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Blowup stability of wave maps without symmetry

Roland Donninger, Frederick Moscatelli

Abstract

We study wave maps from $(1+d)$-dimensional Minkowski space into the $d$-sphere without any symmetry assumptions. There exists an explicit self-similar blowup solution and we prove that this solution is asymptotically stable under small perturbations of the initial data. The proof is fully rigorous and requires no numerical input whatsoever.

Blowup stability of wave maps without symmetry

Abstract

We study wave maps from -dimensional Minkowski space into the -sphere without any symmetry assumptions. There exists an explicit self-similar blowup solution and we prove that this solution is asymptotically stable under small perturbations of the initial data. The proof is fully rigorous and requires no numerical input whatsoever.
Paper Structure (26 sections, 22 theorems, 289 equations)

This paper contains 26 sections, 22 theorems, 289 equations.

Table of Contents

  1. Introduction
  2. Related results
  3. Symmetries
  4. Intrinsic formulation
  5. Similarity coordinates
  6. Notation
  7. Outline of the proof
  8. Linear stability
  9. Definitions
  10. First order formulation
  11. Properties of the free generator
  12. Full linear problem
  13. Properties of the perturbation
  14. Spectrum of the unperturbed generator
  15. Growth bound for the unperturbed generator
  16. ...and 11 more sections

Key Result

Theorem 1.1

For every $d \geq 3$ there exists $k_0 = k_0(d)$ such that the following holds for every $k \geq k_0$. There exist constants $0 < c_\ast,\delta_\ast \leq 1$ and $\varepsilon_\ast > 0$ such that for all $(F,G) \in C^\infty(\overline{\mathbb{B}_2^d},\mathbb{R}^{d+1})^2$ with there exist $(T_\ast,X_\ast,\Theta_\ast) \in [1-\delta_\ast,1+\delta_\ast] \times \overline{\mathbb{B}_{\delta_\ast}^d} \time

Theorems & Definitions (46)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 2.1
  • ...and 36 more