Skip to main content
SpringerLink
Log in
Book cover

Continuous and Distributed Systems pp 181–187Cite as

Topological Properties of Strong Solutions for the 3D Navier-Stokes Equations

  • Chapter
  • First Online:

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 211))

Abstract

In this chapter we give a criterion for the existence of global strong solutions for the 3D Navier-Stokes system for any regular initial data.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Temam, R.: Navier-Stokes Equations. North-Holland, Amsterdam (1979)

    MATH  Google Scholar 

  2. Sohr, H.: The Navier-Stokes Equations. An Elementary Functional Analytic Approach. Verlag, Birkh\({\ddot{\rm a}}\)user (2001)

    Google Scholar 

  3. Zgurovsky, M.Z., Kasyanov, P.O., Kapustyan, O.V., Valero, J., Zadoianchuk, N.V.: Evolution Inclusions and Variation Inequalities for Earth Data Processing III. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  4. Ponce, G., Rascke, R., Sideris, T., Titi, E.: Global stability of large solutions to the 3D Navier-Stokes equations. Commun. Math. Phys. 159, 329–341 (1994)

    Article  MATH  Google Scholar 

  5. Kasyanov, P.O., Toscano, L., Zadoianchuk, N.V.: A criterion for the existence of strong solutions for the 3D Navier-Stokes equations. Appl. Math. Lett. 26(1), 15–17 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Serrin, J.: The initial value problem for the Navier-Stokes equations. In: Langer, R.E. (ed.) Nonlinear Problems, pp. 69–98. University of Wisconsin Press, Madison (1963)

    Google Scholar 

  7. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988)

    Book  MATH  Google Scholar 

  8. Melnik, V.S., Toscano, L.: On weak extensions of extreme problems for nonlinear operator equations. Part I. Weak solutions. J. Automat. Inf. Scien. 38, 68–78 (2006)

    Google Scholar 

  9. Kapustyan, O.V., Kasyanov, P.O., Valero, J.: Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system. J. Math. Anal. Appl. 373, 535–547 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cronin, J.: Fixed Points and Topological Degree in Nonlinear Analysis. American Mathematical Society, Providence (1964)

    MATH  Google Scholar 

Download references

Acknowledgments

The authors thank Professors J.M. Ball, V.V. Chepyzhov, and M.Z. Zgurovsky for useful suggestions during the preparation of this manuscript. The first author was partially supported by the Ukrainian State Fund for Fundamental Researches under grants GP/F44/076, GP/F49/070, and by the NAS of Ukraine under grant 2273/13.

Author information

Authors and Affiliations

  1. Institute for Applied System Analysis, National Technical University of Ukraine “Kyiv Polytechnic Institute”, Peremogy ave., 37, build, 35, Kyiv,  03056, Ukraine

    Pavlo O. Kasyanov

  2. Department of Mathematics and Applications R. Caccioppoli, University of Naples “Federico II”, via Claudio 21, 80125 , Naples, Italy

    Luisa Toscano

  3. Department of Computational Mathematics, Taras Shevchenko National University of Kyiv, Volodimirska Street 64, Kyiv,  03601, Ukraine

    Nina V. Zadoianchuk

Authors
  1. Pavlo O. Kasyanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luisa Toscano
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nina V. Zadoianchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pavlo O. Kasyanov .

Editor information

Editors and Affiliations

  1. National Academy of Science, National Technical University of Ukraine, Kiev, Ukraine

    Mikhail Z. Zgurovsky

  2. Lomonosov Moscow State University, Moscow, Russia

    Victor A. Sadovnichiy

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Kasyanov, P.O., Toscano, L., Zadoianchuk, N.V. (2014). Topological Properties of Strong Solutions for the 3D Navier-Stokes Equations. In: Zgurovsky, M., Sadovnichiy, V. (eds) Continuous and Distributed Systems. Solid Mechanics and Its Applications, vol 211. Springer, Cham. https://doi.org/10.1007/978-3-319-03146-0_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-03146-0_13

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-03145-3

  • Online ISBN: 978-3-319-03146-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation