Archive for Rational Mechanics and Analysis

, Volume 233, Issue 1, pp 385–407 | Cite as

On Convergence of Arbitrary D-Solution of Steady Navier–Stokes System in 2D Exterior Domains

  • Mikhail V. Korobkov
  • Konstantin PileckasEmail author
  • Remigio Russo


We study solutions to stationary Navier–Stokes system in a two dimensional exterior domain. We prove that any such solution with a finite Dirichlet integral converges to a constant vector at infinity uniformly. No additional conditions (on symmetry or smallness, etc.) are assumed. In the proofs we develop the ideas of the classical papers of Gilbarg and Weinberger (Ann Sc Norm Pisa (4) 5:381–404, 1978) and Amick (Acta Math 161:71–130, 1988).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors are grateful to the anonymous referee for many useful remarks and suggestions. M. Korobkov was partially supported by the Ministry of Education and Science of the Russian Federation (Grant 14.Z50.31.0037) and by the Russian Federation for Basic Research (Project Numbers 18-01-00649 and 17-01-00875). The research of K. Pileckas was funded by the Grant No. S-MIP-17-68 from the Research Council of Lithuania.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Amick C.J.: On Leray’s problem of steady Navier–Stokes flow past a body in the plane. Acta Math. 161, 71–130 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Amick C.J.: On the asymptotic form of Navier–Stokes flow past a body in the plane. J. Differ. Equ. 91, 149–167 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Engelking R.: General Topology, 2nd edn. Sigma Series in Pure Mathematics, Vol. 6. Heldermann Verlag. Berlin (1989)Google Scholar
  4. 4.
    Finn R., Smith D.R.: On the stationary solutions of the Navier–Stokes equations in two dimensions. Arch. Ration. Mech. Anal. 25, 26–39 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fujita H.: On the existence and regularity of the steady-state solutions of the Navier–Stokes equation. J. Fac. Sci. Univ. Tokyo(1A) 9, 59–102 (1961)zbMATHGoogle Scholar
  6. 6.
    Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady–State Problems. Springer, 2011Google Scholar
  7. 7.
    Galdi G.P., Sohr H.: On the asymptotic structure of plane steady flow of a viscous fluid in exterior domains. Arch. Ration. Mech. Anal. 131, 101–119 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gilbarg D., Weinberger H.F.: Asymptotic properties of Leray’s solution of the stationary two–dimensional Navier–Stokes equations. Russ. Math. Surv. 29, 109–123 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gilbarg D., Weinberger H.F.: Asymptotic properties of steady plane solutions of the Navier–Stokes equations with bounded Dirichlet integral. Ann. Sc. Norm. Pisa (4) 5, 381–404 (1978)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Korobkov M.V., Pileckas K., Russo R.: The existence of a solution with finite Dirichlet integral for the steady Navier–Stokes equations in a plane exterior symmetric domain. J. Math. Pures Appl. 101, 257–274 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Korobkov, M.V., Pileckas, K., Russo, R.: On the steady Navier–Stokes equations in 2D exterior domains, arXiv:1711.02400
  12. 12.
    Ladyzhenskaia, O.A.: The Mathematical Theory of Viscous Incompressible Fluid. Gordon and Breach, 1969Google Scholar
  13. 13.
    Leray J.: Étude de diverses équations intégrales non linéaire et de quelques problèmes que pose l’hydrodynamique. J. Math. Pures Appl. 12, 1–82 (1933)zbMATHGoogle Scholar
  14. 14.
    Malý J., Swanson D., Ziemer W.P.: The Coarea formula for Sobolev mappings. Trans. AMS 355(2), 477–492 (2002)CrossRefzbMATHGoogle Scholar
  15. 15.
    Pileckas K., Russo R.: On the existence of vanishing at infinity symmetric solutions to the plane stationary exterior Navier–Stokes problem. Math. Ann. 352, 643–658 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Russo A.: A note on the two–dimensional steady-state Navier–Stokes problem. J. Math. Fluid Mech., 11, 407–414 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Sazonov, L.I.: On the asymptotic behavior of the solution of the two-dimensional stationary problem of the flow past a body far from it. (Russian) Mat. Zametki 65, 246–253, 1999; translation in Math. Notes 65, 246–253, 1999Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Mikhail V. Korobkov
    • 1
    • 2
  • Konstantin Pileckas
    • 3
    Email author
  • Remigio Russo
    • 4
  1. 1.School of Mathematical SciencesFudan UniversityShanghaiChina
  2. 2.Voronezh State UniversityVoronezhRussia
  3. 3.Faculty of Mathematics and InformaticsVilnius UniversityVilniusLithuania
  4. 4.Dipartimento di Matematica e FisicaUniversità degli studi della Campania “Luigi Vanvitelli”CasertaItaly

Personalised recommendations