Lane formation by side-stepping - Burger, Martin; Hittmeir, Sabine; Ranetbauer, Helene; Wolfram, Marie-Therese (SIAM Journal on Mathematical Analysis, 2016)
In this paper we study a system of nonlinear partial differential equations, which describes the evolution of two pedestrian groups moving in opposite directions. The pedestrian dynamics are driven by aversion and cohesion, i.e., the tendency to follow individuals from their own group and step aside in the case of contraow. We start with a two-dimensional lattice-based approach, in which the transition rates reect the described dynamics, and derive the corresponding PDE system by formally passing to the limit in the spatial and temporal discretization. We discuss the existence of special stationary solutions, which correspond to the formation of directional lanes and prove existence of global in time bounded weak solutions. The proof is based on an approximation argument and entropy inequalities. Furthermore, we illustrate the behavior of the system with numerical simulations.
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Keywords: Cross diffusion; Diffusion; Global existence of solutions; Size exclusion