Global regularity of the regularized Boussinesq equations with zero diffusion

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.

Keywords

Boussinesq equations, Leray-$\alpha$ model, global regularity

2010 Mathematics Subject Classification

Primary 35Q35, 76D03. Secondary 35Q86.

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This work is supported by the National Natural Science Foundation of China (No. 11701232), by the Natural Science Foundation of Jiangsu Province (No. BK20170224), and by the Qing Lan Project of Jiangsu Province.

Received 4 July 2019

Published 7 July 2020