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

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