A granular model for crowd motion and pedestrian flow

TL;DR

This work analyzes a macroscopic crowd motion model where density $u$ evolves under a transport term and a degenerate diffusion driven by a $p$-Laplacian-type operator, formalized as $∂_t u - ∇·(|∇v|^{p-2}∇v - u V) = f u$ with $0≤u≤1$ and $u∈Sign^+(v)$. Existence and uniqueness of nonnegative weak solutions are established for $p>2$ using nonlinear semigroup methods and a renormalized, doubling-variables approach, including an $L^1$-contraction principle. The paper then proves that as $p→∞$, these solutions converge to a variational solution of a congested crowd motion problem characterized by $|∇v|≤1$ and a complementary relation $m(|∇v|-1)=0$, encoded in a variational inequality with test functions constrained by $|∇ξ|≤1$. This results in a rigorous link between degenerate $p$-Laplacian crowd models and granular/sandpile-like congestion dynamics, offering a solid foundation for constrained macroscopic pedestrian-flow modeling and potential extensions to tangential-gradient formulations.

Abstract

We study a granular model for congested crowd motion and pedestrian flow. Our approach is based on an approximation through a Hele-Shaw type equation involving a degenerate operator of $p$-Laplacian type and a linear drift, for which we prove existence and uniqueness using nonlinear semigroup methods and the doubling variables technique. Our main result shows that, as $p \to \infty$, the weak solutions of the $p-$problem converge to a variational solution of the congested crowd motion problem.

Figures (1)

• Figure 1: Toy pedestrian-cubes model

Theorems & Definitions (23)

• Definition 2.1
• Theorem 2.1
• Definition 2.2
• Theorem 2.2
• Remark 2.1
• Proposition 3.1: Renormalised formulation
• Proposition 3.2: Kato's inequality
• proof
• Theorem 3.1
• proof