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

This article is cited in 33 scientific papers (total in 33 papers)

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

Full text: PDF file (494 kB)
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English version:
Sbornik: Mathematics, 2001, 192:1, 11–47

Bibliographic databases:

UDC: 517.9
MSC: Primary 35B21; Secondary 34C29
Received: 27.04.2000

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
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    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  mathscinet  zmath  isi  elib
    2. M. I. Vishik, V. V. Chepyzhov, “Trajectory and Global Attractors of Three-Dimensional Navier–Stokes Systems”, Math. Notes, 71:2 (2002), 177–193  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  mathnet  crossref  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  isi
    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  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi  elib
    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.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    9. M. I. Vishik, V. V. Chepyzhov, “Attractors of dissipative hyperbolic equations with singularly oscillating external forces”, Math. Notes, 79:4 (2006), 483–504  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    10. Zelik S., “Global averaging and parametric resonances in damped semilinear wave equations”, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 1053–1097  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    12. Guo Boling, Han Yongqian, “Attractors of derivative complex Ginzburg-Landau equation in unbounded domains”, Front. Math. China, 2:3 (2007), 383–416  crossref  mathscinet  zmath  elib  scopus  scopus
    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  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    14. Zhao Caidi, Zhou Shengfan, Li Yongsheng, “Trajectory attractor and global attractor for a two-dimensional incompressible non-Newtonian fluid”, J. Math. Anal. Appl., 325:2 (2007), 1350–1362  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    15. M. I. Vishik, V. Pata, V. V. Chepyzhov, “Time averaging of global attractors for nonautonomous wave equations with singularly oscillating external forces”, Dokl. Math., 78:2 (2008), 689–692  mathnet  crossref  mathscinet  zmath  zmath  isi  elib  elib  scopus
    16. Chepyzhov V.V., Pata V., Vishik M.I., “Averaging of nonautonomous damped wave equations with singularly oscillating external forces”, J. Math. Pures Appl. (9), 90:5 (2008), 469–491  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    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  crossref  mathscinet  isi  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    22. M. I. Vishik, V. V. Chepyzhov, “Trajectory attractors of equations of mathematical physics”, Russian Math. Surveys, 66:4 (2011), 637–731  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    23. T. Tachim Medjo, “Averaging of a 3D primitive equations with oscillating external forces”, Applicable Analysis, 2012, 1  crossref  mathscinet  isi  scopus  scopus  scopus
    24. Medjo T.T., “Averaging of the planetary 3D geostrophic equations with oscillating external forces”, Applied Mathematics and Computation, 218:10 (2012), 5910–5928  crossref  zmath  isi  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    27. T. Medjo, “Averaging of an homogeneous two-phase flow model with oscillating external forces”, Discrete Contin. Dyn. Syst., 32:10 (2012), 3665–3690  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    28. T. Medjo, “Averaging of a multi-layer quasi-geostrophic equations with oscillating external forces”, CPAA, 13:3 (2013), 1119  crossref  mathscinet  isi  scopus  scopus
    29. Mark Vishik, Sergey Zelik, “Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit”, CPAA, 13:5 (2014), 2059  crossref  mathscinet  zmath  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  isi  scopus  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  isi  scopus  scopus  scopus
    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  crossref  mathscinet  isi  scopus
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