Smoothing does not give a selection principle for transport equations with bounded autonomous fields

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.