On channels of energy for the radial linearised energy critical wave equation in the degenerate case

TL;DR

This paper develops a weaker form of channels of energy estimates for the linearised energy critical wave equation in even dimensions, enabling the proof of the soliton resolution conjecture in six dimensions.

Contribution

It introduces a modified channel of energy estimate in even dimensions, overcoming previous limitations and advancing understanding of wave equation soliton dynamics.

Findings

Established weaker energy estimates in even dimensions

Proved the soliton resolution conjecture in six dimensions

Extended the analysis of wave equations to previously challenging cases

Abstract

Channels of energy estimates control the energy of an initial data from that which it radiates outside a light cone. For the linearised energy critical wave equation they have been obtained in the radial case in odd dimensions, first in $3$ dimensions by Duyckaerts, Kenig and Merle (Camb. J. Math., 2013), then for general odd dimensions by the same authors (Comm. Math. Phys., 2020). We consider even dimensions, for which such estimates are known to fail (C\^ote, Kenig and Schlag, Math. Ann., 2014). We propose a weaker version of these estimates, around a single ground state as well as around a multisoliton. This allows us to prove the soliton resolution conjecture in six dimensions (Collot, Duyckaerts, Kenig and Merle, arXiv preprint 2201.01848, 2022 versions 1 and 2).