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Self-similar blowup from arbitrary data for supercritical wave maps with additive noise

Irfan Glogić, Martina Hofmanová, Eliseo Luongo

TL;DR

This work proves that additive corotational noise can drive self-similar blowup in energy-supercritical wave maps with positive probability for arbitrary corotational data. The authors combine a Da Prato–Debussche decomposition with a self-similar change of variables and a Lyapunov–Perron fixed-point argument to establish nonlinear stability of the self-similar blowup profile under stochastic forcing. They further show an irreducibility/controllability property of the stochastic forcing, ensuring the solution can reach neighborhoods in state space with positive probability, which together imply probabilistic blowup via the explicit self-similar profile ${\mathcal U}_T$. The results highlight that noise can reinforce certain singular behaviors in nonlinear wave dynamics, contributing to the broader discourse on regularization-by-noise and stochastic instability in supercritical PDEs, with precise framework built in radial Sobolev spaces and corotational symmetry.

Abstract

We consider stochastically perturbed wave maps from $\mathbb{R}^{1+d}$ into $\mathbb{S}^d$, in all energy-supercritical dimensions $d \geq 3$. We show that corotational non-degenerate Gaussian additive noise leads to self-similar blowup with positive probability for any corotational initial data. The same result without noise is conjectured, but unknown, for large data.

Self-similar blowup from arbitrary data for supercritical wave maps with additive noise

TL;DR

This work proves that additive corotational noise can drive self-similar blowup in energy-supercritical wave maps with positive probability for arbitrary corotational data. The authors combine a Da Prato–Debussche decomposition with a self-similar change of variables and a Lyapunov–Perron fixed-point argument to establish nonlinear stability of the self-similar blowup profile under stochastic forcing. They further show an irreducibility/controllability property of the stochastic forcing, ensuring the solution can reach neighborhoods in state space with positive probability, which together imply probabilistic blowup via the explicit self-similar profile . The results highlight that noise can reinforce certain singular behaviors in nonlinear wave dynamics, contributing to the broader discourse on regularization-by-noise and stochastic instability in supercritical PDEs, with precise framework built in radial Sobolev spaces and corotational symmetry.

Abstract

We consider stochastically perturbed wave maps from into , in all energy-supercritical dimensions . We show that corotational non-degenerate Gaussian additive noise leads to self-similar blowup with positive probability for any corotational initial data. The same result without noise is conjectured, but unknown, for large data.

Paper Structure

This paper contains 17 sections, 13 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Wave maps with additive noise
  3. The main results
  4. On regularization by noise phenomena
  5. Brief history of blowup for (deterministic) wave maps
  6. Outline of the proof of \ref{['main_thm']}
  7. Notation
  8. Preliminaries
  9. Analysis in physical variables: well-posedness and stochastic convolution
  10. Analysis in self-similar variables: linearized dynamics
  11. Stable self-similar blowup profile for stochastic corotational wave maps
  12. Change of Variables
  13. Analysis of the perturbed system
  14. Irreducibility of the stochastic corotational wave maps
  15. Approximate controllability
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $n\geq 5$ and $s,k>0$ such that Under HP_noise_1, for each $\mathcal{F}_0$-measurable random variable $\mathbf{u}_0=(u_0,\hat{u}_{0})$ with values in $\mathcal{H}^{s,k}_{rad}$, there exist a strictly positive stopping time $\tau^*(\mathbf{u}_0):\Omega\rightarrow (0,+\infty]$ and a progressively measurable process $\mathbf{u}:\Omeg Moreover, $\tau^*(\mathbf{u}_0)$ and $\mathbf u$ are unique in

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.2
  • proof
  • Definition 2.3
  • proof : Proof of \ref{['Thm_well_posed']}.
  • ...and 18 more