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

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