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Optimal blowup stability for three-dimensional wave maps

Roland Donninger, David Wallauch

TL;DR

The paper proves the asymptotic stability of the explicit corotational self-similar blowup $U_*^T$ for the 3+1 dimensional wave maps equation into $b S^3$, under perturbations small in the fractional Sobolev norm at the critical level. It achieves this by recasting the problem in similarity coordinates, linearizing around the self-similar profile, and developing a spectral/semigroup framework for the resulting radial equation in five dimensions. A detailed resolvent construction and a suite of Strichartz estimates in fractional Sobolev spaces, together with a projection that isolates the unstable eigenmode, enable a fixed-point argument to control the nonlinear evolution in a norm that captures the self-similar blowup profile. The work advances the understanding of blowup stability in energy-supercritical geometric wave equations by obtaining optimal, fractional-regularity Strichartz estimates in the corotational setting. These techniques provide a robust analytic blueprint for addressing similar stability problems in dispersive geometric PDEs with self-similar blowup.

Abstract

We study corotational wave maps from $(1+3)$-dimensional Minkowski space into the three-sphere. We establish the asymptotic stability of an explicitly known self-similar wave map under perturbations that are small in the critical Sobolev space. This is accomplished by proving Strichartz estimates for a radial wave equation with a potential in similarity coordinates. Compared to earlier work, the main novelty lies with the fact that the critical Sobolev space is of fractional order.

Optimal blowup stability for three-dimensional wave maps

TL;DR

The paper proves the asymptotic stability of the explicit corotational self-similar blowup for the 3+1 dimensional wave maps equation into , under perturbations small in the fractional Sobolev norm at the critical level. It achieves this by recasting the problem in similarity coordinates, linearizing around the self-similar profile, and developing a spectral/semigroup framework for the resulting radial equation in five dimensions. A detailed resolvent construction and a suite of Strichartz estimates in fractional Sobolev spaces, together with a projection that isolates the unstable eigenmode, enable a fixed-point argument to control the nonlinear evolution in a norm that captures the self-similar blowup profile. The work advances the understanding of blowup stability in energy-supercritical geometric wave equations by obtaining optimal, fractional-regularity Strichartz estimates in the corotational setting. These techniques provide a robust analytic blueprint for addressing similar stability problems in dispersive geometric PDEs with self-similar blowup.

Abstract

We study corotational wave maps from -dimensional Minkowski space into the three-sphere. We establish the asymptotic stability of an explicitly known self-similar wave map under perturbations that are small in the critical Sobolev space. This is accomplished by proving Strichartz estimates for a radial wave equation with a potential in similarity coordinates. Compared to earlier work, the main novelty lies with the fact that the critical Sobolev space is of fractional order.
Paper Structure (27 sections, 67 theorems, 446 equations)

This paper contains 27 sections, 67 theorems, 446 equations.

Table of Contents

  1. Introduction
  2. Discussion
  3. Stability of blowup
  4. Optimality
  5. Symmetry
  6. Maximal domain of existence
  7. Supercriticality
  8. Related results
  9. Outline of the proof
  10. Transformations and Semigroup theory
  11. Semigroup theory
  12. Spectral analysis of $\textup{L}$
  13. ODE analysis
  14. Preliminary transformations
  15. Construction of fundamental systems
  16. ...and 12 more sections

Key Result

Theorem 1.1

There exist constants $\delta_0,M>0$ such that the following holds. Let $F: \mathbb R^3\to \mathbb S^3\subset\mathbb R^4$ and $G: \mathbb R^3\to \mathbb R^4$ be given by for smooth, radial functions $f,g: \mathbb R^3\to\mathbb R$. Assume further that $\delta\in [0,\delta_0]$ and Then there exists a $T\in [1-\delta,1+\delta]$ and a unique smooth wave map $U: \Omega_{T}\to \mathbb S^3\subset\mathb

Theorems & Definitions (116)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 106 more