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A globally stable self-similar blowup profile in energy supercritical Yang-Mills theory

Roland Donninger, Matthias Ostermann

Abstract

This paper is concerned with the Cauchy problem for an energy-supercritical nonlinear wave equation in odd space dimensions that arises in equivariant Yang-Mills theory. In each dimension, there is a self-similar finite-time blowup solution to this equation known in closed form. It will be proved that this profile is stable in the whole space under small perturbations of the initial data. The blowup analysis is based on a recently developed coordinate system called hyperboloidal similarity coordinates and depends crucially on growth estimates for the free wave evolution, which will be constructed systematically for odd space dimensions in the first part of this paper. This allows to develop a nonlinear stability theory beyond the singularity.

A globally stable self-similar blowup profile in energy supercritical Yang-Mills theory

Abstract

This paper is concerned with the Cauchy problem for an energy-supercritical nonlinear wave equation in odd space dimensions that arises in equivariant Yang-Mills theory. In each dimension, there is a self-similar finite-time blowup solution to this equation known in closed form. It will be proved that this profile is stable in the whole space under small perturbations of the initial data. The blowup analysis is based on a recently developed coordinate system called hyperboloidal similarity coordinates and depends crucially on growth estimates for the free wave evolution, which will be constructed systematically for odd space dimensions in the first part of this paper. This allows to develop a nonlinear stability theory beyond the singularity.

Paper Structure

This paper contains 39 sections, 30 theorems, 217 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Motivation and related research
  3. Statement of the main result
  4. Notation and coordinate systems
  5. Inequalities
  6. Derivatives
  7. Function spaces
  8. Operators
  9. Coordinate systems
  10. Strategy of the proof
  11. The wave equation in hyperboloidal similarity coordinates
  12. Energy estimates
  13. Half-waves
  14. The wave equation on the half-line
  15. Radial wave evolution
  16. ...and 24 more sections

Key Result

Theorem \oldthetheorem

Let $d\geq 7$ be an odd integer. Fix hyperboloidal similarity coordinates see Fig_HSC. For $R\geq \frac{1}{2}$ consider the region see Fig_OmegaTb. Let $k\in\mathbb{N}$, $k\geq \frac{d+1}{2}$. There are positive constants $M_{0},\delta_{0},\varepsilon,\omega_{0} > 0$ such that for any $0<\delta\leq\delta_{0}$ and $M\geq M_{0}$ the following holds.

Figures (5)

  • Figure 1: Hyperboloidal similarity coordinates on $\mathbb{R}^{1,1}$. The hyperboloids correspond to the level sets of $s$. The radial lines are level sets of $y$. These coordinates cover the whole complement of the future light cone at $(T,0)$.
  • Figure 2: The solutions constructed in \ref{['YMglobThm']} are smoothly defined on the grey shaded region $\Omega_{T,R}$. The solid lines have slope $b$ and may be adjusted to come arbitrarily close to the dashed boundary of the future light cone at $(T,0)$. Together with the curved lines they bound the region $\Omega_{T,R} \setminus \eta_{T}( [s_{0},\infty)\times\mathbb{B}_{R} )$.
  • Figure 3: Portrait of the spectrum for the linearized evolution $\mathbf{L}$. The shaded region is part of the resolvent set of the linearized evolution $\mathbf{L}$ and the isolated unstable eigenvalue $1$ is the only point in the spectrum contained in the positive half-plane.
  • Figure 4: Illustration for the preparation of initial data in the plane. The zigzag line marks the support for the perturbations of initial data, the emerging dashed lines indicate their domain of influence. The other dashed lines delimit the domain of local existence for the classical Cauchy evolution. The grey shaded region depicts the union $\Lambda_\varepsilon$ of these two domains. By construction and the choice $s_{0} = \log(-\tfrac{h(0)}{1+2\varepsilon})$, the initial hyperboloid $y\mapsto(T + \mathrm{e}^{-s_{0}}h(y),\mathrm{e}^{-s_{0}}y)$ lies within $\Lambda_\varepsilon$ for all $T\in\overline{\mathbb{B}_\varepsilon(1)}$. The solid hyperboloid is drawn for $T=1$, the dotted hyperboloids for $T=1\pm\varepsilon$, respectively.
  • Figure 5: This is a depiction of the region where we establish the stable hyperboloidal evolution.

Theorems & Definitions (88)

  • Theorem \oldthetheorem
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 78 more