Propagating phase boundaries: formulation of the problem and existence via the Glimm scheme

TL;DR

This paper establishes the existence and stability of propagating phase boundaries in a hyperbolic-elliptic system modeling elastic materials, using Glimm's scheme and incorporating entropy, kinetic, and nucleation conditions.

Contribution

It introduces a new framework for admissible weak solutions with phase boundary conditions and proves their existence and stability via Glimm's scheme.

Findings

Proves L1 continuous dependence for the Riemann problem.

Establishes existence of propagating phase boundaries.

Demonstrates stability of solutions with respect to initial data.

Abstract

In this paper we consider the hyperbolic-elliptic system of two conservation laws that describes the dynamics of an elastic material governed by a non-monotone strain-stress function. Following Abeyaratne and Knowles, we propose a notion of admissible weak solution for this system in the class of functions with bounded variation. The formulation includes an entropy inequality, a kinetic relation (imposed along any subsonic phase boundary), and an nucleation criterion (for the appearance of new phase boundaries). We prove the L1 continuous dependence of the solution to the Riemann problem. Our main result yields the existence and the stability of propagating phase boundaries. The proof is based on Glimm's scheme and in particular on techniques going back to Glimm-Lax. In order to deal with the kinetic relation, we prove a result of pointwise convergence for the phase boundary.