On viscosity-continuous solutions of the Euler and Navier–Stokes equations with a Navier-type boundary condition

Bellout, Hamid; Neustupa, Jiří; Penel, Patrick

doi:10.1016/j.crma.2009.09.007

Partial Differential Equations/Mathematical Problems in Mechanics

On viscosity-continuous solutions of the Euler and Navier–Stokes equations with a Navier-type boundary condition
[Solutions continues en la viscosité pour les équations d'Euler ou de Navier–Stokes avec des conditions aux limites de type Navier]

Présenté par : Gérard Iooss

Bellout, Hamid ¹ ; Neustupa, Jiří ² ; Penel, Patrick ³

¹ Northern Illinois University, Department of Mathematical Sciences, De Kalb, IL 60115, USA
² Czech Technical University, Faculty of Mechanical Engineering, Department of Technical Mathematics, Karlovo nám. 13, 121 35 Praha 2, Czech Republic
³ Université du Sud Toulon-Var, département de mathématique et laboratoire SNC, BP 20132, 83957 La Garde cedex, France

Comptes Rendus. Mathématique, Tome 347 (2009) no. 19-20, pp. 1141-1146.

Résumé
Abstract

Pourvu que les données soient suffisamment régulières, il existe $T_{0} > 0$ , $ν^{*} > 0$ et ${u^{ν}}_{0 ⩽ ν < ν^{*}}$ famille unique de solutions fortes, locales en temps sur $(0, T_{0})$ et dépendant continûment de ν, pour les problèmes d'Euler ou de Navier–Stokes. Ces solutions vérifient des conditions aux limites de type celles de Navier.

Provided the initial velocity and the external body force are sufficiently smooth, there exist $T_{0} > 0$ , $ν^{*} > 0$ and a unique continuous family of strong solutions $u^{ν}$ ( $0 ⩽ ν < ν^{*}$ ) of the Euler or Navier–Stokes initial–boundary value problem on the time interval $(0, T_{0})$ . The solutions of the Navier–Stokes problem satisfy a Navier-type boundary condition.

Reçu le : 2009-06-06
Accepté le : 2009-09-09
Publié le : 2009-09-30

DOI : 10.1016/j.crma.2009.09.007

Affiliations des auteurs :

Bellout, Hamid ¹ ; Neustupa, Jiří ² ; Penel, Patrick ³

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     title = {On viscosity-continuous solutions of the {Euler} and {Navier{\textendash}Stokes} equations with a {Navier-type} boundary condition},
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Bellout, Hamid; Neustupa, Jiří; Penel, Patrick. On viscosity-continuous solutions of the Euler and Navier–Stokes equations with a Navier-type boundary condition. Comptes Rendus. Mathématique, Tome 347 (2009) no. 19-20, pp. 1141-1146. doi : 10.1016/j.crma.2009.09.007. http://www.numdam.org/articles/10.1016/j.crma.2009.09.007/

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