Induction hardening of steel with restrained Joule heating and nonlinear law for magnetic induction field: Solvability

We study a coupled system of Maxwell’s equations with nonlinear heat equation. We start with derivation of a mathematical model assuming a nonlinear dependency of magnetic field H on magnetic induction B. Moreover the electric conductivity σσ is supposed to be temperature reliant. The coupling between the thermal and electromagnetic equations is provided via the Joule heating term and σσ. Next we use time discretization based on the Rothe’s method to provide energy estimates for discretized system. We prove the existence of a weak solution {u,B} to this coupled system with controlled Joule heating term.