Flows generated by divergence free vector fields with compact support

TL;DR

This paper investigates the existence and uniqueness of measure-preserving flows generated by divergence-free vector fields with compact support, providing counterexamples in both autonomous and non-autonomous cases across different dimensions.

Contribution

It presents the first known counterexamples demonstrating non-uniqueness and non-existence of such flows under specified conditions.

Findings

Counterexamples in 3D autonomous case showing non-uniqueness

Counterexamples in 2D non-autonomous case showing non-existence

Insights into limitations of flow theory for divergence-free vector fields

Abstract

We are concerned with the theory of existence and uniqueness of flows generated by divergence free vector fields with compact support. Hence, assuming that the velocity vector fields are measurable, bounded, and the flows in the Euclidean space are measure preserving, we show two counterexamples of uniqueness/existence for such flows. First we consider the autonomous case in dimension 3, and then, the non autonomous one in dimension 2.