# A Non-Stationary Model of the Incompressible Viscoelastic Kelvin-Voigt Fluid of Non-Zero Order in the Magnetic Field of the Earth

A.O. Kondyukov, T.G. SukachevaWe investigate the Cauchy-Dirichlet problem for a system of Oskolkov equations of nonzero order. The considered mathematical model describes the flow of an incompressible viscoelastic Kelvin-Voigt fluid in the magnetic field of the Earth. The model takes into account that the fluid is subject to various external influences, which depend on both the coordinate of the point in space and the time. The first part of the paper presents the known results obtained by the authors earlier and based on the theory of solvability of the Cauchy problem for semilinear nonautonomous Sobolev type equations. In the second part, we reduce the considered mathematical model to an abstract Cauchy problem. In the third part, we prove the main result that is the theorem on the existence and uniqueness of the solution. Also, we establish the conditions for the existence of quasi-stationary semitrajectories, and describe the extended phase space of the model under study. In this paper, we summarize our results for the Oskolkov system that simulates the motion of a viscoelastic incompressible Kelvin-Voigt fluid of zero order in the magnetic field of the Earth.Full text

- Keywords
- magnetohydrodynamics; Sobolev type equations; extended phase space; incompressible viscoelastic fluid.
- References
- 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.A. Steklova), 1988, no. 179, pp. 126-164. (in Russian)

2. Hide R. On Planetary Atmospheres and Interiors. Providence, American Mathematical Society, 1971.

3. Sukacheva T.G., Kondyukov A.O. Phase Space of a Model of Magnetohydrodynamics. Differential Equations, 2015, vol. 51, no. 4, pp. 502-509. DOI: 10.1134/S0012266115040072

4. Kadchenko S.I., Kondyukov A.O. Numerical Study of a Flow of Viscoelastic Fluid of Kelvin-Voigt Having Zero Order in a Magnetic Field. Journal of Computational and Engineering Mathematics, 2016, vol. 3, no. 2, pp. 40-47. DOI: 10.14529/jcem1602005

5. Sukacheva T.G., Kondyukov A.O. Phase Space of a Model of Magnetohydrodynamics of Nonzero Order. Differential Equations, 2017, vol. 53, no. 8, pp. 1054-1061. DOI: 10.1134/S0012266117080109

6. Kondyukov A.O. Generalized Model of Incompressible Viscoelastic Fluid in the Earth's Magnetic Field. Bulletin of the South Ural State University. Series: Mathematical. Mechanics. Physics, 2016, vol. 8, no. 3, pp. 13-21. (in Russian)

7. Kondyukov A.O., Sukacheva T.G., Kadchenko S.I., Ryazanova L.S. Computational Experiment for a Class of Mathematical Models of Magnetohydrodynamics. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2017, vol. 10, no. 1, pp. 149-155. DOI: 10.14529/mmp170110

8. Sviridyuk G.A., Sukacheva T.G. [Phase Spaces of a Class of Operator Equations]. Differential Equations, 1990, vol. 26, no. 2, pp. 250-258. (in Russian)

9. Sviridyuk G.A., Sukacheva T.G. [The Cauchy Problem for a Class of Semilinear Equations of Sobolev Type]. Sibirskii matematicheskii zhurnal, 1990, vol. 31, no. 5, pp. 109-119. (in Russian)

10. Sviridyuk G.A. On the General Theory of Operator Semigroups. Russian Mathematical Surveys, 1994, vol. 49, no. 4, pp. 45-74. DOI: 10.1070/RM1994v049n04ABEH002390

11. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, Koln, Tokyo, VSP, 2003.

12. Matveeva O.P., Sukacheva T.G. Matematicheskie modeli vyazkouprugikh neszhimaemykh zhidkostey nenulevogo poryadka [The Mathematical Models of a Viscoelastic Incompressible Fluid of Nonzero Order]. Chelyabinsk, Publishing Center of South Ural State University, 2014. (in Russian)

13. 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)

14. Henry D. Geometric Theory of Semilinear Parabolic Equations. Series: Lecture Notes in Mathematics, Vol. 840. Berlin, Springer, 1981.

15. Sviridyuk G.A. On a Model of Weakly Viscoelastic Fluid. Russian Mathematics (Izvestiya VUZ. Matematika), 1994, vol. 38, no. 1, pp. 59-68. (in Russian)