Limits of Riemann Solutions to the Relativistic Euler Systems for Chaplygin Gas as Pressure Vanishes

Abstract

Vanishing pressure limits of Riemann solutions to relativistic Euler system for Chaplygin gas are identified and analyzed in detail. Unlike the polytropic or barotropic gas case, as the parameter decreases to a critical value, the two-shock solution converges firstly to a delta shock wave solution to the same system. It is shown that, as the parameter decreases, the strength of the delta shock increases. Then as the pressure vanishes ultimately, the solution is nothing but the delta shock wave solution to the zero pressure relativistic Euler system. Meanwhile, the two-rarefaction wave solution and the solution containing one-rarefaction wave and one-shock wave tend to the vacuum solution and the contact discontinuity solution to the zero pressure relativistic Euler system, respectively.

1. Introduction

The relativistic fluid dynamics plays a basic and significant role in many physics fields, such as astrophysics, cosmology, and nuclear physics [1]. The Euler system of conservation laws of energy and momentum in special relativity reads (𝜌𝑐²+𝑝)𝑣²𝑐²(𝑐²−𝑣²)+𝜌_𝑡+(𝜌𝑐²+𝑝)𝑣𝑐²−𝑣²_𝑥=0,(𝜌𝑐²+𝑝)𝑣𝑐²−𝑣²_𝑡+(𝜌𝑐²+𝑝)𝑣²𝑐²−𝑣²+𝑝_𝑥=0. (1) Here 𝜌,𝑝 represent the proper mass-energy density and the pressure, respectively, constant 𝑐 is the light speed, the sonic speed 𝑝^′(𝜌) should be not more than the light speed 𝑐, and 𝑣 is the particle speed satisfying the relativistic constraint 𝑣²<𝑐². In this paper, we consider the limit behavior of Riemann solutions of (1) for Chaplygin gas as pressure vanishes.

Formally, system (1) in the Newtonian limit reduces to the classical isentropic Euler equations for compressible fluids as 𝑣/𝑐→0: 𝜌_𝑡+𝜌𝑣_𝑥=0,𝜌𝑣_𝑡+𝜌𝑣²+𝑝_𝑥=0. (2) Thus system (1) can also be viewed as the relativistic generalization of system (2). System (2) and its generalized equation have been investigated intensively and widely as the typical system of nonlinear hyperbolic conservation laws [2–8]. One can refer to [9–11] for more systematic results on the systems of nonlinear hyperbolic conservation laws. For Chaplygin gas, Brenier obtained the Riemann solutions to system (2), in which there appears concentration phenomenon for some certain initial data [12]. That is, for Chaplygin gas, there exists a unique Riemann solution to system (2) involving the so-called delta shock wave in some cases. Chaplygin gas dynamics was widely studied recently, and there are some interesting and important results, especially for Riemann problems. We refer to [13, 14] and the references cited therein for more related results.

System (1) can be formally transformed into the following model as the pressure vanishes: 𝜌𝑐²−𝑣²_𝑡+𝜌𝑣𝑐²−𝑣²_𝑥=0,𝜌𝑣𝑐²−𝑣²_𝑡+𝜌𝑣²𝑐²−𝑣²_𝑥=0. (3) We call system (3) the zero pressure relativistic Euler system. In fact, it can be viewed as a relativistic version of the transport equations: 𝜌_𝑡+𝜌𝑣_𝑥=0,𝜌𝑣_𝑡+𝜌𝑣²_𝑥=0, (4) which can be used to describe the motion process of free particles sticking under collision in the low temperature and the information of large-scale structures in the universe [15, 16]. System (4) has been investigated extensively in the past two decades. In [17], Sheng and Zhang obtained the Riemann solutions of (4) involving delta shock wave or vacuum. The delta shock wave is characterized by the location, propagation speed, and weight which is the mass of concentrated particles. This shows that the delta shock can be regarded as the galaxies in the universe or the concentration of particles. We can also see [18–20] for the related results about the delta shock wave.

As for system (1), we recommend [21–23] for some results, in which the elementary nonlinear waves have been analyzed. The Riemann solutions and the BV weak solutions of Cauchy problem for system (1) under the equation of state 𝑝=𝜎²𝜌 (𝜎 is the sound speed satisfying 0<𝜎<𝑐) were obtained analytically by Smoller and Temple [24]. Then in [25], Chen generalized their results with the general equation of state 𝑝=𝑝(𝜌) satisfying 𝑝^′(𝜌)>0 and 𝑝^′′(𝜌)≥0. Based on the results in [25], Hsu et al. [26] proved the existence of global weak solutions to (1) with a more realistic equation of state and 𝐿^∞ initial data containing vacuum state in the framework of compensated compactness. The existence of entropy solutions for problems without vacuum state was established by LeFloch and Yamazaki [27]. More results about entropy solutions to system (1) can be found in [28–31]. We also refer to [32] for a multidimensional piston problem and [33] for the blow-up of solutions. For Chaplygin gas, Cheng and Yang [34] considered the Riemann problem of system (1). The solutions are a bit different from those for polytropic gas. System (1) in this case is linearly degenerate. Thus there appear five kinds of solutions, in which four cases involve contact discontinuities and another contains delta shock for some certain initial data.

In the present paper, we focus on the limit of Riemann solutions to system (𝜌𝑐²+𝜀𝑝)𝑣²𝑐²(𝑐²−𝑣²)+𝜌_𝑡+(𝜌𝑐²+𝜀𝑝)𝑣𝑐²−𝑣²_𝑥=0,(𝜌𝑐²+𝜀𝑝)𝑣𝑐²−𝑣²_𝑡+(𝜌𝑐²+𝜀𝑝)𝑣²𝑐²−𝑣²+𝜀𝑝_𝑥=0, (5) as pressure vanishes with the pressure function for Chaplygin gas: 1𝑝=−𝜌. (6) It is clear to see that the Chaplygin equations can be mathematically expressed as an isentropic gas dynamics system with a negative pressure and can be used to depict some dark-energy models in cosmology [35].

The same problem to Euler system for isothermal case was carried out in [36]. Li proved that when temperature drops to zero, the solution containing two shock waves converges to the delta shock solution to transport equations (4) and the solution containing two rarefaction waves converges to the solution involving vacuum to system (4). Instead of isothermal case, Chen and Liu [37] considered the formation of delta shock and vacuum state of the Riemann solutions to Euler system for polytropic gas in which they took the equation of state as 𝑃=𝜀𝑝 for 𝑝=𝜌^𝛾/𝛾(𝛾>1). Then, in [38, 39], Yin and Sheng extended the results above to system (1). In [40], Mitrović and Nedeljkov considered the generalized pressureless gas dynamics model with a scaled pressure term: 𝜌_𝑡+𝜌𝑔(𝑢)_𝑥=0,𝜌𝑢_𝑡+𝜌𝑢𝑔(𝑢)+𝜀𝑝𝜌_𝑥=0, (7) where 𝑝=𝜅𝜌^𝛾 for 1<𝛾<3 and 𝑔 is a nondecreasing function. They extended the results in [37] to the system (7) and found that the delta shock wave appears as the limit of the solution involving two shock waves as 𝜀 goes to zero. We also refer to [41, 42] for the related results.

In this paper, we mainly describe the limit of Riemann solutions to system (5)-(6) as pressure vanishes. Unlike the cases we mentioned above, system (5)-(6) is linearly degenerate. That is, there appears delta shock in Riemann solutions. Thus, it is natural for us to guess that different structures or components of Riemann solutions in this case may directly cause some difference and interest during the process of vanishing pressure limit. This motivates us to do this work.

We will show that, as 𝜀 drops to a certain critical value 𝜀₁>0 which only depends on the given Riemann initial data (𝑣_±,𝜌_±), the solution involving two shock waves converges to a delta shock wave of the same system (5)-(6). When 𝜀 continues to decrease, we find that the strength of the delta shock wave increases. Eventually, when 𝜀 drops to zero, the delta shock wave solution is exactly the solution to system (3). Thus, we find that the process of delta shock wave formation is obviously different from those in [36, 37] and so forth. Meanwhile, any Riemann solution involving two rarefaction waves converges to the vacuum solution to system (3). Furthermore, the limit of the solution involving one rarefaction wave 𝑅₁ (or 𝑅₂) and one shock wave 𝑆₂ (or 𝑆₁) is just the contact discontinuity connecting the two constant states (𝑣_±,𝜌_±).

The organization of this paper is as follows: in Sections 2 and 3, we give some results on the Riemann solutions to system (5)-(6) and system (3). In Section 4, we study the limit of Riemann solutions involving two shocks to system (5)-(6) as pressure vanishes when 𝑣₋>𝑣₊. In Section 5, we investigate the limit of solution containing two rarefaction waves to system (5)-(6) when 𝑣₋<𝑣₊. In Section 6, we analyze the limit of solution when 𝑣₋=𝑣₊.

2. Riemann Problems for System (5)-(6)

In this section, we mainly consider the solutions of (5)-(6) with initial data: 𝑣,𝜌(0,𝑥)=𝑣_±,𝜌_±,(±𝑥>0). (8) As mentioned in the introduction, the process without the parameter 𝜀 has been done in [34] for system (1) of Chaplygin gas. However, we provide the Riemann solution of (5)-(6) for concreteness.

2.1. Elementary Waves and Riemann Problems

Noticing the relativistic constraint |𝑣|<𝑐 and 𝜀𝑝^′≤𝑐, we see that the physically relevant region for solutions is ∣|𝑣|√Λ=𝑣,𝜌<𝑐,𝜌≥𝜀𝑐, (9) which is obviously different from that for polytropic and barotropic gas. The eigenvalues of system (5) are 𝜆^𝜀₁=𝑐²𝑣−𝜀𝑝^′𝑐²−𝑣𝜀𝑝^′,𝜆^𝜀₂=𝑐²𝑣+𝜀𝑝^′𝑐²+𝑣𝜀𝑝^′ (10) and the corresponding right eigenvectors are 𝑟₁=((−1/(𝑐²−𝑣²)),(𝜀𝑝^′/(𝜀𝑝+𝜌𝑐²)))^𝑇, 𝑟₂=((1/(𝑐²−𝑣²)),(𝜀𝑝^′/(𝜀𝑝+𝜌𝑐²)))^𝑇. Thus the fact of ∇𝜆^𝜀_𝑖⋅𝑟_𝑖=0(𝑖=1,2) shows that both the characteristic fields are linearly degenerate. The Riemann problem (5) and (8) for 𝜉=𝑥/𝑡 can be reduced to −𝜉𝜌𝑐²+𝜀𝑝𝑣²𝑐²𝑐²−𝑣²+𝜌_𝜉+𝜌𝑐²+𝜀𝑝𝑣𝑐²−𝑣²_𝜉=0,−𝜉𝜌𝑐²+𝜀𝑝𝑣𝑐²−𝑣²_𝜉+𝜌𝑐²+𝜀𝑝𝑣²𝑐²−𝑣²+𝜀𝑝_𝜉()=𝑣=0,𝑣,𝜌±∞_±,𝜌_±. (11)

For smooth solutions, system (11) provides either the general solutions (constant states) (𝑣,𝜌)(𝜉)=const(𝜌>√𝜀/𝑐) or the singular solution 𝑐𝜉=²𝜌𝑣±√𝜀𝜌𝑐²±√=𝑐𝜀𝑣²𝜌₋𝑣₋±√𝜀𝜌₋𝑐²±√𝜀𝑣₋. (12) Given a state (𝑣₋,𝜌₋), the rarefaction wave curves in the phase plane are the sets of states that can be connected on the right by a 1-rarefaction or a 2-rarefaction wave in the form: 1-rarefaction wave curve 𝑅₁(𝑣₋,𝜌₋): 𝜉=𝜆^𝜀₁=𝑐²𝜌𝑣−√𝜀𝜌𝑐²−√=𝑐𝜀𝑣²𝜌₋𝑣₋−√𝜀𝜌₋𝑐²−√𝜀𝑣₋,𝜌<𝜌₋; (13) 2-rarefaction wave curve 𝑅₂(𝑣₋,𝜌₋): 𝜉=𝜆^𝜀₂=𝑐²𝜌𝑣+√𝜀𝜌𝑐²+√=𝑐𝜀𝑣²𝜌₋𝑣₋+√𝜀𝜌₋𝑐²+√𝜀𝑣₋,𝜌>𝜌₋. (14)

For a bounded discontinuity at 𝜉=𝜎, the Rankine-Hugoniot condition reads −𝜎𝜌𝑐²+𝜀𝑝𝑣²𝑐²𝑐²−𝑣²+𝜌+𝜌𝑐²+𝜀𝑝𝑣𝑐²−𝑣²=0,−𝜎𝜌𝑐²+𝜀𝑝𝑣𝑐²−𝑣²+𝜌𝑐²+𝜀𝑝𝑣²𝑐²−𝑣²+𝜀𝑝=0, (15) where 𝜎 is the velocity of the discontinuity. The Lax entropy conditions imply that 𝜌>𝜌₋(1-shock),𝜌<𝜌₋(2-shock). (16) Given a state (𝑣₋,𝜌₋), the shock wave curves in the phase plane are the sets of states that can be connected on the right by a 1-shock or a 2-shock wave in the form 1-shock wave curve 𝑆₁(𝑣₋,𝜌₋): 𝜎^𝜀₁=𝑐²𝜌𝑣−√𝜀𝜌𝑐²−√=𝑐𝜀𝑣²𝜌₋𝑣₋−√𝜀𝜌₋𝑐²−√𝜀𝑣₋,𝜌>𝜌₋; (17) 2-shock wave curve 𝑆₂(𝑣₋,𝜌₋): 𝜎^𝜀₂=𝑐²𝜌𝑣+√𝜀𝜌𝑐²+√=𝑐𝜀𝑣²𝜌₋𝑣₋+√𝜀𝜌₋𝑐²+√𝜀𝑣₋,𝜌<𝜌₋. (18)From (13)-(14) and (17)-(18), we can see that the rarefaction wave curves and the shock wave curves are coincident in the phase plane, which actually correspond to contact discontinuities. For convenience, we still call them rarefaction waves and shock waves, denoted by 𝑅 and 𝑆, respectively, although each of them degenerates to a characteristic.

Given state (𝑣₋,𝜌₋), we draw the curves (13)-(14) and (17)-(18) for 𝜌>√𝜀/𝑐 in the phase plane; see Figure 1. The curves 𝑆₁ and 𝑅₂ have asymptotic lines 𝑣=𝑣₁ and 𝑣=𝑣₂, respectively, where 𝑣₁=𝑐²𝜌₋𝑣₋−√𝜀𝜌₋𝑐²−√𝜀𝑣₋,𝑣₂=𝑐²𝜌₋𝑣₋+√𝜀𝜌₋𝑐²+√𝜀𝑣₋. (19) The curves 𝑆₂ and 𝑅₁ have singularity points (−𝑐,√𝜀/𝑐) and (𝑐,√𝜀/𝑐), respectively. Moreover, starting from the point (𝑣₃,𝜌₋) where 𝑣₃ will be shown below, we draw the contact discontinuity curve (12) with the positive sign, which have the asymptotic line 𝑣=𝑣₁ and the singularity point (−𝑐,√𝜀/𝑐), with 𝑣₃ satisfying 𝑐²𝜌₋𝑣₃+√𝜀𝜌₋𝑐²+√𝜀𝑣₃=𝑐²𝜌₋𝑣₋−√𝜀𝜌₋𝑐²−√𝜀𝑣₋. (20) Thus the region Λ can be divided into five regions I,II,III,IV, and V, as shown in Figure 1.

Figure 1 

Wave curves in the phase plane.

For any given right state (𝑣₊,𝜌₊), there exists a solution of (5)-(6) and (8) when (𝑣₊,𝜌₊)∈(I∪II∪III∪IV)(𝑣₋,𝜌₋) of which configurations are as follows: (1)𝑣₊,𝜌₊∈I𝑣₋,𝜌₋∶𝑅₁+𝑅₂,(2)𝑣₊,𝜌₊∈II𝑣₋,𝜌₋∶𝑅₁+𝑆₂,(3)𝑣₊,𝜌₊∈III𝑣₋,𝜌₋∶𝑆₁+𝑅₂,(4)𝑣₊,𝜌₊∈IV𝑣₋,𝜌₋∶𝑆₁+𝑆₂. (21) For the remaining case (𝑣₊,𝜌₊)∈V, we need to seek a nonclassical solution, which will be considered in the next subsection.

2.2. Delta Shock Wave Solution

In order to construct the unique global Riemann solution, we need to consider the case when (𝑣₊,𝜌₊)∈V, in which we have 𝜆^𝜀₁₊𝑐∶=²𝜌₊𝑣₊−√𝜀𝜌₊𝑐²−√𝜀𝑣₊<𝜆^𝜀₂₊𝑐∶=²𝜌₊𝑣₊+√𝜀𝜌₊𝑐²+√𝜀𝑣₊<𝜆^𝜀₁₋𝑐∶=²𝜌₋𝑣₋−√𝜀𝜌₋𝑐²−√𝜀𝑣₋<𝜆^𝜀₂₋𝑐∶=²𝜌₋𝑣₋+√𝜀𝜌₋𝑐²+√𝜀𝑣₋. (22) It means that characteristic lines from initial data will overlap in a domain Ω shown in Figure 2. Thus singularity must happen in Ω. The singularity is impossible to be a jump with finite amplitude; that is, there is no piecewise smooth and bounded solution. Hence, a weighted 𝛿-measure solution should be constructed.

Figure 2 

Characteristic lines for constant states (𝑣±,𝜌±).

Now let us give the definition of a 𝛿-shock wave type solution for system (5) with (6), in which we use the concept introduced in [43].

Suppose that Γ={𝛾_𝑖∣𝑖∈𝐼} is a graph in the closed upper half-plane {(𝑥,𝑡)∣𝑥∈ℝ,𝑡∈[0,+∞)}⊂ℝ² containing smooth arcs 𝛾_𝑖,𝑖∈𝐼, and 𝐼 is a finite set. Let 𝐼₀ be a subset of 𝐼 such that an arc 𝛾_𝑘 for 𝑘∈𝐼₀ starts from the points of the 𝑥-axis, and let Γ₀={𝑥⁰_𝑘∣𝑘∈𝐼₀} be the set of initial points of arc 𝛾_𝑘,𝑘∈𝐼₀. Let us consider 𝛿-shock wave type initial data (𝑣⁰(𝑥),𝜌⁰(𝑥)), where 𝜌⁰(𝑥)=𝜌₀(𝑥)+𝑤⁰𝛿Γ₀, (23)𝑣⁰, 𝜌₀∈𝐿^∞(ℝ;ℝ), 𝑤⁰𝛿(Γ₀)=∑_𝑘∈𝐼0𝑤⁰_𝑘𝛿(𝑥−𝑥⁰_𝑘), and 𝑤⁰_𝑘 are constants for 𝑘∈𝐼₀. Furthermore, the pressure 𝑝=−1/𝜌 in (5) is a nonlinear term with respect to 𝜌, and 𝑝⁰(𝑥,𝑡) is defined by −1/𝜌₀, where the delta measure does not contribute [12].

Definition 1. A pair of distributions (𝑣(𝑥,𝑡),𝜌(𝑥,𝑡)) and a graph Γ, where 𝜌(𝑥,𝑡) and 𝑝(𝑥,𝑡) have the form 𝜌(𝑥,𝑡)=𝜌(𝑥,𝑡)+𝑤^𝜀()𝛿(Γ)()1𝑥,𝑡,𝑝𝑥,𝑡=−𝜌(),𝑥,𝑡 (24)𝑣,𝜌∈𝐿^∞(ℝ×ℝ₊;ℝ), 𝑤^𝜀(𝑥,𝑡)𝛿(Γ)=∑_𝑖∈𝐼𝑤^𝜀_𝑖(𝑥,𝑡)𝛿(𝛾_𝑖), and 𝑤^𝜀_𝑖(𝑥,𝑡)∈𝐶(Γ) for 𝑖∈𝐼, is called a generalized 𝛿-shock wave type solution of system (5) with the initial data (𝑣⁰(𝑥),𝜌⁰(𝑥)) if the integral identities ₀^+∞+∞_−∞𝜀𝑝+𝜌𝑐²𝑣²𝑐²𝑐²−𝑣²+𝜌𝜙_𝑡+𝜀𝑝+𝜌𝑐²𝑣𝑐²−𝑣²𝜙_𝑥𝑑𝑥𝑑𝑡+_{𝑖∈𝐼𝛾𝑖}𝑤^𝜀_𝑖(𝑥,𝑡)𝑐²𝑐²−𝑣²_𝛿𝜕𝜙𝜕𝑙𝑑𝑙+^+∞_−∞𝜀𝑝⁰+𝜌₀𝑐²𝑣⁰²𝑐²𝑐²−𝑣⁰²+𝜌₀𝜙(𝑥,0)𝑑𝑥+_𝑘∈𝐼0𝑤⁰_𝑘𝑐²𝑐²−𝑣⁰_𝛿𝑥⁰_𝑘²𝜙𝑥⁰_𝑘,0=0,_{0+∞+∞−∞}𝜀𝑝+𝜌𝑐²𝑣𝑐²−𝑣²𝜙_𝑡+𝜀𝑝+𝜌𝑐²𝑣²𝑐²−𝑣²+𝜀𝑝𝜙_𝑥𝑑𝑥𝑑𝑡+_{𝑖∈𝐼𝛾𝑖}𝑤^𝜀_𝑖(𝑥,𝑡)𝑐²𝑣_𝛿𝑐²−𝑣²_𝛿𝜕𝜙+𝜕𝑙𝑑𝑙_+∞−∞𝜀𝑝⁰+𝜌₀𝑐²𝑣⁰𝑐²−𝑣⁰²(𝑥)𝜙(𝑥,0)𝑑𝑥+_𝑘∈𝐼0𝑤⁰_𝑘𝑐²𝑣⁰_𝛿𝑥⁰_𝑘𝑐²−𝑣⁰_𝛿𝑥⁰_𝑘²𝜙𝑥⁰_𝑘,0=0 (25) hold for any test functions 𝜙(𝑥,𝑡)∈𝒟(ℝ×ℝ₊), where 𝜕𝜙/𝜕𝑙 is the tangential derivative on the graph Γ, ∫_𝛾𝑖𝑑𝑙 is a line integral along the arc 𝛾_𝑖, 𝑣_𝛿(𝑥,𝑡) is the velocity of the 𝛿-shock wave, and 𝑣⁰_𝛿(𝑥⁰_𝑘)=𝑣_𝛿(𝑥⁰_𝑘,0),  𝑘∈𝐼₀.

Under the above definition, it is not difficult to get the following result.

Theorem 2. When (𝑣₊,𝜌₊)∈𝑉, for the Riemann problem (5)-(6) and (8), there is a 𝛿-shock wave solution (𝑣(𝑥,𝑡),𝜌(𝑥,𝑡)) of the form 𝑣(𝑥,𝑡)=𝑣₋+[𝑣]𝐻((𝑡,𝜌()𝑥−𝑥))𝑥,𝑡=𝜌₋+𝜌𝐻(𝑥−𝑥(𝑡))+𝑤^𝜀(𝑥,𝑡)𝛿(𝑥−𝑥(𝑡)), (26) which satisfies the integral identities (25) in the sense of Definition 1, where Γ={(𝑥,𝑡)∣𝑥=𝑥(𝑡)=𝜎_𝛿𝑡,𝑡≥0}, 𝜌(𝑥,𝑡)=𝜌₋+[𝜌]𝐻(𝑥−𝑥(𝑡)), _Γ𝑤^𝜀()()𝑥,𝑡𝜕𝜙𝑥,𝑡=𝜕𝑙^∞₀𝑤^𝜀(𝑥(𝑡))(𝑥(𝑡)),𝑡𝑑𝜙,𝑡,𝑑𝑡 (27) and 𝐻(𝑥) is the Heaviside function.

The proof of this theorem is very similar to the Theorem 6 in [43], so we omit its proof.

Definition 3. Suppose that Ω⊂ℝ×ℝ₊ is a region cut by a smooth curve Γ={(𝑥,𝑡)∣𝑥=𝑥(𝑡)} into a left and right hand parts Ω_±={(𝑥,𝑡)∣±(𝑥−𝑥(𝑡))>0}, (𝑣(𝑥,𝑡),𝜌(𝑥,𝑡)) is a generalized 𝛿-shock wave solution of system (5), and functions 𝜌(𝑥,𝑡),𝑣(𝑥,𝑡) are smooth in Ω_± and have one-side limits 𝜌_±,𝑣_± on the curve Γ. Then the generalized Rankine-Hugoniot conditions for 𝛿-shock wave are 𝑑𝑥(𝑡)𝑑𝑡=𝜎_𝛿,𝑑𝑤𝑑𝑡^𝜀(𝑥(𝑡),𝑡)𝑐²𝑐²−𝜎_𝛿²=𝜎_𝛿𝜌𝑐²+𝜀𝑝𝑣²𝑐²𝑐²−𝑣²−+𝜌𝜌𝑐²+𝜀𝑝𝑣𝑐²−𝑣²,𝑑𝑤𝑑𝑡^𝜀(𝑥(𝑡),𝑡)𝜎_𝛿𝑐²𝑐²−𝜎_𝛿²=𝜎_𝛿𝜌𝑐²+𝜀𝑝𝑣𝑐²−𝑣²−𝜌𝑐²+𝜀𝑝𝑣²𝑐²−𝑣²+𝜀𝑝, (28) with initial data 𝑤^𝜀(𝑥(0),0)=0.

To guarantee uniqueness, the generalized entropy condition 𝜆^𝜀₂₊≤𝜎_𝛿≤𝜆^𝜀₁₋ (29) should be satisfied, which means that all the characteristic lines on both sides of the delta shock wave are not out-coming.

Denote 𝐸=𝜌𝑐²+𝜀𝑝𝑣²𝑐²𝑐²−𝑣²+𝜌,𝐹=𝜌𝑐²+𝜀𝑝𝑣𝑐²−𝑣²,𝐺=𝜌𝑐²+𝜀𝑝𝑣²𝑐²−𝑣²+𝜀𝑝. (30) Thus from (28) and (29), we have, for 𝐸≠0, that 𝜎_𝛿=𝐹+𝐹²−𝐸𝐺𝐸,𝑤^𝜀(𝑥(𝑡),𝑡)=𝐹²−𝐸𝐺1−𝐹−𝐹²−𝐸𝐺𝑐𝐸²𝑡, (31) and for 𝐸=0, 𝜎_𝛿=𝐺,𝑤2𝐹_𝜀(𝑥(𝑡))𝐺,𝑡=𝐹1−2𝐹𝑐²𝑡. (32)

3. Riemann Problems for System (3)

In this section, we show some results briefly on Riemann problems to system (3) with initial data (8). The idea is similar with that in [17], so we omit the details here.

The system has a double eigenvalue 𝜆=𝑣 and only one right eigenvector 𝑟=(1,0)^⊤. System (3) is obviously linearly degenerate by ∇𝜆⋅𝑟=0. We seek the self-similar solution (𝑣,𝜌)(𝑡,𝑥)=(𝑣,𝜌)(𝜉), 𝜉=𝑥/𝑡, for which the Riemann problem can be transformed into the infinity boundary value problem: 𝜌−𝜉𝑐²−𝑣²_𝜉+𝜌𝑣𝑐²−𝑣²_𝜉=0,−𝜉𝜌𝑣𝑐²−𝑣²_𝜉+𝜌𝑣²𝑐²−𝑣²_𝜉()=𝑣=0,𝑣,𝜌±∞_±,𝜌_±. (33) For the case 𝑣₋<𝑣₊, the solution consists of two contact discontinuities plus a vacuum state between them and can be expressed as 𝜉)=𝑣𝑣,𝜌(⁻,𝜌₋,−∞<𝜉<𝑣₋,(𝑣(𝜉),0),𝑣₋≤𝜉≤𝑣₊,𝑣₊,𝜌₊,𝑣₊<𝜉<+∞, (34) where 𝑣(𝜉) is an arbitrary smooth function satisfying 𝑣(𝑣₋)=𝑣₋ and 𝑣(𝑣₊)=𝑣₊.

For the case 𝑣₋>𝑣₊, singularity must happen. According to the procedure in Section 2, a 𝛿-shock wave type solution can be constructed. We omit the details here and just give a description of the solution because of the similarity. With the definitions in Section 2, for the case 𝑣₋>𝑣₊, one can construct a 𝛿-shock wave type solution with this form 𝑣=𝑣₋+[𝑣]𝐻(𝑥−𝑥(𝑡)),𝜌=𝜌₋+𝜌𝐻(𝑥−𝑥(𝑡))+𝑤(𝑥(𝑡),𝑡)𝛿(𝑥−𝑥(𝑡)), (35) where 𝑥(𝑡) and 𝑤(𝑥,𝑡) should satisfy the generalized Rankine-Hugoniot condition: 𝑑𝑥𝑑𝑡=𝑣_𝛿,𝑑𝑤(𝑥(𝑡))𝑑𝑡,𝑡𝑐²−𝑣²_𝛿=𝑣_𝛿𝜌𝑐²−𝑣²−𝜌𝑣𝑐²−𝑣²,𝑑𝑤(𝑥(𝑡))𝑣𝑑𝑡,𝑡_𝛿𝑐²−𝑣²_𝛿=𝑣_𝛿𝜌𝑣𝑐²−𝑣²−𝜌𝑣²𝑐²−𝑣². (36) Here [ℎ]=ℎ(𝑡,𝑥(𝑡)+0)−ℎ(𝑡,𝑥(𝑡)−0) is the jump of ℎ across discontinuity.

In order to ensure the uniqueness of the Riemann solution, the generalized entropy condition should be proposed as 𝑣₊<𝑑𝑥𝑑𝑡<𝑣₋, (37) which means that all the characteristic lines on either side of a delta shock run into the line of the delta shock in the (𝑥,𝑡)-plane; this is to say that a delta shock is an overcompressive shock. From (36) and (37), we determine that 𝑣_𝛿=𝑣₊𝜌₊/𝑐²−𝑣²₊+𝑣₋𝜌₋/𝑐²−𝑣²₋𝜌₊/𝑐²−𝑣²₊+𝜌₋/𝑐²−𝑣²₋,𝑥(𝑡)=𝑣_𝛿𝑤(𝑥(𝑡))=𝑡,,𝑡𝜌₊𝑐²−𝑣²₊𝜌₋𝑐²−𝑣²₋𝑣₋−𝑣₊𝑐²−𝑣²_𝛿𝑡. (38)

4. Limit of Riemann Solutions to (5)-(6) for 𝑣₋>𝑣₊

In this section, we deal with the limit behavior of Riemann solutions to system (5)-(6) in the case (𝑣₊,𝜌₊)∈IV∪V(𝑣₋,𝜌₋) (see Figure 3(a)) as pressure vanishes. Let us assume that 𝑣₋>𝑣₊, 𝜌_±>√𝜀/𝑐 and then we divide our discussion into two parts. First, we identify the formation of delta shock wave in the case (𝑣₊,𝜌₊)∈IV(𝑣₋,𝜌₋) and compare them with the delta shock solution to (5)-(6). Then, we show how the strength and propagation speed of the delta shock wave change along with the values of 𝜀, when 𝜀 becomes smaller and smaller and ultimately goes to zero. Here two critical values 𝜀₀ and 𝜀₁ for 𝜀 should be introduced, which will play very important roles in the following discussion.

(a) The phase plane

(b) The (𝑥,𝑡)-plane

Figure 3 

Riemann solutions when (𝑣+,𝜌+)∈IV(𝑣−,𝜌−).

Lemma 4. If 𝑣₋>𝑣₊, then there exist 𝜀₀,𝜀₁>0 such that (𝑣₊,𝜌₊)∈𝐼𝑉(𝑣₋,𝜌₋) when 𝜀₁<𝜀<𝜀₀; (𝑣₊,𝜌₊)∈𝑉(𝑣₋,𝜌₋) when 0<𝜀<𝜀₁.

Proof. It follows from (17) and (18) that all possible states (𝑣,𝜌) which can be connected on the right to the left state (𝑣₋,𝜌₋) by a 1-shock wave 𝑆₁ or a 2-shock wave 𝑆₂ should satisfy 𝑆₁∶𝜌𝑣−√𝜀𝜌𝑐²−√=𝜌𝜀𝑣⁻𝑣₋−√𝜀𝜌₋𝑐²−√𝜀𝑣₋,𝜌>𝜌₋,𝑆₂∶𝜌𝑣+√𝜀𝜌𝑐²+√=𝜌𝜀𝑣⁻𝑣₋+√𝜀𝜌₋𝑐²+√𝜀𝑣₋,𝜌<𝜌₋. (39) The states that can be connected on the right to the left state (𝑣₋,𝜌₋) directly by a delta shock wave 𝑆_𝛿 should satisfy 𝑆_𝛿∶𝜌𝑣+√𝜀𝜌𝑐²+√=𝜌𝜀𝑣⁻𝑣₋−√𝜀𝜌₋𝑐²−√𝜀𝑣₋. (40)
If 𝜌₊≠𝜌₋ and (𝑣₊,𝜌₊)∈IV∪V(𝑣₋,𝜌₋), then we observe that both (𝜌𝑣−√𝜀)/(𝜌𝑐²−√𝜀𝑣) and (𝜌𝑣+√𝜀)/(𝜌𝑐²+√𝜀𝑣) are monotone increasing with 𝑣. Also we can see intuitively from Figure 3(a) together with (39) that 𝜀 should satisfy 𝜌₊𝑣₊−√𝜀𝜌₊𝑐²−√𝜀𝑣₊<𝜌₋𝑣₋−√𝜀𝜌₋𝑐²−√𝜀𝑣₋,if𝜌₊>𝜌₋,𝜌₊𝑣₊+√𝜀𝜌₊𝑐²+√𝜀𝑣₊<𝜌₋𝑣₋+√𝜀𝜌₋𝑐²+√𝜀𝑣₋,if𝜌₊<𝜌₋, (41) which imply that 𝑣₋−𝑣₊𝜀+𝜌₋−𝜌₊𝑐²−𝑣₊𝑣₋√𝜀−𝑐²𝜌₊𝜌₋𝑣₋−𝑣₊<0. (42) Thus we can take √𝜀₀=𝜌₋−𝜌₊𝑣₊𝑣₋−𝑐²2𝑣₋−𝑣₊+𝜌₋−𝜌₊²𝑐²−𝑣₊𝑣₋²+4𝑐²𝜌₊𝜌₋𝑣₋−𝑣₊²2𝑣₋−𝑣₊. (43) Obviously we have (𝑣₊,𝜌₊)∈IV∪V(𝑣₋,𝜌₋) if 𝜀<𝜀₀.
Furthermore, if 𝜌₊≠𝜌₋ and (𝑣₊,𝜌₊)∈V(𝑣₋,𝜌₋), then we can also see intuitively from Figure 3(a) together with (40) that 𝜀 should satisfy 𝜌₊𝑣₊+√𝜀𝜌₊𝑐²+√𝜀𝑣₊<𝜌₋𝑣₋−√𝜀𝜌₋𝑐²−√𝜀𝑣₋, (44) which is equivalent to 𝑣₋−𝑣₊𝜀−𝜌₋+𝜌₊𝑐²−𝑣₊𝑣₋√𝜀+𝑐²𝜌₊𝜌₋𝑣₋−𝑣₊>0. (45) Taking √𝜀₁=𝜌₋+𝜌₊𝑐²−𝑣₊𝑣₋2𝑣₋−𝑣₊−𝜌₋+𝜌₊²𝑐²−𝑣₊𝑣₋²−4𝑐²𝜌₊𝜌₋𝑣₋−𝑣₊²2𝑣₋−𝑣₊, (46) we have (𝑣₊,𝜌₊)∈V(𝑣₋,𝜌₋) if 𝜀<𝜀₁. It is not difficult to show that 𝜀₁<𝜀₀.
In particular, if 𝜌₊=𝜌₋, then (𝑣₊,𝜌₊)∈IV∪V(𝑣₋,𝜌₋) for any 𝜀>0, and moreover (𝑣₊,𝜌₊)∈V(𝑣₋,𝜌₋) if 𝑣₋−𝑣₊𝜀−2𝜌₊𝑐²−𝑣₊𝑣₋√𝜀+𝑐²𝜌²₊𝑣₋−𝑣₊>0. (47) Thus we can take √𝜀₁𝑐∶=²−𝑣₊𝑣₋−𝑐²−𝑣²₊𝑐²−𝑣²₋𝑣₋−𝑣₊𝜌₊ (48) and arbitrary 𝜀₀ which only needs to satisfy 𝜀₀>𝜀₁ in this special situation.

Lemma 4 shows that the shock wave curves 𝑆₁ and 𝑆₂ become steeper when 𝜀 decreases. There is no delta shock wave in the Riemann solution of (5)-(6) for a fluid with strong pressure. As pressure decreases, delta shock wave occurs in the Riemann solution. Hence we divide our discussion into two parts according to different ranges of 𝜀.

4.1. Formation of Delta Shocks

In this subsection, we discuss the situation 𝜀₁<𝜀<𝜀₀, namely (𝑣₊,𝜌₊)∈IV(𝑣₋,𝜌₋). When 𝑣₋>𝑣₊, the Riemann solution to system (5)-(6) with initial data (8) consists of two constant states (𝑣_±,𝜌_±), an intermediate state (𝑣^𝜀_∗,𝜌^𝜀_∗) and two shock wave curves 𝑆₁,𝑆₂ (see Figure 3(a)). Then, we have 𝑆₁∶𝜎^𝜀₁=𝑐²𝜌^𝜀_∗𝑣^𝜀_∗−√𝜀𝜌^𝜀_∗𝑐²−√𝜀𝑣^𝜀_∗=𝑐²𝜌₋𝑣₋−√𝜀𝜌₋𝑐²−√𝜀𝑣₋=𝜆^𝜀₁₋,𝜌^𝜀_∗>𝜌₋;𝑆₂∶𝜎^𝜀₂=𝑐²𝜌^𝜀_∗𝑣^𝜀_∗+√𝜀𝜌^𝜀_∗𝑐²+√𝜀𝑣^𝜀_∗=𝑐²𝜌₊𝑣₊+√𝜀𝜌₊𝑐²+√𝜀𝑣₊=𝜆^𝜀₂₊,𝜌^𝜀_∗>𝜌₊. (49) Equations (49) can be easily transformed into the following form: 𝑆₁∶𝜎^𝜀₁=𝜆^𝜀₁₋,𝑣^𝜀_∗=𝑐²𝜌^𝜀_∗𝜆^𝜀₁₋+√𝜀𝜌^𝜀_∗𝑐²+√𝜀𝜆^𝜀₁₋,𝜌^𝜀_∗>𝜌₋;𝑆₂∶𝜎^𝜀₂=𝜆^𝜀₂₊,𝑣^𝜀_∗=𝑐²𝜌^𝜀_∗𝜆^𝜀₂₊−√𝜀𝜌^𝜀_∗𝑐²−√𝜀𝜆^𝜀₂₊,𝜌^𝜀_∗>𝜌₊. (50)(51) It is not difficult to derive from (50) and (51) that 𝜆^𝜀₁₋−𝜆^𝜀₂₊𝑐²𝜀𝜌^𝜀_∗²𝜆+21−^𝜀1−𝜆^𝜀₂₊𝑐²√𝜀𝜌^𝜀_∗+𝜆^𝜀₁₋−𝜆^𝜀₂₊=0. (52) For given 𝜌_±>0, taking limit 𝜀→𝜀₁ in (52), we have lim_𝜀→𝜀1𝜆1−^𝜀1−𝜆^𝜀₂₊𝑐²√𝜀𝜌^𝜀_∗=0, (53) in which we use the fact that 𝜆^𝜀1₁₋=𝜆^𝜀1₂₊. Noticing that lim_𝜀→𝜀1𝜆^𝜀₁₋,𝜆^𝜀₂₊≠𝑐, we obtain lim_𝜀→𝜀1√𝜀𝜌^𝜀_∗=0. (54) It follows from Lemma 4 that √𝜀₁>0 is a constant if two constant states (𝑣_±,𝜌_±) are given. Hence, we deduce that lim_𝜀→𝜀1𝜌^𝜀_∗=∞. (55) Thus we have the following result.

Lemma 5. Set 𝜎=𝜆^𝜀1₁₋=𝜆^𝜀1₂₊. Then, one has lim_𝜀→𝜀1𝑣^𝜀_∗=lim_𝜀→𝜀1𝜎^𝜀₁=lim_𝜀→𝜀1𝜎^𝜀₂=𝜎,lim_𝜀→𝜀1^{𝜎𝜀2𝑡}_{𝜎𝜀1𝑡}𝜌^𝜀_∗𝑐²𝑐²−𝑣^𝜀_∗²=𝜎𝜀𝑑𝑥₁𝑝+𝜌𝑐²𝑣²𝑐²𝑐²−𝑣²−𝜀+𝜌¹𝑝+𝜌𝑐²𝑣𝑐²−𝑣²𝑡. (56)(57)

Proof. Letting 𝜀→𝜀₁ in (50), we have lim_𝜀→𝜀1𝑣^𝜀_∗=lim_𝜀→𝜀1𝑐²𝜌^𝜀_∗𝜆^𝜀₁₋+√𝜀𝜌^𝜀_∗𝑐²+√𝜀𝜆^𝜀₁₋. (58) Combining (54) and (58), one can obtain lim_𝜀→𝜀1𝑣^𝜀_∗=lim_𝜀→𝜀1𝑐²𝜆^𝜀₁₋+√𝜀/𝜌^𝜀_∗𝑐²+𝜆^𝜀₁₋√𝜀/𝜌^𝜀_∗=𝜎. (59) From (50) and (51), we see that lim_𝜀→𝜀1𝜎^𝜀₁=lim_𝜀→𝜀1𝜎^𝜀₂=𝜎. (60) Thus from (60) the two shocks 𝑆₁ and 𝑆₂ will coalesce together when 𝜀 arrives at 𝜀₁. Using the Rankine-Hugoniot condition (15) for both 𝑆₁ and 𝑆₂, we have 𝜎^𝜀₁𝜌^𝜀_∗𝑐²−𝑣^𝜀_∗²𝑐²−𝜀𝑣^𝜀_∗²𝜌^𝜀_∗²𝑐²−𝜌₋𝑐²−𝑣²₋𝑐²−𝜀𝑣²₋𝜌²₋𝑐²=𝜌^𝜀_∗𝑣^𝜀_∗𝑐²−𝑣^𝜀_∗²𝑐²−𝜀𝜌^𝜀_∗²−𝜌₋𝑣₋𝑐²−𝑣²₋𝑐²−𝜀𝜌²₋,𝜎^𝜀₂𝜌₊𝑐²−𝑣²₊𝑐²−𝜀𝑣²₊𝜌²₊𝑐²−𝜌^𝜀_∗𝑐²−𝑣^𝜀_∗²𝑐²−𝜀𝑣^𝜀_∗²𝜌^𝜀_∗²𝑐²=𝜌₊𝑣₊𝑐²−𝑣²₊𝑐²−𝜀𝜌²₊−𝜌^𝜀_∗𝑣^𝜀_∗𝑐²−𝑣^𝜀_∗²𝑐²−𝜀𝜌^𝜀_∗², (61) which yields 𝜌^𝜀_∗𝑐²−𝑣^𝜀_∗²𝑐²−𝜀𝑣^𝜀_∗²𝜌^𝜀_∗²𝑐²𝜎^𝜀₁−𝜎^𝜀₂+𝜌₊𝜎^𝜀₂𝑐²−𝑣²₊𝑐²−𝜀𝑣²₊𝜌²₊𝑐²−𝜌₋𝜎^𝜀₁𝑐²−𝑣²₋𝑐²−𝜀𝑣²₋𝜌²₋𝑐²=𝜌₊𝑣₊𝑐²−𝑣²₊𝑐²−𝜀𝜌²₊−𝜌₋𝑣₋𝑐²−𝑣²₋𝑐²−𝜀𝜌²₋. (62) Letting 𝜀→𝜀₁, we have lim_𝜀→𝜀1𝜎^𝜀₂−𝜎^𝜀₁𝜌^𝜀_∗𝑐²𝑐²−𝑣^𝜀_∗²𝜌=𝜎𝑐²−𝑣²𝑐²−𝜀₁𝑣²𝜌²𝑐²−𝜌𝑣𝑐²−𝑣²𝑐²−𝜀₁𝜌²𝜀=𝜎¹𝑝+𝜌𝑐²𝑣²𝑐²𝑐²−𝑣²−𝜀+𝜌¹𝑝+𝜌𝑐²𝑣𝑐²−𝑣². (63) According to lim_𝜀→𝜀1^{𝜎𝜀2𝑡}_{𝜎𝜀1𝑡}𝜌^𝜀_∗𝑐²𝑐²−𝑣^𝜀_∗²𝑑𝑥=lim_𝜀→𝜀1𝜎^𝜀₂−𝜎^𝜀₁𝜌^𝜀_∗𝑐²𝑐²−𝑣^𝜀_∗²𝑡. (64) Equation (57) can be easily obtained.

It can be concluded from Lemma 5 that the two shock waves 𝑆₁ and 𝑆₂ will coincide when 𝜀 tends to 𝜀₁ (see Figure 3(b)). We give the following result which provides a very nice depiction of the limit in the case 𝑣₋>𝑣₊. This theorem is similar to that in [37].

Theorem 6. Let 𝑣₋>𝑣₊. For each fixed 𝜀∈(𝜀₁,𝜀₀), assume that (𝑣^𝜀,𝜌^𝜀) is a solution containing two shocks 𝑆₁ and 𝑆₂ of (5)-(6) with initial data (8), constructed in Section 3. Then, (𝑣^𝜀,𝜌^𝜀) converges in the sense of distributions as 𝜀→𝜀₁, and the limit functions 𝜌𝑐²/(𝑐²−𝑣²) and 𝜌𝑣𝑐²/(𝑐²−𝑣²) are the sums of a step function and a 𝛿-measure with weights 𝜎𝜀₁𝑝+𝜌𝑐²𝑣²𝑐²𝑐²−𝑣²−𝜀+𝜌¹𝑝+𝜌𝑐²𝑣𝑐²−𝑣²𝜎𝜀𝑡,₁𝑝+𝜌𝑐²𝑣𝑐²−𝑣²−𝜀₁𝑝+𝜌𝑐²𝑣²𝑐²−𝑣²+𝜀₁𝑝𝑡, (65) respectively, which form a delta shock solution of (5)-(6) when 𝜀=𝜀₁.

Proof. Let 𝜉=𝑥/𝑡. Then for each fixed 𝜀>0, the Riemann solution is determined by 𝑣^𝜀,𝜌^𝜀𝜉)=𝑣(⁻,𝜌₋,as𝜉<𝜎^𝜀₁,𝑣^𝜀_∗,𝜌^𝜀_∗,as𝜎^𝜀₁<𝜉<𝜎^𝜀₂,𝑣₊,𝜌₊,as𝜉>𝜎^𝜀₂, (66) which satisfies the following weak formulations: ^∞_−∞𝜌^𝜀𝑐²𝑐²−(𝑣^𝜀)²𝜓(𝜉)𝑑𝜉+^∞_−∞𝜌^𝜀𝑐⁴+𝜀𝑝𝜌^𝜀𝑣^𝜀2𝑐²𝑐²−(𝑣^𝜀)²𝜉𝜓^′(𝜉)𝑑𝜉−^∞_−∞𝜌^𝜀𝑐²+𝜀𝑝𝜌^𝜀𝑣^𝜀𝑐²−(𝑣^𝜀)²𝜓^′(𝜉)+𝑑𝜉_∞−∞𝜀𝑝𝜌^𝜀𝑣^𝜀2𝑐²𝑐²−(𝑣^𝜀)²𝜓(𝜉)𝑑𝜉=0,_∞−∞𝜌^𝜀𝑐²𝑣^𝜀𝑐²−(𝑣^𝜀)²𝜓(𝜉)𝑑𝜉+^∞_−∞𝜌^𝜀𝑐²+𝜀𝑝𝜌^𝜀𝑣^𝜀𝑐²−(𝑣^𝜀)²𝜉𝜓^′(𝜉)−𝑑𝜉_∞−∞𝜌^𝜀𝑣^𝜀2+𝜀𝑝𝜌^𝜀𝑐²𝑐²−(𝑣^𝜀)²𝜓^′(𝜉)+𝑑𝜉_∞−∞𝜀𝑝𝜌^𝜀𝑣^𝜀𝑐²−(𝑣^𝜀)²𝜓(𝜉)𝑑𝜉=0 (67)(68) for any test function 𝜓∈𝐶^∞₀(−∞,∞). The second integral on the left side of (67) can be decomposed into ^𝜎𝜀1_−∞+^𝜎𝜀2_𝜎𝜀1+^∞_𝜎𝜀2𝜌^𝜀𝑐⁴+𝜀𝑝𝜌^𝜀𝑣^𝜀2𝑐²𝑐²−(𝑣^𝜀)²𝜉𝜓^′(𝜉)𝑑𝜉, (69) which equals 𝜌₋𝑐⁴+𝜀𝑝₋𝑣²₋𝑐²𝑐²−𝑣²₋𝜎^𝜀₁𝜓𝜎^𝜀₁−𝜌₊𝑐⁴+𝜀𝑝₊𝑣²₊𝑐²𝑐²−𝑣²₊𝜎^𝜀₂𝜓𝜎^𝜀₂−𝜌₋𝑐⁴+𝜀𝑝₋𝑣²₋𝑐²𝑐²−𝑣²₋^𝜎𝜀1_−∞𝜓(𝜉)−𝜌𝑑𝜉₊𝑐⁴+𝜀𝑝₊𝑣²₊𝑐²𝑐²−𝑣²₊^∞_𝜎^𝜀₂𝜓(𝜉)−𝜌𝑑𝜉_𝜀∗𝑐⁴+𝜀𝑝^𝜀_∗𝑣^𝜀_∗𝑐²𝑐²−𝑣^𝜀_∗^{2𝜎𝜀2𝜎𝜀}₁𝜓(𝜉)+𝜌𝑑𝜉_𝜀∗𝑐⁴+𝜀𝑝^𝜀_∗𝑣^𝜀_∗𝑐²𝑐²−𝑣^𝜀_∗²𝜎^𝜀₂𝜓𝜎^𝜀₂−𝜎^𝜀₁𝜓𝜎^𝜀₁. (70) The third term on the left side of (67) can be calculated by −^𝜎𝜀1_−∞+^𝜎𝜀2_𝜎𝜀1+^∞_𝜎𝜀2𝜌^𝜀𝑐²+𝜀𝑝𝜌^𝜀𝑣^𝜀𝑐²−(𝑣^𝜀)²𝜓^′(𝜉)𝜌𝑑𝜉=−⁻𝑐²+𝜀𝑝₋𝑣₋𝑐²−𝑣²₋𝜓𝜎^𝜀₁+𝜌₊𝑐²+𝜀𝑝₊𝑣₊𝑐²−𝑣²₊𝜓𝜎^𝜀₂−𝜌^𝜀_∗𝑐²+𝜀𝑝^𝜀_∗𝑣^𝜀_∗𝑐²−𝑣^𝜀_∗²𝜓𝜎^𝜀₂−𝜓𝜎^𝜀₁, (71) and the last term in (67) yields 𝜀𝑝𝜌^∞_−∞^𝜀𝑣^𝜀2𝑐²𝑐²−(𝑣^𝜀)²𝜓(𝜉)=𝑑𝜉_{𝜎𝜀1−∞}+^𝜎𝜀2_𝜎𝜀1+^∞_𝜎𝜀2𝜀𝑝𝜌^𝜀𝑣^𝜀2𝑐²𝑐²−(𝑣^𝜀)²𝜓(𝜉)=𝑑𝜉𝜀𝑝₋𝑣²₋𝑐²𝑐²−𝑣²₋^𝜎𝜀1_−∞𝜓(𝜉)+𝑑𝜉𝜀𝑝₊𝑣²₊𝑐²𝑐²−𝑣²₊^∞_𝜎^𝜀₂𝜓(𝜉)+𝑑𝜉𝜀𝑝^𝜀_∗𝑣^𝜀_∗²𝑐²𝑐²−𝑣^𝜀_∗^{2𝜎𝜀2𝜎𝜀}₁𝜓(𝜉)𝑑𝜉. (72) Combining expressions (67), (70)–(72), we obtain ^∞_−∞𝜌^𝜀𝑐²𝑐²−(𝑣^𝜀)²𝜓(𝜉)𝜌𝑑𝜉=−⁻𝑐⁴+𝜀𝑝₋𝑣²₋𝑐²𝑐²−𝑣²₋𝜎^𝜀₁𝜓𝜎^𝜀₁+𝜌₊𝑐⁴+𝜀𝑝₊𝑣²₊𝑐²𝑐²−𝑣²₊𝜎^𝜀₂𝜓𝜎^𝜀₂+𝜌₋𝑐²𝑐²−𝑣²₋^𝜎𝜀1_−∞𝜓(𝜉)𝜌𝑑𝜉+⁺𝑐²𝑐²−𝑣²₊^∞_𝜎^𝜀₂𝜓(𝜉)+𝜌𝑑𝜉_𝜀∗𝑐²𝑐²−𝑣^𝜀_∗^{2𝜎𝜀2𝜎𝜀}₁𝜓(𝜉)−𝜌𝑑𝜉_𝜀∗𝑐⁴+𝜀𝑝^𝜀_∗𝑣^𝜀_∗𝑐²𝑐²−𝑣^𝜀_∗𝜎^𝜀₂𝜓𝜎^𝜀₂−𝜎^𝜀₁𝜓𝜎^𝜀₁+𝜌₋𝑐²+𝜀𝑝₋𝑣₋𝑐²−𝑣²₋𝜓𝜎^𝜀₁−𝜌₊𝑐²+𝜀𝑝₊𝑣₊𝑐²−𝑣²₊𝜓𝜎^𝜀₂+𝜌^𝜀_∗𝑐²+𝜀𝑝^𝜀_∗𝑣^𝜀_∗𝑐²−𝑣^𝜀_∗²𝜓𝜎^𝜀₂−𝜓𝜎^𝜀₁. (73) Taking the limit 𝜀→𝜀₁ in (73), observing that 𝜓∈𝐶^∞₀(−∞,∞) and 𝑝(𝜌^𝜀_∗) is bounded, together with the fact lim_𝜀→𝜀1𝑣^𝜀_∗=lim_𝜀→𝜀1𝜎^𝜀₁=lim_𝜀→𝜀1𝜎^𝜀₂=𝜎, we have lim_𝜀→𝜀1^∞_−∞𝜌^𝜀𝑐²𝑐²−(𝑣^𝜀)²(𝜉)−𝜌₀𝑐²𝑐²−𝑣²₀(𝜓(𝜉)=𝜎𝜀𝜉−𝜎)𝑑𝜉₁𝑝+𝜌𝑐²𝑣²𝑐²𝑐²−𝑣²−𝜀+𝜌₁𝑝+𝜌𝑐²𝑣𝑐²−𝑣²𝜓(𝜎), (74) where 𝑎₀(𝜉)=𝑎₋+[𝑎]𝐻(𝜉−𝜎) and 𝐻 is the Heaviside function. With the same reason as above, the limit of the second and the last integral on the left side of equality (68) as 𝜀→𝜀₁ is −^𝜎_−∞𝜌₋𝑣₋𝑐²𝑐²−𝑣²₋𝜓(𝜉)𝑑𝜉−^∞_𝜎𝜌₊𝑣₊𝑐²𝑐²−𝑣²₊𝜓(𝜉)(𝜎𝑑𝜉−𝜎𝜓)𝜌𝑐²+𝜀₁𝑝𝑣𝑐²−𝑣². (75) As done to (71), due to the fact that 𝑝(𝜌^𝜀_∗) is bounded and lim_𝜀→𝜀1𝜎^𝜀₁=lim_𝜀→𝜀1𝜎^𝜀₂=𝜎, the third integral on the left side of equality (68) converges to 𝜓(𝜎)𝜌𝑣²+𝜀₁𝑝𝑐²𝑐²−𝑣² (76) as 𝜀→𝜖₁. Then it follows from (68), (75), and (76) that lim_𝜀→𝜀1^∞_−∞𝜌^𝜀𝑣^𝜀𝑐²𝑐²−(𝑣^𝜀)²(𝜉)−𝜌₀𝑣₀𝑐²𝑐²−𝑣²₀(𝜓(𝜉)=𝜎𝜀𝜉−𝜎)𝑑𝜉₁𝑝+𝜌𝑐²𝑣𝑐²−𝑣²−𝜀₁𝑝+𝜌𝑐²𝑣²𝑐²−𝑣²+𝜀₁𝑝𝜓(𝜎). (77)
Finally, we study the limits of 𝜌^𝜀𝑐²/(𝑐²−(𝑣^𝜀)²) and 𝜌^𝜀𝑣^𝜀𝑐²/(𝑐²−(𝑣^𝜀)²) as 𝜀→𝜀₁ by tracing the time-dependence of weights of the 𝛿-measure. Let 𝜙(𝑥,𝑡)∈𝐶^∞₀((−∞,∞)×[0,∞)) and set 𝜙(𝜉,𝑡)∶=𝜙(𝜉𝑡,𝑡). Then we have lim_𝜀→𝜀1^∞₀^∞_−∞𝜌^𝜀𝑐²𝑐²−(𝑣^𝜀)²𝑥𝑡𝜙(𝑥,𝑡)𝑑𝑥𝑑𝑡=lim_𝜀→𝜀1^∞₀𝑡^∞_−∞𝜌^𝜀𝑐²𝑐²−(𝑣^𝜀)²(𝜉)𝜙(𝜉,𝑡)𝑑𝜉𝑑𝑡. (78) On the other hand, lim_𝜀→𝜀1^∞_−∞𝜌^𝜀𝑐²𝑐²−(𝑣^𝜀)²(𝜉)𝜙()=𝜉,𝑡𝑑𝜉_∞−∞𝜌₀𝑐²𝑐²−𝑣²₀()𝜙()+𝜎𝜀𝜉−𝜎𝜉,𝑡𝑑𝜉₁𝑝+𝜌𝑐²𝑣²𝑐²𝑐²−𝑣²−𝜀+𝜌₁𝑝+𝜌𝑐²𝑣𝑐²−𝑣²𝜙(𝜉,𝑡)=𝑡^−1∞_−∞𝜌₀𝑐²𝑐²−𝑣²₀()𝜙()+𝜎𝜀𝑥−𝜎𝑡𝑥,𝑡𝑑𝑥₁𝑝+𝜌𝑐²𝑣²𝑐²𝑐²−𝑣²−𝜀+𝜌₁𝑝+𝜌𝑐²𝑣𝑐²−𝑣²𝜙(𝜎𝑡,𝑡). (79) Combining (78) and (79) together, we obtain lim_𝜀→𝜀1^∞₀^∞_−∞𝜌^𝜀𝑐²𝑐²−(𝑣^𝜀)²𝑥𝑡𝜙()=𝑥,𝑡𝑑𝑥𝑑𝑡_{∞0∞−∞}𝜌₀𝑐²𝑐²−𝑣²₀()𝜙()+𝑥−𝜎𝑡𝑥,𝑡𝑑𝑥𝑑𝑡_∞0𝑡𝜎𝜀₁𝑝+𝜌𝑐²𝑣²𝑐²𝑐²−𝑣²−𝜀+𝜌₁𝑝+𝜌𝑐²𝑣𝑐²−𝑣²𝜙(𝜎𝑡,𝑡)𝑑𝑡. (80) With the same reason as before, we deduce that lim_𝜀→𝜀1^∞₀^∞_−∞𝜌^𝜀𝑣^𝜀𝑐²𝑐²−(𝑣^𝜀)²𝑥𝑡𝜙()=𝑥,𝑡𝑑𝑥𝑑𝑡_{∞0∞−∞}𝜌₀𝑣₀𝑐²𝑐²−𝑣²₀()𝜙()+𝑥−𝜎𝑡𝑥,𝑡𝑑𝑥𝑑𝑡_∞0𝑡𝜎𝜀₁𝑝+𝜌𝑐²𝑣𝑐²−𝑣²−𝜀₁𝑝+𝜌𝑐²𝑣²𝑐²−𝑣²+𝜀₁𝑝𝜙(𝜎𝑡,𝑡)𝑑𝑡. (81) Thus the results have been obtained.

Comparing the above results with (28), we can see that the quantities 𝜎_𝛿, 𝜔^𝜀(𝑡) and the limits of 𝑣^𝜀_∗,𝜎^𝜀₁, and 𝜎^𝜀₂ should be consistent with (31) as proposed for the Riemann solutions to (5)-(6) for 𝐸≠0 when we take 𝜀=𝜀₁. If 𝐸=0, then the assert is obviously true. Thus, as 𝜀→𝜀₁ it uniquely determines that the limit of the Riemann solutions to system (5)-(6) in the case (𝑣₊,𝜌₊)∈IV(𝑣₋,𝜌₋) is just the delta shock solution of (5)-(6) in the case (𝑣₊,𝜌₊)∈𝑆_𝛿, where the curve 𝑆_𝛿 is actually the boundary between the regions IV(𝑣₋,𝜌₋) and V(𝑣₋,𝜌₋).

4.2. Limit Behavior of Delta Shock Wave

In this subsection, we continue to discuss the situation 0<𝜀<𝜀₁, in which (𝑣₊,𝜌₊)∈V(𝑣₋,𝜌₋) with 𝑣₋>𝑣₊ and 𝜌_±>√𝜀/𝑐. In this case, the Riemann solution to system (5)-(6) contains a delta shock wave besides two constant states (𝑣_±,𝜌_±). We want to observe the behavior of strength and propagation speed of the delta shock wave when 𝜀 decreases and finally tends to zero.

If 0<𝜀<𝜀₁, the Riemann solution to system (5)-(6) contains a delta shock wave whose strength and propagation speed can be expressed by (31) for 𝐸≠0. Rewrite 𝜎_𝛿 in (31) as 𝜎_𝛿=𝜌₊𝑣₊𝑐²−𝑣²₊𝑐²−𝜀𝜌²₊−𝜌₋𝑣₋𝑐²−𝑣²₋𝑐²−𝜀𝜌²₋+𝜌₊𝜌₋𝛼(𝜀)𝑐²−𝑣²₊𝑐²−𝑣²₋×𝜌₊𝑐²−𝑣²₊𝑐²−𝜀𝑣²₊𝜌²₊𝑐²−𝜌₋𝑐²−𝑣²₋𝑐²−𝜀𝑣²₋𝜌²₋𝑐²⁻¹, (82) where 𝛼(𝜀)=𝑣₋−𝑣₊²𝑐²−𝜀𝜌₊𝜌₋²1−𝜀𝜌₊−1𝜌₋²𝑐²−𝑣₊𝑣₋². (83) It can be derived from (31), (82), and (83) that 𝑑𝑤(𝑡)𝑐𝑑𝜀𝑐²−𝜎²²_𝛿<0, (84) which means the strength of the delta shock wave increases when 𝜀 decreases. Furthermore, taking the limit 𝜀→0 in (82) leads to lim_𝜀→0𝜎_𝛿=𝑣₊𝜌₊/𝑐²−𝑣²₊+𝑣₋𝜌₋/𝑐²−𝑣²₋𝜌₊/𝑐²−𝑣²₊+𝜌₋/𝑐²−𝑣²₋. (85) Thus one can easily deduce from (31) and (85) that lim_𝜀→0𝑤^𝜀(𝑡)=𝜌₊𝑐²−𝑣²₊𝜌₋𝑐²−𝑣²₋𝑣₋−𝑣₊𝑐²−𝑣²_𝛿𝑡. (86) Thus, the limit values of 𝜎_𝛿 and 𝑤^𝜀(𝑡) are identical with (38). For the special case 𝐸=0, the same result can be derived from (32).

From the above discussion, we can see that the limit of the strength and propagation speed of the delta shock wave in the Riemann solution to system (5)-(6) is in accordance with that of system (4) with the same initial data (8). That is to say, the delta shock solution to system (5)-(6) converges to the delta shock solution to system (4) as pressure vanishes.

Combining the results of the above two subsections, we conclude that the two shock waves of the Riemann solution to system (5)-(6) become steeper as 𝜀 decreases and then coincide with a delta shock wave at a certain critical value 𝜀₁. As 𝜀 continues to drop, the strength of this delta shock becomes stronger and stronger. In the end, as 𝜀 goes to zero, the delta shock solution is nothing but the Riemann solution to the zero-pressure relativistic Euler system (4).

5. Limit of Riemann Solutions to (5)-(6) for 𝑣₋<𝑣₊

In this section, we will show the limit behavior of rarefaction waves in the Riemann solutions to (5)-(6) and (8) when 𝜀 tends to zero in the case (𝑣₊,𝜌₊)∈I(𝑣₋,𝜌₋) with 𝑣₋<𝑣₊ and 𝜌_±>√𝜀/𝑐. See Figure 4(a).

(a) The phase plane

(b) The (𝑥,𝑡)-plane

Figure 4 

Riemann solutions when (𝑣+,𝜌+)∈I(𝑣−,𝜌−).

Lemma 7. If 𝑣₋<𝑣₊, then there exists 𝜀₀>0 such that (𝑣₊,𝜌₊)∈𝐼(𝑣₋,𝜌₋) for 0<𝜀<𝜀₀.

Proof. It follows from (13) and (14) that all possible states (𝑣,𝜌) which can be connected on the right side to the left state (𝑣₋,𝜌₋) by a 1-rarefaction wave 𝑅₁ or a 2-rarefaction wave 𝑅₂ should satisfy 𝑅₁∶𝜌𝑣−√𝜀𝜌𝑐²−√=𝜌𝜀𝑣⁻𝑣₋−√𝜀𝜌₋𝑐²−√𝜀𝑣₋,𝜌<𝜌₋,𝑅₂∶𝜌𝑣+√𝜀𝜌𝑐²+√=𝜌𝜀𝑣⁻𝑣₋+√𝜀𝜌₋𝑐²+√𝜀𝑣₋,𝜌>𝜌₋. (87) Particularly, if 𝜌₊=𝜌₋, then (𝑣₊,𝜌₊)∈I(𝑣₋,𝜌₋) for any 𝜀>0. Thus we can take arbitrary 𝜀₀>0 in this special situation.
Let 𝜌₊≠𝜌₋ and (𝑣₊,𝜌₊)∈I(𝑣₋,𝜌₋). Observing that (𝜌𝑣−√𝜀)/(𝜌𝑐²−√𝜀𝑣) and (𝜌𝑣+√𝜀)/(𝜌𝑐²+√𝜀𝑣) are both monotone increasing with 𝑣, we can see intuitively from Figure 4(a) together with (87) that 𝜀 satisfies 𝜌₊𝑣₊−√𝜀𝜌₊𝑐²−√𝜀𝑣₊>𝜌₋𝑣₋−√𝜀𝜌₋𝑐²−√𝜀𝑣₋,if𝜌₊<𝜌₋,𝜌₊𝑣₊+√𝜀𝜌₊𝑐²+√𝜀𝑣₊>𝜌₋𝑣₋+√𝜀𝜌₋𝑐²+√𝜀𝑣₋,if𝜌₊>𝜌₋, (88) which imply that 𝑣₊−𝑣₋𝜀+𝜌₋−𝜌₊𝑐²−𝑣₊𝑣₋√𝜀−𝑐²𝜌₊𝜌₋𝑣₊−𝑣₋<0. (89) Thus we can take √𝜀₀𝜌∶=⁻−𝜌₊𝑣₊𝑣₋−𝑐²2𝑣₊−𝑣₋+𝜌₋−𝜌₊²𝑐²−𝑣₊𝑣₋²+4𝑐²𝜌₊𝜌₋𝑣₊−𝑣₋²2𝑣₊−𝑣₋. (90) Obviously we have (𝑣₊,𝜌₊)∈I(𝑣₋,𝜌₋) if 𝜀<𝜀₀.

Lemma 7 implies that the rarefaction wave curves 𝑅₁ and 𝑅₂ become steeper when 𝜀 goes to zero. As 𝑣₋<𝑣₊, by Lemma 7, for any given 𝜀∈(0,𝜀₀), the Riemann solution to system (5)-(6) with initial data (8) consists of two constant states (𝑣_±,𝜌_±), an intermediate state (𝑣^𝜀_∗,𝜌^𝜀_∗), and two rarefaction curves 𝑅₁,𝑅₂. See Figure 4(a). Then, it follows from (13) and (14) that 𝑅₁∶𝜆^𝜀₁=𝑐²𝜌𝑣−√𝜀𝜌𝑐²−√=𝑐𝜀𝑣²𝜌₋𝑣₋−√𝜀𝜌₋𝑐²−√𝜀𝑣₋=𝜆^𝜀₁₋,𝜌^𝜀_∗≤𝜌≤𝜌₋,𝑅₂∶𝜆^𝜀₂=𝑐²𝜌𝑣+√𝜀𝜌𝑐²+√=𝑐𝜀𝑣²𝜌₊𝑣₊+√𝜀𝜌₊𝑐²+√𝜀𝑣₊=𝜆^𝜀₂₊,𝜌^𝜀_∗≤𝜌≤𝜌₊. (91) From (91), we can derive 𝜆^𝜀₂₊−𝜆^𝜀₁₋𝜀𝜌^𝜀_∗²+2𝜆^𝜀₁₋𝜆^𝜀₂₊−𝑐²√𝜀𝜌^𝜀_∗+𝜆^𝜀₂₊−𝜆^𝜀₁₋𝑐²=0. (92) So 𝜌^𝜀_∗ can be expressed as 𝜌^𝜀_∗=√𝜀𝜆^𝜀₂₊−𝜆^𝜀₁₋𝑐²−𝜆^𝜀₁₋𝜆^𝜀₂₊+𝑐²−𝜆^𝜀₁₋²𝑐²−𝜆^𝜀₂₊². (93) It is observed that the propagation speed of 𝑅₁ increases with the decrease of 𝜀, while the propagation speed of 𝑅₂ decreases with the decrease of 𝜀. Thus from lim_𝜀→0𝜆^𝜀₁₋=𝑣₋,lim_𝜀→0𝜆^𝜀₂₊=𝑣₊, (93) yields lim_𝜀→0𝜌^𝜀_∗=0,lim_𝜀→0𝜆^𝜀₁=𝑣₋,lim_𝜀→0𝜆^𝜀₂=𝑣₊. (94) The above identities assert that as 𝜀 drops to zero, 𝜌^𝜀_∗ vanishes and two rarefaction waves 𝑅₁ and 𝑅₂ become two contact discontinuities connecting the constant states (𝑣_±,𝜌_±) and the vacuum (𝜌_∗=0), with the speeds 𝑣₋ and 𝑣₊, respectively; see Figure 4(b). From the above discussion, we can summarize our results as follows.

Theorem 8. In the case of 𝑣₋<𝑣₊, as 𝜀 drops to zero, the Riemann solution of (5)-(6) with initial data (8) converges to a vacuum solution, which is exactly the corresponding Riemann solution to system (4) with the same initial data.

6. Limit of Riemann Solutions to (5)-(6) for 𝑣₋=𝑣₊

In this section, we consider the limit of Riemann solutions to system (5)-(6) with initial data (8) when 𝑣₋=𝑣₊. In this case, the Riemann solutions contain a 1-rarefaction wave 𝑅₁ and a 2-shock wave 𝑆₂ for 𝜌₊<𝜌₋ or a 1-shock wave 𝑆₁ and a 2-rarefaction wave 𝑅₂ for 𝜌₊>𝜌₋. Particularly, if 𝜌₊=𝜌₋, the solution is a constant state (𝑣₋,𝜌₋).

For the case 𝜌₊<𝜌₋, we have (𝑣₊,𝜌₊)∈II(𝑣₋,𝜌₋) obviously, as shown in Figure 5(a), the Riemann solution of (5)-(6) has the construction as 𝑣^𝜀,𝜌^𝜀𝜉)=𝑣(⁻,𝜌₋,−∞<𝜉≤𝜆^𝜀₁₋,𝑅₁,𝜆^𝜀₁₋≤𝜉≤𝜆^𝜀_1∗,𝑣^𝜀_∗,𝜌^𝜀_∗,𝜆^𝜀_1∗≤𝜉<𝜎^𝜀₂,𝑣₊,𝜌₊,𝜎^𝜀₂<𝜉<∞, (95) where 𝜎^𝜀₂ is the propagation speed of 𝑆₂ and 𝑅₁ consists of the states (𝑣,𝜌) satisfying (13).

(a) The phase plane

(b) The (𝑥,𝑡)-plane

Figure 5 

Riemann solutions when (𝑣+,𝜌+)∈II(𝑣−,𝜌−).

By (13) and (18), the intermediate state (𝑣^𝜀_∗,𝜌^𝜀_∗) between 𝑅₁ and 𝑆₂ satisfies 𝜌^𝜀_∗𝑣^𝜀_∗−√𝜀𝜌^𝜀_∗𝑐²−√𝜀𝑣^𝜀_∗=𝜌₋𝑣₋−√𝜀𝜌₋𝑐²−√𝜀𝑣₋𝜌_𝜀∗,𝑣^𝜀_∗+√𝜀𝜌^𝜀_∗𝑐²+√𝜀𝑣^𝜀_∗=𝜌₊𝑣₊+√𝜀𝜌₊𝑐²+√𝜀𝑣₊, (96)(97) with 𝜌₊<𝜌^𝜀_∗<𝜌₋. Letting 𝜀→0 in (96), one can immediately get lim_𝜀→0𝑣^𝜀_∗=𝑣₋, noticing the fact that 𝜌^𝜀_∗ is bounded. Meanwhile, we have lim_𝜀→0𝜎^𝜀₂=lim_𝜀→0𝑐²𝜌₊𝑣₊+√𝜀𝜌₊𝑐²+√𝜀𝑣₊=𝑣₋,lim_𝜀→0𝜆^𝜀_1∗=lim_𝜀→0𝑐²𝜌^𝜀_∗𝑣^𝜀_∗−√𝜀𝜌^𝜀_∗𝑐²−√𝜀𝑣^𝜀_∗=𝑣₋,lim_𝜀→0𝜆^𝜀₁₋=lim_𝜀→0𝑐²𝜌₋𝑣₋−√𝜀𝜌₋𝑐²−√𝜀𝑣₋=𝑣₋. (98) From identities (98), as 𝜀 goes to zero, we conclude that the rarefaction wave 𝑅₁ and shock wave 𝑆₂ converge to one contact discontinuity with the propagation speed 𝑣₋, which connects the constant states (𝑣_±,𝜌_±); see Figure 5(b).

For the case 𝜌₊>𝜌₋, (𝑣₊,𝜌₊)∈III(𝑣₋,𝜌₋), as shown in Figure 6(a), the Riemann solution of (5)-(6) has the construction as 𝑣^𝜀,𝜌^𝜀𝜉)=𝑣(⁻,𝜌₋,−∞<𝜉<𝜎^𝜀₁,𝑣^𝜀_∗,𝜌^𝜀_∗,𝜎^𝜀₁<𝜉≤𝜆^𝜀_2∗,𝑅₂,𝜆^𝜀_2∗≤𝜉≤𝜆^𝜀₂₊,𝑣₊,𝜌₊,𝜆^𝜀₂₊≤𝜉<∞, (99) where 𝜎^𝜀₁ is the propagation speed of 𝑆₁ and 𝑅₂ consists of the states (𝑣,𝜌) satisfying (14). By (14) and (17), the intermediate state (𝑣^𝜀_∗,𝜌^𝜀_∗) between 𝑅₂ and 𝑆₁ satisfies (96) and (97) with 𝜌₋<𝜌^𝜀_∗<𝜌₊. Similarly, we have lim_𝜀→0𝜆^𝜀_2∗=lim_𝜀→0𝜆^𝜀₂₊=lim_𝜀→0𝜎^𝜀₁=𝑣₋. (100) Thus as 𝜀 goes to zero, the rarefaction wave 𝑅₂ and shock wave 𝑆₁ converge to a contact discontinuity with the propagation speed 𝑣₋, which connects the constant states (𝑣_±,𝜌_±); see Figure 6(b). From the above discussion, we can summarize our results as follows.

(a) The phase plane

(b) The (𝑥,𝑡)-plane

Figure 6 

Riemann solutions when (𝑣+,𝜌+)∈III(𝑣−,𝜌−).

Theorem 9. In the case of 𝑣₋=𝑣₊, as 𝜀 drops to zero, the Riemann solution of (5)-(6) with initial data (8) converges to a contact discontinuity connecting the constant states (𝑣_±,𝜌_±), which is exactly the corresponding Riemann solution to system (4) with the same initial data.

Acknowledgments

The authors would like to thank the anonymous referees for their profitable comments and suggestions. Gan Yin’s research was partially supported by National Natural Science Foundation of China (11101348).