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.
In this paper, we consider the isentropic Euler equations with the flux perturbation:where denotes the velocity, denotes the pressure, and denotes the density satisfying . Here, 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 . We also call this system the zero-pressure gas dynamics. For more details, readers can see [4–6].
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  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 [10–16]. Furthermore, readers can also see  for the perturbed system of generalized pressureless gas dynamics model and  for the case of triangular conservation law system arising from “generalized pressureless gas dynamics model.” In addition, Yang and Liu  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.
Recently, Wang et al.  considered the isentropic Euler equations for van der Waals gas:where denotes the molecular excluded volume satisfying , denotes a positive constant, and denotes the adiabatic exponent with . It is easy for us to find that when , 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. Therefore, 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. This 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 .
In this paper, we rigorously prove that as , 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 simultaneously, which means that our work can also be regarded as the extension of that in [8, 19].
The 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.
2. 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 are arbitrary constants and then we construct Riemann solutions of this system.
For equations (1) and (3), the eigenvalues and corresponding right eigenvectors arerespectively. By direct calculation, we can obtain that , which implies that both eigenvalues are genuinely nonlinear. Thus, the elementary waves of this system contain rarefaction waves and shock waves.
Now, we consider the smooth solution of (6) and get either the constant state solution or the backward rarefaction waveor the forward rarefaction wave
From these two conditions, we find that the velocity of the backward rarefaction wave is monotonically decreasing with respect to , while the forward one is increasing. Furthermore, integrating the second equations of (8) and (9), respectively, by yields
In the -plane, we call the curve of the second equation of (12) the backward rarefaction wave curve, which is monotonically decreasing with respect to by a direct calculation from the second equation of (8). Further, for this curve, we obtain that . In fact, ; then, the integral is convergent according to Cauchy criterion. Therefore, this curve and the line 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 from the second equation of (9). Since we have proved in  that and it is obvious to find , we obtain that from the second equation of (13).
Now, we turn to a bounded discontinuity at . For equations (1) and (3), the Rankine–Hugoniot compatibility conditionshold, where denotes the jump of function across the discontinuity. By solving (14), in terms of the stability conditionsfor , andfor , we obtain the backward shock and the forward shock as follows:
In the -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 obtainwhich means that the backward shock curve is monotonically decreasing. Similarly, we have for the forward shock wave, which implies that the forward shock curve is monotonically increasing. Further, a direct calculation gives that for the backward shock curve and for the forward shock curve, which implies that this forward shock wave curve and the straight line have an intersection point .
Based on the aforementioned analysis, we conclude that the elementary waves of equations (1) and (3) contain two rarefaction waves ( and ) and two shock waves ( and ). Using the curves of these elementary waves, we can divide the -plane into four regions , , , and for any fixed left state . Furthermore, for any fixed right state , we can construct a unique Riemann solution, no matter which of the four regions the right state belongs to. Specifically, when , the Riemann solution can be constructed with two shocks ( and ) besides a nonvacuum constant state in between. When , the Riemann solution can be constructed with two rarefaction waves ( and ) besides an intermediate constant state which may be a constant-density solution (). The discussion for the other two cases and is trivial, so in this paper we only consider the limit process for the cases and .
3. Formation of Delta Shocks and Vacuums as
3.1. Formation of Delta Shocks
3.1.1. Limit Behavior of the Riemann Solutions as
For the case , we assume that is the intermediate state. Then the left and right states of the backward shock (the forward shock ) are and ( and ), respectively. Thus, we havefor , andfor . Here () denotes the speed of .
Now, we are ready to present our main results for the formation of delta shocks.
Theorem 1. Let and . For any fixed , suppose that is a Riemann solution of equations (1) and (3) with (4) containing two shocks and . Then as , and converge, in the sense of distributions, to the sums of a step function and a -measure with weightsrespectively, which just form the delta shock solution of system (2) with (4).
Before we prove this theorem, some lemmas should be presented. The proof process of these lemmas is similar to that in , so we do not repeat it here.
Lemma 1. .
Lemma 3. Set ; then .
Lemma 5. For above quantity , we can obtain
On the basis of the above results, we find that as , the velocity of backward shock , the velocity of forward shock , and the intermediate velocity of equations (1) and (3) tend to the same quantity , which is proposed for the delta shock solution of system (2), and becomes singular simultaneously.
3.1.2. Proof of Theorem 1
Now, we give the rigorous proof of Theorem 1.
Proof of Theorem 1.
Step 1. Set . By the two-shock Riemann solutionwe can obtain the weak formulationsfor any test function .
Step 2. Here, we prove that the limit functions of and are the sums of a step function and a -measure. We rewrite the first term on the left of (27) and obtainWith the analysis of integration by part, noticing Lemmas 3 and 4, we haveA combination of (29) and (30) gives thatwhereSimilarly, we calculate the second integral on the left side of (27), noticing Lemmas 1–3, and obtain thatSubstituting (31) and (33) into (27), we havefor any .
In the same way, from (26), we can get thatfor any , where
Step 3. We turn to prove the weights of the -measures. For any test function , by (34), we obtainwhich together with definition (2.3) in  yields thatwhereSimilarly, from (35), we obtainand then we get thatwhereThis completes the proof of Theorem 1.
3.2. Formation of Vacuums
In this subsection, the limit of Riemann solutions of equations (1) and (3) is considered in the case with as . For this case, we assume that is the intermediate state. Then the left and right states of the backward rarefaction wave (the forward rarefaction wave ) are and ( and ), respectively. Then, for the backward rarefaction wave , we have
For the forward rarefaction wave , we have
Now, we can get the following theorem.
Theorem 2. Let and . For any fixed , suppose that is a Riemann solution of equations (1) and (3) with (4) containing two rarefaction waves and . Then, there exists a positive constant , and the constant-density solution occurs in the solution when . As , and become two contact discontinuities, which connect the vacuum and constant states .
Proof. Set . Since is on the curve , we obtainWhen , the constant-density state does not appear in the solution, which means that there exists a positive constant and when the Riemann solution only contains two rarefaction waves ( and ) and the two given constant states . Meanwhile, when , the constant-density state occurs, which means that there exists a positive constant , and when the intermediate state between the backward rarefaction wave and the forward rarefaction wave is just a constant-density state.
We set , . Sincewith being convergent according to Cauchy criterion, we can prove that the integral is uniformly convergent in , and then the function is continuous with respect to . In addition, it is easy to get that . Thus, there exists , such that .
Therefore, when , the intermediate state becomes a constant state withHere,Then, it is obvious to get . Furthermore, we get thatBased 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 () and two contact discontinuities () on either side. This completes the proof of Theorem 2.
4. Numerical Results
In this section, we simulate the formation process of delta shocks and vacuums. In order to discretize equations (1) and (3), we use the fifth-order weighted essentially nonoscillatory scheme and third-order Runge-Kutta method with cells. For the sake of convenience, we take .
To illustrate the formation of the delta shock, we take the following initial data:
Here, we begin with , , and and then choose , , and , and finally we choose , , and . The corresponding numerical results at 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 , , and become smaller and smaller, the locations of and get closer and closer, and the intermediate density increases dramatically, and at the same time the velocity turns to be a step function. When , , and vanish, two shocks and coincide with each other and a delta shock wave develops, while the velocity remains a step function.
For the formation of vacuum states, we take the following initial data:
In this case, we start with , , and and then choose , , and , and finally we choose , , and . The corresponding numerical results at are shown in Figures 4–6, which describe the formation of a vacuum state in the Riemann solution containing two rarefaction waves ( and ) 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 , , and decrease, the locations of and get closer and closer, and tends to zero leading to an inside vacuum state as , and at the same time the velocity tends to a linear function. These numerical results illustrate that Riemann solution of equations (1) and (3) consisting of and converges to the vacuum solution of system (2) including a vacuum state as .
To sum up, all of above numerical simulations completely support the theoretical analysis.
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
All authors read and approved the final manuscript.
This work was supported by the Natural Science Foundation of Hebei Province (Grant no. A2019105110), the Science and Technology Foundation of Hebei Education Department (Grant no. BJ2020203), and the Scientific Research Foundation of Tangshan Normal University (Grant no. 2020C02).
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