Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion

Abstract

This article is concerned with the blow-up criterion for smooth solutions of three-dimensional Boussinesq equations with zero diffusion. It is shown that if the velocity field $u(x,t)$ satisfies \begin{eqnarray*} u\in L^p(0,T_1;B^r_{q,\infty}(R^3)),\quad \frac{2}{p}+\frac{3}{q}=1+r,\quad \frac{3}{1+r}< q \leq \infty, \quad -1 < r \leq 1, \end{eqnarray*} then the solution can be continually extended to the interval $(0,T)$ for some $T>T_1$.