The Morse-Sard Theorem for $W^{n,1}$ Sobolev functions on $\mathbb R^n$ and applications in fluid mechanics

The Morse-Sard Theorem for $W^{n,1}$ Sobolev functions on $\mathbb R^n$ and applications in fluid mechanics

Fri, 03/05
17:00 		Mikhail Korobkov (Sobolev Institute of Mathematics, Novosibirsk) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
The Morse-Sard Theorem for $ W^{n,1} $ Sobolev functions on $ \mathbb R^n $ and applications in fluid mechanics
The talk is based on the joint papers [Bourgain J., Korobkov M.V. and Kristensen~J.: Journal fur die reine und angewandte Mathematik (Crelles Journal). DOI: 10.1515/crelle-2013-0002]  and  [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.