Volume 11, no. 4Pages 67 - 77
Phase Space of the Initial-Boundary Value Problem for the Oskolkov System of Highest OrderA.O. Kondyukov, T.G. Sukacheva
In recent decades, the theory of Sobolev type equations is actively studied in various aspects. The application of the semigroup approach to the theory of singular Sobolev type equations has received a deep and wide development in the works of the scientific direction headed by G.A. Sviridyuk. This work is adjacent to this scientific direction. The first initial-boundary value problem for the Oskolkov system is investigated. In our case, the system simulates a plane-parallel incompressible Kelvin-Voigt fluid of the higher order. This problem has an advantage, since the phase space for the above system can be described completely at any values of the parameter that characterizes the elastic properties of the liquid. This article is devoted to presenting of this fact. The study is carried out within the framework of the theory of semi-linear autonomous Sobolev type equations on the basis of the concepts of a relatively spectrally bounded operator and a quasi-stationary trajectory. Full text
- Sobolev type equations; phase space; quasi-stationary trajectories; Oskolkov systems; incompressible viscoelastic Kelvin-Voigt fluid.
- 1. Oskolkov A.P. Initial-Boundary Value Problems for the Equations of the Motion of the Kelvin-Voight and Oldroyd Fluids. Proceedings of the Steklov Institute of Mathematics (Trudy Matematicheskogo instituta imeni V.А. Steklova), 1988, no. 179, pp. 126-164. (in Russian)
2. Sviridyuk G.A. On a Model of the Dynamics of an Incompressible Viscoelastic Fluid. Russian Mathematics (Izvestiya VUZ. Matematika), 1994, vol. 38, no. 1, pp. 59-68. (in Russian)
3. Oskolkov A.P. On a Quasilinear Parabolic System with a Small Parameter Approximating the Navier-Stokes System. Zapiski nauchnykh seminarov POMI, 1980, vol. 96, pp. 233-236. (in Russian)
4. Sviridyuk G.A. On the Variety of Solutions of a Certain Problem of an Incompressible Viscoelastic Fluid. Differential Equations, 1988, vol. 24, no. 10, pp. 1846-1848.
5. Sviridyuk G.A., Sukacheva T.G. Phase Spaces of a Class of Operator Equations. Differential Equations, 1990, vol. 26, no. 2, pp. 250-258.
6. Sviridyuk G.A., Sukacheva T.G. The Cauchy Problem for a Class of Semilinear Equations of Sobolev Type. Siberian Mathematical Journal, 1990, vol. 31, no. 5, pp. 109-119. (in Russian)
7. Sviridyuk G.A. On the General Theory of Semigroups of Operators. Russian Mathematical Surveys (Uspekhi Matematicheskikh Nauk), 1994, vol. 49, no. 4, pp. 47-74. (in Russian)
8. Sviridyuk G.A., Fedorov V.E. Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, VSP, 2003. DOI: 10.1515/9783110915501
9. Sviridyuk G.A. Quasistationary Trajectories of Semilinear Dynamical Equations of Sobolev Type. Russian Academy of Sciences. Izvestiya Mathematics, 1993, vol. 57, no. 3, pp. 192-207. (in Russian)
10. Sviridyuk G.A. Phase Spaces of Semilinear Equations of Sobolev Type with Relatively Strongly Sectorial Operators. Algebra and Analysis, 1994, vol. 6, no. 5, pp. 216-237. (in Russian)
11. Sviridyuk G.A., Yakupov M.M. The Phase Space of the Initial-Boundary Value Problem for the Oskolkov System. Differential Equations, 1996, vol. 32, no. 11, pp. 1538-1543.
12. Kondyukov A.O., Sukacheva T.G. Phase Space of the Initial-Boundary Value Problem for the Oskolkov System of Nonzero Order. Computational Mathematics and Mathematical Physics, 2015, vol. 55, no. 5, pp. 823-829. DOI: 10.7868/S0044466915050130
13. Leng S. Introduction to Differentiable Manifolds. N.Y., Springer, 2002.
14. Borisovich Yu.G., Zvyagin V.G., Sapronov Y.I. Nonlinear Fredholm Mappings and Leray-Schauder Theory. Russian Mathematical Surveys, 1977, vol. 32, no. 4, pp. 3-54. (in Russian)
15. Manakova N.A. Zadachi optimal'nogo upravleniya dlya uravnenij sobolevskogo tipa [Optimal Control Problems for Sobolev Type Equations]. Chelyabinsk, Publishing center of SUSU, 2012. (in Russian)
16. Sagadeeva M.A. Dihotomii reshenij linejnyh uravnenij sobolevskogo tipa [Dichotomies of Solutions of Linear Sobolev Type Equations]. Chelyabinsk, Publishing center of SUSU, 2012. (in Russian)
17. Zamyshlyaeva A.A. Linejnye uravneniya sobolevskogo tipa vysshego poryadka [Linear Sobolev Type Equations of Higher Order]. Chelyabinsk, Publishing center of SUSU, 2012. (in Russian)
18. Zagrebina S.A. Ustojchivye i neustojchivye mnogoobraziya reshenij polulinejnyh uravnenij sobolevskogo tipa [Stable and Unstable Manifolds of Solutions of Semilinear Sobolev Type Equations]. Chelyabinsk, Publishing center of SUSU, 2016. (in Russian)
19. Zagrebina S.A. A Multipoint Initial-Final Value Problem for a Linear Model of Plane-Parallel Thermal Convection in Viscoelastic Incompressible Fluid. tBulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 3, pp. 5-22. (in Russian) DOI: 10.14529/mmp140301
20. Zamyshlyaeva A.A., Bychkov E.V. The Cauchy Problem for the Sobolev Type Equetion of Higher Order. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2018, vol. 11, no. 1, pp. 5-14. DOI: 10.14529/mmp180101
21. Keller A.V. On the Computational Efficiency of the Algorithm of the Numerical Solution of Optimal Control Problems for Models of Leontieff Type. Journal of Computational and Engineering Mathematics, 2015, vol. 2, no. 2, pp. 39-59. DOI: 10.14529/jcem150205
22. Sviridyuk G.A., Manakova N.A. The Barenblatt-Zheltov-Kochina Model with Additive White Noise in Quasi-Sobolev Spaces. Journal of Computational and Engineering Mathematics, 2016, vol. 3, no. 1, pp. 61-67. DOI: 10.14529/jcem160107