On global regularity for a model 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. Attention is focused on the case that the temperature equation is a pure transport equation without regularizing the velocity in the nonlinear term. We establish the global regularity for the regularized Boussinesq equations with zero diffusion in the critical case $\alpha+\beta=\frac{1}{2} + \frac{n}{4}$ and $\beta \geq \frac{1}{2}$. In addition, a regularity criterion via the temperature is also established for the critical case $\alpha+\beta=\frac{1}{2} + \frac{n}{4}$ and $0 \lt \beta \lt \frac{1}{2}$.

Keywords

Boussinesq equations, fractional dissipation, global regularity

2010 Mathematics Subject Classification

35B65, 35Q35, 76D03

Full Text (PDF format)

Received 15 September 2020

Received revised 8 January 2022

Accepted 17 January 2022

Published 14 September 2022