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 bywhere 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 equationsThe 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 equationsand conservation law of energyEquations (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 gaswhere 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 inequalityIn 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 iswhere , , and denote the position, weight, and propagation speed of -shock wave and satisfyas , andas .
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 asandAnalogously, we obtain shock wave curveandAnd 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).
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 dataAlgebraic manipulation of (28) and (29) shows thatas , andas . 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 satisfyingorand from (33) and (34), it is found that And, then, we takeAnd 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 lettingIt is shown that as .
This lemma shows that there is no -shock wave as . When , the Riemann solution is satisfyingandWe discuss the limits of solutions for system (1)–(2) by the following lemmas.
Lemma 2. .
Proof. Utilizing the expressions and , we findSetting in (44), suppose that ; we get the equalityFrom (45) and , it is found thatwhich 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) thatAs 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 expressionFrom (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 concludewhich 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 asSetting 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 thatwhere Taking the limit on the right of (65), we getSubstituting (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 withThe proof of Theorem 7 is completed.
By using (30)–(31) and noting thatas , andas .
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 satisfyingorAlgebraic manipulation of (96) and (97) givestakingAccordingly, we see that when
And so, for any given , the Riemann solution can be expressed aswhich satisfiesThen utilizing (101), we findand also from the above identity, we conclude that And so, the phenomena of cavitation do occur as tend to zero. Next, According toandwe 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).
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).