$T_5$ configurations and hyperbolic systems

TL;DR

This paper investigates the structure of differential inclusions related to hyperbolic conservation laws, demonstrating that certain complex configurations ($T_5$) are absent, extending previous results about $T_4$ configurations.

Contribution

It proves the non-existence of $T_5$ configurations in the differential inclusion associated with hyperbolic systems, advancing the understanding of their geometric properties.

Findings

No $T_5$ configurations in the differential inclusion

Extension of previous results on $T_4$ configurations

Deeper insight into the structure of entropy solutions

Abstract

In this paper we study the rank-one convex hull of a differential inclusion associated to entropy solutions of a hyperbolic system of conservation laws. This was introduced in Section 7 of [Kirchheim, M\"uller, \v{S}ver\'ak, 2003] and many of its properties have already been shown in [Lorent, Peng, 2019]-[Lorent, Peng, 2020]. In particular, in [Lorent, Peng 2020] it is shown that the differential inclusion does not contain any $T_4$ configurations. Here we continue that study by showing that the differential inclusion does not contain $T_5$ configurations.