A global regularity result for the 2D Boussinesq equations with critical dissipation

Abstract

This paper examines the global regularity problem on the two-dimensional incompressible Boussinesq equations with fractional dissipation, given by $\Lambda^\alpha u$ in the velocity equation and by $\Lambda^\beta \theta$ in the temperature equation, where $\Lambda=\sqrt{-\Delta}$ denotes the Zygmund operator. We establish the global existence and smoothness of classical solutions when $(\alpha,\beta)$ is in the critical range: $\alpha>\frac{\sqrt{1777}-23}{24} =0.798103..$, $\beta>0$ and $\alpha+ \beta =1$. This result improves the previous work of Jiu, Miao, Wu and Zhang \cite{JMWZ} which obtained the global regularity for $\alpha> \frac{23-\sqrt{145}}{12} \approx 0.9132$, $\beta>0$ and $\alpha+ \beta =1$.