Error estimates for the Ginzburg-Landau approximation

  • Guido Schneider
Original Papers


Modulation equations play an essential role in the understanding of complicated dynamical systems near the threshold of instability. Here we look at systems defined over domains with one unbounded direction and show that the Ginzburg-Landau equation dominates the dynamics of the full problem, locally, at least over a long time-scale. As an application of our approximation theorem we look here at Bénard's problem. The method we use involves a careful handling of critical modes in the Fourier-transformed problem and an estimate of Gronwall's type.


Dynamical System Error Estimate Essential Role Mathematical Method Critical Mode 
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  1. [CH82]
    S.-N. Chow and J. K. Hale,Methods of Bifurcation Theory. Springer, Berlin 1982.Google Scholar
  2. [CE90]
    P. Collet and J.-P. Eckmann,The time dependent amplitude equation for the Swift-Hohenberg problem. Comm. Math. Phys.132, 139–153 (1990).Google Scholar
  3. [diPES71]
    R. C. di Prima, W. Eckhaus and L. A. Segal,Nonlinear wave-number interaction in near-critical two-dimensional flows. J. Fluid Mech.49, 705–744 (1971).Google Scholar
  4. [Eck92]
    W. Eckhaus,The Ginzburg-Landau equation is an attractor. Preprint 746, University Utrecht (1992).Google Scholar
  5. [He81]
    D. Henry,Geometric theory of semilinear parabolic equations. Lect. Notes in Maths 840, Springer, Berlin 1981.Google Scholar
  6. [IMD89]
    G. Iooss, A. Mielke and Y. Demay,Theory of steady Ginzburg-Landau equation, in hydrodynamic stability problems. Eur. J. Mech., B/Fluids,8, nr. 3, 229–268 (1989).Google Scholar
  7. [KSM92]
    P. Kirrmann, G. Schneider and A. Mielke,The validity of modulation equations for extendedsystems with cubic nonlinearities. Proc. Royal Soc. Edinburgh122A, 85–91 (1992).Google Scholar
  8. [Mi92]
    A. Mielke,Reduction of PDE's on domains with several unbounded directions. J. Appl. Math. Phys. (ZAMP)43 449–470 (1992).Google Scholar
  9. [NW69]
    A. Newell and J. Whitehead,Finite bandwidth, finite amplitude convection. J. Fluid Mech.38 279–303 (1969).Google Scholar
  10. [Sch92a]
    G. Schneider,Die Gültigkeit der Ginzburg-Landau-Approximation. Dissertation, Universität Stuttgart 1992.Google Scholar
  11. [Sch92b]
    G. Schneider,A new estimate for the Ginzburg-Landau approximation on the real axis. J. Nonlin. Sci., to appear 1994.Google Scholar
  12. [VI92]
    A. Vanderbauwhede and G. Iooss,Center-manifold in infinite dimensions. Dynamics Reported1, 125–162. Springer-Verlag, Berlin 1992.Google Scholar
  13. [vH91]
    A. van Harten,On the validity of Ginzburg-Landau's equation. J. Nonlin. Sci.1, 397–422 (1991).Google Scholar
  14. [vH92]
    A. van Harten,Decay to the Ginzburg-Landau manifold. Lecture held in Stuttgart, July 1992.Google Scholar

Copyright information

© Birkhäuser Verlag 1994

Authors and Affiliations

  • Guido Schneider
    • 1
  1. 1.Institut für Angewandte MathematikUniversität HannoverHannoverGermany

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