Nagel Rainer, Rhandi Abdelaziz
Arbeitsbereich Funktionalanalysis, Mathematisches Institut, Auf der Morgenstelle 10, 72076 Tübingen, Germany.
Dipartimento di Ingegneria dell'Informazione, Ingegneria Elettrica e Matematica Applicata, Università degli Studi di Salerno, Via Giovanni Paolo II 132, 84084 Fisciano, Italy.
Philos Trans A Math Phys Eng Sci. 2020 Nov 27;378(2185):20190610. doi: 10.1098/rsta.2019.0610. Epub 2020 Oct 19.
Most dynamical systems arise from partial differential equations (PDEs) that can be represented as an abstract evolution equation on a suitable state space complemented by an initial or final condition. Thus, the system can be written as a Cauchy problem on an abstract function space with appropriate topological structures. To study the qualitative and quantitative properties of the solutions, the theory of one-parameter operator semigroups is a most powerful tool. This approach has been used by many authors and applied to quite different fields, e.g. ordinary and PDEs, nonlinear dynamical systems, control theory, functional differential and Volterra equations, mathematical physics, mathematical biology, stochastic processes. The present special issue of Philosophical Transactions includes papers on semigroups and their applications. This article is part of the theme issue 'Semigroup applications everywhere'.
大多数动力系统源自偏微分方程（PDEs），这些方程可表示为合适状态空间上的抽象演化方程，并辅以初始条件或最终条件。因此，该系统可写成具有适当拓扑结构的抽象函数空间上的柯西问题。为研究解的定性和定量性质，单参数算子半群理论是一个非常强大的工具。许多作者都采用了这种方法，并将其应用于截然不同的领域，例如常微分方程和偏微分方程、非线性动力系统、控制理论、泛函微分方程和沃尔泰拉方程、数学物理、数学生物学、随机过程。《哲学汇刊》的本期特刊包含关于半群及其应用的论文。本文是“半群应用无处不在”主题特刊的一部分。
