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Mode stability of blow-up for wave maps in the absence of symmetry

Max Weissenbacher, Herbert Koch, Roland Donninger

TL;DR

The paper proves mode stability for the explicit self-similar blow-up solution of the three-dimensional wave maps equation into $S^3$ without symmetry restrictions. It builds a symmetry-aware framework that decomposes the linearized equation into symmetry-equivariant radial ODEs via a Clebsch–Gordan basis for $so(3)$, then removes symmetry-generated modes with a SUSY transformation and analyzes the resulting equations in standard forms (hypergeometric and Heun). A rigorous quasi-solution method (Costin–Donninger–Glogić) is then used to exclude the existence of any nontrivial smooth mode solutions with nonnegative real part of the growth rate, except those arising from the symmetry group. Consequently, the blow-up profile is mode-stable modulo its symmetries, establishing a key linear stability ingredient toward nonlinear stability without symmetry assumptions. The work extends the co-rotational results and provides a robust framework for understanding self-similar blow-up in geometric wave equations, with potential implications for continuation past blow-up and nonlinear stability analyses.

Abstract

The wave maps equation in three spatial dimensions with a spherical target admits an explicit blow-up solution. Numerical studies suggest this solution captures the generic blow-up behaviour in the backward light cone of the singularity. In this work, we establish the mode stability of this blow-up solution in the backward light cone of the blow-up point without any assumptions on the symmetries of the perturbation. We classify all smooth mode solutions for growth rates $λ$ with $\mathrm{Re} \, λ\geq 0$ and demonstrate that the blow-up solution is stable up to the mode solutions arising from the symmetry group of the wave maps equation. Our proof relies on a decomposition of the linearised wave maps equation into a tractable system of symmetry-equivariant ordinary differential equations (ODEs), utilising the representation theory of the stabiliser of the blow-up solution. We then use the quasi-solution method of Costin-Donninger-Glogić to show the absence of non-zero smooth solutions for the resulting system of ODEs.

Mode stability of blow-up for wave maps in the absence of symmetry

TL;DR

The paper proves mode stability for the explicit self-similar blow-up solution of the three-dimensional wave maps equation into without symmetry restrictions. It builds a symmetry-aware framework that decomposes the linearized equation into symmetry-equivariant radial ODEs via a Clebsch–Gordan basis for , then removes symmetry-generated modes with a SUSY transformation and analyzes the resulting equations in standard forms (hypergeometric and Heun). A rigorous quasi-solution method (Costin–Donninger–Glogić) is then used to exclude the existence of any nontrivial smooth mode solutions with nonnegative real part of the growth rate, except those arising from the symmetry group. Consequently, the blow-up profile is mode-stable modulo its symmetries, establishing a key linear stability ingredient toward nonlinear stability without symmetry assumptions. The work extends the co-rotational results and provides a robust framework for understanding self-similar blow-up in geometric wave equations, with potential implications for continuation past blow-up and nonlinear stability analyses.

Abstract

The wave maps equation in three spatial dimensions with a spherical target admits an explicit blow-up solution. Numerical studies suggest this solution captures the generic blow-up behaviour in the backward light cone of the singularity. In this work, we establish the mode stability of this blow-up solution in the backward light cone of the blow-up point without any assumptions on the symmetries of the perturbation. We classify all smooth mode solutions for growth rates with and demonstrate that the blow-up solution is stable up to the mode solutions arising from the symmetry group of the wave maps equation. Our proof relies on a decomposition of the linearised wave maps equation into a tractable system of symmetry-equivariant ordinary differential equations (ODEs), utilising the representation theory of the stabiliser of the blow-up solution. We then use the quasi-solution method of Costin-Donninger-Glogić to show the absence of non-zero smooth solutions for the resulting system of ODEs.

Paper Structure

This paper contains 30 sections, 11 theorems, 160 equations, 5 tables.

Table of Contents

  1. Introduction
  2. The blow up solution
  3. Symmetries of the wave maps equation
  4. Self-similar coordinates
  5. Mode stability
  6. Related work
  7. The co-rotational case
  8. Nonlinear stability
  9. Well-posedness and regularity
  10. Overview
  11. Acknowledgments
  12. Symmetries and decoupling the equation
  13. Mode solutions generated by symmetries
  14. Clebsch--Gordan decomposition
  15. Decoupling the linearised equation
  16. ...and 15 more sections

Key Result

Theorem 1

Assume $\mathrm{Re}\, \lambda \geq 0$ and $\Psi$ is a mode solution as in Definition def:modesoln. Then $\lambda \in \{0,1\}$. The space of mode solutions with growth rate $\lambda=1$ is four-dimensional and spanned by the functions $\{ \Psi_{1,0}, \Psi_{0,1}^1, \Psi_{0,1}^2, \Psi_{0,1}^3 \}$, where where $e_i$ denote the standard Euclidean basis of $\mathbb{R}^3$. The space of mode solutions with

Theorems & Definitions (30)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 20 more