Existence of nonclassical solutions in a Pedestrian flow model
Introduction
This paper is concerned with existence and qualitative properties of nonclassical solutions of a Cauchy problem for a scalar conservation law motivated by pedestrian flow. Consider the equation for the conservation of mass where denotes the pedestrian density at time and at point . The function is the flow of pedestrians, i.e. , being the pedestrians’ speed. In the case of car traffic, assuming that leads to the usual Lighthill–Whitham and Richards (LWR) model. Here, we also postulate that is a function of , but with two key differences with respect to the LWR model: the fundamental diagram is qualitatively different and we assign nonclassical solutions to (1.1), so that the usual standard theory of conservation laws does not apply.
The particular nonclassical solutions chosen here are justified by their providing a description of the dynamics of crowds. Indeed, we show below that the present model describes two well known phenomena: the fall in the efficiency of a door due to the rise of panic and Braess’ paradox, namely the increase in a door efficiency due to a suitable obstacle in front of the door.
Pedestrian models are currently under intense investigation in the specialized literature, see [16], [17], [19], [18], [20], [21], [24], [25], [26] and the references therein. Often, these models are of a microscopic nature, i.e. they postulate rules for the individual behavior and then consider many individuals, as in [16], [19], [18], [24], [25], [26].
On the contrary, here we study a continuum, or macroscopic, model, as for example in [12], [20], [21]. The use of continuum models in the context of pedestrian flows is not justified, a priori, by the number of individuals, obviously far lower than the typical number of molecules in fluid dynamics. However, the availability of reliable continuum models allows one to state and possibly solve optimal management problems.
Standard solutions to (1.1) enjoy the “Maximum Principle”, i.e. if the initial data satisfies the bounds for all , then the corresponding solution satisfies the same bounds, i.e. for all and . This property holds also in the multidimensional case, see [14, Theorem 6.2.2] or [22, Chapter IV, Theorem 2.1, (a)]. The maximum principle prevents any increase in the initial maximal density, contrary to crowd dynamics, where a sort of “overcompression” is a well known phenomenon often causing major incidents.
If a boundary constraint is present, then the maximum principle may fail. Nevertheless, unilateral constraints may not describe the well known phenomenon of the fall in a door outflow typical of panic situations, see [9] for a study of unilateral constraint for conservation laws.
In [12], the overcompressed densities are introduced next to the standard ones . Under usual circumstances, varies in , while the rise of panic forces to enter . To describe the rise of panic, suitable nonclassical solutions to (1.1) are selected in [12], where the corresponding Riemann problem is completely solved under mild assumptions on the flow function . The Riemann solver so defined yields solutions that do not satisfy the Maximum Principle. A key role in this framework is played by the fundamental diagram, i.e. the curve vs. . The one postulated in [12] has been recently experimentally confirmed in [17].
Note that the present analytical structure does not fit in with the results on nonclassical solutions to conservation laws in [22, Chapter III, Section 4]. Indeed, the flow function here considered needs not be polynomial. Moreover, it can be qualitatively approximated only through 4th order polynomials, while the theory in [22] applies to 3rd order polynomials.
In Section 2 we state the assumptions on and define the Riemann solver for (1.1). Here, we consider a class of flow functions slightly smaller than that in [12] in order to reduce the number of cases to be considered, while always comprising the qualitative behavior experimentally observed in [17]. Section 3 is devoted to the existence of nonclassical solutions to the Cauchy problem for (1.1) generated by this Riemann solver. In Section 4 we show how the model proposed in [12] is able to describe the Braess’ paradox. Technical proofs are deferred to the final Section 6.
Section snippets
The model and the Riemann problem
The following properties of are assumed, see Fig. 1. First, the Lipschitz continuity of is a minimal regularity requirement:
(Q.1) .
The flow vanishes if and only if the density is either zero or maximal:
(Q.2) if and only if .
Concavity is a standard technical assumption to avoid mixed waves:
(Q.3) The restrictions and are strictly concave.
Hence, there exists a unique and a unique such that and
The Cauchy problem
This section is concerned with the Cauchy problem for the Eq. (1.1). The availability of a Riemann solver allows one to tackle the Cauchy problem through wave front tracking, see [4], [7].
Let and consider the Cauchy problem associated to the Eq. (1.1) with initial condition Recall that a function is a weak solution to the Cauchy problem (1.1), (2.1), (2.2), (3.1) if the initial condition (3.1) holds and
Doors efficiency and Braess’ Paradox
Consider a group of people that need to leave a corridor through a door. We show below that the model introduced in the preceding section is able to describe the well known phenomenon of the fall in the door through-flow when panic arises, see also [12, Section 3]. This qualitative feature of the solutions to (1.1) as constructed in Theorem 3.4 agrees with various experimental observation, see for instance [16], [25] and the references therein.
Then, within the framework of (1.1), we show that
Conclusion
This paper presents a continuum, or macroscopic, model for the description of the flow of human crowds. Its key features are the use of nonclassical shocks and a flow function with two humps, see Fig. 1.
Analytically, it provides an example of a scalar conservation law solved with nonclassical shocks and in which bounds on the total variation of approximate solutions allow one to prove the global existence of weak solutions without restrictions on the size of the total variation. continuous
Technical proofs
Following (Q.1)–(Q.8), is not necessarily defined at . Let denote the set of all piecewise linear and continuous functions .
Lemma 6.1 Assume thatThen, there exists such that for any there exists and satisfying (2.2)and (3.5).
Proof Introduce the function by . By definition, and . Furthermore, by (6.1), we have . Therefore, there exists such that for
Acknowledgments
The authors of [17] are acknowledged for having kindly provided the data for Fig. 7. The second author worked on this paper while visiting the University of Warsaw supported by the Grant of Ministry of Science and Higher Education, Nr. N201 033 32/2269, and by INdAM.
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Present address: Institute of Mathematics of PAN, ul. S’niadeckich 8, 00-950 Warszawa, Poland.