Analysis Seminar 2020/21

The Analysis and Differential Equations Seminar takes place on Thursdays at 2.15 via zoom.
If you have any queries, or if you would like to be on our e-mail list, please contact the organisers Karsten Matthies and Monica Musso.

Spring 2021

Date Speaker Title/Abstract
11 Feb No seminar
Unit boards/interviews
18 Feb Reto Buzano
Queen Mary University of London and Università di Torino
A Local Singularity Analysis for the Ricci Flow
The Ricci Flow is the most famous and most successful geometric flow, having led to resolutions of the Poincaré and Geometrisation Conjectures, as well as proofs of the Differentiable Sphere Theorem and the Generalised Smale Conjecture. For many of these applications, it is important to understand precisely how singularities form along the flow - which is a notoriously difficult task, in particular in dimensions strictly greater than three. In this talk, we explain a recently developed refined singularity analysis for the Ricci Flow, investigating curvature blow-up rates locally. We first introduce general definitions of Type I and Type II singular points and show that these are indeed the only possible types of singular points in a Ricci Flow. In particular, near any singular point the Riemannian curvature tensor has to blow up at least at a Type I rate, generalising a result previously obtained together with Enders and Topping under a global Type I assumption. We also prove analogous results for the Ricci tensor, as well as a localised version of Sesum’s result, namely that the Ricci curvature must blow up near every singular point of a Ricci flow, again at least at a Type I rate. If time permits, we will also see some applications of the theory to Ricci flows with bounded scalar curvature. This is joint work with Gianmichele Di Matteo.
25 Feb No seminar
4 Mar Thierry Gallay
Université Grenoble Alpes
Arnold's variational principle and its application to the stability of viscous planar vortices
We revisit the variational approach to nonlinear stability of planar flows, which was developed by V. I. Arnold around 1965. In particular, we study the coercivity properties of the quadratic form that describes the second variation of the energy at a radially symmetric vortex with strictly decreasing vorticity profile. We also show that this quadratic form can be used to obtain a new proof of nonlinear stability for the Lamb-Oseen vortices, which are self-similar solutions of the two-dimensional Navier-Stokes equations. This talk is based on joint work with V. Sverak.
11 Mar Javier Gòmez Serrano
Brown University and Universitat de Barcelona
Symmetry in stationary and uniformly rotating solutions of fluid equations
In this talk, I will discuss characterizations of stationary or uniformly-rotating solutions of 2D Euler and other similar equations. The main question we want to address is whether every stationary/uniformly-rotating solution must be radially symmetric. Based on joint work with Jaemin Park, Jia Shi and Yao Yao.
18 Mar Josephine Evans
University of Warwick
Long time behaviour (hypocoercivity) of a kinetic model for bacterial chemotaxis
This talk is based on a joint work with Havva Yoldas in Lyon. We look at a run and tumble model for bacterial movement in the presence of a chemical attractant. The model is kinetic in the sense that it is posed on the density of bacteria in position and velocity space and has a free transport term. The model describes microscopic dynamics and has an equation similar to Keller-Segel as its macroscopic/hydrodynamic limit. I will compare the equation to similar PDE appearing in the kinetic theory of gasses. The long -time behaviour of such models is usually approached via hypocoercivity techniques; I will discuss the additional challenges in using hypocoercivity techniques with the run and tumble model. I will also discuss the unusual confinement mechanism in the run and tumble model and why it is challenging to capture.
25 Mar Alberto Enciso
CSIC Madrid
Uniqueness and convexity of Whitham’s highest cusped wave
Whitham’s equation is a nonlinear, nonlocal, very weakly dispersive shallow water wave model in one space dimension. In this talk we are concerned with non-smooth traveling wave solutions to this equation, which are often referred to as waves of extreme form. Their existence was conjectured by Whitham in 1967 and established by Ehrnström and Wahlén just a few years ago, who proved that there is a monotone traveling wave featuring a cusp of exactly C1/2 Hölder regularity at the origin. Our objetive in this talk is to show that there is only one monotone traveling wave of extreme form and that, as widely believed in the community, its profile is in fact convex between crest and trough. This can be understood as the counterpart, in the case of the Whitham equation, of the landmark results on the uniqueness and convexity of traveling water waves of extreme form. The talk is based on joint work with Javier Gòmez-Serrano (Barcelona and Brown) and Bruno Vergara (Barcelona).
Easter break
15 Apr No seminar

22 Apr Tobias Weth
Goethe-Universität Frankfurt
Spiraling solutions of nonlinear Schrödinger equations
I will report on recent results on a new family of solutions to a class stationary nonlinear Schrädinger equations. These solutions are spiraling in the sense that they are not axially symmetric but invariant under screw motion, i.e., they share the symmetry properties of a helicoid. In addition to existence results, we provide information on the shape of spiraling solutions, which depends on the parameter value representing the rotational slope of the underlying screw motion. Our results complement a related analysis of Del Pino, Musso and Pacard for the Allen-Cahn equation, whereas the nature of results and the underlying variational structure are completely different. This is joint work with Oscar Agudelo and Joel Kübler.
29 Apr Veronique Fischer
University of Bath
Towards semi-classical analysis for sub-elliptic operators
In this talk, I will discuss the development of semi-classical analysis for sub-elliptic operators such as sub-Laplacians. For an elliptic operator, this is well understood as the tools and methods to study e.g. quantum ergodicity or Schrödinger equations have become well established over the past fifty years. They rely on the pseudo-differential theory, and in the elliptic case the space of principal symbols is commutative. The aim of this talk is to present my approach to define similar tools for sub-Laplacians, leading to more non-commutative concepts.

Autumn 2020

Date Speaker Title/Abstract
8 Oct Shrish Parmeshwar
University of Bath
Global-in-Time Solutions to the N-Body Euler-Poisson System
We investigate the N-Body compressible Euler-Poisson system, modelling multiple stars interacting with each other via Newtonian gravity. If we prescribe initial data so that each star expands indefinitely, one might expect that two of them will collide in finite time due to their expansion, and the influence of gravity. In this talk we show that there exists a large family of initial positions and velocities for the system such that each star can expand for all time, but no two will touch in finite time. To do this we use a scaling mechanism present in the compressible Euler system, and a careful analysis of how the gravitational interaction between stars affects their dynamics.
22 Oct Jarrod Williams
University of Bath
Elliptic structures in the Gauss-Codazzi-Mainardi equations, with applications to General Relativity
Motivated by the problem of constructing “initial data” for the Cauchy problem in General Relativity, we discuss a certain mixed-order elliptic reduction of the Gauss-Codazzi-Mainardi (G-C-M) equations for an embedded Riemannian 3-manifold. Using this formulation, one can hope to construct perturbative solutions of the G-C-M equations for which particular components of the ambient Weyl curvature are prescribed at the outset. We show that this can be done by a simple Implicit Function Theorem argument when the background belongs to a certain family of closed hyperbolic manifolds. Moreover, the space of solutions in this case is shown to admit an explicit parametrisation via an elliptic complex.
5 Nov Antonio J. Fernández
University of Bath
Non-homogeneous Gagliardo-Nirenberg-type inequalities and a biharmonic NLS
The aim of this talk is twofold. On one hand, we investigate some non-homogeneous Gagliardo-Nirenberg-type inequalities. Special attention will be paid to the method used to prove such estimates. On the other hand, we analyse the standing waves for a fourth-order Schrödinger equation with mixed dispersion that minimize the associated energy the L2-norm (the mass) is kept fixed. The talk is based on a joint work with Louis Jeanjean, Rainer Mandel and Mihai Mariş
19 Nov Miles Wheeler
University of Bath
Existence and non-existence of solitary water waves
3 Dec No seminar

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