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Open PDE & Analysis Seminar

Prof Camillo de Lellis (IAS Princeton)

Flows of vector fields: classical and modern

Friday 29th May 2020, 2pm (Paris time)

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.

BBB link

This is the link to the talks: link or the url https://bbb2.imo.universite-paris-saclay.fr/b/nic-m3v-7dt.

Organisers

Thomas Alazard (ENS Paris-Saclay)         Nicolas Burq (Orsay)         Iván Moyano (Nice)