Past 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
27 October 2014
17:00
Paolo Baroni
Abstract

We consider the two-phase Stefan problem with p-degenerate diffusion, p larger than two, and we prove continuity up to the boundary for weak solutions, providing a modulus of continuity which we conjecture to be optimal. Since our results are proven in the form of a priori estimates for appropriate regularized problems, as corollary we infer the existence of a globally continuous weak solution for continuous Cauchy-Dirichlet datum.

  • Partial Differential Equations Seminar
20 October 2014
17:00
Igor Pazanin
Abstract
Our goal is to present recent results on the stationary motion of incompressible viscous fluid with a pressure-dependent viscosity. Under general assumptions on the viscosity-pressure relation (satisfi ed by the Barus formula and other empiric laws), fi rst we discuss the existence and uniqueness of the solution of the corresponding boundary value problem. The main part of the talk is devoted to asymptotic analysis of such system in thin domains naturally appearing in the applications. We address the problems of fluid flow in pipe-like domains and also study the behavior of a lubricant flowing through a narrow gap. In each setting we rigorously derive new asymptotic model describing the e ffective flow. The key idea is to conveniently transform the governing problem into the Stokes system with small nonlinear perturbation.
This is a joint work with Eduard Marusic-Paloka (University of Zagreb).
  • Partial Differential Equations Seminar
13 October 2014
17:00
Guido de Phillippis
Abstract

      I will show uniqueness result for BV solutions of scalar conservation laws with discontinuous flux in several space dimensions. The proof is based on the notion of kinetic solution and on a careful analysis of the entropy dissipation along the discontinuities of the flux.
 

  • Partial Differential Equations Seminar

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