Shock Reflection-Diffraction, Nonlinear Conservation Laws of Mixed Type, and von Neumann s Conjectures 1

Transcription

1 Contents Preface xi I Shock Reflection-Diffraction, Nonlinear Conservation Laws of Mixed Type, and von Neumann s Conjectures 1 1 Shock Reflection-Diffraction, Nonlinear Partial Differential Equations of Mixed Type, and Free Boundary Problems 3 2 Mathematical Formulations and Main Theorems The potential flow equation Mathematical problems for shock reflection-diffraction Weak solutions of Problem and Problem Structure of solutions: Regular reflection-diffraction configurations Existence of state (2) and continuous dependence on the parameters Von Neumann s conjectures, Problem (free boundary problem), and main theorems Main Steps and Related Analysis in the Proofs of the Main Theorems Normal reflection Main steps and related analysis in the proof of the sonic conjecture Main steps and related analysis in the proof of the detachment conjecture Appendix: The method of continuity and fixed point theorems. 65 II Elliptic Theory and Related Analysis for Shock Reflection-Diffraction 67 4 Relevant Results for Nonlinear Elliptic Equations of Second Order Notations: Hölder norms and ellipticity Quasilinear uniformly elliptic equations

2 vi CONTENTS 4.3 Estimates for Lipschitz solutions of elliptic boundary value problems Comparison principle for a mixed boundary value problem in a domain with corners Mixed boundary value problems in a domain with corners for uniformly elliptic equations Hölder spaces with parabolic scaling Degenerate elliptic equations Uniformly elliptic equations in a curved triangle-shaped domain with one-point Dirichlet condition Basic Properties of the Self-Similar Potential Flow Equation Some basic facts and formulas for the potential flow equation Interior ellipticity principle for self-similar potential flow Ellipticity principle for self-similar potential flow with slip condition on the flat boundary III Proofs of the Main Theorems for the Sonic Conjecture and Related Analysis Uniform States and Normal Reflection Uniform states for self-similar potential flow Normal reflection and its uniqueness The self-similar potential flow equation in the coordinates flattening the sonic circle of a uniform state Local Theory and von Neumann s Conjectures Local regular reflection and state (2) Local theory of shock reflection for large-angle wedges The shock polar for steady potential flow and its properties Local theory for shock reflection: Existence of the weak and strong state (2) up to the detachment angle Basic properties of the weak state (2) and the definition of supersonic and subsonic wedge angles Von Neumann s sonic and detachment conjectures Admissible Solutions and Features of Problem Definition of admissible solutions Strict directional monotonicity for admissible solutions Appendix: Properties of solutions of Problem for largeangle wedges

3 CONTENTS vii 9 Uniform Estimates for Admissible Solutions Bounds of the elliptic domain Ω and admissible solution ϕ in Ω Regularity of admissible solutions away from Γ shock Γ sonic {P 3 } Separation of Γ shock from Γ sym Lower bound for the distance between Γ shock and Γ wedge Uniform positive lower bound for the distance between Γ shock and the sonic circle of state (1) Uniform estimates of the ellipticity constant in Ω \ Γ sonic Regularity of Admissible Solutions away from the Sonic Arc Γ shock as a graph in the radial directions with respect to state (1) Boundary conditions on Γ shock for admissible solutions Local estimates near Γ shock The critical angle and the distance between Γ shock and Γ wedge Regularity of admissible solutions away from Γ sonic Regularity of the limit of admissible solutions away from Γ sonic Regularity of Admissible Solutions near the Sonic Arc The equation near the sonic arc and structure of elliptic degeneracy Structure of the neighborhood of Γ sonic in Ω and estimates of (ψ, Dψ) Properties of the Rankine-Hugoniot condition on Γ shock near Γ sonic C 2,α estimates in the scaled Hölder norms near Γ sonic The reflected-diffracted shock is C 2,α near P Compactness of the set of admissible solutions Iteration Set and Solvability of the Iteration Problem Statement of the existence results Mapping to the iteration region Definition of the iteration set The equation for the iteration Assigning a boundary condition on the shock for the iteration Normal reflection, iteration set, and admissible solutions Solvability of the iteration problem and estimates of solutions Openness of the iteration set Iteration Map, Fixed Points, and Existence of Admissible Solutions up to the Sonic Angle Iteration map Continuity and compactness of the iteration map Normal reflection and the iteration map for θ w = π Fixed points of the iteration map for θ w < π 2 are admissible solutions Fixed points cannot lie on the boundary of the iteration set

4 viii CONTENTS 13.6 Proof of the existence of solutions up to the sonic angle or the critical angle Proof of Theorem 2.6.2: Existence of global solutions up to the sonic angle when u 1 c Proof of Theorem 2.6.4: Existence of global solutions when u 1 > c Appendix: Extension of the functions in weighted spaces Optimal Regularity of Solutions near the Sonic Circle Regularity of solutions near the degenerate boundary for nonlinear degenerate elliptic equations of second order Optimal regularity of solutions across Γ sonic IV Subsonic Regular Reflection-Diffraction and Global Existence of Solutions up to the Detachment Angle Admissible Solutions and Uniform Estimates up to the Detachment Angle Definition of admissible solutions for the supersonic and subsonic reflections Basic estimates for admissible solutions up to the detachment angle Separation of Γ shock from Γ sym Lower bound for the distance between Γ shock and Γ wedge away from P Uniform positive lower bound for the distance between Γ shock and the sonic circle of state (1) Uniform estimates of the ellipticity constant Regularity of admissible solutions away from Γ sonic Regularity of Admissible Solutions near the Sonic Arc and the Reflection Point Pointwise and gradient estimates near Γ sonic and the reflection point The Rankine-Hugoniot condition on Γ shock near Γ sonic and the reflection point A priori estimates near Γ sonic in the supersonic-away-from-sonic case A priori estimates near Γ sonic in the supersonic-near-sonic case A priori estimates near the reflection point in the subsonic-nearsonic case A priori estimates near the reflection point in the subsonic-awayfrom-sonic case

5 CONTENTS ix 17 Existence of Global Regular Reflection-Diffraction Solutions up to the Detachment Angle Statement of the existence results Mapping to the iteration region Iteration set Existence and estimates of solutions of the iteration problem Openness of the iteration set Iteration map and its properties Compactness of the iteration map Normal reflection and the iteration map for θ w = π Fixed points of the iteration map for θ w < π 2 are admissible solutions Fixed points cannot lie on the boundary of the iteration set Proof of the existence of solutions up to the critical angle Proof of Theorem 2.6.6: Existence of global solutions up to the detachment angle when u 1 c Proof of Theorem 2.6.8: Existence of global solutions when u 1 > c V Connections and Open Problems The Full Euler Equations and the Potential Flow Equation The full Euler equations Mathematical formulation I: Initial-boundary value problem Mathematical formulation II: Boundary value problem Normal reflection Local theory for regular reflection near the reflection point Von Neumann s conjectures Connections with the potential flow equation Shock Reflection-Diffraction and New Mathematical Challenges Mathematical theory for multidimensional conservation laws Nonlinear partial differential equations of mixed elliptic-hyperbolic type Free boundary problems and techniques Numerical methods for multidimensional conservation laws Bibliography 794 Index 815