Dan J. Hill: Going around in circles: Developing a functional analytic framework for radial solutions to PDEs

Abstract:

Understanding spatial localisation in higher dimensions remains an open problem in many contexts. There has been recent progress in studying localised axisymmetric and dihedral patterns by expressing the problem in polar coordinates and utilising theory from radial spatial dynamics. However, such techniques are often insufficient for obtaining rigorous existence results.

In order to further develop analytic tools—such as centre-manifold reductions—for radial PDE systems, we first need to establish the basic theory of radial function spaces. In contrast to general nonautonomous PDEs, radial PDEs possess highly structured nonautonomous terms and explicit smoothness conditions at the origin. We utilise this structure in developing our framework of radial function spaces, noting the connection between our functions spaces and Bessel functions and Hankel transforms.

In this talk I will first motivate the need for radial function spaces with two examples of fully localised waves/patterns, before then introducing our radial function spaces. This will include highlighting various properties of each space, as well as their associated nonautonomous differential operators. Following this, I will conclude by briefly discussing future applications of this analytic framework.

This work is in collaboration with Mark Groves (Saarland).