Local in time strong solvability of the non-steady Navier–Stokes equations with Navier's boundary condition and the question of the inviscid limit

Jiří Neustupa; Patrick Penel

doi:10.1016/j.crma.2010.09.021

Partial Differential Equations/Mathematical Problems in Mechanics

Local in time strong solvability of the non-steady Navier–Stokes equations with Navier's boundary condition and the question of the inviscid limit
[Solutions fortes vérifiant des conditions aux limites de Navier pour les équations de Navier–Stokes non stationnaires, et la question de leur limite inviscide]

Présenté par : Gérard Iooss

Jiří Neustupa ¹ ; Patrick Penel ²

¹ Czech Technical University, Faculty of Mechanical Engineering, Department of Technical Mathematics, Karlovo nám. 13, 121 35 Praha 2, Czech Republic
² Université du Sud Toulon-Var, Département de Mathématique et Laboratoire SNC, BP 20132, 83957 La Garde cedex, France

Comptes Rendus. Mathématique, Volume 348 (2010) no. 19-20, pp. 1093-1097.

Résumé
Abstract

Dans cette Note, nous démontrons l'existence locale en temps de solutions fortes pour les équations de Navier–Stokes descriptives de fluides visqueux incompressibles, dans un domaine borné de $R^{3}$ , général et suffisamment régulier, avec des conditions aux limites non homogènes de Navier bien choisies. Ces solutions sont construites avec la même structure remarquable d'approximation de la solution du problème d'Euler que celles obtenues avec des conditions d'imperméabilité généralisées ou des conditions de type celles de Navier : structure permettant de traiter complètement la question de la limite inviscide.

In this Note, we prove the existence of strong solutions to the Navier–Stokes equations for incompressible viscous fluids in a general regular bounded domain of $R^{3}$ on a “short” time interval $(0, T_{0})$ , independent of the viscosity and of the friction between the fluid and the boundary. The solutions to the Navier–Stokes problem satisfy the inhomogeneous Navier's boundary condition and they reveal a remarkable structure of approximation of the solution to the Euler problem, which enables us to solve completely the question of the inviscid limit of the family of obtained solutions on the time interval $(0, T_{0})$ .

Reçu le : 2010-08-31
Accepté le : 2010-09-24
Publié le : 2010-10-01

DOI : 10.1016/j.crma.2010.09.021

Affiliations des auteurs :

Jiří Neustupa ¹ ; Patrick Penel ²

¹ Czech Technical University, Faculty of Mechanical Engineering, Department of Technical Mathematics, Karlovo nám. 13, 121 35 Praha 2, Czech Republic
² Université du Sud Toulon-Var, Département de Mathématique et Laboratoire SNC, BP 20132, 83957 La Garde cedex, France

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     author = {Ji\v{r}{\'\i} Neustupa and Patrick Penel},
     title = {Local in time strong solvability of the non-steady {Navier{\textendash}Stokes} equations with {Navier's} boundary condition and the question of the inviscid limit},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1093--1097},
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     number = {19-20},
     year = {2010},
     doi = {10.1016/j.crma.2010.09.021},
     language = {en},
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Jiří Neustupa; Patrick Penel. Local in time strong solvability of the non-steady Navier–Stokes equations with Navier's boundary condition and the question of the inviscid limit. Comptes Rendus. Mathématique, Volume 348 (2010) no. 19-20, pp. 1093-1097. doi : 10.1016/j.crma.2010.09.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.09.021/

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