×

The Ginzburg-Landau manifold is an attractor. (English) Zbl 0797.35070

Summary: The Ginzburg-Landau modulation equation arises in many domains of science as a (formal) approximate equation describing the evolution of patterns through instabilities and bifurcations. Recently, for a large class of evolution PDE’s in one space variable, the validity of the approximation has rigorously been established in the following sense: Consider initial conditions of which the Fourier-transforms are scaled according to the so-called clustered mode-distribution. Then the corresponding solutions of the “full” problem and the G-L equation remain close to each other on compact intervals of the intrinsic Ginzburg-Landau time-variable. In this paper the following complementary result is established. Consider small, but arbitrary initial conditions. The Fourier-transforms of the solutions of the “full” problem settle to clustered mode-distribution on time-scales which are rapid as compared to the time-scale of evolution of the Ginzburg-Landau equation.

MSC:

35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
35K55 Nonlinear parabolic equations
35B32 Bifurcations in context of PDEs
76E30 Nonlinear effects in hydrodynamic stability
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] P. Collet and J.-P. Eckmann. The time-dependant amplitude equation for the Swift-Hohenberg problem.Commun. Math. Physics 132, 139–153 (1990). · Zbl 0756.35096 · doi:10.1007/BF02278004
[2] R. C. DiPrima, W. Eckhaus, and L. A. Segel. Nonlinear wave-number interactions at near-critical flows.J. Fluid Mech. 49, 705–744 (1971). · Zbl 0229.76039 · doi:10.1017/S0022112071002337
[3] A. Doelman.On nonlinear evolution of patterns. Thesis, Dept. of Mathematics, Utrecht (1990).
[4] W. Eckhaus. On modulation equations of the Ginzburg-Landau type.Proceedings of ICIAM ’91. · Zbl 0797.35070
[5] A. van Harten. On the validity of Ginzburg-Landau’s equation.J. Nonlin. Sci. 1, 397–422 (1991). · Zbl 0795.35112 · doi:10.1007/BF02429847
[6] G. Iooss, A. Mielke, and Y. Demay. Theory of steady Ginzburg-Landau equations in hydrodynamic stability problems.Eur. J. Mech. B/Fluids 3, 229–268 (1989). · Zbl 0675.76060
[7] G. Iooss and A. Mielke. Bifurcating time-periodic solutions of Navier-Stokes equations in infinite cylinders.J. Nonlin. Sci. 1, 107–146 (1991). · Zbl 0797.76010 · doi:10.1007/BF01209150
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.