Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation

Francis Ribaud; Stéphane Vento

doi:10.3934/dcds.2017019

Article Contents

2017, Volume 37, Issue 1: 449-483. Doi: 10.3934/dcds.2017019

This issue Previous Article Zero sequence entropy and entropy dimension Next Article Monotone dynamical systems: Reflections on new advances & applications

Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation

Francis Ribaud^1, and
Stéphane Vento^2,

1.
Université Paris-Est Marne-la-Vallée, Laboratoire d'Analyse et de Mathématiques Appliquées (UMR 8050), 5 Bd Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France
2.
Université Paris 13 Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), 99, avenue Jean-Baptiste Clément, F-93 430 Villetaneuse, France

Received: December 31, 2015

Revised: August 31, 2016

Published: September 2017

The second author is partially supported by the french ANR project GEODISP

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Abstract

We show that the initial value problem associated to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation

$ u_t-D_x^α u_{x} + u_{xyy} = uu_x,\,\,\,\,\,\, (t,x,y)∈\mathbb{R}^3, 1≤ α≤ 2,$

is locally well-posed in the spaces $E^s$, $s > \frac{2}{\alpha } - \frac{3}{4}$, endowed with the norm $\|f{{\|}_{{{E}^{s}}}}=\|{{\left\langle {{\left| \xi \right|}^{\alpha }}+{{\mu }^{2}} \right\rangle }^{s}}\hat{f}{{\|}_{{{L}^{2}}({{\mathbb{R}}^{2}})}}.$ As a consequence, we get the global well-posedness in the energy space$E^{1/2}$ as soon as $α>\frac 85$. The proof is based on the approach of the short time Bourgain spaces developed by Ionescu, Kenig and Tataru [10] combined with new Strichartz estimates and a modified energy.

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Mathematics Subject Classification: Primary:35A01, 35Q53;Secondary:35Q60.

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References

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