A logarithmically improved regularity criterion for the Boussinesq equations in a bounded domain

The paper is concerned with the regularity of solutions of the Boussinesq equations for incompressible fluids without heat conductivity. The main goal is to prove a regularity criterion in terms of the vorticity for the initial boundary value problem in a bounded domain $$\Omega$$ of $$\mathbb {R}^{3}$$ with Navier-type boundary conditions and we prove that if $$\begin{aligned} \int _{0}^{T}\frac{\left\| \omega (\cdot ,t)\right\| _{BMO(\Omega )}}{ \log \left( e+\left\| \omega (\cdot ,t)\right\| _{BMO(\Omega )}\right) }dt<\infty , \end{aligned}$$ where $$\omega :=$$ curl u is the vorticity, then the unique local in time smooth solution of the 3D Boussinesq equations can be prolonged up to any finite but arbitrary time.