Solvability via viscosity solutions for a model of phase transitions driven by configurational forces

TL;DR

This paper introduces a viscosity solution framework to establish the existence of weak solutions for a one-dimensional PDE model of phase transitions driven by configurational forces, addressing multi-dimensional challenges.

Contribution

It proposes a novel approach using viscosity solutions to prove existence of solutions in one dimension for a complex phase transition model, extending prior methods.

Findings

Existence of weak solutions in one dimension established.

Viscosity solutions enable analysis of multi-dimensional problems.

Framework paves the way for future multi-dimensional investigations.

Abstract

In the present article, we are interested in an initial boundary value problem for a coupled system of partial differential equations arising in martensitic phase transition theory of elastically deformable solid materials, e.g., steel. This model was proposed and investigated in previous work by Alber and Zhu in which the weak solutions are defined in a standard way, however the key technique is not applicable to multi-dimensional problem. Intending to solve this multi-dimensional problem and to investigate the sharp interface limits of our models, we thus define weak solutions in a different way by using the notion of viscosity solution, then prove the existence of weak solutions to this problem in one space dimension, yet the multi-dimensional problem is still open.