Forthcoming events in this series


Mon, 07 Nov 2016

16:00 - 17:00
L4

Equilibrium measure for a nonlocal dislocation energy

Lucia Scardia
(University of Bath)
Abstract

In this talk I will present a recent result on the characterisation of the equilibrium measure for a nonlocal and non-radial energy arising as the Gamma-limit of discrete interacting dislocations.

Mon, 31 Oct 2016

16:30 - 17:30
L4

High Ericksen number and the dynamical creation of defects in nematics

Arghir Zarnescu
(Basque Center for Applied Mathematics)
Abstract


We consider the Beris-Edwards model of liquid crystal dynamics. We study a non-dimensionalisation and regime suited for the study of defect patterns, that amounts to a combined high Ericksen and high Reynolds  number regime. 
We identify some of the flow mechanisms responsible for the appearance of localized gradients that increase in time.
This is joint work with Hao Wu (Fudan).
 

Mon, 24 Oct 2016

16:00 - 17:00
L4

Chern-Gauss-Bonnet formulas for singular non-compact manifold

Reto Buzano
(Queen Mary University London)
Abstract

A generalisation of the classical Gauss-Bonnet theorem to higher-dimensional compact Riemannian manifolds was discovered by Chern and has been known for over fifty years. However, very little is known about the corresponding formula for complete or singular Riemannian manifolds. In this talk, we explain a new Chern-Gauss-Bonnet theorem for a class of manifolds with finitely many conformally flat ends and singular points. More precisely, under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the singularities, we obtain a Chern-Gauss-Bonnet type formula with error terms that can be expressed as isoperimetric deficits. This is joint work with Huy Nguyen. 

Mon, 17 Oct 2016

16:00 - 17:00
L4

The geometry of constant mean curvature surfaces in Euclidean space

Giuseppe Tinaglia
(King's College London)
Abstract

In this talk I will begin by reviewing classical geometric properties of constant mean curvature surfaces, H>0, in R^3. I will then talk about several more recent results for surfaces embedded in R^3 with constant mean curvature, such as curvature and radius estimates. Finally I will show applications of such estimates including a characterisation of the round sphere as the only simply-connected surface embedded in R^3 with constant mean curvature and area estimates for compact surfaces embedded in a flat torus with constant mean curvature and finite genus. This is joint work with Meeks.

Mon, 10 Oct 2016

16:00 - 17:00
L4

Homogenization of thin structures in nonlinear elasticity - periodic and non-periodic

Igor Velcic
(University of Zagreb)
Abstract

We will give the results on the models of thin plates and rods in nonlinear elasticity by doing simultaneous homogenization and dimensional reduction. In the case of bending plate we are able to obtain the models only under periodicity assumption and assuming some special relation between the periodicity of the material and thickness of the body. In the von K\'arm\'an regime of rods and plates and in the bending regime of rods we are able to obtain the models in the general non-periodic setting. In this talk we will focus on the derivation of the rod model in the bending regime without any assumption on periodicity.

Mon, 13 Jun 2016

16:00 - 17:00
L4

Enhancement of propagation in reaction-diffusion equations by a line of fast diffusion

Laurent Dietrich
(OxPDE, University of Oxford)
Abstract

we study a new mechanism of reaction-diffusion involving a line with fast diffusion, proposed to model the influence of transportation networks on biological invasions. 
We will be interested in the existence and uniqueness of traveling waves solutions, and especially focus on their velocity. We will show that it grows as the square root of the diffusivity on the line, generalizing and showing the robustness of a result by Berestycki, Roquejoffre and Rossi (2013), and provide a characterization of the growth ratio thanks to an hypoelliptic (a priori) degenerate system. 
Finally we will take a look at the dynamics and show that the waves attract a large class of initial data. In particular, we will shed light on a new mechanism of attraction which enables the waves to attract initial data with size independent of the diffusion on the line : this is a new result, in the sense than usually, enhancement of propagation has to be paid by strengthening the assumptions on the size of the initial data for invasion to happen.

Mon, 16 May 2016

16:00 - 17:00
L4

"Null mean curvature" flow and marginally outer trapped surfaces

Theodora Bourni
(Freie Universität Berlin)
Abstract
In this talk we discuss a new second order parabolic evolution equation

for hypersurfaces in space-time initial data sets, that generalizes mean

curvature flow (MCF). In particular, the 'null mean curvature' - a

space-time extrinsic curvature quantity - replaces the usual mean

curvature in the evolution equation defining MCF.  This flow is motivated

by the study of black holes and mass/energy inequalities in general

relativity. We present a theory of weak solutions using the level-set

method and  outline a natural application of the flow as a parabolic

approach to finding outermost marginally outer trapped surfaces (MOTS),

which play the role of quasi-local black hole boundaries in general

relativity. This is joint work with Kristen Moore.
Mon, 09 May 2016

16:00 - 17:00
L4

The wrinkling of a twisted ribbon

Ethan O'Brien
(Courant Institute)
Abstract

We explore a specific system in which geometry and loading conspire to generate fine-scale wrinkling. This system -- a twisted ribbon held with small tension -- was examined experimentally by Chopin and Kudrolli 
[Phys Rev Lett 111, 174302, 2013].

There is a regime where the ribbon wrinkles near its center. A recent paper by Chopin, D\'{e}mery, and Davidovitch models this regime using a von-K\'{a}rm\'{a}n-like 
variational framework [J Elasticity 119, 137-189, 2015]. Our contribution is to give upper and lower bounds for the minimum energy as the thickness tends to zero. Since the bounds differ by a thickness-independent prefactor, we have determined how the minimum energy scales with thickness. Along the way we find estimates on Sobolev norms of the minimizers, which provide some information on the character of the wrinkling. This is a joint work with  Robert V. Kohn in Courant Institute, NYU.

Mon, 02 May 2016

16:00 - 17:00
L4

Square Functions and the Muckenhoupt Weight Classes of Elliptic Measures

Bernd Kirchheim
(Universität Leipzig)
Abstract

We give a new characterization of the property that the elliptic measure
belongs to the infinity weight Muckenhoupt class
in terms of a Carleson measure property of bounded solutions.
This is joint work with C.Kenig, J.Pipher and T.Toro

Mon, 25 Apr 2016

16:00 - 17:00
L4

The decay of solutions of Maxwell-Klein-Gordon equations

Shiwu Yang
(Cambridge)
Abstract

It has been shown that there are global solutions to 
Maxwell-Klein-Gordon equations in Minkowski space with finite energy 
data. However, very little is known about the asymptotic behavior of the 
solution. In this talk, I will present recent progress on the decay 
properties of the solutions. We show the quantitative energy flux decay 
of the solutions with data merely bounded in some weighted energy space. 
The results in particular hold in the presence of large total charge. 
This is the first result that gives a complete and precise description 
of the global behavior of large nonlinear fields.
 

Mon, 07 Mar 2016

16:00 - 17:00
L4

Macroscopic transport: ballistic, diffusive, super diffusive

Stefano Olla
(Ceremade)
Abstract

In acoustic materials (non null sound velocity), there is a clear separation of scale between the relaxation to mechanical equilibrium, governed by Euler equations, and the slower relaxation to thermal equilibrium, governed by heat equation if thermal conductivity is finite. In one dimension in acoustic systems, thermal conductivity is diverging and the thermal equilibrium is reached by a superdiffusion governed by a fractional heat equation. In non-acoustic materials it seems that there is not such separation of scales, and thermal and mechanical equilibriums are reached at the same time scale, governed by a Euler-Bernoulli beam equation. We prove such macroscopic behaviors in chains of oscillators with dynamics perturbed by a random local exchange of momentum, such that energy and momentum are conserved. (Works in collaborations with T. Komorowski).

Fri, 04 Mar 2016

12:00 - 13:00
L1

The effect of domain shape on reaction-diffusion equations

Henri Berestycki
(EHESS)
Abstract

I will discuss some reaction-diffusion equations of bistable type motivated by biology and medicine. The aim is to understand the effect of the shape of the domain on propagation or on blocking of advancing waves. I will first describe the motivations of these questions and present a result about the existence of generalized “transition waves”. I will then discuss various geometric conditions that lead to either blocking, or partial propagation, or complete propagation. These questions involve new qualitative results for some non-linear elliptic and parabolic partial differential equations. I report here on joint work with Juliette Bouhours and Guillemette Chapuisat.

Mon, 29 Feb 2016

16:00 - 17:00
L4

Crystallization Results for Optimal Location Problems

David Bourne
(Durham University)
Abstract

While it is believed that many particle systems have periodic ground states, there are few rigorous crystallization results in two and more dimensions. In this talk I will show how results by the Hungarian geometer László Fejes Tóth can be used to prove that an idealised block copolymer energy is minimised by the triangular lattice. I will also discuss a numerical method for a broader class of optimal location problems and some conjectures about minimisers in three dimensions. This is joint work with Mark Peletier, Steven Roper and Florian Theil. 

Mon, 22 Feb 2016

16:00 - 17:00
L4

The hydrodynamic limit of the parabolic Ginzburg-Landau equation

Matthias Kurzke
(University of Nottingham)
Abstract

The Ginzburg-Landau functional serves as a model for the formation of vortices in many physical contexts. The natural gradient flow, the parabolic Ginzburg-Landau equation, converges in the limit of small vortex size and finite number of vortices to a system of ODEs. Passing to the limit of many vortices in this ODE, one can derive a mean field PDE, similar to the passage from point vortex systems to the 2D Euler equations. In the talk, I will present quantitative estimates that allow us to directly connect the parabolic GL equation to the limiting mean field PDE. In contrast to recent work by Serfaty, our work is restricted to a fairly low number of vortices, but can handle vortex sheet initial data in bounded domains. This is joint work with Daniel Spirn (University of Minnesota).

Sat, 20 Feb 2016

16:00 - 17:00

TBA

Piotr Mucha
(Warsaw)
Mon, 15 Feb 2016

16:00 - 17:00
L4

Flowing to minimal surfaces

Melanie Rupflin
(OxPDE, University of Oxford)
Abstract

For maps from surfaces there is a close connection between the area of the surface parametrised by the map and its Dirichlet energy and this translates also into a relation for the corresponding critical points. As such, when trying to find minimal surfaces, one route to take is to follow a suitable gradient flow of the Dirichlet energy. In this talk I will introduce such a flow which evolves both a map and a metric on the domain in a way that is designed to change the initial data into a minimal immersions and discuss some question concerning the existence of solutions and their asymptotic behaviour. This is joint work with Peter Topping.

Mon, 08 Feb 2016

16:00 - 17:00
L4

Pseudo-differential operators on Lie groups

Veronique Fischer
(University of Bath)
Abstract
In this talk, I will present some recent developments in the theory of pseudo-differential operators on Lie groups. First I will discuss why `reasonable' Lie groups are the interesting manifolds where one can develop global symbolic pseudo-differential calculi. I will also give a brief overview of the analysis in the context of Lie groups. I will conclude with some recent works developing pseudo-differential calculi on certain classes of Lie groups.
Mon, 01 Feb 2016

16:00 - 17:00
L4

Limits of $\alpha$-harmonic maps

Tobias Lamm
(Karlsruhe Institute of Technology)
Abstract

I will discuss a recent joint work with A. Malchiodi (Pisa) and M. Micallef (Warwick) in which we show that not every harmonic map can be approximated by a sequence of $\alpha$-harmonic maps.

Mon, 25 Jan 2016

16:00 - 17:00
L4

Global well-posedness of the axisymmetric Navier-Stokes equations in the exterior of an infinite cylinder

Ken Abe
(Kyoto and Oxford)
Abstract
We consider the initial-boundary value problem of the Navier-Stokes equations for axisymmetric initial data with swirl in the exterior of an infinite cylinder, subject to the slip boundary condition. We construct global solutions and give an upper bound for azimuthal component of vorticity in terms of the size of cylinder. The proof is based on the Boussinesq system. We show that the system is globally well-posed for axisymmetric data without swirl.
Mon, 18 Jan 2016

16:00 - 17:00
L4

Nonlocal self-improving properties

Tuomo Kuusi
(Aalto University)
Abstract

The classical Gehring lemma for elliptic equations with measurable coefficients states that an energy solution, which is initially assumed to be $H^1$ - Sobolev regular, is actually in a better Sobolev space space $W^{1,q}$ for some $q>2$. This a consequence of a self-improving property that so-called reverse Hölder inequality implies. In the case of nonlocal equations a self-improving effect appears: Energy solutions are also more differentiable. This is a new, purely nonlocal phenomenon, which is not present in the local case. The proof relies on a nonlocal version of the Gehring lemma involving new exit time and dyadic decomposition arguments. This is a joint work with G. Mingione and Y. Sire. 

Mon, 07 Dec 2015

16:00 - 17:00
L4

Biaxiality in liquid crystals at low temperatures (Please note Week 9)

Duvan Henao
(Pontificia Universidad Católica de Chile)
Abstract

We study the low-temperature limit in the Landau-de Gennes theory for liquid crystals. We prove that for minimizers for orientable Dirichlet data tend to be almost uniaxial but necessarily contain some biaxiality around the singularities of a limiting harmonic map. In particular we prove that around each defect there must necessarily exist a maximal biaxiality point, a point with a purely uniaxial configuration with a positive order parameter, and a point with a purely uniaxial configuration with a negative order parameter. Estimates for the size of the biaxial cores are also given.

This is joint work with Apala Majumdar and Adriano Pisante.

Wed, 02 Dec 2015

16:00 - 17:00
L1

Global well-posedness of the energy critical Maxwell-Klein-Gordon equation

Sung-jin Oh
(UC Berkeley)
Abstract

The massless Maxwell-Klein-Gordon system describes the interaction between an electromagnetic field (Maxwell) and a charged massless scalar field (massless Klein-Gordon, or wave). In this talk, I will present a recent proof, joint with D. Tataru, of global well-posedness and scattering of this system for arbitrary finite energy data in the (4+1)-dimensional Minkowski space, in which the PDE is energy critical.

Mon, 16 Nov 2015

16:00 - 17:00
L2

The Stokes-Fourier equations as scaling limit of the hard sphere dynamics

Laure Saint-Raymond
(Ecole Normale Superieure)
Abstract
In his sixth problem, Hilbert asked for an axiomatization of gas dynamics, and he suggested to use the Boltzmann equation as an intermediate description between the (microscopic) atomic dynamics and (macroscopic) fluid models. The main difficulty to achieve this program is to prove the asymptotic decorrelation between the local microscopic interactions, referred to as propagation of chaos, on a time scale much larger than the mean free time. This is indeed the key property to observe some relaxation towards local thermodynamic equilibrium.

 

This control of the collision process can be obtained in fluctuation regimes. In a recent work with I. Gallagher and T. Bodineau, we have established a long time convergence result to the linearized Boltzmann equation, and eventually derived the acoustic and incompressible Stokes equations in dimension 2. The proof relies crucially on symmetry arguments, combined with a suitable pruning procedure to discard super exponential collision trees.
Mon, 16 Nov 2015

15:00 - 16:00
L2

Magnitudes of compact sets in euclidean spaces: an application of analysis to the theory of enriched categories

Tony Carbery
(University of Edinburgh)
Abstract

Leinster and Willerton have introduced the concept of the magnitude of a metric space, as a special case as that of an enriched category. It is a numerical invariant which is designed to capture the important geometric information about the space, but concrete examples of ts values on compact sets in euclidean space have hitherto been lacking. We discuss progress in some conjectures of Leinster and Willerton.

Mon, 09 Nov 2015

16:00 - 17:00
L5

Instance optimality for the maximum strategy

Lars Diening
(University of Osnabruck)
Abstract

We study the adaptive finite element approximation of the Dirichlet problem $-\Delta u = f$ with zero boundary values using newest vertex bisection. Our approach is based on the minimization of the corresponding Dirichlet energy. We show that the maximums strategy attains every energy level with a number of degrees of freedom, which is proportional to the optimal number. As a consequence we achieve instance optimality of the error. This is a joint work with Christian Kreuzer (Bochum) and Rob Stevenson (Amsterdam).

Mon, 02 Nov 2015

16:00 - 17:00
L5

Sharp Trace-Sobolev inequalities of order 4

Antonio Ache
(Princeton University)
Abstract

We establish sharp Sobolev inequalities of order four on Euclidean $d$-balls for $d$ greater than or equal to four. When $d=4$, our inequality generalizes the classical second order Lebedev-Milin inequality on Euclidean $2$-balls. Our method relies on the use of scattering theory on hyperbolic $d$-balls. As an application, we charcaterize the extremals of the main term in the log-determinant formula corresponding to the conformal Laplacian coupled with the boundary Robin operator on Euclidean $4$-balls. This is joint work with Alice Chang. 

Mon, 19 Oct 2015

16:00 - 17:00
L5

The tangential touch problem for fully nonlinear elliptic operators

Emanuel Indrei
(Carnegie Mellon Univeristy)
Abstract
The tangential touch problem in elliptic theory consists of exposing the dynamics of the free boundary near the fixed boundary in obstacle problems. The solution of this problem is discussed for fully nonlinear elliptic operators in two dimensions.
Based on joint work with Andreas Minne.
Mon, 08 Jun 2015

17:00 - 18:00
L4

Shock Development in Spherical Symmetry

Andre Lisibach
(ETH Zurich)
Abstract

The general problem of shock formation in three space dimensions was solved by Christodoulou in 2007. In his work also a complete description of the maximal development of the initial data is provided. This description sets up the problem of continuing the solution beyond the point where the solution ceases to be regular. This problem is called the shock development problem. It belongs to the category of free boundary problems but in addition has singular initial data because of the behavior of the solution at the blowup surface. In my talk I will present the solution to this problem in the case of spherical symmetry. This is joint work with Demetrios Christodoulou.

Mon, 01 Jun 2015

17:00 - 18:00
L4

Uniqueness of the Leray-Hopf solution for a dyadic model

Nikolai Filonov
(Steklov Institute of Mathematics)
Abstract

We consider the system of nonlinear differential equations
\begin{equation}
(1) \qquad
\begin{cases}
\dot u_n(t) + \lambda^{2n} u_n(t) 
- \lambda^{\beta n} u_{n-1}(t)^2 + \lambda^{\beta(n+1)} u_n(t) u_{n+1}(t) = 0,\\
u_n(0) = a_n, n \in \mathbb{N}, \quad \lambda > 1, \beta > 0.
\end{cases}
\end{equation}
In this talk we explain why this system is a model for the Navier-Stokes equations of hydrodynamics. The natural question is to find a such functional space, where one could prove the existence and the uniqueness of solution. In 2008, A. Cheskidov proved that the system (1) has a unique "strong" solution if $\beta \le 2$, whereas the "strong" solution does not exist if $\beta > 3$. (Note, that the 3D-Navier-Stokes equations correspond to the value $\beta = 5/2$.) We show that for sufficiently "good" initial data the system (1) has a unique Leray-Hopf solution for all $\beta > 0$.

Mon, 25 May 2015

17:00 - 18:00
L4

ODE solutions for fractional Laplacian equations in conformal geometry

Maria del Mar Gonzalez
(Universitat Polytecnica de Catalunya)
Abstract

We look at the construction of radial metrics with an isolated singularity for the constant fractional curvature equation. This is a semilinear, non-local equation involving the fractional Laplacian, and appears naturally in conformal geometry. 

Mon, 18 May 2015

17:00 - 18:00
L4

The Existence Theorems and the Liouville Theorem for the Steady-State Navier-Stokes Problems

Mikhail Korobkov
(Sobolev Institute of Mathematics)
Abstract

In the talk we present a survey of recent results (see [4]-[6]) on the existence theorems for the steady-state Navier-Stokes boundary value problems in the plane and axially symmetric 3D cases for bounded and exterior domains (the so called Leray problem, inspired by the classical paper [8]). One of the main tools is the Morse-Sard Theorem for the Sobolev functions $f\in W^2_1(\mathbb R^2)$ [1] (see also [2]-[3] for the multidimensional case). This theorem guaranties that almost all level lines of such functions are $C^1$-curves besides the function $f$ itself could be not $C^1$-regular.

Also we discuss the recent Liouville type theorem for the steady-state Navier-Stokes equations for  axially symmetric 3D solutions in the absence of swirl (see [1]).

References

  1.  Bourgain J., Korobkov M. V., Kristensen J., On the Morse-Sard property and level sets of Sobolev and BV functions, Rev. Mat. Iberoam.,  29 , No. 1, 1-23  (2013).
  2. Bourgain J., Korobkov M. V., Kristensen J., On the Morse-Sard property and level sets of $W^{n,1}$ Sobolev functions on $\mathbb R^n$, Journal fur die reine und angewandte Mathematik (Crelles Journal) (Online first 2013).
  3. Korobkov M. V., Kristensen J., On the Morse-Sard Theorem for the sharp case of Sobolev mappings, Indiana Univ. Math. J., 63, No. 6, 1703-1724  (2014).
  4. Korobkov M. V., Pileckas K., Russo R., The existence theorem for steady Navier-Stokes equations in the axially symmetric case, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 14, No. 1, 233-262  (2015).
  5. Korobkov M. V., Pileckas K., Russo R., Solution of Leray's problem for stationary Navier-Stokes equations in plane and axially symmetric spatial domains,  Ann. of Math., 181, No. 2, 769-807  (2015).
  6. Korobkov M. V., Pileckas K., Russo R., The existence theorem for the steady Navier-Stokes problem in exterior axially symmetric 3D domains, 2014, 75 pp., http://arXiv.org/abs/1403.6921.
  7. Korobkov M. V., Pileckas K., Russo R., The Liouville Theorem for the Steady-State Navier-Stokes Problem for Axially Symmetric 3D Solutions in Absence of Swirl, J. Math. Fluid Mech. (Online first 2015).
  8. Leray J., Étude de diverses équations intégrals nonlinéaires et de quelques problèmes que pose l'hydrodynamique, J. Math. Pures Appl., 9, No. 12, 1- 82 (1933).
Mon, 11 May 2015

17:00 - 18:00
L4

Lipschitz Regularity for Inner Variational PDEs in 2D

Tadeusz Iwaniec
(Syracuse)
Abstract

I will present a joint work with Leonid Kovalev and Jani Onninen. The proofs are  based on topological arguments (degree theory)  and the properties  of planar  quasiconformal mappings. These new ideas  apply well to inner variational equations of conformally invariant energy integrals; in particular, to the Hopf-Laplace equation for the Dirichlet integral.

Mon, 09 Mar 2015

17:00 - 18:00
L4

Sobolev inequalities in arbitrary domains

Andrea Cianchi
(Università degli Studi di Firenze)
Abstract

A theory of Sobolev inequalities in arbitrary open sets in $R^n$ is offered. Boundary regularity of domains is replaced with information on boundary traces of trial functions and of their derivatives up to some explicit minimal order. The relevant Sobolev inequalities involve constants independent of the geometry of the domain, and exhibit the same critical exponents as in the classical inequalities on regular domains. Our approach relies upon new representation formulas for Sobolev functions, and on ensuing pointwise estimates which hold in any open set. This is a joint work with V. Maz'ya.

Mon, 02 Mar 2015

17:00 - 18:00
L4

Kinetic formulation for vortex vector fields

Radu Ignat
(Université Toulouse 3)
Abstract

We will focus on vortex gradient fields of unit-length. The associated stream function solves the eikonal equation, more precisely it is the distance function to a point. We will prove a kinetic formulation characterizing such vector fields in any dimension.
 

Mon, 23 Feb 2015

17:00 - 18:00
L4

A prirori estimates for the relativistic free boundary Euler equations in physical vacuum

Mahir Hadzic
(King's College London)
Abstract
We consider Euler equations on a fixed Lorentzian manifold. The fluid is initially supported on a compact domain and the boundary between the fluid and the vacuum is allowed to move. Imposing the so-called physical vacuum boundary condition, we will explain how to obtain a priori estimates for this problem. In particular, our functional framework allows us to track the regularity of the free boundary. This is joint work with S. Shkoller and J. Speck.
Mon, 16 Feb 2015

17:00 - 18:00
L5

The random paraxial wave equation and application to correlation-based imaging

Josselin Garnier
(Université Paris Diderot)
Abstract

We analyze wave propagation in random media in the so-called paraxial regime, which is a special high-frequency regime in which the wave propagates along a privileged axis. We show by multiscale analysis how to reduce the problem to the Ito-Schrodinger stochastic partial differential equation. We also show how to close and solve the moment equations for the random wave field. Based on these results we propose to use correlation-based methods for imaging in complex media and consider two examples: virtual source imaging in seismology and ghost imaging in optics.

Mon, 09 Feb 2015

17:00 - 18:00
L4

Global existence of solutions of the Ericksen-Leslie system for the Oseen-Frank model

Min-Chun Hong
(The University of Queensland)
Abstract

The dynamic flow of liquid crystals is described by the Ericksen-Leslie system. The Ericksen-Leslie system is a system of  the Navier-Stokes equations coupled with the gradient flow for the Oseen-Frank model,   which generalizes the heat flow for harmonic maps  into the $2$-sphere.   In this talk, we will outline a proof of global existence of solutions of the Ericksen-Leslie system for a general Oseen-Frank  model in 2D.

Mon, 02 Feb 2015

17:00 - 18:00
L4

Unique Continuation, Carleman Estimates, and Blow-up for Nonlinear Wave Equations

Arick Shao
(Imperial College London)
Abstract

In this talk, we consider two disparate questions involving wave equations: (1) how singularities of solutions of subconformal focusing nonlinear wave equations form, and (2) when solutions of (linear and nonlinear) wave equations are determined by their data at infinity. In particular, we will show how tools from solving the second problem - a new family of global nonlinear Carleman estimates - can be used to establish some new results regarding the first question. Previous theorems by Merle and Zaag have established both upper and lower bounds on the local H¹-norm near noncharacteristic blow-up points for subconformal focusing NLW. In our main result, we show that this H¹-norm cannot concentrate along past timelike cones emanating from the blow-up point, i.e., that a significant amount of the action must occur near the corresponding past null cones.

These are joint works with Spyros Alexakis.

Mon, 26 Jan 2015

17:00 - 18:00
L4

Stability and minimality for a nonlocal variational problem

Nicola Fusco
(Università di Napoli Frederico II)
Abstract

We discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. We show that critical configurations  with positive second variation are local minimizers of the nonlocal area functional and, in fact, satisfy a quantitative isoperimetric inequality with respect to sets  that are  $L^1$-close.  As an application, we address the global and local minimality of certain lamellar configurations.

Mon, 19 Jan 2015

17:00 - 18:00
L4

Carleman Estimates and Unique Continuation for Fractional Schroedinger Equations

Angkana Ruland
(University of Oxford)
Abstract
In this talk I present Carleman estimates for fractional Schroedinger
equations and discuss how these imply the strong unique continuation
principle even in the presence of rough potentials. Moreover, I show how
they can be used to derive quantitative unique continuation results in
the setting of compact manifolds. These quantitative estimates can then
be exploited to deduce upper bounds on the Hausdorff dimension of nodal
domains (of eigenfunctions to the investigated Dirichlet-to-Neumann maps).
Mon, 01 Dec 2014

17:00 - 18:00
L6

Functions of bounded variation on metric measure spaces

Panu Lahti
(Aalto University)
Abstract

Functions of bounded variation, abbreviated as BV functions, are defined in the Euclidean setting as very weakly differentiable functions that form a more general class than Sobolev functions. They have applications e.g. as solutions to minimization problems due to the good lower semicontinuity and compactness properties of the class. During the past decade, a theory of BV functions has been developed in general metric measure spaces, which are only assumed to be sets endowed with a metric and a measure. Usually a so-called doubling property of the measure and a Poincaré inequality are also assumed. The motivation for studying analysis in such a general setting is to gain an understanding of the essential features and assumptions used in various specific settings, such as Riemannian manifolds, Carnot-Carathéodory spaces, graphs, etc. In order to generalize BV functions to metric spaces, an equivalent definition of the class not involving partial derivatives is needed, and several other characterizations have been proved, while others remain key open problems of the theory.

 

Panu is visting Oxford until March 2015 and can be found in S2.48

Mon, 24 Nov 2014

15:30 - 16:30
L2

Bifurcations in mathematical models of self-organization

Pierre Degond
(Imperial College London)
Abstract

We consider self-organizing systems, i.e. systems consisting of a large number of interacting entities which spontaneously coordinate and achieve a collective dynamics. Sush systems are ubiquitous in nature (flocks of birds, herds of sheep, crowds, ...). Their mathematical modeling poses a number of fascinating questions such as finding the conditions for the emergence of collective motion. In this talk, we will consider a simplified model first proposed by Vicsek and co-authors and consisting of self-propelled particles interacting through local alignment.
We will rigorously study the multiplicity and stability of its equilibria through kinetic theory methods. We will illustrate our findings by numerical simulations.

Mon, 17 Nov 2014

17:00 - 18:00
L6

Dynamics in anti-de Sitter spacetimes

Claude Warnick
(University of Warwick)
Abstract

When solving Einstein's equations with negative cosmological constant, the natural setting is that of an initial-boundary value problem. Data is specified on the timelike conformal boundary as well as on some initial spacelike (or null) hypersurface. At the PDE level, one finds that the boundary data is typically prescribed on a surface at which the equations become singular and standard energy estimates break down. I will discuss how to handle this singularity by introducing a renormalisation procedure. I will also talk about the consequences of different choices of boundary conditions for solutions of Einstein’s equations with negative cosmological constant.

Mon, 10 Nov 2014

16:00 - 17:00
L1

Stability of the Kerr Cauchy horizon

Jonathan Luk
(University of Cambridge)
Abstract

The celebrated strong cosmic censorship conjecture in general relativity in particular suggests that the Cauchy horizon in the interior of the Kerr black hole is unstable and small perturbations would give rise to singularities. We present a recent result proving that the Cauchy horizon is stable in the sense that spacetime arising from data close to that of Kerr has a continuous metric up to the Cauchy horizon. We discuss its implications on the nature of the potential singularity in the interior of the black hole. This is joint work with Mihalis Dafermos.