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Penn Mathematics Colloquium

Wednesday, April 1, 2020 - 3:30pm

Camillo De Lellis

IAS Princeton


University of Pennsylvania


Tea will be served at 3:00 PM in the Math Lounge, DRL, 4th floor.

Consider a (possibly time-dependent) vector field V on the Euclidean space. The classical Cauchy-Lipschitz (also named Picard-Lindeloef) Theorem states that, if the vector field V is Lipschitz in space, for every initial datum x there is a unique trajectory f starting at x at time t=0 and solving the ODE  f'(t) = V(t, f(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 1980's 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 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.