Paper detail

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

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 $Ω=Ω_1\setminus\barΩ_2, \;\barΩ_2\subset Ω_1$. We prove that this problem has a solution if the flux $\F$ of the boundary value through $\partialΩ_2$ 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.