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
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

@article{CRMATH_2019__357_2_115_0,
     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},
}

TY  - JOUR
AU  - Paul Acevedo
AU  - Chérif Amrouche
AU  - Carlos Conca
AU  - Amrita Ghosh
TI  - Stokes and Navier–Stokes equations with Navier boundary condition
JO  - Comptes Rendus. Mathématique
PY  - 2019
SP  - 115
EP  - 119
VL  - 357
IS  - 2
PB  - Elsevier
DO  - 10.1016/j.crma.2018.12.002
LA  - en
ID  - CRMATH_2019__357_2_115_0
ER  -

%0 Journal Article
%A Paul Acevedo
%A Chérif Amrouche
%A Carlos Conca
%A Amrita Ghosh
%T Stokes and Navier–Stokes equations with Navier boundary condition
%J Comptes Rendus. Mathématique
%D 2019
%P 115-119
%V 357
%N 2
%I Elsevier
%R 10.1016/j.crma.2018.12.002
%G en
%F CRMATH_2019__357_2_115_0

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/

References
Cited by

[1] C. Amrouche; A. Rejaiba $L^{p}$ -theory for Stokes and Navier–Stokes equations with Navier boundary condition, J. Differ. Equ., Volume 256 (2014) no. 4, pp. 1515-1547

[2] C. Amrouche; N. Seloula $L^{p}$ -theory for vector potentials and Sobolev's inequalities for vector fields: application to the Stokes equations with pressure boundary conditions, Math. Models Methods Appl. Sci., Volume 23 (2013) no. 1, pp. 37-92

[3] H. Beirão Da Veiga Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions, Adv. Differ. Equ., Volume 9 (2004) no. 9–10, pp. 1079-1114

[4] L.C. Berselli An elementary approach to the 3D Navier–Stokes equations with Navier boundary conditions: existence and uniqueness of various classes of solutions in the flat boundary case, Discrete Contin. Dyn. Syst., Ser. S, Volume 3 (2010) no. 2, pp. 199-219

[5] C. Conca On the application of the homogenization theory to a class of problems arising in fluid mechanics, J. Math. Pures Appl., Volume 64 (1985) no. 1, pp. 31-75

[6] M. Giaquinta; G. Modica Nonlinear systems of the type of the stationary Navier–Stokes system, J. Reine Angew. Math., Volume 330 (1982), pp. 173-214

[7] D. Medková One problem of the Navier type for the Stokes system in planar domains, J. Differ. Equ., Volume 261 (2016) no. 10, pp. 5670-5689

[8] C.L.M.H. Navier Mémoire sur les lois du mouvement des fluides, Mém. Acad. Sci. Inst. Fr. (2) (1823), pp. 389-440

[9] D. Serre Équations de Navier–Stokes stationnaires avec données peu régulières, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 10 (1983) no. 4, pp. 543-559

[10] V.A. Solonnikov; V.E. Ščadilov A certain boundary value problem for the stationary system of Navier–Stokes equations, Tr. Mat. Inst. Steklova, Volume 125 (1973), pp. 196-210 (235. Boundary value problems of mathematical physics, 8)

Cited by Sources:

Comments - Policy