Georgia Institute of Technology College of Sciences

In the simplest form, our result gives a characterization of bounded,divergence-free vector fields on the plane such that the Cauchyproblem for the associated continuity equation has a unique boundedsolution (in the sense of distribution).Unlike previous results in this directions (Di Perna-Lions, Ambrosio,etc.), the proof does not rely on regularization, but rather on adimension-reduction argument which allows us to prove uniqueness usingwell-known one-dimensional results (it is indeed a variant of theclassical method of characteristics).Note that our characterization is not given in terms of functionspaces, but using a qualitative property which is completelynon-linear in character, namely a suitable weak formulation of theSard property.This is a joint work with Giovanni Alberti (University of Pisa) andStefano Bianchini (SISSA, Trieste).