Delta Shocks and Vacuums to the Isentropic Euler Equations with the Flux Perturbation for van der Waals Gas

In this paper, we study the isentropic Euler equations with the flux perturbation for van der Waals gas, in which the density has both lower and upper bounds due to the introduction of the flux approximation and the molecular excluded volume. First, we solve the Riemann problem of this system and construct the Riemann solutions. Second, the formation mechanisms of delta shocks and vacuums are analyzed for the Riemann solutions as the pressure, the flux approximation, and the molecular excluded volume all vanish. Finally, some numerical simulations are demonstrated to verify the theoretical analysis.


Introduction
In this paper, we consider the isentropic Euler equations with the flux perturbation: where u denotes the velocity, P denotes the pressure, and ρ denotes the density satisfying ρ ≥ 2ϵ. Here, ϵ is a small positive perturbation parameter. When the flux perturbation and pressure both vanish, system (1) turns to be the transport equations: which can describe the formation of large-scale structures in the universe [1,2] and the motion of free particles which stick under collision [3]. We also call this system the zeropressure gas dynamics. For more details, readers can see [4][5][6].
In recent years, a lot of research work and achievement for the formation mechanisms of delta shocks and vacuums have been done by many scholars. Li [7] started the research on the isentropic Euler equations for perfect fluids in 2001, and then Chen and Liu [8,9] made an in-depth study on isentropic and nonisentropic Euler equations for polytropic gas in 2003 and 2004. In [8,9], formation mechanisms of delta shocks and vacuums were discussed as the polytropic gas pressure vanishes. Following that, the results were extended to various systems for different gas state equations in [10][11][12][13][14][15][16]. Furthermore, readers can also see [17] for the perturbed system of generalized pressureless gas dynamics model and [18] for the case of triangular conservation law system arising from "generalized pressureless gas dynamics model." In addition, Yang and Liu [19] considered the limit behaviors of Riemann solutions of system (1) for polytropic gas as the flux approximation and the polytropic gas pressure both vanish. Flux perturbation can be regarded as the external shear force that causes the deformation of fluid particles and can be used to control some dynamic behaviors of fluid. Further, the flux approximation including the pressure perturbation portion is a more general physical consideration.
where b denotes the molecular excluded volume satisfying 0 ≤ bρ ≪ 1, k denotes a positive constant, and c denotes the adiabatic exponent with 1 < c < 2. It is easy for us to find that when b � 0, the state equation (3) just corresponds to the ideal gas. As mentioned in, at a high pressure or a low temperature, the behavior of real gas is not consistent with the ideal polytropic gas model but accords with the van der Waals gas one. erefore, it is natural for us to explore the limit behavior of Riemann solutions of system (1) with the van der Waals gas (3) as the corresponding pressure vanishes.
Compared to the works in [8,19], the flux approximation and the molecular excluded volume are considered simultaneously in this paper, which makes the density has both lower and upper bounds. Further, we find that if only the flux approximation and the pressure tend to zero at the same time and the molecular excluded volume does not tend to zero, then the density always has an upper bound, so only vacuums may be generated, and delta shocks may not occur.
is is the motivation for us to study the case where all of the pressure, the flux approximation, and the molecular excluded volume tend to zero simultaneously; that is, the triple parameters ϵ, k, b ⟶ 0.
In this paper, we rigorously prove that as ϵ, k, b ⟶ 0, any Riemann solution to the perturbation isentropic Euler equations (1) for van der Waals gas (3) containing two shocks converges to the delta shock solution of system (2) and any Riemann solution of equations (1) and (3) containing two rarefaction waves converges to the vacuum solution of system (2). By theory analysis, it is found that the introduction of the flux perturbation and the van der Waal gas pressure does not affect the formation of delta shocks and vacuums when the triple parameters ϵ, k, b ⟶ 0 simultaneously, which means that our work can also be regarded as the extension of that in [8,19]. e remainder of this paper is arranged as follows. In Section 2, we solve the Riemann problem of equations (1) and (3). In Section 3, the limit behaviors of Riemann solution of equations (1) and (3) are considered as the flux perturbation, the van der Waals gas pressure, and the molecular excluded volume all vanish. In Section 4, some numerical results are shown to verify the theoretical analysis of the formation of delta shocks and vacuum states.
For the details of delta shocks and vacuums for the transport equations (2), we will not repeat them here. Readers can refer to Section 2 in [8,11].

Riemann Solutions to the Perturbation Euler Equations for van der Waals Gas
In this section, we solve the Riemann problem of the perturbation isentropic Euler equations (1) for the van der Waals gas (3) with Riemann initial data: where (u ± , ρ ± ) are arbitrary constants and then we construct Riemann solutions of this system. For equations (1) and (3), the eigenvalues and corresponding right eigenvectors are respectively. By direct calculation, we can obtain that ∇λ l · r l → ≠ 0 (l � 1, 2), which implies that both eigenvalues λ l (l � 1, 2) are genuinely nonlinear. us, the elementary waves of this system contain rarefaction waves and shock waves.
We look for the self-similar solution (u, ρ)(ξ) (ξ � x/t) of equations (1) and (3) with (4) and obtain the two-point boundary value problem as follows: Now, we consider the smooth solution of (6) and get either the constant state solution (u, ρ)(ξ) � constant, or the backward rarefaction wave or the forward rarefaction wave By (8) and (9), we calculate that when ϵ is small enough, From these two conditions, we find that the velocity of the backward rarefaction wave λ 1 is monotonically decreasing with respect to ρ, while the forward one λ 2 is increasing. Furthermore, integrating the second equations of (8) and (9), respectively, by In the (u, ρ)-plane, we call the curve of the second equation of (12) the backward rarefaction wave curve, which is monotonically decreasing with respect to ρ by a direct calculation from the second equation of (8). Further, for this curve, we obtain that )ds is convergent according to Cauchy criterion. erefore, this curve and the line ρ � 2ϵ have an intersection point ( Similarly, we call the curve of the second equation of (13) the forward rarefaction wave curve, which is monotonically increasing with respect to ρ from the second equation of (9). Since we have proved in [20] that (13). Now, we turn to a bounded discontinuity at ξ � σ. For equations (1) and (3), the Rankine-Hugoniot compatibility conditions hold, where [F] � F − F − denotes the jump of function F across the discontinuity. By solving (14), in terms of the stability conditions for λ 1 , and for λ 2 , we obtain the backward shock S 1 and the forward shock S 2 as follows: S 2 :

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In the (u, ρ)-plane, we call the curve of the second equations of (17) and (18) the backward shock curve (forward shock curve). From the second equation of (17), we can obtain which means that the backward shock curve is monotonically decreasing. Similarly, we have u ρ > 0 for the forward shock wave, which implies that the forward shock curve is monotonically increasing. Further, a direct calculation gives that lim ρ⟶ 2ϵb)) c )ϵ − 1 for the forward shock curve, which implies that this forward shock wave curve and the straight line ρ � 2ϵ have an intersection point Based on the aforementioned analysis, we conclude that the elementary waves of equations (1) and (3) contain two rarefaction waves (R 1 and R 2 ) and two shock waves (S 1 and S 2 ). Using the curves of these elementary waves, we can for any fixed left state (u − , ρ − ). Furthermore, for any fixed right state (u + , ρ + ), we can construct a unique Riemann solution, no matter which of the four regions the right state belongs to. Specifically, when (u + , ρ + ) ∈ S 1 S 2 (u − , ρ − ), the Riemann solution can be constructed with two shocks (S 1 and S 2 ) besides a nonvacuum constant state in between. When (u + , ρ + ) ∈ R 1 R 2 (u − , ρ − ), the Riemann solution can be constructed with two rarefaction waves (R 1 and R 2 ) besides an intermediate constant state which may be a constantdensity solution (ρ � 2ϵ). e discussion for the other two cases (u + , ρ + ) ∈ R 1 S 2 (u − , ρ − ) and (u + , ρ + ) ∈ S 1 R 2 (u − , ρ − ) is trivial, so in this paper we only consider the limit process for the cases (u + , ρ + ) ∈ S 1 S 2 (u − , ρ − ) and (u + , ρ + ) ∈ R 1 R 2 (u − , ρ − ).

Formation of Delta Shocks and
Vacuums as ε, k, b ⟶ 0

Formation of Delta Shocks.
In this subsection, the formation of delta shocks is considered in the Riemann solutions of equations (1) and (3) in the case

Limit Behavior of the Riemann Solutions as
en the left and right states of the backward shock S 1 (the forward shock S 2 ) are (u − , ρ − ) and (u Π * , ρ Π * ) ((u Π * , ρ Π * ) and (u + , ρ + )), respectively. us, we have for S 1 , and for S 2 . Here σ Π i (i � 1, 2) denotes the speed of S i . Now, we are ready to present our main results for the formation of delta shocks. Theorem 1. Let u − > u + and (u + , ρ + ) ∈ S 1 S 2 (u − , ρ − ). For any fixed ϵ, k, b > 0, suppose that (u Π , ρ Π ) is a Riemann solution of equations (1) and (3) with (4) containing two shocks S 1 and S 2 . en as ϵ, k, b ⟶ 0, ρ Π and ρ Π u Π converge, in the sense of distributions, to the sums of a step function and a δ-measure with weights respectively, which just form the delta shock solution of system (2) with (4).
Before we prove this theorem, some lemmas should be presented. e proof process of these lemmas is similar to that in [10], so we do not repeat it here.

Lemma 4.
lim ϵ,k,b⟶0 Lemma 5. For above quantity σ, we can obtain On the basis of the above results, we find that as ϵ, k, b ⟶ 0, the velocity of backward shock σ Π 1 , the velocity of forward shock σ Π 2 , and the intermediate velocity u Π * of equations (1) and (3) tend to the same quantity σ, which is proposed for the delta shock solution of system (2), and ρ Π * becomes singular simultaneously.
Step 2. Here, we prove that the limit functions of ρ Π u Π and ρ Π are the sums of a step function and a δ-measure. We rewrite the first term on the left of (27) and obtain (28) With the analysis of integration by part, noticing Lemmas 3 and 4, we have lim ϵ,k,b⟶0 (30)

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A combination of (29) and (30) gives that lim ϵ,k,b⟶0 where Similarly, we calculate the second integral on the left side of (27), noticing Lemmas 1-3, and obtain that lim ϵ,k,b⟶0 Substituting (31) and (33) into (27), we have for any ϕ ∈ C 1 0 (− ∞, +∞). In the same way, from (26), we can get that Step 3. We turn to prove the weights of the δ-measures. For any test function ψ ∈ C ∞ 0 (R + × R), by (34), we obtain lim ϵ,k,b⟶0 which together with definition (2.3) in [11] yields that where Similarly, from (35), we obtain and then we get that is completes the proof of eorem 1.

Formation of Vacuums. In this subsection, the limit of Riemann solutions of equations (1) and (3) is considered in
For this case, we assume that (u Π * , ρ Π * ) is the intermediate state. en the left and right states of the backward rarefaction wave R 1 (the forward rarefaction wave R 2 ) are (u − , ρ − ) and (u Π * , ρ Π * ) ((u Π * , ρ Π * ) and (u + , ρ + )), respectively. en, for the backward rarefaction wave R 1 , we have For the forward rarefaction wave R 2 , we have Now, we can get the following theorem.
When u − < u + < A a , the constant-density state does not appear in the solution, which means that there exists a positive constant a 1 and when u − < u + < A a 1 the Riemann solution only contains two rarefaction waves (R 1 and R 2 ) and the two given constant states (u ± , ρ ± ). Meanwhile, when A a < u + , the constant-density state occurs, which means that there exists a positive constant a 2 , and when A a 2 < u + the intermediate state between the backward rarefaction wave R 1 and the forward rarefaction wave R 2 is just a constantdensity state.
We set g(a) � with ρ − 0 �������������� � (((s + 2a 2 ) c− 2 )/s)ds being convergent according to Cauchy criterion, we can prove that the integral ))ds is uniformly convergent in a, and then the function g(a) is continuous with respect to a. In addition, it is easy to get that g(a 1 )g(a 2 ) < 0.
(48) en, it is obvious to get lim ϵ,k,b⟶0 ρ Π * � 0. Furthermore, we get that lim ϵ,k,b⟶0 Based on above analysis, we conclude that the limit solution for this case is a solution to transport equations (2) with (4), containing a vacuum state (ρ � 0) and two contact discontinuities (ξ � x/t � u ± ) on either side. is completes the proof of eorem 2.

Numerical Results
In this section, we simulate the formation process of delta shocks and vacuums. In order to discretize equations (1) and

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(3), we use the fifth-order weighted essentially nonoscillatory scheme and third-order Runge-Kutta method with 150 × 150 cells. For the sake of convenience, we take c � 1.4.
To illustrate the formation of the delta shock, we take the following initial data: Here, we begin with ϵ � 0.1, k � 1, and b � 0.05 and then choose ϵ � 0.05, k � 0.09, and b � 0.001, and finally we choose ϵ � 0.00001, k � 0.00001, and b � 0.00005. e corresponding numerical results at t � 2.0 are listed in Figures 1-3, which present the formation process of a delta shock in the two-shock solution of equations (1) and (3) as the van der Waals gas pressure, the molecular excluded volume, and the flux approximation all vanish. From Figures 1-3, it is easy to find that when the values of ϵ, k, and b become smaller and smaller, the locations of S 1 and S 2 get closer and closer, and the intermediate density ρ Π * increases dramatically, and at the same time the velocity turns to be a step function. When ϵ, k, and b vanish, two shocks S 1 and S 2 coincide with each other and a delta shock wave develops, while the velocity remains a step function.
In this case, we start with ϵ � 0.1, k � 1, and b � 0.05 and then choose ϵ � 0.05, k � 0.09, and b � 0.001, and finally we choose ϵ � 0.00001, k � 0.00001, and b � 0.00005. e corresponding numerical results at t � 2.0 are shown in Figures 4-6, which describe the formation of a vacuum state in the Riemann solution containing two rarefaction waves (R 1 and R 2 ) and a nonvacuum intermediate state for equations (1) and (3) as the van der Waals gas pressure, the molecular excluded volume, and the flux approximation vanish. Figures 4-6 show that as ϵ, k, and b decrease, the locations of R 1 and R 2 get closer and closer, and ρ Π * tends to zero leading to an inside vacuum state as ϵ, k, b ⟶ 0, and at the same time the velocity tends to a linear function. ese numerical results illustrate that Riemann solution of equations (1) and (3) consisting of R 1 and R 2 converges to the vacuum solution of system (2) including a vacuum state (ρ � 0) as ϵ, k, b ⟶ 0.

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To sum up, all of above numerical simulations completely support the theoretical analysis.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this article.