Homogenization of a Doubly Nonlinear Stefan-Type Problem

Abstract

Temperature and phase evolution in phase transitions are represented here by coupling the energy balance equation with a multivalued constitutive relation between the density of internal energy and the temperature, and with a nonlinear conduction law. This doubly nonlinear problem generalizes the classical Stefan model. Existence of a weak solution is proved via time discretization, a priori estimates, and passage to the limit. A medium exhibiting periodic oscillations in space is then considered; as the oscillation period vanishes, two-scale convergence (in the sense of Nguetseng) to a corresponding two-scale homogenized problem is proved. The latter is shown to be equivalent to a coarse-scale model. The cases of Fourier's law with either temperature- or phase-dependent conductivity are also treated.