Abstract
Motivated by the critical dissipative quasigeostrophic equation, we prove that driftdiffusion 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 quasigeostrophic equation with initial $L^2$ data and critical diffusion $(\Delta)^{1/2}$ are locally smooth for any space dimension.

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