On the analytic Birkhoff normal form of the Benjamin–Ono equation and applications. (March 2022)

• Record Type: Journal Article
• Authors: Gérard, Patrick; Kappeler, Thomas; Topalov, Petar
• Is Part Of: Nonlinear analysis. Volume 216(2022)
• Publication Date: 2022-03
• Subjects: primary 37K15 -- secondary 47B35; Benjamin–Ono equation -- Analytic Birkhoff normal form -- Well-posedness -- Solution map -- Nowhere locally uniformly continuous maps; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112687
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we prove that the Benjamin–Ono equation admits an analytic Birkhoff normal form in an open neighborhood of zero in H 0 s ( T, R ) for any s > − 1 / 2 where H 0 s ( T, R ) denotes the subspace of the Sobolev space H s ( T, R ) of elements with mean 0. As an application we show that for any − 1 / 2 < s < 0, the flow map of the Benjamin–Ono equation S 0 t : H 0 s ( T, R ) → H 0 s ( T, R ) is nowhere locally uniformly continuous in a neighborhood of zero in H 0 s ( T, R ) .