Quasi-static evolution and congested crowd transport

TL;DR

This paper establishes the convergence of solutions from the porous medium equation with superharmonic drift to Hele-Shaw and congested crowd motion models, revealing the evolution of initial patches as Hele-Shaw solutions.

Contribution

It demonstrates the convergence of porous medium solutions to Hele-Shaw and crowd motion models using viscosity solutions and gradient flow structures, providing new insights into their relationship.

Findings

Porous medium solutions converge to Hele-Shaw solutions as the exponent tends to infinity.

Porous medium solutions also converge to congested crowd motion solutions.

A comparison principle for the Wasserstein-based minimizing movement scheme is established.

Abstract

We consider the relationship between Hele-Shaw evolution with drift, the porous medium equation with superharmonic drift, and a congested crowd motion model originally proposed by [MRS]- [MRSV]. We first use viscosity solutions to show that the porous medium equation solutions converge to the Hele- Shaw solution as the exponent tends to infinity. Next, using of the gradient flow structure of both the porous medium equation and the crowd motion model, we prove that the porous medium equation solutions also converge to the congested crowd motion. Combining these results lets us deduce that in the case where the initial data to the crowd motion model is given by a patch, or characteristic function, the solution evolves as a patch that is the unique solution to the Hele-Shaw problem. While proving our main results we also obtain a comparison principle for solutions to the minimizing movement scheme based on the Wasserstein metric, of independent interest.