Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations

Abstract

The Riemann problems for the Euler system of conservation laws of energy and momentum in special relativity as pressure vanishes are considered. The Riemann solutions for the pressureless relativistic Euler equations are obtained constructively. There are two kinds of solutions, the one involves delta shock wave and the other involves vacuum. The authors prove that these two kinds of solutions are the limits of the solutions as pressure vanishes in the Euler system of conservation laws of energy and momentum in special relativity.

Delta wave formation and vacuum state in vanishing pressure limit for system of conservation laws to relativistic fluid dynamics

“…System (1.2) in the Newtonian limit reduces to the transport equations in gas dynamics, to which Sheng and Zhang [23] solved the Riemann problem constructively and showed that there are only two kinds of solutions, the one involves delta wave, and the other involves vacuum. The same conclusions have been verified by the authors for system (1.2) [32]. The study on pressure vanishing limit was started from Li [13] in 2001, and then Chen and Liu [3,4] in 2003, 2004.…”

Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for polytropic gases

“…In [34], in the case v − < v + , the authors obtained a solution (ρ, v)(ξ ) consisting of two contact discontinuities and a vacuum state, besides two constant states (ρ ± , v ± ), which can be written as this form…”

“…In this paper, we focus on the prototypical pressure function for polytropic gases: P (ρ, ε) = εp(ρ), p(ρ) = ρ γ , (2) where γ > 1, and the sound speed p (ρ) is less than the light speed c. For the isothermal gases γ = 1, the results have been obtained in [34].…”

Non-relativistic global limits of entropy solutions to the extremely relativistic Euler equations

“…Due to the high complexity of the system itself, current efforts have been made mainly to the corresponding reduced 2 × 2 systems, involving either the conserved equations of baryon numbers and momentum or the conserved equations of momentum and energy(see [2][3][4][5]7,8,[10][11][12][13][14][15]19,[21][22][23][24][25]27,35,36] etc., and the references cited therein).…”