# Solution of Leray’s problem for stationary Navier-Stokes equations in plane and axially symmetric spatial domains

### Abstract

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. (For axially symmetric domains, data is assumed to be axially symmetric as well.) 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.

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## Authors

Mikhail V. Korobkov

Sobolev Institute of Mathematics Novosibirsk State University, Novosibirsk, Russia

Konstantin Pileckas

Faculty of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania

Remigio Russo

Department of Mathematics and Physics, Second University of Naples, Italy