A regularity criterion for the dissipative quasi-geostrophic equations

Abstract

We establish a regularity criterion for weak solutions of the dissipative quasi-geostrophic equations (with dissipation $(-\mathrm{\Delta }{)}^{\gamma /2}$, $0<\gamma ⩽1$). More precisely, we show that if $\theta \in L_t^{r_0}((0,T);B_{p,\infty }^{\alpha }({\mathbb{R}}^2))$ with $\alpha =\frac{2}{p}+1-\gamma +\frac{\gamma }{r_0}$ is a weak solution of the 2D quasi-geostrophic equation, then $\theta$ is a classical solution in $(0,T]\times {\mathbb{R}}^2$. This result extends the regularity result of Constantin and Wu [P. Constantin, J. Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. I. H. Poincaré – AN (2007), doi:10.1016/j.anihpc.2007.10.001] to scaling invariant spaces.