May  2010, 9(3): 667-684. doi: 10.3934/cpaa.2010.9.667

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

1. 

Universidade Estadual de Campinas, Campinas, CEP 13083-970, Brazil

2. 

Universidad Nacional de Colombia, Sede Medellín, Medellín, A.A. 3840, Colombia

Received  April 2009 Revised  September 2009 Published  January 2010

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.
Citation: Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the stability problem for the Boussinesq equations in weak-$L^p$ spaces. Communications on Pure & Applied Analysis, 2010, 9 (3) : 667-684. doi: 10.3934/cpaa.2010.9.667
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