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Sbornik: Mathematics, 2001, Volume 192, Issue 1, Pages 11–47
DOI: https://doi.org/10.1070/sm2001v192n01ABEH000534
(Mi sm534)
 

This article is cited in 47 scientific papers (total in 47 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
References:
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.
Received: 27.04.2000
Bibliographic databases:
UDC: 517.9
MSC: Primary 35B21; Secondary 34C29
Language: English
Original paper language: Russian
Citation: M. I. Vishik, V. V. Chepyzhov, “Averaging of trajectory attractors of evolution equations with rapidly oscillating terms”, Sb. Math., 192:1 (2001), 11–47
Citation in format AMSBIB
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\by M.~I.~Vishik, V.~V.~Chepyzhov
\paper Averaging of trajectory attractors of~evolution equations with rapidly oscillating terms
\jour Sb. Math.
\yr 2001
\vol 192
\issue 1
\pages 11--47
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Linking options:
  • https://www.mathnet.ru/eng/sm534
  • https://doi.org/10.1070/sm2001v192n01ABEH000534
  • https://www.mathnet.ru/eng/sm/v192/i1/p13
  • This publication is cited in the following 47 articles:
    1. Hujun Yang, Xiaoling Han, Caidi Zhao, “Existence and homogenization of trajectory statistical solutions for the 3D incompressible Hall-MHD equations”, DCDS-S, 2024  crossref
    2. Andrew Comech, Alexander Komech, Mikhail Vishik, Trends in Mathematics, Partial Differential Equations and Functional Analysis, 2023, 259  crossref
    3. Kuanysh A. Bekmaganbetov, Gregory A. Chechkin, Vladimir V. Chepyzhov, “Application of Fatou's Lemma for Strong Homogenization of Attractors to Reaction–Diffusion Systems with Rapidly Oscillating Coefficients in Orthotropic Media with Periodic Obstacles”, Mathematics, 11:6 (2023), 1448  crossref
    4. K. A. Bekmaganbetov, V. V. Chepyzhov, G. A. Chechkin, “Strong convergence of attractors of reaction-diffusion system with rapidly oscillating terms in an orthotropic porous medium”, Izv. Math., 86:6 (2022), 1072–1101  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    5. K. A. Bekmaganbetov, A. M. Toleubai, G. A. Chechkin, “Ob attraktorakh 2D sistemy Nave–Stoksa v srede s anizotropnoi peremennoi vyazkostyu i periodicheskimi prepyatstviyami”, Kraevye zadachi matematicheskoi fiziki i smezhnye voprosy teorii funktsii. 50, Zap. nauchn. sem. POMI, 519, POMI, SPb., 2022, 10–34  mathnet
    6. Chunjiao Han, Yi Cheng, Ranzhuo Ma, Zhenhua Zhao, “Average Process of Fractional Navier–Stokes Equations with Singularly Oscillating Force”, Fractal Fract, 6:5 (2022), 241  crossref
    7. Hujun Yang, Xiaoling Han, Caidi Zhao, “Homogenization of Trajectory Statistical Solutions for the 3D Incompressible Micropolar Fluids with Rapidly Oscillating Terms”, Mathematics, 10:14 (2022), 2469  crossref
    8. K. A. Bekmaganbetov, A. M. Toleubai, G. A. Chechkin, “Attractors of the Navier–Stokes Equations in a Two-Dimensional Porous Medium”, J Math Sci, 262:3 (2022), 246  crossref
    9. K. A. Bekmaganbetov, V. V. Chepyzhov, G. A. Chechkin, “On attractors of reaction–diffusion equations in a porous orthotropic medium”, Dokl. Math., 103:3 (2021), 103–107  mathnet  crossref  crossref  zmath  elib
    10. K. A. Bekmaganbetov, V. V. Chepyzhov, G. A. Chechkin, “Homogenization of Attractors of Reaction–Diffusion System with Rapidly Oscillating Terms in an Orthotropic Porous Medium”, J Math Sci, 259:2 (2021), 148  crossref
    11. 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  crossref  mathscinet  zmath  isi
    12. Bekmaganbetov K.A. Chechkin G.A. Chepyzhov V.V., ““Strange Term” in Homogenization of Attractors of Reaction-Diffusion Equation in Perforated Domain”, Chaos Solitons Fractals, 140 (2020), 110208  crossref  mathscinet  isi
    13. 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  zmath  isi  scopus
    14. 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  zmath  isi  scopus  scopus  scopus
    15. 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
    16. Christian Kuehn, Applied Mathematical Sciences, 191, Multiple Time Scale Dynamics, 2015, 583  crossref
    17. 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
    18. 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
    19. T. Medjo, “Averaging of a multi-layer quasi-geostrophic equations with oscillating external forces”, CPAA, 13:3 (2013), 1119  crossref  mathscinet  isi  scopus  scopus
    20. T. Tachim Medjo, “Averaging of a 3D primitive equations with oscillating external forces”, Applicable Analysis, 2012, 1  crossref  mathscinet  isi  scopus  scopus  scopus
    Citing articles in Google Scholar: Russian citations, English citations
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