Concentration Phenomenon of Riemann Solutions for the Relativistic Euler Equations with the Extended Chaplygin Gas
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Concentration Phenomenon of Riemann Solutions for the Relativistic Euler Equations with the Extended Chaplygin Gas Yunfeng Zhang1 · Meina Sun1
Received: 8 June 2019 / Accepted: 22 June 2020 © Springer Nature B.V. 2020
Abstract The solutions of the Riemann problem for the isentropic relativistic Euler equations with the extended Chaplygin gas are constructed completely for all the possible cases. The asymptotic limits of solutions to the Riemann problem for the relativistic Euler equations are captured in detail when the equation of state of extended Chaplygin gas becomes the one of Chaplygin gas. It is shown that the formations of delta shock wave solution and two-contact-discontinuity solution are derived and analyzed rigorously during the limiting process. Keywords Isentropic relativistic Euler equations · Extended Chaplygin gas · Delta shock wave · Riemann problem Mathematics Subject Classification (2010) 35L65 · 35L67 · 76N15
1 Introduction The relativistic fluid dynamics equations are well-known in various astrophysical phenomena. At present, there exists an amazing amount of physical literature in diverse relativistic fluid dynamics models. In fact, only a few mathematical results of relativistic fluid dynamics equations have been presented such as in [1–4] because the structures of models are very complicated and difficult to be dealt with. The present paper is devoted to the study of the isentropic relativistic Euler equations consisting of conservation laws of energy and momentum in special relativity given by [4, 5] ⎧ 2 v2 v ⎨ ( p(ρ)+ρc + ρ)t + ((p(ρ) + ρc2 ) c2 −v 2 )x = 0, c2 c2 −v 2 (1.1) ⎩ ((p(ρ) + ρc2 ) v ) + ((p(ρ) + ρc2 ) v2 + p(ρ)) = 0, x c2 −v 2 t c2 −v 2 This work is partially supported by Shandong Provincial Natural Science Foundation (ZR2019MA019).
B M. Sun
[email protected]
1
School of Mathematics and Statistics Science, Ludong University, Yantai, Shandong Province, 264025, P.R. China
Y. Zhang, M. Sun
in which ρ and v are the density of proper-energy and the speed of particle respectively as well as the constant c stands for the speed of light. Hereafter the equation of state (abbreviated by EoS) p(ρ) is given by the two-order form of the extended Chaplygin gas [6, 7] as follows: B (1.2) p(ρ) = A1 ρ + A2 ρ 2 − , ρ where A1 , A2 and B are positive parameters. √It is evident that the condition A1 + 2A2 ρ + B 2 < c must hold because the sound speed p (ρ) is less than the light speed c. The more ρ2 general extended Chaplygin gas with EoS in the form p(ρ) = nk=1 Ak ρ k − ρBα has been recently introduced by Pourhassan and Kahya [6], which recovers some other known Chaplygin gases by choosing the parameters Ak (k = 1, . . . , n) and α suitably, such as p(ρ) = − Bρ for the Chaplygin gas, p(ρ) = − ρBα for the generalized Chaplygin gas and p(ρ) = Aρ − ρBα for the modified Chaplygin gas. To comprehend the formation mechanism of singularities, as is well known, it is essential to get explicit solutions. In this article, we take the Riemann-type initial condition for Eqs. (1.1)–(1.2) given
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