  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.
