  This paper contains two main contributions. First, it provides optimal
stability estimates for advection-diffusion equations in a setting in which the
velocity field is Sobolev regular in the spatial variable. This estimate is
formulated with the help of Kantorovich--Rubinstein distances with logarithmic
cost functions. Second, the stability estimates are extended to the
advection-diffusion equations with velocity fields whose gradients are singular
integrals of $L^1$ functions entailing a new well-posedness result.
