A non-stationary model of the incompressible viscoelastic Kelvin-Voigt fluid of non-zero order in the magnetic field of the Earth

We 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.