Dissipative quasi-geostrophic equations with Lp data

We seek solutions of the initial value problem for the 2D dissipative quasi-geostrophic (QG) equation with L-p initial data. The 2D dissipative QG equation is a two dimensional model of the 3D incompressible Navier-Stokes equations. We prove global existence and uniqueness of regular solutions for the dissipative QG equation with sub-critical powers. For the QG equation with critical or super-critical powers, we establish explicit global L-p bounds for its solutions and conclude that any possible finite time singularity must occur in the first order derivative.