Abstract
This is a survey about the theory of density-constrained evolutions in theWasserstein space developed by B. Maury, the author, and their collaborators as a model for crowd motion. Connections with microscopic models and other PDEs are presented, as well as several time-discretization schemes based on variational techniques, together with the main theorems guaranteeing their convergence as a tool to prove existence results. Then, a section is devoted to the uniqueness question, and a last one to different numerical methods inspired by optimal transport.
Highlights
Modeling the behavior of human crowds when individuals are an obstacle for each other is a natural issue in applied mathematics, connected in a broad sense to many subdisciplines including, for instance, game theory and fluid mechanics
The applications to crowd motion modeling are the starting point and the motivation but, due to the meta-model nature of the subject, most of the paper will be devoted to the mathematical analysis of the abstract evolution PDEs which are motivated by these models
∇t,xφ||2, which is solved by iteratively repeating three steps: for fixed A and m, finding the optimal φ; for fixed φ and m find the optimal A; update m by going in the direction of the gradient descent, i.e. replacing m with m − r(A − ∇t,xφ)
Summary
Modeling the behavior of human crowds when individuals are an obstacle for each other is a natural issue in applied mathematics, connected in a broad sense to many subdisciplines including, for instance, game theory (when individuals are considered to be rational agents) and fluid mechanics (when they are described as the particles of a fluid). The present paper is concerned with a sort of meta-model introduced by B Maury and his collaborators approximately ten years ago, first in a microscopic setting and in a macroscopic framework in collaboration with the author. Calling it meta-model instead of model means that it is not really concerned with the behavioral choices of the individuals but only with a rigorous mathematical handling of the contacts between them. The applications to crowd motion modeling are the starting point and the motivation but, due to the meta-model nature of the subject, most of the paper will be devoted to the mathematical analysis of the abstract evolution PDEs which are motivated by these models
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