We study the advection equation along vector fields singular at the initial time. More precisely, we prove that for divergence-free vector fields in $L^1 loc ((0, T ];BV(\T^d;\R^d))\cap L^2((0,T)\times\T^d;\R^d)$, there exists a unique vanishing diffusivity solution. This class includes the vector field constructed by Depauw, for which there are infinitely many distinct bounded solutions to the advection equation.