INdAM Unit

We present a model for the flow of pedestrians that describes features typical of this flow, such as the fall due to panic in the outflow of people through a door. The mathematical techniques essentially depend on the use of nonclassical shocks in scalar conservation laws.

Consider the Cauchy problem for a strictly hyperbolic 2×2 system of conservation laws in one space dimension: {ie1-01} assuming that each characteristic field is either linearly degenerate or genuinely nonlinear. This paper develops a new algorithm, based on wave-front tracking, which yields a Cauchy sequence of approximate solutions, converging to a unique limit depending continuously on the initial data. The solutions that we obtain constitute a semigroup S, defined on a set {ie1-02} of integrable functions with small total variation. For some Lipschitz constant L, we have the estimate {ie1-03}

We introduce a sort of semilinear structure on subsets of the family of semigroups defined on a metric space. The key step is the definition of the sum of two semigroups, which is here achieved by means of the classical operator splitting technique. No linearity assumption is required, since the whole construction is in a metric space.

Consider the general scalar balance law ∂ t u + Divf (t, x, u) = F (t, x, u) in several space dimensions. The aim of this note is to estimate the dependence of its solutions from the flow f and from the source F . To this aim, a bound on the total variation in the space variables of the solution is obtained. This result is then applied to obtain well posedness and stability estimates for a balance law with a non local source. 2000 Mathematics Subject Classification: 35L65.

The main result of this note is the existence of nonclassical solutions to the Cauchy problem for a scalar conservation law modeling pedestrian flow. From the physical point of view, the main assumption of this model was recently experimentally confirmed in . From the analytical point of view, this model is an example of a conservation law in which nonclassical solutions have a physical motivation and a global existence result for the Cauchy problem with large data is available.

Consider the initial -boundary value problem for a Temple system of balance laws. Aim of this paper is to prove the well posedness of this problem for large times and without requiring the total variation of the initial data be small. 2000 Mathematics Subject Classification: 35L50, 35L60 Key words and phrases: Balance Laws, Initial boundary value problem for conservation laws.

In this paper we present the macroscopic model for pedestrian flows proposed by Colombo and Rosini [10] and show its main properties. In particular, this model is able to properly describe the movements of crowds, even after panic has arisen. Furthermore, it is able to reproduce the so called Braess' paradox for pedestrians. From the mathematical point of view, it provides one of the few examples of non classical shocks motivated by real problems, for which a global existence result is available. Finally, its assumptions were experimentally confirmed by an empirical study of a crowd crush on the Jamarat Bridge in Mina, Saudi Arabia, near Mecca, see .