The uniqueness of transonic shocks in supersonic flow past a 2-D wedge

J. Math. Anal. Appl. 437 (2016) 194–213

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

The uniqueness of transonic shocks in supersonic flow past a 2-D wedge Beixiang Fang a , Wei Xiang b,∗ a b

Department of Mathematics, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China Department of Mathematics, City University of Hong Kong, Hong Kong, China

a r t i c l e

i n f o

Article history: Received 31 August 2015 Available online 21 December 2015 Submitted by D. Wang Keywords: Uniqueness Transonic shocks 2-D wedge Potential flow equation

a b s t r a c t We have proved the uniqueness of transonic shocks in steady supersonic flows past a slightly perturbed two-dimensional infinite wedge, under appropriate conditions on the downstream subsonic flows. We formulate it to a mathematical problem of the uniqueness of solutions of nonlinear partial differential equations of hyperbolicelliptic mixed type with a free boundary. By working on several elliptic equations of physical quantities separately, we obtain a priori estimates of them, and then prove the uniqueness without assumptions on high regularity. Moreover, uniform estimates on the ellipticity and the positive lower bound of the speed are achieved under a geometrical condition on the wedge. The mathematical ideas and techniques developed here will also be useful for other related problems involving similar analytical difficulties. © 2015 Elsevier Inc. All rights reserved.

Contents 1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Transonic shocks in supersonic flow past against a 2-D wedge . . . . . . . . . . . . . . 1.2. The mathematical formulation of the uniqueness problem . . . . . . . . . . . . . . . . . 1.3. Main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1. A uniqueness theorem for general perturbation . . . . . . . . . . . . . . . . . . 1.3.2. A uniqueness theorem for perturbation satisfying a geometric assumption 1.4. Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. A priori estimates for transonic shock solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. L∞ estimates for θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. L∞ estimates for p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. A priori estimates for the flow field and the shock-front . . . . . . . . . . . . . . . . . . 3. Uniqueness for general perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Uniform estimates and uniqueness for perturbation satisfying a geometric assumption . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

* Corresponding author. E-mail addresses: [email protected] (B. Fang), [email protected] (W. Xiang). http://dx.doi.org/10.1016/j.jmaa.2015.11.067 0022-247X/© 2015 Elsevier Inc. All rights reserved.

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Fig. 1.1. The attached shock-front ahead of a wedge.

1. Introduction 1.1. Transonic shocks in supersonic flow past against a 2-D wedge In this paper we study the uniqueness problem of transonic shocks in steady potential flow past a two-dimensional wedge. It is well-known that a shock-front appears when a supersonic flow past a straight wedge, see Fig. 1.1. In the case that the vertex angle of the wedge is less than some critical value, the shock-front attaches to the tip of the wedge. Moreover, for a given incoming supersonic flow, the flow field behind the shock-front can be determined by solving the Rankine–Hugoniot conditions. It has been shown, for instance in [6,12], that all admissible states behind the shock-front, which satisfies both R–H conditions and the entropy condition, form a loop on the velocity plane. The loop is called shock polar. A part of the shock polar is inside the sonic circle, thus the shock solutions are categorized as supersonic shocks—if the flow behind the shock-front is supersonic, and transonic shocks—if the flow behind the shock-front is subsonic. For a given wedge whose angle is less than the critical value depending on the incoming flow, there are two admissible shock solutions, within which the stronger one is always transonic, and the weaker one may be either supersonic or transonic. It is a longstanding open problem in multidimensional conservation to pick out the right solution from these two entropy solutions. Courant and Friedrichs wrote in their classic monograph Supersonic Flow and Shock Waves [6, p. 317–318] that “· · · · · · , then two oblique shock fronts are possible through which the flow is turned through the angle θK , a weak and a strong one. The question arises which of the two actually occurs. · · · · · · , the problem of determining which of the possible shocks occurs cannot be formulated and answered without taking the boundary conditions at infinity into account. · · · · · · If the pressure prescribed there is below an appropriate limit, the weak shock occurs in the corner. If, however, the pressure at the downstream end is sufficiently high, a strong shock may be needed for adjustment. Under appropriate circumstances this strong shock may begin just in the corner and thus, of the two possibilities mentioned, the one giving a strong shock may actually occur. ······ All statements made here are conjectures so far. While there is little doubt that they are in general correct, they should be supported, if possible, by detailed theoretical investigation.” Thanks to great efforts made by many mathematicians in the past decades, up to now, we already have deep understanding for the supersonic shock solutions, see, for instance, [1–3,5,11,13,15,17,18] and references therein. Compared with them, to our best knowledge, the mathematical theory for the transonic shocks is far away from satisfied. Some progress has been made for the two-dimensional steady flows in Chen–Fang [4], Fang [7], Yin–Zhou [16] and, recently, Fang–Liu–Yuan [8]. In particular, in [4,7,16], it was proved that the transonic shock is conditionally stable under perturbation of the upstream flow and/or perturbation of wedge boundary. In [8], it was proved that the piece-wise constant transonic shock solution is the unique solution when the wedge is straight provided appropriate conditions at the vertex and downstream at the

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infinity. The conditions prescribed in it are helpful to understand the conjecture above made by Courant and Friedrichs. However, it is not clear whether the uniqueness still holds true or not when the wedge is not straight. In this paper, we are going to investigate the uniqueness problem of the transonic shocks for curved wedge, and to try to figure out sufficient conditions which ensure the uniqueness of the transonic shock. 1.2. The mathematical formulation of the uniqueness problem We use the potential flow equations to describe the motion of the fluid, which read  ∂x (ρu) + ∂y (ρv) = 0, ∂x v − ∂y u = 0,

(1.1)

 γ γ 1 2 u + v2 + ργ−1 = κ := , 2 γ−1 γ−1

(1.2)

with the Bernoulli’s law:



where (u, v) is the velocity field, ρ is the density, γ is the adiabatic exponent, the pressure p = ργ , and κ is the Bernoulli constant. By the Bernoulli’s law (1.2), we immediately obtain 1 − γ−1 γ−1 2 q ρ= 1− , 2γ   γ−1 2γ  2γ  1 − ργ−1 = q2 = 1−p γ , γ−1 γ−1



(1.3) (1.4)

 1 where q = u2 + v 2 2 is the speed. Let c = γργ−1 be the local sonic speed. Then it is well-known that the potential equations (1.1) are hyperbolic for supersonic flow with q > c, while

elliptic for subsonic flow 2γ with q < c. By virtue of the Bernoulli’s law, there is a critical sonic speed cso := γ+1 such that the flow is supersonic if and only if q > cso . Assume q > 0 and let θ be the angle between the velocity and x-axis, then we have

u = q cos θ,

v = q sin θ,



and by introducing U = (p, θ) , the equations (1.1) can also be formulated in the matrix form: A (U ) ∂x U + B (U ) ∂y U = 0, where, with M =

(1.5)

q being the Mach number, c

⎡ sin θ 2 ⎢ A (U ) = ⎣ ρq − cos θ

− cos θ

⎤

⎥ , ρq 2 sin θ ⎦ M2 − 1

⎡ cos θ − 2 ⎢ B (U ) = ⎣ ρq − sin θ

Assume a shock front appears at   Γs := (x, y) ∈ R2 : x = ϕs (y) , y > 0 .

− sin θ

⎤

⎥ . ρq 2 cos θ ⎦ − 2 M −1

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Fig. 1.2. The shock polar on u–v plane. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Fig. 1.3. The shock polar on θ–p plane. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Then the Rankine–Hugoniot conditions on the shock front read [ρu] + ϕs [ρv] = 0,

(1.6)

[v] − ϕs [u] = 0,

(1.7)

which can be rewritten as H (U ) := [ρu] [u] + [ρv] [v] = 0, ϕs =

[v] . [u]

(1.8) (1.9)

  0 Let U− = p0− , 0 be a uniform incoming supersonic flow ahead of the shock front. Then (1.8) determines a loop on u–v plane (see, for instance, [6,12]), which is called shock polar, covering all admissible states behind the shock front (see Fig. 1.2). One can observe that the loop is divided by the sonic circle into two parts: the part outside the sonic circle represents supersonic shock solutions with supersonic flow behind the shock front; the other part inside the sonic circle represents transonic shock solutions with subsonic flow behind the shock front. It is worth of pointing out that the shock polar is sometimes translated as a loop, also v called shock polar, in θ–p plane with θ := arctan (see Fig. 1.3). u Now assume a wedge is placed in the flow field with its boundary at   Γw := (x, y) ∈ R2 : y = ϕw (x) , x > 0 , and the rigidity assumption holds on the wedge θ = arctan (∂x ϕw ) := θw (x) .

(1.10)

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Via the shock polar, one can immediately observe that if the wedge is straight and ¯ θw ≡ θ,

(1.11)

0 with θ¯ less than some critical value θ∗ depending on the incoming flow U− , then there exist two admissible shock solutions, represented by the points A and B, both satisfying R–H conditions and the entropy condition (see Fig. 1.2). Within these two solutions, the stronger one represented by the point A is always transonic, while the weaker one represented by the point B can be supersonic or transonic, depending on the value ¯ In case θ¯ < θsonic , where θsonic is the angle corresponding to the intersection point of the sonic circle of θ. and the shock polar, the weaker one B is supersonic. While in case θsonic < θ¯ < θ∗ , B is transonic. Hence, the transonic shock solutions can be divided into two parts: strong transonic shock solutions (the blue part of the shock polar) and weak transonic shock solutions (the red part of the shock polar), see Fig. 1.2 or Fig. 1.3. In [8], it was proved that the piece-wise constant transonic shock solution can be uniquely determined if appropriate conditions at the vertex and downstream infinity are prescribed. In this paper, we are going to study the uniqueness problem described below in case that the wedge is not straight. 0 Uniqueness Problem of Transonic Shock Solutions. Given U− and θw (x). Assume θw (x) is not a constant function and for any x > 0,

θw (x) < θ∗ ,

(1.12)

0 where θ∗ is the critical angle corresponding to U− . Prescribe sufficient conditions at the vertex of the wedge and/or downstream infinity such that there 0 exists at most one transonic shock solution (U− , U+ (x, y); ϕs (y)), satisfying that 1 (i). U+ ∈ C 0 (D ) ∩ Cloc (D \ {O}) satisfies the potential equations (1.1) in D , where

  D := (x, y) ∈ R2 : x > ϕs (y) , y > ϕw (x) is the flow field between the shock front and the wedge; (ii). U+ satisfies the R–H conditions (1.8)–(1.9) on the shock front Γs and the rigidity assumption (1.10) on the wedge Γw ; (iii). U+ is subsonic in D , that is, q+ (x, y) :=



(u+ )2 + v+ )2 < c+ (x, y) :=

γργ−1 + ,

∀(x, y) ∈ D .

1.3. Main theorems In this paper, we shall solve the Uniqueness Problem in case that the wedge is a small perturbation of a straight wedge. The idea is comparing any two transonic solutions and showing that they must coincide. For this purpose, we need to establish a priori estimates for U+ . Since the potential equations are elliptic for subsonic flows in D , the key estimates are L∞ estimates for U+ and we are looking for conditions which are sufficient to establish these estimates. In this paper, we shall prescribe such conditions for two cases: one case is that the perturbation of the wedge is arbitrary, the other is that the perturbation satisfies a geometric assumption. It is natural to impose the following condition on the downstream flow at infinity, with (r, ω) the polar coordinate:

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(H1). The limit of U+ exists as r → ∞, uniformly with respect to ω. As a direct consequence, we need that the limit of θw (x) exists as x → ∞. Assume lim θw (x) = θ¯∞ .

x→∞

(1.13)

Then the state of the downstream flow at infinity is (¯ p∞ , θ¯∞ ) and q¯∞ < cso . ¯ Moreover, for given θ ∈ (θsonic , θ∗ ), both the weaker shock and the stronger shock are transonic shocks, therefore, in order to obtain the uniqueness of the solution, we need one more condition to figure out which one we should choose. It turns out that we should impose different conditions for the strong transonic shocks and for the weak ones: ¯ be the state of the flow behind the shock-front. ¯ := (¯ (H2). Let U p, θ) 0 ¯ ) is a strong transonic shock solution, we impose a state condition at the vertex (i). In case that (U− ; U of the wedge: ¯strong , U+ (0, 0) = (¯ pO , θ¯O ) := U

(1.14)

 0  ¯strong a stronger transonic shock solution; with U− ; U 0 ¯ ) is a weak transonic shock, we impose a state condition at downstream infinity: ; U (ii). In case that (U− ¯weak , U+ (∞) := lim U+ (x, y) = (¯ p∞ , θ¯∞ ) := U r→∞

(1.15)

 0  ¯weak a weaker transonic shock solution. with U− ; U 1.3.1. A uniqueness theorem for general perturbation ¯ We further assume that there We firstly consider the case that θw (x) is a slight general perturbation of θ. is no stationary point in D . Then we have ∀(x, y) ∈ D .

0 < q+ (x, y) < cso ,

(1.16)

Thus, by the assumptions that U+ ∈ C 0 (D ) and that q¯∞ < cso , we have 0 < inf q+ (x, y) ≤ sup q+ (x, y) < cso . D

(1.17)

D

Therefore, it is reasonable to further assume that there exists a positive constant ε0 , such that psonic + ε0 ≤ p+ (x, y) ≤ 1 − ε0 ,

(1.18)

γ  γ−1 2 where psonic = . Then the potential equations are uniformly elliptic in D . γ+1 With the above assumptions, we can obtain the following uniqueness theorem:



Theorem 1.1. Suppose that (H1), (H2), and (1.18) hold. Then there exists a sufficiently small constant ¯ and ε0 , such that for any 0 < σ ≤ σ0 , if the wedge boundary satisfies that σ0 > 0, depending on U   θw − θ¯(0) ≤ σ, 2,α

(1.19)

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 0     then there exists at most one transonic shock solution U− ; U+ ; ϕs such that U+ = (p+ , θ+ ) ∈ C 0 D ∩   C 1 D \ {O} solves the potential equations (1.5) in D , and satisfies the boundary conditions (1.10) on the wedge Γw , the R–H conditions (1.8)–(1.9) on the shock-front Γs . Moreover, the flow field U+ satisfies the estimate   ¯ (0) ≤ Cσ, U+ − U 2,α

(1.20)

¯ and ε0 , where the constant C depends on U (0) ||u||2+α,D

:=

2 

sup

k=0 P ∈D , |β|=k

rPk |Dβ u(P )| +

sup P,Q∈D , |β|=2

2+α rP,Q

Dβ u(P ) − Dβ u(Q) , |P − Q|α

where rP is the distance between the point P and the origin O, and rP,Q = min{rP , rQ }. The shock-front ϕs satisfies the estimate (−1)

ϕs − ϕ¯s 3,α ≤ Cs σ,

(1.21)

¯ and ε0 . where the constant Cs depends on U Remark 1.1. The conditions proposed here is weaker than ones that are used to guarantee the existence in [4,7,16]. The key step in proving Theorem 1.1 is to establish the L∞ estimate for U+ . We firstly establish an L∞ estimate for θ by employing the ellipticity of the equations and carefully analyzing the R–H conditions. The arguments here are similar as the analysis in [8], just with some modification for the case of weaker transonic shocks. Then we establish an L∞ estimate for p, controlled by the perturbation of wedge boundary, via a maximum (minimum) value argument, employing the Hopf-type lemma proved by Finn and Gilbarg (see Lemma 7 in [9]). Once the L∞ estimate for U+ is established, we can apply the classical regularity theory for elliptic equations to obtain a priori estimates (1.20) and (1.21) for U+ and ϕs . Then by comparing two transonic solutions, we can show that they should coincide if the perturbation of the wedge is sufficiently small, which yields the uniqueness of the transonic shock solution. 1.3.2. A uniqueness theorem for perturbation satisfying a geometric assumption In proving Theorem 1.1, it is very important to establish the L∞ estimate for the pressure p such that it can be controlled by the perturbation of wedge boundary. For this purpose, we impose the uniform subsonic condition (1.18) on the flow behind the shock-front. This is not a satisfactory condition because the constant ε0 is given in advance. In case that the wedge is convex, namely, the wedge boundary satisfies that  θw (x) ≥ 0,

(1.22)

then we do not need such a strong condition. Indeed, we are going to further prove the following theorem. Theorem 1.2. Suppose that (H1), (H2), (1.16), and (1.22) hold. Then there exist σ0 > 0, positive constants pˆmin > psonic and pˆmax < 1, such that for any wedges given as in Theorem 1.1, the elliptic equation is strict and uniform, i.e., pˆmin ≤ p+ (x, y) ≤ pˆmax

(1.23)

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where pˆmin and pˆmax does not depend on the specific subsonic flows and σ ∈ (0, θ0 ]. Then Theorem 1.1 holds without the uniform subsonic assumption (1.18). Moreover, if a subsonic–sonic flow is approximated by a sequence of sonic flows, then there must be a sonic point on the shock-front. Remark 1.2. The advantage of the geometric assumption (1.22) is to yield a lower bound for the pressure p, which is independent of ε0 . Therefore, we do not need the uniform parameter ε0 in advance as in (1.18). 1.4. Organization of the paper This paper is organized as follows. In Section 2, we firstly establish the L∞ estimates for the flow angle θ and the pressure p, and then apply the classical elliptic theory to obtain their a priori estimates with weighted Hölder norms. With help of these estimates, in Section 3, we prove the Uniqueness Theorem 1.1 via a contraction argument, in case the wedge is slightly perturbed. In Section 4, we further prove Theorem 1.2 for the case that the perturbation of the wedge is convex. 2. A priori estimates for transonic shock solutions  0  In this section, we are going to establish a priori estimates for a transonic solution U− ; U+ ; ϕs . For the sake of simplicity of notations, from now on, we drop the subscript ‘+ ’ and let U+ = (p, θ)T in D .  0  Proposition 2.1. Suppose that U− ; U+ ; ϕs is a piecewise smooth transonic shock solution corresponding to the wedge boundary Γw . Assume that sup |θw (x)| < θ∗ . x≥0

Suppose that (H1), (H2), and (1.16) hold, then we have the estimate of θ inf θw (x) ≤ θ (x, y) ≤ sup θw (x) ,

x>0

x>0

for all (x, y) ∈ D .

(2.1)

Moreover, if (1.18) holds, then ¯ (0) ¯ (0) ||θ − θ|| 2+α,D ≤ C θw (x) − θ 2+α,Γw .

(2.2)

 In addition, for any σ ∈ (0, σ0 ], such that xθw (x) L∞ (R+ ) ≤ σ, then for any given constant p¯,  sup |p − p¯| ≤ 2(sup |p − p¯| + C||xθw (x)||L∞ (R+ ) ), D

(2.3)

Γs

and (0)

(0)

¯ ||p − p¯||2+α,D ≤ C θw (x) − θ 2+α,Γw .

(2.4)

Remark 2.1. Due to the relation between (p, θ) and (u, v), one can immediately obtain the estimates for (u, v) from (2.2)–(2.4). 2.1. L∞ estimates for θ We firstly establish an L∞ estimate for θ, using the matrix formulation (1.5) of the potential equations. A direct computation yields that the two eigenvalues of the equations (1.5) are

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202

√ M 2 sin θ cos θ + M 2 − 1 , λ+ = (M 2 − 1) cos2 θ − sin2 θ

√ M 2 sin θ cos θ − M 2 − 1 λ− = . (M 2 − 1) cos2 θ − sin2 θ

(2.5)

Obviously, these two eigenvalues are real numbers if and only if M > 1, that is, if and only if the flow is supersonic. Thus the potential equations (1.5) are hyperbolic if M > 1; and if M < 1, that is, the flow is subsonic, the eigenvalues are a pair of conjugate complex numbers, which indicates that (1.5) is elliptic. Hence, for the flow behind the shock-front, the eigenvalues are a pair of conjugate complex numbers: √ M 2 sin θ cos θ ± i 1 − M 2 := λR ∓ λI i, λ± = − (1 − M 2 ) cos2 θ + sin2 θ

(2.6)

where M 2 sin θ cos θ , (1 − M 2 ) cos2 θ + sin2 θ √ 1 − M2 . λI = (1 − M 2 ) cos2 θ + sin2 θ

λR = −

Then by multiplying the associating left eigenvectors  ± to the equations (1.5) and separating the real and imaginary part, we obtain via a direct computation ∂x p + λR ∂y p − ρq 2 (∂x θ + λR ∂y θ) +

ρq 2

∂y θ = 0,

(2.7)

1 − M2 ∂y p = 0. θ + sin2 θ

(2.8)

(1 −

M 2 ) cos2

(1 −

M 2 ) cos2

θ + sin2 θ

The equations (2.7) and (2.8) can be reformulated as Dθ = KDp, 

(2.9)



where Dθ = (∂x θ, ∂y θ) , Dp = (∂x p, ∂y p) , and K = [kij ]2×2

 1 M 2 sin θ cos θ = 2 ρq 1 − M 2 cos2 θ

  1 − 1 − M 2 sin2 θ = 2 K0 , −M 2 sin θ cos θ ρq

or equivalently, Dp = K −1 Dθ,

(2.10)

with K

−1

  = kij 2×2 = −

 2 ρq 2 M sin θ cos θ 1 − M 2 1 − M 2 cos2 θ

  ρq 2 − 1 − M 2 sin2 θ =− K0 . 2 −M sin θ cos θ 1 − M2

Then we can deduce two second order elliptic equations with principle part for p or θ, respectively. ∇ × (KDp) = 0,   ∇ × K −1 Dθ = 0. Then we can establish L∞ estimates for θ.

(2.11) (2.12)

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Lemma 2.1. Suppose 0 < q < cso in D and the hypothesis (H1) and (H2) hold. Assume that sup |θw (x)| < θ∗ . x≥0

Then we have the L∞ estimate (2.1) for θ. In proving Lemma 2.1, we need employ the Hopf-type lemma proved by Finn and Gilbarg (see Lemma 7 in [9]) and we cite it below for convenient of the reader: Lemma 2.2 (Finn–Gilbarg). Let the coefficients of the system −vy = aux + b uy ,

vx = b ux + cuy , 

 2

 = 4ac − (b + b ) ≥ δ > 0

(2.13) (2.14)

be Hölder continuous in the closure of a simply connected domain Ω ⊂ R2 with C 1,α boundary (0 < α < 1), and u, v ∈ C 1 (Ω) is a solution to (2.13). Suppose there is a Q ∈ ∂Ω satisfying u(x) > u(Q) (u(x) < u(Q)) for all x ∈ Ω, then there holds ∂u (Q) > 0, ∂n

(

∂u (Q) < 0) ∂n

(2.15)

where n is the inward drawn normal of the curve ∂Ω at Q. Proof of Lemma 2.1. We shall prove Lemma 2.1 via a maximum principle argument. Since (2.12) is a uniform elliptic equation in divergence form for θ, by applying the strong maximum principle, we immediately obtain that θ cannot attain its supremum and its infimum in D , otherwise θ should be a constant, which contradicts with the boundary condition (1.10) on the wedge. Then we are going to prove, by the method of contradiction, that θ cannot attain its supremum and its infimum on the shock front Γs .  Assume, firstly, that θ attains its supremum θmax at a point Q ∈ Γs . Let τ = (τ1 , τ2 ) be the unit tangential vector of Γs at Q directing to infinity and τ ⊥ is the unit outer normal. Then we have, at Q, ∂τ θ = τ  Dθ = τ  KDp = 0,

(2.16)

and, by employing the Hopf-type Lemma 2.2 for uniformly elliptic equations, that   ∂τ ⊥ θ = τ ⊥ Dθ > 0. Moreover, we also have ∂τ p = τ  Dp = τ  K −1 Dθ = −

ρq 2 τ  K0 Dθ. 1 − M2

Since 

τ K0 τ

⊥



M 2 sin θ cos θ = (τ1 , τ2 ) 1 − M 2 cos2 θ

    −τ2 − 1 − M 2 sin2 θ τ1 −M 2 sin θ cos θ 2

= −1 + M 2 (τ1 sin θ − τ2 cos θ) ≤ −1 + M 2 < 0,

(2.17)

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we obtain, via (2.16) and (2.17), that, at Q, ∂τ p = −

ρq 2 τ  K0 Dθ > 0. 1 − M2

(2.18)

On the other hand, differentiating the boundary condition (1.8) along the shock front Γs , we obtain that at Q, 0 = ∂τ H = ∂p H · ∂τ p + ∂θ H · ∂τ θ = ∂p H · ∂τ p. 

If ∂p H = 0, that is, recalling the shock polar in θ–p plane, θ (Q) = θmax = ±θ∗ with U∗± = (p∗ , ±θ∗ ) the transonic shock solution for the critical angle, then we should have ∂τ p = 0, which contradicts with (2.18). Thus, we have ∂p H = 0, which yields that θ (Q) = θmax = θ∗ and p (Q) = p∗ . Then we are going to show that this is also impossible. To this aim, we need the state condition of the flow at the vertex of the wedge or 0 ¯ ) is a strong transonic at the downstream infinity, according that whether the background solution (U− ; U shock solution or a weak transonic shock solution. 0 ¯ ) is a strong transonic shock solution, by (H2), we have that (1.14) holds with ; U (i). In case that (U− p(O) = p¯O > p∗ = p(Q). Then there must exist a point Qs on the shock-front Γs such that θ (Qs ) = θmax = θ∗ and p > p∗ for all the points on Γs between O and Qs . Hence, we have ∂τ p ≤ 0 at Qs , which also contradicts with (2.18). This contradiction indicates that θ cannot attain its supremum on Γs . 0 ¯ (ii). In case that (U− ; U ) is a weak transonic shock solution, by (H2), we have that (1.15) holds with p(∞) = p¯∞ < p∗ = p(Q). Then there must exist a point Qw on the shock-front Γs such that θ (Qw ) = θmax = θ∗ and p < p∗ for all the points on Γs between Qw and infinity. Hence, we also have ∂τ p ≤ 0 at Qw , which also contradicts with (2.18). This contradiction indicates that θ cannot attain its supremum on Γs . 

˜ ∈ Γs . Let τ = (τ1 , τ2 ) be the unit tangential Then we assume that θ attains its infimum θmin at a point Q ⊥ ˜ ˜ vector of Γs at Q directing to infinity and τ is the unit outer normal. Then we have, at Q, ∂τ θ = τ  Dθ = τ  KDp = 0,

(2.19)

and, by employing the Hopf-type Lemma 2.2 for uniformly elliptic equations, that   ∂τ ⊥ θ = τ ⊥ Dθ < 0.

(2.20)

˜ Again, since τ  K0 τ ⊥ < 0, we obtain, via (2.19) and (2.20), that, at Q, ∂τ p = −

ρq 2 τ  K0 Dθ < 0. 1 − M2

(2.21)

On the other hand, differentiating the boundary condition (1.8) along the shock front Γs , we obtain that ˜ at Q, 0 = ∂τ H = ∂p H · ∂τ p + ∂θ H · ∂τ θ = ∂p H · ∂τ p.   ˜ = θmin = ±θ∗ , then we should have ∂τ p = 0, which contradicts with (2.21). Thus, If ∂p H = 0, that is, θ Q     ˜ = θmin = −θ∗ and p Q ˜ = p∗ . Analogous to the arguments we have ∂p H = 0, which yields that θ Q for the case of supremum, we need to show that this is also impossible. Indeed, (2.21) implies that p is strictly decreasing at any infimum point on the shock-front Γs . Therefore, by the Rankine–Hugoniot

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conditions and the shock polar, whenever θ attains its infimum θmin = −θ∗ , towards the infinity, the value of θ should run into the interval (−θ∗ , −θsonic ), which contradicts with the asymptotic hypothesis (H1): lim θ (x, y) = lim θw (x) > 0. This contradiction indicates that θ cannot attain its infimum on Γs . r→∞

x→∞

Concluding the above arguments, we obtain the a priori estimate (2.1) for θ. 2

2.2. L∞ estimates for p Based on the L∞ estimate (2.1) for θ, we can consider the estimate for p. Indeed, we immediately obtain that along Γs , since (p, θ) satisfy the jump condition H(p, θ) = 0, pmin ≤ p ≤ pmax ,

along Γs ,

(2.22)

where (pmin , pmax ) is the smallest and the biggest value of p satisfying that there exists a θw ∈ [inf x>0 θw (x), supx>0 θw (x)] such that H(p, θw ) = 0. Then via another maximal principle argument for p, we obtain the following L∞ estimate of p: Lemma 2.3. Suppose that (H1), (H2), and (1.18) hold. Then there exists a sufficiently small constant σ0 > 0, ¯ and ε0 , such that for any 0 < σ ≤ σ0 , if the wedge boundary satisfies that depending on U  xθw (x) L∞ (R+ ) ≤ σ,

then we have the estimate  sup |p − p¯| ≤ 2(sup |p − p¯| + C||xθw (x)||L∞ (R+ ) ). D

(2.23)

Γs

Proof. For the given solutions, from (2.7)–(2.8), taking derivatives and using the fact that θxy = θyx , we obtain a uniformly second order elliptic equation for p as follows: 2 

∂j (aij ∂i p) = 0,

(2.24)

i,j=1

where (∂1 , ∂2 ) = (∂x , ∂y ), and a11 =

(1 − M 2 ) cos2 θ + sin2 θ , ρq 2

a12 = a21 = − a22 =

M 2 sin θ cos θ , ρq 2

(1 − M 2 ) + M 4 sin2 θ cos2 θ , ρq 2 [(1 − M 2 ) cos2 θ + sin2 θ]

with the property that 1 aij ξi ξj = 2 ρq



(1 − M 2 )ξ22 I1 (ξ1 , ξ2 ) + (1 − M 2 ) cos2 θ + sin2 θ

 ≥ λ(1 − M 2 )(ξ12 + ξ22 ),

where

I1 (ξ1 , ξ2 ) = [

(1 − M 2 ) cos2 θ + sin2 θξ12 −

M 2 sin θ cos θ (1 − M 2 ) cos2 θ + sin2 θ

ξ2 ]2

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and the constant λ > 0 is universal. Noted that ∂j (aij ) =

∂aij ∂aij ∂j p + ∂j θ, ∂p ∂θ

again by employing (2.7)–(2.8) to replace ∂j (θ) by ∇p, we have ∂j (aij ) = (2.24) can be rewritten as 2 

2

k=1 bik ∂k p.

Therefore, equation

(aij ∂ij p + bij ∂i p∂j p) = 0.

(2.25)

i,j=1

Next along Γw , taking tangential derivatives along Γw yields an oblique boundary condition for p as β · ∇p = g

along Γw ,

(2.26)

where β1 = − sin(θw (x)), β2 = cos(θw (x)) and β · ν > C −1 > 0 for inner normal direction ν of Γs , and  g = −ρq 2 cos(θw (x))θw (x). Now, we can show the L∞ bounds of p. First consider wb = C1 − C2 eμω , where ω = arctan xy which is bounded. Then aij wb,ij + bij wb,i wb,j C2 μ2 eμω (a11 sin2 ω − 2a12 sin ω cos ω + a22 cos2 ω) r2 C2 μeμω + (−2a11 sin ω cos ω + 2a22 sin ω cos ω + 2a12 cos2 ω − 2a12 sin2 ω) r2 C 2 μ2 e2μω (b11 sin2 ω − 2b12 sin ω cos ω + b22 cos2 ω) + 2 2 r C2 μeμω ≤− [μ(λ − C2 eμω Λ) − Λ] ≤ −¯ c<0 r2

=−

if μ large and C2 small depending on Λ/λ. Next along Γw , βi wb,i = −

γ¯ C2 μeμω β · ν ≤ − < 0, r r

where γ¯ depends on the elliptic ratio Λ/λ. Finally choose C1 large such that wb ≥ w ¯ > 0. Actually, we can choose C2 small enough such that C1 = 1 and w ¯ = 1/2. Let p = vwb , then v satisfies equation (aij vi )j +

aij wb,j aij wb,i aij wb,ij + bij wb,i wb,j vi + vj + v=0 wb wb wb

with boundary condition along Γw , β · ∇v +

β · ∇wb g v= . wb wb

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Obviously, v cannot obtain its maximum point at the interior point of D , and if it obtains its maximum point at x0 ∈ Γw , then v(x0 ) =

g β · ∇v g |r g| . − wb ≤ ≤ β · ∇wb β · ∇wb β · ∇wb γ¯

Thus from the bounds of p along Γs and the uniform bounds of wb , we have that max |v| ≤ max |v| + C|rg|∞ , ¯ D

Γs

then sup |p| ≤ 2(sup |p| + C|rg|∞ ), ¯ D

Γs

where constant C depends only on the elliptic ratio Λ/λ. Using p − p¯ to replace p, where p¯ is a constant, and noticing that the equation and boundary conditions for p − p¯ is the same as that for p, we finally obtain that sup |p − p¯| ≤ 2(sup |p − p¯| + C|rg|∞ ),

(2.27)

 sup |p − p¯| ≤ 2(sup |p − p¯| + C||xθw (x)||L∞ (R+ ) ).

(2.28)

¯ D

Γs

so

¯ D

Γs

By (2.22), supΓs |p − p¯| depends only on θ, therefore, by (2.1), supΓs |p − p¯| essentially depends only on the wedge angle θw (x) and the given constant p¯. We proved the L∞ estimate of p. 2 2.3. A priori estimates for the flow field and the shock-front Based on the L∞ -estimates of p and θ, we can work on their higher regularity estimates and then derive the a priori estimate for the shock-front as follows. Lemma 2.4. Under the assumptions of Theorem 1.1, there exists a constant σ0 > 0, depending on the  0  ¯ ; θ¯ , such that for any 0 < σ ≤ σ0 , if (1.19) holds, then for unperturbed transonic shock solution U− ; U any constants u ¯ and v¯, we have (0) (0) ¯ (0) ||u − u ¯||2+α,D + ||v − v¯||2+α,D ≤ C θw (x) − θ 2+α,Γw .

(2.29)

Moreover, if Γs = {(x, y) | x = ϕs (y)}, then ¯ ||ϕs − ϕ¯s ||2+α,R+ ≤ C θw (x) − θ 2+α,Γw , (0)

(0)

(2.30)

[v] where ϕ¯s = [u] , with the solution (u, v) in the subsonic region is replaced by (¯ u, v¯). The same estimate is valid to the case that Γs = {(x, y) | y = ϕs (x)}.

Proof. It is convenient to study it based on the potential flow equation of second order directly. From the second equation of (1.1), we have that ϕx = u and ϕy = v. Replace ρ by ∇ϕ by employing the Bernoulli’s law (1.3), the potential flow equation of second order is

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(c2 − ϕ2x )ϕxx − 2ϕx ϕy ϕxy + (c2 − ϕ2y )ϕyy = 0,

(2.31)

with the boundary condition ϕν = 0,

on Γw

and [ϕ] = [ρϕν ] = 0,

on Γs ,

where the bracket [f ] denotes the difference of function f on the shock Γs . Taking the derivatives on the potential flow equation with respect to x, we then have the equation to u = ϕx as follows: (c2 − u2 )uxx − 2uvuxy + (c2 − v 2 )uyy + bu11 u2x + bu12 ux uy + bu22 u2y = 0

(2.32)

where bu11 = −

[2c2 + (γ − 1)u2 − (γ + 1)v 2 ]u , c2 − v 2

bu12 = −

[(3 − γ)u2 − (γ + 1)v 2 ]v , c2 − v 2

bu22 = −

2(c2 + γv 2 )u . c2 − v 2

Now the boundary condition for u along Γw is  β1u ux + β2u uy − (c2 − v 2 )θw (x)u = 0,

(2.33)

where β1u = −(2c2 − u2 − v 2 ) sin2 θw cos θw , β2u = [(c2 − v 2 )(cos2 θw − sin2 θw ) + 2uv sin θw cos θw ] cos θw . For inner normal ν w = (− sin θw , cos θw ), we easily have that (β1u , β2u ) · ν w ≥ C −1 > 0. √ From (2.1) and (2.23), |∇ϕ| = u2 + v 2 ≤ C, where the constant C only depends on the wedge. By employing the results in [14], we have the C α regularity of ∇ϕ. Then by classic Schauder estimate away from the wedge corner O and infinity, for any constant u ¯, we have (2.29) locally. And also for any point P ∈ D , there exists a radius r which only depends on dist{P, O} linearly, such that Br (P ) ⊂ D¯ , we can also treat this r as the distance of P and O for simplicity. Then from (6.50) in [10], choosing β = 0, and noticing that along Γw , r ∼ x, we have the estimate (2.29) for u. Similarly, we can prove the similar result to v = ϕy , that is the estimate (2.29). Finally, we are going to estimate the free boundary Γs . Without loss of the generality, let us assume that Γs = {(x, y) | x = ϕs (y)}, because otherwise we can assume that Γs = {(x, y) | y = ϕs (x)}. By the Rankine–Hugoniot condition (1.9) and the estimates (2.29), one can easily prove that for the constant vector (¯ u, v¯), the estimate (2.30) holds. 2

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3. Uniqueness for general perturbation Based on the a priori estimates obtained in the last section, now we can study the uniqueness of transonic shocks in supersonic flow past a 2-D wedge. Actually, we will prove the following lemma. Lemma 3.1. Let the constant σ0 > 0 be as in Lemma 2.3, and for any 0 < σ ≤ σ0 , (1.19) holds. If there are two solutions (p1 , θ1 ) and (p2 , θ2 ), both of which satisfy the assumptions of Theorem 1.1, then the two solutions are equal to each other, i.e., (p1 , θ1 ) = (p2 , θ2 ). Proof. First since there is a free boundary, i.e., Γs , so we need introduce a coordinate transformation to flatten the boundary by x = x − ϕs (y),

y  = y − ϕw (x).

(3.1)

Notice that if ϕw (x) and |rg|∞ small, then by the estimates obtained above, ϕs (y) is small. So this transformation is well-defined. Then the domain D becomes the first quadrant, while Γs becomes a straight line x = 0, and Γw becomes another straight line y  = 0. In addition, the relation between (x, y) and (x , y  ) is ∂x = 1, ∂x

∂x [v] = −ϕs (y) = − ; ∂y [u]

∂y  = −ϕw (x) , ∂x

∂y  = 1. ∂y

(3.2)

If ϕw (x) is small enough, then the Jacobi J = 1 − ϕs (y)ϕw (x) > 0, so the mapping (x, y) to (x , y  ) is well-defined and reversible. In (x , y  )-coordinate, (2.7) and (2.8) becomes ∂x θ =

(1 − M 2 )ϕs (y) + [1 − λR ϕs (y)][ϕw (x) − λR ][(1 − M 2 ) cos2 θ + sin2 θ]2 ∂x p ρq 2 [(1 − M 2 ) cos2 θ + sin2 θ][1 − ϕs (y)ϕw (x)] −

∂y  θ =

(1 − M 2 ) + [ϕw (x) − λR ]2 [(1 − M 2 ) cos2 θ + sin2 θ]2 ∂y  p ρq 2 [(1 − M 2 ) cos2 θ + sin2 θ][1 − ϕs (y)ϕw (x)]

(3.3)

(1 − M 2 )[ϕs (y)]2 + [1 − λR ϕs (y)]2 [(1 − M 2 ) cos2 θ + sin2 θ]2 ∂x p ρq 2 [(1 − M 2 ) cos2 θ + sin2 θ][1 − ϕs (y)ϕw (x)] −

(1 − M 2 )ϕs (y) + [1 − λR ϕs (y)][ϕw (x) − λR ][(1 − M 2 ) cos2 θ + sin2 θ]2 ∂x p. ρq 2 [(1 − M 2 ) cos2 θ + sin2 θ][1 − ϕs (y)ϕw (x)]

 Then we have the first order systems for p and θ in the domain D = {(x , y  ) x ≥ 0, y  ≥ 0} as 

∂x θ + A12 ∂x p + A22 ∂y p = 0, ∂y θ − A11 ∂x p − A12 ∂y p = 0

where A11 =

(1 − M 2 )[ϕs (y)]2 + [1 − λR ϕs (y)]2 [(1 − M 2 ) cos2 θ + sin2 θ]2 ; ρq 2 [(1 − M 2 ) cos2 θ + sin2 θ][1 − ϕs (y)ϕw (x)]

A12 = − A22 =

(1 − M 2 )ϕs (y) + [1 − λR ϕs (y)][ϕw (x) − λR ][(1 − M 2 ) cos2 θ + sin2 θ]2 ; ρq 2 [(1 − M 2 ) cos2 θ + sin2 θ][1 − ϕs (y)ϕw (x)]

(1 − M 2 ) + [ϕw (x) − λR ]2 [(1 − M 2 ) cos2 θ + sin2 θ]2 . ρq 2 [(1 − M 2 ) cos2 θ + sin2 θ][1 − ϕs (y)ϕw (x)]

For any (ξ1 , ξ2 ) ∈ R2 ,

(3.4)

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Aij ξi ξj =

(1 − M 2 )(ϕs (y)ξ1 − ξ2 )2 + I2 (ξ1 , ξ2 ) , ρq 2 [(1 − M 2 ) cos2 θ + sin2 θ][1 − ϕs (y)ϕw (x)]

where I2 (ξ1 , ξ2 ) = [(1 − λR ϕs (y))ξ1 − (ϕw (x) − λR )ξ2 ]2 [(1 − M 2 ) cos2 θ + sin2 θ]2 , so λ|ξ|2 ≤ Aij ξi ξj ≤ Λ|ξ|2 , where constants 0 < λ < Λ depend on the positive lower bound and the upper bound of ϕs (y), and here ϕw (x) is small enough. For the ellipticity, notice that if we denote these system as Ljk (x, D)uk = 0, where uk = (θ, p), then L11 = ∂x ,

L12 = A12 ∂x + A22 ∂y ,

L21 = ∂y ,

L22 = −A11 ∂x − A12 ∂y .

If let λ = (λ1 , λ2 ), then  λ |detL(x, λ)| = |  1 λ2

 A12 λ1 + A22 λ2  | = Aij λi λj . −A11 λ1 − A12 λ2 

So detL(x, λ) is not zero for any real non-zero λ. It means the system we derived above is elliptic. ¯ is the constant solution to the case that ϕw (x) = 0, From now on, assume that xϕw (x) small and (¯ p, θ) and also assume that (p1 , θ1 ) and (p2 , θ2 ) are two different solutions to this wedge problem with same boundary conditions. Let p = p1 − p2 and θ = θ1 − θ2 , then from the a priori estimates obtained in last ¯ 2+α are small, while |∇(p, θ)| is small and decays to 0 at infinity. section, |pi − p¯|2+α and |θi − θ| ¯ and subtract the two systems which Now linearize the system around the background solution (¯ p, θ) (pi , θi ) satisfies respectively, then  ¯ p¯)∂x p + A22 (θ, ¯ p¯)∂y p = f1 ∂x θ + A12 (θ, ¯ p¯)∂x p − A12 (θ, ¯ p¯)∂y p = f2 ∂y θ − A11 (θ,

(3.5)

where f1 = (A12 (θ1 , p1 ) − A12 (θ2 , p2 ))∂x p + (A22 (θ1 , p1 ) − A22 (θ2 , p2 ))∂y p ¯ p¯) − A12 (θ1 , p1 ))∂x p + (A22 (θ, ¯ p¯) − A22 (θ1 , p1 ))∂y p + (A12 (θ, and f2 = −(A11 (θ1 , p1 ) − A11 (θ2 , p2 ))∂x p − (A12 (θ1 , p1 ) − A12 (θ2 , p2 ))∂y p ¯ p¯) − A11 (θ1 , p1 ))∂x p + (A12 (θ, ¯ p¯) − A12 (θ1 , p1 ))∂y p + (A11 (θ, ¯ p¯)| is small, so |fi | is bounded by |∇θ| + |∇p| with small coefficients. Then Notice that |∇p| and |(pi , θi ) − (θ, by the standard local estimates of elliptic system with constant coefficients, we have (0)

(0)

(0)

(0)

p1 − p2 1+α,D + θ1 − θ2 1+α,D ≤ C( p1 − p2 1+α,D + θ1 − θ2 1+α,D ),

(3.6)

¯ So p1 − p2 where the small constant  depending on the smallness of the L∞ bounds of pi − p¯ and θi − θ. and θ1 − θ2 can only be identically 0, if  small enough such that C < 1. Thus we prove that there exists at most one transonic shocks in supersonic flow past a 2-D wedge if the perturbation of the wedge is small and also if the state at the infinity is a constant as assumed in (H1) and (H2). 2

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4. Uniform estimates and uniqueness for perturbation satisfying a geometric assumption In this subsection, we are going to study the uniform estimates which only depend on the behavior of the subsonic flows along the shock and at the infinite, and then to analyze the position of sonic points. Actually what we will show is that Lemma 4.1. Under the assumptions of Theorem 1.2, there exists a constant σ0 > 0, depending on the  0  ¯ ; θ¯ , such that for any 0 < σ ≤ σ0 , if unperturbed transonic shock solution U− ; U  xθw (x) L∞ (R+ ) ≤ σ,

then there exist constants 0 < qˆmin < qˆmax < cso which does not depend on the wedge and σ, such that qˆmin ≤ q ≤ qˆmax .

(4.1)

Moreover, if we have two solutions (p1 , θ1 ) and (p2 , θ2 ) such that psonic < pi < 1, for i = 1 or 2, then the two solutions are equal to each other, i.e., (p1 , θ1 ) = (p2 , θ2 ). Proof. First, consider the upper bound. From the Bernoulli law (1.2), the upper bounds of q will be controlled if the lower bounds of p is obtained. From the boundary condition (2.26) and the geometric assumption that  θw (x) ≥ 0, the local minimum value of p cannot be obtained on Γw . In fact, if the local minimum value of p is achieved at P ∈ Γw , then by the Hopf’s lemma, pν > 0 and pτ = 0 at P , where ν is the inner normal.  So β · ∇p > 0 at P . It is impossible due to (2.26) and the assumption that θw (x) ≥ 0. Now from the equation, we have that p ≥ minΓs p, that means q ≤ max q. Γs

By the theory of shock polar, there exists a constant qˆmax depending only on θw (x), such that q ≤ qˆmax < qcr . Remarkably, we have also proved an interesting fact that if a subsonic–sonic flow is approximated by a sequence of sonic flows, then there must be a sonic point at the shock Γs . Next consider that there is no stagnant point in the transonic flow. It will be done by showing that the horizontal velocity u is strictly positive. Obviously in the supersonic region, along Γs and at the infinity, u > 0. In the subsonic domain D , by the equation (2.32), u cannot obtain its minimum value inside the subsonic domain. Along Γw , u > 0 at the corner and at the infinity from Theorem 1.1, so if u < 0 at a point in Γw , then there must be a point in Γw , such that u = 0. By the slip boundary condition of (u, v) along the wedge, v = 0 at this point. So q = 0 at the same point too. It contradicts to the assumption (1.16). So we finally have that in the subsonic region u > 0.  Now we are going to show that if θw (x) ≥ 0, then u has a uniform lower positive bound which only depends on the ellipticity and the lower and upper bounds of the wedge angle θw (x). From the existence of qˆmax < qcr , the lower positive bound will only depend on inf x>0 θw (x) and supx>0 θw (x). (n) Actually, if it is not true, then we can construct a sequence θw with the corresponding solutions u(n) (n) such that the lower bound of u goes to zero. From the regularity, the sequence has a limit θw with the corresponding solution u, the lower bound of which is 0 at Γw . It is a contradiction to the fact that u cannot

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obtain its non-positive minimum value along Γw , by the Hopf’s lemma with the boundary condition (2.33)  (x) ≥ 0. So we have and the assumption that θw q ≥ qˆmin > 0,  where qˆmin only depends on inf x>0 θw (x) and supx>0 θw (x), if θw ≥ 0. Therefore, by (1.4), we know that the constants pˆmin and pˆmax exist, and only depends on the incoming supersonic flow and the upper and the lower bounds of the wedge angle. Finally, we need to show that the two constants does not depend on σ. Obviously, qˆmax does not depend on σ when σ0 is small enough. For qˆmin , we can easily run another contradiction argument with the Hopf’s lemma as done in the precedent paragraph to show that it does not depend on σ too. So with the additional assumption on the wedge, we proved that the flow are strictly subsonic, i.e., there exists ε0 > 0, such that psonic + ε0 ≤ p ≤ 1 − ε0 . Then we can adopt the uniqueness result for the general wedges to prove the uniqueness stated in this lemma. It completes the proof of this lemma. 2

As said in the proof, we actually proved the following interesting fact about possible sonic points. Remark 4.1. If the wedge satisfies the geometric assumption listed in Lemma 4.1, and if a subsonic–sonic flow is approximated by a sequence of sonic flows, then there must be a sonic point at the shock Γs . Acknowledgments The research of Beixiang Fang was supported in part by National Natural Science Foundation of China under Grant Nos. 11031001 and 11371250, Shanghai Jiao Tong University’s Chenxing SMC-B Project, the Shanghai Committee of Science and Technology (Grant No. 15XD1502300), and Shanghai Pujiang Program 12PJ1405200. The research of Wei Xiang was supported in part by the EPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE (EP/E035027/1), the CityU Start-Up Grant for New Faculty 7200429(MA), and by the General Research Fund of Hong Kong under GRF/ECS Grant 9048045 (CityU 21305215). This work started during Beixiang Fang’s visit to Oxford Center for Nonlinear Partial Differential Equations from March, 2012 to February, 2013, which is supported by the State Scholarship Fund of China Scholarship Council under File No. 2011831005. References [1] S.-X. Chen, Existence of local solution to supersonic flow past a three-dimensional wing, Adv. in Appl. Math. 13 (1992) 273–304. [2] S.-X. Chen, Asymptotic behavior of supersonic flow past a convex combined wedge, Chin. Ann. Math. Ser. B 19 (1998) 255–264. [3] S.-X. Chen, Global existence of supersonic flow past a curved convex wedge, J. Partial Differ. Equ. 11 (1998) 73–82. [4] S.-X. Chen, B. Fang, Stability of transonic shocks in supersonic flow past a wedge, J. Differential Equations 233 (2007) 105–135. [5] G.-Q. Chen, Y.-Q. Zhang, D.-W. Zhu, Existence and stability of supersonic Euler flows past Lipschitz wedges, Arch. Ration. Mech. Anal. 181 (2006) 261–310. [6] R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Springer-Verlag, New York, 1948. [7] B.-X. Fang, Stability of transonic shocks for the full Euler system in supersonic flow past a wedge, Math. Methods Appl. Sci. 29 (2006) 1–26. [8] B.-X. Fang, L. Liu, H.R. Yuan, Global uniqueness of transonic shocks in two-dimensional steady compressible Euler flows, Arch. Ration. Mech. Anal. 207 (2013) 317–345. [9] R. Finn, D. Gilbarg, Asymptotic behavior and uniquenes of plane subsonic flows, Comm. Pure Appl. Math. 10 (1957) 23–63. MR0086556. [10] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, 3rd ed., Springer-Verlag, Berlin, Heidelberg, New York, and Tokyo, 1997. [11] C.-H. Gu, A method for solving the supersonic flow past a curved wedge, Fudan J. (Nature Sci.) 7 (1962) 11–14.

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