Об одном классе уравнений соболевского типа
Ключевые слова:
уравнения соболевского типа, несжимаемая вязкоупругая жидкости, относительно p-секториальные операторы, расширенное фазовое пространство.Аннотация
Статья содержит обзор работ Т.Г. Сукачевой и ее учеников в области исследования моделей несжимаемых вязкоупругих жидкостей Кельвина - Фойгта в рамках теории полулинейных уравнений соболевского типа. Основное внимание уделяется нестационарному случаю ввиду его большей общности. Идея исследования демонстрируется на примере нестационарной задачи термоконвекции для модели Осколкова нулевого порядка. Вначале изучается абстрактная задача Коши для полулинейного неавтономного уравнения соболевского типа, а затем соответствующая начально-краевая задача рассматривается как конкретная интерпретация этой задачи. Доказана теорема существования единственного решения указанной задачи, являющегося квазистационарной полутраекторией, и получено описание ее расширенного фазового пространства. Подобным образом могут быть исследованы и другие задачи гидродинамики, например, линеаризованные модели Осколкова, задача Тейлора, а также некоторые модели, описывающие движение несжимаемых вязкоупругих жидкостей Кельвина - Фойгта в магнитном поле Земли.Библиографические ссылки
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