Weak–Strong Uniqueness of Dissipative Measure-Valued Solutions for Polyconvex Elastodynamics

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.

Quasiconvex Elastodynamics: Weak‐Strong Uniqueness for Measure‐Valued Solutions

“…The problem of extending Ball's seminal result to quasiconvex functions remains an important open problem in elastostatics, and we will not be concerned with it here. For polyconvex energies and the evolution problem, the existence and weak-strong uniqueness of measurevalued solutions for the initial boundary value problem on the flat torus has been shown by Demoulini, Stuart, and Tzavaras in [14] and [15], respectively. In particular, the weak-strong uniqueness result in [15] employs the relative entropy method and the convexity of the energy for an enlarged system whose involutions make it equivalent to (1.1).…”

“…The aim of this paper is to study the question of weak-strong uniqueness for measure-valued solutions to system (1.1) in .0; T / Q under the assumption of (strong) quasiconvexity for the stored-energy function W . The question of weak-strong uniqueness is then natural as the minimal requirement for any notion of solution, namely that it must agree with the classical solution whenever the latter exists and has gained much attention in recent years; see [8,15]. The question of weak-strong uniqueness is then natural as the minimal requirement for any notion of solution, namely that it must agree with the classical solution whenever the latter exists and has gained much attention in recent years; see [8,15].…”

“…0 C ; see Ball [2]. In particular, the weak-strong uniqueness result in [15] employs the relative entropy method and the convexity of the energy for an enlarged system whose involutions make it equivalent to (1.1). For polyconvex energies and the evolution problem, the existence and weak-strong uniqueness of measurevalued solutions for the initial boundary value problem on the flat torus has been shown by Demoulini, Stuart, and Tzavaras in [14] and [15], respectively.…”

“…The defined measure-valued solutions assume two additional properties compared to standard definitions (see, e.g., [15]), which are natural in the sense that any reasonable approximating system will fulfill these requirements. For these solutions, a weak-strong uniqueness result is established for stored-energy functions W , which are strongly quasiconvex.…”

“…For these solutions, a weak-strong uniqueness result is established for stored-energy functions W , which are strongly quasiconvex. The defined measure-valued solutions assume two additional properties compared to standard definitions (see, e.g., [15]), which are natural in the sense that any reasonable approximating system will fulfill these requirements. On the one hand, we assume that the measure-valued solutions are generated by a sequence of spatial gradients.…”