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

Nous étudions dans cette Note les équations de Zakharov–Kuznetsov 2D généralisées tu+Δxu+ukxu=0 pour k2. Il est établi que le problème de Cauchy peut être résolu par une méthode itérative dans les espaces de Sobolev Hs(R2) pour s>1/4 si k=2, s>5/12 si k=3 et s>12/k si k4.

In this Note we study the generalized 2D Zakharov–Kuznetsov equations tu+Δxu+ukxu=0 for k2. By an iterative method we prove the local well-posedness of these equations in the Sobolev spaces Hs(R2) for s>1/4 if k=2, s>5/12 if k=3 and s>12/k if k4.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2012.05.007

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
@article{CRMATH_2012__350_9-10_499_0,
     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},
     publisher = {Elsevier},
     volume = {350},
     number = {9-10},
     year = {2012},
     doi = {10.1016/j.crma.2012.05.007},
     language = {en},
}
TY  - JOUR
AU  - Francis Ribaud
AU  - Stéphane Vento
TI  - A Note on the Cauchy problem for the 2D generalized Zakharov–Kuznetsov equations
JO  - Comptes Rendus. Mathématique
PY  - 2012
SP  - 499
EP  - 503
VL  - 350
IS  - 9-10
PB  - Elsevier
DO  - 10.1016/j.crma.2012.05.007
LA  - en
ID  - CRMATH_2012__350_9-10_499_0
ER  - 
%0 Journal Article
%A Francis Ribaud
%A Stéphane Vento
%T A Note on the Cauchy problem for the 2D generalized Zakharov–Kuznetsov equations
%J Comptes Rendus. Mathématique
%D 2012
%P 499-503
%V 350
%N 9-10
%I Elsevier
%R 10.1016/j.crma.2012.05.007
%G en
%F CRMATH_2012__350_9-10_499_0
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/

[1] A.V. Faminskii The Cauchy problem for the Zakharov–Kuznetsov equation, Differential Equations, Volume 31 (1995) no. 6, pp. 1002-1012

[2] L.G. Farah; F. Linares; A. Pastor A note on the 2D generalized Zakharov–Kuznetsov equation: local, global, and scattering results, 2011 | arXiv

[3] C.E. Kenig; G. Ponce; L. Vega Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle, Comm. Pure Appl. Math., Volume 46 (1993) no. 4, pp. 527-620

[4] F. Linares; A. Pastor Well-posedness for the two-dimensional modified Zakharov–Kuznetsov equation, SIAM J. Math. Anal., Volume 41 (2009) no. 4, pp. 1323-1339

[5] F. Linares; A. Pastor Local and global well-posedness for the 2D generalized Zakharov–Kuznetsov equation, J. Funct. Anal., Volume 260 (2011) no. 4, pp. 1060-1085

[6] F. Ribaud; S. Vento Well-posedness results for the 3D Zakharov–Kuznetsov equation (preprint) | arXiv

[7] S. Vento Well-posedness for the generalized Benjamin–Ono equations with arbitrary large initial data in the critical space, Int. Math. Res. Not. IMRN (2) (2010), pp. 297-319

[8] V.E. Zakharov; E.A. Kuznetsov On three dimensional solitons, Sov. Phys. JETP, Volume 39 (1974), pp. 285-286

Cité par Sources :

Commentaires - Politique