Geometric analysis of hyperbolic differential equations : an introduction /

Geometric analysis of hyperbolic differential equations : an introduction /

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Bibliographic Details
Author / Creator:Alinhac, S. (Serge)
Imprint:Cambridge, UK ; New York : Cambridge University Press, 2010.
Description:ix, 118 p. ; 23 cm.
Language:English
Series:London Mathematical Society lecture note series ; 374
London Mathematical Society lecture note series ; 374.
Subject:
Nonlinear wave equations.
Differential equations, Hyperbolic.
Quantum theory.
Geometry, Differential.
Differential equations, Hyperbolic.
Geometry, Differential.
Nonlinear wave equations.
Quantum theory.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/8114407
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ISBN:9780521128223 (pbk.)
0521128226 (pbk.)
Notes:Includes bibliographical references and index.
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"--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.
Table of Contents:
  • Preface
  • 1. Introduction
  • 2. Metrics and frames
  • 2.1. Metrics, duality
  • 2.2. Optical functions
  • 2.3. Null frames
  • 3. Computing with frames
  • 3.1. Metric connection
  • 3.2. Submanifolds
  • 3.3. Hessian and d'Alembertian
  • 3.4. Frame coefficients
  • 4. Energy inequalities and frames
  • 4.1. The energy-momentum teasor
  • 4.2. Deformation teasor
  • 4.3. Energy inequality formalism
  • 4.4. Energy
  • 4.5. Interior teems and positive fields
  • 4.6. Maxwell equations
  • 4.6.1. Duality
  • 4.6.2. Energy formalism
  • 5. The good components
  • 5.1. The problem
  • 5.2. An important remark
  • 5.3. Ghost weights and improved standard energy inequalities
  • 5.4. Conformal inequalities
  • 6. Pointwise estimates and commutations
  • 6.1. Pointwise decay and conformal inqualities
  • 6.2. Commuting fields in the scalar case
  • 6.3. Modified Lorentz fields
  • 6.4. Commuting fields for Maxwell equaticus
  • 7. Frames and curvature
  • 7.1. The curvature tensor
  • 7.2. Optical functions and curvature
  • 7.3. Transport equations
  • 7.4. Elliptic systems
  • 7.5. Mixed transport-elliptic systems
  • 8. Nonlinear equations, a priori estimates and induction
  • 8.1. A simple ODE example
  • 8.2. Local existence theory
  • 8.3. Blowup criteria
  • 8.4. Induction on time for PDFs
  • 9. Applications to some quasilinear hyperbolic problems
  • 9.1. Quasilnear wave equations satisfying the null condition
  • 9.2. Quasilinear wave equations
  • 9.3. Low regularity well-posedness for quasilinear wave equations
  • 9.4. Stability of Minkowski spacetime (first version)
  • 9.5. L 2 conjecture on the curvature
  • 9.6. Stability of Minkowsld spacetime (second version)
  • 9.7. The formation of black holes
  • References
  • Index

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