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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.

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 [

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:

System (

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 [

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 [

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 (

We will deal with the Riemann problem (

This paper is arranged as follows. In Section

This section briefly reviews the delta shock solution of (

System (

For the case

For the case

Let

Under the entropy condition (

For convenience, we now consider a special case when a delta shock wave is emitted at the beginning with the initial data

If

While if

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

In this case, two delta shock waves

According to what has been discussed in Section

We have

At the intersection

In view of

Thus, the result of interaction of two delta shock waves is still a single delta shock wave. This fact can be formulated as

In this situation, a delta shock wave

Since the propagating speed of

Therefore, by solving (

It is clear that

At

The conclusion of this case is that the delta shock wave will penetrate over the whole vacuum region between two contact discontinuities. This fact is expressed as

Similar to Case 2, there are a delta shock wave, two contact discontinuities, and a vacuum near

The delta shock wave

In this situation, both the contact discontinuities with a vacuum state in between are emitted from

In order to verify the validity of the interactions of delta shock waves and vacuum states mentioned in Section

To discretize the system, we employ the second-order nonoscillatory central schemes [

We take the initial data as follows:

Numerical results of

Numerical results of

Numerical results of

We observe from Figures

We choose the following initial data

Numerical results of

Numerical results of

Numerical results of

From Figures

The initial data are

Numerical results of

Numerical results of

Numerical results of

Figures

We select the initial data to be

Numerical results of

Numerical results of

Numerical results of

From Figures

To sum up, all of the above numerical results clearly reveal the interactions of delta shock waves and vacuum states discussed in Section

The authors declare that they have no competing interests.

This work is supported by the National Natural Science Foundation of China (No. 11501488), the Scientific Research Foundation of Xinyang Normal University (No. 0201318), and Nan Hu Young Scholar Supporting Program of XYNU.