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

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  • 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.
    Mathematics Subject Classification: Primary: 35Q35, 76D03; Secondary: 76M05.


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