Analysis Seminar 2014/2015

We share the time slot with the Applied Analysis Reading Group (AARG), which meets in the weeks indicated below.

If you have any queries, or if you would like to be on our e-mail list, please contact the organiser K. Matthies.

Spring 2015

Date Speaker Title/Abstract
5 Feb Elaine Crooks
Swansea University
Self-similar fast-reaction limits for reaction-diffusion systems on unbounded domains
This talk will present a unified approach to characterising fast-reaction limits of systems of either two reaction-diffusion equations, or one reaction-diffusion equation and one ordinary differential equation, on unbounded domains, motivated by a number of different models of fast chemical reactions in which either both reactants are mobile or one is mobile and the other immobile. Under appropriate conditions on the initial data, we show that solutions of four classes of problem each converge in the fast-reaction limit $k \to \infty$ to a self-similar limit profile that has one of four forms, depending on how many of the components diffuse and on whether the spatial domain is a half line or the whole line $\mathbb{R}$. For fixed $k$, long-time convergence to these same self-similar profiles is also established, thanks to a scaling argument of Kamin. This is joint work with Danielle Hilhorst.
12 Feb AARG
19 Feb Eugenie Hunsicker
Loughborough University
Extended harmonic forms and HI Hodge Theory
26 Feb AARG
05 Mar Michael Herrmann
University of Münster
Hysteresis and phase transitions in spatially discrete systems
We consider a discrete forward-backward diffusion lattice in one space dimension and study the formation and propagation of phase interfaces on large spatial and temporal scales. In the first part we allow for a general bistable nonlinearity and investigate the effective dynamics on a formal and heuristic level. In particular, we identify a hysteretic free boundary problem that governs the evolution of both standing and moving phase interfaces. The second part is then devoted to rigorous results for a special case. This is joint work with Michael Helmers (University of Bonn).
12 Mar AARG
19 Mar Jonathan Bennett
University of Birmingham
Control of Fourier multipliers by maximal functions
We describe a natural framework within which a wide variety of Fourier multiplier operators may be "controlled" by relatively simple geometrically-defined maximal operators. The maximal operators that arise are of some independent interest, involving fractional averages and tangential approach regions, along with more novel "improper fractional averages" and "escape" regions. Some applications are given to the theory of oscillatory integrals and dispersive PDE.
26 Mar AARG
16 Apr Marios Stamatakis
University of Bath
Hydrodynamic limits and condensing zero range processes
Condensing zero range processes are interacting particle systems with zero range interaction exhibiting phase separation at densities above a finite critical density. We prove the hydrodynamic limit of mean zero condensing zero range processes with bounded local jump rate for sub-critical initial profiles, i.e: for initial profiles that are everywhere strictly below the critical density. The proof is based on H.T. Yau's relative entropy method and is made possible by a generalisation of the one block estimate. A closed hydrodynamic equation for such zero range processes when starting from a general initial profile is not known. We can obtain though a non-closed hydrodynamic description by proving relative compactness results for the processes induced by two important but not conserved quantities at the microscopic level, the empirical diffusion rate $\mu_N$ and the empirical current $W_N$. Denoting by $\simga_N$ the empirical density, all limit points of the sequence of laws of the triple $(\mu_N;\sigma_N;W_N), N \in \Bbb{N}$, are concentrated on trajectories $(\mu;\sigma;W)$ satisfying the continuity equation $\partial_t \mu = \Delta \sigma(t) = -div W(t)$ in the sense of distributions. Finally we give some regularity results on the limiting triples $(\mu;\sigma;W)$.
23 Apr AARG
30 Apr Filippo Cagnetti
University of Sussex
The rigidity problem for symmetrization inequalities
Steiner symmetrization is a very useful tool in the study of isoperimetric inequality. This is also due to the fact that the perimeter of a set is less or equal than the perimeter of its Steiner symmetral. In the same way, in the Gaussian setting, it is well known that Ehrhard symmetrisation does not increase the Gaussian perimeter. We will show characterisation results for equality cases in both Steiner and Ehrhard perimeter inequalities. We will also characterize rigidity of equality cases. By rigidity, we mean the situation when all equality cases are trivially obtained by a translation of the Steiner symmetral (or, in the Gaussian setting, by a reflection of the Ehrhard symmetral). We will achieve this through the introduction of a suitable measure-theoretic notion of connectedness, and through a fine analysis of the barycenter function for a special class of sets. These results are obtained in collaboration with Maria Colombo, Guido De Philippis, and Francesco Maggi.
4 Jun Dmitry Pelinovsky
McMaster University
Stability of Dirac solitons
We shall review orbital stability of solitons in the integrable Dirac equation known as the massive Thirring model. Orbital stability is obtained by using two different methods: one method relies on the higher-order energy conservation and the other one relies on the Backlund transformation. We shall also give numerical and analaytical results on the transverse stability of Dirac solitons under symmetry-breaking perturbations.
11 Jun Max Fathi
Lyon
Quantitative rates of convergence to the hydrodynamic limit
[AARG seminar] Deriving a hydrodynamic limit consists in rigorously obtaining some macroscopic equation (for example, the heat equation) as the scaling limit of a large system of interacting particles. In 2009, Grunewald, Otto, Villani and Westdickenberg proposed a new method to obtain quantitative rates of convergence to the hydrodynamic limit in Wasserstein distance for random systems. These results can then be combined with dimension-free functional inequalities to obtain local Gibbs behavior of the system, with quantitative bounds on the relative entropy. This talk will be partly based on joint works with H. Duong (Warwick) and G. Menz (Stanford).
16 Jun Michel Bellieud
LMGC, University of Montpellier
Homogenisation of stratified elastic composites with high contrast
[Joint Analysis/CNM/Applied Seminar ] We study the asymptotic behaviour of a stratified linear elastic medium wherein possibly very thin and stiff layers alternate with much softer ones. Higher order terms, characterising bending effects, may appear in the limit equations, depending on the order of magnitude of the elastic moduli in the stiffer phase. A part of this work is set in the context of non-periodic (or stochastic) homogenisation.
2 Jul Olivier Kneuss
Rio de Janeiro
On the prescribed Jacobian inequality $det \nabla u \geq f$ in Sobolev spaces in the plane.
Abstract

Autunm 2014

Date Speaker Title/Abstract
2 Oct Sergey Naboko
Kent/St. Petersburg
On the examples of Jacobi Operators structure
Some examples of Hermitian Jacobi Matrices to be discussed. Relation between the asymptotic behavior of the orthogonal polynomials and the spectral structure of the corresponding Jacobi Operators is the key instrument of the whole analysis. No preliminary knowledge of Jacobi operators or orthogonal polynomials is required for the talk.
8 Oct AARG
16 Oct Alexander Kiselev
St. Petersburg State University
An inverse spectral problem on quantum graphs: reconstruction of matching conditions at graph vertices
We will discuss one of the possible inverse spectral problems for quantum graphs. A quantum graph we study is a compact finite metric graph with an associated second-order differential operator defined on it. The matching conditions at graph vertices which reflect the graph connectivity are used to specify the domain of the corresponding operator. The class of matching conditions we allow is the following: at each graph vertex the coupling is assumed to be of either delta or delta-prime type. It has to be noted that the standard, or Kirchhoff, matching conditions are a particular case of delta-type coupling when all coupling constants zero out. The inverse spectral problem we have in mind is this: does the spectrum of the operator on a graph (be it a Laplace or Schrodinger operator) uniquely determine matching at graph vertices? This type of inverse spectral problem is not as well-studied as, say, the inverse spectral problem of reconstructing the graph connectivity and metric properties based on the spectrum of a Laplace of Schrodinger operator on it. It turns out however that the mathematical apparatus we develop in order to study the former inverse problem can in fact be used in the study of the latter one. In the simplest case of graph Laplacians, we derive a series of trace formulae which link together matching conditions of two operators under the assumption that their spectra coincide counting multiplicities. Thus necessary conditions of isospectrality of two graph Laplacians are obtained. Under the additional restriction that graph edge lengths are rationally independent, we are able to obtain necessary and sufficient conditions of the named isospectrality. It turns out that it can only occur in simplest graphs (e.g., chains or pure cycles). The results in the case of Schrodinger operators appear less complete. We will argue however that in the case of infinitely smooth edge potentials one can advance virtually as far as in the case of graph Laplacians using more or less the same mathematical toolbox.
24 Oct AARG
30 Oct Simon Chandler-Wilde
On Spectral Inclusion Sets and Computing the Spectra and Pseudospectra of Bounded Linear Operators
We derive inclusion sets for the (pseudo)spectrum of an infinite Jacobi matrix $A$, understood as a bounded linear operator on $L^2(Z)$. The first inclusion set is the union of certain pseudospectra of $n\times n$ principal submatrices of $A$. In a second version we work with lower bounds on $n \times \infty$ and $\infty \times n$ submatrices of $A-\lambda I$ , which effectively leads to the study of related $n \times n$ matrices. The latter procedure is motivated by work on second order spectra of Davies, and Davies and Plum, and on higher-order versions by Hansen. Our third set not only yields an upper bound but also an approximation of the (pseudo)spectrum of $A$ as $n \to\infty$. We discuss extensions of our results to band operators of higher bandwidth and band-dominated operators, and to matrices $A$ with operator entries. We illustrate the results by computing spectral inclusion sets for a particular random operator studied by Feinberg and Zee. This is joint work with Marko Lindner (Hamburg-Harburg), Ratchanikorn Chonchaiya (Reading), and Brian Davies (KCL).
6 Nov AARG
13 Nov Antonio Segatti
University of Pavia
Analysis of a variational model for nematic shells
In this talk I will introduce and discuss the analytic features of an elastic surface energy recently introduced by G. Napoli and L.Vergori to model thin films of nematic liquid crystals. The talk will be focused on the following points: i) using methods of the calculus of variations, we establish a relation between the existence of minimizers and the topology of the surface; ii) we prove, by a Ginzburg-Landau approximation, the well-posedness of the gradient flow of the energy; iii) in the case of a parametrized axisymmetric torus we obtain a stronger characterization of global and local minimizers, which we supplement with numerical experiments. This is a joint work with M. Snarski and M. Veneroni.
20 Nov AARG
27 Nov Amit Einav
University of Cambridge
On the Subadditivity of the Entropy on the Sphere
It is an interesting well known fact that the relative entropy with respect to the Gaussian measure on $\mathbb{R}^N$ satisfies a simple subadditivity property. Namely, if $\Pi_1^{(i)}(F_N)$ is the first marginal of the density function F_N in the i-th variable then $$\sum_{i=1}^N H(\Pi_1^{(i)}(F_N) | \gamma_1) \leq H(F_N | \gamma_N),$$ where $\gamma_k$ is the standard Gaussian on $\mathbb{R}^k$. Surprisingly enough, when one tries to achieve a similar result on $\mathbb{S}^{N-1}(\sqrt{N})$ a factor of 2 appears in the right hand side of the inequality (a result due to Carlen, Lieb and Loss), and the constant is sharp. Besides a deviation from the simple equivalence of ensembles principle in equilibrium Statistical Mechanics, this entropic inequality on the sphere has interesting ramifications in other fields, such as Kinetic Theory. In this talk we will present conditions on the density function F_N, on the sphere, under which we can get an ‘almost’ subaditivity property; i.e. the factor 2 can be replaced with a factor of $1+\epsilon_N$, with $\epsilon_N$ given explicitly and going to zero. The main tools to be used in order to proved this result are an entropy conservation extension of F_N to $\mathbb{R}^N$ together with comparison of appropriate transportation distances such as the entropy, Fisher information and Wasserstein distance between the marginal of the original density and that of the extension. Time permitting, we will give an example, one that arises naturally in the investigation of the so-called Kac Model, to many families of functions that satisfy these conditions.
4 Dec AARG
11 Dec James Robinson
University of Warwick
Room change: Chancellors' Building 3.5
A characterisation of local existence for semilinear heat equations in Lebesgue spaces
joint work with Robert Laister (University of the West of England), Mikolaj Sierzega (Warwick), and Alejandro Vidal-Lopez (Xi'an Jiaotong-Liverpool University) We consider the nonlinear heat equation $u_t-\Delta u=f(u)$ with $u(0)=u_0$, with Dirichlet boundary conditions on a bounded domain $\Omega\subset{\mathbb R}^d$. We assume that $f\colon[0,\infty)\to[0,\infty)$ is continuous and non-decreasing. We show that if $q\in(1,\infty)$ then the equation has a local solution bounded in $L^q(\Omega)$ for all initial data in $L^q(\Omega)$ if and only if $\limsup_{s\to\infty}s^{-(1+2q/d)}f(s)<\infty$; and that if in addition $f(s)/s$ is non-decreasing then the equation has a local solution bounded in $L^1(\Omega)$ for all $u_0\in L^1(\Omega)$ if and only if $\int_1^\infty s^{-(2+2/d)}f(s)\, d s<\infty$. Our proofs are also valid for the case $\Omega={\mathbb R}^d$.

Previous Analysis Seminars

This seminar series was previously called PDE seminar and the programme since the academic year 2010/1 can be found here.