Viscous Flows in Domains with a Multiply Connected Boundary

  • V. V. Pukhnachev
Part of the Advances in Mathematical Fluid Mechanics book series (AMFM)


In this paper we consider stationary Navier-Stokes equations in a bounded domain with a boundary, which has several connected components. The velocity vector is given on the boundary, where the fluxes differ from zero on its components. In the general case, the solvability of this problem has been an open question up to now. We provide a survey of previous results, which deal with partial versions of the problem. We construct an a priori estimate of the Dirichlet integral for a velocity vector in the case when the flow has an axis of symmetry and a plane of symmetry perpendicular to it, which also intersects each component of the boundary. Having available this estimate, we prove an existence theorem for the axially symmetric problem in a domain with a multiply connected boundary. We consider also the problem in a curvilinear ring and formulate a conditional result concerning its solvability.

Mathematics Subject Classification (2000)

35Q30 76N10 76D05 


Navier—Stokes equations incompressible liquid multiply connected boundary Dirichlet integral a priori estimate 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Leray, J. Étude de diverses équations intégrales nonlinéaires et de quelques problèmes que pose l’hydrodynamique, J. Math. Pure Appl., 12 (1933), pp. 1–82.zbMATHMathSciNetGoogle Scholar
  2. [2]
    Hopf, E., Ein allgemeiner Endlichkeitssatz der Hydrodynamik, Math. Ann., 117 (1941), pp. 764–775.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Ladyzhenskaya, O.A., The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.zbMATHGoogle Scholar
  4. [4]
    Takeshita, A., A remark on Leray’s inequality, Pacific J. Math., 157 (1993), pp. 151–158.zbMATHMathSciNetGoogle Scholar
  5. [5]
    Finn, R., On steady-state solutions of the Navier-Stokes equations. III, Acta Math., 105 (1961), pp. 197–244.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Galdi, G.P., On the existence of steady motions of a viscous flow with non-homogeneous conditions, Le Matematiche, 66 (1991), pp. 503–524.MathSciNetGoogle Scholar
  7. [7]
    Galdi, G.P., An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. 2, Springer, 1994.Google Scholar
  8. [8]
    Borchers, W. and Pileckas, K., Note on the flux problem for stationary NavierStokes equations in domains with multiply connected boundary, Acta Appl. Math., 37 (1994), pp. 21–30.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Amick, Ch. J., Existence of solutions to the nonhomogeneous steady Navier-Stokes equations, Indiana Univ. Math. J., 33 (1984), pp. 817–830.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Fujita, H., On stationary solutions to Navier-Stokes equations in symmetric plane domains under general out-flow conditions, Proceedings of International Conference on Navier-Stokes Equations, Theory and Numerical Methods, June 1997, Varenna Italy, Pitman Research Notes in Mathematics, 388, pp. 16–30.Google Scholar
  11. [11]
    Morimoto, H., A remark on the existence of 2-D steady Navier-Stokes flow in bounded symmetric domain under general outflow condition, J. Math. Fluid Mech., 9 (2007), pp. 411–418.zbMATHCrossRefMathSciNetADSGoogle Scholar
  12. [12]
    Fujita, H. and Morimoto, H., A remark on the existence of the Navier-Stokes flow with non-vanishing outflow conditions, GAKUTO Internat. Ser. Math. Sci. Appl., 10 (1997), pp. 53–61.MathSciNetGoogle Scholar
  13. [13]
    Morimoto, H., Stationary Navier-Stokes equations under general outflow condition, Hokkaido Math. J., 24 (1995), pp. 641–648.zbMATHMathSciNetGoogle Scholar
  14. [14]
    Morimoto, H. and Ukai, S., Perturbation of the Navier-Stokes equations in an annular domain with non-vanishing outflow condition, J. Math. Sci., Univ. Tokyo, 3 (1996), pp. 73–82.zbMATHMathSciNetGoogle Scholar
  15. [15]
    Fujita, H., Morimoto, H. and Okamoto, H. Stability analysis of the Navier-Stokes equations flows in annuli, Math. Methods in the Appl. Sciences, 20 (1997), pp. 959–978.Google Scholar
  16. [16]
    Hopf, E., On nonlinear partial differential equations, in Lecture Series of the Symposium on Partial Differential Equations, Berkeley, 1955, The University of Kansas, 1–29, 1957.Google Scholar
  17. [17]
    Solonnikov, V.A. and Shchadilov, V.E., A certain boundary value problem for the stationary system of Navier-Stokes equations, Trudy Math. Inst. AN SSSR, 125 (1973), pp. 196–210 (in Russian).Google Scholar
  18. [18]
    Fujita, H., On the existence and regularity of the steady-state solutions of the NavierStokes equations, J. Fac. Sci., Univ. Tokyo, Sec. 1, 9 (1961), pp. 59–102.zbMATHGoogle Scholar
  19. [19]
    Kochin, N.E., Kibel, I.A. and Roze, N.V. Theoretical Hydrodynamics, Vol. 2, Wiley, New York, 1964.Google Scholar
  20. [20]
    Von Mises, R., Bemerkungen zur Hydrodynamik, Z. Angew. Math. Mech., 7 (1927), pp. 425–429.Google Scholar
  21. [21]
    Kronrod, A.S., On functions of two variables, Uspekhi Matematicheskikh Nauk, 5.1 (1950), pp. 24–134 (in Russian).MathSciNetGoogle Scholar
  22. [22]
    Vishik, M.I. and Lyusternik, L.A., Regular degeneration and boundary layer for linear differential equations with a small parameter, Uspekhi Matematicheskikh Nauk, 12.5 (1957), pp. 3–122 (in Russian).MathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • V. V. Pukhnachev
    • 1
  1. 1.Lavrentyev Institute of HydrodynamicsNovosibirskRussia

Personalised recommendations