Generalized Wasserstein distance and its application to transport equations with source

TL;DR

This paper introduces a generalized Wasserstein distance for measures with different masses, explores its properties, and applies it to prove existence and uniqueness of solutions for a measure-dependent transport equation with source.

Contribution

It extends the Wasserstein distance to measures with varying mass and applies this to analyze a measure-dependent transport equation with source.

Findings

Generalized Wasserstein distance metrizes weak convergence.

Existence and uniqueness of solutions for the transport equation are established.

The approach handles measure-dependent vector fields and sources.

Abstract

In this article, we generalize the Wasserstein distance to measures with different masses. We study the properties of such distance. In particular, we show that it metrizes weak convergence for tight sequences. We use this generalized Wasserstein distance to study a transport equation with source, in which both the vector field and the source depend on the measure itself. We prove existence and uniqueness of the solution to the Cauchy problem when the vector field and the source are Lipschitzian with respect to the generalized Wasserstein distance.