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Mat. Sb., 2003, Volume 194, Number 9, Pages 3–30 (Mi msb765)  

This article is cited in 11 scientific papers (total in 11 papers)

Approximation of trajectories lying on a global attractor of a hyperbolic equation with exterior force rapidly oscillating in time

M. I. Vishik, V. V. Chepyzhov

Institute for Information Transmission Problems, Russian Academy of Sciences

Abstract: A quasilinear dissipative wave equation is considered for periodic boundary conditions with exterior force $g(x,t/\varepsilon)$ rapidly oscillating in $t$. It is assumed in addition that, as $\varepsilon\to0+$, the function $g(x,t/\varepsilon)$ converges in the weak sense (in $L_{2,w}^{\mathrm{loc}}(\mathbb R,L_2(\mathbb T^n))$ to a function $\overline g(x)$ and the averaged wave equation (with exterior force $\overline g(x)$ has only finitely many stationary points $ż_i(x), i= 1,…,N\}$, each of them hyperbolic. It is proved that the global attractor $\mathscr A_\varepsilon$ of the original equation deviates in the energy norm from the global attractor $\mathscr A_0$ of the averaged equation by a quantity $C\varepsilon^\rho$, where $\rho$ is described by an explicit formula. It is also shown that each piece of a trajectory $u^\varepsilon(t)$ of the original equation lying on $\mathscr A_\varepsilon$ that corresponds to an interval of time-length $C\log(1/\varepsilon)$ can be approximated to within $C_1\varepsilon^{\rho_1}$ by means of finitely many pieces of trajectories lying on unstable manifolds $M^u(z_i)$ of the averaged equation, where an explicit expression for $\rho_1$ is provided.


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English version:
Sbornik: Mathematics, 2003, 194:9, 1273–1300

Bibliographic databases:

UDC: 517.9
MSC: Primary 35B41, 34C29; Secondary 35L70
Received: 21.03.2003

Citation: 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”, Mat. Sb., 194:9 (2003), 3–30; Sb. Math., 194:9 (2003), 1273–1300

Citation in format AMSBIB
\by M.~I.~Vishik, V.~V.~Chepyzhov
\paper Approximation of trajectories lying on a~global attractor of a~hyperbolic equation with exterior force rapidly oscillating in time
\jour Mat. Sb.
\yr 2003
\vol 194
\issue 9
\pages 3--30
\jour Sb. Math.
\yr 2003
\vol 194
\issue 9
\pages 1273--1300

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    This publication is cited in the following articles:
    1. 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
    2. 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
    3. Yu. A. Goritsky, “Explicit construction of attracting integral manifolds for a dissipative hyperbolic equation”, J. Math. Sci. (N. Y.), 143:4 (2007), 3239–3252  mathnet  crossref  mathscinet  elib
    4. 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
    5. 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
    6. 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
    7. M. I. Vishik, S. V. Zelik, V. V. Chepyzhov, “Regular attractors and nonautonomous perturbations of them”, Sb. Math., 204:1 (2013), 1–42  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. Zelik S.V., Chepyzhov V.V., “Regular Attractors of Autonomous and Nonautonomous Dynamical Systems”, Dokl. Math., 89:1 (2014), 92–97  crossref  mathscinet  zmath  isi  elib  scopus
    9. 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
    10. Chepyzhov V.V. Conti M. Pata V., “Averaging of Equations of Viscoelasticity With Singularly Oscillating External Forces”, J. Math. Pures Appl., 108:6 (2017), 841–868  crossref  mathscinet  zmath  isi  scopus
    11. 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
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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