Skip to main content
SpringerLink
Log in

The generalized Proudman–Johnson equation and its singular perturbation problems

  • Original Paper
  • Area 2
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

We consider the generalized Proudman–Johnson equation with an external force. By varying the Reynolds number \(R\) and another nondimensional parameter \(\alpha \), branching stationary solutions are computed numerically for the global picture of bifurcations of the equation. Asymptotic behavior of solutions as the Reynolds number tends to zero or infinity is also studied by a combination of heuristic analysis and the asymptotic expansion. In doing so, singular perturbation problems of new type are derived and analyzed. As a consequence, through the asymptotic analysis argument, the peculiarity of two dimensional Navier–Stokes flows related to the unimodality is re-confirmed.

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

Access this article

Log in via an institution

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
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27

Similar content being viewed by others

References

  1. Batchelor, G.K.: On steady laminar flow with closed streamlines at large Reynolds number. J. Fluid Mech. 1, 177–190 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  2. Budd, C., Dold, B., Stuart, A.: Blowup in a partial differential equation with conserved first integral. SIAM J. Appl. Math. 53, 718–742 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chandler, G.J., Kerswell, R.R.: Invariant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow. J. Fluid Mech. 722, 554–595 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen, X., Okamoto, H.: Global existence of solutions to the Proudman–Johnson equation. Proc. Jpn. Acad. 76, 149–152 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen, X., Okamoto, H.: Global existence of solutions to the generalized Proudman–Johnson equation. Proc. Jpn. Acad. 78, 136–139 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cho, C.-H., Wunsch, M.: Global and singular solutions to the generalized Proudman–Johnson equation. J. Diff. Eq. 249, 392–413 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gallet, B., Young, W.R.: A two-dimensional vortex condensate at high Reynolds number. J. Fluid Mech. 715, 359–388 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Hinch, E.J.: Perturbation Methods. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  9. Ikeda, H., Kondo, K., Okamoto, H., Yotsutani, S.: On the global branches of the solutions to a nonlocal boundary-value problem arising in Oseen’s spiral flows. Commun. Pure Appl. Anal. 2, 373–382 (2003)

    MathSciNet  Google Scholar 

  10. Ikeda, H., Mimura, M., Okamoto, H.: A singular perturbation problem arising in Oseen’s spiral flows. Jpn. J. Indust. Appl. Math. 18, 393–403 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kim, S.-C.: On Prandtl–Batchelor theory of a cylindrical eddy: asymptotic study. SIAM J. Appl. Math. 58, 1394–1413 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kim, S.-C., Okamoto, H.: Vortices of large scale appearing in the 2D stationary Navier–Stokes equations at large Reynolds numbers. Jpn. J. Ind. Appl. Math. 27, 47–71 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kim, S.-C., Okamoto, H.: The generalized Proudman–Johnson equation at large Reynolds numbers. IMA J. Appl. Math. 78, 379–403 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  14. Okamoto, H.: Well-posedness of the generalized Proudman–Johnson equation without viscosity. J. Math. Fluid Mech. 11, 46–59 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Okamoto, H.: A nonlinear boundary-value problem with an integral constraint. In: Proceedings of Far-From-Equilibrium Dynamics. RIMS Kôkyûroku Bessatsu, vol. B31, pp. 93–115 (2012)

  16. Okamoto, H., Shōji, M.: The Mathematical Theory of Bifurcation of Permanent Progressive Water-Waves, World Scientific, Singapore (2001)

  17. Okamoto, H., Zhu, J.: Some similarity solutions of the Navier–Stokes equations and related topics. Taiwan. J. Math. 4, 65–103 (2000)

    MATH  MathSciNet  Google Scholar 

  18. Sarria, A., Saxton, R.: Blow-up of solutions to the generalized inviscid Proudman–Johnson equation. J. Math. Fluid Mech. 5, 493–523 (2013)

  19. Schlichting, H., Gersten, K.: Boundary Layer Theory. Springer, Berlin, Heidelberg (2000)

Download references

Acknowledgments

S.-C. Kim is supported by the Korea Research Foundation Grant funded by the Korean Government (NRF-2013R1A1A2004815). H. Okamoto is partially supported by JSPS Grant 24244007.

Author information

Authors and Affiliations

  1. Department of Mathematics, Chung-Ang University, 221 Heukseok-dong, Dongjak-ku, Seoul, 156-756, Korea

    Sun -Chul Kim

  2. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Hisashi Okamoto

Authors
  1. Sun -Chul Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hisashi Okamoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hisashi Okamoto.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kim, S.C., Okamoto, H. The generalized Proudman–Johnson equation and its singular perturbation problems. Japan J. Indust. Appl. Math. 31, 541–573 (2014). https://doi.org/10.1007/s13160-014-0146-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-014-0146-4

Keywords

Mathematics Subject Classification

Search

Navigation