The 2D incompressible Boussinesq equations with general critical dissipation

TL;DR

This paper proves the global regularity of 2D incompressible Boussinesq equations with a broad class of critical fractional dissipation, extending previous results to a new range of dissipation parameters.

Contribution

It establishes the global regularity for the Boussinesq equations with general critical dissipation where the sum of dissipation exponents equals one, for a specific range of lpha.

Findings

Global regularity for lpha + eta=1 with 0.9132<lpha<1.

Extension of previous results for lpha=1 and lpha=0 cases.

Utilization of generalized critical surface quasi-geostrophic equation regularity.

Abstract

This paper aims at the global regularity problem concerning the 2D incompressible Boussinesq equations with general critical dissipation. The critical dissipation refers to $\alpha +\beta=1$ when $\Lambda^\alpha \equiv (-\Delta)^{\frac{\alpha}{2}}$ and $\Lambda^\beta$ represent the fractional Laplacian dissipation in the velocity and the temperature equations, respectively. We establish the global regularity for the general case with $\alpha+\beta=1$ and $0.9132\approx \alpha_0<\alpha<1$. The cases when $\alpha=1$ and when $\alpha=0$ were previously resolved by Hmidi, Keraani and Rousset \cite{HKR1,HKR2}. The global existence and uniqueness is achieved here by exploiting the global regularity of a generalized critical surface quasi-gesotrophic equation as well as the regularity of a combined quantity of the vorticity and the temperature.