# The Vanishing Pressure Limit of Riemann Solutions to the Non-Isentropic Euler Equations for Generalized Chaplygin Gas

Authors: Qixia Ding; Lihui Guo
Journal: Advances in Mathematical Physics
 (2019)

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## 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 densityρ and the internal energy H 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.


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## Body

## 1. Introduction

The compressible Euler equations of non-isentropic fluids in one dimension are given by(1)ρt+ρux=0,ρut+ρu2+Px=0,ρu22+Ht+ρu22+H+Pux=0,where ρ is the density, u the velocity, P the pressure, H=ρe≥0 the internal energy, and e the internal energy per unit mass. In order to analyze the limit behavior of solutions, we assume P is a known function P(ρ)=εp(ρ) of the density ρ for sufficiently small ε>0. The generalized Chaplygin gas is a natural extension of Chaplygin gas, and its equation of state is as follows:(2)pρ=-1ρα,0<α≤1.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 equations(3)ρt+ρux=0,ρut+ρu2+Pρx=0.The 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 pressureP vanishes, (1) converge to transport equations(4)ρt+ρux=0,ρut+ρu2x=0,and conservation law of energy(5)ρu22+Ht+ρu32+Hux=0.Equations (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 δ-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 T→0 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 P→0. 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:(6)u,ρ,Hx,0=u-,ρ-,H-,x<0,u+,ρ+,H+,x>0,in which ρ±>0 and H±>0. 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 δ-shock wave. In 2004, Chen and Liu [31] studied non-isentropic Euler equations for polytropic gas(7)pρ=γ-1ρE-ρu22,where ρE 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 inequality(8)ρu2t+ρu3x≥0.In one-dimensional case, we need to prove the following conclusions:( 1 ) For the case of u+<u-, Riemann solution of system (1)–(2) tends to a δ-shock wave solution when ε reduces to a certain value ε2, and the δ-shock wave tends to a δ-shock solution of zero-pressure flow (4) as ε→0.( 2 ) For the case of u+>u-, Riemann solution of system (1)–(2) converges to solution involving vacuum state of system (4) as ε→0.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 Section2, 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:(9)u,ρx,0=u-,ρ-,x<0,u+,ρ+,x>0,where ρ±>0; see [18] for the details.In the case ofu+>u-, the Riemann solution of system (4) is expressed by(10)uξ,ρξ=u-,ρ-,-∞<ξ≤u-,uξ,0,u-≤ξ≤u+,u+,ρ+,u+≤ξ<+∞.In the case ofu+<u-, the Riemann solution is(11)u,ρx,t=u-,ρ-,x<xt,σ,wtδx-xt,x=xt,u+,ρ+,x>xt,where x(t), w(t), and σ denote the position, weight, and propagation speed of δ-shock wave and satisfy(12)xt=σt,σ=ρ+u++ρ-u-ρ++ρ-,wt=ρ+ρ-u--u+t,as [ρ]=ρ+-ρ-≠0, and(13)xt=σt,σ=u++u-2,wt=ρ-u--ρ+u+t,as [ρ]=ρ+-ρ-=0.


## 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(14)ℵ=u,ρ,H∣ρ>0,H≥εα+1ρα,-∞<u<+∞. Three real and distinct eigenvalues for system (1)–(2) are (15)λ1=u-εαρ-α+1/2,λ2=u,λ3=u+εαρ-α+1/2,with the associated right eigenvectors(16)r1→=ρ,-εαρ-α+1/2,H-ερ-αT,r2→=0,0,1T,r3→=ρ,εαρ-α+1/2,H-ερ-αT. By calculation, we obtain(17)∇λi·ri→≠0i=1,3,∇λ2·r2→=0. 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 curveR in the space (u,ρ,H), passing through (u-,ρ-,H-), can be expressed as(18)Rε←u-,ρ-,H-:λ1=u-εαρ-α+1/2,u-u-=2εαα+1ρ-α+1/2-2εαα+1ρ--α+1/2,ρ-H-εα+1ρ-α=ρH--εα+1ρ--α,ρ<ρ-,and(19)Rε→u-,ρ-,H-:λ3=u+εαρ-α+1/2,u-u-=2εαα+1ρ--α+1/2-2εαα+1ρ-α+1/2,ρ-H-εα+1ρ-α=ρH--εα+1ρ--α,ρ>ρ-.Analogously, we obtain shock wave curve(20)Sε←u-,ρ-,H-:σ1ε=u--ερ1/ρ-α-1/ραρ-ρ-ρ-,u=u--ε1ρ--1ρ1ρ-α-1ρα,ρ-H-ε21ρ-α+1ρα=ρH--ε21ρ-α+1ρα,ρ>ρ-,and(21)Sε→u-,ρ-,H-:σ3ε=u+ερ-1/ρ-α-1/ραρρ-ρ-,u=u--ε1ρ--1ρ1ρ-α-1ρα,ρ-H-ε21ρ-α+1ρα=ρH--ε21ρ-α+1ρα,ρ<ρ-.And a contact discontinuity satisfies(22)Jεu-,ρ-,H-:σ2ε=u=u-,ρ=ρ-,H≠H-.The projections of curvesRε→←, Sε→←, Jε, and Sδε divide the phase plane (u,ρ) into five regions, as shown in Figure 1. Consequently, one can construct the global solutions of system (1)–(2).Figure 1
The phase plane (u,ρ).Whenu--ερ--α+1/2<u++ερ+-α+1/2, the structures of solutions can be expressed in the following form.(1)
The projection of(u+,ρ+,H+) onto (u,ρ) phase plane (u+,ρ+)∈I(u-,ρ-)(23)u-,ρ-,H-+Rε←+u∗ε,ρ∗ε,H∗1ε+Jε+u∗ε,ρ∗ε,H∗2ε+Rε→+u+,ρ+,H+;(2)
The projection of(u+,ρ+,H+) onto (u,ρ) phase plane (u+,ρ+)∈II(u-,ρ-)(24)u-,ρ-,H-+Rε←+u∗ε,ρ∗ε,H∗1ε+Jε+u∗ε,ρ∗ε,H∗2ε+Sε→+u+,ρ+,H+;(3)
The projection of(u+,ρ+,H+) onto (u,ρ) phase plane (u+,ρ+)∈III(u-,ρ-)(25)u-,ρ-,H-+Sε←+u∗ε,ρ∗ε,H∗1ε+Jε+u∗ε,ρ∗ε,H∗2ε+Rε→+u+,ρ+,H+;(4)
The projection of(u+,ρ+,H+) onto (u,ρ) phase plane (u+,ρ+)∈IV(u-,ρ-)(26)u-,ρ-,H-+Sε←+u∗ε,ρ∗ε,H∗1ε+Jε+u∗ε,ρ∗ε,H∗2ε+Sε→+u+,ρ+,H+; where (u∗ε,ρ∗ε,H∗1ε) and (u∗ε,ρ∗ε,H∗2ε) denote the intermediate states.When the projection of(u+,ρ+,H+) onto (u,ρ) phase plane (u+,ρ+) belongs to V(u-,ρ-), the solution is(27)u-,ρ-,H-+δSε+u+,ρ+,H+. The δ-shock wave satisfies the following generalized R-H jump conditions:(28)dxεtdt=uδt,dwεtdt=uδtρ-ρu,dwεtuδtdt=uδtρu-ρu2-ερα,dwεtuδ2t/2+hεtdt=uδtρu22+H-ρu22+H-εραu,with initial data(29)xε,uδ,wε,hεt=0=0,0,0,0.Algebraic manipulation of (28) and (29) shows that(30)xεt=uδtt,uδt=ρ+u+-ρ-u-+wεt′ρ+-ρ-,wεt=ρ+ρ-u--u+2+ε1ρ--1ρ+1ρ+α-1ρ-αt,as [ρ]=ρ+-ρ-≠0, and(31)xεt=uδtt,uδt=u++u-2,wεt=ρ-u--ρ+u+t,as [ρ]=ρ+-ρ-=0. Moreover, the δ-shock wave satisfies the δ-entropy condition(32)λ1+<λ2+<λ3+≤uδt≤λ1-<λ2-<λ3-.


## 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 ε→0. In the phase plane, we discuss the two cases of u+<u- and u+>u-.


### 4.1. Caseu+<u-

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 thatu+<u-, then there exists ε1>ε2>0 such that the projection of the state (u+,ρ+,H+) onto the (u,ρ) phase plane (u+,ρ+)∈IV(u-,ρ-) as ε∈(ε2,ε1); (u+,ρ+)∈V(u-,ρ-) as ε∈(0,ε2].Proof.
As a consequence ofu+<u-, we can easily conclude that (u+,ρ+)∈(IV∪V)(u-,ρ-) as ρ+=ρ-.
Whenρ+≠ρ-, we assume (u+,ρ+)∈IV(u-,ρ-), which can be joined with (u-,ρ-) by Sε← or Sε→ satisfying(33)u+<u--ερ--α-ρ+-αρ+ρ-ρ+-ρ-ρ+-ρ-,ρ+>ρ-,or(34)u+<u-+ερ--α-ρ+-αρ+ρ-ρ+-ρ-ρ+-ρ-,ρ+<ρ-,and from (33) and (34), it is found that(35)ε<ρ-α+1ρ+α+1u--u+2ρ+α-ρ-αρ+-ρ-, And, then, we take(36)ε1=ρ-α+1ρ+α+1u--u+2ρ+α-ρ-αρ+-ρ-.And so, (u+,ρ+)∈(IV∪V)(u-,ρ-) as ε<ε1.When(u+,ρ+)∈V(u-,ρ-), which can be joined with (u-,ρ-) directly by a delta shock wave curve satisfying the inequality,(37)u--ερ--α+1/2≥u++ερ+-α+1/2.From the above inequality, we obtain the expression(38)ε≤u--u+ρ--α+1/2+ρ+-α+1/22, letting(39)ε2=u--u+ρ--α+1/2+ρ+-α+1/22.It is shown that (u+,ρ+)∈V(u-,ρ-) as 0<ε≤ε2.This lemma shows that there is noδ-shock wave as ε>ε2. When 0<ε<ε1, the Riemann solution is(40)u-,ρ-,H-+Sε←+u∗ε,ρ∗ε,H∗1ε+Jε+u∗ε,ρ∗ε,H∗2ε+Sε→+u+,ρ+,H+, satisfying(41)Sε←:σ1ε=u--ερ∗ε1/ρ-α-1/ρ∗εαρ-ρ∗ε-ρ-,u∗ε=u--ε1ρ--1ρ∗ε1ρ-α-1ρ∗εα,ρ-H∗1ε-ε21ρ-α+1ρ∗εα=ρ∗εH--ε21ρ-α+1ρ∗εα,ρ∗ε>ρ-,(42)Jε:σ2ε=u∗ε,H∗1ε≠H∗2ε,and(43)Sε→:σ3ε=u++ερ∗ε1/ρ∗εα-1/ρ+αρ+ρ+-ρ∗ε,u∗ε=u++ε1ρ+-1ρ∗ε1ρ+α-1ρ∗εα,ρ∗εH+-ε21ρ+α+1ρ∗εα=ρ+H∗2ε-ε21ρ+α+1ρ∗εα,ρ∗ε>ρ+.We discuss the limits of solutions for system (1)–(2) by the following lemmas.Lemma 2.
l i m ε → ε 2 ρ ∗ ε = + ∞.Proof.
Utilizing the expressions(4.6)2 and (4.8)2, we find(44)u--u+=ε1ρ--1ρ∗ε1ρ-α-1ρ∗εα+ε1ρ+-1ρ∗ε1ρ+α-1ρ∗εα.Setting ε→ε2 in (44), suppose that limε→ε2ρ∗ε=A∈(max⁡(ρ-,ρ+),+∞); we get the equality(45)u--u+=ε21ρ--1A1ρ-α-1Aα+ε21ρ+-1A1ρ+α-1Aα.From (45) and A>ρ±, it is found that(46)u--u+<ε2ρ-α+1+ε2ρ+α+1,which contradicts with (39). And so, we conclude that limε→ε2ρ∗ε=+∞.Lemma 3.
The intermediate internal energyH∗1ε and H∗2ε is unbounded as ε→ε2, that is,(47)limε→ε2H∗1ε=+∞,limε→ε2H∗2ε=+∞.Proof.
It is derived from the third equation in (41) that(48)limε→ε2H∗1ε-ε21ρ∗εα+1ρ-α=limε→ε2ρ∗ερ-H--ε21ρ∗εα+1ρ-α.As a consequence of limε→ε2ρ∗ε=+∞, it is noted that limε→ε2H∗1ε=+∞.Similarly, by (43)3, one can conclude that limε→ε2H∗2ε=+∞.Lemma 4.
(49) l i m ε → ε 2 u ∗ ε = l i m ε → ε 2 σ 1 ε = l i m ε → ε 2 σ 2 ε = l i m ε → ε 2 σ 3 ε = u - ρ - α + 1 / 2 + u + ρ + α + 1 / 2 ρ - α + 1 / 2 + ρ + α + 1 / 2 ≔ u δ .Proof.
In light of Lemma2, from the second equation in (41), we obtain the expression(50)limε→ε2u∗ε=u--limε→ε2ε1ρ--1ρ∗ε1ρ-α-1ρ∗εα=u-ρ-α+1/2+u+ρ+α+1/2ρ-α+1/2+ρ+α+1/2.From (42), we directly deduce that limε→ε2σ2ε=limε→ε2u∗ε=uδ. Note that the first expression in (41) yields(51)limε→ε2σ1ε=u--limε→ε2ερ-α+1=u-ρ-α+1/2+u+ρ+α+1/2ρ-α+1/2+ρ+α+1/2. We similarly obtain that limε→ε2σ3ε=uδ.Lemma 5.
(52) l i m ε → ε 2 ∫ σ 1 ε t σ 3 ε t ρ ∗ ε d x = ρ - ρ + α + 1 / 2 + ρ + ρ - α + 1 / 2 ρ - α + 1 / 2 + ρ + α + 1 / 2 u - - u + t .Proof.
To confirm this, we compute using (41) and (43) that (53)limε→ε2ρ∗εσ2ε-σ1ε=limε→ε2ερ-ρ∗ε1/ρ-α-1/ρ∗εαρ∗ε-ρ-=limε→ε2ερ-1-α=ρ-ρ+α+1/2ρ-α+1/2+ρ+α+1/2u--u+. Similarly, one has(54)limε→ε2ρ∗εσ3ε-σ2ε=ρ+ρ-α+1/2ρ-α+1/2+ρ+α+1/2u--u+. Next integrating ρ∗ε from σ1εt to σ3εt with respect to x and setting ε→ε2, we derive (55)limε→ε2∫σ1εtσ3εtρ∗εdx=limε→ε2ρ∗εσ2ε-σ1εt+limε→ε2ρ∗εσ3ε-σ2εt=ρ-ρ+α+1/2+ρ+ρ-α+1/2ρ-α+1/2+ρ+α+1/2u--u+t.Lemma 6.
(56) l i m ε → ε 2 ∫ σ 1 ε t σ 3 ε t ρ ∗ ε u ∗ ε 2 2 + H ∗ ε d x = u δ ρ u 2 2 + H - ρ u 2 2 + H - ε 2 ρ α u t ,where(57)H∗ε=H∗1ε,σ1ε<xt<σ2ε,H∗2ε,σ2ε<xt<σ3ε.Proof.
Using the Rankine-Hugoniot conditions of (1) for Rε←+Jε and Jε+Rε→, we obtain the following forms:(58)σ1ερ∗εu∗ε22+H∗1ε-ρ-u-22-H-=ρ∗εu∗ε22+H∗1ε-ερ∗εαu∗ε-ρ-u-22+H--ερ-αu-,σ2ερ∗εu∗ε22+H∗2ε-ρ∗εu∗ε22-H∗1ε=ρ∗εu∗ε22+H∗2ε-ερ∗εαu∗ε-ρ∗εu∗ε22+H∗1ε-ερ∗εαu∗ε,σ3ερ+u+22+H+-ρ∗εu∗ε22-H∗2ε=ρ+u+22+H+-ερ+αu+-ρ∗εu∗ε22+H∗2ε-ερ∗εαu∗ε.Set ε→ε2 in (58), to conclude(59)limε→ε2ρ∗εu∗ε22+H∗1εσ2ε-σ1ε+ρ∗εu∗ε22+H∗2εσ3ε-σ2ε=uδρu22+H-ρu22+H-ε2ραu,which implies that(60)limε→ε2∫σ1εtσ3εtρ∗εu∗ε22+H∗εdx=uδρu22+H-ρu22+H-ε2ραut. The proof is completed.In light of the foregoing lemmas, two kinds of shock wavesSε← and Sε→ will coincide as ε→ε2. Additionally, delta shock wave with Dirac delta function in both the density ρ and the internal energy H appears.Theorem 7.
For any givenε>0, we suppose that the Riemann solution for system (1)–(2) is (ρε,ρεuε,ρε(uε)2/2+Hε), which converges to a delta shock wave solution as ε→ε2, and the limit functions ρ, u, and H are sums of a step function and a delta function with weights(61)t1+uδ2uδρ-ρu,t1+uδ2uδρu-ρu2-ε2ρα,t1+uδ2uδρu22+H-ρu22+H-ε2ραu, where (62)uδ=u-ρ-α+1/2+u+ρ+α+1/2ρ-α+1/2+ρ+α+1/2.Proof.
(I) Settingξ=x/t, the Riemann solution is described in the following form:(63)uε,ρε,Hεξ=u-,ρ-,H-,-∞<ξ<σ1ε,u∗ε,ρ∗ε,H∗1ε,σ1ε<ξ<σ2ε,u∗ε,ρ∗ε,H∗2ε,σ2ε<ξ<σ3ε,u+,ρ+,H+,σ3ε<ξ<+∞,where u∗ε,ρ∗ε,H∗1ε, and H∗2ε are constants that only depend on ε.
The solution (63) satisfies the definition of weak solution for an arbitrary ψ∈C0∞(R):(64)∫-∞+∞ξ-uερεψ′dξ+∫-∞+∞ρεψdξ=0,(65)∫-∞+∞ξ-uερεuεψ′dξ+∫-∞+∞ρεuεψdξ=-∫-∞+∞ερεαψ′dξ,(66)∫-∞+∞ξ-uερεuε22+Hεψ′dξ+∫-∞+∞ρεuε22+Hεψdξ=-∫-∞+∞ερεαuεψ′dξ.(II) The first integral in identity (64) can be written as(67)∫-∞σ1ε+∫σ3ε+∞ξ-uερεψ′ξ+∫σ1εσ3εξ-uερεψ′dξ.Setting ε→ε2 in the first term for (67), we see (68)limε→ε2∫-∞σ1εξ-u-ρ-ψ′dξ+∫σ3ε∞ξ-u+ρ+ψ′dξ=limε→ε2u+ρ+ψσ3ε-u-ρ-ψσ1ε+limε→ε2ρ-σ1εψσ1ε-ρ+σ3εψσ3ε-limε→ε2∫-∞σ1ερ-ψdξ-limε→ε2∫σ3ε+∞ρ+ψdξ=ρu-uδρψuδ-∫-∞∞ρ0ξ-uδψdξ,where ρ0(ξ)=ρ-+[ρ]χ(ξ) and χ(ξ) is the characteristic function. In virtue of ψ(ξ)∈C0∞(-∞,+∞),limε→ε2u∗ε=limε→ε2σiε=uδ,(i=1,3), we get (69)limε→ε2∫σ1εσ3εξ-u∗ερ∗εψ′dξ=limε→ε2ρ∗εσ3ε-σ1εσ3εψσ3ε-σ1εψσ1εσ3ε-σ1ε-u∗εψσ3ε-ψσ1εσ3ε-σ1ε-1σ3ε-σ1ε∫σ1εσ3εψdξ=ρu-uδρuδψuδ′-uδψ′uδ-ψuδ)=0.Insert the expressions (68) and (69) into (64) to obtain(70)limε→ε2∫-∞∞ρεψdξ=uδρ-ρuψuδ+∫-∞∞ρ0ξ-uδψdξ.(III) Next, we turn our attention on the the momentum equation (65). Analogously, it is found that(71)limε→ε2∫-∞∞ξ-uερεuεψ′dξ=ρu2-uδρuψuδ-∫-∞∞ρ0u0ξ-uδψdξ,where (72)ρ0u0ξ=ρ-u-+ρuχξ. Taking the limit on the right of (65), we get(73)limε→ε2∫-∞∞-ερεαψ′dξ=ε2ραψuδ.Substituting (71) and (73) into (65), we obtain the equality(74)limε→ε2∫-∞+∞ρεuεψdξ=uδρu-ρu2-ε2ραψuδ+∫-∞+∞ρ0u0ξ-uδψdξ.(IV) Subsequently, by utilizing the energy equation (66), we discover (75)limε→ε2∫-∞+∞ξ-uερεuε22+Hεψ′dξ=ρu22+Hu-uδρu22+Hψuδ-∫-∞uδρ-u-22+H-ψdξ-∫uδ+∞ρ+u+22+H+ψdξ=ρu22+Hu-uδρu22+Hψuδ-∫-∞+∞ρ0u022+H0ξ-uδdξ,where(76)ρ0u022+H0ξ=ρ-u-22+H-+ρu22+Hχξ. By taking the limit for the right side of (66) note that(77)limε→ε2∫-∞+∞-ερεαuεψ′dξ=ε2ραuψuδ. Going back to the identity (66), we find(78)limε→ε2∫-∞+∞ρεuε22+Hεψdξ=uδρu22+H-ρu22+H-ε2ραuψuδ+∫-∞+∞ρ0u022+H0ξ-uδψdξ.(V) From now on, we study the limits of ρε, ρεuε, and ρε(uε)2/2+Hε as ε→ε2.
For anyψ(x,t)∈C0∞(R×R+), it is noted that (79)limε→ε2ρε,ψ=limε→ε2∫0+∞∫-∞+∞ρεxtψx,tdxdt=limε→ε2∫0+∞t∫-∞+∞ρεξψξt,tdξdt. Utilizing (70), thereby obtain the equality (80)limε→ε2∫-∞+∞ρεξψξt,tdξ=∫-∞+∞ρ0ξ-uδψξt,tdξ+uδρ-ρuψuδt,t=1t∫-∞+∞ρ0x-uδtψx,tdx+uδρ-ρuψuδt,t. One can be calculated by using the above two expressions to obtain(81)limε→ε2ρε,ψ=∫0+∞∫-∞+∞ρ0x-uδtψx,tdxdt+∫0+∞tuδρ-ρuψuδt,tdt. According to definition of the measure solution [34], one has(82)∫0+∞tuδρ-ρuψuδt,tdt=w0tδS,ψx,t, with(83)w0t=t1+uδ2uδρ-ρu. Analogously, one can see that(84)limε→ε2ρεuε,ψ=∫0+∞∫-∞+∞ρ0u0x-uδtψx,tdxdt+w1tδS,ψx,t, with (85)w1t=t1+uδ2uδρu-ρu2-ε2ρα. And(86)limε→ε2ρεuε22+Hε,ψ=∫0+∞∫-∞+∞ρ0u022+H0x-uδtψx,tdxdt+w2tδS,ψx,t, with(87)w2t=t1+uδ2uδρu22+H-ρu22+H-ε2ραu.The proof of Theorem 7 is completed.By using (30)–(31) and noting that(88)limε→0wεt=ρ+ρ-u--u+t=wt,(89)limε→0uδt=ρ+u++ρ-u-ρ++ρ-=σ,as [ρ]=ρ+-ρ-≠0, and(90)uδt=u++u-2=σ,wεt=ρ-u--ρ+u+t=wt,as [ρ]=ρ+-ρ-=0.Consequently, we conclude that limit of theδ-shock wave solution to (1) with (6) is δ-shock wave solution of the transport equations (4).Theorem 8.
The pair(u,ρ) is a measure solution in the sense of distributions to the transport equations (4) if it satisfies(91)ρu2t+ρu3x≥0.Proof.
In the sense of distributions, owing toρε(uε)2/2+Hεt+ρε(uε)3/2+Hεuεx=0, then(92)ρεuε2t+ρεuε3x=-2Htε+Hεuεx. Furthermore, for any given nonnegative test function φ∈C0∞((-∞,+∞)×[0,+∞)):(93)ρεuε2t+ρεuε3x,ψ=2Hε,ψt+Hεuε,ψx. By applying Green’s formulas, we compute (94)limε→ε2∫0+∞∫-∞+∞Hεψt+Hεuεψxdxdt=limε→ε2∫0+∞σ1εH∗1ε-H--H∗1εu∗ε-H-u-ψσ1εt,tdt+limε→ε2∫0+∞σ2εH∗2ε-H∗1ε-H∗2εu∗ε-H∗1εu∗εψσ2εt,tdt+limε→ε2∫0+∞σ3εH+-H∗2ε-H+u+-H∗2εu∗εψσ3εt,tdt=∫0+∞uδH+-H--H+u+-H-u-ψuδt,tdt=∫0+∞u--u+αH++1-αH-ψuδt,tdt>0, for α=ρ-α+1/2/ρ-α+1/2+ρ+α+1/2∈(0,1):(95)uδ=u-ρ-α+1/2+u+ρ+α+1/2ρ-α+1/2+ρ+α+1/2=αu-+1-αu+. Through above discussions, we verify the consistency of entropy.


### 4.2. Caseu+>u-

This subsection mainly shows the formation of vacuum state of limit solutions for system (1)–(2) as the pressure decreases.Lemma 9.
In caseu-<u+, then there exists ε~>0 such that the projection of (u+,ρ+,H+) onto the (ρ,u) phase plane (u+,ρ+) lies in I(u-,ρ-) as ε∈(0,ε~).Proof.
In view ofu-<u+, as ρ+=ρ-, the conclusion is true for ε>0. Then, we analyze ρ+≠ρ-, supposing that (u+,ρ+)∈I(u-,ρ-), which can be joined with (u-,ρ-) by Rε← or Rε→ satisfying(96)u+>u-+2εαα+1ρ+-α+1/2-ρ--α+1/2,ρ+<ρ-,or(97)u+>u--2εαα+1ρ+-α+1/2-ρ--α+1/2,ρ+>ρ-.Algebraic manipulation of (96) and (97) gives(98)ε<α+12ρ-α+1ρ+α+1u--u+24αρ-α+1/2-ρ+α+1/22,taking(99)ε~=α+12ρ-α+1ρ+α+1u--u+24αρ-α+1/2-ρ+α+1/22.Accordingly, we see that (u+,ρ+)∈I(u-,ρ-) when 0<ε<ε~.
And so, for any givenε∈(0,ε~), the Riemann solution can be expressed as(100)u-,ρ-,H-+Rε←+u∗ε,ρ∗ε,H∗1ε+Jε+u∗ε,ρ∗ε,H∗2ε+Rε→+u+,ρ+,H+,which satisfies(101)u∗ε=u--2εαα+1ρ--α+1/2+2εαα+1ρ∗ε-α+1/2,ρ∗ε<ρ-,u∗ε=u++2εαα+1ρ+-α+1/2-2εαα+1ρ∗ε-α+1/2,ρ∗ε<ρ+.Then utilizing (101), we find(102)ρ∗ε=α+14εαu+-u-+12ρ+-α+1/2+ρ--α+1/2-2/α+1,and also from the above identity, we conclude that(103)limε→0ρ∗ε=0. And so, the phenomena of cavitation do occur as ε tend to zero. Next, According to(104)λ1=u-εαρ-α+1/2,u=u--2εαα+1ρ--α+1/2+2εαα+1ρ-α+1/2and(105)λ3=u+εαρ-α+1/2,u=u++2εαα+1ρ+-α+1/2-2εαα+1ρ-α+1/2,we see that(106)limε→0λ1=u-,limε→0λ3=u+.In light of the foregoing analysis, it is seen that the limit of the Riemann solution for system (1)–(2) as ε tends to zero and is a double-contact-discontinuity solution of transport equations (4), as shown in Figure 3.


## 4.1. Caseu+<u-

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 thatu+<u-, then there exists ε1>ε2>0 such that the projection of the state (u+,ρ+,H+) onto the (u,ρ) phase plane (u+,ρ+)∈IV(u-,ρ-) as ε∈(ε2,ε1); (u+,ρ+)∈V(u-,ρ-) as ε∈(0,ε2].Proof.
As a consequence ofu+<u-, we can easily conclude that (u+,ρ+)∈(IV∪V)(u-,ρ-) as ρ+=ρ-.
Whenρ+≠ρ-, we assume (u+,ρ+)∈IV(u-,ρ-), which can be joined with (u-,ρ-) by Sε← or Sε→ satisfying(33)u+<u--ερ--α-ρ+-αρ+ρ-ρ+-ρ-ρ+-ρ-,ρ+>ρ-,or(34)u+<u-+ερ--α-ρ+-αρ+ρ-ρ+-ρ-ρ+-ρ-,ρ+<ρ-,and from (33) and (34), it is found that(35)ε<ρ-α+1ρ+α+1u--u+2ρ+α-ρ-αρ+-ρ-, And, then, we take(36)ε1=ρ-α+1ρ+α+1u--u+2ρ+α-ρ-αρ+-ρ-.And so, (u+,ρ+)∈(IV∪V)(u-,ρ-) as ε<ε1.When(u+,ρ+)∈V(u-,ρ-), which can be joined with (u-,ρ-) directly by a delta shock wave curve satisfying the inequality,(37)u--ερ--α+1/2≥u++ερ+-α+1/2.From the above inequality, we obtain the expression(38)ε≤u--u+ρ--α+1/2+ρ+-α+1/22, letting(39)ε2=u--u+ρ--α+1/2+ρ+-α+1/22.It is shown that (u+,ρ+)∈V(u-,ρ-) as 0<ε≤ε2.This lemma shows that there is noδ-shock wave as ε>ε2. When 0<ε<ε1, the Riemann solution is(40)u-,ρ-,H-+Sε←+u∗ε,ρ∗ε,H∗1ε+Jε+u∗ε,ρ∗ε,H∗2ε+Sε→+u+,ρ+,H+, satisfying(41)Sε←:σ1ε=u--ερ∗ε1/ρ-α-1/ρ∗εαρ-ρ∗ε-ρ-,u∗ε=u--ε1ρ--1ρ∗ε1ρ-α-1ρ∗εα,ρ-H∗1ε-ε21ρ-α+1ρ∗εα=ρ∗εH--ε21ρ-α+1ρ∗εα,ρ∗ε>ρ-,(42)Jε:σ2ε=u∗ε,H∗1ε≠H∗2ε,and(43)Sε→:σ3ε=u++ερ∗ε1/ρ∗εα-1/ρ+αρ+ρ+-ρ∗ε,u∗ε=u++ε1ρ+-1ρ∗ε1ρ+α-1ρ∗εα,ρ∗εH+-ε21ρ+α+1ρ∗εα=ρ+H∗2ε-ε21ρ+α+1ρ∗εα,ρ∗ε>ρ+.We discuss the limits of solutions for system (1)–(2) by the following lemmas.Lemma 2.
l i m ε → ε 2 ρ ∗ ε = + ∞.Proof.
Utilizing the expressions(4.6)2 and (4.8)2, we find(44)u--u+=ε1ρ--1ρ∗ε1ρ-α-1ρ∗εα+ε1ρ+-1ρ∗ε1ρ+α-1ρ∗εα.Setting ε→ε2 in (44), suppose that limε→ε2ρ∗ε=A∈(max⁡(ρ-,ρ+),+∞); we get the equality(45)u--u+=ε21ρ--1A1ρ-α-1Aα+ε21ρ+-1A1ρ+α-1Aα.From (45) and A>ρ±, it is found that(46)u--u+<ε2ρ-α+1+ε2ρ+α+1,which contradicts with (39). And so, we conclude that limε→ε2ρ∗ε=+∞.Lemma 3.
The intermediate internal energyH∗1ε and H∗2ε is unbounded as ε→ε2, that is,(47)limε→ε2H∗1ε=+∞,limε→ε2H∗2ε=+∞.Proof.
It is derived from the third equation in (41) that(48)limε→ε2H∗1ε-ε21ρ∗εα+1ρ-α=limε→ε2ρ∗ερ-H--ε21ρ∗εα+1ρ-α.As a consequence of limε→ε2ρ∗ε=+∞, it is noted that limε→ε2H∗1ε=+∞.Similarly, by (43)3, one can conclude that limε→ε2H∗2ε=+∞.Lemma 4.
(49) l i m ε → ε 2 u ∗ ε = l i m ε → ε 2 σ 1 ε = l i m ε → ε 2 σ 2 ε = l i m ε → ε 2 σ 3 ε = u - ρ - α + 1 / 2 + u + ρ + α + 1 / 2 ρ - α + 1 / 2 + ρ + α + 1 / 2 ≔ u δ .Proof.
In light of Lemma2, from the second equation in (41), we obtain the expression(50)limε→ε2u∗ε=u--limε→ε2ε1ρ--1ρ∗ε1ρ-α-1ρ∗εα=u-ρ-α+1/2+u+ρ+α+1/2ρ-α+1/2+ρ+α+1/2.From (42), we directly deduce that limε→ε2σ2ε=limε→ε2u∗ε=uδ. Note that the first expression in (41) yields(51)limε→ε2σ1ε=u--limε→ε2ερ-α+1=u-ρ-α+1/2+u+ρ+α+1/2ρ-α+1/2+ρ+α+1/2. We similarly obtain that limε→ε2σ3ε=uδ.Lemma 5.
(52) l i m ε → ε 2 ∫ σ 1 ε t σ 3 ε t ρ ∗ ε d x = ρ - ρ + α + 1 / 2 + ρ + ρ - α + 1 / 2 ρ - α + 1 / 2 + ρ + α + 1 / 2 u - - u + t .Proof.
To confirm this, we compute using (41) and (43) that (53)limε→ε2ρ∗εσ2ε-σ1ε=limε→ε2ερ-ρ∗ε1/ρ-α-1/ρ∗εαρ∗ε-ρ-=limε→ε2ερ-1-α=ρ-ρ+α+1/2ρ-α+1/2+ρ+α+1/2u--u+. Similarly, one has(54)limε→ε2ρ∗εσ3ε-σ2ε=ρ+ρ-α+1/2ρ-α+1/2+ρ+α+1/2u--u+. Next integrating ρ∗ε from σ1εt to σ3εt with respect to x and setting ε→ε2, we derive (55)limε→ε2∫σ1εtσ3εtρ∗εdx=limε→ε2ρ∗εσ2ε-σ1εt+limε→ε2ρ∗εσ3ε-σ2εt=ρ-ρ+α+1/2+ρ+ρ-α+1/2ρ-α+1/2+ρ+α+1/2u--u+t.Lemma 6.
(56) l i m ε → ε 2 ∫ σ 1 ε t σ 3 ε t ρ ∗ ε u ∗ ε 2 2 + H ∗ ε d x = u δ ρ u 2 2 + H - ρ u 2 2 + H - ε 2 ρ α u t ,where(57)H∗ε=H∗1ε,σ1ε<xt<σ2ε,H∗2ε,σ2ε<xt<σ3ε.Proof.
Using the Rankine-Hugoniot conditions of (1) for Rε←+Jε and Jε+Rε→, we obtain the following forms:(58)σ1ερ∗εu∗ε22+H∗1ε-ρ-u-22-H-=ρ∗εu∗ε22+H∗1ε-ερ∗εαu∗ε-ρ-u-22+H--ερ-αu-,σ2ερ∗εu∗ε22+H∗2ε-ρ∗εu∗ε22-H∗1ε=ρ∗εu∗ε22+H∗2ε-ερ∗εαu∗ε-ρ∗εu∗ε22+H∗1ε-ερ∗εαu∗ε,σ3ερ+u+22+H+-ρ∗εu∗ε22-H∗2ε=ρ+u+22+H+-ερ+αu+-ρ∗εu∗ε22+H∗2ε-ερ∗εαu∗ε.Set ε→ε2 in (58), to conclude(59)limε→ε2ρ∗εu∗ε22+H∗1εσ2ε-σ1ε+ρ∗εu∗ε22+H∗2εσ3ε-σ2ε=uδρu22+H-ρu22+H-ε2ραu,which implies that(60)limε→ε2∫σ1εtσ3εtρ∗εu∗ε22+H∗εdx=uδρu22+H-ρu22+H-ε2ραut. The proof is completed.In light of the foregoing lemmas, two kinds of shock wavesSε← and Sε→ will coincide as ε→ε2. Additionally, delta shock wave with Dirac delta function in both the density ρ and the internal energy H appears.Theorem 7.
For any givenε>0, we suppose that the Riemann solution for system (1)–(2) is (ρε,ρεuε,ρε(uε)2/2+Hε), which converges to a delta shock wave solution as ε→ε2, and the limit functions ρ, u, and H are sums of a step function and a delta function with weights(61)t1+uδ2uδρ-ρu,t1+uδ2uδρu-ρu2-ε2ρα,t1+uδ2uδρu22+H-ρu22+H-ε2ραu, where (62)uδ=u-ρ-α+1/2+u+ρ+α+1/2ρ-α+1/2+ρ+α+1/2.Proof.
(I) Settingξ=x/t, the Riemann solution is described in the following form:(63)uε,ρε,Hεξ=u-,ρ-,H-,-∞<ξ<σ1ε,u∗ε,ρ∗ε,H∗1ε,σ1ε<ξ<σ2ε,u∗ε,ρ∗ε,H∗2ε,σ2ε<ξ<σ3ε,u+,ρ+,H+,σ3ε<ξ<+∞,where u∗ε,ρ∗ε,H∗1ε, and H∗2ε are constants that only depend on ε.
The solution (63) satisfies the definition of weak solution for an arbitrary ψ∈C0∞(R):(64)∫-∞+∞ξ-uερεψ′dξ+∫-∞+∞ρεψdξ=0,(65)∫-∞+∞ξ-uερεuεψ′dξ+∫-∞+∞ρεuεψdξ=-∫-∞+∞ερεαψ′dξ,(66)∫-∞+∞ξ-uερεuε22+Hεψ′dξ+∫-∞+∞ρεuε22+Hεψdξ=-∫-∞+∞ερεαuεψ′dξ.(II) The first integral in identity (64) can be written as(67)∫-∞σ1ε+∫σ3ε+∞ξ-uερεψ′ξ+∫σ1εσ3εξ-uερεψ′dξ.Setting ε→ε2 in the first term for (67), we see (68)limε→ε2∫-∞σ1εξ-u-ρ-ψ′dξ+∫σ3ε∞ξ-u+ρ+ψ′dξ=limε→ε2u+ρ+ψσ3ε-u-ρ-ψσ1ε+limε→ε2ρ-σ1εψσ1ε-ρ+σ3εψσ3ε-limε→ε2∫-∞σ1ερ-ψdξ-limε→ε2∫σ3ε+∞ρ+ψdξ=ρu-uδρψuδ-∫-∞∞ρ0ξ-uδψdξ,where ρ0(ξ)=ρ-+[ρ]χ(ξ) and χ(ξ) is the characteristic function. In virtue of ψ(ξ)∈C0∞(-∞,+∞),limε→ε2u∗ε=limε→ε2σiε=uδ,(i=1,3), we get (69)limε→ε2∫σ1εσ3εξ-u∗ερ∗εψ′dξ=limε→ε2ρ∗εσ3ε-σ1εσ3εψσ3ε-σ1εψσ1εσ3ε-σ1ε-u∗εψσ3ε-ψσ1εσ3ε-σ1ε-1σ3ε-σ1ε∫σ1εσ3εψdξ=ρu-uδρuδψuδ′-uδψ′uδ-ψuδ)=0.Insert the expressions (68) and (69) into (64) to obtain(70)limε→ε2∫-∞∞ρεψdξ=uδρ-ρuψuδ+∫-∞∞ρ0ξ-uδψdξ.(III) Next, we turn our attention on the the momentum equation (65). Analogously, it is found that(71)limε→ε2∫-∞∞ξ-uερεuεψ′dξ=ρu2-uδρuψuδ-∫-∞∞ρ0u0ξ-uδψdξ,where (72)ρ0u0ξ=ρ-u-+ρuχξ. Taking the limit on the right of (65), we get(73)limε→ε2∫-∞∞-ερεαψ′dξ=ε2ραψuδ.Substituting (71) and (73) into (65), we obtain the equality(74)limε→ε2∫-∞+∞ρεuεψdξ=uδρu-ρu2-ε2ραψuδ+∫-∞+∞ρ0u0ξ-uδψdξ.(IV) Subsequently, by utilizing the energy equation (66), we discover (75)limε→ε2∫-∞+∞ξ-uερεuε22+Hεψ′dξ=ρu22+Hu-uδρu22+Hψuδ-∫-∞uδρ-u-22+H-ψdξ-∫uδ+∞ρ+u+22+H+ψdξ=ρu22+Hu-uδρu22+Hψuδ-∫-∞+∞ρ0u022+H0ξ-uδdξ,where(76)ρ0u022+H0ξ=ρ-u-22+H-+ρu22+Hχξ. By taking the limit for the right side of (66) note that(77)limε→ε2∫-∞+∞-ερεαuεψ′dξ=ε2ραuψuδ. Going back to the identity (66), we find(78)limε→ε2∫-∞+∞ρεuε22+Hεψdξ=uδρu22+H-ρu22+H-ε2ραuψuδ+∫-∞+∞ρ0u022+H0ξ-uδψdξ.(V) From now on, we study the limits of ρε, ρεuε, and ρε(uε)2/2+Hε as ε→ε2.
For anyψ(x,t)∈C0∞(R×R+), it is noted that (79)limε→ε2ρε,ψ=limε→ε2∫0+∞∫-∞+∞ρεxtψx,tdxdt=limε→ε2∫0+∞t∫-∞+∞ρεξψξt,tdξdt. Utilizing (70), thereby obtain the equality (80)limε→ε2∫-∞+∞ρεξψξt,tdξ=∫-∞+∞ρ0ξ-uδψξt,tdξ+uδρ-ρuψuδt,t=1t∫-∞+∞ρ0x-uδtψx,tdx+uδρ-ρuψuδt,t. One can be calculated by using the above two expressions to obtain(81)limε→ε2ρε,ψ=∫0+∞∫-∞+∞ρ0x-uδtψx,tdxdt+∫0+∞tuδρ-ρuψuδt,tdt. According to definition of the measure solution [34], one has(82)∫0+∞tuδρ-ρuψuδt,tdt=w0tδS,ψx,t, with(83)w0t=t1+uδ2uδρ-ρu. Analogously, one can see that(84)limε→ε2ρεuε,ψ=∫0+∞∫-∞+∞ρ0u0x-uδtψx,tdxdt+w1tδS,ψx,t, with (85)w1t=t1+uδ2uδρu-ρu2-ε2ρα. And(86)limε→ε2ρεuε22+Hε,ψ=∫0+∞∫-∞+∞ρ0u022+H0x-uδtψx,tdxdt+w2tδS,ψx,t, with(87)w2t=t1+uδ2uδρu22+H-ρu22+H-ε2ραu.The proof of Theorem 7 is completed.By using (30)–(31) and noting that(88)limε→0wεt=ρ+ρ-u--u+t=wt,(89)limε→0uδt=ρ+u++ρ-u-ρ++ρ-=σ,as [ρ]=ρ+-ρ-≠0, and(90)uδt=u++u-2=σ,wεt=ρ-u--ρ+u+t=wt,as [ρ]=ρ+-ρ-=0.Consequently, we conclude that limit of theδ-shock wave solution to (1) with (6) is δ-shock wave solution of the transport equations (4).Theorem 8.
The pair(u,ρ) is a measure solution in the sense of distributions to the transport equations (4) if it satisfies(91)ρu2t+ρu3x≥0.Proof.
In the sense of distributions, owing toρε(uε)2/2+Hεt+ρε(uε)3/2+Hεuεx=0, then(92)ρεuε2t+ρεuε3x=-2Htε+Hεuεx. Furthermore, for any given nonnegative test function φ∈C0∞((-∞,+∞)×[0,+∞)):(93)ρεuε2t+ρεuε3x,ψ=2Hε,ψt+Hεuε,ψx. By applying Green’s formulas, we compute (94)limε→ε2∫0+∞∫-∞+∞Hεψt+Hεuεψxdxdt=limε→ε2∫0+∞σ1εH∗1ε-H--H∗1εu∗ε-H-u-ψσ1εt,tdt+limε→ε2∫0+∞σ2εH∗2ε-H∗1ε-H∗2εu∗ε-H∗1εu∗εψσ2εt,tdt+limε→ε2∫0+∞σ3εH+-H∗2ε-H+u+-H∗2εu∗εψσ3εt,tdt=∫0+∞uδH+-H--H+u+-H-u-ψuδt,tdt=∫0+∞u--u+αH++1-αH-ψuδt,tdt>0, for α=ρ-α+1/2/ρ-α+1/2+ρ+α+1/2∈(0,1):(95)uδ=u-ρ-α+1/2+u+ρ+α+1/2ρ-α+1/2+ρ+α+1/2=αu-+1-αu+. Through above discussions, we verify the consistency of entropy.


## 4.2. Caseu+>u-

This subsection mainly shows the formation of vacuum state of limit solutions for system (1)–(2) as the pressure decreases.Lemma 9.
In caseu-<u+, then there exists ε~>0 such that the projection of (u+,ρ+,H+) onto the (ρ,u) phase plane (u+,ρ+) lies in I(u-,ρ-) as ε∈(0,ε~).Proof.
In view ofu-<u+, as ρ+=ρ-, the conclusion is true for ε>0. Then, we analyze ρ+≠ρ-, supposing that (u+,ρ+)∈I(u-,ρ-), which can be joined with (u-,ρ-) by Rε← or Rε→ satisfying(96)u+>u-+2εαα+1ρ+-α+1/2-ρ--α+1/2,ρ+<ρ-,or(97)u+>u--2εαα+1ρ+-α+1/2-ρ--α+1/2,ρ+>ρ-.Algebraic manipulation of (96) and (97) gives(98)ε<α+12ρ-α+1ρ+α+1u--u+24αρ-α+1/2-ρ+α+1/22,taking(99)ε~=α+12ρ-α+1ρ+α+1u--u+24αρ-α+1/2-ρ+α+1/22.Accordingly, we see that (u+,ρ+)∈I(u-,ρ-) when 0<ε<ε~.
And so, for any givenε∈(0,ε~), the Riemann solution can be expressed as(100)u-,ρ-,H-+Rε←+u∗ε,ρ∗ε,H∗1ε+Jε+u∗ε,ρ∗ε,H∗2ε+Rε→+u+,ρ+,H+,which satisfies(101)u∗ε=u--2εαα+1ρ--α+1/2+2εαα+1ρ∗ε-α+1/2,ρ∗ε<ρ-,u∗ε=u++2εαα+1ρ+-α+1/2-2εαα+1ρ∗ε-α+1/2,ρ∗ε<ρ+.Then utilizing (101), we find(102)ρ∗ε=α+14εαu+-u-+12ρ+-α+1/2+ρ--α+1/2-2/α+1,and also from the above identity, we conclude that(103)limε→0ρ∗ε=0. And so, the phenomena of cavitation do occur as ε tend to zero. Next, According to(104)λ1=u-εαρ-α+1/2,u=u--2εαα+1ρ--α+1/2+2εαα+1ρ-α+1/2and(105)λ3=u+εαρ-α+1/2,u=u++2εαα+1ρ+-α+1/2-2εαα+1ρ-α+1/2,we see that(106)limε→0λ1=u-,limε→0λ3=u+.In light of the foregoing analysis, it is seen that the limit of the Riemann solution for system (1)–(2) as ε 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 u+<u-, Riemann solution of system (1)–(2) tends to a δ-shock wave solution when ε reduces to a certain value ε2, and the δ-shock wave tends to a δ-shock solution of zero-pressure flow (4) as ε→0 (see Figure 2). In the case of u+>u-, Riemann solution of system (1)–(2) converges to solution involving vacuum state of system (4) as ε→0 (see Figure 3).Figure 2
The physical plane(x,t).Figure 3
The physical plane(x,t).



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*Source: 5253717-2019-07-03.xml*