Sobolev Institute of Mathematics

In the talk we present a survey of recent results (see [4]-[6]) on the existence theorems for the steady-state Navier-Stokes boundary value problems in the plane and axially symmetric 3D cases for bounded and exterior domains (the so called Leray problem, inspired by the classical paper [8]). One of the main tools is the Morse-Sard Theorem for the Sobolev functions $f\in W^2_1(\mathbb R^2)$ [1] (see also [2]-[3] for the multidimensional case). This theorem guaranties that almost all level lines of such functions are $C^1$-curves besides the function $f$ itself could be not $C^1$-regular.

Also we discuss the recent Liouville type theorem for the steady-state Navier-Stokes equations for  axially symmetric 3D solutions in the absence of swirl (see [1]).

References

1.  Bourgain J., Korobkov M. V., Kristensen J., On the Morse-Sard property and level sets of Sobolev and BV functions, Rev. Mat. Iberoam.,  29 , No. 1, 1-23  (2013).
2. Bourgain J., Korobkov M. V., Kristensen J., On the Morse-Sard property and level sets of $W^{n,1}$ Sobolev functions on $\mathbb R^n$, Journal fur die reine und angewandte Mathematik (Crelles Journal) (Online first 2013).
3. Korobkov M. V., Kristensen J., On the Morse-Sard Theorem for the sharp case of Sobolev mappings, Indiana Univ. Math. J., 63, No. 6, 1703-1724  (2014).
4. Korobkov M. V., Pileckas K., Russo R., The existence theorem for steady Navier-Stokes equations in the axially symmetric case, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 14, No. 1, 233-262  (2015).
5. Korobkov M. V., Pileckas K., Russo R., Solution of Leray's problem for stationary Navier-Stokes equations in plane and axially symmetric spatial domains,  Ann. of Math., 181, No. 2, 769-807  (2015).
6. Korobkov M. V., Pileckas K., Russo R., The existence theorem for the steady Navier-Stokes problem in exterior axially symmetric 3D domains, 2014, 75 pp., http://arXiv.org/abs/1403.6921.
7. Korobkov M. V., Pileckas K., Russo R., The Liouville Theorem for the Steady-State Navier-Stokes Problem for Axially Symmetric 3D Solutions in Absence of Swirl, J. Math. Fluid Mech. (Online first 2015).
8. Leray J., Étude de diverses équations intégrals nonlinéaires et de quelques problèmes que pose l'hydrodynamique, J. Math. Pures Appl., 9, No. 12, 1- 82 (1933).

In the talk we discuss some results of [1]. We apply our previous methods [2] to geometry and to the mappings with bounded distortion.

\textbf{Theorem 1}.  Let $v:\Omega\to\mathbb{R}$ be a $C^1$-smooth function on a domain (open connected set) $\Omega\subset\mathbb{R}^2$. Suppose

$$ (1)\qquad \operatorname{Int} \nabla v(\Omega)=\emptyset. $$

Then $\operatorname{meas}\nabla v(\Omega)=0$.

Here $\operatorname{Int}E$ is the interior of ${E}$, $\operatorname{meas} E$ is the Lebesgue measure of ${E}$. Theorem 1 is a straight consequence of the following two results.

\textbf{ Theorem 2 [2]}.  Let $v:\Omega\to\mathbb{R}$ be a $C^1$-smooth function on a domain $\Omega\subset\mathbb{R}^2$. Suppose (1) is fulfilled. Then the graph of $v$ is a normal developing surface. 

Recall that a $C^1$-smooth manifold $S\subset\mathbb{R}^3$ is called  a normal developing surface [3] if for any $x_0\in S$ there exists a straight segment $I\subset S$ (the point $x_0$ is an interior point of $I$) such that the tangent plane to $S$ is stationary along $I$.

\textbf{Theorem 3}.  The spherical image of any $C^1$-smooth normal developing surface $S\subset\mathbb{R}^3$ has the area (the Lebesgue measure) zero.

Recall that the spherical image of a surface $S$ is the set $\{\mathbf{n}(x)\mid x\in S\}$, where $\mathbf{n}(x)$ is the unit normal vector to $S$ at the point~$x$. From Theorems 1--3 and the classical results of A.V. Pogorelov (see [4, Chapter 9]) we obtain the following corollaries Corollary 4. Let the spherical image of a $C^1$-smooth surface $S\subset\mathbb{R}^3$ have no interior points. Then this surface is a surface of zero extrinsic curvature in the sense of Pogorelov.

\textbf{ Corollary 5}. Any $C^1$-smooth normal developing surface $S\subset\mathbb{R}^3$ is a surface of zero extrinsic curvature in the sense of Pogorelov.

\textbf{Theorem 6}. Let $K\subset\mathbb{R}^{2\times 2}$ be a compact set and the topological dimension of $K$ equals 1. Suppose there exists $\lambda> 0$ such that $\forall A,B\in K, \, \, |A-B|^2\le\lambda\det(A-B).$

Then for any Lipschitz mapping $f:\Omega\to\mathbb R^2$ on a domain $\Omega\subset\mathbb R^2$ such that $\nabla f(x)\in K$ a.e. the identity $\nabla f\equiv\operatorname{const}$ holds.

Many partial cases of Theorem 6 (for instance, when $K=SO(2)$ or $K$ is a segment) are well-known (see, for example, [5]).

The author was supported by the Russian Foundation for Basic Research (project no. 08-01-00531-a).

[1] {Korobkov M.\,V.,} {``Properties of the $C^1$-smooth functions whose gradient range has topological dimension~1,'' Dokl. Math., to appear.}

[2] {Korobkov M.\,V.} {``Properties of the $C^1$-smooth functions with nowhere dense gradient range,'' Siberian Math. J., \textbf{48,} No.~6, 1019--1028 (2007).}

[3] { Shefel${}'$ S.\,Z.,} {``$C^1$-Smooth isometric imbeddings,'' Siberian Math. J., \textbf{15,} No.~6, 972--987 (1974).}

[4] {Pogorelov A.\,V.,} {Extrinsic geometry of convex surfaces, Translations of Mathematical Monographs. Vol. 35. Providence, R.I.: American Mathematical Society (AMS). VI (1973).}

[5] {M\"uller ~S.,} {Variational Models for Microstructure and Phase Transitions. Max-Planck-Institute for Mathematics in the Sciences. Leipzig (1998) (Lecture Notes, No.~2. http://www.mis.mpg.de/jump/publications.html).}