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Penn Mathematics Colloquium

Wednesday, March 3, 2021 - 3:30pm

Camillo De Lellis



University of Pennsylvania


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