Mathematics for various disciplines Dynamics on diffeomorphism groups: shocks of the Burgers equation and hydrodynamical instability

2007-04-12 16:30 - 2007-04-12 17:30
Graduate School of Mathematical Sciences the University of Tokyo, Room #056
Boris Khesin (University of Toront)
We describe a simple relation between curvatures of the group of volume-preserving diffeomorphisms (responsible for Lagrangian instability of idealfluids via Arnold's approach) and the generation of shocks for potentialsolutions of the inviscid Burgers equation (important in mass transport). For this we characterize focal points of the group of volume-preservingdiffeomorphism, regarded as a submanifold in all diffeomorphisms and thecorresponding conjugate points along geodesics in the Wasserstein space ofdensities. Further, we consider the non-holonomic optimal transport problem, related to the following non-holonomic version of the classical Moser theorem: given abracket-generating distribution on a manifold two volume forms of equal totalvolume can be isotoped by the flow of a vector field tangent to this distribution.