Limits of Riemann Solutions to the Relativistic Euler Systems for Chaplygin Gas as Pressure Vanishes

Abstract

Vanishing pressure limits of Riemann solutions to relativistic Euler system for Chaplygin gas are identified and analyzed in detail. Unlike the polytropic or barotropic gas case, as the parameter decreases to a critical value, the two-shock solution converges firstly to a delta shock wave solution to the same system. It is shown that, as the parameter decreases, the strength of the delta shock increases. Then as the pressure vanishes ultimately, the solution is nothing but the delta shock wave solution to the zero pressure relativistic Euler system. Meanwhile, the two-rarefaction wave solution and the solution containing one-rarefaction wave and one-shock wave tend to the vacuum solution and the contact discontinuity solution to the zero pressure relativistic Euler system, respectively.

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

“…Further, they also obtained the same results for the Euler equations for nonisentropic fluids in [10]. See Yin and Sheng [38] the result of the relativistic Euler equations for polytropic gas, Yin and Song [39,40] for Chaplygin gas, Sheng, Wang and Yin [28] for the isentropic Euler equations for the generalized Chaplygin gas, Yang and Wang [37] for modified Chaplygin gas, Shen [26] for the isentropic magnetogasdynamics equations, Shen and Sun [ 27] for the perturbed Aw-Rascle model, Mitrovic and Nedeljkov [21] for the generalized pressureless gas dynamics model with a scaled pressure term, etc.…”

“…The typical feature of the Chaplygin gas dynamics is that the δ-shocks appear in non-zero pressure cases. The Riemann problem was solved for the nonrelativistic case by Brenier [4], relativistic case by Cheng and Yang [11], followed by its vanishing pressure limit problem by Yin and Song [40].…”

“…For the Chaplygin gas [40], both the characteristic fields are linearly degenerate, thus the classical elementary waves only involves contact discontinuities. The rarefaction wave curves and the shock wave curves are actually coincided to the socalled contact discontinuities, and the δ-shocks appear in non-zero pressure cases.…”

“…Unlike the polytropic gas case, considering the unbounded singularity in Riemann solutions, the limits of Riemann solutions here are expected to behave some difference and more interesting points during the process of vanishing pressure limit. The mathematical difficulties caused by this fact as compared to the Chaplygin gas case by Yin and Song [40] or their later work [39] are that the shock wave curves are given by…”

“…However, for the Chaplygin gas case (α = 1 in (1.2)) by Yin and Song [40], the shock wave curves are given by the following explicit formulas…”