A remark on selection of solutions for the transport equation

We prove that for bounded, divergence-free vector fields in \(L^1_\textrm{loc}((0,+\infty );BV_\textrm{loc}(\mathbb {R}^d;\mathbb {R}^d))\), regularisation by convolution of the vector field selects a single solution of the transport equation for any locally integrable initial datum. We recall the vector field constructed by Depauw in (C R Math Acad Sci Paris 337:249–252, 2003), 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.