A PDE model for unidirectional flows: stationary profiles and asymptotic behaviour

Annalisa Iuorio (1) , Gaspard Jankowiak (1) , Peter Szmolyan (2) , Marie-Therese Wolfram (3, 1)

1 RICAM - Johann Radon Institute for Computational and Applied Mathematics
2 Institute for Analysis and Scientific Computing [Wien]
3 WMI - Warwick Mathematics Institute

In this paper, we investigate the stationary profiles of a convection-diffusion model for unidirectional pedestrian flows in domains with a single entrance and exit. The inflow and outflow conditions at both the entrance and exit as well as the shape of the domain have a strong influence on the structure of stationary profiles, in particular on the formation of boundary layers. We are able to relate the location and shape of these layers to the inflow and outflow conditions as well as the shape of the domain using geometric singular perturbation theory. Furthermore, we confirm and exemplify our analytical results by means of computational experiments. MSC 2020: 34B16, 34E13, 34E15, 35B25, 35B40, 65L11, 76A30.

Mots clés: pedestrian dynamics; geometric singular perturbation theory; non linear boundary value problem; stationary states; Burgers' equation; dimension reduction; MSC 2020: 34B16; 34E13; 34E15; 35B25; 35B40; 65L11; 76A30 pedestrian dynamics

Domaines: Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]

Annalisa Iuorio, Gaspard Jankowiak, Peter Szmolyan, Marie-Therese Wolfram. A PDE model for unidirectional flows: stationary profiles and asymptotic behaviour. 2020. ⟨hal-02981091v2⟩