Boris P Andreianov
email: boris.andreianov@lmpt.univ-tours.fr

AND Giuseppe Maria Coclite
email: giuseppemaria.coclite@uniba.it

Carlotta Donadello
email: carlotta.donadello@univ-fcomte.fr

WELL-POSEDNESS FOR A MONOTONE SOLVER FOR TRAFFIC JUNCTIONS

Keywords: 2010 Mathematics Subject Classification. 90B20, 35L65 Traffic model, networks, conservation laws

.

Introduction

We consider a junction consisting of m incoming and n outgoing roads. Incoming roads are parametrized by x R ¡ while outgoing road by x R in such a way that the junction is always located at x 0.

We describe the evolution of traffic on each road by a scalar conservation law of the form (1) ρ h,t f h pρ h q x 0, for h 1, . . . , m n, where ρ h is the density of vehicles and f h is the flux on the h-th road. For notational simplicity we call Ω h the spatial domain of the density ρ h . Everywhere in the paper we use the index i for the m incoming roads and j for the n outgoing roads (then Ω i R ¡ for all i 1, . . . , m and Ω j R for all j m 1, . . . , m n), see Figure 1. The fluxes f h , h 1, . . . , m n, differ in general as each road may have different maximal capacities and speed limitations. However, we assume that each flux f h is bell-shaped (unimodal), Lipschitz and non-degenerate nonlinear i.e. it satisfies the conditions (F) for all h, f h Lip pr0, Rs; R q with }f I h } V ¤ L h , f h p0q 0 f h pRq, and there exists ρh s0, Rr such that f I h pρq pρ h ¡ ρq ¡ 0 for a.e. ρ r0, Rs, (NLD) for all h, f I h is not constant on any non-trivial subinterval of r0, Rs.

The fundamental postulate of our approach is that any physically relevant solution of the problem has to satisfy, as minimal requirement, the conservation of the total density at the junction. The intuitive way to express this condition is to say that for a.e. t R

(2)

m i1 f i ρ i pt, 0 ¡ q ¨ m n jm 1 f j ρ j pt, 0 q ¨.
Garavello and the second author, in [START_REF] Coclite | Vanishing viscosity for traffic on networks[END_REF], considered the Cauchy problem at the junction and established the existence of weak solutions obtained as limit of vanishing viscosity approximations. In [START_REF] Coclite | Traffic flow on a road network[END_REF], uniqueness for such solutions was only proved in the special case m n and f h f h I for all h, h I t1, . . . , m nu. The present paper naturally completes those results as Figure 1. A junction consisting of m incoming and n outgoing roads.

we obtain the uniqueness of the vanishing viscosity limit for any number of roads. Our approach relies upon a partial generalization of the recent results on scalar conservation laws with discontinuous flux obtained by the first authors and his collaborators, see in particular [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF][START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF]. Let us mention in passing that a large part of the concepts and results of [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF][START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] can be generalized to conservation laws on networks. However, a systematic generalization of the theory of L 1 -dissipative germs is beyond the scope of the present paper: we focus on characterization of solutions to the concrete problem [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF] originating from the vanishing viscosity regularization of [START_REF] Coclite | Vanishing viscosity for traffic on networks[END_REF], and on well-posedness in this framework. Our presentation is essentially self-contained. Let us only mention that in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF][START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF], the authors provide general results, indirectly exploiting some insight from [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF], for a junction whose traffic is described by Hamilton-Jacobi equations. Remark 1. For readers acquainted with the discontinuous-flux theory, let us indicate that we characterize the admissibility of solutions at the junction in terms of the "vanishing viscosity germ" G V V (cf. [START_REF] Andreianov | On vanishing viscosity approximation of 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]) which is introduced under the form that was put forward in [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF] (Definition [START_REF] Crandall | Monotone difference approximations for scalar conservation laws[END_REF]). Note that we give three equivalent definitions of admissible solutions, different definitions being useful for different purposes (meaning of the junction admissibility condition, proof of uniqueness, proof of existence).

We provide the interpretation of G V V in terms of Oleinik-like inequalities of [START_REF] Diehl | A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients[END_REF] (Lemma 2.2). We prove that this germ is L 1 -dissipative, complete and maximal (Lemma 2.5, Lemma 2.7 and Lemma 2.8, respectively). We prove the suitable Kato inequality (64) which leads to the L 1contraction property of the admissible solutions (Proposition 3.1), stability and uniqueness.

To justify existence and the relation to the vanishing viscosity regularization of [START_REF] Coclite | Vanishing viscosity for traffic on networks[END_REF], following [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] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF] we introduce a family of adapted entropies at the junction (Definition 2.10). We put forward the Godunov finite volume scheme inspired by [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF] and justify its convergence and existence of admissible solutions. In addition, we link the definition of (a part of) the germ G V V to the existence of vanishing viscosity profiles (Corollary 1) and identify the admissible solutions with vanishing viscosity limits (Theorem 4.1).

A second important remark is that our uniqueness result is by no means a result on the uniqueness tout court of solutions of the Cauchy problem on a traffic junction. It is well known in the literature that different Riemann Solvers can be used at junctions, depending on the physical situation one aims at describing, see [START_REF] Coclite | Traffic flow on a road network[END_REF][START_REF] Garavello | Traffic flow on networks, volume 1 of AIMS Series on Applied Mathematics[END_REF][START_REF] Holden | A mathematical model of traffic flow on a network of unidirectional roads[END_REF], the recent survey [START_REF] Bressan | Flows on networks: recent results and perspectives[END_REF] and references therein. Let us point out that the definitions and results of Section 2 (starting from § 2.2) and Section 3 can be adapted in a straightforward way to the study of solutions corresponding to Riemann Solvers at the junction which verify the order-preservation property (increasing the Riemann datum on any of the roads results in pointwise increase of the solution on the whole network) and the Lipschitz continuity properties of the corresponding Godunov fluxes, cf. the last paragraph of Remark 3. However, the order-preservation property of Riemann solvers at junctions is not satisfied by most of the models proposed in the literature.

1.1. Preliminaries. We assume that the reader is acquainted with the notion of entropy solution to scalar conservation laws introduced by Kruzhkov [START_REF] Kružhkov | First order quasilinear equations with several independent variables[END_REF]. This notion is suitable for describing admissibility of solutions to (1) away from the junction. But we recall, first, the formulation of the Bardos-LeRoux-Nédélec boundary condition for conservation laws in terms of the Godunov numerical flux, which will be instrumental for the definition of admissible solutions at the junction and for the existence proof. Second, we recall that entropy solutions of non linearly degenerate scalar conservation laws admit boundary traces in the strong L 1 sense.

1.1.1. Godunov's flux. Let u be the entropy solution to the scalar conservation law with Lipschitz continuous flux (3)

u t f puq x 0, pt, xq R ¢ R corresponding to the Riemann initial condition (4) u 0 pxq 5 a, if x 0, b, if x ¡ 0.
One calls Godunov flux the function which associates to the couple pa, bq the value f pupt, 0 ¡ qq f pupt, 0 qq (the two values are equal due to the Rankine-Hugoniot condition). The analytical expression, see for example [START_REF] Holden | Front tracking for hyperbolic conservation laws[END_REF], is given by [START_REF] Andreianov | Entropy conditions for scalar conservation laws with discontinuous flux revisited[END_REF] Gpa, bq

5 min sra,bs f psq if a ¤ b, max srb,as f psq if a ¥ b.
In the sequel, we denote by f a G, resp. f b G, the partial derivative of the Godunov flux G with respect to the first, resp. to the second argument.

The Godunov flux can be used for convergent numerical approximation of (3) by an explicit finite difference / finite volume scheme. This follows from the fact that G satisfies the following two basic properties, shared with several other numerical fluxes as for example Rusanov and Lax-Friedrichs (see, e.g., [START_REF] Crandall | Monotone difference approximations for scalar conservation laws[END_REF]):

Consistency: for all a r0, Rs, Gpa, aq f paq; Monotonicity and Lipschitz continuity: There exists L ¡ 0 such that for all pa, bq r0, Rs 2 we have

(6) 0 ¤ f a Gpa, bq ¤ L, ¡L ¤ f b Gpa, bq ¤ 0. 1.1.2.
A formulation of the Bardos-LeRoux-Nédélec boundary condition. In our setting, the main interest in using Godunov flux is related to the following observation (see [START_REF] Dubois | Boundary conditions for nonlinear hyperbolic systems of conservation laws[END_REF], see also [START_REF] Andreianov | Well-posedness of general boundary-value problems for scalar conservation laws[END_REF] for a review on this topic). Consider the initial and boundary value problem (IBVP)

u t f puq x 0, for pt, xq in R ¢ R ¡ upt, 0q u b ptq, up0, xq u 0 pxq,
and assume that u is a Kruzhkov entropy solution in the interior of the half plane R ¢ R ¡ .

Then u satisfies the boundary condition in the sense of Bardos-LeRoux-Nédélec (see [START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF]) if and only if its trace γuptq upt, 0 ¡ q satisfies f pγuptqq Gpγuptq, u b ptqq. where pa, bq is an interval of R. Assume that u L V pR ¢ pa, bqq satisfies the Kruzhkov entropy inequalities (see [START_REF] Kružhkov | First order quasilinear equations with several independent variables[END_REF] and (11) below). Assume that the Lipschitz flux f in (3) is non linearly degenerate in the sense that f I is not identically zero on any interval (which follows from pNLDq). Then (see [START_REF] Panov | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF], see also [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF]) the function upt, ¤q possesses one-sided limits: e.g., one can define upt, b ¡ q : γuptq where pγuqp¤q is the strong trace of u on R ¢ tbu in the L 1 loc sense: for all ξ DpR q, (8) lim

kÑ0 1 k » R » b b¡k ξptq|upt, xq ¡ γuptq| dx dt 0.
Notice that this property permits to extend the above interpretation of the Bardos-LeRoux-Nédélec boundary condition for problem [START_REF] Audusse | Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies[END_REF] to the case of general L V initial and boundary data, beyond the classical BV framework.

1.1.4. Functional framework. Throughout the paper, we are interested in L V solutions of (1).

We will denote dy Γ the graph pictured in Figure 1 and use the slightly abusive notation L V pR ¢Γ; r0, Rs m n q for pm nq-uplets pρ 1 , . . . , ρ m , ρ m 1 , . . . , ρ m n q of functions such that ρ i L V pR ¢R ¡ ; r0, Rsq for i t1, . . . , mu and ρ j L V pR ¢R ; r0, Rsq for j tm 1, . . . , m nu.

Similarly, L V pΓ; r0, Rs m n q will denote the space of r0, Rs-valued initial data on the graph Γ.

1.2. The notion of admissible solution and the outline of the paper. Our goal is to re-visit and complement the work [START_REF] Coclite | Vanishing viscosity for traffic on networks[END_REF], which studies vanishing viscosity limits for problem [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF]. The property of being a vanishing viscosity limit can be seen as a specific admissibility condition for a weak solution of (1), which boils down to the standard Kruzhkov entropy conditions on each of the roads Ω h , h t1, . . . , m nu; a specific "coupling" condition at the junction, whose description is the main object of the present paper.

An intermediate significant result of our work is the intrinsic characterization of the vanishing viscosity limits for (1): this is done either in terms of the Riemann solver at the junction, or in terms of m n Dirichlet problems on Ω h , h t1, . . . , m nu coupled by a simple transmission condition, or in terms od "adapted" entropy inequalities.

The notion of solution we aim at using is roughly speaking the following. We consider ρ pρ 1 , . . . , ρ m n q in L V pR ¢ Γ; r0, Rs m n q as an admissible solution if, first, for any h t1, . . . , m nu, ρ h is a weak entropy solution in the sense of Kruzhkov in the interior of Ω h . Second, recalling that ρ i (resp., ρ j ) admits a strong trace ρ i p¤, 0 ¡ q γ i ρ i p¤q (resp., ρ j p¤, 0 q γ j ρ j p¤q) at x 0, i.e.

(

) lim kÑ0 ¡ 1 k » R » 0 k ξptq|ρ i pt, xq ¡ γ i ρ i ptq| dx dt 0, for i 1, . . . , m, lim kÑ0 1 k » R » k 0 ξptq|ρ j pt, xq ¡ γ j ρ j ptq| dx dt 0, for j m 1, . . . , m n, 9
we require that the pm nq-uple of traces satisfies condition (2) for a.e. t R and, moreover, for a.e. t R the values of the traces "coincide up to boundary layers". This choice is made in accordance with the fact that the vanishing viscosity approximation of (1) prescribes, for every viscosity parameter ε ¡ 0, the coincidence of all ρ ε h pt, ¤q, h 1, . . . , m n, at x 0; and that taking the limit ε Ñ 0 relaxes this condition analogously to the way in which the boundary condition in [START_REF] Audusse | Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies[END_REF] is relaxed.

In order to give a more precise statement, which is the aim of this section, we need to introduce some notation. In Section 2, we will reformulate the problem in two different forms, suitable for proving the uniqueness and the existence, respectively. 1.2.1. The junction as a collection of IBVPs. Given an initial condition u 0 L V pΓ; r0, Rs m n q, u 0 pu 0 1 , . . . , u 0 m n q, we look for a function ρ pρ 1 , . . . , ρ m n q in L V pR ¢ Γ; r0, Rs m n q such that for any h t1, . . . , m nu, ρ h is a weak entropy solution of the initial and boundary value problem (IBVP) (10) ρ h,t f h pρ h q x 0, on s0, T r¢Ω h , ρ h pt, 0q v h ptq, on s0, T r,

ρ h p0, xq u 0 h pxq, on Ω h ,
where the set of boundary conditions v : R Ñ r0, Rs m n is to be fixed in the sequel so to guarantee that, in particular, the conservativity condition (2) holds. Let us stress that at this point, different choices are possible, and each choice reflects a modeling assumption at the junction.

Definition 1.1. We say that the function ρ h is an entropy weak solution of the initial and boundary values problem [START_REF] Bressan | Flows on networks: recent results and perspectives[END_REF] if

For any test function ξ in DpR ¢Ω h ; R q, ξ| fΩ h 0, and for any k r0, Rs there holds

(11) » R » Ω h t|ρ h ¡ k|ξ t q h pρ h , kqξ x u dx dt » Ω h |u 0 h pxq ¡ k|ξp0, xq dx ¥ 0, q h pu, kq : signpu ¡ kqpf h puq ¡ f h pkqq being the Kruzhkov entropy flux associated to f h .
For a.e. t R , γ h ρ h ptq satisfies the boundary condition in the sense of Bardos-Le Roux-Nédélec (BLN), which we express under the form (cf. Section 1.1)

f i pγ i ρ i q G i pγ i ρ i , v i q if i t1, . . . , mu; (12)
f j pγ j ρ j q G j pv j , γ j ρ j q if j tm 1, . . . , m nu, [START_REF] Coclite | Vanishing viscosity for traffic on networks[END_REF] where G i and G j are the Godunov fluxes associated to f i and f j respectively.

In order to describe the solutions of (1) which can be obtained as vanishing viscosity limit, we postulate that the artificial Dirichlet values v h at the junction need to be the same for all h: [START_REF] Coclite | Traffic flow on a road network[END_REF] for all h t1, . . . , m nu, for a.e. t R v h ptq pptq.

We refer to [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF][START_REF] Andreianov | Entropy conditions for scalar conservation laws with discontinuous flux revisited[END_REF] for detailed motivations, in the discontinuous-flux setting. The criterion for the choice of p is the conservativity condition (2); due to [START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF] and [START_REF] Coclite | Vanishing viscosity for traffic on networks[END_REF], we can now express it in the form [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF]

m i1 Gpγ i ρ i ptq, pptqq m n jm 1
Gppptq, γ j ρ j ptqq, for a.e. t R .

Observe that formally, ( 14) and ( 15) close the coupled system of IBVP's [START_REF] Bressan | Flows on networks: recent results and perspectives[END_REF]. Definition 1.2. Given an initial condition u 0 L V pΓ; r0, Rs m n q, we call ρ pρ 1 , . . . , ρ m n q in L V pR ¢ Γ; r0, Rs m n q an admissible solution of (1) associated with u 0 if there exists a function p in L V pR ; r0, Rsq such that for any h t1, . . . , m nu ρ h is a solution of IBVP [START_REF] Bressan | Flows on networks: recent results and perspectives[END_REF] in the sense of Definition 1.1 with v h , h t1, . . . , m nu chosen to fulfill [START_REF] Coclite | Traffic flow on a road network[END_REF], and such that ρ, p fulfill (15). 1.2.2. Outline of the remaining Sections. We will reformulate Definition 1.2 in Section 2, both in terms of the Riemann solver at the junction and in terms of adapted entropy inequalities that (unlike the "per road" Kruzhkov entropy inequalities [START_REF] Bürger | A family of numerical schemes for kinematic flows with discontinuous flux[END_REF]) account for the admissibility of ρ at the junction. We will establish well-posedness of problem [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF] in the frame of the so defined admissible solutions in Section 3. Finally, we will justify the adequacy of this definition of admissibility for intrinsic characterization of vanishing viscosity limits in Section 4.

Equivalent formulations of admissibility and the underlying Riemann solver at the junction

Observe that in the special case where m n and f h f for all h t1, . . . , 2mu the constant vector function k pk, . . . , kq R 2m satisfies the conditions above with pptq k. This kind of stationary solution is employed in [START_REF] Coclite | Vanishing viscosity for traffic on networks[END_REF] to construct a family of Kruzhkov like entropies.

In general, however, other stationary solutions may be of interest. For example in the case m n 1 all vectors k pk 1 , k 2 q such that k 1 and k 2 are respectively the left and the right state of a Kruzhkov admissible jump are admissible stationary solutions to the problem. In what follows, we introduce the vanishing viscosity germ which will be identified later on with the set of all possible stationary admissible solutions to (1) on R ¢ Γ constant on each road of Γ. This definition will permit us to describe the Riemann solver and the associated fluxes at the junction defined in Lemma 2.4.

Definition of the vanishing viscosity germ.

In this section we describe the stationary admissible solutions of (1) that are constant on each road of Γ. Because of the analogy with the discontinuous-flux setting of [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF][START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] we will use similar notation and terminology (cf. Remark 1 for a brief summary).

Definition 2.1. We call vanishing viscosity germ the subset of r0, Rs m n defined by ( 16) u pu 1 , . . . , u m n q : hp r0, Rs such that

G V V
m i1 G i pu i , pq m n jm 1 G j pp, u j q G i pu i , pq f i pu i q, G j pp, u j q f j pu j q, di, j D G G G G F G G G G E .
It is immediate to see that u G V V if and only if, seen as a vector function in R ¢ Γ Ñ r0, Rs m n , u provides a solution of (1) in the sense of Definition 1.2 such that each component u h of u is constant both in t and x. Following [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF], see also [START_REF] Diehl | A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients[END_REF], we can characterize G V V by a set of inequalities reminiscent of the celebrated Oleinik condition for scalar conservation laws. Here and in the following we use the notation Ira, bs to indicate the closed interval rminta, bu, maxta, bus in R. u pu 1 , . . . , u m n q : hp r0, Rs such that

m i1 G i pu i , pq m n jm 1 G j pp, u j q, ds Iru i , ps pp ¡ u i qpf i psq ¡ f i pu i qq ¥ 0, for i 1, . . . , m, ds Irp, u j s pu j ¡ pqpf j psq ¡ f j pu j qq ¥ 0, for j m 1, . . . , m n D G G G G G G F G G G G G G E .
Proof. Actually, the value p in [START_REF] Diehl | A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients[END_REF] coincides with the value p in [START_REF] Crandall | Monotone difference approximations for scalar conservation laws[END_REF]. One only needs to show that

ds Iru i , ps : pu i ¡ pqpf i psq ¡ f i pu i qq ¥ 0 ô G i pu i , pq f i pu i q,
for any i. This readily comes from the definition of the Godunov flux, as the relation G i pu i , pq f i pu i q rewrites as ( 18)

f i pu i q min sru i ,ps f i psq, if pp ¡ u i q ¥ 0, f i pu i q max srp,u i s f i psq, if pp ¡ u i q ¤ 0.
The proof for the j-index case is analogous.

In order to exhibit the key properties of G V V , we start with the following technical lemma which is crucial for the existence theory. It relies upon the monotonicity properties of the Godunov fluxes G h , h t1, . . . , m nu; we defer to Remark 3 for its interpretation in terms of the Godunov fluxes for the junction. Lemma 2.3. Given u pu 1 , . . . , u m n q in r0, Rs m n , consider the problem

(19) find p u r0, Rs such that m i1 G i pu i , p u q m n jm 1 G j pp u , u j q. (i) The set P u of solutions of (19) is non-empty.
(ii) The values pG i pu i , p u qq it1,...,mu , pG j pp u , u j qq jtm 1,...,m nu do not depend on the choice of the value p u P u .

Example 1. To give an example, consider a junction consisting of two incoming and one outgoing roads, on which the traffic is described through the following flux functions:

(20)

f h pρq ¡hρ 2 h, for h 1, 2, 3.
Remark that, for the reader's convenience, we consider now ρ r¡1, 1s, f h p¡1q f h p1q 0 and ρh 0. This does not change our results but allows for cleaner computations. As above we call G h the Godunov flux corresponding to f h and u h is the constant initial condition on the h-th road. If u h $ 0, let ûh $ u h be the only solution to f h pu h q f h pû h q. By using the standard Riemann Solver, see [START_REF] Holden | Front tracking for hyperbolic conservation laws[END_REF], it is easy to check that the values of G i pu i , ¤q, i 1, 2, as functions of p, are the following

If u i ¤ 0, then G i pu i , pq f i pu i q for all p ¤ ûi and G i pu i , pq f i ppq for all p ¥ ûi ; If u i ¥ 0, then G i pu i , pq f i p0q for all p ¤ 0 and G i pu i , pq f i ppq for all p ¥ 0.
Similarly, the values of G 3 p¤, u 3 q, as a function of p are If u 3 ¤ 0, then G 3 pp, u 3 q f 3 ppq for all p ¤ 0 and G 3 pp, u 3 q f 3 p0q for all p ¥ 0; If u 3 ¥ 0, then G 3 pp, u 3 q f 3 ppq for all p ¤ û3 and G 3 pp, u 3 q f 3 pu 3 q for all p ¥ û3 . One can check that if we take, as an example, u 1 ¡ 1{2, u 2 1{4 and u 3 1{6, then all the values of p between r¡ 1{6, 0s satisfy the relation

(21) G 1 pu 1 , pq G 2 pu 2 , pq G 3 pp, u 3 q,
and that for all these values of p, the collection G ¦ p uq of fluxes at the junction will be given by

G ¦ p uq ¡ G 1 pu 1 , pq, G 2 pu 2 , pq, G 3 pp, u 3 q © p1{2, 2, 5{2q.
Remark that, as explained in the paper [START_REF] Lebacque | The Godunov scheme and what it means for first order traffic flow models[END_REF], the functions G i pu i , ¤q and G j p¤, u j q are closely related to the equilibrium supply/demand functions introduced in the work by Lebacque and his collaborators. In particular we have that the equilibrium demand function ∆ i of the i-th incoming road and the equilibrium supply function Σ j of the j-th outgoing road can be defined as

(22) u i Þ Ñ ∆ i pu i q max p tG i pu i , pqu, u j Þ Ñ Σ j pu j q max p tG j pp, u j qu.
Now, we prove Lemma 2.3.

Proof.

(i) Given u we define the functions Φ in u and Φ out u from r0, Rs to R by

(23) Φ in u : p Þ Ñ m i1 G i pu i , pq, Φ out u : p Þ Ñ m n jm 1 G j pp, u j q.
A quick direct calculation gives us ( 24)

G i pu i , 0q max sr0,u i s f i psq ¥ 0, G i pu i , Rq min sru i ,Rs f i psq 0, G j p0, u j q min sr0,u j s f j psq 0, G j pR, u j q max sru j ,Rs f i psq ¥ 0, so that (25) Φ in u p0q ¥ 0 Φ out u p0q and Φ in u pRq 0 ¤ Φ out u pRq.
The existence of at least one solution p u of ( 19) is ensured by the continuity of Φ in u ¡ Φ out u on r0, Rs and the intermediate value theorem.

(ii) It may happen that there exist several values of p such that Φ in u ppq and Φ out u ppq coincide.

From the Lipschitz continuity and monotonicity property of Godunov flux (see § 1.1.1) we have

(26) di t1, . . . , mu f b G i pu i , pq ¤ 0, dj tm 1, . . . , m nu f a G j pp, u j q ¥ 0, which means in particular that Φ in u ¡ Φ out u
is non-strictly decreasing. Therefore, the set P u of solutions of ( 19) is a closed sub-interval of r0, Rs. Since, moreover, each term of the sums

defining Φ in u ¡ Φ out u
has the same monotonicity, we find that all these terms are constant on

P u . Next, consider the map G ¦ : r0, Rs m n Þ Ñ R m n which is well defined, thanks to (ii): (27) G ¦ p uq ¡ G ¦ 1 p uq, . . . , G ¦ m n p uq © , G ¦ i p uq : G i pu i , p u q, i t1, . . . , mu, G ¦ j p uq : G j pp u , u j q, j tm 1, . . . , m nu and the map F ¦ : r0, Rs m n Þ Ñ R defined by (28)
F ¦ p uq : m i1 G ¦ i p uq m n jm 1 G ¦ j p uq.
Lemma 2.4. With the above definitions, the following properties hold.

(i) For each i t1, . . . , mu, the map u Þ Ñ G ¦ i p uq fulfills f u i G ¦ i ¤ L i and dh t1, . . . , m nu, h $ i : f u h G ¦ i ¤ 0. For each j tm 1, . . . , m nu, the map u Þ Ñ G ¦ j p uq fulfills f u j G ¦ j ¥ ¡L j and dh t1, . . . , m nu, h $ j : f u h G ¦ j ¥ 0. (ii) The map u Þ Ñ F ¦ p uq fulfills di t1, . . . , mu : f u i F ¦ ¥ 0 and dj tm 1, . . . , m nu : f u j F ¦ ¤ 0.
The above differential inequalities should be understood in the sense of distributions, e.g.,

"f u i F ¦ ¥ 0" means that F ¦ is non-decreasing in the variable u i .
Proof.

(i) Without loss of generality, we can fix the normalization p u : min P u . Observe that (29) the map u Þ Ñ p u is monotone non-decreasing in each component u h , h t1, . . . , m nu. Indeed, let u ¤ v in the sense of the per component order: for all h t1, . . . , m nu u h ¤ v h . Bearing in mind formula [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF], the monotonicity properties of the Godunov fluxes G i p¤, pq and G j pp, ¤q and the definition of

p u yield 0 Φ in u pp u q ¡ Φ out u pp u q ¤ Φ in v pp u q ¡ Φ out v pp u q.
Hence the monotonicity of Φ in v ¡ Φ out v exhibited in the proof of Lemma 2.3(ii) and the normalization of p u ensure that p u ¤ p v , which proves [START_REF] Tartar | Nonlinear analysis and mechanics: Heriot-watt symposium[END_REF].

Now the monotonicity claims of (i) are immediate from the monotonicity properties of G h and from [START_REF] Tartar | Nonlinear analysis and mechanics: Heriot-watt symposium[END_REF]. To prove the one-sided Lipschitz continuity properties of G ¦ h claimed in (i), let us focus on h 1. The other cases are proved in the same way. Given u pu 1 , u 2 , . . . , u m n q and v pv 1 , u 2 , . . . , u m n q with v 1 ¡ u 1 , we have

G ¦ 1 p vq ¡ G ¦ 1 p uq G 1 pv 1 , p v q ¡ G 1 pu 1 , p u q G 1 pv 1 , p v q ¡ G 1 pu 1 , p v q G 1 pu 1 , p v q ¡ G 1 pu 1 , p u q ¤ G 1 pv 1 , p v q ¡ G 1 pu 1 , p v q ¤ L 1 pv 1 ¡ u 1 q,
which proves the claim.

(ii) The monotonicity properties of F ¦ readily stem from [START_REF] Tartar | Nonlinear analysis and mechanics: Heriot-watt symposium[END_REF]. If we look at the dependence of F ¦ in u i , i t1, . . . , mu, it is enough to represent F ¦ with the last expression in [START_REF] Panov | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF] and combine [START_REF] Tartar | Nonlinear analysis and mechanics: Heriot-watt symposium[END_REF] with the monotonicity of G j p¤, bq, j tm 1, . . . , m nu. If we look at the dependence of F ¦ in u j , j tm 1, . . . , m nu, then we represent F ¦ with the first expression in [START_REF] Panov | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF] and use the monotonicity of G i pa, ¤q, i t1, . . . , mu. Lemma 2.4(i) we can also prove that G ¦ i (resp., G ¦ j ) is monotone non-decreasing (resp., non-increasing) in the argument u i (resp., u j ), and therefore it is L i (resp., L j ) Lipschitz continuous. Indeed, it is enough to represent, e.g., G ¦ 1 by the following expression derived from [START_REF] Eymard | Finite volume methods[END_REF]:

Remark 2. Actually, in the context of

G ¦ 1 pp u q ¡ m i2 G i pu i , p u q m n jm 1 G j pp u , u j q,
and exploit [START_REF] Tartar | Nonlinear analysis and mechanics: Heriot-watt symposium[END_REF]. However, we do not stress these finer properties of dependence of G ¦ i on u i because they are not essential for the subsequent analysis. Now, we are ready to explore the crucial "dissipativity" properties of G V V . For any h t1, . . . , m nu let q h : r0, Rs 2 Ñ R denote the Kruzhkov entropy flux (30)

q h pu, vq signpu ¡ vqpf h puq ¡ f h pvqq. Lemma 2.5. For any k 1 , k 2 in G V V with k pk 1 , . . . , k m n q, 1, 2, there holds (31) ∆p k 1 , k 2 q : m i1 q i pk 1 i , k 2 i q ¡ m n jm 1 q j pk 1 j , k 2 j q ¥ 0.
Inequality (31) will be exploited in this work to prove that the solutions we consider satisfy a generalized Kato's inequality. The L 1 -contraction property and uniqueness will follow.

Proof. To increase readability we use the shorter notation s h signpk 1 h ¡k 2 h q, for h 1, . . . , m n, and we adopt the convention signp0q 0. Up to reordering the incoming and outgoing roads we can assume that s i ¥ 0 for i t1, . . . , αu, s i ¡1 for i α 1, s i ¤ 0 for i tα 1, . . . , mu, s j ¥ 0 for j tm 1, . . . , βu, s j ¡1 for i β 1 and s i ¤ 0 for i tβ 1, . . . , m nu.

Intuitively we deal with three cases. Actually, the third case is more general and includes the first two.

Case 1: Assume that s i s j ¥ 0 for all pi, jq t1, . . . , mu ¢ tm 1, . . . , m nu. Case 2: Assume that s i s j ¤ 0 for all pi, jq t1, . . . , mu ¢ tm 1, . . . , m nu.

Case 3: Let α $ 1, m and β $ m 1, m n. Case 1 This is the easiest case. The definition of G V V implies that (32) m i1 q i pk 1 i , k 2 i q ¡ m n jm 1 q j pk 1 j , k 2 j q ¨m i1 f i pk 1 i q ¡ f i pk 2 i q ¨© m n jm 1 f j pk 1 j q ¡ f j pk 2 j q ¨m i1 G i pk 1 i , p 1 q © m n jm 1 G j pp 1 , k 1 j q © m i1 G i pk 2 i , p 2 q ¨m n jm 1 G j pp 2 , k 2 j q 0,
where we call p 1 and p 2 the values of p associated respectively to k 1 and k 2 .

Case 2

(33) m i1 q i pk 1 i , k 2 i q ¡ m n jm 1 q j pk 1 j , k 2 j q ¨m i1 f i pk 1 i q ¡ f i pk 2 i q ¨¨m n jm 1 f j pk 1 j q ¡ f j pk 2 j q ¨2 m i1 G i pk 1 i , p 1 q © 2 m i1 G i pk 2 i , p 2 q.
Assume, to fix the ideas, that s i ¡ 0, and s j 0. Then the expression above writes as

(34) 2F ¦ p k 1 q ¡ 2F ¦ p k 2 q
with the notation [START_REF] Panov | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF], and this expression is for sure non negative thanks to the monotonicity properties of the function F ¦ established in Lemma 2.4(ii).

Case 3

In this case we define the vector w, whose components satisfy w h mintk

q i pk 1 i , k 2 i q m i1 s i G i pk 1 i , p 1 q ¡ G i pk 2 i , p 2 q α i1 s i G i pk 1 i , p 1 q ¡ G i pw i , p 2 q ¨ m iα 1 s i G i pw i , p 1 q ¡ G i pk 2 i , p 2 q ¥ α i1 s i G i pk 1 i , p 1 q ¡ G i pw i , p w q ¨ m iα 1 s i G i pw i , p w q ¡ G i pk 2 i , p 2 q m i1 signpk 1 i ¡ w i q G i pk 1 i , p 1 q ¡ G i pw i , p w q ¨ m i1 signpk 2 i ¡ w i q G i pk 2 i , p 2 q ¡ G i pw i , p w q Φ in k 1 pp 1 q ¡ Φ in w pp w q ¡ m iα 1 G i pk 1 i , p 1 q ¡ G i pk 2 i , p w q ¨ Φ in k 2 pp 2 q ¡ Φ in w pp w q ¡ α i1 G i pk 2 i , p 2 q ¡ G i pk 2 i , p w q ¨.
Analogous computations on the sum involving the outgoing roads give a similar result. Therefore we obtain

m i1 q i pk 1 i , k 2 i q ¡ m n jm 1 q j pk 1 j , k 2 j q ¥ ¡ m iα 1 G i pk 1 i , p 1 q ¡ G i pk i i , p w q ¨¡ α i1 G i pk 2 i , p 2 q ¡ G i pk 2 i , p w q ¨ m n jβ 1 G j pk 1 j , p 1 q ¡ G i pk 1 j , p w q ¨ β jm 1 G j pk 2 j , p 2 q ¡ G j pk 2 j , p w q ¨.
Each of the four summands in the right hand side is positive, therefore the desired inequality holds.

2.2. Riemann problem at the junction. In this section we discuss the Riemann solver at the junction associated to the vanishing viscosity limit, i.e. to the Definition 1.2 of admissible solution. A very general definition of Riemann solver at junctions is provided in [10, Def. 2.3].

In the framework of the present paper, such definition specializes as follows Definition 2.6. The Riemann solver associated to the vanishing viscosity limit is a function

(35) RS : r0, Rs m n Ñ r0, Rs m n , RSp u 0 q u,
with the following properties:

(1) There exists p r0, Rs such that f i pu i q G i pu 0 i , pq if i t1, . . . , mu;

(36)

f j pu j q G j pp, u 0 j q if j tm 1, . . . , m nu, ( 37
)
where G i and G j are the Godunov fluxes associated to f i and f j respectively, and

°m i1 f i pu i q °m n jm 1 f j pu j q. (2)
The consistency condition RS pRSp u 0 qq RSp u 0 q holds for all u 0 in r0, Rs m n .

It is clear from the definition above that G V V coincides with the set of equilibria, see [10, Def. 2.5], for the Riemann Solver obtained as vanishing viscosity limit. The next lemma claim that a Riemann solver in the sense of Definition 2.6 exists.

Lemma 2.7. For any given initial condition u 0 in r0, Rs m n there exists a self-similar function ρ pρ 1 , . . . , ρ m n q in L V pR ¢ Γ; r0, Rs m n q which is an admissible entropy solution of the Riemann problem at the junction in the sense of Definition 1.2. In particular, this means that the vector γρ of traces of this solution at the junction belongs to G V V .

Proof. Given u 0 in r0, Rs m n we apply Lemma 2.3 and find p : ρ h,t f h pρ h q x 0, on s0, T r¢Ω h , ρ h pt, 0q p, on s0, T r,

p u 0 r0, Rs such that °m i1 G i pu 0 i , pq °m n jm 1 G j pp, u 0 j q.
ρ h p0, xq u 0 h , on Ω h .
Call ρ h the Kruzhkov entropy weak solution to (38) and γ h ρ h its (strong) trace at fΩ h , satisfying the boundary condition in the sense of Bardos-Le Roux-Nédélec. Because the solution is unique and the problem is invariant under the scaling pt, xq Þ Ñ pct, cxq for all c ¡ 0, the solution is self-similar, i.e., each of the components ρ h depends only on the ratio x t . To conclude the proof, it is enough to observe that (39)

G h pγ h ρ h , pq f h pγ h ρ h q G h pu 0 h , pq, if h ¤ m, G h pp, γ h ρ h q f h pγ h ρ h q G h pp, u 0 h q, otherwise
, because in this case γρ fulfills the definition of G V V with p p u 0 and consequently, one sees that ρ is an admissible solution of the Riemann problem at the junction. Equalities (39) follow from the observations of [START_REF] Dubois | Boundary conditions for nonlinear hyperbolic systems of conservation laws[END_REF]. For the sake of completeness, let's point out that, e.g., for all i t1, . . . , mu, (40)

γ i ρ i 5 argmin ru 0 i ,ps f i , if u 0 i ¤ p, argmax rp,u 0 i s f i , if u 0 i ¥ p.
With this value γ i ρ i , the classical Riemann problem with endstates u 0 i and γ i ρ i is solved only with the waves of negative speed while the classical Riemann problem with endstates γ i ρ i and p is solved only with the waves of positive speed, so that the classical Riemann problem with endstates u 0 i and p is solved by juxtaposition of the two. The definition of the Godunov flux in § 1.1.1 ensures that the values

f i pγ i ρ i q, G i pu 0 i , γ i ρ i q, G i pγ i ρ i , pq and G i pu 0 i , pq coincide.
Remark 3. Given u r0, Rs m n , let p u be defined by [START_REF] Eymard | Finite volume methods[END_REF]. According to the above proof, the self-similar admissible solution ρ of the Riemann problem fulfills, with the notation [START_REF] Panov | On sequences of measure-valued solutions of a first-order quasilinear equation[END_REF],

f i pγ i ρ i ptqq G i pu i , p u q G ¦ i p
uq for all i t1, . . . , mu, f j pγ j ρ j ptqq G j pp u , u j q G ¦ j p uq for all j t1 m, . . . , m nu (recall that while p u may not be unique, the above flux values are uniquely defined). We see that given u, the collection of values G ¦ p uq R m n defines, road per road, the Godunov flux at the junction associated with the Riemann solver at the junction described by Lemma 2.7 (the flux is outgoing from Ω i for i t1, . . . , mu and incoming into Ω j for j tm 1, . . . , m nu).

Lemma 2.4(i) states that each flux G ¦

h is one-sided Lipschitz with respect to u h and monotone with respect to u h I for all h I $ h. Observe that the last equality in [START_REF] Panov | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF] expresses the conservation property at junction. These properties will permit us to formulate a monotone, in the sense of [START_REF] Crandall | Monotone difference approximations for scalar conservation laws[END_REF], conservative finite volume scheme for approximation of admissible solutions of (1).

Lemma 2.8. Let u pu 1 , . . . , u m n q in r0, Rs m n be such that the following family of inequalities holds (41)

d k pk 1 , . . . , k m n q G V V : ∆p u, kq m i1 q i pu i , k i q ¡ m n jm 1 q j pu j , k j q ¥ 0.
Then u is in G V V . Moreover, being understood that the fluxes f h , h t1, . . . , m nu, fulfill the condition (F), the conclusion "

u G V V " still holds if in (41), the condition " k G V V " is replaced by the condition " k G o V V ", where G o V V is the subset of G V V described by (42) G o V V
k pk 1 , . . . , k m n q G V V : hp r0, Rs such that m i1 G i pk i , pq m n jm 1 G j pp, k j q, ds Irk i , psztk i u pp ¡ k i qpf i psq ¡ f i pk i qq ¡ 0, for i 1, . . . , m, ds Irp, k j sztk j u pk j ¡ pqpf j psq ¡ f j pk j qq ¡ 0, for j m 1, . . . , m n D G G G G G G G F G G G G G G G E .
Proof. First, let us prove the result under the assumption (41). Take u pu 1 , . . . , u m n q as initial condition for a Riemann problem at the junction. Then consider the associated solution v pv 1 , . . . , v m n q and the traces γv, as in the proof on Lemma 2.7. We know that γv is in G V V , and therefore by the assumption (41), (43)

m i1 q i pu i , γ i v i q ¡ m n jm 1
q j pu j , γ j v j q ¥ 0.

By construction we have (see the proof of Lemma 2.7) f h pγ

h v h q G h pu h , pq G h pγ h v h , pq if h ¤ m, f h pγ h v h q G h pp, u h q G h pp, γ h v h q if h ¡ m. Moreover, by maximum principle, γ h v h
in between u h and p. This means that (44)

f i pu i q ¡ f i pγ i v i q f i pu i q ¡ G i pu i , pq 5 f i pu i q ¡ min sru i ,ps f i psq ¥ 0 if u i ¤ p, f i pu i q ¡ max srp,u i s f i psq ¤ 0 if u i ¥ p, then f i pu i q ¡ f i pγ i v i q is non negative when u i ¤ γ i v i and non positive when u i ¥ γ i v i . This means that (45) m i1 q i pu i , γ i v i q ¡ m i1 |f i pu i q ¡ f i pγ i v i q|.
In the same way we show that (46) m n jm 1 q j pu j , γ j v j q m n jm 1 |f j pu j q ¡ f j pγ j v j q|.

The sum of non positive terms can be non negative only if all the terms vanish. Therefore for any h t1, . . . , m nu we have f h pu h q f h pγ h v h q G h pu h , pq. In view of [START_REF] Crandall | Monotone difference approximations for scalar conservation laws[END_REF], this shows that v belongs to G V V . Now, let us prove the last claim of the lemma. It is easily seen from the comparison of ( 17) and (42) that due to assumption (F) on the shape of the fluxes, the difference between the subsets G V V and G o V V of r0, Rs m n consists in m n-uplets k for which at least one of the following pm nq events occurs:

pA i q f i pk i q f i ppq and k i p; pB j q f j pk j q f j ppq and k j ¡ p.

Indeed, if, for instance, there holds (47) ds Irk 1 , ps : pp ¡ u 1 qpf 1 psq ¡ f 1 pk 1 qq ¥ 0 and hs 0 Irk 1 , psztk 1 u s.t. f 1 ps 0 q f 1 pk 1 q, then the shape assumption (F) tells us that s 0 p and, moreover, k 1 p.

For the sake of being definite, assume that among pA i q it1,...,mu ,pB j q jtm 1,...,m nu the only event that occurs is pA 1 q, namely, f 1 pk 1 q f 1 ppq, k 1 p but neither pA i q, i t2, . . . , mu nor pB j q, j tm 1, . . . , m nu occur.

Observe that in this case, the pm nq-uplet k I : pp, k 2 , . . . , k m n q belongs to G o V V . Indeed, it belongs to G V V since it corresponds to the same value p, while the event (47) does not occur any more since for k I 1 p, Irk I 1 , psztk I 1 u is empty. Moreover, whatever be u 1 r0, Rs, we have

q 1 pu 1 , k 1 q signpu 1 ¡ k 1 qpf 1 pu 1 q ¡ f 1 pk 1 qq ¥ signpu 1 ¡ pqpf 1 pu 1 q ¡ f 1 ppqq q 1 pu 1 , pq.
Consequently, ∆p u, kq ¥ ∆p u, k I q ¥ 0, where we have used the assumption of the last claim of the lemma and the fact that k

I G o V V .
The general case is fully analogous, so that we find ∆p u, kq ¥ 0 not only for k G o V V , but for all k G V V . We are reduced to the first claim of the lemma. This ends the proof.

2.3.

Reformulation of admissibility in terms of traces at the junction. Now, we are ready to give an alternative formulation of admissibility for (1). To this end, recall (see Section 1.1) that local Kruzhkov entropy solutions admit boundary traces in the strong L 1 sense. Therefore the following definition makes sense. Definition 2.9. Given m n fluxes f h satisfying (F) and an initial condition u 0 in r0, Rs m n , we call ρ pρ 1 , . . . , ρ m n q in L V pR ¢ Γ; r0, Rs m n q a G V V -entropy solution of (1) associated with u 0 if the first item of Definition 1.1 holds, i.e., the Kruzhkov entropy inequalities [START_REF] Bürger | A family of numerical schemes for kinematic flows with discontinuous flux[END_REF] hold for any h t1, . . . , m nu; for a.e. t in R ,the vector γρptq : pγ 1 ρ 1 ptq, . . . , γ m n ρ m n ptqq of traces at the junction belongs to G V V . This formulation will lead to the uniqueness proof. Before turning to the proof of equivalence of Definitions 1.2 and 2.9, we propose another reformulation, which will be useful for proving existence of solutions.

Adapted entropies and another reformulation of admissibility.

Recall that, except very special cases like n m with f h f for all h t1, . . . , 2mu, we cannot expect that constants (seen as k pk, . . . , kq, k r0, Rs) be solutions of (1). Moreover, the above analysis provides us with a wide set of stationary, constant per road solutions associated to k G V V . Therefore, following [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 ¢ 2 conservation laws with L 8 data[END_REF] it is natural to express global (including junction) admissibility in terms of adapted entropy inequalities, where the constants of the Kruzhkov-like formulation proposed in [START_REF] Coclite | Vanishing viscosity for traffic on networks[END_REF] are replaced by the stationary solutions associated with states in G V V . This leads to the following definition. Definition 2.10. Given m n fluxes f h satisfying (F) and an initial condition u 0 in r0, Rs m n , we call ρ pρ 1 , . . . , ρ m n q in L V pR ¢ Γ; r0, Rs m n q an adapted entropy solution of (1) associated with u 0 if The first item of Definitions 1.1, 2.9 holds, i.e., the Kruzhkov entropy inequalities [START_REF] Bürger | A family of numerical schemes for kinematic flows with discontinuous flux[END_REF] hold for any h t1, . . . , m nu. For any k G V V (which should be seen as a road-wise constant solution to the Riemann problem at the junction), ρ satisfies the adapted entropy inequality on the network, namely, for any non negative test function ξ Dps0, Vr¢Rq (48)

m n ḩ1 ¢» R » Ω h t|ρ h ¡ k h |ξ t q h pρ h , k h qξ x u dx dt ¥ 0.
Remark 4. We can consider as a first example the case in which m n 1 and f 1 f 2 f . This example is not totally trivial, as the family of adapted entropies we consider is a priori larger than the family of standard Kruzhkov entropies, because the latter correspond to the case k pk, kq and G V V contains, in addition, all vectors k pk 1 , k 2 q such that k 1 and k 2 are respectively the left and the right state of a stationary Kruzhkov admissible jump. Analogously, in the case m n and f h f for all h t1, . . . , 2mu our definition seems more restrictive than the one given in the paper [START_REF] Coclite | Vanishing viscosity for traffic on networks[END_REF]. In fact, the two approaches give the same result, for reasons similar to those that permit to reduce the second claim of Lemma 2.8 to its first claim.

Remark 5. In the references devoted to the theory of conservation laws with discontinuous flux (case n m 1), see in particular [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF][START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF], the adapted entropy inequality is sometimes written in a differential form, which is equivalent to the integral form (48) for junctions with one incoming and one outgoing road. The integral form (48) is the appropriate expression of adapted entropy inequalities in the case of general junctions.

Equivalence of the three formulations of admissibility.

As we already announced, the three definitions of solution admissibility actually describe one and the same notion.

Theorem 2.11. Definitions 1.2, 2.9 and 2.10 are equivalent. Moreover, in Definition 2.10, the set of adapted entropies can be restricted to k G o V V , without changing the resulting notion of solution.

Proof. We only need to establish equivalence between the second items of the three definitions for functions ρ satisfying the Kruzhkov inequalities [START_REF] Bürger | A family of numerical schemes for kinematic flows with discontinuous flux[END_REF] (which is a common condition for all of them).

We prove that Definition 2.10 (even in the weaker version, where the choice of k for adapted entropy inequalities is restricted to G o V V ) implies Definition 2.9: Let ρ be a solution in the sense of Definition 2.10. We consider a non negative test function ξ DpR q and, for every α ¡ 0,

χ α C V c pRq, such that (49) 0 ¤ χ α ¤ 1, |χ I α | ¤ 1 α , χ α pxq 5 1, if |x| ¤ α, 0, if |x| ¥ 2α.
With the test function ξχ α , inequality (48) becomes (50)

m n ḩ1 » R » Ω h t|ρ h ¡ k h |ξ I ptqχ α pxq q h pρ h , k h qξptqχ I α pxqu dx dt m n ḩ1 » Ω h |u 0 h pxq ¡ k h |ξp0qχ α pxq dx ¥ 0. As α Ñ 0 we get (51) ¡ » R £ m i1 q i pγ i ρ i , k i q ¡ m n jm 1 q j pγ j ρ j , k j q ξptq dt ¥ 0.
Localization then yields, for a.e. t R , ∆p γρptq, kq ¥ 0, in the notation of (41). By Lemma 2.8, (52)

γρptq pγ 1 ρ 1 ptq, . . . , γ m n ρ m n ptqq G V V , a.e. t ¥ 0,
and then ρ is a solution in the sense of Definition 2.9.

Moreover, if we require that adapted entropy inequalities of Definition 2.10 hold only with k G o V V , we still get the same conclusion due to the last claim of Lemma 2.8. This point justifies the last claim of the theorem.

We prove that Definition 2.9 implies Definition 2.10:

Let ρ be a solution in the sense of Definition 2.9. For every non negative test function ξ Dps0, Vr¢Rq, we get (53)

m n ḩ1 » R » Ω h t|ρ h ¡ k h |ξ t pq h pρ h , k h qq ξ x u dx dt ¡ » R ¢ m i1 q i pγ i ρ i ptq, k i q ¡ m n j1 q j pγ j ρ j ptq, k j q ξpt, 0q dt ¥ 0. Since (54) γρptq pγ 1 ρ 1 ptq, . . . , γ m n ρ m n ptqq G V V , a.e. t ¥ 0, we have (55) m i1 q i pγ i ρ i ptq, k i q ¥ m n j1 q j pγ j ρ j ptq, k j q, a.e. t ¥ 0.
Therefore, ρ is a solution in the sense of Definition 2.10.

We prove that Definition 2.9 implies Definition 1.2:

Let ρ be a solution in the sense of Definition 2.9. Since (56)

γρptq pγ 1 ρ 1 ptq, . . . , γ m n ρ m n ptqq G V V , a.e. t ¥ 0, there exists p L V pR , r0, Rsq such that (15) holds for a.e. t ¥ 0. Let us point out that p is indeed measurable. This results, first, from the measurability of the trace vector γρ : R Ñ r0, Rs m n ; and second, given γρptq, from the definition of pptq by equation ( 15) for which we can systematically take the smallest solution (cf. the proof of Lemma 2.4). Then ρ is a solution in the sense of Definition 1.2.

Finally, Definition 1.2 implies Definition 2.9. This follows from the definition of G V V in [START_REF] Crandall | Monotone difference approximations for scalar conservation laws[END_REF], in view of ( 12),(13).

Well-posedness of (1) in the frame of admissible solutions

The goal of this section is to prove the following result. Theorem 3.1. For any given initial condition u 0 pu 0 1 , . . . , u 0 m n q in L V pΓ; R m n q the problem (1) admits one and only one solution ρ in L V pR ¢ Γ; r0, Rs m n q in the sense of the equivalent Definitions 1.2, 2.9, 2.10.

Moreover, such solutions depend continuously on the initial data in the localized L 1 sense. If ρ and ρ are the admissible solutions corresponding respectively to the initial conditions u 0 and v 0 , then for all M ¡ 0 and t M {L, where

L maxt}f I h } L V pr0,Rs;Rq | h 1, . . . , m nu, ( 57
) m i1 » 0 ¡pM¡Ltq |ρ i pt, xq ¡ ρi pt, xq| dx m n jm 1 » M ¡Lt 0 |ρ j pt, xq ¡ ρj pt, xq| dx ¤ m i1 » 0 ¡M |u 0 i pxq ¡ v 0 i pxq| dx m n jm 1 » M 0 |u 0 j pxq ¡ v 0 j pxq| dx.
In particular, the map that associates to u 0 the unique corresponding admissible profile ρptq, is non-expansive w.r.t. the L 1 distance for all t ¡ 0.

We justify the uniqueness of a solution admissible in the sense of Definitions 1.2, 2.9, 2.10 using Definition 2.9. The existence of an admissible solution (which is also justified in Section 4 using the more technical vanishing viscosity method) is proved using Definition 2.10, on the basis of a straightforward finite volume approximation with Godunov fluxes (including the Godunov fluxes at the junction discussed in Remark 3). [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF]. The goal of this section is to prove the following stability result Proposition 3.1. For any given initial condition u 0 pu 0 1 , . . . , u 0 m n q in L V pΓ; R m n q there exists at most one G V V -entropy solution ρ in L V pR ¢ Γ; r0, Rs m n q.

Uniqueness of admissible solutions of

Moreover, if ρ and ρ are the G V V -entropy solutions corresponding respectively to the initial conditions u 0 and v 0 , (57) holds and in particular, the L 1 -contraction estimate holds whenever the right-hand side is finite:

(58) m i1 }ρ i ptq ¡ ρi ptq} L 1 pR ¡ ;Rq m n jm 1 }ρ j ptq ¡ ρj ptq} L 1 pR ;Rq ¤ m i1 u 0 i ¡ v 0 i L 1 pR ¡ ;Rq m n jm 1 u 0 j ¡ v 0 j L 1 pR ;Rq .
Proof. The first part of the proof is devoted to establish a Kato's type inequality for G V V -entropy solutions.

Let ξ be a test function in DpR ¢ R; R q and define, for k ¡ 0, (59)

ξ k pt, xq ξpt, xq min 4 1, p|x| ¡ kq k B .
Then, as ρ h and ρh are Kruzhkov entropy weak solutions in the interior of Ω h , by a standard doubling of variable argument we get (60)

m n ḩ1 ¢ ¡ » R » Ω h t|ρ h ¡ ρh |ξ k,t q h pρ h , ρh qξ k,x u dx dt ¡ » Ω h |u 0 h pxq ¡ v 0 h pxq|ξ k p0, xq dx ¤ 0.
An explicit computation shows that (61)

ξ k,x pt, xq ξ x pt, xq min 4 1, p|x| ¡ kq k B 1 k ξpt, xqp1 sk,2kr pxq ¡ 1 s¡2k,¡kr pxqq, then (60) rewrites as
(62)

m n ḩ1 ¢ ¡ » R » Ω h 4 |ρ h ¡ ρh |ξ k,t q h pρ h , ρh qξ x min 4 1, p|x| ¡ kq k BB dx dt m n ḩ1 1 k ¢» R » Ω h s¡2k,¡kr q h pρ h , ρh qξ dx dt ¡ » R » Ω h sk,2kr q h pρ h , ρh qξ dx dt m n ḩ1 ¢ ¡ » Ω h |u 0 h pxq ¡ v 0 h pxq|ξ k p0, xq dx ¤ 0.
We have that

(63) lim kÑ0 m n ḩ1 1 k ¢» R » Ω h s¡2k,¡kr q h pρ h , ρh qξ dx dt ¡ » R » Ω h sk,2kr q h pρ h , ρh qξ dx dt » R £ m i1 q i pγ i ρ i , γ i ρi q ¡ m n jm 1
q j pγ j ρ j , γ j ρj q ξpt, 0q dt, and this term is positive due to the positivity assumption on ξ and the fact that γρ and γ ρ are in G V V . Therefore by taking the limit as k tends to 0 of the whole left hand side in (62) one can conclude that (64)

m n ḩ1 ¢ ¡ » R » Ω h t|ρ h ¡ ρh |ξ t q h pρ h , ρh qξ x u dx dt ¡ » Ω h |u 0 h pxq ¡ v 0 h pxq|ξp0, xq dx ¤ 0,
which is Kato's inequality adapted to our setting. The conclusion of the proof is classical, following [START_REF] Kružhkov | First order quasilinear equations with several independent variables[END_REF]. One takes ξpt, xq approximating the characteristic function of the trapezoid tpt, xq R ¢ R; |x| ¤ M Ltu where L max ht1,...,m nu max ur0,Rs |f I h puq| is the appropriate Lipschitz constant and M is a nonnegative parameter. For a.e. fixed t R , this yields for M ¡ Lt the inequality (57).

As M Ñ V, we find (58) as soon as its right-hand side is finite.

A finite volume numerical scheme and existence of an admissible solution.

In this section we describe a finite volume numerical scheme for the junction based on Godunov fluxes. Discretizing a fixed initial datum, we prove convergence of the discrete solutions to the unique admissible solution. We stress that there exist other numerical schemes which can be used in practice to approximate admissible solutions of (1): we refer to [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF] for the case m n 1. The choice of Godunov's fluxes is motivated by the fact that this scheme is wellbalanced, i.e., all admissible stationary solutions are scheme's exact solutions. The proofs of our theoretical results are easier in this setting. For other schemes based on monotone numerical fluxes (e.g., Rusanov flux), convergence to an admissible solution can also be proved, but the stationary solutions should replaced by numerical profiles (cf. Section 4 where analogous viscous profiles are constructed).

We fix a space step ∆x. For Z and h t1, . . . , m nu, set x h : ∆x. We consider the uniform spatial mesh on each

Ω h ¤ ¤¡1 px i , x i 1 q on Ω i , for i ¤ m, ( 65
) ¤ ¥0 px j , x j 1 q on Ω j , for j ¥ m 1, (66)
so that the position of the junction x 0 on the road Ω h corresponds to x h 0 . Then we fix a time step ∆t satisfying the CFL condition

(67) ∆t ¤ max h tL h u 2 ∆x,
where

L h is the Lipschitz constant of f h .
At all cell interfaces x h for $ 0 we consider the standard Godunov flux G h corresponding to the flux f h . At the junction x h 0 we take on each road Ω h the Godunov flux G ¦ h corresponding to the admissible solution of the Riemann problem at the junction, see Remark 3. More precisely, let u 0 pu 0 1 , . . . , u 0 m n q be an initial condition in L V pΓ; r0, Rs m n q. We initialize our scheme by discretizing the initial conditions (68)

u h,0 k 1 2 1 ∆x » x h 1 x h u 0 h pxq dx, for all h 1, . . . , m n and for ¤ ¡1 if h ¤ m, ¥ 0 if h ¥ m 1. Then ¡ u h,s 1 1 2 © h, is obtained from ¡ u h,s
G i pu i,s ¡ 1 2 , p s q ¡ m n jm 1 G j pp s , u j,s 1 2 q 0, i.e., setting (70) u ¦,s : ¡ u 1,s ¡ 1 2 , . . . , u m,s ¡ 1 2 , u m 1,s 1 2 , u m n,s 1 2
© we take p s : p u ¦,s according to the notation of Lemma 2.3. (II) We compute (71)

u h,s 1 1 2 u h,s 1 2 ¡ ∆t ∆x ¡ F h,s 1 ¡ F h,s © , where (72)
G h pu h,s ¡ 1 2 , u h,s 1 2 q if h ¤ m and ¤ ¡1 or h ¥ m 1 and ¥ 1, G h pu h,s ¡ 1 2 , p s q G ¦ h p u ¦,s q if h ¤ m and 0, G h pp s , u h,s 1 2 q G ¦ h p u ¦,s q if h ¥ m 1 and 0.
The stage (II) is a standard marching scheme, up to the specific definition of the fluxes for 0. The stage (I) is implicit: once per time step, we have to find a zero of a scalar nonlinear function. Moreover, the nonlinear term in (I) is monotone and continuous (see Lemma 2.3 and its proof) but it is not everywhere differentiable, since G h are not everywhere differentiable. In practice, the value p s can be efficiently computed using the regula falsi method, cf. [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF]. We introduce the notation

Γ discr ¡ t1, . . . , mu ¢ t Z, ¤ ¡1u © ¤ ¡ tm 1, . . . , m nu ¢ t Z, ¥ 0u © ,
for the set of all degrees of freedom at a fixed time step; the notation

U s ¡ u h,s 1 2 © ph, qΓ discr
for the set of all the unknowns of the scheme at time step s; and the notation S ∆x u 0 to indicate the piecewise constant function corresponding to the discrete solution

¡ u h,s 1 2
© h, ,s pU s q s : S ∆x u 0 pu 1,∆x , . . . , u m,∆x , u m 1,∆x , . . . , u m n,∆x q where (73)

u i,∆x şN, ¤¡1 u i,s 1 2
1 Ω i s , 1r pxq1 ss,s 1r ptq, i t1, . . . , mu, u j,∆x şN, ¥0

u j,s 1 2
1 Ω j s , 1r pxq1 ss,s 1r ptq, j tm 1, . . . , m nu.

Lemma 3.2. The above finite volume numerical scheme is well-balanced, i.e., every element of G V V corresponds to a stationary, constant per road solution of the scheme.

Proof. Let k G V V and denote by k ∆x the associated discrete function with entries u h,∆x k h , h t1, . . . , m nu. Then S ∆x k ∆x k ∆x , i.e., k ∆x is a stationary solution of the scheme.

Indeed, consider for instance an incoming road i. Obviously, for all ¡1 the first iteration of (71) initialized with constant initial values u i,0

1 2 k i yields u i,1 1 2 k i ¡ ∆t ∆x pG i pk i , k i q ¡ G i pk i , k i qq k i .
Moreover, due to (69), by definition of G ¦ i and of G V V we have G i pk i , k i q f pk i q Gpk i , p k q G ¦ i p kq, so that for ¡1 we still find

u i 1,¡ 1 2 k i ¡ ∆t ∆x pG i pk i , k i q ¡ G ¦ i p kqq k i .
Similarly, we find that (69), (71) preserves the constant value k j on the jth outgoing road.

Remark 6. Due to the classical properties of Godunov fluxes recalled in § 1.1.1, to the CFL condition 67 and to Lemma 2.4(i) (see also Remark 3), the scheme is monotone in the following sense:

(74)

ds N dph, q Γ discr : u h,s 1

1 2 H h 1 2 pU s q for some functions H h 1 2
that are monotone non-decreasing with respect to each of the arguments (actually each of these functions depends on at most pm n 1q entries of U s ). This implies in particular the order-preservation property

(75) dph, q Γ discr : u h,s 1 2 ¥ ûh,s 1 2 ñ dph, q Γ discr : u h,s 1 1 2 ¥ ûh,s 1 1 2
.

Since, moreover, the scheme is locally conservative by definition (due, in particular, to the condition (69)), and because of the Lipschitz continuity of G h for all h, the scheme is conservative. It follows by the Crandall-Tartar Lemma (see, e.g., [START_REF] Crandall | Monotone difference approximations for scalar conservation laws[END_REF]) that the scheme is L 1 in the sense that the discrete analogue of (58) (with ρ replaced by S ∆x u 0 ) and the initial condition u 0 replaced by the discretized initial condition) holds true. We need a bit more specific property, which is the numerical counterpart of the Kato inequalities (64), in order to justify convergence to an admissible solution.

Slightly extending the usual formalism (see [START_REF] Eymard | Finite volume methods[END_REF]), let t, resp., u denote (component per component, in the case of vector-valued arguments) the maximum, resp. the minimum operation on real scalars, vectors or sequences: e.g., pk 1 , k 2 , k 3 qup k1 , k2 , k3 q pmintk 1 , k1 u, mintk 2 , k2 u, mintk 3 , k3 uq. Proposition 3.2. For any initial conditions u 0 , ˆ u 0 in L V pΓ; r0, Rs m n q the corresponding discrete solutions of the scheme (68)-(72) satisfy discrete Kato inequalities. Namely, for ph, q Γ discr , let Q h rU s , Û s s be defined by

Q h rU s , Û s s G h ¡ u h,s ¡ 1 2 tû h,s ¡ 1 2 , u h,s 1 2 tû h,s 1 2 © ¡ G h ¡ u h,s ¡ 1 2 uû h,s ¡ 1 2 , u h,s 1 2 uû h,s 1 2 © , $ 0, Q h 0 rU s , Û s s G ¦ h p u ¦,s t ˆ u ¦,s q ¡ G ¦ h p u ¦,s u ˆ u ¦,s q.
Then for all ξ Dps0, Vr¢Rq such that ξ ¥ 0 and f x ξ 0 on r¡∆x{2, ∆x{2s, setting

ξ s 1 2 : ξps∆t, p 1 2 q∆xq we have ¡ m i1 V ş1 ∆t ¸ ¤¡1 ∆x|u i,s 1 2 ¡ ûi,s 1 2 | ξ s 1 1 2 ¡ ξ s 1 2 ∆t ¡ m i1 V ş1 ∆t ¸ ¤¡1 ∆xQ i rU s , Û s s ξ s 1 1 2 ¡ ξ s 1 ¡ 1 2 ∆x ¡ m n jm 1 V ş1 ∆t ¸ ¥0 ∆x|u j,s 1 2 ¡ ûj,s 1 2 | ξ s 1 1 2 ¡ ξ s 1 2 ∆t ¡ m n jm 1 V ş1 ∆t ¸ ¥1 ∆xQ j rU s , Û s s ξ s 1 1 2 ¡ ξ s 1 ¡ 1 2 ∆x ¤ 0.
Observe that in Proposition 3.2, we limit our attention to test functions constant in a neighbourhood of the junction. Thanks to this precaution, borrowed from [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF], and to the conservativity of the Riemann solver at the junction the junction 0 does not contribute to the "QrU, Û sf x ξ" term of the discrete Kato inequality.

Proof. The argument is essentially classical in the context of monotone finite volume schemes. First, we state the "per cell contraction principle": for all ph, q Γ discr (76)

|u h,s 1 1 2 ¡ ûh,s 1 1 2 | ¡ |u h,s 1 2 ¡ ûh,s 1 2 | ∆t Q h 1 rU s , Û s s ¡ Q h rU s , Û s s ∆x ¤ 0,
which readily follows from the observation that for all a, b R, |a ¡ b| atb ¡ aub and from the monotonicity of H h 1 2 in (74):

|u h,s 1 1 2 ¡ ûh,s 1 1 2 | u h,s 1 1 2 tû h,s 1 1 2 ¡ u h,s 1 1 2 uû h,s 1 1 2 , u h,s 1 1 2 tû h,s 1 1 2 H h 1 2 pU s qtH h 1 2 p Û s q ¤ H h 1 2 pU s t Û s q, u h,s 1 1 2 uû h,s 1 1 2 H h 1 2 pU s quH h 1 2 p Û s q ¥ H h 1 2 pU s u Û s q, H h 1 2 pU s t Û s q ¡ H h 1 2 pU s u Û s q |u h,s 1 2 ¡ ûh,s 1 2 | ¡ ∆t ∆x ¡ Q h 1 rU s , Û s s ¡ Q h rU s , Û s s © ,
where the formula (71) and the definition of Q h rU s , Û s s are used in the last line to express the function H h

m i1 G ¦ h p kt ˆ kq ¡ m n jm 1 G ¦ h p kt ˆ kq 0 m i1 G ¦ h p ku ˆ kq ¡ m n jm 1 G ¦ h p ku ˆ kq.
It remains to multiply the ph, q's inequality in (76) by the nonnegative quantity ∆t∆xξ s 1 1 2 and sum up; the sum is finite since the support of ξ is compact. Paying attention to the fact that

ξ s ¡ 1 2 ξps∆t, 0q ξ s 1 2
, from the definition of Q h 0 rU s , Û s s we see that for all s, (77)

m i1 ∆xQ i 0 rU s , Û s sξ s 1 ¡ 1 2 ¡ m n jm 1 ∆xQ j 0 rU s , Û s sξ s 1 1 2 0.
Using the Abel transformation (discrete summation by parts) on each road with respect to the time superscripts s and to the space subscripts , bearing in mind (77) and the fact that for all Z, ξ 0 1 2 0 by the choice of ξ, we derive the required discrete Kato inequality. Theorem 3.3. Given an initial datum u 0 L V pΓ; r0, Rs m n q, the numerical scheme (68)-(72) converges to the unique admissible (in the sense of the equivalent Definitions 1.2, 2.9, 2.10) solution ρ of (1), namely, S ∆x u 0 Ñ ρ as ∆x 0, subject to the CFL restriction (67) on ∆t. This ensures, in particular, existence of an admissible solution of (1) for every L V pΓ; r0, Rs m n q initial datum.

Proof. The proof follows the lines of the proof given in [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF] (see also [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF]). Let us only provide a sketch of the key arguments.

We start with compactly supported BV initial data. The compactness, in the sense of the a.e. convergence, of the family

¡ S ∆x u 0 © ∆xp0,1q
of discrete solutions is obtained with the BV loc technique introduced in [START_REF] Bürger | A family of numerical schemes for kinematic flows with discontinuous flux[END_REF]. It relies upon the monotonicity and the Crandall-Tartar lemma (see Remark 6).

The compactness permits to define ρ as a limit of some sequence of discrete solutions S ∆xr u 0 , ∆ r Ñ 0. At the end of the proof, having proved convergence to the unique admissible solution with datum u 0 , by the classical argument we can bypass the extraction of a sequence and get convergence of S ∆x u 0 , ∆x Ñ 0. It is classical (see [START_REF] Eymard | Finite volume methods[END_REF]) to derive that for every h t1, . . . , m nu, ρ h fulfills the first property required in Definitions 1.2, 2.9, 2.10, namely, the Kruzhkov entropy inequalities (11) hold. In order to justify that ρ is an admissible solution, we just need to assess the second property in Definition 2.10, i.e., the adapted entropy inequalities (48) that involve the junction.

The main ingredient of the proof is the discrete Kato inequality proved in Proposition 3.2, where we choose ˆ u 0 k with k G V V . Observe that by Lemma 3.2, we have S ∆xr ˆ u 0 k; passing to the limit as ∆ r Ñ 0, we will indeed derive the adapted entropy inequalities (48) for ρ and complete the proof for compactly supported data of bounded variation.

Following [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF], observe that test functions whose x-derivative vanishes near x 0 are dense (e.g., in the C 1 topology) in Dps0, Vr¢Rq. Starting with the inequalities of Proposition 3.2, with the same arguments as in the previous step (see [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF][START_REF] Eymard | Finite volume methods[END_REF]) we pass to the limit and find (48) first for such specific test functions ξ, and then (by density) for general test functions.

Finally, as in [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF][START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF], in two steps we extend the convergence result to general L V data u 0 . Extension to L 1 L V data follows by the density of compactly supported BV data in L 1 topology, with the help of L 1 contractivity of both the admissible solution semigroup and the discrete solutions semigroups. Extension to L V data is due to the property of finite domain of dependence (57) and to its discrete counterpart that follows from (71) and from the CFL condition.

Vanishing viscosity limits are admissible solutions of (1)

This section is devoted to the proof that the solutions obtained as limit of vanishing viscosity approximations are admissible solutions in the sense of Definitions 1.2, 2.9, 2.10, and are therefore unique. The result follows from the combination of two ingredients: the construction of suitably many vanishing viscosity profiles, and the L 1 contraction property known for the vanishing viscosity approximation. This ensures that vanishing viscosity solutions satisfy a family of adapted entropy inequalities which is sufficiently large to fit the last claim of Theorem 2.11. Proposition 4.1. For any k in G o V V , there exists p r0, Rs and ρ ε pρ ε 1 , . . . , ρ ε m n q in L V pΓ; r0, Rs m n q such that for all h t1, . . . , m nu, ρ ε h solves the ODE problem (78)

6 9 8 9 7 f h pρ ε h q x εpρ ε h q xx , in Ω h ρ ε h p0q p, lim xΩ h , |x|Ñ V ρ ε h pxq k h . Proof.
We take the value p r0, Rs that ensures that k G o V V , according to the definition (42) of G o V V . We consider here the case h ¤ m, the other case being analogous. Let us integrate both sides of (78) on s ¡ V, xs (79)

εpρ ε i q x pxq F i pρ ε i pxqq, where F i : ρ r0, Rs Þ Ñ f i pρq ¡ f i pk i q
, being understood that one should have pρ ε i q x Ñ 0 as x Ñ ¡V. Observe that by the definition of G o V V , k i is the only zero of F i on Irk i , ps. Assume, for the sake of being definite, that k i p (the case k i ¡ p is analogous, while in the case k i p we have the obvious solution ρ ε i k).

Then, the map

P :sk i , ps Ñs ¡ V, 0s, r Þ Ñ ¡ » p r ε f i psq ¡ f i pk i q ds is
well defined and strictly increasing, i.e., it admits the inverse P ¡1 satisfying P ¡1 p0q p, lim xÑ¡V P ¡1 pxq k i , lim xÑ¡V pP ¡1 q I pxq 0. We find that ρ ε i solves (79) if and only if (80)

¡ x » 0 x ερ ε i,x f i pρ ε i q ¡ f i pk i q dx ¡Ppρ ε i q,
i.e. ρ ε i P ¡1 pxq is the required solution.

Following [START_REF] Coclite | Vanishing viscosity for traffic on networks[END_REF], we consider the following parabolic regularization of the initial boundary value problem (1) (81) 6 9 9 9 9 9 9 9 9 9 8 9 9 9 9 9 9 9 9 9 7

ρ ε h,t f pρ ε h q x ερ ε h,xx , t ¡ 0, x Ω h , h t1, ..., m nu, ρ ε h pt, 0q ρ ε h Ipt, 0q, t ¡ 0, h, h I t1, ..., m nu, m °i1 f pρ ε i pt, 0qq ¡ ερ ε i,x pt, 0q m n °jm 1 f pρ ε j pt, 0qq ¡ ερ ε j,x pt, 0q ¨, t ¡ 0, ρ ε h p0, xq u 0 h,ε pxq,
x Ω h , h t1, ..., m nu, where ε ¡ 0. Note that, in the spirit of (2), the third and fourth lines of (81) give the mass conservation at the junction, since the sum of the incoming parabolic fluxes is equal to the sum of the outgoing parabolic ones. On the approximated initial conditions we assume that

u 0 h,ε W 2,1 pΩ h q C V pΩ h q, 0 ¤ u 0 h,ε ¤ R, u 0 h,ε ÝÑ u 0 h ,
a.e. and in L p pΩ h q, 1 ¤ p V, as ε Ñ 0,

u 0 h,ε L 1 pΩ h q ¤ u 0 h L 1 pΩ h q , pu 0 h,ε q x L 1 pΩ h q ¤ T V pu 0 h q, ε pu 0 h,ε q xx L 1 pΩ h q ¤ C 0 , ( 82
)
for each ε ¡ 0, h t1, ..., m nu, where C 0 is a positive constant independent on ε, h.

First, notice that profiles constructed in Proposition 4.1 are solutions of (81). The obvious scaling property of these profiles, that we will denote k ε in the sequel, ensures the convergence of k ε pxq k 1 p x ε q Ñ k as ε Ñ 0, for all x $ 0. This readily yields a wide family of vanishing viscosity limits.

Corollary 1. Any k G o

V V can be obtained as the limit in the L 1 loc sense, as ε Ñ 0, of a family k ε of stationary solutions of (81).

In general, using the theory of semigroups the authors of [START_REF] Coclite | Vanishing viscosity for traffic on networks[END_REF] proved the existence of a unique solution ρ ε of (81) such that (83) ρ ε h Cpr0, Vr; L 2 pΩ h qq L 1 loc ps0, Vr; W 2,1 pΩ h qq, ε ¡ 0, h t1, . . . , m nu, in particular (84) pρ ε h q t L 1 loc ps0, Vr, L 1 pΩ h qq, ε ¡ 0, h t1, . . . , m nu.

Moreover, if we have two different initial conditions pr u 0 1,ε , . . . , r u 0 m n,ε q, pu 0 1,ε , . . . , u 0 m n,ε q for (81) satisfying (82), then the corresponding solutions to (81) pr ρ ε 1 , . . . , r ρ ε m n q, pρ ε 1 , . . . , ρ ε m n q are stable in the following sense

m n ḩ1 }ρ ε h pt, ¤q ¡ r ρ ε pt, ¤q} L 1 pΩ h q ¤ m n ḩ1 u 0 h,ε ¡ r u 0 h,ε L 1 pΩ h q , ( 85
)
for every t ¥ 0.

The compactness argument of [START_REF] Coclite | Vanishing viscosity for traffic on networks[END_REF] is based on the compensated compactness theory [START_REF] Tartar | Nonlinear analysis and mechanics: Heriot-watt symposium[END_REF] and the following a priori estimates T V pu 0 h q, (89) for every t ¥ 0 and ε ¡ 0. The main result in [START_REF] Coclite | Vanishing viscosity for traffic on networks[END_REF] shows that there exist a sequence tε u N p0, Vq, ε Ñ 0 and a solution ρ of (1), in the sense of Definition 1.1, such that ρ ε h ÝÑ ρ h , a.e. and in L p loc pR ¢ Ω h q, 1 ¤ p V,

0 ¤ ρ ε h ¤ R,
for every h t1, . . . , m nu, where ρ ε is the corresponding solution of (81).

Remark 7. Actually these results were proved in [START_REF] Coclite | Vanishing viscosity for traffic on networks[END_REF] in the case there all the functions f h coincide, h 1, . . . , m n, and moreover the strict concavity of the flux function is assumed. Extension to different fluxes f h on different roads is straightforward. The strict concavity assumption can be replaced by the nonlinearity assumption pNLDq: e.g., the strong precompactness result of [START_REF] Panov | On sequences of measure-valued solutions of a first-order quasilinear equation[END_REF] can be used on each road in the place of the compensated compactness method.

Here we improve the result of [START_REF] Coclite | Vanishing viscosity for traffic on networks[END_REF] showing the following:

Theorem 4.1. Assume (82). Let t ρ ε u ε¡0 be the family of solutions of (81). We have that (91) ρ ε h ÝÑ ρ h , a.e. and in L p loc pR ¢ Ω h q, 1 ¤ p V, where ρ is the unique admissible solution of (1) in the sense of Definitions 1.2, 2.9, 2.10.

Proof. Let t ρ ε u N be the solutions of (81) converging to ρ as in (90). According to Theorem 2.11, it is enough to justify that ρ satisfies the per road Kruzhkov inequalities [START_REF] Bürger | A family of numerical schemes for kinematic flows with discontinuous flux[END_REF] for all k r0, Rs and the adapted entropy inequalities (48) for all k G o V V . The first claim is classical, see, e.g., [START_REF] Coclite | Vanishing viscosity for traffic on networks[END_REF]. We only need to justify the second claim.

According to Corollary 1, given k G o V V there exist k ε stationary viscosity profiles converging to k, as ε Ñ 0. Now, arguing as in [13, p. 1773] but inserting in addition a non-negative test function ξ Dps0, Vr¢Rq, we find the following Kato inequality:

(92)

m n ḩ1 ¢» R » Ω h t|ρ ε h ¡ k ε h |ξ t q h pρ ε h , k ε h qξ x ε |ρ ε h ¡ k ε h | x ξ x u dx dt ¥ 0.
Let us give the details of the calculation. Bearing in mind the regularity (83),(84) of solutions and the Lipschitz regularity of f h , it is a classical matter to obtain the per road Kato inequalities:

(93

) » R » Ω h p|ρ ε h ¡ k ε h |ξ t q h pρ ε h , k ε h qξ x ε |ρ ε h ¡ k ε h | x ξ
x q dx dt ¥ 0, h t1, . . . , m nu, for all ξ Dps0, Vr¢pRzt0uqq. Moreover, bearing in mind the existence of strong traces of q h pρ ε h , k ε h q ε |ρ ε h ¡k ε h | x as x Ñ 0, using the truncations defined by (59), with the same argument as in the proof of Proposition 3.1 we can generalize (93) to test functions ξ not necessarily vanishing near the junction; the appropriate boundary terms appear. Summing up the resulting inequalities, and using the conservativity conditions contained in (81) both for solutions ρ ε and k ε , we find

0 ¥ ¡ m n ḩ1 » R » Ω h p|ρ ε h ¡ k ε h |ξ t q h pρ ε h , k ε h qξ x ε |ρ ε h ¡ k ε h | x ξ x q dx dt m ḩ1 » R pq h pρ ε h pt, 0q, k ε h qξpt, 0q ¡ ε |ρ ε h pt, 0q ¡ k ε h | x ξpt, 0qq dt ¡ m n ḩm 1 » R pq h pρ ε h pt, 0q, k ε h qξpt, 0q ¡ ε |ρ ε h pt, 0q ¡ k ε h | x ξpt, 0qq dt ¡ m n ḩ1 » R » Ω h p|ρ ε h ¡ k ε h |ξ t q h pρ ε h , k ε h qξ x ε |ρ ε h ¡ k ε h | x ξ x q dx dt » R sign pρ ε h pt, 0q ¡ k ε h q £ m i1 pf i pρ ε i pt, 0qq ¡ ε ρ ε i,x pt, 0qq ¡ m n jm 1 pf j pρ ε j pt, 0qq ¡ ε ρ ε j,x pt, 0qq ξpt, 0q dt ¡ £ m i1 f i pk ε h q ¡ m n jm 1 f j pk ε j q » R sign pρ ε h pt, 0q ¡ k ε h q ξpt, 0q dt ¡ m n ḩ1 » R » Ω h p|ρ ε h ¡ k ε h |ξ t q h pρ ε h , k ε h qξ x ε |ρ ε h ¡ k ε h | x ξ x q dx dt.
Passing to the limit in (92) as ε Ñ 0, keeping in mind the second term of estimate (88), we find inequality (48). This concludes the proof.

Lemma 2 . 2 .

 22 The vanishing viscosity germ G V V coincides with the subset of r0, Rs m n defined by

  find p s in r0, Rs s.t. m i1

1 2 .

 12 Second, observe that for all k, ˆ k we have by the conservativity property underlying the definition of G ¦ (see Remark 3):

  It is important to notice that the vector w does not belong (in general) to G V V . From the proof of Lemma 2.4 (see (29)) we deduce that p w ¤ mintp 1 , p 2 u where we denoted p

	1 h , k 2 h u.

s : p k s , s 1, 2. We have, using the notation (

23

) and the fact that w i k 1 i for i t1, . . . , mu and w j k 2 j for j tm 1, . . . , m nu, m i1

  Then we consider the h initial boundary value problems

	with constant data	6 9 8
	(38)	9 7

  h t1, . . . , m nu,L 2 pΩ h q 2|n ¡ m| max h }f h } W 1,V p0,Rq t, 1 pΩ h q ¤ pm nqC 0 max h }f I h } L V p0,Rq

			¤	m n ḩ1	h,ε u 0	2
		m n ḩ1	ρ ε h,t pt, ¤q						m n ḩ1
	(86)
	(87)	m n ḩ1	}ρ ε h pt, ¤q} L 1 pΩ h q ¤	m n ḩ1	u 0 h	L 1 pΩ h q ,
	(88)	m n ḩ1	}ρ ε h pt, ¤q} 2 L 2 pΩ h q 2ε	» t 0	£ m n ḩ1	ρ ε h,x ps, ¤q	2 L 2 pΩ h q	ds

L

The second author is member of the Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The work on this paper was supported by the French ANR project CoToCoLa. The third author also acknowlegdes support from the Université de Franche-Comté (projet support EC 2014) and from the Région Franche-Comté (projet mobilité sortante a.a. 2015-16). She would like to thank the Mathematics' Department of the University of Padova for the warm hospitality during the a.y. 2015-16.