Evolution in measure spaces with Wasserstein distance

In this chapter, we provide a fairly general mathematical setting for the nonlinear transport equation analyzed in Chap. 6 (namely Eqs. (5.1) and (6.6)). More precisely, we study the evolution of solutions in measures spaces endowed with the Wasserstein distance and its generalizations. Moreover, we illustrate the connections between the Wasserstein distance, the transport equation and optimal transportation problems in the sense of Monge-Kantorovich. We also deal with numerical schemes for the transport equation in measure spaces and prove convergence of a Lagrangian scheme to the unique solution, when the discretization parameters approach zero. Convergence of an Eulerian scheme is then achieved under more strict hypotheses. Both schemes are discretizations of the push-forward formula defined by the transport equation as in Chap. 6. As a by-product, we obtain existence and uniqueness of the solution under general assumptions. All the results of convergence are proved with respect to the Wasserstein distance. We first show that the total variation distance is not natural for such equations, since we lose uniqueness of the solution. Then transport equations with sources are considered. In this case the solution does not conserve its total mass, thus we can not directly use the classical Wasserstein distance. For this we introduce a generalized Wasserstein distance, which allows mass creation/destruction and has interest in itself as distance among measures with different total mass.