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Mat. Sb., 1996, Volume 187, Number 5, Pages 15–58 (Mi msb126)  

This article is cited in 31 scientific papers (total in 31 papers)

Averaging principle for dissipative dynamical systems with rapidly oscillating right-hand sides

A. A. Ilyin


Abstract: We consider two-dimensional Navier–Stokes equations and a damped non-linear hyperbolic equation. We suppose that the right-hand sides of these equations have the form $f(\omega t)$, $\omega \gg 1$. We suppose also that $f$ has an average. The main result of the paper is proof of a global averaging theorem on the convergence of attractors of non-autonomous equations to the attractor of the average autonomous equation as $\omega \to \infty$.

DOI: https://doi.org/10.4213/sm126

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English version:
Sbornik: Mathematics, 1996, 187:5, 635–677

Bibliographic databases:

UDC: 517.9
MSC: 34G20, 35Q30, 35L70
Received: 09.11.1994

Citation: A. A. Ilyin, “Averaging principle for dissipative dynamical systems with rapidly oscillating right-hand sides”, Mat. Sb., 187:5 (1996), 15–58; Sb. Math., 187:5 (1996), 635–677

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    This publication is cited in the following articles:
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    2. “Time-averaging under fast periodic forcing of parabolic partial differential equations: Exponential estimates”, Journal of Differential Equations, 174:1 (2001), 133–180  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    3. 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
    4. Gao, HJ, “Dynamics of the thermohaline circulation under wind forcing”, Discrete and Continuous Dynamical Systems-Series B, 2:2 (2002), 205  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    5. 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
    6. Gao, HJ, “Dynamics of quasi-geostrophic fluid motion with rapidly oscillating Coriolis force”, Nonlinear Analysis-Real World Applications, 4:1 (2003), 127  crossref  mathscinet  zmath  isi  scopus  scopus
    7. “Exponentially small splitting of homoclinic orbits of parabolic differential equations under periodic forcing”, Discrete and Continuous Dynamical Systems, 9:3 (2003), 585–602  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    8. Cheban, D, “Recurrent motions and global attractors of non-autonomous Lorenz systems”, Dynamical Systems-An International Journal, 19:1 (2004), 41  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
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    11. Chepyzhov, VV, “Integral manifolds and attractors with exponential rate for nonautonomous hyperbolic equations with dissipation”, Russian Journal of Mathematical Physics, 12:1 (2005), 17  mathscinet  zmath  isi  elib
    12. Chepyzhov, VV, “On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging”, Discrete and Continuous Dynamical Systems, 12:1 (2005), 27  crossref  mathscinet  zmath  isi  elib
    13. 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
    14. Zelik, S, “Global averaging and parametric resonances in damped semilinear wave equations”, Proceedings of the Royal Society of Edinburgh Section A-Mathematics, 136 (2006), 1053  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
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    17. V. V. Chepyzhov, M. I. Vishik, “Non-autonomous 2D Navier–Stokes System with Singularly Oscillating External Force and its Global Attractor”, J Dyn Diff Equat, 19:3 (2007), 655  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    18. 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  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    19. Chepyzhov, VV, “Averaging of nonautonomous damped wave equations with singularly oscillating external forces”, Journal de Mathematiques Pures et Appliquees, 90:5 (2008), 469  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    20. V V Chepyzhov, V Pata, M I Vishik, “Averaging of 2D Navier–Stokes equations with singularly oscillating forces”, Nonlinearity, 22:2 (2009), 351  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    21. Yan X., “Dynamical behaviour of non-autonomous 2D Navier–Stokes equations with singularly oscillating external force”, Dyn. Syst., 26:3 (2011), 245–260  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    22. 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
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    24. Gao P., Li Y., “Averaging Principle For the Schrodinger Equations”, Discrete Contin. Dyn. Syst.-Ser. B, 22:6 (2017), 2147–2168  crossref  mathscinet  zmath  isi  elib  scopus
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  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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