Nonclassical interactions portrait in a macroscopic pedestrian flow model

Abstract

In this paper we describe the main characteristics of the macroscopic model for pedestrian flows introduced in [R.M. Colombo, M.D. Rosini, Pedestrian flows and non-classical shocks, Math. Methods Appl. Sci. 28 (13) (2005) 1553–1567] and recently sperimentally verified in [D. Helbing, A. Johansson, H.Z. Al-Abideen, Dynamics of crowd disasters: An empirical study, Phys. Rev. E (Statistical, Nonlinear, and Soft Matter Physics) 75 (4) (2007) 046109]. After a detailed study of all the possible wave interactions, we prove the existence of a weighted total variation that does not increase after any interaction. This is the main ingredient used in [R.M. Colombo, M.D. Rosini, Existence of nonclassical Cauchy problem modeling pedestrian flows, technical report, Brescia Department of Mathematics, 2008] to tackle the Cauchy problem through wave front tracking, see [A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford Lecture Ser. Math. Appl., vol. 20, Oxford Univ. Press, Oxford, 2000, The one-dimensional Cauchy problem; A. Bressan, The front tracking method for systems of conservation laws, in: C.M. Dafermos, E. Feireisl (Eds.), Handbook of Differential Equations; Evolutionary Equations, vol. 1, Elsevier, 2004, pp. 87–168; R.M. Colombo, Wave front tracking in systems of conservation laws, Appl. Math. 49 (6) (2004) 501–537]. From the mathematical point of view, this model is one of the few examples of conservation laws in which nonclassical solutions have a physical motivation, see [P.G. Lefloch, Hyperbolic Systems of Conservation Laws, Lectures Math. ETH Zürich, Birkhäuser, Basel, 2002, The theory of classical and nonclassical shock waves], and an existence result is available.