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

Motivated by the critical dissipative quasi-geostrophic equation, we prove that drift-diffusion equations with L 2 initial data and minimal assumptions on the drift are locally Hlder continuous. As an application we show that solutions of the quasi-geostrophic equation with initial L 2 data and critical diffusion . / 1=2 are locally smooth for any space dimension.