  The main issue addressed in this paper concerns an extension of a result by
Z. Zhang who proved, in the context of the homogeneous Besov space
$\dot{B}_{\infty ,\infty }^{-1}(\mathbb{R}% ^{3})$, that, if the solution of
the Boussinesq equation (\ref% {eq1.1}) below (starting with an initial data in
$H^{2}$) is such that $% (\nabla u,\nabla \theta )\in L^{2}\left(
0,T;\dot{B}_{\infty ,\infty }^{-1}(% \mathbb{R}^{3})\right)$, then the solution
remains smooth forever after $T$. In this contribution, we prove the same
result for weak solutions just by assuming the condition on the velocity $u$
and not on the temperature $\theta$.
