Well-posedness and dispersive decay  of small data solutions  for the Benjamin-Ono equation

Ifrim, Mihaela; Tataru, Daniel

doi:10.24033/asens.2388

Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation
[Problème de Cauchy pour Benjamin-Ono et décroissance dispersive presque globale]

Ifrim, Mihaela ; Tataru, Daniel

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 1, pp. 297-335.

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Résumé
Abstract

Cet article représente une première étape vers la compréhension du comportement en temps long pour l'équation de Benjamin-Ono. Tandis que ce problème est à la fois complètement intégrable et globalement bien posé en $L^{2}$ , beaucoup moins semble être connu en ce qui concerne son comportement en temps long. Nous montrons ici que pour de données petites et localisées, les solutions ont une dynamique dispersive presque globalement en temps. Un autre objectif est de revoir la théorie $L^{2}$ pour Benjamin-Ono et de fournir une approche plus simple et autonome.

This article represents a first step toward understanding the long time dynamics of solutions for the Benjamin-Ono equation. While this problem is known to be both completely integrable and globally well-posed in $L^{2}$ , much less seems to be known concerning its long time dynamics. Here, we prove that for small localized data the solutions have (nearly) dispersive dynamics almost globally in time. An additional objective is to revisit the $L^{2}$ theory for the Benjamin-Ono equation and provide a simpler, self-contained approach.

DOI : 10.24033/asens.2388

Classification : 35Q55, 37K10.
Keywords: Benjamin-Ono, dispersion, renormalization, normal forms
Mot clés : Benjamin-Ono, dispersion, renormalization, formes normales

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     author = {Ifrim, Mihaela and Tataru, Daniel},
     title = {Well-posedness and dispersive decay  of small data solutions  for the {Benjamin-Ono} equation},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {297--335},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 52},
     number = {1},
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     url = {http://www.numdam.org/articles/10.24033/asens.2388/}
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Ifrim, Mihaela; Tataru, Daniel. Well-posedness and dispersive decay  of small data solutions  for the Benjamin-Ono equation. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 1, pp. 297-335. doi : 10.24033/asens.2388. http://www.numdam.org/articles/10.24033/asens.2388/

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