A regularity criterion for the dissipative quasi-geostrophic equations

We establish a regularity criterion for weak solutions of the dissipative quasi-geostrophic equations (with dissipation (−Δ)γ/2, 0<γ⩽1). More precisely, we show that if with is a weak solution of the 2D quasi-geostrophic equation, then θ is a classical solution in (0,T]×R2. 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.