# Flows of vector fields: classical and modern

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Camillo De Lellis
(Institute for Advanced Study, Princeton)

created by gelli on 26 Mar 2020

modified on 03 Apr 2020

8 apr 2020
-- 16:00
[open in google calendar]

https://meet.google.com/uoi-ivsu-fae

**Abstract.**

Consider a (possibly time-dependent) vector field $v$ on the Euclidean space. The classical Cauchy-Lipschitz (also named Picard-Lindel\"of) 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 looses 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 {\em 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.