Oskolkov Models and Sobolev-Type Equations

Данная статья посвящена обзору работ, выполненных автором совместно со своими учениками и посвященных исследованию различных моделей Осколкова. Их отличительной особенностью является использование полугруппового подхода, лежащего в основе метода фазового пространства, широко применяемого в теории уравнений соболевского типа. Приведены различные модели несжимаемой вязкоупругой жидкости, описываемые уравнениями Осколкова. В качестве примеров рассмотрены вырожденная задача магнитогидродинамики, задача термоконвекции и задача Тейлора. Разрешимость соответствующих начально-краевых задач исследуется в рамках теории уравнений соболевского типа, основанной на теории относительно p-секториальных операторов и вырождающихся полугрупп операторов. Доказана теорема существования единственного решения, являющегося квазистационарной полутраекторией, и получено описание расширенного фазового пространства. Основы теории разрешимости уравнений соболевского типа были заложены профессором Г.Свиридюком. Затем эта теория вместе с различными приложениями была успешно развита его последователями.

Ключевые слова

системы Осколкова; уравнения соболевского типа; фазовое пространство; несжимаемая вязкоупругая жидкость.

Литература

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