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

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 of 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.