Instructor: Stamatis Dostglou

Description: The course will present the mathematical theory of homogeneous
turbulence from the viewpoint of statistical mechanics.

Since the work of G.I. Taylor and A.N. Kolmogorov, homogeneous turbulence (i.e. turbulent fluid flow statistically invariant under translations) has been one of the more successful approximations of real flows when trying to understand the laws of fluid motion away from boundaries.

The thrust of the course will be the detailed proof of existence of homogeneous fluid flows or, equivalently, the existence of homogeneous measures on parabolic function spaces supported by generalized solutions of the Navier-Stokes equations. This is work of M.J. Vishik and A.V. Fursikov (and later C. Foias and R. Temam).

We shall need (and present) measure theory on function spaces, weak and characteristic convergence of measures, evolution equations in function spaces, basic theory of Navier-Stokes equations, Galyerkin approximations, Sobolev and parabolic embeddings, generalized solutions, non-linear estimates, mollifications. The course will be as self-contained as possible, so this is a good opportunity to cover many standard analytic tools and see them in action.

The main pedagogical aim is to have you ready for research on this field by the course's end.

Prerequisites: Math 7700/7900 (old 310/311) or equivalent required. Math 8420/8421 (old 404/405) desirable.