Existence of weak solutions to the time-dependent MHD equations coupled to the heat equation with nonlocal radiation boundary conditions

We study the coupling of the system of magnetohydrodynamics to the heat equation in the context of an application to crystal growth. The heat sources are given by the dissipation of current that occurs in the fixed conductors of the system. According to Boussinesq’s model, the dissipative heating is neglected in the fluid. We take into account the natural interface conditions for the magnetic field, and nonlocal radiation boundary conditions for the heat flux. We prove the existence of a weak solution with a defect measure, concentrated in a singular set.