Continuity in time of solutions of a phase-field model

TL;DR

This paper proves the temporal continuity and stability of solutions in a phase-field model describing phase changes in materials with heat-conductive walls, under certain initial conditions.

Contribution

It establishes the continuity in time of temperature and phase variables and their dependence on initial data in a phase-field model with heat-conductive boundaries.

Findings

Temperature and phase variables are continuous in time in specified function spaces.

Solutions depend continuously on initial data and boundary conditions.

Results apply to Lipschitz domains in 2D and 3D.

Abstract

A phase field model proposed by G. Caginalp for the description of phase changes in materials is under consideration. It is assumed that the medium is located in a container with heat conductive walls that are not subjected to phase changes. Therefore, the temperature variable is defined both in the medium and wall regions, whereas the phase variable is only considered in the medium part. The case of Lipschitz domains in two and three dimensions is studied. We show that the temperature and phase variables are continuous in time functions with values in $L^2$ and $H^1$, respectively, provided that the initial values of them are from $L^2$ and $H^1$, respectively. Moreover, continuous dependence of solutions on the initial data and boundary conditions is proved.