Homogenization of trajectory attractors of Ginzburg-Landau equations with randomly oscillating terms

Gregory A. Chechkin; Vladimir V. Chepyzhov; Leonid S. Pankratov

doi:10.3934/dcdsb.2018145

Article Contents

2018, Volume 23, Issue 3: 1133-1154. Doi: 10.3934/dcdsb.2018145

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Homogenization of trajectory attractors of Ginzburg-Landau equations with randomly oscillating terms

1.
Baku branch of M.V. Lomonosov Moscow State University, Universitetskaya st., 1, Xocasan, Binagadi district, Baku, AZ 1144, Azerbaijan
2.
M.V. Lomonosov Moscow State University, Moscow, 119991, Russian Federation
3.
Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetniy 19, Moscow 127051, Russian Federation
4.
Voronezh State University, Universitetskaya sq. 1, Voronezh 394018, Russian Federation
5.
Laboratory of Fluid Dynamics and Seismic (RAEP 5top100), Moscow Institute of Physics and Technology, Institutskiy 9, Dolgoprudny, Moscow Region 141700, Russian Federation

* Corresponding author: G. A. Chechkin
* Corresponding author: G. A. Chechkin

Received: March 2017

Revised: September 2017

Early access: February 2018

Published: July 2018

Work of GAC was supported in part by the Committee of Science of the Ministry of Education and Science of the Republic of Kazakhstan (grant no. AP05131707) and by the Russian Foundation for Basic Research (projects 18-01-00046). This research of VVC was supported by the Ministry of Education and Science of the Russian Federation (grant 14.Z50.31.0037). The work of LSP was partially supported by Russian Science Foundation (grant no. 18-11-00148).

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Abstract

We consider complex Ginzburg-Landau (GL) type equations of the form:

${\partial _t}u = (1 + \alpha i)\Delta u + R{\mkern 1mu} u + (1 + \beta i)|u{|^2}u + g,$

where $R$, $β$, and $g$ are random rapidly oscillating real functions. Assuming that the random functions are ergodic and statistically homogeneous in space variables, we prove that the trajectory attractors of these systems tend to the trajectory attractors of the homogenized equations whose terms are the average of the corresponding terms of the initial systems.

Bibliography: 52 titles.

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Mathematics Subject Classification: Primary: 35B40, 35B41, 35B45, 35Q30.

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Figure 1. Attractors of the Ginzburg-Landau Equations

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