Electroconvection in a Magnetic Field

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 $L^{\mathrm{\infty }}$ data. We show that for strong enough magnetic fields, $L^{\mathrm{\infty }}$-small solutions are smooth globally in time and they obey surface quasigeostrophic equations in the limit of infinite magnetic field strength.