Nonhomogeneous boundary value problems for stationary Navier–Stokes equations in a multiply connected bounded domain

Hideo Kozono and Taku Yanagisawa

Vol. 243 (2009), No. 1, 127–150

DOI: 10.2140/pjm.2009.243.127

Abstract

We consider the stationary Navier–Stokes equations on a multiply connected bounded domain Ω in ℝⁿ for n = 2,3 under nonhomogeneous boundary conditions. We present a new sufficient condition for the existence of weak solutions. This condition is a variational estimate described in terms of the harmonic part of solenoidal extensions of the given boundary data; we prove it by using the Helmholtz–Weyl decomposition of vector fields over Ω satisfying adequate boundary conditions. We also study the validity of Leray’s inequality for various assumptions about the symmetry of Ω.

Keywords

stationary Navier–Stokes equations, nonhomogeneous boundary value problems, Helmholtz–Weyl decomposition

Mathematical Subject Classification 2000

Primary: 35Q30

Milestones

Received: 23 January 2009

Revised: 16 May 2009

Accepted: 22 May 2009

Published: 1 November 2009