Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition

Jean-Pierre Raymond

doi:10.3934/dcdsb.2010.14.1537

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2010, Volume 14, Issue 4: 1537-1564. Doi: 10.3934/dcdsb.2010.14.1537

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Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition

Jean-Pierre Raymond^1,

1.
Université de Toulouse & CNRS, Institut de Mathématiques, UMR 5219, 31062 Toulouse Cedex 9

Received: August 31, 2009

Revised: January 31, 2010

Published: May 2010

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Abstract

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with a nonhomogeneous divergence condition. We also prove the existence of global weak solutions to the 3D Navier-Stokes equations when the divergence is not equal to zero. These equations intervene in control problems for the Navier-Stokes equations and in fluid-structure interaction problems.

Keywords:

Navier-Stokes equations,
nonhomogeneous divergence condition,
nonhomogeneous Dirichlet boundary condition.

Mathematics Subject Classification: Primary: 35Q30, 76D05, 76D03; Secondary: 76D07.

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