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A scattering theory construction of dynamical solitons in 3d

Istvan Kadar

Abstract

We study the energy critical wave equation in 3 dimensions around a single soliton. We obtain energy boundedness (modulo unstable modes) for the linearised problem. We use this to construct scattering solutions in a neighbourhood of timelike infinity ($i_+$), provided the data on null infinity ($\scri$) decay polynomially. Moreover, the solutions we construct are conormal on a blow-up of Minkowski space. The methods of proof also extend to some energy supercritical modifications of the equation.

A scattering theory construction of dynamical solitons in 3d

Abstract

We study the energy critical wave equation in 3 dimensions around a single soliton. We obtain energy boundedness (modulo unstable modes) for the linearised problem. We use this to construct scattering solutions in a neighbourhood of timelike infinity (), provided the data on null infinity () decay polynomially. Moreover, the solutions we construct are conormal on a blow-up of Minkowski space. The methods of proof also extend to some energy supercritical modifications of the equation.
Paper Structure (50 sections, 47 theorems, 246 equations, 2 figures)

This paper contains 50 sections, 47 theorems, 246 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview of the introduction
  3. Review of previous works and motivation
  4. General relativity
  5. Dispersive equations:
  6. Main theorem
  7. Backwards problem
  8. Obstructions to boundedness: unstable and zero modes
  9. Continuous vs discrete modulation
  10. Unstable mode and codimension
  11. Dyadic iteration
  12. Remarks on related problems
  13. Supercritical problem
  14. Forward problem and assumed decay rates
  15. Blow up at infinity
  16. ...and 35 more sections

Key Result

Theorem 1.1

(Rough form of main theorem, see main theorem) Given scattering data $\psi^{dat}(u,\omega)$ ($u\in\mathbb{R}$, $\omega\in S^2$) that fall off at a polynomial rate, there exists a solution $\phi$ to (1.1) in the region $t-r\gg 1$ with the following properties: firstly, $\phi(t,x)-W(x)$ decays to $0$

Figures (2)

  • Figure 1: Penrose diagram of the region where the solution is constructed in dafermos_scattering_2013 (a) with a limit taken as $\tau_2\to\infty$. Scattering constructions in \ref{['main theorem']} ($I$), \ref{['thm:trivial_scattering']} ($II$) (b), $\tilde{\Sigma}$ denotes constant $t$ hypersurfaces.
  • Figure 2: A compactification of Minkowski space on which scattering solution $\phi$ is conormal

Theorems & Definitions (106)

  • Theorem 1.1
  • Corollary 1.1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Remark 1.4: Decay rates
  • Remark 1.5: Conormal structure
  • Remark 1.6: Codimension
  • Remark 1.7: Uniqueness
  • Theorem 1.8
  • Remark 1.9
  • ...and 96 more