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

TL;DR

This paper investigates the steady Navier-Stokes equations with nonhomogeneous boundary conditions in a multiply connected domain, proving existence of solutions when the boundary flux is nonnegative using Bernoulli law and maximum principles.

Contribution

It establishes the existence of solutions for steady Navier-Stokes equations with nonhomogeneous boundary flux in multiply connected domains, utilizing Bernoulli law and maximum principles.

Findings

Solutions exist if boundary flux is nonnegative.

Use of Bernoulli law for weak Euler solutions.

Application of maximum principle for total head pressure.

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.