AIMS Mathematics, 2017, 2(2): 336-347. doi: 10.3934/Math.2017.2.336

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Logarithmically improved regularity criteria for the Boussinesq equations

1 Department of Mathematics, University of Mostaganem, Box 227, Mostaganem, 27000, Algeria
2 Dipartimento di Mathematicae Informatica, Universit `a di Catania Viale Andrea Doria, 6, 95125 Catania, Italy

In this paper, logarithmically improved regularity criteria for the Boussinesq equations are established under the framework of Besov space $\overset{.}{B}_{\infty ,\infty }^{-r}$. We prove the solution $(u,\theta )$ is smooth up to time $T>0$ provided that \begin{equation} \int_{0}^{T}\frac{\left\Vert u(\cdot ,t)\right\Vert _{\overset{.}{B} _{\infty ,\infty }^{-r}}^{\frac{2}{1-r}}}{\log (e+\left\Vert u(t,.)\right\Vert _{\overset{.}{B}_{\infty ,\infty }^{-r}})}dt<\infty \end{equation} for some $0\leq r<1$ or \begin{equation} \left\Vert u(\cdot ,t)\right\Vert _{L^{\infty }(0,T;\overset{.}{B}_{\infty ,\infty }^{-1}(\mathbb{R}^{3}))}<<1. \end{equation} This result improves some previous works.
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