Advertisement

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)

Abstract

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  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.MathSciNetCrossRefGoogle Scholar
  3. [C-E 87]
    P., Collet, J.P., Eckmann. Helvetica Phys.Acta 60, 969, 1987.MathSciNetGoogle 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.MathSciNetGoogle 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.MathSciNetCrossRefGoogle 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.CrossRefGoogle Scholar
  14. [K-Z]
    L. Kramer, W. Zimmermann. Physica 16D, 221–232, 1985.Google Scholar
  15. [Mi 86a]
    A. Mielke. J.Diff.Equ. 65, 68–88, 1986.CrossRefGoogle Scholar
  16. [Mi 86b]
    A. Mielke. J.Diff.Equ. 65, 89–116, 1986.CrossRefGoogle Scholar
  17. [Mi 88a]
    A. Mielke. Math.Meth.Appl.Sci. 10, 51–66, 1988.CrossRefGoogle Scholar
  18. [Mi 88b]
    A. Mielke. Arch.Rat.Mech.Anal. 102, 205–229, 1988.CrossRefGoogle Scholar
  19. [N-W]
    A. Newell, J. Whitehead. J.Fluid Mech. 38, 2, 279–303, 1969.CrossRefGoogle Scholar
  20. [Se]
    L.A. Segel. J.Fluid Mech. 38,1, 203–224, 1969.CrossRefGoogle 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

Personalised recommendations