Anomalous dissipation for the d-dimensional Navier–Stokes equations

Jinlu Li; Yanghai Yu; Weipeng  Zhu

doi:10.5802/crmath.709

Article de recherche - Équations aux dérivées partielles

Anomalous dissipation for the d-dimensional Navier–Stokes equations
[Dissipation anomale pour les équations de Navier–Stokes à d dimensions]

Jinlu Li ¹ ; Yanghai Yu ² ; Weipeng Zhu ³

¹ School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou 341000, China
² School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, China
³ School of Mathematics, Foshan University, Foshan, Guangdong 528000, China

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 345-351.

Abstract (VO)
Résumé

The purpose of this paper is to study the vanishing viscosity limit for the d-dimensional Navier–Stokes equations in the whole space:

{\begin{cases} \partial_{t} u^{ε} + u^{ε} \cdot \nabla u^{ε} - ε Δ u^{ε} + \nabla p^{ε} = 0, \\ div u^{ε} = 0. \end{cases}

We aim to present a simple rigorous example of initial data which generates the corresponding solutions of the Navier–Stokes equations for which the dissipation rate of the kinetic energy is bounded away from zero.

L’objectif de cet article est d’étudier la limite de disparition de la viscosité pour les équations de Navier–Stokes à d dimensions dans l’espace entier :

{\begin{cases} \partial_{t} u^{ε} + u^{ε} \cdot \nabla u^{ε} - ε Δ u^{ε} + \nabla p^{ε} = 0, \\ div u^{ε} = 0. \end{cases}

Nous visons à présenter des exemples simples et rigoureux de données initiales qui génèrent les solutions correspondantes des équations de Navier–Stokes pour lesquelles le taux de dissipation de l’énergie cinétique est borné loin de zéro.

Reçu le : 2024-05-29
Révisé le : 2024-11-07
Accepté le : 2024-12-05
Publié le : 2025-05-20

DOI : 10.5802/crmath.709

Classification : 35Q35, 76D09
Keywords: Navier–Stokes equations, inviscid limit
Mots-clés : Équations de Navier–Stokes, limite inviscide

Affiliations des auteurs :

Jinlu Li ¹ ; Yanghai Yu ² ; Weipeng Zhu ³

¹ School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou 341000, China
² School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, China
³ School of Mathematics, Foshan University, Foshan, Guangdong 528000, China

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2025__363_G4_345_0,
     author = {Jinlu Li and Yanghai Yu and Weipeng  Zhu},
     title = {Anomalous dissipation for the d-dimensional {Navier{\textendash}Stokes} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {345--351},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.709},
     language = {en},
}

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Jinlu Li; Yanghai Yu; Weipeng  Zhu. Anomalous dissipation for the d-dimensional Navier–Stokes equations. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 345-351. doi : 10.5802/crmath.709. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.709/

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