Nonexistence of $T_4$ configurations for hyperbolic systems and the Liu entropy condition

TL;DR

This paper proves that for a broad class of hyperbolic conservation law systems satisfying Liu's entropy condition, the constitutive set does not contain $T_4$ configurations, impacting solution construction methods.

Contribution

It generalizes previous nonexistence results of $T_4$ configurations to a wide class of systems, based on shock curve analysis and Liu entropy condition.

Findings

Nonexistence of $T_4$ configurations for systems satisfying Liu's condition.

Applicable to all well-known 2x2 hyperbolic systems with Liu entropy.

Provides hypotheses to rule out $T_4$ configurations in general systems.

Abstract

We study the constitutive set $\mathcal{K}$ arising from a $2\times 2$ system of conservation laws in one space dimension, endowed with one entropy and entropy-flux pair. The convexity properties of the set $\mathcal{K}$ relate to the well-posedness of the underlying system and the ability to construct solutions via convex integration. Relating to the convexity of $\mathcal{K}$, in the particular case of the $p$-system, Lorent and Peng [Calc. Var. Partial Differential Equations, 59(5):Paper No. 156, 36, 2020] show that $\mathcal{K}$ does not contain $T_4$ configurations. Recently, Johansson and Tione [arXiv e-prints, page arXiv:2208.10979, August 2022] showed that $\mathcal{K}$ does not contain $T_5$ configurations. In this paper, we provide a substantial generalization of these results, based on a careful analysis of the shock curves for a large class of $2\times 2$ systems. In particular, we provide several sets of hypothesis on general systems which can be used to rule out the existence of $T_4$ configurations in the constitutive set $\mathcal{K}$. In particular, our results show the nonexistence of $T_4$ configurations for every well-known $2\times 2$ hyperbolic system of conservation laws which verifies the Liu entropy condition.