Geometric Analysis of Hyperbolic Differential Equations: An Introduction

by S. Alinhac

• ISBN13: 9780521128223
• ISBN10: 0521128226
• Edition: 1st
• Format: Paperback
• Copyright: 2010-06-28
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

Preface	p. vii
Introduction	p. 1
Metrics and frames	p. 8
Metrics, duality	p. 8
Optical functions	p. 12
Null frames	p. 12
Computing with frames	p. 17
Metric connection	p. 17
Submanifolds	p. 20
Hessian and d'Alembertian	p. 21
Frame coefficients	p. 24
Energy inequalities and frames	p. 29
The energy-momentum teasor	p. 29
Deformation teasor	p. 31
Energy inequality formalism	p. 33
Energy	p. 34
Interior teems and positive fields	p. 35
Maxwell equations	p. 41
Duality	p. 41
Energy formalism	p. 43
The good components	p. 45
The problem	p. 45
An important remark	p. 46
Ghost weights and improved standard energy inequalities	p. 47
Conformal inequalities	p. 53
Pointwise estimates and commutations	p. 57
Pointwise decay and conformal inqualities	p. 58
Commuting fields in the scalar case	p. 59
Modified Lorentz fields	p. 61
Commuting fields for Maxwell equaticus	p. 63
Frames and curvature	p. 65
The curvature tensor	p. 65
Optical functions and curvature	p. 67
Transport equations	p. 69
Elliptic systems	p. 71
Mixed transport-elliptic systems	p. 75
Nonlinear equations, a priori estimates and induction	p. 77
A simple ODE example	p. 77
Local existence theory	p. 80
Blowup criteria	p. 81
Induction on time for PDFs	p. 84
Applications to some quasilinear hyperbolic problems	p. 88
Quasilnear wave equations satisfying the null condition	p. 89
Quasilinear wave equations	p. 96
Low regularity well-posedness for quasilinear wave equations	p. 99
Stability of Minkowski spacetime (first version)	p. 102
L2 conjecture on the curvature	p. 106
Stability of Minkowsld spacetime (second version)	p. 108
The formation of black holes	p. 113
References	p. 114
Index	p. 117
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