Abstract:
The article surveys the works of T. G. Sukacheva and her students studying the models of incompressible viscoelastic Kelvin–Voigt fluids in the framework of the theory of semilinear Sobolev-type equations. We focus on the unstable case because of greater generality. The idea is illustrated by an example: the non-stationary thermoconvection problem for the order 0 Oskolkov model. Firstly, we study the abstract Cauchy problem for a semilinear nonautonomous Sobolev-type equation. Then, we treat the corresponding initial-boundary value problem as its concrete realization. We prove the existence and uniqueness of a solution to the stated problem. The solution itself is a quasi-stationary semi-trajectory. We describe the extended phase space of the problem. Other problems of hydrodynamics can also be investigated in this way: for instance, the linearized Oskolkov model, Taylor's problem, as well as some models describing the motion of an incompressible viscoelastic Kelvin–Voigt fluid in the magnetic field of the Earth.
\Bibitem{SukKon14}
\by T.~G.~Sukacheva, A.~O.~Kondyukov
\paper On a class of Sobolev-type equations
\jour Vestnik YuUrGU. Ser. Mat. Model. Progr.
\yr 2014
\vol 7
\issue 4
\pages 5--21
\mathnet{http://mi.mathnet.ru/vyuru234}
\crossref{https://doi.org/10.14529/mmp140401}
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This publication is cited in the following 7 articles:
D. Z. Dhumd, Shatha A. Haddad, “ONSET OF DOUBLE-DIFFUSIVE CONVECTION WITH A KELVIN–VOIGT FLUID OF VARIABLE ORDER”, Special Topics Rev Porous Media, 15:3 (2024), 1
N. V. Kalenova, A. M. Romanenkov, Springer Proceedings in Earth and Environmental Sciences, Physical and Mathematical Modeling of Earth and Environment Processes—2022, 2023, 413
S. A. Zagrebina, A. S. Konkina, “The non-classical models of mathematical physics the multipoint initial-final value condition”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 15:1 (2022), 60–83
B. Straughan, “Instability thresholds for thermal convection in a Kelvin–Voigt fluid of variable order”, Rend. Circ. Mat. Palermo, II. Ser, 71:1 (2022), 187
Alevtina V. Keller, Minzilia A. Sagadeeva, Springer Proceedings in Mathematics & Statistics, 325, Semigroups of Operators – Theory and Applications, 2020, 263
A. O. Kondyukov, T. G. Sukacheva, S. I. Kadchenko, L. S. Ryazanova, “Computational experiment for a class of mathematical models of magnetohydrodynamics”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 10:1 (2017), 149–155
N. A. Manakova, G. A. Sviridyuk, “Neklassicheskie uravneniya matematicheskoi fiziki. Fazovye prostranstva polulineinykh uravnenii sobolevskogo tipa”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 8:3 (2016), 31–51
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