N El Khatib
email: nader.elkhatib@lau.edu.lb

N Forcadel

M Zaydan

HOMOGENIZATION OF A MICROSCOPIC PEDESTRIANS MODEL ON A CONVERGENT JUNCTION

Keywords: Mathematics Subject Classification. 35D40, 90B20, 35B27, 35F20, 45K05 Specified homogenization, Hamilton-Jacobi equations, pedestrians, non-local operators, Slepčev formulation, viscosity solutions, microscopic model on junction

In this paper, we establish a rigorous connection between a microscopic and a macroscopic pedestrians model on a convergent junction. At the microscopic level, we consider a "follow the leader" model far from the junction point and we assume that a rule to enter the junction point is imposed. At the macroscopic level, we obtain the Hamilton-Jacobi equation with a flux limiter condition at x = 0 introduced in Imbert and Monneau [Ann. Sci. l' École Normale Supér. 50 (2017) 357-414]. To obtain our result, we inject using the "cumulative distribution functions" the microscopic model into a non-local PDE. Then, we show that the viscosity solution of the non-local PDE converges locally uniformly towards the solution of the macroscopic one.

Introduction

In the literature, several papers proposed mathematical models of pedestrian flows based on partial differential equations, see [7-9, 26, 31, 34]. Macroscopic pedestrian models are effective for describing pedestrian flows at large scales but from the modeling point of view, since the dynamics of each pedestrian can't be described individually, they are hardly justifiable and incapable to predict all possible situations (for example panic during an evacuation). At the opposite, microscopic models are strong from the modeling point of view but complicated to implement at large scales. Among the works modeling pedestrians at the micro scale, we refer to [START_REF] Gazis | Nonlinear follow-the-leader models of traffic flow[END_REF][START_REF] Helbing | Social force model for pedestrian dynamics[END_REF][START_REF] Helbing | Generalized force model of traffic dynamics[END_REF][START_REF] Hoogendoorn | Simulation of pedestrian flows by optimal control and differential games[END_REF][START_REF] Rosini | Microscopic and macroscopic models for vehicular and pedestrian flows, in Order, Disorder and Criticality: Advanced Problems of Phase Transition Theory[END_REF].

A rigorous way to justify these macroscopic models is to establish a connection between them and microscopic ones, we refer for example to [START_REF] Aw | Derivation of continuum traffic flow models from microscopic follow-the-leader models[END_REF][START_REF] Di Francesco | Follow-the-leader approximations of macroscopic models for vehicular and pedestrian flows[END_REF][START_REF] Francesco | Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit[END_REF][START_REF] Helbing | From microscopic to macroscopic traffic models[END_REF][START_REF] Tordeux | From traffic and pedestrian follow-the-leader models with reaction time to first order convection-diffusion flow models[END_REF].

The "follow-the-leader" (FTL) can be used in modeling at the microscopic scale, the traffic and the pedestrian flows, see [START_REF] Rosini | Microscopic and macroscopic models for vehicular and pedestrian flows, in Order, Disorder and Criticality: Advanced Problems of Phase Transition Theory[END_REF]. In this paper, we consider the FTL model on a convergent junction and we establish a micromacro connection. More precisely, we consider a microscopic pedestrians model, and prove that the cumulative distribution function on each branch will converge to the solution of a Hamilton-Jacobi equation studied by Imbert and Monneau in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF] and then in [START_REF] Barles | Flux-limited and classical viscosity solutions for regional control problems[END_REF][START_REF] Lions | Well-posedness for multi-dimensional junction problems with Kirchoff-type conditions[END_REF]. We obtain our convergence result by using the theory of viscosity solution (see [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF][START_REF] Crandall | Viscosity solutions of Hamilton-Jacobi equations[END_REF]). The first homogenization result in this framework was obtained in [START_REF] Lions | Homogenization of Hamilton-Jacobi equations[END_REF] and since then, several works considered this problem. For more details, the reader can refer to [START_REF] Achdou | Effective transmission conditions for Hamilton-Jacobi equations defined on two domains separated by an oscillatory interface[END_REF][START_REF] Forcadel | Homogenization of some particle systems with two-body interactions and of the dislocation dynamics[END_REF][START_REF] Forcadel | Homogenization of accelerated Frenkel-Kontorova models with n types of particles[END_REF][START_REF] Galise | A junction condition by specified homogenization and application to traffic lights[END_REF].

Presentation of our results

Since the global strategy to get the convergence result is similar to that in [START_REF] Forcadel | Homogenization of a discrete model for a bifurcation and application to traffic flow[END_REF][START_REF] Forcadel | Homogenization of second order discrete model with local perturbation and application to traffic flow[END_REF][START_REF] Forcadel | Specified homogenization of a discrete traffic model leading to an effective junction condition[END_REF], we focus on the presentation of the new ideas and we refer the reader to [START_REF] Forcadel | Homogenization of a discrete model for a bifurcation and application to traffic flow[END_REF] for some technical proofs. The control of oscillations of the solution is an important result that allows us to determine bounds on the gradient of the limit solution. In this paper, we provide a shorter proof (compared to [START_REF] Forcadel | Homogenization of a discrete model for a bifurcation and application to traffic flow[END_REF]) of this result using the idea introduced in [START_REF] Barles | Interior gradient bounds for the mean curvature equation by viscosity solutions methods[END_REF] and the definition of the security distance. In addition, we obtain better gradient estimates far from zero. A similar (but global) result is also needed when constructing the correctors. In this case, we should compare the components of the solution of an approximated problem on two different branches. We use here the definition of a new distance d 0 which can be seen as a security distance while crossing the junction point. We present our results in the case of a convergent junction with two incoming roads and one outgoing one to simplify the presentation of our work and lessen the article. In fact, we will see that in the case of a convergent junction with N branches (see Appendix A), the mathematical results can be easily extended but need additional "cumulative distribution functions" and additional partial differential equations.

Main result

The microscopic model

We consider a microscopic pedestrians model on a convergent junction with two incoming roads and one outgoing road. The junction is defined by J = J 1 ∪ J 2 ∪ J 3 where J 1 and J 2 are the incoming roads and isometric to R -and J 3 is the outgoing road and isometric to R + . The branches of the junction are glued at the point 0. The point 0 is called the junction point and for all k = 1, 2, 3, we define J * k = J k \ {0}. To be more precise, the definitions of the branches are given by

J k = (-∞, 0] • e k for k = 1, 2, J 3 = [0, +∞) • e 3 ,
with e 1 , e 2 and e 3 being three different unit vectors in R 2 (see Fig. 1).

Let x, y ∈ J with x = x k • e k , y = y l • e l : the distance d(x, y) in J is defined by

d(x, y) = |x k -y k | if k = l, |x k | + |y l | if k = l.
Before writing the mathematical model, we describe its purpose. Purpose of the model. The microscopic model on the junction J describes the following behavior: pedestrians on J 1 and J 2 enter the branch J 3 in a periodic manner. Each one of them knows its turn to enter and will not violate the imposed rule. Such behavior is observed in many real life situations. For example, during the COVID 19 pandemic, some supermarkets decided to reduce the number of customers shopping inside the supermarket and a security agent was organizing the customers flow into the supermarket. Usually, in such situations, we observe the formation of many people lines that converge to the entrance. In order to simplify the presentation of our model, we assume that we have two lines of people and the security agent lets them enter in an alternating way, that is one from J 1 and then one from J 2 then one from J 1 , etc.

Pedestrian's initial distribution. We denote by W i (t) the position of pedestrian i at time t ≥ 0. For i ∈ Z, we define the following sets:

L 1 = {i ∈ Z, i = -(2k + 1), k ∈ N} , L 2 = {i ∈ Z, i = -2k, k ∈ N} . (2.1)
We assume that at time t = 0, the position

W i (0) ∈ J * 1 (resp. J * 2 , J 3 ) if i ∈ L 1 (resp. i ∈ L 2 , i ∈ N * )
. Moreover, we have for all i, j ∈ Z,

d(W i (0), W j (0)) > 0.
We refer to Figure 2 which describes pedestrian's initial distribution on the J. The leaders. Let W i (t) be the position of the pedestrian i at time t ≥ 0. For i ∈ Z, we have

W i (t) ∈ J ⇐⇒ W i (t) = U i (t) • e k , (2.2)
with

     U i (t) ≥ 0 and k = 3 if i ∈ N * , (U i (t) < 0 if k = 1) or (U i (t) ≥ 0 if k = 3) if i ∈ L 1 , (U i (t) < 0 if k = 2) or (U i (t) ≥ 0 if k = 3) if i ∈ L 2 .
We denote by d i (t) the distance to its leader. For t ≥ 0, the distance d i is defined by

d i (t) = U i+1 (t) -U i (t) if i ∈ N Figure 3. Schematic representation of φ.
and for i ∈ Z -N,

d i (t) = U i+2 (t) -U i (t) if W i (t) ∈ J * 1 ∪ J * 2 , d(0, W i (t)) ≥ R 1 U i+1 (t) -U i (t) if (W i (t) ∈ J 3 ) or if (W i (t) ∈ J * 1 ∪ J * 2 , d(0, W i (t)) < R 4 )
where R 1 and R 4 are two positive constants defined in (2.3). This means that far before the junction point (resp. after the junction point), the leader is W i+2 (resp. W i+1 ). As in the "follow-the-leader" model, the velocity of pedestrian i is a distance of d i (t). To ensure a continuous passage from (U i+2 (t)

-U i (t)) to (U i+1 (t) -U i (t)), we introduce a function φ. The function φ : R × R × R → R is defined as follow, for R 1 > R 2 > R 3 > R 4 , φ(x, a, b) =                      a if x ≤ -R 1 , a -min(a, b) R 2 -R 1 (x -R 1 ) + a if -R 1 < x ≤ -R 2 , min(a, b) if -R 2 < x ≤ -R 3 , b -min(a, b) R 3 -R 4 (x + R 4 ) + b if -R 3 < x ≤ -R 4 , b if x > -R 4 (2.3)
where a, b ∈ R (see Fig. 3).

Optimal velocity functions. We assume different optimal velocity functions on each branch. The velocity on branch J 1 (resp.

J 2 , J 3 ) is V 1 (resp. V 2 , V 3
). We make the following assumptions.

Assumptions (A)

(A1) For i = 1, 2, 3, the function V i : R → R + is Lipschitz continuous, non-negative and non-decreasing on R.

(A2) There exists h 0 ∈ (0, +∞) such that for all h ≤ h 0 , V i (h) = 0. (A3) There exists h max > h 0 such that for all h ≥ h max , V i (h) = V max . (A4) There exists a strictly negative real number p 3 (resp. p 1 , p 2 ) such that the function

p → pV 3 (-1/p) (resp. p → pV 1 (-3/2p), p → pV 2 (-3/p)) is decreasing on [-1/h 0 , p 3 ) (resp. [-3/2h 0 , p 1 ) ,[-3/h 0 , p 2 ) ) and increasing on [p 3 , 0) (resp. [p 1 , 0) ,[p 2 , 0) ).
The behavior near the junction point. Let i, j ∈ Z -N * : if i < j, then W j will enter the junction point before W i . As we mentioned above, we assume that pedestrians respect the imposed rule: one passes the junction point coming from J 1 , and then another one coming from J 2 . This is the most restrictive assumption of the paper. This means that W 0 enters first, then W 1 , then W 2 , etc. To model this phenomena, we introduce the function ω : R 3 → R defined by

ω(x, y, p) = α(x) + (1 -α(x))β(y, p) (2.4)
with α being a lipschitz continuous function defining the zone in which the pedestrian

W i ∈ J * 1 (resp. W i ∈ J *
2 ) starts spotting the position of W i+1 on the other branch J 2 (resp. J 1 ) . It's defined in the following way:

α(x) = 1 if x < -R 3 -r or x > r, 0 if -R 3 < x < 0 (2.5) where 0 < r < R 1 -R 3 .
The function β is a lipschitz continuous function describing the fact that if pedestrian W i+1 is after the junction point, its influence on W i disappears since collision is not possible in this case. It's defined by

β(y, p) = 1 if y > r , ζ(p) if y ≤ 0 , (2.6)
where ζ is a lipschitz function modeling the fact that W i+1 will enter the junction point before W i ,

ζ(p) = 1 if p > d 0 + r, 0 if p ≤ d 0 , (2.7)
with

d 0 > 0. (2.8)
Remark 2.1 (Comments on ω). The role of the function ω is to manage the order of entrance through the junction point.

Let i ∈ L 1 . If W i (t) is too close to the junction, that is -R 3 < W i (t) < 0, then it will stop (ω = 0) if W i+1 ∈ J * 2 and if |W i (t)| -|W i+1 (t)| < d 0 .
The distance d 0 can be interpreted as a security distance when crossing the junction point. Remark also that when W i+1 enters the branch J 3 i.e. W i+1 (t) > r, the influence of ω disappears (ω = 1). Finally, let us remark that the definition of the function α on [-R 3 , 0] means that the influence of ω begins directly when the driver's velocity tends to depend on the distance to W i+1 (see Fig. 4).

We are now ready to write our model. We assume U 0 (0) > -R 4 and U 1 (0) > r.

(2.9) Our model is described by the following differential equations,

     U i (t) = V 3 (U i+1 (t) -U i (t)) if i ∈ N, U i (t) = F (U i (t), U i+1 (t), V 1 (U i+2 (t) -U i (t)), V 3 (|U i+1 (t)| -U i (t))) if i ∈ L 1 , U i (t) = F (U i (t), U i+1 (t), V 2 (U i+2 (t) -U i (t)), V 3 (|U i+1 (t)| -U i (t))) if i ∈ L 2 \{0}, with F (x, y, a, b) = φ (x, a, b) • ω (x, y, |x| -|y|) .
Using the definition of φ and ω (see (2.3) and (2.4)), we can reformulate our model in the following way,

U i (t) = F (U i (t), U i+1 (t), V 1 (U i+2 (t) -U i (t)), V 3 (|U i+1 (t)| -U i (t))) if i ∈ L 1 ∪ N * , U i (t) = F (U i (t), U i+1 (t), V 2 (U i+2 (t) -U i (t)), V 3 (|U i+1 (t)| -U i (t))) if i ∈ L 2 .
(2.10)

Remark 2.2. Inequalities in (2.9) allow us to simplify the presentation. In fact if one of the two conditions in (2.9) is violated, we are not able to rewrite the model using only two differential equations as in (2.10). Note that the following equality does not necessarily hold

V 3 (U 1 (t) -U 0 (t)) = F (U 0 (t), U 1 (t), V 2 (U 2 (t) -U 0 (t)), V 3 (U 1 (t) -U 0 (t))) .

The macroscopic model

Let k 0 = 1/h 0 . For k = 1, 2, 3, we define H k : R → R by

H 1 (p) =        -p -2k 0 for p < -2k 0 , -V 1 -2 p |p| for -2k 0 ≤ p ≤ 0, p for p > 0, (2.11)
H 2 (p) =        -p -2k 0 for p < -2k 0 , -V 2 -2 p |p| for -2k 0 ≤ p ≤ 0, p for p > 0, (2.12)
and

H 3 (p) =        -p -k 0 for p < -k 0 , -V 3 -1 p |p| for -k 0 ≤ p ≤ 0, p for p > 0.
(2.13)

Note that for all k = 1, 2, 3, the function H k is continuous, coercive and because of (A4), there exists a unique point p k ≤ 0 such that

H k is decreasing on (-∞, p k ), H k is increasing on (p k , +∞).
(2.14)

We denote by

H 0 = max k∈{1,2,3} H k 0 (2.15)
with

H k 0 = min p∈R H k (p). (2.16)
We introduce now the definition of the gradient of a function defined on J. If x ∈ J, then we define

u x (x) = ∂ i u(x) if x ∈ J * i , (∂ 1 u(0), ∂ 2 u(0), ∂ 3 u(0)) if x = 0,
with ∂ i u(x) the derivative of the function u with respect to x ∈ J i . We denote by J * = J\{0}. The macroscopic model is the Hamilton-Jacobi equation with flux limiting condition at the junction point as considered by Imbert and Monneau in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF] and given by

   u 0 t + H k (u 0 x ) = 0 for (t, x) ∈ (0, +∞) × J * k , u 0 t + F A u 0 x = 0 for (t, x) ∈ (0, +∞) × {0}, u 0 (0, x) = u 0 (x), (2.17)
where A is the flux limiter and F A is defined by

F A (p 1 , p 2 , p 3 ) = max A, H + 1 (p 1 ), H + 2 (p 2 ), H - 3 (p 3 ) , (2.18)
with

H - k (p) = H k (p) if p ≤ p k , H k (p k ) if p ≥ p k , and
H + k (p) = H k (p k ) if p ≤ p k , H k (p) if p ≥ p k . (2.19)
Remark 2.3. In [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF], authors showed that equation (2.17) is equivalent (deriving in space) to the model introduced in [START_REF] Lebacque | Modelling vehicular traffic flow on networks using macroscopic models[END_REF].

The micro-macro connection

To establish the micro-macro connection, we need to introduce two cumulative functions ρ and σ. Unlike the case of simple road (like in [START_REF] Forcadel | Specified homogenization of a discrete traffic model leading to an effective junction condition[END_REF]), the pedestrians are not well ordered. In fact, if i ∈ L 1 and j ∈ L 2 , and i < j, we don't necessarily have U i (t) < U j (t). However, we know that if i, j ∈ L 1 ∪ N * , and i < j, then

U i (t) < U j (t).
(2.20)

Similarly, (2.20)

holds if i, j ∈ L 2 ∪ N * . We have ρ(t, U i (t)) = -i (resp. σ(t, U i (t)) = -i) for i ∈ L 1 ∪ N * (resp. i ∈ L 2 ∪ N * ).
The functions ρ and σ are defined as follows,

ρ (t, y) = -   i≥1 H(y -U i (t)) + G(y -U -1 (t)) + 2 i=-2k-1,k∈N * G(y -U i (t))   (2.21)
and

σ (t, y) = -   i≥1 H (y -U i (t)) + 2 i=-2k,k∈N * G(y -U i (t)   , (2.22)
with

H(x) = 1 if x > 0 0 if x ≤ 0.
and

G(x) = 0 if x > 0 -1 if x ≤ 0.
The main result of this paper is given by the following theorem.

Theorem 2.4 (Junction condition by homogenization: application to pedestrians flow). Assume (A) and that at initial time, we have

U i (0) ≤ U i+1 (0) -h 0 if i ∈ N , U i (0) ≤ U i+2 (0) -h 0 if i ∈ Z -N . (2.23)
We also assume that there exists R > 0 such that

     U i+2 (0) -U i (0) = h 1 if i ∈ L 1 and U i (0) < -R, U i+2 (0) -U i (0) = h 2 if i ∈ L 2 and U i (0) < -R, U i+1 (0) -U i (0) = h 3 if i ∈ N * and U i (0) ≥ R.
(

2.24)

with h 1 , h 2 , h 3 ≥ h 0 . We define two functions u 0 and v 0 as follows

u 0 (x) = - 2 h 1 x1 {x<0} - 1 h 3 x1 {x≥0} , v 0 (x) = - 2 h 2 x1 {x<0} - 1 h 3 x1 {x≥0} .
Let ε > 0 and χ ε : R + × J → R be the function defined by

χ ε (t, x) =      ρ ε (t, -d(0, x)) if x ∈ J * 1 , σ ε (t, -d(0, x)) if x ∈ J * 2 , ρ ε (t, d(0, x)) if x ∈ J 3 .
(2.25)

with ρ ε (t, x) = ερ t ε , x ε , σ ε (t, x) = εσ t ε , x ε .
We define

u 0 (x) =      u 0 (-d(0, x)) if x ∈ J * 1 , v 0 (-d(0, x)) if x ∈ J * 2 , u 0 (d(0, x)) if x ∈ J 3 .
Then there exists a unique A ∈ [H 0 , 0] such that the function χ ε converges towards the unique solution u 0 of (2.17).

Remark 2.5. The proof of this result will be done through three steps.

1) Inject the cumulative distributions function into a non-local PDE.

2) Couple the PDE with suitable initial conditions and prove the convergence to the solution of (2.17).

3) Deduce Theorem 2.4.

The non-local equation

In this section, we first define the non-local operators and then show that (ρ, σ, τ ) is a viscosity solution of a non-local PDE. To do this, we use the function φ(x, a, b) = -φ(x, -a, -b) where φ is defined in (2.3). We remark that φ(x, a, •) and φ(x, •, b) are non-decreasing functions which is a crucial property to obtain the comparison principle (see Prop. 4.2).

Definition of the non-local operators

In this subsection, we give the definition of the non-local operators. To do this, we first introduce the following functions. Let a ∈ R. We define

E a (z) = 0 if z ≥ a 1 if z < a, Ẽa (z) = 0 if z > a 1 if z ≤ a, (3.1)
and

F (z) = 0 if z > -1 -1 if z ≤ -1, F (z) = 0 if z ≥ -1 -1 if z < -1. (3.2)
Let U, V : R → R. We define the following non-local operators: for i = 1, 2,

M i (U )(x) = +∞ 0 V i (z)E -2 (U (x + z) -U (x)) dz -V max , (3.3)
N (U, [V ])(x) = z≥x V 3 (|z| -x)E -1 (V (z) -U (x)) dz -V max , (3.4)
and

K(U, [V ])(x) = R ω z (x, z, |x| -|z|)F (V (z) -U (x)) dz + 1, (3.5)
with ω defined in (2.4). Using the non-local operators defined above, we introduce

R 1 (x, U, [V ]) (x) = φ (x, M 1 (U )(x), N (U, [V ])(x)) K(U, [V ])(x), (3.6) R 2 (x, V, [U ]) (x) = φ (x, M 2 (V )(x), N (V, [U ])(x)) • K(V, [U ])(x). ( 3
                   M 1 (ρ(t, •)) (U i (t)) = -V 1 (U i+2 (t) -U i (t)) if i ∈ L 1 , M 2 (σ(t, •)) (U i (t)) = -V 2 (U i+2 (t) -U i (t)) if i ∈ L 2 \{0}, N (ρ(t, U i (t)), [σ(t, •)])(U i (t)) = -V 3 (|U i+1 (t)| -U i (t)) if i ∈ L 1 ∪ N * , N (σ(t, U i (t)), [ρ(t, •)]) (U i (t)) = -V 3 (|U i+1 (t)| -U i (t)) if i ∈ L 2 , K(ρ(t, U i (t)), [σ(t, •)])(U i (t)) = ω (U i (t), U i+1 (t), |U i (t)| -|U i+1 (t)|) if i ∈ L 1 ∪ N * , K (σ(t, U i (t)), [ρ(t, •)]) (U i (t)) = ω (U i (t), U i+1 (t), |U i (t)| -|U i+1 (t)|) if i ∈ L 2 .
Remark that in the definition of M i , the variable z is positive since it models the distance between U i and its leader whose position is on the same branch or on J 3 (looking ahead). In the definition of N , the variable z models the leader on the other branch, U i+1 . We take z > x since for z ≤ x, the velocity of U i is zero due to the function ω. In the definition of K the variable z ∈ R since it's modeling the pedestrian U i+1 .

In the same way, we define Mi , Ñ and K replacing E a and F respectively by Ẽa and F and then we get the definition of R1 and R2 . Finally, we can easily remark that

-V max ≤ R 1 (x, U, [V ]) (x), R 2 (x, V, [U ]) (x) ≤ 0. (3.8)
For ε > 0, we define the following non-local operators: for i = 1, 2,

M ε i (U )(x) = +∞ 0 V i (z)E -2 (U (x + εz) -U (x)) dz -V max (3.9) N ε (U, [V ])(x) = z≥ x ε V 3 |z| - x ε E -1 (V (εz) -U (x)) dz -V max , (3.10)
and

K ε (U, [V ])(x) = R ω z x ε , z, x ε -|z| F (V (εz) -U (x)) dz + 1. (3.11)
We then define

R ε 1 x ε , U, [V ] (x) = φ x ε , M ε 1 (U )(x), N ε (U, [V ])(x) • K ε (U, [V ])(x), (3.12) R ε 2 x ε , V, [U ] (x) = φ x ε , M ε 2 (V )(x), N ε (V, [U ])(x) • K ε (V, [U ])(x). (3.13)
In the same way, we define M ε i , Ñ ε and Kε replacing E a and F respectively by Ẽa and F and then we get the definition of Rε 1 and Rε 2 .

The non-local PDE

In this subsection, we prove that (ρ, σ) is a discontinuous viscosity solutions of the following non-local PDE,

ρ t (t, x) + R 1 (x, ρ(t, x), [σ(t, .)]) (x)|ρ x (t, x)| = 0 for (t, x) ∈ R + × R , σ t (t, x) + R 2 (x, σ(t, x), [ρ(t, .)]) (x)|σ x (t, x)| = 0 for (t, x) ∈ R + × R , (3.14)
with R 1 and R 2 defined in (3.6) and (3.7). We use the definition of viscosity solutions introduced in [START_REF] Slepčev | Approximation schemes for propagation of fronts with nonlocal velocities and Neumann boundary conditions[END_REF]. This definition allows to have a stability result for the non-local term. We refer to Proposition 4.2 of [START_REF] Forcadel | Homogenization of some particle systems with two-body interactions and of the dislocation dynamics[END_REF] for the corresponding stability result. We give now the definition of viscosity solutions of (3.14).

Definition 3.2 (Viscosity solutions for (3.14)). Let u, v : (0, +∞) × R → R be upper semi-continuous (resp. lower semi-continuous) functions. We say that (u, v) is a viscosity sub-solution (resp. super-solution) of (3.14) on (0, +∞) × R, if we have the following:

1) if ϕ ∈ C 1 ([0, +∞) × R
) such that u -ϕ reaches a maximum (resp. a minimum) at the point (t, x), we have

ϕ t (t, x) + R 1 (x, u(t, x), [v(t, .)]) |ϕ x (t, x)| ≤ 0, (resp. ϕ t (t, x) + R1 (x, u(t, x), [v(t, .)]) |ϕ x (t, x)| ≥ 0) . 2) If ϕ ∈ C 1 ([0, +∞) × R
) such that v -ϕ reaches a maximum (resp. a minimum) at the point (t, x), we have

ϕ t (t, x) + R 2 (x, v(t, x), [u(t, .)]) |ϕ x (t, x)| ≤ 0, (resp. ϕ t (t, x) + R2 (x, v(t, x), [u(t, .)]) |ϕ x (t, x)| ≥ 0) .
We say that (u, v) is a viscosity solution of (3.14) if (u * , v * ) and (u * , v * ) are respectively a sub-solution and a super-solution of (3.14).

We have the following theorem.

Theorem 3.3. The function (ρ, σ) is a discontinuous viscosity solution of (3.14). Conversely, if (u, v) are bounded and continuous viscosity solution of (3.14) satisfying for some time T > 0, and for all t ∈ (0, T )

u(t, •), v(t, •) are decreasing,
then the points U i (t), defined by

u(t, U i (t)) = -i for i ∈ L 1 ∪ N * , v(t, U i (t)) = -i for i ∈ L 2 ∪ N * ,
are viscosity solutions of (2.10) on (0, T ) .

The proof of Theorem 3.3 is an easy adaptation of the proof of Theorem 7.1 in [START_REF] Forcadel | Homogenization of a discrete model for a bifurcation and application to traffic flow[END_REF] using the following lemma.

Lemma 3.4. Let (U i ) i∈Z be the solution of (2.10). Then we have

U i (t) = -R 1 (U i (t), ρ(t, U i (t)), [σ(t, •)]) (U i (t)) if i ∈ L 1 ∪ N * , U i (t) = -R 2 (U i (t), σ(t, U i (t)), [ρ(t, •)]) (U i (t)) if i ∈ L 2 . (3.15)
Proof. We skip the detailed proof of this technical lemma since it can be obtained by simple calculations using the definition of the non-local operators. We refer to the Proof of Lemma 7.2 in [START_REF] Forcadel | Homogenization of a discrete model for a bifurcation and application to traffic flow[END_REF].

Results for the non-local PDE with initial conditions

In this section, we consider the following PDE:

         u t (t, x) + R 1 (x, u(t, x), [v(t, .)]) (x)|u x (t, x)| = 0 for (t, x) ∈ R + × R , v t (t, x) + R 2 (x, v(t, x), [u(t, .)]) (x)|v x (t, x)| = 0 for (t, x) ∈ R + × R , u(0, x) = u 0 (x) for x ∈ R, v(0, x) = v 0 (x) for x ∈ R. (4.1)
We will give first some classical results like the comparison principle and existence via Perron's method, then we will prove a gradient estimate result far from the junction point. The initial conditions u 0 and v 0 are lipschitz continuous functions. In addition, to control the gradient of the limit solution of (2.17), we need the following assumption. Let k 0 = 1/h 0 .

(A0) The functions u 0 and v 0 are non-increasing. Let R 1 the positive parameter appearing in the definition (2.3) of φ. We assume that for y < x < -R 1 ,

-2k 0 (x -y) -2 ≤ u 0 (x) -u 0 (y), -2k 0 (x -y) -2 ≤ v 0 (x) -v 0 (y),
and for x > y > 0,

-k 0 (x -y) -1 ≤ u 0 (x) -u 0 (y).
Remark 4.1. The constante k 0 is the maximal pedestrian's density. The initial conditions u 0 and v 0 in Theorem 2.4 satisfy condition (A0). The definition of u 0 and v 0 is crucial to get the homogenization result in Theorem 2.4.

Recall some existence and uniqueness results for (4.1)

The definition of viscosity solutions for (4.1) is the same as in Definition 3.2 for t > 0 and for t = 0, we will add the following inequality for the sub-solution (resp. super-solution), u(0, x) ≤ u 0 (x) and v(0, x) ≤ v 0 (x), (resp. u(0, x) ≥ u 0 (x) and v(0, x) ≥ v 0 (x)).

We begin first with the comparison principle. Proposition 4.2 (Comparison principle for (4.1)). Assume (A). Let (u, v) be a visocsity sub-solution of (4.1) and (û, v) be a viscosity super-solution of (4.1) in the sense introduced of Definition 3.2. Let us also assume that there exists a constant K > 0 such that for all t > 0,

u(t, x) ≤ u 0 (x) + Kt and -û(t, x) ≤ -u 0 (x) + Kt for x ∈ R, v(t, x) ≤ v 0 (x) + Kt and -v(t, x) ≤ -v 0 (x) + Kt for x ∈ R.
Then we have for all t > 0,

u(t, x) ≤ û(t, x) for x ∈ R, v(t, x) ≤ v(t, x) for x ∈ R.
Proof. The proof is very similar to the one in [START_REF] Forcadel | Homogenization of second order discrete model with local perturbation and application to traffic flow[END_REF] and uses the monotony of E -2 , F, φ and w, so we skip it.

In the proof of convergence (see Sect. 6), we need a comparison principle on bounded sets. For a given point (t 0 , x 0 ) ∈ (0, T ) × R and for r, R > 0, we define the set

P r,R (t 0 , x 0 ) = (t 0 -r, t 0 + r) × (x 0 -R, x 0 + R). (4.2)
u ≤ û, v ≤ v outside P r,R , then u ≤ û, v ≤ v in P r,R .
Proof. The proof of this theorem is similar to the one of Proposition 4.2, so we skip it.

Lemma 4.4 (Existence of barriers for (4.1)). Assume (A0) and (A). Let K = 2V max k 0 . We define

(u + (t, x), v + (t, x)) = (Kt + u 0 (x), Kt + v 0 (x)), (u -(t, x), v -(t, x)) = (u 0 (x), v 0 (x)).
Then (u + , v + ) and (u -, v -) are respectively super and sub solution of (4.1).

Proof. The proof is very simple. We just use the bounds of R 1 and R 2 (see (3.8)) and the gradient bounds (A0).

Applying Perron's method (see [START_REF] Alvarez | Viscosity solutions of nonlinear integro-differential equations[END_REF] or [START_REF] Imbert | A non-local regularization of first order Hamilton-Jacobi equations[END_REF] to see how to apply Perron's method for problems with non-local terms), joint to the comparison principle, we obtain the following result.

Gradient estimates

In the following theorem, we obtain gradient estimates for the solution of (4.1). We use a localization argument in order to get precise gradient bounds on each branch. Theorem 4.6 (Control of the oscillations). Assume (A0)-(A). Let T > 0 and (u, v) be the solution of (4.1) on (0, T ) given in Theorem 4.5. For all t > 0, the functions u(t, •) and v(t, •) are non-increasing. Let x < -R 1 and h > 0 small enough such that x + h < -R 1 and h < h 0 . For all t ∈ (0, T ), we have

-2k 0 h -2 ≤ u(t, x + h) -u(t, x), (4.4
)

-2k 0 h -2 ≤ v(t, x + h) -v(t, x). (4.5)
Let x > 0 and 0 < h < h 0 . For all t ∈ (0, T ), we have

-k 0 h -1 ≤ u(t, x + h) -u(t, x). (4.6)
Proof. The proof of monotony is similar to the one in Theorem 4.10 of [START_REF] Forcadel | Specified homogenization of a discrete traffic model leading to an effective junction condition[END_REF]. We will only do the proof of (4.4) since the other inequalities can be proved in the same way. We use the technique introduced in [START_REF] Barles | Interior gradient bounds for the mean curvature equation by viscosity solutions methods[END_REF] for the proof of local gradient estimates. Let x 0 < -R 1 and δ > 0 small enough such that

h + 2δ < h 0 , x 0 + h + δ < -R 1 . (4.7)
We will prove for all t ∈ (0, T ), y ∈ B δ (x 0 ) and x ∈ B δ (x 0 + h), that

-2k 0 (x -y) -2 -L δ (y -x 0 ) 2 -L δ (x -x 0 -h) 2 ≤ u(t, x) -u(t, y) (4.8)
with L δ = KT δ 2 and K defined in (4.3). In particular, taking y = x 0 and x = x 0 + h, we obtain (4.4). To prove (4.8), we introduce

∆ = [0, T ) × B δ (x 0 + h) × B δ (x 0 )
and consider the following supremum

M = sup (t,x,y)∈∆ u(t, y) -u(t, x) -2k 0 (x -y) -2 -L δ (x -x 0 -h) 2 -L δ (y -x 0 ) 2 .
We want to prove that M ≤ 0. We argue by contradiction and assume that M > 0.

Step 1: the test function. For η > 0 small, we define

ϕ(t, x, y) = u(t, y) -u(t, x) -2k 0 (x -y) -2 -L δ (x -x 0 -h) 2 -L δ (y -x 0 ) 2 - η T -t .
Since ϕ is continuous, it reaches a maximum on ∆ at a point that we denote by ( t, x, ȳ). Classically we have for η small enough,

0 < M 2 ≤ ϕ( t, x, ȳ).
Step 2: t > 0. By contradiction, assume first that t = 0. Then we have

η T < u 0 (ȳ) -u 0 (x) -2k 0 (x -ȳ) -2 ≤ 0,
where we used (A0).

Step 3: |x -x 0 -h| = δ and |ȳ -x 0 | = δ. By contradiction, assume that |x -x 0 -h| = δ. Using the barriers and (A0), we get that

0 < η T ≤ u 0 (ȳ) -u 0 (x) -2k 0 (x -ȳ) -2 -KT -L δ δ 2 ≤ K 1 T -L δ δ 2 = 0
where we used the definition of L δ . In the same way, we have |ȳ -

x 0 | = δ.
Step 4: using of the equation. By doing a duplication of the time variable and passing to the limit in this duplication parameter, we get that

η (T -t) 2 ≤ M1 (u( t, •)) (x) • |-2k 0 -2L δ (x -x 0 -h)| -M 1 (u( t, •)) (ȳ) • |-2k 0 + 2L δ (ȳ -x 0 )| (4.9)
where we used the fact that φ(x, a, b) = a and ω(x, y, p) = 1 for x < -R 1 . We claim that

M 1 (u( t, •)) (ȳ) = 0. ( 4.10)
In fact if (4.10) is true we will obtain a contradiction in (4.9) since M1 ≤ 0. The definition of M 1 can be written for z ≥ h 0 since for z < h 0 , we have V 1 = 0. We have

M 1 (u( t, •)) (ȳ) = +∞ h0 V 1 (z)E -2 (u( t, ȳ + z) -u( t, ȳ))dz -V max .
Using that ϕ( t, x, ȳ) > 0, we have

u( t, x) -u( t, ȳ) < -2.
If z > h 0 , using (4.7), we have

ȳ + z > x 0 -δ + h 0 > x 0 + δ + h > x.
Since u(t, •) is non-increasing, we get for z > h 0 ,

u( t, ȳ + z) -u( t, ȳ) ≤ u( t, x) -u( t, ȳ) < -2
which implies E -2 (u( t, ȳ + z) -u( t, ȳ)) = 1 and then (4.10). This ends the proof.

Construction of correctors

In this section, we construct the corrector far and near the junction point, which allow us to use the perturbed test method introduced by Evans in [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF].

Corrector far from the junction point

Proposition 5.1 (Homogenization far from the junction point). Assume (A). For p ≤ 0, we define the following non-local operator:

M p i (U )(x) = +∞ 0 V i (z)E -2 (U (x + z) -U (x) + pz) dz -V max , (5.1)
M p (U, [V ])(x) = +∞ 0 V 3 (z)E -1 (V (x + z) -U (x) + pz) dz -V max (5.2)
with E a defined in (3.1).

1) For every p ≤ 0, there exists a unique λ 1 ∈ R such that there exists a viscosity solution u of

M p 1 (u)(x)|p + u x | = λ 1 x ∈ R u is bounded.
(5.3)

2) For every p ≤ 0, there exists a unique λ 2 ∈ R such that there exists a viscosity solution v of

M p 2 (v)(x)|p + v x | = λ 2 x ∈ R , v is bounded.
(5.4)

3) For every p ≤ 0, there exists a unique λ 3 ∈ R such that there exists a viscosity solution (u, v) of

     M p (u(x), [v])(x)|p + u x | = λ 3 x ∈ R , M p (v(x), [u])(x)|p + v x | = λ 3 x ∈ R . u, v are bounded.
(5.5)

In particular, λ i = H i (p) with H i defined in (2.11),(2.12) and (2.13).

Proof. We can easily verify that

               u = 0 is a solution of (5.3) with λ 1 = -|p|V 1 -2 p v = 0 is a solution of (5.4) with λ 2 = -|p|V 2 -2 p (u, v) = (0, 0) is a solution of (5.5) with λ 3 = -|p|V 3 -1 p .
The uniqueness of λ i is a classical result and the reader can refer to [START_REF] Lions | Homogenization of Hamilton-Jacobi equations[END_REF] for the proof.

Correctors at the junction point

In this subsection, we will construct correctors near the junction point. To do this, we will consider the following equation: for λ ∈ R, we consider a viscosity solution (u, v) of

R 1 (x, u(x), [v]) (x)|u x | = λ x ∈ R, R 2 (x, v(x), [u]) (x)|v x | = λ x ∈ R (5.6)
where the non-local operators R 1 and R 2 are defined in (3.6) and (3.7). We now prove the existence result of the corrector. For A ≥ H 0 with H 0 defined in (2.15), we introduce the real numbers p 1 -, p 2 -and p 3 + satisfying

H 1 (p 1 -) = H - 1 (p 1 -) = A, H 2 (p 2 -) = H - 2 (p 2 -) = A, H 3 (p 3 + ) = H + 3 (p 3 + ) = A.
Theorem 5.2 (Existence of a global corrector for the junction). Assume (A). i) (General properties) There exists a constant Ā ∈ [H 0 , 0] such that there exists a solution (u, v) of (5.6) with λ = A and such that there exists a constant C and a globally Lipschitz continuous function m such that

|u(x) -m(x)| ≤ C, |v(x) -m(x)| ≤ C.
(5.7)

ii) (Rescaling ) For ε > 0, we set

u ε (x) = εu x ε , v ε (x) = εv x ε .
Then (along a subsequence ε n → 0), we have the following convergence locally uniformly:

u ε -→ ε→0 U and v ε -→ ε→0 V with U and V satisfying    |U (x) -U (y)| ≤ C|x -y| for all x, y ∈ R, H 1 (U x ) = A for all x < 0, H 3 (U x ) = A
for all x > 0, (5.8) and

   |V (x) -V (y)| ≤ C|x -y| for all x, y ∈ R, H 2 (V x ) = A for all x < 0, H 3 (V x ) = A
for all x > 0.

(5.9)

In particular, we have (with

U (0) = V (0) = 0) U (x) = p 3 + x1 {x>0} + p 1 -x1 {x<0} , (5.10) V (x) = p 3 + x1 {x>0} + p 2 -x1 {x<0} . ( 5

Proof of Theorem 5.2

This subsection contains the proof of Theorem 5.2. Let R > R 1 (R 1 the constant appearing in (2.3)), we want to find λ R , such that there exists a solution (u R , v R ) of

G 1 R x, u R (x), [v R ], u R x = λ R if x ∈ [-l R , l R ], G 2 R x, v R (x), [u R ], v R x = λ R if x ∈ [-l R , l R ],
(5.12)

with

G 1 R (x, U, [V ], q) = ψ - R (x)ψ + R (x)φ (x, M 1 (U )(x), N (U, [V ])(x)) K(U, [V ])(x)|q| + (1 -ψ - R (x)) • H - 1 (q) + (1 -ψ + R (x)) • H + 3 (q), (5.13) G 2 R (x, V, [U ], q) = ψ - R (x)ψ + R (x)φ (x, M 2 (V )(x), N (V, [U ])(x)) K(V, [U ])(x)|q| + (1 -ψ - R (x)) • H - 1 (q) + (1 -ψ + R (x)) • H + 3 (q), (5.14)
and

ψ + R , ψ - R ∈ C ∞ , ψ ± R : R → [0, 1], with ψ - R (x) = ψ + R (-x), ψ + R ≡ 1 on [-∞, R] 0 outside [R + 1, +∞), and ψ + R non-increasing, and l R >> R + 1 max(h max , R 3 + r).
To each operator G i R , we associate Gi R which is defined in the same way but replacing the non-local operators M i , N and K by Mi , Ñ and K.

G1 R (x, û(x), [v], ûx ) ≥ λ if x ∈ (0, l R ], G2 R (x, v(x), [û], vx ) ≥ λ if x ∈ (0, l R ],
and for ε > 0

G 1 R (x, u(x), [v], u x ) ≤ λ -ε if x ∈ (0, l R ], G 2 R (x, v(x), [u], v x ) ≤ λ -ε if x ∈ (0, l R ]. If u(0) ≤ û(0), v(0) ≤ v(0), then for all x ∈ [0, l R ] , we have u(x) ≤ û(x), v(x) ≤ v(x).
2) Let us consider the following problem,

G1 R (x, û(x), [û], ûx ) ≥ λ if x ∈ [-l R , -R 1 ),
and for ε > 0

G 1 R (x, u(x), [u], u x ) ≤ λ -ε if x ∈ [-l R , -R 1 ). If u(x) ≤ û(x) for x ∈ [-R 1 -h max , -R 1 ], then for all x ∈ [-l R , -R 1 ] , we have u(x) ≤ û(x).
3) Let us consider the following problem,

G2 R (x, v(x), [v], ŵx ) ≥ λ if x ∈ [-l R , -R 1 ),
and for ε > 0

G 2 R (x, v(x), [v], v x ) ≤ λ -ε if x ∈ [-l R , -R 1 ). If v(x) ≤ v(x) for x ∈ [-R 1 -h max , -R 1 ],
then for all x ∈ [-l R , -R 1 ] , we have

v(x) ≤ v(x).
Proof. The proof of this proposition can be done using classical arguments of viscosity solutions theory, (see [START_REF] Barles | An introduction to the theory of viscosity solutions for first-order Hamilton-Jacobi equations and applications[END_REF][START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF]. Let us remark that the initial comparison between sub-solution and super-solution in points 2)-3) is given on the interval [-R 1 -h max , -R 1 ]. This condition will ensure, when doing the proof, that the maximum points (x, y) satisfy x + z, y + z < -R 1 for z ∈ [0, h max ].

Proposition 5.4 (Existence of correctors).

There exists a unique λ R ∈ R such that there exists a solution (u R , v R ) of (5.12). Moreover, there exists a constant C , and a Lipschitz continuous function m R such that

       H 0 ≤ λ R ≤ 0, |m R (x) -m R (y)| ≤ C|x -y| for x ∈ [-l R , l R ], |u R (x) -m R (x)| ≤ C for x ∈ [-l R , l R ], |v R (x) -m R (x)| ≤ C for x ∈ [-l R , l R ].
(5.15)

Proof. In order to construct a corrector on the truncated domain, we will consider the approximated problem

δu δ + G 1 R x, u δ (x), [v δ ], u δ x = 0 if x ∈ [-l R , l R ], δv δ + G 2 R x, v δ (x), [u δ ], v δ x = 0 if x ∈ [-l R , l R ].
(5.16)

Step 1: Construction of the solution.

Lemma 5.5. There exists a unique viscosity solution (u δ , v δ ) of (5.16) such that

     0 ≤ u δ (x) ≤ |H 1 0 | δ , 0 ≤ v δ (x) ≤ |H 2 0 | δ .
(5.17)

Proof. We can easily show that, (u -, v -) = (0, 0) and (u

+ , v + ) = |H 1 0 | δ , |H 2 0 | δ
are respectively sub-solution and super-solution of (5.16). Using the comparison principle and the Perron's method, we construct a unique viscosity solution (u δ , v δ ) satisfying (5.17).

Step 2: A new super-solution.

Lemma 5.6. Let b 0 = max(g 0 , k 0 ) and g 0 = 1/d 0 with d 0 defined in (2.7). Let

(u ++ , v ++ ) = -2b 0 (x -l R ) + |H 3 0 | δ , -2b 0 (x -l R ) + |H 3 0 | δ .
Then (u ++ , v ++ ) is a super-solution of (5.16).

Proof. Since x ∈ [-l R , l R ], we have u ++ , v ++ ≥ |H 3 0 | δ .
In addition, using the definition of H - k , we remark that for k = 1, 2, 3,

H k (-2b 0 ) ≥ H k (-2k 0 ) = H - k (-2k 0 ) ≥ 0.
We recall that for i = 1, 2, 3, we have

V i (h) = 0 if h ≤ h 0 , and ω(x, y, p) = 0 for x ∈ [-R 3 , 0], y < 0 and p ≤ d 0 . (5.18)
We claim that

M 1 (u ++ )(x) = 0 if x ≤ -R 3 , M 2 (v ++ )(x) = 0 if x ≤ -R 3 , K(u ++ (x), [v ++ ])(x) = 0 K(v ++ (x), [u ++ ])(x) = 0 if x ∈ (-R 3 , -d 0 ], N (u ++ (x), [v ++ ])(x) = 0 N (v ++ (x), [u ++ ])(x) = 0 if x ≥ 0.
In addition, on (-d 0 , 0), we have

K(u ++ (x), [v ++ ])(x) = 0 K(v ++ (x), [u ++ ])(x) = 0 if h0 2 < d 0 , N (u ++ (x), [v ++ ])(x) = 0 N (v ++ (x), [u ++ ])(x) = 0 if h0 2 ≥ d 0 .
We prove only four equalities since others are done in the same way. First, let x ≤ -R 3 ,

M 1 (u ++ )(x) + V max = +∞ 0 V 1 (z)E -2 u ++ (x + z) -u ++ (x) dz = +∞ h0 V 1 (z)E -2 (-2b 0 z) dz.
For z ≥ h 0 , -2b 0 z ≤ -2. Using the definition of E -2 (see (3.1)), we get

M 1 (u ++ )(x) = hmax h0 V 1 (z)dz -V max = 0. Second, let x ∈ (-R 3 , -d 0 ], K(u ++ (x), [v ++ ])(x) -1 = R ω z (x, z, |x| -|z|)F v ++ (z) -u ++ (x) dz = R ω z (x, z, |x| -|z|)F (-2b 0 (z -x)) dz = R - ω z (x, z, |x| -|z|)F (-2b 0 (z -x)) dz + R + ω z (x, z, |x| -|z|)F (-2b 0 (z -x)) dz.
Let z ≤ 0: using (5.18), we remark that we can assume that z ≥ x + d 0 . In that case, we have

-2b 0 (z -x) ≤ -2b 0 d 0 = -2 < -1.
The last inequality is also true if we take z > 0. Using the definition of F (see (3.2)), we obtain

K(u ++ (x), [v ++ ])(x) = - 0 x+d0 ω z (x, z, |x| -|z|)dz - R + ω z (x, z, |x| -|z|)dz + 1 = -ω(x, 0, |x|) + ω(x, x + d 0 , d 0 ) -1 + ω(x, 0, |x|) + 1 = 0
where we used in the last line the fact that ω(x, z, p) = 1 if z 0. Finally, let x ∈ (-d 0 , 0): if h0 2 ≤ d 0 , then using (5.18), we evaluate K(u

++ (x), [v ++ ])(x) only for z ∈ R + since x + d 0 > 0 > z. If z ≥ 0, then -2b 0 (z -x) ≤ 2b 0 x ≤ -2b 0 h0 2 ≤ -1, and we get K(u ++ (x), [v ++ ])(x) = - R + ω z (x, z, |x| -|z|)dz = -1 + ω(x, 0, |x|) + 1 = 0
where we used that |x| = -x < d 0 . Assume now h0 2 > d 0 . We have

N (u ++ (x), [v ++ ])(x) + V max = z≥x V 3 (|z| -x)E -1 v ++ (z) -u ++ (x) dz = z≥x V 3 (|z| -x)E -1 (-2b 0 (z -x)) dz = 0>z≥x V 3 (-z -x)E -1 (-2b 0 (z -x)) dz + z≥0 V 3 (z -x)E -1 (-2b 0 (z -x)) dz.
We evaluate the above integral only if z ≥ 0 since if x ≤ z < 0, we get -z -x ≤ -2x < 2d 0 < h 0 and so V 3 (-z -x) = 0. If z ≥ 0 and z -x ≥ h 0 , we get that -2b 0 (z -x) ≤ 2b 0 h 0 ≤ -1 and using the definition of E -1 , we get

N (u ++ (x), [v ++ ])(x) = z≥x+h0 V 3 (z -x)dz -V max = 0.
Recalling the definition of G 1 R and G 2 R (see (5.13) and (5.14)) and using the lower bound of u ++ and v ++ , we deduce that (u ++ , v ++ ) is a super-solution of the approximated problem for x ∈ (-l R , l R ). Let us now treat the boundary conditions. Consider a test function ϕ touching u ++ from below at x = -l R (resp. x = l R ). We have that

δϕ(-l R ) + H - 1 (ϕ x (-l R )) ≥ H - 1 (u ++ x (-l R )) = H - 1 (-2b 0 ) ≥ 0, (resp. δϕ(l R ) + H + 3 (ϕ x (l R )) ≥ δu ++ (l R ) + H 3 0 ≥ 0).
We can argue in the same way for the function v ++ . This implies that (u ++ , v ++ ) is a super-solution of the approximated problem for x ∈ [-l R , l R ]. Using the stability of viscosity super-solution, we can construct a new super-solution,

(min(u + , u ++ ), min(v + , v ++ )).
The comparison principle then implies that

0 ≤ u δ (x) ≤ min(u + , u ++ ), 0 ≤ v δ (x) ≤ min(v + , v ++ ). (5.19)
Remark 5.7. We need the non-constant new super-solution to ensure (after passing to the limit in δ and R) that the flux limiter A satisfies A ≥ H 0 .

Step 3: bounds for the solution.

Lemma 5.8. Let (u δ , v δ ) be the solution of (5.16). The functions u δ and v δ are non-increasing and we have the following bounds,

-1 ≤ u δ (x) -v δ (x) ≤ 1 if 0 ≤ x ≤ l R , (5.20)
and

-1 ≤ u δ (x) -v δ (x) ≤ 1 if -R 3 ≤ x ≤ 0, . (5.21)
Proof. The proof of monotony is similar to that in Lemma 5.5 of [START_REF] Forcadel | Specified homogenization of a discrete traffic model leading to an effective junction condition[END_REF] so we skip it.

We will do the proof of the lower bound in (5.21) (the proof of the upper bound is treated in the same way).

Let -R 3 < z 0 < 0, a > 0 small and L a > |H 2 0 | δa 2 . We will prove that for x ∈ B a (z 0 ),

u δ (x) -v δ (x) ≥ -1 -L a (x -z 0 ) 2 .
If the above inequality is true, then taking x = z 0 , we obtain the first inequality in (5.21). For ε > 0, we define

M ε = sup x,y∈Ba(z0) v δ (y) -u δ (x) -1 -L a (x -z 0 ) 2 - (x -y) 2 2ε .
We have that M ε ≥ M > 0 and since the functions v δ and u δ are continuous, M ε is reached at a point (x, ȳ). Classically, we have

0 < M ε < |H 2 0 | δ - (x -ȳ) 2 2ε
which implies |x -ȳ| → 0 as ε → 0. If |x -z 0 | = a, and using the definition of L a , we obtain

0 < |H 2 0 | δ -L a a 2 ≤ 0
which gives a contradiction and implies that |x -z 0 | = a and |ȳ -z 0 | = a for ε small enough. Writing the sub-solution viscosity inequality, we get

δ < δv δ (ȳ) ≤ -G 2 R ȳ, v δ (ȳ), [u δ ],
x -ȳ ε .

Using the definition of G 2 R for ȳ ∈ (-R 3 , 0), we can see that we obtain a contradiction if we prove that

K v δ (ȳ), [u δ ] (ȳ) = R ω z (ȳ, z, |ȳ| -|z|) F u δ (z) -v δ (ȳ) dz + 1 = 0. (5.22)
Recalling that ω (x, y, p) = 0 if x ∈ (-R 3 , 0), y < 0 and p < 0, we can evaluate the above integral for z ≥ 0. If we prove u δ (z) -v δ (ȳ) < -1 for all z ≥ ȳ, we will get

+∞ ȳ ω z (ȳ, z, |ȳ| -|z|) F u δ (z) -v δ (ȳ) dz = -1
which implies (5.22). Using the supremum M ε , we have

u δ (x) -v δ (ȳ) < -1
and since |x -ȳ| → 0 as ε → 0, we have for ε small enough

u δ (ȳ) -v δ (ȳ) < -1.
Using the fact that u δ is non-increasing, we have for z ≥ ȳ,

u δ (z) -v δ (ȳ) ≤ u δ (ȳ) -v δ (ȳ) < -1
which gives the desired result.

The proof of (5.20) is similar. The difference that in this case (always to prove the lower bound), we prove N v δ (ȳ), [u δ ] (ȳ) = 0.

We claim that v δ (z) -u δ (ȳ) < -1 for all z > ȳ + d 0 . If the preceding inequality is true, we get F v δ (z) -u δ (ȳ) = -1 and then

K u δ (ȳ), [v δ ] (ȳ) = ω (ȳ, ȳ + d 0 , d 0 ) -1 + 1 = 0.
We distinguish two cases:

Case 1: x < z. Using the supremum, we have

u δ (x) -u δ (ȳ) < -2.
The fact that u δ is non-increasing implies

u δ (z) -u δ (ȳ) < -2.
(5.26)

Combining (5.26) and Lemma 5.8, we get v δ (z) -u δ (ȳ) < -1.

Case 2: x ≥ z. Using the supremum, we have N ≥ u δ (z) -u δ (x) -2b 0 (x -z) -2, which implies

u δ (z) -u δ (ȳ) ≤ -2b 0 (z -ȳ) ≤ -2b 0 d 0 ≤ -2. (5.27)
Combining (5.27) and Lemma 5.8, we get v δ (z) -u δ (ȳ) < -1 which gives us the desired result. If ȳ + d 0 ≥ 0, in that case, we only evaluate the integral for z ∈ [0, r] since the other one is zero. We proceed as above assuming that x < z and x ≥ z. We deduce that K u δ (ȳ), [v δ ] (ȳ) = 0 and similarly (5.25). This ends the proof of the lemma.

Step 5: Proof of Proposition 5.4. As in [START_REF] Forcadel | Homogenization of a discrete model for a bifurcation and application to traffic flow[END_REF], we construct a Lipschitz continuous function m δ , such that there exists a constant C, (independent of R and δ) such that for x, y ∈

[-l R , l R ],    |m δ (x) -m δ (y)| ≤ C|x -y|, |u δ (x) -m δ (x)| ≤ C, |v δ (x) -m δ (x)| ≤ C
(5.28) and then using the half relaxed limit method (for δ → 0) joint to Perron method, we can construct a solution (u R , v R ) of (5.12) and a lipschitz function m R satisfying (5.15). The first inequality in (5.15) is a consequence of (5.19). The uniqueness of λ R is classical so we skip it. This ends the proof of Proposition 5.4. Proposition 5.10 (First definition of the flux limiter). The following limit exists (up to a subsequence)

A = lim R→+∞ λ R .
(5.29)

Moreover, we have

H 0 ≤ λ R , A ≤ 0, We deduce that u -satisfies for -l R ≤ x < -R 1 , ψ - R (x)M 1 (u -)(x)|u - x | + (1 -ψ - R (x)) • H - 1 (u - x ) = λ R -µ.
Let x 0 < -R 1 and C = (2b 0 -p 1 µ )h max + 2. Using Proposition 5.4, we have for

x ∈ [x 0 -h max , x 0 ], u R (x) -u R (x 0 ) ≥ p 1 µ (x -x 0 ) -C.
Using the comparison principle (Prop. 5.3, point 2), we deduce that the above inequalities are true for all x ≤ x 0 . Finally, if we choose γ 0 < |p 1 -p1 -| (with p 1 defined in (2.15)), then

H 1 (p 1 -+ γ) = H - 1 (p 1 -+ γ),
and we can choose µ > 0 such that

p µ 1 = p1 -+ γ
and this ends the proof.

Proof of Theorem 5.2. The proof of Theorem 5.2 can be deduced by all the above results. We first pass to the limit as R → +∞ then we study the uniform local convergence of the rescaled functions, see [START_REF] Forcadel | Homogenization of a discrete model for a bifurcation and application to traffic flow[END_REF].

Proof of Convergence

In this section, we will show how to obtain our main result Theorem 2.4. Let ε > 0 and

u ε (t, x) = εu t ε , x ε , v ε (t, x) = εv t ε , x ε ,
with (u, v) the unique viscosity solution of (4.1). Then (u ε , v ε ) is the unique viscosity solution for (t,

x) ∈ R + × R of            u ε t + R ε 1 x ε , u ε (t,•) ε , v ε (t,•) ε (x)|u ε x | = 0, v ε t + R ε 2 x ε , v ε (t,•) ε , u ε (t,•) ε (x)|v ε x | = 0, u ε (0, x) = u 0 (x), v ε (0, x) = v 0 (x). (6.1)
First, we will prove the following convergence result. Theorem 6.1 (Junction condition by homogenization). Assume (A) and (A0). For ε > 0, let (u ε , v ε ) be the solution of (6.1). Let χ ε defined by

χ ε (t, x) =      u ε (t, -d(0, x)) if x ∈ J * 1 , v ε (t, -d(0, x)) if x ∈ J * 2 , u ε (t, d(0, x)) if x ∈ J 3 . (6.2)
Using (6.9), and the definition of the non-local operator K ε (see (3.11)) implies that on the ball P r,r ( t, ȳ), the function (u ε , v ε ) is a solution in the viscosity sense of

   u ε t + N ε u ε (t,•) ε , v ε (t,•) ε (y)|u ε y | = 0, v ε t + N ε v ε (t,•) ε , u ε (t,•)] ε (y)|v ε y | = 0.
We will prove that for (t, y) ∈ P r,r ( t, ȳ), we have

φt + Ñ ε φ(t, •) ε , φ(t, •) ε (y)|ϕ y | ≥ 0.
For all (t, y) ∈ P r,r ( t, ȳ), we have for r small enough φt (t, y)

+ Ñ ε φ ε (t, •), φ ε (t, •) (y) • | φε y | = φt ( t, ȳ) + o r (1) + Ñ ε φ ε (t, •), φ ε (t, •) (y) • | φy | = θ + o r (1) + Ñ ε φ ε (t, •), φ ε (t, •) (y) • |p| -H 3 (p) =: ∆, (6.10)
where we used that φy ( t, ȳ) = ∂ 3 ϕ( t, x) = p and (6.7). If p = 0, then using that H 3 (0) = 0, we obtain that ∆ = θ + o r (1) > 0 for r small small enough. Assume now that p < 0. For all z ∈ [h 0 , h max ], and for ε and r small enough we have that

φ(t, y + εz) -φ(t, y) ε = φ(t, y + εz) -φ(t, y) ε ≤ pz + o r (1) + cε,
where we used the fact that φ ∈ C 1 and that z ∈ [h 0 , h max ]. Now using the fact that Ẽ-1 is decreasing we have

Ẽ-1 (pz + cε + o r (1)) ≤ Ẽ-1 φ(t, y + εz) -φ(t, y) ε .
We deduce that

Ñ ε φ ε (t, •), φ ε (t, •) (y) + V max = z≥ y ε V 3 |z| - y ε Ẽ-1 φ(t, εz) -φ(t, y) ε dz = +∞ 0 V 3 (z) Ẽ-1 φ(t, y + εz) -φ(t, y) ε dz (6.11) ≥ +∞ 0 V 3 (z) Ẽ-1 (pz + o r (1) + cε) dz (6.12) = (-1-cε-or(1))/p 0 V 3 (z) Ẽ-1 (pz + o r (1) + cε) dz + +∞ (-1-cε-or(1))/p V 3 (z) Ẽ-1 (pz + o r (1) + cε) dz (6.13) = 0 + +∞ (-1-cε-or(1))/p V 3 (z)dz = V max -V 3 -1 -cε -o r (1) p (6.14)
where we used in the last line the fact that pz + o r (1) + cε < -1 if z > -a-cε-or (1) p . Injecting (6.14) in (6.10) and choosing ε and r small enough, we obtain

∆ ≥ θ + o r (1) + |p| • -V 3 -a -cε -o r (1) p + V 3 -1 p ≥ θ + o r (1) -||V 3 || ∞ • (cε + o r (1)) ≥ θ 2 ,
where we used assumption (A1) for the second line.

Getting a contradiction. By definition, we have for ε small enough and using (6.8),

u ε ≤ φ -η outside P r,r ( t, ȳ), v ε ≤ φ -η outside P r,r ( t, ȳ). (6.15)
Using the comparison principle on bounded subsets for (6.1) (Thm. 4.3), we get

u ε ≤ φ -η on P r,r ( t, ȳ), v ε ≤ φ -η on P r,r ( t, ȳ). (6.16)
Passing to the limit as ε → 0, we get u ≤ φ -η and v ≤ φ -η on P r,r ( t, ȳ) and this contradicts the fact that z( t, x) = ϕ( t, x).

Second case: x = 0: In that case, we have that

ϕ t ( t, x) + F A (∂ 1 ϕ( t, 0), ∂ 2 ϕ( t, 0), ∂ 3 ϕ( t, 0)) = θ (6.17)
and using Theorem 2.7 in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF], we can assume that ϕ has a specific form given by

ϕ(t, x) = g(t) + p 1 -x1 {x∈J * 1 } + p 2 -x1 {x∈J * 2 } + p 3 + x1 {x∈J * 3 } (6.18)
with g ∈ C 1 (0, T ). In particular, (6.17) can be replaced by

g t ( t) + A = θ. (6.19)
Let us consider the solution (u, v, w) of (5.6) provided by Theorem 5.2, and let us denote by ϕ ε (t, y) = g(t) + u ε (t, y) on P r,2r ( t, 0), ϕ(t, ye 1 1 {y≤0} + ye 3 1 {y≥0} ) outside P r,2r ( t, 0), (6.20) and Ψ ε (t, y) = g(t) + v ε (t, y) on P r,2r ( t, 0), ϕ(t, ye 2 1 {y≤0} + ye 3 1 {y≥0} ) outside P r,2r ( t, 0). (6.21)

We will prove that (ϕ ε , Ψ ε ) is a viscosity super-solution on P r,r ( t, 0) of (6.1). Let h be a test function touching ϕ ε from below at (t 1 , x 1 ) ∈ P r,r ( t, 0), so we have This implies that (using (6.19) and taking r small enough)

u x 1 ε = 1 ε (h(t
h t (t 1 , x 1 ) + R1 x 1 ε , u x 1 ε , [v] (x 1 ) • |h x (t 1 , x 1 )| ≥ A + g (t 1 ) ≥ θ 2 .
Now for ε small enough such that εh max ≤ r, we deduce from the previous inequality and using the fact that the non-local operators are of bounded support, that h t (t 1 , x 1 ) + Rε

1 x 1 ε , ϕ ε (t, x) ε , ψ ε (t, •) ε (x) • |h x (t 1 , x 1 )| ≥ θ 2 .
The two last super-solution inequalities are obtained in same way using the correctors and (6.19).

Getting the contradiction. Using (6.8) and (6.18), we have for ε small enough on P r,2r ( t, 0)\P r,r ( t, 0), u ε ≤ g(t) + p1 -y1 {y<0} + p3 + y1 {y>0} -η, v ε ≤ g(t) + p2 -y1 {y<0} + p3 + y1 {y>0} -η.

Using the fact that u ε → U and v ε → V with U, V defined in (5.11) we have for ε small enough on P r,2r ( t, 0)\P r,r ( t, 0)

u ε ≤ ϕ ε - η 2 and v ε ≤ Ψ ε - η 2 .
Combining this with the particular form of the test function given in (6.18), we get that outside P r,r ( t, 0),

u ε ≤ ϕ ε - η 2 and v ε ≤ Ψ ε - η 2 .
By the comparison principle on bounded subsets (Thm. 4.3) the previous inequality holds in P r,r ( t, 0). Passing to the limit as ε → 0 and evaluating the inequality in ( t, 0), we obtain u( t, 0) ≤ ϕ( t, 0) -η 2 and v( t, 0) ≤ ϕ( t, 0) -η 2 which is a contradiction.

We define N functions u 1 0 , u 2 0 , ..., u N 0 by

u i 0 (x) = - N j=1 nj ni h i x1 {x<0} - 1 h N +1 x1 {x≥0} .
Let ε > 0 and χ ε : R + × J → R be the function defined by

χ ε (t, x) = (ρ 1 i ) ε (t, -d(0, x)) if x ∈ J * i , i = 1, ..., N , (ρ 1 1 ) ε (t, d(0, x)) if x ∈ J N +1 with (ρ 1 i ) ε (t, x) = ερ 1 i t ε , x ε .
We define

u 0 (x) = u i 0 (-d(0, x)) if x ∈ J * i , i = 1, ..., N , u 1 0 (d(0, x)) if x ∈ J N +1 .
Then there exists a unique A ∈ [H 0 , 0] such that the function χ ε converges towards the unique solution u 0 of    u 0 t + H i (u 0 x ) = 0 for (t, x) ∈ (0, +∞) × J * i , u 0 t + F A u 0 x = 0 for (t, x) ∈ (0, +∞) × {0}, u 0 (0, x) = u 0 (x), where for i = 1, ...N , the Hamiltonian H i is given by In blue (resp. pink and red): pedestrians U i whose leader's label at the initial time is i + 1 (resp. i + 2 and i + 3). The definition of the function ρ 1 1 (resp. ρ 2 1 and ρ 1 2 ) depends on pedestrians colored in blue (resp. pink and red).

H i (p) =              -p --
Example if N = 2 and n 1 = 2, n 2 = 1, see Figure A.2. We give the definition of the cumulative functions. We need to introduce three cumulative functions ρ 1 1 , ρ 2 1 and ρ 1 2 . The need of three functions arises from the different type of leaders: far before zero, on branch J 1 , the leader can be U i+1 or U i+2 and on branch J 2 , the leader is U i+3 .

Figure 1 .

 1 Figure 1. The junction J.

Figure 2 .

 2 Figure 2. Initial distribution of pedestrians.

Figure 4 .

 4 Figure 4. The role of ω.

Theorem 4 . 3 (

 43 Comparison principle on bounded sets for (4.1)). Assume (A). If (u, v) and (û, v) are respectively sub-solution and super-solution of (4.1) on the open set P r,R such that

Theorem 4 . 5 (

 45 Existence and uniqueness of viscosity solutions for (4.1)). Assume (A0) and (A). Then, there exists a unique continuous solution of (4.1) which satisfies |u(t, x) -u 0 (x)| ≤ Kt, and |v(t, x) -v 0 (x)| ≤ Kt.(4.3)

. 11 )

 11 iii)(Uniqueness of the flux limiter A) We define the following set of functions S = {(u, v) s.t. ∃Lipschitz function m and C ≥ 0 satisfying (5.7)} . Then we have A = inf {λ ∈ R : ∃ (u, v) ∈ S solution of (5.6)} .

Proposition 5 . 3 (

 53 Comparison principle). Let λ ∈ R, 1) Let us consider the following problem,

  |p| for -k 0 ≤ p ≤ 0. p for p > 0.For p ∈ R N +1 , the function F A (p) is defined byF A (p) = max A, max i=1,..,N H + i (p i ), H -N +1 (p N +1 ) .

Figure A. 2 .

 2 Figure A.2.In blue (resp. pink and red): pedestrians U i whose leader's label at the initial time is i + 1 (resp. i + 2 and i + 3). The definition of the function ρ 1 1 (resp. ρ 2 1 and ρ 1 2 ) depends on pedestrians colored in blue (resp. pink and red).

HH

  -U i (t)) + (-1 + H(y -U -1 (t))) + (-1 + H(y -U -2 (t)) (y -U i (t)) + 4(-1 + H (y -U -4 (t))) (y -U i (t)) + i=-3k,k∈N * 3(-1 + H (y -U i (t)))

  1 , x 1 ) -g(t 1 )) ,

	and
					u(y) ≥	1 ε	(h(εs, εy) -g(εs)) ,
	for (s, y) in a neighbourhood of	t 1 ε	,	x 1 ε	. Therefore, we have
	h t (t 1 , x 1 ) -g (t 1 ) + R1	x 1 ε	, u	x 1 ε	, [v] (x

1 ) • |h x (t 1 , x 1 )| ≥ A.

Acknowledgements. The second author was supported by the European Union with the European regional development fund (ERDF, 18P03390/18E01750/18P02733) and by the Normandie Regional Council via the M2SiNUM project.

https://www.edpsciences.org/en

Step 4: control of oscillations. Lemma 5.9. Let (u δ , v δ ) be a solution of (5.16). Let g 0 = 1/d 0 and b 0 = max(g 0 , k 0 ). We have that -2b 0 (x -y) -2 ≤ u δ (x) -u δ (y) ≤ 0 for -l R ≤ y ≤ x ≤ l R , -2b 0 (x -y) -2 ≤ v δ (x) -v δ (y) ≤ 0 for -l R ≤ y ≤ x ≤ l R .

(5.23)

Proof. We will only prove the first inequality in (5.23). To prove the other inequality, we proceed similarly. We define

We want to prove that N ≤ 0. Assume by contradiction that N > 0. Classicaly, N is reached at a point (x, ȳ) with x ≥ ȳ. Moreover x = ȳ since N > 0. Writing the viscosity inequalities, we get

We claim that

If we prove (5.24), using the definition of G 1 R , we get

Using that H - 1 (-2b 0 ) ≥ 0, H + 3 (-2b 0 ) ≤ 0 and that ψ -and ψ + are respectively non-decreasing and nonincreasing, we get a contradiction.

Let us now prove (5.24). Using the definition of φ (see (2.3)), we obtain (5.24) if we prove:

(5.25)

We will only show the third equality of (5.25) since the others can be treated in the same way and are even simpler. Recalling the definition of ω, and assuming that ȳ + d 0 < 0, we can write,

with H 0 defined in (2.15).

Proof. This results is a direct consequence of the following bound on λ R which is independent of R (see the barriers in (5.19)),

Proposition 5.11 (Control of the slopes). Assume that R is big enough. Let (u R , v R ) be the solution of (5.12) given by Proposition 5.4. We also assume that up to a sub-sequence A = lim R→+∞ λ R > H 0 . Then there exists γ 0 > 0 such that for every γ ∈ (0, γ 0 ), there exists a constant C such that for x > 0 and h ≥ 0,

(5.30)

)

Proof. We only prove (5.31) since the proof for (5.30) and (5.32) is similar. For µ > 0 small enough, we denote by p 1 µ the real number such that

Using that H 0 < λ R ≤ 0, we deduce that p 1 µ exists, is unique and satisfies -2k 0 ≤ p 1 µ ≤ 0 for µ small enough. We define for -

We have that

We also have for

Then there exists A ∈ [H 0 , 0] such that χ ε converges locally uniformly to the unique viscosity solution u 0 of (2.17) with the initial condition u 0 defined by

Remark 6.2. The same result remains true if we take v ε on J 3 .

The proof of Theorem 2.4 can be obtained using the convergence result in Theorem 6.1. The detailed proof is an easy adaptation of the proof of Theorem 2.6 in [START_REF] Forcadel | Homogenization of a discrete model for a bifurcation and application to traffic flow[END_REF].

Theorem 6.1 gives also gradient estimates for the solution of (2.17).

Corollary 6.3. Assume (A0)-(A). Let u 0 be the unique solution of (2.17) then u 0 is Lipschitz continuous and we have,

Proof of Corollary 6.3. The lipschitz continuity of u 0 was proved in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF]. Let x, y ∈ J * 1 s.t x = x 1 • e 1 and y = (x 1 + h) • e 1 with x 1 + h < 0 and h > 0 small enough. For ε > 0 small enough, we have x 1 + h < -εR 1 . Using Theorem 4.6, we have

Using Theorem 6.1 and passing to the limit as ε → 0, we obtain the desired result. We proceed similarly on J * 2 and J * 3 by using respectively (4.5) and (4.6).

We will do now the Proof of Theorem 6.1.

Proof of Theorem 6.1. We first define for a given point (t 0 , x 0 ) ∈ (0, T ) × J and for r, R > 0, the set

We introduce

We then define the following functions

and

(6.6)

We will prove that z and z are respectively sub and super solutions of (2.17). In this case, the comparison principle will imply that z ≤ z and by construction, we have that z ≤ z. Hence, we will get that

where u 0 is the unique viscosity solution of (2.17). Let us prove that z is a sub-solution of (2.17) (the proof for z is similar and we skip it). We argue by contradiction and assume that there exists a test function ϕ ∈ C 1 ((0, T ) × J) (in the sense of Thm. 2.7 in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF]), and a point ( t, x) ∈ (0, +∞) × J such that

where

We denote by ȳ = sign(x)d(0, x) with sign : J → R, defined by

Up to changing ϕ at infinity, we can assume that for ε small enough, we have

First case: x = 0. We only consider the case where x ∈ J * 3 since the other cases can be treated in the same way. Let p = ∂ 3 ϕ( t, x) which according to Corollary 6.3 satisfies p ≤ 0 and

Let r small enough so that ȳ -r > 0. We define a new test function φ : [0, T ] × R → R given by φ(t, y) = ϕ(t, ye 3 ) for y > 0.

For (t, y) ∈ P r,r ( t, ȳ) (see (4.2)), ε small enough and by definition of φ and ω, we have that Appendix A. Appendix: extension

Our result (micro to macro) can be extended to the case of a convergent junction with N incoming roads with an imposed rule near the junction point. The strategy is always the same: inject using the "cumulative distribution functions" the ODE into a non-local PDE and then obtain the homogenization result in the framework of viscosity solutions. The number of these functions depends on the number of branches and the different type of leaders far from the junction point. To be more precise, we consider a junction J with N incoming roads and one outgoing road as in We assume that the imposed rule is the following: n 1 pedestrians enter from J 1 then n 2 pedestrians from J 2 ,..., then n N pedestrians from J N and this phenomena will be repeated. We write a microscopic model following the same ideas of (2.10). Far before the junction point, the velocity of each pedestrian depends on the distance to its leader at time t = 0. We also add the function ω which models pedestrian's respect to the imposed rule. After the junction point, each pedestrian W i will follow W i+1 . We write the model using the variables U i defined by

In this case, the scheme to obtain our micro-macro result is the following: 1) Let i = 1, .., N . Using the pedestrians located at t = 0 on J i ∪ J N +1 , we define n i ∈ N * distribution functions ρ 1 i , ρ 2 i , ..., ρ ni i . 2) We inject these function into a system of non-local PDE. The number of equations of this system is n 1 + n 2 + ... + n N . 3) Following the ideas of our work, we can prove the following theorem.

Theorem A.1. We denote by lead(i) the leader of pedestrian i at time t = 0. We assume that U i (0) ≤ U lead(i) (0) -h 0 .

We define h 1 , h 2 , ..., h N +1 ≥ h 0 . We also assume that there exists R > 0 such that U lead(i) (0) -U i (0) = h k if W i (0) ∈ J k , U i (0) < -R with k = 1, .., N U i+1 (0) -U i (0) = h k+1 if W i (0) ∈ J N +1 , U i (0) > R.


Christian Mougin

Nathalie Cheviron

Cédric Repinçay

Mickael Hedde

Guillermina Hernandez Raquet

Cédric Repincay

Guillermina Hernandez-Raquet

Earthworms highly increase ciprofloxacin mineralization in soils

Keywords: Antibiotics, Fluoroquinolone, Persistence, Bioturbation, Ecotoxicological assessment, Pig slurry

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Fluoroquinolones are a major class of antibiotics world-wide used for human and veterinary medicine. Excreted from cattle and humans in urine and faeces, they enter the environment directly through grazing livestock or indirectly through the use of manure, slurry and sewage sludge as fertilizers of agricultural soils [START_REF] Thiele-Bruhn | Pharmaceutical antibiotic compounds in soils-a review[END_REF][START_REF] Ding | Effect of antibiotics in the environment on microbial populations[END_REF]. Runoff and wastewater are then responsible for the contamination of aquatic ecosystems. There is a growing concern about the potential impact of antibiotic residues on the environment, as well as on human health. Among fluoroquinolones, ciprofloxacin is widely detected in the environment following its own use, or as the main metabolite of enrofloxacin.

The effects and fate of ciprofloxacin on the soil ecosystem are poorly known, thus underlining the ecotoxicological relevance of this antibiotic. The quinolone has been shown to be genotoxic, and [START_REF] Mersch-Sundermann | DNA damage caused by antibiotic drugs: quinolones[END_REF] discussed early possible mutagenic effects in eukaryotic cells.

Effects of ciprofloxacin on several microbial biomarkers have been reviewed by [START_REF] Ding | Effect of antibiotics in the environment on microbial populations[END_REF]. Recently, [START_REF] Girardi | Biodegradation of ciprofloxacin in water and soil and its effects on the microbial communities[END_REF] reported an inhibition of soil respiration, as well as a shift in microbial abundance and diversity of microbial communities after application of ciprofloxacin, although at high amounts of contamination. By contrast, considering realistic levels of antibiotic contamination of pig slurry used to amend French soils, we failed to demonstrate any significant effects of the antibiotic on the microbial community, as well as on enzymatic biomarkers of the white-rot fungus Trametes versicolor maintained in spiked liquid culture [START_REF] Igel-Egalon | Impact of antibiotics from pig slurry on soil microbial communities, including the basidiomycete Trametes versicolor[END_REF].

For many years, ciprofloxacin has been measured in aquatic and terrestrial environments, and its concentration in soils ranges from 6 to 750 µg kg -1 according to the type of soil and way of contamination [START_REF] Golet | Determination of fluoroquinolone antibacterial agents in sewage sludge and sludge-treated soil using accelerated solvent extraction followed by solid-phase extraction[END_REF][START_REF] Kemper | Veterinary antibiotics in the aquatic and terrestrial environment[END_REF][START_REF] Martinez-Carballo | Environmental monitoring study of selected veterinary antibiotics in animal manure and soils in Austria[END_REF]. The antibiotic appears poorly biodegradable and adsorbs onto soil particles with respect to their physic-chemical properties and mineralogy [START_REF] Girardi | Biodegradation of ciprofloxacin in water and soil and its effects on the microbial communities[END_REF][START_REF] Vasudevan | pH-dependent ciprofloxacin sorption to soils: interaction mechanisms and soil factors influencing sorption[END_REF][START_REF] Wu | Adsorption and intercalation of ciprofloxacin on montmorillonite[END_REF]). As a consequence, it is hypothesized that ciprofloxacin can easily accumulate into the soil. Recently, [START_REF] Wen | Effect of complexation on the accumulation and elimination kinetics of cadmium and ciprofloxacin in the earthworm Eisenia fetida[END_REF] reported the accumulation and elimination of ciprofloxacin in the redworm Eisenia fetida grown in quartz sands spiked with an aqueous solution of antibiotics. Ciprofloxacin was also measured in several subcellular fractions obtained from the earthworms.

Here, we assessed the fate of 14 C-labelled ciprofloxacin entering an agricultural soil according to several scenarios of contamination. The first originality of our approach was to include in microcosms soil-dwelling macroorganisms well known for their ability in bioturbation. They are intended to model a realistic scenario of soil-plant-water system (SPWS). Our second originality was the use of relevant species obtained from French soil (and not from rotting vegetation or compost as Eisenia) maintained in their real growing environment, the soil. We hypothesized that the efficiency of microbial communities for biotransformation of recalcitrant chemicals can be enhanced by the engineering role of earthworms. To our knowledge, no other study reported the action of earthworms on the fate of ciprofloxacin has been published.

Materials and methods

Chemicals and reagents

Chemicals and solvents were of analytical grade. All chemicals were purchased from Sigma-Aldrich (France). [2-14 C]-ciprofloxacin (555 MBq mmol -1 ) has been obtained from Moravek Biochemicals and Radiochemicals (Isobio, Fleurus, Belgium). Solvents were purchased from Carlo-Erba (Val de Reuil, France).

Soil characteristics

The silt loam (Luvisol, FAO classification) was collected in the 10-20 cm layer of an experimental site (La Cage, INRA, Versailles, France). It comprised 21.6 % sand, 61.1 % silt and 17.3 % clay. Its content in organic carbon was 0.98 %, and its pHwater was 7.1. Its water holding capacity (WHC) was 15.64 ± 0.21 % at pF = 2.5 and 6.79 ± 0.21 % at pF = 4.2. The soil was roughly homogenized, sieved at 2 mm, and used without storage for the preparation of the soil-plant-water systems (SPWSs).

Pig slurry sampling and characterization

The pig slurry, referred to here as green slurry, was from the quarantine livestock building of the IFIP (French Pig Institute) at Romillé (Brittany, France). The animals were cross-bred from Large White and Landrace and weighed 40-100 kg (fattening pigs). They were fed an unchanging cereal formula consisting of wheat (44 %), barley (30 %), soybeans (18 %), molasses (2 %) and amino acids and minerals (6 %). They did not receive antibiotics.

The pig slurry resulting from about three months of manure accumulation from 20 pigs was homogenized in a pit (about 25 m3) using a submersible pump for about 20 min. Then, 20 L of slurry was collected and stored at 4 °C for 1 day until use.

The slurry pH (7.5), dry matter content (26.2 ± 0.1 g kg -1 ), chemical oxygen demand (36 ± 2 g O2 L - 1 ), total Kjeldahl nitrogen (3.05 ± 0.03 g kg -1 ) and total ammonia-N (2.3 ± 0.1 g kg -1 ) were determined according to standard methods (APHA 1992). All parameters were analysed in triplicate. These data are in agreement with previous reports on pig slurry characterization in French husbandry [START_REF] Burton | Manure management, treatment strategies for sustainable agriculture[END_REF].

Design of the soil-plant-water system design Here, we used the soil-plant-water system (SPWS) developed by our team [START_REF] Ghanem | Fate of herbicides and nonylphenol in soilplant-water systems amended with contaminated sewage sludge[END_REF], and set-up according to [START_REF] Igel-Egalon | Impact of antibiotics from pig slurry on soil microbial communities, including the basidiomycete Trametes versicolor[END_REF]. Four experimental treatments have been prepared: non-amended control soil, soil amended with slurry spiked with low and high amounts of antibiotic and finally, soil spread with acetonic solution of the antibiotic.

Three days before the beginning of the experiments, 1 kg dry soil fractions were prepared and moistened to 50 % of WHC. They were added to each SPWS as lower layers. For the control soil, 1 kg was similarly prepared and added to the SPWS as upper layer. Concerning the slurry-spiked treatments, 25 mL pig slurry was spread on 1 kg of soil which was added to the SPWS. That amount of slurry corresponded to the equivalent dose of 30 T dw ha -1 . The low-spiked and the highspiked treatments received pig slurry previously spiked with labelled ciprofloxacin (42 and 420 kBq, respectively) to ensure a final concentration of 25 and 250 µg kg -1 dry soil in the upper layer, respectively. Finally, in the fourth SPWS, the upper layer of soil was spread with an acetonic solution of labelled ciprofloxacin (420 kBq) to ensure a final concentration of 250 µg kg -1 dry soil.

Afterwards, total soil was moistened to 80 % of WHC and the SPWSs were closed. The four SPWSs were incubated for 168 days under 16 h light at 20 °C and 8 h darkness at 18 °C. Soil moisture was checked every week and adjusted.

After 56 days of incubation, three seeds of wheat were sowed in each SPWS. After 84 days of incubation, four mature specimens of oligochaeta (two Aporrectodea caliginosa Savigny 1826 and two Aporrectodea longa Savigny 1826), previously collected from the same plot as the soil used to fill the SPWSs, were introduced in each SPWS. They were still alive and active at the end of the experiment.

Analytical procedures

We measured the mineralization, the partitioning and the leaching of antibiotic in the soil and the associated biological response in terms of soil microbial activity and transfer in wheat.

Regarding antibiotic mineralization, a stream (0.5 L min -1 ) of wet air was continuously flushed through the SPWSs to allow 14 CO2 trapping in NaOH solutions. NaOH solutions were changed every 7 days and 14 CO2 was determined by liquid scintillation counting. In addition, two soil cores (1.5 cm diameter and 20 cm depth) were extracted to determine extractable and bound 14 C in the layers of the soil and soil/slurry mixtures at the beginning of the experiment, and after 84 days of incubation. At the end of the experiment (168 days of incubation), soils corresponding to the location to the previous upper and lower layers were separated, and samples were collected in each of them to measure the influence of earthworm bioturbation. Ultrasonic extraction was performed to recover radioactive compounds from 10 g fresh soil samples during 20 min, in the presence of 40 mL acidified acetonitrile (ACN) [START_REF] Martinez-Carballo | Environmental monitoring study of selected veterinary antibiotics in animal manure and soils in Austria[END_REF]). The suspension in ACN was kept overnight and shaken during 1 h prior to centrifugation. After removal of the extract, another sequence comprising addition of ACN, shaking and centrifugation was performed. The two extracts were finally pooled and filtered. To assess the leaching of ciprofloxacin, water was sprayed after 70 days of incubation onto the soils to mimic 20-mm rainfalls and to allow recovery of leachates.

Wheat seedlings were harvested and dried after a 28-day period of growth (84 days of incubation).

The radioactivity was measured in all liquid fractions (extracts, leachates) by liquid scintillation counting. Non-extractable radioactivity (soil, soil/slurry samples and seedlings) was determined by combustion, followed by liquid scintillation counting.

After 28, 56 and 84 days of incubation, dehydrogenase activity (DHA; E.C. 1.1.1.1), indicating the level of global metabolic process in the soil, was measured according to an adapted protocol from [START_REF] Schaefer | L'activité déshydrogénasique comme mesure de l'activité biologique globale des sols[END_REF]. That activity was intended to evidence a possible effect of the antibiotic on the soil microbial activity. One millilitre distilled water and 250 µL aqueous solution of 1,3,5triphenyltetrazolium chloride (TTC) at 40 g L -1 were mixed with 1 g fresh soil in 30-mL centrifuge tubes. Controls were prepared without TTC. Incubation was performed at 37 °C for 16 h. After that period, 5 mL ethanol was added to the mixture which was shaken for 5 min in the dark. The medium was incubated at 37 °C for 2 h, and shaken after 1 h. Finally, TTC was added to the control and all the mixtures were centrifuged 5 min at 6,000g. The absorbance of the supernatant (red colour in case of formazan formed) was measured at 485 nm. Results were expressed in mU g -1 dry soil.

Availability of ciprofloxacin in the soil solution

The amounts of ciprofloxacin in the soil solution were determined using the protocol of [START_REF] Gaillardon | Influence of soil moisture on short-term adsorption of diuron and isoproturon by soil[END_REF] adapted by [START_REF] Kollmann | Impact of nonylphenol surfactants on fungi following the application of sewage sludge on agricultural soils[END_REF]. Aliquots of slurry and soil (250 µL + 750 mg) or soil alone (1 g) were spiked with labelled (4.2 kBq) ciprofloxacin corresponding to 2.5 lg antibiotic, the highest amount was used for SPWS incubation. After adding the chemical, the solvent (acetone) was left to evaporate for 30 min. The chemicals were then absorbed onto the slurry and soil for 24 h at 4 °C before these spiked aliquots were mixed with 9 g non-spiked soil. Soil/slurry and soil samples (10 g equivalent dry matter) were placed in 5-cm diameter Petri dishes to give a 3-4 mm thick layer, and moistened dropwise to achieve 80 % of WHC. Although earthworms have not been added, the dishes were incubated in similar conditions than SPWSs, and radioactivity determined according to the above protocols. It was hypothesized that no transformation of the antibiotic occurred during that period.

Concentrations of ciprofloxacin in the soil solution were determined at the beginning of the incubation (just after soil spiking) and after 84 and 168 days of incubation. Two superposed 42.5 mm diameter glass micro fibber filters GF/A (Whatman) were laid on the soil surface and a slight pressure was applied for 10 s to favour wetting of the filters. The upper filter was then recovered.

The volume of soil solution and the dissolved radioactivity retained in the filter were determined by weighing and liquid scintillation counting.

Results and discussion

Mineralization of ciprofloxacin in soils

The mineralization of ciprofloxacin was monitored in the four soil-plant-water systems (SPWSs) (Fig. 1). No mineraliza-tion occurred in the SPWS containing the reference soil. By contrast, traces of mineralization (less than 0.01 % of initial amount of radioactivity) were detected during the first week of incubation in the three SPWSs containing 14 C-ciprofloxacin. That situation corresponded likely to the mineralization of impurities contained in the commercial solution of labelled ciprofloxacin (purity of 98.5 %). Then, mineralization increased very slightly to a maximal value of 0.019 % of initial radioactivity at day 84 for the lower spiking amount, and 0.007-0.010 % for the highest amount of contamination. Interestingly, earthworm introduction in the soil at day 84 led to a sevenfold increase in antibiotic mineralization, reaching 0.058-0.080 % of initial radioactivity after the next 84-day period, depending on the SPWS considered. Several explanations can be provided concerning that last result. Earthworm bioturbation can affect the fate of contaminants by changing their mobility and sorption to the soil in modifying the quality and dynamics of soil organic matter [START_REF] Ngo | The effect of earthworms on carbon storage and soil organic matter compo-sition in tropical soil amended with compost and vermicompost[END_REF]. It can also provide hot spots of microbial activity [START_REF] Monard | Combined effect of bioaug-mentation and bioturbation on atrazine degradation in soil[END_REF]. Finally, earth-worms can be efficient degraders directly by their own enzymatic equipment [START_REF] Achazi | Cytochrome P450 and dependent activities in unexposed and PAH-exposed terrestrial annelids[END_REF], or indirectly because of the microflora present in their intestine [START_REF] Shin | Anaerobic biotransformation of dinitrotoluene isomers by Lactococcus lactis subsp. lactis strain 27 isolated from earthworm intestine[END_REF]. However, ciprofloxacin mineralization remained low. Repeated application of the antibiotic (i.e. by slurry amendment) is exceeding its degradation rates, ensuring the accumulation of the antibiotic on certain soil layers [START_REF] Kemper | Veterinary antibiotics in the aquatic and terrestrial environment[END_REF].

Dehydrogenase activity values measured in the soil samples along the incubation ranged from 0.03 to 0.12 mU g -1 dry soil. They were not enhanced after wheat growth, indicating the lack of effect of the rhizosphere. They were in the range of values previously reported in the similar soil [START_REF] Igel-Egalon | Impact of antibiotics from pig slurry on soil microbial communities, including the basidiomycete Trametes versicolor[END_REF]) spiked with lower amounts of antibiotics (at the ng kg -1 level). Here, no clear relationship has been evidenced between dehydrogenase activity, the type of soil treatment or the presence of ciprofloxacin. Our results evidenced the persistence of a microbial activity of the soil in the presence of antibiotic, and the poor mineralization did not seem linked with any toxic effect on soil microorganisms. Similar low levels of mineralization rates have been recently reported in soil using the OECD assay 307 [START_REF] Girardi | Biodegradation of ciprofloxacin in water and soil and its effects on the microbial communities[END_REF]. It was reported to be sensitive to the contamination rate. Taken together, these results reveal a poor ability of soil endogenous microflora to mineralize ciprofloxacin. To our knowledge, microbial or fungal strains able to metabolize ciprofloxacin have not been identified from environmental media to date.

Distribution and transfer of labelled carbon among the compartments of the system

The mass-balance analysis of labelled carbon was per-formed in the two SPWSs treated with the highest amount of ciprofloxacin at the beginning of the experiment, and then after 84 and 168 days of incubation (Table 1). Just after soil spiking, labelled carbon in the two SPWSs was mainly associated with the solid phase of the soil (77-78 % of initial labelled carbon applied), and only 21 % of ciprofloxacin could be extracted. That result confirmed the strong and rapid sorption of the antibiotic onto the soil particles, without effect of the type of spiking. These initial values are in accordance with recent soil studies using radiochemicals [START_REF] Girardi | Biodegradation of ciprofloxacin in water and soil and its effects on the microbial communities[END_REF].

After 84 days of incubation including the simulated rainfall event, a more detailed analysis showed that the main fraction of radioactive carbon was stabilized in the upper soils, probably as nonextractable residues. In parallel, the amount of extracted labelled carbon decreased to less than 0.6 % of initial radioactivity in the upper layers of soil. Nevertheless, a part of the radioactivity was measured in the lower layers of soil, showing a significant mobility of the antibiotic after the rainfall. Radioactive material was not detected in the extracts of that layer of soil. The strong sorption of antibiotics is described as governed by soil organic matter, soil minerals and soil pH [START_REF] Thiele-Bruhn | Pharmaceutical antibiotic compounds in soils-a review[END_REF]. In our experiment, mobility of ciprofloxacin within the two layers of soil may be closely linked with mobility of dissolved organic matter (here endogenous and exogenous) or minerals. Clays of the studied soil contained 60 % smectite (montmorillonite) with a high CEC (100-150 cmol+ kg -1 ) and the pH of the soil was 7.1. These properties were reported by [START_REF] Wu | Adsorption and intercalation of ciprofloxacin on montmorillonite[END_REF] and [START_REF] Vasudevan | pH-dependent ciprofloxacin sorption to soils: interaction mechanisms and soil factors influencing sorption[END_REF] to ensure a strong adsorption of the zwitterionic form of ciprofloxacin on the minerals.

After 168 days of incubation, earthworm activity resulted in a strong mixing of the upper and lower layers in the SPWSs, as evidenced by the observation of developed burrow systems through the glass wall of the experimental devices and a final packing of the soils. Analysis of the radioactivity confirmed that observation. 51.7 and 38.4 % of the radioactivity were detected in the soil fractions corresponding to the previous upper and lower layers of the soil amended with the spiked slurry. These values were 48.0 and 43.2 % in the case of the soil spread with the antibiotic. Radioactive carbon was never detected in the organic extracts obtained after 168 days of incubation of soils. The lack of published data concerning the action of earthworms on the fate of antibiotics prevents the comparison of our innovative results with other studies.

We studied the mobility of labelled carbon to soil leachates after rainfall simulation at 70 days of incubation, and to wheat seedlings during the 56-84-day period (Table 1). Radioactivity was detected in low amounts when expressed in % of the applied dose of labelled carbon, in leachates from the spiked soils. When expressed as concentrations of equivalent ciprofloxacin, the values were 0.46 and 0.16 µg L -1 corresponding to 1.39 and 0.48 nM in leachates. Ciprofloxacin was also absorbed by wheat seedlings during their early growing period. All the wheat seeds were germinated 7 days after sowing, and the seedlings provided after 28 days of growth identical dry bio-mass whatever the soil treatment (around 70 mg dry biomass). Expressed as % of the applied dose or calculated as concentrations (µg kg -1 dry biomass), values of absorbed radioactivity were very low, suggesting a weak transfer to that higher plant.

Finally, total amounts of labelled carbon measured ranged from 88 to 100 % of initial amounts applied. They were around 90 % after 161 days. Experimental shifts ( 14 CO2 trapping, soil combustion…) may occur, but we cannot exclude any bioaccumulation of labelled carbon in earthworms. Finally, because the amounts of radioactive compounds in the liquid fractions were very low and prevented any qualitative analysis using HPLC coupled to radioactivity detection, we attributed labelled carbon to non-transformed ciprofloxacin.

Availability of ciprofloxacin in the soil solution That experiment was performed to evidence variations of ciprofloxacin availability in the soil solution with respect to time, in the SPWSs spiked with the highest amount of 14 C-chemical (Fig. 2). Because of the previous results demonstrating the recalcitrance of the antibiotic, we hypothesize that labelled carbon corresponded to untransformed ciprofloxacin. Just after soil spiking, ciprofloxacin concentration measured in the soil solution was 0.57 ± 0.07 µg L -1 when it was provided by the slurry, and 0.33 ± 0.07 µg L -1 when it was directly spread. That difference between the soil solutions from two soils with different ways of contamination was not observed in the massbalance analysis of SPWSs based on solvent extraction. In that previous experiment, using the % of labelled carbon extracted at the beginning of the experiment, we were able to calculate a concentration of ciprofloxacin reaching 6 lg L -1 in the organic extracts. Taken together, these results allow us to conclude that only a weak fraction of the solvent extractable antibiotic was available in the soil solution. Then, amounts of ciprofloxacin were 0.41 ± 0.03 and 0.36 ± 0.02 µg L -1 after 168 days of incubation. Statistical analysis using two-way ANOVAs showed significant crossed effects (P = 0.05) of the type of contamination and length of the incubation on ciprofloxacin availability in the soil solution.

Our results demonstrated the low availability of ciprofloxacin for soil organisms. When expressed in molarity, average antibiotic concentrations in the soil solutions ranged from 0.81 to 1.90 nM.

These values can be used to design further experiments for the assessment of antibiotic toxicity on (micro)-organisms in liquid cultures.

Conclusion

From the present study based on the use of radiolabelled ciprofloxacin, it is concluded that the antibiotic introduced in a French Luvisol through spiked pig slurry or direct spreading undergoes a very weak microbial mineralization under aerobic conditions, and may stabilize in the soil as nonextractable residues. Nevertheless, its mineralization and distribution in soils were affected by the action of earth-worms through their engineering activity. Taken together, our results suggest that the action of these macroorganisms may be studied in more details, because they change the exposure of soil organisms to ciprofloxacin, and, in turn, the possible eco-toxicological impact of the antibiotic.

Table 1 Mass-balance analysis at the beginning of the experiment, and after 84 and 168 days of incubation, in the compartments of our soil-plant-water systems filled with soil amended with slurry spiked with the high amounts of antibiotic, or soil spread with the high amount of antibiotic The results are expressed as means of triplicate, with standard deviation lower than 10 % of the mean

Fig. 1

 1 Fig. 1 Mineralization of [2-14 C]-ciprofloxacin in control soil (filled circle), soil amended with slurry spiked with the (open circle) and high (filled inverted triangle) amounts of antibiotic, and soil spread with the high amount of antibiotic (open triangle) during the 168 days of incubation. Wheat seedlings were allowed to grow between 56 and 84 days of incubation. Earthworms have been introduced in the soils after 84 days of incubation. They increased the mineralization rate of the antibiotic

Acknowledgments

The present study, as a part of the research programme DIPERPHA, was supported by the French ANR pro-gramme SEST and by ADEME. We are grateful to Dr. Patrick Dabert (GERE unit, CEMAGREF, Rennes, France) for providing and analysing the samples of green pig slurry. The authors thank C. Marrauld (PESSAC Department, INRA, Versailles, France) for performing dehydrogenase activity measurements in soils.