HAL Id: hal-03211700
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Preprint submitted on 9 Aug 2021
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Micro to macro traffic model: the case of a convergent junction
Nader El Khatib, Nicolas Forcadel, Mamdouh Zaydan
To cite this version:
Nader El Khatib, Nicolas Forcadel, Mamdouh Zaydan. Micro to macro traffic model: the case of a convergent junction. 2021. �hal-03211700v2�
Micro to macro traffic model: the case of a convergent junction
N. El Khatib
1, N. Forcadel
2, M. Zaydan
1July 28, 2021
Abstract
This paper is a continuation of the work [18] where authors established a connection between a microscopic and a macroscopic traffic model on a divergent junction. We prove the same result in the case of a convergent one (two incoming roads and one outgoing road) which is much more difficult since we have to define priority rules at the junction point. At the end of the paper, we briefly describe how we extend this result for the case of a general junction with several incoming and outgoing roads.
AMS Classification: 35D40, 90B20, 35B27, 35F20, 45K05.
Keywords: specified homogenization, Hamilton-Jacobi equations, traffic flow, non-local opera- tors, Slepčev formulation, viscosity solutions, microscopic model on junction.
1 Introduction
Several papers proposed partial differential equations to model the traffic on networks, see [30, 8, 22, 24]. Macroscopic traffic model are effective for describing the traffic at large scales but from modeling point of view, since the dynamics of each driver can’t be described individually, they are hardly justifiable and incapable to predict all traffic situations (for example free or congested flow). At the opposite, microscopic model are strong from modeling point of view but complicated to implement in order to describe the traffic at large scales. Among works modeling traffic at the micro scale, we refer to [4, 25, 33, 9], the last one appeared recently and considered the network case.
A rigorous way to justify these macroscopic models is to establish a connection between them and microscopic ones. Micro to macro connection on a simple road can be found in several paper, we refer for example to [3, 14, 23, 13, 7, 10]. The case of network is more recent and very few results are available. Let us cite for example [20, 19] in which the authors consider a very simple junction with one incoming and one outgoing road (with a local perturbation at the origin). We also like to mention [18] where the case of a divergent junction (one incoming road and several outgoing roads) is considered.
In this paper, we will establish a micro-macro connection on a junction. Since this is the novelty of the paper (see the paragraph below), we will present our results in details for the case of a covergent junction. More precisely, we will consider a microscopic traffic model on a convergent junction, and will 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 [27]
and in [6, 32]. At the end of this paper, we will show how to extend the results in the case of a
1Lebanese American University, Department of computer science and mathematics, Byblos campus, P.O Box 36, Byblos, Lebanon
2Normandie Univ, INSA de Rouen, LMI (EA 3226 - FR CNRS 3335), 76000 Rouen, France, 685 Avenue de l’Université, 76801 St Etienne du Rouvray cedex. France
general junction. We obtain our convergence result by using the theory of viscosity solution (see [12, 11]). The first homogenization result in this framework was obtained in [31] and since then, several works considered this problem, like [17, 16, 21, 1].
Presentation of our results. Since the global strategy to get the convergence result is similar to that in [18, 20, 19], we will focus on the presentation of the new ideas and we will refer to [18]
when the proof of a result is similar. In [18], authors considered a periodic distribution of cars on a divergent junction and obtained at the macroscopic scale a Hamilton-Jacobi equation with a flux limiting condition at the junction point as in [27]. We will extend this result and consider a periodic distribution of cars on a convergent junction. 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 [18]) of this result using the idea introduced in [29] 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 will use here the definition of a new distance d0 which can be seen as a security distance while crossing the junction point. We will present our results in the case of a convergent junction with two incoming roads and one outgoing road to simplify the presentation of our work and lessen the article. In fact, we will see that in the case of a general junction (see Appendix A), the mathematical results can be easily extended but need more "cumulative distribution functions" and more partial differential equations.
2 Main result
2.1 The microscopic model
We consider a microscopic traffic model on a convergent junction with two incoming roads and one outgoing road. The junction is defined byJ =J1∪J2∪J3whereJ1 andJ2are the incoming roads and isometric toR− andJ3 is the outgoing road and isometric toR+. 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 defineJk∗=Jk\ {0}. To be more precise, the definitions of the branches are given by
Jk = (−∞,0]·ek fork= 1,2, J3= [0,+∞)·e3, with e1, e2ande3 three different unit vectors inR2.
Letx, y∈J withx=xk·ek, y=yl·el: the distanced(x, y) inJ is defined by d(x, y) =
(|xk−yk| ifk=l,
|xk|+|yl| ifk6=l.
Before writing the ordinary differential equations, we will first give the ideas of our model.
1) Vehicles distribution: We will assume a periodic distribution: two vehicles enter from the branchJ1 and then one vehicle fromJ2.
2) Vehicles labels: Cars that are initially located onJ3 (resp. onJ1∗∪J2∗) have positive labels (resp. strictly negative labels). On each branch, vehicle’s positions is strictly increasing with respect to their labels. In particular, ifi, j ∈Zwithi < j <0, then carj will pass by the junction point before cari.
3) Velocity dependence: the velocity of a car far before the junction point depends on its initial leader on the same branch. Approaching the junction point, the velocity of cari∈Z\Nwill depend on the distance to "the last car assigned to pass by zero before it", i.e car with label i+ 1. To write our proposed model, we will need to introduce two continuous functions: one modeling the transition of the micro-model before and after the point zero, and the other one modeling the order of junction point entering.
4) Notations: The position of cariat timet >0 is denoted byWi(t) and its velocity isWi′(t).
We define
L1={−1} ∪ {−2−3k, k∈N}, L2={−4−3k, k∈N},
L3={−3k, k∈N∗}.
(2.1)
If i∈ L1∪ L2 (respL3), then Wi(0)∈J1∗ (resp. Wi(0)∈J2∗). We will write a model that satisfies:
– Ifi∈N, then Wi will circulate onJ3. Its leader is alwaysWi+1. – Ifi∈ L1, thenWi will go fromJ1to J3. Its leader is alwaysWi+1.
– Ifi∈ L2, thenWi will go fromJ1to J3. Its leader onJ1 isWi+2 and is Wi+1 onJ3. – Ifi∈ L3, thenWi will go fromJ2to J3. Its leader onJ2 isWi+3 and is Wi+1 onJ3. Fori∈Z, we have
Wi(t)∈J ⇐⇒Wi(t) =Ui(t)·ek (2.2) with
Ui(t)≥0 andk= 3 ifi∈N,
(Ui(t)<0 ifk= 1) and (Ui(t)≥0 ifk= 3) ifi∈ L1∪ L2, (Ui(t)<0 ifk= 2) and (Ui(t)≥0 ifk= 3) ifi∈ L3.
Using the above equivalence (2.2), we will consider in the rest of the paper the variables Ui. Our model is described by the following differential equations,
Ui′(t) = ¯φ(Ui(t), V1(Ui+1(t)−Ui(t)), V3(Ui+1(t)−Ui(t))) ifi∈ L1∪N, Ui′(t) =F(Ui(t), Ui+1(t), V1(Ui+2(t)−Ui(t)), V3(|Ui+1(t)| −Ui(t))) ifi∈ L2, Ui′(t) =F(Ui(t), Ui+1(t), V2(Ui+3(t)−Ui(t)), V3(|Ui+1(t)| −Ui(t))) ifi∈ L3,
(2.3)
with
F(x, y, a, b) = ¯φ(x, a, b)·ω(x, y,|x| − |y|),
where ¯φandω are continuous functions modeling respectively the transition of velocities and the junction point entering order. Their definitions are given below in (2.4) and (2.5).
Remark 2.1. The role of the function φ¯ is to model the change of optimal velocities on each branch of the junction. Far before the junction point, the optimal velocity is V1 (resp. V2) on J1
(resp. on J2) and after this point, the optimal velocity is V3. The role of the function ω is to avoid cars collision at the junction point. The function ω ensures that if car Ui is on the branch J1 (resp. J2), then the vehicleUi+1 located onJ2 (resp. J1) will pass by the junction point before Ui. Let us remark that in the first equation of (2.3)we do not integrateω since the leader’s label does not change and no collision can occur.
Remark 2.2. We could consider a more general vehicles distribution (like 2 vehicles entering from J1 and then two from J2, etc..). Since the simplest case is to consider one vehicle from J1
and then one fromJ2, we decided to consider the above distribution to avoid the simplest case and the complex cases.
We will now give some assumptions concerning the model and the dynamics.
Assumptions (A)
Assumptions on the velocities
(A1) For i = 1,2,3, the function Vi : R → R+ is Lipschitz continuous, non-negative and non-decreasing onR.
(A2) There existsh0∈(0,+∞) such that for allh≤h0,Vi(h) = 0.
(A3) There existshmax> h0 such that for allh≥hmax,Vi(hmax) =Vmax.
(A4) There exists a realp3 (resp. p1, p2) such that the functionp7→pV3(−1/p) (resp. p7→
pV1(−3/2p),p7→pV2(−3/p)) is decreasing on [−1/h0, p3) (resp. [−3/2h0, p1) ,[−3/h0, p2) ) and increasing on [p3,0) (resp. [p1,0) ,[p2,0) ).
Assumptions on the transition function
(A5) The function ¯φ:R×R×R→Ris defined as follow, forR1> R2> R3,
φ(x, a, b) =¯
a ifx≤ −R1,
a−min(a, b) R2−R1
(x−R1) +a if−R1< x≤ −R2, min(a, b) if−R2< x≤ −R3, b−min(a, b)
R3
x+b if−R3< x≤0,
b ifx >0.
(2.4)
Figure 1: Schematic representation of ¯φ
Assumptions on the functionsω
(A6) The function ω:R3→Ris defined by
ω(x, y, p) =α(x) + (1−α(x))β(y, p) (2.5) withαa lipshitz function defining the zone in which the car Ui∈J1∗ (resp. Ui∈J2∗) starts spotting the position ofUi+1on the other branchJ2(resp. J1) . It’s defined in the following way,
α(x) =
(1 if x <−R3−ror x > r,
0 if −R3< x <0. (2.6)
The functionβ is a lipshitz function describing the fact that if carUi+1 is after the junction point, its influence onUi disappears since collision is not possible in this case. It’s defined by
β(y, p) =
(1 ify > r ,
ζ(p) ify≤0 , (2.7)
whereζis a lipshitz function modeling the fact thatUi+1will enter the junction point before Ui,
ζ(p) =
(1 ifp > d0+r,
0 ifp≤d0, (2.8)
with
r, d0>0 and r < R1−R3. (2.9) Remark 2.3 (Comments onω). The role of the functionω is to manage the order of entering the junction point. Let i∈ L2. IfUi(t) is too close to the junction, i.e. −R3< Ui(t)<0, then it will stop (ω= 0) if Ui+1∈J2∗ and if
|Ui(t)| − |Ui+1(t)|< d0.
The distance d0 can be interpreted as a security distance when crossing the junction point.
Remark also that when Ui+1 enters the branch J3 i.e. Ui+1(t)> r, the influence of ω disappears (ω = 1). Finally, let us remark that the definition of the function αon [−R3,0]means that the influence ofω begins directly when the driver’s velocity tends to depend on the distance to Ui+1.
2.2 The macroscopic model.
Letk0= 1/h0. Fork= 1,2,3, we defineHk:R→Rby
H1(p) =
−p−3k0
2 forp <−3k0
2 ,
−V1
−3 2p
|p| for −3k0
2 ≤p≤0,
p forp >0,
(2.10)
Figure 2: The role ofω.
H2(p) =
−p−3k0 forp <−3k0,
−V2
−3 p
|p| for −3k0≤p≤0,
p forp >0,
(2.11)
and
H3(p) =
−p−k0 forp <−k0,
−V3
−1 p
|p| for −k0≤p≤0,
p forp >0.
(2.12)
Note that for allk= 1,2,3, the functionHk is continuous, coercive and because of (A4), there exists a unique pointpk≤0 such that
Hk is decreasing on (−∞, pk),
Hk is increasing on (pk,+∞). (2.13)
We denote by
H0= max
k∈{1,2,3}H0k (2.14)
with
H0k= min
p∈RHk(p). (2.15)
We introduce now the definition of the gradient of a function defined onJ. Ifx∈J, then we define
ux(x) =
(∂iu(x) ifx∈Ji∗, (∂1u(0), ∂2u(0), ∂3u(0)) ifx= 0,
with∂iu(x) the derivative of the functionuwith respect tox∈Ji. We denote byJ∗=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 [27] and given by
u0t+Hk(u0x) = 0 for (t, x)∈(0,+∞)×Jk∗, u0t+FA u0x
= 0 for (t, x)∈(0,+∞)× {0}, u0(0, x) =u0(x),
(2.16)
whereAis the flux limiter andFA is defined by FA(p1, p2, p3) = max
A, H+1(p1), H+2(p2), H−3(p3)
, (2.17)
with
H−k(p) =
H(p) ifp≤pk,
H(pk) if p≥pk, and H+k(p) =
H(pk) if p≤pk,
H(p) ifp≥pk. (2.18) Remark 2.4. In [28], authors showed that equation (2.16)is equivalent (deriving in space) to the trafic model introduced in [30].
2.3 Micro-macro connection.
To establish the micro-macro connection, we need to introduce three cumulative functionsρ, σand τ. The need of three functions arises from the different type of leaders: far before zero, on branch J1, the leader can beUi+1 orUi+2 and on branchJ2, the leader isUi+3. We haveρ(t, Ui(t)) =−i (resp. σ(t, Ui(t)) = −i, τ(t, Ui(t)) = −i) for i ∈ L1∪N (resp. i ∈ L2∪N, i ∈ L3∪N). The functionsρ,σandτ are defined as follows,
ρ(t, y) =−
X
i≥0
H(y−Ui(t)) + (−1 +H(y−U−1(t))) + (−1 +H(y−U−2(t))
− X
i=−2−3k,k∈N∗
3 (−1 +H(y−Ui(t))), (2.19)
σ(t, y) =−
X
i≥0
H(y−Ui(t)) + 4(−1 +H(y−U−4(t)))
− X
i=−4−3k,k∈N∗
3(−1 +H(y−Ui(t))), (2.20)
and
τ(t, y) =−
X
i≥0
H(y−Ui(t)) + X
i=−3k,k∈N∗
3(−1 +H(y−Ui(t)))
, (2.21)
with
H(x) =
1 ifx >0 0 ifx≤0.
The main result of this paper is given by the following theorem.
Theorem 2.5 (Junction condition by homogenization: application to traffic flow). Assume (A) and that at initial time, we have,
Ui(0)≤Ui+1(0)−h0 if i∈ L1∪N, Ui(0)≤Ui+2(0)−h0 if i∈ L2 , Ui(0)≤Ui+3(0)−h0 if i∈ L3 .
(2.22)
We also assume that there exists R >0 such that
Ui+1(0)−Ui(0) =h1 ifi∈ L1 andUi(0)<−R, Ui+2(0)−Ui(0) =h1 ifi∈ L2 andUi(0)<−R, Ui+3(0)−Ui(0) =h2 ifi∈ L3 andUi(0)<−R, Ui+1(0)−Ui(0) =h3 ifi∈NandUi(0)≥R,
(2.23)
Figure 3: In blue (resp. pink and red): vehiclesUi whose leader’s label at the initial time isi+ 1 (resp. i+ 2 andi+ 3). The definition of the functionρ(resp. σandτ) depends on vehicles colored in blue (resp. pink and red).
withh1, h2, h3≥h0. We define two functions u0 andw0 by
u0(x) =− 3 2h1
x1{x<0}− 1 h3
x1{x≥0}, w0(x) =−3
h2
x1{x<0}− 1 h3
x1{x≥0}. Let ε >0 andχε:R+×J →Rbe the function defined by
χε(t, x) =
ρε(t,−d(0, x)) if x∈J1∗, τε(t,−d(0, x)) if x∈J2∗, σε(t, d(0, x)) if x∈J3
(2.24)
with
ρε(t, x) =ερ t
ε,x ε
, σε(t, x) =εσ t
ε,x ε
, τε(t, x) =ετ t
ε,x ε
. We define
u0(x) =
u0(−d(0, x)) ifx∈J1∗, w0(−d(0, x)) ifx∈J2∗, u0(d(0, x)) ifx∈J3.
Then there exists a uniqueA∈[H0,0]such that the functionχε converges towards the unique solution u0 of (2.16).
Remark 2.6. The proof of this result will be divided into 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 (2.16).
3) Deduce Theorem 2.5.
3 The non-local equation
In this section, we will first define the non-local operators and then show that (ρ, σ, τ) is a viscosity solutions of a non-local PDE. To do this, we will use the functionφ(x, a, b) =−φ(x,¯ −a,−b) defined where ¯φdefined in (2.4). We remark thatφ(x, a,·) andφ(x,·, b) are non-decreasing functions which is a crucial property to obtain the comparison principle (see Proposition 4.2).
3.1 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. Leta∈R, we define
Ea(z) =
(0 ifz≥a 1 ifz < a,
E˜a(z) =
(0 ifz > a
1 ifz≤a, (3.1)
and
F(z) =
(0 ifz >−1
−1 ifz≤ −1,
F˜(z) =
(0 ifz≥ −1
−1 ifz <−1. (3.2) LetU, V, W :R→R. We define the following non-local operators: fora= 1,2,3 andi= 1,2,3,
Mi,a(U,[V])(x) = Z +∞
0
Vi′(z)E−a(V(x+z)−U(x))dz−Vmax, (3.3)
N(U,[V])(x) = Z
z≥x
V3′(|z| −x)E−1(V(z)−U(x))dz−Vmax, (3.4) and
K(U,[V])(x) = Z
R
ωz(x, z,|x| − |z|)F(V(z)−U(x))dz+ 1, (3.5) with ωdefined in (2.5). Using the non-local operators defined above, we introduce
R1,1(x, U,[V]) (x) =φ(x, M1,1(U,[V])(x), M3,1(U,[V])(x)), (3.6) R1,2(x, V,[U],[W]) (x) =φ(x, M1,2(V,[U])(x), N(V,[W])(x))·K(V,[W])(x), (3.7) R2,1(x, W,[U]) (x) =φ(x, M2,3(W,[W])(x), N(W,[U])(x))·K(W,[U])(x). (3.8) Remark 3.1 (Comments on the non-local operators definition.). Using the non-local operators defined above, we can inject the ODE (2.3)into a non-local PDE (see Lemma 3.3). In particular, we have that
M1,1(ρ(t, Ui(t)),[σ(t,·)]) (Ui(t)) =−V1(Ui+1(t)−Ui(t)) if i∈ L1, M3,1(ρ(t, Ui(t)),[σ(t,·)]) (Ui(t)) =−V3(Ui+1(t)−Ui(t)) if i∈ L1, M1,2(σ(t, Ui(t)),[ρ(t,·)])(Ui(t)) =−V1(Ui+2(t)−Ui(t)) if i∈ L2∪N, M2,3(τ(t, Ui(t)),[τ(t,·)]) (Ui(t)) =−V2(Ui+3(t)−Ui(t)) if i∈ L3, N(σ(t, Ui(t)),[τ(t,·)])(Ui(t)) =−V3(|Ui+1(t)| −Ui(t)) if i∈ L2∪N, N(τ(t, Ui(t)),[ρ(t,·)]) (Ui(t)) =−V3(|Ui+1(t)| −Ui(t)) if i∈ L3, K(σ(t, Ui(t)),[τ(t,·)])(Ui(t)) =ω(Ui(t), Ui+1(t),|Ui(t)| − |Ui+1(t)|) if i∈ L2∪N, K(τ(t, Ui(t)),[ρ(t,·)]) (Ui(t)) =ω(Ui(t), Ui+1(t),|Ui(t)| − |Ui+1(t)|) if i∈ L3.
Remark that in the definition of Mi,a, the variable z is positive since it models the distance between Ui and its leader whose position is on the same branch or on J3 (looking ahead). The different values ofa= 1,2,3 represent the label ofUi’s leader which can be i+ 1, i+ 2 andi+ 3.
In the definition ofN, the variablez models the leader on the other branch,Ui+1. We takez > x since forz≤x, the velocity of Ui is zero due to the presence of the functionω1. In the definition of K the variable z ∈Rsince it’s modeling the car Ui+1. In the notationRl,k, the labels l andk represent respectively the branches where the initial leader (resp. leader after the junction point) is located att= 0.
In the same way, we define ˜Mi,a,N˜ and ˜K replacingEa andF respectively by ˜Ea and ˜F and then we get the definition of ˜R1,1,R˜1,2 and ˜R2,1. Finally, we can easily remark that
−Vmax≤R1,1(x, U,[V]) (x), R1,2(x, V,[U],[W]) (x), R2,1(x, W,[U]) (x)≤0. (3.9) Forε >0, we define the following non-local operators: fora= 1,2,3, andi= 1,2,3,
Mi,aε (U,[V])(x) = Z +∞
0
Vi′(z)E−a(V(x+εz)−U(x))dz−Vmax (3.10)
Nε(U,[V])(x) = Z
z≥xε
V3′
|z| −x ε
E−1(V(εz)−U(x))dz−Vmax, (3.11) and
Kε(U,[V])(x) = Z
R
ωz
x ε, z,
x ε − |z|
F(V(εz)−U(x))dz+ 1. (3.12) We then define
Rε1,1x
ε, U,[V]
(x) =φx
ε, M1,1ε (U,[V])(x), M3,1ε (U,[V])(x)
, (3.13)
Rε1,2x
ε, V,[U],[W]
(x) =φx
ε, M1,2ε (V,[U])(x), Nε(V,[W])(x)
·Kε(V,[W])(x), (3.14) Rε2,1x
ε, W,[U]
(x) =φx
ε, M2,3ε (W,[W])(x), Nε(W,[U])(x)
·Kε(W,[U])(x). (3.15) In the same way, we define ˜Mi,aε ,N˜εand ˜KεreplacingEa andF respectively by ˜Ea and ˜F and then we get the definition of ˜R1,1ε ,R˜ε1,2 and ˜Rε2,1.
3.2 The non-local PDE
In this subsection, we will prove that (ρ, σ, τ) is a discontinuous viscosity solutions of the following non-local PDE,
ρt(t, x) +R1,1(x, ρ(t, x),[σ(t, .)]) (x)|ρx(t, x)|= 0 for (t, x)∈R+×R, σt(t, x) +R1,2(x, σ(t, x),[ρ(t, .)],[τ(t, .)]) (x)|σx(t, x)|= 0 for (t, x)∈R+×R, τt(t, x) +R2,1(x, τ(t, x),[ρ(t, .)]) (x)|τx(t, x)|= 0 for (t, x)∈R+×R,
(3.16)
with R1,1, R1,2 and R2,1 defined in (3.6),(3.7) and (3.8). We use the definition of viscosity solutions introduced in [34]. This definition allows to have a stability result for the non-local term. We refer to [16, Proposition 4.2] for the corresponding stability result. We give now the definition of viscosity solutions of (3.16).
Definition 3.2 (Viscosity solutions for (3.16)). Let u, v, w : (0,+∞)×R → R be upper semi- continuous (resp. lower semi-continuous) functions. We say that (u, v, w) is a viscosity sub- solution (resp. super-solution) of (3.16) on(0,+∞)×R, if we have the following:
1) ifϕ∈C1([0,+∞)×R)such thatu−ϕreaches a maximum (resp. a minimum) at the point (t, x), we have
ϕt(t, x) +R1,1(x, u(t, x),[w(t, .)])|ϕx(t, x)| ≤0, (resp. ϕt(t, x) + ˜R1,1(x, u(t, x),[w(t, .)])|ϕx(t, x)| ≥0) .
2) Ifϕ∈C1([0,+∞)×R)such thatv−ϕreaches a maximum (resp. a minimum) at the point (t, x), we have
ϕt(t, x) +R1,2(x, v(t, x),[u(t, .)],[w(t, .)])|ϕx(t, x)| ≤0, (resp. ϕt(t, x) + ˜R1,2(x, v(t, x),[u(t, .)],[w(t, .)])|ϕx(t, x)| ≥0).
3) Ifϕ∈C1([0,+∞)×R)such thatw−ϕreaches a maximum (resp. a minimum) at the point (t, x), we have
ϕt(t, x) +R2,1(x, w(t, x),[u(t, .)])|ϕx(t, x)| ≤0, (resp. ϕt(t, x) + ˜R2,1(x, w(t, x),[u(t, .)])|ϕx(t, x)| ≥0) .
We say that(u, v, w)is a viscosity solution of (3.16)if(u∗, v∗, w∗)and(u∗, v∗, w∗)are respectively a sub-solution and a super-solution of (3.16).
We have the following theorem.
Theorem 3.3. The function(ρ, σ, τ) is a discontinuous viscosity solution of (3.16). Conversely, if(u, v, w)are bounded and continuous viscosity solution of (3.16)satisfying for some timeT >0, and for all t∈(0, T)
u(t,·), v(t,·), w(t,·)are decreasing, then the points Ui(t), defined by
u(t, Ui(t)) =−i fori∈ L1∪N, v(t, Ui(t)) =−i fori∈ L2∪N, w(t, Ui(t)) =−i fori∈ L3∪N, are solutions of (2.3)on (0, T).
The proof of Theorem 3.3 is an easy adaptation of the proof of Theorem 7.1 in [18] using the following lemma.
Lemma 3.4. Let (Ui)i∈Z be the solution of (2.3). Then we have
Ui′(t) =−R1,1(Ui(t), ρ(t, Ui(t)),[σ(t,·)]) (Ui(t)) if i∈ L1, Ui′(t) =−R1,2(Ui(t), σ(t, Ui(t)),[ρ(t,·)],[τ(t,·)](Ui(t))) if i∈ L2∪N, Ui′(t) =−R2,1(Ui(t), τ(t, Ui(t)),[ρ(t,·)](Ui(t))) if i∈ L3.
(3.17)
Proof. We will 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 [18].
4 Results for the non-local PDE with initial conditions
In this section, we will consider the following PDE:
ut(t, x) +R1,1(x, u(t, x),[v(t, .)]) (x)|ux(t, x)|= 0 for (t, x)∈R+×R, vt(t, x) +R1,2(x, v(t, x),[u(t, .)],[w(t, .)]) (x)|vx(t, x)|= 0 for (t, x)∈R+×R, wt(t, x) +R2,1(x, w(t, x),[u(t, .)]) (x)|wx(t, x)|= 0 for (t, x)∈R+×R,
u(0, x) =u0(x) forx∈R,
v(0, x) =v0(x) forx∈R,
w(0, x) =w0(x) forx∈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 conditionsu0, v0 andw0are lipschitz continuous functions. In addition, to control the gradient of the limit solution of (2.16), we need the following assumption. Letk0=−1/h0.
(A0) The functionsu0, v0 andw0 are non-increasing. LetR1the positive parameter appearing in the definition (2.4) ofφ. We assume that fory < x <−R1,
(−3k
0
2 (x−y)−1≤v0(x)−u0(y),
−3k0(x−y)−3≤w0(x)−w0(y), and forx > y >0,
−k0(x−y)−1≤v0(x)−u0(y).
Remark 4.1. The constante−k0 is the maximal cars density. The initial conditions u0 andw0
(with v0 =u0) in Theorem 2.5 satisfy condition (A0). The definition of u0 andw0 is crucial to get the homogenization result in Theorem 2.5.
4.1 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)≤u0(x),v(0, x)≤v0(x) andw(0, x)≤w0(x), (resp. u(0, x)≥u0(x),v(0, x)≥v0(x) andw(0, x)≥w0(x)).
We begin first with the comparison principle.
Proposition 4.2 (Comparison principle for (4.1)). Assume (A). Let (u, v, w) be a visocsity sub- solution of (4.1) and (ˆu,ˆv,w)ˆ be a viscosity super-solution of (4.1) in the sense introduced of Definition 3.2. Let us also assume that there exists a constantK >0 such that for allt >0,
u(t, x)≤u0(x) +Kt and −u(t, x)ˆ ≤ −u0(x) +Kt for x∈R, v(t, x)≤v0(x) +Kt and −v(t, x)ˆ ≤ −v0(x) +Kt for x∈R, w(t, x)≤w0(x) +Kt and −w(t, x)ˆ ≤ −w0(x) +Kt for x∈R. Then we have for allt >0,
u(t, x)≤u(t, x)ˆ for x∈R, v(t, x)≤ˆv(t, x) for x∈R, w(t, x)≤w(t, x)ˆ for x∈R.
Proof. The proof is very similar to the one in [19] and uses the monotony ofE, F, φk and wk, so we skip it.
In the proof of convergence (see Section 6), we need a comparison principle on bounded sets.
For a given point (t0, x0)∈(0, T)×Rand forr, R >0, we define the set
Pr,R(t0, x0) = (t0−r, t0+r)×(x0−R, x0+R). (4.2) Theorem 4.3 (Comparison principle on bounded sets for (4.1)). Assume (A). If (u, v, w) and (ˆu,v,ˆ w)ˆ are respectively sub-solution and super-solution of (4.1)on the open set Pr,R such that
u≤uˆ v≤vˆ w≤wˆ outsidePr,R, then
u≤uˆ v≤vˆ w≤wˆ onPr,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 K1 = 3
2Vmaxk0 and K2= 3Vmaxk0. We define
(u+(t, x), v+(t, x), w+(t, x)) = (K1t+u0(x), K1t+v0(x), K2t+w0(x)), (u−(t, x), v−(t, x), w−(t, x)) = (u0(x), v0(x), w0(x).
Then (u+, v+, w+)and(u−, v−, w−)are respectively super and sub solution of (4.1).
Proof. The proof is very simple. We just use the bounds ofR1,1, R1,2andR2,1 (see (3.9)) and the gradient bounds (A0).
Applying Perron’s method (see [2] or [26] to see how to apply Perron’s method for problems with non-local terms), joint to the comparison principle, we obtain the following result.
Theorem 4.5(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)−u0(x)| ≤K1t, |v(t, x)−v0(x)| ≤K1t and |w(t, x)−w0(x)| ≤K2t. (4.3)
4.2 Gradient estimates
Theorem 4.6 (Control of the oscillations). Assume (A0)-(A). Let T > 0 and (u, v, w) be the solution of (4.1) on (0, T) given in Theorem 4.5. For all t >0, the functions u(t,·), v(t,·) and w(t,·) are non-increasing. Let x < −R1 and h > 0 small enough such that x+h < −R1 and h < h0. For allt∈(0, T), we have
−3k0
2 h−1≤v(t, x+h)−u(t, x), (4.4)
−3k0h−3≤w(t, x+h)−w(t, x). (4.5) Let x >0and0< h < h0. For allt∈(0, T), we have
−k0h−1≤v(t, x+h)−u(t, x). (4.6)
Proof. The proof of monotony is similar to the one in [20, Theorem 4.10]. We will only do the proof of (4.4) since the other inequalities can be proved in the same way. We will use the technique introduced in [29] for the proof of local gradient estimates. Letx0<−R1andδ >0 small enough such that
(h+ 2δ < h0,
x0+h+δ <−R1. (4.7)
We will prove for allt∈(0, T),y∈Bδ(x0) andx∈Bδ(x0+h), that
−3k0
2 (x−y)−1−Lδ(y−x0)2−Lδ(x−x0−h)2≤v(t, x)−u(t, y) (4.8) withLδ = K1T
δ2 andK1 defined in (4.3). In particular, takingy=x0 andx=x0+h, we obtain (4.4). To prove (4.8), we introduce
∆ = [0, T)×Bδ(x0+h)×Bδ(x0) and consider the following supremum
M = sup
(t,x,y)∈∆
u(t, y)−v(t, x)−3k0
2 (x−y)−1−Lδ(x−x0−h)2−Lδ(y−x0)2
.
We want to prove thatM ≤0. We argue by contradiction and assume that M >0.
Step 1: the test function. Forη >0 small, we define ϕ(t, x, y) =u(t, y)−v(t, x)−3k0
2 (x−y)−1−Lδ(x−x0−h)2−Lδ(y−x0)2− η T−t. Sinceϕis continuous, it reaches a maximum on ∆ at a point that we denote by (¯t,x,¯ y). Classically¯ we have forη small enough,
0< M
2 ≤ϕ(¯t,x,¯ y).¯
Step 2: t >¯ 0. By contradiction, assume first that ¯t= 0. Then we have η
T < u0(¯y)−v0(¯x)−3k0
2 (¯x−y)¯ −1≤0, where we used (A0).
Step 3: |¯x−x0−h| 6=δ and |¯y−x0| 6=δ . By contradiction, assume that|¯x−x0−h|=δ.
Using the barriers and (A0), we get that
0< η
T ≤u0(¯y)−v0(¯x)−3k0
2 (¯x−y)¯ −1−K1T −Lδδ2
≤K1T−Lδδ2= 0
where we used the definition ofLδ. In the same way, we have|¯y−x0| 6=δ.
Step 4: utilization 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 ≤M˜1,2 v(¯t,x),¯ [u(¯t,·)]
(¯x)·
−3k0
2 −2Lδ(¯x−x0−h)
−M1,1 u(¯t,y),¯ [v(¯t,·)]
(¯y)·
−3k0
2 + 2Lδ(¯y−x0)
(4.9) where we used the fact that ¯φ(x, a, b) =aandω(x, y, p) = 1 forx <−R1. We claim that
M1,1 u(¯t,y),¯ [v(¯t,·)]
(¯y) = 0. (4.10)
In fact if (4.10) is true we will obtain a contradiction in (4.9) since ˜M1,2 ≤0. The definition of M1,1 can be written forz≥h0since for z < h0, we haveV1′ = 0. We have
M1,1 u(¯t,y),¯ [v(¯t,·)]
(¯y) = Z +∞
h0
V1′(z)E−1(v(¯t,y¯+z)−u(¯t,¯y))dz−Vmax. Using thatϕ(¯t,x,¯ y)¯ >0, we have
v(¯t,x)¯ −u(¯t,y)¯ <−1.
Ifz > h0, using (4.7), we have
¯
y+z > x0−δ+h0> x0+δ+h >¯x.
Sincev(t,·) is non-increasing, we get forz > h0,
v(¯t,y¯+z)−u(¯t,y)¯ ≤v(¯t,x)¯ −u(¯t,y)¯ <−1
which impliesE−1(v(¯t,y¯+z)−u(¯t,y)) = 1 and then (4.10). This ends the proof.¯
5 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 [15].
5.1 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:
Mi,ap (U,[V])(x) = Z +∞
0
Vi′(z)E−a(V(x+z)−U(x) +pz)dz−Vmax, (5.1) with Ea defined in (3.1).
1) For everyp≤0, there exists a uniqueλ1∈Rsuch that there exists a viscosity solution(u, v) of
M1,1p (u(x),[v])(x)|p+ux|=λ1 x∈R , M1,2p (v(x),[u])(x)|p+vx|=λ1 x∈R , u, vare bounded.
(5.2)
2) For everyp≤0, there exists a uniqueλ2∈Rsuch that there exists a viscosity solutionwof (M2,3p (w(x),[w])(x)|p+wx|=λ2 x∈R,
wis bounded. (5.3)
3) For every p ≤ 0, there exists a unique λ3 ∈ R such that there exists a viscosity solution (u, v, w)of
M3,1p (u(x),[v])(x)|p+ux|=λ3 x∈R, M3,1p (v(x),[w])(x)|p+vx|=λ3 x∈R, M3,1p (w(x),[u])(x)|p+wx|=λ3 x∈R, u, v, w are bounded.
(5.4)
In particular, λi =Hi(p)with Hi defined in (2.10),(2.11) and (2.12).
Proof. We can easily verify that
(u, v) =
0,1 2
is a solution of (5.2) withλ1=−|p|V1
−3 2p
w= 0 is a solution of (5.3) withλ2=−|p|V2
−3 p
(u, v, w) = (0,0,0) is a solution of (5.4) withλ3=−|p|V3
−1 p
. The uniqueness ofλi is a classical result and the reader can refer to [31] for the proof.
5.2 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: forλ∈R, we consider a viscosity solution (u,v,w) of
R1,1(x,u(x),[v]) (x)|ux|=λ x∈R, R1,2(x,v(x),[u],[w]) (x)|vx|=λ x∈R, R2,1(x,w(x),[u]) (x)|wx|=λ x∈R,
(5.5)
where the non-local operatorsR1,1, R1,2 and R2,1 are defined in (3.6),(3.7) and (3.8). We now prove the existence result of the corrector. ForA ≥H0 with H0 defined in (2.14), we introduce the real numbersp1−, p2− andp3+satisfying
H1(p1−) =H−1(p1−) =A, H2(p2−) =H−2(p2−) =A, H3(p+3) =H+3(p3+) =A.
Theorem 5.2 (Existence of a global corrector for the junction). Assume (A).
i) (General properties) There exists a constant A¯ ∈ [H0,0] such that there exists a solution (u,v,w) of (5.5) with λ = A and such that there exists a constant C and a globally Lipschitz continuous functionmsuch that
|u(x)−m(x)| ≤C, |v(x)−m(x)| ≤C, |w(x)−m(x)| ≤C. (5.6) ii) (Rescaling ) Forε >0, we set
uε(x) =εu x
ε
, vε(x) =εv x
ε
, wε(x) =εw x
ε .
Then (along a subsequenceεn→0), we have the following convergence locally uniformly: uε−→
ε→0U, vε−→
ε→0U andwε−→
ε→0W withU andW satisfying
|U(x)−U(y)| ≤C|x−y| for allx, y ∈R, H1(Ux) =A for allx <0, H3(Ux) =A for allx >0,
(5.7) and
|W(x)−W(y)| ≤C|x−y| for all x, y∈R, H2(Wx) =A for all x <0, H3(Wx) =A for all x >0.
(5.8)
In particular, we have (with U(0) =W(0) = 0)
U(x) =p3+x1{x>0}+p1−x1{x<0}, (5.9) W(x) =p3+x1{x>0}+p2−x1{x<0}. (5.10) iii)(Uniqueness of the flux limiter A) We define the following set of functions
S={(u,v,w)s.t. ∃Lipschitz functionmand C≥0 satisfying (5.6)}. Then we have
A= inf{λ∈R:∃ (u,v,w)∈ S solution of (5.5) }.
5.3 Proof of Theorem 5.2
This subsection contains the proof of Theorem 5.2. Let R > R1 (R1 the constant appearing in (2.4)), we want to findλR, such that there exists a solution (uR, vR, wR) of
G1R x, uR(x),[vR], uRx
=λR ifx∈[−lR, lR], G2R x, vR(x),[uR],[wR], vxR
=λR ifx∈[−lR, lR], G3R x, wR(x),[uR], wxR
=λR ifx∈[−lR, lR],
(5.11) with
G1R(x, U,[V], q) =ψR−(x)ψR+(x)φ(x, M1,1(U,[V])(x), M3,1(U,[V])(x))|q|+
+ (1−ψR−(x))·H−1(q) + (1−ψ+R(x))·H+3(q), (5.12)
G2R(x, V,[U],[W], q) =ψR−(x)ψ+R(x)φ(x, M1,2(V,[U])(x), N(V,[W])(x))K(V,[W])(x)|q|+
+ (1−ψR−(x))·H−1(q) + (1−ψ+R(x))·H+3(q), (5.13) G3R(x, W,[U], q) =ψ−R(x)ψR+(x)φ(x, M2,3(W,[W])(x), N(W,[U])(x))K(W,[U])(x)|q|+
+ (1−ψ−R(x))·H−2(q) + (1−ψR+(x))·H+3(q), (5.14) andψ+R, ψ−R∈C∞,ψR± :R→[0,1], withψ−R(x) =ψR+(−x),
ψ+R ≡
1 on [−∞, R]
0 outside [R+ 1,+∞), andψR+non-increasing,
andlR>> R+ 1>>max(hmax, R3+r). To each operatorGiR, we associate ˜GiR which is defined in the same way but replacing the non-local operatorsMi,a, N andK by ˜Mi.a,N˜ and ˜K.
