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