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Spectral theory and self-similar blowup in wave equations

Roland Donninger

Abstract

This is an expository article that describes the spectral-theoretic aspects in the study of the stability of self-similar blowup for nonlinear wave equations. The linearization near a self-similar solution leads to a genuinely nonself-adjoint operator which is difficult to analyze. The main goal of this article is to provide an accessible account to the only known method that is capable of providing sufficient spectral information to complete the stability analysis. The exposition is based on a mini course given at the Summer School on Geometric Dispersive PDEs in Obergurgl, Austria, in September 2022.

Spectral theory and self-similar blowup in wave equations

Abstract

This is an expository article that describes the spectral-theoretic aspects in the study of the stability of self-similar blowup for nonlinear wave equations. The linearization near a self-similar solution leads to a genuinely nonself-adjoint operator which is difficult to analyze. The main goal of this article is to provide an accessible account to the only known method that is capable of providing sufficient spectral information to complete the stability analysis. The exposition is based on a mini course given at the Summer School on Geometric Dispersive PDEs in Obergurgl, Austria, in September 2022.
Paper Structure (27 sections, 25 theorems, 153 equations)

This paper contains 27 sections, 25 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Wave maps
  3. Corotational wave maps and singularity formation
  4. The mode stability problem
  5. Similarity coordinates
  6. Mode solutions
  7. Solution of the mode stability problem
  8. Fuchsian classification
  9. Frobenius analysis
  10. Supersymmetric removal
  11. Transformation to standard Heun form
  12. The recurrence relation
  13. Properties of the coefficients
  14. The quasi-solution
  15. Functional analytic setup
  16. ...and 12 more sections

Key Result

Theorem 3.1

The blowup solution $u^T$ is mode stable.

Theorems & Definitions (53)

  • Definition 2.1
  • Theorem 3.1
  • Theorem 3.2
  • proof : Idea of proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • Lemma 3.5
  • proof
  • Definition 3.6
  • ...and 43 more