On the analytic Birkhoff normal form of the Benjamin–Ono equation and applications. (March 2022) - Gérard, Patrick; Kappeler, Thomas; Topalov, Petar
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 ) .