*Accepted Paper*

**Inserted:** 1 dec 2020

**Last Updated:** 9 mar 2021

**Journal:** To appear in Annales Mathe'matiques du Que'bec

**Year:** 2020

**Notes:**

In honor of Sasha Shnirelman on occasion of his 75th birthday

**Abstract:**

We give an example of a bounded divergence free autonomous vector field in $\mathbb R^3$ (and of a nonautonomous bounded divergence free vector field in $\mathbb R^2$) and of a bounded initial data for which the Cauchy problem for the corresponding transport equation has 2 distinct solutions. We then show that both solutions are limits of classical solutions of transport equations for appropriate smoothings of the vector fields and of the initial data.

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