# Flows of vector fields: classical and modern – Camillo De Lellis (Institute for Advanced Study, Princeton, and University of Zuerich)

#### Abstract

Consider a (possibly time-dependent) vector field $v$ on the Euclidean space. The classical Cauchy-Lipschitz (also named Picard-Lindelöf) Theorem states that, if the vector field $v$ is Lipschitz in space, for every initial datum $x$ there is a unique trajectory $\gamma$ starting at $x$ at time $0$ and solving the ODE $\dot{\gamma} (t) = v (t, \gamma (t))$. The theorem loses its validity as soon as $v$ is slightly less regular. However, if we bundle all trajectories into a global map allowing $x$ to vary, a celebrated theory put forward by DiPerna and Lions in the 80es show that there is a unique such flow under very reasonable conditions and for much less regular vector fields. A long-standing open question is whether this theory is the byproduct of a stronger classical result which ensures the uniqueness of trajectories for {\emph almost every} initial datum. I will give a complete answer to the latter question and draw connections with partial differential equations, harmonic analysis, probability theory, and Gromov’s $h$-principle.

In order to follow the online exposition it is enough for the audience to connect with the website address https://meet.google.com/uoi-ivsu-fae few minutes before the time planned for the talk. (it should work even if you don’t have a unipi or a google email address).

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