On flows generated by vector fields with compact support

Abstract

We are concerned with the theory of the 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 of such flows. First we consider the autonomous case in dimension 3 and then the non-autonomous one in dimension 2.