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

Abstract

Motivated by the critical dissipative quasi-geostrophic equation, we prove that drift-difiusion equations 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 difiusion (¡¢)1=2, are locally smooth for any space dimension.