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 80's shows 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 *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.

### Penn Mathematics Colloquium

Wednesday, March 3, 2021 - 3:30pm

#### Camillo De Lellis

IAS