# Well-posedness for a monotone solver
for traffic junctions

Boris P. Andreianov  Laboratoire de Mathématiques et Physique Théorique CNRS
UMR7350
Université de Tours
Parc de Grandmont
37200 Tours France<EMAIL_ADDRESS>,  Giuseppe Maria
Coclite  Dipartimento di Matematica
Università degli Studi di Bari
Via Orabona 4
70125 Bari
Italy<EMAIL_ADDRESS>and  Carlotta Donadello  Laboratoire de
Mathématiques CNRS UMR6623
Université de Franche-Comté
16 route de Gray
25030 Besançon Cedex
France<EMAIL_ADDRESS>
###### Abstract.

In this paper we aim at proving well-posedness of solutions obtained as
vanishing viscosity limits for the Cauchy problem on a traffic junction where
$m$ incoming and $n$ outgoing roads meet. The traffic on each road is governed
by a scalar conservation law $\rho_{h,t}+f_{h}(\rho_{h})_{x}=0$, for
$h\in\\{1,\ldots,m+n\\}$. Our proof relies upon the complete description of
the set of road-wise constant solutions and its properties, which is of some
interest on its own. Then we introduce a family of Kruzhkov-type adapted
entropies at the junction and state a definition of admissible solution in the
same spirit as in [17, 15, 2, 1, 4].

###### Key words and phrases:

Traffic model, networks, conservation laws.

###### 2010 Mathematics Subject Classification:

90B20, 35L65

The second 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).
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.

##  1\. Introduction

We consider a junction consisting of $m$ incoming and $n$ outgoing roads.
Incoming roads are parametrized by $x\in\mathbbm{R}_{-}$ while outgoing road
by $x\in\mathbbm{R}_{+}$ 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) $\rho_{h,t}+f_{h}(\rho_{h})_{x}=0,\qquad\text{for }h=1,\ldots,\,m+n,$

where $\rho_{h}$ is the density of vehicles and $f_{h}$ is the flux on the
$h$-th road. For notational simplicity we call $\Omega_{h}$ the spatial domain
of the density $\rho_{h}$. Everywhere in the paper we use the index $i$ for
the $m$ incoming roads and $j$ for the $n$ outgoing roads (then
$\Omega_{i}=\mathbbm{R}_{-}$ for all $i=1,\ldots,m$ and
$\Omega_{j}=\mathbbm{R}_{+}$ for all $j=m+1,\ldots,m+n$), see Figure 1. The
fluxes $f_{h}$, $h=1,\ldots,m+n$, 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

  1. (F)

for all $h$, $f_{h}\in\mathbf{Lip}\left([0,R];\mathbbm{R}_{+}\right)$ with
$\|f_{h}^{\prime}\|_{\infty}\leq L_{h}$, $f_{h}(0)=0=f_{h}(R)$,
and there exists $\bar{\rho}_{h}\in\left]0,R\right[$ such that
$f_{h}^{\prime}(\rho)\leavevmode\nobreak\ (\bar{\rho}_{h}-\rho)>0$ for a.e.
$\rho\in[0,R]$,

  2. (NLD)

for all $h$, $f^{\prime}_{h}$ is not constant on any non-trivial subinterval
of $[0,R]$.

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. $t\in\mathbbm{R}_{+}$

(2)
$\sum_{i=1}^{m}f_{i}\left(\rho_{i}(t,0^{-})\right)=\sum_{j=m+1}^{m+n}f_{j}\left(\rho_{j}(t,0^{+})\right).$

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

Garavello and the second author, in [13], considered the Cauchy problem at the
junction and established the existence of weak solutions obtained as limit of
vanishing viscosity approximations. In [14], uniqueness for such solutions was
only proved in the special case $m=n$ and $f_{h}=f_{h^{\prime}}$ for all
$h,h^{\prime}\in\\{1,\ldots,m+n\\}$. The present paper naturally completes
those results as 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 [1, 4]. Let us
mention in passing that a large part of the concepts and results of [1, 4] can
be generalized to conservation laws on networks. However, a systematic
generalization of the theory of $\mathbf{L}^{1}$-dissipative germs is beyond
the scope of the present paper: we focus on characterization of solutions to
the concrete problem (1) originating from the vanishing viscosity
regularization of [13], and on well-posedness in this framework. Our
presentation is essentially self-contained. Let us only mention that in [24,
23], the authors provide general results, indirectly exploiting some insight
from [4], 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” $\mathcal{G}_{VV}$ (cf. [4, 3]) which is
introduced under the form that was put forward in [1] (Definition 16). 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 $\mathcal{G}_{VV}$ in terms of Oleinik-like
inequalities of [17] (Lemma 2.2). We prove that this germ is
$\mathbf{L}^{1}$-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 $\mathbf{L}^{1}$-contraction property of the admissible solutions
(Proposition 3.1), stability and uniqueness.

To justify existence and the relation to the vanishing viscosity
regularization of [13], following [7, 12, 4] we introduce a family of adapted
entropies at the junction (Definition 2.10). We put forward the Godunov finite
volume scheme inspired by [1] and justify its convergence and existence of
admissible solutions. In addition, we link the definition of (a part of) the
germ $\mathcal{G}_{VV}$ 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 [20, 14, 21], the recent survey [10] 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 [25]. 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 $\mathbf{L}^{1}$ sense.

####  1.1.1. Godunov’s flux

Let $u$ be the entropy solution to the scalar conservation law with Lipschitz
continuous flux

(3) $u_{t}+f(u)_{x}=0,\qquad(t,x)\in\mathbbm{R}_{+}\times\mathbbm{R}$

corresponding to the Riemann initial condition

(4) $u_{0}(x)=\begin{cases}a,&\quad\text{if }x<0,\\\ b,&\quad\text{if
}x>0.\end{cases}$

One calls Godunov flux the function which associates to the couple $(a,b)$ the
value $f(u(t,0^{-}))=f(u(t,0^{+}))$ (the two values are equal due to the
Rankine-Hugoniot condition). The analytical expression, see for example [22],
is given by

(5) $G(a,b)=\begin{cases}\min_{s\in[a,b]}f(s)\quad\text{if }a\leq b,\\\
\max_{s\in[b,a]}f(s)\quad\text{if }a\geq b.\end{cases}$

In the sequel, we denote by $\partial_{a}G$, resp. $\partial_{b}G$, 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., [16]):

  * •

Consistency: for all $a\in[0,R]$, $G(a,a)=f(a)$;

  * •

Monotonicity and Lipschitz continuity: There exists $L>0$ such that for all
$(a,b)\in[0,R]^{2}$ we have

(6) $0\leq\partial_{a}G(a,b)\leq L,\qquad\qquad-L\leq\partial_{b}G(a,b)\leq
0.$

####  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 [18], see also [6] for a review on this topic).
Consider the initial and boundary value problem (IBVP)

(7) $\begin{cases}u_{t}+f(u)_{x}=0,\qquad\text{for }(t,x)\>\text{in
}\mathbbm{R}_{+}\times\mathbbm{R}_{-}\\\ u(t,0)=u_{b}(t),\\\
u(0,x)=u_{0}(x),\end{cases}$

and assume that $u$ is a Kruzhkov entropy solution in the interior of the half
plane $\mathbbm{R}_{+}\times\mathbbm{R}_{-}$. Then $u$ satisfies the boundary
condition in the sense of Bardos-LeRoux-Nédélec (see [9]) if and only if its
trace $\gamma u(t)=u(t,0^{-})$ satisfies $f(\gamma u(t))=G(\gamma
u(t),u_{b}(t))$.

####  1.1.3. Strong boundary traces of local entropy solutions

Consider (3) locally, in $\mathbbm{R}_{+}\times(a,b)$ where $(a,b)$ is an
interval of $\mathbbm{R}$. Assume that
$u\in\mathbf{L}^{\infty}(\mathbbm{R}_{+}\times(a,b))$ satisfies the Kruzhkov
entropy inequalities (see [25] and (11) below). Assume that the Lipschitz flux
$f$ in (3) is non linearly degenerate in the sense that $f^{\prime}$ is not
identically zero on any interval (which follows from $(NLD)$). Then (see [28],
see also [4]) the function $u(t,\cdot)$ possesses one-sided limits: e.g., one
can define $u(t,b^{-}):=\gamma u(t)$ where $(\gamma u)(\cdot)$ is the strong
trace of $u$ on $\mathbbm{R}_{+}\times\\{b\\}$ in the $\mathbf{L}^{1}_{loc}$
sense: for all $\xi\in\mathcal{D}(\mathbbm{R}_{+})$,

(8) $\displaystyle\lim_{k\to
0^{+}}\frac{1}{k}\int_{\mathbbm{R}_{+}}\int_{b-k}^{b}\xi(t)|u(t,x)-\gamma
u(t)|\,dx\,dt=0.$

Notice that this property permits to extend the above interpretation of the
Bardos-LeRoux-Nédélec boundary condition for problem (7) to the case of
general $\mathbf{L}^{\infty}$ initial and boundary data, beyond the classical
$\mathbf{BV}$ framework.

####  1.1.4. Functional framework

Throughout the paper, we are interested in $\mathbf{L}^{\infty}$ solutions of
(1). We will denote dy $\Gamma$ the graph pictured in Figure 1 and use the
slightly abusive notation
$\mathbf{L}^{\infty}(\mathbbm{R}_{+}\times\Gamma;[0,R]^{m+n})$ for
$(m+n)$-uplets $(\rho_{1},\ldots,\rho_{m},\rho_{m+1},\ldots,\rho_{m+n})$ of
functions such that
$\rho_{i}\in\mathbf{L}^{\infty}(\mathbbm{R}_{+}\times\mathbbm{R}_{-};[0,R])$
for $i\in\\{1,\ldots,m\\}$ and
$\rho_{j}\in\mathbf{L}^{\infty}(\mathbbm{R}_{+}\times\mathbbm{R}_{+};[0,R])$
for $j\in\\{m+1,\ldots,m+n\\}$. Similarly,
$\mathbf{L}^{\infty}(\Gamma;[0,R]^{m+n})$ will denote the space of
$[0,R]$-valued initial data on the graph $\Gamma$.

###  1.2. The notion of admissible solution and the outline of the paper

Our goal is to re-visit and complement the work [13], which studies vanishing
viscosity limits for problem (1). 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 $\Omega_{h}$,
$h\in\\{1,\ldots,m+n\\}$;

  * •

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 $\Omega_{h}$, $h\in\\{1,\ldots,m+n\\}$ 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 $\vec{\rho}=(\rho_{1},\ldots,\rho_{m+n})$ in
$\mathbf{L}^{\infty}(\mathbbm{R}_{+}\times\Gamma;[0,R]^{m+n})$ as an
admissible solution if, first, for any $h\in\\{1,\ldots,m+n\\}$, $\rho_{h}$ is
a weak entropy solution in the sense of Kruzhkov in the interior of
$\Omega_{h}$. Second, recalling that $\rho_{i}$ (resp., $\rho_{j}$) admits a
strong trace $\rho_{i}(\cdot,0^{-})=\gamma_{i}\rho_{i}(\cdot)$ (resp.,
$\rho_{j}(\cdot,0^{+})=\gamma_{j}\rho_{j}(\cdot)$) at $x=0$, i.e.

(9) $\displaystyle\lim_{k\to
0^{-}}\frac{1}{k}\int_{\mathbbm{R}_{+}}\int_{k}^{0}\xi(t)|\rho_{i}(t,x)-\gamma_{i}\rho_{i}(t)|\,dx\,dt=0,$
$\displaystyle\qquad\text{for }i=1,\ldots,m,$ $\displaystyle\lim_{k\to
0^{+}}\frac{1}{k}\int_{\mathbbm{R}_{+}}\int_{0}^{k}\xi(t)|\rho_{j}(t,x)-\gamma_{j}\rho_{j}(t)|\,dx\,dt=0,$
$\displaystyle\qquad\text{for }j=m+1,\ldots,m+n,$

we require that the $(m+n)$-uple of traces satisfies condition (2) for a.e.
$t\in\mathbbm{R}_{+}$ and, moreover, for a.e. $t\in\mathbbm{R}_{+}$ 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 $\varepsilon>0$, the coincidence of
all $\rho^{\varepsilon}_{h}(t,\cdot)$, $h=1,\ldots,m+n$, at $x=0$; and that
taking the limit $\varepsilon\to 0$ relaxes this condition analogously to the
way in which the boundary condition in (7) 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
$\vec{u}_{0}\in\mathbf{L}^{\infty}(\Gamma;[0,R]^{m+n})$,
$\vec{u}_{0}=(u^{0}_{1},\ldots,u^{0}_{m+n})$, we look for a function
$\vec{\rho}=(\rho_{1},\ldots,\rho_{m+n})$ in
$\mathbf{L}^{\infty}(\mathbbm{R}_{+}\times\Gamma;[0,R]^{m+n})$ such that for
any $h\in\\{1,\ldots,m+n\\}$, $\rho_{h}$ is a weak entropy solution of the
initial and boundary value problem (IBVP)

(10) $\begin{cases}\rho_{h,t}+f_{h}(\rho_{h})_{x}=0,&\qquad\text{on
}]0,T[\times\Omega_{h},\\\ \rho_{h}(t,0)=v_{h}(t),&\qquad\text{on }]0,T[,\\\
\rho_{h}(0,x)=u^{0}_{h}(x),&\qquad\text{on }\Omega_{h},\end{cases}$

where the set of boundary conditions $\vec{v}:\mathbbm{R}_{+}\to[0,R]^{m+n}$
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 $\rho_{h}$ is an entropy weak solution of the initial
and boundary values problem (10) if

  * •

For any test function $\xi$ in
$\mathcal{D}(\mathbbm{R}_{+}\times\Omega_{h};\mathbbm{R}_{+})$,
$\xi|_{\partial\Omega_{h}}=0$, and for any $k\in[0,R]$ there holds

(11)
$\int_{\mathbbm{R}_{+}}\int_{\Omega_{h}}\left\\{|\rho_{h}-k|\xi_{t}+q_{h}(\rho_{h},k)\xi_{x}\right\\}\,dx\,dt+\int_{\Omega_{h}}|u^{0}_{h}(x)-k|\xi(0,x)\,dx\geq
0,$

$q_{h}(u,k):=\operatorname{sign}(u-k)(f_{h}(u)-f_{h}(k))$ being the Kruzhkov
entropy flux associated to $f_{h}$.

  * •

For a.e. $t\in\mathbbm{R}_{+}$, $\gamma_{h}\rho_{h}(t)$ satisfies the boundary
condition in the sense of Bardos-Le Roux-Nédélec (BLN), which we express under
the form (cf. Section 1.1)

(12) $\displaystyle f_{i}(\gamma_{i}\rho_{i})=G_{i}(\gamma_{i}\rho_{i},v_{i})$
$\displaystyle\quad\text{if }i\in\\{1,\ldots,m\\};$ (13) $\displaystyle
f_{j}(\gamma_{j}\rho_{j})=G_{j}(v_{j},\gamma_{j}\rho_{j})$
$\displaystyle\quad\text{if }j\in\\{m+1,\ldots,m+n\\},$

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 $h$:

(14) $\text{for all $h\in\\{1,\ldots,m+n\\}$, for a.e.
$t\in\mathbbm{R}_{+}$}\;\;v_{h}(t)=p(t).$

We refer to [1, 5] for detailed motivations, in the discontinuous-flux
setting. The criterion for the choice of $p$ is the conservativity condition
(2); due to (12) and (13), we can now express it in the form

(15)
$\sum_{i=1}^{m}G(\gamma_{i}\rho_{i}(t),p(t))=\sum_{j=m+1}^{m+n}G(p(t),\gamma_{j}\rho_{j}(t)),\quad\text{for
a.e. }t\in\mathbbm{R}_{+}.$

Observe that formally, (14) and (15) close the coupled system of IBVP’s (10).

######  Definition 1.2.

Given an initial condition
$\vec{u}_{0}\in\mathbf{L}^{\infty}(\Gamma;[0,R]^{m+n})$, we call
$\vec{\rho}=(\rho_{1},\ldots,\rho_{m+n})$ in
$\mathbf{L}^{\infty}\left(\mathbbm{R}_{+}\times\Gamma;[0,R]^{m+n}\right)$ an
admissible solution of (1) associated with $\vec{u}_{0}$ if there exists a
function $p$ in $\mathbf{L}^{\infty}(\mathbbm{R}_{+};[0,R])$ such that for any
$h\in\\{1,\ldots,m+n\\}$ $\rho_{h}$ is a solution of IBVP (10) in the sense of
Definition 1.1 with $v_{h}$, $h\in\\{1,\ldots,m+n\\}$ chosen to fulfill (14),
and such that $\vec{\rho}$, $p$ 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 (11)) account for the
admissibility of $\vec{\rho}$ at the junction. We will establish well-
posedness of problem (1) 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.

##  2\. 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
$h\in\\{1,\ldots,2m\\}$ the constant vector function
$\vec{k}=(k,\ldots,k)\in\mathbbm{R}^{2m}$ satisfies the conditions above with
$p(t)=k$. This kind of stationary solution is employed in [13] 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
$\vec{k}=(k_{1},k_{2})$ 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
$\mathbbm{R}_{+}\times\Gamma$ constant on each road of $\Gamma$. This
definition will permit us to describe the Riemann solver and the associated
fluxes at the junction defined in Lemma 2.4.

###  2.1. Definition of the vanishing viscosity germ

In this section we describe the stationary admissible solutions of (1) that
are constant on each road of $\Gamma$. Because of the analogy with the
discontinuous-flux setting of [1, 4] 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 $[0,R]^{m+n}$ defined by

(16) $\mathcal{G}_{VV}=\left\\{\begin{aligned}
\vec{u}=(u_{1},\ldots,u_{m+n}):&\>\exists p\in[0,R]\>\text{such that}\\\
&\sum_{i=1}^{m}G_{i}(u_{i},p)=\sum_{j=m+1}^{m+n}G_{j}(p,u_{j})\\\
&\>G_{i}(u_{i},p)=f_{i}(u_{i}),\quad G_{j}(p,u_{j})=f_{j}(u_{j}),\>\forall
i,\,j\end{aligned}\right\\}.$

It is immediate to see that $\vec{u}\in\mathcal{G}_{VV}$ if and only if, seen
as a vector function in $\mathbbm{R}_{+}\times\Gamma\to[0,R]^{m+n}$, $\vec{u}$
provides a solution of (1) in the sense of Definition 1.2 such that each
component $u_{h}$ of $\vec{u}$ is constant both in $t$ and $x$.

Following [4], see also [17], we can characterize $\mathcal{G}_{VV}$ by a set
of inequalities reminiscent of the celebrated Oleinik condition for scalar
conservation laws. Here and in the following we use the notation $I[a,b]$ to
indicate the closed interval $[\min\\{a,b\\},\max\\{a,b\\}]$ in $\mathbbm{R}$.

######  Lemma 2.2.

The vanishing viscosity germ $\mathcal{G}_{VV}$ coincides with the subset of
$[0,R]^{m+n}$ defined by

(17) $\left\\{\begin{aligned} \vec{u}=&(u_{1},\ldots,u_{m+n}):\>\exists
p\in[0,R]\>\text{such that}\\\
&\sum_{i=1}^{m}G_{i}(u_{i},p)=\sum_{j=m+1}^{m+n}G_{j}(p,u_{j}),\\\ &\forall
s\in I[u_{i},p]\quad(p-u_{i})(f_{i}(s)-f_{i}(u_{i}))\geq 0,\>\text{for
}i=1,\ldots,m,\\\ &\forall s\in
I[p,u_{j}]\quad(u_{j}-p)(f_{j}(s)-f_{j}(u_{j}))\geq 0,\>\text{for
}j=m+1,\ldots,m+n\end{aligned}\right\\}.$

###### Proof.

Actually, the value $p$ in (17) coincides with the value $p$ in (16). One only
needs to show that

$\forall s\in I[u_{i},p]\colon\quad(u_{i}-p)(f_{i}(s)-f_{i}(u_{i}))\geq
0\Leftrightarrow G_{i}(u_{i},p)=f_{i}(u_{i}),$

for any $i$. This readily comes from the definition of the Godunov flux, as
the relation $G_{i}(u_{i},p)=f_{i}(u_{i})$ rewrites as

(18) $\displaystyle f_{i}(u_{i})=\min_{s\in[u_{i},p]}f_{i}(s),\quad\text{if
}(p-u_{i})\geq 0,$ $\displaystyle
f_{i}(u_{i})=\max_{s\in[p,u_{i}]}f_{i}(s),\quad\text{if }(p-u_{i})\leq 0.$

The proof for the $j$-index case is analogous. ∎

In order to exhibit the key properties of $\mathcal{G}_{VV}$, we start with
the following technical lemma which is crucial for the existence theory. It
relies upon the monotonicity properties of the Godunov fluxes $G_{h}$,
$h\in\\{1,\ldots,m+n\\}$; we defer to Remark 3 for its interpretation in terms
of the Godunov fluxes for the junction.

######  Lemma 2.3.

Given $\vec{u}=(u_{1},\ldots,u_{m+n})$ in $[0,R]^{m+n}$, consider the problem

(19) $\text{find $p_{\vec{u}}\in[0,R]$ such
that\;\;}\sum_{i=1}^{m}G_{i}(u_{i},p_{\vec{u}})=\sum_{j=m+1}^{m+n}G_{j}(p_{\vec{u}},u_{j}).$

  * (i)

The set $\mathcal{P}_{\vec{u}}$ of solutions of (19) is non-empty.

  * (ii)

The values $(G_{i}(u_{i},p_{\vec{u}}))_{i\in\\{1,\ldots,m\\}}$,
$(G_{j}(p_{\vec{u}},u_{j}))_{j\in\\{m+1,\ldots,m+n\\}}$ do not depend on the
choice of the value $p_{\vec{u}}\in\mathcal{P}_{\vec{u}}$.

######  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) $f_{h}(\rho)=-h\rho^{2}+h,\qquad\text{for }h=1,2,3.$

Remark that, for the reader’s convenience, we consider now $\rho\in[-1,1]$,
$f_{h}(-1)=f_{h}(1)=0$ and $\bar{\rho}_{h}=0$. 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 $u_{h}\neq 0$, let $\hat{u}_{h}\neq u_{h}$ be
the only solution to $f_{h}(u_{h})=f_{h}(\hat{u}_{h})$. By using the standard
Riemann Solver, see [22], it is easy to check that the values of
$G_{i}(u_{i},\cdot)$, $i=1,\,2$, as functions of $p$, are the following

  * •

If $u_{i}\leq 0$, then $G_{i}(u_{i},p)=f_{i}(u_{i})$ for all
$p\leq\hat{u}_{i}$ and $G_{i}(u_{i},p)=f_{i}(p)$ for all $p\geq\hat{u}_{i}$;

  * •

If $u_{i}\geq 0$, then $G_{i}(u_{i},p)=f_{i}(0)$ for all $p\leq 0$ and
$G_{i}(u_{i},p)=f_{i}(p)$ for all $p\geq 0$.

Similarly, the values of $G_{3}(\cdot,u_{3})$, as a function of $p$ are

  * •

If $u_{3}\leq 0$, then $G_{3}(p,u_{3})=f_{3}(p)$ for all $p\leq 0$ and
$G_{3}(p,u_{3})=f_{3}(0)$ for all $p\geq 0$;

  * •

If $u_{3}\geq 0$, then $G_{3}(p,u_{3})=f_{3}(p)$ for all $p\leq\hat{u}_{3}$
and $G_{3}(p,u_{3})=f_{3}(u_{3})$ for all $p\geq\hat{u}_{3}$.

One can check that if we take, as an example, $u_{1}=-\sqrt{1/2}$, $u_{2}=1/4$
and $u_{3}=\sqrt{1/6}$, then all the values of $p$ between $[-\sqrt{1/6},0]$
satisfy the relation

(21) $G_{1}(u_{1},p)+G_{2}(u_{2},p)=G_{3}(p,u_{3}),$

and that for all these values of $p$, the collection $\vec{G}^{*}(\vec{u})$ of
fluxes at the junction will be given by

$\vec{G}^{*}(\vec{u})=\Bigl{(}G_{1}(u_{1},p),G_{2}(u_{2},p),G_{3}(p,u_{3})\Bigr{)}=(1/2,2,5/2).$

Remark that, as explained in the paper [26], the functions
$G_{i}(u_{i},\cdot)$ and $G_{j}(\cdot,u_{j})$ 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
$\Delta_{i}$ of the $i$-th incoming road and the equilibrium supply function
$\Sigma_{j}$ of the $j$-th outgoing road can be defined as

(22) $u_{i}\mapsto\Delta_{i}(u_{i})=\max_{p}\\{G_{i}(u_{i},p)\\},\qquad
u_{j}\mapsto\Sigma_{j}(u_{j})=\max_{p}\\{G_{j}(p,u_{j})\\}.$

Now, we prove Lemma 2.3.

###### Proof.

(i) Given $\vec{u}$ we define the functions $\Phi^{in}_{\vec{u}}$ and
$\Phi^{out}_{\vec{u}}$ from $[0,R]$ to $\mathbbm{R}$ by

(23)
$\Phi^{in}_{\vec{u}}:p\mapsto\sum_{i=1}^{m}G_{i}(u_{i},p),\qquad\Phi^{out}_{\vec{u}}:p\mapsto\sum_{j=m+1}^{m+n}G_{j}(p,u_{j}).$

A quick direct calculation gives us

(24) $\displaystyle G_{i}(u_{i},0)=\max_{s\in[0,u_{i}]}f_{i}(s)\geq 0,$
$\displaystyle\quad G_{i}(u_{i},R)=\min_{s\in[u_{i},R]}f_{i}(s)=0,$
$\displaystyle G_{j}(0,u_{j})=\min_{s\in[0,u_{j}]}f_{j}(s)=0,$
$\displaystyle\quad G_{j}(R,u_{j})=\max_{s\in[u_{j},R]}f_{i}(s)\geq 0,$

so that

(25) $\Phi^{in}_{\vec{u}}(0)\geq 0=\Phi^{out}_{\vec{u}}(0)\quad\text{and
}\Phi^{in}_{\vec{u}}(R)=0\leq\Phi^{out}_{\vec{u}}(R).$

The existence of at least one solution $p_{\vec{u}}$ of (19) is ensured by the
continuity of $\Phi^{in}_{\vec{u}}-\Phi^{out}_{\vec{u}}$ on $[0,R]$ and the
intermediate value theorem.

(ii) It may happen that there exist several values of $p$ such that
$\Phi^{in}_{\vec{u}}(p)$ and $\Phi^{out}_{\vec{u}}(p)$ coincide. From the
Lipschitz continuity and monotonicity property of Godunov flux (see § 1.1.1)
we have

(26) $\forall i\in\\{1,\ldots,m\\}\;\;\partial_{b}G_{i}(u_{i},p)\leq
0,\;\;\forall j\in\\{m+1,\ldots,m+n\\}\;\;\partial_{a}G_{j}(p,u_{j})\geq 0,$

which means in particular that $\Phi^{in}_{\vec{u}}-\Phi^{out}_{\vec{u}}$ is
non-strictly decreasing. Therefore, the set $\mathcal{P}_{\vec{u}}$ of
solutions of (19) is a closed sub-interval of $[0,R]$. Since, moreover, each
term of the sums defining $\Phi^{in}_{\vec{u}}-\Phi^{out}_{\vec{u}}$ has the
same monotonicity, we find that all these terms are constant on
$\mathcal{P}_{\vec{u}}$. ∎

Next, consider the map $\vec{G}^{*}:[0,R]^{m+n}\mapsto\mathbbm{R}^{m+n}$ which
is well defined, thanks to (ii):

(27)
$\vec{G}^{*}(\vec{u})=\Bigl{(}G_{1}^{*}(\vec{u}),\ldots,G_{m+n}^{*}(\vec{u})\Bigr{)},\;\;\begin{array}[]{l}G^{*}_{i}(\vec{u}):=G_{i}(u_{i},p_{\vec{u}}),\;i\in\\{1,\ldots,m\\},\\\\[2.0pt]
G^{*}_{j}(\vec{u}):=G_{j}(p_{\vec{u}},u_{j}),\;j\in\\{m+1,\ldots,m+n\\}\end{array}$

and the map $F^{*}:[0,R]^{m+n}\mapsto\mathbbm{R}$ defined by

(28)
$F^{*}(\vec{u}):=\sum_{i=1}^{m}G^{*}_{i}(\vec{u})\equiv\sum_{j=m+1}^{m+n}G^{*}_{j}(\vec{u}).$

######  Lemma 2.4.

With the above definitions, the following properties hold.

  * (i)

For each $i\in\\{1,\ldots,m\\}$, the map $\vec{u}\mapsto G^{*}_{i}(\vec{u})$
fulfills

$\partial_{u_{i}}G^{*}_{i}\leq L_{i}\;\;\;\text{and}\;\;\;\forall
h\in\\{1,\ldots,m+n\\},\,h\neq i\colon\;\;\partial_{u_{h}}G^{*}_{i}\leq 0.$

For each $j\in\\{m+1,\ldots,m+n\\}$, the map $\vec{u}\mapsto
G^{*}_{j}(\vec{u})$ fulfills

$\partial_{u_{j}}G^{*}_{j}\geq-L_{j}\;\;\;\text{and}\;\;\;\forall
h\in\\{1,\ldots,m+n\\},\,h\neq j\colon\;\;\partial_{u_{h}}G^{*}_{j}\geq 0.$

  * (ii)

The map $\vec{u}\mapsto F^{*}(\vec{u})$ fulfills

$\forall i\in\\{1,\ldots,m\\}\colon\;\;\partial_{u_{i}}F^{*}\geq
0\;\;\;\text{and}\;\;\;\forall
j\in\\{m+1,\ldots,m+n\\}\colon\;\;\partial_{u_{j}}F^{*}\leq 0.$

The above differential inequalities should be understood in the sense of
distributions, e.g., “$\partial_{u_{i}}F^{*}\geq 0$” means that $F^{*}$ is
non-decreasing in the variable $u_{i}$.

###### Proof.

(i) Without loss of generality, we can fix the normalization
$p_{\vec{u}}:=\min\mathcal{P}_{\vec{u}}$. Observe that

(29) $\text{the map $\vec{u}\mapsto p_{\vec{u}}$ is monotone non-decreasing in
each component $u_{h}$, $h\in\\{1,\ldots,m+n\\}$}.$

Indeed, let $\vec{u}\leq\vec{v}$ in the sense of the per component order: for
all $h\in\\{1,\ldots,m+n\\}$ $u_{h}\leq v_{h}$. Bearing in mind formula (23),
the monotonicity properties of the Godunov fluxes $G_{i}(\cdot,p)$ and
$G_{j}(p,\cdot)$ and the definition of $p_{\vec{u}}$ yield

$0=\Phi^{in}_{\vec{u}}(p_{\vec{u}})-\Phi^{out}_{\vec{u}}(p_{\vec{u}})\leq\Phi^{in}_{\vec{v}}(p_{\vec{u}})-\Phi^{out}_{\vec{v}}(p_{\vec{u}}).$

Hence the monotonicity of $\Phi^{in}_{\vec{v}}-\Phi^{out}_{\vec{v}}$ exhibited
in the proof of Lemma 2.3(ii) and the normalization of $p_{\vec{u}}$ ensure
that $p_{\vec{u}}\leq p_{\vec{v}}$, which proves (29).

Now the monotonicity claims of (i) are immediate from the monotonicity
properties of $G_{h}$ and from (29). 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
$\vec{u}=(u_{1},u_{2},\ldots,u_{m+n})$ and
$\vec{v}=(v_{1},u_{2},\ldots,u_{m+n})$ with $v_{1}>u_{1}$, we have

$\displaystyle G^{*}_{1}(\vec{v})-G^{*}_{1}(\vec{u})$
$\displaystyle=G_{1}(v_{1},p_{\vec{v}})-G_{1}(u_{1},p_{\vec{u}})$
$\displaystyle=G_{1}(v_{1},p_{\vec{v}})-G_{1}(u_{1},p_{\vec{v}})+G_{1}(u_{1},p_{\vec{v}})-G_{1}(u_{1},p_{\vec{u}})$
$\displaystyle\leq G_{1}(v_{1},p_{\vec{v}})-G_{1}(u_{1},p_{\vec{v}})\leq
L_{1}(v_{1}-u_{1}),$

which proves the claim.

(ii) The monotonicity properties of $F^{*}$ readily stem from (29). If we look
at the dependence of $F^{*}$ in $u_{i}$, $i\in\\{1,\ldots,m\\}$, it is enough
to represent $F^{*}$ with the last expression in (28) and combine (29) with
the monotonicity of $G_{j}(\cdot,b)$, $j\in\\{m+1,\ldots,m+n\\}$. If we look
at the dependence of $F^{*}$ in $u_{j}$, $j\in\\{m+1,\ldots,m+n\\}$, then we
represent $F^{*}$ with the first expression in (28) and use the monotonicity
of $G_{i}(a,\cdot)$, $i\in\\{1,\ldots,m\\}$. ∎

######  Remark 2.

Actually, in the context of 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 (19):

$G^{*}_{1}(p_{\vec{u}})=-\sum_{i=2}^{m}G_{i}(u_{i},p_{\vec{u}})+\sum_{j=m+1}^{m+n}G_{j}(p_{\vec{u}},u_{j}),$

and exploit (29). 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
$\mathcal{G}_{VV}$. For any $h\in\\{1,\ldots,m+n\\}$ let
$q_{h}:[0,R]^{2}\to\mathbbm{R}$ denote the Kruzhkov entropy flux

(30) $q_{h}(u,v)=\operatorname{sign}(u-v)(f_{h}(u)-f_{h}(v)).$

######  Lemma 2.5.

For any $\vec{k}_{1}$, $\vec{k}_{2}$ in $\mathcal{G}_{VV}$ with
$\vec{k}_{\ell}=(k_{1}^{\ell},\ldots,k_{m+n}^{\ell})$, $\ell=1,\,2$, there
holds

(31)
$\Delta(\vec{k}_{1},\vec{k}_{2}):=\sum_{i=1}^{m}q_{i}(k_{i}^{1},k_{i}^{2})-\sum_{j=m+1}^{m+n}q_{j}(k_{j}^{1},k_{j}^{2})\geq
0.$

Inequality (31) will be exploited in this work to prove that the solutions we
consider satisfy a generalized Kato’s inequality. The
$\mathbf{L}^{1}$-contraction property and uniqueness will follow.

###### Proof.

To increase readability we use the shorter notation
$s_{h}=\operatorname{sign}(k_{h}^{1}-k_{h}^{2})$, for $h=1,\ldots,m+n$, and we
adopt the convention $\operatorname{sign}(0)=0$. Up to reordering the incoming
and outgoing roads we can assume that $s_{i}\geq 0$ for
$i\in\\{1,\ldots,\alpha\\}$, $s_{i}=-1$ for $i=\alpha+1$, $s_{i}\leq 0$ for
$i\in\\{\alpha+1,\ldots,m\\}$, $s_{j}\geq 0$ for $j\in\\{m+1,\ldots,\beta\\}$,
$s_{j}=-1$ for $i=\beta+1$ and $s_{i}\leq 0$ for
$i\in\\{\beta+1,\ldots,m+n\\}$.

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

Case 1:

Assume that $s_{i}s_{j}\geq 0$ for all
$(i,j)\in\\{1,\ldots,m\\}\times\\{m+1,\ldots,m+n\\}$.

Case 2:

Assume that $s_{i}s_{j}\leq 0$ for all
$(i,j)\in\\{1,\ldots,m\\}\times\\{m+1,\ldots,m+n\\}$.

Case 3:

Let $\alpha\neq 1,\,m$ and $\beta\neq m+1,\,m+n$.

Case 1 This is the easiest case. The definition of $\mathcal{G}_{VV}$ implies
that

(32)
$\displaystyle\sum_{i=1}^{m}q_{i}(k_{i}^{1},k_{i}^{2})-\sum_{j=m+1}^{m+n}q_{j}(k_{j}^{1},k_{j}^{2})=\pm\sum_{i=1}^{m}\left(f_{i}(k_{i}^{1})-f_{i}(k_{i}^{2})\right)\mp\sum_{j=m+1}^{m+n}\left(f_{j}(k_{j}^{1})-f_{j}(k_{j}^{2})\right)$
$\displaystyle=\pm\sum_{i=1}^{m}G_{i}(k_{i}^{1},p_{1})\mp\sum_{j=m+1}^{m+n}G_{j}(p_{1},k_{j}^{1})\mp\sum_{i=1}^{m}G_{i}(k_{i}^{2},p_{2})\pm\sum_{j=m+1}^{m+n}G_{j}(p_{2},k_{j}^{2})=0,$

where we call $p_{1}$ and $p_{2}$ the values of $p$ associated respectively to
$\vec{k}_{1}$ and $\vec{k}_{2}$.

Case 2

(33)
$\displaystyle\sum_{i=1}^{m}q_{i}(k_{i}^{1},k_{i}^{2})-\sum_{j=m+1}^{m+n}q_{j}(k_{j}^{1},k_{j}^{2})=\pm\sum_{i=1}^{m}\left(f_{i}(k_{i}^{1})-f_{i}(k_{i}^{2})\right)\pm\sum_{j=m+1}^{m+n}\left(f_{j}(k_{j}^{1})-f_{j}(k_{j}^{2})\right)$
$\displaystyle=\pm 2\sum_{i=1}^{m}G_{i}(k_{i}^{1},p_{1})\mp
2\sum_{i=1}^{m}G_{i}(k_{i}^{2},p_{2}).$

Assume, to fix the ideas, that $s_{i}>0$, and $s_{j}<0$. Then the expression
above writes as

(34) $2F^{*}(\vec{k}^{1})-2F^{*}(\vec{k}^{2})$

with the notation (28), 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 $\vec{w}$, whose components satisfy
$w_{h}=\min\\{k_{h}^{1},\,k_{h}^{2}\\}$. It is important to notice that the
vector $\vec{w}$ does not belong (in general) to $\mathcal{G}_{VV}$. From the
proof of Lemma 2.4 (see (29)) we deduce that
$p_{\vec{w}}\leq\min\\{p_{1},p_{2}\\}$ where we denoted
$p_{s}:=p_{\vec{k}^{s}}$, $s=1,2$. We have, using the notation (23) and the
fact that $w_{i}=k_{i}^{1}$ for $i\in\\{1,\ldots,m\\}$ and $w_{j}=k_{j}^{2}$
for $j\in\\{m+1,\ldots,m+n\\}$,

$\displaystyle\sum_{i=1}^{m}q_{i}(k_{i}^{1},k_{i}^{2})=$
$\displaystyle\sum_{i=1}^{m}s_{i}\left(G_{i}(k_{i}^{1},p_{1})-G_{i}(k_{i}^{2},p_{2})\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{\alpha}s_{i}\left(G_{i}(k_{i}^{1},p_{1})-G_{i}(w_{i},p_{2})\right)+\sum_{i=\alpha+1}^{m}s_{i}\left(G_{i}(w_{i},p_{1})-G_{i}(k_{i}^{2},p_{2})\right)$
$\displaystyle\geq$
$\displaystyle\sum_{i=1}^{\alpha}s_{i}\left(G_{i}(k_{i}^{1},p_{1})-G_{i}(w_{i},p_{\vec{w}})\right)+\sum_{i=\alpha+1}^{m}s_{i}\left(G_{i}(w_{i},p_{\vec{w}})-G_{i}(k_{i}^{2},p_{2})\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{m}\operatorname{sign}(k_{i}^{1}-w_{i})\left(G_{i}(k_{i}^{1},p_{1})-G_{i}(w_{i},p_{\vec{w}})\right)$
$\displaystyle+\sum_{i=1}^{m}\operatorname{sign}(k_{i}^{2}-w_{i})\left(G_{i}(k_{i}^{2},p_{2})-G_{i}(w_{i},p_{\vec{w}})\right)$
$\displaystyle=$
$\displaystyle\Phi^{in}_{\vec{k}^{1}}(p_{1})-\Phi^{in}_{\vec{w}}(p_{\vec{w}})-\sum_{i=\alpha+1}^{m}\left(G_{i}(k_{i}^{1},p_{1})-G_{i}(k_{i}^{2},p_{\vec{w}})\right)$
$\displaystyle+\Phi^{in}_{\vec{k}^{2}}(p_{2})-\Phi^{in}_{\vec{w}}(p_{\vec{w}})-\sum_{i=1}^{\alpha}\left(G_{i}(k_{i}^{2},p_{2})-G_{i}(k_{i}^{2},p_{\vec{w}})\right).$

Analogous computations on the sum involving the outgoing roads give a similar
result. Therefore we obtain

$\displaystyle\sum_{i=1}^{m}q_{i}(k_{i}^{1},k_{i}^{2})-\sum_{j=m+1}^{m+n}q_{j}(k_{j}^{1},k_{j}^{2})$
$\displaystyle\geq-\sum_{i=\alpha+1}^{m}\left(G_{i}(k_{i}^{1},p_{1})-G_{i}(k_{i}^{i},p_{w})\right)-\sum_{i=1}^{\alpha}\left(G_{i}(k_{i}^{2},p_{2})-G_{i}(k_{i}^{2},p_{w})\right)$
$\displaystyle\quad+\sum_{j=\beta+1}^{m+n}\left(G_{j}(k_{j}^{1},p_{1})-G_{i}(k_{j}^{1},p_{w})\right)+\sum_{j=m+1}^{\beta}\left(G_{j}(k_{j}^{2},p_{2})-G_{j}(k_{j}^{2},p_{w})\right).$

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)
$\mathcal{RS}:[0,R]^{m+n}\to[0,R]^{m+n},\qquad\mathcal{RS}(\vec{u}_{0})=\vec{u},$

with the following properties:

  1. (1)

There exists $p\in[0,R]$ such that

(36) $\displaystyle f_{i}(u_{i})=G_{i}(u^{0}_{i},p)\quad\text{if
}i\in\\{1,\ldots,m\\};$ (37) $\displaystyle
f_{j}(u_{j})=G_{j}(p,u^{0}_{j})\quad\text{if }j\in\\{m+1,\ldots,m+n\\},$

where $G_{i}$ and $G_{j}$ are the Godunov fluxes associated to $f_{i}$ and
$f_{j}$ respectively, and
$\sum_{i=1}^{m}f_{i}(u_{i})=\sum_{j=m+1}^{m+n}f_{j}(u_{j})$.

  2. (2)

The consistency condition
$\mathcal{RS}\left(\mathcal{RS}(\vec{u}_{0})\right)=\mathcal{RS}(\vec{u}_{0})$
holds for all $\vec{u}_{0}$ in $[0,R]^{m+n}$.

It is clear from the definition above that $\mathcal{G}_{VV}$ 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 $\vec{u}_{0}$ in $[0,R]^{m+n}$ there exists a
self-similar function $\vec{\rho}=(\rho_{1},\ldots,\rho_{m+n})$ in
$\mathbf{L}^{\infty}\left(\mathbbm{R}_{+}\times\Gamma;[0,R]^{m+n}\right)$
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
$\vec{\gamma\rho}$ of traces of this solution at the junction belongs to
$\mathcal{G}_{VV}$.

###### Proof.

Given $\vec{u}_{0}$ in $[0,R]^{m+n}$ we apply Lemma 2.3 and find
$p:=p_{\vec{u}_{0}}\in[0,R]$ such that
$\sum_{i=1}^{m}G_{i}(u_{i}^{0},p)=\sum_{j=m+1}^{m+n}G_{j}(p,u_{j}^{0})$. Then
we consider the $h$ initial boundary value problems with constant data

(38) $\begin{cases}\rho_{h,t}+f_{h}(\rho_{h})_{x}=0,&\qquad\text{on
}]0,T[\times\Omega_{h},\\\ \rho_{h}(t,0)=p,&\qquad\text{on }]0,T[,\\\
\rho_{h}(0,x)=u^{0}_{h},&\qquad\text{on }\Omega_{h}.\end{cases}$

Call $\rho_{h}$ the Kruzhkov entropy weak solution to (38) and
$\gamma_{h}\rho_{h}$ its (strong) trace at $\partial\Omega_{h}$, 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
$(t,x)\mapsto(ct,cx)$ for all $c>0$, the solution is self-similar, i.e., each
of the components $\rho_{h}$ depends only on the ratio $\frac{x}{t}$. To
conclude the proof, it is enough to observe that

(39) $\displaystyle G_{h}(\gamma_{h}\rho_{h},p)$
$\displaystyle=f_{h}(\gamma_{h}\rho_{h})=G_{h}(u_{h}^{0},p),\quad\text{if
}h\leq m,$ $\displaystyle G_{h}(p,\gamma_{h}\rho_{h})$
$\displaystyle=f_{h}(\gamma_{h}\rho_{h})=G_{h}(p,u_{h}^{0}),\quad\text{otherwise},$

because in this case $\vec{\gamma\rho}$ fulfills the definition of
$\mathcal{G}_{VV}$ with $p=p_{\vec{u}_{0}}$ and consequently, one sees that
$\vec{\rho}$ is an admissible solution of the Riemann problem at the junction.
Equalities (39) follow from the observations of [18]. For the sake of
completeness, let’s point out that, e.g., for all $i\in\\{1,\ldots,m\\}$,

(40) $\gamma_{i}\rho_{i}=\left\\{\begin{aligned}
&\text{argmin}_{[u^{0}_{i},p]}f_{i},\quad\text{if }u^{0}_{i}\leq p,\\\
&\text{argmax}_{[p,u^{0}_{i}]}f_{i},\quad\text{if }u^{0}_{i}\geq
p.\end{aligned}\right.$

With this value $\gamma_{i}\rho_{i}$, the classical Riemann problem with
endstates $u^{0}_{i}$ and $\gamma_{i}\rho_{i}$ is solved only with the waves
of negative speed while the classical Riemann problem with endstates
$\gamma_{i}\rho_{i}$ 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 $f_{i}(\gamma_{i}\rho_{i})$,
$G_{i}(u^{0}_{i},\gamma_{i}\rho_{i})$, $G_{i}(\gamma_{i}\rho_{i},p)$ and
$G_{i}(u^{0}_{i},p)$ coincide. ∎

######  Remark 3.

Given $\vec{u}\in[0,R]^{m+n}$, let $p_{\vec{u}}$ be defined by (19). According
to the above proof, the self-similar admissible solution $\vec{\rho}$ of the
Riemann problem fulfills, with the notation (27),

$\displaystyle
f_{i}(\gamma_{i}\rho_{i}(t))=G_{i}(u_{i},p_{\vec{u}})=G^{*}_{i}(\vec{u})$
$\displaystyle\;\;\text{for all}\;i\in\\{1,\ldots,m\\},$ $\displaystyle
f_{j}(\gamma_{j}\rho_{j}(t))=G_{j}(p_{\vec{u}},u_{j})=G^{*}_{j}(\vec{u})$
$\displaystyle\;\;\text{for all}\;j\in\\{1+m,\ldots,m+n\\}$

(recall that while $p_{\vec{u}}$ may not be unique, the above flux values are
uniquely defined). We see that given $\vec{u}$, the collection of values
$\vec{G}^{*}(\vec{u})\in\mathbbm{R}^{m+n}$ 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 $\Omega_{i}$ for
$i\in\\{1,\ldots,m\\}$ and incoming into $\Omega_{j}$ for
$j\in\\{m+1,\ldots,m+n\\}$).

Lemma 2.4(i) states that each flux $G^{*}_{h}$ is one-sided Lipschitz with
respect to $u_{h}$ and monotone with respect to $u_{h^{\prime}}$ for all
$h^{\prime}\neq h$. Observe that the last equality in (28) expresses the
conservation property at junction. These properties will permit us to
formulate a monotone, in the sense of [16], conservative finite volume scheme
for approximation of admissible solutions of (1).

######  Lemma 2.8.

Let $\vec{u}=(u_{1},\ldots,u_{m+n})$ in $[0,R]^{m+n}$ be such that the
following family of inequalities holds

(41)
$\forall\vec{k}=(k_{1},\ldots,k_{m+n})\in\mathcal{G}_{VV}\colon\;\;\;\Delta(\vec{u},\vec{k})=\sum_{i=1}^{m}q_{i}(u_{i},k_{i})-\\!\sum_{j=m+1}^{m+n}q_{j}(u_{j},k_{j})\geq
0.$

Then $\vec{u}$ is in $\mathcal{G}_{VV}$. Moreover, being understood that the
fluxes $f_{h}$, $h\in\\{1,\ldots,m+n\\}$, fulfill the condition (F), the
conclusion “$\vec{u}\in\mathcal{G}_{VV}$” still holds if in (41), the
condition “$\vec{k}\in\mathcal{G}_{VV}$” is replaced by the condition
“$\vec{k}\in\mathcal{G}_{VV}^{o}$”, where $\mathcal{G}_{VV}^{o}$ is the subset
of $\mathcal{G}_{VV}$ described by

(42) $\mathcal{G}_{VV}^{o}=\left\\{\begin{aligned}
\vec{k}=&(k_{1},\ldots,k_{m+n})\in\mathcal{G}_{VV}:\>\exists
p\in[0,R]\>\text{such that}\\\
&\sum_{i=1}^{m}G_{i}(k_{i},p)=\sum_{j=m+1}^{m+n}G_{j}(p,k_{j}),\\\ &\forall
s\in
I[k_{i},p]\setminus\\{k_{i}\\}\quad(p-k_{i})(f_{i}(s)-f_{i}(k_{i}))>0,\>\text{for
}i=1,\ldots,m,\\\ &\forall s\in
I[p,k_{j}]\setminus\\{k_{j}\\}\quad(k_{j}-p)(f_{j}(s)-f_{j}(k_{j}))>0,\>\text{for
}j=m+1,\ldots,m+n\end{aligned}\right\\}.$

###### Proof.

First, let us prove the result under the assumption (41). Take
$\vec{u}=(u_{1},\ldots,u_{m+n})$ as initial condition for a Riemann problem at
the junction. Then consider the associated solution
$\vec{v}=(v_{1},\ldots,v_{m+n})$ and the traces $\vec{\gamma v}$, as in the
proof on Lemma 2.7. We know that $\vec{\gamma v}$ is in $\mathcal{G}_{VV}$,
and therefore by the assumption (41),

(43)
$\sum_{i=1}^{m}q_{i}(u_{i},\gamma_{i}v_{i})-\sum_{j=m+1}^{m+n}q_{j}(u_{j},\gamma_{j}v_{j})\geq
0.$

By construction we have (see the proof of Lemma 2.7)
$f_{h}(\gamma_{h}v_{h})=G_{h}(u_{h},p)=G_{h}(\gamma_{h}v_{h},p)$ if $h\leq m$,
$f_{h}(\gamma_{h}v_{h})=G_{h}(p,u_{h})=G_{h}(p,\gamma_{h}v_{h})$ if $h>m$.
Moreover, by maximum principle, $\gamma_{h}v_{h}$ in between $u_{h}$ and $p$.
This means that

(44)
$f_{i}(u_{i})-f_{i}(\gamma_{i}v_{i})=f_{i}(u_{i})-G_{i}(u_{i},p)=\begin{cases}f_{i}(u_{i})-\min_{s\in[u_{i},p]}f_{i}(s)\geq
0&\quad\text{if }u_{i}\leq p,\\\ f_{i}(u_{i})-\max_{s\in[p,u_{i}]}f_{i}(s)\leq
0&\quad\text{if }u_{i}\geq p,\end{cases}$

then $f_{i}(u_{i})-f_{i}(\gamma_{i}v_{i})$ is non negative when
$u_{i}\leq\gamma_{i}v_{i}$ and non positive when $u_{i}\geq\gamma_{i}v_{i}$.
This means that

(45)
$\sum_{i=1}^{m}q_{i}(u_{i},\gamma_{i}v_{i})=-\sum_{i=1}^{m}|f_{i}(u_{i})-f_{i}(\gamma_{i}v_{i})|.$

In the same way we show that

(46)
$\sum_{j=m+1}^{m+n}q_{j}(u_{j},\gamma_{j}v_{j})=\sum_{j=m+1}^{m+n}|f_{j}(u_{j})-f_{j}(\gamma_{j}v_{j})|.$

The sum of non positive terms can be non negative only if all the terms
vanish. Therefore for any $h\in\\{1,\ldots,m+n\\}$ we have
$f_{h}(u_{h})=f_{h}(\gamma_{h}v_{h})=G_{h}(u_{h},p)$. In view of (16), this
shows that $\vec{v}$ belongs to $\mathcal{G}_{VV}$.

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 $\mathcal{G}_{VV}$ and
$\mathcal{G}_{VV}^{o}$ of $[0,R]^{m+n}$ consists in $m+n$-uplets $\vec{k}$ for
which at least one of the following $(m+n)$ events occurs:

  * $(A_{i})$

$f_{i}(k_{i})=f_{i}(p)$ and $k_{i}<p$;

  * $(B_{j})$

$f_{j}(k_{j})=f_{j}(p)$ and $k_{j}>p$.

Indeed, if, for instance, there holds

(47) $\forall s\in I[k_{1},p]\colon\;(p-u_{1})(f_{1}(s)-f_{1}(k_{1}))\geq
0\;\;\text{and}\;\;\exists s_{0}\in
I[k_{1},p]\setminus\\{k_{1}\\}\;\,\text{s.t.}\;\,f_{1}(s_{0})=f_{1}(k_{1}),$

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
$(A_{i})_{i\in\\{1,\ldots,m\\}}$,$(B_{j})_{j\in\\{m+1,\ldots,m+n\\}}$ the only
event that occurs is $(A_{1})$, namely, $f_{1}(k_{1})=f_{1}(p)$, $k_{1}<p$ but
neither $(A_{i})$, $i\in\\{2,\ldots,m\\}$ nor $(B_{j})$,
$j\in\\{m+1,\ldots,m+n\\}$ occur.

Observe that in this case, the $(m+n)$-uplet
${\vec{k}}^{\prime}:=(p,k_{2},\ldots,k_{m+n})$ belongs to
$\mathcal{G}_{VV}^{o}$. Indeed, it belongs to $\mathcal{G}_{VV}$ since it
corresponds to the same value $p$, while the event (47) does not occur any
more since for $k_{1}^{\prime}=p$,
$I[k_{1}^{\prime},p]\setminus\\{k_{1}^{\prime}\\}$ is empty. Moreover,
whatever be $u_{1}\in[0,R]$, we have

$q_{1}(u_{1},k_{1})=\operatorname{sign}(u_{1}-k_{1})(f_{1}(u_{1})-f_{1}(k_{1}))\geq\operatorname{sign}(u_{1}-p)(f_{1}(u_{1})-f_{1}(p))=q_{1}(u_{1},p).$

Consequently, $\Delta(\vec{u},\vec{k})\geq\Delta(\vec{u},\vec{k}^{\prime})\geq
0,$ where we have used the assumption of the last claim of the lemma and the
fact that ${\vec{k}}^{\prime}\in\mathcal{G}_{VV}^{o}$.

The general case is fully analogous, so that we find
$\Delta(\vec{u},\vec{k})\geq 0$ not only for $\vec{k}\in\mathcal{G}_{VV}^{o}$,
but for all $\vec{k}\in\mathcal{G}_{VV}$. 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 $\mathbf{L}^{1}$ sense. Therefore the
following definition makes sense.

######  Definition 2.9.

Given $m+n$ fluxes $f_{h}$ satisfying (F) and an initial condition
$\vec{u}_{0}$ in $[0,R]^{m+n}$, we call
$\vec{\rho}=(\rho_{1},\ldots,\rho_{m+n})$ in
$\mathbf{L}^{\infty}\left(\mathbbm{R}_{+}\times\Gamma;[0,R]^{m+n}\right)$ a
$\mathcal{G}_{VV}$-entropy solution of (1) associated with $\vec{u}_{0}$ if

  * •

the first item of Definition 1.1 holds, i.e., the Kruzhkov entropy
inequalities (11) hold for any $h\in\\{1,\ldots,m+n\\}$;

  * •

for a.e. $t$ in $\mathbbm{R}_{+}$,the vector
$\vec{\gamma\rho}(t):=(\gamma_{1}\rho_{1}(t),\ldots,\gamma_{m+n}\rho_{m+n}(t))$
of traces at the junction belongs to $\mathcal{G}_{VV}$.

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.

###  2.4. Adapted entropies and another reformulation of admissibility

Recall that, except very special cases like $n=m$ with $f_{h}=f$ for all
$h\in\\{1,\ldots,2m\\}$, we cannot expect that constants (seen as
$\vec{k}=(k,\ldots,k)$, $k\in[0,R]$) be solutions of (1). Moreover, the above
analysis provides us with a wide set of stationary, constant per road
solutions associated to $\vec{k}\in\mathcal{G}_{VV}$. Therefore, following [8,
7] 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 [13] are replaced by the stationary solutions
associated with states in $\mathcal{G}_{VV}$. This leads to the following
definition.

######  Definition 2.10.

Given $m+n$ fluxes $f_{h}$ satisfying (F) and an initial condition
$\vec{u}_{0}$ in $[0,R]^{m+n}$, we call
$\vec{\rho}=(\rho_{1},\ldots,\rho_{m+n})$ in
$\mathbf{L}^{\infty}\left(\mathbbm{R}_{+}\times\Gamma;[0,R]^{m+n}\right)$ an
adapted entropy solution of (1) associated with $\vec{u}_{0}$ if

  * •

The first item of Definitions 1.1, 2.9 holds, i.e., the Kruzhkov entropy
inequalities (11) hold for any $h\in\\{1,\ldots,m+n\\}$.

  * •

For any $\vec{k}\in\mathcal{G}_{VV}$ (which should be seen as a road-wise
constant solution to the Riemann problem at the junction), $\vec{\rho}$
satisfies the _adapted entropy inequality_ on the network, namely, for any non
negative test function $\xi\in\mathcal{D}(]0,+\infty[\times\mathbbm{R})$

(48)
$\sum_{h=1}^{m+n}\left(\int_{\mathbbm{R}_{+}}\int_{\Omega_{h}}\left\\{|\rho_{h}-k_{h}|\xi_{t}+q_{h}(\rho_{h},k_{h})\xi_{x}\right\\}\,dx\,dt\right)\geq
0.$

######  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 $\vec{k}=(k,k)$
and $\mathcal{G}_{VV}$ contains, in addition, all vectors
$\vec{k}=(k_{1},k_{2})$ 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 $h\in\\{1,\ldots,2m\\}$
our definition seems more restrictive than the one given in the paper [13]. 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 [4, 1], 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.

###  2.5. 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
$\vec{k}\in\mathcal{G}_{VV}^{o}$, without changing the resulting notion of
solution.

###### Proof.

We only need to establish equivalence between the second items of the three
definitions for functions $\vec{\rho}$ satisfying the Kruzhkov inequalities
(11) (which is a common condition for all of them).

$\bullet$ We prove that Definition 2.10 (even in the weaker version, where the
choice of $\vec{k}$ for adapted entropy inequalities is restricted to
$\mathcal{G}_{VV}^{o}$) implies Definition 2.9:

Let $\vec{\rho}$ be a solution in the sense of Definition 2.10. We consider a
non negative test function $\xi\in\mathcal{D}(\mathbbm{R}_{+})$ and, for every
$\alpha>0$, $\chi_{\alpha}\in\mathbf{C}_{c}^{\infty}(\mathbbm{R})$, such that

(49) $0\leq\chi_{\alpha}\leq
1,\qquad|\chi_{\alpha}^{\prime}|\leq\frac{1}{\alpha},\qquad\chi_{\alpha}(x)=\begin{cases}1,&\text{if
$|x|\leq\alpha$},\\\ 0,&\text{if $|x|\geq 2\alpha$}.\end{cases}$

With the test function $\xi\chi_{\alpha}$, inequality (48) becomes

(50)
$\displaystyle\sum_{h=1}^{m+n}\int_{\mathbbm{R}_{+}}\int_{\Omega_{h}}\left\\{|\rho_{h}-k_{h}|\xi^{\prime}(t)\chi_{\alpha}(x)+q_{h}(\rho_{h},k_{h})\xi(t)\chi_{\alpha}^{\prime}(x)\right\\}\,dx\,dt$
$\displaystyle+\sum_{h=1}^{m+n}\int_{\Omega_{h}}|u^{0}_{h}(x)-k_{h}|\xi(0)\chi_{\alpha}(x)\,dx\geq
0.$

As $\alpha\to 0$ we get

(51)
$-\int_{\mathbbm{R}_{+}}\left(\sum_{i=1}^{m}q_{i}(\gamma^{i}\rho_{i},k_{i})-\sum_{j=m+1}^{m+n}q_{j}(\gamma^{j}\rho_{j},k_{j})\right)\xi(t)\,dt\geq
0.$

Localization then yields, for a.e. $t\in\mathbbm{R}_{+}$,
$\Delta(\vec{\gamma\rho}(t),\vec{k})\geq 0$, in the notation of (41). By Lemma
2.8,

(52)
$\vec{\gamma\rho}(t)=(\gamma_{1}\rho_{1}(t),\ldots,\gamma_{m+n}\rho_{m+n}(t))\in\mathcal{G}_{VV},\qquad\text{a.e.
$t\geq 0$},$

and then $\vec{\rho}$ 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 $\vec{k}\in\mathcal{G}_{VV}^{o}$, we still get the same
conclusion due to the last claim of Lemma 2.8. This point justifies the last
claim of the theorem.

$\bullet$ We prove that Definition 2.9 implies Definition 2.10:

Let $\vec{\rho}$ be a solution in the sense of Definition 2.9. For every non
negative test function $\xi\in\mathcal{D}(]0,+\infty[\times\mathbbm{R})$, we
get

(53)
$\displaystyle\sum_{h=1}^{m+n}\int_{\mathbbm{R}_{+}}\int_{\Omega_{h}}\left\\{|\rho_{h}-k_{h}|\xi_{t}+\left(q_{h}(\rho_{h},k_{h})\right)\xi_{x}\right\\}\,dx\,dt$
$\displaystyle-\int_{\mathbbm{R}_{+}}\biggr{(}\sum_{i=1}^{m}q_{i}(\gamma_{i}\rho_{i}(t),k_{i})-\sum_{j=1}^{m+n}q_{j}(\gamma_{j}\rho_{j}(t),k_{j})\biggr{)}\xi(t,0)\,dt\geq
0.$

Since

(54)
$\vec{\gamma\rho}(t)=(\gamma_{1}\rho_{1}(t),\ldots,\gamma_{m+n}\rho_{m+n}(t))\in\mathcal{G}_{VV},\qquad\text{a.e.
$t\geq 0$},$

we have

(55)
$\sum_{i=1}^{m}q_{i}(\gamma_{i}\rho_{i}(t),k_{i})\geq\sum_{j=1}^{m+n}q_{j}(\gamma_{j}\rho_{j}(t),k_{j}),\qquad\text{a.e.
$t\geq 0$}.$

Therefore, $\vec{\rho}$ is a solution in the sense of Definition 2.10.

$\bullet$ We prove that Definition 2.9 implies Definition 1.2:

Let $\vec{\rho}$ be a solution in the sense of Definition 2.9. Since

(56)
$\vec{\gamma\rho}(t)=(\gamma_{1}\rho_{1}(t),\ldots,\gamma_{m+n}\rho_{m+n}(t))\in\mathcal{G}_{VV},\qquad\text{a.e.
$t\geq 0$},$

there exists $p\in\mathbf{L}^{\infty}(\mathbbm{R}_{+},[0,R])$ such that (15)
holds for a.e. $t\geq 0$. Let us point out that $p$ is indeed measurable. This
results, first, from the measurability of the trace vector
$\vec{\gamma\rho}:\mathbbm{R}_{+}\to[0,R]^{m+n}$; and second, given
$\vec{\gamma\rho}(t)$, from the definition of $p(t)$ by equation (15) for
which we can systematically take the smallest solution (cf. the proof of Lemma
2.4). Then $\vec{\rho}$ is a solution in the sense of Definition 1.2.

$\bullet$ Finally, Definition 1.2 implies Definition 2.9.

This follows from the definition of $\mathcal{G}_{VV}$ in (16), in view of
(12),(13). ∎

##  3\. 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 $\vec{u}_{0}=(u^{0}_{1},\ldots,u^{0}_{m+n})$
in $\mathbf{L}^{\infty}(\Gamma;\mathbbm{R}^{m+n})$ the problem (1) admits one
and only one solution $\rho$ in
$\mathbf{L}^{\infty}(\mathbbm{R}_{+}\times\Gamma;[0,R]^{m+n})$ 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 $\mathbf{L}^{1}$ sense. If $\vec{\rho}$ and $\vec{\hat{\rho}}$ are
the admissible solutions corresponding respectively to the initial conditions
$\vec{u}_{0}$ and $\vec{v}_{0}$, then for all $M>0$ and $t<M/L$, where
$L=\max\\{\|f_{h}^{\prime}\|_{\mathbf{L}^{\infty}([0,R];\mathbbm{R})}\,|\,h=1,\ldots,m+n\\}$,

(57)
$\displaystyle\sum_{i=1}^{m}\int_{-(M-Lt)}^{0}|\rho_{i}(t,x)-\hat{\rho}_{i}(t,x)|\,dx+\sum_{j=m+1}^{m+n}\int_{0}^{M-Lt}|\rho_{j}(t,x)-\hat{\rho}_{j}(t,x)|\,dx$
$\displaystyle\qquad\qquad\leq\sum_{i=1}^{m}\int_{-M}^{0}|u^{0}_{i}(x)-v^{0}_{i}(x)|\,dx+\sum_{j=m+1}^{m+n}\int_{0}^{M}|u^{0}_{j}(x)-v^{0}_{j}(x)|\,dx.$

In particular, the map that associates to $\vec{u}_{0}$ the unique
corresponding admissible profile $\vec{\rho}(t)$, is non-expansive w.r.t. the
$\mathbf{L}^{1}$ 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).

###  3.1. Uniqueness of admissible solutions of (1)

The goal of this section is to prove the following stability result

######  Proposition 3.1.

For any given initial condition $\vec{u}_{0}=(u^{0}_{1},\ldots,u^{0}_{m+n})$
in $\mathbf{L}^{\infty}(\Gamma;\mathbbm{R}^{m+n})$ there exists at most one
$\mathcal{G}_{VV}$-entropy solution $\rho$ in
$\mathbf{L}^{\infty}(\mathbbm{R}_{+}\times\Gamma;[0,R]^{m+n})$.

Moreover, if $\vec{\rho}$ and $\vec{\hat{\rho}}$ are the
$\mathcal{G}_{VV}$-entropy solutions corresponding respectively to the initial
conditions $\vec{u}_{0}$ and $\vec{v}_{0}$, (57) holds and in particular, the
$\mathbf{L}^{1}$-contraction estimate holds whenever the right-hand side is
finite:

(58)
$\displaystyle\sum_{i=1}^{m}{\left\|\rho_{i}(t)-\hat{\rho}_{i}(t)\right\|}_{\mathbf{L}^{1}(\mathbbm{R}_{-};\mathbbm{R})}$
$\displaystyle+\sum_{j=m+1}^{m+n}{\left\|\rho_{j}(t)-\hat{\rho}_{j}(t)\right\|}_{\mathbf{L}^{1}(\mathbbm{R}_{+};\mathbbm{R})}$
$\displaystyle\leq\sum_{i=1}^{m}{\left\|u^{0}_{i}-v^{0}_{i}\right\|}_{\mathbf{L}^{1}(\mathbbm{R}_{-};\mathbbm{R})}+\sum_{j=m+1}^{m+n}{\left\|u^{0}_{j}-v^{0}_{j}\right\|}_{\mathbf{L}^{1}(\mathbbm{R}_{+};\mathbbm{R})}.$

###### Proof.

The first part of the proof is devoted to establish a Kato’s type inequality
for $\mathcal{G}_{VV}$-entropy solutions.

Let $\xi$ be a test function in
$\mathcal{D}(\mathbbm{R}_{+}\times\mathbbm{R};\mathbbm{R}_{+})$ and define,
for $k>0$,

(59) $\xi_{k}(t,x)=\xi(t,x)\min\left\\{1,\frac{(|x|-k)^{+}}{k}\right\\}.$

Then, as $\rho_{h}$ and $\hat{\rho}_{h}$ are Kruzhkov entropy weak solutions
in the interior of $\Omega_{h}$, by a standard doubling of variable argument
we get

(60)
$\sum_{h=1}^{m+n}\left(-\int_{\mathbbm{R}_{+}}\\!\\!\int_{\Omega_{h}}\\!\\!\left\\{|\rho_{h}-\hat{\rho}_{h}|\xi_{k,t}+q_{h}(\rho_{h},\hat{\rho}_{h})\xi_{k,x}\right\\}\,dx\,dt-\int_{\Omega_{h}}\\!\\!|u^{0}_{h}(x)-v^{0}_{h}(x)|\xi_{k}(0,x)\,dx\right)\leq
0.$

An explicit computation shows that

(61)
$\xi_{k,x}(t,x)=\xi_{x}(t,x)\min\left\\{1,\frac{(|x|-k)^{+}}{k}\right\\}+\frac{1}{k}\xi(t,x)(\mathds{1}_{]k,2k[}(x)-\mathds{1}_{]-2k,-k[}(x)),$

then (60) rewrites as

(62)
$\displaystyle\sum_{h=1}^{m+n}\left(-\int_{\mathbbm{R}_{+}}\int_{\Omega_{h}}\left\\{|\rho_{h}-\hat{\rho}_{h}|\xi_{k,t}+q_{h}(\rho_{h},\hat{\rho}_{h})\xi_{x}\min\left\\{1,\frac{(|x|-k)^{+}}{k}\right\\}\right\\}\,dx\,dt\right)$
$\displaystyle+\sum_{h=1}^{m+n}\frac{1}{k}\left(\int_{\mathbbm{R}_{+}}\int_{\Omega_{h}\cap]-2k,-k[}\\!\\!q_{h}(\rho_{h},\hat{\rho}_{h})\xi\,dx\,dt-\int_{\mathbbm{R}_{+}}\int_{\Omega_{h}\cap]k,2k[}\\!\\!q_{h}(\rho_{h},\hat{\rho}_{h})\xi\,dx\,dt\right)$
$\displaystyle+\sum_{h=1}^{m+n}\left(-\int_{\Omega_{h}}|u^{0}_{h}(x)-v^{0}_{h}(x)|\xi_{k}(0,x)\,dx\right)\leq
0.$

We have that

(63) $\displaystyle\lim_{k\to
0^{+}}\sum_{h=1}^{m+n}\frac{1}{k}\left(\int_{\mathbbm{R}_{+}}\int_{\Omega_{h}\cap]-2k,-k[}\\!\\!q_{h}(\rho_{h},\hat{\rho}_{h})\xi\,dx\,dt-\int_{\mathbbm{R}_{+}}\int_{\Omega_{h}\cap]k,2k[}\\!\\!q_{h}(\rho_{h},\hat{\rho}_{h})\xi\,dx\,dt\right)$
$\displaystyle=\int_{\mathbbm{R}_{+}}\left(\sum_{i=1}^{m}q_{i}(\gamma^{i}\rho_{i},\gamma^{i}\hat{\rho}_{i})-\sum_{j=m+1}^{m+n}q_{j}(\gamma^{j}\rho_{j},\gamma^{j}\hat{\rho}_{j})\right)\xi(t,0)\,dt,$

and this term is positive due to the positivity assumption on $\xi$ and the
fact that $\vec{\gamma\rho}$ and $\vec{\gamma\hat{\rho}}$ are in
$\mathcal{G}_{VV}$. Therefore by taking the limit as $k$ tends to $0$ of the
whole left hand side in (62) one can conclude that

(64)
$\sum_{h=1}^{m+n}\left(-\int_{\mathbbm{R}_{+}}\int_{\Omega_{h}}\left\\{|\rho_{h}-\hat{\rho}_{h}|\xi_{t}+q_{h}(\rho_{h},\hat{\rho}_{h})\xi_{x}\right\\}\,dx\,dt-\int_{\Omega_{h}}|u^{0}_{h}(x)-v^{0}_{h}(x)|\xi(0,x)\,dx\right)\leq
0,$

which is Kato’s inequality adapted to our setting.

The conclusion of the proof is classical, following [25]. One takes $\xi(t,x)$
approximating the characteristic function of the trapezoid
$\\{(t,x)\in\mathbbm{R}_{+}\times\mathbbm{R};\;|x|\leq M+Lt\\}$ where
$L=\max_{h\in\\{1,\ldots,m+n\\}}\max_{u\in[0,R]}|f_{h}^{\prime}(u)|$ is the
appropriate Lipschitz constant and $M$ is a nonnegative parameter. For a.e.
fixed $t\in\mathbbm{R}_{+}$, this yields for $M>Lt$ the inequality (57).

As $M\to+\infty$, we find (58) as soon as its right-hand side is finite. ∎

###  3.2. 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 [1] for the case
$m=n=1$. The choice of Godunov’s fluxes is motivated by the fact that this
scheme is well-balanced, 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 $\Delta x$. For $\ell\in\mathbbm{Z}$ and
$h\in\\{1,\ldots,m+n\\}$, set $x^{h}_{\ell}:=\ell\Delta x$. We consider the
uniform spatial mesh on each $\Omega_{h}$

(65)
$\displaystyle\bigcup_{\ell\leq-1}(x^{i}_{\ell},x^{i}_{\ell+1})\qquad\text{on
}\Omega_{i},\quad\text{for }i\leq m,$ (66) $\displaystyle\bigcup_{\ell\geq
0}(x^{j}_{\ell},x^{j}_{\ell+1})\qquad\text{on }\Omega_{j},\quad\text{for
}j\geq m+1,$

so that the position of the junction $x=0$ on the road $\Omega_{h}$
corresponds to $x^{h}_{0}$. Then we fix a time step $\Delta t$ satisfying the
CFL condition

(67) $\Delta t\leq\frac{\max_{h}\\{L_{h}\\}}{2}\Delta x,$

where $L_{h}$ is the Lipschitz constant of $f_{h}$.

At all cell interfaces $x^{h}_{\ell}$ for $\ell\neq 0$ we consider the
standard Godunov flux $G_{h}$ corresponding to the flux $f_{h}$. At the
junction $x_{0}^{h}$ we take on each road $\Omega_{h}$ the Godunov flux
$G_{h}^{*}$ corresponding to the admissible solution of the Riemann problem at
the junction, see Remark 3. More precisely, let
$\vec{u_{0}}=(u^{0}_{1},\ldots,u^{0}_{m+n})$ be an initial condition in
$\mathbf{L}^{\infty}(\Gamma;[0,R]^{m+n})$. We initialize our scheme by
discretizing the initial conditions

(68) $u_{k+\frac{1}{2}}^{h,0}=\frac{1}{\Delta
x}\int_{x^{h}_{\ell}}^{x^{h}_{\ell+1}}u^{0}_{h}(x)\,dx,$

for all $h=1,\ldots,m+n$ and for $\ell\leq-1$ if $h\leq m$, $\ell\geq 0$ if
$h\geq m+1$.

Then $\left(u_{\ell+\frac{1}{2}}^{h,s+1}\right)_{h,\ell}$ is obtained from
$\left(u_{\ell+\frac{1}{2}}^{h,s}\right)_{h,\ell}$ through the following
scheme

  1. (I)

We solve

(69) $\text{find $p^{s}$ in $[0,R]$
s.t.}\;\;\sum_{i=1}^{m}G_{i}(u_{-\frac{1}{2}}^{i,s},p^{s})-\sum_{j=m+1}^{m+n}G_{j}(p^{s},u_{\frac{1}{2}}^{j,s})=0,$

i.e., setting

(70)
$\vec{u}^{\,*,s}:=\Bigl{(}u_{-\frac{1}{2}}^{1,s},\ldots,u_{-\frac{1}{2}}^{m,s},u_{\frac{1}{2}}^{m+1,s},u_{\frac{1}{2}}^{m+n,s}\Bigr{)}$

we take $p^{s}:=p_{\vec{u}^{\,*,s}}$ according to the notation of Lemma 2.3.

  2. (II)

We compute

(71) $u_{\ell+\frac{1}{2}}^{h,s+1}=u_{\ell+\frac{1}{2}}^{h,s}-\frac{\Delta
t}{\Delta x}\left(\mathcal{F}^{h,s}_{\ell+1}-\mathcal{F}^{h,s}_{\ell}\right),$

where

(72)
$\mathcal{F}^{h,s}_{\ell}=\begin{cases}G_{h}(u_{\ell-\frac{1}{2}}^{h,s},u_{\ell+\frac{1}{2}}^{h,s})\quad&\quad\text{if
}h\leq m\>\text{ and }\ell\leq-1\>\text{ or }h\geq m+1\>\text{ and }\ell\geq
1,\\\ G_{h}(u_{-\frac{1}{2}}^{h,s},p^{s})\equiv
G^{*}_{h}(\vec{u}^{\,*,s})&\quad\text{if }h\leq m\>\text{ and }\ell=0,\\\
G_{h}(p^{s},u_{\frac{1}{2}}^{h,s})\equiv
G^{*}_{h}(\vec{u}^{\,*,s})&\quad\text{if }h\geq m+1\>\text{ and
}\ell=0.\end{cases}$

The stage (II) is a standard marching scheme, up to the specific definition of
the fluxes for $\ell=0$. 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. [1].

We introduce the notation

$\Gamma_{\rm{discr}}=\Bigl{(}\\{1,\ldots,m\\}\times\\{\ell\in\mathbbm{Z},\;\ell\leq-1\\}\Bigr{)}\bigcup\Bigl{(}\\{m+1,\ldots,m+n\\}\times\\{\ell\in\mathbbm{Z},\;\ell\geq
0\\}\Bigr{)},$

for the set of all degrees of freedom at a fixed time step; the notation

$U^{s}=\Bigl{(}u^{h,s}_{\ell+\frac{1}{2}}\Bigr{)}_{(h,\ell)\in\Gamma_{\rm{discr}}}$

for the set of all the unknowns of the scheme at time step $s$; and the
notation $\mathcal{S}^{\Delta x}\vec{u_{0}}$ to indicate the piecewise
constant function corresponding to the discrete solution
$\left(u_{\ell+\frac{1}{2}}^{h,s}\right)_{h,\ell,s}=(U^{s})_{s}$:
$\mathcal{S}^{\Delta x}\vec{u_{0}}=(u^{1,\Delta x},\ldots,u^{m,\Delta
x},u^{m+1,\Delta x},\ldots,u^{m+n,\Delta x})$ where

(73) $\displaystyle u^{i,\Delta
x}=\sum_{s\in\mathbbm{N},\>\ell\leq-1}u_{\ell+\frac{1}{2}}^{i,s}\mathds{1}_{\Omega_{i}\cap]\ell,\ell+1[}(x)\mathds{1}_{]s,s+1[}(t),$
$\displaystyle\quad i\in\\{1,\ldots,m\\},$ $\displaystyle u^{j,\Delta
x}=\sum_{s\in\mathbbm{N},\>\ell\geq
0}u_{\ell+\frac{1}{2}}^{j,s}\mathds{1}_{\Omega_{j}\cap]\ell,\ell+1[}(x)\mathds{1}_{]s,s+1[}(t),$
$\displaystyle\quad j\in\\{m+1,\ldots,m+n\\}.$

######  Lemma 3.2.

The above finite volume numerical scheme is well-balanced, i.e., every element
of $\mathcal{G}_{VV}$ corresponds to a stationary, constant per road solution
of the scheme.

###### Proof.

Let $\vec{k}\in\mathcal{G}_{VV}$ and denote by $\vec{k}^{\Delta x}$ the
associated discrete function with entries $u^{h,\Delta x}\equiv k_{h}$,
$h\in\\{1,\ldots,m+n\\}$. Then $\mathcal{S}^{\Delta x}\vec{k}^{\Delta
x}=\vec{k}^{\Delta x}$, i.e., $\vec{k}^{\Delta x}$ is a stationary solution of
the scheme. Indeed, consider for instance an incoming road $i$. Obviously, for
all $\ell<-1$ the first iteration of (71) initialized with constant initial
values $u^{i,0}_{\ell+\frac{1}{2}}=k_{i}$ yields

$u^{i,1}_{\ell+\frac{1}{2}}=k_{i}-\frac{\Delta t}{\Delta
x}(G_{i}(k_{i},k_{i})-G_{i}(k_{i},k_{i}))=k_{i}.$

Moreover, due to (69), by definition of $G^{*}_{i}$ and of $\mathcal{G}_{VV}$
we have

$G_{i}(k_{i},k_{i})=f(k_{i})=G(k_{i},p_{\vec{k}})=G^{*}_{i}(\vec{k}),$

so that for $\ell=-1$ we still find

$u^{i}_{1,-\frac{1}{2}}=k_{i}-\frac{\Delta t}{\Delta
x}(G_{i}(k_{i},k_{i})-G^{*}_{i}(\vec{k}))=k_{i}.$

Similarly, we find that (69), (71) preserves the constant value $k_{j}$ on the
$j$th 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) $\forall
s\in\mathbbm{N}\;\forall(h,\ell)\in\Gamma_{\rm{discr}}\colon\;\;\;\;u^{h,s+1}_{\ell+\frac{1}{2}}=H^{h}_{\ell+\frac{1}{2}}(U^{s})$

for some functions $H^{h}_{\ell+\frac{1}{2}}$ that are monotone non-decreasing
with respect to each of the arguments (actually each of these functions
depends on at most $(m+n+1)$ entries of $U^{s}$). This implies in particular
the order-preservation property

(75)
$\forall(h,\ell)\in\Gamma_{\rm{discr}}\colon\;\;\;\;u^{h,s}_{\ell+\frac{1}{2}}\geq\hat{u}^{h,s}_{\ell+\frac{1}{2}}\;\;\;\Rightarrow\;\;\;\forall(h,\ell)\in\Gamma_{\rm{discr}}\colon\;\;\;\;u^{h,s+1}_{\ell+\frac{1}{2}}\geq\hat{u}^{h,s+1}_{\ell+\frac{1}{2}}.$

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., [16]) that the scheme is $\mathbf{L}^{1}$-contractive
in the sense that the discrete analogue of (58) (with $\vec{\rho}$ replaced by
$\mathcal{S}^{\Delta x}\vec{u_{0}}$) and the initial condition $\vec{u}_{0}$
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 [19]), let $\top$, resp., $\bot$
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.,

$(k_{1},k_{2},k_{3})\bot(\hat{k}_{1},\hat{k}_{2},\hat{k}_{3})=(\min\\{k_{1},\hat{k}_{1}\\},\min\\{k_{2},\hat{k}_{2}\\},\min\\{k_{3},\hat{k}_{3}\\}).$

######  Proposition 3.2.

For any initial conditions $\vec{u_{0}},\hat{\vec{u}}_{0}$ in
$\mathbf{L}^{\infty}(\Gamma;[0,R]^{m+n})$ the corresponding discrete solutions
of the scheme (68)–(72) satisfy _discrete Kato inequalities_. Namely, for
$(h,\ell)\in\Gamma_{\rm{discr}}$, let $Q^{h}_{\ell}[U^{s},\hat{U}^{s}]$ be
defined by

$\displaystyle
Q^{h}_{\ell}[U^{s},\hat{U}^{s}]=G_{h}\Bigl{(}u^{h,s}_{\ell-\frac{1}{2}}\top\hat{u}^{h,s}_{\ell-\frac{1}{2}},u^{h,s}_{\ell+\frac{1}{2}}\top\hat{u}^{h,s}_{\ell+\frac{1}{2}}\Bigr{)}-G_{h}\Bigl{(}u^{h,s}_{\ell-\frac{1}{2}}\bot\hat{u}^{h,s}_{\ell-\frac{1}{2}},u^{h,s}_{\ell+\frac{1}{2}}\bot\hat{u}^{h,s}_{\ell+\frac{1}{2}}\Bigr{)},\;\;\ell\neq
0,$ $\displaystyle
Q^{h}_{0}[U^{s},\hat{U}^{s}]=G^{*}_{h}(\vec{u}^{\,*,s}\top\hat{\vec{u}}^{\,*,s})-G^{*}_{h}(\vec{u}^{\,*,s}\bot\hat{\vec{u}}^{\,*,s}).$

Then for all $\xi\in\mathcal{D}(]0,+\infty[\times\mathbbm{R})$ such that
$\xi\geq 0$ and $\partial_{x}\xi=0$ on $[-\Delta x/2,\Delta x/2]$, setting
$\xi^{s}_{\ell+\frac{1}{2}}:=\xi(s\Delta t,(\ell+\frac{1}{2})\Delta x)$ we
have

$\displaystyle-$ $\displaystyle\sum_{i=1}^{m}\sum_{s=1}^{+\infty}\Delta
t\sum_{\ell\leq-1}\Delta
x|u^{i,s}_{\ell+\frac{1}{2}}-\hat{u}^{i,s}_{\ell+\frac{1}{2}}|\frac{\xi^{s+1}_{\ell+\frac{1}{2}}-\xi^{s}_{\ell+\frac{1}{2}}}{\Delta
t}$ $\displaystyle-\sum_{i=1}^{m}\sum_{s=1}^{+\infty}\Delta
t\sum_{\ell\leq-1}\Delta
xQ^{i}_{\ell}[U^{s},\hat{U}^{s}]\frac{\xi^{s+1}_{\ell+\frac{1}{2}}-\xi^{s+1}_{\ell-\frac{1}{2}}}{\Delta
x}$ $\displaystyle-\sum_{j=m+1}^{m+n}\sum_{s=1}^{+\infty}\Delta
t\sum_{\ell\geq 0}\Delta
x|u^{j,s}_{\ell+\frac{1}{2}}-\hat{u}^{j,s}_{\ell+\frac{1}{2}}|\frac{\xi^{s+1}_{\ell+\frac{1}{2}}-\xi^{s}_{\ell+\frac{1}{2}}}{\Delta
t}$ $\displaystyle-\sum_{j=m+1}^{m+n}\sum_{s=1}^{+\infty}\Delta
t\sum_{\ell\geq 1}\Delta
xQ^{j}_{\ell}[U^{s},\hat{U}^{s}]\frac{\xi^{s+1}_{\ell+\frac{1}{2}}-\xi^{s+1}_{\ell-\frac{1}{2}}}{\Delta
x}\leq 0.$

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 [1], and to the conservativity of the Riemann solver at the
junction the junction $\ell=0$ does not contribute to the
“$Q[U,\hat{U}]\partial_{x}\xi$” 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
$(h,\ell)\in\Gamma_{\rm{discr}}$

(76)
$\frac{|u^{h,s+1}_{\ell+\frac{1}{2}}-\hat{u}^{h,s+1}_{\ell+\frac{1}{2}}|-|u^{h,s}_{\ell+\frac{1}{2}}-\hat{u}^{h,s}_{\ell+\frac{1}{2}}|}{\Delta
t}+\frac{Q^{h}_{\ell+1}[U^{s},\hat{U}^{s}]-Q^{h}_{\ell}[U^{s},\hat{U}^{s}]}{\Delta
x}\leq 0,$

which readily follows from the observation that for all $a,b\in\mathbbm{R}$,
$|a-b|=a\top b-a\bot b$ and from the monotonicity of
$H^{h}_{\ell+\frac{1}{2}}$ in (74):

$\displaystyle|u^{h,s+1}_{\ell+\frac{1}{2}}-\hat{u}^{h,s+1}_{\ell+\frac{1}{2}}|=u^{h,s+1}_{\ell+\frac{1}{2}}\top\hat{u}^{h,s+1}_{\ell+\frac{1}{2}}-u^{h,s+1}_{\ell+\frac{1}{2}}\bot\hat{u}^{h,s+1}_{\ell+\frac{1}{2}},$
$\displaystyle
u^{h,s+1}_{\ell+\frac{1}{2}}\top\hat{u}^{h,s+1}_{\ell+\frac{1}{2}}=H^{h}_{\ell+\frac{1}{2}}(U^{s})\top
H^{h}_{\ell+\frac{1}{2}}(\hat{U}^{s})\leq
H^{h}_{\ell+\frac{1}{2}}(U^{s}\top\hat{U}^{s}),$ $\displaystyle
u^{h,s+1}_{\ell+\frac{1}{2}}\bot\hat{u}^{h,s+1}_{\ell+\frac{1}{2}}=H^{h}_{\ell+\frac{1}{2}}(U^{s})\bot
H^{h}_{\ell+\frac{1}{2}}(\hat{U}^{s})\geq
H^{h}_{\ell+\frac{1}{2}}(U^{s}\bot\hat{U}^{s}),$ $\displaystyle
H^{h}_{\ell+\frac{1}{2}}(U^{s}\top\hat{U}^{s})-H^{h}_{\ell+\frac{1}{2}}(U^{s}\bot\hat{U}^{s})=|u^{h,s}_{\ell+\frac{1}{2}}-\hat{u}^{h,s}_{\ell+\frac{1}{2}}|-\frac{\Delta
t}{\Delta
x}\Bigl{(}Q^{h}_{\ell+1}[U^{s},\hat{U}^{s}]-Q^{h}_{\ell}[U^{s},\hat{U}^{s}]\Bigr{)},$

where the formula (71) and the definition of $Q^{h}_{\ell}[U^{s},\hat{U}^{s}]$
are used in the last line to express the function $H^{h}_{\ell+\frac{1}{2}}$.

Second, observe that for all $\vec{k},\hat{\vec{k}}$ we have by the
conservativity property underlying the definition of $\vec{G}^{*}$ (see Remark
3):

$\sum_{i=1}^{m}G^{*}_{h}(\vec{k}\top\hat{\vec{k}})-\sum_{j=m+1}^{m+n}G^{*}_{h}(\vec{k}\top\hat{\vec{k}})=0=\sum_{i=1}^{m}G^{*}_{h}(\vec{k}\bot\hat{\vec{k}})-\sum_{j=m+1}^{m+n}G^{*}_{h}(\vec{k}\bot\hat{\vec{k}}).$

It remains to multiply the $(h,\ell)$’s inequality in (76) by the nonnegative
quantity $\Delta t\Delta x\xi^{s+1}_{\ell+\frac{1}{2}}$ and sum up; the sum is
finite since the support of $\xi$ is compact. Paying attention to the fact
that $\xi^{s}_{-\frac{1}{2}}=\xi(s\Delta t,0)=\xi^{s}_{\frac{1}{2}}$, from the
definition of $Q^{h}_{0}[U^{s},\hat{U}^{s}]$ we see that for all $s$,

(77) $\sum_{i=1}^{m}\Delta
xQ^{i}_{0}[U^{s},\hat{U}^{s}]\xi^{s+1}_{-\frac{1}{2}}-\sum_{j=m+1}^{m+n}\Delta
xQ^{j}_{0}[U^{s},\hat{U}^{s}]\xi^{s+1}_{\frac{1}{2}}=0.$

Using the Abel transformation (discrete summation by parts) on each road with
respect to the time superscripts $s$ and to the space subscripts $\ell$,
bearing in mind (77) and the fact that for all $\ell\in\mathbbm{Z}$,
$\xi^{0}_{\ell+\frac{1}{2}}=0$ by the choice of $\xi$, we derive the required
discrete Kato inequality. ∎

######  Theorem 3.3.

Given an initial datum
$\vec{u_{0}}\in\mathbf{L}^{\infty}(\Gamma;[0,R]^{m+n})$, the numerical scheme
(68)–(72) converges to the unique admissible (in the sense of the equivalent
Definitions 1.2, 2.9, 2.10) solution $\vec{\rho}$ of (1), namely,
$\mathcal{S}^{\Delta x}\vec{u_{0}}\to\vec{\rho}$ as $\Delta x=0$, subject to
the CFL restriction (67) on $\Delta t$. This ensures, in particular, existence
of an admissible solution of (1) for every
$\mathbf{L}^{\infty}(\Gamma;[0,R]^{m+n})$ initial datum.

###### Proof.

The proof follows the lines of the proof given in [1] (see also [4]). Let us
only provide a sketch of the key arguments.

$\bullet$ We start with compactly supported $\mathbf{BV}$ initial data. The
compactness, in the sense of the a.e. convergence, of the family
$\Bigl{(}\mathcal{S}^{\Delta x}\vec{u_{0}}\Bigr{)}_{\Delta x\in(0,1)}$ of
discrete solutions is obtained with the $\mathbf{BV}_{loc}$ technique
introduced in [11]. It relies upon the monotonicity and the Crandall-Tartar
lemma (see Remark 6).

The compactness permits to define $\vec{\rho}$ as a limit of some sequence of
discrete solutions $\mathcal{S}^{\Delta x_{r}}\vec{u_{0}}$, $\Delta_{r}\to 0$.
At the end of the proof, having proved convergence to the unique admissible
solution with datum $\vec{u_{0}}$, by the classical argument we can bypass the
extraction of a sequence and get convergence of $\mathcal{S}^{\Delta
x}\vec{u_{0}}$, $\Delta x\to 0$.

$\bullet$ It is classical (see [19]) to derive that for every
$h\in\\{1,\ldots,m+n\\}$, $\rho_{h}$ 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 $\vec{\rho}$ 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.

$\bullet$ The main ingredient of the proof is the discrete Kato inequality
proved in Proposition 3.2, where we choose $\hat{\vec{u}}_{0}=\vec{k}$ with
$\vec{k}\in\mathcal{G}_{VV}$. Observe that by Lemma 3.2, we have
$\mathcal{S}^{\Delta x_{r}}\hat{\vec{u}}_{0}\equiv\vec{k}$; passing to the
limit as $\Delta_{r}\to 0$, we will indeed derive the adapted entropy
inequalities (48) for $\rho$ and complete the proof for compactly supported
data of bounded variation.

Following [1], observe that test functions whose $x$-derivative vanishes near
$x=0$ are dense (e.g., in the $\mathbf{C}^{1}$ topology) in
$\mathcal{D}(]0,+\infty[\times\mathbbm{R})$. Starting with the inequalities of
Proposition 3.2, with the same arguments as in the previous step (see [19, 1])
we pass to the limit and find (48) first for such specific test functions
$\xi$, and then (by density) for general test functions.

$\bullet$ Finally, as in [4, 1], in two steps we extend the convergence result
to general $\mathbf{L}^{\infty}$ data $\vec{u_{0}}$. Extension to
$\mathbf{L}^{1}\cap\mathbf{L}^{\infty}$ data follows by the density of
compactly supported $\mathbf{BV}$ data in $\mathbf{L}^{1}$ topology, with the
help of $\mathbf{L}^{1}$ contractivity of both the admissible solution
semigroup and the discrete solutions semigroups. Extension to
$\mathbf{L}^{\infty}$ 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. ∎

##  4\. 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 $\mathbf{L}^{1}$ 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 $\vec{k}$ in $\mathcal{G}_{VV}^{o}$, there exists $p\in[0,R]$ and
$\vec{\rho^{\varepsilon}}=(\rho^{\varepsilon}_{1},\ldots,\rho^{\varepsilon}_{m+n})$
in $\mathbf{L}^{\infty}(\Gamma;[0,R]^{m+n})$ such that for all
$h\in\\{1,\ldots,m+n\\}$, $\rho^{\varepsilon}_{h}$ solves the ODE problem

(78)
$\begin{cases}f_{h}(\rho^{\varepsilon}_{h})_{x}=\varepsilon(\rho^{\varepsilon}_{h})_{xx},\quad\text{
in }\>\Omega_{h}\\\ \rho^{\varepsilon}_{h}(0)=p,\\\
\lim_{x\in\Omega_{h},\,|x|\to+\infty}\rho^{\varepsilon}_{h}(x)=k_{h}.\end{cases}$

###### Proof.

We take the value $p\in[0,R]$ that ensures that
$\vec{k}\in\mathcal{G}_{VV}^{o}$, according to the definition (42) of
$\mathcal{G}_{VV}^{o}$. We consider here the case $h\leq m$, the other case
being analogous. Let us integrate both sides of (78) on $]-\infty,x]$

(79)
$\varepsilon(\rho^{\varepsilon}_{i})_{x}(x)=F_{i}(\rho^{\varepsilon}_{i}(x)),\;\;\text{where}\;F_{i}:\rho\in[0,R]\mapsto
f_{i}(\rho)-f_{i}(k_{i}),$

being understood that one should have $(\rho^{\varepsilon}_{i})_{x}\to 0$ as
$x\to-\infty$. Observe that by the definition of $\mathcal{G}_{VV}^{o}$,
$k_{i}$ is the only zero of $F_{i}$ on $I[k_{i},p]$. 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 $\rho^{\varepsilon}_{i}=k$). Then,
the map

$P:]k_{i},p]\to]-\infty,0],\;\;r\mapsto-\int_{r}^{p}\frac{\varepsilon}{f_{i}(s)-f_{i}(k_{i})}\,ds$

is well defined and strictly increasing, i.e., it admits the inverse $P^{-1}$
satisfying $P^{-1}(0)=p$, $\lim_{x\to-\infty}P^{-1}(x)=k_{i}$,
$\lim_{x\to-\infty}(P^{-1})^{\prime}(x)=0$. We find that
$\rho^{\varepsilon}_{i}$ solves (79) if and only if

(80)
$-x=\int_{x}^{0}\frac{\varepsilon\rho^{\varepsilon}_{i,x}}{f_{i}(\rho^{\varepsilon}_{i})-f_{i}(k_{i})}\,dx=-P(\rho^{\varepsilon}_{i}),$

i.e. $\rho^{\varepsilon}_{i}=P^{-1}(x)$ is the required solution. ∎

Following [13], we consider the following parabolic regularization of the
initial boundary value problem (1)

(81)
$\begin{cases}\rho^{\varepsilon}_{h,t}+f(\rho^{\varepsilon}_{h})_{x}=\varepsilon\rho^{\varepsilon}_{h,xx},&\quad
t>0,\,x\in\Omega_{h},\,h\in\\{1,...,m+n\\},\\\
\rho^{\varepsilon}_{h}(t,0)=\rho^{\varepsilon}_{h^{\prime}}(t,0),&\quad
t>0,\,h,\,h^{\prime}\in\\{1,...,m+n\\},\\\
\sum\limits_{i=1}^{m}\left(f(\rho_{i}^{\varepsilon}(t,0))-\varepsilon\rho^{\varepsilon}_{i,x}(t,0)\right)&{}\textstyle\\\
=\sum\limits_{j=m+1}^{m+n}\left(f(\rho_{j}^{\varepsilon}(t,0))-\varepsilon\rho^{\varepsilon}_{j,x}(t,0)\right),&\quad
t>0,\\\ \rho^{\varepsilon}_{h}(0,x)=u_{h,\varepsilon}^{0}(x),&\quad
x\in\Omega_{h},\,h\in\\{1,...,m+n\\},\end{cases}$

where $\varepsilon>0$. 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

(82)
$\begin{split}u^{0}_{h,\varepsilon}\in\mathbf{W}^{2,1}(\Omega_{h})\cap\mathbf{C}^{\infty}(\Omega_{h}),&\qquad
0\leq u^{0}_{h,\varepsilon}\leq R,\\\ u^{0}_{h,\varepsilon}\longrightarrow
u^{0}_{h},&\qquad\text{a.e. and in $\mathbf{L}^{\text{\rm$p$}}(\Omega_{h})$,
$1\leq p<\infty$},\quad\text{as $\varepsilon\to 0$},\\\
{\left\|u^{0}_{h,\varepsilon}\right\|}_{\mathbf{L}^{1}(\Omega_{h})}\leq{\left\|u^{0}_{h}\right\|}_{\mathbf{L}^{1}(\Omega_{h})},&\qquad{\left\|(u^{0}_{h,\varepsilon})_{x}\right\|}_{\mathbf{L}^{1}(\Omega_{h})}\leq
TV(u^{0}_{h}),\qquad\varepsilon{\left\|(u^{0}_{h,\varepsilon})_{xx}\right\|}_{\mathbf{L}^{1}(\Omega_{h})}\leq
C_{0},\end{split}$

for each $\varepsilon>0,\,h\in\\{1,...,m+n\\}$, where $C_{0}$ is a positive
constant independent on $\varepsilon,\,h$.

First, notice that profiles constructed in Proposition 4.1 are solutions of
(81). The obvious scaling property of these profiles, that we will denote
$\vec{k}^{\varepsilon}$ in the sequel, ensures the convergence of
$\vec{k}^{\varepsilon}(x)=\vec{k}^{1}(\frac{x}{\varepsilon})\to\vec{k}$ as
$\varepsilon\to 0$, for all $x\neq 0$. This readily yields a wide family of
vanishing viscosity limits.

######  Corollary 1.

Any $\vec{k}\in\mathcal{G}_{VV}^{o}$ can be obtained as the limit in the
$\mathbf{L}^{1}_{loc}$ sense, as $\varepsilon\to 0$, of a family
$\vec{k}^{\varepsilon}$ of stationary solutions of (81).

In general, using the theory of semigroups the authors of [13] proved the
existence of a unique solution $\vec{\rho^{\varepsilon}}$ of (81) such that

(83)
$\rho_{h}^{\varepsilon}\in\mathbf{C}([0,\infty[;\mathbf{L}^{2}(\Omega_{h}))\cap\mathbf{L}^{1}_{loc}(]0,+\infty[;\mathbf{W}^{2,1}(\Omega_{h})),\qquad\varepsilon>0,\quad
h\in\\{1,\ldots,m+n\\},$

in particular

(84)
$(\rho_{h}^{\varepsilon})_{t}\in\mathbf{L}^{1}_{loc}(]0,+\infty[,\mathbf{L}^{1}(\Omega_{h})),\qquad\varepsilon>0,\quad
h\in\\{1,\ldots,m+n\\}.$

Moreover, if we have two different initial conditions

$(\widetilde{u}_{1,\varepsilon}^{0},\ldots,\widetilde{u}_{m+n,\varepsilon}^{0}),\qquad(u_{1,\varepsilon}^{0},\ldots,u_{m+n,\varepsilon}^{0})$

for (81) satisfying (82), then the corresponding solutions to (81)

$(\widetilde{\rho}_{1}^{\varepsilon},\ldots,\widetilde{\rho}_{m+n}^{\varepsilon}),\qquad(\rho_{1}^{\varepsilon},\ldots,\rho_{m+n}^{\varepsilon})$

are stable in the following sense

(85)
$\begin{split}\sum\limits_{h=1}^{m+n}&{\left\|\rho^{\varepsilon}_{h}(t,\cdot)-\widetilde{\rho}^{\varepsilon}(t,\cdot)\right\|}_{\mathbf{L}^{1}(\Omega_{h})}\leq\sum\limits_{h=1}^{m+n}{\left\|u_{h,\varepsilon}^{0}-\widetilde{u}^{0}_{h,\varepsilon}\right\|}_{\mathbf{L}^{1}(\Omega_{h})},\end{split}$

for every $t\geq 0$.

The compactness argument of [13] is based on the compensated compactness
theory [29] and the following a priori estimates

(86) $\displaystyle 0\leq\rho^{\varepsilon}_{h}\leq R,\qquad
h\in\\{1,\ldots,m+n\\},$ (87)
$\displaystyle\sum\limits_{h=1}^{m+n}{\left\|\rho^{\varepsilon}_{h}(t,\cdot)\right\|}_{\mathbf{L}^{1}(\Omega_{h})}\leq\sum\limits_{h=1}^{m+n}{\left\|u^{0}_{h}\right\|}_{\mathbf{L}^{1}(\Omega_{h})},$
(88)
$\displaystyle\sum\limits_{h=1}^{m+n}{\left\|\rho^{\varepsilon}_{h}(t,\cdot)\right\|}_{\mathbf{L}^{2}(\Omega_{h})}^{2}+2\varepsilon\int_{0}^{t}\left(\sum\limits_{h=1}^{m+n}{\left\|\rho^{\varepsilon}_{h,x}(s,\cdot)\right\|}_{\mathbf{L}^{2}(\Omega_{h})}^{2}\right)ds$
$\displaystyle\qquad\qquad\qquad\leq\sum\limits_{h=1}^{m+n}{\left\|u_{h,\varepsilon}^{0}\right\|}_{\mathbf{L}^{2}(\Omega_{h})}^{2}+2|n-m|\max_{h}{\left\|f_{h}\right\|}_{\mathbf{W}^{1,\infty}(0,R)}t,$
(89)
$\displaystyle\sum\limits_{h=1}^{m+n}{\left\|\rho^{\varepsilon}_{h,t}(t,\cdot)\right\|}_{\mathbf{L}^{1}(\Omega_{h})}\leq(m+n)C_{0}+\max_{h}{\left\|f_{h}^{\prime}\right\|}_{\mathbf{L}^{\infty}(0,R)}\sum\limits_{h=1}^{m+n}TV(u^{0}_{h}),$

for every $t\geq 0$ and $\varepsilon>0$.

The main result in [13] shows that there exist a sequence
$\\{\varepsilon_{\ell}\\}_{\ell\in\mathbbm{N}}\subset(0,\infty)$,
$\varepsilon_{\ell}\to 0$ and a solution $\vec{\rho}$ of (1), in the sense of
Definition 1.1, such that

(90)
$\displaystyle\rho^{\varepsilon_{\ell}}_{h}\longrightarrow\rho_{h},\qquad\textrm{a.e.
and in }\mathbf{L}^{p}_{loc}(\mathbbm{R}_{+}\times\Omega_{h}),\,1\leq
p<\infty,$

for every $h\in\\{1,\ldots,m+n\\}$, where $\vec{\rho^{\varepsilon_{\ell}}}$ is
the corresponding solution of (81).

######  Remark 7.

Actually these results were proved in [13] in the case there all the functions
$f_{h}$ coincide, $h=1,\ldots,m+n$, 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 $(NLD)$: e.g., the strong precompactness result of
[27] can be used on each road in the place of the compensated compactness
method.

Here we improve the result of [13] showing the following:

######  Theorem 4.1.

Assume (82). Let $\\{\vec{\rho^{\varepsilon}}\\}_{\varepsilon>0}$ be the
family of solutions of (81). We have that

(91) $\rho^{\varepsilon}_{h}\longrightarrow\rho_{h},\qquad\textrm{a.e. and in
}\mathbf{L}^{\text{\rm$p$}}_{loc}(\mathbbm{R}_{+}\times\Omega_{h}),\,1\leq
p<\infty,$

where $\vec{\rho}$ is the unique admissible solution of (1) in the sense of
Definitions 1.2, 2.9, 2.10.

###### Proof.

Let $\\{\vec{\rho\,}^{\varepsilon_{\ell}}\\}_{\ell\in\mathbbm{N}}$ be the
solutions of (81) converging to $\vec{\rho}$ as in (90). According to Theorem
2.11, it is enough to justify that $\vec{\rho}$ satisfies the per road
Kruzhkov inequalities (11) for all $k\in[0,R]$ and the adapted entropy
inequalities (48) for all $\vec{k}\in\mathcal{G}_{VV}^{o}$. The first claim is
classical, see, e.g., [13]. We only need to justify the second claim.

According to Corollary 1, given $\vec{k}\in\mathcal{G}_{VV}^{o}$ there exist
$\vec{k}^{\varepsilon}$ stationary viscosity profiles converging to $\vec{k}$,
as $\varepsilon\to 0$. Now, arguing as in [13, p. 1773] but inserting in
addition a non-negative test function
$\xi\in\mathcal{D}(]0,+\infty[\times\mathbbm{R})$, we find the following Kato
inequality:

(92)
$\sum_{h=1}^{m+n}\left(\int_{\mathbbm{R}_{+}}\int_{\Omega_{h}}\left\\{|\rho^{\varepsilon_{\ell}}_{h}-k^{\varepsilon_{\ell}}_{h}|\xi_{t}+q_{h}(\rho^{\varepsilon_{\ell}}_{h},k^{\varepsilon_{\ell}}_{h})\xi_{x}+{\varepsilon_{\ell}}|\rho_{h}^{\varepsilon_{\ell}}-k^{{\varepsilon_{\ell}}}_{h}|_{x}\xi_{x}\right\\}\,dx\,dt\right)\geq
0.$

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)
$\int_{\mathbbm{R}_{+}}\\!\\!\int_{\Omega_{h}}\left(|\rho_{h}^{\varepsilon_{\ell}}-k^{{\varepsilon_{\ell}}}_{h}|\xi_{t}+q_{h}(\rho^{\varepsilon_{\ell}}_{h},k^{{\varepsilon_{\ell}}}_{h})\xi_{x}+{\varepsilon_{\ell}}|\rho_{h}^{\varepsilon_{\ell}}-k^{{\varepsilon_{\ell}}}_{h}|_{x}\xi_{x}\right)\,dx\,dt\geq
0,$

$h\in\\{1,\ldots,m+n\\}$, for all
$\xi\in\mathcal{D}(]0,+\infty[\times(\mathbbm{R}\setminus\\{0\\}))$. Moreover,
bearing in mind the existence of strong traces of
$q_{h}(\rho^{\varepsilon_{\ell}}_{h},k^{{\varepsilon_{\ell}}}_{h})+{\varepsilon_{\ell}}|\rho_{h}^{\varepsilon_{\ell}}-k^{{\varepsilon_{\ell}}}_{h}|_{x}$
as $x\to 0$, 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 $\xi$
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
$\vec{\rho\,}^{\varepsilon_{\ell}}$ and $\vec{k}^{\varepsilon_{\ell}}$, we
find

$\displaystyle 0\geq$
$\displaystyle-\sum_{h=1}^{m+n}\int_{\mathbbm{R}_{+}}\\!\\!\int_{\Omega_{h}}\left(|\rho_{h}^{\varepsilon_{\ell}}-k^{{\varepsilon_{\ell}}}_{h}|\xi_{t}+q_{h}(\rho^{\varepsilon_{\ell}}_{h},k^{{\varepsilon_{\ell}}}_{h})\xi_{x}+{\varepsilon_{\ell}}|\rho_{h}^{\varepsilon_{\ell}}-k^{{\varepsilon_{\ell}}}_{h}|_{x}\xi_{x}\right)\,dx\,dt$
$\displaystyle+\sum_{h=1}^{m}\int_{\mathbbm{R}_{+}}\left(q_{h}(\rho^{\varepsilon_{\ell}}_{h}(t,0),k^{{\varepsilon_{\ell}}}_{h})\xi(t,0)-{\varepsilon_{\ell}}|\rho_{h}^{\varepsilon_{\ell}}(t,0)-k^{{\varepsilon_{\ell}}}_{h}|_{x}\xi(t,0)\right)\,dt$
$\displaystyle-\sum_{h=m+1}^{m+n}\int_{\mathbbm{R}_{+}}\left(q_{h}(\rho^{\varepsilon_{\ell}}_{h}(t,0),k^{{\varepsilon_{\ell}}}_{h})\xi(t,0)-{\varepsilon_{\ell}}|\rho_{h}^{\varepsilon_{\ell}}(t,0)-k^{{\varepsilon_{\ell}}}_{h}|_{x}\xi(t,0)\right)\,dt$
$\displaystyle=$
$\displaystyle-\sum_{h=1}^{m+n}\int_{\mathbbm{R}_{+}}\\!\\!\int_{\Omega_{h}}\left(|\rho_{h}^{\varepsilon_{\ell}}-k^{{\varepsilon_{\ell}}}_{h}|\xi_{t}+q_{h}(\rho^{\varepsilon_{\ell}}_{h},k^{{\varepsilon_{\ell}}}_{h})\xi_{x}+{\varepsilon_{\ell}}|\rho_{h}^{\varepsilon_{\ell}}-k^{{\varepsilon_{\ell}}}_{h}|_{x}\xi_{x}\right)\,dx\,dt$
$\displaystyle+\int_{\mathbbm{R}_{+}}\mathrm{sign}\left(\rho_{h}^{\varepsilon_{\ell}}(t,0)-k^{{\varepsilon_{\ell}}}_{h}\right)\left(\sum_{i=1}^{m}(f_{i}(\rho^{\varepsilon_{\ell}}_{i}(t,0))-{\varepsilon_{\ell}}\rho_{i,x}^{\varepsilon_{\ell}}(t,0))\right.$
$\displaystyle\qquad\left.-\sum_{j=m+1}^{m+n}(f_{j}(\rho^{\varepsilon_{\ell}}_{j}(t,0))-{\varepsilon_{\ell}}\rho_{j,x}^{\varepsilon_{\ell}}(t,0))\right)\xi(t,0)\,dt$
$\displaystyle-\left(\sum_{i=1}^{m}f_{i}(k^{{\varepsilon_{\ell}}}_{h})-\sum_{j=m+1}^{m+n}f_{j}(k^{\varepsilon_{\ell}}_{j})\right)\int_{\mathbbm{R}_{+}}\mathrm{sign}\left(\rho_{h}^{\varepsilon_{\ell}}(t,0)-k^{{\varepsilon_{\ell}}}_{h}\right)\xi(t,0)\,dt$
$\displaystyle=$
$\displaystyle-\sum_{h=1}^{m+n}\int_{\mathbbm{R}_{+}}\\!\\!\int_{\Omega_{h}}\left(|\rho_{h}^{\varepsilon_{\ell}}-k^{{\varepsilon_{\ell}}}_{h}|\xi_{t}+q_{h}(\rho^{\varepsilon_{\ell}}_{h},k^{{\varepsilon_{\ell}}}_{h})\xi_{x}+{\varepsilon_{\ell}}|\rho_{h}^{\varepsilon_{\ell}}-k^{{\varepsilon_{\ell}}}_{h}|_{x}\xi_{x}\right)\,dx\,dt.$

Passing to the limit in (92) as $\varepsilon_{\ell}\to 0$, keeping in mind the
second term of estimate (88), we find inequality (48). This concludes the
proof. ∎

## References

  * [1] Andreianov, B., and Cancès, C. On interface transmission conditions for conservation laws with discontinuous flux of general shape.  J. Hyperbolic Differ. Equ. 12, 2 (2015), 343–384.
  * [2] Andreianov, B., Goatin, P., and Seguin, N. Finite volume schemes for locally constrained conservation laws.  Numerische Mathematik 115 (2010), 609–645.
  * [3] Andreianov, B., Karlsen, K. H., and Risebro, N. H. On vanishing viscosity approximation of conservation laws with discontinuous flux.  Netw. Heterog. Media 5, 3 (2010), 617–633.
  * [4] Andreianov, B., Karlsen, K. H., and Risebro, N. H. A theory of $L^{1}$-dissipative solvers for scalar conservation laws with discontinuous flux.  Arch. Ration. Mech. Anal. 201, 1 (2011), 27–86.
  * [5] Andreianov, B., and Mitrović, D. Entropy conditions for scalar conservation laws with discontinuous flux revisited.  Ann. Inst. H. Poincaré Anal. Non Linéaire 32, 6 (2015), 1307–1335.
  * [6] Andreianov, B., and Sbihi, K. Well-posedness of general boundary-value problems for scalar conservation laws.  Trans. Amer. Math. Soc. 367, 6 (2015), 3763–3806.
  * [7] Audusse, E., and Perthame, B. Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies.  Proc. Roy. Soc. Edinburgh Sect. A 135, 2 (2005), 253–265.
  * [8] Baiti, P., and Jenssen, H. K. Well-posedness for a class of $2\times 2$ conservation laws with $L^{\infty}$ data.  J. Differential Equations 140, 1 (1997), 161–185.
  * [9] Bardos, C., Leroux, A. Y., and Nedelec, J.-C. First order quasilinear equations with boundary conditions.  Communications in partial differential equations 4, 9 (1979), 1017–1034.
  * [10] Bressan, A., Čanić, S., Garavello, M., Herty, M., and Piccoli, B. Flows on networks: recent results and perspectives.  EMS Surv. Math. Sci. 1, 1 (2014), 47–111.
  * [11] Bürger, R., García, A., Karlsen, K. H., and Towers, J. D. A family of numerical schemes for kinematic flows with discontinuous flux.  J. Engrg. Math. 60, 3-4 (2008), 387–425.
  * [12] Bürger, R., Karlsen, K. H., and Towers, J. D. An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections.  SIAM J. Numer. Anal. 47, 3 (2009), 1684–1712.
  * [13] Coclite, G. M., and Garavello, M. Vanishing viscosity for traffic on networks.  SIAM J. Math. Anal. 42, 4 (2010), 1761–1783.
  * [14] Coclite, G. M., Garavello, M., and Piccoli, B. Traffic flow on a road network.  SIAM J. Math. Anal. 36, 6 (2005), 1862–1886.
  * [15] Colombo, R. M., and Goatin, P. A well posed conservation law with a variable unilateral constraint.  J. Differential Equations 234, 2 (2007), 654–675.
  * [16] Crandall, M. G., and Majda, A. Monotone difference approximations for scalar conservation laws.  Math. Comp. 34, 149 (1980), 1–21.
  * [17] Diehl, S. A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients.  J. Hyperbolic Differ. Equ. 06, 01 (2009), 127–159.
  * [18] Dubois, F., and LeFloch, P. Boundary conditions for nonlinear hyperbolic systems of conservation laws.  J. Differential Equations 71, 1 (1988), 93–122.
  * [19] Eymard, R., Gallouët, T., and Herbin, R. Finite volume methods.  In Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII. North-Holland, Amsterdam, 2000, pp. 713–1020.
  * [20] Garavello, M., and Piccoli, B. Traffic flow on networks, volume 1 of AIMS Series on Applied Mathematics.  American Institute of Mathematical Sciences (AIMS), Springfield, MO (2006).
  * [21] Holden, H., and Risebro, N. H. A mathematical model of traffic flow on a network of unidirectional roads.  SIAM J. Math. Anal. 26, 4 (1995), 999–1017.
  * [22] Holden, H., and Risebro, N. H. Front tracking for hyperbolic conservation laws, second ed., vol. 152 of Applied Mathematical Sciences.  Springer, Heidelberg, 2015.
  * [23] Imbert, C., and Monneau, R. Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks.  preprint HAL (2015).
  * [24] Imbert, C., Monneau, R., and Zidani, H. A Hamilton-Jacobi approach to junction problems and application to traffic flows.  ESAIM Control Optim. Calc. Var. 19, 1 (2013), 129–166.
  * [25] Kružhkov, S. N. First order quasilinear equations with several independent variables.  Mat. Sb. (N.S.) 81 (123) (1970), 228–255.
  * [26] Lebacque, J.-P. The Godunov scheme and what it means for first order traffic flow models.  In Internaional symposium on transportation and traffic theory (1996), pp. 647–677.
  * [27] Panov, E. Y. On sequences of measure-valued solutions of a first-order quasilinear equation.  Mat. Sb. 185, 2 (1994), 87–106.
  * [28] Panov, E. Y. Existence of strong traces for quasi-solutions of multidimensional conservation laws.  J. Hyperbolic Differ. Equ. 4, 4 (2007), 729–770.
  * [29] Tartar, L. Nonlinear analysis and mechanics: Heriot-watt symposium.  In Compensated compactness and applications to partial differential equations, vol. IV. Pitman, Boston, 1979, pp. 317–345.