PDEs and dynamical systems

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.

The primary short to medium term impacts of this proposal will be academic.

However, the topics have possible applications in fluid turbulence and mathematical modelling more generally, including in the understanding and development of numerical methods.

The subjects included are closely related to the PhD topics of a number of recently graduated PhD students, and continued work with them will help support their transition from postgraduate students to postdoc positions or permanent academic jobs.