Global Smooth Solutions to a Thermodynamically Consistent Model of Phase-Field Type in Higher Space Dimensions

Abstract

In a recent paper, Penrose and Fife proposed a thermodynamically consistent model of phase-field type for phase transition phenomena such as liquid-solid phase transitions. The model is based on the observation that the second law of thermodynamics postulates that the entropy functional cannot decrease along solution paths. It turns out that the resulting field equations form a system of partial differential equations which is considerably more difficult than the phase-field equations studied by Caginalp and others. For one space dimension global existence, uniqueness, and asymptotic behaviour of smooth solutions of the initial-boundary value problem with Neumann boundary data have been obtained in a recent paper by Zheng. In this paper, we prove global existence and uniqueness in three space dimensions for the initial-boundary value problem with a third boundary condition.