Communications in Mathematical Sciences

Volume 21 (2023)

Number 5

A Wasserstein norm for signed measures, with application to non-local transport equation with source term

Pages: 1279 – 1301

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n5.a4

Authors

Benedetto Piccoli (Department of Mathematical Sciences, Rutgers University, Camden, New Jersey, U.S.A.)

Francesco Rossi (Dipartimento di Matematica, Università degli Studi di Padova, Padova, Italy)

Magali Tournus (Aix-Marseille Université, CNRS, Marseille, France)

Abstract

We introduce an optimal transportation interpretation of the Kantorovich norm on the space of signed Radon measures with finite mass, based on the generalized Wasserstein distance for measures with different masses. With this new interpretation, we obtain new topological properties for this norm. We use these tools to prove existence and uniqueness for solutions to non-local transport equations with source terms, when the initial condition is a signed measure.

Keywords

Wasserstein distance, transport equation, signed measures, Kantorovich duality

2010 Mathematics Subject Classification

28A33, 35A01

Received 2 December 2021

Received revised 30 September 2022

Accepted 30 September 2022

Published 30 August 2023