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This article is cited in 29 scientific papers (total in 29 papers)
Attractors of equations of non-Newtonian fluid dynamics
V. G. Zvyagin, S. K. Kondrat'ev Voronezh State University
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.
Received: 22.06.2014
Citation:
V. G. Zvyagin, S. K. Kondrat'ev, “Attractors of equations of non-Newtonian fluid dynamics”, Uspekhi Mat. Nauk, 69:5(419) (2014), 81–156; Russian Math. Surveys, 69:5 (2014), 845–913
Linking options:
https://www.mathnet.ru/eng/rm9615https://doi.org/10.1070/RM2014v069n05ABEH004918 https://www.mathnet.ru/eng/rm/v69/i5/p81
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Abstract page: | 1062 | Russian version PDF: | 370 | English version PDF: | 15 | References: | 83 | First page: | 62 |
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