Vol. 243, No. 1, 2009

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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
Abstract

We consider the stationary Navier–Stokes equations on a multiply connected bounded domain Ω in n 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
Authors
Hideo Kozono
Mathematical Institute
Tohoku University
Sendai 980-8578
Japan
Taku Yanagisawa
Department of Mathematics
Nara Women’s University
Nara 630-8506
Japan