Inserted: 1 dec 2020
Last Updated: 9 mar 2021
Journal: To appear in Annales Mathe'matiques du Que'bec
In honor of Sasha Shnirelman on occasion of his 75th birthday
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.