Homogenization of second order discrete model with local perturbation and application to traffic flow

The goal of this paper is to derive a traffic flow macroscopic model from a second order microscopic model with a local perturbation. At the microscopic scale, we consider a Bando model of the type following the leader, i.e the acceleration of each vehicle depends on the distance of the vehicle in front of it. We consider also a local perturbation like an accident at the roadside that slows down the vehicles. After rescaling, we prove that the "cumulative distribution functions" of the vehicles converges towards the solution of a macroscopic homogenized Hamilton-Jacobi equation with a flux limiting condition at junction which can be seen as a LWR (Lighthill-Whitham-Richards) model.

1. Introduction. The modelling and simulation of traffic flow is a challenging task in particular in order to design infrastructure. Indeed, there are some examples in which the construction of a new infrastructure did not improve the traffic. For example, in Stuttgart, Germany, after investments into the road network in 1969, the traffic situation did not improve until a section of newly build road was closed for traffic again (see [22]). This is known as the Braess' paradox. In the past years, a lot of work has been done concerning the modelling and simulation of traffic flows problems.
Traffic flow can be modelled at different scales depending on the level of details one wants to observe: the microscopic scale (describes the dynamics of each of the vehicles), the macroscopic scale (describes the dynamics of the density of vehicles) and the mesoscopic scale (describes the dynamics of the density of vehicles but the car-to-car interactions are not lost).
Microscopic models are considered more justifiable because the behaviour of every single vehicle can be described with high precision whereas macroscopic models are based on assumptions which are less verifiable. Another way to justify macroscopic models is to derive them from microscopic models by rescaling arguments.
The problem of deriving macroscopic models from microscopic ones has already been studied for models of the type following the leader (i.e. the velocity or the acceleration of each vehicle depends only on the distance to the vehicle in front of it). We refer for example to [3,8,16,18,23] where the authors rescaled the empirical measure and obtained a scalar conservation law (LWR model). In particular, passing from microscopic to macroscopic model for second-order models was instead investigated in [3,15], where the Aw-Rascle model is derived as the limit of a second order follow-the-leader model.
In this paper we establish a connection between a car-following model and a fluid-dynamic model. This result is a generalization of the results of [13] to a second order microscopic model. We consider a second order microscopic model of followthe-leader type with a local perturbation. In such model, the whole traffic flow is determined by the dynamics of the very first vehicle (the leader ). We will establish a connection between this second order discrete model and a macroscopic model equivalent to a LWR model. The idea is to rescale the microscopic model, which describes the dynamics of each vehicle individually, in order to get a macroscopic model which describes the dynamics of density of vehicles.
The model we study here is similar to the one considered in [12], but in our work, as in [13], we assume that there is a local perturbation (located at the origin for example) that slows down the vehicles and we want to understand how this local perturbation influences the macroscopic dynamics. Due to this perturbation, it is natural to get an Hamilton-Jacobi equation with a junction condition at the origin and an effective flux limiter. Further, our result is stronger than the one in [13] because our microscopic model is a second order model which is more realistic than the first order model considered in the last paper. From a mathematical point of view the fact of considering a second order model presents many technical difficulties. First, we need to consider a system of two non-local PDEs instead of a single equation [11,12]. Moreover, the two functions that we consider have to satisfy certain properties that derive from the physical characteristics of the microscopic model and those properties need to be proven for the system of non-local PDEs which is more complicated in the case of a second order model than in the case of a first order model.
Paper organization. The paper is organized as follows. In Section 2, we present the microscopic model for which we will present an homogenization result. In Section 3, we inject the system of ODEs into a system of PDEs and we present our main results. Section 3.3 is dedicated to the definition of the non-local operators which appear in the PDEs given in Section 3. In Section 4, we introduce the notion of viscosity solutions for the considered problems and give stability, existence and uniqueness results. In Sections 5 and 6 we present the correctors necessaries for the proof of convergence which is located in Section 7. Section 8 contains the proof of existence of correctors for the junction, where we use the idea developed in [1,14] and in the lectures of Lions at the "College de France" [25], which consists in constructing correctors on truncated domains. In Section 9 we show the link between the system of ODEs and the system of PDEs which proof is in Appendix B. Finally in Appendix A we analyse the properties of the microscopic model.

2.
A first main result. In this paper, we are interested in a second order microscopic model that can simulate the presence of a local perturbation. In order to do that, we considered a modified version of the model introduced by Bando [4]. More precisely, we consider a "follow-the-leader" model of the following form where U j denotes the position of the j-th vehicle,U j its velocity andÜ j its acceleration. The function φ simulates the presence of a local perturbation located at the origin and we denote by r its radius of influence. In this model, a ∈ R and V : R → R represent respectively the drivers sensitivity and the optimal velocity function. We make the following assumptions on V , φ and on the coefficient a. Assumption (A).
Remark 2.1 (Remark on (A6)). In the case φ = 0 on an open interval (therefore φ 0 = 0) all the vehicles left of the perturbation would come to a full stop. This case lacks any interest and therefore we can assume that φ 0 > 0.
Remark 2.2 (Remark on (A7)). Assumption (A7) yields that for all (b, x) ∈ R 2 , the function is non-decreasing. This result is particularly important later in the paper because it implies that the systems we consider later in this work are monotone in the sense of Ishii and Koike [21], which will imply the uniqueness of the solution we consider.
As we said in the introduction, in order to obtain an homogenization result for (11), we will inject the system of ODEs into a system of PDEs. To do so, we proceed as in [10,13] by introducing the rescaled "cumulative distribution function", which is the primitive of the rescaled empirical measure, defined by, with The macroscopic model. We define H : R → R, by H is decreasing on (−∞, p 0 ) H is increasing on (p 0 , +∞), and we denote by We want to show that the rescaled "cumulative distribution function" converges to the solution of the following macroscopic model.
where A has to be determined and F A is defined by with The initial condition u 0 is a function that satisfies with f (ε) → 0 as ε → 0. According to [19], for all A ∈ R, there exists a unique solution u 0 of (7).
Remark 2.3. We notice that in the case of traffic flow, (7) is equivalent (deriving in space) to a LWR model (see [24,26]) with a flux limiting condition at the origin. In fact, the fundamental diagram of the model is pV (1/p) and u 0 x corresponds to the density of vehicles.
Passage from a microscopic to a macroscopic model. The main result of this paper is the following convergence result.
Theorem 2.4 (Passage from a microscopic to a macroscopic model). Assume (A). There exists a unique A ∈ [H 0 , 0] such that the function ρ ε defined by (2) converges locally uniformly towards the unique solution of (7).

Main results.
3.1. Injecting the system of ODEs into a system of PDEs. In the rest of the paper, we will work with an equivalent formulation of (1). We borrow the idea from [9,11,12] and consider for all j ∈ Z, Using this new function, we obtain the following system of ODEs equivalent to (1) for all j ∈ Z, for all t ∈ (0, +∞),

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In Appendix A, we give some properties of system (11), such as maximal velocities of the vehicles and minimal and maximal distance between two consecutive vehicles. We now introduce the "cumulative distribution function" for (Ξ j ) j , defined by Under assumption (A), (ρ ε , σ ε ) is a discontinuous viscosity solution (see Theorem 3.3) of the following non-local equation, for all (t, x) ∈ (0, +∞) × R, The definition of M ε and L ε is postponed to the next section. We submit equation (13) to the following initial condition. For all x ∈ R, We also assume that the initial condition satisfies the following assumption.
(A0) (Gradient bound). Let k 0 = 1/h 0 . The functions u 0 and ξ ε 0 are Lipschitz continuous functions, such that and Remark 3.1. The initial conditions u 0 and ξ ε 0 are "regular" functions such that for all ε > 0 we have with f (ε), g(ε) → 0 as ε goes to 0. For ε = 1, the conditions on the gradients translate the fact that at the initial time there is at least h 0 meters between two consecutive vehicles. In the rest of the paper we are interested in the behaviour of ρ ε and σ ε as ε goes to 0. This in fact translates to studying the behaviour of the traffic as the number of vehicles per unit length goes to infinity. For ε = 1 condition (17) translate the fact that at initial time the velocity of the vehicles must be bounded so the ordering of the vehicles is kept. The fact that ξ ε 0 depends on ε comes from the rescaling. In fact, given that σ ε is the "cumulative distribution function" of (Ξ j ) j which are defined using the velocity of the vehicles, an ε appears multiplying the velocity when rescaling (see [12,Remark 1.2]). Therefore, ξ ε 0 tends to u 0 as ε goes to zero. Finally, to simplify the notations, we denote by ξ 0 = ξ ε 0 for ε = 1. 3.2. Convergence result. Theorem 2.4 is a consequence of the following theorems. The proof of Theorem 3.2 is postponed until Section 7 and the proof of Theorem 3.3 is postponed until Section 9.
The following theorem ensures that when we use (7) we only evaluate the function H in the interval [−k 0 , 0]. The proof of Theorem 3.5 is postponed until Section 7.

3.3.
Definition of the non-local operators. In this section, we clarify equation (13). We will give the definition of M and L, and then the definition of M ε and L ε . To do this, we first introduce the following functions.
For x, p ∈ R, we then define the following non-local operators

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with D = h max + 3V max /(2α) + 2r/φ 0 (see Appendixes A and B for more details on where the constant D comes from). We can now define L p . For x, y ∈ R, In the same way, we defineM p ,K p andÑ p by replacing E, F and I respectively bỹ E,F andĨ. Similary, For p = 0, we define and Remark 3.6 (Remarks on the non-local operators). First let us notice that the domain of integration in the non-local operators is bounded by a constant D := h max + 3V max /(2α) + 2r/φ 0 , this comes from the fact that the velocities of the vehicles as well as the distance between two consecutive vehicles from model 11 are bounded (see Appendix A). In particular, there exists a constant M 0 > 0 (independent of p), such that we have the following bounds on the non-local operators, Finally, we would like to point out that given the fact that the function V is non-decreasing (assumption (A2)) and that the function F ≥ 0 and therefore K(Σ, Finally, we introduce for ε > 0, and The bounds provided by Remark 3.6 remain valid for the non-local operators depending on ε > 0. Remark 3.7 (Lagrangian formulation). Another way to treat this problem is to consider a Lagrangian formulation, like in [12], considering the functions, u(t, y) = U y (t) and ξ(t, y) = Σ y (t).
The couple (u, ξ) satisfies for all (t, y) . We note that the system we obtain is much more simple. Nevertheless, the difficulty with this formulation is that the function φ is evaluated at u(t, y) and not at a physical point of the road. At the macroscopic scale, we then expect to get a junction condition located at u = 0. The notion of junction in this case is not well defined and this is why we use the formulation (13)(where the perturbation function is evaluated at a point of the road). This will allow us to use the results of Imbert and Monneau [19] concerning quasi-convex Hamiltonians with a junction condition.
4. Viscosity solutions. This section is devoted to the definition and useful results for viscosity solutions of the problems considered in this paper. The reader is referred to the user's guide of Crandall, Ishii, Lions [6] and the book of Barles [5] for an introduction to viscosity solutions. In order to give a general definition, we will give the definition of viscosity solutions for the following equation, with p ∈ R, and for all (t, x) ∈ (0, +∞) × R, with ψ : R → [0, 1] a Lipschitz continuous function. We also use the following notations for the upper and lower semi-continuous envelopes of a locally bounded function u:  (33)). Let T > 0. Let u : R + × R → R and ξ : R + × R → R be upper semi-continuous (resp. lower semi-continuous) functions. We say that (u, ξ) is a viscosity sub-solution (resp. super-solution) of (33) on [0, T ] × R if u(0, x) ≤ u 0 (x) and ξ(0, x) ≤ ξ 0 (x) (resp. u(0, x) ≥ u 0 (x) and ξ(0, x) ≥ ξ 0 (x)) and for all (t, x) ∈ (0, T ) × R, and for any test function ϕ ∈ C 1 ((0, T )×R) such that u−ϕ attains a local maximum (resp. a local minimum) at the point (t, x), we have and for all (t, x) ∈ (0, T ) × R and any test function ϕ ∈ C 1 ((0, T ) × R) such that ξ − ϕ attains a local maximum (resp. a local minimum) at the point (t, x), we have We say that (u, ξ) is a viscosity solution of (33) if (u * , ξ * ) and (u * , ξ * ) are respectively a sub-solution and a super-solution of (33). Proposition 4.2 (Stability result for (33)). Let (u n , ξ n ) be a sequence of uniformly bounded upper semi-continuous functions (resp. lower semi-continuous) and let (p n ) n be such that p n → p. We assume that (u n , ξ n ) is a sub-solution (resp. a super-solution) of (33) with p = p n . Let (u, ξ) = (lim sup * u n , lim sup * ξ n ) (resp. (u, ξ) = (lim inf * u n , lim inf * ξ n )). Then (u, ξ) (resp. (u, ξ)) is a sub-solution (resp. a super-solution) of (33).
Proof. The proof is classical and we refer to [10]. The only point to note is that both Hamiltonians in (33) are monotone with respect to the non-local operators (this is a consequence of assumption (A7) for the non-local operator K p ).

4.2.
Viscosity solutions for (7). The theory of viscosity solutions for Hamilton-Jacobi equations on networks was recently treated in several papers. We give here some results for viscosity solutions of (7) that will be used in the rest of paper and we refer to [19] for the general theory and for the proofs.  (7)). An upper semi-continuous (resp. lower semi-continuous) function u : [0, +∞) × R → R is a viscosity sub-solution (resp. super-solution) of (7) if u(0, x) ≤ u 0 (x) (resp. u(0, x) ≥ u 0 (x)) and for all (t, x) ∈ J ∞ and for all ϕ ∈ C 1 (J ∞ ) such that u ≤ ϕ (resp. u ≥ ϕ) in a neighbourhood of (t, x) ∈ J ∞ and u(t, x) = ϕ(t, x), We say that a function u is a viscosity solution of (7) if u * and u * are respectively a sub-solution and a super-solution of (7).We refer to this solution as A-flux-limited solution.
Now we recall an equivalent definition (Theorem 2.5 in [19]) for sub and super solution at the junction. We will also consider the following problem, u t + H(u x ) = 0 for t ∈ (0, T ) and x ∈ R\{0}.
(34) Theorem 4.5 (Equivalent definition for sub/super-solutions). Let H given by (4) and consider A ∈ [H 0 , +∞) with H 0 defined in (6). Given arbitrary solutions let us fix any time independent test function φ 0 (x) satisfying Given a function u : (0, T ) × R → R, the following properties hold true.
1. If u is an upper semi-continuous sub-solution of (34) satisfying then u is a H 0 -flux limited sub-solution. 2. Given A > H 0 and t 0 ∈ (0, T ), if u is an upper semi-continous sub-solution of (34) satisfying (36) and if for any test function ϕ touching u from above at (t 0 , 0) with for some ψ ∈ C 1 (0, +∞), we have then u is a A-flux limited sub-solution at (t 0 , 0). 3. Given t 0 ∈ (0, T ), if u is a lower semi-continuous super-solution of (34) and if for any test function ϕ satisfying (37) touching u from above at (t 0 , 0) we have then u is a A-flux limited super-solution at (t 0 , 0).

4.3.
Existence and uniqueness of viscosity solution for the general nonlocal problem. We consider (33) with p = 0 and we recall that for p = 0, our equation is Lemma 4.6 (Existence of barriers for (38)). Assume (A) and (A0). There exists a constant K 1 > 0 such that are respectively sub-solution and super-solution of (38).

SPECIFIED HOMOGENIZATION OF A DISCRETE MODEL FOR TRAFFIC FLOW 1447
Proof. We define K 1 = M 0 k 0 . Let us prove that (u + , ξ + ) is a super-solution of (38).
In fact, we have that where we have used Remark 3.6 for the second inequality. Similarly, using that K ≥ 0 and K 1 ≥ 2||V || ∞ k 0 , we have that  . Let (u, ξ) and (v, ζ) be respectively a sub-solution and a super-solution of (38). We also assume that there exists a constant C > 0 such that for all (t, If Proof. Let us introduce We want to prove that M ≤ 0. We argue by contradiction by assuming that M > 0.
Step 1. test functions. We introduce the following test functions with η, γ small parameters, and A a constant to be chosen later. We denote by Φ(t, x, y) = max (ϕ(t, x, y), ψ(t, x, y)). Using (40) and (41) we have that We have a similar result for ψ which yields that lim |x|,|y|→+∞ Using the fact that our test functions are upper semi continuous, we can see that Φ reaches a maximum at some finite point that we denote by (t, Classicaly we have for η and γ small enough, Step 2.t > 0 for ε small enough. By contradiction, let us assume that Φ reaches its maximum fort = 0. Let us for instance assume that Φ(t,x,ȳ) = ϕ(t,x,ȳ). In this case, we have Therefore, η T < k 0 |x −ȳ| and for ε small enough we get a contradiction. In the same way, we get a contradiction if we assume that φ(t,x,ȳ) = ψ(t,x,ȳ).
Step 3. utilisation of the equation in the case Φ(t,x,ȳ) = ϕ(t,x,ȳ). By duplication of the time variable and passing to the limit we have that there exist two real numbers a, b ∈ R such that (43) and (44), we obtain where we have used the fact that M (u(t,x), [ξ(t, ·)])(x) is finite according to Remark 3.6. We distinguish two cases.
In this case, taking γ going to zero in (45) yields a contradiction.
Case 2. there exists a constant C ε > 0 such that for any γ small enough we have, Changing variables in (45) we can write We define This implies that However, from Remark 3.6, we have that for γ small enough We deduce that for γ small enough, Then for γ small enough (46) implies Choosing A = 4α, we get a contradiction.
Step 4. utilisation of equation in the case Φ(t,x,ȳ) = ψ(t,x,ȳ). By duplication of the time variable and passing to the limit, we have that there exist two real numbers a, b ∈ R such that (48) and (49), we obtain that We recall that we defined L andL using K and V (see (21) and (22)). Therefore, we can see that the right part of inequality (50) is finite (using Remark 3.6). We distinguish two cases.
In this case, taking γ to zero in (50) yields a contradiction.
Case 2. there exists a constant C ε > 0, such that for any γ small enough we have To simplify, we introduce As above, we can provẽ

SPECIFIED HOMOGENIZATION OF A DISCRETE MODEL FOR TRAFFIC FLOW 1451
We have that where we have used for the first inequality the monotonicity (see Remark 2.2). The monotonicity of V yields the second inequality. The third and the final inequalities come from the definition of L and the fact that φ is a Lipschitz function and V is bounded. Finally, combining (51) with (50), we obtain Taking A = 2 (α + 2 ||V || ∞ ||φ || ∞ ), we get a contradiction in (52). The proof of Proposition 4.7 is now complete.
We now give a comparison principle on bounded sets, to do this, we define for a given point (t 0 , x 0 ) ∈ (0, T ) × R and for r, R > 0, the set

Proposition 4.8 (Comparison principle on bounded sets for (38)). Assume (A).
Let (u, ξ) be a sub-solution of (38) and let (v, ζ) be a super-solution of (38) on the open set Q r,R ⊂ (0, T ) × R. We assume that u and ξ (resp. v and ζ) are upper semi-continuous (resp. lower semi-continuous) on Q r,R . Also assume that Applying Perron's method (see [20, Proof of Theorem 6], [2] or [17] 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 4.9 (Existence and uniqueness of viscosity solutions for (38)). Assume (A0) and (A). Then, there exists a unique solution (u, ξ) of (38). Moreover, the functions u and ξ are continuous and there exists a constant K 1 > 0 such that 4.4. Control of the oscillations for (38). We now present a theorem that provides a control on the oscillations in space of the solution of (38). This is a very important theorem, first because it will allow us to prove Theorem 3.5 and also because it presents some of the arguments we use later to build the correctors at the junction. and with k 0 defined in (A0).
Proof. We use the following notation, Proof of the upper bound. We introduce We want to prove that M ≤ 0. We argue by contradiction and assume that M > 0.
Step 1. the test functions. For η, γ > 0 small parameters, we define We denote by Φ(t, x, y) = max (ϕ(t, x, y), ψ(t, x, y)). For x ≥ y, using (53) and (A0) we have Therefore, we deduce lim |x|,|y|→+∞ Since ϕ, ψ are upper semi continuous, Φ reaches a maximum on Ω at a point that we denote by (t,x,ȳ). Classically we have for η and γ small enough Step 2.t > 0 andx >ȳ. By contradiction, assume first thatt = 0. For instance, we assume that Φ(t,x,ȳ) = ϕ(t,x,ȳ). In this case, we have that where we have used the fact that u 0 is non increasing, and we get a contradiction.

SPECIFIED HOMOGENIZATION OF A DISCRETE MODEL FOR TRAFFIC FLOW 1453
Step 3. utilisation of the equation in the case Φ(t,x,ȳ) = ϕ(t,x,ȳ). By duplication of the time variable and passing to the limit we get that η where we have used the fact thatM ( Taking γ to zero, we get a contradiction in (56).
Step 4. utilisation of equation in the case Φ(t,x,ȳ) = ψ(t,x,ȳ). By duplication of the time variable and passing to the limit we get that where we have used the bounds on L andL (see Remark 3.6). Taking γ to zero, we get a contradiction.
Proof of the lower bound. In order to prove our result, we will use the following lemma which proof is postponed.
Now we would like to prove that for all ε > 0, In fact, if (58) is true, then taking ε to 0 and using (57) we directly obtain the lower inequalities in (54) and (55). We argue by contradiction and assume that M > 0.
Step 1. the test function. For η, γ > 0 small parameters, we define Using (A0) and (53), we obtain that Therefore, we have that for (t, x, y) ∈ Ω lim |x|,|y|→+∞ Since ϕ is upper semi continuous, ϕ reaches a maximum on Ω at a point that we denote by (t,x,ȳ). Classically we have for η and γ small enough Step 2.t > 0 andx >ȳ. By contradiction, assume first thatt = 0. Using (A0), we get a contradiction writing that If we assume thatx =ȳ then, using the fact that ϕ(t,x,ȳ) > 0, we get that This inequality yields a contradiction.
Step 3. utilisation of the equation. By duplication of the time variable and passing to the limit we get that where we have used the fact thatM (u(t,x), [ξ(t, ·)])(x) ≤ 0. We replace L by its definition (26) and using (27) Ifȳ + z ≥x, then using that u(t, .) is non increasing, we get that Ifȳ + z <x, using the fact that ϕ(t,x,ȳ + z) ≤ ϕ(t,x,ȳ), we obtain Using Lemma 4.11, we get that u(t,ȳ + z) − ξ(t,ȳ) < −1. We deduce that I(u(t,ȳ + z) − ξ(t,ȳ)) = 0 for z ≥ h 0 and so N ( We now turn to the proof of Lemma 4.11.
Proof of Lemma 4.11. The proof is divided into several steps.
Step 1. proof of the lower bound. We introduce We want to prove that M ≤ 0 and argue by contradiction by assuming that M > 0.

SPECIFIED HOMOGENIZATION OF A DISCRETE MODEL FOR TRAFFIC FLOW 1455
Step 1.2.t > 0 for ε small enough. We assume by contradiction thatt = 0. We have that Taking ε small enough, we get a contradiction.
Step 1.3. utilisation of equation. By duplication of the time variable and passing to the limit, we get that where we have used the fact that V ≥ 0. We claim that Indeed, for z > |x −ȳ|, using that ξ(t, ·) is non increasing and that ϕ(t,x,ȳ) > 0, we have that Therefore, using the definition of E we obtain that (for ε small enough such that Similarly, using the fact that u(t, ·) is non increasing, for all z > |x −ȳ|, we have that Therefore, This ends the proof of (61). Injecting (61) into (60), we get that Taking A = 4α, we get a contradiction for γ small enough.
Step 2. proof of the upper bound. We introduce We want to prove that M ≤ 0. We argue by contradiction and assume that M > 0. Let η, γ be small parameters. We consider Classically, ϕ reaches a maximum on [0, T ] × R × R at (t,x,ȳ) and we have the following result for η and γ small enough As in the previous Step 1.2, we get thatt > 0. By duplication of the time variable and passing to the limit we then get that where we have used the fact thatM ≤ 0 and (27). We want to prove that In fact for all z ≥ h 0 , we have thatx + z >ȳ for ε small enough, so using that ϕ(t,x,ȳ) > 0 we get that Proof. We claim that (w, χ) = 0, − p α V −1 p is a solution of (64) for λ = −|p|V −1 p .
-If p = 0, the result is obvious.
then we have

SPECIFIED HOMOGENIZATION OF A DISCRETE MODEL FOR TRAFFIC FLOW 1457
we recall that D = h max + 3V max /(2α) + 2r/φ 0 . Similarly, for all z > 0, we have Finally, by definition we have that First, notice that thanks to assumption (A7), for all p ∈ [−k 0 , 0), we have 1 In this case, we have Finally, using (65), (66), and (67), we obtain our desired result.
this implies that Combining this result to (66), we obtain Using (65) and (68), we obtain our desired result. The uniqueness of λ is classical and we skip it. The proof is now complete.
6. Correctors for the junction. In order to obtain an homogenization result, we need to find the effective flux-limiter. That is why we consider the following cell problem: find λ ∈ R such that there exists a solution (w, χ) of the following Hamilton-Jacobi equation, for x ∈ R, In this section we present a result of existence of correctors for the junction, which will be used for the proof of Theorem 3.2. We use the following notation: given A ∈ R, A ≥ H 0 , we define two real numbers p − and p + defined by Given the form of H, there exists only one couple of real numbers satisfying (70).
ii) (Bound from below at infinity) If A > H 0 , then there exists a γ 0 > 0 such that for every γ ∈ (0, γ 0 ), we have for all x ≥ r + V max /α and h ≥ 0, and for x ≤ −r − V max /α and h ≥ 0, then (up to a sub-sequence ε n → 0) we have that w ε and χ ε converge locally uniformly towards a function W which satisfies In particular, we have (with W (0) = 0), The proof of this theorem is postponed until Section 8.

7.
Proof of convergence. This section is devoted to the proof of Theorem 3.2 which relies on the existence of correctors provided by Proposition 5.1 and Theorem 6.1. We will use the following lemmas, the first one being a direct consequence of Theorem 4.9.
Lemma 7.1. (Barriers uniform in ε). Assume (A0) and (A). There exist a constant K 1 > 0 such that for all t ≥ 0 and x ∈ R, we have The following lemma is a direct consequence of Theorem 4.10.
Lemma 7.2. (Uniform gradient bound). Assume (A0) and (A). Then the solution (u ε , ξ ε ) of (13) satisfies for all t ≥ 0, for all x, y ∈ R, x ≥ y, Before passing to the proof of Theorem 3.2, let us mention that Theorem 3.5 is a direct consequence of this result joint to Theorem 3.2.
We now turn to the proof of Theorem 3.2.

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Proof of Theorem 3.2. We introduce and v = max u, ξ , v = min u, ξ .
We want to prove that v is a sub-solution of (7) and that v is a super-solution of (7). Indeed, in this case, the comparison principle will imply that v ≤ v. But by construction v ≤ v, hence v = v = u 0 , the unique solution of (7). This implies that u = u = ξ = ξ = u 0 and so u ε and ξ ε converge locally uniformly to u 0 . To prove that v is a sub-solution of (7), we argue by contradiction and assume that there is a point (t,x) ∈ (0, +∞) × R and a test function ϕ ∈ C 1 (J ∞ ) such that where We can assume that for ε small enough (up to changing ϕ at infinity), we have u ε , ξ ε ≤ ϕ − η outside Qr ,r (t,x).
(79) Using Lemmas 7.1 and 7.2 we get that the functions u and ξ satisfy for all t > 0, and for x ≥ y We distinguish two cases.
Case 1.x = 0. We only consider the casex > 0, since the other case (x < 0) is treated in the same way. We define p = ϕ x (t,x), that according to (81), satisfies We chooser small enough so thatx − 2r > 0. We introduce We have the following lemma.
Proof of Lemma 7.3. For all (t, x) ∈ Qr ,r (t,x), we have forr small enough where we have used for the first equality the regularity of the test function ϕ and the fact that the non-local operatorM ε is bounded (see Remark 3.6) and (78) for the second equality. If p = 0, we obtain directly our result. We then assume that p ∈ [−k 0 , 0). For all D ≥ z ≥ 0, and for ε andr small enough we have that where we have used the fact that ϕ ∈ C 1 . Now using the fact thatẼ is non increasing, we havẽ Moreover, we have that We deduce that Using (84),(85) and the definition of H, we have forr and ε small enough, We now prove the second inequality in (83). Let us notice that for ε small enough, using the fact that the non-local operatorK ε is bounded (see Remark 3.6) and the definition of φ, we have that For all (t, x) ∈ Qr ,r (t,x), we have forr small enough

SPECIFIED HOMOGENIZATION OF A DISCRETE MODEL FOR TRAFFIC FLOW 1461
If p = 0, we obtain directly our result. We then assume that p ∈ [−k 0 , 0). For all D ≥ z ≥ 0, and for ε andr small enough we have that Now, using the fact thatF is non increasing, we have that which yields that We now computeÑ ε ψ ε ε (t, x), ϕ ε (t, ·) (x). As above, and using the fact thatĨ is non decreasing, we havẽ We notice that thanks to assumption (A7), for all p ∈ [−k 0 , 0) we have Then, where we have used the definition ofL ε for the second inequality, (86) combined with assumption (A7) (see Remark 2.2) for the third inequality, (88) combined with the fact that V is non-decreasing for the fourth inequality and the fact V is a Lipschitz continuous function for the last inequality.
In particular, by definition of D, we have −1/p ≥ h max for ε andr small enough. Then using (86) and the definition ofÑ ε , we obtaiñ Using assumption (A7) (see Remark 2.2) and the previous inequalities, we get, using the definition ofL ε , that Therefore, we have where we have used assumption (A4) (V (h) = V max ∀h ≥ h max ) and that −1/p ≥ h max . This ends the proof of Lemma 7.3.
Case 2.x = 0. Using Theorem 37, we may assume that the test function has the following form where g is a C 1 function defined on (0, +∞). The last line in condition (78) then becomes Let us consider (w, ζ) the solution of (69) provided by Theorem 6.1. We define We have the following lemma, Lemma 7.4. (ϕ ε , ψ ε ) satisfies in the viscosity sence, forr and ε small enough on Qr ,r (t, 0) , Proof of Lemma 7.4. Let h be a test function touching ϕ ε from below at (t 1 , x 1 ) ∈ Qr ,r (t, 0), so we have for y in a neighbourhood of x 1 ε . Since w does not depend on time, we have that h t (t 1 , x 1 ) = g (t 1 ).
Using that (w, ζ) is a solution of (69), we then deduce that Let f be a test function touching ψ ε from below at (t 2 , x 2 ) ∈ Qr ,r (t, 0). We have for y in a neighbourhood of x 2 ε . Since ζ does not depend on time, we have that Therefore, using that (w, ζ) is a solution of (69), we get Now for ε small enough such that εD ≤r, we deduce from the previous inequality and using the fact that we consider non-local operators with bounded support, that we have Getting the contradiction. We have that for ε small enough Using the fact that w ε , ζ ε → W with W (x) =p − x1 {x<0} +p + x1 {x>0} (see Theorem 6.1), we deduce that for ε small enough, we have Combining this with (90) and (91), we get that u ε + η 2 ≤ ϕ ε and ξ ε + η 2 ≤ ϕ ε outside Qr ,r (t, 0).

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8. Proof of the existence of correctors at the junction. This section contains the proof of Theorem 6.1. We proceed as in [13,14] and we will construct correctors on a truncated domain and then pass to the limit as the size of the domain goes to infinity. For l ∈ (r, +∞), r << l and r ≤ R << l we want to find λ l,R ∈ R such that there exists a solution (w l,R , χ l,R ) of As in the previous sections, to G 1,2 R we associateG 1,2 R which is defined in the same way but we replace the non-local operators M and L respectively byM andL. 8.1. Comparison principle for a truncated problem. Proposition 8.1 (Comparison principle on a truncated domain). Let us consider the following problem for r < l 1 < l 2 and λ ∈ R, with l 2 >> R.
Proof. Like in [13], the only new difficulty to prove this proposition is the comparison at l 2 . But since near l 2 , the system decouples itself, we can prooceed as in [14,Proposition 4.1].
We have a similar result if we exchange the boundary conditions, that is to say for l 1 < l 2 < −r and if for all x ∈ [l 2 , l 2 + D], v(x) ≤ u(x) and ζ(x) ≤ ξ(x), and the following conditions are imposed at x = l 1 ,

8.2.
Existence of correctors on a truncated domain.

Proposition 8.3 (Existence of correctors on a truncated domain).
There exists a constant λ l,R ∈ R such that there exists a solution (w l,R , χ l,R ) of (94) for which there exists a constant C (depending only on k 0 ) and a Lipschitz continuous function m l,R , such that with H 0 defined in (6).
Proof. Classically, we consider the approximated truncated cell problem, (101) Step 1. construction of barriers. Using that (0, 0) and (C 0 /δ, C 0 /δ) are respectively obvious sub and super-solution of (101), with C 0 = | min p∈R H 0 (p)| = −H 0 and that we have a comparison principle, we deduce that there exists a continuous viscosity solution (v δ , ζ δ ) of (101) which satisfies Step 2. control of the oscillations of v δ and ζ δ .
Proof of Lemma 8.4. In the rest of the proof we will use the following notation Ω = (x, y) ∈ [−l, l] 2 s.t. x ≥ y .

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Proof of the upper inequality. We want to prove that We argue by contradiction and assume that M > 0. Since v δ and ζ δ are continuous and x, y belong to a compact, M is reached for a finite point that we denote by (x,ȳ) ∈ Ω. Given that M > 0, we deduce thatx =ȳ. Therefore, we can use the viscosity inequalities for (101). Let us for instance assume that M = v δ (x) − v δ (ȳ), the other case is similar so we skip it. We distinguish 3 cases: -If (x,ȳ) ∈ (−l, l), we have For every value ofx,ȳ we obtain a contradiction, therefore M ≤ 0.
Proof of the lower inequalities. In order to proof these inequalities, we will use the following lemma which proof is postponed.
In order to prove (103), using Lemma 8.5 it is sufficient to prove that We argue by contradiction and assume that M > 0. Since Ω is compact and v δ and ζ δ are continuous, M is reached for a finite point that we denote by (x,ȳ) ∈ Ω.
-Ifx = l andȳ = −l, we obtain and so, we get δM ≤ 0. For every value ofx,ȳ ∈ [−l, l] we get a contradiction, therefore we have M ≤ 0. This ends the proof of Lemma 8.4.
Step 3. construction of a Lipschitz estimate. We want to construct a Lipschitz continuous function m δ , such that there exists a constant C > 0 (independent of l and R) such that We define m δ as an affine function in each interval of the form [ih 0 , (i + 1)h 0 ], with i ∈ Z, such that Since m δ and v δ are non-increasing, and |v δ ( (113) and for all x, y ∈ [−l, l], Now using Lemma 8.5, we have Choosing C = max(2k 0 , 3), we obtain (112).
By stability of viscosity solutions, we have that (w l,R −2C, χ l,R −2C) and (w l,R , χ l,R ) are respectively a sub-solution and a super-solution of (94), and w l,R − 2C ≤ w l,R and χ l,R − 2C ≤ χ l,R .
The uniqueness of λ l,R is classical so we skip it. This ends the proof of Proposition 8.3.
Proof of Lemma 8.5. We separate the proof in two parts. This proof uses the vertex test function of the work of Imbert and Monneau [19,Theorem 3.2] to treat the comparison between v δ and ζ δ near −l and l. In fact, we consider that we have a network composed of a single branch with two nodes (one in −l and the other in l). Near −l we consider an outgoing branch and near l we consider an incoming branch. Step We want to prove that We argue by contradiction and assume that M > 0. Given that v δ and ζ δ are continuous, M is reached at a finite point that we denote byx ∈ [−l, l]. We distinguish 3 cases according to the position ofx in the interval [−l, l]. Case 1.x ∈ (−l, l). We define for ε a small parameter, Since [−l, l] is compact and v δ and ζ δ are continuous functions, the function ϕ reaches a maximum at a finite point that we denote by (x ε , y ε ) ∈ [−l, l]. If we denote M ε = ϕ(x ε , y ε ), by classical arguments, we have that We can also prove that Furthermore, for ε small enough we have x ε , y ε ∈ (−l, l), and using the viscosity inequalities we obtain with p ε = (x ε − y ε )/ε. Combining these inequalities and using the definition of M , we obtain that

SPECIFIED HOMOGENIZATION OF A DISCRETE MODEL FOR TRAFFIC FLOW 1471
where we have replaced G 1 R andG 2 R by their definitions, used the fact that by definition H is a Lipschitz function and that that V ≥ 0 for the second inequality, used Remark 3.6 for the third inequality and (116) for the last inequality.
We will compute the right part of the inequality in different steps. 1-Concerning the local operator.
where we have used the regularity of ψ R for the first inequality, used the fact that by definition of H, we have |H| ≤ V max |p| for the second inequality and used (117) for the last inequality.
Case 2.x = l. In this case, we use the vertex test function introduced by Imbert and Monneau. We refer to [19] for a detailed description of the vertex test function, but for the readers convenience we recall the properties that we used to complete this proof. The vertex test function G γ is associated to the single Hamiltonian H. We fix γ = δM/2. It satisfies the following properties.
with for all x ∈ [−l, l] and p ∈ R, We introduce the following test function, for ε > 0 a small parameter, which like before reaches a maximum at a finite point (x ε , y ε ) ∈ [−l, l] and (116) remains true.
Using the viscosity equations, we have that Using the definition of M and combining the previous inequalities, we get that where we have used (116) combined with the fact that both H and H + are Lipschitz continuous for the second inequality. Using the compatibility condition on the gradient of the vertex test function (124) we obtain and given that γ = δM /2, we get a contradiction for ε small enough.
Case 3.x = −l. This case is exactly like the previous one with the exception that the vertex test function must be adapted to treat the junction at −l. In particular, (125) is replaced by We skip the rest of the computation for this case.

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Step 2. proof of ζ δ (x) − v δ (x) ≤ 1. We want to prove that We argue by contradiction and assume that M > 0. Give that v δ and ζ δ are continuous, M is reached at a finite point that we denote byx ∈ [−l, l]. We distinguish 2 cases according to the position ofx in the interval [−l, l].
Case 1.x ∈ (−l, l). We define for ε a small parameter, Using the same arguments as before, the test function reaches a maximum at a finite point that we denote by (x ε , y ε ) ∈ [−l, l]. If we denote M ε = ϕ(x ε , y ε ) (116) and (117) remain valid.
For ε small enough we have x ε , y ε ∈ (−l, l), and using the viscosity inequalities we get that Combining these inequalities and using the definition of M , we obtain where we have replaced G 2 R andG 1 R by their definition and used (27) and that M ≤ 0. We will compute the right part of (126) in different steps.
1-Concerning the local operator. Like before, we have 2-Concerning the non-local operator N . We claim that To prove this, it suffices to prove that for all z ≥ h 0 , we have Since |x ε − y ε | → 0 as ε goes to 0, we have for all z ≥ h 0 and ε small enough that where we have used the fact that v δ is decreasing for the first inequality and the fact that M ε > 0 for the second inequality. This implies that Injecting (127) and (128) in (126), we obtain δM ≤ o ε (1), and we get a contradiction for ε small enough.
Case 2.x = l ofx = −l. Proceeding like in the previous step we obtain directly a contradiction by using the properties of the vertex test function.
This ends the proof of Lemma 8.5.
Proposition 8.6 (First definition of the flux limiter). The following limits exists (up to some sub-sequence), Moreover, we have Proof. This proposition is a direct consequence of (100). and Similarly, for all x ≤ −r − D and h ≥ 0, and Proof. We only do the proof of (130)-(131), since the proof of (132)-(133) is similar and we skip it. For µ > 0, small enough, we denote by p + µ the real number defined by Using that we deduce that p + µ exists for µ small enough and p + µ ∈ [−k 0 , 0). Let us now consider    w + = p + µ x, Let us consider (w, χ) = 0, − p + µ α V −1 p + µ the correctors provided by Proposition 5.1 for p = p + µ . Given the definition of w + and χ + , we get )(x). In particular this implies that Finally, given that the non-local operator K is bounded by D (see Remark 3.6), we have for all x ∈ (r + D, l] Combining the previous results, we can see that the restriction of (w + , χ + ) to (r + D, l] satisfies Let us introduce, for some x 0 ∈ (r + D, l], with C > 0 the constant provided by Proposition 8.3. Then we have where we have used the fact that p + µ ∈ [−k 0 , 0) and ||V || ∞ ≤ V max . Using that (g, h) is a solution of (98) and (u, v) is a solution of (99) (with ε 0 = µ), joint to the comparison principle (Proposition 8.1), up to changing the value of the constant C, we get that This implies that for all h ≥ 0, and for all x ∈ (r + D, l), Finally, if we choose γ 0 < |p 0 − p + |, then we have Choosing µ > 0 such that p + µ = p + − γ. We obtain (130)-(131).
Proof of Theorem 6.1. The proof is performed in two steps.
Step 1. proof of i) and ii). We want to pass to the limit as l → +∞ and then as R → +∞ on the solution of (94) given by Proposition 8.3. Using (8.3), there exists l n → +∞, such that m ln,R − m ln,R (0) → m R as n → +∞, the convergence being locally uniform. We also define Thanks to (8.3), we know that these limits are finite and satisfy By stability of viscosity solutions (w R − 2C, χ R − 2C) and (w R , χ R ) are respectively a sub-solution and a super-solution of Therefore, using Perron's method, we can construct a solution (w R , χ R ) of (139), with m R , A R , w R and χ R satisfying Using Proposition 8.7, if A > H 0 , we know that there exists a γ 0 > 0 and a constant C > 0 such that for all γ ∈ (0, γ 0 ), for all x ≥ r + D, and h ≥ 0, Similarly, for all x ≤ −r − D and h ≥ 0, Proceeding like before, we pass to the limit as R → +∞ in order to build a solution (w, χ) of (69) with λ = A that satisfies (71), (72) and (73).

SPECIFIED HOMOGENIZATION OF A DISCRETE MODEL FOR TRAFFIC FLOW 1477
Step 2. proof of iii). Let us now consider the rescaled functions w ε = εw(x/ε) and χ ε (x) = εχ(x/ε). Using (71), we have that Therefore, there exists a subsequence ε n → 0 as n → +∞, such that w εn , χ εn → W locally uniformly as n → +∞, with W (0) = 0. Proceeding as in the proof of convergence (Section 7), away from the junction point, we have that W satisfies This proves (74). Let us now prove (75). For x < 0, we have for all γ ∈ (0, γ 0 ), if A > H 0 , where we have used (73). Therefore, we have W x = p − for x < 0, this equality where we have used (72). Therefore, we have that W Combining these results, we obtain (75). We argue by contradiction and assume that there exists λ ∈ E such that λ < A. We denote by (v λ , ζ λ ) a solution of (69) associated to λ. Arguing as in the proof of Theorem 6.1, Step 2, we deduce that the functions v ε λ (x) = εv λ x ε and ζ ε λ (x) = εζ λ x ε (144) have a limit W λ (with W λ (0) = 0) which satisfies H(W λ x ) = λ for x = 0. This means that for all x > 0, we have Similarly, for all x < 0, we have These inequalities imply that for all γ > 0, there exists a constantC γ such that Using Theorem 6.1 (ii), we have for γ small enough, v λ ≤ w and ζ λ ≤ χ for |x| ≥R.
This implies that there exists a constant CR such that for all x ∈ R, we have v λ (x) < w(x) + CR and ζ λ (x) < χ(x) + CR.
Let us now introduce two functions (u, ξ) and (u λ , ξ λ ), defined by Both functions are solutions of (13) (with ε = 1) and Using the comparison principle (Proposition 4.7), we obtain Passing to the limit as t goes to infinity, we get A ≤ λ, which is a contradiction.
9. Link between the system of ODEs and the PDE. This section is devoted to the proof of Theorem 3.3, which is a direct application of our convergence result, Theorem 3.2 joint to the following result.
Theorem 9.1. For ε = 1, (ρ, σ) defined by (2) and (12) is a discontinuous viscosity solution of the following equation The proof of Theorem 9.1 is given in Appendix B. Let us use Theorem 9.1 to do the proof of Theorem 3.3.
Proof of Theorem 3.3. We recall that in Theorem 3.3 we have u 0 (x) = ξ ε 0 (x) = −x/h. Let us begin by proving that for all x ∈ R and for all ε > 0, we have with f (ε), g(ε) → 0 as ε goes to 0. Let us define a piece-wise affine function v satisfying Given that for all U i+1 (0) − U i (0) ≥ h 0 , we notice that v is k 0 -Lipschitz continuous and by definition of ρ 1 (0, x), we have Let us consider the integer i 0 ∈ N defined by i 0 = sup {i ∈ Z, s.t. U i (0) ≤ −R}.
Using the assumption that for all i ∈ Z such that Appendix A. Analysis of system of ODEs. In this section we present some properties of the solution We couple system (153) with an initial condition (U i (0), Ξ i (0)) i that satisfy the following assumption. (A0') (Initial conditions for (153)). For all i ∈ Z, Proof. Let us consider the equation satisfied by Step 1. proof of the upper bound in (154). Using assumptions (A1), (A4), and (A6), we notice that Ξ i − U i is a sub-solution oḟ By comparison, we have Step 2. proof of the lower bound in (154). Using assumptions (A1), (A3), and (A6), we notice that Ξ i − U i is a super-solution oḟ By comparison, we have This ends the proof of Proposition A.1.
Proposition A.2 (Conservation of the order in (153)). Assume (A) and (A0'), then the solution (U i , Ξ i ) i of (153) satisfies for all i ∈ Z, In particular, using Proposition A.1, this result implies that

SPECIFIED HOMOGENIZATION OF A DISCRETE MODEL FOR TRAFFIC FLOW 1481
Proof. We will prove that for all δ > 0 small, we have Then passing to the limit as δ goes to 0 we will obtain (157). Let δ > 0, we argue by contradiction and assume there exists a time Using Proposition A.1, in particular that U j ≤ Ξ j , and assumption (A7) combined with Remark 2.2, we have that Using again Proposition A.1, in particular that the functions (U i ) i are non-decreasing in time, we obtain that which is a contradiction. This ends the proof of Proposition A.2.
Proposition A.3 (Maximal distance between two vehicles). Assume (A) and (A0'), then the solution (U i , Ξ i ) i of (153) satisfies for all i ∈ Z, In particular, using Proposition A.1, we have that for all i ∈ Z, Proof. We will prove that for all δ > 0 small, we have for all i ∈ Z, Passing to the limit in the previous inequality as δ goes to 0, we will obtain (159). Let δ > 0, we argue by contradiction and assume there exists a time Let us consider j ∈ Z such that U j+1 (t * ) − U j (t * ) = h max + 2r φ 0 + 3V max 2α + δ. By continuity and (A0'), there exists a time t 0 ∈ [0, t * ) such that and We distinguish three cases.
In order to compare the distance U j+1 − U j when U j is inside the perturbation, we consider the worst case scenario where the vehicle j advances at a speed V max φ 0 and j + 1 advances at a speed V max , until U j ≥ r (meaning that the vehicle j is outside the perturbation). To be more exact, we notice that the couple (U j , Ξ j ) is a super-solution of the following system Computing the solution of (164) we get By comparison, we obtain that Lett = 1 V max φ 0 V max φ 0 2α + r − U j (t 0 ) + t 0 . Using (166), we have that U j (t) ≥ r.
We now prove thatt < t * . In fact, for all t ∈ [t 0 ,t], we have where we have used Proposition A.1 for the first line. From the previous inequality and the definition of t * , we deduce thatt < t * . The couple (U j , Ξ j ) satisfies for all t ∈ [t, t * ], with

SPECIFIED HOMOGENIZATION OF A DISCRETE MODEL FOR TRAFFIC FLOW 1483
We can easily compute the explicit form of the solution of (168), Using Proposition A.1, for all t ∈ [t, t * ], we have that Therefore, combining the previous results, we have for all t ∈ [t, t * ] where we have used Proposition A.1 for the second and third inequality and we have used (168) for the last inequality. The previous inequality remains valid for t = t * which gives us a contradiction. Case 2. U j (t 0 ) > r. In this case, the couple (U j , Ξ j ) satisfies system (167) for all t ∈ (t 0 , t * ], with the following initial conditions U j (t 0 ) = U j+1 (t 0 ) − h max and 0 ≤ Ξ j (t 0 ) − U j (t 0 ) ≤ V max α .
As above, the explicit solution of (167)-(170) has the following form, Arguing as above, we will obtain U j+1 (t * ) − U j (t * ) ≤ h max + V max 2α which is a contradiction. Case 3. U j (t 0 ) < −r. We treat this case in 3 steps.
Step 1. left of the perturbation. We denote bŷ For all t ∈ [t 0 ,t], the couple (U j , Ξ j ) satisfies (167)-(170) and therefore has the same form as the one presented in Case 2. In particular, for all t ∈ [t 0 ,t], we have This implies thatt < t * .
Step 2. inside the perturbation. In the interval [t, t * ], the couple (U j , Ξ j ) satisfies (163) with the following initial condition Computing the solution of (172), and by comparison, for all t ∈ [t, t * ], we have Using (166), we have that U j (t) ≥ r.
We now prove thatt < t * . We recall that U j (t) = −r. In fact, for all t ∈ [t,t], we have where we have used Proposition A.1 for the first line. From the previous inequality and the definition of t * , we deduce thatt < t * .
Step 3. right of the perturbation. In the interval [t, t * ], the couple (U j , Ξ j ) satisfies (167), with the following initial condition Proceeding like before, we can prove that for all t ∈ [t, t * ], we have which gives us a contradiction for t = t * . This ends the proof of Proposition A.3.
Appendix B. Proof of Theorem 9.1. Before we give the proof of Theorem 9.1, we need the following result.
Proof. We drop the time dependence to simplify the presentation. Let j ∈ Z.