On the Cauchy problem for the generalized Benjamin–Ono equation with small initial data

Luc Molinet; Francis Ribaud

doi:10.1016/j.crma.2003.09.012

Partial Differential Equations

On the Cauchy problem for the generalized Benjamin–Ono equation with small initial data

Presented by: Yves Meyer

Luc Molinet ¹; Francis Ribaud ²

¹ L.A.G.A., Institut Galilée, Université Paris-Nord, 93430 Villetaneuse, France
² Université de Marne-La-Vallée, équipe d'analyse et de mathématiques appliquées, 5, bd. Descartes, Champs-sur-Marne, 77454 Marne-La-Vallée cedex 2, France

Comptes Rendus. Mathématique, Volume 337 (2003) no. 8, pp. 523-526.

Abstract
Résumé

We prove global well-posedness results for small initial data in $H^{s} (ℝ), s > s_{k}$ , and in ${\dot{ℬ}}_{2}^{s_{k}, 1} (ℝ)$ , s_k=1/2−1/k, for the generalized Benjamin–Ono equation $\partial_{t} u + ℋ \partial_{x}^{2} u + \partial_{x} (u^{k + 1}) = 0, k ⩾ 4$ . We also consider the cases k=2,3.

Nous montrons que l'équation de Benjamin–Ono généralisée $\partial_{t} u + ℋ \partial_{x}^{2} u + \partial_{x} (u^{k + 1}) = 0, k ⩾ 4$ , est globalement bien posée dans $H^{s} (ℝ), s > s_{k}$ , et dans ${\dot{ℬ}}_{2}^{s_{k}, 1} (ℝ), s_{k} = 1 / 2 - 1 / k$ , pour les données petites. Nous considérons également les cas k=2,3.

Received: 2003-06-04
Accepted: 2003-09-15
Published online: 2003-10-15

DOI: 10.1016/j.crma.2003.09.012

Author's affiliations:

Luc Molinet ¹; Francis Ribaud ²

¹ L.A.G.A., Institut Galilée, Université Paris-Nord, 93430 Villetaneuse, France
² Université de Marne-La-Vallée, équipe d'analyse et de mathématiques appliquées, 5, bd. Descartes, Champs-sur-Marne, 77454 Marne-La-Vallée cedex 2, France

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     author = {Luc Molinet and Francis Ribaud},
     title = {On the {Cauchy} problem for the generalized {Benjamin{\textendash}Ono} equation with small initial data},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {523--526},
     publisher = {Elsevier},
     volume = {337},
     number = {8},
     year = {2003},
     doi = {10.1016/j.crma.2003.09.012},
     language = {en},
}

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Luc Molinet; Francis Ribaud. On the Cauchy problem for the generalized Benjamin–Ono equation with small initial data. Comptes Rendus. Mathématique, Volume 337 (2003) no. 8, pp. 523-526. doi : 10.1016/j.crma.2003.09.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.09.012/

References
Cited by

[1] H.A. Biagioni; F. Linares Ill-posedness for the derivative Schrödinger and generalized Benjamin–Ono equations, Trans. Amer. Math. Soc., Volume 353 (2001), pp. 3649-3659

[2] C.E. Kenig; G. Ponce; L. Vega On the generalized Benjamin–Ono equations, Trans. Amer. Math. Soc., Volume 342 (1994), pp. 155-172

[3] L. Molinet, F. Ribaud, On the generalized Benjamin–Ono equation with small initial data, Preprint

[4] L. Molinet, F. Ribaud, On the Cauchy problem for the generalized Korteweg–de Vries equation, Comm. Partial Differential Equations, in press

[5] F. Planchon Self-similar solutions and semi-linear wave equations in Besov spaces, J. Math. Pures Appl., Volume 79 (2000), pp. 809-820

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