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

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.


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 lim →0 (, ) = 0, where  is a sufficiently small positive parameter.Meanwhile, the pressure  satisfies the following equation of state: where  > 0,  > 0 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 Λ cold dark matter (ΛCDM) model mentioned in [2][3][4][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][8][9][10][11][12].
If  = 0, 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  = 0,  = 1 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][18][19][20][21][22][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 2 Advances in Mathematical Physics 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.
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).

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.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:

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].
(2) For  − <  + , the structure of Riemann solution is (3) For  − =  + , the structure of Riemann solution is

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  − ≤  + .

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)).
Lemma 2. As  − >  + and  As the pressure vanishes, from the second equation of ( 24)- (25), the limits of the speeds of   1 and   2 are as follows: lim (b) The (, ) plane Equation ( 27)- (28) shows the two shock waves   1 and   2 are merged into one shock wave with speed   () (see Figure 2(b)).By using the first equation of Rankine-Hugoniot conditions (10) for both   1 and   2 , we find Considering ( 28)-( 29 From above analysis and Lemma 2, we have Theorem 3. Theorem 3.For  − >  + , as  → 0, 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  − ≤  + .
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.

The Formation of Discontinuity as the Pressure Vanishes.
In this subsection, we are concerned with the formation of discontinuity for  − ≤  + , as  → 0. Our argument is divided into three parts.
Lemma 4 shows that the phenomenon of vacuum occurs as  − <  + and  → 0. Next, we introduce the formation of discontinuity for  − <  + .

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.