A quantitative theory for the continuity equation

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.