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.
Citation: |
Y. Amirat
, O. Bodart
, G. A. Chechkin
and A. L. Piatnitski
, Boundary homogenization in domains with randomly oscillating boundary, Stochastic Processes and their Applications, 121 (2011)
, 1-23.
![]() ![]() |
|
V. I. Arnol'd and A. Avez, Ergodic Problems of Classical Mechanics, W. A. Benjamin Inc., New York - Amsterdam, 1968.
![]() ![]() |
|
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992.
![]() ![]() |
|
N. S. Bakhvalov and G. P. Panasenko, Averaging Processes in Periodic Media, Mathematics and its Applications (Soviet Series), 36. Kluwer Academic Publishers Group, Dordrecht, 1989.
![]() ![]() |
|
K. A. Bekmaganbetov
, G. A. Chechkin
, V. V. Chepyzhov
and A. Yu. Goritsky
, Homogenization of Trajectory Attractors of 3D Navier-Stokes system with Randomly Oscillating Force, Discrete and Continuous Dynamical Systems. Series A (DCDS-A), 37 (2017)
, 2375-2393.
![]() ![]() |
|
K. A. Bekmaganbetov
, G. A. Chechkin
and V. V. Chepyzhov
, Homogenization of Random Attractors for Reaction-Diffusion Systems, C R Mécanique, 344 (2016)
, 753-758.
doi: 10.1016/j.crme.2016.10.015.![]() ![]() |
|
A. Bensoussan, J. -L. Lions and G. Papanicolau, Asymptotic Analysis for Periodic Structures, Corrected reprint of the 1978 original [MR0503330]. AMS Chelsea Publishing, Providence, RI, 2011.
![]() ![]() |
|
G. D. Birkhoff
, Proof of the ergodic theorem, Proc Natl Acad Sci USA, 17 (1931)
, 656-660.
![]() |
|
N. N. Bogolyubov and Ya. A. Mitropolski, Asymptotic Methods in the Theory of Non-Linear Oscillations, International Monographs on Advanced Mathematics and Physics, Hindustan Publishing Corp., Delhi, Gordon & Breach Science Publishers, New York, 1961.
![]() ![]() |
|
A. Bourgeat
, I. D. Chueshov
and L. Pankratov
, Homogenization of attractors for semilinear parabolic equations in domains with spherical traps, Comptes rendues de l'Académie des Sciences, série I, 329 (1999)
, 581-586.
![]() ![]() |
|
A. Bourgeat and L. Pankratov, Homogenization of reaction-diffusion equations in domains with "traps". In : Proceedings of the International Conference "Porous Media: Physics, Modelling, Simulation", Ed. A. Dmitrievsky, M. Panfilov, World Scientific, Singapore-New Jersey-London-Hong Kong, 2000,267-278.
![]() |
|
L. Boutet de Monvel
, I.D. Chueshov
and E. Ya. Khruslov
, Homogenization of attractors for semilinear parabolic equations on manifolds with complicated microstructure, Ann. Mat. Pura Appl., 172 (1997)
, 297-322.
![]() ![]() |
|
F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183. Springer, New York, 2013.
![]() ![]() |
|
G. A. Chechkin, A. L. Piatnitski and A. S. Shamaev, Homogenization. Methods and Applications, Translated from the 2007 Russian original by Tamara Rozhkovskaya. Translations of Mathematical Monographs, 234. American Mathematical Society, Providence, RI, 2007.
![]() ![]() |
|
G. A. Chechkin
, T. P. Chechkina
, C. D'Apice
and U. De Maio
, Homogenization in Domains Randomly Perforated Along the Boundary, Discrete Continuous Dynam. Systems -B, 12 (2009)
, 713-730.
![]() ![]() |
|
G. A. Chechkin
, T. P. Chechkina
, C. D'Apice
, U. De Maio
and T. A. Mel'nyk
, Asymptotic Analysis of a Boundary Value Problem in a Cascade Thick Junction with a Random Transmission Zone, Applicable Analysis, 88 (2009)
, 1543-1562.
![]() ![]() |
|
G. A. Chechkin
, T. P. Chechkina
, C. D'Apice
, U. De Maio
and T. A. Mel'nyk
, Homogenization of 3D Thick Cascade Junction with the Random Transmission Zone Periodic in One direction, Russ. J. Math. Phys., 17 (2010)
, 35-55.
![]() ![]() |
|
G. A. Chechkin
, C. D'Apice
, U. De Maio
and A. L. Piatnitski
, On the Rate of Convergence of Solutions in Domain with Random Multilevel Oscillating Boundary, Asymptotic Analysis, 87 (2014)
, 1-28.
![]() ![]() |
|
G. A. Chechkin
, T. P. Chechkina
, T. S. Ratiu
and M. S. Romanov
, Nematodynamics and Random Homogenization, Applicable Analysis, 95 (2016)
, 2243-2253.
doi: 10.1080/00036811.2015.1036241.![]() ![]() ![]() |
|
V. V. Chepyzhov
, A. Yu. Goritski
and M. I. Vishik
, Integral manifolds and attractors with exponential rate for nonautonomous hyperbolic equations with dissipation, Russ. J. Math. Phys., 12 (2005)
, 17-39.
![]() ![]() |
|
V. V. Chepyzhov
, V. Pata
and M. I. Vishik
, Averaging of nonautonomous damped wave equations with singularly oscillating external forces, J. Math. Pures Appl., 90 (2008)
, 469--491.
![]() ![]() |
|
V. V. Chepyzhov
and M. I. Vishik
, Trajectory attractors for reaction-diffusion systems, Topol. Methods Nonlinear Anal. J.Julius Schauder Center, 7 (1996)
, 49-76.
![]() ![]() |
|
V. V. Chepyzhov
and M. I. Vishik
, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997)
, 913-964.
![]() ![]() |
|
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002.
![]() ![]() |
|
V. V. Chepyzhov
and M. I. Vishik
, Global attractors for non-autonomous Ginzburg-Landau equation with singularly oscillating terms, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 29 (2005)
, 123-148.
![]() ![]() |
|
I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Kharkov, AKTA, 1999.
![]() ![]() |
|
I. D. Chueshov
and L. S. Pankratov
, Upper semicontinuity of attractors of semilinear parabolic equations with asymptotically degenerating coefficients, Mat. Fiz., Analiz, Geom., 6 (1999)
, 158-181.
![]() ![]() |
|
I. D. Chueshov
and L. S. Pankratov
, Averaging of attractors of nonlinear hyperbolic equations with asymptotically degenerate coefficients, Sb. Math., 190 (1999)
, 1325-1352.
![]() ![]() |
|
I. D. Chueshov
and B. Schmalfuß
, Averaging of attractors and inertial manifolds for parabolic PDE with random coefficients, Advanced Nonlinear Studies, 5 (2005)
, 461-492.
![]() ![]() |
|
Yu. A. Dubinskiĭ
, Weak convergence in nonlinear elliptic and parabolic equations, Sb. Math., 67(109) (1965)
, 609-642.
![]() ![]() |
|
N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, John Wiley & Sons, Inc., New-York, 1988.
![]() ![]() |
|
M. Efendiev
and S. Zelik
, Attractors of the reaction-diffusion systems with rapidly oscillating coefficients and their homogenization, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002)
, 961-989.
![]() ![]() |
|
B. Fiedler
and M. I. Vishik
, Quantitative homogenization of global attractors for reaction-diffusion systems with rapidly oscillating terms, Asymptotic Anal., 34 (2003)
, 159-185.
![]() ![]() |
|
J. M. Ghidaglia
and B. Héron
, Dimension of the attractors associated to the Ginzburg-Landau partial differential equation, Physica D, 28 (1987)
, 282-304.
![]() ![]() |
|
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.
![]() ![]() |
|
J. K. Hale
and S. M. Verduyn Lunel
, Averaging in infinite dimensions, J. Int. Eq. Appl., 2 (1990)
, 463-494.
![]() ![]() |
|
A. A. Ilyin
, Averaging principle for dissipative dynamical systems with rapidly oscillating right-hand sides, Sb. Math., 187 (1996)
, 635-677.
![]() ![]() |
|
V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994.
![]() ![]() |
|
E. Ya. Khruslov and L. S. Pankratov, Homogenization of boundary problems for GinzburgLandau equation in weakly connected domains, In: Spectral Operator Theory and Related Topics, edited by V. A. Marchenko, AMS Providence, 19 (1994), 233-268.
![]() ![]() |
|
J. -L. Lions, Quelques Méthodes de Résolutions des Problémes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.
![]() ![]() |
|
J. -L. Lions and E. Magenes, Problémes Aux Limites Non Homogénes at Applications, volume 1, Dunod, Gauthier-Villars, Paris, 1968.
![]() ![]() |
|
V. A. Marchenko and E. Ya. Khruslov, Homogenization of Partial Differential Equations, Progress in Mathematical Physics, 46. Birkhäuser Boston, Inc., Boston, MA, 2006.
![]() ![]() |
|
A. Mielke
, The complex Ginzburg-Landau equation on large and unbounded domains: sharper bounds and attractors, Nonlinearity, 10 (1997)
, 199-222.
![]() ![]() |
|
L. Pankratov
, Homogenization of Ginzburg-Landau heat flow equation in a porous medium, Applicable Analysis, 69 (1998)
, 31-45.
![]() ![]() |
|
E. Sánchez-Palencia, Homogenization Techniques for Composite Media, Lecture Notes in Physics, 272. Springer-Verlag, Berlin, 1987.
![]() ![]() |
|
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, 2nd edition. Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997.
![]() |
|
M. I. Vishik
and V. V. Chepyzhov
, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms, Sb. Math., 192 (2001)
, 11-47.
![]() ![]() |
|
M. I. Vishik
and V. V. Chepyzhov
, The nonautonomous Ginzburg-Landau equation and its attractors, Sb. Math., 196 (2005)
, 791-815.
![]() ![]() |
|
M. I. Vishik
and V. V. Chepyzhov
, Trajectory attractors of equations of mathematical physics, Russian Math. Surveys, 66 (2011)
, 637-731.
![]() ![]() |
|
M. I. Vishik
and B. Fiedler
, Quantitative averaging of global attractors of hyperbolic wave equations with rapidly oscillating coefficients, Russian Math. Surveys, 57 (2002)
, 709-728.
![]() ![]() |
|
S. V. Zelik
, The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and it's dimension, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 24 (2000)
, 1-25.
![]() ![]() |
|
V. V. Zhikov
, On two-scale convergence, Journal of Mathematical Sciences, 120 (2003)
, 1328-1352.
![]() ![]() |
Attractors of the Ginzburg-Landau Equations