Weak entropy solutions to a model in induction hardening, existence and weak-strong uniqueness

Abstract

In this paper, we investigate a model describing induction hardening of steel. The related system consists of an energy balance, an ordinary differential equation (ODE) for the different phases of steel, and Maxwell’s equations in a potential formulation. The existence of weak entropy solutions is shown by a suitable regularization and discretization technique. Moreover, we prove the weak-strong uniqueness of these solutions, i.e. that a weak entropy solutions coincides with a classical solution emanating form the same initial data as long as the classical one exists. The weak entropy solution concept has advantages in comparison to the previously introduced weak solutions, e.g. it allows to include free energy functions with low regularity properties corresponding to phase transitions.