Shock Reflection-Diffraction and Nonlinear Partial

Report no. OxPDE-11/09

Shock Reflection-Diffraction and Nonlinear Partial Differential Equations of Mixed Type

by Gui-Qiang G. Chen Mathematical Institute, University of Oxford

Mikhail Feldman Department of Mathematics, University of Wisconsin

Oxford Centre for Nonlinear PDE Mathematical Institute, University of Oxford Radcliffe Observatory Quarter, Gibson Building Annexe Woodstock Road Oxford, England OX2 6HA Email: [email protected]

March 2011

Shock Reflection-Diffraction and Nonlinear Partial Differential Equations of Mixed Type Gui-Qiang G. Chen and Mikhail Feldman A BSTRACT. We present our recent results on the existence and regularity of shock reflectiondiffraction configurations by a wedge up to the sonic wedge angle, which is the von Neumann sonic conjecture. The problem is first formulated as a boundary value problem for a second-order nonlinear partial differential equation of mixed hyperbolic-elliptic type in an unbounded domain. Then the boundary value problem is reduced into a one-phase free boundary problem for a nonlinear second-order degenerate elliptic equation with the free boundary meeting the degenerate curve on which the ellipticity of the equation fails. The key steps to achieve the results are the a priori estimates of admissible solutions of the shock reflection-diffraction problem, which yield the existence theory.

1. Shock Reflection-Diffraction Problems Shock reflection-diffraction by a straight-sided wedge is one of the most fundamental multidimensional shock wave phenomena. When a plane shock hits the wedge head on, a self-similar shock of reflection-diffraction moves outward as the original shock moves forward in time. Such problems not only arise in many important physical situations, but also are fundamental in the mathematical theory of multidimensional conservation laws since their solutions are building blocks and asymptotic attractors of general solutions to the multidimensional Euler equations for compressible fluids (cf. CourantFriedrichs [9], von Neumann [27, 28], Glimm-Majda [12], and Morawetz [24]). The complexity of reflection-diffraction configurations was first reported by Ernst Mach [23] in 1878, and experimental, computational, and asymptotic analysis has shown that various patterns of reflected shocks may occur, including regular and Mach reflection (cf. [2, 11, 12, 13, 14, 15, 16, 17, 19, 21, 24, 26, 27, 28]). Most of the fundamental issues for shock reflection-diffraction have not been understood, such as the transition between different patterns, especially for the potential flow equation used widely in aerodynamics. 1991 Mathematics Subject Classification. Primary: 35M12, 3502, 35L65, 35L67, 35J70, 35R35, 76H05, 76N10; Secondary: 35M13, 76L05. Key words and phrases. Nonlinear partial differential equations, mixed elliptic-hyperbolic type, shock reflection-diffraction, transonic shock, von Neumann sonic conjecture, free boundary, existence, regularity, a priori estimates. Gui-Qiang G. Chen’s research was supported in part by the National Science Foundation under Grants DMS-0935967 and DMS-0807551, the Natural Science Foundation of China under Grant NSFC-10728101, the UK EPSRC Science and Innovation Award to the Oxford Centre for Nonlinear PDE (EP/E035027/1), and the Royal Society–Wolfson Research Merit Award (UK). Mikhail Feldman’s research was supported in part by the National Science Foundation under Grants DMS0800245, DMS-0500722, and DMS-0354729. 1

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GUI-QIANG G. CHEN AND MIKHAIL FELDMAN

Therefore, it becomes essential to establish a global theory for the existence and regularity of shock reflection-diffraction configurations in order to understand fully shock reflectiondiffraction phenomena. The Euler equations for potential flow consist of the conservation law of mass and the Bernoulli law, and take the form: (1.1) (1.2)

ρt + div (ρ ∇Φ) = 0, 1 1 Φt + |∇Φ|2 + ργ−1 = const., 2 γ−1

for (t, x) ∈ R+ ×R2 := [0, ∞)×R2 , where ∇ = (∂x1 , ∂x2 ), ρ(t, x) is the density, Φ(t, x) is the velocity potential, and the constant γ > 1 is the adiabatic exponent of the fluid under consideration. Self-similar solutions are the solutions of the form: x (1.3) Φ(t, x) = tψ(ξ, η), ρ(t, x) = ρ(ξ, η) with (ξ, η) = ∈ R2 . t Then the function φ = ψ − 12 (ξ 2 + η 2 ), called the pseudo-velocity potential, satisfies the following equation of second order: ( ) (1.4) div ρ(|Dφ|2 , φ)Dφ + 2ρ(|Dφ|2 , φ) = 0, with

( ) 1 1 ρ(|Dφ|2 , φ) = ρ0γ−1 − (γ − 1)(φ + |Dφ|2 ) γ−1 , 2 where ρ0 is a given constant and D = (∂ξ , ∂η ). Equation (1.4) is of mixed elliptichyperbolic type. It is elliptic if and only if the flow is pseudo-subsonic (subsonic, for short): |Dφ| < c(|Dφ|2 , φ, ρ0 ) (1.5)

with c(|Dφ|2 , φ) := ρ(|Dφ|2 , φ)

γ−1 2

as the sonic speed of the flow, which is equivalent to √ 2 |Dφ| < c∗ (φ, ρ0 ) := (ργ−1 − (γ − 1)φ). γ+1 0 Equation (1.4) is hyperbolic if and only if the flow is pseudo-supersonic (supersonic, for short): |Dφ| > c∗ (φ, ρ0 ).

A shock is a discontinuity in the pseudo-velocity Dφ. That is, if Ω+ and Ω− := Ω\Ω+ are two nonempty open subsets of Ω ⊂ R2 and S := ∂Ω+ ∩ Ω is a C 1 –curve where Dφ 1,1 has a jump, then φ ∈ Wloc (Ω) ∩ C 1 (Ω± ∪ S) ∩ C 2 (Ω± ) is a global weak solution of (1.4) in Ω if and only if φ satisfies equation (1.4) in Ω± and the Rankine-Hugoniot conditions on S: (1.6) (1.7)

[Dφ · τ ]S = 0 ⇐⇒ [φ]S = 0, [ ] ρ(|Dφ|2 , φ)Dφ · ν S = 0,

where [·]S is the jump across S, and ν and τ are the unit normal and tangent vector to S, respectively. A shock is called transonic if the equation changes type from hyperbolic to elliptic across the shock in the flow direction. When a flat shock, orthogonal to the flow direction and separating two uniform states (0) and (1), i.e., the states with constant velocity and density, hits a symmetric wedge W := {|x2 | < x1 tan θw , x1 > 0} at time t = 0 head on, it experiences a reflection-diffraction process. Let (ρ0 , Φ0 (t, x)) and (ρ1 , Φ1 (t, x)) be the densities and the velocity potential

SHOCK REFLECTION-DIFFRACTION AND NONLINEAR PDE OF MIXED TYPE

3

functions of states (0) and (1), respectively, where Φ0 (t, x) = 0, Φ1 (t, x) = u1 x1 , and ρ1 > ρ0 > 0 and u1 > 0 are the constants that are related through the Rankine-Hugoniot condition (1.7) by √ 2(ρ1 − ρ0 )(ργ−1 − ργ−1 ) 0 1 (1.8) u1 = . (γ − 1)(ρ1 + ρ0 ) Let Λ = R2 \ W . In the self-similar coordinates (ξ, η), the pseudo-velocity potential functions corresponding to states (0) and (1) are 1 φ0 (ξ, η) = − (ξ 2 + η 2 ), 2 1 φ1 (ξ, η) = − (ξ 2 + η 2 ) + u1 (ξ − ξ0 ), 2

(1.9) (1.10) where

ξ0 =

(1.11)

ρ1 u1 >0 ρ1 − ρ0

is the coordinate of the incoming shock in the (ξ, η)-plane. Also, by symmetry, it suffices to consider the shock reflection-diffraction problem in the upper half-plane {η > 0} ∩ Λ, with the condition ∂ν φ = 0 on {η = 0} ∩ Λ. Then the problem can be formulated as Boundary Value Problem (BVP). Seek a solution φ of equation (1.4) in the selfsimilar domain Λ ∩ {η > 0} with the slip boundary condition: Dφ · ν|∂{Λ∩{η>0}} = 0

(1.12)

and the asymptotic boundary condition at infinity: φ − φ¯ → 0

(1.13)

when ξ 2 + η 2 → ∞,

where ν is the unit normal to ∂{Λ ∩ {η > 0}}, { φ0 for ξ > ξ0 , η > ξ tan θw , φ¯ = φ1 for ξ < ξ0 , η > 0, and (1.13) holds in the sense that lim ∥φ − φ∥C(Λ\BR (0)) = 0. R→∞

Note that the slip boundary condition (1.12) implies ρDφ · ν|∂Λ = 0.

(1.14)

Also, conditions (1.12) and (1.14) are equivalent if ρ ̸= 0. Since, for the solutions under consideration, ρ ̸= 0 always holds, we use condition (1.14) instead of (1.12) in the definition of weak solutions for Problem (BVP). Condition (1.14) is the conormal condition for equation (1.4). This yields the following definition. 1,1 D EFINITION 1.1. A function φ ∈ Wloc (Λ) is called a weak solution of Problem (BVP) if φ satisfies (1.13) and the following: ) ( (i) ργ−1 − (γ − 1) φ + 12 |Dφ|2 ≥ 0 a.e. in Λ; 0 (ii) (ρ(|Dφ|2 , φ), ρ(|Dφ|2 , φ)|Dφ|) ∈ (L1loc (Λ))2 ; (iii) For every ζ ∈ Cc∞ (R2 ), ∫ ( ) ρ(|Dφ|2 , φ)Dφ · Dζ − 2ρ(|Dφ|2 , φ)ζ dξdη = 0. Λ

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GUI-QIANG G. CHEN AND MIKHAIL FELDMAN

R EMARK 1.1. Note that, since ζ does not need to be zero on ∂Λ, the integral identity in the definition is a weak form of equation (1.4) and the boundary condition (1.14) as the trace. R EMARK 1.2. From Definition 1.1, we can see the following fact: If B ⊂ R2 is an open set and φ is a weak solution of Problem (BVP) satisfying φ ∈ C 2 (B∩Λ)∩C 1 (B∩Λ), then φ satisfies equation (1.4) in B ∩ Λ, the boundary condition (1.14) on B ∩ ∂Λ \ {0}, and Dφ(0) = 0 in the classical sense. R EMARK 1.3. For Problem (BVP), since φ1 does not satisfy the slip boundary condition (1.12), the solution must differ from φ1 in {ξ < ξ0 } ∩ Λ, and thus a shock reflectiondiffraction by the wedge vertex occurs, which is one of the key points of the problem.

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F IGURE 1. Regular reflection-diffraction

2. The von Neumann Sonic Conjecture and Main Theorems In this section, we describe our recent results on the regular shock reflection-diffraction problem for the wedge angle θw larger than the angle θs when state (2) becomes sonic, by which the sonic conjecture proposed by von Neumann in [27] in 1943 (also see [28]) is solved for potential flow (cf. Fig. 1). More specifically, for a regular shock reflection-diffraction pattern, the solution φ consists of three uniform states (0), (1), (2), and a nonuniform state in the domain Ω, and the equation is hyperbolic in the respective regions of states (0), (1), and (2), and is elliptic in Ω. The regular reflection-diffraction is characterized by the property that the reflection point P0 is on the wedge boundary. A necessary condition for the existence of such a pattern is the existence of the local regular shock reflection configuration at P0 , i.e., the existence of a straight line P0 P1 and a uniform state (2) such that the Rankine-Hugoniot conditions are satisfied along P0 P1 and the slip boundary condition (1.12) is satisfied along the wedge. State (2) behind P0 P1 is of the form: (2.1)

1 φ2 (ξ, η) = − (ξ 2 + η 2 ) + u2 (ξ − ξ0 ) + (η − ξ0 tan θw )u2 tan θw , 2

SHOCK REFLECTION-DIFFRACTION AND NONLINEAR PDE OF MIXED TYPE

5

with the constant velocity u2 > 0. The parameters of state (2), i.e., the velocity (u2 , u2 tan θw ) and the density ρ2 , and the direction of P0 P1 are obtained by solving a system of algebraic equations, consisting of the Rankine-Hugoniot conditions (1.6)–(1.7) for states (1) and (2) on S. Depending on the parameters of states (0) and (1), this system is solvable for θw sufficiently close to π2 (see [5, 6]) and is not solvable for θw sufficiently small (and then Mach s c reflection is expected). That is, given states (0) and (1), there exist θw and θw satisfying π c s c π 0 < θw ≤ θw < 2 such that state (2) exists for all θw ∈ (θw , 2 ), and state (2) is supersonic c s s π , θw ). Here, near the reflection point P0 for θw ∈ (θw , 2 ) and is subsonic for θw ∈ (θw a solution in a region is supersonic (resp. subsonic) if equation (1.4) is hyperbolic (resp. elliptic) in that region. Thus, a local supersonic (resp. subsonic) regular reflection exists s π c s for θw ∈ (θw , 2 ) (resp. θw ∈ (θw , θw )). The following conjecture was proposed by von Neumann [27] in 1943 (also see [28]): The Sonic Conjecture (von Neumann 1943): There exists a global regular shock reflection-diffraction configuration with the structure as in Fig. 1 for all wedge angles which admit a local supersonic regular s π reflection, i.e., for all θw ∈ (θw , 2 ). This transition conjecture is based on the argument that the transition between the regular reflection and Mach reflection occurs when the corner-generated signals cannot catch-up with the reflection point P0 . Hence, as long as the flow Mach number behind the reflected shock is larger than 1, i.e., the flow is supersonic, the reflection point is isolated from the corner-generated signals, which cannot reach it. Also see Lock-Dewey [22]. In Chen-Feldman [5, 6], the global existence and stability of solutions to the boundary value problem for a large-angle wedge were first established. Furthermore, the optimal regularity of regular shock reflection-diffraction solutions near the sonic circle to be C 1,1 was established in Bae-Chen-Feldman [1]. In Chen-Feldman [7], we have further established the following theorems. s T HEOREM 2.1. Let u1 ≤ c1 . Let θw ∈ (0, π2 ) be such that state (2) exists and is s π , 2 ), and the parameter u2 in supersonic at the point P0 = P0 (θw ) for each θw ∈ (θw s π (2.1) for state (2) depends continuously on θw on the interval [θw , 2 ]. Then there exists 1 s s π α = α(ρ0 , ρ1 , γ, θw ) ∈ (0, 2 ) such that, when θw ∈ [θw , 2 ), there exists a global selfsimilar solution of the shock reflection-diffraction problem:

x |x|2 Φ(t, x) = t φ( ) + t 2t

for

x ∈ Λ, t > 0, t

with

( ) 1 1 − (γ − 1)(Φt + |∇x Φ|2 ) γ−1 , ρ(t, x) = ργ−1 0 2 where φ is the solution of Problem (BVP) and satisfies that, for (ξ, η) = xt , ¯ φ ∈ C ∞ (Ω) ∩ C 1,α (Ω), (2.2)

  φ0 φ1 φ=  φ2

for ξ > ξ0 and η > ξ tan θw , for ξ < ξ0 and above the reflected shock P0 P1 P2 , in P0 P1 P4 ,

φ is C 1,1 across the part P1 P4 of the sonic circle including the endpoints P1 and P4 , and the reflected shock P0 P1 P2 is C 2 at P1 and C ∞ except P1 . Also, the relative interior of the reflected shock P0 P1 P2 lies in {η > ξ tan θw , η > 0}, i.e., is separated from the wedge and from the symmetry line {η = 0}. Moreover, φ satisfies the following:

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GUI-QIANG G. CHEN AND MIKHAIL FELDMAN

(i) equation (1.4) is strictly elliptic in Ω \ P1 P4 : |Dφ| < c∗ (φ, ρ0 )

(2.3) (ii) φ2 ≤ φ ≤ φ1

in Ω \ P1 P4 ;

in Ω.

R EMARK 2.1. In fact, φ is an admissible solution in the sense of Definition 3.1 below and thus satisfies further properties listed in Definition 3.1. Now we address the other case u1 > c1 . In this case, the results of Theorem 2.1 hold s from the wedge angle π2 until either θw or until the shock hits the wedge vertex P3 at some a s π wedge angle θw ∈ (θw , 2 ). s T HEOREM 2.2. Let u1 > c1 . Let θw ∈ (0, π2 ) be as in Theorem 2.1. Then there exists 1 s π s ∈ [θw , 2 ) and α ∈ (0, 2 ) depending only on (ρ0 , ρ1 , γ, θw ) such that the results of a π a s Theorem 2.1 hold for each wedge angle θw ∈ (θw , 2 ). If θw > θw , then, for the wedge a angle θw = θw , there exists an attached solution, i.e., a solution of Problem (BVP) with the properties as in Theorem 2.1 except that P2 = P3 , and the reflected shock P0 P1 P2 is Lipschitz up to its endpoints and is C 2 at P1 and C ∞ except P1 and P2 .

a θw

R EMARK 2.2. The attached shock case does occur in the Mach shcok reflectiondiffraction configurations as observed; see Fig. 238, page 144, in [26]. Since the solution φ of Problem (BVP) constructed in Theorems 2.1–2.2 is C 1,1 across P1 P4 and satisfies properties (i)–(ii) of Theorem 2.1 and some further estimates, then the regularity results of Bae-Chen-Feldman [1, Theorem 4.2] apply. T HEOREM 2.3. The solution φ in Theorems 2.1–2.2 satisfies the following: (i) φ is C 2,α up to the arc P1 P4 away from the point P1 for any α ∈ (0, 1). That is, for any α ∈ (0, 1) and any given (ξ0 , η0 ) ∈ P1 P4 \ {P1 }, there exist K < ∞ ∗ and d > 0, depending only on ρ0 , ρ1 , γ, α, θw , and dist((ξ0 , η0 ), P1 P2 ), so that ∥φ∥2,α;Bd (ξ0 ,η0 )∩Ω ≤ K; (ii) For any (ξ0 , η0 ) ∈ P1 P4 \ {P1 }, lim (ξ,η)→(ξ0 ,η0 ) (ξ,η)∈Ω

(Drr φ − Drr φ2 ) =

1 , γ+1

where (r, θ) are the polar coordinates with center at (u2 , u2 tan θw ); (iii) D2 φ has a jump across P1 P4 : For any (ξ0 , η0 ) ∈ P1 P4 \ {P1 }, lim (ξ,η)→(ξ0 ,η0 ) (ξ,η)∈Ω

Drr φ −

lim (ξ,η)→(ξ0 ,η0 ) (ξ,η)∈Λ\Ω

Drr φ =

1 ; γ+1

(iv) The limit lim (ξ,η)→P1 D2 φ does not exist. (ξ,η)∈Ω

3. Admissible Solutions and Their A Priori Estimates The key steps to establish the main theorems above in Chen-Feldman [7] are a priori estimates of admissible solutions to Problem (BVP). Most of these estimates such as the regularity across the sonic arc P1 P4 and the monotonicity cones are optimal. In this section, we present these main estimates.

SHOCK REFLECTION-DIFFRACTION AND NONLINEAR PDE OF MIXED TYPE

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3.1. Definition of Admissible Solutions. Let γ > 1, ρ1 > ρ0 > 0, and u1 > 0 be the constants satisfying (1.8), and let ξ0 be defined by (1.11). Let the incident shock S = {ξ = ξ0 } hits the wedge at the point P0 = (ξ0 , ξ0 tan θw ), and let state (0) and state (1) ahead of and behind S be given by (1.9) and (1.10), respectively. Then it can be shown s s π that there exists θw ∈ (0, π2 ) such that, when a wedge angle θw ∈ (θw , 2 ), there exists a state (2) of form (2.1) such that (i) u2 > 0; (ii) The entropy condition ρ2 > ρ1 holds; (iii) The Rankine-Hugoniot condition (1.7) holds along the line S1 = {φ1 = φ2 }; (iv) State (2) is supersonic at the point P0 . R EMARK 3.1. From (i) and θw ∈ (0, π2 ), the velocity vector (u2 , v2 ) := (u2 , u2 tan θw ) is not parallel to (u1 , 0). This last fact implies that the set {φ1 = φ2 } =: S1 is a line. Moreover, P0 ∈ S1 since φ1 (P0 ) = φ2 (P0 ) from (1.10), (2.1), and the definition of P0 . s π R EMARK 3.2. It can be shown that, for any wedge angle θw ∈ (θw , 2 ), the line S1 intersects the sonic circle of state (2) at two points. Denote by P1 the point of intersection which is closer to P0 than the other intersection point. Then P1 ∈ Λ. Note that state (2) is supersonic along P0 P1 .

R EMARK 3.3. Using the fact that state (2) is supersonic along P0 P1 by Remark 3.2 and the sonic circle of state (2) is Bc2 (u2 , v2 ), it follows that ηP1 > v2 . R EMARK 3.4. If θw is an admissible angle, we define √ • The point P4 = (q2 +c2 )(cos θw , sin θw ), where q2 = u22 + v22 = cosu2θw . That is, P4 is the “upper” point of intersection of the sonic circle of state (2) with the wedge line {η = ξ tan θw }. From the definitions, ξP1 < ξP4 . • The line segment Γwedge := P3 P4 ⊂ {η = ξ tan θw }. • The arc Γsonic , which is the “upper” arc P1 P4 of the sonic circle of state (2), that is, Γsonic := {(ξ, fsonic (ξ)) : ξ ∈ [ξP1 , ξP4 ]}, √ with fsonic (ξ) := v2 + c22 − (ξ − u2 )2 . R EMARK 3.5. If the wedge angle θw ∈ (0, π2 ) is admissible, then the condition ρ2 > ρ1 implies that the half-plane {φ1 > φ2 } contains the point (u1 , 0), which is the center of the sonic circle of state (1). Now we introduce the notion of admissible solutions of Problem (BVP). D EFINITION 3.1. Let γ > 1, ρ1 > ρ0 > 0, and u1 > 0 be the constants satisfying s (1.8). Fix the wedge angle θw ∈ [θw , π2 ). A function φ ∈ C 0,1 (Λ) is called an admissible solution for regular reflection-diffraction if φ is a solution of Problem (BVP) satisfying the following: (i) There exists the (shock) curve Γshock := P1 P2 with endpoints P1 and P2 , where P2 = (ξP2 , 0) with ( ) (3.1) ξP2 < min 0, u1 − c1 , ξP2 ≤ ξP1 , and the curve Γshock satisfies

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(3.2)

GUI-QIANG G. CHEN AND MIKHAIL FELDMAN

• For C1 = ∂Bc1 (u1 , 0) that is the sonic circle of state (1), ) ( Γshock ⊂ Λ \ Bc1 (u1 , 0) ∩ {ξP2 ≤ ξ ≤ ξP1 }; − − ext • Denote by Γext shock the curve Γshock := Γshock ∪Γshock ∪{P2 }, where Γshock ext is the reflection of Γshock with respect to {ξ2 = 0}. Then Γshock is C ∞ at its relative interior (including P2 ) in the sense that, for any P in the relative ∞ interior of Γext shock , there exist r > 0, f ∈ C (R), and the orthonormal 2 coordinate system (S, T ) in R such that

Γshock ∩ Br (P ) = {S = f (T )} ∩ Br (P ). Denote the line segment Γsym := P2 P3 . From (3.1)–(3.2), it follows that Γshock , Γsonic , Γsym , and Γwedge do not have common points except their endpoints P1 , P2 , P3 , and P4 . Thus, Γshock ∪ Γsym ∪ Γwedge ∪ Γsonic is a closed curve without self-intersections. Denote by Ω the open bounded domain restricted by this curve. Note that Ω ⊂ Λ, ∂Ω = Γshock ∪ Γsym ∪ Γwedge ∪ Γsonic , and ∂Ω ∩ ∂Λ = Γsym ∪ Γwedge . (ii) φ satisfies (2.2) and (3.3)

φ ∈ C 0,1 (Λ) ∩ C 1 (Λ \ P0 P1 P2 ) ∩ C ∞ (Ω \ (Γsonic ∪ {P3 })) ∩ C 1,α (Ω), s where α ∈ (0, 1) depends only on θw . (iii) equation (1.4) is strictly elliptic in Ω \ Γsonic :

|Dφ| < c∗ (φ, ρ0 ) in Ω \ Γsonic .

(3.4)

(iv) φ2 ≤ φ ≤ φ1 in Ω. (v) Let eS1 be a unit vector parallel to S1 = {φ1 = φ2 } and oriented such that eS1 · eξ < 0. That is, eS 1 =

(3.5)

P1 − P0 (v2 , u1 − u2 ) . = −√ |P1 − P0 | (u1 − u2 )2 + v22

Then (3.6)

∂eS1 (φ1 − φ) ≤ 0,

∂η (φ1 − φ) ≤ 0

in Ω.

R EMARK 3.6 (Γsym ∪ {P2 } are the interior points). Let Ω− (resp. Γ− sonic ) be the reflection of Ω (resp. Γsonic ) with respect to {η = 0} and Ωext := Ω ∪ Ω− ∪ Γsym ,

− Γext sonic := Γsonic ∪ Γsonic .

ext ext Then Γext , where Γext be the even shock ⊂ Ω shock is defined in (i) of Definition 3.1. Let φ ext ext extension of φ into Ω , i.e., φ (ξ, ±η) = φ(ξ, η) for η > 0. Using ∂ν ϕ = 0 on Γsym , 1,α (Ωext ). The function we obtain φext ∈ C 2 (Ωext \ (Γext sonic ∪ {P3 })) ∩ C

ξ 2 + η2 ≡ φext − φext 0 2 satisfies that ϕext is the even extension of ϕ into Ωext so that ϕext := φext +

1,α ϕext ∈ C 2 (Ωext \ (Γext (Ωext ). sonic ∪ {P3 })) ∩ C

Then it is easy to check by an explicit calculation that φext and ϕext satisfy equation (1.4) and its corresponding non-divergence form in Ωext , and that the equations are strictly ext elliptic in Ωext \ Γext satisfies sonic . Moreover, φ φext = φ1 ,

ρ(|Dφext |2 , φext )Dφext · ν = ρ1 Dφ1 · ν

on Γext shock .

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R EMARK 3.7 (The velocity jump across Γshock ). Condition (3.2) implies that φ1 is supersonic on Γshock . Then (iii) of Definition 3.1 implies that Dφ ̸= Dφ1 on Γshock . Similar argument with use of ξP2 < u1 − c1 from (3.1) implies a gradient jump at P2 . Also, using (3.3), Dφ(P1 ) = Dφ2 (P1 ) ̸= Dφ1 (P1 ). Thus, Dφ ̸= Dφ1

(3.7)

on Γshock .

R EMARK 3.8 (The cone of monotonicity directions). The properties in (3.6) imply that, if φ is an admissible solution of Problem (BVP) in the sense of Definition 3.1, then ∂e (φ1 − φ) ≤ 0 in Ω

(3.8)

for all e ∈ Cone(eS1 , eξ2 ), e ̸= 0,

where Cone(e, g) = {ae + bg : a, b ≥ 0}

(3.9)

for e, g ∈ R2 \ {0}.

R EMARK 3.9 (The shock does not intersect with the wedge and the sonic circle of state (1)). The property Γshock ⊂ Λ \ Bc1 (u1 , 0) of (3.2) implies that the shock does not intersect with the wedge and the sonic circle of state (1), and Bc1 (u1 , 0) ∩ Λ ⊂ Ω.

(3.10)

We also note the following property: R EMARK 3.10 (The directions of pseudo-velocities on the shock: the entropy condition). If φ is an admissible solution of Problem (BVP) satisfying properties (i)–(iii) of Definition 3.1, then (3.11)

∂ν φ1 > ∂ν φ > 0

on Γshock ,

where ν is the unit normal to Γshock interior to Ω. From now on through this section, we always assume that φ is an admissible solution of Problem (BVP) in the sense of Definition 3.1. 3.2. The Strict Monotonicity Cone for φ1 − φ and Its Geometric Consequences. First we prove that (3.12)

∂e (φ1 − φ) < 0

in Ω, for all e ∈ Cone0 (eS1 , eη ),

where Cone0 (eS1 , eη ) is the interior of the cone Cone(e, g) defined by (3.9) for e, g ∈ R2 \ {0}. This implies that the shock is graph for a cone of directions. That is, let e ∈ Cone0 (eS1 , eη ). Let e⊥ be orthogonal to e and oriented so that e⊥ · eS1 < 0. Let |e| = |e⊥ | = 1. Let (S, T ) be the coordinates with respect to the basis (e, e⊥ ). Also, denote (SPk , TPk ) the (S, T )-coordinates of the point Pk , k = 1, . . . , 4, and note that TP2 < TP1 < TP4 . Then there exists fe,shock ∈ C 1,α (R) such that (i) Γshock = {S = fe,shock (T ) : T ∈ (TP2 , TP1 )} and Ω ⊂ {S < fe,shock (T ) : T ∈ R}. (ii) In the (S, T )–coordinates, P1 = (fe,shock (TP1 ), TP1 ) and P2 = (fe,shock (TP2 ), TP2 ).

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GUI-QIANG G. CHEN AND MIKHAIL FELDMAN

(iii) For any P ∈ Γshock , there exists r > 0 such that (P − Cone0 (eS1 , eη )) ⊂ {S < fe,shock (T ) : T ∈ R}, (P − Cone0 (eS1 , eη )) ∩ Br (P ) ⊂ Ω, (P + Cone0 (eS1 , eη )) ⊂ {S > fe,shock (T ) : T ∈ R}, (P + Cone0 (eS1 , eη )) ∩ Ω = ∅. That is, at P , the cone Cone0 (eS1 , eη ) is above the graph of fe,shock , and the cone −Cone0 (eS1 , eη ) is below the graph of fe,shock . (iv) In the (S, T )–coordinates, the region Ω has the following form: There exists fe,sonic ∈ C([TP1 , TP4 ]) ∩ C ∞ ([TP1 , TP4 )) and linear functions Le,wedge and Le,sym such that Γsonic = {S = fe,sonic (T ) : T ∈ (TP1 , TP4 )}, Γwedge = {S = Le,wedge (T ) : T ∈ (TP3 , TP4 )}, Γsym = {S = Le,sym (T ) : T ∈ (TP2 , TP3 )}, so that fe,shock (T ) > max(Le,wedge (T ), Le,sym (T )) fe,sonic (T ) > max(Le,wedge (T ), Le,sym (T ))

for T ∈ (TP2 , TP1 ), for T ∈ (TP1 , TP4 ),

Λ = {(S, T ) ∈ R2 : T ∈ R, S > max(Le,wedge (T ), Le,sym (T ))}, { } S < fe,shock (T ) for T ∈ (TP2 , TP1 ) Ω = (S, T ) ∈ Λ : T ∈ (TP2 , TP4 ), . S < fe,sonic (T ) for T ∈ (TP1 , TP4 ) (v) The tangent directions to Γshock are between the directions of the line S1 and {teη , : t ∈ R}, which are the tangent lines to Γshock at the points P1 and P2 respectively. That is, for any T ∈ (TP2 , TP1 ), eS 1 · e eξ 2 · e ′ ′ ′ −∞ < = fe,shock (TP1 ) ≤ fe,shock (T ) ≤ fe,shock (TP2 ) = < ∞. eS1 · e⊥ eξ2 · e⊥ 3.3. The Monotonicity Cone for φ − φ2 and Its Consequences. Then we prove (3.13)

∂e (φ − φ2 ) ≥ 0

in Ω, for all e ∈ Cone0 (eS1 , −ν wedge ),

where Cone0 (eS1 , −ν wedge ) is defined by (3.9) and ν wedge is the unit normal to Γwedge , interior to Ω. As its consequence, we conclude that, in the local coordinates (x, y) with x the normal directional coordinate into Ω with respect to the sonic arc Γsonic , ∂x (φ − φ2 ) > 0 in a uniform neighborhood of the sonic arc Γsonic . 3.4. Uniform Estimate of the Size of Ω, the Lipschitz Norm of the Potential, and the Density from Above and Below. Then we prove the estimates on diam (Ω) and on s s π sup |φ| in Ω. One of the main difficulties is that, if θw and θw ∈ [θw , 2 ) are sufficiently s small, then θw (θw ) < 0, i.e., the ray S1+ = {P0 + t(P2 − P0 ) : t > 0} s π for such θw does not intersect with the ξ-axis. That is, there exists θ˜w ∈ [θw , 2 ) such that, ˜ for θw = θw , the line S1 = {φ1 = φ2 } is parallel to the ξ-axis with limθw →θ˜w + ξ˜ = −∞,

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˜ 0) is the point of intersection of S + with the ξ-axis. Thus, diam (Ω) cannot be where (ξ, 1 bounded by a uniform bound of coordinates of the points of intersection of S1+ with the ξ-axis. Then we obtain that there exists C > 0 such that, if φ is an admissible solution of Problem (BVP) in the sense of Definition 3.1, then (3.14)

Ω ⊂ BC (0),

(3.15)

∥φ∥C 0,1 (Ω) ≤ C,

(3.16)

aρ1 ≤ ρ ≤ C in Ω

(3.17)

ρ1 < ρ ≤ C

1 2 ) γ−1 > 0, γ+1 on Γshock ∪ {P3 }.

with a =

(

3.5. Uniform Positive Lower Bound for the Distance between the Shock and the Sonic Circle of State (1). There exists C < ∞ such that 1 (3.18) dist(Γshock , Bc1 (O1 ) > C s , π2 ), where O1 = (u1 , 0) is for any admissible solution of Problem (BVP) with θw ∈ [θw the center of the sonic circle of state (1). Estimate (3.18) is crucial, especially since it is used for the ellipticity estimate in §3.7 below. 3.6. Uniform Positive Lower Bound for the Distance between the Shock and the Wedge, and Separation of the Shock and the Symmetry Line. We first use the C ∞ regularity of fshock to obtain the following estimates: Separation of the Shock and the Symmetry Line: There exists µ > 0 depending only on the data such that c1 ′ fshock (ξ) > µ on {ξ ∈ (ξP2 , min(ξP1 , 0)) : fshock (ξ) ∈ (0, )} 2 π s for any admissible solution φ of Problem (BVP) with θw ∈ [θw , 2 ), where fshock is defined by (3.19)

Γshock = {(ξ, fshock (ξ)) : ξ ∈ [ξP2 , ξP1 ]}.

As a corollary, we conclude that, for any admissible solution φ of Problem (BVP) with s θw ∈ [θw , π2 ), c1 fshock (ξ) ≥ min( , µ(ξ − ξP2 )) for all ξ ∈ [ξP2 , min(ξP1 , 0)]. 2 In order to consider Γshock up to the point P2 , it is convenient to consider the solutions extended by the even reflection into {η < 0} as in Remark 3.6 and to consider also the extended domain Ωext and shock Γext shock defined in that remark. Furthermore, we have Uniform Positive Lower Bound for the Distance between the Shock and the Wedge: Assume that u1 < c1 . Then there exists C < ∞ such that 1 dist(Γshock , Γwedge ) > C ∗ for any admissible solution of Problem (BVP) with θw ∈ [θw , π2 ). s π If u1 ≥ c1 , then, for each b ∈ [θw , 2 ], we denote by Sb the set of all admissible solutions for θw ∈ (b, π2 ). Also, denote ξ(P2 , φ) the ξ-coordinate of the point P2 for the

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GUI-QIANG G. CHEN AND MIKHAIL FELDMAN

a admissible solution φ. Note that ξ(P2 , φ) < 0. Now we define the number θw in Theorem 2.2 as a θw :=

We show that such that

a θw


0 depending only on (ρ0 , ρ1 , γ, θw ) such that, if φ s is an admissible solution of Problem (BVP) with θw ∈ [θw , π2 ), then

M 2 (ξ, η) ≤ 1 − µ dist((ξ, η), Γsonic )

for all (ξ, η) ∈ Ω(φ).

To achieve this, the result of §3.5 on the positive lower bound on the distance of the shock and the sonic circle of state (1) is crucially used, since it allows to estimate that state (1) is “sufficiently hyperbolic” on the other side of the shock. Write equation (1.4) in the form (3.20)

div A(Dφ, φ, ξ, η) + B(Dφ, φ, ξ, η) = 0,

with p = (p1 , p2 ) ∈ R2 , z ∈ R, where (3.21) A(p, z, ξ, η) ≡ A(p, z) := ρ(|p|2 , z)p,

B(p, z, ξ, η) ≡ B(p, z) := 2ρ(|p|2 , z)

with the function ρ(|p|2 , z) defined by (1.5). We restrict to such (p, z) that (1.5) is defined, i.e., satisfying ργ−1 − (γ − 1)(z + 12 |p|2 ) ≥ 0. Also, for a subset B ⊂ R2 and the function 0 1 φ ∈ C (B), denote (3.22)

E(φ, B) := {(p, z, ξ, η) : (ξ, η) ∈ B, z = φ(ξ, η), p = Dφ(ξ)}.

s As a corollary, we conclude that there exists C > 0 depending only on (ρ0 , ρ1 , γ, θw ) π s such that, if φ is an admissible solution of Problem (BVP) with θw ∈ [θw , 2 ) and B ⊂ Ω satisfies dist(B, Γsonic ) ≥ d > 0, then equation (1.4) is elliptic with constants (C, Cd ) for the solution φ on B. That is, using notation (3.20),

(3.23)

2 ∑ d 2 |κ| ≤ Aipj (p, z, ξ, η)κi κj ≤ C|κ|2 C i,j=1

for all (p, z, ξ, η) ∈ E(φ, B) and κ = (κ1 , κ2 ) ∈ R2 .

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3.8. Regularity Estimates. With the geometry of Ω and the ellipticity controlled, we can conclude the regularity estimates. We perform the regularity estimates separately away from the sonic arc where the equation is uniformly elliptic and near the sonic arc where the ellipticity degenerates. Regularity Estimates away from the Sonic Arc: Away from the ε-neighborhood of the sonic arc, we have the uniform ellipticity estimates for admissible solutions with the constants independent of the solution and the wedge angle. Also, in order to avoid the difficulties related to the corner at the point P2 of intersection of the shock and the symmetry line, we extend the elliptic domain Ω by reflection over the symmetry line and use the even extension of the solution. From the form of the potential flow equation (1.4), it follows that (1.4) is satisfied in the extended domain, and the Rankine-Hugoniot conditions (1.6)–(1.7) are satisfied on the extended shock. Now the boundary part Γsym lies in the interior of the (extended) Ω, and P2 is the interior point of the (extended) shock. Then the C 2,α –estimates in the interior of Ω and near Γwedge (away from the corner P3 ) follow from the standard elliptic theory, where we use the homogeneous Neumann boundary condition on Γwedge , the uniform estimate of the distance between the shock and the wedge, and the Lipschitz estimates of the solution. Also, we obtain the C 1,α –estimates near the corner P3 for sufficiently small α > 0 by using the results of Lieberman [20]. For the estimates of the shock curve and the solution near the shock (away from the ε-neighborhood of the sonic arc), we use the partial hodograph transform. For that, we first show that the function ϕ¯ := φ1 − φ is uniformly monotone in a uniform neighborhood of the shock in the radial direction with respect to the center of the sonic circle O1 of state (1), i.e., there exist δ, σ > 0 such that ∂r ϕ¯ ≤ −δ in Nσ (Γshock ) ∩ Ω. Note that ϕ¯ = 0 on Γshock by the Rankine-Hugoniot condition (1.6), and ϕ¯ > 0 in Ω since φ is an admissible solution of Problem (BVP). Thus we rewrite the potential flow equation (1.4) and the gradient jump Rankine-Hugoniot condition (1.7) in terms of ϕ¯ and in the polar coordinates centered at O1 . Then, working in the subdomain (Nσ (Γshock ) ∩ Ω) \ Nε (Γsonic ), we perform the partial hodograph transform in the radial direction. Thus we obtain an elliptic equation in a fixed domain with flat boundary and an oblique derivative condition on that boundary. Also, the lower bound on the size of domain and the Lipschitz estimate of the unknown function follows from the strict radial monotonicity and ¯ Therefore, we obtain the C k –estimates of the problem in the Lipschitz estimates of ϕ. the hodograph coordinates, for each k = 2, 3, . . . , which implies the C k –estimates of the shock curve and φ near the shock in the original coordinates, away from Nε (Γsonic ). Regularity Estimates near the Sonic Arc, i.e., in Nε (Γsonic ) ∩ Ω for Sufficiently Small ε > 0. Near the sonic arc, it is convenient to work in the coordinates flattening the sonic arc. We consider the polar coordinates (r, θ) with respect to the center O2 of the sonic circle of state (2), note that Γsonic is an arc of the circle r = c2 , and define (x, y) = (c2 − r, θ − θw ). Then, within Nε (Γsonic ), we have Ω ⊂ {x > 0, y > 0},

Γsonic = Ω ∩ {x = 0},

We perform the estimates in terms of function: ψ = φ − φ2 .

Γwedge = Ω ∩ {y = 0}.

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Note that ψ(0, y) ≡ 0 since φ is an admissible solution, i.e., φ = φ2 on Γsonic . Writing the potential flow equation (1.4) in terms of ψ in the (x, y)-coordinates, then the fact that the equation is elliptic in Ω with the ellipticity constant proportional to the distance to the sonic arc Γsonic implies that there exists δ > 0 so that, for each admissible solution, 2−δ x in Nε (Γsonic ) ∩ Ω. ψx ≤ 1+γ Also, from the structure of the monotonicity cone of ψ, we show that ψx ≥ 0. Thus, |ψx | ≤ Cx. Using again the monotonicity cone of ψ, we obtain |ψy | ≤ Cx. Now, since |Dψ| ≤ Cx, we obtain the following degenerate ellipticity structure of (1.4) (in terms of ψ) on any admissible solution ψ near Γsonic : (3.24) 2 2 2 A11 (Dψ, ψ, x)∂xx ψ + 2A12 (Dψ, ψ, x)∂xy ψ + A22 (Dψ, ψ, x)∂yy ψ + A(Dψ, ψ, x) = 0 with smooth (Aij , A)(p, z, x) (independent of y) so that (3.25) λ(ξ 2 +η 2 ) ≤ A11 (p, z, x)

ξη ξ2 1 +2A12 (p, z, x) √ +A22 (p, z, x)η 2 ≤ (ξ 2 +η 2 ) x λ x

for (p, z) = (Dψ, ψ)(x, y) and for all (x, y) ∈ Nε (Γsonic ) ∩ Ω. Also, A(0, 0, x) = 0, i.e., the equation is homogeneous. We use (3.24) and (3.25) for the estimates in the C 2,α –norms weighted and scaled depending on x, which reflect the ellipticity structure. One way to define these norms is (x0 ,y0 ) following: √ √For any (x0 , y0 x) ∈ Nε (Γsonic ) ∩ Ω, let R(x ,y ) = (x0 − d, x0 + d) × (y0 − d, y0 − d), where d = 100 > 0. Rescale ψ from R 0 0 ∩ Ω to the (portion of) the unit square, i.e., define the rescalsed function: √ 1 ψ (x0 ,y0 ) (S, T ) = 2 ψ(x0 + dS, y0 + d T ) d √ in Q(x0 ,y0 ) := {(S, T ) ∈ (−1, 1)2 : (x0 + dS, y0 + d T ) ∈ Ω}. The “parabolic” norm (par) of ∥ψ∥2,α,Nε (Γsonic )∩Ω is the supremum over (x0 , y0 ) ∈ Nε (Γsonic ) ∩ Ω of the norms (par)

∥ψ (x0 ,y0 ) ∥C 2,α (Q(x0 ,y0 ) ) . Note that the estimate in the norm ∥ · ∥2,α,Nε (Γsonic )∩Ω implies the C 1,1 –estimate in Nε (Γsonic ) ∩ Ω. (par) In order to estimate ∥ψ∥2,α,Nε (Γsonic )∩Ω , we need to obtain the C 2,α –estimates of the rescaled functions ψ (x0 ,y0 ) . By a standard covering argument, it suffices to consider three cases: (i) The interior rectangle R(x0 ,y0 ) ⊂ Ω; (ii) The rectangle centered on the wedge (x0 , y0 ) ∈ Γwedge ∩ Nε (Γsonic ); (iii) The rectangle centered on the shock (x0 , y0 ) ∈ Γshock ∩ Nε (Γsonic ). The gradient estimates |Dψ| ≤ Cx and the condition ψ(0, y) ≡ 0 imply |ψ| ≤ Cx2 . Thus, ∥ψ (x0 ,y0 ) ∥L∞ (Q(x0 ,y0 ) ) ≤ C

for all (x0 , y0 ) ∈ Nε (Γsonic ) ∩ Ω.

Also, writing equation (3.24) in terms of the rescaled function ψ (x0 ,y0 ) and using the ellipticity structure (3.25), we see that ψ (x0 ,y0 ) satisfies a uniformly elliptic homogeneous equation in Q(x0 ,y0 ) , with the ellipticity constants and higher norms of the “coefficients” (x ,y ) independent of (x0 , y0 ). Then the C 2,α –estimates of ψ (x0 ,y0 ) in the “half-square” Q 1 0 0 2

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follow in Case (i) from the interior elliptic estimates. In Case (ii), in addition to the equation, we use the boundary condition ∂ν ψ = 0 on Γwedge , which holds under rescaling. In Case (iii), since we do not have the estimates of the shock curve, we use the partial hodograph transform. For that, we show by an explicit calculation that the function ϕ¯0 := φ1 − φ2 satisfies

1 in Nε (Γsonic ) ∩ Ω. C From this, recalling that |D(φ−φ2 )| = |Dψ| ≤ Cx and reducing ε if necessary, we obtain that ϕ¯ = φ1 − φ satisfies 1 ∂y ϕ¯ ≥ in Nε (Γsonic ) ∩ Ω. C ¯ y)) = y. Then we perform the partial hodoThus a function v is defined such that v(x, ϕ(x, graph transform in the y-direction for ϕ¯ in R(x0 ,y0 ) for (x0 , y0 ) ∈ Γshock , thus obtaining ¯ y)). Then an equation in terms of the unknown function v in the variables (s, t) = (x, ϕ(x, Γshock = {ϕ¯ = 0} is transformed into the line {t = 0}, and the Rankine-Hugoniot gradient jump condition (1.7) implies a boundary condition on that line. Also, we perform the partial hodograph transform in the y-direction for the “background solution” ϕ¯0 in R(x0 ,y0 ) , thus obtaining an equation in terms of an unknown function v0 . Then the line S1 = {φ1 = φ2 } ≡ {ϕ¯0 = 0} is mapped to {t = 0}, and the Rankine-Hugoniot gradient jump condition (1.7) for (φ1 , φ2 ) on S1 determines the boundary condition on {t = 0}. We estimate the size and shape of the common domain for v and v0 , and write the equation in this common domain and the boundary condition on the {t = 0}–boundary part in terms of w = v − v0 . This function w has the properties similar to the properties of ψ above, i.e., |Dw| ≤ Cx0 , |w| ≤ Cx20 , and the equation for w has the ellipticity structure similar to (3.25), with x0 replacing x in the scaling part. Then we can rescale w similar to the rescaling of ψ above, and the rescaled function satisfies a uniformly elliptic equation with the ellipticity constants (and the other properties) independent of (x0 , y0 ) ∈ Γshock ∩ Nε (Γsonic ). The rescaled boundary condition has “almost tangential structure” with the constants uniform with respect to (x0 , y0 ) ∈ Γshock ∩ Nε (Γsonic ). This allows to obtain the C 2,α –estimates for w. Transforming back, we obtain the scaled C 2,α –estimates for the shock curve within (x ,y ) (x ,y ) R 1 0 0 and the estimate of ψ (x0 ,y0 ) in the “half-square” Q 1 0 0 in Case (iii). ∂y ϕ¯0 ≥

2

2

4. Existence of the Regular Shock Reflection-Diffraction Configuration Once a priori estimates are proved, there are several ways to obtain the existence of the regular shock reflection-diffraction configuration. In this section, we employ the a priori estimates for a degree theory to give the outline of the existence proof. In order to apply the degree theory, the iteration set should be bounded and open in an appropriate function space (actually, in its product with the parameter space, i.e., the angle interval), and the fixed points of the iteration map should not occur on the boundary of the iteration set. We choose this function space according to the norms and the other quantities in the a priori estimates. Then the a priori estimates allow to conclude that the fixed points cannot occur on the boundary of the iteration set, if the bounds defining the iteration set are chosen appropriately large (or small, depending on the context and the a priori estimates).

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In our case, there is an extra issue of connecting admissible solutions with the normal reflection solution in the setup convenient for applying the degree theory. That is, we use the strict monotonicity properties of the admissible solutions (proved as a part of the a priori estimates). Indeed, these strict monotonicity properties can be made uniform for the wedge angles away from π2 (and away from the appropriate parts of the boundary of the elliptic region), by using the compactness of the set of admissible solutions which is a corollary of the a priori estimates. However, they become nonstrict when the wedge angle is π2 , i.e., at the normal reflection. Then, for the angles near π2 , we use the following facts: (i) All solutions of the free boundary problem for the angles near π2 and sufficiently close to the normal reflection are admissible solutions; (ii) All admissible solutions converge to the normal reflection solution as the wedge angle tends to π2 . From (i) and (ii), we can derive estimates similar to [6] for the admissible solutions and the “approximate” solutions for the angles near π2 . Then, for the wedge angle θw near π2 , the iteration set Kθw is a small neighborhood of the normal reflection solution, where the norms used and the size of neighborhood are related to the estimates of [6]. For the wedge angle θw away from π2 , the iteration set Kθw is defined by the bounds in the appropriate norms (related to the a priori estimates) and by the lower bounds of certain directional derivatives, corresponding to the strict monotonicity properties, so that the actual solution cannot be on the boundary of the iteration set according to the a priori estimates. These two definitions are combined in one setup, with the bounds depending continuously on the wedge angle. Also, since the elliptic domain depends on the solution, a mapping is defined depending on the wedge angle θw and the “approximate solution”, which maps its elliptic domain to the unit square, and the iteration set K is defined in terms of the functions on the unit square. This defines the iteration set K = ∪θw ∈[θws , π2 ] Kθw × {θw }. The iteration map F is defined as follows: Given a wedge angle θw and a function u from the iteration set (i.e., defined on the unit square), define the corresponding “elliptic domain” and set up a boundary value problem for an elliptic equation which is degenerate near the sonic arc. Let u ˆ be its solution, expressed as a function on the unit square in such a way that the gain in the regularity of the solution is preserved. Then the iteration map is defined by F(u, θw ) = u ˆ. The boundary value problem in the definition of F is defined so that, at the fixed point u = u ˆ, its solutions satisfies the potential flow equation (1.4), with the ellipticity cutoff in a small neighborhood of the sonic arc, and both the RankineHugoniot conditions (1.6)–(1.7) on the shock. Also, the other properties, including the inequalities φ2 ≤ φ ≤ φ1 and the monotonicity properties, required in the definition of admissible solutions are known for the wedge angles away from both π2 and the appropriate parts of the boundary of the elliptic domain. Then we prove the following facts: (i) Any fixed point is an admissible solution. For that, we remove the ellipticity cutoff and prove the inequalities and monotonicity properties mentioned above for the regions and the wedge angles where they are not readily known. The fact that these estimates need to be proved only in the localized regions is crucial and is made possible by using the uniform bounds and monotonicity properties which are a part of a priori estimates. (ii) The iteration map is compact, by using the gain in the regularity for the solution of the iteration boundary value problem.

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(iii) The iteration set is open, by using the elliptic estimates to show the existence of a solution for the iteration boundary value problem. (iv) The fixed points of the iteration map cannot occur on the boundary of the iteration set, by using the a priori estimates. (v) The normal reflection solution unormal is in the iteration set, from the definition of the iteration set. Now the degree theory provides that the index i(F(·, θw ), Kθw ) of the iteration map s π does not depend on the wedge angle θw ∈ [θw , 2 ]. π It remains to show that i(F(·, 2 ), K π2 ) ̸= 0. At θw = π2 , the only fixed point is the normal reflection. Then it suffices to show that the kernel of the linear map π Du F(unormal , ) − I 2 is trivial. This is proved by showing that the condition ( ) π Du F(unormal , ) − I V = 0 2 is equivalent to the statement that V is a solution of a linear homogeneous boundary value problem for a degenerate elliptic equation. We show that the solution V = 0 of this problem is unique. s π Thus, i(F(·, θw ), Kθw ) = const. ̸= 0 for all θw ∈ [θw , 2 ]. Then the fixed points exist s π for all θw ∈ [θw , 2 ], and the fixed points are admissible solutions of the shock reflectiondiffraction problem. This completes the proof of the existence theory of admissible solutions with the indicated regularity properties for the shock reflection-diffraction problem. For further details, see Chen-Feldman [7]. References [1] M. Bae, G.-Q. Chen, and M. Feldman, Regularity of solutions to shock reflection problem, Invent. Math. 175 (2009), 505–543. [2] G. Ben-Dor, Shock Wave Reflection Phenomena, 2nd Edition, Springer-Verlag: New York, 2007. [3] T. Chang and G.-Q. Chen, Diffraction of a planar shock along a compressible corner, Acta Mathematica Scientia, 6 (1986), 241–257. [4] G.-Q. Chen and M. Feldman, Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type, J. Amer. Math. Soc. 16 (2003), 461–494. [5] G.-Q. Chen and M. Feldman, Potential theory for shock reflection by a large-angle wedge, Proc. Nat. Acad. Sci. U.S.A. 102 (2005), 15368–15372. [6] G.-Q. Chen and M. Feldman, Global solutions to shock reflection by large-angle wedges for potential flow, Annals of Mathematics, 171 (2010), 1019–1134. [7] G.-Q. Chen and M. Feldman, Mathematics of Shock Reflection-Diffraction and von Neumann’s Conjectures, Research Monograph, Preprint, March 2011. [8] G.-Q. Chen and M. Feldman, Shock reflection-diffraction and multidimensional conservation laws, In: Hyperbolic Problems: Theory, Numerics and Applications, Proc. Sympos. Appl. Math., Vol. 67, Amer. Math. Soc.: Providence, RI, 2010, pp. 25–52. [9] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Springer-Verlag: New York, 1948. [10] V. Elling and T.-P. Liu, The ellipticity principle for steady and selfsimilar polytropic potential flow, J. Hyper. Diff. Eqs. 2 (2005), 909–917. [11] I. M. Gamba, R. Rosales, and E. Tabak, Constraints for formation of singularities for the small disturbance transonic flow equations, Comm. Pure Appl. Math. 52 (1999), 763–779. [12] J. Glimm and A. Majda, Multidimensional Hyperbolic Problems and Computations, Springer-Verlag: New York, 1991. [13] K. G. Guderley, The Theory of Transonic Flow, Pergamon Press: Oxford-London-Paris-Frankfurt; AddisonWesley Publishing Co. Inc.: Reading, Mass. 1962. [14] E. Harabetian, Diffraction of a weak shock by a wedge, Comm. Pure Appl. Math. 40 (1987), 849–863.

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W ISCONSIN , M ADISON , WI 53706-1388, USA