Abstract:
This survey describes a version of the trajectory-attractor\linebreak[4] method, which is applied to study the limit asymptotic behaviour of solutions of equations of non-Newtonian fluid dynamics. The trajectory-\linebreak[4]attractor method emerged in papers of the Russian mathematicians Vishik and Chepyzhov and the American mathematician Sell under the condition that the corresponding trajectory spaces be invariant under the translation semigroup. The need for such an approach was caused by the fact that for many equations of mathematical physics for which the Cauchy initial-value problem has a global (weak) solution with respect to the time, the uniqueness of such a solution has either not been established or does not hold. In particular, this is the case for equations of fluid dynamics. At the same time, trajectory spaces invariant under the translation semigroup could not be constructed for many equations of non-Newtonian fluid dynamics. In this connection, a different approach to the construction of trajectory attractors for dissipative systems was proposed in papers of Zvyagin and Vorotnikov without using invariance of trajectory spaces under the translation semigroup and is based on the topological lemma of Shura–Bura. This paper presents examples of equations of non-Newtonian fluid dynamics (the Jeffreys system describing movement of the Earth's crust, the model of motion of weak aqueous solutions of polymers, a system with memory) for which the aforementioned construction is used to prove the existence of attractors in both the autonomous and the non-autonomous cases. At the beginning of the paper there is also a brief exposition of the results of Ladyzhenskaya on the existence of attractors of the two-dimensional Navier–Stokes system and the result of Vishik and Chepyzhov for the case of attractors of the three-dimensional Navier–Stokes system.
Bibliography: 34 titles.
Keywords:
trajectory spaces, trajectory and global attractors of autonomous systems, uniform attractors of non-autonomous systems.
This publication is cited in the following 30 articles:
Mikhail Turbin, Anastasiia Ustiuzhaninova, “Trajectory and Global Attractors for the Kelvin–Voigt Model Taking into Account Memory along Fluid Trajectories”, Mathematics, 12:2 (2024), 266
A. S. Ustyuzhaninova, “Ravnomernye attraktory modeli Bingama”, Izv. vuzov. Matem., 2024, no. 8, 65–80
A. S. Ustiuzhaninova, “Uniform Attractors for the Bingham Model”, Russ Math., 68:8 (2024), 56
Russian Math. Surveys, 78:4 (2023), 635–777
V. G Zvyagin, A. S Ustyuzhaninova, “Obratnye attraktory modeli Bingama”, Differentsialnye uravneniya, 59:3 (2023), 374
V. G. Zvyagin, A. S. Ustiuzhaninova, “Pullback Attractors of the Bingham Model”, Diff Equat, 59:3 (2023), 377
A. V. Zvyagin, “Uniform Attractors for Non-Autonomous Systems of Nonlinearly Viscous Fluid”, Lobachevskii J Math, 44:3 (2023), 956
V. G. Zvyagin, M. V. Turbin, “Existence of attractors for approximations to the Bingham model and their convergence to the attractors of the initial model”, Siberian Math. J., 63:4 (2022), 699–714
M. V. Turbin, A. S. Ustiuzhaninova, “Convergence of attractors for an approximation to attractors of a modified Kelvin–Voigt model”, Comput. Math. Math. Phys., 62:2 (2022), 325–335
V. G. Zvyagin, M. V. Kaznacheev, “Attraktory dlya avtonomnoi modeli dvizheniya nelineino-vyazkoi zhidkosti”, Materialy Voronezhskoi vesennei
matematicheskoi shkoly
«Sovremennye metody teorii kraevykh
zadach. Pontryaginskie chteniya–XXX».
Voronezh, 3–9 maya 2019 g. Chast 2, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 191, VINITI RAN, M., 2021, 74–91
Ustiuzhaninova A.S., “Uniform Attractors For the Modified Kelvin-Voigt Model”, Differ. Equ., 57:9 (2021), 1165–1176
Polyakov D.M., Zvyagin A., “Dissipative Solvability of Jeffreys-Oldroyd-Alpha Model”, Topol. Methods Nonlinear Anal., 57:2 (2021), 465–488
Ashyralyev A., Zvyagin V., Turbin M., “The Convergence of Approximation Attractors to Attractors For Bingham Model With Periodical Boundary Conditions on Spatial Variables”, AIP Conference Proceedings, 2325, eds. Ashyralyev A., Ashyralyyev C., Erdogan A., Lukashov A., Sadybekov M., Amer Inst Physics, 2021, 020026
Aleksandr Boldyrev, Victor Zvyagin, Springer Proceedings in Mathematics & Statistics, 351, Functional Analysis in Interdisciplinary Applications—II, 2021, 3
V. G. Zvyagin, M. V. Kaznacheev, “Attractors of an autonomous model of nonlinear viscous fluid”, Dokl. Math., 101:2 (2020), 126–128
A. S. Boldyrev, V. G. Zvyagin, “Attractors for model of viscoelastic media with memory motion in non-autonomous case”, Lobachevskii J. Math., 40:7, SI (2019), 918–937
A. V. Zvyagin, V. G. Zvyagin, D. M. Polyakov, “Dissipative solvability of an alpha model of fluid flow with memory”, Comput. Math. Math. Phys., 59:7 (2019), 1185–1198
A. S. Boldyrev, V. G. Zvyagin, “Attractors for weak solution of a regularized problem of viscoelastic fluids motion with memory in non-autonomous case”, Russian Math. (Iz. VUZ), 62:7 (2018), 63–67
V. G. Zvyagin, “About the solvability of initial-boundary value problems for a viscoelastic model with memory”, International Conference on Analysis and Applied Mathematics (ICAAM 2018), AIP Conf. Proc., 1997, eds. A. Ashyralyev, A. Lukashov, M. Sadybekov, Amer. Inst. Phys., 2018, UNSP 020017-1
A. Zvyagin, “Attractors for model of polymer solutions motion”, Discret. Contin. Dyn. Syst., 38:12, SI (2018), 6305–6325
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