# American Institute of Mathematical Sciences

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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