Geometric Analysis of Hyperbolic Differential Equations: An Introduction (London Mathematical Society Lecture Note Series, Series Number 374)

Its Self-contained Presentation And 'do-it-yourself' Approach Make This The Perfect Guide For Graduate Students And Researchers Wishing To Access Recent Literature In The Field Of Nonlinear Wave Equations And General Relativity. It Introduces All Of The Key Tools And Concepts From Lorentzian Geometry (metrics, Null Frames, Deformation Tensors, Etc.) And Provides Complete Elementary Proofs. The Author Also Discusses Applications To Topics In Nonlinear Equations, Including Null Conditions And Stability Of Minkowski Space. No Previous Knowledge Of Geometry Or Relativity Is Required--provided By Publisher. The Field Of Nonlinear Hyperbolic Equations Or Systems Has Seen A Tremendous Development Since The Beginning Of The 1980s. We Are Concentrating Here On Multidimensional Situations, And On Quasilinear Equations Or Systems, That Is, When The Coefficients Of The Principal Part Depend On The Unknown Function Itself. The Pioneering Works By F. John, D. Christodoulou, L. Hörmander, S. Klainerman, A. Majda And Many Others Have Been Devoted Mainly To The Questions Of Blowup, Lifespan, Shocks, Global Existence, Etc. Some Overview Of The Classical Results Can Be Found In The Books Of Majda [42] And Hörmander [24]. On The Other Hand, Christodoulou And Klainerman [18] Proved In Around 1990 The Stability Of Minkowski Space, A Striking Mathematical Result About The Cauchy Problem For The Einstein Equations. After That, Many Works Have Dealt With Diagonal Systems Of Quasilinear Wave Equations, Since This Is What Einstein Equations Reduce To When Written In The So-called Harmonic Coordinates. The Main Feature Of This Particular Case Is That The (scalar) Principal Part Of The System Is A Wave Operator Associated To A Unique Lorentzian Metric On The Underlying Space-time--provided By Publisher. S. Alinhac. Includes Bibliographical References (p. 114-116) And Index. Cover......Page 1 Title......Page 5 Copyright......Page 6 Contents......Page 7 Preface......Page 9 1 Introduction......Page 13 2.1 Metrics, duality......Page 20 2.3 Null frames......Page 24 3.1 Metric connexion......Page 29 3.2 Submanifolds......Page 32 3.3 Hessian and d'Alembertian......Page 33 3.4 Frame coefficients......Page 36 4.1 The energy-momentum tensor......Page 41 4.2 Deformation tensor......Page 43 4.3 Energy inequality formalism......Page 45 4.4 Energy......Page 46 4.5 Interior terms and positive fields......Page 47 4.6.1 Duality......Page 53 4.6.2 Energy formalism......Page 55 5.1 The problem......Page 57 5.2 An important remark......Page 58 5.3 Ghost weights and improved standard energy inequalities......Page 59 5.4 Conformal inequalities......Page 65 6 Pointwise estimates and commutations......Page 69 6.1 Pointwise decay and conformal inequalities......Page 70 6.2 Commuting fields in the scalar case......Page 71 6.3 Modified Lorentz fields......Page 73 6.4 Commuting fields for Maxwell equations......Page 75 7.1 The curvature tensor......Page 77 7.2 Optical functions and curvature......Page 79 7.3 Transport equations......Page 81 7.4 Elliptic systems......Page 83 7.5 Mixed transport-elliptic systems......Page 87 8.1 A simple ODE example......Page 89 8.2 Local existence theory......Page 92 8.3 Blowup criteria......Page 93 8.4 Induction on time for PDEs......Page 96 9 Applications to some quasilinear hyperbolic problems......Page 100 9.1 Quasilinear wave equations satisfying the null condition......Page 101 9.2 Quasilinear wave equations......Page 108 9.3 Low regularity well-posedness for quasilinear wave equations......Page 111 9.4 Stability of Minkowski spacetime (first version)......Page 114 9.5 L2 conjecture on the curvature......Page 118 9.6 Stability of Minkowski spacetime (second version)......Page 120 9.7 The formation of black holes......Page 125 References......Page 126 Index......Page 129 "The field of nonlinear hyperbolic equations or systems has seen a tremendous development since the beginning of the 1980s. We are concentrating here on multidimensional situations, and on quasilinear equations or systems, that is, when the coefficients of the principal part depend on the unknown function itself. The pioneering works by F. John, D. Christodoulou, L. Hörmander, S. Klainerman, A. Majda and many others have been devoted mainly to the questions of blowup, lifespan, shocks, global existence, etc. Some overview of the classical results can be found in the books of Majda [42] and Hörmander [24]. On the other hand, Christodoulou and Klainerman [18] proved in around 1990 the stability of Minkowski space, a striking mathematical result about the Cauchy problem for the Einstein equations. After that, many works have dealt with diagonal systems of quasilinear wave equations, since this is what Einstein equations reduce to when written in the so-called harmonic coordinates. The main feature of this particular case is that the (scalar) principal part of the system is a wave operator associated to a unique Lorentzian metric on the underlying space-time"--Résumé de l'éditeur "Its self-contained presentation and 'do-it-yourself' approach make this the perfect guide for graduate students and researchers wishing to access recent literature in the field of nonlinear wave equations and general relativity. It introduces all of the key tools and concepts from Lorentzian geometry (metrics, null frames, deformation tensors, etc.) and provides complete elementary proofs. The author also discusses applications to topics in nonlinear equations, including null conditions and stability of Minkowski space. No previous knowledge of geometry or relativity is required"--Résumé de l'éditeur "Its self-contained presentation and 'do-it-yourself' approach make this the perfect guide for graduate students and researchers wishing to access recent literature in the field of nonlinear wave equations and general relativity. It introduces all of the key tools and concepts from Lorentzian geometry (metrics, null frames, deformation tensors, etc.) and provides complete elementary proofs. The author also discusses applications to topics in nonlinear equations, including null conditions and stability of Minkowski space. No previous knowledge of geometry or relativity is required"-- Résumé de l'éditeur Machine generated contents note: Preface; 1. Introduction; 2. Metrics and frames; 3. Computing with frames; 4. Energy inequalities and frames; 5. The good components; 6. Pointwise estimates and commutations; 7. Frames and curvature; 8. Nonlinear equations, a priori estimates and induction; 9. Applications to some quasilinear hyperbolic problems; References; Index.