In this article, we study some fundamental properties of nonlinear waves and the Riemann problem of Euler’s relativistic system when the constitutive equation for energy is that of Synge for a monatomic rarefied gas or its generalization for diatomic gas. These constitutive equations are the only ones compatible with the relativistic kinetic theory for massive particles in the whole range from the classical to the ultra-relativistic regime. They involve modified Bessel functions of the second kind and this makes Euler’s relativistic system rather complex. Based on delicate estimates of the Bessel functions, we prove: (i) a limit on the speed of sound of $$1{/}\sqrt{3}$$ 1 / 3 times the speed of light (which a fortiori implies subluminality, that is causality), (ii) the genuine non-linearity of the acoustic waves, (iii) the compatibility of Rankine–Hugoniot relations with the second law of thermodynamics (entropy growth through all Lax shocks), and (iv) the unique resolvability of the initial value problem of Riemann (if we include the possibility of vacuum as in the non-relativistic context).

中文翻译：

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具有合成能的大质量粒子相对论气体中的非线性双曲波

在本文中，我们研究了非线性波的一些基本性质和欧拉相对论系统的黎曼问题，当能量的本构方程是单原子稀薄气体的 Synge 方程或其对双原子气体的推广时。这些本构方程是唯一与大质量粒子在从经典到超相对论体系的整个范围内的相对论动力学理论兼容的方程。它们涉及第二类修正贝塞尔函数，这使得欧拉的相对论系统相当复杂。基于对贝塞尔函数的精细估计，我们证明：(i) 声速的极限为 $$1{/}\sqrt{3}$$ 1 / 3 倍光速（更何况这意味着亚光速，这是因果关系），（ii）声波的真正非线性，