A note on global regularity results for 2D Boussinesq equations with fractional dissipation

TL;DR

This paper establishes new conditions for the global regularity of 2D Boussinesq equations with fractional dissipation, extending previous results by refining the parameter ranges for dissipation exponents.

Contribution

The paper provides an improved global regularity criterion for 2D Boussinesq equations with fractional dissipation, expanding the known parameter ranges for regular solutions.

Findings

Global regularity holds for specific fractional dissipation parameters.

Improves upon previous regularity results by extending parameter ranges.

Uses energy methods and commutator estimates for proof.

Abstract

In this paper we study the Cauchy problem for the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation and thermal diffusion. Invoking the energy method and several commutator estimates, we get the global regularity result of the 2D Boussinesq equations as long as $1-\alpha<\beta< \min\Big\{\frac{\alpha}{2},\,\, \frac{3\alpha-2}{2\alpha^{2}-6\alpha+5}, \,\,\frac{2-2\alpha}{4\alpha-3}\Big\}$ with $0.77963\thickapprox\alpha_{0}<\alpha<1$. As a result, this result is a further improvement of the previous two works \cite{MX,YXX}.