On the critical dissipative quasi-geostrophic equation

The 2D quasi-geostrophic (QG) equation is a two dimensional model of the 3D incompressible Euler equations. When dissipation is included in the model, then solutions always exist if the dissipation's wave number dependence is super-linear. Below this critical power, the dissipation appears to be ins ufficient. For instance, it is not known if the critical dissipative QG equ ation has global smooth solutions for arbitrary large initial data. In this paper we prove existence and uniqueness of global classical solutions of t he critical dissipative QG equation for initial data that have small L-infi nity norm. The importance of an L-infinity smallness condition is due to th e fact that L-infinity is a conserved norm for the non-dissipative QG equat ion and is non-increasing on all solutions of the dissipative QG, irrespect ive of size.