The studies of transonic shock solutions for inviscid compressible flows in different kinds of nozzles had a long history and had obtained many important new achievements during the past twenty years. Courant and Friedrichs [1] had described the transonic shock phenomena in a de Laval nozzle whose cross section decreases first and then increases. It was observed in experiment that if the upcoming flow becomes supersonic after passing through the throat of the nozzle, to match the prescribed appropriately large exit pressure, a shock front intervenes at some place in the diverging part of the nozzle and the gas is compressed and slowed down to subsonic speed.

In this paper, we will consider similar transonic shock phenomena occurring in a flat nozzle when the fluid is exerted with an external force. The 2-D steady compressible isentropic Euler system with external force are of the form

$ \begin{eqnarray} \begin{cases} \partial_{x_1} (\rho u_1)+ \partial_{x_2}(\rho u_2)=0, \\ \partial_{x_1} (\rho u_1^2+P(\rho))+ \partial_{x_2}(\rho u_1 u_2)=\rho \partial_{x_1} \Phi, \\ \partial_{x_1} (\rho u_1u_2)+ \partial_{x_2}(\rho u_2^2+P(\rho))=\rho \partial_{x_2} \Phi, \end{cases} \end{eqnarray} $

(1.1)

where $ (u_1, u_2)={\bf u}:\mathbb{R}^2\rightarrow \mathbb{R}^2 $ is the unknown velocity filed and $ \rho:\mathbb{R}^2\rightarrow \mathbb{R} $ is the density, and $ \Phi(x_1, x_2) $ is a given potential function of external force. For the ideal polytropic gas, the equation of state is given by $ P(\rho)=\rho^{\gamma} $, here $ \gamma (1<\gamma<3) $ is a positive constant.

To this end, let's firstly focus on the 1-D steady compressible flow with external force on an interval $ I=[L_0, L_1] $, which is governed by

$ \begin{eqnarray} \begin{cases} (\bar{\rho} \bar{u})'(x_1)=0, \\ \bar{\rho} \bar{u} \bar{u}'+ \frac{d}{dx_1} P(\bar{\rho})= \bar{\rho} \bar{f}(x_1), \\ \bar{\rho}(L_0)=\rho_0>0, \ \ \bar{u}(L_0)= u_0>0, \end{cases} \end{eqnarray} $

(1.2)

where we assume that the flow state at the entrance $ x_1=L_0 $ is supersonic, meaning that $ u_0^2>c^2(\rho_0)=\gamma \rho_0^{\gamma-1} $.

Then it is easy to derive from (1.2) that if the external force satisfies

$ \begin{eqnarray} \bar{f}(x_1)>0, \ \ \forall L_0<x_1<L_1, \end{eqnarray} $

(1.3)

then the problem (1.2) has a global supersonic solution $ (\bar{\rho}^-, \bar{u}^-) $ on $ [L_0, L_1] $. If one prescribes a sufficiently large end pressure at $ x_1=L_1 $, a shock will form at some point $ x_1=L_s\in (L_0, L_1) $ and the gas is compressed and slowed down to subsonic speed, the gas pressure will increase to match the given end pressure. Mathematically, one looks for a shock $ x_1=L_s $ and smooth functions $ (\bar{\rho}^{\pm}, \bar{u}^{\pm}, \bar{P}^{\pm}) $ defined on $ I^+=[L_{s}, L_1] $ and $ I^-=[L_0, L_s] $ respectively, which solves (1.2) on $ I^{\pm} $ with the jump at the shock $ x_1=L_{s}\in (L_0, L_1) $ satisfying the physical entropy condition $ [\bar{P}(L_{s})]=\bar{P}^+(L_s)-\bar{P}^-(L_s)>0 $ and the Rankine-Hugoniot conditions

$ \begin{eqnarray} \begin{cases} [{\bar \rho} {\bar u}](L_s)=0, \\ [{\bar\rho} {\bar u}^2+P({\bar\rho})](L_s)=0. \end{cases} \end{eqnarray} $

(1.4)

and also the boundary conditions

$ \begin{eqnarray} &&\bar{\rho}(L_0)=\rho_{0}, \ \bar{u}(L_0)=u_{0}>0, \end{eqnarray} $

(1.5)

$ \begin{eqnarray} &&\bar{P}(L_1)= P_{e}. \end{eqnarray} $

(1.6)

We will show that there is a unique transonic shock solution to the 1-D Euler system when the end pressure $ P_e $ lies in a suitable interval. Such a problem will be solved by a shooting method employing the monotonicity relation between the shock position and the end pressure.

Lemma 1.1   Suppose that the initial state $ (u_0, \rho_0) $ at $ x_1=L_0 $ is supersonic and the external force $ f $ satisfying (1.3), there exists two positive constants $ P_0, P_1>0 $ such that if the end pressure $ P_e\in (P_1, P_0) $, there exists a unique transonic shock solution $ (\bar{u}^-, \bar{\rho}^-) $ and $ (\bar{u}^{+}, \bar{\rho}^{+}) $ defined on $ I^-=[L_0, L_s) $ and $ I^+=(L_s, L_1) $ respectively, with a shock located at $ x_1=L_{s}\in (L_0, L_1) $. In addition, the shock position $ x_1=L_s $ increases as the exit pressure $ P_{e} $ decreases. Furthermore, the shock position $ L_s $ approaches to $ L_1 $ if $ P_{e} $ goes to $ P_1 $ and $ L_s $ tends to $ L_0 $ if $ P_{e} $ goes to $ P_0 $. \end{lemma}

Remark 1   Lemma 1.1 shows that the external force helps to stabilize the transonic shock in flat nozzles and the shock position is uniquely determined.

The one dimensional transonic shock solution $ (\bar{u}^{\pm}, \bar{\rho}^{\pm}) $ with a shock occurring at $ x_1= L_s $ constructed in Lemma 1.1 will be called the background solution in this paper. The extension of the subsonic flow $ (\bar u^+(x_1), \bar\rho^+(x_1)) $ of the background solution to $ L_s-\delta_0<x_1<L_1 $ for a small positive number $ \delta_0 $ will be denoted by $ (\hat u^+(x_1), \hat\rho^+(x_1)) $.

It is natural to further consider the structural stability of this transonic shock flows. For simplicity, we only investigate the structural stability under suitable small perturbations of the end pressure. Therefore, the supersonic incoming flow is unchanged and remains to be $ (\bar{u}^-(x_1), 0, \bar{\rho}^-(x_1)) $.

Assume that the possible shock curve $ \Sigma $ and the flow behind the shock are denoted by $ x_1=\xi(x_2) $ and $ (u_1^+, u_2^+, P^+)(x) $ respectively (See Figure 1). Let $ \Omega^+=\{(x_1, x_2): \xi(x_2)<x_1<L_1, -1<x_2<1\} $ denotes the subsonic region of the flow. Then the Rankine-Hugoniot conditions on $ \Sigma $ gives

$ \begin{eqnarray} \begin{cases} [\rho u_1]- \xi'(x_2) [\rho u_2]=0, \\ [\rho u_1^2+P]- \xi'(x_2) [\rho u_1 u_2]=0, \\ [\rho u_1 u_2]- \xi'(x_2) [\rho u_2^2+P]=0. \end{cases} \end{eqnarray} $

(1.7)

In addition, the pressure $ P $ satisfies the physical entropy conditions

$ \begin{eqnarray} P^+(x)>P^-(x) \quad\rm{on}\, \Sigma. \end{eqnarray} $

(1.8)

Since the flow is tangent to the nozzle walls $ x_2=\pm 1 $, then

$ \begin{eqnarray} u_2^+(x_1, \pm 1)=0. \end{eqnarray} $

(1.9)

The end pressure is perturbed by

$ \begin{eqnarray} P^+(L_1, x_2)=P_e+\epsilon P_{ex}(x_2), \end{eqnarray} $

(1.10)

due to some technical reasons, we may readily suppose that $ P_{ex}(x_2)=P_e^{\frac{1}{\gamma}}\hat P_{ex}(x_2)\in $ $ C^{2, \alpha}([-1, 1]) $ $ (\alpha\in (0, 1)) $ satisfies the compatibility conditions

$ \begin{eqnarray} \hat P_{ex}'(\pm 1)=0. \end{eqnarray} $

(1.11)

The main results for the structural stability of transonic shock flows with external force can be stated as follows.

Theorem 1.2   Under the assumptions on the external force and the exit pressure, there exists a constant $ \epsilon_0>0 $ such that for all $ \epsilon\in (0, \epsilon_0] $, the system (1.1), (1.7)–(1.10) has a unique transonic shock solution $ (u_1^+(x), u_2^+(x), P^+(x);\xi(x_2)) $ which admits the following properties:

(ⅰ) The shock $ x_1=\xi(x_2)\in C^{3, {\alpha}}([-1, 1]) $, and satisfies

$ \begin{eqnarray} \|\xi(x_2)-L_s\|_{C^{3, {\alpha}}([-1, 1])}\leq C\epsilon, \end{eqnarray} $

(1.12)

where the positive constant $ C $ only depends on the background solution, the exit pressure and $ {\alpha} $.

(ⅱ) The velocity and pressure in subsonic region $ (u_1^+, u_2^+, P^+)(x)\in C^{2, \alpha}(\bar\Omega^+) $, and there holds

$ \begin{eqnarray} \|(u_1^+, u_2^+, P^+)(x)-(\hat u, 0, \hat P)\|_{C^{2, {\alpha}}(\bar\Omega^+)}\leq C\epsilon, \end{eqnarray} $

(1.13)

where $ \Omega^+=\{(x_1, x_2): \xi(x_2)<x_1<L_1, -1<x_2<1\} $ is the subsonic region and $ (\hat u, 0, \hat P)=(\hat u(x_1), 0, P(\hat\rho(x_1))) $ is the extended background solution.

Remark 2    It is well-known that steady Euler equations are hyperbolic-elliptic coupled in subsonic region, several different decomposition of hyperbolic and elliptic modes have been developed in [2, 3, 4]. Here we resort to a different decomposition based on the deformation and curl of the velocity developed in [5] for three dimensional steady Euler systems. The idea in that decomposition is to rewrite the density equation as a Frobenius inner product of a symmetric matrix and the deformation matrix by using the Bernoulli's law. The vorticity is resolved by an algebraic equation of the Bernoulli's function.

Remark 3    An interesting issue that deserves a further remark is when using the deformation-curl decomposition to deal with the transonic shock problem, the end pressure boundary condition becomes nonlocal since it involves the information arising from the shock front. However, this nonlocal boundary condition reduces to be local after introducing the potential function.

Remark 4    The structural stability of the transonic shock flow constructed in Lemma 1.1 under small perturbations of supersonic incoming flows and nozzle walls can be proved similarly.

The details of proofs for Lemma 1.1 and Theorem 1.2 can be found in [6].