# The Dijkgraaf-Witten invariant is a partial case of the photography method

Vassily Olegovich Manturov  Moscow Institute of Physics and Technology, Moscow
141700, Russia
Nosov Magnitogorsk State Technical University, Zhilyaev Laboratory of
mechanics of gradient nanomaterials, 38 Lenin prospect, Magnitogorsk, 455000,
Russian Federation
<EMAIL_ADDRESS>and  Zheyan Wan  Yanqi Lake Beijing Institute of
Mathematical Sciences and Applications, Beijing 101408, China
<EMAIL_ADDRESS>
###### Abstract.

In [9, 10] the first named author discovered a very general principle (called
the photography principle) which allows one:

a) To solve various equations (like pentagon equation)

b) To construct invariants of manfolds.

The advantage of that principle is that it deals with a very general notion of
data and data transmission which may be of any kind.

In the present paper, we show that the definition of the Dijkgraaf-Witten
invariants of manifolds can be thought of as an evidence of the above
principle.

Keywords: Dijkgraaf-Witten invariants, photography method

MSC 2020: 20F36, 13F60, 57K20, 57K31

##  1\. Introduction

A very general method, the photography method for solving the pentagon
equation, and other equations, was proposed by the first named author in [9].
Roughly speaking, if one has several states (say, triangulations of a
pentagon), some data for each state (e.g., edge lengths or triangle areas) and
a rule (formula) for translating data from one state to another (say, Ptolemy
equation), then after returning to the initial state, we obtain the same data.

Saying more standardly, this means that the photography method gives a
solution to some equations in terms of “data” whatever this data means (in
geometrical setting one can have lengths, angles, areas, volumes, etc.); even
the amount of data may change, so, it is sometimes hard to say what we mean by
an “equation”.

We state the photography method as follows: “We associate some data to each of
the triangulations of an $n$-manifold. Then we use some data transmission law:
if we know all for one triangulation, then we should know all for an adjacent
triangulation, hence for any other triangulation.” We may use this method to
solve equations or construct invariants.

For $n\geq 3$, $n$-manifolds can be described as equivalence classes of
triangulations modulo Pachner moves [11, 12]. With a triangulation, one can
associate some data and some data transmission laws: some elements in a group
$G$ corresponding to 1-simplices and the values of a group $n$-cocycle
$\phi\in H^{n}(G,A)$ ($A$ is an abelian group) corresponding to $n$-simplices,
and the way these values are changing under moves.

Taking this together, we can get to an equation (a system of equations) needed
to get a state-sum associated to triangulations, which will be invariant
under, say, (2-3)-Pachner moves and (1-4)-Pachner moves. This equation is
exactly the $n$-cocycle condition.

There are two versions of the photography method. One of them is for solving
equations [10] and the other one is for constructing invariants [7]. In [10]
we solve the pentagon equation in two ways: one by using the realisability of
edge-lengths as lambda-lengths in the hyperbolic plane and Ptolemy relation,
the other one by using the areas. In [7] we construct invariants of manifolds
by using the following observation. If we have $(n+1)$ points (say, 5) and all
edges between them, and for two $n$-tuples of points some relations hold, then
we can restore the last edge so that the relations hold for all possible
$n$-tuples. For such relations, one can take realisability of edge-lengths as
edge-lengths of a simplex in some Euclidean space.

The invariants we have constructed before [7] and here are not exactly
“Turaev-Viro invariants” [13] or Dijkgraaf-Witten invariants [3].

For example, in the paper [9] instead of taking a finite palette of colours
and solving the Biedenharn-Elliot equations, we guess how to get a solution.
The main guess is the “geometrical” observation that “if four quadrilaterals
ABCD and ABCE are inscribed then any of ABDE, ACDE, BCDE is inscribed”.

Similarly, in [10, Sec. 4] we used “areas” of triangles in order to get some
matrices satisfying pentagon identities. But these “areas” are not areas in
some proper sense: they are just variables which satisfy the only condition:
the sum of “areas” of a whole polyhedron equals the sum of its constituent
parts. Then we take it more abstractly, and get a solution to the pentagon
identity suggested by Korepanov in [8].

In the present paper a similar guess answers the question “How to get
Dijkgraaf-Witten-like invariants”? We take a triangulation and associate data
to its 1-dimensional edges and top-dimensional cells in order to get an
invariant under Pachner moves. The data we want to associate to edges are the
edge lengths, and the data associate to top-dimensional simplices are
“volumes”. When we try to figure out what we really need from volumes, we see
that this should be nothing but the cocycle condition (1). A crucial
observation explaining the “naturality” of the Dijkgraaf-Witten invariants is
Lemma 3.1 which allows to axiomatize volumes as cocycles.

The paper is organized as follows. In Sec. 2, we review the definition of the
Dijkgraaf-Witten invariants. In Sec. 3, we show that the Dijkgraaf-Witten
invariants are a partial case of the photography method. In Sec. 4, we list
some directions of further research.

###  1.1. Acknowledgements

The authors are grateful to L.A.Grunwald, L.H.Kauffman, Seongjeong Kim, and
V.G.Turaev for helpful discussion and comments. The study was supported by the
grant of Russian Science Foundation (No. 22-19-20073 dated March 25, 2022
“Comprehensive study of the possibility of using self-locking structures to
increase the rigidity of materials and structures”).

##  2\. Dijkgraaf-Witten invariants

In the present section, we review the Dijkgraaf-Witten invariants for
$n$-manifolds. For $n=3$, in [3] Dijkgraaf and Witten gave a combinatorial
definition for Chern-Simons with finite gauge groups using 3-cocycles of the
group cohomology. This was generalised to higher dimensional manifolds in [4].
We follow the description in [1], see also [14] (we use the Pachner moves).

Let $T$ be a triangulation of an oriented closed $n$-manifold $M$, with $a$
vertices and $m$ $n$-simplices. Give an ordering to the set of vertices. Let
$G$ be a finite group. Let $\phi:\\{\text{oriented edges}\\}\to G$ be a map
such that

  1. (1)

for any triangle with vertices $v_{0},v_{1},v_{2}$ of $T$, $\phi(\langle
v_{0},v_{2}\rangle)=\phi(\langle v_{0},v_{1}\rangle)\phi(\langle
v_{1},v_{2}\rangle)$, where $\langle v_{i},v_{j}\rangle$ denotes the oriented
edge with endpoints $v_{i}$ and $v_{j}$, and

  2. (2)

$\phi(-e)=\phi(e)^{-1}$.

Let $\alpha:G^{\times n}\to A$,
$(g_{1},g_{2},\dots,g_{n})\mapsto\alpha[g_{1}|g_{2}|\cdots|g_{n}]\in A$, be an
$n$-cocycle valued in a multiplicative abelian group $A$. The $n$-cocycle
condition is

(1)
$\displaystyle\alpha[g_{2}|g_{3}|\cdots|g_{n+1}]\alpha[g_{1}g_{2}|g_{3}|\cdots|g_{n+1}]^{-1}\alpha[g_{1}|g_{2}g_{3}|\cdots|g_{n+1}]\cdots{}$
$\displaystyle\cdot\alpha[g_{1}|g_{2}|\cdots|g_{n}g_{n+1}]^{(-1)^{n}}\alpha[g_{1}|g_{2}|\cdots|g_{n}]^{(-1)^{n+1}}=1.$

Then the Dijkgraaf-Witten invariant is defined by

(2) $\displaystyle
Z_{M}=\frac{1}{|G|^{a}}\sum_{\phi}\prod_{i=1}^{m}W(\sigma_{i},\phi)^{\epsilon_{i}}.$

Here $a$ denotes the number of the vertices of the given triangulation,
$W(\sigma,\phi)=\alpha[g_{1}|g_{2}|\cdots|g_{n}]$ where $\phi(\langle
v_{0},v_{1}\rangle)=g_{1}$, $\phi(\langle v_{1},v_{2}\rangle)=g_{2}$, $\dots$,
$\phi(\langle v_{n-1},v_{n}\rangle)=g_{n}$, for the $n$-simplex
$\sigma=|v_{0}v_{1}v_{2}\cdots v_{n}|$ with the ordering
$v_{0}<v_{1}<v_{2}<\cdots<v_{n}$, and $\epsilon=\pm 1$ according to whether or
not the orientation of $\sigma$ with respect to the vertex ordering matches
the orientation of $M$.

##  3\. Dijkgraaf-Witten invariants are a partial case of the photography
method

We associate some data to each of the triangulations of an $n$-manifold. We
may associate data to edges and $n$-simplices. Then we use some data
transmission law: if we know all for one triangulation, then we should know
all for an adjacent triangulation, hence for any other triangulation. The data
are the elements of $G$ associated to the edges and the values of an
$n$-cocycle on the $n$-simplices and we consider the state sum defined by (2).
The elements of $G$ associated to the edges of a triangle satisfy the product
rule. The data transmission law is the $n$-cocycle condition. Two adjacent
triangulations are connected by a Pachner move.

Figure 1. Pachner 2-3 moves. The tetrahedra 0134 and 1234 on the left becomes
the tetrahedra 0123, 1234, and 0124 on the right. Figure 2. Pachner 1-4 moves.
The tetrahedron 0123 on the left becomes the tetrahedra 0124, 0234, 0134, and
1234 on the right.

For $n=3$, the Pachner moves are the Pachner 2-3 and 1-4 moves, see Fig. 1 and
Fig. 2. The Pachner 2-3 move does not create a new vertex. The values of the
3-cocycle on the initial 2 3-simplices and the final 3 3-simplices satisfy the
3-cocycle condition. The 3-cocycle condition yields the invariance of the
state sum (2) under the Pachner 2-3 moves. The Pachner 1-4 move creates a new
vertex. Hence there is an additional factor $\frac{1}{|G|}$ in the state sum
(2). Also there is a new variable on an additional edge which runs through the
elements of $G$. For each choice of the new variable, the values of the
3-cocycle on the initial 3-simplex and the final 4 3-simplices satisfy the
3-cocycle condition. Hence each summand in $\frac{1}{|G|}\sum$ equals the
initial value of the 3-cocycle. Therefore the state sum (2) is invariant under
the Pachner 1-4 moves.

The cocycle condition can be formulated in terms of volumes. One crucial
observation is the following obvious lemma.

######  Lemma 3.1.

Given an $(n+1)$-simplex $\Delta^{n+1}$ with vertices belonging to the space
${\mathbb{R}}^{n+1}$. Then the sum of volumes of the faces of codimension 1 of
a projection of $\Delta^{n+1}$ onto the subspace ${\mathbb{R}}^{n}$ with
appropriate signs is zero.

Say, for $n=2$ one can consider a quadrilateral whose area can be split into
two triangles in two ways (a quadrilateral is a projection of a tetrahedron),
hence, the sum of two areas is equal to the sum of the other two areas. We can
articulate it as: the “area” function on $n$-simplices (in the latter case,
$2$-simplices) gives zero on the “boundary” of an $(n+1)$-simplex
($3$-simplex).

In Lemma 3.1, we showed that a volume is just a cocycle. Though a cocycle need
not be a volume, we may try to axiomatize volumes further and further to get
to cocycles finally. Therefore, $n$-cocycles can be treated as volumes of
simplices in dimension $n$.

######  Theorem 3.2.

The Dijkgraaf-Witten invariants are a partial case of the photography method.

###### Proof.

We associate weights to $n$-simplices. These weights satisfy the cocycle
condition which means that the sum of weights of a boundary is zero. We
checked the cocycle condition. This means that we checked that the sum of
volumes is zero.

For example, we consider the case $n=3$. Regarding the values of cocycles as
volumes up to a sign corresponding to the orientation, the cocycle condition
may be interpreted as the invariance of the sum of volumes, see Fig. 1 and
Fig. 2. In the Pachner 2-3 move, the sum of the volumes of the tetrahedra
0123, 0124, and 0234 equals the sum of the volumes of the tetrahedra 0134 and
1234. This is exactly the cocycle condition (up to a sign)

$\alpha[g|h|k]+\alpha[gh|k|l]+\alpha[g|h|kl]=\alpha[g|hk|l]+\alpha[h|k|l].$

In the Pachner 1-4 move, the volume of the tetrahedron 0123 equals the sum of
the tetrahedra 0124, 0234, 0134, and 1234. This is exactly the cocycle
condition (up to a sign)

$\alpha[g|h|k]=\alpha[g|h|kl]+\alpha[gh|k|l]+\alpha[g|hk|l]+\alpha[h|k|l].$

Therefore, the data satisfy the data transmission law (the $n$-cocycle
condition) for two adjacent triangulations connected by a Pachner move. The
photography method constructs the Dijkgraaf-Witten invariants.

∎

Recall that in [10, Sec. 4], we took areas. They are not really areas in some
Euclidean space nor hyperbolic space. They are formal variables, maybe from
some field. So, we can predict a solution by looking at some geometry in some
proper sense. Say, area of the quadrilateral equals sum of areas of
constituent parts, and then we can check it by hand. Here we do almost the
same. We take volumes, but volumes can not be negative, hence we take
cocycles. The only thing we need is that: a cocycle is linear and its
coboundary is 0. This is what is guaranteed by volumes, but this may go beyond
the volume in the strict sense.

##  4\. Further research

The authors are excited with the paper by Kevin Walker [15]. In that paper, he
presents a “universal” state-sum invariant. In our opinion, one of the
greatest advantages of that paper is that it deals with cell decompositions/
handle slides and handle cancellations rather than with Pachner moves.

But there are a lot of similarities between that work and our work: we
associate some data to cells, and this data is naturally changed under moves.

The next natural task is the Crane-Yetter invariant, which we shall do in a
subsequent publication.

It is important to calculate our invariants. Calculating Dijkgraaf-Witten
invariants is hard and some special cases were calculated in [2, 5, 6, 16, 17,
18, 19].

## References

  * [1] J.S. Carter, L.H. Kauffman, and M. Saito, Structures and Diagrammatics of Four Dimensional Topological Lattice Field Theories, Adv. Math. 146, 39–100 (1999)
  * [2] M. de Wild Propitius. Topological interactions in broken gauge theories. PhD thesis, University of Amsterdam, 1995.
  * [3] R. Dijkgraaf and E. Witten, Topologcal gauge theories and group cohomology, Comm. Math. Phys. 129 (1990), 393–429.
  * [4] D.S. Freed, Higher algebraic structures and quantization, Comm. Math. Phys. 159(2) (1994), 343–398.
  * [5] Y. Hu, Y. Wan, and Y.-S. Wu. Twisted quantum double model of topological phases in two dimensions. Physical Review B, 87(12):125114, 2013.
  * [6] H.-L. Huang, Z. Wan, and Y. Ye. Explicit cocycle formulas on finite abelian groups with applications to braided linear Gr-categories and Dijkgraaf–Witten invariants. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 150(4), 1937-1964.
  * [7] L. Kauffman, V. O. Manturov, I. M. Nikonov, S. Kim, Photography principle, data transmission, and invariants of manifolds, arXiv:2307.03437
  * [8] I.G. Korepanov, Private communication to V.O. Manturov, 2019.
  * [9] V.O. Manturov, I. Nikonov, The groups $\Gamma_{n}^{4}$, braids, and 3-manifolds, arXiv: 2305.06316.
  * [10] V.O.Manturov, Z.Wan, The photography method: solving pentagon, hexagon, and other equations, arXiv:2305.11945
  * [11] U. Pachner: Konstruktionsmethoden und das kombinatorische Homoomorphieproblem fur Triangulationen kompakter semilinearer Mannigfaltigkeiten, Abh Math Sem Univ Hamburg 57, 69-85 (1986).
  * [12] U. Pachner: P.L. Homeomorphic Manifolds are Equivalent by Elementary Shellings, Europ J Combinatorics 12, 129-145 (1991).
  * [13] V. G. Turaev, O. Ya. Viro, State sum invariants of 3-manifolds and quantum $6j$-symbols, _Topology_ 31:4 (1992), 865–902.
  * [14] M. Wakui, On Dijkgraaf-Witten invariants for 3-manifolds, Osaka J. Math. 29 (1992), 675–696.
  * [15] K. Walker, A universal state sum, arXiv: 2104.02101
  * [16] Y. Wan, J. C. Wang, and H. He. Twisted gauge theory model of topological phases in three dimensions. Physical Review B, 92(4):045101, 2015.
  * [17] J. Wang, K. Ohmori, P. Putrov, Y. Zheng, Z. Wan, M. Guo, H. Lin, P. Gao, and S.-T. Yau. Tunneling topological vacua via extended operators:(Spin-) TQFT spectra and boundary deconfinement in various dimensions. Progress of Theoretical and Experimental Physics, 2018(5):053A01, 2018.
  * [18] J. C. Wang and X.-G. Wen. Non-Abelian string and particle braiding in topological order: Modular SL(3,Z) representation and (3+1)-dimensional twisted gauge theory. Physical Review B, 91(3):035134, 2015.
  * [19] X.-G. Wen. Exactly soluble local bosonic cocycle models, statistical transmutation, and simplest time-reversal symmetric topological orders in 3+1 dimensions. Physical Review B, 95(20):205142, 2017.


# Stability estimates for nonlocal balance laws arising in traffic modelling

Felisia Angela Chiarello
Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica
Via Vetoio, Ed. Coppito 1 67100 L’Aquila AQ, Italy
<EMAIL_ADDRESS>and  Harold Deivi Contreras
GIMNAP-Departamento de Matemáticas, Universidad del Bío-Bío, Concepción,
Chile,
CI2MA-Universidad de Concepción, Casilla 160-C, Concepción, Chile.
<EMAIL_ADDRESS>
###### Abstract.

This paper focuses on the proof of the stability of entropy weak solutions of
a nonlocal balance law modelling vehicular traffic flow on a road with on- and
off-ramps. The stability is obtained with respect to a kernel function in the
source term. We get an estimate of the $\mathbf{L^{1}}-$dependence of the
solution with respect to the initial datum, the on-ramp rate, the off-ramp
rate and the mentioned kernel function. We also show numerical experiments in
order to perform an optimization problem in traffic flow with on-ramps.

##  1\. Introduction

###  1.1. Aim

In traffic flow modeling, nonlocal conservation laws are intended to describe
the behaviour of drivers that adapt their velocity with respect to what
happens in front of them, see [1, 3, 4, 7, 8]; the classical LWR (Lighthill-
Whitham [10] and Richards [13]) is not able to model this type of situation.
For this reason, in order to generalise the LWR model in such a way to make it
able to describe more realistic settings, in [2] a nonlocal balance law
intended to model vehicular traffic flow on a main road wit on- and off-ramps
is introduced and it is given by

(1.1) $\rho_{t}+(\rho
v(\rho*\omega_{\eta}))_{x}=S_{\mathrm{on}}(\cdot,\cdot,\rho,\rho*\omega_{\eta,\delta})-S_{\mathrm{off}}(\cdot,\cdot,\rho),\quad
x\in\mathbb{R},$

where $S_{\mathrm{on}}$ and $S_{\mathrm{off}}$ represents the traffic flow
entering and exiting through an on- and off-ramp respectively, and the
convolution term in $S_{\mathrm{on}}$ is defined as follows

$(\rho*\omega_{\eta,\delta})(t,x)=\int_{x-\eta+\delta}^{x+\eta+\delta}\rho(t,y)\omega_{\eta,\delta}(y-x)\mathrm{d}y,$

with $\eta\in[0,1]$ and $\delta\in[-\eta,\eta].$ Here, the parameter $\eta$
represents the length of the support of the kernel function
$\omega_{\eta,\delta}$, while $\delta$ is the point at which the maximum of
the kernel is attained. This choice of the kernel function models the fact
that drivers on the on-ramp can see what happens on backward and forward on
the main road.
It is well known that on-ramp merging has a great impact on traffic
efficiency, if one concentrates on highway networks the reduction of the
capacity is often due to on- and off-ramps, for this reason in this paper we
are interested to study the dependence of solutions to (1.1) on the
convolution kernel given in the source term $S_{\mathrm{on}}$. The strategies
that we employ are inspired by the results obtained in [2, 5]. In particular,
we adopt the results about existence and uniqueness to (1.1) presented in [2]
and we propose the analytical study of the dependence of solutions to (1.1) on
the kernel function in the source term.

###  1.2. Related work

In literature, the LWR model has been extended to include on and off-ramps by
means of source and sink terms, see [6, 9, 11, 12, 14, 15]. Starting from the
model introduced in [11], in [2] the authors introduce the new nonlocal
balance law (1.1), in which the velocity function depends on a weighted mean
of the downstream traffic density, in order to model the behaviour of drivers
that adapt their velocity with respect to what happens in front of them;
moreover the nonlocal source term is intended to model the fact that drivers
on the on-ramp can see what happens on backward and forward on the main road.
Regarding the aferomentioned model the existence and uniqueness of entropy
weak solution has been proved, specifically, approximating the problem using
an Upwind-type numerical scheme and providing compactness estimates for the
sequence of approximate solutions that allows to get the convergence and thus,
the existence of solution. Furthemore, by using Kružkov’s doubling of
variables technique the authors get an $\mathbf{L^{1}}$ stability estimate
depending on initial data, the on-ramp rate and the off-ramp rate. On the
other hand, motivated by control and optimisation problems, in [5] the authors
study a conservation law with nonlocal flux function given by

(1.2)
$\displaystyle\partial_{t}\rho+\partial_{x}\left(f(t,x,\rho)v(\rho\ast\omega)(t,x)\right)=0.$

The authors obtain an estimate of the dependence of the solution to (1.2) with
respect to the kernel function, the velocity and the initial datum. The
stability is obtained from an entropy condition through the doubling of
variables technique.

###  1.3. Outline of this paper.

This work is organized as follows: In Section 2 we recall the mathematical
model on which we will focus our study as well as the definitions of weak and
entropy weak solution. In section 3 we present the main results of this work,
i.e. the $\mathbf{L^{1}}-$ Lipschitz continuous dependence of the solutions to
(1.1) on to the initial datum, the on-ramp rate, the off-ramp rate and the
kernel function in the source term. Finally, in Section 4 we present numerical
examples in order to illustrate the behaviour of solutions when some
parameters of the kernel function in the source term are varying.

##  2\. Mathematical model

Let us consider the equation (1.1) with terms $S_{\mathrm{on}}$ and
$S_{\mathrm{off}}$ defined as

(2.1) $\displaystyle S_{\mathrm{on}}(t,x,\rho,\rho*\omega_{\eta,\delta})$
$\displaystyle=$
$\displaystyle\mathbf{1}_{\mathrm{on}}(x)q_{\mathrm{on}}(t)\left(1-\frac{\rho}{\rho_{\max}}\right)\left(1-\frac{\rho*\omega_{\eta,\delta}}{\rho_{\max}}\right),$
(2.2) $\displaystyle S_{\mathrm{off}}(t,x,\rho)$ $\displaystyle=$
$\displaystyle\mathbf{1}_{\mathrm{off}}(x)q_{\mathrm{off}}(t)\frac{\rho}{\rho_{\max}},$

with $\rho_{\max}=1$ for the sake of simplicity, and we also endow the
nonlocal traffic reaction model (1.1)-(2.1)-(2.2) with an initial condition as
follows

(2.3)
$\displaystyle\rho(x,0)=\rho_{0}(x)\in\left(\mathbf{L^{1}}\cap\mathbf{BV}\right)(\mathbb{R};[0,{\rho_{\max}}]).$

Let us assume the following assumptions on the parameters of model (1.1).

######  Assumptions 2.1.

We assume

  * ($i$)

$q_{\mathrm{on}}^{\mathrm{ramp}}\in\mathbf{L^{\infty}}(\mathbb{R}^{+};\mathbb{R}^{+}),q_{\mathrm{off}}^{\mathrm{ramp}}\in\mathbf{L^{\infty}}(\mathbb{R}^{+};\mathbb{R}^{+}).$

  * ($ii$)

$v\in\mathbf{C^{2}}([0,{\rho_{\max}}];\mathbb{R}^{+}),\ v^{\prime}(\rho)\leq
0,\,\rho\in[0,\rho_{\max}]$.

  * ($iii$)

$\omega_{\eta}\in\mathbf{C^{1}}([0,\eta];\mathbb{R}^{+})\ \textup{with}\
\omega^{\prime}_{\eta}(x)\leq 0,\
\int_{0}^{\eta}\omega_{\eta}(x)\mathrm{d}x=1,\ \forall\eta>0$.

  * ($iv$)

$\omega_{\eta,\delta}\in(\mathbf{C^{1}}\cap\mathbf{L^{1}})([\delta-\eta,\delta+\eta];\mathbb{R}^{+})\
\textup{with}\ \omega^{\prime}(x)_{\eta,\delta}\geq 0\ \text{for}\
x\in[\delta-\eta,0]$, $\omega^{\prime}(x)_{\eta,\delta}\leq 0\ \text{ for
}x\in[0,\delta+\eta]$, and
$\int_{\delta-\eta}^{\delta+\eta}\omega_{\eta,\delta}(x)\mathrm{d}x=1,\
\forall\eta>0.$

We consider solutions in a weak sense as follows

######  Definition 2.1.

Let
$\rho_{0}\in(\mathbf{L^{1}}\cap\mathbf{BV})(\mathbb{R};[0,{\rho_{\max}}]).$ We
say that
$\rho\in\mathbf{C}([0,T];\mathbf{L^{1}}(\mathbb{R};[0,{\rho_{\max}}])),$ with
$\rho(t,\cdot)\in\mathbf{BV}(\mathbb{R};[0,{\rho_{\max}}])$ for $t\in[0,T]$,
is a weak solution to (1.1)-(2.3) if for any
$\varphi\in\mathbf{C_{c}^{1}}([0,T[\times\mathbb{R};\mathbb{R})$

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}}\left(\rho\varphi_{t}+\rho
V\varphi_{x}\right)\mathrm{d}x\mathrm{d}t+\int_{0}^{T}\int_{\Omega_{\mathrm{on}}}S_{\mathrm{on}}\varphi\mathrm{d}x\mathrm{d}t$
$\displaystyle-\int_{0}^{T}\int_{\Omega_{\mathrm{off}}}S_{\mathrm{off}}\varphi\mathrm{d}x\mathrm{d}t+\int_{\mathbb{R}}\rho_{0}(x)\varphi(0,x)\mathrm{d}x=0,$

where $V(t,x)=v((\rho*\omega)(t,x))$.

Furthermore, we consider entropy weak solutions in Kružkov sense, i.e.

######  Definition 2.2.

Let
$\rho_{0}\in(\mathbf{L^{1}}\cap\mathbf{BV})(\mathbb{R};[0,{\rho_{\max}}]).$ We
say that
$\rho\in\mathbf{C}([0,T];\mathbf{L^{1}}(\mathbb{R};[0,{\rho_{\max}}])),$ with
$\rho(t,\cdot)\in\mathbf{BV}(\mathbb{R};[0,{\rho_{\max}}])$ for $t\in[0,T]$,
is a entropy weak solution to (1.1) with initial datum $\rho_{0}$ if for any
$\varphi\in\mathbf{C_{c}^{1}}([0,T[\times\mathbb{R};\mathbb{R})$ and for all
$k\in\mathbb{R}$

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}}\left(|\rho-k|\varphi_{t}+\left|\rho-k\right|V\varphi_{x}-\operatorname{sgn}(\rho-k)kV_{x}\varphi\right)\mathrm{d}x\mathrm{d}t$
$\displaystyle+\int_{0}^{T}\int_{\Omega_{\mathrm{on}}}\operatorname{sgn}(\rho-k)S_{\mathrm{on}}\varphi\mathrm{d}x\mathrm{d}t-\int_{0}^{T}\int_{\Omega_{\mathrm{off}}}\operatorname{sgn}(\rho-k)S_{\mathrm{off}}\varphi\mathrm{d}x\mathrm{d}t$
$\displaystyle+\int_{\mathbb{R}}|\rho_{0}-k|\varphi(0,x)\mathrm{d}x\geq 0,$

where $\Omega_{\mathrm{on}}$ and $\Omega_{\mathrm{off}}$ are the spatial
positions of on-ramp and off-ramp on the main road, respectively.

##  3\. Main Result

Before giving the main results of this work, we first recall the main Theorem
in [2, Theorem 2.1].

######  Theorem 3.1.

Let
$\rho_{0}\in\left(\mathbf{L^{1}}\cap\mathbf{BV}\right)\left(\mathbb{R};[0,1]\right)$.
Let the Assumptions 2.1 hold. Then, for all $T>0$, the problem (1.1) has a
unique solution
$\rho\in\mathbf{C^{0}}\left([0,T];\mathbf{L^{1}}(\mathbb{R};[0,1])\right)$ in
the sense of Definition 2.2. Moreover, the following estimates hold: for any
$t\in[0,T]$

$\begin{array}[]{ll}\left\|\rho(t)\right\|_{\mathbf{L^{1}}(\mathbb{R})}\leq\mathcal{R}_{1}(t),\\\
0\leq\rho(t,x)\leq 1,\\\ TV(\rho(t))\leq
e^{t\mathcal{H}}\left(TV(\rho_{0})+t\mathcal{Q}_{T}\right),\end{array}$

where

(3.1) $\
\begin{split}\mathcal{R}_{1}&=\left\|\rho_{0}\right\|_{\mathbf{L^{1}}(\mathbb{R})}+\left\|q_{\mathrm{on}}^{\mathrm{ramp}}(\cdot)\right\|_{\mathbf{L^{1}}([0,t])}-\min_{x\in\Omega_{\mathrm{on}}}\left\|q_{\mathrm{on}}^{\mathrm{ramp}}(\cdot)\rho(\cdot,x)\right\|_{\mathbf{L^{1}}([0,t])}\\\
&-\min_{x\in\Omega_{\mathrm{off}}}\left\|q_{\mathrm{off}}^{\mathrm{ramp}}(\cdot)\rho(\cdot,x)\right\|_{\mathbf{L^{1}}([0,t])}\\\
\end{split}$ (3.2) $\displaystyle\mathcal{Q}_{T}$ $\displaystyle=$
$\displaystyle{2}\left(\left\|q_{\mathrm{on}}\right\|_{\mathbf{L^{\infty}}([0,T])}+\left\|q_{\mathrm{off}}\right\|_{\mathbf{L^{\infty}}([0,T])}\right)$
(3.3) $\displaystyle\mathcal{H}$ $\displaystyle=$ $\displaystyle
2\left\|q_{\mathrm{on}}\right\|_{\mathbf{L^{\infty}}([0,T])}+\left\|q_{\mathrm{off}}\right\|_{\mathbf{L^{\infty}}([0,T])}+\omega_{\eta}(0)\mathcal{L}$
(3.4) $\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\left(\|v\|_{\mathbf{L^{\infty}}([0,1])}+\|v^{\prime}\|_{\mathbf{L^{\infty}}([0,1])}\right).$

The following theorem is the main result of this work and it states the
$\mathbf{L^{1}}$\- Lipschitz continuous dependence of solutions to (1.1) on to
the initial datum, the on-ramp rate, the off-ramp rate and the kernel
function.

######  Theorem 3.2.

Let $\rho$ and $\tilde{\rho}$ be two solutions to problem (1.1) in the sense
of Definition 2.2, with initial data $\rho_{0},\
\tilde{\rho}_{0}\in\mathbf{L^{1}}\cap\mathbf{BV}\left(\mathbb{R};[0,1]\right)$,
with on-ramp rates $q_{\mathrm{on}}$, $\tilde{q}_{\mathrm{on}}$, off-ramp
rates $q_{\mathrm{off}}$, $\tilde{q}_{\mathrm{off}}$ and kernel functions
${\omega}_{\eta,\delta},\,$ $\tilde{\omega}_{\eta,\delta}$, respectively.
Assume $v\in\mathbf{C^{2}}\left([0,1],\mathbb{R}^{+}\right)$. Then, for a.e.
$t\in[0,T]$,

$\displaystyle\left\|\rho(t)-\tilde{\rho}(t)\right\|_{\mathbf{L^{1}}(\mathbb{R})}$
$\displaystyle\leq$ $\displaystyle
e^{\mathcal{C}T}\bigg{(}\left\|\rho_{0}-\tilde{\rho}_{0}\right\|_{\mathbf{L^{1}}(\mathbb{R})}+{L}\left(\left\|q_{\mathrm{on}}-\tilde{q}_{\mathrm{on}}\right\|_{\mathbf{L^{1}}([0,t])}+\left\|q_{\mathrm{off}}-\tilde{q}_{\mathrm{off}}\right\|_{\mathbf{L^{1}}([0,T])}\right)$
$\displaystyle+r(T)\|\omega_{\eta,\delta}-\tilde{\omega}_{\eta,\delta}\|_{\mathbf{L^{1}}(\mathbb{R})}\bigg{)},$

where $r(T)$ depends on the $\mathbf{L^{1}}-$norms of initial conditions,
expected inflow flow of the on-ramp and expected output flow of the off-ramp
and $\mathcal{C}$ is defined as in [2, (3.24)].

###### Proof.

Since $\rho$ and $\tilde{\rho}$ are entropy weak solutions to (1.1) then

$\displaystyle\left\\{\begin{array}[]{lcc}\rho_{t}+(\rho
V(t,x))_{x}=S_{\mathrm{on}}\left(t,x,q_{\mathrm{on}},\rho,R_{\mathrm{on}}\right)-S_{\mathrm{off}}\left(t,x,q_{\mathrm{off}},\rho\right)\\\
\rho(0,x)=\rho_{0}(x),\end{array}\right.$
$\displaystyle\left\\{\begin{array}[]{lcc}\tilde{\rho}_{t}+(\tilde{\rho}\tilde{V}(t,x))_{x}={S}_{\mathrm{on}}\left(t,x,\tilde{q}_{\mathrm{on}},\tilde{\rho},\tilde{R}_{\mathrm{on}}\right)-S_{\mathrm{off}}\left(t,x,\tilde{q}_{\mathrm{off}},\rho\right)\\\
\tilde{\rho}(0,x)=\tilde{\rho}_{0}(x),\end{array}\right.$

in a distributional sense, where $V(t,x)=v((\rho\ast\omega_{\eta})(t,x)),$
$R_{\mathrm{on}}=(\rho\ast\omega_{\eta,\delta})(t,x)$,
$\tilde{V}(t,x)=v((\tilde{\rho}\ast\omega_{\eta})(t,x)),$
$\tilde{R}_{\mathrm{on}}=(\tilde{\rho}\ast\tilde{\omega}_{\eta,\delta})(t,x)$.
Following the argument in [2] and using Kružkov’s doubling of variables
technique we get

(3.7)
$\displaystyle\left\|\rho(T,\cdot)-\tilde{\rho}(T,\cdot)\right\|_{\mathbf{L^{1}}(\mathbb{R})}$
$\displaystyle\leq$
$\displaystyle\left\|\rho_{0}-\tilde{\rho}_{0}\right\|_{\mathbf{L^{1}}(\mathbb{R})}+\int_{0}^{T}\int_{\Omega_{\mathrm{on}}}\left|\tilde{\mathcal{S}}_{\mathrm{on}}\right|\mathrm{d}x\mathrm{d}t+\int_{0}^{T}\int_{\Omega_{\mathrm{off}}}\left|\tilde{\mathcal{S}}_{\mathrm{off}}\right|\mathrm{d}x\mathrm{d}t$
$\displaystyle+\int_{0}^{T}\int_{\mathbb{R}}\left|\mathcal{V}\right|\left|\partial_{x}\rho(t,x)\right|\mathrm{d}x\mathrm{d}t+\int_{0}^{T}\int_{\mathbb{R}}\left|\mathcal{V}_{x}\right|\left|\rho(t,x)\right|\mathrm{d}x\mathrm{d}t,$

where

$\displaystyle\tilde{\mathcal{S}}_{\mathrm{on}}$ $\displaystyle=$
$\displaystyle
S_{\mathrm{on}}\left(t,x,q_{\mathrm{on}},\rho,R_{\mathrm{on}}\right)-S_{\mathrm{on}}\left(t,x,\tilde{q}_{\mathrm{on}},\tilde{\rho},\tilde{R}_{\mathrm{on}}\right),$
$\displaystyle\tilde{\mathcal{S}}_{\mathrm{off}}$ $\displaystyle=$
$\displaystyle
S_{\mathrm{off}}\left(t,x,q_{\mathrm{on}},\rho\right)-S_{\mathrm{off}}\left(t,x,\tilde{q}_{\mathrm{on}},\tilde{\rho}\right),$
$\displaystyle\mathcal{V}$ $\displaystyle=$ $\displaystyle v(R)-v(P),$
$\displaystyle\mathcal{V}_{x}$ $\displaystyle=$
$\displaystyle\partial_{x}v(R)-\partial_{x}v(P).$

Except for the second term, all terms appearing at the right-hand side of
(3.7) are computed as in [2, Theorem 3.1],

$\displaystyle\int_{0}^{T}\int_{\Omega_{\mathrm{off}}}\left|\mathcal{S}_{\mathrm{off}}\right|\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq$
$\displaystyle{\left\|{q}_{\mathrm{off}}\right\|_{\mathbf{L^{\infty}}([0,T])}}\int_{0}^{T}\left\|\rho(t,\cdot)-\tilde{\rho}(t,\cdot)\right\|_{\mathbf{L^{1}}(\mathbb{R})}\mathrm{d}t+{L}\left\|q_{\mathrm{off}}-\tilde{q}_{\mathrm{off}}\right\|_{\mathbf{L^{1}}([0,T])},$
$\displaystyle\int_{0}^{T}\int_{\mathbb{R}}\left|\mathcal{V}\right|\left|\partial_{x}\rho(t,x)\right|\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq$
$\displaystyle\omega_{\eta}(0)\left\|v^{\prime}\right\|_{\mathbf{L^{\infty}}([0,1])}\sup_{t\in[0,T]}TV(\rho(t,\cdot))\int_{0}^{T}\left\|\rho(t,\cdot)-\tilde{\rho}(t,\cdot)\right\|_{\mathbf{L^{1}}(\mathbb{R})}\mathrm{d}t,$
$\displaystyle\int_{0}^{T}\int_{\mathbb{R}}\left|\mathcal{V}_{x}\right|\left|\rho(t,x)\right|\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq$
$\displaystyle\mathcal{W}\int_{0}^{T}\left\|\rho(t,\cdot)-\tilde{\rho}(t,\cdot)\right\|_{\mathbf{L^{1}}(\mathbb{R})}\mathrm{d}t.$

Regarding the second term in (3.7), we have

$\displaystyle\int_{0}^{T}\int_{\Omega_{\mathrm{on}}}\left|\tilde{\mathcal{S}}_{\mathrm{on}}\right|\mathrm{d}x\mathrm{d}t$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\Omega_{\mathrm{on}}}\left|S_{\mathrm{on}}\left(t,x,q_{\mathrm{on}},\rho,R_{\mathrm{on}}\right)-S_{\mathrm{on}}\left(t,x,\tilde{q}_{\mathrm{on}},\tilde{\rho},\tilde{R}_{\mathrm{on}}\right)\right|\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq$
$\displaystyle\int_{0}^{T}\int_{\Omega_{\mathrm{on}}}\left(\left|\tilde{\mathcal{S}}_{\mathrm{on}}^{1}\right|+\left|\tilde{\mathcal{S}}_{\mathrm{on}}^{2}\right|+\left|\tilde{\mathcal{S}}_{\mathrm{on}}^{3}\right|\right)\mathrm{d}x\mathrm{d}t,$

where

$\displaystyle\tilde{\mathcal{S}}_{\mathrm{on}}^{1}$ $\displaystyle=$
$\displaystyle
S_{\mathrm{on}}\left(t,x,q_{\mathrm{on}},\rho,R_{\mathrm{on}}\right)-S_{\mathrm{on}}\left(t,x,q_{\mathrm{on}},\rho,\tilde{R}_{\mathrm{on}}\right),$
$\displaystyle\tilde{\mathcal{S}}_{\mathrm{on}}^{2}$ $\displaystyle=$
$\displaystyle
S_{\mathrm{on}}\left(t,x,q_{\mathrm{on}},\rho,\tilde{R}_{\mathrm{on}}\right)-S_{\mathrm{on}}\left(t,x,q_{\mathrm{on}},\tilde{\rho},\tilde{R}_{\mathrm{on}}\right),$
$\displaystyle\tilde{\mathcal{S}}_{\mathrm{on}}^{3}$ $\displaystyle=$
$\displaystyle
S_{\mathrm{on}}\left(t,x,q_{\mathrm{on}},\tilde{\rho},\tilde{R}_{\mathrm{on}}\right)-S_{\mathrm{on}}\left(t,x,\tilde{q}_{\mathrm{on}},\tilde{\rho},\tilde{R}_{\mathrm{on}}\right).$

We bound $\tilde{\mathcal{S}}_{\mathrm{on}}^{2}$ and
$\tilde{\mathcal{S}}_{\mathrm{on}}^{3}$ writing

$\displaystyle\int_{0}^{T}\int_{\Omega_{\mathrm{on}}}\left|\tilde{\mathcal{S}}_{\mathrm{on}}^{2}\right|\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq$
$\displaystyle{\left\|q_{\mathrm{on}}\right\|_{\mathbf{L^{\infty}}([0,T])}}\int_{0}^{T}\left\|\rho(t,\cdot)-\tilde{\rho}(t,\cdot)\right\|_{\mathbf{L^{1}}(\mathbb{R})}\mathrm{d}t,$
$\displaystyle\int_{0}^{T}\int_{\Omega_{\mathrm{on}}}\left|\tilde{\mathcal{S}}_{\mathrm{on}}^{3}\right|\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq$
$\displaystyle\int_{0}^{T}\int_{\Omega_{\mathrm{on}}}\left|q_{\mathrm{on}}-\tilde{q}_{\mathrm{on}}\right|\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq$
$\displaystyle{L}\left\|q_{\mathrm{on}}-\tilde{q}_{\mathrm{on}}\right\|_{\mathbf{L^{1}}([0,T])}.$

Next, we are going to bound $\tilde{\mathcal{S}}_{\mathrm{on}}^{1}$ term,

$\displaystyle\left|\tilde{\mathcal{S}}_{\mathrm{on}}^{1}\right|$
$\displaystyle=$
$\displaystyle\left|\mathbf{1}_{\mathrm{on}}q_{\mathrm{on}}\left(1-\rho\right)\left(\left(1-R_{\mathrm{on}}\right)-\left(1-\tilde{R}_{\mathrm{on}}\right)\right)\right|$
$\displaystyle\leq$
$\displaystyle{\left\|q_{\mathrm{on}}\right\|_{\mathbf{L^{\infty}}([0,T])}}\left|\tilde{R}_{\mathrm{on}}-R_{\mathrm{on}}\right|,$

thus

$\displaystyle\int_{0}^{T}\int_{\Omega_{\mathrm{on}}}\left|\tilde{\mathcal{S}}_{\mathrm{on}}^{1}\right|\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq$
$\displaystyle{\left\|q_{\mathrm{on}}\right\|_{\mathbf{L^{\infty}}([0,T])}}\int_{0}^{T}\int_{\Omega_{\mathrm{on}}}\left|\tilde{R}_{\mathrm{on}}-R_{\mathrm{on}}\right|\mathrm{d}x\mathrm{d}t,$

and observe that

$\displaystyle\int_{\Omega_{\mathrm{on}}}\left|{R}_{\mathrm{on}}-\tilde{R}_{\mathrm{on}}\right|\mathrm{d}x$
$\displaystyle=$
$\displaystyle\int_{\Omega_{\mathrm{on}}}\left|\omega_{\eta,\delta}\ast\rho-\tilde{\omega}_{\eta,\delta}\ast\tilde{\rho}\right|\mathrm{d}x$
$\displaystyle=$
$\displaystyle\int_{\Omega_{\mathrm{on}}}\left|\omega_{\eta,\delta}\ast\rho-\omega_{\eta,\delta}\ast\tilde{\rho}+\omega_{\eta,\delta}\ast\tilde{\rho}-\tilde{\omega}_{\eta,\delta}\ast\tilde{\rho}\right|\mathrm{d}x$
$\displaystyle\leq$
$\displaystyle\int_{\Omega_{\mathrm{on}}}\left|\omega_{\eta,\delta}\ast\rho-\omega_{\eta,\delta}\ast\tilde{\rho}\right|+\left|\omega_{\eta,\delta}\ast\tilde{\rho}-\tilde{\omega}_{\eta,\delta}\ast\tilde{\rho}\right|\mathrm{d}x$
$\displaystyle=$
$\displaystyle\int_{\Omega_{\mathrm{on}}}\left|\omega_{\eta,\delta}\ast\left(\rho-\tilde{\rho}\right)\right|+\int_{\Omega_{\mathrm{on}}}\left|\left(\omega_{\eta,\delta}-\tilde{\omega}_{\eta,\delta}\right)\ast\tilde{\rho}\right|\mathrm{d}x$
$\displaystyle\leq$
$\displaystyle\left\|\omega_{\eta,\delta}\right\|_{\mathbf{L^{1}}(\Omega_{\mathrm{on}})}\left\|\rho-\tilde{\rho}\right\|_{\mathbf{L^{1}}(\Omega_{\mathrm{on}})}+\left\|\omega_{\eta,\delta}-\tilde{\omega}_{\eta,\delta}\right\|_{\mathbf{L^{1}}(\Omega_{\mathrm{on}})}\left\|\tilde{\rho}\right\|_{\mathbf{L^{1}}(\Omega_{\mathrm{on}})}$
$\displaystyle\leq$
$\displaystyle\left\|\omega_{\eta,\delta}\right\|_{\mathbf{L^{1}}(\mathbb{R})}\left\|\rho-\tilde{\rho}\right\|_{\mathbf{L^{1}}(\mathbb{R})}+\left\|\omega_{\eta,\delta}-\tilde{\omega}_{\eta,\delta}\right\|_{\mathbf{L^{1}}(\mathbb{R})}\left\|\tilde{\rho}\right\|_{\mathbf{L^{1}}(\mathbb{R})}$
$\displaystyle\leq$
$\displaystyle\left\|\rho-\tilde{\rho}\right\|_{\mathbf{L^{1}}(\mathbb{R})}+\left\|\omega_{\eta,\delta}-\tilde{\omega}_{\eta,\delta}\right\|_{\mathbf{L^{1}}(\mathbb{R})}\mathcal{R}_{1}(t),$

since $\int_{\mathbb{R}}\omega_{\eta,\delta}(x)\mathrm{d}x=1.$ Here
$\mathcal{R}_{1}$ is defined as in (3.1). Then,

$\displaystyle\int_{0}^{T}\int_{\Omega_{\mathrm{on}}}\left|\tilde{\mathcal{S}}_{\mathrm{on}}^{1}\right|\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq$
$\displaystyle{\left\|q_{\mathrm{on}}\right\|_{\mathbf{L^{\infty}}([0,T])}}\int_{0}^{T}\left\|\rho(t,\cdot)-\tilde{\rho}(t,\cdot)\right\|_{\mathbf{L^{1}}(\mathbb{R})}\mathrm{d}t$
$\displaystyle+\int_{0}^{T}\left\|\omega_{\eta,\delta}-\tilde{\omega}_{\eta,\delta}\right\|_{\mathbf{L^{1}}(\mathbb{R})}\mathcal{R}_{1}(t)\mathrm{d}t$
$\displaystyle=$
$\displaystyle\left\|q_{\mathrm{on}}\right\|_{\mathbf{L^{\infty}}([0,T])}\int_{0}^{T}\left\|\rho(t,\cdot)-\tilde{\rho}(t,\cdot)\right\|_{\mathbf{L^{1}}(\mathbb{R})}\mathrm{d}t$
$\displaystyle+\left\|\omega_{\eta,\delta}-\tilde{\omega}_{\eta,\delta}\right\|_{\mathbf{L^{1}}(\mathbb{R})}\int_{0}^{T}\mathcal{R}_{1}(t)\mathrm{d}t.$

Therefore, the inequality (3.7) is equivalent to following inequality

$\displaystyle\left\|\rho(T,\cdot)-\tilde{\rho}(T,\cdot)\right\|_{\mathbf{L^{1}}(\mathbb{R})}$
$\displaystyle\leq$
$\displaystyle\left\|\rho_{0}-\tilde{\rho}_{0}\right\|_{\mathbf{L^{1}}(\mathbb{R})}+{L}\left(\left\|q_{\mathrm{on}}-\tilde{q}_{\mathrm{on}}\right\|_{\mathbf{L^{1}}([0,t])}+\left\|q_{\mathrm{off}}-\tilde{q}_{\mathrm{off}}\right\|_{\mathbf{L^{1}}([0,t])}\right)$
$\displaystyle+\mathcal{C}\int_{0}^{T}\left\|\rho(t,\cdot)-\tilde{\rho}(t,\cdot)\right\|_{\mathbf{L^{1}}(\mathbb{R})}\mathrm{d}t+\left\|\omega_{\eta,\delta}-\tilde{\omega}_{\eta,\delta}\right\|_{\mathbf{L^{1}}(\mathbb{R})}\int_{0}^{T}\mathcal{R}_{1}(t)\mathrm{d}t$
$\displaystyle=$
$\displaystyle\left\|\rho_{0}-\tilde{\rho}_{0}\right\|_{\mathbf{L^{1}}(\mathbb{R})}+{L}\left(\left\|q_{\mathrm{on}}-\tilde{q}_{\mathrm{on}}\right\|_{\mathbf{L^{1}}([0,t])}+\left\|q_{\mathrm{off}}-\tilde{q}_{\mathrm{off}}\right\|_{\mathbf{L^{1}}([0,t])}\right)$
$\displaystyle+\mathcal{C}\int_{0}^{T}\left\|\rho(t,\cdot)-\tilde{\rho}(t,\cdot)\right\|_{\mathbf{L^{1}}(\mathbb{R})}\mathrm{d}t+r(T)\left\|\omega_{\eta,\delta}-\tilde{\omega}_{\eta,\delta}\right\|_{\mathbf{L^{1}}(\mathbb{R})}.$

An application of Gronwall’s Lemma to the above inequality completes the
proof. ∎

##  4\. Numerical examples

In this Section we use [2, Algorithm 3.1] in order to compute approximate
solutions to (1.1)-(2.3) with the source term (2.1). We simulate an
optimization problem in traffic merging, which consists in finding the optimal
value $\delta$ keeping fixed $\eta$ in the kernel function
$\omega_{\eta,\delta}.$ According to the meaning of that kernel function, this
is equivalent to find where a drivers located on the on-ramp should look in
order to avoid the congestion on the main road. To this end, we consider one
on-ramp with length $L=0.1$ located from $x=1.0$ until $x=1.1$, and we
consider the following kernel functions

$\displaystyle\omega_{\eta}(x)$ $\displaystyle:=$ $\displaystyle
2\frac{\eta-x}{\eta^{2}}\chi_{[0,\eta]}(x),$
$\displaystyle\omega_{\eta,\delta}(x)$ $\displaystyle:=$
$\displaystyle\frac{1}{\eta^{6}}\frac{16}{5\pi}\left(\eta^{2}-(x-\delta)^{2}\right)^{5/2}\chi_{[-\eta+\delta,\eta+\delta]}(x),$

for convective and reactive terms, respectively, with $\eta=0.5$ and
$\delta\in[-\eta,\eta]$, for $x\in[-1,4]$ at simulated time $T=6$ and velocity
function given by $v(\rho)=1-\rho$. We also consider $\Delta x=1/200$, a
constant initial condition $\rho_{0}(x)=0.3$, and the rate of the on-ramp
given by $q_{\mathrm{on}}(t)=1.2$. Following [5], as a metric of traffic
congestion we consider the two following functionals

$\displaystyle
J(T)=\int_{0}^{T}\mathrm{d}|\partial_{x}\rho(t,\cdot)|\mathrm{d}t,$
$\displaystyle\Psi(T;a,b)=\int_{0}^{T}\int_{a}^{b}\varphi(\rho(t,x))\mathrm{d}x\mathrm{d}t,$

where

$\displaystyle\varphi(r)=\left\\{\begin{array}[]{lcc}0,&r<0.75,\\\
10r-7.5,&0.75\leq r\leq 0.85,\\\ 1,&0.85<r\leq 1.\end{array}\right.$

The functional J defined measures the integral with respect to time of the
spatial total variation of the traffic density while the functional $\Psi$
measures the traffic congestion in the interval $[a,b]=[-1,4]$. Figure 1 shows
the values of the functionals $J$ and $\Psi$ when we vary the value of
$\delta,$ keeping fixed $\eta$. We can observe that the minimum value of
$\Psi$ is given when $\delta=0.1$. In Figure 2 we compare the solutions of
(1.1)-(2.3) for different values of $\delta$, keeping fixed $\eta.$ Namely, we
consider $\delta\in\\{-0.5,0.1,0.5\\}$ and $\eta=0.5$. We can see that the
solution for $\delta=0.1$ produces a smaller queue and therefore less increase
of the density on the main road when vehicles enter through the on-ramp than
the other values considered for $\delta$.

Figure 1. Left: Functional $J$ with $\eta=0.5$ and $\delta\in[-\eta,\eta]$.
Right: Functional $\Psi$ with $\eta=0.5$ and $\delta\in[-\eta,\eta]$ Figure 2.
Solution of (1.1)-(2.3) for $\eta=0.5$ and varying
$\delta\in\\{-0.5,0.1,0.5\\}$

##  5\. Conclusions

In this work we proved the $\mathbf{L^{1}}-$stability of entropy weak
solutions of a scalar nonlocal balance law with nonlocal source term arising
in traffic modelling with on- and off-ramps introduced in [2]. We reached an
estimate of the dependence of the solution with respect to the kernel function
in the source term, the on-ramp rate, the off-ramp rate and the initial datum.
The stability has been obtained from the entropy condition through the
doubling of variables technique. Finally, following [5], we have shown
numerical simulations illustrating the dependencies above for two cost
functionals derived from traffic flow applications.

## Acknowledgments

FAC 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). HDC are supported by the INRIA Associated Team “Efficient numerical
schemes for non-local transport phenomena” (NOLOCO; 2018–2020). HDC was
partially supported by the National Agency for Research and Development, ANID-
Chile through Scholarship Program, Doctorado Becas Chile 2021, 21210826 and by
Anillo project ACT210030. The authors would like to thank Luis Miguel Villada
for helpful discussions about this work.

## References

  * [1] S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numerische Mathematik, 132 (2016), pp. 217–241.
  * [2] F. A. Chiarello, H. D. Contreras, and L. M. Villada, Nonlocal reaction traffic flow model with on-off ramps, Networks and Heterogeneous Media, 17 (2022), p. 203.
  * [3] F. A. Chiarello, J. Friedrich, P. Goatin, S. Göttlich, and O. Kolb, A non-local traffic flow model for 1-to-1 junctions, European Journal of Applied Mathematics, 31 (2020), pp. 1029–1049.
  * [4] F. A. Chiarello and P. Goatin, Non-local multi-class traffic flow models, Networks & Heterogeneous Media, 14 (2019), p. 371.
  * [5] F. A. Chiarello, P. Goatin, and E. Rossi, Stability estimates for non-local scalar conservation laws, Nonlinear Analysis: Real World Applications, 45 (2019), pp. 668–687.
  * [6] M. L. Delle Monache, J. Reilly, S. Samaranayake, W. Krichene, P. Goatin, and A. M. Bayen, A pde-ode model for a junction with ramp buffer, SIAM Journal on Applied Mathematics, 74 (2014), pp. 22–39.
  * [7] J. Friedrich, S. Göttlich, and E. Rossi, Nonlocal approaches for multilane traffic models, Communications in Mathematical Sciences, 19 (2021), pp. 2291–2317.
  * [8] J. Friedrich, O. Kolb, and S. Göttlich, A godunov type scheme for a class of lwr traffic flow models with non-local flux, Networks & Heterogeneous Media, 13 (2018), pp. 531–547.
  * [9] D. Helbing, A. Hennecke, V. Shvetsov, and M. Treiber, Master: macroscopic traffic simulation based on a gas-kinetic, non-local traffic model, Transportation Research Part B: Methodological, 35 (2001), pp. 183–211.
  * [10] M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), pp. 317–345.
  * [11] G. Lipták, M. Pereira, B. Kulcsár, M. Kovács, and G. Szederkényi, Traffic reaction model, arXiv preprint arXiv:2101.10190, (2021).
  * [12] G. Liu, A. S. Lyrintzis, and P. G. Michalopoulos, Modelling of freeway merging and diverging flow dynamics, Applied mathematical modelling, 20 (1996), pp. 459–469.
  * [13] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), pp. 42–51.
  * [14] T. Tie-Qiao, H. Hai-Jun, and S. Hua-Yan, Effects of the number of on-ramps on the ring traffic flow, Chinese Physics B, 19 (2010), p. 050517.
  * [15] T. Tie-Qiao, H. Hai-Jun, S. Wong, G. Zi-You, and Z. Ying, A new macro model for traffic flow on a highway with ramps and numerical tests, Communications in Theoretical Physics, 51 (2009), p. 71.