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The zero dispersion limit for the Benjamin–Ono equation on the line
[Limite à faible dispersion pour l’équation de Benjamin–Ono sur la droite]
Patrick Gérard1
1 Université Paris–Saclay, Laboratoire de Mathématiques d’Orsay, CNRS, UMR 8628, 91405 Orsay, France
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 619-634.

Nous identifions la limite à faible dispersion d’une solution de l’équation de Benjamin-Ono sur la droite correspondant à toute donnée initiale de carré intégrable et bornée. Nous en déduisons un principe du maximum et une propriété de régularisation locale pour cette limite. La démonstration est fondée sur une formule explicite pour l’équation de Benjamin–Ono et sur la combinaison de calculs dans le cas particulier de données initiales rationnelles avec des arguments d’approximation. Nous étudions également le cas particulier d’une donnée initiale égale à la fonction caractéristique d’un intervalle de longueur finie, et démontrons que cette limite à faible dispersion ne vérifie pas la propriété de semi-groupe.

We identify the zero dispersion limit of a solution of the Benjamin–Ono equation on the line corresponding to every initial datum in L 2 (ℝ)∩L ∞ (ℝ). We infer a maximum principle and a local smoothing property for this limit. The proof is based on an explicit formula for the Benjamin–Ono equation and on the combination of calculations in the special case of rational initial data, with approximation arguments. We also investigate the special case of an initial datum equal to the characteristic function of a finite interval, and prove the lack of semigroup property for this zero dispersion limit.

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Accepté le :
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DOI : 10.5802/crmath.575
Classification : 37K15, 47B35

Patrick Gérard 1

1 Université Paris–Saclay, Laboratoire de Mathématiques d’Orsay, CNRS, UMR 8628, 91405 Orsay, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Patrick Gérard. The zero dispersion limit for the Benjamin–Ono equation on the line. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 619-634. doi : 10.5802/crmath.575. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.575/

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