A steady Navier–Stokes model for compressible fluid with partially strong solutions

Muzereau, Olivier; Neustupa, Jiří; Penel, Patrick

doi:10.1016/j.crma.2010.04.001

Mathematical Analysis/Partial Differential Equations

A steady Navier–Stokes model for compressible fluid with partially strong solutions
[Un modèle stationaire de Navier–Stokes pour fluide incompressible avec des solutions partiellement fortes]

Présenté par : Gérard Iooss

Muzereau, Olivier ¹ ; Neustupa, Jiří ² ; Penel, Patrick ¹

¹ Université du Sud Toulon-Var, département de mathématique et laboratoire SNC, BP 20132, 83957 La Garde cedex, France
² Mathematical Institute, Czech Academy of Sciences, Žitná. 25, 115 67 Praha 1, Czech Republic

Comptes Rendus. Mathématique, Tome 348 (2010) no. 11-12, pp. 619-624.

Résumé
Abstract

Dans cette Note, nous démontrons l'existence de solutions partiellement fortes pour les équations de Navier–Stokes stationnaires descriptives de fluides visqueux barotropiques compressibles, dans un domaine borné de $R^{3}$ , avec des conditions aux limites d'imperméabilité généralisée ${curl}^{k} u \cdot n = 0$ , $k = 0, 1, 2$ . Nous utilisons le libellé « solution partiellement forte » pour dire que les propriétés de régularité de la partie à divergence nulle du champ des vitesses et de la pression effective associée sont typiques d'une solution forte, tandis que les propriétés de régularité de la densité et de la partie complémentaire du champ des vitesses (gradient d'un potentiel) sont typiques d'une solution faible.

In this Note, we prove the existence of a partially strong solution to the steady Navier–Stokes equations for viscous barotropic compressible fluids, in a bounded simply connected domain of $R^{3}$ with the prescribed generalized impermeability conditions ${curl}^{k} u \cdot n = 0$ , $k = 0, 1, 2$ on the boundary. We call the solution “partially strong” because only the divergence-free part of the velocity field and the associated effective pressure have regularity typical for strong solution, while the density and the gradient part of the velocity have regularity typical for weak solution.

Reçu le : 2010-03-16
Accepté le : 2010-04-08
Publié le : 2010-05-21

DOI : 10.1016/j.crma.2010.04.001

Affiliations des auteurs :

Muzereau, Olivier ¹ ; Neustupa, Jiří ² ; Penel, Patrick ¹

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     title = {A steady {Navier{\textendash}Stokes} model for compressible fluid with partially strong solutions},
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Muzereau, Olivier; Neustupa, Jiří; Penel, Patrick. A steady Navier–Stokes model for compressible fluid with partially strong solutions. Comptes Rendus. Mathématique, Tome 348 (2010) no. 11-12, pp. 619-624. doi : 10.1016/j.crma.2010.04.001. http://www.numdam.org/articles/10.1016/j.crma.2010.04.001/

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