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 Holder continuous. As an application we show that solutions of the quasi-geostrophic equation with initial L^2 data and critical diffusion (-\Delta)^{1/2}, are locally smooth for any space dimension.