On the existence of steady-state solutions to the Navier-Stokes system for large fluxes

Antonio Russo; Giulio Starita

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2008)

  • Volume: 7, Issue: 1, page 171-180
  • ISSN: 0391-173X

Abstract

top
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.

How to cite

top

Russo, Antonio, and 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 7.1 (2008): 171-180. <http://eudml.org/doc/272294>.

@article{Russo2008,
abstract = {In this paper we deal with the stationary Navier-Stokes problem in a domain $\Omega $ with compact Lipschitz boundary $\partial \Omega $ and datum $a$ in Lebesgue spaces. We prove existence of a solution for arbitrary values of the fluxes through the connected components of $\partial \Omega $, 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 $\Omega $ bounded.},
author = {Russo, Antonio, Starita, Giulio},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {1},
pages = {171-180},
publisher = {Scuola Normale Superiore, Pisa},
title = {On the existence of steady-state solutions to the Navier-Stokes system for large fluxes},
url = {http://eudml.org/doc/272294},
volume = {7},
year = {2008},
}

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
PB - Scuola Normale Superiore, Pisa
VL - 7
IS - 1
SP - 171
EP - 180
AB - In this paper we deal with the stationary Navier-Stokes problem in a domain $\Omega $ with compact Lipschitz boundary $\partial \Omega $ and datum $a$ in Lebesgue spaces. We prove existence of a solution for arbitrary values of the fluxes through the connected components of $\partial \Omega $, 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 $\Omega $ bounded.
LA - eng
UR - http://eudml.org/doc/272294
ER -

References

top
  1. [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. Zbl0814.76029MR1308742
  2. [2] R.M. Brown and Z. Shen, Estimates for the Stokes operator in Lipschitz domains, Indiana Univ. Math. J.44 (1995), 1183–1206. Zbl0858.35098MR1386766
  3. [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, Zbl0887.35118MR1602756
  4. [4] H. Fujita, H. Morimoto and H. Okamoto, Stability analysis of Navier–Stokes flows in annuli, Math. Methods Appl. Sci.20 (1997), 959–978, Zbl0881.76023MR1458528
  5. [5] G. P. Galdi, On the existence of steady motions of a viscous flow with nonhomogeneous boundary conditions, Matematiche (Catania) 46 (1991), 503–524. Zbl0780.76018MR1228739
  6. [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. Zbl0949.35004MR1284206
  7. [7] M. Mitrea and M. Taylor, Navier–Stokes equations on Lipschitz domains in Riemannian manifolds, Math. Ann.321 (2001), 955–987. Zbl1039.35079MR1872536
  8. [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. [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. Zbl0941.35063MR1616335
  10. [10] R. Russo, On the existence of solutions to the stationary Navier–Stokes equations, Ricerche Mat.52 (2003), 285–348. Zbl1121.35104MR2091520
  11. [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. Zbl1099.76017MR2205151
  12. [12] Z. Shen, A note on the Dirichlet problem for the Stokes system in Lipschitz domains, Proc. Amer. Math. Soc.123 (1995), 801–811. Zbl0816.35106MR1223521
  13. [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. Zbl0971.35059MR1789922

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.