Abstract:
Consider in the phase space of classical mechanics a Radon
measure that is a probability density carried by the graph of a
Lipschitz continuous (or even less regular) vector field. We study the
structure of the push-forward of such a measure by a Hamiltonian flow.
In particular, we provide an estimate on the number of folds in the
support of the transported measure that is the image of the initial
graph by the flow. We also study in detail the type of singularities in
the projection of the transported measure in configuration space
(averaging out the momentum variable). We study the conditions under
which this projected measure can have atoms, and give an example in
which the projected measure is singular with respect to the Lebesgue
measure and diffuse. We discuss applications of our results to the
classical limit of the Schrödinger equation. Finally we present various
examples and counterexamples showing that our results are sharp.
(work in collaboration with Peter Markowich and
Thierry Paul) |