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

TL;DR

This paper proves the global well-posedness and regularity of solutions to a class of advection-diffusion equations relevant in magneto-geostrophic dynamics, using advanced De Giorgi techniques.

Contribution

It introduces a novel application of De Giorgi methods to establish global regularity for active scalar equations in geophysical fluid dynamics.

Findings

Hölder continuity of weak solutions established

Global regularity proven for magnetostrophic turbulence models

Applicable to equations with divergence-free drift in BMO^{-1}

Abstract

We use De Giorgi techniques to prove H\"older continuity of weak solutions to a class of drift-diffusion equations, with $L^2$ initial data and divergence free drift velocity that lies in $L_{t}^{\infty}BMO_{x}^{-1}$. 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 Earth's fluid core.