u0 ) vpRR 1 ; (2.4.15) r r pR r2 p! 1 1 1 1 2 1 ve1 ✏1 order error, Euler: 1 verr ver v e!! + (u e! ) + ; (2.4.16) r r2 r2 r2 1 1 1 1 1 0 2 ✏1 order error, BL: u1p vp! 1 + u1e vp! 1 + vp1 ver1 + vp1 vpR 1 vpR 2 vp!! + 2 u1p! (2.4.17) r r r r r 1 0 + 2 vp ; (2.4.18) r 62 1 1 ✏3/2 order error: v . (2.4.19) r2 p!! 2.4.1 Prandtl-0 Layer We obtain the Prandtl-0 layer equations from the ✏0 order angular error, equation (2.4.1), 1 and the ✏ order radial error, equation (2.4.7) . We also enforce the divergence free condi- tion. Thus, after dropping the subscripts, the Prandtl-0 layer equations are the following: (u0e + u)u! + r(ve1 + v)uR + P! = ruRR , (2.4.20) p u! + ✏v + rvR = 0, PR = 0. The boundary conditions we take are: u(!, R0 ) = ub ue ; u(0, R) = u ¯0 (R); and v(!, R0 ) = ve1 (!, R0 ). As will be shown rigorously in Theorem 2.4.1, the Prandtl-0 profiles u, v decay sufficiently rapidly, and so evaluating the equation above at R = 1 yields P! = 0. This implies the Prandtl pressure is constant when coupled with PR = 0. We rewrite the first equation as: (ue + u) u! + rv + R0 ve1 (!, R0 ) uR = R0 uRR + e0 + e1 + e2 (2.4.21) ) (ue + u) u! + (rv R0 v(!, R0 )) uR = R0 uRR + e0 + e1 + e2 . Here we have defined: p p e0 := ✏(R R0 )u0er (r)u! + ✏(R R0 )@r (rve1 (!, r))uR , (2.4.22) p e1 := ✏(R R0 )uRR , (2.4.23) 63 Z R Z R p p e2 := u! ✏ u0er (r(⌘)) u0er (r) d⌘ + uR ✏ @r (r(✓)ve1 (!, r(✓))) @r (rve1 (!, r))d✓ R0 R0 Z R Z ⌘ Z R Z ⌘ = ✏u! u0err (r(✓))d✓d⌘ + ✏uR @r2 (r(✓)ve1 (!, r(✓)))d✓d⌘. (2.4.24) R0 R R0 R e2 is high order in ✏, as will be demonstrated in a later section. e0 and e1 will be put into the Prandtl-1 layer, and so we are left with solving (ue +u)u! +(rv R0 v(!, R0 ))uR = R0 uRR together with the divergence free condition and the boundary conditions described above. Z 1 Z 1 To satisfy the divergence free condition, we take rv(!, R) = @R (rv) = u! , which R R ensures the profile v decays at 1. We have the following: ¯0 } Theorem 2.4.1. Suppose min{ub , ue + u c0 > 0. Then for ✓0 sufficiently small, there exists a unique solution u0p (!, R) such that: sup ||Rn/2 @!k u0p ||L2 (R+ ) + ||Rn/2 @!k @R u0p ||L2 (0,✓0 ),L2 (R+ )  C. (2.4.25) [0,✓0 ] Corollary 2.4.2. j sup ||Rn/2 @!k @R [u0p , vp0 ]||L2 (R+ )  C. (2.4.26) [0,✓0 ] Proof. The proof follows exactly as in (GN17), using the von-Mises transformation. Let Z R 0 ue := ue (R0 ), and define ⌘(!, R) = (ue + u0p (!, y))dy, ↵(!, ⌘) = ue + u0p (!, R(⌘)), where R0 we’ve used that for each fixed !, the transformation (!, R) ! (!, ⌘(!, R)) is invertible by appealing to the maximum principle. We now compute: ⌘R = ue + u0p (!, R) = ↵, Z R Z R ⌘! = u0p! = @R (r(y)v)dy = R0 v(!, R0 ) rv(!, R), R0 R0 u0p! = ↵! + ↵⌘ ⌘! = ↵! + ↵⌘ (R0 v(!, R0 ) rv(!, R)), u0pR = ↵⌘ ⌘R = ↵↵⌘ . 64 Inserting these identities into our equation: ↵↵! + ↵↵⌘ (R0 v(!, R0 ) rv(!, R)) + (rv R0 v(!, R0 ))↵↵⌘ = R0 ↵↵⌘2 + ↵2 ↵⌘⌘ (2.4.27) ) ↵! = R0 (↵↵⌘ )⌘ . On the parabolic boundary of our domain, ↵(0, ⌘) = ue + u0p (0, R(⌘)), and ↵(!, 0) = ue + u0p (!, 0) = ub , both of which are strictly positive by assumption. Using the Parabolic maximum principle, ↵ C > 0 for some constant C, and therefore our equation is non- degenerate. The equation (2.4.27) along with the boundary conditions is identical (apart from the constant R0 ) to that in (GN17), and so the rest of the proof follows in the same manner. e0 , e1 will be solved for in the Prandtl-1 layer, and so the contribution to the angular error is e2 , given in (2.4.24). 2.4.2 Euler-1 Layer The equations for the Euler-1 Layer arise from equations (2.4.2) and (2.4.10) together with the divergence free condition. We drop the subscript for u, v and P within this section, with the understanding that the unknowns appearing are that of the Euler-1 layer. The equations read: u0e 2 0 u0e u0 1 v! u u + Pr = 0, u! + u0er v + e v + P! = 0, u! + v + rvr = 0. r r e r r r (2.4.28) 65 As described in (2.2.12) - (2.2.18), the boundary conditions are as follows: v(!, R0 ) = vp0 (!, R0 ), v(0, r) = Vb0 (r), v(✓0 , r) = Vb1 (r). (2.4.29) We go to the vorticity formulation in order to eliminate the pressure term: u0e 2 0 0 = @r u0e u! + ru0er v + u0e v + P! @! ( v! u u + Pr ) r r e = u0er u! + u0e u!r + u0er v + ru0err v + ru0er vr + u0e vr + u0er v + P!r u0e 2 v!! + u0e u! Pr! r r u0e 2 = u0e u!r + ru0err v + u0e vr + u0er v v!! + u0e u! r r 0 0 0 0 u0e 2 = ue ( 2vr rvrr ) + ruerr v + ue vr + uer v v!! + u0e u! ✓ r◆ r 2 = u0e r v + ru0err + u0er v + u0e 2vr v := u0e Lv, (2.4.30) r where we have defined the linear operator L through equation (2.4.30) for ease of nota- tion, and since u0e > 0, 0 = Lv if and only if 0 = u0e Lv. Define the following boundary layer corrector: ✓ ◆ ! vp0 (!, 0) ! vp0 (!, 0) B(!, r) = 1 V b0 (r) + Vb1 (r). (2.4.31) ✓0 vp0 (0, 0) ✓0 vp0 (✓0 , 0) Due to the compatibility conditions, B satisfies the same boundary conditions as v. Define ✓ ◆ ✓ ◆ ru0err u0er 2 F (!, r) = r B+ + B+ 2Br B = LB (2.4.32) u0e u0e r Since B and all of its derivatives decay exponentially fast, and since by assumption |@r (Vb0 Vb1 )| . ✓0 we have ||hrik F ||W k,p  C where C independent of ✓0 . Next, for a 66 cuto↵ function supported on [0, 1], define w ✓0 ! Eb (!, r) = ( )F (0, r) + ( )F (✓0 , r), for ✏ << ✓0 . (2.4.33) ✏ ✏ 1 Since each di↵erentiation of the cuto↵ function gives ✏ , it is easy to see that ||hrin @!k Eb ||Lq  k+ q1 C✏ . We also record for future use that @!k Eb |!=0,✓0 = 0. Consider w = v B, then Lw = Lv LB. In (2.4.67), v solves Lv = Eb instead of Lv = 0 and the error made by this is accounted for in (2.4.72). Therefore ✓ 0 ◆ ✓ ◆ u u0er 2 Lw = r w+ r err + w+ 2wr w = Eb F := f ; w = 0 on @⌦ (2.4.34) u0e u0e r Since B is arbitrarily high regularity, obtaining estimates for w suffices to obtain esti- mates for v. Eb = F on {! = 0, ✓0 } implies f |!=0,✓0 = 0. 2.4.2.1 H 1 Estimates Multiplying (2.4.34) by w and integrating by parts yields: Z Z Z Z Z Z ✓ 0 ◆ wr w!! 2 2 ruerr u0er r(wrr )w 2wwr = fw + w + 0 w2 r r2 r u0e ue Z Z 2 Z Z 2 Z Z 2 w w w ) rwr2 + !  N ( ¯) f 2r + ¯ + ||u0err + u0er rm ||1 r r r Z Z Z Z 2 w  N ( ¯) f 2 r + ( ¯ + ✓02 ) ! . (2.4.35) r We have used the rapid decay of the derivatives of u0e . For ✓0 sufficiently small, we obtain Z Z Z Z 2 Z Z w!2 w 2 rwr + + . f 2 r, where the constant does not depend on small ✓0 . r r We obtain weighted estimates, ||rn w||H 1 , for n 1 by testing the above equation against 67 rn w: Z Z Z Z Z Z Z Z Z Z ru0 u0er n rn+1 wr2 + rn 1 w!2 = f wrn ( err + )r w + 2 w r wr n + 2 w2 rn 1 u0e u0e Z Z Z Z Z Z . f 2 rn+1 + ¯ w 2 r n 1 + ✓0 w!2 rn 1 ) Z Z Z Z wr2 rn+1 + w!2 rn 1 2 n 1 +w r . f 2 rn+1 . (2.4.36) With H 1 estimates in hand, we can establish existence and uniqueness. Again to ease notation, we define ✓ ◆ ✓ ◆ ˜ := u0 u0er 2 Lw r w=f r err + w+ w + 2wr := g. (2.4.37) u0e u0e r Lemma 2.4.3. There exists a unique solution w 2 H 1 to Lw = f in ⌦, w|@⌦ = 0. Proof. First, consider the following problem posed on the bounded domain ⌦N = {! 2 (0, ✓0 ), R 2 (R0 , R0 + N )}: ˜ (N ) = g on ⌦N , Lw w(N ) |@⌦N = 0, (2.4.38) where @⌦N includes an additional boundary component, D = {R = R0 + N }. It is clear that w(N ) obeys H 1 estimates given above uniformly in N , so once each w(N ) has been constructed, we can send N ! 1. We first note the following version of the fundamental positivity estimate: Z Z Z Z Z Z w 0 2 w w rwr2 = r|@r ( u )| = r|@r ( 0 )u0e + 0 u0er |2 u0e e ue ue Z Z Z Z Z Z ✓ ◆ w (u0 )2 w 2 = r|@r ( 0 )|2 (u0e )2 + rw2 er + ru 0 0 u @ e er r ( ) ue (u0e )2 u0e 68 Z Z Z Z ✓ ◆ w 2 0 2 u0err u0 = r|@r ( )| (ue ) r + er w2 (2.4.39) u0e ue0 u0e which, due to an identical calculation to (2.2.36), yields: Z Z Z Z Z Z ✓ ◆ u0err u0 rwr2 . rwr2 + r 0 + er w2 . (2.4.40) ue u0e RR R R ⇣ u0err u0 ⌘ ˜ Define the bilinear forms K[w, '] := rw · r'rdrd! + r u0 + uer0 w', and e e RR 2 ˜ K[w, '] := K[w, '] ˜ r w'. Using the positivity estimate, it is clear that K satisfies the hypothesis of Lax-Milgram, but due to lack of coercivity we cannot directly apply Lax- Milgram to K. RR RR RR Note K[w, '] = ˜ f ' () K[w, '] = f' + 2 () w = T 1 (f + 2r w) r w' where T 1 ˜ () w is the solution operator to K T 1 2w ( r ) =T 1 (f ). T 1 is compact and self-adjoint, so the Fredholm alternative applied to the operator I (2 r· ) enables us to T 1 RR conclude there either exists a unique solution to the original problem K[w, '] = f ' or there exists a nontrivial kernel. The latter option is ruled out by the H 1 estimates given above. 2.4.2.2 H 2 -H 4 Estimates Our starting point is equation (2.4.37). The boundary layer corrector is defined such that: w!! = 0 on {! = 0, ✓0 }. (2.4.41) Di↵erentiating (2.4.37) in ! gives the third-order equation with Neumann boundary conditions: r w! = g! , w!! |@⌦ = 0. (2.4.42) 69 Applying the multiplier w! and noting that g|!=0,✓0 = 0 yields: Z Z 2 Z Z Z Z Z Z 2 w!! w!! 2 rwr! + . g! w! = gw!! . ||rw||H 1 + ||rf ||L2 + ¯ . r r (2.4.43) Applying the weighted multiplier rn w! gives weighted estimates inductively. By using (2.4.37) to express wrr in terms of the rest, we have the full H 2 estimate: ||rn w||H 2  C. (2.4.44) We di↵erentiate the equation (2.4.42) again in !, giving the Dirichlet problem for w!! : r w!! = g!! , w!! |@⌦ = 0. (2.4.45) Multiplying (2.4.45) by w!! gives: Z Z Z Z w2 2 rw!!r + !!!  ||w||2H 2 g! w!!!  C + ||f! rn ||2L2 . ✏ 1 . (2.4.46) r Weighted estimates are obtained inductively via the multiplier rn w!! . The estimate for wrr! is obtained via equation (2.4.42), and the estimate for wrrr is obtained by di↵eren- tiating equation (2.4.37) in r to write wrrr in terms of the other third order terms. This gives: ||rn w||H 3 . ✏ 1/2 . (2.4.47) Evaluating (2.4.45) at ! = 0, ✓0 gives the boundary condition w!!!! = rF!! on {! = 0, ✓0 }. (2.4.48) 70 Di↵erentiating (2.4.45) gives the fifth-order equation: r w!!! = g!!! , (2.4.49) to which we apply the multiplier w!!! : Z Z Z Z Z Z Z Z 2 w!!!!! w!!! 2 w!!!! rwrr!!! w!!! = rwr!!! + (2.4.50) r r Z !=✓0 1 w!!! w!!!! (2.4.51) !=✓0 r !=0 For the boundary terms, we estimate: Z !=✓0 Z !=✓0 Z Z 1 w!!! w!!!! dr = w!!! F!! dr = w!!!! F!! + w!!! F!!! (2.4.52) r !=0 !=0 1 . ¯|| w!!!! ||2L2 + ||w||H 3 + C (2.4.53) r On the right-hand side of (2.4.49), we have up to harmless factors: Z Z ⇣ ⌘ Z Z Z !=✓0 w!!! f!!! + w!!! + w!!!r  w!!!! f!! + w!!! F!! dr (2.4.54) !=0 1 + ¯||w!!!r ||2L2 + ||w||2H 3  ¯||w!!!r ||2L2 + ¯|| w!!!! ||2L2 + ✏ 3 . (2.4.55) r Again, these calculations may be repeated using the weighted multiplier, rn w!!! , to obtain weighted estimates. Putting the above calculations together gives: Z Z rn 1 2 w!!!! + rn+1 wr!!! 2 .✏ 3 . (2.4.56) 71 The estimate for wrr!! can be obtained from (2.4.45), the estimate for wrrr! may be obtained by di↵erentiating (2.4.42) in r, and finally the estimate for wrrrr may be obtained by di↵erentiating (2.4.37) twice in r, ultimately yielding: ||rn w||H 4 . ✏ 3/2 . (2.4.57) 2.4.2.3 W k,q Estimates W k,q estimates are obtained using the framework of Agmon-Douglis-Nirenberg, (ADN59), where k  4 and q 2 (1, 1). To do so, cover the interior of the boundary {r = R0 } using one open set, U c , the boundaries {! = 0, ✓0 } using U a , U b and the interior of the domain using U d . Let a,b,c,d denote the partition of unity associated to this covering, and wa,b,c,d := a,b,c,d w. wa,b,c,d satisfy the equation: X ✓ 0 ◆ ✓ ◆ w!! u u0er 2 X r wX = X rwrr wrX = X f + r err + w X + 2w X r + w r u0e u0e r X X X X 2 X !! 2r r wr r rr w 3 r w ! w! w =: f X , X = a, b, c, d. (2.4.58) r r The estimates for wd follow from the standard interior W k,q estimates: ||wd ||W 2,q (U d ) . ||f ||Lq (⌦) + ||w||W 1,q (⌦)  ||f ||Lq (⌦) + ||w||H 2 (⌦)  C(✓0 ), (2.4.59) 1+ q1 ||wd ||W 3,q (U d ) . ||f! ||Lq (⌦) + ||w||W 2,q (⌦)  C(✓0 )✏ , (2.4.60) 2+ q1 ||wd ||W 4,q (U d ) . ||f!! ||Lq (⌦) + ||w||W 3,q (⌦)  C(✓0 )✏ . (2.4.61) C(✓0 ) is a constant that could depend poorly on ✓0 . The weighted estimates, ||rn wd ||W k,q (U d ) , are obtained by using the weighted H k estimates. wc is supported away from the corners of the domain, and so we can repeat a similar analysis as for wc , remaining 72 cognizant of the boundary condition @!k wc |r=R0 = 0 for k 0. It remains to estimate wa,b which follow by taking odd angular extensions across the boundaries {! = 0} and {! = ✓0 }. For concreteness, we proceed to treat the wa case, with the wb estimate being identical. Define: ˜ a (!, r) = w wa ( !, r) for ! 2 ( ✓0 , 0), ˜ a = wa for ! 2 (0, ✓0 ) w Applying a cuto↵ function: ✓0 ✓0 ¯ a = (!)w w ˜a , supp( ) ⇢ ( , ) ⇥ (R0 , 1) (2.4.62) 2 2 ¯ a satisfies the boundary-value problem: ensures that w !! ! ¯a = r w r ˜ a ) = (!)f˜a + ( (!)w ˜a + 2 w ˜!a , w (2.4.63) r r ¯ a |R=R0 = w w ¯ a |!= ✓0 ,✓0 = 0, (2.4.64) where f˜c is the odd angular extension of f c . Moreover, w ¯ a 2 W 4,q whenever wa 2 W 4,q by the condition w = w!! = 0 at ! = 0. Applying the standard W 2,q estimates to the boundary-value problem in (2.4.63) gives: ¯ a ||W 2,q  C(✓0 ). ||w (2.4.65) ¯ a vanishes on a neighborhood We can di↵erentiate the equation (2.4.63) in ! twice as w of {! = ✓0 , ✓0 }, and repeatedly apply the W 2,q estimates. The full W 3,q estimate is then recovered using the same procedure as in estimate (2.4.47), and the full W 4,q estimate on wc is recovered using the same procedure as in estimate (2.4.57). Combined with the estimates on wc,d we have established: 73 Lemma 2.4.4 (W k,q estimates, q 2 (1, 1)). 1+ q1 2+ q1 ||w||W 2,q . C(✓0 ), ||w||W 3,q . C(✓0 )✏ , ||w||W 4,q . C(✓0 )✏ , (2.4.66) where C(✓0 ) depends poorly on ✓0 . 2.4.2.4 Construction of Euler-1 Layers The Euler-1 layers are defined to solve: ✓ ◆ 2 1 u0e r ve1 + ru0err + u0er ve1 + u0e 2vr v = u0e Eb , (2.4.67) r e Z ! u1e (!, r) = u1e (0, r) @r (rve1 )d✓. (2.4.68) 0 The pressure Pe1 is defined to solve equation with Pr in (2.4.28) exactly: Pe1 (!, r) = R1 2 0 u0e r r ue u r v! . The error made in the P! equation in (2.4.28) is estimated as: u0e u! + ru0er v + u0e v + P! (2.4.69) Z 1 ✓ ◆ 2 0 2u0e = u0e u! + ru0er v + u0e v u u! + u0e Eb + u0e (rvrr ) + 3u0e vr + ru0err u0er v. r r e r By direct computation: @r (u0e u! ) = u0er v ru0er vr 2u0e vr ru0e vrr , (2.4.70) @r (ru0er v) = u0er v + ru0err v + ru0er vr , @r (u0e v) = u0er v + u0e vr . Each of the three terms above is known to be in H 1 , and therefore decay at infinity. We 74 can write: Z 1 u0e u! + ru0er v + u0e v = @r (u0e u! + ru0er v + u0e v). (2.4.71) r Matching these terms with those in the integral in equation (2.4.69), the only term remaining is: Z 1 u0e (✓)Eb (!, ✓)d✓. (2.4.72) r This represents the second contribution to the angular error. The estimates for the Euler-1 layer are summarized: Theorem 2.4.5 (Euler-1 Profile Estimates). 1 3 ||rn ve1 ||1 + ||rn ve1 ||H 2  C, ||rn ve1 ||H 3  C✏ 2 , ||rn ve1 ||H 4  C✏ 2 ; (2.4.73) 1+ q1 2+ q1 ||rn ve1 ||W 2,q  C(✓0 ), ||rn ve1 ||W 3,q  C(✓0 )✏ , ||rn ve1 ||W 4,q  C(✓0 )✏ , (2.4.74) where C(✓0 ) could depend poorly on ✓0 , for q = (1, 1). By definition of u1e , we have: ||rn u1e ||H 1  C, ||rn u1e ||1  C. (2.4.75) Proof. Only the uniform bound on u1e must be proven. To do so, we use the following: Z ! 2 Z ✓0 2 |u1e (!, r)|2 = @r (rv)(✓, r)d✓ . |@r (rv)| (✓, r)d✓ ) 0 0 Z ✓0 Z ✓0 2 2 sup u1e (!, r)  |@r (rv)| (✓, r)d✓ ) ||u1e ||21  sup |@r (rv)|2 (✓, r)d✓. [0,✓0 ] 0 r2[R0 ,1) 0 (2.4.76) 75 Z ✓0 2 Calling '(r) = |@r (rv)| d✓, the Sobolev embedding in R1 gives: 0 Z Z Z Z sup |'(r)|  ||'||L1 + ||@r '||L1 . 2 |@r (rv)| + |@rr (rv)|2 . ||rn ve1 ||H 2  C. r2[R0 ,1) (2.4.77) We can proceed inductively to obtain weighted estimates on ||rn u1e ||1 . These estimates are independent of small ✓0 . 2.4.3 Prandtl-1 Layer 2.4.3.1 Galerkin Formulation and a-Priori Estimates In this subsection, we solve for the Prandtl-1 layers, u1p , vp1 . For this subsection, we drop the subscripts on u1p , vp1 . The starting point is a modification of equation (2.4.3): (u1e + u)u0p! + (u0e + u0p )u! + u0p u1e! + rvp0 u0er + r(vp0 + ve1 )uR + ru0pR v+ (2.4.78) p (u0e + u0p )vp0 + u0p ve1 + Pp! 1 + E0 + E1 + ✏u0er v = ruRR + u0pR , where we have included E0 and E1 , the contributions from the Prandtl-0 layer construc- tion: E0 := (R R0 )u0er (r)u0p! + (R R0 )@r (rve1 (!, r))u0pR , E1 := (R R0 )u0pRR . (2.4.79) The boundary conditions are: vp1 (!, R0 ) = 0; u1p (!, R0 ) = u1e (!, R0 ); u1p (0, R) = u ¯1 (R). In 76 1/2 order to solve the ✏ order error in the radial equation, we take: Z 1 Z (u0p )2 u0p 1 u0p u0p! u0p! PP1 = + 2u0e dt ) PP1 ! = 2 + 2u0e dt. (2.4.80) R r(t) r(t) R r(t) r(t) By the rapid decay of the u0p terms, we have |@ ↵ Pp! 1 |R n for arbitrarily large n and for any multi-index ↵. Let u0 (!, R) = u0e (r) + u0p (w, R). We define F = u1e u0p! u0p u1e! rvp0 u0er u0 vp0 u0p ve1 + u0R PP1 ! , (2.4.81) which gives: u0! u + u0 u! + r(vp0 + ve1 )uR + ru0R v ruRR = F E1 E0 . (2.4.82) We take @R of the above equation, use the divergence-free condition, and divide by u0 to obtain: 1 0 1 1 @RR (rv) + ru v @R (ruRR ) = 0 (FR (E0 + E1 )R ) (2.4.83) u0 RR u0 u 1 u0 u + u0! uR + @R r(vp0 + ve1 )uR := G. u0 !R Now we take @! of the above equation and again use the divergence free condition: 1 0 1 1 @RR (rv! ) + ru v! + 0 @R (r@RRR (rv)) v = G! @! ( 0 ru0RR )v (2.4.84) u0 RR u ✓ ◆ u 1 +@! @R (ruRR ) . u0 With an eye towards obtaining a weak formulation of the equation together with a-priori 77 estimates, we rewrite the third term on the left-hand side as follows: 1 3 2 1 2 p 1 3 2 1 2 0 @R r@R (rv) =r@R ( 0 @R (rv)) + ✏ 0 @R (rv) + r@R ( 0 )@R (rv) (2.4.85) u u u u 1 3 2r@R ( 0 )@R (rv). u Therefore, our equation now becomes: 1 0 2 1 2 p 1 3 @RR (rv! ) + 0 ruRR v! + r@R ( 0 @R (rv)) + ✏ 0 @R (rv) u u u 2 1 2 1 3 1 r@R ( 0 )@R (rv) 2r@R ( 0 )@R (rv) = G! @! ( 0 ru0RR )v u ✓ ◆ u u 1 + @! @R (ruRR ) = (2.4.86.1) (2.4.86.9), (2.4.86) u0 and so we must obtain a-priori estimates for the equation: 1 0 2 1 2 @RR (rv! ) + 0 ruRR v! + r@R ( 0 @R (rv)) = fR + g. (2.4.87) u u Lemma 2.4.6. There exists a unique solution v to Equation (2.4.87) on the domain (0, ✓0 )⇥ (R0 , R0 + N ) subject to the boundary conditions v, vR = 0 at {R = R0 , R0 + N } and the initial condition v = v¯0 at {! = 0}. This solution v satisfies the following estimate, uniform in N : Z Z Z Z Z Z Z |@R (rv! )|2 + sup r|@RR (rv)|2 . f2 + g 2 hR R0 i 3 (2.4.88) Z + r|@RR (rv)|2 . !=0 Z N Z N u0RR Proof. Define an inner product by [[v, w]] := @R (rv)@R (rw) + 0 rvw. All of R0 R0 u the properties of inner-product follow from the properties of the integral, aside from non R degeneracy. Supposing [[v, v]] = 0, by the positivity estimate, (2.2.33), |@R (rv)|2 = 0 ) |@R (rv)| = 0, coupled with the fact that v = vR = 0 at {R = R0 , R0 + N } implies v = 0. 78 Let ej represent an orthonormal basis for H 1 with respect to this inner product. The weak formulation of (2.4.87) reads: Z Z Z 1 [[v! , ej ]] + @RR (rv)@RR (rej ) = f ejR + gej . (2.4.89) u0 k X Define v k (!, R) = aik (!)ei (R). Inserting this into the weak formulation above en- i=1 ables us to solve the corresponding ODE for the coefficients aik . Multiplying the weak formulation by @! aik and summing over i yields: Z Z Z 1 [[v!k , v!k ]] + @RR (rv!k )@RR (rv k ) = k f v!R + gv!k . (2.4.90) u0 In order to pass to the limit as k ! 1, we must obtain a-priori estimates for the above equation. We relabel v k by v for this purpose. In order to obtain the desired a-priori estimate, we multiply equation (2.4.87) by rv! and integrate by parts: Z Z 1 2 0 2 @RR (rv! )rv! + r uRR v! |@R (rv! )|2 (2.4.91) u0 by the Positivity estimate, (2.2.33). Next, we have Z Z 1 Z 1 2 1 2 1 r p 1 @R ( 0 @R (rv))r2 v! = @! 0 |@RR (rv)|2 + 2 ✏ @RR (rv)@R (rv! ). (2.4.92) u 2 R0 u R0 u0 On the right-hand-side, we have: Z Z Z Z Z fR rv! + grv!  f2 + g 2 hR R0 i 3 + |@R (rv! )|2 . (2.4.93) 79 Using Gronwall’s Inequality and integrating yields: Z Z Z 1 Z Z Z Z Z |@R (rv! )|2 + sup r|@RR (rv)|2 . f2 + g 2 hR R0 i 3 + r|@RR (rv)|2 . R0 !=0 (2.4.94) Next, the following high regularity estimate is established: Lemma 2.4.7. Z Z Z Z Z Z Z Z 1 |@R (rv!! )| + sup r|@RR (rv! )| . 2 f +2 2 2 g hR 3 R0 i + r|@RR (rv(0, ·))|2 R0 Z Z Z Z Z 1 2 2 3 + f! + g! hR R0 i + r|@RR (rv! (0, ·))|2 . (2.4.95) R0 Proof. Di↵erentiating the equation (2.4.87) once in ! yields: ✓ ◆ ✓ 0 ◆ ✓ ✓ ◆ ◆ u0RR uRR 1 @RR (rv!! ) + rv ! + rv !! + r@ RR @ ! @ RR (rv) (2.4.96) u0 ! u0 u0 ✓ ◆ 1 + r@RR @RR (rv! ) = fR! + g! . u0 In order to obtain a-priori estimates of the equation (2.4.96), we multiply the above equation by rv!! , remaining cognizant of the boundary condition that vR = 0 ) v!!R = v!R = 0 at {R = R0 , R0 + N }: Z " ✓ ◆ ✓ 0 ◆ ✓ ✓ ◆ ◆ u0RR uRR 1 @RR (rv!! ) + rv! + rv!! + r@RR @! @RR (rv) (2.4.97) u0 ! u0 u0 ✓ ◆# Z  1 + r@RR @RR (rv! ) rv!! = fR! + g! rv!! . u0 80 First, applying the positivity estimate (2.2.33), Z ✓ ◆ Z u0RR @RR (rv!! )rv!! + r2 v!! 2 |@R (rv!! )|2 . (2.4.98) u0 For the second term in (2.4.96), we have: Z ✓ ◆ ✓Z ◆1/2 ✓Z ◆1/2 ✓Z ◆1/2 ✓Z ◆1/2 u0RR r2 v! v!!  R n v!2 R n v!! 2  |@R (v! )|2 |@R (v!! )|2 u0 ! Z Z  N ( ¯) |@R (v! )|2 + ¯ |@R (v!! )|2 . (2.4.99) For the fourth term in (2.4.96), we have: Z ✓ ✓ ◆ ◆ Z ✓ ◆ 1 p 1 r@RR @! 0 @ RR (rv) rv !! = 2 ✏ @RR (rv)@R (rv!! ) u u0 ! Z ✓ ◆ 1 + r @RR (rv)@RR (rv!! ) = (2.4.100.1) + (2.4.100.2). (2.4.100) u0 ! For the fifth term in (2.4.96), we have: Z ✓ ◆ Z 1 2 1 @RR @ RR (rv ! ) r v !! = @RR (rv! )@RR (r2 v!! ) u0 u0 Z 1 p = 0 @RR (rv! ) 2 ✏@R (rv!! ) + r@RR (rv!! ) u Z Z ✓ ◆ Z 1 2 1 2 1 p = @! 0 |@RR (rv! )| 0 |@RR (rv! )| + 0 @RR (rv! )(2 ✏@R (rv!! ) u u ! u (2.4.101) = (2.4.101.1) + (2.4.101.2) + (2.4.101.3). Finally, for the right-hand side, Z Z Z Z Z fR! rv!! + g! rv!!  f!2 + g!2 hR R0 i 3 + @R (rv!! )2 . (2.4.102) 81 (2.4.101.3) and (2.4.100.1) are estimated through Young’s inequality. For (2.4.100.2) , we write: @RR (rv)@RR (rv!! ) = @! [@RR (rv)@RR (rv! )] @RR (rv! ). Applying Gronwall then yields: Z Z Z Z Z Z Z Z 1 |@R (rv!! )|2 +sup r|@RR (rv! )|2 . f 2+ g 2 hR R0 i3 + r|@RR (rv(0, ·))|2 + Z Z Z Z Z 1 Z R 0 f!2 + g!2 hR R0 i3 + r|@RR (rv! (0, ·))|2 + @RR (rv)@RR (rv! ) Z R0 !=✓0 @RR (rv)@RR (rv! ). !=0 The final two terms are estimated through Young’s inequality, where we recall Lemma Z 2.4.6 to estimate N |@RR (rv)|2 . !=0,✓0 We now obtain the weighted variants of the above two lemmas. Notationally, depict the weight by pm (R) = hR R0 i m . Lemma 2.4.8. Z Z Z Z Z Z Z Z pm |@R (rv! )|2 + sup pm r|@RR (rv)|2 . pm f 2 + g 2 pm+3 + rpm |@RR (rv)|2 ; !=0 (2.4.103) and Z Z Z Z Z Z Z Z Z pm |@! @R (rv! )| + sup pm r|@RR (rv! )| . 2 2 pm f + 2 2 g pm+3 + pm f!2 Z Z Z Z + g!2 pm+3 + rpm |@RR (rv! )|2 + rpm |@RR (rv)|2 . (2.4.104) !=0 !=0 82 We will proceed in several steps to establish Lemma 2.4.8. As these are a-priori estimates and we eventually plan to send N ! 1, we work in the domain R 2 (R0 , 1). Define w(R) = (R R0 )m on [R0 , R0 + M ] and w(R) = M m for R M + R0 . Note also that w(R0 ) = 0, which eliminates boundary contributions from {r = R0 }. Z Z Z Z Z Z Claim 1. sup w(R)|@RR (rv)|2 + rw(R)|@R3 (rv)|2 . pm |@RR (rv)|2 + f 2 pm + Z Z !=0 g 2 pm Proof. Multiplying equation (2.4.87) by w(R)@RR (rv) yields: Z " # Z ru0RR 2 1 2 @RR (rv! ) v! r@R ( 0 @R (rv)) w@RR (rv) = (fR + g)w@RR (rv). (2.4.105) u0 u Integrating (2.4.105) by parts yields: Z Z Z Z 2 @R (wr) 3 2 rw 3 @! w|@RR (rv)| + @R (rv)@R (rv) + |@ (rv)|2  (f 2 + g 2 )w + J, u0 u0 R (2.4.106) where J contains: Z Z Z ru0RR N 3 2 n J= v! w@RR (rv) + R |@R (rv)||@R (rv)| + R |@RR (rv)|2 . (2.4.107) u0 p Since @R (rw) = (m + 1) ✏w + R0 m(R R0 ) m 1 : Z Z p Z @R (wr) 3 2 (m + 1) ✏w 3 2 (R R0 )m 1 3 2 0 @ R (rv)@ R (rv) = 0 @ R (rv)@ R (rv) + mR 0 @R (rv)@R (rv) u u u0 Z Z Z Z p 1 3 p 1 2 (R R0 )m 1 3 (R R0 ) m 1 . ✏ 0 |@R (rv)|2 + ✏ 0 |@R (rv)|2 + 0 |@R (rv)|2 + 2 |@R (rv)|2 . u u u u0 (2.4.108) The first two terms above can be absorbed into (2.4.106), and the third and fourth 83 terms above have been estimated inductively. Applying Gronwall and integrating yields the desired lemma. Z Z Z Z Z Z Claim 2. sup w|@RR (rv! )| + 2 3 rw|@R (rv! )|2 . 2 pm |@RR (rv)| + f 2 pm + Z Z Z Z Z Z Z !=0 2 g pm + pm |@RR (rv! )|2 + f!2 pm + g!2 pm . !=0 Proof. We apply the multiplier w(R)@RR (rv! ) to the di↵erentiated equation (2.4.96): Z (Equation 2.4.96) ⇥ ( w(R)@RR (rv! )) . (2.4.109) On the left-hand-side, we have: Z Z Z rw 3 @R (rw) 3 @! w|@RR (rv! )|2 + |@ (rv! )|2 + 2 @R (rv! )@R (rv! ) + J + I. (2.4.110) u0 R u0 Z ✓✓ ◆ ◆ Z ✓ ◆ 1 1 Here J = @R @ RR (rv) @ R (rw@ RR (rv ! ))+ @RR (rv! )@R (rw@RR (rv! )) u0 ! u0 R R 2 and I  v!!R + |@RR (rv! )|2 . J and the third term of (2.4.110) can be treated through p Young’s inequality and induction after noticing that @R (rw) = (m + 1) ✏w + R0 m(R R0 ) m 1 as in the previous lemma. The desired result now follows from Gronwall. Z Z Z Z Z Z Claim 3. w|@R (rv! )|2 + sup wr|@R 2 (rv)|2 . rpm |@RR (rv)|2 + f 2 pm + Z Z !=0 g 2 pm+3 . r Proof. By Claim 2, the quantity @R u0 @RR (rv) is integrable and so integrating equation 84 (2.4.87) from R to 1 to yield: Z 1 ⇣r ⌘ Z 1 ru0RR 2 p 1 2 @R (rv! ) + v ! + @ R @ (rv) ✏ 0 @R (rv) = f + g. (2.4.111) R u0 u0 R u R Multiplying the left-hand side of 2.4.111 by w(R)@R (rv! ) gives: Z " Z 1 ⇣r ⌘ p 1 # ru0RR 2 2 @R (rv! ) v! @ R @ (rv) + ✏ 0 @R (rv) w@R (rv! ) R u0 u0 R u Z Z r 2 = w|@R (rv! )|2 + 2 @ (rv)w(R)@R (rv! ) + J, (2.4.112) u0 R where Z Z p Z Z 1 r 2 ✏ ru0RR J= @ (rv)w0 (R)@R (rv! ) + @ (rv)w@R (rv! ) 0 RR w@R (rv! ) v! . u0 R u R u0 (2.4.113) J can be controlled by Hardy’s inequality, Young’s inequality, and induction for the p R rw0 = ✏mw + R0 m(R R0 )m 1 factor in the ur0 @R 2 (rv)w0 (R)@R (rv! ) term, as in the previous lemma. The result then follows from Gronwall. Z Z Z Z Z Z Claim 4. w(R)|@R (rv!! )| +sup w(R)@RR (rv! ) . 2 pm |@RR (rv)| + 2 f 2 pm + Z Z Z Z Z Z Z !=0 g 2 pm + pm |@RR (rv! )|2 + f!2 pm + g!2 pm+3 !=0 Proof. We di↵erentiate equation (2.4.111) in ! to obtain: Z 1 ✓ ◆ ⇣r ⌘ ✓ ◆ ru0RR 2 p 1 2 @R (rv!! ) + @! v! + @R! @R (rv) ✏@! @ (rv) (2.4.114) R u0 u 0 u0 R 85 Z 1 = f! + g! . R Multiplying (2.4.114) by w(R)@R (rv!! ) yields on the left-hand-side: Z " Z 1 ◆ ✓ ⇣r ⌘ p ✓ ◆# ru0RR 2 1 2 @R (rv!! ) @! v! @R! @ (rv) + ✏@! @ (rv) w@R (rv!! ) R u0 u0 R u0 R Z Z Z 2 r 2 rw0 (R) = w|@R (rv!! )| + @! w|@ RR (rv ! )| + @RR (rv! )@R (rv!! ) + J (2.4.115) u0 u0 = (2.4.115.1) + (2.4.115.2) + (2.4.115.3) + J, where Z ✓ ◆ Z ✓ ◆ 1 p 1 J= rw0 (R)@RR (rv)@R (rv!! ) + ✏ @RR (rv)w@R (rv!! ) u0 ! u0 ! Z Z Z 1 ✓ 0 ◆ p w ruRR + ✏ @RR (rv! )@R (rv!! ) + w@R (rv!! ) @! v! u0 R u0 Z ✓✓ ◆ ◆ 1 @R rw(R)@RR (rv) . (2.4.116) u0 ! p Lastly, we use that rw0 (R) = R0 m(R R0 ) m 1 + ✏mw(R) to estimate (2.4.115.3) in- ductively, as in the previous lemma. The result of the claim now follows from an application of Gronwall. Proof of Lemma. The estimates in the claims above are uniform in M , enabling us to send M ! 1 to obtain: Z Z Z Z Z Z Z (R R0 )m |@R (rv! )|2 + sup (R R0 )m r|@RR (rv)|2 . pm f 2 + g 2 pm+3 Z + rpm |@RR (rv)|2 , (2.4.117) !=0 86 and Z Z Z Z Z Z Z (R R0 )m |@! @R (rv! )|2 + sup (R R0 )m r|@RR (rv! )|2 . pm f 2 + g 2 pm+3 Z Z Z Z Z Z 2 2 2 + pm f ! + g! pm+3 + rpm |@RR (rv! )| + rpm |@RR (rv)|2 . !=0 !=0 (2.4.118) Writing pm . 1+(R R0 )m , it follows by pairing (2.4.117) - (2.4.118) with the unweighted estimates in (2.4.88) - (2.4.95), we can upgrade the weights on the left hand sides of (2.4.117) and (2.4.118) to pm = hR R0 im , thereby establishing our lemma. 2.4.3.2 Boundary Estimates In the following theorem, we use the stream function formulation to give estimates of v and v! on the boundary {! = 0}. Z 1 k+1 Lemma 2.4.9. rn (R R0 )m |@R (rv(0, t)) |2 dt  C + ||rn/2+1 (R R0 )m/2 u ¯1 ||2H k+3 , R0 Z 1 and rn (R k+1 R0 )m |@R (rv! )|2 . C + ||rn/2+1 (R R0 )m/2 u ¯1 ||2H k+3 + ||(R R0 m 1 R0 ) ue!! (0, r(·))||2H k . Proof. Our starting point is equation (2.4.82). Define the stream function (!, R) = Z R Z R Z R u(!, ✓)d✓, so that ! = u! = @✓ (r(✓)v)d✓ = rv, and R = u. Now de- R0 R0 R0 fine w= u0R + u0 R, (2.4.119) 87 so that: w! = u0!R u0R ! + u0! R + u0 R! = u0!R u0R ( rv) + u0! u + u0 u! (2.4.120) = u0!R + F E1 E0 + ruRR r(vp0 + ve1 )uR . Z 1 According to the definitions of E0 , E1 , F in (2.4.79) and (2.4.81), r n R m @R k F (0, t)2 + R0 E0 (0, t)2 + E1 (0, t)2 dt  C, where the constant C depends on the previously constructed profiles. Now we estimate the boundary data of w in terms of u ¯1 using (2.4.119): Z 1 Z 1 Z 1 Z 1 Z 1 w(0, t)2 dt  (u0R )2 2 + ¯21 dt . (u0 )2 u ¯21 dt; and u @tk w(0, t)2  ||¯ u1 ||2H k , R0 R0 R0 R0 R0 (2.4.121) where we have used the rapid decay of u0R for Hardy’s inequality. We use (2.4.120) to do so similarly for w! : Z 1 w! (0, t)2 dt  C + ||r¯ u1 ||2H 2 ; and (2.4.122) R0 Z 1 rn (R R0 )m @tk w! (0, t)2 dt  C + ||rn/2+1 (R R0 ) m u ¯1 ||2H k+2 . (2.4.123) R0 Z R 0 w(!, t) Now we use (2.4.119) to express: =u dt. Di↵erentiating, taking the L2 R0 (u0 )2 norm of both sides, and using Hardy yields: Z k+1 rn (R R0 )m |@R @! |2  ||rn/2 (R R0 )m/2 w(0, ·)||2H k (R0 ,1) + ||rn/2 (R R0 )m/2 w! (0, ·)||2H k (R0 ,1) . This yields: Z 1 Z 1 k+1 k+1 2 rn (R R0 )m |@R (rv(0, t)) |2 dt = rn (R R0 ) m @ R ( ! (0, t)) dt (2.4.124) R0 R0 88  ||rn/2 (R R0 )m/2 w(0, ·)||2H k + ||rn/2 (R R0 )m/2 w! (0, ·)||2H k  C + ||rn/2+1 (R R0 )m/2 u ¯1 ||2H k+3 . We now estimate the H k of v! norm on the boundary {! = 0}. In particular, we start with: Z ! Z R ! R 0 w 0 w w rv! = !! = @!! u 0 2 ) @R (rv! ) = @!! uR + 0 ) R0 (u ) (u0 )2 u Z 1 Z Z Z rn (R R0 )m |@R (rv! )|2 . w2 + w!2 + rn (R R0 )m w!! 2 R0 Z 1 . C(1 + ||r¯ 2 u1 ||H 2 ) + rn (R R0 )m w!! 2 . (2.4.125) R0 Di↵erentiating (2.4.120): w!! = u0!!R u0!R ! + F! (E0 + E1 )! + ruRR! r@! vp0 + ve1 uR r(vp0 + ve1 )uR! (2.4.126) = (2.4.126.1) + (2.4.126.2) + (2.4.126.3) + (2.4.126.4) + (2.4.126.5) + (2.4.126.6) + (2.4.126.7). We estimate each of these terms: Z Z (2.4.126.1) + (2.4.126.2) + (2.4.126.4)  u¯1 + (R R0 ) n !2 + C(u0p , vp0 ); 2 Z Z Z Z Z (2.4.126.3) = F!2  (u1e u0p! )2 + (u1e u0p!! )2 + (u0p! u1e! )2 + r2 (vp! 0 u0er )2 Z Z Z Z Z Z Z + (u0! vp0 )2 + (u0 vp! 0 2 ) + (u0! vp0 )2 + (u0 vp! 0 2 ) + (u0p! ve1 )2 + (u0p ve! 1 2 ) + (u0R! )2 Z Z 0 2 1 2 1 2 + (up ) (ue!! ) + (Pp! ) ; Z Z (2.4.126.5) (2.4.126.7)  r2+n (R R0 )m u2RR! + r2+n (R R0 )m u2R + r2+n (R R0 )m u2!R . All of the terms in (2.4.126.3) above are bounded by a constant C, using the H 2 estimate 89 on ve1 , aside from the u1e!! term. For (2.4.126.5) - (2.4.126.7), we use the divergence free condition. This establishes the desired result. 2.4.3.3 Construction of Prandtl Layer Solutions R R0 We first define v¯(!, R) = v(!, R) R0 (R R0 )u1e! (w, R0 ). Here is a cuto↵ function 1 1 near 0. Then v¯(!, R0 ) = v(w, R0 ) = 0, and v¯R (!, R0 ) = vR (!, R0 ) R0 ue! (!, R0 ) = 0. Since the equation (2.4.87) above is linear, we easily find: ✓ ◆ 1 0 1 @RR (r¯ v! ) + ru v¯! + r@RR @RR (r¯ v) = fR + g, (2.4.127) u0 RR u0 where f and g are defined: 1 p 1 2 2 1 1 f = @! ( 0 )ruRR ✏ 0 @R (rv) + 2r@R (rv)@R ( 0 ) + 0 @! r(vp0 + ve1 )uR (2.4.128) u ✓ ◆ u u u 1 + @! uR r(vp0 + ve1 ) = (f.1) (f.5), u0 1 1 p 1 2 2 1 g= @! ( 0 ru0RR )v @R! ( 0 )ruRR ✏@R ( 0 )@R (rv) 2r@R 2 (rv)@R ( 0) ✓ u◆ u ✓ ◆u u 1 0 1 1 0 1 @R @! r(vp + ve )uR @!R uR r(vp + ve ) u0 u0 1 1 1 + @! ( 0 )(FR E1R E0R ) + 0 (FR! E1R! E0R! ) @! ( 0 ) u0!R u + u0! uR u u u 1 0 0 R R0 1 @! u!R u + u! uR ) @RR (r (R R0 )ue!! (!, R0 )) u0 R0 ✓ ✓ ◆◆ 1 0 R R0 1 1 R R0 1 + 0 ruRR ue!! + r@RR @RR r ue! . (2.4.129) u R0 u0 R0 90 Z Z Z 1 v |||2 = We define |||¯ v! )|2 + sup |@R (r¯ v )|2 . According to estimate r|@RR (r¯ [0,✓0 ] u0 (2.4.88): Z Z Z Z Z 1 r v |||2  |||¯ f2 + g 2 R3 + v )|2 . |@RR (r¯ (2.4.130) R0 u0 (0, R) Using the definition of v¯, we record the following consequence of the divergence free condition: ✓ ◆2 ✓ ◆2 R R0 R R0 u2! . ✏¯ 2 v + 2 (u1e! )2 + r2 v¯R 2 +r 2 2 (u1e! )2 + ( 0 )2 (u1e! )2 , R0 R0 (2.4.131) ✓ ◆2 ✓ ◆2 2 R R0 0 1 R R0 u2!R  ✏¯2 vR + @R (r¯ vR ) + @R r u1e! +r ue! + ✏@R (R R0 )u1e! (!, R0 ) . R0 R0 (2.4.132) We must now give estimates on the terms in f in terms of |||¯ v ||| in order to apply the contraction mapping principle. First, we relate u and v to |||¯ v |||: Z Z Z Z Z Z ✓ ◆ 2 Z 2 2 R R0 vRR  v¯RR + (u1e! )2 @RR  ✓0 sup v |2 + ||u1e! ||2L2 (0,✓0 ) |¯ R0 v |||2 + ||u1e! ||2L2 ,  ✓0 |||¯ (2.4.133) Z Z Z Z Z Z ✏v 2  v 2 + ✏||u1e! ||2L2  ✓02 ✏¯ ✏v!2 + ✏||u1e! ||2L2  ✓02 |||¯ v |||2 + ✏||u1e! ||2L2 , (2.4.134) Z Z Z Z Z Z Z Z u2R  ¯21R + ✓0 u u2!R  C + v )|2 + ||u1e! ||2L2  C + ✓0 |||¯ |@RR (¯ v |||2 + ||u1e! ||2L2 , (2.4.135) Z Z Z Z Z 1 u2! . ||u1e! ||2L2 + v )|2  ||u1e! ||2L2 + ✓0 |@R (r¯ |@R (r¯ v )(0, R)|2 (2.4.136) R0 Z Z + ✓02 v! )|2 . ||u1e! ||2L2 + ✓0 |||¯ |@R (r¯ v |||2 , Z Z Z Z Z u2  u¯21 + ✓0 u2! which has been estimated above; (2.4.137) 91 Z Z Z Z |v!R |2  v! )|2 + ||u1e ||2L2 , |@R (r¯ (2.4.138) Z Z Z Z Z Z |u!! |2 = |@R (rv! )|2  ||u1e! ||2L2 + v! )|2 . |@R (r¯ (2.4.139) Now we turn to the terms in f , which are given in (2.4.128): (f.1) For ruRR we use equation (2.4.82): Z Z 1 |@! ( 0 )|r2 |uRR |2 Z Z u u0 p  | 0! 2 |2 |u0! u + u0 u! + ✏u0 v + r(vp0 + ve1 )uR + ru0R v + E1 + E0 F |2 |u | := (f.1.1) (f.1.8). Each term in this equation is bounded by C + C✓0 |||¯ v |||: Z Z Z Z ✓ Z ◆ ✓Z 1 ◆ (f.1.1) : R n 2 u  R n 2 u(0, R) + ✓0 u2! . ✓0 |¯ 2 u1 | + ||u1e! ||2L2 (0,✓0 ) + |||¯ v ||| 2 , R0 Z Z Z Z Z Z (f.1.2) : R n 2 u!  R n ✏¯ vR |2 . ||u1e! ||2L2 (0,✓0 ) + v 2 + (u1e! )2 + |¯ 2 vRR . Z ||u1e! ||2L2 (0,✓0 ) + ✓0 sup |vRR |2 , Z Z Z Z Z Z (f.1.3) : R n ✏v 2  R n ✏¯v2 + R n ✏|u1e! |2 , Z Z Z Z Z Z Z Z (f.1.4) : (vp0 + ve1 )2 u2R  u2R  u¯21R + ✓0 u2!R ✓Z Z Z ◆  C + ✓0 v )|2 + |u1e! |2 , |@RR (r¯ Z Z Z Z Z n 2 (f.1.5) : R v  R v¯ + |u1e! |2 , n 2 Z Z n (f.1.6) - (f.1.8) : The forcing terms decay rapidly and therefore  R |F E1 E0 |2  C. Z Z Z Z ✓ ◆ 1 2 2 2 2 R R0 1 (f.2) ✏ | 0 | |@R (rv)|  ✏|||¯ v ||| + ✏ |@RR ue! |2 , u R0 Z Z Z Z Z Z ✓ ◆ 2 2 2 1 2 n 2 n R R0 1 (f.3) r |@R (rv)| |@R ( 0 )|  R |@RR (r¯ v) | + R |@RR ue! |2 , u R0 Z Z Z Z 1 2 (f.4) | 0 | |@! r(vp + ve )uR | . ||ve! ||1 0 1 2 1 u2R , u 92 Z Z Z Z Z Z Z Z 1 2 2 2 0 (f.5) |@! ( )| uR r (vp + ve1 )2 . R n 2 uR |vp0 |2 + R n 2 uR (ve1 )2 . u2R . u0 Z Z Combining (f.1) (f.5) with (2.4.133) shows that f 2 . C+✓0 |||¯ v |||2 +||u1e! ||2L2 (r=R0 ) . RR 2 We now give estimates on g hR R0 i3 , recalling the definition (2.4.129): Z Z n (g.1) R v 2 + u2RR + |@RR (rv)|2 + u2! + u2R + u2 + u2R!  C+✓0 |||¯ v |||2 +||u1e! ||2L2 (r=R0 ) , Z Z ✓ ◆ Z Z Z Z 1 2 1 (g.2) |@R | |@! + | + r(vp0 ve1 )uR 2 |@!R ( 0 )uR r(vp0 + ve1 )|2  u2R : u0 u RR RR 2 Here, we have used: R n |ve! 1 2 2 | uR  ||R n ve! 1 ||1 uR , and ||R n 1 ve! ||1  ||R n ve1 ||H 3  ✏n ||rm ve1 ||H 3  C. Z Z Z Z 1 2 2 1 (g.3) |@! ( 0 )| |FR E1R E0R | + | 0 |2 |FR! E1R! E0R! |2  C by Euler u u H 2 bounds, Z Z ✓ ◆ R R0 (g.4) |@R (r@R (R R0 )u1e!! (!, R0 ) )|2 : R0 Z Z Z Z Z Z Z u1e!! (!, R0 )2  @r (u1e!! )2 = u1e!! u1e!!r = 1 ve! 1 + rver! 1 ve!r 1 + @r (rver! )  ||rn ve1 ||W 2,p ||rn ve1 ||W 3,q . ✏  for  arbitrarily small. Z ✓0 Z ✓0 The boundary terms in (2.4.129) can be estimated by |u1e! |2 + |u1e!! |2  0 0 2 C✏ . Z 1 1 2 2 Finally, it remains to give estimates on r @ (r¯ v (0, R)) from (2.4.130): R0 u0 R Z 1 Z 1 Z 1 ✓ ◆ 1 2 2 2 R R0 r 0 @R v (0, R))  (r¯ r|@RR (rv)| + r|@RR |2 |u1e! |2 R0 u R0 R0 R0  C + ||r3/2 u ¯1 ||2H 4 + |u1e! (0, R0 )|2  C + ||¯ u1 ||H 4 . Therefore, we can close the contraction mapping argument. Because the estimates are uniform in N , we can let N ! 1 to obtain a global in R solution. We can repeat this same 93 argument with the weights of Rm . Indeed, the only terms above sensitive to a weight are contained in f , and we treat them here: Z Z Z Z Z Z Z Rm (vp0 + ve1 )2 u2R . Rm u2R  C + Rm |@RR r¯ v |2 + |u1e! |2 , (2.4.140) Z Z Z Z Z 1 2 m 2 m ✏ | 0 | R |@RR (rv)|  ✏ v )| + ✏ |u1e! |2 , R |@RR (r¯ 2 (2.4.141) u Z Z Z Z Z Z 1 | 0 |2 Rm |@! (r(vp0 + ve1 )uR )|2 . Rm u2R + Rm u2R! . (2.4.142) u Again through contraction mapping, this establishes: Z Z Z Lemma 2.4.10. hR R0 im |@R (rv! )|2 + sup hR R0 im r|@RR (rv)|2  ✏  for  > 0, arbitrarily small. Uniform estimates are obtained via the calculation: ✓Z ! ◆2 Z ✓ u(!, R)2 = |¯ u1 (R)|2 + @R (rv) u1 | 2 +  |¯ |@R (rv)|2 0 0 Z ✓0 Z R C+ @RR (rv)@R (rv) 0 R0 Z Z Z Z C+ R n |@R (rv)|2 + Rn |@RR (rv)|2 . (2.4.143) Similarly, using the fact that v(!, R0 ) = 0, we have Z R Z 1 Z 1 v(!, R)2 = vvR  R m |v|2 dR + Rm |vR |2 R0 R0 R0 Z 1 Z ✓0 Z 1 m  R R3 ||vRR ||2L2 + C + |vR! |2 . (2.4.144) R0 0 R0  This yields: ||v||1 + ||u||1  C✏ . We now obtain higher-regularity estimates. The 94 starting point is (2.4.95), and as such we define Z Z Z |||v|||2 = |@R (rv!! )|2 + sup r|@RR (rv! )|2 . (2.4.145) Since f and g have already been estimated, we compute f! and g! : 1 1 p 1 2 2 1 f! = @!! ( 0 )ruRR + @! ( 0 )ruRR! ✏ 0 @R (rv! ) + 2r@R (rv! )@R ( 0 ) u u u ✓ ◆ u 2 1 1 0 1 1 + 2r@R (rv)@!R ( 0 ) + 0 @!! r(vp + ve )uR + @!! uR r(vp0 + ve1 ) u u u0 ✓ ◆ ✓ ◆ ✓ ◆ 1 0 1 1 0 1 1 + @! u!R r(vp + ve ) + @! uR r@! (vp + ve ) + @! (r(vp0 + ve1 )uR ). u0 u0 u0 ! (2.4.146) 1 0 1 1 1 g! = @!2 ( ru )v @! ( 0 ru0RR )v! @R! ( 0 )ru!RR @R!! ( 0 )ruRR u0 RR u u u p 1 2 p 1 2 2 2 1 ✏@!R ( 0 )@R (rv) ✏@R ( 0 )@R (rv! ) 2r@R (rv! )@R ( 0) u u ✓ ◆ u✓ ◆ 2 1 1 1 2r@R (rv)@RR! ( 0 ) @!R 0 @! r(vp0 + ve1 )uR @R @!! r(vp0 + ve1 )uR u u u0 ✓ ◆ ✓ ◆ 1 0 1 1 @!!R 0 uR r(vp + ve ) @!R @! uR r(vp0 + ve1 ) u u0 ✓ ◆ 1 1 + @! @! ( 0 )(FR E1R E0R ) + 0 (FR! E1R! E0R! ) u u 1 1 1 @!! ( 0 ) u0!R u + u0! uR 2@! ( 0 )@! u0!R u + u0! uR 0 @!! u0!R u + u0! uR ) u u ✓ ◆ u R R0 1 0 R R0 1 @RR (r (R R0 )u1e!!! (!, R0 )) + @! 0 ru RR ue!! R0 u R0 ✓ ✓ ◆◆ 1 R R0 1 1 R R0 1 + 0 ru0RR ue!!! + r@!RR @ RR r u e! . (2.4.147) u R0 u0 R0 We now treat the f! terms in (2.4.146) Z Z Z Z Z Z 1 2 f!2  R n u2RR + u2RR! + |@RR (rv! |2 + u2R + u2R! + | | |@!! (ve1 uR )|2 u0 Z Z 1 2 +✏ | | |@RR (rv! )|2 . (2.4.148) u0 95 All of these terms can be estimated in terms of the norm, with uRR! being estimated by taking @! of Equation (2.4.82). We now address g! , bearing in mind that all of these terms are accompanied by rapid decay: Z Z Z Z g!2 hR R0 i m  R N v 2 + v!2 + u2!RR + u2RR + |@RR (rv)|2 + |@RR (rv! )|2 + u2R Z Z ✓ ◆ ! 2 N 1 2 2 1 1 + R |v!! | uR + @ ! @! ( 0 )(FR E1R E0R ) + 0 (FR! E1R! E0R! ) + u2 . u u (2.4.149) All of the above can be estimated in terms of |||¯ v |||2 . The most delicate boundary terms Z in g! is: |u1e!!! |2  C(✓0 )✏ 2 , and the most delicate interior term is: r=R0 Z Z Z Z Z Z 1 |@R ( 0 )|2 |ve!! 1 |2 |uR |2  ||R n 1 ve!! ||21 R n 2 uR  ||R n 1 2 ve ||H 4 R n 2 uR u Z Z  ✏n ||rm ve1 ||2H 4 u2R . (2.4.150) The latter boundary term in estimate (2.4.104) has been estimated before. Therefore, we must estimate the v! boundary term: Z Z ✓ ✓ ◆◆ R R0 1 rhR R0 im |@RR (r¯ v! )|2 = rhR R0 im |@RR r@! v + ue! (0, R0 ) |2 !=0 R0 Z!=0 Z R R0 1  rhR R0 im |@RR (rv! ) |2 + rRm |@RR ( ue!! (0, R0 ))|2 !=0 !=0 R0  C + ||r3/2 Rm/2 u ¯1 ||2H 3 + ||R m 1 ue!! (0, r(·))||H 1 + |u1e!! (0, R0 )|2 3/2  C(✓0 )✏ , (2.4.151) where for the final inequality we use the same calculation as in (GN17, page 25). There- fore, we can close the contraction mapping argument, and again let N ! 1 yielding: Z Z Z Lemma 2.4.11. hR R0 im |@R (rv!! )|2 + sup hR R0 im r|@RR (rv! )|2  C(✓0 )✏ 2 96 where the constant C(✓0 ) could depend poorly on small ✓0 . 2.4.3.4 Cuto↵ Prandtl Layers We define our Prandtl-1 layers (u1p , vp1 ) by cutting o↵ the previously constructed layers (u, v): Z R p p 0 p u1p (!, R) := ( ✏(R R0 ))u + ✏ ( ✏(R R0 )) u, (2.4.152) R0 p vp1 (!, R) := ( ✏(R R0 ))v. Here, is a standard cuto↵ function which equals 1 on [0, 1]. (u1p , vp1 ) satisfy the diver- gence free condition: @! u1p + @R (rvp1 ) = 0. We also have the following estimate for u1p : Z R p 0 p p | ✏ ( ✏(R R0 )) u1p |  ✏(R R0 ) 0 ||u1p ||1  C✏  . (2.4.153) R0 We have the following lower-order estimates of u1p and vp1 : Z Z Z Z n 1 r (vp!! )2 ✏ 1/2 Rm (vp!!R 1 )2  ✏ 5/2 , (2.4.154) Z Z Z Z rn (vp! 1 2 ) ✏ 1/2 Rm (vp!R 1 )2  ✏ 1/2  , (2.4.155) Z Z Z Z rn (vpR 1 2 ) ✏ 1/2 Rm (vpRR 1 )2  ✏ 1/2  , (2.4.156) Z Z Z Z Z Z Z p1 ✏ rn |u1p! |2 ⇡ |@R (rvp1 )|2 = @R (rvp1 )@RR (rvp1 ) R Z Z 1/2 ✏ |@R (rv)||@RR (rv)| ✓Z Z Z Z ◆ ✏ 1/2 m R |@R (rv)| +2 R |@RR (rv)| . ✏ m 2 1/2  , (2.4.157) Z Z Z Z Z ! Z Z Z Z rn |u1pR |2 ⇡ |¯ uR (R) + u1p!R (✓, R)|2  C + |u1p!R |2 = C + |@RR (rvp1 )|2 0  ✏ , (2.4.158) 97 Z Z Z Z Z 1 rn |u1p |2 ⇡ u1 | 2 + |¯ |u1p! |2  ✏ 2  , (2.4.159) Z Z Z Z R0 + p1✏ Z Z R0 + p1✏ 1 rn |vp1 |2 ⇡ |vp1 |2 . ||vp1 ||2L1 dRd! . ✏ 2  . (2.4.160) R0 R0 The weight of rn can be added in because on the support of vp1 and u1p , r is bounded uniformly. For the same reason, these weights can be added in for the uniform estimates. We summarize the results of the Prandtl-1 layer construction in the following: Theorem 2.4.12. Let u1p , vp1 denote the cuto↵ Prandtl layers described above. Then Z Z Z Rm |@R (rvp! 1 )|2 + sup Rm r|@RR (rvp1 )|2  C(✓0 )✏  for  > 0 arbitrarily small, Z Z Z Rm |@R (rvp!! 1 )|2 + sup Rm |@RR (rvp! 1 )|2  C(✓0 )✏ 2 , and ||rn u1p , vp1 ||1  C(✓0 )✏  , where C(✓0 ) is a constant which could depend poorly on small ✓0 . 2.4.4 Estimates on Profile Errors 2.4.4.1 Angular Errors, Ru : There are three lower order error terms from the profile constructions, which come from (2.4.22) for the Prandtl-0 layer and (2.4.72) for the Euler-1 layer and for the Prandtl-1 layer by inserting (2.4.152) into the equation (2.4.78): Z 1 0 1/2 ✏ (e2 ) + ✏ ( Eb (!, ✓)d✓) + ✏1/2 (Prandtl-1 contribution) = (2.4.161.1) + (2.4.161.2) + (2.4.161.3). r (2.4.161) 98 First, we have (2.4.161.1) = ||rm e2 ||L2  ||rm u0pR Rn ||L2 ||@r2 (rve1 )||L2 + ✏||rm u0p! Rn ||L2 ||u0err ||1  C✏. (2.4.162) Next, we have: Z Z ⇣Z 1 ⌘2 Z Z ⇣Z 1 ⌘2 1 r(R)m Eb (!, ✓)d✓ dRd! = ✏ 2 rm Eb (!, ✓)d✓ drd! r(R) r Z 1 ! 1 .✏ 2 ( )2 d! . ✏ 2 . (2.4.163) ✏ To estimate (2.4.161.3), we write: Prandtl-1 angular error contribution =: Z R ! Z R ! Z R ! p p p u0! u1p + ✏ 0 u1p +u0 u1p! + ✏ 0 u1p! +r(vp0 +ve1 ) u1pR +2 ✏ 0 1 up +✏ 00 u1p + Z ! p R p 00 1 ru0R vp1 r u1pRR + ✏ up + 2 ✏ 0 u1pR + ✏3/2 000 u1p + 2✏ 00 + ✏ 0 u1pR (F E1 E0 ) ! Z R Z R p 0 = (1 )(F E0 E1 ) + ✏ u0! u1p +u 0 u1p! + 2(vp0 + ve1 )u1p + 2ru1pR + Z ! Z ! R R 00 3/2 000 ✏ (vp0 + ve1 ) u1p + 3u1p +✏ u1p . We give estimates on each of the terms in the above expression: Z Z rm (1 )2 |F E0 E1 | 2  ✏ n , (2.4.164) Z Z Z R Z Z p p 1 ✏ rm | 0 ( ✏·)|2 |(vp0 + ve1 )u1p + u1pR + u0! u1p |2  ✏1  | 0 ( ✏·)|2 R n  ✏2  , (2.4.165) Z Z Z R Z Z p p ✏ | 0 ( ✏·)|2 |u0 |2 | u1p! |2 = ✏ | 0 ( ✏·)|2 |u0 |2 |vp1 |2  ✏1/2  , (2.4.166) Z Z " Z ! Z ! #2 R R m 00 3/2 000 r ✏ (vp0 + ve1 ) u1p + 3u1p +✏ u1p 99 Z Z Z !2 Z Z p R p 2 2 .✏ 2 00 ( ✏·) u1p r m . ✏2  00 ( ✏·) R2 r m Z Z p . ✏1  00 ( ✏·)2  ✏1/2  . (2.4.167) Now, we must estimate the higher order terms from the expansions (2.4.1) - (2.4.7) : ✏1 -order error: Z Z Z Z rm |u1e + u1p |2 |u1e! + u1p! )|2  ||u1e + u1p ||21 |u1e + u1p! |2 . ✏ 1/2  , (2.4.168) Z Z Z Z rm |vp0 + ve1 |2 |u1er |2  ||rm (vp0 + ve1 )||21 |u1er |2 . ✏ 1/2 , (2.4.169) Z Z Z Z m 1 2 1 2 1 2 r |vp | |upR |  ||vp ||1 |u0er + u1pR |2  ✏ 1/2  , (2.4.170) Z Z Z Z 1 p ✏ rm (u0e + u0p )2 |vp1 |2 . ✏  (u0e + u0p )2 . ✏  1/2 , (2.4.171) Z Z rm (u1e + u1p )2 (vp0 + ve1 )2 . ✏ 1/2  , (2.4.172) Z Z rm |Pp! 2 2 | : (2.4.173) We define: Z 1 1 2 PP2 = u0e vp! 0 + u0p @! (vp0 + ve1 ) + (vp0 + ve1 )vpR 0 (u1 u0 ) 0 vpRR dt R r(t) r(t) e p Z R0 + p1 Z R0 + p1✏ ✏ 2 1 0 1 1 0 u u dt u u dt. R r(t) p p R r(t) p e Using the rapid decay of Prandtl-0 profiles, after taking @! , the first integral is bounded n by R for arbitrarily large n. We must therefore treat the @! of second and third integrals: Z R0 + p1✏ Z R0 + p1✏ Z R0 + p1✏ 2 1 0 2 1 0 1 1 0 u u dt + u u + u u . R r(t) p! p R r(t) p p! R r(t) p! e 100  n The middle term above is bounded by ✏ R . For the first and third terms, we use the divergence free condition and integrate by parts, bearing in mind that vp1 |R0 + p1 = 0 due to ✏ the cuto↵: Z R0 + p1✏ Z R0 + p1✏ 2 2 0 | @t (r(t)vp1 )u0p dt| = | r(t)vp1 @t ( u )dt + vp1 u0p | R r(t) R r(t) p Z 1 ✏  |u0p | + |r(t)vp1 |t n dt . ✏  |u0p | + R n , R Z R0 + p1✏ Z R0 + p1✏ 2 1 0 u0e (r(t)) | u u (r(t))dt| = |vp1 u0e (r) + r(t)vp1 @t ( )dt|. R r(t) p! e R r(t) We place the terms above into integrals, the most delicate being: Z Z Z !2 R0 + p1✏ u0 rm r(t)vp1 @t ( e )dt dRd! R r(t) Z Z Z !2 R0 + p1✏ R0 + p1✏ u0 (r(t)) .✏  |@t ( e )|dt dRd! R0 R r(t) Z Z Z !2 R0 + p1✏ R0 + p1✏ p u0er (r(t)) p u0 (r(t)) .✏  ✏| | + ✏| e 2 |dt dRd! R0 R0 r(t) r(t) Z Z Z R0 +1 ! 2 R0 + p1✏ u0e .✏  |uer (r)| + | |dr dRd! . ✏  0 1/2 . R0 R0 r Z Z rm |u0err |2 + |u0er |2 + |u1pR |2 + |u0p!! |2 + |u0p |2 . ✏ 1/2 , (2.4.174) Z Z Z Z Z 1 (u0e )2 2+ 1 dr r dR  ||ue ||1 0 2 dR . ✏ 1/2 ✏ 1/2 if < 1. (2.4.175) r4 r2 R0 r 2 We emphasize that estimate (2.4.175) is the most delicate and requires the weight pa- rameter of r2+ to be strictly less than 1. 101 ✏3/2 -order error: Z Z Z Z rm |vp1 |2 |u1er + u1e + u1p |2 + rm |u1e + u1p + vp! 0 1 2 + ve! | .✏  1/2 , (2.4.176) Z Z rm |u1err + u1er + u1e!! |2  ✏ 1/2 ||rm u1e ||2H 2  ✏ 1/2 ||rm ve1 ||2H 3  ✏ 3/2 , (2.4.177) Z Z Z Z rm |u1p!! |2 ⇡ |@R (rv! )|2  ✏  , (2.4.178) ✏2 -order error: Z Z rm |vp! 1 2 | .✏ 1/2  . (2.4.179) 2.4.4.2 Radial Errors, Rv : The higher-order radial contributions must be estimated: ✏1/2 order error: Z Z Z Z Z Z rm |u1e ve! 1 2 | + rm |ve1 |2 |ver 1 2 | + rm |u1e |4 Z Z Z Z m 1 0 1 0 1 2 + r ue vp! + up @! (vp + ve ) + rm |vp1 |2 |vpR 0 2 | +  ✏  1/2 , (2.4.180) Z Z 2 rm |vp! 1 2 0 | |ue + u0p |2 + |vp0 |2 |ver 1 2 | + |vp0 |2 |vpR 1 2 | + |u1e |2 |u1p |2 + |u1p |4 + |ve1 |2 |vpR 1  1/2 ✏ , (2.4.181) Z Z rm |vpR 0 2 | + |u0p! |2 + |vpRR 1 |2  ✏  , (2.4.182) ✏ order error: Z Z rm |verr 1 1 + ver 1 + ve!! + u1e! + ve1 |2  ✏ 1/2 ||ve1 ||2H 2 , (2.4.183) 102 Z Z rm |u1p vp! 1 + u1e vp! 1 + vp1 ver 1 + vp1 vpR 1 1 + vpR 0 + vp!! + u1p! + vp0 |2  ✏  1/2 , (2.4.184) ✏3/2 order error: Z Z rm |vp!! 1 |2  ✏ 5/2 . (2.4.185) We have established the main result of this section, Theorem 2.2.10. 2.5 Energy Estimates In this section, the energy estimates in Theorem 2.2.11 are proven. We will need to work with the restricted domain ⌦N = (0, ✓0 ) ⇥ (R0 , R0 + N ), and obtain estimates which are uniform in N . The boundary conditions u = v = 0 on {(!, R) : R = R0 + N } are enforced. Step I: Laplacian and Lower Order Terms By the divergence free condition: ✓ p ◆ ✏ ✏ ✏ ✏ ✏ 0= 2 u!! + 2 u!! = 2 u!! + @! v vR r r r r r ✏ ✏3/2 ✏ = u!! v! v!R , (2.5.1) r2 r2 r ✓ p ◆ ✏ u! 0= vRR + vRR = vRR + @R v r r p ✏ ✏ u!R p u! = vRR vR + 2 v + ✏ 2. (2.5.2) r r r r 103 In this step the terms in (2.2.23 - 2.2.24) which involve (u, v) but do not depend on the profiles, (us , vs ), are treated. After adding in (2.5.1) - (2.5.2) these terms are summarized: p ✏ 2✏ ✏ 3 3/2 ✏ uRR uR 2 u!! + 2 u 2 ✏ v! v!R , (2.5.3) rp r r p r r 2 ✏ ✏ 3 ✏ ✏ u!R 2vRR vR v!! + 2 u! + 2 2 v . (2.5.4) r r2 r r r As described in the introduction, we proceed to multiply (2.5.3) by r1+ u and (2.5.4) by ✏r1+ v: Z Z " p # ✏ 2✏ ✏ 3 3/2 ✏ uRR uR u!! + 2 u ✏ v! v!R ⇥ r1+ u r r2 r r2 r Z Z " p p # 2 ✏ ✏ 3 ✏ ✏ u!R + 2vRR vR 2 v!! + 2 u! + 2 2 v ⇥ ✏r1+ v. (2.5.5) r r r r r Integrating by parts the terms in (2.5.5) yields: Z Z p Z Z ✏ 2✏ ✏u (uRR uR 2 u!! + 2 )ur1+ = u2R r1+ + 2u2! r 1+ + ✏ur 1+ (2.5.6) r r r Z 1+ 2✏ r uu! + J1 , !=✓0 Z Z p Z Z 2 ✏vR ✏v!! ✏v 1+ 2 1+ ( 2vRR + 2 )✏vr = 2✏vR r + ✏2 v!2 r 1+ (2.5.7) r r2 r2 Z + 2✏2 v 2 r 1+ ✏2 r 1+ vv! + J2 , !=✓0 Z Z Z Z Z Z Z Z 3/2 1+ ✏ u!R vr ✏ v!R ur = 2✏ uR v ! r + ✏ r v! u (2.5.8) Z Z ✏uR vr = J3 ✏uR vr , !=✓0 !=✓0 104 Z Z Z Z Z Z p where J1 = ✏ uuR r , J2 = C✏3/2 vvR r , and J3 = 2✏ uR v ! r + Z Z ✏3/2 r 1+ v! u. Putting the remaining terms into J4 yields: Z Z Z Z 3 3 1+ 1+ J4 = 3✏ 2 v! ur + 3✏ 2 vu! r . (2.5.9) Letting J = J1 + J2 + J3 + J4 , it is easy to see that: p |J|  C(✏, ✓0 )||v||2B , C(✓0 , ✏) ⇠ O(✓0 , ✏). (2.5.10) Using the stress-free boundary condition, (2.2.30), the boundary term from (2.5.7) can- cels with that of (2.5.8): Z Z 2 1+ ✏ r vv! ✏uR vr = 0. (2.5.11) !=✓0 !=✓0 The only remaining boundary contribution is then that of (2.5.6): Z 1+ 1 = 2✏ uu! r . (2.5.12) !=✓0 The remaining interior terms from (2.5.6 - 2.5.7) are then: Z Z I1 = u2R r1+ + 2u2! r 1+ + ✏ur 1+ 2 1+ + 2✏vR r + ✏2 v!2 r 1+ + 2✏2 v 2 r 1+ . (2.5.13) Summarizing the calculations from this step, we have: Z Z r1+ u ⇥ (2.5.3) + ✏r1+ v ⇥ (2.5.4) = I1 + 1 + J, (2.5.14) 105 where 1 is defined in (2.5.12), J is estimated in (2.5.10), and the interior terms I1 are defined in (2.5.13). Step II: Profile Terms In this step, we treat the terms from (2.2.23 - 2.2.25) which contain us , vs . For clarity, we display these terms here: p p 1 1 ✏ ✏ us u! + us! u + usR v + vs uR + vs u + us v, (2.5.15) r r r r 1 1 2 1 us v ! + vs! u + vs vR + vsR v p us u. (2.5.16) r r r ✏ Applying our multiplier to (2.5.15 - 2.5.16) yields: Z Z " p p # 1 1 ✏ ✏ r1+ u⇥ us u! + us! u + usR v + vs uR + vs u + us v r r r r Z Z p p = us u! ur + us! u2 r + usR uvr +1 + vs uR ur +1 + ✏vs r u2 + ✏us r uv . ✓0 ||v||2B + ✓0 ||u||2A , (2.5.17) Z Z " # 1+ 1 1 2 1 ✏r v⇥ us v! + vs! u + vs vR + vsR v p us u r r r ✏ Z Z p = ✏us r vv! + ✏vs! r uv + ✏vs vR vr1+ + ✏vsR v 2 r1+ 2 ✏us uvr p . C(✓0 , ✏)||v||2B where C(✓0 , ✏) ⇠ O(✓0 , ✏). (2.5.18) We have used the following uniform bounds on the profiles, recalling that us (!, R) = p u0e (r) + u0p (!, R) + ✏u1e (!, r), and vs (!, R) = vp0 (!, R) + ve1 (!, r). ||us ||1  C, (2.5.19) 106 ||vs rn ||1  ||vP0 rn ||1 + ||ve1 rn ||1  C + ||ve1 rn ||H 2  C, (2.5.20) Z 1 sup u2sR rn (R R0 )  C as shown below in (2.5.27), (2.5.21) !2[0,✓0 ] R0 p p ||vsR rn ||1  ||rn vpR 0 ||1 + ✏||rn ver 1 ||1  C + ✏||rn ve1 ||H 3  C, (2.5.22) ||vs! rn ||1  C(✓0 ) by weighted Euler W 2,q bounds in estimate (2.4.74), (2.5.23) p p ||us! rn ||1  ||u0P ! rn ||1 + ✏||u1e! rn ||1  C + ✏||rn @r (rv)||1 , (2.5.24) p p C+ ✏||rn ve1 ||1 + ✏||rn ver 1 ||1  C, (2.5.25) p ||usR rn ||1  ✏||rn u0er ||1 + ||u0p rn ||1 + ✏||rn u1er ||1  C + ✏||rn ve1 ||H 3  C. (2.5.26) p Of these, us (!, R) = u0e (r)+u0p (!, R)+ ✏u1e (!, r) is the term which cannot absorb factors of r due to the outer Euler flow u0e being bounded below away from zero by assumption. p RR For this reason the estimate for ✏ us uvr in (2.5.17) is the most sensitive to the weight and forces the loss of one factor of r in the energy estimate. The estimate ||u1er ||1 . ||ve1 ||H 3 from (2.5.26) is given below: ✓Z ! ◆2 Z ! Z ✓0 2 2 u1e (!, r) = @r (rv)(✓, r)d✓ ) u1er (!, r)  ✓0 2 |@rr (rv)(✓, r)| d✓  ✓0 |@rr (rv)|2 d✓, 0 0 0 Z ✓0 ) sup |u1er (!, r)|2  ✓0 |@rr (rv)|2 (✓, r)d✓ := ✓0 '(r) ) ||u1er ||1  ✓0 sup |'(r)|. ! 0 r Since ' : [R0 , 1) ! R, according to the 1-dimensional Sobolev embedding: Z 1 Z 1 Z 1 Z ✓0 Z 1 Z ✓0 sup |'|  ||'||W 1,1  |'(r)|dr + |@r '(r)|dr  |@rr (rv)|2 d✓dr + |@rrr (rv)|2 drd✓. r R0 R0 R0 0 R0 0 Combining these gives ||u1er ||1 . ||rn ve1 ||H 3 for a constant independent of small ✓0 . We can repeat the argument for a weight of rn . 107 The profile estimates in (2.5.19) - (2.5.26) are all independent of small ✓0 , with the exception of ||vs! ||1 . Any time ||vs! ||1 appears in our analysis, it must be accompanied by a power of ✏ to overcome the potentially poor dependence on ✓0 . For the uniform bounds, this follows from the fact that the corresponding H 2 estimates were independent of small Z 1 ✓0 . For the sup u2sR rn (R R0 ) estimate, we have: !2[0,✓0 ] R0 Z 1 Z 1 sup (R R0 )rn u2sR  C(u0e , u0p ) + sup (R R0 )rn ✏2 |u1er (r)|2 dR [0,✓0 ] R0 R0 Z 1  C + ✏ sup rn+1 |u1e (r)|2 dr  C + ✏✓0 1 ||rn+1 ve1 ||H 2 . (2.5.27) R0 Step III: Pressure Term Applying our multiplier to the two pressure terms in (2.2.23 - 2.2.25) yields: Z Z Z Z P! ur + PR vr1+ Z Z Z Z Z Z Z p = P u! r (1 + ) ✏vP r r1+ P vR + P ur !=✓0 Z Z Z p = P ✏vr + P ur . (2.5.28) !=✓0 Using the stress-free boundary condition at {! = ✓0 }, which is P r = 2✏u! , this boundary contribution cancels the remaining boundary term 1 from (2.5.12): Z Z 1+ P ur 2✏ uu! r = 0. (2.5.29) !=✓0 !=✓0 The interior term is estimated as follows: Z Z ✓Z Z ◆ 12 ✓Z Z ◆ 12 p 2 2 | P ✏vr |  P r ✏v r  ✓0 ||v||B ||P ||L2⇤, . (2.5.30) 108 Step IV: Right-Hand Side Z Z Z Z Z Z Z Z f r1+ u  N ( ¯) f 2 r2+ + ¯ u2 r  N ( ¯) f 2 r2+ + ✓02 ||v||2B (2.5.31) Z Z Z Z Z Z Z Z ✏ gr 1+ v  N ( ¯)✏ 2 2+ g r +¯ 2 ✏v r  N ✏ g 2 r2+ + ✓02 ||v||2B . (2.5.32) Putting the calculations in (2.5.14), (2.5.17 - 2.5.18), (2.5.30), (2.5.31 - 2.5.32), together yields Theorem 2.2.11. 2.6 Positivity Estimate In this section, we establish the positivity estimate given in Theorem 2.2.13. First, we must prove the positivity calculation in the generality we require, that is, using the weight of r : Proof of Lemma 2.2.12 for general . Z Z Z Z Z Z rv rv rv r |@R (rv)|2 = us )|2 = r |@R ( r |@R ( )us + usR ( )|2 (2.6.1) us us us Z Z Z Z ✓ ◆2 Z Z rv rv rv = r u2s |@R ( )|2 + r u2sR + r us usR @R (( )2 ) us us us Z Z Z Z Z Z ✓ ◆2 rv 2 2 1 rv 2 p rv = r |@R ( )| us r ( ) ✏us usR r us usRR , us us us Z Z Z Z Z Z Z Z us 2 rv rv rv 2 2 rv 2 2 r |@R (rv )| = r |us @R ( ) + usR ( ))|2 . r |@R ( )| us + r ( ) usR us us us us us Z Z rv 2 2 . r |@R ( )| us , (2.6.2) us 109 where we used the Fundamental Theorem of Calculus, the hypothesis that min us > 0, and estimate (2.5.27) for the rapid decay of usR . Lastly, we deal with the term: Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 p 1 2 2 usR ✏r r v . ✏v 2 r u2sR v 2 r2+ us ✓ Z ◆1/2 ✓Z Z ◆1/2 . ✓0 ||v||B sup u2sR (R n R0 )r dR 2 vR . ✓0 ||v||2B . (2.6.3) Estimates (2.6.1) - (2.6.3) immediately imply the desired positivity. Step I: Positive Profile Terms r2 v r1+ v As discussed in the introduction, we apply the multiplier (r @R ( ), ✏@! ( )) to the us us system (2.2.23 - 2.2.25). Here we treat the three profile terms which enable us to obtain the necessary control of ||v||B . Explicitly, we are computing: Z Z ⇣ ⌘ ✓ 2 ◆ Z Z ✓ ◆ us r v us v! 1+ v u! + vusR r @R ✏ r @! . (2.6.4) r us r us We now compute each of the terms in (2.6.4) individually: Z Z Z Z ✓ 2 ◆ 1+ r2 v 1+ r v us r u! @ R ( )= us r ( @R (rv))@R us us Z Z Z Z Z Z p u sR = ✏r v@R (rv) r |@R (rv)|2 + r rv@R (rv) us Z Z Z Z Z Z p 1 usR = ✏r v@R (rv) r |@R (rv)|2 + r @R ((rv)2 ). (2.6.5) 2 us 110 For the first term in (2.6.5), we estimate: Z Z ✓Z Z ◆ 12 ✓Z Z ◆ 12 p | ✏r v@R (rv)|  ✓0 ✏v!2 r r |@R (rv)|2  ✓0 ||v||2B . (2.6.6) The second term in (2.6.4) is: Z Z Z Z ✓ ◆ r2 v p rv @R (rv) 2 usR vusR r @R ( )= vusR r ✏ +r r v 2 us us us us Z Z ✓ ◆ Z Z p rv usR 1 usR = vusR r ✏ r2 v 2 + r @R (rv)2 . us us 2 us (2.6.7) Combining (2.6.5) - (2.6.7), integrating by parts the final terms in both (2.6.5) and (2.6.7), and recalling estimate (2.6.3) yields: Z Z ⇣ ⌘ ✓ 2 ◆ Z Z us r v usRR u! + vusR r @R  ✓0 ||v||2B r |@R (rv)|2 r (rv)2 . (2.6.8) r us us The third term in (2.6.4) is: Z Z ✓ ◆ Z Z Z Z v us! ✏ r us v ! @ ! ( ) = ✏ r v!2 + ✏ r vv! = (2.6.9.1) + (2.6.9.2). us us (2.6.9) We estimate (2.6.9.2): Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 us! ✏ r vv!  ✓0 ||us! ||1 ✏v!2 r ✏v!2 r . ✓0 ||v||2B . (2.6.10) us 111 Putting (2.6.8 - 2.6.10) together and using the positivity estimate (2.2.33) yields: Z Z Z Z r|@R (rv)|2 + ✏ rv!2  (2.6.4) + ✓0 ||v||2B . (2.6.11) Step II: Remaining Profile Terms We now apply the multiplier to the remaining profile terms and provide bounds on each term, keeping in mind the estimates on the profiles we proved in (2.5.19 - 2.5.26). For reference, we include the specific profile terms from equations (2.2.23 - 2.2.25) that we treat in this step: p p 1 ✏ ✏ us! u + vs uR + vs u + us v, and (2.6.12) r r r 1 2 1 vs! u + vs vR + vsR v p us u. (2.6.13) r r ✏ Applying the multiplier to (2.6.12 - 2.6.13) yields: Z Z ✓ p p ◆ ✓ 2 ◆ 1 ✏ ✏ r v us! u + vs uR + vs u + us v r @ R (2.6.14) r r r us Z Z ✓ ◆ ✓ 1+ ◆ 1 2 1 r v ✏ vs! u + vs vR + vsR v p us u @ ! . (2.6.15) r r ✏ us We estimate each of the eight terms appearing in (2.6.14) - (2.6.15) individually: Z Z ✓ ◆ Z Z ✓ ◆ r2 v 1 1+ p rv @R (rv) 2 1 us! ur @R = r us! u ✏ +r + r v@R ( ) us us us us ✓Z Z ◆1/2 ✓Z Z ◆1/2 ✓Z Z ◆1/2 ! . ✓0 r u2! r |@R (rv)|2 + ✏r v 2 Z . ✓0 ||v||2B , where the constant C = ||us! ||1 + sup u2sR (R R0 )rn . (2.6.16) 112 Z Z ✓ ◆ Z Z ✓ ◆ r2 v p rv @R (rv) 1 v s uR r @ R = v s uR r ✏ +r + r2 v@R ( ) us us us us ✓Z Z ◆1/2 ✓Z Z ◆1/2 ✓Z Z ◆1/2 ! . r1+ u2R ✏v 2 r + r |@R (rv)|2 Z . ✓0 ||v||B ||u||A , where the constant is: C = ||vs r ||1 + sup n u2sR rn (R R0 ). (2.6.17) Z Z ✓ ◆ Z Z ✓ ◆ p 1 r2 v p p rv 1+@R (rv) 1 ✏vs ur @R = ✏vs r u ✏ +r + r2 v@R ( ) us us us us ✓Z Z ◆1/2 ✓Z Z ◆1/2 ✓Z Z ◆1/2 ! . ✏u r 2 r |@R (rv)|2 + r ✏v 2 Z p . ✏✓0 ||v||2B , where the constant C = ||vs ||1 + sup u2sR (R R0 ). (2.6.18) Z Z ✓ ◆ Z Z p ✓ ◆ p 1 r2 v ✏ p rv @R (rv) 1 ✏us vr @R = us vr ✏ +r + r2 v@R ( ) (2.6.19) us r us us us = (2.6.19.1) + (2.6.19.2) + (2.6.19.3). Z Z Z Z 2 (2.6.19.1) = ✏r v  ✓0 ✏r v!2 , Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 p (2.6.19.2) = ✏r v@R (rv)  ✓0 ✏r v!2 r |@R (rv)|2 , Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 p 1 2 1+ 2 n 2 (2.6.19.3) = ✏v r us @R ( )  ✓0 ✏r v! r usR rv us ✓Z Z ◆1/2 ✓ Z ◆ ✓Z Z ◆1/2 2 n 2 2  ✓0 ✏r v! sup r usR (R R0 ) |@R (rv)| . Thus, (2.6.19) . ✓0 ||v||2B . We now individually compute the terms in (2.6.15): Z Z ✓ ◆ v! vus! p ✏r vs! u  ✏ (||vs! ||1 + ||us! ||1 ) ✓0 ||v||2B . (2.6.20) us u2s p Here again ✏||vs! ||1 is a good term despite the potentially poor dependence of ||vs! ||1 113 on ✓0 . Z Z ✓ ◆ v! vus! p ✏r1+ vs vR 2  ✏ (||rvs ||1 + ||rus! ||1 ) ||v||2B , (2.6.21) us us Z Z ✓ ◆ v ! vu s! ✏r1+ vsR v  (||rvsR ||1 + ||rus! ||1 ) ✓0 ||v||2B , (2.6.22) us u2s Z Z ✓ ◆ ✓Z Z ◆1/2 ✓Z Z ◆1/2 p v! vus! 2 2 2 ✏r us u  r u ✏r v! us u2s ✓Z Z ◆1/2 ✓Z Z ◆1/2 + ||us! ||1 r u2 ✏r v 2 . ✓0 ||v||2B . (2.6.23) Terms (2.6.19) and (2.6.23) require precision with regards to the weight r in our mul- tipliers, as the profile us cannot absorb any factors of r. The results of this step are summarized: p (2.6.14) + (2.6.15) . C(✓0 , ✏)||v||2B + N ||u||2A , C(✓0 , ✏) ⇠ O(✓0 , ✏). (2.6.24) Step III: Laplacian and Lower Order Terms In this step, we treat the terms which contain (u, v) and do not depend on the profiles us , vs in equations (2.2.23) - (2.2.24). For clarity, these terms are summarized here: p ✏ ✏ ✏ 2 3/2 uRR ur u!! + 2 u ✏ v! , and (2.6.25) pr r2 rp r2 ✏ ✏ 2 ✏ ✏ vRR vR 2 v!! + 2 u! + 2 v. (2.6.26) r r r r Applying the multiplier then yields: Z Z Z Z r2 v r1+ v (Equation 2.6.25) ⇥ r @R ( ) ✏ (Equation 2.6.27) ⇥ @! ( )= us us Z Z ✓ p ◆ ✏ ✏ ✏ 2 3/2 r2 v uRR ur u !! + u ✏ v ! r @ R ( ) (2.6.27) r r2 r2 r2 us 114 Z Z ✓ p p ◆ ✓ 1+ ◆ ✏ ✏ 2 ✏ ✏ r v ✏ vRR vR v !! + u ! + v @! . (2.6.28) r r2 r2 r2 us We proceed to individually estimate all of the terms appearing in (2.6.27 - 2.6.28). Z Z Z Z ✓ ◆ r2 v p rv r 1 uRR r @R ( )= uRR r ✏ + @R (rv) + r2 v@R ( ) us us us us = (2.6.29.1) + (2.6.29.2) + (2.6.29.3). (2.6.29) Z Z Z Z Z Z p r1+ v r v p r (2.6.29.1) = ✏ uR @ R ( )=✏ uR + ✏ uR @R (rv) us us us Z Z p 1 p + ✏ uR r1+ v@R ( )  ✏||u||A ||v||B , us Z Z ✓ ◆ Z Z ✓ ◆ 1+ @ R (rv) 1+ u! (2.6.29.2) = uR @ R r = uR @ R r us us Z Z Z Z Z Z p r u! u !R 1 = (1 + ) ✏ uR uR r1+ r1+ uR u! @R ( ) us us us = (2.6.29.2.1) + (2.6.29.2.2) + (2.6.29.2.3). ✓Z Z ◆1/2 ✓Z Z ◆1/2 p p (2.6.29.2.1) . ✏ r u2R r u2! . ✏||u||A ||v||B , Z Z Z Z ✓ ◆ Z 1 1 1 1 1 1 2 (2.6.29.2.2) = r1+ @! u2R = + r1+ @! u2R r1+ u , 2 us 2 us 2 !=✓0 us R Z Z Z Z Z Z 1 p 1 1 (2.6.29.3) = uR @R (r2+ v@R ( )) = (2 + ) ✏ uR r1+ v@R ( ) + uR vR r2+ @R ( )+ us us us Z Z ✓ ◆ 1 r2+ uR v@RR . us For the last term in (2.6.29.3), we use: Z 1 Z 1 sup rm (R R0 )u2sRR  C + ✏3 sup rm (R R0 )|u1err |2 dR R0 R0 Z  C + ✏2 sup rm |u1err |2 dr  C + ✏2 ✓0 1 ||rn ve1 ||2H 3  C + ✏✓0 1 , (2.6.30) Z 1 Z sup rm (R R0 )|usR |  C + ✏ sup rm (R R0 )|u1e |4 dR  C + C(✓0 )✏3 . 4 4 (2.6.31) R0 115 Z 1 1 2 Summarizing, we have shown (2.6.29) . ¯||v||2A + N ( ¯)||u||2A r1+ u . The 2 !=✓0 us R next term in (2.6.25) is: Z Z ✓ ◆ p 1+ p rv r 2 1 ✏uR r ✏ + @R (rv) + r v@R ( ) us us us ✓Z Z ◆1/2 ✓Z Z ◆1/2 ✓Z Z ◆1/2 ✓Z Z ◆1/2 p p  ✏ r1+ 2 uR ✏r v 2 + ✏ 1+ 2 r uR |@R (rv)| 2 ✓Z Z ◆1/2 ✓Z Z ◆1/2 p + ✓0 ||usR rn ||1 u2R r1+ ✏v!2 r . (✓0 + ✏)||u||A ||v||B . (2.6.32) Next, Z Z Z Z Z 1 r2 v ✏ r2 v u! r2 v ✏ u!! @R ( )= u! @ ! @ R ( ) ✏ @R ( ). (2.6.33) r2 us r2 us !=✓0 r 2 us We treat the interior term in (2.6.33), and place the boundary terms in the boundary contribution, , which will be treated in the next subsection: Z Z Z Z ✓ ◆ ✏ r2 v ✏ p rv @R (rv) 1 u! @ ! @ R ( )= u! @ ! ✏ +r + r2 v@R ( ) (2.6.34) r2 us r2 us us us = (2.6.34.1) + (2.6.34.2) + (2.6.34.3), Z Z Z Z ✓ ◆ ✏3/2 3 1 p (2.6.34.1) = u! v ! r 1 +✏ 2 1 r u! v@! . ✏||v||2B , us us Z Z Z Z Z Z ✓ ◆ ✏ 1 ✏ 1 2 1 ✏ 2 1 1 (2.6.34.2) = u! u!! r = @ ! u! r = u! @ ! r us 2 us 2 us Z Z ✏ 1 ✏ 1 u2 r 1 . ✏||v||2B u2 r 1 , 2 !=✓0 ! us 2 !=✓0 ! us Z Z ✓ ◆ Z Z ✓ ◆ 1 1 (2.6.34.3) = ✏u! r v! @R +✏ r u! v@!R . us us Z 1 2 For the final term in (2.6.34.3) , we have used: ✏ sup (R R0 )|@!R ( )| . C, which [0,✓0 ] us can be estimated in a similar way as (2.6.30) - (2.6.31), to obtain: 116 Z Z ✓ ◆ ✓Z Z ◆1/2 ✓Z Z ◆1/2 1 p 1 2 2 p ✏ r u! v@!R  ✏ u2! r ✏|@!R ( )| v r . ✏||v||2B . us us (2.6.35) Z Z ✏ 1 p u! r2 v This establishes term (2.6.34) . u2 r ✏||v||2B 1 ✏ @ R ( ). 2 !=✓0 ! us !=✓0 r2 us We now treat the second-order terms from (2.6.28): Z Z ✓ ◆ 2 1 v! us! ✏ r v!! v 2 = (2.6.36.1) + (2.6.36.2), (2.6.36) us us Z Z Z Z 1 ✏2 1 (2.6.36.1) = ✏2 r 1 v!! v! = r 1 @! v!2 = us 2 us Z Z ✓ ◆ Z Z ✏2 1 2 1 ✏2 1 2 1 ✏2 1 r v! @! + r v! r 1 v!2 , 2 us 2 !=✓0 us 2 !=0 us Z Z ✓ ◆ Z 2 1 us! 2 us! (2.6.36.2) = ✏ r v! @! v 2 ✏ r 1 v! v 2 us !=✓0 us Z Z Z Z ✓ ◆ Z 2 1 2 us! 2 1 1 2 us! =✏ r v! 2 + ✏ r v! v@!! ✏ r 1 v! v 2 . us us !=✓0 us The middle term in (2.6.36.2) requires the following estimate on the profiles: Z Z sup (R R0 )|us!! |2 dR  C + ✏ sup |u1e!! |2 (r)(R R0 )dR Z  C + sup |u1e!! (r)|2 rdr  C + ✓0 1 ||v||2H 3  ✓0 1 ✏ 1 , (2.6.37) Z Z 4 sup |us! | (R R0 )dR  C + ✏ sup |u1e! |4 (r)(R R0 )dR 2 ✓Z Z Z Z ◆  C + ✏✓0 1 rn |u1e! |4 + rn |u1e!! |4  C + ✏✓0 1 ||ve1 ||W 2,4 . (2.6.38) 117 Z Z p ✏2 1 ✏2 1 Thus, we have term (2.6.36) . ✏||v||2B + r 1 v!2 r 1 2 v! Z 2 !=✓ 0 u s 2 !=0 us 2 1 us! ✏ r v! v 2 . The final second-order term from (2.6.28) is: !=✓0 us Z Z ✓ ◆ 1+ v! us! ✏ r vRR v 2 = (2.6.39.1) + (2.6.39.2), (2.6.39) us us Z Z ✓ ◆ Z Z ✓ p ◆ v! (1 + ) ✏r v! r1+ v!R 1 (2.6.39.1) = ✏ vR @R r1+ = ✏ vR + + r1+ @R ( )v! us us us us Z p ✏ 1 . ✏||v||2B r1+ vR 2 , !=✓0 2 us Z Z ✓ ◆ Z Z Z Z us! 2 us! us! (2.6.39.2) = ✏ vR @R r1+ v 2 = ✏ r1+ vR 2 + (1 + )✏ 3/2 vvR r 2 us us us Z Z ✓ ◆ 1 +✏ r1+ vR v@!R . us Z For the final term in (2.6.39.2), we use that ✏ sup rm (R R0 )|us!R |2  C. Thus, [0,✓0 ] Z p ✏ 1+ 2 1 (2.6.39) . ✏||v||2B r vR . Finally, we have the low-order terms from (2.6.27) !=✓0 2 us - (2.6.28): Z Z ✓ ◆ ✓ ◆ ✏ p rv r 1 2✏3/2 p rv r 1 u ✏ + @R (rv) + r2 v@R ( ) v ! ✏ + @ R (rv) + r 2 v@ R ( ) r2 us us us r2 us us us Z Z ✓ ◆ Z Z ✓ ◆ v! us! v! us! + ✏3/2 r vR v 2 2 ✏3/2 r 1 u! v 2 us us us us Z Z ✓ ◆ v! us! ✏2 r 1 v v 2 . C(✏, ✓0 )||v||2B + N ||u||2A , (2.6.40) us us where C(✏, ✓0 ) ! 0 as either ✓0 ! 0 or ✏ ! 0. Summarizing the results of this step: p (2.6.27) + (2.6.28)  C(✏, ✓0 )||v||2B + N ||u||2A + , C(✓0 , ✏) ⇠ O(✓0 , ✏), (2.6.41) 118 where contains the boundary terms from (2.6.29), (2.6.34), (2.6.36), (2.6.39): Z Z Z Z 1 1 2 ✏ 1 2 r2 v ✏2 v!2 = r1+ u u r 1 ✏ u! r 2 @R ( )+ r 1 2 !=✓0 us R 2 !=✓0 us ! !=✓0 us 2 !=✓0 us Z Z Z us! ✏2 1 ✏ 1 2 ✏2 vv! r 1 r 1 2 v! r +1 v . (2.6.42) !=✓0 u2s 2 !=0 us 2 !=✓0 us R Step IV: Pressure Terms In this step, we apply our multiplier to the pressure terms from (2.2.23 2.2.25), which immediately yield: Z Z ✓ ◆ Z Z Z Z Z Z 1 r2 v 1+ v! r2 v r2 v P! r @R PR r @ ! ( ) = P! r 1 @ R ( ) PR r 1 @! ( ) us us us us Z Z Z ✓ ◆ 1 r2 v p v! us! = r P @R ( )+( 1) ✏r P v . (2.6.43) !=✓0 us us u2s The interior term is estimated: Z Z ✓ ◆ ✓Z Z ◆1/2 p v! us! 2 | 1|| ✏r P v || 1| r P ||v||B . (2.6.44) us u2s We estimate the boundary term from (2.6.43) using the stress-free boundary condition in (2.2.30): Z Z Z Z r2 v 1 r 1 2 3/2 1 u! v 1 P := r P @R ( ) = 2✏ u! + 2✏ r + 2✏ r u! v@R ( ) !=✓0 us !=✓0 us !=✓0 us !=✓0 us Z Z r 1 2  ✏ u! + N ✏v 2 r 1 . (2.6.45) !=✓0 us !=✓0 119 Step V: Boundary Terms We rewrite the Boundary contributions of the Navier Stokes terms, starting with (2.6.42): Z Z Z Z 1 1 2 ✏ 1 2 r2 v ✏2 v!2 = r1+ u u r 1 ✏ u! r 2 @R ( )+ r 1 2 !=✓0 us R 2 !=✓0 us ! !=✓0 us 2 !=✓0 us Z Z Z us! ✏2 1 2 1 ✏ 1 2 ✏2 vv! 2 r 1 r v! r +1 vR !=✓0 us 2 !=0 us 2 !=✓0 u s Z Z Z Z ✏ r 1 2 2 r2 v us! ✏2 r 1 = u ✏ r u! @ R ( ) ✏2 r 1 vv! 2 v!2 2 !=✓0 us ! !=✓0 us !=✓0 us 2 !=0 us Z +1 ✏ r 2 vR (2.6.46) 2 !=✓0 us Z 1 Z Z Z ✏ r 1 u! v 1 us! = u2! +✏ 3/2 r ✏ r u! v@R ( ) ✏ 2 r 1 vv! 2 !=✓0 us !=✓0 us !=✓0 u s !=✓0 u2s Z +1 Z ✏ r ✏2 v 2 2 vR r 1 !. (2.6.47) 2 !=✓0 us 2 !=0 us In the equality yielding (2.6.46), the stress-free condition from (2.2.30) was used. Using p p the divergence free condition: u! ✏v = rvR ) r2 vR 2 = u2! + ✏v 2 2 ✏vu! , we can rewrite two of the terms in (2.6.47): Z 1 Z +1 Z Z ✏ r ✏ r ✏2 r 1 3 r 1 u2! 2 vR = v2 + ✏ 2 vu! . (2.6.48) 2 !=✓0 us 2 !=✓0 us 2 !=✓0 us !=✓0 us R We also note that the !=0 in (2.6.47) is of a beneficial sign, and so we only keep treating R the !=✓0 contributions. Summarizing, we have: Z Z Z Z 1 u! v 1 us! ✏2 r 1 = ✏3/2 r ✏ r u! v@R ( ) ✏2 r 1 vv! v2 !=✓0 us !=✓0 us !=✓0 u2s 2 !=✓0 us Z 1 Z 3 r ✏2 2 1 v! + ✏2 vu! r . (2.6.49) !=✓0 us 2 !=0 us Z Z Z We now note that: ✏v 2 r 1  ✓02 ✏v!2 r 1 . Using Young’s inequality we can !=✓0 120 Z Z absorb all of the terms in (2.6.49) and (2.6.45) into either ✏u2! r 1 or ✏v 2 r 1 !=✓0 !=✓0 terms except for the third term in (2.6.49), which we now estimate: Z Z Z us! us! us! ✏2 r 1 2 vv! = ✏ r u R v 2 = ✏ u@R (r v 2 ) !=✓0 us !=✓0 us !=✓0 us Z Z Z ✓ ◆ p u s! u s! 1 = ✏ u r 1 ✏v 2 ✏ r uvR 2 + ✏ r uv@!R (2.6.50) !=✓0 us !=✓0 us !=✓0 us = (2.6.50.1) + (2.6.50.2) + (2.6.50.3). First, we have: ✓Z ◆ 12 ✓Z ◆ 12 ✓Z Z ◆ 12 ✓Z Z ◆ 12 3 (2.6.50.1) . ✏ 2 u 2 v 2  ✏✓02 u2! ✏v!2 . ✏✓02 ||v||2B . !=✓0 !=✓0 p For the second term, we use the divergence free condition vR = ✏ 1r v 1 r u! , yielding: Z Z 1 us! us! (2.6.50.2) = ✏3/2 r uv +✏ r 1 uu! !=✓0 u2s !=✓0 u2s ✓Z ◆ 12 ✓Z ◆ 12 ✓Z ◆ 12 ✓Z ◆ 12 .✏ 3/2 u 2 v 2 +✏ u 2 u2! !=✓0 !=✓0 !=✓0 !=✓0 ✓Z Z ◆ 12 ✓Z Z ◆ 12 ✓Z Z ◆ 12 ✓Z ◆ 12 . ✏✓02 u2! ✏v!2 + ✏✓0 u2! u2! !=✓0 Z Z Z Z . ✏✓02 ||v||2B + ¯✓0 ✏u2! + N ( ¯)✓0 ✏ u2! . ¯✓0 ✏u2! + ✏✓0 ||v||2B . !=✓0 !=✓0 For the third term, we have: ✓ ◆ ✓Z Z ◆ 12 ✓Z Z ◆ 12 p 1 (2.6.50.3) . ✏||@!R ||1 u2! r ✓0 ✏v!2 r . ✓0 ||v||2B . (2.6.51) us We have used that ✓ ◆ 1 ||@!R ||1 . ||us!R ||1 + ||us! ||1 ||usR ||1 . C + ✏||ve1 ||H 4 . ✏ 1/2 . us 121 The results of this step may be summarized as: Z 1 r p | |+| P| . ✏ u2! + C(✓0 , ✏)||v||2B , C(✓0 , ✏) ⇠ O(✓0 , ✏). (2.6.52) !=✓0 us Step VI: Right-Hand Side Z Z Z Z ✓ ◆ r2 v p rv 1 1 f r @R ( )= fr ✏ + r@R (rv) + r2 v@R ( ) us us us us Z Z Z Z Z Z  N ( ¯) f 2 r2+ + ¯ ✏v 2 r + ¯ r |@R (rv)|2 , (2.6.53) Z Z Z Z Z Z v! ✏ gr1+  N ( ¯)✏ g 2 r2+ + ¯ ✏v!2 r , (2.6.54) us Z Z Z Z Z Z us! ✏ gr 1+ v 2  N ( ¯) 2 2+ ✏g r +¯ ✏v 2 r . (2.6.55) us Placing the above steps together finishes the proof of Theorem 2.2.13. 2.7 Pressure Estimate Given P (!, R) 2 L2 (⌦N ), there is a corresponding scaled p(!, r) = P (!, R), whose domain p is ⌦p✏N = (0, ✓0 ) ⇥ (R0 , R0 + ✏N ). Abusing notation, denote the Euclidean counterpart to p(!, r) by p(x, y). RR Definition 2.7.1. L20 denotes the mean-zero subspace of L2 : q0 2 L20 i↵ q0 dxdy = RR q0 rdrd! = 0. Claim 5 (Mean-zero solvability of div). Denote the annular domain ⌦✓0 = (0, ✓0 ) ⇥ 122 (R0 , R0 + ✓0 ). For each q0 2 L20 (⌦✓0 ), there exists a vector field v0 2 H01 (⌦✓0 ) such that div(v0 ) = q0 ,||v0 ||H01 (⌦✓0 )  C0 ||q0 ||L2 (⌦✓0 ) , where C0 is independent of small ✓0 . Proof. The solvability of div : H01 ! L20 is well known (see (Orl98, pp. 26-28)). The important point for our analysis is that the constant C0 is independent of small ✓0 . This ⇣ ⌘ diam(⌦✓0 ) 2 is guaranteed by (Gal11, p. 162, estimate III.3.4), in which it is shown C0 . R , where R is the radius of a ball BR ⇢ ⌦✓0 with respect to which ⌦✓0 is starlike. In our case, R ⇡ diam(⌦✓0 ). The claim is proven. Claim 6. For each q 2 L2 (⌦✓0 ), there exists a vector field v such that v = 0 on {R = R0 , R0 + ✓0 }, {! = 0}, and ||v||H 1 (⌦✓0 )  C||q||L2 (⌦✓0 ) , where C is independent of small ✓0 . ⇣R R ⌘ Proof. Similar to (Orl98, page 27), define the mean-zero function q0 = q ⌦ ✓0 qrdrd! div(w), ⇣ ⌘ where w = 6✓0 4 (r R0 )(✓0 r + R0 )!, 0 . By direct computation, Z Z div(w)rdrd! = 1, ||div(w)||L2 . ✓0 1 , ||w||H 1 . ✓0 1 . (2.7.1) ⌦ ✓0 Denoting by v0 the vector field guaranteed by Claim 5 for the function q0 , ⇣Z Z ⌘ ||v0 ||H 1 . ||q0 ||L2 . ||q||L2 + |q|rdrd! ||div(w)||L2 . ||q||L2 + ||q||L2 ✓0 ✓0 1 . ||q||L2 . (2.7.2) The factor of ✓ in the final inequality in (2.7.2) arises from Holder’s inequality: Z Z ⇣Z ✓0 Z R0 +✓0 ⌘ 12 |q|rdrd!  ||q||L2 rdrd! . ||q||L2 ✓0 . (2.7.3) ⌦ ✓0 0 R0 ⇣R R ⌘ The desired vector field is now v = v0 + qrdrd! w. Clearly v vanishes on the 123 required components of the boundary, and we have: ⇣Z Z ⌘ ||v||H 1  ||v0 ||H 1 + |q|r ||w||H 1 . ||q||L2 + ||q||L2 ✓0 ✓0 1 = ||q||L2 . (2.7.4) The claim is proven. Claim 7. There exists a vector field F(x, y) = (f (x, y), g(x, y)) such that div(F) = p(x, y), and: ||F||H 1 (⌦p✏N )  C||p||L2 (⌦p✏N ) , (2.7.5) where the constant C is independent of ✏, N , small ✓0 , and where F vanishes on the p Dirichlet portions of the boundary {R = R0 }, {! = 0}, {R = R0 + ✏N }. Proof. Divide the domain ⌦p✏N into ⌦k = {0, ✓0 } ⇥ {R0 + k✓0 , R0 + (k + 1)✓0 }. By Claim 6, there exists a vector field, Fk , such that div(Fk ) = p on ⌦k , Fk (!, R0 + ✓0 k) = Fk (!, R0 + (k + 1)✓0 ) = Fk (0, r) = 0, and ||Fk ||H 1 (⌦k )  C||p||L2 (⌦k ) , where C does not depend on X X X small ✓0 . Define F = Fk . Then ||F||H 1 (⌦) = ||Fk ||H 1 (⌦k )  C ||p||L2 (⌦k ) = k k k C||p||L2 , and div(F) = p. Finally F satisfies the required boundary conditions. The claim is proven. The vector field F can be expressed in the polar coordinate basis e✓ and er , and as functions of !, r, in which case div(F) = r ! + r + r = p(!, r). Converting estimate (2.7.5) to polar coordinates reads: Z Z ✓ 2 2 ◆ Z Z 2 + 2 + ! + ! + 2 r + 2 r rdrd! . p2 (!, r)rdrd!. (2.7.6) ⌦p✏N r2 r2 ⌦p✏N p 1 Define a(!, R) = (!, r) and ✏b(!, R) = (!, r), so a! (!, R) = ! (!, r), p✏ aR (!, R) = 124 p r (!, r). Also, ✏b! (!, R) = ! (!, r), bR (!, R) = r (!, r). Note that this scaling is the same as the Prandtl scaling. Scaling all of the terms in (2.7.6) yields: Z Z ✓ ◆ Z Z a2! b2! 1 2 a2 + ✏b2 + + ✏ + a + b 2 rdRd! . P (!, R)2 rdRd!, (2.7.7) ⌦N r2 r2 ✏ R R ⌦N and the scaled divergence equation: a! p b + ✏ + bR = P. (2.7.8) r r The admissible weights in (2.7.7) must be generalized to r for 2 [0, 1]. The next claim shows this is possible as long as a small error is made in the scaled divergence equation (2.7.8). Claim 8. Given P 2 L2 (⌦N ) there exists a vector field A1 = (a, b) such that Z Z ✓ ◆ Z Z a2! b2! 1 2 a2 + ✏b2 + + ✏ + a + b 2 r dRd! . P (!, R)2 r dRd!, (2.7.9) ⌦N r2 r2 ✏ R R ⌦N where the constant is independent of N, ✏, and small ✓0 . A1 vanishes on {R = R0 , R0 + N }, {! = 0}, and satisfies the scaled divergence equation: a! p b ⇣1 ⌘p 1 + ✏ + bR = P + ✏br . (2.7.10) r r 2 1 Proof. Given P , define P¯ = P r 2 2 . Applying the procedure culminating in estimate (2.7.7) a, ¯b) satisfying: to P¯ gives a vector field (¯ Z Z ✓ ¯b2 ◆ Z Z Z Z ¯ ¯2! a 1 2 ¯2 a2 2 ! ¯ + ✏b + 2 + ✏ 2 + a¯R + bR rdRd! . P¯ 2 r = P 2 r , (2.7.11) ⌦N r r ✏ ⌦N ⌦N 125 and ¯! p ¯b ¯ a 1 + ✏ + bR = P¯ = P r 2 2 . (2.7.12) r r , ¯br 2 1 1 Define A1 = (a, b) = (¯ ar 2 2 2 ). One readily computes the scaled divergence of A1 to check equation (2.7.10), as well as the desired estimates (2.7.9). It is also clear that a, ¯b) vanishes on those components. A1 vanishes on {R = R0 , R0 + N }, {! = 0} because (¯ The claim is proven. We now test against our equation against the multiplier (ar , ✏r b) in several steps. Step I: Pressure Terms Applying the the multiplier (ar , ✏r b) to the terms in equation (2.2.23 - 2.2.25) containing the pressure, P , yields: Z Z Z Z Z Z Z Z Z Z Z 1 1 1 1 p P! ar + PR br = P a! r + P ar P r bR r ✏bP !=✓0 Z Z Z Z Z 3 p = P 2 r + (1 ) ✏br 1 P + P ar 1 . (2.7.13) 2 !=✓0 We have used the relation (2.7.10). Using (2.7.9), the middle term above can be esti- mated: Z Z ✓Z Z ◆ 12 ✓Z Z ◆ 12 1 p |3 | |P r 1 ✏b| . |1 | P 2r ✏b2 r 2 2 ✓Z Z ◆ 12 ✓Z Z ◆ 12 Z Z . |1 |✓0 P 2r ✏b2! r 2 . (1 )✓0 P 2r . (2.7.14) 126 Step II: Laplacian Terms and Lower Order Terms In order to obtain the proper boundary cancellation, we use the representation of the Lapla- cian given in (2.5.3), (2.5.4). Applying our multiplier then yields: Z Z " p # ✏ 2✏ ✏ 3 3/2 ✏ uRR uR u!! + 2 u ✏ v! v!R ar r r2 r r2 r Z Z " p p # 2 ✏ ✏ 3 ✏ ✏ u!R + 2vRR vR 2 v!! + 2 u! + 2 2 v ✏br . (2.7.15) r r r r r We successively treat each term in (2.7.15), starting with the important, high order terms: Z Z Z Z Z Z 1 p uRR r a = r uR a R + r ✏auR ✓Z Z ◆1/2 ✓Z Z ◆1/2 Z Z p  r a2R r u2R + r 1 ✏auR ✓Z Z ◆1/2 ✓Z Z ◆1/2 Z Z p p  ✏ P (!, R)2 r r u2R + r 1 ✏auR p  ✏||P ||L2⇤ , ||u||A , (2.7.16) Z Z Z Z Z 2 2 2 2 ✏r au!! = 2 ✏r a ! u! 2✏ r au! !=✓0 ✓Z Z ◆1/2 ✓Z Z ◆1/2 Z ✏ a2! r 2 u2! r 2 2✏ r 2 au! !=✓0 Z 2  ✏||P ||L2⇤ , ||v||B 2✏ r au! , (2.7.17) !=✓0 Z Z Z Z Z Z 3/2 1 2 ✏vRR r b = 2 ✏r vR bR + 2✏ r bvR ✓Z Z ◆1/2 ✓Z Z ◆1/2 Z Z 2 ✏ r vR r b2R + 2✏3/2 r 1 bvR  ✏||v||B ||P ||L2⇤ , , (2.7.18) 127 Z Z Z Z Z ✏2 v!! br 2 = ✏2 v ! b! r 2 ✏2 v! br 2 !=✓0 ✓Z Z ◆1/2 ✓Z Z ◆1/2 Z ✏ ✏v!2 r 2 ✏b2! r 2 ✏2 r 2 v! b, !=✓0 Z  ✏||v||B ||P ||L2⇤ , ✏2 r 2 v! b, (2.7.19) !=✓0 Z Z Z Z Z 1 1 1 ✏ u!R br =✏ r u R b! ✏ r uR b !=✓0 ✓Z Z ◆1/2 ✓Z Z ◆1/2 Z p  ✏ r u2R ✏b2! r 2 ✏ uR br 1 !=✓0 Z p 1  ✏||u||A ||P ||L2⇤ , ✏ uR br . (2.7.20) !=✓0 The remaining terms can be estimated through Young’s inequality: Z Z p 1 2 ✏uR ar + ✏aur 3✏3/2 r 2 v! a + ✏v!R ar 1 + 2✏3/2 bvR r 1 + 3✏3/2 bu! r 2 + ✏2 r 2 bv  C(✏, ✓0 )||P ||L2⇤, (||u||A + ||v||B ) , (2.7.21) where C(✓0 , ✏) ! 0 as ✏, ✓0 ! 0. Summarizing the results from this step, Z Z (Equation 2.5.3) ⇥ ar + (Equation 2.5.4) ⇥ br  C(✓0 , ✏)||P ||L2⇤, (||u||A + ||v||B ) Z Z Z 2✏ r 2 au! ✏2 r 2 v! b ✏ uR br 1 , !=✓0 !=✓0 !=✓0 (2.7.22) where C(✓0 , ✏) ! 0 as either argument ! 0. 128 Step III: Boundary Contributions We now treat the boundary contributions from the previous two steps. From (2.7.13) and (2.7.22), all of the boundary terms are: Z Z Z Z 1 ✏ uR br ✏2 r 2 v! b 2✏ r 2 au! + P ar 1 = 0. (2.7.23) !=✓0 !=✓0 !=✓0 !=✓0 We have used the stress free boundary conditions from (2.2.30). Step IV: Profile Terms In this step, we give estimates on the terms from (2.2.23) - (2.2.24) which depend on the profiles us , vs . Precisely, we successively treat: Z Z " p p # 1 1 ✏ ✏ us u! + us! u + usR v + vs uR + vs u + us v ar (2.7.24) r r r r Z Z " # 1 1 2 1 + us v! + vs! u + vs vR + vsR v p us u ✏br . (2.7.25) r r r ✏ Throughout the following estimates, we use the bounds on the profiles us , vs that were proven in (2.5.19) - (2.5.26). Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 us u! ar 1  ✓0 ||us ||1 u2! r a2! r 2 . ✓0 ||v||B ||P ||L2⇤ , , (2.7.26) Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 1 us! uar  ||us! ||1 u2 r r 2 2 a ✓Z Z ◆1/2 ✓Z Z ◆1/2  ✓02 u2! r r 2 2 a!  ✓02 ||v||B ||P ||L2⇤ , , (2.7.27) Z Z ✓ Z ◆1/2 ✓Z Z ◆1/2 ✓Z Z ◆1/2 usR var  ✓0 sup u2sR rn (R R0 ) r 2 vR a2! r 2 129 . ✓0 ||v||B ||P ||L2⇤ , , (2.7.28) Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 vs uR r a  ||vs r||1 2 r uR a2 r 2 . ✓0 ||u||A ||P ||L2⇤ , , (2.7.29) Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 p p p ✏vs uar 1 . ✏ u2 r a2 r 2 . ✓0 ✏||v||B ||P ||L2⇤ , , (2.7.30) Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 p ✏us var 1 . v2 r a2 r 2 . ✓0 ||v||B ||P ||L2⇤ , , (2.7.31) Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 ✏ us v! br 1 = ||us ||1 ✏v!2 r ✏b r2 2 . ✓0 ||v||B ||P ||L2⇤ , , (2.7.32) Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 p p ✏ vs! ubr 1 . ✏ u2 r ✏b2 r 2 . ✏✓02 ||v||B ||P ||L2⇤ , , (2.7.33) Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 p p ✏ vs vR br  ✏ r2+ vR 2 ✏b2 r 2 . ✏✓0 ||v||B ||P ||L2⇤ , , (2.7.34) Z Z ✏ vsR vbr  ✓0 ||v||B ||P ||L2⇤ , , (2.7.35) Z Z ✓Z Z ◆1/2 ✓Z Z ◆1/2 p 2 ✏us ubr 1 . u r 2 2 ✏b r 2  ✓02 ||v||B ||P ||L2⇤ , . (2.7.36) We summarize the results of this step: Z Z (Equation 2.5.15) ⇥ ar + (Equation 2.5.16) ⇥ ✏br . C(✓0 , ✏)||P ||L2⇤, (||u||A + ||v||B ) , (2.7.37) where C(✓0 , ✏) ! 0 as either ✓0 , ✏ ! 0. Remark. The choice of weight on the multiplier (ar , ✏br ) is delicate in the sense that it is “critical” in several of the profile estimates given above. Specifically, in calculation (2.7.26), ||us ||L1 is unable to absorb any factors of r, and so the weight after applying the multiplier 1 (which in this case is us u! ar ) must exactly match those of ||v||B and ||P ||L2⇤, . 130 Step V: Right-Hand-Side Finally, we apply our multiplier (ar , ✏br ) to the right hand side of the system (2.2.23) - (2.2.25), which immediately yields: Z Z Z Z Z Z Z Z f ar + ✏gr b . f 2 r + ✏g 2 r + ¯ r ✏b2 + ¯ r a2 Z Z Z Z . f 2r + ✏g 2 r + ¯||P ||2L2⇤ , . (2.7.38) Putting the above steps together yields the desired Pressure estimate in Theorem 2.2.14. 2.8 Linearized Existence and Uniqueness for Navier- Stokes Remainders In this section, we prove Theorem 2.2.15. The full estimate for the linear problem, given in (2.2.48) is uniform in N , ✏, and small ✓0 . It was established in Corollary 2.3.8 that the spaces X and B endowed with their respective norms are Banach spaces, and so we can establish existence and uniqueness of the linear problem by applying Schaefer’s Fixed Point Theorem. We must apply the fixed point theorem for each fixed N , obtaining a solution uN , v N . We can subsequently send N ! 1 as estimate (2.2.48) is uniform in N . For the following discussion, we fix an ✏ and an N < 1. Denote the Prandtl-layer version of the Stokes operator as S⇤ , so S⇤ [u, v, P ] = (f, g) () [u, v, P ] satisfy the linear system: p ✏ ✏ ✏ 2 3/2 1 u uRR ur u!! + 2 u ✏ v! + P! = f˜, (2.8.1) r p r2 r p r2 r v ✏ ✏ 2 ✏ ✏ 1 vRR vR v!! + 2 u! + 2 v + PR = g˜, (2.8.2) ✏ r r2 r r ✏ 131 u! + @R (rv) = 0. (2.8.3) Claim 9. S⇤ 1 : L2 (⌦N ) ! L2 (⌦N ) is compact. 1 Proof. The usual Stokes solution operator, S , is compact on bounded domains from L2 ! L2 . Suppose we take a sequence {fn , gn } 2 L2⇤, (⌦N ) which is uniformly bounded in the norm. Then define [f¯, g¯]n (!, r) = [fn , gn ](!, R(r)), which are uniformly bounded p in L2 (⌦N 0 ) where N 0 = R0 + ✏N . This implies [¯ un , v¯n ] = S(fn , gn ) has a subsequence which converges in L2 (⌦N 0 ). Define [un (!, R), vn (!, R)] = [u(!, r), p1✏ v(!, r)]. un , vn solve the Stokes ⇤ operator, so [un , vn ] = S⇤ 1 (fn , gn ). Moreover, by scaling, un , vn also have a convergent subsequence. Therefore S⇤ 1 is compact on L2 (⌦N ). We will need a version of Korn’s Inequality using the polar coordinate basis, for which we adapt the proof given in (Cia10). Claim 10 (Lions’ Lemma). Let U be a bounded, open set with Lipschitz boundary. Suppose @! a distribution u 2 D0 (U ) has ru = r u, @r u 2H 1 (U ). Then u 2 L2 (U ). u! Proof. Expressing r = cos !uy sin !ux and ur = cos !ux + sin !uy , and the obvious 1 @! 1 inverse relationships, we have that (ux , uy ) 2 H () r u, @r u 2H . From here, the claim follows from Lions’ Lemma, found in (Cia10, Thm 1.1). Definition 2.8.1. Let e(¯ u, v¯) be the symmetric gradient: 0 1 u ¯! 1 v ¯! B r 2 r +u ¯r C e(¯ u, v¯) = @ A. (2.8.4) 1 v ¯! 2 r +u ¯r v¯r Denote the norm ||¯ u, v¯||E = ||¯ u, v¯)||L2 + || u¯r!2 , v¯r!2 ||L2 . u, v¯||L2 + ||e(¯ 132 Claim 11 (Korn-type Inequality). For the solutions (u, v), we have the variant of Korn’s inequality: u, v¯||H 1 (U ) . ||¯ ||¯ u, v¯||E , (2.8.5) where the constant depends on the domain, U , but is independent of small . Remark. As noted in (Cia10), these Korn-type inequalities do not make any restrictions on the behavior of (u, v) on the boundary @U . Proof. Using standard arguments, E is a Banach space. We now show that E (U ) coincides with H 1 (U ). Clearly, H 1 (U ) ⇢ E (U ) continuously: ||¯ u, v¯||E  C||¯ u, v¯||H 1 (U ) where the constant is independent of small . The reverse direction is delicate and requires a use of u, v¯||E < 1, so eij 2 L2 ) reij 2 H Lions’ Lemma. We suppose ||¯ 1 . The components of r2 (¯ u, v¯) can be expressed in terms of reij via: u ¯!! v¯!r u¯!r u ¯! u ¯! v¯!! @! u ¯r! v¯! v¯!r 2 , v¯rr , , = @r ( ) + 2 , 2 = 2 e12 (¯ u, v¯) ,u ¯rr = 2@r (e12 (¯ u, v¯)) + 2 r r r r r r r r r r2 Thus, r2 (¯ u, v¯) 2 H 1 u, v¯) 2 L2 , so (¯ , so by Lions’ Lemma, r(¯ u, v¯) 2 H 1 . The identity map i : H 1 (⌦) ,! E is continuous and bijective, with bound independent of small , and 1 therefore by Banach’s inverse mapping the inverse map i : E ,! H 1 is also bounded. 1 By observing that ||1||E = ||1||H 1 = ||1||L2 (U ) , we see that the operator norm ||i ||op independent of small . Claim 12. "Z Z Z Z # 2 u2! 2 2 2 2 v! 2 2 2 ✏u r + ✏ + uR r + ✏ v r+✏ + ✏vR r r r Z Z Z Z Z Z u2! 2 v! 2 .2 2 2 2 (✏u + ✏ v )r + 2✏ 2 2 + 2✏vR r + uR r + ✏ + 2✏uR v! . (2.8.6) r r 133 Proof. Expanding the definition of eij in (2.8.5), multiplying by 2 yields, and absorbing the || u¯r!2 , v¯r!2 ||L2 term to the left-hand-side gives: Z Z Z Z Z Z ¯2! u v¯2 u, v¯||2H 1 . 2 2||¯ u2 + v¯2 ) + (¯ 2 vr2 r + ! + u + 2¯ ¯2r r + 2 u ¯r v¯! . (2.8.7) r r Given our solutions (u, v), we define p u ¯(!, r) = u(!, R), v¯(!, r) = ✏v(!, R). (2.8.8) Scaling back to (!, R) coordinates yields the desired result: "Z Z Z Z # 2 u2! 2 2 2 2 v! 2 2 2 ✏u r + ✏ + uR r + ✏ v r+✏ + ✏vR r r r Z Z Z Z Z Z u2 v2 .2 (✏u2 + ✏2 v 2 )r + 2✏ ! + 2✏vR2 r + u2R r + ✏2 ! + 2✏uR v! . (2.8.9) r r Proof of Theorem 2.2.15. Consider the following map T (u, v) = (T 1 (u, v), T 2 (u, v)), p p 1 1 ✏ ✏ T 1 (u, v) := f [ us u! + us! u + usR v + vs uR + vs u + us v] + u, (2.8.10) r r r r 1 1 2 1 v T 2 (u, v) := g [ us v ! + vs! u + vs vR + vsR v p us u] + . r r r ✏ ✏ T is a bounded, affine map from H⇤1 ! L2⇤ . Our solution is a fixed point of S⇤ 1 T , which is a compact map from H⇤1 ! H⇤1 . Let u , v denote solutions to: p ✏ ✏ ✏ 2 3/2 P (1 )u uRR uR u + u ✏ v! + ! r r2 !! r2 r2 r 134 p p 1 1 ✏ ✏ = f [ us u! + us! u + usR v + vs uR + vs u + us v ], (2.8.11) r r r r p p v ✏ ✏ 2 ✏ ✏ P (1 ) vRR v v + 2 u! + 2 v + R ✏ r R r2 !! r r ✏ 1 1 2 1 = g [ us v! + vs! u + vs vR + vsR v p us u ]. (2.8.12) r r r ✏ We obtain bounds uniform in in the following way. Select some 0 < 0 << ✏ < 1. For 0   1, the energy, positivity, and pressure estimates in the previous sections can be repeated to obtain uniform bounds (where the size will depend on 0 ). For 0  < 0 << ✏, we must only perform an energy estimate by applying the multiplier (ru, ✏rv) to (2.8.11 - 2.8.12). For the upcoming calculation we drop the superscript on u , v . Z Z ✓ p ◆ Z Z Z ✏ u!! u2 uu! uRR uR 2✏ 2 ru = u2R r + 2✏ ! 2✏ , (2.8.13) r r r !=✓0 r Z Z p Z Z 2 Z ✏ v!! 2 2 v! vv! ( 2vRR vR ✏)✏vr = 2✏ v R r + ✏ ✏2 , (2.8.14) r r2 r !=✓0 r Z Z Z Z 2✏2 v 2 + ✏u2 (1 ) (u2 + v 2 )r + , (2.8.15) r Z Z ✓Z Z 2 ◆ 12 ✓Z Z 2 ◆ 12 3/2 v! u p 2 v! u 3✏  ✏ ✏ , (2.8.16) r r r Z Z ✓Z Z " ◆ 12 ✓Z Z ◆ 1 ✓Z Z ◆ 12 # 3/2 u! v 3/2 p 2 u2! 2 2 vR 3 ✏ ✏ vvR  ✏ v r ✏ + ✏ , (2.8.17) r r r Z Z Z Z Z ✏v!R u ✏u!R v = 2✏ uR v ! ✏uR v, (2.8.18) !=✓0 Z Z Z P! u + PR rv = P u. (2.8.19) !=✓0 The boundary contributions from (2.8.13, 2.8.14, 2.8.18, 2.8.19) cancel, using the same calculation as (2.5.11) and (2.5.29): 135 Z Z Z Z uu! vv! Pu ✏uR v 2✏ ✏2 = 0. (2.8.20) !=✓0 !=✓0 !=✓0 r !=✓0 r Summarizing the interior terms, and applying (2.8.6) Z Z u2! v2 2✏2 v 2 + ✏u2 u2R r + 2✏ + 2✏vR2 r + ✏2 ! + 2✏uR v! + (1 )(u2 + v 2 )r + +J r r r "Z Z Z Z # Z Z 2 u2! 2 2 2 2 v! 2 2 (1 ) 2 ✏u r + ✏ + uR r + ✏ v r+✏ + ✏vR r + (u2 + v 2 )r + J, r r 2 (2.8.21) where Z Z ⇣ ✓Z Z ◆ 12 ✓Z Z ◆ 12 ✓Z Z ◆ 12 ✓Z Z ◆ 12 u! v v! u ⌘ p 2 2 v! u2 u2 v2 J = 3✏ 3/2  ✏ ✏ +✏ ✏ ! . r r r r r r (2.8.22) On the right-hand-side, we have Z Z " p p # 1 1 ✏ ✏ f [ us u! + us! u + usR v + vs uR + vs u + us v] ur r r r r Z Z Z Z u2 . f 2r + u2 r + ! + u2R r + v 2 r, (2.8.23) r Z Z " # 1 1 2 1 g [ us v! + vs! u + vs vR + vsR v p us u] ✏vr r r r ✏ Z Z Z Z 2 v! . g2 r + + u2 r + v R 2 r + v 2 r, (2.8.24) r 136 and therefore since 0   0 << ✏, we have Z Z Z Z ⇣ ⌘ Z Z ⇣ ⌘ (u2 + v 2 )r + |r✏ u|2 + ✏|r✏ v|2 r . f 2 + ✏g 2 r, (2.8.25) uniformly in . An application of Schaefer’s fixed point theorem then shows there exists a solution uN , v N in the function space with the weak norm which we estimated (weak in the sense of weights and in ✏), and by linearity of our equation, applying the estimate above implies uniqueness of this solution. With existence and uniqueness in the weak space in hand, we can bootstrap: each (uN , v N ) is also an element of the space X(⌦N ), B(⌦N ) because ⌦N is a bounded domain. Because this solution is an element of X, B, we can apply the strong estimate [2.2.48] which is uniform in ✏, N , and small ✓0 . We can therefore send N ! 1 to obtain a global in R solution which also obeys estimate (2.2.48). This argument results in the proof of Theorem 2.2.15. 2.9 High Regularity Estimates In this section we prove Lemma 2.2.16, the high regularity estimate for the solution u, v to the problem (2.2.49 - 2.2.52). Notationally, we will keep the M , where M > 0, as a large negative exponent for ✏, not taking care to rename di↵erent exponents as it is inconsequential to the estimate we are proving. Proof of Lemma 2.2.16. We rescale via: p ✓0 u ¯(!, r) = u(✓0 !, ✓0 R(r)); v¯(!, r) = ✏v(✓0 !, ✓0 R(r)); P¯ (!, r) = P (✓0 !, ✓0 R(r)). (2.9.1) ✏ 137 The equation satisfied by the normalized profiles is: u ¯ u ¯!! u ¯ 2 P! ✓2 u ¯rr ✓0 2 + ✓02 2 2 ✓0 v¯! + = 0 f˜ := f¯(!, r), (2.9.2) r r r r r ✏ v¯ v!! 2 ✓02 ✓ 2 ¯r = p0 g˜ := g¯(!, r). v¯rr ✓0 + ✓ 0 u ¯ ! + v ¯ + P (2.9.3) r r2 r2 r2 ✏ H˙ ⇤2 Estimates Using the standard regularity theory for Stokes equation, we can obtain H˙ 2 estimates for u ¯ and v¯ away from the corners of ⌦. Let (!, r) denote a cuto↵ function which is supported near the corners of the domain in such a way that (!, r) 2 supp( ) ) r R0  1. Define 1 (!, r) =1 ¯1 , v¯1 , P¯1 := (!, r pR✏ 0 ). Then the equation for u 1 ¯, v¯, P¯ is given by: · u u ¯1 2 P¯1! ¯ 2 1 ¯ u ¯1 v¯1! + = 1f 2r 1 · r¯ u 1u ¯ 1! v ¯+ 1! P r2 r 2 r r2 r2 u ¯1 2 2✓0 (1 ✓02 ) v¯1! , (2.9.4) r2 r2 2 1 ¯ + 2 ✓0 @ ! ( v¯1 + ¯1! + 2 v¯1 + P¯1r = u 1g ¯ 2r 1 · r¯ v 1v ¯+ @r ( 1 )P 1 )¯ u1 r2 r r2 v¯1 2 2✓0 + (1 ✓02 ) +( )¯ u1! . (2.9.5) r2 r2 p As (!, r) 2 supp r 1 [ supp 1, we have r R0  ✏, and so by the standard Stokes estimate (with inhomogeneous divergence, see Remark 2.9): 2 1 u ¯1 2 2✓0 ||2r 1 · r¯ u 1u ¯ 1! v ¯+ 1! P ¯ (1 ✓02 ) v¯1! ||L2 . ✏ 1 ||f¯, g¯||L2 , r2 r2 r2 r2 (2.9.6) || 2r 1 · r¯ v 1v ¯+ @r ( ¯ + 2 ✓0 @ ! ( 1 )P 1 )¯ u1 + (1 ✓02 ) v¯1 +( 2 2✓0 u1! ||L2 . ✏ )¯ 1 ||f¯, g¯||L2 . r2 r2 r2 (2.9.7) ⇣ ⌘ The ✏ 1 in estimate (2.9.6 - 2.9.7) arises from 1 = (!, r pR✏ 0 ) . Thus using the 138 standard H˙ 2 Stokes estimate: ⇣ p ⌘ ||¯ v1 ||H˙ 2 + ||P¯1 ||H˙ 1 . ✏ u1 ||H˙ 2 + ||¯ 1 ||f¯||L2 + ||¯ g ||L2 . ✏ M ||f˜||L2⇤ + g ||L2⇤ , (2.9.8) ✏||˜ for some potentially large power M . We have used the calculation, according to the definition of f¯ in (2.9.2) Z Z Z Z Z Z p ||f¯||2L2 = f¯2 rdrd! = ✏ f¯2 rdRd! = ✓02 ✏ 3/2 f˜2 rdRd! = ✓02 ✏ 3/2 ||f˜||2L2⇤ , (2.9.9) and analogously for g. Defining the corresponding profiles in Prandtl variables by in- 1 verting (2.9.1) above: u1 (✓0 !, ✓0 R) = u ¯1 (!, r), v1 (✓0 !, ✓0 R) = p v¯ (!, r), P1 (✓0 !, ✓0 R) ✏ 1 = ✏ ¯ ✓0 P1 (!, r), we have: p ||u1 , v1 ||H˙ 2  ✏ M u1 , v¯1 ||H˙ 2 . ✏ ||¯ M ||f˜, g ||L2⇤ . ✏˜ (2.9.10) ⇤ Since M can be arbitrarily large, the quantity appearing on the left of (2.9.10) above is independent of ✏. We can also obtain weighted estimates by repeating the above analysis 1 1 ¯1 r 2 + 2 , v¯1 r 2 + 2 : for the equation for u ⇣ p ⌘ ||r1/2+ /2 (u1 , v1 ) ||H˙ 2 . ✏ M ||r1/2+ /2 f˜, g ||L2⇤ . ✏˜ (2.9.11) ⇤ For the weighted estimate (2.9.11), we use that r  C on supp @ k 1, k 1. This establishes the desired estimate for u1 , v1 . Remark. We apply the Stokes estimate with inhomogeneous divergence because the 1 1 u1 r 2 + 2 , v¯1 r 2 + 2 ) has divergence: weighted, cuto↵ vector field (¯ @! 1 + r2+2 1 1 1 1 ⇣ u ⌘ 1 (r 2 2 u ¯1 ) + v¯1 + r 2 + 2 v¯1r + r 2 2 v¯1 = r 2 + 2 1! + 1r v + r2 2 v¯1 . (2.9.12) r r r 139 M Therefore, a term ✏ ||u, v||L2⇤, appears on the right-hand side of the estimate (2.9.8), p which is in turn controlled by ✏ M ||f˜, ✏˜ g ||L2⇤,2+ by our uniform energy estimates. H 3/2 Estimates: Let 2 = (!, r pR✏ 0 ), so p (!, r) 2 supp( 2) )r R0  ✏ and R R0  1. (2.9.13) u2 , v¯2 , P¯2 ) = Define (¯ 2 · (u, v, P ). By (OS95), the Stokes problem has an H 3/2 estimate: ⇣ p ⌘ u2 , v¯2 ||H 3/2 + ||P¯2 ||H 1/2 . ✏ ||¯ M ||f˜||L2⇤ + g ||L2⇤ . ✏||˜ (2.9.14) We must relate the H 3/2 norm of u ¯2 , v¯2 to that of its scaled counterpart [u2 , v2 ](!, R), given again by inverting transformation in equation (2.9.1). Claim 13. Define the transformation : R2 ! R2 to be defined in polar coordinates via (!, R) = (!, r). Denote by u(!, R) = u ¯(!, r) = u ¯ . Then ||u||H 3/2 = ||¯ u ||H 3/2 . ||¯ u||H 3/2 . Here the constant depends on the derivatives of (which in turn depend on ✏). Proof. The transformation : (!, R) ! (!, r) is bijective and has derivatives which are bounded above and below as a map from ⌦ ⇢ R2 ! ⌦ ⇢ R2 . Indeed, in the polar coordinate basis: 0 1 0 1 1 1 C B 1 0 C ! B R R r (!, R) = @ 2 A=@ p A. 2 R ! R 0 ✏ 1 From here it is easy to see that is Bilipschitz, and | det |, | det | are bounded 140 above and below, keeping in mind that the coordinate basis are functions of !. We may decompose ||u||H 3/2 = ||u||H 1 + ||u||H 1/2 . Using the definition of weak derivative and an approximation argument, it is easy to see the usual chain rule holds, namely ru(x) = uT ( (x))D (x). Moreover, since the derivatives of r¯ are bounded above and below, we have by the change of variables formula: ||u||H 1 = ||¯ u ||H 1 . ||¯ u||H 1 . We must now treat the H 1/2 portion. As shown in (NPV12), the H 1/2 norm of any function u is equivalent to the Gagliardo semi norm, which is defined as follows (where we calculate n + sp = 2 + 12 2 = 3): Z Z 2 |u(x) u(y)|2 [u] = dxdy. ⌦ ⌦ |x y|3 Therefore, Z Z 2 2 |¯ u( (x)) u ¯( (y))|2 [u] = [¯ u ] = dxdy ⌦ ⌦ |x y|3 Z Z u(x0 ) u |¯ ¯(y 0 )|2 = 1 (x0 ) 1 (y 0 )|3 | det D | 1 dx0 dy 0 ⌦ ⌦ | Z Z |¯u(x0 ) u ¯(y 0 )|2  || det D ||1 1 (x0 ) 1 (y 0 )|3 dx0 dy 0 ⌦ ⌦ | Z Z u(x0 ) u |¯ ¯(y 0 )|2 0 0 . dx dy = [¯ u] 2 . ⌦ ⌦ |x0 y 0 |3 We have used that | det D | is bounded above and below and that is Bilipschitz. By using Claim 13, we have: ⇣ p ⌘ ||u2 , v2 ||H 3/2 . ✏ M u2 , v¯2 ||H 3/2 . ✏ ||¯ M ||f˜, g ||L2⇤ . ✏˜ 141 We can arbitrarily weight these norms due to (2.9.13). This concludes the proof of Lemma 2.2.16. We are now able to control the high-regularity quantities appearing in || · ||Z given in equation (2.2.20): Proof of Theorem 2.2.17. We first address the uR term in term (2.2.20). Using interpolated Holder and the uniform energy estimates from estimate (2.2.48): p ||uR ||L2q  ||uR ||✓L2 ||uR ||1L4 ✓ . ||f˜, g ||✓L2 ✏˜ ||uR ||1L4 ✓ . (2.9.15) ⇤,q+↵ ⇤,1+ ↵ ⇤,2+ 2↵ ⇤,2+ ⇤,2+ 2↵ q q q We have used that ↵/q  /2 in order to apply the uniform energy estimates. Here ✓ = ✓( 0 ), where ✓ ! 1 as 0 ! 0. To estimate the L4⇤ term (2.9.15), we decompose u = u1 + u2 as in Lemma 2.2.16: ||uR ||L4  ||u1R ||L4 + ||u2R ||L4 . ||u1R ||L4 + ||u2R ||L4 (2.9.16) ⇤,2+ 2↵ ⇤,2+ 2↵ ⇤,2+ 2↵ ⇤,2+ 2↵ q q q q p  ||u1R ||L4 + ||u2 ||H 3/2  ✏ M ||f˜, ✏g||L2⇤,2+ . (2.9.17) ⇤,2+ 2↵ q In the second inequality of (2.9.16), we have used that R is order 1 on the support of u2 and so L4 is equivalent to L4⇤ . In the first inequality in (2.9.17), we have used the H 1/2 ,! L4 embedding in R2 . In the second inequality in (2.9.17), we used Lemma 2.2.16 to estimate ||u2 ||H 3/2 . By observing u1 |r=R0 = u1! |r=R0 = 0, and u1 |!=0 = u1R |!=0 = 0, we use estimate (2.3.13) for u1R to yield: 1 1 p ||u1R ||L4 . || 1 u||X || 1 u||H 2 ˙2 2 .✏ M ||f˜, ✏g||L2⇤,2+ . (2.9.18) ⇤,2+ 2↵ ⇤,2+ q 142 Inserting (2.9.17) into (2.9.15) and multiplying by ✏ 4 : p ✏ 4 ||uR ||L2q  ✏4 M (1 ✓) ||f˜, g ||L2⇤,2+ . ✏˜ (2.9.19) ⇤,q+↵ 0 We can take small enough such that 4 M (1 ✓) > 0. The u! and vR terms in (2.2.20) are treated in an identical manner, after observing the relevant boundary conditions are respected by the cuto↵ quantity: v1R |r=R0 = v1 |r=R0 = 0, v1 |!=0 = v1R |!=0 = 0, u1 |r=R0 = u1! |r=R0 = 0, and u1 |!=0 = u1! |!=0 = 0. We now treat v! , which is slightly di↵erent from the previous terms. Recall from Theorem 2.2.17 that the parameter > 0. Through interpolated Holder, we have: p p p p p || ✏v! ||L2q  || ✏v! ||✓L2 || ✏v! ||1L4 ✓  ||f˜, g ||✓L2 ✏˜ || ✏v! ||1L4 ✓ . (2.9.20) ⇤, 2 ⇤,2+ 2 ⇤, ⇤, ⇤, q q q Again we decompose v = v1 + v2 where v2 is supported near the corner of the domain as in Lemma 2.2.16, and so: p p p p p || ✏v! ||L4 2  || ✏v1! ||L4 2 + || ✏v2! ||L4 2 . || ✏v1! ||L4 2 +✏ M ||f˜, g ||L2⇤,2+ . ✏˜ ⇤, ⇤, ⇤, ⇤, q q q q (2.9.21) p We cannot immediately apply Lemma 2.3.14 to the || ✏v1! ||L4 2 term because ⇤, q v1 = 1 (!, R)v does not necessarily satisfy the stress-free boundary condition at {! = ✓0 }. However, v1! (✓0 , R) = 1 (✓0 , R)v! (✓0 , R) + 1! (✓0 , R)v(✓0 , R), and so a trivial modification of the proof of Lemma 2.3.14 yields the required result. 143 2.10 Nonlinear Existence and Uniqueness for Navier- Stokes Remainders We now apply contraction mapping on the space Z. For this section, call L the linear operator in the linearized problem appearing in equation (2.2.23). 0 Theorem 2.10.1. Suppose L¯ u, v¯ = f (u, v), g(u, v). Select the > 0 guaranteed by Theorem 0 1+ [2.2.17], and let p = 0 , the Holder conjugate to q = 1 + 0 . Then for 2 (0, 1) sufficiently close to 1, and for 2 +  < 12 , we have h 1 i  ||¯ u, v¯||Z  C(us , vs ) 1 + ✏ 2 2 ||u, v||Z + ✏ 2 ||u, v||2Z . (2.10.1) Thus, the solution operator to the nonlinear problem (2.2.23) - (2.2.30) maps the ball of radius 2C(us , vs ) in Z to itself. q Proof. We apply Theorem 2.2.17, after choosing the parameter = 2p and subsequently p 1 in the interval 1 2p u, v¯||Z . ||f˜, ✏˜  < 1, which yields ||¯ g ||L2⇤,2+ . f˜, g˜ are given by: ✓ p p ◆ 1 1 ✏ ✏ f˜ = f ✏4 us u¯! + us! u¯ + usR v¯ + vs u ¯R + vs u ¯+ us v¯ , (2.10.2) r r r r ✓ ◆ 1 1 2 1 g˜ = g ✏4 us v¯! + vs! u ¯ + vs v¯R + vsR v¯ p us u ¯ , (2.10.3) r r r ✏ where f, g are defined in (2.2.26) - (2.2.27). f, g are used to estimate the ||u||X , ||v||B components of ||u, v||Z according to the linear estimate (2.2.48), and the profile terms in (2.10.2) - (2.10.3) are required to estimate the high-regularity components of ||u, v||Z , ac- cording to Theorem 2.2.17. The factor of ✏ 4 accompanies these profile terms because ✏ 2 was used in the definition of the norm || · ||Z , while only a factor of ✏ 4 was required in p Theorem 2.2.17. We now proceed to estimate ||f˜, ✏˜ g ||L2⇤,2+ in terms of ||u, v||Z . 144 From Theorem 2.2.10, we have: ✓Z Z Z Z ◆ 1 2 1 ✏ Ru2 r2+ dRd! + ✏ Rv2 r2+ dRd! ✏ 2 1 3/2  ✏ = ✏2 2  . (2.10.4) p Here we use that 2 +  < 12 . Next we have ✏Ru,p , as defined in (2.2.28): Z Z Z Z ✏ r (u1p )2 u2!  ||u1p ||2L1 ✏ u2! r  C(u1p )✏1  |||u, v|||2Z , (2.10.5) Z Z ✓ Z ◆Z Z ✏ r (up! ) u . ✏ sup (up! ) (R R0 ) 1 2 2 1 2 u2R  ✏1  |||u, v|||2Z , (2.10.6) Z Z ✓ Z ◆Z Z ✏ r (upR ) v . sup (R R0 )(upR ) 2+ 1 2 2 1 2 ✏vR2  ✏1  |||u, v|||2Z , (2.10.7) Z Z Z Z ✏ r (vp ) uR . ✏||vp ||1 2+ 1 2 2 1 2 u2R  ✏1  |||u, v|||2Z , (2.10.8) Z Z Z Z 2 ✏2 r vp1 u2 . ✏2 ||vp1 ||2L1 u2  ✏2  |||u, v|||2Z , (2.10.9) Z Z Z Z 2 ✏2 r u1p v 2 . ✏||u1p ||21 ✏v 2  ✏1  |||u, v|||2Z . (2.10.10) p We now estimate ✏Rv,p , as defined in (2.2.29): Z Z Z Z ✏ r (u1p )2 v!2 . ||u1p ||21 ✏v!2  ✏  |||u, v|||2Z , (2.10.11) Z Z 1 2 2 ✏ r (vp! ) u  ✏||u||2L1 ||vp! 1 ||2L2  ✏  |||u, v|||2Z , (2.10.12) Z Z Z Z ✏ r2+ (vp1 )2 vR 2 . ✏||vp1 ||21 2 vR  ✏1  |||u, v|||2Z , (2.10.13) Z Z Z Z r (up ) u . ||up ||1 1 2 2 1 2 u2  ✏  |||u, v|||2Z , (2.10.14) Z Z ✓ Z ◆Z Z ✏ r (vpR ) v . ✏ sup (R R0 )(vpR ) 2+ 1 2 2 1 2 2 vR ✏  |||u, v|||2Z . (2.10.15) 1 1 1 For (2.10.12) we use that ||vp! ||2L2  ✏ 2  and ||u||2L1 . ✏ 2 ||u||2Z , according to 145 (2.3.10). For (2.10.15) we have used the bound: Z Z p ✓Z Z Z Z ◆ 1 2 p 1 2 ✏ 1 2 1 ✏  ✏ sup (R R0 )(vpR )  ✏ sup vpR  (vpR ) + (vpR! )2  . ✓0 ✓0 Summarizing the linear components of f, g, 1 p p p 1 ✏ 2 ||Ru , ✏Rv ||L2⇤,2+ + ✏||Ru,p , ✏Rv,p ||L2⇤,2+ . C(us , vs )+✏ 2 2  ||u, v||Z . (2.10.16) 0 For the nonlinear terms we recall the definition of || · ||Z in (2.2.20), where q = 1 + , q q q p= q 1, and 0 < p ↵ 2 . We also recall the low regularity embeddings in Lemmas 2.3.9 and 2.3.10. Z Z ✓Z Z ◆ q1 ✓Z Z ◆ p1 ✏ 2 +1 r u2 u2! ✏ 2 +1 u2q ! 2p u r p .✏ +1 ||u||4Z , (2.10.17) Z Z ✓Z Z ◆ p1 ✓Z Z ◆ q1 ✏ 2 +1 r 2+ v 2 u2R ✏ 2 +1 2p p 1 v r r p u2q Rr q+↵ . ✏ ||u, v||4Z . (2.10.18) 0 ↵ 1 Here the inequality holds because we have selected > 0 so that q p. Z Z ✓Z Z ◆ 12 ✓Z Z ◆ 12 ✏2 +2 r u2 v 2  ✏ 2 +2 r2 v 4 u4  ✏2 +1 ||u, v||4Z . (2.10.19) Nonlinear terms in g: Z Z ✓Z Z ◆ p1 ✓Z Z ◆ q1 1 q ✏2 +1 r u2 v!2  ✏2 +1 u2p r p r 2 r 2p v!2q  ✏ ||u, v||4Z , (2.10.20) 146 Z Z ✓Z Z ◆ p1 ✓Z Z ◆ q1 2q q+↵ ✏ 2 +1 r 2+ v 2 vR 2 ✏ 2 +1 2p p 1 v r r p vR r . ✏ ||u, v||4Z , (2.10.21) Z Z ✏2 u4 r . ✏2 ||u||4Z . (2.10.22) Combined with (2.10.16), we now have: p 1 ||f, ✏g||L2⇤,2+ . C(us , vs ) + ✏ 2 2  ||u, v||Z + ✏ 2 ||u, v||2Z . (2.10.23) We now provide estimates for the profile terms in (2.10.2) - (2.10.3). Z Z Z Z ✏2 ¯2!  ||us ||1 ✏ 2 r us u ¯2! r , u (2.10.24) Z Z Z Z 2 2 2 2 ✏2 us! u¯ r  ||us! ||1 ✏ ¯2 r , u (2.10.25) Z Z Z Z ✏2 r2+ vs2 u ¯2R  ✏ 2 ||vs r||21 ¯2R r , u (2.10.26) Z Z Z Z ✏1+ 2 r vs2 u ¯2  ✏1+ 2 ||vs ||21 ¯2 r , u (2.10.27) Z Z Z Z ✏1+ 2 2 2 r us v¯  ✏ ||us ||1 2 v2 r , ✏¯ (2.10.28) Z Z ✓ Z ◆Z Z ✏2 u2sR r2+ v¯2  ✏ 2 sup u2sR r2 (R R0 ) 2 v¯R , (2.10.29) Z Z Z Z ✏1+ 2 r u2s v¯!2  ✏ 2 ||us ||21 ✏r v¯!2 , (2.10.30) Z Z Z Z 1+ 2 2+ 2 2 1+ 2 2 ✏ r vs v¯R  ✏ ||vs ||1 r2+ v¯R 2 , (2.10.31) Z Z Z Z ✏1+ 2 r2+ vsR 2 v¯2  ||vsR r||21 ✏ 2 ✏r v¯2 , (2.10.32) Z Z Z Z ✏2 r u2s u ¯2  ||us ||21 ✏ 2 r u ¯2 , (2.10.33) Z Z Z Z 1+ 2 2 2 2 1+ 2 ✏ r vs! u ¯  ||vs! ||1 ✏ r u ¯2 . (2.10.34) p Thus, (2.10.24) + ... + (2.10.34) . ✏ 2  ||f, ✏g||2L2 . This concludes the proof. ⇤,2+ 147 Corollary 2.10.2. For sufficiently close to 1, the solution operator of the nonlinear equation is a contraction map on the space Z, satisfying: h u1 ||¯ ¯2 , v¯1 u v¯2 ||Z C(us , vs ) ✏ 2 ||u1 , v 1 ||Z + ||u2 , v 2 ||Z ||u1 u2 , v 1 v 2 ||Z 1 i + ✏ 2 2  ||u1 u2 , v 1 v 2 ||Z . (2.10.35) By applying the contraction mapping theorem, we have proven Theorem 2.2.18 and therefore the main result, Theorem 2.2.3. Chapter Three Global-in-x Prandtl Layers over a Moving Boundary 149 3.1 Abstract In this three-part monograph, we prove that steady, incompressible Navier-Stokes flows posed over the moving boundary, y = 0, can be decomposed into Euler and Prandtl flows in the inviscid limit globally in [1, 1)⇥[0, 1), assuming a sufficiently small velocity mismatch. Sharp decay rates and self-similar asymptotics are extracted for both Prandtl and Eulerian layers. We then develop a functional framework to capture precise decay rates of the remain- ders, and prove the corresponding embedding theorems by establishing weighted estimates for their higher order tangential derivatives. These tools are then used in conjunction with a third order energy analysis, which in particular enables us to control the nonlinearity vuy globally. 3.2 Introduction We consider the steady, incompressible Navier-Stokes equations in two dimensions: U N S UX NS + V N S UYN S + PX NS = ✏ U NS, (3.2.1) U N S VXN S + V N S VYN S + PYN S = ✏ V N S , (3.2.2) NS UX + VYN S = 0, (3.2.3) in the domain, ⌦ = [1, 1) ⇥ R+ . (3.2.4) The boundary Y = 0 is moving with velocity ub > 0. The no-slip boundary conditions are placed on this portion of the boundary: U N S (X, 0) = ub = 1 , V N S (X, 0) = 0. (3.2.5) 150 The boundary conditions at X = 1 will be prescribed explicitly in the text. We take X = 1 for convenience (this enables us to replace weights of (1 + x)k with xk ). Throughout this paper, we assume that prescribed Euler flow is the shear flow: (U E , V E ) = (1, 0). (3.2.6) We are interested in the limit as ✏ ! 0. Formally, one expects that the solutions to Navier-Stokes equations in (3.2.1) - (3.2.3) converges to the Euler shear flow in (3.2.6). This does not happen, however, due to the mismatch at the boundary Y = 0, between the no-slip condition enforced for Navier-Stokes, (3.2.5), and U E (X, Y = 0) = 1. To account for the mismatch at the boundary, Prandtl in 1904 proposed a thin fluid boundary layer which connects the velocity of ub to the Euler velocity of 1. The Prandtl hypothesis is that the Navier-Stokes solutions can be decomposed, up to leading order in ✏, as the sum of the prescribed Euler flow and a boundary layer, the latter of which corrects the disparity at the boundary between Euler and Navier-Stokes: p p U N S = 1 + u0p + h.o.t(✏), V NS = 0 + ✏vp0 + ✏ve1 + h.o.t(✏).1 (3.2.7) The contribution of this paper is to validate the boundary layer theory, equations (3.2.7), in the domain ⌦, which in particular implies that the tangential variable can be taken in [1, 1) if the mismatch is sufficiently small : UE U N S |Y =0 = 1 ub = 1 (1 )= << 1. (3.2.8) 1 Here, “h.o.t” is an acronym for “higher order terms.” 151 Boundary Layer Expansion We will work with scaled, boundary layer variables: Y x = X, y=p . (3.2.9) ✏ The scaled Navier-Stokes unknowns are then given by: V N S (X, Y ) U ✏ (x, y) = U N S (X, Y ), V ✏ (x, y) = p , P ✏ (x, y) = P N S (X, Y ). (3.2.10) ✏ These unknowns satisfy the following system: U ✏ Ux✏ + V ✏ Uy✏ + Px✏ = Uyy ✏ ✏ + ✏Uxx , (3.2.11) Py✏ U ✏ Vx✏ + V ✏ Vy✏ + ✏ = Vyy ✏ + ✏Vxx , (3.2.12) ✏ Ux✏ + Vy✏ = 0. (3.2.13) Note that the steady Prandtl system is obtained by considering the leading order in " of the above system (3.2.11) - (3.2.13). We start with the following asymptotic expansion: n X i i n U ✏ (x, y) = 1 + u0p + ✏ 2 uie + ✏ 2 uip + ✏ 2 + u(x, y), (3.2.14) i=1 n X1 i i n n V ✏ (x, y) = ✏ 2 vpi + ✏ 2 vei+1 + ✏ 2 vpn + ✏ 2 + v(x, y), (3.2.15) i=0 Xn i i i+1 n P ✏ (x, y) = ✏ 2 Pei + ✏ 2 Ppi + ✏i Pei,a + ✏ 2 Ppi,a + ✏ 2 + P (x, y). (3.2.16) i=1 Here, 2 [0, 14 ). We will use the word “profiles” to refer to the terms which appear in the expansions (3.2.14) - (3.2.15), excluding the remainders, [u, v, P ]. All of the profiles with 152 subscript-e are functions of Eulerian variables, (x, Y ), whereas all terms with subscript-p are functions of boundary layer variables, (x, y). Here, [uip , vpi ] are boundary layers to be constructed. The number of intermediate layers, n, is dependent on universal constants. The pressures Pei , Ppi are the pressures associated with the i0 th Euler and Prandtl layers, respectively. We will show that Ppi = 0, that is the leading-order pressure in the boundary layers is zero. The pressures PPi,a , Pe1,a are auxiliary pressures, which are higher-order, whose purpose is to capitalize on the gradient structure of our problem (see 3.7.33). After these layers are constructed, the Navier-Stokes remainders [u, v, P ] are then constructed. Let us now designate names for the partial expansions: i 1 X i X i X i X j j i j j u(i) s := 1 + ✏ 2 ujp + ✏ 2 uje , ¯(i) u (i) 2 i s := us + ✏ up = 1 + ✏ 2 ujp + ✏ 2 uje (3.2.17) j=0 j=1 j=0 j=1 i 1 X i X i X i X j j 1 i j j 1 vs(i) := ✏ 2 vpj + ✏2 2 vej , v¯s(i) := vs(i) + ✏ 2 vpi = ✏ 2 vpj + ✏2 2 vej , (3.2.18) j=0 j=1 j=0 j=1 For the Pressure expansion: i 1 X i 1 X i X i X j j+1 j Ps(i) := ✏ 2 Ppj + ✏ 2 Ppj,a + ✏ 2 Pej + ✏j Pej,a . (3.2.19) j=1 j=1 j=1 j=1 i X i X i X i X j j j+1 P¯s(i) := ✏ 2 Ppj + ✏ 2 Pej + ✏j Pej,a + ✏ 2 Ppj,a . (3.2.20) j=1 j=1 j=1 j=1 We insert the expansions (3.2.14) - (3.2.16) into (3.2.11) - (3.2.13) and collect a heirarchy of equations in powers of ✏. Doing so yields the linearized Prandtl-equations: ⇣ ⌘ (1 + u0p )uipx + u(i) i (i) i 0 i sx up + vs upy + upy vp vpi (x, 0) + Ppx i = uipyy + f (i) , (3.2.21) uip (x, 0) = uie (x, 0), lim uip (x, y) = 0, uip (1, y) = Ui (y). (3.2.22) y!1 153 and the Euler equations: uiex + Pex i = 0, i vex i + PeY = 0, uiex + veY i = 0. (3.2.23) The forcing term f (i) will be defined precisely in (3.7.47). These equations are derived rigorously in the analysis leading up to equations (3.5.5), (3.6.21), (3.7.19), and (3.7.48). Boundary Data: The no-slip boundary condition at the boundary {y = 0} is the most important, and must be enforced at each order in ✏, which gives: u0p (x, 0) = , uie (x, 0) + uip (x, 0) = 0 for i 1, u(x, 0) = 0, (3.2.24) vpi 1 (x, 0) + vei (x, 0) = 0 for i 1, vpn (x, 0) = 0, v(x, 0) = 0. (3.2.25) The in-flow (x = 1) boundary conditions for the leading order boundary layer, u0p , is: u0p (1, y) = U0 (y), U0 (0) = , lim U0 (y) = 0. (3.2.26) y!1 We assume the rapid decay of the profile: ||hyim @yj U0 (y)||L1  C(m, j), for any m, j 0. (3.2.27) We will in addition assume the following smallness condition: ||hyim @yj U0 (y)||L1  O( ; j, m) for any m 0, j = 0, 1, 2. (3.2.28) 154 The in-flow (x = 1) boundary conditions for the boundary-layer profiles are: uip (1, y) = Ui (y), u(1, y) = 0, v(1, y) = 0, for 1  i  n 1. (3.2.29) Here, Ui (y), for i  1  n 1, will be prescribed to be rapidly decaying in y, so: ||hyim uip (1, y)||L1 = ||hyim Ui (y)||L1  C(i, m) for 1  i  n 1. (3.2.30) n For the final Prandtl layer, unp , which occurs at order " 2 , the in-flow data is determined through the analysis and is not explicitly prescribed. This is due to retaining that [unp , vpn ] are divergence free, while cutting o↵ vpn for large values of y. The reader is invited to turn to equation (3.7.142) and corresponding discussion for details regarding this matter. We will enforce ||hyim Un (y)||L1  C(m). However Un is an auxiliary in-flow, which is used to construct [unp , vpn ]. That is: unp (1, y) 6= Un (y), and the in-flow velocity, unp (1, y), is given by an implicit, bounded profile which decays as y ! 1. We will need several compatibility conditions on the in-flow data for the Prandtl layers. The first of these is: U0 (0) = , @yy U0 (y) = 0, Ui (0) = uie (1, 0). (3.2.31) However, we shall also need higher-order compatibility conditions on the Ui (y) at y = 0, (see for instance Remarks 3.4.2, 3.6.2) which we refrain from depicting explicitly here. Finally, the boundary conditions of the boundary-layer profiles as y ! 1 are: lim [uip (x), vpi (x)] = lim [u(x), v(x)] = 0 for all x 1, and all 0  i  n. (3.2.32) y!1 y!1 These boundary conditions are known as the “matching condition”, and physically cor- respond to the Navier-Stokes flow matching the outer Euler flow away from the boundary, 155 n 1 y = 0. According to our construction, we will have rapid matching up to order " 2 : lim [hyiN uip (x), hyiN vpi (x)] = 0 for all x 1, for 1  i  n 1. (3.2.33) y!1 At the highest-order in ", we enforce the matching condition: n n lim " 2 [unp (x, y), vpn (x, y)] = lim " 2 + [u(x, y), v(x, y)] = 0 for all x 1. (3.2.34) y!1 y!1 Let us now turn to the Euler flows. At leading order, we have the prescription: [u0e , ve0 ] = [1, 0]. The higher-order Euler flows will be described starting in Section 3.5 and Subsection 3.7.1. The higher-order Euler flows are obtained as suitable Poisson extensions of the Y = 0 boundary data, vpi 1 (x, 0) (see (3.2.24)), which depend on the constructed Prandtl layers. For these higher-order Euler flows, we do not prescribe the in-flow data, uie (1, Y ). Rather, the in-flow conditions are obtained through the analysis, so we state: n X i Eulerian In-Flow =1 + ✏ 2 uie (1, ·). (3.2.35) i=1 Main Result: In order to state our main result, we need to introduce the norm Z in which we control the remainder solutions, [u, v]: Definition 3.2.1. The norm Z is defined through: 1 p 1 ||u, v||Z :=||u, v||X1 \X2 \X3 + ✏N2 ||u, v||Y2 + ✏N3 ||u, v||Y3 + ✏N4 ||ux 4 , "vx 2 ||L1 p 3 5 1 + ✏N5 sup || "vx x 2 , ux x 4 ||L1 + "N6 sup ||uy x 2 ||L2y x 20 x 20 hZ 1 p i 12 + ✏ N7 x4 || "vxx ||2L1 y dx . (3.2.36) 20 156 Here, Ni , are large numbers which will be specified in (3.9.103) - (3.9.105). They depend only on universal constants. The parameter n from (3.2.14) - (3.2.15) will be taken much larger than any of the Ni . The norms || · ||Xi are energy norms defined in (3.9.3) - (3.9.5). The norms || · ||Yi are elliptic norms defined in (3.9.6) - (3.9.7). For the purposes of stating the main result, we can refrain from being too specific with regards to the definitions of these norms. The essential point that we will record concerns the uniform component: 1 p 1 ✏N4 ||ux 4 , "vx 2 ||L1  ||u, v||Z , (3.2.37) The main result of this paper is: Theorem 3.2.2. Suppose the the outer Euler flow is prescribed with u0e = 1. Suppose the boundary and in-flow data are specified satisfying the conditions outlined in (3.2.24) - (3.2.34). Then there exists an n depending on only universal constants such that the asymptotic expansions in (3.2.14) - (3.2.16) are valid globally on the domain ⌦, for 0  < 14 , so long as the mismatch between the Eulerian boundary trace and the motion of the boundary, , and the viscosity, ", are taken sufficiently small relative to universal constants, and " << . The remainders, [u, v], in the expansions (3.2.14) - (3.2.15) are uniquely determined in the space Z: 1 ||u, v||Z . " 4  , (3.2.38) where  is any fixed constant such that +  < 14 . n Because 2 is large relative to N4 in (3.2.37), we immediately find: Corollary 3.2.3 (Inviscid L1 Convergence). Under the hypothesis of Theorem 3.2.2, there 157 exists a unique Navier-Stokes solution [U N S , V N S , P N S ] on ⌦ such that: 1 1 sup U N S (X, Y ) 1 u0p (X, y) X 4 . ✏ 2 , (3.2.39) (X,Y )2⌦ p p 1 sup V N S (X, Y ) ✏vp0 (X, y) ✏ve1 (X, Y ) X 2 . ✏. (3.2.40) (X,Y )2⌦ Existing Literature: Let us first discuss the issue of establishing wellposedness of the Prandtl equation, which becomes an issue in the unsteady setting (in contrast to the steady setting of the present paper). This program was initiated in the classic works (OS99), (Ole67), in which, under the monotonicity assumption Uy" (t = 0) > 0, globally regular solutions are constructed on the [0, L] ⇥ R+ , where L is sufficiently small, and local solutions are constructed for arbitrary, but finite L. This was extended in (XZ04), in which global weak solutions were constructed for arbitrary L, under both monotonicity and favorable outer-Euler pressure (@x P E (t, x)  0 for t 0) assumptions. From a physical standpoint, the monotonicity and favorable pressure assumptions men- tioned above are stabilizing and in particular prevent boundary layer separation. This phe- nomena was known to Prandtl, see Figure 2 in (Pra04). More recently, it was announced in (DM18) that a proof of boundary layer separation in the steady setting has been obtained. The main tool used both in (Ole67) and (XZ04) is the Crocco transform. Still under monotonicity hypothesis, local wellposedness was obtained in (AYXY15) and (MW15), nei- ther works using the Crocco transform. (AYXY15) use energy methods coupled with a Nash- Moser iteration, and (MW15) use energy methods applied to a good unknown which enjoys crucial cancellation properties. Generalizing to multiple monotonicity regions, (KMVW14) have shown the Prandtl equation is locally well-posed, if an analyticity assumption is made on the complement of the monotonicity regions. 158 Indeed, when the assumption of monotonicity is removed, the wellposedness results are largely in the analytic or Gevrey setting. The reader should consult (SC98a) - (SC98b), (KMVW13), (LCS03), (IV16), and (GVM13) for some results in this direction. In the Sobolev setting without monotonicity, the equations are linearly and nonlinearly ill-posed (see (GVD10) and (GVN12)). A finite-time blowup result was obtained in (EE97) when the outer Euler flow is taken to be zero, in (KVW15) for a particular, periodic outer Euler flow, and in (HH03) for both the inviscid and viscous Prandtl equations. The above discussion is not comprehensive: we refer the reader to the review articles, (E00), (GJT16) and references therein for a more thorough review of the wellposedness theory. The question with which we are concerned is the validity of the asymptotic expansion (3.2.14) - (3.2.16) in the inviscid limit. Let us first discuss unsteady flows. Local-in-time convergence is established in (SC98a), (SC98b) in the analyticity framework, in (GVMM16) in the Gevrey setting, and in (Mae14) when the initial vorticity distribution is supported away from the boundary. The reader should see also (Asa91), (MT08) for related results. p Despite the boundary layer classically having thickness ", an interesting criteria was given in (Kat84) which points to phenomena occurring in a sub-layer of size ✏. There are also several linear and nonlinear instability results (for instance, (Gre00), (GGN16b), (GGN16a), (GGN15), (GN11)) which show the invalidity of Prandtl’s expansion generically in Sobolev spaces in the unsteady setting. For steady flows, there are very few validity results. (GN17) is the first result in this direction, establishing validity of the boundary layer expansion for steady state flows in a rectangular domain over a moving boundary. Geometric e↵ects of the boundary were subsequently considered in (Iye17a). The crucial idea in (GN17) was the use of a positivity estimate, which is coupled with energy estimates and elliptic estimates. Both of these results are local in the tangential variable. In the present work, we prove validity of the boundary layer expansion globally in the tangential variable, x, in the setting of small data. One preliminary piece of our analysis is to obtain the asymptotics of the Prandtl layer, u0p . 159 This has nontrivial dynamics due to the mismatched boundary conditions, u0p (y = 0) = , while limy!1 u0p (y) = 0. These asymptotics were first studied in (Ser67, pg. 493, Inequality 5) using maximum principle techniques and are valid for large data. The result in (Ser67) gives that the di↵erence between u0p and a Gaussian“front” solution to the heat equation (call it w) is o(1) in x, uniformly in y. Under the hypothesis of small data, we sharpen these asymptotics in the following sense: first, w is shown to belong to a higher-order Sobolev space H k (m), where this weight is in the self-similar variable z = py . Second, we obtain x rates of decay of w in x in various norms. We now detail the main difficulties and ideas behind our analysis. Sharp Decay of Profiles (Chapter I): The key issue that we must capture in our analysis is the decay as x ! 1 of various quan- tities. More specifically, a central difficulty is to control contributions from the nonlinearity V ✏ Uy✏ . Let us now introduce the equations for the remainders, [u, v, P ]: Py ✏u + Su + Px = f, ✏v + Sv + = g, ux + vy = 0. (3.2.41) ✏ Here, the terms Su , Sv contain the linearizations of [u, v] around the previously con- (n) (n) structed profiles, [¯ us , v¯s ]. These terms, together with f, g are specifically defined in (3.8.5) - (3.8.6). To organize this discussion, let us record the heuristic: n Difficult Contributions from V " Uy" = u ¯(n) ¯s(n) uy + " 2 + vuy . sy v + v (3.2.42) (n) First, let us discuss u ¯sy v from (3.2.42), which will motivate the crucial decay rates appearing in (3.2.48). Applying the scaled multiplier of (u, ✏v) to the system (3.2.41) requires RR 0 controlling the large convective term, upy uv. We do not have the ability to create 160 a derivative through the Poincare inequality, and so we trade factors of x and y in the following manner: Z Z 1 1 u vx 2 1 u0py uv  ||u0py y 2 x 2 ||2L1 || ||L2 || ||L2  O( )||uy ||L2 ||vy x 2 ||L2 . (3.2.43) y y y The first crucial observation we make is the identification of a self-similar front, ⇤ ( px ), which bridges the boundary conditions: u0p (x, 0) = , u0p (x, 1) = 0. Then, temporarily y identifying u0p ⇡ ⇤ ( px ), (3.2.43) will be satisfied: Requirement 1 (Self-Similarity of Prandtl profiles). 1 ⇣ y ⌘ 1 0 0 ||y 2 x 2 u0py ||L1 = ||y 2 x ⇤ p ||L1 = ||z 2 ⇤ (z)||L1  O( ). (3.2.44) x Summarizing the energy estimate that we obtain: p 1 ||uy ||2L2  O( )||{ "vx , vy }x 2 ||2L2 + Forcing Terms. (3.2.45) 1 Above, the key point is the loss of weight, x 2 . The next ingredient is recovering this weight in the Positivity estimate, (see Proposition 3.10.2), which is summarized: p 1 ||{ "vx , vy }x 2 ||2L2 . ||uy ||2L2 + Forcing Terms. (3.2.46) In order to prove (3.2.46), we must apply the weighted multilplier vy x. Referring to the final two terms in (3.2.42), this gives (temporarily ignoring factors of ✏): Z Z Z Z 1 1 v¯s(n) uy · vy x + vs(n) , v}x 2 ||L1 ||uy ||L2 ||vy x 2 ||L2 . vuy · vy x  ||{¯ (3.2.47) The latter two L2 quantities are controlled by the left-hand sides of (3.2.46) - (3.2.47). 161 From this, we obtain the requirements: Requirement 2 (Uniform Decay). 1 v¯s(n) + v ⇠ x 2 , as x ! 1. (3.2.48) We emphasize that this requirement is inflexible, and we cannot sacrifice even a loga- rithmic factor of x here. The main contribution of Chapter I is the construction of profiles (n) uip , vpi , uie , vei which satisfy the requirement of vs in (3.2.48). The most difficult task is to obtain the estimate (3.2.48) for the Eulerian profiles, ve1 , as these are solutions to elliptic boundary value problems in which the boundary condition exhibits exactly the required 1 decay rate, |ve1 (x, 0)|  x 2 . We refer the reader to the crucial Proposition 3.5.2 in which we introduce novel techniques centered around the explicit integral representation of ve1 , 1 enabling us to prove the required decay, |ve1 | . x 2 . Note that we construct the expansions, (3.2.14) - (3.2.16) for any n 2 N, which is required as discussed in the paragraph following (3.2.52). Our ability to do this relies on the Cauchy-Riemann structure of the Eulerian profiles, which we use in Lemma 3.7.33. The Norm Z (Chapter II): Let us now turn to the v term in Requirement (3.2.48): the main contribution of Chapter II 1 is to prove the required decay estimate |v| . x 2 by using the crucial norm Z (see Lemmas 3.9.15, 3.9.17). The challenge is to extract this precise uniform decay information from the energy norms that are controlled. Bearing in mind the inflexibility of (3.2.48), obtaining the decay for v using the norms Xi is an extremely delicate matter, in which key quantities must overcome the critical Hardy inequality. To see this, we first use the Hy1 (R+ ) ,! L1 y (R+ ) 162 Sobolev embedding (ignoring factors of "): 1 1 1 sup ||vx 2 ||L1 y  sup ||v||L2 2 · sup ||vy x||L2 2 . (3.2.49) y y x 1 x 1 x 1 For the first quantity on the right-hand side above, we write: Z Z @x v 2 dy = 2vvx dy. (3.2.50) p 1 1 Recall now the quantities, ||uy , "vx x 2 , vy x 2 ||2L2 , which are controlled on the left-hand xy sides (3.2.45), (3.2.46) and constitute the energy norm X1 . The right-hand side above fails 1 to be x-integrable, precisely because of criticality of Hardy’s inequality with power x 2 in L2 : Z Z 1 1 1 | vvx dy dx|  ||vx 2  ||vx x 2 ||2L2xy . ||L2xy ||vx x 2 ||L2xy @ (3.2.51) To avert this, we move to higher-order derivatives, which invokes the full strength of the 3 norm Z. Indeed, suppose we knew vx ⇠ x 2 , then coupled with the boundary condition 1 v ! 0 as x ! 1, this would immediately imply v ⇠ x 2 . Establishing the decay rate, 3 ||vx ||L1 y  x 2 , then becomes the goal, which requires us to go to third-order energy estimates (thus explaining the presence of X1 , X2 , X3 in the norm Z). Our main uniform estimates, given in Lemmas 3.9.15, 3.9.17 are given by the following sequence: 1 3 1 1 ||vx 2 ||L1 xy . ||vx x 2 ||L1 xy . sup ||vx x||L2 2 · sup ||vxy x2 ||L2 2 . ||u, v||X1 \X2 \X3 . (3.2.52) y y x 1 x 1 The key point is that the quantities ||vx x||L2y and ||vxy x2 ||L2y appearing above do not face issues of Hardy-criticality present in (3.2.51), which can be seen in Lemma 3.9.15. 163 Applying @xk to the system creates singularities near the corner at (1, 0). To handle this we cuto↵ near the boundary, x = 1, when performing higher order energy estimates. Cuto↵ functions interact poorly with nonlinearities, and so we need to supplement energy estimates with elliptic estimates which retain additional control of [u, v] near x = 1 (though not all the way up to the boundary, x = 1). These are characterized by the norms || · ||Yi in (3.2.36). These Yi norms are controlled by invoking the elliptic theory, which in turn requires sacrificing factors of ". For this, we require a high power of "n to accompany nonlinear terms, which in turn requires us to go to high order expansions in (3.2.14) - (3.2.16). Existence and Uniqueness (Chapter III): Upon proving our main a-priori estimate, Theorem 3.8.1, we prove existence and uniqueness of a solution in Z. As in (3.2.43), applying the multiplier u produces the nonlinearity vuy ·u, which is a perfect derivative and therefore vanishes. This cancellation property is destroyed upon taking di↵erences, and so we cannot rely on a standard application of the contraction mapping theorem. A sequence of auxiliary, approximate systems are then carefully designed to produce enough compactness enabling us to show existence of a solution in the space Z. The uniqueness in Z is a more delicate matter, again due to a lack of the perfect derivative structure. For this, we take the di↵erence of two solutions in Z and repeat the energy analysis with weaker weights. The reader is referred to Lemma 3.17.1, in which the weights must be selected carefully in a small interval below those weights appearing in the energy analysis, for instance in (3.2.45), (3.2.46). These methods are carried out in Chapter III. Notation and Important Parameters There are three important parameters in this paper: ", , and n (see (3.2.14) - (3.2.16)). The notation A . B means A  CB, where C is some constant which is independent of small 164 , ", and large n. Constants denoted by O( ) or O(") satisfy O( ), O(") ! 0 as , " ! 0, respectively. Given any parameter, say p, constants denoted by C(p) mean those constants which depend (perhaps poorly) on large values of p. Given two parameters, and , for instance, we will write O( ; ) to denote a constant which depends on and , but such that for fixed , can be made small to make the constant small, for instance ⇥ . We define R ||f ||pLpy := f (x, y)p dy. When unspecified, || · ||Lp means the Lp norm of two-variables. We define here the di↵erential operators: "u := (@xx + "@yy )u for any profile u; [uie , vei ] := (@xx + @Y Y )[uie , vei ]. (3.2.53) The variable z will denote a self-similar variable, so typically z = py or z = p⌘ . x x Finally, the word “profiles” refers to terms in the expansion (3.2.14) - (3.2.15), excluding the remainders [u, v], and “profile terms” refers to the linearizations in Su , Sv , defined in (3.8.6). Part I: Construction of Profiles 165 166 3.3 Overview of Profile Constructions The purpose of this chapter is to construct each of the profiles appearing in the expansions (3.2.14) - (3.2.16), with the exception of the final terms, [u, v, P ]. This results of this chapter (n) (n) are used extensively in Chapter II. Let us first introduce the following notation for u ¯s , v¯s : n X n X j j uP R := ✏ 2 ujp , uE R := 1 + ✏ 2 uje , ¯(n) uR := u s = uP E R + uR , (3.3.1) j=0 j=1 Xn n X j j 1 P vR := ✏ 2 vpj , E vR := ✏2 2 vej , vR := v¯s(n) = vR P E + vR . (3.3.2) j=0 j=1 We shall also have occasion to further split uP R to distinguish the final layer via: n X j n uP,n R 1 = ✏ 2 ujp , so that uP P,n R = uR 1 + ✏ 2 unp . (3.3.3) j=0 Inserting the expansion (3.2.14) - (3.2.16) into the scaled NS equations, (3.2.11) - (3.2.13), motivates the following definition: Definition 3.3.1. The n’th remainder is denoted by: Ru,n := ¯(n) ✏u s ¯(n) +u s u¯(n) ¯s(n) u sx + v ¯(n) ¯ (n) sy + Psx , (3.3.4) @y ¯ (n) Rv,n := ¯s(n) ✏v ¯(n) +u s v (n) ¯sx + v¯s(n) v¯sy (n) + P . (3.3.5) ✏ s The main result of this chapter is: Theorem 3.3.2. Let n 2 2 N. Let , " be sufficiently small relative to universal constants, and " << . Let the boundary and in-flow data from (3.2.24) - (3.2.35) be prescribed. Then there exist Prandtl profiles [ujp , vpj , Ppj ] for j = 1, ..., n, Euler profiles [uje , vej , Pej ] for j = 1, ..., n, and auxiliary pressures [Ppj,a , Pej,a ] for j = 1, ..., n such that for Ru,n , Rv,n as defined in (3.3.4) - (3.3.5), and for any 2 [0, 14 ), n 2, and for n = 1 10,000 ,  > 0 167 arbitrarily small, the following remainder estimate holds for any k 0: n p 1 3 ✏ 2 @xk Ru,n + ✏@xk Rv,n  C(n, )✏ 4  x k 2 +2 n , (3.3.6) n p p 1 5 ✏ 2 || ✏@xk Ru,n , ✏@xk Rv,n ||L2y  C(n, )✏ 4  x k 4 +2 n + . (3.3.7) The following bounds hold on [uR , vR ] by construction, for any [k, j, m] 0, so long as n is sufficiently large relative to m. j 1 ||@xk @yj vR P m k+ 2 + 2 z x ||L1  C(k, j, m) if k 1, (3.3.8) j 1 ||@yj vR P m 2+2 z x ||L1  C(j, m) if j 2, (3.3.9) j 1 ||@yj vR P m 2+2 z x ||L1  O( ; m, j) if j = 0, 1. (3.3.10) j ||@xk @yj uP m k+ 2 Rz x ||L1  C(k, j, m) for k > 1, j 0 (3.3.11) ||@x uP m R z x||L1  O( ; m), (3.3.12) ||@x @yj uP m R z x||L1  C(m, j) for j 1 (3.3.13) ||@yj uP,n R 1 j m y z ||L1  O( ; m, j), for 0  j  2, (3.3.14) ||@yj uP,n R 1 j m y z ||L1  C(m, j), for j > 2, (3.3.15) 1 ||@yj unp y j x 2 n ||L1  C(n, j) for all j 0, (3.3.16) 1 ||@xk @Yj vR E k+j+ 2 x ||L1  C(k, j) for k + j > 0, (3.3.17) 1 p ||@xk @Yj uE Rx k+j+ 2 ||L1  "C(k, j) for k + j > 0 (3.3.18) 1 ||@xk vR E k x 2 Y ||L1  C(k, j) for k 1, (3.3.19) 1 3 ||{uE R E 1, vR E }x 2 , vRY x 2 ||L1  O( ). (3.3.20) The profiles uR , vR from (3.3.1) - (3.3.2) arise as coefficients in the linearized problem for the Navier-Stokes remainders [u, v, P ], which is to be analyzed in Chapter II. The es- timates obtained in (3.3.8) - (3.3.18) are therefore essential to the analysis of Chapter II. In particular, we invite the reader to compare the requirements discussed in (3.2.44) and 168 (3.2.48) with the estimates we prove in (3.3.8) - (3.3.20). The structure of this chapter is as follows: (Step 1) Construction of the zeroeth-order Prandtl layers, [u0p , vp0 ] (Section 3.4): The distin- guishing feature of u0p are the mismatched boundary conditions, as seen from (3.4.5). As shown in Proposition 3.4.2, this contributes a “front”-profile which looks similar to e , defined in (3.4.11). The decay rates for Prandtl profiles from estimates (3.3.8) - (3.3.18) are dictated by this front profile. For the zeroeth layer, this is formalized in Proposition 3.4.7, Corollaries 3.4.8 and 3.4.9. It is essential that we obtain estimates weighted in the self-similar variable, z = py , as can be seen from (3.3.8) - (3.3.20) x above. (Step 2) Construction of Euler-1 layers, [u1e , ve1 ] (Section 3.5): Given the boundary conditions, 1 which are known to satisfy the progressive estimate: |@xk ve1 (x, 0)| . x 2 k , we must 1 obtain the sharp uniform estimate |@xk ve1 (x, Y )| . x 2 k . This is a delicate matter, as these profiles are elliptic, and as indicated in (3.2.48), the required decay rates must be obtained exactly. We introduce a method to obtain the required pointwise estimates using directly the Poisson integral formulation, which is carried out in Proposition 3.5.2. This section is a major contribution of Chapter I. (Step 3) Construction of Prandtl-1 layer, [u1p , vp1 ] (Section 3.6): This profile is controlled using coupled energy and positivity estimates, given in Lemmas 3.6.4 and Lemma 3.6.5. (Step 4) Construction of Intermediate Euler and Prandtl layers (Section 3.7):. The essential mechanism here is as follows: as one consider linearizations for [ujp , vpj ] for j > 1, one encounters terms which scale poorly in z = py , due to Euler-Euler interactions. x However, due to the Cauchy-Riemann structure present in the Euler profiles (see (3.5.7)), we may introduce auxiliary pressures Ppi,a , PEi,a which creates cancellations of all terms which are “purely-Eulerian”. This is seen in (3.7.32) - (3.7.33). (Step 5) Construction of Final Prandtl layer, [unp , vpn ] (Subsection 3.7.3): The final Prandtl layer 169 satisfies the boundary condition vpn (x, 0) = 0, and so has a contribution as y " 1. We cut-o↵ this layer in the region z  p1 , which honors the parabolic scaling of the " Prandtl layers. This is a generalization of the cut-o↵ used in (GN17). It is delicate to ensure these cuto↵ layers obey desirable estimates, which is done in Lemma 3.7.23. It is also delicate to ensure that this process contributes an error that satisfies estimates (3.3.6) - (3.3.7) above. This is proven in Lemma 3.7.22. 3.4 Asymptotics of Prandtl Layer, u0p : Our starting point is the leading order terms from (3.2.21), which yields the following system for the Prandtl layer, [u0p , vp0 ]: ⇣ ⌘ ⇣ ⌘ 1 + u0p u0px + vp0 + ve1 (x, 0) u0py = u0pyy , u0px + vpy 0 = 0, (3.4.1) u0p (x, 0) = , vp0 (x, 0) = ve1 (x, 0), u0p (1, y) = U0 (y). (3.4.2) As shown in (GN17, pg. 9), taking [u0p , vp0 ] to solve the system (3.4.1) - (3.4.2) creates an error: Z y Z y p 00 0 Ru,0 := ✏u0pxx + 1 ✏yveY u0py + ✏u0py 1 veY Y dy dy (3.4.3) 0 y0 This Ru,0 contribution is higher-order in ", and so will be accounted for as a forcing term in the construction of the next Prandtl layer, [u1p , vp1 ] (see (3.6.23)). After introducing the von-Mises coordinates, Z y ⇣ ⌘ ⌘= 1 + u0p dy 0 , (3.4.4) 0 170 the equation for u0p becomes parabolic, with x being the time-like variable: u0px = @⌘ ((1 + u0p )u0p⌘ ), u0p (x, 0) = , u0p (x, 1) = 0, u0p (1, ⌘) = U0 , (3.4.5) Via the maximum principle, as in (GN17), 1 + u0p 1 . (3.4.6) For the analysis in Section 3.4, it is convenient to introduce the shifted unknown q = u0p + . (3.4.7) The shifted unknown then satisfies the IBVP: ⇣ ⌘ qx = @⌘ (1 + q)q⌘ , q(x, 0) = 0, q(x, 1) = , q(1, ⌘) = u0p (1, ⌘) + . (3.4.8) 3.4.1 Existence of Front Profile The dynamics of the solution to equations (3.4.1) - (3.4.2), or equivalently, (3.4.8), are governed by a self-similar “front”, in the sense of (BKL94). This is due to the mismatch in boundary conditions at y = 0 and y = 1, as seen from (3.4.8). Inserting a self-similar ⌘ anzatz ⇤ (z) = ⇤ ( px ) into equation (3.4.8) gives the following ODE: 2 z 00 0 0 (1 + ⇤) ⇤ + ⇤ + ⇤ = 0, ⇤ (0) = q(x, 0) = 0, ⇤ (1) = q(x, 1) = . (3.4.9) 2 Here the 0 denotes @z , where z is the self-similar variable for ⇤. As the profile ⇤ that 171 we seek is a nonlinear variant of the Gaussian error function, we study: = ⇤ e , (3.4.10) where e is the Gaussian front profile with value at +1: Z z t2 e (z) = p e 4 dt. (3.4.11) ⇡ 0 Let us record the following: Lemma 3.4.1. With e being defined as (3.4.11), e ( p⌘x ) is an explicit solution of the heat equation, which bridges two distinct boundary conditions at y = 0 and y = 1: ⇣ ⌘ ⌘ @x @⌘⌘ e ( p ) = 0, e (0) = 0, e (1) = . (3.4.12) x Proof. We first record the identities: @z 1 @z z =p , = . (3.4.13) @⌘ x @x 2x Di↵erentiating (3.4.11) gives the identities: z2 z z2 z 0 e0 (z) = p e 4 , e00 (z) = p e 4 = e (z). (3.4.14) ⇡ 2x ⇡ 2x One then checks that: @x e ( p⌘x ) = @⌘⌘ e ( p⌘x ) is equivalent to (3.4.14). The boundary condition at 0 is trivial from (3.4.11), and the boundary condition at 1 arises from: e (1) = R 1 t2 p ⇡ 0 e 4 dt = . The lemma is proven. 172 Using the heat equation for e , coupled with (3.4.10) we obtain: ⇣ ⌘ 2 z 00 0 0 1 + + + = 2 0 e0 |e0 |2 e00 e 00 e e00 , 2 (0) = (1) = 0. (3.4.15) The first task is to obtain existence of a solution, , to the above boundary value problem in a suitable Sobolev space.2 Proposition 3.4.2. For sufficiently small, there exists a unique solution to the equation (3.4.15) in Hw2 (R+ ) satisfying || ||Hw2 . , where Hw2 is a weighted variant of H 2 which is formally defined in (3.4.24) Proof. For this argument, fix r > 0. We will eventually let r ! 1. Our starting point is to establish existence of solutions to the linear operator ⇣ z ⌘ @z2 @z . (3.4.16) 2 We recall the identity given in (BKL94): z2 ⇣ z ⌘ z2 ⇣ z2 1⌘ e 8 @z2 @z e 8 = @z2 + + , (3.4.17) 2 16 4 which in turn implies ⇣ z ⌘ @z2 @z = f, (0) = (r) = 0 (3.4.18) 2 1 2 Itis clear by rescaling z ! (1 ) 2 z, we can replace the factor of 1 in front of 00 by simply 1. This rescaling would change the main linear operator, (1 ) 00 + z2 0 to 00 + z2 0 . For notational ease, then, we work simply with the 00 instead of (1 ) 00 . The actual self-similar variable, then, is really (1 ) p⌘x , but as (1 ) is near 1, this causes no confusion in the analysis to follow. 173 if and only if ⇣ z2 1⌘ ˜ 2 2 @z2 + + ˜ (r) = f , ˜(r) (0) = ˜(r) (r) = 0, ˜(r) = e z8 (r) , z f˜ = e 8 f. (3.4.19) 16 4 The subscripts are included to emphasize the domain, (0, r) ⇢ R. Consider: Z Z ⇣ z2 r r 1⌘ ˜ B[ ˜(r) , v] := @z ˜(r) · @z v + + (r) v, H01 (0, r) ⇥ H01 (0, r) ! R. (3.4.20) 0 0 16 4 B is clearly bounded and coercive on H01 (0, r). One obtains the existence of a unique H 1 weak solution to the system (3.4.19), and correspondingly to (3.4.18) from the Lax-Milgram Lemma. Via elliptic regularity, this implies H 2 regularity which, in original unknowns, translates to the existence of a unique (r) 2 H 2 (0, r) for each f 2 L2 (0, r) such that: || (r) ||H 2 (0,r)  C(r)||f ||L2 (0,r) , and (r) (0) = (r) (r) = 0. (3.4.21) Let us rewrite the nonlinear problem (3.4.15) as a fixed point to: 00 z 0 00 (r) (r) = f( (r) ) + (e + ) (r) , (r) (0) = (r) (r) = 0, (3.4.22) 2 where 00 0 2 0 2 f( (r) ) = (r) (r) + (r) + 2e (r) + e0 + e00 (r) + e e00 . (3.4.23) Define now the norm: Z Z 2 2 00 2 0 || (r) ||Hw2 := | (r) | + (1 + z 2 ) (r) , (r) , (3.4.24) 174 and the parameter: n 1 1 o R( ) = max ||e ||L1 , ||e0 ||L1 , ||e00 ||L1 , ||(1 + z 2 ) 2 e00 ||L2 , ||e0 (1 + z 2 ) 2 ||L2 (3.4.25) Via a standard integration by parts argument applied to (3.4.22), we obtain the stabi- lizing estimate: 2 || (r) ||Hw2  R( )4 + R( )2 || 2 (r) ||Hw2 + || 4 (r) ||Hw2 , (3.4.26) which proves that if || (r) ||Hw 2  R( ), then 2 || (r) ||Hw2  R( )4 + 2R( )2 || 2 (r) ||Hw2 (3.4.27) Selecting small enough then ensures 3R( )4 < R( )2 , ensuring that (r) = N( (r) ) 2 BR( ) ⇢ Hw2 whenever (r) 2 BR( ) ⇢ Hw2 . (3.4.28) The second step is to prove the nonlinear map N is a contraction map on BR( ) ⇢ Hw2 . As such, label the pairs: z ⇣ ⌘ 00 00 0 (i,r) (i,r) = f( (i,r) ) + e + (r) , (i,r) (0) = (i,r) (r) = 0, i = 1, 2. (3.4.29) 2 Taking di↵erences yields and performing a standard integration by parts argument gives: 2 || 1,r 2,r ||Hw2  R( )2 || 1,r 2 2,r ||Hw2 , (3.4.30) 175 By the contraction mapping theorem there exists a unique solution to the nonlinear problem (3.4.22) - (3.4.23). Moreover, as this solution lies in the ball BR( ) , it obeys the estimate || (r) ||Hw 2 (0,r)  R( ), (3.4.31) uniformly in r. Therefore, we may let r ! 1 to obtain a solution to the problem (3.4.15). We may repeat the procedure in the above theorem with any weight, giving: ||hziM ||H 2  O( ; M ). (3.4.32) By further di↵erentiating equation (3.4.15), we can obtain ||hziM ||H k  O( ; M, k). (3.4.33) ⇤ Our front from here on will be denoted as = + e . We will abuse notation and depict ⇤ ⇤ (z) = (x, ⌘). (3.4.34) We have the following corollary to the preceding analysis: ⇤ Corollary 3.4.3. The front obeys the following bounds, for any m, k, l 0: ⇣ ⌘ l 1 ⇤ k 2 + 2p ||z m @⌘l @xk ||Lp⌘  O( )x (3.4.35) ⇤ Proof. This follows immediately from writing = + (e ), the definition of e 176 in (3.4.11), the bounds in (3.4.33), and then the chain rule together with the identities (3.4.13). Let us now record the following fact: 0 Corollary 3.4.4. A solution ⇤ to the system (3.4.9) must satisfy ⇤ (0) > 0. 0 Proof. Define uf = ⇤ , vf = ⇤. By (3.4.35), |uf |  c0 for some constant c0 . The system (3.4.9) is then: u0f = vf , (1 + uf )vf0 + |vf |2 + z2 vf = 0. The invariant set = {uf = C, vf = 0}, for c0  C  c0 contains equilibria. The solution ⇤ corresponds to an orbit with initial condition (uf (0) = 0, vf (0)) and final condition (uf (1) = , vf (1)). By ODE uniqueness, the trajectory cannot cross the set unless at ( , 0). Thus, if vf (0)  0, vf (z)  0 for all z, which violates uf (1) = . Thus, we must have vf (0) > 0. 3.4.2 Zeroeth Prandtl Layer, u0p We define the remainder, w, via: ⇤ ⇤ ⇤ w=q (·), g := w(1, ·) = q(1, ·) (·) = U0 (1, ·) + (·). (3.4.36) The initial data in (3.4.36) are clearly rapidly decaying after recalling (3.2.26) - (3.2.28), via the relation: g = U0 (e ) . Moreover, the following smallness is obtained, again by recalling the assumptions (3.2.26) - (3.2.28): ||hyim @yj g||L1  O( ; j, m) for any m, and j = 0, 1, 2. (3.4.37) 177 ⇤ The equation satisfied by w is the following (after substituting q = w + ): ⇣ ⌘ ⇤ wx = (1 )w⌘⌘ + @⌘ qw⌘ + w@⌘ = (1 )w⌘⌘ + K, (3.4.38) w(x, 0) = 0, w(x, 1) = 0, w(1, ⌘) = g(⌘), (3.4.39) where ⇤ ⇤ ⇤ K = w⌘2 + 2 ⌘ w⌘ + ww⌘⌘ + w⌘⌘ + w ⌘⌘ . (3.4.40) Let us first make the following basic observation regarding our initial data: Lemma 3.4.5 (Compatibility of Initial Data). Suppose the compatibility conditions in (3.2.31) are enforced. Then the initial data of wx respects the boundary condition wx (1, 0) = 0. Proof. This follows upon evaluating (3.4.1) at y = 0 and (3.4.8) at ⌘ = 0, x = 1. Remark (Higher Order Compatibility). This same calculation for higher-order compatibility conditions at x = 1, y = 0 can be made. We omit displaying them explicitly, but will feel free to assume higher-order compatibility of the initial data at y = 0 as needed. Let us now define a series of norms in which we control the remainder, w: Z Z 1 Z ||w||2Q(0,0) := sup w 2 , w⌘2 x, w⌘⌘ 2 x2 + w⌘2 , w⌘⌘ 2 2 x, w⌘⌘⌘ x2 . (3.4.41) x 1 1 We shall need the general, di↵erentiated and weighted variant of the above norms: Z 2 2 2 ||w||2Q( 0 ,k) := sup @xk w x2k 2 0 , @xk w⌘ x2k+1 2 0 , @xk w⌘⌘ x2k+2 2 0 x 1 Z 1Z 2 2 2 + @xk w⌘ x2k 2 0 , @xk w⌘⌘ x2k+1 2 0 , @xk w⌘⌘⌘ x2k+2 2 0 . (3.4.42) 1 178 Remark. One should take note of several points. First, each application of @x to the system 1 (3.4.38) adds enhanced decay of x , which is reflected in the norms above. This is because ⇤ of the behavior of the profiles under the application of @x , see estimate (3.4.35). ⇤ Second, upon controlling w in the Q-norms in (3.4.41), the dynamics of q = w + will ⇤ 1 be dominated by those of the front (so long as the parameter 0 < 4, which we will enforce). The small parameter 0 arises for technical reasons when performing weighted estimates, but should be ignored for the unweighted estimates. The final point is that upon heuristically identifying wx ⇡ w⌘⌘ through the equation (3.4.38), one sees that the norms Q( 0 , k) have an iterative structure: Z Z |@xk+1 w|2 x2(k+1) d⌘ ⇡ |@xk w⌘⌘ |2 x2k+2 d⌘, (3.4.43) the quantity on the left-hand side above being in ||w||Q(0,k+1) and the quantity on the right-hand side above being in ||w||Q(0,k) (upon taking sup in x). The reason for this structure is that the equation (3.4.38) is quasilinear. Through standard Sobolev interpolation, it is clear that: Lemma 3.4.6. For any 0 0, and k 0, 1 3 sup ||@xk wx 4 0 +k ||L1 ⌘ + sup ||@xk w⌘ x 4 0 +k ||L1 ⌘ . ||w||Q( 0 ,k) . (3.4.44) x x Proof. As the profile w decays at ⌘ ! 1 for each fixed x, we have: Z 1 1 1 1 |@xk w|2 x 2 2 0 +2k = 2 @xk w@xk w⌘ x 2 2 0 +2k d⌘ 0 . ||wxk 0 ||L2⌘ ||w⌘ xk 0+ 2 ||L2⌘ ⌘ . ||w||2Q( 0 ,k) . (3.4.45) 179 A similar computation works for the @xk w⌘ term in (3.4.44). We now give the energy estimates for w: Proposition 3.4.7. For any 0 > 0, and m 0, w satisfies the following estimate: ||z m w||Q( 0 ,0)  O( ; m, 0 ), (3.4.46) ||z m w||Q( 0 ,k)  C(k, m, 0 ), for k > 0. (3.4.47) Remark. Notice that the weights, z = p⌘ , honor the parabolic scaling of (3.4.38), and are x propagated by the linear flow. Proof. The proof proceeds in several steps. Step 1: Multiplier M = w: Applying the multiplier M = w to the system (3.4.38) generates the following positive terms: Z ⇣ ⌘ Z Z @x 2 wx (1 )w⌘⌘ ·w = w + (1 ) w⌘2 . (3.4.48) 2 Next, we must treat the nonlinear and linearized terms in K. The boundary condition w(x, 0) = 0 enables us to integrate by parts the quasilinear term: Z Z ⇣Z ⌘ w⌘2 · w + ww⌘⌘ · w = . ||w||L1 ww⌘2 ⌘ w 2 ⌘ , (3.4.49) Z Z Z Z Z ⇤ ⇤ ⇤ ⇤ 2 ⇤ 2 w⌘⌘ · w + ⌘⌘ w · w + 2 w ⌘ ⌘ · w = ⌘⌘ w + w⌘ Z 2 Z w  || ⇤⌘⌘ ⌘ 2 ||L1 + || ⇤ || L 1 ||w || ⌘ L⌘ 2 2  O( ) w⌘2 . (3.4.50) ⌘ ⌘2 180 ⇤ Above, we have used Corollary 3.4.3 to absorb two factors of ⌘ to into ⌘⌘ . We have also used the Hardy inequality, which is available as w(x, 0) = 0. Summarizing, we have: Z Z h iZ @x w2 + w⌘2 . O( ) + ||w||L1 ⌘ w⌘2 . (3.4.51) Step 2: Multiplier M = w⌘⌘ x Next, we will apply the multiplier M = w⌘⌘ x. This generates the positive terms: Z ⇣ ⌘ Z Z Z @x @x (1 )@⌘⌘ w · w⌘⌘ x = w⌘2 x w⌘2 + (1 ) 2 w⌘⌘ x. (3.4.52) 2 Let us now turn to the terms in K: Z Z ⇣ ⌘ 1 1 w⌘2 w⌘⌘ x + 2 ww⌘⌘ x  ||w, w⌘ x 2 ||L1 ⌘ ||w⌘ ||2L2⌘ + ||x 2 w⌘⌘ ||2L2⌘ , (3.4.53) Z Z Z ⇤ ⇤ 2 ⇤ ⌘ w ⌘ w ⌘⌘ x + w ⌘⌘ x + ⌘⌘ ww⌘⌘ x 1 1 ⇣ 1 ⌘ ⇤  || , ⌘x 2 ⇤⌘⌘ , x 2 ⇤⌘ ||L1 ||w⌘ ||2L2⌘ + ||w⌘⌘ x 2 ||2L2⌘ ⇣ 1 ⌘  O( ) ||w⌘ ||2L2⌘ + ||w⌘⌘ x 2 ||2L2⌘ . (3.4.54) Summarizing, we have: Z Z ⇣ ⌘Z Z 1 1 @x w⌘2 x + 2 w⌘⌘ x . 1 + ||w⌘ x ||L1 ⌘ 2 w⌘2 + ||w, w⌘ x ||L1 2 ⌘ 2 w⌘⌘ x. (3.4.55) 181 Step 3: Multiplier wx x2 The third step is to take @x of the equation (3.4.38). Doing so gives the system: ⇣ ⌘ @x (1 )@⌘⌘ wx = @x K, wx (x, 0) = 0, (3.4.56) ⇤ ⇤ ⇤ wx (1, ·) = (1 )g⌘⌘ + g⌘2 + 2 ⌘ g⌘ + gg⌘⌘ + g⌘⌘ + ⌘⌘ g. (3.4.57) First, we will record that ||h⌘im wx (1, ⌘)||L1 . O( ; m), for any m 0, by (3.4.57) and (3.2.26) - (3.2.28). Let us now apply the multiplier M = wx x2 to the system (3.4.56). Again, we will remain cognizant of the boundary condition wx (x, 0) = 0 when integrating by parts. Doing so gives the positive terms: Z ⇣ ⌘ Z Z Z 2 @x (1 )@⌘⌘ wx · wx x = @x wx2 x2 wx2 x + (1 ) 2 w⌘x x2 . (3.4.58) Next, we come to the nonlinearity in Kx , where integrating by parts as necessary: Z w⌘ w⌘x wx x2 + wx w⌘⌘ wx x2 + ww⌘⌘x wx x2 1 1  ||w⌘ x 2 ||L1 ⌘ ||w⌘x x||L2⌘ ||wx x 2 ||L2⌘ + ||w||L1 ||w⌘x x||2L2⌘ , (3.4.59) Z Z ⇤ 2 ⇤ 2 ⌘x w ⌘ w x x + ⌘ w⌘x wx x ⇤ ⇤  || ⌘x x⌘||L1 ⌘ ||w⌘ ||L2⌘ ||wx⌘ x||L2⌘ + || ⌘ ⌘||L1 ⌘ ||w⌘x x||2L2⌘ ,  O( )||w⌘ ||L2⌘ ||wx⌘ x||L2⌘ + O( )||w⌘x x||2L2⌘ (3.4.60) Z Z ⇤ 2 ⇤ x w ⌘⌘ w x x + w⌘⌘x wx x2 1 1 ⇤ ⇤ ⇤  || x x||L1 ⌘ ||w⌘⌘ x 2 ||L2⌘ ||wx x 2 ||L2⌘ + || , ⌘ ⌘||L1 ⌘ ||w⌘x x||2L2⌘ , 1 1  O( )||w⌘⌘ x 2 ||L2⌘ ||wx x 2 ||L2⌘ + O( )||w⌘x x||2L2⌘ (3.4.61) Z Z ⇤ 2 ⇤ 2 2 ⌘⌘x ww x x + ⌘⌘ wx x 3 1 1 ⇤ ⇤ 2 2  || ⌘⌘x ⌘x ||L1 2 ⌘ ||wx x 2 ||L2⌘ ||w⌘ ||L2⌘ + || ⌘⌘ x||L1 ||wx x ||L2 182 1 1  O( )||wx x 2 ||L2⌘ ||w⌘ ||L2⌘ + O( )||wx x 2 ||2L2⌘ . (3.4.62) Summarizing this piece: Z Z h iZ 1 @x wx2 x2 + 2 w⌘x x2 . O( ) + ||w, w⌘ x 2 ||L1 ⌘ 2 w⌘x x2 h 1 iZ Z + 1 + O( ) + ||w, w⌘ x ||L1 2 ⌘ wx x + O( ) w⌘2 , w⌘⌘ 2 2 x. (3.4.63) With these estimates in hand, we may now apply a standard continuous induction ar- gument to conclude the global existence of w satisfying (3.4.46) with k = m = 0 = 0. Step 4: Weighted Estimates 2 We now apply the weighted multiplier M = x 0 z m w to the system (3.4.38). First, we have the following positive terms: Z ⇣ ⌘ Z ⇣m ⌘Z @x wx w⌘⌘ · wz 2m x 2 0 = w2 z 2m x 2 0 + + 0 w2 z 2m x 1 2 0 2 2 Z Z + w⌘2 z 2m x 2 0 m(2m 1) w2 z 2m 2 x 1 2 0 . (3.4.64) The last term above has been estimated inductively at the m 1’th iterate. Next, we address the terms in K: Z ⇣ ⌘ w⌘2 + ww⌘⌘ · wz 2m x 0 . ||w||L1 ⌘ ||w⌘ z m x 0 ||2L2⌘ , (3.4.65) Z ⇣ ⌘ 2 ⇤ ⌘ w⌘ + ⇤ w⌘⌘ + w ⇤ ⌘⌘ · z 2m x 2 0 w . ||⌘ ⇤ ⌘, ⌘ 2 ⇤ ⌘⌘ ||L1 ⌘ ||w⌘ z m x 0 ||2L2⌘ . (3.4.66) 183 Summarizing: Z Z Z Z @x w2 z 2m x 2 0 + w⌘2 z 2m x 2 0 + w2 z 2m x 1 2 0 . w2 z 2m 2 x 1 2 0 . (3.4.67) By integrating in x: Z Z Z Z Z sup w2 z 2m x 2 0 + w⌘2 z 2m x 2 0 + w2 z 2m x 1 2 0 x 1 Z Z Z . w2 y 2m + w2 z 2m 2 x 1 2 0  O( , m, 0 ). (3.4.68) x=1 The next step is to apply the multiplier M = w⌘⌘ x1 2 0 z 2m . First, this gives: Z ⇣ ⌘ @x @⌘⌘ w · w⌘⌘ z 2m x1 2 0 = Z Z Z @x m w⌘2 z 2m x1 2 0 + w⌘⌘ 2 2m 1 2 0 z x + w⌘2 z 2m x 2 0 2 2 Z Z 1 2 0 2 2m 2 0 1 w⌘ z x + 2m w⌘ wx z 2m 1 x 2 2 0 . (3.4.69) 2 Next, let us turn to the nonlinear terms in K: Z ⇣ ⌘ ⇣ 1 w⌘2 + ww⌘⌘ · w⌘⌘ z 2m x1 2 0  ||w, w⌘ x 2 ||L1 ⌘ ||w⌘ z m x 0 ||2L2⌘ ⇥ 1 ⌘ ||w⌘⌘ z m x 2 0 ||2L2⌘ , (3.4.70) and the linearized terms: Z ⇣ ⌘ ⇤ ⇤ ⇤ ⌘ w⌘ + w⌘⌘ + ⌘⌘ w · w⌘⌘ z 2m x1 2 1 1 1 ⇤ ⇤ ⇤  ||x 2 ⌘ , x ⌘ ⌘⌘ , 2 ||L1 ||w⌘ x 0 z m ||L2⌘ ||w⌘⌘ z m x 2 0 ||L2⌘ . (3.4.71) 184 Summarizing, Z Z Z Z 1 @x w⌘2 z 2m x1 2 0 + 2 2m 1 2 0 w⌘⌘ z x + w⌘2 z 2m x 2 0 . w⌘ wx z 2m 1 x2 2 0 (3.4.72) Z Z 1 . w⌘2 z 2m x 2 0 + C wx2 x1 2 0 z 2m 2 . (3.4.73) 10, 000 Integrating in x: Z Z 1 Z Z 1 Z sup w⌘2 z 2m x1 2 0 + 2 2m 1 2 w⌘⌘ z x 0 + w⌘2 z 2m x 2 0 x 1 1 1 Z Z Z . w⌘2 y 2m + wx2 x1 2 0 z 2m 2  O( ; 0 , m). (3.4.74) x=1 The final step is to apply the multiplier M = wx x2 2 0 z 2m to the system (3.4.56). This generates the following terms: Z ⇣ ⌘ Z Z @x @x @⌘⌘ wx · wx x2 2 0 z 2m = wx2 x2 2 z 2m + w⌘x 2 x2 2 0 z 2m 0 2 ⇣m ⌘Z Z + 1+ 0 wx2 x1 2 0 z 2m m(2m 1) wx2 x1 2 0 z 2m 2 . (3.4.75) 2 We turn to the nonlinearities contained in Kx , for which we integrate by parts as needed: Z ⇣ ⌘ w⌘ w⌘x + wx2 w⌘⌘ + ww⌘⌘x w · wx x2 2 0 z 2m (3.4.76) 1 ⇣ 1 1 1 ⌘ . ||w, w⌘ x 2 ||L1 ||wx x 2 0 z m ||2L2⌘ + ||wx x 2 0 zm 1 2 ||L2⌘ + ||wx x 2 0 z m ||2L2⌘ . ⇤ Next, we come to the linearizations around the front profile, : Z Z ⇤ 2 2 ⇤ ⌘x w⌘ wx x 0 z 2m + ⌘ w⌘x wx x 2 2 0 z 2m ⇣ ⌘ ⇤ 2m ⇤ 2m  || ⌘x ⌘xz , ⌘ ⌘z ||L1 ||w⌘ ||2L2⌘ + ||wx⌘ x||2L2⌘ , (3.4.77) 185 Z ⇣ ⌘ 1 1 ⇤ x w⌘⌘ wx x 2 2 0 z 2m  ||z 2m x ⇤x ||L1 ||w⌘⌘ x 2 0 ||2L2⌘ + ||wx x 2 0 ||2L2⌘ , (3.4.78) Z Z Z 3 ⇤ ⇤ ⇤ wx⌘⌘ wx x2 2 0 2m z = ⌘ wx wx⌘ x 2 2 0 2m z + 2m wx wx⌘ x 2 2 0 z 2m 1 ⇣ 1 ⌘ . ||⌘ ⇤ 2m ⌘z , ⇤ ||L1 ||xwx⌘ ||2L2⌘ + ||x1 0 wx⌘ z m ||2L2⌘ + ||z m 1 wx x 2 0 ||2L2⌘ , (3.4.79) Z Z ⇤ 2 2 ⇤ ⌘⌘x wwx x 0 z 2m + 2 2 2 ⌘⌘ wx x 0 z 2m ⇣ ⌘ . ||z 2m x⌘ 2 ⇤ ⌘⌘x , z 2m 2 ⇤ ⌘ ⌘⌘ ||L1 ||w⌘ ||2L2⌘ + ||wx⌘ x||2L2⌘ . (3.4.80) Summarizing the results of this multiplier: Z Z Z @x wx2 x2 2 0 z 2m + 2 w⌘x x2 2 0 z 2m . wx2 x1 2 0 z 2m 2 Z Z + wx2 x1 2 0 z 2m 2 + O( ) w⌘2 , w⌘⌘ 2 2 x, wx⌘ x2 , (3.4.81) Again, we may integrate in x to obtain: Z Z 1 Z sup wx2 x2 2 0 z 2m + 2 w⌘x x2 2 0 z 2m x 1 1 Z Z 1Z Z 1 Z . wx (1)2 y 2m + wx2 x1 2 0 z 2m + wx2 x1 2 0 z 2m 2 x=1 1 1 Z 1Z 1 + O( ) w⌘2 , w⌘⌘ 2 2 x 2 , wx⌘ x. (3.4.82) 1 The third and fourth terms on the right-hand side are controlled by ||w||2Q , and the first term is controlled by the initial data. For the second term, we must use the equation (3.4.38): 1 1 ⇣ ⌘ 1 ⇤ ⇤ ⇤ ||wx x 2 0 z m ||L2  ||w⌘⌘ x 2 0 z m ||L2 + || w⌘2 + 2 ⌘ w⌘ + ww⌘⌘ + w⌘⌘ + w ⌘⌘ x2 0 z m ||L2 1 1 ⇤ 12 ⇤ ⇤ 1 1  ||w⌘⌘ x 2 0 z m ||L2 + ||w⌘ x 2 , ⌘ x , w, , ⌘⌘ ⌘x ||L1 ||w⌘ x 2 0 z m , w⌘⌘ x 2 0 z m ||L2 1 . ||w⌘⌘ x 2 0 z m , w⌘ x 0 z m ||L2 . (3.4.83) 186 As the majorizing terms above have been controlled, we can conclude: Z Z 1 Z sup wx2 x2 2 0 z 2m + 2 w⌘x x2 2 0 z 2m  O( ). (3.4.84) x 1 1 By subsequently relating wx to w⌘⌘ and w⌘x to w⌘⌘⌘ , we have shown that ||z m w||Q( 0 ,0)  O( , m, 0 ), which is the k = 0 case of (3.4.46). We may upgrade the previous set of estimates to higher order x derivatives by iterating Steps 2, 3, and 4. The mechanism for doing so ⇤ 1 is that each application of @x to the profile terms produces an extra factor of x , and 1 similarly each time @x hits z, one extra factor of x is produced by (3.4.13). As this is similar to the previous steps, we omit the details. We remark that controlling higher order derivatives does not require smallness of the initial datum (hence, the smallness assumption (3.2.28) is only for j = 0, 1, 2). We now give the following corollaries: Corollary 3.4.8 (u0p Asymptotics). For any m 0, 2  p  1, l 1 k 2 + 2p ||z m @yl @xk u0p ||Lpy  C(m, l, k)x for 2k + l > 2, (3.4.85) l 1 k 2 + 2p ||z m @yl @xk u0p ||Lpy  O( ; m, l, k)x for 2k + l  2. (3.4.86) Proof. We start with the representation of u0p via (3.4.36): ⇣ ⌘ ||z m @⌘l @xk u0p ||Lp⌘ = ||z m @⌘l @xk +e ||Lp⌘ + ||z m @⌘l @xk w||Lp⌘ . (3.4.87) The first quantity on the right-hand side above is controlled by Corollary 3.4.3. For the second quantity on the right-hand side, we simply use (3.4.46) with 0 < 0 < 14 , coupled Ry with the standard Sobolev interpolation. Recall now: ⌘ = 0 1 + u0p (x, ✓)d✓, from which we 187 have the equivalence: (1 )y  ⌘  (1+ )y. Moreover, we have the Jacobians of the change 1 of coordinates are given by: ⌘ 0 (y) = 1 + u0p , y 0 (⌘) = 1+u0p , both of which are bounded and bounded away from zero. Therefore, we can transfer (3.4.87) to (x, y) coordinates. We may give bounds for vp0 : Corollary 3.4.9 (Estimates for vp0 ). We have the following bounds for v, for any m 0: 1 ||z m @xk vp0 (x, ·)||L2y  C(k, m)x k 4 for k 1, (3.4.88) 1 ||z m vp0 (x, ·)||L2y  O( ; m)x 4 , (3.4.89) 1 ||z m @xk vp0 (x, ·)||L1 y  C(k, m)x k 2 for k 1, (3.4.90) 1 ||z m vp0 (x, ·)||L1 y  O( ; m)x 2 . (3.4.91) Proof. First, we give L2 via the Hardy inequality, which is available for m 0 as @xk v(x, y = 1) = 0: 1 ||z m @xk vp0 (x, ·)||L2y  ||z m y@xk vpy 0 (x, ·)||L2y = ||z m y@xk u0px (x, ·)||L2y . x k 4 . (3.4.92) We have used estimate (3.4.85) for k > 0, and one obtains the smallness of O( ) for k = 0 by applying (3.4.86). The L1 estimate then follows via standard Sobolev interpolation. Indeed, as limy!1 vp0 (x, y) = 0, with rapid decay for fixed x, one writes: Z 1 |y m @xk vp0 (x, y)|2 = 2(y m @xk vp0 )@y (y m @xk vp0 ) dy 0 y Z 1 Z 1 0 = 2y m @xk vp0 my m 1 @xk vp0 dy + 2(y m @xk vp0 )y m @xk vpy 0 dy 0 . (3.4.93) y y 188 Dividing both sides by xm then gives: 1 1 1 1 ||z m @xk vp0 (x, ·)||L1 y . ||z m @xk vp0 (x)||L2 2 ||z m @xk vpy 0 (x)||L2 2 + x 4 ||z m 2 @xk vp0 ||L2y , (3.4.94) y y from which the desired results follow upon consultation with (3.4.92) and (3.4.85) - (3.4.86). 3.5 Euler-1 Layer 3.5.1 Derivation of Equations In this section we construct the next order in the expansion, [u1e , ve1 , Pe1 ]. The analysis will p be taking place in Euler coordinates, that is (x, Y ), where Y = ✏y as in (3.2.9). We make the notational convention that di↵erential operators applied to [u1e , ve1 , Pe1 ] will always be in (x, Y ) coordinates, so for example u1e := @x2 u1e + @Y2 u1e . Let us now expand the partial expansion in order to find the lowest-order terms which we then take to solve the Euler-1 equation. The first equation is: 3 ⇣ p ⌘⇣ p ⌘ (1) ✏ us + u(1) (1) (1) (1) (1) s usx + vs usy + Psx = 0 ✏ up ✏ 2 u1e + u¯0s + ✏u1e u0px + ✏u1ex ⇣ ⌘⇣ ⌘ p + vp0 + ve1 u0py + ✏u1eY + ✏Pex 1 1,a + ✏Pex . (3.5.1) Removing now the terms found in equation (3.4.1), and retaining the lowest-order purely Euler terms gives: p h 1 1 i ✏ uex + Pex = 0. (3.5.2) 189 The remaining terms from the above expansion are placed into the next order, which is discussed in detail in equations (3.6.1) - (3.6.8). Let us now turn to the second equation: @y (1) ⇣ p 1 ⌘⇣ 0 ⌘ (1) ✏ vs + u(1) s v (1) sx + v (1) (1) s v sy + P = ✏ v 0 ✏ v 1 + u ¯ 0 + ✏u v + v 1 ✏ s p e s e px ex ⇣ ⌘⇣ p ⌘ p 1,a + vp0 + ve1 vpy 0 1 + ✏veY 1 + PeY + ✏PeY . (3.5.3) Taking the order-1 purely Eulerian terms gives: 1 1 vex + PeY = 0. (3.5.4) The remaining terms are contributed to the next-order error in (3.6.1) - (3.6.8). Putting (3.5.2) - (3.5.4) together with the divergence-free condition yields the following system for the Euler-1 layer: u1ex + Pex 1 = 0, 1 vex 1 + PeY = 0, u1ex + veY 1 = 0, in ⌦. (3.5.5) with prescribed boundary data ve1 (x, 0) = vp0 (x, 0). (3.5.6) Without loss of generality, taking Pe1 = u1e , gives the div-curl system 1 vex u1eY = 0, u1ex + veY 1 = 0. (3.5.7) Notice that these equations are the Cauchy-Riemann equations for ve1 = Re(f ), u1e = 190 Im(f ), f holomorphic on ⌦. Then by taking the curl of the equation, ve1 = 0 in ⌦, ve1 (x, 0) = vp0 (x, 0). (3.5.8) The following boundary decay rates follow from (3.4.90) - (3.4.91): 1 1 @xk vp0 (x, 0)  C(k)x 2 k , vp0 (x, 0)  O( )x 2 (3.5.9) 3.5.2 Uniform Decay Estimates (m) 0 (m) Motivated by (3.5.9), let us define for this section f := x 10 ER+ vp , where ER+ is the m’th order Sobolev extension operator on the half-line R+ (see (AF03, Chapter 5)). Here, m is selected as large as required by the remainder of our analysis. Then, f agrees with vp0 on [0, 1), and is cut-o↵ past x  10. Then by standard Sobolev extension theory, ||f ||H m  C(m)||vp0 ||H m . Our setup for this subsection is: 1 1 v = 0 on H, v(x, 0) = f (x), @xk f (x)  x k 2 , f (x)  O( )x 2 . (3.5.10) Using the Poisson Kernel, we have: Z 1 Z 1 Y v(x, Y ) = PY ⇤ f = PY (x t)f (t)dt = f (t)dt. (3.5.11) 1 1 Y 2 + (x t)2 We will need the following pointwise estimates on the Poisson Kernel: Lemma 3.5.1. For x > 0, k 0, and 0  s  k, we have: |@xk PY (x)| . x s 1 Y (k s) . (3.5.12) 191 Proof. First, let us address the k = 0 case. Indeed, Yx Y 2 + x2 xPY (x) = 22  2  1. (3.5.13) Y +x Y + x2 For k 1, we record the basic identities: k X 2 (k @xk PY (x) = cj,k Y x2j (Y 2 + x2 ) 2 +1) j if k even, (3.5.14) j=0 k 1 X 2 k+1 @xk PY (x) = cj,k Y x2j+1 (Y 2 + x2 ) 2 1 j if k odd. (3.5.15) j=0 For the k even case, we multiply (3.5.14) by xk+1 , use the binomial formula, and then apply Young’s inequality for products in the following manner: k X 2 s+1 k s k (k |x Y @ x PY (x)| = | cj,k Y 1+k s 2j+s+1 x (Y 2 + x2 ) 2 +1) j | j=0 k X 2 Y 1+k s 2j+s+1 x . cj,k k+2+2j j=0 Y + xk+2+2j k k+2+2j X Y (1+k s)( k+2+2j 2 + x(s+1+2j)( s+1+2j ) 1+k s ) . . 1, (3.5.16) j=0 Y k+2+2j + xk+2+2j where the Young’s conjugates are: 1+k s s + 1 + 2j 1= + . (3.5.17) k + 2 + 2j k + 2 + 2j Performing the same calculation using (3.5.15), we have: k 1 X 2 k+1 s+1 k s k |x Y @ x PY (x)| = | cj,k Y 1+k s s+2j+2 x (Y 2 + x2 ) 2 1 j | j=0 192 k 1 X 2 Y 1+k s s+2j+2 x . cj,k k+3+2j j=0 Y + xk+3+2j k 1 k+2j+3 X s)( k+2j+3 2 Y (1+k + x(s+2j+2)( s+2j+2 ) 1+k s ) . cj,k . 1. (3.5.18) j=0 Y k+3+2j + xk+3+2j Here, the Young’s conjugates are: 1+k s s + 2j + 2 1= + . (3.5.19) k + 2j + 3 k + 2j + 3 This proves (3.5.12). We now prove the following uniform estimates for the ve1 (x, Y ), which now drop super- scripts and denote by v for notational ease: Proposition 3.5.2. Let v be the Poisson extension defined via (3.5.11). Then v satisfies the following bounds: 1 1 sup @xk v(x, Y )  C(k)x k 2 , sup v(x, Y )  O( )x 2 . (3.5.20) Y 0 Y 0 Proof. The Y = 0 case is clear by (3.5.9), and so we must treat the case when Y > 0. In this regime, several qualitative facts are available for use. In particular, PY is smooth, has unit mass, and is positive everywhere. As is standard, we shall exploit these qualitative facts in order to extract quantitative bounds independent of Y > 0. Fixing any ↵ > 0, we shall split the integral (3.5.11) into three pieces: Z x 1+↵ Z x 1+↵ Z 1 v(x, Y ) = PY (x t)f (t)dt + PY (x t)f (t)dt + PY (x t)f (t)dt x x 1 1+↵ 1+↵ = I1 + I2 + I3 . (3.5.21) 193 1 For the sake of concreteness, fix: ↵ = 10,000 . and " will be taken small relative to this universal constant, ↵. Let us first turn to I3 : Z 1 1 1 x 2 I3 = PY (x t)x 2 f (t)dt (3.5.22) x 1+↵ Z 1 Z 1 1 1 1  PY (x t){x 2 |t| }f (t)dt + 2 PY (x t)|t| 2 f (t)dt (3.5.23) x x 1+↵ 1+↵ Z 1 ⇣ x ⌘ 12 Z 1 1 1  PY (x t){ 1}|t| 2 f (t)dt + ||f x 2 ||L1 PY (x t)dt (3.5.24) x 1+↵ |t| 1 ⇣ x ⌘ 12 Z 1 1  sup 1 ||f x 2 ||L1 PY (x t)dt + O( )  O( ). (3.5.25) t x 1+↵ |t| 1 R where we have used for each fixed Y > 0, PY 0 and PY = 1. By symmetry, I1 works the same way. Therefore, we are left with I2 , where the main mechanism are pointwise estimates on PY . According to (3.5.12), with k = s = 0, 1 sup PY (x t) . , (3.5.26) x x 1+↵ t 1+↵ x x x as x t > 0 int the region 1+↵ t 1+↵ . Thus, we have: Z x Z x Z x 1+↵ 1 1+↵ O( ) 1+↵ 1 I2  PY (x t)|f (t)|dt . |f (t)|dt  |1 + t| 2 dt x 1+↵ x x 1+↵ x x 1+↵ 1  O( )x 2 . (3.5.27) Combining (3.5.25) - (3.5.27), we may conclude that 1 x 2 v  O( ). (3.5.28) We now need to establish this procedure for higher-order x derivatives for v. We continue with the I1 , I2 , I3 splitting used above. Upon di↵erentiating (3.5.21) with respect to x, let 194 us treat each term Ii individually. Z x 1 x x 1+↵ @ x I1 = PY (x + )f ( )+ @x PY (x t)f (t)dt 1+↵ 1+↵ 1+↵ 1 Z x 1 x x 1+↵ = PY (x + )f ( ) @t PY (x t)f (t)dt 1+↵ 1+↵ 1+↵ 1 Z 1+↵x = B1 + PY (x t)f 0 (t)dt. (3.5.29) 1 Here, 1 x x x x B1 = PY (x + )f ( ) PY (x + )f ( ). (3.5.30) 1+↵ 1+↵ 1+↵ 1+↵ 1+↵ Note now in a similar manner to (3.5.12), 2+↵ x 1 1 3 B 1 . PY ( x)|f ( )| . O( )x 2 .x 2 . (3.5.31) 1+↵ 1+↵ x Using the same method as evaluating (3.5.25): Z x 1+↵ 3 x2 PY (x t)f 0 (t)dt 1 Z x 1+↵ Z x 1+↵ 3 3 3 0  PY (x t)|t| f (t) dt + 2 PY (x t) {x 2 |t| 2 } f 0 (t) dt 1 1 Z 1 Z x 1+↵ 3 3 3  PY (x t)|t| f 0 (t) dt + 2 PY (x t) {x 2 |t| 2 } f 0 (t) dt 1 1 Z x 1+↵ ⇣ x ⌘ 32 3 3  ||x 2 f 0 ||L1 + PY (x t) { 1} |t| 2 f 0 (t) dt 1 |t| ⇣ x ⌘ 32 Z 1 3 3 . ||x f ||L1 + 2 0 sup { 1} ||x f ||L1 2 0 PY (x t)dt  C. (3.5.32) t x 1+↵ |t| 1 195 Next, we turn to I2 , which after di↵erentiating gives: 1 x x 1 x x @ x I2 = PY (x )f ( )+ PY (x + )f ( ) 1+↵ 1+↵ 1+↵ 1+↵ 1+↵ 1+↵ Z 1+↵ x + @x PY (x t)f (t)dt. (3.5.33) x 1+↵ Via (3.5.12) with k = 1, s = 0, 1 1 sup @x PY (x t)  sup . . (3.5.34) Y >0,t2[ x x 1+↵ , 1+↵ ] Y >0,t2[ x x 1+↵ , 1+↵ ] (x t)2 x2 For I2 , this then facilitates the bound: Z x Z x 1+↵ 1 1+↵ 1 3 @x PY (x t)f (t)dt . (1 + t) 2 dt . x 2 . (3.5.35) x 1+↵ x2 x 1+↵ For the @x I3 term, we must integrate by parts in the following manner: Z 1 1 x x @ x I3 = P (x )f ( )+ @x PY (x t)f (t)dt 1+↵ 1+↵ 1+↵ x 1+↵ Z 1 = B3 + PY (x t)f 0 (t)dt. (3.5.36) x 1+↵ Here, B3 contains the boundary terms: 1 x x x x B3 = PY (x )f ( ) PY (x )f ( ) 1+↵ 1+↵ 1+↵ 1+↵ 1+↵ 1 ↵ x ↵ x = PY ( x)f ( ) PY ( x)f ( ). (3.5.37) 1+↵ 1+↵ 1+↵ 1+↵ 1+↵ Again, via direct calculation using the definition of PY , it is clear that: 3 B3 . x 2 . (3.5.38) 196 We may estimate the integral in (3.5.36) in identical fashion to (3.5.32). This procedure can be iterated for higher-order derivatives. To see this, let us start with the splitting given in (3.5.21). We will apply @xk , where k 2: @xk v(x, Y ) = @xk I1 + @xk I2 + @xk I3 . (3.5.39) Our starting point is the expression (3.5.29), from which it becomes clear that applying @xk to I1 , for generic constants c↵ > 0 (which can change from term to term), we have: k X1 Z x 1+↵ @xk I1 = cj @xj PY (c↵ x)@xk 1 j f (c↵ x) + PY (x t)@tk f (t)dt. (3.5.40) j=0 1 Using (3.5.12), we estimate: 1 1 |@xj PY (c↵ x)@xk 1 j f (c↵ x)|  c↵ x j 1 x (k 1 j) 2 .x k 2 . (3.5.41) For the integration term in (3.5.40), we follow (3.5.32) to give: Z x 1+↵ 1 xk+ 2 PY (x t)@tk f (t)dt (3.5.42) 1 Z x 1+↵ Z x 1+↵ k+ 12 1 1  PY (x t)|t| @tk f (t) dt + PY (x t) {xk+ 2 |t|k+ 2 } @tk f (t) dt 1 1 Z 1 Z x 1+↵ k+ 12 1 1  PY (x t)|t| @tk f (t) dt + PY (x t) {xk+ 2 |t|k+ 2 } @tk f (t) dt 1 1 Z x 1+↵ ⇣ x ⌘k+ 12 1 1  ||xk+ 2 @xk f ||L1 + PY (x t) { 1} |t|k+ 2 @tk f (t) dt 1 |t| ⇣ x ⌘k+ 12 Z 1 k+ 12 k+ 12 . ||x @xk f ||L1 + sup { 1} ||x @xk f ||L1 PY (x t)dt  C. t x 1+↵ |t| 1 (3.5.43) By symmetry, the same estimate applies to I3 , and so we turn to I2 . Referring to (3.5.33), 197 one sees the following expression: k X1 Z x 1+↵ @xk I2 = cj @xj PY (c↵ x)@xk 1 j f (c↵ x) + @xk PY (x t)f (t) dt, (3.5.44) x j=0 1+↵ where c↵ > 0 denotes generic constants which depend on ↵, not on , ", and are strictly positive. Referring to (3.5.41), it remains to estimate the integral term above: Z x 1+↵ Z x 1+↵ | @xk PY (x t)f (t) dt|  |@xk PY (x t)||f (t)| dt x x 1+↵ 1+↵ Z x 1+↵ 1 . sup |@xk PY (x t)| |(1 + t)| 2 dt x x x 1+↵ t 1+↵ 1+↵ 1 1 .x k 1 x2 = x k 2 . (3.5.45) This then proves the desired result. By repeating the above proof, we actually have the following: Corollary 3.5.3. Consider boundary values g(x) satisfying: @xk g(x)  x k w , where w 2 (0, 1). (3.5.46) Then the Poisson extension satisfies: sup @xk {PY ⇤ g} . x k w . (3.5.47) Y We now give an estimate more suitable to obtaining L2 in Y estimates. 198 Lemma 3.5.4. For any k 1, and 0  k s  1, we have: 1 @xk v(x, Y ) . x s 2 Y (k s) . (3.5.48) Proof. By Lemma 3.5.2, the claim is only meaningful for Y x, so make this assumption to begin. Again, we start with the splitting (3.5.21). By symmetry it suffices to consider I2 , I3 . Applying @xk , we have the expression (for generic constants c↵ > 0): k X1 Z x 1+↵ @xk I2 = cj,k,↵ @xj PY (c↵ x)@xk 1 j f (c↵ x) + @xk PY (x t)f (t)dt, (3.5.49) x j=0 1+↵ First, 1 |@xj PY (c↵ x)@xk 1 j f (c↵ x)| . Y (k s) x j 1+(k s) x (k 1 j) 2 1 .Y (k s) x s 2 . (3.5.50) Above, we have used (3.5.12) for the case j 1. For the j = 0 case we use the assumption that Y x, and the following estimate on PY , which is valid so long as k s  1: Y Y 1 (k s) 1+(k s) PY (x t) =  = Y x . (3.5.51) Y 2 + (x t)2 Y2 Y For the integral term in (3.5.49), via (3.5.12), sup @xk PY (x t) . Y (k s) x s 1 . (3.5.52) Y 0 x x t2[ 1+↵ , 1+↵ ] 199 From here, the estimate on the integral in (3.5.49) follows: Z x 1+↵ Z x 1+↵ 1 1 @x PY (x t)f (t)dt . Y (k s) x s 1 (1 + t) 2 dt . Y (k s) x s 2 . (3.5.53) x x 1+↵ 1+↵ For the I3 contribution, we apply @xk and integrate by parts in t, reads: k X1 Z 1 @ x I3 = cj,k,↵ @xj PY (c↵ x)@xk 1 j f (c↵ x) + PY (x t)@tk f (t)dt. (3.5.54) x j=0 1+↵ First, as shown in (3.5.50) 1 |@xj PY (c↵ x)@xk 1 j f (c↵ x)| . Y (k s) x s 2 . (3.5.55) Turning to the integration in (3.5.54), our point of view on the kernel: Y Y 1 PY (x t) =  = . (3.5.56) Y 2 + (x t)2 Y2 Y Then in the regime Y x, we have (since k s  1): Z 1 Z 1 1 1 1 k+ 12 PY (x t)@tk f (t)dt  t k 2 dt  x x 1+↵ Y x 1+↵ Y 1 ⇣ Y ⌘1 (k s) k+ 12 (k s) s 1  x Y x 2 . (3.5.57) Y x This establishes the desired result. Lemma 3.5.5 (Interpolation Estimates). We have the following decay estimates: 1 3 xk+ 2 ||@Yk v(x, ·)||L1 Y  C, x 2 ||@Y v(x, ·)||L1 Y  O( ). (3.5.58) 200 Proof. We treat the case when k = 1, with the extension for higher k values being obvious. This follows via standard interpolation arguments: fix an x, and consider vY (x, ·) as a function of the Y variable, defined on the half-space Y 0. Then via Gagliardo-Nirenberg interpolation estimates, and via harmonicity vxx = vY Y , 3 ⇣ 1 ⌘ 12 ⇣ 5 ⌘ 12 x 2 ||vY (x, ·)||L1 Y . x 2 ||v(x, ·)|| 1 L Y x 2 ||v xx (x, ·)|| L 1 Y . (3.5.59) Importantly, via Remark 4, Page 125 in (Nir59), we need not include a lower-order term on the right-hand side of (3.5.59), which is due to the fact that our domain, for each fixed x, is the half-space Y 0. Taking the supremum over x and applying (3.5.20) yields the desired result. The smallness of O( ) for k = 1 case is guaranteed due to the v term in (3.5.59). Proposition 3.5.6 (Euler Correctors). Suppose [u1e , ve1 ] solves the boundary value problem: Z 1 ve1 = 0, ve1 (x, 0) = vp0 (x, 0), u1e = 1 veY (x0 , Y )dx0 , for x 1. (3.5.60) x Then the following bounds holds for any k, j 0: 1 h 1 3 i sup @xk @Yj ve1 xk+j+ 2  C, sup u1e , ve1 x 2 + u1ex , veY 1 x 2  O( ). (3.5.61) Y 0 Y 0 Proof. We shall take ve1 = v from the above lemmas. For the u1e profile, we define for x 1: Z 1 u1e (x, Y ) := 1 veY (x0 , Y )dx0 . (3.5.62) x 201 From here it is clear that the Cauchy-Riemann equations hold: Z 1 Z 1 u1ex = 1 veY , u1eY = 1 veY Y = 1 vexx 1 = vex . (3.5.63) x x 1 We have used that vex (x, Y ) ! 0 as x ! 1. From Lemma 3.5.5, it is then clear that: Z 1 Z 1 Z 1 3 1 u1e = u1ex dx0 = 1 veY dx0  O( ) |x0 | 2 dx0  O( )x 2 . (3.5.64) x x x Remark (In-flow Conditions). Note that we do not solve for the Euler correctors given an arbitrary in-flow condition at x = 1. Rather, we take [u1e , ve1 , Pe1 ] to be the explicit profiles obtained by applying the Poisson extension to f . In this sense, the set-up we consider here is distinct from the construction of Euler profiles in (GN17). 3.6 Prandtl Layer 1 3.6.1 Derivation of Linearized Prandtl Equations: In this section, we will construct the Prandtl correctors, [u1p , vp1 , Pp1 ]. Let us now obtain the equations that they will satisfy. We do this in a manner which can be easily generalized in the next section. We expand the nonlinear terms: ⇣ p 1 p 1 ⌘⇣ (0) p 1 p ⌘ ¯(1) u s u¯(1) sx = u¯(0) s + ¯sx + ✏uex + ✏u1px ✏ue + ✏up u p p p p ¯(0) =u s u¯(0) sx + ✏u(1) 1 sx up + ✏u(1) 1 1 1 s upx + ✏up upx + ✏u1ex u0p + ✏u1e u0px p + ✏u1ex + ✏u1e u1ex , (3.6.1) ⇣ p ⌘⇣ p ⌘ v¯s(1) u ¯(1) 0 1 sy = vp + ve + ✏vp1 u0py + ✏u1eY + ✏u1py 202 p p (1) 1 = vs(1) u0py + ✏u(1) 1 sy vp + ✏vs upy + ✏vp1 u1py + ✏vp0 u1eY + ✏ve1 u1eY , (3.6.2) ⇣ p p ⌘⇣ 0 p 1 ⌘ ¯(1) u s v (1) ¯sx = 1 + u0p + ✏u1e + ✏u1p vpx 1 + vex + ✏vpx p p p ¯(0) =u 0 s vpx + 1 ✏vpx u(1) s + (1) 1 ✏vsx up + ✏u1p vpx 1 + u0p vex 1 + ✏u1e vpx 0 1 p + vex + ✏u1e vex 1 , (3.6.3) ⇣ p ⌘⇣ p 1 p 1 ⌘ v¯s(1) v¯sy (1) = vp0 + ve1 + ✏vp1 vpy0 + ✏veY + ✏vpy p p p p = vp0 vpy 0 + 1 (1) ✏vpy vs + (1) 1 ✏vsy vp + ✏vp1 vpy 1 + ✏vp0 veY 1 + ve1 vpy 0 + ✏ve1 veY 1 . (3.6.4) Let us now denote: u,1 p p u,1 RE := ve1 u0py + ✏u1ex u0p + ✏u1e u0px + ✏vp0 u1eY , EE := ✏ve1 u1eY + ✏u1e u1ex , (3.6.5) v,1 p p v,1 p p RE = u0p vex 1 + ✏u1e vpx 0 + ✏vp0 veY 1 + ve1 vpy 0 , EE := ✏u1e vex 1 + ✏ve1 veY 1 . (3.6.6) The purpose of the terms above is to separate the Euler-Prandtl terms and the pure- Euler terms. The pure-Euler terms are harmful to our analysis, because they scale di↵erently than the Euler-Prandtl terms. From a practical point of view, their presence prevents the application of weights of the form z = py , and therefore obstructs self-similarity. It turns x out that all pure-Euler terms are of “gradient-type”, see (3.7.32). Thus, by introducing appropriate potential functions in the pressure expansion, we can force these terms to vanish identically. With this notation, we may write the Navier-Stokes expansion as: ¯(1) ✏u ¯(1) +u ¯(1) ¯s(1) u ¯(1) ¯ (1) ¯(0) ¯(0) ¯(0) ¯s(0) u ¯(0) ¯ (0) s s u sx + v sy + Px = ✏u s +u s u sx + v sy + Px p h 1 (1) 1 (1) 1 (1) 1 (1) 1 p 1 1 + ✏ ✏ up + us upx + usx up + usy vp + vs upy + ✏up upx p i 3 p u,1 u,1 1,a + ✏vp1 u1py + Ppx 1 1,a + ✏Ppx + RE + EE ✏ 2 u1e + ✏PEx + ✏(u1ex + Pex 1 ). (3.6.7) 203 The normal equation is expanded via: (1) (0) P¯y P¯y ¯s(1) ✏v ¯(1) +u s v ¯ (1) sx + v ¯ (1) (1) s v ¯ sy + = ✏ v ¯ (0) s + u ¯ (0) (0) s v ¯ sx + v ¯ (0) (0) s v ¯ sy + ✏ ✏ p h 1 (1) 1 (1) 1 (1) 1 (1) 1 p 1 1 + ✏ ✏ vp + us vpx + vsx up + vs vpy + vsy vp + ✏up vpx 1 i p Ppy @y v,1 v,1 @y 1,a @y p 1 + ✏vp1 vpy 1 + + ✏Pp1,a + RE + EE ✏ ve1 + ✏PE + ( 1 ✏Pe + vex ). ✏ ✏ ✏ ✏ (3.6.8) (0) (0) (1) (1) In the above expressions, the [¯ us , v¯s ], [us , vs ] are known at this stage, and the u1p , vp1 are unknowns, to be constructed in this step. Similarly for the pressures, the the Pe1 is known, but the auxiliary pressures PE1,a , Pp1,a are to be defined in this section. Finally, as u,1 v,1 u,1 v,1 can be seen from the definitions in (3.6.5), (3.6.6), the RE , RE , EE , EE are all knowns. Let us now simplify the above expressions. First, via the construction of [u0p , vp0 ] we may write: Ru,0 := ¯(0) ✏u s ¯(0) +u s u¯(0) ¯s(0) u sx + v ¯(0) ¯ (0) sy + Px + ve upy 1 0 Z y Z y0 p 1 0 00 0 = ✏u0pxx + ✏yveY upy + ✏u0py 1 veY Y dy dy . (3.6.9) 0 y u,1 This then accounts for the lowest order term from RE in (3.6.5), allowing us to redefine: u,1 p p RE = ✏u1ex u0p + ✏u1e u0px + ✏vp0 u1eY . (3.6.10) Let us now define the auxiliary Euler pressure: 1⇣ 1 2 2⌘ PE1,a := ue + ve1 , (3.6.11) 2 so that, combined with the equations we have taken for [u1e , ve1 ], we have: 204 Lemma 3.6.1. With PE1,a as in (3.6.11), and EE u,1 v,1 , EE as in (3.6.5) - (3.6.6), @y 1,a @x ✏PE1,a + EE u,1 = v,1 ✏P + EE = 0. (3.6.12) ✏ E @Y Remark. Denoting by r✏ := (@x , p " ), the above lemma reads: r" "PE1,a + (EE u,1 v,1 , EE ) = 0. (3.6.13) Thus, the purely Eulerian terms are of gradient structure, which is then exploited with the introduction of our auxiliary pressure. Proof. The proof follows by direct calculation and an appeal to (3.5.5): "@x PE1,a = "u1e u1ex "ve1 vex 1 = "u1e u1ex "ve1 u1eY = u,1 EE . (3.6.14) Similarly, @ p p pY "PE1,a = "u1e u1eY "ve1 veY 1 = "u1e vex 1 "ve1 veY 1 = v,1 EE . (3.6.15) " The claim has been proven. Corollary 3.6.2. All “purely Eulerian” terms from the expansions (3.6.7), (3.6.8) vanish identically. That is: u,1 3 1,a p EE ✏2 u1e + ✏PEx + ✏(u1ex + Pex 1 ) = 0, (3.6.16) v,1 @y 1,a @y p 1 EE ✏ ve1 + ✏P + ( 1 ✏Pe + vex ) = 0. (3.6.17) ✏ E ✏ 205 Proof. First, by harmonicity of [u1e , ve1 ], we have: 3 ✏2 u1e = ✏ ve1 = 0. (3.6.18) Next, regarding the final terms in both (3.6.7) and (3.6.8), by the equations (3.5.2) and (3.5.4), we see that these terms drop out: u1ex + Pex 1 = 0, 1 vex 1 + PeY = 0. (3.6.19) Coupled with (3.6.12), this establishes the desired claim. In the above calculation, we are using crucially the Cauchy-Riemann structure of [u1e , ve1 ] in 3.5.7; harmonicity alone does not suffice here. Next, define the auxiliary Prandtl pressure: Z 1 h i Z 1 v,1 Pp1,a := ¯s(0) ✏v + ¯(0) u s v (0) ¯sx + v¯s(0) v¯sy (0) + RE . (3.6.20) y y Combining all this, we take our Prandtl-1 equation to be: ⇣ ⌘ u1pyy + (1 + u0p )u1px + u(1) 1 0 1 sx up + upy vp vp1 (x, 0) + vs(1) u1py + Ppx 1 = f (1) , (3.6.21) together with the divergence free condition u1px + vpy 1 = 0, and the boundary conditions: u1p (x, 0) = 0, lim [u1p (x, y), vp1 (x, y)] = 0, vp1 (x, 0) = ve2 (x, 0), u1p (1, y) = U1 (y). y!1 (3.6.22) 206 Here, the forcing term is defined as: 1 h i u,1 f (1) := ✏ 2 Ru,0 + RE 1,a + ✏Ppx . (3.6.23) The boundary contribution of vp1 (x, 0) in (3.6.21) again arises from the calculation: Z y Z y0 p ve2 (x, Y )u0py = ve2 (x, 0)u0py + ✏yu0py veY 2 + ✏u0py 2 veY Y 0 y Z y Z y0 p = vp1 (x, 0)u0py + ✏yu0py veY 2 + ✏u0py 2 veY Y. (3.6.24) 0 y Consolidating (3.6.18), (3.6.19), (3.6.12), (3.6.20), (3.6.21), (3.6.24) with the expressions (3.6.7) and (3.6.8) then shows that the following remainder is contributed: Z Z p h p y y0 p Ru,1 = ✏ ✏u1pxx + 2 ✏yveY u0py + ✏u0py 2 veY 1 Y + (✏ueY + ✏u1py )vp1 (3.6.25) 0 y p i + ✏(u1e + u1p )u1px , p h p p i Rv,1 = ✏ 1 ✏ vp + u(1) 1 (1) 1 (1) 1 (1) 1 s vpx + vsx up + vs vpy + vsy vp + ✏u1p vpx 1 + ✏vp1 vpy 1 . (3.6.26) Let us emphasize the boundary condition at y ! 1 for vp1 means that we will define: Z 1 vp1 (x, y) = u1px (x, y 0 )dy 0 . (3.6.27) y We may evaluate the equation (3.6.21) at y = 1 to see that the pressure term drops out. That is the pressure in the Prandtl layer is constant, so we may WLOG take Pp1 = 0. 207 3.6.2 Global in x Existence and Decay: The first step is to homogenize the boundary conditions by introducing the new unknowns: Z 1 u = u1p + (y)u1e (x, 0), v = vp1 vp1 (x, 0) + u1ex (x, 0)I (y), I (y) = (✓)d✓, (3.6.28) y where (y) is a cuto↵ function satisfying: Z 1 (0) = 1, @yk (0) = 0, (y)dy = 0. (3.6.29) 0 The mean-zero condition is meant to ensure that v(0) = 0. The homogenized profiles now satisfy the following system: (1 + u0p )ux uyy + P(u, v) = f (1) + J ; u(x, 0) = v(x, 0) = 0, lim u(x, y) = 0. (3.6.30) y!1 along with the divergence free condition ux + vy = 0. Here: P := u(1) 0 (1) sx u + upy v + vs uy , (3.6.31) 00 1 J := u(1) s (y)u1ex (x, 0) ue (x, 0) (y)u(1) 1 sx ue (x, 0) u0py I (y)u1ex (x, 0) + vs(1) 0 (y)u1e (x, 0). (3.6.32) We note that v(x, y) does not vanish as y ! 1 due to the definition in (4.9.80). The essential feature of v(x, y) that will be in use is that v(x, 0) = 0. Examining (3.6.31), one observes that v is always accompanied by u0p , which decays rapidly in y for each fixed x. 208 Let us now define the norm in which we shall control the Prandtl solutions: Z Z X1 Z ||u||2P (X1 , ) := sup { u2 x 2 + u2y x1 2 }+ {u2 x 1 2 + u2y x 2 + u2x x1 2 }. 1xX1 1 (3.6.33) We shall also need the following, di↵erentiated version, of the above Prandtl-layer norm: Z ||u||2Pk (X1 , ) := sup { |@xk u|2 x2k 2 + |@xk uy |2 x2k+1 2 } (3.6.34) 1xX1 Z X1 Z + {|@xk u|2 x2k 1 2 + |@xk uy |2 x2k 2 + |@xk+1 u|2 x2k+1 2 }. 1 Through the Hy1 ,! L1 y embedding, it is clear that: 1 sup xk+ 4 ||@xk u||L1  ||u||Pk (X1 , ) . (3.6.35) 1xX1 Finally, we shall write the global norms as: ||u||P ( ) := ||u||P (1, ) , ||u||Pk ( ) := ||u||Pk (1, ) . (3.6.36) Remark. A comparison of the norms Pk with the estimates valid for [u0p , vp0 ] in (3.4.86), 1 (3.4.91) show that u1p is roughly “x 4 -better” than u0p . There is no front-profile as in the case of [u0p , vp0 ], which is because the present boundary condition, u1p (x, 0) = u1e (x, 0) decays as x ! 1, due to (3.5.64), in contrast with the boundary condition for u0p (x, 0) = . The secondary reason is because f (1) in (3.6.23) contains derivative and nonlinear terms from the previous layers, which enhances the decay in x. The first step is to give the following estimates on the forcing terms, which capitalize on the structure of these terms either having many derivatives (thereby enhancing decay in x) or are of product form (also enhancing decay in x): 209 Lemma 3.6.3 (Forcing Estimates). For any k, m 0, and arbitrary N > 0 1 z m @xk J  C(k, m)hyi N x k 2 , (3.6.37) 5 ||z m @xk f (1) ||L2y  C(k, m)hxi k 4 . (3.6.38) Proof. The estimate for the J terms is direct, thanks to Proposition 3.5.6, estimate (3.5.61). For f (1) , in consultation with the definition (3.6.23), we start with the Ru,0 , defined in (3.6.9): p p 7 ✏||u0pxx ||L2y . ✏x 4 , (3.6.39) 1 1 5 ||yu0py veY 1 ||L2y  ||x 4 yu0py ||L2y sup veY 1 x4 . x 4 , (3.6.40) y Z y Z y0 p p p 7 ✏||u0py 1 vexx dy 00 dy 0 ||L2y  ✏||y 2 u0py ||L2y ||vexx 1 ||L1 y . ✏x 4 . (3.6.41) 0 y u,1 Second, we control RE via: 1 u,1 p ✏ 2 ||RE ||L2y  ||u1ex ||L1 y ||u0p ||L2y + ||u1e ||L1 y ||u0px ||L2y + ✏||vp0 ||L2y ||u1eY ||L1 y 5 .x 4 . (3.6.42) Next, according to (3.6.20), we have: 3 ⇣ ⌘ 3 1,a v,1 ||Ppx ||L2y  ||hyi 2 + @x ¯s(0) ✏v ¯(0) +u s v (0) ¯sx + v¯s(0) v¯sy (0) ||L1 y + ||hyi 2 + REx ||L1 y . (3.6.43) 3  3 For each of these terms, the ability to trade x 4 + 2 for y 2 + is in constant use: 3 3 3 ||hyi 2 + 0 ✏ vpx ||L1 y + ||hyi 2 + @x {¯ u(0) s v (0) ¯sx }||L1 y + ||hyi 2 + @x {¯ vs(0) v¯sy (0) }||L1 y 3 3 p 3 p + ||hyi 2 + @x {u0p vex 1 }||L1 y + ||hyi 2 + @x { ✏u1e vpx 0 }||L1 y + ||hyi 2 + @x { ✏vp0 veY 1 }||L1 y 210 3 3 + ||hyi 2 + @x {ve1 vpy 0 }||L1 y .x 2 . (3.6.44) Above, we are using the established estimates in (3.4.86), (3.4.91) for the Prandtl-0 profiles, and (3.5.61) for the Euler-1 profiles. The desired estimates are proven for m = 0. For general m the estimates work in an identical manner, after noticing that powers of z play no role when accompanied by [u0p , vp0 ], which appear in every term above. The above lemma relies crucially on Corollary 3.6.2 in order to apply the weight z m . We now give the following energy estimate: Lemma 3.6.4. Let > 0 and fix any X1 > 1. Then: Z Z X1 Z Z X1 Z 2 sup x u2 + x 1 2 u2 + x 2 u2y x2[1,X1 ] 1 1 Z X1 Z . O( ; ) vy2 x1 2 + C( ). (3.6.45) 1 The constant above depends poorly on small . Remark. The need for this > 0 is to avoid certain x-integrations being critical. We make the notational convention that we will not rename di↵erent values for (for instance 2 ), as we can always redefine to be smaller. However, within a single calculation (for instance the upcoming proof) we fix a > 0. 2 Proof. Applying the multiplier M = ux to the system (3.6.30) gives the following terms: Z ⇣ ⌘ Z Z Z Z (1+u0p )@x @yy u·ux 2 & @x (1+u0p )x 2 2 u + x 1 2 2 u + x 2 u2y u0px u2 x 2 . (3.6.46) The constant in the above estimate depends poorly on , as there is a factor of accom- 211 R 1 2 panying the x u2 term. The final term above then gets placed into the contributions from P, to which we now turn (see the definition in (3.6.31)): Z ⇣Z Z ⌘ 2 P · ux  ||xu(1) sx , yu 0 , py syv (1) x|| 1 u2 x 1 2 + vy2 x1 2 ⇣Z Z ⌘ 2 1 2  O( ) u x + vy2 x1 2 . (3.6.47) Above, we have used the Prandtl-0 bounds in (3.4.86), which crucially provides the smallness of {u0px , u0py } in terms of O( ). No smallness of Eulerian profiles is required due to p the extra factor of ". The key estimate which forces a loss of @x derivative is the following: Z Z v 1 1 u0py v · ux 2  ||u0py y||L1 ux 2  O( )||vy x 2 ||L2y ||ux 2 ||L2y . (3.6.48) y The structure of the key estimate above, (3.6.48), is omnipresent in our analysis: we use the y-absorption of u0py to produce a vy term via Hardy’s inequality (which is valid as 1 v(x, 0) = 0). This then forces a loss of x 2 in the decay, which must then be regained in the next lemma. Again, the estimate ||yu0py ||L1  O( ) arises from (3.4.86). Next, we arrive at the forcing terms. First, via (3.6.37): Z Z 1 1 N u J · ux 2 . hyi N x 2 |u|x 2 . ||x 2 hyi 2 ||L2y ||x N ||L2y hyi 2 1  ||uy x ||2L2y + Cx 1 2 . (3.6.49) 100, 000 Upon taking an integration in dx, the majorizing terms above are finite. Next, according to (3.6.38), via Young’s inequality: Z Z 1 1 3 1 f (1) · ux 2  ||f (1) x 2 ||L2y ||ux 2 ||L2y  Cx 2 + u2 x 1 2 dy. 100, 000 212 Placing these estimates together: Z Z Z Z @x x 2 u2 + x 1 2 u2 + x 2 u2y . O( ) u2x x1 2 +x 1 2 . Integrating above from x = 1 to x = X1 yields: Z Z X1 Z Z X1 Z 2 X1 u2 (X1 )dy + x 1 2 u2 + x 2 u2y 1 1 Z X1 Z . O( ) vy2 x1 2 + C. (3.6.50) 1 Finally, we reason as follows: the second and third terms on the left-hand side above are positive, which then gives for this X1 : Z Z X1 Z X1 2 u2 (X1 )dy . O( ) vy2 x1 2 + C. (3.6.51) 1 For any X2 2 [1, X1 ], the same estimate holds, namely: Z Z X2 Z Z X1 Z X2 2 u2 (X2 )dy . O( ) vy2 x1 2 + C . O( ) vy2 x1 2 + C. (3.6.52) 1 1 Above, we have used the monotonicity of the right-hand side as X2 < X1 . This allows R us to replace in (3.6.50) the first term on the left-hand side with sup1xX1 x 2 u2 dy. This then gives the desired estimate in (3.6.45). We now recover the vy term on the right-hand side of (3.6.45) via: 213 Lemma 3.6.5. Let > 0, and fix any X1 > 1. Then: Z Z X1 Z Z X1 Z sup u2y x1 2 + u2x x1 2 . C + O( ) u2 x 1 2 1xX1 1 1 Z X1 Z + u2y x 2 . (3.6.53) 1 Proof. We now apply the multiplier M = ux x1 2 to the system in (3.6.30). Doing so yields the following positive terms: Z ⇣ ⌘ Z Z Z (1 + u0p )@x @yy u · ux x 1 2 & @x u2y x1 2 u2y x 2 + u2x x1 2 . (3.6.54) Note crucially that the middle term in the above estimate, ||uy x ||2L2 , has been esti- y mated in (3.6.45) upon taking an x-integration. To close this sequence of estimates, there- 1 fore, it is crucial that this small parameter O( ) is attached to the ||vy x 2 ||2L2 term in (3.6.45). Next, we come to the profile terms, P (see (3.6.31) for the definition): Z ⇣ ⌘ (1) 12 1 1 P · ux x1 2  ||xu(1) 0 sx , yupy , vs x ||L1 ||ux 2 ||2L2y + ||ux x 2 ||2L2y ⇣ 1 1 ⌘  O( ) ||ux 2 ||2L2y + ||ux x 2 ||2L2y . (3.6.55) We have used estimates (3.4.86), (3.4.91), and (3.5.61), which provide the smallness of O( ). Next, we come to f (1) , for which we use the estimate in (3.6.38) (with k = 0): Z 1 1 3 1 1 f (1) ux x1 2  ||x 2 f (1) ||L2y ||ux x 2 ||L2y  Cx 2 + ||ux x 2 ||2L2y . (3.6.56) 100, 000 Piecing the above estimates together gives: Z Z @x u2y x1 2 + u2x x1 2 214 Z Z Z  u2y x 2 + O( ) u2 x 1 2 + Cx 1 + J · ux x1 2 . (3.6.57) Now, we take an integration from x = 1 to x = X1 : Z Z X1 Z Z X1 Z u2y (X1 )X11 2 + u2x x1 2 . u2y x 2 1 1 Z X1 Z Z X1 Z + O( ) u2 x 1 2 +C + J · ux x1 2 . (3.6.58) 1 1 The last step is to come to the terms in J , from (3.6.32). The most delicate term here requires successive integration by parts: Z X1 Z 00 (y)x1 2 u1e (x, 0)ux (3.6.59) 1 Z X1 Z Z 00 00 = (y)u@x {u1e (x, 0)x1 2 } u1e (1, 0)u (y)dy 1 x=1 Z 00 + X11 2 u1e (X1 , 0) (y)u(X1 , y)dy (3.6.60) x=X1 Z X1 Z 00 = (y)u@x {u1e (x, 0)x1 2 }+C 1 Z X11 2 u1e (X1 , 0) 0 (y)uy (X1 , y)dy (3.6.61) x=X1 Z X1 Z Z 0 1  (y)uy @x {u1e (x, 0)x1 2 }+C + X 1 2 u2y (X1 ) (3.6.62) 1 100, 000 x=X1 1 1 1 . ||uy x ||L2 || 0 (y)x 2 ||L2 +C C + ||uy x ||2L2 . (3.6.63) 100, 000 1 Going from (3.6.61) to (3.6.62), we have used |u1e (X1 , 0)| . X1 2 , according to (3.5.61). 3 Going from (3.6.62) to (3.6.63), we have used |u1ex (x, 0)| . x 2 , also according to (3.5.61). R Finally, we have absorbed the boundary contribution at x = X1 , u2y X11 2 into the left- hand side of (3.6.58). A consultation with (3.6.32) shows that we can perform a similar calculation with the remaining terms in J because these terms either have one extra x- (1) 1 derivative, or are accompanied by vs , which contributes additional decay of x 2 . This 215 then gives: Z Z X1 Z Z X1 Z Z X1 Z u2y (X1 )X11 2 + u2x x1 2 . u2y x 2 + O( ) u2 x 1 2 + C. 1 1 1 R Using a similar line of reasoning as in the Lemma 3.6.4, we can replace the u2y (X1 )X11 2 with the supremum over all x 2 [1, X1 ], thereby yielding the desired result. Consolidating the results of the previous two lemmas and applying contraction mapping: Corollary 3.6.6. For , ✏ sufficiently small relative to , a solution to the Prandtl system in (3.6.30) satisfies the following a-priori estimate in the space P : ||u||2P (X1 , ) . C( ). (3.6.64) For , ✏ sufficiently small, there exists a unique solution to (3.6.30), satisfying ||u||P (X1 )  C( ). The constant above in (3.6.64) is independent of X1 , and so we can immediately send X1 ! 1, thereby yielding a global solution on x 2 [1, 1) satisfying: ||u||P  C( ). It is possible to successively di↵erentiate the system in @x , and re-apply the previous 1 estimates, noting that the added x derivative adds a factor of x to each term above, enabling us to enhance the multiplier to [xk 2 @xk u, x1+k 2 @xk+1 u]. This is the reason for the di↵erentiated version of the Prandtl-norm in (3.6.34). To do so, we simply need: Lemma 3.6.7 (Initial Conditions). For each k 0, the initial data @xk u(1, y) is of order and decays rapidly in y. 216 Proof. This follows from using the equation (3.6.21) to write: vp1 (1, y) vp1 (1, 0) u1px (1, y) = U1yy (y) + u(1) 0 sx (1, y)U1 (y) + yupy (1, y) y + vs(1) (1, y)U1y (y) + f (1) (1, y). (3.6.65) We may now multiply by (1 + y)n , use the inequality || v(1,y) y ||L 1 . ||vy (1, y)||L1 for functions v satisfying v(x, 0) = 0 (here we take v = vp1 (1, y) vp1 (1, 0)), and use ||hyin U1 (y)u0px (1, y)||L1  O( )||u0px (1, y)||L1 , to obtain: ||hyin u1px (1, y)||L1 y  C. It is clear that the same procedure may be applied to higher order x-derivatives. Remark. [Higher Order Compatibility] In order to apply the procedure described, we require high-order compatibility condition such that the initial data of u1px (1, 0) = 0, thereby honor- ing the boundary condition. These condition can be ascertained inductively from (3.6.65). For instance: 0 = u1px (1, 0) = U1yy (0) + u(1) (1) sx (1, 0)U1 (0) + vs U1y (0) + f (1) (1, 0). (3.6.66) We suppose these compatibility conditions for large k. Repeating the previous set of estimates after applying the self-similar weight z M , and upon applying @xk to the system, we arrive at the following: Lemma 3.6.8. Given any > 0, let , ✏ be sufficiently small relative to . Consider the system given in (3.6.21), together with the boundary conditions (3.6.22). Let all derivatives of the prescribed data U1 (y) be exponentially decaying in its argument. Then there exists a unique, global in x solution [up , vp ], satisfying: ||z M @xk u||Pk ( )  C(M, k, ). (3.6.67) 217 It is now our task to extract similar estimates for the profiles u1p , vp1 from (3.6.67). First, Lemma 3.6.9. For any m, k 0, > 0, Z sup z 2m |@xk u1p |2 x2k 2  C(M, m, k, ). (3.6.68) x 1 1 Proof. This follows by writing u1p = u (y)u1e (x, 0), and using @xk u1e (x, 0) . x k 2 . Next, we may give the following uniform decay estimate for vp1 : Corollary 3.6.10 (Uniform Estimates for vp1 ). For any m 0, 3 ||z m @xk vp1 ||L1 y .x k 4+ C( , m, k). (3.6.69) Proof. First, according to (3.6.27), we have the rapid decay vp1 ! 0 as y ! 1. By the divergence-free condition, and the trace inequality, and for any small  > 0, Z 1 Z 1 2 vp1 1 vp1 1 vp1 . vp1 vpy  1  y2  1 vpy  || 1  ||L2y ||y 2  1 vpy ||L2y (3.6.70) y 0 y2 y2 1 1 1 1 1 1  ||y 2 + vpy 1 ||L2y ||y 2  1 vpy ||L2y = ||x 4 + z 2 + vpy 1 ||L2y ||x 4  z2  1 vpy ||L2y (3.6.71) 1 3 1+ 1+ 2 +2  x2 x x =x . (3.6.72) We have used the  > 0 to avoid the critical Hardy inequality. The Hardy inequality 1 we have used (with power y 2 + vp1 ) relies on the vanishing of vp1 at y = 1. From here the desired bound follows for m = 0, k = 0, and k, m 1 works analogously. The final ingredient we will need is to understand the connection between the norms we have controlled, Pk , and the quantities u1py , u1pyy . This is the content of the following: 218 Lemma 3.6.11. 3 1 x4 ||z m u1py ||L1 y + x2 ||z m u1py ||L2y + x1 ||z m u1pyy ||L2y  C < 1. (3.6.73) Proof. First, let us record: Z 2 Z Z x1 2 z 2m u1py  x1 2 z 2m u2y + x1 2 z 2m 2 |u1e |2 (x, 0)  ||z m u||P ( ) + C. (3.6.74) Via the equation (3.6.21), we have: x1 ||u1pyy ||L2y  x1 ||u1px ||L2y + x1 ||P||L2y + x1 ||f (1) ||L2y . ||u||P1 ( ) + ||u||P ( ) + C. (3.6.75) We are using the decay rates established for u0p in (3.4.86), and the pointwise decay of the Euler profiles established in (3.5.48) and the relations in (3.6.68). Let us give P in detail, x1 ||u(1) 1 (1) sx up ||L2y  ||xusx ||L1 x ||u1p ||L2y  O( )||u||P + O( ), (3.6.76) ⇣ ⌘ vp1 (x, y) vp1 (x, 0) x1 ||u0py vp1 (x, y) vp1 (x, 0) ||L2y  x1 ||yu0py ||L1 || ||L2y y  O( )x1 ||u1px ||L2y  O( )||u||P1 , (3.6.77) 1 1 x1 ||vs(1) u1py ||L2y  ||x 2 vs(1) ||L1 x 2 ||u1py ||L2y  O( )||u||P + O( ). (3.6.78) For the final line we have used (3.6.74). Next, via Sobolev interpolation in the y direction, 3 ⇣ ⌘ 12 ⇣ 1 ⌘ 12 1 1 x4 ||u1py ||L1  x1 ||u1pyy ||L2y x2 ||u1py ||L2y  C||u||P + x ||u1pyy ||L2y . y 100 (3.6.79) 219 Coupled with the u1pyy estimate in (3.6.75), this then establishes the desired bounds. The weighted estimate in z follows analogously. We will select now, 3 = 1 = . (3.6.80) 3n ⇥ 10, 000 Summarizing the results of this section: Proposition 3.6.12 (Prandtl-1 Layer Bounds). Given any n 2 N, let 1 be as in (3.6.80), and let , ✏ be sufficiently small relative to n and 1. Consider the system given in (3.6.21), together with the boundary conditions (3.6.22). Let all derivatives of the prescribed data U0 (y) be exponentially decaying in its argument. Then there exists a unique, global in x solution [u1p , vp1 ], satisfying: 3 ||z M @xk u1p ||Pk ( 1) + xk+ 4 1 ||z m @xk vp1 ||L1 y  C(M, k, n), for any k, m 0. (3.6.81) 3.7 Intermediate Layers We now construct intermediate layers i = 2 through i = n 1. This is achieved inductively, starting with the construction of the Euler Layer, uie , vei . Let us fix the parameters: 3i i = for i = 2, ..., n. (3.7.1) 3n ⇥ 10, 000 The reason for this selection will be seen in (3.11.55). 220 3.7.1 Construction of Euler Layer, [uie , vei ] (i 1) (i 1) For this step in the construction, we suppose that [¯ us , v¯s ] have already been con- structed. The inductive hypothesis on the 1, ..., i 1 Prandtl profiles are as follows: ||z m @xk ujp ||Pk ( j)  C(k, m, n), for j = 1, ..., i 1, (3.7.2) where [ujp , vpj ] satisfy the system given in (3.7.48) for 2  j  i 1, and for i = 2, that [u1p , vp1 ] satisfy (3.6.21). In order to obtain the equations for [uie , vei ], we expand the nonlinear terms including the new Euler terms: ⇣ i ⌘⇣ i ⌘ u(i) (i) s usx = u ¯(i s 1) + ✏ 2 uie ¯(i u sx 1) + ✏ 2 uiex i i ¯(i =u s 1) (i 1) u ¯sx + ✏ 2 u ¯(i sx 1) i ¯(i ue + ✏ 2 u s 1) i uex + ✏i uie uiex , (3.7.3) ⇣ i 1 ⌘⇣ p i ⌘ vs(i) u(i) sy = v¯s(i 1) + ✏ 2 vei u ¯(i sy 1) + ✏✏ 2 uieY i 1 p i = v¯s(i 1) u ¯(i sy 1) +✏ 2 u ¯(i sy 1) i ve + ✏✏ 2 v¯s(i 1) i ueY + ✏i vei uieY . (3.7.4) ⇣ i ⌘⇣ i 1 ⌘ u(i) (i) s vsx = u ¯(i s 1) + ✏ 2 ui e v ¯ (i) sx + ✏ 2 vi ex i i 1 1 ¯(i =u s 1) (i 1) (i 1) i v¯sx + ✏ 2 v¯sx ue + ✏ 2 u ¯(i s 1) i vex + ✏i 2 uie vex i , (3.7.5) ⇣ i 1 ⌘⇣ i ⌘ vs(i) vsy (i) = v¯s(i 1) + ✏ 2 vei v¯sy(i 1) i + ✏ 2 veY i 1 i 1 = v¯s(i 1) (i 1) v¯sy +✏ 2 (i v¯sy 1) i ve + ✏ 2 v¯s(i 1) i veY + ✏i 2 vei veY i . (3.7.6) We will now define several terms: Definition 3.7.1. The i 1’th remainder is denoted by: i 1 Ru,i 1 := ¯(i ✏u s 1) ¯(i +u s 1) (i 1) u ¯sx + v¯s(i 1) (i 1) u ¯sy + P¯sx (i 1) +" 2 vei u0py , (3.7.7) @y ¯ (i Rv,i 1 := ¯s(i 1) ✏v ¯(i +u s 1) (i 1) v¯sx + v¯s(i 1) (i 1) v¯sy + P 1) . (3.7.8) ✏ s 221 We will also split the Euler-Euler interaction terms and the Euler-Prandtl terms via: Definition 3.7.2. i hX i 1 j i i hX i 1 j i u,i RE := ✏ 2 uiex ✏ 2 ujp + ✏ 2 uie ✏ 2 ujpx j=0 j=0 i p hX i 1 j i i 1 hX i 1 j i + ✏2 ✏uieY ✏ 2 vpj + ✏ 2 vei ✏ 2 ujpy , (3.7.9) j=0 j=1 i 1 X i 1 X i 1 j i j v,i i RE := ✏ 2 vex ✏ 2 ujp + ✏ 2 uie j ✏ 2 vpx j=0 j=0 i 1 X i 1 X i 1 j p i 1 j +✏ 2 vei j ✏ 2 vpy + ✏✏ 2 i veY ✏ 2 vpj , (3.7.10) j=0 j=0 h i i i 1 X j i i 1 X j Eeu,i := ✏i uie uiex + vei uieY + ✏ 2 uiex ✏ 2 uje + ✏ 2 uie ✏ 2 ujex j=1 j=1 p i hX i 1 j 1 i i 1 hX i 1 1 j i + ✏✏ 2 uieY ✏ 2 vej + ✏ 2 vei ✏ 2 ✏ 2 ujeY , (3.7.11) j=1 j=1 1 h i i hX i 1 j 1 i i 1 hX i 1 j i Eev,i := ✏i 2 uie vex i + vei veY i + ✏ 2 uie j ✏ 2 vex i + ✏ 2 vex ✏ 2 uje j=1 j=1 i 1 X i 1 X p i 1 j 1 j p i 1 j 1 + ✏✏ 2 vei ✏ 2 veY + ✏✏ 2 i veY ✏ 2 vej . (3.7.12) j=1 j=1 Remark. The natural definition of the remainder term, Ru,i 1 should be: Ru,i 1 “=” ¯(i ✏u s 1) ¯(i +u s 1) (i 1) u ¯sx + v¯s(i 1) (i 1) u ¯sy + P¯sx (i 1) , (3.7.13) u,i i 1 and the natural definition of RE would contain the lowest-order term, " 2 vei u0py . How- u,i ever, for i < n, the quantity of interest in (3.7.22) is the sum, Ru,i 1 + RE , that is contributed to the next order (see the definition of the forcing in (3.7.31)). Thus, for con- i 1 venience (see calculation 3.7.56 and (3.7.58)), we add and subtract one factor " 2 vei u0py , which explains the definitions of (3.7.7) and (3.7.9). This, however, will not be done for i = n. 222 The Navier-Stokes expansion reads: (i) ✏ us + u(i) (i) (i) (i) (i) s usx + vs usy + Psx = ¯(i ✏u s 1) ¯(i +u s 1) (i 1) u ¯sx + v¯s(i 1) (i 1) u ¯sy + P¯sx (i 1) i i i i 1 ✏ 2 +1 uie + ✏ 2 u ¯(i sx 1) i ue ¯(i + ✏2 u s 1) i uex + ✏i uie uiex + ✏ 2 ¯(i u sy 1) i ve p i i + ✏✏ 2 v¯s(i 1) uieY + ✏i vei uieY + ✏ 2 Pex i + ✏i Pex 1,a (3.7.14) i h i i u,i = " 2 uiex + Pex i + " 2 +1 uie + Ru,i 1 + RE + Eeu,i + "i Pex i,a . (3.7.15) For the normal equation, the expansions read: (i) @y (i) @y ¯ (i ✏ vs + u(i) (i) (i) (i) s vsx + vs vsy + P = ¯s(i 1) ✏v ¯(i +u 1) (i 1) v¯sx + v¯s(i 1) (i 1) v¯sy + P 1) ✏ s s ✏ s i+1 i i 1 1 i 1 ✏ 2 vei + ✏ 2 v¯sx (i 1) i ue +✏ 2 ¯(i u s 1) i vex + ✏i 2 uie vex i +✏ 2 (i v¯sy 1) i ve i 1 1,a i 1 1 + ✏ 2 v¯s(i 1) veY i + ✏i 2 vei veY i i + ✏ 2 PeY + ✏i 2 PeY (3.7.16) i 1 h i i+1 1 i i v,i = " 2 vex + PeY + " 2 vei + Rv,i 1 + RE + Eev,i + "i 2 i PeY . (3.7.17) From here, we simply read o↵ the highest order terms that are “purely-Eulerian”. All of the remaining terms will be treated in the next subsection. In (3.7.14), these are at order i i 1 ✏ 2 , and in (3.7.16), these are at order ✏ 2 : i h i i 1 h i ✏ 2 uiex + Pex i = 0, ✏ 2 i vex i + PeY =0 (3.7.18) When paired with the divergence-free condition, we arrive at the equations that are taken for the [uie , vei ], which are the Cauchy-Riemann equations: uiex + Pex i = 0, i vex i + PeY = 0, uiex + veY i = 0. (3.7.19) The boundary conditions for the i0 th Euler layer is vei (x, 0) = vpi 1 (x, 0). According to 223 the inductive hypothesis, the decay rate of this boundary condition is: 1 1 1 1 vei (x, 0) = vpi 1 (x, 0)  ||vpi 1 (x, y)||L2 2 ||vpy i 1 i 1 (x, y)||L2 2  ||yvpy i 1 (x, y)||L2 2 ||vpy (x, y)||L2 2 y y y y 3  C( , n)x 4+ i 1 , for i 2. (3.7.20) A comparison of (3.7.20) to (3.5.9) shows that the decay rate of the boundary condition has improved, enabling us to improve the Euler decay rates. Indeed, as the system (3.7.19) is the identical system to the first Euler layer (and is in particular the Cauchy-Riemann equations), we may simply repeat the analysis given there to conclude: Proposition 3.7.3 (Euler-i Layer). Let i 2. The i0 th Euler layer, defined by the Cauchy- Riemann equations (3.7.19) taken with boundary conditions (3.7.20) satisfy the enhanced decay rates: 3 3 xk+m+ 4 i 1 @xk @Ym vei + x 4 i 1 uie  C(k, m, n). (3.7.21) Proof. This follows by repeating the arguments in Section 3.5 with the enhanced boundary condition (3.7.20). 1 Remark. It is not possible to enhance the rates (3.7.21) much more (for instance, past x for k = m = 0. This is due to the restriction of w 2 (0, 1) in Corollary 3.5.3. We have the simplified expression for (3.7.15), (3.7.17) Lemma 3.7.4. According to definitions in (3.7.7) - (3.7.12), one has (i) u,i ✏ us + u(i) (i) (i) (i) (i) s usx + vs usy + Psx = R u,i 1 + RE + Eeu,i + "i Pex i,a , (3.7.22) (i) @y (i) v,i 1 ✏ vs + u(i) (i) (i) (i) s vsx + vs vsy + P = Rv,i 1 + RE + Eev,i + "i 2 i PeY . (3.7.23) ✏ s Proof. By construction in Proposition 3.7.3, the uie , vei are harmonic, and so the uie , vei terms vanish identically. This then accounts for all of the terms in (3.7.14) - (3.7.16). 224 3.7.2 Construction of Prandtl Layer, [uip , vpi ] (i) (i) For this step in the construction, we suppose that [us , vs ] have been constructed. The inductive hypothesis on these profiles are that the following remainders (according to the definitions in (3.7.7) for Ru,i 1 and (3.7.8) for Rv,i 1 ) have been accumulated: i 1 h p i 1 X j Ru,i 1 =✏ 2 ✏uipxx1 + ✏yu0py veY i + uipx 1 " 2 {uje + ujp } j=1 Z y Z y0 ⇣X i 1 ⌘i j p + ✏u0py i veY 00 0 i Y dy dy + vp 1 ✏ 2 {ujpy + ✏ujeY } , (3.7.24) 0 y j=1 i 1 h Rv,i 1 =✏ 2 i 1 ✏ vp + u(i s 1) i 1 vpx (i + vsx 1) i 1 up + vs(i 1) i 1 vpy (i + vsy 1) i 1 vp i 1 i 1 i +✏ 2 uip 1 vpx i 1 +✏ 2 vpi 1 vpy i 1 . (3.7.25) The induction will start at i = 2, and so (3.7.24) - (3.7.25) should be compared to (3.6.26) for this case and to (3.7.50) for the general i case. The relevant profile estimates, according to Propositions 3.5.6, 3.7.3 and 3.6.12 which hold inductively are: 1 1 xk+m+ 2 |@xk @Ym ve1 | + x 2 |u1e |  C(k, m, n), (3.7.26) 3 3 xk+m+ 4 j 1 @xk @Ym vej + x 4 j 1 uje  C(k, m, n) for j = 2, .., i, (3.7.27) ||z m @xk ujp ||Pk ( j)  C(k, m, n) for j = 1, ..., i 1. (3.7.28) (i) (i) i (i) (i) (i) i 1 (i) By writing u ¯s = us + ✏ 2 up , v¯s = vs + ✏ 2 vp , and expanding the Navier-Stokes equations, we obtain the two expansions: i+1 ¯(i) "u s +u¯(i) s u ¯(i) sx + v ¯s(i) u ¯(i) ¯ (i) sy + Psx = (i) (i) (i) (i) (i) (i) ✏ us + us usx + vs usy + Psx + ✏ 2 Ppx i,a i h i i i i (i) i (i) i (i) i (i) i i i i i i + ✏2 " up + usx up + us upx + usy vp + vs upy + ✏ 2 up upx + ✏ 2 vp upy + Ppx , (3.7.29) 225 and: (i) @y ¯ (i) Psy i 1 ¯s(1) ✏v ¯(i) +u s v (i) ¯sx + v¯s(i) v¯sy (i) + P = (i) ✏ vs + u(i) + ✏ 2 Ppyi,a (i) (i) (i) s vsx + vs vsy + ✏ s ✏ i h i i Ppyi i i (1) i (i) i (i) i (i) i i i i i + ✏2 " vp + us vpx + vsx up + vsy vp + vs vpy + ✏ 2 up vpx + ✏ 2 vp vpy + . ✏ (3.7.30) Define the forcing term to be those terms from (3.7.29) which do not appear in the bracket: i i+1 ✏ 2 f (i) := (i) ✏ us + u(i) (i) (i) (i) (i) s usx + vs usy + Psx + ✏ 2 i,a Ppx u,i i+1 = Ru,i 1 + RE + Eeu,i + ✏i Pex 1,a +✏ 2 i,a Ppx , (3.7.31) where we have used (3.7.22) to simplify the expression. The first step, here, is to intro- duce a potential Pressure which eliminates the “purely” Eulerian terms from above: Definition 3.7.5. The i’th auxiliary Euler pressure, Pe1,a , is defined by: i 1 X i 1 X 2 j i j i 1 i 1 i 2 Pe1,a := ✏ 2 vei vej ✏ 2 uie uje v u . (3.7.32) j=1 j=1 2 e 2 e We may now check that: Lemma 3.7.6. With the definition above (3.7.32), Pe1,a serves as a gradient potential to eliminate the purely-Eulerian terms, [Eeu,i , Eev,i ] from the expansion ⇣ @y ⌘ i 1,a ⇣ u,i v,i ⌘ @x , ✏ Pe + Ee , Ee = 0. (3.7.33) ✏ Proof. By scaling, we will write: ⇣ @y ⌘ i 1,a ⇣ @Y ⌘ @x , ✏ Pe = @x , p ✏i Pe1,a . (3.7.34) ✏ ✏ 226 We will go term by term through the definition in (3.7.32), starting with: ⇣ i 1 X j i ⌘ i 1 X j+i ⇣ ⌘ i 1 X j+i ⇣ ⌘ i " @x " 2 vei vej = " 2 i vex vej + vei vex j = " 2 uieY vej + vei ujeY . j=1 j=1 j=1 (3.7.35) Next, ⇣ i 1 X j i ⌘ i 1 X j+i ⇣ ⌘ i " @x " 2 uie uje = " 2 uiex uje + uie ujex . (3.7.36) j=1 j=1 Third, ⇣ 1 i2 1 i 2⌘ "i @ x |v | |u | = "i vei vex i "i uie uiex = "i vei uieY "i uie uiex . (3.7.37) 2 e 2 e Comparing these expressions, (3.7.35) - (3.7.37) to the expression (3.7.11), one observes the exact cancellation: u,i @x "i Pe1,a + EE = 0. (3.7.38) @Y Next, we will move to the p " terms: @ ⇣X j i i 1 ⌘ i 1 X i+j 1 ⇣ ⌘ pY "i " 2 vei vej = " 2 2 i veY j vej + vei veY . (3.7.39) " j=1 j=1 Next, @ ⇣X j i i 1 ⌘ i 1 X i+j 1 ⇣ ⌘ pY "i " 2 uie uje = " 2 2 uieY uje + uie ujeY " j=1 j=1 227 i 1 X i+j 1 ⇣ ⌘ i = " 2 2 vex uje + uie vex j . (3.7.40) j=1 Third, @ ⇣ ⌘ 1 1 1 1 pY "i |vei |2 |uie |2 = "i 2 vei veY i "i 2 uie uieY = "i 2 vei veY i "i 2 uie vex i . (3.7.41) " Comparing these expressions, (3.7.39) - (3.7.41) to the expression (3.7.11), one observes the exact cancellation: @ pY "i Pe1,a + EE v,i = 0. (3.7.42) " This establishes the desired result, (3.7.33). Remark. One should notice the essential role played by the Cauchy-Riemann equations, ujeY = vex j in the equalities above. Let us now turn to the terms outside of the bracket in (3.7.30), which we also simplify via (3.7.23) and subsequently via (3.7.33): (i) (i) Psy i 1 ✏ vs + u(i) (i) (i) (i) s vsx + vs vsy + i,a + ✏ 2 Ppy ✏ v,i 1 1,a i 1 = Rv,i 1 + RE + Eev,i + ✏i 2 PeY +✏ 2 i,a Ppy v,i i 1 = Rv,i 1 + RE +✏ 2 i,a Ppy . (3.7.43) Motivated by this, define the auxiliary Pressure via: 228 Definition 3.7.7. The i’th auxiliary Prandtl pressure, PPi,a is defined via: Z 1 i+1 ✏ 2 PPi,a := ✏ Rv,i 1 v,i + RE . (3.7.44) y Immediately from this definition, we have: Lemma 3.7.8. According to the Definition 3.7.44, the following identity holds: (i) (i) Psy i 1 ✏ vs + u(i) (i) (i) (i) s vsx + vs vsy + i,a + ✏ 2 Ppy = 0. (3.7.45) ✏ Proof. By direct calculation from (3.7.44), i 1 " 2 PPi,a = Rv,i 1 v,i RE . (3.7.46) Combined with (3.7.43) then implies the desired result. We are now able to rewrite the forcing for our equation, which was defined in (3.7.31): i h i+1 i u,i f (i) = ✏ 2 Ru,i 1 + RE +✏ 2 i,a Ppx . (3.7.47) i The ✏ 2 factor arises as we are in the i0 th order of the construction. Let us briefly comment on the orders of the three-terms on the right-hand side of (3.7.47). According to i u,i (3.7.24), it is evident that Ru,i 1 is order ✏ 2 . According to (3.7.9), it is clear that RE is i order ✏ 2 . According to (3.7.44) coupled with (3.7.25) and (3.7.10), it is clear that PPi,a is order 1. Thus, all of the terms in (3.7.47) are order 1 or higher. Reading o↵ from (3.7.29) - (3.7.30), the system we will now be considering is: ⇣ ⌘ (1 + u0p )uipx + u(i) i (i) i 0 i sx up + vs upy + upy vp vpi (x, 0) + Ppx i = uipyy + f (i) , (3.7.48) 229 uip (x, 0) = uie (x, 0), lim uip (x, y) = 0, uip (1, y) = Ui (y), i Ppy = 0. (3.7.49) y!1 i Again, by evaluating the equation at y = 1 gives Ppx = 0. Coupled with the normal i equation Ppy = 0 then shows that Ppi = 0. After this construction, Ru,i , Rv,i contain the terms from (3.7.29) - (3.7.30) which were omitted in the construction of [uip , vpi ]: Lemma 3.7.9. For each i, with [Ru,i , Rv,i ] defined as in (3.7.7) - (3.7.8), with [uip , vpi ] taken to solve the system (3.7.48), the following identities hold: h Z y Z y0 u,i i p i+1 i+1 00 0 R =✏ 2 ✏uipxx + ✏yu0py veY + ✏u0py veY Y dy dy (3.7.50) 0 y i X j p i X j i + vpi ✏ 2 {ujpy + ✏ujeY } + uipx " 2 {uje + ujp } , j=1 j=1 i h i i i Rv,i = ✏ 2 i ✏ vp + u(i) s v i px + v (i) i u sx p + v (i) i s v py + v (i) (i) v sy p + ✏ 2 ui v i p px + ✏ 2 vi vi p py . (3.7.51) Proof. Starting with the definition in equation (3.7.7), we have: i Ru,i := ¯(i) ✏u s ¯(i) +u s u¯(i) ¯s(i) u sx + v ¯(i) ¯ (i) 2 i+1 0 sy + Psx + " ve upy (3.7.52) i+1 (i) = + u(i) ✏ us (i) (i) (i) s usx + vs usy + Psx + ✏ (i) 2 P i,a px i h i (i) i (i) i (i) i (i) i + ✏2 " up + usx up + us upx + usy vp + vs upy i i i i + ✏ 2 uip uipx + ✏ 2 vpi uipy + Ppxi + " 2 vei+1 u0py (3.7.53) i i h = " 2 f (i) + ✏ 2 i (i) i (i) i (i) i " up + usx up + us upx + usy vp + vs upy (i) i i i i i + ✏ 2 uip uipx + ✏ 2 vpi uipy + Ppxi + " 2 vei+1 u0py (3.7.54) i i h = " 2 f (i) + ✏ 2 uipyy + u(i) i 0 i 0 i sx up + (1 + up )upx + upy vp + vs upy (i) i i X i i j + ✏ 2 uip uipx + ✏ 2 vpi uipy + Ppx i "uipxx + " 2 {uje + ujp }uipx j=1 i X j p i i + " 2 {ujpy + "ujeY }vpi + " 2 vei+1 u0py (3.7.55) j=1 i i h = " 2 f (i) + ✏ 2 uipyy + u(i) i 0 i 0 i (i) i sx up + (1 + up )upx + upy vp + vs upy 230 i X i i j + ✏ 2 uip uipx + ✏ 2 vpi uipy + Ppx i "uipxx + " 2 {uje + ujp }uipx j=1 i X j p i i h p + " 2 {ujpy + "ujeY }vpi + " 2 vpi (x, 0)u0py + i+1 "yu0py veY j=1 Z y Z y0 i i+1 00 0 + "u0py veY Y dy dy (3.7.56) 0 y i h i X j i X j p p = "2 "uipxx + " 2 {uje + ujp }uipx + " 2 {ujpy + "ujeY }vpi + i+1 "yu0py veY j=1 j=1 Z y Z y0 i i+1 00 0 + "u0py veY Y dy dy . (3.7.57) 0 y For the calculation in (3.7.53), we have used the expansion (3.7.29). For the calculation in (3.7.54), we have used (3.7.31). For the calculation in (3.7.55), we have simply rearranged terms. For the calculation in (3.7.56), we have used: h Z y Z y0 i i i p i+1 i+1 00 0 " 2 vei+1 u0py =" 2 vpi (x, 0)u0py + "yu0py veY + "u0py veY Y dy dy . (3.7.58) 0 y Finally, for the calculation in (3.7.57), we used (3.7.48). The identity (3.7.50) then follows. For the Rv,i contribution, we combine (3.7.45) with the expression (3.7.30), and finally with the condition that Ppi = 0 from (3.7.49). A comparison with (3.7.24) - (3.7.25) shows that this closes the construction. We now give estimates on the forcing term based on the inductively assumed decay rates and reg- ularity in (3.7.2), together with the [u0p , vp0 ] bounds in (3.4.86) and the [u1e , ve1 ] bounds in (3.5.61). Lemma 3.7.10 (Forcing Estimates). Let i 2. For any m, k 0, with f (i) as in (3.7.47), 5 ||z m @xk f (i) ||L2y  C(k, m, , n)x k 4 +2 i 1 . (3.7.59) 231 u,i Proof. We start with the RE terms, which are defined in (3.7.9): i 1 X i 1 X j j 3 3 5 ✏ 2 ||uiex ujp ||L2y  ✏ 2 ||uiex x 2 ||L1 x 2 ||ujp ||L2y . x 4 , (3.7.60) j=0 j=0 i 1 X i 1 X j j 1 1 5 ||uie ✏ 2 ujpx ||L2y . ✏ 2 ||uie x 2 ||L1 ||x 2 ujpx ||L2y  x 4 , (3.7.61) j=0 j=0 i 1 X i 1 X j+1 j+1 3 3 7 ✏ 2 ||uieY vpj ||L2y . ✏ 2 ||uieY x 2 ||L1 x 2 ||z m vpj ||L2y . x 4 , (3.7.62) j=0 j=0 i 1 X i 1 X i 1 j 3 3 5 ✏ 2 ✏ 2 ||vei ujpy ||L2y . ||vei x 4 i 1 ||L1 ||x 4+ i 1 ujpy ||L2y  x 4 +2 i 1 . (3.7.63) j=1 j=1 In (3.7.63), we have used the enhanced Eulerian decay rate in (3.7.21). Next, we have the Prandtl contributions from Ru,i 1 , according to (3.7.24): p p 7 ✏||uipxx1 ||L2y  ✏x 4+ i 1 , (3.7.64) 1 1 5 ||yu0py veY i ||L2y  ||x 4 yu0py ||L2y x 4 ||veY i ||L1 y . x 4, (3.7.65) Z y Z y0 p 0 00 p 2 3 0 3 p 7 ✏||u0py i veY Y dy dy ||L2y  i ✏||y x 4 upy ||L2y x 4 ||veY Y ||L1 y  ✏x 4 , (3.7.66) 0 y j 1 j 1 5 ✏ 2 ||vpi 1 ujpy ||L2y  ✏ 2 x 4 +2 i 1 ,j 1 (3.7.67) j j ✏ 2 ||vpi 1 ujeY ||L2y  ✏ 2 x 2+2 i 1 ,j 1, (3.7.68) j p 5 " 2 ||uipx 1 {uje + ujp }||L2y . "x 4 +2 i 1 ,j 1. (3.7.69) i,a We now move to the term Ppx . For this we note: i i+1 p i 1 3 ⇣ ⌘ i,a v,i ✏ 2 ✏ 2 ||Ppx ||L2y  ✏✏ 2 ||hyi 2 + @x Rv,i 1 + RE ||L1 y . (3.7.70) We now turn to evaluating the right-hand side of (3.7.70). Let us comment that it is essential at this stage that the pressure Pe1,a was introduced to eliminate all of the purely 3 Eulerian contributions, as those terms cannot handle the weight of y 2 + . Each term below 232 3 has at least one Prandtl ([uip , vpi ]) factor, and so we may exchange the weight of y 2 + for 3  decay, x 4 + 2 . For the calculations below, we simply pair this with the decay rates in (3.7.2): 3 ⇣ ⌘ 3 ⇣ ⌘ 3 ||y 2 + @x vex i ujp ||L1y + ||y 2 + @ x u i j e px ||L1 v y . x 2 +2 i 1 , (3.7.71) 3 ⇣ ⌘ 3 ||y 2 + @x vei vpy j ||L1 y . x 2 +2 i 1 (3.7.72) 3 ⇣ ⌘ ||y 2 + @x veY i vpj ||L1 y . x 2+2 i 1 , (3.7.73) 3 ⇣ ⌘⌘ 5 ||y 2 + @x ✏ vpi 1 ||L1 y . x 4 +2 i 1 , (3.7.74) 3 ⇣ ⌘ 3 ⇣ ⌘ 3 ||y 2 + @x u(i s 1) i 1 vpx ||L1 y + ||y 2 + @x vsx (i 1) i 1 up ||L1y .x 2 +2 i 1 , (3.7.75) 3 ⇣ ⌘ 3 ⇣ ⌘ 3 ||y 2 + @x vs(i 1) vpyi 1 ||L1 y + ||y 2 + @x vsy (i 1) i 1 vp ||L1y .x 2 +2 i 1 , (3.7.76) 3 ⇣ i 1 ⌘ i 1 5 ||y 2 + @x ✏ 2 uip 1 vpx i 1 + vpi 1 vpy i 1 ||L1 y . ✏ 2 x 4 +2 i 1 . (3.7.77) This proves the claim for k = 0, m = 0. The general case can be obtained in an identical fashion, upon noticing that powers of z will not influence the estimates when accompanied by Prandtl profiles, which are present in each term above. Indeed, the boundary condition in (3.7.48) has improved, referring to (3.7.21): 3 uie (x, 0)  x 4+ i 1 , (3.7.78) according to (3.7.21). We can expect enhanced decay due to the enhanced decay of the boundary data. This is the content of the following: Proposition 3.7.11 (Construction of i0 th Prandtl Layer). For i 2, there exists a solution [uip , vpi ] to the system (3.7.48), satisfying for any m, k 0: 1 ||z m x 4 @xk uip ||Pk ( i) + xk+1 i ||z m @xk vpi ||L1 y  C(k, m, n). (3.7.79) 233 This will be proven in several steps. We homogenize the boundary conditions by defining: Z 1 u = up + (y)uie (x, 0), v = vp vp (x, 0) + unex (x, 0)I (y), I (y) = (✓)d✓, y where is a localized cut-o↵ function selected to have mean-zero. These profiles satisfy: (1 + u0p )ux uyy + P(u, v) = f (i) + J , (3.7.80) where 00 i J := u(i) s (y)uiex (x, 0) ue (x, 0) (y)u(i) n sx ue (x, 0) u0py I (y)uiex (x, 0) + vs(i) 0 (y)uie (x, 0). (3.7.81) We will record the estimate on J , which follows directly from (3.7.78): 3 |J | . hyi N x 4+ i 1 . (3.7.82) We introduce the stream function, Z 1 (x, y) := u(x, y 0 )dy 0 , y = u, x =v v(x, 1). (3.7.83) y The stream function satisfies the following system: ⇣ ⌘ Z 1 Z 1 Z 1 Z 1 @x @yy = u0p ux + P+ f (n) + J, (3.7.84) y y y y Z 1 y (x, 0) = u(x, 0) = 0, (x, 1) = 0, (1, y) = Un (y 0 )dy 0 . (3.7.85) y 234 For technical reasons (due to certain integrals being critical), it is necessary to start with the stream-function formulation in order to obtain the desired, enhanced decay. Lemma 3.7.12. The stream function solving the system (3.7.84) satisfies: Z Z Z Z Z 1 3 1 2 2 2 2 2 sup x 2 i + x 2 i + u2x 2 i x Z Z 3 . C + O( ) vy2 x 2 2 i . (3.7.86) 1 2 Proof. Applying the multiplier M = x 2 i to the above system yields the following terms on the left-hand side: Z Z Z Z ⇣ ⌘ 1 3 1 1 @x x 2 2 i 2 + x 2 2 i 2 + x 2 2 i 2 y . @x @yy · x 2 2 i . (3.7.87) For the first term on the right-hand side of (3.7.84), we will now give the bound via Hardy’s inequality, and (3.4.86): Z Z 1 Z 1 1 3 1 x 2 2 i u0p ux dy 0 dy  ||x 4 i ||L2y ||x 4 u0p ux dy 0 ||L2y y y 3 1 3 3  ||x 4 i ||L2y ||x 4 i yu0p ux ||L2y  O( )||x 4 i ||L2y ||x 4 i ux ||L2y . (3.7.88) Next, let us address the profile terms in P. For the first term from P, we will split: Z Z 1 1 2 0 x 2 i u(i) sx u dy dy (3.7.89) y i 1Z X Z 1 i Z X Z 1 1 j 1 j = x 2 2 i " 2 ujpx u dy 0 dy + x 2 2 i " 2 ujex u dy 0 dy j=0 y j=1 y For the first term above, we apply the Hardy inequality in y (it suffices to consider 235 j = 0): Z Z 1 Z 1 1 3 1 x 2 2 i u0px u dy 0 dy  ||x 4 i ||L2y ||x 4 i u0px u dy 0 ||L2y y y 3 1 1  ||x 4 i ||L2y ||x 2 yu0px ||L1 ||x 4 i u||L2y 3 1  O( )||x 4 i ||2L2y + O( )||x 4 i u||2L2y . (3.7.90) We have used the smallness which is guaranteed by (3.4.86). For the second term in y!1 (3.7.89), we will integrate by parts in y and recall ! 0, (again it suffices to consider j = 1): Z Z 1 Z Z Z 1 1 2 p 1 p 1 x 2 i "u1ex @y = x 2 2 i "u1ex 2 + x 2 2 i "u1exY . (3.7.91) y y The first term in (3.7.91): Z 1 p p 3 p 3 x 2 2 i "u1ex 2  "||u1ex x||L1 ||x 4 i ||2L2y . "||x 4 i ||2L2y . (3.7.92) For the second term in (3.7.91), we apply Hardy in y and appeal to estimate (3.5.48) with k = 2, s = 1: Z Z 1 1 2 p 3 x 2 i "u1exY  "||u1exY Y x||L1 ||x 4 i ||2L2y y p 3 p 3 = 1 "||vexx Y x||L1 ||x 4 i ||2L2y . "||x 4 i ||2L2y . (3.7.93) Let us now move to the second term in P. One integration by parts in y produces: Z Z 1 Z Z Z 1 1 1 1 x 2 2 i vs(i) uy dy 0 dy = x 2 2 i vs(i) y + x 2 2 i u(i) 0 sx udy dy. y y (3.7.94) 236 For the second term in (3.7.94), one notices this is exactly the term on the left-hand side of (3.7.89). For the first term in (3.7.94), we estimate: Z ⇣ ⌘ 1 1 3 1 2 x 2 i vs(i) y  ||vs(i) x 2 ||L1 ||x 4 i ||2L2y + ||x 4 i 2 y ||L2y ⇣ 3 1 ⌘  O( ) ||x 4 i ||2L2y + ||x 4 i 2 y ||L2y . (3.7.95) We have used the smallness guaranteed by (3.4.91). The final term from P is: Z Z 1 Z 1 1 3 1 x 2 2 i u0py vdy 0 dy  ||x 4 i ||L2y ||x 4 i u0py v||L2y y y 3 1  ||x 4 i ||L2y x 4 i ||yu0py v||L2y 1 3 3  ||y 2 x 2 u0py ||L1 ||x 4 i ||L2y ||vy x 4 i ||L2y 3 3  O( )||x 4 i ||L2y ||vy x 4 i ||L2y . (3.7.96) Consolidating the previous estimates, taking small enough relative to i (and conse- quently relative to large n, according to (3.7.1)) and applying Young’s inequality gives: Z Z Z 1 3 1 2 2 2 2 @x x 2 i + x 2 i + x 2 2 i 2 y Z Z Z 1 h i 3 1 . O( ) vy2 x 2 2 i + x 2 2 i f (i) + J . (3.7.97) y For the forcing terms, we appeal to the bounds in (3.7.59), coupled with (3.7.82): Z 1 Z 1 1 3 1 ||x 2 2 i f (i) dy 0 ||L2y  ||x 4 i ||L2y ||x 4 i f (i) dy 0 ||L2y (3.7.98) y y 3 1  ||x 4 i ||L2y ||x 4 i yf (i) ||L2y (3.7.99) 3 3  ||x 4 i ||L2y ||x 4 i zf (i) ||L2y (3.7.100) 3 1  ||x 4 i ||L2y x 2 i (3.7.101) 237 1 3 0  ||x 4 i ||2L2y + Cx 1 , (3.7.102) 100, 000 0 where > 0. We have used (3.7.1) via: i 2 i 1 =3 i 1 2 i 1 = i 1, to calculate: 3 3 5 1 1 ||x 4 i zf (i) ||L2y . x 4 i x 4 +2 i 1 = Cx 2 i +2 i 1 = Cx 2 i 1 . (3.7.103) Next, through Young’s inequality Z Z 1 Z 1 1 3 | x 2 2 i J dy 0 | . x 2 2 i hyi N x 4+ i 1 y 1 3 0 . ||x 4 i ||2L2y + Cx 1 , (3.7.104) 100, 000 0 where > 0. Inserting these into (3.7.97), relabeling y = u, and integrating in x, one obtains the desired result in (3.7.86). From here, we may repeat the calculations in Lemmas 3.6.4 - 3.6.5: Lemma 3.7.13. The solutions [u, v] to the system (3.7.48) satisfy the following inequality: Z Z Z Z Z Z Z 1 1 3 1 sup 2 u x 2 2 i + u2y x 2 2 i . C + O( ) vy2 x 2 2 i + u2 x 2 2 i . (3.7.105) x 1 2 Proof. We multiply both sides of equation (3.7.80) by ux 2 i and integrate by parts. This gives on the left-hand side: Z ⇣ ⌘ Z Z 1 1 1 (1 + u0p )@x @yy u · ux 2 2 i & @x (1 + u0p )u2 x 2 2 i + u2y x 2 2 i Z Z 1 1 u0px u2 x 2 2 i (1 + u0p )u2 x 2 2 i . (3.7.106) 238 One now sees that it is crucial to have controlled the final term in (3.7.106), which is the purpose of (3.7.97). Moving next to the sequence of terms in P(u, v): Z Z 1 1 1 | u(i) 2 2 sx u x 2 i |  ||u(i) sx x||L1 u2 x 2 2 i  O( )||ux 4 i ||2L2y , (3.7.107) Z Z (i) 1 vsy 2 1 | vs(i) uy ux 2 2 i |=| u x2 2 i | 2 1 1 . ||u(i) sx x||L1 ||ux 4 i ||2L2y  O( )||ux 4 i ||2L2y , (3.7.108) Z 1 v 3 1 | u0py uvx 2 2 i | . ||u0py y||L1 || x 4 i ||L2y ||ux 4 i ||L2y y 3 1 . O( )||vy x 4 i ||L2y ||ux 4 i ||L2y . (3.7.109) We can summarize the above terms by writing: Z 1 3 1 | P · ux 2 i |  O( )||vy x 4 i ||2L2y + O( )||ux 4 i ||2L2y . (3.7.110) Note that the smallness in the above estimates is guaranteed by (3.4.86), (3.4.91), and (3.5.61). We now move to the forcing terms, f and J : Z 1 3 1 1 0 1 | f (i) · ux 2 2 i |  ||f (i) x 4 i ||L2y ||ux 4 i ||L2y . x 2 ||ux 4 i ||L2y 1 0 1 1  Cx + ||ux 4 i ||2L2y . (3.7.111) 100, 000 Next, we use the Hardy inequality: Z Z 1 3 1 | J · ux 2 2 i |. hyi N x 4+ i 1 |u|x 2 2 i 1 0 u 1 . ||hyi N +1 ||L2y x 2 || ||L2y x 4 i y 1 0 1 1  Cx + ||uy x 4 i ||2L2y . (3.7.112) 100, 000 239 0 Combining the above estimates together, one obtains, for some > 0 small, Z Z Z Z 1 1 0 3 1 @x (1 + u0p )u2 x 2 2 i + u2y x 2 2 i . Cx 1 + O( ) vy2 x 2 2 i + u2 x 2 2 i , (3.7.113) and so integrating in x gives the desired result. The third step in establishing the desired bounds is: Lemma 3.7.14. For solutions [u, v] to the system in (3.7.80) one has the following positivity estimate: Z Z Z Z Z Z Z 3 3 1 1 sup u2y x 2 2 i + u2x x 2 2 i .C+ u2y x 2 2 i + O( ) u2 x 2 2 i . x (3.7.114) 3 2 Proof. We apply the multiplier ux x 2 i to the system (3.7.80). This gives on the left-hand side: Z ⇣ ⌘ Z Z Z 3 3 3 1 (1 + u0p )@x @yy u · ux x 2 2 i & (1 + u0p )u2x x 2 2 i + @x u2y x 2 2 i u2y x 2 2 i (3.7.115) Let us now move to the terms in P: Z 3 1 3 | u(i) sx uux x 2 2 i |  ||u(i) sx x||L1 ||ux 4 i ||L2y ||ux x 4 i ||L2y 1 3 . O( )||ux 4 i ||L2y ||ux x 4 i ||L2y , (3.7.116) Z 3 1 1 3 | vs(i) uy ux x 2 2 i |  ||vs(i) x 2 ||L1 ||uy x 4 ||L2y ||ux x 4 i ||L2y 1 3  O( )||uy x 4 i ||L2y ||ux x 4 i ||L2y , (3.7.117) 240 Z 3 v 3 3 | u0py ux vx 2 2 i | . ||u0py y||L1 || x 4 i ||L2y ||ux x 4 i ||L2y y 3 . O( )||vy x 4 i ||2L2y . (3.7.118) We can summarize this contribution via: Z 3 3 1 1 2 | P · ux x 2 i |  O( )||ux x 4 i ||2L2y + O( )||uy x 4 i ||2L2y + O( )||ux 4 i ||2L2y . (3.7.119) Again, the smallness is guaranteed by (3.4.86), (3.4.91), and (3.5.61). Next, the forcing is given in the same way, due to the choice of i relative to i 1: Z 3 3 3 | f (i) · ux x 2 2 i |  ||f (i) x 4 i ||L2y ||ux x 4 i ||L2y 1 0 3 0 1 3 .x 2 ||ux x 4 i ||L2y . x 1 + ||ux x 4 i ||2L2y . (3.7.120) 100, 000 We now integrate up in x up to some fixed point X1 : Z Z X1 Z 3 2 3 u2y (X1 )X12 i + u2x x 2 2 i (3.7.121) 1 Z X1 Z Z X1 Z Z X1 Z 1 1 3 .C+ u2y x 2 2 i + O( ) u2 x 2 2 i + J · ux x 2 2 i . 1 1 1 The last part is to control the J term above: Z X1 Z 3 00 2 (y)x 2 i uie (x, 0)ux (3.7.122) 1 Z X1 Z Z 3 00 00 = (y)u@x {uie (x, 0)x 2 2 i } uie (0, 0)u (y)dy 1 x=1 Z 3 2 00 + X12 i uie (X1 , 0) (y)u(X1 , y)dy (3.7.123) x=X1 241 Z X1 Z 3 00 = (y)u@x {uie (x, 0)x 2 2 i }+C 1 Z 3 2 C X12 i uie (X1 , 0) 0 (y)uy (X1 , y)dy (3.7.124) x=X1 Z X1 Z Z 0 3 1 3 2 i 2  (y)uy @x {uie (x, 0)x 2 2 }+C + i X2 uy (X1 ) (3.7.125) 1 100, 000 x=X1 1 1 1 1 1  C||uy x 4 i ||L2 || 0 (y)x 2 i ||L2 +C C + ||uy x 4 i ||2L2 . (3.7.126) 100, 000 R 3 2 We have absorbed the x boundary contribution, u2y X12 i into the left-hand side of (3.7.121). For the other terms in J , we can perform a similar calculation. As X1 is arbitrary, this then implies the desired result, (3.7.114). Proof of Proposition 3.7.11. Consolidating estimates the three estimates (3.7.86), (3.7.105), and (3.7.114), and subsequently taking small enough, one obtains: Z 1 1 3 2 2 sup x 2 i +u2 x 2 2 i + u2y x 2 2 i x Z Z 3 1 1 3 + 2 x 2 2 i + u2 x 2 2 i + u2y x 2 2 i + u2x x 2 2 i . O( ; n). (3.7.127) It is a standard matter now to obtain weighted in z estimates, and to also successively di↵erentiate the system in x and repeat the previously established bounds. This procedure establishes the desired result fpr uip in (3.7.79). To estimate vpi , we take: Z 1 vpi = uipx dy 0 , (3.7.128) y and repeat the procedure as in estimate (3.6.69). 242 3.7.3 Final Prandtl Layer We now construct the final Prandtl layer, [unp , vpn ]. This final, n’th layer is slightly di↵erent because vpn will be taken to satisfy the boundary condition: vpn (x, 0) = 0. According to (3.7.48), the system for the n’th layer is: (1 + u0p )upx + u(n) (n) 0 n sx up + vs upy + upy vp + Ppx = upyy + f (n) , (3.7.129) [up (x, 0), vp (x, 0)] = [ une (x, 0), 0], lim up (x, y) = 0, up (1, y) = Un (y), (3.7.130) y!1 n Ppy = 0, upx + vpy = 0. (3.7.131) As usual from the Prandtl layers, by evaluating the equation at y = 1, it is clear that the leading order Prandtl pressure is constant, that is Ppn = 0. Once up , vp are constructed to solve (3.7.129), we shall cut them o↵, thereby defining [unp , vpn ]. The relevant remainders are given in (3.3.4) - (3.3.5), which we recall here for convenience: Definition 3.7.15. The n’th remainder is denoted by: Ru,n := ¯(n) ✏u s ¯(n) +u s u¯(n) ¯s(n) u sx + v ¯(n) ¯ (n) sy + Psx , (3.7.132) @y ¯ (n) Rv,n := ¯s(n) ✏v ¯(n) +u s v (n) ¯sx + v¯s(n) v¯sy (n) + P . (3.7.133) ✏ s Remark. We refer the reader to Remark 3.7.1. In this case, there is no n + 1’th Euler construction, which explains the definition in (3.7.132). This is seen as the remainder which is contributed to the next order in the specification of system (3.8.1) - (3.8.7). 243 Using similar arguments to (3.7.50) - (3.7.51), we have the following errors: n h ⇣X n j p ⌘ n X j i Ru,n = ✏ 2 ✏unpxx + vpn ✏ 2 {ujpy + ✏ujeY } + unpx ✏ 2 {uje + ujp } + E (n) , (3.7.134) j=1 j=1 n h n n i Rv,n = ✏ 2 v ✏ p n + u (n) n s v px + v (n) n u sx p + v (n) n s v py + v (n) n v sy p + ✏ 2 un v n + ✏ 2 v n v n p px p py , (3.7.135) Here E (n) = E (n) (up , vp ) is an error term created by cutting o↵ these layers, which will be defined in (3.7.155). One can repeat the procedure used to construct the previous Prandtl layers to conclude: Proposition 3.7.16. Let n be defined according to (3.7.1). The Prandtl layer, up con- structed to satisfy the equation (3.7.129) together with the boundary conditions in (3.7.130), satisfies: 1 ||x 4 z m @xk up ||Pk ( n)  C(k, m, n). (3.7.136) Let us summarize the decay rates which result from this construction, by recalling the definition of our Prandtl norms, Pk , given in (3.6.33) - (3.6.34): Corollary 3.7.17. For any k, j, M 0, the profiles up satisfy the following decay rates: j 1 j 1 xk+ 2 + 2 n ||z M @xk @yj up ||L1 y + xk+ 2 + 4 n ||z M @xk @yj up ||L2y  C(k, j, M, n). (3.7.137) Ry In order to satisfy vp (x, 0) = 0, vp is obtained from up via: vp (x, y) = 0 upx (x, y 0 ) dy 0 . This is distinct from (3.6.27) and (3.7.128), and distinguishes the final, n th Prandtl layer from the previous layers. Then, Corollary 3.7.18. vp obeys the following uniform decay estimate: xk+1 n ||@xk vp ||L1 y  C(k, n). (3.7.138) 244 Proof. Using the boundary condition vp (x, 0) = 0, one can write: 2 Z y Z 1 vp 1 vp 1 vp . vp vpy  1 y 2 + vpy  || 1 ||L2y ||y 2 + vpy ||L2y (3.7.139) 0 0 y 2 + y 2 + 1 1 1 1 1 1   ||y 2 vpy ||L2y ||y 2 + vpy ||L2y = ||x 4  z2  vpy ||L2y ||x 4 + z 2 + vpy ||L2y (3.7.140) 1 3 1+ 1+ 2 +2 n  x2 x n x n =x . (3.7.141) We have used the  > 0 to avoid the critical Hardy inequality. The Hardy inequality we 1  1 have used (with power y 2 vp ) relies on the vanishing of vpn at y = 0. Next, we introduce a cuto↵ function which honors the scaling, z = py : x Z h ⇣p p 1 y ⌘ ⌘i vpn := ( ✏z)vp , unp := @y ✏p vp (x0 , y) dx0 . (3.7.142) x x0 py y0 Here, we make the notational convention for integrands: z 0 = x0 or z 0 = p x where 0 denotes the integration variable. It is clear, then, that @x unp + @y vpn = 0. The following boundary conditions are also clear: [unp , vpn ] ! 0 as y ! 1, [unp , vpn ]|y=0 = [up , vp ]|y=0 . (3.7.143) Remark. This cut-o↵ honors the parabolic scaling of the Prandtl layers for all x > 0. The cut-o↵ introduced in (GN17) was in the region y p1 . Locally in x, z is equivalent to y, " so these cut-o↵s agree locally in x. We must now record two properties about the cut-o↵ layers. First, the uniform estimates of the cut-o↵ layers remain unchanged: Lemma 3.7.19. The following pointwise decay bounds hold for the cut-o↵ layers, for any 245 k 0: @xk vpn . C(k, n)x k 1+ n , (3.7.144) 3 @xk vpy n . C(k, n)x k 2+ n , (3.7.145) @xk unpy . C(k, n)x k 1+ n , (3.7.146) 1 @xk unp . C(k, n)x k 2+ n . (3.7.147) Proof. First, p ✏ 3 n |vpy | p 0 |vp | + |vpy | . x 2+ n . (3.7.148) x Next, Z 1 h i Z 1 Z 1 p Z 1 p ✏ ✏ unpy = @yy ( ✏z 0 )vp dx0  00 vp + 2 p vpy + vpyy x x x0 x x0 x .x 1+ n . (3.7.149) Finally, for unp , Z 1 Z 1 p 1 0 1 unp  ✏p |vp | + |vpy |  x 2+ n . (3.7.150) x x0 x By di↵erentiating successively in x, the result for k 1 follows in the same manner. Lemma 3.7.20 (L2y Estimates). The following L2y decay bounds hold for the cut-o↵ layers, for any k 0: j 1 ||@xk @yj unp ||L2y . C(k, , n)x k 2 4+ n . (3.7.151) 246 Proof. In order to compute the L2 norms, we must trade in the following way: 1  1 1  o( ) " 4 + 2 hyi 2 + x 4 2 . 1, (3.7.152) where o( ) means or any derivative of , the essential feature being that z  p1 . " Using this: Z 1 p Z 1 " 1 ||unp ||L2y  || p 0 vp ||L2y + || vpy ||L2y . x 4+ n . (3.7.153) x x0 x For unpy , again according to the expression in (3.7.149), and the decay rates in (3.7.137): Z 1 Z 1 p Z 1 ✏ 00 " 3 ||unpy ||L2y . || vp ||L2y + || p vpy ||L2y + ||vpyy ||L2y . x 4+ n . (3.7.154) x x0 x x0 x It is now clear we can repeat these calculations for higher x and y derivatives, thereby obtaining the desired result. The next task is to control the error made by cutting o↵ the layers. This error is obtained by inserting the new, cut-o↵ layers into the equation (3.7.129): E (n) := (1 + u0p )unpx + u(n) n (n) n 0 n sx up + vs upy + upy vp unpyy f (n) . (3.7.155) We will proceed to expand (3.7.155), term-by-term: p ✏ 0 (1 + u0p )unpx = (1 + (1 + u0p )upx u0p ) p vp , (3.7.156) x Z 1 p p ✏ 0 0 ✏z u(n) n (n) (n) sx up = usx up + usx p vp + up , (3.7.157) x x0 x0 247 Z 1 p p ✏ 00 ✏ 0 ✏ 0 vs(n) unpy = vs(n) upy + vs(n) vp + 2 p vpy p zupy , (3.7.158) x x0 x0 x0 u0py vpn = u0py vp , (3.7.159) Z 1 p p ✏ 0z ✏ 00 0 ✏ unpyy = upyy upyy + C1 vpy + C2 p vpyy x x0 2 x0 x0 3 ✏2 000 + 3 vp . (3.7.160) (x0 ) 2 Let us provide justification to the above expressions, first turning to (3.7.157). Applying the definition in (3.7.142) yields: p Z 1 Z 1 ✏ 0 u(n) n sx up = u(n) sxp (n) vp + usx vpy x x0 x Z 1 p Z 1 ✏ 0 = u(n) sx p vp u(n)sx upx dx0 x 0 x x Z 1h p p i ✏ 0 (n) z ✏ 0 (n) = usx p vp + usx 0 up dx0 + u(n) sx up , (3.7.161) x x0 x the final equality following from an integration by parts in x. Similarly, for (3.7.158): Z 1 ⇣ ⌘ vs(n) unpy = vs(n) @yy vp dx0 x ⇣✏ Z 1 p ⌘ 00 0 ✏ = vs(n) 0 v p + 2 p vpy + vpyy dx0 x x x0 Z 1 p Z 1 ✏ 00 0 ✏ = vs(n) 0 v p + 2 v 0 py v (n) s upxy dx0 x x x x Z 1h p i ✏ 00 0 ✏ p 0 = vs(n) 0 vp + 2 0 vpy ✏ 0 zupy dx0 + vs(n) upy . (3.7.162) x x x x This same computation is performed for (3.7.160): Z 1 ⇣ ⌘ unpyy = @y3 vp dx0 x Z 1 h " 32 p i Z 1 000 " 00 " 0 = 3 vp + C1 0 vpy + C2 p vpyy + vpyyy x (x0 ) 2 x x0 x 248 Z 1 h " 32 p i Z 1 000 " "00 = 3 vp + C1 0 vpy + C2 p vpyy 0 upxyy x (x0 ) 2 x x0 x Z h " 32 p i Z h 1 000 " " 1 z p i 0 = 3 vp + C1 0 00 vpy + C2 p vpyy 0 0 upyy " dx x (x0 ) 2 x x0 x 2x0 + upyy . (3.7.163) Summing the above terms together, we arrive at our expression: p p Z 1 p Z 1 (n) ✏ ✏ 0 0 0 ✏z E = (1 + u0p ) p vp + p u(n) (n) vp + usx sx up (3.7.164) x x x 0 x x0 Z 1 p p Z 1p ✏ 00 ✏ 0 ✏ 0 ✏ 0z + vs(n) 0 vp + 2 p vpy 0 zupy 0 upyy (3.7.165) x x x 0 x x x 2 Z 1h p 3 i ✏ ✏ ✏2 + C2 p vpyy + C1 0 00 vpy + 0 3 000 vp + (1 )f (n) . (3.7.166) x x0 x (x0 ) 2 Lemma 3.7.21. For  > 0 arbitrarily small, the error, E (n) obeys the following bound: 1 3 1 5 E (n)  C(n, )✏ 4  x 2 + n + , ||E (n) ||L2y  C(n, )✏ 4  x 4 + n + . (3.7.167) Proof. We will proceed in order from (3.7.164) - (3.7.166), and we will focus on the L2y estimates, as the uniform estimates are straight-forward. First, p ✏ p 1 1 1 ||(1 + u0p ) p 0 (z)vp ||L2y . ✏||hyi 2 + hyi 2  0 vp p ||L2y x x p 1 1 p 1 1  . ✏||hyi 2 + 0 x 2 vp ||L1 y . ✏||z 2 + 0 x 4+ 2 vp ||L1 y 1 5  . ✏4  x 4+ 2 + n . (3.7.168) The essential characteristic in the above calculation is the ability to pay a factor of 1  1  ✏4+ 2 x 4 2 in order to obtain an L2y quantity due to the presence of the cut-o↵ function 0 (see 3.7.152). In similar manner, we have: Z 1 p Z 1 p ✏ 0 ✏ 1 ||u(n) sx p vp dx0 ||L2y  ||u(n) sx ||L1 || p 0 vp ||L1 y ||hyi 2  ||L2y dx0 x x0 x x0 249 Z 1 1  1  1  1  1 .x 1 (x0 ) 1+ n ✏4 2 x 4+ 2 ||✏ 4 + 2 x 4 2 hyi 2 + 0 ||L1 x 1 5  .✏ 4 + x 4 + n+ 2 . (3.7.169) The third term on line (3.7.164) can be estimated analogously. Coming now to the first term in (3.7.165), Z 1 Z 1 ✏ 00 3  3  1  1  1 ||vs(n) vp ||L2y  ||vs(n) ||L1 ✏4 2 (x0 ) 4+ 2 (x0 ) 1+ n ||✏ 4 + 2 x 4 2 hyi 2 + 00 ||L1 x x x 3  5  . ✏4 2 x 4+ 2 + n . (3.7.170) The second term in (3.7.165): Z 1 p Z 1 ✏ 0 1  1  3 ||vs(n) p vpy ||L2y  ||vs(n) ||L1 ✏4 2 (x0 ) 4+ 2 (x0 ) 2+ n dx0 x x0 x 1  5  . ✏4 2 x 4 + n+ 2 . (3.7.171) Moving to the third, which is slightly di↵erent due to the weight z, but this causes no harm as zupy is known to be in L1 y according to (3.7.136): Z 1 p Z 1 p ✏ 1 ✏ 3 p 5 ||vs(n) 0 zupy ||L2y . x 2 (x0 ) 4+ n dx0 . ✏x 4 n . (3.7.172) x x0 x x0 Next, we move to the unpyy contributions: Z 1 p " " z p p 5 ||C1 00 vpy + C2 p vpyy 0 0 upyy "||L2y . "x 4+ n , (3.7.173) x x0 x0 2x 0 5 7 5 according to the rates, ||upyy zx 4 n , vpyy x 4 n , vpy x 4 n ||L2y  C from (3.7.137). Fi- 250 nally, for the vp term in (3.7.166), one must use the tradeo↵ in (3.7.152), to obtain: Z 1 3 Z 1 "2 5  1  3 || 3 000 vp ||L2y dx0 . " 4 2 (x0 ) 4 + 2 (x0 ) 2 ||vp ||L1 y x (x0 ) 2 x Z 1 5  1  3 5  5  . "4 2 (x0 ) 4 + 2 (x0 ) 2 (x0 ) 1+ n . "4 2 x 4+ 2 + n . x (3.7.174) The f (n) term can then be estimated upon observing that f (n) is exponentially small in the support of 1 : ||(1 )f (n) ||L2y = ||(1 )z N z N f (n) ||L2y N N 5 N 5 . ✏ 2 ||z N f (n) ||L2y . ✏ 2 x 4+ n 1 .✏2x 4+ n , (3.7.175) for any N large, according to estimate (3.7.59). This now concludes the proof of (3.7.167). By construction, one has the following: Lemma 3.7.22 (Remainder Estimates). For Ru,n , Rv,n as defined in (3.7.135), and for any 2 [0, 14 ), n 2, and for n as in (3.7.1),  > 0 arbitrarily small, n p 1 3 ✏ 2 @xk Ru,n + ✏@xk Rv,n . C(n, )✏ 4  x k 2 +2 n , (3.7.176) n p p 1 5 ✏ 2 || ✏@xk Ru,n , ✏@xk Rv,n ||L2y . C(n, )✏ 4  x k 4 +2 n + . (3.7.177) Proof. This follows from (3.7.167) and those bounds established in (3.7.136). We shall proceed term by term from (3.7.135). First, ⇣ ⌘ ⇣ 0 p ⌘ "1 ||unpxx ||L2y = "1 ||@x ||L2y = "1 ||@x p vp "vp + vyp ||L2y y x 00 0 p 0 p " p 0 = "1 ||" 3 zvp + 3 vp + p "vpx + " zvpy + vpxy ||L2y (3.7.178) x2 x2 x x 251 = (3.7.178.1) + ...(3.7.178.5). First, p ⇣ 1 1  1  ⌘ 1  5  |(3.7.178.1)| . "1 || "z · hyi 2 + x 4 2 "4+ 2 00 ||L1 y "4 2 ||vp ||L1 x 4+ 2 5  9  . "4 2 x 4+ 2 + n . (3.7.179) It is clear that (3.7.178.1), (3.7.178.2), and (3.7.178.3) are identical, and so we move to: p p " " 5 |(3.7.178.4)| . "1 ||zvpy ||L2y . "1 x 4+ n , (3.7.180) x x and finally: 9 |(3.7.178.5)| = "1 || vpxy ||L2y . "1 x 4+ n . (3.7.181) We next move to the second term in Ru,n , for which we immediately use (3.7.152) n n 1  1  "2 ||vpn unpy ||L2y  " 2 4 2 x 4 + 2 ||vpn ||L1 y ||unpy ||L1 y n 1  1  n 1  7  . "2 4 2 x4+ 2 x 2+2 n = "2 4 2 x 4 + 2 +2 n . (3.7.182) where we have used (3.7.144), (3.7.146). Similarly, for j 1, one has: 1 1 1 1 1 3 "2 ||vpn ujpy ||L2y  " 2 ||vpn ||L1 ||ujpy ||L2y . " 2 x 1+ n x 2+ n . "2 x 2 +2 n (3.7.183) where we have used the specification of the norm || · ||P given in (3.6.33). For the term 252 vpn ujeY , we calculate, for  > 0: 1 j 1 j 1 j ✏2+2 ||vpn ujeY ||L2y  ✏ 4 + 2 ||vpn ||L1 y ||ujeY ||L2Y . C(n)✏ 4 + 2 x 1+ n x 1+ . (3.7.184) The last estimate follows from applying L2Y to the Eulerian self-similar estimate, (3.5.48): 1 u1eY = vex 1  C()x 1+ Y 2  . (3.7.185) The next term in Ru,n is, for j 1, for which we use (3.6.81) for the ujp term, (3.7.21) (n) for the uje term, and (3.7.151) for the upx term in L2 : j j 1 1  1 5 px (ue + up )||L2y . " x ||" 2 u(n) ||unpx ||L2y . " 4 j j 4+ n 4+ n 4+ n 2 2 x x . (3.7.186) The final term in Ru,n is the error term, E (n) , which has been controlled in (3.7.167). We may now move to Rv,n . In so doing, we first note that "vpxx n can be estimated identically to (3.7.187). Second, using (3.7.151), we have: 1 1 7 "2 n ||vpyy ||L2y . " 2 x 4+ n . For the L2 bound on the vpx n term in (3.7.135), we employ (3.7.152): 1 1 1 s vpx ||L2y . ✏ ||u(n) n n n ✏2 2 ||vpx ||L2y  ✏ 2 ||1z p1 vpx ||L2y ✏ 1 1 1 1 2+ 2+  ✏2 ||1z p1 x ||L2y = ✏ 2 ||1z p1 x n (1 + y) 2 + (1 + y) 2  ||L2y ✏ ✏ 1 1 1  7  1  ✏2 ||1z p1 (1 + y) 2 + x 4 2 ||L1 y x 4 + n+ 2 ||(1 + y) 2  ||L2y ✏ 1 1 7  1 7   ✏2 ||1z p1 (1 + z) 2 + ||L1 y x 4 + n+ 2  ✏4  x 4 + n+ 2 . (3.7.187) ✏ 253 The next two terms: 1 p p 3 1 "2 (n) n ||vsx up ||L2y  (n) "||vsx ||L1 ||unp ||L2y . "x 2 x 4 , (3.7.188) 1 1 7 "2 ||vs(n) vpy n ||L2y  " 2 x 4 +2 n . (3.7.189) Now, by using the trade-o↵ in (3.7.152): 1 1 1  1 7  "2 (n) n ||vsy vp ||L2y  " 4  (n) ||vsy ||L1 y ||vpn ||L1 y x4+ 2 . "4  x 4 +2 n + 2 . (3.7.190) We will now move to: n p n + 12 n + 12 9 "2 "||unp vpx n ||L2y . " 2 ||unp ||L2y ||vpx n ||L1 y . "2 x 4 +2 n , (3.7.191) n p n + 12 9 "2 "||vpn vpy n ||L2y . " 2 x 4 +2 n . (3.7.192) where we have used (3.7.144) and (3.7.151). This completes all of the terms in (3.7.135), thereby establishing (3.7.177). The uniform estimates follow in an analogous manner. For the energy estimates in Section 3.10, we will need to retain the self-similarity of unp (the ability to absorb factors of z). As the profiles unp , vpn are higher order in ✏, we will be happy to pay factors of ✏ in order to retain this ability: Lemma 3.7.23. The following point-wise decay estimates holds for any m 0, m z m @xk vpn . ✏ 2 C(k, n, m)x k 1+ n , (3.7.193) m 3 z m @xk vpy n .✏ 2 C(k, n, m)x k 2+ n , (3.7.194) 1 y j @yj unp . C(k, n, m)x 2+ n , (3.7.195) 254 m z m @xk unpy . ✏ 2 C(k, n, m)x 1+ n k , for k 1. (3.7.196) Proof. Via the definition: p m m z m |vpn |  z m | ( ✏z)vp |  ✏ 2 |vp | . ✏ 2 x 1+ n . (3.7.197) Next, applying @y yields: p v m 3 z m |vpy n | = z m ✏| 0 pp | + z m |vpy | . ✏ 2 x 2+ n . (3.7.198) x The estimate for unpy is more complicated. The m = 0 case was treated in (3.7.147). Suppose m = 1. Then, Z 1 Z 1 Z p Z 1 0 ✏ 00 ✏ 0 yunpy =y @yy ( vp )dx = yvp + p yvpy + yvpyy (3.7.199) x x x0 x x0 x = I1 + I2 + I3 . (3.7.200) We have: Z 1 Z 1 |vp | p 3 p 1 |I1 |  || 00 z||L1 ✏ p dx0  ✏ (x0 ) 2+ dx0 . ✏x 2+ n . (3.7.201) x 0 x x Next, Z 1 Z 1 p p 3 p 1 |I2 |  ✏ |zvpy |dx . 0 0 ✏(x0 ) 2 dx0 . ✏x 2+ n . (3.7.202) x x 255 Lastly, Z 1 Z 1 1 1 |y vpyy |dx0  (x0 ) 2 zvpyy dx0 . x 2+ n . (3.7.203) x x Piecing these estimates together gives: 1 y|unpy | . x 2+ n , (3.7.204) as desired. Higher y-derivatives of unp follow an identical calculation. Let us now move to x-derivatives of unpy : p ✏ 00 ✏ 0 unpyx = @yy ( vp ) = vp + p vpy + vpyy . (3.7.205) x x From here, once can estimate: p ✏ ✏ z m |unpxy | . | 00 |z m |vp | + p | 0 |z m |vpy | + z m |vpyy | x x m 2+ ✏ 2 x n . (3.7.206) We are now ready to conclude Chapter I: Proof of Theorem 3.3.2. Consolidating Corollaries 3.4.8 - 3.4.9, Propositions 3.5.6, 3.6.12, 3.7.3, 3.7.11, Lemma 3.7.22, and Lemma 3.7.23 gives Theorem 3.3.2. Part II: a-Priori Estimates in k · kZ 256 257 3.8 Overview of a-priori Z-Norm Estimates We will now consider the system satisfied by the remainders, [u, v, P ] as defined in (3.2.14) - (3.2.16). Expanding the Navier-Stokes equations in (3.2.11) - (3.2.13), one obtains: ✏u + Su (u, v) + Px = f (u, v), (3.8.1) Py ✏v + Sv (u, v) + = g(u, v), (3.8.2) ✏ ux + vy = 0, (3.8.3) which are taken together with the boundary conditions: [u, v]|{y=0} = [u, v]|{x=1} = lim [u, v] = lim [u, v] = 0. (3.8.4) y!1 x!1 The terms in equations (3.8.1) - (3.8.2) are defined: n n f (u, v) := ✏ 2 Ru,n + N u (u, v), g(u, v) := ✏ 2 Rv,n + N v (u, v), (3.8.5) Su (u, v) := uR ux + uRx u + vR uy + uRy v, Sv (u, v) := uR vx + vRx u + vR vy + vRy v, (3.8.6) n n n n N u (u, v) := ✏ 2 + uux + ✏ 2 + vuy , N v (u, v) := ✏ 2 + uvx + ✏ 2 + vvy . (3.8.7) The forcing terms, Ru,v , Rv,n are given by the expressions in (3.7.134) - (3.7.135), and have been controlled in Theorem 3.3.2. The coefficients uR , vR in Su (u, v), Sv (u, v) given in (3.8.6) have been defined in (3.3.1) - (3.3.2), and controlled according to the estimates in Theorem 3.3.2. Due to the eventual need to perform a fixed-point argument, we will actually consider the slightly generalized system, where f, g above are replaced by: n n n f (u, u ¯, v¯) = ✏ 2 Ru,n + ✏ 2 + u ¯u¯x + ✏ 2 + v¯uy , (3.8.8) 258 n n n g(¯ u, v¯) = ✏ 2 Rv,n + ✏ 2 + u ¯v¯x + ✏ 2 + v¯v¯y . (3.8.9) When u = u ¯, v = v¯ (which corresponds to a fixed point of an appropriately defined map), one obtains the actual system of interest, with f as in (3.8.5). For technical purposes, in this chapter we will consider the system (3.8.1) - (3.8.3) on the domain: ⌦N := {(x, y) : x > 0, 0 < y < N }. (3.8.10) All estimates will be made independent of N , allowing us to eventually send N ! 1. When working on this domain, we will use the boundary conditions: [u, v]|{y=0} = [u, v]|{x=1} = [u, v]|{y=N } = lim [u, v] = 0. (3.8.11) x!1 The norms which we will work in are defined in the next section, Section 3.9, starting with (3.9.3) - (3.9.8). We invite the reader to read these definitions at this point. The main result of this chapter is then: Theorem 3.8.1 (Complete Z Estimate). Fix any N > 0. Let , ✏ be sufficiently small based on universal constants, with ✏ << , and n 2 N sufficiently large relative to universal constants. Let Ni be parameters in the definition of Z, given in (3.9.8). Let 2 (0, 14 ), and u, v¯||Z(⌦N )  1. Then [u, v] 2 Z(⌦N ), solutions parameter  > 0 arbitrarily small. Suppose ||¯ to the system (3.8.1) - (3.8.3), (3.8.5) - (3.8.9), with boundary conditions (3.8.11), obey the following a-priori estimate: 1 n ⇣ ⌘ ||u, v||2Z(⌦N ) . ✏ 4  + ✏2 !(Ni ) u, v¯||4Z(⌦N ) , ||¯ (3.8.12) for some fixed function !(Ni ), where the constants above are independent of N . 259 Let us give a brief overview of the steps to prove Theorem 3.8.1. (Step 1) The Space Z (Section 3.9): We introduce the various components of the crucial norm Z. The energy norms, X1 , X2 , X3 are to be controlled using energy and positivity estimates (in the next steps). The elliptic norms, Y2 , Y3 , provide additional controls near the boundary, x = 1, and are related to the Xi norms in subsection 3.9.1. Most 1 importantly, there are uniform-type norms, defined in (3.9.16), of which the ||vx 2 ||L1 is the most crucial ingredient. These are controlled in Lemmas 3.9.15, 3.9.17. The relation between all of the norms is summarized in Theorem 3.9.20. This analysis of Z is the main novelty of Chapter II. (Step 2) Energy Estimates: The lowest-order energy estimates are performed in Proposition 3.10.1, for which the sharp profile estimates from (3.3.8) - (3.3.20) are essential. An 1 examination of Proposition 3.10.1 shows that the energy estimate loses a factor of x 2 which must be recovered in the next step. (Step 3) Positivity Estimates: The lowest-order positivity estimates are performed in Proposi- tion 3.10.2. Here again, the estimates from (3.3.8) - (3.3.18) are used in an essential way. In particular, the estimate (3.10.66) forces the requirement shown in (3.2.48). (Step 4) Higher-Order Energy/ Positivity Estimates: In Propositions 3.10.4, 3.10.6, 3.10.7, 3.10.8, we control the higher-order energy norms X2 , X3 from definitions (3.9.4) - (3.9.5). This is achieved by applying appropriate iterations of @x to the system (3.8.1) - (3.8.3). (Step 5) Nonlinear Estimates: The estimates of the nonlinearities present in (f, g) from (3.8.8) - (3.8.9) is performed in Section 3.11, in particular in Lemma 3.11.1. Here, there 1 are several delicate matters. First, the sharp decay of ||x 2 v||L1 , which is present in (3.9.8) due to Lemma 3.9.17, is essential in order to control the term v¯u ¯y . This estimate is (3.11.8). Second, estimate (3.11.5) capitalizes on a cancellation structure which enables us to control a term that would be otherwise out of reach. Third, the 260 top-order nonlinear terms in estimates (3.11.20) and (3.11.24) are controlled by using mixed-norms. 3.9 The Function Space Z In this section, we define and analyze the high-order, weighted norm, Z, on which we will close our nonlinear analysis. The functional framework of this section is required in order to follow the calculations in the upcoming sections. First, we need to define the energy norms in which we obtain energy estimates, and also several auxiliary norms that will supplement the energy norms. To define these, first define the cut-o↵ functions: 8 > > <0 for 1  x  3 , 2 ⇣3 (x) = (3.9.1) > > :1 for x 2. 8 > > <0 for 1  x  50 + 50(k 2), ⇢k (x) = (3.9.2) > > :1 for x 60 + 50(k 2). The energy norms are defined as follows: p 1 ||u, v||2X1 := ||uy ||2L2 + ||{ ✏vx , vy }x 2 ||2L2 (3.9.3) p 3 ||u, v||2X2 := ||uxy · ⇢2 x||2L2 + ||{ ✏vxx , vxy } · (⇢2 x) 2 ||2L2 , (3.9.4) p 5 ||u, v||2X3 := ||uxxy · (⇢3 x)2 ||2L2 + ||{ ✏vxxx , vxxy } · (⇢3 x) 2 ||2L2 . (3.9.5) Definition 3.9.1. The norms Y2 , Y3 are strengthenings of X2 , X3 near the boundary, x = 1, and defined through: p 3 ||u, v||2Y2 := ||uxy x||2L2 + ||{ ✏vxx , vxy }x 2 ||2L2 + ||uyy ||L2 (x2000) , (3.9.6) p 5 ||u, v||2Y3 := ||uxxy · ⇣3 x2 ||2L2 + ||{ ✏vxxx , vxxy } · ⇣3 x 2 ||2L2 . (3.9.7) 261 Definition 3.9.2. The norm Z is defined through: 1 p 1 ||u, v||Z :=||u, v||X1 \X2 \X3 + ✏N2 ||u, v||Y2 + ✏N3 ||u, v||Y3 + ✏N4 ||ux 4 , "vx 2 ||L1 p 3 5 1 + ✏N5 sup || "vx x 2 , ux x 4 ||L1 + "N6 sup ||uy x 2 ||L2y x 20 x 20 hZ 1 p i 12 + ✏ N7 x4 || "vxx ||2L1 y dx . (3.9.8) 20 Here, Ni , are some large numbers which will be specified in (3.9.103) - (3.9.105). They depend only on universal constants. It will be understood that Ni are much larger than any of the quantities Mi appearing in the forthcoming lemmas, and that n is much larger than any of the Ni , Mi . Now that the norms we will be working in have been specified, we will define correspond- ing spaces. First, some basic notations: Definition 3.9.3. For any open set U ⇢ ⌦, the space C01 (U ) denotes the space of smooth functions with compact support in U . We will also use C01 (U ) to denote the space of smooth vector fields with compact support in U , which will be clear from context. The 1 space C0,D (U ) denotes the space of smooth, divergence-free vector fields with compact support in U . When the set U is clear from context, we shall suppress it. Definition 3.9.4. Given any open set U ⇢ ⌦, the space Z(U ) is defined to be space of 1 those divergence-free vector fields which lie in the closure of C0,D (U ) under the norm X1 , such that ||u, v||Z < 1. Definition 3.9.5. Given any open set U ⇢ ⌦, the space (X1 \ X2 \ X3 )(U ) is defined to 1 be space of those divergence-free vector fields which lie in the closure of C0,D (U ) under the norm X1 , such that ||u, v||X1 \X2 \X3 < 1. Due to the weights in the norm Z and the energy norms X1 \X2 \X3 , there is no “H = W Theorem” generically available, which would equate the density of smooth functions with compact support with the class of functions satisfying ||u, v||Z < 1. This is the reason 262 must specify that: 1 ||·||X1 Z(U ) ⇢ C0,D (U ). (3.9.9) We will first record the boundary behavior of Z(⌦)-vector fields: Lemma 3.9.6. For [u, v] 2 Z(⌦), the following boundary conditions are satisfied: [u, v]|x=1 = [u, v]|y=0 = lim [u, v] = lim [u, v] = 0. (3.9.10) x!1 y!1 Proof. The boundary conditions at x = 1 and y = 0 are satisfied by the density specification (3.9.9). The boundary conditions as x ! 1 is satisfied by the definition of norm Z, (3.9.8) because the decay is encoded (with rates) in the norm. The boundary conditions as y ! 1 1 is enforced because [u, v] 2 Z(⌦) implies that [u, v] 2 H(xA) , and candidacy in such a Sobolev space automatically encodes decay as y ! 1. The above lemma shows that Z(⌦) is a suitable space to obtain solutions to our boundary-value problem, after a consultation with (3.8.4). That X1 \ X2 \ X3 also en- codes the boundary conditions from (3.9.10) is less obvious, and is proven in Lemma 3.9.16. Lemma 3.9.7. For any open set U ⇢ ⌦, Z(U ) and (X1 \ X2 \ X3 )(U ) are Banach spaces. Proof. This follows from standard arguments. For the energy norm, X1 \ X2 \ X3 , we have Hilbertian structure. Given two divergence- free vector-fields: u := (u, v), a := (a, b), (3.9.11) 263 define the following inner-products: Z Z Z Z Z Z hu, aiX1 := uy a y + ux ax x + "vx bx x, (3.9.12) Z Z Z Z Z Z hu, aiX2 := uxy axy (⇢2 x)2 + uxx axx (⇢2 x)3 + "vxx bxx (⇢2 x)3 , (3.9.13) Z Z Z Z Z Z hu, aiX3 := uxxy axxy (⇢3 x)4 + uxxx axxx (⇢3 x)5 + "vxxx bxxx (⇢3 x)5 , (3.9.14) hu, aiX1 \X2 \X3 := hu, aiX1 + hu, aiX2 + hu, aiX3 . (3.9.15) Temporarily accepting Lemma 3.9.16, we have: Lemma 3.9.8. With the inner product (3.9.15), (X1 \ X2 \ X3 )(⌦) is a Hilbert space. Proof. The inner-product (3.9.15) is non-degenerate due to the boundary conditions [u, v]|y=0 . The remaining inner-product axioms are straightforward to check, which cou- pled with the completeness in Lemma 3.9.7 gives the result. Notationally, it sometimes is convenient to refer at once to all of the “uniform-type” quantities in the norm Z, so we designate the following notation: Definition 3.9.9. The norm U is defined by: 1 p 1 p 3 5 ||u, v||U := ✏N4 ||ux 4 , "vx 2 ||L1 + ✏N5 sup || "vx x 2 , ux x 4 ||L1 x 20 1 hZ 1 p i 12 + "N6 sup ||uy x 2 ||L2y + ✏N7 4 x || "vxx ||2L1 y dx . (3.9.16) x 20 20 264 3.9.1 Elliptic Estimates and the Spaces Yi We will first utilize elliptic theory in order to obtain basic H k estimates for our solution, which hold generically for Stokes-type equations. These elliptic estimates are meant to supplement the energy estimates that we will perform in Section 3.10 which are meant to control the energy norms, X1 \ X2 \ X3 . In particular, the estimates from this section cannot replace the energy estimates for two reasons: (1) They scale poorly in ✏, and (2) One cannot extract sharp enough global-in-x information using this procedure. Their purpose is to provide more controls near the boundary x = 1, which is reflected in the norms Y2 , Y3 . To see this one should compare the support of the cut-o↵ functions in our energy norms, ⇢k , with the support of ⇣3 in Y3 . For the set of calculations in this subsection, there are many di↵erent cuto↵ functions which arise in addition to the important cuto↵s, ⇢k , ⇣k which have already been defined. Remark (Notational Convention). To simplify notations, given a cut-o↵ function , we introduce the notation o( ) to mean either or any of its derivatives, or any positive powers of or any of its derivatives. Roughly speaking, any “variant” of is denoted by o( ), the essential feature being o( ) is supported in the same (or similar) region. The first step will be to provide some controls of ||u, v||H˙ 2 . Lemma 3.9.10 (H 2 Regularity). Let [u, v], [¯ u, v¯] 2 X1 be solutions to (3.8.1) - (3.8.3), with f, g as in (3.8.8) - (3.8.9). For some M2 , perhaps large, dependent only on universal constants: sup ||u, v||L1 y + ||u, v||H˙ 2 (x2000) x2000 h n ⇣ ⌘ i .✏ M2 C + ✏ 2 + ||¯ u, v¯||L1 ||¯ u, v¯||X1 + ||u, v||X1 + ||u, v||X1 . (3.9.17) 265 Proof. Notationally, we will not take care to rename generic constants M2 within this proof. We rescale the system back to Eulerian coordinates via: p 1 u ˜(x, Y ) := u(x, y), v˜(x, Y ) := ✏v(x, y), P˜ (x, Y ) = P (x, y), (3.9.18) ✏ which, by rescaling (3.8.1) - (3.8.3), yields the Stokes-sytem, for some M2 perhaps large: h i h i ˜ + P˜x = ✏ u M2 f˜ + S˜u , v˜ + P˜Y = ✏ M2 g˜ + S˜v , u ˜x + v˜Y = 0. (3.9.19) Above, f˜, g˜, S˜u , S˜v have been also rescaled to Eulerian coordinates. By considering the RY stream function = 0 u ˜, we obtain the following biharmonic problem: h i h i 2 = F := ✏ M2 {@Y f˜ + S˜u @x g˜ + S˜v }, (3.9.20) (x, 0) = Y (x, 0) = 0, (1, Y ) = x (1, Y ) = 0. (3.9.21) We now introduce the partition of unity, { m }m 1 of the region [1, 2000] ⇥ [0, 1). The specifications of these cut-o↵ functions are as follows. First, define: 8 8 > > > > 1 for m 1  Y  m + 1, > > > > <1 for 1  x  2000, < (a) (b) (x) = m (Y )= 0 for Y m + 2, (3.9.22) > > > > :0 for x 4000, > > > > :0 for Y  m 2. for all m 1. Then define: (a) (b) m (x, Y )= (x) m (Y ). (3.9.23) 266 The purpose of selecting such a partition is to localize near x = 1, in such a way that the region between 1 and the support of ⇢2 is captured (see the definition in (3.9.2)). Denote by ¯m = m Then: 2 ¯m = mF +[ 2 , m] , ¯m (x, 0) = @Y ¯m (x, 0) = ¯m (1, Y ) = @x ¯m (1, Y ) = 0. (3.9.24) Here the commutator is defined as: 2 [ , ] := YYYY +6 YY YY +4 YYY Y +4 Y YYY +6 xx xx +4 xxx x +4 x xxx + xxxx + xxY Y +2 YY xx +2 YYx x +2 x xY Y +2 xxY y +2 y xxY +4 xY xY . (3.9.25) According to (BR80), Theorems 1 and 2, with k = 1, coupled with Figure 2, P. 562 in (BR80) with “C/C” boundary conditions, we have the following H 3 estimate for : || ¯m ||H 3 . || m F ||H 1 + ||[ 2 , m] ||H 1 . (3.9.26) We will first address the commutator terms in (3.9.26). First, consider the terms in (3.9.25) which have three derivatives on , which are denoted by o(@ 3 ). Using the definition 1 of H , for any compactly supported test function ↵(x, Y ) 2 H01 , 3 2 2 ho( m )o(@ ),↵(x, y)iH 1 ,H 1 0 = ho(@ m )o(@ ), ↵i + ho( m )o(@ ), @↵i 2  ||o( m )o(@ )||L2 ||↵||H01 . (3.9.27) 267 Then taking the sup over all ||↵||H01 = 1, we obtain: ||o( m )o(@ 3 )||H 1 . ||o( m )o(@ 2 )||L2 . ||o( m )o(@)[˜ u, v˜]||L2 .✏ M2 ||o( m )]{u, v}||X1 . (3.9.28) Next, turn to the terms in (3.9.25) which has two derivatives on : 2 2 M2 ||o( m )o(@ )||H 1  ||o( m )o(@ )||L2  " ||o( m )]{u, v}||X1 . (3.9.29) Next, we must address those terms in (3.9.25) which has one derivatives on . For this, we use the Poincare inequality in x direction: ||o( m )@ ||H 1  ||o( m )@ ||L2  ||o( m ){u, v}||L2  ||ux , vx ||L2 (x2000) M2  ||o( m ){ux , vx }||L2 " ||o( m ){u, v}||X1 . (3.9.30) For the zeroeth order terms in , we must argue as follows: Z x Z x 1 ||o( m) ||H 1 . ||o( m) || L2 = ||o( m) v|| L2 . ||o( m ) v||L2 0 x 0 ." 1 ||o( m ){u, v}||X1 . (3.9.31) Summarizing, then, ||[ 2 , m] ||H 1 .✏ M2 ||o( m ){u, v}||X1 , (3.9.32) It remains, then, to analyze the majorizing terms in F in estimate (3.9.26). Up to redefining M2 in (3.9.20), we may scale back to Prandtl coordinates (x, y). By integrating 268 by parts against compactly supported test functions, we have: h i || m · [@y (f + Su ) @x (g + Sv )]||H 1 . ||o( m) f + Su + g + Sv ||L2 . (3.9.33) First, we’ll start with: h i ||o( m ) · Su ||L2 = ||o( m ) u u R x + u Rx u + v u R y + u Ry v ||L2 3  ||uR , xuRx , vR , uP E Ry y, uRY x ||L1 ||o( 2 m ){ux , uy , vy , vx }||L2 . ||o( m ){u, v}||X1 . (3.9.34) Similarly, h i ||o( m) · Sv ||L2 = ||o( m) uR vx + vRx u + vR vy + vRy v ||L2 (3.9.35)  ||uR , xvRx , vR , vRy y||L1 ||o( m ){vx , ux , vy }||L2 . ||o( m ){u, v}||X1 . Next, we come to the nonlinear terms: n h i ✏ 2 + ||o( m) · u¯u ¯v¯x + v¯v¯y ||L2 ¯x + v¯uy + u n  ✏ 2 + ||o( u, v¯}||L1 ||o( m ){¯ m ){¯ ux , v¯x , uy }||L2 n ⇣ ⌘ . ✏ 2 + ||o( u, v¯}||L1 m ){¯ ||o( u, v¯}||X1 m ){¯ + ||o( m ){u, v}||X1 . (3.9.36) Finally, we have the forcing terms in f, g for which we cite Lemma 3.7.22, 1 X u,n ||o( m ){R , Rv,n }||L2  C. (3.9.37) m=1 269 Upon taking summation in m: X X X X 2 || ||H 3 (x2000)  || m ||H 3  || m ||H 3  || m F ||H 1 + ||[ , m] ||H 1 m m m m hX X ." M2 ||o( m ){R u,n , Rv,n }||L2 + ||o( m ){u, v}||X1 m m n ⇣ ⌘i + " 2 + ||o( m ){||¯ u, v¯}||L1 ||o( m ){¯u, v¯}||X1 + ||o( m ){u, v}||X1 h n ⇣ ⌘i . " M2 C + ||{u, v}||X1 + " 2 + ||¯u, v¯||L1 ||{¯u, v¯}||X1 + ||{u, v}||X1 . For the L1 component of our desired claim, we simply use the standard H 2 embedding. Corollary 3.9.11. Suppose ||¯ u, v¯||Z  1. There exists a universal constant M2 such that for any selection of N2 , N4 , n, , the following estimate holds: n ✏N2 ||u, v||Y2 . ✏N2 M2 + ✏ 2 + +N2 M2 N4 ||¯ u, v¯||2Z ⇣ n ⌘ + ✏N2 M2 + " 2 + +N2 M2 N4 ||u, v||X1 + ✏N2 ||u, v||X2 . (3.9.38) We will now bootstrap the above elliptic regularity, away from the boundary x = 1 (thereby avoiding the corners of our domain). Lemma 3.9.12 (Third-Order Elliptic Regularity). For some M3 0 perhaps large, but independent of n, and supposing ||¯ u, v¯||Z  1, we have: p h n i ||{ux , ✏vx } · ⇣3 ||H˙ 2 (x1000) . ✏ M3 C + ||u, v||X1 \Y2 + " 2 + N4 N2 ||u, v||X1 \Y2 M3 2N2 2N4 + n 2+ +✏ u, v¯||2Z . ||¯ (3.9.39) 270 Proof. Start with the system in (3.9.20), and di↵erentiate once in x: h i h i 2 x = Fx := " M2 {@Y f˜x + @x S˜u @x g˜x + @x S˜v }, @x (x, 0) = @x Y (x, 0) = 0. (3.9.40) We shall now define a new cut-o↵ function, via: 8 > > >0 for 1  x  3 , > > > 2 < (2,a) (x) = 1 for 2  x  1000, (3.9.41) > > > > > > :0 for 2000  x. Then, referring back to (3.9.22), (2) (2,a) (b) m (x, Y ) := (x) m (Y ) (3.9.42) (2) The collection { m } is meant to fill the gap between ⇣3 and ⇢3 (see the definitions in (2) (3.9.1), (3.9.2)). We will consider the unknown m x, which satisfies the system: ⇣ ⌘ 2 (2) (2) 2 (2) m x = m Fx +[ , m ] x, (3.9.43) where the commutator expression is given by (3.9.25). Then via the standard, local in x, H 3 estimate, one has: || (2) m x ||H 3 . || (2) m Fx ||H 1 + ||[ 2 , (2) m ] x ||H 1 . (3.9.44) 271 2 (2) Again, as [ , m ] is localized in x and contains three-derivatives of x, we easily obtain: ||[ 2 , (2) m ] x ||H 1 ." M3 ||o( (2) m ){u, v}||Y2 \X1 . (3.9.45) We will now evaluate the Fx term above. By selecting the exponent M3 large enough, one can rescale f˜x , g˜x to fx , gx and @x S˜u , @x S˜v to Su , Sv , which we automatically do. Then, h i (2) M3 (2) (2) || m Fx ||H 1 " || m fxy ||H 1 + || m gxx ||H 1 h i M3 (2) (2) " ||o( m )fx ||L2 + ||o( m )gx ||L2 . (3.9.46) Then, according to the definition (3.8.8) - (3.8.9), one has for the nonlinear terms: n (2) ||o( m ) · Nxu ||L2 = ✏ 2 + ||o( (2) m )[¯ uu ¯2x + v¯x uy + v¯uxy ]||L2 ¯xx + u n h  ✏ 2 + ||o( (2) u||L1 ||o( (2) m )¯ m )¯uxx ||L2 + ||o( (2) ux ||2L4 m )¯ i (2) + ||o( m )¯vx ||L4 ||o( (2) m )uy ||L4 + ||o( (2) m )¯v ||L1 ||o( (2) m )uxy ||L2 n h  ✏ 2 + ||o( (2) m )¯u||L1 ||o( (2) m )¯uxx ||L2 + ||o( (2) m )¯ux ||L2 ||o( (2) m )¯ux ||H˙ 1 1 1 1 1 (2) + ||o( m )¯vx ||L2 2 ||o( (2) m )¯vx ||H 2 (2) 2 (2) ˙ 1 ||o( m )uy ||L2 ||o( m )uy ||H 2 ˙1 i + ||o( (2) m )¯ v || L 1 ||o( (2) m )u || xy L 2 n h i . ✏ 2 + " 2N2 2N4 ||o( (2) m ){¯u, v¯}||2Z + " 2N2 2N4 ||o( (2) m ){u, v}||X1 \Y2 . (3.9.47) Above, we have used the assumption that ||¯ u, v¯||Z  1 in the term: 1 1 1 1 (2) ||o( m )¯vx ||L2 2 ||o( (2) m )¯vx ||H 2 (2) 2 (2) ˙ 1 ||o( m )uy ||L2 ||o( m )uy ||H 2 ˙1 1 1 N2 (2) (2) " ||¯ v ||Z ||o( m )uy ||L2 ||o( m )uy ||H 2 2 ˙1 1 1 N2 (2) (2) " ||o( m )uy ||L2 ||o( m )uy ||H 2 2 ˙1 272 1 1 N2 (2) N2 " ||o( m )uy ||X1 " 2 ||o( (2) m )u||Y2 . 2 (3.9.48) (2) In the final estimate, we have used that is supported on a strictly smaller region in x than the region over which we have controlled uyy (compare (3.9.23) to the uyy estimate in (3.9.17)). Similarly, n (2) v ||o( m )Nx ||L2 = ✏ 2 + ||o( (2) m )[¯ uv¯xx + u ¯x v¯x + v¯x v¯y + v¯v¯xy ]||L2 n h  ✏ 2 + ||o( (2) u||L1 ||o( (2) m )¯ vxx ||L2 + ||o( (2) m )¯ ux ||L4 ||o( m )¯ (2) m )¯vx ||L4 i (2) + ||o( m )¯vx ||L4 ||o( (2) m )¯vy ||L4 + ||o( (2) v ||L1 ||o( m )¯ (2) m )¯vxy ||L2 n h i  ✏2+ " 2N2 2N4 ||o( (2) u, v¯}||2Z . m ){¯ (3.9.49) Next, according to estimates (3.7.177), one has: X (2) u,n (2) v,n ||o( m )Rx , o( m )Rx ||L2  C. (3.9.50) m 1 Taking summation in m and scaling back to Prandtl coordinates gives the desired result upon comparing the supports of 2 and ⇢2 : X (2) (2) || ||H 3  || m ||H 3 m X n X ." M3 ||o( (2) m )u, v||Y2 \X1 + ✏2+ " 2N2 2N4 ||o( (2) u, v¯}||2Z m ){¯ m m n X X 2+ N2 N4 ||o( (2) ||o( (2) u,n (2) v,n +✏ " m ){u, v}||X1 \Y2 + m )Rx , o( m )Rx ||L2 m m h n n i ." M3 C + ||u, v||Y2 \X1 + " 2+ 2N2 2N4 u, v¯||2Z + " 2 + ||¯ N2 N4 ||u, v||X1 \Y2 . (3.9.51) 273 Corollary 3.9.13. There exists a universal constant M3 such that for any selection of N2 , N3 , N4 , n, , so long as ||u, v||Z  1, we have: h i ✏N3 ||u, v||Y3 . ✏N3 M3 C + ||u, v||X1 \Y2 + "N3 ||u, v||X3 n M3 2N2 2N4 + n + "2+ N2 N4 M3 +N3 ||u, v||X1 \Y2 + ✏N3 2+ u, v¯||2Z . (3.9.52) ||¯ We now upgrade the previous estimate to a fourth order, weighted estimate. This fourth order estimate is not included in our norm, (3.9.8). This will simply be used in order to justify rigorously one of the integrations by parts in the energy estimates (in particular, calculation (3.10.95)). This is the reason our right-hand side below in (3.9.53) need not be depicted explicitly; we only need the qualitative information that this quantity is finite. Lemma 3.9.14 (Fourth-Order Elliptic Regularity). Suppose ||¯ u, v¯||Z  1. Solutions [u, v, P ] 2 Z to the system (3.8.1) - (3.8.3), with forcing terms as in (3.8.8) - (3.8.9) satisfy the following fourth-order estimate: p ||{uxx , "vxx } · x||H˙ 2 (x 20) < 1. (3.9.53) 2 Proof. Taking two derivatives of (3.9.20), one has ( xx ) = Fxx . Define a partition of unity of {x 20}, { m }m 0 , such that 0 is supported on Y 1, and all of the m, m 1 are localized near the boundary Y = 0 in the region Y 2 [0, 2] and near x = m. Then: 2 2 ( m x xx ) =[ , m x] xx +x m Fxx , (3.9.54) 2 2 ( 0 x xx ) =[ , 0 x] xx +x 0 Fxx , (3.9.55) where we refer the reader to the commutator expression in (3.9.25). For equation (3.9.55), one uses the standard, interior H˙ 3 estimate for the Bi-Laplacian: || 0 x xx ||H ˙3 . ||[ 2 , 0 x] xx ||H 1 + ||x 0 Fxx ||H 1 . (3.9.56) 274 For equation (3.9.54), one uses the boundary H 3 estimate which gives: || m x xx ||H 3 . ||[ 2 , m x] xx ||H 1 + ||x m Fxx ||H 1 . (3.9.57) Let us write the expression for Fxx that we will read from, referring to (3.9.20) (we will rename the power of ") h i Fxx = " M4 @xxY f˜ + @xxY S˜u @xxx g˜ @xxx S˜v . (3.9.58) 1 As usual, by sacrificing powers of ", it suffices to evaluate the H norm of the above expression in Prandtl coordinates. We will start with Su : ||{ m, 0 }x@xxY S˜u ||H 1 . ||{ m, ˜ 0 }x@xx Su ||L2 + ||{o(@y m , @y ˜ 0 )x@xx Su }||L2 . (3.9.59) Similarly for Sv , one obtains: ||o( m, 0 )@xxx xSv ||H 1  ||o( m, 0 )@xx Sv ||L2 + ||o(@x m , @x 0 )@xx xSv ||L2 + ||o( m, 0 )x@xx Sv ||L2 . (3.9.60) Computing two derivatives of Su , Sv , one obtains: @xx Su = uRxxx u + 2uRxx ux + uRx uxx + uRxx ux + 2uRx uxx + uR uxxx + uRyxx v + 2uRyx vx + uRy vxx + vRxx uy + 2vRx uxy + vR uxxy , (3.9.61) @xx Sv = uRxx vx + 2uRx vxx + uR vxxx + vRxxx u + 2vRxx ux + vRx uxx + vRxx vy + 2vRx vxy + vR vxxy + vRxxy v + 2vRxy vx + vRy vxx . (3.9.62) 275 From here, given 0, m are supported on x 20, it is easy to see that: h i ||o( m, 0) x@xx Su , x@xx Sv ||L2 (x 20) . ||o( m, 0 ){u, v}||Z . (3.9.63) Next, turning to the expressions in f, g: n u,n fxx = " 2 Rxx +u ¯u¯xxx + 3¯ ux u ¯xx + v¯xx uy + 2¯ vx uxy + v¯uxxy , (3.9.64) n v,n gxx = " 2 Rxx +u ¯xx v¯x + u ¯v¯xxx + 2¯ ux v¯xx + v¯xx v¯y + v¯v¯xxy + 2¯ vx v¯xy . (3.9.65) From here, using (3.7.177), and the definition the norm Z in (3.9.8), one observes: h i ||o( m, 0) x@xx f, x@xx g ||L2 (x 20) . ||o( m, 2 0 ){u, v}||Z + ||o( m, u, v¯}||2Z . 0 ){¯ (3.9.66) Summarizing this: X X X ||x m Fxx ||H 1 . ||o( m ){u, v}||Z + ||o( 2 m ){u, v}||Z m 0 m 0 m 0 X + ||o( u, v¯}||Z m ){¯ < 1. (3.9.67) m 0 We now move to the commutator terms from (3.9.56) - (3.9.57), which we write out, denoting by generically either m or 0: 2 [ , x ]vx = @xyy (x )vxx + @xxy (x )vxy + @x (x )@xyy vx + x@yy vxxx + @y (x )@xxy vx + @xx (x )vxyy + @xy (x )vxxy 4 X 4 X + @xk (x )@x4 k vx + x@yk @y4 k vx . (3.9.68) k=1 k=1 276 Examining the commutator terms, one observes that the worst term is when all deriva- tives fall on the cut-o↵, and none on either x or vx . Such a term arises, for instance, when k = 4 in the final summation above in (3.9.68). In this case, one uses that the partition of unity is selected such that @y 0, m is a bounded region in y: ||@y4 { m, 0} · xvx ||H 1  ||@y4 { m, 0} · xvx ||L2 . ||o( m ), o(@ 0 )xvxy ||L2 . (3.9.69) It is straightforward to check that all terms in the commutator can be estimated either in this manner or directly using the definitions of Z in (3.9.8). Thus, taking summation over the partition of unity again gives: X ||[ 2 , m x] xx ||H 1 . ||u, v||Z < 1. (3.9.70) m 0 Combining (3.9.56), (3.9.57), (3.9.67), and (3.9.70), the lemma is proven. 3.9.2 Embedding Theorems for the Space Z The next task is to pinpoint the interplay between the uniform quantities in the norm Z and the norms Yi . Let us start with: Lemma 3.9.15. For > 0 arbitrarily small, h p p i sup || " x 1 ||2L2y + ||ux ||2L2y + || "vx ||2L2y . C( )||u, v||2X1 , (3.9.71) x 1 h 1 i p sup ||uy x 2 ||2L2y + ||ux x||2L2y + sup || ✏vx x||2L2y . ||u, v||2X1 \Y2 , (3.9.72) x 1 x 20 h 3 p i sup ||uxy x 2 ||2L2y + ||{vxy , "vxx }x2 ||2L2y . ||u, v||2Y2 \Y3 . (3.9.73) x 20 277 The constant C( ) " 1 as # 0. Finally, for [u, v] 2 Z, we have the following property: sup ||{vxxx , uxxy }x||L2y < 1, (3.9.74) x 20 Remark. The estimate (3.9.74) is required in order to rigorously justify one integration by parts in our energy estimates, in particular calculation (3.10.100), and is not required as part of the norm Z, which is why we do not characterize the right-hand side. The most important parts of this lemma are the first three estimates, (3.9.71) - (3.9.73). Proof. First, take a di↵erentiation of: Z Z Z @x u2 x 2 dy = 2 uux x 2 dy 2 u2 x 1 2 dy, (3.9.75) which upon an integration in x, and recalling u(1, y) = 0, yields: Z Z x1 Z Z x1 Z u2 x1 2 dy = 2 uux x 2 dx dy 2 u2 x 1 2 dx dy. (3.9.76) 1 1 Taking absolute values, applying Holder’s inequality, and taking the supremum in x1 : Z 1 1 1 1 sup u2 x 2 . ||ux 2 ||L2 ||ux x 2 ||L2 + ||ux 2 ||2L2 . ||ux x 2 ||2L2 . ||u, v||2X1 . (3.9.77) x 1 Here the factor of > 0 is required to avert the critical Hardy inequality, which occurs 1 with weight x 2 in L2 . The stream function estimate follows similarly: Z Z Z 2 2 2 2 3 @x x =2 vx (2 ) x , 278 and so integration gives: Z 3 1 1 sup 2 x 2  || vx 2 ||L1 + || x 2 2 ||2L2 . ||vx 2 2 ||2L2 . ||vx x 2 2 ||2L2 x 1 . ||u, v||2X1 . (3.9.78) The estimate for v in (3.9.71) works in a similar fashion. We will now move to first-order estimates in (3.9.72). The uy estimate follows easily after a di↵erentiation: Z Z Z @x u2y x = 2 uy uxy x + u2y . (3.9.79) Taking an integration in x, and recalling that uy (1, y) = 0, then gives: Z sup u2y x . ||uy ||L2 ||uxy x||L2 + ||uy ||2L2 . ||u, v||2X1 \Y2 . (3.9.80) x 1 Next, Z Z Z @x u2x x2 = 2 ux uxx x2 + 2 u2x x, (3.9.81) so taking an x-integration and using that ux (1, y) = 0, we have: Z 1 3 1 sup u2x x2 . ||ux x 2 ||L2 ||uxx x 2 ||L2 + ||ux x 2 ||2L2 . ||u||2X1 \Y2 . x 1 Let us now introduce new cut-o↵ functions, for k = 2, 3: 8 > > <0 for 1  x  3 + 6(k 2), ⌘k (x) = (3.9.82) > > :1 for x 6 + 6(k 2). 279 R Then ⌘2 , ⌘3 are “in-between” ⇣3 and ⇢2 , ⇢3 . Consider now the quantity ✏vx2 x2 ⌘2 (x): Z Z Z Z @x ✏vx2 x2 ⌘2 (x) = 2 ✏vx2 x⌘2 (x) + 2 "vx vxx x2 ⌘2 + "vx2 x2 ⌘20 (x). (3.9.83) As ⌘2 vanishes on [1, 3], we can take the integration up from x = 1: Z Z p 1 p 3 sup "vx2 x2  sup ✏vx2 x2 ⌘2 (x) . || "vx x 2 ||2L2 + || "vxx x 2 ||2L2 , (3.9.84) x 20 x 1 where we have used that |⌘20 (x)x2 | . 1. We now move to the second-order estimates in (3.9.73), starting with: Z Z Z Z @x u2xy x3 ⌘3 = 2 uxy uxxy x3 ⌘3 + u2xy 3x2 ⌘3 + u2xy x3 ⌘30 . (3.9.85) Using that ⌘3 uxy (1, y) = 0, we have: Z sup u2xy x3 ⌘3 . ||uxy x||2L2 + ||uxy x||L2 ||uxxy x2 ⌘3 ||L2 . ||u, v||2Y2 \Y3 . (3.9.86) x 1 Above, we have used that ⌘3  ⇣3 . The final calculation is: Z Z Z Z @x 2 vxy x4 ⌘3 = C vxy vxxy x4 ⌘3 + C 2 vxy x3 ⌘3 + C 2 vxy x2 ⌘30 . (3.9.87) Integrating up from x = 1, Z 3 5 sup 2 vxy x4 ⌘3 . ||vxy x 2 ||2L2 + ||vxxy x 2 ⌘3 ||2L2 . ||u, v||2Y2 \Y3 . (3.9.88) x 1 p It is clear that "vxx works in an identical manner to (3.9.87), and also in an identical 280 manner, one obtains (3.9.74) by pairing with (3.9.53). We will now record the following about the x ! 1 behavior of elements in (X1 \ X2 \ X3 )(⌦): Lemma 3.9.16. Suppose [u, v] 2 X1 \ X2 \ X3 (⌦). Then the following boundary conditions are automatically enforced: [u, v]|x=1 = [u, v]|y=0 = lim [u, v] = lim [u, v] = 0 (3.9.89) x!1 y!1 Proof. All follow as in Lemma 3.9.6 aside from the condition at x ! 1. For this, we use: 1 1 1 1 2 ||ux 4 ||L1 y  ||ux ||L2 2 ||uy x 2 ||L2 2 < 1, (3.9.90) y y 1 1 1 2 ||vx 2 ||L1 y  ||vx ||L2 2 ||vy x||L2 2 < 1, (3.9.91) y y according to the estimates in (3.9.71). Thus, we can conclude that [u, v] ! 0 as x ! 1. The estimates (3.9.90) - (3.9.91) yield uniform decay of [u, v] as x ! 1. However, the factor of x is too weak to close our nonlinear analysis (see estimate (3.11.8)). This in turn is caused by Hardy’s inequality being critical (see (3.9.77)). In the next lemma, we use higher-order decay estimates to avoid this criticality: Lemma 3.9.17 (Uniform Embeddings). For [u, v] 2 X1 \ X2 \ X3 (⌦), we have: 1 p 1 sup ||ux 4 , "vx 2 ||L1 y . ||u, v||X1 \Y2 \Y3 , (3.9.92) x 1 5 p 3 sup ||ux x 4 , "vx x 2 ||L1 y . ||u, v||X1 \Y2 \Y3 . (3.9.93) x 20 281 Proof. We will first turn to the first-order estimates in (3.9.93). These follow from the evolution estimates in the previous lemma via standard Sobolev interpolation: 5 1 3 1 ||ux x 4 ||L1 y  ||ux x||L2 2 ||uxy x 2 ||L2 2 . (3.9.94) y y Similarly, 3 1 1 ||vx x 2 ||L1 y  ||vx x||L2 2 ||vxy x2 ||L2 2 . (3.9.95) y y The result now follows after taking the supremum in x 20 and appealing to the evolution estimates above. Next, we address (3.9.92). For x  20, we may simply appeal to the uniform estimates in estimate (3.9.17). For v, the result follows immediately from: Z 1 Z 1 h i 3 3 3 1 v(x, y) = vx (x0 , y)dx0  ||vx x 2 ||L1 y (x0 ) 2 dx0 = sup ||vx x 2 ||L1 y x 2 . x x x 20 (3.9.96) We have used the qualitative boundary condition that v ! 0 as x ! 1, which is available due to Lemma 3.9.16. For the u estimate follows in the same manner: Z 1 h i 5 5 5 1 u(x, y)  ||ux x 4 ||L1 y (x0 ) 4 dx0 . sup ||ux x 4 ||L1 y x 4 . (3.9.97) x x 20 Now taking the sup in x, y of both of the above inequalities yields the desired result. We now turn to controlling the top-order uniform-type norm: Lemma 3.9.18 (Top Order Uniform Embedding). For [u, v] 2 X1 \ Y2 \ Y3 , one has the 282 following mixed-norm estimate: Z 1 Z 1 p 7 x4 || "vxx ||2L1 y dx + x 2 ||uxx ||2L1 y dx . ||u, v||2X1 \Y2 \Y3 . (3.9.98) 20 20 Proof. For x 20, we start with: p 3p 1 5p 1 x2 || "vxx ||L1 y  ||x 2 "vxx ||L2 2 ||x 2 "vxxy ||L2 2 . y y Taking square on both sides: p 3p 5p x4 || "vx ||2L1 y  ||x 2 "vxx ||2L2y + ||x 2 "vxxy ||2L2y , (3.9.99) so integrating Z 1 p x4 || "vxx ||2L1 y dx  ||u, v||2X1 \Y2 \Y3 . (3.9.100) 20 For the uxx estimate, we have: 7 3 1 1 x 4 ||uxx ||L1 y  ||x 2 uxx ||L2 2 ||x2 uxxy ||L2 2 , (3.9.101) y y Squaring both sides and taking an x-integration gives: Z 1 Z 1 Z 1 7 3 x 2 ||uxx ||2L1 y  ||x 2 uxx ||2L2y + ||x2 uxxy ||2L2y 20 20 20 . ||u, v||2X1 \Y2 \Y3 . (3.9.102) 283 We shall make the following selections, given arbitrary constants M2 , M3 0: N2 is selected such that N2 M2 = 100; (3.9.103) N3 is selected such that N3 M3 = 2N2 ; (3.9.104) Nk = N3 for k = 4, 5, 6, 7, (3.9.105) n sufficiently large relative to Ni , Mi . (3.9.106) We now consolidate the above series of lemmas, coupled with (3.9.38) and (3.9.52). Corollary 3.9.19. With N4 , ..., N7 as in (3.9.105), we have: ||u, v||U . "max{N2 ,N3 } ||u, v||X1 \Y2 \Y3 . (3.9.107) Theorem 3.9.20 (Z embedding). Suppose ||¯ u, v¯||Z  1. For appropriate choices of N2 , ...N7 , based only on universal constants, there exists a universal constant !(Ni ) such that: n ||u, v||Z . ✏100 + ||u, v||X1 \X2 \X3 + " 2 + !(Ni ) u, v¯||2Z . ||¯ (3.9.108) Proof. According to (3.9.107), it suffices to treat the Y2 , Y3 terms in || · ||Z . For this we simply use the selections in (3.9.103) - (3.9.106) to rewrite (3.9.52) n "N3 ||u, v||Y3 . "100 + "100 ||u, v||X1 \X3 + "N2 ||u, v||Y2 + " 2 + !(Ni ) u, v¯||2Z . ||¯ (3.9.109) The condition (3.9.106) allows us to control, in estimate (3.9.52): n "2+ N2 N4 M3 +N3 ||u, v||X1 \Y2  "N2 +100 ||u, v||X1 \Y2 , (3.9.110) 284 n for instance, so long as: 2 + N2 N4 M3 + N3 > N2 + 100. There are many such criteria which arise over the course of our analysis, and so we retain the generality as stated in (3.9.106). Next, rewriting (3.9.38) in a similar fashion, one has: n "N2 ||u, v||Y2 . "100 + "100 ||u, v||X1 \X2 + " 2 + !(Ni ) u, v¯||2Z . ||¯ (3.9.111) Summing these two estimates yields the desired result. 3.9.3 Function Space, Z(⌦N ) We will have occasion to consider, Z(⌦N ), where ⌦N is defined in (3.8.10), and N is some large but finite, fixed number. Due to the boundedness in the y-direction of ⌦N : Lemma 3.9.21. For [u, v] 2 Z(⌦N ), p || ||L2 (⌦N ) + ||{ ✏v, vy }x||L2 (⌦N )  C(N )||u, v||Z(⌦N ) , (3.9.112) where C(N ) depends poorly on large N . Proof. This follows from the Poincare inequality, as both = v = vy = 0 on y = 0: ||vx||L2  C(N )||vy x||L2  C(N )||vyy x||L2 = ||uxy x||L2y . C(N )||u, v||Z(⌦N ) . (3.9.113) Similarly, || ||L2  C(N )|| y ||L2 = ||u||L2  C(N )||uy ||L2  C(N )||u||Z(⌦N ) . (3.9.114) 285 Lemma 3.9.22. For [u, v] 2 Z(⌦N ), h 3 1 1 i sup x 2 ||v, vy ||L2y + x 2 || ||L2y + x 2 ||u||L2y + x2 ||vx ||L2y  C(N )||u, v||Z(⌦N ) , (3.9.115) x 20 where C(N ) depends poorly on large N . Proof. By applying the Poincare inequality twice in the y-direction, 3 3 3 ||vx 2 ||L2y  C(N )||vy x 2 ||L2y  C(N )||uxy x 2 ||L2y . ||u, v||Z . (3.9.116) The final inequality following from (3.9.73). Similarly, ||vx x2 ||L2y  C(N )||vxy x2 ||L2y . ||u, v||Z . (3.9.117) Finally, by repeating all of the calculations which culminated in Theorem 3.9.20 on the domain ⌦N , the following estimate holds: Theorem 3.9.23 (Z(⌦N ) embedding). Fix any N > 0, large. Suppose ||¯ u, v¯||Z(⌦N )  1. For appropriate choices of N2 , ...N7 , based only on universal constants, there exists a universal constant !(Ni ) such that we have: n ||u, v||Z(⌦N ) . ✏100 + ||u, v||X1 \X2 \X3 (⌦N ) + " 2 + !(Ni ) u, v¯||2Z(⌦N ) . ||¯ (3.9.118) The constants in the above estimate are independent of N . 286 3.10 Navier-Stokes Remainders: Energy Estimates In this section, we shall obtain a family of energy and positivity estimates for the system in (3.8.1) - (3.8.3). As mentioned in the prior section, we seek a solution [u, v] 2 Z(⌦), where Z(⌦) is defined precisely in equation (3.9.4). Our point of view for this section, then, is to obtain a-priori estimates under the assumption that [u, v] 2 Z. Such a solution necessarily encodes decay rates of the solutions and their derivatives (see, for instance, (3.9.71) - (3.9.73), (3.9.92), and (3.9.93)). We shall give a-priori estimates on the domain ⌦N , as defined in (3.8.10), which are independent of N , allowing us to take N ! 1. For this purpose, we take the boundary conditions shown in (3.8.11). RR Remark (Notational Convention). For Section 3.10, all integrations and norms, || · ||, without further specification of domains are over ⌦N . Going to the vorticity formulation of (3.8.1) - (3.8.3): ⇣ ⌘ ⇣ Py ⌘ @y ✏u + Px + S u ✏@x ✏v + + Sv = ✏ ⇣ ⌘ ⇣ ⌘ @y ✏ u + Su ✏@x ✏ v + Sv = f y ✏gx . (3.10.1) Define the stream function through Z y (x, y) = u(x, y 0 )dy 0 , x = v, y = u, (3.10.2) 0 and note via the boundary conditions (3.8.4) and (3.8.11), |y=0,y=N = |x=1 = x |x=1 = y |y=0,y=N = 0. (3.10.3) 287 To see that |y=N = 0, we can write: Z N Z N @x = ux (x, y 0 )dy 0 = vy (x, y 0 )dy 0 = v(x, N ) v(x, 0) = 0. (3.10.4) 0 0 Next, the boundary condition (1, y) = 0 enables us to evaluate the corner: (1, N ) = 0. Thus, coupling these two facts yields (x, N ) = 0. Next, we record the observation: Z x Z x (x, y) = (x, y) (1, y) = @x (x0 , y)dx0 = v(x0 , y)dx0 , (3.10.5) 1 1 and so taking absolute values, and supremum in y yields: Z x Z x 1 1 || (x0 )||L1 y  ||v(x0 )||L1 y dx0  (x0 ) 2 dx0 . x 2 . (3.10.6) 1 1 We also will have occassional need for the auxiliary domain: ⌦N M := {0 < x < M, 0 < y < N }. (3.10.7) 3.10.1 Energy Estimates We now give the energy estimates on [u, v]. Let us first introduce the notation: Z Z Z Z W1,E = | f · u| + "|g||v|, (3.10.8) Z Z Z Z W1,P = |f ||vy |x + "|g||vx |x, (3.10.9) W1 = W1,E + W1,P . (3.10.10) Proposition 3.10.1. Let ✏ << and , " be sufficiently small relative to universal constants. Let [u, v] 2 Z(⌦N ) be solutions to the system (3.8.1) - (3.8.3) on the domain ⌦N . Then 288 these solutions satisfy the a-priori energy estimate: p p 1 1 || ✏ux , uy ||2L2  O( )|| ✏vx x 4 , vy x 2 ||2L2 + W1,E . (3.10.11) The constant in the above estimate is independent of N . Remark. Note carefully in (3.10.8) that we do not bring the absolute value inside of the RR integration for the f u term, which is important to treat term (3.11.5). For the remaining terms above in W1 , we place the absolute values inside the integration for convenience (in terms of comparison with other terms that arise, see for instance (3.10.160) - (3.10.161)). Proof. The vorticity equation (3.10.1) is multiplied by the stream function and we proceed to integrate by parts. According to the definition of the norm Z in (3.9.8) and the estimate (3.9.112), all integrands appearing in this estimate will be L1 (⌦N ), and so all applications of Fubini are justified. First, we will treat the highest order terms: Z Z ⇣ ⌘ Z Z @y ✏u + ✏@x ✏v = ✏u y ✏ ✏v x (3.10.12) Z Z Z Z = ✏ uu ✏ ✏ vv (3.10.13) Z Z Z Z hZ i = |r✏ u|2 + ✏ |r✏ v|2 lim "ux u + "2 vx v (3.10.14) M !1 x=M Z Z Z Z = |r✏ u|2 + ✏ |r✏ v|2 . (3.10.15) For the limiting integrals over x = M above, we have used the bounds from (3.9.112) to conclude: Z 1 3 2 M !1 | "uux |  "||ux 2 ||L2y ||ux x 2 ||L2y M ! 0, (3.10.16) x=M Z 3 7 M !1 | "2 vvx |  "2 ||vx 2 ||L2y ||vx x2 ||L2y M 2 ! 0. (3.10.17) x=M 289 Let us justify rigorously the integration by parts found in line (3.10.12). To isolate the corners, define Cr1,2 to be solid balls of radius r centered at the two corners, (1, 0), and (1, N ). Then, Z Z ⇣ ⌘ Z Z ⇣ ⌘ 2 Z Z X ⇣ ⌘ @y ✏u = @y ✏u + @y ✏u (3.10.18) ⌦N [2i=1 Cri i=1 Cri Z Z Z Z Z ⇣ ⌘ = ✏ uu ✏ u dS + @y ✏u . (3.10.19) ⌦N [2i=1 Cri @Cri Cri First, as we know ✏ uu 2 L1 (⌦N ), we have: Z Z Z Z lim ✏ uu = ✏ uu. (3.10.20) r!0 ⌦N [2i=1 Cri Next, we appeal to the classical expansion in the vicinity of a corner point in (BR80), Page 57, equation (5.5), from which it follows that: |u, v| . r, | | . r, |r2 u, r2 v| . r 1 in Cri . (3.10.21) Then the boundary integral from (3.10.19) can be controlled via: Z Z Z 1 r!0 ✏u  ✏u  rr dS  r ! 0. (3.10.22) @Cri @Cri @Cri Finally, we arrive at the interior term from the corners in (3.10.19). For this, the expan- sion in (BR80), Page 57, equation (5.5) implies that: |r3 [u, v]| + |r3 | . r 2 +u ˜ 2 L2 . ˜, where u (3.10.23) 290 Using this gives: Z Z ⇣ ⌘ Z Z Z Z @y ✏u dxdy . r 2 r dxdy + |˜ u|rdxdy Cri Cri Cri Z Z 2 r!0  r rrdrd! + ||˜ u||L2 (Cri ) ||r||L2 (Cri ) ! 0. (3.10.24) Cri Summarizing, we have shown the validity of the integration by parts Z Z Z Z @y ( ✏ u) = ✏ uu. (3.10.25) This calculation works generically at the corners (it was not specific to the particular derivatives involved, just the order of them), and so we will avoid repeating it each time. We now turn to the v-terms in (3.10.12), for which we write: Z Z Z Z @x ✏v = lim @x ✏v ⌦N Cri M !1 ⌦N Cri M hZ Z Z Z i = lim ✏ vv + ✏v dy ✏v dy M !1 ⌦N Cri x=M @Cri M Z Z Z Z = ✏ vv + lim ✏v dy ✏v dy. (3.10.26) ⌦N Cri M !1 x=M @Cri For the x = M boundary term in (3.10.26), we have: Z 1 3 2 M !1 ✏v  || ✏ v||L2y || ||L2y . || x 2 ||L2y || " vx 2 ||L2y M ! 0, (3.10.27) x=M according to (3.9.71), (3.9.73), and (3.9.115). Subsequently, sending r ! 0 as above gives the desired identity: Z Z Z Z @x ✏v = ✏ vv. (3.10.28) 291 Next, we come to the profile terms, Su , from the equation (3.8.1). We refer the reader to the definition of Su , which is in (3.8.6). Z Z ⇣ ⌘ Z Z Z Z @ y Su = Su y = Su u Z Z h i = uR ux + uRx u + vR uy + uRy v u. (3.10.29) RR The first three of these terms in Su u are handled through an integration by parts: Z Z Z Z Z Z Z 2 2 uR 2 uR ux u + uRx u + vR uuy = uRx u + lim u = vRy u2 . M !1 x=M 2 (3.10.30) 1 Above, we have used the estimate for ||ux 2 ||L2y in (3.9.115). The term on the right-hand side of (3.10.30) is handled via: Z Z Z Z ⇣ ⌘ p vRy u2 = P vRy + E ✏vRY u2 P u p E 3 u  ||vRy y 2 ||L1 || ||2L2 + ✏||vRY x 2 ||L1 || 3 ||2L2 y x4 1  O( )||uy ||2L2 + O( )||ux x 4 ||2L2 . (3.10.31) In (3.10.31), we have first used the profile estimates in (3.3.10) and (3.3.20), and subse- quently the Hardy inequality which is available (for exponents of x which are not equal to 1 RR 2 ) as u(1, y) = u(x, 0) = 0. The large convective term in Su u, (3.10.29), is given by: Z Z Z Z n p uRy uv = {uP,n Ry 1 + ✏ 2 unp + ✏uE RY }uv. (3.10.32) 292 First, by estimate (3.3.14), with j = 1, m = 1, we have: Z Z 1 1 u vx 2 1 uP,n Ry 1 uv  ||y 2 x 2 uP,n Ry 1 ||L1 || ||L2 || ||L2  O( )||uy ||L2 ||vy x 2 ||L2 . (3.10.33) y y Second, according to (3.3.16) with j = 1: Z Z 1 n n 1 vx 2 ✏ 2 unpy uv  ✏ 2 ||unpy yx 2 n ||L1 ||ux 1+ n ||L2 || ||L2 y n 1 n 1 1  C(n)✏ 2 ||ux x n ||L2 ||vy x 2 ||L2 . C(n)✏ 2 ||ux x 2 ||L2 ||vy x 2 ||L2 . (3.10.34) Third, according to (3.3.18): Z Z p 3 p v u ✏uE E RY uv  ||uRY x ||L1 || 2 ||L2 || ✏ 3 ||L2 3 x x4 4 p 1 p 1  ✏||ux x 4 ||L2 || ✏vx x 4 ||L2 . (3.10.35) In (3.10.33), we have used the Hardy inequality in the y direction: v 1 v 1 1 || x 2 ||L2 = |||| x 2 ||L2y ||L2x  ||vy x 2 ||L2 , (3.10.36) y y which is available as v|y=0 = 0. We have also used the Hardy inequality in x direction, which is available as v|x=1 = 0. Rigorously, turning to (3.10.34): u u u || ||L2 = |||| 1 ||L2x ||L2y  |||| ||L2x ||L2y . ||||ux (x 1) ||L2x ||L2y (3.10.37) x1 x (x 1)1 . ||||ux x ||L2x ||L2y = ||ux x ||L2 . (3.10.38) 293 Summarizing, Z Z p 1 1 Su u  O( )||uy ||2L2 + O( )|| ✏vx x 4 , vy x 2 ||2L2 . (3.10.39) 1 The important mechanism in controlling (3.10.33) is the ability to trade a factor of y 2 x 2 which is absorbed by the Prandtl profiles, uP,n Ry 1 , according to (3.3.16) with j = 1, m = 2. 1 This creates two y derivatives, uy and vy x , both of which are order 1. The next step is to 2 control the profile terms Sv : Z Z Z Z Z Z Z ✏ @ x Sv =✏ Sv v " lim Sv =✏ Sv v. (3.10.40) M !1 x=M By inspecting (3.9.115), one sees easily that the above limit vanishes. We now treat the interior terms from (3.10.40), which we expand for convenience: Z Z Z Z ⇣ ⌘ Sv ✏v = uR vx + vRx u + vR vy + vRy v ✏v. (3.10.41) The first, third, and fourth terms above in (3.10.41) are given via the following calcula- tions: Z Z Z Z Z " ✏uR vx v + ✏vR vvy + ✏vRy v 2 = ✏ vRy v 2 + lim uR v 2 M !1 x=M 2 Z Z =✏ vRy v 2 (3.10.42) P p 3 p 1  ✏||vRy y 2 ||L1 ||vy ||2L2 + E ✏||vRY x 2 ||L1 || ✏vx x 4 ||2L2 (3.10.43) p p 1 . "||vy ||2L2 + "|| "vx x 4 ||2L2 . (3.10.44) 3 The M limit vanishes by the estimate for ||vx 2 ||L2y in (3.9.115). We have also used 294 estimates (3.3.10) and (3.3.17) for the profiles. Next, Z Z p 3 1 p 1 ✏vRx uv  ✏||x 2 vRx ||L1 ||ux x 4 ||L2 || ✏vx x 4 ||L2 . (3.10.45) Again, we have used (3.3.8) and (3.3.17) for the profiles. Thus, Z Z p p 1 1 Sv ✏v . ✏|| "vx x 4 , vy x 4 ||2L2 . (3.10.46) On the right-hand side of equation (3.10.1), we have: Z Z ⇣ ⌘ Z Z Z Z Z Z fy ✏gx = f y + ✏g x = f u + ✏gv. (3.10.47) First, we will note that each term in the above integration by parts is in L1 (⌦N ), which follows from the definitions in (3.8.5), and a consultation with the definition of Z in (3.9.8): n n ⇣ ⌘ fy = ✏ 2 Ryu,n + ✏ 2 + uy ux + uuxy + vy uy + vuyy , (3.10.48) n n ⇣ ⌘ gx = ✏ 2 Rxv,n + ✏ 2 + ux vx + uvxx + vx vy + vvxy . (3.10.49) It remains to justify the boundary terms resulting from the x-integration by parts, at x ! 1, in (3.10.47). This, however, follows just as in (3.10.40) by inspecting the decay rates in (3.9.115). A comparison with (3.10.8) then gives the desired result. Combining all of the previous estimates proves (3.10.11). 3.10.2 Positivity Estimate We now give the following Positivity estimate: 295 Proposition 3.10.2. Let ✏ << and , " be sufficiently small relative to universal constants. Then [u, v] 2 Z(⌦N ) solutions to the system (3.8.1) - (3.8.3) satisfy the following estimate: Z Z ⇣p ⌘ 1 ✏2 vx2 dy + lim u2y x dy + || ✏vx , vy x 2 ||2L2 . ||uy ||2L2 + W1 . (3.10.50) x=1 M !1 x=M Remark (Selection of Multiplier). There is a distinction between the Positivity estimate from (GN17), Page 31, and the present case. In the case of (GN17), the profiles, uR , were vy not assumed small, and so they required the normalized multiplier uR " uvRx . In our case as the profiles are assumed size , we need not normalize by a factor of uR . On the other hand, we need to capture precise behavior at x ! 1, which is the reason our multiplier is (vy "vx ) · x, or equivalently in the vorticity formulation, v · x. Proof. We apply the multiplier x x = xv to the equation (3.10.1), and we will subsequently take the following integration: Z Z lim Equation (3.10.1) · xv dy dx. (3.10.51) M !1 ⌦N M The purpose of specifying the order of integration is for the terms in (3.10.54), which are not automatically in L1 (⌦N ) prior to integrating by parts in y. All other terms are in L1 (⌦N ) according to the norm Z, (3.9.8), and the estimate (3.9.112). Thus, with the exception of the term in (3.10.54), the limiting procedure above can (and will) be omitted, and Fubini can be justified. Second Order Terms First, we will treat the highest order terms. Let us begin with: Z Z Z Z @y ✏ uvx = (✏uxx + uyy )vy x. (3.10.52) 296 We will note that the uxx term on the right-hand side above is in L1 (⌦N ), away from the corners. This follows from the definition of Z, and (3.9.112). Thus, the forthcoming applications of Fubini are justified: Z Z Z Z Z Z ⇣ ⌘ ✏ ✏uxx vy x = ✏vxy vy x = @x vy2 x 2 Z Z Z Z Z ✏ ✏ ✏ = vy2 lim 2 vy x = vy2 . (3.10.53) 2 M !1 2 x=M 2 The above limit vanishes according to (3.9.115). For the uyy term in (3.10.52), we can integrate by parts again in y, due to (3.10.51), to obtain: Z Z Z Z uyy vy x dy dx = uy vyy x dy dx. (3.10.54) ⌦N M ⌦N M Now, the right-hand side of (3.10.54) is in L1 (⌦N ) according to our norm Z. Interchang- ing the order of integration: Z Z Z Z ⇣ u2 ⌘ Z Z Z y u2y u2y uyy vy x dy dx = @x x dx dy = dx dy + x. ⌦N M ⌦N M 2 ⌦N M 2 x=M 2 (3.10.55) As the solid integrals on the right-hand side of (3.10.55) is known to be in L1 (⌦N ), we can pass to the limit M ! 1 (and also drop the notation dx dy when the order no longer matters): Z Z Z Z Z u2y lim @y uyy · vx dy dx = + lim u2y x. (3.10.56) M !1 ⌦N M 2 M !1 x=M The limit above appears with a good sign, and therefore contributes to the left-hand side of the desired estimate in (3.10.50). We have omitted the delicate limiting process near the corners in this calculation, as this process is identical to that of (3.10.19). We shall now 297 examine the term: Z Z Z Z ⇣ ⌘ ✏@x ✏ vvx dy dx = ✏ ✏vxxx + vxyy vx dy dx (3.10.57) A direct computation then gives: Z Z Z Z Z ✏2 vxxx vx = ✏2 vxx @x (vx) + ✏2 lim vxx vxdy (3.10.58) M !1 x=M Z Z = ✏2 vxx @x (vx) (3.10.59) The x = M term in (4.10.19) is controlled by using (3.9.73) and (3.9.115): Z 3 3 3 vxx vx  ||vxx x 2 ||L2y ||vx 2 ||L2y M ! 0. (3.10.60) x=M Again, we omit displaying the limiting process which handles the corners of the domain, because this is identical to (3.10.19). We now turn to the next integration by parts for the interior term in (4.10.19). The first observation is that both vxx vx x and vxx v are in L1 (⌦N ) by inspection of the norm Z. Therefore, we are justified in the integration by parts: Z Z Z Z ✏2 vxx vx x ✏2 vxx v Z Z Z h ✏2 Z Z i 3✏2 2 ✏2 ✏2 = vx + v2 lim vx2 xdy + vvx dy 2 2 x=1 x M !1 2 x=M 2 x=M Z Z Z 3✏2 2 ✏2 = vx + v2 , (3.10.61) 2 2 x=1 x where we have appealed to estimates (3.9.115) to show the limits above vanish. We must now treat the vxyy term from (3.10.57): Z Z Z Z Z Z ⇣ v2 ⌘ y ✏vxyy vx = ✏ vyx vy x = ✏ @x x 2 298 Z Z Z Z Z ✏ ✏ ✏ = vy2 lim vy2 x = vy2 . (3.10.62) 2 M !1 2 x=M 2 Again, we appeal to (3.9.115) to show the limit vanishes. Summarizing, then: Z Z Z Z Z Z Z 3✏2 ✏ 2 ✏2 ✏@x ✏ vx = vx2 + v + vx2 . (3.10.63) 2 2 y 2 x=1 Profile Terms, Su : Next, we treat the Su profile terms, for which we refer the reader to the expressions in (3.8.6). Integrating by parts in y: Z Z Z Z Z Z Z Z @y Su vx = Su v y x = uR ux v y x + R3 = uR vy2 x + R3 . (3.10.64) We give estimates on R3 , starting with: Z Z Z Z uRx uxvy = {uP E Rx + uRx }uxvy 1 1 3 1  ||yx 2 uP 2 E Rx ||L1 ||uy ||L2 ||vy x ||L2 + ||uRx x ||L1 ||ux ||L2 ||vy x ||L2 2 2 p 1  O( )||uy ||2L2 + "||vy x 2 ||2L2 . (3.10.65) Above, we have used estimate (3.3.12) with m = 1, and (3.3.18). Next, by using (3.3.10), we have: Z Z 1 1 1 vR uy vy x  ||vR x 2 ||L1 ||uy ||L2 ||vy x 2 ||L2  O( )||uy ||L2 ||vy x 2 ||L2 . (3.10.66) The estimate (3.10.66) is significant in that it essentially determines the rate of decay, 1 x , that must be satisfied exactly by the profiles, vR . The next profile term, according to 2 299 (3.3.14) with j = 1, m = 0, estimates (3.3.16), and (3.3.18), is: Z Z Z Z p uRy vvy x = {uP Ry + ✏uE RY }vvy x 1 3 1 p  ||yuP 2 2 E Ry ||L1 ||vy x ||L2 + ||uRY x ||L1 ||vy x ||L2 || ✏vx ||L2 2 2 1 p p  O( )||vy x 2 ||2L2 + ✏|| ✏vx ||2L2 . (3.10.67) Summarizing, we have: Z Z Z Z 1 p p @y Su vx = uR vy2 x + R3 , R3  O( )||uy ||2L2 + O( )||vy x 2 ||2L2 + ✏|| ✏vx ||2L2 . (3.10.68) Profile Terms, Sv : Next, we treat the Sv profile terms, for which we refer the reader to (3.8.6). The first step is to integrate by parts in x: Z Z Z Z Z Z Z ✏@x Sv vx = ✏ Sv xvx + ✏ Sv v lim "Sv vx M !1 x=M Z Z Z Z =✏ Sv xvx + ✏ Sv v. (3.10.69) The limit above is easily seen to vanish using (3.9.115). We now treat the first term on the right-hand side of (3.10.69). We start with the following term which enables control p over ✏vx : Z Z Z Z ✏uR vx2 x min uR ✏vx2 x, (3.10.70) The second term from Sv can be controlled according to estimate (3.3.8) and (3.3.17), 300 via: Z Z p 3 p 1 ✏ vRx uvx x  ✏||x 2 vRx ||L1 || ✏vx x 2 ||L2 ||ux ||L2 p p 1  ✏|| ✏vx x 2 ||L2 ||ux ||L2 (3.10.71) Next, we come to the third term in Sv , where again we use (3.3.10) and (3.3.17): Z Z p 1 p 1 ✏ vR vy vx x  ✏||vR ||L1 ||vy x 2 ||L2 || ✏vx x 2 ||L2 p 1 p 1  ✏O( )||vy x 2 ||L2 || ✏vx x 2 ||L2 (3.10.72) The fourth profile term in Sv is controlled according to estimates (3.3.10) and (3.3.20) by: Z Z p P 1 p 1 p E 3 p 1 ✏ vRy vvx x  ✏||vRy y||L1 ||vy x 2 ||L2 || ✏vx x 2 ||L2 + ✏||vRY x 2 ||L1 || ✏vx x 2 ||2L2 p 1 p p 1  ✏O( )||vy x 2 ||2L2 + "O( )|| ✏vx x 2 ||2L2 (3.10.73) Next, we note that the second interior term on the right-hand side of (3.10.69) is exactly that contained in (3.10.46), which yields: Z Z Z Z ✏@x Sv vx = ✏uR vx2 x + R4 , p 1 R4  O( )||{ ✏vx , vy }x 2 ||2L2 + O( )||uy ||2L2 . (3.10.74) 301 Right-Hand Side On the right-hand side, we have Z Z ⇣ ⌘ Z Z Z Z ⇣ ⌘ fy ✏gx vx = f vy x + ✏ g vx x + v . (3.10.75) We now justify both the integration by parts above. First, let us turn to the fy term, which, according to (3.8.5) contains the forcing terms Ru,n and the nonlinearity N u : ⇣ n ⌘ fy vx = ✏ 2 @y Ru,n + @y {uux + vuy } vx ⇣ n ⌘ = ✏ 2 @y Ru,n + uy ux + uuxy + vy uy + vuyy vx. (3.10.76) From here it is easy to see that [fy vx, ✏gx vx] 2 L1 (⌦N ). Therefore, it remains to treat the x-integration by parts boundary terms at x = 1, for which we simply appeal to (3.9.115) in an identical fashion to (3.10.69). Combining the previous estimates proves (3.10.50). 3.10.3 Second Order Bounds In this part, we obtain second order control of the solution to the system (3.8.1) - (3.8.3). To do so, we consider the di↵erentiated system in vorticity form: ⇣ ⌘ ⇣ Py ⌘ @xy ✏u + Px + S u ✏@xx ✏v + + Sv = ✏ ⇣ ⌘ ⇣ ⌘ @xy ✏ u + Su ✏@xx ✏ v + Sv = fxy ✏gxx . (3.10.77) We will now repeat the Energy and Positivity estimates from the previous section, with 302 higher order multipliers. One should briefly recall the definition of the cut-o↵ function ⇢2 from (3.9.2). Define our weight via: w2 = ⇢2 x. (3.10.78) The essential property of this weight is that: Lemma 3.10.3 (Almost Linear Property). |@xk w2 |  @xk x for k = 0, 1, and |@xk w2 | . x M , for k 2, for any M. (3.10.79) Remark. This property of the weight distinguishes it from a generic weight approximating the function x in that all of the nonlinear fluctuations are in an order-1 region around x = 1. This structure is needed in (3.10.138), and distinguishes the second order energy estimates from the third-order estimates. Define: Z Z W2,E = |fx ||ux ||⇢22 x2 | + ✏|gx ||vx ||⇢22 x2 |. (3.10.80) Z Z W2,P = |fx ||vxy ||⇢32 x3 | + ✏|gx ||vxx ||⇢32 x3 |}, (3.10.81) W2 = W2,E + W2,P . (3.10.82) Proposition 3.10.4 (Second-Order Energy Estimate). Let ✏ << , and , " be sufficiently small relative to universal constants. Then solutions [u, v] 2 Z(⌦N ) to the system (3.8.1) - (3.8.3) satisfy the following energy estimate: p p 3 ||uxy w2 ||2L2 + ✏||{vxy , ✏vxx }w2 ||2L2  O( )||{ ✏vxx , vxy }w22 ||2L2 + ||u, v||2X1 + W1 + W2,E . (3.10.83) Remark. Note the presence of absolute values inside the integration in the definition of W2 for the f term, unlike in W1 . This will be important for calculation (3.11.5). 303 Remark (Degenerate Weights near x = 1). Due to our weight, w2 , degenerating near the boundary x = 1, we cannot say that w22 . w23 . It is imperative that we retain control of the p non-degenerate weight of w on the left-hand side of (3.10.83) for the terms {vxy , ✏vxx }. Remark. We will continue to justify rigorously each integration by parts, as we have not cut-o↵ as x ! 1. Starting with the second-order positivity estimate (see (3.10.173)), it becomes possible to work with cut-o↵s in x, and so the rigorous justifications of Fubini and vanishing boundary contributions at x = 1 become automatic. However, for the present calculation, as in the first-order positivity estimate, we take integrations in the order: Z Z lim · dy dx. (3.10.84) M !1 ⌦N M We also refer the reader to the results on the auxiliary space Z(⌦N ) in Subsection 3.9.3, which will be cited in the forthcoming calculation, for some formal justifications. Proof of Proposition. We apply the multiplier vw22 to the system (3.10.77). We shall drop the subscript-2 from w2 , for the proof, with the understanding that w = w2 for this calcu- lation. Integration by parts several times gives the highest order terms: Z Z ⇣ ⌘ ⇣ ⌘ 2 {@xy ✏u ✏@xx ✏ v } · vw Z Z Z Z Z Z Z Z 2 2 2 2 2 2 = uxy w + ✏uxx w + ✏vxy w + ✏2 vxx 2 w 2 + J0 , (3.10.85) where |J0 | . ✏||u, v||2X1 . To see this, let us first start with the ✏u terms: Z Z Z Z 2 2 @xy ( ✏ u)vw = ✏ ux ux w Z Z ⇣ ⌘ Z Z = ✏uxx @x ux w2 + u2xy w2 (3.10.86) Z Z Z Z 2 2 2 = (✏uxx + uyx )w + 2 ✏uxx ux w@x w (3.10.87) Z Z Z Z = (✏u2xx + u2yx )w2 ✏u2x @x2 w2 . (3.10.88) 304 We will now examine the weight in (3.10.88). Indeed, |@x2 w2 | = |2w@x2 w + 2(@x w)2 | . 1, (3.10.89) according to (3.10.79). Therefore, Z Z Z Z ✏u2x @x2 w2 . ✏u2x  ✏||u||2X1 . (3.10.90) Due to the cuto↵ function, ⇢2 , in w, there are no boundary terms at x = 1 when integrating by parts in the x direction. We shall now provide some formalities. First, due to the order of integration in (3.10.84), one can integrate by parts twice in y for the following term: Z Z Z Z Z Z @x @y uyy · vw2 dy dx = @x uyy vy w2 dy dx = uxy vyy w2 ⌦M N ⌦M N ⌦M N Z Z = u2xy w2 . (3.10.91) ⌦M N The final quantity above is known to be in L1 (⌦N ) according to our norm Z, and therefore we can take the limit: Z Z Z Z lim u2xy w2 = u2xy w2 . (3.10.92) M !1 ⌦M N We next turn to the integration in x in (3.10.86). One easily checks that the integrand "uxxx ux w2 2 L1 (⌦N ) for u 2 Z, and so the following calculation is justified: Z Z Z Z Z "uxxx ux w2 = "uxx @x (ux w2 ) + lim "uxx ux w2 . (3.10.93) M !1 x=M 305 For the limit, we use (3.9.72) - (3.9.73): Z 1 M !1 | uxx ux w2 | . ||uxx x||L2y ||ux x||L2y . M ! 0. (3.10.94) x=M The x-integration in (3.10.87) works in an identical manner. Let us turn to the ✏v term: Z Z Z Z 2 2 ✏@xx ( ✏ v)vw = ✏@x ( ✏ v)@x (vw ) (3.10.95) Z Z Z Z 2 2 2 2 = (✏ vxx + ✏vxy )w + ✏2 v 2 @x4 w2 ✏2 vx2 @x2 w2 ✏vy2 @x2 w2 . The final three integrations above are estimated as: Z Z Z Z ✏2 vx2 @x2 w2  ✏2 vx2 . ✏||v||2X1 , ||@x2 w2 ||L1 (3.10.96) Z Z Z Z 2 Z Z 2v 2 2 4 2 2 4 2 ✏ v @x w  ||x @x w ||L1 ✏ 2 . ✏2 vx2 . ✏||v||2X1 , (3.10.97) x Z Z Z Z ✏vy2 @x2 w2 . ||@x2 w2 ||L1 ✏vy2 . ✏||v||2X1 . (3.10.98) Let us now give formal justifications for (3.10.95). First, we note the following bound: Z Z | "@xx ("vxx + vyy )vw2 | . ||(vxxxx + vxxyy )w||L2 ||vw||L2 . ||{uxx , vxx }⇣4 x||H˙ 2 ||vw||L2 < 1. (3.10.99) For the final estimate, we have used the elliptic regularity established in (3.9.53) and the estimate found in (3.9.112). This then justifies: Z Z Z Z Z 2 2 2 "@xx ( " v)vw = "@x ( " v)@x (vw ) + lim "@x " vvw (3.10.100) M !1 Z Z 2 = "@x ( " v)@x (vw ). (3.10.101) 306 For the above limit, one must use (3.9.115) together with the estimate (3.9.74). The re- maining integrations in (3.10.95) are justified in the standard way, as in (3.10.93) - (3.10.94) with the aid of (3.9.112) - (3.9.115). In so doing, one computes limits of the following types: Z 3 7 | vxx v@x w2 |  ||vxx x2 ||L2y ||vx 2 ||L2y M 2 M 2, (3.10.102) x=M Z | vxx vx w2 |  ||vxx x2 ||L2y ||vx x2 ||L2y M 4 M 2, (3.10.103) x=M Z | vx2 w  ||vx x2 ||2L2y M 4 M, (3.10.104) x=M Z 3 7 | vx v@x2 w2 |  ||vx x2 ||L2y ||vx 2 ||L2y M 2 , (3.10.105) x=M Z 3 | v 2 @x3 w2 |  ||vx 2 ||2L2y M 3 . (3.10.106) x=M All of these terms vanish upon taking M ! 1. This completes the formal justification of all of the integrations thus far. For the profile terms, calculations which are analogous to the lowest-order case yield: Z Z p 3 {@yx Su ✏@xx Sv } · vw2 . ||u, v||2X1 + O( )||{ ✏vxx , vxy }w 2 ||2L2 . For completeness, we include all details here. We will now be working through the following set of terms, referring to the definition in (3.8.6): Z Z Z Z @yx Su · vw2 = @ x Su · u x w 2 Z Z h i = @x uR ux + uRx u + uRy v + vR uy · ux w2 . (3.10.107) We begin with: Z Z Z Z Z Z 2 | @x (uR ux )ux w | = | uRx u2x w2 + uR uxx ux w2 | Z Z Z Z Z uRx 2 2 uR 2 uR 2 2 =| u w u @x w2 + lim u w | 2 x 2 x M !1 x=M 2 x 307 1  ||uRx x, uR ||L1 ||@x w||L1 ||ux x 2 ||2L2 . ||u||2X1 . (3.10.108) Above, we have used the estimate (3.3.12) and (3.3.18) for the uE Rx term. The limit above has vanished according to (3.9.115). Next, Z Z Z Z @x (uRx u)ux w2 = uRxx uux w2 + uRx u2x w2 (3.10.109) We must break up the profile term, uRxx , into Euler and Prandtl. For the uP Rxx term we use (3.3.11) with k = 2, j = 0, m = 1, and subsequently the Hardy inequality in y: Z Z 3 u 1 | uP 2 P Rxx uux w |  ||uRxx yx ||L1 || ||L2 ||ux x ||L2 2 2 y 1 . ||uy ||L2 ||ux x 2 ||L2 . ||u||2X1 . (3.10.110) For the uE Rxx term, we use (3.3.18), followed by the Hardy inequality in x: Z Z 5 u 1 p Rxx uux w ||  ||uRxx x ||L1 || ||L2 ||ux x ||L2 . uE 2 E | 2 2 "||u||2X1 , (3.10.111) x Z Z 1 | uRx u2x w2 |  ||uRx x||L1 ||ux x 2 ||2L2  O( )||u||2X1 . (3.10.112) The next profile term from (3.10.107) is the most delicate convective term: Z Z Z Z Z Z @x (uRy v) · ux w2 = uRxy vux w2 + uRy vx ux w2 (3.10.113) For the first term in (3.10.113), we bound, according to (3.3.11) for the Prandtl contri- bution, coupled with the Hardy inequality in y: Z Z v 1 1 | uP 2 P Rxy vux w |  ||uRxy xy||L1 || x ||L2 ||ux x ||L2 2 2 y 308 1 1 Rxy xy||L1 ||vy x ||L2 ||ux x ||L2 . ||u, v||X1 ,  ||uP 2 2 2 (3.10.114) and (3.3.18) for the Euler contribution, followed by the Hardy inequality in x direction: Z Z p p 5 v 1 | ✏uE 2 RxY vux w |  ✏||uE RxY x ||L1 || ||L2 ||ux x ||L2 2 2 x p p 1 p . ✏|| ✏vx ||L2 ||ux x 2 ||L2 . ✏||u, v||2X1 . (3.10.115) For the second term in (3.10.113), we bound by using profile estimates (3.3.14), (3.3.16), (3.3.18): Z Z Z Z ⇣ ⌘ 2 p uRy vx ux w = uP Ry + ✏uE RY vx ux w 2 vx 3 1 3 p 1  ||uP Ry y||L1 || w 2 ||L2 ||ux x 2 ||L2 + ||uE RY x ||L1 || ✏vx ||L2 ||ux x ||L2 2 2 y 3 p  O( )||vxy w 2 ||L2 ||u||X1 + "||u, v||2X1 . (3.10.116) Implicit in the above calculation is the fact that both the profile terms uRy and the terms from X1 are controlled on the full domain, ⌦N , and therefore do not demand any of the “nondegeneracy” of the cut-o↵ weight w near x = 1. The next profile term from Su is: Z Z ⇣ ⌘ Z Z Z Z @ x v R uy · ux w 2 = vRx uy ux w2 + vR uxy ux w2 (3.10.117) We estimate by using (3.3.8) - (3.3.10), and the Eulerian estimates in (3.3.17) and (3.3.20): Z Z 3 1 vRx uy ux w2  ||vRx x 2 ||L1 ||uy ||L2 ||ux x 2 ||L2 . ||u, v||2X1 , (3.10.118) Z Z 1 1 vR uxy ux w2  ||vR x 2 ||L1 ||uxy w||L2 ||ux x 2 ||L2 309 . ||u, v||2X1 + O( )||uxy w||2L2 . (3.10.119) Let us now summarize the Su contribution: Z Z ⇣ ⌘ 3 @x Su · ux w2 . ||u, v||2X1 + O( )||vxy w 2 ||2L2 + O( )||uxy w||2L2 . (3.10.120) The uxy term in the above estimate is absorbed to the left-hand side of (3.10.88), by taking sufficiently small. The next task is to move to the four profile terms in Sv , which we now do, and recall the terms for convenience: Z Z ✏@xx {uR vx + vRx u + vR vy + vRy v}vw2 . (3.10.121) Let us begin by giving some formal justification to the initial x-integration by parts which will be required to treat the above set of terms. First, one observes using (3.9.112) and the definition of Z in (3.9.8) that all terms are in L1 (⌦N ). Therefore, an integration by parts in x would contribute the following integration in the limit: Z h i lim ✏@x uR vx + vRx u + vR vy + vRy u vw2 dy M !1 Z h = lim ✏ uRx vx + uR vxx + vRxx u + vRx ux + vRx vy M !1 i + vR vxy + vRxy u + vRy ux vw2 dy. (3.10.122) We shall estimate each term above, with the aid of the profile estimates in (3.3.10) - (3.3.17), and also the Z(⌦N ) estimates in Subsection 3.9.3, (3.9.115). Z 3 9 | uRx vx v|  ||uRx x||L1 ||vx x2 ||L2y ||vx 2 ||L2y M 2 , (3.10.123) Z 3 7 | uR vxx v|  ||vxx x2 ||L2y ||vx 2 ||L2y M 2 , (3.10.124) 310 Z 5 1 3 9 | vRxx uv|  ||vRxx x 2 ||L1 ||ux 2 ||L2y ||vx 2 ||L2y M 2 , (3.10.125) Z 3 3 3 9 | vRx ux v|  ||vRx x 2 ||L1 ||ux x 2 ||L2y ||vx 2 ||L2 M 2 , (3.10.126) Z 1 3 | vR vxy v|  ||vR x 2 ||L1 ||vxy x2 ||L2y ||vx 2 ||L2y M 4 , (3.10.127) Z 1 3 | vRxy uv|  ||vRxy x2 ||L1 ||ux 2 ||L2y ||vx 2 ||L2y M 4 , (3.10.128) Z 3 3 4 | vRy ux v|  ||vRy x||L1 ||ux x 2 ||L2y ||vx 2 ||L2y M . (3.10.129) From here, it is clear that the limit above in (3.10.122) is zero. With this formal justifi- cation in hand, we continue with the a-priori estimate. For the first term from (3.10.121), Z Z Z Z ✏@xx (uR vx ) · vw =2 ✏@x (uR vx ) · {vx w2 + 2vww0 } Z Z = ✏{uRx vx + uR vxx } · {vx w2 + 2vww0 } . (3.10.130) Let us individually treat each term in (3.10.130). First, by combining (3.3.12) and (3.3.18): Z Z p 1 ✏uRx vx2 w2 . ||uRx x||L1 || ✏vx x 2 ||2L2  O( )||v||2X1 . (3.10.131) Next, Z Z Z Z Z ✏ 2 " ✏uR vxx vx w2 = vx @x (uR w2 ) + lim uR vx2 w2 2 M !1 x=M 2 Z Z Z Z ✏ 2 = vx uRx w2 + ✏vx2 uR ww0 2 p 1 . ||uRx w, w0 ||L1 || ✏vx x 2 ||2L2 . ||v||2X1 . (3.10.132) 311 The limit above vanishes according to (3.9.115). Next, we shall split uRx = uP E Rx + uRx : Z Z Z Z Z Z ✏uRx vx vww0  ✏uP 0 Rx vx vww + ✏uE Rx vx vww 0 p 1 p 1 v  ✏||uP Rx yx ||L1 || ✏vx x ||L2 || ||L2 2 2 y 3 p 1 p v + ||uE Rx x ||L1 || ✏vx x ||L2 || ✏ ||L2 2 2 x p 1 p 1  ✏||uPRx yx ||L1 || ✏vx x ||L2 ||vy ||L2 2 2 3 p 1 p + ||uE Rx x ||L1 || ✏vx x ||L2 || ✏vx ||L2 2 2  O( )||v||2X1 . (3.10.133) Above, we have used (3.3.12) (with m = 1), coupled with Hardy inequalities in y and x, and (3.3.18) for the Euler term. The fourth term in (3.10.130) is by far the most delicate: Z Z Z Z ⇣ ⌘ Z ✏uR vxx v@x w2 = ✏vx @x uR v@x w2 + lim "uR vx v@x w2 M !1 x=M Z Z = ✏vx {uRx v@x w2 + uR vx @x w2 + uR v@x2 w2 } = I1 + I2 + I3 . (3.10.134) We justify the above integration. It is clear, according to (3.9.112), that vxx v@x w2 2 L1 (⌦N ), and so the boundary contribution at x = 1 is: Z 3 7 M !1 | "uR vx v@x w2 |  ||vx x2 ||L2y ||vx 2 ||L2y M 2 M ! 0. (3.10.135) x=M I1 follows similarly to (3.10.133). For I2 , we have: Z Z p 1 |I2 | = | "uR vx2 @x w2 | . ||@x w||L1 ||uR ||L1 || "vx x 2 ||2L2 . ||v||2X1 . (3.10.136) 312 We will estimate I3 . For this, we note that due to the almost-linear structure of our weight, |@x3 w2 | . x K , for any K, (3.10.137) and so (again with the aid of (3.9.115)): Z Z Z Z Z ✏ 2⇣ ⌘ ✏uR vx v@x2 w2 = v uRx @x2 w2 + uR @x3 w2 + lim "uR v 2 @x2 w2 2 M !1 x=M Z Z ✏ 2⇣ ⌘ = v uRx @x2 w2 + uR @x3 w2 . (3.10.138) 2 For the uRx term above in (3.10.138), we shall spit into Euler and Prandtl components, and use estimates (3.3.12) (with m = 2), (3.3.18): Z Z v 2 Rx @x w | . ✏||uRx y ||L1 || ||L2  ✏||uRx y ||L1 ||vy ||L2 . ✏O( )||v||X1 , ✏v 2 uP 2 2 P 2 P 2 2 2 | y (3.10.139) Z Z 3 p v 2 p p 1 Rx @x w | . ||uRx x ||L1 || ✏ 3 ||L2 . ✏v 2 uE "|| ✏vx x 4 ||2L2 . O( )||v||2X1 . 2 2 E | 2 x4 (3.10.140) For the uR term in (3.10.138) above, the structure of our weight, (3.10.137) is important: Z Z p | ✏v 2 @x3 w2 | . || ✏vx ||2L2 . ||v||2X1 . (3.10.141) For the second term from (3.10.121), we integrate by parts again to arrive at: Z Z ✏@x (vRx u) · {vx w2 + 2vww0 } Z Z = ✏{vRxx u + vRx ux } · {vx w2 + 2vww0 } (3.10.142) 313 Of these, we estimate, according to (3.3.8) - (3.3.10) and (3.3.17): Z Z 5 u v p 1 p 1 ✏ vRxx uvww0  ✏||vRxx x 2 ||L1 || 3 ||L2 || 3 ||L2 . ✏||ux x 4 ||L2 || ✏vx x 4 ||L2 , x 4 x 4 (3.10.143) Z Z p 5 u p 1 p ✏ vRxx uvx w2  ✏||vRxx x 2 ||L1 || ||L2 || ✏vx x 2 ||L2 . "||u, v||2X1 , (3.10.144) x Z Z p 3 p 1 p ✏ vRx ux vx w2  "||vRx x 2 ||L1 ||ux ||L2 || "vx x 2 ||L2 . "||u, v||2X1 , (3.10.145) Z Z 3 1 v p 1 p ✏ vRx ux vww0  ✏||vRx x 2 ||L1 ||ux x 2 ||L2 || ||L2 . ✏||ux x 2 ||L2 || ✏vx ||L2 . (3.10.146) x The third term from (3.10.121) is given by: Z Z Z Z ✏ @xx (vR vy )vw2 = ✏@x (vR vy ){vx w2 + 2ww0 v} Z Z = ✏{vRx vy + vR vxy }{vx w2 + 2ww0 v}. (3.10.147) We estimate term by term, appealing to estimate (3.3.8) and (3.3.17): Z Z p 3 p 1 p ✏vRx vy vx w2 . ✏||vRx x 2 ||L1 ||vy ||L2 || ✏vx x 2 ||L2 . ✏||v||2X1 , (3.10.148) Z Z 3 1 v p 1 p ✏vRx vy vww0 . ✏||vRx x 2 ||L1 ||vy x 2 ||L2 || ||L2 . ✏||vy x 2 ||L2 || ✏vx ||L2 x p . ✏||v||2X1 . (3.10.149) Next, appealing to estimate (3.3.10) (with k = 0, j = 0, 1, m = 0, 1), and the Euler estimate in (3.3.20), coupled with the Hardy inequality in x and in y directions: Z Z Z Z ✏vR vxy vww0 = ✏vx {vR vy + vRy v}ww0 1 p 1 p P 1 p 1 v . ||vR x 2 ||L1 || ✏vx x 2 ||L2 ||vy ||L2 + ✏||vRy yx 2 ||L1 || ✏vx x 2 ||L2 || ||L2 y p E 3 p 1 p v + "||vRY x 2 ||L1 || ✏vx x 2 ||L2 || ✏ ||L2 x 314 1 p 1 p P 1 p 1 . ||vR x 2 ||L1 || ✏vx x 2 ||L2 ||vy ||L2 + ✏||vRy yx 2 ||L1 || ✏vx x 2 ||L2 ||vy ||L2 p E 3 p 1 p + "||vRY x 2 ||L1 || ✏vx x 2 ||L2 || ✏vx ||L2 . O( )||u, v||2X1 . (3.10.150) Upon integrating by parts and appealing to (3.3.10) (with k = 0, j = 1, m = 0) and (3.3.20): Z Z Z Z ✏ p p 1 ✏vR vxy vx w2 = vRy vx2 w2 . ✏||xvRy ||L1 || ✏vx x 2 ||2L2  O( )||v||2X1 . 2 (3.10.151) The final contribution from Sv in (3.10.121) is: Z Z Z Z ✏ @xx (vRy v) · vw2 = ✏ {vRxxy v + 2vRxy vx + vRy vxx }vw2 (3.10.152) For the first term, we split: Z Z v P ✏vRxxy v 2 w2  ||vRxxy P x2 y 2 ||L1 || ||2L2 . ||vy ||2L2 . ||u, v||2X1 , (3.10.153) y Z Z 3 7 v p p 1 p E ✏vRxxy v 2 w2  ✏ 2 ||vRxxY E x 2 ||L1 || 3 ||2L2 . ✏|| "vx x 4 ||2L2 . ✏||u, v||2X1 . x 4 (3.10.154) For (3.10.153), we appeal to estimate (3.3.8) (with k = 2, j = 1, m = 2), and for (3.10.154), we appeal to estimate (3.3.17). For the next term in (3.10.152), again with the P E splitting vR = vR + vR , by using estimate (3.3.8), we have: Z Z P p 3 v 1 p 1 | "vRxy vx vw2 |  P "||vRxy yx 2 ||L1 || x 2 ||L2 || "vx x 2 ||L2 y p P 3 1 p 1  "||vRxy yx 2 ||L1 ||vy x 2 ||L2 || "vx x 2 ||L2 315 p  "||u, v||2X1 . (3.10.155) For the Euler contribution, by (3.3.17), we have: Z Z 3 E p 5 p v p 1 | " 2 vRxY vvx w2 |  E "||vRxY x 2 ||L1 || " ||L2 || "vx x 2 ||L2 x p E 5 p p 1  "||vRxY x 2 ||L1 || "vx ||L2 || "vx x 2 ||L2 p  "||v||2X1 . (3.10.156) The third term in (3.10.152) we split into Euler and Prandtl components, and use (3.3.10) (with j = 1, m = 1), and (3.3.17): Z Z Z Z Z Z ✏vRy vxx vw2  P ✏vRy vxx vw2 + E ✏vRy vxx vw2 p P 1 v 1 p 3  ✏||vRy yx 2 ||L1 || x 2 ||L2 || ✏vxx w 2 ||L2 y p E 3 p 3 p v + "||vRY x 2 ||L1 || ✏vxx w 2 ||L2 || ✏ ||L2 x p p 1 p 3  "||{vy , ✏vx }x 2 ||L2 || ✏vxx w 2 ||L2 p p 3  "||v||X1 || ✏vxx w 2 ||L2 . (3.10.157) Summarizing the contributions from Sv : Z Z p 3 ✏@xx {Sv }vw2 . ||u, v||2X1 + O( )|| ✏vxx w 2 ||2L2 (3.10.158) On the right-hand side, we have: Z Z Z Z 2 @xy f · vw = fx · vy w 2 , (3.10.159) Z Z Z Z ✏ @xx g · vw2 = ✏ gx · {vx w2 + 2vww0 } 316 Z Z Z Z =✏ g x vx w 2 ✏ g{vx ww0 + v@xx (w2 )}. (3.10.160) We estimate the g term: Z Z Z Z Z Z | "gvx ww0 + gv@xx (w2 )|  " |g||vx ||ww0 | + " |g||v||@xx (w2 )| Z Z  "|g||vx ||w| + "|g||v|  W1 . (3.10.161) A formal point: one can easily check that @xx g · vw2 and gx {vx w2 + v@x w2 } 2 L1 (⌦N ), and so the term at x = 1 which is contributed as a result of integrating by parts in x is: Z lim | "@x gvw2 | M !1 x=M Z ⇣ ⌘ n n = lim | " " 2 Rxv,n + " 2 + (ux vx + uvxx + vx vy + vvxy ) vw2 |. (3.10.162) M !1 x=M We can estimate each term, using (3.7.177), (3.9.115), for arbitrarily small constants  > 0, and n as in (3.7.1) Z 9 3 15 | Rxv,n · vw2 | . ||Rxv,n x 4 2 n  ||L2y ||vx 2 ||L2y M 4 +2 n + M 2, (3.10.163) x=M Z 3 3 3 9 | ux vx vw2 | . ||vx x 2 ||L1 ||ux x 2 ||L2y ||vx 2 ||L2y . M 2 M 2, (3.10.164) x=M Z 3 7 | uvxx vw2 | . ||u||L1 ||vxx x2 ||L2y ||vx 2 ||L2y M 2 M 2, (3.10.165) x=M Z 3 3 3 9 | vx vy vw2 |  ||vx x 2 ||L1 ||vy x 2 ||L2y ||vx 2 ||L2y M 2 M 2, (3.10.166) x=M Z 1 3 | vvxy vw2 |  ||vx 2 ||L1 ||vxy x2 ||L2y ||vx 2 ||L2y  M 4 M 2. (3.10.167) x=M It is clear that all of these vanish as M ! 1. This justifies the integration by parts in (3.10.160). The second terms in (3.10.160) are part of (3.10.10). The desired estimate is established. 317 Next, we come to the second-order positivity estimate. It is necessary to apply ap- proximations to the actual weights we would like to control in order to avoid boundary contributions from x = 1. To this end, let us define (x) to be the standard mollifier, with the usual properties of unit mass, positivity, and support in B(0, 1). Next, let (x) be a standard cuto↵ function, equal to 1 inside [1, 2], and equal to zero on the interval [3, 1). Also, recall the definition of ⇢2 provided in equation (3.9.2). Then let us define 1 ⇣ x ⌘ aL (x) := min{x, L}, L (x) := , (3.10.168) (L/2) (L/2) ⇣ ⌘ ⇣ x ⌘ w2,L (x) := aL ⇤ L ⇢2 (x). (3.10.169) 10L The relevant properties of this weight are summarized: Lemma 3.10.5. The weight wL satisfies: L 3L w2,L (x) = x, for 60  x  , w2,L (x) = L for  x  10L, (3.10.170) 2 2 w2,L (x) = 0 for x 30L and 1  x  50, xk 1 k @x w2,L  C independent of L. (3.10.171) Proof. All are clear by basic properties of convolutions. Heuristically, the property (3.10.171) ensures that the weight w2,L (x) behaves like x in the sense that each additional derivative eliminates one factor of the weight. We record: lim w2,L (x) = x⇢2 (x) = w2 (x), for all x 1. (3.10.172) L!1 Proposition 3.10.6 (Second-Order Positivity). For , ✏ sufficiently small relative to uni- 318 versal constants and " << , solutions [u, v] 2 Z(⌦N ) to the system (3.8.1) - (3.8.3) satisfy: p 3 p ||{vxy , ✏vxx }w22 ||2L2 . ||uxy w2 ||2L2 + ✏||{vxy , ✏vxx }w2 ||2L2 + ||u, v||2X1 + W1 + W2 . (3.10.173) 3 Proof. We apply the multiplier vx w2,L . We will drop the subscript-2 from w2,L and simply use wL for this calculation. Upon integrating by parts, the highest-order terms are: Z Z ⇣ ⌘ Z Z 33 2 0 @xy uyy · vx wL = u2xy wL wL , (3.10.174) 2 Z Z ⇣ ⌘ Z Z 3 3✏ 2 0 @xy ✏uxx · vx wL = u2xx wL wL , (3.10.175) 2 Z Z Z Z Z Z 2 0 ✏2 @x4 v · vx wL 3 =C ✏2 vxx 2 wL wL + ✏vx2 @x3 {wL 3 }, (3.10.176) Z Z Z Z 3 2 2 0 ✏@xxyy v · vx wL =C ✏vxy wL wL . (3.10.177) Above, we have used the property (3.10.171): Z Z Z Z ✏vx2 @x3 {wL 3 }  ||@x3 wL 3 ||L1 ✏vx2 . ||v||2X1 . (3.10.178) Similarly, we have: Z Z 2 0 2 0 0 p ✏2 vxx 2 wL 2 wL + ✏vxy wL wL  ||wL ||L1 ✏||{ ✏vxx , vxy }wL ||2L2 . (3.10.179) Thus, all of the terms on the right-hand sides of (3.10.174) - (3.10.177) appear on the right-hand side of (3.10.173), which in turn are controlled by the Energy Estimate, see (3.10.83). We will now work through the profile terms, contained in Su , which we write below upon using definition (3.8.6): Z Z 3 @xy {uR ux + uRx u + uRy v + vR uy } · vx wL (3.10.180) 319 First, we have the main profile terms: Z Z ⇣ ⌘ Z Z 3 3 @xy uR ux · vx wL = @x (uR ux )vxy wL Z Z 3 = uRx ux vxy wL + uR u2xx wL 3 . (3.10.181) As usual, this term retains control of the main term: Z Z Z Z uR u2xx wL 3 & min |uR | u2xx wL 3 . (3.10.182) The other term in (3.10.181) may be estimated by recalling (3.3.12) and (3.3.18), via: Z Z 1 3 3 uRx ux vxy wL  ||uRx x||L1 ||ux x 2 ||L2 ||vxy wL2 ||L2 3  O( )||u, v||2X1 + O( )||vxy wL2 ||2L2 . (3.10.183) The second term on the right-hand side above, in (3.10.183), can be absorbed by the main positive term, (3.10.182) by taking small enough. Next, we have the remaining profile terms from Su : Z Z 3 @xy {uRx u + uRy v + vR uy } · vx wL (3.10.184) For the first term in (3.10.184), we will integrate by parts in y, expand the product, and use Young’s inequality: Z Z Z Z 3 3 @xy {uRx u} · vx wL = {uRxx u + uRx ux }vxy wL 3 3 5 3  ||uP 2 2 2 E Rxx yx ||L1 ||uy ||L2 ||vxy wL ||L2 + ||x uRxx ||L1 ||ux ||L2 ||vxy wL ||L2 2 1 3 + ||uRx x||L1 ||ux x 2 ||L2 ||vxy wL2 ||L2 320 3 1 3 . ||u, v||X1 ||vxy wL2 ||L2  ||vxy wL2 ||2L2 + C||u, v||2X1 . (3.10.185) 100, 000 We have used (3.3.11), (3.3.12), and the Euler estimates from (3.3.18). The next term from (3.10.184) is the convection term, which we start by integrating by parts: Z Z Z Z 3 3 @xy (uRy v)vx wL = @x (uRy v)vxy wL Z Z Z Z 3 3 = uRxy vvxy wL uRy vx vxy wL Z Z Z Z 3 1 = uRxy vvxy wL + uRyy vx2 wL 3 (3.10.186) 2 = (3.10.186.1) + (3.10.186.2). First, by applying (3.3.11) (with k = j = m = 1) and the Euler estimate in (3.3.18), and subsequently the Hardy inequality in both the y and x directions, v 1 3 5 p v 3 (3.10.186.1)  ||uP 2 2 E Rxy xy||L1 || x ||L2 ||vxy wL ||L2 + ||uRxY x ||L1 || ✏ ||L2 ||vxy wL ||L2 2 2 y x 1 3 5 p 3  ||uP 2 2 E Rxy xy||L1 ||vy x ||L2 ||vxy wL ||L2 + ||uRxY x ||L1 || ✏vx ||L2 ||vxy wL ||L2 2 2 3 1 3 . ||v||X1 ||vxy wL2 ||L2  ||vxy wL2 ||2L2 + C||v||2X1 . (3.10.187) 100, 000 Next, by applying (3.3.14), and (3.3.16) with j = 2 and (3.3.18), we have: vx 32 2 5 p 1 (3.10.186.2)  ||y 2 uP Ryy ||L1 || wL ||L2 + ||uE 2 2 RY Y x ||L1 || ✏vx x ||L2 2 y 3 5 p 1  ||y 2 uP 2 2 E 2 2 Ryy ||L1 ||vxy wL ||L2 + ||uRY Y x ||L1 || ✏vx x ||L2 2 3 p  O( )||vxy wL2 ||2L2 + "||u, v||2X1 . (3.10.188) 321 The final term from (3.10.184), upon integrating by parts once in y, is: Z Z ⇣ ⌘ Z Z Z Z 3 3 3 @x vR uy · vxy wL  vRx uy vxy wL + vR uxy vxy wL 1 ⇣ 3 ⌘ 3 3 . ||vR x 2 ||L1 ||uxy wL ||2L2 + ||vxy wL2 ||2L2 + ||vRx x 2 ||L1 ||uy ||L2 ||vxy wL2 ||L2 ⇣ 1 ⌘ 3 . O( )||uxy wL ||2L2 + O( ) + ||vxy wL2 ||2L2 + C||u, v||X1 . 100, 000 (3.10.189) We have used (3.3.10) and (3.3.20) to estimate vR , in which we crucially retain the smallness from O( ). We have used (3.3.8), (3.3.17), followed by Young’s inequality for the vRx term. Summarizing: Z Z 3 @xy {uRx u + uRy v + vR uy } · vx wL (3.10.190) ⇣ 1 ⌘⇣ 3 ⌘ . O( )||u, v||2X1 + O( ) + ||vxy wL2 ||2L2 + O( )||uxy wL ||2L2 . 100, 000 The middle term on the right-hand side above gets absorbed into (3.10.182). We will now come to the profile terms from the normal equation, Sv . For convenience, we display the terms we will be reading from here, according to the definition in (3.8.6): Z Z h i 3 ✏ @xx uR vx + vRx u + vRy v + vR vy · vx wL . (3.10.191) The main term upon integrating by parts once in x and expanding the product is: Z Z Z Z 3 3 2 0 ✏@xx (uR vx ) · vx wL = ✏{uR vxx + uRx vx } · {vxx wL + 3vx wL wL }. (3.10.192) 322 The first term above yields the desired positivity, namely: Z Z Z Z 2 ✏uR vxx 3 wL & min |uR | 2 ✏vxx 3 wL . (3.10.193) Let us now turn to the remaining terms above in (3.10.192). First an integration by parts gives: Z Z Z Z 3 ✏uR vxx vx @x wL = ✏vx2 @x {uR @x wL 3 } Z Z = ✏vx2 {uRx @x wL 3 + uR @x2 wL 3 } p 1 . ||uRx x||L1 || ✏vx x 2 ||2L2 . O( )||v||2X1 , (3.10.194) where we have used the estimate |@x2 wL 3 | . wL . x, and also (3.3.12) and (3.3.18), both of which guarantee the smallness of O( ). Still from (3.10.192), by using (3.3.11) - (3.3.12) and (3.3.18), we have: Z Z Z Z 3 ✏ 2 3 ✏uRx vx vxx wL = v @x {uRx wL } 2 x Z Z Z Z ✏ 2 3 ✏ 2 3 = vx uRxx wL v uRx @x wL 2 2 x p 1 . ||uRxx x2 , uRx x||L1 || ✏vx x 2 ||2L2 . ||v||2X1 . (3.10.195) Last from (3.10.192), Z Z p 1 ✏vx2 uRx wL wL . ||uRx x||L1 || ✏vx x 2 ||2L2 . O( )||v||2X1 . 2 0 (3.10.196) Above, we have used (3.3.12) and (3.3.18). We now move to the second term in Sv , for 323 which we integrate by parts once in x and expand the resulting product: Z Z Z Z 3 3 ✏@xx (vRx u) · vx wL = ✏@x (vRx u)@x {vx wL } Z Z 3 3 3 3 = ✏vRxx uvxx wL ✏vRx ux vxx wL ✏vRxx uvx @x wL ✏vRx ux vx @x wL = (3.10.197.1) + ... + (3.10.197.4). (3.10.197) First, by (3.3.8) and (3.3.17): p 5 p 3 u (3.10.197.1)  ✏||vRxx x 2 ||L1 || ✏vxx wL2 ||L2 || ||L2 x p 5 p 3 . ✏||vRxx x 2 ||L1 || ✏vxx wL2 ||L2 ||ux ||L2 p p 3 . ✏|| ✏vxx wL2 ||L2 ||u||X1 . (3.10.198) Second, p 3 p 3 p p 3 (3.10.197.2) . ✏||vRx x 2 ||L1 ||ux ||L2 || ✏vxx wL2 ||L2 . ✏||u||X1 || ✏vxx wL2 ||L2 . (3.10.199) Third, p 5 u p 1 p (3.10.197.3) . ✏||vRxx x 2 ||L1 || ||L2 || ✏vx x 2 ||L2 . ✏||u, v||2X1 . (3.10.200) x Fourth, p 3 1 p p (3.10.197.4) . ✏||vRx x 2 ||L1 ||ux x 2 ||L2 || ✏vx ||L2 . ✏||u, v||2X1 . (3.10.201) We have used estimate (3.3.8) and (3.3.17) in the above calculations. Let us now move 324 to the third term in Sv for which we integrate by parts once in x and expand the product: Z Z Z Z 3 3 ✏@xx (vRy v)vx wL = ✏ @x (vRy v)@x (vx wL ) Z Z 3 3 2 0 = ✏vRxy vvxx wL ✏vRy vx vxx wL 3✏vRxy vvx wL wL 2 0 3✏vRy vx2 wL wL (3.10.202) = (3.10.202).1 + ... + (3.10.202).4. We shall treat term by term above, starting with: Z Z 3 P 3 E 3 |(3.10.202).1| = | ✏vRxy vvxx wL ✏ 2 vRxY vvxx wL | p P 3 v 1 p 3  ✏||vRxy yx 2 ||L1 || x 2 ||L2 || ✏vxx wL2 ||L2 y p E 5 p 3 p v + ✏||vRxY x 2 ||L1 || ✏vxx wL2 ||L2 || ✏ ||L2 x p P 3 1 p 3  ✏||vRxy yx 2 ||L1 ||vy x 2 ||L2 || ✏vxx wL2 ||L2 p E 5 p 3 p + ✏||vRxY x 2 ||L1 || ✏vxx wL2 ||L2 || ✏vx ||L2 p p 3 . ✏||v||X1 || ✏vxx wL2 ||L2 . (3.10.203) Above, we have used (3.3.8) with k = 1, j = 1, m = 1, and the Euler bounds in (3.3.17). Next, according to estimate (3.3.10), (3.3.17): p 1 p 3 p 3 |(3.10.202).2|  ||vRy x||L1 || ✏vx x 2 ||L2 || ✏vxx wL2 ||L2 . O( )||v||X1 || ✏vxx wL2 ||L2 . (3.10.204) Third, also using the estimates from (3.3.8) and (3.3.17), p P 3 v 1 p 1 p E 5 p v p 1 |(3.10.202).3|  ✏||vRxy yx 2 ||L1 || x 2 ||L2 || ✏vx x 2 ||L2 + ✏||vRxY x 2 ||L1 || ✏ ||L2 || ✏vx x 2 ||L2 y x p 1 p 1 p p 1 p  "||vy x 2 ||L2 || ✏vx x 2 ||L2 + ✏|| ✏vx x 2 ||2L2 . "||v||2X1 . (3.10.205) 325 Fourth, again by estimate (3.3.10) and (3.3.17): p 1 |(3.10.202).4| . ||vRy x||L1 || ✏vx x 2 ||2L2  O( )||v||2X1 . (3.10.206) We now move to the fourth term in (3.10.191), for which we integrate by parts once and distribute the product: Z Z Z Z 3 3 ✏@xx (vR vy )vx wL = ✏@x (vR vy )@x (vx wL ) Z Z 3 3 2 0 = ✏vRx vy vxx wL ✏vR vxy vxx wL 3✏vRx vy vx wL wL 2 0 3✏vR vxy vx wL wL (3.10.207) = (3.10.207).1 + ... + (3.10.207).4. First, by (3.3.8), (3.3.10) and (3.3.17): p 3 1 p 3 |(3.10.207).1|  ✏||vRx x 2 ||L1 ||vy x 2 ||L2 || ✏vxx wL2 ||L2 , (3.10.208) p 1 p 3 3 |(3.10.207).2|  ✏||vR x 2 ||L1 || ✏vxx wL2 ||L2 ||vxy wL2 ||L2 , (3.10.209) p 3 1 p 1 |(3.10.207).3|  ✏||vRx x 2 ||L1 ||vy x 2 ||L2 || ✏vx x 2 ||L2 . (3.10.210) For the fourth term, we integrate by parts in y and again appeal to (3.3.10) and (3.3.17): Z Z 3✏ p 1 |(3.10.207).4| = | vRy vx2 wL wL | . ||vRy x||L1 || ✏vx x 2 ||2L2 . 2 0 (3.10.211) 2 Summarizing the last three terms from the Sv contribution: Z Z 3 ✏@xx {vRx u + vR vy + vRy v} · vx wL p 3 3  O( )|| ✏vxx wL2 ||2L2 + O( )||vxy wL2 ||2L2 + O( )||u, v||2X1 . (3.10.212) 326 Finally, on the right-hand side, we have: Z Z Z Z 3 3 @xy f · vx wL = fx uxx wL , (3.10.213) Z Z Z Z 3 3 2 0 ✏@xx g · vx wL = ✏gx {vxx wL + 3vx wL wL }. We estimate the term: Z Z Z Z 2 0 | "gx vx wL wL |  "|gx ||vx |(⇢2 x)2  W2 . (3.10.214) Taking the limit as L ! 1, and appealing to the Monotone Convergence Theorem then completes the calculation. 3.10.4 Third Order Bounds In this step, we obtain third-order bounds for our solution. We must repeat the above calculations to the twice-di↵erentiated system: ⇣ ⌘ ⇣ Py ⌘ @y @x2 ✏u + Px + S u ✏@x @x2 ✏v + + Sv = ✏ ⇣ ⌘ ⇣ ⌘ @y @x2 ✏ u + Su 2 ✏@x @x 2 ✏ v + Sv = @ y @ x f ✏@x @x2 g, (3.10.215) To state our energy estimate, we will recall the definition of ⇢3 from (3.9.2) and the definitions of aL , L given in (3.10.168). We will then define the weights: ⇣ ⌘ ⇣ x ⌘ w3 = ⇢3 (x)x, w3,L (x) := aL ⇤ L ⇢3 (x). (3.10.216) 10L 327 Remark (Selection of cut-o↵s, ⇢k ). As k increases from 2 to 3, the supports of ⇢k shift away from x = 1. The purpose is so that when obtaining the third-order estimate, the second- order terms should have a non-degenerate estimate in the support of ⇢3 , which is achieved so long as ⇢3 is supported sufficiently far to the right of x = 1 as compared to ⇢2 . Remark. It is worth emphasizing again the distinguishing feature of the second-order esti- mate, which vanishes for the third-order estimate. This is estimate (3.10.138) where factors of v (no derivative) appear. This subsequently forces the weight w to be “almost-linear” in the sense of (3.10.137). As soon as the order is upgraded to third-order, this problem vanishes, enabling us to apply the weights w3,L which vanish at x = 1. See (3.10.245) in the forthcoming estimate to contrast with (3.10.138). Let us now define the third-order forcing term: Z Z W3,E = |fxx ||uxx |w34 + ✏|gxx ||vxx |w34 , (3.10.217) Z Z W3,P = |fxx ||uxxx |w35 + ✏|gxx ||vxxx |w35 , (3.10.218) W3 = W3,E + WE,P . (3.10.219) We are now ready to state our energy estimate: Proposition 3.10.7 (Third-Order Energy Estimate). Let , ✏ be sufficiently small relative to universal constants, and " << . Then solutions [u, v] 2 Z(⌦N ) to the system (3.8.1) - (3.8.3) satisfy: p ||@x2 uy w32 ||2L2 + ✏||@x2 { ✏vx , vy }w32 ||2L2 2 X p 5 . O( )||@x2 { ✏vx , vy }w32 ||2L2 + ||u, v||2X1 \X2 + Wj + W3,E . (3.10.220) j=1 4 Proof. We shall apply the multiplier vx w3,L to the system (3.10.215). Notationally, we drop the subscript-3 from the weight, and simply call wL the weight appearing in (3.10.216) for 328 this calculation. We start with the highest-order terms, first from ✏ u: Z Z ⇣ ⌘ Z Z @y @x2 uyy ✏uxx · vx wL4 = ✏ uxx · uxx wL 4 Z Z ⇣ ⌘ Z Z 2 2 4 1 = uxxy + ✏uxxx wL ✏u2xx @x2 wL 4 . (3.10.221) 2 Next, the terms from ✏ v: Z Z ✏2 @x @x2 vxx vx wL 4 Z Z Z Z = ✏2 @x2 vxx vxx wL 4 ✏2 @x2 vxx vx @x wL 4 Z Z 1 2 2 2 4 = ✏2 vxxx 2 4 wL ✏ v @ w + C✏2 vx2 @x4 wL 4 . (3.10.222) 2 xx x L Finally, Z Z ✏@x @x2 vyy ·vx wL 4 Z Z = ✏@x2 vyy vxx wL 4 c✏@x2 vyy vx @x wL 4 Z Z 2 4 2 = ✏vxxy wL ✏vxy @x2 wL 4 . (3.10.223) For the final integrations from (3.10.221) - (3.10.223), we use the inequalities |@x2 wL 4 | . wL 2 , |@x4 wL 4 | . C, (3.10.224) to estimate: Z Z p | ✏u2xx @x2 wL 4 | . || ✏uxx wL ||2L2 . ✏||u||2X2 , (3.10.225) Z Z p | ✏2 vxx 2 @x2 wL 4 | . ✏|| ✏vxx wL ||2L2 . ✏||v||2X2 , (3.10.226) 329 Z Z p | ✏2 vx2 @x4 wL 4 | . ✏|| ✏vx ||2L2 . ✏||v||2X1 , (3.10.227) Z Z | 2 ✏vxy @x2 wL 4 | . ✏||uxx wL ||2L2  ✏||u||2X2 . (3.10.228) This then leaves from (3.10.221) - (3.10.223) the three terms on the left-hand side of (3.10.220). We now move to the profile terms contained in Su , which we will display here for convenience upon integrating by parts in y and using the divergence-free condition: Z Z h i 4 @y @xx uR ux + uRx u + vR uy + uRy v · vx wL Z Z h i 4 = @xx uR ux + uRx u + vR uy + uRy v · uxx wL . (3.10.229) First, we will expand via the product rule: Z Z 2 X Z Z 4 @xx (uR ux )uxx wL = ck @xk uR @x2 k 4 ux · uxx wL . (3.10.230) k=0 For k = 0, we integrate by parts and appeal to (3.3.12), (3.3.17): Z Z ⇣ ⌘ Z Z 2 4 | uR @x |uxx | wL | =| |uxx |2 @x (uR wL 4 )| Z Z ⇣ ⌘ =| |uxx |2 uRx wL 4 4 + cuR @x wL | 3 . ||uRx x, uR ||L1 ||uxx wL2 ||2L2 . ||u||2X2 . (3.10.231) For k > 0, we directly estimate using (3.3.11) and (3.3.17): Z Z | @xk uR @x2 k 4 ux · uxx wL | 5 k 3 . ||@xk uR xk ||L1 ||@x2 k ux wL2 ||L2 ||uxx x 2 ||L2 . ||u||2X1 \X2 . (3.10.232) 330 Let us now move to the second term from (3.10.229): Z Z 2 X Z Z @x2 (uRx u) · 4 uxx wL = ck @xk uRx @x2 k 4 u · uxx wL (3.10.233) k=0 For the k = 2 case, that is when all derivatives avoid the u term, we have to employ the splitting @x3 uR = @x3 uP 3 E R + @x uR , and treat each piece separately. First, by (3.3.11): Z Z 5 u 3 | @x3 uP 4 3 P R u · uxx wL |  ||@x uR yx ||L1 || ||L2 ||uxx wL ||L2 2 2 y 3 . ||uy ||L2 ||uxx wL2 ||L2 . ||u||X1 ||u||X2 . (3.10.234) Next, for the uE R contribution, by (3.3.18): Z Z 3 E 72 u 3 | @x3 uE 4 R u · uxx wL |  ||@x uR x ||L2 || ||L2 ||uxx wL ||L2 2 x 3 . ||ux ||L2 ||uxx wL2 ||L2 . ||u||X1 ||u||X2 . (3.10.235) For (3.10.233) when k < 2, we have by (3.3.11), (3.3.18): Z Z | @xk uRx @x2 k 4 u · uxx wL | 3 3  ||@xk uRx x1+k ||L1 ||@x2 k ux 2 k ||L2 ||uxx x 2 ||L2  ||u||X2 k ||u||X2 . (3.10.236) We will now move to the third term from (3.10.229): Z Z 2 X Z Z 4 @xx (vR uy ) · uxx wL = ck @xk vR @x2 k 4 uy uxx wL k=0 2 X 1 3 2 k  ||@xk vR xk+ 2 ||L1 ||@x2 k uy w L ||L2 ||uxx x 2 ||L2 k=0 331 2 X . O( )||@x2 uy wL 2 ||L2 ||u, v||X2 + ||u||X3 k ||u||X2 . (3.10.237) k=1 Above we have used estimate (3.3.10) and (3.3.20) to obtain the smallness of O( ) for the vR term, and estimate (3.3.8) for the remaining terms. We must take small enough to absorb the top order term above into (3.10.221). Finally, the fourth convective term from (3.10.229) is expanded into, Z Z 2 X Z Z @x2 (uRy v) · 4 uxx wL = ck @xk uRy @x2 k 4 v · uxx wL (3.10.238) k=0 We now split uR = uP E R + uR . First, according to (3.3.11), (3.3.13), (3.3.14), and (3.3.16): Z Z | @xk uP 2 Ry @x k 4 v · uxx wL | @x2 k v 5 3  ||@xk uP k Ry yx ||L1 || x2 k ||L2 ||uxx x 2 ||L2 y 5 3 . ||@xk uP k 2 Ry yx ||L1 ||@x k vy x 2 k ||L2 ||uxx x 2 ||L2 2 X 5 . O( )||@x2 vy x 2 ||L2 ||u||X2 + ||u, v||X3 k ||u||X2 . (3.10.239) k=1 Let us remark that term (3.10.239), when k = 0, requires the top order norm, ||v||X3 in order to control, and so the smallness of O( ) is essential above. This smallness is obtained n via (3.3.14) and (3.3.16), the latter of which is accompanied by " 2 . Next, we treat the Eulerian contribution via estimate (3.3.18): Z Z p | ✏ @xk uE 2 RY @x k 4 v · uxx wL | k+ 32 p 3  ||@xk uE RY x ||L1 || ✏@x2 k vx1 k ||L2 ||uxx x 2 ||L2 p . "||u, v||X2 k ||u||X2 . (3.10.240) 332 For the previous calculation, in the event that k = 2, we have used the Hardy inequality: p v p || ✏ ||L2 . || ✏vx ||L2 . ||v||X1 . (3.10.241) x We are now ready to move to the terms from Sv , which we depict here for convenience: Z Z h i ✏@x @x2 uR vx + vRx u + vR vy + vRy v · vx wL 4 (3.10.242) We will now work through the first term: Z Z 4 ✏@x @xx (uR vx )vx wL Z Z ⇣ ⌘ 4 4 = ✏@xx (uR vx ) vxx wL + vx @x wL 2 X X 2 Z Z = ck,i ✏@xk uR @x2 k vx · @xi v@x2 i wL 4 . (3.10.243) k=0 i=1 First, we will treat the k = 0 terms from (3.10.243), using estimates (3.3.12) and (3.3.18): Z Z Z Z Z Z ⇣ ⌘ 4 1 2 4 2 4 4 ✏uR vxxx · vxx wL = ✏vxx @x (uR wL ) =C ✏vxx uRx wL + uR @ x w L 2 p 3 . ||uR , uRx x||L1 || ✏vxx wL2 ||2L2 . ||v||2X2 . (3.10.244) The second k = 0 term, again using (3.3.12) and (3.3.18): Z Z Z Z 4 4 ✏uR vxxx vx @x wL = ✏vxx · @x {uR vx @x wL } Z Z 4 2 4 = ✏vxx uRx vx @x wL ✏vxx uR @ x w L ✏vxx uR vx @x2 wL 4 (3.10.245) ⇣ p 3 p 1 ⌘ . ||uR , uRx x||L1 || ✏vxx wL2 ||2L2 + || ✏vx x 2 ||2L2 (3.10.246) 333 . ||v||2X1 \X2 . (3.10.247) Remark. The third term in line (3.10.245) is the term which has improved for the higher- order energy estimates from the second order energy estimate, allowing us to use the cut-o↵ weight wL . Indeed, reading this term with k = 2 shows that a factor of v appears, which cannot be controlled in L2 . Next, we move to the k > 0 terms from (3.10.243), for which (3.3.11), (3.3.12), and (3.3.18) are in constant use: Z Z | ✏@xk uR @x2 k 4 vx · vxx wL | p 5 p 3 . ||@xk uR xk ||L1 || ✏@x3 k vx 2 k ||L2 || ✏vxx x 2 ||L2 . ||v||X3 k ||v||X2 , (3.10.248) Z Z | ✏@xk uR @x2 k 4 vx · vx @x w L | p 5 p 1 . ||@xj uR xj ||L1 || ✏@x3 k vx 2 k ||L2 || ✏vx x 2 ||L2 . ||v||X3 k ||v||X1 . (3.10.249) We will now approach the second term from (3.10.242), for which we use (3.3.8) and (3.3.17): Z Z ✏@x @x2 (vRx u)@x vwL 4 2 Z Z X = ✏@xk vRx @x2 k 4 u · vxx wL ✏@xk vRx @x2 k 4 u · vx @ x w L k=0 2 X p 3 ⇣ p 3  ✏||@xk vRx xk+ 2 ||L1 ||@x2 k ux1 k ||L2 || ✏vxx wL2 ||L2 k=0 p 1 ⌘ + || ✏vx wL2 ||L2  ||u, v||2X1 \X2 . (3.10.250) 334 Above, in the event when k = 2, we have used the Hardy inequality to obtain control || ux ||L2 . ||ux ||L2 . We will now approach the third term from (3.10.242), where again we use (3.3.8), (3.3.10), and (3.3.17): Z Z | ✏@x @x2 (vR vy )@x vwL4 | Z Z =| ✏@x2 (vR vy )vxx wL 4 + @x2 (vR vy )vx @x wL 4 | 2 X Z Z ⇣ ⌘ =| ck ✏@xk vR @x2 k 4 vy vxx wL 4 + vx @x wL | k=0 2 X p 5 ⇣ p 3 p ⌘ 1 k 1 . ✏||@xk vR x 2 +k ||L1 ||@x2 k vy wL2 ||L2 || ✏vxx wL2 ||L2 + || ✏vx x 2 ||L2 k=0 5 . ||v||2X1 \X2 + "||@x2 vy wL2 ||2L2 . (3.10.251) We move to the fourth term from (3.10.242): Z Z ✏@x @x2 (vRy v)vx wL 4 Z Z ⇣ ⌘ = ✏@x2 (vRy v) vxx wL 4 4 + Cvx @x wL 2 X Z Z ⇣ ⌘ = ck ✏@xk vRy @x2 k 4 v vxx wL 4 + Cvx @x wL . (3.10.252) k=0 P E We must split vR = vR + vR . First, we treat the Prandtl component by using estimates (3.3.8) - (3.3.10): Z Z ⇣ ⌘ | ✏@xk vRy P @x2 k 4 v vxx wL 4 + Cvx @x wL | p 1 kv 5 k p 3 p 3  ✏||@xk vRy P yx 2 +k ||L1 ||@x2 wL2 ||L2 || ✏vxx wL2 , ✏vx x 2 ||L2 y p 1 5 k p 3 p 3  ✏||@xk vRy P yx 2 +k ||L1 ||@x2 k vy wL2 ||L2 || ✏vxx wL2 , ✏vx x 2 ||L2 p p 5  ✏||u, v||2X1 \X2 + "||@x2 vy wL2 ||L2 ||u, v||X1 \X2 . (3.10.253) 335 Now, fix k < 2. The Eulerian contribution is controlled via (3.3.17): Z Z ⇣ ⌘ p ✏ ✏@xk vRY E @x2 k 4 v vxx wL 4 + Cvx @x wL p 3 p p 3 p 1  ✏||@xk vRY E x 2 +k ||L1 || ✏@x2 k vx1 k ||L2 || ✏vxx x 2 , ✏vx x 2 ||L2 p . ✏||v||X2 k ||v||X1 \X2 . (3.10.254) For k = 2, we must employ the Hardy inequality in addition to (3.3.17) to conclude: Z Z ⇣ ⌘ p ✏ ✏@x2 vRY E 4 v vxx wL 4 + Cvx @x wL p 3 p p 3 p 1  ✏||@x2 vRY E x 2 +2 ||L1 || ✏vx 1 ||L2 || ✏vxx x 2 , ✏vx x 2 ||L2 p 3 p p 3 p 1  ✏||@x2 vRY E x 2 +2 ||L1 || ✏vx ||L2 || ✏vxx x 2 , ✏vx x 2 ||L2 p 3  ✏||@x2 vRY E x 2 +2 ||L1 ||v||X1 ||v||X1 \X2 . (3.10.255) The final task is to address the right-hand side: Z Z Z Z 4 4 @xxy f · vx wL = fxx uxx wL , (3.10.256) Z Z Z Z 4 4 3 0 ✏ @xxx g · vx wL = ✏gxx {vxx wL + vx 4wL wL } Z Z 4 3 0 4 = ✏gxx vxx wL + C✏gx {vxx wL wL + vx @xx {wL }}. (3.10.257) We estimate the terms: Z Z Z Z | 3 0 "gx {vxx wL 4 wL + vx @xx {wL }}| . "|gx |{|vxx |wL3 2 + |vx |wL } Z Z  "|gx |{|vxx |w33 + |vx |w32 } Z Z  "|gx |{|vxx |w23 + |vx |w22 }  W2 . (3.10.258) 336 A comparison now with the definitions in (3.10.219) gives the desired result upon taking L ! 1. We now give the third-order positivity estimate, which is the final estimate for our linear analysis: Proposition 3.10.8 (Third Order Positivity Estimate). Let , ✏ be sufficiently small relative to universal constants. Then solutions [u, v] 2 Z(⌦N ) to the system (3.8.1) - (3.8.3) satisfy: p 5 p ||{ ✏vxxx , vxxy }w32 ||2L2 . ||{uxxy , ✏uxxx , ✏vxxx }w32 ||2L2 + ||u, v||2X1 \X2 + W1 + W2 + W3 . (3.10.259) 5 Proof. We apply the multiplier vxx w3,L . As usual, we shall drop the subscript-3 for this calculation, with the understanding that w3,L = wL .The highest-order terms are: Z Z 5 {@yxx ( + ✏@xxx ( ✏ v)} · vxx wL ✏ u) Z Z Z Z 5 = uyxx @x wL + C✏ u2xxx @x wL 5 + C✏2 vxxx 2 5 @x wL Z Z + C✏2 2 vxx @x3 {wL5 }. (3.10.260) Next, we have the main profile term from Su , which we will list here for convenience: Z Z h i 5 @y @xx uR ux + uRx u + vR uy + uRy v · vxx wL Z Z h i 5 = @xx uR ux + uRx u + vR uy + uRy v · uxxx wL . (3.10.261) 337 We will now treat the first term from (3.10.261): Z Z Z Z ⇣ ⌘ 5 5 @xx (uR ux ) · uxxx wL = uR uxxx + 2uRx uxx + uRxx ux · uxxx wL Z Z Z Z & min uR 2 5 uxxx wL + 5 uRx uxx uxxx wL 5 + uRxx ux uxxx wL . (3.10.262) Bounds on the final two integrals on the right-hand side above follow from (3.3.11), (3.3.18): Z Z 3 5 5 | uRx uxx uxxx wL |  ||uRx x||L1 ||uxx wL2 ||L2 ||uxxx wL2 ||L2 1 5  C||u, v||2X2 + ||uxxx wL2 ||L2 , (3.10.263) 100, 000 Z Z 1 5 5 | uRxx ux uxxx wL |  ||uRxx x2 ||L1 ||ux x 2 ||L2 ||uxxx wL2 ||L2 1 5  C||u, v||2X1 + ||uxxx wL2 ||2L2 . (3.10.264) 100, 000 The uxxx terms on the right-hand side above can be absorbed by the first term in (3.10.262). Here, we use that in the support of ⇢3 , C1 x  ⇢2 (x)  C2 x so that uxx from (3.10.262), for instance, does not require any of the degeneration from wL . The next term from (3.10.261) is: Z Z Z Z ⇣ ⌘ 5 5 @xx (uRx u) · uxxx wL | =| uRxxx u + 2uRxx ux + uRx uxx · uxxx wL . We shall now split uR = uP E P R + uR . First, the uR contribution, via estimate (3.3.11): Z Z 1 @x3 k u 5 | @xk uP i 5 k P k R @x · uxxx wL |  ||@x uR x 2 y||L1 || x3 k ||L2 ||uxxx wL2 ||L2 y 1 5  ||@xk uP Rx k 2 y||L1 ||@x3 k uy x3 k ||L2 ||uxxx wL2 ||L2 5 1 5 . ||u||X4 k ||uxxx wL2 ||L2  C||u, v||2X1 \X2 + ||uxxx wL2 ||2L2 , 100, 000 (3.10.265) 338 for k = 3, 2. It is worth distinguishing the k = 1 case, although the calculation is identical, by appealing to estimate (3.3.12): Z Z 1 5 | uP 5 P 2 Rx uxx uxxx wL |  ||uRx yx ||L1 ||uxxy wL ||L2 ||uxxx wL ||L2 2 2 5 2  O( )||uxxy wL ||L2 ||uxxx wL2 ||L2 , (3.10.266) because now both majorizers are part of the X3 norm: uxxy has been estimated in the energy estimate, and uxxx appears in (3.10.262). We may then absorb the uxxx term from above into (3.10.262) by taking sufficiently small. Next, the Eulerian contribution, for which we use (3.3.18), first with k = 1, 2: Z Z 1 1 5 | @xk uE 3 R @x k 5 u · uxxx wL |  ||@xk uE Rx k+ 2 ||L1 ||@x3 k ux3 k 2 ||L2 ||uxxx wL2 ||L2 p 5  "||u||X3 k ||uxxx wL2 ||L2 . (3.10.267) For the k = 3 case, we must add an extra step via Hardy’s inequality: Z Z 7 u 5 | uE 5 E Rxxx uuxxx wL |  ||uRxxx x ||L1 || ||L2 ||uxxx wL ||L2 2 2 x p 5 p 5 . "||ux ||L2 ||uxxx wL2 ||L2 . "||u||X1 ||uxxx wL2 ||L2 . (3.10.268) Next, we have the convection term (we set k = 0, 1, 2 below): Z Z 2 Z Z X 5 @xx (uRy v) · uxxx wL = @xk uRy @x2 k 5 v · uxxx wL k=0 2 Z Z ⇣ X p E ⌘ 2 = @xk uP Ry + ✏uRY @x k 5 v · uxxx wL . (3.10.269) k=0 339 First, the Prandtl contributions, using estimates (3.3.13): Z Z @x2 k v 3 k 1 5 | @xk uP 2 Ry @x k 5 v · uxxx wL | . ||@xk uP k Ry yx ||L1 || wL 2 ||L2 ||uxxx wL2 ||L2 y 1 5 3 k . ||@xk uP k 2 Ry yx ||L1 ||@x k vy wL 2 ||L2 ||uxxx wL2 ||L2 . Again, we distinguish the k = 0 case above, both majorizing terms are in X3 , and so we must use the smallness of O( ) to absorb into the (3.10.262) positive term. In particular, appealing to estimate (3.3.14), (3.3.16), we have: 5 5 5 5 ||uP 2 2 2 2 Ry y||L1 ||@x vy wL ||L2 ||uxxx wL ||L2  O( )||@x vy wL ||L2 ||uxxx wL ||L2 . 2 2 (3.10.270) For k > 0, we have by using (3.3.11) - (3.3.13) and then Young’s inequality: 1 5 5 3 k ||@xk uP k 2 Ry yx ||L1 ||@x k vy w L 2 ||L2 ||uxxx wL2 ||L2 . ||u, v||X1 \X2 ||uxxx wL2 ||L2 1 5  C||u, v||2X1 \X2 + ||uxxx wL2 ||2L2 . (3.10.271) 100, 000 For the Euler contributions uE R , we estimate for the k = 0, 1 cases, using (3.3.18): Z Z p k+1+ 12 p 2 k 1 5 | ✏@xk uE 2 RY @x k 5 vuxxx wL | . ||@xk uE RY x ||L1 || ✏@x2 k vwL 2 ||L2 ||uxxx wL2 ||L2 p 5 . "||v||X2 k ||uxxx wL2 ||L2 . (3.10.272) For the k = 2 case, we must additionally use the Hardy inequality: Z Z p 7 p v 5 RY xx vuxxx wL | . ||uRY xx x ||L1 || ✏ ||L2 ||uxxx wL ||L2 ✏uE 5 E | 2 2 x 7 p 5 . ||uE RY xx x ||L1 || ✏vx ||L2 ||uxxx wL ||L2 2 2 p 5 . "||v||X1 ||uxxx wL2 ||L2 . (3.10.273) 340 The final term in Su is easily estimated directly, where k = 0, 1, 2, by applying estimates (3.3.8) - (3.3.10), and (3.3.17), (3.3.20), crucially obtaining the factor O( ) when k = 0: Z Z 1 5 5 @xx (vR uy ) · uxxx wL . ||@xk vR xk+ 2 ||L1 ||@x2 k 2 k uy w L ||L2 ||uxxx wL2 ||L2 (3.10.274) 5 1 5 . O( )||@x2 uy wL 2 ||L2 ||uxxx wL2 ||L2 + ||uxxx wL2 ||2L2 + C||u, v||2X1 \X2 . 100, 000 We now address the profile terms from Sv . We shall record these below for convenience: Z Z h i 5 ✏@x @xx uR vx + vRx u + vR vy + vRy v · vxx wL Z Z h i h i 5 5 = ✏@xx uR vx + vRx u + vR vy + vRy v · vxxx wL + vxx @x wL . (3.10.275) Let us start with the first term above, which yields the desired positivity: Z Z 5 5 ✏@xx (uR vx ) · (vxxx wL + vxx @x wL ) Z Z 2 X X 3 2 5 = ✏uR vxxx wL + ✏@xk uR @x3 k v · @xi v@x3 i wL 5 k=1 i=2 Z Z 2 X X 3 2 5 min uR ✏vxxx wL + ✏@xk uR @x3 k v · @xi v@x3 i wL 5 . (3.10.276) k=1 i=2 We estimate the summation on the right-hand side above, by using (3.3.11) - (3.3.12), (3.3.18) Z Z | ✏@xk uR @x3 k v · @xi v@x3 i wL 5 || p 1 p i 1  ||@xk uR xk ||L1 || ✏@x3 k vx3 k 2 ||L2 || ✏@xi vwL 2 ||L2 1 p 5 . ||v||X3 k ||v||Xi  C||u, v||2X1 \X2 + || "vxxx wL2 ||2L2 100, 000 (3.10.277) 341 When i = 3, one obtains the top norm X3 above, and must be absorbed into the positive term from (3.10.276), which is done via Young’s inequality. The next profile term follows P E by using the bounds in (3.3.8) - (3.3.10) for vR and (3.3.17) for vR : Z Z 5 5 | ✏@xx (vRx u) · (vxxx wL + vxx @x wL )| 2 X X 3 Z Z  | ✏@xk+1 vR @x2 k u@xi v@x3 i wL 5 | k=0 i=2 2 X 3 p X 3 p i 1 . ✏ ||@xk+1 vR xk+ 2 ||L1 ||@x2 k ux1 k ||L2 || ✏@xi v · wL 2 ||L2 k=0 i=2 p p p 5 . ✏||u, v||2X1 \X2 + "|| "@x3 vwL2 ||2L2 . (3.10.278) Above, we have used the Hardy inequality in the case when k = 2, for the term: u || ||L2 . ||ux ||L2 . ||u||X1 . (3.10.279) x The third profile term requires a splitting into Euler and Prandtl components: Z Z 5 5 ✏@xx (vRy v) · (vxxx wL + vxx @x wL ) 3 Z Z 2 X X = ✏@xk vRy @x2 k v · @xi v@x3 i wL 5 k=0 i=2 3 Z Z ⇣ 2 X X p E ⌘ 2 = ✏@xk vRy P + ✏vRY @x k v · @xi v@x3 i wL 5 (3.10.280) k=0 i=2 First, according to (3.3.8) - (3.3.10), Z Z | ✏@xk vRy P @x2 k v@xi v@x3 i wL 5 | p 1 @x2 k v 3 k 1 p i 1  ✏||@xk vRy P yxk+ 2 ||L1 || wL 2 ||L2 || ✏@xi vwL 2 ||L2 y p 1 3 k 1 p i 1 . ✏||@xk vRy P yxk+ 2 ||L1 ||@x2 k vy wL 2 ||L2 || ✏@xi vwL 2 ||L2 342 p p p 5 p 5 . ✏||u, v||2X1 \X2 + "|| "@x3 vwL2 ||2L2 + "||@x2 vy wL2 ||2L2 . (3.10.281) Next, according to (3.3.17), Z Z 3 | ✏ 2 @xk vRY E @x2 k v@xi v@x3 i wL 5 | p 3 p 1 k p i 1  ✏||@xk vRY E xk+ 2 ||L1 || ✏@x2 k vwL ||L2 || ✏@xi vwL 2 ||L2 p p p 5 . ✏||u, v||2X1 \X2 + "|| "@x3 vwL ||2L2 . 2 (3.10.282) Above, we have used the Hardy inequality in the case when k = 2 for the term: p v p || ✏ ||L2 . || ✏vx ||L2 . ||v||X1 . (3.10.283) x The final profile term is handled by appealing to estimates (3.3.8) - (3.3.10) and (3.3.17): Z Z ⇣ ⌘ 5 5 | ✏@xx (vR vy ) · vxxx wL + vxx @x wL | 2 X X 3 Z Z " | ✏@xk vR @x2 k vy @xi v@x3 i wL | k=0 i=2 2 X X 3 1 1 1 3 k i " ||@xk vR xk+ 2 ||L1 ||@x2 k vy wL 2 ||L2 ||@xi vwL 2 ||L2 k=0 i=2 p 5 p p 5 . ||u, v||2X1 \X2 + "||@x2 vy wL2 ||2L2 + "|| "@x3 vwL2 ||2L2 . (3.10.284) Finally, we have the right-hand side: Z Z Z Z 5 5 @xxy f · vxx wL = fxx · vxxy wL , (3.10.285) Z Z Z Z 5 5 4 0 ✏@xxx g · vxx wL = ✏gxx {vxxx wL + 5vxx wL wL }. (3.10.286) 343 We estimate: Z Z Z Z Z Z 4 0 4 | "gxx vxx wL wL |  "|gxx ||vxx |wL  "|gxx ||vxx |w34  W3 . (3.10.287) We end by taking L ! 1. A comparison with (3.10.219) shows that our claim is proven. Summarizing, then, the conclusions of the linear analysis: Theorem 3.10.9 (Linear Estimates). Let ", be sufficiently small relative to universal constants, and let " << . Then [u, v] 2 Z, solutions to the system (3.8.1) - (3.8.3), (3.8.5) - (3.8.9), with boundary conditions (3.8.11), satisfy the following a-priori estimate: ||u, v||2X1 \X2 \X3 . W1 + W2 + W3 . (3.10.288) 3.11 Nonlinear Analysis 3.11.1 a-priori Estimate of Nonlinearities In this subsection, we exhibit control of the right-hand side of (3.10.288). We continue to consider the system (in a manner independent of N ): h i9 ¯x + v¯uy > n n "u + S u + Px = " 2 Ru,n + " 2 + u ¯u > > > > h i = Py in ⌦N n n v,n + (3.11.1) "v + Sv + =" 2 R +" 2 u ¯v¯x + v¯v¯y , " > > > > > ; ux + vy = 0. That is, f = f (u, u ¯, v¯) and g = g(¯ u, v¯), as in (3.8.8), (3.8.9). Notationally, we continue 344 RR to depict integration over ⌦N by and similarly, norms without further specification are taken over ⌦N . For the forthcoming calculation, we refer the reader to the definitions of W i , in equations (3.10.10), (3.10.82), (3.10.219). 1 Lemma 3.11.1. Suppose ||¯ u, v¯||Z  1. For 0  < 4, and fixed parameters  > 0 arbitrarily small, for , ✏ sufficiently small, n sufficiently large: 1 1 W1 + W2 + W3 . ✏ 4  + ✏4  ||u, v||2X1 \X2 \X3 n n !(Ni ) + ✏2 ||u, v||2Z + ✏ 2 !(Ni ) u, v¯||4Z , ||¯ (3.11.2) where !(Ni ) is a function which depends only on universal constants and Ni in the definition of norm Z. Proof. For clarity of exposition, the order in which we treat the terms are as follows: we will first treat the nonlinear terms, N u , arising from f in W1 , W2 , W3 , then those nonlinear terms, N v , arising from g, and finally the forcing terms, Ru,n , Rv,n , in both f and g. Turning first to W1 , equation (3.10.10), we start with the term: Z Z Z Z ⇣ ⌘ n Nu · u = "2+ u ¯u¯x + v¯uy · u. (3.11.3) First, Z Z n n 1 1 u "2+ u ¯u¯x · u . " 2 + ||¯ ux 4 ||L1 ||¯ ux x 2 ||L2 || 3 ||L2 (3.11.4) x4 n 1 1 1 . " 2 + ||¯ ux 4 ||L1 ||¯ ux x 2 ||L2 ||ux x 4 ||L2 n . "2+ !(Ni ) u, v¯||2Z ||u, v||Z . ||¯ For the next term in (3.11.3), we must use the structure of the nonlinearity by integrating 345 by parts once in y: Z Z Z Z n n u2 " 2 + v¯uy · u = " 2 + v¯y (3.11.5) 2 n 1 1 3 . " 2 + ||ux 4 ||L1 ||¯ vy x 2 ||L2 ||ux 4 ||L2 n 1 1 1 . " 2 + ||ux 4 ||L1 ||ux x 4 ||L2 ||¯ vy x 2 ||L2 n . "2+ !(Ni ) ||u, v||2Z ||¯ u, v¯||Z . Note carefully that no absolute values were included on this term in the definition of W1 , equation (3.10.10). We then move to the next term from (3.10.10), which we display below: Z Z Z Z n |N u ||vy |x  " 2 + |¯ uu¯x + v¯uy ||vy |x. (3.11.6) The nonlinear terms here are treated via: Z Z n n 1 1 " 2 + |¯ uu¯x vy x|  " 2 + ||¯ u||L1 ||¯ ux x 2 ||L2 ||vy x 2 ||L2 , n . "2+ !(Ni ) u, v¯||2Z ||u, v||Z , ||¯ (3.11.7) Z Z n n 1 1 " 2 + |¯ v uy vy x|  " 2 + ||¯ v x 2 ||L1 ||uy ||L2 ||vy x 2 ||L2 n . "2+ !(Ni ) u, v¯||Z ||u, v||2Z . ||¯ (3.11.8) Staying with N u , we will move to W2 in (3.10.82): Z Z h i |@x N u | |ux |⇢22 x2 + |uxx |⇢32 x3 Z Z h i n = " 2 + |@x (¯ uu¯x + v¯uy )| |ux |⇢22 x2 + |uxx |⇢32 x3 . (3.11.9) 346 First, we will expand: Z Z ⇣ ⌘ n " 2 + |@x u ¯x ||ux |⇢22 x2 ¯u Z Z Z Z n n + 2 2 2 = " 2 |¯ ux ||ux |⇢2 x + " 2 + |¯ uu¯xx ||ux |⇢22 x2 n 5 1 n 1 3  " 2 + ||ux x 4 ||L1 (x 20) ||¯ ux x 2 ||2L2 + " 2 + ||¯ u||L1 ||ux x 2 ||L2 ||¯ uxx x 2 ||L2 n  "2+ !(Ni ) u, v¯||2Z . ||u, v||Z ||¯ (3.11.10) Above, we have used that the support of ⇢2 is when x 50. Next, Z Z ⇣ ⌘ Z Z n n " 2 + |@x u ¯x ||uxx |⇢32 x3 = ¯u " 2 + |{¯ u2x + u ¯u¯xx }| · |uxx |⇢32 x3 5 1 3 1 3 3  ||¯ ux x 4 ||L1 (x 20) ||¯ ux x 2 ||L2 ||uxx x 2 ||L2 + ||¯ ux 4 ||L1 ||¯ uxx x 2 ||L2 ||uxx x 2 ||L2 n . "2+ !(Ni ) u, v¯||2Z . ||u, v||Z ||¯ (3.11.11) Next, the second nonlinearity in (3.11.9): Z Z ⇣ ⌘ Z Z n n " 2+ |@x v¯uy ||ux |⇢22 x2 = " 2 + |{¯ vx uy + v¯uxy }||ux |⇢22 x2 n 3 1 n 1 1  " 2 + ||¯ vx x 2 ||L1 (x 20) ||uy ||L2 ||ux x 2 ||L2 + " 2 + ||¯ v x 2 ||L1 ||uxy x||L2 ||ux x 2 ||L2 n . "2+ !(Ni ) u, v¯||Z ||u, v||2Z . ||¯ (3.11.12) For this same nonlinearity: Z Z n " 2 + |{¯ v uxy + v¯x uy }||uxx |⇢32 x3 n 1 3 n 3 3  " 2 + ||¯ v x 2 ||L1 ||uxy x||L2 ||uxx x 2 ||L2 + " 2 + ||¯ vx x 2 ||L1 (x 20) ||uy ||L2 ||uxx x 2 ||L2 n . "2+ !(Ni ) u, v¯||Z ||u, v||2Z . ||¯ (3.11.13) 347 We’ll now move to the N u terms in W3 (see (3.10.219)), which are the most delicate. These terms are: Z Z h i @xx N u |uxx ⇢43 x4 + |uxxx |2 ⇢53 x5 Z Z ⇣ ⌘ h i n = " 2 + @xx u¯u¯x + v¯uy · |uxx |⇢43 x4 + |uxxx |2 ⇢53 x5 . (3.11.14) First, Z Z ⇣ ⌘ Z Z n n " 2+ |@xx u ¯u¯x ||uxx |⇢43 x4 = " 2 + |{¯ uu¯xxx + 3¯ ¯xx }||uxx |⇢43 x4 ux u n 5 3  " 2 + ||¯ u||L1 ||¯ uxxx x 2 ||L2 (x 20) ||uxx x 2 ||L2 n 5 3 3 + " 2 + ||¯ ux x 4 ||L1 (x 20) ||¯ uxx x 2 ||L2 ||uxx x 2 ||L2 n . "2+ !(Ni ) u, v¯||2Z ||u, v||Z . ||¯ (3.11.15) Above, we have used that ⇢3 is supported in a strict subset of {x 20}, which in turn is supported in a strict subset of ⇣3 , which appears in the norm Z (see (3.9.1) - (3.9.2)). Referring back to (3.11.14), for this same nonlinear term: Z Z n " 2 + |{¯ uu¯xxx + 3¯ ¯xx }||uxxx |⇢53 x5 ux u n 5 5  " 2 + ||¯ u||L1 ||¯ uxxx x 2 ||L2 (x 20) ||uxxx x 2 ||L2 (x 20) n 5 3 5 2+ ||¯ ux x ||L1 (x +" 20) ||¯ uxx x ||L2 ||uxxx x 2 ||L2 (x 4 2 20) n " 2+ !(Ni ) u, v¯||2Z ||u, v||Z . ||¯ (3.11.16) We now move to the final nonlinearity contributed by N u , which is the @xx {vuy } term 348 from (3.11.14). This term is the most delicate to control. First, expanding yields: Z Z ⇣ ⌘ Z Z n n " 2 + |@xx v¯uy ||uxx |⇢43 x4 = " 2 + |{¯ vx uxy + v¯xx uy }||uxx |⇢43 x4 v uxxy + 2¯ (3.11.17) First, Z Z n n 1 3 " 2 + |¯ v uxxy ||uxx |⇢43 x4  " 2 + ||¯ v x 2 ||L1 ||uxxy x2 ||L2 (x 20) ||uxx x 2 ||L2 n  "2+ !(Ni ) u, v¯||Z ||u, v||2Z . ||¯ (3.11.18) Second, Z Z n n 3 3 " 2 + |¯ vx uxy ||uxx |⇢43 x4  " 2 + ||¯ vx x 2 ||L1 (x 20) ||uxy x||L2 ||uxx x 2 ||L2 n . "2+ !(Ni ) ||u, v||2Z ||¯ u, v¯||Z . (3.11.19) Finally, Z Z h i hZ i 12 n n 1 3 " 2+ vxx uy ||uxx |⇢43 x4 |¯ " 2+ sup ||uy x ||L2y ||uxx x ||L2 2 2 x4 ||¯ vxx ||2L1 y x 20 x=20 n ." 2+ !(Ni ) u, v¯||Z ||u, v||2Z . ||¯ (3.11.20) Turning back to (3.11.14), for this same nonlinearity, it remains to treat: Z Z n " 2 + |{¯ vx uxy + v¯xx uy }||uxxx |⇢53 x5 . v uxxy + 2¯ (3.11.21) 349 First, Z Z n n 1 5 " 2 + |¯ v uxxy ||uxxx |⇢53 x5  " 2 + ||¯ v x 2 ||L1 ||uxxy x2 ||L2 (x 20) ||uxxx x 2 ||L2 (x 20) n . "2+ !(Ni ) u, v¯||Z ||u, v||2Z . ||¯ (3.11.22) Second, Z Z n n 3 5 " 2 + |¯ vx uxy ||uxxx |⇢53 x5  " 2 + ||¯ vx x 2 ||L1 (x 20) ||uxy x||L2 ||uxxx x 2 ||L2 (x 20) n . "2+ !(Ni ) u, v¯||Z ||u, v||2Z . ||¯ (3.11.23) Last, Z Z n " 2 + |¯ vxx uy ||uxxx |⇢53 x5 n 1 5 hZ 1 i 12  " 2 + sup ||uy x 2 ||L2y ||uxxx x 2 ||L2 (x 20) x4 ||¯ vxx ||2L1 dx x 20 x=20 n ." 2+ !(Ni ) u, v¯||Z ||u, v||2Z . ||¯ (3.11.24) We now move to the nonlinear terms, N v , from g, defined in (3.8.9), starting with W1 , equation (3.10.10): Z Z h i Z Z h n ih i "|N v | |v| + |vx |x = " " 2 + |¯ uv¯x | + |¯ v v¯y | |v| + |vx |x . (3.11.25) First, Z Z n n 1 1 3 "2+ +1 uv¯x | · |v| . " 2 + |¯ +1 ||¯ ux 4 ||L1 ||¯ vx x 2 ||L2 ||x 4 v||L2 n 1 p 1 p 1 . " 2 + ||¯ ux 4 ||L1 || "¯ vx x 2 ||L2 || "vx x 4 ||L2 , 350 n . "2+ !(Ni ) u, v¯||2Z ||u, v||Z , ||¯ (3.11.26) Z Z n n 1 1 "2+ +1 v v¯y | · |v| . " 2 + |¯ +1 ||¯ v x 2 ||L1 ||¯ vy x 2 ||L2 ||vx ||L2 n . "2+ !(Ni ) u, v¯||2Z ||u, v||Z . ||¯ (3.11.27) Next, Z Z n n 1 1 "2+ +1 uv¯x ||vx |x  " 2 + |¯ +1 ||¯ u||L1 ||¯ vx x 2 ||L2 ||vx x 2 ||L2 , n . "2+ !(Ni ) u, v¯||2Z ||u, v||Z , ||¯ (3.11.28) Z Z n n 1 1 1 "2+ +1 v v¯y ||vx |x  " 2 + |¯ +1 ||¯ v x 2 ||L1 ||¯ vy x 2 ||L2 ||vx x 2 ||L2 , n . "2+ !(Ni ) u, v¯||2Z . ||u, v||Z ||¯ (3.11.29) We will now move to the nonlinear terms, N v , arising from W2 , (see equation (3.10.82)), which are summarized here: Z Z h i "|@x N v | |vx |⇢22 x2 + |vxx |⇢32 x3 Z Z h ih i n = " 2 + +1 |@x u ¯v¯x + v¯v¯y | |vx |⇢22 x2 + |vxx |⇢32 x3 (3.11.30) We will go through (3.11.30) term by term, starting with: Z Z ⇣ ⌘ Z Z n n "2+ +1 |@x u¯v¯x ||vx |⇢22 x2 = "2+ +1 |{¯ ¯v¯xx }| · |vx |⇢22 x2 ux v¯x + u n 5 1 1  "2+ +1 ||¯ ux x 4 ||L1 (x 20) ||¯ vx x 2 ||L2 ||vx x 2 ||L2 n 3 1 + "2+ +1 ||¯ u||L1 ||¯ vxx x 2 ||L2 ||vx x 2 ||L2 , n  "2+ !(Ni ) u, v¯||2Z ||u, v||Z . ||¯ (3.11.31) 351 Staying with this nonlinearity, Z Z n "2+ +1 |{¯ ¯v¯xx }| · |vxx |⇢32 x3 ux v¯x + u n 5 1 3  "2+ +1 ||¯ ux x 4 ||L1 (x 20) ||¯ vx x 2 ||L2 ||vxx x 2 ||L2 n 3 3 + "2+ +1 ||¯ u||L1 ||¯ vxx x 2 ||L2 ||vxx x 2 ||L2 n . "2+ !(Ni ) u, v¯||2Z ||u, v||Z . ||¯ (3.11.32) We now move to the vvy nonlinearity, still in term (3.11.30), which we expand: Z Z ⇣ ⌘ Z Z n n "2+ +1 |@x v¯v¯y ||vx |⇢22 x2 = "2+ +1 vx v¯y + v¯v¯xy }||vx |⇢22 x2 |{¯ n 3 1 1  "2+ +1 ||¯ vx x 2 ||L1 (x 20) ||¯ vy x 2 ||vx x 2 ||L2 n 1 1 3 + "2+ +1 ||¯ v x 2 ||L1 ||vx x 2 ||L2 ||¯ vxy x 2 ||L2 n . "2+ !(Ni ) u, v¯||2Z ||u, v||Z . ||¯ (3.11.33) For this same nonlinear term: Z Z n "2+ +1 vx v¯y + v¯v¯xy }| · |vxx |⇢32 x3 |{¯ n 3 1 3  "2+ +1 ||¯ vx x 2 ||L1 (x 20) ||¯ vy x 2 ||L2 ||vxx x 2 ||L2 n 1 3 3 + "2+ +1 ||¯ v x 2 ||L1 ||¯ vxy x 2 ||L2 ||vxx x 2 ||L2 n . "2+ !(Ni ) u, v¯||2Z . ||u, v||Z ||¯ (3.11.34) We now move to the highest-order terms, which we read from (3.10.219), and summarize here: Z Z h i n "2+ +1 |@xx N v | |vxx |⇢43 x4 + |vxxx |⇢53 x5 352 Z Z h ih i n = "2+ +1 |@xx u¯v¯x + v¯v¯y | |vxx |⇢43 x4 + |vxxx |⇢53 x5 . (3.11.35) We shall expand: Z Z ⇣ ⌘ n "2+ +1 |@xx u¯v¯x | · |vxx |⇢43 x4 Z Z n = " 2 + +1 |{¯ uv¯xxx + 2¯ ¯xx v¯x }| · |vxx |⇢43 x4 . ux v¯xx + u (3.11.36) First, Z Z n n 5 3 "2+ +1 uv¯xxx | · |vxx |⇢43 x4  " 2 + |¯ +1 ||¯ u||L1 ||¯ vxxx x 2 ||L2 (x 20) ||vxx x 2 ||L2 n . "2+ !(Ni ) u, v¯||2Z ||u, v||Z . ||¯ (3.11.37) Next, Z Z n n 5 3 3 "2+ +1 ux v¯xx | · |vxx |⇢43 x4  " 2 + |¯ +1 ||¯ ux x 4 ||L1 (x 20) ||¯ vxx x 2 ||L2 ||vxx x 2 ||L2 n  "2+ !(Ni ) u, v¯||2Z ||u, v||Z . ||¯ (3.11.38) Third, Z Z n n 3 3 3 "2+ +1 |¯ ¯xx ||vxx |⇢43 x4  " 2 + vx u +1 ||¯ vx x 2 ||L1 (x 20) ||¯ uxx x 2 ||L2 ||¯ vxx x 2 ||L2 n . "2+ !(Ni ) u, v¯||2Z ||u, v||Z . ||¯ (3.11.39) Turning back to (3.11.35), for this same nonlinearity, we will now treat: Z Z n "2+ +1 |{¯ uv¯xxx + 2¯ ¯xx v¯x }||vxxx |⇢53 x5 ux v¯xx + u (3.11.40) 353 First, Z Z n n 5 5 "2+ +1 uv¯xxx ||vxxx |⇢53 x5  " 2 + |¯ +1 ||¯ u||L1 ||¯ vxxx x 2 ||L2 (x 20) ||vxxx x 2 ||L2 (x 20) n . "2+ !(Ni ) u, v¯||2Z ||u, v||Z . ||¯ (3.11.41) Second, Z Z n n 5 3 5 "2+ +1 ux v¯xx ||vxxx |⇢53 x5  " 2 + |¯ +1 ||¯ ux x 4 ||L1 (x 20) ||¯ vxx x 2 ||L2 ||vxxx x 2 ||L2 (x 20) n . "2+ !(Ni ) u, v¯||2Z ||u, v||Z . ||¯ (3.11.42) Third, Z Z n n 3 3 5 "2+ +1 uxx v¯x ||vxxx |⇢53 x5  " 2 + |¯ +1 ||¯ vx x 2 ||L1 (x 20) ||¯ uxx x 2 ||L2 ||vxxx x 2 ||L2 (x 20) n . "2+ !(Ni ) u, v¯||2Z ||u, v||Z . ||¯ (3.11.43) Turning to (3.11.35), we now approach the final nonlinear term in g (see definition (3.8.9)), which is the v¯v¯y term. First, we will expand: Z Z ⇣ ⌘ n "2+ +1 |@xx v¯v¯y | · |vxx |⇢43 x4 Z Z n = " 2 + +1 |{¯vxx v¯y + v¯x v¯xy + v¯v¯xxy }| · |vxx |⇢43 x4 . (3.11.44) First, Z Z n n 5 3 3 "2+ +1 vxx v¯y | · |vxx |⇢43 x4  " 2 + |¯ +1 ||¯ ux x 4 ||L1 (x 20) ||¯ vxx x 2 ||L2 ||vxx x 2 ||L2 n . "2+ !(Ni ) u, v¯||2Z ||u, v||Z . ||¯ (3.11.45) 354 Second, Z Z n n 3 3 3 "2+ +1 vx v¯xy ||vxx |⇢43 x4  " 2 + |¯ +1 ||¯ vx x 2 ||L1 (x 20) ||¯ vxy x 2 ||L2 ||¯ vxx x 2 ||L2 n . "2+ !(Ni ) u, v¯||2Z ||u, v||Z . ||¯ (3.11.46) Third, Z Z n n 1 5 3 "2+ +1 v v¯xxy ||vxx |⇢43 x4  " 2 + |¯ +1 ||¯ v x 2 ||L1 ||¯ vxxy x 2 ||L2 (x 20) ||vxx x 2 ||L2 n . "2+ !(Ni ) u, v¯||2Z ||u, v||Z . ||¯ (3.11.47) Again turning to (3.11.35), for this same nonlinearity, we must also treat: Z Z n "2+ +1 vxx v¯y + v¯x v¯xy + v¯v¯xxy }| · |vxxx |⇢53 x5 . |{¯ (3.11.48) First, Z Z n "2+ +1 vxx v¯y ||vxxx |⇢53 x5 |¯ n 5 3 5  "2+ +1 ||¯ ux x 4 ||L1 (x 20) ||¯ vxx x 2 ||L2 ||vxxx x 2 ||L2 (x 20) n " 2+ !(Ni ) u, v¯||2Z ||u, v||Z . ||¯ (3.11.49) Second, Z Z n "2+ +1 vx v¯xy ||vxxx |⇢53 x5 |¯ n 3 3 5  "2+ +1 ||¯ vx x 2 ||L1 (x 20) ||¯ vxy x 2 ||L2 ||vxxx x 2 ||L2 (x 20) n ." 2+ !(Ni ) u, v¯||2Z ||u, v||Z . ||¯ (3.11.50) 355 Finally, Z Z n "2+ +1 v v¯xxy ||vxxx |⇢53 x5 |¯ n 1 5 5  "2+ +1 ||¯ v x 2 ||L1 ||¯ vxxy x 2 ||L2 (x 20) ||vxxx x 2 ||L2 (x 20) n . "2+ !(Ni ) u, v¯||2Z ||u, v||Z . ||¯ (3.11.51) This now concludes all of the nonlinear terms in Wi . The final task is to control the Ru,n , Rv,n terms in f, g. Via Lemma 3.7.22, for  arbitrarily small and n as in (3.7.1), we 0 1 choose now = 10,000 . Then, we have: Z Z Z Z n n n 1 0 1 0 1 ✏ 2 Ru,n u + ✏ 2 Ru,n vy x . ✏ 2 ||Ru,n x 2 + ||L2 ||x 2 u, x 2 vy ||L2 n 1 0  C(n)✏ 2 ||||Ru,n ||L2y x 2 + ||L2x ||u, v||X1 (3.11.52) 1 3 0  4 +2 n + +  C(n)✏ 4 ||x ||L2x ||u, v||X1 (3.11.53) 1   C(n)✏ 4 ||u, v||X1 . (3.11.54) 1 In (3.11.52), we have used the Hardy inequality with power x 2 , which is admissible as u(1, y) = 0. Upon citing (3.7.1), one has: 3 0 3 2 1 ||x 4 +2 n + + ||L2x || = ||x 4 + 10,000 + 10,000 + ||L2x < 1. (3.11.55) Next, again via Lemma 3.7.22, we have: Z Z ⇣ ⌘ n n p 1 0 1 0 p 1 p ✏1 2 |Rv,n | |v| + |vx |x  ✏ 2 || ✏Rv,n x 2 + ||L2 ||x 2 ✏v, x 2 ✏vx ||L2 1 . ✏4  ||u, v||X1 . (3.11.56) 356 Summarizing these forcing terms: Z Z n ✏ 2 Ru,n u + ✏Rv,n v + Ru,n vy x + ✏Rv,n vx x 1 1    ✏4 + ✏4 ||u, v||2X1 . (3.11.57) Similarly, for higher order terms, Z Z n ✏ 2 |Rxu,n |{|ux |⇢22 x2 + |uxx |⇢32 x3 } 1 5 0 3 1 3 3 . ✏4  ||x 1 4+ +0 x 2 ||L2x ||ux x 2 , uxx ⇢22 x 2 ||L2 1 . "4  ||u, v||X1 \X2 . (3.11.58) and Z Z n u,n ✏ 2 |Rxx |{|uxx |⇢43 x4 + |uxxx |⇢53 x5 } 1 5 0 5 3 3 5 5 . ✏4  ||x 2 4+ + x 2 ||L2x ||uxx ⇢32 x 2 , uxxx ⇢32 x 2 ||L2 1 . ✏4  ||u, v||X1 \X2 \X3 . (3.11.59) For the terms from g: Z Z n ✏ 2 ✏|Rxv,n |{|vx |⇢22 x2 + |vxx |⇢32 x3 } 1 5 0 3 p 1 p 3 3 .✏ 4  ||x 1 4+ + x 2 ||L2x || ✏vx x 2 , ✏vxx ⇢22 x 2 ||L2 1 ." 4  ||u, v||X1 \X2 . (3.11.60) 357 and Z Z n v,n ✏ 2 ✏|Rxx |{|vxx |⇢43 x4 + |vxxx |⇢53 x5 } 1 5 0 5 p 3 3 p 5 5 . ✏4  ||x 2 4+ + x 2 ||L2x || ✏vxx ⇢32 x 2 , ✏vxxx ⇢32 x 2 ||L2 1 . ✏4  ||u, v||X1 \X2 \X3 . (3.11.61) Combining all of the estimates we have established, we have proven Lemma 3.11.1. We may now prove the main result, Theorem 3.8.1: Proof of Theorem 3.8.1. The starting point is estimate (3.9.118): n ||u, v||2Z . ✏100 + ✏ 2 + !(Ni ) u, v¯||4Z + ||u, v||2X1 \X2 \X3 ||¯ (3.11.62) n . ✏100 + ✏ 2 + !(Ni ) u, v¯||4Z + W1 + W2 + W3 ||¯ (3.11.63) n 1 1 . ✏100 + ✏ 2 + !(Ni ) u, v¯||4Z + ✏ 4 ||¯  + ✏4  ||u, v||2X1 \X2 \X3 n n !(Ni ) + ✏2 ||u, v||2Z + ✏ 2 !(Ni ) u, v¯||4Z , ||¯ (3.11.64) so absorbing the ||u, v|| terms to the left-hand side gives: 1 n ⇣ ⌘ ||u, v||2Z . ✏ 4  + ✏2 !(Ni ) u, v¯||4Z . ||¯ (3.11.65) Part III: Existence and Uniqueness 358 359 3.12 Overview of Existence and Uniqueness With the main estimate, (3.11.65), in hand, we will prove existence and uniqueness of solutions to the nonlinear system specified in (3.8.1) - (3.8.3), which is defined on the domain ⌦, with boundary conditions given in (3.8.4), and f, g as in (3.8.5). The main result of this chapter is: 1 Theorem 3.12.1. For ", sufficiently small, " << ,  > 0 small, and 0  < 4, there exists a unique solution [u, v] 2 Z(⌦) to the system (3.8.1) - (3.8.3), (3.8.4), (3.8.5) satisfying the bound: 1 ||u, v||Z(⌦) . C(uR , vR )" 4  . (3.12.1) The main result, Theorem 3.2.2 follows immediately from Theorem 3.12.1. The proof of this theorem proceeds in several steps, which we now outline: (Step 1) Linear existence of solutions to weighted Stokes system, defined as follows: 2 ✏ + ↵A( ) = Fy ✏Gx on ⌦N , F, G 2 L2 (⌦N ), (3.12.2) |y=0,N = y |y=0,N = 0, and |x=1 = x |x=1 = 0, (3.12.3) lim [ x, y] = 0. (3.12.4) x!1 where ↵ > 0, and h A( ) = x2m yy x 2m+2 @x ( x x2m+2 ) + yyyy x2m+4 ⇣ ⌘ ⇣ ⌘i + @x yyx x2m+4 + @xx xx x2m+4 . (3.12.5) Here, m > 0 is sufficiently large, and can remain temporarily unspecified. The scaled 2 Bilaplacian is defined as ✏ := @y4 + ✏@y2 @x2 + ✏2 @x4 . The right-hand sides, F, G, should 360 1 be thought of as generic elements satisfying Fy "Gx 2 H . Upon introducing appropriate function spaces, we define the weak formulation of (3.12.2) - (3.12.3) in (3.13.6). Depicting the weak-solution operator to the above system by S↵ 1 (see 3.13.12 for a precise definition), Step 1 amounts to studying the solvability of S↵ = Fy "Gx . The boundary conditions as x ! 1 in (3.12.4) are selected in order to be consistent with (3.8.4). However, due to the terms in A( ), the weak solution, [ , u, v] exhibits rapid decay as x ! 1. (Step 2) Linear existence of compact perturbations to S↵ . Define the maps: h n i T [ ] := @y uR xy uRx y (vR + " 2 + v¯) yy + uRy x h i "@x uR xx vRy y + vR xy + vRy x , (3.12.6) n T0 [ ] := T [ ] + " 2 + @y [¯ v yy ], (3.12.7) h i Ta [ ] := uR xy uRx y vR yy + uRy x , (3.12.8) h i Tb [ ] := uR xx vRy y + vR xy + vRy x . (3.12.9) T has a dependence on v¯, so to be precise we will sometimes write T [ ; v¯]. When there is no danger of confusion, we simply write T [ ]. The map T0 [ ] is defined to match the profile terms, Su (u, v), Sv (u, v) (see the definition in (3.8.6)), when they are written in terms of the stream function, . We have defined the notation Ta , Tb so that we can write T0 = @y Ta "@x Tb . In this step, we are interested in establishing solvability of the system: S ↵ + T [ ] = Fy "Gx on ⌦N , (3.12.10) [ = x ]|x=1 =[ = y ]|y=0 =[ = y ]|y=N = lim [ x, y] = 0. (3.12.11) x!1 The essence of the arguments in this step is that upon applying S↵ 1 to both sides above, S↵ 1 T is seen as a compact perturbation of the identity. Despite ⌦N being unbounded in the x-direction, the required compactness arises from the weights, w, 361 present in A( ) above in (3.12.5). The solution of (3.12.10) is known to decay rapidly as x ! 1, due to the presence of A( ). This is captured in estimate (3.14.67). (Step 3) Nonlinear existence of auxiliary system: we first invite the reader to refer back to (3.8.5) and (3.8.8) - (3.8.9) for the definitions of f and g. Given this and the definition of T in (3.12.6), we define: n n n f˜(¯ u, v¯) := " 2 Ru,n + " 2 + u ¯u¯x , ¯, v¯) = f˜(¯ so that f (u, u u, v¯) + " 2 + v¯uy . (3.12.12) The aim of this step is to obtain existence of solutions (which we now index by ↵ and N for clarity) to the nonlinear system: S↵ ↵,N + T[ ↵,N ; v ↵,N ] = f˜y (u↵,N , v ↵,N ) + "gx (u↵,N , v ↵,N ) on ⌦N . (3.12.13) This existence is obtained in the unit ball of Z(⌦N ) via Schaefer’s fixed point theorem. (Step 4) Nonlinear existence of solutions to the system (3.8.1) - (3.8.3), with f, g as in (3.8.5): By re-applying the analyses in Sections 3.9 - 3.10 and in Lemma 3.11.1, one obtains the 1 uniform-in-(↵, N ) estimate: ||u↵,N , v ↵,N ||Z(⌦N ) . O( )" 4  , which then enables the passage to weak limits in the space X1 \ X2 \ X3 . The weak limit is denoted by [u, v], and is demonstrated to satisfy a weak formulation of system (3.8.1) - (3.8.3), see (3.16.10) for this formulation. Moreover, [u, v] 2 X1 \X2 \X3 , gives enough regularity to upgrade immediately to a strong solution of (3.8.1) - (3.8.3). Remark. To establish existence, we rely on compactness methods as opposed to ap- plying a contraction mapping. The essential reason for this is seen by examining calculation (3.11.5), in which the structure is not preserved under taking di↵erences. Remark. It is important to establish nonlinear existence of the auxiliary system before establishing nonlinear existence of the system (3.8.1) - (3.8.3), as opposed to jumping from linear existence of (3.8.1) - (3.8.3) to nonlinear existence because the compactness methods we rely on require the weights from ↵A( ). 362 (Step 5) Nonlinear uniqueness for solutions to the system (3.8.1) - (3.8.3), with f, g as in (3.8.5): In order to prove uniqueness, we re-apply the estimates in Sections 3.9 - 3.10 b with weights that are weaker by x , where b < 1, but is arbitrarily close to 1. This step is necessary (with the weaker weight) due again to the calculation in (3.11.5), whose structure is destroyed upon considering di↵erences. 3.13 Step 1: Invertibility of Weighted Stokes Operator, S↵ In this step, we study the system (3.12.2) - (3.12.3). We remind the reader that still, all integrations and all norms are taken over ⌦N unless otherwise specified. There is an abuse of notation here; should be indexed by ↵ and N , but this will not cause any confusion for this step, as we view both ↵ and N as fixed. Our intention of this subsection is to exhibit solvability of the system (3.12.2) in the space Z(⌦N ). Denote by 1 (x) a cut-o↵ function satisfying (refer to (3.9.1) for the definition of ⇣3 ): 12 11 1 = 1 on x , 1 = 0 for 1  x  . (3.13.1) 10 10 We define higher-order cut-o↵s similar to (3.13.1), satisfying the following property: support k ⇢{ k 1 = 1}. Define the following auxiliary norms via: Z Z || ||2Hw2 := 2 2m + |r |2 x2m+2 + |r2 |2 x2m+4 x (3.13.2) Z Z 2 || ||2Hw3 := || ||Hw2 + r2 x x2m+4 , (3.13.3) Z Z 2 2 2 || ||Gk := || ||Hwk + @yk , for any bounded subset B ⇢ ⌦N ,k = 0, ...3, (3.13.4) w,B B Z Z 2 2 2 2 2 k 2 || ||Hwk := || ||Hw3 + k r @x x2m+4 , for k 4. (3.13.5) 363 We will also call Gkw,loc (⌦N ) the space such that || ||Gkw,B  C(B) for all compact subsets B. Define the weak formulation of (3.12.2) to be: Z Z hZ Z Z Z Z Z i r2✏ : r2✏ + ↵ x2m + r · r x2m+2 + r2 : r2 x2m+4 = hFy ✏Gx , iH 1 ,H 1 for all 2 C01 (⌦N ), where 2 Hw2 (⌦N ). (3.13.6) Above, r2 is the Hessian matrix, and the inner product between two matrices is given by A : B = trace(AB). We will need one more norm: k X ||(F, G)||H 1 := ||@xj {Fy "Gx }||H 1 . (3.13.7) k j=0 Relevant spaces are defined here: Definition 3.13.1. Hw2 (⌦N ) is defined to be the closure of C01 (⌦N ) under the norm ||·||Hw2 . Hwk (⌦N ) for k 3 consists of the subspace of Hw2 (⌦N ) whose Hwk (⌦N ) norm is finite. Note that Hw3 (⌦N ) does not contain all of the third derivatives of ; it is missing @y3 , which is the reason for the norm, || · ||Gw,B . Remark. There is a distinction between Hw2 (⌦N ), and Hwk (⌦N ) in that: ||·||H 2 ||·||H 3 Hw2 (⌦N ) = C01 (⌦N ) w but for k 3, Hw3 (⌦N )⇢ = C01 (⌦N ) Z w . (3.13.8) Due to the weights, there is no “H = W ” theorem generically for Hwk (⌦N ). Lemma 3.13.2. For 2 Hw2 (⌦N ), the following boundary conditions are satisfied: |y=0,N = y |y=0,N = |x=1 = x |x=1 =0 (3.13.9) Proof. If 2 Hw2 (⌦N ), obtain a sequence (n) such that || (n) ||Hw2 ! 0. The claim now follows by the standard boundedness properties of the trace operator. 364 Lemma 3.13.3. Hw2 (⌦N ) as defined in Definition 3.13.1 is a Banach space. Proof. Consider the auxiliary space: n o 2 H0,w (⌦N ) = : r , r2 exist in the weak sense, and || ||Hw2 (⌦N ) < 1 . (3.13.10) 2 Through standard arguments, H0,w (⌦N ) is a Banach space. Suppose { (n) } is a Cauchy sequence in Hw2 (⌦N ). Then { (n) 2 } is Cauchy in H0,w (⌦N ), and so there exists a limit point (n) n!1 (n) (n) such that: || ||Hw2 ! 0. As 2 Hw2 (⌦N ), we may find a sequence { m }m 1 (n) m!1 (n) such that || m (n) ||Hw2 ! 0, where m 2 C01 (⌦N ). In particular, define, for each (n) (n) n (n) n!1 n, by selecting m large enough: || ||Hw2 < 2 . Thus, || ||Hw2 ! 0, ||·||H 2 proving that 2 C01 w . This establishes the desired result. Lemma 3.13.4. Endowed with the inner product, Z Z h , 'iHw2 := 'x2m + r · r'x2m+2 + r2 : r2 'x2m+4 , (3.13.11) Hw2 is a Hilbert Space. The inner product in (3.13.11) induces the norm defined in (3.13.2). Proof. One easily verifies the standard axioms of an inner-product for (3.13.11). Non- degeneracy of (3.13.11) is obtained via the boundary conditions in (3.12.3). Completeness is then obtained via Lemma 3.13.3 Definition 3.13.5. The ↵ Stokes operator is defined through: S↵ = Fy "Gx for 2 Hw2 (⌦N ), Fy "Gx 2 H 1 (⌦N ), if and only if (3.13.6) holds. (3.13.12) 365 It is our aim to study the invertibility of S↵ : 1 Lemma 3.13.6. Given Fy "Gx 2 H (⌦N ), there exists a unique weak solution 2 Hw2 (⌦N ) satisfying (3.13.6). Such a weak solution satisfies the energy inequality: 1 1 || ||2Hw2 . ||Fy ✏Gx ||2H 1 = ||S↵ ||2H 1 . (3.13.13) ↵ ↵ Proof. Define: Z Z hZ Z B[ , ] := r2✏ : r2✏ + ↵ x2m Z Z Z Z i + r · r x2m+2 + r2 : r2 x2m+4 . (3.13.14) It is immediate to see that B is bilinear, bounded, and coercive on Hw2 (⌦N ). Next, Fy "Gx act as bounded linear functionals on Hw2 (⌦N ) through the pairing: hFy ✏Gx , iHw 2 ,H 2 := hFy ✏Gx , iH 1 ,H 1 . This follows from: hFy ✏Gx , iH 1 ,H 1  w ||Fy ✏Gx ||H 1 || ||Hw2 . The existence of 2 Hw2 (⌦N ) a solution to (3.13.6) is then a stan- dard application of the Lax-Milgram Lemma to the Hilbert Space Hw2 (⌦N ). The energy identity above follows from density of C01 (⌦N ) in Hw2 (⌦N ), which enables us to replace with in (3.13.6). The above lemma then says that S↵ 1 : H 1 (⌦N ) ! Hw2 (⌦N ) is well-defined. Our intention now is to upgrade regularity. 1 Lemma 3.13.7. Given Fy "Gx 2 H (⌦), the unique weak solution in Hw2 (⌦N ) guaran- teed by Lemma 3.13.6 is in Hw3 (⌦N ) and satisfies: 1 1 || ||2Hw3 . ||Fy ✏Gx ||2H 1 = ||S↵ ||2H 1 . (3.13.15) ↵ ↵ Proof. As our weak solutions are only in Hw2 (⌦N ), we must formally use di↵erence quotients within the weak formulation (3.13.6) to upgrade to Hw3 (⌦N ). However, we will generate the 366 Hw3 estimate via di↵erentiating (3.12.2), with the understanding that everything that is done can be formalized through the use of di↵erence quotients in the standard manner. As such, we take @x of the system (3.12.2), which gives: 2 " x + ↵A( x) + [@x , ↵A] = @x (Fy ✏Gx ), (3.13.16) where h [@x , ↵A] = ↵ 2mx2m 1 (2m + 2)x2m+1 yy (2m + 2)@x ( xx 2m+1 ) ⇣ ⌘ + (2m + 4)x2m+3 yyyy + (2m + 4)@x yyx x 2m+3 ⇣ ⌘i + (2m + 4)@xx xx x2m+3) . (3.13.17) Let 1 be as above in (3.13.1). Define the quantities: x ˜1 = 1 1, ⇢M (x) = 1 (x) ˜1 ( ), which implies xk @xk ⇢M (x)  2. (3.13.18) M We now test the above equation, (3.13.16), against the multiplier ⇢M x. Doing so first gives from the Bilaplacian: Z Z Z Z h i 2 " x · ⇢M x = ⇢M "| xxy |2 + "2 | xxx |2 + | xyy |2 Z Z h i Z Z + c0 ⇢00M "| xy |2 + "2 | xx |2 + c1 @x4 ⇢M · | x| 2 , (3.13.19) for constants c0 , c1 . Next, we have the terms coming from A: Z Z ↵A( · ⇢M x) x Z Z &↵ [ 2 2m xx + 2 xy x 2m+2 + 2 xx x 2m+4 + 2 xyy x 2m+4 367 2 2m+4 2 2m+4 + xxy x + xxx x ]⇢M || ||2Hw2 . (3.13.20) Through a direct integration by parts, the commutator contains lower order terms: Z Z 1 [@x , ↵A] · x ⇢M . || ||2Hw2 . ||Fy ✏Gx ||2H 1 . (3.13.21) ↵ For detailed proofs of calculations (3.13.20) and (3.13.21), we refer the reader to (3.14.99) - (3.14.106). Finally, on the right-hand side of (3.13.16), we have: h@x (Fy ✏Gx ), x ⇢M i H 2 ,H 2  ||Fy ✏Gx ||H 1 ||⇢M xx ||H 1 . (3.13.22) We can send M ! 1 so that the weight ⇢M " 1, resulting in Z Z 2 1 1 r2 x x2m+4 . ||Fy ✏Gx ||2H 1 . (3.13.23) ↵ For the region 1  x  20, and 0  y  N , we apply the standard H˙ 2 (⌦N ) estimate for solutions, u↵ , v ↵ Stokes’ equation near corners (see (BR80), Theorems 1 and 2, and Figure 2, P. 562 also in (BR80) with “C/C” boundary conditions). Formally, fix another cut-o↵ function, 2 (x, y) localized near the corner (1, 0) (the identical argument can be given for the other corner, (1, N )). First, by calculation, we have: ⇣ ⌘ 2 2 2 ✏ 2 = 2 ✏ +[ ✏, ] , (3.13.24) where the expression for the commutator is given explicitly: 3 [ ✏, 2] = 4@y 2 @y + 4@y3 2 @y + 6@y2 2 2 @y + 2✏@x2 @y2 2 + 2✏@ 2 2 2 @x 368 + 4✏@x @y2 2 @x + 2✏@x2 2 2 @y + 4✏@x 2 2 @x @y + 4✏@x2 @y 2 @y 2 + 4✏@y 2 @x @y + 8✏@xy 2 @xy + 6✏2 @x2 2 2 @x + ✏2 @x4 2 + 4✏2 @x3 2 @x + 4✏2 @x 3 2 @x . (3.13.25) The salient feature of (3.13.25) will be: 3 [ ✏, 2] = o( 2@ ), (3.13.26) where this is short-hand notation for containing up to three -derivatives, and localized by 2 (or any derivative of 2 which is also localized). Localizing (3.12.2) using 2: || 2 ||H 3 . || 2 (Fy ✏Gx )||H 1 + ||o( 2@ 3 )||H 1 . || 2 (Fy ✏Gx )||H 1 . (3.13.27) Combining (3.13.27) and (3.13.23) gives the desired result. Lemma 3.13.8. Fix any bounded set B ⇢ ⌦N . Then we have: 1 || ||2G3 . C(B) ||Fy ✏Gx ||2H 1 , (3.13.28) w,B ↵ where the constant C(B) depends on B. Proof. This argument proceeds identically to the calculation from the previous lemma which resulted in (3.13.27) by simply replacing 2 with cut-o↵ functions localized to each interval x 2 [M, M + 1]. The dependence on B in the constant in (3.13.28) arises from the weights x2m , x2m+2 , x2m+4 appearing in the equation (3.12.2) through A( ). 369 The above lemmas roughly show that S↵ 1 gains four derivatives. By repeating this procedure for higher-order x-derivatives, we can upgrade to higher-regularity: Lemma 3.13.9. Given (F, G) 2 H2 1 , the unique weak solution guaranteed by Lemma 3.13.6 satisfies: 1 || |2Hw5 . ||(F, G)||2H 1 (3.13.29) ↵ 2 For (F, G) 2 H2 1 , we can upgrade weak solutions to strong solutions: Lemma 3.13.10. Given (F, G) 2 H2 1 , the unique weak solution guaranteed by Lemma 3.13.6 is a strong solution of (3.12.2). Proof. An integration by parts of the weak formulation (3.13.6), justified according to the previous lemma, is equivalent to the equation (3.12.2) being satisfied pointwise on ⌦N . The boundary conditions at x = 1, y = 0, y = N are satisfied by Lemma 3.13.2. The boundary condition at x ! 1 comes from the norms, (3.13.2), which when applying with k = 5, imply that up to four derivatives of vanish rapidly at x ! 1. 3.14 Step 2: Compact Perturbations, S↵ + T [ ] For this step, we invite the reader to refer back to the specification of T [ ], given in (3.12.6), and the system that we will focus on, given in (3.12.10). Note that T [ ] contains a loss of n three-derivatives for . Note also the presence of the term " 2 + v¯. We will now need some compactness lemmas. Lemma 3.14.1. Fix two weights, w1 = xm1 , and w2 = xm2 , where m2 > m1 0. Then, one has the following compact embedding: 1 Hloc (⌦N ) \ L2w2 (⌦N ) ,!,! L2w1 (⌦N ). (3.14.1) 370 Proof. Consider a family of functions {f n } defined on ⌦N such that: Z Z sup fn2 w22 < 1, (3.14.2) n 1 and such that fn 2 Hloc (⌦N ), uniformly in n. By taking Sobolev extensions across @⌦N , and subsequently cutting o↵ in the y and negative x directions, we can assume {fn } are defined on R2 , compactly supported in the y direction and negative x direction. Fix any 0 > 0. Since m2 > m1 , there exists a compact set K = K( 0 ) such that: 0 sup ||fn ||L2w (K c )  . (3.14.3) n 1 2 0 On K, by Rellich compactness, there exists a subsequence (depending on ) such that 0 lim sup ||fnj fnk ||L2 (K)  . (3.14.4) j,k!1 2 ⇥ diam(K)m1 Then, 0 lim sup ||fnj fnk ||L2w (K)  . (3.14.5) j,k!1 1 2 Combining the above two estimates, 0 lim sup ||fnj fnk ||L2w  . (3.14.6) 1 j,k!1 0 n Taking successively =2 and applying a diagonalization argument gives the result. Lemma 3.14.2. Let the weight, x2m , in the expression for A( ), equation (3.12.5), be 371 selected for any m > 0. Then the map S↵ 1 T is well-defined and compact H 2 (⌦N ) ! H 2 (⌦N ). Proof. According to (3.13.28), this follows from the compactness of G3w,loc (⌦N ) ,!,! H 2 (⌦N ), which in turn follows from (3.14.1). The lemma is proven. We are now ready to study system (3.12.10). The first task is to obtain an energy estimate to the inhomogeneous problem: Lemma 3.14.3. Suppose 2 H 2 (⌦N ) is a solution to (3.12.10), where (F, G) 2 H2 1 , and ||¯ u, v¯||Z  1. Then obeys the following energy estimate: Z Z p 1 1 ||uy ||2L2 + ↵|| ||2Hw2 . O( )|| ✏vx x , vy x 2 2 ||2L2 + F u + "|G||v|. (3.14.7) 1 Proof. Supposing there existed such a , we would have T [ ] 2 H (⌦N ), and so by (3.13.15), we know 2 Hw3 (⌦N ). By bootstrapping this regularity, we obtain that: 2 Hw5 (⌦N ). (3.14.8) We would like to apply the multiplier to the equation (3.12.10) in order to repeat the energy estimate from Proposition 3.10.1. Select test functions, (n) 2 C01 (⌦N ), which satisfy: (n) || ||Hw2 ! 0. (3.14.9) This is possible according to the density of C01 (⌦N ) in Hw2 in Definition 3.13.1. Multi- (n) plying (3.12.10) by , then gives on the left-hand side: Z Z ⇣ ⌘ Z Z 2 (n) (n) " +T · +↵ A( ) . (3.14.10) 372 First, we shall use (3.12.7) to write: Z Z ⇣ ⌘ Z Z ⇣ ⌘ Z Z n 2 (n) 2 (n) " + T[ ] = " + T0 [ ] " 2 + @y (¯ v uy ) · (n) Z Z ⇣ ⌘ Z Z n 2 (n) = " + T0 [ ] + " 2 + (¯ v uy ) · (n) y . (3.14.11) According to (3.14.9), we pass to limits in the following terms: Z Z Z Z Z Z 2 (n) ( " + T0 [ ]) = r2" : r2" (n) + T0 [ ] (n) Z Z Z Z n!1 ! |r2" |2 + T0 [ ] . (3.14.12) We have used: Z Z Z Z (n) (n) | T0 [ ] T0 [ ] | = (@y Ta [ ] @x Tb [ ]) · ( ) (n)  ||Ta [ ], Tb [ ]||L2 || ||H 1 (n) n!1  || ||Hw5 || ||H 1 ! 0, (3.14.13) according to (3.14.8) and the definition in equation (3.12.7). The integration on the right-hand side of (3.14.12) arises exactly from the energy estimates, Proposition 3.10.1, in particular, terms (3.10.12), (3.10.29), (3.10.41), and so we may write: Z Z ⇣ ⌘ p 1 1 | lim 2 " + T0 [ ] (n) | & ||uy ||2L2 O( )|| ✏vx x 2 , vy x 2 ||2L2 . (3.14.14) n!1 We may pass to the limit in the final term of (3.14.11) due to the calculation: Z Z (n) (n) | v¯uy ( y y )|  ||¯ v ||L1 ||uy ||L2 || y y ||L2 N4 (n) n!1 " ||¯ v ||Z ||uy ||L2 || y y ||L2 ! 0. (3.14.15) 373 Upon passing to the limit, we integrate by parts: Z Z Z Z Z Z n n (n) n!1 n "2+ " 2 + (¯ v uy ) · y ! " 2 + (¯ v uy ) · u = v¯y u2 (3.14.16) 2 From here, we estimate identically as in (3.11.5): Z Z n "2+ n 1 n 1 | v¯y u2 |  " 2 + !(Ni ) u, v¯||Z ||vy x 2 ||2L2  " 2 + ||¯ !(Ni ) ||vy x 2 ||2L2 . (3.14.17) 2 (n) It remains to treat (3.14.10), for which we use the compact support of to justify the integration by parts: Z Z Z Z (n) (n) 2m (n) 2m+2 ↵A( ) · = x +r ·r x + r2 : r2 (n) 2m+4 x . (3.14.18) Passing to the limit, according to (3.14.9): Z Z Z Z (n) 2 2m lim ↵A( ) =↵ x + |r |2 x2m+2 + |r2 |2 x2m+4 , (3.14.19) n!1 On the right-hand side, we have: Z Z Z Z Z Z (n) (n) n!1 Fy · = F y ! F u, (3.14.20) Z Z Z Z Z Z (n) (n) n!1 "Gx · = "G · x ! "Gv. (3.14.21) Consolidating the previous estimates gives the desired estimate, (3.14.7). The task now is to estimate the right-hand side of (3.14.7) in terms of the left-hand 374 side using the smallness of O( ). We refer the reader to Proposition 3.10.2, whose proof we follow closely. We will point out the subtle di↵erences: Lemma 3.14.4. Suppose 2 H 2 (⌦N ) is a solution to (3.12.10), where (F, G) 2 H2 1 , and u, v¯||Z  1. Suppose the weight w = x2m from equation (3.12.5) is selected such that m is ||¯ sufficiently large relative to universal constants. Then obeys: Z Z p 1 ||{ ✏vx , vy }x 2 ||2L2 . ||uy ||2L2 + ↵|| ||2Hw2 + |F ||ux |x + "|G||v| + "|G||vx |x. (3.14.22) Proof. We will repeat the positivity estimate of Proposition 3.10.2. To do so, we apply the 2 multiplier x x L,↵ to (3.12.10). Here, is a normalized cut-o↵ function equal to 1 on [1, 2] and 0 on [3, 1), and ↵ ↵k L,↵ (x) := ( x), so that @xk L,↵ = (k) . (3.14.23) L Lk Such a cut-o↵ function was not present in Proposition 3.10.2. The necessity of it is due to the terms arising from A( ). The presence of this cut-o↵ function enables us to justify all integrations by parts in the x-direction. For our fixed ↵ > 0, we will eventually send 2 L ! 1. Applying the multiplier x x L,↵ to (3.12.10), gives on the left-hand side: Z Z ⇣ ⌘ Z Z 2 2 2 T[ ] + " · x x L,↵ +↵ A( ) · x x L,↵ (3.14.24) Z Z ⇣ ⌘ Z Z n 2 = T0 [ ] + " + " 2 + @y [¯ v uy ] · 2 x x L,↵ +↵ A( ) · 2 x x L,↵ . We will first focus on the first two integrands above in (3.14.24), which appeared in Proposition 3.10.2. The obstacle to repeating the calculations exactly as in Proposition 2 3.10.2 is the presence of the cut-o↵ function, L,↵ , in the multiplier. The essential idea is 2 this: when no derivative falls on L,↵ , the estimate will be the same as the corresponding 2 term in Proposition 3.10.2. When at least one derivative falls on L,↵ , we may use the factors of ↵ obtained from the scaling in (3.14.23) to absorb the new terms into the left- 375 hand side of (3.14.7). Let us start with the profile terms from T0 [ ], for which we refer the reader to estimates (3.10.64) - (3.10.68). We will transfer all of the terms to velocity formulation so as to remain consistent with estimates (3.10.64) - (3.10.68). Z Z Z Z 2 2 @y Su · vx L,↵ = Su u x x L,↵ Z Z = [uR ux + uRx u + vR uy + uRy v]ux x 2L,↵ Z Z Z Z & u2x x 2L,↵ | [uRx u + vR uy + uRy v]ux x 2 L,↵ |. (3.14.25) We will treat the three terms on the right-hand side above, using (3.3.12) and (3.3.18) starting with: Z Z Z Z 2 uRx uxux L,↵ = {uP E Rx + uRx }uxux 2 L,↵ 1 1  ||yx 2 uP Rx ||L1 ||uy ||L2 ||vy x 2 L,↵ ||L2 3 u 1 + ||uE Rx x ||L1 || 2 L,↵ ||L2 ||ux x 2 L,↵ ||L2 x 1 ↵  O( )||uy ||2L2 + O( )||ux x 2 2 L,↵ ||L2 + ||u||2L2 L 1 ↵  O( )||uy ||2L2 + O( )||ux x 2 2 L,↵ ||L2 + || ||2Hw2 . (3.14.26) L Above, we have used the Hardy inequality: u ↵ || L,↵ ||L2 . ||@x (u L,↵ )||L2  ||ux L,↵ ||L2 + ||u 0 L,↵ ||L2 x L ↵ . ||ux L,↵ ||L2 + || ||Hw2 . (3.14.27) L Next, by (3.3.10), (3.3.20), we have: Z Z 1 1 2 v R uy v y x L,↵  ||vR x 2 ||L1 ||uy ||L2 ||vy x 2 L,↵ ||L2 1  O( )||uy ||L2 ||vy x 2 L,↵ ||L2 . (3.14.28) 376 Next, by (3.3.14), (3.3.16), (3.3.18) we have: Z Z Z Z 2 p uRy vux x L,↵ = {uP Ry + ✏uE RY }vvy x 2 L,↵ 1  ||yuP Ry ||L1 ||vy x 2 2 L,↵ ||L2 p 3 1 p v + ✏||uE RY x ||L1 ||vy x 2 2 L,↵ ||L2 || ✏ L,↵ ||L2 x 1  ||yuP Ry ||L1 ||vy x 2 2 L,↵ ||L2 p 3 1 p + ✏||uE RY x ||L1 ||vy x 2 2 L,↵ ||L2 || ✏vx L,↵ ||L2 ↵p 3 1 p 0 + ✏||uE RY x ||L1 ||vy x 2 2 L,↵ ||L2 || ✏v L,↵ ||L2 L 1 2 p p 2 ↵  O( )||vy x 2 L,↵ ||L2 + ✏O( )|| ✏vx L,↵ ||L2 + O( ) || |2Hw2 . L (3.14.29) Summarizing the previous four terms: Z Z 1 ↵ @y Su · vx 2 L,↵ & ||vy x 2 2 L,↵ ||L2 || ||2Hw2 L p p O( )||uy ||2L2 O( ) "|| "vx 2 L,↵ ||L2 . (3.14.30) We will now move to the profile terms from Sv , which are located starting from estimate (3.10.69). First, Z Z Z Z 2 2 2 2↵ 0 "@x Sv · vx L,↵ = "Sv · [vx x L,↵ +v L,↵ + vx L,↵ L,↵ ]. (3.14.31) L Referring to definition (3.8.5), consider the term uR vx in Sv , which is the most delicate profile term: Z Z Z Z ↵ 0 "uR vx2 x 2 L,↵ + "uR vx v[ L,↵ + 2x L,↵ L,↵ ]. (3.14.32) L 377 The first term above in (3.14.32) gives positivity: Z Z Z Z "uR vx2 x 2 L,↵ & "vx2 x 2 L,↵ . (3.14.33) We will treat the second term on the right-hand side of (3.14.32): Z Z Z Z 2 v2 ↵ 0 "uR vvx L,↵ = " [uRx 2L,↵ + 2uR L,↵ L,↵ ], (3.14.34) 2 L Z Z Z Z ↵ 0 v2 ↵ 0 "uR vvx x L,↵ L,↵ = " [uRx x L,↵ L,↵ L 2 L ↵ 0 ↵2 00 + uR L,↵ L,↵ + uR x 2 L,↵ L,↵ ]. (3.14.35) L L The first term on the right-hand side of (3.14.34) yields: Z Z p p 1 ↵ | "v 2 uRx 2 L,↵ | . "|| L,↵ { "vx , vy }x 2 ||2L2 + || ||2Hw2 , (3.14.36) L in nearly an identical manner to estimate (3.10.43). We now estimate: Z Z Z Z uR 2 ↵ ↵ ↵ | " v 0 L,↵ | . "v 2 . || ||2Hw2 . (3.14.37) 2 L L L This same estimate can be performed for all the terms in (3.14.35). Consolidating these bounds: Z Z p p 1 ↵ "vx2 x 2 L,↵ . (3.14.32) + "|| L,↵ { "vx , vy }x 2 ||2L2 + || ||Hw2 . (3.14.38) L It remains now to treat the remaining three terms in Sv . The second, third, and fourth 378 terms from Sv can be controlled in the same manner as in (3.10.71) - (3.10.73): Z Z 2 p 3 p 1 ↵ ✏| vRx uvx x L,↵ |  ✏||x 2 vRx ||L1 || ✏vx x 2 L,↵ ||L2 ||ux L,↵ ||L2 + || ||2Hw2 . L (3.14.39) Z Z 2 p 1 p 1 ↵ ✏ vR vy vx x L,↵  ✏||vR ||L1 ||vy x 2 L,↵ ||L2 || ✏vx x 2 L,↵ ||L2 + || ||2Hw2 . (3.14.40) L Z Z 2 p P 1 p 1 ✏ vRy vvx x L,↵  ✏||vRy y||L1 ||vy x 2 L,↵ ||L2 || ✏vx x 2 L,↵ ||L2 p E 3 p 1 2 ↵ + ✏||vRY x 2 ||L1 || ✏vx x 2 L,↵ ||L2 + || ||2Hw2 . L (3.14.41) We now turn back to (3.14.31), addressing the second term in the bracket for the final three profile terms from Sv : Z Z 2 [vRx u + vR vy + vRy v] · "v L,↵ . (3.14.42) First, through the Hardy inequality and (3.3.8), (3.3.17): Z Z 2 p 3 u p v | "vRx uv L,↵ |  "||x 2 vRx ||L1 || 3 L,↵ ||L2 || " 3 L,↵ ||L2 x 4 x4 p h 1 2 p 1 2 ↵ p 1 0 2 i  " ||ux x 4 L,↵ ||L2 + || "vx x 4 L,↵ ||L2 + ||{u, "vx 4 L,↵ ||L2 L p h 1 2 p 1 2 ↵ i  " ||ux x 4 L,↵ ||L2 + || "vx x 4 L,↵ ||L2 + || ||2Hw2 , (3.14.43) L 1 so long as w = xm is selected larger than x 4 , which is true by the assumption of this lemma. Next, through an integration by parts and (3.3.10), (3.3.20): Z Z Z Z 2 1 [vR vy + vRy v]"v L,↵ = vRy v 2 " 2 L,↵ P  ✏||vRy y 2 ||L1 ||vy 2 L,↵ ||L2 2 p E 3 p 1 2 ↵ + ✏||vRY x 2 ||L1 || ✏vx x 4 L,↵ ||L2 + || ||2Hw2 . (3.14.44) L 379 The final task for the Sv profile contributions is the third term from (3.14.31): Z Z 2↵ 0 ↵ | "[vRx u + vR vy + vRy v] · vx L,↵ L,↵ |  || ||2Hw2 , (3.14.45) L L so long as w = xm is selected larger than x, which is true by assumption of this lemma. Let us consolidate all of the calculations from Sv : Z Z Z Z p p 1 ↵ "@x Sv · vx 2 L,↵ & "vx2 x 2 L,↵ "|| L,↵ { "vx , vy }x 2 ||2L2 || ||2Hw2 . L (3.14.46) 2 It now remains to come to those terms contributed by " into (3.14.24). We will follow closely the calculations from (3.10.52) - (3.10.63) in Proposition 3.10.2. We will again omit the justifications near the corners of our domain as these are identical to Proposition 3.10.2. Again, we will write these terms in the velocity form, to remain consistent with the calculations in (3.10.52) - (3.10.63). First, Z Z Z Z u2y ↵ | uyy ux x 2 L,↵ | =| [ 2 L,↵ + 2x 0 L,↵ L,↵ ]| . ||uy ||2L2 . (3.14.47) 2 L Next, Z Z Z Z u2x ↵ ↵ "uxx ux x 2 L,↵ | =| " [ 2 L,↵ + 2x 0 L,↵ L,↵ ] . "||ux 2 L,↵ ||L2 + " || ||2Hw2 . 2 L L (3.14.48) We now move to the terms from " v, starting with: Z Z Z Z ↵ 0 | "2 vxxx · vx 2 L,↵ | =| "2 vxx · [vx x 2 L,↵ +v 2 L,↵ + 2vx L,↵ L,↵ ]| L p 2 ↵  "|| "vx L,↵ ||L2 + || ||2Hw2 . (3.14.49) L 380 Finally, we have: Z Z Z Z 2 2 | "vxyy vx L,↵ | =| "vxy vy x L,↵ | Z Z ↵ =| "vy2 [ 2 L,↵ + 2x L,↵ L,↵ ]| L ↵ . "||vy 2 L,↵ ||L2 + || ||2Hw2 . (3.14.50) L By combining calculations (3.14.30), (3.14.46), (3.14.47) - (3.14.50), and absorbing rele- vant terms to the left-hand side below, we have: p 1 ↵ ||{ ✏vx , vy }x 2 2 L,↵ ||L2 . ||uy ||2L2 + || ||2Hw2 Z Z L Z Z n 2 +↵ A( ) · x x L,↵ + " 2 + v¯uy ux x 2 L,↵ Z Z + Fy · vx 2L,↵ "Gx · vx 2L,↵ . (3.14.51) Via direct integration by parts, which is justified due to the presence of the cut-o↵ function in x, we compute: Z Z ↵ A( ) 2 x x L,↵ . ↵|| ||2Hw2 . (3.14.52) Let us compute each term in A( ) to verify (3.14.52), referring to the definition in (3.12.5), starting with: Z Z Z Z ↵ | ↵ x2m vx 2 L,↵ | =| 2 @x [x2m+1 2 L,↵ ]| 2 Z Z ↵ 2 ↵ 0 =| [Cx2m 2 L,↵ + 2x2m+1 L,↵ L,↵ ]| (3.14.53) 2 L  ↵|| ||2Hw2 . (3.14.54) 381 ↵ For the second term in (3.14.53), we have used: | L x L,↵ | . 1. Next, let us turn to: Z Z 2m+2 2m+4 2 ↵ ( yy x + yyyy x )vx L,↵ Z Z =↵ uy vx2m+3 2 L,↵ ↵uy uxy x2m+5 2 L,↵ Z Z =↵ uux x2m+3 2 L,↵ + ↵u2y @x (x2m+5 2 L,↵ ) Z Z .↵ u2 x2m+2 + u2y x2m+4 . ↵|| ||2Hw2 . (3.14.55) Next, Z Z Z Z 2m+2 2 ↵ @x ( xx )vx =↵ L,↵ vx2m+2 @x (vx 2 L,↵ ) Z Z ↵ 0 =↵ vx2m+2 [vx x 2L,↵ + v 2 L,↵ + 2vx L,↵ L,↵ ] L Z Z .↵ v 2 x2m+2 . ↵|| ||2Hw2 . (3.14.56) Next, Z Z Z Z 2m+4 2 ↵ @x ( yyx x )vx =↵ L,↵ @x ( xy x2m+4 )vy x 2L,↵ Z Z Z Z =↵ @x (ux x2m+4 )ux x 2L,↵ . ↵ u2x x2m+4 . ↵|| ||2Hw2 . (3.14.57) The final term in A( ) is: Z Z Z Z 2m+4 2 2m+4 2 ↵ @xx ( xx x )vx L,↵ =↵ xx x @xx [vx L,↵ ] Z Z =↵ vx x2m+4 [vxx x 2 L,↵ + 2vx @x (x L,↵ ) + v@xx (x L,↵ )] . ↵|| ||2Hw2 . (3.14.58) 382 This concludes all the terms in A( ), according to (3.12.5). Estimating the next term in (3.14.51) exactly as in (3.11.8) yields: Z Z n n 1 | " 2 + v¯uy ux x 2 L,↵ |  "2+ !(Ni ) ||¯ u, v¯||Z ||uy ||L2 ||ux x 2 L,↵ ||L2 n 1 n  "2+ !(Ni ) ||ux x 2 2 L,↵ ||L2 + "2+ !(Ni ) ||uy ||2L2 . (3.14.59) Finally, we come to the right-hand side: Z Z Z Z [Fy "Gx ] · vx L,↵ = F ux x + "G@x [vx L,↵ ] L,↵ Z Z Z Z ↵ 0  |F ||ux |x + | "G[vx x L,↵ + v L,↵ + vx( ) L,↵ ]| L Z Z  |F ||ux |x + "|G||vx |x + "|G||v|., (3.14.60) Inserting the previous few calculations into estimate (3.14.51) gives: p 1 ↵ ||{ ✏vx , vy }x 2 2 L,↵ ||L2 . ||uy ||2L2 + || ||2 2 LZ Z Hw + ↵|| ||2Hw2 + |F ||ux |x + "|G|[|vx |x + |v|]. (3.14.61) We now send L ! 1, and appeal to Monotone Convergence Theorem, as L,↵ " 1 to establish the desired result. Having understood the inhomogeneous problem: Lemma 3.14.5. For (F, G) 2 H2 1 , and ||¯ u, v¯||Z  1, there exists a unique weak solution 2 Hw2 (⌦N ) to the system (3.12.10). 383 Proof. We apply S↵ 1 to both sides of (3.12.10), which is valid as the right-hand side and 1 therefore the left-hand side is assumed to be in at least H (⌦N ), thereby yielding: ⇣ ⌘ + S↵ 1 T = S ↵ 1 Fy "Gx . (3.14.62) We will study the equation (3.14.62) as an equality in the space H 2 (⌦N ). According to the Fredholm alternative, which is available according to Lemma 3.14.2, there either exists a unique solution 2 H 2 (⌦N ) to the system (3.14.62), or a non-trivial solution 2 H 2 (⌦N ) to: + S↵ 1 T = 0 () S↵ = T . (3.14.63) Therefore, coupling (3.14.22) with (3.14.7), taking F = G = 0, we have: p p 1 1 || ✏ux , uy ||2L2 + ↵|| ||2Hw2 + || ✏vx x 2 , vy x 2 ||2L2  0, (3.14.64) implying , u, v = 0. Thus, by the Fredholm alternative, there exists a unique solution 2 H 2 (⌦N ) to (3.14.62). Rearranging (3.14.62): ⇣ ⌘ = S ↵ 1 Fy "Gx T , (3.14.65) 1 where Fy "Gx T 2H (⌦N ), and so an application of (3.13.13) shows that 2 Hw2 (⌦N ). This concludes the proof. Lemma 3.14.6. Let be the unique Hw2 (⌦N ) weak solution from Lemma 3.14.5. Then for (F, G) 2 H2 1 , 2 Hw5 (⌦N ). 384 1 Proof. T 2H (⌦N ), and so S↵ = T + Fy "Gx 2 H 1 (⌦N ), which implies that 2 Hw3 (⌦N ) according to (3.13.15). Iterating this regularity then gives 2 Hw5 (⌦N ). We now introduce more notation, which is more suitable for the velocities: L↵,¯v [u, v] = (f˜, g) () S↵ + T [ ; v¯] = f˜y "gx . (3.14.66) Summarizing the established results, we have: Corollary 3.14.7. For f˜, g 2 H2 1 (⌦N ), ↵ > 0, and ||¯ u, v¯||Z(⌦N )  1, the map L↵,¯v [u, v] is invertible, where 1 ˜ 1 N 4 N v : (f , g) 2 H2 (⌦ ) ! [u, v] 2 Hw (⌦ ). L↵,¯ (3.14.67) 1 ˜ Moreover, the boundary conditions (3.12.3) are satisfied by [u, v] = L↵,¯ v [f , g]. It is now our intention to repeat the second and third order energy and positivity esti- mates from Section 3.10, with our new system (3.12.10). For this, we will need to understand several calculations. First, we introduce some norms: || ||2J 2 := || ||2Hw2 , (3.14.68) Z Z || ||2J k+2 := |@xk |2 (⇢k+1 )2k x2m+2k + |r@xk |2 ⇢2k k+1 x 2m+2k+2 (3.14.69) + |r2 @xk |2 ⇢2k k+1 x 2m+2k+4 for k 1. (3.14.70) The reader is referred to the definitions of ⇢k provided in (3.9.2). The essential di↵erence between these J k -norms and the Hwk norms introduced in (3.13.2) are the growing weights of x which each application of @x , which mimics the structure of the energy norms, Xk , in (3.9.3). 385 Lemma 3.14.8. Z Z k X1 A(@xk ) · @xk x2k 2 2k L,↵ ⇢k+1 & || L,↵ ||2J k+2 || ||2J i+2 (3.14.71) i=0 Proof. Referring to (3.12.5), the first term is: Z Z |@xk |2 x2m x2k 2 2k L,↵ ⇢k+1 . (3.14.72) The next terms, via an integration by parts in y: Z Z Z Z @xk yy x 2m+2 · @xk x2k ⇢2k k+1 2 L,↵ = |@xk y |2 x2m+2k+2 ⇢2k k+1 2 L,↵ , (3.14.73) Z Z Z Z @xk yyyy x 2m+4 · @xk x2k ⇢2k k+1 2 L,↵ = |@xk yy |2 x2m+2k+4 ⇢2k k+1 2 L,↵ . (3.14.74) Next, Z Z @x [(@xk )yyx x2m+4 ] · @xk x2k 2L,↵ ⇢2k k+1 Z Z = @xk+1 y x2m+4 · @x [@xk y x2k 2L,↵ ⇢2k k+1 ] (3.14.75) Z Z Z Z 2 2m+2k+2 2(k 1) & |@xk+1 y |2 x2m+2k+4 2L,↵ ⇢2k k+1 |@xk y| x ⇢k (3.14.76) Z Z k X1 & |@xk+1 2 2m+2k+4 y| x 2 2k L,↵ ⇢k+1 || ||J i+2 . (3.14.77) i=0 Above, we have used the calculation: 2↵ 0 @x [x2k 2 2k L,↵ ⇢k+1 ] = 2kx2k 1 2 2k L,↵ ⇢k+1 + x2k 2k L,↵ L,↵ ⇢k+1 L 2k 1 0 + x2k 2 L,↵ 2k⇢k+1 ⇢k+1 . (3.14.78) 386 For the second term on the right-hand side of (3.14.78), we estimate: 2↵ L x L,↵ . 1. For the third term on the right-hand side, we use that the support of ⇢0k+1 is localized in x. We also use that: support(⇢k ) ⇢ {⇢k 1 = 1}. Next, we integrate by parts twice in x to obtain: Z Z @xx [(@xk )xx x2m+4 ] · @xk x2k 2L,↵ ⇢2k k+1 Z Z = @xk+2 x2m+4 @xx [@xk x2k 2L,↵ ⇢2k k+1 ] (3.14.79) Z Z Z Z = |@xk+2 |2 x2m+2k+4 2L,↵ ⇢2k k+1 + @xk+2 x2m+4 @xk+1 @x [x2k 2 2k L,↵ ⇢k+1 ] Z Z + @xk+2 x2m+4 @xk @xx [x2k 2L,↵ ⇢2kk+1 ]. (3.14.80) The final two terms on the right-hand side of (3.14.80) are estimated through further integrations by parts: Z Z Z Z | @xk+2 x2m+4 @xk+1 @x [x2k 2 2k L,↵ ⇢k+1 ] + @xk+2 x2m+4 @xk @xx [x2k 2 2k L,↵ ⇢k+1 ]| . || ||2J k+1 . (3.14.81) Finally, Z Z Z Z @x [(@xk )x x 2m+2 ]· @xk x2k 2L,↵ ⇢2k k+1 = @xk+1 x2m+2 @x [@xk x2k 2 2k L,↵ ⇢k+1 ] Z Z = |@xk+1 |2 x2m+2+2k 2L,↵ ⇢2k k+1 (3.14.82) Z Z + @xk+1 x2m+2 @xk @x [x2k 2 2k L,↵ ⇢k+1 ] (3.14.83) Z Z & |@xk+1 |2 x2m+2+2k 2 2k L,↵ ⇢k+1 || ||2J k+1 . (3.14.84) Piecing all of the above estimates together yields the desired bound. 387 Lemma 3.14.9. Z Z k X | A(@xk ) · @xk+1 x2k+1 2 2k+1 L,↵ ⇢k+1 | . || ||2J i+2 . (3.14.85) i=0 Proof. Again, referring to definition (3.12.5), we will proceed term by term, starting with the following, for which we integrate by parts once: Z Z Z Z 2k+1 1 2k+1 | @xk x2m · @xk+1 x2k+1 2 L,↵ ⇢k+1 | = |@xk |2 @x [x2m+2k+1 2 L,↵ ⇢k+1 ] 2 (3.14.86) Let us expand the product rule above: 2k+1 2k+1 ↵ 0 2k+1 @x [x2m+2k+1 2 L,↵ ⇢k ] = Cx2m+2k 2 L,↵ ⇢k+1 + Cx2m+2k+1 L,↵ L,↵ ⇢k+1 L + x2m+2k+1 2 2k 0 L,↵ ⇢k+1 ⇢k+1 . x2m+2k ⇢2k k+1 . (3.14.87) This, the term (3.14.86) can be controlled via: Z Z k X |(3.14.86)| . k+1 . |@xk |2 x2m+2k ⇢2k || ||2J i+2 . (3.14.88) i=0 The second term in (3.12.5) is treated via: Z Z @x (@xk+1 x2m+2 ) · @xk+1 x2k+1 2L,↵ ⇢2k+1 k+1 Z Z = [ @xk+2 x2m+2 C@xk+1 x2m+1 ] · @xk+1 x2k+1 2L,↵ ⇢2k+1k+1 Z Z = |@xk+1 |2 @x [x2m+2k+3 2L,↵ ⇢2k+1 k+1 ] C|@xk+1 |2 x2m+2k+2 2L,↵ ⇢2k+1 k+1 (3.14.89) Z Z k X . k+1 . |@xk+1 |2 x2m+2k+2 ⇢2k || ||J i+2 (3.14.90) i=0 388 We have expanded the product in the first term on the right-hand side of (3.14.89): 2k+1 2k+1 ↵ 0 2k+1 @x [x2m+2k+3 2 L,↵ ⇢k+1 ] = Cx2m+2k+2 2 L,↵ ⇢k+1 + Cx2m+2k+3 ( ) L,↵ L,↵ ⇢k+1 L 2k+1 2k 0 + Cx2m+2k+3 2 L,↵ k+1 ⇢k+1 ⇢k+1 . x2m+2k+2 ⇢2k k+1 . (3.14.91) Next, we have: Z Z Z Z 2k+1 2k+1 @xk yy x 2m+2 · @xk+1 x2k+1 2 L,↵ ⇢k+1 = |@xk y| 2 @x [x2m+2k+3 2 L,↵ ⇢k+1 ] (3.14.92) We will expand the product rule above: 2k+1 2k+1 ↵ 0 2k+1 @x [x2m+2k+3 2 L,↵ ⇢k+1 ] = Cx2m+2k+2 2 L,↵ ⇢k+1 + C x2m+2k+3 L,↵ L,↵ ⇢k+1 L + Cx2m+2k+3 2 2k 0 L,↵ ⇢k+1 ⇢k . x2k+2m+2 ⇢2k k+1 . (3.14.93) Pk Inserting this above yields: |(3.14.92)| . i=0 || ||2J i+2 . Next, after two integrations by parts in y, and one in x: Z Z Z Z @xk yyyy x 2m+4 · @xk+1 x2k+1 ⇢2k+1 k+1 2 L,↵ = |@xk yy | 2 @x [x2m+2k+5 ⇢2k+1 k+1 2 L,↵ ] (3.14.94) Expanding the product rule above yields: 0 @x [x2m+2k+5 ⇢2k+1 k+1 2 L,↵ ] = Cx2m+2k+4 ⇢2k+1 k+1 2 L,↵ + Cx2m+2k+5 ⇢2k k+1 ⇢k+1 2 L,↵ ↵ + Cx2m+2k+5 ⇢2k+1 k+1 0 L,↵ L,↵ . x2m+2k+4 ⇢2k k+1 (3.14.95) L 389 Pk Inserting above yields: |(3.14.94)| . i=0 || ||2J i+2 . The next term from A( ) in defini- tion (3.12.5) is: Z Z @x [(@xk )yyx x2m+4 ] · @xk+1 x2k+1 ⇢2k+1 k+1 2 L,↵ Z Z = @xk+1 y x2m+4 · @x [@xk+1 y x2k+1 ⇢2k+1 k+1 2 L,↵ ] Z Z = @xk+1 y x2m+4 · @xk+2 y x2k+1 ⇢2k+1 k+1 2 L,↵ Z Z |@xk+1 y |2 x2m+4 @x [x2k+1 ⇢2k+1 k+1 2 L,↵ ] Z Z = |@xk+1 y |2 @x [x2m+2k+5 ⇢2k+1 k+1 2 L,↵ ] Z Z |@xk+1 y |2 x2m+4 @x [x2k+1 ⇢2k+1 k+1 2 L,↵ ] Z Z k X . |@xk+1 y| 2 2m+2k+4 2k x ⇢k+1 . || ||2J i+2 . (3.14.96) i=0 The final term from A( ) in definition (3.12.5) is: Z Z @xx (@xk+2 x2m+4 ) · @xk+1 x2k+1 ⇢2k+1 k+1 2 L,↵ (3.14.97) Z Z = @xk+2 x2m+4 · @xx [@xk+1 x2k+1 ⇢2k+1 k+1 2 L,↵ ] Z Z = @xk+2 x2m+4 · @xk+3 x2k+1 ⇢2k+1 k+1 2 L,↵ Z Z |@xk+2 |2 x2m+4 · @x [x2k+1 ⇢2k+1 k+1 2 L,↵ ] Z Z |@xk+1 |2 @xxx [x2m+2k+5 ⇢2k+1 k+1 2 L,↵ ] k X . || ||2J i+2 . (3.14.98) i=0 This concludes the proof of the desired estimate, (3.14.85). 390 Lemma 3.14.10. Z Z k X1 | [@xk , A] · @xk x2k 2 2k L,↵ ⇢k+1 | . || ||2J i+2 . (3.14.99) i=0 Proof. To keep notations simple, we will prove the k = 1 case, with the k 2 cases following identically. We will proceed term by term from the commutator expression in (3.13.17). First, Z Z Z Z 1 x2m 1 · xx 2 2 2 L,↵ ⇢2 = | |2 @x [x2m+1 2 2 L,↵ ⇢2 ] 2 Z Z . x . || ||2J 2 . 2 2m (3.14.100) Next, Z Z Z Z 2m+1 2 2 2 2 2m+3 2 2 yy x · xx L,↵ ⇢2 = y @x [x L,↵ ⇢2 ] Z Z . 2 2m+2 yx . || ||2J 2 . (3.14.101) Next, Z Z Z Z @x ( xx 2m+1 )· xx 2 2 2 L,↵ ⇢2 . 2 2m+2 xx . || ||2J 2 . (3.14.102) We will now move to the high order terms, starting with: Z Z Z Z 2m+3 yyyy x · x x2 2 2 L,↵ ⇢2 = yy yyx x 2m+5 2 2 L,↵ ⇢2 Z Z 1 = | yy | 2 @x [x2m+5 2 2 L,↵ ⇢2 ] . || ||2J 2 . (3.14.103) 2 391 Next, again integrating by parts several times: Z Z 2m+3 2 2 2 @x [ yyx x ]· xx L,↵ ⇢2 Z Z = 2 xy [x 2m+3 @x [x2 2L,↵ ⇢22 ] @x [x2m+5 2 2 L,↵ ⇢2 ]] . || ||2J 2 . (3.14.104) The final term from A( ), which after integrating by parts several times in the same way as above, Z Z Z Z @xx [ xx x 2m+3 ]· xx 2 2 2 L,↵ ⇢2 . || ||2J 2 . (3.14.105) This concludes the proof of (3.14.99). Lemma 3.14.11. Z Z k X | [@xk , A] · @xk+1 x2k+1 2 2k+1 L,↵ ⇢k+1 | . || ||J i+2 . (3.14.106) i=0 Proof. This estimate proceeds in the same manner as those from (3.14.99), with the ad- justment that the extra derivative in the multiplier from (3.14.106) is accounted for by the increment in order on the right-hand sides of (3.14.106) versus (3.14.99). Indeed, let us take the highest order term from the commutator, [@x , A] : Z Z Z Z | @xx ( xx x 2m+3 )· xx x 3 2 3 L,↵ ⇢2 | . | xxx | 2 2m+6 2 x 3 L,↵ ⇢2 + || ||2J 2 . (3.14.107) The first term on the right-hand side above can be controlled by || ||2J 3 , as can be seen from a comparison to (3.14.70) with k = 1. The remaining terms work identically. 392 Using the above calculations, we may repeat the energy and positivity estimates, for k 1: Lemma 3.14.12 (k + 1’th order Auxiliary Energy Estimate). Let k = 1, 2. Then, k X1 p 1 k+ 1 ||@xk uy · (⇢k+1 x)k ||2L2 + ↵|| ||2J k+2 . ↵ || ||2J i+2 + O( )||@xk { "vx , vy }xk+ 2 ⇢k+12 ||2L2 i=0 k X + W1 + Wi+1 . (3.14.108) i=1 Proof. We apply the operator @xk to the system (3.12.10): k " @x + @xk T [ ] + ↵A(@xk ) + ↵[@xk , A] = @xk {Fy "Gx }. (3.14.109) We subsequently apply the multiplier @xk x2k ⇢2k k+1 2 L,↵ : Z Z k [ " @x + @xk T [ ] + ↵A(@xk ) + ↵[@xk , A] ] · @xk x2k ⇢2k k+1 2 L,↵ Z Z = [@xk {Fy "Gx }] · @xk x2k ⇢2k 2 k+1 L,↵ . (3.14.110) The desired estimate now follows using similar calculations as in Lemma 3.14.3. Lemma 3.14.13 (k + 1’th order Auxiliary Positivity Estimate). k X k X p 1 k+ 1 ||@xk { "vx , vy }xk+ 2 ⇢k+12 ||2L2 . ||@xk uy · (⇢k+1 x)k ||2L2 + ↵ || ||2J i+2 + W1 + Wi+1 . i=0 i=1 (3.14.111) 393 Proof. We apply the multiplier @xk+1 x2k+1 ⇢2k+1 k+1 2 L,↵ to the system (3.14.109): Z Z [ k " @x + @xk T [ ] + ↵A(@xk ) + ↵[@xk , A] ] · @xk+1 x2k+1 ⇢2k+1 k+1 2 L,↵ Z Z = [@xk {Fy "Gx }] · @xk+1 x2k+1 ⇢2k+1 k+1 2 L,↵ . (3.14.112) The desired estimate now follows using similar calculations as in Lemma 3.14.4. 3.15 Step 3: Nonlinear Existence of Auxiliary Systems For this subsection, it is necessary to be more precise with notation; we will index solutions by (↵, N ) and also specify domains over which norms are being taken. We shall also transi- tion our right-hand sides from being generic (F, G) to being the particular right-hand sides of interest, (f˜, g) as defined in (3.12.12). Our intention now is to study the map, M ↵ : M ↵ [¯ u↵,N , v¯↵,N ] = [u↵.N , v ↵,N ] () L↵,¯v↵,N [u↵,N , v ↵,N ] = f˜y (¯ u↵,N , v¯↵,N ) u↵,N , v¯↵,N ) "gx (¯ 1 () [u↵,N , v ↵,N ] = L↵,¯ v ↵,N {f˜y (¯ u↵,N , v¯↵,N ) u↵,N , v¯↵,N )}. "gx (¯ (3.15.1) which corresponds to the system written in vorticity form: 2 " ↵,N + ↵A( ↵,N ) + T( ↵,N ; v¯↵,N ) = f˜y (¯ u↵,N , v¯↵,N ) u↵,N , v¯↵,N ) "gx (¯ on ⌦N . (3.15.2) A fixed point of (3.15.2) corresponds to the desired solution of (3.12.13). By repeating 394 the analysis in Section 3.9,the energy and positivity estimates in Section 3.10, and finally the estimates on Wi in Lemma 3.11.1 for the system, one obtains u↵,N , v¯↵,N ||Z(⌦N )  1. Fix any open set B ⇢ ⌦N . Let ↵ > 0 Lemma 3.15.1. Suppose ||¯ ↵,N and N >> 1. Solutions , or equivalently [u↵,N , v ↵,N ], to the system (3.15.2) satisfy the following estimates, independent of N , where !(Ni ) is based on universal constants: "N0 C(B)|| ↵,N ||H 5 (B) + ||u↵,N , v ↵,N ||Z(⌦N ) (3.15.3) n . ✏100 + ||u↵,N , v ↵,N ||X1 \X2 \X3 (⌦N ) + " 2 + !(Ni ) u↵,N , v¯↵,N ||2Z(⌦N ) . ||¯ The following energy and positivity estimates hold: ↵|| ↵,N 2 ||Hw4 (⌦N ) + ||u↵,N , v ↵,N ||2X1 \X2 \X3 (⌦N ) . W1 + W2 + W3 , (3.15.4) Finally, one has: ↵,N 2 ↵|| ||Hw4 (⌦N ) + "N0 C(B)|| ↵,N 2 ||H 5 (B) + ||u↵,N , v ↵,N ||2Z(⌦N ) 1 n . "4  + "2 !(Ni ) u↵,N , v¯↵,N ||4Z(⌦N ) . ||¯ (3.15.5) All constants appearing in the above estimates are independent of (↵, N ). Proof of Estimate (3.15.3). This follows by repeating the proofs of elliptic regularity in Subsection 3.9.1, namely Lemmas 3.9.10 and 3.9.12, to the new system, (3.15.2). The only new term in (3.15.2) as compared to 3.8.1 - (3.8.3), (3.8.8) - (3.8.9) are ↵A( ). The proof of Lemmas 3.9.10 and 3.9.12 then follows identically, as these are local-in-x estimates, which are una↵ected by the weights in A( ). At this point, one repeats the estimates in Subsection 3.9.2, which hold independent of any equation. 395 Proof of Estimate 3.15.4. This follows from Lemmas 3.14.12 - 3.14.13, and subsequently comparing || · ||J k with || · ||Hwk . Proof of Estimate 3.15.5. This follows by repeating the proof of Lemma 3.11.1. Motivated by (3.15.5), we define the notation: ||u↵,N , v ↵,N ||F (⌦N ) := ↵|| ↵,N 2 ||Hw4 (⌦N ) + ||u↵,N , v ↵,N ||2Z(⌦N ) . (3.15.6) Lemma 3.15.2 (Properties of M ↵ ). Fix any ↵ > 0 and any N > 0, and ,  > 0 arbitrarily small. (1) M ↵ : BZ (1) ⇢ Z(⌦N ) ! BZ (1) ⇢ Z(⌦N ), where BZ (1) is the unit ball in Z(⌦N ); (2) M ↵ is continuous and compact as an operator on BZ (1). (3) There exists a fixed point, [u↵,N , v ↵,N ] = M ↵ [u↵,N , v ↵,N ] in BZ (1). 1 (4) The fixed point satisfies, ||u↵,N , v ↵,N ||Z(⌦N ) . " 4  , independent of ↵, N . (3.15.7) Proof. The outline of this proof is as follows. The map M ↵ is shown to be well-defined in the appropriate domains and codomains, according to (1) above. Continuity of M ↵ is investigated by considering di↵erences, and compactness of M ↵ is obtained using our compactness lemmas above. One then applies a fixed point argument to prove (3) and (4). u, v¯] 2 Z(⌦N ). This implies that (f˜, g) 2 H2 1 , so by (3.14.67), the (1) Suppose [¯ map M ↵ is well-defined on Z(⌦N ). The property (3.9.9) is verified according to Lemma 396 3.13.6, and the definition of Hw2 (⌦N ), Definition 3.13.1, which ensures that [u↵ , v ↵ ] are con- 1 ||·||X1 tained in C0,D . Supposing the pre-images are contained in the unit ball of Z(⌦N ), u↵,N , v¯↵,N ||Z(⌦N )  1, one has estimate (3.15.5), which implies that M ↵ (¯ ||¯ u, v¯) 2 BZ (1). (2) To check continuity of the map M ↵ on BZ (1), suppose: u↵,N M ↵ [¯ i , v¯i↵,N ] = [u↵,N i , vi↵,N ] for i = 1, 2, (3.15.8) where u↵,N ||¯ i , v¯i↵,N ||Z  1. (3.15.9) Define the notation for the di↵erences, [ˆ ¯, u ˆ ¯] = [ ¯2↵,N ¯, vˆ ¯↵,N , u 1 ¯↵,N 2 ¯↵,N u 1 , v¯2↵,N v¯1↵,N ], (3.15.10) [ ˆ, u ˆ, vˆ] = [ ↵,N 2 ↵,N 1 , u↵,N 2 u↵,N 1 , v2↵,N v1↵,N ]. (3.15.11) By consulting (3.15.2), one then obtains the following system satisfied by the di↵erences: 2 ˆ + ↵A( ˆ) + T ( ˆ) = f˜y (u↵,N , u ¯↵,N , v¯↵,N ) f˜y (u↵,N ¯↵,N ,u , v¯1↵,N ) " 2 2 2 1 1 u↵,N "gx (¯ 2 , v¯2↵,N ) + "gx (¯ u↵,N 1 , v¯1↵,N ). (3.15.12) We may then repeat the estimates which resulted in (3.15.3) - (3.15.5) to obtain: 1 u, vˆ||2Z(⌦N ) . ||ˆ ˆ¯, vˆ¯||2Z(⌦N ) . ||u (3.15.13) ↵2 The only non-trivial calculation when repeating the estimates which resulted in (3.15.3) - (3.15.5) is to handle the nonlinearity (3.11.5) under taking di↵erences. For this, we first 397 write: v¯2↵,N u↵,N 2y v¯1↵,N u↵,N 1y = v¯2↵,N u↵,N 2y v¯2↵,N u↵,N ¯2↵,N u↵,N 1y + v 1y v¯1↵,N u↵,N 1y = v¯2↵,N u ˆy + vˆ¯u↵,N 1,y . (3.15.14) Repeating calculation (3.11.5) then yields: Z Z Z Z n n " 2+ v2↵,N u↵,N [¯ 2y v¯1↵,N u↵,N 1y ] ·u ˆ= " 2 + [¯v2↵,N uˆy + vˆ¯u↵,N 1,y ] · u ˆ Z Z n+ Z Z "2 ↵,N n = ˆ2 v¯2y u + " 2 + vˆ¯u↵,N 1,y u ˆ 2 (3.15.15) For the first term in the right-hand side above, we give the same estimate as in (3.11.5), which shows: Z Z n " 2 + 2 ↵,N n n | ˆ v¯2y | . " 2 + u !(Ni ) ||ˆ u↵,N u||2Z(⌦N ) ||¯ 2 , v¯2↵,N ||Z(⌦N ) . " 2 + !(Ni ) u||2Z(⌦N ) . ||ˆ 2 (3.15.16) This then gets absorbed into the left-hand side of (3.15.13). For the second term on the right-hand side above, we estimate: Z Z n n 1 1 | ¯u↵,N " 2 + vˆ ˆ|  " 2 + ||vˆ¯x 2 ||L1 ||u↵,N 1,y u m 1,y x ||L2 ||ˆ ux m 2 ||L2 n . "2+ !(Ni ) ||vˆ¯||Z(⌦N ) ||u↵,N 1 ||Hw2 (⌦N ) ||ˆ u||Z(⌦N ) n 1 . "2+ !(Ni ) ||vˆ¯||Z(⌦N ) ||u↵,N ||F (⌦N ) ||ˆ u||Z(⌦N ) ↵ 1 n 1 . "2+ !(Ni ) ˆ ||v¯||Z(⌦N ) ||ˆ u||Z(⌦N ) ↵ n 1 . "2( 2 + !(Ni )) u||2Z(⌦N ) + 2 ||vˆ¯||2Z(⌦N ) , ||ˆ (3.15.17) ↵ 398 where we have used (3.15.9) coupled with (3.15.5) to conclude that: ||u↵,N 1 ||F (⌦N ) . 1  "4 . The weight, xm , arises from the definition (3.12.5), and consequently in (3.13.2). The first term on the right-hand side of (3.15.17) is absorbed into the left-hand side of (3.15.13), whereas the second term contributes to the right-hand side of (3.15.13). All of the remaining calculations which produced (3.15.5) can be repeated in a similar fashion. Estimate (3.15.13) then implies the continuity of M ↵ on BZ (1). The modulus of continuity 1 of M ↵ is ↵2 , which prevents M ↵ from being a contraction map. Nevertheless, continuity is retained for all ↵ > 0. We now turn to compactness. According to Lemma 3.14.2, (3.15.5) shows that M ↵ (BZ (1)) is compactly embedded in BZ (1) so long as m is sufficiently large. (3 and 4) Consider the family of solutions: [u↵,N , v ↵,N ] = M ↵ [u↵,N , v ↵,N ], for 0   1. (3.15.18) By (3.15.1) and linearity of L↵ 1 , this occurs if and only if [u↵,N , v ↵,N ] = L↵ 1 { f˜y (u↵,N , v ↵,N ) " gx (u↵,N , v ↵,N )}. (3.15.19) By repeating the estimates which culminated in (3.15.5), one sees the uniform in bound: 1 ||u↵,N , v ↵,N ||2Z(⌦N ) . " 4  . (3.15.20) Thus, Schaefer’s fixed point theorem applied to the convex subset BZ (1) ⇢ Z(⌦N ) produces a fixed point, [u↵,N , v ↵,N ] 2 BZ (1). The estimate it obeys follows from (3.15.5). 399 3.16 Step 4: Nonlinear Existence We now need to pass to the limit as ↵ ! 0 and as N ! 1. The fixed point of the system (3.15.2), from Lemma 3.15.2 satisfies the following integral identity for any 2 C01 (⌦N ): Z Z hZ Z Z Z r2✏ N,↵ : r2✏ + ↵ N,↵ x2m + r N,↵ · r x2m+2 ⌦N ⌦ N ⌦ N Z Z i Z Z 2 N,↵ 2 2m+4 + r :r x + Su · y + "Sv · x N ⌦N Z Z⌦ h i n + " 2 Ru,n · y "Rv,n · x ⌦N Z Z h i n = " 2+ uN,↵ uN,↵ v N,↵ uN,↵ + "uN,↵ vxN,↵ + "v N,↵ vyN,↵ . (3.16.1) x y y y x x ⌦N First, we shall pass to the limit as ↵ ! 0, fixing an N . To do so, we first use (3.15.7) to obtain a weak subsequential limit point: uN,↵ * uN , weakly in (X1 \ X2 \ X3 )(⌦N ). (3.16.2) It is now our task to pass to the limit in the equation, (3.16.1), along the subsequence ↵ ! 0. Given a test-function, denote by U to be the support of . As U is bounded, we have Poincare inequalities available: hZ Z Z Z Z Z i N,↵ 2m N,↵ 2m+2 ↵| x + r ·r x + r2 N,↵ : r2 x2m+4 | ⌦N ⌦N ⌦N h i N,↵ N,↵  C( )↵ || ||L2 (U ) + ||r ||L2 (U ) + ||r2 N,↵ ||L2 (U ) ↵!0  C( )↵||ruN,↵ , rv N,↵ ||L2 (U )  C( )↵||uN,↵ , v N,↵ ||Z(⌦N ) ! 0. (3.16.3) For all of the linear terms, we use the weak convergence in (X1 \ X2 \ X3 )(⌦N ): Z Z Z Z lim r2" ↵,N : r2" Su (uN,↵ , v N,↵ ) y + "Sv (uN,↵ , v N,↵ ) · x ↵!0 ⌦N ⌦N 400 Z Z Z Z = r2" N : r2" Su (uN , v N ) y + "Sv (uN , v N ) · x. (3.16.4) ⌦N ⌦N Finally, we turn to the nonlinear terms for which we integrate by parts: Z Z Z Z uN,↵ uN,↵ x y + v N,↵ uN,↵ y y = |uN,↵ |2 xy uN,↵ v N,↵ yy , (3.16.5) ⌦N ⌦N Z Z Z Z uN,↵ vxN,↵ x + v N,↵ vyN,↵ x = |v N,↵ |2 xy uN,↵ v N,↵ xx . (3.16.6) ⌦N ⌦N Fixing a compactly supported , we can localize the integrations above to U . On this set, the weak convergence of uN,↵ * uN implies strong convergence in L2 . Thus, Z Z h i | |uN,↵ |2 uN,↵ uN + uN,↵ uN |uN |2 xy U . ||uN,↵ uN ||L2 (U ) ||uN,↵ ||L2 (U ) + ||uN ||L2 (U ) ||uN,↵ uN ||L2 (U ) . (3.16.7) The right-hand side converges to zero. The same bound works for all of the other nonlin- ear terms. Thus, the weak limit [uN , v N ] or equivalently N satisfies the weak formulation: Z Z Z Z r2✏ N : r2✏ Su (uN , v N ) · y + "Sv (uN , v N ) · x ⌦N ⌦N Z Z h i n + " 2 Ru,n · y "Rv,n · x ⌦N Z Z h i n = " 2+ uN uN v N uN + "uN vxN + "v N vyN . (3.16.8) x y y y x x ⌦N The weak limit [uN , v N ] must satisfy the bound: 1 ||uN , v N ||(X1 \X2 \X3 )(⌦N ) . C(uR , vR )" 4  , (3.16.9) independent of N . We may now repeat this exact procedure with the subsequential N 401 limit: denote by [u, v] and the subsequential (X1 \ X2 \ X3 )(⌦)-weak limit as N ! 1, guaranteed by (3.16.9). One then passes to the limit in the equation (3.16.8) to obtain: Z Z Z Z r2✏ : r2✏ Su (u, v) · y + "Sv (u, v) · x ⌦ ⌦ Z Z h i n + " 2 Ru,n · y "Rv,n · x ⌦ Z Z h i n = " 2+ uux vuy + "uvx + "vvy , (3.16.10) y y x x ⌦ with the limit satisfying: 1 ||u, v||(X1 \X2 \X3 )(⌦) . " 4  . (3.16.11) We now state the main existence result: Theorem 3.16.1. For ", sufficiently small,  > 0 small, and 0  < 14 , there exists a solution to the system (3.8.1) - (3.8.3), (3.8.4), (3.8.5) satisfying: 1 ||u, v||Z(⌦) . C(uR , vR )" 4  . (3.16.12) Proof. Estimate (3.16.11) implies enough regularity to integrate by parts identity (3.16.1) to: Z Z h i 2 " + @ y Su "@x Sv @y f + "@x g · = 0, (3.16.13) ⌦ which then implies that the PDE is satisfied pointwise in ⌦. The boundary conditions (3.8.4) are satisfied by elements in (X1 \ X2 \ X3 )(⌦), according to Lemma 3.9.16. From here, one repeats the embedding theorems in Section 3.9 which give estimate (3.16.12). That this is possible for those embeddings in Subsection 3.9.2 is straightforward to see, as these did not require [u, v] to satisfy any equations. Let us then turn to Subsection 3.9.1. 402 We must repeat the proofs of Lemmas 3.9.10 and 3.9.12 to the nonlinear system, (3.8.1) - (3.8.3), with f, g as in (3.8.5). This amounts to replacing [¯ u, v¯] with [u, v] in Lemmas 3.9.10 and 3.9.12, and foregoing the assumption that ||¯ u, v¯||Z  1. A nearly identical proof to Lemma 3.9.10 then yields: sup ||u, v||L1 y + ||u, v||H˙ 2 (x2000) . " M2 . (3.16.14) x2000 One now bootstraps the estimate in Lemma 3.9.12 in the identical manner. This gives estimate (3.16.12). We have verified that [u, v] 2 Z(⌦) satisfies (3.8.1) - (3.8.3), (3.8.4), (3.8.5). 3.17 Step 5: Uniqueness In this final subsection, we prove uniqueness of the solution [u, v] from Theorem 3.16.1. Suppose there existed two solutions, [u1 , v1 ] and [u2 , v2 ] to the system in (3.8.1) - (3.8.3), (3.8.4), (3.8.5). Define: u ˆ = u1 u2 , vˆ = v1 v2 , Pˆ = P1 P2 . (3.17.1) Then the new unknowns satisfy: n h i ✏ u ˆ + S u (ˆ u , v ˆ ) + ˆ P x = ˆ f := " 2+ u1 u1x u2 u2x + v1 u1y v2 u2y , (3.17.2) Pˆy n h i ✏v ˆ+ Sv (ˆ u, vˆ) + = gˆ := " 2 + u1 v1x u2 v2x + v1 v1y v2 v2y , (3.17.3) ✏ together with the divergence-free condition, u ˆx + vˆy = 0, and also satisfy the boundary 403 conditions: {ˆ u, vˆ}|{y=0} = {ˆ u, vˆ}|{x=1} = 0. (3.17.4) Going to vorticity, h i h i n n h @y "u ˆ + Su (ˆ u, vˆ) "@x " v + S v (ˆ u , v ˆ ) = " 2 @y u1 u1x i h io u2 u2x + v1 u1y v2 u2y "@x u1 v1x u2 v2x + v1 v1y v2 v2y . (3.17.5) We shall repeat the basic energy and positivity estimates using a slightly weaker weight. It is convenient to work with the weak formulation, which is given in (3.16.10). Then, u ˆ, vˆ satisfy the following: Z Z Z Z Z Z h i r2✏ ˆ : r2✏ + u, vˆ) · "Sv (ˆ x u, vˆ) · Su (ˆ y = fˆ y + "ˆ g x , (3.17.6) for all 2 C01 (⌦). We make the notational convention that Z Z Z Z := . (3.17.7) ⌦ Lemma 3.17.1. There exists a 0 < b < 1, sufficiently close to 0, depending only on universal constants, such that for , " sufficiently small and " << << b, the solutions u, vˆ] 2 Z to the system (3.17.2) - (3.17.3) with boundary conditions (3.17.4) satisfy the [ˆ following estimate: p 1 p 1 b||{ˆ u, v }x "ˆ b 2 ||2L2 + ||ˆ uy x b 2 ||L2 . O( )||{ "ˆ vx , vˆy }x 2 b 2 ||L2 + W1,E,b , (3.17.8) 404 where Z Z W1,E,b := fˆu ˆx 2b + "ˆ g vˆx 2b g ˆx 2b"ˆ 2b 1 , (3.17.9) Z Z W1,P,b := fˆu ˆ x x1 2b g vˆx x1 + "ˆ 2b , (3.17.10) W1,b = W1,E,b + W1,P,b . (3.17.11) Proof. The estimate will follow upon applying the multiplier ˆ · x 2b to the system in (3.17.5). To work rigorously, we will apply approximate multipliers, and work with the un , vˆ(n) , ˆ(n) ] 2 C01 (⌦), such that: weak formulation given in (3.17.6). Fix [ˆ X u(n) , vˆ(n) ] ! [ˆ 1 [ˆ u, vˆ], (3.17.12) where X1 is defined in (3.9.3). Within the notation of (3.17.6), = ˆ(n) x 2b . The u, vˆ] 2 Z(⌦). That existence of the sequence specified in (3.17.12) is guaranteed by [ˆ is compactly supported in (x, y) follows from the representations: Z y Z x ˆ(n) = ˆ(n) = u vˆ(n) . (3.17.13) 0 0 Let us first treat the second-order terms: Z Z Z Z r2" ˆ : r2" = r2" ˆ : r2" ( ˆ(n) x 2b ) Z Z ⇣ ⌘ ⇣ ⌘ = ˆyy ˆ(n) x 2b + 2" ˆxy @x ˆ(n) x 2b + "2 ˆxx @xx ˆ(n) x 2b . (3.17.14) yy y The first two terms from (3.17.14) above are: Z Z ˆyy ˆ(n) x 2b + 2" ˆxy ˆxy (n) x 2b + 2" ˆxy ˆy(n) @x x 2b yy 405 Z Z = u ˆ(n) ˆy u y x 2b + 2"ˆ ˆ(n) ux u x x 2b 2"ˆ ˆ(n) @x x ux u 2b . (3.17.15) We shall take the limit as n ! 1 above. According to the definition (3.9.3), the convergence in (3.17.12) implies: Z Z Z Z | u u(n) ˆy (ˆ y u ˆy )x 2b |+| u u(n) ˆx (ˆ x u ˆx )x 2b | Z Z n!1 +| u u(n) ˆx (ˆ u ˆ)@x x 2b | ! 0. (3.17.16) Expanding the third term from (3.17.14), Z Z ⇣ ⌘ Z Z h i " ˆxx @xx ˆ(n) x 2 2b = "2 vˆx · vˆx(n) x 2b v (n) @x x + 2ˆ 2b + ˆ(n) @xx x 2b . (3.17.17) By referring to the definition of X1 in (3.9.3) and (3.17.12), we may pass to the limit: Z Z Z Z n!1 Equation (3.17.15) ! u2y + 2"ˆ [ˆ u2x ]x 2b "ˆ ˆ@x x 2b ux u Z Z Z 2 2 2b 2 2 2b = [ˆ uy + 2"ˆux ]x + b(2b + 1)"ˆu x + "b lim ˆ2 x 1 2b u M !1 x=M Z Z = u2y + 2"ˆ [ˆ u2x ]x 2b + b(2b + 1)"ˆ u2 x 2 2b , (3.17.18) and: Z Z n!1 Equation (3.17.17) ! "2 vˆx2 x 2b 4b"2 vˆx vˆx 1 2b + 2b(2b + 1)"2 vˆx ˆx 2 2b . (3.17.19) Integrating by parts the final two terms above in (3.17.19), and referring to estimate 406 (3.9.71), Z Z Z Z Z 4b "2 vˆx vˆx 1 2b = 2b"2 vˆ2 @x x 1 2b 2b lim "2 vˆ2 x 1 2b M !1 x=M Z Z = 2b(1 + 2b) "2 vˆ2 x 2 2b , (3.17.20) and similarly, to treat the final term in (3.17.19), we appeal to the estimates in (3.9.71): Z Z Z Z Z Z "2 vˆx ˆx 2 2b = "2 vˆ2 x 2 2b "2 vˆ ˆ@x x 2 2b Z + lim "2 vˆ ˆx 2 2b (3.17.21) M !1 x=M Z Z Z Z (2b + 3)(2b + 2) 2 ˆ2 4 2b = "2 vˆ2 x 2 2b + " x 2 Z 2b + 2 2 ˆ2 3 2b + lim " x (3.17.22) M !1 x=M 2 Z Z Z Z (2b + 3)(2b + 2) 2 ˆ2 4 2b = "2 vˆ2 x 2 2b + " x . (3.17.23) 2 Therefore, summarizing the highest order calculation: Z Z Z Z r2" ˆ : r2 ( ˆ(n) x 2b )& u2y x 2b + 2"ˆ [ˆ u2x + "2 vˆx2 ]x 2b " Z Z ["2 vˆ2 + "ˆu2 ]x 2 2b + "2 ˆ2 x 4 2b (3.17.24) Z Z Z Z Z Z & ˆ2y x 2b C u "2 vˆx2 x 2b C u2x x "ˆ 2b . (3.17.25) To go from (3.17.24) to (3.17.25), we have used the Hardy inequality in the x-direction. We will now address the profile terms arising from Su (ˆ u, vˆ) in the weak formulation (3.17.6), whose definition has been given in (3.8.5): Z Z h i uR u ˆx + uRx u ˆ y · @y ˆ + uRy vˆ + vR u Z Z h i = uR u ˆx + uRx u ˆy · @y ˆ(n) x ˆ + uRy vˆ + vR u 2b 407 Z Z h i = uR u ˆx + uRx u ˆ + uRy vˆ + vR u ˆ(n) x ˆy · u 2b . (3.17.26) We will first pass to the limit in (3.17.26), using the definition of X1 in (3.9.3), which gives: Z Z h i n!1 2b (3.17.26) ! uR u ˆx + uRx u ˆy · u ˆ + uRy vˆ + vR u ˆx . (3.17.27) We proceed to treat each term in (3.17.27), starting with: Z Z Z Z Z 2b @x ⇣ ⌘ uR u ˆx u ˆx = ˆ2 uR x 2b + lim u ˆ2 x 2b u 2 M !1 x=M Z Z ⇣ ⌘ = ˆ2 uRx x 2b 2buR x 2b 1 u Z Z Z Z & ||uRx x||L1 ˆ2 x 2b 1 + 2b min uR u ˆ2 x u 2b 1 Z Z &b ˆ2 x 2b 1 , u (3.17.28) according to estimates (3.3.12), (3.3.17), so long as is taken small relative to b. For the M -limit above, we have used estimate (3.9.71), which is valid so long as b > 0. For the second term in (3.17.27), we again appeal to estimates (3.3.12), (3.3.18): Z 1 1 | ˆ2 x uRx u 2b | . ||uRx x||L1 ||ˆ ux 2b 2 ||2L2 . O( )||ˆ ux b 2 ||2L2 . n For the third term, we shall split uR = unR 1,p + " 2 unpR + uE R . First, we apply estimate (3.3.14): Z Z 1 u ˆ vˆ 1 | uP,n Ry 1 vˆu ˆx 2b |  ||y 2 x 2 uP,n Ry 1 ||L1 || x b ||L2 || x 2 b ||L2 y y 1 b b  O( )||ˆ uy x ||L2 ||ˆ vy x 2 ||L2 . (3.17.29) 408 Next, for n as in (3.7.1), according to estimate (3.3.16), Z Z n 1 vˆ 1 | uP,n Ry vˆu ˆx 2b |  " 2 ||unpy yx 2 n ||L1 ||ˆ ux 1+ n b ||L2 || x 2 b ||L2 y n 1 n 1 . " 2 ||ˆ ux x n b ||L2 ||ˆ vy x 2 b ||L2 . " 2 O( )||ˆ vy x 2 b 2 ||L2 . (3.17.30) Finally, the Eulerian contribution is handled by an application of (3.3.18): Z Z p p 3 u ˆ vˆ | "uE RY u ˆvˆx 2b | "||uE RY x ||L1 || 2 3 ||L2 || 3 ||L2 x 4 +b x 4 +b p 1 p 1 . "||ˆ ux x 4 b ||L2 || "ˆ vx x 4 b ||L2 . (3.17.31) The fourth term from (3.17.26), upon using estimate (3.3.12) and (3.3.18), reads: Z Z Z Z vRy 2 1 1 | vR u ˆy u ˆx 2b |=| u ˆ x 2b | . ||uRx x||L1 ||ˆ ux 2 b 2 ||L2 . O( )||ˆ ux 2 b 2 ||L2 . 2 (3.17.32) Summarizing these calculations, 1 1 |(3.17.27)| & b||ˆ ux 2 b 2 ||L2 O( )||ˆ ux 2 b 2 ||L2 O( )||ˆ uy x b 2 ||L2 p 1 b 2 O( )||{ "vx vˆy }x 2 ||L2 1 p 1 & b||ˆ ux 2 b 2 ||L2 O( )||{ "vx , vy }x 2 b 2 ||L2 . (3.17.33) We have absorbed the u ˆy terms into (3.17.24), and taken sufficiently small relative to b. We shall now address the profile terms from Sv : Z Z Z Z h i u, vˆ) · ˆx = "Sv (ˆ ˆ + vR vˆy + vRy vˆ ⇥ " uR vˆx + vRx u h i vˆ(n) x 2b 2b ˆ(n) x 2b 1 . (3.17.34) 409 We may take n ! 1 above due to the definition of X1 from (3.9.3) and (3.17.12): Z Z h i h i n!1 (3.17.34) ! ˆ + vR vˆy + vRy vˆ · vˆx " uR vˆx + vRx u 2b 2b ˆx 2b 1 . (3.17.35) We will now proceed to treat each term in (3.17.35). The first profile term, uR vx is the most delicate: Z Z "uR vˆx [ˆ vx 2b 2b ˆx 2b 1 ]. (3.17.36) First, Z Z Z Z Z 2b @x ⇣ ⌘ "uR 2 "uR vˆx vˆx = v2 "ˆ uR x 2b + lim vˆ x 2b 2 M !1 x=M 2 Z Z Z Z uRx 2b = v2 "ˆ x + b"uR vˆ2 x 2b 1 . (3.17.37) 2 The M -limit above vanishes due to (3.9.71). Staying with the term (3.17.36): Z Z Z Z ⇣ ⌘ 2b "uR vˆx ˆx 2b 1 = 2b v @x uR ˆx 2b 1 "ˆ Z Z Z Z = 2b"uRx vˆ ˆx 2b 1 + 2b"uR vˆ2 x 2b 1 Z Z 2b(2b + 1)"uR ˆvˆx 2b 2 (3.17.38) Z Z Z Z = 2b"uRx vˆ ˆx 2b 1 + 2b"uR vˆ2 x 2b 1 Z Z + b(2b + 1)" ˆ2 uRx x 2b 2 Z Z b(2b + 1)(2b + 2)"uR ˆ2 x 2b 3 . (3.17.39) Combining the positive terms in (3.17.39) and (3.17.37), the total positive contribution 410 RR is 3b"uR vˆ2 x 2b 1 . For the final term in (3.17.39), we will now give the estimate: Z Z Z Z @x uR ˆ2 x 2b 3 = uR ˆ2 x 2b 2 2b + 2 Z Z Z Z 2 uRx ˆ2 = uR ˆvˆx 2b 2 + x 2b 2 2b + 2 2b + 2 h1 1 3 1 4 1 1 i  ||uR2 ˆx b 2 ||2L2 + 2 ||uR2 vˆx b 2 ||2L2 2 2 (2b + 2) Z Z ||uRx x||L1 sup |uR | + uR ˆ2 x 2b 3 . (3.17.40) 2b + 2 inf |uR | By collecting terms and rearranging, we obtain: h 1 ||uRx x||L1 sup |uR | i 12 ˆ 3 2 1 1 b 1 ||uR x 2 ||2L2  2 ||uR2 vˆx b 2 ||2L2 . (3.17.41) 2 2b + 2 inf |uR | (2b + 2) This then implies: 1 3 1 4 1 1 ||uR2 ˆx b 2 ||2L2  2 ||uR2 vˆx b 2 ||2L2 . (3.17.42) 1 O( ) (2b + 2) Inserting this into (3.17.39), one arrives at: Z Z | b(2b + 1)(2b + 2)"uR ˆ2 x 2b 3 | Z Z 1 4b(2b + 1)(2b + 2)  "uR vˆ2 x 1 2b 1 O( ) (2b + 2)2 Z Z 5b  v 2 x 1 2b , uR "ˆ (3.17.43) 2 so long as b is sufficiently close to 0, by the following calculation: (2b + 1)(2b + 2) 1 lim 2 = . (3.17.44) b!0 (2b + 2) 2 411 Thus, taking b sufficiently small, and recalling the positive contributions from (3.17.39) and (3.17.37), we have: Z Z Z Z Z Z 5b b 3b "uR vˆ2 x 2b 1 "uR vˆ2 x 2b 1 = "uR vˆ2 x 2b 1 . (3.17.45) 2 2 The remaining terms from (3.17.37) and (3.17.39) are then estimated in terms of (3.17.45) using the smallness of O( ). Summarizing, we have established control over: Z Z h i Z Z "uR vˆx · vˆx 2b 2b ˆx 2b 1 & v2 x b"ˆ 1 2b , (3.17.46) for a constant independent of small and b. We will now move to the second term from (3.17.34), for which we recall estimates (3.3.8) and (3.3.17): Z Z h i p 3 u ˆ p vˆ | ˆ · vˆx "vRx u 2b 2b ˆx 2b 1 | "||vRx x 2 ||L1 || 3 b ||L2 || " 3 b ||L2 x 4 x4 p 1 b p 1 b  "||ˆ ux x 4 ||L2 || "ˆ vx x 4 ||L2 . (3.17.47) For the third term from (3.17.34), we use Young’s inequality and estimates (3.3.10), (3.3.20): Z Z h i | "vR vˆy vˆx 2b 2b ˆx 2b 1 | 1 h 1 p 1 p 3 i  ||vR x 2 ||L1 ||ˆ v x b 2 ||2L2 + || " ˆx b 2 ||2L2 vy x 2 b ||2L2 + || "ˆ h 1 p 1 p 3 i  O( ) ||ˆ vy x 2 b ||2L2 + || "ˆv x b 2 ||2L2 + || " ˆx b 2 ||2L2 . (3.17.48) For the final term from (3.17.34), we use Young’s inequality and estimates (3.3.10), 412 (3.3.20): Z Z h i | "vRy vˆ · vˆx 2b 2b ˆx 2b 1 | (3.17.49) h p 1 p 3 i . ||vRy x||L1 || "ˆ vx b 2 ||2L2 + b|| " ˆx b 2 ||2L2 . Summarizing these last few terms, we obtain: Z Z h i p 1 1 |(3.17.35)| & b"ˆ vx 1 2b + O( ) ||{ˆ u, "v}x 2 b 2 ||L2 + ||ˆ vy x 2 b 2 ||L2 . (3.17.50) The final task is to turn to the right-hand side. Reading from (3.17.6), and (3.17.2) - (3.17.3): Z Z Z Z fˆ · y g· + "ˆ x = fˆ · u ˆ(n) x 2b + "ˆ v (n) x 2b + ˆ(n) @x x 2b ] g · [ˆ Z Z n!1 ! fˆ · u ˆ(n) x 2b + "ˆ v x 2b + ˆ@x x 2b ], g · [ˆ (3.17.51) where we have passed to the limit using again the definition of X1 from (3.9.3). Com- bining (3.17.24), (3.17.33), (3.17.50), and (3.17.51), one obtains the desired result, estimate (3.17.8). We now repeat the positivity estimate, with a correspondingly weaker weight in order to close the above energy estimate. We refer the reader to Proposition 3.10.2 for a comparison. Lemma 3.17.2. Fix any 0 < b < 1. Let , " be sufficiently small relative to universal u, vˆ] 2 Z solutions to (3.17.2) - (3.17.3) with boundary constants, and " << . Then for [ˆ 413 conditions (3.17.4) satisfy the following estimate: p 1 p 1 ||{ˆ ux , vx }x 2 "ˆ b 2 ||L2 . ||ˆ uy x b 2 ||L2 + ||{ "ˆ ˆ}x v, u 2 b 2 ||L2 + W1,P,b . (3.17.52) Proof. The estimate will follow upon applying the multiplier vˆx1 2b to the system (3.17.5). In order to proceed formally, we must start with the weak formulation given in (3.17.6), and select the test function: X = vˆ(n) x1 2b , u(n) , vˆ(n) ] ! [ˆ 1 [ˆ u, vˆ], (3.17.53) where X1 is defined in (3.9.3). Turning to the weak formulation in (3.17.6), we will first expand the second-order terms: Z Z Z Z r2" ˆ : r2" = r2" ˆ : r2" vˆ(n) x1 2b Z Z ⇣ ⌘ ⇣ ⌘ = ˆyy vˆ(n) x1 2b + 2" ˆxy @x vˆy(n) x1 2b + "2 ˆxx @xx vˆ(n) x1 2b yy Z Z ⇣ ⌘ ⇣ ⌘ (n) 1 2b = u ˆy vˆyy x vy @x vˆy(n) x1 + 2"ˆ 2b + "2 vˆx @xx vˆ(n) x1 2b (3.17.54) We first arrive at the first two terms from (3.17.54): Z Z (n) 1 2b (n) 1 2b u ˆy vˆyy x 2"ˆ ux vˆxy x ux vˆy(n) x 2"ˆ 2b Z Z = ˆ(n) u y @x [ˆuy x1 2b ] u(n) 2"ˆ ux x1 x @x [ˆ 2b ] ux vˆy(n) x 2"ˆ 2b . (3.17.55) Referring to the definition of X1 in (3.9.3), according to (3.17.53), we may pass to the limit as n ! 1, and appeal to the estimates in (3.9.71) and (3.9.72), to obtain: Z Z n!1 (3.17.55) ! u uy x1 ˆy @x [ˆ 2b ] 2"ˆ ux x1 ux @x [ˆ 2b ] 2"ˆ ux vˆy x 2b 414 Z Z Z h1 i (1 2b) = ˆ2y x u 2b u2x x + (1 + 2b)"ˆ 2b + lim ˆ2y x1 u 2b u2x x1 "ˆ 2b 2 M !1 x=M 2 Z Z (1 2b) = ˆ2y x u 2b u2x x + (1 + 2b)"ˆ 2b . (3.17.56) 2 Again referring to the definition in (3.9.3), the third term from (3.17.54) is treated by: Z Z ⇣ ⌘ Z Z ⇣ ⌘ "2 vˆx @xx vˆ(n) x1 2b = "2 vˆxx @x vˆ(n) x1 2b Z Z ⇣ ⌘ Z Z Z Z n!1 2 1 2b ! " vˆxx @x vˆx = "2 vˆxx vˆx x1 2b "2 vˆxx vˆ(1 2b)x 2b . (3.17.57) Integrating by parts the first term on the right-hand side of (3.17.57), and appealing to estimate (3.9.72): Z Z Z Z Z 1 2b "2 2 1 "2 vˆxx vˆx x1 2b = "2 vx |2 x |ˆ 2b lim vˆ x 2b 2 M !1 x=M 2 x Z Z 1 2b = "2 vx |2 x |ˆ 2b . (3.17.58) 2 Integrating by parts the second term on the right-hand side of (3.17.57), and again appealing to estimates (3.9.71) - (3.9.73) for the M -limit below: Z Z Z Z h i Z 2 2b 2 2b " vˆxx vˆ(1 2b)x = " (1 2b)ˆ vx @x vˆx + lim "2 (1 2b)ˆ vx vˆx 2b M !1 x=M Z Z Z Z = "2 (1 2b)ˆ 2 vx x 2b "2 2b(1 2b)ˆ vx vˆx 2b 1 Z Z Z Z = "2 (1 vx2 x 2b + 2b)ˆ "2 b(1 2b)ˆ v 2 @x x 2b 1 lim "2 b(1 v2 x 2b)ˆ 2b 1 M !1 Z Z Z Z = "2 (1 vx2 x 2b)ˆ 2b "2 b(1 v2 x 2b)(2b + 1)ˆ 2b 2 . (3.17.59) 415 Combining the above estimates: Z Z 3 (3.17.57) = (1 2b)"2 vˆx2 x 2b b(2b + 1)(1 2b)"2 vˆ2 x 2b 2 . (3.17.60) 2 Hence, summarizing (3.17.55) - (3.17.59): Z Z Z Z Z Z | lim r2" ˆ : r2" | = | r2" ˆ : r2" (ˆ v x1 2b )| . ["2 vˆx2 + "ˆ u2x + u ˆ2y ]x 2b . n!1 (3.17.61) We will now turn to the profile terms from Su , which upon consultation with (3.17.6), the definition in (3.9.3), and (3.17.53), read: Z Z h i uR u ˆx + uRx u ˆ + uRy vˆ + vR u ˆ(n) ˆy · u x x 1 2b Z Z h i n!1 ! uR uˆx + uRx u ˆ + uRy vˆ + vR u ˆ x x1 ˆy · u 2b . (3.17.62) We now turn our attention to (3.17.62). The first term yields the desired positivity: Z Z Z Z ˆ2x x1 uR u 2b & min uR ˆ2x x1 u 2b . (3.17.63) Next, by (3.3.12), (3.3.18): Z Z 1 1 | uRx u ˆuˆ x x1 2b |  ||uRx x||L1 ||ˆ ux x 2 b ||L2 ||ˆ ux 2 b ||L2 1 b 2  O( )||ˆ ux x 2 ||L2 . (3.17.64) Next, we shall split uR = uP E R + uR , and use estimate (3.3.14) and (3.3.16) for (3.17.65) 416 below and (3.3.18) for (3.17.66) below: Z Z 1 | uP Ry v ˆuˆ x x1 2b |  ||yuP Ry ||L1 ||ˆ vy x 2 b 2 ||L2 , (3.17.65) Z Z p 3 p 1 1 | "uE RY v ˆuˆ x x1 2b |  ||uE RY x ||L1 || "ˆ 2 vx 2 b ||L2 ||ˆ ux x 2 b ||L2 p h p 1 1 i . " || "ˆ vx 2 b 2 ||L2 + ||ˆ ux x 2 b 2 ||L2 . (3.17.66) For the fourth term from (3.17.62), by estimates (3.3.10) and (3.3.20): Z Z 1 1 | vR u ˆ x x1 ˆy u 2b |  ||vR x 2 ||L1 ||ˆ uy x b ||L2 ||ˆ ux x 2 b ||L2 1 b b  O( )||ˆ uy x ||L2 ||ˆ ux x 2 ||L2 . (3.17.67) Summarizing the last four calculations: Z Z h 1 |(3.17.62)| & ˆ2x x1 u 2b O( ) ||ˆ ux 2 b 2 ||L2 p 1 i b 2 b 2 + ||ˆ uy x ||L2 + || "ˆ vx 2 ||L2 . (3.17.68) The final three terms appearing on the right-hand side above all appear on the right- hand side of estimate (3.17.52). Turning now to the profile terms, from Sv , for which we read (3.17.6) with = vˆ(n) x1 2b , appeal to (3.9.3) and (3.17.53), giving ultimately: Z Z h i v (n) x1 2b ] ˆ + vR vˆy + vRy vˆ · @x [ˆ " uR vˆx + vRx u Z Z h i n!1 ! " uR vˆx + vRx u v x1 ˆ + vR vˆy + vRy vˆ · @x [ˆ 2b ]. (3.17.69) We will treat each term in (3.17.69). For the first term from (3.17.69): Z Z ⇣ ⌘ "uR vˆx vˆx x1 2b + (1 2b)ˆ vx 2b 417 Z Z Z Z 1 2b = "uR vˆx2 x1 2b + b(1 2b)uR "ˆv2 x 1 2b "uRx vˆ2 x 2b 2 Z Z Z Z & vx2 x1 "ˆ 2b +b v 2 x 1 2b . "ˆ (3.17.70) Above we have used (3.3.12) and (3.3.18). For the second term, we integrate by parts: Z Z Z Z ⇣ ⌘ Z 1 2b 1 2b "vRx u ˆ@x [ˆ vx ]= ˆ · vˆx "@x vRx u + lim ˆvˆx1 vRx u 2b M !1 x=M Z Z = ˆvˆx1 "vRxx u 2b ˆ x x1 "vRx vˆu 2b p 3 5 h 1 b 2  "||vRx x 2 , vRxx x 2 ||L1 ||ˆ ux 2 ||L2 1 p 1 i b 2 b 2 + ||ˆ ux x 2 ||L2 + || "ˆ vx 2 ||L2 . (3.17.71) The above M limit vanishes according to estimates (3.9.71), and we have used estimates (3.3.8) and (3.3.17). For the third term, we recall estimates (3.3.10), (3.3.20): Z Z "vR vˆy vˆx x1 2b + c0 "vR vˆy vˆx 2b p 1 h 1 p 1 p 1 i b 2 b 2 b 2  "||vR x 2 ||L1 ||ˆ vy x 2 ||L2 + || "ˆ vx x 2 ||L2 + || "ˆ vx 2 ||L2 . (3.17.72) For the fourth term, we integrate by parts and appeal to (3.9.71), (3.3.8) - (3.3.10), and (3.3.17): Z Z Z Z Z 1 2b 1 2b "vRy vˆ · @x [ˆ vx ]= "@x [vRy vˆ] · vˆx + lim "vRy vˆ2 x1 2b M !1 x=M Z Z = "vRxy vˆ2 x1 2b "vRy vˆx vˆx1 2b h p 1 p 1 i  ||vRy x, vRxy x2 ||L1 || "ˆ vx x 2 b 2 ||L2 + || "ˆ vx 2 b 2 ||L2 . (3.17.73) 418 Summarizing these four terms, Z Z Z Z |(3.17.69)| & vx2 x1 "ˆ 2b +b v2 x "ˆ 1 2b h 1 1 p 1 p 1 i b b 2 b b 2 O( ) ||ˆ ux 2 ,u ˆx x 2 ||L2 + || "ˆ vx 2 , "ˆ vx x 2 ||L2 . (3.17.74) On the right-hand side, appealing again to (3.9.3), (3.17.53), and the definitions of fˆ, gˆ in (3.17.2) - (3.17.3), one obtains: Z Z h i fˆu ˆ(n) x x 1 2b + gˆ vˆx(n) x1 2b + (1 v (n) x 2b)ˆ 2b Z Z h i n!1 ! fˆu ˆ x x1 2b + gˆ vˆx x1 2b + (1 2b)ˆ vx 2b , (3.17.75) Placing the above estimates together yields the estimate (3.17.52). We will now introduce some notation, which is a natural adaptation of what is found b in Section 3.9 to the weaker weight of x . The reader should recall the definitions of the cuto↵ functions introduced in (3.9.1) - (3.9.2). The energy norms are defined as follows: p 1 ||u, v||2X1,b := ||uy x b 2 ||L2 + ||{ ✏vx , vy }x 2 b 2 ||L2 (3.17.76) p 3 3 ||u, v||2X2,b := ||uxy · ⇢2 x1 b 2 ||L2 + ||{ ✏vxx , vxy } · ⇢2 x 2 2 b 2 ||L2 , (3.17.77) p 5 5 ||u, v||2X3,b := ||uxxy · ⇢23 x2 b 2 ||L2 + ||{ ✏vxxx , vxxy } · ⇢32 x 2 b 2 ||L2 . (3.17.78) Definition 3.17.3. The norms Y2,b , Y3,b are strengthenings of X2,b , X3,b near the boundary, x = 1, and defined through: p 3 ||u, v||2Y2,b := ||uxy x1 b 2 ||L2 + ||{ ✏vxx , vxy }x 2 b 2 ||L2 + ||uyy ||L2 (x2000) , (3.17.79) p 5 ||u, v||2Y3,b := ||uxxy · ⇣3 x2 b 2 ||L2 + ||{ ✏vxxx , vxxy } · ⇣3 x 2 b 2 ||L2 . (3.17.80) 419 Definition 3.17.4. The norm Zb is defined through: ||u, v||Zb :=||u, v||X1,b \X2,b \X3,b + ✏N2 ||u, v||Y2,b + ✏N3 ||u, v||Y3,b 1 p 1 p 3 5 + ✏N4 ||ux 4 b , "vx 2 b ||L1 + ✏N5 sup || "vx x 2 b , ux x 4 b ||L1 x 20 1 hZ 1 p i 12 + "N6 sup ||uy x 2 b ||L2y + ✏N7 4 b x || "vxx ||2L1 y dx . (3.17.81) x 20 20 Next, we record the second and third order versions of the energy and positivity esti- mates, which mimic Propositions 3.10.4, 3.10.6, 3.10.7, 3.10.8. We will omit most details, and record only those di↵erences which arise. Lemma 3.17.5 (Second-Order Energy Estimate). Fix any 0 < b < 1. Let , " be sufficiently u, vˆ] 2 Z solutions to (3.17.2) small relative to universal constants, and " << . Then for [ˆ - (3.17.3): p 3 3 uxy ⇢2 x1 ||ˆ b 2 ||L2 . O( )||{ "ˆ vxx , vˆxy }⇢22 x 2 b 2 ||L2 u, vˆ||2X1,b + W1,b + W2,E,b , (3.17.82) + ||ˆ where (recall the definition of ⇢2 from (3.9.2)): Z Z Z Z W2,E,b := fˆx u ˆx ⇢22 x2 2b + gx vˆx ⇢22 x2 "ˆ 2b , (3.17.83) Z Z Z Z W2,P,b = fˆx u ˆxx ⇢32 x3 2b + gx vˆxx ⇢32 x3 "ˆ 2b , (3.17.84) W2,b := W2,E,b + W2,P,b . (3.17.85) Proof. Di↵erentiating the weak formulation gives: Z Z Z Z r2✏ ˆx : r2 u, vˆ) · @x Su (ˆ y u, vˆ) · + "@x Sv (ˆ x ✏ Z Z h i n = "2+ @x fˆ y + "@x g ˆ x , (3.17.86) 420 For the second-order energy estimate, we select = ⇢22 vˆ(n) x2 2b , where: 1 X u(n) , vˆ(n) ] 2 C0,D [ˆ , u(n) , vˆ(n) ] ! [ˆ 1 [ˆ u, vˆ]. (3.17.87) Let us turn to the highest-order terms: Z Z Z Z r2" ˆx : r2" = r2" vˆ : r2" ⇢22 vˆ(n) x2 2b Z Z = vˆyy ⇢22 vˆyy (n) 2 x 2b + 2"ˆ vy(n) ⇢22 x2 vxy @x [ˆ 2b ] + "2 vˆxx @xx [⇢22 vˆ(n) x2 2b ] Z Z = ˆxy ⇢22 u u ˆ(n) xy x 2 2b + 2"ˆ vy(n) ⇢22 x2 vxy @x [ˆ 2b ] + "2 vˆxx @xx [⇢22 vˆ(n) x2 2b ] Z Z = uxy ⇢22 x2 @x [ˆ 2b u(n) ]ˆ y vxxy vˆy(n) ⇢22 x2 2"ˆ 2b "2 vˆxxx @x [⇢22 vˆ(n) x2 2b ] (3.17.88) One now checks according to the definition (3.9.3), that (3.17.87) suffices to pass to the limit in the above identity, which upon integrating by parts in x yields: Z Z n!1 (3.17.88) ! uxy ⇢22 x2 @x [ˆ 2b ]ˆ uy vxxy vˆy ⇢22 x2 2"ˆ 2b "2 vˆxxx @x [⇢22 vˆx2 2b ] Z Z = u2xy + "ˆ [ˆ u2xx + "2 vˆxx 2 ]⇢22 x2 2b + J. where |J| = |c0 "2 vˆx2 @xx (⇢22 x2 2b ) + c1 "2 vˆ2 @x4 (⇢22 x2 2b )| . ||u, v||2X1,b . From here, re- peating the calculations in Proposition 3.10.4 gives the desired result, where the required integrations by parts are justified upon using that b > 0, combined with the estimates in (3.9.71) - (3.9.73). These justifications are analogous to those in Lemma 3.17.1, and so we omit the details. Lemma 3.17.6 (Second-Order Positivity Estimate). Fix any 0 < b < 1. Let , " be suffi- 421 u, vˆ] 2 Z solutions to ciently small relative to universal constants, and " << . Then for [ˆ (3.17.2) - (3.17.3): p 3 3 ||{ "ˆ vxx , vˆxy }⇢22 x 2 b 2 ||L2 . ||ˆ uxy ⇢2 x1 b 2 ||L2 u, vˆ||2X1,b + W1,b + W2,b . + ||ˆ (3.17.89) Proof. We start again with the weak formulation in (3.17.86). Fix a large 0 < L < 1. We (n) then make the selection: = vˆx · ⇢32 x3L 2b , where, referring to (3.10.168), the weight xL is ⇣ ⌘ ⇣ ⌘ x defined via: xL := aL ⇤ L 10L . Define the domain: ⌦L := {x : 3 < x < 50L + 100}, so that vˆx · ⇢32 x3L 2b = 0 on ⌦C ˆ(n) is selected according to: L . The sequence v 1 H 1 (⌦L ) u(n) [ˆ ˆx(n) ] 2 C0,D x ,v (⌦L ), u(n) [ˆ ˆx(n) ] x ,v ! [ˆ ux , vˆx ]. (3.17.90) The existence of such a sequence is guaranteed due to the standard Sobolev space theory, because we are now in the un-weighted setting. It is now straightforward to repeat all estimates in Proposition 3.10.6 using the test function . Upon doing so, we pass to the limit first as n ! 1, and then as L ! 1 to obtain the desired estimate. Lemma 3.17.7 (Third-Order Energy Estimate). Fix any 0 < b < 1. Let , " be sufficiently u, vˆ] 2 Z solutions to (3.17.2) small relative to universal constants, and " << . Then for [ˆ - (3.17.3): 2 X p 5 5 uxxy ⇢23 x2 ||ˆ b 2 ||L2 . O( )||{ "ˆ vxxx , vˆxxy }⇢32 x 2 b 2 ||L2 u, vˆ||2X1,b \X2,b + + ||ˆ Wi,b + W3,E,b , i=1 (3.17.91) where Z Z Z Z W3,E,b := fˆxx u ˆxx ⇢43 x4 2b + gxx vˆxx ⇢43 x4 "ˆ 2b , (3.17.92) 422 Z Z Z Z W3,P,b := fˆxx u ˆxxx ⇢53 x5 2b + gxx vˆxxx ⇢53 x5 "ˆ 2b , (3.17.93) W3,b := W3,E,b + W3,P,b . (3.17.94) Proof. The first step is to di↵erentiate the weak formulation (3.17.86) yet again, which formally takes place using di↵erence quotients, yielding: Z Z Z Z r2✏ ˆxx : r2✏ u, vˆ) · @xx Su (ˆ y u, vˆ) · x + "@xx Sv (ˆ Z Z h i n = "2+ @xx fˆ y + "@xx gˆ x , (3.17.95) (n) Fix any L large, finite. The selection of test function is now := vˆx ⇢43 x4L 2b , where the sequence: 1 H 1 (⌦L ) u(n) [ˆ xx , v (n) ˆxx ] 2 C0,D (⌦L ), u(n) [ˆ xx , v (n) ˆxx ] ! [ˆ uxx , vˆxx ]. (3.17.96) From here, repeating the estimates given in Proposition 3.10.7, and sending n ! 1 and then L ! 1 gives the desired result. Lemma 3.17.8 (Third-Order Positivity Estimate). Fix any 0 < b < 1. Let , " be suffi- u, vˆ] 2 Z solutions to ciently small relative to universal constants, and " << . Then for [ˆ (3.17.2) - (3.17.3): 3 X p 5 5 ||{ "ˆ vxxx , vˆxxy }⇢32 x 2 b 2 ||L2 . ||ˆ uxxy ⇢23 x2 b 2 ||L2 u, vˆ||2X1,b \X2,b + + ||ˆ Wi,b . (3.17.97) i=1 Proof. Again, fix any L large, finite. The selection of the test function is now := 423 (n) vˆxx ⇢53 x5L 2b u(n) , vˆ(n) ] is selected according to: , where the sequence [ˆ 1 H 1 (⌦L ) u(n) [ˆ xx , v (n) ˆxx ] 2 C0,D (⌦L ), u(n) [ˆ xx , v (n) ˆxx ] ! [ˆ uxx , vˆxx ]. (3.17.98) From here, repeating the estimates in Proposition 3.10.8, and sending n ! 1 and then L ! 1 gives the desired result. Piecing together the above set of estimates, Proposition 3.17.9. Let , " be sufficiently small relative to universal constants, and " << u, vˆ] 2 Z solutions to (3.17.2) - (3.17.3): << b. Then for [ˆ u, vˆ||2X1,b \X2,b \X3,b . W1,b + W2,b + W3,b , ||ˆ (3.17.99) where Wi,b have been defined in (3.17.11), (3.17.85), (3.17.94). By repeating the analysis in Section 3.9, one has: Lemma 3.17.10. Let , " be sufficiently small relative to universal constants, and " << u, vˆ] 2 Z solutions to (3.17.2) - (3.17.3): << b. Then for [ˆ n u, vˆ||2Zb . " 2 + ||ˆ !(Ni ) u, vˆ||4Zb + ||ˆ ||ˆ u, vˆ||2X1,b \X2,b \X3,b . (3.17.100) Due to (3.17.99), we will now turn to estimating Wi,b Lemma 3.17.11. Let W1,b , W2,b , W3,b be as in (3.10.10), (3.10.82), (3.10.219). Then: n |W1,b + W2,b + W3,b | . C(b)" 2 + !(Ni ) u, vˆ||2Zb , ||ˆ (3.17.101) 424 where C(b) " 1 as b # 0. Proof. We will work with the expression: n h i fˆ = " 2 + u(1) u(1) x u(2) u(2) x +v (1) (1) uy v (2) u(2) y n h i = "2+ u ˆu(1) x + u (2) u ˆ x + v ˆ u (1) y + v (2) u ˆ y , (3.17.102) n h i gˆ = " 2 + u(1) vx(1) u(2) vx(2) + v (1) vy(1) v (2) vy(2) n h i = "2+ u ˆvx(1) + u(2) vˆx + vˆvy(1) + v (2) vˆy . (3.17.103) RR Concerning W1,b , let us bring particular attention to the following term from |fˆ| · 2b |ˆ u|x : Z Z n " 2 + [ˆ v u(1) y +v (2) ˆy ] · |ˆ u u|x 2b n 1 1  " 2 + ||ˆ vx 2 b ||L1 ||u(1) y ||L2 ||ˆ ux 2 b ||L2 n 1 1 + " 2 + ||v (2) x 2 ||L1 ||ˆ uy x b ||L2 ||ˆ ux 2 b ||L2 n h 1 1  " 2 + ||ˆ vx 2 b ||L1 ||u(1) y ||L2 ||ˆ ux x 2 b ||L2 1 1 i + ||v (2) x 2 ||L1 ||ˆ uy x b ||L2 ||ˆ ux x 2 b ||L2 n  C(b)" 2 + !(Ni ) ||u(i) , v (i) ||Z ||ˆ u, vˆ||2Zb . (3.17.104) 2b The above term requires the weight of x , b > 0, in order to apply the Hardy inequality. Indeed, this was not required for the existence proof (see calculation (3.11.5)), because the structure of vuy · u enabled us to integrate by parts, unlike in the present situation. The remaining terms in W1,b , and all terms in W2,b , W3,b are treated nearly identically to the Lemma 3.11.1, and so we omit repeating those calculations. 425 Corollary 3.17.12. Fix 0 < b < 1 sufficiently small, relative to universal constants. Sup- pose ", are sufficiently small, such that " << << b. Then u ˆ, vˆ = 0. Proof. Combining estimate (3.17.101) and (3.17.100) with estimate (3.16.12) yields: n u, vˆ||2Zb . C(b)" 2 + ||ˆ !(Ni ) u, vˆ||2Zb . ||ˆ (3.17.105) For " sufficiently small, this then implies ||ˆ u, vˆ||Zb = 0. Upon consultation with the norm Zb , and (3.17.4), this implies that u ˆ, vˆ = 0. Remark. We have controlled the second and third order energy norms, (3.17.77) - (3.17.78) R R (1) in order to treat the term u|x 2b , which appears in (3.17.104). This term forces vˆuy |ˆ 1 us to control ||ˆ vx 2 b ||L1 . One cannot get around placing this term in L1 (for instance by (1) integrating by parts from uy ) because this produces suboptimal decay rates, according to (3.9.92) - (3.9.93). This then establishes Theorem 3.12.1, and controlling [u, v] 2 Z then immediately es- tablishes the main result, Theorem 3.2.2. Chapter Four Steady Prandtl Layers over a Moving Boundary: Nonshear Flows 427 4.1 Abstract In this article we establish the validity of Prandtl layer expansions around Euler flows which are not shear. The presence of non-shear flows at the leading order creates a singularity of O( p1" ). A new y-weighted positivity estimate is developed to control this leading-order growth at the far field. 4.2 Introduction We consider the steady, incompressible Navier-Stokes equations on the domain ⌦ = (0, L) ⇥ (0, 1). The boundary consists of three components, Y = 0, x = 0, and x = L. The system reads: 9 U N S UxN S + V N S UYN S + PxN S = ✏ U N S > > > > > = NS NS NS NS NS NS in ⌦ (4.2.1) U Vx + V V Y + PY = ✏ V > > > > > NS NS Ux + VY = 0. ; The system above is taken together with the no-slip boundary condition on Y = 0, which in addition is assumed to be moving with velocity ub > 0. The boundary conditions at x = 0, L are inflow and outflow conditions, to be prescribed specifically in the article. We are interested in the asymptotic behavior of solutions to (4.2.1) as " ! 0. Such asymptotics must capture the formation of boundary layers, which we now describe in generality. Suppose an outer Euler flow is prescribed: [u0e (x, Y ), ve0 (x, Y ), Pe0 (x, Y )], (4.2.2) 428 satisfying the Euler equations: 9 u0e u0ex + ve0 u0eY + Pex 0 = 0> > > > > = 0 0 0 0 0 in ⌦ (4.2.3) ue vex + ve veY + PeY = 0> > > > > 0 0 uex + veY = 0, ; together with the no penetration boundary conditions at Y = 0, Y ! 1: ve0 |Y =0 = ve0 |Y !1 = 0. (4.2.4) Generically there is a mismatch between the boundary velocity u0e (x, 0) and ub , indicating that one should not expect solutions of (4.2.1) to converge to [u0e , ve0 ] in the L1 norm. Rather, it was proposed in 1904 by Ludwig Prandtl that one should expect the formation of boundary layers, which can be expressed mathematically as an asymptotic expansion: Y p U N S (x, Y ) = u0e (x, Y ) + u0p (x, p ) + O( "), (4.2.5) " p Y p V N S (x, Y ) = ve0 (x, Y ) + "vp0 (x, p ) + "ve1 (x, Y ) + O("), " Y p P N S (x, Y ) = Pe0 (x, Y ) + Pp0 (x, p ) + O( "). " The flows considered under the present setup are elliptic. Thus, a mathematical formu- lation of validating the expansion (4.2.5) is to assume boundary data are prescribed so that the expansions (4.2.5) are valid at the boundaries, x = 0, L, and to then prove that they must be valid in the interior of the domain, ⌦. Under the setup described above, (4.2.5) has been justified rigorously for shear flows in (GN17). Our aim in this article is to generalize the results to non-shear flows that are “sufficiently close to shear”, to be made rigorous by assumption (4.2.25) in our main result. 429 As is evident from (4.2.5), such a generalization is a leading order e↵ect, which when scaled to Prandtl variables creates a singularity of O( p1" ); this is evident in the specification of (4.2.14) below. Let us briefly highlight the physical importance of developing a method to handle non- shear Eulerian flows. A classical setup from fluid mechanics deals with horizontal flows past a rotating disk, see for instance (SG00). Such a flow is non-shear, as in the set-up considered here. In the simpler case when the flows are actually circular (and therefore shear), as opposed to horizontal, in the presence of a rotating disk, the article of (Iye17a) develops machinery to handle the geometry of the boundary. The present article can be viewed as a first step in studying non-shear flows, without adding the complexities of a curved boundary. Boundary Layer Expansions Y We will work with scaled, boundary layer variables y = p " , and consider the scaled Navier- Stokes unknowns: V N S (x, Y ) U " (x, y) = U N S (x, Y ), V " (x, y) = p P " (x, y) = P N S (x, Y ). (4.2.6) " In the new unknowns, the system (4.2.1) becomes: U " Ux" + V " Uy" + Px" = Uyy " " + "Uxx , (4.2.7) Py" U " Vx" + V " Vy" + " = Vyy " + "Vxx (4.2.8) " Ux" + Vy" . (4.2.9) 430 We start with the following expansions: p p 1 1 U " = u0e + u0p + "u1e + "u1p + " 2 + u := us + " 2 + u, (4.2.10) v0 p 1 1 V " = pe + vp0 + ve1 + "vp1 + " 2 + v = vs + " 2 + v, (4.2.11) " p p 1 1 P " = Pe0 + Pp0 + "Pe1 + "Pp1 + "Pp2 + " 2 + P = Ps + " 2 + P. (4.2.12) We are prescribed the Euler flow: [u0e , ve0 , Pe0 ]. (4.2.13) Importantly, the fact that u0e is not shear means that it can have an x-dependence. This in turn implies that ve0 and Pe0 are nonzero. Our analysis does not assume a sign condition for @x Pe0 . Due to the x-dependence of u0e , it is natural that in the scaled, Prandtl variable, there is a singularity of O( p1" ), (see below, equation (4.2.14)). We will construct the remaining terms in [us , vs , Ps ], as defined by (4.2.10) - (4.2.12), in Appendix 4.9. We will specify the particular equations satisfied by each of the terms in [us , vs ] in Appendix 4.9. Let us explicitly write the form of vs : v0 p vs = pe + vp0 + ve1 + "vp1 . (4.2.14) " As can be seen from above, the presence of nonzero ve0 creates a leading order singularity of O( p1" ), which is the main difficulty that must be addressed by our analysis. The main part of the article will be to construct and control the final term in the expan- sion, [u, v, P ], which we term the “remainders”. The equations satisfied by the remainders [u, v, P ] are specified in (4.2.30) - (4.2.32). 431 We now discuss the boundary data of each term above. The key point is that the no slip condition on Y = 0 must be enforced at each order in the expansion: u0e (x, 0) + u0p (x, 0) = ub , u1p (x, 0) = u1e (x, 0), u(x, 0) = 0 (4.2.15) ve0 (x, 0) = 0, ve1 (x, 0) = vp0 (x, 0), vp1 (x, 0) = 0, v(x, 0) = 0. (4.2.16) The boundary data at x = 0 must be specified for the Prandtl layers as follows: u0p (x, 0) = u0p0 (y), u1p (x, 0) = u1p0 (y). (4.2.17) The equations for uip are di↵usion equations, and so need to only be prescribed initial data at x = 0. vpi are then recovered via the divergence free condition, and therefore do not need in-flow boundary conditions. We will assume that uip0 are smooth and exponentially decaying. In contrast, the Euler layers, [u1e , ve1 ] satisfy an elliptic system, and we must prescribe boundary data at both x = 0, L. We do so at the level of the stream function, where r? 1 = [u1e , ve1 ]: 1 1 1 1 (0, Y ) = 0 (Y ), (L, Y ) = L (Y ). (4.2.18) These are also assumed smooth and rapidly decaying, and in addition must satisfy a compatibility condition which we call “well-prepared” boundary data defined in Definition 4.9.6. Finally, we can describe the boundary data for the remainders, [u, v, P ]: [u, v]|x=0 = [a0 (y), b0 (y)], [u, v]|y=0 = [u, v]|y!1 = 0, (4.2.19) 432 P 2"ux |x=L = aL (y), uy + "vx |x=L = bL (y). (4.2.20) The boundary condition at x = 0 allows the prescription of in-flow data. The boundary conditions at x = L in (4.2.20) is known as the (inhomogeneous) stress-free boundary condition, and corresponds to evaluating the Cauchy stress tensor at the boundary x = L. We will provide assumptions on the boundary data: p |@yk aL | . "hyi N , |@yk {a0 , b0 , bL }| . hyi N , supp{a0 , b0 , aL , bL } ⇢ {y 1}. (4.2.21) for sufficiently large k, N . Main Theorem In order to state our result, we must introduce the norm in which will control the solution. Define our X norm to be: p p ||u, v||X := ||uy · y||L2 + || "ux · y||L2 + ||vy , "vx ||L2 n p o p + || uyy , "uxy , "uxx · y||L2 + " 2 ||u, "v||L1 + ||u, v||B , (4.2.22) where the boundary norm is given by: p p ||u, v||B := ||uy · y, "ux · y||L2 (x=L) + || "ux ||L2 (x=L) . (4.2.23) We will also have to define the space, X , for which we refer the reader to Appendix B, equation 4.10.9. 433 Theorem 4.2.1. Consider an Euler flow [u0e (x, Y ), ve0 (x, Y )] satisfying the following hy- pothesis: 0 < c0  u0e  C0 < 1, (4.2.24) ve0 || ||L1 << 1, and (4.2.25) Y ||Y k rm ve0 ||L1 < 1 for sufficiently large k, m 0, (4.2.26) ||Y k rm u0e ||L1 < 1 for sufficiently large k 0, m 1. (4.2.27) Let the interval L be sufficiently small relative to universal constants. Suppose in addition that the boundary data described above are prescribed, assumed to be smooth and rapidly decaying in their arguments, satisfy the assumptions (4.2.21), and satisfy the compatibility conditions given in Definition 4.9.6. Then the remainder solutions [u, v, P ] exist in the space X and satisfy the estimate: ||u, v||X . 1. (4.2.28) Corollary 4.2.2. In the inviscid limit, we have the convergence: p ||U N S u0e u0p ||L1 + ||V N S ve0 ||L1  ". (4.2.29) Remark. The Euler flows which satisfy the assumptions of (4.2.24) - (4.2.27) are plentiful, see Proposition 4.9.1 in Appendix 4.9. Let us place this result in the context of recent developments in the boundary layer theory. We will restrict to stationary, two dimensional flows. A central task in this setting is to establish validity of an expansion of the type (4.2.5), and this is considered to be one of the most challenging open problems in fluid mechanics. It has been achieved in the setting of a moving boundary in (GN17), (Iye17a), (Iye16). The method introduced by (GN17) relies on establishing a crucial positivity estimate which gives o(1) control over the remainder 434 p quantity ||vy , "vx ||L2 . The flows considered in those works were all shear flows, and the aim of the present result is to generalize (in particular the result of (GN17)) to the case of non-shear flows. As can be seen in the expansion (4.2.11), this is a leading order e↵ect, and therefore requires a new y-weighted estimate. Within the stationary, two dimensional setting, the recent work of (DM18) addresses the related question of blowup of the Prandtl equation in the presence of an unfavorable pressure gradient. For unsteady flows, the validity of an asymptotic expansion of the form (4.2.5) has been established in the analyticity framework, (Asa91), (SC98a), (SC98b), in the Gevrey setting in (GVMM16), for initial vorticity bounded away from the origin in (Mae14), and for special flows in (MT08). Giving a more exhaustive survey of results in the unsteady setting would lead us astray, and so we refer the reader to the review articles of (E00), (GJT16), and (MM17) and the references therein. Overview of Proof Let us introduce the system satisfied by the remainders. For our discussion, we will consider the linearized version of system (4.9.105) - (4.9.107) and regard f, g and generic elements of L2 . "u + Su + Px = f, (4.2.30) Py "v + Sv + = g, (4.2.31) " ux + vy = 0, (4.2.32) together with the inhomogeneous boundary conditions: [u, v]|y=0 = [u, v]|x=0 = [u, v]|y!1 = 0, (4.2.33) P 2"ux |x=L = aL (y), {uy + "vx }|x=L = bL (y). (4.2.34) 435 In actuality, f, g contain the nonlinear components. Also note that we can, up to redefin- ing aL , bL , reduce the boundary data from (4.2.19) - (4.2.20) to the homogenized boundary data (4.2.33) - (4.2.34). This is proven in Lemma 4.9.14. We provide relevant definitions below: S u = us ux + usx u + vs uy + usy v, S v = us vx + vsx u + vs vy + vsy v, (4.2.35) 1 h i 1 h i N u (u, v) = " 2 + uux + vuy , N v (u, v) = " 2 + uvx + vvy , (4.2.36) 1 1 f =" 2 Ru,1 + N u + Lb1 , g=" 2 Rv,1 + N v + Lb2 . (4.2.37) Here, Ru , Rv are high order profile remainders which are defined specifically in (4.9.15), (4.9.24) and estimated in (4.9.123), and Lb1 , Lb2 arise from homogenizing the boundary data, are defined in (4.9.117) and estimated in (4.9.118). The important consideration for the purposes of this discussion is the rough specification of [us , vs ], which we can write: p us ⇡ u0e + u0p + O( "), (4.2.38) ve0 vs ⇡ p + O(1). (4.2.39) " Here, the Prandtl layer, u0p is rapidly decaying in the Prandtl variable, y. The main idea is to close a y-weighted estimate which can control the O( p1" ) contribution from vs . The first estimate in our scheme is the basic energy estimate: p p ||uy ||2L2 . O(L)||vy , "vx ||2L2 + ||f, "g||2L2 + C(a0 , b0 , aL , bL ). (4.2.40) This estimate is standard, and is obtained by applying (u, "v) to the system (4.2.30) - p (4.2.31). The main coercive term is the " , which yields control over ||uy , "vy , "vx ||2L2 . The O( p1" ) singular term from vs does not play a role at the level of energy estimates because a factor of @y hits each instance of vs upon applying the multiplier (u, "v). Nevertheless, the 436 energy estimate is too weak to close as a standalone estimate due to large convective terms. For instance, one considers: Z Z ⇣ p p ⌘ | usy uv| = | "u0eY + u0py + O( ") uv|  O(L)||ux ||L2 ||vy ||L2 . (4.2.41) Thus, it is required that vy be controlled at O(1), which is a famous difficulty in the boundary layer theory. This is the content of the next step, which generates the following positivity estimate: p p v0 p ||vy , "vx ||2L2 + || "ux ||L2 (x=L) .||uy ||2L2 + || e ||L1 ||uy · y, "vy · y||2L2 Y p 2 + ||f, "g||L2 + C(a0 , b0 , aL , bL ). (4.2.42) The above estimate is generating by applying [@y uvs , "@x uvs ] to the system (4.2.30) - (4.2.32). Here, O(ve0 ) is a constant that can be made small according to the assumption in (4.2.25). This estimate was introduced in the context of shear flows by (GN17), and crucially p utilizes the multiplier uvs which is able to generate coercivity over ||vy , "vx ||2L2 . We refer the reader to the article of (GN17) for more details, but emphasize that the significant v0 p di↵erence when addressing non-shear flows is the term || Ye ||L1 ||uy · y, "vy · y||2L2 . The key difficulty for our analysis is the loss of one y-weight on the right-hand side due to this term, which is the leading order e↵ect of the non-shear flow. Specifically, consider the term vs uy appearing in S u , as seen from definition (4.2.35), and recall that according to (4.2.39) the v0 leading order of vs ⇡ pe . " The outcome then becomes: Z Z Z v0 v0 vy2 . pe uy vy = p e yuy vy " "y ve0  || ||L1 ||uy · y||L2 ||vy ||L2 . (4.2.43) Y Our first main contribution of this paper is to develop the following y-weighted estimate 437 which controls the term ||uy · y||L2 term from (4.2.42): p p p ||{uyy , "uxy , "uxx } · y||2L2 + ||{uy , "ux } · y||2L2 + ||{uy , "ux } · y||L2 (x=L) p p . || "ux ||L2 (x=L) + ||uy ||2L2 + ||vy , "vx ||2L2 + Forcing Terms + C(a0 , b0 , aL , bL ). (4.2.44) The key idea is to first apply @y to the system (4.2.35). Let us extract the main terms coming from S u : @y S u = us uxy + vs uyy + usyy v. (4.2.45) We now introduce a mixed weight multiplier uy y 2 · 1 x, where the 1 x can take advantage of @x integrating by parts: Z Z Z 2 usx 2 2 us 2 2 us uxy · uy y (1 x) = u y (1 x) + u y 2 y 2 y Z us 2 2 + uy y (1 L), (4.2.46) x=L 2 Z Z Z vsy 2 2 vs uyy · uy y 2 (1 x) = uy y (1 x) vs yu2y (1 x). (4.2.47) 2 Summing (4.2.46) - (4.2.47), using usx + vsy = 0, and the smallness given in (4.2.25), we have: Z Z Z us 2 2 us 2 2 (4.2.46) + (4.2.47) & uy y vs yu2y (1 x) + uy y (1 L) 2 x=L 2 Z Z us 2 2 us 2 2 & u y + uy y (1 L). (4.2.48) 2 y x=L 2 At the level of the convection, the additional @y is necessary to generate an additional 438 p factor of ": Z Z usyy v · uy y 2 (1 x) ⇡ "u0eY Y v · uy y 2 (1 x) (4.2.49) v  ||Y 2 u0eY Y ||L1 || ||L2 ||uy · y||L2 . ||vy ||L2 ||uy y||L2 . y The above series of estimates closes by using the smallness of L and ve0 . Let us make a few remarks. Although the purpose of the weighted estimate, (4.2.44), is to capture behavior for large y, we cannot introduce a cut-o↵ function that avoids the y = 0 boundary into the multiplier for instance by selecting uy y 2 (1 x) (y). This is because the higher-order terms arising from @y " will generate local terms which cannot be controlled. Apart from from the weighted estimate, (4.2.44), a second novelty of our analysis is that we treat a large class of inhomogeneous boundary data at x = 0, x = L, as is shown in (4.2.19) - (4.2.20). This level of generality is important and very physical: it corresponds to taking measurements of the fluid at the inflow and outflow edges, x = 0 and x = L, and taking these values as inputs. The technique for treating these boundary conditions is based on Lemma 4.9.14, proved in Appendix 4.9. We first construct an auxiliary divergence free vector field which attains the boundary data from (4.2.19) at {x = 0}. Using this auxiliary vector field to homogenize then creates a boundary contribution at x = L, which cannot be removed by a further homogenization due to the need to preserve the divergence-free condition. This has the e↵ect of contributing several new boundary terms from x = L into the positivity estimate, (4.2.42), which must then be controlled. A third novelty of our analysis is to develop a scaled, weighted version of Korn’s inequality to close the above scheme of estimates. Such an estimate is needed due to the higher order contributions which are created in order to perform estimate (4.2.44). In particular, the estimate we prove is a coercivity estimate of the form: Z h i u2yy + 4"u2xy + "2 u2xx 2"uyy uxx y 2 · (1 x) (4.2.50) 439 Z h i & u2yy + "u2xy + "2 u2xx y 2 · (1 x) Acceptable Contributions. Notation Within lemmas, we will use X ⇠ O(LHS) and X ⇠ O(RHS) to mean X can be controlled, up to a universal constant, by the left-hand side (or right-hand side, respectively) of the lemma we are proving. Quantities denoted by O(L) refer to those which can be made small by making L small, and quantities denoted by O(ve0 ) refer to those which can be made small according to the smallness assumptions in (4.2.25). 4.3 Energy Estimate We will now give the basic energy estimate. The reader should recall the properties of the profiles, given in Appendix 4.9, in particular Lemma 4.9.16, and the space X , as defined by the norm introduced in (4.2.22), and the definition in (4.10.9). Proposition 4.3.1. Solutions [u, v, P ] 2 X , as defined by (4.10.9), to the system (4.2.30) - (4.2.32), with the boundary conditions (4.2.33) - (4.2.34), satisfy the following estimate: Z us ⇣ 2 ⌘ p ||uy ||2L2 + u + "v 2 . O(L)||vy , "vx ||2L2 + R1 + ||aL , bL ||2L2 , (4.3.1) x=L 2 where: Z R1 := f · u + "g · v. (4.3.2) Proof. This follows upon applying (u, "v) to the system (4.2.30) - (4.2.32). First, we will 440 write the " terms in the following way: "u = uyy + 2"uxx + "vxy , "v = 2vyy + "@x {uy + "vx }. (4.3.3) Using the above representation, we now integrate by parts: Z Z Z uyy · u 2"uxx · u "vxy u Z Z Z Z 2 2 = uy + 2"ux + "vx uy 2"ux u, (4.3.4) x=L Z Z 2vyy · "v "@x {uy + "vx } · v Z Z Z Z = + 2"vy2 + "2 vx2 + "uy vx "vbL (y), (4.3.5) x=L Z Z Z Px · u + Py v = P u, (4.3.6) x=L For (4.3.4) and (4.3.5) we have used the boundary conditions from (4.2.34). First, we will estimate the interior term from (4.3.5): Z p p p h p i | "uy vx |  "|| "vx ||L2 ||uy ||L2  " ||uy ||2L2 + || "vx ||2L2 . (4.3.7) Next, the boundary term from (4.3.5): Z p | "vbL |  ||"v||L2 (x=L) ||bL ||L2 (x=L)  O(L)"|| "vx ||2L2 + ||bL ||2L2 . (4.3.8) x=L We combine the boundary term from (4.3.4) and (4.3.6) by invoking the stress free boundary condition, in (4.2.34): Z Z {P 2"ux } · u = aL (y) · u x=L x=L  ||aL ||L2 (x=L) ||u||L2 (x=L) 441  ||aL ||2L2 (x=L) + O(L)||ux ||2L2 . (4.3.9) We will now move to the terms from S u , as defined in (4.2.35): Z Z h i Su · u = us ux + usx u + vs uy + usy v · u (4.3.10) The most difficult convective term from S u is: Z p | usy uv|  O(L)||usy · y||L1 ||vy , "vx ||2L2 . We have used the estimate (4.9.124) with k = 1. The remaining profile terms: Z Z Z us 2 {us ux + usx u + vs uy }u = u + usx u2 2 Zx=L us 2 & u O(L)||usx ||L1 ||ux ||2L2 . (4.3.11) x=L 2 Notice that crucially, the vs ⇠ O( p1" )ve0 singular term is accompanied by a factor of @y which cancels the singularity. We now move to the profile terms from S v , as defined in (4.2.35): Z {us vx + vsx u + vs vy + vsy v}"v Z Z Z 1 =+ us v 2 + "vsx uv + "vsy v 2 2 Zx=L 1 p p &+ us v 2 || "vsx ||L1 O(L)||ux ||2 || "vx ||2 x=L 2 p + ||vsy ||L1 O(L)|| "vx ||2L2 . (4.3.12) Above, we have again used that @y vs O(1), according to (4.9.124) with k = 1. This 442 concludes the proof. 4.4 Positivity Estimate For the positivity estimate, we must work with the new unknown: v = . (4.4.1) us Note that this quantity is well-defined because us > 0, according to (4.9.125). We first establish the equivalence: Lemma 4.4.1. For any function v satisfying v|y=0 = 0 = v|x=0 = 0, and defined through (4.4.1), the following estimate is valid: p p ||vy , "vx ||2L2 . || y, " 2 x ||L2 (4.4.2) Proof. The proof forwards directly from: Z Z Z ⇣ ⌘2 vy2 = |@y {us }| = 2 usy + us y Z Z Z . u2sy 2 + u2s 2 y . ||yusy ||2L1 u2s 2 y. (4.4.3) We have used above that |y=0 = 0, according to the assumptions of the lemma. We have also used estimate (4.9.124) with k = 1. Next, Z Z Z ⇣ ⌘2 vx2 = |@x {us }|2 = usx + us x 443 Z Z Z . u2sx 2 + u2s 2 x . 2 x. (4.4.4) Above, we have used that |x=0 = 0, according to the assumptions of the lemma. This concludes the proof. According to the above lemma, it suffices to control ||r" ||L2 , to which we now turn: Proposition 4.4.2 (Positivity Estimate). Solutions [u, v, P ] 2 X , defined in (4.2.22), (4.10.9), to the system (4.2.30) - (4.2.32), with the boundary conditions (4.2.33) - (4.2.34) satisfy: Z p 1 2 p || y, " 2 x ||L2 + "vy .||uy ||2L2 + O(ve0 )||uy · y, "vy · y||2L2 x=L us aL + R2 + ||bL , @y bL , p ||2L2 (x=L) . (4.4.5) " where: Z R2 := f· y + "g · x. (4.4.6) Proof. We will apply to the system (4.2.30) - (4.2.32) the multiplier: [ y , +" x ]. (4.4.7) Our analysis consists of a series of steps, which we now detail: Step 1: Su Profile Terms 444 Referring to the definition of S u in (4.2.35), we have via the divergence-free condition: Su = us vy + usy v + vs uy + usx u = u2s y + vs uy + usx u. (4.4.8) We gain: Z Z u2s y · y = u2s 2 y. (4.4.9) Referring to the definition of vs in (4.2.11), the main convective term is: Z Z ⇣ 0⌘ Z v p v s uy · y = pe uy y + {vp0 + ve1 + "vp1 }uy y. (4.4.10) " The lowest order term is the most dangerous: Z v0 ve0 | p e yuy y|  || ||L1 ||uy · y||L2 || y ||L2 "y Y h i  O(ve0 ) ||uy · y||2L2 + || y ||2L2 . (4.4.11) v0 Here we need the small parameter || Ye ||L1 . For the higher-order contributions: Z ⇣ v0 ⌘ v0 | vs pe uy y|  ||vs pe ||L1 ||uy ||L2 || y ||L2 " " 2  || y ||L2 + N ||uy ||2L2 . (4.4.12) Finally, the last term from (4.4.8) Z | usx u · y|  O(L)||usx ||L1 ||ux ||L2 || y ||L2 . O(L)|| 2 y ||L2 . (4.4.13) 445 Step 2: Sv Profile Terms Referring to the definition of S v in (4.2.35), here we will be treating: Z Z ⇣ ⌘ Sv · "@x {uy 2 w} = us vx + vsx u + vs vy + vsy v · "@x {uy 2 w} (4.4.14) First: Z Z Z us v x · " x = "u2s 2 x + "us usx x Z Z Z & "u2s 2 x O(L) "u2s x2 & "u2s 2 x, (4.4.15) Z p p | vsx u · " x|  O(L)|| "vsx ||L1 ||ux ||L2 || " x ||L2 . (4.4.16) v0 Next, we will use the smallness of || Ye ||L1 : Z v0 v0 p p | pe vy · " x|  || p e ||L1 || "vy · y||L2 || " x ||L2 " "y h p i  O(ve0 ) || "vy · y||2L2 + O(LHS) , (4.4.17) Z v0 p v0 p | {vs pe }vy · " x |  "||vs pe ||L1 ||vy ||L2 || " x ||L2 " " p  "O(LHS), (4.4.18) Z p p | vsy v · " x|  O(L)||vsy ||L1 || "vx ||L2 || " x ||L2 . (4.4.19) Step 3: Pressure Terms Z Z Z Z Z Px · y + Py · x = P y = 2"ux y aL y x=L x=L x=L Z Z Z vy2 1 =+ 2" 2"ux v@y { } aL y x=L us x=L us x=L Z vy2 p p =+ 2" O(L)||usy ||L1 || "vx ||L2 || "vy ||L2 (x=L) x=L us 446 Z aL y. (4.4.20) x=L The above term crucially yields control over the boundary term appearing in (4.4.5). We must estimate the contribution: Z aL p | aL y|  || p ||L2 (x=L) || " y ||L2 (x=L) x=L " aL p . N || p ||2L2 (x=L) + || " y ||2L2 (x=L) , (4.4.21) " the latter of which can be absorbed into (4.4.20). Step 4: Vorticity Terms We will now move to the vorticity terms from (4.2.30) - (4.2.32), where the stress-free boundary condition shown in (4.2.34) will be used repeatedly. Z Z vy usy + uyy y = uyy · uyy v 2 us us Z vy usy = uy @y { } + uy @y {v 2 } us us Z Z Z vyy 1 usy = uy 2uy vy @y { } + uy v@y 2 us us us Z 2 Z 2 Z uy 1 uy 1 = @x + 2uy vy @y { } 2 us x=L 2us us Z usy + uy v@y { 2 } us Z 2 h i uy & ||usx , usy , yu2sy , yusyy ||L1 N ||uy ||2L2 + ||vy ||2L2 . (4.4.22) x=L 2us The boundary term above, as with all boundary terms from this set of calculations, will 447 be put into (4.4.28), and subsequently estimated. Next: Z Z Z + "uxx y = "ux xy + "ux y x=L Z Z vy usy = "ux @x { v 2 }+ "ux y us u x=L Z Zs Z " 2 1 " 2 1 = u @x { } + ux "ux vy @x 2 x us x=L 2u s u s Z Z Z usy usy + "ux vx 2 + "ux v@x { 2 } + "ux y . (4.4.23) us us x=L The boundary terms are estimated as in: Z Z " 2 + u + "ux y x=L 2us x x=L Z Z 1 usy = "u2x + "ux v 2 x=L 2us x=L us Z 1 p p  "u2x + O(L)||usy ||L1 || "vx ||L2 || "ux ||L2 (x=L) , (4.4.24) x=L 2us the final term above being absorbed into (4.4.20) using the smallness of L. The bulk terms are estimated via: Z Z Z Z " 2 1 1 usy usy | ux @x { }| + | "ux vy @x | + | "ux vx 2 | + | "ux v@x 2 | 2 us us us us p h p i  "||usx , usy , usxy ||L1 ||ux ||2L2 + || "vx ||2L2 . (4.4.25) Next: Z Z "vyy x =+ "vy xy Z vx usx =+ "vy @y { v} us u2s Z ⇣v xy usy usx usx ⌘ =+ "vy vx 2 @y { 2 }v vy us us us u2s 448 Z Z Z " usx 2 " 2 usy =+ 2 v y + v y "vx vy 2 2 us x=L 2us us Z Z usx u sx "vvy @y { 2 } "vy2 2 us us Z h i " 2 p p & vy ||usx , usy , usxy ||L1 | O(L)|| "vx ||2L2 + "||vy ||2L2 . (4.4.26) x=L 2us Finally: Z Z Z 2 2 vx 1 " vxx x = " vxx "2 vxx v@x us us Z 2 Z " 1 "2 2 =+ @x vx2 vx 2 us x=L 2us Z Z "2 2 1 + vx + "2 vx2 @x 2u s u s Zx=0 Z 2 1 1 + " vvx @xx { } "2 vvx @x { } us x=L us Z Z Z "2 2 "2 2 1 & vx vx "2 vvx @x { } x=0 2us x=L 2us x=L us p "||usx , usxx ||L1 O(L)|| "vx ||2L2 . (4.4.27) Collecting the highest order x = L boundary contributions from (4.4.22), (4.4.24), (4.4.26), (4.4.27): Z Z Z Z u2y " 2 " 2 "2 2 + u + v v x=L 2us x=L 2us x x=L 2us y x=L 2us x Z Z u2y "2 2 =+ v x=L 2us x=L 2us x Z Z u2y (uy + "vx uy ) 2 =+ x=L 2us x=L 2us Z Z u2y (bL uy ) 2 =+ x=L 2us x=L 2us Z Z Z Z u2y b2L u2y 1 =+ + bL u y x=L 2us x=L 2us x=L 2us x=L us Z Z b2L bL = u@y { } x=L 2us x=L us . ||bL , @y bL ||2L2 (x=L) + ||u||2L2 (x=L) 449 . ||bL , @y bL ||2L2 (x=L) + O(L)||ux ||2L2 . (4.4.28) The final boundary term from (4.4.27) can be estimated via: Z Z 1 1 | "2 vvx @x { }| = | "vuy @x { }| x=L us us Z Z 1 1 = | "vy u@x { }| + | "vu@xy { }| us us p 1 p  "||usx , y@xy { }||L1 || "vy ||L2 (x=L) O(L)||ux ||L2 (4.4.29) us The boundary contribution from (4.4.29) can be absorbed into (4.4.20). This concludes the proof. 4.5 Weighted Estimates In this section, we will bootstrap to the weighted estimates described in (4.2.44). By di↵er- entiating the system (4.2.30) - (4.2.32), we have: " uy + Pxy + @y Su = @y f (4.5.1) Pyy " vy + + @y Sv = @y g, (4.5.2) " where: @y Su = us uxy + vs uyy + usyy v + usxy u (4.5.3) @y Sv = us vxy + vs vyy + usy vx + vsxy u + vsx uy + 2vsy vy + vsyy v. (4.5.4) 450 We will now prove the main weighted estimate. The reader should keep in mind Lemma 4.9.16 which will be in constant use. Proposition 4.5.1. Consider [u, v, P ] 2 X solutions to (4.2.30) - (4.2.32), with the bound- ary conditions (4.2.33) - (4.2.34). Such a solution satisfies the following estimate: n p o n p o || uyy , "uxy , "uxx · y||2L2 + || uy , "ux · y||2L2 n p o p + || uy , "ux · y||2L2 (x=L) . ||uy ||2L2 + ||vy , "vx ||2L2 p + || "ux ||2L2 (x=L) + ||{aL , @y aL , bL , @y bL }hyi2 ||2L2 (x=L) + R3 , (4.5.5) where: Z R3 := @y f · @y {uy 2 w} "@y g · @x {uy 2 w}. (4.5.6) Proof. We will apply the weighted multiplier: h i @y {uw(x)y 2 }, "@x {uw(x)y 2 } , (4.5.7) where w(x) = 1 x. The analysis proceeds in several steps which we will now detail. Step 1: Positive Profile Terms We will now generate the positive quantities on the left-hand side of (4.5.5), by consid- ering from (4.5.3) - (4.5.4) the following terms: Z ⇣ ⌘ Z ⇣ ⌘ us uxy + vs uyy · @y {uwy 2 } " us vxy + vs vyy · @x {uwy 2 }. (4.5.8) 451 First from (4.5.8): Z Z Z us uxy · @y {uy 2 w} = us uxy uy y 2 w + 2us uxy uyw Z Z Z usx 2 2 us 2 2 us 2 2 = u y w+ u y + uy y w 2 y 2 y x=L 2 Z 2ux @y {us uy}w Z Z Z usx 2 2 us 2 2 us 2 2 = uy y w + uy y + uy y w 2 2 x=L 2 Z Z Z 2ux us uy yw 2ux usy uyw 2ux us uw. (4.5.9) The final three terms above are estimated: Z | 2us ux uy yw|  ||uy y||2L2 + N ||ux ||2L2 , (4.5.10) Z | 2usy ux uyw|  O(L)||usy y||L1 ||ux ||2L2 , (4.5.11) Z | 2us uux w|  O(L)||ux ||2L2 . (4.5.12) Next from (4.5.8): Z Z Z vs uyy @y {uy 2 w} = vs uyy uy y 2 w + 2vs uyy uwy Z Z Z vsy 2 2 2 = u y w vs yuy w 2vsy uuy yw 2 y Z Z 2wvs uuy 2vs yu2y w Z Z vsy 2 2 = uy y w 3vs yu2y w 2 Z Z 2vsy yuuy w + wvsy u2 . (4.5.13) 452 The final two terms above are estimated: Z | 2vsy yuuy w|  O(L)||vsy ||L1 ||yuy ||L2 ||ux ||L2 , (4.5.14) Z | vsy u2 w|  O(L)||vsy ||L1 ||ux ||2L2 . (4.5.15) Summing (4.5.9) - (4.5.13): Z Z us us 2 2 (4.5.9) + (4.5.13) & { y2 3vs yw}u2y + u y w O(RHS). (4.5.16) 2 x=L 2 y We will consider the vs term above. At leading order: Z Z v0 ve0 2 2 v0 pe ywu2y = y wuy |·| || e ||L1 ||uy y||2L2 . (4.5.17) " Y Y v0 Here we use that || Ye ||L1 is taken sufficiently small by assumption (4.2.25) to absorb into the positive contribution from (4.5.16.) The higher order contributions can be estimated: Z v0 v0 | {vs pe }yu2y w|  ||vs pe ||L1 ||uy · y||L2 ||uy ||L2 " " . ||uy · y||2L2 + N ||uy ||2L2 . (4.5.18) Ultimately this yields: Z Z us 2 2 us 2 2 (4.5.9) + (4.5.13) & y uy + u y O(RHS). (4.5.19) 2 x=L 2 y We now move to the positive terms from @y Sv : Z Z Z "us vxy · @x {uy 2 w} = + "us vxy vy y 2 w + "us vxy uy 2 453 Z Z Z usx 2 2 vy2 2 us 2 2 = v y "w + "us y + " v y w 2 y 2 x=L 2 y Z Z 2 2 "vy @x {us uy } + us vy uy " x=L Z Z Z usx 2 2 vy2 us = vy y "w + "us y 2 + " vy2 y 2 w 2 2 2 Z Z Z x=L "vy usx uy 2 "vy us ux y 2 + us vy uy 2 " x=L Z Z Z usx 2 2 3 us = vy y "w + " us vy2 y 2 + " vy2 y 2 w 2 2 x=L 2 Z Z "vy usx uy 2 + us vy uy 2 ". (4.5.20) x=L We will estimate the final two terms from (4.5.20): Z p | "usx vy uy 2 |  O(L)||usx ||L1 || "ux y||2L2 , (4.5.21) Z p p | "us vy uy 2 |  || "vy y||L2 (x=L) || "uy||L2 (x=L) . (4.5.22) x=L Finally, from above: Z p p || "uy||2L2 (x=L) = "u2 y 2  O(L)|| "ux y||2L2 . (4.5.23) x=L Next: Z Z Z "vs vyy @x {uwy 2 } = + "vs vyy vy wy 2 + "vs vyy uy 2 Z Z vy2 = " @y {vs wy 2 } "vy @y {vs uy 2 } 2 Z Z Z vsy 2 2 = " vy y w "vs ywvy2 "vsy vy uy 2 2 Z Z 2 "vs vy uy y 2"vs vy uy. (4.5.24) 454 We will estimate three of the terms above: Z p | "vsy vy uy 2 |  O(L)||vsy ||L1 || "vy y||2L2 , (4.5.25) Z p p | 2"vs vy uy|  O(L)|| "vs ||L1 || "vy y||L2 ||ux ||L2 , (4.5.26) Z Z Z ve0 v0 | "vs vy uy y |  | " p vy uy y | + | "{vs pe }vy uy y 2 | 2 2 (4.5.27) " " p  ||ve0 "y||L1 ||vy ||L2 ||uy · y||L2 p v0 p + "||vs pe ||L1 || "vy y||L2 ||uy y||L2 " p . ||ve0 · Y ||L1 ||vy ||L2 ||uy y||L2 + "O(LHS) p . ||uy · y||2L2 + N ||vy ||2L2 + "O(LHS). (4.5.28) The ||uy y||2L2 term can be absorbed into the positive contribution from (4.5.20), whereas the ||vy ||2L2 term is O(RHS). Thus, summing (4.5.24) and (4.5.20) yields: Z ⇣ ⌘ 3 (4.5.24) + (4.5.20) & us y 2 vs yw "vy2 O(RHS). (4.5.29) 2 We must now examine the vs term above: Z p p p {vp0 + "vp1 }yw"vy2  ||vp0 y, "vp1 y||L1 || "vy ||2L2 , (4.5.30) Z p | "ve1 ywvy2 |  ||ve1 Y ||1 "||vy ||2L2 , (4.5.31) Z p | "ve0 ywvy2 |  ||ve0 · Y ||L1 ||vy ||2L2 , (4.5.32) all of which are acceptable contributions according to the right-hand side of (4.5.5). Summarizing this set of calculations: Z Z us 2 ⇣ 2 ⌘ us ⇣ 2 2 ⌘ (4.5.8) & y uy + "vy2 + uy y + "vy2 O(RHS). (4.5.33) 2 x=L 2 455 Step 2: Remaining Profile Terms We now extract the remaining terms from (4.5.3) - (4.5.4): Z Z Z usyy v · @y {uy 2 w} = usyy vuy y 2 w + usyy vu2yw v  ||usyy y 2 ||L1 || ||L2 ||uy y||L2 y v + O(L)||usyy y 2 ||L1 || ||L2 ||ux ||L2 y  ||usyy y 2 ||L1 ||vy ||L2 ||uy y||L2 + O(L)||usyy y 2 ||L1 ||vy ||L2 ||ux ||L2 , h i  ||usyy y 2 ||L1 N ||vy ||2L2 + ||uy y||L2 + O(L)||usyy y 2 ||L1 ||vy ||L2 ||ux ||L2 , (4.5.34) all of which are acceptable contributions. Note that we have used estimate (4.9.124) to absorb y 2 into usyy . Next: Z Z Z 2 2 usxy u · @y {uy w} = usxy uuy y w + usxy u2 2yw h i |·| ||usxy y||L1 O(L) ||ux ||2L2 + ||uy y||L2 , (4.5.35) which is an acceptable contribution by taking L << 1. Next, we move to the terms from @y Sv according to (4.5.4), starting with: Z Z Z "usy vx @x {uwy 2 } = "usy vx ux wy 2 + "usy vx uy 2 p p  ||usy y||L1 || "ux y||L2 || "vx ||L2 h p p i  ||usy y||L1 || "ux y||2L2 + N || "vx ||2L2 , (4.5.36) 456 which is seen to be an acceptable contribution according to (4.5.33). Next: Z h i p "vsxy u ux wy 2 uy 2  O(L)||vsxy ||L1 || "ux y||2L2 , (4.5.37) Z h i p "vsx uy ux wy 2 uy 2  || "vsx Y ||L1 ||uy y||L2 ||ux ||L2 , (4.5.38) p h i . || "vsx Y ||L1 ||uy y||2L2 + N ||ux ||2L2 , Z h i 2"vsy vy ux wy 2 uy 2 . ||vsy Y 2 ||L1 ||ux ||2L2 , (4.5.39) Z h i h p i p "vsyy v ux wy 2 uy 2  O(L)||vsyy y||L1 || "vx ||2L2 + || "ux y||2L2 . (4.5.40) Step 3: Vorticity Terms We record the following identities: " uy = uyyy 2"uxxy "vxyy , (4.5.41) " vy = 2vyyy "@x {uyy + "vxy }. (4.5.42) In the forthcoming calculations, we provide estimates on the vorticity terms: Z Z n o 2 " uy · @y {uy w} = uyyy 2"uxxy "vxyy · @y {uy 2 w}, (4.5.43) Z Z n o + " vy · "@x {uy 2 w} = 2vyyy + "@x {uyy + "vxy } · "@x {uy 2 w}. (4.5.44) Starting with the first term from (4.5.43): Z Z uyyy · @y {uwy 2 } = + uyy @y2 {uy 2 w} (4.5.45) Z Z Z 2 2 = uyy y w + 4uyy uy yw + 2uyy uw Z Z = + u2yy y 2 w 4 u2y w 457 Z &+ u2yy y 2 O(RHS). (4.5.46) We must provide the rigorous justification of the integration found in (4.5.45). The delicate calculation occurs near x = L, y = 0 corner, for which we use the regularity theory 1 in (OS95), which yields the asymptotic behavior: 1 1 3 |u| . r 2 , |ux , uy | . r 2 , |D2 u| . r 2 , (4.5.47) where r is the distance to the corner. Defining Cr to be a solid ball of radius r around the corner, we have: Z Z Z uyyy · @y {uwy 2 } = uyyy · @y {uwy 2 } uyyy · @y {uwy 2 }. (4.5.48) ⌦ Cr Cr 3 First, the expansions in (OS95) show that uyyy r 2 2 L2 . Therefore, taking limit as r ! 0, the latter term in (4.5.48) vanishes, and it remains to treat the former term: Z Z Z uyyy · @y {uwy 2 } = + uyy · @yy {uwy 2 } uyy · @y {uwy 2 } dS. (4.5.49) ⌦ Cr ⌦ Cr @Cr For the surface integral, we use the expansions from (4.5.47), and that y  r: Z Z Z 3 1 uyy @y {uwy 2 } dS  r 2 r 2 y2  1 ! 0. (4.5.50) @Cr @Cr @Cr 1 One 1 applies Theorem 4.1 in (OS95) with = 1 + , q = q1 = 2, h = and h1 = 2 + to obtain 3 1 1 1 1 = 2 . Theorem 4.1 gives ||r 2 u, r 2 Du, r 2 D 2 u|| L2 < 1. One can then bootstrap this regularity to 1 obtain ||r 2 +k D 2+k u||L2 < 1. Standard Sobolev embedding arguments give the pointwise asymptotics in (4.5.47). 458 We now move to the second term from (4.5.43): Z Z Z 2"uxxy · @y {uy 2 w} = + 2"uxy @xy {uy 2 w} 2"uxy @y {uy 2 w} x=L Z Z 2 2 = + 2"uxy y + 4"uxy ux yw Z Z Z 2 2"uxy uy y 4"uxy uy 2"uxy @y {uy 2 w} x=L Z Z & 2"uxy y 2 2 2"uxy @y {uy 2 w} x=L O(RHS) "O(LHS), (4.5.51) where we have used the following estimates: Z Z 4"uxy ux yw = 2"u2x w, (4.5.52) Z Z 2 2"uxy uy y = "u2y y 2  " · (4.5.33)  "O(LHS), (4.5.53) x=L Z Z Z h i 4"uxy uy = + 4"ux u + 4"ux uy y  " ||ux ||2L2 + ||uy y||2L2 . (4.5.54) Next, the third term from (4.5.43): Z Z "vxyy @y {uy 2 w} = + "vxy @yy {uy 2 w} Z h i =+ "vxy uyy y 2 w + 2uw + 4uy yw Z Z Z 2 = "uxx uyy y w 2"vx uy w 4"uxx uy yw Z Z Z = "uxx uyy y 2 w 2"vx uy w + 4"ux uxy yw Z 4"ux uy yw x=L Z = "uxx uyy y 2 w + O(RHS) + "O(LHS). (4.5.55) 459 where we have estimated: Z p p | 4"ux uy yw| . "|| "ux ||L2 (x=L) ||uy y||L2 (x=L) , (4.5.56) Zx=L p p | 2"vx uy w|  "||uy ||L2 || "vx ||L2 , (4.5.57) Z Z 4"uxy ux yw = 2"u2x w. (4.5.58) We now come to the first term from (4.5.44): Z "@x {uyy + "vxy } · @x {uy 2 w} Z Z = {uyy + "vxy } · @xx {uy 2 w} + "{uyy + "vxy } · @x {uy 2 w} x=L Z h i Z = {uyy + "vxy } · uxx y 2 w 2ux y 2 + "@y bL · @x {uy 2 w} x=L Z Z Z Z = "uyy uxx wy + " uxx y w + 2"uyy ux y + 2"2 vxy ux y 2 2 2 2 2 2 Z + "@y bL · @x {uy 2 w} Z Zx=L = "uyy uxx wy + "2 u2xx y 2 w + "O(RHS) + "O(LHS) 2 (4.5.59) p + ||@y bL hyi2 ||2L2 (x=L) + "||u, "ux ||2L2 (x=L) . We have estimated: Z Z Z 2 2 + 2"uyy ux y = 2"uy uxy y 4"uy ux y Z Z = "u2y y 2 4"uy ux y, x=L  "||uy y||2L2 (x=L) + "||uy y||L2 ||ux ||L2 h i . " O(LHS) + O(RHS) , (4.5.60) Z Z Z + 2"2 vxy ux y 2 = 2"2 uxx ux y 2 = "2 u2x y 2  "O(LHS). (4.5.61) x=L 460 Next from (4.5.42): Z Z Z + 2vyyy · "@x {uwy 2 } = 2"vyyy vy wy 2 2"vyyy uy 2 Z Z Z Z 2 2 2 = + 2"vyy y + 4"vyy vy wy + 2"vyy uy y + 4"vyy uy Z Z Z 2 2 = 2"vyy y w 2"vy2 w 2"uxy uy y 2 Z Z 4"vy uy y 4"vy u Z h i & 2"vyy2 2 y w " O(LHS) + O(RHS) , (4.5.62) where we have estimated the following terms: Z Z 2 2"uxy uy y = "u2y y 2 , (4.5.63) x=L Z | 4"vy uy y|  "||uy y||L2 ||vy ||L2 , (4.5.64) Z | 4"vy u|  "O(L)||ux ||2L2 . (4.5.65) We can now collect the estimates from (4.5.46), (4.5.51), (4.5.55), (4.5.59), (4.5.62) to get: Z Z " uy · @y {uy 2 w} + "vy · @x {uy 2 w} Z h i Z & u2yy + 4"u2xy + "2 u2xx 2"uxx uyy y 2 w 2"uxy @y {uy 2 w} x=L O(RHS) "O(LHS). (4.5.66) We now have the Pressure contributions: Z Z Z Pyx · @y {uy 2 w} + Pyy · @x {uy 2 w} = Py · @y {uy 2 w}. (4.5.67) x=L 461 Using Py 2"uxy = aL (y) on x = L, the boundary term above can be combined with that in (4.5.66) yielding: Z ⇣ ⌘ Z Py 2"uxy · @y {uy 2 w} = @y aL · @y {uy 2 w} x=L x=L  ||@y aL · hyi2 ||L2 (x=L) ||uy y, u||L2 (x=L) (4.5.68) . N ||@y aL · hyi2 ||2L2 (x=L) + ||uy y, u||2L2 (x=L) , (4.5.69) the latter of which can be absorbed into the left-hand side of our estimate, specifically the positive contribution of (4.5.33). Thus, summing (4.5.67) with (4.5.66) yields: Z Z " uy · @y {uy w} + "vy · @x {uy 2 w} 2 Z Z + Pyx · @y {uy 2 w} + Pyy · @x {uy 2 w} Z n o & u2yy + 4"u2xy + "2 u2xx 2"uxx uyy y 2 w O(RHS) "O(LHS) Z n o & u2yy + "u2xy + "2 u2xx y 2 , (4.5.70) where the final inequality follows from (4.5.71). This concludes the proof. 4.5.1 The Korn’s Inequality Lemma 4.5.2. For any functions [u, v] 2 X , the following estimate is valid: Z h i u2yy + 4"u2xy + "2 u2xx 2"uyy uxx y 2 w(x) dx dy Z h i & u2yy + "u2xy + "2 u2xx y 2 w(x) dx dy p "||uy y, "ux y||2L2 ||uy , ux ||2L2 . (4.5.71) 462 Proof. We would like to apply the Korn inequality to generate positive terms: Z h i u2yy + 4"u2xy + "2 u2xx 2"uyy uxx y 2 w(x) dx dy. (4.5.72) We will first rescale to original Eulerian coordinates, so as to ensure all estimates are independent of L: p x " p X= , Y = y, U (X, Y ) = u(x, y), V (X, Y ) = "v(x, y). (4.5.73) L L Define also wL (X) = 1 LX. This gives the following relations: L L2 UX = Lux , UY = p uy , UY Y = uyy , (4.5.74) " " p p L2 VX = "Lvx , VY = Lvy , VXX = L2 "vxx , VY Y = p vyy . (4.5.75) " It is clear that: Z h i p (4.5.72) = " UY2 Y + 4UXY 2 2 + UXX 2UY Y UXX Y 2 wL (X) dX dY. (4.5.76) We will define: p p U (1) := UY Y wL , V (1) := VY Y wL . (4.5.77) (1) p p UY = UY Y Y wL + UY wL , (4.5.78) (1) p 1 UX = UXY Y wL + UY Y p L, (4.5.79) 2 wL (1) p p VY = V Y Y Y w L + VY w L (4.5.80) 463 (1) p 1 VX = VXY Y wL + VY Y p L. (4.5.81) 2 wL We will now calculate: Z Z p " UY2 wL dX dY = u2y w dx dy, (4.5.82) Z 2 Z p 2 2L " UY Y dX dY = " u2y y 2 dx dy, (4.5.83) wL Z Z p " VY2 wL dX dY = "vy2 w dx dy, (4.5.84) Z Z p L2 " VY2 Y 2 dX dY = "2 vy2 y 2 dx dy, (4.5.85) wL Z Z p " " |U (1) |2 dX dY = 2 u2y y 2 w dx dy, (4.5.86) L Z Z p (1) 2 "2 " |V | dX dY = 2 vy2 y 2 w dx dy, (4.5.87) L Thus: Z Z p p (1) " UY2 Y 2 Y wL = " |UY |2 + C, Z Z p 2 p (1) " UXY Y 2 wL = " 4|UX |2 + C, Z Z p 2 p (1) " UXX Y 2 wL = " |VX |2 + C, Z Z p 2 p (1) (1) 2 " UY Y UXX Y wL = " 2UY VX + C, where: p |C| .N · ||uy , ux ||L2 + " · ||uy y, "ux y||2L2 Z p (1) (1) (1) + " |UY |2 + |VX |2 + |UX |2 . (4.5.88) 464 According to this, we can write: Z p h (1) (1) (1) (1) (1) i (4.5.72) & " |UY |2 + 4|UX |2 + |VX |2 2UY VX |C|. (4.5.89) By adding and subtracting (4.5.86) - (4.5.87) and up to redefining C, we have: Z h i p (1) (1) (1) (1) (1) (4.5.72) & " |UY |2 + 4|UX |2 + |VX |2 2UY VX dX dY Z h i p + " |U (1) |2 + |V (1) |2 dX dY |C|. An application of Korn’s inequality yields: Z p (1) (1) (1) (4.5.72) & " |UY |2 + |UX |2 + |VX |2 p ||uy , ux ||2L2 + "||uy y, "ux y||2L2 Z n o & u2yy + "u2xy + "2 u2xx y 2 w(x) dy dx p ||uy , ux ||2L2 + "||uy y, "ux y||2L2 . (4.5.90) This concludes the proof. 4.5.2 Summary of L2 Estimates: Let us now consolidate the L2 -based estimates, by combining (4.3.1), (4.4.5), (4.5.5). First, we will define the following L2 based norm: p p ||u, v||X1 := ||uy · y||L2 + || "ux · y||L2 + ||vy , "vx ||L2 465 n p o + || uyy , "uxy , "uxx · y||L2 . (4.5.91) Recalling the boundary norm given in (4.2.23), accumulating estimates (4.5.5), (4.3.1), v0 and (4.4.5), and taking 0 < L << || Ye ||L1 << 1 gives: ||u, v||2X1 + ||u, v||2B . R1 + R2 + R3 . (4.5.92) 4.6 Uniform Estimates We will now obtain L1 estimates for solutions [u, v] to the system (4.2.30) - (4.2.32), which are based on bootstrapping estimates that are valid for the Stokes operator. Lemma 4.6.1. Solutions [u, v] 2 X to the system (4.2.30) - (4.2.32), with the boundary conditions (4.2.33) - (4.2.34) satisfy the following uniform estimate: p n p " 4 ||u, ¯ .C( , L) ||u, "v||H 1 + C(aL , bL ) "v||L1 (⌦) p p o + ||Su , "Sv ||L2 + ||f, "g||L2 . (4.6.1) Proof. The proof follows from (GN17), Lemma 4.1. Note that the estimate up to the ¯ is guaranteed according to (AF03), P. 98, Equation 9, as our domain ⌦ boundary, L1 (⌦), satisfies the strong local Lipschitz property, as defined by (AF03), P. 66. We emphasize that for our analysis, it is important to obtain the uniform control on the boundary x = L, due for instance, to the nonlinear contributions from (4.7.6). We now relate the right-hand side above to our norms. 466 Lemma 4.6.2. For any functions, [u, v] 2 X , the following estimate holds: p p 1 p p ||u, "v||H 1 + ||Su , "Sv ||L2 + ||" 2 {Ru , "Rv }||L2 + ||Lb1 , "Lb2 ||L2 p + ||N u (¯ u, v¯), u, v¯)||L2 . 1 + ||u, v||X1 + ||¯ "N v (¯ u, v¯||2X . (4.6.2) p 1 p Proof. The estimates on ||u, "v||H 1 , ||" 2 {Ru , "Rv }||L2 follow trivially, the latter from p (4.9.123). The estimates on ||Lb1 , "Lb2 ||L2 , as defined in (4.9.116) - (4.9.117), follow from (4.9.118). Next, referring to the definition of Su in (4.2.35), and the estimates in (4.9.124), ||Su ||L2 = ||us ux + usx u + vs uy + usy v||L2 ve0 v0  ||us , usx , , vs pe , usy y||L2 ||ux , uy y||L2 . (4.6.3) Y " Similarly, referring to the definition of Sv given in (4.2.35) and the estimates (4.9.124): p p ⇣ ⌘ || "Sv ||L2  || " us vx + vsx u + vs vy + vsy v ||L2 p p p  ||us , "vsx , "vs , vsy ||L2 || "vx ||L2 . (4.6.4) Referring to the definitions of the nonlinearities given in (4.2.36): 1 ||N u (¯ u, v¯)||L2 = " 2 + ||¯uu¯x + v¯u¯y ||L2 n p o  " 2 ||" 2 u ¯, "¯ v ||L1 ||¯ ¯y ||L2 , ux , u (4.6.5) p ⇣ ⌘ || "N v (¯u, v¯)||L2  ||"1+ u ¯v¯x + v¯v¯y ||L2 1 p 1 1  " 2 + 2 ||" 2 u vx ||L2 + " 2 + 2 ||" 2 + 2 v¯||L1 ||¯ ¯||L1 || "¯ vy ||L2 . (4.6.6) The above estimates imply the result. 467 Combining (4.6.1), (4.6.2) with the definition of (f, g) given in (4.2.36) - (4.2.37), together with relevant definitions in (4.9.15), (4.9.24), and (4.9.117) gives the following: Corollary 4.6.3. Solutions [u, v] 2 X to the system (4.2.30) - (4.2.32), with the boundary conditions (4.2.33) - (4.2.34) satisfy the following uniform estimate: p h i " 2 ||u, "v||L1 . " 4 + " 4 ||u, v||X1 + ||¯ u, v¯||2X . (4.6.7) Combining with (4.5.92), we have now controlled the full X norm: Corollary 4.6.4. Solutions [u, v] 2 X to the system (4.2.30) - (4.2.32), with the boundary conditions (4.2.33) - (4.2.34) satisfy the following estimate: ||u, v||2X . " 2 + R1 + R2 + R3 + " 2 ||¯ u, v¯||4X . (4.6.8) It remains to control Ri , which we now expand by recalling (4.3.2), (4.4.6), (4.5.6), and (4.2.37): Z h i 1 R1 = " 2 Ru,1 + N u + Lb1 · u Z h i 1 + " " 2 Rv,1 + N v + Lb2 v, (4.6.9) Z h i 1 R2 = " 2 Ru,1 + N u + Lb1 · y Z h i 1 + " " 2 Rv,1 + N v + Lb2 · x (4.6.10) Z h i 1 R3 = " 2 @y Ru,1 + @y N u + @y Lb1 · @y {uy 2 w} Z h i 1 " " 2 @y Rv,1 + @y N v + @y Lb2 · @x {uy 2 w}. (4.6.11) We now turn to controlling these quantities. 468 4.7 Nonlinearities We now provide estimates on R1 , R2 , R3 , as displayed in (4.6.9) - (4.6.11). We will first estimate the nonlinear terms, N u , N v , which are in turn defined in (4.2.36). Because we will eventually perform a contraction mapping argument, we will consider N u (¯ u, v¯), N v (¯ u, v¯), u, v¯] 2 X . We have: where [¯ 1 n o @y N u (¯ u, v¯) = " 2 + u ¯u¯xy + v¯u ¯yy , (4.7.1) 1 n o @y N v (¯ u, v¯) = " 2 + u ¯v¯xy + v¯y2 + v¯v¯yy . ¯y v¯x + u (4.7.2) The first step is to provide estimates on the nonlinear contributions from R3 , as defined in (4.5.6). For this we have: u, v¯] 2 X , the following estimate holds: Lemma 4.7.1. For any vector fields [u, v], [¯ Z Z | @y N u (¯ u, v¯) · @y {uwy 2 }| + | u, v¯) · "@x {uwy 2 }| . " 2 ||¯ @y N v (¯ u, v¯||2X ||u, v||X . (4.7.3) Proof. Turning to the first term from (4.7.1), we will expand via the product rule: Z 1 "2+ u ¯u¯xy · @y {uy 2 w} Z Z 1 1 + = " 2 u ¯u¯xy · uy y w + " 2 + u 2 ¯u¯xy · u2yw Z Z 1 1 = "2+ u ¯u¯xy uy y 2 w "2+ u ¯y u ¯x u2yw Z Z 1 1 "2+ u ¯u ¯x uy 2yw "2+ u¯uu¯x 2w p  " 2 ||" 2 u ¯||L1 || "¯ uxy y||L2 ||uy y||L2 1 + " 2 + 2 ||" 2 u||L1 ||¯ uy y||L2 ||¯ ux ||L2 1 + " 2 + 2 ||" 2 u ¯||L1 ||uy y||L2 ||¯ ux ||L2 1 + " 2 + 2 ||" 2 u||L1 ||¯ ux ||2L2 . (4.7.4) 469 Turning to the second term from (4.7.1): Z 1 " 2 + v¯u ¯yy · @y {uy 2 w} Z Z 1 1 + = " 2 ¯yy uy y w + " 2 + v¯u v¯u 2 ¯yy u2yw 1 h i  " 2 ||" 2 + 2 v¯||L1 ||¯ uyy y||L2 ||uy y||L2 + O(L)||ux ||L2 . (4.7.5) We will now turn to the first term from (4.7.2), which is the most delicate because v¯x cannot accept any weights of y, according to our norm X , (4.2.22). As a result, we must rely on an integration by parts in x: Z Z Z 3 3 3 "2+ u ¯y v¯x @x {uy 2 w} = "2+ u ¯y v¯x ux y 2 w "2+ u ¯y v¯x uy 2 Z Z 3 3 = " 2 + v¯u ¯xy ux y 2 w " 2 + v¯u¯y uxx y 2 w Z Z 3 3 + 2 " 2 v¯u ¯ y ux y "2+ u ¯y v¯x uy 2 Z 3 + " 2 + v¯u ¯ y ux y 2 x=L Z Z 3 3 + = " 2 v¯u ¯xy ux y 2 w " 2 + v¯u¯y uxx y 2 w Z Z Z 3 3 3 + 2 + " 2 v¯u ¯ y ux y + " 2 v¯ux u¯y y + " 2 + v¯u¯ 2 uxy y 2 Z Z 3 3 " 2 + v¯u¯ uy y 2 + " 2 + v¯u ¯ y ux y 2 x=L x=L 1 p p  " 2 ||" 2+2 v¯||L1 || "uxy y||L2 || "ux y||L2 1 + " 2 ||" 2 + 2 v¯||L1 ||¯ uy y||L2 ||"uxx y||L2 1 1 p + " 2 + 2 ||" 2 + 2 v¯||L1 ||¯ uy y||L2 || "ux y||L2 1 1 p + " 2 + 2 ||" 2 + 2 v¯||L1 || "ux y||L2 ||¯ uy y||L2 1 p p + " 2 ||" 2 + 2 v¯||L1 || "ux y||L2 || "uxy y||L2 1 1 p + " 2 + 2 ||" 2 + 2 v¯||L1 ||¯ uy y||L2 (x=L) || "uy||L2 (x=L) 1 1 p + " 2 + 2 ||" 2 + 2 v¯||L1 ||¯ uy y||L2 (x=L) || "ux y||L2 (x=L) . (4.7.6) 470 Note that for the above term, (4.7.6), it is imperative to obtain control of v on the boundary x = L, as shown in estimate (4.6.1). We now move to the second term from (4.7.2): Z Z Z 3 3 3 "2+ u ¯v¯xy @x {uy 2 w} = "2+ u ¯v¯xy ux y 2 w "2+ u ¯v¯xy uy 2 p  " 2 ||" 2 u ¯||L1 ||"¯ uxx y||L2 || "ux y||L2 p + " 2 ||" 2 u||L1 ||"uxx y||L2 || "ux y||L2 . (4.7.7) Now we turn to the third term from (4.7.2): Z Z Z 3 3 3 " 2 + v¯y2 @x {uy 2 w} = " 2 + v¯y2 ux y 2 w " 2 + v¯y2 uy 2 Z Z 3 3 = " 2 + v¯v¯yy ux y 2 w " 2 + v¯v¯y uxy y 2 w Z Z 3 3 + " 2 v¯v¯y ux 2yw " 2 + v¯y2 uy 2 1 p p  " 2 ||" 2 + 2 v¯||L1 || "¯ vyy y||L2 || "ux y||L2 1 p p + " 2 ||" 2 + 2 v¯||L1 || "¯ ux y||L2 || "uxy y||L2 1 1 p + " 2 + 2 ||" 2 + 2 v¯||L1 || "¯ ux y||L2 ||ux ||L2 1 p + " 2 + 2 ||" 2 u||L1 || "¯ ux y||2L2 . (4.7.8) Now we turn to the fourth, final term from (4.7.2): Z Z Z 3 3 3 " 2+ 2 v¯v¯yy @x {uwy } = " 2+ v¯v¯yy ux wy 2 " 2 + v¯v¯yy uy 2 1 p p . " 2 ||" 2 + 2 v¯||L1 || "¯ uxy y||L2 || "ux y||L2 . (4.7.9) These estimates conclude the proof of the desired result, estimate (4.7.3). 471 We will now come to the nonlinear contributions to the energy estimates, which are contained in (4.6.9): u, v¯] 2 X , the following estimate holds: Lemma 4.7.2. For any vector fields [u, v], [¯ Z | N u (¯ u, v¯) · u + "N v (¯ u, v¯||2X ||u, v||X . u, v¯) · v|  " 2 ||¯ (4.7.10) Proof. We turn to the definitions of N u , N v which are given in (4.2.36). From there, the following calculations follow: Z 1 1 1 "2+ | u ¯u¯x · u|  " 2 + 2 ||" 2 u ux ||2L2 . " 2 + 2 ||¯ ¯||L1 ||¯ u, v¯||2X ||u, v||X , (4.7.11) Z 1 p "2+ | ¯y · u|  " 2 ||" 2 v¯u ux ||L2 ||uy ||L2 . " 2 ||¯ v ||L1 ||¯ "¯ u, v¯||2X ||u, v||X , (4.7.12) Z 1 1 p p 1 "2+ | ¯v¯x · "v|  " 2 + 2 ||" 2 u u vx ||L2 || "vx ||L2 . " 2 + 2 ||¯ ¯||L1 || "¯ u, v¯||2X ||u, v||X , (4.7.13) Z 1 1 1 p 1 " 2+ | vy ||L2 . " 2 + 2 ||¯ v¯v¯y · "v|  " 2 + 2 ||" 2 + 2 v¯||L1 || "vx ||L2 ||¯ u, v¯||2X ||u, v||X . (4.7.14) The desired result follows from these calculations. We will now provide nonlinear estimates arising from the positivity estimate, in particular we must evaluate the contributions of the nonlinearity in (4.6.10): u, v¯] 2 X , the following estimate holds: Lemma 4.7.3. For any vector fields [u, v], [¯ Z Z | N u (¯ u, v¯) · y| + | N v (¯ u, v¯) · " x| u, v¯||2X ||u, v||X .  " 2 ||¯ (4.7.15) Proof. We again turn to the definitions of N u , N v from (4.2.36): Z 1 1 "2+ | u ¯u¯x · y|  " 2 + 2 ||" 2 u ¯||L1 ||¯ ux ||L2 || y ||L2 , (4.7.16) 472 Z 1 1 "2+ | ¯y · v¯u y|  " 2 ||" 2 + 2 v¯||L1 ||¯ uy ||L2 || y ||L2 , (4.7.17) Z 1 p p "2+ | ¯v¯x · " u x|  " 2 ||" 2 u ¯||L1 || "¯ vx ||L2 || " x ||L2 , (4.7.18) Z 1 1 p "2+ | v¯v¯y · " x|  " 2 ||" 2 + 2 v¯||L1 ||¯ vy ||L2 || " x ||L2 . (4.7.19) This concludes the proof. 4.8 Forcing Recall the definitions given in (4.9.15) and (4.9.24), and the definitions given in (4.9.116) - (4.9.117). The purpose of the following estimates is to estimate the contributions of the forcing terms Ru,1 , Lb1 , Rv,1 , Lb2 into R1 , R2 , R3 , as shown in (4.6.9) - (4.6.11) Thus, we will analyze the forcing contributions: Lemma 4.8.1. For any vector fields [u, v] 2 X , the following estimates hold: Z n o Z n o 1 1 | " 2 Ru,1 · u + "Rv,1 · v | + | " 2 Ru,1 · y + "R v,1 x | Z Z 1 1 + | " 2 @y Ru,1 · @y (uy 2 w)| + | " 2 "@y Rv,1 @x {uy 2 w}| Z Z Z b b b + | L1 · u + "L2 v| + | L1 · y + "Lb2 x | Z Z + | @y Lb1 · @y {uy 2 w}| + | "@y Lb2 · @x {uy 2 w}| 1 . (C(a0 , b0 , aL , bL ) + " 4 )||u, v||X . (4.8.1) Proof. We recall estimate (4.9.123) from the Appendix, which we then directly use. First, we start with the contributions to R1 , shown in (4.6.9): Z n o 1 1 p p " 2 Ru,1 · u + "Rv,1 · v  " 2 ||Ru,1 , "Rv,1 ||L2 ||u, "v||L2 1 3 p " 2 " 4 O(L)||ux , "vx ||L2 . (4.8.2) 473 We now move the contributions from R2 , shown in (4.6.10): Z n o 1 1 3 p " 2 Ru,1 · y + "Rv,1 x " 2 " 4 || y, " x ||L2 . (4.8.3) Next, we move to the higher order quantities from R3 , shown in (4.6.11): Z Z Z 1 1 1 " 2 @y Ru,1 · @y (uy 2 w) = " 2 @y Ru,1 · uy y 2 w + " 2 @y Ru,1 · u2yw 1 h i " ||@y Ru,1 y||L2 ||uy y||L2 + O(L)||ux ||L2 2 1 3 h i  " 2 " 4 ||uy y||L2 + O(L)||ux ||L2 , (4.8.4) Z Z h i 1 1 " 2 "@y Rv,1 @x {uy 2 w} = " 2 "@y Rv,1 ux y 2 w uy 2 1 p p " 2 || "@y Rv,1 y||L2 || "ux y||L2 1 3 p " 2 " 4 || "ux y||L2 . (4.8.5) The estimates on Lb1 , Lb2 contributions follow directly from estimate (4.9.118). This concludes the proof. Combining (4.7.3), (4.7.10), (4.7.15), and (4.8.1): Corollary 4.8.2. For R1 , R2 , R3 defined as in (4.4.6), (4.5.6), (4.3.2), we have: h 1 i |R1 + R2 + R3 | . C(a0 , b0 , aL , bL ) + " 4 ||u, v||X + " 2 ||u, v||2X + " 2 ||¯ u, v¯||4X . (4.8.6) Combining the above estimate with (4.5.92) and (4.6.7), and performing Young’s in- equality for the product C(a0 , b0 , aL , bL )||u, v||X above to absorb ||u, v||2X to the left-hand side of (4.8.7), we have now established the main a-priori estimate: 474 Theorem 4.8.3 (X -Estimate). Solutions [u, v] 2 X to the system (4.2.30) - (4.2.32), with the boundary conditions (4.2.33) - (4.2.34) satisfy the following estimate: 1 ||u, v||2X . C(a0 , b0 , aL , bL ) + " 4 u, v¯||4X . + " 2 ||¯ (4.8.7) With the main a-priori estimate in hand, we give the formal arguments leading to existence of a solution in X in Appendix 4.10. In particular, Theorem 4.8.3 coupled with Proposition 4.10.2 gives the main result, Theorem 4.2.1. 4.9 Construction of Profiles 4.9.1 Specification of Ru Define: Ru := U " @x U " + V " @y U " + @x P " @yy U " "@xx U " , (4.9.1) @y " Rv := U " @x V " + V " @y V " + P @yy V " "@xx V " . (4.9.2) " In this subsection, we will specify the equations we shall take for Ru . We will first expand the nonlinear terms in the following manner: ⇣ p p 1 ⌘ U " @x U " = u0e + u0p + "u1e + "u1p + " 2 + u ⇥ ⇣ p p 1 ⌘ u0ex + u0px + "u1ex + "u1px + " 2 + ux = {u0e (x, 0) + u0p }u0px + u0ex (x, 0)u0p + {u0e u0e (x, 0)}u0px + {u0ex u0ex (x, 0)}u0p p h i p h + " u0p u1ex + u1e u0px + " {u0e u0e (x, 0)}u1px 475 p h 0 i + " {ue (x, 0) + u0p }u1px + {u0ex (x, 0) + u0px }u1p i h + {u0ex u0ex (x, 0)}u1p + " (u1e + u1p )u1px i h p ⇣ ⌘ i + u1ex u1p + u0e u0ex + " u1ex u0e + u1e u0ex + "u1e u1ex 1 + " 2 + {us ux + usx u} + "1+2 uux . (4.9.3) ⇣ v0 p 1 ⌘ V " @y U " = pe + vp0 + ve1 + "vp1 + " 2 + v " ⇣p p 1 ⌘ ⇥ "u0eY + u0py + "u1eY + "u1py + " 2 + uy ⇣ ⌘ 0 = yveY (x, 0) + vp0 + ve1 (x, 0) u0py p h i + " {vp0 + yveY 0 (x, 0) + ve1 (x, 0)}u1py + u0py vp1 p h p i + " vp0 (u0eY + "u1eY ) h v0 i ⇣ p ⌘ + pe 0 yveY (x, 0) u0py + "vp1 u0eY + "u1eY + "vp1 u1py " h i p h i + ve0 Y veY 0 (x, 0) u1py + " ve1 ve1 (x, 0) u1py h i p + ve1 ve1 (x, 0) Y veY 1 u0py + "yveY 1 u0py h p ⇣ ⌘ i + ve0 u0eY + " ve0 u1eY + ve1 u0eY + "ve1 u1eY 1 + " 2 + {usy v + vs uy } + "1+2 vuy . (4.9.4) Inserting into the system (4.9.1) gives the following expansion: n Ru = {u0e (x, 0) + u0p }u0px + u0ex (x, 0)u0p + {yveY 0 (x, 0) + vp0 + ve1 (x, 0)}u0py o 0 + Ppx u0pyy (4.9.5) p n + " {u0e (x, 0) + u0p }u1px + {u0ex (x, 0) + u0px }u1p o 0 + {yveY (x, 0) + vp0 + ve1 (x, 0)}u1py + u0py vp1 u1pyy + Ppx 1 F1 (4.9.6) 476 p h 0 1 i + " ue uex + u0ex u1e + ve0 u1eY + ve1 u0eY + Pex 1 (4.9.7) h i + R˜u,1 + "Ppx 1 2 + "2+ u u " u + S (u, v) + Px + N (u, v) . (4.9.8) We will define: p 1 h v0 i F1 :=vp0 {u0eY + "u1eY } + p pe 0 yveY (x, 0) u0py " " 1 h i 1 + yveY u0py + p {u0e u0e (x, 0)}u0px + {u0ex u0ex (x, 0)}u0p " + u0p u1ex + u1e u0px , (4.9.9) ⇣ ⌘ R˜u,1 :=" 2 vp1 u1eY + "vp1 u1py + ve0 Y veY 3 0 (x, 0) u1py ⇣ ⌘ p h i + ve1 ve1 (x, 0) Y veY 1 (x, 0) u0py + " ve1 ve1 (x, 0) u1py p p + "{u0e u0e (x, 0)}u1px + "u1p {u0ex u0ex (x, 0)} h i + " (u1e + u1p )u1px + u1ex u1p + u0eY vp1 3 h p i + "u0pxx + " 2 u1pxx + " u1e u1ex + ve1 u1eY + u0e + " u1e , (4.9.10) 1 ⇣ ⌘ N u (u, v) := " 2 + uux + vuy , (4.9.11) 1 ⇣ ⌘ N v (u, v) := " 2 + uvx + vvy , (4.9.12) S u (u, v) := us ux + usx u + vs uy + usy v, (4.9.13) S v (u, v) := us vx + vsx u + vs vy + vsy v. (4.9.14) Equations (4.9.5) - (4.9.7) define the equations for our approximate layers, as seen in (4.9.53), (4.9.59), and (4.9.74), thereby contributing the final line, (4.9.8) into the remainder ˜ u,1 to Ru,1 , which accounts for the fact that equation, (4.9.105). We must actually modify R the layers [u1p , vp1 ] are cuto↵ at y ! 1: p ˜ u,1 + Ru,1 := R "Rpu + "Ppx 2 , (4.9.15) 477 where: Rpu := {u0e (x, 0) + u0p }u1px + {u0ex (x, 0) + u0px }u1p + {yveY 0 (x, 0) + vp0 }u1py + u0py vp1 u1pyy + Ppx 1 F1 . (4.9.16) We are then left with: 1 h i "2+ "u + S u (u, v) + Px + N u (u, v) = Ru,1 . (4.9.17) 4.9.2 Specification of Rv We turn now to the simplification of (4.9.2). 1 h i P0 py 1 Ppy Rv = p u0e vex 0 + ve0 veY 0 + PeY 0 + + p (4.9.18) " " " h i + u0e vex 1 + u1e vex0 + ve0 veY 1 + veY0 ve1 + PeY1 (4.9.19) ⇣ p p ⌘ 0 + vpx u0e + u0p + "u1e + "u1p v0 ⇣ p ⌘ ⇣ p ⌘ + pex u0p + "u1p + vex 1 u0p + "u1p " 0 v 0 ⇣ p 1 p 1 ⌘ + pe vpy + ve0 vpy 1 + vp0 veY 0 + vpy0 + "veY + "vpy " p 1 + ve1 (vpy 0 + "vpy ) + " vp0 + Ppy 2 (4.9.20) p 1 ⇣ 0 p p ⌘ + "vpx ue + u0p + "u1e + "u1p + " vp1 p ⇣ 0 p 1 p 1 ⌘ + "vp1 veY + vpy 0 + "veY + "vpy hp p p i + " ve0 + " ve1 + "u1e vex 1 + "ve1 veY 1 1 h Py i + "2+ " v + S v (u, v) + + N v (u, v) (4.9.21) " We shall make the identifications so that (4.9.18) and (4.9.19) vanish by using these 478 equations to define the construction of the approximate layers in (4.9.53), (4.9.60), and (4.9.74). We then define Pp2 via (4.9.20): Z 1 ⇣ ⌘ v0 ⇣ ⌘ p p p Pp2 = 0 vpx u0e + u0p + "u1e + "u1p + pex u0p + "u1p y " ⇣ p ⌘ 0 v 0 ⇣ p 1 p 1 ⌘ 1 + vex u0p + "u1p + pe vpy + ve0 vpy 1 + vp0 veY 0 0 + vpy + "veY + "vpy " p + ve1 (vpy 0 + "vpy 1 ) + " vp0 . (4.9.22) This choice enforces the vanishing of line (4.9.20). We are then left with: 1 h Py i "2+ v " v + S (u, v) + + N v (u, v) = Rv,1 , (4.9.23) " where p 1 ⇣ 0 p p ⌘ Rv,1 := "vpx ue + u0p + "u1e + "u1p + " vp1 p ⇣ 0 p 1 p 1 ⌘ + "vp1 veY 0 + vpy + "veY + "vpy hp p p i + " ve0 + " ve1 + "u1e vex 1 + "ve1 veY 1 . (4.9.24) This defines the second equation for the remainder, as seen in (4.9.106). 4.9.3 Construction of Layers We are prescribed the Euler flow [u0e , ve0 , Pe0 ]. The first task is to verify that there exists Euler flows satisfying assumptions (4.2.24) - (4.2.27): Proposition 4.9.1. There exists a nontrivial set of Euler flows, [u0e , ve0 , Pe0 ] satisfying as- sumptions (4.2.24) - (4.2.27). 479 We will start with the shear flow U0 (Y ), satisfying the following hypothesis: c0  U0  C0 , (4.9.25) U0 smooth, with rapidly decaying derivatives, (4.9.26) @Y U0 0, (4.9.27) U0 = 1 in a neighborhood of 0. (4.9.28) RY Such a shear has stream function 0 (Y ) = 0 U0 . Such a stream function has the following asymptotics: 0 0 |Y =0 = 0, 0 |x=0 = 0 |x=L = 0 (Y ), lim = U1 2 (c0 , C0 ). (4.9.29) Y !1 Y Note that assumption (4.9.25) implies c0 Y  0  C0 Y . To define our final Euler flow, we must first solve for an perturbative stream function, , using the following elliptic equation: = @ Y U0 + fe ( 0 + ), |x=0 = A0 (Y ), x=L = AL (Y ), |Y =0 = 0, |Y !1 = 0. (4.9.30) We will assume the following conditions on fe and the boundary data A0,L : 0  fe  << 1, (4.9.31) |@ k fe (x + a)| . |@ k fe (x)| for a 0, (4.9.32) fe 2 C 1 (R), rapidly decaying in it’s argument, (4.9.33) fe supported in a neighborhood away from 0 , (4.9.34) 0  A0 , AL  ⇥ L10 , (4.9.35) 480 |@Yk {A0 , AL }|  ⇥ L10 (4.9.36) A0 , AL 2 C 1 (R+ ), rapidly decaying in it’s argument, (4.9.37) A0 , AL supported in a neighborhood away from 0. (4.9.38) A0 6= AL . (4.9.39) It is straightforward to see that the set of admissible fe , A0 , AL is nonempty. First, via hypothesis (4.9.27) and (4.9.31), we have  0, so that via the maximum principle and assumption on the boundary data (4.9.35): 0. (4.9.40) Lemma 4.9.2. Assume (4.9.25) - (4.9.28) and the assumptions (4.9.31) - (4.9.39) are satisfied. For 0 < L << << 1, the following energy estimate holds: ||Y k ||H 1  Ck O( ). (4.9.41) Proof. Define: L x x B(x, Y ) = A0 (Y ) + AL (Y ). (4.9.42) L L B is smooth and all derivatives are order by the assumptions (4.9.36) on A0,L . Define now ¯ = B, which satisfies: ¯= B + @ Y U 0 + fe ( 0 + ), ¯|@⌦ = 0. (4.9.43) 481 An energy estimate coupled with Poincare’s inequality gives: Z Z ⇣ ⌘ Z |r ¯|2 = B + @ Y U 0 · ¯ + fe ( 0 + )· ¯  O( , L)|| ¯x ||L2 + ||fe ( 0 + )||L2 O(L)|| ¯x ||L2 . (4.9.44) We now use (4.9.40) together with assumptions (4.9.32) and (4.9.25) to estimate: ||fe ( 0 + )||L2  ||fe ( 0 )||L2  ||fe (c0 Y )||L2  O( ). (4.9.45) This concludes the proof. We now upgrade to weighted estimates, and higher regularity: Lemma 4.9.3. Assume (4.9.25) - (4.9.28) and the assumptions (4.9.31) - (4.9.39) are satisfied. For 0 < L << << 1, the following energy estimate holds: ||Y m @xj @Yk ||L2  Cm,k,j for any k, m, j 0. (4.9.46) (1) Proof. The first step is to di↵erentiate (4.9.30) in Y . Defining := @Y , this produces: (1) = @Y2 U0 + fe0 ( 0 + )(@Y 0 + (1) ), (1) (1) Y |Y =0 = 0, |x=0,L = @Y A0,L , (4.9.47) where we have evaluated (4.9.30) using the condition (4.9.28), (4.9.34), and (4.9.38) to obtain the Neumann boundary condition above. A homogenization procedure and energy estimate nearly identical to (4.9.43) - (4.9.44) produces: Z | xY |2 + | YY |2 . O( ). (4.9.48) 482 By using now the equation(4.9.30), we also obtain xx in L2 . Note crucially that ||@Y U0 ||L2  O(L) due to the integration in the x-direction, which prevents us from requir- ing a smallness condition on @Y U0 . One can iterate this procedure for higher derivatives. It is also straightforward to obtain weighted in Y estimates, using hypothesis (4.9.26), (4.9.33), and (4.9.37) to absorb weights of Y . This concludes the proof of (4.9.46). E E Proof of Proposition 4.9.1. If we define := 0 + , then solves: E E E E = fe ( ), (0, Y ) = 0 + A0 (Y ), (L, Y ) = 0 + AL (Y ), E E (x, Y ) Y !1 (x, 0) = 0, ! U1 . (4.9.49) Y Solutions to such elliptic equations solve the 2D Euler equations (see (CS12)) by setting: 1 u0e = @Y E , ve0 = @x E = @x , Pe0 = |r E 2 | + Fe ( E ), Fe0 = fe . (4.9.50) 2 E We view as a O( )-perturbation to the shear flow (U0 (Y ), 0) for which = 0, which is therefore achieved by setting fe = A0 = AL = 0. Note that the property (4.9.39) creates 1 the x-dependence, for if A0 = AL , one could solve (4.9.30) for as just a function of Y , creating another shear flow. All properties (4.2.24) - (4.2.27) are easily verified, where the crucial smallness is obtained through the use of (4.9.46): 0 vE 0 || ||L1  ||veY ||L1 = || xY ||L1  || xY ||H 2  O( ). (4.9.51) Y This concludes the proof of the proposition. 483 We will now abandon the particular construction of Proposition 4.9.1, and consider any flow satisfying the assumptions of the paper, namely (4.2.24) - (4.2.27). Similar to the above considerations, there exists a function fe such that: 1 u0eY 0 vex = we0 = 0 = fe ( 0 ), Pe0 = |r 0 2 | + Fe ( 0 ), Fe0 = fe . (4.9.52) 2 Our assumptions (4.2.24) - (4.2.27) guarantee the following: Z Y c0  u0e  C0 ) 0 = u0e ⇠ Y, 0 coupled with we0 is bounded and decaying in Y implies that fe together with derivatives are bounded and decaying, which we state now as a lemma: Lemma 4.9.4. Define fe to satisfy the equalities in (4.9.52). The assumptions on [u0e , ve0 ] stated in (4.2.24) - (4.2.27) imply that fe together with sufficiently many derivatives is bounded and decaying in its argument. The above lemma is in spirit a converse to Proposition 4.9.1, which will be convenient for later constructions (see specifically equation (4.9.62)). In accordance with (4.9.5) and (4.9.18), we will take the following system for the leading order Prandtl layer: {u0e (x, 0) + u0p }u0px + u0ex (x, 0)u0p + {yveY 0 (x, 0) + vp0 + ve1 (x, 0)}u0py 0 + Ppx u0pyy , 0 Ppy = 0, (4.9.53) u0p (x, 0) = ub u0e (x, 0), u0p (0, y) = u0p,0 (y), vp0 (x, 0) = ve1 (x, 0). (4.9.54) 484 Remark. By rewriting the system (4.9.53) for the unknowns: ¯ := u0e (x, 0) + u0( x, y), u 0 v¯ = yveY (x, 0) + vp0 (x, y) + ve1 (x, 0), (4.9.55) we obtain: Z 1 u ¯u¯x + v¯u ¯y ¯yy = u0e (x, 0)u0ex (x, 0), u v¯ = u ¯x , y ¯|y=0 = ub , u ¯|y=1 = u0e (x, 0). u (4.9.56) By evaluating equation (4.2.3) at Y = 0, we see that u0e u0ex |Y =0 = 0 Pex |Y =0 . Note that we do not demand any sign condition on this forcing term. For the system (4.9.53), we have: Proposition 4.9.5. There exists a unique solution, [u0p , vp0 ], to the system (4.9.53), satis- fying the following: sup ||y M @xj @yk {u0p , vp0 }||L2y . C(M, k, j). (4.9.57) x Moreover, the following profile is strictly positive: u0p + u0e & 1. (4.9.58) Proof. The proof follows via an appropriate von-Mises transformation, an application of the standard parabolic maximum principle, and energy estimates in a very similar manner to (GN17). We therefore omit the proof. 485 We will next move to the first Euler layer, which in accordance to (4.9.7) and (4.9.19) is obtained via the following system: u0e u1ex + u0ex u1e + ve0 u1eY + ve1 u0eY + Pex 1 = 0, (4.9.59) u0e vex 1 + u1e vex 0 + ve0 veY 1 0 + veY ve1 + PeY 1 = 0, (4.9.60) u1ex + veY 1 = 0. (4.9.61) By going to the stream function formulation, where r? 1 = [u1e , ve1 ], we have: 1 = fe0 ( 0 ) 1 , (4.9.62) Z x 1 x (x, 0) = ve1 (x, 0) = vp0 (x, 0), ) 1 (x, 0) = 1 + vp0 (x0 , 0) dx0 , (4.9.63) 0 1 1 1 1 (0, y) = 0 (y), (L, y) = L (y). (4.9.64) We assume the data in (4.9.63), (4.9.64) are well-prepared in the following sense: 1 Definition 4.9.6 (Well Prepared Boundary Data). There exists a value of Y Y |Y =0 which 1 is given by evaluating equation (4.9.62) on Y = 0 and using (4.9.63): YY (x, 0) = 1 xx (x, 0) fe0 ( 0 ) 1 (x, 0). The value of 1 YY (x, 0)|x=0 should equal @Y Y 1 0 |Y =0 . Similarly, 1 1 YY (x, 0)|x=0 = @Y Y L |Y =0 . If this is the case, we say the boundary data are well-prepared up to order 2. The generalization to order k is obtained by repeating the above procedure. By standard elliptic regularity, one has: 1 Lemma 4.9.7. Assuming well-prepared boundary data, there exists a solution to the system (4.9.62) - (4.9.64), satisfying the following estimate: ||Y m 1 ||H k .k,m 1. (4.9.65) 486 Proof. Introduce the corrector: 1 x 1 1 x L (Y ) 1 B(x, Y ) = (1 ) 0 (Y ) (x, 0) + 1 (L, 0) (x, 0). (4.9.66) L L By definition, B is regular and decays exponentially fast in Y . Homogenizing: ¯= 1 B, (4.9.67) we have: ¯ f 0( 0 )¯ = B f 0( 0 )B, ¯|@⌦ = 0. (4.9.68) As our boundary data are well-prepared according to Definition 4.9.6, we may take @Y2 of the system and repeat the procedure. In particular: @Y2 1 fe0 ( 0 )@Y2 1 = 2f 00 ( 0 ) 0 1 y Y + f 000 ( 0 )| 0 2 1 Y| + f 00 ( 0 ) 0 YY 1 . (4.9.69) One may define the new corrector B analogously to (4.9.66) and perform standard elliptic estimates to conclude that: ||Y m { 1 YYY , 1 Y Y X, 1 YY }||L2 . 1. (4.9.70) 1 By Hardy inequality, as all derivatives of decay as Y ! 1, we can conclude: ||Y m 1 xY ||L2 . 1. (4.9.71) 487 From equation (4.9.62), it is clear that: || 1 m xx Y ||L2 . 1. (4.9.72) We have thus obtained all H 2 quantities. Taking @Y of (4.9.62) enables us to estimate 1 xxY and taking @x of (4.9.62) enables us to estimate xxx , giving the full H 3 estimate. Next, we can conclude that: || 1 Y ||L1 ([0,1])  || 1 Y ||H 1 ((0,L))  || 1 ||H 3 . 1. (4.9.73) This enables us to iterate the procedure. In accordance to the (4.9.6) and (4.9.18), we will take the following system for the Prandtl-1 layer: u0 upx + u0x up + v 0 upy + u0y vp upyy = F1 , 1 Ppy = 0, (4.9.74) upx + vpy = 0, u1p (x, 0) = u1e (x, 0), u1p (0, y) = u1p0 (y), (4.9.75) u0 := u0e (x, 0) + u0p , v 0 := yveY 0 (x, 0) + vp0 + ve1 (x, 0). (4.9.76) Here, vp will be recovered via: Z y vp = u1px dy 0 . (4.9.77) 0 488 We will homogenize in the following manner: define such that: Z 1 (0) = 1, @yk (0) = 0 for k 1, dy = 0. (4.9.78) 0 Then define: Z 1 u = u1p + (y)u1e (x, 0), v = vp1 + u1ex (x, 0)I (y), I (y) = . (4.9.79) y The new unknowns, [u, v] satisfy the following system: u0 ux + u0x u + v 0 uy + u0y v uyy = F1 + H1 , (4.9.80) H1 := u0 u1ex (x, 0) + u0x u1e (x, 0) + v 0 0 u1e (x, 0) + u0y I u1ex (x, 0) 00 1 ue (x, 0). (4.9.81) We recall the definition of F1 given in (4.9.9). Furthermore, we have the following estimate on the forcing: Lemma 4.9.8. For any m, k, j 0, the following estimate for F1 holds: ||hyim @yk @xj F1 ||2L2 .m,k,j 1. (4.9.82) Proof. The proof follows directly due to the smoothness and rapid decay properties of [u0p , vp0 ]. Lemma 4.9.9. Solutions [u, v] as defined in (4.9.79) to the problem (4.9.80) satisfy the following estimate: sup ||u||2L2y + ||uy ||2L2  C(u1p0 ) + O(L)||ux ||2L2 . (4.9.83) x2[0,L] 489 Proof. One applies u to the above system, (4.9.80), and integrates: Z ⇣ ⌘ Z u0 ux + u0x u + v 0 uy + u0y v · u = {F1 + H1 } · u. (4.9.84) The result follows upon integrating in x and estimating: Z Z Z Z Z Z | u0y uv| + | u0x u2 | + | {F1 + H1 }u| h i  C(u1p0 ) + O(L)||u0y · y, u0x ||L1 ||ux ||2L2 + ||F1 , H1 ||2L2 . (4.9.85) Lemma 4.9.10. Solutions [u, v] as defined in (4.9.79) to the problem (4.9.80) satisfy the following estimate: ||ux ||2L2 + sup ||uy ||2L2y . C(u1p0 ) + ||uy ||2L2 + C(ve0 )||yuy ||2L2 . (4.9.86) Proof. Introduce v = u0 . Then the system becomes: u0 vy + u0y v + u0x u + v 0 uy uyy = |u0 |2 y + u0x u + v 0 uy uyy . Multiplying by y and integrating in y yields: Z Z Z Z |u0 |2 y2 + uyy y = |u0 |2 y2 uy yy Z Z Z Z uxy 1 1 = |u0 |2 2 y + uy 0 2uy vy @y 0 uy v@y2 0 u u u Z Z Z Z @ x 1 2 1 1 = |u0 |2 y2 + u u2y @x { 0 } 2 uy v y @ y 0 2 u0 y u u Z 1 uy v@y2 0 . (4.9.87) u 490 Upon integrating further in x, the final three terms above are estimated: Z Z Z Z Z Z 1 1 1 | u2y @x { } 2 uy v y @ y uy v@y2 | u0 u0 u0 . ||uy ||2L2 + ||vy ||2L2 . (4.9.88) The remaining terms, upon integrating in x: Z Z | u0x u · y|  O(L)|| 2 y ||L2 , (4.9.89) Z Z | v 0 uy · y|  C(ve0 )||yuy ||L2 || y ||L2 . (4.9.90) The right-hand side is estimated simply using Holder’s inequality. Lemma 4.9.11 (Weighted Estimates). Solutions [u, v] as defined in (4.9.79) to the problem (4.9.80) satisfy the following estimate: ||{uy , uyy } · y (y)||2L2 . 1 + ||ux ||2L2 + ||uy ||2L2 . (4.9.91) Proof. Applying @y to the system gives: u0 uxy + u0xy u + v 0 uyy + u0yy v uyyy = @y {F1 + H1 }. (4.9.92) We apply the multiplier uy y 2 · (1 x). The main positive terms are: Z Z u0 uxy · uy y 2 (1 x) + v 0 uyy · uy y 2 (1 x) Z Z 0 Z 0 @x 0 2 2 u 2 2 ux 2 2 = u uy y (1 x) + uy y u y (1 x) 2 2 2 y 491 Z Z Z 2 vy0 2 2 uy 2 0 uy y (1 x) u2y v 0 y(1 x) y (1 x) 2 2 Z Z 0 Z @x u 2 2 = u0 u2y y 2 (1 x) + uy y u2y v 0 y(1 x) 2 2 Z 2 uy 2 y (1 x) 0 . (4.9.93) 2 0 Upon taking integration in x from 0 to X⇤ , we obtain using the smallness of veY (x, 0): Z X⇤ Z (4.9.93) & u0 u2y y 2 (1 X⇤ ) + ||uy y ||2L2 ||uy ||2L2 . (4.9.94) 0 x=X⇤ The remaining terms, upon integrating in x from [0, X⇤ ]: Z | u0xy u · uy y 2 (1 x)|  ||u0xy y 2 ||L1 ||uy ||2L2 , (4.9.95) Z | u0yy v · uy y 2 (1 x)|  ||u0yy y 2 ||L1 ||uy ||L2 ||vy ||L2 , (4.9.96) Z uyyy uy y 2 (1 x) & ||uyy y ||2L2 ||uy ||2L2 , (4.9.97) Z | @y {F1 + H1 } · uy y 2 (1 x)| . 1 + ||uy ||2L2 . (4.9.98) Placing the above series of estimates together closes the basic estimate for u1p . It is possible to take @xk and repeat with weights y m . We omit these details. Summarizing: Lemma 4.9.12. For any k, m 0, solutions [u, v] as defined in (4.9.79) to the problem (4.9.80) satisfy the following estimate: sup ||y m @xk up ||L2y + ||y m @xk upy ||L2 + ||vp ||L1 . C(k, m). (4.9.99) x Proof. Only the vp estimate remains to be proven, for which we appeal to Hardy (as vp |y=0 = 492 0): Z y vp2 = vp vpy  ||vpy ||L2y ||yvpy ||L2 = ||upx ||L2y ||yupx ||L2 . 1, (4.9.100) 0 the final estimate following from the up estimates in (4.9.99). The final task is to cut-o↵ the Prandtl-1 layer: Z 1 p p 0 p p u1p = ( "y)up " ( "y) up (x, s) ds, vp1 = ( "y)vp (4.9.101) y It is clear that the divergence free structure is preserved and the same estimates from (4.9.99) hold. The error created by such a cut-o↵ layer is: Z 1 p 0 0 p Rpu = (1 )F1 + "u vp "u0x 0 up y Z 1 p p + 2 "v 0 0 up "v 0 00 up + 3 " 0 upy y Z 1 3 00 000 + 3" up " 2 up . (4.9.102) y Rpu then contributes into Ru,1 , according to (4.9.15). Lemma 4.9.13. The remainder Rpu defined in (4.9.102) satisfies the following estimate: 1 ||Rpu ||L2 + ||y@y Rpu ||L2 . " 4 . (4.9.103) Proof. All follow via the estimates in (4.9.99) aside from F1 , for which we must use the rapid decay and that support of 1 is on y p1 . Next, the term with vp , we must use: " p p 1 || "u0 0 vp ||L2  "||vp ||L1 ||1||L2 (y p1 )  "4 . (4.9.104) " 493 Upon applying y@y , an identical calculation yields the desired result. 4.9.4 Remainder System Collecting the constructions above, according to (4.9.17) and (4.9.23), the remainders [u, v, P ] are to satisfy the following system: 1 "u + S u (u, v) + Px = N u (u, v) + " 2 Ru,1 := f0 , (4.9.105) Py 1 "v + S v (u, v) + = N v (u, v) + " 2 Rv,1 := g0 . (4.9.106) " ux + vy = 0, (4.9.107) together with the boundary conditions: [u, v]|x=0 = [a0 (y), b0 (y)], [u, v]|y=0 = [u, v]|y!1 = 0, (4.9.108) {uy + "vx }|x=L = bL (y), {P 2"ux }|x=L = aL (y). (4.9.109) The definitions of Su , Sv , N u , N v are given in (4.9.11) - (4.9.14). The assumptions on [a0 , b0 , aL , bL ] are given in (4.2.21). Up to renaming aL , bL , it is possible to, without loss of generality, consider the following simplification: 1 "u ¯ + S u (¯ u, v¯) + Px = N u (¯ u, v¯) + " 2 Ru,1 := f, (4.9.110) Py 1 "v ¯+ S v (¯ u, v¯) + = N v (¯ u, v¯) + " 2 Rv,1 := g. (4.9.111) " u ¯x + v¯y = 0, (4.9.112) 494 together with homogenized boundary conditions: [¯ u, v¯]|x=0 = [0, 0], [¯ u, v¯]|y=0 = [¯ u, v¯]|y!1 = 0, (4.9.113) {¯ vx }|x=L = ¯bL (y), uy + "¯ {P ux }|x=L = a 2"¯ ¯L (y). (4.9.114) The reason is: Lemma 4.9.14. If the assumptions in (4.2.21) are satisfied, [u, v] solves the system (4.9.105) - (4.9.107) with boundary conditions (4.9.108) - (4.9.109) if and only if [¯ u, v¯] = [u u0 , v v0 ] solves (4.9.105) - (4.9.107) with (4.9.113) - (4.9.114), and with modifying [f0 , g0 ] to [f, g] as defined by: f := f0 + Lb1 , g := g0 + Lb2 , (4.9.115) where: Lb1 :=us u0x + usx u0 + vs u0y + usy v0 1 ⇣ ⌘ + " 2 + u0 ux + uu0x + v0 uy + u0y v , 1 ⇣ ⌘ + " 2 + u0 u0x + v0 u0y (4.9.116) Lb2 :=us v0x + vsx u0 + vs v0y + vsy v0 1 ⇣ ⌘ + " 2 + u0 vx + uv0x + v0 vy + v0y v 1 ⇣ ⌘ + " 2 + u0 v0x + v0 v0y . (4.9.117) where [u0 , v0 ] are defined below in (4.9.119). Finally, we have the following estimate: n p p o ||hyiN Lb1 , "Lb2 , @y Lb1 , "@y Lb2 ||L2 . 1. (4.9.118) 495 Proof. We will define the following auxiliary profiles: u0 = a0 (y) x@y b0 (y), v0 = b0 (y). (4.9.119) It is clear that [u0 , v0 ] is a divergence free vector field, that achieves the boundary conditions at x = 0, y = 0, y ! 1. It is also clear that [u0 , v0 ] are order-1, and decay rapidly in y. Consider now the di↵erence: u ¯=u u0 , v¯ = v v0 . (4.9.120) At x = L, the following boundary conditions are satisfied: a ¯L := P 2"¯ux |x=L = P 2"ux 2"@y b0 (y) = aL (y) 2"@y b0 (y), (4.9.121) ⇣ ⌘ ¯bL := u vx |x=L = uy + "vx |x=L @y u0 = bL @y u0 . ¯y + "¯ (4.9.122) It is clear that [¯ u, v¯] will achieve the boundary conditions in (4.9.109), with [aL , bL ] aL , ¯bL ], and that [¯ replaced by [¯ aL , ¯bL ] satisfy the required assumptions, (4.2.21). Finally, the new profiles [¯ u, v¯] satisfy the new system (4.9.105) - (4.9.107) with f, g defined in (4.9.115) according to a standard linearization. The estimate in (4.9.118) follows from the definitions (4.9.116) - (4.9.117), together with (4.9.119). Remark (Notation). Due to this lemma, we can restrict to considering (4.9.110) - (4.9.114), and we will rename [¯ u, v¯] to [u, v] to help simplify notation. Proposition 4.9.15. For [Ru,1 , Rv,1 ] defined as in (4.9.15), (4.9.24), we have: p p 3 ||Ru,1 , "Rv,1 ||L2 + ||hyi@y {Ru,1 , "Rv,1 }||L2  " 4 . (4.9.123) Proof. We will start with Ru,1 , as defined in (4.9.15). The estimate on Rpu follows from 496 p recalling the prefactor of " given in (4.9.15) coupled with (4.9.103). The estimate on 2 "Ppx follows upon noticing that each term in the definition (4.9.22) exhibits rapid decay, and therefore |Pp2 |  hyi M ˜ u,1 , as defined in . We can thus move to the terms from R (4.9.10), and Rv,1 as defined in (4.9.24). Combining estimates (4.9.57), (4.9.65), and (4.9.99) immediately implies the desired result. We will also record here the following, which will be in constant use throughout the paper: Lemma 4.9.16 (Uniform Estimates of Profiles). With [us , vs ] defined as in (4.2.10) - (4.2.11), for any k, j 0, we have: ||y k @yk @xj us , y k @yk+1 @xj vs ||L1 . 1. (4.9.124) Moreover, we have the strict positivity: us & 1. (4.9.125) Proof. Using the definitions provided in (4.2.10), we see that: p p @yk us = @yk {u0e + u0p + "u1e + "u1p } (4.9.126) k p p = " 2 {@Yk u0e , "@Yk u1e } + @yk {u0p , "u1p }. (4.9.127) p k Multiplying by y k and using " y k = Y k gives the desired result. An analogous compu- tation can be made using the definition (4.2.11), and finally iterates of @x do not contribute p factors of ", which is why they do not enhance the weight of y. Finally, the positivity in (4.9.125) follows from (4.9.58) and the uniform estimates on u1e , u1p found in (4.9.65), (4.9.99). 497 4.10 Existence and Uniqueness of Remainder The main result of this appendix is the following: Proposition 4.10.1 (Linear Existence). Given (f, g) 2 L2 , and given (aL , bL ) satisfying the assumptions (4.2.21), for L sufficiently small, there exists a unique solution to the linear problem (4.2.30) - (4.2.32), together with boundary conditions (4.2.33) -(4.2.34). Proposition 4.10.2 (Nonlinear Existence). Given boundary data satisfying assumptions (4.2.21), there exists a unique solution [u, v] 2 X to the full nonlinear problem (4.9.110) - (4.9.114). We will define the operator: S↵,m [u, v, P ] := "u + Px 10↵@y {hyi2m uy 1 (y)} Py "v + 2↵@y {hyi2m vy 1 (y)} " ↵@x {hyi2m {uy + "vx } 1 (y)}. (4.10.1) defined always on divergence free vector fields, together with the boundary conditions: [u, v]|x=0 = [u, v]|y=0 = [u, v]|y!1 = 0, P 2"ux |x=L = aL (y), uy + "vx |x=L = bL (y). (4.10.2) Here 1 a cuto↵ function which is equal to 0 on [0, 1) and 1 on (2, 1). Strictly, S↵,m must return a four-tuple, with the first two components being (4.10.1), and the final two components including (aL , bL ). Fix another cut-o↵ function 2 (y) = 0 on [0, 10) and 2 =1 498 on (20, 1). Define now the norms: n p o ||u, v||2Hm 1 := || uy , "vy , "vx hyim ||2L2 , (4.10.3) n p o ||u, v||2Hm 2 := || uyy , "vyy , "vxx hyim 2 ||2L2 . (4.10.4) Notationally, we will refer to the m = 0 norm as simply H 1 . Define now the space: n C0,S := (', ) 2 C 1 : compactly supported in y, (4.10.5) o supported away from x = 0, and @x ' + @y = 0 . We will define: 1 ||·||H 1 Hm := C0,S m , (4.10.6) and the scaled, symmetric gradient via: 0 1 p B "ux uy + "vx C D" = @ p A. (4.10.7) uy + "vx "vy Recalling the definition of || · ||X from (4.2.22), our ultimate space X is defined via: ||·||H 1 X0 := C0,s , (4.10.8) X := {[u, v] 2 X0 : ||u, v||X < 1}. (4.10.9) 499 Define the weak formulation of (4.10.1) to be: Z Z Z Z D" u · D" ' + "D" v · D" + aL ' "bL x=L x=L Z ⇣ ⌘ +↵ 10uy · 'y + 2"vy · y + {"2 vx + "uy } · x 1 (y)hyi 2m Z Z 2m ↵ "y 1 bL (y) = f · ' + "g · , (4.10.10) x=L for all (', ) 2 C0,S . Lemma 4.10.3. Given (f, g) 2 L2 , and boundary values (aL , bL ) satisfying the assumptions 1 (4.2.21), there exists a weak solution [u, v, P ] 2 Hm satisfying the estimate: p na o L 1 . ||f, ↵||u, v||2Hm "g||2L2 + || p , bL y 2m ||2L2 (x=L) . (4.10.11) " Proof. The existence of solutions follows directly from Lax-Milgram. We must verify the Bilinear form in (4.10.10) is coercive: Z Z B[(u, v), (', )] := D" u · D" ' + "D" v · D" (4.10.12) Z ⇣ ⌘ +↵ 10uy · 'y + 2"vy · y + {"2 vx + "uy } · x 1 (y)hyi 2m . This is immediate, apart from the cross term, to which we first appeal to the density: Z Z (n) 2m n!1 2m "uy x 1 hyi ! "uy vx 1 hyi Z Z 1 1 |·| "2 vx2 1 hyi2m + u2y 1 hyi 2m , (4.10.13) 2 2 which explains the constants of 10 appearing in (4.10.1). We view the terms: Z Z Z Z f · ' + "g · aL ' + "bL + ↵ "y 2m 1 bL , (4.10.14) x=L x=L x=L 500 1 as a functional on H . We must thus estimate the following boundary term: Z | ↵"y 2m 1 bL |  ↵||bL hyi2m ||L2 (x=L) ||" ||L2 (x=L) x=L . ||bL hyi2m ||2L2 (x=L) + ↵2 O(L)||" 2 x ||L2 , (4.10.15) R the latter term being absorbed into the positive contributions from |D" v|2 using the smallness of L and ↵. Next, we must estimate the boundary terms: Z aL p | aL '|  || p ||L2 (x=L) || "'||L2 (x=L) x=L " aL 2 p . || p ||L2 (x=L) + O(L)|| "'x ||2L2 , (4.10.16) " Z | "bL |  ||bL ||L2 (x=L) ||" ||L2 (x=L) x=L  ||bL ||2L2 (x=L) + O(L)||" 2 x ||L2 . (4.10.17) the latter terms in both of the above calculations can be absorbed into the positive R contributions from |D" v|2 . 1 It is clear that each solution [u, v] 2 Hm is automatically in X0 . We will now bootstrap 2 to Hm solutions. 1 Lemma 4.10.4. Solutions [u, v] 2 Hm to the system (4.10.1) satisfy: p na o L 2 . ||f, ↵||u, v||Hm "g||L2 + || p , bL y 2m ||2L2 (x=L) . (4.10.18) " Moreover, such solutions are strong solutions, which satisfy the boundary conditions of (4.10.2). Proof. This follows formally from di↵erentiating (4.10.1) in y and applying the multiplier uy , 501 with the help of the cut-o↵ function 2 in (4.10.4) to avoid the corners. Rigorously, one needs to work with di↵erence quotients within the weak formulation (4.10.10). We demonstrate 1 this now for the main weighted term. Given [u, v] 2 Hm , there exists a sequence 'n , n 1 Hm such that: ['n , n ] ! [u, v] by the density (4.10.6). Denote by Dh the di↵erence quotient u(x,y+h) u(x,y) in the y-direction: Dh u(x, y) = h . We will select the multiplier D h Dh ' to apply the weak formulation (4.10.10): Z Z n o ym 1 (y)uy D h Dh '(n) y = Dh ym 1 (y)uy Dh '(n) y . (4.10.19) (n) L2 By definition of di↵erence quotient, for each fixed h, hyim Dh 'y ! hyim Dh uy . Simi- larly, for each fixed h, hyim Dh uy 2 L2 . Thus, for each fixed h, we can take n ! 1: Z n o n!1 (4.10.19) ! Dh ym 1 (y)uy D h uy . (4.10.20) Next taking limits in h gives: Z h!0 (4.10.20) ! @y {y m 1 (y)uy } · uyy . (4.10.21) Performing similar calculations for each of the terms yields the desired result. The boundary conditions (4.10.2) are satisfied by integrating by parts (4.10.1) against a test function, justified as [u, v] are strong solutions, and comparing the boundary terms with (4.10.10). Near the boundary y = 0, standard Stokes theory (applicable due to cuto↵ 1 (y)) implies: 502 Lemma 4.10.5. Solutions [u, v] to the system (4.10.1) satisfy the following estimate: p na o L ||u, v||23 . ||f, "g||2L2 + || p , bL y 2m ||2L2 (x=L) . (4.10.22) 2 Hloc " To summarize, we have established that: 3 i2 1 Corollary 4.10.6. For m, ↵ > 0, the map S↵,m : [L2 ]⇥2 ⇥ [L2 (x = L)]⇥2 ! [Hm 2 \ Hloc 2 is well defined, and returns a solution to the system (4.10.1) which satisfies the boundary conditions specified by the third and fourth inputs of S↵,m . We now define: T [u, v] =us ux + usx u + vs uy + usy v us vx + vsx u + vs vy + vsy v. (4.10.23) We will study: S↵,m [u, v] + T [u, v] = (f, g) ) h i h i 1 1 [u, v] + S↵,m T [u, v], aL , bL = S↵,m f, g, aL , bL (4.10.24) as an equality in H 1 ⇥ H 1 . Lemma 4.10.7. For m > 0, we have the following compact embedding: 3 2 Hm \ Hloc 2 ⇢⇢ H 1 . (4.10.25) Proof. The proof follows from a standard argument, see for instance (Iye17a, P. 145, Lemma 13.1). 503 1 As a direct consequence, S↵,m T is a compact operator on H 1 . An application of the Fredholm Alternative shows that to produce an H 1 solution of (4.10.24), we must rule out nontrivial solutions to the homogeneous problem, which occurs when f = g = aL = bL = 0. For this purpose, we give a-priori estimates of the problem (4.10.24), under the hypothesis that [u, v] 2 H 1 . For such functions, we automatically know that [u, v] 2 Hm 2 due to (4.10.18). 2 Lemma 4.10.8 (Energy Estimates). Solutions [u, v] 2 Hm to the system (4.10.1) satisfy the following energy estimate: p p ||uy , "ux , "vx ||2L2 + ↵||{uy , "vy , "vx } · y m ||2L2 p na o L . O(L)||ux , "vx ||2L2 + R1 + || p , bL y 2m ||2L2 (x=L) . (4.10.26) " Proof. This follows upon testing the system (4.10.24) against ['(n) , (n) ], where the sequence 1 Hm ['(n) , (n) ] ! [u, v], and repeating the energy estimate in Proposition 4.3.1. 2 Lemma 4.10.9 (Positivity Estimates). Let m = 1. Then solutions [u, v] 2 Hm to the system (4.10.1) satisfy the following estimate: p p p ||vy , "vx ||2L2 + || "ux ||2L2 (x=L) . ||uy ||2L2 + O(ve0 )||uy · y, "vy y||2L2 p + ↵||{uy , "vy , "vx } · y m ||2L2 + R2 na o L + || p , bL , @y bL y 2m ||2L2 (x=L) . (4.10.27) " Proof. We must perform estimates on the new, weighted quantities appearing from (4.10.1). Temporarily omitting the prefactor of 10, we have: Z Z hv 2m v usy i y +↵ @y {uy y } · @y = ↵ uy y 2m @y v 2 (4.10.28) us us u Z Z s vyy usy = ↵ uy y 2m + ↵ uy y 2m vy 2 us us 504 Z usy ↵ uy y 2m v@y { 2 } us Z Z 2m u xy usy = +↵ uy y + ↵ uy y 2m vy 2 us us Z u sy ↵ uy y 2m v@y { 2 } us Z Z 1 1 = ↵ u2y y 2m @x { } + ↵u2y y 2m us 2u s Z Z x=L u sy u sy + ↵ uy y 2m vy 2 ↵ uy y 2m v@y { 2 }. (4.10.29) us us The boundary contribution above is positive, whereas the other terms can all be esti- mated by the ↵ term in (4.10.27). We need to justify the integration by parts leading to 2 the equality in (4.10.28). For this we notice that our solution is in Hm , and so both the left and right-hand sides of (4.10.28) are in L1 . This then justifies the following limit: Z Z M v v @y {uy y 2m } · @y = lim @y {uy y 2m } · @y us M !1 y=0 us h Z Z M 2m v vi = lim uy y @yy + uy y 2m @y M !1 0 us y=M us Z v = uy y 2m @yy , (4.10.30) us where the limit of the boundary contribution vanishes as ||uy y m ||2L2 and ||vy y m ||L2x are x Hy1 functions, according to the definition of 2 Hm . We similarly have: Z v 2↵@y { 1 y 2m vy } · "@x { } us Z ⇣v 2m xy 1 1 1⌘ = 2↵ y "vy + vy @x + vx @y + v@xy . (4.10.31) us us us us Finally: Z v ↵"@x { y 2m {uy + "vx }} · @x { } us 505 Z Z v 1 = ↵" y 2m uxy @x { } ↵"2 vxx vx y 2m us us Z 1 "2 ↵vxx v@x y 2m . (4.10.32) us The latter two terms in (4.10.32) are estimated according to standard calculations. For the first term: Z Z v v ↵" y uxy @x { } = ↵"ux @y { y 2m @x { }} 2m us us Z hv xy 1 1 1i = "↵ux y 2m + vx @y + vy @x { } + v@xy us us us us Z v + "↵ux @x @y { y 2m }. (4.10.33) us For the final term above, we must use that m = 1: Z v p p "↵ux @x @y { y 2m }  ↵|| "ux y||L2 || "vx ||L2 . (4.10.34) us The remaining terms can all be estimated similarly to estimate (4.4.5). We first recall the definition of R1 given in (4.3.2). 2 Lemma 4.10.10 (Weighted Estimate). Solutions [u, v] 2 Hm to the system (4.10.1) satisfy the following estimate: n p o p p || uyy , "uxy , "uxx · y||2L2 + ||{uy , "ux }y||2L2 + ||{uy , "ux }y||2L2 (x=L) n p o p + ↵|| uyy , "uxy , "uxx · y m+1 ||2L2 . ↵||{uy , "vy , "vx } · y m ||2L2 p + ||vy , "vx ||2L2 + ||uy ||2L2 + ||{aL , @y aL , bL , @y bL }hyi2 ||2L2 (x=L) p + || "ux ||2L2 (x=L) + R1 . (4.10.35) 506 Proof. For this step, we can apply a cut-o↵ N (y) = ( Ny ), and take N ! 1. Due to the cut-o↵, there is no need to justify contributions from y = 1. Consider the new term: Z ↵ @yy {y 2m (y)uy } · @y {uy 2 w(x)} N (y) Z 1 0 = +↵ @y {y 2m (y)uy } · @y {uy 2 w} (y) N N Z 2m+2 2 = +↵ Ny uyy + ↵O(||uy ||2Hm 1 ). (4.10.36) Analogous calculations can be performed for the remaining ↵ terms from (4.10.1). For the remaining terms from (4.10.24), one can repeat the proof of Proposition 4.5.1 with the additional cuto↵ term N (y). We omit repeating those details. Putting the above estimates, (4.10.26), (4.10.27), (4.10.35) together gives the following uniform in ↵ estimate: n p o p ||u, v||2X + ↵|| uyy , "uxy , "uxx · y m+1 ||2L2 + ↵||{uy , "vy , "vx } · y m ||2L2 . R1 + R2 + R3 + C(aL , bL ). (4.10.37) Taking the forcing f = g = aL = bL = 0 (thus Ri = 0), we can apply the Fredholm Alternative to conclude that there exists an H 1 solution [u, v] to the problem (4.10.24). Such a solution is automatically H 2 by (4.10.18), and so is a strong solution. The final task is to establish a solution to our original system (4.9.105) - (4.9.107), which can be achieved as a weak limit in X as ↵ ! 0 using the uniform in ↵ estimate (4.10.37). This then proves Proposition 4.10.1. Proposition 4.10.2 then follows upon applying the Contraction Mapping Theorem when coupled with the main X -estimate in Theorem 4.8.3. Chapter Five Stationary Inviscid Limit to Shear Flows 508 5.1 Abstract This is based on a joint work with Chunhui Zhou (see (IZ17)). We establish a density result for certain stationary shear flows, µ(y), that vanish at the boundaries of a horizontal channel. We construct stationary solutions to 2D Navier-Stokes that are "-close in L1 to the given shear flow. Our construction is based on a coercivity estimate for the Rayleigh operator, R[v], which is based on a decomposition made possible by the vanishing of µ at the boundaries. 5.2 Introduction We are considering 2D, stationary flows on the strip: ⌦ = (0, L) ⇥ (0, 2). (5.2.1) We consider an Euler shear flow: u0 = (µ(y), 0). (5.2.2) Let u" solve the Navier-Stokes equations: 9 " " " "> > u · ru + rP = " u > > > = " . (5.2.3) r·u =0 > > > > > u" |y=0 = 0, u" |y=2 = ub ; Here ub 0 denotes the velocity of the boundary at {y = 2}. Our main result, Theorem 509 5.2.1 treats the non-moving case of ub = 0. We are interested in the asymptotic behavior of u" as " ! 0. In the presence of boundaries, the vanishing viscosity asymptotics are a major open problem in fluids made challenging due to the mismatch between the no-slip condition u" |@⌦ = 0 and the no penetration condition typically satisfied by Euler flows: u0 · n = 0. This mismatch is typically rectified by the presence of Prandtl’s boundary layer (see (GN17), (Iye17a), (Iye16), (Iye17b) for relevant results in the 2D stationary setting). In this article, we will consider Euler flows that themselves satisfy no-slip: µ(0) = 0, (5.2.4) for which there is no leading order boundary layer. Denote now the asymptotic expan- sion: 0 1 0 1 3 3 3 " 1 1 2 2 + Bu C Bµ + "ue + "up + " 2 ue + " 2 up + " 2 uC u" := @ A = @ A. (5.2.5) 3 3 3 v" "ve1 + " 2 vp1 + " 2 ve2 + "2 vp2 + " 2 + v We denote: 3 3 us := µ + "u1e + "u1p + " 2 u2e + " 2 u2p , (5.2.6) 3 3 vs := "ve1 + " 2 vp1 + " 2 ve2 + "2 vp2 . (5.2.7) We impose the boundary conditions: [u, v]|x=0 = [u, v]|y=0 = [u, v]|y=2 = 0, (5.2.8) @y u + @x v = 0, P = 2"@x u. (5.2.9) 510 The system satisfied by [u, v] is: 9 " u + Su + @x P = f := N1 (u, v) + Fu > > > > > = " v + Sv + @y P = g := N2 (u, v) + Fv > in ⌦. (5.2.10) > > > > @x u + @y v = 0 ; We have defined: Su := us ux + usx u + usy v + vs uy , Sv := us vx + vsx u + vs vy + vvsy , (5.2.11) 3 ⇣ ⌘ 3 ⇣ ⌘ N1 := " 2 + u@x u + v@y u , N2 := " 2 + u@x v + v@y v , (5.2.12) and Fu , Fv are defined in (5.5.55). Let us now define several norms in which we will control the solution: p p ||u, v||E := || "ru||L2 + || "rv||L2 , (5.2.13) p ||u, v||P := || us rv||L2 , (5.2.14) p ||u, v||X := ||u, v||E + " 2 || "{u, v}||1 (5.2.15) We introduce here the notation: y˜ = y · (2 y). (5.2.16) The main theorems we prove are the following: Theorem 5.2.1. Let ub 0 in (5.2.3). Let µ(y) 2 C 1 ([0, 2]) be a given function, satisfying the conditions: µ(0) = 0, µ(2) = ub , (5.2.17) 511 @yj µ(0) = @yj µ(2) = 0 for 2  j  N0 , (5.2.18) @y µ(0) > 0, |@y µ(2)| > 0. (5.2.19) where N0 < 1 and large but unspecified.1 Let also standard compatibility conditions at the corners of ⌦ be prescribed for the layers in us .2 Then there exists a unique solution, u" satisfying the Navier-Stokes equations, (5.2.3), such that: ||u" µ||1 + ||v " ||1  c0 (µ)". (5.2.20) The constant c0 (µ) satisfies: µ000 c0 (µ) . || ||W 100,1 . (5.2.21) µ Our ultimate interest is motivated by Yudovich’s ninth problem, (Yud03). Classical ex- periments starting with Reynolds have shown that unsteady flows in a 2D channel that start near Couette or Poiseulle flow do not converge to these flows. This indicates the existence of infinitely many stationary solutions to Navier-Stokes “near” Couette or Poiseulle. Establish- ing the existence of these solutions is an open problem. Our second result, Corollary 5.2.2, produces stationary solutions sufficiently close to Couette, assuming x 2 [0, L], L << 1, and a moving boundary at y = 2. Corollary 5.2.2. Let any ↵ > 0 be prescribed, which could depend on ". Let µ ˜ be prescribed to satisfy the vanishing conditions: @yk µ ˜|y=0 = @yk µ ˜|y=2 = 0 for 0  k  N0 . There exists a unique solution, u" to (5.2.3) with ub = 2 such that: ⇣ ⌘ ||u" µ(y) ||1 + ||v " ||1 . ↵". y + ↵˜ (5.2.22) 1 We have selected not to optimize N0 . The optimal N0 is likely between 4 and 10. 2 We omit stating the precise form of these compatibility conditions here. They can be found in (5.5.34), (5.5.39). 512 Proof. One can obtain this by applying Theorem 5.2.1 with µ(y) = y + ↵˜ µ(y), where µ ˜ vanishes at high order near y = 0, 2. In this case, the constant c0 (µ) . ↵. Remark. The requirement of ub = 2 is so that the no-slip condition is satisfied by the Couette flow. We do not use this motion of the boundary anywhere in the proof. The present article is structured as follows: the construction of the approximate layers, us , vs , in the expansion (5.2.5) is performed in the Appendix. The main analysis in Sections 5.3, 5.4 is centered around the system (5.2.10). 513 5.3 Linear Estimates We will analyze the system (5.2.10). The reader is urged to consult Lemma 5.5.3 for relevant properties of the linearizations, us , and the forcing terms, f, g. 5.3.1 Energy Estimate Proposition 5.3.1. For any ✓ > 0, solutions [u, v] to (5.2.10) satisfy: p ||u, v||2E + || us {u, v}||2L2 (x=L) . C(✓)" ✓ ||u, v||2P + R1 , (5.3.1) where: Z Z R1 := f ·u+ "g · v. (5.3.2) Proof. Apply [u, v] to (5.2.10). The coercive quantities are: Z Z Z " u⇥u " v⇥v+ rP · u Z h i Z = " @yy u 2@xx u @xy v ⇥ u + @x P u h Z i Z + " 2@yy v @x {@y u + @x v} ⇥ v + @y P v Z h i = " |@y u2 + |@x v|2 + 4|@y v|2 + 2@x v@y u Z h i & " |ru|2 + |rv|2 . (5.3.3) Above, we have used the stress-free boundary condition in (5.2.9). We now have the 514 convection terms: Z Z Z 1 [us ux + usx u + vs uy ] · u = usx u2 + us u2 , (5.3.4) 2 x=L Z Z Z 2 1 [us vx + vs vy + vsy v] · v = vsy v + us v 2 . (5.3.5) 2 x=L We estimate the two bulk terms above using (5.5.59) - (5.5.61): Z Z Z p p p | usx u2 | + | vsy v 2 |  | y [u2 + v 2 ]|  "˜ "O(L)|| us rv||22 . (5.3.6) We now move to: Z p p | vsx uv|  "|| y˜ux ||2 || y˜vx ||2 , (5.3.7) again by using (5.5.61). For the usy v convection term, we first handle the leading order contribution from µ, and we must take care to avoid the critical Hardy inequality: Z Z h 1 1 19 19 i | µ0 uv| = | µ0 uv (y  )+ ( y ) + (y ) |. 10 10 10 10 For the interior contributions: Z 1 19 p p | µ0 uv ( y )|  O(L)|| us @x u||2 || us @x v||2 (5.3.8) 10 10 1 1 The y  10 contribution is exactly analogous to the y 10 , and so we treat the former. 1 Let ˜ denote a fattened relative to (y  10 ). Fix an ! > 0 small. Z 1 ( 12 ! 1 ! | µ0 uv (y  )|  ||µ0 ||1 ||y 2) v ˜||2 ||y 2 2 u ˜||2 (5.3.9) 10 515 We estimate each L2 term above individually. Z Z 1+! 2 @y ! 2 y v ˜= {y }v ˜ ! Z ! Z y y! 2 0 = 2v@y v ˜ v ˜ ! ! 1 ( 12 ! p 1 p  ||y 2) v ˜||2 || us @y v||2 + O(L)|| us @x v||22 . (5.3.10) ! ! Next from (5.3.9): 1 ! 1 u ✓(!) 1 ✓(!) ||y 2 2 u ˜||2 . ||y 2 u ˜||2 || ˜||2 y p h p p i✓(!) 1 ✓(!) . " ✓(!) || us @x u||2 || "@y u ˜||2 + || us @x u ˜||2 (5.3.11) Inserting (5.3.10) and (5.3.11) into (5.3.9), one obtains for small  > 0 |(5.3.9)|  ||u, v||2E + N "0 ||u, v||2P . (5.3.12) For the higher-order contributions, we use the estimate (5.5.60), and subsequently split: Z Z p | [usy µ0 ]uv|  "|u||v|[ + c + + ] = (5.3.13.1) + (5.3.13.2) + (5.3.13.3). (5.3.13) + c + Here, (y) = (2 y) and =1 , where: 8 > < 1 on y  (y) = (5.3.14) > : 0 on y 2 516 Terms (5.3.13.1) and (5.3.13.3) are identical. We estimate: p p up p |(5.3.13.1)| . O(L)|| " ˜ ||2 || us ux ||2 , (5.3.15) y p " p |(5.3.13.2)| . || us rv||22 . (5.3.16) We now estimate: Z p up @y || " ˜ ||22 = " {y 1 }u2 ˜ y 1 Z Z 1 1 2 ˜0 = "y 2u@y u ˜ + "y u p up p " p  || " ˜ ||2 || "@y u||2 + 3 || us @x u||22 . (5.3.17) y 1 We may thus take = " 4 and insert (5.3.17) into (5.3.15) to conclude. 5.3.2 Positivity Estimate Proposition 5.3.2. Solutions [u, v] to (5.2.10) satisfy, for any  > 0: p p ||u, v||2P + || "@x v||2L2 (x=0) + || "@x u||2L2 (x=L) . "1  ||u, v||2E + R2 , (5.3.18) where: Z Z R2 := f· @y v + g · @x v. (5.3.19) 517 Proof. We will apply the multiplier M := ( @y v, @x v) to the system (5.2.10). This gives: Z ⇣ ⌘ Z us @y v + v@y us · @y v + us @x v · @x v Z h i Z u 2 2 syy 2 = us |@y v| + |@x v| + v 2 Z Z usyy us |rv|2 || ||1 v 2 2 Z & us |rv|2 . (5.3.20) We have used the splitting: Z Z v2 = v2 [ + c ] 2 Z Z L 2  @y {y}v 2 | |@x v| + | Z Z Z 0 L2  us |@x v| + | y2v@y v | + | yv 2 | 2 Z p p p L2 . us |@x v|2 + || us @y v||2 ⇥ LHS Z p p p =L = L us |@x v|2 + L|| us @y v||2 ⇥ LHS. (5.3.21) For the vorticity terms, repeated integration by parts gives: Z Z Z " u · @y v " v · @x v + rP · M Z Z " 2 "h ⇣ = |@x v| + |@y u|2 |@x u|2 + |@y v|2 2 x=0 2 x=L ⌘i Z 2 |@x v| + P @x u Z Z x=L " = |@x v|2 + 2"|@x u|2 , (5.3.22) 2 x=0 x=L where we have used the Stress-Free boundary condition from (5.2.9). We now come to 518 the remaining linearized terms from (5.2.10): Z ⇣ ⌘ Z ⇣ ⌘ | u@x us + vs @y u · @y v| + | u@x vs + vs @y v + v@y vs · @x v| . " ⇥ LHS of (5.3.18) + "1  ||u, v||2E , where we have used the Poincare inequality and the estimates in (5.5.59) - (5.5.61). Finally, the right-hand side of (5.3.18) follows from the definition of R2 . Lemma 5.3.3. For any ✓ > 0, p p h i "✓ || "u, "v||1  C✓ ||u, v||E + ||f, g||2 . (5.3.23) Proof. We omit the proof, this is found in (GN17) using interpolation arguments and esti- mates for the Stokes operator on domains with corners. As a direct corollary to (5.3.1), (5.3.18), and taking ✓ = 4 in (5.3.23): Corollary 5.3.4. ||u, v||2X . R1 + R2 + " 2 ||f, g||22 . (5.3.24) 519 5.4 Evaluation of Right-Hand Sides We first provide the nonlinear estimates: Lemma 5.4.1. With N1 , N2 defined as in (5.2.12): Z Z 3 3 | " 2 + N1 · [u + @x u]| + | " 2 + N2 · [v + @x v]| h i + " 4 ||N1 , N2 ||2  " 2 ||u, v||3X + ||u, v||2X . (5.4.1) Proof. We compute directly: Z 3 | " 2 + [u@x u + v@y u] · [u + @x u]| p p  " 2 || "" 2 {u, v}||1 || "r{u, v}||22 . " 2 ||u, v||3X . (5.4.2) Similarly: Z 3 | " 2 + [u@x v + v@y v] · [v + @x v]| p p  " 2 || "" 2 {u, v}||1 || "r{u, v}||22 . " 2 ||u, v||3X . (5.4.3) Finally: 3 h i ||N1 , N2 ||2  " 2 + ||u{@x u, @x v}||2 + ||v{@y u, @y v}||2 . (5.4.4) Lemma 5.4.2. With Fu , Fv defined as in (5.5.55), for any > 0: Z Z 3 3 " 2 Fu · [u + @x u] + " 2 Fv · [v + @x v] 520 µ000  ||u, v||2X + C(us , vs )c0 ( ), µ 3 1 3 µ000 " 4 ||" 2 {Fu , Fv }||2 . " 2 4 c0 ( ). (5.4.5) µ Proof. First recall the decomposition of Fu , Fv given in (5.5.56). We first estimate: Z Z 1 1 c " 2 T1 · u = "2 T1 · u[ + ]. (5.4.6) For the nonlocal part, we use estimate (5.5.62): Z 1 c 1 1 p | "2 T1 u | 2 "2 ||T1 ||2 || us @x u||2 . (5.4.7) For the local component, we integrate by parts in y: Z Z 0 1 1 | " 2 T1 u (y)| = " 2 | y@y uT1 + yu@y T1 + yuT1 | p 1 p p  " || "@y u||2 ||T1 ||2 + " 2 ||@y T1 ||2 || us @x u||2 1 1 p + 2 "2 ||T1 ||2 || us @x u||2 . (5.4.8) 1 The same estimates can be used for " 2 T2 · v. We now come to the higher order terms, in which the non-local contributions are estimated via: Z Z 1 c 1 c 1 1 p | "2 T1 · @ x u + "2 T2 · @ x v | 2 "2 ||T1 ||2 || us rv||2 . (5.4.9) We now focus on the T1 localized contributions individually. First: Z p p 1 vp2 "2 | µ0 vp2 @y v |" ||µ0 , ||1 || us @y v||2 , (5.4.10) Y Z 1 1 v1 p "2 | @Y u2p ve1 @y v |  "2 ||@Y u2p ||1 || e ||2 || us @y v||2 , (5.4.11) y 521 Z h i 1 1 p "2 | µ 0 vp2,0 + 3 0 @Y u2,0 p @y v|  " 2 ||µ 0 vp2,0 + 3 0 @Y u2,0 p ||1 || us @y v||2 . (5.4.12) The remaining terms in T1 are handled by integrating by parts in y and proceeding as in (5.4.8): Z h i 1 "2 u1e @x u1e + ve1 @y u1e u1e · @y v Z h i 1 = "2 @y u1e @x u1e + ve1 @y u1e u1e ·v | {z } m1 Z h i 0 1 " 2 u1e @x u1e + ve1 @y u1e u1e · v = (5.4.13.1) + (5.4.13.2). (5.4.13) First: Z 1 1 1 p |(5.4.13.1)| = | "2 m1 v|  " 2 ||m1 ||2 ||v||2  " 2 ||m1 ||2 || us rv||2 . (5.4.14) Second: 1 3 p |(5.4.13.2)|  " 2 2 ||u1e @x u1e + ve1 @y u1e u1e ||2 || us @x v||2 (5.4.15) We now consider the localized contributions from T2 , for which we apply estimate (5.5.62): Z 1 T2 p "2 | T2 · @x v|  || ||2 || us @x v||2 . (5.4.16) y˜ We now make the selection of = "10 , and << 1 sufficiently small, which closes all 522 5 of the above estimates. Finally, the O(" 2 ) are handled easily via: Z Z 3 " 2 | [Fu T1 ] · [u + @x u] + [Fv T2 ] · [v + @x v]| Z . "1 · [u + @x u + v + @x v]| 1 p . "2 || "rv||2 . (5.4.17) We now obtain our complete nonlinear estimate: Corollary 5.4.3. Solutions [u, v] to the system (5.2.10) satisfy: µ000 ||u, v||2X . C(us , vs )c0 ( ) + " 2 ||u, v||3X . (5.4.18) µ From here, the main result, Theorem 5.2.1 follows from a straightforward application of the contraction mapping theorem. 523 5.5 Construction of Layers We start with the asymptotic expansions: 3 3 3 u" := µ + "u1e + "u1p + " 2 u2e + " 2 u2p + " 2 + u, (5.5.1) 3 3 3 v " := "ve1 + " 2 vp1 + " 2 ve2 + "2 vp2 + " 2 + v, (5.5.2) 3 3 3 P " := "Pe1 + " 2 Pe2 + "[Pp1 + "Pp1,a ] + " 2 Pp2 + " 2 + P (5.5.3) 5.5.1 Formal Asymptotic Expansion Here the Eulerian profiles are functions of (x, y), whereas the boundary layer profiles are functions of (x, Y ), where: 8 > 2 y > < Y+ := p" if 1  y  2, Y = (5.5.4) > > y : Y := p if 0  y  1. " Due to this, we break up the boundary layer profiles into two components, one supported near y = 0 and one supported near y = 2: uip (x, Y ) = ui, i,+ p (x, Y ) + up (x, Y+ ). (5.5.5) As a notational convention, we use: @Y uip := @Y ui, p @Y+ ui,+ p . (5.5.6) 524 The purpose of such a convention is to obtain the chain rule: 1 @y uip = p @Y uip . (5.5.7) " Let us set the following notations: 3 3 u"E := µ + "u1e + " 2 u2e , " vE := "ve1 + " 2 ve2 , (5.5.8) 3 3 u(2) 1 1 2 2 2 2 s := µ + "ue + "up + " ue + " up , (5.5.9) 3 3 vs(2) := "ve1 + " 2 vp1 + " 2 ve2 + "2 vp2 , (5.5.10) 3 3 Ps(2) := P " := "Pe1 + " 2 Pe2 + "[Pp1 + "Pp1,a ] + " 2 Pp2 . (5.5.11) Using the expansions (5.5.1) - (5.5.3), we will first expand out the purely Euler terms: h 3 i h 3 i u"E @x u"E = µ + "u1e + " 2 u2e · "u1ex + " 2 u2ex 5 3 5 = "µu1ex + "2 u1e u1ex + " 2 u2e u1ex + " 2 µu2ex + " 2 u1e u2ex + "3 u2e u2ex (5.5.12) h 3 i h 3 i " vE @y u"E = "ve1 + " 2 ve2 · µ0 + "u1ey + " 2 u2ey 5 3 5 = "µ0 ve1 + "2 ve1 u1ey + " 2 ve1 u2ey + " 2 µ0 ve2 + " 2 ve2 u1ey + "3 ve2 u2ey (5.5.13) h 3 i h 3 i u"E @x vE " = µ + "u1e + " 2 u2e · "vex 1 2 + " 2 vex 3 5 5 1 2 = "µvex + µ" 2 vex + "2 u1e vex 1 + " 2 u1e vex 2 + " 2 u2e vex 1 + "3 u2e vex 2 , (5.5.14) h 3 i h 3 i v " @y vE " = "ve1 + " 2 ve2 · "vey 1 2 + " 2 vey 525 5 5 = "2 ve1 vey 1 + " 2 ve1 vey 2 + " 2 ve2 vey 1 + "3 ve2 vey 2 . (5.5.15) 3 @x PE" = "Pex 1 2 + " 2 Pex , (5.5.16) 3 @y PE" = "Pey 1 2 + " 2 Pey (5.5.17) 5 " u"E = "µ00 (y) + "2 u1e + " 2 u2e , (5.5.18) 5 " " vE = "2 ve1 + " 2 ve2 . (5.5.19) We now expand: 5 u(2) (2) " " 2 1 1 2 1 1 2 1 2 s @x us =uE @x uE + " up uex + " up upx + " up uex 5 5 5 + "µu1px + "2 u1e u1px + " 2 u2e u1px + " 2 u2p u1ex + " 2 u2p u1px 3 5 + "3 u2p u2ex + "3 u2p u2px + " 2 µu2px + " 2 u1e u2px 5 + " 2 u1p u2px + "3 u2e u2px . (5.5.20) 3 5 0 1 vs(2) @y u(2) " " 2 1 1 2 1 1 3 1 2 s =vE @y uE + " µ vp + " vp uey + " vp upY + " vp uey 2 3 5 + " 2 ve1 u1pY + "2 ve2 u1pY + "2 µ0 vp2 + "3 u1ey vp2 + " 2 vp2 u1pY 3 5 5 + " 2 vp2 u2ey + "3 vp2 u2pY + " 2 ve2 u2pY + " 2 vp1 u2pY + "2 ve1 u2pY . (5.5.21) 5 5 3 u(2) (2) " " 2 1 1 2 1 1 2 1 2 1 s @x vs =uE @x vE + " up vex + " up vpx + " up vex + " µvpx 2 5 + " 2 u1e vpx 1 + "3 u2e vpx 1 + "2 µvpx 2 + "3 u1e vpx 2 526 5 7 + "3 u1p vpx 2 + " 2 u2e vpx 2 + " 2 u2p vpx 2 + "3 u2p vex 2 5 + "3 u2p vpx 1 1 2 + " 2 vex up . (5.5.22) 5 5 vs(2) @y vs(2) =vE " " @y vE + " 2 vp1 vey 1 + " 2 vp1 vpY 1 + "3 vp1 vey 2 + "2 ve1 vpY 1 5 7 7 + " 2 ve2 vpY 1 + "3 vey 1 2 vp + "3 vpY 1 vp2 + " 2 vp2 vey 2 + " 2 vp2 vpY 2 5 + " 2 ve1 vpY 2 + "3 vp1 vpY 2 + "3 ve2 vpY 2 . (5.5.23) Finally, we have the linear terms: 3 @x Ps = @x PE" + " 2 Ppx 2 1 + "Ppx + "2 Ppx 1,a , (5.5.24) p 1,a3 @y Ps = @y PE" + "PpY 2 + 1 "PpY + " 2 PpY (5.5.25) 5 3 " u" = " u"E + "2 u1pxx + " 2 u2pxx + "u1pY Y + " 2 u2pY Y , (5.5.26) 5 3 " v " = " vE " 1 + " 2 vpxx 1 + " 2 vpY 3 2 2 2 Y + " vpxx + " vpY Y . (5.5.27) 5.5.2 Euler Equations The equations satisfied by the Euler layers are obtained by collecting the O(") order terms from (5.5.12) - (5.5.19), and is now shown: 9 > > µ@x u1e + µ0 ve1 + @x Pe1 00 = µ (y) > > > > > > µ@x ve1 + @y Pe1 = 0, > = (5.5.28) > > @x u1e + @y ve1 = 0, > > > > > > > ; ve1 |x=0 = ve1 |y=0 = ve1 |y=2 = ve1 |x=L = 0. 527 By going to the vorticity formulation, we arrive at the following problem: Z x µ ve1 + µ00 ve1 = µ000 (y), ve1 |@⌦ = 0, u1e := 1 vey . (5.5.29) 0 We will make the assumptions that: µ00 µ000 , vanish at high order at y = 0, 2. (5.5.30) µ µ According to (5.5.30), we divide (5.5.29) by µ to obtain: µ00 1 µ000 ve1 + ve = , ve1 |@⌦ = 0. (5.5.31) µ µ By evaluating (5.5.31) at y = 0, 2 and recalling (5.5.30), it is clear that @yy ve1 |y=0,2 = 0. 3 The system satisfied by the second Euler layer is obtained by collecting the O(" 2 ) terms from (5.5.12) - (5.5.19), and is shown here: 9 > > µ@x u2e + µ0 ve2 + @x Pe2 =0 > > > > > > µ@x ve2 + @y Pe2 = 0, > = (5.5.32) > > @x u2e + @y ve2 = 0, > > > > > > > ; ve2 |x=0 = ve2 |x=L = ve2 |y=2 = 0, ve2 |y=0 = vp1 |Y =0 . Going to vorticity produces the system: µ ve2 + µ00 ve2 = 0, ve2 |y=0,2 = vp1 |y=0,2 . (5.5.33) We will assume high-order compatibility conditions on the data ve2 |x=0,L with ve2 |y=0,2 at the four corners of the domain, ⌦. The first of these conditions at the corner x = 0, y = 0 528 is as follows: µ00 1 @yy ve1 |x=0 (0) = @yy ve1 |y=0 (0) = v |y=0 . (5.5.34) µ p The remaining compatibility conditions may be derived in the same manner. These will contribute higher order terms, which are the O("2 ) terms from (5.5.12) - (5.5.19): p p C1,u :="2 [u1e @x u1e + "u2e @x u1e + "u1e @x u2e + "u2e @x u2e ]+ h p p i "2 ve1 @y u1e + "ve2 @y u1e + "ve1 @y u2e + "ve2 @y u2e 5 "2 u1e "2 u2e , (5.5.35) 5 5 C1,v :="2 u1e @x ve1 + " 2 u1e @x ve2 + " 2 u2e @x ve1 + "3 u2e @x ve2 5 5 + "2 ve1 @y ve1 + " 2 ve2 @y ve1 + " 2 ve1 @y ve2 + "3 ve2 @y ve2 5 "2 ve1 "2 ve2 . (5.5.36) The following follow from standard elliptic theory: Lemma 5.5.1. Assuming (5.5.30) and compatibility conditions for both ve1 , ve2 for arbitrary order as in (5.5.34), there exist unique solutions, ve1 , ve2 to (5.5.28) and (5.5.32) that are regular: µ000 |@xl @ym {uie , vei }| . c0 ( ) ⇥ Cl,k for i = 1, 2. (5.5.37) µ 529 5.5.3 Boundary Layer Equations Collecting the O(") terms from (5.5.20)- (5.5.27): 9 µ@x u1,0, @Y Y u1,0, = 0, @Y Pp1,0, = 0, > > p p > > > > = u1,0, p |x=0 =, u1,0, p |Y =0 = u1e |y=0 , u1,0, p |Y !1 =0 (5.5.38) Z 1 > > > > vp1,0, = @x u1,0, . > > p ; Y Here we must assume the compatibility condition: u1,0, p (0, Y )|Y =0 = u1e |y=0 , @Y2 u1,0, p (0, Y )|Y =0 = 0. (5.5.39) We will also assume higher order compatibility conditions that can be obtained by dif- ferentiating the above system and reading the resulting equalities. Note that we construct p u1,0, p , vp1,0, on (0, L) ⇥ (0, 1). We now cut-o↵ these layers and make a O( ")-order error: p p p Z x p "Y 1,0, " 0 "Y "Y 1,0, u1, p = ( )u ( ) v 1,0, , vp1, := ( )v (5.5.40) 100 p 100 100 0 100 p u1,0,+ , v 1,0,+ , u1,+ , v 1,+ are defined analogously, and we omit these details. We then define: u1,0 1,0, p := up + u1,0,+ p , vp1 := vp1,0, + vp1,0,+ , (5.5.41) u1p := u1, p + u1,+ p , vp1 := vp1, + vp1,+ . (5.5.42) Note that due to the cut-o↵ in (5.5.40), u1p , vp1 is smooth. The contributions to the next 530 layer are: C2,u :="2 @x Pp1,a + "2 u1e @x u1p + "2 u1p @x u1e + "2 u1p @x u1p 3 3 5 + " 2 ve1 @Y u1p + " 2 vp1 µ0 + " 2 vp1 @y u1e + "2 vp1 @Y u1p h 5⇣ ⌘ i "2 @xx u1p + " 2 u2e u1px + u1p u2ex + "2 ve2 u1pY + "3 vp1 u2ey + Ccut 1 . (5.5.43) 1 Here Ccut is the error introduced by the cut-o↵ functions in (5.5.40): p p 1 Ccut := "µ 0 vp1,0 + 3 " 0 @Y u2,0 p Z 1 3 + 3" 00 u2,0 p " 2 000 u2,0 p , (5.5.44) Y Define the auxiliary pressure via: Z 1 h p Pp1,a := µ@x vp1 + "u1e @x vp1 + "u1p @x ve1 + "u1p @x vp1 Y p + "ve1 @Y vp1 + "vp1 @y ve1 + "vp1 @Y vp1 @Y Y vp1 "@xx vp1 3 ⇣ 5 5 ⌘i + " 2 "3 u2e vpx 1 + " 2 u1p vex 2 + " 2 ve2 vpy 1 + "3 vp1 vey 2 . (5.5.45) With such a choice, C2,v := 0. (5.5.46) 3 Collecting the O(" 2 ) terms from (5.5.20) - (5.5.27), the system satisfied by the second 531 boundary layers is: 9 3 > > µ@x u2,0 p @Y Y u2,0 p = f2 := " 2 C2,u , @Y Pp2 = 0, @x u2,0 p + @ v Y p 2,0 = 0, > > > > > > u2,0 u2,0, u2e |y=0 , u2,0,+ 2 > > p |x=0 =, p |Y =0 = p |y=2 = ue |y=2 , = (5.5.47) u2,0, |Y !1 = 0, u2,0,+ |Y ! 1 = 0 > > p p > > Z Y Z > > Y+ > > > > vp2,0, = @x u2,0, p , v 2,0,+ p = @x u2,0,+ p . ; 0 2 Note that in the same manner as in u1p , vp1 , we have two boundary layer variables, Y , Y+ . We compactify the notation in (5.5.47) to simultaneously address both. Define the cut-o↵ layer via: p p p Z x p "Y 2,0 " 0 "Y "Y 2,0 u2p := ( )u ( ) vp2 , vp2 := ( )v . (5.5.48) 100 p 100 100 0 100 p Lemma 5.5.2. Assume high order compatibility conditions in the sense of (5.5.39) for both u1p , u2p . There exist unique solutions to (5.5.38) and (5.5.47) that are regular and satisfy the following estimates: µ000 |Y m @xk @Yl {u1p , vp1 }|  c0 ( ) ⇥ Cm,k,l for any k, l, m 0, (5.5.49) µ µ000 |Y m @xk @Yl u2p |  c0 ( ) ⇥ Cm,k,l for any k, l, m 0, (5.5.50) µ µ000 |@xk vp2 |  c0 ( ) ⇥ Ck for any k 0. (5.5.51) µ Proof. These follow from standard heat equation estimates. The following are the errors contributed to the next layer: 5 p p 3 C3,u :=" 2 [u1e + u1p + "u2p + "u2e ]@x u2p + " 2 u2p ["@x u1e + "@x u1p 3 3 h 3 3 i + " 2 @x u2p + " 2 @x u2e ] + "@Y u2p "ve1 + " 2 vp1 + " 2 ve2 + "2 vp2 532 h p 3 i 5 3 + "2 vp2 µ0 + "@y u1e + "@Y u1p + " 2 @y u2e " 2 @xx u2p + " 2 Ccut , (5.5.52) h 3 i 3 h 3 3 i C3,v :="2 @x vp2 µ + "u1e + "u1p + " 2 u2e + " 2 u2p "@x ve1 + " 2 @x vp1 + " 2 @x ve2 7 h 3 i + " 2 u2p @x vp2 + "2 vp2 "@y ve1 + "@y vp1 + " 2 @y ve2 3 h 3 3 i 7 + " 2 @Y vp2 "ve1 + " 2 vp1 + " 2 ve2 + " 2 vp2 @y vp2 . (5.5.53) Here Ccut is the error contributed by cutting o↵ the layers: p p Ccut :=(1 )f2 + "µ 0 vp2,0 + 3 " 0 @Y u2,0 p Z 1 3 00 2,0 + 3" up " 2 000 u2,0 p . (5.5.54) Y The total contributions to the remainder forcing is: Fu := C1,u + C3,u , Fv := C1,v + C3,v . (5.5.55) We can break up the forcing contribution into: 5 5 Fu = Tu,"2 + O(" 2 ), Fv = Tv,"2 + O(" 2 ), (5.5.56) where the terms at O("2 ) are the following: h i Tu,"2 := "2 u1e @x u1e + ve1 @y u1e u1e + @Y u2p ve1 + µ0 vp2 + µ 0 vp2,0 + 3 0 @Y u2,0 p (5.5.57) | {z } T1 h i Tv,"2 := "2 u1e @x ve1 + ve1 @y ve1 ve1 + µ@x vp2 . (5.5.58) | {z } T2 Summarizing the above constructions: 533 Lemma 5.5.3. The following estimates are satisfied by us , vs : p |@x us | + |@y vs | + |us µ|  min{O( ")˜ y , O(")}, (5.5.59) p |@y us µ0 | . ", (5.5.60) |@xl vs | . "˜ y for l 0, (5.5.61) The following are satisfied by Tu,"2 , Tv,"2 : T2 µ000 ||T1 , ||2 . c0 ( ), (5.5.62) y˜ µ µ000 1 ||@y T1 , @y T2 ||2 . c0 ( )⇥" 4 . (5.5.63) µ T2 Proof. 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