Forthcoming events in this series


Thu, 31 May 2018

12:00 - 13:00
L5

Boundary layers in periodic homogenization

Christophe Prange
(University of Bordeaux)
Abstract

This talk is concerned with quantitative periodic homogenization in domains with boundaries. The quantitative analysis near boundaries leads to the study of boundary layers correctors, which have in general a nonperiodic structure. The interaction between the boundary and the microstructure creates geometric resonances, making the study of the asymptotics or continuity properties particularly challenging. The talk is based on work with S. Armstrong, T. Kuusi and J.-C. Mourrat, as well as work by Z. Shen and J. Zhuge

Thu, 24 May 2018

12:00 - 13:00
L5

The Cauchy problem in General Relativity and Kaluza Klein spacetimes

Zoe Wyatt
(University of Edinburgh and MIGSAA)
Abstract

In this talk I will start with a brief overview of the Cauchy problem for the Einstein equations of general relativity, and in particular the nonlinear stability of the trivial Minkowski solution in wave gauge as shown by Lindblad and Rodnianski. I will then discuss the Kaluza Klein spacetime of the form $R^{1+3} \times K$ where $K$ is the $n-$torus with the flat metric.  An interesting question to ask is whether this solution to the Einstein equations, viewed as an initial value problem, is stable to small perturbations of the initial data. Motivated by this problem, I will outline how the proof of stability in a restricted class of perturbations in fact follows from the work of Lindblad and Rodnianski, and discuss the physical justification behind this restriction. 

Wed, 16 May 2018

16:00 - 17:00
L6

On some connections between domain geometry and blow-up type in a nonlinear heat equation

Mikołaj Sierżęga
(University of Warsaw)
Abstract

The Fujita equation $u_{t}=\Delta u+u^{p}$, $p>1$, has been a canonical blow-up model for more than half a century. A great deal is known about the singularity formation under a variety of conditions. In particular we know that blow-up behaviour falls broadly into two categories, namely Type I and Type II. The former is generic and stable while the latter is rare and highly unstable. One of the central results in the field states that in the Sobolev subcritical regime, $1<p<\frac{n+2}{n-2}$, $n\geq 3$, only type I is possible whenever the domain is \emph{convex} in $\mathbb{R}^n$. Despite considerable effort the requirement of convexity has not been lifted and it is not clear whether this is an artefact of the methodology or whether the geometry of the domain may actually affect the blow-up type. In my talk I will discuss how the question of the blow-up type for non-convex domains is intimately related to the validity of some Li-Yau-Hamilton inequalities.

Thu, 10 May 2018
12:00
L4

Untangling of trajectories for non-smooth vector fields and Bressan's Compactness Conjecture

Paolo Bonicatto
(Universität Basel)
Abstract

Given $d \ge 1$, $T>0$ and a vector field $\mathbf b \colon [0,T] \times \mathbb R^d \to \mathbb R^d$, we study the problem of uniqueness of weak solutions to the associated transport equation $\partial_t u + \mathbf b \cdot \nabla u=0$ where $u \colon [0,T] \times \mathbb R^d \to \mathbb R$ is an unknown scalar function. In the classical setting, the method of characteristics is available and provides an explicit formula for the solution of the PDE, in terms of the flow of the vector field $\mathbf b$. However, when we drop regularity assumptions on the velocity field, uniqueness is in general lost.
In the talk we will present an approach to the problem of uniqueness based on the concept of Lagrangian representation. This tool allows to represent a suitable class of vector fields as superposition of trajectories: we will then give local conditions to ensure that this representation induces a partition of the space-time made up of disjoint trajectories, along which the PDE can be disintegrated into a family of 1-dimensional equations. We will finally show that if $\mathbf b$ is locally of class $BV$ in the space variable, the decomposition satisfies this local structural assumption: this yields in particular the renormalization property for nearly incompressible $BV$ vector fields and thus gives a positive answer to the (weak) Bressan's Compactness Conjecture. This is a joint work with S. Bianchini.
 

Fri, 27 Apr 2018
12:00
L4

Is dispersion a stabilizing or destabilizing mechanism? Landau-damping induced by fast background flows

Edriss Titi
(Texas A&M University)
Abstract

In this talk I will present a unified approach for the effect of fast rotation and dispersion as an averaging mechanism for, on the one hand, regularizing and stabilizing certain evolution equations, such as the Navier-Stokes and Burgers equations. On the other hand, I will  also present some results in which large dispersion acts as a destabilizing mechanism for the long-time dynamics of certain dissipative evolution equations, such as the Kuramoto-Sivashinsky equation. In addition, I will present some new results concerning two- and three-dimensional turbulent flows with high Reynolds numbers in periodic domains, which exhibit ``Landau-damping" mechanism due to large spatial average in the initial data.

Thu, 26 Apr 2018

12:00 - 13:00
L4

Estimates for the Green's function of the discrete Bilaplacian

Florian Schweiger
(University of Bonn)
Abstract

We consider the discrete Bilaplacian on a cube in two and three dimensions with zero boundary data and prove estimates for its Green's function that are sharp up to the boundary. The main tools in the proof are Caccioppoli estimates and a compactness argument which allows one to transfer estimate for continuous PDEs to the discrete setting. One application of these estimates is to understand the so-called membrane model from statistical physics, and we will outline how these estimates can be applied to understand the phenomenon of entropic repulsion. We will also describe some connections to numerical analysis, in particular another approach to these estimates based on convergence estimates for finite difference schemes.

Thu, 08 Mar 2018

12:00 - 13:00
L5

Interfaces in a spatial population model

Marcel Ortgiese
(University of Bath)
Abstract

We consider the  symbiotic branching model, which describes a spatial population consisting of two types in terms of a coupled system of stochastic PDEs. One particularly important special case is Kimura's stepping stone model in evolutionary biology. Our main focus is a description of the interfaces between the types in the large scale limit of the system. As a new tool we will introduce a moment duality, which also holds for the limiting model. This also has implications for a classification of entrance laws of annihilating Brownian motions.

Wed, 07 Mar 2018

14:00 - 15:00
L4

Uniform energy distribution for a non-local isoperimetric problem

Katarína Bellová
(Universität Leipzig)
Abstract

For energy functionals composed of competing short- and long-range interactions, minimizers are often conjectured to form essentially periodic patterns on some intermediate lengthscale. However,  not many detailed structural results or proofs of periodicity are known in dimensions larger than 1. We study a functional composed of  the attractive, local interfacial energy of charges concentrated on a hyperplane and the energy of the electric field generated by these charges in the full space, which can be interpreted as a repulsive, non-local functional of the charges. We follow the approach of Alberti-Choksi-Otto and prove that the energy of minimizers of this functional is uniformly distributed  on cubes intersecting the hyperplane, which are sufficiently large with respect to the intrinsic lengthscale.

This is a joint work with A. Julia and F. Otto.

Fri, 02 Mar 2018

12:00 - 13:00
C3

On the Existence of $C^{1,1}$ Isometric Immersions of Some Negatively Curved Surfaces

Siran Li
(Rice University)
Abstract

In this talk we discuss the recent proof for the existence of $C^{1,1}$ isometric immersions of several classes of negatively curved surfaces into $\R^3$, including the Lobachevsky plane, metrics of helicoid type and a one-parameter family of perturbations of the Enneper surface. Our method, following Chen--Slemrod--Wang and Cao--Huang--Wang, is to transform the Gauss--Codazzi equations into a system of hyperbolic balance laws, and prove the existence of weak solutions by finding the invariant regions. In addition, we provide further characterisation of the $C^{1,1}$ isometrically immersed generalised helicoids/catenoids established in the literature.

Thu, 01 Mar 2018
12:00
L3

Potentials for A-quasiconvexity

Bogdan Raita
(Oxford University)
Abstract

Many problems arising in Physics can be posed as minimisation of energy functionals under linear partial differential constraints. For example, a prototypical example in the Calculus of Variations is given by functionals defined on curl-free fields, i.e., gradients. Most work done subject to more general constraints met significant difficulty due to the lack of associated potentials. We show that under the constant rank assumption, which holds true of almost all examples of constraints investigated in connection with lower-semicontinuity, linear constraints admit a potential in frequency space. As a consequence, the notion of A-quasiconvexity, which involves testing with periodic fields leading to difficulties in establishing sufficiency for weak sequential lower semi-continuity, can be tested against compactly supported fields. We will indicate how this can simplify the general framework.

Wed, 28 Feb 2018

12:00 - 13:00
L4

On the Geometric Regularity Criteria for Incompressible Navier--Stokes Equations

Siran Li
(Rice University)
Abstract

We present some recent results on the regularity criteria for weak solutions to the incompressible Navier--Stokes equations (NSE) in 3 dimensions. By the work of Constantin--Fefferman, it is known that the alignment of vorticity directions is crucial to the regularity of NSE in $\R^3$.  In this talk we show a boundary regularity theorem for NSE on curvilinear domains with oblique derivative boundary conditions. As an application, the boundary regularity of incompressible flows on balls, cylinders and half-spaces with Navier boundary condition is established, provided that the vorticity is coherently aligned up to the boundary. The effects of  vorticity alignment on the $L^q$, $1<q<\infty$ norm of the vorticity will also be discussed.

Thu, 22 Feb 2018
12:00
L3

Stability of toroidal nematics

Epifanio Virga
(Università di Pavia)
Abstract

When nematic liquid crystal droplets are produced in the form or tori (or such is the shapes of confining cavities), they may be called toroidal nematics, for short. When subject to degenerate planar anchoring on the boundary of a torus, the nematic director acquires a natural equilibrium configuration within the torus, irrespective of the values of Frank's elastic constants. That is the pure bend arrangement whose integral lines run along the parallels of all inner deflated tori. This lecture is concerned with the stability of such a universal equilibrium configuration. Whenever its stability is lost, new equilibrium configurations arise in pairs, the members of which are symmetric and exhibit opposite chirality. Previous work has shown that a rescaled saddle-splay constant may be held responsible for such a chiral symmetry breaking. We shall show that that is not the only possible instability mechanism and, perhaps more importantly, we shall attempt to describe the qualitative properties of the equilibrium nematic textures that prevail when the chiral symmetry is broken.

Thu, 15 Feb 2018
12:00
L5

Stability, mixing, and stochastics in hydrodynamics

Michele Coti Zelati
(Imperial College, London)
Abstract

The basic mathematical models that describe the behavior of fluid flows date back to the eighteenth century, and yet many phenomena observed in experiments are far from being well understood from a theoretical viewpoint. For instance, especially challenging is the study of fundamental stability mechanisms when weak dissipative forces (generated, for example, by molecular friction) interact with advection processes, such as mixing and stirring. The goal of this talk is to have an overview on recent results on a variety of aspects related to hydrodynamic stability, such as the stability of vortices and laminar flows, the enhancement of dissipative force via mixing, and the statistical description of turbulent flows.

Thu, 08 Feb 2018
15:00
L4

Non-existence and Non-uniqueness in the Kinetic Theory of Non-spherical Particles

Mark Wilkinson
(Heriot-Watt University, Edinburgh)
Abstract

The Boltzmann equation is a well-studied PDE that describes the statistical evolution of a dilute gas of spherical particles. However, much less is known — both from the physical and mathematical viewpoints — about the Boltzmann equation for non-spherical particles. In this talk, we present some new results on the non-existence and non-uniqueness of weak solutions to the initial-boundary value problem for N non-spherical particles which have importance for the Boltzmann equation.

We present work which was done jointly with L. Saint-Raymond (ENS Lyon), and also with P. Palffy-Muhoray (Kent State), E. Virga (Pavia) and X. Zheng (Kent State).

Thu, 01 Feb 2018

12:00 - 13:00
L5

Growth Model for Tree Stems and Vines

Michele Palladino
(Penn State University)
Abstract

In this talk, we propose a model describing the growth of tree stems and vine, taking into account also the presence of external obstacles. The system evolution is described by an integral differential equation which becomes discontinuous when the stem hits the obstacle. The stem feels the obstacle reaction not just at the tip, but along the whole stem. This fact represents one of the main challenges to overcome, since it produces a cone of possible reactions which is not normal with respect to the obstacle. However, using the geometric structure of the problem and optimal control tools, we are able to prove existence and uniqueness of the solution for the integral differential equation under natural assumptions on the initial data.

Thu, 25 Jan 2018
12:00
L5

Blowup phenomena in nonlocal and nonlinear conservation laws

Grzegorz Karch
(University of Wrocław)
Abstract


Recent results on viscous conservation laws with nonlocal flux will be presented. Such models contain, as a particular example, the celebrated parabolic-elliptic Keller-Segel model of chemotaxis. Here, global-in-time solutions are constructed under (nearly) optimal assumptions on the size of radial initial data. Moreover, criteria for blowup of solutions in terms of their local concerntariotions will be derived.

Thu, 18 Jan 2018
12:00
L5

Nonlinear cross-diffusion systems and gradient flows

Maria Bruna
(Oxford University)
Abstract

We will discuss nonlinear cross-diffusion models describing cell motility of two distinct populations. The continuum PDE model is derived systematically from a stochastic discrete model consisting of impenetrable diffusing spheres. In this talk, I will outline the derivation of the cross-diffusion model, discuss some of its features such as the gradient-flow structure, and show numerical results comparing the discrete stochastic system to the derived model.

Thu, 30 Nov 2017

12:00 - 13:00
L4

McKean–Vlasov problems with contagion effects

Sean Ledger
(University of Bristol)
Abstract

I will introduce a McKean—Vlasov problem arising from a simple mean-field model of interacting neurons. The equation is nonlinear and captures the positive feedback effect of neurons spiking. This leads to a phase transition in the regularity of the solution: if the interaction is too strong, then the system exhibits blow-up. We will cover the mathematical challenges in defining, constructing and proving uniqueness of solutions, as well as explaining the connection to PDEs, integral equations and mathematical finance.

Thu, 23 Nov 2017

12:00 - 13:00
L4

Weak limits of Sobolev homeomorphisms

Daniel Campbell
(Universität Erlangen-Nürnberg)
Abstract


In the study of variational models for non-linear elasticity in the context of proving regularity we are led to the challenging so-called Ball-Evan's problem of approximating a Sobolev homeomorphism with diffeomorphisms in its Sobolev space. In some cases however we are not able to guarantee that the limit of a minimizing sequence is a homeomorphism and so the closure of Sobolev homeomorphisms comes into the game. For $p\geq 2$ they are exactly Sobolev monotone maps and for $1\leq p<2$ the monotone maps are intricately related to these limits. In our paper we prove that monotone maps can be approximated by diffeomorphisms in their Sobolev (or Orlicz-Sobolev) space including the case $p=1$ not proven by Iwaniec and Onninen.
 

Thu, 09 Nov 2017

12:00 - 13:00
L4

Two-dimensional pseudo-gravity model: particles motion in a non-potential singular force field

Dan Crisan
(Department of Mathematics, Imperial College London)
Abstract

I will describe a simple macroscopic model describing the evolution of a cloud of particles confined in a magneto-optical trap. The behavior of the particles is mainly driven by self--consistent attractive forces. In contrast to the standard model of gravitational forces, the force field does not result from a potential; moreover, the nonlinear coupling is more singular than the coupling based on the Poisson equation.  In addition to existence of uniqueness results of the model PDE, I will discuss the convergence of the  particles description towards the solution of the PDE system in the mean field regime.

Thu, 02 Nov 2017

12:00 - 13:00
L4

Acoustic and electromagnetic transmission problems

Euan Spence
(University of Bath)
Abstract

In this talk I will discuss acoustic and electromagnetic transmission problems; i.e. problems where the wave speed jumps at an interface. I will focus on what is known mathematically about resonances and trapped waves (e.g. When do these occur? When can they be ruled out? What do we know in each case?). This is joint work with Andrea Moiola (Pavia).

Thu, 26 Oct 2017
12:00
L4

The Cauchy problem for the Landau-Lifshitz-Gilbert equation in BMO and self-similar solutions

Susana Gutierrez
(University of Birmingham)
Abstract

The Landau-Lifshitz-Gilbert equation (LLG) is a continuum model describing the dynamics for the spin in ferromagnetic materials. In the first part of this talk we describe our work concerning the properties and dynamical behaviour of the family of self-similar solutions under the one-dimensional LLG-equation.  Motivated by the properties of this family of self-similar solutions, in the second part of this talk we consider the Cauchy problem for the LLG-equation with Gilbert damping and provide a global well-posedness result provided that the BMO norm of the initial data is small.  Several consequences of this result will be also given.

Thu, 19 Oct 2017

12:00 - 13:00
L4

Maximal Hypersurfaces with boundary conditions

Ben Lambert
(University College London)
Abstract

We construct maximal surfaces with Neumann boundary conditions in Minkowski space using mean curvature flow. In particular we find curvature conditions on a boundary manifold so that mean curvature flow may be shown to exist for all time, and give conditions under which the maximal hypersurfaces are stable under the flow.

Thu, 12 Oct 2017
12:00
L4

A new flocking model through body attitude coordination

Sara Merino Aceituno
(Imperial College)
Abstract

We present a new model for multi-agent dynamics where each agent is described by its position and body attitude: agents travel at a constant speed in a given direction and their body can rotate around it adopting different configurations. Agents try to coordinate their body attitudes with the ones of their neighbours. This model is inspired by the Vicsek model. The goal of this talk will be to present this new flocking model, its relevance and the derivation of the macroscopic equations from the particle dynamics.

Fri, 06 Oct 2017

16:00 - 17:00
C5

Some recent results in several complex variables and complex geometry

Xiangyu Zhou
(Chinese Academy of Science Beijing)
Abstract

After recalling some backgrounds and motivations, we'll report some recent results on the optimal L^2 extensions and multiplier ideal sheaves, with emphasizing the close relations between SCV and PDE.