Measure solutions to piston problem for compressible fluid flow of generalized chaplygin gas

We consider the Euler equations of one-dimensional compressible fluid flow in the following form:

where $t\geq 0$ is a time variable, and $x\in R$ is a space variable; $\rho$, $u$ and $P$ represent the density of mass, the velocity and the pressure of the fluid, respectively. System (1.1) was first given by Earnshaw [1], and is regarded as the one-dimensional compressible fluid flow [2]. In addition, System (1.1) has other different physical backgrounds. For instance, it is a scaling limit system of Newtonian dynamics with long-range interaction for a continuous distribution of mass in $R$ [3] and also a hydrodynamic limit for the Vlasov equation [4].

There have been many studies on the Euler system of one-dimensional compressible fluid flow with different equation of state. Diperna [5] established the existence of global weak solution to the Cauchy problem for the case of $1<\gamma<3$ by using Glimm’s scheme under the equation of state $P(\rho)=A\rho^{\gamma}$, where $A>0$ and $\gamma>1$. Based on this result, Li [6] solved the same problems in the case $-1<\gamma<1$. Lu [7] and Cheng [8] proved analytically an existence theorem for the global entropy solutions in the case of $\gamma>3$ and $1<\gamma<3$ respectively, by means of the theory of compensated compactness coupled with some basic ideas of kinetic formulation. For the negative pressure laws, Cheng et al. [9] studied the Riemann problem for the 1D compressible fluid flow of a Chaplygin gas and solved constructively the existence and uniqueness of delta shock solutions, and the model of generalized Chaplygin gas was also investigated by Pang et al [10].

In this work, we are concerned with generalized Chaplygin gas, the equation of state is as follows:

A distinctive feature of the generalized Chaplygin gas is that it has a negative pressure with the positive sound speed. The (1.2) with $\gamma=1$ is also called the pure Chaplygin gas, which was introduced by Chaplygin [11], Tsien [12] and von Karman [13] as a mathematical approximation for calculating the lifting force on a wing of an airplane in aerodynamics. The generalized Chaplygin gas is usually used to describe the current accelerated expansion of the universe and the evolution of the perturbations of energy density [14,15].

The piston problem, serving as an important typical physical model in mathematical fluid dynamics, has been studied extensively in the past decades [16-19]. Qu and Yuan [20] considered the piston problem for compressible Euler equations

with the Chaplygin gas. They defined a Radon measure solution of piston problem within a convenient space of distributions that contains discontinuous functions and Dirac measures. Roughly speaking, the Radon measure solution is a solution such that at least one of the state variables has a Dirac delta function with the boundary of the domain as its support. Moreover, Gao, Qu and Yuan [21] established the equivalence of free piston and delta shock, for the one-dimensional pressureless compressible Euler equations, which helps to understand the physics of delta shocks, but also provides a way to reduce the fluid–solid interaction problem. Recently, Fan et.al [22] investigated the piston problem for system (1.3) with the generalized Chaplygin gas (1.2), where the Radon measure solutions are constructed to deal with the concentration of mass on the piston. For the theory of piston problem and Radon measure solution, readers can refer to [16, 20-23] for more details.

To our knowledge, no literature investigates the piston problem for model (1.1) with a negative pressure law (1.2). Motivated by the above discussions, we will study the piston problem of system (1.1) and (1.2). First, let us describe the piston problem as follows. Suppose there is a piston which can move leftward or rightward in an infinite long and thin tube extending along the horizontal x-axis. The tube is filled with gas, which is enclosed by the piston on the right hand side and is uniform and static initially. The goal of piston problem is to study what happens if the piston moves?

For the convenience of treating this question, we insert (1.2) into (1.1) to give the following compressible fluid flow with the generalized Chaplygin gas

where $A=\frac{s\gamma}{1+\gamma}$, $\alpha=\gamma+1$. (1.4) is totally equivalent to (1.1) and (1.2).

Suppose initially the piston lies at $x=0$, and the gas fills the domain $\{x<0\}$. We assume the gas is static and uniform with given state

The piston moves with a given constant speed $v_{0}$ and the trajectory of the piston is $x=v_{0}t~{}~{}(t\geq 0)$. Then, the time-space domain we consider is

On the trajectory of the piston, we impose the impermeable condition

The piston problem considered in this paper is to find a solution of (1.4) in the domain $\Omega_{t}$, satisfying (1.5) and (1.7).

Recall that the local sonic speed in generalized Chaplygin gas is given by

Then, the Mach number $M_{0}$ of the piston with respect to the gas is defined by

where $v_{0}$ is the move speed of piston, $c_{0}$ is as in (1.8) for $\rho=\rho_{0}$. In this paper, we will show that if the piston moves into the gas with $M_{0}\in\left(0,\sqrt{2}(1+\gamma)^{-\frac{1}{2}}\right)$, or recedes from the gas with any constant speed, there exists integral weak solutions, consisting of shocks or rarefaction waves respectively. However, there is no piecewise constant integral weak solution if the piston moves into the gas with $M_{0}\geq\sqrt{2}(1+\gamma)^{-\frac{1}{2}}$, and then a concept of Radon measure solution is proposed to understand the compressible fluid flow equations when the unknowns are measures, which also demenstrates the necessity of considering general measure solutions in the study of piston problems of systems of hyperbolic conservation laws.

The outline of this paper is organized as follows. In Section 2, we reformulate the piston problem in a convenient way by shifting the coordinates system to move with the piston with Galilean transformation, and present a definition of measure solutions. The main results of piston problem are Theorem 2.1 and Theorem 2.2 given at the end of this section. In Section 3, we give detailed proof to Theorem 2.1. First, we construct an integral weak solution consisting of a shock wave when the piston rushes into the gas and Mach number less than $\sqrt{2}(1+\gamma)^{-\frac{1}{2}}$. Then, we find a solution containing a weighted Dirac measure supported on the piston if $M_{0}\geq\sqrt{2}(1+\gamma)^{-\frac{1}{2}}$, which also justified the concept of measure solutions we proposed. In Section 4, we rigorously prove Theorem 2.2. We construct a self-similar solution containing rarefaction waves as the piston recedes from the gas, and find the occurrence of vacuum state when Mach number tends to infinity.

This section is devoted to the proof of Theorem 2.2. Let $U(x,t)=(\rho(x,t),u(x,t))$. The system (1.4) has two eigenvalues [10]

with two associated right eigenvectors

satisfying

where $A=\frac{s\gamma}{1+\gamma}$, $\alpha=\gamma+1$, $s=\frac{1}{\gamma M_{0}^{2}}$. Thus, all of the characteristic fields are genuinely nonlinear and the Riemann solutions of the system (1.4) contain the first family rarefaction wave $R_{1}(U)$ and the second family rarefaction wave $R_{2}(U)$.

Now, we can as well construct a solution of the form $U(t,x)=V\left(\frac{x}{t}\right)$. For any fixed $0<M_{0}<\infty$, we suppose the solution is composed of two constant states $V_{0}=(1,-1)$, $V_{1}=(\rho_{1},0)$, and a rarefaction wave $V_{m}$ connecting them:

where $U(t,x)=V(\frac{x}{t})$. Replacing $\rho_{0},u_{0}$ by $\rho_{0}=1,u_{0}=-1$ and the second equation of (4.5), we calculate that

Then, substituting $A=\frac{s\gamma}{1+\gamma},~{}\alpha=1+\gamma,~{}s=\frac{1}{\gamma M_{0}^{2}}$ into (4.6) gives

Taking (4.7) into the first equation of (4.5), one obtains

where $0<\gamma<1$ and $0<M_{0}<\infty$. The density $\rho$ will be regard as a function of $\eta$. From (4.8), we have

Combining (4.7) with (4.9), we obtain

Considering $V_{0}=(1,-1)$, by the first equation of (4.5), we obtain $\eta_{-1}=\lambda_{1}(V_{0})=1-M_{0}^{-1}$. Moreover, for state $V_{1}=(\rho_{1},0)$, $u(\eta)$ satisfies the boundary condition $u(\eta_{0})=0$, which deduces $\eta_{0}=-\frac{\gamma+1}{2}-\frac{1}{M_{0}}$. Based on the above discussion and (4.9), we conclude that there has $V_{m}\left(\frac{x}{t}\right)=(\rho(\eta),u(\eta))$, satisfying

and

Consequently, (4.4) gives a rarefaction wave solution for system (1.1) with (2.4) and (2.12) in the domain $\Omega$.

If the Mach number $M_{0}\rightarrow\infty$, we obtain $\eta_{-1}\rightarrow-1$, $\eta_{0}\rightarrow-\frac{1+\gamma}{2}$, $P_{0}\rightarrow 0$, $P_{1}\rightarrow 0$.

For arbitrarily given $\eta$ satisfying $\eta_{-1}<\eta\leq\eta_{0}$. If $M_{0}\rightarrow\infty$, we have $-1<\eta\leq-\frac{1+\gamma}{2}$. By this and (4.9), we obtain

Substituting $u(\eta_{0})=0$ into (4.7) yields $\rho(\eta_{0})=\left[\frac{M_{0}(\gamma+1)+2}{2}\right]^{-\frac{2}{\gamma+1}}$, it is easy to get

Based on above analysis and (4.11) and (4.12), we conclude that the vacuum state occurs in the front of the piston and the right discontinuity line $x=-\frac{1+\gamma}{2}t$ vanishes if $M_{0}\rightarrow\infty$. Moreover, the rarefaction wave $\eta=\frac{x}{t}\in[\eta_{-1},\eta_{0}]$ degenerates discontinuity line $x=-t$, the limit equations for high Mach number are pressureless Euler equations.

Analogously, for the second rarefaction wave $R_{2}(U(x,t))$, we can deduce the self-similar solution as follow:

Similar calculations lead to

and

where $-1+M_{0}^{-1}\leq\eta\leq-\frac{\gamma+1}{2}+M_{0}^{-1}$. Since $u_{1}=0$, we have

There are several different results. If $0<M_{0}<\frac{2}{1+\gamma}$, then $\rho_{1}>1$. Moreover, we have $\rho_{1}<0$ for $0<\gamma<1$ and $M_{0}>\frac{2}{1+\gamma}$. If $M_{0}\rightarrow\frac{2}{1+\gamma}$, we have

The above result contradicts with the requirement of $0\leq\rho_{1}<\rho_{0}=1$. Therefore, the second rarefaction wave $R_{2}(U(x,t))$ is not a physical solution as the piston recedes from the gas.

Remark 4.1. we remark that the limiting equations for compressible fluid flow of generalized chaplygin gas under the Mach number $M_{0}\rightarrow\infty$ is the pressureless Euler equations. The rarefaction wave solutions of compressible fluid flow equations for piston problems at $x=-t$ degenerate into contact discontinuity.

This research is supported by the Department of Education of Fujian Province in China (No. JAT210254) and the Minnan Normal University (No. KJ2021020).