On an Elliptic Free Boundary Problem and Subsonic Jet Flows for a Given Surrounding Pressure

TL;DR

This paper studies subsonic jet flows in a convergent nozzle governed by a free boundary elliptic PDE, establishing existence, uniqueness, and regularity results depending on nozzle length and flow parameters.

Contribution

It introduces a well-posedness framework for subsonic jet flows with free boundaries, identifying conditions for existence and uniqueness based on flow speed and nozzle length.

Findings

Existence of a unique subsonic jet flow for certain nozzle lengths

Identification of a minimal speed and velocity potential difference for flow space

Optimal regularity and properties of the flows

Abstract

This paper concerns compressible subsonic jet flows for a given surrounding pressure from a two-dimensional finitely long convergent nozzle with straight solid wall, which are governed by a free boundary problem for a quasilinear elliptic equation. For a given surrounding pressure and a given incoming mass flux, we seek a subsonic jet flow with the given incoming mass flux such that the flow velocity at the inlet is along the normal direction, the flow satisfies the slip condition at the wall, and the pressure of the flow at the free boundary coincides with the given surrounding pressure. In general, the free boundary contains two parts: one is the particle path connected with the wall and the other is a level set of the velocity potential. We identify a suitable space of flows in terms of the minimal speed and the maximal velocity potential difference for the well-posedness of the problem. It is shown that there is an optimal interval such that there exists a unique subsonic jet flow in the space iff the length of the nozzle belongs to this interval. Furthermore, the optimal regularity and other properties of the flows are shown.

Equations

$$\frac{\partial }{\partial x}(\rho u)+\frac{\partial }{\partial y}(\rho v)=0,\frac{\partial }{\partial x}(P+\rho u^2)+\frac{\partial }{\partial y}(\rho uv)=0,\frac{\partial }{\partial x}(\rho uv)+\frac{\partial }{\partial y}(P+\rho v^2)=0,$$

$$\displaystyle\rho(q^{2})=\Big{(}1-\displaystyle\frac{\gamma-1}{2}q^{2}\Big{)}^{1/(\gamma-1)},\quad 0<q<\sqrt{2/{(\gamma-1)}}.$$

$$\text{mbox}div(\rho (|\nabla \phi {|}^2)\nabla \phi )=0,$$

$$-{\int }_{{\Gamma }_{\mathrm{in}}}\rho (|\nabla \phi (x,y){|}^2)\nabla \phi (x,y)\cdot \nu (x,y)dl=m,$$

$$\phi (x,y)=0,$$

$$\nabla \phi (x,y)\cdot \nu (x,y)=0,$$

$$P(\rho (|\nabla \phi (x,y){|}^2))=P_e,$$

$$\phi (x,y)=\xi ,$$

$$\rho (c_l^2){\int }_{c_l}^{c_e}\frac{\rho (q^2)+2q^2{\rho }^{\prime }(q^2)}{q{\rho }^2(q^2)}dq=1.$$

$$-\tan \vartheta <W_{[R]}^{\prime }(x)<0,W_{[R]}^{\prime \prime }(x)>0,x\in (-R\cos \vartheta ,x_{[R]}),$$

$$0<J_{[R]}^{\prime }(y)<\tan \vartheta ,J_{[R]}^{\prime \prime }(y)>0,y\in (0,y_{[R]}).$$

$$c_l<|\nabla {\phi }_{[R]}(x,y)|<c_e,-\frac{\partial {\phi }_{[R]}}{\partial x}(x,y)\tan \vartheta <\frac{\partial {\phi }_{[R]}}{\partial y}(x,y)<0,(x,y)\in {\Omega }_{[R]}.$$

$$\underset{\stackrel{R\to R_1}{R\in [\hat{R},R_{\ast }]\cap [\hat{R},R_0)}}{\lim }{\Omega }_{[R]}={\Omega }_{[R_1]},$$

$$\underset{\stackrel{R\to R_1}{R\in [\hat{R},R_{\ast }]\cap [\hat{R},R_0)}}{\lim }\nabla {\phi }_{[R]}(x,y)=\nabla {\phi }_{[R_1]}(x,y)\text{mbox}uniformlyfor(x,y)\in {\Omega }_{[R_1]}.$$

$$\frac{\partial \phi }{\partial x}=u=q\cos \theta ,\frac{\partial \phi }{\partial y}=v=q\sin \theta ,\frac{\partial \psi }{\partial x}=-\rho v=-\rho q\sin \theta ,\frac{\partial \psi }{\partial y}=\rho u=\rho q\cos \theta ,$$

$$\frac{\partial \theta }{\partial \psi }+\frac{\rho (q^2)+2q^2{\rho }^{\prime }(q^2)}{q{\rho }^2(q^2)}\frac{\partial q}{\partial \phi }=0,\frac{1}{q}\frac{\partial q}{\partial \psi }-\frac{1}{\rho (q^2)}\frac{\partial \theta }{\partial \phi }=0$$

$$\frac{{\partial }^{2}A(q)}{\partial {\phi }^2}+\frac{{\partial }^{2}B(q)}{\partial {\psi }^2}=0,$$

$$A(q)={\int }_{c_{\ast }}^q\frac{\rho (s^2)+2s^2{\rho }^{\prime }(s^2)}{s{\rho }^2(s^2)}ds,B(q)={\int }_{c_{\ast }}^q\frac{\rho (s^2)}{s}ds,0<q<\sqrt{2/(\gamma -1)}.$$

$$x(s)=-R_0\cos \frac{s}{R_0},y(s)=R_0\sin \frac{s}{R_0},s\in [0,R_0\vartheta ],$$

$${\int }_0^m\frac{1}{q(0,\psi )\rho (q^2(0,\psi ))}d\psi =R_0\vartheta .$$

$$\frac{\partial A(q)}{\partial \phi }(0,\psi )=-\frac{\partial \theta }{\partial \psi }(0,\psi )=\frac{1}{R_0}{S}_{\mathrm{in}}^{\prime }(\psi )=\frac{1}{R_{0}q(0,\psi )\rho (q^2(0,\psi ))},\psi \in (0,m).$$

$$\frac{{\partial }^{2}A(q)}{\partial {\phi }^2}+\frac{{\partial }^{2}B(q)}{\partial {\psi }^2}=0,$$

$$\frac{\partial A(q)}{\partial \phi }(0,\psi )=\frac{1}{R_{0}q(0,\psi )\rho (q^2(0,\psi ))},$$

$$\frac{\partial B(q)}{\partial \psi }(\phi ,0)=0,$$

$$\frac{\partial B(q)}{\partial \psi }(\phi ,m)=0,$$

$$q(\phi ,m)=c_e,$$

$$q(\xi ,\psi )=c_e,$$

$${\int }_0^m\frac{1}{q(0,\psi )\rho (q^2(0,\psi ))}d\psi =R_0\vartheta ,$$

$$\frac{d^2}{d{\phi }^2}A(\hat{q}(\phi ))=0,\phi \in (0,\zeta ),$$

$$\frac{d}{d\phi }A(\hat{q}(0))=\frac{1}{R_0\hat{q}(0)\rho ({\hat{q}}^2(0))},\hat{q}(\zeta )=c_e,$$

$$\frac{m}{\hat{q}(0)\rho ({\hat{q}}^2(0))}=R_0\vartheta ,$$

$$\displaystyle\hat{q}(\varphi)=A^{-1}\big{(}A(c_{e})-{{\vartheta}}(\hat{\zeta}-\varphi)/m\big{)},\quad\varphi\in[0,\hat{\zeta}].$$

$$\displaystyle\hat{\varphi}(x,y)=\displaystyle\int^{R_{0}}_{\sqrt{x^{2}+y^{2}}}\hat{h}\Big{(}\displaystyle\frac{m}{{\vartheta}r}\Big{)}dr,\quad(x,y)\in\overline{\hat{\Omega}},$$

$$\displaystyle\hat{\Omega}=\big{\{}(x,y)\in\mathbb{R}^{2}:\hat{R}<\sqrt{x^{2}+y^{2}}<R_{0},x<0,0<y<-x\tan{\vartheta}\big{\}},$$

$$\frac{{\partial }^{2}Q}{\partial {\phi }^2}+\frac{{\partial }^2(b(\phi ,\psi )Q)}{\partial {\psi }^2}=0,$$

$$\frac{\partial Q}{\partial \phi }(0,\psi )=-\frac{h(\psi )}{R_0}Q(0,\psi ),Q(\xi ,\psi )=0,$$

$$\frac{\partial Q}{\partial \psi }(\phi ,0)=0,\frac{\partial Q}{\partial \psi }(\phi ,m)=0,$$

$$\displaystyle b(\varphi,\psi)=\displaystyle\int_{0}^{1}\displaystyle\frac{B^{\prime}}{A^{\prime}}\big{(}A^{-1}(tA(q_{1}(\varphi,\psi))+(1-t)A(q_{2}(\varphi,\psi)))\big{)}dt,\quad(\varphi,\psi)\in(0,\hat{\zeta})\times(0,m),$$

Full text

Keywords: Free boundary, Jet flow, Mixed boundary conditions.

2010 MR Subject Classification: 35R35 76N10 35J66

1 Introduction

In this paper we study the compressible subsonic jet flows for a given surrounding pressure. Such problems arise naturally in physical experiments and engineering designs ([7]), and have received much attention for a long time. Many examples and numerical results can be found in the monographs [7, 8, 10, 20]. In general, compressible subsonic jet flows are governed by elliptic free boundary problems. The first rigorous mathematical theory was established until 1980’s. H. W. Alt, L. A. Caffarelli and A. Friedman developed a variational approach to solve free boundary problem for elliptic equations in [1, 2, 3], which can be applied to the subsonic jet flow problems. In these works, the free boundary and the solution are obtained together by solving a minimum problem with free boundary. Furthermore, the solution on the free boundary must take the extreme value so that the variational approach works. Recently, L. L. Du et al [12, 14] used this variational approach to study impinging subsonic jets and collision of two subsonic flows. There are also many works on irrotational and rotational subsonic flows past profiles or in nozzles, which are formulated as fixed boundary problems, and we refer to [5, 6, 11, 13, 15, 16, 18, 28, 34, 35, 36] and the references therein. Continuous subsonic-sonic flows in convergent nozzles was studied in [29, 30, 31, 32, 33], where the flows are governed by free boundary problems of a degenerate elliptic equation, and the sonic curve is a free boundary where the flow velocity is along the normal direction.

It was shown in [1, 2, 3] that for a nozzle satisfying some assumptions, if the mass flux of the flow is prescribed, then there exists a unique subsonic jet flow which is infinitely long and whose pressure on the free boundary is a constant. Here, the free boundary is the particle path connected with the wall of the nozzle. Furthermore, the pressure of the subsonic jet flow on the free boundary is a constant which is determined by the flow to be found. That is to say, the surrounding pressure cannot be given in advance in these problems. In many physical problems, the surrounding pressure should be a known constant. So a natural question is how to formulate a subsonic jet flow whose pressure on the free boundary coincides with the given surrounding pressure. In this paper, we study subsonic jet flows for a given surrounding pressure from a finitely long convergent nozzle with straight solid wall. At the inlet, the incoming mass flux is a given constant and the flow angle is prescribed, which is different from [1, 2, 3] where the boundary condition at the inlet is to prescribe the stream function. If the pressure of the flow in the nozzle is greater than the surrounding pressure, it is expected that there is an accelerating subsonic jet flow whose pressure on the free boundary coincides with the given surrounding pressure. In general, one part of the free boundary should be the particle path connected with the wall of the nozzle as in [1, 2, 3]. Because the surrounding pressure is given in advance, the subsonic jet flow may be located in a bounded domain. The outlet of the subsonic jet flow is another part of the free boundary, and one should prescribe another boundary condition on the outlet except for the coincidence between the pressure of the flow and the given surrounding pressure. As in [33], this boundary condition is prescribed that the flow velocity at the outlet is along the normal direction.

Two-dimensional steady compressible fluids satisfy the Euler system:

[TABLE]

where $(u,v)$, $P$ and $\rho$ represent the velocity, pressure and density of the flow, respectively. The flow is assumed to be isentropic so that $P=P(\rho)$ is a smooth function. In particular, for a polytropic gas with adiabatic exponent $\gamma>1$, $P(\rho)=\rho^{\gamma}/\gamma$ is the normalized pressure. Assume further that the flow is irrotational. Then the density $\rho$ is expressed in terms of the speed $q$ according to the Bernoulli law ([7])

[TABLE]

The sound speed $c$ is defined as $c^{2}=P^{\prime}(\rho)$. At the sonic state, it is $c_{*}=\sqrt{2/{(\gamma+1)}}$, which is critical in the sense that the flow is subsonic ($q<c$) when $q<c_{*}$, sonic ($q=c$) when $q=c_{*}$, and supersonic ($q>c$) when $q>c_{*}$. It is well-known that the above Euler system can be transformed into the full potential equation ([7])

[TABLE]

where $\varphi$ is a velocity potential with $\nabla\varphi=(u,v)$, and $\rho$ is given by (1.1).

Assume that the nozzle is located symmetrically with respect to the $x$-axis with the vertex being $(0,0)$ and the angle at the vertex being $2{\vartheta}\in(0,\pi)$. In this paper we only consider the upper part of the subsonic jet flow due to the symmetry. The inlet of the nozzle $\Gamma_{\rm in}$ is the arc centered at $(0,0)$ with radius $R_{0}>0$. The upper wall of the nozzle $\Gamma_{\rm w}$ ends at $(-R\cos{\vartheta},R\sin{\vartheta})$ for $R\in(0,R_{0})$. For given constants $P_{e}>0$ and $m>0$, we seek a subsonic jet flow in $\Omega$, which is bounded by $\Gamma_{\rm in}$, $\Gamma_{\rm w}$, the $x$-axis, the particle path connected with the upper wall $\Gamma_{\rm ws}$ and the outlet of the flow $\Gamma_{\rm out}$, such that the incoming mass flux at $\Gamma_{\rm in}$ is $m$, the flow velocity at $\Gamma_{\rm in}$ and $\Gamma_{\rm out}$ is along the normal direction, the flow satisfies the slip condition at $\Gamma_{\rm w}\displaystyle\cup\Gamma_{\rm ws}$ and the $x$-axis, and the pressure of the flow at $\Gamma_{\rm ws}\displaystyle\cup\Gamma_{\rm out}$ is $P_{e}$, where $\Gamma_{\rm ws}$ and $\Gamma_{\rm out}$ are free. Such a subsonic jet flow problem is formulated as the following free boundary problem

[TABLE]

where $\nu$ is the unit outer normal to $\partial\Omega$, $\xi$ is a free constant, and $\Gamma_{\rm ws}\displaystyle\cup\Gamma_{\rm out}$ is a free boundary. As to the Dirichlet boundary conditions (1.5) and (1.8), it means that the flow velocity at $\Gamma_{\rm in}$ and $\Gamma_{\rm out}$ is along the normal direction. Hence both $\Gamma_{\rm in}$ and $\Gamma_{\rm out}$ are level sets of the velocity potential, and the value at $\Gamma_{\rm in}$ is normalized to be zero without loss of generality.

For the subsonic jet flow problem (1.3)–(1.8), its pressure coincides with the given surrounding pressure at the free boundary. In general, this free boundary contains two parts: one is the particle path connected with the wall and the other is a level set of the velocity potential. Both the velocity potential and the stream function do not take the extreme value on the whole free boundary. Hence the variational approach by H. W. Alt, L. A. Caffarelli and A. Friedman cannot be applied to this free boundary problem. As far as we know, there are no studies on such problems although elliptic free boundary problems have been studied extensively (see, e.g., [9, 19, 27] and the references therein). Indeed, it is hard to solve the problem (1.3)–(1.8) in the physical plane since the characteristics of the two parts of the free boundary are completely different. It is noted that $\Gamma_{\rm ws}$ and $\Gamma_{\rm out}$ are two segments in the potential-stream coordinates. So, we study the problem (1.3)–(1.8) in the potential plane since the shape but not the precise location of the free boundary is known in advance in the potential plane. However, for the subsonic jet flow problem in the potential plane, the flow satisfies a nonlinear Robin boundary condition at the inlet which causes a crucial difficulty for its well-posedness as in [33]. We need to choose a suitable space of solutions to ensure its well-posedness. Besides this difficulty, there are three new ones completely different from [33]. First of all, the subsonic jet flow problem in this paper is not a perturbed problem and there are no background solutions, while [33] concerns the structural stability of a symmetric flow. The other two new difficulties are that the free boundary contains two different parts, and mixed Dirichlet-Neumann boundary conditions are prescribed on a segment. For these difficulties, we need some new estimates and techniques completely different from [33], such as the optimal Hölder estimates for subsonic jet flows, the continuous dependence of subsonic jet flows with respect to the free boundary, the precise properties of the free boundary.

To solve the subsonic jet flow problem in the potential plane, we first study the fixed boundary problem, for which mixed Dirichlet-Neumann boundary conditions are prescribed on a segment. We use a duality argument to show the uniqueness of the solution to the fixed boundary problem. For the dual problem, mixed Dirichlet-Neumann boundary conditions are prescribed on a segment, and it may be ill-posed due to the boundary condition at the inlet. We prove the dual problem is well-posed under some restrictions on the upper bound for the velocity potential difference and the lower bound for the speed. As to the existence of solutions to the fixed boundary problem, we prescribe a Neumann boundary condition instead of the nonlinear Robin one at the inlet and then use a fixed point argument to get solutions. To do so, we also need some restrictions on the upper bound for the velocity potential difference and the lower bound for the speed. Since mixed Dirichlet-Neumann boundary conditions are prescribed on a segment, the regularity of solutions is weak at the joint point of the Dirichlet and Neumann data. Although there are many studies on mixed boundary value problems for elliptic equations (see, e.g., [4, 17, 22, 23, 24, 25] and the references therein), the optimal regularity of solutions to this problem is unknown yet. In this paper, we use a series of elaborate estimates to get the optimal Hölder continuity of solutions. Summing up, by restricting a suitable upper bound for the velocity potential difference, we show the well-posedness and the optimal regularity of solutions to the fixed boundary problem in a suitable space of solutions where there is a lower bound for the speed. With the well-posedness of the fixed boundary problem, together with the continuous dependence of its solutions, we can solve the subsonic jet flow problem in the potential plane. We identify a suitable space of flows in terms of the minimal speed and the maximal velocity potential difference, by which we get a complete classification of the nozzles whether there are subsonic jet flows or not. By studying the precise properties of the subsonic jet flows, one can transform them into the physical plane. The main result of this paper is that: For $P_{e}=P(\rho(c_{e}^{2}))\in(P(\rho(c_{*}^{2})),1/\gamma)$ and ${R_{0}}{{\vartheta}}{c_{l}}\rho(c_{l}^{2})<m<{R_{0}}{{\vartheta}}c_{e}\rho(c_{e}^{2})$ ($c_{l}\in(0,c_{e})$ is a constant), there exists a positive constant $R_{*}\in(\hat{R},R_{0}]$ with $\hat{R}={m}/{({\vartheta}c_{e}\rho(c_{e}^{2}))}$, such that the problem (1.3)–(1.8) admits a unique solution $(\varphi,\Omega)$ in the suitable space if $R\in[\hat{R},R_{*}]\displaystyle\cap[\hat{R},R_{0})$, while there is not such a solution if $R\in[0,\hat{R})\displaystyle\cup(R_{*},R_{0})$. It is noted that the bounds of the incoming mass flux are needed. For the regularity of the subsonic jet flow, it is proved that $\varphi\in C^{1,\alpha}(\overline{\Omega})$ and $\Gamma_{\rm w}\displaystyle\cup\Gamma_{\rm ws}\in C^{1+\alpha}$ for each exponent $\alpha\in(0,1/2)$, which are almost optimal. As to the geometry of the free boundary, it is shown that both $\Gamma_{\rm ws}$ and $\Gamma_{\rm out}$ are strictly convex, whose tangent lines are located at the same side as the flow. In this paper, for a class of nozzles, a given surrounding pressure and a given incoming mass flux, we solve a subsonic jet flow problem in a suitable space, which is identified for the well-posedness of the problem, and we get a complete classification of the nozzles whether there are subsonic flows in this space or not. However, there may be other subsonic jet flows not in this space, which will be dealt with in our forthcoming study. In particular, it is shown that if the angle of the nozzle is small, there is an infinite long subsonic or sonic jet flow whose free boundary is particle path connected the wall and the infinity.

The paper is arranged as follows. In $\S\,2$, we state the main results of the paper and formulate the subsonic jet flow problem into a free boundary problem in the potential plane. It is proved in $\S\,3$ that the fixed boundary problem is well-posed. Subsequently, the free boundary problem is solved in $\S\,5$.

2 Main results and formulation in the potential plane

In this section, we first state the main results of the paper (well-posedness, nonexistence and properties of solutions). Then we formulate the subsonic jet flow problem in the potential plane and introduce the spaces of solutions.

2.1 Main results

Definition 2.1

For $P_{e}>0$, $m>0$ and $0<R<R_{0}$, $(\varphi,\Omega)$ is said to be a solution to the free boundary problem (1.3)–(1.8), if $\Gamma_{\rm ws},\Gamma_{\rm out}\in C^{1}$ such that $\Gamma_{\rm w}\displaystyle\cap\Gamma_{\rm ws}=(-R\cos{\vartheta},R\sin{\vartheta})$ and $\Gamma_{\rm out}$ connects the $x$-axis and $\Gamma_{\rm ws}$, and $\varphi\in C^{2}(\overline{\Omega}\setminus\{\Gamma_{\rm w}\displaystyle\cap\Gamma_{\rm ws},\Gamma_{\rm ws}\displaystyle\cap\Gamma_{\rm out}\})\displaystyle\cap H^{2}(\Omega)\displaystyle\cap C^{1}(\overline{\Omega})$ with $0<\displaystyle\inf_{\Omega}|\nabla\varphi|\leq\displaystyle\sup_{\Omega}|\nabla\varphi|<c_{*}$ such that (1.3)–(1.8) hold, where $\Omega$ is the domain bounded by $\Gamma_{\rm in}$, $\Gamma_{\rm w}$, the $x$-axis, $\Gamma_{\rm ws}$ and $\Gamma_{\rm out}$.

The main results of the paper are the following theorems.

Theorem 2.1

For $P(\rho(c_{*}^{2}))<P_{e}<1/\gamma$ and ${R_{0}}{{\vartheta}}{c_{l}}\rho(c_{l}^{2})<m<{R_{0}}{{\vartheta}}c_{e}\rho(c_{e}^{2})$, there exists a constant $R_{*}\in(\hat{R},R_{0}]$ depending only on $R_{0}$, ${\vartheta}$, $P_{e}$, $m$ and $\gamma$, such that the problem (1.3)–(1.8) admits a unique solution $(\varphi_{[R]},\Omega_{[R]})$ with $\displaystyle\inf_{\Omega_{[R]}}|\nabla\varphi_{[R]}|\geq{c_{l}}$ and $\displaystyle\sup_{\Omega_{[R]}}\varphi_{[R]}\leq R_{0}{c_{l}}$ if $R\in[\hat{R},R_{*}]\displaystyle\cap[\hat{R},R_{0})$, while there is not such a solution if $R\in[0,\hat{R})\displaystyle\cup(R_{*},R_{0})$, where $c_{e}\in(0,c_{*})$ such that $P(\rho(c_{e}^{2}))=P_{e}$, $\hat{R}={m}/{({\vartheta}c_{e}\rho(c_{e}^{2}))}$, and ${c_{l}}\in(0,c_{e})$ is the root to

[TABLE]

Remark 2.1

The bounds of the incoming mass flux and restrictions of the solution in Theorem 2.1 are needed in this paper (see the argument in $\S$ 2.3 and $\S$ 2.4 below). Furthermore, Theorem 2.1 gives a complete classification of the nozzles whether there are subsonic flows in this space or not.

Theorem 2.2

Assume that $P(\rho(c_{*}^{2}))<P_{e}<1/\gamma$ and ${R_{0}}{{\vartheta}}{c_{l}}\rho(c_{l}^{2})<m<{R_{0}}{{\vartheta}}c_{e}\rho(c_{e}^{2})$, $(\varphi_{[R]},\Omega_{[R]})$ with $\displaystyle\inf_{\Omega_{[R]}}|\nabla\varphi_{[R]}|\geq{c_{l}}$ and $\displaystyle\sup_{\Omega_{[R]}}\varphi_{[R]}\leq R_{0}{c_{l}}$ is the solution to the problem (1.3)–(1.8) for $R\in[\hat{R},R_{*}]\displaystyle\cap[\hat{R},R_{0})$.

(i)* $\Gamma_{\rm ws}$ and $\Gamma_{\rm out}$ can be regarded as the graphs of $y=W_{[R]}(x)\,(-R\cos{\vartheta}\leq x\leq x_{[R]})$ and $x=J_{[R]}(y)\,(0\leq y\leq y_{[R]})$, respectively, where $W_{[R]}\in C^{\infty}(-R\cos{\vartheta},x_{[R]})\displaystyle\cap C^{1,1}((-R\cos{\vartheta},x_{[R]}])\displaystyle\cap C^{1,\alpha}([-R\cos{\vartheta},x_{[R]}])$ for each exponent $\alpha\in(0,1/2)$, $J_{[R]}\in C^{\infty}([0,y_{[R]}))\displaystyle\cap C^{1,1}([0,y_{[R]}])$, and*

[TABLE]

Moreover, $\Gamma_{\rm ws}=\emptyset$ if and only if $R=\hat{R}$.

(ii)* $\varphi_{[R]}\in C^{\infty}(\overline{\Omega}_{[R]}\setminus\{(-R\cos{\vartheta},R\sin{\vartheta}),(x_{[R]},y_{[R]})\})\displaystyle\cap C^{1,1}(\overline{\Omega}_{[R]}\setminus\{(-R\cos{\vartheta},R\sin{\vartheta})\})\displaystyle\cap C^{1,\alpha}(\overline{\Omega}_{[R]})$ for each exponent $\alpha\in(0,1/2)$, and*

[TABLE]

(iii)* If $R_{*}<R_{0}$ additionally, then $\displaystyle\sup_{\Omega_{[R_{*}]}}\varphi_{[R_{*}]}=R_{0}{c_{l}}$.*

(iv)* For each $R_{1}\in[\hat{R},R_{*}]\displaystyle\cap[\hat{R},R_{0})$,*

[TABLE]

(v)* For $R_{1},R_{2}\in[\hat{R},R_{*}]\displaystyle\cap[\hat{R},R_{0})$ with $R_{1}<R_{2}$, $\displaystyle\sup_{\Omega_{[R_{1}]}}\varphi_{[R_{1}]}<\displaystyle\sup_{\Omega_{[R_{2}]}}\varphi_{[R_{2}]}$.*

Remark 2.2

$\alpha\in(0,1/2)$* in Theorem 2.2 is almost optimal.*

2.2 Formulation in the potential plane

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

[TABLE]

where $\theta$ is the flow angle. The full potential equation (1.2) can be reduced to the following Chaplygin equations ([7]):

[TABLE]

in the potential-stream coordinates $(\varphi,\psi)$. And the coordinate transformations between the two coordinate systems are valid at least in the absence of stagnation points. Eliminating $\theta$ from (2.1) yields the following second-order quasilinear equation

[TABLE]

where

[TABLE]

Here, $B(\cdot)$ is strictly increasing in $(0,\sqrt{2/{(\gamma-1)}})$, while $A(\cdot)$ is strictly increasing in $(0,c_{*}]$ and strictly decreasing in $[c_{*},\sqrt{2/{(\gamma-1)}})$. We use $A^{-1}(\cdot)$ to denote the inverse function of A(\cdot)\big{|}_{(0,c_{*})} in this paper.

Assume that $(-R_{0},0)$ in the physical plane is transformed into the origin in the potential plane without loss of generality. Then $\Omega$ is transformed into $(0,\xi)\times(0,m)$. Rewrite $\Gamma_{\rm in}$ as

[TABLE]

where $s$ is the arc length of $\Gamma_{\rm in}$. Denote the coordinate transformation from $\{0\}\times[0,m]$ to $\Gamma_{\rm in}$ by $S_{\rm in}(\psi)$. Then $S_{\rm in}(0)=0$, $S_{\rm in}(m)=R_{0}{\vartheta}$, and $S^{\prime}_{\rm in}(\psi)={1}/{(q(0,\psi)\rho(q^{2}(0,\psi)))}$ for $\psi\in[0,m]$. Hence

[TABLE]

It follows from the first equation in (2.1) that

[TABLE]

Assume that the velocity potential at $\Gamma_{\rm w}\displaystyle\cap\Gamma_{\rm ws}$ is $\zeta$. It follows from (1.6) and the second equation in (2.1) that \displaystyle\frac{\partial B(q)}{\partial\psi}(\cdot,0)\Big{|}_{(0,\xi)}=\displaystyle\frac{\partial B(q)}{\partial\psi}(\cdot,m)\Big{|}_{(0,\zeta)}=0. Furthermore, (1.7) yields q(\cdot,m)\big{|}_{(\zeta,\xi)}=q(\xi,\cdot)\big{|}_{(0,m)}=c_{e}, where $c_{e}\in(0,c_{*})$ such that $P(\rho(c_{e}^{2}))=P_{e}$. Therefore, the subsonic jet flow problem (1.3)–(1.8) is formulated in the potential plane as the following free boundary problem

[TABLE]

where $0<c_{e}<c_{*}$, $m>0$ and $\zeta>0$ are given constants, while $(q,\xi)$ is the solution.

Definition 2.2

For $0<c_{e}<c_{*}$, $m>0$ and $\zeta>0$, $(q,\xi)$ is said to be a solution to the free boundary problem (2.3)–(2.9), if $q\in C^{2}([0,\xi]\times[0,m]\setminus\{(\zeta,m),(\xi,m)\})$ with $0<\displaystyle\inf_{(0,\xi)\times(0,m)}q\leq\displaystyle\sup_{(0,\xi)\times(0,m)}q<c_{*}$ and $\xi\geq\zeta$ such that (2.3)–(2.9) hold.

Solutions, subsolutions and supersolutions to the fixed boundary problem (2.3)–(2.8) can be defined similarly. Although this fixed boundary problem is uniformly elliptic, the well-posedness, regularity and continuous dependence of solutions are unknown yet.

2.3 Bounds of the incoming mass flux

If $\xi=\zeta$, then the problem (2.3)–(2.9) is symmetric, and it can be simplified into

[TABLE]

where $\zeta>0$ is free. The free boundary problem (2.10)–(2.12) admits a solution if and only if $0<m<{R_{0}}{{\vartheta}}c_{e}\rho(c_{e}^{2})$.

Lemma 2.1

For $0<c_{e}<c_{*}$ and $0<m<{R_{0}}{{\vartheta}}c_{e}\rho(c_{e}^{2})$, the free boundary problem (2.10)–(2.12) admits a unique solution $\hat{q}$ with $\zeta=\hat{\zeta}={m(A(c_{e})-A({c_{m}}))}/{\vartheta}$, where ${c_{m}}\in(0,c_{e})$ is the unique root to ${R_{0}}{{\vartheta}}{c_{m}}\rho(c_{m}^{2})=m$. More precisely, the solution is

[TABLE]

In physical plane, it is

[TABLE]

where $\hat{h}$ is the inverse function of $q\rho(q^{2})$ in $q\in(0,c_{*})$, and $\hat{R}={m}/{({\vartheta}c_{e}\rho(c_{e}^{2}))}$.

It is noted that $\hat{q}$ solves the fixed boundary problem (2.3)–(2.8) with $\xi=\zeta=\hat{\zeta}$. Assume that $q_{1}$ and $q_{2}$ are two solutions to this problem. Set $Q=A(q_{1})-A(q_{2})$. Then $Q$ solves

[TABLE]

where

[TABLE]

Its dual problem is

[TABLE]

If $q_{1}$ and $q_{2}$ are small perturbations of $\hat{q}$, then $h$ is a small perturbation of $1/{{c_{m}}}$. Therefore, the eigenvalue problem for (2.13)–(2.15) is a small perturbation of

[TABLE]

where $\lambda$ is a constant. If $\hat{\zeta}\geq{R_{0}}{c_{m}}$, it is clear that the problem (2.16)–(2.18) admits a nonpositive eigenvalue. If $\hat{\zeta}<{R_{0}}{c_{m}}$, one can prove that there is not a nontrivial solution to the problem (2.16)–(2.18) for each $\lambda\leq 0$ (the proof can be found in Proposition 3.3). So, to prove that $\hat{q}$ is the unique solution to the fixed boundary problem (2.3)–(2.8) with $\xi=\zeta=\hat{\zeta}$ by a duality argument, it is reasonable to restrict $\hat{\zeta}<{R_{0}}{c_{m}}$, which is equivalent to

[TABLE]

where $c_{l}$ is the constant given in Theorem 2.1.

Remark 2.3

If $\hat{\zeta}\geq{R_{0}}{c_{m}}$, it is unknown whether the solution to the fixed boundary problem (2.3)–(2.8) with $\xi=\zeta=\hat{\zeta}$ is unique or not. As mentioned above, $\hat{\zeta}<{R_{0}}{c_{m}}$ is a reasonable restriction when one proves the uniqueness by a duality argument.

Remark 2.4

${c_{l}}\in(0,c_{e})$* given in Theorem 2.1 satisfies $\rho(c_{l}^{2})(A(c_{e})-A({c_{l}}))=1$. Note that $\rho(q^{2})(A(c_{e})-A(q))$ is strictly decreasing in $q\in(0,c_{e})$ and $\displaystyle\lim_{q\to 0^{+}}\rho(q^{2})(A(c_{e})-A(q))=+\infty$. Hence ${c_{l}}$ is well defined.*

2.4 Spaces of solutions to the fixed and free boundary problems

In order to solve the free boundary problem (2.3)–(2.9), we first show the well-posedness of the fixed boundary problem (2.3)–(2.8) and then determine the free boundary by (2.9).

The uniqueness of the solution to the fixed boundary problem (2.3)–(2.8) will be proved by a duality argument in the paper. As shown in $\S$ 2.3, $m$ should satisfy (2.19). For such $m$, the solution $\hat{q}$ to the free boundary problem (2.10)–(2.12) satisfies $\displaystyle\inf_{(0,\hat{\zeta})}\hat{q}>c_{l}$. Hence we choose

[TABLE]

as a space of solutions to the problem (2.3)–(2.8). A similar duality argument as in $\S$ 2.3 shows that a sufficient condition for the uniqueness of the solution to the problem (2.3)–(2.8) in ${\mathscr{J}}$ is

[TABLE]

The existence of solutions in ${\mathscr{J}}$ to the problem (2.3)–(2.8) will be proved by a fixed point argument as follows: For a given ${g}\in C^{1,\alpha}([0,m])$ satisfying $c_{l}\leq\displaystyle\inf_{(0,m)}{g}\leq\displaystyle\sup_{(0,m)}{g}<c_{*}$, we solve

[TABLE]

and then define a mapping by {\mathscr{T}}({g})=q(0,\cdot)\big{|}_{[0,m]}. The problem (2.3)–(2.8) admits a solution if ${\mathscr{T}}$ has a fixed point. It is noted that

[TABLE]

are super and sub solutions to the problem (2.21)–(2.25), respectively. If $\zeta$ and $\xi$ satisfy (2.20), the comparison principle yields

[TABLE]

By this estimate and other ones, one can prove that ${\mathscr{T}}$ admits a fixed point.

Summing up, we will show the well-posedness of the fixed boundary problem (2.3)–(2.8) in ${\mathscr{J}}$ for $\zeta$ and $\xi$ satisfying (2.20). It can be checked that (2.20) holds for the symmetric flow in Lemma 2.1 if and only if $m$ satisfies (2.19). Hence, we will solve the free boundary problem (2.3)–(2.9) in the space

[TABLE]

for ${R_{0}}{{\vartheta}}{c_{l}}\rho(c_{l}^{2})<m<{R_{0}}{{\vartheta}}c_{e}\rho(c_{e}^{2})$ and $0<\zeta<R_{0}c_{l}$.

3 Well-posedness of the fixed boundary problem

In this section, we prove the well-posedness of the fixed boundary problem (2.3)–(2.8) in ${\mathscr{J}}$, where $0<c_{e}<c_{*}$, $m>0$, and $\zeta$ and $\xi$ satisfy (2.20).

3.1 Linear elliptic problem with mixed Dirichlet-Neumann boundary conditions

Assume that $m>0$, $0<\zeta\leq\xi$, $b\in C^{1}((0,\xi)\times(0,m))\displaystyle\cap H^{1}((0,\xi)\times(0,m))$ satisfying

[TABLE]

where $b_{1}\leq b_{2}$ are positive constants. Consider the following problem

[TABLE]

where $\omega\in C^{1}([0,\xi]\times[0,m])$ and $h\in L^{\infty}(0,m)$.

Definition 3.1

A function $U\in C^{2}((0,\xi)\times(0,m))\displaystyle\cap H^{1}((0,\xi)\times(0,m))\displaystyle\cap C([0,\xi]\times[0,m])$ is said to be a subsolution (supersolution, solution) to the problem (3.2)–(3.6), if

[TABLE]

for any nonnegative $\eta\in H^{1}((0,\xi)\times(0,m))\displaystyle\cap C([0,\xi]\times[0,m])$ with $\eta=0$ on $(\zeta,\xi)\times\{m\}\displaystyle\cup\{\xi\}\times(0,m)$, and U(\cdot,m)\big{|}_{(\zeta,\xi)}\leq(\geq,=)0 and U(\xi,\cdot)\big{|}_{(0,m)}\leq(\geq,=)0 hold.

Lieberman has proved the well-posedness of general linear mixed boundary value problems for elliptic equations in weighted Hölder spaces in [23, 24]. For the problem (3.2)–(3.6), we can show its optimal Hölder continuity by suitable super and sub solutions in the following proposition. Moreover, the solution is still Hölder continuous if $b\in C([0,\xi]\times[0,m])$ is relaxed to that the oscillation of $b$ near $(\zeta,m)$ is suitably small (see Proposition 3.2 below).

Proposition 3.1

Assume that $m>0$, $0<\zeta\leq\xi$, $b\in C^{1}((0,\xi)\times(0,m))\displaystyle\cap H^{1}((0,\xi)\times(0,m))\displaystyle\cap C([0,\xi]\times[0,m])$ satisfying (3.1), $\omega\in C^{1}([0,\xi]\times[0,m])$ and $h\in L^{\infty}(0,m)$.

(i)* If $\underline{U},\overline{U}\in C^{2}((0,\xi)\times(0,m))\displaystyle\cap H^{1}((0,\xi)\times(0,m))\displaystyle\cap C([0,\xi]\times[0,m])$ are sub and super solutions to the problem (3.2)–(3.6), respectively, then $\underline{U}\leq\overline{U}$ in $(0,\xi)\times(0,m)$.*

(ii)* The problem (3.2)–(3.6) admits a unique solution $U\in C^{2}((0,\xi)\times(0,m))\displaystyle\cap H^{1}((0,\xi)\times(0,m))\displaystyle\cap C^{\alpha}([0,\xi]\times[0,m])$ for each exponent $\alpha\in(0,1/2)$. Furthermore,*

[TABLE]

where $M>0$ is a constant depending only on $m$, $\zeta$, $\xi$, $\alpha$, $b$, $\|\omega\|_{L^{\infty}((0,\xi)\times(0,m))}$ and $\|h\|_{L^{\infty}(0,m)}$.

Proof. It is clear that the comparison principle holds for the problem (3.2)–(3.6). As to the existence, it is assumed that $\xi>\zeta$. The proof for the case $\xi=\zeta$ is simpler. For each positive integer $n$, choose $h_{n}\in C^{2}([0,m])$ such that $\|h_{n}-h\|_{L^{2}(0,m)}\leq 1/n$ and $\|h_{n}\|_{L^{\infty}(0,m)}\leq\|h\|_{L^{\infty}(0,m)}$, and set

[TABLE]

where $\delta=1/3\displaystyle\min\{m,\xi-\zeta\}$. Consider the following problem

[TABLE]

where $G_{n}=\{(\varphi,\psi)\in\mathbb{R}^{2}:0<\varphi<\xi,0<\psi<f_{n}(\varphi)\}$. The classical theory on elliptic equations ([21]$\S\,6.7$) yields that the problem (3.8)–(3.13) admits a unique solution $U_{n}\in C^{2}(\overline{G}_{n})$ such that

[TABLE]

where $M_{1}>0$ is a constant depending only on $m$, $\zeta$, $\xi$, $b$, $\|\omega\|_{L^{\infty}((0,\xi)\times(0,m))}$ and $\|h\|_{L^{\infty}(0,m)}$.

We now prove that $U_{n}$ is uniformly Hölder continuous. For a given exponent $\alpha\in(0,1/2)$, set

[TABLE]

where

[TABLE]

[TABLE]

$\zeta_{n}=\zeta+\delta/n$, $\lambda=\sqrt{b(\zeta,m)}$ and $\mu=1+1/{(2\alpha)}$. Then $\tilde{U}_{n}\in C^{2}(\overline{G}_{n})$ and for $(\varphi,\psi)\in\overline{G}_{n}$,

[TABLE]

[TABLE]

[TABLE]

Therefore

[TABLE]

Since $\alpha\in(0,1/2)$ and $b\in C([0,\xi]\times[0,m])$ satisfying (3.1), it follows from the choice of $\mu$ and $\lambda$ that there exist two positive constants $\tau_{1}$ and $\tau_{2}$, which depend only on $\alpha$ and $b$, such that

[TABLE]

The definition of $\beta$ yields $\sin\beta\leq-\sin(1/2-1/\mu)\pi<0$ in $(0,\xi)\times(0,m)$, which, together with (3.1) and (3.1), leads to

[TABLE]

From $b\in C([0,\xi]\times[0,m])$ and the choice of $\lambda$, there exists two constants $\tau_{3}\in(0,\tau_{1}]$ and $\tau_{4}>0$, which depend only on $\alpha$ and $b$, such that

[TABLE]

Using the comparison principle, together with (3.17) and (3.14), one gets that

[TABLE]

where $M_{2}>0$ and $\tau>0$ are suitable constants depending only on $m$, $\zeta$, $\xi$, $\alpha$, $b$, $\|\omega\|_{L^{\infty}((0,\xi)\times(0,m))}$ and $\|h\|_{L^{\infty}(0,m)}$. From (3.14) and (3.18), one can prove that the problem (3.2)–(3.6) admits a solution $U\in H^{1}((0,\xi)\times(0,m))$ satisfying

[TABLE]

where $M_{3}>0$ is a constant depending only on $m$, $\zeta$, $\xi$, $\alpha$, $b$, $\|\omega\|_{L^{\infty}((0,\xi)\times(0,m))}$ and $\|h\|_{L^{\infty}(0,m)}$. From (3.19), the Schauder theory and the Hölder estimates for elliptic equations ([21]$\S\,6.7\&8.10$), one can get that $U\in C^{2}((0,\xi)\times(0,m))\displaystyle\cap C^{\alpha}([0,\xi]\times[0,m])$ satisfying (3.7). $\hfill\Box$

Remark 3.1

If $b$ is a positive constant, then $\tilde{U}_{n}$ with $\alpha=1/2$ in the proof of Proposition 3.1 solves the homogeneous equation of (3.2). Hence the Hölder continuity in Proposition 3.1 is almost optimal.

Proposition 3.2

Assume that $m>0$, $0<\zeta\leq\xi$, $b\in C^{1}((0,\xi)\times(0,m))\displaystyle\cap H^{1}((0,\xi)\times(0,m))$ satisfying (3.1). There exist an exponent $\alpha\in(0,1/2)$ and a constant $\sigma>0$, depending only on ${b_{1}}$ and ${b_{2}}$, such that if the oscillation of $b$ near $(\zeta,m)$ is not greater than $\sigma$, then for $\omega\in C^{1}([0,\xi]\times[0,m])$ and $h\in L^{\infty}(0,m)$, the problem (3.2)–(3.6) admits a unique solution $U\in C^{2}((0,\xi)\times(0,m))\displaystyle\cap H^{1}((0,\xi)\times(0,m))\displaystyle\cap C^{\alpha}([0,\xi]\times[0,m])$.

Proof. The proof is similar to Proposition 3.1 and one needs only to construct a suitable supersolution to the problem (3.8)–(3.13). Set

[TABLE]

where $\alpha\in(0,1)$ is a constant to be determined,

[TABLE]

[TABLE]

$\zeta_{n}=\zeta+\delta/n$ and $\lambda=\sqrt{b(\zeta,m)}$. Direct calculations give that for $(\varphi,\psi)\in\overline{G}_{n}$,

[TABLE]

Choose $\alpha={b_{1}^{2}}/{(36b_{2}^{2})}$ and $\sigma={3\alpha(2-\alpha)b_{1}^{2}}/{(2b_{2})}$. If there exists a positive constant $\tau$ such that

[TABLE]

then

[TABLE]

Subsequently, one can prove the proposition similarly to Proposition 3.1. $\hfill\Box$

3.2 Comparison principle

Proposition 3.3** (Comparison principle)**

Assume that $0<c_{e}<c_{*}$, $m>0$, and $\zeta$ and $\xi$ satisfy (2.20). Let $q_{1},q_{2}\in C^{2}([0,\xi]\times[0,m]\setminus\{(\zeta,m),(\xi,m)\})\displaystyle\cap H^{1}((0,\xi)\times(0,m))\displaystyle\cap{\mathscr{J}}$ be sub and super solutions to the problem (2.3)–(2.8), respectively. Then $q_{1}\leq q_{2}$ in $(0,\xi)\times(0,m)$.

Proof. The proof is based on a duality argument. Set

[TABLE]

Then, $b\in C^{2}([0,\xi]\times[0,m]\setminus\{(\zeta,m),(\xi,m)\})\displaystyle\cap H^{1}((0,\xi)\times(0,m))\displaystyle\cap C([0,\xi]\times[0,m])$ satisfying $b_{1}\leq b\leq b_{2}$ in $(0,\xi)\times(0,m)$ with some positive constants $b_{1}\leq b_{2}$, and $h\in C^{2}([0,m])$ satisfying $1/{c_{*}}\leq h\leq 1/{c_{l}}$ in $(0,m)$.

For each nonpositive $\omega\in C^{1}([0,\xi]\times[0,m])$, consider the problem

[TABLE]

We prove the well-posedness of the problem (3.20)–(3.24) by the contraction mapping principle. Set {\mathscr{C}}=\big{\{}{\mathscr{U}}\in C([0,m]):{\mathscr{U}}\geq 0\mbox{ in }(0,m)\big{\}}. For each ${\mathscr{U}}\in{\mathscr{C}}$, it follows from Proposition 3.1 that the problem

[TABLE]

admits a unique solution $0\leq U\in C^{2}((0,\xi)\times(0,m))\displaystyle\cap H^{1}((0,\xi)\times(0,m))\displaystyle\cap C([0,\xi]\times[0,m])$. Therefore, we can define a mapping $J$ from ${\mathscr{C}}$ to itself by J({\mathscr{U}})=U(0,\cdot)\big{|}_{[0,m]}. For ${\mathscr{U}}_{1},{\mathscr{U}}_{2}\in{\mathscr{C}}$, one has J({\mathscr{U}}_{1})-J({\mathscr{U}}_{2})=\tilde{U}(0,\cdot)\big{|}_{[0,m]} with $\tilde{U}\in C^{2}((0,\xi)\times(0,m))\displaystyle\cap H^{1}((0,\xi)\times(0,m))\displaystyle\cap C([0,\xi]\times[0,m])$ solving

[TABLE]

It follows from Proposition 3.1 that the problem

[TABLE]

admits a unique solution $0\leq V\in C^{2}((0,\xi)\times(0,m))\displaystyle\cap H^{1}((0,\xi)\times(0,m))\displaystyle\cap C([0,\xi]\times[0,m])$. It is noted that

[TABLE]

are super and sub solutions to the problem (3.30)–(3.34), respectively. Proposition 3.1 shows that $J$ is a contraction mapping if $V$ satisfies

[TABLE]

Set

[TABLE]

Then $\overline{V}$ is a supersolution to the problem (3.35)–(3.39), and Proposition 3.1 leads to

[TABLE]

If $\xi<{R_{0}}c_{l}$, then (3.41) yields (3.40). Turn to the other case that $0<\zeta<\xi=R_{0}c_{l}$. It is noted that both $V$ and $\overline{V}$ solve (3.35). The Hopf Lemma yields that $V(0,\psi)<\overline{V}(0,\psi)=1$ for $\psi\in[0,m]$, which and (3.41) imply (3.40). Summing up, if $\zeta$ and $\xi$ satisfy (2.20), then $J$ is a contraction mapping. Therefore, $J$ admits a unique fixed point, and there exists a unique solution $U\in C^{2}((0,\xi)\times(0,m))\displaystyle\cap H^{1}((0,\xi)\times(0,m))\displaystyle\cap C([0,\xi]\times[0,m])$ to the problem (3.20)–(3.24). Furthermore, it follows from the classical theory on elliptic equations ([21]Theorems 6.24 and 6.30) that $U\in C^{2}([0,\xi]\times[0,m]\setminus\{(\zeta,m),(\xi,m)\})$.

For a positive integer $n$, let $0\leq\eta_{n}\in C^{2}(\mathbb{R}^{2})$ such that

[TABLE]

[TABLE]

It follows from the definition of sub and super solutions that

[TABLE]

Letting $n\to\infty$ leads to

[TABLE]

which completes the proof due to the arbitrariness of $\omega$. $\hfill\Box$

Below we prove the following result for (2.2) similar to the Hopf Lemma.

Lemma 3.1

Assume that $G\subset\mathbb{R}^{2}$ is a circle. Let $q_{1},q_{2}\in C^{2}(\overline{G})$ with $0<\displaystyle\inf_{G}q_{k}\leq\displaystyle\sup_{G}q_{k}<c_{*}\,(k=1,2)$ be sub and super solutions to

[TABLE]

respectively. If $q_{1}=q_{2}$ at a point $P_{0}\in\partial G$ and $q_{1}<q_{2}$ on $\overline{G}\setminus\{P_{0}\}$, then $\displaystyle\frac{\partial q_{1}}{\partial\nu}>\displaystyle\frac{\partial q_{2}}{\partial\nu}$ at $P_{0}$, where $\nu$ is the outer normal to $\partial G$.

Proof. Assume that $G$ is a circle centered at the origin with radius $r$. Denote $G^{*}=\{(\varphi,\psi)\in G:\varphi^{2}+\psi^{2}>r^{2}/4\}$, and set $Q=A(q_{1})-A(q_{2})$ on $\overline{G^{*}}$. Then $Q\in C^{2}(\overline{G^{*}})$ is a subsolution to the following linear equation

[TABLE]

where

[TABLE]

It holds that $b\in C^{2}(\overline{G^{*}})$ and $b_{1}\leq b\leq b_{2}$ in ${G^{*}}$ with some positive constants $b_{1}\leq b_{2}$. Similar to the proof of Proposition 3.3, one can show that the comparison principle holds for the problem of (3.43) with Dirichlet boundary condition. Set

[TABLE]

where $\beta$ is a positive constant to be determined. Direct calculations show that for $(\varphi,\psi)\in{G^{*}}$,

[TABLE]

Therefore, there exists a suitable constant $\beta>0$ such that

[TABLE]

Since $q_{1}<q_{2}$ on $\overline{G}\setminus\{P_{0}\}$, the comparison principle leads to

[TABLE]

which, together with $q_{1}=q_{2}$ at $P_{0}$, yields that $\displaystyle\frac{\partial q_{1}}{\partial\nu}>\displaystyle\frac{\partial q_{2}}{\partial\nu}$ at $P_{0}$. $\hfill\Box$

3.3 Existence of solutions

Proposition 3.4

Assume that $0<c_{e}<c_{*}$, $m>0$, and $\zeta$ and $\xi$ satisfy (2.20). There is a solution $q\in C^{\infty}([0,\xi]\times[0,m]\setminus\{(\zeta,m),(\xi,m)\})\displaystyle\cap H^{1}((0,\xi)\times(0,m))\displaystyle\cap C^{0,1}([0,\xi]\times[0,m]\setminus\{(\zeta,m)\})$ to the problem (2.3)–(2.8) such that

[TABLE]

Proof. It is assumed that $\xi>\zeta$, and the proof for the case $\xi=\zeta$ is simpler. For each positive integer $n$, consider the following problem

[TABLE]

where $f_{n}$, $G_{n}$ and $\delta$ are defined in the proof of Proposition 3.1. Set

[TABLE]

where $\alpha\in(0,1)$ and $M_{1},M_{2}>0$ are constants to be determined. For each ${g}\in{\mathscr{C}}$, consider the problem

[TABLE]

It is noted that

[TABLE]

are super and sub solutions to the problem (3.53)–(3.58), respectively. By a standard fixed point argument and the theory on elliptic equations (see, e.g., [21]$\S\,15.5$ and [26]), one can show that the problem (3.53)–(3.58) admits a unique solution ${\mathscr{Q}}\in C^{\infty}(G_{n})\displaystyle\cap C^{2}(\overline{G}_{n})$ satisfying

[TABLE]

Thanks to the Harnack inequality, there exist two constants $\tilde{\alpha}\in(0,1)$ and $\tilde{M}_{1}>0$ depending only on $m$, $\zeta$, $R_{0}$, $c_{e}$ and $\gamma$, such that

[TABLE]

The Schauder theory gives

[TABLE]

where $\tilde{M}_{2},\tilde{M}_{3}>0$ are constants depending only on $m$, $\zeta$, $R_{0}$, $c_{e}$, $\gamma$ and $M_{1}$, while $\tilde{M}_{3}$ also on $M_{2}$. Multiplying (3.53) by ${\mathscr{Q}}-c_{e}$ and then integrating over $G_{n}$ by parts, one gets from (3.54)–(3.59) that

[TABLE]

where $\tilde{M}_{4}$ is a positive constant depending only on $m$, $\xi$, $R_{0}$, $c_{e}$ and $\gamma$.

Denote ${E}(s)={A}({B}^{-1}(s))$ for $s<0$. Then ${E}^{\prime}(s)>0$, ${E}^{\prime\prime}(s)<0$ and ${E}^{\prime\prime\prime}(s)<0$ for $s<0$. Set $w=\displaystyle\frac{\partial{B}({\mathscr{Q}})}{\partial\varphi}$ and $z=\displaystyle\frac{\partial{B}({\mathscr{Q}})}{\partial\psi}$ on $\overline{G}_{n}$. Then $w,z\in C^{\infty}(G_{n})\displaystyle\cap C^{1}(\overline{G}_{n})$ solve

[TABLE]

where $p_{k}\in L^{\infty}(G_{n})\,(1\leq k\leq 6)$ are defined by

[TABLE]

and

[TABLE]

The comparison principle yields $w\geq 0$ and $z\geq 0$ on $\overline{G}_{n}$. Hence

[TABLE]

Now take $\alpha=\tilde{\alpha}$, $M_{1}=\tilde{M}_{1}$ and $M_{2}=\tilde{M}_{2}$. It follows from (3.59)–(3.61) and (3.63) that one can define a mapping $J$ from ${\mathscr{C}}$ to itself by J({g})={\mathscr{Q}}(0,\cdot)\big{|}_{[0,m]}. It follows from (3.61) that $J$ is compact. One can prove the continuity of $J$ by using its compactness and the uniqueness result for the problem (3.53)–(3.58) (see, e.g., [33]Proposition 4.7). The Schauder fixed point theorem yields that $J$ admits a fixed point. Hence there exists a solution $q_{n}\in C^{\infty}(G_{n})\displaystyle\cap C^{2}(\overline{G}_{n})$ to the problem (3.47)–(3.52). Furthermore, it follows from (3.62), (3.59) and (3.63) that $\|q_{n}\|_{H^{1}(G_{n})}\leq\tilde{M}_{4}$, and

[TABLE]

Then one can get by a standard limit process that the problem (2.3)–(2.8) admits a solution $q\in H^{1}((0,\xi)\times(0,m))$ satisfying

[TABLE]

The Schauder theory shows that $q\in C^{\infty}([0,\xi]\times[0,m]\setminus\{(\zeta,m),(\xi,m)\})\displaystyle\cap C^{0,1}([0,\xi]\times[0,m]\setminus\{(\zeta,m)\})$. Lemma 3.1, (2.3)–(2.8) and (3.65) imply (3.44) and

[TABLE]

It suffices to verify (3.45) and (3.46) for $(\varphi,\psi)\in(0,\xi)\times(0,m)$. Set $\tilde{w}=\displaystyle\frac{\partial{B}(q)}{\partial\varphi}$ and $\tilde{z}=\displaystyle\frac{\partial{B}(q)}{\partial\psi}$ in $(0,\xi)\times(0,m)$. Then $\tilde{w},\tilde{z}\in C^{\infty}([0,\xi]\times[0,m]\setminus\{(\zeta,m),(\xi,m)\})$ solve

[TABLE]

where $\tilde{p}_{k}\in C^{\infty}((0,\xi)\times(0,m))\,(1\leq k\leq 6)$ are defined similarly as $p_{k}$. The strong maximum principle and (3.65) yield $\tilde{w}>0$ and $\tilde{z}>0$ in $(0,\xi)\times(0,m)$. $\hfill\Box$

The following proposition shows the optimal Hölder continuity of solutions to the problem (2.3)–(2.8) obtained in Proposition 3.4.

Proposition 3.5

Assume that $0<c_{e}<c_{*}$, $m>0$, and $\zeta$ and $\xi$ satisfy (2.20). Let $q$ be the solution to the problem (2.3)–(2.8) obtained in Proposition 3.4. Then $q\in C^{\alpha}([0,\xi]\times[0,m])$ for each exponent $\alpha\in(0,1/2)$, and there exists a constant $M>0$ depending only on $m$, $\zeta$, $\xi$, $R_{0}$, $c_{e}$, $\gamma$ and $\alpha$ such that $\|q\|_{C^{\alpha}([0,\xi]\times[0,m])}\leq M$.

Proof. For $\tau\in(0,\delta]$, set $\Omega_{\tau}=\{(\varphi,\psi)\in\mathbb{R}^{2}:(\varphi-\zeta_{n})^{2}+(\psi-m)^{2}<\tau^{2},\psi<m\}$. Similar to the proof of Proposition 3.3, one can prove that the comparison principle holds for the following problem

[TABLE]

where $n>\delta/\tau$, $\tilde{c}\in[c_{l},c_{e}]$ is a constant, and $g\in C(\overline{G}_{n})$. Note that $A(q_{n})$ is a supersolution to the problem (3.66)–(3.69) if $A(q_{n})\geq g$ on $\partial\Omega_{\tau}\displaystyle\cap G_{n}$, where $q_{n}\in C^{\infty}(G_{n})\displaystyle\cap C^{2}(\overline{G}_{n})$ is the solution to the problem (3.47)–(3.52).

We construct subsolutions to the quasilinear problem (3.66)–(3.69) similar to the ones in the proof of Proposition 3.2. It is noted that $({B^{\prime}}/{A^{\prime}})^{\prime}>0$ in $(0,c_{*})$. For $\tilde{Q}\in C^{2}(\overline{\Omega}_{\tau}\displaystyle\cap\overline{G}_{n})$ satisfying $A(c_{l}/2)\leq\tilde{Q}\leq A(c_{e})$ in $\Omega_{\tau}\displaystyle\cap G_{n}$, one has

[TABLE]

[TABLE]

Due to (3.71), as shown in the proof of Proposition 3.2, there exist an exponent ${\alpha_{0}}\in(0,1/2)$ and a constant $\sigma>0$, which depend only on $c_{e}$ and $\gamma$, such that for

[TABLE]

satisfying

[TABLE]

it holds that

[TABLE]

where $\check{M}>0$ is a constant,

[TABLE]

[TABLE]

It follows from (3.3) and (3.74) that

[TABLE]

Since $({B^{\prime}}/{A^{\prime}})^{\prime}>0$ and $A^{\prime\prime}<0$ in $(0,c_{*})$, one has

[TABLE]

where

[TABLE]

It is clear that $\sin\beta\leq-1/2$ on $[0,\xi]\times[0,m]$. Therefore, if $\tau\leq{\check{M}}^{-1/{\alpha_{0}}}\tau_{0}$, then (3.73) holds, where \tau_{0}=\displaystyle\min\big{\{}\delta,{(c_{l}/2)^{1/{\alpha_{0}}}}/{\overline{r}},{(\sigma/\mu_{0})^{1/{\alpha_{0}}}}/{\overline{r}}\big{\}}. Summing up, for $\tilde{c}\in[c_{l},c_{e}]$, $\check{M}>0$ and $\tau\leq{\check{M}}^{-1/{\alpha_{0}}}\tau_{0}$, $\check{Q}$ given by (3.72) is a subsolution to the problem (3.66)–(3.69) if $\check{Q}\leq g$ on $\partial\Omega_{\tau}\displaystyle\cap G_{n}$.

Below, we get a lower barrier function of $q_{n}$ at $(\zeta,m)$ by using a sequence of subsolutions of the form (3.72) to the problem (3.66)–(3.69). For $n>\delta/{\tau_{0}}$, set

[TABLE]

where {c_{0}}=\displaystyle\min\big{\{}c_{e},A^{-1}\big{(}A(c_{l})+(\underline{r}\tau_{0})^{\alpha_{0}}/2\big{)}\big{\}}. Then, the above discussion shows that $\check{Q}_{0}$ is a subsolution to

[TABLE]

It follows from (3.47), (3.50), (3.51) and (3.64) that $A(q_{n})$ is a supersolution to the problem (3.75)–(3.78). Therefore,

[TABLE]

Take $\tau_{1}=2^{-1/{\alpha_{0}}}\tau_{0}$. For $n>\delta/{\tau_{1}}$, set

[TABLE]

where c_{1}=\displaystyle\min\big{\{}c_{e},A^{-1}\big{(}A({c_{0}})+(\underline{r}\tau_{1})^{\alpha_{0}}/2\big{)}\big{\}}. Then, $\check{Q}_{1}$ is a subsolution to

[TABLE]

while $A(q_{n})$ is a supersolution to this problem due to (3.47), (3.50), (3.51) and (3.79). Therefore,

[TABLE]

Take $\tau_{2}=2^{-2/{\alpha_{0}}}\tau_{0}$. For $n>\delta/{\tau_{2}}$, set

[TABLE]

where c_{2}=\displaystyle\min\big{\{}c_{e},A^{-1}\big{(}A(c_{1})+(\underline{r}\tau_{2})^{\alpha_{0}}\big{)}\big{\}}. Then, $\check{Q}_{2}$ is a subsolution to

[TABLE]

while $A(q_{n})$ is a supersolution to this problem due to (3.47), (3.50), (3.51) and (3.80). Therefore,

[TABLE]

Repeating the above process, one gets that for each positive integer $k$ and each $n>\delta/{\tau_{k}}$,

[TABLE]

where c_{k}=\displaystyle\min\big{\{}c_{e},A^{-1}\big{(}A(c_{k-1})+2^{k-2}(\underline{r}\tau_{k})^{\alpha_{0}}\big{)}\big{\}}. Note that $2^{k-2}(\underline{r}\tau_{k})^{\alpha_{0}}=(\underline{r}\tau_{0})^{\alpha_{0}}/4$ for $k=1,2,\cdots$. Therefore, there exists a positive integer $k_{0}$ depending only on $c_{e}$ and $\gamma$ such that $c_{k_{0}}=c_{e}$. Hence for each $n>\delta/{\tau_{k_{0}}}$,

[TABLE]

According to (3.64) and (3.81), one can prove that $q\in{C^{\alpha_{0}}([0,\xi]\times[0,m])}$ and

[TABLE]

where $q$ is the solution to the problem (2.3)–(2.8) obtained in Proposition 3.4, and $M_{0}>0$ is a constant depending only on $m$, $\zeta$, $\xi$, $R_{0}$, $c_{e}$ and $\gamma$. Set $Q=A(q)$ on $[0,\xi]\times[0,m]$. As the proof of (3.3), $Q$ is a supersolution to the linear equation

[TABLE]

For each exponent $\alpha\in(0,1/2)$, it follows from (3.44), (3.64), (3.82) and Proposition 3.1 that

[TABLE]

where $\tilde{M}>0$ is a constant depending only on $m$, $\zeta$, $\xi$, $R_{0}$, $c_{e}$, $\gamma$ and $\alpha$. The Hölder estimates for elliptic equations ([21]$\S\,8.10$) and (3.83) complete the proof of the proposition. $\hfill\Box$

Remark 3.2

The regularity in Proposition 3.5 is almost optimal.

4 Free boundary problem

In this section, we solve the free boundary problem (2.3)–(2.9) in ${\mathscr{S}}$ for $0<c_{e}<c_{*}$, ${R_{0}}{{\vartheta}}{c_{l}}\rho(c_{l}^{2})<m<{R_{0}}{{\vartheta}}c_{e}\rho(c_{e}^{2})$ and $0<\zeta<R_{0}c_{l}$. To do so, we need the continuous dependence of solutions to the problem (2.3)–(2.8) together with its well-posedness in $\S\,3$.

4.1 Continuous dependence of solutions

Proposition 4.1

Assume that $0<c_{e}<c_{*}$ and $m>0$. For given $\zeta_{0}$ and $\xi_{0}$ satisfying $0<\zeta_{0}<R_{0}c_{l}$ and $\zeta_{0}\leq\xi_{0}\leq R_{0}c_{l}$, it holds that

[TABLE]

where $q[\zeta,\xi]\in C^{2}([0,\xi]\times[0,m]\setminus\{(\zeta,m),(\xi,m)\})\displaystyle\cap H^{1}((0,\xi)\times(0,m))\displaystyle\cap{\mathscr{J}}$ is the solution to the fixed boundary problem (2.3)–(2.8).

Proof. It is assumed that $\xi_{0}>\zeta_{0}$, and the proof for the case $\xi_{0}=\zeta_{0}$ is similar. For convenience, we use $M_{i}\,(1\leq i\leq 6)$ to denote generic constants depending only on $m$, $\zeta_{0}$, $\xi_{0}$, $R_{0}$, $c_{e}$ and $\gamma$. Furthermore, a parenthesis after a generic constant means that this constant depends also on the variables in the parentheses.

First we prove the continuous dependence on $\xi$. Fix $(\zeta_{0}+\xi_{0})/2\leq\xi_{1}<\xi_{2}\leq R_{0}c_{l}$. Denote $q_{k}=q[\zeta_{0},\xi_{k}]$ for $k=1,2$. It follows from Propositions 3.3 and 3.4 that

[TABLE]

Set $Q=A(q_{1})-A(q_{2})$ on $[0,\xi_{1}]\times[0,m]$. Then $0\leq Q\in C^{2}([0,\xi_{1}]\times[0,m]\setminus\{(\zeta_{0},m),(\xi_{1},m)\})\displaystyle\cap H^{1}((0,\xi_{1})\times(0,m))\displaystyle\cap C([0,\xi_{1}]\times[0,m])$ solves

[TABLE]

where

[TABLE]

Consider its dual problem

[TABLE]

It follows from the proof of Proposition 3.3 that the problem (4.3)–(4.7) admits a unique nonpositive solution $U\in C^{2}([0,\xi_{1}]\times[0,m]\setminus\{(\zeta_{0},m),(\xi_{1},m)\})\displaystyle\cap H^{1}((0,\xi_{1})\times(0,m))\displaystyle\cap C([0,\xi_{1}]\times[0,m])$. The classical theory on elliptic equations ([21]Theorem 8.33) yields that

[TABLE]

Multiplying the equation of $Q$ by $U$ and then integrating over $(0,\xi_{1})\times(0,m)$ by parts, one gets from (4.1), (4.2) and (4.8) that

[TABLE]

For each $\eta\in C^{2}([0,\xi_{1}]\times[0,m])$ vanishing near $\{0\}\times[0,m]\displaystyle\cup\{(\zeta_{0},m)\}\displaystyle\cup\{\xi_{1}\}\times[0,m]$, it follows from (4.9) and the classical theory on elliptic equations ([21, Theorem 8.12]) that $\|\eta Q\|_{H^{2}((0,\xi_{1})\times(0,m))}\leq M_{1}(\eta)(\xi_{2}-\xi_{1})^{1/2}$, which leads to

[TABLE]

Set

[TABLE]

Then $0\leq Q\in C^{\infty}([0,\tilde{\zeta}]\times[0,m])$ solves

[TABLE]

Similar to the proof of Proposition 3.3, one can show that the comparison principle holds for the problem (4.11)–(4.13). Set

[TABLE]

Then

[TABLE]

[TABLE]

Hence $\bar{Q}$ is a supersolution to the problem (4.11)–(4.13). Thus

[TABLE]

which, together with (4.1), (4.10) and Proposition 3.5, leads to that

[TABLE]

Turn to the continuous dependence on $\zeta$. Fix $\zeta_{0}/2\leq\zeta_{1}<\zeta_{2}\leq(\zeta_{0}+\xi_{0})/2$. Denote $\check{q}_{k}=q[\zeta_{k},\xi_{0}]$ for $k=1,2$ and $\check{Q}=A(\check{q}_{1})-A(\check{q}_{2})$. For each exponent $\alpha\in(0,1/2)$, $\check{Q}\in C^{2}([0,\xi_{0}]\times[0,m]\setminus\{(\zeta_{1},m),(\zeta_{2},m),(\xi_{0},m)\})\displaystyle\cap H^{1}((0,\xi_{0})\times(0,m))\displaystyle\cap C^{\alpha}([0,\xi_{0}]\times[0,m])$ solves

[TABLE]

where

[TABLE]

Consider its dual problem

[TABLE]

which admits a unique solution $\check{U}\in C^{2}([0,\xi_{0}]\times[0,m]\setminus\{(\zeta_{1},m),(\xi_{0},m)\})\displaystyle\cap H^{1}((0,\xi_{0})\times(0,m))\displaystyle\cap C^{\alpha}([0,\xi_{0}]\times[0,m])$ from the proof of Proposition 3.3. For $(\varphi,\psi)\in(0,\xi_{0})\times(0,m)$ satisfying $(\varphi-(\zeta_{1}+\zeta_{2})/2)^{2}+(\psi-m)^{2}\leq(\zeta_{2}-\zeta_{1})^{2}$, it follows from the Hölder estimates for elliptic equations ([21]$\S\,8.11$) that

[TABLE]

Choose $0\leq\omega\in C^{1}([0,\xi_{0}]\times[0,m])$ such that for $(\varphi,\psi)\in(0,\xi_{0})\times(0,m)$,

[TABLE]

[TABLE]

Multiplying the equation of $\check{Q}$ by $\omega\check{U}$ and then integrating over $(0,\xi_{0})\times(0,m)$ by parts, one gets that

[TABLE]

A similar argument as (4.14) leads to $\displaystyle\lim_{\zeta_{2}-\zeta_{1}\to 0}\|\check{q}_{1}-\check{q}_{2}\|_{L^{\infty}((0,\xi_{0})\times(0,m))}=0$. $\hfill\Box$

4.2 Solvability and properties of solutions

Using Propositions 3.3, 3.4, 3.5, 4.1 and Lemma 3.1, one can prove the following three lemmas.

Lemma 4.1

Assume that $0<c_{e}<c_{*}$ and ${R_{0}}{{\vartheta}}{c_{l}}\rho(c_{l}^{2})<m<{R_{0}}{{\vartheta}}c_{e}\rho(c_{e}^{2})$.

(i)* For $0<\zeta<R_{0}c_{l}$, there is at most one solution to the problem (2.3)–(2.9) in ${\mathscr{S}}$.*

(ii)* For $\hat{\zeta}<\zeta<R_{0}c_{l}$, there is not a solution to the problem (2.3)–(2.9) in ${\mathscr{S}}$.*

Proof. For $0<\zeta<R_{0}c_{l}$, assume that $(q_{1},\xi_{1}),(q_{2},\xi_{2})\in{\mathscr{S}}$ are two solutions to the problem (2.3)–(2.9). We first prove $\xi_{1}=\xi_{2}$ by contradiction. If not, it is assumed that $\xi_{1}<\xi_{2}$ without loss of generality. Then, Propositions 3.3, 3.4, 3.5 lead to $q_{2}(\xi_{1},\psi)<c_{e}=q_{1}(\xi_{1},\psi)$ for $\psi\in(0,m)$. It follows from this estimate, Proposition 3.3 and Lemma 3.1 that $q_{2}(0,\psi)<q_{1}(0,\psi)$ for $\psi\in[0,m]$, which contradicts (2.9). Hence $\xi_{1}=\xi_{2}$. And Proposition 3.3 shows $q_{1}=q_{2}$.

For $\hat{\zeta}<\zeta<R_{0}c_{l}$, assume that $(q,\xi)\in{\mathscr{S}}$ is a solution to the problem (2.3)–(2.9). Then $\xi\geq\zeta>\hat{\zeta}$. Propositions 3.3, 3.4, 3.5 and Lemma 2.1 lead to $q(\hat{\zeta},\psi)<c_{e}=\hat{q}(\hat{\zeta})$ for $\psi\in(0,m)$, which, together with Proposition 3.3 and Lemma 3.1, shows that $q(0,\psi)<\hat{q}(0)$ for $\psi\in[0,m]$. Hence

[TABLE]

which contradicts (2.9). $\hfill\Box$

Lemma 4.2

Assume that for $\zeta_{0}\in(0,\hat{\zeta}]$, the problem (2.3)–(2.9) with $\zeta=\zeta_{0}$ admits a solution $(q_{0},\xi_{0})\in{\mathscr{S}}$ with $\xi_{0}<R_{0}c_{l}$. Then, there exists $\tau\in(0,\zeta_{0})$, such that for each $\zeta\in(\zeta_{0}-\tau,\zeta_{0})$, the problem (2.3)–(2.9) admits a solution $(q,\xi)\in{\mathscr{S}}$.

Proof. For $\zeta=\zeta_{0}$, denote $q_{1},q_{2}\in{\mathscr{J}}$ to be the solutions to the problem (2.3)–(2.8) with $\xi=R_{0}c_{l}$ and $\xi=(\xi_{0}+R_{0}c_{l})/2$, respectively. Then, Propositions 3.3, 3.4, 3.5 and Lemma 3.1 yield $c_{l}\leq q_{1}(0,\psi)<q_{2}(0,\psi)<q_{0}(0,\psi)$ for $\psi\in[0,m]$. Due to Propositions 3.3, 3.4, 3.5 and 4.1, there exists $\tau\in(0,\zeta_{0})$, such that for $\zeta\in(\zeta_{0}-\tau,\zeta_{0})$, the solution $q_{+,\zeta}\in{\mathscr{J}}$ to the problem (2.3)–(2.8) with $\xi=(\xi_{0}+R_{0}c_{l})/2$ satisfies $c_{l}\leq q_{+,\zeta}(0,\psi)<q_{0}(0,\psi)$ for $\psi\in[0,m]$. Hence

[TABLE]

For $\zeta\in(\zeta_{0}-\tau,\zeta_{0})$, Propositions 3.3, 3.4, 3.5 and Lemma 3.1 show that the solution $q_{-,\zeta}\in{\mathscr{J}}$ to the problem (2.3)–(2.8) with $\xi=\xi_{0}$ satisfies $q_{-,\zeta}(0,\psi)>q_{0}(0,\psi)$ for $\psi\in[0,m]$, which yields

[TABLE]

For $\zeta\in(\zeta_{0}-\tau,\zeta_{0})$, it follows from (4.15), (4.16), Propositions 3.4, 3.5 and 4.1 that there exists $\xi\in(\xi_{0},R_{0}c_{l})$ such that the problem (2.3)–(2.9) admits a solution $(q,\xi)\in{\mathscr{S}}$. $\hfill\Box$

Lemma 4.3

Assume that for $\zeta_{0}\in(0,\hat{\zeta})$, the problem (2.3)–(2.9) with $\zeta=\zeta_{0}$ admits a solution $(q_{0},\xi_{0})\in{\mathscr{S}}$. Then, for each $\zeta\in(\zeta_{0},\hat{\zeta})$, the problem (2.3)–(2.9) admits a solution $(q,\xi)\in{\mathscr{S}}$.

Proof. Give $\zeta\in(\zeta_{0},\hat{\zeta})$. Denote $q_{1},q_{2}\in{\mathscr{J}}$ to be the solutions to the problem (2.3)–(2.8) with $\xi=\xi_{0}$ and $\xi=\hat{\zeta}$, respectively. Then, Propositions 3.3, 3.4, 3.5 and Lemmas 2.1, 3.1 yield $c_{l}\leq q_{1}(0,\psi)<q_{0}(0,\psi)$ and $q_{2}(0,\psi)>\hat{q}(0)>c_{l}$ for $\psi\in[0,m]$. Hence

[TABLE]

Therefore, Propositions 3.4, 3.5 and 4.1 show that there exists $\xi\in(\hat{\zeta},\xi_{0})$ such that the problem (2.3)–(2.9) admits a solution $(q,\xi)\in{\mathscr{S}}$. $\hfill\Box$

We are ready to prove the following existence and nonexistence results for the free boundary problem (2.3)–(2.9) in ${\mathscr{S}}$.

Theorem 4.1

For $0<c_{e}<c_{*}$ and ${R_{0}}{{\vartheta}}{c_{l}}\rho(c_{l}^{2})<m<{R_{0}}{{\vartheta}}c_{e}\rho(c_{e}^{2})$, there exists a constant $\zeta_{*}\in[0,\hat{\zeta})$, depending only on $R_{0}$, ${\vartheta}$, $c_{e}$ and $\gamma$, such that the problem (2.3)–(2.9) admits a unique solution $(q[\zeta],\xi[\zeta])\in C^{2}([0,\xi]\times[0,m]\setminus\{(\zeta,m),(\xi,m)\})\displaystyle\cap H^{1}((0,\xi)\times(0,m))\displaystyle\cap{\mathscr{S}}$ if $\zeta\in[\zeta_{*},\hat{\zeta}]\displaystyle\cap(0,\hat{\zeta}]$, while there is not such a solution if $\zeta\in(0,\zeta_{*})\displaystyle\cup(\hat{\zeta},R_{0}c_{l})$, where ${c_{l}}$ and $\hat{\zeta}$ are given in Theorem 2.1 and Lemma 2.1, respectively. If $\zeta_{*}>0$ additionally, then $\xi[\zeta_{*}]=R_{0}c_{l}$.

Proof. Set $\zeta_{*}=\displaystyle\inf{\mathscr{E}}$, where

[TABLE]

It follows from Lemmas 4.1, 4.2, 4.3 and 2.1 that

[TABLE]

If $\zeta_{*}>0$ additionally, Proposition 4.1 and Lemma 4.2 show that $\zeta_{*}\in{\mathscr{E}}$ and $\xi[\zeta_{*}]=R_{0}c_{l}$. $\hfill\Box$

Solutions to the problem (2.3)–(2.9) have the following properties.

Theorem 4.2

Assume that $0<c_{e}<c_{*}$ and ${R_{0}}{{\vartheta}}{c_{l}}\rho(c_{l}^{2})<m<{R_{0}}{{\vartheta}}c_{e}\rho(c_{e}^{2})$. For $\zeta\in[\zeta_{*},\hat{\zeta}]\displaystyle\cap(0,\hat{\zeta}]$, let $(q[\zeta],\xi[\zeta])\in{\mathscr{S}}({c_{l}})$ be the unique solution to the problem (2.3)–(2.9), where $\zeta_{*}$, ${c_{l}}$ and $\hat{\zeta}$ are given in Theorem 4.1, Theorem 2.1 and Lemma 2.1, respectively.

(i)* $q[\zeta],\theta[\zeta]\in C^{\infty}([0,\xi]\times[0,m]\setminus\{(\zeta,m),(\xi,m)\})\displaystyle\cap H^{1}((0,\xi)\times(0,m))\displaystyle\cap C^{0,1}([0,\xi]\times[0,m]\setminus\{(\zeta,m)\})\displaystyle\cap C^{\alpha}([0,\xi]\times[0,m])$ for each exponent $\alpha\in(0,1/2)$, and*

[TABLE]

where $\theta[\zeta]$ is the flow angle of $q[\zeta]$.

(ii)* $(q[\hat{\zeta}],\xi[\hat{\zeta}])=(\hat{q},\hat{\zeta})$ with $\hat{q}$ given in Lemma 2.1.*

(iii)* If $\zeta_{*}>0$ additionally, then $\xi[\zeta_{*}]=R_{0}{c_{l}}$.*

(iv)* For each $\zeta_{0}\in[\zeta_{*},\hat{\zeta}]\displaystyle\cap(0,\hat{\zeta}]$,*

[TABLE]

(v)* For $\zeta_{1},\zeta_{2}\in[\zeta_{*},\hat{\zeta}]\displaystyle\cap(0,\hat{\zeta}]$ with $\zeta_{1}<\zeta_{2}$, it holds*

[TABLE]

where $L[\zeta]$ is the length of the upper wall of the nozzle in the physical plane for the flow $(q[\zeta],\xi[\zeta])$.

Proof. Theorem 4.1, Propositions 3.3, 3.4, 3.5 and 4.1, Lemmas 2.1 and 3.1 yield (i)–(iv) directly, and it suffices to verify (v). Give $\zeta_{1},\zeta_{2}\in[\zeta_{*},\hat{\zeta}]\displaystyle\cap(0,\hat{\zeta}]$ with $\zeta_{1}<\zeta_{2}$. For convenience, denote $q_{k}=q[\zeta_{k}]$ and $\xi_{k}=\xi[\zeta_{k}]$ for $k=1,2$. If $\xi_{1}\leq\xi_{2}$, then (i) leads to $q_{2}(\xi_{1},\psi)<c_{e}=q_{1}(\xi_{1},\psi)$ for $\psi\in(0,m)$. This estimate, Proposition 3.3 and Lemma 3.1 show that $q_{2}(0,\psi)<q_{1}(0,\psi)$ for $\psi\in[0,m]$, which contradicts (2.9). Hence $\xi_{1}>\xi_{2}$, which yields the first estimate in (4.17). It follows from $\xi_{1}>\xi_{2}$ and (i) that

[TABLE]

Set $\Gamma_{+}=[\zeta_{1},\zeta_{2})\times\{m\}$, $\Gamma_{-}=\{\xi_{2}\}\times[0,m]$, and G_{\pm}=\big{\{}(\varphi,\psi)\in[0,\xi_{2}]\times[0,m]:\pm q_{1}(\varphi,\psi)>\pm q_{2}(\varphi,\psi)\big{\}}. Then, (i) and (4.20) show that $G\pm$ are relatively open sets in $[0,\xi_{2}]\times[0,m]$ and $\Gamma_{\pm}\subset G_{\pm}$. We claim that $G_{\pm}$ are connected. If not, let $G_{\pm}^{*}$ be a maximal subdomain of $G_{\pm}$ such that $\Gamma_{\pm}\displaystyle\cap G_{\pm}^{*}=\emptyset$. Note that $q_{1}$ and $q_{2}$ satisfy the same boundary conditions on $\partial((0,\xi_{2})\times(0,m))\setminus(\Gamma_{+}\displaystyle\cup\Gamma_{-})$, $q_{1}=q_{2}$ on $\partial G_{\pm}^{*}\displaystyle\cap(0,\xi_{2})\times(0,m)$, and $G_{\pm}^{*}$ satisfy the interior cone condition. Then, the comparison principle, which can be proved similar to Proposition 3.3, leads to $\pm q_{1}\leq\pm q_{2}$ in $G_{\pm}^{*}$, which contradicts $G_{\pm}^{*}\subset G_{\pm}$. Hence

[TABLE]

Next we show that

[TABLE]

by contradiction. If not, from (4.21) and $\Gamma_{+}\subset G_{+}$, there exists a $C^{1}$ simple curve ${\mathscr{L}}_{+}^{*}\subset(0,\xi_{2})\times[0,m]\displaystyle\cup\{(0,0),(\xi_{2},0)\}$ from $[0,\xi_{2}]\times\{0\}$ to $\Gamma_{+}$ such that $q_{1}\geq q_{2}$ on ${\mathscr{L}}_{+}^{*}$ and ${\mathscr{G}}_{+}$ satisfies the interior cone condition, where {\mathscr{G}}_{+}=\big{\{}(\varphi,\psi)\in(0,\xi_{2})\times(0,m):(\varphi,\psi)\mbox{ is located at the left side of }{\mathscr{L}}_{+}^{*}\big{\}}. Using the comparison principle, which can be proved in a similar way as Proposition 3.3, one gets that $q_{1}\geq q_{2}$ on $\overline{\mathscr{G}}_{+}$. Then, Lemma 3.1 leads to $q_{1}(0,\psi)>q_{2}(0,\psi)$ for $\psi\in(0,m)$, which contradicts (2.9). Similarly, one can prove that

[TABLE]

If not, from (4.21) and $\Gamma_{-}\subset G_{-}$, there exists a $C^{1}$ simple curve ${\mathscr{L}}_{-}^{*}\subset(0,\xi_{2}]\times(0,m)\displaystyle\cup[0,\zeta_{1}]\times\{m\}$ from $\Gamma_{-}$ to $[0,\zeta_{1}]\times\{m\}$ such that $q_{1}\leq q_{2}$ on ${\mathscr{L}}_{-}^{*}$ and ${\mathscr{G}}_{-}$ satisfies the interior cone condition, where {\mathscr{G}}_{-}=\big{\{}(\varphi,\psi)\in(0,\xi_{2})\times(0,m):(\varphi,\psi)\mbox{ is located at the left side of }{\mathscr{L}}_{-}^{*}\big{\}}. Using the comparison principle and Lemma 3.1, one gets that $q_{1}(0,\psi)<q_{2}(0,\psi)$ for $\psi\in(0,m)$, which contradicts (2.9). It follows from (4.21)–(4.23) that there exists a suitable constant $\tau\in(0,m)$ such that

[TABLE]

which, together with Lemma 3.1, leads to (4.18) and (4.19). Note that

[TABLE]

So, the second estimate in (4.17) follows from the first estimate in (4.17) and (4.18). $\hfill\Box$

Transforming Theorems 4.1 and 4.2 into the physical plane, one can get Theorem 2.1 and 2.2.