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 Hölder 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.