Regularity of solutions to a model for solid-solid phase transitions driven by configurational forces

TL;DR

This paper investigates the regularity of solutions to a one-dimensional model for solid-solid phase transitions driven by configurational forces, focusing on the mathematical challenges posed by weighted gradient terms.

Contribution

It extends previous existence results by analyzing the regularity of weak solutions in one dimension with higher initial data regularity, addressing the role of weighted gradient terms.

Findings

Regular solutions are shown to exist in one dimension with $H^2$ initial data.

The analysis reveals the influence of the gradient term acting as a weight.

The proof highlights the challenges of weighted degeneracy in phase transition models.

Abstract

In a previous work, we prove the existence of weak solutions to an initial-boundary value problem, with $H^1(\Omega)$ initial data, for a system of partial differential equations, which consists of the equations of linear elasticity and a nonlinear, degenerate parabolic equation of second order. Assuming in this article the initial data is in $H^2(\Omega)$, we investigate the regularity of weak solutions that is difficult due to the gradient term which plays a role of a weight. The problem models the behavior in time of materials with martensitic phase transitions. This model with diffusive phase interfaces was derived from a model with sharp interfaces, whose evolution is driven by configurational forces, and can be thought to be a regularization of that model. Our proof, in which the difficulties are caused by the weight in the principle term, is only valid in one space dimension.