# Smooth approximation is not a selection principle for the transport equation with rough vector field

## TL;DR

This paper demonstrates that smooth approximation of rough vector fields in the transport equation does not necessarily select a unique solution, showing the existence of multiple limits and challenging the assumption of convergence to a single solution.

## Contribution

The authors provide a counterexample with a vector field admitting infinitely many flows, illustrating that smooth approximations do not guarantee solution uniqueness.

## Key findings

- Existence of a vector field with infinitely many flows
- Smooth approximations can lead to different limit solutions
- Selection principle via smooth approximation fails in this context

## Abstract

In this paper we analyse the selection problem for weak solutions of the transport equation with rough vector field. We answer in the negative the question whether solutions of the equation with a regularized vector field converge to a unique limit, which would be the selected solution of the limit problem. To this aim, we give a new example of a vector field which admits infinitely many flows. Then we construct a smooth approximating sequence of the vector field for which the corresponding solutions have subsequences converging to different solutions of the limit equation.