Analysis of two stationary magnetohydrodynamics systems of equations including Joule heating

We study the coupling of the equations of steady-state magnetohydrodynamics (MHD) with the heat equation when the buoyancy effects due to temperature differences in the flow as well as Joule effect and viscous heating are (all) taken into account. Two models for the gravity force are considered: the first one is the well-known Boussinesq approximation; in the second one density is assumed to be constant except in the gravity force, where it is assumed to be a non-increasing function of the temperature. The equations are posed in a bounded three-dimensional domain. We give existence results of weak solutions to both models under certain conditions on the data. We also give some uniqueness results.