A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations

Yoshikazu Giga

doi:10.3934/dcdss.2013.6.1277

Article Contents

2013, Volume 6, Issue 5: 1277-1289. Doi: 10.3934/dcdss.2013.6.1277

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A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations

Yoshikazu Giga^1,

1.
Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Tokyo 153-8914

Received: November 2011

Revised: January 2012

Published: October 2013

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Abstract

We construct a Poiseuille type flow which is a bounded entire solution of the nonstationary Navier-Stokes and the Stokes equations in a half space with non-slip boundary condition. Our result in particular implies that there is a nontrivial solution for the Liouville problem under the non-slip boundary condition. A review for cases of the whole space and a slip boundary condition is included.

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Mathematics Subject Classification: Primary: 35Q30; Secondary: 35B53, 76D05.

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