On weak solutions to the stationary MHD-equations coupled to heat transfer with nonlocal radiation boundary conditions

Abstract

We study the coupling of the stationary system of magnetohydrodynamics to the heat equation. Coupling occurs on the one hand from temperature-dependent coefficients and from a temperature-dependent force term in the Navier-Stokes equations. On the other hand, the heat sources are given by the dissipation of current in the electrical conductors, and of viscous stresses in the fluid. We consider a domain occupied by several different materials, and have to take into account interface conditions for the electromagnetic fields. Since we additionally want to treat high-temperatures applications, we also take into account the effect of heat radiation, which results in nonlocal boundary conditions for the heat flux. We prove the existence of weak solutions for the coupled system, under the assumption that the imposed velocity at the boundary of the fluid remains sufficiently small. We prove a uniqueness result in the case of constant coefficients and small data. Finally, we discuss the regularity issue in a simplified setting.