# Analysis Seminar 2018/19

The Analysis and Differential Equations Seminar takes place on Thursdays at 2.15 in 4W 1.7 (Wolfson Lecture Theatre).
If you have any queries, or if you would like to be on our e-mail list, please contact the organisers Mark Haskins, Karsten Matthies and Monica Musso.

## Spring 2019

Date Speaker Title/Abstract
Imperial College
Diffusion and mixing in incompressible flows
We study diffusion and mixing in different fluid dynamics models, mainly related to incompressible flows. In this setting, mixing is a purely advective effect which causes a transfer of energy to high frequencies. Mixing acts to enhance the dissipative forces, giving rise to what we refer to as enhanced dissipation. We understand by this the identification of a decay time-scale, which is faster than the purely diffusive one. We will give a general quantitative criterion that links mixing rates (in terms of decay of negative Sobolev norms) to enhanced dissipation time-scales. Applications include passive scalar evolution in both planar and radial settings, fractional diffusion, linearized two-dimensional Navier-Stokes equations, and even simple examples in kinetic theory.
7 Feb John Toland
University of Bath
Localizing weak convergence in L
This lecture will review the Yosida-Hewitt identification of the dual of L as a class of finitely additive measures and examine its consequences. One of these, a necessary and sufficient test for weak convergence of sequences, will be illustrated by examples. Another, in the setting of a locally compact Hausdorff topological space X, is the equivalence between weak convergence on X, and weak convergence at every point in the one-point compactification of X. This leads to a relation with the essential range of an L function u at a point x. Finally the relationship between the the Yosida-Hewitt representation (a finitely additive measure) of a bounded functional on L and the Riesz Representation (a countably additive measure) of its restriction to continuous functions will be touched upon, time allowing. The lecture is designed for graduate students or anyone such as the speaker who is mystified by the dual of L
14 Feb No seminar
No seminar
21 Feb Melanie Rupflin
University of Oxford
Sharp eigenvalue estimates on degenerating surfaces
We consider the first non-zero eigenvalue λ1 of the Laplacian on hyperbolic surfaces for which one disconnecting collar degenerates and show that the gradient of λ1 is given essentially explicitly in terms of the dual of the differential of the degenerating length coordinate. As a corollary we obtain sharp error estimates on λ1 that improve previous results of Burger and Schoen-Wolpert-Yau and provide new information on the second term in the polyhomogenous expansion of λ1. The presented results are joint work with Nadine Grosse.
28 Feb José Luis Rodrigo
University of Warwick
Results motivated by the study of the evolution of isolated vortex lines for 3D Euler
In the study of an isolated vortex line for 3D Euler one is trying to make sense of the evolution of a curve, where the vorticity (a distribution in this case) is supported, and tangential to the curve. This idealised vorticity generates a velocity field that is too singular (like the inverse of the distance to the curve and therefore not in L^2) and making rigorous sense of the evolution of the curve remains a fundamental problem. In the talk I will present examples of simple globally divergence-free velocity fields for which an initial delta function in one point (in 2D, with analogous results in 3D) becomes a delta supported on a set of Hausdorff dimension 2. In this examples the velocity does not correspond to an active scalar equation. I will also present a construction of an active scalar equation in 2D, with a milder singularity than that present in Euler for which there exists an an initial data given by a point delta becomes a one dimensional set. These results are joint with C. Fefferman and B. Pooley. These are examples in which we have non-uniqueness for the evolution of a singular "vorticity". For the Surface Quasi-Geostrophic equation, an equation with great similarities with 3D Euler, the evolution of a sharp front is the analogous scenario to a vortex line for 3D Euler. I will describe a geometric construction using "almost-sharp" fronts than ensure the evolution according to the equation derived heuristically. This part is joint work with C. Fefferman.
7 Mar Marie-Therese Wolfram
University of Warwick
On asymptotic gradient flow structures of PDE models with excluded volume effects
joint work with M. Bruna (Oxford), M. Burger (Münster) and H. Ranetbauer (Vienna)
We discuss the analysis of a cross-diffusion PDE system for mixtures of hard spheres, which was derived by Bruna and Chapman (J Chem Phys 137, 2012) from a stochastic system of interacting Brownian particles using the methods of matched asymptotics. While the system has a gradient flow structure in the symmetric case of all particles having the same size and diffusivity, this is not valid in general. For the general case, we introduce the concept of an asymptotic gradient flow structure and show how it can be used to study the behavior close to equilibrium. To gain further insights into the dynamics of asymptotic gradient flows, we study the system in the special case of two specific species – namely diffusing (Brownian) and immobile (obstacle) particles. In this case the cross-diffusion system reduces to a single nonlinear nonlinear Fokker--Planck equation, which again has no full gradient flow structure. However it can be interpreted as an asymptotic gradient flow for different entropy and mobility pairs. We discuss several possible such pairs and present global in time existence results as well as study the long time behavior of the corresponding full gradient flow equation. Furthermore we illustrate the dynamics of the different equations with numerical simulations.
14 Mar Matteo Cozzi
University of Bath
Regularity and rigidity results or nonlocal minimal graphs
Nonlocal minimal surfaces are hypersurfaces of Euclidean space that minimize the fractional perimeter, a geometric functional introduced in 2010 by Caffarelli, Roquejoffre \& Savin in connection with phase transition problems displaying long-range interactions. In this talk, I will introduce these objects, describe the most important progresses made so far in their analysis, and discuss the most challenging open questions. I will then focus on the particular case of nonlocal minimal graphs and present some recent results obtained on their regularity and classification in collaboration with X. Cabr\'e (ICREA \& UPC, Barcelona), A. Farina (Universit\'e de Picardie, Amiens), and L. Lombardini (UWA, Perth).
21 Mar Van Tien Nguyen
NYU Abu Dhabi
Singularity formation in the energy-supercritical harmonic heat flow and wave maps
Abstract in pdf
28 Mar Alexander V. Sobolev
UCL
Venue 6W1.2: On the Schatten-von Neumann properties of some pseudo-differential operators
We obtain explicit estimates for quasi-norms of pseudo-differential operators in the Schatten-von Neumann classes S_q with 0< q < 1. The estimates are applied to derive semi-classical bounds for operators with smooth or non-smooth symbols.
4 Apr Kevin Hughes
University of Bristol
Discrete harmonic analysis
We will discuss recent work on two discrete harmonic analysis problems. The first problem is discrete restriction theory related to Waring’s problem. Waring’s problem is a difficult problem in analytic number theory, but surprisingly recent progress relies little on arithmetic. The second problem concerns averages over integer points on spheres. In contrast to the first problem, the second problem is very sensitive to the arithmetic structure of the integers.
11 Apr Dimitrios Roxanas
University of Bath
Global solutions to the focusing energy-critical heat equation in R4
The main theme of this talk is the long-time behaviour of the focusing energy-critical nonlinear heat equation ut − Δ u − |u|2u = 0, in R4. We will see that solutions emanating from initial data with energy and kinetic energy below those of the stationary solution are global and decay to zero. We show that global solutions dissipate to zero, building on a refined small data theory and L2−dissipation, expanding on ideas that have previously been applied to the Navier-Stokes system. To rule out the possibility of blow-up we argue by resorting to the concentration-compactness plus rigidity’’ approach of Kenig and Merle for dispersive equations. We exploit the dissipation but our proof does not rely on maximum/comparison principles. This is joint work with Stephen Gustafson.
18 Apr Gregory Seregin
University of Oxford
Type I blow ups and Liouville Type Theorems for Navier-Stokes equations
In the talk, the relationship between existence of singularities of weak solutions to the Navier-Stokes equations and Liouville type theorems for mild bounded ancient solutions to the same equations are going to be discussed. Technically, mild bounded ancient solutions can be derived rescaling around a potential singular point. Those solutions can be used to rule out or to construct singularities of type I.

## Autunm 2018

Date Speaker Title/Abstract
4 Oct David Lafontaine
University of Bath
About wave and Schrödinger equations in the exterior of many strictly convex obstacles
In order to study the non-linear Schrödinger and wave equations, it is crucial to understand the decay of solutions of the associated linear equations. When a trapped trajectory exists, a loss is unavoidable for a first family of a-priori estimates of the linear flow: the so-called smoothing estimates. In contrast, we will show that in the exterior of many strictly convex obstacles, the estimates of space-time norms of solutions, known as Strichartz estimates, hold with no loss with respect to the flat case, as soon as the dynamic of the trapped trajectories is sufficiently unstable. Finally, if time permits, we will say a word about the associated non-linear equations: if the geometry does not induce too much concentration of energy, we expect that the solutions behave linearly in large times.
11 Oct Carlos Roman
Max-Planck-Institut für Mathematik in den Naturwissenschaften Leipzig, Germany
On the 3D Ginzburg-Landau model of superconductivity.
The Ginzburg-Landau model is a phenomenological description of superconductivity. A crucial feature is the occurrence of vortices (similar to those in fluid mechanics, but quantized), which appear above a certain value of the strength of the applied magnetic field called the first critical field. We are interested in the regime of small ɛ, where ɛ>0 is the inverse of the Ginzburg-Landau parameter. In this regime, the vortices are at main order codimension 2 topological singularities. In this talk I will present a quantitative 3D vortex approximation construction for the Ginzburg-Landau functional, which provides an approximation of vortex lines coupled to a lower bound for the energy, optimal to leading order, analogous to the 2D ones, and valid for the first time at the ɛ-level. I will then apply these results to provide a sharp estimate of the first critical field and describe the behavior of global minimizers for the 3D Ginzburg-Landau functional below and near this value.
18 Oct Matthew Gursky
Notre Dame
The singular Yamabe problem and some fully nonlinear generalizations
I will begin with an overview of the work of Loewner-Nirenberg on constructing complete conformal metrics of constant negative scalar curvature on domains in Euclidean space, and the question of regularity up to the boundary. I will then describe some fully nonlinear generalizations of their work, the geometric content of these generalizations, and some boundary/interior regularity issues that arise in their study.
25 Oct Sofía Ortega Castillo
Cimat, visiting University of Bath
Strong pseudoconvexity in Banach spaces
I will talk about strong pseudoconvexity as a relevant notion to understand the local geometry of domains of holomorphy with C^2 boundary in the finite-dimensional Euclidean spaces over the complex plane. It is also desirable to comprehend the geometry near the boundary of domains of holomorphy in arbitrary Banach spaces, but it may be out of reach to have two degrees of differentiability of a simple example such as the ball of a Banach space, because its norm may lack differentiability. I will justify a suitable notion of strong pseudoconvexity for bounded domains without C^2-smooth boundary, that will rely on an adequate generalization of a strict plurisubharmonic function to the case when it is not differentiable. I will also mention examples and counterexamples for these extended notions. I will also discuss special solutions to the Cauchy-Riemann equations on strongly pseudoconvex domains without C^2-smooth boundary.
1 Nov
No seminar
8 Nov Yang Li
Imperial College
The Dirichlet problem for maximal graphs of higher codimension
Maximal submanifolds in Lorentzian type ambient spaces are the formal analogues of minimal submanifolds in Euclidean spaces, which arise naturally in adiabatic problems for G2 manifolds. We obtain general existence and uniqueness results for the Dirichlet problem of graphical maximal submanifolds in any codimension, which stand in sharp contrast to the analogous problem for graphical minimal submanifolds.
15 Nov Neshan Wickramasekera
University of Cambridge
Regularity of stable CMC hypersurfaces
22 Nov Lucia Scardia
Heriot-Watt University
Equilibrium measures for nonlocal energies: The effect of anisotropy
Nonlocal energies are continuum models for large systems of particles with long-range interactions. Under the assumption that the interaction potential is radially symmetric, several authors have investigated qualitative properties of energy minimisers. But what can be said in the case of anisotropic kernels? Motivated by the example of dislocation interactions in materials science, we pushed the methods developed for nonlocal energies beyond the case of radially symmetric potentials, and discovered surprising connections with random matrices and fluid dynamics.
29 Nov Julian Braun
University of Warwick
The Thermodynamic Limit of Transition Rates of Crystalline Defects
We consider a point defect in a homogeneous crystalline solid. We can show that the formation free energy as well as the transition rate between two stable configurations converge as the number of atoms tends to infinity and characterise the limit. Both cases can be reduced to a careful renormalisation analysis of the vibrational entropy difference, which is achieved by identifying an underlying spatial decomposition of said entropy. (This is joint work with Hong Duong and Christoph Ortner.)
6 Dec Veronique Fischer
University of Bath
Quantum ergodicity and sub-Riemannian geometries
In this talk, I will discuss pseudo-differential theory in the context of sub-Riemannian geometries. The relevant underlying structure of these geometries is the one of nilpotent Lie groups. I will present the progress towards semi-classical analysis there. I will finish the talk with my hopes for its development, especially regarding quantum ergodicity in this context.
13 Dec Peter Topping
University of Warwick
Ricci flow and Ricci limit spaces
Venue: 4E3.5 A sequence of Einstein manifolds, or more generally a sequence of manifolds with Ricci curvature bounded below, will converge (after passing to a subsequence) to a limit. This `Ricci limit space’ is a limit of smooth objects, but can itself be singular, and it is a central question to understand how singular it can be. I will describe some new developments in Ricci flow theory that give good insight into this, including a proof that 3D Ricci limit spaces are locally bi-Holder homeomorphic to smooth manifolds, which solves more than an old conjecture of Anderson-Cheeger-Colding-Tian in this dimension. The essential issue is to try to smooth the Ricci limit space with Ricci flow, which is essentially a parabolic PDE, but attempting to do so raises a lot of new well-posedness issues for the equation. No knowledge of Ricci flow will be assumed. Joint work with Miles Simon and Andrew McLeod.