A geometric perspective on hydrodynamic PDE
Channel: Alexander Schmeding
Date: 2020-06-09

This talk was given in the analysis seminar at the university of Delft.

Abstract: The motion of a rigid body, can be described as a differential equation on the configuration space of the system which turns out to be a finite-dimensional Lie group.
Building on this observation, Arnold postulated that the Euler equation governing an incompressible fluid in a domain can be described as a differential equation on its configuration space, the group of volume preserving diffeomorphisms. Indeed as Ebin and Marsden showed, one can reformulate this partial differential equation as an ordinary differential equation, for the price of switching to an infinite dimensional manifold. Using geometric techniques local wellposedness of the Euler equation can then be established. We were recently able to apply this circle of ideas to stochastic versions of the Euler equation (joint with M. Maurelli (Milano) and K. Modin (Chalmers, Gothenburg)). In the talk I will give an introduction to these topics together with an overview on these new developments.  So we will neither supposes familiarity with infinite-dimensional manifolds and their geometry nor with stochastic analysis.