Global regularity of the regularized Boussinesq equations with zero diffusion

Abstract

. In this paper, we consider the n -dimensional regularized incompressible Boussinesq equations with a Leray-regularization through a smooth- ing kernel of order α in the quadratic term and a β -fractional Laplacian in the velocity equation. We prove the global regularity of the solution to the n dimensional logarithmically supercritical Boussinesq equations with zero diffu- sion. As a direct corollary, we obtain the global regularity result for the regularized Boussinesq equations with zero diffusion in the critical case α + β = 12 + n 4 . Therefore, our results settle the global regularity case previously mentioned in the literatures.