Analysis Seminar 2025/26

This is work in collaboration with David Bourne and two former PhD students. We have used semi-discrete optimal transport to give new proofs of existence and regularity of weak solutions to a system of equations, both incompressible and compressible, modelling large-scale atmospheric flows.  This method also yields the first fully 3D numerical simulations, and several new theoretical side results. More generally, I can discuss the formulation of a variety of geophysical fluid problems using optimal transport, yielding naturally the proof of physically conjectured properties.

In 1974, Loewner and Nirenberg established that any smooth bounded Euclidean domain admits a conformally flat metric which is complete in the interior and has constant negative scalar curvature. Generalisations to compact manifolds with boundary, asymptotic expansions of solutions and other related problems have since received significant attention from many authors (Aviles, McOwen, Mazzeo etc.) In this talk I will discuss recent work with Luc Nguyen on generalisations of the Loewner-Nirenberg problem, in which one replaces the scalar curvature with other curvature quantities involving the Schouten tensor. These problems involve solving fully nonlinear, non-uniformly elliptic PDEs with infinite boundary data. In particular, I will discuss our recent existence results which yield solutions on arbitrary compact manifolds with boundary, and I will also discuss some surprising non-uniqueness phenomena in the upper half space, demonstrating that the 'expected' Liouville theorems do not always hold.

I will report on some recent works about a new model of liquid crystal flows enjoying a geometric free boundary condition. After recalling some classical models on liquid crystal flows, I will explain how to incorporate a free boundary condition, in relation with harmonic mappings with partially free boundary. I will develop on some results about those mappings and how they can be used to derive some regularity and partial regularity results for this new liquid crystal system. I will explain also how parabolic gluing can be exploited to construct some blow-up solutions.

The existence of localised two-dimensional patterns has been observed and studied in numerous experiments and simulations: ranging from optical solitons, to patches of desert vegetation, to fluid convection. And yet, our mathematical understanding of these emerging structures remains extremely limited beyond one-dimensional examples.

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.