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 \((t_0,t]\).