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Think Global, Act Local: A Mathematician's Guide to Inducing Localised Patterns
You can join this seminar online: https://teams.microsoft.com/l/meetup-join/19%3ameeting_OTkwNTUzZjYtODU0ZC00ZDIxLThjZWItMjIwZTkxZTRjMjY4%40thread.v2/0?context=%7b%22Tid%22%3a%22cc95de1b-97f5-4f93-b4ba-fe68b852cf91%22%2c%22Oid%22%3a%22e6ced614-5673-458c-832d-5d4ada66f593%22%7d
In this talk I will discuss how adding a compact region of spatial heterogeneity to a PDE model can not only induce the emergence of fully localised 2D patterns, but also allows us to rigorously prove and characterise their bifurcation. The idea is inspired by experimental and numerical studies of magnetic fluids and tornados, where our compact heterogeneity corresponds to a local spike in the magnetic field and temperature gradient, respectively. In particular, we obtain local bifurcation results for fully localised patterns both with and without radial or dihedral symmetry, and rigorously continue these solutions to large amplitude. Notably, the initial bifurcating solution (which can be stable at bifurcation) varies between a radially-symmetric spot and a ‘dipole’ solution as the width of the spatial heterogeneity increases.
This work is in collaboration with David J.B. Lloyd and Matthew R. Turner (both University of Surrey).
Date:
16 October 2025, 12:00
Venue:
Mathematical Institute, Woodstock Road OX2 6GG
Venue Details:
L3
Speaker:
Dan J. Hill (University of Oxford, Mathematical Institute)
Organiser:
Alain Goriely (University of Oxford)
Organiser contact email address:
Kirkham@maths.ox.ac.uk
Part of:
Industrial & Applied Mathematics Seminar
Booking required?:
Not required
Cost:
Free
Audience:
Public
Editor:
Nicola Kirkham
