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

1. Introduction

The inhomogeneous modified Chaplygin gas equations have the following form:

[[rho].sub.t] + [([rho]u).sub.x] = 0,

[([rho]u).sub.t] + [([rho][u.sup.2] + P).sub.x] = [beta][rho], (1)

where [beta] is a constant and [rho], u denote the density and the velocity, respectively. The scalar pressure P = P([rho], [alpha]) is the modified Chaplygin gas pressure P = [alpha]p satisfying [lim.sub.[alpha][right arrow]0]]P([rho], [alpha]) = 0, where [alpha] is a sufficiently small positive parameter. Meanwhile, the pressure p satisfies the following equation of state:

p = A[rho] - B/[rho], (2)

where A > 0, B > 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 A cold dark matter (ACDM) 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 [beta] = 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 A = 0, B = 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 [17-23]. For inhomogeneous Chaplygin gas equations, Shen [24] studied Riemann problem by introducing a new velocity:

v (x, t) = u(x, t) - [beta]t, (3)

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 [alpha] [right arrow] 0, system (1)-(2) becomes the inhomogeneous pressureless Euler system:

[[rho].sub.t] + [([rho]u).sub.x] = 0, [([rho]u).sub.t] + [([rho][u.sup.2]).sub.x] = [beta][rho]. (4)

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 (l)-(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:

[[rho].sub.t] + [([rho](v + [beta]t)).sub.x] = 0, [([rho]v).sub.t] + [([rho]v (v + [beta]t) + [alpha] (A[rho] - B/[rho])).sub.x] = 0. (5)

We consider the Riemann initial value as follows:

(v, [rho])(x, 0) = ([u.sub.[+ or -]], [[rho].sub.[+ or -]]), [+ or -] x [greater than or equal to] 0, (6)

where [[rho].sub.[+ or -]] [greater than or equal to] 0 and [u.sub.[+ or -]] are given constants. Letting f([rho], [alpha]) = [square root of ([alpha](A + B/[[rho].sup.2]))], the two eigenvalues of system (5) are [[lambda].sub.1] = v + [beta]t - f([rho], [alpha]), [[lambda].sub.2] = v + [beta]t + f([rho], [alpha]). For arbitrary positive constants A, B, system (5) is strictly hyperbolic. The corresponding right eigenvectors are [??].sub.1] = [(-[rho], f([rho], [alpha])).sup.T], [[??].sub.2] = [([rho], f([rho], [alpha])).sup.T]. By simple calculation, we obtain that [nabla][[lambda].sub.i] x [[??].sub.i] = A[alpha]/f([rho], [alpha]) [not equal to] 0, (i = 1, 2), which implies that both the characteristic fields are genuinely nonlinear.

Given a state ([u.sub.-], [[rho].sub.-]) in the phase plane, the curve of backward rarefaction wave is

[mathematical expression not reproducible] (7)

and the corresponding curve of forward rarefaction wave is

[mathematical expression not reproducible] (8)

where

[mathematical expression not reproducible] (9)

Denote the propagating speed of the bounded discontinuity x = x(t) as [sigma](t) = x'(t). The Rankine-Hugoniot conditions read

[mathematical expression not reproducible] (10)

For convenience, we write h([rho], [[rho].sub.-], [alpha]) = [square root of (([alpha]/[rho][[rho].sub.-])(A + B/[rho][[rho].sub.-]))]. Then for a given state ([u.sub.-], [[rho].sub.-]), from (10), we obtain two kinds of shock wave curves, i.e., the backward shock wave curve,

[mathematical expression not reproducible] (11)

and the forward shock wave curve,

[mathematical expression not reproducible] (12)

In the phase plane, given a state ([u.sub.-], [[rho].sub.-]), the curves of [R.sup.[alpha].sub.j]([u.sub.-], [[rho].sub.-]) and [S.sup.[alpha].sub.j]([u.sub.-], [[rho].sub.-]), (j = 1, 2) divide the phase plane into four regions.

When ([u.sub.+], [[rho].sub.+]) [member of] ([R.sup.[alpha].sub.1][R.sup.[alpha].sub.2] [union] [R.sup.[alpha].sub.1][S.sup.[alpha].sub.2] [union] [S.sup.[alpha].sub.1][R.sup.[alpha].sub.2] [union] [S.sup.[alpha].sub.1][S.sup.[alpha].sub.2])([u.sub.-], [[rho].sub.-]) (see Figure 1), the Riemann solutions of system (5) are

(1) ([u.sub.+], [[rho].sub.+]) [member of] [R.sup.[alpha].sub.1][R.sup.[alpha].sub.2] ([u.sub.-], [[rho].sub.-]) : ([u.sub.-], [[rho].sub.-]) + [R.sup.[alpha].sub.1] + ([v.sub.*[alpha]], [[rho].sub.*[alpha]]) + [R.sup.[alpha].sub.2] + ([u.sub.+], [[rho].sub.+]);

(2) ([u.sub.+], [[rho].sub.+]) [member of] [R.sup.[alpha].sub.1][S.sup.[alpha].sub.2] ([u.sub.-], [[rho].sub.-]) : ([u.sub.-], [[rho].sub.-]) + [R.sup.[alpha].sub.1] + ([v.sub.*[alpha]], [[rho].sub.*[alpha]]) + [S.sup.[alpha].sub.2] + ([u.sub.+], [[rho].sub.+]);

(3) ([u.sub.+], [[rho].sub.+]) [member of] [S.sup.[alpha].sub.1][R.sup.[alpha].sub.2] ([u.sub.-], [[rho].sub.-]) : ([u.sub.-], [[rho].sub.-]) + [S.sup.[alpha].sub.1] + ([v.sub.*[alpha]], [[rho].sub.*[alpha]]) + [R.sup.[alpha].sub.2] + ([u.sub.+], [[rho].sub.+]);

(4) ([u.sub.+], [[rho].sub.+]) [member of] [S.sup.[alpha].sub.1][S.sup.[alpha].sub.2] ([u.sub.-], [[rho].sub.-]) : ([u.sub.-], [[rho].sub.-]) + [S.sup.[alpha].sub.1] + ([v.sub.*[alpha]], [[rho].sub.*[alpha]]) + [S.sup.[alpha].sub.2] + ([u.sub.+], [[rho].sub.+]),

where ([v.sub.*[alpha]], [[rho].sub.*[alpha]]) denotes the intermediate state. By using (3), we obtain the Riemann solutions of (1)-(2) as follows:

(1) ([u.sub.+], [[rho].sub.+]) [member of] [R.sup.[alpha].sub.1][R.sup.[alpha].sub.2]([u.sub.-], [[rho].sub.-]) : ([u.sub.-] + [beta]t, [[rho].sub.-]) + [R.sup.[alpha].sub.1] + ([v.sub.*[alpha]] + [beta]t, [[rho].sub.*[alpha]]) + [R.sup.[alpha].sub.2] + ([u.sub.+] + [beta]t, [[rho].sub.+]);

(2) ([u.sub.+], [[rho].sub.+]) [member of] [R.sup.[alpha].sub.1][S.sup.[alpha].sub.2]([u.sub.-], [[rho].sub.-]) : ([u.sub.-] + [beta]t, [[rho].sub.-]) + [R.sup.[alpha].sub.1] + ([v.sub.*[alpha]] + [beta]t, [[rho].sub.*[alpha]]) + [S.sup.[alpha].sub.2] + ([u.sub.+] + [beta]t, [[rho].sub.+]);

(3) ([u.sub.+], [[rho].sub.+]) [member of] [S.sup.[alpha].sub.1][R.sup.[alpha].sub.2]([u.sub.-], [[rho].sub.-]) : ([u.sub.-] + [beta]t, [[rho].sub.-]) + [S.sup.[alpha].sub.1] + ([v.sub.*[alpha]] + [beta]t, [[rho].sub.*[alpha]]) + [R.sup.[alpha].sub.2] + ([u.sub.+] + [beta]t, [[rho].sub.+]);

(4) ([u.sub.+], [[rho].sub.+]) [member of] [S.sup.[alpha].sub.1][S.sup.[alpha].sub.2]([u.sub.-], [[rho].sub.-]) : ([u.sub.-] + [beta]t, [[rho].sub.-]) + [S.sup.[alpha].sub.1] + ([v.sub.*[alpha]] + [beta]t, [[rho].sub.*[alpha]]) + [S.sup.[alpha].sub.2] + ([u.sub.+] + [beta]t, [[rho].sub.+]);

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 [u.sub.-] > [u.sub.+], the Riemann solution of system (4) has the following form:

[mathematical expression not reproducible] (13

[mathematical expression not reproducible] (14)

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:

[mathematical expression not reproducible] (15)

where x(t), [omega](t), and [u.sub.[delta]](t), respectively, denote the location, weight, and the velocity of delta shock wave, and (x(0), w(0)) = (0,0).

Letting [v.sub.[delta]] = ([square root of ([[rho].sub.u])][u.sub.-] + [square root of ([[rho].sub.+])][u.sub.+])/ ([square root of ([[rho].sub.-])] + [square root of ([[rho].sub.+])]), by simple calculation, we obtain

[omega](t) = [square root of ([[rho].sub.-][[rho].sub.+])] ([u.sub.-] - [u.sub.+]) t,

[u.sub.[delta]] (t) = [v.sub.[delta]] + [beta]t,

x(t) = [v.sub.[delta]]t + 1/2 [beta][t.sup.2], (16)

for [[rho]] [not equal to] 0 and

[omega](t) = -[[rho]u] t,

[u.sub.[delta]] (t) = 1/2 ([u.sub.+] + [u.sub.-]) + 1/3 [beta]t,

x (t) = 1/2 ([u.sub.+] + [u.sub.-])t + 1/6[beta][t.sup.2], (17)

for [[rho]] = 0.

(2) For [u.sub.-] < [u.sub.+], the structure of Riemann solution is

([u.sub.-] + [beta]t, [[rho].sub.-]) + [J.sub.1] + Vac + [J.sub.2] + ([u.sub.+] + [beta]t, [[rho].sub.+]). (18)

(3) For [u.sub.-] = [u.sub.+], the structure of Riemann solution is

([u.sub.-] + [beta]t, [[rho].sub.-]) + [J.sup.*] + ([u.sub.+] + [beta]t, [[rho].sub.+]). (19)

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 [u.sub.-] > [u.sub.+], the occurrence of vacuum for [u.sub.-] < [u.sub.+], and the formation of discontinuity for [u.sub.-] < [u.sub.+].

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 [u.sub.-] > [u.sub.+] (see Figure 2(a)).

Lemma 1. Assume [u.sub.-] > [u.sub.+], then there exists certain value [[alpha].sub.1] > 0, such that ([u.sub.+], [[rho].sub.+]) [member of] [S.sup.[alpha].sub.1][S.sup.[alpha].sub.2] ([u.sub.-], [[rho].sub.-]) when 0 < [alpha] < [[alpha].sub.1].

Proof. If [u.sub.-] > [u.sub.+] and ([u.sub.+], [[rho].sub.+]) [member of] [S.sup.[alpha].sub.1][S.sup.[alpha].sub.2] ([u.sub.-], [[rho].sub.-]), we obtain (see Figure 2(a)):

[u.sub.+] < [u.sub.-] - h([[rho].sub.+], [[rho].sub.-], [alpha]a) ([[rho].sub.+] - [[rho].sub.-]), [[rho].sub.+] > [[rho].sub.-], (20)

[u.sub.+] < [u.sub.-] + h([[rho].sub.+], [[rho].sub.-], [alpha]a) ([[rho].sub.+] - [[rho].sub.-]), [[rho].sub.+] > [[rho].sub.-]. (21)

From (20)-(21) and (2), we have

[square root of ([alpha])] [absolute value of (h ([[rho].sub.+], [[rho].sub.-], 1) ([[rho].sub.+] - [[rho].sub.-]))] < [u.sub.-] - [u.sub.+]. (22)

From (22), we derive that

[alpha] < [([u.sub.-] - [u.sub.+]).sup.2]/[h.sup.2] ([[rho].sub.+], [[rho].sub.-], 1) [([[rho].sub.+] - [[rho].sub.- ]).sup.2]. (23)

Let [[alpha].sub.1] = [([u.sub.-] - [u.sub.+]).sup.2]/[h.sup.2]([[rho].sub.+], [[rho].sub.-], 1)[([[rho].sub.+] - [[rho].sub.-]).sup.2], then ([u.sub.+], [[rho].sub.+]) [member of] [S.sup.[alpha].sub.1][S.sup.[alpha].sub.2] ([u.sub.-], [[rho].sub.-]) when [u.sub.-] > [u.sub.+] and 0 < [alpha] < [[alpha].sub.1]. It is obvious that this conclusion is also true for [[rho].sub.+] = [[rho].sub.-], The proof is completed.

For ([u.sub.+], [[rho].sub.+]) [member of] [S.sup.[alpha].sub.1][S.sup.[alpha].sub.2] ([u.sub.-], [[rho].sub.-]), the Riemann solution of (5)(6) contains two shock waves satisfying

[mathematical expression not reproducible], (24)

[mathematical expression not reproducible], (25)

where ([v.sub.*[alpha]], [[rho].sub.*[alpha]]) is the intermediate state (see Figure 2(a)). From the first equation of (24)-(25), we obtain

[u.sub.-] - [u.sub.+] = h ([[rho].sub.*[alpha]], [[rho].sub.-], [alpha]) ([[rho].sub.*[alpha]] - [[rho].sub.-]) + h([[rho].sub.*[alpha]], [[rho].sub.+], [alpha])([[rho].sub.*[alpha]] - [[rho].sub.+]), [[rho].sub.*[alpha]] > [[rho].sub.[+ or -]]. (26)

From (26), we can derive that [lim.sub,[alpha][right arrow]0][[rho].sub.*[alpha]] = +[infinity] and [lim.sub,[alpha][right arrow]0][alpha][[rho].sub.*[alpha]] = [[rho].sub.+][[rho].sub.-] [([u.sub.-] - [u.sub.+]).sup.2] /A[([[square root of ([[rho].sub.+])] + [[square root of ([[rho].sub.-])]).sup.2]. From the above analysis, we can obtain Lemma 2 as follows.

Lemma 2. As [u.sub.-] > [u.sub.+] and ([u.sub.+], [[rho].sub.+]) [member of] [S.sup.[alpha].sub.1][S.sup.[alpha].sub.2] ([u.sub.-], [[rho].sub.-]), [lim.sub.[alpha][right arrow]0][[rho].sub.*[alpha]] = +[infinity] and = [[rho].sub.+][[rho].sub.-][([u.sub.-] - [u.sub.+]).sup.2] / A[([[square root of ([[rho].sub.+])] + [[square root of ([[rho].sub.-])]).sup.2], i.e., the phenomenon of concentration occurs.

Next, we consider the formation of delta shock waves. Firstly, we analyze the velocity of intermediate state [u.sub.*]. It follows from (3), (24) and Lemma 2 that

[mathematical expression not reproducible] (27)

As the pressure vanishes, from the second equation of (24)-(25), the limits of the speeds of [S.sup.[alpha].sub.1] and [S.sup.[alpha].sub.2] are as follows:

[mathematical expression not reproducible] (28)

Equation (27)-(28) shows the two shock waves [S.sup.[alpha].sub.1] and [S.sup.[alpha].sub.2] are merged into one shock wave with speed [u.sub.[delta]](t) (see Figure 2(b)).

By using the first equation of Rankine-Hugoniot conditions (10) for both [S.sup.[alpha].sub.1] and [S.sup.[alpha].sub.2], we find

[mathematical expression not reproducible] (29)

Considering (28)-(29), we obtain

[mathematical expression not reproducible] (30)

From (30), we obtain that

[mathematical expression not reproducible] (31)

From above analysis and Lemma 2, we have Theorem 3.

Theorem 3. For [u.sub.-] > [u.sub.+], as [alpha] [right arrow] 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 [u.sub.-] > [u.sub.+] as the pressure vanishes. In the following, we study the convergence of the Riemann solutions to system (1)-(2) for [u.sub.-] [less than or equal to] [u.sub.+].

3.2. The Occurrence of Vacuum as the Pressure Vanishes. In this subcase, as the pressure vanishes, we study the occurrence of vacuum when ([u.sub.+], [[rho].sub.+]) [member of] [R.sup.[alpha].sub.1][R.sup.[alpha].sub.2] ([u.sub.-], [[rho].sub.-]) and [u.sub.-] < [u.sub.+] (see Figure 3(a)).

Suppose that ([u.sub.+], [[rho].sub.+]) [member of] [R.sup.[alpha].sub.1][R.sup.[alpha].sub.2] ([u.sub.-], [[rho].sub.- ]), from (7)-(8) and (9), one can derive that

[u.sub.+] > [u.sub.-] - g ([[rho].sub.-], [[rho].sub.+], [alpha]), [[rho].sub.+] < [[rho].sub.-], (32)

[u.sub.+] > [u.sub.-] + g ([[rho].sub.-], [[rho].sub.+], [alpha]), [[rho].sub.+] > [[rho].sub.-]. (33)

According to (32)-(33), we obtain that

[alpha] < [([u.sub.+] - [u.sub.-]/f ([[rho].sub.+], 1) + f ([[rho].sub.-], 1) + k ([[rho].sub.+], [[rho].sub.- ])).sup.2], (34)

where k([[rho].sub.+], [[rho].sub.-]) = In(([square root of (A[[rho].sup.2.sub.+] + B)] + [square root of (A[[rho].sub.+])])/ ([square root of (A[[rho].sup.2.sub.+] + B)] + [square root of (A[[rho].sub.-])]). Letting [[alpha].sub.0] = [(([u.sub.+] - [u.sub.-])/(f([[rho].sub.+], 1) + f([[rho].sub.-], 1) + k([[rho].sub.+], [[rho].sub.-]))).sup.2], the right state ([u.sub.+], [[rho].sub.+]) [member of] [R.sup.[alpha].sub.1][R.sup.[alpha].sub.2] ([u.sub.-], [[rho].sub.-]) as 0 < [alpha] < [[alpha].sub.0]. Next, we analyze the formation of vacuum when ([u.sub.+], [[rho].sub.+]) [member of] [R.sup.[alpha].sub.1][R.sup.[alpha].sub.2] ([u.sub.-], [[rho].sub.-]).

Lemma 4. As [u.sub.+] > [u.sub.-] and ([u.sub.+], [[rho].sub.+]) [member of] [R.sup.[alpha].sub.1][R.sup.[alpha].sub.2] ([u.sub.-], [[rho].sub.-]), [lim.sub.[alpha][right arrow]0][[rho].sub.*[alpha]] = 0, i.e., the phenomenon of vacuum occurs.

Proof. From the above analysis, there exists a constant [[alpha].sub.0], such that ([u.sub.+], [[rho].sub.+]) [member of] [R.sup.[alpha].sub.1][R.sup.[alpha].sub.2]([u.sub.-], [[rho].sub.-]) when [alpha] [member of] (0, [[alpha].sub.0]). From (7)-(8), we obtain that v = [u.sub.-] - g([[rho].sub.-], [rho], [alpha]) on [R.sup.[alpha].sub.1]([[rho].sub.-], [u.sub.-]) and v = [u.sub.+] + g([[rho].sub.+], [rho], [alpha]) on [R.sup.[alpha].sub.2]([[rho].sub.+], [u.sub.+]), which derive that

[usub.-] - [u.sub.+] = g([[rho].sub.+], [rho], [alpha]) + g([[rho].sub.-], [rho], [alpha]). (35)

If [lim.sub.[alpha][right arrow]0][[rho].sub.*[alpha]] = [c.sub.0] [member of] (0, min([[rho].sub.-], [[rho].sub.+])), as [alpha] tends to zero, from (35), we have [u.sub.-] = [u.sub.+], which leads to a contradiction. Therefore, [lim.sub.[alpha][right arrow]0][[rho].sub.*[alpha]] = 0, 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 [u.sub.-] [less than or equal to] [u.sub.+], as [alpha] [right arrow] 0. Our argument is divided into three parts.

Case 1. [J.sub.1] + Vac + [J.sub.2] for [u.sub.-] < [u.sub.+] and [alpha] [right arrow] 0.

Lemma 4 shows that the phenomenon of vacuum occurs as [u.sub.-] < [u.sub.+] and [alpha] [right arrow] 0. Next, we introduce the formation of discontinuity for [u.sub.-] < [u.sub.+].

When ([u.sub.+], [[rho].sub.+]) [member of] [R.sup.[alpha].sub.1][R.sup.[alpha].sub.2]([u.sub.-], [[rho].sub.-]), from (7)-(8), we have

[mathematical expression not reproducible] (36)

and

[mathematical expression not reproducible] (37)

It follows from (36)-(37) that the two rarefaction waves [R.sup.[alpha].sub.1] and [R.sup.[alpha].sub.2] turn into two contact discontinuities as the pressure vanishes for [u.sub.-] < [u.sub.+] (see Figure 3(b)). We summarize our conclusion as follows.

Theorem 5. For [u.sub.-] < [u.sub.+], as [alpha] [right arrow] 0, the two rarefaction waves of system (1)-(2) converge to two contact discontinuities which are the corresponding Riemann solution of system (4).

Case 2. [J.sup.*] for [u.sub.-] = [u.sub.+] and [alpha] [right arrow] 0.

We have studied the convergence of the Riemann solution to system (1)-(2) for [u.sub.-] > [u.sub.+] and [u.sub.-] < [u.sub.+] as the pressure vanishes. Next, we discuss the situation for [u.sub.-] = [u.sub.+]. We divide our discussion into two parts for ([u.sub.+], [[rho].sub.+]) [member of] [R.sup.[alpha].sub.1][S.sup.[alpha].sub.2]([u.sub.-], [[rho].sub.-]) (see Figure 4(a)) and ([u.sub.+], [[rho].sub.+]) [member of] [S.sup.[alpha].sub.1][R.sup.[alpha].sub.2]([u.sub.-], [[rho].sub.-]) (see Figure 5(a)).

When ([u.sub.+], [[rho].sub.+]) [member of] [R.sup.[alpha].sub.1][S.sup.[alpha].sub.2]([u.sub.-], [[rho].sub.-]), from (7) and (25), we deduce that

[mathematical expression not reproducible], (38)

which implies that the rarefaction wave [R.sub.1]([[rho].sub.-], [u.sub.-]) and the shock wave [S.sub.2]([[rho].sub.+], [u.sub.+]) become a contact discontinuity as the pressure vanishes (see Figure 4(b)).

Similarly, from (8) and (24), we know that

[mathematical expression not reproducible], (39)

which means that the rarefaction wave [R.sup.[alpha].sub.2]([[rho].sub.+], [u.sub.+]) and the shock wave [S.sup.[alpha].sub.1]([[rho].sub.-], [u.sub.-]) turn into a contact discontinuity as [alpha] [right arrow] 0 (see Figure 5(b)).

We have considered the convergence of the Riemann solution of system (1)-(2) for [u.sub.-] = [u.sub.+], as the pressure vanishes. We summarize our main results as follows.

Theorem 6. For [u.sub.-] = [u.sub.+], as [alpha] [right arrow] 0, the Riemann solutions of system (1)-(2) converge to a contact discontinuity [J.sup.*] 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.

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

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).

References

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Yongqiang Fan, (1) Lihui Guo (iD), (1) and Gan Yin (2)

(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; lihguo@126.com

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

Academic Editor: Claudio Dappiaggi

Caption: Figure 1: The phase plane (v, [rho]).

Caption: Figure 2: Riemann solution when ([u.sub.+], [[rho].sub.+]) [member of] [S.sup.[alpha].sub.1][S.sup.[alpha].sub.2]([u.sub.-], [[rho].sub.-]).

Caption: Figure 3: Riemann solution when ([u.sub.+], [[rho].sub.+]) [member of] [R.sup.[alpha].sub.1][R.sup.[alpha].sub.2]([u.sub.-], [[rho].sub.-]).

Caption: Figure 4: Riemann solution when ([u.sub.+], [[rho].sub.+]) [member of] [R.sup.[alpha].sub.1][S.sup.[alpha].sub.2]([u.sub.-], [[rho].sub.-]).

Caption: Figure 5: Riemann solution when ([u.sub.+], [[rho].sub.+]) [member of] [S.sup.[alpha].sub.1][R.sup.[alpha].sub.2]([u.sub.-], [[rho].sub.-]).
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Title Annotation:Research Article
Author:Fan, Yongqiang; Guo, Lihui; Yin, Gan
Publication:Advances in Mathematical Physics
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2018
Words:5250
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