- Research
- Open Access
The asymptotic limits of Riemann solutions for the isentropic extended Chaplygin gas dynamic system with the vanishing pressure
- Meizi Tong^{1},
- Chun Shen^{1}Email author and
- Xiuli Lin^{2}
- Received: 24 July 2018
- Accepted: 12 September 2018
- Published: 24 September 2018
Abstract
We construct the solutions to the Riemann problem for the isentropic extended Chaplygin gas dynamic system for all kinds of situations by the phase plane analysis. We investigate the asymptotic limits of solutions to this problem in detail when the pressure given by the state equation of the system becomes the one of pressureless gas. During the process of vanishing pressure, the two-shock Riemann solution tends to a delta shock solution, whereas the two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution with a vacuum state.
Keywords
- Extended Chaplygin gas
- Pressureless gas
- Delta shock wave
- Vacuum state
- Riemann problem
MSC
- 35L65
- 35L67
- 76N15
1 Introduction
The Chaplygin gas model was first proposed by Chaplygin [1] as a model in aerodynamics with the equation of state in the form \(p=-B/\rho\) with constant \(B>0\). It was discovered in [2] that the equation of state for the Chaplygin gas was very suitable to describe the dark energy in the universe from the viewpoint of string theory. To be in more agreement with observational data, it was extended to the generalized Chaplygin gas [3] with \(p=-B/\rho^{\alpha}\) and the modified Chaplygin gas [4] with \(p=A \rho-B/\rho^{\alpha}\), where \(A,B>0\) and \(0<\alpha\leq1\). The equation of state for the modified Chaplygin gas contains two terms; the first term gives an ordinary fluid obeying a linear barotropic equation of state, and the second one is related to some power of the inverse of energy density. However, there are other barotropic fluids with quadratic or higher-order equation of state. To overcome this drawback, the modified Chaplygin gas has been further generalized by Pourhassan and Kahya [5] to the extended Chaplygin gas with the equation of state in the form \(p= \sum_{k=1} ^{n} A_{k}\rho^{k} -B/\rho^{\alpha}\). It is clear that all the Chaplygin gas models mentioned are particular cases of the extended Chaplygin gas model; see [6–9] for the related studies of the extended Chaplygin gas model.
The formation of δ-shock wave and vacuum state to the Riemann problem (1.2) and (1.3) was considered initially for the isothermal [21] and isentropic [10] situations by the vanishing pressure limit approach. The result was further extended to the generalized pressureless gas dynamics model in [22], the isentropic magneto-gas-dynamics model in [23], and the Aw–Rascle model in [24]. Recently, the limiting relations of Riemann solutions from a variety of Chaplygin gas dynamic systems to the pressureless gas dynamic system have been extensively studied. More precisely, Sheng, Wang, and Yin [25] considered the asymptotic limits \(B\rightarrow0\) of Riemann solutions for the generalized Chaplygin gas \(p=-B/\rho^{\alpha}\). Furthermore, Chen and Sheng [26] have made a step further by considering the isentropic magneto-gas-dynamics Euler system for the generalized Chaplygin gas as the vanishing magnetic field. Yang and Wang made a step further by considering the modified Chaplygin gas \(p=A \rho-B/\rho^{\alpha}\) with \(\alpha =1\) in [27] and \(0<\alpha<1\) in [28]. The vanishing pressure limit problem for the Chaplygin gas \(p=-B/\rho\) and the generalized Chaplygin gas \(p=-B/\rho^{\alpha}\) with source term [29] has also been considered by Guo, Li, and Yin [30, 31]. Li and Shao [32] have generalized these results to the relativistic Euler system with the generalized Chaplygin gas \(p=-B/\rho^{\alpha}\). Also see [33–35] for the other related study about the formation of δ-shock wave. In this work, we make a step further by extending these results to the relatively more general pressure term \(p= \sum_{k=1} ^{n} A_{k}\rho^{k} -B/\rho\). In a recent paper [36], we have also considered the limiting relations of Riemann solutions from the isentropic extended Chaplygin gas dynamic system to the isentropic Chaplygin gas dynamic system when only the limit \(A_{k}\ (k=1,\ldots,n) \rightarrow0\) is taken in system (1.1). We also refer to [37–43] for other related works for the isentropic Chaplygin gas dynamic system.
The arrangement of this paper is as follows. In Sect. 2, we simply restate the solutions to the Riemann problem for the pressureless gas dynamic system (1.2) for self-consistency. In Sect. 3, we investigate in detail the basic properties of the isentropic extended Chaplygin gas dynamic system (1.1) and consequently construct the solutions to the Riemann problem (1.1) and (1.3) for all kinds of Riemann initial data. In Sect. 4, we make a step further by investigating the limit relations of Riemann solutions from the isentropic extended Chaplygin gas dynamic system (1.1) to the pressureless gas dynamic system (1.2) as the limit \(A_{k}\) (\(k=1,\ldots,n\)), \(B\rightarrow0\) is taken. The formation of δ-shock wave and vacuum state can be observed during the process of vanishing pressure limit.
2 Preliminaries
Definition 2.1
For completeness, we give a more general definition of δ-shock wave solution suggested by Danilov et al. [45, 46]. Suppose that \(\Gamma=\{\gamma_{i}|i\in I\}\) is a graph in the upper half-plane with the requirement \((x,t)\in R \times R_{+}\), which is comprised of Lipschitz continuous curves \(\gamma_{i}\) for \(i \in I\), where I is a finite index set. Subsequently, \(I_{0}\) is a subset of I such that the curves \(\gamma _{i}\) for \(i \in I_{0}\) start from the points on the x-axis. By \(\Gamma_{0}=\{x_{k}^{0}|k\in I_{0}\}\) we denote the set of initial points of \(\gamma_{k}\) for \(k\in I_{0}\). As is shown further, the solutions of the initial value problem for the pressureless gas dynamic system (1.2) with the δ-measure initial data are defined in the distributional sense.
Definition 2.2
3 The Riemann problem for the isentropic extended Chaplygin gas dynamic system (1.1)
4 The limits of Riemann solutions from system (1.1) to system (1.2) as \(A_{k}\) (\(k=1,\ldots,n\)), \(B \rightarrow0\)
In this section, we focus ourselves on the asymptotic limits of solutions to the Riemann problem (1.1) and (1.3) as all the parameters \(A_{k}\) (\(k=1,\ldots,n\)) and B tend to zero. Let the left state \((u_{-},\rho_{-})\) be fixed. If the limit \(A_{k}\) (\(k=1,\ldots,n\)), \(B\rightarrow0\) is taken, then it can be seen from (3.9), (3.10), (3.25), and (3.26) that all the wave curves \(R_{1}(u_{-},\rho_{-})\), \(R_{2}(u_{-},\rho_{-})\), \(S_{1}(u_{-},\rho _{-})\), and \(S_{2}(u_{-},\rho_{-})\) tend to the line \(u=u_{-}\) in the half-upper \((u,\rho)\) phase space. We further observe that the regions II and III disappear in the limit \(A_{k}\) (\(k=1,\ldots,n\)), \(B\rightarrow0\) situation. Hence, if \(u_{+}< u_{-}\), then the right state \((u_{+},\rho_{+})\) is located in the region IV for sufficiently small \(A_{k}\) (\(k=1,\ldots,n\)) and B. Otherwise, if \(u_{+}>u_{-}\), then the right state \((u_{+},\rho _{+})\) is located in the region I for sufficiently small \(A_{k}\) (\(k=1,\ldots,n\)) and B. In the special situation, if \(u_{+}=u_{-}\), then the right state \((u_{+},\rho_{+})\) is located in the region II for \(0\leq\rho_{+}<\rho_{-}\) or in the region III for \(\rho_{+}>\rho_{-}\) when all the parameters \(A_{k}\) (\(k=1,\ldots,n\)) and B are sufficiently small.
In what follows, we give some lemmas to show the limiting behaviors of the solutions to the Riemann problem (1.1) and (1.3) consisting of two shock waves for the situation \(u_{+}< u_{-}\) when \(A_{k}\) (\(k=1,\ldots,n\)), \(B\rightarrow0\).
Lemma 4.1
Proof
Moreover, we have the following limiting relations of mass and momentum in the limit \(A_{k}\) (\(k=1,\ldots,n\)), \(B\rightarrow0\) situation, which can be described by the following lemma.
Lemma 4.2
Proof
Remark 4.1
It can be concluded from the lemmas that if \(u_{+}< u_{-}\), then the shock waves \(S_{1}\) and \(S_{2}\) coincide with each other, and the intermediate density \(\rho_{*}\) becomes the singular measure of Dirac mass for the solution to the Riemann problem (1.1) and (1.3) in the limit \(A_{k}\) (\(k=1,\ldots,n\)), \(B\rightarrow0\) situation, which is just the δ-shock wave solution (2.6) associated with (2.7) to the Riemann problem for the pressureless gas dynamic system (1.2) with the same Riemann initial data (1.3) as in Sect. 2.
In what follows, for the situation \(u_{+}< u_{-}\), we use the following theorem to describe the formation of singularity of solution in the sense of distributions, which is similar to the result of Theorem 3.1 in [10].
Theorem 4.3
Proof
Theorem 4.4
When \(u_{-}< u_{+}\), if the solution to the Riemann problem (1.1) and (1.3) consists of two rarefaction waves for sufficiently small \(A_{k}\) (\(k=1,\ldots,n\)) and B, then the limit of solution as \(A_{k}\) (\(k=1,\ldots,n\)), \(B\rightarrow0\) is a two-contact-discontinuity solution with a vacuum state of the form (2.1), which is identical with that for the pressureless gas dynamic system (1.2).
Proof
Finally, we are dedicating to investigating the special situation \(u_{-}=u_{+}\), in which the formation of contact discontinuity may be illustrated well by using the following theorem.
Theorem 4.5
When \(u_{-}=u_{+}\), the limit of solution to the Riemann problem (1.1) and (1.3) as \(A_{k}\) (\(k=1,\ldots,n\)), \(B\rightarrow0\) is only a contact discontinuity connecting the two constant states \((u_{-},\rho_{-})\) and \((u_{+},\rho_{+})\) directly, which is identical with that for the pressureless gas dynamic system (1.2).
Proof
Declarations
Acknowledgements
The authors are very grateful to two anonymous referees for critical comments, which greatly improved the presentation of the manuscript.
Availability of data and materials
Not applicable.
Funding
This work is partially supported by Natural Science Foundation of China (11441002), Shandong Provincial Natural Science Foundation (ZR2014AM024), and STPF of Shandong Province (J17KA161).
Authors’ contributions
The authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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