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.

中文翻译：

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感应淬火模型的弱熵解、存在性和弱-强唯一性

在本文中，我们研究了一个描述钢的感应淬火的模型。相关系统由能量平衡、钢不同相的常微分方程 (ODE) 和潜在公式中的麦克斯韦方程组成。弱熵解的存在通过适当的正则化和离散化技术来证明。此外，我们证明了这些解的弱-强唯一性，即只要经典解存在，弱熵解与从相同初始数据产生的经典解相吻合。与之前介绍的弱解相比，弱熵解概念具有优势，例如，它允许包含与相变相对应的具有低规律性的自由能函数。