# Mean field limit for one dimensional opinion dynamics with Coulomb
interaction and time dependent weights

Immanuel Ben-Porat Mathematical Institute, University of Oxford, Oxford OX2
6GG, UK<EMAIL_ADDRESS>José A. Carrillo Mathematical
Institute, University of Oxford, Oxford OX2 6GG, UK<EMAIL_ADDRESS>Sondre T. Galtung Department of Mathematical Sciences, Norwegian University of
Science and Technology – NTNU, 7491 Trondheim, Norway.
<EMAIL_ADDRESS>
###### Abstract

The mean field limit with time dependent weights for a 1D singular case, given
by the attractive Coulomb interactions, is considered. This extends recent
results [1, 8] for the case of regular interactions. The approach taken here
is based on transferring the kinetic target equation to a Burgers-type
equation through the distribution function of the measures. The analysis
leading to the stability estimates of the latter equation makes use of Kruzkov
entropy type estimates adapted to deal with nonlocal source terms.

##  1 Introduction

###  1.1 General background

In this paper, we are concerned with analyzing the mean field limit of the
following system of $2N$ ODEs

$\begin{cases}\dot{x}_{i}^{N}(t)=\frac{1}{N}\stackrel{{\scriptstyle[}}{{j}}=1]{N}{\sum}m_{j}^{N}(t)\partial_{x}V(x_{j}(t)-x_{i}(t)),&x_{i}^{N}(0)=x_{i}^{0,N}\\\
\dot{m}_{i}^{N}(t)=\psi_{i}^{N}(\mathbf{x}_{N}(t),\mathbf{m}_{N}(t)),&m_{i}^{N}(0)=m_{i}^{0,N}.\end{cases}$
(1.1)

The notation is as follows: the unknowns $x_{i}\in\mathbb{R}$ and
$m_{i}\in\mathbb{R}$ are referred to as the opinions and weights respectively.
The evolution of the opinions is given in terms of the weights and a function
$V:\mathbb{R}\rightarrow\mathbb{R}$ which is called the interaction. The
evolution of the weights is given by means of functions
$\psi_{i}^{N}:\mathbb{R}^{N}\times\mathbb{R}^{N}\rightarrow\mathbb{R}$ where
we apply the notation

$\mathbf{x}_{N}(t)\coloneqq(x_{1}^{N}(t),...,x_{N}^{N}(t)),\qquad\mathbf{m}_{N}(t)\coloneqq(m_{1}^{N}(t),...,m_{N}^{N}(t)).$

How the system (1.1) originates from real-life phenomena is beyond the scope
of this work. Just to mention a few works which explain how this system models
phenomena in biology and the social sciences, we refer to [1],[13] and [16].
The system (1.1) is a weighted version of the first order $N$-body problem
(simply by taking all the weights to be identically equal to $1$), to which we
now briefly draw our attention to. By now, the mean field limit of the
$N$-body problem

$\dot{x}_{i}^{N}(t)=\frac{1}{N}\stackrel{{\scriptstyle[}}{{j}}=1]{N}{\sum}\nabla
V(x_{j}(t)-x_{i}(t)),\qquad x_{i}^{N}(0)=x_{i}^{0,N}$ (1.2)

is fairly well understood, even in the case of interactions with strong
singularities near the origin—we will comment more about this later on. This
mean field limit is understood in terms of the empirical measure which is
defined by

$\mu_{N}(t)\coloneqq\frac{1}{N}\stackrel{{\scriptstyle[}}{{i}}=1]{N}{\sum}\delta_{x_{i}(t)}.$

Thanks to the work of Dobrushin [7], and assuming $\nabla V$ is Lipschitz, it
is possible to show that $\mu_{N}(t)$ converges to the solution $\mu$ of the
Vlasov equation

$\partial_{t}\mu(t,x)-\mathrm{div}(\mu\nabla
V\star\mu)(t,x)=0,\qquad\mu(0,\cdot)=\mu^{in}$ (1.3)

with respect to the Wasserstein metric (provided this is true initially of
course). When time dependent weights are included the problem is rendered
difficult, since now the candidate for the empirical measure is

$\mu_{N}(t)\coloneqq\frac{1}{N}\stackrel{{\scriptstyle[}}{{i}}=1]{N}{\sum}m_{i}(t)\delta_{x_{i}(t)}$

and formal considerations (see Proposition 15 in [7]) suggest the following
transport equation with self-consistent source term as the target equation

$\partial_{t}\mu(t,x)-\mathrm{div}(\mu\nabla
V\star\mu)(t,x)=h[\mu](t,x),\qquad\mu(0,\cdot)=\mu^{in}.$ (1.4)

Here $h[\mu]$ is the self-consistent source term which arises from the
inclusion of weights, and for the moment we do not specify it. Already at the
level of the well posedness theory of the target equation (1.4) and the system
(1.1) some care is needed, since apriori it is not entirely that the solution
stays a probability density for all times. The well-posedness of equations
(1.1) and (1.4) as well as the weighted mean field limit has been successfully
established in [1] and [8] in arbitrary dimension and for interactions with
Lipschitz gradient. The approach in [8] is based on stability estimates for
the Wasserstein distance, which imply both well-posedness and mean field
limit, whereas the approach in [1] recovers to some extent the mean field
limit obtained in [8], and is based on the graph limit method. We refer also
to [14] and [15] for more details about the graph limit regime and its link
with the mean field limit. Other works which consider weighted opinion
dynamics are [10], in which the weights are taken to be time-independent, but
may vary from one opinion to another, and [16] which serves as a general
survey.

###  1.2 Main results

All the existing literature reviewed so far concerns arbitrary dimension and
relatively well behaved potentials in terms of regularity, typically with at
least locally Lipschitz gradient. It is the aim of this work to investigate
how to overcome the challenges created due to singular potentials in 1D. In
particular, we consider the attractive 1D Coulomb interaction
$V(x)=\left|x\right|$ (so that $\partial_{x}V=\mathrm{sgn}(x)$, with the
convention $\mathrm{sgn}(0)=0$). In addition, we add further limitations on
the equation governing the weights, namely we take

$\psi_{i}^{N}(\mathbf{x}_{N},\mathbf{m}_{N})=\frac{1}{N}\stackrel{{\scriptstyle[}}{{j}}=1]{N}{\sum}m_{i}m_{j}S(x_{j}-x_{i})$
(1.5)

where $S\in C_{0}^{\infty}(\mathbb{R})$ is assumed to be odd. This parity
condition turns out to be important for the purpose of guaranteeing
perservation of the total mass. For these choices, the system (2.1) takes the
form

$\begin{cases}\dot{x}_{i}^{N}(t)=\frac{1}{N}\stackrel{{\scriptstyle[}}{{j}}=1]{N}{\sum}m_{j}^{N}(t)\operatorname*{sgn}(x_{j}(t)-x_{i}(t)),&x_{i}^{N}(0)=x_{i}^{0,N},\\\
\dot{m}_{i}^{N}(t)=\frac{1}{N}\stackrel{{\scriptstyle[}}{{j}}=1]{N}{\sum}m_{i}(t)m_{j}(t)S(x_{j}(t)-x_{i}(t)),&m_{i}^{N}(0)=m_{i}^{0,N}.\end{cases}$
(1.6)

In this case, the mean field equation takes the form

$\partial_{t}\mu(t,x)-\partial_{x}\left(\mu(t,x)\int^{x}_{-\infty}\mu(t,y)dy-\mu(t,x)\int^{\infty}_{x}\mu(t,y)dy\right)=\mu(t,x)S\star\mu(t,x),\qquad\mu(0,x)=\mu^{in}.$

When the kernel $S$ is odd one expects that $\mu$ stays a probability density
for all times and therefore the equation formally transforms to

$\partial_{t}\mu(t,x)-\partial_{x}\left(\mu(t,x)\left(2\int^{x}_{-\infty}\mu(t,y)dy-1\right)\right)=\mu(t,x)S\star\mu(t,x),\qquad\mu(0,x)=\mu^{in}.$
(1.7)

Setting $F(t,x)=-\frac{1}{2}+\int^{x}_{-\infty}\mu(t,y)dy$ and integrating
both sides of equation (1.7), we arrive (still at the formal level) at the
following non local Burgers type equation for $F$

$\partial_{t}F+\partial_{x}(A(F))=\mathbf{S}[F](t,x)$ (1.8)

where $A(F)\coloneqq-F^{2}$ and

$\mathbf{S}[F](t,x)\coloneqq F(t,x)(\phi\star
F)(t,x)-\int^{x}_{-\infty}F(t,z)(\partial_{z}\phi\star F)(t,z)dz,\
\phi\coloneqq\partial_{x}S.$

Of course, the definition of $\mathbf{S}[F]$ is motivated by integrating by
parts the expression
$\int^{x}_{-\infty}\partial_{z}F(t,z)(S\star\partial_{z}F)(t,x)dz$. The idea
of analyzing the equation for the primitive of $\mu$ stems from the work [4]
which studies the homogenous Burgers equation. The advantage of the equation
(1.8) in comparision to (1.7) is that the flux term is local, which raises the
hope that a well posedness theory is in reach. We refer to [6] for a well
posedness theory in the case of non-local fluxes (yet with no source term). By
closely adapting the method introduced in [11] (which has its roots in the
classical Kruzkov entropy) we are able to simultaneously prove the $BV$-well
posedness of the equation (1.8), and derive the unique solution of (1.8) in
the limit as $N\rightarrow\infty$ of the primitive of the empirical measure
(shifted by $-\frac{1}{2}$ in order to ensure proper cancellation), namely

$F_{N}(t,x)\coloneqq-\frac{1}{2}+\frac{1}{N}\stackrel{{\scriptstyle[}}{{k}}=1]{N}{\sum}m_{k}(t)H(x-x_{k}(t))$

where $H$ is the Heaviside function. We stress that the 1D settings are
essential for the analysis to be carried out properly. In the context of
higher dimensions, we mention the recent breakthrough [18], in which the
author succeeded in deriving the equation (1.3) from the system (1.2) for
interaction which may have even stronger singularities than Coulomb. The
modulated energy strategy in [18] is well suited for higher dimensions, which
raises the interesting question whether this strategy can be extended to the
framework of the present work in which time dependent weights are considered.

In Section 2 we fix some notation, prove the well posedness of the system
(1.1) for the case of the 1D attractive Coulomb interaction and record other
basic properties of the solutions of the ODE system. Section 3 is aimed at
constructing a discretized flux and showing that $F_{N}$ satisfies the Burgers
equation associated with this flux. Section 4 is devoted towards proving
Theorem 4.1, the main result of this work. The mean field limit and the
$BV$-well posedness of equation (1.8) are the content of Theorem 4.1. The main
differences which put our work in variance with [11] are reflected in the
well-posedness for the system (1.1), the handling of the source term, and the
verification that $F_{N}$ is an entropy solution of (a discretized version of)
equation (1.8).

##  2 Preliminaries

Consider the weighted $N$-body problem

$\begin{cases}\dot{x}_{i}^{N}(t)=\frac{1}{N}\underset{1\leq j\leq N:j\neq
i}{\sum}m_{j}^{N}(t)\mathrm{sgn}(x_{j}^{N}(t)-x_{i}^{N}(t)),&x_{i}(0)=x_{i}^{0,N},\\\
\dot{m}_{i}^{N}(t)=\psi_{i}^{N}(\mathbf{x}_{N}(t),\mathbf{m}_{N}(t)),&m_{i}(0)=m_{i}^{0,N}.\end{cases}$
(2.1)

For the purpose of establishing well posedness of the system (2.1) on short
times it is not strictly necessary to take $\psi_{i}^{N}$ to be of the form
(1.5). However, in order to have well-posedness for all times this special
form, and specifically the oddness of $S$ seems to be needed at least to some
extent. The hypotheses that we impose on $\psi_{i}^{N}$ in the sequel are
identical to those specified in [1] and are recalled below.

Hypothesis (H1). There is a constant $L>0$ such that for all
$\left(\mathbf{x}_{N},\mathbf{y}_{N},\boldsymbol{\mathbf{m}}_{N},\mathbf{p}_{N}\right)\in\mathbb{R}^{4N}$
the inequalities

$\left|\psi_{i}^{N}(\mathbf{x}_{N},\boldsymbol{\mathbf{m}}_{N})-\psi_{i}^{N}(\mathbf{y}_{N},\boldsymbol{\mathbf{m}}_{N})\right|\leq
L\left|\mathbf{x}_{N}-\mathbf{y}_{N}\right|,$

$\left|\psi_{i}^{N}(\mathbf{x}_{N},\boldsymbol{\mathbf{m}}_{N})-\psi_{i}^{N}(\mathbf{x}_{N},\boldsymbol{\mathbf{p}}_{N})\right|\leq
L\left|\boldsymbol{\mathbf{m}}_{N}-\mathbf{p}_{N}\right|$

hold. In addition, there is a constant $C>0$ such that and for each
$\left(\mathbf{x}_{N},\boldsymbol{\mathbf{m}}_{N}\right)\in\mathbb{R}^{4N}$ it
holds that

$\left|\psi_{i}^{N}(\mathbf{x}_{N},\boldsymbol{\mathbf{m}}_{N})\right|\leq
C(1+\underset{1\leq k\leq N}{\max}\left|m_{k}\right|).$ (2.2)

We shall hereafter supress the dependence on $N$ of the opinions/weights
whenever this dependence is irrelevant. Since we wish to prove well posedness
on a time interval which does not shrink to zero as $N\rightarrow\infty$, a
natural assumption to impose is that the opinions are seperated initially,
i.e.

$\forall N\in\mathbb{N}:x_{1}^{\mathrm{in}}<...<x_{N}^{\mathrm{in}}.$ (2.3)

###  2.1 Fillipov solutions

Before embarking on the task of establishing the stability estimates, we must
first prove the existence of a solution to the Equation (1.6) on some time
interval $[0,T]$, where $T$ is independent of $N$ (because eventually we wish
to take $N\rightarrow\infty$). It is the aim of this section to explain how
this is achieved- in fact we will prove existence for arbitrarily large times.
First, let us be lucid about the notion of solution that we work with. We
start by examining the more general ODE 2.1 and we specify later the stage in
which the special form 1.5 comes into play.

######  Definition 2.1.

A classical solution of the system (2.1) on $[0,T$] is an absolutely
continuous curve
$t\mapsto(\mathbf{x}_{N}(t),\boldsymbol{\mathbf{m}}_{N}(t))\in\mathrm{AC}([0,T];\mathbb{R}^{2N})$
such that the system (2.1) is satisfied for a.e. $t\in[0,T]$. Equivalently,
for all $t\in[0,T]$ it holds that

$\displaystyle x_{i}^{N}(t)$
$\displaystyle=x_{i}^{0,N}+\int^{t}_{0}\frac{1}{N}\underset{1\leq j\leq
N:j\neq
i}{\sum}m_{j}^{N}(\tau)\operatorname*{sgn}\left(x_{j}^{N}(\tau)-x_{i}^{N}(\tau)\right)d\tau,$
$\displaystyle m_{i}^{N}(t)$
$\displaystyle=m_{i}^{0,N}+\int^{t}_{0}\psi_{i}^{N}(\mathbf{x}_{N}(\tau),\mathbf{m}_{N}(\tau))d\tau.$

In order to prove existence of solutions in the sense of Definition 2.1 it is
convenient to use the machinery of differential inclusions as developed by
Fillipov [9]. The special form (1.6) is irrelevant for what concerns the
abstract theory of Fillipov. First we must review some basic definitions and
facts from convex analysis. Let us recall how to view the system (2.1) as a
differential inclusion. Let $\Omega\subset\mathbb{R}^{l}$ be some domain.
Given a vector field $f:\Omega\rightarrow\mathbb{R}^{l}$ whose set of
discontinuities is $D$ we assign to it a set valued map
$\mathcal{S}[f]:\Omega\rightarrow 2^{\mathbb{R}^{l}}$ defined as

$\mathcal{S}[f](X)\coloneqq\left\\{\underset{X^{\ast}\notin
D}{\underset{X^{\ast}\rightarrow X}{\lim}}f(X^{\ast})\right\\}.$

With this notation define the set valued map $\mathcal{F}[f]:\Omega\rightarrow
2^{\mathbb{R}^{l}}$by

$\mathcal{F}[f](X)\coloneqq\overline{\mathrm{co}}\mathcal{S}[f](X),$ (2.4)

where $\mathrm{co}$ stands for the convex hull.

######  Remark 2.1.

In the context of system (2.1) we take $f(\mathbf{x}_{N},\mathbf{m}_{N})$ to
be the vector

$\left(\frac{1}{N}\underset{1\leq j\leq
N}{\sum}m_{1}\mathrm{sgn}(x_{j}-x_{1}),\ldots,\frac{1}{N}\underset{1\leq j\leq
N}{\sum}m_{N}\mathrm{sgn}(x_{j}-x_{N}),\psi_{1}^{N}(\mathbf{x}_{N},\mathbf{m}_{N}),\ldots,\psi_{N}^{N}(\mathbf{x}_{N},\mathbf{m}_{N})\right).$
(2.5)

Next, we recall the notion of upper-semicontinuity.

######  Definition 2.2.

For each closed sets $A,B\subset\Omega$ let $\beta(A,B)\coloneqq\sup_{a\in
A}\inf_{b\in B}\left|a-b\right|$. A function $\mathcal{F}:\Omega\rightarrow
2^{\mathbb{R}^{l}}$ is said to be upper-semicontinuous at $p\in\Omega$ if
$\underset{p^{\prime}\rightarrow
p}{\lim}\beta(\mathcal{F}(p),\mathcal{F}(p^{\prime}))=0$. If $\mathcal{F}$ is
upper-semicontinuous for each $p\in\Omega$ then we say that it is upper-
semicontinuous.

On the other hand, recall the notion of a solution to a differential inclusion

######  Definition 2.3.

Let $\mathcal{F}:\Omega\rightarrow 2^{\mathbb{R}^{l}}$ be a set valued map. A
solution on $[\underline{T},\overline{T}]$ to the differential inclusion

$\dot{X}(t)\in\mathcal{F}(X(t))$ (2.6)

is an absolutely continuous map
$X:[\underline{T},\overline{T}]\rightarrow\mathbb{R}^{l}$ such that (2.6)
holds for a.e. $t\in[\underline{T},\overline{T}]$.

The existence theory of differential inclusions has been developed by several
different groups (Just to mention a few, see [2],[3] and [9]). We follow [9],
and by no means aim to give an exhaustive overview of this rich theory. The
following theorem is the main tool that we need to get existence for the
opinion dynamics.

######  Theorem 2.1.

([9], Theorem 1, Page 77) Let $\mathcal{F}:\mathbb{R}^{l}\rightarrow
2^{\mathbb{R}^{l}}$ be such that

1\. $\mathcal{F}(x)\neq\emptyset$ for all $x\in\Omega$.

2\. $\mathcal{F}(x)$ is compact and convex for all $x\in\Omega$.

3\. $\mathcal{F}$ is upper-semicontinuous.

Then for any $\overline{T}>0$ and any $X_{\underline{T}}\in\mathbb{R}^{l}$
there exist a solution (in the sense of Definition 2.3) to the differential
inclusion

$\dot{X}(t)\in\mathcal{F}(X(t)),\ X(\underline{T})=X_{\underline{T}}$

on the time interval $[\underline{T},\overline{T}]$.

Here is a convenient criteria for upper semi-continuity

######  Lemma 2.1.

([9], Lemma 3, Page 67) Let $f:\Omega\rightarrow\mathbb{R}^{l}$ be piecewise
continuous and let $\mathcal{F}[f]$ be defined as in (2.4). Then
$\mathcal{F}[f]$ is upper-semicontinuous.

We can now gather all of the above reminders to get the following important
conclusion

######  Corollary 2.1.

Let $\psi_{i}^{N}$ satisfy (H1). Let
$f:\mathbb{R}^{2N}\rightarrow\mathbb{R}^{2N}$ be given by (2.5). For any
$\overline{T}>0$ and any
$(\mathbf{x}_{N}^{\underline{T}},\mathbf{m}_{N}^{\underline{T}})\in\mathbb{R}^{2N}$
there exist a solution to the differential inclusion

$(\dot{\mathbf{x}}_{N},\dot{\mathbf{m}}_{N})\in\mathcal{F}[f](\mathbf{x}_{N}(t),\mathbf{m}_{N}(t)),\qquad(\mathbf{x}_{N}(\underline{T}),\mathbf{m}_{N}(\underline{T}))=(\mathbf{x}_{N}^{\underline{T}},\mathbf{m}_{N}^{\underline{T}})$

on $[\underline{T},\overline{T}]$.

###### Proof.

That $\mathcal{S}[f](p)$ is compact is an immediate consequence of the easy
observation that $\mathcal{S}[f]$ is finite set-valued map. For the same
reason $\mathcal{F}[f](p)$ is the set of all convex combinations of the points
of $\mathcal{S}[f](p),$ and is therefore compact (see, e.g., page 62 in [9]).
By Lemma 2.1, $\mathcal{F}[f]$ is upper-semicontinuous, so that by Theorem 2.1
the claim follows. ∎

It is important to observe that when $f$ is continuous, the map
$\mathcal{F}[f]$ is single valued and as a result the corresponding
differential inclusion reduces to an ODE in the sense of Definition 2.1. This
observation, as well as the special structure which arises from the attractive
interaction and forces opinions to stick together, allows us to construct
solutions to the system (2.1) in the sense of Definition 2.1. We now return to
the form 1.6. The following proposition is the main result of this section.

######  Proposition 2.1.

Let assumption (2.3) hold and suppose $m_{i}^{\mathrm{in}}>0$ and
$\frac{1}{N}\stackrel{{\scriptstyle[}}{{i}}=1]{N}{\sum}m_{i}^{\mathrm{in}}=1$.
Let $S$ be odd, continuous and bounded. For each $\overline{T}>0$ there exist
a solution on $[0,\overline{T}]$ to the system (1.6) (in the sense of
Definition (2.1)).

Proof. Step 1(a). Preservation of total mass. Let $\overline{T}>0$ and let
$(\mathbf{x}_{N}(t),\mathbf{m}_{N}(t))$ be some solution on $[0,\overline{T}]$
given by Corollary 2.1. For each $1\leq i\leq 2N$ let
$\mathcal{\mathfrak{f}}_{i}:\mathbb{R}^{2N}\rightarrow\mathbb{R}^{2N}$ be the
vector field whose $i$-th component identifies with the $i$-th component of
$f$ and is $0$ in all other components, so that
$f=\stackrel{{\scriptstyle[}}{{i}}=1]{2N}{\sum}\mathcal{\mathfrak{f}}_{i}$. By
subadditivity of $\mathcal{F}[f]$ we have

$\mathcal{F}[f]\subset\stackrel{{\scriptstyle[}}{{i}}=1]{2N}{\cup}\mathcal{F}[\mathcal{\mathfrak{f}}_{i}].$

Since the functions $\psi_{i}^{N}$ are continuous,
$\mathcal{F}[\mathcal{\mathfrak{f}}_{i}]=\left\\{\mathcal{\mathfrak{f}}_{i}\right\\}$
for each $N+1\leq i\leq 2N$, so that the weights are governed by an ODE,
namely for a.e. $t\in[0,\overline{T}]$ one has

$\dot{m}_{i}(t)=\psi_{i}^{N}(\mathbf{x}_{N}(t),\mathbf{m}_{N}(t)),\qquad
m_{i}(0)=m_{i}^{\mathrm{in},N}.$

We claim that the total mass is preserved. Indeed, using that $S$ is odd, let
us compute

$\displaystyle\frac{1}{N}\stackrel{{\scriptstyle[}}{{k}}=1]{N}{\sum}\dot{m}_{k}(t)$
$\displaystyle=\frac{1}{N^{2}}\stackrel{{\scriptstyle[}}{{k}}=1]{N}{\sum}\stackrel{{\scriptstyle[}}{{j}}=1]{N}{\sum}m_{k}(t)m_{j}(t)S(x_{j}(t)-x_{k}(t))=-\frac{1}{N^{2}}\stackrel{{\scriptstyle[}}{{k}}=1]{N}{\sum}\stackrel{{\scriptstyle[}}{{j}}=1]{N}{\sum}m_{k}(t)m_{j}(t)S(x_{k}(t)-x_{j}(t))$
$\displaystyle=-\frac{1}{N^{2}}\stackrel{{\scriptstyle[}}{{j}}=1]{N}{\sum}m_{j}(t)\stackrel{{\scriptstyle[}}{{k}}=1]{N}{\sum}m_{k}(t)S(x_{k}(t)-x_{j}(t))=-\frac{1}{N}\stackrel{{\scriptstyle[}}{{j}}=1]{N}{\sum}\dot{m}_{j}(t).$

As a result we get

$\frac{d}{dt}\frac{1}{N}\stackrel{{\scriptstyle[}}{{k}}=1]{N}{\sum}m_{k}(t)=0,$

so that

$\frac{1}{N}\stackrel{{\scriptstyle[}}{{k}}=1]{N}{\sum}m_{k}(t)=\frac{1}{N}\stackrel{{\scriptstyle[}}{{k}}=1]{N}{\sum}m_{k}^{\mathrm{in}}=1.$

Step 1(b).Weights remain positive. We claim that $m_{i}(t)>0$ for all
$t\in[0,\overline{T}]$. Indeed using preservation of total mass we obtain

$\left|\frac{d}{dt}\frac{1}{2}\log\left(m_{i}^{2}+\varepsilon^{2}\right)\right|=\left|\frac{m_{i}(t)\dot{m_{i}}(t)}{m_{i}^{2}+\varepsilon^{2}}\right|=\left|\frac{m_{i}^{2}(t)}{(m_{i}^{2}(t)+\varepsilon^{2})N}\stackrel{{\scriptstyle[}}{{j}}=1]{N}{\sum}m_{j}(t)S(x_{j}(t)-x_{i}(t))\right|\leq\left\|S\right\|_{\infty},$

hence

$-\left\|S\right\|_{\infty}\leq\frac{1}{2}\frac{d}{dt}\log\left(m_{i}^{2}(t)+\varepsilon^{2}\right)\leq\left\|S\right\|_{\infty}.$

Integration in time yields the for all $t\in[0,\overline{T}]$ the estimate

$-2\left\|S\right\|_{\infty}t+\log\left((m_{i}^{\mathrm{in}})^{2}+\varepsilon^{2}\right)\leq\log\left(m_{i}^{2}+\varepsilon^{2}\right)\leq
2\left\|S\right\|_{\infty}t+\log\left((m_{i}^{\mathrm{in}})^{2}+\varepsilon^{2}\right),$

and consequently

$\left((m_{i}^{\mathrm{in}})^{2}+\varepsilon^{2}\right)\exp(-2\left\|S\right\|_{\infty}t)\leq
m_{i}^{2}(t)+\varepsilon^{2}\leq\left((m_{i}^{\mathrm{in}})^{2}+\varepsilon^{2}\right)\exp(2\left\|S\right\|_{\infty}t).$

Letting $\varepsilon\rightarrow 0$ gives

$(m_{i}^{\mathrm{in}})^{2}\exp(-2\left\|S\right\|_{\infty}t)\leq
m_{i}^{2}(t)\leq(m_{i}^{\mathrm{in}})^{2}\exp(2\left\|S\right\|_{\infty}t).$

In particular $m_{i}(t)$ does not vanish on $[0,\overline{T}]$, which by
continuity and the assumption $m_{i}^{\mathrm{in}}>0$ implies $m_{i}(t)>0$ for
all $t\in[0,\overline{T}]$ with the estimate

$m_{i}^{\mathrm{in}}\exp(-\left\|S\right\|_{\infty}t)\leq m_{i}(t)\leq
m_{i}^{\mathrm{in}}\exp(\left\|S\right\|_{\infty}t).$

Step 2\. Construction of a classical solution on $[0,\overline{T}]$. The idea
is to apply an iteration argument which terminates after at most $N$ steps.
This iteration proceeds as follows. The assumption that $\forall i\neq
j:x_{i}^{0}\neq x_{j}^{0}$ implies that $\forall i\neq j:x_{i}(t)\neq
x_{j}(t)$ on some sufficiently (possibly $N$-dependent) short time interval
(by continuity). As a result the solution is classical on some short time- by
the same argument at the beginning of Step 1(a). Let

$T_{1}^{\ast}=\sup\left\\{0<T\leq\overline{T}\left|\forall t\in[0,T),\forall
i\neq j:x_{i}(t)\neq x_{j}(t)\right.\right\\}.$

If $T_{1}^{\ast}=\overline{T}$ we are done. If $T_{1}^{\ast}<\overline{T}$,
consider the set of all collisions at time $T_{1}^{\ast}$ with the $i$-th
particle, i.e.

$J_{1}^{i}\coloneqq\left\\{j\neq
i\left|x_{i}(T_{1}^{\ast})=x_{j}(T_{1}^{\ast})\right.\right\\},$

and set

$f_{i}^{1}(\mathbf{x}_{N},\mathbf{m}_{N})\coloneqq\frac{1}{N}\underset{j\notin
J_{1}^{i}}{\sum}m_{j}\mathrm{sgn}(x_{j}-x_{i}),$

and

$f^{1}\coloneqq(f_{1}^{1},...,f_{N}^{1}).$

Obviously $J_{1}^{i}=J_{1}^{j}$ iff $j\in J_{1}^{i}$. From the same
consideration as before the solution of the differential inclusion

$\dot{\mathbf{x}}_{N,1}\in\mathcal{F}[f^{1}](\mathbf{x}_{N,1}(t)),\
\mathbf{x}_{N,1}(T_{1}^{\ast})=\mathbf{x}_{N}(T_{1}^{\ast})$

satisfies $x_{i}(t)\neq x_{j}(t)$ for all $1\leq i\leq N$ and
$j\in\left[N\right]\setminus J_{1}^{i}$ with $i\neq j$, and is classical (both
of these conclusions hold on some short time of course). Let

$T_{2}^{\ast}=\sup\left\\{T_{1}^{\ast}<T\leq\overline{T}\left|\forall
t\in[T_{1}^{\ast},\overline{T}),\forall 1\leq i\leq N,\forall
j\in\left[N\right]\setminus J_{1}^{i},i\neq j:x_{i}(t)\neq
x_{j}(t)\right.\right\\}.$

If $T_{2}^{\ast}=\overline{T}$, then consider the curve

$\mathbf{y}_{N}(t)=\left\\{\begin{array}[]{c}\mathbf{x}_{N}(t),\
t\in[0,T_{1}^{\ast}]\\\ \mathbf{x}_{N,1}(t)\
t\in(T_{1}^{\ast},\overline{T}]\end{array}\right..$

In this case we make use of the attractive structure, which implies that
opinions which collided remain so, in order to show that $\mathbf{y}_{N}(t)$
is a solution:

######  Claim 2.1.

The curve $(\mathbf{y}_{N}(t),\mathbf{m}_{N}(t))$ is a solution to the system
(1.6) on $[0,\overline{T}]$.

Proof. The curve $\mathbf{y}_{N}(t)$ solves the Equation (1.6) on
$[0,T_{1}^{\ast}]$. To see that $\mathbf{y}_{N}(t)$ solves the equation on
$(T_{1}^{\ast},\overline{T}]$ we need to explain why particles that collided
at time $T_{1}^{\ast}$ remain collided for all $t>T_{1}^{\ast}$. Indeed,
suppose on the contrary that $x_{i_{0}}(T_{1}^{\ast})=x_{j_{0}}(T_{1}^{\ast})$
for some $i_{0}\neq j_{0}$, but

$\tau\coloneqq\mathrm{inf}\left\\{T_{1}^{\ast}<t\leq\overline{T}\left|x_{i_{0}}(t)\neq
x_{j_{0}}(t)\right.\right\\}>0,$

so that $x_{i_{0}}(\tau)=x_{j_{0}}(\tau)$. We may assume with no loss of
generality that $x_{i_{0}}(t)-x_{j_{0}}(t)>0$ on some sufficiently small time
interval $(\underline{\tau},\overline{\tau})\subset(\tau,\overline{T}]$. Let

$\tau_{\ast}\coloneqq\inf\left\\{\left.\tau<\tau^{\prime}<\underline{\tau}\right|\forall
t\in(\tau^{\prime},\overline{\tau}):x_{i_{0}}(t)-x_{j_{0}}(t)>0\right\\},$

so that $x_{i_{0}}(t)-x_{j_{0}}(t)>0$ for all
$t\in(\tau_{\ast},\overline{\tau})$ and
$x_{i_{0}}(\tau_{\ast})=x_{j_{0}}(\tau_{\ast})$. On the other hand, since
$J_{1}^{i_{0}}=J_{1}^{j_{0}}$, for a.e. $t\in(\tau_{\ast},\overline{\tau})$ we
compute that

$\displaystyle\dot{x}_{i_{0}}(t)-\dot{x}_{j_{0}}(t)$
$\displaystyle=\frac{1}{N}\underset{j\notin
J_{1}^{i_{0}}}{\sum}m_{j}(t)(\mathrm{sgn}(x_{j}(t)-x_{i_{0}}(t))-\mathrm{sgn}(x_{j}(t)-x_{j_{0}}(t)))$
(2.7) $\displaystyle=\frac{1}{N}\underset{j\notin
J_{1}^{i_{0}}:x_{i_{0}}(t)\geq x_{j}\geq
x_{j_{0}}(t)}{\sum}m_{j}(t)(\mathrm{sgn}(x_{j}-x_{i_{0}})-\mathrm{sgn}(x_{j}-x_{j_{0}}))\leq
0,$

which implies that for all $t\in(\tau_{\ast},\overline{\tau})$

$x_{i_{0}}(t)-x_{j_{0}}(t)\leq
x_{i_{0}}(\tau_{\ast})-x_{j_{0}}(\tau_{\ast})=0,$

which is absurd. Remark that Inequality (2.7) is thanks to the fact the the
weights are positive (step 1(b)). Therefore $x_{i_{0}}(t)=x_{j_{0}}(t)$ for
all $t>T_{1}^{\ast}$ which shows that for all $T_{1}^{\ast}<t\leq\overline{T}$
it holds that

$\frac{1}{N}\underset{j\notin
J_{1}^{i_{0}}}{\sum}\mathrm{sgn}(x_{j}(t)-x_{i}(t))=\frac{1}{N}\stackrel{{\scriptstyle[}}{{j}}=1]{N}{\sum}\mathrm{sgn}(x_{j}(t)-x_{i}(t)),$

so that $\mathbf{y}_{N}(t)$ is a solution to the system (1.1) on
$(T_{1}^{\ast},\overline{T}]$ as well.

$\square$

If $T_{2}^{\ast}<\overline{T}$, then we continue according to the algorithm
described above, which must terminate after at most $N$ steps, thereby
yielding an absolutely continuous curve $\mathbf{y}_{N}(t)$ on
$[0,\overline{T}]$. The same argument demonstrated in Claim (2.1) shows that
$\mathbf{y}_{N}(t)$ is a solution to the system (1.6) .

$\square$

We finish this section by observing a few elementary properties of solutions
to (1.6). First we stress that any classical solution on some given time
interval to the system (1.6) must satisfy the “sticky opinions property”,
namely that opinions that collide stay collided. As a result there exist a
finite partition of the time interval into sub-intervals on each of which no
new collisions occur. Moreover, opinions preserve the initial ordering. This
is summarized in the following

######  Proposition 2.2.

Let the assumptions of Proposition 2.1 hold. Let
$(\mathbf{x}_{N}(t),\mathbf{m}_{N}(t))$ be a solution to system (1.6) (in the
sense of Definition (2.1)) on some time interval $[0,\overline{T}]$. There
exist $0=T_{0}<T_{1}<...<T_{k-1}<T_{k}=\overline{T}$ ($1\leq k\leq N$) such
that for any given $1\leq l\leq k$ it holds that for any given $1\leq i,j\leq
N$ either $\forall t\in[T_{l-1},T_{l}):x_{i}(t)=x_{j}(t)$ or $\forall
t\in[T_{l-1},T_{l}):x_{i}(t)\neq x_{j}(t)$. Moreover, $x_{1}(t)\leq...\leq
x_{N}(t),\ t\in[0,\overline{T}]$.

A careful examination of the argument in Proposition 2.1 reveals that the
proof of Proposition 2.2 is in fact implicitly included in that of Proposition
2.1, and is therefore omitted. This is just to clarify that the ”sticky
opinions property” is not some kind of an extra assumption. In addition, for
the mean field limit it is important to observe that

$F_{N}(t,x)\coloneqq-\frac{1}{2}+\frac{1}{N}\stackrel{{\scriptstyle[}}{{k}}=1]{N}{\sum}m_{k}(t)H(x-x_{k}(t)),$
(2.8)

becomes constant for any $x$ outside some interval, which is a consequence of
preservation of total mass- this is precisely the place where we use the
assumption that $S$ is odd. This would allow to obtain stability estimates
globally in $L^{1}$ (rather only in $L_{\mathrm{loc}}^{1}$) and is recorded in
the following simple

######  Lemma 2.2.

Let the assumptions of Proposition 2.1 hold. Suppose also there is some
$\overline{X}$ such that for all $N\in\mathbb{N}$ and $1\leq i\leq N$ it holds
that $\left|x_{i}^{\mathrm{in}}\right|\leq\overline{X}$. Then, there exist
some
$\overline{R}=\overline{R}(\left\|S\right\|_{\infty},\overline{X},\overline{T})>0$
such that for any $F_{N}(t,\pm x)=\pm\frac{1}{2}$ for all
$t\in[0,\overline{T}]$ and $\left|x\right|>\overline{R}$.

###### Proof.

Thanks to the equation for the opinions, the bound for the masses in step 1 of
Proposition 2.1 and the assumption on the initial opinions, we have

$\left|x_{i}(t)\right|\leq\left|x_{i}^{\mathrm{in}}\right|+\frac{1}{N}\underset{1\leq
j\leq N:j\neq
i}{\sum}\int^{t}_{0}\left|m_{j}^{N}(\tau)\right|d\tau\leq\overline{X}+\overline{M}\overline{T},$

for some $\overline{M}=\overline{M}(\left\|S\right\|_{\infty},\overline{T})$.
Set
$\overline{R}=\overline{R}(\overline{T},\overline{X},\overline{M})\coloneqq\overline{X}+\overline{M}\overline{T}$.
Then according, for all $x\geq\overline{R}$ it holds that

$H(x-x_{k}(t))=1$

while for all $x<-\overline{R}$ it holds that

$H(x-x_{k}(t))=0.$

Keeping in mind step 1 of Proposition 2.1 we conclude that for all $x>R$ and
$t\in[0,\overline{T}]$

$F_{N}(t,x)=-\frac{1}{2}+\frac{1}{N}\sum_{k=1}^{N}m_{k}(t)=-\frac{1}{2}+1=\frac{1}{2},$

while for all $x<-\overline{R}$ and $t\in[0,\overline{T}]$

$F_{N}(t,x)=-\frac{1}{2}+0=-\frac{1}{2}.$

∎

##  3 The discretized version of the Burgers-type equation

In this section we construct a discretization of the flux by means of the
weights, and then show that $F_{N}$ is an entropy solution for the
corresponding Burgers like equation– that is equation (1.8), with the only
difference being that the flux $A$ is replaced by a discretized approximation
thereof. As usual, we assume that the opinion are ordered increasingly, i.e.,

$x_{1}^{0}<x_{2}^{0}<\cdots<x_{N}^{0}.$

Throughout this section we always work under the assumptions of Proposition
2.1 and take $t\mapsto(\mathbf{x}_{N}(t),\boldsymbol{\mathbf{m}}_{N}(t))$ to
be a solution of the system 1.6. For each $0\leq i\leq N$ we set

$\theta_{i}(t)\coloneqq\left\\{\begin{array}[]{c}-\frac{1}{2}+\frac{1}{N}\stackrel{{\scriptstyle[}}{{j}}=1]{i}{\sum}m_{j}(t),\
\ 1\leq i\leq N,\\\ -\frac{1}{2},\ \ i=0.\end{array}\right.$

Note that since the weights are positive for all times (step 1 in Proposition
2.1 ) the $\theta_{i}$ are ordered increasingly

$-\frac{1}{2}\equiv\theta_{0}(t)<\cdots<\theta_{N}(t)\equiv\frac{1}{2}.$

The discretized flux, denoted $A_{N}(t,x)$, is defined for each $t$ to be the
(unique) continuous, piecewise linear function with break points only at
$\left(\theta_{i}(t)\right)_{i=1}^{N-1}$ such that
$A(t,\theta_{i}(t))=A(\theta_{i}(t))$. In other words for each
$x\in(\theta_{i-1}(t),\theta_{i}(t))$

$A_{N}(t,x)\coloneqq\left(\frac{A(\theta_{i}(t))-A(\theta_{i-1}(t))}{\theta_{i}(t)-\theta_{i-1}(t)}\right)(x-\theta_{i}(t))+A(\theta_{i}(t)).$

Although the $A_{N}$ are time dependent, the following simple lemma shows that
they approximate the time independent flux $A$.

######  Lemma 3.1.

For each $t\in[0,T]$ and each $x\in[-\frac{1}{2},\frac{1}{2}]$ it holds that

$\left|A_{N}(t,x)-A(x)\right|\leq\frac{2\overline{M}\mathrm{Lip}(\left.A\right|_{[-\frac{1}{2},\frac{1}{2}]})}{N}.$

where $\overline{M}$ is as in Lemma 2.2.

###### Proof.

Keep $t\in[0,T]$ fixed. Let $x\in[-\frac{1}{2},\frac{1}{2}]$, and pick $1\leq
i\leq N$ such that $x\in[\theta_{i-1}(t),\theta_{i}(t)]$. By the above formula
for $A_{N}$ we get

$\left|A_{N}(t,x)-A(x)\right|\leq\left|\frac{A(\theta_{i}(t))-A(\theta_{i-1}(t))}{\theta_{i-1}(t)-\theta_{i}(t)}\left(x-\theta_{i}(t)\right)\right|+\left|A(\theta_{i}(t))-A(x)\right|.$

Clearly

$\left|\frac{x-\theta_{i}(t)}{\theta_{i-1}(t)-\theta_{i}(t)}\right|\leq 1,$

so that

$\displaystyle\left|A_{N}(t,x)-A(x)\right|$
$\displaystyle\leq\left|A(\theta_{i}(t))-A(\theta_{i-1}(t))\right|+\left|A(\theta_{i}(t))-A(x)\right|$
$\displaystyle\leq
2\mathrm{Lip}(\left.A\right|_{[-\frac{1}{2},\frac{1}{2}]})\left|\theta_{i}(t)-\theta_{i-1}(t)\right|\leq\frac{2\overline{M}\mathrm{Lip}(\left.A\right|_{[-\frac{1}{2},\frac{1}{2}]})}{N}.$

∎

The definition of entropy solutions originated in the celebrated work [12]. We
recall the notion of entropy solution for conservation laws with non-local
source terms, which will be the the notion of solution that we will use for
what concerns the mean field limit and the well posedness for the Cauchy
problem of the Burgers equation.

######  Definition 3.1.

Let $A\in C([0,T];\mathrm{Lip}[-\frac{1}{2},\frac{1}{2}])$. A function $F\in
BV([0,T]\times\mathbb{R})$ is called an entropy solution of the equation

$\partial_{t}F+\partial_{x}(A(t,F))=\mathbf{S}[F](t,x)$

with initial data $F^{\mathrm{in}}\in BV(\mathbb{R})$ if

(1) $\forall t\in[0,T],\forall x\geq R:F(t,\pm x)=\pm\frac{1}{2}$.

(2) The map $x\mapsto F(t,x)$ is non-decreasing for any $t\in[0,T]$.

(3) $F(t,\cdot)$ converges to $F^{\mathrm{in}}$ as $t\to 0^{+}$ in the sense
of distributions.

(4) For any $\chi\in C_{0}^{\infty}((0,T)\times\mathbb{R})$ and any
$\alpha\in\mathbb{R}$ it holds that

$\int_{0}^{T}\int_{\mathbb{R}}\left(\left|F-\alpha\right|\chi_{t}+\mathrm{sgn}(F-\alpha)\left(A(F)-A(\alpha)\right)\partial_{x}\chi+\mathrm{sgn}(F-\alpha)\chi\mathbf{S}[F](t,x)\right)dxdt\geq
0.$ (3.1)

######  Remark 3.1.

A particular byproduct of the requirements (1) and (2) is that for any
$t\in[0,T]$ the total variation $\left|\partial_{x}F(t,\cdot)\right|$ is a
probability measure.

######  Remark 3.2.

We emphasize that we can keep the classical definition of the convolution for
the term $\mathbf{S}[F]$ unlike in [11]. That is, if
$\mathrm{supp}(S)\subset[-r,r]$ we take
$R=\max\left\\{r,\overline{R}\right\\}$($\overline{R}$ as in Lemma 2.2) and
realizing that

$\phi\star F(t,z)\coloneqq\int_{-2R}^{2R}\phi(z-\zeta)F(t,\zeta)d\zeta,$

due to the definition of $S\star F$, $\phi=\partial_{z}S$, and the fact that
$F$ is constant outside the interval $[-2R,2R]$. It is now evident that
$\phi\star F,\partial_{z}\phi\star F\in C_{0}([0,T]\times\mathbb{R})$, which
enables to make sense of the last term in the left hand side of inequality
(3.1).

Next we claim that $F_{N}$ is an entropy solution to our Burgers equation but
with a discretized flux,. i.e. the equation

$\partial_{t}F_{N}+\partial_{x}(A_{N}(t,F_{N}))=\mathbf{S}[F_{N}](t,x).$ (3.2)

To verify this we need to check that

i. $F_{N}$ is a classical solution on finitely many regions which form a
disjoint partition of the whole domain $(0,T)\times\mathbb{R}$

ii. The Rankine-Hugoniot condition

iii. The Oleinik conditions.

It is classical that the verification of i-iii imply the the integral
inequality (3.1)- see Appendix 5 for more details about this implication. We
start by verifying point 1\. Let us recall that by Proposition 2.2 we know
there exist finitely many times $0=T_{0}<T_{1}<...<T_{k-1}<T_{k}=T$ such that
on each $(T_{j-1},T_{j})$ collision does not occur. More specifically, we know
that for each $1\leq j\leq k$ there is a disjoint partition of
$\left[N\right]$ into subsets
$I_{1}^{j},...,I_{m_{j}}^{j}\subset\left[N\right]$ (in brief
$\stackrel{{\scriptstyle[}}{{i}}=1]{m_{j}}{\sqcup}I_{i}^{j}=\left[N\right]$)
such that:

(a) Given $1\leq i\leq m_{j}$ it holds that $x_{\alpha}(t)=x_{\beta}(t)$ for
each $\alpha,\beta\in I_{i}^{j}$ and each $t\in[T_{j-1},T_{j})$.

(b) For each $1\leq i<i^{\prime}\leq m_{j}$ it holds that
$x_{\alpha}(t)<x_{\beta}(t)$ for each $\alpha\in I_{i}^{j},\beta\in
I_{i^{\prime}}^{j}$ and each $t\in[T_{j-1},T_{j})$.

For each $1\leq j\leq k$ and $1\leq i\leq m_{j}$ we let $i^{\ast}(j)$ be the
maximal index inside $I_{i}^{j}$, i.e.

$i^{\ast}(j)\coloneqq\max\left\\{r\left|r\in I_{i}^{j}\right.\right\\}.$

With this notation we have the following simple

######  Lemma 3.2.

Let $F_{N}$ be given by (2.8).For each $1\leq j\leq k$ and $1\leq i\leq m_{j}$
set

$V_{L}^{i,j}\coloneqq\left\\{(t,x)\left|t\in[T_{j-1},T_{j}),x_{(i-1)^{\ast}(j)}(t)\leq
x<x_{i^{\ast}(j)}(t)\right.\right\\}$

and

$V_{R}^{i,j}\coloneqq\left\\{(t,x)\left|t\in[T_{j-1},T_{j}),x_{i^{\ast}(j)}(t)<x\leq
x_{(i+1)^{\ast}(j)}(t)\right.\right\\}.$

(with the convention $x_{0^{\ast}}=-\infty$ and
$x_{(m_{j}+1)^{\ast}}=+\infty$. Then, for each $1\leq j\leq k$ and $1\leq
i\leq m_{j}$ , $F_{N}$ is a classical solution on both $V_{L}^{i,j}$ and
$V_{R}^{i,j}$.

###### Proof.

For any $(t,x)\in V_{L}^{i,j}$ we have

$F_{N}(t,x)=-\frac{1}{2}+\frac{1}{N}\stackrel{{\scriptstyle[}}{{r}}=1]{(i-1)^{\ast}(j)}{\sum}m_{r}(t)=\theta_{(i-1)^{\ast}(j)}(t)$

and for any $(t,x)\in V_{R}^{i,j}$ we have

$F_{N}(t,x)=-\frac{1}{2}+\frac{1}{N}\stackrel{{\scriptstyle[}}{{r}}=1]{i^{\ast}(j)}{\sum}m_{r}(t)=\theta_{i^{\ast}(j)}(t).$

In particular, note that $A_{N}(t,F_{N}(t,x))$ is constant in $x$ on both
regions, so that its $x$-derivative vanishes. In addition for each
$x_{(i-1)^{\ast}(j)}(t)\leq x<x_{i^{\ast}(j)}(t)$ and $t\in[T_{j-1},T_{j})$ a
routine calculation shows that

$\displaystyle\mathbf{S}[F_{N}](t,x)$
$\displaystyle=\frac{1}{N^{2}}\underset{l,r}{\sum}H(x-x_{l}(t))m_{l}(t)m_{r}(t)S(x_{l}(t)-x_{r}(t))$
$\displaystyle=\frac{1}{N^{2}}\stackrel{{\scriptstyle[}}{{r}}=1]{N}{\sum}\stackrel{{\scriptstyle[}}{{l}}=1]{(i-1)^{\ast}(j)}{\sum}m_{l}(t)m_{r}(t)S(x_{l}(t)-x_{r}(t)),$

while

$\displaystyle\frac{d}{dt}F_{N}(t,x)$
$\displaystyle=\frac{1}{N^{2}}\stackrel{{\scriptstyle[}}{{l}}=1]{(i-1)^{\ast}(j)}{\sum}\stackrel{{\scriptstyle[}}{{r}}=1]{N}{\sum}m_{l}(t)m_{r}(t)S(x_{l}(t)-x_{r}(t))$
$\displaystyle=\frac{1}{N^{2}}\stackrel{{\scriptstyle[}}{{r}}=1]{N}{\sum}\stackrel{{\scriptstyle[}}{{l}}=1]{(i-1)^{\ast}(j)}{\sum}m_{l}(t)m_{r}(t)S(x_{l}(t)-x_{r}(t))=\mathbf{S}[F_{N}].$

This shows that $F_{N}$ is a classical solution on $V_{L}^{i,j}$. The same
calculation shows that $F_{N}$ is a classical solution on $V_{R}^{i,j}$. ∎

We move to verify conditions (ii) and (iii).

######  Lemma 3.3.

The function $F_{N}$ is an entropy solution to Equation (3.2) in the sense of
Definition (3.1)

###### Proof.

Condition (1) is from Lemma 2.2 and that $x\mapsto F_{N}(t,x)$ is non-
decreasing is immediate from the fact that the weights are positive, which
gives condition (2). Validating the requested weak inequality is slightly
longer and rests upon the Rankine-Hugoniot and Oleinik conditions (which as
already remarked, are recapped in Appendix 5). With the same notation of Lemma
3.2, keep $1\leq j\leq k$ and $1\leq i\leq m_{j}$ fixed and consider the curve

$\Gamma_{i,j}\coloneqq\left\\{(t,x_{i^{\ast}(j)}(t))\left|t\in[T_{j-1},T_{j})\right.\right\\}.$

The Rankine-Hugoniot Condition. We wish to show that

$\frac{A(t,F_{N,L}^{i,j}(t))-A(t,F_{N,R}^{i,j}(t))}{F_{N,L}^{i,j}(t)-F_{N,R}^{i,j}(t)}=\dot{x}_{i^{\ast}(j)}(t).$

To make the equations a bit lighter let us abbreviate
$i^{\ast}(j)=\overline{i},(i-1)^{\ast}(j)=\underline{i}$. It is clear that

$F_{N,L}^{i,j}(t)=-\frac{1}{2}+\frac{1}{N}\stackrel{{\scriptstyle[}}{{r}}=1]{\underline{i}}{\sum}m_{r}(t)=\theta_{\underline{i}}(t)$

and

$F_{N,R}^{i,j}(t)=-\frac{1}{2}+\frac{1}{N}\stackrel{{\scriptstyle[}}{{r}}=1]{\overline{i}}{\sum}m_{r}(t)=\theta_{\overline{i}}(t).$

Since $\frac{1}{N}\stackrel{{\scriptstyle[}}{{r}}=1]{N}{\sum}m_{r}(t)=1$ we
may rewrite

$F_{N,L}^{i,j}(t)=\frac{1}{2}-\frac{1}{N}\underset{r>\underline{i}}{\sum}m_{r}(t).$

Therefore, using that
$\frac{1}{N}\stackrel{{\scriptstyle[}}{{j}}=1]{N}{\sum}m_{j}(t)=1$ and the
fact that the initial order is preserved, the following identites hold

$\displaystyle\frac{A_{N}(t,F_{N,L}^{i,j}(t))-A_{N}(t,F_{N,R}^{i,j}(t))}{F_{N,L}^{i,j}(t)-F_{N,R}^{i,j}(t)}$
$\displaystyle=\frac{\theta_{\overline{i}}^{2}(t)-\theta_{\underline{i}}^{2}(t)}{\theta_{\underline{i}}(t)-\theta_{\overline{i}}(t)}=-(\theta_{\overline{i}}(t)+\theta_{\underline{i}}(t))$
$\displaystyle=1-\frac{1}{N}\overset{\overline{i}}{\underset{r=1}{\sum}}m_{r}(t)-\frac{1}{N}\overset{\underline{i}}{\underset{r=1}{\sum}}m_{r}(t)$
$\displaystyle=1-\left(1-\frac{1}{N}\underset{r>\overline{i}}{\sum}m_{r}(t)\right)-\frac{1}{N}\overset{\underline{i}}{\underset{r=1}{\sum}}m_{r}(t)=\frac{1}{N}\underset{r>\overline{i}}{\sum}m_{r}(t)-\frac{1}{N}\overset{\underline{i}}{\underset{r=1}{\sum}}m_{r}(t)$
$\displaystyle=\frac{1}{N}\underset{r>\overline{i}}{\sum}\mathrm{sgn}(x_{r}(t)-x_{\overline{i}}(t))m_{r}(t)+\frac{1}{N}\overset{\underline{i}}{\underset{r=1}{\sum}}\mathrm{sgn}(x_{r}(t)-x_{\overline{i}}(t))m_{r}(t)$
$\displaystyle\quad+\frac{1}{N}\overset{\overline{i}}{\underset{r=\underline{i}+1}{\sum}}\mathrm{sgn}(x_{r}(t)-x_{\overline{i}}(t))m_{r}(t)$
$\displaystyle=\frac{1}{N}\stackrel{{\scriptstyle[}}{{r}}=1]{N}{\sum}m_{r}(t)\mathrm{sgn}(x_{r}(t)-x_{\overline{i}}(t))=\dot{x}_{\overline{i}}(t).$

The Oleinik Condition. Let
$\theta\in(\theta_{\underline{i}}(t),\theta_{\overline{i}}(t))$. We wish to
show

$\frac{A_{N}(t,\theta)-A_{N}(t,\theta_{\underline{i}}(t)))}{\theta-\theta_{\underline{i}}(t)(t)}\geq\dot{x}_{\overline{i}}(t).$

Note that if we pick $\underline{i}+1\leq m\leq\overline{i}$ such that
$\theta\in[\theta_{m-1}(t),\theta_{m}(t)]$ then since $A_{N}$ is piecewise
linear we clearly have

$\frac{A_{N}(t,\theta)-A_{N}(t,\theta_{\underline{i}}(t)))}{\theta-\theta_{\underline{i}}(t)}\geq\min\left\\{\frac{A_{N}(t,\theta_{m}(t))-A_{N}(t,\theta_{\underline{i}}(t)))}{\theta_{m}(t)-\theta_{\underline{i}}(t)},\frac{A_{N}(t,\theta_{m-1}(t))-A_{N}(t,\theta_{\underline{i}}(t)))}{\theta_{m-1}(t)-\theta_{\underline{i}}(t)}\right\\}.$

Therefore it suffices to check the inequality for $\theta=\theta_{k}(t)$ where
$\underline{i}\leq k\leq\overline{i}$. We have

$\frac{A_{N}(t,\theta_{k}(t))-A_{N}(t,\theta_{\underline{i}}(t)))}{\theta_{k}(t)-\theta_{\underline{i}}(t)}=\frac{A(\theta_{k}(t))-A(\theta_{\underline{i}}(t))}{\theta_{k}(t)-\theta_{\underline{i}}(t)}=-(\theta_{k}(t)+\theta_{\underline{i}}(t))\geq-(\theta_{\overline{i}}(t)+\theta_{\underline{i}}(t))=\dot{x}_{\overline{i}}(t)$

where the inequality is because $\theta_{\overline{i}}(t)\geq\theta_{k}(t)$,
and the last identity is a byproduct of the previous step. ∎

##  4  The Mean Field Limit

In this section we prove well-posedness, stability estimates and mean field
limit for the equation

$\partial_{t}F+\partial_{x}(A(t,F))=\mathbf{S}[F](t,x)$ (4.1)

where

$\mathbf{S}[F](t,x)\coloneqq F(t,x)(\phi\star
F)(t,x)-\int^{x}_{-\infty}F(t,z)(\partial_{z}\phi\star F)(t,z)dz,\
\phi\coloneqq\partial_{x}S.$

This equation has a slightly more general form than the Equation 1.8, since
here the flux is time dependent. As already explained, the strategy of proof
is a modification of the argument in [11], and can be divided into the
following steps

1\. Extraction of a converging subsequence from $F_{N}$ with a limit $F$.

2\. Showing that the limit $F$ obtained in 1. is an entropy solution.

3\. Provided steps 1+2 are successfully established, it remains to prove
stability estimates (from which the remaining parts- that is uniqueness and
mean field limit- would follow). This is one step where our argument differs
from the one in [11], since the source term in question carries a different
form. Some technical modifications appear in step 2 as well.

Before detailing the proof of the plan proposed above, we recall the following
chain rule for $BV$ functions, which will be used in the course of the proof

######  Lemma 4.1.

([5], Lemma A.21) Suppose $W\in BV_{\mathrm{loc}}(\mathbb{R})$ and $f$ is
Lipschitz. Then $f\circ W\in BV_{\mathrm{loc}}(\mathbb{R})$ and

$\left|\frac{d}{dx}f\circ
W\right|\leq\left|f\right|_{\mathrm{Lip}}\left|\frac{d}{dx}W\right|$

in the sense of measures.

In order to treat the source term we will be forced to verify the entropy
inequality only on dense subset of the real line, which is sufficient, as
observed in the following simple

######  Lemma 4.2.

The entropy inequality holds (3.1) iff it holds for some dense set
$D\subset\mathbb{R}$.

###### Proof.

Let $D\subset\mathbb{R}$ be dense and suppose (3.1) holds all $\beta\in D$,
and let $\alpha\in\mathbb{R}$. Then, taking a sequence
$\left(\beta_{k}\right)\subset D$ such that $\beta_{k}\rightarrow\alpha$, we
have that for all $k$

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}}\mathbf{1}_{F>\beta_{k}}\left((F-\beta_{k})\chi_{t}+\left(A(F)-A(\beta_{k})\right)\partial_{x}\chi+\chi\mathbf{S}(F)(t,x)\right)dxdt$
$\displaystyle\quad-\int_{0}^{T}\int_{\mathbb{R}}\mathbf{1}_{F<\beta_{k}}\left((F-\beta_{k})\chi_{t}+\left(A(F)-A(\beta_{k})\right)\partial_{x}\chi+\chi\mathbf{S}(F)(t,x)\right)dxdt.$

As $k\rightarrow\infty$ the first integral tends to

$\int_{0}^{T}\int_{\mathbb{R}}\mathbf{1}_{F\geq\alpha}\left((F-\alpha)\chi_{t}+\left(A(F)-A(\alpha)\right)\partial_{x}\chi+\chi\mathbf{S}(F)(t,x)\right)dxdt,$

whereas the second integral tends to

$-\int_{0}^{T}\int_{\mathbb{R}}\mathbf{1}_{F\leq\alpha}\left((F-\alpha)\chi_{t}+\left(A(F)-A(\alpha)\right)\partial_{x}\chi+\chi\mathbf{S}(F)(t,x)\right)dxdt,$

so that

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}}\left(\left|F-\alpha\right|\chi_{t}+\mathrm{sgn}(F-\alpha)\left(A(F)-A(\alpha)\right)\partial_{x}\chi+\mathrm{sgn}(F-\alpha)\chi\mathbf{S}(F)(t,x)\right)dxdt$
$\displaystyle\quad=\int_{0}^{T}\int_{\mathbb{R}}\mathbf{1}_{F\geq\alpha}\left((F-\alpha)\chi_{t}+\left(A(F)-A(\alpha)\right)\partial_{x}\chi+\chi\mathbf{S}(F)(t,x)\right)dxdt$
$\displaystyle\qquad-\int_{0}^{T}\int_{\mathbb{R}}\mathbf{1}_{F\leq\alpha}\left((F-\beta_{k})\chi_{t}+\left(A(F)-A(\beta_{k})\right)\partial_{x}\chi+\chi\mathbf{S}(F)(t,x)\right)dxdt\geq
0.$

∎

We are now in a good position to prove the main result, which we now state.
Remark that we insist on including time dependency in the flux in order to
enable applying the belowstated stability estimate for the fluxes $A_{N}(t,x)$
and $A(x)$ as defined in Section 3.

######  Theorem 4.1.

Let $S\in C_{0}^{\infty}(\mathbb{R})$ be such $\mathrm{supp}(S)\subset[-r,r]$
for some $r>0$. Let $A(t,x),\widetilde{A}(t,x)\in
C([0,T];\mathrm{Lip}([-\frac{1}{2},\frac{1}{2}]))$. Suppose
$F^{\mathrm{in}}\in BV(\mathbb{R})$ is non-decreasing and there is some $R>0$
such that

$\forall x\geq R:F^{\mathrm{in}}(\pm x)=\pm\frac{1}{2}.$ (4.2)

1\. (Well-posedness) There exist a unique entropy solution (in the sense of
Definition (3.1)) to the problem (4.1).

2\. (Stability) If $F,\widetilde{F}$ are two entropy solutions with initial
datas $F^{\mathrm{in}},\widetilde{F^{\mathrm{in}}}$ respectively satisfying
(4.2) then there is some
$C=C(r,R,\left\|\partial_{z}\phi\right\|_{\infty},\left\|\phi\right\|_{\infty})>0$
such that

$\left\|F(t,\cdot)-\widetilde{F}(t,\cdot)\right\|_{1}\leq
e^{Ct}\left(\left\|F^{\mathrm{in}}-\widetilde{F^{\mathrm{in}}}\right\|_{1}+t\underset{t\in[0,T]}{\sup}\left|A(t,\cdot)-\widetilde{A}(t,\cdot)\right|_{\mathrm{Lip}}\right).$

3\. (Mean field limit) It holds that
$F_{N}-F\underset{N\rightarrow\infty}{\rightarrow}0$ in
$C([0,T];L^{1}(\mathbb{R}))$ provided
$F_{N}^{\mathrm{in}}-F^{\mathrm{in}}\rightarrow 0$ in $L^{1}(\mathbb{R})$.

Proof. Step 1. Extracting a converging subsequence from $F_{N}$. Keep
$t\in[0,T]$ fixed . We know that $F_{N}(t,\cdot)$ is a sequence of non-
decreasing functions with uniformly bounded (with respect to $N$) total
variation on each compact subset. Therefore, by Helly’s selection theorem
there is a subsequence, still labled $F_{N}(t,\cdot)$, and some $F(t,\cdot)\in
L_{\mathrm{loc}}^{1}(\mathbb{R})$ such that
$F_{N}(t,\cdot)\underset{N\rightarrow\infty}{\rightarrow}F(t,\cdot)$ pointwise
a.e. By diagonalization we may find a subsequence, still labled $F_{N}(t)$,
such that $F_{N}(t,\cdot)$ converge to $F(t,\cdot)$ for all $t\in\mathbb{Q}$.
Recall that by Lemma 2.2, as long as $x>R$, $F_{N}(t,\pm x)=\pm\frac{1}{2}$
for all $t$. As a result, the same conclusion is true for the limit
$F(t,\cdot)$, so that we have
$F_{N}(t,\cdot)-F(t,\cdot)\underset{N\rightarrow\infty}{\rightarrow}0$ in
$L^{1}(\mathbb{R})$ for all $t\in\mathbb{Q}$. Next, we upgrade the convergence
to irrational times as well.

$\displaystyle\int_{\mathbb{R}}\left|F_{N}(t,x)-F_{N}(s,x)\right|dx$
$\displaystyle=\int_{\left|x\right|\leq
R}\left|\frac{1}{N}\stackrel{{\scriptstyle[}}{{i}}=1]{N}{\sum}m_{i}(t)\mathbf{1}_{\left\\{x\left|x\geq
x_{i}(t)\right.\right\\}}-\frac{1}{N}\stackrel{{\scriptstyle[}}{{i}}=1]{N}{\sum}m_{i}(s)\mathbf{1}_{\left\\{x\left|x\geq
x_{i}(s)\right.\right\\}}\right|dx$ $\displaystyle\leq\int_{\left|x\right|\leq
R}\left|\frac{1}{N}\stackrel{{\scriptstyle[}}{{i}}=1]{N}{\sum}m_{i}(t)\mathbf{1}_{\left\\{x\left|x_{i}(t)\leq
x\leq x_{i}(s)\right.\right\\}}\right|dx$
$\displaystyle\quad+\int_{\left|x\right|\leq
R}\left|\frac{1}{N}\stackrel{{\scriptstyle[}}{{i}}=1]{N}{\sum}(m_{i}(t)-m_{i}(s))\mathbf{1}_{\left\\{x\left|x\geq
x_{i}(s)\right.\right\\}}\right|dx.$

The first integral is

$\leq\overline{M}\underset{1\leq i\leq
N}{\max}\left|x_{i}(t)-x_{i}(s)\right|\leq\overline{M}^{2}\left|t-s\right|,$

whereas the second integral is

$\leq 2R\underset{1\leq i\leq N}{\max}\left|m_{i}(t)-m_{i}(s)\right|\leq
2RC(1+\overline{M})\left|t-s\right|,$

so that

$\left\|F_{N}(t,\cdot)-F_{N}(s,\cdot)\right\|_{1}\leq c\left|t-s\right|,$
(4.3)

for some suitable constant $c=c(C,\overline{M},R)$. The estimate (4.3) implies
that $F_{N}(t,\cdot)$ is a Cauchy sequence in $L^{1}((-R,R))$ for any
$t\in[0,T]$ . Indeed, keep $t$ fixed and let
$\left\\{t_{k}\right\\}_{k=1}^{\infty}\subset\mathbb{Q}$ such that
$t_{k}\underset{k\rightarrow\infty}{\rightarrow}t$ . Let $\varepsilon>0$. We
estimate

$\displaystyle\left\|F_{N}(t,\cdot)-F_{N+p}(t,\cdot)\right\|_{1}$
$\displaystyle\leq\left\|F_{N}(t,\cdot)-F_{N}(t_{k},\cdot)\right\|_{1}+\left\|F_{N}(t_{k},\cdot)-F(t_{k},\cdot)\right\|_{1}$
$\displaystyle\quad+\left\|F_{N+p}(t,\cdot)-F_{N+p}(t_{k},\cdot)\right\|_{1}+\left\|F_{N+p}(t_{k},\cdot)-F(t_{k},\cdot)\right\|_{1}$
$\displaystyle\leq
2c\left|t-t_{k}\right|+\left\|F_{N}(t_{k},\cdot)-F(t_{k},\cdot)\right\|_{1}+\left\|F_{N+p}(t_{k},\cdot)-F(t_{k},\cdot)\right\|_{1}.$

Pick $k$ large enough so that $\left|t-t_{k}\right|<\varepsilon$ and pick
$N_{0}$ large enough so that for any $N\geq N_{0}$ one has
$\left\|F_{N}(t_{k},\cdot)-F(t_{k},\cdot)\right\|_{1}<\varepsilon.$ Then for
these choices we get

$\left\|F_{N}(t,\cdot)-F_{N+p}(t,\cdot)\right\|\lesssim\varepsilon$

and therefore there is some $\widetilde{F}(t,\cdot)\in L^{1}((-R,R))$ such
that
$F_{N}(t,\cdot)\underset{N\rightarrow\infty}{\rightarrow}\widetilde{F}(t,\cdot)$
in $L^{1}((-R,R))$. In particular, for each $t$, the sequence $F_{N}(t,\cdot)$
is confined in a compact set of $L^{1}((-R,R))$. Therefore the estimate (4.3)
makes the theorem of Arzelá-Ascoli available (Theorem 1.1 in [17] for
example), thereby ensuring the existence of a subsequence, still labled
$F_{N}$, and an element $\widetilde{\widetilde{F}}\in
C([0,T];L_{\mathrm{loc}}^{1}(\mathbb{R}))$ such that
$F_{N}-\widetilde{\widetilde{F}}\underset{N\rightarrow\infty}{\rightarrow}0$
in $C([0,T];L^{1}(\mathbb{R}))$. It is clear that
$\widetilde{\widetilde{F}}\in BV([0,T]\times\mathbb{R})$ and $\forall x\geq
R:\widetilde{\widetilde{F}}(t,\pm x)=\pm\frac{1}{2}$. To minimize cumbersome
notation we shall hereafter denote by $F$ the limit function
$\widetilde{\widetilde{F}}$ that we constructed.

Step 2. The Limit function $F$ is an entropy solution. We wish to show that
the limit function $F$ obtained in step 1. is an entropy solution to the
original equation, which will establish the existence claim. By Lemma 3.3 we
know that

$\int_{0}^{T}\int_{\mathbb{R}}\left|F_{N}-\alpha\right|\chi_{t}+\mathrm{sgn}(F_{N}-\alpha)\left(A_{N}(t,F_{N})-A_{N}(t,\alpha)\right)\partial_{x}\chi+\mathrm{sgn}(F_{N}-\alpha)\chi\mathbf{S}[F_{N}](t,x)dxdt\geq
0,$

and we wish now to pass to the limit as $N\rightarrow\infty$. That

$\int_{0}^{T}\int_{\mathbb{R}}\left|F_{N}-\alpha\right|\chi_{t}\rightarrow\int_{0}^{T}\int_{\mathbb{R}}\left|F-\alpha\right|\chi_{t}$

is due to dominated convergence theorem. In addition, Lemma 3.1 and the
identity $\mathrm{sgn}(a-b)(a^{2}-b^{2})=\left|a-b\right|(a+b)$ entails

$\displaystyle\left|\int_{0}^{T}\int_{\mathbb{R}}\mathrm{sgn}(F_{N}-\alpha)\left(A_{N}(t,F_{N})-A_{N}(t,\alpha)\right)\partial_{x}\chi-\int_{0}^{T}\int_{\mathbb{R}}\mathrm{sgn}(F-\alpha)\left(A(F)-A(\alpha)\right)\partial_{x}\chi\right|$
$\displaystyle\quad\leq\left|\int_{0}^{T}\int_{\mathbb{R}}\mathrm{sgn}(F_{N}-\alpha)\left(A_{N}(t,F_{N})-A_{N}(t,\alpha)\right)\partial_{x}\chi-\int_{0}^{T}\int_{\mathbb{R}}\mathrm{sgn}(F_{N}-\alpha)\left(A(F_{N})-A(\alpha)\right)\partial_{x}\chi\right|$
$\displaystyle\qquad+\left|\int_{0}^{T}\int_{\mathbb{R}}\mathrm{sgn}(F_{N}-\alpha)\left(A(F_{N})-A(\alpha)\right)\partial_{x}\chi-\int_{0}^{T}\int_{\mathbb{R}}\mathrm{sgn}(F-\alpha)\left(A(F)-A(\alpha)\right)\partial_{x}\chi\right|$
$\displaystyle\quad\leq\frac{4\overline{M}\mathrm{Lip}(\left.A\right|_{[-\frac{1}{2},\frac{1}{2}]})}{N}\int_{0}^{T}\int_{\mathbb{R}}\left|\partial_{x}\chi\right|dxdt$
$\displaystyle\qquad+\left|\int_{0}^{T}\int_{\mathbb{R}}\left|F_{N}-\alpha\right|(F_{N}+\alpha)\partial_{x}\chi-\int_{0}^{T}\int_{\mathbb{R}}\left|F-\alpha\right|(F+\alpha)\partial_{x}\chi\right|\underset{N\rightarrow\infty}{\rightarrow}0.$

The last term is slightly more subtle. Observe that

$\int_{-2R}^{2R}\phi(z-\zeta)F_{N}(t,\zeta)d\zeta\underset{N\rightarrow\infty}{\rightarrow}\int_{-2R}^{2R}\phi(z-\zeta)F(t,\zeta)d\zeta,$

so that

$F_{N}(t,x)\phi\star
F_{N}(t,x)\underset{N\rightarrow\infty}{\rightarrow}F(t,x)\phi\star F(t,x)$

pointwise a.e. Recall that $\phi^{\prime}\star F_{N}$ is compactly supported
and therefore dominated convergence is applicable for the second integral as
well which implies

$\int^{x}_{-\infty}F_{N}(t,z)\phi^{\prime}\star
F_{N}(t,z)dz\underset{N\rightarrow\infty}{\rightarrow}\int^{x}_{-\infty}F(t,z)\phi^{\prime}\star
F(t,z)dz.$

By Lemma (4.2) it suffices to verify the entropy condition on dense set. Note
that since $F$ is locally summable it holds that
$\lambda(\left\\{(t,x)\left|F(t,x)=\alpha\right.\right\\})=0$ for a.e.
$\alpha$. In particular there is dense set $D\subset\mathbb{R}$ such that
$\lambda(\left\\{F(t,x)=\alpha\right\\})=0$ for all $\alpha\in D$. Therefore,
since $F_{N}\rightarrow F$ and
$\mathbf{S}[F_{N}](t,x)\rightarrow\mathbf{S}[F](t,x)$ pointwise a.e., we
conclude that for all $\alpha\in D$

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}}\mathrm{sgn}(F_{N}-\alpha)\chi\mathbf{S}[F_{N}](t,x)dxdt$
$\displaystyle=\int_{0}^{T}\int_{\mathbb{R}}\mathbf{1}_{\left\\{F(t,x)\neq\alpha\right\\}}\mathrm{sgn}(F_{N}-\alpha)\chi\mathbf{S}[F_{N}](t,x)dxdt$
$\displaystyle\underset{N\rightarrow\infty}{\rightarrow}\int_{0}^{T}\int_{\mathbb{R}}\mathrm{sgn}(F-\alpha)\chi\mathbf{S}[F](t,x)dxdt$

as desired.

Step 3. Uniqueness and Stability. Let $F,\widetilde{F}$ be two entropy
solutions. Keep $(s,y)\in[0,T]\times\mathbb{R}$ fixed. Then by definition we
have

$\displaystyle 0$
$\displaystyle\leq\iint\left|F(t,x)-\widetilde{F}(s,y)\right|\partial_{t}\chi(t,x,s,y)dxdt$
$\displaystyle\quad+\iint\mathrm{sgn}(F(t,x)-\widetilde{F}(s,y))\left(A(t,F(t,x))-A(t,\widetilde{F}(s,y))\right)\partial_{x}\chi(t,x,s,y)dxdt$
$\displaystyle\quad+\iint\mathrm{sgn}(F(t,x)-\widetilde{F}(s,y))\chi(t,x,s,y)\mathbf{S}[F](t,x)dxdt.$

Exchanging the roles of $F$ and $\widetilde{F}$ we also have the inequality

$\displaystyle 0$
$\displaystyle\leq\iint\left|\widetilde{F}(s,y)-F(t,x)\right|\partial_{s}\chi(t,x,s,y)dyds$
$\displaystyle\quad+\iint\mathrm{sgn}(\widetilde{F}(s,y)-F(t,x))\left(\widetilde{A}(s,\widetilde{F}(s,y))-\widetilde{A}(s,F(t,x))\right)\partial_{y}\chi(t,x,s,y)dyds$
$\displaystyle\quad+\iint\mathrm{sgn}(\widetilde{F}(s,y)-F(t,x))\chi(t,x,s,y)\mathbf{S}[\widetilde{F}](s,y)dyds.$

Integrating both of the above inequalities over the free variables and summing
up gives

$\displaystyle 0$
$\displaystyle\leq\iiiint\left|F(t,x)-\widetilde{F}(s,y)\right|\left(\partial_{t}\chi+\partial_{s}\chi\right)$
$\displaystyle\quad+\iiiint\mathrm{sgn}(F(t,x)-\widetilde{F}(s,y))\left(A(t,F(t,x))-A(t,\widetilde{F}(s,y))\right)\partial_{x}\chi$
$\displaystyle\quad+\iiiint\mathrm{sgn}(\widetilde{F}(s,y)-F(t,x))\left(\widetilde{A}(s,\widetilde{F}(s,y))-\widetilde{A}(s,F(t,x))\right)\partial_{y}\chi$
$\displaystyle\quad+\iiiint\left(\chi\mathrm{sgn}(F(t,x)-\widetilde{F}(s,y))\mathbf{S}[F](t,x)+\chi\mathrm{sgn}(\widetilde{F}(s,y)-F(t,x))\mathbf{S}[\widetilde{F}](s,y)\right)dxdtdyds.$
(4.4)

Now consider the variable change

$\overline{y}=\frac{x-y}{2},\quad\overline{s}=\frac{t-s}{2},\quad\overline{x}=\frac{x+y}{2},\quad\overline{t}=\frac{t+s}{2},$

and take $\chi$ to be of the form

$\displaystyle\chi(t,x,s,y)$
$\displaystyle=b_{\varepsilon}\left(\frac{x-y}{2}\right)b_{\varepsilon}\left(\frac{t-s}{2}\right)g\left(\frac{x+y}{2}\right)h_{\delta}\left(\frac{t+s}{2}\right)$
$\displaystyle=b_{\varepsilon}\left(\overline{y}\right)b_{\varepsilon}\left(\overline{s}\right)g\left(\overline{x}\right)h_{\delta}\left(\overline{t}\right).$

where $b_{\varepsilon},g,h_{\delta}$ are chosen as follows. Keep
$\sigma<\tau\in(0,T)$ fixed. For each $\varepsilon>0,\delta>0$ such that
$0<\varepsilon+\delta<\min(\sigma,T-\tau)$ we define the functions as follows.

1\. $b_{\varepsilon}$ is an approximation of the identity, i.e.
$b_{\varepsilon}=\frac{1}{\varepsilon}\zeta(\frac{x}{\varepsilon})$ for some
radial $0\leq\zeta\in C_{0}^{\infty}(\mathbb{R})$ with
$\mathrm{supp}(\zeta)\Subset B_{1}(0)$.

2\. $g(x)=1$ for all $x\in[-R,R]$ and $g\in C_{0}^{\infty}(\mathbb{R})$.

3\. Let

$h_{\delta}(t)\coloneqq\left\\{\begin{array}[]{c}1,\ \ t\in[\sigma,\tau]\\\
-\frac{1}{\delta}t+\frac{\tau+\delta}{\delta},\ \ t\in[\tau,\tau+\delta]\\\
\frac{1}{\delta}t-\frac{\sigma-\delta}{\delta},\ \
t\in[\sigma-\delta,\sigma]\\\ 0,\ \
t\notin[\sigma-\delta,\tau+\delta]\end{array}\right..$

With this choice we easily observe that $\chi(t,x,s,y)$ is compactly supported
in the variables $(t,x)$ for any fixed $(s,y)$, and viceversa.

######  Claim 4.1.

For each fixed $(s,y)\in(0,T)\times\mathbb{R}$ the function
$(t,x)\mapsto\chi(t,x,s,y)$ is compactly supported in $(0,T)\times\mathbb{R}$.
Vice versa, for each fixed $(t,x)\in(0,T)\times\mathbb{R}$ the function
$(s,y)\mapsto\chi(t,x,s,y)$ is compactly supported in $(0,T)\times\mathbb{R}$.

###### Proof.

That $x\mapsto\chi(t,x,s,y)$ vanishes outside some finite interval is clear.
In addition if $s>2\varepsilon$ then $b_{\varepsilon}(\frac{t-s}{2})=0$ for
all $t\in(0,s-2\varepsilon)$, while if $s\leq 2\varepsilon$ then
$h_{\delta}(\frac{t+s}{2})=0$ then since $\sigma>\varepsilon+\delta$ it
follows that $2\sigma-2\delta-s>0$, so that $h_{\delta}(\frac{t+s}{2})=0$ for
all $t\in(0,2\sigma-2\delta-s)$. Hence $\chi(t,x,s,y)=0$ for all
$t\in(0,\mathbf{1}_{s<2\varepsilon}(s-2\varepsilon)+\mathbf{1}_{s\geq
2\varepsilon}2\sigma-2\delta-s)$ (note that the right end of the interval is
strictly positive). If $s<T-2\varepsilon$ then
$b_{\varepsilon}(\frac{t-s}{2})=0$ for all $t\in(s+2\varepsilon,T)$ while if
$s\geq T-2\varepsilon$ then $2\tau+2\delta-s\leq
2\tau+2\delta+2\varepsilon-T<2T-T=T$ because $\tau<T-(\delta+\varepsilon)$.
Therefore $\chi(t,x,s,y)=0$ for all
$t\in(\mathbf{1}_{s<T-2\varepsilon}(s+2\varepsilon)+\mathbf{1}_{s\geq
T-2\varepsilon}(2\tau+2\delta-s),T)$. To conclude $t\mapsto\chi(t,x,s,y)$
vanishes outside some interval compactly supported in $(0,T)$. Since
$b_{\varepsilon}$ is radial, by symmetry the same argument shows that for any
fixed $(t,x)$ the function $(s,y)\mapsto\chi(t,x,s,y)$ is compactly supported.
∎

In addition, it is readily checked that

$\partial_{t}+\partial_{s}=\partial_{\overline{t}},\quad\partial_{x}+\partial_{y}=\partial_{\overline{x}},\quad\partial_{x}-\partial_{y}=\partial_{\overline{y}}.$

Under this variable change, the second and third term in 4.4 can be grouped
together as follows (to make the equations lighter the arguments are of $F$
and $\widetilde{F}$ are always
$(\overline{t}+\overline{s},\overline{x}+\overline{y})$ and
$(\overline{t}-\overline{s},\overline{x}-\overline{y})$ respectively, and are
implicit )

$\displaystyle\iiiint\mathrm{sgn}(F(t,x)-\widetilde{F}(s,y))\left(A(t,F(t,x))-A(t,\widetilde{F}(s,y))\right)\partial_{x}\chi$
$\displaystyle\qquad+\iiiint\mathrm{sgn}(\widetilde{F}(s,y)-F(t,x))\left(\widetilde{A}(s,\widetilde{F}(s,y))-\widetilde{A}(s,F(t,x))\right)\partial_{y}\chi$
$\displaystyle\quad=\frac{1}{2}\iiiint\mathrm{sgn}(F-\widetilde{F})\left(A(\overline{t}+\overline{s},F)-A(\overline{t}+\overline{s},\widetilde{F})\right)(\partial_{\overline{x}}+\partial_{\overline{y}})\chi$
$\displaystyle\qquad+\frac{1}{2}\iiiint\mathrm{sgn}(\widetilde{F}-F)\left(\widetilde{A}(\overline{t}-\overline{s},\widetilde{F})-\widetilde{A}(\overline{t}-\overline{s},F)\right)(\partial_{\overline{x}}-\partial_{\overline{y}})\chi$
$\displaystyle\quad=\frac{1}{2}\iiiint\mathrm{sgn}(F-\widetilde{F})\left(A(\overline{t}+\overline{s},F)+\widetilde{A}(\overline{t}-\overline{s},F)-A(\overline{t}+\overline{s},\widetilde{F})-\widetilde{A}(\overline{t}-\overline{s},\widetilde{F})\right)\partial_{\overline{x}}\chi$
$\displaystyle\qquad+\frac{1}{2}\iiiint\mathrm{sgn}(F-\widetilde{F})\left(A(\overline{t}+\overline{s},F)-\widetilde{A}(\overline{t}-\overline{s},F)-A(\overline{t}+\overline{s},\widetilde{F}(s,y))+\widetilde{A}(\overline{t}-\overline{s},\widetilde{F}(s,y))\right)\partial_{\overline{y}}\chi$
$\displaystyle\quad=\iiiint\mathrm{sgn}(F-\widetilde{F})\left(A_{+}(\overline{t}+\overline{s},\overline{t}-\overline{s},F)-A_{+}(\overline{t}+\overline{s},\overline{t}-\overline{s},\widetilde{F})\right)\partial_{\overline{x}}\chi$
$\displaystyle\qquad+\iiiint\mathrm{sgn}(F-\widetilde{F})\left(A_{-}(\overline{t}+\overline{s},\overline{t}-\overline{s},F)-A_{-}(\overline{t}+\overline{s},\overline{t}-\overline{s},\widetilde{F})\right)\partial_{\overline{y}}\chi,$

where we have set

$A_{\pm}(t,s,x)\coloneqq\frac{A(t,x)\pm\widetilde{A}(s,x)}{2}.$

The Inequality 4.4 is then transformed to

$\displaystyle 0$ $\displaystyle\leq
I_{\varepsilon,\delta}+II_{\varepsilon,\delta}+III_{\varepsilon,\delta}+IV_{\varepsilon,\delta}$
(4.5)
$\displaystyle\coloneqq\iiiint\left|F(\overline{t}+\overline{s},\overline{x}+\overline{y})-\widetilde{F}(\overline{t}-\overline{s},\overline{x}-\overline{y})\right|\partial_{\overline{t}}\chi(\overline{t},\overline{x},\overline{s},\overline{y})d\overline{t}d\overline{x}d\overline{s}d\overline{y}$
$\displaystyle\quad+\iiiint\mathrm{sgn}(F(\overline{t}+\overline{s},\overline{x}+\overline{y})-\widetilde{F}(\overline{t}-\overline{s},\overline{x}-\overline{y}))\left(A_{+}(\overline{t}+\overline{s},\overline{t}-\overline{s},F)-A_{+}(\overline{t}+\overline{s},\overline{t}-\overline{s},\widetilde{F})\right)\partial_{\overline{x}}\chi(\overline{t},\overline{x},\overline{s},\overline{y})d\overline{t}d\overline{x}d\overline{s}d\overline{y}$
$\displaystyle\quad+\iiiint\mathrm{sgn}(F(\overline{t}+\overline{s},\overline{x}+\overline{y})-\widetilde{F}(\overline{t}-\overline{s},\overline{x}-\overline{y}))\left(A_{-}(\overline{t}+\overline{s},\overline{t}-\overline{s},F)-A_{-}(\overline{t}+\overline{s},\overline{t}-\overline{s},\widetilde{F})\right)\partial_{\overline{y}}\chi(\overline{t},\overline{x},\overline{s},\overline{y})d\overline{t}d\overline{x}d\overline{s}d\overline{y}$
$\displaystyle\quad+\iiiint\chi\mathrm{sgn}(F(\overline{t}+\overline{s},\overline{x}+\overline{y})-\widetilde{F}(\overline{t}-\overline{s},\overline{x}-\overline{y}))\left(\mathbf{S}[F](\overline{t}+\overline{s},\overline{x}+\overline{y})-\mathbf{S}[\widetilde{F}](\overline{t}-\overline{s},\overline{x}-\overline{y})\right)d\overline{t}d\overline{x}d\overline{s}d\overline{y}.$

We handle separately each one of the above integrals.

The integral $I_{\varepsilon,\delta}$. Letting $\varepsilon\rightarrow 0$ we
get the 2D integral

$I_{\delta}\coloneqq\underset{\varepsilon\rightarrow
0}{\lim}I_{\varepsilon,\delta}=\int\int\left|F(\overline{t},\overline{x})-\widetilde{F}(\overline{t},\overline{x})\right|g\left(\overline{x}\right)h_{\delta}^{\prime}\left(\overline{t}\right)d\overline{x}d\overline{t}.$

We can pass to the limit as $\delta\rightarrow 0$ to find that

$\displaystyle I_{\delta}$
$\displaystyle=\int\int\left|F(\overline{t},\overline{x})-\widetilde{F}(\overline{t},\overline{x})\right|g\left(\overline{x}\right)h_{\delta}^{\prime}\left(\overline{t}\right)$
$\displaystyle=\frac{1}{\delta}\int_{\sigma-\delta}^{\sigma}\int\left|F(\overline{t},\overline{x})-\widetilde{F}(\overline{t},\overline{x})\right|g\left(\overline{x}\right)d\overline{x}d\overline{t}-\frac{1}{\delta}\int_{\tau}^{\tau+\delta}\int\left|F(\overline{t},\overline{x})-\widetilde{F}(\overline{t},\overline{x})\right|g\left(\overline{x}\right)d\overline{x}d\overline{t}$
$\displaystyle\underset{\delta\rightarrow
0}{\rightarrow}\int\left|F(\sigma,\overline{x})-\widetilde{F}(\sigma,\overline{x})\right|g\left(\overline{x}\right)d\overline{x}-\int\left|F(\tau,\overline{x})-\widetilde{F}(\tau,\overline{x})\right|g\left(\overline{x}\right)d\overline{x}$
$\displaystyle=\int\left|F(\sigma,\overline{x})-\widetilde{F}(\sigma,\overline{x})\right|d\overline{x}-\int\left|F(\tau,\overline{x})-\widetilde{F}(\tau,\overline{x})\right|d\overline{x},$
(4.6)

where in the last equation we relied on the choice of $g$ as well as fact that
$F-\widetilde{F}$ is $0$ outside $[-R,R]$.

The integral $II_{\varepsilon,\delta}$. Letting $\varepsilon\rightarrow 0$ the
integral becomes

$II_{\delta}=\underset{\varepsilon\rightarrow
0}{\lim}II_{\varepsilon,\delta}=\iint\mathrm{sgn}(F(\overline{t},\overline{x})-\widetilde{F}(\overline{t},\overline{x}))\left(A_{+}(\overline{t},\overline{t},F(\overline{t},\overline{x}))-A_{+}(\overline{t},\overline{t},\widetilde{F}(\overline{t},\overline{x}))\right)g^{\prime}\left(\overline{x}\right)h_{\delta}\left(\overline{t}\right)d\overline{t}d\overline{x}.$

Observe that $g^{\prime}\left(\overline{x}\right)=0$ for all
$\overline{x}\in\mathrm{supp}(A_{+}(\overline{t},\overline{t},F(\overline{t},\cdot))-A_{+}(\overline{t},\overline{t},\widetilde{F}(\overline{t},\cdot)))\subset[-R,R]$,
where the latter inclusion is because of the uniform in time Lipschitz
continuity assumed for $A,\widetilde{A}$. We therefore infer that the above
integral vanishes identically, i.e.

$II_{\delta}=0.$ (4.7)

The integral $III_{\varepsilon,\delta}$. The following claim is a
straightforward adaptation of Lemma 5.4 in [11]

######  Claim 4.2.

The function
$\gamma(\overline{t},F,\widetilde{F})\coloneqq\mathrm{sgn}(F-\widetilde{F})(A_{-}(\overline{t},\overline{t},F)-A_{-}(\overline{t},\overline{t},\widetilde{F}))$
is uniformly in time Lipschitz with respect to the first and second variable
with the bounds

$\left|\gamma(\overline{t},\cdot,\widetilde{F})\right|_{\mathrm{Lip}}\leq\left|A_{-}(\overline{t},\overline{t},\cdot)\right|_{\mathrm{Lip}},\
\left|\gamma(\overline{t},F,\cdot)\right|_{\mathrm{Lip}}\leq\left|A_{-}(\overline{t},\overline{t},\cdot)\right|_{\mathrm{Lip}}.$

Proof. Keep $\overline{t}\in[0,T],\widetilde{F}\in[-\frac{1}{2},\frac{1}{2}]$
fixed. Let $F_{1},F_{2}\in[-\frac{1}{2},\frac{1}{2}]$. If
$\mathrm{sgn}(F_{1}-\widetilde{F})=\mathrm{sgn}(F_{2}-\widetilde{F})$ then we
have

$\left|\gamma(\overline{t},F_{1},\widetilde{F})-\gamma(\overline{t},F_{2},\widetilde{F})\right|=\left|A_{-}(\overline{t},\overline{t},F_{1})-A_{-}(\overline{t},\overline{t},F_{2})\right|\leq\left|A_{-}(\overline{t},\overline{t},\cdot)\right|_{\mathrm{Lip}}\left|F_{1}-F_{2}\right|.$

If $\mathrm{sgn}(F_{1}-\widetilde{F})\neq\mathrm{sgn}(F_{2}-\widetilde{F})$
then

$\displaystyle\left|\gamma(\overline{t},F_{1},\widetilde{F})-\gamma(\overline{t},F_{2},\widetilde{F})\right|$
$\displaystyle=\left|A_{-}(\overline{t},\overline{t},F_{1})-A_{-}(\overline{t},\overline{t},\widetilde{F})+A_{-}(\overline{t},\overline{t},F_{2})-A_{-}(\overline{t},\overline{t},\widetilde{F})\right|$
$\displaystyle\leq\left|A_{-}(\overline{t},\overline{t},\cdot)\right|_{\mathrm{Lip}}(\left|F_{1}-\widetilde{F}\right|+\left|F_{2}-\widetilde{F}\right|)$
$\displaystyle=\left|A_{-}(\overline{t},\overline{t},\cdot)\right|_{\mathrm{Lip}}\left|F_{1}-\widetilde{F}+\widetilde{F}-F_{2}\right|=\left|A_{-}(\overline{t},\overline{t},\cdot)\right|_{\mathrm{Lip}}\left|F_{1}-F_{2}\right|.$

$\square$

We can apply this observation together with Lemma 4.1 in order to integrate by
parts and obtain

$III_{\varepsilon,\delta}=-\iiiint\partial_{\overline{y}}\gamma(t,F,\widetilde{F})\chi.$

Moreover we get the bound

$\displaystyle\left|\iiiint\partial_{\overline{y}}\gamma(t,F,\widetilde{F})\chi\right|$
$\displaystyle\quad\leq\underset{\overline{t}\in[0,T]}{\sup}\left|A_{-}(\overline{t},\overline{t},\cdot)\right|_{\mathrm{Lip}}\iiiint\left(\left|\partial_{1}F(\overline{t}+\overline{s},\overline{x}+\overline{y})\right|+\left|\partial_{1}\widetilde{F}(\overline{t}-\overline{s},\overline{x}-\overline{y})\right|\right)b_{\varepsilon}\left(\overline{y}\right)b_{\varepsilon}\left(\overline{s}\right)g\left(\overline{x}\right)h_{\delta}\left(\overline{t}\right).$

Letting $\varepsilon\rightarrow 0$ and setting
$III_{\delta}\coloneqq\underset{\varepsilon\rightarrow
0}{\lim}III_{\varepsilon,\delta}$ we arrive at

$\left|III_{\delta}\right|\leq\underset{\overline{t}\in[0,T]}{\sup}\left|A_{-}(\overline{t},\overline{t},\cdot)\right|_{\mathrm{Lip}}\iint\left(\left|\partial_{1}F(\overline{t},\overline{x})\right|+\left|\partial_{1}\widetilde{F}(\overline{t},\overline{x})\right|\right)g\left(\overline{x}\right)h_{\delta}\left(\overline{t}\right)d\overline{x}d\overline{t}.$

Letting $\delta\rightarrow 0$ this becomes

$\underset{\delta\rightarrow
0}{\lim}\left|III_{\delta}\right|\leq\underset{\overline{t}\in[0,T]}{\sup}\left|A_{-}(\overline{t},\overline{t},\cdot)\right|_{\mathrm{Lip}}\int_{\sigma}^{\tau}\int\left(\left|\partial_{1}F(\overline{t},\overline{x})\right|+\left|\partial_{1}\widetilde{F}(\overline{t},\overline{x})\right|\right)d\overline{x}d\overline{t}.$

Since for any fixed $\overline{t}$

$\int\left(\left|\partial_{1}F(\overline{t},\overline{x})\right|+\left|\partial_{1}\widetilde{F}(\overline{t},\overline{x})\right|\right)=2,$

we cocnlude

$\underset{\delta\rightarrow 0}{\lim}\left|III_{\delta}\right|\leq
2\underset{\overline{t}\in[0,T]}{\sup}\left|A_{-}(\overline{t},\overline{t},\cdot)\right|_{\mathrm{Lip}}(\tau-\sigma)=\underset{\overline{t}\in[0,T]}{\sup}\left|A(\overline{t},\cdot)-\widetilde{A}(\overline{t},\cdot)\right|_{\mathrm{Lip}}(\tau-\sigma).$
(4.8)

The integral $IV_{\varepsilon,\delta}$. The treatment of this integral
reflects the novelty for what concerns the stability estimate, since the non-
local source term in question stands in variance to the one in [11]. Letting
$\varepsilon\rightarrow 0$ we get the 2D integral

$IV_{\delta}\coloneqq\iint\mathrm{sgn}(F(\overline{t},\overline{x})-\widetilde{F}(\overline{t},\overline{x}))\left(\mathbf{S}[F](\overline{t},\overline{x})-\mathbf{S}[\widetilde{F}](\overline{t},\overline{x})\right)g\left(\overline{x}\right)h_{\delta}\left(\overline{t}\right)d\overline{t}d\overline{x}.$

The integral $IV_{\delta}$ is now to be bounded by 4 terms, each of which is
mastered separately.

$\displaystyle\left|IV_{\delta}\right|$
$\displaystyle\leq\iint\left|\mathbf{S}[F](t,x)-\mathbf{S}[\widetilde{F}](t,x)\right|g\left(x\right)h_{\delta}\left(t\right)dxdt$
$\displaystyle\leq\iint\left|F(t,x)\phi\star
F(t,x)-\widetilde{F}(t,x)\phi\star\widetilde{F}(t,x)\right|g\left(x\right)h_{\delta}\left(t\right)dxdt$
$\displaystyle\quad+\iint\left|\int_{-\infty}^{x}\widetilde{F}(t,z)\partial_{z}\phi\star\widetilde{F}(t,z)-F(t,z)\partial_{z}\phi\star
F(t,z)dz\right|g\left(x\right)h_{\delta}\left(t\right)dxdt$
$\displaystyle\leq\iint\left|\left(F(t,x)-\widetilde{F}(t,x)\right)\phi\star
F(t,x)\right|g\left(x\right)h_{\delta}\left(t\right)dxdt$
$\displaystyle\quad+\iint\left|\widetilde{F}(t,x)\phi\star(F-\widetilde{F})(t,x)\right|g\left(x\right)h_{\delta}\left(t\right)dxdt$
$\displaystyle\quad+\iint\left|\int_{-\infty}^{x}\widetilde{F}(t,z)(\partial_{z}\phi\star(\widetilde{F}-F)(t,z))dz\right|g\left(x\right)h_{\delta}\left(t\right)dxdt$
$\displaystyle\quad+\iint\left|\int_{-\infty}^{x}\partial_{z}\phi\star
F(t,z)(\widetilde{F}-F)(t,z))dz\right|g\left(x\right)h_{\delta}\left(t\right)dxdt\coloneqq\stackrel{{\scriptstyle[}}{{k}}=1]{4}{\sum}J_{k}.$

Estimate on $J_{1}$.

$\left|\left(F(t,x)-\widetilde{F}(t,x)\right)\phi\star F(t,x)\right|\leq
2R\left\|\phi\right\|_{\infty}\left|F(t,x)-\widetilde{F}(t,x)\right|,$

so that

$\displaystyle\left|J_{1}\right|$ $\displaystyle\leq
2R\left\|\phi\right\|_{\infty}\iint\left|F(t,x)-\widetilde{F}(t,x)\right|g\left(x\right)h_{\delta}\left(t\right)dxdt$
$\displaystyle\leq
2R\left\|\phi\right\|_{\infty}\int\left\|F(t,\cdot)-\widetilde{F}(t,\cdot)\right\|_{1}h_{\delta}\left(t\right)dt\underset{\delta\rightarrow
0}{\rightarrow}2R\left\|\phi\right\|_{\infty}\int_{\sigma}^{\tau}\left\|F(t,\cdot)-\widetilde{F}(t,\cdot)\right\|_{1}dt.$

Estimate on $J_{2}$.

$\left|\phi\star(F-\widetilde{F})(t,x)\right|\leq\left\|\phi\right\|_{\infty}\left\|(F-\widetilde{F})(t,\cdot)\right\|_{1},$

so that

$\displaystyle\left|J_{2}\right|$
$\displaystyle\leq\frac{1}{2}\left\|\phi\right\|_{\infty}\iint\left\|(F-\widetilde{F})(t,\cdot)\right\|_{1}g\left(x\right)h_{\delta}\left(t\right)dxdt$
$\displaystyle\leq
2R\left\|\phi\right\|_{\infty}\int\left\|(F-\widetilde{F})(t,\cdot)\right\|_{1}h_{\delta}\left(t\right)dt\underset{\delta\rightarrow
0}{\rightarrow}2R\left\|\phi\right\|_{\infty}\int_{\sigma}^{\tau}\left\|F(t,\cdot)-\widetilde{F}(t,\cdot)\right\|_{1}dt.$

Estimate on $J_{3}$. Since
$\mathrm{supp}(\partial_{z}\phi\star(\widetilde{F}-F))\subset\mathrm{supp}(\phi)+\mathrm{supp}(\widetilde{F}-F)\subset[-(r+R),r+R]$
it follows that

$\displaystyle\left|\int_{-\infty}^{x}\widetilde{F}(t,z)(\partial_{z}\phi\star(\widetilde{F}-F)(t,z))dz\right|$
$\displaystyle\leq\left|\int_{-(r+R)}^{r+R}\widetilde{F}(t,z)(\partial_{z}\phi\star(\widetilde{F}-F)(t,z))dz\right|$
$\displaystyle\leq(r+R)\left\|\partial_{z}\phi\right\|_{\infty}\left\|\widetilde{F}(t,\cdot)-F(t,\cdot)\right\|_{1}.$

Consequently

$\displaystyle\left|J_{3}\right|$
$\displaystyle\leq(r+R)\left\|\partial_{z}\phi\right\|_{\infty}\iint\left\|\widetilde{F}(t,\cdot)-F(t,\cdot)\right\|_{1}g\left(x\right)h_{\delta}\left(t\right)dxdt$
$\displaystyle\leq(r+R)\left\|\partial_{z}\phi\right\|_{\infty}\iint\left\|\widetilde{F}(t,\cdot)-F(t,\cdot)\right\|_{1}g\left(x\right)h_{\delta}\left(t\right)dxdt$
$\displaystyle\leq
4R(r+R)\left\|\partial_{z}\phi\right\|_{\infty}\int\left\|\widetilde{F}(t,\cdot)-F(t,\cdot)\right\|_{1}h_{\delta}\left(t\right)dt$
$\displaystyle\underset{\delta\rightarrow
0}{\rightarrow}4R(r+R)\left\|\partial_{z}\phi\right\|_{\infty}\int_{\sigma}^{\tau}\left\|\widetilde{F}(t,\cdot)-F(t,\cdot)\right\|_{1}dt.$

Estimate on $J_{4}$.

$\left|\int_{-\infty}^{x}\partial_{z}\phi\star
F(t,z)(\widetilde{F}-F)(t,z))dz\right|\leq
2R\left\|\partial_{z}\phi\right\|_{\infty}\left\|\widetilde{F}(t,\cdot)-F(t,\cdot)\right\|_{1}.$

As a result

$\displaystyle\left|J_{4}\right|$ $\displaystyle\leq
2R\left\|\partial_{z}\phi\right\|_{\infty}\iint\left\|\widetilde{F}(t,\cdot)-F(t,\cdot)\right\|_{1}g\left(x\right)h_{\delta}\left(t\right)dxdt$
$\displaystyle\leq
8R^{2}\left\|\partial_{z}\phi\right\|_{\infty}\int\left\|\widetilde{F}(t,\cdot)-F(t,\cdot)\right\|_{1}h_{\delta}\left(t\right)dt$
$\displaystyle\underset{\delta\rightarrow
0}{\rightarrow}8R^{2}\left\|\partial_{z}\phi\right\|_{\infty}\int_{\sigma}^{\tau}\left\|\widetilde{F}(t,\cdot)-F(t,\cdot)\right\|_{1}dt.$

Summarizing

$\left|\underset{\delta\rightarrow 0}{\lim}\underset{\varepsilon\rightarrow
0}{\lim}IV_{\varepsilon,\delta}\right|\leq
C(r,R,\left\|\phi\right\|_{\infty},\left\|\partial_{z}\phi\right\|_{\infty})\int_{\sigma}^{\tau}\left\|F(t,\cdot)-\widetilde{F}(t,\cdot)\right\|_{1}dt.$
(4.9)

Step 4. Conclusion. The combination of (4.6),(4.7),(4.8), (4.9) and (4.5)
yields the inequality

$\displaystyle\underset{t\in[0,T]}{\sup}\left|A(t,\cdot)-\widetilde{A}(t,\cdot)\right|_{\mathrm{Lip}}(\tau-\sigma)$
$\displaystyle+C\int_{\sigma}^{\tau}\left\|F(t,\cdot)-\widetilde{F}(t,\cdot)\right\|_{1}dt$
$\displaystyle+\int\left|F(\sigma,x)-\widetilde{F}(\sigma,x)\right|dx-\int\left|F(\tau,x)-\widetilde{F}(\tau,x)\right|dx\geq
0,$

for some constant
$C=C(r,R,\left\|\partial_{z}\phi\right\|_{\infty},\left\|\phi\right\|_{\infty})$.
In particular we get

$\left\|F(t,\cdot)-\widetilde{F}(t,\cdot)\right\|_{1}\leq
t\underset{t\in[0,T]}{\sup}\left|A(t,\cdot)-\widetilde{A}(t,\cdot)\right|_{\mathrm{Lip}}+\left\|F(0,\cdot)-\widetilde{F}(0,\cdot)\right\|_{1}+C\int_{0}^{t}\left\|F(s,\cdot)-\widetilde{F}(s,\cdot)\right\|_{1}ds,$

which by Gronwall implies

$\left\|F(t,\cdot)-\widetilde{F}(t,\cdot)\right\|_{1}\leq
e^{Ct}\left(\left\|F(0,\cdot)-\widetilde{F}(0,\cdot)\right\|_{1}+t\underset{t\in[0,T]}{\sup}\left|A(t,\cdot)-\widetilde{A}(t,\cdot)\right|_{\mathrm{Lip}}\right),$

as desired.

$\square$

######  Remark 4.1.

Given initial data $F^{\mathrm{in}}$, it is not difficult to construct
explicitly initial weights $m_{i}^{\mathrm{in,}N}$and initial opinions
$x_{i}^{\mathrm{in,}N}$ such that
$\left\|-\frac{1}{2}+\frac{1}{N}\stackrel{{\scriptstyle[}}{{i}}=1]{N}{\sum}m_{i}^{\mathrm{in,}N}H(x-x_{i}^{\mathrm{in,}N})-F^{\mathrm{in}}\right\|_{1}\underset{N\rightarrow\infty}{\rightarrow}0$,
thereby witnessing the fact that the assumption
$\left\|F_{N}^{\mathrm{in}}-F^{\mathrm{in}}\right\|_{1}\underset{N\rightarrow\infty}{\rightarrow}0$
is reasonably typical. See Lemma 5.3 in [11] for a guidance how to do this.

##  5  Appendix A: The Rankine–Hugoniot and Oleinik Conditions Revisited

The Rankine–Hugoniot and Oleinik conditions are a very standard tool in the
theory of conservation laws. However it seems that the literature typically
formulates these conditions for conservation laws with time independent
fluxes, no source terms and when there is only a single curve of
discontinuity. All of these additions are completely harmless, but for the
sake of completeness we revisit the derivation of the entropy inequality
subject in this slightly more general settings. The equation is as usual

$\partial_{t}F+\partial_{x}(A(t,F))=\mathbf{S}[F](t,x).$ (5.1)

######  Proposition 5.1.

(Oleinik condition) Suppose there are times
$0=T_{0}<T_{1}<...<T_{k-1}<T_{k}=T$ such that for each $1\leq j\leq k$ there
are curves $\left\\{(t,s_{i}^{j}(t))\right\\}_{i=1}^{m_{j}}$
($m_{j}\in\mathbb{N}$) such that $F$ is a classical solution to Equation (5.1)
on

$V_{L}^{i,j}\coloneqq\left\\{(t,x)\left|t\in[T_{j-1},T_{j}),s_{i-1}^{j}(t)\leq
x<s_{i}^{j}(t)\right.\right\\}$

and

$V_{R}^{i,j}\coloneqq\left\\{(t,x)\left|t\in[T_{j-1},T_{j}),s_{i}^{j}(t)<x\leq
s_{i+1}^{j}(t)\right.\right\\},$

with the convention $s_{0}^{j}(t)=-\infty$ and $s_{m_{j}+1}^{j}(t)=+\infty$.
For each $t\in[T_{j-1},T_{j})$ let

$F_{L}^{i,j}(t)\coloneqq\underset{x\nearrow s_{i}^{j}(t)}{\lim}F(t,x),\
F_{R}^{i,j}(t)\coloneqq\underset{x\searrow s_{i}^{j}(t)}{\lim}F(t,x).$

Suppose that for each $t\in[T_{j-1},T_{j})$ and each
$\theta\in(F_{L}^{i,j}(t),F_{R}^{i,j}(t))$ it holds that

$\frac{A(t,F_{L}^{i,j}(t))-A(t,F_{R}^{i,j}(t))}{F_{L}^{i,j}(t)-F_{R}^{i,j}(t)}=\dot{s}_{i}^{j}(t)$
(5.2)

and

$\frac{A(t,\theta)-A(t,F_{L}^{i,j}(t))}{\theta-
F_{L}^{i,j}(t)}\geq\dot{s}_{i}^{j}(t).$ (5.3)

Then for each $\alpha\in\mathbb{R}$ $F$ satisfies the entropy inequality

$\iint_{V}\partial_{t}\chi(F-\alpha)\mathrm{sgn}(F-\alpha)+\partial_{x}\chi\left(A(F)-A(\alpha)\right)\mathrm{sgn}(F-\alpha)+\chi\mathrm{sgn}(F-\alpha)\mathbf{S}[F](t,x)dxdt\geq
0.$ (5.4)

Proof. Let us first consider
$\eta:\mathbb{R}\rightarrow\mathbb{R},\psi:\mathbb{R}\rightarrow\mathbb{R}$
where $\eta\in C^{1,1}(\mathbb{R})$ is convex and
$\psi^{\prime}=\eta^{\prime}A^{\prime}$. Then we have that

$\displaystyle\iint\partial_{t}\chi\eta(F)+\partial_{x}\chi\psi(F)+\chi\mathbf{S}[F](t,x)dxdt=$
$\displaystyle\,\stackrel{{\scriptstyle[}}{{j}}=1]{k}{\sum}\stackrel{{\scriptstyle[}}{{i}}=1]{m_{j}}{\sum}\iint_{V_{L}^{i,j}}\left(\partial_{t}\chi\eta(F)+\partial_{x}\chi\psi(F)+\chi\eta^{\prime}(F)\mathbf{S}[F](t,x)\right)dxdt$
$\displaystyle+\stackrel{{\scriptstyle[}}{{j}}=1]{k}{\sum}\stackrel{{\scriptstyle[}}{{i}}=1]{m_{j}}{\sum}\iint_{V_{R}^{i,j}}\left(\partial_{t}\chi\eta(F)+\partial_{x}\chi\psi(F)+\chi\eta^{\prime}(F)\mathbf{S}[F](t,x)\right)dxdt.$

Set
$\Gamma_{i,j}\coloneqq\left\\{(t,s_{i}^{j}(t))\left|t\in[T_{j-1},T_{j})\right.\right\\}$.
Keep $1\leq j\leq k$ and $1\leq i\leq m_{j}$ fixed and take a test function
$\chi\in C_{0}^{\infty}((0,T)\times\mathbb{R})$. For readability, we omit the
indices $i,j$. Using that $F$ is a classical solution on $V_{L},V_{R}$ we get

$\displaystyle\iint_{V_{L}}\left(\partial_{t}\chi\eta(F)+\partial_{x}\chi\psi(F)+\chi\eta^{\prime}(F)\mathbf{S}[F](t,x)\right)dxdt$
$\displaystyle\qquad+\iint_{V_{R}}\left(\partial_{t}\chi\eta(F)+\partial_{x}\chi\psi(F)+\chi\eta^{\prime}(F)\mathbf{S}[F](t,x)\right)dxdt$
$\displaystyle\quad=\iint_{V_{L}}\left(-\chi\partial_{t}\eta(F)-\chi\partial_{x}\psi(F)+\chi\eta^{\prime}(F)\mathbf{S}[F](t,x)\right)dxdt$
$\displaystyle\qquad+\iint_{V_{R}}\left(-\chi\partial_{t}\eta(F)-\chi\partial_{x}\psi(F)+\chi\eta^{\prime}(F)\mathbf{S}[F](t,x)\right)dxdt$
$\displaystyle\qquad+\int_{\Gamma}\left(\chi\eta(F_{L})\nu^{1}+\chi\psi(F_{L})\nu^{2}\right)d\sigma-\int_{\Gamma}\chi\eta(F_{R})\nu^{1}+\chi\psi(F_{R})\nu^{2})d\sigma$
$\displaystyle\quad=\int_{\Gamma}\left(\chi(\psi(F_{L})-\psi(F_{R}))\nu^{1}+\chi(\eta(F_{L})-\eta(F_{R}))\nu^{2}\right)d\sigma.$

Here

$(\nu^{1},\nu^{2})=\frac{1}{\sqrt{1+\dot{s}^{2}}}(-\dot{s},1),$

and the last equality is because of the identity

$\partial_{t}\eta(F)+\partial_{x}\psi(F)+\eta^{\prime}(F)\mathbf{S}[F]=0,$

which is easily derived using that $F$ is a classical solution on each region.
We claim that

$(\eta(F_{L})-\eta(F_{R}))-(\psi(F_{L})-\psi(F_{R}))\dot{s}(t)\geq 0.$

First, we integrate by parts to find that

$\displaystyle\psi(F_{R})-\psi(F_{L})$
$\displaystyle=\int_{F_{L}}^{F_{R}}\eta^{\prime}(y)A^{\prime}(t,y)dy$
$\displaystyle=-\int_{F_{L}}^{F_{R}}\eta^{\prime\prime}(y)(A(t,y)-A(t,F_{L}(t)))dy+\left.\eta^{\prime}(\cdot)(A(t,\cdot)-A(t,F_{L}))\right|_{F_{L}}^{F_{R}}$
$\displaystyle=-\int_{F_{L}}^{F_{R}}\eta^{\prime\prime}(y)(A(t,y)-A(t,F_{L}))dy+\eta^{\prime}(F_{R}(t))(A(t,F_{R}(t))-A(t,F_{L}(t))),$

and so thanks to the assumption (5.3) and the convexity of $\eta$ we have the
inequality

$\displaystyle\psi(F_{L})-\psi(F_{R})$
$\displaystyle=\int_{F_{L}}^{F_{R}}\eta^{\prime\prime}(y)(A(t,y)-A(t,F_{L}))dy-\eta^{\prime}(F_{R})(A(t,F_{R})-A(t,F_{L}))$
(5.5)
$\displaystyle\geq\dot{s}(t)\int_{F_{L}}^{F_{R}}\eta^{\prime\prime}(y)(y-F_{L})dy-\eta^{\prime}(F_{R})(A(t,F_{R})-A(t,F_{L}).$

On the other hand

$\displaystyle\eta(F_{R})-\eta(F_{L})$
$\displaystyle=\int_{F_{L}}^{F_{R}}\eta^{\prime}(y)dy=-\int_{F_{L}}^{F_{R}}\eta^{\prime\prime}(y)(y-F_{L})dy+\left.(y-F_{L})\eta^{\prime}\right|_{F_{L}}^{F_{R}}$
$\displaystyle=-\int_{F_{L}}^{F_{R}}\eta^{\prime\prime}(y)(y-F_{L}(t))dy+(F_{R}-F_{L})\eta^{\prime}(F_{R}).$

Together with inequality (5.5) the last equation entails

$\psi(F_{L})-\psi(F_{R})-\dot{s}(t)(\eta(F_{L})-\eta(F_{R}))\geq\eta^{\prime}(F_{R})\left(\dot{s}(t)(F_{R}-F_{L})-A(t,F_{R})-A(t,F_{L})\right)=0,$

where the last equality is because of the assumption (5.2). It follows that

$\int\partial_{t}\chi\eta(F)+\partial_{x}\chi\psi(F)+\chi\eta^{\prime}(F)\mathbf{S}[F](t,x)dxdt\geq
0.$ (5.6)

To finish, we use a standard approximation argument. For each $\varepsilon>0$
consider the convex function $\mathfrak{s}_{\varepsilon}\in
C^{1,1}(\mathbb{R})$ defined by

$\mathfrak{s}_{\varepsilon}(z)\coloneqq\begin{cases}\frac{1}{2\varepsilon}z^{2},&\left|z\right|\leq\varepsilon,\\\
\left|z\right|-\frac{\varepsilon}{2},&\left|z\right|>\varepsilon.\end{cases}$

and for each $\alpha\in\mathbb{R}$ let

$\eta_{\varepsilon}(z)\coloneqq\mathfrak{s}_{\varepsilon}(z-\alpha).$

It is clear that we have pointwise convergence

$\eta_{\varepsilon}\underset{\varepsilon\rightarrow
0}{\rightarrow}\left|z-\alpha\right|,\
\eta_{\varepsilon}^{\prime}\underset{\varepsilon\rightarrow
0}{\rightarrow}\mathrm{sgn}(z-\alpha).$

Furthermore we can take

$\psi_{\varepsilon}(F)=\int_{\alpha}^{F}\eta_{\varepsilon}^{\prime}A^{\prime}(t,y)dy=\int_{\alpha}^{F}\eta_{\varepsilon}^{\prime}(A(t,y)-A(t,\alpha))^{\prime}dy,$

and as $\varepsilon\rightarrow 0$ we get

$\psi_{\varepsilon}(F)\underset{\varepsilon\rightarrow
0}{\rightarrow}\int_{\alpha}^{F}\mathrm{sgn}(y-\alpha)(A(t,y)-A(t,\alpha))^{\prime}dy=\mathrm{sgn}(F-\alpha)(A(t,F)-A(t,\alpha))$

pointwise. Therefore testing inequality (5.6) with
$\eta_{\varepsilon},\psi_{\varepsilon}$ and taking the limit as
$\varepsilon\rightarrow 0$ yields the asserted inequality (5.4).

$\square$

#### Acknowledgments

IBP and JAC were supported by the EPSRC grant number EP/V051121/1. This work
was also supported by the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for
Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization) of
the European Research Council Executive Agency (ERC) under the European
Union’s Horizon 2020 research and innovation programme (grant agreement No.
883363). STG was supported by the Research Council of Norway through grant no.
286822, Wave Phenomena and Stability — a Shocking Combination (WaPheS).

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