Japan Journal of Industrial and Applied Mathematics

, Volume 35, Issue 3, pp 1065–1083 | Cite as

Unimodal solutions of the generalized Constantin–Lax–Majda equation with viscosity

  • Sun-Chul Kim
  • Tomoyuki Miyaji
  • Hisashi OkamotoEmail author
Original Paper Area 1


Steady-states of the generalized Constantin–Lax–Majda equation with the viscosity and an external force are computed numerically by the spectral method. This equation is regarded as a model for two-dimensional turbulent motion of incompressible viscous fluid. We demonstrate numerically that the equation admits unimodal solutions—solutions with one and only one peak and bottom, if the Reynolds number is sufficiently large. We also report some interesting properties of the spectra of unimodal solutions.


Unimodal solution Bifurcation Large Reynolds number flow 

Mathematics Subject Classification

65N35 76N99 



One of the referees kindly let us know the reference [21] and his/her comments are acknowledged to be very useful.


  1. 1.
    Bae, H., Chae, D., Okamoto, H.: On the well-posedness of various one-dimensional model equations for fluid motion. Nonlinear Anal. 160, 25–43 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Batchelor, G.K.: Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids 12, II-233–II-239 (1969)CrossRefGoogle Scholar
  3. 3.
    Chen, X., Okamoto, H.: Global existence of solutions to the generalized Proudman–Johnson equation. Proc. Jpn. Acad. Ser. A 78, 136–139 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Constantin, P., Lax, P.D., Majda, A.J.: A simple one-dimensional model for the three-dimensional vorticity equation. Commun. Pure Appl. Math. 38, 715–724 (1985)MathSciNetCrossRefGoogle Scholar
  5. 5.
    De Gregorio, S.: On a one-dimensional model for the three-dimensional vorticity equation. J. Stat. Phys. 59, 1251–1263 (1990)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Govaerts, W.J.F.: Numerical methods for bifurcation of dynamical equilibria. SIAM, Philadelphia (2000)CrossRefGoogle Scholar
  7. 7.
    Iudovich, V.I.: Example of the generation of a secondary stationary or periodic flow when there is loss of stability of the laminar flow of a viscous incompressible fluid. J. Appl. Math. Mech. 29, 527–544 (1965)CrossRefGoogle Scholar
  8. 8.
    Julian, P.R., Washington, W.M., Hembree, L., Ridley, C.: On the spectral distribution of large-scale atmospheric kinetic energy. J. Atmos. Sci. 27, 376–387 (1970)CrossRefGoogle Scholar
  9. 9.
    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)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kim, S.-C., Okamoto, H.: The generalized Proudman–Johnson equation at large Reynolds numbers. IMA J. Appl. Math. 78, 379–403 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kim, S.-C., Okamoto, H.: Unimodal patterns appearing in the Kolmogorov flows at large Reynolds numbers. Nonlinearity 28, 3219–3242 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kraichnan, R.: Inertial ranges in two-dimensional turbulence. Phys. Fluid 10, 1417–1423 (1967)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Marchioro, C.: An example of absence of turbulence for any Reynolds number. Commun. Math. Phys. 105, 99–106 (1986)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Matsumoto, T., Sakajo, T.: One-dimensional hydrodynamic model generating a turbulent cascade. Phys. Rev. E 93, 053101 (2016)CrossRefGoogle Scholar
  15. 15.
    Matsumoto, T., Sakajo, T.: Turbulence, Cascade and singularity in a generalization of the Constantin–Lax–Majda equation. arXiv:1707.05205
  16. 16.
    Nastrom, G.D., Gage, K.S., Jasperson, W.H.: Kinetic energy spectrum of large and mesoscale atmospheric processes. Nature 310, 36–38 (1984)CrossRefGoogle Scholar
  17. 17.
    Okamoto, H.: Nearly singular two-dimensional Kolmogorov flows for large Reynolds number. J. Dyn. Differ. Equ. 8, 203–220 (1996)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Okamoto, H.: Models and special solutions of the Navier–Stokes equations. In: Giga Y, Novotoný A (eds.) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer (2018)Google Scholar
  19. 19.
    Okamoto, H., Sakajo, T., Wunsch, M.: On a generalization of the Constantin–Lax–Majda equation. Nonlinearity 21, 2447–2461 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Okamoto, H., Sakajo, T., Wunsch, M.: Steady-states and traveling wave solutions of the generalized Constantin–Lax–Majda equation. Discrete Contin. Dyn. Syst. 34, 3155–3170 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Rutgers, M.A.: Forced 2D turbulence: experimental evidence of simultaneous inverse energy and forward enstrophy cascades. Phys. Rev. Lett. 81, 2244–2247 (1998)CrossRefGoogle Scholar
  22. 22.
    Saffman, P.G.: On the spectrum and decay of random two-dimensional vorticity distributions at large Reynolds number. Stud. Appl. Math. 50, 377–383 (1971)CrossRefGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsChung-Ang UniversitySeoulRepublic of Korea
  2. 2.Meiji Institute for Advanced Study of Mathematical SciencesMeiji UniversityTokyoJapan
  3. 3.Department of MathematicsGakushuin UniversityTokyoJapan

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