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Journal of Nonlinear Science

, Volume 25, Issue 4, pp 827–859 | Cite as

Differential Equations Modeling Crowd Interactions

  • Raul Borsche
  • Rinaldo M. ColomboEmail author
  • Mauro Garavello
  • Anne Meurer
Article

Abstract

Nonlocal conservation laws are used to describe various realistic instances of crowd behaviors. First, a basic analytic framework is established through an ad hoc well-posedness theorem for systems of nonlocal conservation laws in several space dimensions interacting nonlocally with a system of ODEs. Numerical integrations show possible applications to the interaction of different groups of pedestrians and also with other agents.

Keywords

Nonlocal conservation laws Crowd dynamics Car traffic 

Mathematics Subject Classification

35L65 90B20 

1 Introduction

This paper deals with a system composed by several populations and individuals, or agents. The former are described through their macroscopic densities and the latter through discrete points. In analytic terms, this leads to a system of conservation laws coupled with ordinary differential equations. From a modeling point of view, it is natural to encompass also interactions that are nonlocal, in both cases of interactions within the populations and between each population and each individual agent.

Throughout, \(t \in {\mathbb {R}}^+\) is time and the space coordinate is \(x \in {\mathbb {R}}^d\). The number of populations is \(n\), and their densities are \(\rho ^i = \rho ^i (t,x)\), for \(i=1, \ldots , n\). The individuals are described through a vector \(p = p (t)\), with \(p \in {\mathbb {R}}^m\). In the case of \(N\) agents, \(p\) may consist of the vector of each individual position, so that \(m = N\,d\), or else it may contain also each individual speed, so that \(m = 2 \, N \, d\).

Setting \(\rho = (\rho ^1, \ldots , \rho ^n)\), we are thus led to consider the system
$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \partial _t \rho ^i +\nabla _x\cdot \left[ q^i(\rho ^i) \; v^i \left( t, x, \left( {\mathcal {A}}^i \left( \rho (t)\right) \right) (x), p \right) \right] = 0,\\ \displaystyle \dot{p} = F\left( t,p,\left( {\mathcal {B}} \left( \rho (t)\right) \right) (p)\right) , \end{array} \right. \end{aligned}$$
(1.1)
where \({\mathcal {A}}^i\) and \({\mathcal {B}}\) are nonlocal operators, reflecting the fact that the behavior of the members of the population as well as of the agents depends on suitable spatial averages. The function \(v^i\) is related to the speed of the \(i\)th population, and \(F\) yields the evolution of the individuals. We defer to Sect. 2 for the precise definitions and regularity requirements.

Motivations for the study of (1.1) are found, for instance, in Borsche et al. (2010, 2012), Colombo et al. (2012) and Colombo and Lécureux-Mercier (2012a, b), which all provide examples of realistic situations that fall within (1.1). Beside these, system (1.1) also allows to describe new scenarios, and some examples are considered in detail in Sect. 3. There, we limit our scope to \({\mathbb {R}}^2\) (i.e., \(d=2\)) essentially due to visualization problems in higher dimensions. The analytic treatment below, however, is fully established in any spacial dimension.

As a first example, in Sect. 3.1, we study two groups of tourists each following a guide. The two groups are described through the pedestrian model in Colombo et al. (2012) and Colombo and Lécureux-Mercier (2012a, b) and the guides move according to an ODE. Each group follows its guide and interacts with the other group, while both guides need to wait for their respective group.

Section 3.2 is devoted to pedestrians crossing a street at a crosswalk, while cars are driving on the road. The pedestrians’ movement is described as in the previous example, the attractive role of the guides being substituted by a repulsive effect of cars on pedestrians. On the other hand, cars move according to a follow-the-leader model and try to avoid hitting pedestrians. This results in a strong coupling between the ODE and PDE, since the pedestrians cannot cross the street if a car is coming and on the other hand the cars have to stop if there are people on the road.

As a third example, see Sect. 3.3, two groups of hooligans confront with each other. Police officers try to separate the two groups heading toward the areas with the strongest mixing of hooligans. Thus, they move according to the densities of the hooligans, which themselves try to avoid the contact with the police. All examples are illustrated by numerical integrations showing central features of the models.

The current literature offers alternative approaches to the modeling of crowds (Helbing and Molnar 1995; Hughes 2002) and to the interaction between individuals and a continuum (Colombo and Pogodaev 2012). Notably, we recall the so-called multiscale framework, based on measure-valued differential equations, see Cristiani et al. (2011) and Piccoli and Tosin (2009, 2010). There, the interplay between the atomic part and the absolutely continuous part of the unknown measure reminds of the present interplay between the PDE and the ODE. Nevertheless, differently from the cited references, here we exploit the distinct nature of the two equations to assign different roles to agents and crowds.

This paper is organized as follows: In Sect. 2, we give a precise definition of a solution of system (1.1) and state the main analytic results. In Sect. 3, we describe three examples that fit into the above framework and present accompanying numerical integrations. All the technical details, including a sharper version of the main result, are collected in Sect. 4.

2 Analytical Results

In this section, we state some analytical results for solutions of (1.1). Throughout we denote \({\mathbb {R}}^+ = [0, +\infty [\), \(R\) is a positive constant and \(I \subseteq {\mathbb {R}}^+\) is an interval containing 0.

The functions defining problem (1.1) are assumed to satisfy the following assumptions:
(q)

For every \(i \in \left\{ 1, \ldots , n\right\} \), \(q^i \in {\mathbf {C}}^{2}\left( {\mathbb {R}}^+; {\mathbb {R}}^+\right) \) satisfies \(q^i (0) = 0\) and \(q^i(R) = 0\).

(v)

For every \(i \in \left\{ 1, \ldots , n\right\} \), \(v^i \in ({\mathbf {C}}^{2} \cap {\mathbf {L}^\infty }) ({\mathbb {R}}^+ \times {\mathbb {R}}^d \times {\mathbb {R}}^{d} \times {\mathbb {R}}^m; {\mathbb {R}}^d)\).

(F)
The map \(F \in {\mathbf {C}}^{0}({\mathbb {R}}^+ \times {\mathbb {R}}^m \times {\mathbb {R}}^\ell ; {\mathbb {R}}^m)\) is such that
(F.1)
For all compact subset \(K\) of \({\mathbb {R}}^m\), there exists a constant \(L_F > 0\) such that, for every \(t \in {\mathbb {R}}^+\), \(p_1, p_2 \in K\) and \(b_1,b_2 \in {\mathbb {R}}^\ell \),
$$\begin{aligned} {\left\| F(t,p_1,b_1) - F(t,p_2,b_2)\right\| }_{{\mathbb {R}}^m} \leqslant L_F \, \left( {\left\| p_1 - p_2\right\| }_{{\mathbb {R}}^m} + {\left\| b_1 - b_2\right\| }_{{\mathbb {R}}^\ell } \right) . \end{aligned}$$
(F.2)
There exists a map \(C_F \in {{\mathbf {L}}_{loc}^{1}}({\mathbb {R}}^+; {\mathbb {R}}^+)\) such that for all \(t>0\), \(b \in {\mathbb {R}}^\ell \) and \(p \in {\mathbb {R}}^m\)
$$\begin{aligned} {\left\| F(t,p,b)\right\| }_{{\mathbb {R}}^m} \leqslant C_F(t) \, \left( 1 + {{\left\| p\right\| }}_{{\mathbb {R}}^m} + {\left\| b\right\| }_{{\mathbb {R}}^\ell }\right) . \end{aligned}$$
\((\varvec{{\mathcal {A}}})\)
For every \(i \in \left\{ 1, \ldots , n\right\} \), the maps \({\mathcal {A}}^i:{\mathbf {L}^1}({\mathbb {R}}^d; {\mathbb {R}}^n) \rightarrow ({\mathbf {C}}^{2} \cap {\mathbf {W}^{2,1}}) ({\mathbb {R}}^d; {\mathbb {R}}^d)\) are Lipschitz continuous and satisfy \({\mathcal {A}}^i (0) = 0\). In particular, there exists a positive constant \(L_A>0\) such that, for every \(\rho _1, \rho _2 \in {\mathbf {L}^1}({\mathbb {R}}^d; [0,R]^n)\),
$$\begin{aligned} {\left\| {\mathcal {A}}^i (\rho _1) - {\mathcal {A}}^i (\rho _2)\right\| }_{{\mathbf {W}^{2,1}}} + {\left\| {\mathcal {A}}^i (\rho _1) - {\mathcal {A}}^i (\rho _2)\right\| }_{{\mathbf {C}}^{2}} \leqslant L_A {\left\| \rho _1 - \rho _2\right\| }_{{\mathbf {L}^1}}. \end{aligned}$$
\((\varvec{{\mathcal {B}}})\)
The map \({\mathcal {B}}:{\mathbf {L}^1}({\mathbb {R}}^d; {\mathbb {R}}^n) \rightarrow {\mathbf {W}^{1,\infty }}({\mathbb {R}}^m; {\mathbb {R}}^\ell )\) is Lipschitz continuous and satisfies \({\mathcal {B}}(0) = 0\). In particular, there exists a positive constant \(L_B > 0\) such that, for every \(\rho _1, \rho _2 \in {\mathbf {L}^1}({\mathbb {R}}^d; [0,R]^n)\),
$$\begin{aligned} {\left\| {\mathcal {B}} (\rho _1) - {\mathcal {B}} (\rho _2)\right\| }_{{\mathbf {W}^{1,\infty }}} \leqslant L_B {\left\| \rho _1 - \rho _2\right\| }_{{\mathbf {L}^1}}. \end{aligned}$$

We begin with the rigorous definition of solution to (1.1).

Definition 2.1

Fix \(\rho _o \in ({\mathbf {L}^1} \cap {\mathbf {BV}}) ({\mathbb {R}}^d; {\mathbb {R}}^n)\) and \(p_{o} \in {\mathbb {R}}^m\). A vector \(\left( \rho , p\right) \) with
$$\begin{aligned} \rho \in {\mathbf {C}}^{0}\left( I; {\mathbf {L}^1}({\mathbb {R}}^d; {\mathbb {R}}^n)\right) \qquad \text { and } \qquad p \in {\mathbf {W}^{1,1}} (I; {\mathbb {R}}^m) \end{aligned}$$
is a solution to (1.1) with \(\rho (0,x) = \rho _o(x)\) and \(p(0) = p_o\) if
  1. 1.
    For \(i=1, \ldots , n\), the map \(\rho ^i\) is a Kružkov solution to the scalar conservation law
    $$\begin{aligned}&\partial _t \rho ^i + \nabla _x\cdot \left[ q^i(\rho ^i)\, V (t,x) \right] = 0 \quad \text{ where } \\&V (t,x) = v^i \left( t, x, \left( {\mathcal {A}}^i \left( \rho (t)\right) \right) (x), p (t) \right) . \end{aligned}$$
     
  2. 2.
    The map \(p\) is a Caratheodory solution to the ordinary differential equation
    $$\begin{aligned} \dot{p} = {\mathcal {F}} (t, p) \quad \text{ where } \quad {\mathcal {F}} (t,p) = F\left( t, p, \Big ({\mathcal {B}} \left( \rho (t)\right) \Big ) (p)\right) . \end{aligned}$$
     
  3. 3.

    \(\rho (0, x) = \rho _o(x)\) for a.e. \(x \in {\mathbb {R}}^d\).

     
  4. 4.

    \(p(0) = p_{o}\).

     

Above, for the definition of Kružkov solution, we refer to Kružkov (1970, Definition 1). By Caratheodory solution, we mean the solution to the integral equation, see Bressan and Piccoli (2007, Chapter 2). Observe that by (q), the function \((0,p)\), respectively \((R,p)\), solves (1.1) as soon as \(\dot{p} = F (t,p,0)\), respectively \(\dot{p} = F \left( t,p, \left( {\mathcal {B}} (R)\right) (p)\right) \).

We are now ready to state the main result of this work in the case of initial densities in the space \({\mathcal {R}}\) defined by
$$\begin{aligned} {\mathcal {R}}= \left\{ \rho \in ({\mathbf {L}^1} \cap {\mathbf {BV}}) ({\mathbb {R}}^d; [0,R]^n) :\mathrm{spt}\rho \text{ is } \text{ compact }\right\} . \end{aligned}$$
The proof in Sect. 4 relies on the more general Theorem 4.4, which does not require any compactness assumption, but whose statement is less immediate.

Theorem 2.2

Assume that (q), (v), (F), \((\varvec{{\mathcal {A}}})\) and \((\varvec{{\mathcal {B}}})\) hold. Then, there exists a process
$$\begin{aligned} {\mathcal {P}} :\{(t_1,t_2) :t_2 \geqslant t_1 \geqslant 0\} \times {\mathcal {R}} \times {\mathbb {R}}^m \rightarrow {\mathcal {R}} \times {\mathbb {R}}^m \end{aligned}$$
such that
  1. 1.

    for all \(t_1,t_2,t_3 \in {\mathbb {R}}^+\) with \(t_3 \geqslant t_2 \geqslant t_1\), \({\mathcal {P}}_{t_2,t_3} \circ {\mathcal {P}}_{t_1,t_2} = {\mathcal {P}}_{t_1,t_3}\) and \({\mathcal {P}}_{t,t}\) is the identity for all \(t \in {\mathbb {R}}^+\).

     
  2. 2.

    For all \((\rho _o,p_o) \in {\mathcal {R}} \times {\mathbb {R}}^m\), the continuous map \(t \mapsto {\mathcal {P}}_{t_o,t} (\rho _o, p_o)\), defined for \(t \geqslant t_o\), is the unique solution to (1.1) in the sense of Definition 2.1 with initial datum \((\rho _o, p_o)\) assigned at time \(t_o\).

     
  3. 3.
    For any pair \((\rho _o^1,p_o^1), (\rho _o^2,p_o^2) \in {\mathcal {R}} \times {\mathbb {R}}^m\), there exists a function \({\mathcal {L}} \in {\mathbf {C}}^{0} ({\mathbb {R}}^+; {\mathbb {R}}^+)\) such that \({\mathcal {L}} (0)=0\) and, setting \((\rho _i,p_i) (t) = {\mathcal {P}}_{0,t} (\rho _o^i,p_o^i)\),
    $$\begin{aligned} {\left\| \rho _1 (t) - \rho _2 (t)\right\| }_{{\mathbf {L}^1}}\leqslant & {} \left( 1+{\mathcal {L}} (t)\right) \, {\left\| \rho _o^1-\rho _o^2\right\| }_{{\mathbf {L}^1}} + {\mathcal {L}} (t) \, {\left\| p_o^1-p_o^2\right\| }_{{\mathbb {R}}^m} ,\\ {\left\| p_1 (t) - p_2 (t)\right\| }_{{\mathbb {R}}^m}\leqslant & {} {\mathcal {L}} (t) \, {\left\| \rho _o^1-\rho _o^2\right\| }_{{\mathbf {L}^1}} + \left( 1+{\mathcal {L}} (t)\right) \, {\left\| p_o^1-p_o^2\right\| }_{{\mathbb {R}}^m} . \end{aligned}$$
     
  4. 4.
    For any \((\rho _o, p_o) \in {\mathcal {R}} \times {\mathbb {R}}^m\), if \(q_1,q_2\), \(v_1,v_2\) and \(F_1,F_2\) satisfy (q), (v) and (F), then there exists a function \({\mathcal {K}} \in {\mathbf {C}}^{0} ({\mathbb {R}}^+; {\mathbb {R}}^+)\) such that \({\mathcal {K}} (0)=0\) and, calling \((\rho _i, p_i)\) the corresponding solutions,
    $$\begin{aligned} {\left\| \rho _1 (t) - \rho _2 (t)\right\| }_{{\mathbf {L}^1}}\leqslant & {} {\mathcal {K}} (t) \left( {\left\| q_1 - q_2\right\| }_{{\mathbf {W}^{1,\infty }}} + {\left\| v_1 - v_2\right\| }_{{\mathbf {W}^{1,\infty }}} + {\left\| F_1 - F_2\right\| }_{{\mathbf {L}^\infty }} \right) ,\\ {\left\| p_1 (t) - p_2 (t)\right\| }_{{\mathbb {R}}^m}\leqslant & {} {\mathcal {K}} (t) \left( {\left\| q_1 - q_2\right\| }_{{\mathbf {W}^{1,\infty }}} + {\left\| v_1 - v_2\right\| }_{{\mathbf {W}^{1,\infty }}} + {\left\| F_1 - F_2\right\| }_{{\mathbf {L}^\infty }} \right) . \end{aligned}$$
     

3 Examples and Numerical Simulations

In this section, we present three sample applications of system (1.1) that fit into the framework of Theorem 2.2. We consider, in Sect. 3.1, two groups of tourists each following a guide, in Sect. 3.2, pedestrians crossing a road, and in Sect. 3.3, two groups of hooligans fighting each other.

To show qualitative features of the solutions, we numerically integrate (1.1). More precisely, the ODE is solved by means of the explicit forward Euler method, while for the PDE we use a FORCE scheme on a triangular mesh, see Toro (2009, § 18.6). We use the same time step according to a CFL number 0.9 and to the stability bound of the ODE solver. The coupling is achieved by fractional stepping LeVeque (2002, § 19.5). All numerical integrations are based on the same framework.

3.1 Guided Groups

We consider two groups of tourists following their own guide. Members of both groups always aim to stay close to the respective guide, but also try to avoid too crowded places. Moreover, the leaders follow a given circular path and adapt their speed according to the amount of the members of their group. In this setting, we have \(x \in {\mathbb {R}}^d\) with \(d = 2\), \(n = 2\) populations \(\rho ^i (t,x)\) describing the density of the \(i\)th group of tourists, \(p = [p^1,p^2]^T \in {\mathbb {R}}^m\) with \(m = 4\), where \(p^i\) describes the position in \({\mathbb {R}}^2\) of the guide of the \(i\)th group. The density \(\rho ^i\) solves the conservation law
$$\begin{aligned} \partial _t \rho ^i + \nabla _x \cdot \left[ \rho ^i\left( 1 - \rho ^i\right) \left( w^i(p^i-x) - {{\mathcal {A}}}^i(\rho ) \right) \right] = 0 \end{aligned}$$
(3.1)
while the positions \(p^i\) evolves according to the ordinary differential system
$$\begin{aligned} \left\{ \begin{array}{lcc} \dot{p}^i_1(t) &{} = &{} d^i \left( p^i_2(t) - c^i_2\right) (\bar{\eta } * \rho ^i) \left( p^i(t)\right) ,\\ \dot{p}^i_2(t) &{} = &{} - d^i \left( p^i_1(t) - c^i_1\right) (\bar{\eta } * \rho ^i) \left( p^i(t)\right) , \end{array} \right. \qquad i=1,2. \end{aligned}$$
(3.2)
Specific choices of functions and parameters in (3.1) and in (3.2) are provided in Sect. 4.2, so that system (3.1)–(3.2) fits into (1.1) and Theorem 2.2 applies, see Proposition 4.5. As a specific example, we consider the situation identified by the parameters (4.28) and by the functions (4.29). The parameters are chosen in such a way to produce a strong repulsion between members of different groups. The computational domain \([0,1]^2\) is discretized by a triangular mesh with a minimal inner radius of \(h=0.023\). As initial conditions, we choose
$$\begin{aligned} \rho _o^1 \!= \!0.75\, \chi _{[0.5,1.5]\times [0.5,1.5]} \quad \rho _o^2 \!= \!\chi _{ [2.5,3.5]\times [0.5,1.5]} \qquad p_o^1 = [2,3]^T, \quad p_o^2 = [2,2]^T. \end{aligned}$$
In Fig. 1, the solution up to \(T\approx 40\) is shown. The densities of the groups are the blue (for \(i=1\)) and red (for \(i=2\)) regions, whereas the guides are located at the dots of the corresponding color.
Fig. 1

Plots of \(\max \{\rho _1, \rho _2\}\) on the \((x,y)\) plane, where \((\rho _1, \rho _2)\) solve (3.1)–(3.2)–(4.26). The circles are the fixed trajectories of the guides. The blue refers to \(i=1\), while the yellow/red to \(i=2\). The blue guide moves clockwise and the other one counterclockwise (Color figure online)

According to (3.2)–(4.26), the groups walk toward their guides and come into contact at \(t \approx 7.4\). At \(t \approx 20.7\), the blue guide is surrounded by the reds and waits for his group. Meanwhile, the red group bypasses the blues and avoids the congestion. At about \(t \approx 28.3\), the groups are separated, while they meet again at \(t \approx 40.4\).

3.2 Interacting Crowds and Vehicles

We consider two groups of pedestrians crossing a street at a crosswalk, following Borsche et al. (2014) and Borsche and Meurer (2014). The motion of pedestrians is modeled as in Colombo et al. (2012), Colombo and Lécureux-Mercier (2012b) and Etikyala et al. (2014), whereas the dynamics of the cars is described with the classical follow-the-leader model (Gazis et al. 1961). The people near the crosswalk reduce their speed and possibly stop if cars are near to the crosswalk. At the same time, cars slow down and possibly stop as soon as pedestrians are present in front of them.

The density \(\rho ^i(t,x)\), for \(i=1, \ldots , n\), describes the \(i\)th group of pedestrians. The positions of cars along the road are identified by their scalar coordinates \(p^k\), for \(k = 1 \ldots , N\). Without loss of generality, we assume that the road is parallel to the vector \([1, 0]^T\), with width \(2h_R\), i.e., \({\left| x_2 - \bar{x}_2\right| } \leqslant h_R\). Therefore, we have \(d = 2\), \(n = 2\), \(m = N\), \(\ell = N\). The dynamics of pedestrians is described as in Colombo et al. (2012) and Colombo and Lécureux-Mercier (2012b), but modified by the interaction with cars, as in Borsche et al. (2014); Borsche and Meurer (2014), namely:
$$\begin{aligned} \partial _t \rho ^i + \nabla _x\cdot \left[ 2 \rho ^i \; (1-\rho ^i) \; w^i (x,p) \, \left( V^i (x) - {\mathcal {A}}^i (\rho )\right) \right] = 0, \quad i=1,2.\qquad \end{aligned}$$
(3.3)
Here, \(w^i \in {\mathbf {C}}^{2}({\mathbb {R}}^d \times {\mathbb {R}}^m; {\mathbb {R}}^+)\) describes the interaction between the member of the \(i\)th group located at \(x\) and the cars. \({\mathcal {A}}^i\) is chosen as in (4.27) and models the interactions of pedestrians. Finally, the vector field \({V}^i \in {\mathbf {C}}^{2}({\mathbb {R}}^d; {\mathbb {R}}^d)\) stands for the preferred trajectories of the people.
The dynamics of cars along the road is described by the follow-the-leader model
$$\begin{aligned} \left\{ \begin{array}{lcl} \dot{p}^k &{} = &{} g\!\left( \left( {\mathcal {B}} (\rho )\right) (p^k)\right) \; u(p^{k+1} - p^{k}), \qquad k=1, \ldots , m-1,\\ \dot{p}^m &{} = &{} v_L (t), \end{array} \right. \end{aligned}$$
(3.4)
where the nonincreasing function \(g \in {\mathbf {C}}^{2} ({\mathbb {R}}; [0,1])\) describes the slowing down of cars when near to pedestrians. The nondecreasing function \(u \in {\mathbf {C}}^{2} ({\mathbb {R}}; [0,1])\) vanishes on \({\mathbb {R}}^-\) and describes the usual drivers’ behavior in follow-the-leader models. The assigned function \(v_L = v_L (t)\) is the speed of the leader, i.e., of the first vehicle. For simplicity, we assume that the initial position of the first car is after the crosswalk, so that its subsequent dynamics is independent from the crowd.

The present model fits into the framework of Sect. 2, as shown by Proposition 4.30. As a specific example, we consider the spatial domain \({\mathcal {D}} = [0,1] \times [0,1]\), with a road occupying the region \({\mathcal {R}} = [0,1] \times [0.45, 0.55]\) (so that \(\bar{x}_2 = 0.5\) and \(h_R = 0.05\)) and the crosswalk \({\mathcal {C}} = [0.4, 0.6] \times [0.45, 0.55]\), and further details on the choices of the various parameters are in Sect. 4.3. Therefore, pedestrians may walk in \({\mathcal {C}} \cup ({\mathcal {D}} {\setminus } {\mathcal {R}})\), while cars travel along \({\mathcal {R}}\) from left to right. The \(\rho ^1\) population targets the bottom boundary \([0,1] \times \{0\}\), while \(\rho ^2\) points toward the top boundary \([0,1] \times \{1\}\), see Fig. 2. No individual is allowed to cross the road out of the crosswalk. The spatial domain is discretized using a triangular mesh with minimal inner radius \(h=0.003\). This resolution is fine enough to resolve adequately the visual cones of pedestrians and drivers, described by \(\tilde{\eta }_r\) and \(\eta \) in (4.32)–(4.33).

The vector \(V^1 (x)\), respectively \(V^2 (x)\), is chosen with norm 1 and tangent to the geodesic path at \(x\) for the population 1, respectively 2. In general, these vectors can be computed as solutions to the Eikonal equation, and their regularity depends on the geometry of the domain (Sethian 1999).

As initial condition, we place the two groups of pedestrians on the two sides of the road, while cars are equally spaced along the road, so that
$$\begin{aligned} \begin{array}{rcl} \rho _0^1 &{} = &{} \chi _{[0.1,0.9]\times [0.7,0.9]},\\ \rho _0^2 &{} = &{} 0.5 \, \chi _{[0.1,0.9]\times [0.1,0.3]}, \end{array} \qquad \text{ and } \qquad \begin{array}{rcl} p_o^1 &{} = &{} 0.000,\\ p_o^2 &{} = &{} 0.333,\\ p_o^3 &{} = &{} 0.667. \end{array} \end{aligned}$$
In Fig. 2, \(\rho _1\) is blue and \(\rho _2\) is red. Where the two densities are superimposed, only the higher one is shown. Pedestrians start walking toward the crosswalk, and the cars can drive freely. The first car has maximal speed 1, and the other ones adapt their speed according to the distance to their leading car, see Fig. 3. At time \(t\approx 0.2\), the second car is in the middle of the crosswalk and only few pedestrians try to cross the road (Fig. 2, top left). After a car passes the crosswalk, pedestrians restart moving and form lanes while crossing the other group, see Fig. 2, top right. When the third car approaches the crosswalk, the pedestrians in front of the crosswalk stop, while those on the road continue their way (Fig. 2, bottom left). As the street is not cleared immediately, the car almost has to stop (Fig. 3, left). After it passed, pedestrians move and reach their destinations (Fig. 2, bottom right).
Fig. 2

Plots of \(\max \{\rho _1, \rho _2\}\) on the \((x,y)\) plane, where \((\rho _1, \rho _2)\) solve (3.3)–(3.4)–(4.30)–(4.31). The blue population \(\rho _1\) moves downward and the red one, \(\rho _2\), upward. Cars are represented by the red dots along the road. Above: left pedestrians wait until the second car has passed the crosswalk; right pedestrians cross the road and form lanes. Bottom: left pedestrians wait until the third car has passed the crosswalk; right pedestrians cross the road and form lanes (Color figure online)

Fig. 3

Positions, left, and velocities, right, of the cars in the solution to (3.3)–(3.4)–(4.30)–(4.31) as a function of time. The green refers to \(p_3\), the red to \(p_2\) and the blue to \(p_1\) (Color figure online)

3.3 Policemen Separating Conflicting Hooligans

Consider \(n=2\) groups of conflicting hooligans and their interaction with police officers in a \(d=2\) dimensional region. For the hooligans, we use a model of the form
$$\begin{aligned} \partial _t \rho ^i + \nabla _x\cdot \left[ \rho ^i (1-\rho ^i) \left( - w^i(x,p) + {\mathcal {A}}^i (\rho ) \right) \right] = 0, \qquad i=1,2, \end{aligned}$$
(3.5)
where \(\rho ^i\) is the density of the \(i\)th group. Here, \(w^i \in {\mathbf {C}}^{2}({\mathbb {R}}^d \times {\mathbb {R}}^m; {\mathbb {R}}^d)\) describes the preferred direction of the hooligans belonging to the \(i\)th group at position \(x\). This is influenced by the presence of the police officers \(p^1, \ldots , p^N\). The terms \({\mathcal {A}}^1 (\rho ), {\mathcal {A}}^2 (\rho )\) modify the hooligans’ direction according to their distribution in space. The movement of the \(N=4\) police officers is described by the ODEs
$$\begin{aligned} \dot{p}^k = I_k(p) + {\mathcal {B}}_k(\rho ), \qquad k=1, \ldots , N, \end{aligned}$$
(3.6)
where \(p^k = [p^k_1, p^k_2]^T\) denotes the position in \({\mathbb {R}}^2\) of the \(k\)th policeman; so we set \(m = 2N\). The function \(I_k \in {\mathbf {C}}^{0}({\mathbb {R}}^{2N};{\mathbb {R}}^2)\) avoids concentrations of officers at the same place. The term \({\mathcal {B}}_k(\rho )\) models the attraction of the policemen to conflicting groups of hooligans in order to separate them. The present model fits in the framework presented in Sect. 2, as proved in Proposition 4.7, refer to Sect. 4.4 for further details on the parameters’ choices.
In the numerical example, we choose two initially separated opposed groups, while the policemen are at the corners of the domain, according to
$$\begin{aligned} \begin{array}{rcl} \rho _0^1 &{} = &{}0.9 \; \chi _{ [0.25,0.75]\times [0.2,0.5]},\\ \rho _0^2 &{} = &{} 0.7 \; \chi _{ [0.25,0.75]\times [0.5,0.8]}, \end{array} \qquad \text{ and } \qquad \begin{array}{rcl} p_o^1 &{} = &{}[0.1,0.7]^T,\\ p_o^2 &{} = &{}[0.9,0.3]^T,\\ p_o^3 &{} = &{}[0.1,0.4]^T,\\ p_o^4 &{} = &{}[0.9,0.7]^T. \end{array} \end{aligned}$$
(3.7)
In the pictures, the densities of the two groups are plotted separately. The police officers are indicated by green circles. At the beginning, the two groups of hooligans fight in the middle of the domain, while parts of the groups do not take part to the action (Fig. 4, top left). As the conflicting groups mix, policemen try to separate them. The first two officers alone cannot completely succeed (Fig. 4, top right) since on the sides the hooligans keep fighting. Once the other two policemen arrive (Fig. 4, bottom left), the conflict stops. At the end, policemen separate the conflicting parties (Fig. 4, bottom right). This latter configuration appears to be relatively stationary. The solution to the same equations but with no police officers, i.e., \(\varepsilon _3 = \varepsilon _4 = 0\) in (4.36), is displayed in Fig. 5. Note that now the two groups superimpose and the fight does not stop.
Fig. 4

Plots of the solution to (3.5)–(3.6)–(3.7) on the \((x,y)\) plane. In each of the four pairs of diagrams, \(\rho _1\) is on the left and \(\rho _2\) is on the right. Above left two groups of hooligans fight. Above right two officers try to separate the groups. Bottom left the four officers succeed in separating the groups in the central part, but fighting continues on the sides. Bottom right the four officers succeed in separating the two groups

Fig. 5

Plots of the solution to (3.5)–(3.6)–(3.7) on the \((x,y)\) plane, but with \(\varepsilon _3 = \varepsilon _4 = 0\), so that police officers are absent. Note that, differently from the integration shown in Fig. 4, here the two groups superimpose, meaning that a fight takes place. In each of the four pairs of diagrams, \(\rho _1\) is on the left and \(\rho _2\) is on the right

4 Technical Section

This section contains all the technical parts of the paper. It is divided into several paragraphs: The first one deals with the proof of Theorem 2.2, while the others with the proofs and the technicalities of Sect. 3.

4.1 Proof of Theorem 2.2

Denote \(W_d = \int _0^{\pi /2} \left( \cos (\vartheta )\right) ^d {{\mathrm {d}}{\vartheta }}\). For later use, we state here without proof the Grönwall-type lemma used in the sequel.

Lemma 4.1

Let \(T > 0\), \(\delta \in {\mathbf {C}}^{0}\left( [0,T];{\mathbb {R}}^+\right) \), \(\alpha \in {{\mathbf {L}}_{loc}^{\infty }}\left( [0,T];{\mathbb {R}}^+\right) \) and \(\beta \in {{\mathbf {L}}_{loc}^{1}}([0,T];{{{\mathbb {R}}}}^+)\). If \(\displaystyle \delta (t) \leqslant \alpha (t) + \int _0^t \beta (\tau ) \, \delta (\tau ) {{\mathrm {d}}{\tau }}\) for a.e. \(t \in [0, T]\) then,
$$\begin{aligned}&\delta (t) \leqslant \alpha (t) + \int _0^t \alpha (\tau ) \, \beta (\tau ) \, e^{\int _\tau ^t \beta (s)\, {{\mathrm {d}}{s}}} \, {{\mathrm {d}}{\tau }}\leqslant \left( \sup _{\tau \in [0,t]} \alpha (\tau )\right) \; e^{\int _0^t \beta (\tau )\, {{\mathrm {d}}{\tau }}} ,\\&\quad \text {for a.e.}~t \in [0,T]. \end{aligned}$$
The well posedness of the Cauchy problem
$$\begin{aligned} \left\{ \begin{array}{l} \partial _t \rho + \nabla _x \cdot \left( q(\rho ) V(t,x)\right) = 0,\\ \rho (0,x) = \rho _o (x). \end{array} \right. \end{aligned}$$
(4.1)
follows from Lécureux-Mercier (2010, Proposition 2.9).

Lemma 4.2

Assume \(R>0\) and
$$\begin{aligned} q\in & {} {\mathbf {C}}^{2}({\mathbb {R}}^+; {\mathbb {R}}^+) \text { satisfies } q(0) = 0 \text { and } q(R) = 0,\end{aligned}$$
(4.2)
$$\begin{aligned} V\in & {} {\mathbf {C}}^{2} ({\mathbb {R}}^+ \times {\mathbb {R}}^d; {\mathbb {R}}^d) \text { satisfies } \left\{ \begin{array}{l} \nabla _x\cdot V(t, \cdot ) \in {\mathbf {W}^{1,1}} ({\mathbb {R}}^d; {\mathbb {R}}^d),\\ V(t, \cdot ) \in {\mathbf {W}^{1,\infty }} ({\mathbb {R}}^d; {\mathbb {R}}^d), \end{array} \right. \text { for } t \in {\mathbb {R}}^+,\end{aligned}$$
(4.3)
$$\begin{aligned} \rho _o\in & {} {\mathbf {L}^1} ({\mathbb {R}}^d; [0,R]). \end{aligned}$$
(4.4)
Then, there exists a unique Kružkov solution \(\rho \in {\mathbf {C}}^{0} \left( {\mathbb {R}}^+; {\mathbf {L}^1}({\mathbb {R}}^d; [0,R])\right) \) to (4.1) and
$$\begin{aligned} {\left\| \rho (t)\right\| }_{{\mathbf {L}^1}} = {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} \qquad \text{ for } \text{ all } t \in {\mathbb {R}}^+ . \end{aligned}$$
(4.5)
If moreover \(\rho _o \in {\mathbf {BV}}({\mathbb {R}}^d; [0,R])\), then, for every \(t > 0\)
$$\begin{aligned} \mathrm{TV}\left( \rho (t)\right) \leqslant \left( \mathrm{TV}\left( \rho _o\right) + d \, W_d {\left\| q\right\| }_{{\mathbf {L}^\infty }} \int _0^t \int _{{\mathbb {R}}^d} {\left\| \nabla _x \nabla _x\cdot V(\tau ,x)\right\| }_{{\mathbb {R}}^d} \,\mathrm{d}{x} {{\mathrm {d}}{\tau }}\right) e^{\kappa _o\, t},\nonumber \\ \end{aligned}$$
(4.6)
and for every \(0 < t_1 < t_2\),
$$\begin{aligned} \begin{array}{rcl} \displaystyle {\left\| \rho (t_2) - \rho (t_1)\right\| }_{{\mathbf {L}^1}} &{} \leqslant &{} \displaystyle {\left\| q\right\| }_{{\mathbf {L}^\infty }} \int _{t_1}^{t_2} \int _{{\mathbb {R}}^d} {\left| \nabla _x \cdot V(t,x)\right| } \,\mathrm{d}{x} {{\mathrm {d}}{t}}\\ &{}&{} \displaystyle +\, (t_2 - t_1) {\left\| q'\right\| }_{{\mathbf {L}^\infty }} {\left\| V\right\| }_{{\mathbf {L}^\infty }} \sup _{\tau \in [0, t_2]} \mathrm{TV}\left( \rho (\tau )\right) , \end{array} \end{aligned}$$
(4.7)
where \(\kappa _o = (2d + 1) {\left\| q'\right\| }_{{\mathbf {L}^\infty }} {\left\| \nabla _x V\right\| }_{{\mathbf {L}^\infty }}\).
Let \(q_1\), \(q_2\), \(V_1\), \(V_2\) and \(\rho _o^1\), \(\rho _o^2\) satisfy (4.2), (4.3) and (4.4). Call \(\rho _1\), \(\rho _2\) the solutions to
$$\begin{aligned} \left\{ \begin{array}{l} \partial _t \rho _1 + \nabla _x\cdot \left( q_1(\rho _1) V_1(t,x)\right) = 0,\\ \rho _1(0,x) = \rho _o^{1}(x), \end{array} \right. \text { and } \left\{ \begin{array}{l} \partial _t \rho _2 + \nabla _x\cdot \left( q_2(\rho _2) V_2(t,x)\right) = 0,\\ \rho _2(0,x) = \rho _o^{2}(x). \end{array} \right. \nonumber \\ \end{aligned}$$
(4.8)
Then, for every \(t \in {\mathbb {R}}^+\),
$$\begin{aligned}&{\left\| \rho _1(t) - \rho _2(t)\right\| }_{{\mathbf {L}^1}} \leqslant {\left\| \rho _o^{1} - \rho _o^{2}\right\| }_{{\mathbf {L}^1}} e^{\kappa t}\\&\quad + \frac{\kappa _o e^{\kappa _o t} - \kappa e^{\kappa t}}{\kappa _o - \kappa } \left[ \mathrm{TV}(\rho _o^1) + d\, W_d {\left\| q_1\right\| }_{{\mathbf {L}^\infty }} {\left\| \nabla _x \nabla _x\cdot V_1\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} \right] \\&\quad \times \left[ {\left\| q'_2\right\| }_{{\mathbf {L}^\infty }} {\left\| V_1 - V_2\right\| }_{{\mathbf {L}^1}([0,t];{\mathbf {L}^\infty })} + {\left\| q'_1 - q'_2\right\| }_{{\mathbf {L}^\infty }} {\left\| V_1\right\| }_{{\mathbf {L}^1}([0,t];{\mathbf {L}^\infty })} \right] \\&\quad + \left[ {\left\| q_1\right\| }_{{\mathbf {L}^\infty }} {\left\| \nabla _x\cdot (V_1 - V_2)\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} + {\left\| q_1-q_2\right\| }_{{\mathbf {L}^\infty }} {\left\| \nabla _x\cdot V_2\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} \right] e^{\kappa t}\!, \end{aligned}$$
where
$$\begin{aligned} \begin{array}{rcl} \kappa _o &{} = &{} \displaystyle (2d + 1) {\left\| q'_1\right\| }_{{\mathbf {L}^\infty }} {\left\| \nabla _x V_1\right\| }_{{\mathbf {L}^\infty }([0,t],{\mathbf {L}^\infty })},\\ \kappa &{} = &{} \displaystyle {\left\| q_1' \, \nabla _x\cdot V_1 - q_2' \, \nabla _x\cdot V_2\right\| }_{{\mathbf {L}^\infty }}. \end{array} \end{aligned}$$
(4.9)

Proof

The equality (4.5) directly follows from (q) and Kružkov (1970, Theorem 1). The total variation bound (4.6) follows from Lécureux-Mercier (2010, Theorem 2.2). Estimate (4.7) follows from Lécureux-Mercier (2010, Corollary 2.4). The stability estimate follows from Lécureux-Mercier (2010, Proposition 2.9). \(\square \)

Below, we aim at the proof of a version of Theorem 2.2 without any compactness assumption on the support of the initial density. This is possible at the cost of strengthening the assumptions in Sect. 2 introducing the following conditions.
(v \('\))
Assumption (v) holds and for all \(T \in {\mathbb {R}}^+\) and all compact set \(K \subseteq {\mathbb {R}}^m\), there exists a function \({\mathcal {C}}_K \in ({\mathbf {L}^1} \cap {\mathbf {L}^\infty }) ({\mathbb {R}}^d; {\mathbb {R}}^+)\) such that, for \(t \in [0,T]\), \(x \in {\mathbb {R}}^d\), \(A \in {\mathbb {R}}^{d}\) and \(p \in K\)
(F \('\))
The map \(F :{\mathbb {R}}^+ \times {\mathbb {R}}^m \times {\mathbb {R}}^\ell \longrightarrow {\mathbb {R}}^m\) is such that (F.1) and (F.2) hold, together with
(F.3)

For all \(t>0\) and \(b \in {\mathbb {R}}^\ell \), the function \(\begin{array}{ccc} {\mathbb {R}}^m &{} \longrightarrow &{} {\mathbb {R}}^m \\ p &{} \longmapsto &{} F(t,p,b) \end{array}\) is continuous.

(F.4)

For all \(t>0\) and \(p \in {\mathbb {R}}^m\), the function \(\begin{array}{ccc} {\mathbb {R}}^\ell &{} \longrightarrow &{} {\mathbb {R}}^m \\ b &{} \longmapsto &{} F(t,p,b) \end{array}\) is continuous.

(F.5)

For all \(b \in {\mathbb {R}}^\ell \) and \(p \in {\mathbb {R}}^m\), the function \(\begin{array}{ccc} {\mathbb {R}}^+ &{} \longrightarrow &{} {\mathbb {R}}^m \\ t &{} \longmapsto &{} F(t,p,b) \end{array}\) is Lebesgue measurable.

Assumption (F \('\)) ensures that the ordinary differential equation in (1.1) fits into the usual framework of Caratheodory equations, see Filippov (1988, § 1).

Lemma 4.3

Assume that (F \('\)) and \((\varvec{{\mathcal {B}}})\) hold. Fix \(p_o \in {\mathbb {R}}^m\) and \(r \in {\mathbf {C}}^{0}\Big ({\mathbb {R}}^+; {\mathbf {L}^1}({\mathbb {R}}^d; {\mathbb {R}}^n)\Big )\). Then, the problem
$$\begin{aligned} \left\{ \begin{array}{l} \dot{p} = F\left( t, p, \left( {\mathcal {B}}(r)\right) (p)\right) ,\\ p(0) = p_o, \end{array} \right. \end{aligned}$$
admits a unique Caratheodory solution \(p \in {\mathbf {W}^{1,1}} ({\mathbb {R}}^+; {\mathbb {R}}^m)\) and for every \(t>0\)
$$\begin{aligned} {\left\| p(t)\right\| }_{{\mathbb {R}}^m} \leqslant \left( {\left\| p_o\right\| }_{{\mathbb {R}}^m} + \int _0^t C_F(s) \left( 1 + L_B \, {\left\| r (s)\right\| }_{{\mathbf {L}^1}} \right) {{\mathrm {d}}{s}} \right) \exp \left( \int _0^t C_F(s){{\mathrm {d}}{s}}\right) .\nonumber \\ \end{aligned}$$
(4.10)
If \(T > 0\), \(p_o^1,p_o^2 \in {\mathbb {R}}^m\) and \(r_1,r_2 \in {\mathbf {C}}^{0}\left( {\mathbb {R}}^+; {\mathbf {L}^1} ({\mathbb {R}}^d; {\mathbb {R}}^n)\right) \), calling \(p_1,p_2\) the solutions to
$$\begin{aligned} \left\{ \begin{array}{l} \dot{p} = F_1\left( t, p, \left( {\mathcal {B}}(r_1)\right) (p)\right) ,\\ p(0) = p_o^1, \end{array} \right. \qquad \qquad \left\{ \begin{array}{l} \dot{p} = F_2\left( t, p, \left( {\mathcal {B}}(r_2)\right) (p)\right) ,\\ p(0) = p_o^2, \end{array} \right. \end{aligned}$$
for every \(t \in [0,T]\) the following estimate holds
$$\begin{aligned} {\left\| p_1 (t) - p_2 (t)\right\| }_{{\mathbb {R}}^m}\leqslant & {} \displaystyle \left[ {\left\| p_o^1-p_o^2\right\| }_{{\mathbb {R}}^m} + t {\left\| F_1 - F_2\right\| }_{{\mathbf {L}^\infty }} + L_B {\left\| r_1 - r_2\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} \right] \nonumber \\&\times e^{L_F (t + L_B {\left\| r_1\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})})} . \end{aligned}$$
(4.11)

Proof

The existence and uniqueness of the solution follows, for instance, from Bressan and Piccoli (2007, Theorem 2.1.1). Moreover, by (F \('\)) and \((\varvec{{\mathcal {B}}})\),
$$\begin{aligned} {\left\| p(t)\right\| }_{{\mathbb {R}}^m}= & {} {\left\| p_o + \int _0^t F\left( s, p(s), \left( {\mathcal {B}}(r_1(s))\right) \left( p(s)\right) \right) {{\mathrm {d}}{s}}\right\| }_{{\mathbb {R}}^m}\\\leqslant & {} {\left\| p_o\right\| }_{{\mathbb {R}}^m} + \int _0^t C_F(s) \left[ 1 + {\left\| p(s)\right\| }_{{\mathbb {R}}^m} + {\left\| \left( {\mathcal {B}}(r_1(s))\right) \left( p(s)\right) \right\| }_{{\mathbb {R}}^\ell } \right] {{\mathrm {d}}{s}}\\\leqslant & {} {\left\| p_o\right\| }_{{\mathbb {R}}^m} + \int _0^t C_F(s) \left( 1 + L_B \, {\left\| r_1 (s)\right\| }_{{\mathbf {L}^1}} \right) {{\mathrm {d}}{s}} + \int _0^t C_F(s) {\left\| p(s)\right\| }_{{\mathbb {R}}^m} {{\mathrm {d}}{s}}. \end{aligned}$$
By Lemma 4.1, we deduce (4.10). To prove the stability estimate, first note that, given \(T>0\), by (4.10) there exists a compact set \(K \subseteq {\mathbb {R}}^m\) such that \(p_1(t), p_2(t) \in K\) for every \(t \in [0, T]\). Denote with \(L_F\) the constant related to \(K\) in (F \('\)). Using (F \('\)) and \((\varvec{{\mathcal {B}}})\), we get
$$\begin{aligned}&{\left\| p_1 (t) - p_2 (t)\right\| }_{{\mathbb {R}}^m} \leqslant {\left\| p_o^1-p_o^2\right\| }_{{\mathbb {R}}^m}\\&\qquad + \,\int _0^t {\left\| F_1\left( \tau ,p_1 (\tau ), \left( {\mathcal {B}} (r_1)\right) \left( p_1 (\tau )\right) \right) - F_2\left( \tau ,p_1 (\tau ), \left( {\mathcal {B}} (r_1)\right) \left( p_1 (\tau )\right) \right) \right\| }_{{\mathbb {R}}^m} {{\mathrm {d}}{\tau }}\\&\qquad + \,\int _0^t {\left\| F_2\left( \tau ,p_1 (\tau ),\left( {\mathcal {B}} (r_1)\right) \left( p_1 (\tau )\right) \right) - F_2\left( \tau ,p_1 (\tau ),\left( {\mathcal {B}} (r_1)\right) \left( p_2 (\tau )\right) \right) \right\| }_{{\mathbb {R}}^m} {{\mathrm {d}}{\tau }}\\&\qquad + \,\int _0^t {\left\| F_2\left( \tau ,p_1 (\tau ),\left( {\mathcal {B}} (r_1)\right) \left( p_2 (\tau )\right) \right) - F_2\left( \tau ,p_1 (\tau ),\left( {\mathcal {B}} (r_2)\right) \left( p_2 (\tau )\right) \right) \right\| }_{{\mathbb {R}}^m} {{\mathrm {d}}{\tau }}\\&\qquad + \,\int _0^t {\left\| F_2\left( \tau ,p_1 (\tau ),\left( {\mathcal {B}} (r_2)\right) \left( p_2 (\tau )\right) \right) - F_2\left( \tau ,p_2 (\tau ),\left( {\mathcal {B}} (r_2)\right) \left( p_2 (\tau )\right) \right) \right\| }_{{\mathbb {R}}^m} {{\mathrm {d}}{\tau }}\\&\quad \leqslant {\left\| p_o^1-p_o^2\right\| }_{{\mathbb {R}}^m} + t {\left\| F_1 - F_2\right\| }_{{\mathbf {L}^\infty }}\\&\qquad + \,L_F \int _0^t {\left\| \left( {\mathcal {B}}\left( r_1 (\tau )\right) \right) \left( p_1 (\tau )\right) - \left( {\mathcal {B}}\left( r_1 (\tau )\right) \right) \left( p_2 (\tau )\right) \right\| }_{{\mathbb {R}}^\ell } {{\mathrm {d}}{\tau }}\\&\qquad + \,L_F \int _0^t {\left\| \left( {\mathcal {B}}\left( r_1 (\tau )\right) \right) \left( p_2 (\tau )\right) - \left( {\mathcal {B}}\left( r_2 (\tau )\right) \right) \left( p_2 (\tau )\right) \right\| }_{{\mathbb {R}}^\ell } {{\mathrm {d}}{\tau }}\\&\qquad +\, L_F \int _0^t {\left\| p_1 (\tau ) - p_2 (\tau )\right\| }_{{\mathbb {R}}^m} {{\mathrm {d}}{\tau }}\\&\quad \leqslant {\left\| p_o^1-p_o^2\right\| }_{{\mathbb {R}}^m} + t {\left\| F_1 - F_2\right\| }_{{\mathbf {L}^\infty }} + L_B \int _0^t{\left\| r_1 (\tau ) - r_2 (\tau )\right\| }_{{\mathbf {L}^1}} {{\mathrm {d}}{\tau }}\\&\qquad + \,L_F \int _0^t \left( 1+ L_B {\left\| r_1 (\tau )\right\| }_{{\mathbf {L}^1}} \right) {\left\| p_1 (\tau ) - p_2 (\tau )\right\| }_{{\mathbb {R}}^m} {{\mathrm {d}}{\tau }}. \end{aligned}$$
Apply Lemma 4.1 to complete the proof. \(\square \)

Now, we pass to the well posedness of (1.1) without requiring any compactness assumptions on the density.

Theorem 4.4

Assume that (q),  (v \('\)), (F \('\)), \((\varvec{{\mathcal {A}}})\) and \((\varvec{{\mathcal {B}}})\) hold. Then, there exists a process
$$\begin{aligned} {\mathcal {P}} :\{(t_1,t_2) :t_2\geqslant & {} t_1 \geqslant 0\} \times ({\mathbf {L}^1} \cap {\mathbf {BV}}) ({\mathbb {R}}^d; [0,R]^n) \nonumber \\&\times \,{\mathbb {R}}^m \rightarrow ({\mathbf {L}^1} \cap {\mathbf {BV}}) ({\mathbb {R}}^d; [0,R]^n) \times {\mathbb {R}}^m \end{aligned}$$
such that
  1. 1.

    For all \(t_1,t_2,t_3 \in {\mathbb {R}}^+\) with \(t_3 \geqslant t_2 \geqslant t_1\), \({\mathcal {P}}_{t_2,t_3} \circ {\mathcal {P}}_{t_1,t_2} = {\mathcal {P}}_{t_1,t_3}\) and \({\mathcal {P}}_{t,t}\) is the identity for all \(t \in {\mathbb {R}}^+\).

     
  2. 2.

    For all \((\rho _o,p_o) \in ({\mathbf {L}^1} \cap {\mathbf {BV}}) ({\mathbb {R}}^d; [0,R]^n) \times {\mathbb {R}}^m\), the continuous map \(t \mapsto {\mathcal {P}}_{t_o,t} (\rho _o, p_o)\), defined for \(t \geqslant t_o\), is the unique solution to (1.1) in the sense of Definition 2.1 with initial datum \((\rho _o, p_o)\) assigned at time \(t_o\).

     
  3. 3.
    For any pair \((\rho _o^1,p_o^1), (\rho _o^2,p_o^2) \in ({\mathbf {L}^1}\cap {\mathbf {BV}}) ({\mathbb {R}}^d; [0,R]^n) \times {\mathbb {R}}^m\), there exists a function \({\mathcal {L}} \in {\mathbf {C}}^{0} ({\mathbb {R}}^+; {\mathbb {R}}^+)\) such that \({\mathcal {L}} (0)=0\) and, setting \((\rho _i,p_i) (t) = {\mathcal {P}}_{0,t} (\rho _o^i,p_o^i)\),
    $$\begin{aligned} {\left\| \rho _1 (t) - \rho _2 (t)\right\| }_{{\mathbf {L}^1}}\leqslant & {} \left( 1+{\mathcal {L}} (t)\right) \, {\left\| \rho _o^1-\rho _o^2\right\| }_{{\mathbf {L}^1}} + {\mathcal {L}} (t) \, {\left\| p_o^1-p_o^2\right\| }_{{\mathbb {R}}^m} ,\\ {\left\| p_1 (t) - p_2 (t)\right\| }_{{\mathbb {R}}^m}\leqslant & {} {\mathcal {L}} (t) \, {\left\| \rho _o^1-\rho _o^2\right\| }_{{\mathbf {L}^1}} + \left( 1+{\mathcal {L}} (t)\right) \, {\left\| p_o^1-p_o^2\right\| }_{{\mathbb {R}}^m} . \end{aligned}$$
     
  4. 4.
    For any \((\rho _o, p_o) \in ({\mathbf {L}^1}\cap {\mathbf {BV}}) ({\mathbb {R}}^d; [0,R]^n) \times {\mathbb {R}}^m\), if \(q_1,q_2\), \(v_1,v_2\) and \(F_1,F_2\) satisfy (q), (v) and (F), then there exists a function \({\mathcal {K}} \in {\mathbf {C}}^{0} ({\mathbb {R}}^+; {\mathbb {R}}^+)\) such that \({\mathcal {K}} (0)=0\) and, calling \((\rho _i, p_i)\) the corresponding solutions,
    $$\begin{aligned} {\left\| \rho _1 (t) - \rho _2 (t)\right\| }_{{\mathbf {L}^1}}\leqslant & {} {\mathcal {K}} (t) \left( {\left\| q_1 - q_2\right\| }_{{\mathbf {W}^{1,\infty }}} + {\left\| v_1 - v_2\right\| }_{{\mathbf {W}^{1,\infty }}} + {\left\| F_1 - F_2\right\| }_{{\mathbf {L}^\infty }} \right) ,\\ {\left\| p_1 (t) - p_2 (t)\right\| }_{{\mathbb {R}}^m}\leqslant & {} {\mathcal {K}} (t) \left( {\left\| q_1 - q_2\right\| }_{{\mathbf {W}^{1,\infty }}} + {\left\| v_1 - v_2\right\| }_{{\mathbf {W}^{1,\infty }}} + {\left\| F_1 - F_2\right\| }_{{\mathbf {L}^\infty }} \right) . \end{aligned}$$
     

Proof of Theorem 4.4

The proof is divided into various steps.

Introduction of \(X\) and \({\mathcal {T}}\). Fix the initial data \((\rho _o,p_o) \in ({\mathbf {L}^1} \cap {\mathbf {BV}}) ({\mathbb {R}}^d; [0,R]^n) \times {\mathbb {R}}^m\) and a positive \(T < 1\). For positive \(\Delta _\rho , \Delta _p\), define the closed balls
$$\begin{aligned} B_\rho= & {} \left\{ \rho \in {\mathbf {L}^1} ({\mathbb {R}}^d; [0,R]^n) :{\left\| \rho - \rho _o\right\| }_{{\mathbf {L}^1}} \leqslant \Delta _\rho \right\} \quad \text{ and } \nonumber \\ B_p= & {} \left\{ p \in {\mathbb {R}}^m :{\left\| p-p_o\right\| }_{{\mathbb {R}}^m} \leqslant \Delta _p \right\} \end{aligned}$$
and the space
$$\begin{aligned} X = {\mathbf {C}}^{0} \left( [0,T]; B_\rho \times B_p\right) \end{aligned}$$
which is a complete metric space with the distance
$$\begin{aligned} d \left( \left( \rho _1, p_1\right) , \left( \rho _2, p_2\right) \right) = \sup _{t \in [0,T]} {\left\| \rho _1(t) - \rho _2(t)\right\| }_{{\mathbf {L}^1}} + \sup _{t \in [0,T]} {\left\| p_1(t) - p_2(t)\right\| }_{{\mathbb {R}}^m} . \end{aligned}$$
Consider the function \({\mathcal {T}} :X \rightarrow X\), with \({\mathcal {T}}(r,\pi ) = (\rho , p)\), if \((\rho , p)\) is the solution toIn the spirit of Definition 2.1, here by solution we mean that \((\rho ,p) \in {\mathbf {C}}^{0} ([0,T]; {\mathbf {L}^1} ({\mathbb {R}}^d; {\mathbb {R}}^n)\times {\mathbb {R}}^m)\) satisfies \((\rho ,p) (0) = (\rho _o,p_o)\) and for all \(i=1, \ldots , n\), the following inequality holds
$$\begin{aligned}&\displaystyle \int _0^T \int _{{\mathbb {R}}^d} \mathrm{sgn}\left( \rho ^i (t,x) -k\right) \left[ \left( \rho ^i (t,x) -k\right) \partial _t \varphi (t,x)\right. +\left( q^i \left( \rho ^i (t,x)\right) - q^i (k)\right) v^i \nonumber \\&\displaystyle \quad \times \left. \left( t, x, \left( {\mathcal {A}}^i\left( r (t)\right) \right) (x), \pi (t) \right) \nabla _x \varphi (t,x) \right] \,\mathrm{d}{x} {{\mathrm {d}}{t}} \geqslant 0 \end{aligned}$$
(4.13)
for all \(\varphi \in {\mathbf {C}}_c^{1} \big (\big ]0,T\big [\times {\mathbb {R}}^d; {\mathbb {R}}^+\big )\) and for all \(k \in {\mathbb {R}}\), while, for the \(p\) component,
$$\begin{aligned} p (t) = p_o + \int _0^t F\left( \tau , p (\tau ), \left( {\mathcal {B}}\left( r (\tau )\right) \right) \left( p (\tau )\right) \right) {{\mathrm {d}}{\tau }}\end{aligned}$$
(4.14)
for \(t \in [0,T]\). Lemma 4.2 and Lemma 4.3 ensure that (4.12) admits a unique solution.
\({\mathcal {T}}\) is well defined. To bound the \(p\) component, we use (F \('\)), \((\varvec{{\mathcal {B}}})\) and (4.10)
$$\begin{aligned}&{\left\| p (t) - p_o\right\| }_{{\mathbb {R}}^m} \leqslant \int _0^t {\left\| F\left( s, p (s), \left( {\mathcal {B}} \left( r(s)\right) \right) \left( p (s)\right) \right) \right\| }_{{\mathbb {R}}^m} {{\mathrm {d}}{s}}\\&\quad \leqslant \int _0^t C_F (s) \left( 1 + {\left\| p (s)\right\| }_{{\mathbb {R}}^m} + {\left\| \left( {\mathcal {B}} \left( r(s)\right) \right) \left( p (s)\right) \right\| }_{{\mathbb {R}}^\ell } \right) {{\mathrm {d}}{s}}\\&\quad \leqslant \int _0^t C_F (s) \left( 1 + {\left\| p (s)\right\| }_{{\mathbb {R}}^m} + L_B {\left\| r (s)\right\| }_{{\mathbf {L}^1}} \right) {{\mathrm {d}}{s}}\\&\quad \leqslant \left( 1 + L_B \left( {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} + \Delta _\rho \right) \right) \int _0^t C_F (s) {{\mathrm {d}}{s}}\\&\qquad + \int _0^t C_F (s) \left( {\left\| p_o\right\| }_{{\mathbb {R}}^m} + \left( 1 + L_B \left( {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} + \Delta _\rho \right) \right) \int _0^s C_F(\tau ) {{\mathrm {d}}{\tau }} \right) e^{\int _0^s C_F(\tau ) {{\mathrm {d}}{\tau }}} {{\mathrm {d}}{s}} \end{aligned}$$
and the latter term above can be made smaller than \(\Delta _p\) if \(T\) is sufficiently small.
To obtain similar estimates for the \(\rho \) component, we set \(V (t,x) \!=\! v^i\left( t,x,{\mathcal {A}}^i (r),\pi \right) \) for \(i=1, \ldots , n\) and compute
$$\begin{aligned} \nabla _x\cdot V(t,x)= & {} \sum _{j=1}^d \partial _{x_j} v_j^i \left( t, x, \left( {\mathcal {A}}^i \left( r(t)\right) \right) (x), \pi (t) \right) \nonumber \\&+\, \sum _{j,h=1}^d \nabla _{A_h} v_j^i \left( t, x, \left( {\mathcal {A}}^i \left( r(t)\right) \right) (x), \pi (t) \right) \; \partial _{x_j} \left( {\mathcal {A}}^i_h \left( r(t)\right) \right) (x)\nonumber \\= & {} \nabla _x \cdot v^i \left( t, x, \left( {\mathcal {A}}^i \left( r(t)\right) (x)\right) , \pi (t) \right) \nonumber \\&+\partial _{A} v^i \left( t, x, \left( {\mathcal {A}}^i \left( r(t)\right) \right) (x), \pi (t) \right) \cdot \nabla _x \left( {\mathcal {A}}^i\left( r(t)\right) \right) (x), \end{aligned}$$
(4.15)
$$\begin{aligned} \nabla _x \nabla _x\cdot V\left( t,x\right)= & {} \nabla _x \nabla _x \cdot v^i \left( t, x, \left( {\mathcal {A}}^i \left( r(t)\right) \right) (x), \pi (t) \right) \\&+\, 2\nabla _x \cdot \nabla _A v^i \left( t, x, \left( {\mathcal {A}}^i \left( r(t)\right) \right) (x), \pi (t) \right) \cdot \nabla _x \left( {\mathcal {A}}^i \left( r(t)\right) \right) (x)\\&+\, \nabla ^2_A v^i\left( t, x, \left( {\mathcal {A}}^i \left( r(t)\right) \right) (x), \pi (t) \right) \cdot \left[ \nabla _x \left( {\mathcal {A}}^i \left( r(t)\right) \right) (x) \right] ^2\\&+\, \nabla _{A} v^i \left( t, x, \left( {\mathcal {A}}^i \left( r(t)\right) \right) (x), \pi (t) \right) \cdot \nabla ^2_x \left( {\mathcal {A}}^i \left( r(t)\right) \right) (x), \end{aligned}$$
and by (v \('\)) and \((\varvec{{\mathcal {A}}})\), setting \(K = \overline{B (p_o, \Delta _p)}\),
$$\begin{aligned} \int _0^{t} \int _{{\mathbb {R}}^d} {\left| \nabla _x\cdot V(s,x)\right| } \,\mathrm{d}{x}\, {{\mathrm {d}}{s}}\leqslant & {} {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^1}} t + L_A {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} \int _0^t {\left\| r (s)\right\| }_{{\mathbf {L}^1}} {{\mathrm {d}}{s}}\end{aligned}$$
(4.16)
$$\begin{aligned}\leqslant & {} t \left( {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^1}} + L_A {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} \left( {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} + \Delta _\rho \right) \right) ,\end{aligned}$$
(4.17)
$$\begin{aligned} \int _0^t \int _{{\mathbb {R}}^d} {\left\| \nabla _x \nabla _x\cdot V(\tau ,x)\right\| }_{{\mathbb {R}}^d} \,\mathrm{d}{x} {{\mathrm {d}}{\tau }}\leqslant & {} {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^1}} t + 2 L_A t {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} \left( {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} + \Delta _\rho \right) \nonumber \\&+\, L_A^2 t {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} \left( {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} + \Delta _\rho \right) ^2\nonumber \\&+\, L_A t {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} \left( {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} + \Delta _\rho \right) \nonumber \\= & {} \left( {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^1}} + 3 L_A {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} \left( {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} + \Delta _\rho \right) \right. \nonumber \\&\left. +\, L_A^2 {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} \left( {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} + \Delta _\rho \right) ^2 \right) t. \end{aligned}$$
(4.18)
Proceeding to the \(\rho \) component, using (4.7) and (4.6) together with (4.17) and (4.18) above
$$\begin{aligned}&{\left\| \rho (t) - \rho _o\right\| }_{{\mathbf {L}^1}}\nonumber \\&\quad \leqslant \int _0^{t} \int _{{\mathbb {R}}^d} {\left\| q\right\| }_{{\mathbf {L}^\infty }} {\left| \nabla _x\cdot V(s,x)\right| } \,\mathrm{d}{x}\, {{\mathrm {d}}{s}} + t {\left\| q'\right\| }_{{\mathbf {L}^\infty }} {\left\| V\right\| }_{{\mathbf {L}^\infty }} \sup _{\tau \in [0, t]} \mathrm{TV}\left( \rho (\tau , \cdot )\right) \\&\quad \leqslant {\left\| q\right\| }_{{\mathbf {L}^\infty }} \int _0^{t} \int _{{\mathbb {R}}^d} {\left| \nabla _x\cdot V(s,x)\right| } \,\mathrm{d}{x}\, {{\mathrm {d}}{s}}\\&\qquad + t {\left\| q'\right\| }_{{\mathbf {L}^\infty }} {\left\| V\right\| }_{{\mathbf {L}^\infty }} \left( \mathrm{TV}\left( \rho _o\right) \right. \nonumber \\&\left. \qquad + d \, W_d {\left\| q\right\| }_{{\mathbf {L}^\infty }\left( [0,R]\right) } \int _0^t \int _{{\mathbb {R}}^d} {\left\| \nabla _x \nabla _x\cdot V(\tau ,x)\right\| }_{{\mathbb {R}}^d} \,\mathrm{d}{x} {{\mathrm {d}}{\tau }}\right) e^{\kappa _o\, t}\\&\quad \leqslant t \, {\left\| q\right\| }_{{\mathbf {L}^\infty }} \left( {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^1}} \!+ \!L_A {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} \left( {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} \!+ \!\Delta _\rho \right) \right) \!+ \!t {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} {\left\| q'\right\| }_{{\mathbf {L}^\infty }} \mathrm{TV}(\rho _o) e^{\kappa _o t}\\&\qquad + t^2 {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} {\left\| q'\right\| }_{{\mathbf {L}^\infty }} d W_d {\left\| q\right\| }_{{\mathbf {L}^\infty }}\\&\qquad \times \left( {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^1}} \!+ \!3 L_A {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} \left( {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} \!+\! \Delta _\rho \right) \!+ \!L_A^2 {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} \left( {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} \!+ \!\Delta _\rho \right) ^2 \right) e^{\kappa _o t}, \end{aligned}$$
where
$$\begin{aligned} \kappa _o= & {} (2d + 1) {\left\| q'\right\| }_{{\mathbf {L}^\infty }} {\left\| \nabla _x V\right\| }_{{\mathbf {L}^\infty }}\nonumber \\= & {} (2d + 1) {\left\| q'\right\| }_{{\mathbf {L}^\infty }} {\left\| \nabla _x v^i + \nabla _A v^i \cdot \nabla _x {\mathcal {A}}^i\left( r(t)\right) \right\| }_{{\mathbf {L}^\infty }}\nonumber \\\leqslant & {} (2d + 1) {\left\| q'\right\| }_{{\mathbf {L}^\infty }} \left[ {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} + {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} L_A {\left\| r(t)\right\| }_{{\mathbf {L}^1}} \right] \end{aligned}$$
(4.19)
$$\begin{aligned}\leqslant & {} (2d + 1) {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} {\left\| q'\right\| }_{{\mathbf {L}^\infty }} \left[ 1 + L_A \left( {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} + \Delta _\rho \right) \right] . \end{aligned}$$
(4.20)
Hence, for \(T\) sufficiently small, also \({\left\| \rho (t)-\rho _o\right\| }_{{\mathbf {L}^1}} \leqslant \Delta _\rho \) completing the proof that \({\mathcal {T}}\) is well defined.

Notation. In the sequel, for notational simplicity, we introduce the Landau symbol \({\mathcal {O}}(1)\) to denote a bounded quantity, possibly dependent on \(T\) and on the constants in (v \('\)), (F \('\)), \((\varvec{{\mathcal {A}}})\), \((\varvec{{\mathcal {B}}})\) and (q).

\({\mathcal {T}}\) is a contraction. Fix \((r_1, \pi _1), (r_2, \pi _2) \in X\) and denote \((\rho _i, p_i) = {\mathcal {T}} (r_i, \pi _i)\). We now estimate \(d\left( (\rho _1,p_1),(\rho _2,p_2)\right) \). Consider first the \(p\) component. Using Lemma 4.3, we get
$$\begin{aligned} {\left\| p_1 (t) - p_2 (t)\right\| }_{{\mathbb {R}}^m}\leqslant & {} L_B \, {\left\| r_1 - r_2\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} \, e^{L_F (t + L_B {\left\| r_1\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})})}\\\leqslant & {} L_B \, t \, {\left\| r_1 - r_2\right\| }_{{\mathbf {C}}^{0} ([0,t];{\mathbf {L}^1})} \, e^{L_F t (1 + L_B ({\left\| \rho _o\right\| }_{{\mathbf {L}^1}} + \Delta _\rho ))}\\= & {} {\mathcal {O}}(1)\,t \, {\left\| r_1 - r_2\right\| }_{{\mathbf {C}}^{0} ([0,t];{\mathbf {L}^1})}. \end{aligned}$$
Apply now Lemma 4.2 with \(\rho _o^1 = \rho _o = \rho _o^2\), \(q_1 = q = q_2\), \(V_j^i (t,x) = v^i\left( t, x, {\mathcal {A}}^i(r_j(t)(x), \pi _j(t)\right) \) for \(i = 1, \ldots , n\) and \(j=1,2\). Equality (4.15) and (v \('\)) allow to bound \(\kappa \) in (4.9) as follows
$$\begin{aligned} \kappa \leqslant {\left\| q'\right\| }_{{\mathbf {L}^\infty }} {\left\| \nabla _x\cdot (V^i_1-V^i_2)\right\| }_{{\mathbf {L}^\infty }} \leqslant 2 \, {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} \, {\left\| q'\right\| }_{{\mathbf {L}^\infty }} \left( 1 + L_A\left( {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} + \Delta _\rho \right) \right) , \end{aligned}$$
which ensures, together with (4.20)
$$\begin{aligned} \frac{\kappa _o e^{\kappa _o t} - \kappa e^{\kappa t}}{\kappa _o - \kappa }\leqslant & {} \left( 1 + (2d + 1) {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} {\left\| q'\right\| }_{{\mathbf {L}^\infty }} \left[ 1 + L_A \left( {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} + \Delta _\rho \right) \right] t \right) \\&\times \exp \left( (2d + 1) {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} {\left\| q'\right\| }_{{\mathbf {L}^\infty }} \left[ 1 + L_A \left( {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} + \Delta _\rho \right) \right] t \right) \\= & {} {\mathcal {O}}(1). \end{aligned}$$
Using also (4.18), we obtain
$$\begin{aligned}&{\left\| \rho _1^i(t) - \rho _2^i(t)\right\| }_{{\mathbf {L}^1}} \leqslant t \frac{\kappa _o e^{\kappa _o t} - \kappa e^{\kappa t}}{\kappa _o - \kappa } \left[ \mathrm{TV}(\rho _o)\right. \nonumber \\&\qquad + \,\left. d\, W_d {\left\| q\right\| }_{{\mathbf {L}^\infty }} {\left\| \nabla _x \nabla _x\cdot V_1^i\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} \right] {\left\| q'\right\| }_{{\mathbf {L}^\infty }} {\left\| V_1^i - V_2^i\right\| }_{{\mathbf {C}}^{0}([0,t];{\mathbf {L}^\infty })}\\&\qquad + {\left\| q\right\| }_{{\mathbf {L}^\infty }} {\left\| \nabla _x\cdot (V_1^i - V_2^i)\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} e^{\kappa t}\\&\quad = t {\mathcal {O}}(1)\left[ 1 + {\left\| \nabla _x \nabla _x\cdot V_1^i\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} \right] {\left\| V_1^i - V_2^i\right\| }_{{\mathbf {C}}^{0}([0,t];{\mathbf {L}^\infty })}\nonumber \\&\qquad +\, {\mathcal {O}}(1){\left\| \nabla _x\cdot (V_1^i - V_2^i)\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})}. \end{aligned}$$
By (4.18), we get \({\left\| \nabla _x \nabla _x\cdot V_1^i\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} = {\mathcal {O}}(1)\). By (v \('\)) and \((\varvec{{\mathcal {A}}})\), we bound \({\left\| V_1^i - V_2^i\right\| }_{{\mathbf {L}^\infty }}\)
$$\begin{aligned} {\left\| V_1^i - V_2^i \right\| }_{{\mathbf {L}^\infty }}\leqslant & {} {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} \sup _{\tau \in [0,t],x \in {\mathbb {R}}^d} \left[ {\left\| {\mathcal {A}}^i(r_1(\tau ))(x) - {\mathcal {A}}^i(r_2(\tau ))(x) \right\| }_{{\mathbb {R}}^{d}} \right. \nonumber \\&\left. +\, {\left\| \pi _1(\tau ) -\pi _2(\tau )\right\| }_{{\mathbb {R}}^m} \right] \nonumber \\\leqslant & {} L_A t {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} {\left\| r_1 - r_2\right\| }_{{\mathbf {C}}^{0}([0,t];{\mathbf {L}^1})} + {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} {\left\| \pi _1 - \pi _2\right\| }_{{\mathbf {C}}^{0}}\nonumber \\= & {} {\mathcal {O}}(1)\left( t {\left\| r_1 - r_2\right\| }_{{\mathbf {C}}^{0}([0,t];{\mathbf {L}^1})} + {\left\| \pi _1 - \pi _2\right\| }_{{\mathbf {C}}^{0}}\right) . \end{aligned}$$
(4.21)
To estimate \({\left\| \nabla _x\cdot (V_1^i \!- \!V_2^i)\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})}\), we first deal with \({\left| \nabla _x\cdot \left( V_1^i(\tau ,x) \!-\! V_2^i(\tau ,x)\right) \right| }\), which can be estimated by (v \('\)) and \((\varvec{{\mathcal {A}}})\)
$$\begin{aligned}&{\left| \nabla _x\cdot \left( V_1^i(\tau ,x) - V_2^i(\tau ,x)\right) \right| }\\&\quad \leqslant {\left| \nabla _x \cdot v^i\left( \tau , x, {\mathcal {A}}^i \left( r_1(\tau )\right) (x), \pi _1(\tau )\right) - \nabla _x \cdot v^i\left( \tau , x, {\mathcal {A}}^i \left( r_2(\tau )\right) (x), \pi _2(\tau )\right) \right| }\\&\qquad + \,\left| \nabla _A v^i\left( \tau , x, {\mathcal {A}}^i \left( r_1(\tau )\right) (x), \pi _1(\tau )\right) \nabla _x {\mathcal {A}}^i \left( r_1(\tau )\right) (x)\right. \\&\qquad \left. - \,\nabla _{A} v^i\left( \tau , x, {\mathcal {A}}^i \left( r_2(\tau )\right) (x), \pi _2(\tau )\right) \nabla _x {\mathcal {A}}^i\left( r_2(\tau )\right) (x) \right| \\&\quad \leqslant {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} {\left\| {\mathcal {A}}^i \left( r_1(\tau )\right) (x) - {\mathcal {A}}^i \left( r_2(\tau )\right) (x)\right\| }_{{\mathbb {R}}^d} + {\left| {\mathcal {C}}_K(x)\right| }\, {\left\| \pi _1(\tau ) - \pi _2(\tau )\right\| }_{{\mathbb {R}}^m}\\&\qquad + \,{\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} {\left\| \nabla _x {\mathcal {A}}^i (r_1(\tau ))(x) - \nabla _x {\mathcal {A}}^i (r_2(\tau ))(x) \right\| }_{{\mathbb {R}}^{2d}}\\&\qquad +\, {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} {\left\| \nabla _x {\mathcal {A}}^i\left( r_2(\tau )\right) (x)\right\| }_{{\mathbb {R}}^{d\times d}} \, {\left\| \nabla _x {\mathcal {A}}^i (r_1(\tau ))(x) - {\mathcal {A}}^i (r_2(\tau ))(x) \right\| }_{{\mathbb {R}}^{d\times d}}\\&\qquad + \,{\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} {\left\| \nabla _x {\mathcal {A}}^i\left( r_2(\tau )\right) (x)\right\| } _{{\mathbb {R}}^{d\times d}} \, {\left\| \pi _1(\tau ) - \pi _2(\tau )\right\| }_{{\mathbb {R}}^m} . \end{aligned}$$
Therefore, using \((\varvec{{\mathcal {A}}})\),
$$\begin{aligned}&{\left\| \nabla _x\cdot (V_1^i - V_2^i)\right\| }_{{\mathbf {L}^1}([0,t];{\mathbf {L}^1})}\nonumber \\&\quad \le L_A {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} \int _0^t {\left\| r_1(\tau ) - r_2(\tau )\right\| }_{{\mathbf {L}^1}} {{\mathrm {d}}{\tau }}+ {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^1}} \int _0^t {\left\| \pi _1(\tau ) - \pi _2(\tau )\right\| }_{{\mathbb {R}}^m} {{\mathrm {d}}{\tau }}\nonumber \\&\qquad + \,L_A {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} \int _0^t {\left\| r_1(\tau ) - r_2(\tau )\right\| }_{{\mathbf {L}^1}} {{\mathrm {d}}{\tau }}\nonumber \\&\qquad +\, L_A^2 {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} \int _0^t {\left\| r_2(\tau )\right\| }_{{\mathbf {L}^1}} {\left\| r_1(\tau ) - r_2(\tau )\right\| }_{{\mathbf {L}^1}} {{\mathrm {d}}{\tau }}\nonumber \\&\qquad +\, L_A {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} \int _0^t {\left\| r_2(\tau )\right\| }_{{\mathbf {L}^1}} {\left\| \pi _1(\tau ) - \pi _2(\tau )\right\| }_{{\mathbb {R}}^m} \, {{\mathrm {d}}{\tau }}\nonumber \\&\quad \leqslant L_A {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} \left( 2 + L_A\left( {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} + \Delta _\rho \right) \right) \int _0^t {\left\| r_1(\tau ) - r_2(\tau )\right\| }_{{\mathbf {L}^1}} {{\mathrm {d}}{\tau }}\nonumber \\&\qquad +\, \left( {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^1}} + L_A {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }}\left( {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} + \Delta _\rho \right) \right) \int _0^t {\left\| \pi _1(\tau ) - \pi _2(\tau )\right\| }_{{\mathbb {R}}^m} {{\mathrm {d}}{\tau }}\nonumber \\&\quad \le t \, {\mathcal {O}}(1)\left( {\left\| r_1 - r_2\right\| }_{{\mathbf {C}}^{0}([0,t];{\mathbf {L}^1})} + {\left\| \pi _1 - \pi _2\right\| }_{{\mathbf {C}}^{0}} \right) . \end{aligned}$$
(4.22)
Going back to the \(\rho \) components,
$$\begin{aligned}&{\left\| \rho _1^i(t) - \rho _2^i(t)\right\| }_{{\mathbf {L}^1}}\\&\quad \leqslant t {\mathcal {O}}(1)\left[ 1 + {\left\| \nabla _x \nabla _x\cdot V_1^i\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} \right] \! {\left\| V_1^i - V_2^i\right\| }_{{\mathbf {C}}^{0}([0,t];{\mathbf {L}^\infty })} \nonumber \\&\qquad +\, {\mathcal {O}}(1){\left\| \nabla _x\cdot (V_1^i - V_2^i)\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})}\\&\quad \leqslant t \, {\mathcal {O}}(1)\left( {\left\| r_1 - r_2\right\| }_{{\mathbf {C}}^{0}([0,t];{\mathbf {L}^1})} + {\left\| \pi _1 - \pi _2\right\| }_{{\mathbf {C}}^{0}} \right) . \end{aligned}$$
The above estimate ensures that for \(T\) sufficiently small, \({\mathcal {T}}\) is a contraction. Hence, it admits a unique fixed point \((\rho _*, p_*)\), defined on \([0,T]\).

\((\rho _*,p_*)\) is a solution to (1.1) on \([0,T]\). Writing that \((\rho _*,p_*)\) is a fixed point for \({\mathcal {T}}\) in (4.13) and (4.14) shows that \((\rho _*, p_*)\) solves (1.1) in the sense of Definition 2.1 on \([0,T]\).

Global uniqueness. Consider two solutions
$$\begin{aligned} \left( \rho _j, p_j\right) \in {\mathbf {C}}^{0} \left( [0,T_j]; {\mathbf {L}^1} ({\mathbb {R}}^d; [0,R]^n) \times {\mathbb {R}}^m \right) , \quad \text{ for } j = 1,2, \end{aligned}$$
to (1.1) in the sense of Definition 2.1, corresponding to the same initial datum \((\rho _o,p_o) \in ({\mathbf {L}^1} \cap {\mathbf {BV}}) ({\mathbb {R}}^d; [0,R]^n) \times {\mathbb {R}}^m\). Define
$$\begin{aligned} T^*= \sup \left\{ t \in [0,\min \{T_1,T_2\}]:\, \left( \rho _1, p_1\right) (s) = \left( \rho _2, p_2\right) (s)\text { for all } s \in [0,t]\right\} . \end{aligned}$$
Clearly, \(T^*\ge 0\) and \(\left( \rho _1, p_1\right) (T^*) = \left( \rho _2, p_2\right) (T^*)\).

Assume by contradiction that \(T^*< \min \{T_1,T_2\}\) and define \(\left( \rho ^*, p^*\right) = \left( \rho _1, p_1\right) (T^*) = \left( \rho _2, p_2\right) (T^*)\). The previous steps, which can be applied thanks to the bound (4.6), ensure the local existence to problem (1.1) with datum \((\rho ^*, p^*)\) assigned at time \(T^*\). Hence, \((\rho _1, p_1)(t) = (\rho _2, p_2)(t)\) on a full neighborhood of \(T^*\). This contradicts the assumption \(T^*< \min \{T_1, T_2\}\), proving global uniqueness.

\({\mathbf {BV}}\) estimate on \(\rho \) and \({\mathbf {L}^\infty }\) estimate on \(p\). Let \((\rho ,p)\) be the solution to (1.1). By (4.10) and (4.5),
$$\begin{aligned} {\left\| p(t)\right\| }_{{\mathbb {R}}^m} \leqslant \left( {\left\| p_o\right\| }_{{\mathbb {R}}^m} + \left( 1 + L_B \, {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} \right) \int _0^t C_F(s) {{\mathrm {d}}{s}} \right) \exp \int _0^t C_F(s){{\mathrm {d}}{s}} .\nonumber \\ \end{aligned}$$
(4.23)
Call \(K_t\) the closed ball in \({\mathbb {R}}^m\) with radius \( \left[ {\left\| p_o\right\| }_{{\mathbb {R}}^m} + \left( 1 + L_B \, {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} \right) \int _0^t C_F(s) {{\mathrm {d}}{s}} \right] e^{\int _0^t C_F(s){{\mathrm {d}}{s}}}\). By (4.6) and observing that \({\left\| \rho (t)\right\| }_{{\mathbf {L}^1}} = {\left\| \rho _o\right\| }_{{\mathbf {L}^1}}\) for every \(t > 0\)
$$\begin{aligned} \mathrm{TV}\left( \rho (t)\right)\leqslant & {} \mathrm{TV}\left( \rho _o\right) e^{\kappa _t\, t}\\&+\, d \, W_d {\left\| q\right\| }_{{\mathbf {L}^\infty }} t \left( {\left\| {\mathcal {C}}_{K_t}\right\| }_{{\mathbf {L}^1}} + 3 L_A {\left\| {\mathcal {C}}_{K_t}\right\| }_{{\mathbf {L}^\infty }} {\left\| \rho _o\right\| }_{{\mathbf {L}^1}}\right. \nonumber \\&\left. +\, L_A^2 {\left\| {\mathcal {C}}_{K_t}\right\| }_{{\mathbf {L}^\infty }} \left( {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} \right) ^2 \right) e^{\kappa _t t}, \end{aligned}$$
where, by (4.20),
$$\begin{aligned} \kappa _t = (2d+1) {\left\| {\mathcal {C}}_{K_t}\right\| }_{{\mathbf {L}^\infty }} {\left\| q'\right\| }_{{\mathbf {L}^\infty }} (1+ L_A {\left\| \rho _o\right\| }_{{\mathbf {L}^1}}) . \end{aligned}$$
Hence, \(\rho (t) \in {\mathbf {BV}}({\mathbb {R}}^d; [0,R]^n)\) as long as \(\rho \) is defined.
\(\rho \) is Lipschitz continuous in time. Let \((\rho ,p)\) be the solution to (1.1). Fix \(t_1,t_2\) with \(t_1 < t_2\) inside the time interval where \((\rho ,p)\) is defined. Use (4.7), (4.16) and (v \('\)) to obtain
$$\begin{aligned} {\left\| \rho (t_2) - \rho (t_1)\right\| }_{{\mathbf {L}^1}}\leqslant & {} \displaystyle (t_2-t_1) \, {\left\| q\right\| }_{{\mathbf {L}^\infty }} \left( {\left\| {\mathcal {C}}_{K_t}\right\| }_{{\mathbf {L}^1}} + L_A {\left\| {\mathcal {C}}_{K_t}\right\| }_{{\mathbf {L}^\infty }} {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} \right) \\&\displaystyle +\, (t_2 - t_1) {\left\| {\mathcal {C}}_{K_t}\right\| }_{{\mathbf {L}^\infty }} {\left\| q'\right\| }_{{\mathbf {L}^\infty }} \, \sup _{\tau \in [0, t_2]} \mathrm{TV}\left( \rho (\tau )\right) . \end{aligned}$$
This estimate, together with the \({\mathbf {BV}}\) bound above, ensures that on any bounded time interval on which it is defined, \(\rho \) is Lipschitz continuous in time.
\(p\) is uniformly continuous in time. Let \((\rho ,p)\) be the solution to (1.1). Fix \(t_1,t_2\) with \(t_1 < t_2\) inside the time interval where \((\rho ,p)\) is defined. By (F \('\)), (4.23), \((\varvec{{\mathcal {B}}})\) and (4.5), we have
$$\begin{aligned}&{\left\| p (t_2) - p (t_1)\right\| }_{{\mathbb {R}}^m}\\&\quad \leqslant \int _{t_1}^{t_2} {\left\| F\left( \tau , p (\tau ), \left( {\mathcal {B}}\left( \rho (\tau )\right) \right) \left( p (\tau )\right) \right) \right\| }_{{\mathbb {R}}^m} {{\mathrm {d}}{\tau }}\\&\quad \leqslant \int _{t_1}^{t_2} C_F (\tau ) \left( 1 + {\left\| p (\tau )\right\| } + L_B {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} \right) {{\mathrm {d}}{\tau }}\\&\quad \leqslant \int _{t_1}^{t_2} C_F (\tau ) {{\mathrm {d}}{\tau }}\left[ 1 + \left( {\left\| p_o\right\| }_{{\mathbb {R}}^m} + \left( 1 + L_B \, {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} \right) \int _0^{t_2} C_F(s) {{\mathrm {d}}{s}} \right) e^{\int _0^{t_2} C_F(s){{\mathrm {d}}{s}}} \right. \nonumber \\&\qquad \left. +\, L_B {\left\| \rho _o\right\| }_{{\mathbf {L}^1}} \right] , \end{aligned}$$
which shows the uniform continuity of \(p\) on any bounded time interval.
Global existence. Fix the initial datum \((\rho _o,p_o) \in ({\mathbf {L}^1} \cap {\mathbf {BV}}) ({\mathbb {R}}^d; [0,R]^n) \times {\mathbb {R}}^m\). Define
$$\begin{aligned} T_* = \sup \left\{ T > 0 :\begin{array}{l} \exists (\rho , p) \in {\mathbf {C}}^{0} \left( [0,T]; {\mathbf {L}^1} ({\mathbb {R}}^d; [0,R]^n) \times {\mathbb {R}}^m \right) \\ \left( \rho , p\right) \text{ solves } \text{(1.1) } \text{ according } \text{ to } \text{ Definition } \text{(2.1) }\\ \text {with~} \left( \rho , p\right) (0) = \left( \rho _o, p_o\right) \end{array} \right\} . \end{aligned}$$
Assume by contradiction that \(T_*< + \infty \). Then, by the existence and uniqueness proved above, there exists a solution \((\rho _*,p_*)\) to (1.1) with \((\rho _*,p_*)(0) = (\rho _o,p_o)\) which is defined on \([0, T_*[\). By the previous steps, the map \(t \rightarrow (\rho _*,p_*) (t)\) is uniformly continuous on \([0, T_*[\); hence, it can be uniquely extended by continuity to \([0,T_*]\). Call \((\bar{\rho },\bar{p}) = (\rho _*,p_*) (T_*)\). The Cauchy problem consisting of (1.1) with initial datum \((\bar{\rho }, \bar{p})\) assigned at time \(T_*\) still satisfies all conditions to have a unique solution defined also on a right neighborhood of \(T_*\), which contradicts the choice of \(T_*\).
Continuous dependence from the initial datum. Fix a positive \(T\). For \(j = 1, 2\), choose \((\rho _o^j,p_o^j) \in ({\mathbf {L}^1}\cap {\mathbf {BV}}) ({\mathbb {R}}^d;[0,R]^m)\) and call \((\rho _j,p_j)\) the corresponding solution as constructed above. For any \(t \in [0,T]\), by (4.11)
$$\begin{aligned} {\left\| p_1(t) - p_2 (t)\right\| }_{{\mathbb {R}}^m}\leqslant & {} \left( {\left\| p_o^1-p_o^2\right\| }_{{\mathbb {R}}^m} + L_B \, {\left\| \rho _1 - \rho _2\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} \right) e^{L_F t(1 + L_B {\left\| \rho _o^1\right\| }_{{\mathbf {L}^1}})}\nonumber \\\leqslant & {} \left( {\left\| p_o^1-p_o^2\right\| }_{{\mathbb {R}}^m} + {\mathcal {O}}(1){\left\| \rho _1 - \rho _2\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} \right) e^{{\mathcal {O}}(1)t(1+{\left\| \rho _o^1\right\| }_{{\mathbf {L}^1}})}.\nonumber \\ \end{aligned}$$
(4.24)
For \(j = 1, 2\) and \(i= 1, \ldots , n\), define \(V^i_j = v_j^i\left( t,x,\left( {\mathcal {A}}^i (\rho _j \left( t)\right) \right) (x), p_j (t)\right) \) and using (4.22), (4.18), (v \('\)) and \((\varvec{{\mathcal {A}}})\), compute preliminary the following terms
$$\begin{aligned} {\left\| \nabla _x\cdot (V_1^i - V_2^i)\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})}\leqslant & {} {\mathcal {O}}(1)\left( 1+{\left\| \rho _o^2\right\| }_{{\mathbf {L}^1}}\right) \left( {\left\| \rho _1-\rho _2\right\| }_{{\mathbf {L}^1}([0,t],{\mathbf {L}^1})}\right. \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \left. +\, {\left\| p_1-p_1\right\| }_{{\mathbf {L}^1}} \right) ,\nonumber \\ {\left\| \nabla _x \nabla _x\cdot V_1^i\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})}\leqslant & {} t \, {\mathcal {O}}(1)\left( 1 + {\left\| \rho _o^1\right\| }_{{\mathbf {L}^1}} + \left( {\left\| \rho _o^1\right\| }_{{\mathbf {L}^1}} \right) ^2 \right) ,\nonumber \\ {\left\| V_1^i - V_2^i\right\| }_{{\mathbf {L}^1}([0,t];{\mathbf {L}^\infty })}\leqslant & {} {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} \int _0^t \left[ {\left\| {\mathcal {A}}^i\left( \rho _1 (\tau )\right) - {\mathcal {A}}^i\left( \rho _2 (\tau )\right) \right\| }_{{\mathbf {L}^\infty }} \right. \nonumber \\&\left. +\, {\left\| p_1 (\tau ) - p_2 (\tau )\right\| }_{{\mathbb {R}}^m} \right] {{\mathrm {d}}{\tau }}\nonumber \\\leqslant & {} {\left\| {\mathcal {C}}_K\right\| }_{{\mathbf {L}^\infty }} \int _0^t \left( L_A {\left\| \rho \right\| }_1 (\tau ) \right. \nonumber \\&\left. -\, \rho _2 (\tau )_{{\mathbf {L}^1}} + {\left\| p_1 (\tau ) - p_2 (\tau )\right\| }_{{\mathbb {R}}^m} \right) {{\mathrm {d}}{\tau }}\nonumber \\= & {} {\mathcal {O}}(1)\left( {\left\| \rho _1 - \rho _2\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} + {\left\| p_1 - p_2\right\| }_{{\mathbf {L}^1}} \right) .\nonumber \\ \end{aligned}$$
(4.25)
By Lemma 4.2, we get
$$\begin{aligned}&{\left\| \rho _1^i(t) - \rho _2^i(t)\right\| }_{{\mathbf {L}^1}}\\&\quad \leqslant {\left\| \rho _o^{1,i} - \rho _o^{2,i}\right\| }_{{\mathbf {L}^1}} e^{\kappa t} + {\left\| q\right\| }_{{\mathbf {L}^\infty }} {\left\| \nabla _x\cdot (V_1^i - V_2^i)\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} e^{\kappa t}\\&\qquad +\, \frac{\kappa _o e^{\kappa _o t} - \kappa e^{\kappa t}}{\kappa _o - \kappa } \! \left[ \mathrm{TV}(\rho _o^{1,i}) \right. \nonumber \\&\qquad \qquad \qquad \qquad \left. + \, d\, W_d {\left\| q\right\| }_{{\mathbf {L}^\infty }} \! {\left\| \nabla _x \nabla _x\cdot V_1^i\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} \right] \!\! {\left\| q'\right\| }_{{\mathbf {L}^\infty }} \! {\left\| V_1^i - V_2^i\right\| }_{{\mathbf {L}^1}([0,t];{\mathbf {L}^\infty })}\\&\quad = \left( 1+t{\mathcal {O}}(1)\right) {\left\| \rho _o^{1,i} - \rho _o^{2,i}\right\| }_{{\mathbf {L}^1}}\\&\qquad +\, {\mathcal {O}}(1)\left( {\left\| \nabla _x\cdot (V_1^i - V_2^i)\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} \right. \nonumber \\&\qquad \qquad \qquad \qquad \left. + \left[ 1 + {\left\| \nabla _x \nabla _x\cdot V_1^i\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} \right] {\left\| V_1^i - V_2^i\right\| }_{{\mathbf {L}^1}([0,t];{\mathbf {L}^\infty })} \right) \\&\quad = \left( 1+t{\mathcal {O}}(1)\right) {\left\| \rho _o^{1,i} - \rho _o^{2,i}\right\| }_{{\mathbf {L}^1}}\\&\qquad +\, {\mathcal {O}}(1)\left( 1 + {\left\| \rho _o^1\right\| }_{{\mathbf {L}^1}} + \left( {\left\| \rho _o^1\right\| }_{{\mathbf {L}^1}} \right) ^2 \right) \left( {\left\| \rho _1 - \rho _2\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} + {\left\| p_1 - p_2\right\| }_{{\mathbf {L}^1}} \right) . \end{aligned}$$
A further application of Lemma 4.1, using also (4.24), allows to conclude the proof of 3. in Theorem 4.4.
Stability with respect to \(q\). Fix a positive \(T\). For \(j = 1,2\), let \((\rho _j,p_j)\) solve (1.1) with \(q\) replaced by \(q_j\) and with the initial datum \((\rho _o,p_o)\) assigned at time \(t=0\). For \(j = 1, 2\) and \(i= 1, \ldots , n\), define \(V^i_j = v^i\left( t,x,\left( {\mathcal {A}}^i (\rho _j \left( t)\right) \right) (x), p_j (t)\right) \). Using (4.19), (4.9), (4.25), (4.18), (4.22), (4.16) compute preliminary
$$\begin{aligned} \kappa _o= & {} {\mathcal {O}}(1)\, \max _{j=1,2}{\left\| q_j'\right\| }_{{\mathbf {L}^\infty }},\\ \kappa= & {} {\mathcal {O}}(1)\, \max _{j=1,2}{\left\| q_j'\right\| }_{{\mathbf {L}^\infty }},\\ {\left\| V^i_1 \!- \!V^i_2\right\| }_{{\mathbf {L}^1}([0,t];{\mathbf {L}^\infty })}= & {} {\mathcal {O}}(1)\left( {\left\| \rho _1 \!- \!\rho _2\right\| }_{{\mathbf {L}^1}([0,t];{\mathbf {L}^1})} \!+ \!{\left\| p_1 \!- \!p_2\right\| }_{{\mathbf {L}^1}} \right) , \\ {\left\| \nabla _x \nabla _x\cdot V^i_1(\tau , x)\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})}= & {} t \, {\mathcal {O}}(1),\\ {\left\| \nabla _x\cdot \left( V^i_1(\tau ,x) \!- \!V^i_2(\tau ,x)\right) \right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})}= & {} {\mathcal {O}}(1)\left( {\left\| \rho _1\!-\!\rho _2\right\| }_{{\mathbf {L}^1}([0,t];{\mathbf {L}^1})} \!+\! t \,{\left\| p_1 \!- \!p_2\right\| }_{{\mathbf {C}}^{0}} \right) ,\\ {\left\| \nabla _x\cdot V^i_2(\tau ,x)\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})}= & {} t \, {\mathcal {O}}(1). \end{aligned}$$
Apply Lemma 4.3 to obtain
$$\begin{aligned} {\left\| p_1 (t) - p_2 (t)\right\| }_{{\mathbb {R}}^m}\leqslant & {} {\mathcal {O}}(1)\, {\left\| \rho _1 - \rho _2\right\| }_{{\mathbf {L}^1}([0,t];{\mathbf {L}^1})} . \end{aligned}$$
Similarly, using Lemma 4.2,
$$\begin{aligned}&{\left\| \rho _1(t) - \rho _2(t)\right\| }_{{\mathbf {L}^1}}\\&\quad = {\mathcal {O}}(1)\left( {\left\| q_1 - q_2\right\| }_{{\mathbf {W}^{1,\infty }}} + {\left\| V_1 - V_2\right\| }_{{\mathbf {L}^1}([0,t];{\mathbf {L}^\infty })} + {\left\| \nabla _x\cdot (V_1 - V_2)\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} \right) \\&\quad = {\mathcal {O}}(1)\left( {\left\| q_1 - q_2\right\| }_{{\mathbf {W}^{1,\infty }}} + {\left\| \rho _1-\rho _2\right\| }_{{\mathbf {L}^1}([0,t];{\mathbf {L}^1})} + t \, {\left\| p_1 - p_2\right\| }_{{\mathbf {C}}^{0}} \right) . \end{aligned}$$
A final application of Lemma 4.1 completes the proof of this part.
Stability with respect to \(v\). Fix a positive \(T\). For \(j = 1,2\), let \((\rho _j,p_j)\) solve (1.1) with \(v\) replaced by \(v_j\) and with the initial datum \((\rho _o,p_o)\) assigned at time \(t=0\). For \(i= 1, \ldots , n\), define \(V^i_j = v_j^i\left( t,x,\left( {\mathcal {A}}^i (\rho _j \left( t)\right) \right) (x), p_j (t)\right) \). Compute preliminary, using (4.9), (v \('\)), \((\varvec{{\mathcal {A}}})\)
$$\begin{aligned} \kappa _o= & {} {\mathcal {O}}(1),\\ \kappa= & {} {\mathcal {O}}(1),\\ {\left\| V^i_1 - V^i_2\right\| }_{{\mathbf {L}^1}([0,t];{\mathbf {L}^\infty })}= & {} t {\left\| v_1-v_2\right\| }_{{\mathbf {L}^\infty }}\\&+ {\mathcal {O}}(1)\left( {\left\| \rho _1 - \rho _2\right\| }_{{\mathbf {L}^1}([0,t];{\mathbf {L}^1})} + {\left\| p_1 - p_2\right\| }_{{\mathbf {L}^1}} \right) , \\ {\left\| \nabla _x \nabla _x\cdot V^i_1(\tau , x)\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})}= & {} t \, {\mathcal {O}}(1),\\ {\left\| \nabla _x\cdot \left( V^i_1(\tau ,x) - V^i_2(\tau ,x)\right) \right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})}= & {} t {\left\| \nabla _x\cdot (v_1-v_2)\right\| }_{{\mathbf {L}^\infty }}\\&+ {\mathcal {O}}(1)\left( {\left\| \rho _1-\rho _2\right\| }_{{\mathbf {L}^1}([0,t];{\mathbf {L}^1})} + t \,{\left\| p_1 - p_2\right\| }_{{\mathbf {C}}^{0}} \right) , \end{aligned}$$
so that
$$\begin{aligned}&{\left\| \rho _1(t) - \rho _2(t)\right\| }_{{\mathbf {L}^1}}\\&\quad = {\mathcal {O}}(1)\frac{\kappa _o e^{\kappa _o t} - \kappa e^{\kappa t}}{\kappa _o - \kappa } \left[ 1 + {\left\| \nabla _x \nabla _x\cdot V_1\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} \right] {\left\| V_1 - V_2\right\| }_{{\mathbf {L}^1}([0,t];{\mathbf {L}^\infty })}\\&\qquad +\, {\mathcal {O}}(1){\left\| \nabla _x\cdot (V_1 - V_2)\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})}\\&\quad = {\mathcal {O}}(1){\left\| v_1 - v_2\right\| }_{{\mathbf {W}^{1,\infty }}} + {\mathcal {O}}(1)\left( {\left\| \rho _1-\rho _2\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} + {\left\| p_1 - p_2\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbb {R}}^m)} \right) , \end{aligned}$$
and similar to the previous step, applying Lemmas 4.3 and 4.2
$$\begin{aligned} {\left\| p_1 (t) - p_2 (t)\right\| }_{{\mathbb {R}}^m}\leqslant & {} {\mathcal {O}}(1){\left\| \rho _1 - \rho _2\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})},\\ {\left\| \rho _1(t) - \rho _2(t)\right\| }_{{\mathbf {L}^1}}\leqslant & {} {\mathcal {O}}(1){\left\| v_1 - v_2\right\| }_{{\mathbf {W}^{1,\infty }}} + {\mathcal {O}}(1){\left\| \rho _1-\rho _2\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})}, \end{aligned}$$
the proof of the stability with respect to \(v\) is completed.
Stability with respect to \(F\). Apply (4.11) in Lemmas 4.3 and 4.2 to obtain
$$\begin{aligned} {\left\| p_1 (t) - p_2 (t)\right\| }_{{\mathbb {R}}^m}\leqslant & {} {\mathcal {O}}(1)\left( t {\left\| F_1 - F_2\right\| }_{{\mathbf {L}^\infty }} + {\left\| \rho _1 - \rho _2\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} \right) ,\\ {\left\| \rho _1 (t) - \rho _2 (t)\right\| }_{{\mathbf {L}^1}}\leqslant & {} {\mathcal {O}}(1)\left( {\left\| \rho _1 - \rho _2\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbf {L}^1})} + {\left\| p_1 - p_2\right\| }_{{\mathbf {L}^1} ([0,t];{\mathbb {R}}^m)} \right) , \end{aligned}$$
and a further application of Lemma 4.1 completes the proof. \(\square \)

Proof of Theorem 2.2

Define \(K = \overline{B (\mathrm{spt}\rho _o, {\left\| v\right\| }_{{\mathbf {L}^\infty }} T)}\). Note that for any function \(\rho \in {\mathbf {L}^1} ({\mathbb {R}}^d; [0,R]^n)\) with \(\mathrm{spt}\rho \subseteq K\), by \((\varvec{{\mathcal {A}}})\)
$$\begin{aligned} {\left\| {\mathcal {A}} (\rho )\right\| }_{{\mathbf {L}^\infty }({\mathbb {R}}^d; {\mathbb {R}}^d)} \leqslant L_A {\left\| \rho \right\| }_{{\mathbf {L}^1} ({\mathbb {R}}^d; {\mathbb {R}}^n)} \leqslant L_A R \, {\mathcal {L}} ^d(K) . \end{aligned}$$
By (4.10), for all \(t \in [0,T]\), we have \({\left\| p (t)\right\| }_{{\mathbb {R}}^m} \leqslant P\) where
$$\begin{aligned} P = \left( {\left\| p_o\right\| }_{{\mathbb {R}}^m} + \left( 1+L_B \, R \, {\mathcal {L}}^d(K) \right) {\left\| C_F\right\| }_{{\mathbf {L}^1} ([0,T]; {\mathbb {R}})} \right) \exp {\left\| C_F\right\| }_{{\mathbf {L}^1} ([0,T]; {\mathbb {R}})} . \end{aligned}$$
Let \(\chi _x \in {\mathbf {C}}_c^{\infty }({\mathbb {R}}^d; [0,1])\) be such that \(\chi _x (x) =1\) for all \(x \in K\). Similarly, let \(\chi _A \in {\mathbf {C}}_c^{\infty }({\mathbb {R}}^d; [0,1])\) such that \(\chi _A (A) = 1\) for all \(A \in \overline{B \left( 0, L_A \, R \, {\mathcal {L}}^d(K)\right) }\) and let \(\chi _p \in {\mathbf {C}}_c^{\infty }({\mathbb {R}}^m; [0,1])\) be such that \(\chi _p (p) = 1\) for all \(p \in \overline{B (0,P)}\). Then, (v \('\)) is satisfied with
$$\begin{aligned} \tilde{v} (t,x,A,p)= & {} \chi _x (x) \, \chi _A (A) \, \chi _p (p) \, v (t,x,A,p),\\ {\mathcal {C}}_K (x)= & {} \chi _x (x) \, {\left\| v\right\| }_{{\mathbf {C}}^{2} ([0,T] \cup K \times \overline{B (0, L_A R {\mathcal {L}}^d (K))} \times \overline{B (0, P)})} . \end{aligned}$$
By 1. in Definition 2.1, the solution \(t \rightarrow \left( \rho (t), p (t)\right) \) to (1.1) as constructed in Theorem 4.4 is such that \(\mathrm{spt}\rho (t) \subseteq K\) for all \(t \in [0,T]\). Hence, \(t \rightarrow \left( \rho (t), p (t)\right) \) also solves (1.1). \(\square \)

4.2 Technical Details Referring to Sect. 3.1

In Eq. (3.1), we choose the functionsWith reference to (1.1), we write \(F \!=\! [F_1, F_2, F_3, F_4]^T\), \(\left( {{\mathcal {B}}}(\rho )\right) (p) \!= \!\left( [\rho ^1 , \rho ^2 ]^T *\bar{\eta }\right) (p)\) and
$$\begin{aligned} w^i(\xi ) = \varepsilon _i \, \frac{\xi }{ \sqrt{1 + {\left\| \xi \right\| }_{{\mathbb {R}}^2}^4}} \quad \text{ and } \quad {\mathcal {A}}^i(\rho ) = \sum _{j=1}^2 \varepsilon _{ij} \frac{\nabla _x (\rho ^j*\eta )}{\sqrt{1+{\left\| \nabla _x (\rho ^j*\eta )\right\| }_{{\mathbb {R}}^2}^2}} .\nonumber \\ \end{aligned}$$
(4.27)
Here, \(\varepsilon _i\) and \(\varepsilon _{ij}\) are positive constants and \(\eta \in {\mathbf {C}}_c^{2}({\mathbb {R}}^2;{\mathbb {R}}^+)\). Moreover, \(w^i (p^i-x)\) describes the interaction between the member at \(x\) of the \(i\)th population and his/her guide at \(p^i\). The two addends in the nonlocal operator \({\mathcal {A}}^i\) model the interaction among members of the same population, the \(\varepsilon _{ii}\) term, and between the two populations, the \(\varepsilon _{ij}\) term.

We assume that the trajectory \(p^i\) of the \(i\)th guide is constrained to the circumference of radius \(r^i\), centered at the point \(c^i = [c^i_1, c^i_2]^T \in {\mathbb {R}}^2\), and its speed depends on an average density of tourists around its position.

Proposition 4.5

Assume \(\eta , \bar{\eta } \in {\mathbf {C}}_c^{2}({\mathbb {R}}^2; {\mathbb {R}}^+)\). Then, the functions in (4.26)–(4.27) satisfy (v.1), (F), \((\varvec{{\mathcal {A}}})\), \((\varvec{{\mathcal {B}}})\) and (q). In particular, Theorem 2.2 applies to (3.1)–(3.2)–(4.26).

Proof

The verification of (v), (F) and (q), with \(R=1\), is straightforward. Assumption \((\varvec{{\mathcal {A}}})\) holds, since the map \(\nu :{\mathbb {R}}^2 \rightarrow {\mathbb {R}}^2\) defined by \(\nu (x) = \left. x / \sqrt{1+{\left\| x\right\| }^2_{{\mathbb {R}}^2}} \right. \) is of class \({\mathbf {C}}^{3} ({\mathbb {R}}^2; {\mathbb {R}}^2)\) and \({\left\| \nu \right\| }_{{\mathbf {C}}^{3}} < +\infty \). By the standard properties of the convolution product, we deduce that
$$\begin{aligned}&{\left\| {\mathcal {B}}(\rho _1) - {\mathcal {B}}(\rho _2)\right\| }_{{\mathbf {W}^{1,\infty }}}\\&\quad ={\left\| {\mathcal {B}}(\rho _1) - {\mathcal {B}}(\rho _2)\right\| }_{{\mathbf {L}^\infty }} + {\left\| D_p{\mathcal {B}}(\rho _1) - D_p{\mathcal {B}}(\rho _2)\right\| }_{{\mathbf {L}^\infty }}\\&\quad = \sup _{(p^1, p^2) \in {\mathbb {R}}^4} {\left\| \left( \rho ^1_1 *\bar{\eta }(p^1), \rho ^2_1 *\bar{\eta }(p^2)\right) \right\| }_{{\mathbb {R}}^2} \nonumber \\&\qquad + \sup _{(p^1, p^2) \in {\mathbb {R}}^4} {\left\| \left( \rho ^1_1 *D\bar{\eta }(p^1), \rho ^2_1 *D\bar{\eta }(p^2)\right) \right\| }_{{\mathbb {R}}^4}\\&\quad \leqslant {\left\| \bar{\eta }\right\| }_{{\mathbf {W}^{1,\infty }}} {\left\| \rho _1 - \rho _2\right\| }_{{\mathbf {L}^1}}, \end{aligned}$$
which implies \((\varvec{{\mathcal {B}}})\), concluding the proof. \(\square \)
In the simulation presented in Fig. 1, we use the following parametersand the functions

4.3 Technical Details Referring to Sect. 3.2

In (3.3)–(3.4), we set To describe the interaction of cars with pedestrians, we introduce the smooth function \(\beta _{\alpha _1, \alpha _2} \in {\mathbf {C}}^{\infty }({\mathbb {R}}; [0,1])\) with \(\alpha _1 < \alpha _2\) as
The factor \(w^i\), \(i=1,2\), models the slowing down of pedestrians and is defined asHere, \(\hat{\eta }_i\) describes the region considered by each pedestrian in reacting to cars. For instance, cars behind a pedestrian are ignored when at a distance greater than \(r_{vb}\), while cars in front of the pedestrian are considered up to a distance \(r_v\). The term \(\prod _{l=1}^3 \beta _{r_i,r_a}\) stops the pedestrians if a car is located ahead. On the other hand, the term \(\beta _{h_R, h_R + \varepsilon _\gamma }\) allows that pedestrians have priority over cars when crossing the road on the crosswalk.
The convolution kernel in the nonlocal operators \({\mathcal {A}}^1\) and \({\mathcal {A}}^2\) iswith the parameters \(\varepsilon _{11} = 0.1\), \(\varepsilon _{22} = 0.1\), \(\varepsilon _{12} = 0.7\), \(\varepsilon _{21} = 0.7\).
The \(N=3\) cars move according to (3.4) with
$$\begin{aligned} v_L (t)=1 , \qquad g (B) = \beta _{r_j,r_b} (B), \quad \text{ and } \quad u (\xi ) = 1 - \beta _{H, 10H} (\xi )^{K}, \end{aligned}$$
and \(r_j = 0.125\), \(r_b = 0.5\), \(H = 0.167\), \(K = 50\). Drivers slow down if pedestrians are ahead on the road, according to the following convolution kernel in the nonlocal operator \({\mathcal {B}}\) in (4.31): All visual circles are modeled so that pedestrians and cars give more relevance to what they have in front, rather than behind. The visual radius of drivers is such that they react only to people on the road. On the other hand, pedestrians see cars located also far ahead, i.e., their visual radius is large enough that they can stop safely and smoothly.

Proposition 4.6

Assume \(\eta , \bar{\eta } \in {\mathbf {C}}_c^{2}({\mathbb {R}}^2; {\mathbb {R}}^+)\), \(w^i \in {\mathbf {C}}^{2}({\mathbb {R}}^2 \times {\mathbb {R}}^{2N}; {\mathbb {R}}^+)\), \(V^i \in {\mathbf {C}}^{2}\left( {\mathbb {R}}^2; {\mathbb {R}}^2\right) \), \(v_L \in {\mathbf {L}^1} ({\mathbb {R}}^+; {\mathbb {R}}^+)\) and \(g,u \in {\mathbf {C}}^{2}\left( {\mathbb {R}};[0,1]\right) \). Then, the functions defined in (4.30)–(4.31) satisfy (v.1), (F), \((\varvec{{\mathcal {A}}})\), \((\varvec{{\mathcal {B}}})\) and (q). In particular, Theorem 2.2 applies to (3.3)–(3.4).

Proof

Assumption (v) is immediate. The verification of (F) and (q), with \(R=1\), is straightforward. Assumption \((\varvec{{\mathcal {A}}})\) follows in the same way as in the proof of Proposition 4.5. Standard properties of the convolution product permit to verify assumption \((\varvec{{\mathcal {B}}})\). \(\square \)

4.4 Technical Details Referring to Sect. 3.3

In (3.5)–(3.6), we set In (4.35), the operator \({\mathcal {A}}^i\) is composed by two terms describing the attraction, respectively repulsion, between members of the same, respectively different, group. Here, we introduce a preferred density \(\bar{\rho } \in [0,1]\). If the density of one group is lower than \(\bar{\rho }\), then members of that group tend to move toward each other. On the contrary, if the density is bigger than \(\bar{\rho }\), then they tend to disperse. Moreover, the operator \({\mathcal {A}}^i\) also models the fact that one group of hooligans aims at attacking the other group as soon as it feels to be stronger. On the contrary, hooligans of a faction try to avoid the adversaries in case they are less represented.

Proposition 4.7

Let \(N \in {\mathbb {N}}{\setminus } \{0\}\). Assume \(\eta , \bar{\eta } \in {\mathbf {C}}_c^{2}({\mathbb {R}}^2; {\mathbb {R}}^+)\), \(w^i \in {\mathbf {C}}^{2}({\mathbb {R}}^2 \times {\mathbb {R}}^{2N}; {\mathbb {R}}^2)\), \(I_k \in ({\mathbf {C}}^{0} \cap {\mathbf {L}^\infty })({\mathbb {R}}^{2N}; {\mathbb {R}}^2)\). Then, the functions defined in (4.34)–(4.35) satisfy (v.1), (F), \((\varvec{{\mathcal {A}}})\), \((\varvec{{\mathcal {B}}})\) and (q). In particular, Theorem 2.2 applies to (4.34)–(4.35).

Proof

The proofs of (v), (F) and (q) are immediate, with \(R=1\). To prove \((\varvec{{\mathcal {A}}})\), note that the real-valued function \(\varphi (\xi ) = \left. \xi / \sqrt{1+\xi ^2}\right. \) is globally Lipschitz continuous and the map \((r_1,r_2) \rightarrow \varphi (r_1 \, r_2)\) is Lipschitz continuous for \((r_1,r_2) \in [0,1]^2\). The standard properties of the convolution also ensure the Lipschitz continuity and the boundedness of the maps \(\rho \rightarrow \eta * (\rho - \bar{\rho })\) and \(\rho \rightarrow \nabla _x (\eta *\rho )\) in the required norms, proving \((\varvec{{\mathcal {A}}})\). The proof of \((\varvec{{\mathcal {B}}})\) is entirely analogous. \(\square \)

As a specific example, we consider the computational domain \([0,1]^2\), discretized by a triangular mesh with a minimal inner radius \(h=0.0045\) and the following parameters for the strength of the nonlocal interaction terms
$$\begin{aligned} \varepsilon _{11} = 0.5, \qquad \varepsilon _{22} = 0.5 ,\qquad \varepsilon _{12} = 0.5, \qquad \varepsilon _{21} = 0.5 ,\qquad \bar{\varepsilon }_1 = 0.4, \qquad \bar{\rho } = 0.5. \end{aligned}$$
The mutual repulsion of the policemen and their influence onto the hooligans is described byThe inner interactions of the different groups are modeled with the following smoothing kernelsThe parameters \(\varepsilon _{ij},\varepsilon _i\) and \(\bar{\varepsilon }_i\) are chosen so that the interactions within hooligans and within policemen are stronger than the interactions between hooligans and policemen. Additionally, the visual radii are chosen so that hooligans see policemen at a distance higher than that at which they see other hooligans, since they want to avoid any contact with the officers. On the contrary, policemen see hooligans from a distance higher than that at which they see other policemen, because officers aim at separating the two groups.

Notes

Acknowledgments

This work was partially supported by the INDAM–GNAMPA project Conservation Laws: Theory and Applications, by the Graduiertenkolleg 1932 “Stochastic Models for Innovations in the Engineering Sciences” and by the Deutsche Forschungsgemeinschaft (DFG) project “Stochastic Models for Innovations in the Engineering Sciences”.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Raul Borsche
    • 1
  • Rinaldo M. Colombo
    • 2
    Email author
  • Mauro Garavello
    • 3
  • Anne Meurer
    • 1
  1. 1.Fachbereich MathematikTechnische Universität KaiserslauternKaiserslauternGermany
  2. 2.Unità INdAM, c/o DIIUniversità degli studi di BresciaBresciaItaly
  3. 3.Dipartimento di Matematica e ApplicazioniUniversità di Milano BicoccaMilanItaly

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