Dynamics of nonlinear Klein-Gordon equations in low regularity on S^2 - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Annales de l'I.H.P. (C) - Analyse non linéaire Année : 2022

Dynamics of nonlinear Klein-Gordon equations in low regularity on S^2

Résumé

We describe the long time behavior of small non-smooth solutions to the nonlinear Klein-Gordon equations on the sphere S^2. More precisely, we prove that the low harmonic energies (also called super-actions) are almost preserved for times of order ε^−r , where r >> 1 is an arbitrarily large number and ε << 1 is the norm of the initial datum in the energy space H^1 × L^2. Roughly speaking, it means that, in order to exchange energy, modes have to oscillate at the same frequency. The proof relies on new multilinear estimates on Hamiltonian vector fields to put the system in Birkhoff normal form. They are derived from new probabilistic bounds on products of Laplace eigenfunctions that we obtain using Levy's concentration inequality.
Fichier principal
Vignette du fichier
KGonS2.pdf (508.33 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-03334431 , version 1 (03-09-2021)

Identifiants

Citer

Joackim Bernier, Benoît Grébert, Gabriel Rivière. Dynamics of nonlinear Klein-Gordon equations in low regularity on S^2. Annales de l'I.H.P. (C) - Analyse non linéaire, 2022, 40 (5), ⟨10.4171/AIHPC/55⟩. ⟨hal-03334431⟩
68 Consultations
73 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More