The Morse-Sard Theorem for $W^{n,1}$ Sobolev functions on $\mathbb R^n$ and applications in fluid mechanics
Abstract
The talk is based on the joint papers [{\it Bourgain J., Korobkov
M.V. and Kristensen~J.}: Journal fur die reine und angewandte Mathematik
(Crelles
Journal).
DOI: 10.1515/crelle-2013-0002] \ and \
[{\it Korobkov~M.V., Pileckas~K. and Russo~R.}:
arXiv:1302.0731, 4 Feb 2013]
We establish Luzin $N$ and Morse--Sard
properties for functions from the Sobolev space
$W^{n,1}(\mathbb R^n)$. Using these results we prove
that almost all level sets are finite disjoint unions of
$C^1$-smooth compact manifolds of dimension
$n-1$. These results remain valid also within
the larger space of functions of bounded variation
$BV_n(\mathbb R^n)$.
As an application, we study the nonhomogeneous boundary value problem
for the Navier--Stokes equations of steady motion of a viscous
incompressible fluid in arbitrary bounded multiply connected
plane or axially-symmetric spatial domains. We prove that this
problem has a solution under the sole necessary condition of zero total
flux through the boundary.
The problem was formulated by Jean Leray 80 years ago.
The proof of the main result uses Bernoulli's law
for a weak solution to the Euler equations based on the above-mentioned
Morse-Sard property for Sobolev functions.