Past Partial Differential Equations Seminar

31 October 2016
16:30
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).
 

  • Partial Differential Equations Seminar
24 October 2016
16:00
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. 

  • Partial Differential Equations Seminar
17 October 2016
16:00
Giuseppe Tinaglia
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.

  • Partial Differential Equations Seminar
10 October 2016
16:00
Igor Velcic
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.

  • Partial Differential Equations Seminar
13 June 2016
16:00
Laurent Dietrich
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.

  • Partial Differential Equations Seminar
16 May 2016
16:00
Theodora Bourni
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.
  • Partial Differential Equations Seminar
9 May 2016
16:00
Ethan O'Brien
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.

  • Partial Differential Equations Seminar

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