Advances in Mathematical Physics
Volume 2019 (2019), Article ID 5253717, 12 pages
https://doi.org/10.1155/2019/5253717

Research Article

The Vanishing Pressure Limit of Riemann Solutions to the Non-Isentropic Euler Equations for Generalized Chaplygin Gas

Qixia Ding and Lihui Guo

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China

Correspondence should be addressed to Lihui Guo; lihguo@126.com

Received 19 March 2019; Accepted 10 June 2019; Published 3 July 2019

Guest Editor: Tudor Barbu

Copyright © 2019 Qixia Ding and Lihui Guo. 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.

We analyze the appearance of delta shock wave and vacuum state in the vanishing pressure limit of Riemann solutions to the non-isentropic generalized Chaplygin gas equations. As the pressure vanishes, the Riemann solution including two shock waves and possible one contact discontinuity converges to a delta shock wave solution. Both the density and the internal energy simultaneously present a Dirac delta singularity. And the Riemann solution involving two rarefaction waves and possible one contact discontinuity converges to a solution involving vacuum state of the transport equations.

1. Introduction

The compressible Euler equations of non-isentropic fluids in one dimension are given by

where is the density, the velocity, the pressure, the internal energy, and the internal energy per unit mass. In order to analyze the limit behavior of solutions, we assume is a known function of the density for sufficiently small The generalized Chaplygin gas is a natural extension of Chaplygin gas, and its equation of state is as follows:

Compared with the polytropic gas, the essential difference is that the generalized Chaplygin gas owns a negative pressure. As a model of cosmology, the generalized Chaplygin gas is used to describe the unification of dark matter and dark energy; see [1–3] for related results.

In 1991, Smoller [4] considered the Riemann problem of isentropic Euler equations

The Riemann solutions of equations (3) are constructed by the analysis method in phase plane. In 2013, Wang [5] studied the generalized Chaplygin equations with constant initial data and obtained the global Riemann solution involving nonclassical wave (delta shock wave). In 2016, Sun [6] considered system (3) with a source term and constructed the exact solutions which contain the delta shock wave. In 2006, Panov and Shelkovich [7] introduced a new type of weak solution, which was called delta shock wave. Delta shock wave in the physical sense is used to represent the formation of black hole, or the concentration of particles in the universe. As for delta shock waves, see [8–14].

As the pressure vanishes, (1) converge to transport equations

and conservation law of energy

Equations (4) are used to describe the motion of colliding particles at low temperatures and low pressure and the formation of large-scale matter in the universe [15, 16]. The zero-pressure system (4) has been widely studied. In 1994, Bouchut [17] studied the existence of a measure solution and constructed Riemann solutions for transport equations (4). In [18], Sheng and Zhang constructively solved the 1-D and 2-D Riemann problems for transportation equations. In either case, there exist two solutions: one contains -shocks and the other contains vacuum states. In 2001, Li [19] considered the isentropic system (3) with zero temperature. He found that the limit solutions of system (3) as the temperature are the Riemann solutions of (4). In 2015, Sheng [20] analyzed the appearance of concentration and cavitation by studying the limit solutions of system (2)-(3) as . In addition, the limit method is generalized to many gas dynamics systems; the interested readers refer to [21–25].

In 1979, in order to construct the solution of system (4)-(5) for any initial data, Kraiko [26] introduced discontinuities which differ from classical waves and carry mass, impulse, and energy. In 2012, Cheng [27] investigated the Riemann problem of system (4)-(5) with the following constant initial data:

in which and He obtained the Riemann solutions involving vacuum state, delta shock wave, and contact discontinuities. In 2016, Wang [28] solved the Riemann problem of system (4)-(5) with delta initial data and showed the existence of generalized solutions involving delta shock wave. Moreover, he also considered all conceivable interactions of contact vacuum states and delta shock waves. Unlike non-isentropic Chaplygin Euler equations [29], three characteristic fields of non-isentropic generalized Chaplygin Euler equations are not all linearly degenerate. In 2016, Pang [30] investigated the Riemann problem of system (1)–(2) with constant initial data (6). He obtained two kinds of solutions including classical waves and -shock wave. In 2004, Chen and Liu [31] studied non-isentropic Euler equations for polytropic gas

where denotes the total energy, he deduced that the limit solutions of non-isentropic Euler equations tend to the Riemann solutions for system (4). Yang and Liu [32] considered flux-approximation limit of Riemann solutions to non-isentropic Euler equations for (7) in 2015. In this paper, we focus on the limit behavior of Riemann solutions for system (1)–(2) with the initial data (6). As pressure vanishes, we analyze the appearance of concentration and cavitation in limit solutions; both are physical phenomena in fluid dynamics. In the sense of distributions, (5) yields the entropy inequality

In one-dimensional case, we need to prove the following conclusions:

For the case of , Riemann solution of system (1)–(2) tends to a -shock wave solution when reduces to a certain value , and the -shock wave tends to a -shock solution of zero-pressure flow (4) as

For the case of , Riemann solution of system (1)–(2) converges to solution involving vacuum state of system (4) as

These conclusions show that the occurrence of delta shock wave for pressureless Euler equations is caused by the concentration of density, and the formation of vacuum state is caused by the cavitation as the pressure decreases.

Since the vanishing pressure limit system is not strictly hyperbolic, the phenomena of concentration and cavitation in limit solution are considered as resonance between two characteristic fields of the pressureless Euler system (4). Resonance is not only one of the most common and frequent natural phenomena in the universe but also widely used in physics. Theories of resonance can be found in [33].

The plan of this article is as follows. In Section 2, we briefly review the Riemann solutions of (4). In Section 3, we recall the Riemann solutions for system (1)–(2). In Section 4, the vanishing pressure limits of solutions to (1)–(2) are studied. In Section 5, we give corresponding conclusions.

2. Riemann Solution of System (4)

In this section, We briefly recall the solutions of system (4) with the following Riemann initial data:

where ; see [18] for the details.

In the case of , the Riemann solution of system (4) is expressed by

In the case of , the Riemann solution is

where , , and denote the position, weight, and propagation speed of -shock wave and satisfy

as , and

as .

3. Riemann Solutions of System (1)–(2)

In this section, we review the Riemann problem of system (1)–(2); see [30] for the details. The physically correlative region of solutions is

Three real and distinct eigenvalues for system (1)–(2) are

with the associated right eigenvectors

By calculation, we obtain

We conclude that the first and third characteristic fields are genuinely nonlinear, which means that the corresponding elementary waves are the rarefaction waves or the shock waves. Nevertheless, the second one is linearly degenerate and the corresponding elementary wave is a contact discontinuity.

The rarefaction wave curve in the space , passing through , can be expressed as

and

Analogously, we obtain shock wave curve

and

And a contact discontinuity satisfies

The projections of curves , , , and divide the phase plane into five regions, as shown in Figure 1. Consequently, one can construct the global solutions of system (1)–(2).

Figure 1: The phase plane ().

When , the structures of solutions can be expressed in the following form.

(1) The projection of onto phase plane

(2) The projection of onto phase plane

(3) The projection of onto phase plane

(4) The projection of onto phase plane

where and denote the intermediate states.

When the projection of onto phase plane belongs to , the solution is

The -shock wave satisfies the following generalized R-H jump conditions:

with initial data

Algebraic manipulation of (28) and (29) shows that

as , and

as . Moreover, the -shock wave satisfies the -entropy condition

4. Limits of Riemann Solutions for System (1)–(2)

In this section, we analyze the limit behavior of the Riemann solutions for system (1)–(2) as . In the phase plane, we discuss the two cases of and .

4.1. Case

In this subsection, we consider the formation of delta shock wave solution for system (1)–(2) as the pressure decreases.

Lemma 1. In the case that , then there exists such that the projection of the state onto the phase plane as ; as

Proof. As a consequence of , we can easily conclude that as .
When , we assume , which can be joined with by or satisfying

or

and from (33) and (34), it is found that

And, then, we take

And so, as .

When , which can be joined with directly by a delta shock wave curve satisfying the inequality,

From the above inequality, we obtain the expression

letting

It is shown that as .

This lemma shows that there is no -shock wave as . When , the Riemann solution is

satisfying

and

We discuss the limits of solutions for system (1)–(2) by the following lemmas.

Lemma 2. .

Proof. Utilizing the expressions and , we find

Setting in (44), suppose that ; we get the equality

From (45) and , it is found that

which contradicts with (39). And so, we conclude that .

Lemma 3. The intermediate internal energy and is unbounded as , that is,

Proof. It is derived from the third equation in (41) that

As a consequence of , it is noted that .

Similarly, by (43)₃, one can conclude that .

Lemma 4.

Proof. In light of Lemma 2, from the second equation in (41), we obtain the expression

From (42), we directly deduce that Note that the first expression in (41) yields

We similarly obtain that

Lemma 5.

Proof. To confirm this, we compute using (41) and (43) that

Similarly, one has

Next integrating from to with respect to and setting , we derive

Lemma 6.

where

Proof. Using the Rankine-Hugoniot conditions of (1) for and , we obtain the following forms:

Set in (58), to conclude

which implies that

The proof is completed.

In light of the foregoing lemmas, two kinds of shock waves and will coincide as . Additionally, delta shock wave with Dirac delta function in both the density and the internal energy appears.

Theorem 7. For any given , we suppose that the Riemann solution for system (1)–(2) is , which converges to a delta shock wave solution as , and the limit functions , , and are sums of a step function and a delta function with weights

where

Proof. (I) Setting , the Riemann solution is described in the following form:

where , and are constants that only depend on
The solution (63) satisfies the definition of weak solution for an arbitrary

(II) The first integral in identity (64) can be written as

Setting in the first term for (67), we see

where and is the characteristic function. In virtue of , we get

Insert the expressions (68) and (69) into (64) to obtain

(III) Next, we turn our attention on the the momentum equation (65). Analogously, it is found that

where

Taking the limit on the right of (65), we get

Substituting (71) and (73) into (65), we obtain the equality

(IV) Subsequently, by utilizing the energy equation (66), we discover

where

By taking the limit for the right side of (66) note that

Going back to the identity (66), we find

(V) From now on, we study the limits of , , and as
For any , it is noted that

Utilizing (70), thereby obtain the equality

One can be calculated by using the above two expressions to obtain

According to definition of the measure solution [34], one has

with

Analogously, one can see that

with

And

with

The proof of Theorem 7 is completed.

By using (30)–(31) and noting that

as , and

as .

Consequently, we conclude that limit of the -shock wave solution to (1) with (6) is -shock wave solution of the transport equations (4).

Theorem 8. The pair is a measure solution in the sense of distributions to the transport equations (4) if it satisfies

Proof. In the sense of distributions, owing to , then

Furthermore, for any given nonnegative test function

By applying Green’s formulas, we compute

for

Through above discussions, we verify the consistency of entropy.

4.2. Case

This subsection mainly shows the formation of vacuum state of limit solutions for system (1)–(2) as the pressure decreases.

Lemma 9. In case , then there exists such that the projection of onto the phase plane lies in as

Proof. In view of , as , the conclusion is true for . Then, we analyze , supposing that , which can be joined with by or satisfying

or

Algebraic manipulation of (96) and (97) gives

taking

Accordingly, we see that when
And so, for any given , the Riemann solution can be expressed as

which satisfies

Then utilizing (101), we find

and also from the above identity, we conclude that

And so, the phenomena of cavitation do occur as tend to zero. Next, According to

and

we see that

In light of the foregoing analysis, it is seen that the limit of the Riemann solution for system (1)–(2) as tends to zero and is a double-contact-discontinuity solution of transport equations (4), as shown in Figure 3.

5. Discussion

We have considered the vanishing pressure limit of Riemann solutions for system (1)–(2). In the case of , Riemann solution of system (1)–(2) tends to a -shock wave solution when reduces to a certain value , and the -shock wave tends to a -shock solution of zero-pressure flow (4) as (see Figure 2). In the case of , Riemann solution of system (1)–(2) converges to solution involving vacuum state of system (4) as (see Figure 3).

Figure 2: The physical plane .

Figure 3: The physical plane .

Data Availability

There is 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 supported by National Natural Science Foundation of China (11761068, 11401508, and 11461066), China Scholarship Council, and the Natural Science Foundation of Xinjiang (2017D01C053).

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