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Comptes Rendus Mathématique
Volume 357, n° 2
pages 115-119 (février 2019)
Doi : 10.1016/j.crma.2018.12.002
Received : 18 June 2018 ;  accepted : 7 December 2018
Stokes and Navier–Stokes equations with Navier boundary condition
Équations de Stokes et de Navier–Stokes avec la condition de Navier

Paul Acevedo a , Chérif Amrouche b , Carlos Conca c , Amrita Ghosh b, d
a Escuela Politécnica Nacional, Departamento de Matemática, Facultad de Ciencias, Ladrón de Guevara E11-253, P.O. Box 17-01-2759, Quito, Ecuador 
b LMAP, UMR CNRS 5142, Bâtiment IPRA, avenue de l'Université, BP 1155, 64013 Pau cedex, France 
c Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Santiago, Chile 
d Departamento de Matemáticas, Facultad de Ciencias y Tecnología, Universidad del País Vasco, Barrio Sarriena s/n, 48940 Lejona, Vizcaya, Spain 


In this paper, we study the stationary Stokes and Navier–Stokes equations with non-homogeneous Navier boundary condition in a bounded domain   of class   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   (and  ) to the linear problem for all   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  . Finally, we discuss the same questions for the non-linear system.

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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   de classe  , 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   (et  ) pour tout   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  . Finalement, nous étudions les mêmes questions pour le système non linéaire.

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