In this paper we deal with the stationary Navier-Stokes problem in a domain with compact Lipschitz boundary and datum in Lebesgue spaces. We prove existence of a solution for arbitrary values of the fluxes through the connected components of , with possible countable exceptional set, provided is the sum of the gradient of a harmonic function and a sufficiently small field, with zero total flux for bounded.
@article{ASNSP_2008_5_7_1_171_0,
author = {Russo, Antonio and Starita, Giulio},
title = {On the existence of steady-state solutions to the {Navier-Stokes} system for large fluxes},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {171--180},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 7},
number = {1},
year = {2008},
zbl = {1150.76015},
mrnumber = {2413675},
language = {en},
url = {http://archive.numdam.org/item/ASNSP_2008_5_7_1_171_0/}
}
TY - JOUR AU - Russo, Antonio AU - Starita, Giulio TI - On the existence of steady-state solutions to the Navier-Stokes system for large fluxes JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2008 DA - 2008/// SP - 171 EP - 180 VL - Ser. 5, 7 IS - 1 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2008_5_7_1_171_0/ UR - https://zbmath.org/?q=an%3A1150.76015 UR - https://www.ams.org/mathscinet-getitem?mr=2413675 LA - en ID - ASNSP_2008_5_7_1_171_0 ER -
Russo, Antonio; Starita, Giulio. On the existence of steady-state solutions to the Navier-Stokes system for large fluxes. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 1, pp. 171-180. http://archive.numdam.org/item/ASNSP_2008_5_7_1_171_0/
[1] and , Note on the flux problem for stationary incompressible Navier-Stokes equations in domains with a multiply connected boundary, Acta Appl. Math. 37 (1994), 21-30. | MR 1308742 | Zbl 0814.76029
[2] and , Estimates for the Stokes operator in Lipschitz domains, Indiana Univ. Math. J. 44 (1995), 1183-1206. | MR 1386766 | Zbl 0858.35098
[3] and , A remark on the existence of the Navier-Stokes flow with non-vanishing outflow condition, GAKUTO Internat. Ser. Math. Sci. Appl. 10 (1997), 53-61, | MR 1602756 | Zbl 0887.35118
[4] , and , Stability analysis of Navier-Stokes flows in annuli, Math. Methods Appl. Sci. 20 (1997), 959-978, | MR 1458528 | Zbl 0881.76023
[5] , On the existence of steady motions of a viscous flow with nonhomogeneous boundary conditions, Matematiche (Catania) 46 (1991), 503-524. | MR 1228739 | Zbl 0780.76018
[6] , “An Introduction to the Mathematical Theory of the Navier-Stokes Equations”, Vol. I, II, revised edition, Springer Tracts in Natural Philosophy, C. Truesdell (ed.), Vol. 38, Springer-Verlag, 1998. | MR 1284206 | Zbl 0949.35004
[7] and , Navier-Stokes equations on Lipschitz domains in Riemannian manifolds, Math. Ann. 321 (2001), 955-987. | MR 1872536 | Zbl 1039.35079
[8] , Note on the boundary value problem for the Navier-Stokes equations in 2-D domain with general outflow condition (in Japanese), Memoirs of the Institute of Science and Technology, Meiji University 35 (1997), 95-102.
[9] , General outflow condition for Navier-Stokes system, In: “Recent Topics on Mathematical Theory of Viscous Incompressible fluid”, Lectures Notes in Num. Appl. Anal., Vol. 16, 1998, 209-224. | MR 1616335 | Zbl 0941.35063
[10] , On the existence of solutions to the stationary Navier-Stokes equations, Ricerche Mat. 52 (2003), 285-348. | MR 2091520 | Zbl 1121.35104
[11] and , A note on the existence of solutions to the Oseen system in Lipschitz domains, J. Math. Fluid Mech. 8 (2006), 64-76. | MR 2205151 | Zbl 1099.76017
[12] , A note on the Dirichlet problem for the Stokes system in Lipschitz domains, Proc. Amer. Math. Soc. 123 (1995), 801-811. | MR 1223521 | Zbl 0816.35106
[13] V. Sverák and T-P Tsai, On the spatial decay of 3-D steady-state Navier-Stokes flows, Comm. Partial Differential Equations 25 (2000), 2107-2117. | MR 1789922 | Zbl 0971.35059
