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

Dans cette Note on prouve que v L , la solution vitesse de Leray des équations stationnaires, incompressibles, bidimensionnelles de Navier–Stokes, tend à l'infini vers le vecteur imposé v . On montre aussi que la suite (v R i ,p R i ) de solutions de Leray du même problème aux limites dans les domaines bornés Ω R i ,i, converge quasi-uniformément dans Ω ¯ vers (v L ,p L ).

In this Note we prove that v L , the Leray velocity solution to the steady incompressible, two-dimensional Navier–Stokes equations, tends at infinity to the prescribed vector v . We show also that the sequence (v R i ,p R i ) of Leray solutions to the same boundary value problem in the bounded domains Ω R i ,i, converges quasi-uniformly in Ω ¯ to (v L ,p L ).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00127-4
Socolescu, Dan 1

1 Fachbereich Mathematik, Universität Kaiserslautern, Erwin-Schrödinger-Strasse, 67663 Kaiserslautern, Germany
@article{CRMATH_2003__336_9_739_0,
     author = {Socolescu, Dan},
     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},
     publisher = {Elsevier},
     volume = {336},
     number = {9},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00127-4},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/S1631-073X(03)00127-4/}
}
TY  - JOUR
AU  - Socolescu, Dan
TI  - On the convergence at infinity of the Leray solution of the two-dimensional Navier–Stokes equations to the prescribed asymptotic value
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 739
EP  - 744
VL  - 336
IS  - 9
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/S1631-073X(03)00127-4/
DO  - 10.1016/S1631-073X(03)00127-4
LA  - en
ID  - CRMATH_2003__336_9_739_0
ER  - 
%0 Journal Article
%A Socolescu, Dan
%T On the convergence at infinity of the Leray solution of the two-dimensional Navier–Stokes equations to the prescribed asymptotic value
%J Comptes Rendus. Mathématique
%D 2003
%P 739-744
%V 336
%N 9
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/S1631-073X(03)00127-4/
%R 10.1016/S1631-073X(03)00127-4
%G en
%F CRMATH_2003__336_9_739_0
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/

[1] Amick, C.J. On Leray's problem of steady Navier–Stokes flow past a body in the plane, Acta Math., Volume 161 (1988), pp. 71-130

[2] Babenko, K.I. The asymptotic behaviour of a vortex far away from a body in a plane flow of viscous fluid, Prikl. Mat. Mekh., Volume 34 (1970), pp. 911-925

[3] Gilbarg, D.; Weinberger, H. Asymptotic properties of Leray's solution of the stationary two-dimensional Navier–Stokes equations, Uspekhi Mat. Nauk, Volume 29 (1974), pp. 109-122

[4] Gilbarg, D.; Weinberger, H. Asymptotic properties of steady plane solutions of Navier–Stokes equations with bounded Dirichlet integral, Ann. Scuola Norm. Sup. Pisa, Volume 5 (1978), pp. 381-404

[5] Leray, J. Études de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique, J. Math. Pures Appl., Volume 12 (1933), pp. 1-82

[6] Nicolescu, M. Analiza Matematica, Vol. II, Editura Tehnica, Bucuresti, 1958

[7] Pompeiu, D.; Pompeiu, D. Sur une classe de fonctions d'une variable complexe et sur certaines équations intégrales, Rend. Circ. Mat. Palermo, Volume 33 (1912), pp. 108-113

[8] Socolescu, D. On the asymptotic behaviour of solutions with bounded Dirichlet integral to the steady Navier–Stokes equations, C. R. Acad. Sci. Paris, Ser. I, Volume 330 (2000), pp. 427-432

[9] D. Socolescu, On the unique solvability of the Leray problem to the steady two-dimensional Navier–Stokes equations, submitted for publication

[10] Socolescu, D. On the unique solvability of the two-dimensional Poincaré-Stekloff problem for viscous incompressible fluids (Cleja-Tigoiu, S.; Socolescu, D.; Tigoiu, V., eds.), Geometry, Continua and Microstructures, Proceedings of the fifth International Seminar, Sinaia-Romania, September 26–28, 2001

[11] Vekua, I.N. Verallgemeinerte analytische Funktionen, Akademie-Verlag, Berlin, 1963

Cité par Sources :