The zero dispersion limit for the Benjamin–Ono equation on the line
We identify the zero dispersion limit of a solution of the Benjamin–Ono equation on the line corresponding to every initial datum in $L^2(\mathbb{R})\cap L^\infty (\mathbb{R})$. We infer a maximum principle and a local smoothing property for this limit. The proof is based on an explicit formula for...
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| Language: | English |
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Académie des sciences
2024-07-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.575/ |
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| author | Gérard, Patrick |
| author_facet | Gérard, Patrick |
| author_sort | Gérard, Patrick |
| collection | DOAJ |
| description | We identify the zero dispersion limit of a solution of the Benjamin–Ono equation on the line corresponding to every initial datum in $L^2(\mathbb{R})\cap L^\infty (\mathbb{R})$. 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. |
| format | Article |
| id | doaj-art-e346739a7704496f8f8a2c9fb9152113 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-07-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-e346739a7704496f8f8a2c9fb91521132025-02-07T11:21:51ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-07-01362G661963410.5802/crmath.57510.5802/crmath.575The zero dispersion limit for the Benjamin–Ono equation on the lineGérard, Patrick0Université Paris–Saclay, Laboratoire de Mathématiques d’Orsay, CNRS, UMR 8628, 91405 Orsay, FranceWe identify the zero dispersion limit of a solution of the Benjamin–Ono equation on the line corresponding to every initial datum in $L^2(\mathbb{R})\cap L^\infty (\mathbb{R})$. 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.575/ |
| spellingShingle | Gérard, Patrick The zero dispersion limit for the Benjamin–Ono equation on the line Comptes Rendus. Mathématique |
| title | The zero dispersion limit for the Benjamin–Ono equation on the line |
| title_full | The zero dispersion limit for the Benjamin–Ono equation on the line |
| title_fullStr | The zero dispersion limit for the Benjamin–Ono equation on the line |
| title_full_unstemmed | The zero dispersion limit for the Benjamin–Ono equation on the line |
| title_short | The zero dispersion limit for the Benjamin–Ono equation on the line |
| title_sort | zero dispersion limit for the benjamin ono equation on the line |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.575/ |
| work_keys_str_mv | AT gerardpatrick thezerodispersionlimitforthebenjaminonoequationontheline AT gerardpatrick zerodispersionlimitforthebenjaminonoequationontheline |