The Mathematics of Shock Reflection-Diffraction and von Neumann's Conjectures: (AMS-197) (Annals of Mathematics Studies (197))

Cover 1 Title 4 Copyright 5 Contents 6 Preface 12 I Shock Reflection-Diffraction, Nonlinear Conservation Laws of Mixed Type, and von Neumann’s Conjectures 16 1 Shock Reflection-Diffraction, Nonlinear Partial Differential Equations of Mixed Type, and Free Boundary Problems 18 2 Mathematical Formulations and Main Theorems 31 2.1 The potential flow equation 31 2.2 Mathematical problems for shock reflection-diffraction 34 2.3 Weak solutions of Problem 2.2.1 and Problem 2.2.3 38 2.4 Structure of solutions: Regular reflection-diffraction configurations 39 2.5 Existence of state (2) and continuous dependence on the parameters 42 2.6 Von Neumann’s conjectures, Problem 2.6.1 (free boundary problem), and main theorems 43 3 Main Steps and Related Analysis in the Proofs of the Main Theorems 52 3.1 Normal reflection 52 3.2 Main steps and related analysis in the proof of the sonic conjecture 52 3.3 Main steps and related analysis in the proof of the detachment conjecture 70 3.4 Appendix: The method of continuity and fixed point theorems 80 II Elliptic Theory and Related Analysis for Shock Reflection-Diffraction 82 4 Relevant Results for Nonlinear Elliptic Equations of Second Order 84 4.1 Notations: Hölder norms and ellipticity 84 4.2 Quasilinear uniformly elliptic equations 87 4.3 Estimates for Lipschitz solutions of elliptic boundary value problems 120 4.4 Comparison principle for a mixed boundary value problem in a domain with corners 157 4.5 Mixed boundary value problems in a domain with corners for uniformly elliptic equations 160 4.6 Hölder spaces with parabolic scaling 207 4.7 Degenerate elliptic equations 212 4.8 Uniformly elliptic equations in a curved triangle-shaped domain with one-point Dirichlet condition 222 5 Basic Properties of the Self-Similar Potential Flow Equation 231 5.1 Some basic facts and formulas for the potential flow equation 231 5.2 Interior ellipticity principle for self-similar potential flow 237 5.3 Ellipticity principle for self-similar potential flow with slip condition on the flat boundary 242 III Proofs of the Main Theorems for the Sonic Conjecture and Related Analysis 244 6 Uniform States and Normal Reflection 246 6.1 Uniform states for self-similar potential flow 246 6.2 Normal reflection and its uniqueness 253 6.3 The self-similar potential flow equation in the coordinates flattening the sonic circle of a uniform state 254 7 Local Theory and von Neumann’s Conjectures 257 7.1 Local regular reflection and state (2) 257 7.2 Local theory of shock reflection for large-angle wedges 260 7.3 The shock polar for steady potential flow and its properties 263 7.4 Local theory for shock reflection: Existence of the weak and strong state (2) up to the detachment angle 278 7.5 Basic properties of the weak state (2) and the definition of supersonic and subsonic wedge angles 288 7.6 Von Neumann’s sonic and detachment conjectures 294 8 Admissible Solutions and Features of Problem 2.6.1 296 8.1 Definition of admissible solutions 296 8.2 Strict directional monotonicity for admissible solutions 301 8.3 Appendix: Properties of solutions of Problem 2.6.1 for large-angle wedges 320 9 Uniform Estimates for Admissible Solutions 334 9.1 Bounds of the elliptic domain Ω and admissible solution φ in Ω 334 9.2 Regularity of admissible solutions away from Г shock ∪Гsonic ∪{P3} 337 9.3 Separation of Гshock from Гsym 354 9.4 Lower bound for the distance between Гshock and Гwedge 356 9.5 Uniform positive lower bound for the distance between Гshock and the sonic circle of state (1) 369 9.6 Uniform estimates of the ellipticity constant in Ω\Γsonic 384 10 Regularity of Admissible Solutions away from the Sonic Arc 397 10.1 Γshock as a graph in the radial directions with respect to state (1) 397 10.2 Boundary conditions on Γshock for admissible solutions 400 10.3 Local estimates near Γshock 402 10.4 The critical angle and the distance between Γshock and Γwedge 404 10.5 Regularity of admissible solutions away from Γsonic 405 10.6 Regularity of the limit of admissible solutions away from Γsonic 407 11 Regularity of Admissible Solutions near the Sonic Arc 411 11.1 The equation near the sonic arc and structure of elliptic degeneracy 411 11.2 Structure of the neighborhood of Γsonic in Ω and estimates of (Ψ, DΨ) 413 11.3 Properties of the Rankine-Hugoniot condition on Γshock near Γsonic 428 11.4 C2,α –estimates in the scaled Hölder norms near Γsonic 436 11.5 The reflected-diffracted shock is C2,α near P1 446 11.6 Compactness of the set of admissible solutions 449 12 Iteration Set and Solvability of the Iteration Problem 455 12.1 Statement of the existence results 455 12.2 Mapping to the iteration region 455 12.3 Definition of the iteration set 476 12.4 The equation for the iteration 484 12.5 Assigning a boundary condition on the shock for the iteration 500 12.6 Normal reflection, iteration set, and admissible solutions 519 12.7 Solvability of the iteration problem and estimates of solutions 520 12.8 Openness of the iteration set 535 13 Iteration Map, Fixed Points, and Existence of Admissible Solutions up to the Sonic Angle 539 13.1 Iteration map 539 13.2 Continuity and compactness of the iteration map 543 13.3 Normal reflection and the iteration map for θw = π/2 545 13.4 Fixed points of the iteration map for θw c1 577 13.9 Appendix: Extension of the functions in weighted spaces 579 14 Optimal Regularity of Solutions near the Sonic Circle 601 14.1 Regularity of solutions near the degenerate boundary for nonlinear degenerate elliptic equations of second order 601 14.2 Optimal regularity of solutions across Γsonic 614 IV Subsonic Regular Reflection-Diffraction and Global Existence of Solutions up to the Detachment Angle 628 15 Admissible Solutions and Uniform Estimates up to the Detachment Angle 630 15.1 Definition of admissible solutions for the supersonic and subsonic reflections 630 15.2 Basic estimates for admissible solutions up to the detachment angle 632 15.3 Separation of Γshock from Γsym 633 15.4 Lower bound for the distance between Γshock and Γwedge away from P0 633 15.5 Uniform positive lower bound for the distance between Γshock and the sonic circle of state (1) 636 15.6 Uniform estimates of the ellipticity constant 637 15.7 Regularity of admissible solutions away from Γsonic 640 16 Regularity of Admissible Solutions near the Sonic Arc and the Reflection Point 644 16.1 Pointwise and gradient estimates near Γsonic and the reflection point 644 16.2 The Rankine-Hugoniot condition on Γshock near Γsonic and the reflection point 648 16.3 A priori estimates near Γsonic in the supersonic-away-from-sonic case 650 16.4 A priori estimates near Γsonic in the supersonic-near-sonic case 651 16.5 A priori estimates near the reflection point in the subsonic-near-sonic case 671 16.6 A priori estimates near the reflection point in the subsonic-away-from-sonic case 680 17 Existence of Global Regular Reflection-Diffraction Solutions up to the Detachment Angle 705 17.1 Statement of the existence results 705 17.2 Mapping to the iteration region 705 17.3 Iteration set 722 17.4 Existence and estimates of solutions of the iteration problem 740 17.5 Openness of the iteration set 752 17.6 Iteration map and its properties 756 17.7 Compactness of the iteration map 760 17.8 Normal reflection and the iteration map for θw = π/2 762 17.9 Fixed points of the iteration map for w c1 768 V Connections and Open Problems 770 18 The Full Euler Equations and the Potential Flow Equation 772 18.1 The full Euler equations 772 18.2 Mathematical formulation I: Initial-boundary value problem 776 18.3 Mathematical formulation II: Boundary value problem 777 18.4 Normal reflection 783 18.5 Local theory for regular reflection near the reflection point 784 18.6 Von Neumann’s conjectures 792 18.7 Connections with the potential flow equation 796 19 Shock Reflection-Diffraction and New Mathematical Challenges 800 19.1 Mathematical theory for multidimensional conservation laws 800 19.2 Nonlinear partial differential equations of mixed elliptic-hyperbolic type 803 19.3 Free boundary problems and techniques 805 19.4 Numerical methods for multidimensional conservation laws 806 Bibliography 809 Index 830 This book offers a survey of recent developments in the analysis of shock reflection-diffraction, a detailed presentation of original mathematical proofs of von Neumann's conjectures for potential flow, and a collection of related results and new techniques in the analysis of partial differential equations (PDEs), as well as a set of fundamental open problems for further development.Shock waves are fundamental in nature. They are governed by the Euler equations or their variants, generally in the form of nonlinear conservation laws--PDEs of divergence form. When a shock hits an obstacle, shock reflection-diffraction configurations take shape. To understand the fundamental issues involved, such as the structure and transition criteria of different configuration patterns, it is essential to establish the global existence, regularity, and structural stability of shock reflection-diffraction solutions. This involves dealing with several core difficulties in the analysis of nonlinear PDEs--mixed type, free boundaries, and corner singularities--that also arise in fundamental problems in diverse areas such as continuum mechanics, differential geometry, mathematical physics, and materials science. Presenting recently developed approaches and techniques, which will be useful for solving problems with similar difficulties, this book opens up new research opportunities.--Provided by publisher This book offers a survey of recent developments in the analysis of shock reflection-diffraction, a detailed presentation of original mathematical proofs of von Neumann's conjectures for potential flow, and a collection of related results and new techniques in the analysis of partial differential equations (PDEs), as well as a set of fundamental open problems for further development. Shock waves are fundamental in nature. They are governed by the Euler equations or their variants, generally in the form of nonlinear conservation laws--PDEs of divergence form. When a shock hits an obstacle, shock reflection-diffraction configurations take shape. To understand the fundamental issues involved, such as the structure and transition criteria of different configuration patterns, it is essential to establish the global existence, regularity, and structural stability of shock reflection-diffraction solutions. This involves dealing with several core difficulties in the analysis of nonlinear PDEs--mixed type, free boundaries, and corner singularities--that also arise in fundamental problems in diverse areas such as continuum mechanics, differential geometry, mathematical physics, and materials science. Presenting recently developed approaches and techniques, which will be useful for solving problems with similar difficulties, this book opens up new research opportunities