On the analytic Birkhoff normal form of the Benjamin-Ono equation and applications. (English) Zbl 1521.37085
Summary: In this paper we prove that the Benjamin-Ono equation admits an analytic Birkhoff normal form in an open neighborhood of zero in \(H_0^s ( \mathbb{T} , \mathbb{R} )\) for any \(s > - 1 / 2\) where \(H_0^s ( \mathbb{T} , \mathbb{R} )\) denotes the subspace of the Sobolev space \(H^s ( \mathbb{T} , \mathbb{R} )\) 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 \(\mathcal{S}_0^t : H_0^s ( \mathbb{T} , \mathbb{R} ) \to H_0^s ( \mathbb{T} , \mathbb{R} )\) is nowhere locally uniformly continuous in a neighborhood of zero in \(H_0^s ( \mathbb{T} , \mathbb{R} )\).
MSC:
| 37K55 | Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems |
| 35Q51 | Soliton equations |
Keywords:
Benjamin-Ono equation; analytic Birkhoff normal form; well-posedness; solution map; nowhere locally uniformly continuous mapsReferences:
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