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Stability of Spherically Symmetric Wave Maps

Joachim Krieger

TL;DR

The paper proves global regularity for Wave Maps from R^{2+1} into the hyperbolic plane H^2 under small H^{1+μ} (μ>0) perturbations of smooth spherically symmetric data, and also for perturbations near geodesic maps. It leverages a derivative formulation in the Coulomb gauge, sophisticated null-frame spaces, and a bootstrap based on frequency envelopes to control subcritical norms and nonlinear interactions. By exploiting the negative curvature and the refined microlocal analysis, it extends stability results in this energy-critical setting and provides a robust framework (including Moser-type estimates) for handling high-order nonlinearities. The work generalizes Sideris’ results, demonstrates stability near geodesic and spherically symmetric profiles, and contributes techniques potentially applicable to broader large-data perturbations in 2+1 dimensional Wave Maps.

Abstract

We study Wave Maps from R^{2+1} to the hyperbolic plane with smooth compactly supported initial data which are close to smooth spherically symmetric ones with respect to some H^{1+\mu}, \mu>0. We show that such Wave Maps don't develop singularities and stay close to the Wave Map extending the spherically symmetric data with respect to all H^{1+\delta}, \delta<\mu_{0}(\mu). We obtain a similar result for Wave Maps whose initial data are close to geodesic ones. This generalizes a theorem of Sideris for this context.

Stability of Spherically Symmetric Wave Maps

TL;DR

The paper proves global regularity for Wave Maps from R^{2+1} into the hyperbolic plane H^2 under small H^{1+μ} (μ>0) perturbations of smooth spherically symmetric data, and also for perturbations near geodesic maps. It leverages a derivative formulation in the Coulomb gauge, sophisticated null-frame spaces, and a bootstrap based on frequency envelopes to control subcritical norms and nonlinear interactions. By exploiting the negative curvature and the refined microlocal analysis, it extends stability results in this energy-critical setting and provides a robust framework (including Moser-type estimates) for handling high-order nonlinearities. The work generalizes Sideris’ results, demonstrates stability near geodesic and spherically symmetric profiles, and contributes techniques potentially applicable to broader large-data perturbations in 2+1 dimensional Wave Maps.

Abstract

We study Wave Maps from R^{2+1} to the hyperbolic plane with smooth compactly supported initial data which are close to smooth spherically symmetric ones with respect to some H^{1+\mu}, \mu>0. We show that such Wave Maps don't develop singularities and stay close to the Wave Map extending the spherically symmetric data with respect to all H^{1+\delta}, \delta<\mu_{0}(\mu). We obtain a similar result for Wave Maps whose initial data are close to geodesic ones. This generalizes a theorem of Sideris for this context.

Paper Structure

This paper contains 15 sections, 24 theorems, 445 equations.

Table of Contents

  1. Introduction
  2. A priori estimates for spherically symmetric Wave Maps.
  3. The perturbation argument
  4. Precise statement of theorem. Outline of the procedure
  5. Sobolev type spaces
  6. Tataru's null-frame spaces
  7. A modification of the spaces; Moser type estimates
  8. Proof of the Theorem \ref{['Main']}
  9. Proof of Proposition \ref{['bootstrap']}
  10. The proof of theorem \ref{["Main'"]}
  11. The proof of Proposition 3.17.
  12. Estimating the trilinear null-forms.
  13. The quintilinear and higher order terms linear in the perturbation.
  14. The higher order error terms at least quadratic in $\delta\psi$. Completing the proof of Proposition \ref{['refined bootstrap']}.
  15. Appendix: Proof of theorem \ref{['Moser']}.

Key Result

Theorem 1.1

Let n=2,3,\ldots. Then there exists \epsilon>0 such that for smooth initial data ({\bf{x}},{\bf{y}}),\,(\partial_{t}{\bf{x}},\partial_{t}{\bf{y}}):{\mathbf{R}}^{n}\times\{0\}\rightarrow {\mathbf{H}}^{2}\times T{\mathbf{H}}^{2} satisfying there exists a smooth global-in-time Wave Map extending them.

Theorems & Definitions (26)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 16 more