### Penn Mathematics Colloquium

Wednesday, March 3, 2021 - 3:30pm

#### Camillo De Lellis

IAS

Location

University of Pennsylvania

Zoom

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.