On the analytic Birkhoff normal form of the Benjamin–Ono equation and applications

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).