The Einstein-Klein-Gordon Coupled System

Name: The Einstein-Klein-Gordon Coupled System | Global Stability of the Minkowski Solution
Brand: Princeton University Press
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Global Stability of the Minkowski Solution

Alexandru D. Ionescu Benoît Pausader(Autor*in)

Princeton University Press

1. Auflage

Erschienen am 15. März 2022

308 Seiten

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978-0-691-23303-1 (ISBN)

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Inhalt

Cover
Contents
1. Introduction
1.1 The Einstein-Klein-Gordon Coupled System
1.1.1 Wave Coordinates and PDE Formulation of the Problem
1.2 The Global Regularity Theorem
1.2.1 Global Stability Results in General Relativity
1.2.2 Simplified Wave-Klein-Gordon Models
1.2.3 Small Data Global Regularity Results
1.2.4 Assumptions on the Initial Data
1.2.5 The Mini-bosons
1.3 Main Ideas and Further Asymptotic Results
1.3.1 Global Nonlinear Stability
1.3.2 Nonlinear Scattering
1.3.3 Asymptotic Bounds and Causal Geodesics
1.3.4 Weak Peeling Estimates
1.3.5 The ADM Energy and the Linear Momentum
1.3.6 The Bondi Energy
1.3.7 Organization
1.3.8 Acknowledgements
2. The Main Construction and Outline of the Proof
2.1 Setup and the Main Bootstrap Proposition
2.1.1 The Nonlinearities Nhaß and N?
2.1.2 The Fourier Transform and Frequency Projections
2.1.3 Vector-fields
2.1.4 Decomposition of the Metric Tensor
2.1.5 Linear Profiles and the Z-norms
2.1.6 The Main Bootstrap Proposition
2.2 Outline of the Proof
2.2.1 Chapter 3: Preliminary Estimates
2.2.2 Chapter 4: the Nonlinearities Nhaß and N?
2.2.3 Chapter 5: Improved Energy Estimates
2.2.4 Chapter 6: Improved Profile Bounds
3. Preliminary Estimates
3.1 Some Lemmas
3.1.1 General Lemmas
3.1.2 The Phases Fsµv
3.1.3 Elements of Paradifferential Calculus
3.2 Linear and Bilinear Estimates
3.2.1 Linear Estimates
3.2.2 Multipliers and Bilinear Operators
3.2.3 Bilinear Estimates
3.2.4 Interpolation Inequalities
3.3 Analysis of the Linear Profiles
4. The Nonlinearities Nhaß and N?
4.1 Localized Bilinear Estimates
4.1.1 Frequency Localized L2 Estimates
4.1.2 The Classes of Functions Ga
4.2 Bounds on the Nonlinearities Nhaß and N?
4.2.1 The Quadratic Nonlinearities
4.2.2 The Cubic and Higher Order Nonlinearities
4.2.3 Additional Low Frequency Bounds
4.2.4 Additional Bounds on Some Quadratic Nonlinearities
4.3 Decompositions of the Main Nonlinearities
4.3.1 The Variables FL, FL, pL,w jL, OjL,v jkL
4.3.2 Energy Disposable Nonlinearities
4.3.3 Null Structures
4.3.4 The Main Decomposition
5. Improved Energy Estimates
5.1 Setup and Preliminary Reductions
5.1.1 Energy Increments
5.1.2 The Main Spacetime Bounds
5.1.3 Poincaré Normal Forms
5.1.4 Paralinearization of the Reduced Wave Operator
5.2 Pure Wave Interactions
5.2.1 Null Interactions
5.2.2 Non-null Semilinear Terms
5.2.3 Second Symmetrization and Paradifferential Calculus
5.3 Mixed Wave-Klein-Gordon Interactions
6. Improved Profile Bounds
6.1 Weighted Bounds
6.2 Z-norm Control of the Klein-Gordon Field
6.2.1 Renormalization
6.2.2 Improved Control
6.3 Z-norm Control of the Metric Components
6.3.1 The First Reduction
6.3.2 The Nonlinear Terms Rahaß, a ? {1,2,4,6}
6.3.3 Localized Bilinear Wave Interactions
6.3.4 The Nonlinear Terms R3haß
6.3.5 The Nonlinear Terms R5haß
7. The Main Theorems
7.1 Global Regularity and Asymptotics
7.1.1 Global Regularity
7.1.2 Decay of the Metric and the Klein-Gordon Field
7.1.3 Null and Timelike Geodesics
7.2 Weak Peeling Estimates and the ADM Energy
7.2.1 Peeling Estimates
7.2.2 The ADM Energy
7.2.3 The Linear Momentum
7.2.4 Gauge Conditions and Parameterizations
7.3 Asymptotically Optical Functions and the Bondi Energy
7.3.1 Almost Optical Functions and the Friedlander Fields
7.3.2 The Bondi Energy
7.3.3 The interior energy
Bibliography
Index

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