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 $L_{\text{loc}}^1((0,T]; BV(\mathbb{T}^d; \mathbb{R}^d)) \cap L^2((0,T) \times \mathbb{T}^d; \mathbb{R}^d)$, there exists a unique vanishing diffusivity solution. This class includes the vector field constructed by Depauw in [13], for which there are infinitely many distinct bounded solutions to the advection equation.