A Wasserstein norm for signed measures, with application to nonlocal transport equation with source term

TL;DR

This paper introduces a Wasserstein norm for signed measures, enabling the analysis of nonlocal transport equations with source terms, and proves existence and uniqueness of solutions for such equations with signed initial measures.

Contribution

It develops a new Wasserstein norm for signed measures and applies it to establish solution existence and uniqueness for nonlocal transport equations with sources.

Findings

Established a generalized Wasserstein distance for signed measures.

Proved existence and uniqueness of solutions to nonlocal transport equations.

Provided new topological properties of the introduced norm.

Abstract

We introduce the optimal transportation interpretation of the Kantorovich norm on thespace of signed Radon measures with finite mass, based on a generalized Wasserstein distancefor measures with different masses.With the formulation and the new topological properties we obtain for this norm, we proveexistence and uniqueness for solutions to non-local transport equations with source terms, whenthe initial condition is a signed measure.