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

This article is cited in 23 scientific papers (total in 23 papers)

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.


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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
Received: 10.12.2010

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
\by M.~I.~Vishik, V.~V.~Chepyzhov
\paper Trajectory attractors of equations of mathematical physics
\jour Uspekhi Mat. Nauk
\yr 2011
\vol 66
\issue 4(400)
\pages 3--102
\jour Russian Math. Surveys
\yr 2011
\vol 66
\issue 4
\pages 637--731

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    This publication is cited in the following articles:
    1. V. G. Zvyagin, S. K. Kondratyev, “Approximating topological approach to the existence of attractors in fluid mechanics”, J. Fixed Point Theory Appl., 13:2 (2013), 359–395  crossref  mathscinet  zmath  isi  elib
    2. P. Kalita, G. Łukaszewicz, “Attractors for Navier–Stokes flows with multivalued and nonmonotone subdifferential boundary conditions”, Anal. Real World Appl., 19 (2014), 75–88  crossref  mathscinet  zmath  isi
    3. V. G. Zvyagin, S. K. Kondrat'ev, “Attractors of equations of non-Newtonian fluid dynamics”, Russian Math. Surveys, 69:5 (2014), 845–913  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Zhao Caidi, Kong Lei, Liu Guowei, Zhao Min, “The trajectory attractor and its limiting behavior for the convective Brinkman-Forchheimer equations”, Topol. Methods Nonlinear Anal., 44:2 (2014), 413–433  crossref  mathscinet  zmath  isi  elib
    5. V. Chepyzhov, “Trajectory attractors for non-autonomous dissipative 2d Euler equations”, Discrete Contin. Dyn. Syst. Ser. B, 20:3 (2015), 811–832  crossref  zmath  isi  elib
    6. Levakov A.A., Zadvornyi Ya.B., “Stable Sets, Attracting Sets, and Attractors of Semidynamical Systems in Nonlocally Compact Metric Spaces”, 51, no. 7, 2015, 847–856  crossref  mathscinet  zmath  isi
    7. V. V. Chepyzhov, “Approximating the trajectory attractor of the 3D Navier-Stokes system using various $\alpha$-models of fluid dynamics”, Sb. Math., 207:4 (2016), 610–638  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    8. Zvyagin V., Kondratyev S., “Pullback Attractors of the Jeffreys-Oldroyd Equations”, 260, no. 6, 2016, 5026–5042  crossref  mathscinet  zmath  isi
    9. Lukaszewicz G. Kalita P., “Navier–Stokes Equations: An Introduction With Applications”, Navier-Stokes Equations: An Introduction With Applications, Advances in Mechanics and Mathematics, Springer, 2016, 1–390  crossref  mathscinet  isi
    10. Chepyzhov V., Ilyin A., Zelik S., “Strong trajectory and global $\mathbf{W^{1,p}}$-attractors for the damped-driven Euler system in $\mathbb R^2$”, Discrete Contin. Dyn. Syst.-Ser. B, 22:5, SI (2017), 1835–1855  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    12. Sun W., Li Y., “Trajectory Attractor and Global Attractor For Keller-Segel-Stokes Model With Arbitrary Porous Medium Diffusion”, Topol. Methods Nonlinear Anal., 50:2 (2017), 581–602  crossref  mathscinet  zmath  isi
    13. Zvyagin V., “Attractors Theory For Autonomous Systems of Hydrodynamics and Its Application to Bingham Model of Fluid Motion”, Lobachevskii J. Math., 38:4, SI (2017), 767–777  crossref  mathscinet  zmath  isi
    14. Wang Y., Sui M., “Finite Lattice Approximation of Infinite Lattice Systems With Delays and Non-Lipschitz Nonlinearities”, Asymptotic Anal., 106:3-4 (2018), 169–203  crossref  mathscinet  zmath  isi
    15. 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  isi
    16. Chepyzhov V., Ilyin A., Zelik S., “Vanishing Viscosity Limit For Global Attractors For the Damped Navier–Stokes System With Stress Free Boundary Conditions”, Physica D, 376:SI (2018), 31–38  crossref  mathscinet  isi  scopus
    17. Zvyagin A., “Attractors For Model of Polymer Solutions Motion”, Discret. Contin. Dyn. Syst., 38:12, SI (2018), 6305–6325  crossref  isi  scopus
    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  crossref  mathscinet  isi  scopus
    19. Martirosyan D., “Large Deviations For Invariant Measures of the White-Forced 2D Navier-Stokes Equation”, J. Evol. Equ., 18:3 (2018), 1245–1265  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  isi
    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  crossref  isi
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