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A stability theory beyond the co-rotational setting for critical Wave Maps blow up

Joachim Krieger, Shuang Miao, Wilhelm Schlag

Abstract

We exhibit non-equivariant perturbations of the blowup solutions constructed in \cite{KST} for energy critical wave maps into $\mathbb{S}^2$. Our admissible class of perturbations is an open set in some sufficiently smooth topology and vanishes near the light cone. We show that the blowup solutions from \cite{KST} are rigid under such perturbations, including the space-time location of blowup. As blowup is approached, the dynamics agree with the classification obtained in \cite{DJKM}, and all six symmetry parameters converge to limiting values. Compared to the previous work \cite{KMiao} in which the rigidity of the blowup solutions from \cite{KST} under equivariant perturbations was proved, the class of perturbations considered in the present work does not impose any symmetry restrictions. Separation of variables and decomposing into angular Fourier modes leads to an infinite system of coupled nonlinear equations, which we solve for small admissible data. The nonlinear analysis is based on the distorted Fourier transform, associated with an infinite family of Bessel type Schrödinger operators on the half-line indexed by the angular momentum~$n$. A semi-classical WKB-type spectral analysis relative to the parameter $\hbar=\frac{1}{n+1}$ for large $|n|$ allows us to effectively determine the distorted Fourier basis for the entire infinite family. Our linear analysis is based on the global Liouville-Green transform as in the earlier works \cite{CSST, CDST}.

A stability theory beyond the co-rotational setting for critical Wave Maps blow up

Abstract

We exhibit non-equivariant perturbations of the blowup solutions constructed in \cite{KST} for energy critical wave maps into . Our admissible class of perturbations is an open set in some sufficiently smooth topology and vanishes near the light cone. We show that the blowup solutions from \cite{KST} are rigid under such perturbations, including the space-time location of blowup. As blowup is approached, the dynamics agree with the classification obtained in \cite{DJKM}, and all six symmetry parameters converge to limiting values. Compared to the previous work \cite{KMiao} in which the rigidity of the blowup solutions from \cite{KST} under equivariant perturbations was proved, the class of perturbations considered in the present work does not impose any symmetry restrictions. Separation of variables and decomposing into angular Fourier modes leads to an infinite system of coupled nonlinear equations, which we solve for small admissible data. The nonlinear analysis is based on the distorted Fourier transform, associated with an infinite family of Bessel type Schrödinger operators on the half-line indexed by the angular momentum~. A semi-classical WKB-type spectral analysis relative to the parameter for large allows us to effectively determine the distorted Fourier basis for the entire infinite family. Our linear analysis is based on the global Liouville-Green transform as in the earlier works \cite{CSST, CDST}.

Paper Structure

This paper contains 92 sections, 196 theorems, 2982 equations.

Table of Contents

  1. Introduction
  2. Blow up in the co-rotational setting
  3. Beyond the co-rotational setting
  4. Statement of the main theorem
  5. Summary of the sections of the paper
  6. The linear part
  7. The nonlinear part
  8. Description of the key components of the strategy
  9. The distinction between the angular frequencies $|n|\geq 2$ and the exceptional ones $n\in \{0, \pm 1\}$.
  10. Controlling the unstructured smooth part and the structured singular/shock part
  11. Controlling the evolution of the coefficients $c_n$, $n = 0,\,\pm 1$. Modulation theory
  12. Perturbing equivariant blowup solutions
  13. The linear operators governed by the rescaled harmonic map, separation of variables
  14. Symmetries and zero modes
  15. Diagonalizing the linear operators
  16. ...and 77 more sections

Key Result

Theorem \oldthetheorem

Let $\Phi = \left(\right)$, $U = U(t,r)$ be one of the finite time co-rotational blow up solutions constructed in KSTGaoK, with $\nu>0$ sufficiently small, and restricted to the space time slab $(0,t_0]\times \mathbb{R}^2$, where $t_0 = t_0(\nu)$ is sufficiently small. Then there are $\delta_* = \de and such that constitutes a data set for eq:BasicWMS2target1, and furthermore $\delta\Phi_j$, $j =

Theorems & Definitions (383)

  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • ...and 373 more