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Future stability of large-data wave maps in energy-supercritical dimensions

Andras Bonk, Roland Donninger

Abstract

We consider energy-supercritical co-rotational wave maps from Minkowski spacetime to the sphere in odd spatial dimensions. The equation admits an explicit co-rotational self-similar blowup solution, which also induces solutions that blow up in the past. In the region after the blowup the solution treated in this paper is remarkable, as it is smooth forward in time and exhibits less than dispersive decay. We prove nonlinear asymptotic stability of this large-data self-similar solution inside forward light cones. In particular, we identify an open set of initial data close to the explicit solution that give rise to forward-in-time wave maps whose decay is slower than that of generic free waves.

Future stability of large-data wave maps in energy-supercritical dimensions

Abstract

We consider energy-supercritical co-rotational wave maps from Minkowski spacetime to the sphere in odd spatial dimensions. The equation admits an explicit co-rotational self-similar blowup solution, which also induces solutions that blow up in the past. In the region after the blowup the solution treated in this paper is remarkable, as it is smooth forward in time and exhibits less than dispersive decay. We prove nonlinear asymptotic stability of this large-data self-similar solution inside forward light cones. In particular, we identify an open set of initial data close to the explicit solution that give rise to forward-in-time wave maps whose decay is slower than that of generic free waves.
Paper Structure (24 sections, 22 theorems, 163 equations, 3 figures)

This paper contains 24 sections, 22 theorems, 163 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Explicit solutions and symmetry reduction
  3. Solutions after blowup.
  4. Perturbations
  5. Coordinates and geometry
  6. Main results
  7. Discussion
  8. Proof strategy
  9. Related results
  10. Notation
  11. Free wave evolution in FHSC
  12. Connection to the blowup problem
  13. The free evolution is uniformly exponentially stable
  14. Linearised evolution
  15. Generation of linearised semigroup
  16. ...and 9 more sections

Key Result

Theorem 1.4

Let $n, k \in \mathbb{N}$ with There exists $\varepsilon > 0$ such that for all smooth, co-rotational data $(F_1,F_2): \Sigma_0^n \rightarrow \emph{T}\mathbb{S}^n \subset \mathbb{R}^{n+1}\times \mathbb{R}^{n+1}$ of the form with $\widetilde{f}_1,\widetilde{f_2} \in C_{\operatorname{rad}}^\infty(\Sigma_0^1)$ which satisfy $\widetilde{f}_i(t,\lvert {x} \rvert)=f_i(t,x)$ for $i\in \{1,2\}$ with th

Figures (3)

  • Figure 1: Projection of FHSC. The straight lines correspond to $y = \text{const}$, the hyperboloids correspond to $s = \text{const}$. The red hyperboloid $\Sigma_0$ is the lower boundary of the coordinate region and functions as the initial hypersurface for the IVP. The dashed line depicts the light cone with vertex $\left(\frac{1}{2},0 \right)$.
  • Figure 2: Projection of the causal past of the point $(t,x)=\Psi(s_0,y_0)$ up to the initial hyperboloid. The red patches of the hyperboloids $\Sigma_{s'}, \Sigma_0$ are the images of $D_{s'}$ resp. $D_0$ for a given $0 \leq s' < s_0$.
  • Figure 3: Projection of $J^{-}(s_0,y_0)$, i.e. left picture in FHSC. Note that the boundary (dashed) is not given by a straight line but by two logarithmic functions.

Theorems & Definitions (56)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • ...and 46 more