Nonexistence of a class of $T_N$ configurations for a hyperbolic system with one entropy

TL;DR

This paper proves the nonexistence of certain complex geometric configurations, called $T_N$ configurations, within the structure of a specific set related to hyperbolic conservation laws with one entropy, advancing the understanding of their geometric properties.

Contribution

It extends previous results by showing that the set $K_a$ does not contain a class of three-dimensional and two-dimensional $T_N$ configurations for general N, deepening the geometric analysis of entropy solutions.

Findings

No three-dimensional $T_N$ configurations in $K_a$

No two-dimensional $T_N$ configurations in $K_a$ for any N

Advances understanding of the geometric structure of entropy solutions

Abstract

In Kirchheim, M\"{u}ller and \v{S}ver\'{a}k [Studying nonlinear PDE by geometry in matrix space. Geometric analysis and nonlinear partial differential equations, 2003], the authors proposed the program to use the differential inclusion approach to study entropy solutions for systems of conservation laws. In particular, they raised questions concerning the local structure of the rank-one convex hull of a set $K_a\subset\mathbb{R}^{3\times 2}$, which arises from the differential inclusion formulation of a classical $2\times 2$ system of conservation laws (the $p$-system) coupled with one entropy. Recently, this question has been studied extensively by showing that the set $K_a$ does not contain the so-called $T_N$ configurations for $N=4$ and $N=5$. In this paper, we continue this program by showing that the set $K_a$ does not contain a class of three-dimensional $T_N$ configurations, as well as two-dimensional $T_N$ configurations for general $N$.