Mathematische Annalen

, Volume 370, Issue 1–2, pp 727–784 | Cite as

The existence theorem for the steady Navier–Stokes problem in exterior axially symmetric 3D domains

  • Mikhail Korobkov
  • Konstantin PileckasEmail author
  • Remigio Russo


We study the nonhomogeneous boundary value problem for the Navier–Stokes equations of steady motion of a viscous incompressible fluid in a three-dimensional exterior domain with multiply connected boundary. We prove that this problem has a solution for axially symmetric domains and data (without any smallness restrictions on the fluxes). Our main tool is a recent version of the Morse–Sard theorem for Sobolev functions obtained by Bourgain et al. (Rev Mat Iberoam 29(1):1–23, 2013).

Mathematics Subject Classification

35Q30 76D03 76D05 



The authors are deeply indebted to V.V. Pukhnachev for valuable discussions. The research of M. Korobkov was partially supported by the Russian Foundation for Basic Research (Project No. 14-01-00768-a.). M. Korobkov thanks also the Gruppo Nazionale per la Fisica Matematica of the Istituto Nazionale di Alta Matematica for the financial support during his stays in the Department of Mathematics and Physics of the Second University of Naples (Italy). The research of K. Pileckas was funded by the Lithuanian-Swiss cooperation programme to reduce economic and social disparities within the enlarged European Union under the project agreement No. CH-3-ŠMM-01/01. The research of R. Russo was supported by the Research Council of Lithuania (Grant No. VIZ-TYR-130).


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Mikhail Korobkov
    • 1
    • 2
  • Konstantin Pileckas
    • 3
    Email author
  • Remigio Russo
    • 4
  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia
  3. 3.Faculty of Mathematics and InformaticsVilnius UniversityVilniusLithuania
  4. 4.Seconda Università di NapoliCasertaItaly

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