  In this paper we provide a complete analogy between the Cauchy-Lipschitz and
the DiPerna-Lions theories for ODE's, by developing a local version of the
DiPerna-Lions theory. More precisely, we prove existence and uniqueness of a
maximal regular flow for the DiPerna-Lions theory using only local regularity
and summability assumptions on the vector field, in analogy with the classical
theory, which uses only local regularity assumptions. We also study the
behaviour of the ODE trajectories before the maximal existence time. Unlike the
Cauchy-Lipschitz theory, this behaviour crucially depends on the nature of the
bounds imposed on the spatial divergence of the vector field. In particular, a
global assumption on the divergence is needed to obtain a proper blow-up of the
trajectories.
