Nous étudions dans cette Note les équations de Zakharov–Kuznetsov 2D généralisées pour . Il est établi que le problème de Cauchy peut être résolu par une méthode itérative dans les espaces de Sobolev pour si , si et si .
In this Note we study the generalized 2D Zakharov–Kuznetsov equations for . By an iterative method we prove the local well-posedness of these equations in the Sobolev spaces for if , if and if .
Accepté le :
Publié le :
@article{CRMATH_2012__350_9-10_499_0, author = {Ribaud, Francis and Vento, St\'ephane}, 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}, url = {http://www.numdam.org/articles/10.1016/j.crma.2012.05.007/} }
TY - JOUR AU - Ribaud, Francis AU - Vento, Stéphane 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 UR - http://www.numdam.org/articles/10.1016/j.crma.2012.05.007/ DO - 10.1016/j.crma.2012.05.007 LA - en ID - CRMATH_2012__350_9-10_499_0 ER -
%0 Journal Article %A Ribaud, Francis %A Vento, Stéphane %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 %U http://www.numdam.org/articles/10.1016/j.crma.2012.05.007/ %R 10.1016/j.crma.2012.05.007 %G en %F CRMATH_2012__350_9-10_499_0
Ribaud, Francis; Vento, Stéphane. A Note on the Cauchy problem for the 2D generalized Zakharov–Kuznetsov equations. Comptes Rendus. Mathématique, Tome 350 (2012) no. 9-10, pp. 499-503. doi : 10.1016/j.crma.2012.05.007. http://www.numdam.org/articles/10.1016/j.crma.2012.05.007/
[1] The Cauchy problem for the Zakharov–Kuznetsov equation, Differential Equations, Volume 31 (1995) no. 6, pp. 1002-1012
[2] A note on the 2D generalized Zakharov–Kuznetsov equation: local, global, and scattering results, 2011 | arXiv
[3] 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] Well-posedness for the two-dimensional modified Zakharov–Kuznetsov equation, SIAM J. Math. Anal., Volume 41 (2009) no. 4, pp. 1323-1339
[5] Local and global well-posedness for the 2D generalized Zakharov–Kuznetsov equation, J. Funct. Anal., Volume 260 (2011) no. 4, pp. 1060-1085
[6] Well-posedness results for the 3D Zakharov–Kuznetsov equation (preprint) | arXiv
[7] 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] On three dimensional solitons, Sov. Phys. JETP, Volume 39 (1974), pp. 285-286
Cité par Sources :