Past Partial Differential Equations Seminar

16 February 2015
17:00
Josselin Garnier
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.

  • Partial Differential Equations Seminar
9 February 2015
17:00
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.

  • Partial Differential Equations Seminar
2 February 2015
17:00
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.

  • Partial Differential Equations Seminar
26 January 2015
17:00
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.

  • Partial Differential Equations Seminar
19 January 2015
17:00
Angkana Ruland
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 Hausdor ff dimension of nodal
domains (of eigenfunctions to the investigated Dirichlet-to-Neumann maps).
  • Partial Differential Equations Seminar
1 December 2014
17:00
Panu Lahti
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

  • Partial Differential Equations Seminar
24 November 2014
15:30
Pierre Degond
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.

  • Partial Differential Equations Seminar
17 November 2014
17:00
Claude Warnick
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.

  • Partial Differential Equations Seminar
10 November 2014
16:00
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.

  • Partial Differential Equations Seminar

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