On admissible positions of Transonic Shocks for Steady Euler Flows In A 3-D Axisymmetric Cylindrical Nozzle

Given a flat cylindrical nozzle:

$$\mathcal{N}:=\left\{(x_{1},x_{2},x_{3})\in{\mathbb{R}}^{3}:0<x_{1}<L,\ 0\leq x_{2}^{2}+x_{3}^{2}<1\right\},$$

(1.9)

with the entrance $\mathcal{N}_{0}$, the exit $\mathcal{N}_{L}$, and the wall $\mathcal{N}_{w}$ being

$$\begin{split}\mathcal{N}_{0}&:=\mathcal{N}\cap\{(x_{1},x_{2},x_{3})\in{\mathbb{R}}^{3}:x_{1}=0\},\\ \mathcal{N}_{L}&:=\mathcal{N}\cap\{(x_{1},x_{2},x_{3})\in{\mathbb{R}}^{3}:x_{1}=L\},\\ \mathcal{N}_{w}&:=\mathcal{N}\cap\{(x_{1},x_{2},x_{3})\in{\mathbb{R}}^{3}:x_{2}^{2}+x_{3}^{2}=1\},\end{split}$$

it is well-known that there may exist plane normal shocks in it with the shock front being a plane perpendicular to the $x_{1}$-axis (see Figure 1.1). Let $x_{1}=\bar{x}_{\mathrm{s}}$ with $\bar{x}_{\mathrm{s}}\in(0,L)$ be the position of the plane shock front, $\bar{U}_{\mathcal{N}_{-}}:=(\bar{q}_{-},0,0,\bar{p}_{-},\bar{\rho}_{-})$ be the state of the uniform supersonic flow ahead of it, and $\bar{U}_{\mathcal{N}_{+}}:=(\bar{q}_{+},0,0,\bar{p}_{+},\bar{\rho}_{+})$ be the state of the uniform subsonic flow behind it. (In this paper, the subscript “$-$” will be used to denote the parameters ahead of the shock front and the subscript “$+$” behind the shock front.) Then the R-H conditions $\eqref{eq810}$-$\eqref{eq813}$ yield

$$\left\{\begin{aligned} &[\bar{\rho}\bar{q}]=\bar{\rho}_{+}\bar{q}_{+}-\bar{\rho}_{-}\bar{q}_{-}=0,\\ &[\bar{p}+\bar{\rho}\bar{q}^{2}]=(\bar{p}_{+}+\bar{\rho}_{+}\bar{q}_{+}^{2})-(\bar{p}_{-}+\bar{\rho}_{-}\bar{q}_{-}^{2})=0,\\ &[\bar{B}]=\bar{B}_{+}-\bar{B}_{-}=0.\end{aligned}\right.$$

(1.10)

Then direct computations yield that

	$\displaystyle\bar{p}_{+}=\Big{(}(1+{\mu}^{2})\bar{M}_{-}^{2}-{\mu}^{2}\Big{)}\bar{p}_{-},$		(1.11)
	$\displaystyle\bar{q}_{+}={\mu}^{2}\Big{(}\bar{q}_{-}+\frac{2}{\gamma+1}\frac{\bar{c}_{-}^{2}}{\bar{q}_{-}}\Big{)},$		(1.12)
	$\displaystyle\bar{\rho}_{+}=\frac{{\bar{\rho}_{-}\bar{q}_{-}}}{\bar{q}_{+}}=\frac{{\bar{\rho}_{-}\bar{q}_{-}^{2}}}{\bar{c}_{*}^{2}},$		(1.13)

where $c=\sqrt{\displaystyle\frac{\gamma p}{\rho}}$ is the sonic speed, $M=\displaystyle\frac{q}{c}$ is the Mach number, and

$\displaystyle\bar{c}_{*}^{2}:=\bar{q}_{+}\bar{q}_{-}={\mu}^{2}\Big{(}\bar{q}_{-}^{2}+\frac{2}{\gamma+1}\bar{c}_{-}^{2}\Big{)},\,\,\,{\mu}^{2}=\frac{\gamma-1}{\gamma+1},\,\,\,\bar{M}_{-}^{2}=\frac{\bar{q}_{-}^{2}}{\bar{c}_{-}^{2}}.$

(1.14)

Based on the above steady plane normal shock solutions in a flat cylindrical nozzle, this paper is going to study the existence of transonic shocks in an axisymmetric 3-D nozzle, which is a small perturbation of a flat cylindrical nozzle, with a pressure condition at the exit.

Let $(z,r,\varpi)$ be the cylindrical coordinate in $\mathbb{R}^{3}$ such that

$\displaystyle(x_{1},x_{2},x_{3})=(z,r\cos\varpi,r\sin\varpi),$

and the perturbed nozzle is axisymmetric with respect to $x_{1}$-axis:

$$\widetilde{\mathcal{N}}:=\{(z,r,\varpi)\in{\mathbb{R}}^{3}:0<z<L,\ 0<r<1+\varphi(z)\},$$

(1.15)

where, with $\sigma>0$ sufficiently small and $\Theta(\cdot)$ a given function defined on $[0,\ L]$,

$$\varphi(z):=\int_{0}^{z}\tan(\sigma\Theta(\tau))\mathrm{d}\tau.$$

(1.16)

In this paper, we further assume that the states of the fluid in the nozzle are also axisymmetric with respect to $x_{1}$-axis such that $U_{\mathcal{N}}$ are independent of $\varpi$, and

$$-u_{2}\sin\varpi+u_{3}\cos\varpi=0.$$

Then direct computations yield that the 3-D steady Euler equations $\eqref{eq1}$-$\eqref{eq3}$ are reduced to

$$\left\{\begin{array}[]{l}\partial_{z}(r\rho u)+\partial_{r}(r\rho v)=0,\\ \\ \partial_{z}(\rho uv)+\partial_{r}(p+\rho v^{2})+\displaystyle\frac{\rho v^{2}}{r}=0,\\ \\ \partial_{z}(p+\rho u^{2})+\partial_{r}(\rho uv)+\displaystyle\frac{\rho uv}{r}=0,\\ \\ \partial_{z}(r\rho uB)+\partial_{r}(r\rho vB)=0,\end{array}\right.$$

(1.17)

where

$$u=u_{1},\,\,\,v=u_{2}\cos\varpi+u_{3}\sin\varpi.$$

(1.18)

Hence, it suffices to determine $U:=(\theta,p,q,s)$, with $\theta=\arctan\displaystyle\frac{v}{u}$ the flow angle and $q=\sqrt{u^{2}+v^{2}}$ the magnitude of the flow velocity, as functions of variables $(z,r)$.

Moreover, under the cylindrical coordinate, the position of a shock front which is axisymmetric with respect to $x_{1}$-axis can be denoted as $\left\{z=\psi_{\widetilde{\mathcal{N}}}(r)\right\}$, and the Rankine-Hugoniot conditions $\eqref{eq810}$-$\eqref{eq813}$ across it become

	$\displaystyle[\rho u]-\psi_{\widetilde{\mathcal{N}}}^{{}^{\prime}}[\rho v]=0,$		(1.19)
	$\displaystyle[\rho uv]-\psi_{\widetilde{\mathcal{N}}}^{{}^{\prime}}[p+\rho v^{2}]=0,$		(1.20)
	$\displaystyle[p+\rho u^{2}]-\psi_{\widetilde{\mathcal{N}}}^{{}^{\prime}}[\rho uv]=0,$		(1.21)
	$\displaystyle[B]=0.$		(1.22)

In the $(z,r)$-space, let

$$\widetilde{\mathcal{N}}:=\{(z,r)\in\mathbb{R}^{2}:0<z<L,\ 0<r<1+\varphi(z)\},$$

(1.23)

denotes the domain of the nozzle with the boundaries

		$\displaystyle E_{0}=\{(z,r)\in\mathbb{R}^{2}:z=0,\,\,\,0<r<1+\varphi(0)\},$		(1.24)
		$\displaystyle W_{0}=\{(z,r)\in\mathbb{R}^{2}:0<z<L,\,\,\,r=0\},$
		$\displaystyle E_{L}=\{(z,r)\in\mathbb{R}^{2}:z=L,\,\,\,0<r<1+\varphi(L)\},$
		$\displaystyle W_{\varphi}=\{(z,r)\in\mathbb{R}^{2}:0<z<L,\,\,\,r=1+\varphi(z)\},$

in which $E_{0}$ is the entry, $E_{L}$ is the exit, $W_{\varphi}$ is the nozzle wall, and $W_{0}$ denotes the symmetry axis of the nozzle. In this paper, we are going to determine the steady flow pattern with a single shock front in $\widetilde{\mathcal{N}}$, satisfying the slip boundary condition on the nozzle wall $W_{\varphi}$, for given supersonic states at the entry $E_{0}$ and given pressure condition at the exit $E_{L}$. The problem is formulated as a free boundary problem described in detail below.

The Free Boundary Problem FBPC

Let $\alpha\in(\frac{1}{2},1)$. Given $\bar{U}_{-}:=(0,\bar{p}_{-},\bar{q}_{-},\bar{s}_{-})$, $P_{\mathrm{e}}\in\mathcal{C}^{2,\alpha}(\bar{\mathbb{R}}_{+})$, $\Theta\in\mathcal{C}^{2,\alpha}([0,L])$, try to determine the states of the fluid $U$ in the nozzle with a single shock front $\Gamma_{\mathrm{s}}^{\widetilde{\mathcal{N}}}:=\{z=\psi_{\widetilde{\mathcal{N}}}(r)\}$ ( see Figure 1.2) such that:

1. (i).

   The nozzle domain $\widetilde{\mathcal{N}}$ is divided by $\Gamma_{\mathrm{s}}^{\widetilde{\mathcal{N}}}$ into two parts:

   	$\displaystyle\widetilde{\mathcal{N}}_{-}=\{(z,r)\in\mathbb{R}^{2}:0<z<\psi_{\widetilde{\mathcal{N}}}(r),\ 0<r<1+\varphi(z)\},$		(1.25)
   	$\displaystyle\widetilde{\mathcal{N}}_{+}=\{(z,r)\in\mathbb{R}^{2}:\psi_{\widetilde{\mathcal{N}}}(r)<z<L,\ 0<r<1+\varphi(z)\},$		(1.26)

   where $\widetilde{\mathcal{N}}_{-}$ denotes the region of the supersonic flow ahead of the shock front, and $\widetilde{\mathcal{N}}_{+}$ is the region of the subsonic flow behind it;

2. (ii).

   In $\widetilde{\mathcal{N}}_{-}$, the states of the fluid $U=U_{-}(z,r)$, which satisfies the Euler system $\eqref{eq536}$, given supersonic state at the entry of the nozzle

   $$U_{-}=\bar{U}_{-},\quad\text{on}\quad E_{0},$$

   (1.27)

   and the slip boundary condition on the wall of the nozzle

   $$\theta_{-}=\sigma\Theta(z),\quad\text{on}\quad W_{\varphi}\cap\overline{\widetilde{\mathcal{N}}_{-}};$$

   (1.28)

3. (iii).

   In $\widetilde{\mathcal{N}}_{+}$, the states of the fluid $U=U_{+}(z,r)$, which satisfies the Euler system $\eqref{eq536}$, the slip boundary condition on the wall of the nozzle

   $$\theta_{+}=\sigma\Theta(z),\quad\text{on}\quad W_{\varphi}\cap\overline{\widetilde{\mathcal{N}}_{+}},$$

   (1.29)

   and given pressure at the exit of the nozzle

   $$p_{+}=p_{\mathrm{e}}(L,r):=\bar{p}_{+}+\sigma P_{\mathrm{e}}(r),\quad\text{on}\quad E_{L};$$

   (1.30)

4. (iv).

   On the shock front $\Gamma_{\mathrm{s}}^{\widetilde{\mathcal{N}}}$, the Rankine-Hugoniot conditions $\eqref{eq889}$-$\eqref{eq899}$ hold for the states $(U_{-},U_{+})$;

5. (v).

   Finally, on the axis $\Gamma_{2}$, under the assumption of axisymmetric, both $U_{-}$ and $U_{+}$ satisfy

   $$\theta_{-}=0,\quad\partial_{r}(p_{-},q_{-},s_{-})=0,\quad\partial_{r}^{2}\theta_{-}=0,\quad\text{on}\quad W_{0}\cap\overline{\widetilde{\mathcal{N}}_{-}},$$

   (1.31)

   and

   $$\theta_{+}=0,\quad\partial_{r}(p_{+},q_{+},s_{+})=0,\quad\partial_{r}^{2}\theta_{+}=0,\quad\text{on}\quad W_{0}\cap\overline{\widetilde{\mathcal{N}}_{+}}.$$

   (1.32)

This paper will deal with the problem FBPC and establish the existence of the transonic shock solution in the 3-D axisymmetric nozzle by showing the following theorem.

In dealing with the free boundary problem FBPC, one of the key difficulties is determining the position of the shock front. However, there is no information on it since the problem FBPC is going to be solved near the steady plane normal shock solutions and, as pointed out in Remark 1.2, the position of the shock front can be arbitrary in the flat nozzle. This difficulty also arises for 2-D transonic shock problem in an almost flat nozzle and Fang and Xin successfully overcome it in [9] by designing a free boundary problem of the linearized 2-D Euler system based on the background normal shock solution, which provides information on the position of the shock front as long as it is solvable: the free boundary can be regarded as an initial approximating position of the shock front. It turns out that this idea also works for the 3-D axisymmetric case studied in this paper, and an initial approximating position of the shock front can be obtained by solving the free boundary problem of the linearized 3-D axisymmetric Euler system based on the plane normal shocks (see Section 4 for details). Different from the problem for 2-D case in [9], there exist additional 0-order terms and singularities along the symmetric axis in the linearized Euler system for 3-D axisymmetric case. These differences will bring new difficulties in solving the free boundary problem and determining the position of the free boundary. They need further observation and careful analysis which will be done in this paper. Once the initial approximation of the transonic shock solution is obtained, nonlinear iteration process similar as in [9] can be executed which converges to a transonic shock solution to the problem FBPC.

The study on gas flows with shocks in a nozzle plays a fundamental role in the operation of turbines, wind tunnels and rockets. Thanks to steady efforts made by many mathematicians, there have been plenty of results on it from different viewpoints and for different models, for instance, see [1, 2, 3, 4, 5, 6, 7, 9, 16, 20, 21, 22, 23, 24] and references therein. For steady multi-dimensional flows with shocks in a finite nozzle, in order to determine the position of the shock front, Courant and Friedrichs pointed out in [7] that, without rigorous mathematical analysis, additional conditions are needed to be imposed at the exit of the nozzle and the pressure condition is preferred among different possible options. From then on, many mathematicians have been working on this issue and there have been many substantial progresses. In particular, Chen-Feldman proved in [3] the existence of transonic shock solutions in a finite flat nozzle for multi-dimensional potential flows with given potential value at the exit and an assumption that the shock front passes through a given point. Later in [2], with given vertical component of the velocity at the exit, Chen-Chen-Song established the existence of the shock solutions for 2-D steady Euler flows. See also [18] for a recent result for 3-D axisymmetric case. Both existence results are established under the assumption that the shock front passes through a given point, which is employed to deal with the same difficulty as the problem in this paper that the position of the shock front of the unperturbed shock solutions can be arbitrary in the nozzle. Without such an artificial assumption, recently in [9], Fang-Xin establish the existence of the transonic shock solutions in an almost flat nozzle with the pressure condition at the exit, as suggested by Courant-Friedrichs in [7]. It is interesting that the results in [9] indicate that, for a generic nozzle and given pressure condition at the exit, there may exist more than one shock solutions, that is, there may exist more than one admissible positions of the shock front. It should be noted that, in a diverging nozzle which is an expanding angular sector, the position of the shock front can be uniquely determined by the pressure condition at the exit under the assumption that the flow states depend only on the radius (see [7]). And the structural stability of this shock solution under small perturbation of the nozzle boundary as well as the pressure condition at the exit has been established for 2-D case in a series of papers [15], [16] by Li-Xin-Yin and in [6] by Chen. See also [19, 20] for a recent advance towards 3-D axisymmetric case.

The paper is organized as follows. In Section 2, the problem FBPC is reformulated by a modified Lagrange transformation, introduced by Weng-Xie-Xin in [20], which straightens the stream line without the degeneracy along the symmetric axis. Then the free boundary problem for the linearized Euler system is described, which serves to determine the initial approximation. Finally, the main theorem to be proved is given. In Section 3, we shall establish a well-posed theory for boundary value problem of the elliptic sub-system of the linearized Euler system at the subsonic state behind the shock front. It turns out that there exists a solution to the problem if and only if a solvability condition is satisfied for the boundary data. This solvability condition will be employed to determine the position of the free boundary. In Section 4, we prove the existence of the initial approximation by applying the theorem proved in Section 3. Then a nonlinear iteration scheme will be described, starting from the initial approximation, in Section 5. Finally, in Section 6, the nonlinear iteration scheme will be verified to be well-defined and contractive, which concludes the proof for the main theorem.

For the 3-D steady axisymmetric Euler system, the Lagrange transformation is defined as, with $\eta(z,0)=0$,

$$\left\{\begin{aligned} &\xi=z,\\ &\eta=\int_{(0,0)}^{(z,r)}t\rho u(s,t)\mathrm{d}t-t\rho v(s,t)\mathrm{d}s.\end{aligned}\right.$$

(2.1)

It can be easily verified that its Jacobi matrix degenerates along the symmetric axis $r=0$, which yields that it is not invertible. This difficulty can be overcome with a modification introduced in [20] by Weng-Xie-Xin, which will be used in this paper.

The modification is as follows. Let

$$\left\{\begin{aligned} &\tilde{\xi}=\xi,\\ &\tilde{\eta}=\eta^{\frac{1}{2}}.\end{aligned}\right.$$

(2.2)

Then the Jacobian of the modified transformation is

$$\displaystyle\frac{\partial(\tilde{\xi},\tilde{\eta})}{\partial(z,r)}=\left|\begin{matrix}\partial_{z}\tilde{\xi}&\partial_{r}\tilde{\xi}\\ \partial_{z}\tilde{\eta}&\partial_{r}\tilde{\eta}\end{matrix}\right|=\left|\begin{matrix}1&0\\ -\displaystyle\frac{r\rho v}{2\tilde{\eta}}&\displaystyle\frac{r\rho u}{2\tilde{\eta}}\end{matrix}\right|=\displaystyle\frac{r\rho u}{2\tilde{\eta}}.$$

For the background steady plane normal shock solution $\bar{U}_{\pm}$, it is easily seen that

$${\eta}(z,r)=\int_{0}^{r}t\bar{\rho}_{\pm}\bar{q}_{\pm}\mathrm{d}t=\displaystyle\frac{1}{2}\bar{\rho}_{\pm}\bar{q}_{\pm}r^{2},$$

(2.3)

such that $\tilde{\eta}(z,r)=r$, where it is assume that, without loss of generality, $\bar{\rho}_{+}\bar{q}_{+}=\bar{\rho}_{-}\bar{q}_{-}=2$. Therefore, it can be anticipated that, if $U_{\pm}$ is close to the background solution $\bar{U}_{\pm}$, then there exist positive constants $C_{1}$ and $C_{2}$, depending only on $\bar{U}_{\pm}$, such that $C_{1}r\leq\tilde{\eta}(z,r)\leq C_{2}r$, which implies that there exists a constant $C_{3}$, depending on $\bar{U}_{\pm}$, such that the Jacobian of the modified Lagrange transformation satisfies

$$\displaystyle\frac{\partial(\tilde{\xi},\tilde{\eta})}{\partial(z,r)}\geq C_{3}>0,$$

(2.4)

that is, it does not degenerate and is invertible.

Under the modified Lagrange transformation, we have $r(\tilde{\xi},0)=0$ and

$\displaystyle\mathrm{d}z=\mathrm{d}\tilde{\xi},\quad\mathrm{d}r=\displaystyle\frac{2\tilde{\eta}}{r\rho u}\mathrm{d}\tilde{\eta}+\displaystyle\frac{v}{u}\mathrm{d}\tilde{\xi}.$

(2.5)

It follows that

$\displaystyle r=r(\tilde{\xi},\tilde{\eta})=\left(2\int_{0}^{\tilde{\eta}}\displaystyle\frac{2t}{\rho u(\tilde{\xi},t)}\mathrm{d}t\right)^{\frac{1}{2}}.$

(2.6)

In particular, at the background state, one has

$\displaystyle\bar{r}=\Big{(}2\int_{0}^{\tilde{\eta}}\displaystyle\frac{2t}{\bar{\rho}_{\pm}\bar{q}_{\pm}}\mathrm{d}t\Big{)}^{\frac{1}{2}}=\sqrt{\frac{2}{\bar{\rho}_{\pm}\bar{q}_{\pm}}}\tilde{\eta}.$

(2.7)

Then, it follows after direct computations that, under the modified Lagrange transformation, the Euler system $\eqref{eq536}$ becomes

		$\displaystyle\partial_{\tilde{\xi}}\Big{(}\frac{2{\tilde{\eta}}}{r\rho u}\Big{)}-\partial_{\tilde{\eta}}\Big{(}\frac{v}{u}\Big{)}=0,$		(2.8)
		$\displaystyle\partial_{\tilde{\xi}}v+\frac{r}{2{\tilde{\eta}}}\partial_{\tilde{\eta}}p=0,$
		$\displaystyle\partial_{\tilde{\xi}}\Big{(}u+\frac{p}{\rho u}\Big{)}-\frac{r}{2{\tilde{\eta}}}\partial_{\tilde{\eta}}\Big{(}\frac{pv}{u}\Big{)}-\frac{pv}{r\rho u^{2}}=0,$
		$\displaystyle\partial_{\tilde{\xi}}B=0.$

For simplicity of the notations, we drop “ $\tilde{}$ ” hereafter as there is no confusion taking place.

Further computation yields that the system (2.8) can be rewritten as

	$\displaystyle\partial_{\eta}p-\displaystyle\frac{2{\eta}}{r}\cdot\displaystyle\frac{\sin\theta}{\rho q}\partial_{\xi}p+\displaystyle\frac{2{\eta}}{r}\cdot q\cos\theta\cdot\partial_{\xi}\theta=0,$		(2.9)
	$\displaystyle\partial_{\eta}\theta-\displaystyle\frac{2{\eta}}{r}\cdot\displaystyle\frac{\sin\theta}{\rho q}\partial_{\xi}\theta+\displaystyle\frac{2{\eta}}{r}\cdot\displaystyle\frac{\cos\theta}{\rho q}\displaystyle\frac{M^{2}-1}{\rho q^{2}}\partial_{\xi}p+\displaystyle\frac{2{\eta}}{r^{2}}\cdot\displaystyle\frac{\sin\theta}{\rho q}=0,$		(2.10)
	$\displaystyle\rho q\partial_{\xi}q+\partial_{\xi}p=0,$		(2.11)
	$\displaystyle\partial_{\xi}s=0,$		(2.12)

where the equation (2.11) can also be replaced by the following equation in the conservative form:

$$\partial_{{\xi}}{B}=0.$$

(2.13)

Let ${\Gamma}_{\mathrm{s}}:=\{({{\xi}},{{\eta}})\in\mathbb{R}^{2}:{\xi}={\psi}({\eta}),\,\,0<{\eta}<1\}$ be the position of a shock front, under the modified Lagrange transformation, the Rankine-Hugoniot conditions $\eqref{eq889}$-$\eqref{eq899}$ across the shock front become

	$\displaystyle\displaystyle\frac{2\eta}{r}\Big{[}\displaystyle\frac{1}{\rho u}\Big{]}+\psi^{\prime}\Big{[}\displaystyle\frac{v}{u}\Big{]}=0,$		(2.16)
	$\displaystyle[v]-\psi^{\prime}\displaystyle\frac{r}{2\eta}[p]=0,$		(2.17)
	$\displaystyle\Big{[}u+\displaystyle\frac{p}{\rho u}\Big{]}+\psi^{\prime}\displaystyle\frac{r}{2\eta}\Big{[}\displaystyle\frac{pv}{u}\Big{]}=0,$		(2.18)
	$\displaystyle[B]=0.$		(2.19)

Using the equation $\eqref{eq2904}$, we can eliminate the quantity $\psi^{\prime}$ in the equations $\eqref{eq2903}$ and $\eqref{eq2905}$ respectively to obtain

		$\displaystyle G_{1}(U_{+},U_{-}):=\Big{[}\displaystyle\frac{1}{\rho u}\Big{]}[p]+\Big{[}\displaystyle\frac{v}{u}\Big{]}[v]=0,$		(2.20)
		$\displaystyle G_{2}(U_{+},U_{-}):=\Big{[}u+\displaystyle\frac{p}{\rho u}\Big{]}[p]+\Big{[}\displaystyle\frac{pv}{u}\Big{]}[v]=0,$		(2.21)

moreover, we denote

	$\displaystyle G_{3}(U_{+},U_{-}):=\Big{[}\frac{1}{2}q^{2}+i\Big{]}=0,$		(2.22)
	$\displaystyle G_{4}(U_{+},U_{-};\psi^{\prime}):=[v]-\psi^{\prime}\displaystyle\frac{r}{2\eta}[p]=0.$		(2.23)

Under the modified Lagrange transformation, the domain $\widetilde{\mathcal{N}}$ becomes

$\displaystyle{\Omega}=\{({\xi},{\eta})\in\mathbb{R}^{2}:0<{\xi}<L,\,\,\,0<{\eta}<1\},$

(2.24)

and it is separated by a shock front ${\Gamma}_{\mathrm{s}}$ into two parts (see Figure 2.1): the supersonic region and subsonic region, denoted by

$${\Omega}_{-}=\{({\xi},{\eta})\in\mathbb{R}^{2}:0<{\xi}<\psi({\eta}),\,\,\,0<{\eta}<1\},$$

(2.25)

$${\Omega}_{+}=\{({\xi},{\eta})\in\mathbb{R}^{2}:\psi({\eta})<{\xi}<L,\,\,\,0<{\eta}<1\}$$

(2.26)

respectively. The boundaries $E_{0}$, $W_{0}$, $E_{L}$, $W_{\varphi}$ become

		$\displaystyle{\Gamma}_{1}=\{({{\xi}},{{\eta}})\in\mathbb{R}^{2}:{\xi}=0,\,\,\,0<{\eta}<1\},$		(2.27)
		$\displaystyle{\Gamma}_{2}=\{({{\xi}},{{\eta}})\in\mathbb{R}^{2}:0<{\xi}<L,\,\,\,{\eta}=0\},$
		$\displaystyle{\Gamma}_{3}=\{({{\xi}},{{\eta}})\in\mathbb{R}^{2}:{\xi}=L,\,\,\,0<{\eta}<1\},$
		$\displaystyle{\Gamma}_{4}=\{({{\xi}},{{\eta}})\in\mathbb{R}^{2}:0<{\xi}<L,\,\,\,{\eta}=1\}.$

Thus, the free boundary problem FBPC is reformulated as follows under the modified Lagrange transformation.

The Free Boundary Problem FBPL

Try to determine $(U_{-}(\xi,\eta),\ U_{+}(\xi,\eta),\ \psi(\eta))$ in $\Omega$ such that:

1. (i).

   $U_{-}(\xi,\eta)$ satisfies the equations (2.9)-(2.12) in $\Omega_{-}$, and the following boundary conditions:

   	$\displaystyle U_{-}=\bar{U}_{-},$		$\displaystyle\text{on}\,\,\,\Gamma_{1},$		(2.28)
   	$\displaystyle\theta_{-}=\sigma\Theta(\xi).$		$\displaystyle\text{on}\,\,\,\Gamma_{4}\cap\overline{\Omega_{-}}.$		(2.29)

2. (ii).

   $U_{+}(\xi,\eta)$ satisfies the equations (2.9)-(2.12) in $\Omega_{+}$, and the following boundary conditions:

   	$\displaystyle\theta_{+}=\sigma\Theta(\xi),$		$\displaystyle\text{on}\quad\Gamma_{4}\cap\overline{\Omega_{+}},$		(2.30)
   	$\displaystyle p_{+}=p_{\mathrm{e}}(L,\eta):=\bar{p}_{+}+\sigma P_{\mathrm{e}}(r(L,\eta)),$		$\displaystyle\text{on}\quad\Gamma_{3},$		(2.31)

   where

   $$r(L,\eta)=\Big{(}2\int_{0}^{{\eta}}\displaystyle\frac{2t}{\rho u(L,t)}\mathrm{d}t\Big{)}^{\frac{1}{2}}.$$

   (2.32)

3. (iii).

   On the shock front $\Gamma_{\mathrm{s}}$, the Rankine-Hugoniot conditions $\eqref{eq2903}$-$\eqref{eq2906}$ hold for the states $(U_{-},U_{+})$;

4. (iv).

   Finally, on $\Gamma_{2}$, both $U_{-}$ and $U_{+}$ satisfy

   $$\theta_{-}=0,\quad\partial_{\eta}(p_{-},q_{-},s_{-})=0,\quad\partial_{\eta}^{2}\theta_{-}=0,\quad\text{on}\quad\Gamma_{2}\cap\overline{\Omega_{-}},$$

   (2.33)

   and

   $$\theta_{+}=0,\quad\partial_{\eta}(p_{+},q_{+},s_{+})=0,\quad\partial_{\eta}^{2}\theta_{+}=0,\quad\text{on}\quad\Gamma_{2}\cap\overline{\Omega_{+}}.$$

   (2.34)

In this paper, the free boundary problem FBPL is going to be solved near the background solution $\bar{U}_{\pm}$. Once it is solved, the existence of shock solutions to the problem FBPC can be established since the modified Lagrange transformation is invertible.

To solve the free boundary problem FBPL, one of the key step is to obtain information on the position of the shock front. Motivated by the ideas in [9], we are going to design a free boundary problem for the linearized Euler system based on the background shock solution $\bar{U}_{\pm}$, whose solution could serve as an initial approximation.

Assume the initial approximating position of the shock front is

$$\dot{\Gamma}_{\mathrm{s}}=\{(\xi,\eta):\xi=\dot{\xi}_{*},\,\,\,0<\eta<1\},$$

(2.35)

where $0<\dot{\xi}_{*}<L$ is unknown and will be determined later (see Figure 2.2). Then the whole domain $\Omega$ is divided by $\dot{\Gamma}_{\mathrm{s}}$ into two parts:

	$\displaystyle\dot{\Omega}_{-}=\{(\xi,\eta)\in\mathbb{R}^{2}:0<\xi<\dot{\xi}_{*},\,\,\,0<\eta<1\},$		(2.36)
	$\displaystyle\dot{\Omega}_{+}=\{(\xi,\eta)\in\mathbb{R}^{2}:\dot{\xi}_{*}<\xi<L,\,\,\,0<\eta<1\},$		(2.37)

where $\dot{\Omega}_{-}$ is considered as the initial approximating domain of the supersonic flow ahead of the shock front, and $\dot{\Omega}_{+}$ is the subsonic flow behind the shock front. Let $\dot{U}_{-}=(\dot{\theta}_{-},\dot{p}_{-},\dot{q}_{-},\dot{s}_{-})^{T}$ defined in $\dot{\Omega}_{-}$ satisfies the linearized Euler system at the supersonic state $\bar{U}_{-}$ below which will be taken as the initial approximation for the supersonic flow ahead of the shock front:

	$\displaystyle\partial_{\eta}\dot{p}_{-}+2\bar{q}_{-}\partial_{\xi}\dot{\theta}_{-}=0,$		(2.38)
	$\displaystyle\partial_{\eta}\dot{\theta}_{-}+2\displaystyle\frac{\bar{M}_{-}^{2}-1}{\bar{\rho}_{-}^{2}\bar{q}_{-}^{3}}\partial_{\xi}\dot{p}_{-}+\displaystyle\frac{1}{\eta}\dot{\theta}_{-}=0,$		(2.39)
	$\displaystyle\bar{\rho}_{-}\bar{q}_{-}\partial_{\xi}\dot{q}_{-}+\partial_{\xi}\dot{p}_{-}=0,$		(2.40)
	$\displaystyle\partial_{\xi}\dot{s}_{-}=0.$		(2.41)

Moreover, let $\dot{U}_{+}=(\dot{\theta}_{+},\dot{p}_{+},\dot{q}_{+},\dot{s}_{+})^{T}$ defined in $\dot{\Omega}_{+}$ satisfies the linearized Euler system at the subsonic state $\bar{U}_{+}$ below which will be taken as the initial approximation for the subsonic flow behind the shock front:

	$\displaystyle\partial_{\eta}\dot{p}_{+}+2\bar{q}_{+}\partial_{\xi}\dot{\theta}_{+}=0,$		(2.42)
	$\displaystyle\partial_{\eta}\dot{\theta}_{+}+2\displaystyle\frac{\bar{M}_{+}^{2}-1}{\bar{\rho}_{+}^{2}\bar{q}_{+}^{3}}\partial_{\xi}\dot{p}_{+}+\displaystyle\frac{1}{\eta}\dot{\theta}_{+}=0,$		(2.43)
	$\displaystyle\partial_{\xi}\Big{(}\bar{q}_{+}\dot{q}_{+}+\displaystyle\frac{1}{\bar{\rho}_{+}}\dot{p}_{+}+\bar{T}_{+}\dot{s}_{+}\Big{)}=0,$		(2.44)
	$\displaystyle\partial_{\xi}\dot{s}_{+}=0.$		(2.45)