Transport of congestion in two-phase compressible/incompressible flows

TL;DR

This paper investigates the mathematical modeling of crowd dynamics through a two-phase flow system, establishing existence of solutions, approximation methods, and numerical simulation techniques for congested and uncongested regimes.

Contribution

It introduces a novel two-phase model for crowd motion, proving the existence of weak solutions and developing an approximation via compressible Navier-Stokes equations with singular pressure.

Findings

Existence of weak solutions for the two-phase crowd flow model.

Approximation of the model using compressible Navier-Stokes with singular pressure.

Application of the approximation method to numerical simulations in 1D.

Abstract

We study the existence of weak solutions to the two-phase model of crowd motion. The model encompasses the flow in the uncongested regime (compressible) and the congested one (incompressible) with the free boundary separating the two phases. The congested regime appears when the density in the uncongested regime $\varrho^*(t,x)$ achieves a threshold value $\varrho^*(t,x)$ that describes the comfort zone of individuals. This quantity is prescribed initially and transported along with the flow. We prove that this system can be approximated by the fully compressible Navier-Stokes system with a singular pressure, supplemented with transport equation for the congestion density. We also present the application of this approximation for the purposes of numerical simulations in the one-dimensional domain.