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.

A nanopteron is a traveling wave whose profile is the superposition of an exponentially localized “core” term and a small-amplitude periodic “ripple” term. I will discuss the construction of nanopteron traveling wave solutions to a versatile ordinary differential equation that is a singular perturbation of the traveling wave ODE for the Korteweg-de Vries (KdV) equation. The nanopteron problem permits two free variables—the amplitude and the phase shift of the periodic ripple—and I will highlight how each variable separately resolves what is otherwise an overdetermined perturbation problem. This nanopteron construction draws on techniques from bifurcation theory, functional analysis, Fourier analysis, and oscillatory integral theory and generalizes and unifies a variety of approaches by Amick and Toland, Sun and Shen, and Lombardi. More broadly, this perturbed KdV equation has served as a valuable model for singularly perturbed nonlocal problems posed on lattices, and I will sketch how many technical complications in lattices show up transparently here.

The study of sharp upper bounds for Laplacian eigenvalues under area constraints is a classical topic in spectral geometry. A key source of interest lies in the remarkable fact that metrics achieving equality in such bounds correspond to metrics induced by minimal surfaces in spheres. Even more strikingly, analogous connections between eigenvalues and natural geometric structures continue to emerge across diverse settings. In this talk, I will highlight notable examples of these correspondences and discuss their implications for both spectral estimates and geometric analysis.

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.

This talk presents some of my recent joint work with P.-T. Nguyen, Q.-H. Nguyen, and P. Zhang.

In the first part, after reviewing several qualitative and quantitative regularity criteria for the three-dimensional Navier–Stokes system, I will discuss our recent results that extend T. Tao’s theorem from $L^3$ solutions to $\dot B_{p,\infty}^{-1+\frac{3}{p}}$ solutions. In particular, we obtain a new blow-up rate estimate in lower regularity spaces. The proof relies on refined Carleman inequalities and carefully constructed approximate solutions.

In the second part, I will focus on the instantaneous radius of analyticity. After introducing the concept and some classical results, I will describe several new bilinear estimates that enable us to establish an optimal lower bound of radius of analyticity for solutions with $L^p$ initial data.

In this talk, I will discuss the existence of global solutions to a time-fractionally damped nonlinear Jordan—Moore—Gibson—Thompson (JMGT) model. JMGT models arise when considering a class of thermal relaxation laws in the modelling of sound wave propagation through complex media (such as human tissue).

From the analysis point of view, we place ourselves in the so-called critical case (most challenging for JMGT equations), such that the nonlinearity structure and time-fractional kernel must be properly exploited to extract sufficient dissipation to rule out blow-up. Doing so, we obtain a global existence result for small data.

This talk is based on a recent collaboration with B. Said-Houari (University of Sharjah).

We investigate a passive vector field which is transported and stretched by a divergence-free Gaussian velocity field, delta-correlated in time and poorly correlated in space (spatially nonsmooth). Although the advection of a scalar field (Kraichnan's passive scalar model) is known to enjoy regularizing properties, the potentially competing stretching term in vector advection may induce singularity formation. We establish that the regularization effect is actually retained in certain regimes. While this is true in any dimension $d\ge 3$, it notably implies a regularization result for linearized 3D Euler equations with stochastic modeling of turbulent velocities, and for the induction equation in magnetohydrodynamic turbulence. The presentation is based on a joint work with Francesco Grotto and Mario Maurelli.

We present work on the Gross-Pitaevskii equation in the infinite strip $\mathbb{R} \times (0,l)$, for positive width $l>0$, with Neumann boundary conditions. The one dimensional equation on $\mathbb{R}$ admits a family of travelling wave solutions for wave speeds $|c|<\sqrt{2}$, $S_{c}$. We show that for wave speeds $c$ close enough to $0$ and values of $l$ close enough to some critical widths $l_{c,k}$, $k\geq 1$ an integer, there are travelling wave solutions $\Psi_{l,k}$ that bifurcate from $S_{c}$, and depend smoothly on the strip width $l$.