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In this paper we provide regularity results for active scalars that are weaksolutions of almost critical drift-diffusion equations in general surfaces.This includes models of anisotropic non-homogeneous media and the physicallymotivated case of the two-dimensional sphere. Our finest result deals with thecritical surface quasigeostrophic equation on the round sphere.

Continuity of weak solutions of the critical quasigeostrophic equation on the sphere