Perturbation Methods for Abstract Cauchy Problems associated to Evolution Equations

与演化方程相关的抽象柯西问题的摄动方法

• 批准号: 468736785
• 负责人: Dr. Christian Budde
• 金额: --
• 依托单位: Department of Mathematics and Applied Mathematics
• 依托单位国家: 德国
• 项目类别: WBP Fellowship
• 财政年份: 2021
• 资助国家: 德国
• 起止时间: 2020-12-31 至 无数据
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/468736785?language=en
• 关键词: Perturbation; Methods; Abstract; Cauchy; Problems

In this project we study the perturbation theory for abstract Cauchy problems associated to evolution equations. Many well-known partial differential equations modeling physical systems, such as the heat equation, the Schrödinger equation or the wave equation, use temporal change of states. "Evolution equation" is an umbrella term for such equations that can be interpreted as differential laws describing the development of a system or as a mathematical treatment of motion in time. By modeling systems evolving in time, the time variable plays a crucial role, as the equations are built by balancing the change of the system in time against its "spatial" behaviour. By finding a solution of an evolution equation one has the possibility to predict the future of the corresponding physical system which makes it deterministic. In an ideal world, one would be able to find an exact solution in terms of elementary functions for a given equation since the evolution of the system is given explicitly. This would be beneficial for practitioners. However, most of the really interesting equations cannot be treated in such a way. Nonetheless, the study of existence of solutions, their uniqueness, as well as the qualitative properties serve as a partial validation of the used models and also provides a foundation for the numerical analysis of the equation which is important for the practitioners. Perturbation theory is a powerful tool in this regard and allows for a more general and abstract view on such problems. The general idea is to start with a related and simpler problem where the exact solution is already known.On the one hand side, we study autonomous dynamical systems. We will not only consider C_0-semigroups but also non-strongly continuous semigroups or semigroups on vector lattices by means of bi-continuous semigroups and ru-continuous semigroups, respectively. On the other hand side, we also consider time-dependent and hence non-autonomous dynamical systems of different orders.

在这个项目中，我们研究了与演化方程相关的抽象柯西问题的摄动理论。许多著名的模拟物理系统的偏微分方程，如热方程、Schrödinger方程或波动方程，都使用状态的时间变化。“进化方程”是这类方程的总称，它可以被解释为描述系统发展的微分定律，也可以被解释为对时间运动的数学处理。通过对随时间演化的系统建模，时间变量起着至关重要的作用，因为方程是通过平衡系统在时间上的变化与其“空间”行为来建立的。通过找到一个进化方程的解，人们就有可能预测相应物理系统的未来，从而使其具有确定性。在理想世界中，由于系统的演化是明确给出的，因此人们能够用初等函数找到给定方程的精确解。这对从业者是有益的。然而，大多数真正有趣的方程不能用这种方式来处理。尽管如此，对解的存在性、唯一性以及定性性质的研究，作为对所用模型的部分验证，也为方程的数值分析提供了基础，这对实践者来说很重要。摄动理论在这方面是一个强大的工具，它允许对这类问题有一个更一般和抽象的看法。一般的想法是从一个相关的、更简单的问题开始，而这个问题的确切解是已知的。一方面，我们研究自主动力系统。利用双连续半群和鲁连续半群，我们不仅考虑c_0半群，而且考虑向量格上的非强连续半群或半群。另一方面，我们也考虑了不同阶次的时变非自治动力系统。