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 (-Δ) 1/2 are locally smooth for any space dimension.

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