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London Mathematical Society Lecture Note Series : Geometric Analysis of Hyperbolic Differential Equations: An Introduction Series Number 374 - Serge Alinhac

London Mathematical Society Lecture Note Series

Geometric Analysis of Hyperbolic Differential Equations: An Introduction Series Number 374

Paperback Published: 20th May 2010
ISBN: 9780521128223
Number Of Pages: 118

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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 non-linear 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 non-linear equations, including null conditions and stability of Minkowski space. No previous knowledge of geometry or relativity is required

'This book provides an excellent introduction to nonlinear wave equations, and it can be recommended to anyone who wants to access the recent mathematical literature on this subject.' Zentralblatt MATH

Prefacep. vii
Introductionp. 1
Metrics and framesp. 8
Metrics, dualityp. 8
Optical functionsp. 12
Null framesp. 12
Computing with framesp. 17
Metric connectionp. 17
Submanifoldsp. 20
Hessian and d'Alembertianp. 21
Frame coefficientsp. 24
Energy inequalities and framesp. 29
The energy-momentum teasorp. 29
Deformation teasorp. 31
Energy inequality formalismp. 33
Energyp. 34
Interior teems and positive fieldsp. 35
Maxwell equationsp. 41
Dualityp. 41
Energy formalismp. 43
The good componentsp. 45
The problemp. 45
An important remarkp. 46
Ghost weights and improved standard energy inequalitiesp. 47
Conformal inequalitiesp. 53
Pointwise estimates and commutationsp. 57
Pointwise decay and conformal inqualitiesp. 58
Commuting fields in the scalar casep. 59
Modified Lorentz fieldsp. 61
Commuting fields for Maxwell equaticusp. 63
Frames and curvaturep. 65
The curvature tensorp. 65
Optical functions and curvaturep. 67
Transport equationsp. 69
Elliptic systemsp. 71
Mixed transport-elliptic systemsp. 75
Nonlinear equations, a priori estimates and inductionp. 77
A simple ODE examplep. 77
Local existence theoryp. 80
Blowup criteriap. 81
Induction on time for PDFsp. 84
Applications to some quasilinear hyperbolic problemsp. 88
Quasilnear wave equations satisfying the null conditionp. 89
Quasilinear wave equationsp. 96
Low regularity well-posedness for quasilinear wave equationsp. 99
Stability of Minkowski spacetime (first version)p. 102
L2 conjecture on the curvaturep. 106
Stability of Minkowsld spacetime (second version)p. 108
The formation of black holesp. 113
Referencesp. 114
Indexp. 117
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780521128223
ISBN-10: 0521128226
Series: London Mathematical Society Lecture Notes
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 118
Published: 20th May 2010
Country of Publication: GB
Dimensions (cm): 22.61 x 14.99  x 0.76
Weight (kg): 0.18