Nonexistence of T4 configurations for hyperbolic systems and the Liu entropy condition

Abstract

We study the constitutive set $K$ 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 $K$ relate to the well-posedness of the underlying system and the ability to construct solutions via convex integration. Relating to the convexity of $K$, in the particular case of the p-system, Lorent and Peng (2020) [21] show that $K$ does not contain $T_4$ configurations. Recently, Johansson and Tione (2024) [14] showed that $K$ does not contain $T_5$ configurations.

In this paper, we provide a substantial generalization of Lorent-Peng, based on a careful analysis of the shock curves for a large class of $2\times 2$ systems. We provide several sets of hypotheses on general systems which can be used to rule out the existence of $T_4$ configurations in the constitutive set $K$. In particular, our results show the nonexistence of $T_4$ configurations for every well-known $2\times 2$ hyperbolic system of conservation laws for which both families of shocks verify the Liu entropy condition.