Flows of vector fields: classical and modern

Flows of vector fields: classical and modern 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 γ starting at x at time 0 and solving the ODE γ˙(t)=v(t,γ(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.