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 Mat. Sb., 2001, Volume 192, Number 1, Pages 13–50 (Mi msb534)

Averaging of trajectory attractors of evolution equations with rapidly oscillating terms

M. I. Vishik, V. V. Chepyzhov

Institute for Information Transmission Problems, Russian Academy of Sciences

Abstract: Evolution equations containing rapidly oscillating terms with respect to the spatial variables or the time variable are considered. The trajectory attractors of these equations are proved to approach the trajectory attractors of the equations whose terms are the averages of the corresponding terms of the original equations. The corresponding Cauchy problems are not assumed here to be uniquely soluble. At the same time if the Cauchy problems for the equations under consideration are uniquely soluble, then they generate semigroups having global attractors. These global attractors also converge to the global attractors of the averaged equations in the corresponding spaces.
These results are applied to the following equations and systems of mathematical physics: the 3D and 2D Navier–Stokes systems with rapidly oscillating external forces, reaction-diffusion systems, the complex Ginzburg–Landau equation, the generalized Chafee–Infante equation, and dissipative hyperbolic equations with rapidly oscillating terms and coefficients.

DOI: https://doi.org/10.4213/sm534

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English version:
Sbornik: Mathematics, 2001, 192:1, 11–47

Bibliographic databases:

UDC: 517.9
MSC: Primary 35B21; Secondary 34C29

Citation: M. I. Vishik, V. V. Chepyzhov, “Averaging of trajectory attractors of evolution equations with rapidly oscillating terms”, Mat. Sb., 192:1 (2001), 13–50; Sb. Math., 192:1 (2001), 11–47

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Chepyzhov V.V., Vishik M.I., “Global attractor and its perturbations for a dissipative hyperbolic equation”, Russ. J. Math. Phys., 8:3 (2001), 251–266
2. M. I. Vishik, V. V. Chepyzhov, “Trajectory and Global Attractors of Three-Dimensional Navier–Stokes Systems”, Math. Notes, 71:2 (2002), 177–193
3. Chepyzhov V.V., Vishik M.I., “Non-autonomous 2D Navier–Stokes system with a simple global attractor and some averaging problems”, ESAIM Control Optim. Calc. Var., 8 (2002), 467–487
4. M. I. Vishik, V. V. Chepyzhov, “Approximation of trajectories lying on a global attractor of a hyperbolic equation with exterior force rapidly oscillating in time”, Sb. Math., 194:9 (2003), 1273–1300
5. A. M. Rekalo, I. D. Chueshov, “Global attractor of a contact parabolic problem in a thin two-layer domain”, Sb. Math., 195:1 (2004), 97–119
6. Chepyzhov V.V., Goritsky A.Yu., Vishik M.I., “Integral manifolds and attractors with exponential rate for nonautonomous hyperbolic equations with dissipation”, Russ. J. Math. Phys., 12:1 (2005), 17–39
7. Chepyzhov V.V., Vishik M.I., Wendland W.L., “On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging”, Discrete Contin. Dyn. Syst., 12:1 (2005), 27–38
8. Guo Boling, Huang Daiwen, “Existence of weak solutions and trajectory attractors for the moist atmospheric equations in geophysics”, J. Math. Phys., 47:8 (2006), 083508, 23 pp.
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10. Zelik S., “Global averaging and parametric resonances in damped semilinear wave equations”, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 1053–1097
11. Vishik M.I., Chepyzhov V.V., “The global attractor of the nonautonomous 2D Navier–Stokes system with singularly oscillating external force”, Dokl. Math., 75:2 (2007), 236–239
12. Guo Boling, Han Yongqian, “Attractors of derivative complex Ginzburg-Landau equation in unbounded domains”, Front. Math. China, 2:3 (2007), 383–416
13. Chepyzhov V.V., Vishik M.I., “Non-autonomous 2D Navier–Stokes system with singularly oscillating external force and its global attractor”, J. Dynam. Differential Equations, 19:3 (2007), 655–684
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17. Chepyzhov V.V., Pata V., Vishik M.I., “Averaging of 2D Navier–Stokes equations with singularly oscillating forces”, Nonlinearity, 22:2 (2009), 351–370
18. Bloemker D., Han Y., “Asymptotic Compactness of Stochastic Complex Ginzburg-Landau Equation on an Unbounded Domain”, Stochastics and Dynamics, 10:4 (2010), 613–636
19. T. Medjo, “A non-autonomous 3D Lagrangian averaged Navier–Stokes-$\alpha$ model with oscillating external force and its global attractor”, CPAA, 10:2 (2010), 415
20. Medjo T.T., “AVERAGING OF A 3D LAGRANGIAN AVERAGED Navier–Stokes-alpha MODEL WITH OSCILLATING EXTERNAL FORCES”, Commun Pure Appl Anal, 10:4 (2011), 1281–1305
21. Medjo T.T., “Non-autonomous planetary 3D geostrophic equations with oscillating external force and its global attractor”, Nonlinear Anal Real World Appl, 12:3 (2011), 1437–1452
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23. T. Tachim Medjo, “Averaging of a 3D primitive equations with oscillating external forces”, Applicable Analysis, 2012, 1
24. Medjo T.T., “Averaging of the planetary 3D geostrophic equations with oscillating external forces”, Applied Mathematics and Computation, 218:10 (2012), 5910–5928
25. Medjo T.T., “Non-autonomous 3D primitive equations with oscillating external force and its global attractor”, Discrete and Continuous Dynamical Systems, 32:1 (2012), 265–291
26. Medjo T.T., “A non-autonomous two-phase flow model with oscillating external force and its global attractor”, Nonlinear Analysis-Theory Methods & Applications, 75:1 (2012), 226–243
27. T. Medjo, “Averaging of an homogeneous two-phase flow model with oscillating external forces”, Discrete Contin. Dyn. Syst., 32:10 (2012), 3665–3690
28. T. Medjo, “Averaging of a multi-layer quasi-geostrophic equations with oscillating external forces”, CPAA, 13:3 (2013), 1119
29. Mark Vishik, Sergey Zelik, “Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit”, CPAA, 13:5 (2014), 2059
30. Medjo T.T., “Pullback Attractors For the Multi-Layer Quasi-Geostrophic Equations of the Ocean”, Nonlinear Anal.-Real World Appl., 17 (2014), 365–382
31. Bekmaganbetov K.A. Chechkin G.A. Chepyzhov V.V. Goritsky A.Yu., “Homogenization of trajectory attractors of 3D Navier–Stokes system with randomly oscillating force”, Discret. Contin. Dyn. Syst., 37:5 (2017), 2375–2393
32. Chechkin G.A. Chepyzhov V.V. Pankratov L.S., “Homogenization of Trajectory Attractors of Ginzburg-Landau Equations With Randomly Oscillating Terms”, Discrete Contin. Dyn. Syst.-Ser. B, 23:3 (2018), 1133–1154
33. Bekmaganbetov K.A. Chechkin G.A. Chepyzhov V.V., “Weak Convergence of Attractors of Reaction-Diffusion Systems With Randomly Oscillating Coefficients”, Appl. Anal., 98:1-2, SI (2019), 256–271
34. Bekmaganbetov K.A. Chechkin G.A. Chepyzhov V.V., “Strong Convergence of Trajectory Attractors For Reaction-Diffusion Systems With Random Rapidly Oscillating Terms”, Commun. Pure Appl. Anal, 19:5 (2020), 2419–2443
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