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

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

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