Existence of a solution for a thermoelectric model with several phase changes and a Carathéodory thermal conductivity

We present an existence result for the time domain eddy current model for electromagnetic field, coupled with a Stefan problem for temperature. We have extended the existence result proved by Bermúdez et al. (2005) [40] to the case of materials with several phase changes and thermal conductivity depending on both position and temperature.The new proof starts from a totally implicit time discretization of a truncated system. The thermal part can be rewritten as a variational inequality of the second kind. Then, a priori estimates independent of the truncation parameter are obtained for the solution of the truncated problem, using a technique that adapts the method of Boccardo and Gallouët (1989)  [28] to the case of materials with several phase changes.