Crowd Dynamics Through Conservation Laws

Abstract

We consider several macroscopic models, based on systems of conservation laws, for the study of crowd dynamics. All the systems considered here contain nonlocal terms, usually obtained through convolutions with smooth functions, used to reproduce the visual horizon of each individual. We classify the various models according to the physical domain (the whole space ${\mathbb {R}}^N$ or a bounded subset), to the terms affected by the nonlocal operators, and to the number of different populations we aim to describe. For all these systems, we present the basic well posedness and stability results.

Appendices

Appendices

1.1 Regular Entropy Solutions for IBVP Problems

In this appendix we briefly recall the concept of regular entropy solutions for an initial boundary value problem. To this aim, fix T > 0, an open and bounded subset Ω of ${\mathbb {R}}^N$, and let us consider the system

(7.1)

where satisfies, for every , $u(t) \in {\mathbf {C}}^{2} ( \Omega ; {\mathbb {R}}^N)$ and ${\left \|u(t)\right \|}_{{\mathbf {C}}^{2}\left (\Omega ; {\mathbb {R}}^N\right )} \le M$ for a suitable positive constant M. The definition of a boundary entropy–entropy flux pair is as follows.

Definition A.1 ([28, Definition 4.1], [60, Definition 2])

The pair of functions is said a boundary entropy–entropy flux pair for (7.1) if:

• 1. the function z↦H(z, w) is convex for every $w \in {\mathbb {R}}$;

• 2. the equality $\partial _z Q\left (t, x, z, w\right ) = \left (\partial _z H\left (z, w\right )\right ) u\left (t, x\right )$ holds for every , $x \in \overline \Omega$, and $z, w \in {\mathbb {R}}$;

• 3. the equalities $H\left (w, w\right ) = 0$, $Q\left (t, x, w, w\right ) = 0$, and $\partial _z H\left (w, w\right ) = 0$ hold for every , $x \in \overline \Omega$, and $w \in {\mathbb {R}}$.

It is now possible to state the definition of regular entropy solution.

Definition A.2 ([57, Definition 3.3])

A regular entropy solution to (7.1) is a function such that, for every boundary entropy–entropy flux pair $\left (H, Q\right )$, in the sense of Definition A.1, for every $k \in {\mathbb {R}}$ and for every $\varphi \in {\mathbf {C}}_c^{1} ({\mathbb {R}} \times {\mathbb {R}}^N; {\mathbb {R}}^+)$, it holds

where ${\mathcal {H}}^{N-1}$ denotes the Hausdorff measure of dimension N − 1.

1.2 List of Symbols

C ^{0, 1}(A; B):

with A and B subsets of normed vector spaces, is the set of functions defined on A, with values in B, that are Lipschitz continuous on A.

C ^k(A; B):

with A and B subsets of normed vector spaces, is the set of functions defined on A, with values in B, whose k-derivatives are continuous on A.

${\mathbf {C}}_c^{k} (A;B)$ :

with A and B subsets of normed vector spaces, is the set of compactly supported functions defined on A, with values in B whose k-derivatives are continuous on A.

$\overline {I}$ :

is the closure of the set I.

:

is the interior of the set I.

L ^p(A; B):

with p ≥ 1, $A \subseteq {\mathbb {R}}^n$ and $B \subseteq {\mathbb {R}}^m$, is the set of measurable functions f defined on A, with values in B, such that ${\left |f\right |}^p$ is Lebesgue integrable on A.

L ^∞(A; B):

with $A \subseteq {\mathbb {R}}^n$ and $B \subseteq {\mathbb {R}}^m$, is the set of measurable functions f defined on A, with values in B, essentially bounded.

${\mathbb {R}}^+$ :

is the set $\left [0, +\infty \right [$ of 0 and all positive real numbers.

:

is the set $\left ]0, +\infty \right [$ of all strictly positive real numbers.

$\mathbb S^{N-1}$ :

is the unit sphere in ${\mathbb {R}}^N$.

$\mathop {\mathrm {spt}} \rho$ :

is the support of the function ρ.

W ^{1, p}(A; B):

with 1 ≤ p ≤∞, $A \subseteq {\mathbb {R}}^n$ and $B \subseteq {\mathbb {R}}^m$, is the Sobolev space of functions defined in A with values in B whose first weak derivative is in L ^p.