# 2022-23

**Organisers:** Jo Evans, Marie-Therese Wolfram, Jose Rodrigo, Charlie Elliott

**Time & place:** B3.02 Tuesdays 12-1

The PDEA seminar will be held in a hybrid format this year - speakers either come in person or join us via MS teams. You can do the same - come in person or watch it on teams ;-)

Recently, Terence Tao used a new quantitative approach to infer that certain ‘slightly supercritical’ quantities for the Navier–Stokes equations must become unbounded near a potential blow-up time. In this talk I’ll discuss a new strategy for proving quantitative bounds for the Navier–Stokes equations, as well as applications to behaviours of potentially singular solutions. This talk is based upon joint work with Christophe Prange (CNRS, Cergy Paris Université).

We consider one-dimensional chains and multi-dimensional networks of harmonic oscillators coupled to two Langevin heat reservoirs at different temperatures. Each particle interacts with its nearest neighbours by harmonic potentials and all individual particles are confined by harmonic potentials, too. In previous works we investigated the sharp N-particle dependence of the spectral gap of the associated generator in different physical scenarios and for different spatial dimensions. In this talk I will present new results on the behaviour of the spectral gap when considering longer-range interactions in the same model. In particular, depending on the strength of the longer-range interaction, there are different regimes appearing where the gap drastically changes behaviour but even the hypoellipticity of the operator breaks down. This is a joint work with Simon Becker (ETH)

Antiferromagnetic spin systems are magnetic lattice systems in which the exchange interaction

between two spins favors anti-alignment. Such systems are said to be geometrically frustrated, if

due to the geometry of the lattice no spin configuration can simultaneously minimize all pairwise

interactions. As a consequence of that, ground states of frustrated spin systems may exhibit

nontrivial patterns and give rise to unconventional magnetic order. An example of such a system

is the antiferromagnetic XY-model on the triangular lattice (AFXY). Here the frustration leads

to a concentration of energy at a surface scaling which can be tracked via the so-called chirality

of the spin field. In this talk we are concerned with the discrete-to-continuum variational analysis

of the AFXY in an energetic regime where the system cannot overcome the energetic barrier of

a chirality transition and instead finitely many vortex-type singularities emerge in the continuum

limit.

This is joint work with M. Cicalese (TU Munich), L. Kreutz (CMU Pittsburgh), and G. Orlando

(Politecnico Bari).

The seminar focuses on the study of a class of partial differential equations on graphs, motivated by application in data science. More precisely, we will discuss graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance, using Benamou--Brenier formulation. The graph continuity equation uses an upwind interpolation to define the density along the edges; while this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of “vertices” is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-interaction energy the nonlocal nonlocal-interaction equation (NLNLIE), and develop existence theory as curve of maximal slope. Furthermore, we establish a discrete-to-continuum convergence result with respect to the number of vertices. On a slightly different perspective, by means of a classical fixed-point argument we can show existence and uniqueness of solutions to a class of nonlocal continuity equations on graphs. We consider general interpolation functions, which give rise to a variety of different dynamics, e.g., the nonlocal interaction dynamics coming from a solution-dependent velocity field. Our analysis reveals structural differences with the more standard Euclidean space, as some analogous properties rely on the interpolation chosen. The talk is based on works in collaboration with F. S. Patacchini (IFP Energies Nouvelles), A. Schlichting (University of Münster), and D. Slepcev (Carnegie Mellon University).

We consider the Euler equations for incompressible fluids in 3-dimension. A classical question that goes back to Helmholtz is to describe the evolution of vorticities with a high concentration around a curve. The work of Da Rios in 1906 states that such a curve must evolve by the so-called "binormal curvature flow". Existence of true solutions whose vorticity is concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings, and of a helical filament, associated to a translating-rotating helix. In this talk I will consider the case of two vortex rings interacting between each other, the so-called leapfrogging. The results are in collaboration with J. Davila (U. of Bath), M. del Pino (U. of Bath) and J. Wei (U. of British Columbia).

The kinetic wave equation is expected to provide a kinetic model for (weak) turbulence. To exploit this model to better understand turbulence, the challenge is twofold: on the one hand, understand the conditions of validity of the model, and on the other hand, understand the dynamics it predicts. I will present results in both directions, relying in particular on work with C. Collot, H. Dietert, G. Dubach, B. Harrop-Griffiths.