Remarks on a nonhomogeneous model of magnetohydrodynamics

Abstract

This paper is devoted to a model of magnetohydrodynamics described by a parabolic system of partial differential equations coupling the nonhomogeneous incompressible Navier--Stokes equations and Maxwell's equations. In the case of two-dimensional flows, we prove global regularity results under the assumption that the fluids' viscosities are close enough to their average. On the other hand, a more detailed description of the interface and of the regularity of the third component of the magnetic field is given when the fluids have the same viscosity.