Comptes Rendus
Mathematical analysis/Partial differential equations
Stokes and Navier–Stokes equations with Navier boundary condition
[Équations de Stokes et de Navier–Stokes avec la condition de Navier]
Comptes Rendus. Mathématique, Volume 357 (2019) no. 2, pp. 115-119.

In this paper, we study the stationary Stokes and Navier–Stokes equations with non-homogeneous Navier boundary condition in a bounded domain ΩR3 of class C1,1 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 W1,p(Ω) (and W2,p(Ω)) to the linear problem for all 1<p< 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. 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 ΩR3 de classe C1,1, 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 W1,p(Ω) (et W2,p(Ω)) pour tout 1<p< 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. Finalement, nous étudions les mêmes questions pour le système non linéaire.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.12.002

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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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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