Mathematical Justification of Steady Ginzburg-Landau Equation Starting from Navier-Stokes

  • Gérard Iooss
  • Alexander Mielke
  • Yves Demay
Part of the NATO ASI Series book series (NSSB, volume 237)


Many classical hydrodynamical stability problems deal with flows in a very long domain. This is often theoretically modelized by an infinite domain, which simplifies the linear analysis. Here we consider cases of cylindrical domains of one or two dimensional bounded cross-section Ω. Examples of such a situation are i) the Taylor-Couette problem of the flow between two concentric rotating cylinders, where the section is a 2-dimensional annulus, ii) the Bénard convection problem of a liquid heated from below in a long box, and where the section is a rectangle. In both of these problems there are two very important symmetries. First, the problem is invariant under translations parallel to the generatrices of the cylinder, second the problem is invariant under the reflection symmetry through any cross-sectional plane.


Periodic Solution Normal Form Rayleigh Number Imaginary Axis Center Manifold 


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  1. [A-K]
    C.J.Amick, K.Kirchgässner. Arch.Rat.Mech.Anal. (to appear).Google Scholar
  2. [C-E 86]
    P. Collet, J. P. Eckmann. Comm.Math.Phys. 107, 39–92, 1986.MathSciNetADSCrossRefGoogle Scholar
  3. [C-E 87]
    P., Collet, J.P., Eckmann. Helvetica Phys.Acta 60, 969, 1987.MathSciNetMATHGoogle Scholar
  4. [C-S]
    P., Coullet, E.A., Spiegel. SIAM J.Applied Math. 43, 774–819, 1983.Google Scholar
  5. [C-R]
    P.Coullet, D.Repaux. Instabilities and Nonequilibrium Structures. E.Tirapegui, D.Villaroel ed.,179–195,Reidel, 1987.CrossRefGoogle Scholar
  6. [Ec]
    W.Eckhaus. Studies in nonlinear stability theory. Springer tracts in Nat. Philo. Vol.6,1965.Google Scholar
  7. [E-T-B-C-I]
    C.Elphick, E.Tirapegui, M.E.Brachet, P.Coullet, G.Iooss. Physica 29D, 95–127, 1987.MathSciNetMATHGoogle Scholar
  8. [He]
    D.Henry. Springer Lecture Notes in Math.840,1981.Google Scholar
  9. [Io 71]
    G.Iooss. Arch.Rat.Mech.Anal. 40,3,166–208,1971.MathSciNetMATHCrossRefGoogle Scholar
  10. [I-M-D]
    G.Iooss, A.Mielke, Y.Demay. Eur.J.Mech.B,1989 (to appear).Google Scholar
  11. [Jul]
    V.I.Iudovich. Dokl.Akad.Nauk. SSSR, 161,5,1037–1040, 1965.Google Scholar
  12. [Ka]
    T. Kato. Perturbation theory for linear operators. Springer Verlag, Berlin, 1966.MATHGoogle Scholar
  13. [Ki]
    K. Kirchgässner. J.Diff.Equ. 45, 113–127, 1982.MATHCrossRefGoogle Scholar
  14. [K-Z]
    L. Kramer, W. Zimmermann. Physica 16D, 221–232, 1985.MATHGoogle Scholar
  15. [Mi 86a]
    A. Mielke. J.Diff.Equ. 65, 68–88, 1986.MathSciNetMATHCrossRefGoogle Scholar
  16. [Mi 86b]
    A. Mielke. J.Diff.Equ. 65, 89–116, 1986.MathSciNetMATHCrossRefGoogle Scholar
  17. [Mi 88a]
    A. Mielke. Math.Meth.Appl.Sci. 10, 51–66, 1988.MathSciNetMATHCrossRefGoogle Scholar
  18. [Mi 88b]
    A. Mielke. Arch.Rat.Mech.Anal. 102, 205–229, 1988.MathSciNetMATHCrossRefGoogle Scholar
  19. [N-W]
    A. Newell, J. Whitehead. J.Fluid Mech. 38, 2, 279–303, 1969.MathSciNetADSMATHCrossRefGoogle Scholar
  20. [Se]
    L.A. Segel. J.Fluid Mech. 38,1, 203–224, 1969.ADSMATHCrossRefGoogle Scholar
  21. [Va]
    A.Vanderbauwhede. Dynamics Reported 2, 1989 (to appear).Google Scholar

Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • Gérard Iooss
    • 1
  • Alexander Mielke
    • 2
  • Yves Demay
    • 1
  1. 1.Laboratoire de MathématiquesU.A.CNRS 168, Université de Nice, Parc ValroseNiceFrance
  2. 2.Mathematisches Institut AUniversität StuttgartStuttgart 80Germany

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