Viscous Regularization of Delta Shock Wave Solution for a Simplified Chromatography System

Abstract

The delta shock wave for a simplified chromatography system is obtained in the Riemann solution when −1<𝑣₋<0<𝑣₊. In fact, the result in this paper is the reasonable generalization of the result in Sun (2011) with 𝑣₋=0<𝑣₊ which is exactly the critical situation for −1<𝑣₋<0<𝑣₊. The self-similar viscosity vanishing approach is also used to check the delta shock wave solution.

1. Introduction

The theory of nonlinear chromatography system is the foundation of chromatography separation process and also plays an important role in the modern industry. Recently, the delta shock wave has been captured numerically and experimentally by Mazzotti et al. [1, 2] in the Riemann solutions for the local equilibrium model of two-component nonlinear chromatography. Since then, extensive attention has been drawn on the formation of delta shock wave solution to all kinds of chromatography systems [3–6].

In [5], Sun considered the Riemann problem for a simplified chromatography system: 𝑣_𝑡+𝑣1+𝑣_𝑥𝑤=0,_𝑡+𝑤1+𝑣_𝑥=0, (1) with the initial data (𝑣,𝑤)(𝑥,0)=𝑣_±,𝑤_±,±𝑥>0, (2) where 𝑣_± and 𝑤_± are all given constants. She discovered that a delta shock wave solution also appears in the situation 𝑣₋=0<𝑣₊ by employing the self-similar viscosity vanishing approach.

In the present paper, we extend the result from the critical situation 𝑣₋=0<𝑣₊ in [5] to the general situation −1<𝑣₋<0<𝑣₊ and discover that the delta shock wave also appears in the Riemann solution to (1) and (2). Actually, in the situation −1<𝑣₋<0<𝑣₊, the Riemann solution to (1) and (2) is a delta shock wave connecting two constant states (𝑣_±,𝑤_±) whose propagation speed and strength are, respectively, 1𝜎=1+𝑣₋1+𝑣₊(𝑡)=𝑤,𝛽⁻𝑣₊−𝑤₊𝑣₋𝑡1+𝑣₋1+𝑣₊. (3) On the other hand, in the situation 𝑣₋=0<𝑣₊ in [5], the propagation speed and strength of delta shock wave are, respectively, 1𝜎=1+𝑣₊(𝑡)=𝑤,𝛽⁻𝑣₊𝑡1+𝑣₊. (4) If we let 𝑣₋→0+ in the situation −1<𝑣₋<0<𝑣₊, then the result is identical with that for the situation 𝑣₋=0<𝑣₊ in [5]. Thus, one can see that the situation 𝑣₋=0<𝑣₊ is the critical one, where the delta shock wave can be obtained, and the result in this paper is the reasonable generalization of the result in [5].

Similar to [5], we also consider the following viscous regularization system: 𝑣_𝑡+𝑣1+𝑣_𝑥=𝜀𝑡𝑣_𝑥𝑥,𝑤_𝑡+𝑤1+𝑣_𝑥=0, (5) which enables us to see that the delta shock wave is the reasonable choice of the Riemann solution to (1) and (2) when −1<𝑣₋<0<𝑣₊.

In addition, we emphasize that our result here is also reasonable in the sense of chromatography engineer since a simplified chromatography system can also be derived from the other nonlinear chromatography system: 𝜕_𝑡𝑢₁+𝜕_𝑥𝑢₁1−𝑢₁+𝑢₂𝜕=0,_𝑡𝑢₂+𝜕_𝑥𝑢₂1−𝑢₁+𝑢₂=0, (6) by introducing the change of variables 𝑣=𝑢₂−𝑢₁,𝑤=𝑢₁+𝑢₂, (7) which is different from the derivation in [5]. Here 𝑢₁, 𝑢₂ are the nonnegative functions of the variables (𝑥,𝑡)∈𝑅×𝑅₊ to denote the Langmuir-like and the anti-Langmuir-like components, respectively, which express transformations of the concentrations of two solutes. It is clear that the delta measure appears in 𝑤 for the system (1) is equivalent to saying that the delta measures appear in both the state variables 𝑢₁ and 𝑢₂ for the system (6). It is easily derived that the strengths of the delta measures in 𝑢₁ and 𝑢₂ are equal to each other. In other words, the phenomena of singular concentrations in the two components 𝑢₁ and 𝑢₂ appear simultaneously for the system (6).

The system (1) is the simplest example for systems of Temple class [7]. In other words, the shock curves coincide with the rarefaction curves in the phase plane. Compared with general systems of conservation laws, well-posed results for systems of Temple class [8] are available for a much larger class of initial data due to the above feature. Furthermore, thanks to the particular structure, wave interactions for the systems of Temple class can also be dealt with completely [9–12], which enables us to solve the Cauchy problem for arbitrary data of bounded variation.

The self-similar viscosity vanishing approach was first proposed by Dafermos [13] to construct Riemann solutions for broad classes of 2×2 systems that include the equations of isentropic, Lagrangian, and gas dynamics. This method was widely used in [14] for the admissibility of weak solutions of the Riemann problem for hyperbolic systems of conservation laws, and the admissibility criterion was called the wave-fan criterion in [15]. Special attention was also paid in [6, 16–22] to the formation of the delta shock waves in the Riemann solutions for some systems of hyperbolic conservation laws.

There exist numerous excellent papers for the related equations and results about the measure-valued solutions for hyperbolic systems of conservation laws. The well-known examples are the transport equations [21, 23–27] (or called the pressureless Euler equations) and the Chaplygin gas dynamics equations [28, 29] which have been extensively studied. For the other related equations and results, we can refer to [30–35] and the related references therein.

The paper is organized as follows. In Section 2, the Riemann problem of (1) and (2) is considered. In particular, the delta shock wave does occur in the Riemann solution when −1<𝑣₋<0<𝑣₊. In Section 3, we show that there exists a smooth solution for the regularized system (5) with the initial data (2). Then, by letting 𝜀→0, we prove that the limit of the regularized solution is exactly the corresponding Riemann solution to (1) and (2) when −1<𝑣₋<0<𝑣₊.

2. The Riemann Problem for (1) and (2) in the Phase Space 𝑣>−1

In this section, we generalize the Riemann problem for (1) and (2) from the phase space (𝑣,𝑤)∈[0,∞)×(−∞,∞) in [5] to the phase space (𝑣,𝑤)∈(−1,∞)×(−∞,∞). Compared with the result in [5] that there are three kinds of the Riemann solutions to (1) and (2) in the phase space (𝑣,𝑤)∈[0,∞)×(−∞,∞), there exist six kinds of the Riemann solutions to (1) and (2) in the phase space (𝑣,𝑤)∈(−1,∞)×(−∞,∞)(Figure 1). In what follows, we first depict the properties of (1) and then list the Riemann solutions to (1) and (2) in the phase space (𝑣,𝑤)∈(−1,∞)×(−∞,∞) case by case. Finally, the generalized Rankine-Hugoniot relations of delta shock wave are derived in detail. We can also refer to [3, 4, 12] for the related results about the other nonlinear chromatography systems.

(a) 𝑅+𝐽 when 𝑣−>𝑣+>0

(b) 𝑆+𝐽 when 0<𝑣−<𝑣+

(c) 𝑅1+𝑅2 when −1<𝑣+<0<𝑣−

(d) 𝐽+𝑆 when −1<𝑣−<𝑣+<0

(e) 𝐽+𝑅 when −1<𝑣+<𝑣−<0

(f) 𝛿𝑆 when −1<𝑣−<0<𝑣+

Figure 1 

The Riemann solutions to (1) and (2) in all the situations.

The characteristic eigenvalues of (1) are 𝜆₁=1/(1+𝑣)² and 𝜆₂=1/(1+𝑣), respectively. Thus, it is clear to see that (1) is strictly hyperbolic in the phase space 𝑣>0 and −1<𝑣<0 and is nonstrictly hyperbolic on the line 𝑣=0. The corresponding right characteristic vectors are 𝑟₁=(𝑣,𝑤)^𝑇 and 𝑟₂=(0,1)^𝑇, respectively. Thus 𝜆₁ is genuinely nonlinear for 𝑣≠0, and 𝜆₂ is always linearly degenerate. Therefore, the associated waves are rarefaction waves 𝑅 or shock waves 𝑆 for the characteristic field 𝜆₁ and contact discontinuities 𝐽 for the characteristic field 𝜆₂. The Riemann invariants along the characteristic fields are 𝑤/𝑣 and 𝑣, respectively.

Now we are in a position to deliver the Riemann solutions to (1) and (2) according to the value of 𝑣 as follows.(1)If 𝑣₋>𝑣₊>0, then the Riemann solution of (1) and (2) can be expressed as 𝑅+𝐽:()=𝑣𝑣,𝑤)(𝑥,𝑡₋,𝑤₋,𝑥<𝜆₁𝑣₋𝑡,𝑡𝑥𝑤−1,⁻𝑣₋𝑡𝑥−1,𝜆₁𝑣₋𝑡≤𝑥≤𝜆₁𝑣₊𝑣𝑡,₊,𝑣₊𝑤₋𝑣₋,𝜆₁𝑣₊𝑣𝑡<𝑥<𝜏𝑡,₊,𝑤₊,𝑥>𝜏𝑡, (8)in which the propagation speed of 𝑅 is 𝜉=1/(1+𝑣)² for 𝑣 from 𝑣₋ to 𝑣₊ such that 0<𝜉<1 and that of 𝐽 is 𝜏=1/(1+𝑣₊).(2)If 0<𝑣₋<𝑣₊, then the Riemann solution of (1) and (2) can be expressed as 𝑆+𝐽: ()=𝑣𝑣,𝑤)(𝑥,𝑡⁻,𝑤₋𝑣,𝑥<𝜎𝑡,₊,𝑣₊𝑤₋𝑣₋𝑣,𝜎𝑡<𝑥<𝜏𝑡,₊,𝑤₊,𝑥>𝜏𝑡, (9) in which the propagation speed of 𝑆 is 𝜎=1/((1+𝑣₋)(1+𝑣₊)) and that of 𝐽 is 𝜏=1/(1+𝑣₊). It is clear to see that 0<𝜎<𝜏<1.(3)If −1<𝑣₊<0<𝑣₋, then the Riemann solution of (1) and (2) can be expressed as 𝑅₁+𝑅₂: ()=𝑣𝑣,𝑤)(𝑥,𝑡₋,𝑤₋,𝑥<𝜆₁𝑣₋𝑡,𝑡𝑥𝑤−1,⁻𝑣₋𝑡𝑥−1,𝜆₁𝑣₋𝑡≤𝑥<𝑡,𝑡𝑥𝑤−1,⁺𝑣₊𝑡𝑥−1,𝑡<𝑥≤𝜆₁𝑣₊𝑣𝑡,₊,𝑤₊,𝑥>𝜆₁𝑣₊𝑡. (10) It should be emphasized that the state (𝑣,𝑤) in 𝑅₁ is determined by 𝑤/𝑣=𝑤₋/𝑣₋ which is on the left of the line 𝑥=𝑡; otherwise, the state (𝑣,𝑤) in 𝑅₂ is determined by 𝑤/𝑣=𝑤₊/𝑣₊ which is on the right of the line 𝑥=𝑡.(4)If −1<𝑣₋<𝑣₊<0, then the Riemann solution of (1) and (2) can be expressed as 𝐽+𝑆: ()=𝑣𝑣,𝑤)(𝑥,𝑡⁻,𝑤₋𝑣,𝑥<𝜏𝑡,₋,𝑣₋𝑤₊𝑣₊𝑣,𝜏𝑡<𝑥<𝜎𝑡,₊,𝑤₊,𝑥>𝜎𝑡, (11) in which the propagation speed of 𝐽 is 𝜏=1/(1+𝑣₋) and that of 𝑆 is 𝜎=1/(1+𝑣₋)(1+𝑣₊). Here we notice that 1<𝜏<𝜎.(5)If −1<𝑣₊<𝑣₋<0, then the Riemann solution of (1) and (2) can be expressed as 𝐽+𝑅:()=𝑣𝑣,𝑤)(𝑥,𝑡₋,𝑤₋𝑣,𝑥<𝜏𝑡,₋,𝑣₋𝑤₊𝑣₊,𝜏𝑡<𝑥<𝜆₁𝑣₋𝑡,𝑡𝑥𝑤−1,⁺𝑣₊𝑡𝑥−1,𝜆₁𝑣₋𝑡≤𝑥≤𝜆₁𝑣₊𝑣𝑡,₊,𝑤₊,𝑥>𝜆₁𝑣₊𝑡, (12)in which the propagation speed of 𝐽 is 𝜏=1/(1+𝑣₋)>1 and that of 𝑅 is 𝜉=1/(1+𝑣)²>1 for 𝑣 from 𝑣₋ to 𝑣₊.(6)For the case −1<𝑣₋<0<𝑣₊, motivated by the result in [22], we consider that the Riemann solution to (1) and (2) is a delta shock wave 𝛿𝑆 connecting the two constant states (𝑣₋,𝑤₋) and (𝑣₊,𝑤₊). In order to deal with it, we need the following definition of a delta shock wave solution in the framework introduced in [36] and used in [12, 37].

Let us suppose that Γ={𝛾_𝑖∣𝑖∈𝐼} is a graph in the closed upper half-plane {(𝑥,𝑡)∣(𝑥,𝑡)∈(−∞,∞)×[0,∞)}, containing Lipschitz continuous arcs 𝛾_𝑖 with 𝑖∈𝐼, in which 𝐼 is a finite index set. Let us also suppose that 𝐼₀ is the subset of 𝐼 containing all indices of arcs linking to the 𝑥-axis and Γ₀={𝑥⁰_𝑘∣𝑘∈𝐼₀} is the set of initial points of the arcs 𝛾_𝑘 with 𝑘∈𝐼₀.

Definition 1. Let (𝑣,𝑤) be a pair of distributions, where 𝑤 is represented in the form 𝑤(𝑥,𝑡)=𝑤(𝑥,𝑡)+𝛼(𝑥,𝑡)𝛿(Γ), (13) in which 𝑣,𝑤∈𝐿^∞(𝑅×𝑅₊) and the singular part should be defined by 𝛼(𝑥,𝑡)𝛿(Γ)=_𝑖∈𝐼𝛼_𝑖(𝑥,𝑡)𝛿𝛾_𝑖. (14) Let us consider the initial data of the form (𝑣,𝑤)(𝑥,0)=𝑣₀(𝑥),𝑤₀(𝑥)+_𝑘∈𝐼0𝛼_𝑘𝑥⁰_𝑘,0𝛿𝑥−𝑥⁰_𝑘, (15) in which 𝑣₀,𝑤₀∈𝐿^∞(𝑅); then the earlier pair of distributions (𝑣,𝑤) are called a generalized delta shock wave solution of the simplified chromatography system (1) with the initial data (15) if the following integral identities _𝑅+𝑅𝑣𝜓_𝑡+𝑣𝜓1+𝑣^𝑥𝑑𝑥𝑑𝑡+_𝑅𝑣₀(𝑥)𝜓(𝑥,0)𝑑𝑥=0,_𝑅+𝑅𝑤𝜓_𝑡+𝑤𝜓1+𝑣^𝑥𝑑𝑥𝑑𝑡+_{𝑖∈𝐼𝛾𝑖}𝛼_𝑖()()𝑥,𝑡𝜕𝜓𝑥,𝑡+𝜕𝑙_𝑅𝑤₀(𝑥)𝜓(𝑥,0)𝑑𝑥+_𝑘∈𝐼0𝛼_𝑘𝑥⁰_𝑘,0𝜓𝑥⁰_𝑘,0=0 (16) hold for all test functions 𝜓∈𝐶^∞_𝑐(𝑅×𝑅₊), in which 𝜕𝜓(𝑥,𝑡)/𝜕𝑙 denotes the tangential derivative of a function 𝜓 on the graph 𝛾_𝑖 and ∫_𝛾𝑖 denotes the line integral along the arc 𝛾_𝑖.

With the above definition, similar to that in [5], we can also construct the Riemann solution to (1) and (2) for the situation −1<𝑣₋<0<𝑣₊ in the following theorem.

Theorem 2. For the case −1<𝑣₋<0<𝑣₊, the Riemann problem (1) and (2) has the piecewise smooth Riemann solution in the form ()=𝑣𝑣,𝑤)(𝑥,𝑡⁻,𝑤₋𝑣,𝑥<𝜎𝑡,_𝛿(𝑡)𝛿()𝑣,𝛽𝑥−𝜎𝑡,𝑥=𝜎𝑡,₊,𝑤₊,𝑥>𝜎𝑡, (17) where 𝑣_𝛿=𝑣₋+𝑣₊+𝑣₋𝑣₊1,𝜎=1+𝑣_𝛿=11+𝑣₋1+𝑣₊,𝛽(𝑡)=𝑤₋𝑣₊−𝑤₊𝑣₋𝑡1+𝑣₋1+𝑣₊. (18) The measure solution (17) with (18) also satisfies the generalized Rankine-Hugoniot condition as follows: 𝑑𝑥(𝑡)𝑑𝑡=𝜎,𝑑𝛽[𝑤]−𝑤𝑑𝑡=𝜎,𝜎[𝑣]=𝑣1+𝑣.1+𝑣 (19)

Proof. Let us check that the 𝛿-measure solution (17) with (18) obeys (1) in the sense of distributions. In other words, we only need to check that (17) with (18) satisfies ^∞₀^∞_−∞𝑣𝜓_𝑡+𝑣𝜓1+𝑣^𝑥𝑑𝑥𝑑𝑡=0,_{∞0∞−∞}𝑤𝜓_𝑡+𝑤𝜓1+𝑣^𝑥𝑑𝑥𝑑𝑡=0. (20)
Actually, we can verify the first equation in (20) as follows: ^∞₀^∞_−∞𝑣𝜓_𝑡+𝑣𝜓1+𝑣^𝑥=𝑑𝑥𝑑𝑡_{∞0𝜎𝑡−∞}𝑣₋𝜓_𝑡+𝑣₋1+𝑣₋𝜓_𝑥+𝑑𝑥𝑑𝑡_{∞0∞𝜎𝑡}𝑣₊𝜓_𝑡+𝑣₊1+𝑣₊𝜓_𝑥=𝑑𝑥𝑑𝑡_∞00−∞𝑣₋𝜓_𝑡𝑑𝑥𝑑𝑡+^∞₀₀^𝜎𝑡𝑣₋𝜓_𝑡+𝑑𝑥𝑑𝑡_{∞0𝜎𝑡−∞}𝑣₋𝜓_𝑥1+𝑣₋+𝑑𝑥𝑑𝑡_{∞0∞𝜎𝑡}𝑣₊𝜓_𝑡+𝑣₊1+𝑣₊𝜓_𝑥=𝑑𝑥𝑑𝑡_{∞0+∞𝑥/𝜎}𝑣₋𝜓_𝑡𝑑𝑡𝑑𝑥+^∞₀^𝜎𝑡_−∞𝑣₋𝜓_𝑥1+𝑣₋+𝑑𝑥𝑑𝑡_{∞00𝑥/𝜎}𝑣₊𝜓_𝑡𝑑𝑡𝑑𝑥+^∞₀^∞_𝜎𝑡𝑣₊𝜓_𝑥1+𝑣₊𝑑𝑥𝑑𝑡=−^∞₀𝑣₋𝜓𝑥𝑥,𝜎𝑑𝑥+^∞₀𝑣₋1+𝑣₋𝜓()+𝜎𝑡,𝑡𝑑𝑡_∞0𝑣₊𝜓𝑥𝑥,𝜎𝑑𝑡−^∞₀𝑣₊1+𝑣₊𝜓(𝜎𝑡,𝑡)𝑑𝑡=0, (21) in which the fact that 𝜓(𝑥,𝑡) is compactly supported in the region 𝑅×𝑅₊ has been used.
Consequently, for the second equation of (20), we have ^∞₀^∞_−∞𝑤𝜓_𝑡+𝑤𝜓1+𝑣^𝑥=𝑑𝑥𝑑𝑡_{∞0𝜎𝑡−∞}𝑤₋𝜓_𝑡+𝑤₋𝜓_𝑥1+𝑣₋+𝑑𝑥𝑑𝑡_{∞0∞𝜎𝑡}𝑤₊𝜓_𝑡+𝑤₊𝜓_𝑥1+𝑣₊+𝑑𝑥𝑑𝑡_∞0𝛽(𝑡)𝜓_𝑡(𝜎𝑡,𝑡)+𝜎𝜓_𝑥()=𝜎𝑡,𝑡𝑑𝑡_{∞00𝜎𝑡}𝑤₋𝜓_𝑡𝑑𝑥𝑑𝑡+^∞₀^𝜎𝑡_−∞𝑤₋𝜓_𝑥1+𝑣₋+𝑑𝑥𝑑𝑡_{∞0∞𝜎𝑡}𝑤₊𝜓_𝑡𝑑𝑥𝑑𝑡+^∞₀^∞_𝜎𝑡𝑤₊𝜓_𝑥1+𝑣₊+𝑑𝑥𝑑𝑡_∞0𝛽(𝑡)𝜓_𝑡(𝜎𝑡,𝑡)+𝜎𝜓_𝑥()=𝜎𝑡,𝑡𝑑𝑡_∞0𝑤₊−𝑤₋𝜓𝑥𝑥,𝜎+𝑑𝑥_∞0𝑤₋1+𝑣₋−𝑤₊1+𝑣₊𝜓()+𝜎𝑡,𝑡𝑑𝑡_∞0𝛽(𝑡)()=𝑤𝑑𝜓𝜎𝑡,𝑡₊−𝑤₋1+𝑣₋1+𝑣₊+𝑤₋1+𝑣₋−𝑤₊1+𝑣₊−𝛽^′(𝑡×_∞0)𝜓(𝜎𝑡,𝑡)𝑑𝑡=0, (22) in which the changes of variables and integration by parts have been used.
Through the earlier verification, we can conclude that (17) with (18) is indeed the piecewise smooth Riemann solution to (1) and (2) in the sense of distributions.

Remark 3. If we take the limit 𝑣₋→0 in the Riemann solution (17) with (18), then we can obtain the result which is identical with that in [5]. In other words, the Riemann solution to (1) and (2) for 0=𝑣₋<𝑣₊ can be seen as the critical situation for −1<𝑣₋<0<𝑣₊. Thus, the result in [5] can be seen as the special situation in this paper.

Remark 4. If we take the limit 𝑣₋→−1 in the Riemann solution (17) with (18), then both the propagation speed and strength of the delta shock wave will tend to +∞. In fact, both 𝜆₁=1/(1+𝑣)² and 𝜆₂=1/(1+𝑣) tend to +∞ if we let 𝑣→−1 on the right. Thus, the line 𝑣=−1 in the phase space (𝑣,𝑤) can be seen as a singular curve.

Remark 5. The Riemann solution to (1) and (2) can be constructed simply if we take the Riemann initial data (2) in the phase space (𝑣,𝑤)∈(−∞,−1)×(−∞,∞). However, it is difficult to construct the global Riemann solution to (1) and (2) if we require the Riemann initial data (2) in the phase space (𝑣,𝑤)∈(−∞,∞)×(−∞,∞). In this situation, we at least need to make some criterions on the line 𝑣=−1.

3. Viscous Regularization of Delta Shock Wave Solution

In this section, we are only interested in the viscous regularization of delta shock wave solution to the Riemann problem (1) and (2). Thus, we confine our attention only to the situation −1<𝑣₋<0<𝑣₊.

Performing the self-similar transformation 𝜉=𝑥/𝑡, we get the boundary value problem: −𝜉𝑣_𝜉+𝑣1+𝑣_𝜉=𝜀𝑣_𝜉𝜉,−𝜉𝑤_𝜉+𝑤1+𝑣_𝜉=0, (23) with the boundary conditions (𝑣,𝑤)(±∞)=𝑣_±,𝑤_±. (24)

By observing (23), we know that the solution of the first equation in (23) does not rely on the second one in (23). So, we can first consider the first equation in (23), namely, −𝜉𝑣_𝜉+𝑣1+𝑣_𝜉=𝜀𝑣_𝜉𝜉,𝑣(±∞)=𝑣_±. (25)

With the similar deduction and calculation in Theorem 4.1 in [5], we can also obtain the same conclusion as follows.

Theorem 6. For each fixed 𝜀>0, there exists a unique and monotonic smooth solution 𝑣^𝜀(𝜉) of the boundary value problem (25) for −1<𝑣₋<0<𝑣₊.

Now, we turn our attention to the second equation in (23) and consider the existence of the solution 𝑤(𝜉). Based on Theorem 6, also with the similar deduction and calculation in Theorem 4.2 in [5], we can obtain the following theorem.

Theorem 7. For each fixed 𝜀>0, assume that 𝑤(𝜉) is a weak solution of the following boundary value problem: −𝜉𝑤_𝜉+𝑤1+𝑣^𝜀_𝜉𝑤()=0,±∞=𝑤_±, (26) where 𝑣^𝜀(𝜉) is the solution of (25).
Then, one has 𝑤^𝜀(𝜉)∈𝐿¹(−∞,+∞) which can be expressed as𝑤^𝜀(𝜉)=𝑤₋exp^𝜉_−∞𝑣̇^𝜀(𝑠)1+𝑣^𝜀(𝑠)−𝑠(1+𝑣^𝜀(𝑠))²𝑑𝑠𝑓𝑜𝑟𝜉<𝜉^𝜀_𝛼,𝑤₊exp_𝜉^+∞−𝑣̇^𝜀(𝑠)1+𝑣^𝜀(𝑠)−𝑠(1+𝑣^𝜀(𝑠))²𝑑𝑠𝑓𝑜𝑟𝜉>𝜉^𝜀_𝛼, (27)where 𝜉^𝜀_𝛼 is the unique fixed point of 𝜉=1/(1+𝑣^𝜀(𝜉)).

It follows from Theorem 6 that 𝑣^𝜀(𝜉) is strictly monotonically increasing with respect to 𝜉 when −1<𝑣₋<0<𝑣₊, which enables us to see that both 1/(1+𝑣^𝜀(𝜉)) and 1/(1+𝑣^𝜀(𝜉))² are strictly monotonically decreasing with respect to 𝜉 when −1<𝑣₋<0<𝑣₊. Thus, there exist unique fixed points of the following equations, and we use 𝜉^𝜀_𝛼 and 𝜉^𝜀_𝛽 to express as 𝜉^𝜀_𝛼=11+𝑣^𝜀𝜉^𝜀_𝛼,𝜉^𝜀_𝛽=11+𝑣^𝜀𝜉^𝜀_𝛽², (28)(29) respectively. Consequently, we take the limits 𝜉_𝛼=lim_𝜀→0⁺𝜉^𝜀_𝛼 and 𝜉_𝛽=lim_𝜀→0⁺𝜉^𝜀_𝛽, which will play very critical roles in the following discussion.

Lemma 8. For any 𝜂>0, draws the conclusion that lim_𝜀→0+𝑣^𝜀(𝜉)=𝑣₋,𝑓𝑜𝑟𝜉<𝜉_𝛽𝑣−𝜂,₊,𝑓𝑜𝑟𝜉>𝜉_𝛽+𝜂 (30) is in uniform in the earlier intervals.

Proof. It is easy to take 𝜀>0 sufficiently small such that 𝜉^𝜀_𝛽<𝜉_𝛽+𝜂/4 for 𝜉_𝛽=lim_𝜀→0⁺𝜉^𝜀_𝛽. If we take 𝜉₁=𝜉_𝛽+𝜂/2, then it is clear that 𝜉^𝜀_𝛽<𝜉₁−𝜂/4. From Theorem 6, we know that 𝑣̇^𝜀(𝜉)>0 if −1<𝑣₋<0<𝑣₊. Then (25) can be written as 1𝜀1−𝜉+(1+𝑣^𝜀(𝜉))²=𝑣̈^𝜀(𝜉)𝑣̇^𝜀(𝜉). (31) By integrating (31) over (𝜉₁,𝜉), we arrive at 𝑣̇^𝜀(𝜉)=𝑣̇^𝜀𝜉₁exp^𝜉_𝜉11𝜀1−𝑠+(1+𝑣^𝜀(𝑠))²𝑑𝑠. (32) Integrating (32) over (𝜉₁,+∞), we get 𝑣₊−𝑣^𝜀𝜉₁=𝑣̇^𝜀𝜉_1𝜉^+∞₁exp^𝜉_𝜉11𝜀1−𝑠+(1+𝑣^𝜀(𝑠))²𝑑𝑠𝑑𝜉. (33)
Because 𝑣^𝜀(𝜉) is monotonically increasing in the real axis, we have 𝑣̇^𝜀(𝜉₁)≥0 and 𝑣₋≤𝑣^𝜀(𝑠)≤𝑣₊. Thus, one can conclude that𝑣₊−𝑣₋≥𝑣₊−𝑣^𝜀𝜉₁=𝑣̇^𝜀𝜉_1𝜉^+∞₁exp^𝜉_𝜉11𝜀1−𝑠+(1+𝑣^𝜀(𝑠))²≥𝑣̇𝑑𝑠𝑑𝜉_𝜀𝜉_1𝜉^+∞₁1exp𝜀−12𝜉−𝜉₁𝜉+𝜉₁+𝜉−𝜉₁1+𝑣₊²=𝑣̇𝑑𝜉_𝜀𝜉₁₀^+∞1exp𝜀−12𝜁²1+𝜁1+𝑣₊²−𝜉₁≥𝑣̇𝑑𝜁_𝜀𝜉₁⋅𝜀𝐶₁ (34)for 0<𝜀<1 and 𝐶₁ is a constant independent of 𝜀, in which the change of variables 𝜁=𝜉−𝜉₁ has been used. So, we can get 0≤𝑣̇^𝜀(𝜉₁)≤(𝑣₊−𝑣₋)/(𝜀𝐶₁). Together with (32), we know that 𝑣̇^𝜀(𝜉)≤𝑣₊−𝑣₋𝜀𝐶₁exp^𝜉_𝜉11𝜀1−𝑠+(1+𝑣^𝜀(𝑠))²𝑑𝑠. (35) In view of (29), we have 1−𝑠+(1+𝑣^𝜀(𝑠))²=1(1+𝑣^𝜀(𝑠))²−11+𝑣^𝜀𝜉^𝜀_𝛽²−𝑠−𝜉^𝜀_𝛽, (36) noticing that𝜉^𝜀_𝛽+𝜂/4<𝜉₁<𝑠, by applying the mean value theorem, and then there exists 𝜉₂∈(𝜉^𝜀_𝛽,𝑠) such that 1−𝑠+(1+𝑣^𝜀(𝑠))²=−2𝑣̇^𝜀𝜉₂1+𝑣^𝜀𝜉₂³−1𝑠−𝜉^𝜀_𝛽𝜂≤−4. (37) So, we get the following inequality: 𝑣̇^𝜀(𝜉)≤𝑣₊−𝑣₋𝜀𝐶₁−𝜂exp4𝜀𝜉−𝜉₁, (38) which means that 𝑣̇^𝜀(𝜉)→0 uniformly on 𝜉>𝜉_𝛽+𝜂.
For any 𝜉>𝜉_𝛽+𝜂, we have 𝑣₊−𝑣^𝜀(𝜉)=_𝜉^+∞𝑣̇^𝜀(𝑠)𝑑𝑠. (39) Obviously, we know that lim_𝜀→0⁺𝑣^𝜀(𝜉)=𝑣₊ uniformly on 𝜉>𝜉_𝛽+𝜂 in view of (38).
Now, we try to prove thatlim_𝜀→0⁺𝑣^𝜀(𝜉)=𝑣₋ uniformly on 𝜉<𝜉_𝛽−𝜂. Similarly, it is easy to take 𝜀>0 sufficiently small such that 𝜉^𝜀_𝛽>𝜉_𝛽−𝜂/4. If we take 𝜉₃=𝜉_𝛽−𝜂/2, then we also have 𝜉^𝜀_𝛽>𝜉₃+𝜂/4. By integrating (25) over (𝜉,𝜉₃), we have 𝑣̇^𝜀(𝜉)=𝑣̇^𝜀𝜉₃exp^𝜉_𝜉31𝜀1−𝑠+(1+𝑣^𝜀(𝑠))²𝑑𝑠. (40) Integrating (25) over (−∞,𝜉₃), we have 𝑣^𝜀𝜉₃−𝑣₋=𝑣̇^𝜀𝜉₃^𝜉3_−∞exp^𝜉_𝜉31𝜀1−𝑠+(1+𝑣^𝜀(𝑠))²𝑑𝑠𝑑𝜉. (41) Thus, we get the following result:𝑣₊−𝑣₋≥𝑣^𝜀𝜉₃−𝑣₋=𝑣̇^𝜀𝜉₃^𝜉3_−∞exp^𝜉_𝜉31𝜀1−𝑠+(1+𝑣^𝜀(𝑠))²≥𝑣̇𝑑𝑠𝑑𝜉_𝜀𝜉₃^𝜉3_−∞1exp𝜀−12𝜉−𝜉₃𝜉+𝜉₃+𝜉−𝜉₃1+𝑣₋²=𝑣̇𝑑𝜉_𝜀𝜉₃⁰_−∞1exp𝜀−12𝜁²1+𝜁1+𝑣₋²−𝜉₃≥𝑣̇𝑑𝜁_𝜀𝜉₃⋅𝜀𝐶₂ (42)for 0<𝜀<1 and 𝐶₂ is a constant independent of 𝜀, in which the change of variables 𝜁=𝜉−𝜉₃ has also been used. Thus we have 𝑣̇^𝜀(𝜉₃)≤(𝑣₊−𝑣₋)/(𝜀𝐶₂), together with (40), and we also know that 𝑣̇^𝜀(𝜉)≤𝑣₊−𝑣₋𝜀𝐶₂exp^𝜉_𝜉31𝜀1−𝑠+(1+𝑣^𝜀(𝑠))²𝑑𝑠. (43) Similarly, by applying the mean value theorem, then there exists 𝜉₄∈(𝑠,𝜉^𝜀_𝛽) such that 1−𝑠+(1+𝑣^𝜀(𝑠))²=−2𝑣̇^𝜀𝜉₄1+𝑣^𝜀𝜉₄³−1𝑠−𝜉^𝜀_𝛽≥𝜂4. (44) Combining (43) together with (44), we get 𝑣̇^𝜀(𝜉)≤𝑣₊−𝑣₋𝜀𝐶₂−𝜂exp𝜉4𝜀³−𝜉, (45) which implies that 𝑣̇^𝜀(𝜉)→0 uniformly on 𝜉<𝜉_𝛽−𝜂.
Thus, for any 𝜉<𝜉_𝛽−𝜂, we have 𝑣^𝜀(𝜉)−𝑣₋=^𝜉_−∞𝑣̇^𝜀(𝑠)𝑑𝑠. (46) Obviously, we can see that lim_𝜀→0⁺𝑣^𝜀(𝜉)=𝑣₋ uniformly on 𝜉<𝜉_𝛽−𝜂 in view of (45).

Lemma 9. Let 𝜉_𝛼 and 𝜉_𝛽 be defined as above. Then, one has 𝜉_𝛼=𝜉_𝛽=11+𝑣₋1+𝑣₊. (47)

Proof. Take any 𝜙(𝜉)∈𝐶^∞₀(𝜉₁,𝜉₂) where 𝜉₁<𝜉_𝛽<𝜉₂. It follows from (25) that ^𝜉2_𝜉1𝜀𝑣_𝜉𝜉⋅𝜙(𝜉)𝑑𝜉=^𝜉2_𝜉1−𝜉𝑣_𝜉+𝑣1+𝑣_𝜉𝜙(𝜉)𝑑𝜉. (48)
By employing integration by parts and applying the fact that 𝜙 is compactly supported in (𝜉₁,𝜉₂), we have ^𝜉2_𝜉1𝜀𝑣𝜙^′′(𝜉)𝑑𝜉=^𝜉2_𝜉1𝑣𝜙(𝜉)+𝜉𝜙^′(𝜉)−𝑣𝜙1+𝑣^′(𝜉)𝑑𝜉. (49) Taking the limit 𝜀→0 in (49), it can be obtained from Lemma 8 that ^𝜉𝛽_𝜉1𝑣₋𝜙(𝜉)+𝜉𝜙^′(𝜉)−𝑣₋1+𝑣₋𝜙^′(𝜉)+𝑑𝜉_{𝜉2𝜉𝛽}𝑣₊𝜙(𝜉)+𝜉𝜙^′(𝜉)−𝑣₊1+𝑣₊𝜙^′(𝜉)𝑑𝜉=0. (50)
Through simplifying (50), we get 𝜙𝜉_𝛽⋅𝑣₋𝜉_𝛽−𝑣₋1+𝑣₋−𝑣₊𝜉_𝛽+𝑣₊1+𝑣₊=0, (51) which leads to 𝜉_𝛽=1/(1+𝑣₋)(1+𝑣₊) for the arbitrary of 𝜙(𝜉).
Let us turn back to calculate 𝜉_𝛼; we have 1+𝑣₊𝜉_𝛽=11+𝑣₋≥𝜉_𝛼=lim_𝜀→0+𝜉^𝜀_𝛼=lim_𝜀→0+11+𝑣^𝜀𝜉^𝜀_𝛼≥11+𝑣₊=1+𝑣₋𝜉_𝛽. (52)
It is clear that 1/(1+𝑣₋)>𝜉_𝛽>1/(1+𝑣₊) for −1<𝑣₋<0<𝑣₊. Furthermore, if we assume that 1/(1+𝑣₋)>𝜉_𝛼>𝜉_𝛽, through taking 𝜂=(𝜉_𝛼−𝜉_𝛽)/2>0 here, then we can take 𝜀>0 sufficiently small such that 𝜉^𝜀_𝛼>𝜉_𝛼−𝜂=𝜉_𝛽+𝜂 for 𝜉_𝛼=lim_𝜀→0⁺𝜉^𝜀_𝛼. It can be derived from Lemma 8 that lim_𝜀→0⁺𝑣^𝜀(𝜉^𝜀_𝛼)=𝑣₊. Thus, we arrive at 𝜉_𝛼=1/(1+𝑣₊) by applying (52), which contradicts the assumption 1/(1+𝑣₋)>𝜉_𝛼>𝜉_𝛽.
On the other hand, if we assume that 𝜉_𝛽>𝜉_𝛼>1/(1+𝑣₊), by taking 𝜂=(𝜉_𝛽−𝜉_𝛼)/2>0 now, then we can also take 𝜀>0 sufficiently small such that 𝜉^𝜀_𝛼<𝜉_𝛼+𝜂=𝜉_𝛽−𝜂. It can be derived from Lemma 8 that lim_𝜀→0⁺𝑣^𝜀(𝜉^𝜀_𝛼)=𝑣₋. Thus, we arrive at 𝜉_𝛼=1/(1+𝑣₋) by applying (52), which also contradicts the assumption 𝜉_𝛽>𝜉_𝛼>1/(1+𝑣₊).
From the above discussion, we can get the relation 𝜉_𝛼=𝜉_𝛽=1/(1+𝑣₋)(1+𝑣₊). The proof is completed.

Lemma 10. For any 𝜂>0, one can obtain lim_𝜀→0+𝑤^𝜀(𝜉)=𝑤₋,𝑓𝑜𝑟𝜉<𝜉_𝛽𝑤−𝜂,₊,𝑓𝑜𝑟𝜉>𝜉_𝛽+𝜂, (53) uniformly in the above intervals.

The process of this proof is similar to that of Lemma 5.3 in [5], and thus we omit it.

Theorem 11. Let −1<𝑣₋<0<𝑣₊ and let (𝑣^𝜀(𝜉),𝑤^𝜀(𝜉)) be the solution of (23) and (24). Then, the limit of (𝑣^𝜀(𝜉),𝑤^𝜀(𝜉)) as 𝜀→0 is (17) with (18), namely, the corresponding Riemann solution of (1) and (2).

Proof. To prove it, we should consider the limit behavior of 𝑤^𝜀(𝜉) in the neighborhood of 𝜉=𝜎=1/(1+𝑣₋)(1+𝑣₊). We take 𝜙(𝜉)∈𝐶^∞₀(𝜉₁,𝜉₂) with 𝜉₁<𝜎<𝜉₂ and satisfy 𝜙(𝜉)=𝜙(𝜎) on the neighborhood of 𝜉=𝜎 such as 𝑁(𝜇)=[𝜎−𝜇,𝜎+𝜇] for sufficiently small 𝜇>0, which is called the sloping test functions in [22].
Then, by (26), we have ^𝜉2_𝜉1−𝜉𝑤^𝜀_𝜉+𝑤^𝜀1+𝑣^𝜀_𝜉𝜙(𝜉)=𝑑𝜉_𝜉2𝜉1𝑤^𝜀𝜙+𝑤^𝜀1𝜉−1+𝑣^𝜀𝜙^′𝑑𝜉=0. (54)
It follows from 𝜙(𝜉)=𝜙(𝜎) for 𝜉∈𝑁(𝜇) that lim_𝜀→0+^𝜉2_𝜉1𝑤^𝜀1𝜉−1+𝑣^𝜀𝜙^′=𝑑𝜉_{𝜉𝜎−𝜇1}𝑤₋1𝜉−1+𝑣₋𝜙^′+𝑑𝜉_{𝜉2𝜎+𝜇}𝑤₊1𝜉−1+𝑣₊𝜙^′𝑑𝜉=𝑤₋1𝜎−𝜇−1+𝑣₋𝜙(𝜎)−𝑤_−𝜉^𝜎−𝜇₁𝜙𝑑𝜉−𝑤₊1𝜎+𝜇−1+𝑣₊𝜙(𝜎)−𝑤₊^𝜉2_𝜎+𝜇𝜙𝑑𝜉. (55)
By taking the limit 𝜇→0 above, we have lim_𝜀→0+^𝜉2_𝜉1𝑤^𝜀1𝜉−1+𝑣^𝜀𝜙^′=[𝑤]+𝑤𝑑𝜉−𝜎𝜙(𝜎)−1+𝑣_𝜉2𝜉1𝑤₋𝐻(𝜎−𝜉)+𝑤₊𝐻(𝜉−𝜎)𝜙(𝜉)𝑑𝜉, (56) in which 𝐻 denotes the Heaviside function.
Combining (54) together with (56), we have lim_𝜀→0+^𝜉2_𝜉1𝑤^𝜀−𝑤₋𝐻(𝜎−𝜉)−𝑤₊𝐻()𝜙(𝜉)=𝜎[𝑤]−𝑤𝜉−𝜎𝑑𝜉𝜙(𝜎)1+𝑣 (57) for all sloping test functions 𝜙(𝜉)∈𝐶^∞₀[𝜉₁,𝜉₂]. With the same technique in [5, 22], one can see that the conclusion can also be drawn for any test function 𝜙(𝜉)∈𝐶₀[𝜉₁,𝜉₂].
By taking the limits 𝜉₁→𝜎− and 𝜉₂→𝜎+ in (57), we can obtain 𝑤(𝜉)=lim_𝜀→0+𝑤^𝜀(𝜉)=𝑤₋𝐻(𝜎−𝜉)+𝑤₊𝐻()+𝜎[𝑤]−𝑤𝜉−𝜎𝛿().1+𝑣𝜉−𝜎 (58) Thus the limit of 𝑤^𝜀(𝜉) is exactly the corresponding Riemann solution (17) with (18) for 𝑤 when −1<𝑣₋<0<𝑣₊.
Finally, the value of 𝑣(𝜉) at the discontinuity point 𝜉=𝜎 should be calculated. Then, (26) can be changed into the following form: ^𝜉2_𝜉1𝑤⋅𝜙+𝜉𝜙^′−𝜙^′1+𝑣𝑑𝜉=0. (59)
By substituting (58) into (59), through a simple calculation, one can arrive at _𝜉^𝜎−₁𝜙+𝜉𝜙^′−𝜙^′1+𝑣𝑑𝜉+^𝜉2_𝜎+𝜙+𝜉𝜙^′−𝜙^′+𝜎[𝑤]−𝑤1+𝑣𝑑𝜉𝜙(𝜎)1+𝑣+𝜎𝜙^′(𝜎)−𝜙^′(𝜎)1+𝑣_𝛿=0; (60) namely, 𝑤₋1𝜎−1+𝑣₋𝜙(𝜎)−𝑤₊1𝜎−1+𝑣₊𝜙(𝜎)+𝜎[𝑤]−𝑤𝜙(𝜎)1+𝑣+𝜎𝜙^′(𝜎)−𝜙^′(𝜎)1+𝑣_𝛿=0. (61) Since 𝜙 is arbitrary, one can obtain 𝜎=1/(1+𝑣_𝛿)=1/(1+𝑣₋)(1+𝑣₊), which implies that 𝑣_𝛿=𝑣₋+𝑣₊+𝑣₊𝑣₋.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (11001116, 11271176) and the Project of Shandong Provincial Higher Educational Science and Technology Program (J11LA03, J12LI01) and Shandong Provincial Natural Science Foundation (ZR2010AL012).