On vanishing diffusivity selection for the advection equation

We study the advection equation along vector fields singular at the initial time. More precisely, we prove that for divergence-free vector fields in $L_{loc}^1((0,T];BV({\mathbb{T}}^d;{\mathbb{R}}^d))\cap L^2((0,T)\times {\mathbb{T}}^d;{\mathbb{R}}^d)$, 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.