Giuseppe M Coclite

Carlotta Donadello

VANISHING VISCOSITY ON A STAR-SHAPED GRAPH UNDER GENERAL TRANSMISSION CONDITIONS AT THE NODE

Keywords: 2010 Mathematics Subject Classification. 35L65, 35R02 . conservation laws, vanishing viscosity, networks, transmission conditions at nodes. 1

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Introduction

We consider a family of scalar conservation laws defined on an oriented graph Γ consisting of m incoming and n outgoing edges Ω , = 1, . . . m + n joining at a single vertex. Incoming edges are parametrized by x ∈ (-∞, 0] while outgoing edges by x ∈ [0, ∞) in such a way that the junction is always located at x = 0. We use the index i, i = 1, . . . , m, to refer to incoming edges and j, j = m + 1, . . . , m + n for the outgoing ones.

On the edge Ω we introduce a scalar conservation law, describing the evolution of a density ρ . Then on the incoming edges we have eq:basicL eq:basicL (1.1)

∂ t ρ i + ∂ x f i (ρ i ) = 0, t > 0, x < 0, i = 1, ..., m,
and on the outgoing ones eq:basicR eq:basicR (1.2) ∂ t ρ j + ∂ x f j (ρ j ) = 0, t > 0, x > 0, j = m + 1, ..., m + n.

The fluxes f 1 , ..., f m+n , differ in general, however we assume that they are bell-shaped (unimodal), Lipschitz and non-degenerate nonlinear, i.e.

ass:f (H.1) for each ∈ {1, ..., m + n}, f ∈ C 2 ([0, 1]), f (0) = f (1) = 0, f ≥ 0, and there exist ρ ∈ (0, 1) such that f (ρ) (ρ -ρ) > 0 for every ρ ∈ [0, 1] \ {ρ };

ass:f-gn (H.2) for any ∈ {1, ..., m + n}, |{ρ : f (ρ) = 0}| = 0. We augment (1.1) and (1.2) with the initial conditions eq:init eq:init (1.3) ρ i (0, x) = ρ i,0 (x), x < 0, i = 1, ..., m, ρ j (0, x) = ρ j,0 (x),

x > 0, j = m + 1, ..., m + n, assuming that (H.3) ass:init ρ 1,0 , ..., ρ m,0 ∈ L 1 (-∞, 0) ∩ BV (-∞, 0), ρ m+1,0 , ..., ρ m+n,0 ∈ L 1 (0, ∞) ∩ BV (0, ∞) and 0 ≤ ρ 1,0 , ..., ρ m+n,0 ≤ 1. Finally, we introduce the necessary conservation assumption at the node, which transforms our family of independent equations into a single problem m i=1 f i (ρ i (t, 0-)) = m+n j=m+1 f j (ρ j (t, 0+)) for a.e. t ≥ 0. Questions related to existence, uniqueness and stability of solutions for problems of this kind have been extensively investigated in recent years, mainly in relation with traffic modeling. The interested reader can refer to [START_REF] Bressan | Flows on networks: recent results and perspectives[END_REF][START_REF] Garavello | Traffic flow on networks[END_REF] for an overview of the subject. Here our point of view is different, as we do not focus on a specific model. We consider a parabolic regularization of the problem, similarly to what has been done in [START_REF] Coclite | Vanishing viscosity for traffic on networks with degenerate diffusivity[END_REF][START_REF] Coclite | Vanishing viscosity for traffic on networks[END_REF], but instead of enforcing a continuity condition at the node for the regularized solutions, we introduce a more general set of transmission conditions on the parabolic fluxes.

In this work we adopt the following definition of weak solution for the problem (1.1), (1.2), and (1.3). We stress that this definition is for sure not sufficient to ensure uniqueness. On the contrary it fix somehow a minimal set of properties that any reasonable solution is expected to satisfy, see [START_REF] Andreianov | Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network[END_REF] and references therein for a more detailed discussion on this point.

:weaksol Definition 1.1. Let ρ 1 , ..., ρ m : [0, ∞) × (-∞, 0] → R and ρ m+1 , ..., ρ m+n : [0, ∞) × [0, ∞) → R be functions. We say that (ρ 1 , ...., ρ m+n ) is a weak solution of (1.1), (1.2), and

(1.3) if def:reg (D.1) f 1 (ρ 1 ), ..., f m (ρ m ) ∈ BV loc ((0, ∞)×(-∞, 0)) and f m+1 (ρ m+1 ), ..., f m+n (ρ m+n ) ∈ BV loc ((0, ∞)× (0, ∞)); ef:entrL (D.
2) for every i ∈ {1, ..., m}, every c ∈ R and every nonnegative test function

ϕ ∈ C ∞ (R × (-∞, 0)) with compact support ˆ∞ 0 ˆ0 -∞ (|ρ i -c|∂ t ϕ + sign (ρ i -c) (f i (ρ i ) -f i (c))∂ x ϕ) dtdx + ˆ0 -∞ |ρ i,0 (x) -c|ϕ(0, x)dx ≥ 0; ef:entrR (D.3) for every j ∈ {m + 1, ..., m + n}, every c ∈ R and every nonnegative test function ϕ ∈ C ∞ (R × (0, ∞)) with compact support ˆ∞ 0 ˆ∞ 0 (|ρ j -c|∂ t ϕ + sign (ρ j -c) (f j (ρ R ) -f j (c))∂ x ϕ) dtdx + ˆ∞ 0 |ρ j,0 (x) -c|ϕ(0, x)dx ≥ 0; def:flux (D.4) m i=1 f i (ρ i (t, 0-)) = m+n j=m+1 f j (ρ j (t, 0+
)) for a.e. t ≥ 0.

In [START_REF] Coclite | Vanishing viscosity for traffic on networks[END_REF] the authors approximated (1.1), (1.2), and (1.3) in the following way q:oldCLeps q:oldCLeps (1.4)

                               ∂ t ρ i,ε + ∂ x f i (ρ i,ε ) = ε∂ 2 xx ρ i,ε , t > 0, x < 0, i, ∂ t ρ j,ε + ∂ x f j (ρ j,ε ) = ε∂ 2 xx ρ j,ε , t > 0, x > 0, j, ρ i,ε (t, 0) = ρ j,ε (t, 0), t > 0, i, j, m i=1 (f i (ρ i,ε (t, 0)) -ε∂ x ρ i,ε (t, 0)) = m+n j=m+1 (f j (ρ j,ε (t, 0)) -ε∂ x ρ j,ε (t, 0)), t > 0, ρ i,ε (0, x) = ρ i,0,ε (x), x < 0, i, ρ j,ε (0, x) = ρ j,0,ε (x),
x > 0, j, where i ∈ {1, ..., m} and j ∈ {m + 1, ..., m + n} and ρ i,0,ε , ρ j,0,ε are smooth approximations of ρ i,0 , ρ j,0 . In this setting they showed that

ρ i,ε → ρ i a.e. in (0, ∞) × (-∞, 0) and in L p loc ((0, ∞) × (-∞, 0)), 1 ≤ p < ∞, as ε → 0 for every i, ρ j,ε → ρ j a.e. in (0, ∞) × (0, ∞) and in L p loc ((0, ∞) × (0, ∞)), 1 ≤ p < ∞,
as ε → 0 for every j, where (ρ 1 , ...., ρ m+n ) is a weak solution of (1.1), (1.2), (1.3), in the sense of Definition 1.1.

In this paper we modify the transmission condition of (1.4) and inspired by [START_REF] Guarguaglini | Global smooth solutions for a hyperbolic chemotaxis model on a network[END_REF] we consider the following viscous approximation of (1.1), (1.2), and (1.3) eq:eps eq:eps (1.5)

                   ∂ t ρ i,ε + ∂ x f i (ρ i,ε ) = ε∂ 2 xx ρ i,ε , t > 0, x < 0, i, ∂ t ρ j,ε + ∂ x f j (ρ j,ε ) = ε∂ 2 xx ρ j,ε , t > 0, x > 0, j, f i (ρ i,ε (t, 0)) -ε∂ x ρ i,ε (t, 0) = β i (ρ 1,ε (t, 0), ...., ρ m+n,ε (t, 0)),
t > 0, i, f j (ρ j,ε (t, 0)) -ε∂ x ρ j,ε (t, 0) = β j (ρ 1,ε (t, 0), ...., ρ m+n,ε (t, 0)), t > 0, j, ρ i,ε (0, x) = ρ i,0,ε (x),

x < 0, i, ρ j,ε (0, x) = ρ j,0,ε (x),

x > 0, j, where, of course, etabalance etabalance (1.6)

m i=1 β i (ρ 1,ε (t, 0), . . . , ρ m+n,ε (t, 0)) = m+n j=m+1 β j (ρ 1,ε (t, 0), . . . , ρ m+n,ε (t, 0)).
The additional assumptions we make on the functions β and on the initial conditions ρ ,0,ε are postposed to the next section.

The main result of the paper is the following.

ρ i,ε k -→ ρ i ,
a.e. and in L p loc ((0, ∞) × (-∞, 0)), 1 ≤ p < ∞, i ∈ {1, .., m}, q:comp3-th q:comp3-th (1.7)

ρ j,ε k -→ ρ j ,
a.e. and in L p loc ((0, ∞) × (0, ∞)), 1 ≤ p < ∞, j ∈ {m + 1, .., m + n}, q:comp4-th q:comp4-th (1.8)

f 1 (ρ 1 ), ..., f m (ρ m ) ∈ BV ((0, ∞) × (-∞, 0)), f m+1 (ρ m+1 ), ..., f m+n (ρ m+n ) ∈ BV ((0, ∞) × (0, ∞)),
eq:BV eq:BV (1.9)

where (ρ 1,ε k , ..., ρ m+n,ε k ) is the corresponding solution of (1.5).

It worth mentioning that a complete characterization of the limit solution obtained from (1.4) as ε → 0 is given in [START_REF] Andreianov | Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network[END_REF], where the authors adapt to a star shaped graph setting some ideas and techniques originally developed for conservation laws with discontinous flux, see in particular [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF][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].

At the moment we are not able to formulate a similar characterization of the limit of (1.5). In general, however, the limits coming from parabolic regularization subject to the two different kinds of transmission conditions are different.

To show this consider the simple case of a junction with one incoming and one outgoing edges. So we have the conservation law

:basicL1 :basicL1 (1.10) ∂ t ρ 1 + ∂ x f 1 (ρ 1 ) = 0, t > 0, x < 0,
on the incoming edge and

:basicR1 :basicR1 (1.11) ∂ t ρ 2 + ∂ x f 2 (ρ 2 ) = 0, t > 0, x > 0,
on the outgoing one. Assume that

f 1 (0) = f 1 (1) = f 2 (0) = f 2 (1) = 0, f 1 , f 2 < 0, there exists 0 < ρ < ρ < 1 and G > 0 such that f 1 (ρ) = f 2 (ρ) = G(ρ -ρ)
. q:ass-ex q:ass-ex (1.12) Consider the simplified version of (1.5) q:eps-ex q:eps-ex (1.13)

               ∂ t ρ 1,ε + ∂ x f 1 (ρ 1,ε ) = ε∂ 2 xx ρ 1,ε , t > 0, x < 0, ∂ t ρ 2,ε + ∂ x f 2 (ρ 2,ε ) = ε∂ 2 xx ρ 2,ε , t > 0, x > 0, f 1 (ρ 1,ε (t, 0)) -ε∂ x ρ 1,ε (t, 0) = f 2 (ρ 2,ε (t, 0)) -ε∂ x ρ 2,ε (t, 0) = G(ρ 1,ε -ρ 2,ε ), t > 0, ρ 1,ε (0, x) = ρ, x < 0, ρ 2,ε (0, x) = ρ, x > 0.
The unique solution of (1.13) is

(1.14) ρ 1,ε (•, •) = ρ, ρ 2,ε (•, •) = ρ, ε > 0.
Therefore, as ε → 0 we get the solution of (1.10)-(1.11)

(1.15) ρ 1 (•, •) = ρ, ρ 2 (•, •) = ρ.
This stationary solution is not admissible in the sense of the classical vanishing viscosity germ, see [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF]Sec. 5], as it consists of a nonclassical shock. However, when dealing with conservation laws with discontinuous flux, it is well known that infinitely many L 1 contractive semigroups of solutions exist, also in relation with different physical applications. In particular, when the right and left fluxes are bell-shaped, as we assume in condition (H.1), each of those notions of admissible solution is uniquely determined by the choice of a (A, B)-connection, see [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF][START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF][START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF][START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF] for precise definitions and exemples. In the exemple above the couple (ρ, ρ) is a connection. It is worth noticing that entropy solutions admissible in the sense of a (A, B)-connection can be obtained as limits of a sequence of parabolic approximations made with adapted viscosities but a classical condition of continuity at the interface, see [4, Sec. 6.2] for a general result, but also [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF][START_REF] Kaasschieter | Solving the buckley-leverett equation with gravity in a heterogeneous porous medium[END_REF] for an application to the Buckley-Leverett equation.

It is difficult, however, to establish a direct equivalence between the aforementioned results and the one we put forward in this paper. In particular, in the present case we miss information on the boundary layers at the parabolic level and we do not know how the transmission conditions we impose on the parabolic fluxes translates into a condition for the hyperbolic problem.

The paper is organized as follows: Section 2 contains the precise list of assumptions on the initial and transmission conditions in the parabolic problem (1.5). In Section 3 we present the proofs of all necessary a priori estimates on (1.5). Finally, in Section 4 we detail the proof of Theorem 1.1.

Initial and transmission conditions for the parabolic problem sec:1

The initial conditions ρ ,0 , = 1, . . . , m + n, on the hyperbolic problem (1.1), (1.2), and (1.3) satisfy (H.3).

Once the functions ρ ,0 are fixed, we impose on (1.5) initial conditions ρ ,0,ε such that

ρ i,0,ε ∈ C ∞ ((-∞, 0]) ∩ L 1 (-∞, 0), ρ j,0,ε ∈ C ∞ ([0, ∞)) ∩ L 1 (0, ∞), ε > 0, ρ i,0,ε → ρ i,0 a.e. in (-∞, 0) and in L p loc (-∞, 0), 1 ≤ p < ∞, as ε → 0, ρ j,0,ε → ρ j,0 a.e. in (0, ∞) and in L p loc (0, ∞), 1 ≤ p < ∞, as ε → 0, 0 ≤ ρ i,0,ε , ρ j,0,ε ≤ 1, ε > 0, ρ i,0,ε L 1 (-∞,0) ≤ ρ i,0 L 1 (-∞,0) , ρ j,0,ε L 1 (0,∞) ≤ ρ j,0 L 1 (0,∞) , ε > 0, ρ i,0,ε L 2 (-∞,0) ≤ ρ i,0 L 2 (-∞,0) , ρ j,0,ε L 2 (0,∞) ≤ ρ j,0 L 2 (0,∞) , ε > 0, ∂ x ρ i,0,ε L 1 (-∞,0) ≤ T V (ρ i,0 ), ∂ x ρ j,0,ε L 1 (0,∞) ≤ T V (ρ j,0 ), ε > 0, ε ∂ x ρ i,0,ε L 1 (-∞,0) , ε ∂ 2 xx ρ j,0,ε L 1 (0,∞) ≤ C, ε > 0,
eq:asseps eq:asseps (2.1)

for some constant C > 0 independent on ε.

The functions β appearing in the trasmission conditions in (1.5) take the form (2.2)

β i (ρ 1,ε (t, 0), ...., ρ m+n,ε (t, 0)) = m+n j=m+1 G i,j (ρ i,ε (t, 0), ρ j,ε (t, 0)) + ε m h=1 K i,h (ρ i,ε (t, 0), ρ h,ε (t, 0)) - m+n h=1 K h,i (ρ h,ε (t, 0), ρ i,ε (t, 0)) ;
for i ∈ {1, . . . , m}, and for j ∈ {m + 1, . . . , m + n}

(2.
3)

β j (ρ 1,ε (t, 0), ...., ρ m+n,ε (t, 0)) = m i=1 G i,j (ρ i,ε (t, 0), ρ j,ε (t, 0)) + ε m+n h=m+1 K h,j (ρ h,ε (t, 0), ρ j,ε (t, 0)) - m+n h=1 K j,h (ρ j,ε (t, 0), ρ h,ε (t, 0)) .
The functions G i,j (u, v) ∈ C ∞ (R 2 ), i ∈ {1, . . . , m}, j ∈ {m+1, . . . , m+n}, and

K h, (u, v) ∈ C ∞ (R 2 ), h, ∈ {1, . . . , m + n}, satisfy ∂ v G i,j (•, •) ≤ 0 ≤ ∂ u G i,j (•, •), G i,j (0, 0) = G i,j (1, 1) = 0, ∂ u K h, (•, •) ≤ 0 ≤ ∂ v K h, (•, •), K h, (0, 0) = K h, (1, 1)
= 0. eq:assG eq:assG (2.4) In particular, (2.4) implies

(sign (u) -sign (v))∇G i,j (•, •) • (u, v) ≥ 0, u, v ∈ R, (sign (u) -sign (v))∇K h, (•, •) • (u, v) ≤ 0, u, v ∈ R, (sign u -u -sign v -v )(G i,j (u, v) -G i,j (u , v )) ≥ 0, u, u , v, v ∈ R, (sign u -u -sign v -v )(K h, (u, v) -K h, (u , v )) ≤ 0, u, u , v, v ∈ R, (χ (-∞,0) (u) -χ (-∞,0) (v))G i,j (u, v) ≤ 0, u, v ∈ R, (χ (-∞,0) (u) -χ (-∞,0) (v))K h, (u, v) ≥ 0, u, v ∈ R,
eq:assG' eq:assG' (2.5)

where χ (-∞,0) is the characteristic function of the set (-∞, 0). This specific form of transmission conditions is reminiscent of the parabolic transmission conditions considered in [START_REF] Guarguaglini | Global smooth solutions for a hyperbolic chemotaxis model on a network[END_REF][START_REF] Bretti | A hyperbolic model of chemotaxis on a network: a numerical study[END_REF], which were originally inspired from the Kedem-Katchalsky conditions for membrane permeability introduced in [START_REF] Kedem | Thermodynamic analysis of permeability of biological membranes to non-electrolytes[END_REF] eq:assG'' eq:assG'' (2.6)

K h, (u, v) = c h, (u -v),
for some constants c h, > 0. Our conditions are more general and in particular we can notice that the function K h, above satisfies eq:assL2 eq:assL2 (2.7)

K h, (u, v)(u -v) ≥ 0,
that allows the authors in [START_REF] Guarguaglini | Global smooth solutions for a hyperbolic chemotaxis model on a network[END_REF] to get the L 2 conservation (see Lemma 3.3 below). We can observe that the equality (1.6) holds as

m i=1 β i (ρ 1,ε (t, 0), ...., ρ m+n,ε (t, 0)) = m i=1 m+n j=m+1 G i,j (ρ i,ε (t, 0), ρ j,ε (t, 0)) + ε m i=1 m h=1 K i,h (ρ i,ε (t, 0), ρ h,ε (t, 0)) - m+n h=1 K h,i (ρ h,ε (t, 0), ρ i,ε (t, 0)) = m i=1 m+n j=m+1
(G i,j (ρ i,ε (t, 0), ρ j,ε (t, 0)) -εK j,i (ρ j,ε (t, 0), ρ i,ε (t, 0)))

eq:CD1 eq:CD1 (2.8)

and analogously m+n j=m+1

β j (ρ 1,ε (t, 0), ...., ρ m+n,ε (t, 0)) = m+n j=m+1 m i=1 (G i,j (ρ i,ε (t, 0), ρ j,ε (t, 0)) -εK j,i (ρ j,ε (t, 0), ρ i,ε (t, 0)))
.

eq:CD2 eq:CD2 (2.9)

A priori estimates stimates

This section is devoted to establish a priori estimates, uniform with respect to ε, which are necessary toward the proof of our main convergence result in the next section.

For every ε > 0, let (ρ 1,ε , ..., ρ m+n,ε ) be a solution of (1.5) satisfying (2.1).

m:linfty Lemma 3.1 (L ∞ estimate). We have that q:linfty q:linfty (3.1) 0 ≤ ρ i,ε , ρ j,ε ≤ 1, i, j.

Proof. Consider the function

η(ξ) = -ξχ (-∞,0) (ξ). Since η (ξ) = -χ (-∞,0) (ξ), using (2.4) we obtain d dt   m i=1 ˆ0 -∞ η(ρ i,ε )dx + m+n j=m+1 ˆ∞ 0 η(ρ j,ε )dx   = m i=1 ˆ0 -∞ η (ρ i,ε )∂ t ρ i,ε dx + m+n j=m+1 ˆ∞ 0 η (ρ j,ε )∂ t ρ j,ε dx = - m i=1 ˆ0 -∞ χ (-∞,0) (ρ i,ε )∂ t ρ i,ε dx - m+n j=m+1 ˆ∞ 0 χ (-∞,0) (ρ j,ε )∂ t ρ j,ε dx = m i=1 ˆ0 -∞ χ (-∞,0) (ρ i,ε )∂ x (f i (ρ i,ε ) -ε∂ x ρ i,ε )dx + m+n j=m+1 ˆ∞ 0 χ (-∞,0) (ρ j,ε )∂ x (f j (ρ j,ε ) -ε∂ x ρ j,ε )dx = m i=1 χ (-∞,0) (ρ i,ε (t, 0))(f i (ρ i,ε (t, 0)) -ε∂ x ρ i,ε (t, 0)) - m+n j=m+1 χ (-∞,0) (ρ j,ε (t, 0))(f j (ρ j,ε (t, 0)) -ε∂ x ρ j,ε (t, 0)) + m i=1 ˆ0 -∞ ∂ x ρ i,ε (f i (ρ i,ε ) -ε∂ x ρ i,ε )dδ {ρ i,ε =0} ≤0 + m+n j=m+1 ˆ∞ 0 ∂ x ρ j,ε (f j (ρ j,ε ) -ε∂ x ρ j,ε )dδ {ρ j,ε =0} ≤0 ≤ m+n j=m+1 m i=1 χ (-∞,0) (ρ i,ε (t, 0)) -χ (-∞,0) (ρ j,ε (t, 0)) • • (G i,j (ρ i,ε (t, 0), ρ j,ε (t, 0)) -εK j,i (ρ j,ε (t, 0), ρ i,ε (t, 0))) ≤ 0,
where δ {ρ i,ε =0} and δ {ρ j,ε =0} are the Dirac deltas concentrated on the sets {ρ i,ε = 0} and {ρ j,ε = 0}, respectively and we apply [6, Lemma 2]. Integrating over (0, t) and using (2.1) we get

0 ≤ m i=1 ˆ0 -∞ η(ρ i,ε (t, x))dx + m+n j=m+1 ˆ∞ 0 η(ρ j,ε (t, x))dx ≤ m i=1 ˆ0 -∞ η(ρ i,0,ε )dx + m+n j=m+1 ˆ∞ 0 η(ρ j,0,ε )dx = 0
and then ρ i,ε , ρ j,ε ≥ 0, i, j, that proves the lower bounds in (3.1). The upper bounds in (3.1) can be proved in the same way using the function ξ → (ξ -1)χ (1,∞) (ξ).

lm:l1 Lemma 3.2 (L 1 estimate). We have that

m i=1 ρ i,ε (t, •) L 1 (-∞,0) + m+n j=m+1 ρ j,ε (t, •) L 1 (0,∞) ≤ m i=1 ρ i,0 L 1 (-∞,0) + m+n j=m+1 ρ j,0 L 1 (0,∞) , t ≥ 0.
eq:l1 eq:l1 (3.2)

Proof. Thanks to (1.5), (2.8), (2.9), and (3.1), we have that

d dt   m i=1 ˆ0 -∞ |ρ i,ε |dx + m+n j=m+1 ˆ∞ 0 |ρ j,ε |dx   = d dt   m i=1 ˆ0 -∞ ρ i,ε dx + m+n j=m+1 ˆ∞ 0 ρ j,ε dx   = m i=1 ˆ0 -∞ ∂ t ρ i,ε dx + m+n j=m+1 ˆ∞ 0 ∂ t ρ j,ε dx = - m i=1 ˆ0 -∞ ∂ x (f i (ρ i,ε ) -ε∂ x ρ i,ε )dx - m+n j=m+1 ˆ∞ 0 ∂ x (f j (ρ j,ε ) -ε∂ x ρ j,ε )dx = - m i=1
β i (ρ 1,ε (t, 0), . . . , ρ m+n,ε (t, 0)) + m+n j=m+1 β j (ρ 1,ε (t, 0), . . . , ρ m+n,ε (t, 0)) = 0.

Integrating over (0, t) and using (2.1) we get (3.2).

lm:l2 Lemma 3.3 (L 2 estimate). We have that

m i=1 ρ i,ε (t, •) 2 L 2 (-∞,0) + m+n j=m+1 ρ j,ε (t, •) 2 L 2 (0,∞) + 2ε ˆt 0   m i=1 ∂ x ρ i,ε (s, •) 2 L 2 (-∞,0) + m+n j=m+1 ∂ x ρ j,ε (s, •) 2 L 2 (0,∞)   ds ≤ m i=1 ρ i,0 2 L 2 (-∞,0) + m+n j=m+1 ρ j,0 2
L 2 (0,∞) + 2 m+n =1 β L ∞ ((0,1) m+n ) + m i=1 f i L 1 (0,1) t,
eq:l2 eq:l2 (3.3)

for every t ≥ 0.

Proof. Thanks to (1.5), we have that

d dt   m i=1 ˆ0 -∞ ρ 2 i,ε 2 dx + m+n j=m+1 ˆ∞ 0 ρ 2 j,ε 2 dx   = m i=1 ˆ0 -∞ ρ i,ε ∂ t ρ i,ε dx + m+n j=m+1 ˆ∞ 0 ρ j,ε ∂ t ρ j,ε dx = - m i=1 ˆ0 -∞ ρ i,ε ∂ x (f i (ρ i,ε ) -ε∂ x ρ i,ε )dx - m+n j=m+1 ˆ∞ 0 ρ j,ε ∂ x (f j (ρ j,ε ) -ε∂ x ρ j,ε )dx = - m i=1 ρ i,ε (t, 0)(f i (ρ i,ε (t, 0)) -ε∂ x ρ i,ε (t, 0)) + m+n j=m+1 ρ j,ε (t, 0)(f j (ρ j,ε (t, 0)) -ε∂ x ρ j,ε (t, 0)) + m i=1 ˆ0 -∞ ∂ x ˆρi,ε (t,x) 0 f i (ξ)dξ dx + m+n j=m+1 ˆ∞ 0 ∂ x ˆρj,ε (t,x) 0 f j (ξ)dξ dx -ε m i=1 ˆ0 -∞ (∂ x ρ i,ε ) 2 dx -ε m+n j=m+1 ˆ∞ 0 (∂ x ρ j,ε ) 2 dx = m i=1 ρ j,ε (t, 0)β i (ρ 1,ε (t, 0), ...., ρ m+n,ε (t, 0)) - m+n j=m+1 ρ i,ε (t, 0))β j (ρ 1,ε (t, 0), ...., ρ m+n,ε (t, 0)) + m i=1 ˆρi,ε (t,0) 0 f i (ξ)dξ - m+n j=m+1 ˆρj,ε (t,0) 0 f j (ξ)dξ ≤0 -ε m i=1 ˆ0 -∞ (∂ x ρ i,ε ) 2 dx -ε m+n j=m+1 ˆ∞ 0 (∂ x ρ j,ε ) 2 dx ≤ m+n =1 β L ∞ ((0,1) m+n ) + m i=1 f i L 1 (0,1) -ε m i=1 ˆ0 -∞ (∂ x ρ i,ε ) 2 dx -ε m+n j=m+1 ˆ∞ 0 (∂ x ρ j,ε ) 2 dx.
Integrating over (0, t) and using (2.1) we get (3.3).

lm:BVt Lemma 3.4 (BV estimate). We have that

m i=1 ∂ t ρ i,ε (t, •) L 1 (-∞,0) + m+n j=m+1 ∂ t ρ j,ε (t, •) L 1 (0,∞) ≤(m + n)C + m i=1 f i L ∞ (0,1) T V (ρ i,0 ) + m+n j=m+1
f j L ∞ (0,1) T V (ρ j,0 ), eq:BVt eq:BVt (3.4) for every t ≥ 0.

Proof. From (1.5) we get

∂ 2 tt ρ i,ε + ∂ x (f i (ρ i,ε )∂ t ρ i,ε ) = ε∂ 3 txx ρ i,ε , ∂ 2 tt ρ j,ε + ∂ x (f j (ρ j,ε )∂ t ρ j,ε ) = ε∂ 3 txx ρ j,ε , f i (ρ i,ε (t, 0))∂ t ρ i,ε (t, 0) -ε∂ 2 tx ρ i,ε (t, 0) = m+n j=m+1 ∇G i,j (ρ i,ε (t, 0), ρ j,ε (t, 0)) • (∂ t ρ i,ε (t, 0), ∂ t ρ j,ε (t, 0)) + ε m h=1 ∇K i,h (ρ i,ε (t, 0), ρ h,ε (t, 0)) • (∂ t ρ i,ε (t, 0), ∂ t ρ h,ε (t, 0)) -ε m+n h=1 ∇K h,i (ρ h,ε (t, 0), ρ i,ε (t, 0)) • (∂ t ρ h,ε (t, 0), ∂ t ρ i,ε (t, 0)), f j (ρ j,ε (t, 0))∂ t ρ j,ε (t, 0) -ε∂ 2 tx ρ j,ε (t, 0) = m i=1 ∇G i,j (ρ i,ε (t, 0), ρ j,ε (t, 0)) • (∂ t ρ i,ε (t, 0), ∂ t ρ j,ε (t, 0)) + ε m+n h=m+1 ∇K h,j (ρ h,ε (t, 0), ρ j,ε (t, 0)) • (∂ t ρ h,ε (t, 0), ∂ t ρ j,ε (t, 0)) -ε m+n h=1 ∇K j,h (ρ j,ε (t, 0), ρ h,ε (t, 0)) • (∂ t ρ i,ε (t, 0), ∂ t ρ h,ε (t, 0)).
Thanks to (2.5), we have that

d dt   m i=1 ˆ0 -∞ |∂ t ρ i,ε |dx + m+n j=m+1 ˆ∞ 0 |∂ t ρ j,ε |dx   = m i=1 ˆ0 -∞ ∂ 2 tt ρ i,ε sign (∂ t ρ i,ε ) dx + m+n j=m+1 ˆ∞ 0 ∂ 2 tt ρ j,ε sign (∂ t ρ j,ε ) dx = - m i=1 ˆ0 -∞ sign (∂ t ρ i,ε ) ∂ x (f i (ρ i,ε )∂ t ρ i,ε -ε∂ 2 tx ρ i,ε )dx - m+n j=m+1 ˆ∞ 0 sign (∂ t ρ j,ε ) ∂ x (f j (ρ j,ε )∂ t ρ j,ε -ε∂ 2 tx ρ j,ε )dx = - m i=1 sign (∂ t ρ i,ε (t, 0)) (f i (ρ i,ε (t, 0))∂ t ρ i,ε (t, 0) -ε∂ 2 tx ρ i,ε (t, 0)) + m+n j=m+1 sign (∂ t ρ j,ε (t, 0)) (f j (ρ j,ε (t, 0))∂ t ρ j,ε (t, 0) -ε∂ 2 tx ρ j,ε (t, 0)) + 2 m i=1 ˆ0 -∞ ∂ 2 tx ρ i,ε (f i (ρ i,ε )∂ t ρ i,ε -ε∂ 2 tx ρ i,ε )dδ {∂tρ i,ε =0 } ≤0 + 2 m+n j=m+1 ˆ0 -∞ ∂ 2 tx ρ j,ε (f j (ρ j,ε )∂ t ρ j,ε -ε∂ 2 tx ρ j,ε )dδ {∂tρ j,ε =0 } ≤0 ≤ - m i=1 m+n j=m+1 (sign (∂ t ρ i,ε (t, 0)) -sign (∂ t ρ j,ε (t, 0))) × × ∇G i,j (ρ i,ε (t, 0), ρ j,ε (t, 0)) • (∂ t ρ i,ε (t, 0), ∂ t ρ j,ε (t, 0)) + ε m i=1 m+n j=m+1 (sign (∂ t ρ i,ε (t, 0)) -sign (∂ t ρ j,ε (t, 0))) × × ∇K j,i (ρ i,ε (t, 0), ρ j,ε (t, 0)) • (∂ t ρ i,ε (t, 0), ∂ t ρ j,ε (t, 0)) ≤ 0,
where δ {∂tρ i,ε =0} and δ {∂tρ j,ε =0} are the Dirac deltas concentrated on the sets {∂ t ρ i,ε = 0} and {∂ t ρ j,ε = 0}, respectively and we apply [6, Lemma 2].

Integrating over (0, t) and using (2.1), (3.1) we get

m i=1 ∂ t ρ i,ε (t, •) L 1 (-∞,0) + m+n j=m+1 ∂ t ρ j,ε (t, •) L 1 (0,∞) ≤ m i=1 ∂ t ρ i,ε (0, •) L 1 (-∞,0) + m+n j=m+1 ∂ t ρ j,ε (0, •) L 1 (0,∞) = m i=1 ε∂ 2 xx ρ i,0,ε -∂ x f i (ρ i,0,ε ) L 1 (-∞,0) + m+n j=m+1 ε∂ 2 xx ρ j,0,ε -∂ x f j (ρ j,0,ε ) L 1 (0,∞) ≤ m i=1 ε ∂ 2 xx ρ i,0,ε L 1 (-∞,0) + f i (ρ i,0,ε ) L ∞ (-∞,0) ∂ x ρ i,0,ε L 1 (-∞,0) + m+n j=m+1 ε ∂ 2 xx ρ j,0,ε L 1 (0,∞) + f j (ρ j,0,ε ) L ∞ (0,∞) ∂ x ρ j,0,ε L 1 (0,∞) ≤ (m + n)C + m i=1 f i L ∞ (0,1) T V (ρ i,0 ) + m+n j=m+1 f j L ∞ (0,1) T V (ρ j,0 ), that is (3.4).
lm:stab Lemma 3.5 (Stability estimate). Let (ρ 1,ε , ..., ρ m+n,ε ) and (ρ 1,ε , ..., ρ m+n,ε ) be two solutions of (1.5).

The following estimate holds

m i=1 ρ i,ε (t, •) -ρ i,ε (t, •) L 1 (-∞,0) + m+n j=m+1 ρ j,ε (t, •) -ρ j,ε (t, •) L 1 (0,∞) ≤ m i=1 ρ i,0,ε -ρ i,0,ε L 1 (-∞,0) + m+n j=m+1 ρ j,0,ε -ρ j,0,ε L 1 (0,∞) , t ≥ 0.
eq:stab eq:stab (3.5)

Proof. From (1.5) we get

∂ t (ρ i,ε -ρ i,ε ) + ∂ x (f i (ρ i,ε ) -f i (ρ i,ε )) = ε∂ 2 xx (ρ i,ε -ρ i,ε ), ∂ t (ρ j,ε -ρ j,ε ) + ∂ x (f j (ρ j,ε ) -f j (ρ j,ε )) = ε∂ 2 xx (ρ j,ε -ρ j,ε
). Thanks to (1.5), (2.5), and (3.1), we have that

d dt   m i=1 ˆ0 -∞ |ρ i,ε -ρ i,ε |dx + m+n j=m+1 ˆ∞ 0 |ρ j,ε -ρ j,ε |dx   = m i=1 ˆ0 -∞ sign ρ i,ε -ρ i,ε ∂ t (ρ i,ε -ρ i,ε )dx + m+n j=m+1 ˆ∞ 0 sign ρ j,ε -ρ j,ε ∂ t (ρ j,ε -ρ j,ε )dx = - m i=1 ˆ0 -∞ sign ρ i,ε -ρ i,ε ∂ x ((f i (ρ i,ε ) -f i (ρ i,ε )) -ε∂ x (ρ i,ε -ρ i,ε ))dx - m+n j=m+1 ˆ∞ 0 sign ρ j,ε -ρ j,ε ∂ x ((f j (ρ j,ε ) -f j (ρ j,ε )) -ε∂ x (ρ j,ε -ρ j,ε ))dx = - m i=1 m+n j=m+1 [sign ρ i,ε (t, 0) -ρ i,ε (t, 0) -sign ρ j,ε (t, 0) -ρ j,ε (t, 0) ]× × [G i,j (ρ i,ε (t, 0), ρ j,ε (t, 0)) -G i,j (ρ i,ε (t, 0), ρ j,ε (t, 0))] + ε m i=1 m+n j=m+1 [sign ρ i,ε (t, 0) -ρ i,ε (t, 0) -sign ρ j,ε (t, 0) -ρ j,ε (t, 0) ]× × [K j,i (ρ i,ε (t, 0), ρ j,ε (t, 0)) -G i,j (ρ i,ε (t, 0), ρ j,ε (t, 0))] + 2 m i=1 ˆ0 -∞ ∂ x (ρ i,ε -ρ i,ε )((f i (ρ i,ε ) -f i (ρ i,ε )) -ε∂ x (ρ i,ε -ρ i,ε ))dδ {ρ i,ε =ρ i,ε } ≤0 + 2 m+n j=m+1 ˆ∞ 0 ∂ x (ρ j,ε -ρ j,ε )((f i (ρ j,ε ) -f i (ρ j,ε )) -ε∂ x (ρ j,ε -ρ j,ε ))dδ {ρ j,ε =ρ j,ε } ≤0 ≤ 0,
where we use [6, Lemma 2] and we denote by δ {ρ i,ε =ρ i,ε } and δ {ρ j,ε =ρ j,ε } respectively the Dirac deltas concentrated on the sets {ρ i,ε = ρ i,ε } and {ρ j,ε = ρ j,ε }.

Integrating over (0, t) we get (3.5).

Proof of Theorem 1.1 ompactness

The well-posedness of smooth solutions for (1.5) can be proved following the argument used in [START_REF] Coclite | Vanishing viscosity for traffic on networks[END_REF]Theorem 1.2] to establish the well-posedness of smooth solutions for (1.4). Indeed, the existence of a linear semigroup of solutions in the linear case (i.e., when f ≡ 0) is shown in [START_REF] Guarguaglini | Global smooth solutions for a hyperbolic chemotaxis model on a network[END_REF]. Then the Duhamel Formula, estimates similar to the ones in the previous section and a fixed point argument lead to the result.

The main result of this section is the following.

lm:comp Lemma 4.1. Let (ρ 1,ε , ..., ρ m+n,ε ) be the solution of (1.5). There exist a sequence {ε k } k∈N ⊂ (0, ∞), ε k → 0, and m + n maps ρ 1 , ..., ρ m+n such that ρ 1 , ..., ρ m ∈ L 1 ((0, ∞) × (-∞, 0)) ∩ L ∞ ((0, ∞) × (-∞, 0)), eq:comp1 eq:comp1 (4.1) ρ m+1 , ..., ρ m+n ∈ L 1 ((0, ∞) × (0, ∞)) ∩ L ∞ ((0, ∞) × (0, ∞)), eq:comp1.1 eq:comp1.1 (4.2) 0 ≤ ρ ≤ 1, ∈ {1, ..., m + n}, eq:comp2 eq:comp2 (4.3) ρ i,ε k -→ ρ i , a.e. and in L p loc ((0, ∞) × (-∞, 0)), 1 ≤ p < ∞, i ∈ {1, .., m}, eq:comp3 eq:comp3 (4.4) ρ j,ε k -→ ρ j , a.e. and in L p loc ((0, ∞) × (0, ∞)), 1 ≤ p < ∞, j ∈ {m + 1, .., m + n}. eq:comp4 eq:comp4 (4.5)

Moreover, we have that

m i=1 ρ i (t, •) L 1 (-∞,0) + m+n j=m+1 ρ j (t, •) L 1 (0,∞) eq:comp5 eq:comp5 (4.6) ≤ m i=1 ρ i,0 L 1 (-∞,0) + m+n j=m+1 ρ j,0 L 1 (0,∞) , m i=1 ρ i (t, •) 2 L 2 (-∞,0) + m+n j=m+1 ρ j (t, •) 2 L 2 (0,∞)
eq:comp6 eq:comp6 (4.7)

≤ m i=1 ρ i,0 2 L 2 (-∞,0) + m+n j=m+1 ρ j,0 2 L 2 (0,∞) + 2 m+n =1 β L ∞ ((0,1) m+n ) + m i=1 f i L 1 (0,1) t, m i=1 T V (f i (ρ i (t, •))) + m+n j=m+1 T V (f j (ρ j (t, •))) eq:comp7 eq:comp7 (4.8) = m i=1 ∂ t ρ i (t, •) M(-∞,0) + m+n j=m+1 ∂ t ρ j (t, •) M(0,∞) ≤ (m + n)C + m i=1 f i L ∞ (0,1) T V (ρ i,0 ) + m+n j=m+1 f j L ∞ (0,1) T V (ρ j,0 ).
Thanks to the genuine nonlinearity of f 1 , ..., f m+n , we can use the Tartar compensated compactness method [START_REF] Tartar | Nonlinear analysis and mechanics: Heriot-watt symposium[END_REF] to obtain strong convergence of a subsequence of viscosity approximations. The notation R can stand for (0, ∞) or (-∞, 0). lem:CC Theorem 4.1 (Tartar). Let {v ν } ν>0 be a family of functions defined on (0, ∞) × R such that

v ν L ∞ ((0,T )×R) ≤ M T , T, ν > 0,
and the family

{∂ t η(v ν ) + ∂ x q (v ν )} ν>0 is compact in H -1 loc ((0, ∞) × R), for every convex η ∈ C 2 (R), where q = f η . Then there exist a sequence {ν n } n∈N ⊂ (0, ∞), ν n → 0, and a map v ∈ L ∞ ((0, T ) × R), T > 0, such that v νn -→ v a.e. and in L p loc ((0, ∞) × R), 1 ≤ p < ∞.
The following compact embedding of Murat [START_REF] Murat | L'injection du cône positif de H -1 dans W -1, q est compacte pour tout q < 2[END_REF] is useful.

em:Murat Theorem 4.2 (Murat). Let Ω be a bounded open subset of R N , N ≥ 2. Suppose the sequence {L n } n∈N of distributions is bounded in W -1,∞ (Ω). Suppose also that L n = L 1,n + L 2,n ,
where {L 1,n } n∈N lies in a compact subset of H -1 loc (Ω) and {L 2,n } n∈N lies in a bounded subset of L 1 loc (Ω). Then {L n } n∈N lies in a compact subset of H -1 loc (Ω). Proof of Lemma 4.1. Let us fix i ∈ {1, ..., m} and prove the lemma for the incoming edges, as the proof for the outgoing ones is analogous.

Let η : R → R be any convex C 2 entropy function, and let q i : R → R be the corresponding entropy flux defined by q i = η f i . By multiplying i-th equation in (1.5) by η (ρ i,ε ) and using the chain rule, we get omp-proof1 omp-proof1 (4.9)

∂ t η(ρ i,ε ) + ∂ x q i (ρ i,ε ) = ε∂ 2 xx η(ρ i,ε ) L 1,ε -εη (ρ i,ε ) (∂ x ρ i,ε ) 2 L 2,ε . We claim that L 1,ε -→ 0 in H -1 ((0, T ) × (-∞, 0)), T > 0, as ε → 0, {L 2,ε } ε is uniformly bounded in L 1 ((0, T ) × (-∞, 0)), T > 0. claim1-new claim1-new (4.
ε∂ x η(ρ i,ε ) L 2 ((0,T )×(-∞,0)) ≤ √ ε η L ∞ (0,1) √ ε∂ x ρ i,ε L 2 ((0,∞)×(-∞,0)) ≤ √ ε η L ∞ (0,1)   m i=1 ρ i,ε,0 L 2 (-∞,0) + m+n j=m+1 ρ j,ε,0 L 2 (0,∞) + 2 m+n =1 β L ∞ ((0,1) m+n ) + m i=1 f i L 1 (0,1) T   → 0, εη (ρ i,ε ) (∂ x ρ i,ε ) 2 L 1 ((0,T )×(-∞,0)) ≤ η L ∞ (0,1)   m i=1 ρ i,ε,0 2 L 2 (-∞,0) + m+n j=m+1 ρ j,ε,0 2 L 2 (0,∞) +2 m+n =1 β L ∞ ((0,1) m+n ) + m i=1 f i L 1 (0,1) T .
Due to (2.1), (4.10) follows. Therefore, Theorems 4.2 and 4.1 give the existence of a subsequence {ρ i,ε k } k∈N and a limit function ρ i satisfying (4.1) such that as k → ∞

ρ i,ε k -→ ρ i in L p loc ((0, ∞) × (-∞, 0)) for any p ∈ [1, ∞), ρ i,ε k -→ ρ i a.
e. in (0, ∞) × (-∞, 0), eq:convu' eq:convu' (4.11) that guarantees (4.3) and (4.4).

Finally, thanks to Lemmas 3.2, 3.3, and 3.4 we have (4.6), (4.7), and (4.8).

Proof of Theorem 1.1. The first part of the statement related to the convergence of vanishing viscosity approximations has been proved in Lemma 4.1.

Let us fix i ∈ {1, . . . , m} and prove (1.9) for the incoming edges, the case of the outgoing ones is analogous.

Thanks to (3.4) and (4.4), for all ϕ ∈ C ∞ ((0, ∞) × (-∞, 0)) with compact support, we have

ˆ∞ 0 ˆ0 -∞ ρ i ∂ t ϕ dxdt = lim k ˆ∞ 0 ˆ0 -∞ ρ i,ε k ∂ t ϕ dxdt = -lim k ˆ∞ 0 ˆ0 -∞ ∂ t ρ i,ε k ϕ dxdt ≤ ϕ L ∞ ((0,∞)×(-∞,0))   (m + n)C + m i=1
f i L ∞ (0,1) T V (ρ i,0 ) + m+n j=m+1 f j L ∞ (0,1) T V (ρ j,0 )   , therefore eq:BVt-mes eq:BVt-mes (4.12) ∂ t ρ i ∈ M((0, ∞) × (-∞, 0)),

where M((0, ∞) × (-∞, 0)) is the set of all Radon measures on (0, ∞) × (-∞, 0). Moreover, from the equations in (1.1) and (1.2) we have also eq:BVx eq:BVx (4.13) ∂ x f i (ρ i ) ∈ M((0, ∞) × (-∞, 0)).

Clearly (4.12) and (4.13) give (1.9) and so the trace at the junction f (ρ i (t, 0-)) exists for a.e. t > 0.

We prove now that the identity eq:RH eq:RH (4. for every x ≥ 0 and ν ≥ 1. Moreover, for every ν ≥ 1, we define the sequence {r ν } ν∈N\{0} ⊂ C 2 ((-∞, 0]) by writing rν (x) = r ν (-x) for every x ≤ 0. From (1.5) we have that

0 = m i=1 ˆ∞ 0 ˆ0 -∞ ∂ t ρ i,ε k + ∂ x f i (ρ i,ε k ) -ε k ∂ 2 xx ρ i,ε k ϕ(t)r ν (x)dxdt + m+n j=m+1 ˆ∞ 0 ˆ∞ 0 ∂ t ρ j,ε k + ∂ x f j (ρ j,ε k ) -ε k ∂ 2 xx ρ j,ε k ϕ(t)r ν (x)dxdt = - m i=1 ˆ∞ 0 ˆ0 -∞ ρ i,ε k ϕ (t)r ν (x) + f i (ρ i,ε k )ϕ(t)r ν (x) -ε k ∂ x ρ i,ε k ϕ(t)r ν (x) dxdt

junctionFigure 1 .

 1 Figure 1. A junction consisting of m incoming and n outgoing edges.

14 ) m i=1 f

 14i=1 i (ρ i (t, 0-)) = n+m j=m+1 f j (ρ j (t, 0+)) holds for a.e. t > 0; consequently the functions ρ 1 , . . . , ρ m+n provide a solution to (1.1), (1.2), and (1.3) in the sense of Definition 1.1.Let ϕ ∈ C 1 ([0, ∞)), ϕ(0) = 0 with compact support. Consider the sequence {r ν } ν∈N\{0} ⊂ C 2 ([0, ∞)) of cut-off functions satisfying eq:r_nu eq:r_nu (4.15) 0 ≤ r ν (x) ≤ 1, r ν (0) = 1, supp (r ν ) ⊆ 0, 1 ν ,

- m+n j=m+1 ˆ∞ 0

 0 ˆ∞ 0 ρ j,ε k ϕ (t)r ν (x) + f j (ρ j,ε k )ϕ(t)r ν (x) -ε k ∂ x ρ j,ε k ϕ(t)r ν (x) dxdt + m i=1 ˆ∞ 0 (f i (ρ i,ε k (t, 0)) -ε k ∂ x ρ i,ε k (t, 0)) ϕ(t)dt -m+n j=m+1 ˆ∞ 0 (f j (ρ j,ε k (t, 0)) -ε k ∂ x ρ j,ε k (t, 0)) ϕ(t)dt = -m i=1 ˆ∞ 0 ˆ0 -∞ ρ i,ε k ϕ (t)r ν (x) + f i (ρ i,ε k )ϕ(t)r ν (x) -ε k ∂ x ρ i,ε k ϕ(t)r ν (x) dxdt -m+n j=m+1 ˆ∞ 0 ˆ∞ 0 ρ j,ε k ϕ (t)r ν (x) + f j (ρ j,ε k )ϕ(t)r ν (x) -ε k ∂ x ρ j,ε k ϕ(t)r ν (x) dxdt.As k → ∞, due to (3.3), (4.4), and (4.5),0 = -m i=1 ˆ∞ 0 ˆ0 -∞ ρ i ϕ (t)r ν (x) + f i (ρ i )ϕ(t)r ν (x) dxdt -m+n j=m+1 ˆ∞ 0 ˆ∞ 0 ρ j ϕ (t)r ν (x) + f j (ρ j )ϕ(t)r ν (x) dxdt. Finally, sending ν → ∞, 0 = -m i=1 ˆ∞ 0 f i (ρ i (t, 0-))ϕ(t)dt +m+n j=m+1 ˆ∞ 0 f j (ρ j (t, 0+))ϕ(t)dt, that gives (4.14).

Acknowledgments

Both authors wish to thank professor Roberto Natalini for suggesting the problem. The final version of this work was completed while the first author was invited professor at the University of Bourgogne Franche-Comté, projet "mobilité entrente 2018 -MoDySpaPop". The second author acknowledges the support of the Région Bourgogne Franche-Comté, projet 2017-2020 "Analyse mathématique et simulation numérique d'EDP issus de problèmes de contrôle et du trafic routier".

The first author is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

E-mail address: carlotta.donadello@univ-fcomte.fr URL: http://lmb.univ-fcomte.fr/Donadello-Carlotta