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Smooth finite time singularity formation without quantization

Istvan Kadar

Abstract

We revisit the finite time singularity formation of Krieger-Schlag-Tataru [KST09] for the focusing energy critical wave equation in $\mathbb{R}^{3+1}$ from a geometric singular-analytic point of view, following Hintz [Hintz23]. We construct $C^{ν/2-}$ regular approximate solutions that settle down to multiple solitons, shrinking at a rate $t^ν$ with $ν>1$, and approaching the origin on different geodesics $\{x=zt\}$. By fine tuning the velocities, sizes and signs of the solitons, we are able to construct smooth ansätze with any $ν>8$. Using robust energy estimates, the ansätze are corrected to exact solutions.

Smooth finite time singularity formation without quantization

Abstract

We revisit the finite time singularity formation of Krieger-Schlag-Tataru [KST09] for the focusing energy critical wave equation in from a geometric singular-analytic point of view, following Hintz [Hintz23]. We construct regular approximate solutions that settle down to multiple solitons, shrinking at a rate with , and approaching the origin on different geodesics . By fine tuning the velocities, sizes and signs of the solitons, we are able to construct smooth ansätze with any . Using robust energy estimates, the ansätze are corrected to exact solutions.
Paper Structure (43 sections, 52 theorems, 252 equations, 2 figures)

This paper contains 43 sections, 52 theorems, 252 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Singularity formation: scattering versus evolution
  3. Many body problems.
  4. Geometric setup
  5. Main result
  6. Ideas of the proof
  7. Overview:
  8. Acknowledgement:
  9. Geometric and analytic setup
  10. Conormal functions
  11. Unit size coordinate functions
  12. Compactification
  13. Energy spaces
  14. Extra notation and computations
  15. Mapping properties and model operators
  16. ...and 28 more sections

Key Result

Theorem 1.1

Given a finite collection of velocities $\{z\in A\}=A\subset \mathring{B}:=\{x\in\mathbb{R}^3:\left\lvert x\right\rvert<1\}$, scales $\boldsymbol{\lambda}_z\in\mathbb{R}_{\neq0}$, signs $\sigma_z\in\{\pm1\}$ and singularity speed $\nu\in\mathbb{R}_{>1}$, there exists a solution, $\phi\in C^{\nu/2-}$ Moreover, for $\nu>8$ and a well chosen soliton data $(A,\boldsymbol{\lambda},\sigma)$ depending on

Figures (2)

  • Figure 1: We construct the solution in within the lightcone $\mathcal{C}$. The solitons move along the $\{x_z=0\},\{x_{z'}=0\}$ curves, and we also indicated the size of the $\{x_z=0\}$ soliton by dashed line. In the exterior region, we indicated where the solution can be extended using kadar_note_2026. The solution can be regarded as part of the maximal development of some data prescribed on the Cauchy hypersurface $\Sigma$.
  • Figure 2: Depicted is the compactification of $\overline{\mathcal{M}}$, with boundary components $\mathcal{C},i_-,\mathcal{K}_z$ and local respective local coordinates $\{t,x/\left\lvert x\right\rvert\},\{x/t\},\{y_z\}$.

Theorems & Definitions (117)

  • Theorem 1.1
  • Corollary 1.1
  • Proposition 1.1: Irregular solution, \ref{['an:prop:multi']}
  • Remark 1.1: Singularity speed
  • Remark 1.2: Outgoing radiation
  • Remark 1.3: Perturbed equations
  • Remark 1.4: Expansion
  • Proposition 1.2: Smooth quantised, \ref{['an:prop:smooth_quantized']}
  • Remark 1.5: Restriction on the number of solitons
  • Remark 1.6: Codimension stability for smooth solutions
  • ...and 107 more