Global solutions of shock reflection problem for the pressure gradient system

We are concerned with the shock reflection in gas dynamics for the pressure gradient system. Experimental and computational analysis has shown that two patterns of regular reflection may occur: supersonic and subsonic reflection. In this paper we establish the global existence of solutions for both configurations. The ideas and techniques developed here will be useful for the two-dimensional Riemann problems for hyperbolic conservation laws.


1.
Introduction. Shock reflection is a fundamental phenomenon in gas dynamics. When a shock wave passes through an obstacle, the shock reflection occurs generally. E. Mach [25] first studied this problem in 1878. He observed two patterns of shock reflection-diffraction configurations that are now named the regular reflection (RR) and the Mach reflection (MR). In 1943 von Neumann [30] modeled the problem on moving plane shock attacking an inclined ramp. Later, a lot of research works concentrated on the shock reflection mainly by experiments and computations. It is observed that there are various different and complicated shock patterns depending on many factors, including the shape of the obstacle, the parameters of the incident shock, etc., see Courant-Friedrichs [12], Ben [2], Glimm-Majda [14], and also [4,5,9,10,13,16,26,28] and the references cited therein. Mathematical analysis of shock reflection-diffraction configurations involves several core difficulties in the analysis of nonlinear PDEs, including nonlinear PDEs of mixed hyperbolic-elliptic type, free boundary problem for nonlinear degenerate elliptic PDEs, corner singularities and so on. The shock reflection problem governed by full Euler equations or isentropic Euler equations is still open. Many efforts have been made mathematically for the reflection problem via simplified models, including the potential flow equation [1,[7][8][9]26], the unsteady transonic small disturbance equation (UTSD) [18][19][20], the pressure gradient system [32] and the nonlinear wave system [6,21,27]. Moreover, there are many papers devoted to the Mach reflection, see [3,11,17]. Besides the importance of shock reflection on aerodynamics, the shock reflection configurations are core configurations in the structure of global entropy solutions of the two-dimensional Riemann problems of hyperbolic conservation laws, see [22,33].
In this paper, we are concerned with the pressure gradient system in two dimensional space where t ≥ 0 and x = (x 1 , x 2 ) ∈ R 2 are the time and spatial variables. p, (u, v) and E = (u 2 + v 2 )/2 + p are the pressure, the velocity of the fluid and the energy, respectively. The three equations in (1.1) stand for the conservation of momentum along the x 1 -direction and x 2 -direction and the conservation of energy, respectively. When a plane shock S 0 in the (x, t)−coordinates with the left state U 1 = (p 1 , u 1 , 0) and right state U 0 = (p 0 , 0, 0) hits a symmetric wedge head on, it experiences a reflection-diffraction process. Here θ w is the half wedge angle. Since the solid wedge W is symmetric with respect to the axis x 2 = 0, it suffices to consider the shock reflection problem in the upper half-plane {(x 1 , x 2 ) ∈ R 2 : x 2 > 0}. Then we can formulate this problem as follows: Problem 1. (Initial-boundary value problem). Seek a solution of system (1.1) satisfying the initial condition at t = 0: (p, u, v) = (p 0 , 0, 0) for x 2 > x 1 tan θ w , x 1 > 0, (p 1 , u 1 , 0) for x 1 < 0, x 2 > 0, and the slip boundary condition along the wedge boundary ∂W = {x 2 = x 1 tan θ w , x 1 > 0} and the symmetric boundary Γ sym = {x 1 < 0, x 2 = 0}: where ν is the exterior unit normal.
Definition 1.1. (Weak solution). A function w ∈ L ∞ loc (Ω) for an open region Ω ⊂ R 2 is a weak solution of (1.4) in Ω, provided that Ω (A(w(ξ)) · ∇ϕ(ξ) − B(w(ξ))ϕ(ξ)) dξ = 0 for any ϕ ∈ C 1 0 (Ω). We observe that once p is known within a region, the first two equations in (1.4) can be used to obtain (u, v), provided (u, v) are known on a suitable boundary and the region is of suitable structure. This is possible for our problem, thus we shall mainly consider (1.5) for p instead of (1.4).
To connect the two constant states U 0 and U 1 with a single forwards shock S 0 , the initial condition should satisfy the relations u 1 = p 1 − p 0 √ p 10 , p 1 > p 0 , p 10 := p 1 + p 0 2 .
The organization of this paper is as follows. In Section 2, we discuss the normal shock reflection and then derive the necessary condition for the existence of regular shock reflection. We will also make clear the supersonic regular reflection and subsonic regular reflection, and present the main results. In Section 3, we deal with the supersonic reflection. We first introduce a new coordinate system and obtain some basic estimates for the solutions and the domain. Next we consider the regularity of solutions near and away from the sonic arc, which is a degenerate boundary for p. Then we construct the iteration set and use the Leray-Schauder theory to prove the existence of admissible solutions. In Section 4 we prove the existence of solutions for the subsonic regular reflection.
2.1. The Rankine-Hugoniot condition. Consider a piecewise smooth weak solution of (1.4) with a jump in (p, u, v) across a curve S = {ξ = ξ(η)} with the slope σ = ξ (η). Across S it should satisfy the Rankine-Hugoniot condition where the bracket [w] denotes the jump of the quantity w across S. A discontinuity of (p, u, v) on S is called a shock wave if it further satisfies the physical entropy condition: p increases across S.
We can solve (2.1) to obtain contact discontinuity There are two states for shocks, with "+" and "−", which are called weak reflection and strong reflection, respectively. In the sequel, we only focus on the weak reflected shock, denoted by Γ shock . The reason is that the states for weak reflection tend to those for the normal shock reflection as θ w → π 2 −, which will be specified later. From (2.1) and the last two equations in (2.2), we obtain the condition that p should satisfy on Γ shock : where [p] = p − p 1 , andṗ = σp ξ + p η is the tangential differentiation of p along the shock.
In the polar (r, θ) coordinates, the shock becomes dr dθ = −r r 2 − p p =: g(r(θ), θ, p(r, θ)), (2.4) and the condition for p on Γ shock changes to where β = (β 1 , β 2 ) is a function of (p 1 , p, r(θ), r (θ)) with Thus the obliqueness becomes Note that µ becomes zero when r (θ) = 0, that is r 2 = p. When the obliqueness fails, we have Next, we first discuss the normal shock reflection, then derive the necessary conditions for the existence of regular reflection.
2.3. Local existence theory for shock reflection. The necessary condition for the existence of regular shock reflection is stated as follows, whose proof can be found in Zheng [32].
Proposition 1 (Regular reflection of the algebraic portion [32]). There exists a critical angle θ w = θ d w ∈ (0, π/2), depending only on p 1 /p 0 , given by the formula such that, for each θ w ∈ (θ d w , π 2 ) there exist two states (p 2 , u 2 , v 2 ) satisfying the Rankine-Hugoniot condition, given by Here, we will take the 'minus one' or 'physical one' for the pressure p 2 , which tends to p * 2 as θ w → π 2 −. Furthermore, by a direct calculation we have Meanwhile, state (2) is subsonic in the sense that ξ 2 10 + η 2 10 < p 2 for θ w ∈ (θ d w , θ s w ). Later, the angles θ s w and θ d w are called the sonic angle and the detachment angle, respectively.
. Let P 0 be the intersection point of the incident shock with the wedge. For different wedge angles, there are two types of configurations for the regular reflection: supersonic and subsonic reflection. Supersonic reflection corresponds to when the point P 0 is outside the sonic circle for state (2); subsonic reflection corresponds to when P 0 is either on the boundary or in the interior of the sonic circle for state (2), see Figure 1.
In the supersonic regular shock reflection configuration, let P 0 P 1 be the straight shock which intersects the sonic circle C 2 of state (2) at P 1 , and P 1 P 2 be the reflected curved shock. Let P 4 be the intersection point between C 2 and the wedge. There are three uniform states (0)(1)(2), and a non-uniform state in domain Ω = P 1 P 2 P 3 P 4 . The solution is equal to state (0) and (1) ahead of and behind the incident shock S 0 , away from P 0 P 2 P 3 . The solution is equal to state (2) in subregion P 0 P 1 P 4 , which is supersonic. The non-uniform state in Ω is subsonic. We denote the boundary parts of Ω by Γ sonic := P 1 P 4 , Γ shock := P 1 P 2 , Γ sym := P 2 P 3 , Γ wedge := P 3 P 4 including their endpoints and ∂Ω = Γ sonic ∪ Γ shock ∪ Γ sym ∪ Γ wedge .
In the subsonic case, the main difference is that the subregion P 0 P 1 P 4 shrinks into one point P 0 . Hence there are two uniform states(0)(1) and a non-uniform state in Ω. And p should match with the state (2) only at the single point P 0 , and the state is subsonic in Ω.
2.4. The von Neumann conjecture and main theorems. This paper is devoted to solve the von Neumann Detachment Conjecture on shock reflection: There exists a global regular reflection for any angle θ w ∈ (θ d w , π 2 ), i.e. the existence of state (2) implies the existence of a regular reflection configuration. Moreover, the type (supersonic or subsonic) of the reflection configuration is determined by the type of the weak state (2) at point P 0 .
According to different wedge angles, Problem 2 can further be reformulated into two problems as follows: Problem 3 (Supersonic regular shock reflection problem). For a supersonic wedge angle θ w ∈ (θ s w , π 2 ), find a shock wave Γ shock , and a function p, defined in a region Ω such that (i) p satisfies equation (1.5) in Ω; (ii) Equation (1.5) is strictly elliptic in Ω, i.e., ξ 2 + η 2 < p in Ω; (iii) The Rankine-Hugoniot condition and entropy condition are satisfied on Γ shock ; (iv) p = p 2 on Γ sonic ; (v) p ν = 0 on Γ sym ∪ Γ wedge , where ν is the interior unit normal to Ω.
Problem 4 (Subsonic regular shock reflection problem). For a subsonic wedge angle θ w ∈ (θ d w , θ s w ], find a shock wave Γ shock , and a function p, defined in a region Ω such that all conditions in Problem 3 are satisfied except that (iv) is replaced of p = p 2 at P 0 .
In most papers concerning the shock reflection problem, they only considered the supersonic reflection case. Recently, Chen-Feldman [9] and Rigby [27] solved the whole regular shock reflection problem, including subsonic reflection for potential flow and nonlinear wave equations, respectively. In this paper we solve the regular shock reflection problem for the pressure gradient system. We first give a definition of admissible solutions for Problems 3-4. C ase I. For θ w ∈ (θ s w , π 2 ): (i) There exists a shock curve Γ shock = {r = r(θ)} with endpoints P 1 and P 2 , such that (i-1) Curve Γ shock satisfies , where θ 1 is the θ-coordinate of P 1 . (i-3) Curves Γ shock , Γ sonic , Γ wedge and Γ sym do not have common points except for P 1 , P 2 , P 3 and P 4 . Thus, Γ sonic ∪Γ shock ∪Γ sym ∪Γ wedge is a closed curve without self-intersection. Denote by Ω the bounded domain enclosed by this closed curve.
(iv) p 1 < p ≤ p 2 in Ω. C ase II. For θ w ∈ (θ d w , θ s w ]. In this case the subregion P 0 P 1 P 4 shrinks into one point P 0 , and let P 0 = P 1 = P 4 . (i) There exists a shock curve Γ shock = {r = r(θ)} with endpoints P 0 and P 2 , such that (i-1) Curve Γ shock satisfies . (i-3) Curves Γ shock , Γ wedge and Γ sym do not have common points except for P 0 , P 2 and P 3 . Thus, Γ shock ∪ Γ sym ∪ Γ wedge is a closed curve without self-intersection. Denote by Ω the bounded domain enclosed by this closed curve.
Our main result of this paper is stated as follows. The main difficulty is how to link different cases as θ w crosses the sonic angle to the detachment angle continuously. In Section 3 and 4, we will consider Problem 3 and 4, respectively.
3. Existence result for the supersonic regular shock reflection. In this section we consider Problem 3. We first introduce a new coordinate system.
which shifts the origin in the self-similar plane to the center of the sonic circle of state (2). Without confusion of notation, we will always work in the new coordinates without changing the notation (ξ, η).
In the new shifted coordinates, the elliptic domain Ω can be expressed as where ξ = f (η) is the position of the reflected shock Γ shock . Let S 1 : ξ = l(η) be the location of the reflected shock of state (2), which is a straight line, where θ s and (ξ, 0) are the angle and the intersection point between the shock S 1 and η = 0, respectively. Denote by P 1 (ξ 1 , η 1 ) with η 1 > 0 the intersection point of the line S 1 and the sonic circle of state (2), i.e., . From the fact that the reflected shock Γ shock and the straight part of the reflected shock S 1 should match at least up to first-order, we have 3.2. Some basic estimates. In the following, we always assume that (p, r) is an admissible solution of Problem 3 and derive some basic estimates.
3.2.1. Monotonicity of p on Γ shock and convexity of Ω.
Proposition 3. Let (p, r) be an admissible solution for Problem 3. Then p is monotone increasing on Γ shock from P 1 to P 2 , and the elliptic region Ω is convex.
Proof. The monotonicity of p along Γ shock can be proved by contradiction method. Let us examine the C α −function p on Γ shock . Without confusion, we can label the points along Γ shock by their ξ−coordinates, and refer to intervals along Γ shock by the labels. The lack of monotonicity implies that, there exist two points A and B on Γ shock with P 2 < A < B < P 1 , such that p(A) < p(B). We immediately deduce that (a) In (A, P 1 ), there exists C with p( C) = max [A,P1] p; (b) In (P 2 , C), there exists D with p(D) = min [P2, C] p. Then we can find two points C and D on Γ shock with D < C, such that Here property (ii) may not hold with C = C because that p( C) is the maximum value only on the interval [D, P 1 ]. If D > A and there is a point in (A, D) at which p > p( C), then we let C be the point at which p obtains its maximum value in this interval. Otherwise, we let C = C. Then all three properties hold. Now we look at the function p in the domain Ω. The idea is to partition Ω into three subdomains by two curves Γ C and Γ D from C and D to points E and F on Γ wedge ∪ Γ sym , respectively, such that p(E) > p(F ). We want to deduce that there is a point X 0 on Γ wedge at which p achieves a maximum on either the subdomain Ω A or the domain Ω B , which violates the Hopf maximum principle. It suffices to show that p(X 0 ) is the maximum value of p on the boundary of Ω E or Ω F . Let

Figure 2. Hypothetical curves
We want to construct the Lipschitz curves on which p satisfies for some M > 0 and X 1 , X 2 ∈ Ω. Thus on any ball with radius r > 0, we have We construct Γ C as follows. In B R (C)∩Ω, let Y 1 be a point at which p attains its maximum value in B R (C). Then the first segment of Γ C is a straight line from C to Y 1 , and on the segment, we have

Now we continue inductively, forming a sequence of line segments with corners at
Since the domain is finite, the process must terminate after finite steps when we reach a point E ∈ ∂Ω. By construction, Γ C has the properties indicated in (3.1). Similarly, we construct Γ D with termination point F ∈ ∂Ω.
Next we show that the points E and F lie on Γ wedge ∪ Γ sym . First the two curves can not cross each other, because at every point on Finally, E can not lie in the segment [P 2 , D] because this would trap Γ D in a region where p ≤ p(D), which contradicts the fact that D is a local minimum in Ω. Hence E ∈ Γ wedge ∪ Γ sym . Similarly, F must lie on Γ wedge ∪ Γ sym , between P 2 and F . Now we find the final contradiction. Since p(F ) is smaller than p(P 2 ) and p(E), then there must be a point X 0 along the boundary P 2 OE at which p attains its minimum. Assume that X 0 is not the origin, then X 0 can not be a local minimum for the domain Ω by the Hopf maximum principle. However, along the entire boundary of the domain P 2 P 3 ECDP 2 , p ≥ p(X 0 ), which implies that it is a minimum. This is a contradiction. Now if X 0 coincides with the origin, the similar maximum point resembling E can not coincide with the origin. We can find no existence of such maximum point. We conclude that C and D do not exist, and hence C and D do not exist, and p is monotonic on Γ shock . Since Γ shock is vertical at P 2 , and is tangent to S 1 at P 1 , we conclude that p is monotone increasing, pointing from P 1 to P 2 .
Next, we calculate the curvature of Γ shock . Let p τ denote the tangential derivative of p along Γ shock in the direction of increasing θ. By the formula for r (θ) in (2.4), we have which implies the curvature k(θ) of Γ shock is positive. Hence Γ shock is concave, so the domain Ω is convex.
The convexity of Ω leads to the following corollary: be the outer unit normal to Γ wedge , and ν sh the inner unit normal to Γ shock for the admissible solution (p, r). Then for some δ > 0 depending only on the initial data.
2) by the convexity of Ω. 3.2.2. Bounds for the elliptic region Ω and p within Ω. We first note that Lemma 3.1. For any admissible solution (p, r), p can only attain its global maximum and minimum on Γ shock ∪ Γ sonic : Proof. Since equation (1.5) is strictly elliptic in Ω\Γ sonic , we can consider the domain Ω := {P ∈ Ω : dist(P, ∂Ω) > } for > 0, in which (1.5) is uniformly elliptic, and then by sending → 0+ we conclude that the extremum of p cannot be attained in the interior of Ω. Moreover, for any P ∈ Γ sym ∪ Γ wedge , equation (1.5) is uniformly elliptic in some neighborhood of P . Due to p ν = 0 on Γ sym ∪ Γ wedge , the extremum of p over Ω cannot be attained on Γ sym ∪ Γ wedge , unless p is constant in Ω. However, p cannot be a constant. Therefore, the extremum of p must be attained on Γ sonic ∪ Γ shock . Moreover, Proof. Since r (θ) ≤ 0, so r(θ) ≤ r(θ 1 ) = c 2 on Γ shock , (3.3) holds by the structure of the region. Since Γ shock lies outside the sonic circle of state (1), and by the ellipticity of the equation, we have p ≤ r 2 ≤ p 2 , p 1 < r 2 ≤ p, so (3.4) holds. By the explicit formula (2.4) for r (θ), w can obtain (3.5) easily.
The next lemma states the uniform separations of the components of ∂Ω, which implies that we can consider each part of the boundary separately, and piece local estimates together to obtain a global one.
. There exists a positive constant C < ∞ depending only on the initial data such that for any admissible solution with wedge angle Proof. In the polar coordinate, The next lemma is for a uniform positive lower bound for p − p 1 , which is critical for deriving ellipticity and obliqueness estimates in Ω.
. There exists δ > 0 depending only on the initial data and θ * w , such that for any admissible solution (p, r) with the wedge angle θ w ∈ [θ * w , π 2 ), we have p 1 + δ < p ≤ p 2 in Ω, and r > c 1 + δ on Γ shock .
Proof. It suffices to prove p > p 1 +δ. We prove this result by contradiction. Suppose that there exists a sequence of admissible solution (p k , r k ) with wedge angles θ (k) w and points ξ k ∈ Γ shock = {r = r k (θ)|θ ∈ [θ 1 , θ 2 ]} such that p k (P 2 ) → p 1 . We can show that r k converges in C 1,α to some limiting r, and θ (k) w , p k convergence to some θ w , p, respectively. Consider a neighborhood N of P 2 in Ω. We construct a barrier function of the form Via choosing suitable constants A, B, C and β ∈ ( 1 2 , 1) we can prove that the optimal regularity of p is C 1 2 near the shock in N . Next we introduce the coordinates It can be verified that due to the C with small x 0 , we have . There exist λ > 0 and δ > 0 depending only on the initial data and θ * w , such that for any admissible solution (p, r) with the wedge Proof. We first define a smooth approximation to ξ → dist(ξ, Γ sonic ) as follows.
Let C 2 = ∂B c2 (O) be the sonic circle of state (2). Denote by Q another point of intersection of the straight shock S 1 and C 2 except P 1 . Denote by Q = P1+Q which is an approximation of dist(ξ, Γ sonic ). We consider the function p−r 2 −λg(ξ), where λ > 0 will be specified later. Applying the ellipticity principle, we can prove that p−r 2 −λg(ξ) cannot attain a minimum in the interior of Ω. Since p satisfies slip boundary condition on Γ wedge and Γ sym , we can extend Ω and p by even reflection, then the boundary Γ wedge and Γ sym can be treated as the interior of the extended domain. Similarly, we can show that p − r 2 − λg(ξ) can not attain its minimum on Γ wedge ∪ Γ sym . Next, we turn to the shock boundary. Suppose that p − r 2 − λg(ξ) attain its minimum at P min ∈ Γ shock . Then (p − r 2 )(P min ) ≤ λg(P min ) ≤ λC for some C > 0. Using the boundary condition (2.5)-(2.6), we can choose λ sufficiently small, then β · ∇(p − r 2 − λg(ξ))(P min ) < 0, which contradicts the Hopf maximum principle. Thus, p − r 2 − λg(ξ) must attain its minimum on Γ sonic , and (3.8) is proved.
3.3. Regularity away from the sonic arc. In the subsection, we will obtain uniform estimates for admissible solutions away from the sonic arc. Define For the estimates in Ω\Ω with small > 0, we should pay more attention to the wedge vertex P 3 , and the point P 2 , where the obliqueness fails.
3.3.1. Regularity away from P 2 ∪ Γ sonic . Notice that for > 0, equation (1.7) is strictly elliptic in Ω\Ω with constant depending on . For small δ > 0, the boundary condition (2.5)-(2.6) is strictly oblique on Γ shock \B δ (P 2 ) with obliqueness constant depending on δ. Moreover, r (θ) is a Lipschitz function of p and r, with Lipschitz constant depending on δ and initial data. Thus, away from P 3 , we can obtain C 1,α estimate depending on the initial data using classical elliptic theory. Near P 3 , we can use the method in [23] to obtain a C 0,α estimate and then extend it to C 1,α following the method in [24,29]. The regularity away from P 2 ∪ Γ sonic is stated as follows.
. For any , δ > 0, there exist α ∈ (0, 1) and C > 0 depending on , δ, the initial data and θ * w such that, for any admissible solution 3.3.2. Regularity near P 2 . Now we consider the regularity near P 2 , where the obliqueness fails. Because r(θ) is Lipschitz continuous with bound depending only on the initial data, from the expression (2.4) of r(θ), we can easily prove Lemma 3.6. There exists C > 0 depending only on the initial data such that Next we will show an algebraic growth estimate of p on Γ shock in order to obtain the weighted Hölder norms for admissible solutions.
. Then there exists R > 0 depending only on the data and θ * w , such that for any integer m ≥ 1 and Furthermore, there exists 0 < C < ∞ depending on R, θ * w and the initial data such that is either a minimum within Ω ∩ B R (P 2 ) or a saddle point. If X is a minimum point, by the explicit formula (2.6), we have which is a contradiction due to the Hopf maximum principle. It needs more care to handle the saddle case. One can construct a modified function ψ instead of p − p(P 2 ) + M (θ 2 − θ) m , such that ψ can attain a minimum over Γ shock at X. Applying the Hopf maximum principle again one can derive a contradiction. The construction of ψ can be referred to Lemma 3.2 in [21] for details.
We introduce the notation for the weighted Hölder norms. Let Ω ⊂ R 2 be an open bounded set and Σ ⊂ ∂Ω. Set Then, for k ∈ R, α ∈ (0, 1) and m ∈ N, define (k),Σ m,α,Ω . We can now deduce the following estimates: . Let (p, r) be an admissible solution with wedge angle θ w ∈ [θ * w , π 2 ). Then there exists α ∈ (0, 1) depending on the initial data and θ * w such that, for > 0 and β ∈ (0, 1), there exists C < ∞ depending on , β, the initial data and θ * w such that p Proof. Using the structure of Ω and the property r (θ) we can deduce that for any a > 0, the ball B a (ξ 2 +a, η 2 ) is an exterior ball to Ω at P 2 . We may employ Theorem 1.1.15 [15] to deduce that for C < ∞ depending only on M from Proposition 3.7 and the ellipticity and boundedness of the coefficients in B R (P 2 ), that , for R depending on the initial data and θ * w . Since there exists σ > 0 such that by a standard argument with rescaled functions, we can deduce that there exists α ∈ (0, 1), depending only on the ellipticity in B R (P 2 ), such that for any β ∈ (0, 1), there exists C(β) depending only on β, θ * w and the initial data, such that p Combining with the former estimate (3.9), we obtain (3.12).

21)
and where ν is the inner unit normal to Ω.

3.5.
Regularity estimates in the scaled Hölder norms near Γ sonic . We will make a regularity estimate for ϕ = p 2 − p and the shock curve {y = f (x)} near Γ sonic . Since equation (3.15) degenerates on Γ sonic , we will use the parabolic Hölder norms following [9,27]. For α ∈ (0, 1), denote the parabolic distance be an open domain. For any nonnegative integer m, real σ > 0, α ∈ (0, 1), the parabolic Hölder norms, weighted and scaled by the distance to {x 1 = 0} is defined by  Lemma 3.13. There exist σ > 0 and α ∈ (0, 1) depending on the initial data and θ * w , such that the following holds: Let l > 0, = min{σ, l 2 }. If θ w ∈ [θ * w , π 2 ) satisfies θ 1 − θ w ≥ l, and (p, r) is an admissible solution with the wedge angle θ w , then there exists C < ∞ such that ϕ = p 2 − p satisfies ϕ (par) 1,α,Ω ≤ C. (3.22) Proof. First of all, we rescale the domain. For any z 0 = (x 0 , y 0 ) ∈ Ω and r ∈ (0, 1), we define It is obvious that R z0,r ∩ Γ sonic = ∅ for all z 0 ∈ Ω , and R z0,r ⊂ Ω ∩ (s, t) : Next, we observe that the following three types of domains consist an open covering of Ω . Moreover, from (3.17), we see that Hence, for small > 0 It means that Γ wedge and Γ shock can not close to each other. This fact fails if θ w tends to θ s w . We denote Q r = (−r, r) 2 , and for fixed z 0 = (x 0 , y 0 ) ∈ Ω , introduce the new variables (S, T ) defined by Next in order to obtain the estimates in Ω , we consider the three kinds of domains, respectively.
Combining Lemma 3.8 and 3.13, we now deduce the following global estimate of ϕ in the subsonic domain Ω.
Theorem 3.14. Let θ * w ∈ (θ s w , π 2 ), 0 > 0 be as in Lemma 3.11, then there exists α ∈ (0, 1) depending only on the initial data and θ * w , such that for any β ∈ (0, 1) and ∈ (0, 0 ], there exists C < ∞ depending on the initial data, , β, θ * w such that, for any admissible solution (p, r) of Problem 3 with wedge angle θ w ∈ [θ * w , π 2 ), we have 2,α,(0, ) ≤ C. 3.6. Existence of admissible solutions. We will use Leray-Schauder fixed point degree to prove the existence of admissible solutions. The relevant degree theory can be referred to [31]. The main points of the existence argument is as follows. We first construct an invertible mapping which can map the elliptic region Ω to the iteration set Q iter = (0, 1) 2 . Then we define the iteration boundary problem and iteration functions (u, v) corresponding to (p, r) in Q iter . We construct an iteration map I and prove that this map has a fixed point using Leray-Schauder point theory. We invert the fixed point to obtain an admissible solution.
Since our problem is a free boundary problem, the elliptic region Ω of the solution is a priori unknown. We should consider a larger domain for the iteration procedure. To this aim, for fixed wedge angle θ w ∈ (θ s w , π 2 ), we move the straight reflected shock S 1 := {ξ · ν S1 = − √ p 12 } upwards, and let where d > 0 will be defined later. LetP 1 be the closet point of intersection of S 1,d with ∂B c2 (O), and Γ 2 ) c 2 (θ w ). Here we take the boundary {ξ = −2M } to ensure that Q is bounded. It is obvious that Ω ⊂ Q.
In the (x, y)-coordinates, for ∈ (0, 1 ], denote We want to convert the free boundary value problem to fixed boundary problem, so we first construct the coordinate map and then define the related functions in the new coordinates. The construction is the same as that in Lemma 12.2.2 [9] since domain Ω has suitably similar structure. Lemma 3.15. There exist δ > 0 and 0 < C < ∞ depending on the data, a one-to- , and a function η ≡ (θw) : [0, {(s, t)|s ∈ (0, c 2 ), t ∈ (0, η(s))}. For all P ∈ D d, , F 1 (P ) = (x, y − θ w ). In particular, Next we need to construct a function f to approximate Γ shock . we note that Γ shock matches S 1 up to the first order at P 1 , and is vertical to {η = 0} at P 2 .
Then F 2,g sh ( Ω) = (0, 1) 2 =: Q iter , and F 2,g sh is invertible given by Now for an admissible solution (p, r), define u : (3.29) Then from Theorem 3.14 and the construction of the functions u and v, we have Lemma 3.16. Fix θ * w ∈ (θ s w , π 2 ) and let (p, r) be an admissible solution with wedge angle θ w ∈ [θ * w , π 2 ). Then there exist α ∈ (0, 1 2 ) and > 0, such that the functions u and v defined by (3.29) satisfy where C < ∞ depends on θ * w , α and initial data, and the parabolic norms are with respect to s = 0. Besides, (3.30) We will invert the mapping (p, r) → (u, v), i.e. obtain (p, r) from (u, v) to be able to determine our iteration equation. Let Θ = Θ (α,β) be defined by We define (i) A function h sh : (0, 1) → R such that h sh (s * ) = t * , where (s * , t * ) is uniquely Moreover, we require (u, v, θ w ) ∈ Θ satisfy Then we define ϕ on Ω by setting ϕ = u(F −1 (ξ)), and Γ shock = {(r, θ)|r = r(θ), θ 1 ≤ θ ≤ θ 2 }, Since Γ shock is a free boundary, in the iteration process we should modify it to keep the original properties. So we give an oblique cutoff for r(θ) near P 2 . Define We can prove that r arising from this cutoff still defines a suitable shock curve Γ shock , satisfying and e t · ν sh ≤ −δ on Γ shock . In the (s, t)-coordinates, and we then define ϕ from u using g sh (s) instead of g sh (s).
3.6.2. Boundary value problem for the iteration. Consider the iteration problem First, for the oblique boundary condition on Γ shock , we freeze the coefficients at ϕ and r determined by (u, v) as β 1 (ϕ, r) ϕ r + β 2 (ϕ, r) ϕ θ = 0, and now the boundary condition is linearized.
As before, we can prove hence it is uniformly oblique away from P 2 and non-oblique at P 2 . Moreover, 1,α/2,Γ shock \∂Ω ≤ C. For the equation, we use (3.32) in Ω\D eq /10 and freeze the coefficients to linearize it . Then (3.36) is uniformly elliptic in Ω\D eq /10 and the coefficients satisfy Near Γ sonic , the equation and oblique boundary conditions should be paid more attention. We will work in the (x, y) coordinates. For our iteration, we only obtain |ϕ| ≤ N 0 x δ * , which is not sufficient to guarantee the ellipticity of (3.32). We should make a cutoff. Let ς 1 (t) ∈ C ∞ (R) satisfy if |t| < 2c 2 − δ 2 5 , and ς 1 ≥ 0, ς 1 (−t) = −ς 1 (t) on R.

HANCHUN YANG, MEIMEI ZHANG AND QIN WANG
Then the modified equation in the (x, y) coordinates is defined as follows where O i are defined as From the expressions of O i (i = 1, · · · , 5), we have Lemma 3.18. Let eq ∈ (0, 1 ) and (u, v, θ w ) ∈ Θ, then where C < ∞ depends only on the initial data.
Next we solve the iteration problem.
Now we define the iteration map I = I θ on K as where u, v are defined in Proposition 7. Then I is continuous, and i-compact. Moreover, we have Lemma 3.21. Let (p, r) correspond to a fixed point (u, v) of I for some wedge angle θ w . Then, if in the iteration set is chosen sufficiently small depending on the data, ϕ corresponding to p satisfies Using the estimates of Lemma 3.21 we can remove the cutoff functions ς 1 , ζ 1 . Thus we conclude that In order to use the Leray-Schauder fixed point theory to obtain the existence of admissible solutions for all supersonic wedge angles, we need to show that the fixed point degree of the iteration map for θ w = π 2 is non-zero. We first note that when θ w = π 2 , the normal reflection is the unique admissible solution. Therefore, the unique fixed point of I π 2 is (0, 0). From Theorem 3.4.7 in [9], it follows that Ind(I π 2 , K( π 2 )) = 1.
Finally, applying the Leray-Schauder theory, we conclude that I θw has a fixed point for any angle θ w ∈ [θ * w , π 2 ]. As θ * w is arbitrary, we have proven the existence of admissible solutions for all supersonic wedge angles.

4.
Admissible solutions up to the detachment angle. In this section, we consider the subsonic shock reflection when θ w ∈ (θ d w , θ s w ]. In this case the subregion P 0 P 1 P 4 shrinks to one point P 0 , and we let P 1 = P 4 = P 0 . 4.1. Regularity away from the reflected point. First we show the monotonicity of p on Γ shock and convexity of Ω. Lemma 4.1. Let (p, r) be an admissible solution of problem 4. Then p is monotone increasing from P 0 to P 2 on Γ shock , and the elliptic region Ω is convex.
Lemma 4.2. Let ν w denote the outer unit normal to Γ wedge . Then, for any admissible solution to problem 4, denoting by ν sh the inner unit normal to Γ shock , then ν sh · ν w ≤ −δ, where δ > 0 depends only on the initial data.
The estimates for p and the elliptic region Ω are stated as follows.
Lemma 4.3. Let (p, r) be an admissible solution with wedge angle θ w ∈ (θ d w , π 2 ). Then there exists C < ∞ depending only on the initial data such that (iv) The subsonic-away from sonic case θ w ∈ (θ d w , θ s w − δ * ]. In this case the sonic arc has zero length, and the ellipticity is strict. We will obtain C 0,β regularity up to the reflection point for any β ∈ (0, 1).
We note that S 1 near P 1 can be written as {y = f 0 (x)}. We have the following pointwise estimates for admissible solutions.
The coefficients ( β 1 , β 2 ) in the oblique boundary condition satisfy for some C < ∞, where ν is the inner unit normal to Ω.
Next we make clear the estimates for the four different case respectively.
Note that there exists k > 0 depending only on the initial data such that where b so = θ 1 −θ w . We want to prove the algebraic growth bounds for ϕ. Consider equation (3.23) as a quasilinear equation, obtained by freezing the coefficients at ϕ, where the coefficients satisfy for some C < ∞. Furthermore, the boundary condition on Γ shock can be rewritten as a linear condition Mϕ = β(x, y) · ∇ϕ = 0 on Γ shock ∩ ∂Ω , (4.2) and the boundary condition is strictly oblique in the sense that Also, the other boundary conditions are Basing on this, we can construct a suitable supersolution to (4.1) with boundary condition (4.2)-(4.3).
Lemma 4.13. Let m ≥ 2 be integer. There exists δ * > 0, > 0, and C > 0 depending only on the initial data and m such that, for any admissible solution (p, r) with wedge angle θ w ∈ (θ s w , θ s w + δ * ) and ≤ , we have Now using the growth estimates above, we will prove a priori estimates of solutions in the weighted and scaled C 1,α spaces.

Finally we have
Corollary 4. Let δ * > 0 be as in Lemma 4.17. Then there exists α ∈ (0, 1) depending on the data, such that for each β ∈ (0, 1), there exists C < ∞ depending only on the initial data and β such that, for any admissible solution (p, r) with wedge angle θ w ∈ [θ s w − δ * , θ s w ], ϕ ∈ C 1,α (Ω\{P 2 }), ϕ  Case(4). The subsonic-away from sonic case. In this case we have strict ellipticity, and a one-point Dirichlet condition at P 1 . We will attain C 0,β (P 1 ) estimates for each β ∈ (0, 1) using the same argument as that for P 2 . Fix θ * w ∈ (θ d w , θ s w − δ * ] for δ * > 0. We first show that for any A > 0 and integer m ≥ 2, A(y − θ w ) m − ϕ does not achieve a minimum on Γ shock ∩ ∂Ω , Next, Ω satisfies an exterior ball condition at P 1 , so we may use the barrier function to deduce that for any β ∈ (0, 1), there exists C depending only on β, θ * w such that Using the same argument used for P 2 , we find that Lemma 4.18. There exist α ∈ (0, 1) depending only on the data, θ * w and δ * , such that for any β ∈ (0, 1), there exists C < ∞ depending on the data, θ * w , δ * and α, such that for any admissible solution (p, r) with wedge angle

Weighted norm.
There are two difficulties in obtaining the existence of admissible solutions up to the detachment angle. The first one is how to combine the estimates near Γ sonic for ϕ into a unified estimates for u in the iteration set. The second one is how to extend the definition of the iteration set to the subsonic case. We first extend the mapping construction to all wedge angle θ w ∈ (θ d w , π 2 ). In subsonic case, we denote P 0 = P 1 = P 4 and Γ sonic = {P 1 }. Let s(θ w ) := |OP 4 |(θ w ).
Given an admissible solution (p, r), we define g sh : [0, s] → R, F −1 2,g sh : Q iter → Ω as in (3.28), as well as u : Q iter → R and v : (0, 1) → R defined by (3.29). In subsonic case g sh (0) = 0, so the transforation F 2,g sh is now singular at {s = 0}. In order to estimate u, we need a new weighted and scaled norm.
where g sh was obtained from v. Also, we may take δ 3 small enough in the iteration set, depending on the data, so that Γ shock = {r = r(θ)|θ ∈ [θ 1 , θ 2 ]}, and define v by the formula (3.29). Then u and v satisfy u * * ,2 1, α,Q iter ≤ C, v (−3/2−β/2) 2, α/2,(0,1) ≤ C, where C only depends on the data and θ * w . Finally according to the gain-in-regularity, we conclude the proof of existence of admissible solutions up to the detachment angle in the same way as for supersonic wedge angles in Section 3.