Advances in Mathematical Physics

Volume 2017 (2017), Article ID 3276182, 12 pages

https://doi.org/10.1155/2017/3276182

## The Perturbed Riemann Problem for Special Keyfitz-Kranzer System with Three Piecewise Constant States

School of Mathematics and Statistics Science, Ludong University, Yantai, Shandong Province 264025, China

Correspondence should be addressed to Chun Shen; moc.anis@1463nuhcnehs

Received 1 June 2017; Accepted 27 July 2017; Published 29 August 2017

Academic Editor: Pavel Kurasov

Copyright © 2017 Yuhao Jiang and Chun Shen. 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 Riemann problem for a special Keyfitz-Kranzer system is investigated and then seven different Riemann solutions are constructed. When the initial data are chosen as three piecewise constant states under suitable assumptions, the global solutions to the perturbed Riemann problem are constructed explicitly by studying all occurring wave interactions in detail. Furthermore, the stabilities of solutions are obtained under the specific small perturbations of Riemann initial data.

#### 1. Introduction

The Keyfitz-Kranzer system is a pair of conservation laws in the following form [1]:in which is a given function. System (1) arises as a model for the stretched elastic string to describe the propagation of forward longitudinal and transverse waves [1, 2]. It is also used to illustrate certain features of the solar wind in magnetohydrodynamics [3, 4]. Furthermore, it is usually assumed that is a function of , where , which is called the symmetric Keyfitz-Kranzer system and thus widely investigated such as in [3, 5]. Thus, system (1) was also taken as an example for a nonstrictly hyperbolic system of conservation laws. In addition, it should be pointed out that the nonsymmetric Keyfitz-Kranzer systemhas also been proposed and widely investigated in [6–8]; also see [9, 10] about the Riemann problem for some special forms of system (2).

Recently, it has been assumed in [11], where is a function of , that system (1) can be simplified into the formOne may also see [12] where is a function of . It is easily shown that system (3) owns two eigenvalues and ; thus system (3) is also nonstrictly hyperbolic. More precisely, the characteristic field associated with is linearly degenerate and the characteristic field associated with is genuinely nonlinear when or otherwise linearly degenerate when . In order to ensure , the assumptions , , and were made and then the delta shock wave was captured in [11]. In the present paper, we give up the assumption and want to discover some new interesting nonlinear phenomena. More precisely, we take the detailed example , such that system (3) is simplified intoin which and are two given positive constants, which enable us to deal with system (4) in completely explicit forms.

It is easily shown that system (4) has two eigenvalues and ; thus we can deduce that for and otherwise for . It is easy to see that the special form (4) of the Keyfitz-Kranzer system (1) is also nonstrictly hyperbolic in the quarter phase plane. One of the main features of system (4) lies in that the shock curves have the same representation with the rarefaction ones for the -characteristic family in the quarter phase plane thanks to the special form of system (4), which belongs to the so-called Temple class for some hyperbolic systems of conservation laws [13, 14]. Compared with general hyperbolic systems of conservation laws, the well-posed result for Temple class may be achieved in a more general sense of initial data.

In this paper, we want to construct the global solutions to the particular Cauchy problem for system (4) in fully explicit forms when the three piecewise constant states are taken for the initial conditions as follows:in which is arbitrarily small. This type of initial data (5) has been widely used to study the wave interaction problem [15–17] for different hyperbolic systems of conservation laws. It is worthwhile to notice that the initial data (5) may be regarded as a special small perturbation of the corresponding Riemann initial dataThus, the particular Cauchy problem (4) and (5) is usually called the perturbed Riemann problem (or the double Riemann problems) in literature.

The first task of this paper is to construct the solutions to the Riemann problem (4) and (6) when the Riemann initial data (6) lie in the quarter phase plane. More precisely, we find seven different combinations for the solutions to the Riemann problem (4) and (6) according to the choices of Riemann initial data (6). In particular, some composite waves are needed be introduced in the constructions of Riemann solutions. The second task of this paper is to construct the global solutions to the perturbed Riemann problem (4) and (5), which is essential to study various possible interactions of elementary waves for system (4). For simplicity, we restrict ourselves to consider the situation that the condition is satisfied for the initial data (5). Under this assumption, the global solutions to the perturbed Riemann problem (4) and (5) are constructed in fully explicit forms by studying all the wave interactions appearing in the construction processes of solutions. Furthermore, we can see that the solutions of the Riemann problem (4) and (6) are stable under the particular small perturbation (5) of Riemann initial data (6) when the limit is taken in the solutions. In addition, it should be pointed out that the wave interaction problem for the Temple class has been widely investigated recently, such as for the Aw-Rascle model [18, 19], the Chaplygin gas model [20–23], and the chromatography model [24–28].

The paper is organized in the following way. In Section 2, the Riemann problem (4) and (6) is investigated and the Riemann solutions are constructed for seven different cases. In Section 3, under the suitable assumptions, the global solutions to the perturbed Riemann problem (4) and (5) are constructed in fully explicit forms by investigating all appearing wave interactions when the initial data are taken to be three piecewise constant states. In the end, the stability of Riemann solutions is analyzed with respect to the specific small perturbations (5) of the Riemann initial data (6).

#### 2. The Riemann Problem for the Special Keyfitz-Kranzer System (4)

In this section, we are devoted to investigating the Riemann problem for system (4) associated with the Riemann initial data (6). Let us see, for example, [1, 3, 11, 12] about the related Riemann problem for system (4). We can also refer to [15, 29] for the general knowledge about the Riemann problem for hyperbolic systems of conservation laws.

By a simple calculation, it is obtained that there are two eigenvalues for system (4) as follows:We have for ; otherwise we have for . Thus, system (4) is nonstrictly hyperbolic in the quarter phase plane . The corresponding right eigenvectors for system (4) are given, respectively, byIt is easy to get and , in which . The characteristic field is genuinely nonlinear for provided that and is always linear degeneracy for . The waves associated with the first characteristic field will be either shock waves (denoted by ) or rarefaction waves (denoted by ), which are determined by the choice of initial data. The waves associated with the second characteristic field are always contact discontinuities (denoted by ).

For the Riemann problem (4) and (6), it is invariant under uniform stretching of coordinates: with , such that we may consider solutions in the self-similar formBy carrying out the self-similar transformation , the Riemann problem (4) and (6) may be reformulated as the ordinary different equationsassociated with the boundary values at infinity .

For smooth solutions, (10) is equivalent toIt means either constant state or singular solution, which is a wave of the first characteristic family,or a wave of the second characteristic family,For the given left state , by integrating (12), the contact discontinuity curves in the quarter phase plane, which are the state set that can be connected on the right by a contact discontinuity, are as follows:Similarly, by integrating (13), the rarefaction wave curves in the quarter phase plane, which are the state set that can be connected on the right by a rarefaction wave, are as follows:

On the other hand, for a bounded discontinuity at , the Rankine-Hugoniot conditions readin which is the propagation speed of the discontinuity and is the jump across the discontinuity with and . When the two states can be connected by only one discontinuous wave directly, by solving (16), one can see if it is a contact discontinuity corresponding to a wave of the first characteristic family; then they should satisfyOtherwise, if it is a shock wave corresponding to a wave of the second characteristic family, then they should satisfyin which the entropy condition is taken into account.

Using these elementary waves, we are now in a position to construct the solutions to the Riemann problem (4) and (6) by the analysis method in the quarter phase plane. It can be seen from (14), (15), (17), and (18) that if the Riemann initial data (6) satisfy or , then the Riemann solution contains only a single wave. For the other situations, we can construct the following seven different combinations for the solutions to the Riemann problem (4) and (6) according to the choices of Riemann initial data (6).

(1) When , the solution can be expressed by the symbol (see Figure 1), which is given byThe symbol is used to stand for a shock wave followed by a contact discontinuity . In what follows, similar symbols are also used and not explained again without confusion. Here and below, the symbol is used to indicate the propagation speed of the shock wave and is used to indicate the propagation speed of the contact discontinuity.