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.

In this paper we deal with the stationary Navier-Stokes problem in a domain Ω with compact Lipschitz boundary ∂Ω and datum a in Lebesgue spaces. We prove existence of a solution for arbitrary values of the fluxes through the connected components of ∂Ω, with possible countable exceptional set, provided a is the sum of the gradient of a harmonic function and a sufficiently small field, with zero total flux for Ω bounded.

Classification : 76D05, 31B10, 35Q30, 42B20 
@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},
     mrnumber = {2413675},
     zbl = {1150.76015},
     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
SP  - 171
EP  - 180
VL  - 7
IS  - 1
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2008_5_7_1_171_0/
LA  - en
ID  - ASNSP_2008_5_7_1_171_0
ER  - 
%0 Journal Article
%A Russo, Antonio
%A Starita, Giulio
%T On the existence of steady-state solutions to the Navier-Stokes system for large fluxes
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2008
%P 171-180
%V 7
%N 1
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2008_5_7_1_171_0/
%G en
%F ASNSP_2008_5_7_1_171_0
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] W. Borchers and K. Pileckas, 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 | Zbl

[2] R.M. Brown and Z. Shen, Estimates for the Stokes operator in Lipschitz domains, Indiana Univ. Math. J. 44 (1995), 1183-1206. | MR | Zbl

[3] H. Fujita and H. Morimoto, 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 | Zbl

[4] H. Fujita, H. Morimoto and H. Okamoto, Stability analysis of Navier-Stokes flows in annuli, Math. Methods Appl. Sci. 20 (1997), 959-978, | MR | Zbl

[5] G. P. Galdi, On the existence of steady motions of a viscous flow with nonhomogeneous boundary conditions, Matematiche (Catania) 46 (1991), 503-524. | MR | Zbl

[6] G. P. Galdi, “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 | Zbl

[7] M. Mitrea and M. Taylor, Navier-Stokes equations on Lipschitz domains in Riemannian manifolds, Math. Ann. 321 (2001), 955-987. | MR | Zbl

[8] H. Morimoto, 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] H. Morimoto, 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 | Zbl

[10] R. Russo, On the existence of solutions to the stationary Navier-Stokes equations, Ricerche Mat. 52 (2003), 285-348. | MR | Zbl

[11] R. Russo and C. G. Simader, A note on the existence of solutions to the Oseen system in Lipschitz domains, J. Math. Fluid Mech. 8 (2006), 64-76. | MR | Zbl

[12] Z. Shen, A note on the Dirichlet problem for the Stokes system in Lipschitz domains, Proc. Amer. Math. Soc. 123 (1995), 801-811. | MR | Zbl

[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 | Zbl