Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics

TL;DR

This paper establishes the weak-strong uniqueness principle for dissipative measure-valued solutions in polyconvex elastodynamics, demonstrating their agreement with classical solutions and analyzing convergence and initial data attainment.

Contribution

It introduces a notion of dissipative measure-valued solutions for elastodynamics and proves their uniqueness when classical solutions exist, also applying the method to lattice approximations and entropy systems.

Findings

Dissipative measure-valued solutions coincide with classical solutions when they exist.

Short proof of strong convergence of lattice approximations in elastodynamics.

Dissipative measure-valued solutions attain initial data strongly after time averaging.

Abstract

For the equations of elastodynamics with polyconvex stored energy, and some related simpler systems, we define a notion of dissipative measure-valued solution and show that such a solution agrees with a classical solution with the same initial data when such a classical solution exists. As an application of the method we give a short proof of strong convergence in the continuum limit of a lattice approximation of one dimensional elastodynamics in the presence of a classical solution. Also, for a system of conservation laws endowed with a positive and convex entropy, we show that dissipative measure-valued solutions attain their initial data in a strong sense after time averaging.