In this thesis we study the $L^p$ Dirichlet problem for second order divergence-form operators having a BMO antisymmetric part. To be precise, for the divergence-form …
We consider the one-dimensional nonlinear Schroedinger equation (NLS) with focusing point nonlinearity, which replaces the pure power nonlinearity in the standard NLS by inserting a …
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 …
We study the one-dimensional Gross-Pitaevskii equation, a cubic defocusing non-linear Schrodinger equation with nonvanishing boundary conditions. In particular we linearize around the dark solitons, which …
Extending work of Kapouleas and Yang, we construct sequences of closed minimal surfaces embedded in the round unit 3-sphere and converging to a Clifford torus …
The FitzHugh-Nagumo equations are known to admit fast traveling pulses that have monotone tails and arise as the concatenation of Nagumo fronts and backs in …
In this thesis, we study some boundary problems of the Boltzmann equation and the Boltzmann equation with the large external potential.If the gas is contained …
This goal of this thesis is to understand patterns in the Swift{Hohenberg equation. Thepatterns studied are localized, stationary and radially symmetric in dimensions one throughthree. …
Many organisms need to establish spatial orientation during early development. In egg cells (oocytes) of the frog Xenopus laevis, spatial differentiation is achieved by localization …
Planar target patterns are radially symmetric time-periodic structures that connect a core region with a spatially periodic traveling wave in the far field. These patterns …
In this thesis, we study the non-local differential equations. The first part is a comprehensive study of $L_p$ estimates for time fractional wave equations of …
This thesis presents three results in geometric analysis. We first analyze the curve-shortening flow on figure eight curves in the plane. Afterwards, we examine the …
We develop a high-order nonlinear energy method to study the stability of steady states of the Stefan problem with surface tension. There are two prominent …
In this thesis, we investigate the stability properties of a variety of PDE models concerning fluid dynamics, including charged fluids (plasmas). The models which we …
This thesis considers two-dimensional stratified water waves propagating under the force of gravity over an impermeable flat bed and with a free surface. In the …
This dissertation presents new regularity criteria of the incompressible Navier-Stokes equations and derive further results, including extensions of the Ladyzhenskaya-Prodi-Serrin condition in high dimensional critical …
This dissertation is centered around the broad topic of physics-informed deep learning (DL). Specifically, it covers the following three fronts: Algorithms. Recent developments of DL, …
