Local well-posedness and blow-up in the energy space for a class of L 2 critical dispersion generalized Benjamin–Ono equations
Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 6, pp. 853-887.

We consider a family of dispersion generalized Benjamin–Ono equations (dgBO)

u t - x |D| α u+|u| 2α x u=0,(t,x)×,
where |D| α u ˆ=|ξ| α u ˆ and 1α2. These equations are critical with respect to the L 2 norm and global existence and interpolate between the modified BO equation (α=1) and the critical gKdV equation (α=2).First, we prove local well-posedness in the energy space for 1<α<2, extending results in Kenig et al. (1991, 1993) [13,14] for the generalized KdV equations.Second, we address the blow-up problem in the spirit of Martel and Merle (2000) [19] and Merle (2001) [22] concerning the critical gKdV equation, by studying rigidity properties of the dgBO flow in a neighborhood of the solitons. We prove that for α close to 2, solutions of negative energy close to solitons blow up in finite or infinite time in the energy space H α 2 .The blow-up proof requires both extensions to dgBO of monotonicity results for local L 2 norms by pseudo-differential operator tools and perturbative arguments close to the gKdV case to obtain structural properties of the linearized flow around solitons.

Nous considérons une famille dʼéquations de Benjamin–Ono à dispersion généralisée (dgBO)

u t - x |D| α u+|u| 2α x u=0,(t,x)×,
|D| α u ˆ=|ξ| α u ˆ et 1α2. Ces équations sont critiques par rapport à la norme L 2 et à lʼexistence globale et peuvent être vues comme des interpolations entre lʼéquation de Benjamin–Ono généralisée critique (α=1) et lʼéquation de Korteweg–de Vries généralisée critique (α=2).Dʼabord, nous montrons le caractère bien posé de ces équations dans lʼespace dʼénergie pour 1<α<2, étendant les résultats de Kenig et al. (1991, 1993) [13,14] pour les équations de Korteweg–de Vries généralisées.Ensuite, nous étudions le phénomène dʼexplosion dans lʼesprit de Martel et Merle (2000) [19] et Merle (2001) [22] concernant lʼéquation de gKdV critique, en étudiant les propriétés de rigidité du flot de dgBO dans un voisinage des solitons. Nous montrons que pour α proche de 2, les solutions dʼénergie négative proches des solitons explosent en temps fini ou infini dans lʼespace dʼénergie H α 2 .La preuve de ce résultat dʼexplosion est basée dʼune part sur lʼadaptation à dgBO de résultats de monotonie de normes L 2 locales par des méthodes dʼopérateurs pseudo-differentiels et dʼautre part sur des arguments de perturbation pour obtenir des propriétés structurelles du flot linéarisé autour des solitons lorsque lʼéquation est proche de gKdV.

@article{AIHPC_2011__28_6_853_0,
     author = {Kenig, C.E. and Martel, Y. and Robbiano, L.},
     title = {Local well-posedness and blow-up in the energy space for a class of $ {L}^{2}$ critical dispersion generalized {Benjamin{\textendash}Ono} equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {853--887},
     publisher = {Elsevier},
     volume = {28},
     number = {6},
     year = {2011},
     doi = {10.1016/j.anihpc.2011.06.005},
     zbl = {1230.35102},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2011.06.005/}
}
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Kenig, C.E.; Martel, Y.; Robbiano, L. Local well-posedness and blow-up in the energy space for a class of $ {L}^{2}$ critical dispersion generalized Benjamin–Ono equations. Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 6, pp. 853-887. doi : 10.1016/j.anihpc.2011.06.005. http://www.numdam.org/articles/10.1016/j.anihpc.2011.06.005/

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