Skip to main content
Springer Nature Link
Log in

On time periodic solutions, asymptotic stability and bifurcations of Navier-Stokes equations

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

In this article, we investigate the time periodic solutions for two-dimensional Navier-Stokes equations with nontrivial time periodic force terms. Under the time periodic assumption of the force term, the existence of time periodic solutions for two-dimensional Navier-Stokes equations has received extensive attention from many authors. With the smallness assumption of the time periodic force, we show that there exists only one time periodic solution and this time periodic solution is globally asymptotically stable in the \(H^1\) sense. Without smallness assumption of the force term, there is no stability analysis theory addressed. It is expected that when the amplitude of the force term is increasing, the time periodic solution is no longer asymptotically stable. In the last part of the article, we use numerical experiments to study the bifurcation of the time periodic solutions when the amplitude of the force is increasing. Extrapolating to the heating of the earth by the sun, the bifurcation diagram hints that when the earth receives a relatively small amount of solar energy regularly, the time periodic fluid patterns are asymptotically stable; while/when the earth receives too much solar energy even though in a time periodic way, the time periodic pattern of the fluid motions will lose its stability.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Adams, R.A.: Sobolev spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Constantin, P., Foias, C.: Navier-stokes equations. The University of Chicago Press, Chicago (1988)

    MATH  Google Scholar 

  3. Beirao da Veiga, H.: Time-periodic solutions of the Navier-Stokes equations in unbounded cylindrical domains—Lerays problem for periodic flows. Arch. Ration. Mech. Anal. 178, 301–325 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Foias, C., Prodi, G.: Sur le comportement global des solutions non-stationnaires des’equations de navier-stokes en dimension 2. Rend. Sem. Mat. Univ. Padova 39, 1–34 (1967)

    MathSciNet  MATH  Google Scholar 

  5. Galdi, G.P.: An introduction to the mathematical theory of the navier-stokes equations, vol. I. Springer, New York (1994)

    MATH  Google Scholar 

  6. Galdi, G., Sohr, H.: Existence and uniqueness of time-periodic physically reasonable navier-stokes flow past a body. Arch. Ration. Mech. Anal. 172(3), 363–406 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gie, G.-M., Jung, C.-Y.: Vorticity layers of the 2D Navier-Stokes equations with a slip type boundary condition. Asymptot. Anal. 84(1–2), 17–33 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Galdi, G.P.: Existence and uniqueness of time-periodic solutions to the Navier-Stokes equations in the whole plane. Discrete Contin. Dyn. Syst. Ser. S 6(5), 1237–1257 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Galdi, G.P., Silvestre, A.L.: Existence of time-periodic solutions to the Navier-Stokes equations around a moving body. Pac. J. Math. 223(2), 251–267 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hsia, C.-H., Shiue, M.-C.: Time-periodic solutions of the primitive equations for large-scale moist atmosphere: existence and stability. Appl. Anal. 94(9), 1926–1963 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hsia, C.-H., Shiue, M.-C.: On the asymptotic stability analysis and the existence of time-periodic solutions of the primitive equations. Indiana Univ. Math. J. 12(2), 403–441 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Xianpeng, H.: Existence of time-periodic solutions to incompressible Navier-Stokes equations in the whole space. Electron. J. Differ. Equations 2005(104), 6 (2005) (electronic)

  13. Iwabuchi, T., Takada, R.: Time periodic solutions to the Navier-Stokes equations in the rotational framework. J. Evol. Equ. 12(4), 985–1000 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jung CY, Nguyen TB (2015) Semi-analytical time differencing methods for stiff problems. J. Sci. Comput. 63(2), 355–373 (2015)

  15. Jung CY, Nguyen TB (2016) New time differencing methods for spectral methods. J. Sci. Comput. (to appear)

  16. Kagei, Y., Tsuda, K.: Existence and stability of time periodic solution to the compressible Navier-Stokes equation for time periodic external force with symmetry. J. Differ. Equations 258(2), 399–444 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kaniel, S., Shinbrot, M.: A reproductive property of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 24, 363–369 (1967)

    MathSciNet  MATH  Google Scholar 

  18. Kato, H.: Existence of periodic solutions of the Navier-Stokes equations. J. Math. Anal. Appl. 208(1), 141–157 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kozono, H., Mashiko, Y., Takada, R.: Existence of periodic solutions and their asymptotic stability to the Navier-Stokes equations with the Coriolis force. J. Evol. Equ. 14(3), 565–601 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kučera, P.: The time-periodic solutions of the Navier-Stokes equations with mixed boundary conditions. Discrete Contin. Dyn. Syst. Ser. S 3(2), 325–337 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kyed, M.: The existence and regularity of time-periodic solutions to the three-dimensional Navier-Stokes equations in the whole space. Nonlinearity 27(12), 2909–2935 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ladyzhenskaya, O.A.: The boundary value problems of mathematical physics. Springer, New York (1985)

    Book  MATH  Google Scholar 

  23. Lemarié-Rieusset P.G.: On some classes of time-periodic solutions for the Navier-Stokes equations in the whole space. SIAM J. Math. Anal. 47(2), 1022–1043 (2015)

  24. Maremonti, P.: Existence and stability of time-periodic solutions to the Navier-Stokes equations in the whole space. Nonlinearity 4(2), 503–529 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  25. Morimoto, H.: Time periodic Navier-Stokes flow with nonhomogeneous boundary condition. J. Math. Sci. Univ. Tokyo 16(1), 113–123 (2009)

    MathSciNet  MATH  Google Scholar 

  26. Perlerkar, P., Mitra, D.: Spectral-soap. http://code.google.com/p/spectral-soap/

  27. Serrin, J.: A note on the existence of periodic solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 3, 120–122 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  28. Serrin, J.: On the stability of viscous fluid motions. Arch. Ration. Mech. Anal. 3, 1–13 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  29. Takeshita, A.: On the reproductive property of the \(2\)-dimensional Navier-Stokes equations. J. Fac. Sci. Univ. Tokyo Sect. I 16(1969), 297–311 (1970)

  30. Takeshita, A.: On the reproductive property of the \(2\)-dimensional Navier-Stokes equations. J. Fac. Sci. Univ. Tokyo. Sec. IA 16, 297–311 (1970)

    MathSciNet  MATH  Google Scholar 

  31. Taniuchi, Y.: On the uniqueness of time-periodic solutions to the Navier-Stokes equations in unbounded domains. Math. Z. 261(3), 597–615 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Temam, R.: Navier-Stokes equations and nonlinear functional analysis. In: CBMS-NSF Regional Conference Series in Appl. Math., 2nd edn, vol. 66. SIAM, Philadelphia (1995)

  33. Temam, R.: Infinite dimensional dynamical systems in mechanics and physics, 2nd edn. Springer, New York (1997)

  34. Vishik, M.I.: Asymptotic behaviour of solutions of evolutionary equations. In: Radicati, L.A. (ed.) Lezioni Lincee. Cambridge University Press, Cambridge (1992)

Download references

Acknowledgments

This work was supported under the framework of international cooperation program managed by the National Research Foundation of Korea (2015K2A1A2070543) and supported by the National Research Foundation of Korea grant funded by the Ministry of Education (2015R1D1A1A01059837). Chang-Yeol Jung and Thien Binh Nguyen would like to thank authors P. Perlekar and D. Mitra of the open source codes spectral-soap in [26]. Chun-Hsiung Hsia and Ming-Cheng Shiue were partially supported by the Ministry of Science and Technology, Taiwan under grant MOST 104-2628-M-002-007-MY3 and MOST 104-2115-M-009-012-MY2, respectively.

Author information

Authors and Affiliations

  1. Department of Mathematics, Institute of Applied Mathematical Sciences, National Center for Theoretical Sciences, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd, Taipei, 10617, Taiwan

    Chun-Hsiung Hsia

  2. Department of Mathematical Sciences, School of Natural Science, Ulsan National Institute of Science and Technology, UNIST-gil 50, Ulsan, 689-798, Republic of Korea

    Chang-Yeol Jung & Thien Binh Nguyen

  3. Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 300, Taiwan

    Ming-Cheng Shiue

Authors
  1. Chun-Hsiung Hsia
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Chang-Yeol Jung
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Thien Binh Nguyen
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Ming-Cheng Shiue
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Chang-Yeol Jung.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hsia, CH., Jung, CY., Nguyen, T.B. et al. On time periodic solutions, asymptotic stability and bifurcations of Navier-Stokes equations. Numer. Math. 135, 607–638 (2017). https://doi.org/10.1007/s00211-016-0812-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-016-0812-3

Mathematics Subject Classification