Delta Shock Waves for a Linearly Degenerate Hyperbolic System of Conservation Laws of Keyfitz-Kranzer Type

Abstract

This paper is devoted to the study of delta shock waves for a hyperbolic system of conservation laws of Keyfitz-Kranzer type with two linearly degenerate characteristics. The Riemann problem is solved constructively. The Riemann solutions include exactly two kinds. One consists of two (or just one) contact discontinuities, while the other contains a delta shock wave. Under suitable generalized Rankine-Hugoniot relation and entropy condition, the existence and uniqueness of delta shock solution are established. These analytical results match well the numerical ones. Finally, two kinds of interactions of elementary waves are discussed.

1. Introduction

Consider the following hyperbolic system of conservation laws: 𝜌_𝑡+𝜌(𝑢−𝑃)_𝑥=0,𝜌𝑢_𝑡+𝜌𝑢(𝑢−𝑃)_𝑥=0,(1) where 𝑃=𝑃(𝜌) with 𝜌≥0. Model (1) belongs to the nonsymmetric system of Keyfitz-Kranzer type [1, 2] as 𝜌_𝑡+𝜌𝜙𝜌,𝑢₁,𝑢₂,…,𝑢_𝑛𝑥=0,𝜌𝑢_𝑖𝑡+𝜌𝑢_𝑖𝜙𝜌,𝑢₁,𝑢₂,…,𝑢_𝑛𝑥=0,𝑖=1,2,…,𝑛.(2) System (1) was also introduced as a macroscopic model for traffic flow by Aw and Rascle [3], where 𝜌 and 𝑢 are the density and the velocity of cars on the roadway, and the function 𝑃 is smooth and strictly increasing and it satisfies 𝜌𝑃^′′𝜌+2𝑃^′𝜌>0for𝜌>0.(3) Model (1) was also studied by Lu [2]. By using the compensated compactness method, he established the existence of global bounded weak solutions of the Cauchy problem under the following two assumptions on 𝑃(𝜌), respectively: 𝑃(0)=0,lim_𝜌→0𝜌𝑃^′𝜌=0,𝜌𝑃^′′𝜌+2𝑃^′𝜌>0for𝜌>0,lim_𝜌→0𝜌𝑃𝜌=0,lim_𝜌→∞𝜌𝑃𝜌^′≥𝐴,𝜌𝑃^′′𝜌+2𝑃^′𝜌>0for𝜌>0.(4)

One can find that system (1) has two characteristics; one is always linearly degenerate while the other is linearly degenerate or genuinely nonlinear depending on the behaviors of 𝑝(𝜌). In [2, 3], the condition (3) is required, which implies that the second characteristic is genuinely nonlinear. Thus, one interesting topic is the case when both characteristics are linearly degenerate.

These motivate us to consider (1) with 𝜌1𝑃=𝑃=−𝜌,(5) which is the prototype function satisfying 𝜌𝑃^′′𝜌+2𝑃^′𝜌≡0for𝜌>0.(6) For system (1) with (5) or (6), a distinctive feature is that both eigenvalues are linearly degenerate; that is, this is a fully linearly degenerate system. Thus, all the classical elementary waves only consist of contact discontinuities. Moreover, the overlapping of linearly degenerate characteristics may result in the formation of delta shock wave. A delta shock wave is a generalization of an ordinary shock wave. Speaking informally, it is a kind of discontinuity, on which at least one of the state variables may develop an extreme concentration in the form of a Dirac delta function with the discontinuity as its support. It is more compressive than an ordinary shock wave in the sense that more characteristics enter the discontinuity line. From the physical point of view, a delta shock wave represents the process of concentration of the mass and formation of the universe. For related results of delta shock waves, we refer the readers to the papers [4–24] and the references therein.

Putting (5) into (1), we obtain the equivalent system 𝜌_𝑡+𝜌𝑢_𝑥=0,𝜌𝑢_𝑡+𝜌𝑢²+𝑢_𝑥=0.(7) Firstly, we consider the Riemann problem for (7) with initial data )=𝑢𝑢,𝜌(𝑥,𝑡=0⁻,𝜌₋𝑢,𝑥<0,₊,𝜌₊,𝑥>0.(8) By the analysis method in phase plane, when 𝑢₊+(1/𝜌₊)>𝑢₋, we can construct Riemann solution only involving two different (or just one) contact discontinuities. However, for the case 𝑢₊+(1/𝜌₊)≤𝑢₋, we find that the Riemann solution can not be constructed by these classical contact discontinuities and delta shock wave should occur. For delta shock wave, by a definition of measure solutions to (7), we derive the generalized Rankine-Hugoniot relation which describes the relation among the limit states on both sides of the discontinuity, location, propagation speed, weight, and the assignment of the component 𝑢 on its discontinuity relative to the delta shock wave. The entropy condition is an overcompressive one and guarantees the uniqueness of solution. With the generalized Rankine-Hugoniot relation and entropy condition, we solve the delta shock solution uniquely. Secondly, we simulate the obtained Riemann solutions by using N-T scheme [25] and get the numerical results coinciding with the analytical ones. Thirdly, we study two kinds of interactions of delta shock waves and contact discontinuities by considering the initial-value problem with initial data as )=𝑢𝑢,𝜌(𝑥,𝑡=0⁻,𝜌₋𝑢,𝑥<0,_𝑚,𝜌_𝑚𝑢,0<𝑥<𝑎,₊,𝜌₊,𝑥>𝑎,(9) where 𝑎>0 is a constant. In the first kind, the interaction of two contact discontinuities leads to a delta shock wave. In the second one, the delta shock wave vanishes after it interacts with a contact discontinuity. Such two cases are not encountered in the interactions of elementary waves for some other systems, such as the zero-pressure gas dynamics.

This paper is organized as follows. In Section 2, we present some preliminaries and construct the Riemann solution consisting of contact discontinuities by phase plane analysis method. Then, in Section 3, with suitable generalized Rankine-Hugoniot relation and entropy condition, we solve delta shock solution uniquely. Section 4 presents some numerical results of Riemann solutions obtained by using the Nessyahu-Tadmor scheme [25]. Finally, in Section 5, we study two kinds of interactions of delta shock waves and contact discontinuities.

2. Preliminaries and Solutions Involving Contact Discontinuities

In this section, we analyze some basic properties and then solve the Riemann problem (7)-(8) by the analysis in phase plane. The system has two eigenvalues 𝜆₁=𝑢,𝜆₂1=𝑢+𝜌(10) with associated right eigenvectors 𝑟₁⃗=(1,0)^𝑇,𝑟₂⃗=11,𝜌^2𝑇(11) satisfying ∇𝜆_𝑖⋅𝑟_𝑖⃗≡0,𝑖=1,2.(12) Thus, system (7) is fully linearly degenerate. The linear degeneracy also excludes the possibility of rarefaction waves and shock waves.

As usual, we seek the self-similar solution ()=(𝜉)𝑥𝑢,𝜌𝑡,𝑥𝑢,𝜌,𝜉=𝑡,(13) for which system (7) becomes −𝜉𝜌_𝜉+𝜌𝑢_𝜉=0,−𝜉𝜌𝑢_𝜉+𝜌𝑢²+𝑢_𝜉=0,(14) and initial data (8) changes to the boundary condition 𝑢,𝜌(±∞)=𝑢_±,𝜌_±.(15) This is a two-point boundary value problem of first-order ordinary differential equations with the boundary values in the infinity.

For any smooth solution, (14) turns into 1𝑢−𝜉𝜌0𝑢+𝜌−𝜉𝑑𝜌𝑑𝑢=0.(16) It provides either the general solutions (constant states) 𝑢,𝜌=const.𝜌>0(17) or singular solutions 𝜉=𝜆₁=𝑢,𝑑𝑢=0,𝜉=𝜆₁1=𝑢+𝜌1,𝑑𝑢+𝜌=0.(18) Integrating (18) from (𝑢₋,𝜌₋) to (𝑢,𝜌), respectively, one can get that 𝜉=𝑢,𝑢=𝑢₋,1𝜉=𝑢+𝜌,1𝑢+𝜌=𝑢₋+1𝜌₋.(19) Indeed, (19) are contact discontinuities; see (21) and (22) in the following.

For a bounded discontinuous solution at 𝜉=𝜔, the Rankine-Hugoniot relation 𝜌++−𝜔𝜌𝑢=0,−𝜔𝜌𝑢𝜌𝑢²+𝑢=0(20) holds, where [𝑞]=𝑞₋−𝑞 is the jump of 𝑞 across the discontinuous line and 𝜔 is the velocity of the discontinuity. A simple calculation gives 𝜔=𝑢,𝑢=𝑢₋,1(21)𝜔=𝑢+𝜌,1𝑢+𝜌=𝑢₋+1𝜌₋.(22) Obviously, (21) and (22) are two kinds of contact discontinuities, denoted by ^←𝐽 and 𝐽, respectively. In the phase plane, the curves expressed by the second equations in (21) and (22) are called the contact discontinuity curves.

Starting from any point (𝑢₋,𝜌₋), we draw the curve 𝑢=𝑢₋ for 𝜌>0 in the phase plane, which is parallel to the 𝜌-axis. We also draw the curve 𝑢+(1/𝜌)=𝑢₋+(1/𝜌₋) for 𝜌>0, which is monotonically increasing and has the lines 𝜌=0 and 𝑢=𝑢₋+(1/𝜌₋) as its two asymptotic lines. Also, from the point (𝑢₋−(1/𝜌₋),𝜌₋), we draw the curve 𝑢+(1/𝜌)=𝑢₋. Then the phase plane is divided into five regions as follows (shown in Figure 1): I=𝑢,𝜌∣𝑢₋<𝑢<𝑢₋+1𝜌₋,10<𝜌<𝑢₋+𝜌₋⁻¹∪𝑢−𝑢⁻+1𝜌₋,<𝑢<+∞,0<𝜌<+∞II=𝑢,𝜌∣−∞<𝑢<𝑢₋1,0<𝜌<𝑢₋+𝜌₋⁻¹,−𝑢III=𝑢,𝜌∣𝑢₋<𝑢<𝑢₋+1𝜌₋,1𝑢₋+𝜌₋⁻¹,−𝑢<𝜌<+∞IV=𝑢,𝜌∣−∞<𝑢<𝑢₋,1𝑢₋+𝜌₋⁻¹1−𝑢<𝜌<𝑢₋,−𝑢V=𝑢,𝜌∣−∞<𝑢<𝑢₋,1𝑢₋.−𝑢<𝜌<+∞(23)

Figure 1 

Contact discontinuity curves.

By the analysis method in the phase plane, when (𝑢₊,𝜌₊)∈I,II,III,IV, namely, 𝑢₊+(1/𝜌₊)>𝑢₋, one can easily construct the solution consisting of two different (or just one) contact discontinuities (see Figure 2), in which, ^←𝐽∶𝑥=𝑢₋𝑡,𝐽∶𝑥=𝑢₊+1𝜌₊𝑡,(24) the intermediate states (𝑢^∗,𝜌^∗) connecting two contact discontinuities satisfy 𝑢^∗=𝑢₋,𝑢^∗+1𝜌^∗=𝑢₊+1𝜌₊.(25)

Figure 2 

Riemann solution involving two contact discontinuities.

3. Delta Shock Solution

In this section, let us in detail discuss the last case when (𝑢₊,𝜌₊)∈V(𝑢₋,𝜌₋); that is, 𝑢₊+(1/𝜌₊)≤𝑢₋. The characteristic lines from initial data will overlap in a domain Ω={(𝑥,𝑡)∶𝑢₊+(1/𝜌₊)<𝑥/𝑡<𝑢₋} shown in Figure 3. So singularity must happen in Ω. It is easy to know that the singularity is impossible to be a jump with finite amplitude because the Rankine-Hugoniot relation is not satisfied on the bounded jump. In other words, there is no solution which is piecewise smooth and bounded. Motivated by [7, 8, 16, 21, 22], we seek solutions with delta distribution at the jump.

Figure 3 

Riemann solution containing delta shock wave.

Denote by 𝐵𝑀(𝑅) the space of bounded Borel measures on 𝑅, and then the definition of a measure solution of (7) in 𝐵𝑀(𝑅) can be given as follows (see [16, 22, 24]).

Definition 1. A pair (𝑢,𝜌) is called a measure solution to (7) if there hold 𝜌∈𝐿^∞((0,∞),𝐵𝑀(𝑅))∩𝐶(0,∞),𝐻^−𝑠(𝑅),𝑢∈𝐿^∞(0,∞),𝐿^∞(𝑅)∩𝐶(0,∞),𝐻^−𝑠(𝑅)(),,𝑠>0,𝑢ismeasurablewithrespectto𝜌atalmostforall𝑡∈0,∞(26) and (7) is satisfied in measured and distributional senses; that is, 𝐼₁=_𝑅+×𝑅𝜙_𝑡+𝑢𝜙_𝑥𝐼𝑑𝜌𝑑𝑡=0,₂=_𝑅+×𝑅𝑢𝜙_𝑡+𝑢𝜙_𝑥𝑑𝜌𝑑𝑡+_𝑅+×𝑅𝑢𝜙_𝑥𝑑𝑥𝑑𝑡=0(27) for all test function 𝜙∈𝐶^∞₀(𝑅⁺×𝑅).

Definition 2. A two-dimensional weighted delta function 𝑤(𝑠)𝛿_𝐿 supported on a smooth curve 𝐿 parameterized as 𝑡=𝑡(𝑠), 𝑥=𝑥(𝑠)(𝑐≤𝑠≤𝑑) is defined by 𝑤(𝑠)𝛿_𝐿,𝜙(𝑡,𝑥)=^𝑑_𝑐𝑤(𝑠)𝜙(𝑡(𝑠),𝑥(𝑠))𝑑𝑠(28) for all 𝜙∈𝐶^∞₀(𝑅²).

Let us consider the discontinuity of (7) of the form ()=𝑢𝑢,𝜌𝑥,𝑡_𝑙,𝜌_𝑙()(𝑡),𝑢𝑥,𝑡,𝑥<𝑥_𝛿(𝑡)(𝑡)𝛿((𝑡(𝑡),𝑢,𝑤𝑥−𝑥)),𝑥=𝑥_𝑟,𝜌_𝑟(𝑥,𝑡),𝑥>𝑥(𝑡),(29) in which (𝑢_𝑙,𝜌_𝑙)(𝑥,𝑡) and (𝑢_𝑟,𝜌_𝑟)(𝑥,𝑡) are piecewise smooth bounded solutions of (7), for which we also set 𝑑𝑥(𝑡)/𝑑𝑡=𝑢_𝛿(𝑡) for the concentration in 𝜌 needs to travel at the speed of discontinuity (also see [16, 22, 24]). We assert that the discontinuity (29) is a measure solution to (7) if and only if the following relation 𝑑𝑥(𝑡)𝑑𝑡=𝑢_𝛿(𝑡),𝑑𝑤(𝑡)𝑑𝑡=−𝑢_𝛿(𝑡𝜌+,(𝑡)𝑢)𝜌𝑢𝑑𝑤_𝛿(𝑡)𝑑𝑡=−𝑢_𝛿(𝑡)𝜌𝑢+𝜌𝑢²+𝑢(30) is satisfied.

In fact, for any 𝜙∈𝐶^∞₀(𝑅⁺×𝑅), with Green’s formula and taking (7) in mind, one can calculate 𝐼₂=_𝑅+×𝑅𝑢𝜙_𝑡+𝑢𝜙_𝑥𝑑𝜌𝑑𝑡+_𝑅+×𝑅𝑢𝜙_𝑥=𝑑𝑥𝑑𝑡_{0+∞𝑥(𝑡)−∞}𝜌_𝑙𝑢_𝑙𝜙_𝑡+𝜌_𝑙𝑢²_𝑙𝜙_𝑥+𝑑𝑥𝑑𝑡_{0+∞𝑥(𝑡)−∞}𝑢_𝑙𝜙_𝑥+𝑑𝑥𝑑𝑡_{0+∞𝑥(𝑡)+∞}𝜌_𝑟𝑢_𝑟𝜙_𝑡+𝜌_𝑟𝑢²_𝑟𝜙_𝑥+𝑑𝑥𝑑𝑡_{0+∞𝑥(𝑡)+∞}𝑢_𝑟𝜙_𝑥+𝑑𝑥𝑑𝑡_0+∞𝑤(𝑡)𝑢_𝛿(𝑡)𝜙_𝑡+𝑢_𝛿(𝑡)𝜙_𝑥𝑑𝑡=₀^{+∞𝑥(𝑡)}_−∞𝜌_𝑙𝑢_𝑙𝜙_𝑡+𝜌_𝑙𝑢²_𝑙+𝑢_𝑙𝜙_𝑥𝑑𝑥𝑑𝑡−₀^{+∞𝑥(𝑡)}_−∞𝜌_𝑙𝑢_𝑙𝑡+𝜌_𝑙𝑢²_𝑙+𝑢_𝑙𝑥+𝜙𝑑𝑥𝑑𝑡_{0+∞𝑥(𝑡)+∞}𝜌_𝑟𝑢_𝑟𝜙_𝑡+𝜌_𝑟𝑢²_𝑟+𝑢_𝑟𝜙_𝑥−𝑑𝑥𝑑𝑡_{0+∞𝑥(𝑡)+∞}𝜌_𝑟𝑢_𝑟𝑡+𝜌_𝑟𝑢²_𝑟+𝑢_𝑟𝑥+𝜙𝑑𝑥𝑑𝑡_0+∞𝑤(𝑡)𝑢_𝛿(𝑡)𝑑𝜙=𝑑𝑡𝑑𝑡_{0+∞𝑥(𝑡)−∞}𝜌_𝑙𝑢_𝑙𝜙_𝑡+𝜌_𝑙𝑢²_𝑙+𝑢_𝑙𝜙_𝑥+𝑑𝑥𝑑𝑡_{0+∞𝑥(𝑡)+∞}𝜌_𝑟𝑢_𝑟𝜙_𝑡+𝜌_𝑟𝑢²_𝑟+𝑢_𝑟𝜙_𝑥+𝑑𝑥𝑑𝑡_0+∞𝑤(𝑡)𝑢_𝛿(𝑡)𝑑𝜙𝑑𝑡𝑑𝑡=_𝜕Ω−−𝜌_𝑙𝑢_𝑙𝜙𝑑𝑥+𝜌_𝑙𝑢²_𝑙+𝑢_𝑙+𝜙𝑑𝑡_𝜕Ω+−𝜌_𝑟𝑢_𝑟𝜙𝑑𝑥+𝜌_𝑟𝑢²_𝑟+𝑢_𝑟−𝜙𝑑𝑡_0+∞𝑑𝑤(𝑡)𝑢_𝛿(𝑡)=𝑑𝑡𝜙𝑑𝑡_0+∞−𝜌_𝑙𝑢_𝑙𝑑𝑥+𝜌𝑑𝑡^𝑙𝑢²_𝑙+𝑢_𝑙+𝜌_𝑟𝑢_𝑟𝑑𝑥−𝜌𝑑𝑡^𝑟𝑢²_𝑟+𝑢_𝑟−𝜙𝑑𝑡_0+∞𝑑𝑤(𝑡)𝑢_𝛿(𝑡)=𝑑𝑡𝜙𝑑𝑡_0+∞−𝑢_𝛿(𝑡)𝜌𝑢+𝜌𝑢²−(𝑡)𝑢+𝑢𝑑𝑤_𝛿(𝑡)𝑑𝑡𝜙𝑑𝑡,(31) where Ω₋={(𝑥,𝑡)∶𝑡∈𝑅⁺,−∞<𝑥<𝑥(𝑡)}, Ω₊={(𝑥,𝑡)∶𝑡∈𝑅⁺,𝑥(𝑡)<𝑥<+∞}, 𝜕Ω_± is the boundaries of Ω_±, and ∮_𝜕Ω± are the line integral on boundaries 𝜕Ω_±.

Similarly, we have 𝐼₁=_𝑅+×𝑅𝜙_𝑡+𝑢𝜙_𝑥𝑑𝜌𝑑𝑡=₀^+∞−𝑢_𝛿(𝑡𝜌+−(𝑡))𝜌𝑢𝑑𝑤𝑑𝑡𝜙𝑑𝑡.(32) With the above results, we can reach the assertion easily.

Relation (30) reflects the exact relationship among the limit states on two sides of the discontinuity, the weight, propagation speed and the location of the discontinuity. We call it the generalized Rankine-Hugoniot relation of the discontinuity.

In addition, the admissibility (entropy) condition for the discontinuity (29) is 𝜆₂𝜌_𝑟,𝑢_𝑟≤𝑢_𝛿(𝑡)≤𝜆₁𝜌_𝑙,𝑢_𝑙,(33) which means that all characteristics on both sides of the discontinuity are not outcoming and guarantees uniqueness of solution.

Definition 3. A discontinuity which is presented in the form (29) and satisfies (30) and (33) will be called a delta shock wave to system (7), symbolized by 𝛿.

In what follows, the generalized Rankine-Hugoniot relation will in particular be applied to the Riemann problem (7)-(8) for the case (𝑢₊,𝜌₊)∈V(𝑢₋,𝜌₋), which is equivalent to the inequality 𝜆₁𝑢₊,𝜌₊<𝜆₂𝑢₊,𝜌₊≤𝜆₁𝑢₋,𝜌₋<𝜆₂𝑢₋,𝜌₋,(34) that is, 𝑢₊<𝑢₊+1𝜌₊≤𝑢₋<𝑢₋+1𝜌₋.34^′

At this moment, the Riemann solution is a delta shock wave besides two constant states with the form ()=𝑢𝑢,𝜌𝑥,𝑡₋,𝜌₋(𝑡),𝑢,𝑥<𝑥_𝛿(𝑡)(𝑡)𝛿((𝑡(𝑡),𝑢,𝑤𝑥−𝑥)),𝑥=𝑥₊,𝜌₊,𝑥>𝑥(𝑡).(35) To determine 𝑥(𝑡), 𝑤(𝑡), and 𝑢_𝛿(𝑡) uniquely, we solve (30) with initial data 𝑡=0∶𝑥(0)=0,𝑤(0)=0(36) under entropy condition (33) which is, at this moment, 𝑢₊+1𝜌₊≤𝑢_𝛿(𝑡)≤𝑢₋.33^′ From (30) and (36), it follows that 𝑤(𝑡)𝜌𝑥(𝑡)+𝑤(𝑡)𝑢=−𝜌𝑢𝑡,_𝛿(𝑡)=−𝜌𝑢𝑥(𝑡)+𝜌𝑢²+𝑢𝑡.(37) Multiplying the first equation in (37) by 𝑢_𝛿(𝑡) and then subtracting it from the second one, we obtain that 𝜌𝑥(𝑡)𝑢_𝛿(𝑡)−𝜌𝑢𝑢_𝛿(𝑡)𝑡−𝜌𝑢𝑥(𝑡)+𝜌𝑢²+𝑢𝑡=0,(38) or 𝑑𝜌𝑑𝑡2𝑥²(𝑡)−𝜌𝑢𝑥(𝑡)𝑡+𝜌𝑢²+𝑢2𝑡²=0.(39) It gives 𝜌𝑥²(𝑡)−2𝜌𝑢𝑥(𝑡)𝑡+𝜌𝑢²+𝑢𝑡²=0.(40) From (40), we can find that 𝑥^′(𝑡)=𝑢_𝛿(𝑡)∶=𝑢_𝛿 is a constant and 𝑥(𝑡)=𝑢_𝛿𝑡. Then (40) can be rewritten into 𝐻𝑢_𝛿∶=𝜌𝑢²_𝛿−2𝜌𝑢𝑢_𝛿+𝜌𝑢²+𝑢=0.(41) When 𝜌₋=𝜌₊, (41) is just a linear equation of 𝑢_𝛿, and then we have 𝑢_𝛿=𝑢₋+𝑢₊+1/𝜌₋2,𝑥(𝑡)=𝑢₋+𝑢₊+1/𝜌₋𝑡2,𝑤(𝑡)=𝜌₋𝑢₋−𝑢₊𝑡,(42) which satisfies the entropy condition 33^′ obviously.

When 𝜌₋≠𝜌₊, (41) is just a quadratic equation with respect to 𝑢_𝛿, and the discriminant is Δ=4𝜌𝑢²−4𝜌𝜌𝑢²+𝑢=4𝜌₋𝜌₊𝑢₋−𝑢₊𝑢₋+1𝜌₋−𝑢₊−1𝜌₊>0(43) by virtue of (34), and then we can find 𝑢_𝛿=𝜌𝑢−𝜌𝑢²−𝜌𝜌𝑢²+𝑢𝜌,𝑤(𝑡)=𝜌𝑢²−𝜌𝜌𝑢²𝑥(𝑡)=−+𝑢𝑡,𝜌𝑢𝜌𝑢²−𝜌𝜌𝑢²+𝑢𝜌𝑢𝑡,(44)_𝛿=𝜌𝑢+𝜌𝑢²−𝜌𝜌𝑢²+𝑢𝜌,𝑤(𝑡)=−𝜌𝑢²−𝜌𝜌𝑢²𝑥(𝑡)=++𝑢𝑡,𝜌𝑢𝜌𝑢²−𝜌𝜌𝑢²+𝑢𝜌𝑡.(45)

Next, with the help of the entropy condition 33^′, we will choose the admissible solution to Riemann problem (7)-(8) from (44) and (45). Taking into account 34^′, one can calculate that −𝜌𝜆₁𝑢₋,𝜌₋+𝜌𝑢=𝜌₊𝑢₋−𝑢₊−𝜌𝜆>0,₂𝑢₊,𝜌₊+𝜌𝑢=𝜌₋𝑢₋+1𝜌₋−𝑢₊−1𝜌₊𝜌𝜆>0,₁𝑢₋,𝜌₋²−2𝜌𝑢𝜆₁𝑢₋,𝜌₋+𝜌𝑢²+𝑢=−𝜌₊𝑢₋−𝑢₊𝑢₋−𝑢₊−1𝜌₊𝜌𝜆≤0,₂𝑢₊,𝜌₊²−2𝜌𝑢𝜆₂𝑢₊,𝜌₊+𝜌𝑢²+𝑢=𝜌₋𝑢₋−𝑢₊−1𝜌₊𝑢₋+1𝜌₋−𝑢₊−1𝜌₊≥0.(46) Then, for solution (44), we have 𝑢_𝛿−𝜆₂𝑢₊,𝜌₊=𝜌𝑢−𝜌𝑢²−𝜌𝜌𝑢²+𝑢𝜌−𝜆₂𝑢₊,𝜌₊=𝜌𝜆₂𝑢₊,𝜌₊²−2𝜌𝑢𝜆₂𝑢₊,𝜌₊+𝜌𝑢²+𝑢−𝜌𝜆₂𝑢₊,𝜌₊+𝜌𝑢+𝜌𝑢²−𝜌𝜌𝑢²𝑢+𝑢≥0,_𝛿−𝜆₁𝑢₋,𝜌₋=𝜌𝑢−𝜌𝑢²−𝜌𝜌𝑢²+𝑢𝜌−𝜆₁𝑢₋,𝜌₋=𝜌𝜆₁𝑢₋,𝜌₋²−2𝜌𝑢𝜆₁𝑢₋,𝜌₋+𝜌𝑢²+𝑢−𝜌𝜆₁𝑢₋,𝜌₋+𝜌𝑢+𝜌𝑢²−𝜌𝜌𝑢²+𝑢≤0,(47) which imply that the entropy condition 33^′ is valid. One can easily observe especially that, in the two inequalities above, both of the signs “=” are valid if and only if 𝜆₁(𝑢₋,𝜌₋)=𝜆₂(𝑢₊,𝜌₊). In this case, we have 𝑢_𝛿=𝜆₁(𝑢₋,𝜌₋)=𝜆₂(𝑢₊,𝜌₊).

For solution (45), when 𝜌₋<𝜌₊, 𝑢_𝛿−𝜆₂𝑢₊,𝜌₊=𝜌𝑢+𝜌𝑢²−𝜌𝜌𝑢²+𝑢𝜌−𝜆₂𝑢₊,𝜌₊=−𝜌𝜆₂𝑢₊,𝜌₊+𝜌𝑢+𝜌𝑢²−𝜌𝜌𝑢²+𝑢𝜌<0,(48) and when 𝜌₋>𝜌₊, 𝑢_𝛿−𝜆₁𝑢₋,𝜌₋=𝜌𝑢+𝜌𝑢²−𝜌𝜌𝑢²+𝑢𝜌−𝜆₁𝑢₋,𝜌₋=−𝜌𝜆₁𝑢₋,𝜌₋+𝜌𝑢+𝜌𝑢²−𝜌𝜌𝑢²+𝑢𝜌>0.(49) These show that the solution (45) does not satisfy the entropy condition 33^′.

Thus we have proved the following result.

Theorem 4. Let (𝑢₊,𝜌₊)∈𝑉(𝑢₋,𝜌₋). Then Riemann problem (8) for (7) admits one and only one entropy solution in the sense of measures of the form ()=𝑢𝑢,𝜌𝑥,𝑡₋,𝜌₋,𝑥<𝑢_𝛿𝑢𝑡,_𝛿,𝑤(𝑡)𝛿𝑥−𝑢_𝛿𝑡,𝑥=𝑢_𝛿𝑢𝑡,₊,𝜌₊,𝑥>𝑢_𝛿𝑡,(50) where 𝑢_𝛿 and 𝑤(𝑡) are shown in (42) for 𝜌₋=𝜌₊ or (44) for 𝜌₋≠𝜌₊.

At last, associating with the results in Section 2, we can conclude the following.

Theorem 5. For Riemann problem (7)-(8), there exists a unique entropy solution, which contains two different (or just one) contact discontinuities when 𝑢₋<𝑢₊+(1/𝜌₊) and a delta shock wave when 𝑢₋≥𝑢₊+(1/𝜌₊).

4. Numerical Simulations for Riemann Solutions

In this section, by employing the Nessyahu-Tadmor scheme [25] with 300×300 cells and 𝐶𝐹𝐿=0.475, we simulate the obtained Riemann solutions in two cases 𝑢₋<𝑢₊+(1/𝜌₊) and 𝑢₋≥𝑢₊+(1/𝜌₊). We take (𝑢₋,𝜌₋)=(2,1).

For the case 𝑢₋<𝑢₊+(1/𝜌₊), we take (𝑢₊,𝜌₊)=(4,2), and from (24), we can calculate (𝑢^∗,𝜌^∗)=(2,0.4). For the case 𝑢₋≥𝑢₊+(1/𝜌₊), we take (𝑢₊,𝜌₊)=(−1,3,). The numerical results are presented in Figures 4 and 5, respectively.

(a)

(b)

Figure 4 

Numerical results for the case 𝑢−<𝑢++(1/𝜌+) at 𝑡=0.2.

(a)

(b)

Figure 5 

Numerical results for the case 𝑢−≥𝑢++(1/𝜌+) at 𝑡=0.5.

It can be clearly observed that the contact discontinuities develop in Figure 4 while a delta shock wave is developed in Figure 5. All the numerical results are in complete agreement with the theoretical analysis.

5. Two kinds of Interactions of Elementary Waves

In this section, we investigate two kinds of interesting interactions of elementary waves by considering the initial-value problem (9) for system (7). The first case is that the interaction of two contact discontinuities results in a delta shock wave. The second case is that a delta shock wave vanishes after its interaction with a contact discontinuity.

Case (i). Assume that the initial data in (9) satisfy 𝑢₋+1𝜌₋=𝑢_𝑚+1𝜌_𝑚,𝑢_𝑚=𝑢₊.(51) At this moment, a contact discontinuity 𝐽 emits from (0,0) with a speed 𝑢_𝑚+(1/𝜌_𝑚) and a contact discontinuity ^←𝐽 emits from (𝑎,0) with a speed 𝑢_𝑚, respectively. From 𝑢_𝑚+(1/𝜌_𝑚)>𝑢_𝑚, it is known that 𝐽 will overtake ^←𝐽 at a point (𝑥₀,𝑡₀), which can be calculated to be (𝑥₀,𝑡₀)=(𝑎(𝜌_𝑚𝑢_𝑚+1),𝑎𝜌_𝑚). At the moment 𝑡=𝑡₀, a new Riemann problem is formed with (𝑢₋,𝜌₋) and (𝑢₊,𝜌₊) on both sides. We consider the following situation for 𝜌: 1𝜌_𝑚−1𝜌₋≥1𝜌₊,(52) which means that from (51) 𝑢₋≥𝑢₊+1𝜌₊.(53) Then from Theorem 5, it can be concluded that a delta shock wave 𝛿 emits from the moment 𝑡=𝑡₀, whose speed, location, and weight can be obtained by solving (30) under (33) with initial data 𝑡=𝑡₀∶𝑥𝑡₀=𝑥₀,𝑤𝑡₀=0.(54) See Figure 6(a), in which ⓘ=(𝑢_𝑖,𝜌_𝑖)(𝑖=−,𝑚,+).

(a)

(b)

Figure 6 

Two kinds of interactions of elementary waves.

Case (ii). Suppose the initial data (9) in satisfy 𝑢₋≥𝑢_𝑚+1𝜌_𝑚,𝑢_𝑚=𝑢₊.(55) At this time, a delta shock wave 𝛿 emits from (0,0) with the speed 𝜎 satisfying 𝑢₋≥𝜎≥𝑢_𝑚+(1/𝜌_𝑚) and a contact discontinuity ^←𝐽 emits from (𝑎,0) with the speed 𝑢_𝑚. Since 𝜎≥𝑢_𝑚+(1/𝜌_𝑚)>𝑢_𝑚, 𝛿 will interact with ^←𝐽 at a point (𝑥₀,𝑡₀)=(𝑎𝜎/(𝜎−𝑢₊),𝑎/(𝜎−𝑢₊)). As before, at the moment 𝑡=𝑡₀, a new Riemann problem is formed with (𝑢₋,𝜌₋) and (𝑢₊,𝜌₊) on both sides. Let us suppose that the initial data furthermore satisfy 𝑢₋<𝑢₊+1𝜌₊,(56) then from Theorem 5, we can conclude that two contact discontinuities emit form 𝑡=𝑡₀ as follows: ^←𝐽∶𝑑𝑥𝑑𝑡=𝑢₋,𝐽∶𝑑𝑥𝑑𝑡=𝑢₊+1𝜌₊,(57) and the intermediate states (𝑢^∗,𝜌^∗) connecting two contact discontinuities satisfy 𝑢^∗=𝑢₋,𝑢^∗+1𝜌^∗=𝑢₊+1𝜌₊.(58) See Figure 6(b), in which ⊛=(𝑢^∗,𝜌^∗).

Acknowledgments

This paper is partly supported by the National Natural Science Foundation of China (11226191) and the Scientific Research Foundation of Yunnan University (2011YB30). The author would like to thank Professor Hanchun Yang, his advisor, for some valuable and inspiring discussions and suggestions.