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

Flows generated by divergence free vector fields with compact support