A regularity criterion for the 3D Boussinesq equations in homogeneous Besov spaces with negative indices

Abstract

In this paper, we study the regularity criteria for the 3D Boussinesq equations in terms of one partial derivative of the velocity in Besov spaces. More precisely, it is proved that if the velocity $u$ holds $\int_{0}^{T}\| \partial_{3} u\|_{\dot{B}_{\infty,\infty}^{-r}}^{\frac{2}{1-r}}\mbox{d}t<\infty,\ with\ \ 0\leq r<1$, then the solution $(u, \theta)$ is regular on $[0,T]$.