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

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 (LtBMOx-1)-B-infinity. 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 Earth's fluid core. (C) 2011 Elsevier Masson SAS. All rights reserved.