论文标题
$ t_4 $配置的非含量为双曲线系统和LIU熵条件
Nonexistence of $T_4$ configurations for hyperbolic systems and the Liu entropy condition
论文作者
论文摘要
我们研究由$ 2 \ times 2 $在一个空间维度中的$ 2 \ times 2 $的保护法系统产生的本构集$ \ Mathcal {k} $，并带有一个熵和熵 - 频率对。集合$ \ MATHCAL {k} $的凸度属性与基础系统的适合度有关，以及通过凸集成构建解决方案的能力。与$ \ Mathcal {k} $的凸度有关，在$ p $ SYSTEM，LORENT和PENG的特殊情况下[Calc。 var。部分微分方程，59（5）：纸编号156、36、2020]表明$ \ MATHCAL {K} $不包含$ T_4 $配置。最近，Johansson和Tione [Arxiv e-prints，Page arXiv：2208.10979，2022年8月]显示$ \ Mathcal {K} $不包含$ T_5 $配置。 在本文中，我们基于对大型$ 2 \ times 2 $系统的冲击曲线的仔细分析，对这些结果进行了实质性的概括。特别是，我们在通用系统上提供了几组假设，这些假设可用于排除本构设置$ \ Mathcal {K k} $中$ T_4 $配置的存在。特别是，我们的结果表明，对于每一个众所周知的$ 2 \ times 2 $夸张的保护法律，$ T_4 $配置不存在，该系统验证了LIU熵条件。
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.
