11/16/2011
Alessio Figalli
University of Texas-Austin
Di Perna-Lions' theory, with application to semiclassical limits for the Schrodinger equation
At the beginning of the '90, DiPerna and Lions studied in detail the connection between transport equations and ordinary differential equations. In particular, by proving an existence and uniqueness result at the level of the transport equation, they obtained (roughly speaking) existence and uniqueness of solutions for ODEs with Sobolev vector-fields for a.e. initial condition. Ten years later, Ambrosio has been able to extend such a result to BV vector fields. In some recent works we have investigated this theory in a more general setting, which allows us to show the semiclassical convergence of the quantum dynamics to the Liouville dynamics for the linear Schrodinger equations, under very weak regularity assumptions on the potential. In analogy to the classical DiPerna-Lions' theory, the price to pay for allowing singular potential is that the convergence result holds true only for "a.e. initial data", where "a.e." is with respect to a suitable family of reference measures in the space of the initial data. In this expository talk I'll give an overview of these results.