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Stable blowup for supercritical wave maps into perturbed spheres

Roland Donninger, Birgit Schörkhuber, Alexander Wittenstein

TL;DR

This work addresses stable self-similar blowup for energy-supercritical wave maps from $(1+d)$-dimensional Minkowski space into perturbed spheres $S^d_{\epsilon}$ under co-rotational symmetry. The authors reformulate the problem in similarity coordinates, construct self-similar blowup profiles by a perturbative fixed-point argument around the ground state at $\epsilon=0$, and prove nonlinear stability of these blowup solutions under small co-rotational perturbations on the full space via spectral analysis and a robust contraction mapping in an abstract Cauchy problem. The analysis relies on intersection Sobolev spaces, Schauder-type estimates with parameter dependence, and a careful treatment of the symmetry-induced unstable mode to ensure exponential decay on the stable subspace. The results extend prior stability results known for the sphere target to a class of warped-product targets close to the sphere, deepening understanding of blowup dynamics in geometric wave equations and suggesting broader robustness of self-similar blowup phenomena under scale-invariant perturbations.

Abstract

We consider wave maps from $(1+d)$-dimensional Minkowski space, $d\geq3$, into rotationally symmetric manifolds which arise from small perturbations of the sphere $\mathbb S^d$. We prove the existence of co-rotational self-similar finite time blowup solutions with smooth blowup profiles. Furthermore, we show the nonlinear asymptotic stability of these solutions under suitably small co-rotational perturbations on the full space.

Stable blowup for supercritical wave maps into perturbed spheres

TL;DR

This work addresses stable self-similar blowup for energy-supercritical wave maps from -dimensional Minkowski space into perturbed spheres under co-rotational symmetry. The authors reformulate the problem in similarity coordinates, construct self-similar blowup profiles by a perturbative fixed-point argument around the ground state at , and prove nonlinear stability of these blowup solutions under small co-rotational perturbations on the full space via spectral analysis and a robust contraction mapping in an abstract Cauchy problem. The analysis relies on intersection Sobolev spaces, Schauder-type estimates with parameter dependence, and a careful treatment of the symmetry-induced unstable mode to ensure exponential decay on the stable subspace. The results extend prior stability results known for the sphere target to a class of warped-product targets close to the sphere, deepening understanding of blowup dynamics in geometric wave equations and suggesting broader robustness of self-similar blowup phenomena under scale-invariant perturbations.

Abstract

We consider wave maps from -dimensional Minkowski space, , into rotationally symmetric manifolds which arise from small perturbations of the sphere . We prove the existence of co-rotational self-similar finite time blowup solutions with smooth blowup profiles. Furthermore, we show the nonlinear asymptotic stability of these solutions under suitably small co-rotational perturbations on the full space.

Paper Structure

This paper contains 18 sections, 28 theorems, 291 equations.

Table of Contents

  1. Introduction
  2. The main result
  3. Blowup in energy supercritical wave maps - Known results
  4. Reformulation of the problem and outline of the proof
  5. Existence of self-similar blowup profiles
  6. Stability of self-similar blowup
  7. Notation
  8. Similarity variables and functional analytic setup
  9. Reformulation of the problem in similarity coordinates
  10. Time-independent solutions
  11. Intersection Sobolev spaces
  12. Known results for
  13. Existence of blowup solutions into perturbed spheres
  14. Stability analysis
  15. Perturbations of the free evolution for small parameters
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $d \geq 3$. There exists an $\epsilon^* \in \mathbb{R}$, $0 < \epsilon^* \leq \epsilon_0$ such that for every $\epsilon \in \mathbb{R}$ with $\lvert\epsilon\rvert \leq \epsilon^*$, Eq. equation for radial profile admits a self-similar solution $u_{\epsilon}^T \in C^{\infty}([0,T) \times [0,\inft such that its gradient blows up in $r=0$, i.e., The profile $f_{\epsilon}$ can be written as with

Theorems & Definitions (52)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Corollary 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 42 more