On a Nonsymmetric Keyfitz-Kranzer System of Conservation Laws with Generalized and Modified Chaplygin Gas Pressure Law

This paper is devoted to the study of a nonsymmetricKeyfitz-Kranzer systemof conservation lawswith the generalized andmodified Chaplygin gas pressure law, which may admit delta shock waves, a topic of interest. Firstly, we solve the Riemann problems with piecewise constant data having a single discontinuity. For the generalized Chaplygin gas pressure law, the solution consists of three different structures: R + J, S + J, and δ. Existence and uniqueness of delta shock solution are established under the generalized Rankine-Hugoniot relation and entropy condition. For the modified Chaplygin gas pressure law, the structures of solution are R+J and S + J. Secondly, we discuss the limits of Riemann solutions for the modified Chaplygin gas pressure law as the pressure law tends to the generalized Chaplygin gas one. In particular, for some cases, the solution S + J tends to a delta shock wave, and it is different from the delta shock wave for the generalized Chaplygin gas pressure law with the same initial data. Thirdly, we simulate the Riemann solutions and examine the formation process of delta shock wave by employing the Nessyahu-Tadmor scheme. The numerical results are coincident with the theoretical analysis.


Introduction
Nonlinear hyperbolic conservation laws are a fundamental principle in building mathematical models for many natural processes.For them, there exists an important kind of nonclassical solution, that is, delta shock wave.It is a generalization of an ordinary shock.From the mathematical point of view, it is a kind of discontinuity, on which at least one of the state variables contains Dirac delta function with a shock as its support.From the physical point of view, it represents the process of concentration of the mass.The theory of nonlinear hyperbolic conservation laws admitting delta shock waves is interesting and has been extensively developed in the last several years; see [1][2][3][4][5][6][7][8][9][10][11][12][13] and the references cited therein.
For delta shock waves, the nonsymmetric form is more convenient than the symmetric form (see [14]).Model (1) is also a transformation of the traffic flow model introduced by Aw and Rascle [16], where  and  > 0 are the density and velocity of cars on the roadway and  is the velocity offset.
Let us recall the linear degeneracy and genuine nonlinearity of characteristic fields for quasilinear hyperbolic systems.Assume that   is the th eigenvalue of a quasilinear hyperbolic system and   is the corresponding eigenvector.

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Then the th characteristic field is said to be genuinely nonlinear if ∇  ⋅   ̸ = 0, while if, on the other hand, ∇  ⋅   ≡ 0, we call it linearly degenerate (see [17,18]).For (1), one can easily calculate that the eigenvalues are and the corresponding right eigenvectors are Then Therefore the first characteristic field is always linearly degenerate and the second one is linearly degenerate or genuinely nonlinear depending on the behaviors of ().
In general, a solution to system (1) strongly depends on the function ().In [15], it is required that A prototype function satisfying (6) or ( 7) is  =   ,  > 0, with which the second characteristic  2 is genuinely nonlinear.Recently, in [19], Cheng introduced the function which is the prototype function satisfying At this moment, both eigenvalues are linearly degenerate; that is, this is a fully linearly degenerate system.The overlapping of linear degenerate characteristics results in the formation of delta shock waves.
In the present paper, we continue to study the system (1).We extend (8) to the function which does not satisfy (9).For the Riemann problem with initial data with the analysis method in phase plane, we construct solutions which are  +  when  + + (/  + ) ≥  − + (/  − ) and  +  when  − <  + + (/  + ) <  − + (/  − ).However, for the rest case  + + (/  + ) ≤  − , the solution cannot be constructed by classical waves and delta shock waves should occur.Both existence and uniqueness of solutions involving delta shock waves are obtained by solving the generalized Rankine-Hugoniot relation under entropy condition.
We also introduce the function For the Riemann problem with initial data (11), by the analysis in phase plane, only two kinds of classical solutions  +  and  +  are constructed.Noticing that (12) formally becomes (10) as  → 0, we are interested in the topic that the limits of Riemann solutions to (1) and ( 12) are whether or not the Riemann solutions to (1) and (10).It is rigorously shown that, when  → 0, some Riemann solutions containing  and  tend to a delta shock solution.However, its weight and speed are different from those of delta shock solution to (1), (10) with the same initial data (11).Besides, it is also proven that the rest Riemann solutions tend to just the ones of ( 1), (10) with the same initial data (11).
To confirm the theoretical analysis, we simulate the Riemann solutions and the formation process of delta shock waves by using the Nessyahu-Tadmor scheme [20].The numerical results coincide completely with the theoretical analysis.
We remark that the functions (8), (10), and (12) are also the equations of state characterizing the standard, generalized, and modified Chaplygin gas, respectively, where  is the pressure and  the density.As everyone knows, the universe is experiencing an accelerated expansion.Usually, it is thought that the source of this acceleration is attributed to an exotic type of fluid called dark energy.Then people began to search for different candidates of the dark energy.Up to now, some theoretical models have been proposed.Among them, the standard, generalized, and modified Chaplygin gas are plausible (see [21][22][23][24][25]).
This paper is organized as follows.In Sections 2 and 3, we solve the Riemann problem for (1) with ( 10) and (12), respectively.In Section 4, we investigate the limits of solutions to (1), (12), and (11) as  → 0. In Section 5, some numerical results are presented.Section 6 gives a conclusion.(1) with (10) In this section, we construct the solutions to system (1), (10) with initial data (11).The considered system has two eigenvalues:

Riemann Problem for
with corresponding right eigenvectors: satisfying, respectively, Therefore the system is strictly hyperbolic.The first wave family is linearly degenerate and the second one is genuinely nonlinear.
As usual, we seek the self-similar solution: for which (1) with (10) becomes and (11) changes into the infinity boundary condition For any smooth solutions, ( 17) is equivalent to which provides either the general solution (constant state) or singular solution, which is a wave of the first characteristic family, or rarefaction wave, which is a wave of the second characteristic family, Integrate (21) to obtain which is actually a contact discontinuity (see (26)).We also integrate (22) and take the requirement  2 ( − ,  − ) <  2 (, ) into account to obtain For a bounded discontinuity at  = , the Rankine-Hugoniot relation holds, where and in what follows [] =  − −  denotes the jump of of  across the discontinuity.By solving (25), we obtain contact discontinuity, which is a wave of the first characteristic family, or shock wave, which is a wave of the second characteristic family, The stability condition (entropy condition) for shocks can be defined as "three incoming, one outgoing, " which means that three of the characteristic lines on both sides of shock, one  1 and two  2 , are incoming with respect to the shock, while the remaining one,  1 , is outgoing.The stability condition is equivalent to The curves in (, )-plane expressed by the second equation in (24), (26), and (27) are called as the rarefaction wave curve, contact discontinuity curve, and shock wave curve, respectively.For the contact discontinuity curve, we have   = ()/ +1 > 0 and   = −( + 1)/ +2 < 0, which mean that the curve is monotonic increasing and concave; moreover, it can be proved that lim  → 0  = −∞ and lim  → +∞  =  − + (/  − ), which show that the curve has  = 0 and  =  − + (/  − ) as its two asymptotes.Fixing a left state ( − ,  − ), we draw the rarefaction wave curve, contact discontinuity curve, and shock wave curve.Also, we draw the curve  + (/  ) =  − .Then the phase plane is divided into three domains ( − ,  − ), ( − ,  − ) and ( − ,  − ) by the curves  + (/  ) =  − + (/  − ), and  + (/  ) =  − (see Figure 1): Then by the analysis method in phase plane, using the classical elementary waves , , and , one can construct the solutions of Riemann problem in the following cases: (1) when ( + ,  + ) ∈ ( − ,  − ), the solution is  + ; (2) when ( + ,  + ) ∈ ( − ,  − ), the solution is  + .
In these solutions, the intermediate state ( * ,  * ) satisfies However, for the case ( + ,  + ) ∈ ( − ,  − ), one can check that the Riemann solution cannot be constructed by the classical elementary waves.In fact, at this moment, the characteristic lines from initial data will overlap in a domain.So singularity must happen.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.
We define a two-dimensional weighted delta function ()  supported on a smooth curve  parameterized as  = (), and  = () ( ≤  ≤ ) by for all  ∈  ∞ 0 ( 2 ).With this definition, motivated by [4-6, 10, 12, 19], it is well known that the Riemann solution should be the delta shock solution of the form with 1/ defined by (see [10,12,26]) which satisfies the generalized Rankine-Hugoniot relation and the entropy condition where () and  are the weight and velocity of delta shock wave, respectively.
In what follows, we establish the existence and uniqueness of solution to the generalized Rankine-Hugoniot relation (34) under the entropy condition (35).

Riemann Problem for
The corresponding right eigenvectors are Moreover, Thus the system is strictly hyperbolic.The first wave family is linearly degenerate and the second one is genuinely nonlinear.

Limits of Riemann Solutions to (1) with (12)
In this section, we consider the limits of the Riemann solutions to (1), (12), and (11) as  → 0 case by case.
If  + ≤  − ,  0 can be taken as arbitrary nonnegative real number.If  + >  − , by taking  0 determined by we have the conclusion.
For any  0 >  > 0, the solution of Riemann problem ( 1), (11), and ( 12) is a shock  followed by a contact discontinuity  with the intermediate state (  * ,   * ) besides two constant states ( − ,  − ) and ( + ,  + ).The solution can be expressed as where   and   are the propagation speeds of  and , respectively.
For this case, analogous to Lemma 3, we have the following.
Proof.From the second equations in (62) and (63), we easily get
Proof.From (62) and (63), we conclude that lim from which and Lemma 7 we can easily get (68) and (70).Returning to (73), we immediately obtain that lim Similarly, from (74), we can obtain with (3) We study the limits of   and     by tracking the time-dependence of the weights of the -measure as  → 0.
This completes the proof.
One can observe that when  + + (/  + ) <  − , the weight and speed of the delta shock wave in the limit situation are different from the weight and speed of the delta shock wave obtained by solving (1), (10) with the same initial data (11) in Section 2; and only when  + + (/  + ) =  − , they are coincident.
One can observe that all the numerical results are in complete agreement with the theoretical analysis.
One can see clearly from these numerical results that when  decreases, the location of the shock and contact discontinuity becomes closer, and the density of the intermediate state increases dramatically.

Conclusion
The nonlinear hyperbolic systems of conservation laws of Keyfitz-Kranzer type in the symmetric or nonsymmetric form are important models.This paper is concerned with a special nonsymmetric Keyfitz-Kranzer system (1) with an unknown function (), which is often encountered in some problems.Generally, its solution depends strongly on the behaviors of ().It is very necessary to study such a system for different choices of ().
It is well known that, for nonlinear conservation laws, the classical waves consist of rarefaction wave (), shock (), contact discontinuity (), and vacuum (Vac), and the nonclassical ones contain delta shock wave ().For system (1), we find that the waves involved in the Riemann solutions are different for different choices of the function ().This fact is shown in Table 1, where PG, CG, GCG, and MCG denote the pressure law for polytropic gas, standard Chaplygin gas, generalized Chaplygin gas, and modified Chaplygin gas, respectively; ✓ means that the waves will appear in the Riemann solutions, and M is opposite.It is also concluded that only when the domain ( − ,  − ) appears in the phase plane, delta shock wave will develop.
It is seen that the modified Chaplygin gas pressure law can be regarded as a linear combination of the polytropic gas pressure law (the case for  = 1) and the generalized Chaplygin gas pressure law.We are interested in the limits of the Riemann solutions for the modified Chaplygin gas pressure law as parameter  vanishes.As a result, it is in particular proved that, for some initial data, the limit is a delta shock wave.However, it is not coincident with the delta shock solution to the limit system.This is different from the results in [7,8].

Figure 1 :
Figure 1: Phase plane and elementary wave curves.

Figure 2 :
Figure 2: Phase plane and elementary wave curves.

Table 1 :
Waves involved in the Riemann solutions for different ().