Une perturbation hyperbolique des équations de Navier-Stokes

Free access article

Issue		ESAIM: Proc. Volume 21, 2007 Journées d'analyse fonctionnelle et numérique en l'honneur de Michel Crouzeix


Page(s)		65 - 87
DOI		http://dx.doi.org/10.1051/proc:072106
Published online		04 December 2007

ESAIM: Proc., 2007, Vol. 21, pp. 65-87
DOI: 10.1051/proc:072106

Une perturbation hyperbolique des équations de Navier-Stokes

Marius Paicu¹ and Geneviève Raugel²

¹ Univ Paris-Sud, Laboratoire de Mathématiques d'Orsay, Orsay Cedex, F-91405; CNRS, Orsay cedex, F-91405
² CNRS, Laboratoire de Mathématiques d'Orsay, Orsay Cedex, F-91405; Univ Paris-Sud, Orsay cedex, F-91405

(... / Published online: 4 December 2007)

Abstract
In this paper, we consider a hyperbolic perturbation of the Navier-Stokes equations in $\mathbb{R} ^n$ , n=2,3, given by (0.2), which consists in adding the term $\varepsilon u_{tt}$ to the Navier-Stokes equations. In the case n=2, we recall the global existence and uniqueness of mild solutions of (0.2), for initial data in the Hilbert space $H^1(\mathbb{R} ^2)^2 \times L^2(\mathbb{R} ^2)^2$ and appropriate forcing term f, when $\varepsilon>0$ is small enough, that has been proved in [16]. In the three-dimensional case, we prove a global existence result under a smallness condition of the initial data in $H^{1+\delta}(\mathbb{R} ^3)^3 \times H^\delta(\mathbb{R} ^3)^3$ , $\delta >0$ , for an appropriate forcing term f, when $\varepsilon>0$ is small enough. This smallness condition is analogous to the one known for the global existence of strong solutions of the three-dimensional Navier-Stokes equations.

Résumé
Dans cet article, nous considérons l'équation (0.2), qui est une perturbation hyperbolique des équations de Navier-Stokes, par le terme $\varepsilon u_{tt}$ . Dans le cas de la dimension deux d'espace, nous rappelons des résultats d'existence globale et d'unicité des solutions dans $H^1(\mathbb{R} ^2)^2 \times L^2(\mathbb{R} ^2)^2$ , quand $\varepsilon>0$ est suffisamment petit ([16]). Dans le cas de la dimension trois d'espace, pour $\varepsilon>0$ suffisamment petit, nous démontrons l'existence globale de solutions intégrales sous une hypothèse de petitesse sur les données initiales dans $H^{1+\delta}(\mathbb{R} ^3)^3 \times H^\delta(\mathbb{R} ^3)^3$ , $\delta >0$ et des hypothèses adéquates sur le terme de force. Cette hypothèse de petitesse est totalement en accord avec l'hypothèse de petitesse classique pour les équations de Navier-Stokes en dimension trois.

Mathematics Subject Classification. Primary 35Q30, 76D05, 46E35; Secondary 35B65, 35K55.

Key words: Hyperbolic Navier-Stokes equations, global existence, global regularity, comparaison, limiting equation, second order in time equation

© EDP Sciences, ESAIM 2007

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