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1.   Introduction

The Euler system

is usually used to describe the two-dimensional steady isentropic inviscid compressible flow, where $(u,v)$, $p$ and $\rho$ represent the velocity, pressure and density of the flow, respectively, and $p(\rho) = \rho^\gamma/\gamma$ for a polytropic gas with the adiabatic exponent $\gamma>1$ after the nondimensionalization. Suppose that the flow is irrotational, i.e.,

Then the density $\rho$ can be formulated as a function of the flow speed $q = \sqrt{u^2+v^2}$ according to the Bernoulli law ([2]):

The sound speed $c$ is defined as $c^2 = p'(\rho)$. At the sonic state, the flow speed is $c_* = \sqrt{{2}/{(\gamma+1)}}$, which is critical in the sense that the flow is subsonic when $q<c_*$, sonic when $q = c_*$, and supersonic when $q>c_*$. The system (1), (2) can be transformed into the full potential equation

where $\varphi$ is the velocity potential with $\nabla\varphi = (u,v)$, $\rho$ is the function given by (3). It is noted that (4) is elliptic in the subsonic region, degenerate at the sonic state, while hyperbolic in the supersonic region.

Subsonic-sonic flow is one of the most interesting aspects in the mathematical theory of compressible flows. The related problems are usually raised in physical experiments and engineering designs, and there are a lot of numerical simulations and rigorous theory involved in this field (see, e.g., [2,8,15]). Two kinds of subsonic-sonic flows have been intensively studied for decades: the flow past a profile and the flow in a nozzle. The outstanding work [1] by L. Bers proved that there exists a unique two-dimensional subsonic potential flow past a profile provided that the freestream Mach number is less than a critical value and the maximum flow speed tends to the sound speed as the freestream Mach number tends to the critical value. Later, the similar results for multi-dimensional cases were established in [13,9] by G. Dong, R. Finn and D. Gilbarg. These three works did not cover the flow with the critical freestream Mach number. It was shown in [3] based on a compensated compactness framework that the two-dimensional flow with sonic points past a profile may be realized as the weak limit of a sequence of strictly subsonic flows. However, all the subsonic-sonic flows above are obtained in the weak sense and their smoothness and uniqueness are unknown yet, so are the subsonic-sonic flows in an infinitely long nozzle. For a two-dimensional infinitely long nozzle, C. Xie et al. ([22]) proved that there exists a critical value such that a strictly subsonic flow exists uniquely as long as the incoming mass flux is less than the critical value, and a subsonic-sonic flow exists as the weak limit of a sequence of strictly subsonic flows. The multi-dimensional cases were investigated in [24,12,14]. A typical subsonic-sonic flow with precise regularity is a radially symmetric subsonic-sonic flow in a convergent straight nozzle. The structural stability was initially proved in [20] for the case of two-dimensional finitely long nozzle, and some new results can be found in [16,17,18,21,19]. In the recent decade, there are also some studies on rotational subsonic and subsonic-sonic flows, see [4,6,11,7,5,23] and the references therein.

In the present paper, we would like to investigate the subsonic-sonic flow in a class of semi-infinitely long nozzles. Assume precisely that $l_0$, $l_1>0$ and $\alpha\in(0,1)$ are constants, and $f\in C^{2,\alpha}((-\infty,0])$ satisfies

The upper and lower wall of the nozzle are described as

respectively, where $k\in(0,1]$ and

The sonic curve of the flow is a free boundary intersecting the upper wall at the origin, which is chosen as the outlet of the nozzle and is denoted by

It is assumed further that the subsonic-sonic flow satisfies the slip conditions at $\Gamma_{\rm up}$ and $\Gamma_{\rm low}$, and its velocity is along the normal direction at $\Gamma_{\rm out}$. See the following figure for an intuition.

As in [18,21], the subsonic-sonic flow problem can be formulated in the physical plane as

where $(\varphi,S)$ is a solution and $\varOmega_k$ is the semi-infinitely long nozzle bounded by $\Gamma_{\rm up}$, $\Gamma_{\rm low}$ and $\Gamma_{\rm out}$. The problem (7)–(10) is a free boundary problem of a quasilinear degenerate elliptic equation in an unbounded domain, whose degeneracy occurs at the free boundary and is characteristic. As mentioned in Remark 1.6 of [22], one can not require in advance that the flow tends to be uniformly subsonic at the far fields, otherwise, the elliptic problem may be overdetermined. In the paper, we prove that the subsonic-sonic flow in the nozzle is uniformly subsonic at the far fields, and the uniqueness of the flow results from this property. Similar to [20,18,16,17,21], we still solve the problem in the potential plane for the reason that the shape of the sonic curve is unknown in the physical plane while known in the potential plane, and the estimates of the flow speed can be made conveniently. In the potential plane, the subsonic-sonic flow problem (7)–(10) can be transformed into a quasilinear degenerate elliptic problem with free parameters and nonlocal boundary conditions in unbounded domain. The unboundedness of the domain makes the problem more difficulty than the ones in [20,18,16,17,21]. The Schauder fixed point theorem is employed to prove the existence of subsonic-sonic flows. For a given incoming mass flux and flow speed at the upper wall, we solve a fixed boundary problem of a quasilinear degenerate elliptic equation. If the solved incoming mass flux and flow speed at the upper wall are just the given ones, we get the solution. Note that the problem we concerns is in unbounded domain, we get the solution to the fixed boundary problem by taking limits of the sequences of the solutions to the truncated problems. Like that in [21], it seems very hard to construct appropriate super and sub solutions to prove the existence of solutions to truncated problems without sufficiently small $\|(-x)^{-1/2}f''\|_{L^\infty((-l_0,0))}$. The method in [21] is used here: we first solve every regularized truncated problem when the flow speed at the outlet is suitable small and get priori estimates for the average and the derivatives of the solution, then we show the existence of the solution to the regularized truncated problem by use of the preliminaries obtained above, and finally we prove that their limit as the flow speed tends to be sonic at the outlet is a desired solution to the truncated problem. The difficulty here is that in order to get the solution to the fixed boundary problem by taking the limit of the solutions to the truncated problems, we must seek a suitable variation rate $k_0$ such that the solutions to all the truncated problems exit provided that $k\in(0,k_0]$. We overcome this difficulty by constructing complicated super and sub solutions to all the truncated problems. The Harnack's inequality is used to achieve the regularities and the asymptotic behaviors of the solution to the fixed boundary problem. As to the uniqueness of the subsonic-sonic flow, we first fix the free boundaries into fixed ones and transform the nonlocal boundary conditions into common ones by a proper coordinates transformation, and then we estiblish the uniqueness theorem by the energy estimates. Summing up, it is proved in this paper that if $f$ satisfies (5) and (6), then there exists a unique subsonic-sonic flow to the problem (7)–(10) for suitably small $k$, and the flow speed is only $C^{1/2}$ Hölder continuous and the flow acceleration blows up at the sonic curve. Furthermore, the flow is uniformly subsonic at the far fields.

The paper is arranged as follows. In Section 2, we formulate the subsonic-sonic flow problem (7)–(10) in the potential plane. Then in Section 3, we solve the fixed boundary problem of a quasilinear degenerate elliptic equation in an unbounded domain. Finally in Section 4, we establish the well-posedness of the subsonic-sonic flow, and prove that the flow is uniformly subsonic at the far fields.

2.   Formulation of the subsonic-sonic flow problem in the potential plane

Define a velocity potential $\varphi$ and a stream function $\psi$, respectively, by

where $\theta$ is the flow angle. The system (1), (2) can be reduced to the Chaplygin equations ([2]):

in the potential-stream coordinates $(\varphi,\psi)$. The coordinates transformation (11) between the two coordinate systems are valid at least in the absence of stagnation points. Eliminating $\theta$ from (12) yields the following quasilinear equation of second order

where

It is obvious that $B(\cdot)$ is strictly increasing in $(0,\sqrt{2/{(\gamma-1)}})$, while $A(\cdot)$ is strictly increasing in $(0,c_*]$ and strictly decreasing in $[c_*,\sqrt{2/{(\gamma-1)}})$. It follows from [21] that there exist two constants $0<N_1<N_2$ depending only on $\gamma$ such that for $c_*/6\le q\le c_*$,

where ${E} = {A}\circ {B}^{-1}$ and $B^{-1}$ is the inverse function of $B$. We use $A^{-1}(\cdot)$ to denote the inverse function of $A(\cdot)\Big|_{(0,c_*]}$ in this paper. Additionally, the flow angle at the upper and the lower wall are

respectively.

As in [18,21], in order to describe the problem in the potential plane, we denote the flow speed at the upper wall by

then the potential function at the upper wall is expressed by

with

The inverse function of $\Phi_{\rm up}$ is denoted by $X_{\rm up}$. The subsonic-sonic flow problem (7)–(10) can be formulated in the potential plane as follows:

where $(q,m)$ is a solution with $m>0$ being the incoming mass flux. Solutions to the problem (16)–(19) are defined as follows.

Definition 2.1. For $m>0$, a function $q\in L^\infty((-\infty,0)\times(0,m))$ is called a solution to the fixed boundary problem (16)–(19), if

such that the integral equation

holds for any $\xi\in C^2((-\infty,0)\times[0,m])$ which vanishes for large $|\varphi|$ with

The existence of solutions to the problem (16)–(20) will be proved by a fixed point argument. Give $m$ and $Q_{\rm up}$ in advance as follows:

with

while $Q_{\rm up}\in C^{1/4}((-\infty,0])$ satisfies

For such $Q_{\rm up}$, it is clear that $\Phi_{\rm up}$ and $X_{\rm up}$ are well determined. Direct calculations yield that

where $\chi_{[\zeta_0,0]}(\varphi)$ is the characteristic function of the interval $[\zeta_0,0]$, and

For $f\in C^{2,\alpha}((-\infty,0])$ satisfying (5) and (6), it follows from [21] that there exists a constant $\tilde l_0\in(0,l_0)$ depending only on $f'(0)$ and $\|(-x)^{-1/2}f''\|_{L^\infty(-l_0,0)}$ such that $2f'(0)\le f'(x)\le f'(0)/2$ for $x\in[-\tilde l_0,0]$, and hence there exist two constants $0<\tau_1\le\tau_2$ depending only on $\tilde l_0$, $l_0$, $f'(0)$, $\inf_{(-l_0,-\tilde l_0)}f$ and $\sup_{(-l_0,-\tilde l_0)}f$ such that

3.   Fixed boundary problem of a quasilinear degenerate elliptic equation in an unbounded domain

In this section, we deal with the well-posedness of the fixed boundary problem. For the given $m$ and $Q_{\rm up}\in C^{1/4}((-\infty,0])$ satisfy (21) and (22), respectively, we solve the degenerate elliptic problem (16)–(19). Since the problem is in an unbounded domain, we first deal with the truncated problem in $[\zeta_0-n,0]\times[0,m]$ with any sufficient large positive integer $n$, and make some useful compact estimates. Then we solve the problem (16)–(19) by a limit process. The key of the proof is seeking the variation rate $k$, which ensures the solutions to the truncated problems exist, is independent of $n$.

3.1.   Well-posedness of the truncated problem

The truncated problem is written as

Note that (26) is degenerate at $q_n = c_*$, we replace (30) with the following boundary condition

where $c\in[c_*/3,c_*)$ is a constant, and consider the regularized truncated problem (26)–(29), (31). Then we solve the problem (26)–(30) by a limit process.

The proof can be divided into four steps.

Step 1. Well-posedness of the problem (26)–(29), (31) for $c\in[c_*/3,c_*/2]$.

Lemma 3.1. Assume that $n\geq2\delta_4+1$ and $c\in[c_*/3,c_*/2]$. There exists a constant $k_1\in(0,1]$ depending only on $\gamma$, $l_0$, $l_1$, $f(-l_0)$, $\|f'\|_{L^\infty((-l_0,0))}$ and $\|(-x)^{-1/2}f''\|_{L^\infty((-l_0,0))}$, such that if $k\in(0,k_1]$, then the problem (26)–(29), (31) admits a unique solution $q_{n,c}\in C^\infty((\zeta_0-n,0)\times(0,m))\cap C^1([\zeta_0-n,0)\times[0,m])\cap C([\zeta_0-n,0]\times[0,m])$. Furthermore, $q_{n,c}$ satisfies

Proof. The uniqueness result follows from Proposition 3.2 in [20]. Set

For $k\in(0,k_1]$, define

where

Thanks to (13), (14), (23) and (24), direct calculations show that

and

where $\chi_{[-2\delta_4,0]}(\varphi)$ is the characteristic function of the interval $[-2\delta_4,0]$. Therefore, $\overline{q}_{n,c}$ and $\underline{q}_{n,c}$ are a supersolution and a subsolution to the problem (26)–(29), (31), respectively. Thanks to the comparison principle (Proposition 3.2 in [20]) and a standard argument in the classical theory for elliptic equations, one can complete the lemma.

Step 2. A priori estimates of the average of solutions to the problem (26)–(29), (31).

Lemma 3.2. Assume that $n\geq2\delta_4+1$, $c\in[c_*/3,c_*)$ and $q_{n,c}\in C^\infty((\zeta_0-n,0)\times(0,m))\cap C^1([\zeta_0-n,0)\times[0,m])\cap C([\zeta_0-n,0)\times[0,m])$ is a solution to the problem (26)–(29), (31). Then

Furthermore, there exist three constants $k_2\in(0,1]$ and $0<\sigma_1\leq\sigma_2$ depending only on $\gamma$, $\tau_1$, $\tau_2$ and $\|f'\|_{L^\infty((-l_0,0))}$ such that if $k\in(0,k_2]$, then

Proof. The proof is similar to the proof of Lemma 3.2 in [21]. Integrating (26) over $(0,m)$ with respect to $\psi$ and using (28) and (29) show that

And (27) yields that

One gets from (6), (36) and (37) that

and

Thus (34) follows from (38). As in the proof of Lemma 3.2 in [21], it follows from (15) and (39) that

where $O(\cdot)$ depend only on $\|f'\|_{L^\infty((-l_0,0))}$. Using (25), (34) and (40), we can obtain (35).

Step 3. A priori derivative estimates of solutions to the problem (26)–(29), (31).

Lemma 3.3. Assume that $n\geq2\delta_4+1$, $c\in[c_*/3,c_*)$, and $q_{n,c}\in C^\infty((\zeta_0-n,0)\times(0,m))\cap C^1([\zeta_0-n,0)\times[0,m])\cap C([\zeta_0-n,0)\times[0,m])$ is a solution to the problem (26)–(29), (31) satisfying (32) and (33). Then for $k\in(0,1]$,

where $\sigma_3$ and $\sigma_4$ are positive constants depending only on $\gamma$, $l_0$, $l_1$, $f(-l_0)$, $\|f'\|_{L^\infty((-l_0,0))}$ and $\|(-x)^{-1/2}f''\|_{L^\infty((-l_0,0))}$.

Proof. The proof is similar to Proposition 3.2 in [20]. Set

Then $z\in C^\infty((\zeta_0-n,0)\times(0,m))\cap C([\zeta_0-n,0]\times[0,m])$ solves the problem

where $j_{i}\in C^\infty((\zeta_0,0)\times(0,m))\,(1\le i\le4)$ are defined by

and $E$ satisfies (14). It is clear that

Due to (24), one can show that

are a supersolution and a subsolution to the problem (43)–(47), respectively. The comparison principle (Proposition 3.2 in [20]) implies that

Define

It is easy to verify that $\tilde{z}_\pm$ are a supersolution and subsolution to the following problem

respectively. The comparison principle shows that

which, together with (48), leads to (41). Finally, (42) can be proved in the same way as the proof of Proposition 3.2 in [20].

Step 4. Well-posedness of the truncated problem (26)–(30).

Lemma 3.4. Assume that $n\geq2\delta_4+1$. There exists a constant $0<k_3\le\min\{k_1,\,k_2\}$ depending only on $\gamma$, $\tau_1$, $\tau_2$, $l_0$, $l_1$, $f(-l_0)$, $\|f'\|_{L^\infty((-l_0,0))}$ and $\|(-x)^{-1/2}f''\|_{L^\infty((-l_0,0))}$, such that if $k\in(0,k_3]$, then the problem (26)–(30) admits a unique solution $q_n\in C^\infty((\zeta_0-n,0)\times(0,m))\cap C^1([\zeta_0-n,0)\times[0,m])\cap C([\zeta_0-n,0]\times[0,m])$ satisfies

where $0<\sigma_5\leq\sigma_6$ are constants depending only on $\gamma$, $\tau_1$, $\tau_2$, $l_0$, $l_1$, $f(-l_0)$, $\|f'\|_{L^\infty((-l_0,0))}$ and $\|(-x)^{-1/2}f''\|_{L^\infty((-l_0,0))}$.

Proof. The uniqueness result follows from Proposition 3.2 in [20]. For $0<k\le\min\{k_1,k_2\}$, set

It follows from Lemma 3.1 and the comparison principle (Proposition 3.2 in [20]) that $\mathscr{C}_k$ is a nonempty interval. Assume that $c\in\mathscr{C}_k$. For $\varphi\in[\zeta_0-n,0]$, thanks to $c\in\mathscr{C}_k$, (13) and (35), there exists a number $\psi_\varphi\in(0,m)$ such that

which, together with (41), yields

Choose

For $0<k\le k_3$, one gets from $c\in\mathscr{C}_k$, (23) and (52) that

It follows from $c\in\mathscr{C}_k$, (31) and (42) that

Thanks to (53)–(55), one can prove from the comparison principle (Proposition 3.2 in [20]) and the continuous dependence of solutions to the problem (26)–(29), (31) that $\mathscr{C}_k = [c_*/3,c_*)$ for $0<k\le k_3$.

Let $0<k\le k_3$. For $c_*/3\le c_1<c_2<c_*$, the comparison principle (Proposition 3.2 in [20]) gives

Set

Due to (41), (42) and (53), it is clear that $q_n$ is a solution to the problem (26)–(30), and $q_n$ satisfies (49), (50) and the second inequality in (51). For $\varphi\in[\zeta_0-n,0]$, it follows from (35) and (13) that there exists a number $\tilde\psi_\varphi\in(0,m)$ such that

This estimate above and (49) yield

Hence the first inequality in (51) holds for $\sigma_6 = {(\sigma_2/{N_1})}^{1/2}+\sigma_3\delta_2$. Finally, the Schauder theory for elliptic equations shows that $q_n\in C^\infty((\zeta_0-n,0)\times(0,m))\cap C^1([\zeta_0-n,0)\times[0,m])\cap C([\zeta_0-n,0]\times[0,m])$.

3.2.   Well-posedness of the fixed boundary problem

Let us establish the existence of the solution to the problem (16)–(19).

Proposition 1. Assume that $k\in(0,k_3]$, then the problem (16)–(19) admits a solution $q\in C^\infty((-\infty,0)\times(0,m))\cap C^1((-\infty,0)\times[0,m])\cap C((-\infty,0]\times[0,m])$ satisfies

where $\sigma_3$, $\sigma_4$, $\sigma_5$ and $\sigma_6$ are given in Lemmas 3.3 and 3.4. Furthermore,

where

Proof. For any $n>2\delta_4+1$, the truncated problem (26)–(30) admits a unique solution

satisfying (49)–(51). Therefore, there exists a subsequence of $\{q_n\}$ weakly star convergenting to a function $q$ in $L^\infty((-\infty,0)\times(0,m))$, and $q$ satisfies (58). It is not hard to check that $q$ is a solution to the problem (16)–(19), and $q$ satisfies (56)–(58). Finally, the Schauder theory for elliptic equations yields that

Integrating (16) over $(0,m)$ with respect to $\psi$ and using (6), (17) and (18) lead to that

and then there exists some constant $C$ such that

which implies that

It follows from (57) and (62) that

that is,

One can get $C = 0$ by taking $\varphi\to-\infty$ in (63), and then (61) implies that

Therefore, (59) holds.

The solution to the problem (16)–(19) has the following regularity and asymptotic behavior.

Proposition 2. Assume that $q$ is a solution to the problem (16)–(19) satisfying Proposition 1. Then $q\in C^{1/2}([2\zeta_0,0]\times[0,m])$ and

where $\sigma_7$ is a positive constants depending only on $\gamma$, $\tau_1$, $\tau_2$, $l_0$, $l_1$, $f(-l_0)$, $\|f'\|_{L^\infty((-l_0,0))}$ and $\|(-x)^{-1/2}f''\|_{L^\infty((-l_0,0))}$. Moreover, it holds that

and hence

where $q_\infty$ is given in (60), and $\sigma_8,\,\sigma_9>0$ depend only on $\gamma$, $\tau_1$, $\tau_2$, $l_0$, $l_1$, $f(-l_0)$, $\|f'\|_{L^\infty((-l_0,0))}$ and $\|(-x)^{-1/2}f''\|_{L^\infty((-l_0,0))}$.

Proof. Similarly to the proof of Proposition 4.1 in [18], one can prove that $q\in C^{1/2}([2\zeta_0,0]\times[0,m])$ and satisfies (64).

In the remaining of the proof, we use $\mu_i\,(1\leq i\leq11)$ to denote a generic positive constant depending only on $\gamma$, $\tau_1$, $\tau_2$, $l_0$, $l_1$, $f(-l_0)$, $\|f'\|_{L^\infty((-l_0,0))}$ and $\|(-x)^{-1/2}f''\|_{L^\infty((-l_0,0))}$. It follows from (59) that for any $\varphi\in(-\infty,\zeta_0]$, there exists a number $\psi_\varphi\in(0,m)$ such that

which, together with (56), yields

Note that $q\in C^\infty((-\infty,0)\times(0,m))\cap C^1((-\infty,0)\times[0,m])\cap C((-\infty,0]\times[0,m])$ solves

where

Fix integer $n\geq2$. Introducing

and setting

One can verify that

solves

where

Extending the problem (68)–(70) into the domain $[3k^{-1/4}\zeta_0,-k^{-1/4}\zeta_0/2]\times[0,2m]$ yields

where for $(\check{\varphi},\check{\psi}) \in[3k^{-1/4}\zeta_0,-k^{-1/4}\zeta_0/2] \times[(i-1)m/n,im/n]\; (1\leq i\leq2n)$,

Duo to (13), (51) and (67), one gets that

and

It follows from the Hölder continuity estimates for uniformly elliptic equations that there exists a number $\beta\in(0,1)$ such that

which implies

The Schauder estimates on uniformly elliptic equations imply that

Transforming (71) into the $(\varphi,\psi)$ plane, one can get that

Similar to (67), we have from (72) that

Using (73) and the same operation on $q$ leads to that

Then the arbitrariness of $n\geq2$ leads to (65), and hence (66) holds.

Remark 1. Through the similar process of the proof of Proposition 2, one can show that for any positive integer $\lambda$, it holds that

and

where $\sigma_8',\,\sigma_9'>0$ depend only on $\lambda$, $\gamma$, $l_0$, $l_1$, $f(-l_0)$, $\|f'\|_{L^\infty((-l_0,0))}$ and $\|(-x)^{-1/2}f''\|_{L^\infty((-l_0,0))}$.

The solution to the problem (16)–(19) is also unique for small $k$.

Proposition 3. There exists a constant $k_4\in(0,1]$ depending only on $\gamma$, $\tau_1$, $\tau_2$, $l_0$, $l_1$, $f(-l_0)$, $\|f'\|_{L^\infty((-l_0,0))}$ and $\|(-x)^{-1/2}f''\|_{L^\infty((-l_0,0))}$, such that if $k\in(0,k_4]$, then the problem (16)–(19) admits at most one solution $q\in C^\infty((-\infty,0)\times(0,m)) \cap C^1((-\infty,0)\times[0,m]) \cap C((-\infty,0]\times[0,m])$ satisfying (58).

Proof. In the proof, we use $\nu_i\,(1\leq i\leq5)$ to denote a generic positive constant depending only on $\gamma$, $\tau_1$, $\tau_2$, $l_0$, $l_1$, $f(-l_0)$, $\|f'\|_{L^\infty((-l_0,0))}$ and $\|(-x)^{-1/2}f''\|_{L^\infty((-l_0,0))}$. Let $q^{(1)},\,q^{(2)}\in C^\infty((-\infty,0)\times(0,m)) \cap C^1((-\infty,0)\times[0,m]) \cap C((-\infty,0]\times[0,m])$ be two solution to the problem (16)–(19) satisfying (58). Define

Then $w_i\,(i = 1,\,2)$ solves

Set

It is easy to show that $w$ solves

where

Thanks to (56), (58), (64) and (65), direct calculations yield

where $\langle-\varphi\rangle = \min\{-\varphi,-2\zeta_0\}$. Fix $\zeta<2\zeta_0-1$. Multiplying (74) by $-w$, then integrating over $(\zeta,0)\times(0,m)$ by parts and using (75)–(77), we have

which, together with (78)–(80), yields

Then the Hölder's inequality gives

It follows from the Hölder's inequality and Cauchy inequality that

and

Substituting (82)–(84) into (81) to get

Choose $k_4 = 1/(16\nu_5^2+1)$. For any $k\in(0,k_4]$, (85) implies

Taking $\zeta\to-\infty$ in (86) to get

which implies

It follows (77) and (87) that

Therefore, $q^{(1)} = q^{(2)}$.

4.   Well-posedness of the subsonic-sonic flow problem

First we prove the existence of the solution to the problem (16)–(20) by a fixed point argument.

Theorem 4.1. Assume that $f\in C^{2,\alpha}([-l_0,0])$ satisfies (5) and (6). There exists a constant $k_0\in(0,1]$ depending only on $\gamma$, $\tau_1$, $\tau_2$, $l_0$, $l_1$, $f(-l_0)$, $\|f'\|_{L^\infty((-l_0,0))}$ and $\|(-x)^{-1/2}f''\|_{L^\infty((-l_0,0))}$, such that if $k\in(0,k_0]$, then the problem (16)–(20) admits a solution $(q,m)$ satisfying

where

and $\sigma_3$, $\sigma_4$, $\sigma_5$, $\sigma_6$ are given in Lemmas 3.3 and 3.4. Furthermore,

and for any positive integer $\lambda$, it holds that

and

where $\sigma_7$, $\sigma_8'$ and $\sigma_9'$ are given in Proposition 2 and Remark 1. Therefore, the flow is uniformly subsonic at the far fields.

Proof. Choose

For $k\in(0,k_0]$, set

with the norm

For a given $(m,Q_{\rm up})\in\mathscr{Q}$, it is clear that $\Phi_{\rm up}$, $X_{\rm up}$ and $q_\infty$ are well determined, and it follows from Propositions 1–3 that the problem (16)–(19) admits a unique solution $q\in C^\infty((-\infty,0)\times(0,m)) \cap C^1((-\infty,0)\times[0,m]) \cap C((-\infty,0]\times[0,m])$ satisfying (56)–(58) and (66). Set

From (56)–(58), (66) and the choice of $k_0$, it is easy to verify that $(\hat{m},\widehat{Q}_{\rm up})\in\mathscr{Q}$ and

is a self-mapping. Furthermore, one can prove the compactness of $\mathcal{K}$ by using (56)–(58), and the continuity of $\mathcal{K}$ by using its compactness and the uniqueness result for the problem (16)–(19). Therefore, the Schauder fixed point theorem shows that the problem (16)–(20) admits a solution $(q,m)$ such that $q\in C^\infty((-\infty,0)\times(0,m)) \cap C^1((-\infty,0)\times[0,m]) \cap C^{1/2}((-\infty,0]\times[0,m])$ satisfies (88)–(94).

From Theorem 4.1, for $k\in(0,k_0]$, the problem (16)–(20) admits a solution $(q,m)$ satisfying $q\in C^\infty((-\infty,0)\times(0,m)) \cap C^1((-\infty,0)\times[0,m]) \cap C^{1/2}((-\infty,0]\times[0,m])$,

and

where

and $M_1$, $M_2$, $M_3$ are positive constants. Indeed, this solution is also unique if $k$ is suitably small.

Theorem 4.2. Assume that $f\in C^{2,\alpha}([-l_0,0])$ satisfies (5) and (6). There exists a constant $k_0'\in(0,1]$ depending only on $\gamma$, $l_0$, $l_1$, $f(-l_0)$, $\|f'\|_{L^\infty((-l_0,0))}$, $\|(-x)^{-1/2}f''\|_{L^\infty((-l_0,0))}$, $M_1$ and $M_2$, such that if $k\in(0,k_0']$, then there is at most one solution $(q,m)$ to the problem (16)–(20) such that $q\in C^\infty((-\infty,0)\times(0,m)) \cap C^1((-\infty,0)\times[0,m]) \cap C((-\infty,0]\times[0,m])$ and $q$ satisfies (95).

Proof. In the proof, we use $C_i\,(1\leq i\leq5)$ to denote a generic positive constant depending only on $\gamma$, $l_0$, $l_1$, $f(-l_0)$, $\|f'\|_{L^\infty((-l_0,0))}$, $\|(-x)^{-1/2}f''\|_{L^\infty((-l_0,0))}$, $M_1$ and $M_2$. Let $(q^{(1)},m^{(1)})$ and $(q^{(2)},m^{(2)})$ be two solutions to the problem (16)–(20) such that $q^{(i)}\in C^\infty((-\infty,0)\times(0,m^{(i)})) \cap C^1((-\infty,0)\times[0,m^{(i)}]) \cap C((-\infty,0]\times[0,m^{(i)}])$ and satisfies (95) for $i = 1,\,2$. Denote $\Phi_{{\rm up},i}$ and $X_{{\rm up},i}$ to be the associated functions defined in Section 2 corresponding to $q^{(i)}$ for $i = 1,\,2$. For $i = 1,\,2$, introduce the new coordinates transformations

Define

Then $W_i$ satisfies

where

Set

One can verify from that $W$ satisfies

where

It follows from (13), (59), (88) and (90)–(93) that

where

Fix $L>L_0$. Multiplying (100) by $-W$ and then integrating by parts over $(-L,0)\times(0,1)$, one gets from (96)–(99) that

which, together with (23), (90), (101) and (103), yields

Below, let us make estimates on $J_i\,(1\leq i\leq 5)$ and $I_L$ in (107). The following five inequalities are necessary. From the Hölder's inequality and (99), it follows

and

Then from the Cauchy's inequality, (108) and (109), we have

and

It follows from Cauchy's inequality with $\varepsilon$, (102)–(106) and (108)–(112) that

and

where $\varepsilon>0$ is to be determined. Additionally,

Substituting (113)–(118) into (107) to get

Choose $\varepsilon = (4C_5)^{-1}$ and $k_0' = \min\{(16C_5^2+1)^{-1},\,(4C_5+1)^{-1}\}$. For any $k\in(0,k_0']$, (119) implies

Taking $L\to+\infty$ in (120), we obtain that

which shows that

Then $W(x,y) = 0$ follows from (99), and hence $(q^{(1)},m^{(1)}) = (q^{(2)},m^{(2)})$.

In terms of the physical variables, Theorems 4.1 and 4.2 can be transformed as

Theorem 4.3. Assume that $f\in C^{2,\alpha}([-l_0,0])$ satisfies (5) and (6). There exist four constants $\widetilde{k}_0\in(0,1]$ and $\widetilde{M}_1,\,\widetilde{M}_2>0$ depending only on $\gamma$, $\tau_1$, $\tau_2$, $l_0$, $l_1$, $f(-l_0)$, $\|f'\|_{L^\infty((-l_0,0))}$ and $\|(-x)^{-1/2}f''\|_{L^\infty((-l_0,0))}$, such that if $k\in(0,\widetilde{k}_0]$ then the problem (7)–(10) admits a unique solution $(\varphi,S,m)$ satisfying $\varphi\in C^3(\varOmega_k)\cap C^2(\overline{\varOmega}_k\setminus S) \cap C^1(\overline{\varOmega}_k)$, $S\in C^1([-l_1,0])$,

where ${\rm dist}_S(x,y)$ is the distance from $(x,y)$ to $S$ and $\langle x\rangle = \max\{x,\,-l_0\}$. Moreover, for any positive integer $\lambda$, there exists a constant $\widetilde{M}_3>0$ depending only on $\lambda$, $\gamma$, $\tau_1$, $\tau_2$, $l_0$, $l_1$, $f(-l_0)$, $\|f'\|_{L^\infty((-l_0,0))}$ and $\|(-x)^{-1/2}f''\|_{L^\infty((-l_0,0))}$, such that

where

Therefore, the flow is uniformly subsonic at the far fields.