Non-classical Riemann solvers and kinetic relations. II. An hyperbolic–elliptic model of phase-transition dynamics

Abstract

This paper deals with the Riemann problem for a partial differential equation's model arising in phase-transition dynamics and consisting of an hyperbolic–elliptic system of two conservation laws. First of all, we provide a complete description of all solutions of the Riemann problem that are consistent with the mathematical entropy inequality associated with the total energy of the system. Second, following Abeyaratne and Knowles, we impose a kinetic relation to determine the dynamics of subsonic phase boundaries. Based on the requirement that subsonic phase boundaries are preferred whenever available, we determine the corresponding wave curves associated with composite waves (shocks, rarefaction fans, phase boundaries). It turns out that even after the kinetic relation is imposed, the Riemann problem may admit up to two solutions. A nucleation criterion is necessary to select between a solution remaining in a single phase and a solution containing two phase boundaries. Alternatively, a strong assumption on the kinetic relation ensures that the Riemann solution is unique and depends continuously upon its initial data.