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

Abstract

A weak-strong uniqueness result is proved for measure-valued solutions to the system of conservation laws arising in elastodynamics. The main novelty brought forward by the present work is that the underlying stored-energy function of the material is assumed to be strongly quasiconvex. The proof employs tools from the calculus of variations to establish general convexity-type bounds on quasiconvex functions and recasts them in order to adapt the relative entropy method to quasiconvex elastodynamics.

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

“…We correct a gap in the proof of Theorem 5.1 in [5]. Precisely, the proof results in the constant C 1 being dependent on t 0 in a way that cannot be controlled.…”

“…where z k is constructed in Lemma 5.6 in [5]. Next, integrate in time and take the limit k → ∞, using the equiintegrability of (|∇z k | p ) and that (∇z k ) generates (ν F t 0 ,x ) x∈Q , to conclude the proof of Theorem 5.1, i.e.…”

“…We recall that the function W ∈ C 2 (R d×d ) in [5] is required to be strongly quasiconvex with constant c 0 , p-coercive with p-growth. Note that in the notation of [5], G W (F (t, x), ξ) = G(t, x, ξ).…”

$\mathcal{A}$-Quasiconvexity, Gårding Inequalities, and Applications in PDE Constrained Problems in Dynamics and Statics

“…directly apply to elasticity where η(v, F ) = 1 2 |v| 2 + W (F ). However, as |v| 2 is convex, it is immediate to deduce the result assuming (H1)-(H4) on W [24]. Remark 5.…”

“…Then, naturally, it leads to stability and weak-strong uniqueness results for such entropic weak solutions. In [24] and the case of elasticity, it was understood that the crucial Gårding inequality and the subsequent weak-strong uniqueness result can be proved without the assumption of small oscillations, provided the entropy instead satisfies the stronger condition of quasiconvexity 1 .…”