Phase Space of the Initial-Boundary Value Problem for the Oskolkov System of Highest Order

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.

Keywords

Sobolev type equations; phase space; quasi-stationary trajectories; Oskolkov systems; incompressible viscoelastic Kelvin-Voigt fluid.

References

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