On the Boussinesq system: regularity criteria and singular limits

We consider the 3D Boussinesq system and we prove several criteria, not involving the density, for the continuation of smooth solutions. We give particular emphasis to the results in bounded domains, under various boundary conditions. The results we prove are partially known and we collect them in a unified framework, mainly with the perspective of understanding the stabi- lization/smoothing required by numerical methods (especially by large scales methods). In the final section, we also consider the vanishing viscosity/diffusivity limits, proving (locally-in-time) sharp singular limits for smooth solutions of the Cauchy problem