Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation

The critical dissipative quasi-geostrophic equation was proposed by several authors as a toy model to study the regularity of solutions to 3D Navier-Stokes equations. In this work, in collaboration with L. Caffarelli, we prove that drift-diffusion equatons with L2 initial data and minimal assumptions on the drift are locally Holder continuous. As an application we show that solutions of the quasi-geostrophic equation with initial L2 data and critical diffusion (-Delta)^{1/2}, are locally smooth for any space dimension.