I will discuss recent results concerning the uniqueness of Lagrangian particle trajectories associated to weak solutions of the Navier-Stokes equations. In two dimensions, for which the weak solutions are unique, I will present a mcuh simpler argument than that of Chemin & Lerner that guarantees the uniqueness of these trajectories (this is joint work with Masoumeh Dashti, Warwick). In three dimensions, given a particular weak solution, Foias, Guillopé, & Temam showed that one can construct at leaset one trajectory mapping that respects the volume-preserving nature of the underlying flow. I will show that under the additional assumption that $u\in L^{6/5}(0,T;L^\infty)$ this trajectory mapping is in fact unique (joint work with Witek Sadowski, Warsaw).
