Transonic Flows with Shocks Past Curved Wedges for the Full Euler Equations

We establish the existence, stability, and asymptotic behavior of transonic flows with a transonic shock past a curved wedge for the steady full Euler equations in an important physical regime, which form a nonlinear system of mixed-composite hyperbolic-elliptic type. To achieve this, we first employ the coordinate transformation of Euler-Lagrange type and then exploit one of the new equations to identify a potential function in Lagrangian coordinates. By capturing the conservation properties of the Euler system, we derive a single second-order nonlinear elliptic equation for the potential function in the subsonic region so that the transonic shock problem is reformulated as a one-phase free boundary problem for a second-order nonlinear elliptic equation with the shock-front as a free boundary. One of the advantages of this approach is that, given the shock location or quivalently the entropy function along the shock-front downstream, all the physical variables can expressed as functions of the gradient of the potential function, and the downstream asymptotic behavior of the potential function at the infinite exit can be uniquely determined with uniform decay rate. To solve the free boundary problem, we employ the hodograph transformation to transfer the free boundary to a fixed boundary, while keeping the ellipticity of the second-order equations, and then update the entropy function to prove that it has a fixed point. Another advantage in our analysis here is in the context of the real full Euler equations so that the solutions do not necessarily obey Bernoulli's law with a uniform Bernoulli constant, that is, the Bernoulli constant is allowed to change for different fluid trajectories.

1. Introduction. We are concerned with the existence, stability, and asymptotic behavior of steady transonic flows with transonic shocks past curved wedges for the full Euler equations. The two-dimensional steady, full Euler equations for polytropic gases have the form (cf. [14,15,25]):    ∇ · (ρu) = 0, ∇ · (ρu ⊗ u) + ∇p = 0, ∇ · Ä ρu(E + p ρ ) where ∇ = ∇ x is the gradient in x = (x 1 , x 2 ) ∈ R 2 , u = (u 1 , u 2 ) the velocity, ρ the density, p the pressure, and E = 1 2 |u| 2 + e the total energy with internal energy e. Choose density ρ and entropy S as the independent thermodynamical variables. Then the constitutive relations can be written as (e, p, θ) = (e(ρ, S), p(ρ, S), θ(ρ, S)) governed by θdS = de − p ρ 2 dρ, where θ represents the temperature. For a polytropic gas, (1.2) or equivalently, p = p(ρ, S) = κρ γ e S/cv , e = e(ρ, S) = κ γ − 1 ρ γ−1 e S/cv , where R > 0 may be taken to be the universal gas constant divided by the effective molecular weight of the particular gas, c v > 0 is the specific heat at constant volume, γ > 1 is the adiabatic exponent, and κ > 0 is any constant under scaling. The sonic speed of the flow for polytropic gas is The flow is subsonic if |u| < c and supersonic if |u| > c. For a transonic flow, both cases occur in the flow, and then system (1.1) is of mixed-composite hyperbolicelliptic type, which consists of two equations of mixed elliptic-hyperbolic type and two equations of hyperbolic type. System (1.1) is a prototype of general nonlinear systems of conservation laws: where U : R n → R m is unknown, while F : R m → M m×n is a given nonlinear mapping for the m × n matrix space M m×n . For (1.1), we may choose U = (u, p, ρ). The systems with form (1.5) often govern time-independent solutions for multidimensional quasilinear hyperbolic systems of conservation laws; cf. Lax [20,21]. It is well known that, for a steady upstream uniform supersonic flow past a straight-sided wedge whose vertex angle is less than the critical angle, there exists a shock-front emanating from the wedge vertex so that the downstream state is either subsonic or supersonic, depending on the downstream asymptotic condition at infinity (see Appendix B and Fig. 2.1 for the shock polar). The study of two-dimensional steady uniform supersonic flows past a straight-sided wedge can date back to the 1940s (cf. Courant-Friedrichs [15]; also see Prandtl [26] and von Neumann [28]).
For the case of supersonic-supersonic shock (i.e. both states of the shock are supersonic), local solutions around the curved wedge vertex were first constructed by Gu [19], Li [23], Schaeffer [27], and the references cited therein. Global potential solutions are constructed in [6,7,15,29] when the wedge has certain convexity or the wedge is a small perturbation of the straight-sided wedge with fast decay in the flow direction. Furthermore, the stability and uniqueness of entropy solutions in BV containing the strong supersonic-supersonic shock were established in Chen-Li [4].
For the case of supersonic-subsonic shock (i.e. transonic shock-front), the stability of these fronts under a perturbation of the upstream flow, or a perturbation of wedge boundary, has been studied in Chen-Fang [11] for the potential flow and in Fang [16] for the Euler flow with a uniform Bernoulli constant. In particular, the stability of transonic shocks in the steady Euler flows with a uniform Bernoulli constant was first established in the weighted Sobolev norms in Fang [16], even though the downstream asymptotic decay rate of the shock slope at infinity was not derived.
In this paper, one of our main objectives is to deal with the asymptotic behavior of steady transonic flows with a transonic shock past a curved wedge for the full Euler equations, especially the uniform decay rate of the transonic shock slope and the subsonic flows downstream at infinity. For a fixed uniform supersonic state U − 0 , there is an arc on the shock polar corresponding to the subsonic states; see Fig. 2.1. When the wedge angle is less than the critical angle θ c w , the tangential point T corresponding to the critical angle divides arcHS into the two open arcsT S and T H. The nature of these two cases is very different. In this paper, we focus mainly on the stability of transonic shocks in the important physical regimeT S when the wedge angle is between the sonic angle θ s w and the critical angle θ c w > θ s w . To achieve this, we first rewrite the problem in Lagrangian coordinates so that the original streamlines in Eulerian coordinates become straight lines and the curved wedge boundary in Eulerian coordinates becomes a horizontal ray in Lagrangian coordinates. Then we exploit one of the new equations to identify a potential function φ in Lagrangian coordinates. By capturing the conservation properties of the Euler system, we derive a single second-order nonlinear elliptic equation for the potential function φ in the subsonic region as in [1], so that the original transonic shock problem is reformulated as a one-phase free boundary problem for a secondorder nonlinear elliptic equation with the shock-front as a free boundary. One of the advantages of this approach is that, given the location of the shock-front, or equivalently the entropy function A (which is constant along the fluid trajectories) along the shock-front downstream, all the physical variables U = (u, p, ρ) can be expressed as functions of the gradient of φ, and the asymptotic behavior φ ∞ of the potential φ at the infinite exit can be uniquely determined.
To solve the free boundary problem, we have to determine the free boundary, and both the subsonic phase and entropy function defined in the downstream domain with the free boundary as a part of its boundary. We approach this problem by employing the hodograph transformation to transfer the free boundary to a fixed boundary, while keeping the ellipticity of the second order partial differential equations, and then by updating the entropy function A to prove that the updating map for A has a fixed point.
For given entropy function A, we first determine a priori the limit function of the potential function downstream at infinity. Then we solve the second-order elliptic equations for the potential function in the unbounded domain with the fixed boundary conditions and the downstream asymptotic condition at infinity. This is achieved through the fixed point argument by designing an appropriate map. In order to define this map, we first linearize the second-order elliptic equation for the identified potential function φ based on the limit function φ ∞ of φ, solve the linearized problem in the fixed region, and then make delicate estimates of the solutions, especially the corner singularity near the intersection between the fixed shock-front and the wedge boundary. These estimates allow us to prove that the map has a fixed point that is the subsonic solution in the downstream domain. Finally, we prove that the entropy function A is a fixed point via the implicit function theorem.
Since the transformation between the Eulerian and Lagrangian coordinates is invertible, we obtain the existence and uniqueness of solutions of the wedge problem in Eulerian coordinates by transforming back the solutions in Lagrangian coordinates, which are the real subsonic phase for the free boundary problem. The asymptotic behavior of solutions at the infinite exit is also clarified. The stability of transonic shocks and corresponding transonic flows is also established by both employing the transformation from Eulerian to Lagrangian coordinates and developing careful, detailed estimates of the solutions.
Another advantage in our analysis here is in the context of the real full Euler equations so that the solutions do not necessarily obey Bernoulli's law with a uniform Bernoulli constant, i.e., the Bernoulli constant is allowed to change for different fluid trajectories (in comparison with the setup in [9,10,13,16]).
By the closeness assumption of solution U to the uniform flow in the subsonic region, we obtain the asymptotic behavior of U as y 1 → ∞. The asymptotic state U ∞ = (u ∞ , p ∞ , ρ ∞ ) is uniquely determined by state U − of the incoming flow and the wedge angle at infinity.
We remark that, when U + 0 is on arcT H (see Fig. 2.1 below), the nature of the oblique boundary condition near the origin is significantly different from the case when U + 0 is on arcT S. Such a difference affects the regularity of solutions at the origin in general. It requires a further understanding of global features of the problem, especially the global relation between the regularity near the origin and the decay of solutions at infinity, to ensure the existence of a C 1,α solution. A different approach may be required to handle this case, which is currently under investigation.
The organization of this paper is as follows: In §2, we first formulate the wedge problem into a free boundary problem and state the main theorem.
In §3, we reduce the Euler system into a second-order nonlinear elliptic equation in the subsonic region and then reformulate the wedge problem into a one-phase free boundary problem for the second-order nonlinear elliptic equation with the shock-front as a free boundary.
In §4, we use the hodograph transformation to make the free boundary into a fixed boundary, in order to reduce the difficulty of the free boundary. After that, we only need to solve for the unknown entropy function A as a fixed point.
In §5, for a given entropy function A, we solve the reformed fixed boundary value problem in the unbounded domain and determine a priori the downstream asymptotic function of the potential function at infinity. Then, in §6, we prove that the entropy function A is a fixed point via the implicit function theorem, which is one of the novel ingredients in this paper.
In §7, we determine the decay of the solution to the asymptotic state in the physical coordinates.
In §8, we establish the stability of the transonic solutions and transonic shocks under small perturbations of the incoming flows and wedge boundaries. We finally give some remarks for the problem when the downstream state of the background solution is on arcT H in §9. In Appendices, we show two comparison principles and derive a criterion for different arcsT S andT H on the shock polar, which are employed in the earlier sections.
Finally, we remark that the stability of conical shock-fronts in three-dimensional flow has also been studied in the recent years. The stability of conical supersonicsupersonic shock-fronts has been studied in Liu-Lien [24] in the class of BV solutions when the cone vertex angle is small, and Chen [8] and Chen-Xin-Yin [12] in the class of smooth solutions away from the conical shock-front when the perturbed cone is sufficiently close to the straight-sided cone. The stability of three-dimensional conical transonic shock-fronts in potential flow has been established in Chen-Fang [2] with respect to the conical perturbation of the cone boundary and the upstream flow in appropriate function spaces. Also see Chen-Feldman [3].
2. Mathematical Setup and the Main Theorem. In this section, we first formulate the wedge problem into a free boundary problem for the composite-mixed Euler equations, and state the main theorem.
As is well-known, for a uniform horizontal incoming flow U − 0 = (u − 10 , 0, p − 0 , ρ − 0 ) past a straight wedge with half-wedge angle θ 0 , the downstream constant flow can be determined by the Rankine-Hugoniot conditions, that is, the shock polar (see Appendix B and Fig. 2.1). According to the shock polar, the two flow angles are important: One is the critical angle θ c w that ensures the existence of the attached shocks at the wedge vertex, and the other is the sonic angle θ s w for which the downstream fluid velocity at the sonic speed in the direction. When the straight wedge angle θ w is between θ s w and θ c w , there are two subsonic solutions; while the wedge angle θ w is smaller than θ s w , there are one subsonic solution and one supersonic solution. We focus on the subsonic constant state U + 0 = (u + 0 , p + 0 , ρ + 0 ) with u + 0 · (sin θ 0 , − cos θ 0 ) = 0. Then the transonic shock-front S 0 is also straight, described by x 1 = s 0 x 2 . The question is whether the transonic shock solution is stable under a perturbation of the incoming supersonic flow and the wedge boundary.
Assume that the perturbed incoming flow U − is close to U − 0 , which is supersonic and almost horizontal, and the wedge is close to a straight wedge. Then, for any suitable wedge angle (smaller than a critical angle), it is expected that there should be a shock-front which is attached to the wedge vertex. If the subsonicity condition is imposed in the far field downstream after the shock-front, then the flow U between the shock-front and the wedge should be subsonic. Since the upper and lower subsonic regions do not interact with each other, it suffices to study the upper part.
We now use a function b(x 1 ) to describe the wedge boundary: Along the wedge boundary ∂W, the slip condition is naturally prescribed: Let the shock-front S be x 1 = σ(x 2 ) with σ(0) = 0. Then the domain for the subsonic flow is denoted by To be a weak solution of the Euler equations (1.1), the Rankine-Hugoniot conditions should be satisfied along the shock-front: as the free boundary conditions on S, where [ · ] denotes the jump between the quantity of two states across the shock-front.
For a fixed uniform supersonic state U − 0 , there is an arc on the shock polar corresponding to the subsonic states. When the wedge angle is less than the critical angle θ c w > θ s w , the tangential point T corresponding to the critical angle divides arcHS into the two open arcsT S andT H. The nature of these two cases is very different.
In this paper, we analyze the existence, stability, and asymptotic behavior of steady transonic flows with a transonic shock in the important regimeT S for the wedge angle θ w . To state our results, we need the following weighed Hölder norms: For any x, x ′ in a two-dimensional domain E and for a subset P of ∂E, define δ x := min {dist(x, P ), 1} , δ x,x ′ := min{δ x , δ x ′ }, Let α ∈ (0, 1), σ, τ, l ∈ R, and let k be a nonnegative integer. Let k = (k 1 , k 2 ) be an integer-valued vector, where k 1 , k 2 ≥ 0, |k| = k 1 + k 2 , and D k = ∂ k1 x1 ∂ k2 x2 . We define For a function of one variable defined on (0, ∞), we define the Hölder norms with a weight at infinity. The definition above can be reduced to one-dimensional if we keep only the weights at infinity. Then the notation becomes f k,α;(τ );(0,∞) .
We also need the norms with weights at infinity which apply only for the derivatives: Similarly, the Hölder norms for a function of one variable on (0, ∞) with only the weights at infinity are denoted by f * k,α;(τ );(0,∞) . In terms of supersonic flows, we prescribe the initial data: Let Ω − be the domain for the incoming flows defined by For a given shock We now fix parameters α, β ∈ (0, 1) with suitably small β, depending on the background states.
Then we can conclude that there is ε > 0, depending on the background states, such that, when there exists a constant C 0 > 0, independent of ε, and a unique supersonic solution This can be achieved by rewriting the problem as an initial-boundary value problem in the polar coordinates (r, θ) so that system (1.1) is still a hyperbolic system, domain Ω − becomes a half strip with θ time-like and r space-like, the initial data is on {r > 0, θ = 0}, and the boundary data v = 0 is on the characteristic line {r = 0, 0 ≤ θ ≤ arctan(2s 0 )}. This is a standard initial-boundary value problem 8 GUI-QIANG CHEN, JUN CHEN AND MIKHAIL FELDMAN whose almost-global existence of solutions can be obtained as long as ε is sufficiently small.
Assume that the wedge boundary satisfies Then there is ε > 0 such that, when the initial data U 0 and the wedge boundary ∂W = {x 2 = b(x 1 ), b(0) = 0} satisfy (2.9) and (2.11) respectively, there exist a strong transonic shock S : The Rankine-Hugoniot conditions (2.4) hold along the shock-front S; (iii) The slip condition (2.2) holds along the wedge boundary ∂W; (iv) The following estimates hold: where C is a constant depending only on U 0 ± , but independent of ε. Moreover, solution U is unique within the class of transonic solutions such that the left-hand side of estimate (2.12) is less than Cε. Remark 1. Estimate (2.12) implies that the downstream flow and the transonic shock-front are close to the background transonic solution. The subsonic solution U converges to U ∞ at rate |x| −β and the slope of the shock converges to the slope of the background shock at rate |x| −β−1 .
Remark 2. Theorem 2.1 indicates that the asymptotic downstream state U ∞ generally is not a uniform constant state. If the x-coordinates are rotated with angle θ 0 into the new coordinates (x 1 ,x 2 ) so that the unperturbed wedge boundary . In Lagrangian coordinates, y = (y 1 , y 2 ), determined by (3.1) in §3, the asymptotic downstream state is a function of y 2 in general: However, our argument also shows that, in the isentropic case with a constant Bernoulli quantity B (see (3.14)), the asymptotic state must be uniform and equal to the background state. Also see Chen-Chen-Feldman [1].
3. Reduction of the Euler System and Reformulation of the Wedge Problem. In this section, we first reduce the Euler system into a second-order nonlinear elliptic equation and then reformulate the wedge problem into a free boundary problem for the nonlinear elliptic equation with the shock-front as the free boundary.
From the first equation in (1.1), there exists a unique stream function ψ in domain To simplify the analysis, we employ the following coordinate transformation to the Lagrangian coordinates: ® under which the original curved streamlines become straight. In the new coordinates y = (y 1 , y 2 ), we still denote the unknown variables U (x(y)) by U (y) for simplicity of notation. The original Euler equations in (1.1) become the following equations in divergence form: 1 Let T : y 1 =σ(y 2 ) be a shock-front. Then, from the above equations, we can derive the Rankine-Hugoniot conditions along T : The background shock-front now is T 0 : Without loss of generality, we assume that the supersonic solution U − exists in domain D − defined by For a given shock functionσ(y 2 ), let In either the supersonic or subsonic region, x 2 can be solved as a function of y since ψ x2 = ρu 1 = 0. Let x 2 := φ(y) in the subsonic region Dσ and x 2 := φ − (y) in the supersonic region D − σ . Given U − , we can find the corresponding function φ − . We now use function φ(y) to reduce the original Euler system to an elliptic equation in the subsonic region.
Assume that A = A(y 2 ) has been known. Then (u, p, ρ) can be expressed as functions of ∇φ: We now choose (3.4) to derive a second-order nonlinear elliptic equation for φ so that the full Euler system is reduced to this equation in the subsonic region. Set Then we obtain the second-order nonlinear equation for φ: , ∇φ), i = 1, 2, are given by , . Then a careful calculation shows that Thus, the discriminant in the subsonic region with ρu 1 = 0. Therefore, when solution φ is sufficiently close to φ + 0 (determined by the subsonic background state U + 0 ) in the C 1 norm, equation (3.20) is uniformly elliptic, and the Euler system (3.2)-(3.5) is reduced to the elliptic equation (3.20) in domain Dσ, whereσ is the function for the transonic shock.
The boundary condition for φ on the wedge boundary {y 2 = 0} is The condition on T is derived from the Rankine-Hugoniot conditions (3.6)-(3.8). Condition (3.6) is equivalent to the continuity of φ across T : It also givesσ Replacingσ ′ (y 2 ) in (3.7) and (3.8) with (3.28) gives rise to the conditions on T : We will combine the above two conditions into the boundary condition for (3.20) by eliminating A.
By calculation, we have (3.32) Thus, we obtain ] < 0 and u 2− is close to 0. Therefore, both equations (3.29) and (3.30) can be solved for A to obtain A = g 1 (U − , ∇φ) and A = g 2 (U − , ∇φ), respectively. Then we obtain our desired condition on the free boundary (i.e. the shock-front):

Hodograph Transformation and Fixed Boundary Value Problem.
In order to reduce the difficulty of the free boundary, we employ the hodograph transformation to make the shock-front into a fixed boundary. After that, we only need to solve for the unknown function A.
We now extend the domain of y 2 , which is the background potential function. We can extend φ − into We then use the following partial hodograph transformation: so that y 1 is a function of (z 1 , z 2 ): y 1 = ϕ(z 1 , z 2 ). Let Therefore, equation (3.20) becomes Notice that 3) which implies that equation (4.2) is uniformly elliptic, for any solution ϕ that is close to ϕ + 0 (determined by the background solution U + 0 ) in the C 1 norm. Then the unknown shock-front T becomes a fixed boundary, which is the z 2 -axis. Along the z 2 -axis, condition (3.33) is now We also convert condition (3.30) into the z-coordinates: ‹ H(z, A, ϕ, ∇ϕ) along the z 2 -axis.
The condition on the z 1 -axis can be derived from (3.26) as follows: Restricted on z 2 = 0, the coordinate transformation (4.1) becomes Then y 1 can be solved in terms of z 1 so that Let Q be the first quadrant. Then the original wedge problem is now reduced to both solving equation (4.2) for ϕ in the unbounded domain Q with the boundary conditions (4.7) and (4.9) and determining A via (4.8).
This will be achieved by the following fixed point arguments. Consider a Banach space: 1,α;(1+β);(0,∞) < ∞} as defined in (6.3) below. Then we define our iteration map J : X −→ X through the following two steps:

Consider any
for some fixed constant C 0 > 0, where w t = w t (z 2 ) is determined by (6.2) below.

2.
With this ϕ = ϕ A , we solve (4.8) to obtain a uniqueÃ that defines Finally, by the implicit function theorem, we prove that J has a fixed point A − w t in §6, for which A satisfies (4.10).
5. An Elliptic Problem to Determine ϕ in Domain Q. In this section, for given A satisfying (4.10), we solve equation (4.2) for ϕ in the unbounded domain Q with boundary conditions (4.7) and (4.9). Before this, we determine a priori the limit function ϕ ∞ at infinity. 5.1. Determine a priori the limit function ϕ ∞ at infinity. First, we assume that the asymptotic downstream state U ∞ depends only on y 2 , which will be verified later. Then we determine the limit function φ ∞ for φ. From (3.2), we expect the flow direction at infinity is the same as that of the wedge. That is, Then φ ∞ = tan θ 0 y 1 + l(y 2 ).
Replacing φ with φ ∞ in Bernoulli's law (3.16), we obtain From (3.4), we expect that pressure p → p + 0 and then relation (3.15) becomes p + 0 = A(ρ ∞ ) γ so that A = A(y 2 ) and ρ ∞ (y 2 ) = ( p + 0 A(y2) ) 1/γ . Therefore, the above equation becomes This equation gives the expression for l ′ (y 2 ). We can find l(y 2 ) by integration with Then we employ to solve for ϕ ∞ . Also, equation (5.2) restricted on z 2 = 0 gives rise to from which we can solve forb 0 . By the definition of ϕ ∞ , we know that ϕ ∞ satisfies (4.2). That is, where the wedge boundary ∂W is the z 1 -axis. We will omit ∂W in the norm when no confusion arises.
To solve equation (4.2) in the first quadrant Q, we first linearize (4.2) and solve the linearized equation in bounded domains, and then take the limit to obtain a solution in the unbounded domain Q.
For given ϕ such that ϕ − ϕ ∞ ∈ Σ δ , we define a map Q:φ − ϕ ∞ =: Q(ϕ − ϕ ∞ ) in Σ δ as the solution of the linearized equation (5.5) of the nonlinear equation (4.2) at ϕ as described below, and show that there exists a fixed point that is a solution for equation (4.2).
We use a straight line L R := {z 2 = R − z 1 } to cut off Q into a triangular domain Taking the difference of equations (4.2) and (5.3) and linearizing the resulting equation lead to for i, j = 1, 2.
Therefore, we derive the oblique condition: which have all the corresponding bounded Hölder norms. When U + 0 is on arcT S, the direction of ν = (ν 1 , ν 2 ) is Since, on arcT S, C p < 0 from (B.11), we have In particular, if δ is small, then where ν + 20 is the quantity ν 2 for the background subsonic state. This implies that condition (5.10) is uniformly oblique. Set Now, for any function f of (U − , U ), we usef to denote the value at the background states:f = f (U − 0 , U + 0 ). We also omit domain Q R and boundary ∂W in the norms when no confusion arises. In the following estimates, all generic constants C's and c 0 are only dependent on the background states and α, β, but independent of ε, δ. We define the transformation z = Kz, wherez = (z 1 ,z 2 ) is a new coordinate system. Then j=1,2 k ijzj = z i implies i,j=1,2â ij ∂ 2 zizj = ∆z.

GUI-QIANG CHEN, JUN CHEN AND MIKHAIL FELDMAN
Therefore, we compute where c 0 > 0 is a constant depending on the background state and α, β.
Denote any quarter ball B r (0) ∩ Q with radius r by B + r . In B + 2 , The same argument as in (5.14) implies that, at the maximum point P , Similarly, we obtain the corresponding lower bound. Therefore, we conclude |w(z)| ≤ εCr 1+α for any z ∈ B + 2 . (5.18) With estimate (5.18), we can use the scaling technique to obtain the C 1,α estimate for w up to the corner. More precisely, for any point P * ∈ B + 1 with polar coordinates (d * , θ * ), we consider two cases for different values of θ * . 8.33 in [17]), we have where f i is defined in (5.17), C is a constant independent of d * , and the weight of the norm is up to ∂B 2 . Therefore, by (5.18), we conclude Case 2: Boundary estimate for θ * > π 3 or θ * < π 6 . Denote B 3 = Q ∩ B 2d * 3 (P * ). By the Schauder boundary estimate (cf. (4.46) and Theorem 8.33 in [17]), we have Combining Case 1 with Case 2 yields the corner estimate: (5.20) 5.5. C 1,α estimate for v away from corner O. Away from the corner, suppose z * = (z * 1 , z * 2 ) ∈ Q R with polar coordinates (R * , θ * ) for 1 < R * < R 2 . Two cases will be considered below.
Combining the estimates in the two cases above, we obtain the estimate for v in Q R/2 v 1,α;(β,0);Q R/2 ≤ Cε. (5.21) 5.6. C 2,α regularity. For the C 2,α estimates with a weight to the z 1 -axis, we rewrite equation (5.5) into the non-divergence form: with the boundary condition on the z 2 -axis: We use the same scaling as in the C 1,α estimates and consider two cases. One is away from the z 1 -axis and the other is close to the z 1 -axis. We will estimate the The same Schauder interior and boundary estimates as (5.24) applied tov gives rise to v 2,α;(0,0);B ′ * ≤ Cε|z * | −β z * 2 1+α .

5.7.
Uniqueness. Because of the decay ofṽ at infinity, we can obtain the uniqueness ofṽ by the comparison principle as follows: Suppose that v 1 and v 2 are two solutions of (5.5). The difference w = v 1 − v 2 satisfies the same equation and boundary conditions on the z 2 -axis, and w = 0 on the z 1 -axis.
For any small positive constant τ , we let R to be large enough such that |w| ≤ τ on the cutoff boundary L R . Similar to (5.14), we employ Theorem A.1 to obtain |w| ≤ τ in Q R . Let R → ∞ and τ → ∞, we conclude that w ≡ 0, which implies the uniqueness. 5.8. Determination of ϕ as a fixed point. We define a map Q : Σ δ → Σ δ by where the closed set Σ δ is defined in (4.11). We employ the Schauder fixed point theorem to prove the existence of a fixed point for Q. That is, we need to verify the following facts: (i) Σ δ is a compact and convex set in a Banach space B; (ii) Q : Σ δ → Σ δ is continuous in B. Choose the Banach space B as defined in (4.12). Then Σ δ is compact and convex in B.
For the continuity of Q, we make the following contradiction argument. Let w 0 , w n ∈ Σ δ and w n → w 0 in B. Then v n ≡ Q(w n ) in Σ δ and v 0 ≡ Q(w 0 ) in Σ δ . We want to prove that v n → v 0 in B.
Assume that v n v 0 . Then there exist c 0 > 0 and a subsequence {v n k } such that v n k − v 0 B ≥ c 0 . Since {v n k } ⊂ Σ δ is compact in B, we can find another subsequence, again denoted by {v n k }, converging to some v 1 ∈ Σ δ . Then v 0 and v 1 satisfy the same equation (5.5), where ϕ = ϕ ∞ + w 0 , which contradicts with the uniqueness of solutions for (5.5). Therefore, Q is continuous in B.
Thus, we have a fixed point v for Q, which gives a solution ϕ ≡ ϕ ∞ + v for the nonlinear equation (4.2). The solution is unique by applying the same comparison principle as for the linear equation.
6. Determination of the Entropy Function A as a Fixed Point. In this section, we employ the implicit function theorem to prove the existence of a fixed point A.
6.1. Setup for the implicit function theorem for A. Through the shock polar, we can determine the values of U at O, and hence A(0) = A t is fixed, depending on the values of U − (O) and b ′ (0). Then we solve (4.8) to obtain a unique solutioñ A = h(z, ϕ, ∇ϕ) that defines the iteration map. To complete the proof, we need to prove that the iteration map exists and has a fixed point by the implicit function theorem.
In order to employ the implicit function theorem, we need to set up a Banach space for A. To realize this, we perform the following normalization for (A, U − , b). Let where t is a function determined by the Rankine-Hugoniot conditions (3.6)-(3.9).
We now define a linear operator D λ P(λ, ω, µ) and show that it is the partial differential of P with respect to λ. When no confusion arises, we may drop the variables (λ, ω, µ) in D λ P(λ, ω, µ).
We divide the proof into four steps.
1. Definition of a linear operator D λ P(λ, ω, µ). Given δλ ∈ X, we solve the following equation for δϕ: with boundary conditions: Once we have known δϕ, we define where It is easy to see that δλ(0) = 0. Then D λ P(λ, ω, µ) is a linear operator from X to X.
3. Continuity of P and D λ P. It suffices to show the continuity of P at any point (λ * , ω * , µ * ) near (0, 0, 0) by a contradiction argument, since the same argument applies to D λ P.
Therefore, given U − and b, operator P has a fixed point λ, which determines A = w t + λ.

7.
Decay of the Solution to the Asymptotic State in the Physical Coordinates. Now we determine the decay of the solution to the asymptotic state U ∞ in the x-coordinates. We divide the proof into four steps.
Then, changing the variables from y to x (which is bi-Lipschitz) and using (7.8), we obtain that, in the x-coordinates with x 2 = φ(y), This completes the proof for the decay of solution U (x) to the asymptotic state U ∞ .
8. Stability of Solutions. In this section, we prove that the subsonic solutions are stable under small perturbations of the incoming flows and the wedges as stated in Theorem 2.1. We modify operator P into P as follows: We first modify the definitions of the spaces in (6.4)-(6.7) in §6.2 by discarding the constraints: where norms · X and · Z are defined in (6.4) and (6.8), respectively. We still use the same space Y and the related norm as in (6.5) and (6.6).
Let ω = U − − U − 0 , λ = A − A + 0 , and µ = b − b 0 . Given (λ, ω, µ), we define P(λ, ω, µ) in the same way as for P by the end of §5, except that we do not restrict the value of A(0) by (6.1). The restriction for A(0) is essential for the isomorphism of D λ P. To prove the stability, we need to eliminate this restriction so that the differentiability in ω can be achieved in a larger space.
In particular, one may not expect a solution for caseT S is C 1,α ; it is generically only in C α .
For example, in the first quadrant, let domain OAB be the quarter of the unit disc. The oblique direction ν = (−1, −1). Let u = r 1 2 sin( θ 2 ), where (r, θ) are the polar coordinates. In OAB, u satisfies the Laplace equation: ∆u = 0, (9.1) and the boundary conditions: u = 0 on OA, and ∇u · ν = 0 on OB. However, u is Hölder continuous only in C 1 2 . Therefore, it requires a further understanding of global features of the problem, especially the global relation between the regularity near the origin and the decay of solutions at infinity, to ensure the existence of a smooth solution, more regular than the Hölder continuity. A different approach may be required to handle this case.
Then v w achieves its positive maximum on the boundary: By calculation, we have By assumption, we know that Lw w ≤ 0 and Lv w ≥ 0. Therefore, by the weak maximum principle, Theorem 8.1 in [17], we conclude (A.1).
Then v w achieves its positive maximum on the boundary or no greater than 1 in Ω: sup This implies that i,j=1,2 which contradicts (A.5). This completes the proof.