Minicourse: Analysis of the Equations of Incompressible Fluid Flow

This mini-course will center on the fundamental analytical results for the equations of incompressible fluid motion. Part of the aim is to show how the analytical tools learned in other courses (especially functional analysis and elliptic PDE) can be used. I hope this will be of value as a bridge from standard course material toward research using analysis. Although the lectures will have technical details, I intend them to be educational in emphasizing important arguments and techniques, without always being complete.

I expect we will prove basic existence theorems for the Navier-Stokes equations (with viscosity) and the Euler equations (without). This will illustrate a strategy for time-dependent nonlinear equations, but also involves special considerations such as the projection on the subspace of divergence-free vector fields. The Sobolev estimates and related estimates play an important role. Other results have to do with qualitative behavior of solutions. The choice of topics might be influenced by the interest of those who take part.

The most important background is familiarity with concepts and theorems of functional analysis (our Math 242 is more than enough), fundamentals of Sobolev spaces, distributions or generalized functions, and elliptic partial diff'l eq'ns (our Math 282 is more than enough). Familiarity with fluid mechanics is not necessary but would be helpful. (I will be teaching Math 228 and it will provide background and context.) An interested student who has not had all of this might find it informative to see what turns out to be useful for our purpose. I will be glad to discuss the content and the needed background. There is a small chance you could win a million dollars, in which case you should share it with me.