Global well-posedness of the viscous Boussinesq equations

Thomas Y. Hou; Congming Li

doi:10.3934/dcds.2005.12.1

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2005, Volume 12, Issue 1: 1-12. Doi: 10.3934/dcds.2005.12.1

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Global well-posedness of the viscous Boussinesq equations

Thomas Y. Hou^1, and
Congming Li^2,

1.
Department of Applied and Computational Mathematics, California Institute of Technology, Pasadena, CA 91125
2.
Department of Applied Mathematics, University of Colorado at Boulder, Boulder, CO 80309-0524

Received: October 31, 2004

Revised: October 31, 2004

Published: March 2005

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Abstract

We prove the global well-posedness of the viscous incompressible Boussinesq equations in two spatial dimensions for general initial data in $H^m$ with $m\ge 3$. It is known that when both the velocity and the density equations have finite positive viscosity, the Boussinesq system does not develop finite time singularities. We consider here the challenging case when viscosity enters only in the velocity equation, but there is no viscosity in the density equation. Using sharp and delicate energy estimates, we prove global existence and strong regularity of this viscous Boussinesq system for general initial data in $H^m$ with $m \ge 3$.

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Mathematics Subject Classification: 76B03, 76D03, 76D05, 35M10.

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