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

TL;DR

This work characterizes the set of non radiative, energy-critical NLW solutions by showing that near zero the nonlinear radiative data form a $C^1$-submanifold $P(R)$ of the energy space with tangent space $P_L(R)$, and provides a local chart mapping that preserves the linear projection onto $P_L(R)$. It proves a wave-operator type result: for any prescribed radiation field $F$ in $L^2$, there is a unique global nonlinear solution whose radiation matches $F$, with an explicit inversion of the radiation map via the operator $\mathcal{T}$ tied to the Radon transform. The results extend the linear description of radiative data to the nonlinear setting and give a constructive framework for building nonlinear radiative data, advancing the understanding of long-time dynamics and potential soliton-resolution mechanisms.

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.

Theorems & Definitions (14)

• Proposition 1.1: Radiation field and concentration of energy on the light cone, CL24
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Lemma 2.2: CL24
• Corollary 2.3
• proof
• Proposition 2.4
• proof
• proof : Proof of Theorem \ref{['thm:1']}