Abstract
This paper studies the long-term behavior of solutions to the Ginzburg-Landau partial differential equation. For each positive integerm we explicitly produce a sequence of approximate inertial manifoldsℳ m,j ,j = 1, 2,..., of dimensionm and associate with each manifold a thin neighborhood into which the orbits enter with an exponential speed and in a finite time. Of course this neighborhood contains the universal attractor which embodies the large time dynamics of the equations. The thickness of these neighborhoods decreases with increasingm for a fixed orderj; however, for a fixedm no conclusion can be made about the thickness of the neighborhoods associated to two differentj's. The neighborhoods associated to the manifolds localize the universal attractor and provide computabie large time approximations to solutions of the Ginzburg-Landau equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Benjamin, T. B. (1967). Some developments in the theory of vortex breakdown.J. Fluid Mech. 28, 65–84.
Blennerhassett, P. J. (1980). On the generation of waves of wind.Philos. Trans. Roy. Soc. Lond. Ser. A 298, 451–494.
Constantin, P., Foias, C., Nicolaenko, B., and Temam, R. (1988).Integral and Inertial Manifolds for Dissipative Partial Differential Equations, Applied Mathematical Science Service, Vol. 70; Springer-Verlag, New York.
Debussche, A., and Marion, M. (1991). On the construction of families of approximate inertial manifolds.J. Diff. Eq. (in press).
Foias, C., Manley, O., and Temam, R. (1988a). Modefling of the interaction of small and large eddies in two dimensional turbulent flows.Math. Model. Numer. Anal. 22, 93–118.
Foias, C., Nicolaenko, B., Sell, G., and Temam, R. (1988b). Inertial manifolds for the Kuramoto Sivashinsky equation and an estimate of their lowest dimension.J. Math. Pures Appl. 67, 197–226.
Foias, C., Sell, G., and Temam, R. (1988c). Inerital Manifolds for nonlinear evolution equations.J. Diff. Eq. 73, 309–353.
Ghidaglia, J. M., and Héron, B. (1987). Dimension of the attractors associated to the Ginzburg-Landau partial differential equation.Physica 28D, 282–304.
Haken, H. (1983).Advanced Synergetics, Springer-Verlag, New York.
Jauberteau, F., Rosier, C., and Temam, R. (1989/1990). The nonlinear Galerkin method in computational fluid dynamics.Appl. Numer. Math. 6, 361–370.
Maller-Paret, J., and Sell, G. (1988). Inertial manifolds for reaction-diffusion equations in higher space dimension.J. AMS 1, 805–866.
Marion, M. (1989). Approximate inertial manifolds for reaction-diffusion equations in high space dimension.J. Dynam. Diff. Eq. 1, 245–267.
Marion, M., and Temam, R. (1989). Nonlinear Galerkin methods.SIAM J. Num. Anal. 26(5), 1139–1157.
Moon, H. T., Huerre, P., and Redekopp, L. G. (1982). Three frequency motion and chaos in the Ginzburg-Landau equation.Phys. Rev. Lett. 49, 458–460.
Moon, H. T., Huerre, P., and Redekopp, L. G. (1983). Transitions to chaos in Ginzburg-Landau equation.Physica 7D, 135–150.
Newell, A. C., and Whitehead, J. A. (1969). Finite bandwidth, finite amplitude convection.J. Fluid Mech. 38, 279–304.
Nicolaenko, B., Scheureu, B., and Temam, R. (1985). Some global dynamical properties of the Kuramato-Sivashinsky equation: Nonlinear stability and attractors.Physica 16D, 155–183.
Nirenberg, L. (1959). On elliptic partial differential equations.Ann. Sc. Norm. Sup. Pisa 13, 116–162.
Promislow, K. S. (1990). Induced trajectories and approximate inertial manifolds for the Ginzburg-Landau partial differential equation.Physica D 41, 232–252.
Promislow, K. S. (1991). Time analyticity and Gevrey class regularity for the solutions of a class of dissipative partial differential equations.J. Nonlin. Anal. Theory Methods Appl. 16, 959–980.
Stuart, J. T., and Di Prima, R. C. (1978). The Eckhaus and Benjamin-Feir resonance mechanisms.Proc. Roy. Soc. London Ser. A 362, 27–41.
Temam, R. (1988a).Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York.
Temam, R. (1988b). Variétés inertielles approximatives pour les équations de Navier-Stoke bidimensionelle.CRAS 306 (Ser. II), 399–402.
Temam, R. (1988c). Dynamical systems, turbulence and the numerical solution of the Navier-Stokes equation. In Dwoyer, D., and Voigt, R. (eds.),Proceedings of the 11'th International Conference on Numerical Methods in Fluid Dynamics, Williamsburg, June, Lecture Notes in Physics, Springer-Verlag, New York.
Temam, R. (1989). Attractors for the Navier-Stokes equations: Localization and approximation.J. Fac. Sc. Tokyo Sect. IA 36, 626–647.
Titi, E. S. (1988). Une variété approximante de l'attracteur universel des équations de Navier-Stokes, nonlinéaire, de dimension finie.C. R. Acad. Sci. Sér. I Paris 307, 383–385.
Titi, E. S. (1990). On approximate inertial manifolds to the Navier-Stokes equations.J. Math. Anal. Appl. 149(2), 540–557.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Promislow, K., Temam, R. Localization and approximation of attractors for the Ginzburg-Landau equation. J Dyn Diff Equat 3, 491–514 (1991). https://doi.org/10.1007/BF01049097
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01049097