Non-autonomous set-valued dynamical processes: Asymptotics and applications

非自治集值动态过程：渐近学和应用

基本信息

批准号：
169166715
负责人：
Professor Dr. Peter E. Kloeden
金额：
--
依托单位：
Fachbereich 12 - Informatik und Mathematik
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2010
资助国家：
德国
起止时间：
2009-12-31 至 2013-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/169166715?language=en
关键词：
Non autonomous set valued dynamical

项目摘要

The main thrust of this project concerns the long-term dynamical behaviour of non-autonomous setvalued processes on non-linear metric state spaces with different topologies when considered as the domain and the image space. These provide an abstract framework for diverse types of differential equations, difference equations and further problems occurring in modelling. Such nonunique evolution occurs naturally in, for example, control systems, and also when we are not able to establish it for Cauchy problems, such as in the 3-dimensional Navier-Stokes equations. Banach spaces with many standard topologies spaces (like the norm and weak topologies) are often adequate as state spaces, but there are important applications, e.g., involving constraints, where they are not. Improved and more widely applicable conditions sufficient for the existence of attractors are expected by a more general choices of state spaces and topologies on them when considered as the domain or the image spaces. Existence and stability of solutions have a rather introductory character, but are nevertheless essential. This project focuses on the asymptotic properties, i.e. in particular, on global and pullback attractors. The theory of attractors is extended for non-autonomous set-valued processes. Particular emphasis will be given to examples, both to illustrate and motivate the results. The evolution of shapes (i.e. possibly non-smooth sets in the Euclidean space) and their boundaries will also be investigated.

该项目的主要目的是涉及非自治setValed过程在非线性度量状态空间上具有不同拓扑的长期动力学行为，当被视为域名和图像空间时。这些为各种类型的微分方程，差异方程式以及建模中发生的其他问题提供了一个抽象的框架。这种非唯一的演变自然发生在例如控制系统中，而且当我们无法为库奇问题（例如3维Navier-Stokes方程）建立它时。具有许多标准拓扑空间的Banach空间（例如标准和弱拓扑）通常是状态空间足够的，但是存在重要的应用，例如，涉及约束，而不是限制。当将其视为域或图像空间时，可以通过更一般的状态空间和拓扑选择更一般的状态空间和拓扑选择来改善和更广泛的适用条件。解决方案的存在和稳定性具有相当介绍性的特征，但至关重要。该项目侧重于渐近性能，尤其是全球和回调吸引子。吸引子的理论扩展了非自主设置值的过程。示例将特别强调示例，以说明和激励结果。还将研究形状的演变（即欧几里得空间中的非平滑套件）及其边界也将进行研究。