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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 |
${\partial _t}u = (1 + \alpha i)\Delta u + R{\mkern 1mu} u + (1 + \beta i)|u{|^2}u + g,$ |
$R$ |
$β$ |
$g$ |
References:
[1] |
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.
|
[2] |
V. I. Arnol'd and A. Avez, Ergodic Problems of Classical Mechanics, W. A. Benjamin Inc., New York - Amsterdam, 1968. |
[3] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992. |
[4] |
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. |
[5] |
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.
|
[6] |
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. |
[7] |
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. |
[8] |
G. D. Birkhoff,
Proof of the ergodic theorem, Proc Natl Acad Sci USA, 17 (1931), 656-660.
|
[9] |
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. |
[10] |
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.
|
[11] |
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. |
[12] |
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.
|
[13] |
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. |
[14] |
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. |
[15] |
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.
|
[16] |
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.
|
[17] |
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.
|
[18] |
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.
|
[19] |
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. |
[20] |
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.
|
[21] |
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.
|
[22] |
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.
|
[23] |
V. V. Chepyzhov and M. I. Vishik,
Evolution equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997), 913-964.
|
[24] |
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. |
[25] |
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.
|
[26] |
I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Kharkov, AKTA, 1999. |
[27] |
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.
|
[28] |
I. D. Chueshov and L. S. Pankratov,
Averaging of attractors of nonlinear hyperbolic equations with asymptotically degenerate coefficients, Sb. Math., 190 (1999), 1325-1352.
|
[29] |
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.
|
[30] |
Yu. A. Dubinskiĭ,
Weak convergence in nonlinear elliptic and parabolic equations, Sb. Math., 67(109) (1965), 609-642.
|
[31] |
N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, John Wiley & Sons, Inc., New-York, 1988. |
[32] |
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.
|
[33] |
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.
|
[34] |
J. M. Ghidaglia and B. Héron,
Dimension of the attractors associated to the Ginzburg-Landau partial differential equation, Physica D, 28 (1987), 282-304.
|
[35] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. |
[36] |
J. K. Hale and S. M. Verduyn Lunel,
Averaging in infinite dimensions, J. Int. Eq. Appl., 2 (1990), 463-494.
|
[37] |
A. A. Ilyin,
Averaging principle for dissipative dynamical systems with rapidly oscillating right-hand sides, Sb. Math., 187 (1996), 635-677.
|
[38] |
V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. |
[39] |
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. |
[40] |
J. -L. Lions, Quelques Méthodes de Résolutions des Problémes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. |
[41] |
J. -L. Lions and E. Magenes, Problémes Aux Limites Non Homogénes at Applications, volume 1, Dunod, Gauthier-Villars, Paris, 1968. |
[42] |
V. A. Marchenko and E. Ya. Khruslov, Homogenization of Partial Differential Equations, Progress in Mathematical Physics, 46. Birkhäuser Boston, Inc., Boston, MA, 2006. |
[43] |
A. Mielke,
The complex Ginzburg-Landau equation on large and unbounded domains: sharper bounds and attractors, Nonlinearity, 10 (1997), 199-222.
|
[44] |
L. Pankratov,
Homogenization of Ginzburg-Landau heat flow equation in a porous medium, Applicable Analysis, 69 (1998), 31-45.
|
[45] |
E. Sánchez-Palencia, Homogenization Techniques for Composite Media, Lecture Notes in Physics, 272. Springer-Verlag, Berlin, 1987. |
[46] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, 2nd edition. Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. |
[47] |
M. I. Vishik and V. V. Chepyzhov,
Averaging of trajectory attractors of evolution equations with rapidly oscillating terms, Sb. Math., 192 (2001), 11-47.
|
[48] |
M. I. Vishik and V. V. Chepyzhov,
The nonautonomous Ginzburg-Landau equation and its attractors, Sb. Math., 196 (2005), 791-815.
|
[49] |
M. I. Vishik and V. V. Chepyzhov,
Trajectory attractors of equations of mathematical physics, Russian Math. Surveys, 66 (2011), 637-731.
|
[50] |
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.
|
[51] |
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.
|
[52] |
V. V. Zhikov,
On two-scale convergence, Journal of Mathematical Sciences, 120 (2003), 1328-1352.
|
show all references
To the blessed memory of I. D. Chueshov
References:
[1] |
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.
|
[2] |
V. I. Arnol'd and A. Avez, Ergodic Problems of Classical Mechanics, W. A. Benjamin Inc., New York - Amsterdam, 1968. |
[3] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992. |
[4] |
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. |
[5] |
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.
|
[6] |
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. |
[7] |
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. |
[8] |
G. D. Birkhoff,
Proof of the ergodic theorem, Proc Natl Acad Sci USA, 17 (1931), 656-660.
|
[9] |
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. |
[10] |
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.
|
[11] |
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. |
[12] |
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.
|
[13] |
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. |
[14] |
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. |
[15] |
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.
|
[16] |
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.
|
[17] |
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.
|
[18] |
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.
|
[19] |
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. |
[20] |
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.
|
[21] |
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.
|
[22] |
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.
|
[23] |
V. V. Chepyzhov and M. I. Vishik,
Evolution equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997), 913-964.
|
[24] |
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. |
[25] |
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.
|
[26] |
I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Kharkov, AKTA, 1999. |
[27] |
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.
|
[28] |
I. D. Chueshov and L. S. Pankratov,
Averaging of attractors of nonlinear hyperbolic equations with asymptotically degenerate coefficients, Sb. Math., 190 (1999), 1325-1352.
|
[29] |
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.
|
[30] |
Yu. A. Dubinskiĭ,
Weak convergence in nonlinear elliptic and parabolic equations, Sb. Math., 67(109) (1965), 609-642.
|
[31] |
N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, John Wiley & Sons, Inc., New-York, 1988. |
[32] |
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.
|
[33] |
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.
|
[34] |
J. M. Ghidaglia and B. Héron,
Dimension of the attractors associated to the Ginzburg-Landau partial differential equation, Physica D, 28 (1987), 282-304.
|
[35] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. |
[36] |
J. K. Hale and S. M. Verduyn Lunel,
Averaging in infinite dimensions, J. Int. Eq. Appl., 2 (1990), 463-494.
|
[37] |
A. A. Ilyin,
Averaging principle for dissipative dynamical systems with rapidly oscillating right-hand sides, Sb. Math., 187 (1996), 635-677.
|
[38] |
V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. |
[39] |
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. |
[40] |
J. -L. Lions, Quelques Méthodes de Résolutions des Problémes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. |
[41] |
J. -L. Lions and E. Magenes, Problémes Aux Limites Non Homogénes at Applications, volume 1, Dunod, Gauthier-Villars, Paris, 1968. |
[42] |
V. A. Marchenko and E. Ya. Khruslov, Homogenization of Partial Differential Equations, Progress in Mathematical Physics, 46. Birkhäuser Boston, Inc., Boston, MA, 2006. |
[43] |
A. Mielke,
The complex Ginzburg-Landau equation on large and unbounded domains: sharper bounds and attractors, Nonlinearity, 10 (1997), 199-222.
|
[44] |
L. Pankratov,
Homogenization of Ginzburg-Landau heat flow equation in a porous medium, Applicable Analysis, 69 (1998), 31-45.
|
[45] |
E. Sánchez-Palencia, Homogenization Techniques for Composite Media, Lecture Notes in Physics, 272. Springer-Verlag, Berlin, 1987. |
[46] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, 2nd edition. Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. |
[47] |
M. I. Vishik and V. V. Chepyzhov,
Averaging of trajectory attractors of evolution equations with rapidly oscillating terms, Sb. Math., 192 (2001), 11-47.
|
[48] |
M. I. Vishik and V. V. Chepyzhov,
The nonautonomous Ginzburg-Landau equation and its attractors, Sb. Math., 196 (2005), 791-815.
|
[49] |
M. I. Vishik and V. V. Chepyzhov,
Trajectory attractors of equations of mathematical physics, Russian Math. Surveys, 66 (2011), 637-731.
|
[50] |
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.
|
[51] |
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.
|
[52] |
V. V. Zhikov,
On two-scale convergence, Journal of Mathematical Sciences, 120 (2003), 1328-1352.
|

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