Séminaire Analyse non linéaire et EDP

Nous etudions le probleme du transfer de la chaleur dans unfluide incompressible, sous l'approximation de Boussinesq.Nous etudions le comportement des solutions dans laregion parabolique $|x|>!!>t^{1/2}$ : notre analyse montreque certaines normes $L^p$ des solutions, et notamment la norme d'energie,deviennent arbitrairement grandes en temps long.Il s'agit d'un travail en collaboration avec Maria Schonbek (UCSC).

I will present an atomic-to-continuum derivation in nonlinearelasticity. The atomistic model is based on a two-body interactionenergy, with a potential of the Lennard-Jones type. Performing apointwise Taylor expansion, we obtain a continuum model that predictselastic energy, sharp-interface energy and smooth-interface energy. Thisalso gives a method to describe the configuration of the atoms betweentwo consecutive sharp interfaces, which qualitatively agrees withexperiments in Ni-Mn alloys presenting microstructure.

In this talk we consider a stochastically perturbed Allen-Cahn equation. Theclassical Allen-Cahn equation describes phase separation of non-conservedfields. In the so-called sharp interface limit solutions converge tosolutions of mean curvature flow. We consider here additional random effectsin form of a perturbation by a stochastic flow. We present a tightnessresult in the sharp interface limit and discuss the relation to a version ofstochastically perturbed mean curvature flow. (This is joint work withHendrik Weber from Warwick.)

The streamlines of periodic irrotationaltraveling water waves are known to be real-analytic,with exception of the free surface in the case thewave of greatest height which has a corner at the wavecrest (the lateral tangents being at an angle of 2pi/3).The regularity of waves of small and moderate amplitudeis, perhaps surprisingly, little affected by thepresence of vorticity in the flow. This is joint workwith J. Escher.