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

这个项目的主旨关注的长期动态行为的非自治集值过程的非线性度量状态空间与不同的拓扑结构时，被认为是域和图像空间。这些提供了一个抽象的框架，不同类型的微分方程，差分方程和进一步的问题发生在建模。这种非唯一的演化自然发生在例如控制系统中，也发生在我们无法为柯西问题（例如三维Navier-Stokes方程）建立它的时候。具有许多标准拓扑空间（如范数和弱拓扑）的Banach空间通常足以作为状态空间，但也有重要的应用，例如，涉及到约束条件，而这些约束条件并不存在。改进的和更广泛的适用条件充分的吸引子的存在，预计由一个更一般的选择状态空间和拓扑结构时，他们被视为域或图像空间。解决方案的存在和稳定性具有相当的入门性质，但仍然是必不可少的。这个项目的重点是渐近性质，即特别是在全球和拉回吸引子。将吸引子理论推广到非自治集值过程。将特别强调例子，以说明和激励的结果。形状的演变（即可能在欧几里德空间的非光滑集）和它们的边界也将被调查。

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.