Camillo De Lellis

Institute for Advanced Study

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–Lindelöf) 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 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.