Abstract
We study the constitutive set K arising from a 2×2 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 T4 configurations. Recently, Johansson and Tione (2024) [14] showed that K does not contain T5 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 2×2 systems. We provide several sets of hypotheses on general systems which can be used to rule out the existence of T4 configurations in the constitutive set K. In particular, our results show the nonexistence of T4 configurations for every well-known 2×2 hyperbolic system of conservation laws for which both families of shocks verify the Liu entropy condition.
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