Abstract
We study the qualitative dynamics of weak solutions for the modified Kelvin–Voigt model based on the theory of pullback-attractors of trajectory spaces. First, for the studied model, an auxiliary problem is considered, its solvability in the weak sense is proved, and solution estimates are established. Then, on the basis of the obtained estimates of the solutions, a family of trajectory spaces is determined and the existence of trajectory and minimal pullback-attractors of the considered trajectory spaces is proved.
This is a preview of subscription content, log in via an institution to check access.
REFERENCES
Carvalho, A.N., Langa, J.A., Robinson, J.C. Attractors for infinite-dimensional non-autonomous dynamical systems, Appl. Math. Sci. 182 (Springer-Verlag, New York, 2013 ).
Vorotnikov, D. "Asymptotic behaviour of the non-autonomous 3D NavierЦ-Stokes problem with coercive force", J. Diff. Equat. 251 (8), 2209-2225 (2011).
Pavlovskii, V.A. "To the problem of theoretical description of weak fluid solutions of polymers", Dokl. AN SSSR 200 (4), 809-812 (1971) [in Russian].
Amfilokhiev, V.B., Voitkunskii, Ya.I., Mazaeva, N.P., Khodorkovskii, Ya.S. "Flows of polymeric solutions in presence of convective accelerations", Tr. Lenin. korablestroit. inst. 96, 3-9 (1975) [in Russian].
Amfilokhiev, V.B., Pavlovskii, V.A. "Experimental data on laminar-turbulent transition in the flows of polymeric solutions in pipes", Tr. Lenin. korablestroit. inst. 104, 3-5 (1976) [in Russian].
Oskolkov, A.P. "Theory of nonstationary flows of Kelvin–Voigt fluids", J. Soviet Math. 28, 751-758 (1985).
Oskolkov, A.P. "Initial-boundary value problems for equations of motion of Kelvin–Voight fluids and Oldroyd fluids", Proc. Steklov Inst. Math. 179, 137-182 (1989).
Zvyagin, V.G., Turbin, M.V. "The study of initial-boundary value problems for mathematical models of the motion of Kelvin–Voigt fluids", J. Math. Sci. 168 (2), 157-308 (2010).
Zvyagin, V.G., Turbin, M.V. Mathematical problems of hydrodynamics of viscoelastic media (KRASAND (URSS), Moscow, 2012 ) [in Russian].
Turbin, M.V., Ustiuzhaninova, A.S. "The existence theorem for a weak solution to initial-boundary value problem for system of equations describing the motion of weak aqueous polymer solutions", Russ. Math. 63, 54-69 (2019).
Funding
This work is funded by the Russian Foundation for Basic Research (project no. 20-01-00051) and by the Ministry of Science and Higher Education (project no. FZGU-2020-0035).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by V. G. Zvyagin.
Dedicated to 90th anniversary of the birth of Yuri Grigorievich Borisovich
Russian Text © The Author(s), 2021, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2021, No. 5, pp. 98–104.
About this article
Cite this article
Ustiuzhaninova, A.S. Pullback-Attractors for the Modified Kelvin–Voigt Model. Russ Math. 65, 77–82 (2021). https://doi.org/10.3103/S1066369X2105011X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X2105011X