Traveling wave solutions to the free boundary incompressible Navier-Stokes equations

This thesis contains work completed in the area of nonlinear partial differential equations, specifically on the analysis of the Navier-Stokes equations which arises from the study of mathematical fluid dynamics. In joint works with Professor Ian Tice, we demonstrate that a certain type of solutions known as traveling wave solutions are generic under a variety of physical considerations. 

In the first chapter, we provide some background material on the subject and give an overview of the work contained in this thesis. Throughout the thesis we focus on studying a single layer of viscous fluid in a strip-like domain that is bounded below by a flat rigid surface and above by a moving surface. 

In the second chapter, we report the work completed in [50], where we study the effects of inclination and also construct spatially periodic solutions by allowing the fluid cross section to be periodic in various directions in our analysis. An essential component of our analysis is the development of some new functional analytic properties of a scale of anisotropic Sobolev spaces X^s (R^d) introduced in [53], including that these spaces are an algebra in the supercritical regime s > d/2. 

In the third chapter, we report our work on the analysis of the equations when we impose the so-called Navier-slip conditions on the fluid bottom, which posits that the tangential fluid velocity is proportional to the tangential stress experienced by the fluid. We demonstrate that the no-slip solutions constructed in [50, 53] can be recovered by the Navier-slip solutions v by sending the characteristic slip length to zero in the appropriate topology. 

We conclude the thesis by providing a detailed appendix of analysis tools employed throughout our analysis.