The Main Construction and Outline of the Proof

doi:10.23943/princeton/9780691233055.003.0002

- Copyright Page
  - Notes
  - Notes
- Dedication
  - Notes
  - Notes
One Introduction
- Notes
- Notes
- 2.1 Setup and the Main Bootstrap Proposition 2.1 Setup and the Main Bootstrap Proposition
  - 2.1.1 The Nonlinearities $N_{α β}^{h}$ and $N^{ψ}$ 2.1.1 The Nonlinearities $N_{α β}^{h}$ and $N^{ψ}$
    - 2.1.1.1 The quadratic nonlinearities 2.1.1.1 The quadratic nonlinearities
  - 2.1.2 The Fourier Transform and Frequency Projections 2.1.2 The Fourier Transform and Frequency Projections
  - 2.1.3 Vector-fields 2.1.3 Vector-fields
  - 2.1.4 Decomposition of the Metric Tensor 2.1.4 Decomposition of the Metric Tensor
  - 2.1.5 Linear Profiles and the Z-norms 2.1.5 Linear Profiles and the Z-norms
  - 2.1.6 The Main Bootstrap Proposition 2.1.6 The Main Bootstrap Proposition
- 2.2 Outline of the Proof 2.2 Outline of the Proof
  - 2.2.1 Chapter 3: Preliminary Estimates 2.2.1 Chapter 3: Preliminary Estimates
    - 2.2.1.1 Normal forms and the bilinear phases Φ_σμν 2.2.1.1 Normal forms and the bilinear phases Φ_σμν
    - 2.2.1.2 Linear estimates 2.2.1.2 Linear estimates
    - 2.2.1.3 Bilinear estimates 2.2.1.3 Bilinear estimates
    - 2.2.1.4 Bounds on the functions $U^{ℒ h_{α β}}, U^{ℒ ψ}, V^{ℒ h_{α β}}, and V^{ℒ ψ}$ 2.2.1.4 Bounds on the functions $U^{ℒ h_{α β}}, U^{ℒ ψ}, V^{ℒ h_{α β}}, and V^{ℒ ψ}$
  - 2.2.2 Chapter 4: the Nonlinearities $N_{α β}^{h}$ and $N^{ψ}$ 2.2.2 Chapter 4: the Nonlinearities $N_{α β}^{h}$ and $N^{ψ}$
    - 2.2.2.1 Weighted bounds on the nonlinearities $N_{α β}^{h} and N^{ψ}$ 2.2.2.1 Weighted bounds on the nonlinearities $N_{α β}^{h} and N^{ψ}$
    - 2.2.2.2 Energy disposable nonlinear terms 2.2.2.2 Energy disposable nonlinear terms
    - 2.2.2.3 Elliptic consequences of the harmonic gauge condition 2.2.2.3 Elliptic consequences of the harmonic gauge condition
    - 2.2.2.4 Null structures and decompositions of the nonlinearities $N_{α β}^{h} and N^{ψ}$ 2.2.2.4 Null structures and decompositions of the nonlinearities $N_{α β}^{h} and N^{ψ}$
  - 2.2.3 Chapter 5: Improved Energy Estimates 2.2.3 Chapter 5: Improved Energy Estimates
    - 2.2.3.1 Semilinear wave interactions 2.2.3.1 Semilinear wave interactions
    - 2.2.3.2 Low frequencies and the quasilinear terms 2.2.3.2 Low frequencies and the quasilinear terms
    - 2.2.3.3 Wave-Klein-Gordon undifferentiated interactions 2.2.3.3 Wave-Klein-Gordon undifferentiated interactions
  - 2.2.4 Chapter 6: Improved Profile Bounds 2.2.4 Chapter 6: Improved Profile Bounds
    - 2.2.4.1 The weighted bounds (2.1.51) 2.2.4.1 The weighted bounds (2.1.51)
    - 2.2.4.2 The Z-norm bounds on the Klein-Gordon profile 2.2.4.2 The Z-norm bounds on the Klein-Gordon profile
    - 2.2.4.3 The Z-norm bounds on the metric profiles 2.2.4.3 The Z-norm bounds on the metric profiles
- Notes
- Notes
Three Preliminary Estimates
- Notes
- Notes
Four The Nonlinearities Nαβh and Nψ
- Notes
- Notes
Five Improved Energy Estimates
- Notes
- Notes
Six Improved Profile Bounds
- Notes
- Notes
Seven The Main Theorems
- Notes
- Notes
- Bibliography
  - Notes
  - Notes
- Index
  - Notes
  - Notes

Chapter

Two The Main Construction and Outline of the Proof

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https://doi.org/10.23943/princeton/9780691233055.003.0002

Pages

24–47

Published:

March 2022

Abstract

This chapter discusses the main notations and definitions, and states precisely the main bootstrap proposition. This proposition is the key quantitative result leading to global nonlinear stability. It begins by identifying the quadratic components of the nonlinearities N^h_αβ and N^ψ, which play a key role in the nonlinear evolution. The chapter then explains the Fourier transform and the Fourier inversion formula. To identify null structures, it uses a double Hodge decomposition for the metric tensor, which is connected to the work of Arnowitt-Deser-Misner on the Hamiltonian formulation of General Relativity. Ultimately, the goal is to control simultaneously three types of norms: energy norms involving up to three vector-fields; weighted norms on the linear profiles; and the Z-norms on the undifferentiated profiles.

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Contents

Two The Main Construction and Outline of the Proof

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