Communications on Pure and Applied Analysis (CPAA)

On the stability problem for the Boussinesq equations in weak-$L^p$ spaces

Pages: 667 - 684, Volume 9, Issue 3, May 2010      doi:10.3934/cpaa.2010.9.667

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Lucas C. F. Ferreira - Universidade Estadual de Campinas, Campinas, CEP 13083-970, Brazil (email)
Elder J. Villamizar-Roa - Universidad Nacional de Colombia, Sede Medellín, Medellín, A.A. 3840, Colombia (email)

Abstract: We consider the Boussinesq equations in either an exterior domain in $\mathbb{R}^{n}$, the whole space $\mathbb{R}^{n}$, the half space $\mathbb{R}_{+}^{n}$ or a bounded domain in $\mathbb{R}^{n}$, where the dimension $n$ satisfies $n \geq 3$. We give a class of stable steady solutions, which improves and complements the previous stability results. Our results give a complete answer to the stability problem for the Boussinesq equations in weak-$L^{p}$ spaces, in the sense that we only assume that the stable steady solution belongs to scaling invariant class $L_{\sigma }^{(n,\infty)}\times L^{(n,\infty)}$. Moreover, some considerations about the exponential decay (in bounded domains) and the uniqueness of the disturbance are done.

Keywords:  Boussinesq equations, stability, strong solutions.
Mathematics Subject Classification:  Primary: 35Q35, 76D03; Secondary: 76M05.

Received: April 2009;      Revised: September 2009;      Available Online: January 2010.