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: $\alpha >\frac{\sqrt{1777}-23}{24}=\mathrm{0.798103..}$, $\beta >0$ 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 $\alpha +\beta =1$.

arXiv commentThis version fix a minor error in the previous version

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• Potentially Singular Solutions of the 3D Incompressible Euler Equations
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Cited by in corpus (1)

• The global attractor of the 2D Boussinesq equations with fractional Laplacian in Subcritical case