Existence of solution for a density-dependent magnetohydrodynamic equation

Abstract

We prove a global-in-time existence result of a weak solution for a magnetohydrodynamic (MHD) problem set in a bounded domain of $\mathbb{R}^3$. The fluid is supposed to be incompressible but with an unhomogeneous density, viscosity and electrical conductivity. The displacement currents are neglected in the time-dependent Maxwell equations. The model describes in particular the flow of two immiscible fluids in presence of a magnetic field