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

TL;DR

This paper establishes the existence of an analytic Birkhoff normal form for the Benjamin-Ono equation near zero in certain Sobolev spaces, and demonstrates the flow map's lack of local uniform continuity in these spaces.

Contribution

It proves the Benjamin-Ono equation admits an analytic Birkhoff normal form near zero in Sobolev spaces with s > -1/2, and analyzes the flow map properties.

Findings

Existence of an analytic Birkhoff normal form near zero.

Flow map is nowhere locally uniformly continuous for -1/2 < s < 0.

Applicable to Sobolev spaces with mean zero.

Abstract

In this paper we prove that the Benjamin-Ono equation admits an analytic Birkhoff normal form in an open neighborhood of zero in $H^{s}_{0}(\T, \R)$ for any $s>-1/2$ where $H^{s}_{0}(\T, \R)$ 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 $\mathcal{S}_0^t : H^{s}_{0}(\T, \R)\to H^{s}_{0}(\T, \R)$ is nowhere locally uniformly continuous in a neighborhood of zero in $H^{s}_{0}(\T, \R)$.