Channels of Energy for the Linearized Energy Critical Wave Equation in Even Dimensions $N\geq 8$

TL;DR

This work develops exterior energy and channels-of-energy estimates for the linearized energy-critical wave equation around multisolitons in all even dimensions $N\ge 8$, extending prior results from the lower dimensions. It tackles the rise of generalized eigenfunctions of the static operator, requiring high-dimensional projection controls and linear-system techniques to bound the radiative versus nonradiative components. The authors establish parity-dependent bounds linking radiated energy $E_{out}$ to projections in $\dot{H}^1$ and $L^2$ spaces, with localized $Z_{\alpha}$-type norms and a small spectral-gap parameter $\gamma(\lambda)$ governing multisoliton interactions. These advances form a crucial step toward an alternative soliton-resolution approach for radial equivariant wave maps in even dimensions and set the stage for broader applicability to radial energy-critical perturbations.

Abstract

We prove an exterior energy estimate for the linearized energy critical wave equation around a multisoliton for even dimensions $N\geq 8.$ This extends previous work of Collot-Duyckaerts-Kenig-Merle to higher dimensions. During the proof we encounter various additional important technical difficulties compared to lower dimensions. In particular, we need to deal with a number of generalized eigenfunctions of the static operator which increases linearly in $N.$ This makes the analysis of projections onto these eigenfunctions a higher dimensional problem, which requires linear systems to control. This is a crucial ingredient in our upcoming work where we give an alternative proof of the soliton resolution for the wave maps equation based on the method of channels of energy developed by Duyckaerts-Kenig-Merle.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• proof
• Proposition 3.1
• Lemma 4.1
• proof
• Proposition 4.2
• proof