Relative entropy in multi-phase models of 1d elastodynamics: Convergence of a non-local to a local model

TL;DR

This paper demonstrates that solutions of a non-local regularized elastodynamics model converge to those of a local model, using the relative entropy method to handle non-convex energies.

Contribution

It introduces a framework showing convergence of non-local to local models in elastodynamics with non-convex energy, expanding the applicability of relative entropy methods.

Findings

Non-local solutions converge to local solutions in a specific regime.

Relative entropy framework effectively handles non-convex energies.

Method enables stability analysis without requiring convexity.

Abstract

In this paper we study a local and a non-local regularization of the system of nonlinear elastodynamics with a non-convex energy. We show that solutions of the non-local model converge to those of the local model in a certain regime. The arguments are based on the relative entropy framework and provide an example how local and non-local regularizations may compensate for non-convexity of the energy and enable the use of the relative entropy stability theory -- even if the energy is not quasi- or poly-convex.