The sharp interface limit of the Van der Waals-Cahn-Hilliard phase model for fixed and time dependent domains

Abstract

We study the equilibria of liquid--vapor phase transitions of a single substance at constant temperature and relate the sharp interface model of classical thermodynamics to a phase field model that determines the equilibria by the stationary van der Waals--Cahn--Hilliard theory. For two reasons we reconsider this old problem. 1. Equilibria in a two phase system can be established either under fixed total volume of the system or under fixed external pressure. The latter case implies that the domain of the two--phase system varies. However, in the mathematical literature rigorous sharp interface limits of phase transitions are usually considered under fixed volume. This brings the necessity to extend the existing tools for rigorous sharp interface limits to changing domains since in nature most processes involving phase transitions run at constant pressure. 2. Thermodynamics provides for a single substance two jump conditions at the sharp interface, viz. the continuity of the specific Gibbs free energies of the adjacent phases and the discontinuity of the corresponding pressures, which is balanced by the mean curvature. The existing estimates for rigorous sharp interface limits show only the first condition ...