Hausdorff Kolloquium 2019/2020

Date: October 23, 2019 - January 29, 2020 

Venue: Mathematikzentrum, Lipschitz Lecture Hall, Endenicher Allee 60, Bonn

Wednesday, October 23

15:15 Jani Lukkarinen (University of Helsinki): Mathematical puzzles in kinetic theory
16:45 Monica Visan (UCLA)

Wednesday, November 20

15:15 Mike Hopkins (Harvard University)
16:45 TBA

Wednesday, January 29

15:15 TBA
16:45 Philippe Michel (EPF Lausanne)

Abstracts

Jani Lukkarinen (University of Helsinki): Mathematical puzzles in kinetic theory

Kinetic theory and the associated Boltzmann transport equations are one of the few tools which allow to bridge the orders of magnitude from the scale of microscopic dynamical models to their macroscopic transport properties. Although it is often restricted to models whose dynamics are weak perturbations of constant velocity motion, many physical systems offer examples of such behaviour. One case which can be controlled mathematically, is a rarefied gas of hard spheres with elastic collisions, in the Boltzmann-Grad scaling limit. Other less obvious examples are given by weakly perturbed wave motions, including weakly interacting quantum particles and weakly nonlinear discrete wave equations.

In this talk, we discuss mathematical evidence which support the validity of these kinetic theory approximations, as well as their known limitations and modifications. One key step is starting the microscopic dynamics with initial data which is random and sufficiently "chaotic". The importance of this assumption is highlighted in the dynamical formation of a Bose-Einstein condensate: the macroscopic correlations appearing together with the condensate are not compatible with the standard assumptions, and indeed modified kinetic equations are expected to be needed after the formation. This picture is corroborated by the work of Escobedo and Velázquez who prove that the related bosonic kinetic theory has solutions which blow up in a finite (kinetic) time.

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