  We prove that for bounded, divergence-free vector fields in
$L^1_{loc}((0,+\infty);BV_{loc}(R^d;R^d))$, regularisation by convolution of
the vector field selects a single solution of the transport equation for any
integrable initial datum. We recall the vector field constructed by Depauw in
[10], which lies in the above class of vector fields. We show that the
transport equation along this vector field has at least two bounded weak
solutions for any bounded initial datum.
