Uniqueness and Lagrangianity for solutions with lack of integrability of the continuity equation

TL;DR

This paper investigates the uniqueness and Lagrangianity of solutions to the continuity equation with Sobolev vector fields, especially in cases lacking local integrability where classical theories do not apply.

Contribution

It introduces a new principle for proving Lagrangianity of solutions without local integrability, utilizing disintegration along flows and a novel directional Lipschitz extension lemma.

Findings

Established a general criterion for Lagrangian solutions in non-integrable settings

Developed a new method to construct test functions for the continuity equation

Extended the theory of uniqueness and Lagrangianity beyond classical integrability assumptions

Abstract

We deal with the uniqueness of distributional solutions to the continuity equation with a Sobolev vector field and with the property of being a Lagrangian solution, that means transported by a flow of the associated ordinary differential equation. We work in a framework with lack of local integrability of the solution, in which the classical DiPerna-Lions theory of uniqueness and Lagrangianity of distributional solutions does not apply due to the insufficient integrability of the commutator. We introduce a general principle to prove that a solution is Lagrangian: we rely on a disintegration along the unique flow and on a new directional Lipschitz extension lemma, used to construct a large class of test functions in the Lagrangian distributional formulation of the continuity equation.