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

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.

DOI: https://doi.org/10.4213/sm765

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

Bibliographic databases:

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

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
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Related articles on Google Scholar: Russian articles, English articles

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
2. M. I. Vishik, V. V. Chepyzhov, “Attractors of dissipative hyperbolic equations with singularly oscillating external forces”, Math. Notes, 79:4 (2006), 483–504
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
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
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
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
7. M. I. Vishik, S. V. Zelik, V. V. Chepyzhov, “Regular attractors and nonautonomous perturbations of them”, Sb. Math., 204:1 (2013), 1–42
8. Zelik S.V., Chepyzhov V.V., “Regular Attractors of Autonomous and Nonautonomous Dynamical Systems”, Dokl. Math., 89:1 (2014), 92–97
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
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
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
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