Advances in Mathematical Physics

Volume 2016 (2016), Article ID 1783689, 12 pages

http://dx.doi.org/10.1155/2016/1783689

## Interactions of Delta Shock Waves for Zero-Pressure Gas Dynamics with Energy Conservation Law

^{1}The Basic Department, The First Aeronautic Institute of the Air Force, Xinyang 464000, China^{2}College of Mathematics and Information Science, Xinyang Normal University, Xinyang 464000, China

Received 1 April 2016; Accepted 15 May 2016

Academic Editor: Ming Mei

Copyright © 2016 Wei Cai and Yanyan Zhang. 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

We study the interactions of delta shock waves and vacuum states for the system of conservation laws of mass, momentum, and energy in zero-pressure gas dynamics. The Riemann problems with initial data of three piecewise constant states are solved case by case, and four different configurations of Riemann solutions are constructed. Furthermore, the numerical simulations completely coinciding with theoretical analysis are shown.

#### 1. Introduction

As is well known, the system of zero-pressure gas dynamics consisting of conservation laws of mass and momentum, which is also called the transport equations, or Euler equations for pressureless fluids, has been extensively investigated since the 90s of 20th century. It is derived from Boltzmann equations [1] and the flux-splitting scheme of the full compressible Euler equations [2, 3] and can be used to describe the motion process of free particles sticking together under collision [4] and the formation of large-scale structures in the universe [5, 6].

However, we have to mention that, as having no pressure, the energy transport must be taken into account for the considered media. Therefore, it is very necessary to consider the conservation law of energy in zero-pressure gas dynamics. To this end, we study the one-dimensional zero-pressure gas dynamics governed by the conservation laws of mass, momentum, and energy:where and represent the density and velocity, respectively, is the internal energy and assumed to be nonnegative, and is the internal energy per unit mass. The regions in the physical space where and are identified with the vacuum regions of the flow. Here, is considered as an independent variable just for convenience.

System (1) was early studied by Kraiko [7]. In contrast to the traditional zero-pressure gas dynamics system that contains only the conservation laws of mass and momentum, in order to construct the solution of (1) for arbitrary initial data, a new type of discontinuities which are different from the classical ones and carry mass, impulse, and energy are needed. In [8, 9], system (1) was further discussed. Some special integral identities were introduced to define the delta-shock solutions and construct the Rankine-Hugoniot relation for delta shock waves. Moreover, using these integral identities, the balance laws describing mass, momentum, and energy transport from the area outside the delta shock wave front onto its front were derived. What is more, the delta shock wave type solutions for multidimensional zero-pressure gas dynamics with the energy conservation law were defined in [10].

A delta shock wave is a generalization of an ordinary shock wave. Roughly speaking, 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 weighted Dirac delta function with the discontinuity as its support. It is more compressive than an ordinary shock wave and more characteristics enter the discontinuity line. Physically, the delta shock waves describe the process of formation of the galaxies in the universe and the process of concentration of particles. As for delta shock waves, there are numerous excellent papers, see [11–21] and so forth. Nevertheless, compared to these results, a distinctive feature for (1) is that the Dirac delta functions develop in both state variables and simultaneously, which is quite different from those aforementioned, in which only one state variable contains the Dirac delta function. In fact, the theory of delta shock waves with Dirac delta functions developing in both state variables has been established by Yang and Zhang [22, 23] for a class of nonstrictly hyperbolic systems of conservation laws.

In the past over two decades, the investigation of interactions of delta shock waves has been increasingly active. This is important not only because of their significance in practical applications but also because of their basic role as building blocks for the general mathematical theory of quasi-linear hyperbolic equations. And the results on interactions are also touchstones for the numerical schemes. Specifically, Sheng and Zhang [18] discussed the overtaking of delta shock waves and vacuum states in one-dimensional zero-pressure gas dynamics. By solving the two-dimensional Riemann problems for zero-pressure gas dynamics with three constant states, Cheng et al. [24] studied the interactions among delta-shock waves, vacuums, and contact discontinuities. In addition, with the help of a generalized plane wave solution, Yang [25] studied a type of generalized plane delta-shock wave for the -dimensional zero-pressure gas dynamics and investigated the overtaking of two plane delta shocks. For more works on the interactions of delta shock waves, we refer to [26–29] and so forth.

Motivated by the discussions above, in the present paper, we are concerned with the interactions among delta shock waves, vacuum states, and contact discontinuities in solutions. Therefore, we study the Riemann problem of (1) with initial data of three piecewise constant states as follows:where , , are arbitrary constants and , are any two fixed points on -axis.

We will deal with the Riemann problem (1), (2) case by case along with constructing the solutions. For this purpose, it is necessary to consider whether two adjacent waves intersect and interact with each other when constructing the global solution. However, it is often not so easy to see whether two delta shock waves meet and how they interact with each other. Therefore, some technical treatments are needed.

This paper is arranged as follows. In Section 2, the delta shock solution of (1) is reviewed and a general case when the delta shock wave is emitted at the beginning with a nonzero initial data is considered. Section 3 discusses the interactions of the delta shock waves and vacuum states. The Riemann solutions of (1), (2) are constructed globally. Finally, four kinds of numerical simulations coinciding with the theoretical analysis are presented in Section 4.

#### 2. Delta-Shock Solution

This section briefly reviews the delta shock solution of (1) with the initial dataand the detailed study can be found in [30].

System (1) has a triple eigenvalue and two right eigenvectors and . Since for , is linearly degenerate, which means that the elementary waves involve only contact discontinuities. The self-similar solution is constructed by two cases.

For the case , the solution containing two contact discontinuities and a vacuum state besides two constants is expressed aswhere is a smooth function satisfying and .

For the case , the singularity of solutions must develop because of the overlap of characteristic lines. Therefore, the solution involving a delta shock wave is introduced.

Let be the delta shock solution of the formand then the following generalized Rankine-Hugoniot relation holdswhere . In order to ensure the uniqueness, the delta shock wave should satisfy the entropy conditionwhich means that the characteristics on both sides of the discontinuity are in-coming.

Under the entropy condition (7), by solving the ordinary differential equations (6) with the initial data , one has

For convenience, we now consider a special case when a delta shock wave is emitted at the beginning with the initial datasatisfying It yields from (6) and (9) thatOne can check that the delta shock solution (10) satisfies the following: (1) is a monotone function of . (2)If , then While if , then (3)

#### 3. Interactions of Delta Shock Waves

In this section, we analyze the interactions of delta shock waves. To ensure that all the cases are covered completely, according to the relation among , our discussion is divided into four cases:(1);(2);(3);(4).

*Case 1 (). *In this case, two delta shock waves and will be emitted from and , respectively, as shown in Figure 1.