no. 1
Mathematical problems in mechanics
Symmetric solutions to the Leray problem
[Solutions symétriques du problème de Leray]

Présenté par : Gérard Iooss

Pukhnachev, Vladislav1
1 Novosibirsk State University, Lavrentyev Institute of Hydrodynamics, Lavrentyev av., 15, 630090 Novosibirsk, Russia
Comptes Rendus. Mathématique, Tome 355 (2017) no. 1, pp. 113-117.

On considère le problème avec conditions au bord pour les équations de Navier–Stokes stationnaires régissant l'écoulement d'un fluide incompressible dans une couche sphérique. On donne la vitesse au bord. Jean Leray (1933) a démontré la solvabilité de ce problème sous la condition d'un flux nul à travers chacune des composantes connexes du bord. Le problème suivant est à présent ouvert : est-ce qu'une solution du problème avec flux existe sous la seule condition d'un flux total nul ? La note ci-dessous considère le problème de Leray dans une couche sphérique. On obtient une estimation a priori de la solution, sous la condition de symétrie par rapport à un plan. Cette estimation implique la solvabilité du problème.

A stationary boundary-value problem for the Navier–Stokes equations of an incompressible fluid in a domain of a spherical layer type is considered. The velocity vector on the boundary is given. The solvability of this problem was proven by Jean Leray (1933) under an additional condition of a zero flux through each connected component of the flow domain boundary. The following problem is open up to now: does a solution to the flux problem exist if only the necessary condition of a zero total flux is satisfied? The present communication is devoted to the consideration of the Leray problem in a spherical-layer-type domain. An a priori estimate of the solution under the condition of flow symmetry with respect to a plane is obtained. This estimate implies the solvability of the problem.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2016.11.010
Pukhnachev, Vladislav 1

1 Novosibirsk State University, Lavrentyev Institute of Hydrodynamics, Lavrentyev av., 15, 630090 Novosibirsk, Russia 
@article{CRMATH_2017__355_1_113_0,
     author = {Pukhnachev, Vladislav},
     title = {Symmetric solutions to the {Leray} problem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {113--117},
     publisher = {Elsevier},
     volume = {355},
     number = {1},
     year = {2017},
     doi = {10.1016/j.crma.2016.11.010},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2016.11.010/}
}
TY  - JOUR
AU  - Pukhnachev, Vladislav
TI  - Symmetric solutions to the Leray problem
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 113
EP  - 117
VL  - 355
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2016.11.010/
DO  - 10.1016/j.crma.2016.11.010
LA  - en
ID  - CRMATH_2017__355_1_113_0
ER  - 
%0 Journal Article
%A Pukhnachev, Vladislav
%T Symmetric solutions to the Leray problem
%J Comptes Rendus. Mathématique
%D 2017
%P 113-117
%V 355
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2016.11.010/
%R 10.1016/j.crma.2016.11.010
%G en
%F CRMATH_2017__355_1_113_0
Pukhnachev, Vladislav. Symmetric solutions to the Leray problem. Comptes Rendus. Mathématique, Tome 355 (2017) no. 1, pp. 113-117. doi : 10.1016/j.crma.2016.11.010. http://www.numdam.org/articles/10.1016/j.crma.2016.11.010/

[1] Leray, J. Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'Hydrodynamique, J. Math. Pures Appl., 9e série, Volume 12 (1933), pp. 1-82

[2] Fujita, H. On the existence and regularity of the steady-state solutions of the Navier–Stokes equations, J. Fac. Sci., Univ. Tokyo, Sect. 1, Volume 9 (1961), pp. 59-102

[3] Finn, R. On the steady-state solutions of the Navier–Stokes equations. III, Acta Math., Volume 105 (1961) no. 3–4, pp. 197-244

[4] Fujita, H.; Morimoto, H. A remark on the existence of the Navier–Stokes flow with non-vanishing outflow condition, Sapporo, 1995 (GAKUTO Int. Ser. Math. Sci. Appl.), Volume vol. 10, Gakkotosho, Tokyo (1997), pp. 53-61

[5] Korobkov, M.V.; Pileckas, K.; Pukhnachev, V.V.; Russo, R. The flux problem for the Navier–Stokes equations, Russ. Math. Surv., Volume 69 (2014) no. 6, pp. 1065-1122 (translated from Uspekhi Math. Nauk, 69, 6, 2014, pp. 115-176)

[6] Ladyzhenskaya, O.A. The Mathematical Theory of Viscous Incompressible Flow, Gordon and Beach, New York, 1969

[7] Hopf, E. Ein allgemeiner Endlichkeitssatz der Hydrodynamik, Math. Ann., Volume 117 (1941) no. 1, pp. 764-775

[8] Amick, Ch.J. Existence of solutions to the nonhomogeneous steady Navier–Stokes equations, Indiana Univ. Math. J., Volume 33 (1984) no. 6, pp. 817-830

[9] Sazonov, L.I. On the existence of a stationary symmetric solution of the two-dimensional fluid flow problem, Math. Notes, Volume 54 (1993) no. 6, pp. 1280-1283 (translated from Mat. Zametki, 54, 6, 1993, pp. 138-141)

[10] Fujita, H. On stationary solutions to Navier–Stokes equations in symmetric plane domains under general outflow condition, Varenna, 1997 (Pitman Res. Notes Math. Ser.), Volume vol. 388, Longman, Harlow, UK (1998), pp. 16-30

Cité par Sources :