Local Uniqueness of Steady Spherical Transonic Shock-fronts for the Three-Dimensional Full Euler Equations

We establish the local uniqueness of steady transonic shock solutions with spherical symmetry for the three-dimensional full Euler equations. These transonic shock-fronts are important for understanding transonic shock phenomena in divergent nozzles. From mathematical point of view, we show the uniqueness of solutions of a free boundary problem for a multidimensional quasilinear system of mixed-composite elliptic--hyperbolic type. To this end, we develop a decomposition of the Euler system which works in a general Riemannian manifold, a method to study a Venttsel problem of nonclassical nonlocal elliptic operators, and an iteration mapping which possesses locally a unique fixed point. The approach reveals an intrinsic structure of the steady Euler system and subtle interactions of its elliptic and hyperbolic part.


Introduction
The study of the Euler equations for compressible fluids is one of the central topics in the mathematical fluid dynamics, and the analysis of solutions to the system is of particular interest in applications. In particular, in the recent years, important progress has been made in the analysis of transonic shock solutions of the steady potential flow equation and the steady Euler system in multidimensions (cf. [4,5,6,7,8,9,15,21,22] and the references cited therein).
In this paper, we are concerned with the local uniqueness of transonic shock solutions with spherical symmetry for the three-dimensional, steady, full Euler system of polytropic gases. Such a study not only helps us to understand transonic shock phenomena occurred in divergent nozzles, which have many important applications, but also provides new insights for the theory of free boundary problems of partial differential equations of composite-mixed elliptic-hyperbolic type. This problem can be formulated as a free boundary problem for the Euler system in a spherical shell, with the transonic shock-front as a free boundary, which is a graph of a function defined on S 2 (the unit 2-sphere in R 3 ). Therefore, for such a problem, although there exists a system of global Descartes coordinates, it is more convenient to use the local spherical coordinates and the terminology of differential geometry (see Appendix A).
Let r 0 < r 1 be two positive constants. A spherical shell in R 3 centered at the origin is a Riemannian manifold M = [r 0 , r 1 ] × S 2 with a metric G = G ij dx i ⊗ dx j † in local spherical coordinates (see Appendix A). Its boundary ∂M is Σ 0 ∪ Σ 1 with Σ i := {(x 0 , x) ∈ M : x 0 = r i , x = (x 1 , x 2 ) ∈ S 2 }, i = 0, 1, denoting respectively the entry and exit of M. Let p, ρ, and S in M represent the pressure, density, and entropy of gas flow in the manifold respectively. For polytropic gases, p = A(S)ρ γ with the adiabatic exponent γ > 1, and the sonic speed is c = γp/ρ. where div and grad are respectively the divergence and gradient operator in M, and We use U = (p, ρ, u) to denote the state of fluid flow. If U depends only on x 0 and u = u 0 (x 0 )∂ 0 , then (1.1)-(1.3) can be reduced to the following differential equations: .
(1. 6) It has been shown in Yuan [22] that, for equations (1.4)-(1.6), given supersonic data U − b (r 0 ) = (p − b (r 0 ), ρ − b (r 0 ), u − b (r 0 )) on the entry Σ 0 , there exists an interval I such that, if the back pressure p + b (r 1 ) ∈ I, then there exists a unique r b ∈ (r 0 , r 1 ) so that is a transonic shock-front, determined by the Rankine-Hugoniot jump conditions: The flow U − b (x 0 ), x 0 ∈ (r 0 , r b ), ahead of S b is supersonic; U + b (x 0 ), x 0 ∈ (r b , r 1 ), behind of S b is subsonic; and the physical entropy condition p + b (r b ) > p − b (r b ) holds on S b . We call such a spherical transonic shock solution U b := (U − b , U + b ; S b ) to the Euler equations as a background solution. The objective of this paper is to study the uniqueness of transonic shock solutions, if it exists, in a neighborhood of a class of background solutions under the three-dimensional perturbations of the upcoming † In this paper we always use the Einstein summation convention for the Roman indices from 0 to 2 and for the Greek indices from 1 to 2.
supersonic flow U − b at the entry Σ 0 . This class of background solutions especially include those background solutions satisfying the S-Condition defined in §2. In particular, we prove that, for a given back pressure, for sufficiently large upstream pressure and Mach number, the background transonic shock-front itself is locally unique.
The main theorem of this paper is the following: Then there exist ε 0 and C 0 depending only on U b and γ such that, if the upcoming supersonic flow U − on Σ 0 satisfies for some α ∈ (0, 1) and there exists a transonic shock solution U = (U − , U + ; S ψ ) of (1.1)-(1.3) in M satisfying Property (A) below, then this solution is unique. Here Property (A) consists of the three conditions: , (x 1 , x 2 ) ∈ S 2 } is the shock-front; U − is the supersonic flow ahead of S ψ ; and U + is the subsonic flow behind of S ψ . The physical entropy condition: p + | S ψ > p − | S ψ holds on S ψ . Moreover, ψ ∈ C 4,α (S 2 ) satisfies , whereū is the 1-form corresponding to the vector field u via the Riemannian metric of M, and C k,α r denotes the space of r-forms in M (i.e., A r (M)) with C k,α components in local coordinates and the norms are defined in the usual way by partition of unity in the manifold.
As explained in [10,22], the background solutions coincide with the solutions of the steady quasione-dimensional model of flows in divergent nozzles. Therefore, the above uniqueness result will help to understand transonic shock phenomena in divergent nozzles, as well as the effectiveness of the quasi-one-dimensional model [19]. Also see [20] for an explanation of the quasi-one-dimensional model from the viewpoint of flows in Riemannian manifolds and [15] for the stability result of transonic shock-fronts for the two-dimensional case.
Apart from the physical implications, the approach by considering the Euler equations in a Riemannian manifold is of interest itself in mathematics. We note that some studies have been made for conservation laws in general Riemannian manifolds (cf. [1,2,17] and the references cited therein). This approach via differential geometry reveals some intrinsic structures of the steady Euler system, which are valid in general Riemannian manifolds.
Finally, we remark that the existence and uniqueness of supersonic flow U − in M subject to the initial data U − | Σ 0 satisfying (1.7) follow directly from the theory of semi-global classical solutions of the Cauchy problem of quasilinear symmetric hyperbolic systems if ε 0 is sufficiently small (cf. [14,18]). Furthermore, one can obtain where C 1 > 0 and ε 0 > 0 depend solely on U − b (r 0 ) and r 1 . Thus, we focus on showing the uniqueness of ψ and U + below. Indeed, what we obtain here is much more than this: We design an iteration mapping and show that it always has a unique fixed point, and any solution to (1.1)-(1.3) must be a fixed point of this iteration mapping. Therefore, the solution to (1.1)-(1.3) is unique, and then Theorem 1.1 is proved. However, we have not known whether the fixed point of the iteration mapping is a solution to the original problem (1.1)-(1.3), therefore, the existence problem for solutions to (1.1)-(1.3) is still open, though we believe that the ideas and approaches developed here will be useful to establish an existence theorem which is out of the scope of this paper.
For simplicity, we write U + as U from now on. We emphasize here that the supersonic flow U − is defined in the whole M, while, by Proposition 9 in [22], U + b can be extended to [r b − h b , r 1 ] × S 2 and still obeys the Euler system, with h b > 0 depending only on U − b (r 0 ). The rest of this paper is organized as follows. In §2, we derive some elliptic or transport equations, as well as certain equations of exterior differential forms in M and on the shock-front, from the Euler equations (1.1)-(1.3) and the Rankine-Hugoniot jump conditions. Most of the formulas obtained here are also valid in general Riemannian manifolds. Based on these decompositions, in §3, we present an iteration mapping and establish the existence of a unique fixed point, which yields Theorem 1.1. Some facts and notations of differential geometry are shown and described in Appendix A.
We remark in passing that the analysis and results developed here should be straightforward extended to the higher dimensional case, even general Riemannian manifolds for most of them.

Reduction of the Euler system and Rankine-Hugoniot Jump Conditions
In this section we introduce a reduction of the Euler system and analyze the Rankine-Hugoniot jump conditions. 2.1. The Euler Equations in M. We use d to denote the exterior differential operator and D u ω to denote the covariant derivative of a tensor field ω with respect to a vector field u in M; while ∇ u ω is the covariant derivative on S 2 . L u ω is the Lie derivative of ω with respect to u in M. The symbol ∆ represents the Laplacian of forms in M, and ∆ ′ is the Laplacian of forms on S 2 , which are both positive operators (cf. (A.10)). Note that ψ ∈ C 4,α (S 2 ) defines a mapping from S 2 to M by (x 1 , x 2 ) → (ψ(x 1 , x 2 ), x 1 , x 2 ). We use ψ * to denote the pull back of forms and functions induced by this mapping; for example, for a function p ∈ A 0 (M), ψ * p = p| S ψ . The volume 2-form of S 2 is written as vol, and vol 3 is the volume 3-form of M.
In the following, we derive some well-known equations from the Euler system (1.1)-(1.3) which are valid only for C 1 flows (cf. [18]). Since these involve differentiations of (1.1)-(1.3), the solution of the reduced equations might not be a solution to the Euler system (1.1)-(1.3). One point below is to express the relations between these derived equations and the original Euler equations, which may be useful in the future to verify that the solution of these derived equations satisfies the Euler system indeed.
The conservation of mass (1.2) can be written (equivalently for C 1 flows) as for two vector fields u and v, the momentum equations (1.1) become Similarly, the conservation law of energy (1.3) may be written as This is exactly the well-known Bernoulli law.
we have i.e., the invariance of entropy along the flow trajectories for C 1 flows. For a vector field u = u i ∂ i in M, we always useū = u j G ij dx i to denote its corresponding 1-form with respect to the metric G of M. Then (2.2) is equivalent tō . This implies which is a transport equation of vorticity. Moreover, dū| S ψ , the initial value of vorticity on S ψ , expressed in local spherical coordinates, is where g = g αβ dx α ⊗ dx β is the standard metric of S 2 . Let d * be the codifferential operator in M. Using the identity d * ū = −div u, we obtain Let ·, · be the inner product in M of forms and let * be the Hodge star operator, which imply * vol 3 = 1, * 1 = vol 3 , d( * ū) = (div u) vol 3 , α ∧ * β = α, β vol 3 .

Then (2.8) is
Note that, by the equation of state p = A(S)ρ γ , we have the equation of conservation of mass may be written as By this equation and the definition of the Laplacian of forms ∆ = dd * + d * d, we also have The Rankine-Hugoniot Jump Conditions on a Shock-Front. A shock-front S ψ is a hyper-surface in M across which the physical variables have a jump. In our case, it can be expressed as the graph of a mapping ψ : S 2 → M. In local spherical coordinates, we may write The normal vector field and the corresponding normal 1-form of S ψ with respect to M are From (1.1)-(1.3), the Rankine-Hugoniot jump conditions (i.e., the R-H conditions) on S ψ are ⌈G(u, n)ρu + pn⌋| S ψ = 0 ⇐⇒ ⌈n(u)ρū + pn⌋| S ψ = 0, (2.14) ⌈G(u, n)ρ⌋| S ψ = 0 ⇐⇒ ⌈n(u)ρ⌋| S ψ = 0, (2.15) ⌈G(u, n)ρE⌋| S ψ = 0 ⇐⇒ ⌈n(u)ρE⌋| S ψ = 0, (2.16) where ⌈·⌋ denotes the jump of a quantity across S ψ . It is well-known (see [10,11]) that a piecewise C 1 state U = (U − , U + ; S ψ ) is a weak entropy solution of (1.1)-(1.3) if and only if U satisfies these equations in M ± ψ in the classical sense, the R-H conditions along S ψ , as well as the physical entropy Thenn(u) = dψ(u 1 ) − u 0 , and (2.14) can be separated into ψ * ⌈gu 0 + p⌋ = 0, (2.20) where g = g(U, ψ, Dψ) := (ρu 0 −dψ(ρu 1 ))| S ψ is a function on S 2 , with dψ ∈ A 1 (S 2 ) being considered as a 1-form in M; and the expression g(U, ψ, Dψ) means that g depends on U, ψ, and the first-order derivatives of ψ. Then (2.15)-(2.16) become Equations (2.3) and (2.21) indicate that E is a constant along the same trajectory even across the shock-front. Therefore, we may write E| S ψ = E 0 (x), x ∈ S 2 , with E 0 (x) a given function depending only on the supersonic data at the entry. Now, if ψ * ⌈p⌋ = 0 (which is guaranteed later by the physical entropy condition of the background solution), (2.19) yields by using the first equation in (2.21), with and g 0 the higher order term defined below (see Definition 2.2) 1 , which contains U, U − , ψ, and Dψ (the first-order derivatives of ψ). We note that equations (2.20)-(2.22) are equivalent to (2.14)-(2.16). Equation (2.22) also indicates dω = 0. Therefore, we have Remark 2.1. The reason why we distinguish the "position" and the "profile" is that they are determined by different mechanisms: The "profile" is determined by the R-H conditions, while the "position" is determined by the solvability conditions closely related to the conservation of mass.
A higher order term is an expression that contains either (i) U − − U − b and its first-order derivatives; or (ii) the products of ψ p , r p − r b ,Û , and their derivatives DÛ, D 2Û , Dψ, D 2 ψ, and D 3 ψ, where D k u are the k th -derivatives of u in local coordinates.
Next, we linearize the R-H conditions (2.20)-(2.21). We write them equivalently as 1 From now on, we always use gi to denote the higher order terms on S ψ , and fi to denote the higher order terms in M + ψ .
As in [15], where we use "•" as the scalar product of vectors in the phase (Euclidean) space, and By the Taylor expansion, the terms in I i are of higher order. However, The Landau symbol O(|ψ| 2 ) means the terms of order at least to be two of ψ. One can also obtain where h.o.t. represents the higher order terms for short.
We can solve this linear system to obtain

Restriction of the Conservation of Mass and Momentum on the Shock-Front.
We now calculate d * (ψ * ū 1 ) by restricting the equations of conservation of mass and momentum on the shock-front S ψ and obtain a second-order elliptic equation for ψ p .
In local spherical coordinates, we have On the other hand, from the conservation of momentum, where we have set ψ * u 1 := (ψ * u α )∂ α which is a well-defined vector field on S 2 , ∇ ψ * u 1 (ψ * u 0 ) is a higher order term, and Γ 0 αβ are the Christoffel symbols (see Section A.1). Then we obtain d * (ψ * ū1 ) = I + II + III, where, in the term I, we used that ) is a higher order term.
Note that, for the background solution, II = 0. Then the Taylor expansion and the boundary conditions (2.28)-(2.31) yield is the Mach number of the flow behind the transonic shock-front of the background solution.

An Elliptic Equation for
the Pressure in M + ψ . First, we note the following tensor identity: where Du is the covariant differential of the vector field u in a Riemannian manifold, Ric(·, ·) is the Ricci curvature tensor, and C i j (T ) is the contraction on the upper i and lower j indices of a tensor T . In our case, since M is flat, Ric(u, u) ≡ 0. From the Euler equations, we have while direct calculation yields that, for Now the point is that the above right-hand sides may be expressed as functions of ∂ 2 0 p, ∂ 0 p, p, ψ * A(S), and some higher order terms.
Indeed, due to the conservation of momentum, By the equation of state, However, the invariance of entropy implies that Therefore, we have and where L 1 and L 2 do not involve the derivatives of ϕ i , and L 1 (0) = L 2 (0, 0) = 0. We then have by (2.29) and (2.31), where L 3 (0, 0, 0) = 0 and e i = e i (x 0 ), i = 1, · · · , 4, are known functions determined by the background solution: We also recall that t(x 0 ) is monotonically decreasing for the background solution and satisfies the following differential equation (cf. [22]): (2.40) 2.5. The Normalization of M + ψ and Reduced Equations. The above equations and boundary conditions are obtained in M + ψ for the given subsonic flow U and the shock-front ψ satisfying Theorem 1.1. The computations are relatively easy for the sake of rather simple metric G. However, to show the uniqueness of the transonic shock-front and the subsonic flow behind it, we need to set up an iteration mapping to find a newψ ∈ K σ : for a positive constant σ 0 to be specified later, by solving several boundary value problems in M + ψ for any ψ ∈ K σ , and then show that there is a fixed point that is unique. Therefore, it is convenient to introduce a C 4,α -homeomorphism Ψ : (x 0 , x) ∈ M + ψ → (y 0 , y) ∈ Ω := [0, 1] × S 2 by to normalize M + ψ to Ω. We set Ω j = {j} × S 2 , j = 0, 1. Then ∂Ω = Ω 0 ∪ Ω 1 . We will use i to denote the embedding of Ω 0 in Ω. We also define the metric of Ω to bẽ which differs from the metric induced by Ψ only those terms involving Dψ or ψ − r b . Therefore, according to (2.18), we define, for a vector field u in Ω, the correspondingū 1 = u αG αβ dy β . In the following, ∂ i = ∂ ∂y i in Ω for short. Then (2.7) can be written on Ω 0 as where E 1 is a 1-form in Ω depending smoothly on U, ψ, DU, Dψ, and D 2 ψ. Equations (2.22)-(2.23) on Ω 0 are respectively Similarly, (2.28)-(2.31) are transferred to In addition, we have wheref j ,ḡ j differ from f j , g j by some higher order terms due to the facts thatG differs from (Ψ −1 ) * G by the terms involving Dψ or ψ − r b and thatÛ is small. We also note that e i = e i ((r 1 − r b )y 0 + r b ) in (2.55).
2.6. The S-Condition. We now state the S-condition assumed in our main theorem.
Consider the following boundary value problem: where λ n = n(n + 1), n = 0, 1, 2, · · · , and e i = e i (t(y 0 )) with t( The following lemmas show that there exist certain background solutions which satisfy the S-Condition.
Lemma 2.1. For given γ > 1, let U b be a background solution determined by the supersonic is the Mach number of the upstream supersonic flow. Then, for given ρ − b (r 0 ) and the back pressure p + b (r 1 ), when p − b (r 0 ) and M − b (r 0 ) are sufficiently large, U b satisfies the S-Condition. Proof. We divide the proof into three steps.
Step 1. By an analysis of the background solution in [22], it suffices to show that, if κ = r 1 −r b > 0 and ) are rather small, then U b satisfies the S-Condition. We will prove by contradiction: we will first assume that (2.56)-(2.58) has a solution and then lead to a contradiction.
We note that, once t − b (r 0 ) is given, we can solve t(x 0 ) for all x 0 ∈ [r 0 , r 1 ] (i.e., it is independent of p − b (r 0 ), ρ − b (r 0 ), and p + b (r 1 )). This can be seen from (2.40) and the Rankine-Hugoniot condition of the Mach number: 1 Moreover, since p + b (r 1 ) is given, we have the estimate: with C depending only on p + b (r 1 ), r 0 , and r 1 .
For n = 0, 1, 2, we will utilize an energy estimate below to obtain a contradiction if κ is also small.
By a change of the independent variable y → z n : the above equation becomes where ′ is the derivative with respect to z n , α n = α n (y(z n )), and β n = β n (y(z n )). The boundary conditions are Note here that, since n < 3, and |β n |, |α n |, and z * n are bounded by a constant C, On the other hand, This reaches a contradiction when κ < min{ 1 √ C ′′′ , 1 √ 2C ′ }. Note that C ′′′ depends only on r 0 , r 1 , t − b (r 0 ), n, and p + b (r 1 ). Remark 2.3. In Lemma 2.1, for the case that κ = r 1 − r b > 0 is small, we require only p − b (r 0 ) to be large (see [22]).
In the following, we provide some other results on the existence of background solutions that satisfy the S-Condition. For given [r 0 , r 1 ], we note that a background solution U + b is determined by the five parameters (γ, Lemma 2.2. For given γ > 1, ρ + b (r b ) > 0, and σ 0 ∈ (0, 1), there exist a set S 1 ⊂ [0, σ 0 ] of at most countable infinite points and a set S 2 ∈ [r 0 , r 1 ] of at most finite points such that the background solution determined by γ > 1, r b ∈ (r 0 , r 1 ) \ S 2 , p + b (r b ) > 0, ρ + b (r b ) > 0, and t(r b ) ∈ (0, σ 0 ) \ S 1 satisfies the S-Condition.
Note that t(x 0 , 0) ≡ 0. Thus, in this case, (2.56)-(2.58) become By the Hopf maximum principle, we infer a contradiction at y 0 = 0. Therefore, in these cases, f −1 n ({0}) consists of finite points, which implies that , and t(r b ) satisfies the S-Condition.
Proof. As in the above proof, consider v(1) as an analytical function of r b for each n. If the preimage of the zero has infinite points for some n, then the function is identically zero and, by (2.67), there is a contradiction.
This lemma improves somewhat the results of Lemma 2.1, but we do not have an estimate of r 1 − r * here as that in Lemma 2.1.

An Iteration Mapping and Decomposition of the Euler System
In this section we set up an iteration mapping and show that it has a unique fixed point. By the derivations in §2, it is easy to see that any transonic shock solution to (1.1)-(1.3) satisfying those requirements in Theorem 1.1 must be a fixed point of the iteration mapping, which implies its uniqueness claimed in Theorem 1.1. Another motivation to introduce the iteration mapping is for constructing approximate solutions to show the existence of global transonic shock solutions, which require further exploration.
3.1. The Iteration Set. For given ψ ∈ K σ , its position r p and profile ψ p satisfy with constant δ 0 to be chosen later. Given U − satisfying (1.11), for any ψ ∈ K σ andǓ ∈ O δ , we construct a mapping K σ × O δ → K σ × O δ denoted as T (ψ,Ǔ ) = (ψ,Û ) by the following iteration process. Then we show that T has a unique fixed point in K σ × O δ .

3.2.
A Nonlocal Venttsel Problem for the Candidate Pressure in Ω. We first choose ε 0 , σ 0 , and δ 0 small enough such that the formulations in §2 valid. For any ψ ∈ K σ ,Ǔ ∈ O δ , and U − satisfying (1.11), we may express the higher order termsf i andḡ i in terms of U = U + b +Ǔ . By (1.9) and (2.54)-(2.55), we solvep from the following linear nonlocal Venttsel problem: Thanks to Theorem 1.5 in [16] for the Venttsel problem (note that µ 9 < 0) and Theorem 6.6 in [13] for the Dirichlet problem, with the aid of a standard higher regularity argument as in Theorem 6.19 of [13], by considering (r 1 − r b ) 2 e 4p | Ω 0 as a nonhomogeneous term and using interpolation inequalities, for the solutionp ∈ C 3,α (Ω), we have the apriori Schauder estimate with constant C 2 depending only on U b . Next, by Lemma A.4, let u n,m (y) be the eigenfunctions of ∆ ′ on S 2 with respect to the eigenvalues λ n = n(n + 1) ≥ 0, n = 0, 1, 2, · · · . Then p = ∞ n=0 2n+1 m=1 v n,m (y 0 )u n,m (y). Forf 9 = 0 andḡ 8 = 0, each v n,m satisfies the nonlocal differential equation: The S-condition in §2.6 guarantees the nonexistence of a solution to this problem. Thus, v n,m (0) = 0, which impliesp ≡ 0. Therefore, the S-Condition implies the uniqueness of solutions of the Venttsel problem (3.2)-(3.4). Then, by (3.5) and a standard argument of contradiction based on the compactness (cf. Lemma 9.17 in [13]), we have the apriori estimate: for any C 3,α solution of the above nonlocal Venttsel problem. Then, by the method of continuity as carried out in [16], we see that this problem has a unique solutionp ∈ C 3,α (Ω) satisfying estimate (3.12).

3.3.
Update of the Candidate Free Boundary. Once we getp, according to (2.52)-(2.53), we may obtain the new profile of the free boundaryψ p and the positionr p bŷ (3.14) In fact, one may show that S 2ψ p vol = 0 by (3.13) and (3.4); However, we do not need this in the process. We need to improve the regularity ofψ. By (3.4) and (3.13),ψ p satisfies this elliptic equation on S 2 : The right-hand side belongs to C 2,α (S 2 ). Thus, by Theorem 6.19 in [13],ψ =ψ p +r p obviously obeys the estimate: with the aid of (3.12).

3.4.
Solving the Candidate Velocity on Ω 0 . Now we need to solveû on Ω 0 . To this end, by (2.45) and (2.50), we reformulate this problem as where d and d * are respectively the exterior differential and codifferential operator of forms on S 2 . By integrating (3.4) on S 2 , we see that the integration of the right-hand side of (3.18) on S 2 also vanishes. Then, by Lemma A.1, there exists a unique solutionû 1 | Ω 0 ∈ C 3,α 1 (S 2 ) with the estimate: ) Therefore, combining this with (2.46), the velocity on the candidate free boundary is obtained.
3.5. Solving the Candidate Entropy, Density, and Vorticity in Ω. We note that the entropy can be solved according to (2.4) and (2.49) by the following Cauchy problem of the linear transport equations: in Ω, (3.20) By the theory of ordinary differential equations, since u is close to (u 0 ) + b ∂ 0 in C 3,α (Ω), the trajectories of u still fill Ω. We also have the estimate: Hence, we may solve the candidate densityρ + ρ + b by the state function p = A(S)ρ γ . Then .

3.6.
Solving the Lower Regular Candidate Velocity in Ω and on Ω 1 . From (2.5), by subtracting the background solution, we formulate a transport equation of the velocityû in Ω (to distinguish the lower regular velocity obtained here from the candidate velocity in the next subsection, we write u as u l ): (3.25) Here we used that Lû lū With the Cauchy dataû l | Ω 0 as the right-hand side of (3.24), we may uniquely solveû l , particularly its restriction on Ω 1 , i.e.,û l | Ω 1 . We also have an estimate: Solving the Candidate Velocity in Ω. Note thatû l obtained in the above step is only in C 2,α 1 ; it is not our desired candidate velocity. In fact, we will solve the velocityû by the following elliptic equation motivated by (2.12): (3.27) We impose the Dirichlet condition (3.24) on Ω 0 and the Neumann condition on Ω 1 according to (3.25) by (3.28) Note that, in local spherical coordinates, so it does not contain the derivatives ofû l . Therefore, we may solveû (i.e.,û) in Ω by Lemma A.2 to obtain û C 3,α 1 (Ω) ≤ C 2 û l C 2,α 1 (Ω) + dû C 2,α 2 (Ω) + (p,ρ) C 3,α (Ω) + f 12 C 1,α 1 (Ω) + f 11 C 2,α (Ω 1 ) .
Then the mapping T has a unique fixed point if ε 0 is small enough by a simple generalized Banach fixed point theorem. In particular, the uniqueness of the fixed point implies the transonic shock solution of (1.1)-(1.3) satisfying the requirements in Theorem 1.1 is unique, as claimed there. We also note that, although T always has a fixed point as we proved, it is not clear yet whether this fixed point is a solution to (1.1)-(1.3); therefore, in order to obtain the existence result as assumed in Theorem 1.1, it requires further work, which is out of scope of this paper.

Appendix A. Some Notations and Facts of Differential Geometry
In this appendix, we present some notations in differential geometry and some basic facts used above for self-containedness.
A.1. The Metric of M in Local Spherical Coordinates. In local spherical coordinates, for r 0 ≤ x 0 ≤ r 1 , 0 ≤ x 1 < π, −π ≤ x 2 < π, the standard Euclidean metric of M can be written as Hence, √ G := det(G ij ) = (x 0 ) 2 sin x 1 . For the Christoffel symbols, since Γ i jk = Γ i kj , only the following are nonzero: We also use (G ij ) to denote the inverse of the matrix (G ij ), and |u| 2 = G(u, u) = G ij u i u j . In local spherical coordinates, we write the standard metric of S 2 as Therefore, we have √ g := det(g ij ) = sin The nonzero Christoffel symbols are Lemma A.1. There exists a unique ω ∈ A 1 (S 2 ) that solves if χ ∈ A 2 (S 2 ) and ψ ∈ A 0 (S 2 ) satisfy S 2 χ = 0 and S 2 ψ vol = 0.
Proof. Since the first Betti number b 1 of S 2 is 0 (i.e., b 1 = 0), by the Hodge theorem, we can uniquely solve ω via It suffices to show that (A.2) holds. First, since ∆d = d∆, we have In addition, S 2 (dω − χ) = 0 by the Stokes theorem and the assumption. Since the second Betti number b 2 of S 2 is 1, by the Hodge theorem, the space H 2 of harmonic 2-forms on S 2 is onedimensional. Note that vol ∈ H 2 because d * = − * d * and * vol = 1. Therefore, dω − χ = 0. Similarly, we have due to ∆d * = d * ∆, and by the divergence theorem and the assumption. Note that the zero-th Betti number b 0 of S 2 is 1; according to the Hodge theorem, the space H 0 of harmonic functions on S 2 is one-dimensional. One easily sees that 1 ∈ H 0 . Therefore, d * ω − ψ = 0 as desired.
Let ω = ω i dz i . By the Weizenböck formula, holds globally (cf. [12]). Therefore, (A.6)-(A.8) represent three decoupled boundary value problems of the Poisson equations. The uniqueness, existence, and estimates of the solution are then clear.
The following result follows from Lemma 4.6 in [5]. It particularly implies that the norm of a smooth function in a manifold Ω 2 is equivalent to the norm of its pull back in another manifold Ω 1 which is homeomorphic to Ω 2 .
A.3. On the Transport Equations Involving Lie Derivatives in Manifolds. In Section 3.5, we need solve the differential forms from the Cauchy problems of the transport equations involving Lie derivatives. Here we present the basic theorem with a proof. Let M be an n-dimensional closed C ∞ differentiable manifold, M = [0, T ]× M , X a C k+1 -vector field in M which is transverse to Γ t = {t} × M for t ∈ [0, T ], and f a C k -function in M (k is a nonnegative integer). Without loss of generality, we assume that X points to the interior of M when restricted on Γ 0 . We wish to solve a r-form ω (r ≥ 1) in M which satisfies the following problem: Here L is the Lie derivative in M, θ is a given C k r-form in M, and ω 0 is a given point-wise defined r-form of class C k on Γ 0 . We have the following existence and uniqueness results: Lemma A.5. Under the above assumptions, there is a unique r-form ω in M that solves (A.12)-(A.13). In addition, there holds with a positive constant C depending only on f C k (M) and X C k+1 (M) .
For the proof, we first get familiar what (A.12) stands for in a local coordinate chart. Let E be a local coordinate chart of M . ThenẼ = [0, T ] × E is a coordinate chart of M. Iñ E, problem (A.12)-(A.13) is an initial value problem of the transport equations. To see this, for simplicity, suppose that X = X 0 ∂ 0 + X α ∂ α , with X α = 0 for α = 1, · · · , n in M. Since x 0 is a global coordinate, X 0 = dx 0 (X) is a C k -function defined in M. By our assumption, X 0 is positive and bounded away from zero since M is compact.
For the general case that X = X 0 ∂ 0 , equation (A.12) would be a first-order hyperbolic system with n+1 r unknowns. In addition, we can not use the rather simple coordinate charts likeẼ, since the characteristic curves (i.e., integral curves of X) may escapeẼ at some t < T by solving the initial value problem.
We note that the Lie derivative behaves well under the homeomorphisms of differentiable manifolds.
(A. 20) This can be shown by using the Cartan formula L X ω = di X ω + i X dω and the formulae (cf. [12]), where i X ω is the interior product of ω and X. In addition, by Lemma A.3, the C k -norm of a differential form in M is equivalent under C khomeomorphism of the manifold. Thus, to prove Lemma A.5 for the general case, it suffices to straighten the vector field X to the form X 0 ∂ 0 globally by a suitable homeomorphism of M.
Proof. Step 1. LetẼ be a coordinate chart of M as introduced above and X = X i ∂ i with X 0 > 0. Since X 0 is a nonzero function in M, we may define a C k+1 -vector fieldX in M bỹ Step 2. NowX generates a flow φ t in M by theory of ordinary differential equations since M is compact: For any P ∈ M , γ(t) = φ t (P ), t ∈ [0, T ], is a curve in M with initial value γ(0) = (0, P ) ∈ Γ 0 . We then define Φ : M → M by Φ(φ t (P )) = (t, P ), P ∈ M. (A.22) Note that φ t (P ) ∈ Γ t sinceX 0 = 1. We now show that Φ is a homeomorphism.
Step 3. For any Q ∈ M, it is easy to see that there is uniquely a pair (t, P ) with P ∈ M and t ∈ [0, T ] such that φ t (P ) = Q by solving backward the integral curve ofX through Q. So Φ is defined for all the points in M. In addition, Φ is obviously surjective and injective by the uniqueness and existence results of the initial value problem of ordinary differential equations. By continuous dependence on the initial data (0, P ) and t, we see that Φ and Φ −1 are also continuous. IfX ∈ C k+1 , then Φ and Φ −1 are C k+1 -mappings by C k+1 -dependence of solutions of ordinary differential equations on t and initial data. This proves that Φ is a C k+1 -homeomorphism.