Abstract
The initial boundary value problem for a system of partial differential equations modeling the dynamics of Kelvin-Voigt incompressible viscoelastic fluid of higher order in the Earth's magnetic field is studied. Problems of this type arise in the study of the process of rotation of a certain volume of fluid in the Earth's magnetic field. Research of the models of Kelvin–Voigt media has its source in the scientific works by A.P. Oskolkov, who summarizes the system of Navier–Stokes equations and theorems of unique existence of solutions to the corresponding initial boundary value problems. Subsequently, these models are studied by G.A. Sviridyuk and his followers. This model is studied for the first time and summarizes corresponding results for the model of magnetohydrodynamics of the nonzero order. The article deals with local unique solvability of chosen problem in the framework of the theory of autonomous semilinear Sobolev type equations. The main method is the method of phase space. The basic tool is the notion of p -sectorial operator and resolving singular semigroup of operators generated by it. In other words, the semigroup approach is used in the research. Besides the introduction, conclusion and reference list, the article includes three parts. In the first part of the article, the abstract Cauchy problem for semilinear autonomous equation of Sobolev type is presented. Here the concepts of Cauchy problem for Sobolev type equations, the phase space, quasi-stationary semitrajectory are introduced, and the theorems providing necessary and sufficient conditions for the existence of quasi-stationary semitrajectories are presented. In the second part, the Cauchy–Dirichlet problem is considered as a specific interpretation of the abstract problem. In the third part, the existence of a unique solution to the problem, which is a quasi-stationary semitrajectory is proved, and the description of its phase space is obtained. In conclusion, the possible ways of further research are outlined
Highlights
Также надо заметить, что задача (1), (2) входит в круг исследований сред Кельвина–Фойгта, начатых в работах [1, 4], в которых обобщалась система уравнений Навье–Стокса [5, 6] и получены теоремы существования и единственности соответствующих начально-краевых задач
The initial boundary value problem for a system of partial differential equations modeling the dynamics of Kelvin-Voigt incompressible viscoelastic fluid of higher order in the Earth's magnetic field is studied
Oskolkov, who summarizes the system of Navier–Stokes equations and theorems of unique existence of solutions to the corresponding initial boundary value problems
Summary
Описано фазовое пространство задачи Коши–Дирихле для системы уравнений в частных производных, моделирующей движение несжимаемой жидкости Кельвина–Фойгта высшего порядка в магнитном поле Земли. В первой части статьи изложим абстрактную задачу Коши для полулинейного автономного уравнения соболевского типа (все результаты почерпнуты из монографии [12], поэтому будут приведены без доказательств). Назовем локальным решением (далее просто решением) задачи (3), (4) вектор-функцию u ∈C∞ ((0,T );UM ) , которая удовлетворяет уравнению (4) и такая, что u(t) → u0 при t → 0 +. 32] также известно, что решения задачи (3), (4) существуют не для всех u0 ∈UM. Множество B ⊂ UM назовем фазовым пространством уравнения (4), если для любой точки u0 ∈ B существует единственное решение задачи (3), (4), причем u(t) ∈ B.
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