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),
where denotes the intermediate state. By using (3), we obtain the Riemann solutions of (1)-(2) as follows:(1);(2);(3);(4)
2.2. Riemann Problem for (4)
In this subsection, we restate the results on the Riemann solutions to system (4). The detailed results can be referred to in [28].
(1) For , the Riemann solution of system (4) has the following form:in which the delta shock wave is introduced into the Riemann solution (13)-(14). About the definition of delta shock wave, please refer to [31, 34, 39, 40], for example. The delta shock wave satisfies the generalized Rankine-Hugoniot conditions:where , , and , respectively, denote the location, weight, and the velocity of delta shock wave, and .
Letting , by simple calculation, we obtainfor andfor .
(2) For , the structure of Riemann solution is
(3) For , the structure of Riemann solution is
3. Convergence of Riemann Solutions to (1)-(2)
In this section, we consider the convergence of the Riemann solutions to system (1)-(2) in the vanishing pressure. We divide our discussions into three parts: the concentration and delta shock wave for , the occurrence of vacuum for , and the formation of discontinuity for .
3.1. Concentration and Delta Shock Wave as the Pressure Vanishes
In this subcase, we show the phenomenon of the concentration and delta shock wave in the vanishing pressure limit of Riemann solutions to (1)-(2) when (see Figure 2(a)).
(a) The phase plane
(b) The plane
Lemma 1. Assume , then there exists certain value , such that when .
Proof. If and , we obtain (see Figure 2(a)):From (20)-(21) and (2), we haveFrom (22), we derive thatLet , then when and . It is obvious that this conclusion is also true for , The proof is completed.
For , the Riemann solution of (5)-(6) contains two shock waves satisfyingwhere is the intermediate state (see Figure 2(a)). From the first equation of (24)-(25), we obtainFrom (26), we can derive that and From the above analysis, we can obtain Lemma 2 as follows.
Lemma 2. As and , and , i.e., the phenomenon of concentration occurs.
Next, we consider the formation of delta shock waves. Firstly, we analyze the velocity of intermediate state . It follows from (3), (24) and Lemma 2 thatAs the pressure vanishes, from the second equation of (24)-(25), the limits of the speeds of and are as follows:Equation (27)-(28) shows the two shock waves and are merged into one shock wave with speed (see Figure 2(b)).
By using the first equation of Rankine-Hugoniot conditions (10) for both and , we findConsidering (28)-(29), we obtainFrom (30), we obtain thatFrom above analysis and Lemma 2, we have Theorem 3.
Theorem 3. For , as , the two shock waves of system (1)-(2) converge to a delta shock solution which is the corresponding Riemann solution of system (4).
We have considered the convergence of the Riemann solutions to system (1)-(2) for as the pressure vanishes. In the following, we study the convergence of the Riemann solutions to system (1)-(2) for .
3.2. The Occurrence of Vacuum as the Pressure Vanishes
In this subcase, as the pressure vanishes, we study the occurrence of vacuum when and (see Figure 3(a)).
(a) The phase plane
(b) The plane
Suppose that , from (7)-(8) and (9), one can derive thatAccording to (32)-(33), we obtain thatwhere . Letting , the right state as . Next, we analyze the formation of vacuum when .
Lemma 4. As and , , i.e., the phenomenon of vacuum occurs.
Proof. From the above analysis, there exists a constant , such that when . From (7)-(8), we obtain that on and on , which derive thatIf , as tends to zero, from (35), we have , which leads to a contradiction. Therefore, , which means the vacuum occurs as the pressure vanishes.
In Sections 3.1 and 3.2, we discussed the concentration and delta shock wave and the occurrence of vacuum. In the following, we consider the formation of discontinuity.
3.3. The Formation of Discontinuity as the Pressure Vanishes
In this subsection, we are concerned with the formation of discontinuity for , as . Our argument is divided into three parts.
Case 1. for and .
Lemma 4 shows that the phenomenon of vacuum occurs as and . Next, we introduce the formation of discontinuity for .
When , from (7)-(8), we haveandIt follows from (36)-(37) that the two rarefaction waves and turn into two contact discontinuities as the pressure vanishes for (see Figure 3(b)). We summarize our conclusion as follows.
Theorem 5. For , as , the two rarefaction waves of system (1)-(2) converge to two contact discontinuities which are the corresponding Riemann solution of system (4).
Case 2. for and .
We have studied the convergence of the Riemann solution to system (1)-(2) for and as the pressure vanishes. Next, we discuss the situation for . We divide our discussion into two parts for (see Figure 4(a)) and (see Figure 5(a)).
(a) The phase plane
(b) The plane
(a) The phase plane
(b) The plane
When , from (7) and (25), we deduce thatwhich implies that the rarefaction wave and the shock wave become a contact discontinuity as the pressure vanishes (see Figure 4(b)).
Similarly, from (8) and (24), we know thatwhich means that the rarefaction wave and the shock wave turn into a contact discontinuity as (see Figure 5(b)).
We have considered the convergence of the Riemann solution of system (1)-(2) for , as the pressure vanishes. We summarize our main results as follows.
Theorem 6. For , as , the Riemann solutions of system (1)-(2) converge to a contact discontinuity which is the corresponding Riemann solution of system (4).
4. Conclusion
We have considered the convergence of the Riemann solutions to the modified Chaplygin gas equations with a coulomb-like friction term. As the pressure vanishes, the concentration and delta shock wave are concerned. Meanwhile the occurrence of vacuum and formation of discontinuity are studied. We find that the Riemann solutions of (1)-(2) converge to the corresponding Riemann solutions of (4) as the pressure vanishes.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is partially supported by the Natural Science Foundation of Xinjiang (Grant no. 2017D01C053).