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

本课题是在对流体力学问题中产生的偏微分方程进行严密的分析。我将看一个自由边界粘性不可压缩的Navier-Stokes流体在斜面上。Navier-Stokes方程表示流体运动中的动量和质量守恒，在许多物理和工程应用中都很有用。许多流动也有自由的边界，比如海洋和河流的波浪。我最初的目的是证明PDE系统在平行于平面的重力分量引起的剪切流的邻域中存在稳定解。在相关问题上已经做了很多工作，但是这些工作涉及到复杂的各向异性空间，并且只在稳定剪切流解周围的扰动下才建立了解的存在性。我们将采用另一种方法，使用更规范的映射，我们希望这将允许我们使用更传统的椭圆PDE方法来反演线性化系统。然后，我们的目标是使用全局分岔理论的技术构建大型解决方案。这将包括，将线性分析扩展到更一般的解。项目所需的技术和理论将从功能分析、谐波分析和PDE分析中汲取。

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.