# Local well-posedness for a class of singular Vlasov equations

Thomas Chaub

###### Abstract

In this article we study a singular Vlasov system on the torus where the force
field has the smoothness of a (fractional) derivative $D^{\alpha}$ of the
density, where $\alpha>0$. We prove local well-posedness in Sobolev spaces
without restriction on the data. This is in sharp contrast with the case
$\alpha=0$ which is ill-posed in Sobolev spaces for general data.

###### Summary

  1. 1 Introduction and main result
    1. 1.1 Context and notations
    2. 1.2 Sketch of the proof
  2. 2 Beginning of the proof of Theorem 2
    1. 2.1 Preliminary lemmas
    2. 2.2 Setting up the bootstrap
  3. 3 Estimates of the density term
    1. 3.1 Introduction of the $f_{I,J}$
    2. 3.2 Straightening the transport vector field
    3. 3.3 Introduction of the average operator
    4. 3.4 Focus on the operator $K_{G}$
    5. 3.5 Conclusion of the estimates for $\rho_{\varepsilon}$
  4. 4 End of the proof of Theorem 2
  5. 5 Solutions for the limit problem
    1. 5.1 Uniqueness of the solution
    2. 5.2 Existence of the solution

##  1 Introduction and main result

###  1.1 Context and notations

In this article we focus on a class of nonlinear singular Vlasov systems in
the torus $\mathbb{T}^{d}=\mathbb{R}^{d}/2\pi\mathbb{Z}^{d}$, a prototypal
example is

(1.1)
$\left\\{\begin{array}[]{ll}\partial_{t}{f}+v\cdot\nabla_{x}f+E\cdot\nabla_{v}f=0,\\\
\gamma E=-\nabla_{x}V,\\\
(-\Delta_{x})^{\alpha/2}V=\left(\rho-\rho_{0}\right),\\\
f(0,x,v)=f^{0}(x,v),\end{array}\right.$

where the parameter $\gamma$ will be taken to $1$ or $-1$, and the equation
will be respectively called repulsive or attractive. The function $f$ stands
for a distribution function in the domain $\mathbb{T}^{d}\times\mathbb{R}^{d}$
and may represent the distribution of electrons in a plasma, or the density of
stars in stellar dynamics. The density associated with $f$ will be denoted
$\rho=\int_{\mathbb{R}^{d}}f(t,x,v)dv$. The term
$\rho_{0}=\int_{\mathbb{R}^{d}\times\mathbb{T}^{d}}f^{0}(t,x,v)dvdx$
corresponds, in the repulsive setting, to the density of ions. In the
attractive setting, we remove the mean value of $\rho$ as a trick called the
Jeans swindle, which is mathematically relevant [8]. The initial condition,
$f^{0}$, can be taken non-negative, and $\alpha>0$ is a parameter.
The aim of this work is to give a local well-posedness theory in Sobolev
regularity for the general case $\alpha>0$. To introduce our main result, we
shall first introduce some notations.

For $k\in\mathbb{N}$, $r\in\mathbb{N}$, we introduce the weighted Sobolev
norms

${\left\lVert
f\right\rVert}_{\mathcal{H}_{r}^{k}}:=\left(\sum_{|\alpha|+|\beta|\leq
k}\int_{\mathbb{T}^{d}}\int_{\mathbb{R}^{d}}(1+|v|^{2})^{r}|\partial_{x}^{\alpha}\partial_{v}^{\beta}f|^{2}dvdx\right)^{1/2},$

where, for
$\alpha=(\alpha_{1},...,\alpha_{d}),\beta=(\beta_{1},...,\beta_{d})\in\mathbb{N}^{d}$,
we write

$|\alpha|=\sum_{i=1}^{d}\alpha_{i},\quad|\beta|=\sum_{i=1}^{d}\beta_{i},$

$\partial_{x}^{\alpha}:=\partial_{x_{1}}^{\alpha_{1}}...\partial_{x_{d}}^{\alpha_{d}},\quad\partial_{x}^{\beta}:=\partial_{x_{1}}^{\beta_{1}}...\partial_{x_{d}}^{\beta_{d}}.$

We will also use the classic Sobolev spaces. We will write $H_{x,v}^{k}$ (resp
$W^{k,\infty}_{x,v})$ the standard Sobolev space for the norm $L^{2}$ (resp
$L^{\infty}$) for functions depending on $(x,v)$, and $H_{x}^{k}$ for
functions only depending on $x$.

We will use the following notation and convention for the Fourier transform of
a function $U$

$\widehat{U}(k):=(2\pi)^{-d}\int_{\mathbb{T}^{d}}U(x)e^{-ik\cdot x}dx.$

Instead of (1.1), we consider a more general system under the form

(1.2)
$\left\\{\begin{array}[]{ll}\partial_{t}{f}+v\cdot\nabla_{x}f+E\cdot\nabla_{v}f=0,\\\
E=-\nabla_{x}\int_{\mathbb{T}^{d}}U(x-y)\left(\rho(y)-\rho_{0}\right)dy,\\\
f(0,x,v)=f^{0}(x,v),\end{array}\right.$

making the following assumptions for $U$ :

######  Assumptions 1 (A1).

$\widehat{U}(0)=0$ and there exists $\alpha>0$, $C_{\alpha}>0$ such that

(1.3) $\forall k\neq
0,\>k\in\mathbb{Z}^{d},\quad|\widehat{U}(k)|\leq\frac{C_{\alpha}}{|k|^{\alpha}}.$

In particular, by taking $\widehat{U}(k)=\pm\frac{1}{|k|^{\alpha}}$,
$\widehat{U}(0)=0$, we recover (1.1).
Note that we can also consider potentials $U=\displaystyle\sum_{i=1}^{p}U_{i}$
such that each $U_{i}$ satisfies (1.3) for $\alpha_{i}>0$. Then $U$ satisfies
(1.3) with $\alpha=\min(\alpha_{i})_{i\in[1,p]}$. Those types of potentials
may appear in some physical models in gravitation (see the discussion below
about Manev potentials).

We denote by $\lfloor x\rfloor$ the floor value of $x$ and

(1.4) $m_{0}=3+\frac{d}{2}+p_{0},\quad p_{0}=\lfloor\frac{d}{2}\rfloor+1,\quad
r_{0}=\max\left(d,2+\frac{d}{2}\right).$

Our main well-posedness result is the following

######  Theorem 1.

Let $f^{0}\in\mathcal{H}^{m}_{2r}$ with $m>m_{0}$, $2r>r_{0}$. Assuming (A1),
then there exists $T>0$ for which there exists a unique solution of the system
(1.2) with initial condition $f^{0}$ and such that
$f\in\mathcal{C}([0,T],\mathcal{H}^{m-1}_{2r})$, $\rho\in
L^{2}([0,T],H_{x}^{m})$.

Note that we consider both the attractive and repulsive case.

######  Remark 1.1.

Notice that in the above statement the solution $f$ is less regular than the
initial condition $f^{0}$. This was expected because of the nature of the
equation. Nevertheless, the fact that $f$ is in
$\mathcal{C}([0,T],\mathcal{H}^{m-1}_{2r})$ is not optimal. With sharper
estimates in the following proof, when $\alpha\leq 1$, $f$ is expected to be
in $\mathcal{C}([0,T],\mathcal{H}^{m-1+\alpha}_{2r})$, provided that we use
the fractional space $\mathcal{H}^{s}_{2r}$ when $s$ is not an integer (which
has not been defined here). For simplicity, we will only give the proof of
Theorem 1, as the gain of regularity we could obtain does not seem
fundamental.

Our results could be easily extended in the case of $\mathbb{R}^{d}$ instead
of $\mathbb{T}^{d}$ assuming that

$\forall\xi\in\mathbb{R}^{d},\>|\widehat{U}(\xi)|\leq\frac{C_{\alpha}}{1+|\xi|^{\alpha}}.$

The main interest of our theorem is for small values of $\alpha$. Indeed, if
we take $\alpha=2$ in (1.1), the system we obtain is the well-known Vlasov-
Poisson system, and a local well-posedness theory is very easy to get. We even
have global existence for physical dimensions $d\leq 3$ (see [12], [10],
[11]).

For $1\leq\alpha<2$, system (1.1) behaves at least like a Burgers type
equation and local well-posedness theory follows by standard energy estimates.
The case $\alpha=1$ in particular, has physical meaning. It was first
introduced by Manev as a correction of the Newtonian potential. The
interactions between Newtonian and Manev potentials have been studied (see
[9]), but the Manev potential on its own is also interesting. In the
attractive setting, it is called the Pure Stellar dynamic Manev system (PSM)
[2] , and it arises in stellar dynamics.
For $\frac{3}{4}\leq\alpha\leq 1$ local well-posedness was recently obtained
in (1.1) using the additional regularity provided by averaging lemmas in the
whole space in [3]. In addition, they give general conditions such that the
system (1.1), that they call the Vasov-Riesz system because of the
introduction of a Riesz type interaction, have finite-time singularity
formation for solutions.

Thus, the main contribution of this article corresponds to the case
$\alpha\in(0,\frac{3}{4})$. The well-posedness theory of the equation is more
challenging, because the apparent lost derivative in $x$ on the force field
$E$ is no longer compensated by regularization. In the critical case
$\alpha=0$ (which does not enter our framework), named the Vlasov-Dirac-Benney
system, it has been proved that the system is in general ill-posed in Sobolev
spaces [5]. Nevertheless, on the Torus and in the repulsive case, we can
ensure hypotheses to have the local well-posedness of the system. The first
results were obtained in dimension $d=1$ by Bardos and Besse [1]. They proved
that the Vlasov-Dirac-Benney system is locally well-posed provided that for
all $(t,x)$, there exists a function $m(t,x)$ such that $v\mapsto f(t,x,v)$
remains compactly supported and is increasing for $v\leq m(t,x)$, decreasing
for $v\geq m(t,x)$, in other words, one bump shaped functions. More recently,
Han-Kwan and Rousset [6] have proved that the Vlasov-Dirac-Benney system is
locally well-posed in any dimension for Sobolev regularity, provided that for
all $x$, the profile $v\mapsto f^{0}(x,v)$ satisfies a Penrose stability
condition, which means that if we define the Penrose function

(1.5)
$\mathcal{P}(\gamma,\tau,\eta,f)=1-\int_{0}^{+\infty}e^{-(\gamma+i\tau)s}\frac{i\eta}{1+|\eta|^{2}}\cdot(\mathcal{F}_{v}\nabla_{v}f)(\eta
s)ds,\quad\gamma>0,\tau\in\mathbb{R},\eta\in\mathbb{R}^{d}\backslash\\{0\\},$

$v\mapsto f^{0}(x,v)$ must satisfy the following condition, for some $c_{0}>0$

(1.6)
$\underset{(\gamma,\tau,\eta)\in(0,+\infty)\times\mathbb{R}\times\mathbb{R}^{d}\backslash\\{0\\}}{\inf}|\mathcal{P}(\gamma,\tau,\eta,f^{0})|\geq
c_{0}.$

###  1.2 Sketch of the proof

In order to prove Theorem 1, we shall define another system, which is a
regularized Vlasov-Poisson type system, that depends on a new parameter
$\varepsilon>0$

(1.7)
$\left\\{\begin{array}[]{ll}\partial_{t}{f_{\varepsilon}}+v\cdot\nabla_{x}f_{\varepsilon}+E_{\varepsilon}\cdot\nabla_{v}f_{\varepsilon}=0,\\\
E_{\varepsilon}=-\nabla_{x}(U*V_{\varepsilon}),\\\
-\varepsilon^{2}\Delta_{x}V_{\varepsilon}+V_{\varepsilon}=\int_{\mathbb{R}^{d}}f_{\varepsilon}(.,v)dv,\\\
f_{\varepsilon}(0,x,v)=f^{0}_{\varepsilon}(x,v).\end{array}\right.$

Taking formally the limit $\varepsilon\rightarrow 0$, we obtain the system
(1.2), which is exactly the system we want to study. The idea is to use the
well-posedness of the system (1.7) to get a family of functions
$(f_{\varepsilon})$ which satisfy uniform estimates. Then, using compactness
we extract (in a certain sense that will be defined later) a function $f$
which is a solution of the limit system (1.2). After that, we will show
uniqueness for the solution of (1.2) in the class of
$\\{f\in\mathcal{C}([0,T],\mathcal{H}^{m-1}_{2r})$, $\rho\in
L^{2}([0,T],H_{x}^{m})$}.
The first step is to show that we can get a family of functions
$(f_{\varepsilon})$ defined on a time interval $[0,T]$ with $T$ being
independent of $\varepsilon$. To do so, we introduce the following key
quantity

(1.8) $\mathcal{N}_{m,2r}(t,f):={\left\lVert
f\right\rVert}_{L^{\infty}\left([0,t],\mathcal{H}^{m-1}_{2r}\right)}+{\left\lVert\rho\right\rVert}_{L^{2}\left([0,t],H_{x}^{m}\right)},$

and we have the following theorem

######  Theorem 2.

Let $\alpha\in\mathbb{R^{*}_{+}}$. We assume that for all
$\varepsilon\in(0,1]$, $f^{0}_{\varepsilon}\in\mathcal{H}^{m}_{2r}$ with
$m>m_{0}$, $2r>r_{0}$ and that there exists $M_{0}>0$ so that for all
$\varepsilon\in(0,1]$, ${\left\lVert
f^{0}_{\varepsilon}\right\rVert}_{\mathcal{H}_{m}^{2r}}\leq M_{0}$.
Then there exist $T>0$, $R>0$ (independent of $\varepsilon$) and a unique
solution $f_{\varepsilon}\in\mathcal{C}([0,T],\mathcal{H}^{m}_{2r})$ of (1.7)
with

(1.9)
$\underset{\varepsilon\in(0,1]}{\sup}\mathcal{N}_{m,2r}(T,f_{\varepsilon})\leq
R.$

The proof of this theorem will rely on a bootstrap argument following the
strategy of [6]. For most of the proof, we will try to estimate the key
quantity $\mathcal{N}_{m,2r}(t,f_{\varepsilon})$ independently of
$\varepsilon$. Of course, if we could give an estimate of ${\left\lVert
f_{\varepsilon}\right\rVert}_{L^{\infty}\left([0,t],\mathcal{H}^{m}_{2r}\right)}$
independently of $\varepsilon$, the proof would be over, but this cannot be
done in the general case. This is the reason why we choose to work with
${\left\lVert
f_{\varepsilon}\right\rVert}_{L^{\infty}\left([0,t],\mathcal{H}^{m-1}_{2r}\right)}$
(losing one derivative for $f$ is enough to give an estimate that does not
depend on $\varepsilon$), and
${\left\lVert\rho_{\varepsilon}\right\rVert}_{L^{2}\left([0,t],H_{x}^{m}\right)}$.
The fact that it is possible to give an estimate independent of $\varepsilon$
on $\rho$ without loss of derivatives while it is not for $f$ is not trivial
and is fundamental here. This is reminiscent of the results on averaging
lemmas (see [4]). Here though, the situation is a bit different, as we will be
averaging in $v$ but also in time. More precisely, anticipating a bit on the
following proof, let us look at the operator $K$ (defined in Lemma 8),

$K_{G}(F)(t,x)=\int_{0}^{t}\int(\nabla_{x}F)(s,x-(t-s)v)\cdot G(t,s,x,v)dvds.$

There is an apparent loss of one derivative over $x$. However, the key
proposition 4 shows that $K_{G}$ is a bounded operator from
$L^{2}([0,T],H^{\alpha}_{x})$ into $L^{2}([0,T],L^{2}_{x})$ (provided that $G$
is smooth enough and $\alpha\in(0,1)$), and the bound can be estimated up to a
constant by $T^{\alpha/2}$ (actually, the result remains true for $\alpha=0$
[6]). The apparent loss of a derivative in $x$ is somehow compensated by
taking the averages in $v$ and in time.

######  Remark 1.2.

Because of the apparent links with the theory of averaging lemmas [4], we
could try to apply directly those results to prove our theorem. As stated
earlier, a recent work [3] has proved that, when $x$ is in the whole space,
the system (1.1) is locally well-posed up to $\alpha=3/4$. Nevertheless,
standard averaging lemmas cannot cover the whole range $\alpha\in(0,1]$.

Sections 2, 3 and 4 will be dedicated to the proof of Theorem 2. In section 5,
we give the elements to conclude the proof of the main Theorem 1.

##  2 Beginning of the proof of Theorem 2

###  2.1 Preliminary lemmas

We present here some lemmas that will be useful for the proof of Theorem 2.
They have been proved in [6]. We use the notation $[A,B]=AB-BA$ to denote the
commutator between two operators.

######  Lemma 1.

Let $s\geq 0$ and $\chi=\chi(v)$ a non-negative smooth function, with
$|\partial^{\alpha}\chi|\leq C_{\alpha}\chi$ for all
$\alpha\in\mathbb{N}^{d}$, $|\alpha|\leq s$.

  * •

Consider two functions $f=f(x,v)$, $g=g(x,v)$, then for all $k\geq s/2$

(2.1) ${\left\lVert\chi fg\right\rVert}_{H^{s}_{x,v}}\lesssim{\left\lVert\chi
g\right\rVert}_{H^{s}_{x,v}}{\left\lVert
f\right\rVert}_{W^{k,\infty}_{x,v}}+{\left\lVert\chi
f\right\rVert}_{H^{s}_{x,v}}{\left\lVert g\right\rVert}_{W^{k,\infty}_{x,v}}.$

  * •

Consider a function $E=E(x)$ and a function $F=F(x,v)$. Then for all $s_{0}>d$

(2.2) ${\left\lVert\chi EF\right\rVert}_{H^{s}_{x,v}}\lesssim{\left\lVert
E\right\rVert}_{H^{s_{0}}_{x}}{\left\lVert\chi
F\right\rVert}_{H^{s}_{x,v}}+{\left\lVert\chi
F\right\rVert}_{H^{s}_{x,v}}{\left\lVert E\right\rVert}_{H^{s}_{x}}.$

  * •

Consider a vector field $E=E(x)$ and a function $f=f(x,v)$, then for all
$s_{0}>1+d$ and for all $\alpha,$ $\beta\in\mathbb{N}^{d}$ with
$|\alpha|+|\beta|=s\geq 1$

(2.3)
${\left\lVert\chi[\partial^{\alpha}_{x}\partial^{\beta}_{v},E(x)\cdot\nabla_{v}]f\right\rVert}_{L^{2}_{x,v}}\lesssim{\left\lVert
E\right\rVert}_{H^{s_{0}}_{x}}{\left\lVert\chi
f\right\rVert}_{H^{s}_{x,v}}+{\left\lVert\chi
f\right\rVert}_{H^{s}_{x,v}}{\left\lVert E\right\rVert}_{H^{s}_{x}}.$

The main use of this lemma will be for
$\chi(v)=(1+|v|^{2})^{\frac{\sigma}{2}}$ thus yielding estimates for products
in the space $\mathcal{H}^{s}_{\sigma}$.
Let us introduce another product type estimate.

######  Lemma 2.

Consider two functions $f=f(x,v)$, $g=g(x,v)$. Then, for all $s\geq 0$,
$\alpha,\beta\in\mathbb{N}^{2d}$ with $|\alpha|+|\beta|\leq s$, and $\chi(v)$
positive function satisfying $|\partial^{\alpha}\chi|\leq C_{\alpha}\chi$ (as
in the precedent lemma). We have

(2.4)
${\left\lVert\partial^{\alpha}_{x,v}f\partial^{\beta}_{x,v}g\right\rVert}_{L^{2}_{x,v}}\lesssim{\left\lVert\frac{1}{\chi}f\right\rVert}_{L^{\infty}_{x,v}}{\left\lVert\chi
g\right\rVert}_{H^{s}_{x,v}}+{\left\lVert\frac{1}{\chi}f\right\rVert}_{H^{s}_{x,v}}{\left\lVert\chi
g\right\rVert}_{L^{\infty}_{x,v}}.$

Finally, the following lemma is a very useful commutation formula between
$\partial_{x}^{\alpha}\partial_{v}^{\beta}$ and the transport operator
$\mathcal{T}$ defined by

(2.5) $\mathcal{T}=\partial_{t}+v\cdot\nabla_{x}+E\cdot\nabla_{v}.$

######  Lemma 3.

For every $\alpha,$ $\beta\in\mathbb{N}^{d}$, we have for every smooth
functions $f$

(2.6)
$\partial_{x}^{\alpha}\partial_{v}^{\beta}(\mathcal{T}f)=\mathcal{T}(\partial_{x}^{\alpha}\partial_{v}^{\beta}f)+\sum_{i=1}^{d}\mathds{1}_{\beta_{i}\neq
0}\partial_{x_{i}}\partial_{x}^{\alpha}\partial_{v}^{\overline{\beta}^{i}}f+[\partial_{x}^{\alpha}\partial_{v}^{\beta},E\cdot\nabla_{v}]f,$

where $\overline{\beta}^{i}$ is equal to $\beta$ except
$\overline{\beta}^{i}_{i}=\beta_{i}-1$.

###  2.2 Setting up the bootstrap

The proof of the theorem will rely on a bootstrap argument. From standard
energy estimates (that we shall recall later, see the proof of proposition 2),
we first have the following result.

######  Proposition 1.

The system (1.7) is locally well-posed in $\mathcal{H}^{m}_{2r}$ for all $m$
and $r$ satisfying $m>1+d$ and $2r>d/2$. In other words, if
$f_{\varepsilon}^{0}\in\mathcal{H}^{m}_{2r}$, there exists $T>0$ (which
depends on $\varepsilon$) so that there exists a unique
$f_{\varepsilon}\in\mathcal{C}([0,T],\mathcal{H}^{m}_{2r})$ solution of system
(1.7).

Thanks to the previous proposition, we may define a maximal solution
$f_{\varepsilon}\in\mathcal{C}\left([0,T^{*}),\mathcal{H}^{m}_{2r}\right)$. As
a direct consequence, for every
$T<T^{*},\quad\underset{[0,T]}{\sup}{\left\lVert
f_{\varepsilon}\right\rVert}_{\mathcal{H}_{2r}^{m-1}}<+\infty$. In order to
define $\mathcal{N}_{m,2r}(T,f_{\varepsilon})$, we just have to prove that
${\left\lVert\rho_{\varepsilon}\right\rVert}_{L^{2}\left([0,T],H_{x}^{m}\right)}<+\infty$.
But because of the definition of weighted Sobolev norms, and with the use of
the Cauchy-Schwarz inequality, we get that

${\left\lVert\rho_{\varepsilon}\right\rVert}_{H_{x}^{m}}\lesssim{\left\lVert
f_{\varepsilon}\right\rVert}_{\mathcal{H}_{2r}^{m}}<+\infty.$

We have thus shown that the quantity $\mathcal{N}_{m,2r}(T,f_{\varepsilon})$
is well defined and is continuous in $T$ for every $T<T^{*}$. This allows us
to consider, for $R>0$ to be defined later,

$T^{\varepsilon}=\sup\left\\{T\in[0,T^{*}),\mathcal{N}_{m,2r}(T,f_{\varepsilon})\leq
R\right\\}.$

By taking $R$ large enough, we have by continuity that $T^{\varepsilon}>0$. Of
course $T^{\varepsilon}$ depends on $\varepsilon$. We want to show that by
taking $R$ large enough (but independent of $\varepsilon$), $T^{\varepsilon}$
is uniformly bounded from below by some time $T>0$. Only the following two
situations can happen

1\. Either $T^{\varepsilon}=T^{*}$,

2\. Or $T^{\varepsilon}<T^{*}$ and
$\mathcal{N}_{m,2r}(T^{\varepsilon},f_{\varepsilon})=R$.

Let us analyze the first case. If $T^{\varepsilon}=T^{*}=+\infty$, then
$\mathcal{N}_{m,2r}(T^{\varepsilon},f_{\varepsilon})\leq R$ for every $T>0$
and there is nothing to do, so we only have to consider when
$T^{\varepsilon}=T^{*}<+\infty$. Actually, by energy estimates, we can show
that this case is impossible. Indeed, we have the following proposition

######  Proposition 2.

Assume that $T^{\varepsilon}<+\infty$, then for every $f_{\varepsilon}$
solution of (1.7), we have for some $C>0$ independent of $\varepsilon$ the
estimate

$\underset{[0,T^{\varepsilon})}{\sup}{\left\lVert
f_{\varepsilon}(t)\right\rVert}_{\mathcal{H}^{m}_{2r}}^{2}\leq{\left\lVert
f_{\varepsilon}^{0}\right\rVert}_{\mathcal{H}^{m}_{2r}}^{2}\exp\left[C\left(T^{\varepsilon}+\frac{1}{\varepsilon}(T^{\varepsilon})^{\frac{1}{2}}R\right)\right].$

###### Proof.

Let $f_{\varepsilon}$ satisfying (1.7). We thus have
$\mathcal{T}f_{\varepsilon}=0$, and by using (2.6)

$\mathcal{T}(\partial_{x}^{\alpha}\partial_{v}^{\beta}f_{\varepsilon})=-\sum_{i=1}^{d}\mathds{1}_{\beta_{i}\neq
0}\partial_{x_{i}}\partial_{x}^{\alpha}\partial_{v}^{\overline{\beta}^{i}}f_{\varepsilon}-[\partial_{x}^{\alpha}\partial_{v}^{\beta},E_{\varepsilon}\cdot\nabla_{v}]f_{\varepsilon}.$

Taking the scalar product with
$(1+|v|^{2})^{2r}\partial_{x}^{\alpha}\partial_{v}^{\beta}f_{\varepsilon}$ and
summing for every $|\alpha|+|\beta|\leq m$, we obtain for the left hand size

$\sum_{|\alpha|+|\beta|\leq
m}\int_{\mathbb{T}^{d}}\int_{\mathbb{R}^{d}}\mathcal{T}(\partial_{x}^{\alpha}\partial_{v}^{\beta}f_{\varepsilon})\cdot\partial_{x}^{\alpha}\partial_{v}^{\beta}f_{\varepsilon}=\frac{d}{dt}{\left\lVert
f_{\varepsilon}(t)\right\rVert}_{\mathcal{H}^{m}_{2r}}^{2}.$

For the first term of the right hand size, we use the Cauchy-Schwarz
inequality

$\left|\sum_{|\alpha|+|\beta|\leq
m}\int_{\mathbb{T}^{d}}\int_{\mathbb{R}^{d}}\sum_{i=1}^{d}\chi\mathds{1}_{\beta_{i}\neq
0}\partial_{x_{i}}\partial_{x}^{\alpha}\partial_{v}^{\overline{\beta}^{i}}f_{\varepsilon}\cdot\chi\partial_{x}^{\alpha}\partial_{v}^{\beta}f_{\varepsilon}\right|\lesssim{\left\lVert
f_{\varepsilon}\right\rVert}_{\mathcal{H}^{m}_{2r}}^{2}.$

For the second term, we use (2.3) with $s=m,\>\chi(v)=(1+|v|^{2})^{r}$ and
$s_{0}=m$, from which we deduce

${\left\lVert\chi\left[\partial_{x}^{\alpha}\partial_{v}^{\beta},E_{\varepsilon}(x)\cdot\nabla_{v}\right]f_{\varepsilon}\right\rVert}_{L^{2}_{x,v}}\lesssim{\left\lVert
E_{\varepsilon}\right\rVert}_{H_{x}^{m}}{\left\lVert
f_{\varepsilon}\right\rVert}_{\mathcal{H}^{m}_{2r}},$

and thus by using again Cauchy-Schwarz, we obtain

$\left|\int\chi\left[\partial_{x}^{\alpha}\partial_{v}^{\beta},E_{\varepsilon}(x)\cdot\nabla_{v}\right]f_{\varepsilon}\chi\partial_{x}^{\alpha}\partial_{v}^{\beta}f_{\varepsilon}\right|\lesssim{\left\lVert
E_{\varepsilon}\right\rVert}_{H_{x}^{m}}{\left\lVert
f_{\varepsilon}\right\rVert}_{\mathcal{H}^{m}_{2r}}^{2}.$

We have, by elliptic regularity,

${\left\lVert
E_{\varepsilon}\right\rVert}_{H_{x}^{m}}={\left\lVert\nabla_{x}(U*V_{\varepsilon})\right\rVert}_{H_{x}^{m}}\lesssim\frac{1}{\varepsilon}{\left\lVert\rho_{\varepsilon}\right\rVert}_{H_{x}^{m}}.$

Putting all together, we have shown that

$\frac{d}{dt}{\left\lVert
f_{\varepsilon}(t)\right\rVert}_{\mathcal{H}^{m}_{2r}}^{2}\lesssim\left(\frac{1}{\varepsilon}{\left\lVert\rho_{\varepsilon}\right\rVert}_{H_{x}^{m}}+1\right){\left\lVert
f_{\varepsilon}(t)\right\rVert}_{\mathcal{H}^{m}_{2r}}^{2}.$

We integrate between $0$ and $t$ for $t\in[0,T^{\varepsilon})$. For some $C>0$
independent of $\varepsilon$, we get

${\left\lVert
f_{\varepsilon}(t)\right\rVert}_{\mathcal{H}^{m}_{2r}}^{2}\leq{\left\lVert
f_{\varepsilon}^{0}\right\rVert}_{\mathcal{H}^{m}_{2r}}^{2}+C\int_{0}^{t}\left(\frac{1}{\varepsilon}{\left\lVert\rho_{\varepsilon}\right\rVert}_{H_{x}^{m}}+1\right){\left\lVert
f_{\varepsilon}(s)\right\rVert}_{\mathcal{H}^{m}_{2r}}^{2},$

and we use the Gronwall inequality to show

$\displaystyle\underset{[0,T^{\varepsilon})}{\sup}{{\left\lVert
f_{\varepsilon}(t)\right\rVert}}_{\mathcal{H}^{m}_{2r}}^{2}$
$\displaystyle\leq{\left\lVert
f_{\varepsilon}^{0}\right\rVert}_{\mathcal{H}^{m}_{2r}}^{2}+\exp\left[C\left(T^{\varepsilon}+\frac{1}{\varepsilon}(T^{\varepsilon})^{\frac{1}{2}}{\left\lVert\rho_{\varepsilon}\right\rVert}_{L^{2}\left([0,T^{\varepsilon}\right),H_{x}^{m})}\right)\right]$
$\displaystyle\leq{\left\lVert
f_{\varepsilon}^{0}\right\rVert}_{\mathcal{H}^{m}_{2r}}^{2}+\exp\left[C\left(T^{\varepsilon}+\frac{1}{\varepsilon}(T^{\varepsilon})^{\frac{1}{2}}\mathcal{N}_{m,2r}(T^{\varepsilon},f_{\varepsilon})\right)\right].$

Finally, since $\mathcal{N}_{m,2r}(T^{\varepsilon},f_{\varepsilon})\leq R$, we
obtain the expected estimate. ∎

If we use the proposition for $T^{\varepsilon}=T^{*}$, we obtain that

$\underset{[0,T^{*})}{\sup}{\left\lVert
f_{\varepsilon}(t)\right\rVert}_{\mathcal{H}^{m}_{2r}}^{2}\leq{\left\lVert
f_{\varepsilon}^{0}\right\rVert}_{\mathcal{H}^{m}_{2r}}^{2}\exp\left[C\left(T^{*}+\frac{1}{\varepsilon}(T^{*})^{\frac{1}{2}}R\right)\right].$

This means that the solution could be continued beyond $T^{*}$, and thus
contradicts the definition of $T^{*}$, which shows that this case is
impossible.

We then have to consider the remaining case, $T^{\varepsilon}<T^{*}$ and
$\mathcal{N}^{m}_{2r}(T^{\epsilon},f_{\varepsilon})=R$. Choosing $R$ large
enough, the objective is to find some time $T^{\\#}>0$ independent of
$\varepsilon$, such that the equality

$\mathcal{N}^{m}_{2r}(T,f_{\varepsilon})=R,$

cannot hold for any $T\in[0,T^{\\#}]$, which will prove that
$T^{\varepsilon}>T^{\\#}>0$.

We need to estimate $\mathcal{N}^{m}_{2r}(T,f_{\varepsilon})$ for
$T<T^{\varepsilon}$. The easier part is the term ${\left\lVert
f_{\varepsilon}\right\rVert}_{\mathcal{H}^{m-1}_{2r}}$. We cannot use the
previous estimates because they depend on $\varepsilon$ (we used the elliptic
regularity provided by the Poisson equation). Nevertheless, we can still give
estimates based on energy methods, this time being careful that every estimate
must be independent of $\varepsilon$. In this proposition and in the following
of this article, $\Lambda$ will stand for a generic continuous function,
independent of $\varepsilon$, which is non-decreasing with respect to each of
its arguments.

######  Proposition 3.

For $m>2+d$ and $2r>d/2$, $f_{\varepsilon}$ solution of (1.7) satisfies the
estimate

(2.7) $\underset{[0,T]}{\sup}{\left\lVert
f_{\varepsilon}\right\rVert}_{\mathcal{H}^{m-1}_{2r}}\leq{\left\lVert
f_{\varepsilon}^{0}\right\rVert}_{\mathcal{H}^{m-1}_{2r}}\>+T^{\frac{1}{2}}\Lambda(T,R),$

for all $T\in[0,T^{\varepsilon})$.

###### Proof.

Let $\alpha,\beta\in\mathbb{N}^{d}$ such that $|\alpha|+|\beta|=m-1$. We use
(2.6) and, as before, we take the scalar product with
$\left(1+|v|^{2}\right)^{2r}\partial_{x}^{\alpha}\partial_{v}^{\beta}f_{\varepsilon}$,
and we take the sum for every $|\alpha|+|\beta|=m-1$. Like we did previously,
we use (2.3) with $s=m-1$, $\chi(v)=\left(1+|v|^{2}\right)^{r}$ and
$s_{0}=m-1$. We obtain that

(2.8) $\frac{d}{dt}{\left\lVert
f_{\varepsilon}\right\rVert}_{\mathcal{H}^{m-1}_{2r}}^{2}\lesssim{\left\lVert
f_{\varepsilon}\right\rVert}_{\mathcal{H}^{m-1+}_{2r}}^{2}+{\left\lVert
f_{\varepsilon}\right\rVert}_{\mathcal{H}^{m-1}_{2r}}^{2}{\left\lVert
E_{\varepsilon}\right\rVert}_{H_{x}^{m-1}}.$

Integrating in time, we get that there exists a $C>0$ such that

$\underset{[0,T]}{\sup}{\left\lVert
f_{\varepsilon}\right\rVert}_{\mathcal{H}^{m-1}_{2r}}\leq{\left\lVert
f_{\varepsilon}^{0}\right\rVert}_{\mathcal{H}^{m-1}_{2r}}+C\underset{[0,T]}{\sup}{\left\lVert
f_{\varepsilon}\right\rVert}_{\mathcal{H}^{m-1}_{2r}}\left(T+\int_{0}^{T}{\left\lVert
E_{\varepsilon}\right\rVert}_{H_{x}^{m-1}}dt\right).$

We can now use the following estimate, independent of $\varepsilon$

(2.9) ${\left\lVert
E_{\varepsilon}\right\rVert}_{H_{x}^{m-1}}={\left\lVert\nabla_{x}(U*V_{\varepsilon})\right\rVert}_{H_{x}^{m-1}}\lesssim{\left\lVert\rho_{\varepsilon}\right\rVert}_{H_{x}^{m}}.$

And we still have $\mathcal{N}_{m,2r}(T,f_{\varepsilon})\leq R$, (because
$T\in[0,T^{\epsilon}])$. We have thus shown that

$\underset{[0,T]}{\sup}{\left\lVert
f_{\varepsilon}\right\rVert}_{\mathcal{H}^{m-1}_{2r}}\leq{\left\lVert
f_{\varepsilon}^{0}\right\rVert}_{\mathcal{H}^{m-1}_{2r}}+CR(T+T^{\frac{1}{2}}R),$

which concludes the proof of the lemma.

∎

######  Remark 2.1.

As said in the introduction, our estimates could be sharper. This can been
seen in (2.9), which is far from being optimal, because we do not use the
regularity provided by the convolution with $U$. Nevertheless, this
regularization is not needed to prove Theorem 2, thus we chose not to exploit
it for simplicity.

##  3 Estimates of the density term

###  3.1 Introduction of the $f_{I,J}$

Now we need to tackle the second term of
$\mathcal{N}_{m,2r}(T,f_{\varepsilon})$, which is the norm of the density
${\left\lVert\rho_{\varepsilon}\right\rVert}_{L^{2}\left([0,T],H_{x}^{m}\right)}$.
We can try to apply the operator $\partial_{x}^{\alpha}$ to (1.7) with
$|\alpha|=m$ , but this involves commutator terms such as
$\partial_{x}^{\alpha^{\prime}}E_{\varepsilon}\cdot\nabla_{v}\partial_{x}^{\alpha-\alpha^{\prime}}f_{\varepsilon}$
which contain $m$ order derivatives of $f_{\varepsilon}$ when
$|\alpha^{\prime}|=1$, and those cannot be estimated uniformly in
$\varepsilon$. To get rid of this problem, we choose to apply a larger class
of differential operators to (1.7).

######  Definition 3.1.

For $I=(i_{1},...,i_{d}),J=(j_{1},...,j_{d})\in\mathbb{N}^{d},|I|+|J|=m$, we
define

$f_{I,J}:=\partial_{x}^{I}\partial_{v}^{J}f_{\varepsilon}.$

Note that the $(f_{I,J})$ contain all the $\partial^{I}_{x}f_{\varepsilon}$
with $|I|=m$. By applying $\partial_{x}^{I}\partial_{v}^{J}$ to (1.7), we find
that the $(f_{I,J})$ satisfy a differential system, which is the purpose of
the following lemma.

######  Lemma 4.

We assume that $m>3+d$ and $2r>d$. For all $T<T^{\varepsilon}$, and for all
$I,J\in\mathbb{N}^{d}$, we have, for $f_{\varepsilon}$ satisfying (1.7), that
$f_{I,J}$ is solution of

(3.1)
$\mathcal{T}(f_{I,J})+\partial_{x}^{I}\partial_{v}^{J}E_{\varepsilon}\cdot\nabla_{v}f_{\varepsilon}+\mathcal{M}_{I,J}\mathcal{F}=\mathcal{V}_{I,J},$

where

$\mathcal{F}=(f_{I,J})_{I,J\in\mathbb{N}^{d},|I|+|J|=m},\quad\mathcal{M}_{I,J}\mathcal{F}=\sum_{k=1}^{d}\mathds{1}_{i_{k}\neq
0}f_{\hat{I}^{k},\overline{J}^{k}}+\sum_{p=1}^{d}\sum_{k=1}^{d}\mathds{1}_{i_{p}\neq
0}\partial_{x_{p}}E_{\varepsilon}f_{\overline{I}^{p},\hat{J^{k}}},$

with

$\hat{I}^{k}=(i_{1}...i_{k-1},i_{k}+1,i_{k+1}...i_{d}),\quad\overline{I}^{k}=(i_{1}...i_{k-1},i_{k}-1,i_{k+1}...i_{d}),$

and $\mathcal{V}=(\mathcal{V}_{I,J})_{I,J\in\mathbb{N}^{d}}$ is a remainder,
which means that for every $T<T^{\varepsilon}$

${\left\lVert\mathcal{V}\right\rVert}_{L^{2}([0,T],\mathcal{H}^{0}_{r})}\leq\Lambda(T,R).$

We therefore obtain that $\mathcal{F}=(f_{I,J})$ satisfies a system which is
coupled through the linear term $\mathcal{M}_{I,J}\mathcal{F}$. We now prove
this lemma.

###### Proof.

We know that $f_{\varepsilon}$ solves (1.7), so we have

$\partial_{x}^{I}\partial_{v}^{J}(\mathcal{T}(f_{\varepsilon}))=0.$

By using (2.6), we obtain

$\mathcal{T}(f_{I,J})+\partial_{x}^{I}\partial_{v}^{J}E_{\varepsilon}\cdot\nabla_{v}f_{\varepsilon}+\mathcal{M}_{I,J}\mathcal{F}=\mathcal{V}_{I,J},$

with

$\mathcal{V}_{I,J}=\sum_{k=2}^{\min(|I|,m-1)}\sum_{\sigma,|\sigma|=k}C_{I,J,k,\sigma}\>\partial^{\sigma}_{x}E_{\varepsilon}\>\cdot\nabla_{x}\partial^{I-\sigma}_{x}\partial^{J}_{v}f_{\varepsilon}.$

Notice that because $|\sigma|=k\geq 2$, we have in particular that
$1+|I|-|\sigma|+|J|\leq m-1$. Also, in the case $|I|\leq 1$, we simply have
$\mathcal{V}_{I,J}=0$.
We want to estimate
${\left\lVert\mathcal{V}_{I,J}\right\rVert}_{L^{2}([0,T],\mathcal{H}^{0}_{r})}$.
We have that

${\left\lVert\mathcal{V}_{I,J}\right\rVert}_{\mathcal{H}^{0}_{r}}\lesssim{\left\lVert
E_{\varepsilon}\right\rVert}_{H_{x}^{m-1}}{\left\lVert
f_{\varepsilon}\right\rVert}_{\mathcal{H}^{m-1}_{r}}.$

And, because ${\left\lVert
E_{\varepsilon}\right\rVert}_{H_{x}^{m-1}}\lesssim{\left\lVert\rho_{\varepsilon}\right\rVert}_{H_{x}^{m}}$
(estimate independent of $\varepsilon$),

${\left\lVert\mathcal{V}_{I,J}\right\rVert}_{L^{2}([0,T],\mathcal{H}^{0}_{r})}\lesssim{\left\lVert\rho_{\varepsilon}\right\rVert}_{L^{2}([0,T],H_{x}^{m})}{\left\lVert
f_{\varepsilon}\right\rVert}_{L^{\infty}([0,T],\mathcal{H}^{m-1}_{r})},$

and we obtain that

${\left\lVert\mathcal{V}\right\rVert}_{L^{2}([0,T],\mathcal{H}^{0}_{r})}\leq\Lambda(T,R),$

which ends the proof. ∎

###  3.2 Straightening the transport vector field

In the following, we make a change of variable to straighten the vector field

$\partial_{t}+v\cdot\nabla_{x}+E\cdot\nabla_{v}\quad\text{into}\quad\partial_{t}+\Phi(t,x,v)\cdot\nabla_{x},$

where $\Phi$ is defined in the following lemma.

######  Lemma 5.

Let $f_{I,J}$ a solution of (3.1). We consider $\Phi(t,x,v)$ a smooth solution
of the Burgers equation

(3.2) $\partial_{t}\Phi+\Phi\cdot\nabla_{x}\Phi=E_{\varepsilon},$

such that the Jacobian matrix $(\nabla_{v}\Phi)$ is inversible. We define
$g_{I,J}$ by

$g_{I,J}(t,x,v):=f_{I,J}(t,x,\Phi).$

Then $g_{I,J}$ is solution of the equation

(3.3)
$\partial_{t}g_{I,J}+\Phi\cdot\nabla_{x}g_{I,J}+\partial_{x}^{I}\partial_{v}^{J}E_{\varepsilon}\cdot\left(\nabla_{v}f_{\varepsilon}\right)(t,x,\Phi)+\mathcal{M}_{I,J}\mathcal{G}=\mathcal{V}_{I,J}(t,x,\Phi),$

where $\mathcal{G}=(g_{I,J})_{I,J\in\mathbb{N}^{d},|I|+|J|=m}$.

###### Proof.

The proof is based on a simple calculation

$\begin{split}\partial_{t}g_{I,J}+\Phi\cdot\nabla_{x}g_{I,J}+\partial_{x}^{I}\partial_{v}^{J}E_{\varepsilon}\cdot\left(\nabla_{v}f_{\varepsilon}\right)(t,x,\Phi)+\mathcal{M}_{I,J}\mathcal{G}=\mathcal{V}_{I,J}(t,x,\Phi)\\\
+^{t}(\nabla_{v}\Phi)^{-1}\nabla_{v}g_{I,J}\cdot\left(\partial_{t}\Phi+\Phi\cdot\nabla_{x}\Phi-
E_{\varepsilon}\right),\end{split}$

hence the result when (3.2) is satisfied. ∎

######  Remark 3.1.

Writing $\mathcal{J}(t,x,v)=|\det\nabla_{v}\Phi(t,x,v)|$, notice that

(3.4)
$\int_{\mathbb{R}^{d}}g_{I,J}\mathcal{J}dv=\int_{\mathbb{R}^{d}}f_{I,J}dv.$

Because we introduced the function $\Phi$ in our equations, we shall estimate
its Sobolev norms, which is the purpose of the following lemma.

######  Lemma 6.

Suppose that $m>3+d$, there exists $T_{0}$ (depending on $R$ but independent
on $\varepsilon$) such that for all $T<\min(T_{0},T^{\varepsilon})$, there
exists a unique smooth solution on $[0,T]$ of the Burgers equation (3.2) with
initial condition $\Phi|_{t=0}=v$.
Moreover, for all $T<\min(T_{0},T^{\varepsilon})$, we have the following
estimates:

(3.5)
$\underset{[0,T]}{\sup}{\left\lVert\Phi-v\right\rVert}_{W^{k,\infty}_{x,v}}+\underset{[0,T]}{\sup}{\left\lVert\frac{1}{\left(1+|v|^{2}\right)^{\frac{1}{2}}}\partial_{t}\Phi\right\rVert}_{W^{k-1,\infty}_{x,v}}\leq
T^{\frac{1}{2}}\Lambda(T,R),\quad k<m-d/2-1,$

for $|\alpha|\leq m-1$ and $|\beta|\leq m-2$

(3.6)
$\underset{[0,T]}{\sup}\,\underset{v}{\sup}{\left\lVert\partial^{\alpha}_{x,v}\left(\Phi-v\right)\right\rVert}_{L^{2}_{x}}+\underset{[0,T]}{\sup}\,\underset{v}{\sup}{\left\lVert\frac{1}{\left(1+|v|^{2}\right)^{\frac{1}{2}}}\partial^{\beta}_{x,v}\partial_{t}\Phi\right\rVert}_{L^{2}_{x}}\leq
T^{\frac{1}{2}}\Lambda(T,R).$

A very similar lemma (Lemma 11) has been proved in [6], we refer to it for a
complete proof.

Now we have to consider our new equation (3.3), that we rewrite as

(3.7)
$\partial_{t}g_{I,J}+\Phi\cdot\nabla_{x}g_{I,J}+\partial^{I}_{x}\partial^{J}_{v}E_{\varepsilon}\cdot(\nabla_{v}f_{\varepsilon})(t,x,\Phi)+\mathcal{M}_{I,J}\mathcal{G}=S_{I,J},$

where $S_{I,J}(t,x,v)=\mathcal{V}_{I,J}(t,x,\Phi(t,x,v))$.
The next step is to introduce the flow of the equation that we denote by
$X(t,s,x,v),0\leq s,t\leq T$ and is given as the solution of

$\partial_{t}X(t,s,x,v)=\Phi(t,X(t,s,x,v),v),\quad X(s,s,x,v)=x.$

We have to control the Sobolev norms of $X$.

######  Lemma 7.

For all $t,s,\>0\leq s\leq t\leq T$ and $m>3+d$, we write

$X(t,s,x,v)=x+(t-s)\left(v+\tilde{X}(t,s,x,v)\right).$

We have that $\tilde{X}$ satisfies, for $|\alpha|<m-d/2-1$, $|\beta|<m-d/2-2$

(3.8)
$\underset{t,s\in[0,T]}{\sup}{\left\lVert\partial^{\alpha}_{x,v}\tilde{X}(t,s,x,v)\right\rVert}_{L^{\infty}_{x,v}}+\underset{t,s\in[0,T]}{\sup}{\left\lVert\dfrac{1}{(1+|v|^{2})^{\frac{1}{2}}}\partial^{\beta}_{x,v}\partial_{t}\tilde{X}(t,s,x,v)\right\rVert}_{L^{\infty}_{x,v}}\leq
T^{\frac{1}{2}}\Lambda(T,R).$

Furthermore, there exists $\hat{T}_{0}(R)>0$ small enough such that for
$T\leq\min(T_{0},\hat{T}_{0},T^{\varepsilon})$, we get that $x\mapsto
x+(t-s)\tilde{X}(t,s,x,v)$ is a diffeomorphism and that, for $|\alpha|<m-1$,
$|\beta|<m-2$,

(3.9)
$\underset{t,s\in[0,T]}{\sup}\underset{v}{\sup}{\left\lVert\partial^{\alpha}_{x,v}\tilde{X}(t,s,x,v)\right\rVert}_{L^{2}_{x}}+\underset{t,s\in[0,T]}{\sup}\underset{v}{\sup}{\left\lVert\dfrac{1}{(1+|v|^{2})^{\frac{1}{2}}}\partial^{\beta}_{x,v}\partial_{t}\tilde{X}(t,s,x,v)\right\rVert}_{L^{2}_{x}}\leq
T^{\frac{1}{2}}\Lambda(T,R).$

Finally, there exists $\Psi(t,s,x,v)$ such that for $t,s\in[0,T]$ and
$T\leq\min(T_{0},\hat{T}_{0},T^{\varepsilon})$, we have,

(3.10) $X(t,s,x,\Psi(t,s,x,v))=x+(t-s)v,$

with $\Psi$ satisfying the estimates, for $|\alpha|<m-d/2-1$,
$|\beta|<m-d/2-2$

(3.11)
$\underset{t,s\in[0,T]}{\sup}{\left\lVert\partial^{\alpha}_{x,v}(\Psi(t,s,x,v)-v)\right\rVert}_{L^{\infty}_{x,v}}+\underset{t,s\in[0,T]}{\sup}{\left\lVert\dfrac{1}{(1+|v|^{2})^{\frac{1}{2}}}\partial^{\beta}_{x,v}\partial_{t}\Psi(t,s,x,v)\right\rVert}_{L^{\infty}_{x,v}}\leq
T^{\frac{1}{2}}\Lambda(T,R),$

for $|\alpha|<m-1$, $|\beta|<m-2$

(3.12)
$\underset{t,s\in[0,T]}{\sup}\underset{v}{\sup}{\left\lVert\partial^{\alpha}_{x,v}(\Psi(t,s,x,v)-v)\right\rVert}_{L^{2}_{x}}+\underset{t,s\in[0,T]}{\sup}\underset{v}{\sup}{\left\lVert\dfrac{1}{(1+|v|^{2})^{\frac{1}{2}}}\partial^{\beta}_{x,v}\partial_{t}\Psi(t,s,x,v)\right\rVert}_{L^{2}_{x}}\leq
T^{\frac{1}{2}}\Lambda(T,R).$

Once again, a similar lemma (Lemma 13) has already been proved in [6].

In order to control the linear part, we define the tensor $\mathcal{M}$ by
$(\mathcal{M}H)_{I,J}=M_{I,J}H$ and for $0\leq s,t\leq T$,
$x\in\mathbb{T}^{d}$, $v\in\mathbb{R}^{d}$, and we introduce
$\mathfrak{M}(t,s,x,v)$ as the solution of

$\partial_{t}\mathfrak{M}(t,s,x,v)=\mathcal{M}(t,x,\Phi(t,x,v))\mathfrak{M}(t,s,x,v),\quad\mathfrak{M}(s,s,x,v)=I,$

whose existence and uniqueness is guaranteed by the Cauchy-Lipschitz theorem.
By a Gronwall type argument and thanks to Lemma 6, we can show that for
$k<m-2$

(3.13) $\underset{0\leq s,t\leq
T}{\sup}\left({\left\lVert\mathfrak{M}\right\rVert}_{W^{k,\infty}_{x,v}}+{\left\lVert\partial_{t}\mathfrak{M}\right\rVert}_{W^{k,\infty}_{x,v}}+{\left\lVert\partial_{s}\mathfrak{M}\right\rVert}_{W^{k,\infty}_{x,v}}\right)\leq\Lambda(T,R).$

###  3.3 Introduction of the average operator

We define in the next lemma the fundamental operator $K_{G}$ which was
introduced in [6]

######  Lemma 8.

For a smooth function $G(t,s,x,v)$, we define the integral operator $K_{G}$
acting on $F(t,x)$ by

(3.14)
$K_{G}(F)(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}(\nabla_{x}F)(s,x-(t-s)v)\cdot
G(t,s,x,v)dvds.$

For $f_{\varepsilon}$ satisfying (1.7) and $\rho_{\varepsilon}=\int
f_{\varepsilon}dv$, the functions $\partial^{I}_{x}\rho_{\varepsilon}$ with
$|I|=m$ satisfy the equation:

$\partial^{I}_{x}\rho_{\varepsilon}=\sum_{K\in\\{1,...,d\\}^{m}}K_{H_{(K,0),(I,0)}}\left(U*\left(\left(I-\varepsilon^{2}\Delta\right)^{-1}\partial_{x}^{K}\rho_{\varepsilon}\right)\right)+\mathcal{R}_{I,0},$

with

$\begin{split}&H_{(K,L),(I,J)}=\\\
&\mathfrak{M}_{(K,L),(I,J)}(s,t,x,\Psi(s,t,x,v))(\nabla_{v}f)(s,x-(t-s)v,\Psi(s,t,x,v))\mathcal{J}(t,s,\Psi(s,t,x,v))\tilde{\mathcal{J}}(s,t,x,v),\end{split}$
$\mathcal{J}=|det\nabla_{v}\Phi(t,x,\Psi(s,t,x,v))|,\quad\tilde{\mathcal{J}}=|det\nabla_{v}\Psi(s,t,x,v)|,$

and $\mathcal{R}_{I,0}$ satisfies for $T$ small enough, the estimate

${\left\lVert\mathcal{R}_{I,0}\right\rVert}_{L^{2}([0,T],L^{2}_{x})}\lesssim
T^{\frac{1}{2}}\Lambda(T,R).$

###### Proof.

We first introduce the notations

$\eta(t,x,v)=\left(\partial^{I}_{x}\partial^{J}_{v}E_{\varepsilon}(t,x)\cdot\nabla_{v}f_{\varepsilon}(s,x,\Phi(t,x,v))\right)_{I,J},\quad
S=(S_{I,J})_{I,J}.$

This allows us to put the system (3.7) under the following equation satisfied
by $\mathcal{G}$

$\partial_{t}\mathcal{G}(t,x,v)+\Phi\cdot\nabla_{x}\mathcal{G}(t,x,v)+\eta(t,x,v)+\mathcal{M}\mathcal{G}(t,x,v)=S(t,x,v).$

Integrating by respect with the time variable the expression
$\partial_{t}(\mathfrak{M}(t,s,x,v)\mathcal{G}(t,X(t,s,x,v),v))$ we obtain
that

$\begin{split}\mathcal{G}(t,x,v)=\mathfrak{M}(0,t,x,v)\mathcal{G}^{0}(X(0,t,x,v),v)+\int_{0}^{t}\mathfrak{M}(s,t,x,v){S}(s,X(s,t,x,v),v)ds&\\\
-\int_{0}^{t}\mathfrak{M}(s,t,x,v)\eta(s,X(s,t,x,v),v)ds,\end{split}$

with $\mathcal{G}^{0}=(g^{0}_{I,J})_{I,J}$. We multiply the equation by
$\mathcal{J}$ and then with integrate by respect with the variable $v$, which
yields

(3.15)
$\int_{\mathbb{R}^{d}}\mathcal{G}(t,x,v)\mathcal{J}(t,x,v)dv=\mathcal{I}_{0}+\mathcal{I}_{F}-\int_{\mathbb{R}^{d}}\int_{0}^{t}\mathfrak{M}(s,t,x,v)\eta(s,X(s,t,x,v),v)\mathcal{J}(t,x,v)dvds,$

with

$\mathcal{I}_{0}=\int_{\mathbb{R}^{d}}\mathfrak{M}(0,t,x,v)\mathcal{G}^{0}(X(0,t,x,v),v)\mathcal{J}(t,x,v)dv,$
$\mathcal{I}_{F}=\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathfrak{M}(s,t,x,v)S(s,X(s,t,x,v),v)\mathcal{J}(t,x,v)dsdv.$

We want to show that $\mathcal{I}_{0},\mathcal{I}_{F}$ can be considered as
remainders. Let us recall some of the previous estimates on $\Phi$ and
$\mathfrak{M}$

(3.16) $\underset{0\leq s,t\leq
T}{\sup}{\left\lVert\mathfrak{M}(t,s)\right\rVert}_{L^{\infty}_{x,v}}\leq\Lambda(T,R).$
$\underset{[0,T]}{\sup}{\left\lVert\Phi(t)-v\right\rVert}_{W^{1,\infty}_{x,v}}\leq\Lambda(T,R).$

Now we can give the following estimate

$\begin{split}\left|\int\mathfrak{M}(t,0,x,v)\mathcal{G}^{0}(X(0,t,x,v),v)\mathcal{J}(t,x,v)dv\right|&\\\
\leq\underset{0\leq s,t\leq
T}{\sup}{\left\lVert\mathfrak{M}(s,t)\right\rVert}_{L^{\infty}_{x,v}}\underset{0\leq
t\leq
T}{\sup}{\left\lVert\mathcal{J}(t)\right\rVert}_{L^{\infty}_{x,v}}\int|\mathcal{G}^{0}(X(0,t,x,v),v)|dv&\\\
\leq\Lambda(T,R)\sum_{I,J}\int|g^{0}_{I,J}(X(0,t,x,v),v)|dv.\end{split}$

Thus, we have

${\left\lVert\mathcal{I}_{0}\right\rVert}_{L^{2}([0,T],L^{2}_{x})}\leq\Lambda(T,R)\sum_{I,J}{\left\lVert\int_{v}{\left\lVert
g^{0}_{I,J}(X(0,t,\cdot,v),v)\right\rVert}_{L^{2}_{x}}dv\right\rVert}_{L^{2}[0,T]}.$

To estimate the last term, we use the change of variable in $x$,
$y=X(0,t,x,v)+tv=x-t\tilde{X}(0,t,x,v)$ and, thanks to the estimates on
$\tilde{X}$

$\underset{t,s\in[0,T]}{\sup}{\left\lVert\partial_{x}\tilde{X}(t,s,x,v)\right\rVert}_{L^{\infty}_{x,v}}\leq\Lambda(T,R),$

we get that

${\left\lVert
g^{0}_{I,J}(X(0,t,\cdot,v),v)\right\rVert}_{L^{2}_{x}}dv\leq\Lambda(T,R){\left\lVert
g^{0}_{I,J}(\cdot-tv),v\right\rVert}_{L^{2}_{x}}\leq\Lambda(T,R){\left\lVert
g^{0}_{I,J}(\cdot,v)\right\rVert}_{L^{2}_{x}}.$

Then by Cauchy-Schwarz, we deduce

${\left\lVert\mathcal{I}_{0}\right\rVert}_{L^{2}([0,T],L^{2}_{x})}\leq
T^{\frac{1}{2}}\Lambda(T,R)\left(\int_{\mathbb{R}^{d}}\frac{dv}{(1+|v|^{2})^{r}}\right)^{\frac{1}{2}}\sum_{I,J}{\left\lVert
g^{0}_{I,J}\right\rVert}_{\mathcal{H}^{0}_{r}}.$

Again, we can make the following change of variable

$\int_{x,v}|g^{0}_{I,J}(x,v)|^{2}dxdv=\int_{x,v}|f^{0}_{I,J}(x,v)|^{2}\mathcal{J}(0,x,v)dxdv\leq\Lambda(T,R){\left\lVert
f^{0}_{I,J}\right\rVert}_{\mathcal{H}^{0}_{r}}^{2}.$

Finally, we can use the fact that ${\left\lVert
f^{0}_{I,J}\right\rVert}_{\mathcal{H}^{0}_{r}}\leq{\left\lVert
f_{\varepsilon}\right\rVert}_{\mathcal{H}^{m}_{r}}\leq R$ to conclude that

${\left\lVert\mathcal{I}_{0}\right\rVert}_{L^{2}([0,T],L^{2}_{x})}\leq
T^{\frac{1}{2}}\Lambda(T,R).$

Let us show with similar arguments that

${\left\lVert\mathcal{I}_{f}\right\rVert}_{L^{2}([0,T],L^{2}_{x})}\leq
T\Lambda(T,R).$

First, we use once again (3.16) to show that

${\left\lVert\mathcal{I}_{F}\right\rVert}_{L^{2}([0,T],L^{2}_{x})}\leq\Lambda(T,R)\sum_{I,J}{\left\lVert\int_{0}^{t}\int_{v}{\left\lVert
S_{I,J}(s,X(s,t,\cdot,v),v)\right\rVert}_{L^{2}_{x}}dvds\right\rVert}_{L^{2}[0,T]}.$

As we did just above, we use the change of variable in $x$,
$y=X(s,t,x,v)+(t-s)v=x-(t-s)\tilde{X}(s,t,x,v)$, and the estimates on
$\tilde{X}$ to show that

${\left\lVert
S_{I,J}(s,X(s,t,\cdot,v),v)\right\rVert}_{L^{2}_{x}}\leq\Lambda(T,R){\left\lVert
S_{I,J}(s,\cdot,v)\right\rVert}_{L^{2}_{x}}.$

Integrating in $v$, we get by the Cauchy-Schwarz inequality

$\int_{v}{\left\lVert
S_{I,J}(s,X(s,t,\cdot,v),v)\right\rVert}_{L^{2}_{x}}dv\leq\Lambda(T,R){\left\lVert
S_{I,J}(s)\right\rVert}_{\mathcal{H}^{0}_{r}}.$

So we finally arrive at

${\left\lVert\mathcal{I}_{F}\right\rVert}_{L^{2}([0,T],L^{2}_{x})}\leq\Lambda(T,R){\left\lVert\int_{0}^{t}{\left\lVert
S_{I,J}(s)\right\rVert}_{\mathcal{H}^{0}_{r}}\right\rVert}_{L^{2}[0,T]}\leq\Lambda(T,R)T{\left\lVert
S\right\rVert}_{L^{2}([0,T],\mathcal{H}^{0}_{r})}.$

Recalling that $S(t,x,v)=\mathcal{V}(t,x,\Phi(t,x,v))$, we can use one last
change of variable, and the estimates on the derivatives of $\Phi$, to show
that

${\left\lVert
S\right\rVert}_{L^{2}([0,T],\mathcal{H}^{0}_{r})}\leq\Lambda(T,R){\left\lVert\mathcal{V}\right\rVert}_{L^{2}([0,T],\mathcal{H}^{0}_{r})}\leq\Lambda(T,R),$

where the last inequality comes from the fact that $\mathcal{V}$ is a
remainder term, which we proved in lemma 4.
Thus, we have proved that

${\left\lVert\mathcal{I}_{F}\right\rVert}_{L^{2}([0,T],L^{2}_{x})}\leq
T\Lambda(T,R).$

Let us go back to (3.15). Thanks to the results on $\mathcal{I}_{0}$ and
$\mathcal{I}_{F}$, and using (3.4), we have

$\begin{split}&\partial^{I}_{x}\rho_{\varepsilon}=\mathcal{R}_{I,0}-\\\
&\int_{0}^{t}\int\sum_{K}\mathfrak{M}_{(K,0),(I,J)}(\partial^{K}_{x}E)(s,X(s,t,x,v))\cdot(\nabla_{v}f)(s,X(s,t,x,v),\Phi(s,X(s,t,x,v),v))\mathcal{J}(t,x,v)dvds,\end{split}$

with

${\left\lVert\mathcal{R}_{I,0}\right\rVert}_{L^{2}([0,T],L^{2}_{x})}\lesssim
T^{\frac{1}{2}}\Lambda(T,R).$

We can finally use the change of variable $v=\Psi(s,t,x,w)$ provided by lemma
7 to obtain that

$\partial^{I}_{x}\rho_{\varepsilon}=-\int_{0}^{t}\int\sum_{K}(\partial^{K}_{x}E)(s,x-(t-s)v)\cdot
H_{(K,0),(I,0)}(t,s,x,v)dvds+\mathcal{R}_{I,0},$

with

$\begin{split}&H_{(K,L),(I,J)}=\\\
&\mathfrak{M}_{(K,L),(I,J)}(s,t,x,\Psi(s,t,x,v))(\nabla_{v}f)(s,x-(t-s)v,\Psi(s,t,x,v))\mathcal{J}(t,s,\Psi(s,t,x,v))\tilde{\mathcal{J}}(s,t,x,v),\end{split}$

which gives the result. ∎

###  3.4 Focus on the operator $K_{G}$

In order to control the norms of the derivatives of $\rho_{\varepsilon}$, we
have to understand better the operator $K$. Following [6], let us first
introduce a new norm.

######  Definition 3.2.

For $T>0$, we define

${\left\lVert G\right\rVert}_{T,s_{1},s_{2}}=\underset{0\leq t\leq
T}{\sup}\left(\sum_{k}\underset{0\leq s\leq
T}{\sup}\underset{\xi}{\sup}\left((1+|k|)^{s_{2}}(1+|\xi|)^{s_{1}}\left|(\mathcal{F}_{x,v}G)(t,s,k,\xi)\right|\right)^{2}\right)^{\frac{1}{2}}.$

######  Proposition 4.

There exists $C>0$ such that for every $T>0$, $\alpha\in(0,1)$, for every $G$
satisfying ${\left\lVert G\right\rVert}_{T,s_{1},s_{2}}<\infty$ and for all
$s_{1}>1$, $s_{2}>d/2$, we have

${\left\lVert K_{G}(F)\right\rVert}_{L^{2}([0,T],L^{2}_{x})}\leq
C\,T^{\alpha/2}\,{\left\lVert G\right\rVert}_{T,s_{1},s_{2}}{\left\lVert
F\right\rVert}_{L^{2}([0,T],H^{\alpha}_{x})},\quad\forall F\in
L^{2}([0,T],H^{\alpha}_{x}).$

For practical uses, it is convenient to relate the norm ${\left\lVert
G\right\rVert}_{T,s_{1},s_{2}}<\infty$ to a more tractable norm. From [6], we
know that if $p>1+d$, $\sigma>d/2$, we can find $s_{2}>d/2$ et $s_{1}>1$ such
that

${\left\lVert G\right\rVert}_{T,s_{1},s_{2}}\leq\underset{0\leq s,t\leq
T}{\sup}{\left\lVert G(t,s)\right\rVert}_{\mathcal{H}^{p}_{\sigma}}.$

In the expression of the operator $K_{G}$ (3.14), there seems to have of loss
of a derivative in $x$ , but this proposition shows that the operator is
actually continuous from $L^{2}([0,T],H^{\alpha}_{x})$ into
$L^{2}([0,T],L^{2}_{x})$ provided that $G$ is smooth enough. Once again, we
emphasize that this is a key propriety in the proof of Theorem 2. We are able
to gain regularity in $x$ by integrating over $v$ and $t$. This propriety
explains why we can estimate the $H^{m}$ norm of $\rho_{\varepsilon}$ without
loss of derivative.

Proposition 4 remains true for $\alpha=0$, this was proved in [6].

###### Proof.

By using Fourier series in $x$, we write that

$F(t,x)=\sum_{k\in\mathbb{Z}^{d}}\hat{F}_{k}(t)e^{ik\cdot x}.$

By definition of $K_{G}$, we have

$\displaystyle K_{G}F(t,x)$
$\displaystyle=\int_{0}^{t}\sum_{k}\hat{F}_{k}(s)e^{ik\cdot x}\cdot\int
e^{-ik\cdot v(t-s)}G(t,s,x,v)dvds$
$\displaystyle=\int_{0}^{t}\sum_{k}\hat{F}_{k}(s)e^{ik\cdot
x}ik\cdot(\mathcal{F}_{v}G)(t,s,x,k(t-s))ds,$

where $\mathcal{F}_{v}G$ is the Fourier transform of $G(t,s,x,v)$ with respect
to the last variable. By Fourier expanding in the $x$ variable, we deduce that

$K_{G}F(t,x)=\sum_{k}e^{ik\cdot x}\sum_{l}e^{ik\cdot
x}\int_{0}^{t}\hat{F}_{k}(s)ik\cdot(\mathcal{F}_{x,v}G)(t,s,l,k(t-s))ds.$

Changing $l$ into $l+k$ we can rewrite this expression as

$K_{G}F(t,x)=\sum_{l}e^{ik\cdot
x}\left(\sum_{k}\int_{0}^{t}\hat{F}_{k}(s)ik\cdot(\mathcal{F}_{x,v}G)(t,s,l-k,k(t-s))ds\right).$

From the Bessel-Parseval identity, we infer that

${\left\lVert
K_{G}\right\rVert}^{2}_{L^{2}_{x}}=\sum_{l}\left|\sum_{k}\int_{0}^{t}\hat{F}_{k}(s)ik\cdot(\mathcal{F}_{x,v}G)(t,s,l-k,k(t-s))ds\right|^{2}.$

By using Cauchy-Schwarz for $t$ and $k$, we have

$\begin{split}{\left\lVert
K_{G}\right\rVert}^{2}_{L^{2}_{x}}\lesssim\sum_{l}(\sum_{k}\int_{0}^{t}|\hat{F}_{k}(s)&|^{2}|k\cdot(\mathcal{F}_{x,v}G)(t,s,l-k,k(t-s))|ds\\\
&\cdot\sum_{k}\int_{0}^{t}|k\cdot(\mathcal{F}_{x,v}G)(t,s,l-k,k(t-s))|ds),\end{split}$

and by integrating in time, we obtain that

$\begin{split}{\left\lVert
K_{G}\right\rVert}^{2}_{L^{2}([0,T],L^{2}_{x})}&\lesssim\sum_{l}\int_{0}^{T}\int_{0}^{t}\sum_{k}|\hat{F_{k}}(s)|^{2}|k\cdot(\mathcal{F}_{x,v}G)(t,s,l-k,k(t-s))|dsdt\\\
&\cdot\underset{l}{\sup}\underset{t\in[0,T]}{\sup}\int_{0}^{t}\sum_{k}|k\cdot(\mathcal{F}_{x,v}G)(t,s,l-k,k(t-s))|ds)\leq
I\cdot II.\end{split}$

Let us first consider the term $II$. We observe that for all $s_{1}\geq 0$ ,

$\underset{l}{\sup}\underset{t\in[0,T]}{\sup}\int_{0}^{t}\sum_{k}|k\cdot(\mathcal{F}_{x,v}G)(t,s,l-k,k(t-s))|ds\leq$
$\underset{l}{\sup}\underset{t\in[0,T]}{\sup}\sum_{k}\left(\underset{0\leq
s\leq
t}{\sup}\underset{\xi}{\sup}\left[(1+|\xi|)^{s1}|(\mathcal{F}_{x,v}G)(t,s,l-k,\xi)|\right]\int_{0}^{t}\frac{|k|}{(1+|k|(t-s))^{s_{1}}}ds\right).$

We choose $s_{1}>1$. Changing variables, we observe that

$\int_{0}^{t}\frac{|k|}{(1+|k|(t-s))^{s_{1}}}ds\leq\int_{0}^{\infty}\frac{1}{(1+\tau^{s_{1}})}d\tau<\infty.$

It follows that

$II\lesssim\underset{l}{\sup}\underset{t\in[0,T]}{\sup}\sum_{k}\underset{0\leq
s\leq
t}{\sup}\underset{\xi}{\sup}\left[(1+|\xi|)^{s1}|(\mathcal{F}_{x,v}G)(t,s,l-k,\xi)|\right].$

Next we choose $s_{2}>d/2$ so that $\sum\frac{1}{(1+|k|)^{2\,s_{2}}}<\infty$.
By using Cauchy-Schwarz, we deduce

$\begin{split}II&\lesssim\underset{t\in[0,T]}{\sup}\left(\sum_{k}\underset{0\leq
s\leq
t}{\sup}\underset{\xi}{\sup}\left((1+|k|)^{s_{2}}(1+|\xi|)^{s_{1}}|(\mathcal{F}_{x,v}G)(t,s,l-k,\xi)|\right)^{2}\right)^{\frac{1}{2}}\\\
&\lesssim{\left\lVert G\right\rVert}_{T,s_{1},s_{2}}.\end{split}$

Now let us consider the other term $I$. By using Fubini, we have for all
$\alpha\in(0,1)$,

$\displaystyle\sum_{l}\int_{0}^{T}\int_{0}^{t}\sum_{k}|\hat{F_{k}}(s)|^{2}|k\cdot(\mathcal{F}_{x,v}G)(t,s,l-k,k(t-s))|dsdt$
$\displaystyle=\int_{0}^{T}\sum_{k}|\hat{F_{k}}(s)|^{2}\int_{s}^{T}\sum_{l}|k||(\mathcal{F}_{x,v}G)(t,s,l-k,k(t-s))|dtds$
$\displaystyle=\int_{0}^{T}\sum_{k}|\hat{F_{k}}(s)|^{2}(1+|k|^{\alpha})\int_{s}^{T}\sum_{l}\frac{|k|}{(1+|k|^{\alpha})}|(\mathcal{F}_{x,v}G)(t,s,l-k,k(t-s))|dtds$
$\displaystyle={\left\lVert
F\right\rVert}_{L^{2}([0,T],H_{x}^{\alpha})}^{2}\underset{k}{\sup}\underset{0\leq
s\leq
T}{\sup}\int_{s}^{T}\sum_{l}\frac{|k|}{(1+|k|^{\alpha})}|(\mathcal{F}_{x,v}G)(t,s,l-k,k(t-s))|dt.$

As previously, by choosing $s_{1}>1$, $s_{2}>d/2$, we have

$\displaystyle\underset{k}{\sup}\underset{0\leq s\leq
T}{\sup}\int_{s}^{T}\sum_{l}\frac{|k|}{(1+|k|^{\alpha})}|(\mathcal{F}_{x,v}G)(t,s,l-k,k(t-s))|dt$
$\displaystyle\leq\underset{k}{\sup}\underset{0\leq s\leq
T}{\sup}\int_{s}^{T}\frac{|k|}{(1+|k|^{\alpha})(1+|k|(t-s))^{s_{1}}}\sum_{l}\underset{\xi}{\sup}(1+|\xi|)^{s1}|(\mathcal{F}_{x,v}G)(t,s,l-k,\xi)|dt$
$\displaystyle\lesssim\underset{k}{\sup}\underset{0\leq s\leq
T}{\sup}\int_{s}^{T}\frac{|k|}{(1+|k|^{\alpha})(1+|k|(t-s))^{s_{1}}}{\left\lVert
G\right\rVert}_{T,s_{1},s_{2}}.$

To conclude, we have to estimate $\underset{k}{\sup}\underset{0\leq s\leq
T}{\sup}\int_{s}^{T}\frac{|k|}{(1+|k|^{\alpha})(1+|k|(t-s))^{s_{1}}}dt$. We
first use a change of variable

$\displaystyle\underset{k}{\sup}\underset{0\leq s\leq
T}{\sup}\int_{s}^{T}\frac{|k|}{(1+|k|^{\alpha})(1+|k|(t-s))^{s_{1}}}dt$
$\displaystyle\leq\underset{k}{\sup}\frac{1}{(1+|k|^{\alpha})}\underset{0\leq
s\leq T}{\sup}\int_{0}^{|k|(T-s)}\frac{1}{(1+\tau)^{s_{1}}}d\tau$
$\displaystyle\leq\underset{k}{\sup}\frac{1}{(1+|k|^{\alpha})}\int_{0}^{|k|T}\frac{1}{(1+\tau)^{s_{1}}}d\tau.$

Then, by Holder inequality, we have the estimate

$\displaystyle\underset{k}{\sup}\frac{1}{(1+|k|^{\alpha})}\int_{0}^{|k|T}\frac{1}{(1+\tau)^{s_{1}}}d\tau$
$\displaystyle\leq\underset{k}{\sup}\frac{1}{(1+|k|^{\alpha})}\times(|k|T)^{\alpha}\left(\int_{0}^{\infty}\frac{1}{(1+\tau)^{\frac{s_{1}}{1-\alpha}}}d\tau\right)^{1-\alpha}$
$\displaystyle\lesssim T^{\alpha}.$

We have finally shown that

$I\lesssim T^{\alpha}{\left\lVert G\right\rVert}_{T,s_{1},s_{2}}{\left\lVert
F\right\rVert}_{L^{2}([0,T],H_{x}^{\alpha})}^{2},$

which ends the proof. ∎

###  3.5 Conclusion of the estimates for $\rho_{\varepsilon}$

By Lemma 8

$\partial^{I}_{x}\rho_{\varepsilon}=\sum_{K\in\\{1,...,d\\}^{m}}K_{H_{(K,0),(I,0)}}\left(U*\left(\left(I-\varepsilon^{2}\Delta\right)^{-1}\partial_{x}^{K}\rho\right)\right)+\mathcal{R}_{I,0},$

with

${\left\lVert\mathcal{R}_{I,0}\right\rVert}_{L^{2}([0,T],L^{2}_{x})}\lesssim
T^{\frac{1}{2}}\Lambda(T,R).$

Let $\alpha>0$ be given by Assumption (A1). Notice that if $\alpha\geq 1$,
Assumption (A1) will be satisfied for any real inferior to $\alpha$, thus we
can assume in the following that $\alpha<1$.
We get, thanks to Proposition 4

$\displaystyle{\left\lVert
K_{H_{(K,0),(I,0)}}\left(U*\left(\left(I-\varepsilon^{2}\Delta\right)^{-1}\partial_{x}^{K}\rho\right)\right)\right\rVert}_{L^{2}([0,T],L_{x}^{2})}$
$\displaystyle\lesssim
T^{\alpha/2}{\left\lVert\left(U*\left(\left(I-\varepsilon^{2}\Delta\right)^{-1}\partial_{x}^{K}\rho\right)\right)\right\rVert}_{L^{2}([0,T],H_{x}^{\alpha})}$
$\displaystyle\lesssim T^{\alpha/2}R.$

We emphasize that this inequality is independent of $\varepsilon$. We can thus
estimate all of the $\partial^{I}_{x}\rho_{\varepsilon}$ with $|I|\leq m$.
Summing up all of the results we have, we have proved that

${\left\lVert\rho_{\varepsilon}\right\rVert}_{L^{2}([0,T],H_{x}^{m})}\leq
T^{\alpha/2}\Lambda(T,R).$

##  4 End of the proof of Theorem 2

The previous sections have given all the tools to end the proof. Indeed,
Section 3 has been dedicated to show that

${\left\lVert\rho_{\varepsilon}\right\rVert}_{L^{2}([0,T],H_{x}^{m})}\leq
T^{\frac{\alpha}{2}}\Lambda(T,R).$

We have shown at the end of Section 2 that

${\left\lVert
f_{\varepsilon}\right\rVert}_{L^{\infty}([0,T],\mathcal{H}^{m-1}_{2r})}\leq{\left\lVert
f_{\varepsilon}^{0}\right\rVert}_{\mathcal{H}^{m-1}_{2r}}+T^{\frac{1}{2}}\Lambda(T,R),$

from which we deduce that for all $0<T<T^{\varepsilon}$

$\mathcal{N}_{m,2r}(T,f_{\varepsilon})\leq{\left\lVert
f_{\varepsilon}^{0}\right\rVert}_{\mathcal{H}^{m-1}_{2r}}+T^{\frac{1}{2}}\Lambda(T,R)+T^{\frac{\alpha}{2}}\Lambda(T,R).$

Next, we consider $R$ large enough such that

$\frac{1}{2}R>{\left\lVert
f_{\varepsilon}^{0}\right\rVert}_{\mathcal{H}^{m-1}_{2r}}.$

With $R$ being fixed, we can find by continuity a $T^{\\#}$ small enough such
that the previous estimates are satisfied and that for every
$T\in[0,T^{\\#}]$, we have

$T^{\frac{1}{2}}\Lambda(T,R)+T^{\frac{\alpha}{2}}\Lambda(T,R)<\frac{1}{2}R.$

Thus, for every $T\in[0,T^{\\#}]$, $\mathcal{N}_{m,2r}(t,f_{\varepsilon})<R$
and so $T^{\varepsilon}>T^{\\#}$. This means that the time $T^{\varepsilon}$
is uniformly bounded from below by a certain $T^{\\#}$ which is independent of
$\varepsilon$. That concludes the proof of Theorem 2.

##  5 Solutions for the limit problem

Let us recall the limit system, which is the system we want to study in
Theorem 1

$\left\\{\begin{array}[]{ll}\partial_{t}{f}+v\cdot\nabla_{x}f+E\cdot\nabla_{v}f=0,\\\
E=-\nabla_{x}\int_{\mathbb{T}^{d}}U(x-y)\left(\int_{\mathbb{R}^{d}}f(t,y,v)dv\right)dy,\\\
f(0,x,v)=f^{0}(x,v).\end{array}\right.$

This system is a non linear transport equation. But taking the formal limit
$\varepsilon\rightarrow 0$, we have lost the elliptic propriety that we had
for the system (1.7). Proving local well-posedness is thus more challenging,
and we can use Theorem 2 to find a solution of (1.2).

Indeed, this theorem gives us a family of functions $f_{\varepsilon}$
solutions of (1.7) on a time interval $[0,T]$, $T$ being independent of
$\varepsilon$. We can use this family of functions to find a solution of the
limit system. In the following, we will extract a subsequence that will
converge to a certain function $f$. We next have to show that $f$ is actually
a solution of the limit system. We will start by showing the uniqueness, by
similar arguments that we used in Theorem 2.

###  5.1 Uniqueness of the solution

######  Proposition 5.

Let $f_{1}$, $f_{2}\in\mathcal{C}([0,T],\mathcal{H}^{m-1}_{2r})$ with
$m>m_{0}$ and $2r>r_{0}$ be two solutions of (1.2) with the same initial
condition $f^{0}$. We write $\rho_{i}=\int f_{i}dv$ and we suppose that
$\rho_{i}\in L^{2}([0,T],H_{x}^{m})$. Then $f_{1}=f_{2}$ on
$[0,T]\times\mathbb{T}^{d}\times\mathbb{R}^{d}$.

###### Proof.

Let $f=f_{1}-f_{2}$. We denote by $E_{2}$ the force field associated with the
density $\rho_{2}$ and $E$ associated with $\rho=\rho_{1}-\rho_{2}$. We have
that $f$ solves the equation

(5.1)
$\partial_{t}f+v\cdot\nabla_{x}f+E\cdot\nabla_{v}f_{1}+E_{2}\cdot\nabla_{v}f=0.$

Let $\Phi$ satisfying the Burgers equation (3.2) associated with $E_{2}$, with
initial condition $\Phi(0,x,v)=v$. As previously, we write
$g(t,x,\Phi(t,x,v))=f(t,x,v)$. We obtain that $g$ is solution of

$\partial_{t}g+\Phi\cdot\nabla_{x}g+E\cdot\nabla_{v}f_{1}=0.$

Recall that

$\int_{\mathbb{R}^{d}}g(t,x,v)\mathcal{J}(t,x,v)dv=\rho(t,x),$

for $\mathcal{J}(t,x,v)=|\det\nabla_{v}\Phi(t,x,v)|$.
We consider the characteristics $X$ satisfying

$\partial_{t}X(t,s,x,v)=\Phi(t,X(t,s,x,v),v),\quad X(s,s,x,v)=x.$

Following the steps of Lemma 8, we have

$\rho(t,x)=K_{H}(U*\rho),$

with
$H(t,s,x,v)=(\nabla_{v}f_{1})(s,x-(t-s)v,\Psi(t,s,x,v))\mathcal{J}(t,x,\Psi(t,s,x,v))\tilde{\mathcal{J}}(t,s,x,v)$.

We use proposition 4 to show that

$\begin{split}{\left\lVert\rho\right\rVert}_{L^{2}([0,T],L^{2}_{x})}&\leq
CT^{\alpha/2}{\left\lVert(U*\rho)\right\rVert}_{L^{2}([0,T],H^{\alpha}_{x})}\\\
&\leq
CT^{\alpha/2}{\left\lVert\rho\right\rVert}_{L^{2}([0,T],L^{2}_{x})}.\end{split}$

We take $T_{0}$ such that $CT_{0}^{\alpha/2}<1$, and deduce that we must have
$\rho=0$ on $[0,T_{0}]$. Next, we go back to (5.1) on $[0,T_{0}]$

$\partial_{t}f+v\cdot\nabla_{x}f+E_{2}\cdot\nabla_{v}f=0.$

$f$ is solution of an homogeneous transport equation, with initial condition
being $0$. That means that $f=0$ on all $[0,T_{0}]$ . To obtain this result on
$[0,T]$, we make the same reasoning on $[T_{0},T]$ to obtain the result on
$[0,2T_{0}]$ and so on.

∎

###  5.2 Existence of the solution

All is left to do to prove Theorem 1 is to show the existence of solutions.
Let ($f_{\varepsilon}$) be the family of functions solutions of (1.7) with the
initial conditions $f^{0}_{\varepsilon}=f^{0}$. Thanks to Theorem 2, there
exist $T$ and $R$ independent of $\varepsilon$ such that
$f_{\varepsilon}\in\mathcal{C}([0,T],\mathcal{H}^{m}_{2r})$ satisfies

(5.2)
$\underset{\varepsilon\in(0,1]}{sup}\mathcal{N}_{m,2r}(T,f_{\varepsilon})\leq
R.$

We get by (5.2) that $f_{\varepsilon}$ is uniformly bounded in
$\mathcal{C}([0,T],\mathcal{H}^{m-1}_{2r})$. By (1.2), we obtain that
$\partial_{t}f_{\varepsilon}$ is uniformly bounded in
$L^{\infty}([0,T],\mathcal{H}^{m-2}_{2r-1})$. The Ascoli theorem gives the
existence of a function $f\in\mathcal{C}([0,T],L^{2}_{x,v})$ and a sequence
$\varepsilon_{n}$ such that $f_{\varepsilon_{n}}$ converges to $f$ in
$\mathcal{C}([0,T],L^{2}_{x,v})$. By interpolation, we actually have
convergence in $\mathcal{C}([0,T],\mathcal{H}^{m-1-\delta}_{2r-\delta})$ for
all $\delta>0$. By Sobolev embedding, $f_{\varepsilon_{n}}$ converges to $f$
in $L^{\infty}([0,T]\times\mathbb{T}^{d}\times\mathbb{R}^{d})$ and
$\rho_{\varepsilon_{n}}$ converges to $\rho=\int fdv$ in
$L^{2}([0,T],L_{x}^{2})\cap L^{\infty}([0,T]\times\mathbb{R}^{d})$. Thus, the
limit function $f$ solves the system (1.2).

To conclude, we want to apply Proposition 5. We only have to show that
$f\in\mathcal{C}([0,T],\mathcal{H}^{m-1}_{2r})$ and that $\rho\in
L^{2}([0,T],H_{x}^{m})$. By weak compacity arguments, $f\in
L^{\infty}([0,T],\mathcal{H}^{m-1}_{2r})$ and $\rho\in
L^{2}([0,T],H_{x}^{m})$. We then use an energy estimate which have been shown
previously (the formula (2.8)) to show that

$\frac{d}{dt}{\left\lVert f\right\rVert}_{\mathcal{H}^{m-1}_{2r}}^{2}\leq C.$

It follows that $f\in\mathcal{C}([0,T],\mathcal{H}^{m-1}_{2r})$. We can
finally apply Proposition 5 which gives the uniqueness of $f$ and concludes
the proof of Theorem 1.

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