Birkhoff normal forms for Hamiltonian PDEs in their energy space

Joackim Bernier; Benoît Grébert

doi:10.5802/jep.193

Birkhoff normal forms for Hamiltonian PDEs in their energy space
[Formes normales de Birkhoff pour les EDP hamiltoniennes dans l’espace d’énergie]

Joackim Bernier ¹ ; Benoît Grébert ¹

¹ Nantes Université, CNRS, Laboratoire de Mathématiques Jean Leray, LMJL F-44000 Nantes, France

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 681-745.

Résumé
Abstract

On étudie le comportement en temps long des petites solutions d’équations dispersives hamiltoniennes semi-linéaires sur des domaines bornés. Si le système satisfait à une nouvelle condition de non-résonance et à une estimée d’énergie suffisamment forte, on prouve que ses basses super-actions sont quasiment préservées pendant des temps très longs. En d’autres termes cela signifie que, pour échanger de l’énergie, les modes doivent osciller à la même fréquence. La nouveauté de ce résultat est que l’on n’a pas à supposer que les solutions sont particulièrement régulières. Il suffit qu’elles soient dans l’espace d’énergie. On applique notre résultat aux équations de Klein-Gordon en dimension $d = 1$ ainsi qu’aux équations de Schrödinger non linéaires en dimension $d \leq 2$ .

We study the long time behavior of small solutions of semi-linear dispersive Hamiltonian partial differential equations on confined domains. Provided that the system enjoys a new non-resonance condition and a sufficiently strong energy estimate, we prove that its low super-actions are almost preserved for very long times. Roughly speaking, it means that only modes with the same linear frequency will be able to exchange energy in a reasonable time. Contrary to the previous existing results, we do not require the solutions to be particularly regular. They only have to live in the energy space. We apply our result to nonlinear Klein-Gordon equations in dimension $d = 1$ and nonlinear Schrödinger equations in dimension $d \leq 2$ .

Reçu le : 2021-03-12
Accepté le : 2022-03-21
Publié le : 2022-04-04

DOI : 10.5802/jep.193

Classification : 35Q55, 81Q05, 37K55
Keywords: Birkhoff normal forms, dispersive equations, low regularity, Hamiltonian PDE, Sturm–Liouville
Mot clés : Formes normales de Birkhoff, équations dispersives, faible régularité, EDP hamiltonienne, Sturm–Liouville

Affiliations des auteurs :

Joackim Bernier ¹ ; Benoît Grébert ¹

¹ Nantes Université, CNRS, Laboratoire de Mathématiques Jean Leray, LMJL F-44000 Nantes, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Joackim Bernier and Beno{\^\i}t Gr\'ebert},
     title = {Birkhoff normal forms for {Hamiltonian} {PDEs} in their energy space},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {681--745},
     publisher = {\'Ecole polytechnique},
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Joackim Bernier; Benoît Grébert. Birkhoff normal forms for Hamiltonian PDEs in their energy space. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 681-745. doi : 10.5802/jep.193. https://jep.centre-mersenne.org/articles/10.5802/jep.193/

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