Advances in Mathematical Physics

Volume 2018, Article ID 4925957, 11 pages

https://doi.org/10.1155/2018/4925957

## The Perturbed Riemann Problem with Delta Shock for a Hyperbolic System

^{1}School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China^{2}Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China

Correspondence should be addressed to Lijun Pan; moc.361@94101089

Received 30 June 2018; Accepted 26 August 2018; Published 5 September 2018

Academic Editor: Carlo Bianca

Copyright © 2018 Xinli Han and Lijun Pan. 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

In this paper, we study the perturbed Riemann problem with delta shock for a hyperbolic system. The problem is different from the previous perturbed Riemann problems which have no delta shock. The solutions to the problem are obtained constructively. From the solutions, we see that a delta shock in the corresponding Riemann solution may turn into a shock and a contact discontinuity under a perturbation of the Riemann initial data. This shows the instability and the internal mechanism of a delta shock. Furthermore, we find that the Riemann solution of the hyperbolic system is instable under this perturbation, which is also quite different from the previous perturbed Riemann problems.

#### 1. Introduction

In this paper, we are concerned with the following nonstrictly hyperbolic system of conservation laws in the formEquations (1) can be derived from a two-dimensional hyperbolic system of conservation lawsThe above system is the mathematical simplification of Euler equations of gas dynamics. In 1991 and 1994, Yang and Zhang [1] and Tan and Zhang [2] obtained both numerical and analytical solutions to the Riemann problem of system (2). The form of Dirac delta functions supported on shocks was found necessary and was used as parts in the Riemann solutions. We call it a delta shock. A delta shock is the generalization of an ordinary shock. It is more compressive than an ordinary shock in the sense that more characteristics enter the discontinuity line. Mathematically, the delta shocks are new type singular solutions such that their components contain delta functions and their derivatives. Physically, they are interpreted as the process of formation of the galaxies in the universe, or the process of concentration of particles [3].

To investigate the validity of delta shock, in 1994, Tan et al. [4] considered the Riemann problem for the one-dimensional model (1). They found that there exist delta shock as the limit of vanishing viscosity for system (1). As for delta shock, there are numerous excellent papers. We refer readers to [3–14] and the references cited therein. There are still many open and complicated problems in the delta shock theories. Study of this area gives a new perspective in the theory of conservation law systems.

In this paper, we are interested in the internal mechanism and instability of a delta shock. For this purpose, we study system (1) with the following initial data:where and are all bounded functions with the following property:Here and are constants with . The initial value (3) is a perturbation of Riemann initial value (5) at the neighborhood of the origin in the plane. The perturbation on the Riemann initial data is reasonable. For example, error is unavoidable in computation and the error forms a perturbation of the initial data.

We divide our work into two parts according to the presence of delta shock or not. When the delta shock is not involved, the perturbed Riemann problem (1) and (3) is classical. More importantly, there is no delta shock in the corresponding Riemann solutions for the previous work on the perturbed Riemann problem. Therefore, we only pay attention to the perturbed Riemann problem (1) and (3) when the delta shock is involved. To overcome the difficulty caused by delta shock, we adopt the method of characteristic analysis and the local existence and uniqueness theorem proposed by Li Ta-tsien and Yu Wen-ci [15]. We construct the solution to the perturbed problem (1) and (3) locally in time.

Our result shows that a perturbation of initial data may bring essential change when a delta shock appears in the corresponding Riemann solution. A delta shock may turn into a shock and a contact discontinuity. This shows the instability of the delta shock, which allows us to better investigate the internal mechanism of a delta shock. Furthermore, the previous works [15, 16] about the perturbed Riemann problem pay more attention to the stability of the corresponding Riemann solution. In other words, the Riemann solution has a local structure stability with respect to the perturbation of Riemann initial data. A distinctive feature for this paper is that, for some initial data (3), the preceding local structure stability fails. We pay more attention to the differences between Riemann solution and perturbed Riemann solution.

The paper is organized as follows. In Section 2, we present some preliminary knowledge about the hyperbolic system (1). Then, the construction and proof of the solution to the perturbed Riemann problem (1) and (3) with delta shock are presented in Section 3.

#### 2. Preliminaries

In this section, we recall the main properties of system (1) with Riemann initial datawhere and are constants with (see [4, 17, 18] for a more detailed study of the model).

The eigenvalues of the hyperbolic system (1) arewith the corresponding left eigenvectors,and the corresponding right eigenvectors,By a direct calculation,Therefore, is always linearly degenerate; is genuinely nonlinear if and linearly degenerate if .

The Riemann invariants of system (1) along the characteristic fields are

*Definition 1 (see [3, 19]). *A pair of is called a generalized delta shock solution to (1) with the initial data (3) on local time , if there exists a smooth curve and a weight such that and are represented in the following form:in which is the delta function, , and satisfyfor all the test functions . Here is the tangential derivative of the curve , and stands for the tangential derivative of the function on the curve .

*Definition 2. *For an matrix , define and

#### 3. The Perturbed Riemann Problem with Delta Shock

In this section, we construct the perturbed Riemann solutions of hyperbolic system (1) with initial data (3) for local time and investigate the internal mechanism and instability of delta shock. About the perturbed Riemann problem, we have six cases according to the different constructions of the solutions to the corresponding Riemann problem (1) and (5) as follows:(1)When , the Riemann solution is (2)When , the Riemann solution is delta shock (3)When , the Riemann solution is (4)When , the Riemann solution is (5)When , the Riemann solution is (6)When , the Riemann solution is

Here “+” means “followed by”; the capitals , , and denote shock, contact discontinuity, and rarefaction wave, respectively.

Since the perturbed Riemann problem (1) and (3) with no delta shock is classical and well known, it will not be pursued here. In this section, we mainly study the differences between the perturbed Riemann solution and the corresponding Riemann solution. Thus we only consider the perturbed Riemann problem (1) and (3) with delta shock, a.e., .

It is known from classical theory that the classical solution and can be defined in a strip domains and for local time, respectively (see Figure 1). Here and are local smooth solutions to the initial problem (1) with corresponding initial data and on both sides of , respectively. The right boundary of domain is a characteristic ; namely,The left boundary of domain is characteristic ; namely,