Strichartz estimates for geophysical fluid equations using Fourier restriction theory

Corentin Gentil; Côme Tabary

doi:10.5802/crmath.618

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Strichartz estimates for geophysical fluid equations using Fourier restriction theory
[Estimées de Strichartz pour les équations qui décrivent les fluides géophysiques à l’aide de la théorie de la restriction de Fourier]

Corentin Gentil ¹ ; Côme Tabary ¹

¹ Département de Mathématiques et Applications, ENS-PSL, 45 rue d’Ulm 75005 Paris, France

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1155-1181.

Résumé
Abstract

Nous prouvons des estimées de Strichartz pour les semi-groupes associés aux fluides géophysiques non visqueux stratifiés et/ou en rotation en utilisant la théorie de restriction de Fourier. Nous prouvons de nouveaux résultats pour les fluides stratifiés en rotation, et retrouvons des résultats de Koh, Lee, Takada, 2014 pour des fluides en rotation uniquement, et de Lee, Takada, 2017 pour des fluides stratifiés uniquement. Nos estimées de restriction sont obtenues par la méthode de découpage en tranches (Nicola 2009), qui repose sur le théorème bien connu de Tomas–Stein pour les sphères en dimension 2. A notre connaissance, une telle méthode n’a jamais été utilisée dans ce cadre. De plus, lorsque le fluide est stratifié, notre approche donne des estimées optimales, montrant que la méthode de découpage capture toute la courbure disponible des surfaces étudiées.

We prove Strichartz estimates for the semigroups associated to stratified and/or rotating inviscid geophysical fluids using Fourier restriction theory. We prove new results for rotating stratified fluids, and recover results from Koh, Lee, Takada, 2014 for rotation only, and from Lee, Takada, 2017 for stratification only. Our restriction estimates are obtained by the slicing method (Nicola 2009), which relies on the well-known Tomas–Stein theorem for 2-dimensional spheres. To our knowledge, such a method has never been used in this setting. Moreover, when the fluid is stratified, our approach yields sharp estimates, showing that the slicing method captures all the available curvature of the surfaces of interest.

Reçu le : 2023-09-20
Révisé le : 2024-01-15
Accepté le : 2024-01-23
Publié le : 2024-11-05

DOI : 10.5802/crmath.618

Classification : 35Q86
Keywords: Geophysical fluids, Fourier restriction theorems, Strichartz estimates
Mot clés : Fluides géophysiques, théorèmes de restriction de Fourier, estimées de Strichartz

Affiliations des auteurs :

Corentin Gentil ¹ ; Côme Tabary ¹

¹ Département de Mathématiques et Applications, ENS-PSL, 45 rue d’Ulm 75005 Paris, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Corentin Gentil and C\^ome Tabary},
     title = {Strichartz estimates for geophysical fluid equations using {Fourier} restriction theory},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1155--1181},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.618},
     language = {en},
}

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Corentin Gentil; Côme Tabary. Strichartz estimates for geophysical fluid equations using Fourier restriction theory. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1155-1181. doi : 10.5802/crmath.618. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.618/

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