双曲系统的 $T_4$ 构型不存在性与 Liu 熵条件
Nonexistence of $T_4$ configurations for hyperbolic systems and the Liu entropy condition
AI总结：
本文通过分析一大类 $2\times 2$ 守恒律系统的激波曲线，给出排除本构集中 $T_4$ 构型的若干一般性假设，从而将 Lorent-Peng 及 Johansson-Tione 的结果推广到所有满足 Liu 熵条件的著名 $2\times 2$ 双曲系统。
摘要：
我们研究由一维空间中一个 $2\times 2$ 守恒律系统所产生的本构集 $\mathcal{K}$，该系统配有一个熵和熵流对。集合 $\mathcal{K}$ 的凸性性质与底层系统的适定性以及通过凸积分构造解的能力有关。关于 $\mathcal{K}$ 的凸性，在 $p$-系统的特殊情形下，Lorent 和 Peng [Calc. Var. Partial Differential Equations, 59(5):Paper No. 156, 36, 2020] 证明了 $\mathcal{K}$ 不包含 $T_4$ 构型。最近，Johansson 和 Tione [arXiv e-prints, page arXiv:2208.10979, August 2022] 证明了 $\mathcal{K}$ 不包含 $T_5$ 构型。在本文中，我们基于对一大类 $2\times 2$ 系统的激波曲线的细致分析，对这些结果给出了实质性的推广。特别地，我们给出了关于一般系统的若干组假设，这些假设可用于排除本构集 $\mathcal{K}$ 中 $T_4$ 构型的存在性。特别地，我们的结果表明，对于每一个满足 Liu 熵条件的著名 $2\times 2$ 双曲守恒律系统，$T_4$ 构型都不存在。
英文摘要：
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.
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