Electroconvection in a Magnetic Field

TL;DR

This paper studies electroconvection in porous media under strong magnetic fields, showing the existence of global solutions and their relation to surface quasigeostrophic equations as magnetic strength increases.

Contribution

It introduces a new active scalar equation for charge density in electroconvection under magnetic fields and analyzes solution regularity and limiting behavior.

Findings

Global weak solutions exist for the active scalar equation.

Strong magnetic fields lead to smooth, globally existing solutions.

Solutions converge to surface quasigeostrophic equations as magnetic field strength approaches infinity.

Abstract

Electroconvection in a porous medium under a strong transversal magnetic field is described by an active scalar equation for the charge density. The equation has global weak solutions with $L^{\infty}$ data. We show that for strong enough magnetic fields, $L^{\infty}$-small solutions are smooth globally in time and they obey surface quasigeostrophic equations in the limit of infinite magnetic field strength.