Solutions with concentration and cavitation to the Riemann problem for the isentropic relativistic Euler system for the extended Chaplygin gas

Abstract The solutions to the Riemann problem for the isentropic relativistic Euler system for the extended Chaplygin gas are constructed for all kinds of situations by using the method of phase plane analysis. The asymptotic limits of solutions to the Riemann problem for the relativistic extended Chaplygin Euler system are investigated in detail when the pressure given by the equation of state of extended Chaplygin gas becomes that of the pressureless gas. During the process of vanishing pressure, the phenomenon of concentration can be identified and analyzed when the two-shock Riemann solution tends to a delta shock wave solution as well as the phenomenon of cavitation also being captured and observed when the two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution with a vacuum state between them.


Introduction
It is very important to understand the relativistic uid dynamics in the study of various astrophysical phenomena [1], such as the gravitational collapse, the supernova explosion and the formation and acceleration of the universe. Nowadays, there exists a vast amount of literature in various models of relativistic uid dynamics since the fundamental work of Taub [2]. However, only a few analytical theories have been developed such as in [3][4][5][6] due to the complicated structures of various relativistic uid dynamics models. In this present work, we draw our attention on the isentropic Euler system of two conservation laws consisting of energy and momentum in special relativity in the following form [3,[5][6][7] (1.1) Here the unknown state variables ρ(x, t) and v(x, t) stand for the proper-energy density and the particle speed respectively and the unknown function p(ρ) is used to denote scalar pressure which is a function of ρ for the isentropic situation. In addition, the constant c is the speed of light. The system (1.1) was often used to describe the dynamics of plane waves in special relativistic uids in the two-dimensional Minkowski spacetime [3]. In our present study, the equation of state p(ρ) is chosen as the third-order form of the extended Chaplygin gas [8,9] as follows: (1.2) in which A , A , A ≥ and B > . It requires that the speed of sound p (ρ) is less than the speed of light c, such that the condition A + A ρ + A ρ + Bαρ −α− < c is satis ed. The Chaplygin gas with the equation of state given by p(ρ) = − B ρ with the constant B > was rst introduced by Chaplygin [10] as an e ective mathematical approximation to compute the lifting force on a wing of an airplane. The equation of state for the Chaplygin gas is also very suitable to describe the dark energy and the dark matter in the universe within the framework of string theory [11]. In order to be consistent with the observed data, the equation of state was generalized to the form p(ρ) = − B ρ α for the generalized Chaplygin gas [12] and subsequently was further modi ed to the form p(ρ) = Aρ − B ρ α for the modi ed Chaplygin gas [13], in which A, B > and < α ≤ . It is essential to deal with a two uid model about the equation of state for the modi ed Chaplygin gas for the reason that the rst term Aρ gives an ordinary uid obeying a linear barotropic equation of state while the second one − B ρ α is the pressure to some power of the inverse of energy density. However, it is possible to consider the barotropic uid, whose equation of state is quadratic and to even higher orders. In view of the aforementioned facts, the extended Chaplygin gas with the equation of state p(ρ) = n Σ k= A k ρ k − B ρ α has been proposed by Pourhassan and Kahya [8]. It is easy to know that the extended Chaplygin gas recovers all the above Chaplygin gases by selecting A k (k = , . . . , n) and α suitably. It is worthwhile to notice that the third-order form of the extended Chaplygin gas with the equation of state (1.2) has a good agreement with the cosmological parameters such as dark energy density, scale factor and Hubble expansion parameter [9,[14][15][16]. Of course we can also carry out the study for higher n terms of the extended Chaplygin gas, but the e ects of more corrected terms are in nitesimal and are therefore of less importance [9]. Due to the above results, we shall draw our attention on the third-order form of the extended Chaplygin gas with the equation of state (1.2).
It is well known that the explicit solution can help us to understand the formation mechanism of singularities. For this purpose, we restrict ourselves to consider the system (1.1)-(1.2) with the Riemann-type initial data which is taken to be Formally, if we adopt the Newtonian limit (namely the limit v c → is taken), then the system (1.1)-(1.2) becomes the classical isentropic Euler system for the compressible uid in the form which has been widely studied as in [17,18]. On the other hand, if the limit A , A , A , B → is taken, then the system (1.1)-(1.2) turns out to be the following zero-pressure relativistic Euler system (1.5) The system (1.5) is a non-strictly hyperbolic and completely linearly degenerate system, whose elementary wave only involves the contact discontinuity. More speci cally, the solution to the Riemann problem (1.3) and (1.5) is either a delta shock wave solution when v− > v+ or a two-contact-discontinuity solution with a vacuum momentum in special relativity. Furthermore, Yin and Song [43] considered the vanishing pressure limits of solutions to the Riemann problems about the Euler system of conservation laws consisting of baryon numbers and momentum in special relativity for the Chaplygin gas. In addition, Yang and Zhang [44] introduced the ux approximation approach to study the formation of vacuum state and delta shock wave to the Riemann problem for the zero-pressure relativistic Euler system (1.5).
The paper is arranged as follows. In Section 2, we are mainly concerned with the construction of solutions to the Riemann problems for the isentropic relativistic Euler system (1.1) associated with the equation of state (1.2) in detail. In addition, we give a brief description of the solutions to the Riemann problem for the zero-pressure relativistic Euler system (1.5). In Section 3, we shall focus on the vanishing pressure limits of solutions to the Riemann problems from the system (1.1)-(1.2) to the zero-pressure relativistic Euler system (1.5) for the case c > v− > v+ > −c when the limit A , A , A , B → is taken, in which the formation of δshock wave can be observed and analyzed. In Section 4, we turn back to investigate the formation of vacuum state for the case −c < v− < v+ < c when the limit A , A , A , B → is taken.

The Riemann problems for the isentropic and zero-pressure relativistic Euler systems
In this section, we rst illustrate the solutions to the Riemann problem for the isentropic relativistic Euler system (1.1) associated with the equation of state (1.2). Then, we recollect the related results for the zeropressure relativistic Euler system (1.5), whose Riemann solution is a delta shock wave solution when c > v− > v+ > −c or a two-contact-discontinuity solution with a vacuum state between them when −c < v− < v+ < c.

. The Riemann problem for the system (1.1) with the equation of state (1.2)
In this subsection, we shall rst analyze the properties of elementary waves and then construct the solutions to the Riemann problem (1.1)-(1.3) for all kinds of situations. Since the speed of sound p (ρ) is less than the speed of light c, the condition A + A ρ + A ρ + Bαρ −α− < c has to hold. There exist ρ and ρ satisfying < ρ < ρ < ρ < +∞ for the xed A , A , A , B, in which ρ and ρ are determined by In fact, the above ρ and ρ can be calculated numerically when all the coe cients A , A , A , B and α are given. More precisely, we can estimate ρ and ρ simply for su ciently small A , A , A , B. It can be concluded that the following two inequalities hold simultaneously, which enables us to have at least Thus, the physically relevant region of solutions for the xed A , A , A , B is restricted to where the matrixes C and D are given respectively by By means of a direct calculation, we can achieve two real and distinct eigenvalues (2.7) Corresponding to each λ i (i = , ), the right eigenvectors are calculated respectively by (2.8) Thus the system (1.1)-(1.2) is strictly hyperbolic [3,6,46]. Let us introduce the notion ∇ = ( ∂ ∂ρ , ∂ ∂v ), by a direct calculation, then we have As a consequence, the following can be obtained in which Both the characteristic elds of λ and λ are genuinely nonlinear. That being said, we shall show that the elementary waves for each of the two characteristic elds are either rarefaction waves or shock waves [3,6,46]. Let us rst consider the rarefaction wave curves. Both the system (1.1)-(1.2) and the Riemann initial data (1.3) are unchanged under the scalable coordinates: (x, t) → (kx, kt) (k > is a constant). Therefore, we want to solve the self-similar solutions of the form Now, we can use the following boundary value problems of ordinary di erential equations to take the place of the Riemann problem (1.1)-(1.3) as follows: (2.11) For smooth solutions, (2.11) is reduced to If (dρ, dv) = ( , ), then it is easy to get the trivial solution that (ρ, v) is a constant state. Otherwise, if (dρ, dv) ≠ ( , ), by a trivial and tedious calculation, then we can obtain the singular solutions and (2.14) One can obtain ρ < ρ < ρ− < ρ directly from the requirement λ (ρ) > λ (ρ−). Let the left state (ρ−, v−) be xed, then integrating the second equation in (2.13) from ρ− to ρ enables us to obtain the 1-rarefaction wave That is to say, v decreases as ρ increases for the curve R (ρ−, v−). Analogously, due to ρ ξ = > , we can derive the 2-rarefaction wave curve as follows: For the 1-rarefaction wave, owing to a tedious but straightforward calculation for the second equation of (2.13), we have vρρ > for all the ρ < ρ < ρ . In other words, the 1-rarefaction wave curve R is convex in the half-upper (ρ, v) phase plane. Analogously, we can also have vρρ < for all the ρ < ρ < ρ from the second equation in (2.14). That is to say, the 2-rarefaction wave curve R is concave in the half-upper (ρ, v) phase plane.
From now on, we focus our attention on the shock wave curves. The Rankine-Hugoniot conditions are as follows where [ρ] = ρ − ρ− denotes the jump across the discontinuity. We call σ the speed of the discontinuity, where On the other hand, if σ ≠ , by removing σ from (2.17), we can obtain From direct calculation and simpli cation, (2.18) turns out to be For the sake of simplicity, we set To sum up for the given left state (ρ−, v−), the two shock waves are shown respectively as

22) and
(2.23) From either (2.22) or (2.23), a tedious but straightforward computation shows that It is easy to see that vρ < from ρ > ρ > ρ− > ρ for the 1-shock curve and vρ > from ρ < ρ < ρ− < ρ for the 2-shock curve. It follows that v decreases as ρ increases for the curve S (ρ−, v−) while v increases as ρ increases for the curve S (ρ−, v−). Comparing with the 1-rarefaction (or 2-rarefaction) curve, similar convexity (or concavity) are to be found in the 1-shock (or 2-shock) curve. The computation is tedious and trivial and thus the details are omitted here. By combining ( 1)-(1.3) is determined uniquely by the above four regions. More precisely, the solution can be expressed as S + S when (ρ+, v+) ∈ I, R + S when (ρ+, v+) ∈ II, S + R when (ρ+, v+) ∈ III or R + R when (ρ+, v+) ∈ IV respectively. Here and in what follows, the symbol S + S is adopted to represent a 1-shock wave S followed by a 2-shock wave S , etc.
. The Riemann problem for the zero-pressure relativistic Euler system (1.

5)
In this subsection, we shall brie y summarize the solutions to the Riemann problem for the zero-pressure relativistic Euler system (1.5), which have been well described such as in [39,40]. The system (1.5) has the two coincident eigenvalues λ = λ = v, which means that the system (1.5) is non-strictly hyperbolic. The corresponding right eigenvector is As before, if we look for the self-similar solution (ρ, v)(x, t) = (ρ, v)(ξ ), ξ = x t , then the Riemann problem (1.3) and (1.5) is reduced to the boundary value problem of the following system of ordinary di erential equations (2.26) In the case v− < v+, the solutions (ρ, v)(ξ ) including two contact discontinuities with a vacuum state between them can be written as (ρ+, v+), v+ < ξ < +∞. (2.27) Otherwise, in the case v− > v+, a delta shock wave solution is generated due to the overlapping characteristics for the Riemann problem (1.3) and (1.5). It is necessary to introduce the de nition of δ−measure [19,22,45] in order to depict the delta shock wave solution to the Riemann problem (1.3) and (1.5).
For the purpose of completeness, it is necessary to o er the following generalized de nition of delta shock wave solution introduced by Danilov et al. [47][48][49][50]. Let I be a nite index set, then we make the assumption that Γ = {γ i |i ∈ I} is a graph in the upper-half plane (x, t) ∈ R × R+ involving Lipschitz continuous curves γ i for i ∈ I. Later, let I be a subset of I, then the curves γ i with i ∈ I originate from the x−axis. In the end, let Γ = {x k |k ∈ I } be the set of initial points of γ k with k ∈ I .

R). Then, a pair of distributions (ρ, v) is called a δ−shock wave type solution for the system (1.5) with the initial data (2.29) if and only if the following integral identities
Moreover, the delta shock wave solution (2.32) in comparison with (2.33) obeys the generalized Rankine-Hugoniot conditions listed below (2.34) In order to guarantee the uniqueness of solution, it should also obey the over-compressive δ-entropy condition In addition, the above-constructed delta shock wave solution (2.32) in comparison with (2.33) is satis ed with the system (1.5) in the sense of distributions. In other words, the weak form of the system (1.5) as below holds for any test function ϕ(x, t) ∈ C ∞ ((−∞, +∞) × ( , +∞)), in which  (2.38). The process of derivation is completely similar to that for the zero-pressure Euler system in [19], thus the details are omitted here. As a consequence, the existence and uniqueness of delta shock wave solution in the form (2.32) can be checked as in [44] by using the generalized Rankine-Hugoniot conditions (2.34) together with the over-compressive δ-entropy condition (2.35). It is remarkable that the delta shock wave solution (2.32) together with (2.33) are no longer in the space of BV or L ∞ functions. However, the divergences of certain entropy and entropy ux elds are still in the space of Radon measures [22]. It is natural to discuss this problem in the theory of divergence-measure elds and thus the delta shock wave solution (2.32) together with (2.33) can be understood in the form of Tartar-Murat measure solution [51][52][53], in which the velocity must take a value at the point of the jump. A , A , A

The formation of delta shock wave as
It is easy to know from [22,26]  Let (ρ * , v * ) be the intermediate state between two shock waves, we obtain the solution which joins (ρ−, v−) and (ρ * , v * ) by means of the 1-shock wave S with the speed σ and then joins (ρ * , v * ) and (ρ+, v+) by means of the 2-shock wave S with the speed σ . To be more speci c, we have S : and S : .
Proof. Without loss of generality, we suppose lim which enables us to have . (3.8) .
(3.9) Furthermore, we have Squaring both sides of (3.10) and then simplifying, yields This is a quadratic form of M and can be solved as Thus, one has . (3.13) If the negative sign in (3.13) is chosen, then one has . (3.14) Analogously, one also has . (3.15) Let v− and v+ be xed to satisfy c > v− > v+ > −c, then it is easy to know that we can choose ρ− and ρ+ suitably to satisfy either which means that (3.7) is not satis ed. Similarly, if the Riemann initial data (1.3) satisfy c > v− > v+ > −c and (3.17) at the same time, then we can also see that (3.7) is still not satis ed. As a consequence, it can be concluded from the above discussion that the negative sign cannot be chosen in (3.13) for the reason that (3.7) does not always hold for any given Riemann initial data (1.3) On the other hand, if we choose the positive sign in (3.13), then it yields , (3.18) and . (3.19) Thus, it is easy to get which enables us to see that (3.7) is indeed satis ed. In a word, it can be concluded from the above discussion that . (3.20) As a consequence, the limiting relation (3.4) can be established.
such that we have .
On the one hand, it can be clearly seen that On the other hand, it is easy to see that The proof is completed. (3.44) According to De nition 2.1, the last part of (3.44) is equivalent to ω (t)δs , ψ(·, ·) , in which In the same way as before, from (3.43), we also obtain (3.46) in which Thus, the conclusion of Theorem 3.3 can be drawn. A , A , A

The formation of vacuum state as
and R :  (2.27), which is identical with that for the pressureless relativistic Euler system (1.5).
Proof. Taking into account (4.1) and (4.2), we obtain where ρ * ≤ min(ρ−, ρ+). Thus, it is easy to get that Moreover, it is worthwhile to notice that the rarefaction curves R and R turn out to be the lines of contact discontinuities J : v = v− and J : v = v+ in the half-upper (ρ, v) phase plane respectively. Therefore, we take a step further to get lim It is obvious to see that the rarefaction curves R and R tend to the contact discontinuities J and J with the speeds of v− and v+ respectively. The proof is accomplished.