PDEs and dynamical systems

该基金旨在支持在偏微分方程（PDEs）和更广泛的动力系统的各个主题上的持续合作。偏微分方程提供了一种（非常大的）预测模型，用于整个科学领域，它们的数学研究已经很好地建立起来了。研究这样一个数学模型的所有解的集合，并描述它们的行为，是动力系统理论的职权范围。这一建议研究了一些基本模型，例如热方程和模拟流体的Navier-Stokes方程，旨在理解初始热场的结构如何在线性系统中产生看似非线性的行为，以及数值计算如何与抽象的数学处理（在Navier-Stokes方程的情况下）相关联。在更抽象的动力系统设置中，一股将发展出一个适当的框架来处理系统本身（而不仅仅是其状态）随时间变化的情况（所谓的“非自主”动力系统）。另外两个纯数学问题——两者都有应用（一个在动力系统中，一个在偏微分方程中）——寻求用有限的变量集合来重现“有限维集合”，并通过具有“良好”性质的正则函数来近似不规则函数。因此，提议的主题涵盖了许多数学领域，具有潜在的大量应用。

The grant is designed to support continued collaboration in various topics in partial differential equations (PDEs) and, more generally, dynamical systems.PDEs provide one (very large) class of predictive models used throughout the sciences, and their mathematical study is well established. The study of the collection of all solutions of such a mathematical model, and the description of their behaviour, is the remit of the theory of dynamical systems.This proposal investigates some fundamental models, e.g. the heat equation and the Navier-Stokes equations that model fluids, and aims to understand how the structure of the initial heat field can produce seemingly nonlinear behaviour in a linear system, and how numerical computations can be realted to an abstract mathematical treatment (in the case of the Navier-Stokes equations).In the more abstract setting of dynamical systems, one strand will develop an appropriate framework in which to treat situations where the system itself (and not just its state) can change over time (so-called "non-autonomomus" dynamical systems).Two more purely mathematical problems - both of which have applications (one in dynamical systems, one in PDEs) - seek to reproduce "finite-dimensional sets" using a finite collection of variables, and to approximate irregular functions by regular functions with "nice" properties.As such the proposed topics cover a number of mathematical areas with - potentially - a large number of applications.