A Directional Lipschitz Extension Lemma, with Applications to Uniqueness and Lagrangianity for the Continuity Equation

TL;DR

This paper introduces a Lipschitz extension lemma that preserves two different distances simultaneously, enabling new results on the uniqueness and Lagrangianity of solutions to the continuity equation under weaker assumptions.

Contribution

The paper presents a novel Lipschitz extension lemma that relaxes integrability requirements for uniqueness in the DiPerna-Lions theory of continuity equations.

Findings

Established a Lipschitz extension lemma for Euclidean and geodesic distances.

Proved uniqueness of weak solutions under minimal integrability assumptions.

Demonstrated Lagrangianity for solutions with only local integrability.

Abstract

We prove a Lipschitz extension lemma in which the extension procedure simultaneously preserves the Lipschitz continuity for two non-equivalent distances. The two distances under consideration are the Euclidean distance and, roughly speaking, the geodesic distance along integral curves of a (possibly multi-valued) flow of a continuous vector field. The Lipschitz constant for the geodesic distance of the extension can be estimated in terms of the Lipschitz constant for the geodesic distance of the original function. This Lipschitz extension lemma allows us to remove the high integrability assumption on the solution needed for the uniqueness within the DiPerna-Lions theory of continuity equations in the case of vector fields in the Sobolev space $W^{1,p}$, where $p$ is larger than the space dimension, under the assumption that the so-called "forward-backward integral curves" associated to the vector field are trivial for almost every starting point. More precisely, for such vector fields we prove uniqueness and Lagrangianity for weak solutions of the continuity equation that are just locally integrable.