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arXiv · 2103.07981

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

P. Gérard·T. Kappeler·P. Topalov
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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 $H^{s}_{0}(\T, \R)$.

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P. Gérard, T. Kappeler, P. Topalov. 2021-03-14. On the analytic Birkhoff normal form of the Benjamin-Ono equation and applications. https://arxiv.org/abs/2103.07981

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