In the video celebrating his fields medal, Alessio Figalli describes the motion of a cloud by looking at how the different particles move in time under a certain velocity field, which represents the velocity of the air at each point. To understand this phenomenon, a fundamental object is the flow of a given vector field b (namely, the solution X(t) of the ODE X’(t) = b(t, X(t)) from any initial datum x in space).

The classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow provided the vector field is sufficiently smooth. The theorem looses its validity as soon as v is slightly less regular. However, in 1989, Di Perna and Lions introduced a generalized notion of flow, consisting of a suitable selection among the trajectories of the ODE, and they showed existence, uniqueness and stability for this notion of flow for much less regular vector fields. In this talk I will give an overview of the topic and provide a negative answer to the following long-standing open question: are the trajectories of the ODE unique for a.e. initial datum in R^d for vector fields as in Di Perna and Lions theorem?

The result exploits the connection between the notion of flow and an associated PDE, the transport equation, and combines ingredients from probability theory, harmonic analysis, and the “convex integration” method for the construction of nonunique solutions to certain PDEs.