15th Seminar on Mathematics for various disciplines

(COE Partner Seminar, Department of Mathematics, Hokkaido University )

Contents

Outline

Period :
April 12, 2007 (Thursday) 16:30-17:30
Place :
Graduate School of Mathematical Sciences the University of Tokyo, Room #056
Programme :
Boris Khesin (University of Toront)
Dynamics on diffeomorphism groups: shocks of the Burgers equation and hydrodynamical instability.
ABSTRACT:
We describe a simple relation between curvatures of the group of volume-preserving diffeomorphisms (responsible for Lagrangian instability of ideal fluids via Arnold's approach) and the generation of shocks for potential solutions of the inviscid Burgers equation (important in mass transport). For this we characterize focal points of the group of volume-preserving diffeomorphism, regarded as a submanifold in all diffeomorphisms and the corresponding conjugate points along geodesics in the Wasserstein space of densities.

Further, we consider the non-holonomic optimal transport problem, related to the following non-holonomic version of the classical Moser theorem: given a bracket-generating distribution on a manifold two volume forms of equal total volume can be isotoped by the flow of a vector field tangent to this distribution.