P Cardaliaguet

N Forcadel

R Monneau

A class of germs arising from homogenization in traffic flow on junctions

Keywords: AMS Classification: 35L65, 76A30, 35B27, 35R02, 35D40, 35F20 traffic flow models, scalar conservation laws, homogenization on networks. Contents

We consider traffic flows described by conservation laws. We study a 2:1 junction (with two incoming roads and one outgoing road) or a 1:2 junction (with one incoming road and two outgoing roads). At the mesoscopic level, the priority law at the junction is given by traffic lights, which are periodic in time and the traffic can also be slowed down by periodic in time flux-limiters.

). The identification of this germ and of a characteristic subgerm which determines the whole germ, is the first key result of the paper.

The second key result of the paper is the construction of a family of correctors whose values at infinity are related to each element of the characteristic subgerm. This construction is indeed explicit at the level of some mixed Hamilton-Jacobi equations for concave Hamiltonians (i.e. fluxes). The explicit solutions are found in the spirit of representation formulas for optimal control problems.

Introduction

In this section we introduce the problem and the main notations, assumptions and results of the paper. We start with a foreword in which we explain the goal of the paper. Then we introduce the notions of germs and our two main models (mesoscopic and macroscopic). We give our main results and compare them with the literature. We finally describe the organization of the paper.

Foreword

The goal of the paper is to understand and to justify effective junction conditions for macroscopic models of traffic flows arising by homogenization of mescoscopic models. We concentrate here on junctions involving two incoming roads and a single outgoing one (referred later on as 2:1 junctions), or the opposite: one incoming road and two outgoing ones (referred as 1:2 junctions). On each road, the equation satisfied by the density is a scalar conservation law of the form B t ρ `Bx pf pρqq " 0, where the concave flux function f can depend on the road. At the junction point we require of course a Rankine-Hugoniot condition, as well as relations between the incoming and outgoing fluxes, which define what is called a germ. For the mesoscopic model, the germ is an oscillating function of time, which can be interpreted as periodic in time traffic lights (or more generally flux limiters). For instance, for the 1:2 junction, traffic lights regulate the traffic, dispatching the vehicles in one of the two exit branches. For 2:1 junction, the traffic lights give the priority rules.

Looking at long time behavior and on large space scale, we show that the oscillating germ for the mesoscopic model homogenizes in an effective (and homogeneous) germ for the macroscopic model. On the branches, the PDEs satisfied by the densities are the same for the macroscopic model and the mesoscopic model; only the junction condition (the germ) changes. Our homogenization procedure naturally introduces a general class of germs for conservation laws on 1:2 and 2:1 junctions. The guess and the study of those germs (Theorem 2.1) is the first key contribution of this paper. The second key contribution is the rigorous justification of the homogenization by the construction of suitable correctors (Theorem 1.7 for 2:1 junctions and Theorem 1.4 for 1:2 junctions).

For the mesoscopic model, we manage to reduce the junction condition to a 1:1 junction, involving at each time one incoming road and one outgoing road only. 1:1 junctions are well understood and justified [START_REF] Adimurthi | Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux[END_REF][START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF][START_REF] Audusse | Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies[END_REF][START_REF] Baiti | Well-Posedness for a Class of 2 ˆ2 Conservation Laws with L 8 Data[END_REF][START_REF] Garavello | Conservation laws with discontinuous flux[END_REF][START_REF] Towers | Convergence of a difference scheme for conservation laws with a discontinuous flux[END_REF][START_REF] Towers | A difference scheme for conservation laws with a discontinuous flux: the nonconvex case[END_REF]; they are known to arise by homogenization of microscopic models of follow-theleader type [START_REF] Cardaliaguet | Forcadel Microscopic derivation of a traffic flow model with a bifurcation[END_REF][START_REF] Cardaliaguet | From heterogeneous microscopic traffic flow models to macroscopic models[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] Forcadel | Specified homogenization of a discrete traffic model leading to an effective junction condition[END_REF][START_REF] Forcadel | Homogenization of a discrete model for a bifurcation and application to traffic flow[END_REF][START_REF] Galise | A junction condition by specified homogenization and application to traffic lights[END_REF] and there is an equivalence between the approach through the germ theory for 1:1 junctions and the one using Hamilton-Jacobi (HJ) equations on such junctions [START_REF] Cardaliaguet | Conservation law and Hamilton-Jacobi equations on a junction: the convex case[END_REF]. We will make an extensive use of this equivalence (still in the case 1:1) in the construction of correctors. Our new junction conditions (for 2:1 and 1:2 junctions) arise rigorously by mixing these very natural 1:1 junctions. Let us underline that the mesoscopic models we consider possess an L1 ´contraction property, and, as expected, this is also the case for our limit models after homogenization. Note however that, in the literature, there exists some junction models which do not possess this L 1 ´contraction property 1 .

For our mesoscopic models, we use the approach through germs developed in [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF]. This approach, which relies on the notion of trace developed by Panov [START_REF] Panov | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF] (see also [START_REF] Vasseur | Strong Traces for Solutions of Multidimensional Scalar Conservation Laws[END_REF]), consists in requiring that the trace of the solution at the junction belongs to a set, the germ. As recalled in Subsection 1.3, the fact that the germ is "maximal" ensures the uniqueness of the solution to the conservation law and its stability. Existence, on the other hand, comes from the "completeness" of this germ.

As explained above, the paper partially relies (for the construction of correctors) on the formulation of traffic flows in terms of Hamilton-Jacobi on a 1:1 junction. HJ equation on junctions have been discussed in many works [START_REF] Achdou | Hamilton-Jacobi equations constrained on networks[END_REF][START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF][START_REF] Camilli | A comparison among various notions of viscosity solution for Hamilton-Jacobi equations on networks[END_REF][START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF][START_REF] Lions | Viscosity solutions for junctions: well posedness and stability[END_REF][START_REF] Lions | Well-posedness for multi-dimensional junction problems with Kirchoff-type conditions[END_REF][START_REF] Schieborn | Viscosity solutions of Eikonal equations on topological networks[END_REF]; see also the recent monograph [START_REF] Barles | An illustrated guide of the modern approches of Hamilton-Jacobi equations and control problems with discontinuities[END_REF]. The central notion of flux limiters, used throughout this paper, has been developed in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF]. Questions of homogenization in this framework are discussed in [START_REF] Achdou | Hamilton-Jacobi equations on networks as limits of singularly perturbed problems in optimal control: dimension reduction[END_REF][START_REF] Cardaliaguet | Forcadel Microscopic derivation of a traffic flow model with a bifurcation[END_REF][START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF][START_REF] Forcadel | Specified homogenization of a discrete traffic model leading to an effective junction condition[END_REF][START_REF] Forcadel | Homogenization of a discrete model for a bifurcation and application to traffic flow[END_REF][START_REF] Galise | A junction condition by specified homogenization and application to traffic lights[END_REF]. In contrast with the approach developed here, these papers rely on a comparison principle. Homogenization of scalar conservation laws has been less considered in the literature: see [START_REF] Dalibard | Homogenization of non-linear scalar conservation laws[END_REF][START_REF]Homogenization of scalar conservation laws with oscillatory forcing terms[END_REF][START_REF] Serre | Correctors for the homogenization of conservation laws with oscillatory forcing terms[END_REF], and, as far as we know, never for problems on a junction. Now a few words about the techniques of proof are in order. Let us first underline that, for technical reasons, we mainly work throughout the paper in the case of 1:2 junctions; the maybe more interesting problem of junctions of type 2:1 is handled by a simple change of variables in Subsection 4.2. Second, and in contrast with most homogenization results we are aware of on the topic and quoted above, the homogenization does not rely directly on a comparison principle for some Hamilton-Jacobi formulation on the junction: indeed the limit problem cannot naturally be formulated in terms of pure HJ equations with some general comparison principle at the HJ level.

The homogenization must therefore be proved directly at the level of the scalar conservation laws. The construction of correctors for each element of the homogenized germ seems to be a difficult task in general. For this reason we first show the existence of a subset of the germ, called a characteristic subgerm, which determines the whole germ (Lemma 1.5). This characteristic subgerm will be then used to guide the construction of correctors. Indeed, to each element of the characteristic subgerm, we associate a corrector whose values at infinity are given by the values of this element (Theorem 1.6). This construction uses explicit solutions for suitable HJ equations with concave Hamiltonians in the flavor of the Lax-Oleinik formula. The explicit solutions are guessed in the spirit of representation formulas in optimal control theory on junctions [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF]. The proof of homogenization is then achieved thanks to Kato's inequality and germ's theory developed in [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF][START_REF] Musch | Well-posedness theory for nonlinear scalar conservation laws on networks[END_REF].

Note that the mesoscopic model can itself be thought as the limit of a microscropic model taking the form of a follow-the-leader model on a junction, as discussed in [START_REF] Colombo | On the microscopic modeling of vehicular traffic on general networks[END_REF] for instance. However the rigorous derivation of the macroscopic model from a microscopic one seems a very challenging question. Another open problem is the analysis of junctions involving four branches or more, which seems to require new ideas.

Standing notation and assumptions

The following assumptions are in force throughout the paper.

Let R 0 " p´8, 0q ˆt0u be the incoming branch, R j " p0, 8q ˆtju for j " 1, 2 being the outgoing ones. We consider the set R " Ť 2 j"0 R j Y t0u with the topology of three half lines glued together at the origin 0.

Let a j ă b j ă c j for j P t0, 1, 2u. We make the following assumptions on the fluxes for some δ ą 0:

For j P t0, 1, 2u, the flux f j : ra j , c j s Ñ R is of class C 2 , with pf j q 2 ď ´δ ă 0 on ra j , c j s, increasing on ra j , b j s and decreasing on rb j , c j s, with f j pa j q " f j pc j q " 0.

We set f j max :" max ra j ,c j s f j " f j pb j q ą 0 (2)

and define the nondecreasing envelope of f j f j,`p pq :" " f j ppq for p P ra j , b j s f j pb j q for p P rb j , c j s

and its nonincreasing envelope f j,´p pq :" " f j pb j q for p P ra j , b j s f j ppq for p P rb j , c j s.

Throughout the paper, the set I 1 (respectively I 2 ) denotes the time sets on which the branch 1 (resp. the branch 2) is active in the mesoscopic model. The sets I 1 and I 2 form a partition of R, each I k , k " 1, 2, being periodic and of period 1 and locally the union of a finite number of intervals:

I 1 Y I 2 " R, I 1 X I 2 "
H, I j is periodic of period 1 and consists locally in a finite number of intervals, j " 1, 2.

(

The flux limiter in the mesoscopic model is a time dependent map A : R Ñ R, such that A : R Ñ R is piecewise constant, periodic of period 1 and such that 0 ď Aptq ď

" mintf 0 max , f 1 max u on I 1 , mintf 0 max , f 2 max u on I 2 . ( 6
)

Entropy pairs and germs

We now introduce the notion of germs, following [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF][START_REF] Musch | Well-posedness theory for nonlinear scalar conservation laws on networks[END_REF]. Germs define the junction conditions and play a central role in this paper. Let us recall that the pair (entropy, entropy flux) is given, for p, p P R, by ηpp, pq " |p ´p|, q j pp, pq " signpp ´pqpf j ppq ´f j ppqq.

We define the box Q :" ra 0 , c 0 s ˆra 1 , c 1 s ˆra 2 , c 2 s (7)

and the subset of Q satisfying Rankine-Hugoniot condition

Q RH :" P " pp 0 , p 1 , p 2 q P Q, f 0 pp 0 q " f 1 pp 1 q `f 2 pp 2 q ( (8)
Definition 1.1. (dissipation, germ, maximality) i) (Dissipation) For P " pp 0 , p 1 , p 2 q, P " pp 0 , p1 , p2 q P Q, we define the dissipation by Dp P , P q :" q 0 pp 0 , p 0 q ´ q 1 pp 1 , p 1 q `q2 pp 2 , p 2 q

( " IN ´OUT ii) (Germ) Consider a set G Ă Q. We say that G is a germ (for dissipation D) if " G Ă Q RH (Rankine-Hugoniot) Dp P , P q ě 0 for all P , P P G (dissipation) iii) (Maximal set) Let G Ă Q be a set.
We say that G is maximal (for the dissipation D relatively to the box Q) if for every P P Q, we have `Dp P , P q ě 0 for all P P G ˘ùñ P P G.

The mesoscopic problem

We are interested in a problem with one incoming branch and two outgoing ones; a periodic traffic light regulates the traffic, dispatching the vehicles in one of the two exit branches, slowing down the traffic or stopping it at the junction. On the time-intervals I 1 , cars coming from road 0 can enter road 1 only, while on the time-intervals I 2 cars coming from road 0 can enter road 2 only. The traffic can also be limited on the junction by the flux limiter A, which is time dependent, but piecewise constant. For instance, time intervals on which Aptq " 0 correspond to periods where the traffic light stops completely the traffic at the junction. Let ρ j (j " 0, 1, 2) be the density of vehicles. Then ρ " pρ 0 , ρ 1 , ρ 2 q solves piq ρ j P ra j , c j s a.e. on p0, 8q ˆRj , j " 0, 1, 2 piiq B t ρ j `Bx pf j pρ j qq " 0 on p0, 8q ˆRj , j " 0, 1, 2 piiiq pρ 0 pt, 0 ´q, ρ 1 pt, 0 `q, ρ 2 pt, 0 `qq P Gptq for a.e. t P p0, 8q,

∂ t ρ 0 + ∂ x (f 0 (ρ 0 )) = 0 ∂ t ρ 1 + ∂ x (f 1 (ρ 1 )) = 0 ∂ t ρ 2 + ∂ x (f 2 (ρ 2 )) = 0 x > 0 x > 0 x < 0
where the time dependent germ tÞ ÑGptq is the piecewise constant set-valued map given by

Gptq " G Λ k ptq on I k , k " 1, 2, (10)
and G Λ1 ptq " " P " pp 0 , p 1 , p 2 q P Q, ˇˇˇf 2 pp 2 q " 0 minpAptq, f 0,`p p 0 q, f 1,´p p 1 qq "

f 0 pp 0 q " f 1 pp 1 q * , (11)
G Λ2 ptq " " P " pp 0 , p 1 , p 2 q P Q, ˇˇˇf 1 pp 1 q " 0 minpAptq, f 0,`p p 0 q, f 2,´p p 2 qq "

f 0 pp 0 q " f 2 pp 2 q * . (12)
Recall that the assumption on the time intervals I k , k " 1, 2, and the flux limiter A are given in ( 5) and ( 6) respectively. The notation G Λ k is justified in Section 2 below, where we also explain that the G Λ k ptq are maximal germs for each t P R (Lemma 2.2). The germs G Λ1 ptq and G Λ2 ptq are very natural from a traffic flow point of view. Indeed, during the time-interval I 1 (for instance), the flux on the road 2 is null and we consider only a 1:1 junction between the incoming road 0 and the outgoing road 1. In this situation, the description of the germ is well understood and take the above form (see [START_REF] Cardaliaguet | Forcadel Microscopic derivation of a traffic flow model with a bifurcation[END_REF] for the derivation of the junction condition in terms of Hamilton-Jacobi equations and [START_REF] Cardaliaguet | Conservation law and Hamilton-Jacobi equations on a junction: the convex case[END_REF] for a reformulation in term of scalar conservation laws). Let us recall that, for any j " 0, 1, 2, the L 8 map ρ j , being a solution to the scalar conservation law B t ρ j `Bx pf j pρ j qq " 0, has a strong trace (see Theorem A.1) at x " 0 in the sense of Panov [START_REF] Panov | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF], because the fluxes are strongly concave in the sense of (1).

We say that a function v is a standard Krushkov entropy solution of B t v `Bx pf pvqq " 0 on p0, `8q t p0, `8q x with initial condition v, if for every C 1 c pr0, `8q t ˆp0, `8q x q function ϕ ě 0, we have ˆp0,`8qt ˆp0,`8qx |v ´c|ϕ t `tsignpv ´cqu ¨pf pvq ´f pcqqϕ x `ˆt0uˆp0,`8qx |v ´c|ϕ ě 0 for all c P R

The next lemma states that equation ( 9) is well-posed and defines a semigroup of contraction in L 1 .

Lemma 1.2. (Existence, uniqueness, L 1 -contraction on the junction) Assume (1), ( 5) and [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF]. Given an initial condition ρ " pρ j q j"0,1,2 in L 8 pRq with ρj P ra j , c j s a.e., there exists a unique entropy solution to [START_REF] Barles | An illustrated guide of the modern approches of Hamilton-Jacobi equations and control problems with discontinuities[END_REF], in the sense that ρ j is a standard Krushkov entropy solution of B t ρ j `Bx pf j pρ j qq " 0 on p0, 8q ˆRj with ρ j p0, ¨q " ρj a.e., and such that the traces pρ 0 pt, 0 ´q, ρ 1 pt, 0 `q, ρ 2 pt, 0 `qq belong to the set Gptq for a.e. t P p0, 8q.

In addition, if ρ is a solution to (9) associated with the initial condition ρ and ρ 1 is a solution to (9) associated with the initial condition ρ1 , then Kato's inequality holds:

2 ÿ j"0 ˆ8 0 ˆRj |ρ j ´ρj 1 |φ j t `!signpρ j ´ρj 1 q
) ¨pf j pρ j q ´f j pρ j 1 qqB x φ j `2 ÿ j"0 ˆRj |ρ j ´ρ j 1 |φ j p0, xq ě 0 [START_REF] Cardaliaguet | Forcadel Microscopic derivation of a traffic flow model with a bifurcation[END_REF] for any continuous nonnegative test function φ : r0, 8q ˆR Ñ r0, 8q with a compact support and such that φ j :" φ |r0,`8qˆpR j Yt0uq is C 1 for any j " 0, 1, 2.

The proof of Lemma 1.2 is postponed to Subsection 4.1. Let us underline that equation ( 9) almost fits the usual existence and uniqueness framework of conservation laws on a junction, as discussed in [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF], as only one outgoing branch is active at any time.

The macroscopic problem

We expect the limit problem to be of the same form as the mesoscopic problem, but with an autonomous germ G. The limit scalar conservation law should take the form: piq ρ j P ra j , c j s a.e. on p0, 8q ˆRj , j " 0, 1, 2, piiq B t ρ j `pf j pρ j qq x " 0 on p0, 8q ˆRj , j " 0, 1, 2, piiiq pρ 0 pt, 0q, ρ 1 pt, 0q, ρ 2 pt, 0qq P G for a.e. t P p0, 8q,

Here the set G is the limit germ and is the main unknown of our problem. We now define the notion of solution for equation [START_REF] Cardaliaguet | From heterogeneous microscopic traffic flow models to macroscopic models[END_REF], following [START_REF] Fjordholm | Well-posedness and convergence of a finite volume method for conservation laws on networks[END_REF][START_REF] Musch | Well-posedness theory for nonlinear scalar conservation laws on networks[END_REF]. Definition 1.3. (Entropy solution of ( 14)) Given a maximal germ G Ă Q and an initial condition ρ P L 8 pRq such that ρj P ra j , c j s a.e. for j " 0, 1, 2, we say that a map ρ P L 8 pp0, 8q ˆRq is an entropy solution of (14) if, for any j " 0, 1, 2, ρ j is a Kruzkhov entropy solution of ( 14)-(ii) on R j , if its trace at t " 0 is ρ and if, its trace ρp¨, 0q " pρ 0 p¨, 0 ´q, ρ 1 p¨, 0 `q, ρ 2 p¨, 0 `qq at x " 0 belongs to G: ρpt, 0q P G a.e. t ě 0.

Following [START_REF] Fjordholm | Well-posedness and convergence of a finite volume method for conservation laws on networks[END_REF][START_REF] Musch | Well-posedness theory for nonlinear scalar conservation laws on networks[END_REF], and because the germ G is maximal, the last condition in Definition 1.3 is equivalent to the following entropy inequality:

2 ÿ j"0 "ˆ8 0 ˆRj
ηpu j ´ρj qB t φ j `qj pu j , ρ j qB x φ j `ˆR j ηpu j , ρj qφ j p0, xq * ě 0 for any u " pu j q P G and any continuous nonnegative test function φ : r0, 8q ˆR Ñ r0, 8q with a compact support and such that φ j :" φ |r0,`8qˆpR j Yt0uq is C 1 for any j " 0, 1, 2.

Let us also point out that the entropy solution ρ of ( 14) is in C 0 pr0, `8q, L 1 loc pRqq: this is an easy consequence of the classical continuity in L 1 loc of bounded entropy solution of scalar conservation laws on the line (see [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF]Theorem 6.2.2,Lemma 6.3.3]) and of finite speed of propagation arguments.

Main result: the homogenization

We are interested in the homogenization of [START_REF] Barles | An illustrated guide of the modern approches of Hamilton-Jacobi equations and control problems with discontinuities[END_REF]. Namely, given an initial condition ρ0 , we want to understand the behavior as Ñ 0 of the solution ρ " pρ ,0 , ρ ,1 , ρ ,2 q to piq ρ ε,j P ra j , c j s a.e. on p0, 8q ˆRj , j " 0, 1, 2 piiq B t ρ ,j `Bx pf j pρ ,j qq " 0 on p0, 8q ˆRj , j " 0, 1, 2 piiiq pρ ,0 pt, 0q, ρ ,1 pt, 0q, ρ ,2 pt, 0qq P Gpt{ q for a.e. t P p0, 8q, pivq ρ p0, ¨q " ρ0 on t0u ˆR,

Our main homogenization result is the following:

Theorem 1.4. (Homogenization of the 1:2 junction) Assume that (1), ( 5) and (6) hold. Then there exists a maximal germ GΛ Ă Q, such that the following holds true. Let the initial data ρ0 " pρ i 0 q P L 8 pRq be such that ρi 0 P ra i , c i s a.e. for i " 0, 1, 2. Then the solution ρ of (15) converges in L 1 loc pr0, 8q ˆRq to the unique entropy solution ρ to piq ρ j P ra j , c j s a.e. on p0, 8q ˆRj , j " 0, 1, 2 piiq B t ρ j `Bx pf j pρ j qq " 0 on p0, 8q ˆRj , j " 0, 1, 2 piiiq pρ 0 pt, 0q, ρ 1 pt, 0q, ρ 2 pt, 0qq P GΛ for a.e. t P p0, 8q, pivq ρp0, ¨q " ρ0 on t0u ˆR,

Let us point out that Theorem 1.4 itself implies the existence of a solution to [START_REF] Colombo | On the microscopic modeling of vehicular traffic on general networks[END_REF], which is not obvious otherwise. This shows in particular that the germ GΛ is complete in the terminology of [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF][START_REF] Musch | Well-posedness theory for nonlinear scalar conservation laws on networks[END_REF]. The germ GΛ is described in Subsection 2.1.3.

In order to prove the theorem, we need to build suitable correctors of the equation, associated to elements of the germ. For this, the point is that we will not have to do it for all elements of the germ GΛ, but only for a subset of it (which will indeed determine the whole germ GΛ, as we will see later on). This subset, denoted by EΛ, is called a characteristic subgerm and is given in the following expression (where the continuous, nondecreasing maps p 0 Ñ pj p 0 for j " 1, 2 are introduced in (35)):

EΛ :" ! pp 0 , p 1 , p 2 q P Q RH such that one of the following conditions holds:

(i) p j " pj p 0 , j " 1, 2, f 0 pp 0 q " f 0,`p p 0 q ď ´1 0 Aptqdt, (ii) p 2 " c 2 , f 0 pp 0 q " f 0,´p p 0 q " ´1 0 1 I 1 ptqAptqdt " f 1 pp 1 q " f 1,`p p 1 q, (iii) p 1 " c 1 , f 0 pp 0 q " f 0,´p p 0 q " ´1 0 1 I 2 ptqAptqdt " f 2 pp 2 q " f 2,`p p 2 q, (iv) p j " c j , j " 0, 1, 2

) .

(

) 17
Case piq corresponds to a situation in which the traffic is fluid on all branches at the macroscopic level, and fluid on the exit branches at the mesoscopic level. In case piiq, the outgoing branch 2 is completely congested and the traffic is stopped on this branch. The traffic reduces to a classical 1:1 junction, the only difficulty being that the traffic is congested at the macroscopic level on the incoming branch and fluid (but saturated by the flux limiter A) on the outgoing branch 1. Case piiiq is symmetric, exchanging the role of the outgoing roads. The last case, Case pivq, is particularly simple since it corresponds to a situation in which the traffic is completely congested (and the velocity of the traffic is null everywhere).

The following lemma states that the germ GΛ is a sort of closure of EΛ:

Lemma 1.5. (EΛ generates GΛ) Assume that (1), ( 5) and (6) hold. We have EΛ Ă GΛ and EΛ generates GΛ: namely, for any U P Q, ´DpU, Ū q ě 0 @ Ū P EΛ ¯ùñ U P GΛ.

The two main ingredients of the proof of Theorem 1.4 are the correct guess of the effective germ GΛ (with its generation property given in Lemma 1.5) and the construction of a corrector for each element of EΛ: Theorem 1.6. (Existence of correctors with prescribed values at infinity) Assume that (1), ( 5) and (6) hold. For any p " pp 0 , p 1 , p 2 q P EΛ, there exists an entropy solution u p " pu i p q P L 8 pR ˆRq of (9) which is 1-periodic in time and a constant C ą 0 such that for all M ě C

}u 0 p ´p0 } L 8 pRˆp´8,´M qq `}u i p ´pi } L 8 pRˆpM,8qq ď CM ´1, i " 1, 2. (18)
If, in addition, p is as in (i) in the definition (17) of EΛ, then

u 0 p " p 0 on R ˆp´8, ´Cq.
The definition of the germ Gλ, the proof of its maximality as well as the proof of Lemma 1.5 are given in Subsection 2.1.3. The proofs of Theorem 1.4 (convergence part) and Theorem 1.6 are postponed to the last section (Subsection 4.1).

Homogenization for 2:1 junctions

We complete the section by the analysis of homogenization on 2:1 junctions: as already pointed out, this case is more realistic in terms of applications. The junction is now described by the two incoming branches Řj " p´8, 0q ˆtju, j " 1, 2, and the outgoing branch Ř0 " p0, 8q ˆt0u. We set Ř " Ť 2 j"0 Řj Y t0u. The mesoscopic model we are interested in concerns a junction with a periodic traffic light which regulates the traffic. As before the time-interval R is split into the 1´periodic sets I 1 and I 2 , each I k consisting locally in a finite number of intervals. On the time-intervals I 1 , only cars coming from road 1 are allowed to enter the junction and the road 0, while on the time-intervals I 2 only cars coming from road 2 can enter road 0. The traffic is also limited on the junction by a flux limiter A " Aptq. To summarize: (see figure 2). We fix ą 0 a scaling parameter. In this model the scaled densities ρ " pρ ,0 , ρ ,1 , ρ ,2 q solve the conservation law:

$ ' ' ' ' & ' ' ' ' %
$ ' ' & ' ' %
ρ ,j P rǎ j , čj s a.e on. p0, `8q ˆŘ j , j " 0, 1, 2 B t ρ ,j `Bx p f j pρ ,j qq " 0 on p0, `8q ˆŘ j , j " 0, 1, 2 pρ ,0 pt, 0 `q, ρ ,1 pt, 0 ´q, ρ ,2 pt, 0 ´qq P Ǧpt{ q for a.e. t P p0, `8q, ρ p0, ¨q " ρ on t0u ˆŘ.

B t ρ0 `Bx p f 0 p ρ0 qq " 0 The fluxes f j satisfy condition (1) with ǎj , bj , čj in place of a j , b j , c j , and f j,˘a re defined similarly as in (3), (4). The time periodic maximal germ Ǧ of period equal to 1 is given by Ǧptq :"

x ą 0 B t ρ1 `Bx p f 1 p ρ1 qq " 0 x ă 0 x ă 0 B t ρ2 `Bx p f 2 p ρ2 qq " 0
" Ǧ1 ptq on I 1 Ǧ2 ptq on I 2 (20)
and Ǧ1 ptq " tpp 0 , p 1 , p 2 q P Q, f 2 pp 2 q " 0, min Aptq, f 1,`p p 1 q, f 0,´p p 0 q ( " f 1 pp 1 q " f 0 pp 0 qu, Ǧ2 ptq " tpp 0 , p 1 , p 2 q P Q, f 1 pp 1 q " 0, min Aptq, f 2,`p p 2 q, f 0,´p p 0 q ( " f 2 pp 2 q " f 0 pp 0 qu.

As in the previous parts, I 1 and I 2 form a partition of R satisfying [START_REF] Andreianov | Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network[END_REF], and the flux limiter A : R Ñ R is a periodic, piecewise constant map such that (6) holds. Finally the initial condition ρ " p ρj q P L 8 p Řq satisfies ρj P rǎ j , čj s a.e.. Theorem 1.7. (Homogenization of the 2:1 junction) Under the previous assumptions, for any ą 0 there exists a unique entropy solution to [START_REF]Homogenization of scalar conservation laws with oscillatory forcing terms[END_REF] and, as Ñ 0 `the solution pρ q to (19) converges in L 1 loc pr0, 8q ˆRq to the unique entropy solution ρ of the homogenized problem $ ' ' & ' ' % ρj P rǎ j , čj s a.e. on p0, `8q ˆŘ j , j " 0, 1, 2 B t ρj `Bx p f j pρ j qq " 0 on p0, `8q ˆŘ j , j " 0, 1, 2 pρ 0 pt, 0 `q, ρ1 pt, 0 ´q, ρ2 pt, 0 ´qq P G f , Λ for a.e. t P p0, `8q, ρp0, ¨q " ρ on t0u ˆŘ.

(

) 21
where the maximal germ G f , Λ is defined explicitly in (87) below with Λ given in Subsection 2.1.3.

The proof of this theorem is given in Subsection 4.2.

Review of the literature

Conservation laws (CL) on junctions (and their application to traffic flows) have attracted a lot of attention: see for instance the monograph [START_REF] Garavello | Traffic flow on networks[END_REF] and the survey paper [START_REF] Bressan | Flows on networks: Recent results and perspectives[END_REF]. A large part of the literature is concerned with conservation laws on 1:1 junctions, involving one flux function for the incoming road and a possibly different one on the outgoing road, see [START_REF] Adimurthi | Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux[END_REF][START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF][START_REF] Audusse | Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies[END_REF][START_REF] Baiti | Well-Posedness for a Class of 2 ˆ2 Conservation Laws with L 8 Data[END_REF][START_REF] Garavello | Conservation laws with discontinuous flux[END_REF][START_REF] Towers | Convergence of a difference scheme for conservation laws with a discontinuous flux[END_REF][START_REF] Towers | A difference scheme for conservation laws with a discontinuous flux: the nonconvex case[END_REF]. It turns out that the approach through the germ theory for 1:1 junctions is strongly linked with Hamilton-Jacobi (HJ) equations on such junctions (still in the 1:1 case, see [START_REF] Cardaliaguet | Conservation law and Hamilton-Jacobi equations on a junction: the convex case[END_REF]). Combining both approaches gives a rough picture of this 1:1 setting: in a nutshell, the junction condition reduces to a flux limiter (a scalar), the conservation law is an L 1 ´contraction and is equivalent to the HJ approach at the level of the antiderivative. Let us also underline that the Hamilton-Jacobi equation possesses itself an L 8 ´contraction property. In conclusion, this 1:1 framework is now relatively well understood.

The situation is completely different for junctions involving at least 3 branches. Indeed, although many works have been devoted to such junctions (see for instance [START_REF] Andreianov | Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network[END_REF][START_REF] Fjordholm | Well-posedness and convergence of a finite volume method for conservation laws on networks[END_REF][START_REF] Garavello | Conservation laws on complex networks[END_REF][START_REF] Goatin | Comparative study of macroscopic traffic flow models at road junctions[END_REF][START_REF] Holle | Entropy Dissipation at the Junction for Macroscopic Traffic Flow Models[END_REF][START_REF] Musch | Well-posedness theory for nonlinear scalar conservation laws on networks[END_REF][START_REF] Towers | Well-posedness of a model of merging and branching traffic flow[END_REF]), the problem is still poorly understood and the general picture is far from clear. For instance, if the germ approach of [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] has been recently extended to general junctions in [START_REF] Fjordholm | Well-posedness and convergence of a finite volume method for conservation laws on networks[END_REF][START_REF] Musch | Well-posedness theory for nonlinear scalar conservation laws on networks[END_REF] (and we strongly use this extension in the paper), there are still few examples of germs which are maximal and complete; one of the outcome of our paper is to describe a new class of such germs (note however that a particular case was previously discussed in [START_REF] Towers | Well-posedness of a model of merging and branching traffic flow[END_REF]). On the other hand, models involving more than 3 branches seem far richer than the 1:1 set-up: for instance our junction condition (in terms of germs) can be parametrized by a whole family of increasing functions (in contrast with the 1:1 set-up where there is just a single parameter). Another difference with the 1:1 setting is that 2:1 and 1:2 junctions are not always L 1 ´contractions. And last, the equivalence between CL and HJ is lost in general: the limit models for 1:2 and 2:1 junctions discussed in this paper do not seem to fit a HJ framework.

It is interesting to compare our class of germs (that we call here the class of traffic light germs, TLgerms in brief) with some of the known germs in the literature on junctions (see in particular [START_REF] Goatin | Comparative study of macroscopic traffic flow models at road junctions[END_REF]). We only consider 1:2 junctions because a reversed germ is automatically constructed for 2:1 junctions, by reversion transform. In [START_REF] Towers | Well-posedness of a model of merging and branching traffic flow[END_REF], the author defines a germ which is a special case of TL-germs for very special functions satisfying moreover f 0 max " f 1 max `f 2 max with λj pλq " θ j λ for j " 1, 2. In the pioneering work [START_REF] Holden | A mathematical model of traffic flow on a network of unidirectional roads[END_REF], the authors introduced a class of germs, by the maximization of some entropy at the junction. It has been only very recently proved in [START_REF] Holle | Entropy Dissipation at the Junction for Macroscopic Traffic Flow Models[END_REF] that those germs are L 1 -contractant. We do not know what is the relationship between this class of germs and the class of TL-germs, even if the intersection of the two classes is empty or not.

The vanishing viscosity germ studied in [START_REF] Andreianov | Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network[END_REF] can be either or not a TL-germ, depending on the flux functions. For instance, for

f 0 " f , f 1 " α 1 f , f 2 " α 2 f , it is possible to show that the vanishing viscosity germ is a TL-germ if and only if α 1 `α2 ď 1.
Hamilton-Jacobi germs (HJ-germs in brief) were defined in [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF] and studied in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF]. These HJ-germs are the same (going from the HJ level to the level of conservation laws) as the ones defined previously in the monograph [START_REF] Garavello | Traffic flow on networks[END_REF] for divergent junctions, and a single ingoing road. These germs are a particular case of RS 2 germs in [START_REF] Garavello | Conservation laws on complex networks[END_REF], where the authors also show that the total variation of the fluxes is bounded by a constant if it is the case for the initial data. This allows them to show the existence of a solution. The uniqueness seems an open question in general (at least at the direct level of conservation laws). Notice that for N ě 3 branches (like 1:2 junctions), it is easy to check that HJ-germs are never L 1 -contractant germs (see [START_REF] Cardaliaguet | Conservation law and Hamilton-Jacobi equations on a junction: the convex case[END_REF]).

In the monograph [START_REF] Garavello | Traffic flow on networks[END_REF], the authors introduce in particular a germ for 2:1 junctions which is the same (by reversion) as the one called RS 1 in the article [START_REF] Garavello | Conservation laws on complex networks[END_REF] for junctions 1:2. It is defined for f i " f for i " 0, 1, 2, and it is possible to show that it is not in the class of what we call here TL-germs. The existence of a solution is shown in [START_REF] Garavello | Conservation laws on complex networks[END_REF], but the uniqueness seems open. We do not know if these germs have the L 1 -contraction property or not.

Organization of the paper

In Section 2, we provide some key results concerning the germs discovered in this paper. Section 3 is devoted to the construction of correctors. The proof of the main homogenization results, Theorem 1.4 and Theorem 1.7, are given in Section 4.

Germs for divergent 1:junctions

In this section, we introduce a new general class of sets, prove that these sets are maximal germs, and show how the different germs encountered in the main results enter into this general framework.

In contrast with the rest of the paper, in this section we only use a weaker assumption than (1), namely For j P t0, 1, 2u, for a j ă b j ă c j , the function f j : ra j , c j s Ñ R is continuous, increasing on ra j , b j s and decreasing on rb j , c j s, with f j pa j q " f j pc j q " 0, [START_REF] Forcadel | Specified homogenization of a discrete traffic model leading to an effective junction condition[END_REF] and we use the same notation f j,˘a s defined in (3), (4). We start the section with a description of the general class of germs used throughout the paper and explain their main properties. We illustrate this notion by showing that the germs introduced for the mesoscopic model do fit this general framework.

Then we present the germs found through the homogenization procedure and give several examples. We complete the section by the proof of the main properties of our class of germs.

A general family of germs

The main result on germs

In this section we investigate a general class of germs on 1:2 junctions. This family is described through a set of parameters Λ "

! λ0 , λ1 , λ2 , λ1 , λ2 )
satisfying the following conditions

$ ' ' ' ' & ' ' ' ' % λj P r0, f j max s for j " 0, 1, 2 λ0 " λ1 `λ 2
the maps λk : r0, f 0 max s Ñ r0, λk s are continuous nondecreasing for k " 1, 2 λk p0q " 0, λk p λ0 q " λk for k " 1, 2 λ1 pλq `λ 2 pλq " minpλ, λ0 q for λ P r0, f 0 max s.

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) 23
The germ G Λ is defined from Λ as follows:

G Λ :" G f,Λ " $ ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' % P " pp 0 , p 1 , p 2 q P R 3 , ˇˇˇˇˇˇˇˇˇˇˇˇˇa j ď p j ď c j , j " 0, 1, 2 0 ď f j pp j q ď λj , j " 0, 1, 2 f 0 pp 0 q " f 1 pp 1 q `f 2 pp 2 q f k,`p p k q ě λk pf 0,`p p 0 qq, k " 1, 2 , / / / / / / / / . / / / / / / / / - . (24)
Theorem 2.1. (Germ for divergent 1:2 junction) Under assumptions ( 22) and ( 23), let us consider the set G Λ defined in [START_REF] Galise | A junction condition by specified homogenization and application to traffic lights[END_REF].

Then (i) G Λ is a maximal germ, (ii) G Λ is determined by its subset E Λ :" Γ Y tP 1 , P 2 , P 3 u , (25)
where the curve Γ and the points P 1 , P 2 , P 3 are defined below in ( 26) and ( 27) respectively. This means that, for any P P Q,

" Dp P , P q ě 0 @ P P E Λ ı ùñ P P G Λ .
In order to describe the curves Γ and the points P i (for i " 1, 2, 3), let us first introduce the roots of f j,˘p ¨q " λ for j " 0, 1, 2:

" " a j , b j ‰ Q u j `pλq :" r such that f j,`p rq " λ P " 0, f j max ‰ " b j , c j ‰ Q u j ´pλq :" r such that f j,´p rq " λ P " 0, f j max ‰ .
We will also use later the notation u j ˘" pf j,˘q´1 . Then Γ :"

! P " pu 0 `pλq, u 1 `pλ 1 q, u 2 `pλ
2 qq with λ k :" λk pλq for k " 1, 2 and λ P r0, λ0 s )

and

$ ' ' ' ' & ' ' ' ' % P 0 :" pu 0 `p0q, u 1 `p0q, u 2 `p0qq " pa 0 , a 1 , a 2 q P Γ P 3 :" pu 0 ´p0q, u 1 ´p0q, u 2 ´p0qq " pc 0 , c 1 , c 2 q P 1 :" pu 0 ´pλ1 q, u 1 `pλ1 q, u 2 ´p0qq P 2 :" pu 0 ´pλ2 q, u 1 ´p0q, u 2 `pλ2 qq. (27)
Heuristically, the curve Γ corresponds to a situation in which all the branches are fluids, while

$ ' ' ' ' & ' ' '
' % P 0 " p empty road, empty road, empty road q P Γ P 3 " p fully congested, fully congested, fully congested q P 1 " p congested, fluid and saturated, fully congested q P 2 " p congested, fully congested, fluid and saturated q

where "fully congested" means that the road is with a maximal density of vehicles (hence with zero velocity). On the other hand, "fluid and saturated" means that the outgoing road is still fluid, but that we can not increase the flux passing through the junction point.

The proof of Theorem 2.1 is postponed to Subsection 2.2.

Let us now explain how the germs introduced for the mesoscopic model and the homogenized germ introduced for the macroscopic model fit into the framework just described.

Germs in the mesoscopic model

We check here that the sets G Λ k ptq (for t P I k and k " 1, 2) introduced in ( 11) and ( 12) respectively, are of the form [START_REF] Galise | A junction condition by specified homogenization and application to traffic lights[END_REF] for suitable sets Λ k ptq. For t P I 1 , the set Λ 1 ptq " p λ0

1 ptq, λ1 1 ptq, λ2 1 ptq, λ1 1 pt, ¨q, λ2 1 pt, ¨qq is given by $ & % λ0 1 ptq " λ1 1 ptq " Aptq, λ2 1 " 0 λ1
1 pt, λq " minpλ, Aptqq for λ P r0, f 0 max s λ2

1 pt, λq " 0 for λ P r0, f 0 max s.

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) 28
For t P I 2 , the set Λ 2 ptq " p λ0 2 ptq, λ1 2 ptq, λ2 2 ptq, λ1 2 pt, ¨q, λ2 2 pt, ¨qq is defined symmetrically, exchanging the indices 1 and 2:

$ & % λ0 2 ptq " λ2 2 ptq " Aptq, λ1 2 " 0 λ1 2 pt, λq " 0 for λ P r0, f 0 max s λ2 2 pt, λq " minpλ, Aptqq for λ P r0, f 0 max s. (29)
The next lemma claims that the germs G Λ k ptq (for k " 1, 2) associated with the Λ k ptq through definition [START_REF] Galise | A junction condition by specified homogenization and application to traffic lights[END_REF], coincide precisely with the germs G Λ k ptq introduced in ( 11) and ( 12) respectively for the mesoscopic model: Lemma 2.2. (Characterization of the maximal germs G Λ k ) For any k " 1, 2 and any t P I k , the set G Λ k ptq , defined through [START_REF] Galise | A junction condition by specified homogenization and application to traffic lights[END_REF] from the sets Λ k ptq is a maximal germs and coincides with the set G Λ k ptq introduced in (11) (for k " 1) and (12) (for k " 2).

Proof. The proof is elementary. By symmetry, we can only do it for G Λ1ptq for t P I 1 . Notice that Λ 1 ptq satisfies [START_REF] Forcadel | Homogenization of a discrete model for a bifurcation and application to traffic flow[END_REF]. Hence G Λ1ptq is a maximal germ, from Theorem 2.1. If P " pp k q k"0,1,2 belongs to G Λ1ptq or to G Λ1 ptq, we have λ :" f 0 pp 0 q " f 1 pp 1 q P r0, λ0

1 ptqs and then

p 0 P u 0 ˘pλq ( , p 1 P u 1 ˘pλq ( .
This gives 2 ˆ2 cases. Examining all cases in details (it is slightly tedious to do it for both expressions), we can check in both expressions that all cases are possible except the following case which is excluded by both expressions p 0 " u 0 ´pλq, p 1 " u 1 `pλq for λ P r0, λ0 1 ptqq. Hence the two expressions coincide and the lemma holds true.

The homogenized germ in the macroscopic model

We now turn to the homogenized germ. This germ is naturally associated with the correctors introduced in the next section. It happens however that it can be built independently: we present this construction here. We also give several examples in which the germ can be explicitly computed (Propositions 2.6, 2.7 and 2.10).

The homogenized germ GΛ introduced in Theorem 1.4 is defined through the set of parameters

Λ " ! λ0 , λ1 , λ2 , λ1 , λ2 )
by relation ( 24) that we recall:

GΛ :" $ & % P " pp 0 , p 1 , p 2 q P Q RH , ˇˇˇˇˇ0 ď f j pp j q ď λj , j " 0, 1, 2 f k,`p p k q ě λk pf 0,`p p 0 qq, k " 1, 2
, .

-

. ( 30
)
In Λ, the effective limiters λ0 , λ1 , λ2 are given by

$ ' ' ' ' & ' ' ' ' % λk :" ˆ1 0 1 I k ptqAptqdt for k " 1, 2, λ0 :" ˆ1 0 Aptqdt " λ1 `λ 2 ď f 0 max . (31)
For λ P r0, λ0 s, let p 0 " pf 0,`q´1 pλq. Note that p 0 satisfies the inequality

f 0,`p p 0 q " f 0 pp 0 q ď ˆ1 0 Aptqdt " λ0 . (32)
We introduce the 1-periodic map2 F λ " F p 0 : R Ñ r0, f 0 max s as

@t P R F λ ptq " F p 0 ptq " $ & % λ " f 0 pp 0 q if 1 t ´t1 ˆt t1
Apsqds ě f 0 pp 0 q, @t 1 ă t, Aptq otherwise, [START_REF] Karlsen | A note on Front tracking and the Equivalence between Viscosity Solutions of Hamilton-Jacobi Equations And Entropy Solutions of scalar Conservation Laws[END_REF] and set, for k " 1, 2, λk pλq "

ˆ1 0 F λ ptq1 I k ptqdt. (34)
We extend the functions λk up to f 0 max by λk pλq :" λk for λ P r λ0 , f 0 max s, k " 1, 2.

Finally, we set, for k " 1, 2, pk p 0 " u k `pλ k pf 0 pp 0 qqq, @p 0 P ra 0 , b 0 s with f 0 pp 0 q ď λ0 ,

where we recall the notation u k ˘" pf k,˘q´1 . The interpretation of these quantities is the following: we show in Lemma 3.5 below that F p 0 is the flux at the junction x " 0 of the 1´periodic corrector taking value p 0 at ´8 (or, equivalently, having a flux λ " f 0 pp 0 q at ´8). Proposition 3.12 shows that the pk p 0 are the densities at `8 and on the branch k of this corrector. Hence the quantities λk pλq are the fluxes at `8 of the time periodic corrector with a flux λ at ´8.

F λ " λ `A ´pΦ λ q 1 a.e. on R
where Φ λ is a viscosity solution to the following obstacle problem minpΦ λ ´B, pΦ λ q 1 ´λq " 0 on R, B 1 " A such that Φ λ ´B is 1-periodic. Moreover Φ λ is unique for λ P r0, λ0 q and we have tΦ λ " Bu Ă tF λ " λu and tΦ λ ą Bu Ă tF λ " Au.

Finally, we have the following representation for Φ λ :

Φ λ ptq :" sup τ ě0
tBpt ´τ q `λτ u .

See Lemma 2.5 for related results on ψ p 0 " Φ λ ´B.

We then have the following properties Lemma 2.4. (Properties of the fluxes λk ) For k " 1, 2, λk ď f k max and the map λÞ Ñ λk pλq is continuous and nondecreasing on r0, f 0 max s with λ1 pλq `λ 2 pλq " λ @λ P r0, λ0 s

and 0 ď λk pλq ď λk " λk p λ0 q, k " 1, 2, @λ P r0, λ0 s.

Proof.

Step 0: preliminaries. Let us first note for later use that

F p 0 ptq ď Aptq a.e.. (39)
Indeed, let t be a point of continuity of A. Then either F p 0 ptq " Aptq, or 1 t´t1 ´t t1 Apsqds ě f 0 pp 0 q for any t 1 ă t. In this later case,

Aptq " lim t1Ñt ´1 t ´t1 ˆt t1 Apsqds ě f 0 pp 0 q " F p 0 ptq, (40)
which shows [START_REF] Schieborn | Viscosity solutions of Eikonal equations on topological networks[END_REF]. Fix k " 1, 2. On I k , we have Aptq ď f k max by assumption on A. Thus

λk " ˆ1 0 Aptq1 I k ptqdt ď f k max |r0, 1s X I k | ď f k max .
Let us set

@t P R, ψ p 0 ptq " max t1ďt ! ˆt t1 pf 0 pp 0 q ´Apsqqds ) .
We explain in Lemma 2.5 below that ψ p 0 is nonnegative, Lipschitz continuous, 1´periodic and satisfies

ψ 1 p 0 ptq " " f 0 pp 0 q ´Aptq if ψ p 0 ptq ą 0 0 if ψ p 0 ptq " 0 a.e. ( 41
)
and ψ 1 p 0 ptq " f 0 pp 0 q ´Fp 0 ptq a.e..

Moreover, by the definition of ψ p 0 , for any t P R, ψ p 0 ptq " 0 is equivalent to saying that 1 t´t1 ´t t1 Apsqds ě f 0 pp 0 q for any t 1 , and thus, as explained in [START_REF] Serre | Correctors for the homogenization of conservation laws with oscillatory forcing terms[END_REF], one has A ě f 0 pp 0 q a.e. on tψ p 0 " 0u.

Step 1: λk are nondecreasing. Fix k " 1, 2. We now prove that the λk are non decreasing on r0, λ0 s (it is constant on r λ0 , λ 0 max s). If 0 ď λ ď λ ď λ0 , then a 0 ď p 0 :" u 0 `pλq ď u 0 `pλ q ": p0 ď b 0 .

Hence, by the definition of ψ p 0 and ψ p0 , ψ p 0 ď ψ p0 and therefore tψ p 0 ą 0u Ă tψ p0 ą 0u. Recalling ( 41), [START_REF] Towers | A difference scheme for conservation laws with a discontinuous flux: the nonconvex case[END_REF] and the facts that f 0 pp 0 q ď f 0 pp 0 q and that A ě f 0 pp 0 q a.e. on tψ p 0 " 0u, we have F p 0 ptq " f 0 pp 0 q ´ψ1 p 0 ptq " 1 tψ p 0 ą0u ptqAptq `1tψ p 0 "0u ptqf 0 pp 0 q " 1 tψ p0 ą0u ptqAptq `1tψ p0 "0u ptqf 0 pp 0 q `1tψ p0 ą0, ψ p 0 "0u pf 0 pp 0 q ´Aptqq ď 1 tψ p0 ą0u ptqAptq `1tψ p0 "0u ptqf 0 pp 0 q " F p0 ptq.

Recalling [START_REF] Lions | Viscosity solutions for junctions: well posedness and stability[END_REF] then shows that λk is nondecreasing.

Step 2: λk is continuous in r0, λ0 s. We assume that λ n converge to λ in r0, λ0 s. Let p 0,n " u 0 ´pλ n q and p 0 " u 0 ´pλq. Then pp 0,n q converges to p 0 and pψ p 0,n q converges uniformly to ψ p 0 . Using assumption (5), we can write the set I k X r0, 1s into a finite union of disjoint intervals ppt j 1 , t j 2 qq j"1,...,J k up to a set of measure 0. Then [START_REF] Towers | A difference scheme for conservation laws with a discontinuous flux: the nonconvex case[END_REF] shows that

ˆ1 0 F p 0,n ptq1 I k ptqdt " J k ÿ j"1 ˆtj 2 tj 1 p´ψ 1 p 0,n ptq `f 0 pp 0,n qqdt " J k ÿ j"1 pψ p 0,n pt j 1 q ´ψp 0,n pt j 2 q `f 0 pp 0,n qpt j 2 ´tj 1 qq converges to J k ÿ j"1 pψ p 0 pt j 1 q ´ψp 0 pt j 2 q `f 0 pp 0 qpt j 2 ´tj 1 qq " ˆ1 0 F p 0 ptq1 I k dt.
By [START_REF] Lions | Viscosity solutions for junctions: well posedness and stability[END_REF] this shows the continuity of λk in r0, λ0 s.

Step 3: proof of (37) and [START_REF] Panov | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF]. By [START_REF] Lions | Viscosity solutions for junctions: well posedness and stability[END_REF] and [START_REF] Towers | A difference scheme for conservation laws with a discontinuous flux: the nonconvex case[END_REF] we have, for λ P r0, λ0 s, λ1 pλq `λ 2 pλq " ˆ1 0 F p 0 ptqdt " ˆ1 0 pf 0 pp 0 q ´ψ1 p 0 ptqqdt " f 0 pp 0 q " λ, since ψ p 0 is periodic. This is [START_REF] Musch | Well-posedness theory for nonlinear scalar conservation laws on networks[END_REF]. By [START_REF] Schieborn | Viscosity solutions of Eikonal equations on topological networks[END_REF], F p 0 ptq ď Aptq a.e.. Hence by [START_REF] Lions | Viscosity solutions for junctions: well posedness and stability[END_REF], λk pλq ď λk for any λ P r0, λ0 s. For λ " λ0 , we then have λ0 " λ1 `λ 2 ě λ1 p λ0 q `λ 2 p λ0 q " λ0 , which shows that the inequalities λk ě λk p λ0 q are actually equalities. This is [START_REF] Panov | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF]. Let us finally remark that λk pλq " λk , @λ ě λ0 . Hence λk is also continuous in r0, f 0 max s (recall that it is continuous in r0, λ0 s by Step 2).

In the proof of Lemma 2.4 we used the following result: Lemma 2.5 (Analysis of ψ p 0 ). Fix p 0 P ra 0 , b 0 s such that (32) holds and let

@t P R, ψ p 0 ptq:" max t1ďt ! ˆt t1 pf 0 pp 0 q ´Apsqqds ) .
Then ψ p 0 is nonnegative, Lipschitz continuous, 1´periodic and satisfies, a.e., ψ 1 p 0 ptq " f 0 pp 0 q ´Fp 0 ptq " " f 0 pp 0 q ´Aptq if ψ p 0 ptq ą 0 0 if ψ p 0 ptq " 0.

In addition, ψ 1 p 0 ptq `Aptq ´f 0 pp 0 q ě 0 a.e..

Proof. Note, choosing t 1 " t ´1 as a competitor and using [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF], that ψ p 0 ě 0. Moreover, by [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF] and periodicity of A, the maximum in t 1 in the definition of ψ p 0 can be chosen in rt ´1, ts. By periodicity of A, ψ p 0 is 1´periodic. Moreover, as

ψ p 0 ptq " max t1PR ! ˆt t1^t pf 0 pp 0 q ´Apsqqds ) ,
where the integrand is bounded, ψ p 0 is also Lipschitz continuous as the supremum of uniformly Lipschitz continuous quantities.

Let us now compute ψ 1 p 0 . On tψ p 0 " 0u, we have ψ 1 p 0 " 0 a.e.. Let t be a point of derivability of ψ p 0 with ψ p 0 ptq ą 0 and such that t is a point of continuity of A. If t1 is optimal in the definition of ψ p 0 , then t1 ă t because ψ p 0 ptq ą 0. Hence, for |h| ą 0 small, ψ p 0 pt `hq ě ˆt`h t1 pf 0 pp 0 q ´Apsqqds " ψ p 0 ptq `ˆt`h t pf 0 pp 0 q ´Apsqqds, which implies that ψ 1 p 0 ptq " f 0 pp 0 q ´Aptq. So we have proved that

ψ 1 p 0 ptq " " f 0 pp 0 q ´Aptq if ψ p 0 ptq ą 0 0 if ψ p 0 ptq " 0 a.e..
On the other hand equality ψ p 0 ptq " 0 is equivalent to saying that, for any t 1 ă t,

1 t ´t1 ˆt t1 Apsqqds ě f 0 pp 0 q. (43)
Comparing [START_REF] Karlsen | A note on Front tracking and the Equivalence between Viscosity Solutions of Hamilton-Jacobi Equations And Entropy Solutions of scalar Conservation Laws[END_REF] with the previous equality shows that ψ 1 p 0 ptq " f 0 pp 0 q ´Fp 0 ptq a.e.. Finally, we have seen in [START_REF] Serre | Correctors for the homogenization of conservation laws with oscillatory forcing terms[END_REF] that A ě f 0 pp 0 q a.e. on tψ p 0 " 0u, which shows the last claim.

Proof of Theorem 1.4: GΛ is a maximal germ. By Lemma 2.4, Λ satisfies [START_REF] Forcadel | Homogenization of a discrete model for a bifurcation and application to traffic flow[END_REF], which implies by Theorem 2.1 that GΛ is a maximal germ.

Proof of Lemma 1.5. Let us set Γ :"

! U " pu 0 `pλq, u 1 `pλ 1 q, u 2 `pλ
2 qq with λ k :" λk pλq for k " 1, 2 and λ P r0, λ0 s )

" U " pp 0 , p1 p 0 , p2 p 0 q, p 0 P ra 0 , b 0 s with f 0 pp 0 q ď λ0 ( and $ ' ' ' ' & ' ' ' ' % P 0 :" pa 0 , a 1 , a 2 q P Γ P 3 :" pc 0 , c 1 , c 2 q P 1 :" pu 0 ´pλ1 q, u 1 `pλ1 q, c 2 q P 2 :" pu 0 ´pλ2 q, c 1 , u 2 `pλ2 qq Then EΛ defined in [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF] is equal to EΛ " Γ Y tP 1 , P 2 , P 3 u, the curve Γ corresponding to case (i) in [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF], P 1 to case (ii), P 2 to case (iii) and P 3 to case (iv). Therefore EΛ generates GΛ by Theorem 2.1-(ii).

Three explicit examples. We complete this part by three explicit computations. In the first one, there is no flux limiter (and hence no stop); the homogenized germ is then quite straightforward. The second one involves one stop only and no other flux limiter; it shows that the order (stop-road 1-road 2 or stop-road 2-road 1) influences the homogenized germ, even if the flux function is the same for both exit roads. The last one gives a hint of the class of germs that can be reached through our homogenization procedure.

Example 1: the case where the traffic is never limited. We assume that

Aptq " mintf 0 max , f j max u for t P I j , j " 1, 2 [START_REF] Vasseur | Strong Traces for Solutions of Multidimensional Scalar Conservation Laws[END_REF] and that the sets I 1 and I 2 are as simple as possible:

Up to a translation in time, the restriction of

I k to r0, 1s is a single interval. ( 45
)
Under these assumptions, we can compute explicitly Λ.

Proposition 2.6. Assume (44) and (45). Let us set θ k " ˇˇI k X r0, 1s ˇˇ(for k " 1, 2). Then

$ & % λk :" θ k mintf 0 max , f j max u for k " 1, 2, λ0 :" θ 1 mintf 0 max , f 1 max u `θ2 mintf 0 max , f 2 max u.
Letting θ k ˚:" λk λ0 (for k " 1, 2), the curves λ1 , λ2 are given by

$ ' ' ' ' & ' ' ' ' % " λ1 pλq :" maxpθ 1 λ, λ ´λ 2 q λ2 pλq :" minpθ 2 λ, λ2 q ˇˇˇf or λ P r0, λ0 s if θ 2 ě θ 2 ˚,
" λ1 pλq :" minpθ 1 λ, λ1 q λ2 pλq :" maxpθ 2 λ, λ ´λ 1 q ˇˇˇf or λ P r0, λ0 s if

θ 2 ă θ 2 ˚.
Proof. The computation of the λk (k " 0, 1, 2) is immediate. Let us now compute the λk (k " 1, 2). To fix the ideas, we assume that θ 2 ě θ 2 ˚, the other case being treated in a symmetric way. Without loss of generality, we also assume that 0 ă θ 1 ă 1, since otherwise the problem reduces to a problem with a single outgoing road. We set φ k " mintf 0 max , f k max u, k " 1, 2. Note that θ 2 ě θ 2 ˚is equivalent to saying that φ 1 ě φ 2 . Fix λ P r0, λ0 s and let p 0 " u 0 `pλq.

Let us first assume that λ P r0, φ 2 s. Recalling that θ 2 " 1 ´θ1 , we have maxpθ 1 λ, λ ´λ 2 q " θ 1 λ and minpθ 2 λ, λ2 q " θ 2 λ. On the other hand, in this case, the map F p 0 defined in [START_REF] Karlsen | A note on Front tracking and the Equivalence between Viscosity Solutions of Hamilton-Jacobi Equations And Entropy Solutions of scalar Conservation Laws[END_REF] is constant and equal to λ " f 0 pp 0 q. Then, for k " 1, 2, λk pλq " ˆ1 0 F p 0 pλq ptq1 I k ptqdt " θ k λ.

Let us now suppose that λ P pφ 2 , λ0 s. Then maxpθ 1 λ, λ ´λ 2 q " λ ´λ 2 and minpθ 2 λ, λ2 q " λ2 . To compute F p 0 , we assume without loss of generality that I 1 X r0, 1q " r0, θ 1 q while I 2 X r0, 1q " rθ 1 , 1q. Since λ ď λ0 " θ 1 φ 1 `θ2 φ 2 and λ ą φ 2 , we deduce that λ ă φ 1 . Hence the minimum over t 1 of 1 t´t1 ´t t1 pApsq ´λqds is reached for t 1 " ´p1 ´θ1 q " ´θ2 if t P r0, θ 1 q. Then, by [START_REF] Karlsen | A note on Front tracking and the Equivalence between Viscosity Solutions of Hamilton-Jacobi Equations And Entropy Solutions of scalar Conservation Laws[END_REF],

F p 0 ptq " # f 0 pp 0 q if t P r θ 2 pλ´φ 2 q φ 1 ´λ , θ 1 q Aptq otherwise (mod 1), so that λ1 pλq " ˆ1 0 F p 0 pλq ptq1 I 1 ptqdt " ˆθ1 0 F p 0 pλq ptqdt " θ 2 pλ ´φ2 q φ 1 ´λ φ 1 `pθ 1 ´θ2 pλ ´φ2 q φ 1 ´λ qλ " θ 2 pλ´φ 2 q`θ 1 λ " λ´λ 2 , while λ2 pλq " ˆ1 0 F p 0 pλq ptq1 I 2 ptqdt " θ 2 φ 2 " λ2 .
Example 2: one stop followed successively by two exits. Consider now the case where f 1 " f 2 " f , and f 0 may be different. We set A 0 :" maxpf 0 max , f max q. We also assume that for τ 0 " θ 0 , τ 1 " θ 0 `θ1 , τ 2 " θ 0 `θ1 `θ2 " 1 with θ i ą 0, we have Aptq "

$ & % 0 on r0, τ 0 q " I 0 A 0 on rτ 0 , τ 1 q " I 1 A 0 on rτ 1 , τ 2 q " rτ 1 , 1q " I 2

In other worlds, all the incoming vehicles from road 0, go on road j during the time interval I j for j " 1, 2, while they are all stopped at the junction during the time interval I 0 . We then have the following result Proposition 2.7. (Flux computation with one stop followed successively by two exits) Under the previous assumptions, we have for λ P r0, λ0 s $ & % λ1 pλq " mintλpθ 0 `θ1 q, λ1 u λ2 pλq " maxtλθ 2 , λ ´λ 1 u p λ1 `λ 2 qpλq " λ with λ0 :" A 0 pθ 1 `θ2 q, λ1 :"

A 0 θ 1 , λ2 " A 0 θ 2 .
Moreover, if θ 1 " θ 2 , then we have λ1 " λ2 and λ1 ą λ2 on p0, λ0 q with equality at both end points of the interval p0, λ0 q.

Remark 2.8. The result of Proposition 2.7 in the special case θ 1 " θ 2 , means that the order (stop-road 1-road 2) matters with respect to the order (stop-road-2-road 1). The road which receives the traffic just after the stop, will have a higher passing flux than the other one.

After reversion, this corresponds to a convergent 2:1 junction where the outgoing road 0 is congested.

Then road 1 (just after the stop) will evacuate more easily than road 2, its vehicles onto the road 0. This happens because the stop created some free space on road 0 just after the junction. This last interpretation is much more intuitive here.

Proof. For t P r0, 1s, let Bptq " maxp0, A 0 pt ´θ0 qq and extend B to R by Bpt `1q " Bptq `A0 pθ 1 `θ2 q such that B 1 " A. For any λ P r0, λ0 s, where λ0 :" ´r0,1s A " A 0 pθ 1 `θ2 q, define Φ λ as in (36) and t " t λ P pθ 0 , 1s such that λt " Bptq i.e t λ ´θ0 " λθ 0 A 0 ´λ .

We then have, using that tΦ λ " Bu Ă tF λ " λu,

λj pλq " ˆIj F λ " ˆIj XtΦ λ ąBu A `ˆI j XtΦ λ "Bu λ " ˆIj A `ˆI j XtΦ λ "Bu pλ ´Aq.
Since tΦ λ " Bu X r0, 1s " rt λ , 1s, we deduce that λ1 pλq " A 0 θ 1 `ˆrτ1^t λ ,τ1s pλ ´A0 q

Let us set λ ˚" A 0 θ 1 θ 0 `θ1 such that t λ˚" τ 1 . This implies that, for λ P r0, λ ˚s, we have λ1 pλq " A 0 θ 1 `pλ ´A0 qpτ 1 ´tλ q " λτ 1 and then λ1 pλq " minpλτ 1 , λ1 q on r0, λ0 s, with λ1 :" λ ˚τ1 " A 0 θ 1 .

Similarly, using that λ2 pλq " ˆrt λ _τ1,1s pλ ´A0 q `A0 θ 2

we can show that λ2 pλq " maxpλθ 2 , λ ´λ 1 q on r0, λ0 s

This ends the proof.

Remark 2.9. (Bounds on the derivatives of λj ) A natural question is the characterization of the functions λj that can be constructed by homogenization. In fact, the derivative of these fluxes has to be bounded between 0 and 1. More precisely, one can show that 1 ´g2 pλq ě p λ1 q 1 pλq ě g 1 pλq ě 0 a.e. for λ P r0, λ0 s (and symmetrically for λ2 ) with g j pλq :" | tF λ " λu X I j | Moreover g j P L 8 pr0, λ0 sq has a monotone nonincreasing representant in the class of L 8 functions. We can show that this also implies that if there exists some λ 1 P p0, λ0 q such that the derivative vanishes p λj q 1 pλ 1 q " 0 then λj " const on rλ 1 , λ0 s. Moreover each λj is sandwiched in between a concave function and a convex function.

Example 3: concave flux λ1 . We now explain how to compute λ1 from A when A has a particular structure and is assumed to be continuous.

Proposition 2.10. (The case of λ1 concave and A continuous) Given 0 ă t 1 ´t0 ă 1, assume furthermore that A : R Ñ r0, `8q (still 1-periodic) is C 1 , decreasing on rt 0 , t 1 s, and increasing on rt 1 , t 0 `1s. Given λ0 :" ´r0,1s A, consider t0 P rt 0 , t 1 s such that Ap t0 q " λ0 . Assume now that A 1 ă 0 on r t0 , t 1 q and I 1 " r t0 , t 1 s mod. 1

Then up to translate A, we can assume that t0 " 0, and we have

1 ą p λ1 q 1 pλq " " pA |r0,t1s q ´1pλq if λ P pApt 1 q, Ap0qs |I 1 | " t 1 if λ P r0, Apt 1 qq (46)
The function λ1 is C 1 and concave on r0, λ0 s. Moreover λ1 is linear on r0, Apt 1 qs, and C 2 strictly concave on pApt 1 q, λ0 s. We also have p λ1 q 1 p λ0 q " t0 " 0 when 0 " t0 ă t 1 .

Proof. We first notice that for λ P r0, Apt 1 qs, we have F λ " λ and λ1 pλq " |I 1 |λ. For λ P rApt 1 q, Ap t0 qs, we define t λ P r t0 , t 1 s such that Apt λ q " λ. Arguing as in the proof of Proposition 2.7, we have βpλq :" λ1 p λ0 q ´λ 1 pλq "

ˆtλ t0 pA ´λq Because λ Þ Ñ t λ " pA |r t0,t1s q ´1pλq " A ´1pλq is C 1 on pApt 1 q, Ap

pApsq ´Aptqqds

Taking the derivative, and dividing by A 1 ptq ă 0, and up to assume that t0 " 0, we get

p´β 1 q ˝A " Id r t0,t1q
with ´β1 " p λ1 q 1 . This implies that p λ1 q 1 " pA |r t0,t1s q ´1 on pApt 1 q, Ap t0 qs Remark 2.11. 1) Notice that we can also prove a sort of recriprocal result. Given any C 2 concave function λ1 : r0, λ0 s Ñ r0, `8q with p λ1 q 2 ă 0 on p0, λ0 q and λ1 p0q " p λ1 qp λ0 q " 0 ă p λ1 q 1 p0q ă 1, we can cook-up a suitable 1-periodic function A with Apt 1 q " 0. Everything can be done such that λ1 is associated to A as in Proposition 2.10 (except that A is constant on pt 1 , t 0 `1q and possibly discontinuous at t 0 and t 1 ).

2) Notice also that in this remark and in Proposition 2.10, the function A is not piecewise constant, as it is assumed in our homogenization result. Nevertheless, an approximation of such A by a sequence of piecewise constant functions is always possible, and then relation ( 46) is still valid, once it is correctly interpreted (where λ1 is continuous and piecewise linear). Then any concave λ1 as in point 1), can then be obtained as limits of homogenized λ1 of piecewisely approximated functions A.

Proof of Theorem 2.1

This subsection is devoted to the proof of Theorem 2.1. Starting with a lemma describing how the dissipation condition can be violated (Lemma 2.12), we prove that G Λ is maximal and generated by E Λ (Lemma 2.13) and then that it is a germ (Lemma 2.14).

A technical lemma

We consider P " pp 0 , p 1 , p 2 q and P " pp 0 , p1 , p2 q with P, P P Q RH , i.e. such that we have the Rankine-Hugoniot relations " f 0 pp 0 q " f 1 pp 1 q `f 2 pp 2 q f 0 pp 0 q " f 1 pp 1 q `f 2 pp 2 q.

Defining $ & % F 0 :" f pp 0 q ´f pp 0 q, s 0 :" signpp 0 ´p0 q F 1 :" f pp 1 q ´f pp 1 q, s 1 :" signpp 1 ´p1 q F 2 :" f pp 2 q ´f pp 2 q, s 2 :" signpp 2 ´p2 q (47) we get Dp P , P q " s 0 F 0 ´ s 1 F 1 `s2 F 2 ( with F 0 " F 1 `F 2 and s j " 0 implies F j " 0.

Lemma 2.12. (Violated dissipation for divergent 1:2 junction) Let us consider the dissipation

D :" s 0 F 0 ´ s 1 F 1 `s2 F 2 ( with $ & % F 0 " F 1 `F 2 s j P t0, ˘1u
for j " 0, 1, 2 s j " 0 implies F j " 0 for j " 0, 1, 2.

Then D ă 0 if and only if $ ' ' ' ' & ' ' ' ' %
s 0 F 0 ă 0, s 1 " s 2 " s 0 weakly or s 1 F 1 ą 0, s 0 " s 2 " s 1 weakly or s 2 F 2 ą 0, s 0 " s 1 " s 2 weakly where s 1 " s 2 " s 0 weakly ðñ s 0 " 0, s 1 s 2 ě 0, s 0 s 1 ď 0, s 0 s 2 ď 0.

Proof. The proof is technical but elementary. Up to change pF 0 , F 1 , F 2 q in pF 0 , ´F 1 , ´F 2 q, we can assume that D " s 0 F 0 `s1 F 1 `s2 F 2 with F 0 `F 1 `F 2 " 0 and we want to show that D ă 0 if and only if

$ ' ' ' ' & ' ' ' ' % (
0) s 0 F 0 ă 0, s 1 " s 2 " s 0 weakly or (1) s 1 F 1 ă 0, s 0 " s 2 " s 1 weakly or (2) s 2 F 2 ă 0, s 0 " s 1 " s 2 weakly.

Step 1: (0),( 1) or ( 2) imply D ă 0 We only consider the case (0) (the other cases being symmetric). This means that we have s 0 F 0 ă 0, s 0 " 0, s 1 s 2 ě 0, s 0 s 1 ď 0, s 0 s 2 ď 0 and we distinguish several cases. Case 1.a: s 1 " 0 " s 2 . Then F 1 " 0 " F 2 and D " s 0 F 0 ă 0. Case 1.b: s 1 " 0 " s 2 . Then F 1 " 0 and then F 2 " ´F 0 and also s 2 " ´s0 . We get D " 2s 0 F 0 ă 0. Case 1.c: s 1 " 0 " s 2 . This is symmetric to case 1.b. Case 1.d: s 1 " 0, s 2 " 0. Then s 1 " s 2 " ´s0 , and F 1 `F 2 " ´F 0 gives D " 2s 0 F 0 ă 0.

We conclude that D ă 0 in all cases of Step 1.

Step 2: if we do not have (0),( 1) nor ( 2) then D ě 0 If s j F j ě 0 for all j " 0, 1, 2, then D ě 0. Then assume that at least one such term is negative. By symmetry, we can assume that s 0 F 0 ă 0.

Notice also that if all the s j for j " 0, 1, 2 have the same sign (with value in t0, ˘1u), then D " 0 (because F 0 `F 1 `F 2 " 0). Then we can assume that the s j do not have all the same sign. Moreover recall that we don't have (0). Hence we can assume in particular that $ & % s 0 F 0 ă 0 s 0 " 0 and `s1 s 2 ă 0 or s 0 s 1 ą 0 or s 0 s 2 ą 0 s0 , s 1 , s 2 do not have all the same sign.

We distinguish several cases. Case 2.a: s 0 s 1 ą 0. If s 2 " 0, then s 1 " s 0 " ´s2 and F 0 `F 1 " ´F 2 which gives D " 2s 2 F 2 ě 0 because case (2) is also excluded. If s 2 " 0, then F 2 " 0 and F 1 " ´F 0 which implies D " 0. Case 2.b: s 0 s 2 ą 0. This case is symmetric of case 2.a. Case 2.c: s 1 s 2 ă 0. If s 0 " s 1 , then s 0 " s 1 " ´s2 and F 0 `F 1 " ´F 2 . This implies that D " 2s 2 F 2 ě 0, because (2) does not hold. If s 0 " s 2 , then we obtain, in a symmetric way, that D ě 0.

We conclude that D ě 0 in all cases of Step 2. This completes the proof of the lemma.

Maximality

Lemma 2.13. (Maximality of G Λ ) We work under the assumptions of Theorem 2.1. Let us consider a set G Ă Q satisfying the dissipation condition Dp P , P q ě 0 for all P , P P G. Let E Λ :" Γ Y tP 1 , P 2 , P 3 u defined in [START_REF] Garavello | Traffic flow on networks[END_REF].

If E Λ Ă G, then we have G Ă G Λ .
This implies in particular that G Λ is maximal.

Proof. We choose P P G and we will test it with P P Γ Y tP 1 , P 2 , P 3 u using the dissipation condition Dp P , P q ě 0 in order to show that P P G Λ . We write P " pp 0 , p 1 , p 2 q, P " pp 0 , p1 , p2 q

We use notation (47) for the fluxes F j for j " 0, 1, 2.

Step 1: recovering Rankine-Hugoniot condition We choose P :" P 3 . Because for all P P Q " ra 0 , c 0 s ˆra 1 , c 1 s ˆra 2 , c 2 s, we have p j ď pj " c j for all j " 0, 1, 2, we get 0 ď Dp P , P q " F 0 ´pF 1 `F 2 q, f 0 pp 0 q " f 1 pp 1 q `f 2 pp 2 q, which implies

f 0 pp 0 q ´ f 1 pp 1 q `f 2 pp 2 q ( ď 0. (48)
We now choose P :" P 0 . Because for all P P Q, we have p j ě pj " a j for all j " 0, 1, 2, we get

´ F 0 ´pF 1 `F 2 q ( ě 0, f 0 pp 0 q " f 1 pp 1 q `f 2 pp 2 q, which implies f 0 pp 0 q ´ f 1 pp 1 q `f 2 pp 2 q ( ě 0. ( 49
)
Combining ( 48) and (49), we get the Rankine-Hugoniot relation and then P P Q RH .

Step 2: getting flux limiters

Step 2.1: 0 ď f 1 pp 1 q ď λ1 . We set P :" P 1 " pp 0 ´pλ1 q, p 1 `pλ1 q, p 2 ´p0qq. Assume by contradiction that λ 1 :" f 1 pp 1 q ą λ1 " f 1 pp 1 q.

Using Rankine-Hugoniot relation and the facts that f 2 ě 0 and f 2 pp 2 q " 0, we get λ :" f 0 pp 0 q ą λ1 " f 0 pp 0 q.

Using that p1 P ra 1 , b 1 s and that p0 P rb 0 , c 0 s, we deduce that p 1 ą p1 , p 0 ă p0 .

Then we get the table

k " 0 1 2 s k ą 0 ă 0 F k ă 0 ă 0 ď 0 s k F k ă 0 ą 0
with the convention that the boxed inequalities are the known ones, and the unboxed inequalities are the deduced ones.

Hence whatever is the value of s 2 , we deduce from Lemma 2.12 that D ă 0 either from s 0 F 0 ă 0 or from s 1 F 1 ą 0 (depending on the value of s 2 ). Contradiction.

Step 2.2: 0 ď f 2 pp 2 q ď λ2 . Choosing P :" P 2 , we get the result in a symmetric way.

Step 2.3: conclusion From Rankine-Hugoniot relation, we deduce that 0 ď f 0 pp 0 q ď λ0 :" λ1 `λ 2 , which, combining with Steps 2.1 and 2.2, implies the limiters 0 ď f j pp j q ď λj for j " 0, 1, 2.

Step 3: getting key inequalities defining G Λ .

Step 3.1: f 1,`p p 1 q ě λ1 pf 0,`p p 0 qq. Assume by contradiction that f 1,`p p 1 q ă λ1 pf 0,`p p 0 qq.

We choose λ " minp λ0 , f 0,`p p 0 qq and we define P " pp 0 , p1 , p2 q :" pu 0 `pλ q, u 1 `pλ1 q, u 2 `pλ2 qq with λk " λk p λq. This implies in particular that f 0 pp 0 q " λ ě f 0 pp 0 q ": λ.

Hence (recalling that λ1 is nondecreasing) λ 1 :" f 1 pp 1 q ď f 1,`p p 1 q ă λ1 pf 0,`p p 0 qq ď λ1 p λq " λ1 " f 1 pp 1 q " f 1,`p p1 q and then p 1 P ra 1 , b 1 s, p 1 ă p1 .

Then we get the table

k " 0 1 2 s k ą 0 F k ě 0 ą 0 s k F k ą 0
In order to go further, we have to distinguish cases.

Case A: λ ă λ. Then λ " f 0 pp 0 q " minp λ0 , f 0,`p p 0 qq ą f 0 pp 0 q " λ and p0 " u 0 `pλ q ď u 0 `pλ 0 q ă p 0 i.e.

k " 0 1 2

s k ă 0 ą 0 F k ą 0 ą 0 s k F k ă 0 ą 0
Hence whatever is the value of s 2 , we deduce from Lemma 2.12 that D ă 0 either from s 0 F 0 ă 0 or from s 1 F 1 ą 0 (depending on the value of s 2 ). Contradiction. Case B: λ " λ. Then, we have with λ k " f k pp k q and λk " f k pp k q for k " 1, 2 λ 1 ă λ1 and λ 1 `λ2 " λ " λ " λ1 `λ 2 Hence λ 2 ą λ2 i.e. f 2,`p p 2 q ě f 2 pp 2 q ą f 2 pp 2 q " f 2,`p p2 q and then p2 ă p 2 .

We can almost complete the table

k " 0 1 2 s k ą 0 ă 0 F k " 0 ą 0 ă 0 s k F k " 0 ą 0 ą 0
Again we deduce from Lemma 2.12 that D ă 0 using s 2 F 2 ą 0 or s 1 F 1 ą 0 (depending on the sign of s 0 ). Contradiction.

We get a contradiction in all the cases and so f 1,`p p 1 q ě λ1 pf 0,`p p 0 qq.

Step 3.2: f 2,`p p 2 q ě λ2 pf 0,`p p 0 qq. Proceeding symmetrically to Step 3.1, we get the result.

Step 3.3: conclusion Finally, this shows that P P G Λ and completes the proof of the lemma.

Germ property

Lemma 2.14. (Germ property of G Λ ) Under the assumptions of Theorem 2.1, the set G Λ defined by ( 24) is a germ.

Proof. By construction, we have G Λ Ă Q RH , and then we only have to show that3

Dp P , P q ě 0 for all P , P P G Λ .

Assume by contradiction that there exists P , P P G Λ such that Dp P , P q ă 0.

Then from Lemma 2.12, we have two cases. Either s 0 F 0 ă 0 and s 0 " s 1 " s 2 weakly, or (up to exchange the indices 1 and 2), we have s 1 F 1 ą 0 and s 1 " s 0 " s 2 weakly.

Case A: s 0 F 0 ă 0 and s 0 " s 1 " s 2 weakly. Up to exchange P and P , this means that

F 0 ă 0, s 0 " 1, s 1 ď 0, s 2 ď 0,
i.e. p0 ą p 0 , p1 ď p 1 , p2 ď p 2 , f 0 pp 0 q ă f 0 pp 0 q ď λ0 .

Hence p0 ą u 0 ´pλ 0 q.

Recall that f 1,`p p1 q ě λ1 pf 0,`p p0 qq, f 2,`p p2 q ě λ2 pf 0,`p p0 qq and in particular f 0,`p p0 q ě λ0 , f 1,`p p1 q ě λ1 , f 2,`p p2 q ě λ2 where we have used the fact that λk pf 0 max q " λk p λ0 q " λk for k " 1, 2. Therefore, since f k pp k q ď λk , we have

$ & % p1 P u 1 `pλ1 q ( Y ru 1 ´pλ1 q, c 1 s p2 P u 2 `pλ2 q ( Y ru 2 ´pλ2 q, c 2 s. This implies that f 1 pp 1 q ď f 1 pp 1 q, f 2 pp 2 q ď f 2 pp 2 q
and then f 0 pp 0 q " f 1 pp 1 q `f 2 pp 2 q ď f 1 pp 1 q `f 2 pp 2 q " f 0 pp 0 q ă f 0 pp 0 q.

Contradiction.

Case B: s 1 F 1 ą 0 and s 1 " s 0 " s 2 weakly. Up to exchange P and P , this means that

F 1 ą 0, s 1 " 1, s 0 ď 0, s 2 ď 0, i.e. f 1 pp 1 q ą f 1 pp 1 q, p1 ą p 1 , p0 ď p 0 , p2 ď p 2 .
Recall also that $ & % f 1,`p p 1 q ě λ1 pf 0,`p p 0 qq, f 2,`p p 2 q ě pf 0,`p p 0 qq f 1,`p p1 q ě λ1 pf 0,`p p0 qq, f 2,`p p2 q ě λ2 pf 0,`p p0 qq.

Case B.1: p 0 ě u 0 `pλ0 q. Then f 1,`p p 1 q ě λ1 pf 0,`p p 0 qq " λ1 and p 1 ě u 1 `pλ 1 q which implies f 1 pp 1 q ď f 1 pp 1 q.

Contradiction. Case B.2: p 0 ă u 0 `pλ0 q. Hence we have p0 ď p 0 ă u 0 `pλ 0 q and then F 0 ď 0.

Using the fact that F 1 ą 0, we get F 2 ă 0. This implies that

" p1 ą p 1 , p2 ă p 2 f 1 pp 1 q ą f 1 pp 1 q, f 2 pp 2 q ă f 2 pp 2 q.
Hence p 1 ă u 1 `pλ 1 q, p2 ă u 2 `pλ 2 q. Moreover " f 1 pp 1 q " f 1,`p p 1 q ě λ1 pf 0,`p p 0 qq ě λ1 pλq, λ :" f 0 pp 0 q f 2 pp 2 q " f 2,`p p2 q ě λ2 pf 0,`p p0 qq ě λ2 p λq, λ :" f 0 pp 0 q ď λ.

This implies in particular (using λ1 p λq `λ 2 p λq " λ " f 1 pp 1 q `f 2 pp 2 q) that f 1 pp 1 q ď λ1 p λq.

Using the monotonicity of the map λ Þ Ñ λ1 pλq, we get f 1 pp 1 q ď λ1 p λq ď λ1 pλq ď f 1 pp 1 q.

Contradiction with f 1 pp 1 q ą f 1 pp 1 q. This completes the proof of the lemma.

Proof of Theorem 2.1

Proof of Theorem 2.1. The proof of Theorem 2.1 is a straightforward application of Lemma 2.14, which says that G is a germ, and of Lemma 2.13, which proves at the same time its maximality and the fact that it is generated by E Λ " Γ Y tP 1 , P 2 , P 3 u.

Construction of the correctors

In this section, we build a corrector associated to a density at ´8 equal to some p 0 P ra 0 , c 0 s such that

ˆ1 0 Aptqdt ě f 0 pp 0 q. ( 51
)
Let us recall that a corrector is a time-periodic solution to the mesoscopic model ( 9), which is equal to p 0 at ´8. The construction of the corrector relies, on the one hand, on the equivalence between Hamilon-Jacobi equations and conservation laws in one space dimension and, on the other hand, on representation formulas for solutions of Hamilon-Jacobi equations for concave Hamiltonians. We proceed in four steps. We start with a general construction of a periodic in time solution to a Hamilton-Jacobi equation on a half-line p0, `8q, with a periodic Dirichlet condition at x " 0 (Lemma 3.1). We apply this construction to the entry line (road 0) for a junction condition problem (Lemma 3.5). The surprising fact is that this construction can be achieved independently of the outgoing roads 1 and 2. The reason for this is that, in the periodic regime, the flux entering roads 1 and 2 will be at each time the maximal flux coming from road 0: thus no information coming from the outgoing roads is needed to build the solution on the incoming road. Given the flux exiting road 0, one can solve the Hamilton-Jacobi problem on the exit lines 1 and 2 (Lemma 3.10) thanks again to the general construction of Lemma 3.1. In the fourth step we glue the solutions together and show that they form a periodic solution to the conservation law (9) (Proposition 3.12 for the fluid regime and Proposition 3.13 for regimes in which one of the outgoing branches is fully congested).

A periodic solution to a HJ equation on a half-line

In this section, we assume that f : ra, bs Ñ R is a strictly increasing map which is of class C 2 and strongly concave: f 2 ppq ď ´δ ă for any p P ra, bs, for some constant δ ą 0 (52) and ψ : R Ñ R is a Lipschitz continuous map, which is 1´periodic and satisfies ψ 1 ptq P r´f pbq, ´f paqs a.e. t P R.

We consider the Hamilton-Jacobi equation

$ & % piq B
x w P ra, bs a.e. in R ˆp0, `8q, piiq B t w `f pB x wq " 0 for t P R, x ą 0, piiiq wpt, 0q " ψptq for t P R.

(54)

Inspired by the Lax-Oleinik formula and by optimal control on junctions (see for instance [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF]), we can guess a representation of the solution. The following result checks afterwards that the candidate is indeed the unique solution.

where the map ξ : r0, 8q 2 Ñ R is defined by ξps, yq " max pPra,bs ´py `sf ppq @s ě 0, y ě 0.

Proof.

Step 1: Uniqueness of the solution to (54).

We only sketch the proof, arguing as if the two solutions w and w of (54) were smooth: the general case can be treated by standard viscosity techniques. Arguing by contradiction, we assume that sup w´w ą 0.

Then we look at the maximum of wpt, xq ´wpt, xq ´ x 2 (for ą 0 small). At the maximum point pt, xq one gets B t w " B t w and B x w " B x w `2 x with x ą 0 (since w " w at x " 0), so that 0 ě B t w `f pB x wq ´Bt w ´f pB x wq " f pB x w `2 xq ´f pB x wq ą 0, as x ą 0 and f is increasing. This leads to a contradiction.

In order to proceed, we first need to rule out the case in which ψ is constant. In this case the solution to (54) is given by wpt, xq " ψ `p˚x where p ˚P ra, bs is such that f pp ˚q " 0. On the other hand we have by (53) that 0 P rf paq, f pbqs. Using Lemma 3.2 below, one can easily check that the optimal s in the expression of wpt, xq " ψ ´inf sě0 ξps, xq is given by s ˚" x{f 1 pp ˚q and then ξps ˚, xq " ´p˚w which gives the correct expression for w.

From now on we assume that ψ is not constant. We note for later use that this implies that f paq ă 0 and f pbq ą 0 because ´ψ1 P rf paq, f pbqs and ψ is periodic and not constant. We suppose in addition that ψ is of class C 1 and satisfies ψ 1 ptq ă ´f paq for any t P R. This extra condition is removed at the very end of the proof.

Step 2: w is globally Lipschitz continuous on R ˆr0, `8q We first note that the sup in the definition of w is in fact a max, because ψ is bounded and, as sup pPra,bs f ppq is positive,

lim t1Ñ´8 " inf xPr0,Rs ξpt ´t1 , xq * " `8 @R ą 0. ( 56
)
In particular w is uniformly bounded on any strip R ˆr0, Rs. As explained in Lemma 3.2, the map ps, yq Ñ ξps, yq is globally Lipschitz continuous and bounded in C Step 3: w is locally semiconvex in time-space Next we check that w is locally semiconvex in time-space in R ˆp0, `8q: we use this property below to check that w is a solution. This local semiconvexity is not straightforward because wpt, xq is defined as a supremum of an expression on the interval p´8, ts which itself depends on the variable t. In order to overcome this difficulty, we will show that the maximum time t1 t,x in the definition of wpt, xq is indeed strictly less than t (with some bound), which will allow us to replace locally the interval p´8, ts by some smaller interval locally independent on t. For the proof, let us introduce a few notation. Given pt, xq P R ˆp0, 8q, let t1 t,x ď t be a maximum point in the definition of wpt, xq and pt,x P ra, bs be the unique maximum point in the definition of ξpt ´t 1 t,x , xq. We claim that, for any 0 ă ă 1, there exists η ą 0 such that, if x P r , 1{ s, then t1 t,x ď t ´η. Indeed, otherwise, there exists a sequence pt n , x n q such that x n P r , 1{ s, t1 tn,xn ą t n ´1{n. By periodicity we can assume without loss of generality that t n P r0, 1s and converges to some t and that px n q converges to some x P r , 1{ s. Then t1 tn,xn converges to t, which is a maximum point in the definition of wpt, xq, and ptn,xn converges to some p P ra, bs, which is the unique maximum point in the definition of ξp0, xq. As t1 t,x " t is a maximum for wpt, xq, we get by the optimality conditions (using the additional regularity ψ P C 1 ), ψ 1 ptq ě ´Bs ξp0, xq " ´f ppq " ´f paq, because the unique maximizer p of p Ñ ´px on ra, bs is p " a. This contradicts our additional assumption that ψ 1 ă ´f paq and shows that there exists η ą 0 such that, if x P r , 1{ s, then t1 t,x ď t ´η. As a consequence, given pt, xq P R ˆp0, 8q, there exists a neighborhood V of pt, xq and η 1 ą 0 such that, wps, yq " sup

t1ďt´η 1
ψpt 1 q ´ξps ´t1 , yq, y ě x{2 @ps, yq P V.

Note that the upper bound for t 1 in the above problem is now independent of ps, yq. Recalling that ξ is bounded in C 1,1 in r0, 8q ˆrx{2, 8q, this shows the semiconvexity of w in V.

Step 4: w is solution of (54). As f is uniformly concave and w locally semiconvex, w satisfies the equation in (54) in the viscosity sense if and only if it satisfies this equation at any point of differentiability. Let pt, xq P R ˆp0, 8q be a point of differentiability of w. By the envelop theorem (Theorem A.5), for any optimizer t1 t,x ă t for wpt, xq and if pt,x P ra, bs is the unique maximizer for ξpt ´t 1 t,x , xq, we get B x wpt, xq " ´By ξpt ´t 1 t,x , xq " pt,x , B t wpt, xq " ´Bs ξpt ´t 1 t,x , xq " ´f pp t,x q.

Thus B t w `f pB x wq " ´f pp t,x q `f pp t,x q " 0.

This shows that w satisfies the equation in (54) and that B x w P ra, bs a.e.. For the boundary condition, we first note that (choosing t 1 " t as a competitor)

wpt, 0q ě ´ψptq ´ξp0, 0q " ψptq.

Moreover, wpt, 0q " ψp t1 t,0 q ´pt ´t 1 t,0 q max pPra,bs f ppq " ψp t1 t,0 q ´pt ´t 1 t,0 qf pbq.

If, contrary to our claim, we had wpt, 0q ą ψptq, then one would have t1 t,0 ă t and pt ´t 1 t,0 qf pbq ă ψp t1 t,0 q ´ψptq " ´ˆt t1 t,0 ψ 1 psqds ď pt ´t 1 t,0 qf pbq, which is impossible since f pbq ą 0. Hence wpt, 0q " ´ψptq.

Step 5: Conclusion. We finally remove the extra assumption that ψ P C 1 and satisfies ψ 1 ă ´f paq: let pψ n q be a sequence of smooth periodic maps satisfying ´f pbq ď pψ n q 1 ă ´f paq and which converges to ψ (such a sequence exists since ´f pbq ď ψ 1 ď ´f paq a.e.). Let w n be given by (55) for ψ n in place of ψ. Then w n solves the HJ equation for ψ n and, by stability, converges locally uniformly to the unique viscosity solution of the problem with ψ. Note that (54)-(i) holds as well by L 8 ´˚convergence of B x w n to B x w.

It remains to state and check the intermediate lemma.

Lemma 3.2. (Properties of the fundamental solution ξ)

The map ξ defined by ξps, yq " max pPra,bs ´py `sf ppq @s ě 0, y ě 0.

is globally Lipschitz continuous in r0, 8q ˆr0, 8q and bounded in C 1,1 in r0, 8q s ˆr , 8q y for any ą 0. Moreover, ξ is differentiable at any ps, yq with s ą 0 and B y ξps, yq " ´p , B s ξps, yq " f pp s,y q,

where ps,y is the unique point of maximum in the definition of ξps, yq and is given by ps,y "

$ & % pf 1 q ´1py{sq if y{s P pf 1 pbq, f 1 paqq b if y{s ď f 1 pbq a if y{s ě f 1 paq (58)
Proof. As f is increasing and strongly concave, the point of maximum ps,y in the definition of ξps, yq is unique for s ą 0 and y P r0, 8q and given by (58). Thus, by the envelope theorem (Theorem A.5), ξ is differentiable at any ps, yq with s ą 0 and its derivatives are given by (57). As ps,y is bounded, this implies that ξ is globally Lipschitz continuous in r0, 8q ˆr0, 8q.

It remains to show that ps, yq Ñ ps,y is Lipschitz continuous in r0, 8q ˆr , 8q (where ą 0 is fixed). As f is strongly concave, f 1 is decreasing. f is increasing, this implies that f 1 paq ą 0.

Using again that f is strongly concave with f 2 ď ´δ ă 0, we see that ´C0 ď ppf 1 q ´1q 1 ď 0 with C 0 " 1{δ. Let ps, yq, ps 1 , y 1 q P p0, 8q ˆrε, 8q be such that (to fix the ideas) y{s ď y 1 {s 1 , and then ps 1 ,y 1 ď ps,y . The idea consists in using ε in order to control y, y 1 , which will in turn control also s, s 1 in some sense.

Without loss of generality we can also assume that y{s ă f 1 paq since otherwise ps,y " a " ps 1 ,y 1 . We have

|p s 1 ,y 1 ´p s,y | ď C 0 ˇˇˇy 1 s 1 ^f 1 paq ´y s _ f 1 pbq ˇˇˇď C 0 ˆy1 s 1 ^f 1 paq ´y s ˙.
Let us first suppose that y{s, y 1 {s 1 ď f 1 paq. As y, y 1 ě , we get 1{s 1 ď f 1 paq{y 1 ď f 1 paq{ . Hence

|p s 1 ,y 1 ´p s,y | ď C 0 ˆ1 s 1 |y 1 ´y| `y ss 1 |s ´s1 | ˙ď C 0 ˆf 1 paq |y 1 ´y| `pf 1 paqq 2 |s ´s1 | ˙.
Finally, if y 1 {s 1 ě f 1 paq and y{s ă f 1 paq, then

|p s 1 ,y 1 ´p s,y | ď C 0 ˆf 1 paq ´y1 s `y1 s ´y s ˙ď C 0 ˆf 1 paqp1 ´s1 s q `f 1 paq |y 1 ´y| ď C 0 ˆpf 1 paqq 2 |s 1 ´s| `f 1 paq |y 1 ´y| ˙.
This shows that the map ps, yq Ñ ps,y is Lipschitz continuous in p0, 8q ˆr , 8q, and thus on r0, `8q rε, `8q. Therefore ξ is bounded in C 1,1 in this set.

In order to show that the correctors will have the good behavior at infinity, we have to examine carefully the behavior of the solution of the HJ equation at infinity. Lemma 3.3 (behavior of the solution at 8). Assume that conditions (52) and (53) on f and ψ hold and that 0 P ra, bs with f p0q " 0. Then the solution w of (54) is bounded and there exists a constant C ą 0 such that

}B x w} L 8 pRˆpM,8qq ď C M @M ě C. (59)
Remark 3.4. We can actually show that there exists a constant C ą 0 such that

}w ´max ψ} L 8 pRˆpM,8qq ď C M @M ě C.
The bound w ď max ψ follows by comparison, while the other bound is obtained using the uniform concavity of f in the representation formula.

Proof. We can assume without loss of generality that a ă 0 ă b since, if a " 0 or b " 0, then by (53) ψ must be constant and therefore, since f p0q " 0, w " ψ is also constant.

As w `pt, xq " }ψ} 8 and w ´pt, xq " ´}ψ} 8 are respectively time-periodic super-and sub-solution of the equation, we have |w| ď }ψ} 8 by comparison.

We now turn to proof of (59). Given any pt, xq P R ˆp0, `8q a point of differentiability of w, consider some optimizer t1 ď t for wpt, xq and p the optimizer in the definition of ξpt ´t 1 , xq. From the proof of Lemma 3.1, we know that t1 ă t and that B x wpt, xq " p. So, to prove (59), we just need to estimate p.

Recalling Lemma 3.2 again, we have B s ξps, yq " f ppq where p " ´Bx ξps, yq " pf 1 q ´1ppT f 1 paq f 1 pbq py{sqq, T β α pzq " maxpα, minpβ, zqq.

Hence f p0q " 0 implies " B s ξps, yq ă 0 if y{s ą f 1 p0q, B s ξps, yq ą 0 if y{s ă f 1 p0q.
We claim that x{pt ´t 1 `1q ď f 1 p0q. Indeed, otherwise, x{pt ´t 1 q ě x{pt ´t 1 `1q ą f 1 p0q and thus ξp¨, xq is decreasing on rt ´t 1 , t ´t 1 `1s. This implies, as ψ is 1´periodic, that ψpt ´t 1 `1q ´ξpt ´t 1 `1, xq ą ψpt ´t 1 q ´ξpt ´t 1 , xq " wpt, xq, a contradiction because t 1 " t1 ´1 is a competitor in the definition of wpt, xq. Thus x{pt ´t 1 `1q ď f 1 p0q.

In the same way one can check that, if t1 `1 ă t, then x{pt ´t 1 ´1q ě f 1 p0q, using t 1 " t1 `1 as a competitor in the definition of wpt, xq. Let us check that indeed t1 `1 ă t if x is large enough: otherwise, |t ´t 1 | ď 1 and therefore wpt, xq " ψpt ´t 1 q `min pPra,bs tpx ´pt ´t 1 qf ppqu ď }ψ} 8 `}f } 8 `min pPra,bs px " }ψ} 8 `}f } 8 `ax, which yields to a contradiction if x is large enough, because a ă 0 and w is bounded.

The two estimates on x{pt ´t 1 `1q and x{pt ´t 1 ´1q imply that, for x large enough,

|x ´f 1 p0qpt ´t 1 q| ď f 1 p0q,
where f 1 p0q ą 0. Thus, for x large enough, x{pt ´t 1 q is close to f 1 p0q P pf 1 pbq, f 1 paqq and therefore for x large enough

|p| " ˇˇˇp f 1 q ´1 ˆT f 1 paq f 1 pbq ˆx t ´t 1 ˙˙ˇˇˇˇ" ˇˇˇp f 1 q ´1 ˆx t ´t 1 ˙´pf 1 q ´1pf 1 p0qq ˇˇˇď C ˇˇˇx t ´t 1 ´f 1 p0q ˇˇˇď C t ´t 1 ď C x .

Periodic solutions to a HJ equation on the entry line

We build in this part an antiderivative of the corrector on the incoming road R 0 . We suppose here that f 0 satisfies (1) and that the flux limiter A satisfies ( 5) and [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF]. For p 0 P ra 0 , c 0 s such that (51) holds, let f 0 p 0 ppq " f 0 pp `p0 q ´f 0 pp 0 q for p P ra 0 ´p0 , c 0 ´p0 s,

so that f 0 p 0 p0q " 0 and 0 P ra 0 ´p0 , c 0 ´p0 s. We consider the periodic in time viscosity solution w 0 p 0 to the HJ equation $ & % piq B x w 0 P ra 0 ´p0 , c 0 ´p0 s a.e. in R ˆp´8, 0q, piiq B t w 0 `f 0 pB x w 0 q " 0 for t P R, x ă 0 piiiq B t w 0 `mintAptq ´f 0 pp 0 q, f 0,`p B x w 0 qu " 0 for t P R, x " 0 (61) By a solution, we mean that w 0 p 0 is continuous on r0, `8q ˆp´8, 0s and is a viscosity solution in the sense of [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF] to (61)-(i)-(ii)-(iii) on each open interval on which A is constant. It is easy to check that the whole theory developed in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF] generalizes to this simple time-dependent setting. Notice that, if w 0 is a solution of (61), then w 0 `c is also solution for any constant c P R. Still we have the following existence result. Lemma 3.5 (Explicit time-periodic solution in the incoming road). Assume that f 0 satisfies (1) and that (5) and (6) hold. Let p 0 be such that p 0 P ra 0 , b 0 s and (51) holds, or p 0 " pf 0,´q´1 ´´1 0 Apsqds ¯.

Then there exists a bounded, Lipschitz continuous and time-periodic solution w 0 p 0 to (61), with period 1, which is given by the representation formula

w 0 p 0 pt, xq " $ & % max ! 0, max t2ďt tψ p 0 pt 2 q ´ξ0 p 0 pt ´t2 , xqu ) , if p 0 P ra 0 , b 0 s, max t2ďt ! ψ p 0 pt 2 q ´ξ0 p 0 pt ´t2 , xqu ) , if p 0 " pf 0,´q´1 ´´1 0 Apsqds ¯, (62)
where ξ 0 p 0 ps, yq " max pPrb 0 ´p0 ,c 0 ´p0 s ´py `s f 0 p 0 ppq @s ě 0, y ď 0 and

ψ p 0 ptq " $ ' ' & ' ' % max t1ďt "ˆt t1 pf 0 pp 0 q ´Apsqqds * if p 0 P ra 0 , b 0 s, ˆt 0 pf 0 pp 0 q ´Apsqqds if p 0 " pf 0,´q´1 ´´1 0 Apsqds ¯. (63)
In addition, there exists a constant C ą 0 (depending on p 0 ), such that w 0 p 0 pt, xq " 0 for x ď ´C, t P R if p 0 P ra 0 , b 0 s, and }B x w 0 p 0 } L 8 pRˆp´8,M qq ď C M for M ě C if p 0 " pf 0,´q´1 ´´1 0 Apsqds ¯with p 0 P pb 0 , c 0 s. (64) Finally, w 0 p 0 pt, 0q " ψ p 0 ptq @t P R, and, if p 0 P ra 0 , b 0 s, the map xÞ Ñw 0 p 0 pt, xq is nondecreasing on p´8, 0s for any t P R.

Recall that the map ψ p 0 (for p 0 P ra 0 , b 0 s) was introduced in Lemma 2.5 when building the homogenized germ GΛ. Remark 3.6. Notice that in case p 0 P pb 0 , c 0 s, it is possible to construct explicit examples of solutions where B x w 0 p0 pt, xq has no compact support in the space variable x, but tends to zero as x Ñ ´8.

Proof. Note first that, if p 0 " b 0 satisfies (51), then w 0 p 0 " 0 is the solution to (61) because in this (very particular) case, assumption [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] implies that Aptq " f 0 max . From now on we assume that

p 0 ‰ b 0 .
As p 0 is fixed, we remove the subscript p 0 throughout the proof for simplicity of notation. Note that, if p 0 P ra 0 , b 0 q, 0 ă b 0 ´p0 ă c 0 ´p0 and thus the map yÞ Ñξ 0 ps, yq is decreasing on p´8, 0s for any s ě 0. Hence the map xÞ Ñw 0 p 0 pt, xq is nondecreasing on p´8, 0s for any t P R.

Step 1: w 0 is a viscosity solution to the HJ equation (61)-(ii). If p 0 P ra 0 , b 0 q is such that (51) holds, Lemma 2.5 states that the map ψ p 0 is Lipschitz continuous and 1´periodic and we can rewrite w 0 " w 0 p 0 in the form

w 0 pt, xq " max ! 0, w0 pt, xq
) with w0 pt, xq " max t2ďt tψ p0 pt 2 q ´ξ0 pt ´t2 , xqu.

In the case p 0 " pf 0,´q´1 p ´1 0 Apsqdsq, the map ψ p 0 is also Lipschitz continuous and 1´periodic and we set w0 :" w 0 . Our aim is to use Lemma 3.1 to check that w0 is a viscosity solution to the HJ equation ( 61)-(ii). For this we change variable and set ŵ0 pt, xq " w0 pt, ´xq " max t2ďt tψ p 0 pt 2 q ´ξ 0 p pt ´t2 , xqu, t ě 0, x ě 0, where ξ0 ps, yq " max pPr´c 0 `p0 ,´b 0 `p0 s ´py `s f 0 p´pq s ě 0, y ě 0.

Note that the map pÞ Ñ f 0 p´pq is uniformly concave and strictly increasing on r´c 0 `p0 , ´b0 `p0 s. In addition, the maps ψ p 0 defined in (63) is Lipschitz continuous, 1´periodic and satisfies, by Lemma 2.5, ψ 1 p 0 ptq P t0, f 0 pp 0 q ´Aptqu Ă r´f 0 p´p´b 0 `p0 qq, ´f 0 p´p´c 0 `p0 qqs a.e. t P R,

where, for the proof of the inclusion, we used ( 6) and the equality r´f 0 p´p´b 0 `p0 qq, ´f 0 p´p´c 0 `p0 qqs " r´f 0 max `f 0 pp 0 q, f 0 pp 0 qs.

Therefore we can apply Lemma 3.1 which states that ŵ0 is globally Lipschitz continuous, 1´periodic in time, and satisfies the HJ equation (54) in R ˆp0, `8q for f ppq " f 0 p´pq and the boundary condition ŵ0 p¨, 0q " ψ p 0 . This implies that w0 pt, xq " ŵ0 pt, ´xq is a Lipschitz continuous viscosity solution of (61)-(i) and ( 61)-(ii) in R ˆp´8, 0q, with w0 p¨, 0q " ψ p 0 . Assume now that p 0 P ra 0 , b 0 q. As f 0 is concave and the constant map pt, xqÞ Ñ0 is also a solution of (61)-(ii) in R ˆp´8, 0q, w 0 is also a viscosity solution of ( 61)-(ii) in R ˆp´8, 0q. In addition, (61)-(i) holds since 0 P ra 0 ´p0 , c 0 ´p0 s and B x w0 P ra 0 ´p0 , c 0 ´p0 s. Note finally that w 0 p¨, 0q " ψ p 0 p¨q as ψ p 0 ě 0.

Step 2: w 0 bounded and satisfies (64). As f p0q " 0, Lemma 3.3 states that ŵ0 and thus w 0 are bounded. Let us first assume that p 0 P ra 0 , b 0 q. Fix x ă 0 and t 2 ď t. Then min pPrb 0 ´p0 ,c 0 ´p0 s px ´pt ´t2 q f 0 ppq ď pb 0 ´p0 qx `min pPrb 0 ´p0 ,c 0 ´p0 s t´pt ´t2 q f 0 ppqu " pb 0 ´p0 qx ´pt ´t2 qpf 0 max ´f 0 pp 0 qq. pApsq ´f 0 pp 0 qqds ´pt ´t2 qpf 0 max ´f 0 pp 0 qqu is a continuous, periodic function. Hence it is bounded. As p 0 ă b 0 , this shows the existence of C ą 0 such that, for any x ď ´C and t P R, w0 pt, xq ď 0. Therefore (64) holds in this case. In the case p 0 " b 0 , it is easy to see that A " f 0 max and then ψ p 0 " 0, which implies that w 0 p 0 " 0 is solution. Hence (64) holds in this case.

Thus

Finally, we consider the case p 0 " pf 0,´q´1 ´´1 0 Apsqds ¯ą b 0 . Then (64) follows from Lemma 3.3 and a change of variables.

Step 3: w 0 satisfies the boundary condition (61)-(iii). For proving the supersolution property, we just need to check that w 0 p¨, 0q is Lipschitz continuous and satisfies B t w 0 pt, 0q `Aptq ´f 0 pp 0 q ě 0 a.e. (cf. [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF]Theorem 2.11]). Recalling that w 0 p¨, 0q " ψ p 0 p¨q, this inequality is obvious if p 0 " pf 0,´q´1 p ´1 0 Apsqdsq. If p 0 P ra 0 , b 0 q, it holds thanks to Lemma 2.5.

Next we turn to the subsolution property. Assume that ϕpt, xq :" αptq `q0 x is a C 1 test function touching w 0 from above at pt 0 , 0q, where t 0 is a point of continuity of A and with (condition (2.12) in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF]Theorem 2.7])

Apt 0 q ´f 0 pp 0 q " f 0 pq 0 q " f 0,´p q 0 q.

(65)

We have to that α 1 pt 0 q `Apt 0 q ´f 0 pp 0 q ď 0. Without loss of generality, we can assume that αpt 0 q " w 0 pt 0 , 0q. Assume first that p 0 " pf 0,´q´1 p ´1 0 Apsqdsq. Then, for h P R, αpt 0 `hq " ϕpt 0 `h, 0q ě w 0 pt 0 `h, 0q " w 0 pt 0 , 0q`ˆt 0 `h t0 pf 0 pp 0 q´Apsqqds " αpt 0 q`ˆt 0`h t0 pf 0 pp 0 q´Apsqqds, which proves that α 1 pt 0 q " f 0 pp 0 q ´Apt 0 q. We now assume that p 0 P ra 0 , b 0 q. Let t1 ď t 0 be optimal in the definition of ψ p 0 pt 0 q in (63). We claim that t1 ă t 0 . Indeed, otherwise, t1 " t 0 and thus w 0 pt 0 , 0q " 0 " αpt 0 q. So, for any x ă 0, ϕpt 0 , xq " q 0 x ě w 0 pt 0 , xq ě 0, which implies that q 0 ď 0. But (65) says that q 0 P rb 0 ´p0 , c 0 ´p0 s, where b 0 ´p0 ą 0, a contradiction.

As t1 ă t 0 , for any h P R with |h| small, αpt 0 `hq " ϕpt 0 `h, 0q ě w 0 pt 0 `h, 0q " ψ p 0 pt 0 `hq ě ˆt0`h t1 pf 0 pp 0 q ´Apsqqds " w 0 pt 0 , 0q `ˆt0`h t0 pf 0 pp 0 q ´Apsqqds " αpt 0 q `ˆt0`h t0 pf 0 pp 0 q ´Apsqqds.

Hence α 1 pt 0 q `Apt 0 q ´f 0 pp 0 q " 0.

The next step is the computation of the trace f 0,`p B x w 0 p 0 pt, 0 ´q `p0 q, where w 0 p 0 is the solution of (61). The computation of this trace will be useful for gluing the correctors on each branch. Let us note that, as w 0 p 0 is a Lipschitz continuous viscosity solution to (61), Lemma A.3 states that B x w 0 p 0 is a Krushkov entropy solution to the scalar conservation law " piq ρ P ra 0 ´p0 , c 0 ´p0 s a.e. in R ˆp´8, 0q, piiq B t ρ `Bx p f 0 pρqq " 0 for t P R, x ă 0.

Thus B x w 0 p 0 possesses a trace, denoted as B x w 0 p 0 p¨, 0 ´q, at x " 0 (Theorem A.1), in the sense that there exists a set N of measure zero in p´8, 0q such that, for any t 1 ă t 2 , lim εÑ0 sup xPp´ε,0qzN }B x w 0 p 0 p¨, xq ´Bx w 0 p 0 p¨, 0 ´q} L 1 pt1,t2q " 0.

By continuity of f 0,`, we infer the existence of the trace f 0,`p B x w 0 p 0 pt, 0 ´q `p0 q. Lemma 3.7 (Computation of the trace f 0,`p B x w 0 p 0 pt, 0 ´q `p0 q). Under the assumption of Lemma 3.5, let w 0 p 0 be the solution of (61) given in Lemma 3.5. Then f 0,`p B x w 0 p 0 pt, 0 ´q`p 0 q " # f 0 pp 0 q if w 0 p 0 pt, 0q " 0 and p 0 P ra 0 , b 0 q f 0 max if w 0 p 0 pt, 0q ą 0 or p 0 " pf 0,´q´1 ´´1 0 Apsqds ¯or p 0 " b 0 a.e. t P R (67) and B t w 0 p 0 pt, 0q " ´min ! Aptq ´f 0 pp 0 q , f 0,`p B x w 0 p 0 pt, 0 ´qq ) a.e. in R, " f 0 pp 0 q ´Fp 0 ptq a.e. in R,

where the flux F p 0 is defined in (33) for p 0 P ra 0 , b 0 q and by

F p 0 p¨q " Ap¨q if p 0 " pf 0,´q´1 ˆˆ1 0 Apsqds ˙or p 0 " b 0 . ( 69
)
Proof of Lemma 3.7. In the case where p 0 " pf 0,´q´1 ´´1 0 Apsqds ¯or p 0 " b 0 the proof is quite simple. Indeed, in those two cases, we have saturation, i.e. B x w 0 p 0 `p0 P rb 0 , c 0 s a.e., and then f 0,`p B x w 0 p 0 `p0 q " f 0 max , which shows (67). Moreover, we have B t w p 0 pt, 0 ´q " ´ψ1 p 0 ptq " # f 0 pp 0 q ´Fp 0 ptq " 0 " f 0 pp 0 q ´Aptq by Lemma 2.5 and (6) if p 0 " b 0 f 0 pp 0 q ´Aptq by (63) if p 0 " pf 0,´q´1 ´´1 0 Apsqds which shows (68). We now prove the results in the case p 0 P ra 0 , b 0 q.

Step 1: Proof of (67). We first claim that

f 0,`p B x w 0 p 0 pt, xq `p0 q " " f 0 pp 0 q if w 0 p 0 pt, xq " 0 f 0 max if w 0 p 0 pt, xq ą 0 a.e. pt, xq P R ˆp´8, 0q. (70)
To prove (70), let pt, xq P R ˆp´8, 0q be a point of differentiability of the Lipschitz map w 0 p 0 . Then, if w 0 p 0 pt, xq " 0, we get B x w 0 p 0 pt, xq " 0 since w 0 p 0 ě 0, and thus (70) holds in this case. Let us now assume that w 0 p 0 pt, xq ą 0. Let t2 ď t be optimal in the definition of w 0 p 0 in (62). We have already proved (see Step 3 in the proof of Lemma 3.1) that t2 ă t. Then ξ p 0 is differentiable at pt ´t 2 , xq with, by the envelope theorem A.5 used twice, B x w 0 p 0 pt, xq " ´Bx ξ 0 p 0 pt ´t 2 , xq " p P rb 0 ´p0 , c 0 ´p0 s, where p is optimal for ξ 0 p 0 pt ´t 2 , xq. As f 0,`p rb 0 , c 0 sq " tf 0 max u, this shows (70).

Fix t P R. Recalling that xÞ Ñw 0 p 0 pt, xq is nondecreasing and nonnegative, equality w 0 p 0 pt, 0q " 0 implies that w 0 p 0 pt, xq " 0 for any x ď 0. Thus lim xÑ0 ´1tw 0 p 0 pt,xqą0u " 1 tw 0 p 0 pt,0qą0u .

Combining the remark above with (66) and (70) gives (67).

Step 2: proof of (68). We recall that w 0 p 0 p¨, 0q " ψ p 0 p¨q. Thus, by Lemma 2.5, B t w 0 p 0 pt, 0q " ´pAptq ´f 0 pp 0 qq for a.e. t P R with w 0 p 0 pt, 0q ą 0.

On the other hand, if w 0 p 0 pt, 0q ą 0, then by (67) f 0,`p B x w 0 p 0 pt, 0 ´qq " f 0 max ´f 0 pp 0 q ě Aptq ´f 0 pp 0 q, thanks to [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF]. This proves that (68) holds a.e. in tw 0 p 0 p¨, 0q ą 0u. Fix now t P R a point of continuity of A, of derivability of w 0 p 0 p¨, 0q and such that w 0 p 0 pt, 0q " 0 and (67) holds. Then B t w 0 p 0 p¨, 0q " 0 since w 0 p 0 ě 0. As t1 " t is optimal in (63) and A is continuous at t, one necessarily has Aptq ´f 0 pp 0 q ě 0 by optimality, so that by (67) min

!

Aptq ´f 0 pp 0 q , f 0,`p B x w 0 p 0 pt, 0 ´qq

) " min ! Aptq ´f 0 pp 0 q , 0 ) " 0.
This proves the first equality in (68) in tw 0 p 0 p¨, 0q " 0u. The second one is just the last statement of Lemma 2.5 since w 0 p 0 pt, 0q " ψ p 0 ptq.

Periodic solutions to a HJ equation on the exit lines

We proceed with our construction of correctors, now building the correctors on the exit lines. Again we use a representation formula of the solution in terms of a Hamilton-Jacobi equation. Let p 0 P ra 0 , b 0 s satisfying (51) or p 0 " pf 0,´q´1 ´´1 0 Apsqds ¯. Let w 0 p 0 be defined in Lemma We fix j " 1, 2 and assume that f j satisfies [START_REF] Achdou | Hamilton-Jacobi equations constrained on networks[END_REF]. Recalling the definition of the flux F p 0 in ( 33) and (69), we introduce the flux entering the exit-line j (where j " 1, 2) as F j p 0 ptq " F p 0 ptq1 I j ptq " " mintAptq, f 0,`p B x w 0 p 0 pt, 0 ´q `p0 qu if t P I j , 0 otherwise, where the second equality comes from (68) in Lemma 3.7. Let us also recall the definition of pj p 0 introduced in (35) in the case p 0 P ra 0 , b 0 s. Definition 3.8 (The notation pj p 0 ). Given p 0 P ra 0 , b 0 s satisfying (51) or p 0 " pf 0,´q´1 ´´1 0 Apsqds ¯, let pj p 0 P ra j , b j s (for j " 1, 2) be the unique solution to f j pp j p 0 q " f j,`p pj p 0 q " ˆ1 0 F j p 0 ptqdt. Remark 3.9. Note that pj p 0 indeed exists and is unique since, by [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] and the definition of F j p 0 , 0 ď ´1 0 F j p 0 ptqdt ď f j max and f j is one-to-one from ra j , b j s to r0, f j max s.

Let us now set f j p 0 ppq " f j pp `p j p 0 q ´f j pp j p 0 q for p P ra j ´p j p 0 , b j ´p j p 0 s, and ψj p 0 ptq " ´ˆt 0 pF j p 0 psq ´f j pp j p 0 qqds.

Note that ψj p 0 is a 1´periodic, Lipschitz continuous map, satisfying p ψj p 0 q 1 P ´r´f j pp j p 0 q, max f j p 0 s " ´r f j p 0 pa j ´p j p 0 q, f j p 0 pb j ´p j p 0 qs a.e..

Let us consider the time-periodic viscosity solution w j p 0 to the Hamilton-Jacobi equation

$ ' &

' % piq B x w j P ra j ´p j p 0 , b j ´p j p 0 s a.e. in R ˆp0, `8q, piiq B t w j `f j pB x w j q " 0 for t P R, x ą 0, piiiq w j pt, 0q " ψj p 0 ptq for t P R.

(73) Lemma 3.10 (Explicit time-periodic solution in the outgoing roads). Fix j " 1, 2. Assume that f j satisfies (1) and that (5) and (6) hold. Let p 0 P ra 0 , b 0 s satisfying (51) or p 0 " pf 0,´q´1 ´´1 0 Apsqds ānd let ψj p 0 be defined in (71). Then, there exists a unique time-periodic Lipschitz continuous viscosity solution w j p 0 to (73), of time period equal to 1. It is given by

w j p 0 pt, xq " sup t1ďt t ψj p 0 pt 1 q ´ξj p 0 pt ´t1 , xqu, (74)
where the map ξ j p 0 : r0, 8q 2 Ñ R is defined by ξ j p 0 ps, yq " max pPra j ´p j ,b j ´p j s ´py `s f j p 0 ppq @s ě 0, y ě 0.

Finally, there exists a constant C ą 0 such that

}B x w j p 0 } L 8 pRˆpM,8qq ď C M @M ě C. (75)
We have to prove that α 1 pt 0 q `Aptq ď 0. As the map pt, xqÞ Ñαptq `q0 x1 xă0 touches locally W from above on pτ 1 , τ 2 q ˆp´8, 0s at pt 0 , 0q, the map pt, xqÞ Ñαptq `pq 0 ´p0 qx1 xă0 `f 0 pp 0 qpt ´τ1 q touches locally w 0 p 0 from above on R ˆp´8, 0s at pt 0 , 0q. By the equation satisfied w 0 p 0 , this implies that α 1 pt 0 q `f 0 pp 0 q `mintApt 0 q ´f 0 pp 0 q, f 0,`p q 0 ´p0 qu ď 0.

Recalling the definition of f 0 in (60), the inequality above yields α 1 pt 0 q `min ! Apt 0 q, f 0,`p q 0 q ) ď 0, where, because of ( 80) and ( 6), f 0,`p q 0 q " f 0 max ě Apt 0 q. Hence α 1 pt 0 q `Apt 0 q ď 0. This proves that W is a viscosity solution to (79).

We now rely on Lemma A.4, which implies that the trace B x W p¨, tq satisfies

B x W pt, 0q P G a.e. in pτ 1 , τ 2 q,
where G " tpu 0 , u j q P ra 0 , c 0 s ˆra j , c j s, min Aptq, f 0,`p u 0 q, f j,´p u j q ( " f 0 pu 0 q " f j pu j qu.

This is (76).

Step 2: proof of (77). Fix j P t1, 2u and let pτ 1 , τ 2 q Ă RzI j on which A is constant and let W " pW 0 , W j q : R Ñ R be given by # W 0 pt, xq " w j p 0 pt, 0q `aj x ´f j pp j p 0 qpt ´τ1 q in pτ 1 , τ 2 q ˆp´8, 0q, W j pt, xq " w j p 0 pt, xq `p j p 0 x ´f j pp j p 0 qpt ´τ1 q in pτ 1 , τ 2 q ˆp0, `8q,

We claim that W is a viscosity solution of the HJ equation on the 1:1 junction $ ' ' & ' ' % piq B t W 0 `f j pB x W 0 q " 0 for t P pτ 1 , τ 2 q, x ă 0, piiq B t W j `f j pB x W j q " 0 for t P pτ 1 , τ 2 q, x ą 0, piiiq W pt, 0q :" W 0 pt, 0 ´q " W j pt, 0 `q for t P pτ 1 , τ 2 q, pivq B t W pt, 0q `mint0, f j,`p B x W 0 pt, 0 ´qq, f j,´p B x W j pt, 0 `qqu " 0 for t P pτ 1 , τ 2 q.

(82) Indeed, by construction, W is continuous and conditions (ii) and (iii) hold. On pτ 1 , τ 2 q ˆp´8, 0q, we have (in the a.e. sense and thus, by the smoothness of W 0 which is affine, in the viscosity sense) B t W 0 pt, xq `f j pB x W 0 pt, xqq " B t w j p 0 pt, 0q ´f j pp j p 0 q `f j pa j q " 0 since B t w j p 0 pt, 0q " p ψj p 0 q 1 ptq " f j pp j p 0 q as F j p 0 " 0 on pτ 1 , τ 2 q Ă RzI j . Thus piq holds. The same proof shows that B t W pt, 0q " 0, which implies condition pivq. As W is a viscosity solution of (82) we infer from Lemma A.4 that the trace at x " 0 of B x W satisfies B x W pt, 0q P G, where G " tpu 0 , u j q P ra j , c j s 2 , min 0, f j,`p u 0 q, f j,´p u 1 q ( " f j pu 0 q " f j pu j qu " tpu 0 , u j q P ra j , c j s 2 , 0 " f j pu 0 q " f j pu j qu.

This implies (77).

Construction of the correctors

We are now ready to build the correctors, i.e., the time-periodic solutions to (9) with a specific behavior at infinity. Throughout this part, assumptions (1), ( 5) and ( 6) are in force.

with initial condition ρj , where the germ G H,2 is given by G H,2 " tpp H , p 2 q P ra H , c H s ˆra 2 , c 2 s, min f H,`p p H q, f 2,´p p 2 q ( " f H pp H q " f 2 pp 2 qu.

This shows that ρ 2 is an entropy solution of ρ 2 P 2 , c 2 s a.e. on p0, τ 1 q ˆR2 , B t ρ 2 `Bx pf 2 pρ 2 qq " 0 on p0, τ 1 q ˆR2 , ρ 2 pt, 0q P G 2 on p0, τ 1 q ˆtx " 0u , with initial condition ρ2 and with G 2 :" p 2 P R such that p0, p 2 q P G H,2 ( " p 2 P ra 2 , c 2 s, f 2 pp 2 q " 0 ( " a 2 , c 2 ( Therefore ρ " pρ 0 , ρ 1 , ρ 2 q solves (9) on p0, τ 1 q with initial condition ρ.

Step 2: existence on r0, τ k q given the solution on r0, τ k´1 q, k ě 2 Assume that we have built ρ on r0, τ k´1 q. Recall that ρ has a continuous in time representative with values in L 1 loc (see [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF]Theorem 6.2.2]). Let us set ρ :" ρpτ k´1 , ¨q. We can then build as in the previous step a solution ρ " pρ 0 , ρ1 , ρ2 q of (9) on pτ k´1 , τ k q with initial condition ρ at time τ k´1 . It remains to check that the concatenation ρpt, ¨q " " ρpt, ¨q in r0, τ 1 q ρpt, ¨q in rτ 1 , τ 2 q is an entropy solution to (9) on r0, τ 2 q with initial condition ρ. Note that the junction condition ( 9)-(iii) at x " 0 is satisfied because this is the case for ρ on p0, τ k´1 q and for ρ on pτ k´1 , τ k q. It remains to check that ρj is an entropy solution on r0, τ k q ˆRj for any j " 0, 1, 2. The argument is standard and we only sketch it. To fix the ideas, we do the proof for j " 1, the argument for j " 0 and j " 2 being symmetric. Fix a C 1 c pr0, τ k q ˆp0, `8qq function ϕ ě 0. Let θ n : r0, τ k´1 q Ñ r0, 1s be smooth, nonincreasing map, with a compact support and such that θ n Ñ 1 and θ 1 n Ñ 0 uniformly in r0, τ k´1 ´δs for any δ ą 0. As ρ 1 is an entropy solution on r0, τ k´1 q ˆR1 , we have, for all c P R, ˆp0,τ k´1 q ˆp0,`8q |ρ 1 ´c|pϕ t θ n `ϕθ 1 n q` signpρ 1 ´cq ( ¨pf pρ 1 q´f pcqqϕ x θ n `ˆt0uˆp0,`8q |ρ 1 ´c|ϕθ n p0q ě 0.

By the continuity of tÞ Ñρ 1 pt, ¨q in L 1 loc pp0, 8qq, we find, when letting n Ñ 8, ´ˆtτ k´1 uˆp0,`8q |ρ 1 ´c|ϕ`ˆp 0,τ k´1 q ˆp0,`8q |ρ 1 ´c|ϕ t ` signpρ 1 ´cq ( ¨pf pρ 1 q´f pcqqϕ x `ˆt0uˆp0,`8q |ρ 1 ´c|ϕ ě 0.

As ρ1 is an entropy solution on rτ k´1 , τ k q ˆR1 with initial condition ρ 1 , we also have ˆpτ k´1 ,τ k q ˆp0,`8q |ρ 1 ´c|ϕ t ` signpρ 1 ´cq ( ¨pf pρ 1 q ´f pcqqϕ x `ˆtτ k´1 uˆp0,8q |ρ 1 ´c|ϕ ě 0.

Putting together the two previous inequalities proves that ρ 1 is an entropy solution on r0, τ k q ˆR1 with initial condition ρ1 .

Step 4: existence on r0, `8q By induction this proves the existence of a solution of the whole time interval r0, 8q.

Step 5: Kato's inequality [START_REF] Cardaliaguet | Forcadel Microscopic derivation of a traffic flow model with a bifurcation[END_REF] and uniqueness We claim that pρ 0 , ρ 1 , ρ 2 q satisfies Kato's inequality [START_REF] Cardaliaguet | Forcadel Microscopic derivation of a traffic flow model with a bifurcation[END_REF]. Indeed, as the sets G Λ 1 and G Λ 2 introduced in (11) and ( 12) are maximal germs (see Lemma 2.2), we just need to apply Kato's inequality given in [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] on each time interval pτ k , τ k`1 q for k P N and then proceed as above to glue the solution together. The uniqueness of the solution ρ is then an obvious consequence of Kato's inequality.

Proof of Theorem 1.6. Let p P EΛ. The existence of a corrector when p satisfies (i) in the definition [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF] of EΛ is given by Proposition 3.12. The case piiq is the aim of Proposition 3.13. The cases piiiq is symmetric to the case (ii), exchanging the indices 1 and 2. The case pivq is obvious because then one can choose u j p " c j for j " 0, 1, 2.

Proof of Theorem 1.4. Recall that the construction of GΛ and the proof that it is a maximal germ are given in Subsection 2.1.3. We now prove the homogenization. It is known that the sequence pρ q is relatively compact in L 1 loc pp0, `8q ˆRq (Proposition A.2 in the Appendix). Let ρ " pρ i q i"0,1,2 be a limit (in L 1 loc pp0, `8q ˆRq and up to a subsequence) of pρ q. We to check that ρ is the unique solution to [START_REF] Colombo | On the microscopic modeling of vehicular traffic on general networks[END_REF]. By stability, ρ i is an entropy solution on r0, `8q ˆRi and satisfies ρ i P ra i , c i s a.e. on p0, `8q ˆRi for i " 0, 1, 2.

Let p " pp i q P EΛ. By Theorem 1.6 there exists a time-periodic solution u p of ( 9) and C ą 0 such that for M ě C, we have }u 0 p ´p0 } L 8 pRˆp´8,´M qq `}u i p ´pi } L 8 pRˆpM,8qq ď CM ´1, i " 1, 2. (

We set u p pt, xq " u p pt{ , x{ q. Note that the scaled function u p " pu ,k p q is a solution to (15) (without the initial condition). Thus, by Kato's inequality (13), we have 2 ÿ i"0 " ˆ8 0 ˆRi |ρ ,i ´u ,i p |φ i t ` signpρ ,i ´u ,i p q ( ¨pf i pρ ,i q ´f i pu ,i p qqB x φ i `ˆR i |ρ i 0 pxq ´u ,i p p0, xq|φ i p0, xq * ě 0 for any continuous nonnegative test function φ : r0, 8q ˆR Ñ r0, 8q with a compact support and such that φ j :" φ |r0,`8qˆpR j Yt0uq is C 1 for any j " 0, 1, 2. Letting Ñ 0 and recalling (83), which implies that u p converges in L 1 loc to p as Ñ 0, this gives for any test function φ as above:

2 ÿ i"0 "ˆ8 0 ˆRi
|ρ i ´pi |φ i t ` signpρ i ´pi q ( ¨pf i pρ i q ´f i pp i qqB x φ i `ˆR i |ρ i 0 p0, xq ´pi |φ i pxq * ě 0.

Following the argument in [37, Proposition 2.12], this implies that, for a.e. t ě 0, q 0 pp 0 , ρ 0 pt, 0 ´qq ě q 1 pp 1 , ρ 1 pt, 0 `qq `q2 pp 2 , ρ 2 pt, 0 `qq.

This inequality holds for any p P EΛ and for a.e. t ě 0, and we have pρ 0 pt, 0 ´q, ρ 1 pt, 0 `q, ρ 2 pt, 0 `qq P Q for a.e. t ě 0. Therefore Lemma 1.5 implies that ρpt, 0q " pρ i pt, 0qq P GΛ. It follows that ρ solves [START_REF] Colombo | On the microscopic modeling of vehicular traffic on general networks[END_REF], which has a unique solution ρ. Therefore the whole sequence pρ ε q converges to ρ. Moreover the L 8 bound on ρ implies its convergence in L 1 loc pr0, `8q ˆRq.

Proof for a 2:1 junction

The main idea of the proof is to derive Theorem 1.7 from Theorem 1.4 by a simple change of variables, transforming 2:1 junctions into 1:2 junctions.

A general framework for junctions with three roads

We first introduce a general class of germs, defined for fluxes f j for j " 0, 1, 2 satisfying (1). The entropy flux associated to f j is defined for c, c P ra j , c j s as q f j pc, cq :" pf j pcq ´f j pcqqsignpc ´cq and let signpR j q " " `if R j " p0, `8q ´if R j " p´8, 0q with a general set of three roads R " pR 0 , R 1 , R 2 q.

Given f " pf 0 , f 1 , f 2 q, the dissipation for P " pp 0 , p1 , p2 q and P " pp 0 , p 1 , p 2 q is defined by D f,R p P , P q " ´ÿ j"0,1,2 signpR j q ¨qf j pp j , p j q.

Let us now explain how to build germs for the fluxes f j on the junction Ř.

Lemma 4.2. (Germ for a convergent 2:1 junction) Let Ř be defined in (88), and fluxes f j for j " 0, 1, 2 satisfying (1), ( 2), ( 3) and (4). Under assumption [START_REF] Forcadel | Homogenization of a discrete model for a bifurcation and application to traffic flow[END_REF] on Λ, let us consider the set G f ,Λ defined in (87). Then this set G f ,Λ Ă Q is a maximal germ (for dissipation D , Ř) determined by its subset E f ,Λ defined in (86). Recall here that for P " pp 0 , p 1 , p 2 q and P " pp 0 , p1 , p2 q D f , Řp P , P q " q f 1 pp 1 , p 1 q `q f 2 pp 2 , p 2 q ´q f 0 pp 0 , p 0 q " IN ´OUT Proof. Lemma 4.2 follows from Theorem 2.1 for G Λ " G f,Λ . Applying reversion transform (89) for P " pp 0 , p 1 , p 2 q P G f,Λ , which consists here in the transform pP, f q Þ Ñ p´P, f q and using the fact that ´P P G f ,Λ ðñ P P G f,Λ

we see from Lemma 4.1 that G f ,Λ is a germ. Moreover, because for σ P t˘u we have ´uf j σ pλq " u f j ´σ pλq, we see that ´Γf ,Λ " Γ f,Λ ´P f ,Λ,´ " P f,Λ,` for " 1, 2, 3.

Now recall that G f,Λ is determined by E f,Λ . Hence for ´P P Q :" ź j"0,1,2 rǎ j , čj s, we have ´D f , Řp´P , ´P q ě 0 for all ´P P E f ,Λ ¯ùñ ´P P G f ,Λ , which shows that G f ,Λ is determined by the set E f ,Λ . This gives the desired result for dissipation D f , Ř and completes the proof of the lemma.

By this simple change of variables and (90), we have immediately Corollary 4.3. (Reversion of the germ) Given ρ, ρ P L 8 pp0, 8q ˆŘq, let ρ j pt, xq :" ´ρ j pt, ´xq, ρ j, pt, xq :" ´ρ j, pt, ´xq, ρj 0 pxq :" ´ρ j p´xq, for x P R j .

(91)

Given p f j q, let pf j q be given by reversion transform (89). Then ρ solves (19) (with initial data ρ) with germ Ǧp¨q given by (20), if and only if ρ solves (15) (with initial data ρ) with germ Gp¨q given by [START_REF] Bressan | Flows on networks: Recent results and perspectives[END_REF].

In the same way, if Λ satisfies [START_REF] Forcadel | Homogenization of a discrete model for a bifurcation and application to traffic flow[END_REF], then ρ solves (21) for the germ G f ,Λ given by (87), if and only if ρ solves (16) for the germ G Λ " G f,Λ given by (87).

Proof of Theorem 1.7

Proof of Theorem 1.7. The existence and the uniqueness of a solution to [START_REF]Homogenization of scalar conservation laws with oscillatory forcing terms[END_REF] is a consequence of Lemma 1.2 and Corollary 4.3. Given ρ a solution to [START_REF]Homogenization of scalar conservation laws with oscillatory forcing terms[END_REF], let ρ and ρ0 be defined by (91). We know from Corollary 4.3 that ρ solves (15), with pf j q defined by (89), Gp¨q given by [START_REF] Bressan | Flows on networks: Recent results and perspectives[END_REF]. Then Theorem 1.4 says that the pρ q converges in L 1 loc as Ñ 0 `to the solution ρ of ( 16) for the germ GΛ " G f, Λ defined in [START_REF] Holle | Entropy Dissipation at the Junction for Macroscopic Traffic Flow Models[END_REF], where Λ is given in Subsection 2.1.3. Let ρ be defined from ρ by the transform (91). Then, by Corollary 4.3, ρ is a solution of [START_REF] Fjordholm | Well-posedness and convergence of a finite volume method for conservation laws on networks[END_REF] for the germ G f , Λ. This shows that the pρ q converges in L 1 loc as Ñ 0 `to ρ, which is the unique solution to [START_REF] Fjordholm | Well-posedness and convergence of a finite volume method for conservation laws on networks[END_REF] for the germ G f , Λ.

1 . 3 1 . 4 1 . 5 1 . 6 1 . 7 1 . 8 1 . 9

 13141516171819 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Standing notation and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Entropy pairs and germs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The mesoscopic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The macroscopic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Main result: the homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Homogenization for 2:1 junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Review of the literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 * 1 *

 21 0 to branch 1 with limiter Aptq, no traffic entering in branch on the time-interval I 1 , traffic from branch 0 to branch 2 with limiter Aptq, no traffic entering in branch on the time-intervals I 2 .

Figure 1 :

 1 Figure 1: Divergent 1:2 junction

traffic from branch 1 2 * 1 *

 121 to branch 0 with flux limiter Aptq, no traffic exiting branch on the time intervals I 1 , traffic from branch 2 to branch 0 with flux limiter Aptq, no traffic exiting branch on the time intervals I 2

Figure 2 :

 2 Figure 2: Convergent 2:1 junction

Remark 2 .

 2 3. (An obstacle problem)The flux F λ at the junction can be recovered by an obstacle problem. More precisely, one can show that

Lemma 3 . 1 .

 31 (Explicit time-periodic solution of the HJ equation) Under the assumptions (52) and (53) on f and ψ, there exists a unique time-periodic Lipschitz continuous viscosity solution w : R ˆr0, `8q Ñ R to which is of time period equal to 1. It is given by wpt, xq " sup t1ďt ψpt 1 q ´ξpt ´t1 , xq

2 t1 2 t1

 22 w0 pt, xq ď pb 0 ´p0 qx `max t1ďt2ďt t´ˆt pApsq ´f 0 pp 0 qqds ´pt ´t2 qpf 0 max ´f 0 pp 0 qqu.We note that the map tÞ Ñ max t1ďt2ďt t´ˆt

  t0 qs, we see for later use that β is also C 1 , and is moreover nonincreasing. Now for t :" t λ , we have Aptq " λ and

		ˆA´1 pλq
		βpλq "	pApsq ´Aptqqds
		t0
	i.e.	pβ ˝Aqptq "	ˆt t0

  1,1 in r0, 8q ˆr , `8q (for any ą 0), with B y ξps, yq " ´p, B s ξps, yq " f ppq, where p is the unique maximum in the definition of ξps, yq. Since w can be rewritten as

	wpt, xq " sup

t1PR ψpt 1 ^tq ´ξpt ´pt 1 ^tq, xq it is globally Lipschitz continuous on r0, `8q 2 .

CEREMADE, UMR CNRS 7534, Université Paris Dauphine-PSL, Place de Lattre de Tassigny, 75775 Paris Cedex 16, France.

INSA Rouen Normandie, Normandie Univ, LMI UR

3226, F-76000 Rouen, France. 3 CERMICS, Université Paris-Est, Ecole des Ponts ParisTech, 6-8 avenue Blaise Pascal, 77455 Marne-la-Vallée Cedex 2, France.

For instance traffic flows on 1:2 junctions in which the positive proportion of the traffic entering each outgoing road is fixed, are never L 1 ´contractions.

For simplicity we use the same expression F λ and F p 0 although the relationship between λ and p 0 is the equality p 0 " pf 0,`q´1 pλq: the first notation makes more sense in the present section, while the second one will be used throughout Section

on the construction of correctors.

The proof of inequality (50) is a short proof. Still it is quite difficult to guess that proof from scratch (and also the expression of the germ G Λ ) and it needs a lot of tries. Notice that each component of P and P can be either in the nondecreasing (i.e. fluid) or nonincreasing (i.e. congested) part of the flux. A first (tedious) proof was done distinguishing p2 3 q 2 " 64 cases, and using a much more complicate (and equivalent) expression of G Λ . Finally, the proof we give here is easy to follow line by line but is absolutely not intuitive.

In[START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF], the Hamiltonian is coercive. To cover this case, we just have to extend each f j as a concave function on R such that ´f j is coercive. Then using the comparison principle and suitable barriers (built on the initial data), it is quite standard that we can show that Btw j ď 0 for our initial data satisfying f j pBx wj q ě 0. Then using the PDE itself, we can show that the solution satisfies f j pBxw j q ě 0 and then Bxw j P ra j , c j s almost everywhere.

Acknowledgement. This research was partially funded by l'Agence Nationale de la Recherche (ANR), project ANR-22-CE40-0010 COSS. For the purpose of open access, the authors have applied a CC-BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.

Proof. Following Lemma 3.1, w j p 0 defined in (74) is the unique solution to (73) and is Lipschitz continuous because by construction f j p 0 : ra j ´p j p 0 , b j ´p j p 0 s Ñ R is increasing and uniformly concave and ψp 0 satisfies (72). As f j p 0 p0q " 0, Lemma 3.3 implies that w j p 0 is bounded and satisfies (75).

In order make the link with conservation laws, we need the following technical result which will allow us to glue the solutions on the different branches. Fix p 0 as in Lemma 3.10 and let w 0 p 0 , pj p 0 P ra j , b j s and w j p 0 be respectively defined by Lemma 3.5 , Definition 3.8 and Lemma 3.10. The maps w j p 0 being a solution to the HJ equation (61) (for j " 0) and (73) (for j " 1, 2), B x w j p 0 is a solution to the corresponding conservation law (Lemma A.3). Therefore B x w j p 0 has a trace at x " 0 in the sense of Panov (Theorem A.1). Lemma 3.11 (Expression of the flux of the traces). On I j (for j " 1, 2), the trace pB x w 0 p 0 p¨, 0 ´q, B x w j p 0 p¨, 0 `qq satisfies min ! Aptq, f 0,`p B x w 0 p 0 pt, 0 ´q `p0 q, f j,´p B x w j p 0 pt, 0 `q `p j p 0 q

) " f 0 pB x w 0 p 0 pt, 0 ´q `p0 q " f j pB x w j p 0 pt, 0 `q `p j p 0 q a.e.,

while on RzI j , the trace B x w j p 0 p¨, 0 `q satisfies f j pB x w j p 0 pt, 0 `q `p j p 0 q " 0 a.e..

Proof.

Step 1: proof of (76). The main idea is to reduce the problem to a HJ equation on a junction and then to use the equivalence between HJ and conservation law on a simple junction with only two branches given by Lemma A.4. Fix j " 1, 2 and let pτ 1 , τ 2 q Ă I j on which A is constant. We set # W 0 pt, xq " w 0 p 0 pt, xq `p0 x ´f 0 pp 0 qpt ´τ1 q in pτ 1 , τ 2 q ˆp´8, 0q, W j pt, xq " w j p 0 pt, xq `p j p 0 x ´f j pp j p 0 qpt ´τ1 q ´wj p 0 pτ 1 , 0q `w0 p 0 pτ 1 , 0q in pτ 1 , τ 2 q ˆp0, `8q,

We claim that W " pW 0 , W j q is a viscosity solution to the problem on the 1:1 junction (in the sense of [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF]):

piq B t W 0 `f 0 pB x W 0 q " 0 for t P pτ 1 , τ 2 q, x ă 0, piiq B t W j `f j pB x W j q " 0 for t P pτ 1 , τ 2 q, x ą 0, piiiq W pt, 0q :" W 0 pt, 0 ´q " W j pt, 0 `q for t P pτ 1 , τ 2 q, pivq B t W pt, 0q `mintAptq, f 0,`p B x W pt, 0 ´qq, f j,´p B x W pt, 0 `qqu " 0 for t P pτ 1 , τ 2 q.

(79) Indeed, by construction, W pτ 1 , 0 ´q " W pτ 1 , 0 `q. By Lemma 3.7, we have B t W pt, 0 ´q " B t w 0 p 0 pt, 0 ´q ´f 0 pp 0 q " ´Fp 0 ptq " ´F j p 0 ptq a.e. in pτ 1 , τ 2 q, while, by the boundary condition satisfied by w j p 0 and the definition of ψj p 0 in (71), B t W pt, 0 `q " B t w j p 0 pt, 0 `q ´f j pp j p 0 q " p ψj p 0 q 1 ptq ´f j pp j p 0 q " ´F j p 0 ptq a.e. in pτ 1 , τ 2 q.

Thus equality (iii) in (79) holds. Note also that, by the equation satisfied by w 0 p 0 and w j p 0 , (79)-(i) and (79)-(ii) hold. Let us finally check that the junction condition (79)-(iv) holds in the viscosity sense. As, by the definition of F j p 0 , B t W pt, 0q `Aptq " ´F j p 0 ptq `Aptq ě 0 a.e., [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF]Theorem 2.11] implies that W is a supersolution. For the subsolution property, assume that ϕpt, xq :" αptq `q0 x1 xă0 `qj 1 xą0 is a test function touching w 0 from above at pt 0 , 0q, where t 0 P pτ 1 , τ 2 q and with (condition (2.12) in [31, Theorem 2.7])

Aptq " f 0 pq 0 q " f 0,´p q 0 q " f j pq j q " f j,`p q j q.

(80)

The correctors in the fluid case

We build here a corrector when pp 0 , p 1 , p 2 q is as in case (i) of the definition (17) of EΛ.

Proposition 3.12. Assume that p 0 P ra 0 , b 0 s satisfies f 0 pp 0 q ď ´1 0 Aptqdt. Then there exists a bounded solution u p 0 " pu j p 0 q to on R ˆR, which is time-periodic of period 1 and satisfies, for a constant C ą 0 depending on the data and on p 0 , u 0 p 0 pt, xq " p 0 for a.e. on R ˆp0, `8q, j " 1, 2, where w 0 p 0 , pj p 0 P ra j , b j s and w j p 0 are defined in Lemma 3.5, Definition 3.8 and Lemma 3.10 respectively. By construction, u p 0 is bounded and time-periodic of period 1 as w 0 p 0 and w j p 0 are Lipschitz continuous and 1´periodic in time. As w 0 p 0 and w j p 0 solve (61)-(i)-(ii) and ( 73)-(i)-(ii) respectively, u p 0 satisfies ( 9)-(i)-(ii) thanks to the local correspondance between viscosity solution and conservation laws in 1´space dimension recalled in Lemma A.3. The behavior at infinity of u p 0 is a consequence of ( 64) and (75). As for the junction condition ( 9)-(iii), it is proved in Lemma 3.11.

The correctors in the fully congested case

In this part we assume that the second exit road is fully congested (case (ii) in ( 17)): Proposition 3.13. Assume that pp 0 , p 1 , p 2 q P Q satisfies p 2 " c 2 , f 0 pp 0 q " f 0,´p p 0 q " ˆ1 0 1 I 1 ptqAptqdt " f 1 pp 1 q " f 1,`p p 1 q.

Then there exists a bounded solution u " pu j q to (9) on R ˆR, which is time-periodic of period 1 and satisfies, for a constant C ą 0 depending on the data and on p 0 , u 2 " c 2 and

Proof. Let us define a new flux limiter by setting à :" A1 I 1 . We note that p 0 " pf 0,´q´1 ´´1 0 Ãpsqds

¯.

Let us consider w 0 p 0 the solution introduced in Lemma 3.5 and w 1 p0 the solution given for j " 1 in Lemma 3.10 for the the new flux limiter Ã. We set

As w 0 p 0 and w 1 p 0 solve (61)-(i)-(ii) and ( 73)-(i)-(ii) respectively (with flux limiter Ã), pu 0 p 0 , u 1 p 0 , c 2 q satisfies (9)-(i)-(ii) thanks to the local correspondance between viscosity solution and conservation laws in 1´space dimension recalled in Lemma A.3. The behavior at infinity of pu 0 p 0 , u 1 p 0 q is a consequence of (64) and (75). As for the junction condition ( 9)-(iii), it is proved in Lemma 3.11.

Proof of the homogenization

The section is dedicated to the proof of the existence of a solution to the mesoscopic model and of the homogenization for the 1:2 junctions (Subsection 4.1) and for the 2:1 junctions (Subsection 4.2).

Proof for a 1:2 junction

In this part, we prove Lemma 1.2, Theorem 1.6 and Theorem 1.4.

Proof of Lemma 1.2. We show the existence of a solution to [START_REF] Barles | An illustrated guide of the modern approches of Hamilton-Jacobi equations and control problems with discontinuities[END_REF] with initial condition ρ by induction on the time intervals r0, τ k`1 q, k P N, where (rτ k , τ k`1 q) form a partition of r0, `8q such that, for any k P N, A is constant on the interval pτ k , τ q and pτ k , τ k`1 q Ă I i for some i " 1, 2.

Step 1: existence on p0, τ 1 q To fix the ideas we assume here that p0, τ 1 q Ă I 1 , as the case where p0, τ 1 q Ă I 2 can be treated in a symmetric way. Let A denote the (constant) restriction of the flux limiter Ap¨q to p0, τ 1 q.

Let w be an antiderivative of the initial data ρ, i.e. w : R Ñ R is Lipschitz continuous and such that B x w " ρ.

On the time interval r0, τ 1 q we set pρ 0 , ρ 1 , ρ 2 q " pB x w 0 , B x w 1 , B x w 2 q on p0, τ 1 q where pw 0 , w 1 q solves the HJ equation, with a junction condition at x " 0, B t w j `f j pB x w j q " 0 on p0, τ 1 q ˆRj , j " 0, 1, wpt, 0q :" w 0 pt, 0 ´q " w 1 pt, 0 `q on p0, τ 1 q ˆtx " 0u, B t w `mintA, f 0,`p B x w 0 q, f 1,´p B x w 1 qu " 0 on p0, τ 1 q ˆtx " 0u, w j " wj on tt " 0u ˆR, j " 0, 1, and w 2 is the solution to

where the solutions are given by the theory developed in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF] 3 . From Lemma A.4, we know that ρ :" pρ 0 , ρ 1 q is an entropy solution to ρ j P ra j , c j s a.e. on p0, τ 1 q ˆRj , j " 0, 1 B t ρ j `Bx pf j pρ j qq " 0 in p0, τ 1 q ˆRj , j " 0, 1, ρpt, 0q P G 0,1 a.e. on p0, τ 1 q ˆt0u , with initial condition pρ 0 , ρ1 q, where the (maximal) germ G 0,1 is given by G 0,1 :" tpp 0 , p 1 q P ra 0 , c 0 s ˆra 1 , c 1 s, min A, f 0,`p p 0 q, f 1,´p p 1 q ( " f 0 pp 0 q " f 1 pp 1 qu.

Moreover, introducing wH " w 2 p0, 0q, f H " 0 " f H,`, R H :" p´8, 0q, we see that pw H " w 2 p0, 0q, w 2 q is solution to B t w j `f j pB x w j q " 0 on p0, τ 1 q ˆRj , j " H, 2 wpt, 0q " w H pt, 0q " w 2 pt, 0q at p0, τ 1 q ˆtx " 0u , B t w `mintf H,`p B x w H q, f 2,´p B x w 2 qu " 0 at p0, τ 1 q ˆtx " 0u . w j " wj on tt " 0u ˆRj , j " H, 2.

Setting ρH " 0, ρ H " B x w H " 0 and a H " 0 " c H , we see from Lemma A.4 that ρ " pρ H , ρ 2 q is an entropy solution of ρ j P ra j , c j s a.e. on p0, τ 1 q ˆRj , j " H, 2 B t ρ j `Bx pf j pρ j qq " 0 on p0, τ 1 q ˆRj , j " H, 2 ρpt, 0q P G H,2 on p0, τ 1 q ˆtx " 0u ,

We now build associated germs. Let us define the roots u f j ˘of f j,˘p ¨q " λ as

For Λ " p λ0 , λ1 , λ2 , λ1 , λ2 q satisfying (23), and for σ P t˘u, we consider the curve Γ σ f,Λ :" ! P " pu f 0 σ pλq, u f 1 σ pλ 1 q, u f 2 σ pλ 2 qq with λ k :" λk pλq for k " 1, 2 and λ P r0, λ0 s

and the points

We also define

The case σ " `corresponds to the divergent 1:2 junction, while the case σ " ´corresponds to the convergent 2:1 junction.

We consider the following general set (using notation Q RH defined in ( 7) and ( 8))

Germs for 2:1 junctions, by reversion

Consider the convergent 2:1 junction (with an abuse of notation) Ř :" p Ř0 , Ř1 , Ř2 q with " Řj " p´8, 0q for j " 1, 2 Ř0 " p0, `8q

and associated fluxes f j for j " 0, 1, 2 satisfying (1), ( 2), ( 3) and ( 4), with ǎj , bj , čj instead of a j , b j , c j . Similarly, we consider the divergent 1:2 junction denoted by R and defined as (also with an abuse of notation)

We now explain how to transform fluxes p f j q defined on the convergent junction Ř into fluxes pf j q defined on the divergent junction R: we set f j pvq :" f j p´vq with pa j , b j , c j q :" p´č j , ´b j , ´ǎ j q.

(89)

As before we set Q " ra 0 , c 0 s ˆra 1 , c 1 s ˆra 2 , c 2 s and Q " rǎ 0 , č0 s ˆrǎ 1 , č1 s ˆrǎ 2 , č2 s.

Lemma 4.1. (Effect of reversion on the dissipation) For P, P P Q, D f , Řp´P , ´P q " D f,R p P , P q.

Proof. Using q f j p´p j , ´pj q " ´qf j pp j , p j q we deduce that D f , Řp´P , ´P q " D f,R p P , P q.

A Appendix

In this appendix, we collect several results needed throughout the paper.

A.1 Panov's theorem on strong traces

Let T ą 0 and let us consider the following equation B t u `Bx pf puqq " 0 on p0, T q t ˆp0, `8q x (92)

We recall the following result. Assume that f : R Ñ R is continuous and that P L 8 pp0, T q t ˆp0, `8q x q is a standard Krushkov entropy solution of (92) on p0, T q t ˆp0, `8q x . Assume moreover that f satisfies the following nondegeneracy condition:

Then there exists w P L 8 p0, T q and a measurable set N Ă p0, `8q x of measure zero, such that

}up¨, xq ´w} L 1 p0,T q " 0 and we write ess lim p0,`8qQxÑ0

`up¨, xq " w in L 1 p0, T q.

We call w the strong trace of u on the interface p0, T q ˆt0u and we denote it by up¨, 0 `q.

A.2 Local regularity of scalar conservation in one space dimension

We assume for a ă c, δ ą 0,

Given pt, xq P R 2 and R ą 0, let

We are interested in BV estimates of solutions to the the scalar conservation law B t u `Bx pf puqq " 0 in Q R pt, xq.

Proposition A.2. (Local BV bound for a conservation law with a convex flux)

Under assumption (94), there exists a constant C ą 1, depending on c ´a, on }f 1 } 8 and on δ ą 0 (the concavity constant of f ), such that, for any R P p0, 1s and any pt,

Proof. We only sketch the proof, as it is standard (we just did not find a reference giving the formulation above needed in the paper). Without loss of generality we can assume that pt, xq " p0, 0q and a " 0, so that }u} 8 ď c. We abbreviate Q R p0, 0q into Q R . By finite speed of propagation, the restriction of u to Q R{3 depends only on the value of up´R, ¨q in r´R 1 , R 1 s, where R 1 :" 2R}f 1 } 8 . Let us denote by ũ the solution of B t ũ `Bx pf pũqq " 0 in p´8, 8q ˆR starting from ũ0 at time ´R, where ũ0 " up´R, ¨q on r´R 1 , R 1 s and ũ0 " 0 otherwise. Then ũ " u in Q R{3 and ũ satisfies the Lax-Oleinik bound:

in the sense of distributions. Thus, for any smooth test function φ with a compact support in Q R{3 , we have, at least formally, (C denoting a constant depending on c ´a, }f This implies the result.

A.3 Local correspondence: viscosity solutions versus entropy solution

Equivalence between Hamilton-Jacobi equation and scalar conservation laws in one space dimension has been discussed in several papers: see for instance [START_REF] Colombo | Initial data identification in conservation laws and Hamilton-Jacobi equations[END_REF][START_REF] Karlsen | A note on Front tracking and the Equivalence between Viscosity Solutions of Hamilton-Jacobi Equations And Entropy Solutions of scalar Conservation Laws[END_REF] (see also Lemma A.4 below). The following statement can be deduced from these reference combined with a localization argument in the spirit of the proof of Proposition A.2:

. (Local correspondence viscosity solution versus entropy solution)

Let a ă c, δ ą 0 and f : ra, cs Ñ R be C 2 such that f 2 ď ´δ. Let T ą 0 and R ą 0 Let v : Ω Ñ R be a Lipschitz continuous function with Ω :" p0, T q ˆp´R, Rq and B x v P ra, cs a.e. on Ω.

If v is a viscosity solution of B t v `f pB x vq " 0 on Ω then u " B x v is an entropy solution of B t u `Bx f puq " 0 on Ω.

A.4 Correspondence for a junction: viscosity solutions versus entropy solution

Lemma A.4. (Correspondence for a junction: viscosity solution versus entropy solution, [START_REF] Cardaliaguet | Conservation law and Hamilton-Jacobi equations on a junction: the convex case[END_REF]) For i " L, R, let real numbers a i ă b i ă c i and functions f i : ra i , c i s Ñ R be C 2 satisfying pf i q 2 ď ´δ ă 0, increasing on ra i , b i s and decreasing on rb i , c i s and such that f i pa i q " f i pc i q " 0. We define the monotone envelopes f i,`p pq " " f i ppq for p P ra i , b i s f i pb i q for p P rb i , c i s and f i,´p pq " " f i pb i q for p P ra i , b i s f i ppq for p P rb i , c i s Let T ą 0 and a flux limiter A ě 0. Let v " pv L , v R q be a viscosity solution (in the sense of [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF]) of

a.e. on p0, T q ˆt0u u " u 0 a.e. on t0u ˆR with u 0 " B x v 0 and with G A :" pp L , p R q P ra L , c L s ˆra R , c R s, min A, f L,`p p L q, f R,´p p R q ( " f L pp L q " f R pp R q ( .

ii) (A variant)

The result is still true for f i " f i,˘" 0 " a i " b i " c i for i " L, or for i " R or for both.

A.5 The envelope theorem

We recall the following result (which is easy to prove directly). We make the following assumptions on ϕ $ & % the map ϕ is continuous on Ω x ˆYy the map ϕp¨, yq is differentiable on Ω x for each y P Y , with derivative ϕ x p¨, yq the map B x ϕ is continuous on Ω x ˆYy (96)

i) (The directional derivative) For any v P R n , the function h has directional derivative at each point x 0 P Ω which is defined by D v hpx 0 q :" lim εÑ0 `hpx `εvq ´hpxq ε and we have D v hpx 0 q " max y0PArgmax ϕpx0,¨q v ¨Bx ϕpx 0 , y 0 q with Argmax ϕpx 0 , ¨q :" " y 0 P Y, ϕpx 0 , y 0 q " max yPY ϕpx 0 , yq * .

ii) (When h has already a derivative)

Assume that h has a derivative at x 0 P Ω. Then we have B x hpx 0 q " B x ϕpx 0 , y 0 q for all y 0 P Argmax ϕpx 0 , ¨q.

iii) (Existence of a derivative for h) Let x 0 P Ω. If the map v Þ Ñ D v hpx 0 q is linear, then h has a derivative at x 0 . iv) (The basic result) Let x 0 P Ω. If Argmax ϕpx 0 , ¨q " ty 0 u is a singleton, then h has a derivative at x 0 and B x hpx 0 q " B x ϕpx 0 , y 0 q.