Logarithmically improved regularity criteria for the Boussinesq equations

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