Global existence and stability of time-periodic solution to isentropic compressible Euler equations with source term

In this paper, we are concerned with isentropic compressible Euler equations with a nonlinear term:

$$\left\{\begin{aligned} &\partial_{t}\rho+\partial_{x}(\rho u)=0,\\ &\partial_{t}(\rho u)+\partial_{x}(\rho u^{2}+p)=\beta\rho|u|^{\alpha}u,\end{aligned}\right.\quad(t,x)\in[0,+\infty)\times[0,L],$$

(1.1)

where $\rho,u$ and $p$ are the density, velocity and pressure of gas, respectively. The pressure $p(\rho)$ is governed by $p(\rho)=a\rho^{\gamma}$, here the adiabatic exponent $\gamma>1$ and the parameter $a$ is scaled to unity for mathematical convenience. The sound speed $c\geq 0$ is defined by $c^{2}=\partial p/\partial\rho$. And the term $\beta\rho|u|^{\alpha}u$ represents the friction with $\alpha,\beta\in\mathbb{R}$.
In this paper, we assume the initial data are

$\displaystyle(\rho,u)^{\top}|_{t=0}=(\rho_{0}(x),u_{0}(x))^{\top}.$

(1.2)

The boundary conditions are

$\displaystyle(\rho,u)^{\top}|_{x=0}=(\rho_{l}(t),u_{l}(t))^{\top}$

(1.3)

and $\rho_{l}(t),u_{l}(t)$ are periodic functions with a period $P>0$, i.e.

$$\rho_{l}(t+P)=\rho_{l}(t),u_{l}(t+P)=u_{l}(t).$$

In order to obtain the $C^{1}$ solution, the initial and boundary data should satisfy the following compatibility conditions at point $(0,0)$

$$\left\{\begin{aligned} &\rho_{l}^{\prime}(0)+\rho_{0}^{\prime}(0)u_{0}(0)+\rho_{0}(0)u_{0}^{\prime}(0)=0,\\ &\rho_{l}^{\prime}(0)u_{l}(0)+\rho_{l}(0)u_{l}^{\prime}(0)+\rho_{0}^{\prime}(0)u_{0}^{2}(0)+2\rho_{0}(0)u_{0}(0)u_{0}^{\prime}(0)\\ &\quad+p^{\prime}_{0}(0)-\beta\rho_{0}(0)u_{0}^{\alpha+1}(0)=0,\\ &\rho_{0}(0)=\rho_{l}(0),u_{0}(0)=u_{l}(0),\end{aligned}\right.$$

(1.4)

where

$$p^{\prime}_{0}(0)=\gamma\rho_{0}^{\gamma-1}(0)\rho^{\prime}_{0}(0).$$

Because of the widespread application background, the compressible Euler equation with several kinds of source term have been studied extensively and there are fruitful results. For example, we can refer [7, 6, 21, 16] for the research on the existence and stability of the small smooth solution, [2, 3, 9, 8, 23, 26, 5] for the singularity formation of smooth solution and the results on weak solution. In this paper, we are interested in the time-periodic solution of problem (1.1)-(1.3). As far as we know, there are many works on the studies of time-periodic solutions of the partial differential equations such as the viscous fluids equations [1, 11, 15, 17, 18] and the hyperbolic conservation laws [4, 20, 24, 25, 19]. All of the studies mentioned above discuss the time-periodic solutions which are derived by the time-periodic external forces or the piston motion. But there are few works on the time-periodic solutions of the hyperbolic conservations laws derived by the time-periodic boundary condition. In [29], Yuan studied time-periodic supersonic solutions for the isentropic compressible Euler equation (i.e. $\beta=0$) triggered by periodic supersonic boundary condition. For the quasilinear hyperbolic system with a more general time-periodic boundary conditions, Qu showed the existence and stability of the time-periodic solutions around a small neighborhood of $u\equiv 0$ in [22]. Recently, Wei et al. [28] studied the global stability problem for supersonic flows in one dimensional compressible Euler equations with a friction term $-\mu\rho|u|u,\mu>0$.
In this paper, we would like to show global existence and uniqueness of time-periodic supersonic solutions of initial-boundary value problem (1.1)-(1.3) with the general friction term $\beta\rho|u|^{\alpha}u$ by perturbing some supersonic Fanno flow. Different from [28], we consider (1.1)-(1.3) in the form of sound speed and fluid speed. Then the Fanno fluid are considered for some upstream with positive constants state $(c_{-},u_{-})$ at the left side. After analyzing the ODEs carefully, we get the maximal duct length $L_{m}$, exceed which the flow will get chock. Base on the supersonic Fanno flow, we prove the existence of time periodic solution by wave decomposition.
The main results of this paper are:

The rest of the paper is organised as follows. In Section $2$, we construct the Fanno flow. In Section $3$, we present a reformulation of the problem by perturbing the solution around the supersonic Fanno flow and introduce a wave decomposition for the perturbed solution. In Section $4$, we prove global existence and uniqueness of solution under the help of uniform a-priori estimates. In Section $5$, we prove time-periodicity of solutions by the Gronwall’s inequality.

Fanno flow refers to adiabatic flow through a constant area duct where the effect of friction $(i.e.\beta<0)$ is considered. The friction causes the flow properties to change along the duct. For the completeness of our results, we also consider the case $\beta>0$ in this section.
We rewrite the initial-boundary problem (1.1)- (1.3) in terms of the sound speed $c=\sqrt{\gamma}\rho^{\frac{\gamma-1}{2}}$ and the fluid velocity $u$ as follows

$$\left\{\begin{aligned} &c_{t}+c_{x}u+\frac{\gamma-1}{2}cu_{x}=0,\\ &u_{t}+uu_{x}+\frac{2}{\gamma-1}cc_{x}=\beta|u|^{\alpha}u,\\ &(c,u)^{\top}|_{t=0}=(c_{0}(x),u_{0}(x))^{\top},\\ &(c,u)^{\top}|_{x=0}=(c_{l}(t),u_{l}(t))^{\top},\end{aligned}\right.$$

(2.1)

where $c_{0}(x)=\sqrt{\gamma}\rho_{0}^{\frac{\gamma-1}{2}}(x),c_{l}(t)=\sqrt{\gamma}\rho_{l}^{\frac{\gamma-1}{2}}(t)$.
Now, we consider the positive solution $(\tilde{c},\tilde{u})^{\top}$ of the steady flow of system (2.1) which satisfies

$\displaystyle\left\{\begin{aligned} &\tilde{c}^{\prime}\tilde{u}+\frac{\gamma-1}{2}\tilde{c}\tilde{u}^{\prime}=0,\\ &\tilde{u}\tilde{u}^{\prime}+\frac{2}{\gamma-1}\tilde{c}\tilde{c}^{\prime}=\beta\tilde{u}^{1+\alpha},\\ &(\tilde{c},\tilde{u})^{\top}|_{x=0}=(c_{-},u_{-})^{\top},\end{aligned}\right.$

(2.2)

where $u_{-}$ and $c_{-}$ are two positive constants.
First, by $~{}\eqref{a5}_{1}$, we get

$\displaystyle\tilde{c}$

$\displaystyle=c_{-}u_{-}^{\frac{\gamma-1}{2}}\tilde{u}^{-\frac{\gamma-1}{2}}.$

(2.3)

Substituting (2.3) into $\eqref{a5}_{2}$, we have

$\displaystyle\tilde{u}^{-\alpha}\tilde{u}^{\prime}-c_{-}^{2}u_{-}^{\gamma-1}\tilde{u}^{-\gamma-\alpha-1}\tilde{u}^{\prime}=\beta.$

(2.4)

We consider (2.4) by classifying $\alpha$ and $\beta$.

Case 1: $\alpha\neq 1$ and $\alpha\neq-\gamma$.
In this case, (2.4) becomes

$\displaystyle\frac{1}{-\alpha+1}(\tilde{u}^{-\alpha+1})^{\prime}+\frac{1}{\gamma+\alpha}c_{-}^{2}u_{-}^{\gamma-1}(\tilde{u}^{-\gamma-\alpha})^{\prime}=\beta.$

(2.5)

Integrating (2.5) from $0$ to $x$, we get

$\displaystyle\frac{1}{-\alpha+1}\tilde{u}^{-\alpha+1}+\frac{1}{\gamma+\alpha}c_{-}^{2}u_{-}^{\gamma-1}\tilde{u}^{-\gamma-\alpha}=\frac{1}{-\alpha+1}u_{-}^{-\alpha+1}+\frac{1}{\gamma+\alpha}c_{-}^{2}u_{-}^{-1-\alpha}+\beta x.$

(2.6)

Denote the left-hand-side function of (2.6) as $h(s)$, i.e.

$$h(s)=\frac{1}{-\alpha+1}s^{-\alpha+1}+\frac{1}{\gamma+\alpha}c_{-}^{2}u_{-}^{\gamma-1}s^{-\gamma-\alpha},$$

then we deduce

	$\displaystyle h^{\prime}(s)<0,\quad{\rm for~{}}0<s<s_{c};$
	$\displaystyle h^{\prime}(s)>0,\quad{\rm for~{}}s>s_{c},$

where $s_{c}=c_{-}^{\frac{2}{\gamma+1}}u_{-}^{\frac{\gamma-1}{\gamma+1}}$. This means that $h(s)$ gets its minimum at point $s=s_{c}$. On the other hand, from  (2.3), we have $\tilde{c}=c_{-}^{\frac{2}{\gamma+1}}u_{-}^{\frac{\gamma-1}{\gamma+1}}$, when $\tilde{u}=s_{c}=c_{-}^{\frac{2}{\gamma+1}}u_{-}^{\frac{\gamma-1}{\gamma+1}}$. That is, the flow speed equals to the sound speed $(i.e.M=1)$ at the choked point $(s_{c},h(s_{c}))$. See Figure 1 below.

If $\beta>0$ and the upstream is supersonic (i.e. $u_{-}>c_{-}$), $\tilde{u}$ is monotonically increasing by considering (2.6) and $\tilde{u}>u_{-}$. By (2.3), $\tilde{c}$ is monotonically decreasing and $\tilde{c}<c_{-}$. Then, $\tilde{u}>\tilde{c}$. If $\beta>0$ and the upstream is subsonic (i.e. $u_{-}<c_{-}$), $\tilde{u}$ is monotonically decreasing and $\tilde{c}$ is monotonically increasing. Then $\tilde{u}<\tilde{c}$.
When $\beta<0$, from (2.6), $h(s)$ decreases with respect to the length of the duct till arriving its minimum. Therefore, we can get the maximal length of the duct $L_{m}$ for a supersonic or subsonic flow before it gets choked, which is

$\displaystyle L_{m}=\frac{1}{\beta}\Big{(}\frac{1}{-\alpha+1}(s_{c}^{-\alpha+1}-u_{-}^{-\alpha+1})+\frac{1}{\gamma+\alpha}c_{-}^{2}(u_{-}^{\gamma-1}s_{c}^{-\gamma-\alpha}-u_{-}^{-1-\alpha})\Big{)}.$

(2.7)

Case 2: $\alpha=1$ or $\alpha=-\gamma$.
Now, (2.4) is turned into

$\displaystyle(\ln\tilde{u})^{\prime}+\frac{1}{\gamma+1}c_{-}^{2}u_{-}^{\gamma-1}(\tilde{u}^{-\gamma-1})^{\prime}=\beta,\quad{\rm for}~{}~{}\alpha=1,$

(2.8)

and

$\displaystyle\frac{1}{\gamma+1}(\tilde{u}^{\gamma+1})^{\prime}-c_{-}^{2}u_{-}^{\gamma-1}(\ln\tilde{u})^{\prime}=\beta,\quad{\rm for}~{}~{}\alpha=-\gamma.$

(2.9)

Integrating  (2.8) and (2.9) from $0$ to $x$, we get

$\displaystyle\ln\tilde{u}+\frac{1}{\gamma+1}c_{-}^{2}u_{-}^{\gamma-1}\tilde{u}^{-\gamma-1}=\ln u_{-}+\frac{1}{\gamma+1}c_{-}^{2}u_{-}^{-2}+\beta x,\quad{\rm for}~{}~{}\alpha=1,$

(2.10)

and

$\displaystyle\frac{1}{\gamma+1}\tilde{u}^{\gamma+1}-c_{-}^{2}u_{-}^{\gamma-1}\ln\tilde{u}=\frac{1}{\gamma+1}u_{-}^{\gamma+1}-c_{-}^{2}u_{-}^{\gamma-1}\ln u_{-}+\beta x,\quad{\rm for}~{}~{}\alpha=-\gamma.$

(2.11)

Define

$$f(s)=\ln s+\frac{1}{\gamma+1}c_{-}^{2}u_{-}^{\gamma-1}s^{-\gamma-1},$$

and

$$g(s)=\frac{1}{\gamma+1}s^{\gamma+1}-c_{-}^{2}u_{-}^{\gamma-1}\ln s.$$

The functions $f(s)$ and $g(s)$ get their minimums at point $s=s_{c}=c_{-}^{\frac{2}{\gamma+1}}u_{-}^{\frac{\gamma-1}{\gamma+1}}$. Furthermore, we get the maximal length of the duct $L_{m}$ :

$\displaystyle L_{m}=\frac{1}{\beta}\Big{(}\frac{1}{\gamma+1}c_{-}^{2}(u_{-}^{\gamma-1}s_{c}^{-\gamma-1}-u_{-}^{-2})+\ln\frac{s_{c}}{u_{-}}\Big{)},{~{}~{}\rm for~{}~{}}\alpha=1$

(2.12)

and

$\displaystyle L_{m}=\frac{1}{\beta}\Big{(}\frac{1}{\gamma+1}(s_{c}^{\gamma+1}-u_{-}^{\gamma+1})-c_{-}^{2}u_{-}^{\gamma-1}\ln\frac{s_{c}}{u_{-}}\Big{)},{~{}~{}\rm for~{}~{}}\alpha=-\gamma.$

(2.13)

We can get the similar results with the case 1, we omit the details here.
From the above discussion, we have the following Lemma.

This result means that a subsonic flow entering a duct with friction $(\beta<0)$ will have an increase in its Mach number until the flow is choked at $M=1$, i.e. $\tilde{u}=\tilde{c}$. Conversely, the Mach number of a supersonic flow will decrease until the flow is choked. However, if a flow entering a duct with acceleration $(\beta>0)$, the Mach number of a subsonic flow will decrease and the Mach number of a supersonic flow will increase (i.e. accelerating the initial subsonic or supersonic state). It is worthy to be pointed out that the theoretic calculations are consistent with its experiment. Different from the calculations in [28], where the authors consider a differential equation that relates the change in Mach number with respect to the length of the duct ${dM}\over{dx}$, we rewrite the dominating equations by the relations between the sound speed and flow speeds. Fortunately, the resulting equations can be decoupled easily. Therefore, we can show the maximal duct length which makes the flow choke assuming the upstream Mach number is supersonic or subsonic. Thus by Lemma 2.1 and $\tilde{c}=\sqrt{\gamma}\tilde{\rho}^{\frac{\gamma-1}{2}}$, we can directly get Theorem 1.1.
From the result, we observe that no matter what the constant number $\alpha$ is, the Mach number of a supersonic (subsonic) flow will increase (decrease) when $\beta>0$. While when $\beta<0$, the Mach number of a supersonic flow will decrease until the flow is choked; conversely, a subsonic flow will have an increase in its Mach number until the flow is choked.

For the supersonic flow, we should have $u>0$. Then, we can write the system (1.1) as

$\displaystyle\left\{\begin{aligned} &\rho_{t}+\rho_{x}u+\rho u_{x}=0,\\ &u_{t}+uu_{x}+\gamma\rho^{\gamma-2}\rho_{x}=\beta u^{\alpha+1}.\end{aligned}\right.$

(3.1)

Letting

$\displaystyle\rho(t,x)=\bar{\rho}(t,x)+\tilde{\rho}(x),\quad u(t,x)=\bar{u}(t,x)+\tilde{u}(x),$

(3.2)

where $(\bar{\rho}(t,x),\bar{u}(t,x))^{\top}$ is the perturbation of the supersonic Fanno flow. Substituting  (3.2) into  (3.1), we get

$\displaystyle\left\{\begin{aligned} &\bar{\rho}_{t}+\bar{\rho}_{x}u+\rho\bar{u}_{x}+\tilde{\rho}^{\prime}\bar{u}+\bar{\rho}\tilde{u}^{\prime}+\tilde{\rho}^{\prime}\tilde{u}+\tilde{\rho}\tilde{u}^{\prime}=0,\\ &\bar{u}_{t}+u\bar{u}_{x}+\bar{u}\tilde{u}^{\prime}+\tilde{u}\tilde{u}^{\prime}+\gamma\rho^{\gamma-2}\bar{\rho}_{x}+\gamma\rho^{\gamma-2}\tilde{\rho}^{\prime}=\beta(\bar{u}+\tilde{u})^{\alpha+1}.\end{aligned}\right.$

(3.3)

Moreover, the system  (3.3) can be further written into

$\displaystyle\left\{\begin{aligned} &\bar{\rho}_{t}+\bar{\rho}_{x}u+\rho\bar{u}_{x}=-\tilde{\rho}^{\prime}\bar{u}-\bar{\rho}\tilde{u}^{\prime},\\ &\bar{u}_{t}+u\bar{u}_{x}+\gamma\rho^{\gamma-2}\bar{\rho}_{x}=-F(\rho,\tilde{\rho})\bar{\rho}\tilde{\rho}^{\prime}-\bar{u}\tilde{u}^{\prime}-G(u,\tilde{u})\bar{u},\end{aligned}\right.$

(3.4)

where $F(\rho,\tilde{\rho})\bar{\rho}=\gamma(\rho^{\gamma-2}-\tilde{\rho}^{\gamma-2})$, $G(u,\tilde{u})\bar{u}=-\beta[u^{\alpha+1}-\tilde{u}^{\alpha+1}]$ and $F(\rho,\tilde{\rho})$ and $G(u,\tilde{u})$ can be taken the following expressions

$\displaystyle F(\rho,\tilde{\rho})=\gamma(\gamma-2)\int_{0}^{1}(\theta\bar{\rho}+\tilde{\rho})^{\gamma-3}d\theta,\quad G(u,\tilde{u})=-\beta(\alpha+1)\int_{0}^{1}(\theta\bar{u}+\tilde{u})^{\alpha}d\theta.$

(3.5)

We also consider the perturbations of the initial and boundary conditions. The initial data is reformulated as

$\displaystyle t=0:\left\{\begin{aligned} \rho_{0}(x)=\bar{\rho}_{0}(x)+\tilde{\rho}(x),\quad x\in[0,L],\\ \ u_{0}(x)=\bar{u}_{0}(x)+\tilde{u}(x),\quad x\in[0,L],\end{aligned}\right.$

(3.6)

where $L<L_{m}$, and boundary condition is

$\displaystyle x=0:\left\{\begin{aligned} &\rho_{l}(t)=\bar{\rho}_{l}(t)+\tilde{\rho}(0),\quad t\geq 0,\\ &u_{l}(t)=\bar{u}_{l}(t)+\tilde{u}(0),\quad t\geq 0,\end{aligned}\right.$

(3.7)

where $\bar{\rho}_{0},\bar{u}_{0},\bar{\rho}_{l},\bar{u}_{l}$ are $C^{1}$ functions.

Let $\bar{V}=(\bar{\rho},\bar{u})^{\top}$, the system (3.4) can be rewritten as the following quasi-linear form

$$\bar{V}_{t}+A(V)\bar{V}_{x}+D(\tilde{V})\bar{V}=0$$

(3.8)

with the initial data

$\displaystyle\bar{V}|_{t=0}=\bar{V}_{0}=(\bar{\rho}_{0},\bar{u}_{0})^{\top},$

(3.9)

and the boundary condition

$\displaystyle V|_{x=0}=V_{l}$

$\displaystyle=(\rho_{l},u_{l})^{\top},$

(3.10)

where $V(t,x)=\bar{V}(t,x)+\tilde{V}(x)$, and

$$A(V)=\left(\begin{matrix}u&\rho\\ \gamma\rho^{\gamma-2}&u\end{matrix}\right),\quad D(\tilde{V})=\left(\begin{matrix}\tilde{u}^{\prime}&\tilde{\rho}^{\prime}\\ F(\rho,\tilde{\rho})\tilde{\rho}^{\prime}&\tilde{u}^{\prime}+G(u,\tilde{u})\end{matrix}\right).$$

We next introduce a wave decomposition of the solution $\bar{V}$ to the system (3.8). We can easily get the following two eigenvalues of the coefficient matrix $A(V)$

$$\lambda_{1}(V)=u-c,\quad\lambda_{2}(V)=u+c,$$

where $c=\sqrt{\gamma}\rho^{\frac{\gamma-1}{2}}$. The corresponding two right eigenvectors $r_{i},i=1,2$ are

$\displaystyle r_{1}(V)=\frac{1}{\sqrt{\rho^{2}+c^{2}}}(\rho,-c)^{\top},\quad r_{2}(V)=\frac{1}{\sqrt{\rho^{2}+c^{2}}}(\rho,c)^{\top}.$

(3.11)

The left eigenvectors $l_{i}(V),i=1,2$ are determined by

$\displaystyle l_{i}(V)r_{j}(V)\equiv\delta_{ij},\quad r_{i}^{\top}(V)r_{i}(V)\equiv 1,\quad i,j=1,2,$

(3.12)

where $\delta_{ij}$ stands for the Kronecker’s symbol. Then, $l_{i},i=1,2$ have the following expressions

$\displaystyle l_{1}(V)=\frac{\sqrt{\rho^{2}+c^{2}}}{2}(\rho^{-1},-c^{-1}),\quad l_{2}(V)=\frac{\sqrt{\rho^{2}+c^{2}}}{2}(\rho^{-1},c^{-1}),$

(3.13)

which have the same regularity as $r_{i}(V)$.
Let

$\displaystyle m_{i}=l_{i}(V)\bar{V},\quad n_{i}=l_{i}(V)\bar{V}_{x},\quad m=(m_{1},m_{2})^{\top},\quad n=(n_{1},n_{2})^{\top},$

(3.14)

then

	$\displaystyle\bar{V}$	$\displaystyle=\sum_{k=1}^{2}m_{k}r_{k}(V),\quad\frac{\partial\bar{V}}{\partial x}=\sum_{k=1}^{2}n_{k}r_{k}(V),$		(3.15)
	$\displaystyle\frac{\partial\bar{V}}{\partial t}$	$\displaystyle=-D(\tilde{V})\bar{V}-\sum_{k=1}^{2}\lambda_{k}(V)n_{k}r_{k}(V).$		(3.16)

Thus, we have

$$\begin{split}\frac{d\bar{V}}{d_{i}t}&=\frac{\partial\bar{V}}{\partial t}+\lambda_{i}(V)\frac{\partial\bar{V}}{\partial x}\\ &=\sum_{k=1}^{2}(\lambda_{i}(V)-\lambda_{k}(V))n_{k}r_{k}(V)-D(\tilde{V})\bar{V}.\end{split}$$

(3.17)

By (3.12)- (3.17), one has

$$\begin{split}\frac{dm_{i}}{d_{i}t}=&\frac{\partial m_{i}}{\partial t}+\lambda_{i}(V)\frac{\partial m_{i}}{\partial x}\\ =&\sum_{j,k=1}^{2}\Psi_{ijk}(V)n_{j}m_{k}+\sum_{j,k=1}^{2}\tilde{\Psi}_{ijk}(V)m_{j}m_{k}-\sum_{k=1}^{2}\tilde{\tilde{\Psi}}_{ik}(V)m_{k},\end{split}$$

(3.18)

where

	$\displaystyle\Psi_{ijk}(V)$	$\displaystyle=(\lambda_{j}(V)-\lambda_{i}(V))l_{i}(V)r_{j}(V)\cdot\nabla_{V}r_{k}(V),$		(3.19)
	$\displaystyle\tilde{\Psi}_{ijk}(V)$	$\displaystyle=l_{i}(V)D(\tilde{V})r_{j}(V)\cdot\nabla_{V}r_{k}(V),$		(3.20)
	$\displaystyle\tilde{\tilde{\Psi}}_{ik}(V)$	$\displaystyle=\lambda_{i}(V)l_{i}(V)\tilde{V}^{\prime}\cdot\nabla_{V}r_{k}(V)+l_{i}(V)D(\tilde{V})r_{k}(V).$		(3.21)

Similarly, using (3.8) and  (3.12)- (3.17), we also get

$$\begin{split}\frac{dn_{i}}{d_{i}t}=&\frac{\partial n_{i}}{\partial t}+\lambda_{i}(V)\frac{\partial n_{i}}{\partial x}\\ =&\sum_{j,k=1}^{2}\Phi_{ijk}(V)n_{j}n_{k}+\sum_{j,k=1}^{2}\tilde{\Phi}_{ijk}(V)n_{k}-\sum_{k=1}^{2}l_{i}(V)D_{x}(\tilde{V})r_{k}(V)m_{k},\end{split}$$

(3.22)

where the term $D_{x}(\tilde{V})$ makes sense if $\tilde{V}$ is a $C^{2}$ function, and

	$\displaystyle\Phi_{ijk}(V)=$	$\displaystyle(\lambda_{j}(V)-\lambda_{k}(V))l_{i}(V)r_{j}(V)\cdot\nabla_{V}r_{k}(V)$
		$\displaystyle-r_{j}(V)\cdot\nabla_{V}\lambda_{k}(V)\delta_{ik},$		(3.23)
	$\displaystyle\tilde{\Phi}_{ijk}(V)=$	$\displaystyle-\lambda_{k}(V)l_{i}(V)\tilde{V}^{\prime}\cdot\nabla_{V}r_{k}(V)+l_{i}(V)D(\tilde{V})r_{j}(V)\cdot\nabla_{V}r_{k}(V)m_{j}(V)$
		$\displaystyle-l_{i}(V)D(\tilde{V})r_{k}(V)-\tilde{V}^{\prime}\cdot\nabla_{V}\lambda_{k}(V)\delta_{ik}.$		(3.24)

For later use, we rewrite the system  (3.4) by exchanging the variable $t$ and $x$ as follows

$$\bar{V}_{x}+A^{-1}(V)\bar{V}_{t}+A^{-1}(V)D(\tilde{V})\bar{V}=0.$$

Denote $\hat{\lambda}_{i}(V),i=1,2$ are eigenvalues of the matrix $A^{-1}(V)$, $\hat{l}_{i}(V),i=1,2$ and $\hat{r}_{i}(V),i=1,2$ are the left and right eigenvectors respectively. They can be determined in terms of $\lambda_{i}(V),r_{i}(V)$ and $l_{i}(V)$ as follows

$\displaystyle\hat{\lambda}_{i}(V)=\lambda_{i}(V)^{-1},\quad\hat{r}_{i}(V)=r_{i}(V),\quad\hat{l}_{i}(V)=l_{i}(V).$

(3.25)

Therefore, $\hat{r}_{i}(V)$ and $\hat{l}_{i}(V)$ also satisfy (3.12).

Let

$\displaystyle\hat{m}_{i}=\hat{l}_{i}(V)\bar{V},\quad\hat{n}_{i}=\hat{l}_{i}(V)\bar{V}_{t},\quad\hat{m}=(\hat{m}_{1},\hat{m}_{2})^{\top},\quad\hat{n}=(\hat{n}_{1},\hat{n}_{2})^{\top}.$

(3.26)

By applying the similar arguments as in (3.18)- (3.24), we can get

$$\begin{split}\frac{d\hat{m}_{i}}{d_{i}x}=&\frac{\partial\hat{m}_{i}}{\partial x}+\hat{\lambda}_{i}(V)\frac{\partial\hat{m}_{i}}{\partial t}\\ =&\sum_{j,k=1}^{2}\hat{\Psi}_{ijk}(V)\hat{n}_{j}\hat{m}_{k}+\sum_{j,k=1}^{2}\hat{\tilde{\Psi}}_{ijk}(V)\hat{m}_{j}\hat{m}_{k}-\sum_{k=1}^{2}\hat{\tilde{\tilde{\Psi}}}_{ik}(V)\hat{m}_{k}\end{split}$$

(3.27)

with

	$\displaystyle\hat{\Psi}_{ijk}(V)=$	$\displaystyle(\hat{\lambda}_{j}(V)-\hat{\lambda}_{i}(V))\hat{l}_{i}(V)\hat{r}_{j}(V)\cdot\nabla_{V}\hat{r}_{k}(V),$		(3.28)
	$\displaystyle\hat{\tilde{\Psi}}_{ijk}(V)=$	$\displaystyle\hat{\lambda}_{i}(V)\hat{l}_{i}(V)D(\tilde{V})\hat{r}_{j}(V)\cdot\nabla_{V}\hat{r}_{k}(V),$		(3.29)
	$\displaystyle\hat{\tilde{\tilde{\Psi}}}_{ik}(V)=$	$\displaystyle\hat{l}_{i}(V)\tilde{V}^{\prime}\cdot\nabla_{V}\hat{r}_{k}(V)+\hat{\lambda}_{i}(V)\hat{l}_{i}(V)D(\tilde{V})\hat{r}_{k}(V),$		(3.30)

and

$$\begin{split}\frac{d\hat{n}_{i}}{d_{i}x}=&\frac{\partial\hat{n}_{i}}{\partial x}+\hat{\lambda}_{i}(V)\frac{\partial\hat{n}_{i}}{\partial t}\\ =&\sum_{j,k=1}^{2}\hat{\Phi}_{ijk}(V)\hat{n}_{j}\hat{n}_{k}+\sum_{j,k=1}^{2}\hat{\tilde{\Phi}}_{ijk}(V)\hat{n}_{k}-\sum_{k=1}^{2}\hat{l}_{i}(V)(A^{-1}(V)D(\tilde{V}))_{t}\hat{r}_{k}(V)\hat{m}_{k}(V)\end{split}$$

(3.31)

with

	$\displaystyle\hat{\Phi}_{ijk}(V)=$	$\displaystyle(\hat{\lambda}_{j}(V)-\hat{\lambda}_{k}(V))\hat{l}_{i}(V)\hat{r}_{j}(V)\cdot\nabla_{V}\hat{r}_{k}(V)-\hat{r}_{j}(V)\cdot\nabla_{V}\hat{\lambda}_{k}(V)\delta_{ik},$
	$\displaystyle\hat{\tilde{\Phi}}_{ijk}(V)=$	$\displaystyle-\hat{l}_{i}(V)\tilde{V}^{\prime}\cdot\nabla_{V}\hat{r}_{k}(V)+\hat{\lambda}_{i}(V)\hat{l}_{i}(V)D(\tilde{V})\hat{r}_{j}(V)\cdot\nabla_{V}\hat{r}_{k}(V)\hat{m}_{j}(V)$
		$\displaystyle-\hat{\lambda}_{i}(V)\hat{l}_{i}(V)D(\tilde{V})\hat{r}_{k}(V).$

We also provide the wave decomposition of the initial and boundary data as follows

$\displaystyle m_{0}=(m_{10},m_{20})^{\top},\quad n_{0}=(n_{10},n_{20})^{\top}$

(3.32)

with

$$m_{i0}=l_{i}(V_{0})\bar{V}_{0},\quad n_{i0}=l_{i}(V_{0})\bar{V}^{\prime}_{0},$$

and

$\displaystyle\hat{m}_{l}=(\hat{m}_{1l},\hat{m}_{2l})^{\top},\quad\hat{n}_{l}=(\hat{n}_{1l},\hat{n}_{2l})^{\top}.$

(3.33)

with

$$\hat{m}_{il}=\hat{l}_{i}(V_{l})\bar{V}_{l},\quad\hat{n}_{il}=\hat{l}_{i}(V_{l})\bar{V}^{\prime}_{l},$$

where $\bar{V}_{0}$ and $\bar{V}_{l}$ are defined by (3.9) and (3.10) respectively, and

	$\displaystyle V_{0}$	$\displaystyle=(\rho_{0},u_{0})^{\top},\quad\bar{V}^{\prime}_{0}=(\bar{\rho}^{\prime}_{0},\bar{u}^{\prime}_{0})^{\top},$		(3.34)
	$\displaystyle V_{l}$	$\displaystyle=(\rho_{l},u_{l})^{\top},\quad\bar{V}^{\prime}_{l}=(\bar{\rho}^{\prime}_{l},\bar{u}^{\prime}_{l})^{\top}.$		(3.35)