Abstracts

Tobias Barker

Abstract Motivated by an open question posed by Chemin, Zhang and Zhang, I will discuss the dynamics of the 3D incompressible Navier-Stokes equations from 'geometrically constrained' initial data. Joint work with Christophe Prange (Cergy) and Jin Tan (Cergy).

Juan Davila

Abstract We construct overhanging solitary waves with constant vorticity. Although there is numerical evidence for their existence, these almost singular solutions seem difficult to construct using complex variables or bifurcation theory. We employ a method that is similar in spirit to the desingularisation of constant mean curvature surfaces. This is collaboration with Manuel del Pino, Monica Musso and Miles Wheeler (U. of Bath).

Arjen Doelman

Abstract Ecosystem models typically are of reaction-diffusion type, with components representing (various kinds of) vegetation, soil and overland water, etc.. The local topography of the terrain is an important factor, and it appears as a heterogeneous spatially varying term in such models. In this talk we first briefly consider earlier studies of (somewhat) realistic two-component reaction-diffusion models that focus on the persistence under such a heterogeneous effect of (singular) localized structures that exist (and are stable) in the homogeneous limit. However, to build a conceptual understanding of the impact of such a heterogeneity on the patterns exhibited by a reaction-diffusion model, we consider the most simple setting of a scalar bi-stable Allen-Cahn model driven by a small spatially heterogeneous effect. Motivated by the observation that the stationary problem associated to the PDE can be seen as a (weakly) driven planar Hamiltonian system, we consider the question as, “What kind of novel (stable) multiple-front patterns may be initiated by the heterogeneous topographical effects?”, and “What is the impact of the heterogeneity on the dynamics of (multiple) interacting patterns/fronts?” In the case of a spatially periodic topography these questions bring us into the realm of (spatial) horseshoe dynamics, for spatially localized topographies we will show that the decay rates of the topography have a crucial impact on the long-term dynamics of multi-front patterns.

Jean Dolbeault

Abstract Self-similar profiles attract large time solutions of some nonlinear diffusion equations which are deeply connected with classical functional inequalities of Sobolev type. A notion of generalized entropy is the key tool which relates the nonlinear regime to the linearized problem around the asymptotic profile and reduces the analysis to a spectral problem. Estimates can be made constructive. This gives quantitative stability results with explicit constants. Entropy methods will be compared with other direct methods, intended for instance to obtain bounds on the stability constant in the Bianchi-Egnell stability result for the Sobolev inequality.

Thierry Gallay

Abstract The goal of these lectures is to present in a recent result in collaboration with V. Sverak, which is devoted to the vanishing viscosity limit for vortex rings originating from circular filaments. The whole analysis is carried out in the framework of axisymmetric flows without swirl, which is both mathematically convenient and physically relevant for the phenomena we want to study. We first review a few standard properties of the axisymmetric solutions without swirl of the incompressible Navier-Stokes equations. We then explain how to construct an approximate solution of our problem, which is accurate enough in the high Reynolds number regime. Finally, we establish the stability of our approximation using carefully designed energy estimates, which partially rely on Arnold's geometric approach to the stability of stationary flows for the two-dimensional Euler equations.

Claudia Garcia

Abstract In this talk, we will construct a large class of non-trivial (non radial) self-similar solutions of the generalized surface quasi-geostrophic equation. To the best of our knowledge, this is the first rigorous construction of any self-similar solution for these equations. Moreover, the solutions are of spiral type, locally integrable and may have a change of sign. This is a joint work with Javier Gómez-Serrano.

Michael Herrmann

Abstract We study a non-autonomous mean-field model for a system of interacting bistable particles and characterize the stability of traveling wave solutions under certain simplifying assumptions. Our results predict two distinct propagation modes for phase interfaces with hysteresis and explain the dynamical behavior in numerical simulations with different parameters. joint with Barbara Niethammer (University of Bonn)

Gustav Holzegel

Abstract This mini course will be an introduction to the dynamics of linear and non-linear waves in various geometric settings, focussing on (but not restricted to) black hole spacetimes. Starting with linear problems, we will outline the proofs of the relevant decay estimates emphasising the role played by various geometric phenomena associated with black holes. We finally present a novel and unifying framework to prove small data global existence results for quasi-linear wave equations in these geometric settings, which will include the case of the (near) Kerr black hole geometry. The last part will be based on recent joint work with Dafermos, Rodnianski and Taylor, arXiv:2212.14093

John King

Abstract A very widely studied class of asymptotic behaviour will be revisited by formal approaches, focussing on some perhaps underexplored features, particularly in the case of 'fast' diffusion. Aspects of potentially broader relevance will be highlighted.

Camilla Nobili

Abstract In this talk we consider the two-dimensional Boussinesq equations on a bounded domain with Navier-slip boundary conditions. In the first part we will present some recent results concerning scaling laws for the Nusselt number in the case of flat and rough boundaries. In the second part, we will discuss a new result on long-time asymptotics and convergence to hydrostatic equilibrium for the system when the molecular diffusivity is zero.

Jens Rademacher

Abstract Spatial patterns far from homogeneous steady states can sometimes be studied with the help of singular perturbation theory. A singular limit in parameters is typically an infinite temporal or spatial scale separation, such as an infinite diffusion ratio in reaction diffusion systems. The lectures will discuss elements of geometric singular perturbation theory and its application to existence, stability and reduced dynamics of resulting patterns. This in particular entails the role of the singular perturbation in the associated eigenvalue problem. The FitzHugh-Nagumo model will serve as a basic example, and further cases will be outlined with focus on reaction-diffusion systems. Some time will be used for exercises.

Wolfgang Reichel

Abstract see here

Guido Schneider

Abstract We are interested in weakly unstable dissipative systems with a conservation law exhibiting a Turing instability, a Turing-Hopf instability, or a long wave Hopf instability. With the help of a multiple scaling perturbation ansatz a Ginzburg-Landau equation coupled to a scalar conservation law can be derived as an amplitude system for the approximate description of the dynamics of the original pattern forming system near this first instability. We use this approximate description for showing long time existence beyond the natural Ginzburg-Landau time scale of all solutions starting in a small neighborhood of the weakly unstable ground state in the original system.

Philippe Souplet

Abstract

(Joint work with Vuk Milisic, CNRS and Univ.~Brest) We investigate a mathematical model of cell motility involving space dependent adhesion forces mediated by transient elastic linkages. The model is fourth order in space and involves a time delay or memory operator, which takes into account the ``age'' of the linkages through a parameter \(\varepsilon>0\), corresponding to the typical persistence time scale of the memory term. We show that:

For sufficiently small \(\varepsilon>0\), the problem is globally well posed, and its solution \(z_\varepsilon\) converges to one of the steady states as \(t\to\infty\).

When \(\varepsilon>0\), the solution \(z_\varepsilon\) converges in a certain sense to the solution of a limiting Cauchy problem with friction and without delay (in particular the past data for \(t<0 \) is ``forgotten'').

For sufficiently small \(\varepsilon>0\), the time delay problem inherits part of the large time asymptotic properties of the problem without delay.

Martin Taylor

Abstract Spatially homogeneous Friedmann–Lemaitre–Robertson–Walker solutions constitute an infinite dimensional family of cosmological solutions of the Einstein--massless Vlasov system. Each member describes a spatially homogenous universe, filled with massless particles, evolving from a big bang singularity and expanding towards the future at a decelerated rate. I will present a theorem on the future stability of this family to spherically symmetric perturbations.

Peter Topping

Abstract I will explain some very recent work that completes the optimal existence and uniqueness story for the logarithmic fast diffusion equation in 2D. This equation has a close connection with the so-called Ricci flow, which naturally leads us to the right spaces in which to consider solutions and initial data. (No prior knowledge of Ricci flow will be assumed.) The recent progress centres on how to handle uniqueness when the initial data is very rough and the underlying domain is noncompact. I may also explain some geometric applications. Joint work with Hao Yin and Luke Peachey.

Miles Wheeler

Abstract Hollow vortices are bounded regions of constant pressure with finite circulation embedded into an otherwise irrotational flow. This is a classical model for localized vorticity, with connections to both water waves and vortex patches. We prove that non-degenerate configurations of point vortices which steadily translate or rotate can be desingularized into analogous configurations of hollow vortices. Moreover, the resulting local curves of solutions can be extended using global bifurcation theory. As examples, we give what appear to be the first rigorous existence results for rotating hollow vortex pairs and for stationary hollow vortex tripoles. This is joint work with Ming Chen and Samuel Walsh.

Michael Winkler

Abstract Simple models for nutrient-oriented bacterial migration are compared. In particular, the potential of certain cross-degenerate diffusion mechanisms to adequately describe experimentally observed phenomena related to emergence and stabilization of structures are discussed. Resulting mathematical challenges are described and possible approaches outlined, both at levels of basic existence theories and at the stage of qualitative analysis.

Ewelina Zatorska

Abstract I will present our recent results on the 1-dimensional hydrodynamic models of traffic, lubrication and collective behaviour in one space dimension. I will discuss existence results, interesting two-velocity reformulations, singular limits (hard congestion) and long-time behaviour of solutions.