Diffusive-dispersive travelling waves and kinetic relations. II A hyperbolic–elliptic model of phase-transition dynamics

Abstract

We deal here with a mixed (hyperbolic-elliptic) system of two conservation laws modelling phase-transition dynamics in solids undergoing phase transformations. These equations include nonlinear viscosity and capillarity terms. We establish general results concerning the existence, uniqueness and asymptotic properties of the corresponding travelling wave solutions. In particular, we determine their behaviour in the limits of dominant diffusion, dominant dispersion or asymptotically small or large shock strength. As the viscosity and capillarity parameters tend to zero, the travelling waves converge to propagating discontinuities, which are either classical shock waves or supersonic phase boundaries satisfying the Lax and Liu entropy criteria, or else are undercompressive subsonic phase boundaries. The latter are uniquely characterized by the so-called kinetic function, whose properties are investigated in detail here.

Phase transitions and sharp-interface limits for the 1d-elasticity system with non-local energy

“…This is in complete agreement with existence results for the local model (1.5), (1.7) and related systems. We refer to [8,30,42]. Proof of Theorem 3.3: For δ > 0 we define…”

“…Systems with capillarity terms like D ε given by (1.7) have been analyzed by many authors ( [2,8,29,39]), also in the closely related case of liquid-vapour phase transitions in Van-der-Waals fluids ( [3,9,19,20,26,42]). In particular with the ε-scaling given by (1.5), (1.7) solutions of the Cauchy problem for (1.5) have been shown to converge to weak solutions of the Cauchy problem for the sharpinterface limit system (1.1) which contain dynamical phase boundaries ( [29]).…”

The sharp-interface approach for fluids with phase change: Riemann problems and ghost fluid techniques

“…Let us also stress as a striking property of the system (1) that the Clausius-Duhem inequality is not enough to ensure the unique solvability of the Riemann problem (see [23]). To overcome these difficulties we follow the approach developed by LeFloch and co-workers [5,17,24] which relies on the concept of the kinetic relation [1,39,40]. The main existence result for a thermodynamically admissible Riemann problem solution is then presented at the end of Section 3 (Thm.…”

Diffusive–Dispersive Traveling Waves and Kinetic Relations

“…This paper (together with the follow-up papers [4,5]) concerns the effect of vanishing diffusion and dispersion on the solutions of hyperbolic or hyperbolic-elliptic systems of conservation laws. This research is directly motivated by complex models arising in the modeling of phase dynamics in solids or fluids.…”

“…Fluid models including capillarity have for instance been studied by Slemrod [23] and Truskinovsky [24]. In the present paper, we restrict attention to a scalar hyperbolic model, while the forthcoming articles [4,5] will be devoted to an analysis of traveling wave solutions to a system of two conservation laws arising in phase transition dynamics. We refer to [17] for a review on this subject and further references.…”