Some Convergence Results for a Class of Nonlinear Phase-Field Evolution Equations

Two heat diffusion problems in the framework of the parabolic phase-field model are presented. The first problem is related to a single isotropic fluid and the other describes the heat transmission between two different substances in contact. Some known existence and uniqueness results are briefly recalled. Then, an asymptotic analysis of both situations is carried out as the kinetic equation collapses to a temperature-phase relation of Stefan type, in the first case in the whole material, and in the second in only one of the substances. In both cases, a convergence result for the solutions is proved. The second problem shows some more mathematical difficulties that are due to the presence of nontrivial terms on the common boundary. In order to control the latter, some tools are used from the Γ-convergence theory for convex functionals.