In this dissertation, we mainly discuss two topics of partial differential equations in fluid dynamics and kinetic theory: viscous surface wave and diffusive limit. With respect to viscous surface wave, we consider an incompressible viscous flow without surface tension in a finite-depth domain of three dimensions, with a free top boundary and a fixed bottom boundary. The system is governed by the Navier-Stokes equations in this moving domain and the transport equation on the moving boundary. Following the framework of geometric mapping by Y. Guo and I. Tice, we further prove the local well-posedness with general smooth data and give a simpler proof of global well-posedness. Also, we construct a stable numerical scheme to simulate the evolution of this system by discontinuous Galerkin method. With respect to diffusive limit, we revisit the asymptotic analysis of a steady neutron transport equation in a two-dimensional unit disk with one-speed velocity. We disprove the classical boundary layer theory by a concrete counterexample with a different boundary layer expansion with geometric correction. Also, we provide the correct boundary layer construction with both in-flow and diffusive boundary.
Wu, Lei,
"From Kinetic Theory to Fluid Mechanics: Viscous Surface Wave and Hydrodynamic Limit"
(2015).
Applied Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.7301/Z02N50P3
