Advances in Mathematical Physics

Volume 2018, Article ID 3174719, 8 pages

https://doi.org/10.1155/2018/3174719

## The Convergence of Riemann Solutions to the Modified Chaplygin Gas Equations with a Coulomb-Like Friction Term as the Pressure Vanishes

^{1}College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China^{2}Department of Mathematics, Zhejiang University of Science & Technology, Hangzhou, Zhejiang 310023, China

Correspondence should be addressed to Lihui Guo; moc.621@oughil

Received 28 May 2018; Accepted 17 July 2018; Published 1 August 2018

Academic Editor: Claudio Dappiaggi

Copyright © 2018 Yongqiang Fan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper studies the convergence of Riemann solutions to the inhomogeneous modified Chaplygin gas equations as the pressure vanishes. The delta shock waves and vacuum states occur as the pressure vanishes. The Riemann solutions of inhomogeneous modified Chaplygin gas equations are no longer self-similar. It is obviously different from the Riemann solutions of homogeneous modified Chaplygin gas equations. When the pressure vanishes, the Riemann solutions of the modified Chaplygin gas equations with a coulomb-like friction term converge to the Riemann solutions of the pressureless Euler system with a source term.

#### 1. Introduction

The inhomogeneous modified Chaplygin gas equations have the following form:where is a constant and , denote the density and the velocity, respectively. The scalar pressure is the modified Chaplygin gas pressure satisfying , where is a sufficiently small positive parameter. Meanwhile, the pressure satisfies the following equation of state:where , are two positive constants. Modified Chaplygin gas (MCG) model was proposed in [1] by Benaoum in 2002. MCG [2, 3] represents the evaluation of the cosmology starting from the radiation era to the cold dark matter (CDM) model mentioned in [2–5]. As an exotic fluid, the MCG plays an important role in describing the accelerated expansion of the universe. In recent years, some researchers made some studies on the thermal equation of state to MCG and found that it could cool down in some constraints of parameters [6]. To know more interesting results related to MCG, the readers are referred to [7–12].

If , the system (1) becomes the homogeneous modified Chaplygin gas equations. In [13], Yang and Wang considered the formation of delta shock waves and the vacuum states in the solutions of the homogeneous isentropic Euler equations for modified Chaplygin gas when the pressure vanishes.

Letting , in (2), the equation of state is Chaplygin gas which was introduced by Chaplygin [14] in 1904, Tsien [15], and von Karman [16] as a suitable mathematical approximation for calculating the lifting force on a wing of an airplane in aerodynamics. The Chaplygin gas can be used to describe the dark energy. For the Riemann problem of homogeneous Chaplygin gas equations, there are lots of results. We refer the readers to [17–23]. For inhomogeneous Chaplygin gas equations, Shen [24] studied Riemann problem by introducing a new velocity:which was introduced by Faccanoni and Mangeney in [25]. In 2016, Sun [26] studied the non-self-similar Riemann solution of inhomogeneous generalized Chaplygin gas equations. Guo, Li, Pan, and Han [27] considered the Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations with a source term.

As , system (1)-(2) becomes the inhomogeneous pressureless Euler system:Shen [28] considered the Riemann problem of pressureless Euler system (4). Daw and Nedeljkov [29] studied the shadow waves for pressureless gas balance laws.

In this paper, we are concerned with the convergence of the Riemann solutions to the inhomogeneous modified Chaplygin gas equations as the pressure vanishes. Firstly, we give the Riemann solutions of the inhomogeneous modified Chaplygin gas equations. Then, we study the convergence of the Riemann solutions to the modified Chaplygin gas equations with a source term as the pressure vanishes. We find that the Riemann solutions of inhomogeneous modified Chaplygin gas equations converge to the corresponding Riemann solutions of the pressureless Euler system. We mainly use the method of vanishing pressure limits which was introduced by Li [30] and Chen and Liu [31, 32] in which they studied the formation of delta shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations and the concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids. Shen [33] considered the limits of Riemann solutions to the isentropic magnetogasdynamics. Shen and Sun [34] studied the formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model. Sheng, Wang, and Yin [35] studied the vanishing pressure limit of the generalized Chaplygin gas dynamics system. Yin and Sheng [36] considered the delta shocks and vacuum states in vanishing pressure limit of solutions to the relativistic Euler equations for polytropic gases. For inhomogeneous equations, Guo, Li, and Yin [37, 38] considered the vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term and the limit behavior of the Riemann solutions to the generalized Chaplygin gas equations with a source term.

We organize this article as follows: in Section 2, we give some preliminaries which include the consideration of Riemann solutions to system (1)-(2) and the review of Riemann solutions to system (4). In Section 3, we study the convergence of Riemann solutions to system (1)-(2).

#### 2. Some Preliminaries

In this section, we consider the Riemann solutions of the inhomogeneous modified Chaplygin gas equations and briefly review the Riemann solutions of pressureless Euler system with a coulomb-like friction term.

##### 2.1. Riemann Problem for (1)-(2)

In this subsection, we are concerned with the Riemann problem of (1)-(2).

From (3), system (1)-(2) is turned into the following conservation laws:We consider the Riemann initial value as follows:where and are given constants. Letting , the two eigenvalues of system (5) are , . For arbitrary positive constants , system (5) is strictly hyperbolic. The corresponding right eigenvectors are , . By simple calculation, we obtain that , , which implies that both the characteristic fields are genuinely nonlinear.

Given a state in the phase plane, the curve of backward rarefaction wave isand the corresponding curve of forward rarefaction wave iswhere Denote the propagating speed of the bounded discontinuity as . The Rankine-Hugoniot conditions readFor convenience, we write . Then for a given state , from (10), we obtain two kinds of shock wave curves, i.e., the backward shock wave curve,and the forward shock wave curve,

In the phase plane, given a state , the curves of and , divide the phase plane into four regions.

When (see Figure 1), the Riemann solutions of system (5) are(1);(2);(3);(4),