Math 388-02: Analysis of the Equations of Incompressible Fluid Flow (Sep
30-Nov 4)
Instructor: J. T. Beale

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.  If time allows,
I may describe some very interesting recent work of Jian-Guo Liu (in our
department) and Robert Pego.

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 for
perspective.  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
with an interested student.  Oh, there is a small chance you could win a
million dollars.