On an induction–conduction PDEs system in the harmonic regime

We study the existence of weak solutions to a nonlinear coupled parabolic–elliptic system arising in the heating industrial process of a steel workpiece. The unknowns are the electric potential, the magnetic vector potential and the temperature. The different time scales related to the electric potential and the magnetic vector potential versus the temperature lead us to introduce the harmonic regime. This yields to a new system of nonlinear partial differential equations.