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Global regularity of the regularized Boussinesq equations with zero diffusion. (English) Zbl 1446.35147
Summary: In this paper, we consider the $$n$$-dimensional regularized incompressible Boussinesq equations with a Leray-regularization through a smoothing kernel of order $$\alpha$$ in the quadratic term and a $$\beta$$-fractional Laplacian in the velocity equation. We prove the global regularity of the solution to the $$n$$-dimensional logarithmically supercritical Boussinesq equations with zero diffusion. As a direct corollary, we obtain the global regularity result for the regularized Boussinesq equations with zero diffusion in the critical case $$\alpha + \beta = \frac{1}{2} + \frac{n}{4}$$. Therefore, our results settle the global regularity case previously mentioned in the literatures.
##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 35Q86 PDEs in connection with geophysics 86A05 Hydrology, hydrography, oceanography 35B65 Smoothness and regularity of solutions to PDEs
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