On the global regularity of the 2D critical Boussinesq system with $\alpha>2/3$

TL;DR

This paper proves that the 2D critical Boussinesq system with fractional dissipation greater than 2/3 admits globally regular solutions, using a novel change of variables and commutator estimates to improve existing results.

Contribution

Introduces a new change of variables and commutator estimates to establish global regularity for the 2D critical Boussinesq system with b1 > 2/3.

Findings

Global regular solutions exist for b1 > 2/3.

New change of variables improves linear derivative control.

Enhanced commutator estimates may be useful for related problems.

Abstract

This paper examines the question for global regularity for the Boussinesq equation with critical fractional dissipation. The main result states that the system admits global regular solutions for all (reasonably) smooth and decaying data, as long as $\al>2/3$. This significantly improves upon some recent works. The main new idea is the introduction of a new, second generation Hmidi-Keraani-Rousset type, change of variables, which further improves the linear derivative in temperature term in the vorticity equation. This approach is then complemented by new set of commutator estimates, which may be of independent interest.