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

本课题主要研究具有不同拓扑结构的非线性度量状态空间上非自治集值过程的长期动力学行为。它们为不同类型的微分方程式、差分方程式和在建模中出现的其他问题提供了一个抽象的框架。例如，这种非独特的演化自然发生在控制系统中，当我们不能建立柯西问题的这种演化时，例如在三维的Navier-Stokes方程中。具有许多标准拓扑空间(如范数和弱拓扑)的Banach空间通常作为状态空间是足够的，但有一些重要的应用，例如，涉及约束，而它们不是。当考虑区域或像空间时，状态空间和状态空间上的拓扑的更广泛的选择，期望得到更好和更广泛适用的充分存在吸引子的条件。解的存在和稳定性具有相当初级的性质，但仍然是必不可少的。本课题主要研究系统的渐近性质，特别是全局吸引子和拉回吸引子。将吸引子理论推广到非自治集值过程。将特别强调例子，既是为了说明结果，也是为了激励结果。还将研究形状(即欧几里德空间中可能的非光滑集)及其边界的演变。

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.