Regularity of H\"older continuous solutions of the supercritical quasi-geostrophic equation

We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical ($\alpha< 1/2$) dissipation $(-\Delta)^\alpha$ : If a Leray-Hopf weak solution is H\"{o}lder 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 $(t_0,t]$.