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Transonic Shocks In Multidimensional Divergent Nozzles

We establish existence, uniqueness and stability of transonic shocks for steady compressible non-isentropic potential flow system in a multidimensional divergent nozzle with an arbitrary smooth cross-section, for a prescribed exit pressure. The proof is based on solving a free boundary problem for a system of partial differential equations consisting of an elliptic equation and a transport equation. In the process, we obtain unique solvability for a class of transport equations with velocity fields of weak regularity(non-Lipschitz), an infinite dimensional weak implicit mapping theorem which does not require continuous Frechet differentiability, and regularity theory for a class of elliptic partial differential equations with discontinuous oblique boundary conditions.


Introduction
We consider inviscid compressible steady flow of ideal polytropic gas. One of important but difficult subjects in the study of transonic flow is to understand a global feature of flow through a convergent-divergent nozzle so called a de Laval nozzle. According to the quasi-linear approximation in [CFR, Chapter V. section 147], given incoming subsonic flow at the entrance, the flow is always accelerated through the convergent part of the nozzle unless the exit pressure exceeds the pressure at the entrance. The flow pattern through the divergent part, however, varies depending on the exit pressure and the shape of the nozzle. In the divergent part, the flow may remain subsonic all the way to the exit, or it may be accelerated to a supersonic state after passing the throat of the nozzle and have a transonic shock across which the velocity jumps down from supersonic to subsonic. In particular, the approximation implies that if the exit pressure is given by a constant p c satisfying 0 < p min < p c < p max < ∞ for some constants p min and p max , then a transonic shock occurs in the divergent part of the nozzle. One may refer [CFR] for the quasi-linear approximation in curved nozzles. In section 2.4, for any given constant exit pressure p c ∈ (p min , p max ), we rigorously compute the transonic flow of the Euler system in a multidimensional nozzle expanding cone-shaped.
The issues above motivated recent works by several authors on existence and stability of steady transonic shocks in nozzles using the models of potential flow or compressible Euler system [CF1,CF2,CF3,XY1,CH1,YU1,CCF,XY2]. In these works transonic shocks were studied in cylindrical nozzles and its perturbations. A physically natural setup for transonic shock problem is to prescribe the parameters of flow on the nozzle entrance and the pressure on the nozzle exit. In particular, an important issue is to show that the shock location is uniquely determined by these parameters. However, such problem is not well posed in the cylindrical nozzles. Indeed, the flat shock between uniform states can be translated along the cylindrical nozzle, and this transform does not change the flow parameters on the nozzle entrance and the pressure on the nozzle exit. This provides an explicit example of non-uniqueness. Also, for uniform states in cylindrical nozzles, the flow on the nozzle entrance determines the pressure on the nozzle exit for a transonic shock solution. This degeneracy leads to non-existence, i.e. one cannot prescribe an arbitrary (even if small) perturbation of pressure on the nozzle exit. On the other hand, as the de Laval nozzle example suggests, it is natural to study transonic shocks in the diverging nozzles. This removes the translation invariance in the case of Euler system. Recent works by Liu, Yuan [LY], S.-X. Chen [CH3], and by Li, Xin, Yin [LXY] showed well-posedness of transonic shock problem with prescribed flow on entrance and pressure on exit of a divergent nozzle, which are small perturbation of the corresponding parameters of the background solution, in the case of compressible Euler system in dimension two, under some additional restrictions on the perturbed pressure on the exit. In particular, it was assumed that for the prescribed pressure on exit, the normal derivatives on the nozzle walls vanish. While this condition is not physically natural, mathematically it leads to substantial simplifications: it allows to show C 2 regularity of the shock up to the nozzle walls, which allows to work with C 1 vector fields in the transport equations.
In this paper we study transonic shocks in diverging nozzles in any dimension, for the case of perturbed cone-shaped nozzle of arbitrary cross-section. Moreover, we do not make assumptions on the pressure on the nozzle exit, other than the smallness of the perturbation. This requires to consider shocks with regularity deteriorating near the nozzle walls, and then vector fields in transport equations are of low regularity (non-Lipschitz).
Furthermore, we study this problem in the framework of potential flow. A surprising feature of this problem is that it is not well-posed for the "standard" potential flow equation. Namely, for radial solutions in a cone-shaped nozzle, the spherical transonic shock can be placed in any location between the nozzle entrance and exit, without changing the parameters of flow on the nozzle entrance and exit. This follows from a calculation by G.-Q Chen, M. Feldman [CF1,, see more details below. Thus, spherical shocks separating radial flows in the cone-shaped divergent nozzles can be translated along the nozzle for potential flow equation. This is similar to the case of flat shocks separating uniform flows in straight nozzles.
In order to remove this degeneracy, we introduce a new potential flow model, which we call non-isentropic potential flow system. It is based on the full Euler system (as opposed to the "standard" potential flow equation, which is based on the isentropic Euler system), and allows entropy jumps across the shock. The system consists of an equation of second order, which is elliptic in the subsonic regions and hyperbolic in the supersonic regions, and two transport equations.
The main purpose of this paper is to establish the existence, uniqueness and stability of a transonic shock for non-isentropic potential flow system in a multidimensional divergent nozzle with an arbitrary smooth cross-section provided that we smoothly perturb the cone-shaped divergent nozzle, incoming radial supersonic flow and the constant pressure at the exit.
In Section 2, we introduce the non-isentropic potential flow model, and define a transonic shock solution of this new model. Then we state two main theorems of this paper and present a framework of the proof for the theorems. Section 3 is devoted to a free boundary problem for the velocity potential ϕ where the exit normal velocity is fixed . In section 4, we plug a solution ϕ of the free boundary problem in section 3 to a transport equation to find the corresponding pressure p. In section 5, we prove the existence of a transonic shock solution of the non-isentropic potential flow for a fixed exit pressure p ex . The uniqueness of the transonic shock solution is proven in section 6.

Main theorems
2.1. Compressible Euler system and shocks. We study transonic shocks of steady inviscid compressible flow of ideal polytropic gas. Such flow is governed by the conservation of mass, momentum and the conservation of energy. These three conservation laws provide a system of PDEs for the density ρ, velocity u = (u 1 , · · · , u n ) and the pressure p as follows: div(ρ u) = 0 div(ρ u ⊗ u + pI n ) = 0 div ρ u( 1 2 | u| 2 + γp (γ − 1)ρ ) = 0 (2.1) where I n is n × n identity matrix. The quantity c, given by is called the sound speed. The flow type can be classified by the ratio, called the Mach number, of | u| to c. If | u| c > 1 then the flow is said supersonic, if | u| c < 1 then the flow is said subsonic. If | u| c = 1 then the flow is said sonic. Depending on entrance and exit data of flow or a shape of a channel, there may be a jump transition of the flow type across a curve or a surface. Such a transition can be understood in a weak sense as follows.
Suppose that a smooth (n-1)-dimensional surface S divides Ω into two disjoint subsets Ω ± . If (ρ, u, p) ∈ L 1 loc (Ω) is in C 1 (Ω ± ) ∩ C 0 (Ω ± ) then (ρ, u, p) is a weak solution of (2.1) if and only if (ρ, u, p) satisfies (2.1) in each of Ω ± and the Rankine-Hugoniot jump conditions(which is abbreviated as R-H conditions hereafter) [ρ u · ν s ] S = 0 (2.4) [ρ( u · ν s ) u + pν s ] S = 0 (2.5) for a unit normal ν s on S where [F ] S is defined by To study a transonic shock in R n for n ≥ 2, we consider an irrotational flow in which the velocity u has an expression of u = ∇ϕ for a scalar function ϕ. Such a flow is called a potential flow and ϕ is called a (velocity) potential.
2.2. Steady isentropic potential flow equation and radial transonic shocks. A widely used steady potential flow model(see e.g. [CF1,XY1] et al.) consists of the conservation law of mass, the Bernoulli law for the velocity. So it can be written into the following second-order, nonlinear elliptic-hyperbolic equation of mixed type for the velocity potential ϕ : Ω ⊂ R n → R 1 : div(ρ(|∇ϕ| 2 )∇ϕ) = 0, (2.7) where, with a particular choice of the Bernoulli's invariant, the density function ρ(q 2 ) is (2.8) We note that equation (2.7) can be derived formally from the isentropic Euler system. Steady isentropic Euler system is obtained from (2.1) by setting p = κρ γ , where κ > 0 is a constant, and dropping the third line in (2.1). Then equation (2.7) can be obtained by a formal calculation, substituting u = ∇ϕ into the isentropic Euler system, and choosing κ = 1 γ . The second-order nonlinear equation (2.7) is strictly elliptic if |∇ϕ| > c, and is strictly hyperbolic if |∇ϕ| < c (2.9) with c = γp ρ = ρ γ−1 . The elliptic regions of the equation (2.7) correspond to the subsonic flow, and the hyperbolic regions of (2.7) to the supersonic flow.
Note that ∂ r ϕ ± 0 > 0. Thus flow enters Ω through {|x| = r 1 } and exits Ω through {|x| = r 2 }. By part(iii) above, ∇ϕ does not change on the entrance and exit when the position r s of the shock changes within (r 1 , r 2 ). Since u = ∇ϕ determines the density ρ by (2.8) and pressure p = κρ γ , it follows that parameters of the flow on the entrance and exist of the annulus-shaped domain Ω do not determine position r = r s of the shock.
Restricting the annulus-shaped domain Ω to a cone-shaped (nozzle) domain N as in Section 2.4, we see that the the same example shows non-uniqueness of the shock position in the cone-shaped nozzle.
Thus we propose a new potential model, to fix this degeneracy.
2.3. The non-isentropic potential flow system and transonic shocks. The new model, which we call the non-isentropic potential flow system, is obtained by substituting u = ∇ϕ into the the full steady compressible Euler system (2.1). Unknown functions in the new model are the scalar functions (ϕ, ρ, p). Note that second line of (2.1), representing conservation of momentum, consists of n equations. Thus we have overdeterminacy in the case of potential model, i.e. we need to retain one equation from the conservation of momentum equations. For that, we note that the component in the direction of u of the vector determined by the second line of (2.1) represents conservation of entropy along streamlines (in the smooth flow regions). This motivates us to introduce the following non-isentropic potential flow system: (2.13) (2.11) and (2.13) represent the conservation of mass and energy respectively, and (2.12) concerns the conservation of the entropy along each streamline but it allows the entropy to change between streamlines. Since (2.12) is not in a divergence form, we cannot define a shock solution for this new model in sense of (2.3). Instead, we employ all the R-H conditions (2.4)-(2.6) to define a shock solution of the new model as follows: Definition 1 (Shock solutions). (ρ, ϕ, p) is a shock solution of the non-isentropic potential flow in Ω with a shock S if where the Bernoulli's invariant B is defined by An explicit computation shows that, as a second-order PDE for ϕ, (2.14) with (2.17) is strictly elliptic if |∇ϕ| < c, and is strictly hyperbolic if |∇ϕ| > c with c defined in (2.2). Note that this is similar to (2.9).
Let τ s be a tangent vector field on S, then, by (2.5), there holds Remark 2.1. The specific entropy ǫ of ideal polytropic gas satisfies the constitutive relation κ exp(ǫ/c v ) = p ρ γ for constants κ and c v . A straightforward calculation using (2.4),(2.6),(2.19) and (2.20) indicates that the entropy ǫ increases across a shock S for the non-isentropic potential flow model. The entropy jump across a shock plays an essential role for the well-posedness of the problem considered in this paper.
Regarding Λ as a n-1 dimensional submanifold of R n−1 , let x ′ = (x ′ 1 ,· · · ,x ′ n−1 ) be a coordinate system in Λ. Particularly, if Λ S n−1 , then Λ can be described by a single smooth diffeomorphism(e.g. stereographic projection), say, T i.e., there is Λ T ⊂ R n−1 with a smooth boundary so that Λ T can be described by (2.21) Then, for any x ∈ N , there exists unique x ′ satisfying x |x| = T (x ′ ). For convenience, we denote as To simplify notations, we denote as x ′ ∈ Λ instead of x ′ ∈ T Λ hereafter. The coordinates (r, x ′ ) in (2.22) is regarded as a spherical coordinate system in N . For later use, we set To simplify notations, for a function ψ(x) defined for x ∈ N , if x = T (r, x ′ ) then we identify ψ(r, x ′ ) with ψ(x) without any further specification for the rest of paper. Similarly, any function . Furthermore, any function ϕ 0 varying only by the radial variable |x|, we write as ϕ 0 (r). Also, in (r, x ′ ) coordinates, we denote as where ν w is the unit normal on Γ w toward the interior of N . Since (2.11)-(2.13) are invariant under rotations, we expect (ρ 0 , ϕ 0 , p 0 ) to be functions of r only. Then (2.25) automatically holds on Γ w . For a continuously differentiable radial solution (ρ, ϕ, p), if ϕ r = 0 in N , then (2.11)-(2.13) are equivalent to (2.26) (2.26) can be considered as a system of ODEs for (ϕ r , ρ) because of . (2.27) To find (ϕ r , p), we rewrite (2.26) as Fix r s ∈ (r 0 , r 1 ), and let us find a radial transonic shock solution with the shock on {r = r s }. For that purpose, we need to solve (2.28) and (2.29) in two separate regions of supersonic state and subsonic state.
Another important property of the background solutions is the monotonicity of the exit values of (ρ 0 , ∂ r ϕ 0 , p 0 ) depending on the location of shocks r s . In [YU2], it is proven that, for the full Euler system, ρ + 0 (r 1 ; r s ) and p + 0 (r 1 ; r s ) monotonically decrease with respect to r s while ∂ r ϕ + 0 (r 1 ; r s ) monotonically increases. If ∂ r ϕ 0 (r; r s ) = 0 in N , then the non-isentropic potential flow model is equivalent to the full Euler system for smooth flow. So the monotonicity properties of the exit data applies to the background solution (ρ 0 , ϕ 0 , p 0 ) of the non-isentropic potential flow model so we have: Proposition 2.4 ([YU2, Theorem 2.1]). Let us set p min = p + 0 (r 1 ; r 1 ) and p max := p + 0 (r 1 ; r 0 ). Then p min < p max holds. Moreover, for any constant p c ∈ (p min , p max ), there exists unique r * s ∈ (r 0 , r 1 ) so that the background solution (ρ 0 , ϕ 0 , p 0 )(r; r * s ) satisfies p + 0 (r 1 ; r * s ) = p c . Readers can refer [YU2] for the proof although we would like to point out that the following lemma, which will be used later in this paper, plays an important role for the proof of Proposition 2.4.
2.5. Main theorems. According to Proposition 2.4, for any given constant p c ∈ (p min , p max ), there exists a transonic shock solution whose exit pressure is p c . Our goal is to achieve the existence of a transonic shock solution when we smoothly perturb incoming supersonic flow, the exit pressure and the nozzle N . Let N be a nozzle smoothly perturbed from N by a diffeomorphism Ψ : N → R n , and (ρ,φ,p) be a transonic shock solution of a non-isentropic potential flow in N with a shock S. Then S separates N into a supersonic region N − and a subsonic region N + . Let us denote (ρ,φ,p)| N ± as (ρ ± ,φ ± ,p ± ). If the entrance data for (ρ,φ,p) is given so that the Bernoulli's invariant B is a constant at the entrance of the nozzle N then, by (2.4), (2.6) and (2.16), (ρ ± ,φ ± ,p ± ) is a solution to the system of (2.14), (2.15) and where the constant B 0 > 0 is determined by the entrance data. On the shock S, (2.19) and (2.38) imply for a unit normalν s on S.
Our claim is that if a background supersonic solution (ρ − 0 , ϕ − 0 , p − 0 ) and a constant exit pressure p c ∈ (p min , p max ) are perturbed sufficiently small, and also if Ψ is a small perturbation of the identity map, then there exists a corresponding transonic shock solution for a nonisentropic potential flow.
Remark 2.6. Let Ω be a connected subset of N , and Γ be a portion of ∂Ω. For T in (2.22), let us set Then there is a constant C * (N , Ω, n, m, α, k) so that, for any u ∈ C m,α k,Γ (Ω), there holds For a fixed constant data (ρ in , v in , p in ) ∈ R 3 + satisfying v 2 in > γp in ρ in and p c ∈ (p min , p max ), let (ρ 0 , ϕ 0 , p 0 ) be the background solution with (ρ 0 , ∂ r ϕ 0 , p 0 )(r 0 ) = (ρ in , v in , p in ) and p 0 (r 1 ) = p c , and let S 0 = {r = r s } ∩ N be the shock of (ρ 0 , ϕ 0 , p 0 ). Now, we state the main theorems.
Theorem 2 (Uniqueness). Under the same assumptions with Theorem 1, for any α ∈ ( 1 2 , 1), there exists a constant σ 2 > 0 depending on the data in sense of Remark 2.7 so that whenever 0 < σ ≤ σ 2 , the transonic shock solution in Theorem 1 is unique in the class of weak solutions satisfying Theorem 1(a)-(d).
Remark 2.8. By Theorem 1(c), Theorem 2 implies that the transonic shock solutions are stable under a small and smooth perturbation of the incoming supersonic flow, exit pressure and the nozzle.
2.6. Framework of the proof of Theorem 1 and Theorem 2. By the decomposition (2.42)-(2.44), if we solve (2.42) for ϕ, then we can plug it into the transport equation to solve for p, and solve the Bernoulli's law for ρ. To solve (2.43) for p, however, the initial condition on S can be computed by (2.48) and (2.49). But if we also fix p = p ex on Γ ex , then the problem gets overdetermined. So, we set the framework to prove Theorem 1 and 2 as follows.
(Step 1) To prove Theorem 1, we first solve Problem 3 (Free boundary problem for ϕ). Given the incoming supersonic flow Let us emphasize that Lemma 2.5 is the key ingredient to prove Lemma 3.3, and this lemma provides well-posedness of Problem 3. (Step 2) In Section 4, we solve (2.43) for p after plugging ϕ in to (2.43) by the method of characteristics in N + with the initial condition (2.49) on S. However, as we will show later, Dϕ is only in C α up to the boundary of N + so the standard ODE theory is not sufficient to claim solvability of (2.43). For that reason, we need a subtle analysis of Dϕ to overcome this difficulty.
In Section 5, we will show that P(Ψ, ϕ − , p − , ·) : v ex → p ex is locally invertible in a weak sense near the background solution. The weak invertibility is proven by modifying the proof of the right inverse mapping theorem for finite dimensional Banach spaces in [SZ], and this proves Theorem 1. (Step 4) To prove Theorem 2, we need to estimate the difference of two transonic shock solutions in a weaker space than the space of the transonic shock solutions in step 3. This requires a careful analysis of solutions to elliptic boundary problems with a blow-up of oblique boundary conditions at the corners of the nozzle N . See Lemma 6.3.

Problem 3: Free Boundary Problem for ϕ
To solve Problem 3, we fix the boundary condition on the exit Γ ex as where v ex is a small perturbation of the constant v c given by Here, ν ex is the outward unit normal on Γ ex so ν ex isr = x |x| . Let (ρ 0 , ϕ 0 , p 0 ) and r s be as in Theorem 1.
Proposition 3.1. Under the assumptions of Theorem 1(i) and (ii), for any given α ∈ (0, 1), there exist constants σ 3 , δ, C > 0 depending on the data in sense of Remark 2.7 so that whenever then there exists a unique solution ϕ to (2.42), (2.47), (2.48), (2.53), (3.1) with the shock S, and moreover, (a) there exists a function f : We prove Proposition 3.1 by the implicit mapping theorem. For that, we first derive the equation and the boundary conditions for ψ := ϕ − ϕ + 0 .

Nonlinear boundary problem for
where A = (A 1 , · · · , A n ) is defined in (2.45). The boundary condition for ψ on S needs to be computed more carefully.
So we regard (Ψ, ϕ − , p − ) as factors of perturbations of the nozzle, and the incoming supersonic flow.
Lemma 3.3. Let f and ϕ be as in Proposition 3.1 and set (3.8) Moreover, b 1 , µ f and g 1 satisfy, for µ 0 > 0 defined in Lemma 2.5, where the constant λ also only depends on the data as well. Proof.
(3.17) shows how to formulate a boundary condition for ϕ in general.
(3.9) is verified. By Lemma 2.5, µ 0 is a positive constant. We note that h * in (3.16) depends on DΨ, p − , Dψ − in sense of Remark 3.2, and also depends on Dψ. So we should carefully decompose h * in the form of h * = β · Dψ + g so that a weighted Hölder norm of g does not depend on ψ We note that, by (2.47), ν s −r is written as where we set for ξ, η ∈ R n . Then, we can rewrite a 1 as Here, all the quantities are evaluated on (f ( To rewrite a 2 for our purpose, let us consider K 0 − K s first.
is independent of ϕ but q * should be treated carefully.
It is easy to see that a Hölder norm of I 2 is uniformly bounded independent of ϕ. Let us consider I 1 now. By (2.52), a simple calculation provides where a Hölder norm of I 3 is independent of ψ if σ 3 is sufficiently small. By (3.18), (3.23) and (3.24), a 2 is (3.26) By (3.22) and (3.26), we can write h * = a 1 − a 2 as By the definition of g 1 in (3.28) and Proposition 3.1, we have It remains to verify (3.10). From the definition of ν(ξ, η), one can check .22) and (3.25) imply β 1 (0, 0) = β 1 (0, 0) = 0. As vector valued functions of ξ, η ∈ R n , β 1 and β 2 are smooth for ξ, η ∈ B n d (0) for a constant d > 0 sufficiently small. So if σ 3 is sufficiently small, then we obtain (3.29) and this implies By (2.52) and Proposition 3.1, ν s −r satisfies therefore (3.10) is verified provided that we choose σ 3 in Proposition 3.1 sufficiently small depending on the data.
For a sufficiently small R > 0, B (1) is a set of small perturbations of the nozzle, the incoming supersonic flow and the exit boundary condition for ϕ. For two constants for x ′ ∈ Λ where ψ satisfies the following conditions: setting for a constant M > 0 to be determined later; (ii) ψ solves the nonlinear boundary problem: where A in (3.34) is defined as in (3.1). For σ * > 0 sufficiently small, if M in (3.32) is appropriately chosen depending on the data so that (3.33) is a uniformly elliptic boundary problem with oblique boundary conditions, then J is well defined. More precisely, we have the following lemma.
Lemma 3.4 (Well-posedness (3.33)). Fix α ∈ (0, 1) and C * > 0. Then there is a constant M, σ ♯ depending on the data in sense of Remark 2.7 and also depending on C * so that the followings hold: is a uniformly elliptic boundary problem, and it has a unique solution ψ satisfying (3.32), (ii) if σ ♯ is sufficiently small depending on the data and C * , then there is a constant C depending only on the data so that the solution of (3.33) satisfies where ς 1 , ς 2 , ς 4 are defined in Theorem 2.4 and Proposition 3.1.
We will prove Lemma 3.4 by the contraction mapping principle. For that purpose, we first need to study a linear boundary problem related to (3.33). Let σ * > 0 be a small constant to be determined later. For a given constant (2) C * σ * (r s ), and let us set ψ − = ϕ − − ϕ − 0 . Reduce σ * sufficiently small depending on the data so that DΨ is invertible in N , and set For a fixed φ ∈ K f (M ), consider the following linear problem in N + f : The definitions of a jk , b 1 , µ f , g 1 , g 2 , g 3 are given in (3.6), (3.15), (3.28), and (3.34).
Lemma 3.5 (Well posedness of (3.35)). For any given α ∈ (0, 1), and M, C * > 0 is σ * > 0 is chosen sufficiently small depending only on the data, C * and M , then there exists a constant C depending on the data and α so that, for any given f ∈ B (2) To prove Lemma 3.5, we basically repeat the proof of [CF3, Theorem 3.1]. A key ingredient of the proof in [CF3] is the method of continuity and the divergence structure of a linear boundary problem. In our case, however, (3.35) is not in a divergence form because of the derivative boundary condition on S f . For that reason, we need Lemma 3.8 so that (3.35) can be treated as a divergence problem. Before we proceed further, we first note two obvious lemmas for the later use.
Lemma 3.6. For any C * , M > 0, and for f ∈ B (2) where the constant C in (3.37)-(3.40) only depends on the data in sense of Remark 2.7. Thus, for any C * , M > 0, if σ * is chosen sufficiently small depending on the data, C * and M , then the equation of (3.35) is uniformly elliptic in N + f . Proof. (3.37)-(3.40) are obvious by the definitions.
For any given ζ ∈ R n , a ij (x, 0) satisfies So, by (2.34), (2.35) and Remark 2.2, for any C * , M > 0, one can choose σ * depending on the data, C * and M so that (3.35) is uniformly elliptic in N + f . Lemma 3.7. For any C * , M > 0, let ν f be the inward unit normal on S f , and b 1 be defined in (3.28). Then, for any givenf ∈ B (2) (2) where µ 0 is defined in Lemma 2.5 if σ * is chosen sufficiently small depending on the data, C * and M .
Proof. For ν defined in (3.20), we have for a constant C depending only on the data. By (3.38), for any By Lemma 3.6 and 3.7, for any C * , M > 0, choosing σ * sufficiently small depending on the data, C * and M , (3.3) is a uniformly elliptic linear boundary problem with oblique boundary conditions on the boundary.
Proof of Lemma 3.5.
Step 1. Suppose that u solves (3.35) and then we first verify (3.36) by following the method of the proof for [CF3, Theorem 3.1]. First, we apply the change of variables x → y = Φ(x) with Φ defined by This change of variables transforms N + f to N + rs , and rs in the spherical coordinates in (2.22). Then w(y) = u • Φ −1 (y) solves a linear boundary problem of the same structure as (3.35). For simplicity, we write the the equation and the boundary conditions for w as ) through the change of variables x → y = Φ(x) for j, k = 1, · · · , n, and m = 1, 2, 3. Next, we rewrite (3.45) as a conormal boundary problem with coefficients a jk (y, 0) and b 1 (y, 0) =r for the equation and the boundary conditions of w.
We remind that, by (3.6), the matrix [a jk (x, 0)] is Lemma 3.8 (Boundary condition for w on S 0 ). Suppose that w solves (3.45), and let M, C * , σ * be as in Lemma 3.5. For any given α ∈ (0, 1), f ∈ B (2) there is a constant C depending only on the data in sense of Remark 2.7 so that w satisfies Proof. The unit normail on S 0 toward N + rs isr. Then, by (3.46) and the boundary condition of w on S 0 in (3.45), there holds By (3.38) and the definition ofb 1 , (3.48) is easily obtained.
Step 2. Using a weak formulation of the equation for w in (3.49) with (3.46), one can check for (r, x ′ ) is given as in (2.22), and every b lm (x ′ ) is smooth in Λ. Here we setG = ω 1 [ ∂(r,x ′ ) ∂x ] T G. We note that ω 1 is bounded below by a positive constant in Λ, and the linear operator L 0 is uniformly elliptic in N + rs ,and bounded by (3.41) and (2.35).
Step 3. To obtain the desired estimate of u in (3.35), we use (3.49)-(3.52), Lemma 3.8 and the interpolation inequality of Hölder norms to follow the method of the proof for [CF3,Theorem 3.1], and prove the following lemma.
Step 4. Finally, we prove the existence of a solution to (3.45). Consider the following auxiliary problem: for F ∈ C α (N + rs ), g 1 , g 3 ∈ C α (Λ) and g 2 ∈ C α (Γ w,rs ). (3.55) has a unique weak solution because the functional has a minimizer over W 1,2 (N + rs ) with the outward unit normal ν out of ∂N + rs where we write g := g 1 χ S 0 + g 2 χ Γw,r s − g 3 χ Γex with χ D defined by χ D (y) = 1 for y ∈ D, χ D (y) = 0 otherwise for D ⊂ R n .
Remark 3.10. According to the definition of the laplacian △ S n−1 on a sphere S n−1 , we note u) =△ S n−1 u where we replace − △ S n−1 by △ S n−1 in the widely used convention(e.g. [BR]) of △ S n−1 .
Lemma 3.4 is a direct result from Lemma 3.5.
(2) C * σ * (r s ), and define a nonlinear mapping G f by for the constant C in (3.36), then G f maps K f (M ) into itself which is a Banach space. For k = 1, 2, let us set w k = G f (φ k ). By subtracting the boundary problem (3.45) for w 2 from the boundary problem (3.45) for w 1 , one can directly show that there is a constantC depending on the data so that there holds (3.57) So if we reduce σ * > 0 satisfying Lemma 3.6, (3.42) ,(3.43) andCσ * < 1, then the contraction mapping principle applies to G f . Therefore G f has a unique fixed point, say, ψ. Obviously, ψ is a solution to (3.33) and it also satisfies Lemma 3.4(ii). From this, we choose σ ♯ = σ * . We note that the choice of σ ♯ depends on the data( in sense of Remark 2.7) and C * .

Transport Equation for Pressure p
This section is devoted to step 2 in section 2.6. 4.1. The method of characteristics. By the choice of σ 3 in Proposition 3.1, if ϕ is a solution to Problem 3, then ∂ r ϕ is bounded below by a positive constant in N + . So r can be regarded as a time-like variable and then we apply the method of characteristics to solve (2.43) for p. According to (3.2), however, Dϕ is not Lipschitz continuous up to ∂N + . So the unique solvability of (2.43) needs to be established first, and we will prove the following proposition.
To prove Lemma 4.1, it suffices to show that the nonlinear boundary problem (3.33) in the fixed domain N + has a unique solution ψ so that ϕ = ϕ + 0 + ψ satisfies (4.2), and this can be proven by a similar method to the proof of Lemma 3.4 except that the scaling argument in step 4 needs to be performed for any point y 0 ∈ N + \ (S ∪ Γ ex ). So we skip the proof of Lemma 4.1.
Considering r as a time-like variable, it is more convenient to use a changes of variables to transform N + to N + rs keeping (4.2) in the transformed domain N + rs . This requires a new change of variables: with a smooth function χ(r) satisfying χ(r) = 0 if r ≤ r s + r 1 −rs 10 1 if r ≥ r 1 − r 1 −rs 2 and χ ′ (r) ≥ 0 for r > 0.
By (4.2), if ς 1 + ς 2 + ς 4 = σ is small depending on the data in sense of Remark 2.7, and we choose a constant k satisfying We choose x ′ in (2.22) so that (r, x ′ ) forms an orthogonal coordinate system. To specify this choice, we write (r, x ′ ) as (r, ϑ) with ϑ = (ϑ 1 , · · · , ϑ n−1 ) for the rest of paper. And then, we apply the change of variables (4.3) to rewrite (2.43) in N + as where J ϕ is the Jacobian matrix associated with the change of variables in (4.3). Here,r,θ j are unit vectors pointing the positive directions ofr and ϑ j respectively. By Proposition 3.1, Lemma 4.2 and (4.3), we have for ς 1 , ς 2 , ς 4 in Theorem 1 and Proposition 3.1. Thus, reducing σ 3 in Proposition 3.1 if necessary, there is a constant ω 0 > 0 depending on the data in sense of Remark 2.7 so that there holds V ·r ≥ ω 0 > 0 in N + rs . The proof is obtained by a straightforward computation so we skip here.
By Lemma 4.3, we can divide the equation (4.6) by V ·r so that the radial speed of all characteristics associated with V is 1. From now on, we consider the equation We denote (4.9) The initial condition for E is given from (2.49). Setting Q 2 := (DΨ −1 ) T DΨ −1 , E(r, ϑ) must satisfy E(r s , ϑ) = E int (ϑ) for all ϑ ∈ Λ where we set (4.10) Above, f (ϑ) is the location of the transonic shock for the potential ϕ in Proposition 3.1. To simplify notation, we writer as r hereafter. By (4.8) and (4.10), Proposition 4.1 is a direct result of the following lemma.
Lemma 4.5. Fix α ∈ (0, 1), and suppose that a vector valued function W = (1, W 2 , · · · , W n ) satisfies W · ν w = 0 on Γ w,rs = ∂N + rs ∩ Γ w for a unit normal ν w on Γ + w , and for some constant k > 0. Then the initial value problem has a unique solution F satisfying for a constant C > 0 depending on n, Λ, r s , r 1 , α and k.
Now it remains to prove Lemma 4.5. Consider the following ODE systeṁ for (r, ϑ) ∈ N + rs ∪ Γ ex . The slip boundary condition W · ν w = 0 on Γ + w implies that any characteristic curve X does not penetrate out of N + rs . If (4.16) has a unique solution X(t; r, ϑ) for any given (r, ϑ) ∈ N + rs ∪ Γ ex , then we can define a mapping J : N + rs ∪ Γ ex → Λ by J : (r, ϑ) → X(2r − r s ; r, ϑ). (4.17) If J is differentiable then F = F 0 • J is a solution to (4.12).
4.2. The system of ODEs (4.16). (4.16) has a solution for any given (r, ϑ) ∈ (r s , r 1 ] × Λ by the Cauchy-Peano theorem [AM,7.3]. But the standard ODE theory is not sufficient to claim the uniqueness of a solution for (4.16) because W is only in C α (N + rs ). To prove the uniqueness, we will use the fact that DW is integrable along each solution X to (4.16). Denote as X(t; r, ϑ) for a solution to (4.16). Hereafter, we call each X(t; r, ϑ) a characteristic associated with W initiated from (r, ϑ).
Since W ·ν w = 0 on Γ + w , any characteristic, initiated from a point on the wall of N + rs , remains on the wall i.e., if ϑ 2 ∈ ∂Λ, then X (2) lies on Γ w . Then (4.22) implies This verifies the second inequality of (4.19). One can argue similarly to verify the first inequality of (4.19).
Lemma 4.8. Fix α ∈ (0, 1), and let k be as in Lemma 4.5. The mapping J defined in (4.17) is continuously differentiable, and it satisfies for a constant C > 0 depending on n, Λ, r s , r 1 , α and k.
The uniqueness of a solution can be checked easily. If F (1) and F (2) are solutions to (4.12) then F (1) − F (2) is identically 0 along every characteristic X(t; r, ϑ) associated with W , and so F (1) = F (2) holds in N + rs .

Proof of Theorem 1:Existence
To prove Theorem 1, we prove a weak implicit mapping theorem for infinite dimensional Banach spaces.

Weak implicit mapping theorem. Fix a constant R > 0 and set
where p is the solution to (2.43) with ϕ in Proposition 3.1 for the given (Ψ, ϕ − , p − , v ex ). By Proposition 3.1 and 4.1, there exists positive constants σ 3 and C so that whenever 0 < σ ≤ σ 3 , P, defined in (5.2), is well defined. To prove theorem 1, we need to show P −1 (p ex ) = ∅ for any p ex ∈ B (2) σ (p c ) for a sufficiently small constantσ > 0. So, we need the following lemma. Proposition 5.1 (Weak Implicit Mapping Theorem). Let C 1 and C 2 be Banach spaces compactly imbedded in Banach spaces B 1 and B 2 respectively. Also, suppose that there are two Banach spaces C 3 , B 3 with C 3 ⊂ B 3 . For a point (x 0 , y 0 ) ∈ C 1 × C 2 , suppose that a mapping F satisfies the followings: (i) F maps a small neighborhood of (x 0 , y 0 ) in B 1 × B 2 to B 3 , and maps a neighborhood of (x 0 , y 0 ) in iii) F, as a mapping from B 1 × B 2 to B 3 also as a mapping from C 1 × C 2 to C 3 , is Fréchet differentiable at (x 0 , y 0 ), (iv) the partial Fréchet derivative D x F(x 0 , y 0 ), as a mapping from B 1 × B 2 to B 3 also as a mapping from C 1 × C 2 to C 3 , is invertible. Then there is a small neighborhood U 2 (y 0 ) of y 0 in C 2 so that, for any given y ∈ U 2 (y 0 ), there exists x * = x * (y) satisfying F(x * (y), y) = 0.
In [SZ], the author used the Brouwer fixed point theorem to prove the right inverse function theorem for finite dimensional Banach spaces without assuming continuous differentiability of a mapping. As an analogy, we apply the Schauder fixed point theorem to prove Proposition 5.1. The detailed proof is given in Appendix A.
To apply Proposition 5.1 to P, we need to verify that P is continuous, Fréchet differentiable at (Id, ϕ − 0 , p − 0 , v c ) and a partial Fréchet derivative of P at the point is invertible. 5.2. Fréchet differentiability of P at ζ 0 = (Id, ϕ − 0 , p − 0 , v c ). To prove Fréchet differentiability of P at ζ 0 , it suffices to consider how to compute the partial Fréchet derivative of P with respect to v ex at ζ 0 because the other partial derivatives of P can be obtained similarly.
For P defined in (5.2), let us define R and Q by If R and Q are Fréchet differentiable at ζ 0 then the Fréchet derivative DP of P at ζ 0 is given by (5.4) To simplify notations, we denote as D v P, D v Q and D v R respectively for the partial Fréchet derivatives of P, Q and R at ζ 0 with respect to v ex hereafter.
Since the constant C in (4.1),(4.2), (5.14) and (5.21) only depends on the data in sense of Remark 2.7 , combining (5.22) with (5.23), we conclude that the mapping L satisfies (5.10) with o(ε) being independent of w. Thus we conclude D v Q = L. Since it is easier to verify (5.5) so we omit the proof.
Proof. For any w ∈ C 1,α (−α,∂Λ) (Λ), let ψ 0 be a unique solution to (5.19), (5.20). Then there are constants C, C > 0 depending only on the data to satisfy . For a local invertibility of P, by Lemma 5.1, it remains to show that D v P is invertible and P is continuous in a Banach space.
(5.25) By Corollary 5.3, T is compact. Then, by the Fredholm alternative theorem, either D v Pw = 0 has a nontrivial solution w or D v P is invertible. In the following lemma,we show that D v P satisfies the latter hence D v P is invertible.
Proof. The proof is divided into two steps.
Then w has an eigenfunction expansion where every η j is an eigenfunction of the eigenvalue problem with η j L 2 (Λ) = 1. We note that all the eigenvalues of (5.28) are non-negative real numbers. We claim that (5.27) implies c j = 0 for all j ∈ N. Once this is verified, then we obtain w = 0.
For each N ∈ N, let us set w N := N j=1 c j η j and ψ N (r, ϑ) := N j=1 q j (r)η j (ϑ) for one variable functions q j satisfying .
Suppose that c j = 0 for some j ∈ N. Without loss of generality, we assume c j > 0. By the sign of a 1 a 2 considered in Step 1, we have q ′ j (r s ) < 0, q j (r s ) < 0 and q ′ j (r 1 ) > 0. We will derive a contradiction.
(i) If λ j = 0 then there is a constant m 0 so that k 1 q ′ j = m 0 for all r ∈ [r s , r 1 ]. Since k 1 is strictly positive in [r s , r 1 ], we have sgn m 0 = sgn q ′ k (r 1 ) and sgn m 0 = sgn q ′ j (r s ). Then sgn q ′ j (r s ) = sgn q ′ j (r 1 ) must hold. But this contradicts to the assumption of q ′ j (r s ) < 0 < q ′ j (r 1 ). (ii) Assume λ j > 0. By (5.29), we have Then q j (r s ) < 0 implies max [rs,r 1 ] q j > 0 so there exists t 1 ∈ [r s , r 1 ] satisfying q j (t 1 ) = 0. Also, by the intermediate value theorem, there exists t 2 ∈ [r s , r 1 ] satisfying q ′ j (t 2 ) = 0. If t 1 = t 2 then the uniqueness theorem of second-order linear ODEs implies q j ≡ 0 on [r s , r 1 ]. But this is a contradiction. If t 1 = t 2 , assume t 1 < t 2 without loss of generality. Then, by the maximum principle, q j ≡ 0 on [t 1 , t 2 ] so q j ≡ 0 on [r s , r 1 ]. This is a contradiction as well.
Thus every c j must be 0 hence w = 0. By the Fredholm alternative theorem, we finally conclude that D v P is invertible.
The rest of section 6 is devoted to the proof of Lemma 6.1 and Lemma 6.2.
Proof of Lemma 6.2.
Step 1.For convenience, for each j = 1, 2, let us denote as P j , Q j and R j for P(ζ * , v j ), Q(ζ * , v j ) and R(ζ * , v j ) respectively where Q, R are defined in (5.3). Then we can write . We estimate l 1 R 1 , l 2 Q 2 , l 3 separately.
So it suffices to estimate |ψ| 0,α,N + rs . Since N + rs is a cylindrical domain with the cross-section Λ in (r, x ′ )−coordinates in (2.22), there is a constant κ 0 > 0 depending only on n, Λ to satisfy, for any x 0 ∈ N + rs and R > 0, assuming R * < 1 without loss of generality where the constant C depends only on n, α, Λ, r s , r 1 . So, we devote the rest of proof to find a constant M > 0 satisfying (B.4).