Generalized Wasserstein Distance and its Application to Transport Equations with Source

In this article, we generalize the Wasserstein distance to measures with different masses. We study the properties of this 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 a source, in which both the vector field and the source depend on the measure itself. We prove the 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.