GLOBAL EXISTENCE FOR PHASE TRANSITION PROBLEMS VIA A VARIATIONAL SCHEME

Abstract

We construct approximate solutions of the initial value problem for dynamical phase transition problems via a variational scheme in one space dimension. First, we deal with a local model of phase transition dynamics which contains second and third order spatial derivatives modeling the effects of viscosity and surface tension. Assuming that the initial data are periodic, we prove the convergence of approximate solutions to a weak solution which satisfies the natural dissipation inequality. We note that this result still holds for non-periodic initial data. Second, we consider a model of phase transition dynamics with only Lipschitz continuous stress–strain function which contains a non-local convolution term to take account of surface tension. We also establish the existence of weak solutions. In both cases the proof relies on implicit time discretization and the analysis of a minimization problem at each time step.