In the first part of the talk I will give an overview on some recent
advances on the study of the continuity equation when the velocity
field is non-smooth. This kind of equation appears very often in
problems originating from the dynamics of fluids, and the lack of
regularity of the velocity field is due "irregular" physical
behaviours, like shocks or turbulence. I will motivate the need for
the use of geometric measure theory for this kind of analysis, and I
will illustrate the approach based on the notion of renormalized
solutions used by DiPerna-Lions and by Ambrosio to study the Sobolev
and the bounded variation cases, respectively.


In the second part, I will present some results from a project in collaboration
with Giovanni Alberti (University of Pisa) and Stefano Bianchini
(SISSA, Trieste). We focus on the two-dimensional case. In the
simplest form, our result gives a characterization of (bounded,
autonomous and divergence-free) vector fields on the plane such that
uniqueness for the continuity equation holds. The proof relies on a
dimension-reduction argument which reduces the problem to a family of
one-dimensional problems. I will try to convey to the audience some
flavour of the techniques in our proof.