Delta shock wave with Dirac delta function in multiple components for the system of generalized Chaplygin gas dynamics
Pang Boundary Value Problems (2016) 2016:202
DOI 10.1186/s13661-016-0712-6
RESEARCH
Open Access
Delta shock wave with Dirac delta
function in multiple components for the
system of generalized Chaplygin gas
dynamics
Yicheng Pang*
*
Correspondence:
School of Mathematics and
Statistics, Guizhou University of
Finance and Economics, Guiyang,
550025, China
Abstract
We study the Riemann problem for the compressible Euler equations with the
generalized Chaplygin gas. Based on the analysis on the physically relevant region, we
obtain five kinds of exact solutions. It is shown that a delta shock wave with Dirac
delta function in both density and internal energy develops in the exact solutions.
The formation mechanism, generalized Rankine-Hugoniot relation and entropy
condition are clarified for this type of delta shock wave. The numerical results are also
presented to confirm this type of delta shock wave.
Keywords: delta shock wave; generalized Chaplygin gas; Euler equations; Riemann
problem
1 Introduction
The compressible Euler equations are governed by
⎧
⎪
⎨ ρt + (ρu)x = ,
(ρu)t + (ρu + p(ρ, s))x = ,
⎪
⎩
(ρu / + ρe)t + ((ρu / + ρe + p(ρ, s))u)x = ,
(.)
where the variables ρ, u, s, p, e denote density, velocity, specific entropy, pressure, and
specific energy. Both p and e are given functions of ρ and s, satisfying the thermodynamical
constraint
T ds = de + p d ,
ρ
(.)
where T = T(ρ, s) is the temperature. Considerable progress has been made on the Riemann problems or other closely related problems for system (.) with the polytropic gas;
see [–] and the references therein. Here, we concern ourselves with the equation of
state
p(ρ, s) = –Aρ –α ,
(.)
© Pang 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and
indicate if changes were made.
Pang Boundary Value Problems (2016) 2016:202
Page 2 of 20
which is called the generalized Chaplygin gas, where < α ≤ , A > are constants. A substantial difference between the polytropic gas and the generalized Chaplygin gas is that
the latter has a negative pressure with a positive sound speed. The generalized Chaplygin
gas is used as a unified description for the recent accelerated expansion of the universe
and the evolution of the perturbations of energy density. It has also emerged as a unification of dark energy and dark matter. Equation (.) with α = is called a Chaplygin
gas, which was introduced by Chaplygin [] and Tsien [] as a mathematical approximation for calculating the lifting force on a wing of an airplane in aerodynamics. The
reader is referred to [–] for more physical background on the generalized Chaplygin
gas.
Recently, the (generalized) Chaplygin gas has attracted intensive attention. Brenier []
considered the Riemann problem for the isentropic Euler equations
ρt + (ρu)x = ,
(ρu)t + (ρu + p(ρ, s))x = ,
(.)
with the Chaplygin gas, where the solutions with concentration were obtained when the
initial data belong to a certain domain in the phase plane. Guo et al. [] removed this
constraint, and they obtained the delta shock wave solutions. Roughly speaking, the delta
shock solution is a solution such that at least one of the state variables has a Dirac delta
function []. Physically, the delta shock waves are interpreted as the process of formation
of the galaxies in the universe, or the process of concentration of particles. For the theory
of delta shock wave and its related topics, the reader is referred to [] for a more detailed
review. Wang [] constructed the Riemann solutions to system (.) for the generalized
Chaplygin gas, while the formation of a delta shock wave and vacuum state for system (.)
as pressure vanishes was analyzed by Sheng et al. []. In addition, Sun [] dealt with the
Riemann problem of system (.) with the Coulomb-like friction term for the generalized
Chaplygin gas, and the delta shock wave solutions were constructed. However, in contrast
to the extensive investigations on the isentropic Euler equations (.) with the (generalized) Chaplygin gas, little literature contributed to the compressible Euler equations (.)
with the (generalized) Chaplygin gas.
In [], Kraiko studied the system (.) with p(ρ, s) = , and the discontinuities which
would be different from classical ones and carry mass, impulse, and energy, were introduced to construct the solution for arbitrary initial data. Since both density ρ and specific
energy e involve the Dirac delta function, it is difficult to define the product of them. To
avoid this difficulty, Nilsson et al. [, ] denoted the internal energy ρe by a new variable H and showed the processes of concentration of both mass and internal energy on
the delta shock wave front. Subsequently, Cheng [] solved the Riemann problem for (.)
with p(ρ, s) = , where the delta shock wave with a Dirac delta function in both density and
internal energy developed in the solutions. For the studies on the delta shock wave with a
Dirac delta function in multiple state variables, the reader is referred to [, –]. Zhu
and Sheng [] obtained the solutions to the Riemann problem for system (.) with the
Chaplygin gas. We notice that the delta shock wave with a Dirac delta function only in
density issued in the solution. Motivated by the idea in [, ], in contrast to [], we
Pang Boundary Value Problems (2016) 2016:202
Page 3 of 20
consider the compressible Euler equations of the form
⎧
⎪
⎨ ρt + (ρu)x = ,
(ρu)t + (ρu + p(ρ, s))x = ,
⎪
⎩
(ρu / + H)t + ((ρu / + H + p(ρ, s))u)x = ,
(.)
where the state variable H ≥ is the internal energy.
We study the Riemann problem for (.) and (.) with the initial data
(ρ, u, H)(, x) =
(ρ– , u– , H– ), x < ,
(ρ+ , u+ , H+ ), x > ,
(.)
where ρi > , ui , Hi > , i = –, +, are different constants. In a recent paper, the case where
α = was solved. It was found that a delta shock wave with Dirac delta function in both
density and internal energy appeared in the solutions. Meanwhile, the formation mechanism of this kind of delta shock wave results from the overlapping of the linearly degenerate characteristic lines. In this article, we pay attention to the case where < α < .
In the generalized Chaplygin gas < α < , with the thermodynamical constraint (.),
we first conduct the physically relevant region for system (.) and (.). Then, based on the
projections of the classical wave curves onto the (ρ, u)-plane, the Riemann problem is di√
√
vided into two cases. In the case u– – Aρ––(+α)/ < u+ + Aρ+–(+α)/ , by the analysis on the
physically relevant region and the method of characteristic analysis, we obtain four kinds
of exact solutions, (...truncated)
