On the convergence at infinity of the Leray solution of the two-dimensional Navier–Stokes equations to the prescribed asymptotic value

Socolescu, Dan

doi:10.1016/S1631-073X(03)00127-4

Partial Differential Equations/Mathematical Problems in Mechanics

On the convergence at infinity of the Leray solution of the two-dimensional Navier–Stokes equations to the prescribed asymptotic value
[Sur la convergence à l'infini de la solution de Leray des équations bidimensionnelles de Navier–Stokes vers la valeur asymptotique imposée]

Présenté par : Pierre-Louis Lions

Socolescu, Dan ¹

¹ Fachbereich Mathematik, Universität Kaiserslautern, Erwin-Schrödinger-Strasse, 67663 Kaiserslautern, Germany

Comptes Rendus. Mathématique, Tome 336 (2003) no. 9, pp. 739-744.

Résumé
Abstract

Dans cette Note on prouve que $\underset{\sim}{v_{L}}$ , la solution vitesse de Leray des équations stationnaires, incompressibles, bidimensionnelles de Navier–Stokes, tend à l'infini vers le vecteur imposé $\underset{\sim}{v_{\infty}} .$ On montre aussi que la suite $(\underset{\sim}{v_{R_{i}}}, p_{R_{i}})$ de solutions de Leray du même problème aux limites dans les domaines bornés $Ω_{R_{i}}, i \in ℕ,$ converge quasi-uniformément dans $\bar{Ω}$ vers $(\underset{\sim}{v_{L}}, p_{L}) .$

In this Note we prove that $\underset{\sim}{v_{L}}$ , the Leray velocity solution to the steady incompressible, two-dimensional Navier–Stokes equations, tends at infinity to the prescribed vector $\underset{\sim}{v_{\infty}} .$ We show also that the sequence $(\underset{\sim}{v_{R_{i}}}, p_{R_{i}})$ of Leray solutions to the same boundary value problem in the bounded domains $Ω_{R_{i}}, i \in ℕ,$ converges quasi-uniformly in $\bar{Ω}$ to $(\underset{\sim}{v_{L}}, p_{L}) .$

Reçu le : 2002-05-22
Accepté le : 2003-01-14
Publié le : 2003-04-23

DOI : 10.1016/S1631-073X(03)00127-4

Affiliations des auteurs :

Socolescu, Dan ¹

¹ Fachbereich Mathematik, Universität Kaiserslautern, Erwin-Schrödinger-Strasse, 67663 Kaiserslautern, Germany

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     title = {On the convergence at infinity of the {Leray} solution of the two-dimensional {Navier{\textendash}Stokes} equations to the prescribed asymptotic value},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {739--744},
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Socolescu, Dan. On the convergence at infinity of the Leray solution of the two-dimensional Navier–Stokes equations to the prescribed asymptotic value. Comptes Rendus. Mathématique, Tome 336 (2003) no. 9, pp. 739-744. doi : 10.1016/S1631-073X(03)00127-4. http://archive.numdam.org/articles/10.1016/S1631-073X(03)00127-4/

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