On the set of non radiative solutions for the energy critical wave equation

TL;DR

This paper characterizes the structure of small non radiative solutions to the energy critical wave equation, showing they form a manifold and constructing solutions with prescribed radiation fields, advancing understanding of long-term dynamics.

Contribution

It demonstrates that small non radiative solutions form a manifold with a tangent space of linear solutions and constructs nonlinear solutions with arbitrary radiation fields.

Findings

Small non radiative solutions form a manifold.

The tangent space is given by linear non radiative solutions.

Constructed solutions with arbitrary radiation fields.

Abstract

Non radiative solutions of the energy critical non linear wave equation are global solutions $u$ that furthermore have vanishing asymptotic energy outside the lightcone at both $t \to \pm \infty$:\[ \lim_{t \to \pm \infty} \| \nabla_{t,x} u(t) \|_{L^2(|x| \ge |t|+R)} = 0, \]for some $R \> 0$. They were shown to play an important role in the analysis of long time dynamics of solutions, in particular regarding the soliton resolution: we refer to the seminal works of Duyckaerts, Kenig and Merle, see \cite{DKM:23} and the references therein.We show that the set of non radiative solutions which are small in the energy space is a manifold whose tangent space at $0$ is given by non radiative solutions to the linear equation (described in \cite{CL24}). We also construct nonlinear solutions with an arbitrary prescribed radiation field.