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 Uspekhi Mat. Nauk, 2011, Volume 66, Issue 4(400), Pages 3–102 (Mi umn9407)

Trajectory attractors of equations of mathematical physics

M. I. Vishik, V. V. Chepyzhov

Institute for Information Transmission Problems

Abstract: In this survey the method of trajectory dynamical systems and trajectory attractors is described, and is applied in the study of the limiting asymptotic behaviour of solutions of non-linear evolution equations. This method is especially useful in the study of dissipative equations of mathematical physics for which the corresponding Cauchy initial-value problem has a global (weak) solution with respect to the time but the uniqueness of this solution either has not been established or does not hold. An important example of such an equation is the 3D Navier–Stokes system in a bounded domain. In such a situation one cannot use directly the classical scheme of construction of a dynamical system in the phase space of initial conditions of the Cauchy problem of a given equation and find a global attractor of this dynamical system. Nevertheless, for such equations it is possible to construct a trajectory dynamical system and investigate a trajectory attractor of the corresponding translation semigroup. This universal method is applied for various types of equations arising in mathematical physics: for general dissipative reaction-diffusion systems, for the 3D Navier–Stokes system, for dissipative wave equations, for non-linear elliptic equations in cylindrical domains, and for other equations and systems. Special attention is given to using the method of trajectory attractors in approximation and perturbation problems arising in complicated models of mathematical physics.
Bibliography: 96 titles.

Keywords: dynamical systems, trajectory attractors, equations of mathematical physics, ill-posed problems.

DOI: https://doi.org/10.4213/rm9407

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English version:
Russian Mathematical Surveys, 2011, 66:4, 637–731

Bibliographic databases:

UDC: 517.958
MSC: Primary 37-02; Secondary 35-02, 35B41, 35J60, 35K57, 35Q30, 37L30

Citation: M. I. Vishik, V. V. Chepyzhov, “Trajectory attractors of equations of mathematical physics”, Uspekhi Mat. Nauk, 66:4(400) (2011), 3–102; Russian Math. Surveys, 66:4 (2011), 637–731

Citation in format AMSBIB
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This publication is cited in the following articles:
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5. V. Chepyzhov, “Trajectory attractors for non-autonomous dissipative 2d Euler equations”, Discrete Contin. Dyn. Syst. Ser. B, 20:3 (2015), 811–832
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11. 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
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18. Li Ya., Wang R., She L., “Backward Controllability of Pullback Trajectory Attractors With Applications to Multi-Valued Jeffreys-Oldroyd Equations”, Evol. Equ. Control Theory, 7:4 (2018), 617–637
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20. 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
21. Zhao C., Caraballo T., “Asymptotic Regularity of Trajectory Attractor and Trajectory Statistical Solution For the 3D Globally Modified Navier-Stokes Equations”, J. Differ. Equ., 266:11 (2019), 7205–7229
22. Dmitrenko A.V., “the Correlation Dimension of An Attractor Determined on the Base of the Theory of Equivalence of Measures and Stochastic Equations For Continuum”, Continuum Mech. Thermodyn., 32:1 (2020), 63–74
23. 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
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