Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parametrix for wave equations on a rough background II: construction and control at initial time

Jeremie Szeftel

TL;DR

<3-5 sentence high-level summary>Parametrix for wave equations on a rough background II develops a time-zero parametrix construction for the homogeneous wave equation $\square_{\mathbf{g}}\phi=0$ on a rough Einstein vacuum spacetime and proves $L^2$ boundedness results for the associated Fourier integral operators under minimal regularity of the phase and symbol. The work centers on prescribing the initial-phase functions $u_±$ on the initial slice and solving a boundary data system $(f_+,f_-)$ via $M_+ f_+ + M_-f_- = \phi_0$ and $Q_+(\lambda f_+) - Q_-(\lambda f_-) = i\phi_1$, with quantitative $L^2$ control on $\lambda f_\pm$. Core techniques combine dyadic decompositions in frequency and angle with geometric integrations by parts along tangential directions to foliation level sets, together with an almost-orthogonality framework to bound Fourier integral operators with rough phases and symbols. These results advance the analysis needed for the bounded $L^2$ curvature conjecture by enabling robust control of wave propagation on rough backgrounds and contribute broadly to the theory of $L^2$ bounds for FIOs under limited regularity.

Abstract

This is the second of a sequence of four papers \cite{param1}, \cite{param2}, \cite{param3}, \cite{param4} dedicated to the construction and the control of a parametrix to the homogeneous wave equation $\square_{\bf g} φ=0$, where ${\bf g}$ is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes $L^2$ bounds on the curvature tensor ${\bf R}$ of ${\bf g}$ is a major step of the proof of the bounded $L^2$ curvature conjecture proposed in \cite{Kl:2000}, and solved by S. Klainerman, I. Rodnianski and the author in \cite{boundedl2}. On a more general level, this sequence of papers deals with the control of the eikonal equation on a rough background, and with the derivation of $L^2$ bounds for Fourier integral operators on manifolds with rough phases and symbols, and as such is also of independent interest.

Parametrix for wave equations on a rough background II: construction and control at initial time

TL;DR

<3-5 sentence high-level summary>Parametrix for wave equations on a rough background II develops a time-zero parametrix construction for the homogeneous wave equation on a rough Einstein vacuum spacetime and proves boundedness results for the associated Fourier integral operators under minimal regularity of the phase and symbol. The work centers on prescribing the initial-phase functions on the initial slice and solving a boundary data system via and , with quantitative control on . Core techniques combine dyadic decompositions in frequency and angle with geometric integrations by parts along tangential directions to foliation level sets, together with an almost-orthogonality framework to bound Fourier integral operators with rough phases and symbols. These results advance the analysis needed for the bounded curvature conjecture by enabling robust control of wave propagation on rough backgrounds and contribute broadly to the theory of bounds for FIOs under limited regularity.

Abstract

This is the second of a sequence of four papers \cite{param1}, \cite{param2}, \cite{param3}, \cite{param4} dedicated to the construction and the control of a parametrix to the homogeneous wave equation , where is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes bounds on the curvature tensor of is a major step of the proof of the bounded curvature conjecture proposed in \cite{Kl:2000}, and solved by S. Klainerman, I. Rodnianski and the author in \cite{boundedl2}. On a more general level, this sequence of papers deals with the control of the eikonal equation on a rough background, and with the derivation of bounds for Fourier integral operators on manifolds with rough phases and symbols, and as such is also of independent interest.

Paper Structure

This paper contains 65 sections, 21 theorems, 495 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Presentation of the parametrix
  4. Prescription of $u_+$ and $u_-$ on $\Sigma$
  5. The choice of $f_+$ and $f_-$
  6. Regularity assumptions on the phase $u(x,\omega)$
  7. Main results
  8. Boundnessness on $L^2$ for pseudodifferential operators acting on $\mathbb{R}^3$ with rough symbols
  9. Proof of Theorem \ref{['th3']}
  10. The basic computation
  11. Structure of the proof of Theorem \ref{['th3']}
  12. Step 1: decomposition in frequency
  13. Step 2: decomposition in angle
  14. Step 3: control of the diagonal term
  15. Proof of Theorem \ref{['th3']}
  16. ...and 50 more sections

Key Result

Theorem 1.1

Let $(\mathcal{M}, {\bf g})$ an asymptotically flat solution to the Einstein vacuum equations eq:I1 together with a maximal foliation by space-like hypersurfaces $\Sigma_t$ defined as level hypersurfaces of a time function $t$. Let $r_{vol}(\Sigma_t,1)$ the volume radius on scales $\leq 1$ of $\Sigm Then, there exists a small universal constant $\varepsilon_0>0$ such that if $0<\varepsilon<\vareps

Theorems & Definitions (51)

  • Theorem 1.1: Theorem 1.10 in boundedl2
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 2.1
  • ...and 41 more