Existence and uniqueness of maximal regular flows for non-smooth vector fields
Arch. Ration. Mech. Anal.
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.
The paper is available on the
cvgmt preprint server.