an:07229736
Zbl 1446.35147
Ye, Zhuan
Global regularity of the regularized Boussinesq equations with zero diffusion
EN
Dyn. Partial Differ. Equ. 17, No. 3, 245-273 (2020).
00451606
2020
j
35Q35 76D03 35Q86 86A05 35B65
Boussinesq equations; Leray-\( \alpha\) model; global regularity
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.