Global well-posedness for an advection–diffusion equation arising in magneto-geostrophic dynamics

Abstract We use De Giorgi techniques to prove Holder continuity of weak solutions to a class of drift-diffusion equations, with L 2 initial data and divergence free drift velocity that lies in L t ∞ BMO x − 1 . We apply this result to prove global regularity for a family of active scalar equations which includes the advection–diffusion equation that has been proposed by Moffatt in the context of magnetostrophic turbulence in the Earthʼs fluid core.

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