Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation

Abstract

We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical ($\alpha <1/2$) dissipation $(-\mathrm{\Delta }{)}^{\alpha }$: If a Leray–Hopf weak solution is Hölder continuous $\theta \in C^{\delta }({\mathbb{R}}^2)$ with $\delta >1-2\alpha$ on the time interval $[t_0,t]$, then it is actually a classical solution on $(t_0,t]$.