  In this work, we provide stability estimates for the continuity equation with
Sobolev vector fields. The results are inferred from contraction estimates for
certain logarithmic Kantorovich--Rubinstein distances. As a by-product, we
obtain a new proof of uniqueness in the DiPerna--Lions setting. The novelty in
the proof lies in the fact that it is not based on the theory of renormalized
solutions.
