CONCENTRATION AND CAVITATION IN THE VANISHING PRESSURE LIMIT OF SOLUTIONS TO A 3 × 3 GENERALIZED CHAPLYGIN GAS EQUATIONS

. The phenomena of concentration and cavitation are identiﬁed and analyzed by studying the vanishing pressure limit of solutions to the 3 × 3 isentropic compressible Euler equations for generalized Chaplygin gas (GCG) with a small parameter. It is rigorously proved that, any Riemann solution containing two shocks and possibly one-contact-discontinuity of the GCG equations converges to a delta-shock solution of the same system as the parameter decreases to a certain critical value. Moreover, as the parameter goes to zero, that is, the pressure vanishes, the limiting solution is just the delta-shock solution of the pressureless gas dynamics (PGD) model, and the intermediate density between the two shocks tends to a weighted δ -measure that forms the delta shock wave; any Riemann solution containing two rarefaction waves and possibly one contact-discontinuity tends to a two-contact-discontinuity solution of the PGD model, and the nonvacuum intermediate state in between tends to a vacuum state. Finally, some numerical results are presented to exhibit the processes of concentration and cavitation as the pressure decreases.


Introduction
It is universally acknowledged that fluids are substances whose molecular structure offers no resistance to external shear forces: even the smallest force causes deformation of a fluid particle. In most cases of interest, a fluid can be regarded as continuum, thus satisfies the balance laws of density, momentum, and energy. However, if the fluid is isentropic, then just the balance laws of density and momentum should be taken into account. In this situation, the pressure, as a thermodynamic variable, only depends on density. Such dependence is usually called as the state equation, which is possible to be estimated from statistic mechanics or kinetic theory and is usually obtained by laboratory measurement.
On the other hand, as stated in [9], in the compressible fluid flow, if the speed is larger than the sound speed, shocks may form when particles collide. However, as observed numerically in [4][5][6] for gas dynamics in the regime of small pressure: for one case, the particles seem to be more sticky and tend to concentrate at some shock locations which move with the associated shock speeds, and for the other case, the particles seem to be far apart and tend to form cavitation in the region of rarefaction waves. Such phenomena may be regarded as a tendency towards the concentration and cavitation in terms of the density.
One of the main objectives of this paper is to show rigorously that the phenomena of concentration and cavitation in the solutions are fundamental and physical in the following 3 × 3 isentropic compressible Euler equations consisting of three scalar equations, namely, the conservation of mass and two linear momentums, in which ρ > 0 and (u, v) represent the density and velocity of the fluid, respectively, p = p(ρ) stands for the scalar pressure.
In order to analyze the phenomena of concentration and cavitation in solutions, in this paper, the state equation for (1.1) is considered as the generalized Chaplygin gas (GCG) which reflects the relation between pressure p and density ρ, where 0 < α < 1 is constant, > 0 is a small scaling perturbed parameter modeling the strength of pressure p. The GCG (1.3) with the negative pressure and positive sound speed was proposed in [1], where its relation to a scalar-field Lagrangian of a generalized Born-Infeld type was clarified. In present, the cosmological models based on the dynamics of generalized Chaplygin gases have attracted considerable interest [2,21]. Specially, when α = 1, (1.3) is called the Chaplygin gas (CG), which is connected to string theory and was initially introduced by Chaplygin [7], Tsien [26], and von Karman [27] as a suitable mathematical approximation to compute the lifting force on an airplane wing in aerodynamics. Currently, the CG and GCG are regarded as the possible phenomenological models for dark energy and used to describe the accelerating expansion of the universe. It should be claimed that, although the parameter in (1.3) can be considered very small and reflects the strength of the underlying pressure, it does not vanish in general. We propose to include this parameter in hopes of understanding the process of the formation of concentration and cavitation in the isentropic compressible Euler equations (1.1). Particularly, as → 0, that is, the pressure vanishes, the system (1.1) formally becomes the pressureless gas dynamics (PGD) model which is also called the zero-pressure flow model or constant pressure fluid dynamics. The system of zero-pressure type fluid dynamics is usually used to describe some important physical phenomena, such as the motion of free particles sticking together under collision [3] and the formation of large scale structures in the universe [22,30].
The Riemann problem for (1.4) with piecewise constant initial data has been studied by Hu [13], where ρ ± > 0, u ± and v ± are constants. The Riemann solution has been constructed by virtue of the viscosity vanishing approach. Interestingly, a special type of nonlinear singular shock wave called the delta shock wave is found in solution. Meanwhile, the vacuum state also appears. Since the three eigenvalues of the zero-pressure flow (1.4) coincide, the occurrence of delta shock waves and vacuum states as t > 0 may be regarded as a result of resonance among the three characteristic fields. Besides, one can refer to [34] for the interactions of delta shock waves and vacuum states for (1.4). Mathematically, the delta shock wave and vacuum state are used to describe the phenomena of concentration and cavitation, respectively. A delta shock wave is a new kind of discontinuity, on which at least one of the state variables may develop an extreme concentration in the form of a weighted Dirac delta function with the discontinuity as its support. It is more compressive than a traditional shock wave in the sense that more characteristics will enter the discontinuity line. Nowadays, the delta shock wave has become a hot topic in the study of hyperbolic conservation laws. Besides, it is worth recalling that, although the delta shock wave is also obtained in solutions of (1.1) and (1.2) for CG (see [12,25]), the occurrence mechanism of which is essentially different from that of the PGD model (1.4). For the former, the system is strict hyperbolic and owns three kinds of degenerated characteristics. While for the later, the system has three repeated degenerated characteristics and as a result the strict hyperbolicity fails.
In order to rigorously justify the formation of concentration and cavitation as well as the appearance of delta shock wave and vacuum state, an effective approach is the so called vanishing pressure limit method. This method may date back to the beginning of 21st century when Li [16] and Chen and Liu [8,9] studied the asymptotic behavior of solutions to the compressible Euler equations by means of imposing the pressure or temperature drop to zero. Since then, by using this method, a large number of scholars have investigated the formation mechanism of delta shocks and vacuums in various conservation laws and achieved a series of fruitful results, which makes the researches on this subject particularly noticeable.
As for the vanishing pressure limit of solutions to CG and GCG equations, some literatures have been contributed. For example, Sheng et al. [24] studied the vanishing pressure limit of solutions to the isentropic Euler equations for generalized Chaplygin gas; Zhang et al. [33] analyzed the concentration and cavitation in the vanishing pressure limit of solutions to the generalized Chaplygin Euler equations of compressible fluid flow. See also Pan and Han [18] for the Aw-Rascle traffic flow model with CG equation, Yin and Song [29] as well as Li and Shao [15] for the CG and GCG for relativistic fluid, etc. Recently, the authors in [11,17] discussed the vanishing pressure limit of Riemann solutions to the non-isentropic Euler equations for CG and GCG, respectively. Besides, the vanishing pressure limit of solutions to (1.1) with polytropic gas was studied by Zhang [31].
Motivated by the above discussions, we in this paper focus on the phenomena of concentration and cavitation as well as the limit behavior of solutions to the GCG equations (1.1), (1.3) in vanishing pressure. It is shown that, when decreases to a certain critical value, any Riemann solution of (1.1), (1.3) containing two shocks and possibly one-contact-discontinuity converges to a delta-shock solution of the system itself. Moreover, as → 0, the limiting solution is nothing but the delta-shock solution of the PGD model (1.4), and the intermediate density between the two shocks tends to an extreme concentration in the form of a weighted δ-measure that forms the delta shock wave. By contrast, any Riemann solution of (1.1), (1.3) containing two rarefaction waves and possibly one contact-discontinuity tends to a two-contact-discontinuity solution of PGD model (1.4), and the nonvacuum intermediate state in between tends to a vacuum state. These indicate a physical fact in fluid dynamics, that is, as the limit of solution in vanishing pressure, a delta shock wave for the PGD model (1.4) is a result of concentration of density, while a vacuum state is a result of cavitation. It is noticed that, our result is similar to that of Shen and Sun [23], in which the formation of delta shock wave and vacuum state for a 3 × 3 two-phase flow model was systemically studied.
Finally, by the second-order non-oscillatory central schemes [14], we examine the formation processes of delta shock waves and vacuum states with some numerical results as decreases. The numerical simulations are completely coinciding with the theoretical analysis.
This paper is organized as follows. Section 2 restates the Riemann solutions to the PGD model (1.4). In Section 3, we solve the Riemann problem (1.5) for the GCG equations (1.1), (1.3), and discuss the dependence of solutions on the parameter . In Sections 4 and 5, we investigate the limits of solutions to (1.1), (1.3). Section 6 exhibits some numerical results.

Delta shocks and vacuums for the PGD model (1.4)
We will begin with reviewing delta shock waves and vacuum states of (1.4), (1.5). Readers can refer to [13,31] for more details.
From ∇λ · − → r i ≡ 0 (i = 1, 2), we know that the characteristic λ is linearly degenerate. Thus, the elementary waves involve only contact discontinuities. Noting that (1.4) and (1.5) are invariant under the uniform stretching of coordinates: (t, x) → (βt, βx) (β is constant), it follows that if the solution is unique, then the solution must depend on x/t alone. Thus, we look for the self-similar solutions of (1.4) and (1.5) as follows Then, the Riemann problem (1.4) and (1.5) is reduced to the boundary value problem with the boundary condition (ρ, u, v)(±∞) = (ρ ± , u ± , v ± ). As in [13,31], the solution of (1.4), (1.5) can be constructed by two cases. When u − < u + , the solution containing two contact discontinuities and a vacuum state can be shown as in which u(ξ) and v(ξ) are two smooth functions. When u − > u + , one needs to introduce the delta-shock solution since the singularity of solution must develop due to the overlap of characteristic lines. For this purpose, the definition of a weighted delta function supported on a curve is given at first.
Based on this definition, we can introduce a family of delta-shock solutions with the parameter σ to construct the solution of (1.4), which takes the form σ the velocity of the delta shock wave, and χ(x) the characteristic function that is 0 when x < 0 and 1 when x > 0.
Moreover, the the delta-shock solution of (1.4) constructed above satisfies and v has the similar integral identities as above.
Then, we look for a piecewise smooth solution to (1.4) in the form where w(t) is the strength of the delta shock wave, and u δ , v δ are the corresponding values of u and v on the discontinuous curve x = x(t).
The solution (2.8) should obey the generalized Rankine-Hugoniot relation (2.9) and the entropy condition Here, we use dx dt = σ 0 = u δ to denote the propagation speed of delta shock wave for the reason that the concentration of ρ needs to travel at the same propagation speed of the discontinuity.

Solution involving classical waves
The eigenvalues for system (1.1) are with the associated right eigenvectors respectively. Clearly, (1.1) is strictly hyperbolic. Denoted by ∇ = (∂ ρ , ∂ u , ∂ v ), then for 0 < α < 1, one can check that So, λ 1 and λ 3 are genuinely nonlinear for ρ > 0 and λ 2 is always linearly degenerate. As in Section 2, we look for the self-similar solution (ρ, u, v)(t, x) = (ρ, u, v)(ξ) (ξ = x/t), then the Riemann problem (1.1) and (1.5) changes into the boundary value problem For smooth solution, (3.3) can be rewritten as which provides either the general constant solution (ρ, u, v) = Const., or the backward centered rarefaction wave or the forward centered rarefaction wave For a given left state (ρ − , u − , v − ), the possible states (ρ, u, v) that can be connected to (ρ − , u − , v − ) by a backward or forward centered rarefaction wave are symbolized by (3.5) to obtain the backward centered rarefaction wave Similarly, the forward centered rarefaction wave is derived by integrating (3.6) with the requirement λ 3 (ρ + , u + , v + ) > λ 3 (ρ, u, v). Analogously, for a given state (ρ + , u + , v + ), we can obtain For the bounded discontinuity at ξ = ω , it obeys the Rankine-Hugoniot condition , from which we have either a backward shock wave a contact discontinuity or a forward shock wave Similar to the rarefaction waves, for a given state (ρ − , u − , v − ), the possible states (ρ, u, v) that can be connected to (ρ − , u − , v − ) by a backward or forward shock wave are symbolized by Then, by (3.10) and the Lax shock inequalities λ 1 (ρ, u, v) < ω 1 < λ 1 (ρ − , u − , v − ) and ω 1 < λ 2 (ρ, u, v), we obtain the backward shock wave Using (3.11), we get the contact discontinuity curve which is the set of states that can be joined with the left state (ρ − , u − , v − ) by a contact discontinuity. Analogously, using (3.12) and the Lax shock inequalities we obtain the forward shock wave curve Similarly, for a given state (ρ + , u + , v + ), we can obtain Then, by a similar analysis as in [24,25,28], we know that du − . In addition, as done in [24,25,28], we draw a curve S δ as follows which is indeed the projection of the boundary for the region where nonclassical solutions appear. Then, the upper-half (u, ρ)-phase plane is divided into five regions I, II, III, IV and V , as shown in Figure 1. By the analysis method in phase plane, we construct the solution of (1.1) and (1.3). When the projection , the Riemann solution to (1.1) and (1.3) can be expressed in the following form: where (ρ * , u * , v * 1 ) and (ρ * , u * , v * 2 ) are the intermediate states satisfying Figure 2. Analysis of characteristics for delta-shock of (1 .1) and (1.3).
, the Riemann solution to (1.1) and (1.3) admits four kinds of different configurations, which consist of a centered rarefaction wave, a shock wave, and a contact discontinuity besides the constant states.

Delta shock wave solution
When the projection of (ρ + , u , we need to seek a nonclassical solution. In fact, from the above discussions, we know that the nonclassical solution may occur under the condition that . In this situation, we have where (±) = (ρ ± , u ± , v ± ), namely, which means that the characteristic lines from initial data will overlap in the domain Ω, as shown in Figure 2. So singularity must happen in Ω. It is well known that the singularity is impossible to be a jump with finite amplitude, which implies that there is no solution that is piecewise smooth and bounded. Motivated by [19,20,24,25,28], etc., the solution for (1.1) with delta distribution at the jump (i.e., the delta-shock solution) should be constructed, which is, under the Definition 2.1, where S = {(t, σ t) : 0 ≤ t < ∞}, and where 1 ρ α is defined as [24,28] 1 As before, we seek a delta shock wave solution with the discontinuity x = x (t) to (1.1) and (1.3) in the form where w (t) is the strength of the delta shock wave, and u δ , v δ are the corresponding values of u and v on the discontinuous curve x = x (t). (3.21) Proof. We only prove the third one of (3.21). As a matter of fact, for any test function φ ∈ C ∞ 0 (R + × R 1 ), using the previous symbols in (2.7), we have Then, I 3 can be composed as By Green's formulation and integrating by parts, we have Thus, the third equation of (3.21) is obtained by taking I 3 = 0. The second and last equations can be proved similarly. This is the end of the proof.
To guarantee the uniqueness, the delta-shock solution must obey the entropy condition which means that all of the six characteristic lines on both sides of the discontinuity are incoming. A discontinuity satisfying (3.21) and (3.22) is called a delta shock wave to the system (1.1) and (1.3), denoted by δ S , as shown in Figure 2.
Under the entropy condition (3.22), via solving the generalized Rankine-Hugoniot conditions (3.21) with initial data x (0) = 0 and w (0) = 0, we have, by a trivial calculation, for ρ − = ρ + , and  1) and (1.3). A key point is to show how the solution changes along with the values of → 0. Thus, we introduce two critical values 0 and 1 , which will play important roles in the following discussion.
From the above analysis, we observe that, when 0 < < 1 , namely, (ρ + , u + ) ∈ IV , the curves ← − S and − → S will become steeper as is much smaller. At this time, since (ρ + , u + ) ∈ IV , so the solution of (1.1) and (1.3) can be formulated as Here, ω 1 , ω 2 and ω 3 are the propagation velocity of ← − S , J and − → S , respectively. We now give some lemmas to show the limit behavior of solutions to (1.1) as → 0 .
Proof. Together with the second equations of (4.4) and (4.6), we have from which, by taking → 0 on both sides, one has , one has lim → 0 ρ * = +∞. The proof is completed.
Proof. From the velocity relation for ← − S and J, one has . Similarly, we obtain from the velocity relation for J and − → S that lim → 0 ρ * (ω 3 − ω 2 ) = ρ + (σ − u + ). As a result, we have Moreover, it yields from the Rankine-Hugoniot relation (3.9) for which leads to This is the end of the proof.
Lemmas 4.2-4.4 imply that, as → 0 , the two shocks ← − S and − → S and the contact discontinuity J coincide, and the intermediate density ρ * becomes singular. Let us give the following result which gives a very nice depiction of the limit in the case u − > u + . Theorem 4.5. Let u − > u + . For each fixed ∈ ( 0 , 1 ), assume that (ρ , u , v ) is a solution of (1.1) and (1.3) with (1.5), which consists of two shocks ← − S , − → S and a contact discontinuity J as constructed in Section 3. Then, ρ , ρ u and ρ v converge in the sense of distributions as → 0 , and the limit functions ρ, ρu and ρv are the sums of a step function and a Dirac delta function with weights Proof. (i). Set ξ = x/t. For each fixed ∈ ( 0 , 1 ), the solution of (1.1) reads Moreover, for any test function φ ∈ C ∞ 0 (−∞, +∞), the solution (4.8) satisfies (ii). Let us discuss the limits of ρ u , ρ v and ρ depending on ξ. For the first integral on the left of (4.10), we have (4.12) For the first and last terms on the right of (4.12), it can be calculated that where H 2 (ξ − σ) = ρ ± u ± , ±(ξ − σ) > 0. While for the second and third terms on the right of (4.12), the limit of which when → 0 equals (4.14) From (4.13) and (4.14), it follows that We turn to the second term on the left of (4.10), one has Thereby, by substituting (4.15) and (4.16) into (4.10), we immediately obtain that for any test function φ ∈ C ∞ 0 (−∞, +∞). Let us turn to (4.11). For the first term on the left of (4.11), it can be written as The first and last terms on the right of (4.18) can be calculated as where While for the middle two terms on the right of (4.18), noting that Lemma 4.4. So,using (4.18) and (4.19) again, we have (4.20) Taking (4.20) into (4.11) yields that In a similar manner, from (4.9), we can prove that where H 1 (ξ − σ) = ρ ± , ±(ξ − σ) > 0. (iii). Finally, we check the limits of ρ , ρ u and ρ v by tracing the time-dependence of weights of the δ-measure as → 0 . Then, for any ψ(t, x) ∈ C ∞ 0 (R + × R 1 ), we have by using (4.17), in which according to the Definition 2.1. Analogously, we can conclude that . We finish the proof of Theorem 4.5.
Theorem 4.6. If u − > u + and the projection of the state (ρ + , u + , v + ) onto the (ρ, u) phase plane (ρ + , u + ) ∈ V , then the limit of the delta shock wave of (1.1) and (1.3) with (1.5) when → 0 is a delta shock wave solution of the PGD model (1.4) with the same initial data.
Proof. When (ρ + , u + ) ∈ V , the delta shock wave solution of (1.1) and (1.3) with (1.5) is expressed in (3.23) and (3.24). Then, it is easily checked that for ρ − = ρ + , and  [24,29,33], when (ρ + , u + ) ∈ IV , the limit of Riemann solutions to the GCG equations (1.1) and (1.3) as → 0 is nothing but the delta-shock solution of (1.1) and (1.3) itself in the case (ρ + , u + ) ∈ S δ , where the curve S δ is actually the boundary between the regions IV and V . From the results of this section, we conclude that the two shocks ← − S , − → S and possibly a contact discontinuity J coincide as a delta shock wave of (1.1) and (1.3) itself when decreases to a certain critical value 0 . Moreover, as continues to drop and in the end, tends to zero, the delta shock solution is just that of the PGD model (1.4). Form this point of view, our results can be regarded as a generalization of [24,33]. In this section, we analyze the cavitation phenomenon. To this end, we need to discuss the limit behavior of solutions to the Riemann problem (1.1), (1.3) and (1.5) for u − < u + .
We conclude the following theorem.
Theorem 5.2. Let u − < u + . For each fixed ∈ (0,˜ 0 ), assume that (ρ , u , v ) is a solution of (1.1) and (1.3) consisting of two rarefaction waves ← − R , − → R and a contact discontinuity J, as constructed in Section 3. Then, as → 0, the two rarefaction waves and possibly one contact discontinuity become two contact discontinuities connecting the constant states (ρ ± , u ± , v ± ) and vacuum state (ρ = 0), which form a vacuum solution of the PGD model (1.4).

Numerical simulations
We in this section present some representative numerical simulations to show the validity of concentration and cavitation studied in Sections 4 and 5. Many more numerical tests have been performed to make sure that what are presented are not numerical artifacts. To discretize the system, we employ the second-order non-oscillatory central schemes [14] with 150 × 150 cells and CFL=0.475. For convenience, we take α = 0.5 in (1.1) and (1.3).    Then, we get from (4.2) that 0 ≈ 0.0089 for this group of initial data. The numerical results with = 1, = 0.0089 and = 0.001 at t = 0.3 are presented in Figures 3-5 to show the influence of the perturbed parameter on the formation of delta shock wave.
It is clearly observed that, there is no concentration in the solution of (1.1) as = 1, that is, for a fluid with strong pressure, no concentration occurs generally. However, as drops to 0 approximately, the velocities u  and v become step functions and the intermediate density ρ * increases dramatically, namely, the concentration occurs. Finally, as → 0, the concentration of density ρ * leads to a delta shock wave in the limit, and the changes of velocities u and v yield two step functions. Therefore, the delta shock wave of the PGD model (1.4) is really the limit of the delta shock wave of the GCG equations (1.1) as → 0. Case 2. u − < u + . In this situation, the initial data are chosen as Based on this group of initial data, we present some numerical results with = 1 and = 0.003 at t = 0.6 to indicate the influence of the perturbed parameter on the formation of vacuum state, please see Figures 6 and 7. The numerical results in Figures 6 and 7 imply that, as decreases, the intermediate density ρ * infinitely tends to 0, which results in cavitation, and the velocities u and v are closer to two step functions. Thus, the vacuum solution of (1.4) can be obtained as the limit of any solution of (1.1) consisting of two rarefaction waves and possibly one contact discontinuity.
In a word, the numerical results presented above coincide with the theoretical analysis.