Evolution models for mass transportation problems

TL;DR

This paper surveys various mass transportation problems modeled through a dynamic approach involving a cost functional dependent on density and velocity, addressing congestion and concentration effects in different applications.

Contribution

It provides a comprehensive overview of the Benamou-Brenier dynamic formulation for mass transportation problems and its extensions to congestion and concentration scenarios.

Findings

Unified framework for different mass transportation problems

Modeling congestion effects in traffic and crowd simulations

Analysis of concentration phenomena leading to branched structures

Abstract

We present a survey on several mass transportation problems, in which a given mass dynamically moves from an initial configuration to a final one. The approach we consider is the one introduced by Benamou and Brenier in [5], where a suitable cost functional $F(\rho,v)$, depending on the density $\rho$ and on the velocity $v$ (which fulfill the continuity equation), has to be minimized. Acting on the functional $F$ various forms of mass transportation problems can be modeled, as for instance those presenting congestion effects, occurring in traffic simulations and in crowd motions, or concentration effects, which give rise to branched structures.