Abstract
We show that the solutions of the Ginzburg-Landau equation on a periodic interval are of Gevrey-class regularity. Specifically, we prove that the modes of the Fourier decomposition of a solution u decay exponentially. Using this preliminary result, we establish that the N-dimensional linear Galerkin approximation of u converges exponentially fast to uas N goes to infinity. We discuss the influence of the parameters in the Ginzburg-Landau equation on the decay rate of the Fourier modes and on the rate of convergence of the Galerkin approximations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bartuccelli, M., Constantin, P., Doering, C. R., Gibbon, J. D. and Gisselfält, M. (1990) ‘On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation’, Phys. 44D, 421–444.
Bernoff, A. J. (1988) ‘Slowly varying fully nonlinear wavetrains in the Ginzburg-Landau equation’, Phys., 30D, 363–381.
Constantin, P., and Foias, C. (1988) Navier-Stokes Equations, University of Chicago Press, Chicago.
Devulder, C., Marion, M., and Titi, E.S. (1992) ‘On the rate of convergence of the nonlinear Galerkin methods’, Math. of Comp., to appear.
DiPrima, R. C., Eckhaus, W., and Segel, L. A. (1971) ‘Non-linear wave-number interaction in near-critical two-dimensional flows’, J. Fluid Mech., 49, 705–744.
Doelman, A. (1989) ‘Slow time-periodic solutions of the Ginzburg-Landau equation’, Phys., 40D, 156–172.
Doelman, A. (1991) ‘Finite dimensional models of the Ginzburg-Landau equation’, Nonlin., 4, 231–250.
Doelman, A., and Titi, E. S. (1992) ‘Regularity of solutions and the convergence of the linear Galerkin method in the Ginzburg-Landau equation’, in preparation.
Doering, C. R., Gibbon, J. D., Holm, D. D., and Nicolaenko, B. (1988) ‘Low dimensional behaviour in the complex Ginzburg-Landau equation’, Nonlin., 1, 279–310.
Foias, C., and Temam, R. (1989) ‘Gevrey class regularity for the solutions of the Navier-Stokes equations’, J. Funct. An., 87, 359–369.
Friedman, A. (1976) Partial Differential Equations, R. Krieger Pub. Co., New York.
Ghidaglia, J. M., and Héron, B. (1987) ‘Dimension of the attractors associated to the Ginzburg-Landau equation’, Phys., 28D, 282–304.
van Harten, A. (1991) ‘On the validity of the Ginzburg-Landau equation’, J. Nonl. Sc., 1, 397–422.
Hocking, L. M., Stewartson, K., and Stuart, J. T. (1972) ‘A nonlinear stability burst in plane parallel flow’, J. Fluid Mech., 51, 705–735.
Holmes, P. (1986) ‘Spatial structure of time-periodic of the Ginzburg-Landau equation’, Phys., 23D, 84–90.
Iooss, G., Mielke, A., and Demay, Y. (1989) ‘Theory of steady Ginzburg-Landau equation, in hydrodynamic stability problems’, Eur. J. Mech. B/Fluids, 8, 229–268.
Keefe, L. R. (1985) ‘Dynamics of perturbed wave train solutions to the Ginzburg-Landau equation’, Stud. in Appl. Math., 73, 91–153.
Moon, H. T., Huerre, P., and Redekopp, L. G. (1983) ‘Transitions to chaos in the Ginzburg-Landau equation’, Phys., 7D, 135–150.
Newell, A. C. (1974) ‘Envelope equations’ Lect. Appl. Math., Vol. 15, 157–163.
Promislow, K. (1991) ‘The development and numerical implementation of approximate inertial manifolds for the Ginzburg-Landau equation’, Ph.D. thesis, Indiana University.
Rodriguez, J. D., and Sirovich, L. (1990) ‘Low-dimensional dynamics for the complex Ginzburg-Landau equation’, Phys., 43D, 77–86.
Stewartson, K., and Stuart, J. T. (1971) ‘A non-linear instability theory for a wave system in plane Poiseuille flow’, J. Fluid Mech., 48, 529–545.
Temam, R. (1977) Navier-Stokes Equations: Theory and Numerical Solution, Stud. Math. Appl. Vol. 2, North-Holland, Amsterdam.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Doelman, A., Titi, E.S. (1993). Exponential Convergence of the Galerkin Approximation for the Ginzburg-Landau Equation. In: Kaper, H.G., Garbey, M., Pieper, G.W. (eds) Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters. NATO ASI Series, vol 384. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-1810-1_15
Download citation
DOI: https://doi.org/10.1007/978-94-011-1810-1_15
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-4798-2
Online ISBN: 978-94-011-1810-1
eBook Packages: Springer Book Archive