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

2024/06/21 by Raphaël Côte, Côte, Raphaël, Camille Laurent +1
Earth and Planetary Sciences · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Ocean Waves and Remote Sensing

paper · pdf · doi:10.48550/arxiv.2406.14932

openalex publication_date 2024/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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 → ± ∞:lim_{t → ± ∞} ‖ ∇_t,x u(t) ‖_{L²(|x| ≥ |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 \citeDKM: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 \citeCL24). We also construct nonlinear solutions with an arbitrary prescribed radiation field.

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