AIMS Mathematics, 2017, 2(3): 451-457. doi: 10.3934/Math.2017.2.451

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A regularity criterion of weak solutions to the 3D Boussinesq equations

1 Department of Mathematical Science , Faculty of Applied Science, Umm Alqura University, P. O.Box 14035, Makkah 21955, Saudi Arabia
2 Department of Mathematics, University of Mostaganem, Algeria
3 Dipartimento di Mathematica e Informatica, Università di Catania, Viale Andrea Doria, 6, 95125Catania, Italy
4 RUDN University, 6 Miklukho - Maklay St, Moscow, 117198, Russia

In this note, a regularity criterion of weaksolutions to the 3D-Boussinesq equations with respect to Serrin type condition under the framework of Besov space $\overset{.}{B}_{\infty ,\infty}^{r}$. It is shown that the weak solution $(u,\theta )$ is regular on $%(0,T] $ if $u$ satisfies $\int\limits_{0}^{T}{\left\| u\left( \cdot ,t \right) \right\|_{\overset{\cdot R}{\mathop{{{B}_{\infty ,\infty }}}}\,}^{\frac{2}{1+r}}}\ \ dt < \infty ,$ for 0<r<1. This result improves some previous works.
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