Large steady solutions to the free-boundary Navier-Stokes equations

点击翻译按钮获取中文摘要

The project is in the rigorous analysis of partial differential equations arising from fluid mechanical problems. I will be looking at a free boundary viscous incompressible Navier-Stokes fluid on an inclined plane. The Navier-Stokes equations represent conservation of momentum and mass in the context of fluid motion, and are useful in many physics and engineering applications. Many flows also have free boundaries, such as ocean and river waves. My initial aim is to prove existence of steady solutions to the PDE system in a neighbourhood of the shear flow induced by the component of gravity parallel to the plane. Various work has been done on related problems, however these works involve complicated anisotropic spaces and existence of solutions has only been established for perturbations around the steady shear flow solution. We will pursue an alternative approach using a more canonical mapping which we hope will allow us to use more conventional elliptic PDE methods to be able to invert the linearised system. We then aim to construct large solutions using techniques from global bifurcation theory. This will involve, among many other things, extending the linear analysis to more general solutions. The techniques and theory required for the project will be drawn from functional analysis, harmonic analysis, and analysis of PDE.