The Riemann problem for thermoelastic materials with phase change

Abstract

We consider the Riemann problem for a system of conservation laws related to a phase transition problem. The system is nonisentropic and we treat the case where the latent heat is not zero. We study the cases where the initial data are given in the same phase and in the different phases. The role of the entropy condition is studied as well as the kinetic relation and the entropy rate admissibility criterion. We confine our attention to the case where the speeds of phase boundaries are close to zero. This is one interesting case in physics. We discuss the number of phase boundaries consistent with the above criteria and the uniqueness and nonuniqueness issue of the solution to the Riemann problem. r 2004 Elsevier Inc. All rights reserved.

L1 well posedness of Euler equations with dynamic phase boundaries

“…In this section, we firstly summarize the results in [9] concerning the Riemann problems with dynamic phase transitions. The configuration of Riemann problems is essential in the front tracking approximations when a phase boundary collides with a small physical wave.…”

“…In the case where the speeds of phase boundaries are much smaller than those of the forward and backward waves, we have the configuration of the phase boundaries given the following theorem. For the discussion of the cases where v l and v r in the same phase, refer to [9]. Furthermore, except case (4) there is a unique solution satisfying the kinetic relation (1.4) provided that is sufficiently small.…”

“…Furthermore, except case (4) there is a unique solution satisfying the kinetic relation (1.4) provided that is sufficiently small. The details of the proof are available in [9]. In what follows, we choose the initial data such that cases (2) and (3) in Theorem 2.2 will occur.…”

“…1.1). We consider one interesting case in physics [9] where the speed of a phase boundary is close to zero such that a phase boundary moves much slower than any shocks (except for contact discontinuity) or rarefaction waves.…”

“…Section 2 is the preliminary where we summarize the solutions of the Riemann problems discussed in [9] and introduce the Finiteness Condition and Stability Condition. In Section 3, we introduce the front tracking approximation of the initial value problem and state the main theorem on existence.…”