séminaire-edp-BBT

15 Juin 2021

Nicolas Burq (Orsay) –  Almost sure scattering for the one dimensional nonlinear Schrödinger equation

Abstract:  We consider the one-dimensional nonlinear Schrödinger equation with a nonlinearity of degree $p>1$.  On compact manifolds many probability measures are invariant by the flow of the  linear Schrödinger equation (e.g. Wiener measures), and it is sometimes possible to modify them suitably and get  invariant (Gibbs measures)  or  quasi-invariant measures for the non linear problem. On $\mathbb{R}^d$, the large time dispersion shows that the only invariant measure is the $\delta$ measure on the trivial solution $u =0$, and the good notion to track is whether the non linear evolution of the initial measure is well described by the linear (non trivial) evolution. This is precisely what we achieve in this work.  We exhibit measures on the space of initial data for which we describe the non trivial evolution by the linear Schrödinger flow and we show that their nonlinear evolution is absolutely continuous with respect to this linear evolution. Actually, we give precise (and optimal) bounds on the Radon-Nikodym derivatives of these measures with respect to each other and we characterise their $L^p$ regularity.  We deduce from this precise description the global well-posedness of the equation for $p>1$ and scattering for $p>3$ (actually even for $1<p \leq 3$, we get a dispersive property of the solutions and exhibit an almost sure polynomial decay in time of their $L^{p+1}$ norm). To the best of our knowledge, it is the first occurence where the description of quasi-invariant measures allows to get quantitative asymptotics (here scattering properties or decay) for the nonlinear evolution. This is a joint work with L. Thomann (Université de Lorraine)

Site organisateur : Bordeaux (David Lannes, Laurent Michel)

Lien zoom https://u-bordeaux-fr.zoom.us/j/81041813364 (ID de réunion : 810 4181 3364)

11 Mai 2021 – Attention horaire exceptionnel : 15h

Lien zoom https://zoom.us/j/96676815209?pwd=QUxXMVgrQzlFNjBQL1R6RDd0TGNMdz09

Meeting ID: 966 7681 5209

Passcode: 027793

6 Avril 2021

Pierre Degond (IMT) – Topological states in collective dynamics

Résumé : States of matter are characterized by different types of order. Recently, a new notion of order, popularized by the 2016 physics nobel prizes, has emerged : that of topological order. It refers to the global rigidity of the system arising from topological constraints. Recently, topological states has been shown to exist in collective dynamics, which describes systems of self-propelled particles. In this work, we consider a system of self-propelled solid bodies interacting through local full body alignment proposed in a joint work with A. Frouvelle, S. Merino-Aceituno and A. Trescases. In the large-scale limit, this system can be described by hydrodynamic equations with topologically non-trivial explicit solutions. At the particle level, these solutions undergo topological phase transitions towards trivial flocking states. Numerically we show that these transitions require the system to pass through a phase of disorder. To our knowledge, it is the first time that a hydrodynamic model guides the design of topologically non-trivial states and allows for their quantitative analysis and understanding. On the way, we will raise interesting mathematical questions underpinning the analysis of collective dynamics systems.
Joint work with Antoine Diez and Mingye Na (Imperial College London)

Site organisateur : Toulouse (J.M. Bouclet, R. Duboscq, M. Maris, A. Trescases)

Video of the talk : https://www.youtube.com/watch?v=EIBfmIJbnFo