On the flux problem in the theory of steady Navier-Stokes equations with nonhomogeneous boundary conditions

Abstract

We study the nonhomogeneous boundary value problem for Navier--Stokes equations of steady motion of a viscous incompressible fluid in a two--dimensional bounded multiply connected domain $\Omega ={\Omega }_1\setminus {\bar{\Omega }}_2,{\bar{\Omega }}_2\subset {\Omega }_1$. We prove that this problem has a solution if the flux $\F$ of the boundary value through $\partial {\Omega }_2$ is nonnegative. The proof of the main result uses the Bernoulli law for a weak solution to the Euler equations and the one-side maximum principle for the total head pressure corresponding to this solution.

arXiv comment35 pages