The asymptotic behavior of the stochastic Ginzburg-Landau equation with additive noise. (English) Zbl 1139.65007

Summary: We show that the stochastic Ginzburg-Landau equation with additive noise can be solved pathwise and the unique solution generates a random dynamic system. Then we prove that the system possesses a compact random attractor in \(L^{2}(D)\) when the spatial dimension of \(D\) is one and two, respectively.


65C30 Numerical solutions to stochastic differential and integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
35F05 Linear first-order PDEs
Full Text: DOI


[1] Temam, R., Infinite-Dimensional Systems in Mechanics and Physics (1988), Springer-Verlag: Springer-Verlag New York · Zbl 0662.35001
[2] Ginibre, J.; Velo, G., The Cauchy problem in local spaces for the complex Ginzburg-Landau equation, I. Compactness, methods, Physica D, 95, 191-228 (1996) · Zbl 0889.35045
[3] Ghidaglia, J. M.; Héron, B., Dimension of the attractor associated to the Ginzburg-Landau equation, Physica D, 28, 282-304 (1987) · Zbl 0623.58049
[4] Crauel, H.; Flaudoli, F., Attractors for random dynamical systems, Probab. Theory Relat. Fields, 100, 365-393 (1994) · Zbl 0819.58023
[5] Crauel, H.; Debussche, A.; Franco, F., Random attractors, J. Dyn. Diff. Eq., 9, 2, 307-341 (1997) · Zbl 0884.58064
[6] Yang, D., The asymptotic behavior of the stochastic Ginzburg-Landau equation with multiplicative noise, J. Math. Phys., 45, 11, 4064-4076 (2004) · Zbl 1064.37038
[7] Arnold, L., Random dynamical system, Springer Monographs in Mathematics (1998), Springer-Verlag: Springer-Verlag Berlin · Zbl 0906.34001
[8] Gardiner, C. W., Handbooks of Stochastic Methods for Physics, Chemistry and Natural Sciences (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0862.60050
[9] Debussche, A., Hausdorff dimension of random invariant set, J. Math. Pure Appl., 77, 967-988 (1998) · Zbl 0919.58044
[10] Da Prato, G.; Zabczyk, J., Stochastic Equations in Infinite Dimensions (1992), Cambridge University Press · Zbl 0761.60052
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.