Dissipative quasi-geostrophic equations in critical Sobolev spaces: smoothing effect and global well-posedness

Abstract

We study the critical and super-critical dissipative quasi-geostrophic equations in $\bR^2$ or $\bT^2$. Higher regularity of mild solutions with arbitrary initial data in $H^{2-\gamma}$ is proved. As a corollary, we obtain a global existence result for the critical 2D quasi-geostrophic equations with periodic $\dot H^1$ data. Some decay in time estimates are also provided.