Existence of Solutions for a Class of One-Dimensional Models of Pedestrian Evacuations

SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3386-3411, June 2024.
Abstract. Pedestrian evacuation in a corridor can de described mathematically by different variants of the model introduced by R. L. Hughes [Transp. Res. Part B Methodol., 36 (2002), pp. 507–535]. We identify a class of such models for which existence of a solution is obtained via a topological fixed point argument. In these models, the dynamics of the pedestrian density [math] (governed by a discontinuous-flux Lighthill, Whitham, and Richards model [math]) is coupled to the computation of a Lipschitz continuous “turning curve” [math]. We illustrate this construction by several examples, including the Hughes model with affine cost (a variant of the original problem that is encompassed in the framework of El-Khatib, Goatin, and Rosini [Z. Angew. Math. Phys., 64 (2013), pp. 223–251]. Existence holds either with open-end boundary conditions or with boundary conditions corresponding to panic behavior with capacity drop at exits. Other examples put forward versions of the Hughes model with inertial dynamics of the turning curve and with general costs.