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

TL;DR

This paper demonstrates that smoothing vector fields in transport equations does not guarantee a unique solution, providing explicit examples where multiple solutions arise despite boundedness and divergence-free conditions.

Contribution

It introduces explicit examples of bounded divergence-free vector fields where smoothing fails to select a unique solution for the transport equation.

Findings

Existence of multiple solutions for the transport equation with bounded divergence-free fields.

Smoothing the vector fields does not ensure uniqueness of solutions.

Classical solutions can approximate multiple solutions through smoothing.

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.