Nonlinear Hyperbolic Waves in Relativistic Gases of Massive Particles with Synge Energy
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Nonlinear Hyperbolic Waves in Relativistic Gases of Massive Particles with Synge Energy Tommaso Ruggeri , Qinghua Xiao & Huijiang Zhao Communicated by T.-P. Liu
Abstract 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/ 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).
1. Introduction The aim of this paper is to consider the properties of nonlinear waves and the Riemann problem of the Euler system in the relativistic framework. Let V α and T αβ be the particle-particle flux and energy-momentum tensor, respectively [1–5], such that V α := ρu α , T αβ := ph αβ +
e α β u u . c2
(1)
Then, the field equations for a relativistic single fluid are the conservation of particle numbers and energy-momentum tensor in Minkowski space: ∂α V α = 0,
∂α T αβ = 0.
(2)
T. Ruggeri, Q. Xiao & H. Zhao
Here, ρ = nm is the density, n is the particle number, m is the mass in rest frame, u α ≡ (u 0 = c, u i = v i ) is the four-velocity vector, = 1/ 1 − v 2 /c2 is the Lorentz factor, v i is the velocity, h αβ = u α u β /c2 − g αβ is the projector tensor, g αβ is the metric tensor with signature (+, −, −, −), p is the pressure, e = ρ(c2 + ε)
(3)
is the energy–that is, the sum of internal energy (ε is the internal energy density) and the energy in the rest frame, c is the speed of light; ∂α = ∂/∂ x α , x α ≡ (x 0 = ct, x i ) are the space-time coordinates and the greek indices run from 0 to 4 while the Latin indices from 1 to 3 and, as usual, pairs of lower/upper indices indicate summation. For two dimensional space-time case, the system (2) with (1) is expressed as ρc ρcv + ∂x √ = 0, ∂t √ c2 − v 2 c2 − v 2 (e + p)v (e + p)v 2 ∂t + ∂x + p = 0, c2 − v 2 c2 − v 2 (e + p)v 2 (e + p)c2 v ∂t + e + ∂ = 0. (4) x c2 − v 2 c2 − v 2 We need the constitutive equation p ≡ p(ρ, e) to close the system (4). This is usually obtained, in parametric form, through the thermal and caloric equation of state p ≡ p(ρ, T ),
e ≡ e(ρ, T ),
(5)
where T is the temperature.1 Corresponding to (4), for the classical Euler system, there have been enormous
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