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 figalli on 19 Aug 2024

[BibTeX]

Published Paper

Inserted: 9 jun 2014
Last Updated: 19 aug 2024

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

Abstract:

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


Download: