Crowd motion and evolution PDEs under density constraints

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.

New estimates on the regularity of the pressure in density‐constrained mean field games

“…Yet, due to the multiphasic nature of the problem formulated by Brenier, it is in general not possible to translate all the available techniques into such a more complicated setting (see, for instance, where the time‐convexity of the entropy is proven, but, differently from , the same cannot be obtained for other internal energies; analogously, the same results of are also recovered in , and the same algebraic obstruction prevents from generalizing the result to more general energies). On the other hand, the works on density‐constrained MFG (including a first attempt, with a non‐variational model, in ) were inspired by previous works of the second author on crowd motion formulated as a gradient flow with density constraints (see and ), and the present technique seems possible to be applied to such a first‐order (in time) setting. Indeed, as explained in the core of the article, the technique of proof for the regularity of the final pressure is performed exactly as if we had a JKO scheme for a gradient flow (see ).…”

A Hele-Shaw Limit Without Monotonicity

“…The consequent lack of monotonicity significantly complicates the analysis: The derivation of the complementarity condition was only achieved recently [DP20] and the geometric description of the tumor growth still remains to be understood. Similarly, the study of congested crowd motion that involve de-congestion phenomena is of great interest (see [MRCS10], [San18]).…”

A remark on memory effects in constrained fluid systems

“…where G is a possible additional reaction term. This system is really close to macroscopic equations used in the modeling of crowd motion (we refer to [19] and to the review paper [27], in that case η and k are fixed to 1). There, the source term is replaced by an advection term:…”