# Past Partial Differential Equations Seminar

3 November 2008
17:00
Philippe Lauren&ccedil;ot
Abstract
In space dimension 2, it is well-known that the Smoluchowski-Poisson system (also called the simplified or parabolic-elliptic Keller-Segel chemotaxis model) exhibits the following phenomenon: there is a critical mass above which all solutions blow up in finite time while all solutions are global below that critical mass. We will investigate the case of the critical mass along with the stability of self-similar solutions with lower masses. We next consider a generalization to several space dimensions which involves a nonlinear diffusion and show that a similar phenomenon takes place but with some different features.
• Partial Differential Equations Seminar
27 October 2008
17:00
Peter Constantin
Abstract
I will talk about recent work concerning the Onsager equation on metric spaces. I will describe a framework for the study of equilibria of melts of corpora -- bodies with finitely many degrees of freedom, such as stick-and-ball models of molecules.
• Partial Differential Equations Seminar
13 October 2008
17:00
Gregory Seregin
Abstract
In the lecture, I am going to explain a connection between local regularity theory for the Navier-Stokes equations and Liouville type theorems for bounded ancient solutions to these equations.
• Partial Differential Equations Seminar
6 October 2008
17:00
Tristan Rivi&egrave;re
Abstract
The Willmore Functional for surfaces has been introduced for the first time almost one century ago in the framework of conformal geometry (though it's one dimensional version already appears in thework of Daniel Bernouilli in the XVIII-th century). Maybe because of its simplicity and the depth of its mathematical relevance, it has since then played a significant role in various fields of sciences and technology such as cell biology, non-linear elasticity, general relativity...optical design...etc. Critical points to the Willmore Functional are called Willmore Surfaces. They satisfy the so called Willmore Equations introduced originally by Gerhard Thomsen in 1923 . This equation, despite the elegance of it's formulation, is very inappropriate for dealing with analysis questions such as regularity, compactness...etc. We will present a new formulation of the Willmore Euler-Lagrange equation and explain how this formulation, together with the Integrability by compensation theory, permit to solve fundamental analysis questions regarding this functional, which were untill now totally open.
• Partial Differential Equations Seminar
9 June 2008
17:00
James Robinson
Abstract
I will discuss recent results concerning the uniqueness of Lagrangian particle trajectories associated to weak solutions of the Navier-Stokes equations. In two dimensions, for which the weak solutions are unique, I will present a mcuh simpler argument than that of Chemin & Lerner that guarantees the uniqueness of these trajectories (this is joint work with Masoumeh Dashti, Warwick). In three dimensions, given a particular weak solution, Foias, Guillopé, & Temam showed that one can construct at leaset one trajectory mapping that respects the volume-preserving nature of the underlying flow. I will show that under the additional assumption that $u\in L^{6/5}(0,T;L^\infty)$ this trajectory mapping is in fact unique (joint work with Witek Sadowski, Warsaw).
• Partial Differential Equations Seminar
2 June 2008
17:00
Michael S. Vogelius
Abstract
• Partial Differential Equations Seminar
19 May 2008
17:00
• Partial Differential Equations Seminar
12 May 2008
17:00
Elise Fouassier
Abstract
We compute the high frequency limit of the Hemholtz equation with source term, in the case of a refraction index that is discontinuous along a sharp interface between two unbounded media. The asymptotic propagation of energy is studied using Wigner measures. First, in the general case, assuming some geometrical hypotheses on the index and assuming that the interface does not capture energy asymptotically, we prove that the limiting Wigner measure satisfies a stationary transport equation with source term. This result encodes the refraction phenomenon. Second, we study the particular case when the index is constant in each media, for which the analysis goes further: we prove that the interface does not capture energy asymptotically in this case.
• Partial Differential Equations Seminar