A Note on the Cauchy problem for the 2D generalized Zakharov–Kuznetsov equations

Francis Ribaud; Stéphane Vento

doi:10.1016/j.crma.2012.05.007

Partial Differential Equations

A Note on the Cauchy problem for the 2D generalized Zakharov–Kuznetsov equations
[Une Note sur le problème de Cauchy pour les équations de Zakharov–Kuznetsov 2D généralisées]

Présenté par : the Editorial Board

Francis Ribaud ¹ ; Stéphane Vento ²

¹ Laboratoire dʼanalyse et de mathématiques appliquées, Université Paris-est, 5 boulevard Descartes, Champs-Sur-Marne, 77454 Marne-La-Vallée cedex 2, France
² Laboratoire analyse, géométrie et applications, Université Paris 13, Institut Galilée, 99, avenue J.B. Clément, 93430 Villetaneuse, France

Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 499-503.

Abstract (VO)
Résumé

In this Note we study the generalized 2D Zakharov–Kuznetsov equations $\partial_{t} u + Δ \partial_{x} u + u^{k} \partial_{x} u = 0$ for $k ⩾ 2$ . By an iterative method we prove the local well-posedness of these equations in the Sobolev spaces $H^{s} (R^{2})$ for $s > 1 / 4$ if $k = 2$ , $s > 5 / 12$ if $k = 3$ and $s > 1 - 2 / k$ if $k ⩾ 4$ .

Nous étudions dans cette Note les équations de Zakharov–Kuznetsov 2D généralisées $\partial_{t} u + Δ \partial_{x} u + u^{k} \partial_{x} u = 0$ pour $k ⩾ 2$ . Il est établi que le problème de Cauchy peut être résolu par une méthode itérative dans les espaces de Sobolev $H^{s} (R^{2})$ pour $s > 1 / 4$ si $k = 2$ , $s > 5 / 12$ si $k = 3$ et $s > 1 - 2 / k$ si $k ⩾ 4$ .

Reçu le : 2011-11-18
Accepté le : 2012-05-16
Publié le : 2012-05-01

DOI : 10.1016/j.crma.2012.05.007

Affiliations des auteurs :

Francis Ribaud ¹ ; Stéphane Vento ²

¹ Laboratoire dʼanalyse et de mathématiques appliquées, Université Paris-est, 5 boulevard Descartes, Champs-Sur-Marne, 77454 Marne-La-Vallée cedex 2, France
² Laboratoire analyse, géométrie et applications, Université Paris 13, Institut Galilée, 99, avenue J.B. Clément, 93430 Villetaneuse, France

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     author = {Francis Ribaud and St\'ephane Vento},
     title = {A {Note} on the {Cauchy} problem for the {2D} generalized {Zakharov{\textendash}Kuznetsov} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {499--503},
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     year = {2012},
     doi = {10.1016/j.crma.2012.05.007},
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Francis Ribaud; Stéphane Vento. A Note on the Cauchy problem for the 2D generalized Zakharov–Kuznetsov equations. Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 499-503. doi : 10.1016/j.crma.2012.05.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.05.007/

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