A quantitative theory for the continuity equation

TL;DR

This paper introduces a new quantitative approach to the continuity equation with Sobolev vector fields, providing stability estimates and a novel proof of uniqueness without relying on renormalized solutions.

Contribution

It offers a new proof of uniqueness in the DiPerna--Lions setting using contraction estimates for logarithmic Kantorovich--Rubinstein distances.

Findings

Stability estimates for the continuity equation with Sobolev vector fields.

A new proof of uniqueness in the DiPerna--Lions framework.

Connection between contraction estimates and stability in PDEs.

Abstract

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.