On the critical dissipative quasi-geostrophic equation

Abstract

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 insufficient. For instance, it is not known if the critical dissipative QG equation has global smooth solutions for arbitrary large initial data. In this paper we prove existence and uniqueness of global classical solutions of the critical dissipative QG equation for initial data that have small $L^\infty$ norm. The importance of an $L^{\infty}$ smallness condition is due to the fact that $L^{\infty}$ is a conserved norm for the non-dissipative QG equation and is non-increasing on all solutions of the dissipative QG., irrespective of size.