Quantization of time-like energy for wave maps into spheres

TL;DR

This paper investigates the behavior of large energy wave maps into spheres in 2+1 dimensions near singularities, demonstrating a decomposition into solitons and dispersive components with vanishing error.

Contribution

It provides a detailed analysis of the wave maps' dynamics near singularities, extending the understanding of their decomposition into solitons and dispersive parts.

Findings

Decomposition of wave maps into solitons and dispersive parts near singularities.

Asymptotic vanishing of energy dispersion norm on null boundary.

Convergence of the dispersive component to a constant inside the light cone.

Abstract

In this article we consider large energy wave maps in dimension 2+1, as in the resolution of the threshold conjecture by Sterbenz and Tataru, but more specifically into the unit Euclidean sphere, and study further the dynamics of the sequence of wave maps that are obtained by Sterbenz and Tataru after the final rescaling at a first, finite or infinite, time singularity. We prove that, on a suitably chosen sequence of time slices at this scaling, there is a decomposition of the map, up to an error with asymptotically vanishing energy, into a decoupled sum of rescaled solitons concentrating in the interior of the light cone and a term having asymptotically vanishing energy dispersion norm, concentrating on the null boundary and converging to a constant locally in the interior of the cone, in the energy space.