Acta mathematica scientia, Series B >
THE ONE-DIMENSIONAL HUGHES MODEL FOR EDESTRIAN FLOW: RIEMANN-TYPE SOLUTIONS
Received date: 2011-11-03
Online published: 2012-01-20
Supported by
MDF is partially supported by the grant MTM 2011-27739-C04-02 of the Spanish Ministry of Science and Innovation, and supported by the ‘Ramon y Cajal’ Sub-programme (MICINN-RYC) of the Spanish Ministry of Science and Innovation, Ref. RYC-2010-06412.
This paper deals with a coupled system consisting of a scalar conservation law
and an eikonal equation, called the Hughes model. Introduced in [24], this model attempts
to describe the motion of pedestrians in a densely crowded region, in which they are seen
as a ‘thinking’ (continuum) fluid. The main mathematical diffculty is the discontinuous
gradient of the solution to the eikonal equation appearing in the flux of the conservation
law. On a one dimensional interval with zero Dirichlet conditions (the two edges of the
interval are interpreted as ‘targets’), the model can be decoupled in a way to consider
two classical conservation laws on two sub-domains separated by a turning point at which
the pedestrians change their direction. We shall consider solutions with a possible jump
discontinuity around the turning point. For simplicity, we shall assume they are locally
constant on both sides of the discontinuity. We provide a detailed description of the local-
in-time behavior of the solution in terms of a ‘global’ qualitative property of the pedestrian
density (that we call ‘relative evacuation rate’), which can be interpreted as the attitude
of the pedestrians to direct towards the left or the right target. We complement our result
with explicitly computable examples.
Key words： pedestrian flow; nonlocal conservation law; eikonal equation
Debora Amadori , M. Di Francesco . THE ONE-DIMENSIONAL HUGHES MODEL FOR EDESTRIAN FLOW: RIEMANN-TYPE SOLUTIONS[J]. Acta mathematica scientia, Series B, 2012 , 32(1) : 259 -280 . DOI: 10.1016/S0252-9602(12)60016-2
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