#  Concentration profiles in FitzHugh-Nagumo neural networks: A Hopf-Cole
approach

###### Abstract.

In this paper we focus on a spatially extended FitzHugh-Nagumo model with
interactions. In the regime where strong and local interactions dominate, we
quantify how the probability density of neurons concentrates into a Dirac
distribution. Previous work investigating this question have provided relative
bounds in integrability spaces. Using a Hopf-Cole framework, we derive precise
$L^{\infty}$ estimates using subtle explicit sub- and super- solutions which
prove, with rates of convergence, that the blow up profile is Gaussian.

Alain Blaustein 111Institut de Mathématiques de Toulouse, Université Paul
Sabatier, 118 route de Narbonne - F-31062 Toulouse Cedex 9, France. E-mail :
<EMAIL_ADDRESS>Emeric Bouin 222CEREMADE - Université
Paris-Dauphine, UMR CNRS 7534, Place du Maréchal de Lattre de Tassigny, 75775
Paris Cedex 16, France. E-mail<EMAIL_ADDRESS>

Keywords:  .

Mathematics Subject Classification (2010):

##  1\. Introduction

### The model.

Neurons inside the brain are known to interact with each other through the
variation of their membrane potential. One of the main challenges in
neuroscience consists in understanding complex dynamics induced by these
interactions. A quantitative description started with the pioneering work [12]
of A. Hodgkin and A. Huxley, where they built an accurate model to describe
the membrane potential dynamics of a single nerve cell submitted to an
external current. Simplified versions of this model exist, for example
FitzHugh-Nagumo models [11, 16]: these ones keep the main features of Hodgkin-
Huxley models but are more tractable from a mathematical point of view. Based
on FitzHugh-Nagumo equations, we focus on the following model

(1.1)
$\displaystyle\partial_{t}f^{\varepsilon}+\mathrm{div}_{\bm{u}}\left[\mathbf{b}^{\varepsilon}f^{\varepsilon}\right]-\partial^{2}_{v}f^{\varepsilon}=\frac{1}{\varepsilon}\rho^{\varepsilon}_{0}\partial_{v}\left[(v-{\mathcal{V}}^{\varepsilon})f^{\varepsilon}\right].$

Equation (1.1) is obtained as the mean-field limit of a microscopic model of
the following form (the precise expression of $N$ and $A$ will be given later
on), which is a coupled system of stochastic differential equations describing
a neural network of size $n$,

$\left\\{\begin{array}[]{lll}&\displaystyle\mathrm{d}v^{i}_{t}=\left(N(v^{i}_{t})-w^{i}_{t}-\frac{1}{n}\sum_{j=1}^{n}\Phi_{\varepsilon}({\bm{x}}_{i},{\bm{x}}_{j})(v^{i}_{t}-v^{j}_{t})\right)\mathrm{d}t+\sqrt{2}\mathrm{d}B^{i}_{t},\\\\[20.00003pt]
&\displaystyle\mathrm{d}w^{i}_{t}=A\left(v^{i}_{t},w^{i}_{t}\right)\mathrm{d}t,\end{array}\right.$

In the limit $n\rightarrow+\infty$ and with a suitable choice for the
interaction kernel $\Phi_{\varepsilon}$, the empirical measure associated to
the latter system converges to $f^{\varepsilon}$: see for instance [1, 9, 15,
13] for a rigorous proof of this limit in the case a the FitzHugh-Nagumo
system and [7] for a related model in collective dynamics.

In (1.1), $\bm{u}=(v,w)\in{\mathbb{R}}^{2}$, where the first variable $v$
stands for the membrane potential of a neuron and is coupled to an auxiliary
variable $w$ called the adaptation variable. The distribution function
$f^{\varepsilon}:=f^{\varepsilon}(t,{\bm{x}},\bm{u})$ represents the density
of neurons at time $t$, located at position ${\bm{x}}$ in a compact set
$K\subseteq{\mathbb{R}}^{d}$, with a membrane potential $v\in{\mathbb{R}}$ and
an adaptation variable $w\in{\mathbb{R}}$. In the latter equation, the right
hand side accounts for local interactions between neurons and it is re-scaled
by a small parameter $\varepsilon>0$ which describes their strength. It also
displays the macroscopic quantities associated to the network: the spatial
distribution of neurons throughout the network

$\rho_{0}^{\varepsilon}({\bm{x}})=\int_{{\mathbb{R}}^{2}}f^{\varepsilon}(t,{\bm{x}},\bm{u})\mathrm{d}\bm{u},$

and the averaged voltage and adaptation variable at a spatial location
${\bm{x}}$

(1.2)
$\left\\{\begin{array}[]{ll}\displaystyle\rho_{0}^{\varepsilon}\left({\bm{x}}\right){\mathcal{V}}^{\varepsilon}(t,{\bm{x}})&=\displaystyle\int_{{\mathbb{R}}^{2}}v~{}f^{\varepsilon}(t,{\bm{x}},\bm{u})\mathrm{d}\bm{u},\\\\[11.00008pt]
\displaystyle\rho_{0}^{\varepsilon}\left({\bm{x}}\right){\mathcal{W}}^{\varepsilon}(t,{\bm{x}})&=\displaystyle\int_{{\mathbb{R}}^{2}}w~{}f^{\varepsilon}(t,{\bm{x}},\bm{u})\mathrm{d}\bm{u}.\end{array}\right.$

In the sequel, we use the vector notation
$\displaystyle\mathcal{U}^{\varepsilon}=\left(\mathcal{V}^{\varepsilon},\mathcal{W}^{\varepsilon}\right)$.
Notice that $\rho_{0}^{\varepsilon}$ is indeed time-homogeneous, integrating
the mean field equation (1.1) with respect to $\bm{u}\in{\mathbb{R}}^{2}$. In
(1.1), coefficient $\mathbf{b}^{\varepsilon}$ is defined for all
$\displaystyle\left(t,{\bm{x}},\bm{u}\right)\in{\mathbb{R}}^{+}\times
K\times{\mathbb{R}}^{2}$ as

(1.3)
$\mathbf{b}^{\varepsilon}(t,{\bm{x}},\bm{u}):=\begin{pmatrix}\displaystyle
B^{\varepsilon}(t,{\bm{x}},\bm{u})\\\\[8.99994pt] \displaystyle
A(\bm{u})\end{pmatrix}=\begin{pmatrix}N(v)-w-\mathcal{K}_{\Psi}[f^{\varepsilon}]\left(t,{\bm{x}},v\right).\\\\[8.99994pt]
\displaystyle av-bw+c\end{pmatrix},$

where $a$, $c\in{\mathbb{R}}$ and $b>0$. The drift
$N\in{\mathscr{C}}^{2}({\mathbb{R}})$ satisfies

(1.4) $\limsup_{|v|\to+\infty}\frac{N(v)}{{\text{\rm sgn
}}{v}|v|^{p}}<0,\qquad\sup_{|v|\geq
1}\left|\frac{N(v)}{|v|^{p}}\right|<+\infty.$

for some $p\geq 2$, and

(1.5) $\sup_{|v|\geq
1}\left(\left|N^{\prime\prime}(v)\right|+\left|N^{\prime}(v)\right|\right)|v|^{-p^{\prime}}<+\infty,$

for some $p^{\prime}$. A historical choice for $N$ is a confining non-
linearity with the form

$N(v)=v-v^{3},$

but many other cases are possible.

The operator $\mathcal{K}_{\Psi}$, which accounts for the long-range
interactions, is defined as

$\mathcal{K}_{\Psi}[f](t,{\bm{x}},v)=\int_{K\times{\mathbb{R}}^{2}}\Psi({\bm{x}},{\bm{x}}^{\prime})(v-v^{\prime})f(t,{\bm{x}}^{\prime},\bm{u}^{\prime})\mathrm{d}{\bm{x}}^{\prime}\mathrm{d}\bm{u}^{\prime}.$

The connectivity kernel
$\Psi\in{\mathscr{C}}^{0}\left(K_{{\bm{x}}},L^{1}\left(K_{{\bm{x}}^{\prime}}\right)\right)$
satisfies

(1.6) $\sup_{{\bm{x}}^{\prime}\in
K}\int_{K}\left|\Psi({\bm{x}},{\bm{x}}^{\prime})\right|\mathrm{d}{\bm{x}}<\;+\infty,\qquad\sup_{{\bm{x}}\in
K}\int_{K}\left|\Psi({\bm{x}},{\bm{x}}^{\prime})\right|^{r}\mathrm{d}{\bm{x}}^{\prime}<+\infty,$

for some $r>1$. Thanks to this set of assumptions on $\Psi$ our model takes
into account non-symmetric interactions between neurons and authorize $\Psi$
to behave like a power law, a case which is considered in the physical
literature (see [13]). We point out that $\mathcal{K}_{\Psi}$ can be expressed
in terms of the macroscopic quantities

$\mathcal{K}_{\Psi}[f^{\varepsilon}](t,{\bm{x}},v)=\Psi*_{r}\rho^{\varepsilon}_{0}({\bm{x}})v-\Psi*_{r}(\rho^{\varepsilon}_{0}\;{\mathcal{V}}^{\varepsilon})(t,{\bm{x}}),$

where $*_{r}$ is a shorthand for the convolution on the right side of any
function $g$ with $\Psi$

$\Psi*_{r}g({\bm{x}})\;=\int_{K}\Psi({\bm{x}},{\bm{x}}^{\prime})g({\bm{x}}^{\prime})\mathrm{d}{\bm{x}}^{\prime}.$

Before going further, we shall be precise about the notion of solution we
consider for equation (1.1). For this, we suppose, for each $\varepsilon>0$,

(1.7)
$f^{\varepsilon}_{0}\in{\mathscr{C}}^{0}\left(K,L^{1}\left({\mathbb{R}}^{2}\right)\right),\quad
f^{\varepsilon}\geq
0\quad\textrm{and}\quad\int_{{\mathbb{R}}^{2}}f^{\varepsilon}_{0}({\bm{x}},\bm{u})\mathrm{d}\bm{u}\mathrm{d}{\bm{x}}=1,\quad\forall{\bm{x}}\in
K.$

Therefore, $\rho_{0}^{\varepsilon}\in{\mathscr{C}}^{0}\left(K\right)$. We also
suppose that

(1.8) $m_{*}\leq\rho_{0}^{\varepsilon}\leq 1/m_{*},$

for all $\varepsilon>0$ and for some positive constant $m_{*}$ independent of
$\varepsilon$.

######  Remark 1.1.

The uniform lower bound condition on $\rho_{0}^{\varepsilon}$ seems necessary
in our analysis because the problem degenerates when $\rho_{0}^{\varepsilon}$
vanishes as it may be seen on equation (1.1). However the upper bound
condition on $\rho_{0}^{\varepsilon}$ may be relaxed at the cost of loosing
uniform convergence with respect to ${\bm{x}}$ in our main result Theorem 1.4.

On top of that, we assume the following condition: there exists two positive
constants $m_{p}$ and $\mskip
1.5mu\overline{\mskip-1.5mu{m}\mskip-1.5mu}\mskip 1.5mu_{p}$, independent of
$\varepsilon$, such that

(1.9) $\sup_{{\bm{x}}\in
K}\int_{{\mathbb{R}}^{2}}|\bm{u}|^{2(p+p^{\prime})}f^{\varepsilon}_{0}\left({\bm{x}},\bm{u}\right)\mathrm{d}\bm{u}\leq
m_{p},$

and such that

(1.10)
$\int_{K\times{\mathbb{R}}^{2}}|\bm{u}|^{2(p+p^{\prime})r^{\prime}}f^{\varepsilon}_{0}\left({\bm{x}},\bm{u}\right)\mathrm{d}\bm{u}\mathrm{d}{\bm{x}}\leq\mskip
1.5mu\overline{\mskip-1.5mu{m}\mskip-1.5mu}\mskip 1.5mu_{p},$

where $p$, $p^{\prime}$ and $r^{\prime}$ are given in (1.4), (1.5) and (1.6).

######  Definition 1.2.

For all $\varepsilon>0$, we say that $f^{\varepsilon}$ solves (1.1) with
initial condition $f^{\varepsilon}_{0}$ if
$f^{\varepsilon}\in{\mathscr{C}}^{0}\left({\mathbb{R}}^{+}\times
K,L^{1}\left({\mathbb{R}}^{2}\right)\right)$ and for all ${\bm{x}}\in K$,
$t\geq 0$, and
$\displaystyle\varphi\in{\mathscr{C}}_{c}^{\infty}\left({\mathbb{R}}^{2}\right)$
, it holds

$\displaystyle\int_{{\mathbb{R}}^{2}}\varphi(\bm{u})\left(f^{\varepsilon}\left(t,{\bm{x}},\bm{u}\right)-f^{\varepsilon}_{0}\left({\bm{x}},\bm{u}\right)\right)\mathrm{d}\bm{u}$
$\displaystyle=\int_{0}^{t}\int_{{\mathbb{R}}^{2}}\left[\left(\nabla_{\bm{u}}\varphi\cdot\mathbf{b}^{\varepsilon}+\partial_{v}^{2}\varphi\right)f^{\varepsilon}\right](s,{\bm{x}},\bm{u})\mathrm{d}\bm{u}\mathrm{d}s$
$\displaystyle-\frac{\rho_{0}^{\varepsilon}({\bm{x}})}{\varepsilon}\int_{0}^{t}\int_{{\mathbb{R}}^{2}}\left[\partial_{v}\varphi\left(v-{\mathcal{V}}^{\varepsilon}\right)f^{\varepsilon}\right](s,{\bm{x}},\bm{u})\mathrm{d}\bm{u}\mathrm{d}s,$

where $\mathcal{V}^{\varepsilon}$ and $\mathbf{b}^{\varepsilon}$ are given by
(1.2) and (1.1) respectively.

With this notion of solution, equation (1.1) is well-posed, the following
result being proved in [6].

######  Theorem 1.3 ([6]).

For any $\varepsilon>0$, suppose that assumptions (1.4) on $N$, (1.6) on
$\Psi$ and (1.7)-(1.8) on the initial condition are fulfilled and that
$f_{0}^{\varepsilon}$ also verifies

$\left\\{\begin{array}[]{l}\displaystyle\sup_{{\bm{x}}\in
K}\int_{{\mathbb{R}}^{2}}e^{|\bm{u}|^{2}/2}f^{\varepsilon}_{0}({\bm{x}},\bm{u})\mathrm{d}\bm{u}<+\infty,\\\\[15.00002pt]
\displaystyle\sup_{{\bm{x}}\in
K}\int_{{\mathbb{R}}^{2}}\ln{\left[f^{\varepsilon}_{0}({\bm{x}},\bm{u})\right]}f^{\varepsilon}_{0}({\bm{x}},\bm{u})\mathrm{d}\bm{u}<+\infty,\end{array}\right.$

and

$\sup_{{\bm{x}}\in
K}\left\|\nabla_{\bm{u}}\sqrt{f^{\varepsilon}_{0}}\right\|^{2}_{L^{2}({\mathbb{R}}^{2})}<+\infty.$

Then there exists a unique solution $f^{\varepsilon}$ to equation (1.1) with
initial condition $f^{\varepsilon}_{0}$, in the sense of Definition 1.2 which
verifies

$\displaystyle\sup_{(t,{\bm{x}})\in[0,T]\times
K}\int_{{\mathbb{R}}^{2}}e^{|\bm{u}|^{2}/2}f^{\varepsilon}(t,{\bm{x}},\bm{u})\mathrm{d}\bm{u}<+\infty,$

for all $T\geq 0$,

### The question at hand.

The purpose of this article is to go through the mathematical analysis of the
neural network in the regime of strong local interactions, that is when
$\varepsilon\ll 1$. More precisely, we prove that the voltage distribution
concentrates into a Dirac mass and propose a quantitative description of the
concentration profile. First, we highlight this concentration phenomenon with
some formal computations. We look for the leading order in (1.1): in our case,
it is induced by short range interactions between neurons, and as
$\varepsilon\rightarrow 0$, we expect

$(v-\;{\mathcal{V}}^{\varepsilon})f^{\varepsilon}\underset{\varepsilon\rightarrow
0}{\to}0,$

to make sure that no terms are singular. This means that $f^{\varepsilon}$
concentrates towards a Dirac mass centred in ${\mathcal{V}}^{\varepsilon}$
with respect to the $v$-variable, that is

$f^{\varepsilon}(t,{\bm{x}},\bm{u})\underset{\varepsilon\rightarrow
0}{\approx}\delta_{0}\left(v-{\mathcal{V}}^{\varepsilon}(t,{\bm{x}})\right)\otimes
F^{\varepsilon}(t,{\bm{x}},w),$

where $\displaystyle{\mathcal{V}}^{\varepsilon}$ is given by (1.2) and
$F^{\varepsilon}$ is defined as the marginal of $f^{\varepsilon}$ with respect
to the voltage variable

$F^{\varepsilon}\left(t,{\bm{x}},w\right)=\int_{{\mathbb{R}}}f^{\varepsilon}\left(t,{\bm{x}},\bm{u}\right)\mathrm{d}v.$

Multiplying equation (1.1) by $v$ and then integrating over
$\bm{u}\in{\mathbb{R}}^{2}$ (resp. $v\in{\mathbb{R}}$), multiplying the second
line of (1.11) by $w/\rho_{0}^{\varepsilon}$ and integrating with respect to
$w$, one finds that the couple
$\displaystyle({\mathcal{V}}^{\varepsilon},{\mathcal{W}}^{\varepsilon})$
solves the following system

(1.11)
$\left\\{\begin{array}[]{llll}&\displaystyle\partial_{t}{\mathcal{V}}^{\varepsilon}=N({\mathcal{V}}^{\varepsilon})-{\mathcal{W}}^{\varepsilon}-\mathcal{L}_{\rho^{\varepsilon}_{0}}\left[\mathcal{V^{\varepsilon}}\right]+{\mathcal{E}}(f^{\varepsilon}),\\\\[8.00003pt]
&\partial_{t}{\mathcal{W}}^{\varepsilon}=A({\mathcal{V}}^{\varepsilon},{\mathcal{W}}^{\varepsilon}),\end{array}\right.$

where the error term $\displaystyle{\mathcal{E}}\left(f^{\varepsilon}\right)$
is given by

(1.12)
${\mathcal{E}}\left(f^{\varepsilon}\left(t,{\bm{x}},\cdot\right)\right)\;=\;\frac{1}{\rho_{0}^{\varepsilon}({\bm{x}})}\int_{{\mathbb{R}}^{2}}N(v)f^{\varepsilon}\left(t,{\bm{x}},\bm{u}\right)\mathrm{d}\bm{u}-N({\mathcal{V}}^{\varepsilon}),$

and $\displaystyle\mathcal{L}_{\rho_{0}^{\varepsilon}}$ is a non local
operator given by

$\mathcal{L}_{\rho_{0}^{\varepsilon}}\left[{\mathcal{V}}^{\varepsilon}\right]={\mathcal{V}}^{\varepsilon}\Psi*_{r}\rho_{0}^{\varepsilon}-\Psi*_{r}(\rho_{0}^{\varepsilon}{\mathcal{V}}^{\varepsilon})\;.$

All this, in turn, implies that as $\varepsilon$ vanishes,
$\displaystyle\left({\mathcal{V}}^{\varepsilon},{\mathcal{W}}^{\varepsilon}\right)$
converges to the couple
$\displaystyle\left({\mathcal{V}},{\mathcal{W}}\right)$, which solves

(1.13)
$\left\\{\begin{array}[]{l}\displaystyle\partial_{t}{\mathcal{V}}=N({\mathcal{V}})-{\mathcal{W}}-\mathcal{L}_{\rho_{0}}[{\mathcal{V}}],\\\\[8.99994pt]
\displaystyle\partial_{t}{\mathcal{W}}=A({\mathcal{V}},{\mathcal{W}}),\\\\[8.99994pt]
\displaystyle\left({\mathcal{V}}\left(0,\cdot\right),{\mathcal{W}}\left(0,\cdot\right)\right)=\left({\mathcal{V}}_{0},{\mathcal{W}}_{0}\right).\end{array}\right.$

As of the marginal $F^{\varepsilon}$, it has been shown in [10, 5, 6] that it
also converges.

In this article, we refine the latter result by investigating the
concentration profile of the solution $f^{\varepsilon}$ when $\varepsilon$
goes to $0$. For that purpose, we perform the so-called Hopf-Cole transform of
$f^{\varepsilon}$,

(1.14)
$\phi^{\varepsilon}:=\varepsilon\ln\left(\sqrt{\frac{2\pi\varepsilon}{\rho_{0}}}f^{\varepsilon}\right),$

and study the convergence of $\phi^{\varepsilon}$ as $\varepsilon$ goes to
zero. This approach has been widely used in selection-mutation models in
population dynamics (for example), to study concentration phenomena occurring
in the small mutation regime for populations structured by space and trait:
see for instance [3, 14].

### Heuristics and main result.

Let us now formally present the convergence of $\phi^{\varepsilon}$: injecting
ansatz (1.14) in equation (1.1), we find that $\phi^{\varepsilon}$ solves the
following Hamilton-Jacobi equation

(1.15)
$\partial_{t}\phi^{\varepsilon}+\nabla_{\bm{u}}\phi^{\varepsilon}\cdot\mathbf{b}^{\varepsilon}+\varepsilon\mathrm{div}_{\bm{u}}\left[\mathbf{b}^{\varepsilon}\right]-\partial^{2}_{v}\phi^{\varepsilon}-\rho_{0}^{\varepsilon}=\frac{1}{\varepsilon}\left(\partial_{v}\left(\frac{1}{2}\rho_{0}^{\varepsilon}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{2}+\phi^{\varepsilon}\right)\partial_{v}\phi^{\varepsilon}\right)$

Keeping only the leading order in equation (1.15), it yields

$\phi^{\varepsilon}\underset{\varepsilon\rightarrow
0}{\approx}-\frac{\delta_{1}}{2}\,\rho_{0}^{\varepsilon}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{2}\mathds{1}_{v<{\mathcal{V}}^{\varepsilon}}-\frac{\delta_{2}}{2}\rho_{0}^{\varepsilon}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{2}\mathds{1}_{v>{\mathcal{V}}^{\varepsilon}}+c\,,$

for some positive constant $c$ and where $\delta_{i}$ lies in $\\{0,1\\}$.
Provided that the macroscopic quantities $\rho^{\varepsilon}$ and
$\displaystyle{\mathcal{V}}^{\varepsilon}$ converge as $\varepsilon$ goes to
$0$, we expect

$\phi^{\varepsilon}(\cdot,\cdot,v)\underset{\varepsilon\rightarrow
0}{\longrightarrow}\;-\,\frac{\delta_{1}}{2}\,\rho_{0}\left|v-{\mathcal{V}}\right|^{2}\mathds{1}_{v<{\mathcal{V}}}-\frac{\delta_{2}}{2}\rho_{0}\left|v-{\mathcal{V}}\right|^{2}\mathds{1}_{v>{\mathcal{V}}}+c\,.$

Furthermore, since our problem conserves mass, we expect

$\int_{v\in{\mathbb{R}}}\exp{\left(-\,\frac{\delta_{1}}{2\,\varepsilon}\,\rho_{0}\left|v-{\mathcal{V}}\right|^{2}\mathds{1}_{v<{\mathcal{V}}}-\frac{\delta_{2}}{2\,\varepsilon}\rho_{0}\left|v-{\mathcal{V}}\right|^{2}\mathds{1}_{v>{\mathcal{V}}}+\frac{c}{\varepsilon}\right)}\,\mathrm{d}v\,=\,\sqrt{\frac{\rho_{0}}{2\pi\varepsilon}}\,,$

for all $\varepsilon$. This forces $\delta_{1}=\delta_{2}=1$ and $c=0$ and
therefore we obtain

$\phi^{\varepsilon}(\cdot,\cdot,v)\underset{\varepsilon\rightarrow
0}{\longrightarrow}\;-\,\frac{1}{2}\,\rho_{0}\left|v-{\mathcal{V}}\right|^{2}\,.$

This convergence is the object of our main result, Theorem 1.4 below. It will
justify the latter limit and provide explicit convergence rates. Our strategy
consists in performing a Hilbert expansion of $\phi^{\varepsilon}$ with
respect to $\varepsilon$ and to prove that the higher order terms in the
expansion are uniformly bounded with respect to $\varepsilon$ using a
comparison principle (see Section 3 for more details).

######  Theorem 1.4.

Assume (1.4)-(1.10) and the additional assumptions of Proposition 2.2 and
Theorem 1.3. Suppose that there exists a positive constant $C$ independent of
$\varepsilon$ such that the following compatibility assumption holds

(1.16)
$\left\|\mathcal{U}_{0}-\mathcal{U}_{0}^{\varepsilon}\right\|_{L^{\infty}(K)}+\|\rho_{0}-\rho_{0}^{\varepsilon}\|_{L^{\infty}(K)}\leq
C\varepsilon,$

as well as the following set of ”smallness assumptions”

(1.17a)
$\displaystyle\displaystyle\left|\phi^{\varepsilon}_{0}+\frac{1}{2}\rho_{0}\left|v-{\mathcal{V}}\right|^{2}-\varepsilon
n\right|\leq\varepsilon
C\left(1+|\bm{u}|^{2}\right),\quad\forall({\bm{x}},\bm{u})\in
K\times{\mathbb{R}}^{2},$ (1.17b)
$\displaystyle\int_{{\mathbb{R}}^{2}}\left(|v-{\mathcal{V}}^{\varepsilon}_{0}|^{2}+|v-{\mathcal{V}}^{\varepsilon}_{0}|^{p^{\prime}+1}\right)f^{\varepsilon}_{0}(\cdot,\bm{u})\mathrm{d}\bm{u}\leq
C\varepsilon.$

Then the sequence $\left(\phi^{\varepsilon}\right)_{\varepsilon>0}$ of Hopf-
Cole transforms of $\left(f^{\varepsilon}\right)_{\varepsilon>0}$ is well
defined and it converges locally uniformly on ${\mathbb{R}}^{+}\times
K\times{\mathbb{R}}^{2}$ to
$-\frac{1}{2}\rho_{0}\left|v-{\mathcal{V}}\right|^{2}$ with rate
$\varepsilon$. More precisely, there exist two positive constants $C$ and
$\varepsilon_{0}$ such that for all $\varepsilon\leq\varepsilon_{0}$,

$\left|\phi^{\varepsilon}+\frac{1}{2}\rho_{0}\left|v-{\mathcal{V}}\right|^{2}-\varepsilon
n\right|(t,{\bm{x}},\bm{u})\leq\varepsilon
Ce^{Ct}\left(1+|\bm{u}|^{2}\right),\quad\forall(t,{\bm{x}},\bm{u})\in{\mathbb{R}}^{+}\times
K\times{\mathbb{R}}^{2}.$

As a consequence, $f^{\varepsilon}$ converges uniformly to $0$ on the compact
subsets of ${\mathbb{R}}^{+}\times
K\times{\mathbb{R}}^{2}\setminus\left\\{v\neq{\mathcal{V}}\left(t,{\bm{x}}\right)\right\\}$.
In the latter results, constants $C$ and $\varepsilon_{0}$ only depends on the
data of our problem: $f^{\varepsilon}_{0}$ (only through the constants
appearing in assumptions (1.8)-(1.10) and (1.16)-(1.17b)), $N$, $A$ and
$\Psi$.

Before going further into our analysis, let us comment on our result. We first
emphasize that our result deals with uniform convergence with respect to all
variables, which is a great improvement in comparison to former results
obtained in [5, 6], where $L^{1}$, $L^{2}$ and weak convergence estimates were
obtained. We also point out that the present article is in line with [17],
which also goes through the analysis of the convergence of
$\phi^{\varepsilon}$. However, the latter article relies on a compactness
argument and this has two major consequences: first the limit
$-\frac{1}{2}\rho_{0}\left|v-{\mathcal{V}}\right|^{2}$ is not identified and
second no rate of convergence is obtained. To end with, authors in [17] deal
with a spatially homogeneous neural network.

Our result does not yet describe the limiting pointwise dynamics of
$f^{\varepsilon}$ with respect to the adaptation variable $w$: this
corresponds to limit of $f^{\varepsilon}$ on the set of points
$\displaystyle(t,{\bm{x}},\bm{u})\in{\mathbb{R}}^{+}\times
K\times{\mathbb{R}}^{2}$ such that $v={\mathcal{V}}(t,{\bm{x}})$. Indeed, due
to ansatz (1.14), we only treat the limiting dynamics with respect to the fast
variable $v$. This difficulty is quite common in problems which display fast
and slow variables: see [8] for an another instance in a model for a
population structured by trait and spatial location. In order to fill the gap,
one would need to investigate the convergence of $\phi^{\varepsilon}_{1}$
defined as

(1.18)
$\phi^{\varepsilon}_{1}:=\frac{1}{\varepsilon}\left(\phi^{\varepsilon}+\frac{1}{2}\rho_{0}\left|v-{\mathcal{V}}\right|^{2}\right).$

which corresponds to the next term in the expansion of $\phi^{\varepsilon}$
with respect to $\varepsilon$. We provide a partial answer to this concern in
Section 3 by identifying a formal equivalent of $\phi^{\varepsilon}_{1}$ but
the rigorous analysis of the convergence of $\phi^{\varepsilon}_{1}$ with
locally uniform bounds is beyond the scope of this article.

The last comment on our result is that it holds in a perturbative setting.
Indeed, our assumption at time $t=0$ translates on $f^{\varepsilon}_{0}$ as
follows

$f^{\varepsilon}_{0}\left({\bm{x}},\bm{u}\right)\underset{\varepsilon\rightarrow
0}{=}\sqrt{\frac{\rho_{0}({\bm{x}})}{2\pi\varepsilon}}\exp{\left(-\frac{\rho_{0}({\bm{x}})}{2\varepsilon}\left|v-{\mathcal{V}}_{0}\left({\bm{x}}\right)\right|^{2}+O(1)\right)},$

which means that the $f^{\varepsilon}$ is already concentrated at initial
time. This restriction is quite common in articles which follow a Hopf-Cole
transform approach: it is the case in all the references cited above [3, 14,
17, 8]. Let us point out that [17] deals with initial conditions concentrated
with respect to the adaptation variable $w$ as well, a condition which is
lifted in our result. In [6, 5], $f^{\varepsilon}$ is not initially
concentrated and we deal with general initial conditions, however this is
possible since convergences are in $L^{1}$ and $L^{2}$.

### Comments on the stategy.

Let us outline our strategy and the challenges in order to prove Theorem 1.4.
The main difficulty is induced by the drift $N$, which is not Lipschitz
according to assumption (1.4). Indeed, a common technique consists in
obtaining some uniform bounds with respect to $\varepsilon$ on the derivatives
of $\phi^{\varepsilon}$ by applying the Bernstein method (see [2] for a
general setting in which the Bernstein method applies and [3, 17] for some
application in particular contexts) and then to conclude on the convergence of
$\phi^{\varepsilon}$ with a compactness argument. However, this method, does
not seem to apply easily here since $N$ is not uniformly Lischitz and
therefore induces high-order terms with respect to $v$ in the equations on the
derivatives of $\phi^{\varepsilon}$. An alternate mean to carry out the proof
would be to use the method of half-relaxed limits introduced by Barles and
Perthame in [4] which applies without requiring any regularity estimates, at
the cost of loosing continuity and therefore uniqueness in the limit
$\varepsilon\rightarrow 0$. To recover uniqueness, we add the additional
constraint

$\phi(t,{\bm{x}},{\mathcal{V}},w)\,=\,0\,$

on the limiting problem. But proving that the limit provided by the method of
half-relaxed limits satisfies this constraint brings us back to our initial
unsolved problem since it requires regularity estimates on the derivatives of
$\phi^{\varepsilon}$. To bypass these difficulties, we choose another approach
which does not require regularity estimates and which has the advantage of
providing explicit convergence rates: instead of proving uniform estimates on
the derivatives of $\phi^{\varepsilon}$, we prove uniform estimates on the
first term in the expansion of $\phi^{\varepsilon}$ with respect to
$\varepsilon$. This is made possible since this first term takes into account
the non-linear fluctuations induced by $N$. Indeed, these non-linear
variations induced by $N$ are expected to be perturbations of order
$\varepsilon$, as it may be seen rewriting equation (1.15) on
$\phi^{\varepsilon}$ as follows

$\partial_{v}\left(-\frac{1}{2}\rho_{0}^{\varepsilon}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{2}+\varepsilon
n(v)-\phi^{\varepsilon}\right)\partial_{v}\phi^{\varepsilon}+\ldots=0,$

where the correction $n(v)$ is such that $n^{\prime}(v)=N(v)$ and where
”$\ldots$” gathers the lower order terms with respect to $v$ and $w$. Hence,
at least formally as $\varepsilon$ goes to zero, we expect

$\phi^{\varepsilon}\underset{\varepsilon\rightarrow
0}{\approx}-\frac{1}{2}\rho_{0}^{\varepsilon}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{2}+\varepsilon
n(v).$

Therefore, our strategy consists in considering the first term
$\phi^{\varepsilon}_{1}$ in the expansion of $\phi^{\varepsilon}$ with respect
to $\varepsilon$, that is

$\phi^{\varepsilon}=-\frac{1}{2}\rho_{0}^{\varepsilon}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{2}+\varepsilon\phi^{\varepsilon}_{1},$

and to prove that it looks like $n(v)$ plus some uniformly bounded with
respect to $\varepsilon$ lower order terms which are induced by the globally
Lipschitz coefficient in equation (1.15). To do so, we identify a formal
equivalent $\overline{\phi^{\varepsilon}_{1}}$ of $\phi^{\varepsilon}_{1}$ as
$\varepsilon$ goes to zero, which displays $n(v)$ and which depends on
$\varepsilon$ only through the macroscopic quantities
$\left({\mathcal{V}}^{\varepsilon},{\mathcal{W}}^{\varepsilon}\right)$ (see
Section 3). In Lemma 3.1, we look for super and sub-solutions to the equation
solved by $\phi^{\varepsilon}_{1}$ with the form
$\chi_{\pm}=\overline{\phi^{\varepsilon}_{1}}\pm\psi$. Once this is done, we
apply a comparison principle in order to obtain
$\chi_{-}\leq\phi^{\varepsilon}_{1}\leq\chi_{+}$. The last step consists in
proving that the equivalents
$-\frac{1}{2}\rho_{0}^{\varepsilon}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{2}$
and $\overline{\phi^{\varepsilon}_{1}}$, which depend on $\varepsilon$ only
through the macroscopic quantities
$\left({\mathcal{V}}^{\varepsilon},{\mathcal{W}}^{\varepsilon}\right)$ have
convergence and boundedness properties. This is done relying on previous
results which ensure that the macroscopic quantities converge (see Theorem
2.3).

### Comments and perspectives.

Two major perspectives arise from our work. First, our result holds in a
perturbative setting in the sense that we need the initial data to be
concentrated in order for our result to hold true. It would be interesting to
lift this constraint and to treat a general set of initial data without
requiring any well-preparedness condition. To achieve this, one possibility
would be to adapt the strategy adopted in [5], where we introduced a time
dependant scaling in order to take into account the initial layer induced by
the ill-preparedness of the initial data. Second, our result does not describe
the limiting dynamics with respect to the adaptation variable, which is a
”slow variable” in our problem. Indeed, due to our starting point, ansatz
(1.14), our result is doomed to only describe the limiting dynamics with
respect to $v$, the fast variable of our problem. It would be a great but
challenging improvement to carry out the convergence analysis of the first
corrective term in the Hilbert expansion of $\phi^{\varepsilon}$ with respect
to the scaling parameter $\varepsilon$ in order to fill this gap.

### Structure of the paper.

The remaining part of this article is organized as follows: in Section 2, we
prove some regularity estimates for equation (1.1) in order to make our
further computations rigorous: this is the object of Lemma 2.1 and Proposition
2.2. In Theorem 2.3 and Proposition 2.4, we also we recall and prove some
convergence results on the macroscopic quantities
${\mathcal{U}}^{\varepsilon}$ and ${\mathcal{E}}\left(f^{\varepsilon}\right)$.
Then we pass to Section 3, which is dedicated to the proof of Theorem 1.4. The
proof relies on the key Lemma 3.1, in which we construct sub- and super-
solution for equation (1.15) on $\phi^{\varepsilon}$.

##  2\. Preliminary estimates

######  Lemma 2.1.

For all function $\varphi\in{\mathscr{C}}^{2}\left({\mathbb{R}}^{2}\right)$
with polynomial growth of order $r\geq 0$, that is

$\left|\varphi(\bm{u})\right|+\left|\nabla_{\bm{u}}\varphi(\bm{u})\right|+\left|\nabla^{2}_{\bm{u}}\varphi(\bm{u})\right|\underset{|\bm{u}|\rightarrow+\infty}{=}O\left(|\bm{u}|^{r}\right),$

the function
$\displaystyle\left(\left(t,{\bm{x}}\right)\mapsto\int_{{\mathbb{R}}^{2}}\varphi(\bm{u})f^{\varepsilon}\left(t,{\bm{x}},\bm{u}\right)\mathrm{d}\bm{u}\right)$
is continuous and has continuous time derivative over ${\mathbb{R}}^{+}\times
K$. In particular, the macroscopic quantities
$\displaystyle\mathcal{V}^{\varepsilon}$ and
$\displaystyle\mathcal{W}^{\varepsilon}$ given by (1.2) and the error
$\displaystyle{\mathcal{E}}\left(f^{\varepsilon}\right)$ given by (1.12) are
continuous and have continuous time derivatives.

###### Proof.

Take such a function $\displaystyle\varphi$. To simplify notations we write

$\varphi\left(f^{\varepsilon}\right):\left(t,{\bm{x}}\right)\mapsto\int_{{\mathbb{R}}^{2}}\varphi(\bm{u})f^{\varepsilon}\left(t,{\bm{x}},\bm{u}\right)\mathrm{d}\bm{u}.$

We start by proving that $\displaystyle\varphi\left(f^{\varepsilon}\right)$ is
continuous. This is straightforward according to the following estimate
obtained after applying Cauchy-Schwarz inequality

$\left|\varphi\left(f^{\varepsilon}\right)(t,{\bm{x}})-\varphi\left(f^{\varepsilon}\right)(s,\bm{y})\right|\leq\left(\varphi^{2}\left(f^{\varepsilon}\right)(t,{\bm{x}})+\varphi^{2}\left(f^{\varepsilon}\right)(s,\bm{y})\right)^{1/2}\|f^{\varepsilon}\left(t,{\bm{x}}\right)-f^{\varepsilon}\left(s,\bm{y}\right)\|_{L^{1}({\mathbb{R}}^{2})}^{1/2}.$

Indeed, according to Theorem 1.3, $\varphi^{2}\left(f^{\varepsilon}\right)$ is
locally bounded with respect to $(t,{\bm{x}})$ since $f^{\varepsilon}$ has
exponential moments and $\varphi$ has polynomial growth. Therefore, we obtain
the result since $f^{\varepsilon}$ lies in
${\mathscr{C}}^{0}\left({\mathbb{R}}^{+}\times
K,L^{1}\left({\mathbb{R}}^{2}\right)\right)$ according to Definition 1.2.

We now prove that $\varphi\left(f^{\varepsilon}\right)$ has continuous time
derivative. We multiply equation (1.1) by $\varphi$ and integrate with respect
to $\bm{u}$. After an integration by part, this yields

$\partial_{t}\varphi\left(f^{\varepsilon}\right)=\xi_{1}\left(f^{\varepsilon}\right)+\left(\frac{1}{\varepsilon}\rho_{0}^{\varepsilon}{\mathcal{V}}^{\varepsilon}+\Psi*_{r}(\rho^{\varepsilon}_{0}\;{\mathcal{V}}^{\varepsilon})\right)\partial_{v}\varphi\left(f^{\varepsilon}\right)-\left(\frac{1}{\varepsilon}\rho_{0}^{\varepsilon}+\Psi*_{r}\rho^{\varepsilon}_{0}\right)\xi_{2}\left(f^{\varepsilon}\right),$

with

$\left\\{\begin{array}[]{ll}\displaystyle\xi_{1}(\bm{u})=\displaystyle\partial_{v}\varphi(\bm{u})\left(N(v)-w\right)+\partial_{w}\varphi(\bm{u})A(\bm{u})+\partial_{v}^{2}\varphi(\bm{u})\par,\\\\[11.00008pt]
\displaystyle\xi_{2}(\bm{u})=\displaystyle\partial_{v}\varphi(\bm{u})v\par.\end{array}\right.$

Functions
$\displaystyle\xi_{1}\left(f^{\varepsilon}\right),\xi_{2}\left(f^{\varepsilon}\right),\partial_{v}\varphi\left(f^{\varepsilon}\right),{\mathcal{V}}^{\varepsilon}$
are continuous according to the previous step. Furthermore, we obtain that
function $\Psi*_{r}(\rho^{\varepsilon}_{0}\;{\mathcal{V}}^{\varepsilon})$ and
$\Psi*_{r}\rho^{\varepsilon}_{0}$ are continuous using continuity of
${\mathcal{V}}^{\varepsilon}$ and $\rho_{0}^{\varepsilon}$ and the assumption
$\Psi\in{\mathscr{C}}^{0}\left(K_{{\bm{x}}},L^{1}\left(K_{{\bm{x}}^{\prime}}\right)\right)$.
This yields the result. ∎

We prove that when the initial data $f^{\varepsilon}_{0}$ is smooth, the
associated solution $f^{\varepsilon}$ to (1.1) is regular.

######  Proposition 2.2.

Under the assumptions of Theorem 1.3, suppose in addition that
$f^{\varepsilon}_{0}$ lies in
${\mathscr{C}}^{0}\left(K,{\mathscr{C}}^{\infty}_{c}\left({\mathbb{R}}^{2}\right)\right)$
and that $N$ meets the following assumptions

(2.1) $\sup_{|v|\geq
1}\left|N^{\prime}(v)\right||v|^{1-p}<+\infty,\qquad\sup_{|v|\geq
1}\left|N^{\prime\prime\prime}(v)\right||v|^{-p^{\prime}}<+\infty.$

where $p$ is given in assumption (1.4). Then the solution $f^{\varepsilon}$ to
equation (1.1) provided by Theorem 1.3 verifies

$f^{\varepsilon}\in L^{\infty}_{loc}\left({\mathbb{R}}^{+}\times
K,W^{2,1}\left({\mathbb{R}}^{2}\right)\right),\quad\partial_{t}f^{\varepsilon}\in
L^{\infty}_{loc}\left({\mathbb{R}}^{+}\times
K,L^{1}\left({\mathbb{R}}^{2}\right)\right).$

We postpone the proof to Appendix B: it is mainly technical and relies on
moment estimates on the derivatives of $f^{\varepsilon}$.

For self-consistency, we recall a result from [6] about the control of the
macroscopic quantities
$\displaystyle\left({\mathcal{V}}^{\varepsilon},{\mathcal{W}}^{\varepsilon}\right)$
defined by (1.2) and the error term
$\displaystyle\mathcal{E}\left(f^{\varepsilon}\right)$ defined by (1.12). We
also provide uniform estimates with respect to $\varepsilon$ for the moments
of $f^{\varepsilon}$ and for the relative energy given by

$\left\\{\begin{array}[]{l}\displaystyle
M_{q}\left[f^{\varepsilon}\right](t,{\bm{x}}):=\frac{1}{\rho_{0}^{\varepsilon}({\bm{x}})}\int_{{\mathbb{R}}^{2}}|\bm{u}|^{q}\par
f^{\varepsilon}(t,{\bm{x}},\bm{u})\mathrm{d}\bm{u},\\\\[11.00008pt]
\displaystyle
D_{q}\left[f^{\varepsilon}\right](t,{\bm{x}}):=\frac{1}{\rho_{0}^{\varepsilon}({\bm{x}})}\int_{{\mathbb{R}}^{2}}|v-{\mathcal{V}}^{\varepsilon}(t,{\bm{x}})|^{q}f^{\varepsilon}(t,{\bm{x}},\bm{u})\mathrm{d}\bm{u},\end{array}\right.$

where $q\geq 2$.

######  Theorem 2.3 ([6]).

Under assumptions (1.4)-(1.10) and under the additional assumptions of Theorem
1.3, consider the solutions $f^{\varepsilon}$ to (1.1) provided by Theorem 1.3
and the solution ${\mathcal{U}}$ to (1.13). Furthermore, define the initial
macroscopic error as

$\mathcal{E}_{\mathrm{mac}}=\left\|\mathcal{U}_{0}-\mathcal{U}_{0}^{\varepsilon}\right\|_{L^{\infty}(K)}+\|\rho_{0}-\rho_{0}^{\varepsilon}\|_{L^{\infty}(K)}.$

There exists
$\displaystyle(C,\varepsilon_{0})\in\left({\mathbb{R}}^{+}_{*}\right)^{2}$
such that

  1. (1)

for all $\varepsilon\leq\varepsilon_{0}$, it holds

$\left\|\mathcal{U}(t)-\mathcal{U}^{\varepsilon}(t)\right\|_{L^{\infty}(K)}\leq
C\min{\left(e^{Ct}\left(\mathcal{E}_{\mathrm{mac}}+\varepsilon\right),1\right)},\quad\quad\forall
t\in{\mathbb{R}}^{+},$

where ${\mathcal{U}}^{\varepsilon}$ and ${\mathcal{U}}$ are respectively given
by (1.2) and (1.13).

  2. (2)

For all $\varepsilon>0$ and all $q$ in $\displaystyle[2,2(p+p^{\prime})]$ it
holds

$M_{q}[f^{\varepsilon}](t,{\bm{x}})\leq
C,\quad\quad\forall(t,{\bm{x}})\in{\mathbb{R}}^{+}\times K,$

where exponent $p$ is given in assumption (1.4). In particular,
${\mathcal{U}}^{\varepsilon}$, $\partial_{t}\,{\mathcal{U}}^{\varepsilon}$ and
${\mathcal{E}}(f^{\varepsilon})$ are uniformly bounded with respect to both
$\displaystyle(t,{\bm{x}})\in{\mathbb{R}}^{+}\times K$ and $\varepsilon$,
where $\displaystyle\mathcal{E}$ is defined by (1.12).

  3. (3)

For all $\varepsilon>0$ and all $q$ in $\displaystyle[2,2(p+p^{\prime})]$ it
holds

$D_{q}[f^{\varepsilon}](t,{\bm{x}})\leq
C\left[D_{q}[f^{\varepsilon}](0,{\bm{x}})\exp\left(-qm_{*}\frac{t}{\varepsilon}\right)+\varepsilon^{\frac{q}{2}}\right],\quad\forall(t,{\bm{x}})\in{\mathbb{R}}^{+}\times
K.$

In this theorem, constants $C$ and $\varepsilon_{0}$ only depend on $m_{p}$,
$\mskip 1.5mu\overline{\mskip-1.5mu{m}\mskip-1.5mu}\mskip 1.5mu_{p}$, $m_{*}$
(see (1.8)-(1.10) ) and on the data of the problem $N$, $A$ and $\Psi$.

We now complement this result with an additional estimate on the derivative of
the error term, that will be used in a later proof.

######  Proposition 2.4.

For all $\varepsilon>0$ and all $(t,{\bm{x}})\in{\mathbb{R}}^{+}\times K$ we
have

$\left|\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{E}(f^{\varepsilon}\left(t,{\bm{x}},\cdot\right))\right|\leq
C\left[\left(D_{2}+D_{p^{\prime}+1}\right)[f^{\varepsilon}](0,{\bm{x}})\varepsilon^{-1}\exp\left(-2m_{*}\frac{t}{\varepsilon}\right)+1\right],$

which simplifies into

$\left|\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{E}(f^{\varepsilon}\left(t,{\bm{x}},\cdot\right))\right|\leq
C,$

as soon as assumption (1.17b) is fulfilled.

###### Proof.

We compute the derivative of $\mathcal{E}(f^{\varepsilon})$ taking the
difference between equation (1.1) multiplied by $N/\rho_{0}^{\varepsilon}$ and
integrated with respect to $\bm{u}$, and the first line of (1.11) multiplied
by $N^{\prime}({\mathcal{V}}^{\varepsilon})$. After integrating by part with
respect to $v$, it yields

(2.2)
$\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{E}(f^{\varepsilon}\left(t,{\bm{x}},\cdot\right))={\mathcal{A}}+{\mathcal{B}},$

where ${\mathcal{A}}$ and ${\mathcal{B}}$ are given by

$\left\\{\begin{array}[]{l}\displaystyle{\mathcal{A}}=-\frac{1}{\varepsilon}\int_{{\mathbb{R}}^{2}}N^{\prime}(v)\left(v-{\mathcal{V}}^{\varepsilon}(t,{\bm{x}})\right)f^{\varepsilon}(t,{\bm{x}},\bm{u})\mathrm{d}\bm{u},\\\\[11.00008pt]
\displaystyle{\mathcal{B}}=\frac{1}{\rho_{0}^{\varepsilon}}\int_{{\mathbb{R}}^{2}}\left[\left(N^{\prime}B^{\varepsilon}\par+\par
N^{\prime\prime}\right)f^{\varepsilon}\right](t,{\bm{x}},\bm{u})\mathrm{d}\bm{u}-N^{\prime}({\mathcal{V}}^{\varepsilon})\left(B^{\varepsilon}\left(t,{\bm{x}},{\mathcal{U}}^{\varepsilon}\right)+{\mathcal{E}}(f^{\varepsilon})\right).\end{array}\right.$

The main difficulty here consists in estimating the stiffer term
${\mathcal{A}}$: this is what we start with. According to the definition of
${\mathcal{V}}^{\varepsilon}$, we have

${\mathcal{A}}=-\frac{1}{\varepsilon}\int_{{\mathbb{R}}^{2}}\left(N^{\prime}(v)-N^{\prime}\left({\mathcal{V}}^{\varepsilon}\right)\right)\left(v-{\mathcal{V}}^{\varepsilon}\right)f^{\varepsilon}(t,{\bm{x}},\bm{u})\mathrm{d}\bm{u}.$

Therefore, relying on our regularity assumptions on $N$, assumption (1.5) and
item (2) of Theorem 2.3, which ensures that ${\mathcal{V}}^{\varepsilon}$ is
uniformly bounded with respect to both $(t,{\bm{x}})\in{\mathbb{R}}^{+}\times
K$ and $\varepsilon>0$, we obtain some constant $C$ such that

${\mathcal{A}}\leq\frac{C}{\varepsilon}\left(D_{p^{\prime}+1}[f^{\varepsilon}]+D_{2}[f^{\varepsilon}]\right),$

where we also used assumption (1.8) to bound $\rho_{0}^{\varepsilon}$. Hence,
we apply item (3) of Theorem 2.3 and deduce

${\mathcal{A}}\leq
C\left[\left(D_{2}+D_{p^{\prime}+1}\right)[f^{\varepsilon}](0,{\bm{x}})\varepsilon^{-1}\exp\left(-2m_{*}\frac{t}{\varepsilon}\right)+1\right],$

for all $(t,{\bm{x}})\in{\mathbb{R}}^{+}\times K$ and $\varepsilon>0$.

According to assumptions (1.4) and (1.5), ${\mathcal{B}}$ only displays
moments of $f^{\varepsilon}$ up to order $2(p+p^{\prime})$, which are
uniformly bounded with respect to both $(t,{\bm{x}})\in{\mathbb{R}}^{+}\times
K$ and $\varepsilon>0$, according to item (2) of Theorem 2.3. On top of that
both ${\mathcal{U}}^{\varepsilon}$ and
${\mathcal{E}}\left(f^{\varepsilon}\right)$ are also uniformly bounded
according to item (2) in Theorem 2.3. Furthermore, according to assumptions
(1.6) and (1.8) on $\Psi$ and $\rho_{0}^{\varepsilon}$,
$\Psi*_{r}\rho^{\varepsilon}_{0}$ is uniformly bounded as well. Therefore,
there exists a constant $C$ such that

${\mathcal{B}}\leq C,$

for all $(t,{\bm{x}})\in{\mathbb{R}}^{+}\times K$ and $\varepsilon>0$.

We obtain the expected result gathering the estimates obtained on
${\mathcal{A}}$ and ${\mathcal{B}}$. ∎

##  3\. Proof of Theorem 1.4

In this section, we derive uniform $L^{\infty}$ convergence estimates for the
solution $\phi^{\varepsilon}$ to equation (1.15). To do so, our strategy
consists in performing a Hilbert expansion of $\phi^{\varepsilon}$ with
respect to $\varepsilon$ and to prove that the higher order terms are
uniformly bounded with respect to $\varepsilon$. Denote by
$\phi^{\varepsilon}_{1}$ the correction of order $1$ in the expansion of
$\phi^{\varepsilon}$ in the regime of strong interactions

(3.1)
$\phi^{\varepsilon}=-\frac{1}{2}\rho_{0}^{\varepsilon}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{2}+\varepsilon\phi^{\varepsilon}_{1},$

Plugging this ansatz in (1.15), we find that $\phi^{\varepsilon}_{1}$ solves
the following equation

(3.2)
$\displaystyle\partial_{t}\phi^{\varepsilon}_{1}+\nabla_{\bm{u}}\phi^{\varepsilon}_{1}\cdot\mathbf{b}^{\varepsilon}+\mathrm{div}_{\bm{u}}\left[\mathbf{b}^{\varepsilon}\right]-\partial^{2}_{v}\phi^{\varepsilon}_{1}-\left|\partial_{v}\phi^{\varepsilon}_{1}\right|^{2}+\frac{1}{\varepsilon}\,\rho_{0}^{\varepsilon}\left(v-{\mathcal{V}}^{\varepsilon}\right)\partial_{v}\left(\phi^{\varepsilon}_{1}-\overline{\phi^{\varepsilon}_{1}}\right)=0,$

where $\overline{\phi^{\varepsilon}_{1}}$ is given by

$\overline{\phi^{\varepsilon}_{1}}\left(t,{\bm{x}},\bm{u}\right)=n(v)-n({\mathcal{V}}^{\varepsilon})-\left(v-{\mathcal{V}}^{\varepsilon}\right)\left(N({\mathcal{V}}^{\varepsilon})+(w-{\mathcal{W}}^{\varepsilon})+{\mathcal{E}}(f^{\varepsilon})+\frac{1}{2}\Psi*_{r}\rho_{0}^{\varepsilon}\left(v-{\mathcal{V}}^{\varepsilon}\right)\right),$

where $n$ is the primitive of $N$ given in assumption (1.17a). Keeping the
leading order, we expect that $\phi^{\varepsilon}_{1}$ will look like
$\overline{\phi^{\varepsilon}_{1}}$. Therefore, in the following lemma, we
look for sub and super-solutions to equation (3.2) with the form
$\overline{\phi^{\varepsilon}_{1}}+\psi$, where $\psi$ needs to be determined.
This is done in the following lemma, which is the keystone of our analysis.

######  Lemma 3.1.

Consider some positive constant $\alpha_{0}$ and define $\psi$ as follows

$\psi\left(t,{\bm{x}},\bm{u}\right)=\frac{\alpha_{0}}{2}\left|v-{\mathcal{V}}^{\varepsilon}(t,{\bm{x}})\right|^{2}+\frac{\alpha(t)}{2}\left|w-{\mathcal{W}}^{\varepsilon}(t,{\bm{x}})\right|^{2},$

where $\alpha$ is given by

$\alpha(t)=\alpha_{0}e^{2(|a|+b)t}+\frac{1}{|a|+b}\left(e^{2(|a|+b)t}-1\right).$

The following functions

$\chi_{+}=\overline{\phi^{\varepsilon}_{1}}+\psi+m,\qquad\chi_{-}=\overline{\phi^{\varepsilon}_{1}}-\psi-m.$

are respectively super and sub-solutions to equation (3.2), where
$\displaystyle\left(t\mapsto m(t)\right)$ is given by

$m(t)=m_{0}+C\,\exp{\left(6(a+b)t\right)},$

for all $m_{0}\in{\mathbb{R}}$ and where the constant $C$ only depends on
$\alpha_{0}$, the constants in (1.8)-(1.10) and (1.17b), and the data of the
problem $N$, $A$ and $\Psi$.

###### Proof.

The first step of the proof consists in proving that equation (3.2) evaluated
in $\overline{\phi^{\varepsilon}_{1}}$ stays uniformly bounded with respect to
$\varepsilon$. This corresponds to estimating the term ${\mathcal{A}}$ defined
as follows

(3.3)
${\mathcal{A}}\,=\,\partial_{t}\overline{\phi^{\varepsilon}_{1}}+\nabla_{\bm{u}}\overline{\phi^{\varepsilon}_{1}}\cdot\mathbf{b}^{\varepsilon}+\mathrm{div}_{\bm{u}}\left[\mathbf{b}^{\varepsilon}\right]-\partial^{2}_{v}\overline{\phi^{\varepsilon}_{1}}-\left|\partial_{v}\overline{\phi^{\varepsilon}_{1}}\right|^{2}\,.$

To this aim, we compute the derivatives of
$\overline{\phi^{\varepsilon}_{1}}$. We first notice that equation (1.11) on
${\mathcal{V}}^{\varepsilon}$ can be rewritten

(3.4)
$\partial_{t}{\mathcal{V}}^{\varepsilon}=B^{\varepsilon}(t,{\bm{x}},{\mathcal{U}}^{\varepsilon})+{\mathcal{E}}(f^{\varepsilon})\,,$

and that therefore $\overline{\phi^{\varepsilon}_{1}}$ can be rewritten

$\overline{\phi^{\varepsilon}_{1}}\left(t,{\bm{x}},\bm{u}\right)=\int_{{\mathcal{V}}^{\varepsilon}}^{v}\left(B^{\varepsilon}\left(t,{\bm{x}},\tilde{v},w\right)-\partial_{t}{\mathcal{V}}^{\varepsilon}\right)\,\mathrm{d}\tilde{v}\,,$

where $B^{\varepsilon}$ is defined in (1.3). Using these relations, we find

$\left\\{\begin{array}[]{llll}\displaystyle&\nabla_{\bm{u}}\overline{\phi^{\varepsilon}_{1}}&=&\begin{pmatrix}B^{\varepsilon}(t,{\bm{x}},\bm{u})-\partial_{t}{\mathcal{V}}^{\varepsilon}\\\\[8.99994pt]
\displaystyle-\left(v-{\mathcal{V}}^{\varepsilon}\right)\end{pmatrix}\,,\\\\[16.00008pt]
\displaystyle&\partial_{v}^{2}\overline{\phi^{\varepsilon}_{1}}&=&\mathrm{div}_{\bm{u}}\left[\mathbf{b}^{\varepsilon}\right]-\partial_{w}A(u)\,,\\\\[8.00003pt]
\displaystyle&\partial_{t}\,\overline{\phi^{\varepsilon}_{1}}&=&\partial_{t}{\mathcal{V}}^{\varepsilon}\left(\partial_{t}{\mathcal{V}}^{\varepsilon}-B^{\varepsilon}\left(t,{\bm{x}},{\mathcal{V}}^{\varepsilon},w\right)\right)-(v-{\mathcal{V}}^{\varepsilon})\partial_{t}^{2}{\mathcal{V}}^{\varepsilon}\,.\end{array}\right.$

Relying on the latter relations we rewrite ${\mathcal{A}}$ as follows

${\mathcal{A}}\,=\,\partial_{t}{\mathcal{V}}^{\varepsilon}\left(B^{\varepsilon}\left(t,{\bm{x}},\bm{u}\right)-B^{\varepsilon}\left(t,{\bm{x}},({\mathcal{V}}^{\varepsilon},w)\right)\right)-(v-{\mathcal{V}}^{\varepsilon})\partial_{t}^{2}{\mathcal{V}}^{\varepsilon}+\partial_{w}A(u)-(v-{\mathcal{V}}^{\varepsilon})A(u)\,.$

On the one hand, $B^{\varepsilon}$ is given by (1.3), it holds

$B^{\varepsilon}(t,{\bm{x}},\bm{u})-B^{\varepsilon}(t,{\bm{x}},({\mathcal{V}}^{\varepsilon},w))=N(v)-N({\mathcal{V}}^{\varepsilon})-\Psi*_{r}\rho^{\varepsilon}_{0}(v-{\mathcal{V}}^{\varepsilon})\,.$

On the other hand, using relation (3.4), we obtain

$\partial_{t}^{2}{\mathcal{V}}^{\varepsilon}=\partial_{t}\,{\mathcal{U}}^{\varepsilon}\cdot\nabla_{\bm{u}}B^{\varepsilon}(t,{\bm{x}},{\mathcal{U}}^{\varepsilon})+\frac{\mathrm{d}}{\mathrm{d}t}\,{\mathcal{E}}(f^{\varepsilon})\,,$

which according to (1.3) and equation (1.11) on ${\mathcal{W}}^{\varepsilon}$
can be rewritten

$\partial_{t}^{2}{\mathcal{V}}^{\varepsilon}=\partial_{t}\,{\mathcal{V}}^{\varepsilon}\left(N^{\prime}\left({\mathcal{V}}^{\varepsilon}\right)-\Psi*_{r}\rho_{0}^{\varepsilon}\right)-A\left({\mathcal{U}}^{\varepsilon}\right)+\frac{\mathrm{d}}{\mathrm{d}t}\,{\mathcal{E}}(f^{\varepsilon})\,.$

Therefore, ${\mathcal{A}}$ may be expressed as follows

${\mathcal{A}}\,=\,\partial_{t}{\mathcal{V}}^{\varepsilon}\left(N(v)-N\left({\mathcal{V}}^{\varepsilon}\right)\right)-\left(v-{\mathcal{V}}^{\varepsilon}\right)\left(\partial_{t}{\mathcal{V}}^{\varepsilon}N^{\prime}\left({\mathcal{V}}^{\varepsilon}\right)+\left(A(u)-A({\mathcal{U}}^{\varepsilon})\right)+\frac{\mathrm{d}}{\mathrm{d}t}\,{\mathcal{E}}(f^{\varepsilon})\right)\,+\,\partial_{w}A\,.$

On the one hand, $\partial_{t}{\mathcal{V}}^{\varepsilon}$ and
${\mathcal{V}}^{\varepsilon}$ are uniformly bounded according to item (2) in
Theorem 2.3. On the other hand, we apply Proposition 2.4 which ensures that
under the smallness assumption (1.17b), the time derivative of
${\mathcal{E}}\left(f^{\varepsilon}\right)$ is uniformly bounded as well.
Consequently, for all positive $\varepsilon$, it holds

$\left(\left|\partial_{t}{\mathcal{V}}^{\varepsilon}\right|+\left|\,{\mathcal{U}}^{\varepsilon}\right|+\left|\frac{\mathrm{d}}{\mathrm{d}t}\,{\mathcal{E}}(f^{\varepsilon})\right|\right)(t,{\bm{x}})\,\leq\,C\,,\quad\forall\,(t,{\bm{x}})\in{\mathbb{R}}^{+}\times
K\,,$

for some constant $C$ depending only on the constants in assumptions
(1.8)-(1.10), (1.17b) and on the data of the problem $N$, $A$ and $\Psi$.
Therefore, we deduce the following bound for ${\mathcal{A}}$

$\left|{\mathcal{A}}\right|\,\leq\,C\left(1+\left|v-{\mathcal{V}}^{\varepsilon}\right|+\left|v-{\mathcal{V}}^{\varepsilon}\right|^{2}+\left|N(v)-N({\mathcal{V}}^{\varepsilon})\right|\right)+b\,\left|(v-{\mathcal{V}}^{\varepsilon})(w-{\mathcal{W}}^{\varepsilon})\right|\,.$

Then we apply Young’s inequality to bound the crossed term between $v$ and $w$
and use assumption (1.4) to bound $N(v)$. In the end, it yields

(3.5)
$\left|{\mathcal{A}}\right|\,\leq\,C\left(1+\left|v-{\mathcal{V}}^{\varepsilon}\right|^{p}\right)+\frac{1}{2}\,\left|w-{\mathcal{W}}^{\varepsilon}\right|^{2}\,.$

Building on this estimate, we can now pass to the heart of the proof and show
that $\chi_{+}$ and $\chi_{-}$ are respectively super- and sub-solutions to
(3.2). We evaluate equation (3.2) in $\chi_{+}$ and $\chi_{-}$ and obtain

$\displaystyle\partial_{t}\chi_{\pm}+\nabla_{\bm{u}}\chi_{\pm}\cdot\mathbf{b}^{\varepsilon}+\mathrm{div}_{\bm{u}}\left[\mathbf{b}^{\varepsilon}\right]-\partial^{2}_{v}\chi_{\pm}-\left|\partial_{v}\chi_{\pm}\right|^{2}+\frac{1}{\varepsilon}\,\rho_{0}^{\varepsilon}\left(v-{\mathcal{V}}^{\varepsilon}\right)\partial_{v}\chi_{\pm}={\mathcal{A}}\pm{\mathcal{B}}\pm
m^{\prime}(t)-\left|\partial_{v}\psi\right|^{2}\,,$

where ${\mathcal{A}}$ is given by (3.3) and ${\mathcal{B}}$ is given by

(3.6)
${\mathcal{B}}\,=\,\displaystyle\partial_{t}\psi+\nabla_{\bm{u}}\psi\cdot\mathbf{b}^{\varepsilon}-\partial^{2}_{v}\psi-2\,\partial_{v}\psi\,\partial_{v}\overline{\phi^{\varepsilon}_{1}}+\frac{1}{\varepsilon}\,\rho_{0}^{\varepsilon}\left(v-{\mathcal{V}}^{\varepsilon}\right)\partial_{v}\psi\,.$

In order to conclude that $\chi_{+}$ and $\chi_{-}$ are respectively sub- and
super-solutions to (3.2), it is sufficient to prove

${\mathcal{B}}+m^{\prime}(t)-\left|\partial_{v}\psi\right|^{2}-\left|{\mathcal{A}}\right|\geq
0\,.$

Therefore, we focus on proving the latter inequality. To begin with, we have

$\displaystyle{\mathcal{B}}-\left|\partial_{v}\psi\right|^{2}$
$\displaystyle=$
$\displaystyle-\alpha_{0}\left(v-{\mathcal{V}}^{\varepsilon}({\bm{x}})\right)\partial_{t}{\mathcal{V}}^{\varepsilon}+\frac{\alpha^{\prime}(t)}{2}\left|w-{\mathcal{W}}^{\varepsilon}(t,{\bm{x}})\right|^{2}-\alpha(t)\left(w-{\mathcal{W}}^{\varepsilon}(t,{\bm{x}})\right)\partial_{t}{\mathcal{W}}^{\varepsilon}$
$\displaystyle+\alpha_{0}\left(v-{\mathcal{V}}^{\varepsilon}\right)B^{\varepsilon}(t,{\bm{x}},\bm{u})+\alpha(t)\left(w-{\mathcal{W}}^{\varepsilon}\right)A(\bm{u})-\alpha_{0}^{2}\left(v-{\mathcal{V}}^{\varepsilon}\right)^{2}-\alpha_{0}+\frac{\rho_{0}^{\varepsilon}}{\varepsilon}\alpha_{0}\left(v-{\mathcal{V}}^{\varepsilon}\right)^{2}$
$\displaystyle-2\alpha_{0}\left(v-{\mathcal{V}}^{\varepsilon}\right)\left(B^{\varepsilon}(t,{\bm{x}},\bm{u})-\partial_{t}{\mathcal{V}}^{\varepsilon}\right)$
$\displaystyle=$
$\displaystyle\frac{\alpha^{\prime}(t)}{2}\left|w-{\mathcal{W}}^{\varepsilon}(t,{\bm{x}})\right|^{2}-\alpha_{0}\left(v-{\mathcal{V}}^{\varepsilon}\right)\left(B^{\varepsilon}(t,{\bm{x}},\bm{u})-\partial_{t}\,{\mathcal{V}}^{\varepsilon}\right)$
$\displaystyle+\alpha\left(w-{\mathcal{W}}^{\varepsilon}\right)\left(A(\bm{u})-A({\mathcal{U}}^{\varepsilon})\right)-\alpha_{0}+\left(\frac{\rho_{0}^{\varepsilon}}{\varepsilon}\alpha_{0}-\alpha_{0}^{2}\right)\left(v-{\mathcal{V}}^{\varepsilon}\right)^{2}$
$\displaystyle=$
$\displaystyle\frac{\alpha^{\prime}(t)}{2}\left|w-{\mathcal{W}}^{\varepsilon}(t,{\bm{x}})\right|^{2}-\alpha_{0}\left(v-{\mathcal{V}}^{\varepsilon}\right)\left(B^{\varepsilon}(t,{\bm{x}},\bm{u})-B^{\varepsilon}(t,{\bm{x}},{\mathcal{U}}^{\varepsilon})-{\mathcal{E}}(f^{\varepsilon})\right)$
$\displaystyle+\alpha\left(w-{\mathcal{W}}^{\varepsilon}\right)\left(a\left(v-{\mathcal{V}}^{\varepsilon}\right)-b\left(w-{\mathcal{W}}^{\varepsilon}\right)\right)-\alpha_{0}+\left(\frac{\rho_{0}^{\varepsilon}}{\varepsilon}\alpha_{0}-\alpha_{0}^{2}\right)\left(v-{\mathcal{V}}^{\varepsilon}\right)^{2}$
$\displaystyle\geq$
$\displaystyle\left(\frac{\alpha^{\prime}(t)}{2}-\alpha(|a|+b)\right)\left|w-{\mathcal{W}}^{\varepsilon}(t,{\bm{x}})\right|^{2}-\alpha_{0}\left(v-{\mathcal{V}}^{\varepsilon}\right)\left(B^{\varepsilon}(t,{\bm{x}},\bm{u})-B^{\varepsilon}(t,{\bm{x}},{\mathcal{U}}^{\varepsilon})-{\mathcal{E}}(f^{\varepsilon})\right)$
$\displaystyle-\alpha_{0}+\left(\frac{\rho_{0}^{\varepsilon}}{\varepsilon}\alpha_{0}-\alpha_{0}^{2}-\frac{|a|}{4}\alpha(t)\right)\left(v-{\mathcal{V}}^{\varepsilon}\right)^{2}\,.$

where we have used the Young inequality at the last line. Gathering the latter
estimate and (3.5) we obtain

$\displaystyle\displaystyle{\mathcal{B}}+m^{\prime}(t)-\left|\partial_{v}\psi\right|^{2}-\left|{\mathcal{A}}\right|$
$\displaystyle\geq\,$ $\displaystyle
m^{\prime}(t)+\left(\frac{\alpha^{\prime}}{2}-\alpha(|a|+b)\right)\left|w-{\mathcal{W}}^{\varepsilon}(t,{\bm{x}})\right|^{2}-\alpha_{0}\left(v-{\mathcal{V}}^{\varepsilon}\right)\left(B^{\varepsilon}(t,{\bm{x}},\bm{u})-B^{\varepsilon}(t,{\bm{x}},{\mathcal{U}}^{\varepsilon})-{\mathcal{E}}(f^{\varepsilon})\right)$
$\displaystyle-\alpha_{0}+\left(\frac{\rho_{0}^{\varepsilon}}{\varepsilon}\alpha_{0}-\alpha_{0}^{2}-\frac{|a|}{4}\alpha(t)\right)\left|v-{\mathcal{V}}^{\varepsilon}\right|^{2}-C\left(1+\left|v-{\mathcal{V}}^{\varepsilon}\right|^{p}\right)-\left|w-{\mathcal{W}}^{\varepsilon}\right|^{2}\,.$
$\displaystyle\geq\,$ $\displaystyle
m^{\prime}(t)+\left(\frac{\alpha^{\prime}}{2}-\alpha(|a|+b)-1\right)\left|w-{\mathcal{W}}^{\varepsilon}\right|^{2}-\alpha_{0}-\alpha_{0}\left(v-{\mathcal{V}}^{\varepsilon}\right)\left(N(v)-N({\mathcal{V}}^{\varepsilon})\right)$
$\displaystyle+\left(\alpha_{0}\Psi*_{r}\rho^{\varepsilon}_{0}({\bm{x}})+\frac{\rho_{0}^{\varepsilon}}{\varepsilon}\alpha_{0}-\frac{3}{2}\,\alpha_{0}^{2}-\frac{|a|}{4}\alpha(t)-\left(|a|+\frac{b}{2}+\frac{1}{2}\right)-\frac{1}{2}\,\alpha_{0}\right)\left(v-{\mathcal{V}}^{\varepsilon}\right)^{2}$
$\displaystyle+\alpha_{0}\left(v-{\mathcal{V}}^{\varepsilon}\right){\mathcal{E}}(f^{\varepsilon})-C\left(1+\left|v-{\mathcal{V}}^{\varepsilon}\right|^{p}\right)\,,$

where we have used Young inequality and the following relation

$B^{\varepsilon}(t,{\bm{x}},\bm{u})-B^{\varepsilon}(t,{\bm{x}},{\mathcal{U}}^{\varepsilon})=N(v)-N({\mathcal{V}}^{\varepsilon})-(w-{\mathcal{W}}^{\varepsilon})-\Psi*_{r}\rho^{\varepsilon}_{0}({\bm{x}})(v-{\mathcal{V}}^{\varepsilon})\,.$

Now we observe crucially that according to assumption (1.4), it holds

$\left(v-{\mathcal{V}}^{\varepsilon}\right)\left(N(v)-N({\mathcal{V}}^{\varepsilon})\right)\leq
C-\frac{1}{C}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{p+1}\,,$

for some constant $C$ great enough. Furthermore, choosing $\alpha$ such that
$\frac{\alpha^{\prime}}{2}-\alpha(|a|+b)-1=0$, that is

$\alpha(t)=\alpha_{0}\exp{\left(2(|a|+b)t\right)}+\frac{1}{|a|+b}\left(\exp{\left(2(|a|+b)t\right)}-1\right),$

we obtain

$\displaystyle\displaystyle{\mathcal{B}}+m^{\prime}(t)-\left|\partial_{v}\psi\right|^{2}-\left|{\mathcal{A}}\right|$
$\displaystyle\geq\,$ $\displaystyle
m^{\prime}(t)+\frac{1}{C}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{p+1}\,-\,C\,\left(1+\exp{\left(2\,(|a|\,+\,b)\,t\right)}\left|v\,-\,{\mathcal{V}}^{\varepsilon}\right|^{2}\,+\,\left|v\,-\,{\mathcal{V}}^{\varepsilon}\right|^{p}\,\right)\,.$

Then we find that

$\frac{1}{C}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{p+1}\,-\,C\,\left(1+\exp{\left(2\,(|a|\,+\,b)\,t\right)}\left|v\,-\,{\mathcal{V}}^{\varepsilon}\right|^{2}\,+\,\left|v\,-\,{\mathcal{V}}^{\varepsilon}\right|^{p}\,\right)\,\geq\,-\,\tilde{C}\,\exp{\left(\,6\,(|a|\,+\,b)\,t\,\right)}\,,$

and therefore conclude the proof. ∎

We are now able to proceed to the proof of Theorem 1.4. Indeed, relying on
Lemma 3.1 and applying a comparison principle to equation (3.2), we deduce
convergence estimates for the Hopf-Cole transform $\phi^{\varepsilon}$ of
$f^{\varepsilon}$.

###### Proof of Theorem 1.4.

All along this proof, we consider some positive constants $\alpha_{0}$ (to be
determined later on) and we work with the associated quantities $\psi$,
$\chi_{+}$ and $\chi_{-}$ defined in Proposition 3.1. We proceed in three
steps

  1. (1)

we prove that under our set of assumptions, it holds uniformly in
$\varepsilon$

$\chi_{-}(0,{\bm{x}},\bm{u})\leq\phi_{1}^{\varepsilon}(0,{\bm{x}},\bm{u})\leq\chi_{+}(0,{\bm{x}},\bm{u}),\quad\forall({\bm{x}},\bm{u})\in
K\times{\mathbb{R}}^{2},$

where $\chi_{+}$ and $\chi_{-}$ are defined in Proposition 3.1,

  2. (2)

we apply Lemma 3.1 and prove a comparison principle to deduce that the latter
inequality holds for all positive time, that is

$\chi_{-}(t,{\bm{x}},\bm{u})\leq\phi_{1}^{\varepsilon}(t,{\bm{x}},\bm{u})\leq\chi_{+}(t,{\bm{x}},\bm{u}),\quad\forall(t,{\bm{x}},\bm{u})\in{\mathbb{R}}^{+}\times
K\times{\mathbb{R}}^{2},$

  3. (3)

we conclude that $\phi^{\varepsilon}$ converges locally uniformly to
$-\frac{1}{2}\rho_{0}\left|v-{\mathcal{V}}\right|^{2}$.

We start with step (1). According to assumption (1.17a), for all positive
$\varepsilon$, it holds

$n(v)-C\left(1+|\bm{u}|^{2}\right)\leq\left(\phi^{\varepsilon}_{1}+\frac{1}{\varepsilon}\left(-\frac{1}{2}\rho_{0}^{\varepsilon}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{2}+\frac{1}{2}\rho_{0}\left|v-{\mathcal{V}}\right|^{2}\right)\right)(0,{\bm{x}},\bm{u})\leq
n(v)+C\left(1+|\bm{u}|^{2}\right),$

for all $({\bm{x}},\bm{u})\in K\times{\mathbb{R}}^{2}$. On the one hand,
according to assumptions (1.16), (1.8) and (1.9), it holds

$\frac{1}{\varepsilon}\left|-\frac{1}{2}\rho_{0}^{\varepsilon}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{2}+\frac{1}{2}\rho_{0}\left|v-{\mathcal{V}}\right|^{2}\right|(0,{\bm{x}},\bm{u})\leq
C\left(1+|\bm{u}-{\mathcal{U}}^{\varepsilon}_{0}|^{2}\right),$

for all $({\bm{x}},\bm{u})\in K\times{\mathbb{R}}^{2}$, for some constant $C$
depending only on the initial condition $f^{\varepsilon}_{0}$ (only through
the constants appearing in assumptions (1.8), (1.9) and (1.16)). On the other
hand, according to assumptions (1.8) and (1.6),
$\Psi*_{r}\rho^{\varepsilon}_{0}$ is uniformly bounded with respect to both
${\bm{x}}\in K$ and $\varepsilon>0$. On top of that,
$\displaystyle{\mathcal{U}}^{\varepsilon}_{0}$ and
$\displaystyle{\mathcal{E}}\left(f^{\varepsilon}_{0}\right)$ are also
uniformly bounded with respect to both ${\bm{x}}\in K$ and $\varepsilon>0$
according to assumptions (1.4) and (1.9). Therefore, according to the
definition of $\overline{\phi^{\varepsilon}_{1}}$ (see below (3.2)) and since
$N$ is continuous, it holds

$\left|n-\overline{\phi^{\varepsilon}_{1}}\right|(0,{\bm{x}},\bm{u})\leq
C\left(1+|\bm{u}-{\mathcal{U}}^{\varepsilon}_{0}|^{2}\right),$

for all $({\bm{x}},\bm{u})\in K\times{\mathbb{R}}^{2}$, for some constant $C$
depending on the initial condition $f^{\varepsilon}_{0}$ (only through the
constants appearing in assumptions (1.8)-(1.9)) and $N$. Gathering these
considerations and writing

$\phi^{\varepsilon}_{1}-\overline{\phi^{\varepsilon}_{1}}=\frac{1}{\varepsilon}\left(\phi^{\varepsilon}+\frac{1}{2}\rho_{0}\left|v-{\mathcal{V}}\right|^{2}-\varepsilon
n\right)+\frac{1}{\varepsilon}\left(-\frac{1}{2}\rho_{0}\left|v-{\mathcal{V}}\right|^{2}+\frac{1}{2}\rho_{0}^{\varepsilon}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{2}\right)+n-\overline{\phi^{\varepsilon}_{1}},$

we deduce that according to assumption (1.17a), for all positive
$\varepsilon$, it holds

$\overline{\phi^{\varepsilon}_{1}}(0,{\bm{x}},\bm{u})-C\left(1+|\bm{u}-{\mathcal{U}}^{\varepsilon}_{0}|^{2}\right)\leq\phi^{\varepsilon}_{1}(0,{\bm{x}},\bm{u})\leq\overline{\phi^{\varepsilon}_{1}}(0,{\bm{x}},\bm{u})+C\left(1+|\bm{u}-{\mathcal{U}}^{\varepsilon}_{0}|^{2}\right),$

for all $({\bm{x}},\bm{u})\in K\times{\mathbb{R}}^{2}$. Therefore, taking
$\alpha_{0}/2$ and $m(0)$ greater than $C$, we conclude step (1), indeed for
all positive $\varepsilon$ it holds

$\chi_{-}(0,{\bm{x}},\bm{u})\leq\phi_{1}^{\varepsilon}(0,{\bm{x}},\bm{u})\leq\chi_{+}(0,{\bm{x}},\bm{u}),\quad\forall({\bm{x}},\bm{u})\in
K\times{\mathbb{R}}^{2},$

where $\chi_{+}$ and $\chi_{-}$ are given in Proposition 3.1.

Let us now turn to step (2), which consists in proving that the latter
estimates holds true for all positive time. This step relies on two main
ingredients: first, we apply Lemma 3.1 which ensures that $\chi_{-}$ and
$\chi_{+}$ are respectively sub and super-solutions to equation (3.2), then we
apply a comparison argument whose proof is detailed in Appendix A and which is
divided into two steps: first we apply Lemma A.1, which ensures that the
quantities $f_{+}$ and $f_{-}$ defined for all
$(t,{\bm{x}},\bm{u})\in{\mathbb{R}}^{+}\times K\times{\mathbb{R}}^{2}$ as

$f_{\pm}(t,{\bm{x}},\bm{u})=\sqrt{\frac{\rho_{0}({\bm{x}})}{2\pi\varepsilon}}\exp{\left(-\frac{1}{2\,\varepsilon}\,\rho_{0}^{\varepsilon}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{2}+\chi_{\pm}\left(t,{\bm{x}},\bm{u}\right)\right)},$

are respectively super and sub-solutions to equation (1.1).

######  Remark 3.2.

Lemma A.1 indeed applies here since the regularity required on $f_{\pm}$ is
ensured by Lemma 2.1.

We come back to the proof, and notice that according to the previous step, it
holds

$f_{-}(0,{\bm{x}},\bm{u})\leq f^{\varepsilon}_{0}({\bm{x}},\bm{u})\leq
f_{+}(0,{\bm{x}},\bm{u}),\quad\forall({\bm{x}},\bm{u})\in
K\times{\mathbb{R}}^{2}.$

Therefore, according to Lemma A.2, we obtain

$f_{-}(t,{\bm{x}},\bm{u})\leq f_{0}^{\varepsilon}(t,{\bm{x}},\bm{u})\leq
f_{+}(t,{\bm{x}},\bm{u}),\quad\forall(t,{\bm{x}},\bm{u})\in{\mathbb{R}}^{+}\times
K\times{\mathbb{R}}^{2}.$

We deduce that the bound obtained in step (1), propagates through time, that
is, for all positive $\varepsilon$, it holds

$\chi_{-}(t,{\bm{x}},\bm{u})\leq\phi_{1}^{\varepsilon}(t,{\bm{x}},\bm{u})\leq\chi_{+}(t,{\bm{x}},\bm{u}),\quad\forall(t,{\bm{x}},\bm{u})\in{\mathbb{R}}^{+}\times
K\times{\mathbb{R}}^{2}.$

We can now turn to the last step and prove our main result. According to the
definition of $\phi^{\varepsilon}_{1}$ and the result of step (2), it holds

$-\frac{1}{2}\rho_{0}^{\varepsilon}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{2}+\frac{1}{2}\rho_{0}\left|v-{\mathcal{V}}\right|^{2}+\varepsilon\chi_{-}\leq\phi^{\varepsilon}+\frac{1}{2}\rho_{0}\left|v-{\mathcal{V}}\right|^{2}\leq-\frac{1}{2}\rho_{0}^{\varepsilon}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{2}+\frac{1}{2}\rho_{0}\left|v-{\mathcal{V}}\right|^{2}+\varepsilon\chi_{+},$

for all $\displaystyle(t,{\bm{x}},\bm{u})\in{\mathbb{R}}^{+}\times
K\times{\mathbb{R}}^{2}$. On the one hand, relying on item (1) in Theorem 2.3
and since the initial condition $f^{\varepsilon}_{0}$ meets the compatibility
assumption (1.16), there exists two positive constants $C$ and
$\varepsilon_{0}$ such that for all $\varepsilon\leq\varepsilon_{0}$, it holds

$\left|-\frac{1}{2}\rho_{0}^{\varepsilon}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{2}+\frac{1}{2}\rho_{0}\left|v-{\mathcal{V}}\right|^{2}\right|(t,{\bm{x}},\bm{u})\leq\varepsilon
Ce^{Ct}\left(1+|\bm{u}|^{2}\right),$

for all $(t,{\bm{x}},\bm{u})\in{\mathbb{R}}^{+}\times
K\times{\mathbb{R}}^{2}$, where constants $C$ and $\varepsilon_{0}$ only
depends on the data of our problem: $f^{\varepsilon}_{0}$ (only through the
constants appearing in assumptions (1.8)-(1.10) and (1.16)), $N$, $A$ and
$\Psi$. On the other hand, according to assumption (1.8) and (1.6),
$\Psi*_{r}\rho^{\varepsilon}_{0}$ is uniformly bounded with respect to both
${\bm{x}}\in K$ and $\varepsilon>0$. On top of that,
$\displaystyle{\mathcal{U}}^{\varepsilon}$ and
$\displaystyle{\mathcal{E}}\left(f^{\varepsilon}\right)$ are also uniformly
bounded with respect to both $(t,{\bm{x}})\in{\mathbb{R}}^{+}\times K$ and
$\varepsilon>0$ according item (2) in Theorem 2.3. Therefore, according to the
definitions of $\chi_{-}$ and $\chi_{+}$ (see Lemma 3.1) and since $N$ is
continuous, it holds

$\left|\chi_{\pm}-n\right|(t,{\bm{x}},\bm{u})\leq
Ce^{Ct}\left(1+|\bm{u}|^{2}\right),$

for all $(t,{\bm{x}},\bm{u})\in K\times{\mathbb{R}}^{2}$, where the constant
$C$ only depends on the data of our problem: $f^{\varepsilon}_{0}$ (only
through the constants appearing in assumptions (1.8)-(1.10)), $N$, $A$ and
$\Psi$. Hence, we deduce the result: for all $\varepsilon\leq\varepsilon_{0}$
it holds

$\varepsilon\left(n(v)-Ce^{Ct}\left(1+|\bm{u}|^{2}\right)\right)\leq\left(\phi^{\varepsilon}+\frac{1}{2}\rho_{0}\left|v-{\mathcal{V}}\right|^{2}\right)(t,{\bm{x}},\bm{u})\leq\varepsilon\left(n(v)+Ce^{Ct}\left(1+|\bm{u}|^{2}\right)\right),$

for all $\displaystyle(t,{\bm{x}},\bm{u})\in{\mathbb{R}}^{+}\times
K\times{\mathbb{R}}^{2}$, where constants $C$ and $\varepsilon_{0}$ only
depends on the data of our problem: $f^{\varepsilon}_{0}$ (only through the
constants appearing in assumptions (1.8)-(1.10) and (1.16)-(1.17b)), $N$, $A$
and $\Psi$. ∎

## Acknowledgment

The authors thank warmly Francis Filbet all the discussions without which it
would not have been possible to achieve this work, Philippe Laurençot for his
enlightening suggestions and comments and Sepideh Mirrahimi for her precious
explanations regarding Hamilton-Jacobi equations. A.B. gratefully acknowledge
the support of ANITI (Artificial and Natural Intelligence Toulouse Institute).
This project has received support from ANR ChaMaNe No: ANR-19-CE40-0024.

##  Appendix A Comparison principles

The object of this section is to prove a comparison principle for equation
(3.2) in order to complete step (2) in the proof of Theorem 1.4. More
precisely, we prove that if the quantity $\phi^{\varepsilon}_{1}$ defined by
(3.1) verifies

$\chi_{-}(0,{\bm{x}},\bm{u})\leq\phi_{1}^{\varepsilon}(0,{\bm{x}},\bm{u})\leq\chi_{+}(0,{\bm{x}},\bm{u}),\quad\forall({\bm{x}},\bm{u})\in
K\times{\mathbb{R}}^{2},$

where $\chi_{-}$ an $\chi_{+}$ are respectively sub and super-solutions to
(3.2), then the latter estimate propagates through time, that is

$\chi_{-}(t,{\bm{x}},\bm{u})\leq\phi_{1}^{\varepsilon}(t,{\bm{x}},\bm{u})\leq\chi_{+}(t,{\bm{x}},\bm{u}),\quad\forall(t,{\bm{x}},\bm{u})\in{\mathbb{R}}^{+}\times
K\times{\mathbb{R}}^{2}.$

Instead of working directly on equation (3.2), our strategy consists in
proving a comparison principle for the following linearized version of the
kinetic equation (1.1)

(A.1)
$\displaystyle\partial_{t}f+\mathrm{div}_{\bm{u}}\left[\mathbf{b}^{\varepsilon}f\right]-\partial^{2}_{v}f=\frac{1}{\varepsilon}\rho^{\varepsilon}_{0}\partial_{v}\left[(v-{\mathcal{V}}^{\varepsilon})f\right].$

Indeed, it is more convenient to work on equation (A.1) since we can rely on
the decaying properties of solutions to (1.1) provided by Theorem 1.3. From
the comparison principle on equation (A.1), we will easily deduce the expected
result. This approach is made possible since, according to the following
lemma, there is a direct link between sub and super-solutions to equations
(A.1) and (3.2)

######  Lemma A.1.

Consider some fixed $\varepsilon>0$. Under the assumptions of Theorem 1.3,
consider the solution $f^{\varepsilon}$ to equation (1.1) and its associated
macroscopic quantities
$\left({\mathcal{V}}^{\varepsilon},{\mathcal{W}}^{\varepsilon}\right)$
provided by Theorem 1.3. Furthermore, consider a strictly positive function
$f$ such that $f$, $\partial_{t}f$ and $\nabla^{2}_{\bm{u}}f$ lie in
${\mathscr{C}}^{0}\left({\mathbb{R}}^{+}\times K\times{\mathbb{R}}^{2}\right)$
and define $\chi$ as follows

$f(t,{\bm{x}},\bm{u})=\sqrt{\frac{\rho_{0}({\bm{x}})}{2\pi\varepsilon}}\exp{\left(\left(-\frac{1}{2\varepsilon}\rho_{0}^{\varepsilon}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{2}+\chi\right)\left(t,{\bm{x}},\bm{u}\right)\right)},\quad\forall(t,{\bm{x}},\bm{u})\in{\mathbb{R}}^{+}\times
K\times{\mathbb{R}}^{2},$

Then the following statements are equivalent

  1. (1)

$f$ is a super-solution (resp. sub-solution) to (A.1).

  2. (2)

$\chi$ is a super-solution (resp. sub-solution) to (3.2).

More precisely it holds

$\displaystyle\partial_{t}f+\mathrm{div}_{\bm{u}}\left[\mathbf{b}^{\varepsilon}f\right]-\partial^{2}_{v}f-\frac{1}{\varepsilon}\rho^{\varepsilon}_{0}\partial_{v}\left[(v-{\mathcal{V}}^{\varepsilon})f\right]=\left({\mathscr{H}}^{\varepsilon}_{1}+\frac{1}{\varepsilon}{\mathscr{J}}^{\varepsilon}_{1}\right)\left[\phi_{1}\right]f,$

for all $(t,{\bm{x}},\bm{u})\in{\mathbb{R}}^{+}\times
K\times{\mathbb{R}}^{2}$.

###### Proof.

We consider $f$ and $\chi$ as in Lemma A.1 and proceed in two steps. On the
one hand, plugging $f$ in equation (A.1), one has the following relation

$\displaystyle\partial_{t}f+\mathrm{div}_{\bm{u}}\left[\mathbf{b}^{\varepsilon}f\right]-\partial^{2}_{v}f-\frac{1}{\varepsilon}\rho^{\varepsilon}_{0}\partial_{v}\left[(v-{\mathcal{V}}^{\varepsilon})f\right]=\frac{1}{\varepsilon}f\,{\mathcal{A}}\,,$

for all $(t,{\bm{x}},\bm{u})\in{\mathbb{R}}^{+}\times K\times{\mathbb{R}}^{2}$
where ${\mathcal{A}}$ gathers the terms obtained plugging
$\displaystyle\phi\,:=\,-\frac{1}{2}\rho_{0}^{\varepsilon}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{2}+\varepsilon\chi$
into equation (1.15), that is

${\mathcal{A}}\,=\,\partial_{t}\phi+\nabla_{\bm{u}}\phi\cdot\mathbf{b}^{\varepsilon}+\varepsilon\mathrm{div}_{\bm{u}}\left[\mathbf{b}\right]-\partial^{2}_{v}\phi-\rho_{0}^{\varepsilon}-\frac{1}{\varepsilon}\left(\partial_{v}\left(\frac{1}{2}\rho_{0}^{\varepsilon}\left|v-{\mathcal{V}}^{\varepsilon}\right|^{2}+\phi\right)\partial_{v}\phi\right)\,.$

On the other hand, according to computations already detailed at the beginning
of Section 3, ${\mathcal{A}}$ also corresponds to the terms obtained plugging
$\chi$ into equation (3.2), that is

${\mathcal{A}}\,=\,\varepsilon\,\left(\partial_{t}\chi+\nabla_{\bm{u}}\chi\cdot\mathbf{b}^{\varepsilon}+\mathrm{div}_{\bm{u}}\left[\mathbf{b}^{\varepsilon}\right]-\partial^{2}_{v}\chi-\left|\partial_{v}\chi\right|^{2}+\frac{1}{\varepsilon}\,\rho_{0}^{\varepsilon}\left(v-{\mathcal{V}}^{\varepsilon}\right)\partial_{v}\left(\chi-\overline{\phi^{\varepsilon}_{1}}\right)\right)\,,$

for all $(t,{\bm{x}},\bm{u})\in{\mathbb{R}}^{+}\times
K\times{\mathbb{R}}^{2}$. Gathering these two relations, we obtain the result.
∎

It is now left to prove that a comparison principle holds for equation (A.1).
It is the object of the following Lemma

######  Lemma A.2.

Consider some fixed $\varepsilon>0$. Under the assumptions of Theorem 1.3,
consider the solution $f^{\varepsilon}$ to equation (1.1) and its associated
macroscopic quantities
$\left({\mathcal{V}}^{\varepsilon},{\mathcal{W}}^{\varepsilon}\right)$
provided by Theorem 1.3. Furthermore, consider a strictly positive function
$f$ such that $f$, $\partial_{t}f$ and $\nabla^{2}_{\bm{u}}f$ lie in
${\mathscr{C}}^{0}\left({\mathbb{R}}^{+}\times
K\times{\mathbb{R}}^{2}\right)$. Suppose that at initial time, it holds

$f^{\varepsilon}_{0}({\bm{x}},\bm{u})\leq
f(0,{\bm{x}},\bm{u}),\quad\forall({\bm{x}},\bm{u})\in
K\times{\mathbb{R}}^{2},$

and that $f$ is super-solution to equation (A.1), that is

$0\leq\displaystyle\partial_{t}f+\mathrm{div}_{\bm{u}}\left[\mathbf{b}^{\varepsilon}f\right]-\partial^{2}_{v}f-\frac{1}{\varepsilon}\rho^{\varepsilon}_{0}\partial_{v}\left[(v-{\mathcal{V}}^{\varepsilon})f\right]$

for all $(t,{\bm{x}},\bm{u})\in{\mathbb{R}}^{+}\times
K\times{\mathbb{R}}^{2}$. Then it holds

$f^{\varepsilon}(t,{\bm{x}},\bm{u})\leq
f(t,{\bm{x}},\bm{u}),\quad\forall(t,{\bm{x}})\in{\mathbb{R}}^{+}\times
K,\text{a.e. in }\bm{u}\in{\mathbb{R}}^{2}.$

Furthermore, the latter statement is also true if we replace ”super-solution”
by ”sub-solution” and the symbol ”$\geq$” by ”$\leq$”.

###### Proof.

Since the proof of the comparison principle for super and sub-solutions is the
same, we only detail it for super-solutions. Let us first suppose that
$f^{\varepsilon}_{0}$ lies in
${\mathscr{C}}^{0}\left(K,{\mathscr{C}}^{\infty}_{c}\left({\mathbb{R}}^{2}\right)\right)$
so that, according to Proposition 2.2, $f^{\varepsilon}$ is a classical
solution to (1.1). Then it holds

$\displaystyle\partial_{t}(f^{\varepsilon}-f)+\mathrm{div}_{\bm{u}}\left[\mathbf{b}^{\varepsilon}(f^{\varepsilon}-f)\right]-\partial^{2}_{v}(f^{\varepsilon}-f)-\frac{1}{\varepsilon}\rho^{\varepsilon}_{0}\partial_{v}\left[(v-{\mathcal{V}}^{\varepsilon})(f^{\varepsilon}-f)\right]\leq
0$

$a.e.$ with respect to $(t,{\bm{x}},\bm{u})\in{\mathbb{R}}^{+}\times
K\times{\mathbb{R}}^{2}$. Therefore, multiplying the latter relation by
$\displaystyle\mathds{1}_{f^{\varepsilon}\geq f}$, we deduce that in the weak
sense, it holds

$\displaystyle\partial_{t}(f^{\varepsilon}-f)_{+}+\mathrm{div}_{\bm{u}}\left[\mathbf{b}^{\varepsilon}(f^{\varepsilon}-f)_{+}\right]-\partial^{2}_{v}(f^{\varepsilon}-f)_{+}-\frac{1}{\varepsilon}\rho^{\varepsilon}_{0}\partial_{v}\left[(v-{\mathcal{V}}^{\varepsilon})(f^{\varepsilon}-f)_{+}\right]\leq
0$

where $(\cdot)_{+}$ stands for the positive part and is defined by
$(\cdot)_{+}=(|\cdot|+id_{{\mathbb{R}}})/2$. In order to derive the latter
relation, we follow a classical procedure which we do not detail here and
which consists in regularizing the positive part $(\cdot)_{+}$. Then we
integrate the latter relation with respect to time and $\bm{u}$ and obtain

$\int_{{\mathbb{R}}^{2}}(f^{\varepsilon}-f)_{+}(t,{\bm{x}},\bm{u})\mathrm{d}\bm{u}\leq\int_{{\mathbb{R}}^{2}}(f^{\varepsilon}-f)_{+}(0,{\bm{x}},\bm{u})\mathrm{d}\bm{u}.$

The latter computations are justified since $0\leq(f^{\varepsilon}-f)_{+}\leq
f^{\varepsilon}$ and since, according to Theorem 1.3, $f^{\varepsilon}$ has
moments up to any order with respect to $\bm{u}$. Since $f^{\varepsilon}\leq
f$ at time $t=0$, we deduce the result.

To conclude the proof, we lift the condition
$f^{\varepsilon}_{0}\in{\mathscr{C}}^{0}\left(K,{\mathscr{C}}^{\infty}_{c}\left({\mathbb{R}}^{2}\right)\right)$
using the
${\mathscr{C}}^{0}\left(K,L^{1}\left({\mathbb{R}}^{2}\right)\right)$-continuity
of equation (1.1) with respect to the initial data (see [6, Section A.2] for a
detailed proof of this continuity result) and following a density argument. ∎

##  Appendix B Regularity estimates

In this section, we derive regularity estimates for the solution
$f^{\varepsilon}$ to equation (1.1) and therefore prove Proposition 2.2. The
main difficulty here consists in dealing with the contribution due to the non-
linear drift $N$. We manage to bypass this difficulty by estimating the norm
of $f^{\varepsilon}$ and its derivatives in the following weighted $L^{1}$
spaces

$L^{1}(\omega_{q})=\left\\{f:{\mathbb{R}}^{2}\mapsto{\mathbb{R}},\int_{{\mathbb{R}}^{2}}|f|\omega_{q}(\bm{u})\mathrm{d}\bm{u}<+\infty\right\\},$

where $\displaystyle\omega_{q}(\bm{u})=1+|\bm{u}|^{q}$, for $q\geq 2$. The
first step consists in estimating the norm of $f^{\varepsilon}$ in
$L^{1}(\omega_{q})$. This step relies on previous results obtained in [6].
Then we adapt these computations to evaluate the norm of the derivatives of
$f^{\varepsilon}$ in $L^{1}(\omega_{q})$. Indeed, the derivatives of
$f^{\varepsilon}$ solve equation (1.1) with additional source terms whose
contribution can be controlled thanks to the confining properties of the non-
linear drift $N$. Let us outline the strategy in the case of the first order
derivatives: equation (1.1) on $f^{\varepsilon}$ reads as follows

$\partial_{t}f^{\varepsilon}={\mathscr{A}}^{\varepsilon}f^{\varepsilon},$

where operator ${\mathscr{A}}^{\varepsilon}$ is given by

${\mathscr{A}}^{\varepsilon}f=\partial^{2}_{v}f+\frac{1}{\varepsilon}\rho^{\varepsilon}_{0}\partial_{v}\left[(v-{\mathcal{V}}^{\varepsilon})f\right]-\mathrm{div}_{\bm{u}}\left[\mathbf{b}^{\varepsilon}f\right].$

Relying the arguments developed in the proof of [6, Proposition 3.1] it holds

######  Lemma B.1.

Consider some fixed $\varepsilon$ and some $q\geq 2$. Under the assumptions of
Theorem 1.3, consider the operator ${\mathscr{A}}^{\varepsilon}$ associated to
the solution $f^{\varepsilon}$ to equation (1.1). There exists a positive
constant $C$ such that for all ${\bm{x}}\in K$, it holds

$\int_{{\mathbb{R}}^{2}}{\text{\rm sgn
}}{(f)}{\mathscr{A}}^{\varepsilon}\left(f\right)\omega_{q}(\bm{u})\mathrm{d}\bm{u}\leq
C\|f\|_{L^{1}\left(\omega_{q}\right)}+q\int_{{\mathbb{R}}^{2}}\mathds{1}_{|v|\geq
1}\frac{N(v)}{v}|v|^{q}f(\bm{u})\mathrm{d}\bm{u},$

for all function $f$ lying in $W^{2,1}(\omega_{p+q})$.

As a direct consequence of Lemma B.1, we deduce that for any smooth initial
data $f^{\varepsilon}_{0}$, the solution $f^{\varepsilon}$ to equation (1.1)
provided by Theorem 1.3 lies in $L^{\infty}_{loc}\left({\mathbb{R}}^{+}\times
K,L^{1}\left(\omega_{q}\right)\right)$ for all exponent $q\geq 2$. Indeed,
since $N$ meets the confining assumption (1.4) we have:
$\displaystyle\mathds{1}_{|v|\geq 1}N(v)/v\leq C$ for some constant $C$.
Therefore, multiplying equation (1.1) by ${\text{\rm sgn
}}{\left(f^{\varepsilon}\right)}\omega_{q}(\bm{u})$, integrating with respect
to $\bm{u}$ and applying Lemma B.1, we obtain that for all exponent $q$, it
holds

$\frac{\mathrm{d}}{\mathrm{d}t}\|f^{\varepsilon}\|_{L^{1}\left(\omega_{q}\right)}\leq
C\|f^{\varepsilon}\|_{L^{1}\left(\omega_{q}\right)}.$

Hence, applying Gronwall’s lemma and taking the supremum over all ${\bm{x}}\in
K$, we deduce

(B.1)
$\|f^{\varepsilon}(t)\|_{L^{\infty}\left(K,L^{1}\left(\omega_{q}\right)\right)}\leq
e^{Ct}\|f^{\varepsilon}_{0}\|_{L^{\infty}\left(K,L^{1}\left(\omega_{q}\right)\right)},$

for all $t\in{\mathbb{R}}^{+}$. We follow the same strategy for derivatives of
$f^{\varepsilon}$: writing
$\displaystyle(g,h)=\left(\partial_{v}f^{\varepsilon},\partial_{w}f^{\varepsilon}\right)$
and differentiating equation (1.1) with respect to $v$ and $w$ we obtain

(B.2)
$\left\\{\begin{array}[]{ll}\displaystyle\partial_{t}g={\mathscr{A}}^{\varepsilon}g-\left(N^{\prime}-\Psi*_{r}\rho_{0}^{\varepsilon}-\frac{1}{\varepsilon}\rho_{0}^{\varepsilon}\right)g-N^{\prime\prime}f^{\varepsilon}-ah\par,\\\\[11.00008pt]
\displaystyle\partial_{t}h={\mathscr{A}}^{\varepsilon}h+g+bh.\end{array}\right.$

Therefore, $g$ and $h$ solve the same equation as $f^{\varepsilon}$ with
additional an high order term due to the non-linear drift $N$. We control this
additional term thanks to the confining properties of $N$.

###### Proof of Proposition 2.2.

Let us consider some initial data $f^{\varepsilon}_{0}$ lying in
${\mathscr{C}}^{0}\left(K,{\mathscr{C}}^{\infty}_{c}\left({\mathbb{R}}^{2}\right)\right)$
and the associated solution $f^{\varepsilon}$ to (1.1) provided by Theorem
1.3. We start by proving that $f^{\varepsilon}$ lies in $\displaystyle
L^{\infty}_{loc}\left({\mathbb{R}}^{+}\times
K,W^{1,1}\left({\mathbb{R}}^{2}\right)\right)$. We fix some ${\bm{x}}\in K$,
some exponent $q\geq 2$ and integrate with respect to $\bm{u}$ the sum between
the first equation in (B.2) multiplied by ${\text{\rm sgn
}}{(g)}\omega_{q}(\bm{u})$ and the second multiplied by ${\text{\rm sgn
}}{(h)}\omega_{q}(\bm{u})$. According to Lemma B.1, assumptions (1.6) on
$\Psi$, (1.8) on $\rho_{0}^{\varepsilon}$ and (1.5) on $N$, we obtain

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(\|g\|_{L^{1}\left(\omega_{q}\right)}+\|h\|_{L^{1}\left(\omega_{q}\right)}\right)\leq$
$\displaystyle
C\left(\|g\|_{L^{1}\left(\omega_{q}\right)}+\|h\|_{L^{1}\left(\omega_{q}\right)}+\|f^{\varepsilon}\|_{L^{1}\left(\omega_{p^{\prime}}\right)}\right)$
$\displaystyle+\int_{{\mathbb{R}}^{2}}\left(q\mathds{1}_{|v|\geq
1}\frac{N(v)}{v}-N^{\prime}(v)\right)|v|^{q}g(\bm{u})\mathrm{d}\bm{u},$

where $p^{\prime}$ is given in assumption (1.5). Since $N$ meets assumptions
(1.4) and (2.1), we deduce that for $q$ great enough, it holds

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(\|g\|_{L^{1}\left(\omega_{q}\right)}+\|h\|_{L^{1}\left(\omega_{q}\right)}\right)\leq
C\left(\|g\|_{L^{1}\left(\omega_{q}\right)}+\|h\|_{L^{1}\left(\omega_{q}\right)}+\|f^{\varepsilon}\|_{L^{1}\left(\omega_{p^{\prime}}\right)}\right).$

Therefore, applying Gronwall’s lemma, taking the supremum over all
${\bm{x}}\in K$ and replacing
$\displaystyle\|f^{\varepsilon}\|_{L^{1}\left(\omega_{p^{\prime}}\right)}$
according to estimate (B.1), we deduce

$\displaystyle\|g(t)\|_{L^{\infty}\left(K,L^{1}\left(\omega_{q}\right)\right)}+\|h(t)\|$
${}_{L^{\infty}\left(K,L^{1}\left(\omega_{q}\right)\right)}\leq$
$\displaystyle
e^{Ct}\left(\|g_{0}\|_{L^{\infty}\left(K,L^{1}\left(\omega_{q}\right)\right)}+\|h_{0}\|_{L^{\infty}\left(K,L^{1}\left(\omega_{q}\right)\right)}+\|f^{\varepsilon}_{0}\|_{L^{\infty}\left(K,L^{1}\left(\omega_{q}\right)\right)}\right),$

for all time $t\in{\mathbb{R}}^{+}$. As a straightforward consequence, we
obtain the expected result: $\displaystyle f^{\varepsilon}\in
L^{\infty}_{loc}\left({\mathbb{R}}^{+}\times
K,W^{1,1}\left({\mathbb{R}}^{2}\right)\right)$.

We obtain that $\displaystyle f^{\varepsilon}\in
L^{\infty}_{loc}\left({\mathbb{R}}^{+}\times
K,W^{2,1}\left({\mathbb{R}}^{2}\right)\right)$ iterating the same procedure as
before but this time on the derivatives of $g$ and $h$ and using assumption
(2.1), which ensures that $N^{\prime\prime\prime}$ has polynomial growth.

To end with, we obtain $\displaystyle\partial_{t}f^{\varepsilon}\in
L^{\infty}_{loc}\left({\mathbb{R}}^{+}\times
K,L^{1}\left({\mathbb{R}}^{2}\right)\right)$ noticing that according to the
definition of ${\mathcal{A}}^{\varepsilon}$, it holds

$\|{\mathscr{A}}^{\varepsilon}f^{\varepsilon}\|_{L^{\infty}\left(K,L^{1}\left({\mathbb{R}}^{2}\right)\right)}\leq
C\|f^{\varepsilon}\|_{L^{\infty}\left(K,W^{2,1}\left(\omega_{q}\right)\right)},$

for $q$ great enough. Then we apply the previous estimates to the relation

$\partial_{t}f^{\varepsilon}={\mathscr{A}}^{\varepsilon}f^{\varepsilon}.$

∎

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