PDE CDT Lunchtime Seminar

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

Past events in this series
15 November 2018
12:00
Abstract

I will prove the 2d Biot-Savart law for the vorticity being an unbounded measure $\mu$, i.e. such that $\mu(\mathbb{R}^2)=\infty$, and show how can one infer some useful information concerning Kaden's spirals using it. Vorticities being unbounded measures appear naturally in the engineering literature as self-similar approximations of 2d Euler flows, see for instance Kaden's or Prandtl's spirals. Mathematicians are interested in such objects since they seem to be related to the questions of well-posedness of Delort's solutions of the 2d vortex sheet problem for the Euler equation. My talk is based on a common paper with K.Oleszkiewicz, M. Preisner and M. Szumanska.

  • PDE CDT Lunchtime Seminar
Today
12:00
Siran Li
Abstract

Harmonic map equations are an elliptic PDE system arising from the  
minimisation problem of Dirichlet energies between two manifolds. In  
this talk we present some some recent works concerning the symmetry  
and stability of harmonic maps. We construct a new family of  
''twisting'' examples of harmonic maps and discuss the existence,  
uniqueness and regularity issues. In particular, we characterise of  
singularities of minimising general axially symmetric harmonic maps,  
and construct non-minimising general axially symmetric harmonic maps  
with arbitrary 0- or 1-dimensional singular sets on the symmetry axis.  
Moreover, we prove the stability of harmonic maps from $\mathbb{B}^3$  
to $\mathbb{S}^2$ under $W^{1,p}$-perturbations of boundary data, for  
$p \geq 2$. The stability fails for $p <2$ due to Almgren--Lieb and  
Mazowiecka--Strzelecki.

(Joint work with Prof. Robert M. Hardt.)

  • PDE CDT Lunchtime Seminar
29 November 2018
12:00
Jean-Philippe Nicolas
Abstract

This talk will be an introduction to the use of conformal methods in asymptotic analysis in general relativity. We shall consider the explicit example of flat spacetime (Minkowski spacetime). The full conformal compactification will be constructed. For a simple example of a conformally invariant equation (we'll take the wave equation), we shall see how the compactification allows to infer precise informations on the asymptotic behaviour of the solution in all directions, for a certain class of data at any rate. Then, depending on time and questions, I will either describe how a scattering theory can be constructed using the same method or, explain how conformal methods can be used on other asymptotically flat geometries.

  • PDE CDT Lunchtime Seminar
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