Stokes and Navier–Stokes equations with Navier boundary condition

Paul Acevedo; Chérif Amrouche; Carlos Conca; Amrita Ghosh

doi:10.1016/j.crma.2018.12.002

Mathematical analysis/Partial differential equations

Stokes and Navier–Stokes equations with Navier boundary condition

Presented by: the Editorial Board

Paul Acevedo ¹; Chérif Amrouche ²; Carlos Conca ³; Amrita Ghosh ^2,⁴

¹ Escuela Politécnica Nacional, Departamento de Matemática, Facultad de Ciencias, Ladrón de Guevara E11-253, P.O. Box 17-01-2759, Quito, Ecuador
² LMAP, UMR CNRS 5142, Bâtiment IPRA, avenue de l'Université, BP 1155, 64013 Pau cedex, France
³ Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Santiago, Chile
⁴ Departamento de Matemáticas, Facultad de Ciencias y Tecnología, Universidad del País Vasco, Barrio Sarriena s/n, 48940 Lejona, Vizcaya, Spain

Comptes Rendus. Mathématique, Volume 357 (2019) no. 2, pp. 115-119.

Abstract (Original language)
Résumé

In this paper, we study the stationary Stokes and Navier–Stokes equations with non-homogeneous Navier boundary condition in a bounded domain $Ω \subset R^{3}$ of class $C^{1, 1}$ from the viewpoint of the behavior of solutions with respect to the friction coefficient α. We first prove the existence of a unique weak solution (and strong) in $W^{1, p} (Ω)$ (and $W^{2, p} (Ω)$ ) to the linear problem for all $1 < p < \infty$ considering minimal regularity of α, using some inf–sup condition concerning the rotational operator. Furthermore, we deduce uniform estimates of the solutions for large α, which enables us to obtain the strong convergence of Stokes solutions with Navier slip boundary condition to the one with no-slip boundary condition as $α \to \infty$ . Finally, we discuss the same questions for the non-linear system.

Dans cette note, nous étudions les équations stationnaires de Stokes et de Navier–Stokes avec une condition aux limites non homogène de Navier dans un domaine borné $Ω \subset R^{3}$ de classe $C^{1, 1}$ , et envisageons le comportement des solutions par rapport au coefficient de friction α. Nous prouvons, d'abord dans le cas linéaire, l'existence d'une solution faible (et d'une solution forte) unique dans $W^{1, p} (Ω)$ (et $W^{2, p} (Ω)$ ) pour tout $1 < p < \infty$ en supposant α le moins régulier possible et en utilisant une condition inf–sup concernant l'opérateur rotationnel. De plus, nous déduisons des estimations uniformes des solutions pour α grand, qui nous permettent d'obtenir la convergence forte des solutions de Stokes avec la condition de glissement vers les solutions vérifiant la condition d'adhérence lorsque $α \to \infty$ . Finalement, nous étudions les mêmes questions pour le système non linéaire.

Received: 2018-06-18
Accepted: 2018-12-07
Published online: 2018-12-19

DOI: 10.1016/j.crma.2018.12.002

Author's affiliations:

Paul Acevedo ¹; Chérif Amrouche ²; Carlos Conca ³; Amrita Ghosh ^{2, 4}

¹ Escuela Politécnica Nacional, Departamento de Matemática, Facultad de Ciencias, Ladrón de Guevara E11-253, P.O. Box 17-01-2759, Quito, Ecuador
² LMAP, UMR CNRS 5142, Bâtiment IPRA, avenue de l'Université, BP 1155, 64013 Pau cedex, France
³ Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Santiago, Chile
⁴ Departamento de Matemáticas, Facultad de Ciencias y Tecnología, Universidad del País Vasco, Barrio Sarriena s/n, 48940 Lejona, Vizcaya, Spain

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     author = {Paul Acevedo and Ch\'erif Amrouche and Carlos Conca and Amrita Ghosh},
     title = {Stokes and {Navier{\textendash}Stokes} equations with {Navier} boundary condition},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {115--119},
     publisher = {Elsevier},
     volume = {357},
     number = {2},
     year = {2019},
     doi = {10.1016/j.crma.2018.12.002},
     language = {en},
}

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Paul Acevedo; Chérif Amrouche; Carlos Conca; Amrita Ghosh. Stokes and Navier–Stokes equations with Navier boundary condition. Comptes Rendus. Mathématique, Volume 357 (2019) no. 2, pp. 115-119. doi : 10.1016/j.crma.2018.12.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.12.002/

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