Past Partial Differential Equations Seminar

15 February 2016
16:00
Melanie Rupflin
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.

  • Partial Differential Equations Seminar
8 February 2016
16:00
Veronique Fischer
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.
  • Partial Differential Equations Seminar
25 January 2016
16:00
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.
  • Partial Differential Equations Seminar
18 January 2016
16:00
Tuomo Kuusi
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. 

  • Partial Differential Equations Seminar
7 December 2015
16:00
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.

  • Partial Differential Equations Seminar
2 December 2015
16:00
Sung-jin Oh
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.

  • Partial Differential Equations Seminar
16 November 2015
16:00
Laure Saint-Raymond
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.
  • Partial Differential Equations Seminar
16 November 2015
15:00
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.

  • Partial Differential Equations Seminar
9 November 2015
16:00
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).

  • Partial Differential Equations Seminar

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