Consider a (possibly time-dependent) vector field $v$ on the Euclidean space. The classical Cauchy-Lipschitz (also named Picard-Lindel\u00f6f) 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 loses 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 {\\emph 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.<\/p>\n\n\n\n
In order to follow the online exposition it is enough for the audience to connect with the website address https:\/\/meet.google.com\/uoi-ivsu-fae few minutes before the time planned for the talk. (it should work even if you don’t have a unipi or a google email address).<\/p>\n","protected":false},"excerpt":{"rendered":"
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…<\/p>\n