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

TL;DR

This paper develops a local version of the DiPerna-Lions theory for ODEs, establishing existence and uniqueness of maximal regular flows under local assumptions, and analyzes the trajectories' behavior before maximal existence time.

Contribution

It introduces a local regularity framework for the DiPerna-Lions theory, paralleling classical ODE theory, and examines trajectory behavior relative to divergence bounds.

Findings

Existence and uniqueness of maximal regular flows under local assumptions.

Trajectory behavior depends on divergence bounds, with global bounds needed for blow-up analysis.

Provides a local analogy to classical ODE existence and uniqueness results.

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.