Steady Navier–Stokes system with nonhomogeneous boundary conditions in the axially symmetric case

Abstract

The nonhomogeneous boundary value problem for the steady Navier–Stokes equations is studied in a three-dimensional axially symmetric bounded domain with multiply connected Lipschitz boundary. We assume that the boundary value is axially symmetric. Our results imply, in particular, the existence of the solution with arbitrary large fluxes over the connected components of the boundary, provided that all these components intersect the axis of the symmetry. The proof 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.