Calculus of Variations and Geometric Measure Theory

L. Ambrosio - M. Colombo - A. Figalli

Existence and uniqueness of maximal regular flows for non-smooth vector fields

created by ambrosio on 09 Jun 2014
modified by colombom on 17 Jun 2015


Accepted Paper

Inserted: 9 jun 2014
Last Updated: 17 jun 2015

Journal: Arch. Ration. Mech. Anal.
Year: 2015


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.

Tags: GeMeThNES