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The Scientific World Journal
Volume 2014 (2014), Article ID 287256, 11 pages
http://dx.doi.org/10.1155/2014/287256
Research Article

Limit of Riemann Solutions to the Nonsymmetric System of Keyfitz-Kranzer Type

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China

Received 18 January 2014; Accepted 2 March 2014; Published 3 April 2014

Academic Editors: D. Baleanu and H. Jafari

Copyright © 2014 Lihui Guo and Gan Yin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The limit of Riemann solutions to the nonsymmetric system of Keyfitz-Kranzer type with a scaled pressure is considered for both polytropic gas and generalized Chaplygin gas. In the former case, the delta shock wave can be obtained as the limit of shock wave and contact discontinuity when and the parameter tends to zero. The point is, the delta shock wave is not the one of transport equations, which is obviously different from cases of some other systems such as Euler equations or relativistic Euler equations. For the generalized Chaplygin gas, unlike the polytropic or isothermal gas, there exists a certain critical value depending only on the Riemann initial data, such that when drops to , the delta shock wave appears as , which is actually a delta solution of the same system in one critical case. Then as becomes smaller and goes to zero at last, the delta shock wave solution is the exact one of transport equations. Furthermore, the vacuum states and contact discontinuities can be obtained as the limit of Riemann solutions when and , respectively.

1. Introduction

The nonsymmetric system of Keyfitz-Kranzer type can be written as where is a nonlinear function. A more general form of system (1) was first derived as a model for the elastic string by Keyfitz and Kranzer [1].

When , , and , system (1) can be read as

Let ; system (3) can be rewritten as the Aw-Rascle model [2]: where represent the density and the velocity of cars on the roadway, respectively; the state equation is smooth and strictly increasing with The Aw-Rascle model (4) resolves all the obvious inconsistencies and explains instabilities in car traffic flow, especially near the vacuum, that is, for light traffic with few slow drivers. In 2008, Berthelin et al. [3] studied the limit behavior which was investigated by changing into and taking , where is the maximal density which corresponds to a total traffic jam and is assumed to be a fixed constant although it should depend on the velocity in practice. Then, Shen and Sun [4] studied the limit behavior without the constraint of the maximal density, in which the delta shock and vacuum state were obtained through perturbing the pressure suitably.

For the nonsymmetric system of Keyfitz-Kranzer type (3), under the following two assumptions on , Lu [5] established the existence of global bounded weak solutions of the Cauchy problem by using the compensated compactness method. Recently, Lu [6] studied the existence of global entropy solutions to general system of Keyfitz-Kranzer type (3). In 2013, Cheng [7] considered the Riemann problem and two kinds of interactions of elementary waves for system (3) with the state equation for Chaplygin gas:

In this paper, our main purpose is to study the limit behavior of Riemann solutions to the nonsymmetric system of Keyfitz-Kranzer type (3) as the parameter goes to zero. In 2001, Li [8] was concerned with the limits of Riemann solutions to the compressed Euler equations for isothermal gas by letting the temperature go to zero. Then Chen and Liu [9, 10] presented the results of the compressible Euler equations as pressure vanishes. There are many results on the vanishing pressure limits of Riemann solutions; we refer readers to [4, 1113] and the references cited therein for more details.

As the pressure vanishes, system (3) formally transforms into the so-called pressureless gas dynamics model or transport equations: where and stand for the density and the velocity of the gas, respectively. System (8) is also called zero-pressure gas dynamics. It can be derived from zero-pressure isentropic gas dynamics [14]. System (8) is referred to as the adhesion particle dynamics system to describe the motion process of free particles sticking under collision in the low temperature and the information of large-scale structure in the universe [15, 16]. It is easy to see that the delta shock and vacuum do occur in the Riemann solutions of (8); see [17]. We also refer readers to [4, 1823] and the references cited therein for some results on delta shock waves.

By letting be , system (3) can be changed to

In the present paper, we focus on system (9) with equation of state for both polytropic gas and generalized Chaplygin gas. Firstly, we study limit of Riemann solutions to system (9) with the state equation as tends to zero. If , we found that the Riemann solution tends to a delta shock wave solution when . However, the propagating speed and the strength of the delta shock wave in the limit situation are different from the classical results of transport equations (8) with the same Riemann initial data. If , the Riemann solution tends to a two-contact discontinuity solution to the transport equations (8) as . The intermediate state between the two-contact discontinuities is a vacuum state. When , the Riemann solutions converge to one-contact discontinuity solutions of system (8). Then, we investigate system (9) for generalized Chaplygin gas: where is for Chaplygin gas. We find that, as arrives at a certain critical value depending only on the given Riemann initial data , the solution involving one shock and one contact discontinuity converges to a delta shock solution of system (9) and (11). Eventually, when tends to zero, the delta shock wave solution is exactly the solution of transport equations (8). Thus we can see that the process of delta shock wave formation is obviously different from those in [4, 813] and so forth.

The paper is organized as follows. In Section 2, we give some preliminary knowledge for system (8). In Section 3, we present the Riemann solutions to system (9). In Section 4, we display the limit of Riemann solutions to the nonsymmetric system of Keyfitz-Kranzer type (9).

2. The Riemann Solutions of System (8)

In this section, we briefly review the Riemann solutions of (8) with initial data: where , the detailed study of which can be founded in [17].

Transport equations (8) have a double eigenvalue with only one corresponding right eigenvector . By simple calculation, we obtain , which means that system (8) is linearly degenerate.

Given any two constant states , we can constructively obtain the Riemann solutions of (8) and (12) containing contact discontinuities, vacuum, or delta shock wave.

For the case , the solution containing two contact discontinuities and a vacuum state can be expressed as

For the case , we connect the constant states by one contact discontinuity.

For the case , a solution containing a weighted -measure supported on a line will be constructed to connect the constant . So we define the solution in the sense of distributions as follows.

Definition 1. A pair constitutes a solution of (8) in the sense of distributions if it satisfies for any test function .

Moreover, we define a two-dimensional weighted delta functions as follows.

Definition 2. A two-dimensional weighted delta function supported on a smooth curve parameterized as is defined by for all test functions .

With these definitions, one can construct a -measure solution as where and are weight and velocity of the delta shock wave, respectively, satisfying the generalized Rankine-Hugoniot condition: with initial data , where . By simple calculation, we obtain for , and for .

We can also justify that the delta shock wave satisfies the entropy condition: which means that all the characteristics on both sides of the delta shock are incoming.

3. The Riemann Solutions for System (9)

In this section, we analyze some basic properties and solve the Riemann problem for (9).

3.1. The Riemann Solutions for System (9) and (10)

System (9) and (10) have two eigenvalues with corresponding right eigenvectors satisfying

So the 1-characteristic field is genuinely nonlinear, and the 2-characteristic field is always linearly degenerate.

Since (9)-(10) and (12) remain invariant under a uniform expansion of coordinates and , the solution is only connected with . Thus we should seek the self-similar solution Then, the Riemann problem (9)-(10) and (12) can be reduced to with .

For smooth solutions, system (25) can be rewritten as which provides either the general solutions (constant states), or rarefaction wave, which is wave of the first characteristic family, or contact discontinuity, which is of the second characteristic family,

For a bounded discontinuity at , the Rankine-Hugoniot condition holds, where and is the velocity of the discontinuity. From (30), we obtain either shock wave, which is wave of the first characteristic family, or contact discontinuity, which is of the second characteristic family, Here we notice that the shock wave curve and the rarefaction wave curve passing through the same point coincid in the phase plane; that is, (9)-(10) belong to “Temple class” [24].

Through the point , we draw the curve for in the phase plane, which is parallel to the -axis. We denote it by when and when . Through the point , we draw the curve (29) which intersects the -axis at the point , denoted by . Then the phase plane is divided into four regions (see Figure 1). Thus we can construct the Riemann solutions of system (9)-(10) as follows:(1)when , that is, , the solution is ;(2)when , that is, , the solution is ;(3)when , that is, , the solution is ;(4)when , that is, , the solution is .

287256.fig.001
Figure 1: The upper half plane with is divided into 4 regions for both cases and .
3.2. The Riemann Solutions of System (9) and (11)

Systems (9) and (11) have two eigenvalues: with corresponding right eigenvectors: satisfying Thus the 1-characteristic field is genuinely nonlinear and 2-characteristic field is always linearly degenerate as , while both the two characteristic fields are fully linearly degenerate as .

When , we get rarefaction wave and shock wave which can be expressed by or contact discontinuity which can be expressed by

When , through the point , we draw the curve for in the phase plane, denoted by when and when . Through the point , we draw the curve (37) which has two asymptotes and , denoted by . Through the point , we draw the curve (37), which has two asymptotic lines and , denoted by . Then the phase plane is divided into five regions; see Figure 2.

287256.fig.002
Figure 2: The upper half plane with is divided into 5 regions.

For any given , the Riemann solution is showed as follows:(1)when , that is, and , the solution is ;(2)when , that is, and , the solution is ;(3)when , that is, and , the solution is ;(4)when , that is, and , the solution is .The nonvacuum intermediate constant state is given by

When , we introduce a definition of -measure solution, in which we introduce a definition of a generalized solution [19, 20, 22, 25] for system (9) and (11).

Suppose that is a graph in the closed upper half-plane containing smooth arcs , and is a finite set. is subset of such that an arc for starts from the point of the -axis; is the set of initial points of arc .

Consider the -shock wave type initial data , where , , and are constants for . Furthermore, the pressure in (11) is a nonlinear term with respect to defined by .

Definition 3. A pair of distributions and a graph , where and have the form for is called a generalized -shock wave type solution of system (9) with the initial data if the integral identities hold for any test functions , where is the tangential derivative on the graph , is a line integral along the arc , is the velocity of the -shock wave, and .

Theorem 4. When , for the Riemann problem (9), (11), and (12), there is a -shock wave solution with form which satisfies the integral identities (41) in the sense of Definition 3, where , , and is the Heaviside function , .

Suppose that is a region cut by a smooth curve into left and right hand parts ; is a generalized -shock wave solution of system (9) and (11); functions and are smooth in and have one-side limits on the curve . Then the generalized Rankine-Hugoniot conditions for -shock wave are with initial data , where , .

From (44), we obtain as , and as .

We also can justify that the delta shock wave satisfies the entropy condition: which means that all the characteristics on both sides of the delta shock are not outcoming.

When , the detailed study can be found in [7]; we omit it.

Thus, we have obtained the solutions of the Riemann problem for (9).

4. Limit of Riemann Solutions to the Keyfitz-Kranzer Type System

In this section, our main purpose is to consider the limits of the Riemann solutions of (9) and compare them with the corresponding Riemann solutions to transport equations (8). Our discussion depends on the order of and .

4.1. The Limits of Riemann Solutions of (9)-(10)

Firstly, we display the limit of Riemann solution to (9)-(10) for .

Lemma 5. In the case , when , for arbitrary ; when , there exists , such that when .

This lemma shows that the curve becomes steeper as is much small. As , from Lemma 5, we know that when . Then the Riemann solutions of (9)-(10) consist of the rarefaction wave and the contact discontinuity with the intermediate constant state besides the two constant states as this form: where is determined by (21),

When , from (50), and when is small enough to satisfy , we know that a vacuum state appears in the Riemann solutions of (9)-(10). By (21), (49), and (50), it is easy to get that which mean that the rarefaction wave and the contact discontinuity become the contact discontinuities and , respectively, as . Meanwhile the vacuum state will fill up the region between the two contact discontinuities, which is exactly identical with the corresponding Riemann solutions of system (8).

Secondly, when , the Riemann solution contains a shock wave with the propagating speed besides the states for , or a rarefaction wave with the speed for ; see Figure 1. From (31) and (50), we obtain or from (21) and (50), we have

We conclude that, when , the Riemann solution of system (9)-(10) containing one shock wave or one rarefaction wave converges to the contact discontinuity solution of the transport equations (8) as .

Finally, we display the limit of Riemann solutions to (9)-(10) for .

Lemma 6. In the case , when , for arbitrary ; when , there exists , such that when .

From this lemma we know that the contact discontinuity becomes steeper and steeper when decreases; that is, for small . In this case, the Riemann solution of (9)-(10) consists of a shock wave and a contact discontinuity with the intermediate constant state as where is given by (50) and

When , from (50), it is easy to see that By (55), we obtain

From (56)-(57) and we know that and coincide with a new type of nonlinear hyperbolic wave which is called the delta shock wave in [23]. Compared with the corresponding Riemann solutions of (8), it is clear to see that the propagation speed of the delta shock wave here is which is different from that of (8).

From (30), we have which mean that It is obvious that

From (61), we obtain that the strength of the delta shock wave is also different from transport equations (8), which may be due to the different propagation speed of the delta shock wave. For the limit situation of (9)-(10), the characteristics on the left side of the delta shock wave will come into the delta shock wave line while the characteristics on the right side of it will be parallel to it. For transport equations (8), the characteristics on the two sides will come into the delta shock wave curve . So, the Riemann solution of (9)-(10) does not converge to solution of (8) as when .

4.2. The Limit of Riemann Solutions of System (9) and (11)

In this subsection, we deal with the limit behavior of Riemann solutions to system (9) and (11).

Firstly, we display the limit of Riemann solutions to (9) and (11) for .

Lemma 7. For the case , when , for arbitrary ; when , then there exists such that as .

From Lemma 7, we know that the contact discontinuity becomes steeper as becomes smaller and smaller; that is, for small . Then the Riemann solution of (9) and (11) consists of a rarefaction wave and a contact discontinuity with the intermediate constant state besides the two constant states , which has this form: where , are determined by (33) and (38), respectively, and

From (38), we obtain and then a vacuum state appears in the Riemann solution of (9)–(11).

By (33), (38), and (63), we get which mean that the rarefaction wave and the contact discontinuity become the contact discontinuities and , respectively, as . Meanwhile the vacuum state will fill up the region between the two contact discontinuities, which is exactly identical with the corresponding Riemann solution of system (8).

Secondly, when , as done in Section 4.1, it is easy to see that the Riemann solution of (9) and (11) converges to the contact discontinuity of system (8); we omit it.

Finally, we discuss the limit of Riemann solutions of (9) and (11) when .

Lemma 8. If , then there exist such that when ; when .

Proof. When , it is easy to find that for arbitrary directly from Figure 2. On the other hand, when and , see Figure 2 together with (37), we can get that should satisfy , which gives . In one word, for small .
If , should satisfy , , and . From the above inequalities, we obtain when , and when , where The results have been obtained.

When and , the Riemann solution of (9) and (11) consists of a shock wave and a contact discontinuity with the intermediate state besides the two constant states , which is as this form: where , are determined by (38) and (63), respectively, and It is easy to see that For given , letting in (69) yields Hence, we deduce that Thus we have the following result.

Lemma 9. Consider where is given by (63) and (68), and

Proof. Due to (63) and (68), we get Thus it can be seen from (74) that shock wave and contact discontinuity will coalesce together when arrives at .
Using the Rankine-Hugoniot condition for shock and contact discontinuity , we have which implies that It is obvious that The proof is completed.

From Lemma 5, it can be concluded that the shock wave and contact discontinuity will coincide when tends to . On the other hand, for , by substituting into (45), we have

So, we obtain that the quantities , and the limits of and are consistent with (45) as proposed for the Riemann solutions of (9) and (11) for when we take . Otherwise, the assert is obviously true when . Thus, it uniquely determines that the limit of the Riemann solutions to system (9) and (11) when in the case is just the delta shock solution of (9) and (11) in the case , where the curve is actually the boundary between the regions and .

Theorem 10. In the case , for each fixed , assume that is a solution containing the shock wave and contact discontinuity of (9) and (11) with Riemann initial data, constructed in Section 3.2. Then, converges in the sense of distributions, when , and the limit functions and are the sum of step function and a -measure with weights respectively, and then form a delta shock solutions of (9) and (11) when .

Proof. When , let ; then for each fixed , the Riemann solutions are determined by which satisfy for any test function .
The first integral in (81) can be decomposed into The sum of the first and the last terms in (82) is Letting in (83), we have where and is the Heaviside function.
The second term in (82) can be calculated by By , we obtain Then, from (81)1, (84), and (86), we get that holds for any test function .
With the same reason as above, we have
Finally, we study the limits of and as , by tracing the time-dependence of weights of the -measure.
Let and set ; then we obtain On the other hand, By (89) and (90), we get
With the same reason as before, we obtain Thus the result has been obtained.

When and , . So the Riemann solution of (9) and (11) consists of a delta shock wave besides the constant states . We want to observe the behavior of strength and propagation speed of the delta shock wave when decreases and finally tends to zero.

For , letting in (45), we have For the special situation , by (46), we can obtain the same result as above.

From the above discussion, we can conclude that the limit of the strength and propagation speed of the delta shock wave in Riemann solution of system (9) and (11) are in accordance with those of transport equations (8) with the same Riemann initial data. That is to say, the delta shock solution to system (9) and (11) converges to the delta shock solution to transport equations (8) as pressure vanishes.

Combining the results of the above, when , we conclude that the shock wave and a contact discontinuity coincide as a delta shock wave when . As continues to drop and goes to zero eventually, the delta shock solution is nothing but the Riemann solution to transport equations (8).

5. Conclusion

So far, the discussion for limit of Riemann solutions to the nonsymmetric system of Keyfitz-Kranzer type with both the polytropic gas and generalized Chaplygin gas has been completed. From the above analysis, as the pressure vanishes, there appear delta shock wave, vacuum state, and contact discontinuity when , , and , respectively. For the polytropic gas, different from cases of some other systems such as Euler equations or relativistic Euler equations, the delta shock wave is not the one of transport equations as parameter tends to zero. For the generalized Chaplygin gas, the delta shock wave appears as parameter tends to , depending only on the Riemann initial data. Then as becomes smaller and goes to zero at last, the delta shock wave solution is the exact one of transport equations.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This paper is supported by the National Natural Science Foundation of China (no. 11101348).

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