# Derivation of Yudovich solutions of Incompressible Euler from the Vlasov-
Poisson system

Immanuel Ben-Porat Mathematical Institute, University of Oxford, Oxford OX2
6GG, UK<EMAIL_ADDRESS>Mikaela Iacobelli ETH Zürich,
Department of Mathematics, Rämistrasse 101, 8092 Zürich, Switzerland.
<EMAIL_ADDRESS>Alexandre Rege ETH Zürich, Department of
Mathematics, Rämistrasse 101, 8092 Zürich, Switzerland.
<EMAIL_ADDRESS>
###### Abstract

We derive the two dimensional incompressible Euler equation as a quasineutral
limit of the Vlasov-Poisson equation using a modulated energy approach. We
propose a strategy which enables to treat solutions where the gradient of the
velocity is merely $\mathrm{BMO}$, in accordance to the celebrated Yudovich
theorem.

##  1 Introduction

###  1.1 General overview

In this paper we study the derivation of the 2D incompressible Euler equation
as a quasineutral limit from the Vlasov–Poisson system. The Euler equation on
$[0,T]\times\mathbb{T}^{2}$ in vorticity form reads

$\partial_{t}\omega+\mathrm{div}(\omega u)=0,\ \omega=\Delta\psi,\
u=(\nabla\psi)^{\bot}.$ (E)

Here $u:[0,T]\times\mathbb{T}^{2}\rightarrow\mathbb{R}^{2}$ is a vector field
referred to as the velocity field , $\omega$ is a scalar field on
$[0,T]\times\mathbb{T}^{2}$ referred to as the vorticity, and $v^{\bot}$
designates the rotation by $\frac{\pi}{2}$ of a vector $v$, i.e.
$v^{\bot}\coloneqq(-v_{2},v_{1})$. In the quasineutral scaling, the
Vlasov–Poisson system, reads as follows:

$\left\\{\begin{array}[]{lc}\partial_{t}f^{\varepsilon}+\xi\cdot\nabla_{x}f^{\varepsilon}-\nabla\Phi^{\varepsilon}\cdot\nabla_{\xi}f^{\varepsilon}=0,\vspace{0.25
cm}\\\ \rho^{\varepsilon}=1-\varepsilon\Delta\Phi^{\varepsilon},\
\rho^{\varepsilon}(t,x)=\int_{\mathbb{R}^{2}}f(t,x,\xi)d\xi.\end{array}\right.$
(VP)

Here, the unknown is a time dependent probability density function
$f^{\epsilon}(t,x,\xi)$ on $\mathbb{\mathbb{T}}^{2}\times\mathbb{R}^{2}$. The
variables $(x,\xi)\in\mathbb{\mathbb{T}}^{2}\times\mathbb{R}^{2}$ are referred
to as the space variable and velocity variable respectively. From a physical
point of view, the parameter $\varepsilon$ represents the Debye length, and
the quasineutral limits consists in the rigorous study of the above system
when $\varepsilon$ tends to zero.

Typically, in the quasineutral limit one seeks to prove uniform in time weak
convergence of the density $\rho^{\varepsilon}$ to the vorticity $\omega$ in
the limit as $\varepsilon$ goes to zero. This type of limit is a hydrodynamic
limit, as it derives an equation of perfect fluids from a kinetic equation.
The mathematical study of the quasineutral limit was initiated in the works of
Brenier and Grenier [5], [13], which used an approach based on defect
measures. Then in [14] Grenier established the quasineutral limit of
Vlasov–Poisson in the analytic regime. The quasineutral limit was also studied
for magnetized plasmas in Golse and Saint-Raymond [12] and Saint-Raymond [31]
by employing compactness arguments. Soon after, Brenier [4] used a modulated
energy approach to prove convergence of the density and current associated to
the system (VP) to the vorticity and velocity field respectively of Euler’s
equation. The notion of solution considered by Brenier is that of dissipative
solutions which was introduced by Lions in [21]. This result was then extended
by Masmoudi in [24] to more general initial data. More recent works include
[26], in which the author proposes a Lagrangian trajectories point of view on
the problem, [18] which studies a combined mean field quasineutral/gyrokinetic
regime, [25] which explores the mean-field limit of a two dimensional spray
model, and [3, 29] which are the quantum counterpart of the latter work. See
also [7] for the related combined mean field small inertia limit.

###  1.2 Main contribution

The focus in this work is how the amount of regularity that is imposed for the
solution of the limit equation (E) affects the argument leading to the
quasineutral limit. More specifically, in view of the work of Yudovich [34]
about the existence and uniqueness of weak solutions for the Euler equation
(E), it is natural to raise the question of the derivation of this class of
solutions as a quasineutral limit. All the works mentioned so far did not
cover solutions of Yudovich type. To properly understand the difficulty that
emerges when applying the modulated energy approach introduced by Brenier in
[4], let us briefly recall the strategy of the proof. The argument in [4] is
based on a Grönwall estimate for the following time dependent quantity, known
as the modulated energy:

$E_{\varepsilon}(t)\coloneqq\frac{1}{2}\int\left|\xi-u(t,x)\right|^{2}f^{\varepsilon}(t,x,\xi)dxd\xi+\frac{\varepsilon}{2}\int\left|\nabla\Phi^{\varepsilon}\right|^{2}(t,x)dx.$

After some classical manipulations involving the conservation of energy which
can be found in the Appendix of [4], one obtains the following expression for
the time derivative of $E_{\varepsilon}(t)$:

$\displaystyle\dot{E_{\varepsilon}}(t)=-$
$\displaystyle\int_{\mathbb{T}^{2}}\int_{\mathbb{R}^{2}}d(u)(t,x):(\xi-u(t,x))\otimes(\xi-u(t,x))f^{\varepsilon}(t,x,\xi)dxd\xi$
$\displaystyle+\varepsilon\int_{\mathbb{T}^{2}}d(u)(t,x):\nabla\Phi^{\varepsilon}\otimes\nabla\Phi^{\varepsilon}dx$
$\displaystyle+\int_{\mathbb{T}^{2}}A(u)(t,x)\cdot
u(t,x)\left(\rho^{\varepsilon}(t,x)-1\right)dx\coloneqq\stackrel{{\scriptstyle[}}{{k}}=1]{3}{\sum}I_{k}.$

Here, $d(u)$ stands for the symmetric part of the Jacobian matrix of $u$, i.e.
$(d(u))_{ij}\coloneqq\partial_{i}u_{j}+\partial_{j}u_{i}$ and
$A(u)\coloneqq\partial_{t}u+u\cdot\nabla u$. Let us examine the integral
$I_{1}$ for instance. When $\nabla u\in L^{\infty}$, it is noticeable that
$I_{1}$ is controlled by the kinetic part of the energy
$E_{\varepsilon}^{1}(t)$. However, when working with Yudovich solutions, the
boundedness of $\nabla u$ is no longer at our disposal.

The aim of this work is to show how to treat the more singular integrals that
appear in the expression for $\dot{E_{\varepsilon}}(t)$. To this aim we will
introduce a more subtle argument, based on the combination of Fefferman’s
characterization of the dual of the Hardy space $\mathscr{H}^{1}$ as the space
$\mathrm{BMO}$, with some new estimates on the growth with respect to
$\varepsilon$ of the characteristics in Vlasov-type system introduced in [17].
In particular, a large portion of our proof will be devoted to carefully study
how the Hardy norm of terms of the form

$\int_{\mathbb{R}^{2}}(\xi-u(t,x))\otimes(\xi-u(t,x))f^{\varepsilon}(t,x,\xi)d\xi$
(1.1)

grow with respect to $\varepsilon$. In view of the inequality $\left|\int
fg\right|\leq\left\|f\right\|_{\mathrm{BMO}}\left\|g\right\|_{\mathscr{H}^{1}}$
(or slight variants thereof) this will ultimately allow us to conclude and
prove the following:

######  Theorem 1.1 (Main result).

Let the following assumptions on the initial data hold

$\mathbf{H1}$. $0\leq f_{0}^{\varepsilon}$$\in L^{\infty}\cap
L^{1}(\mathbb{T}^{2}\times\mathbb{R}^{2})$ is a probability density.

$\mathbf{H2}.$
$\left\|(1+\left|\xi\right|^{k_{0}})f_{0}^{\varepsilon}\right\|_{L^{\infty}\cap
L^{1}}\leq\overline{M}_{\varepsilon}$ where $k_{0}>2$ and
$\overline{M}_{\varepsilon}=O(\varepsilon^{-\alpha})$ for some $\alpha>0$.

$\mathbf{H3}.$ There is some $\beta>0$ such that
$E_{\varepsilon}(0)=O({\varepsilon}^{\beta})$.

$\mathbf{H4}.$ $\omega_{0}\in L^{\infty}(\mathbb{T}^{2})$.
Let $f^{\varepsilon}$ be the unique solution to (VP) with initial data
$f_{0}^{\varepsilon}$ provided by A.1, and let $\omega$ be the Yudovich
solution to (E) with initial data $\omega_{0}$ provided by 2.3. Then, there is
some
$T_{\ast}=T_{\ast}\left(\left\|\omega_{0}\right\|_{L^{\infty}(\mathbb{T}^{2})},\alpha,\beta\right)>0$
such that

$\underset{t\in[0,T_{\ast}]}{\sup}\left\|\rho^{\varepsilon}(t,\cdot)-\omega(t,\cdot)\right\|_{H^{-1}(\mathbb{T}^{2})}\underset{\varepsilon\rightarrow
0}{\rightarrow}0.$

Let us observe that it is not difficult to construct initial data
$f_{0}^{\varepsilon}$ which verify the assumptions $\mathbf{H1}$-$\mathbf{H3}$
\- we shall elaborate on this in Section 5. The motivation for this work grew
out of a question raised by Rosenzweig in [28], which asks for a combined
quasineutral mean field limit for Yudovich solutions, in the spirit of [32].
Such question already needs to be clarified for the quasineutral limit.
Indeed, the dissipative solutions considered by Brenier [4] are not known to
coincide with Yudovich solutions. In fact, the question of uniqueness for
dissipative solutions with low regularity vorticity is subtle, and is still
not fully understood, see 4.4 in [21] for the precise regularity assumptions
needed in order to guarantee uniqueness of dissipative solutions. Finally, we
stress that the main result (1.1) holds for a short time interval. In order to
be able to extend this result for arbitrary times we expect that we will need
to have at least a logarithmic (in $\varepsilon$) bound on
$\left\|\rho^{\varepsilon}(t,\cdot)\right\|_{\infty}$ \- which is open.

Outline of the paper: In Section 2 we review some preliminaries, and in
particular we include reminders about the general theory for Yudovich
solutions to (E) as well as general harmonic analysis estimates. In Section 3,
we continue with presenting slight refinements of classical estimates for the
maximal function and BMO norms. Section 4 reflects the main novelty of the
present work. In the first part of this section we follow closely the approach
taken in [17] in order to establish quantitative estimates on
$\left\|\rho^{\varepsilon}(t,\cdot)\right\|_{\infty}$ in the case of Vlasov-
Poisson. In the second part, we apply these estimates in order to quantify the
Hardy norm of integrals of the form (1.1) by means of the modulated energy.
These estimates eventually lead to the proof of our main result, which shows
that, for short times, the modulated energy approach can be adapted for
Yudovich solutions, where a priori the gradient of the velocity is just BMO.
In Section 5 we explicitly construct well prepared initial data, i.e. initial
data which simultaneously realizes the asymptotic vanishing of the modulated
energy as well as growth conditions (with respect to $\varepsilon$) on the
initial $L^{\infty}$ norm of the density. Finally, in the Appendix we quickly
recall the well-posedness theory for (VP) on the two-dimensional torus.

##  2 Preliminaries

The aim of this section is to recall basic properties of Yudovich solutions,
following mainly [1] and [23].

###  2.1 The Coulomb potential on $\mathbb{T}^{2}$

Unlike in the planar settings, when working on the torus, the Coulomb
potential does not admit a simple closed formula and therefore we start by
briefly recalling how to properly define the Coulomb potential on the torus.
We fix some general notations. Let
$\mathbb{T}^{2}\coloneqq\left[-\frac{1}{2},\frac{1}{2}\right]^{2},$ and set

$\mathcal{\mathscr{P}}(\mathbb{T}^{2})\coloneqq\left\\{f:\mathbb{R}^{2}\rightarrow\mathbb{R}\left|\forall\alpha\in\mathbb{Z}^{2},\forall
x\in\mathbb{\mathbb{R}}^{2}:f(x+\alpha)=f(x)\right.\right\\},$

$\mathcal{\mathscr{P}}(\mathbb{T}^{2}\times\mathbb{R}^{2})\coloneqq\left\\{f:\mathbb{R}^{2}\times\mathbb{R}^{2}\rightarrow\mathbb{R}\left|\forall\alpha\in\mathbb{Z}^{2},\forall
x,\xi\in\mathbb{\mathbb{R}}^{2}:f(x+\alpha,\xi)=f(x,\xi)\right.\right\\}.$

For each non-negative integer $n\in\mathbb{N}_{0}$ define

$C^{n,\alpha}(\mathbb{T}^{2})\coloneqq
C^{n,\alpha}(\mathbb{R}^{2})\cap\mathcal{\mathscr{P}}(\mathbb{T}^{2}),\
C^{n,\alpha}(\mathbb{T}^{2}\times\mathbb{R}^{2})\coloneqq
C^{n,\alpha}(\mathbb{R}^{2}\times\mathbb{R}^{2})\cap\mathcal{\mathscr{P}}(\mathbb{T}^{2}\times\mathbb{R}^{2}).$

######  Lemma 2.1.

There exists a unique function $V\in
C^{\infty}(\mathbb{R}^{2}\setminus\mathbb{Z}^{2})\cap\mathcal{\mathscr{P}}(\mathbb{T}^{2})$
with the following properties.

(a) For any $\varrho\in\mathcal{\mathscr{P}}(\mathbb{T}^{2})\cap
L^{\infty}(\mathbb{T}^{2})$ with $\int_{\mathbb{T}^{2}}\varrho(x)dx=0$ the
function defined by

$\Phi(x)\coloneqq\int_{\mathbb{T}^{2}}V(x-y)\varrho(y)dy$

is $C^{1}(\mathbb{T}^{2})$ and is the unique solution of

$-\Delta\Phi=\varrho,\ \int_{\mathbb{T}^{2}}\Phi(x)dx=0.$

(b) There is a function $V_{0}\in
C^{\infty}((\mathbb{R}^{2}\setminus\mathbb{Z}^{2})\cup\left\\{\mathbf{0}\right\\})$
such that for all $x\in\mathbb{R}^{2}\setminus\mathbb{Z}^{2}$ it holds that

$V(x)=V_{0}(x)+V_{1}(x),$

where $V_{1}(x)\coloneqq-\frac{1}{2\pi}\log(\left|x\right|)$. The function $V$
is henceforth referred to as the Coulomb potential.

###  2.2 The Yudovich Theorem

By a weak solution to (E) we mean the following.

######  Definition 2.2.

We say that $\omega:[0,T]\times\mathbb{T}^{2}\rightarrow\mathbb{R}$ is a weak
solution to (E) if $\omega\in L^{\infty}([0,T];L^{1}(\mathbb{T}^{2})\cap
L^{p}(\mathbb{T}^{2}))$ for some $p>2$ and for all $\varphi\in
W^{1,\infty}([0,T];C_{0}^{1}(\mathbb{R}^{2}))$ and all $t\in[0,T]$ it holds
that

$\displaystyle\int_{\mathbb{T}^{2}}\omega(t,x)\varphi(t,x)dx=\int_{\mathbb{T}^{2}}\omega_{0}(x)\varphi(0,x)dx+\int_{0}^{t}\int_{\mathbb{T}^{2}}\omega(s,x)\partial_{s}\varphi(s,x)dxds$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{T}^{2}}\nabla\varphi(s,x)\cdot\nabla^{\bot}V\star\omega(s,x)\omega(s,x)dxds.$

In [34], Yudovich obtained existence and uniqueness of such a weak solution to
(E) under the assumption that the initial vorticity is bounded. The periodic
version of Yudovich’s theorem stated here is taken from [8].

######  Theorem 2.3.

Let $\omega_{0}\in L^{\infty}(\mathbb{T}^{2})$. Then there exists a unique
solution $\omega\in L^{\infty}([0,\infty);L^{\infty}(\mathbb{T}^{2})),$ in the
sense of 2.2. Moreover the $L^{p}$ norms of this solution are conserved, i.e.
for all $t\in[0,\infty)$ it holds that

$\left\|\omega(t,\cdot)\right\|_{p}=\left\|\omega_{0}(\cdot)\right\|_{p},\
1\leq p\leq\infty.$

######  Remark 2.4.

It is essential to keep in mind that the boundedness of the vorticity $\omega$
by no means guarantees the boundedness of $\nabla u$, due to the failure of
the Calderon-Zygmund inequality at the end-points exponents $p=1,\infty$ (see
[1, footnote 3, page 294] and reference therein for details). If the vorticity
enjoys some Besov regularity than it can be shown that the associated velocity
field is Lipschitz (Proposition 7.7 in [1]), but the point is that we avoid to
impose any assumption of this sort.

Before going further we recall the definition of the function space
$\mathrm{BMO}(\Omega)$.

######  Definition 2.5.

Let $\Omega=\mathbb{T}^{d}$ or $\Omega=\mathbb{R}^{d}$ and let $f\in
L_{\mathrm{loc}}^{1}(\Omega)$. We shall say that $f$ is of bounded mean
oscillation on $\Omega$ or in brief $\mathrm{BMO}(\Omega)$, if

$\left\|f\right\|_{\mathrm{BMO}(\Omega)}\coloneqq\underset{Q\subset\Omega}{\sup}\frac{1}{\left|Q\right|}\int_{Q}\left|f(x)-\frac{1}{\left|Q\right|}\int_{Q}f\right|dx<\infty.$
(2.1)

The supremum in Equation (2.1) is taken over all cubes $Q\subset\Omega$ with
edges parallel to the axis, and
$\left\|f\right\|_{\mathrm{\mathrm{BMO(}\Omega})}$ is called the
$\mathrm{BMO}$ norm.

The failure of the boundedness of the Calderon-Zygmund type operators from
$L^{\infty}$ to itself is bypassed by the fact that operators of the latter
type are bounded from $L^{\infty}$ to $\mathrm{BMO}$, precisely put we have
the following:

######  Theorem 2.6.

Let $V$ be as in 2.1. Let
$T\omega=T_{kj}\omega\coloneqq\nabla_{k}\nabla_{j}^{\bot}V\star\omega$. Then,
there is a constant $A>0$ such that

$\left\|T\omega\right\|_{\mathrm{BMO}(\mathbb{T}^{2})}\leq
A\left\|\omega\right\|_{L^{\infty}(\mathbb{T}^{2})}$

for all $\omega\in L^{\infty}(\mathbb{T}^{2})$. In particular, if $\omega$ is
as in 2.3 and $u$ is the associated velocity field then

$\underset{t\in[0,T]}{\sup}\left\|\nabla
u(t,\cdot)\right\|_{\mathrm{\mathrm{BMO}(\mathbb{T}^{2})}}\lesssim\left\|\omega(0,\cdot)\right\|_{L^{\infty}(\mathbb{T}^{2})}.$

###### Proof.

Write $V=V_{0}+V_{1}$. Denote by $\varpi$ the extension of $\omega$ by $0$ to
$\mathbb{R}^{2}$. Then

$\nabla_{k}\nabla_{j}^{\bot}V\star\omega(x)=\int_{\mathbb{R}^{2}}\nabla_{k}\nabla_{j}^{\bot}V_{1}(x-y)\mathbf{1}_{\mathbb{T}^{2}}(x-y)\varpi(y)dy+\int_{\mathbb{R}^{2}}\nabla_{k}\nabla_{j}^{\bot}V_{0}(x-y)\mathbf{1}_{\mathbb{T}^{2}}(x-y)\varpi(y)dy.$
(2.2)

Let $\chi$ be a smooth radial compactly supported function such that
$\mathrm{supp}(\chi)\subset B_{\frac{1}{4}}(0)$ and $\chi\equiv 1$ on
$B_{\frac{1}{8}}(0)$. The first integral is

$\displaystyle\int_{\mathbb{R}^{2}}\nabla_{k}\nabla_{j}^{\bot}V_{1}(x-y)\mathbf{1}_{\mathbb{T}^{2}}(x-y)\varpi(y)dy=$
$\displaystyle\int_{\mathbb{R}^{2}}\nabla_{k}\nabla_{j}^{\bot}V_{1}(x-y)\chi(x-y)\mathbf{1}_{\mathbb{T}^{2}}(x-y)\varpi(y)dy$
$\displaystyle+\int_{\mathbb{R}^{2}}\nabla_{k}\nabla_{j}^{\bot}V_{1}(x-y)(1-\chi(x-y))\mathbf{1}_{\mathbb{T}^{2}}(x-y)\varpi(y)dy$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{2}}\nabla_{k}\nabla_{j}^{\bot}V_{1}(x-y)\chi(x-y)\varpi(y)dy$
$+\int_{\mathbb{R}^{2}}\nabla_{k}\nabla_{j}^{\bot}V_{1}(x-y)(1-\chi(x-y))\mathbf{1}_{\mathbb{T}^{2}}(x-y)\varpi(y)dy.$
(2.3)

Thanks to the usual Calderon-Zygmund theory on $\mathbb{R}^{2}$ (see e.g.
[33]) we can estimate the $\mathrm{BMO}(\mathbb{T}^{2})$ norm of first
integral in (2.3) as

$\displaystyle\left\|\int_{\mathbb{R}^{2}}\nabla_{k}\nabla_{j}^{\bot}V_{1}(\cdot-y)\chi(\cdot-y)\varpi(y)dy\right\|_{\mathrm{BMO}(\mathbb{T}^{2})}$
$\displaystyle\leq\left\|\int_{\mathbb{R}^{2}}\nabla_{k}\nabla_{j}^{\bot}V_{1}(\cdot-y)\chi(\cdot-y)\varpi(y)dy\right\|_{\mathrm{BMO}(\mathbb{R}^{2})}$
$\displaystyle\leq
C\left\|\varpi\right\|_{L^{\infty}(\mathbb{R}^{2})}=C\left\|\omega\right\|_{L^{\infty}(\mathbb{T}^{2})}.$

Observe the inequality
$\left\|\cdot\right\|_{\mathrm{BMO}(\mathbb{T}^{2})}\leq
2\left\|\cdot\right\|_{L^{\infty}(\mathbb{T}^{2})}$. Therefore the first
integral is

$\displaystyle\left\|\int_{\mathbb{R}^{2}}\nabla_{k}\nabla_{j}^{\bot}V_{1}(\cdot-y)(1-\chi(\cdot-y))\mathbf{1}_{\mathbb{T}^{2}}(\cdot-y)\varpi(y)dy\right\|_{\mathrm{BMO}(\mathbb{T}^{2})}$
$\displaystyle\leq$ $\displaystyle
2\left\|\int_{\mathbb{R}^{2}}\nabla_{k}\nabla_{j}^{\bot}V_{1}(\cdot-y)(1-\chi(\cdot-y))\mathbf{1}_{\mathbb{T}^{2}}(\cdot-y)\varpi(y)dy\right\|_{L^{\infty}(\mathbb{T}^{2})}$
$\displaystyle\leq$ $\displaystyle
2\log(8)\left\|\varpi\right\|_{L^{\infty}(\mathbb{R}^{2})}=2\log(8)\left\|\omega\right\|_{L^{\infty}(\mathbb{T}^{2})}.$

As for the second integral in (2.2), recall that by Lemma (2.1) we have
$\nabla_{k}\nabla_{j}^{\bot}V_{0}\in L^{\infty}(\mathbb{T}^{2})$. It follows
that

$\displaystyle\left\|\int_{\mathbb{R}^{2}}\mathbf{1}_{\mathbb{T}^{2}}(\cdot-y)\nabla_{k}\nabla_{j}^{\bot}V_{0}(\cdot-y)\varpi(y)dy\right\|_{\mathrm{BMO}(\mathbb{T}^{2})}$
$\displaystyle\leq
2\left\|\int_{\mathbb{R}^{2}}\mathbf{1}_{\mathbb{T}^{2}}(\cdot-y)\nabla_{k}\nabla_{j}^{\bot}V_{0}(\cdot-y)\varpi(y)dy\right\|_{L^{\infty}(\mathbb{T}^{2})}$
$\displaystyle\leq
2\left\|\nabla_{k}\nabla_{j}^{\bot}V_{0}\right\|_{L^{\infty}(\mathbb{T}^{2})}\left\|\omega\right\|_{L^{\infty}(\mathbb{T}^{2})}.$

∎

The following control on several time dependent quantities in terms of the
initial data are classical (and are also recalled in [28] and references
therein)

######  Lemma 2.7.

It holds that
$\left\|u\right\|_{L^{\infty}([0,T];L^{\infty}(\mathbb{T}^{2}))}\lesssim\left\|\omega_{0}\right\|_{L^{\infty}(\mathbb{T}^{2})}$.

Proof. We have

$\displaystyle\left\|\nabla^{\bot}V\star\omega(t,\cdot)\right\|_{L^{\infty}(\mathbb{T}^{2})}\leq\left\|\nabla^{\bot}V_{1}\star\omega(t,\cdot)\right\|_{L^{\infty}(\mathbb{T}^{2})}+\left\|\nabla^{\bot}V_{0}\star\omega(t,\cdot)\right\|_{L^{\infty}(\mathbb{T}^{2})}.$

First

$\displaystyle\left\|\nabla^{\bot}V_{1}\star\omega(t,\cdot)\right\|_{L^{\infty}(\mathbb{T}^{2})}$
$\displaystyle\lesssim\left\|\frac{\boldsymbol{\mathbf{1}}_{\left|\cdot\right|\leq
1}}{\left|\cdot\right|}\star\omega(t,\cdot)\right\|_{L^{\infty}(\mathbb{T}^{2})}+\left\|\frac{\boldsymbol{\mathbf{1}}_{\left|\cdot\right|\geq
1}}{\left|\cdot\right|}\star\omega(t,\cdot)\right\|_{L^{\infty}(\mathbb{T}^{2})}$
$\displaystyle\lesssim\left\|\omega(t,\cdot)\right\|_{L^{\infty}\cap
L^{1}(\mathbb{T}^{2})}=\left\|\omega_{0}\right\|_{L^{\infty}\cap
L^{1}(\mathbb{T}^{2})}.$

Second

$\left\|\nabla^{\bot}V_{0}\star\omega(t,\cdot)\right\|_{L^{\infty}(\mathbb{T}^{2})}\leq\left\|\nabla^{\bot}V_{0}\right\|_{L^{\infty}(\mathbb{T}^{2})}\left\|\omega_{0}\right\|_{L^{1}(\mathbb{T}^{2})}.$

$\square$

##  3 Harmonic Analysis

###  3.1 The maximal function and local BMO spaces

Since we are working on the torus, which is a compact subset of
$\mathbb{R}^{d}$, we need to introduce a local variant of the $\mathrm{BMO}$
space. These spaces were studied in [19], following the pioneering work of
Fefferman-Stein on the topic.

Assumption $\mathbf{D}$. We assume the following on the pair $(\Omega,\mu)$.

1\. $\Omega\subset\mathbb{R}^{d}$ is non-empty and compact.

2\. $\mu$ is a positive Borel measure, finite on compact sets and
$\mathrm{supp}(\mu)\subset\Omega$.

3\. There are some constants $c,C$ such that for all $x\in\Omega$ and $0<r\leq
1$ it holds that

$cr^{d}\leq\mu(B(x,r))\leq Cr^{d}.$

######  Definition 3.1.

Let assumption D hold. We say that $f\in L^{1}(\Omega,d\mu)$ is of local
bounded bounded mean ossiclation on $\Omega$ if

$\left\|f\right\|_{\mathrm{bmo}(\Omega,\mu)}\coloneqq\underset{B=B(x,r),0<r\leq
1,x\in\Omega}{\sup}\underset{c}{\inf}\frac{1}{\mu(B)}\int_{B}\left|f-c\right|d\mu+\underset{B=B(x,1),x\in\Omega}{\sup}\frac{1}{\mu(B)}\int_{B}\left|f\right|d\mu<\infty.$

The space $\mathrm{bmo}(\Omega,\mu)$ consists of all $f\in L^{1}(\Omega,d\mu)$
such that $\left\|f\right\|_{\mathrm{bmo}(\Omega,\mu)}<\infty$ and
$\left\|f\right\|_{\mathrm{\mathrm{bmo(}\Omega})}$ is called the
$\mathrm{bmo}$ norm.

We now define the local maximal function.

######  Definition 3.2.

Let

$S\coloneqq\left\\{\varphi\in
C_{0}^{\infty}(\mathbb{R}^{d})\left|\mathrm{supp}(\varphi)\subset
B(0,1),\max\left\\{\left\|\varphi\right\|_{\infty},\left\|\nabla\varphi\right\|_{\infty}\right\\}\leq
1\right.\right\\}.$

Given $f\in L^{1}(\mathbb{R}^{d},d\mu)$ we define the local maximal function
by

$\mathcal{M}f(x)\coloneqq\underset{\varphi\in S}{\sup}\underset{0<r\leq
1}{\sup}\left|\frac{1}{\left|B(x,r)\right|}\int_{\mathbb{R}^{d}}\varphi\left(\frac{x-y}{r}\right)f(y)d\mu(y)\right|,\
x\in\Omega.$

Equipped with the above definitions we can now formulate the local duality
inequality

######  Theorem 3.3 ([19], Theorem 4.2).

Let assumption D hold. Then, it holds that

$\left|\int_{\Omega}f(x)g(x)d\mu(x)\right|\leq
C\left\|\mathcal{M}f\right\|_{L^{1}(\mu,\Omega)}\left\|g\right\|_{\mathrm{bmo}(\Omega,\mu)}.$

In what follows, we take $\Omega=\mathbb{T}^{2}$ and $\mu$ is defined for any
Borel set $B\subset\mathbb{R}^{2}$ by

$\mu(B)\coloneqq\left|\mathbb{T}^{2}\cap B\right|.$

In order to apply the above result freely for the pair $(\mathbb{T}^{2},\mu)$,
we verify that this pair satisfies assumption D.

######  Lemma 3.4.

The pair $(\mathbb{T}^{2},\mu)$ satisfies assumption D.

###### Proof.

Points 1 and 2 are trivial. The right hand side of the inequality in point 3
is also clear. Let us prove the left hand side inequality. Fix
$x\in\mathbb{T}^{2}$ and $0<r\leq 1$. Given a ball $B(x,r)$ with
$x\in\mathbb{T}^{d}$ and $0<r\leq 1$ let $Q(x,r)$ be the cube prescribing
$B(x,r)$, i.e. the cube with center $x$ and edge length $2r$. Note that the
ratio $\frac{\left|Q\right|}{\left|B\right|}$ is independent of $x,r$, and
therefore it suffices to prove the inequality for cubes with center $x$ and
edge length $2r$. Given such a cube $Q(x,r)$, consider the case where
$Q\cap\mathbb{R}^{2}\setminus\mathbb{T}^{2}\neq\emptyset$ (otherwise there is
nothing to prove). Note that in this case $\partial Q$ intersects
$\partial\mathbb{T}^{2}$ in exactly 2 points, say $z,z^{\prime}$ . If
$z,z^{\prime}$ are both on the same line then the upper\lower or right\left
half of $Q$ is contained in $\mathbb{T}^{2}$, so that

$\left|\mathbb{T}^{2}\cap Q\right|\geq\frac{1}{2}\left|Q\right|=2r^{2}.$

If $z^{\prime}$ is on the line orthogonal to the line of $z$ then a quarter of
$Q$ is contained in $\mathbb{T}^{2}$ so that

$\left|\mathbb{T}^{2}\cap
Q\right|\geq\frac{1}{4}\left|Q\right|=\frac{r^{2}}{2}.$

Finally, consider the case where $z^{\prime}$ is on the line parallel to the
line of $z$. In this case it must be that $\mathbb{T}^{2}\cap Q$ contains a
rectangle with edges of lengths $r$ and $1$. Thus

$\left|\mathbb{T}^{2}\cap Q\right|\geq r\cdot 1\geq r^{2}.$

∎

Next, we notice that $\left\|\cdot\right\|_{\mathrm{bmo}(\mathbb{T}^{2},\mu)}$
is dominated by $\left\|\cdot\right\|_{\mathrm{BMO}(\mathbb{T}^{2})}$ as
observed in the following Lemma.

######  Lemma 3.5.

It holds that

$\left\|f\right\|_{\mathrm{bmo}(\mathbb{T}^{2},\mu)}\leq\left\|f\right\|_{\mathrm{BMO}(\mathbb{T}^{2})}.$

###### Proof.

Given a ball $B(x,r)$ with $x\in\mathbb{T}^{2}$ and $0<r\leq 1$ let
$Q(x,r)\subset\mathbb{T}^{2}$ be the cube prescribing $B(x,r)$, i.e. the cube
with center $x$ and edge length $2r$. As before, consider the where case
$\partial Q$ intersects $\partial\mathbb{T}^{2}$ in exactly 2 points, say
$z,z^{\prime}$ . If $z,z^{\prime}$ are both on the same line, or orthogonal
lines, then enlarge the rectangle $Q\cap\mathbb{T}^{2}$ to a cube
$Q^{\prime}(x,r)$ with edge length $\leq 2r$ and $Q\cap\mathbb{T}^{2}\subset
Q^{\prime}(x,r)$. If $z,z^{\prime}$ are on parallel lines then $r>\frac{1}{2}$
in which case take $Q^{\prime}(x,r)=\mathbb{T}^{2}$. With this choice of
$Q^{\prime}(x,r)$ we see that

$cr^{2}\leq\left|Q^{\prime}(x,r)\right|\leq Cr^{2},\
B(x,r)\cap\mathbb{T}^{2}\subset Q^{\prime}(x,r).$

Therefore

$\displaystyle\underset{c}{\inf}\frac{1}{\mu(B)}\int_{B}\left|f-c\right|d\mu$
$\displaystyle\leq\frac{1}{\mu(B)}\int_{B}\left|f-\frac{1}{\left|Q^{\prime}\right|}\int_{Q^{\prime}}f\right|d\mu$
$\displaystyle\leq\frac{1}{\mu(B)}\int_{Q^{\prime}}\left|f-\frac{1}{\left|Q^{\prime}\right|}\int_{Q^{\prime}}f\right|dx\lesssim\frac{1}{\left|Q^{\prime}\right|}\int_{Q^{\prime}}\left|f-\frac{1}{\left|Q^{\prime}\right|}\int_{Q^{\prime}}f\right|dx,$

which shows that

$\underset{B(x,r),0<r\leq
1,x\in\Omega}{\sup}\underset{c}{\inf}\frac{1}{\mu(B)}\int_{B}\left|f-c\right|d\mu\lesssim\underset{Q\subset\mathbb{T}^{2}}{\sup}\frac{1}{\left|Q\right|}\int_{Q}\left|f-\frac{1}{\left|Q\right|}\int_{Q}f\right|dx\leq\left\|f\right\|_{\mathrm{BMO}(\mathbb{T}^{2})}.$

Since

$\frac{1}{\mu(B(x,1))}\int_{B(x,1)}\left|f\right|d\mu\lesssim\int_{\mathbb{T}^{2}}\left|f\right|dx\lesssim\left\|f\right\|_{\mathrm{BMO}(\mathbb{T}^{2})},$

we have thus proved

$\left\|f\right\|_{\mathrm{bmo}(\mathbb{T}^{2},\mu)}\lesssim_{d}\left\|f\right\|_{\mathrm{BMO}(\mathbb{T}^{2})}.$

∎

The combination of Theorem 3.3, Lemma 3.4 and Lemma 3.5 yields

######  Corollary 3.6.

It holds that

$\left|\int_{\mathbb{T}^{2}}f(x)g(x)dx\right|\leq
C\left\|\mathcal{M}f\right\|_{L^{1}(\mathbb{T}^{2})}\left\|g\right\|_{\mathrm{BMO}(\mathbb{T}^{2})}.$

###  3.2 A Wiener type inequality

The next Lemma is a Wiener type inequality, and its proof follows closely
[35].

######  Lemma 3.7.

Suppose that $g\in L^{1}(\mathbb{T}^{d})$. Then, for each $\eta>0$ it holds
that

$\int_{\mathbb{T}^{d}}\mathcal{M}g(x)dx\lesssim_{d}\eta+\int_{\mathbb{T}^{d}}\left|g\right|(x)\log_{+}\left|g\right|(x)dx+\log(\frac{1}{\eta})\left\|g\right\|_{L^{1}(\mathbb{T}^{d})}.$

###### Proof.

We have

$\displaystyle\int_{\mathbb{T}^{d}}\mathcal{M}g(x)dx$
$\displaystyle=\int_{0}^{\infty}\left|\left\\{x\in\mathbb{T}^{d}:\mathcal{M}g>t\right\\}\right|dt$
$\displaystyle=\frac{1}{2}\int_{0}^{\eta}\left|\left\\{x\in\mathbb{T}^{d}:\mathcal{M}g>2t\right\\}\right|dt+\frac{1}{2}\int_{\eta}^{\infty}\left|\left\\{x\in\mathbb{T}^{d}:\mathcal{M}g>2t\right\\}\right|dt.$

Evidently
$\left|\left\\{x\in\mathbb{T}^{d}:\mathcal{M}g>2t\right\\}\right|\leq 1$, so
that

$\frac{1}{2}\int_{0}^{\eta}\left|\left\\{x\in\mathbb{T}^{d}:\mathcal{M}g>t\right\\}\right|dt\leq\eta.$

We move to handle the second integral. Let us estimate the second integral. We
split $g$ to the parts in which it is smaller and larger than $g$, i.e.
$g=g_{1}+g_{2}$ where $g_{1}\coloneqq
g\mathbf{1}_{\left\\{\left|g\right|>t\right\\}}$ and
$g_{2}=g\mathbf{1}_{\left\\{\left|g\right|\leq t\right\\}}$. Evidently
$\mathcal{M}g_{2}\leq t$ so that

$\left\\{x\in\mathbb{T}^{d}\left|\mathcal{M}g>2t\right.\right\\}\subset\left\\{x\in\mathbb{T}^{d}\left|\mathcal{M}g_{1}>t\right.\right\\}.$

Thus, by the Hardy-Littlewood maximal Inequality we have (Theorem 7.4 in [30])

$\displaystyle\int_{\eta}^{\infty}\left|\left\\{x\in\mathbb{T}^{d}:\mathcal{M}g(x)>2t\right\\}\right|dt$
$\displaystyle\leq\int_{\eta}^{\infty}\left|\left\\{x\in\mathbb{T}^{d}:\mathcal{M}g_{1}(x)>t\right\\}\right|dt\lesssim\int_{\eta}^{\infty}\frac{1}{t}\left(\int_{\left|g\right|>t}\left|g\right|(x)dx\right)dt$
$\displaystyle=\int_{\eta}^{\infty}\int_{\mathbb{T}^{d}}\frac{1}{t}\mathbf{1}_{\left\\{\left|g\right|>t\right\\}}\left|g\right|(x)dxdt=\int_{\mathbb{T}^{d}}\left|g\right|(x)\int_{\eta}^{\infty}\frac{1}{t}\mathbf{1}_{\left\\{\left|g\right|>t\right\\}}dtdx$
$\displaystyle\leq\int_{\mathbb{T}^{d}}\left|g\right|\int_{\eta}^{\max\left\\{\left|g\right|,1\right\\}}\frac{dt}{t}dx=\int_{\mathbb{T}^{d}}\left|g\right|\log_{+}(\left|g\right|)dx-\log(\eta)\left\|g\right\|_{L^{1}(\mathbb{T}^{d})}.$

∎

##  4  The quasineutral limit for Yudovich solutions

###  4.1 Quantitative propagation in time of
$\left\|\rho^{\varepsilon}(t,\cdot)\right\|_{\infty}$ on $\mathbb{T}^{2}$

We first recall a classical velocity moment inequality for bounded
distribution functions and the conservation of energy for the Vlasov–Poisson
system, which we take from [17].

######  Lemma 4.1.

(Lemma 2.3 in [17]) Let $d\geq 1$. Let $0\leq f\in
L^{\infty}(\mathbb{T}^{d}\times\mathbb{R}^{d})$ satisfy for some $k>1$

$M_{k}\coloneqq\int_{\mathbb{T}^{d}\times\mathbb{R}^{d}}\left|\xi\right|^{k}f(x,\xi)dxd\xi<\infty.$

Then the spatial density

$\rho(x)\coloneqq\int_{\mathbb{R}^{d}}f(x,\xi)d\xi$

belongs to $L^{1+\frac{k}{d}}(\mathbb{T}^{d})$ with the estimate

$\left\|\rho\right\|_{1+\frac{k}{d}}\leq
C_{k,d}\left\|f\right\|_{\infty}^{\frac{k}{d+k}}M_{k}^{\frac{d}{d+k}}.$

######  Lemma 4.2.

Let the assumptions of A.1 hold and let $f^{\varepsilon}(t,x,\xi)$ be the
solution to the Cauchy problem

$\left\\{\begin{array}[]{lc}\partial_{t}f^{\varepsilon}+\xi\cdot\nabla_{x}f^{\varepsilon}-\nabla\Phi^{\varepsilon}\cdot\nabla_{\xi}f^{\varepsilon}=0,\
f^{\varepsilon}(0,x,\xi)=f_{0}^{\varepsilon}\\\
\rho^{\varepsilon}-1=-\varepsilon\Delta\Phi^{\varepsilon},\
\rho^{\varepsilon}(t,x)=\int_{\mathbb{R}^{2}}f^{\varepsilon}(t,x,\xi)d\xi.\end{array}\right.$

Then the energy

$\mathcal{E}_{\varepsilon}^{0}(t)\coloneqq\frac{1}{2}\int_{\mathbb{T}^{2}\times\mathbb{R}^{2}}\left|\xi\right|^{2}f^{\varepsilon}(t,x,\xi)dxd\xi+\frac{\varepsilon}{2}\int_{\mathbb{T}^{2}}\left|\nabla\Phi^{\varepsilon}\right|^{2}(t,x)dx$

is conserved, i.e.
$\mathcal{E}_{\varepsilon}^{0}(t)=\mathcal{E}_{\varepsilon}^{0}(0)$.

The combination of 4.1 and 4.2 gives a uniform estimate in time and
$\varepsilon$ on $\left\|\rho^{\varepsilon}(t,\cdot)\right\|_{2}$, i.e.

######  Lemma 4.3.

Let the assumptions $\mathbf{H1}$-$\mathbf{H2}$ hold. Suppose that
$\int_{\mathbb{R}^{2}}\left|\xi\right|^{2}f_{0}^{\varepsilon}(x,\xi)d\xi\leq
M_{2}$. Then

$\left\|\rho^{\varepsilon}(t,\cdot)\right\|_{2}\leq
C\sqrt{M_{2}}\sqrt{\overline{M}_{\varepsilon}},$

where $\overline{M}_{\varepsilon}$ is from $\mathbf{H2}$.

The main purpose of this section is to study the propagation in time of the
$L^{\infty}$ norm of the density of a given a solution to the Vlasov-Poisson
equation (VP). The fact that we are able to get _polynomial_ explicit bounds
in terms of $\varepsilon$ will prove to be crucial for closing the estimate
for the modulated energy. Our approach follows closely that of [17, section
5], adapted to the case of electrons. Define the time dependent quantity

$Q_{\ast}(t)\coloneqq\underset{(x,\xi)}{\sup}\left|\xi-\Xi_{\left(t,x,\xi\right)}(0)\right|,$

where $\Xi_{\left(t,x,\xi\right)}$ is the velocity characteristic of the
Vlasov–Poisson system defined just below at (4.1). The first step is to obtain
a bound on $\left\|\rho^{\varepsilon}(t,\cdot)\right\|_{\infty}$ by means of
$Q_{\ast}(t)$, which is the content of the following lemma.

######  Lemma 4.4.

Let assumptions $\mathbf{H1}$-$\mathbf{H2}$ hold. Let
$f^{\varepsilon}(t,x,\xi)$ be a solution to the Cauchy problem

$\left\\{\begin{array}[]{lc}\partial_{t}f^{\varepsilon}+\xi\cdot\nabla_{x}f^{\varepsilon}-\nabla\Phi^{\varepsilon}\cdot\nabla_{\xi}f^{\varepsilon}=0,\
f^{\varepsilon}(0,x,\xi)=f_{0}^{\varepsilon}\vspace{0.25 cm}\\\
\rho^{\varepsilon}-1=-\varepsilon\Delta\Phi^{\varepsilon},\
\rho^{\varepsilon}(t,x)=\int_{\mathbb{R}^{2}}f^{\varepsilon}(t,x,\xi)d\xi.\end{array}\right.$

Then

$\left\|\rho^{\varepsilon}(t,\cdot)\right\|_{\infty}\leq C(1+Q_{\ast}^{2}(t))$

where

$C=C(\overline{M}_{\varepsilon},k_{0})=\overline{M}_{\varepsilon}c_{k_{0}}.$

Proof. Consider the trajectories

$\left\\{\begin{array}[]{lc}\frac{d}{ds}X_{\left(t,x,\xi\right)}(s)=\Xi_{\left(t,x,\xi\right)}(s),\
X_{\left(t,x,\xi\right)}(t)=x\vspace{0.25 cm}\\\
\frac{d}{ds}\Xi_{\left(t,x,\xi\right)}(s)=\nabla\Phi_{\left(t,x,\xi\right)}^{\varepsilon}(s,X(s)),\
\Xi_{\left(t,x,\xi\right)}(t)=\xi.\end{array}\right.$ (4.1)

We can write the solution $f^{\varepsilon}(t,x,\xi)$ in a Lagrangian manner,
namely

$f^{\varepsilon}(t,x,\xi)=f_{0}^{\varepsilon}(X_{\left(t,x,\xi\right)}(0),\Xi_{\left(t,x,\xi\right)}(0))$

for a.e. $(x,\xi)\in\mathbb{T}^{2}\times\mathbb{R}^{2}$. Therefore, by
assumption we have the estimate

$f^{\varepsilon}(t,x,\xi)\leq\frac{\overline{M}_{\varepsilon}}{1+\left|\Xi_{\left(t,x,\xi\right)}(0)\right|^{k_{0}}},\
(t,x,\xi)\in[0,T]\times\mathbb{T}^{2}\times\mathbb{R}^{2}.$

Furthermore

$\left|\Xi_{\left(t,x,\xi\right)}(0)\right|\geq\left|\left|\xi\right|-\left|\xi-\Xi_{\left(t,x,\xi\right)}(0)\right|\right|\geq\left(\left|\xi\right|-\left|\xi-\Xi_{\left(t,x,\xi\right)}(0)\right|\right)_{+}\geq\left(\left|\xi\right|-Q_{\ast}(t)\right)_{+},$

and as a result we get

$f^{\varepsilon}(t,x,\xi)\leq\frac{\overline{M}_{\varepsilon}}{1+\left(\left|\xi\right|-Q_{\ast}(t)\right)_{+}^{k_{0}}}.$

Integration in $\xi$ yields the estimate

$\rho^{\varepsilon}(t,x)\leq\overline{M}_{\varepsilon}\int_{\mathbb{R}^{2}}\frac{1}{1+\left(\left|\xi\right|-Q_{\ast}(t)\right)_{+}^{k_{0}}}d\xi.$

Noticing that the integrand in the last integral is radial in $\xi$, we can
write

$\int_{\mathbb{R}^{2}}\frac{1}{1+\left(\left|\xi\right|-Q_{\ast}(t)\right)_{+}^{k_{0}}}d\xi=\int_{0}^{\infty}\frac{r}{1+\left(r-Q_{\ast}(t)\right)_{+}^{k_{0}}}dr.$

The estimate for the last integral proceeds as follows. First we split it as

$\int_{0}^{Q_{\ast}(t)}rdr+\int_{Q_{\ast}(t)}^{\infty}\frac{r}{1+\left(r-Q_{\ast}(t)\right)_{+}^{k_{0}}}dr=\frac{1}{2}Q_{\ast}^{2}(t)+\int_{0}^{\infty}\frac{r-Q_{\ast}(t)}{1+r^{k_{0}}}dr.$
(4.2)

Since $k_{0}>2$, the second integral in the right hand side of Equation (4.2)
is bounded by

$\int_{0}^{\infty}\frac{r+Q_{\ast}(t)}{1+r^{k_{0}}}dr\leq
c^{\prime}_{k_{0}}(1+Q_{\ast}(t)).$

To conclude, we arrive at the announced estimate because

$\rho^{\varepsilon}(t,x)\leq\frac{\overline{M}_{\varepsilon}}{2}Q_{\ast}^{2}(t)+\overline{M}_{\varepsilon}c^{\prime}_{k_{0}}(1+Q_{\ast}(t))\leq
C(\overline{M}_{\varepsilon},k_{0})(1+Q_{\ast}^{2}(t)).$

$\square$

The second step is the following functional inequality which ultimately leads
to an estimate on
$\left\|\nabla\Phi^{\varepsilon}\right\|_{L^{\infty}(\mathbb{T}^{2})}$.

######  Lemma 4.5.

Suppose that $\rho^{\varepsilon}\in L^{\infty}(\mathbb{T}^{2})$ and
$\left\|\rho^{\varepsilon}\right\|_{L^{1}(\mathbb{T}^{2})}=1$. Denote

$\Phi^{\varepsilon}(x)\coloneqq\frac{1}{\varepsilon}(-\Delta)^{-1}\left(\rho^{\varepsilon}-1\right).$

Then, for all $l\in[0,1[$ it holds that

$\left\|\nabla\Phi^{\varepsilon}\right\|_{L^{\infty}(\mathbb{T}^{2})}\leq\frac{2}{\varepsilon}\left\|\nabla
V_{0}\right\|_{L^{\infty}(\mathbb{T}^{2})}+\frac{1}{\varepsilon}l\left(\left\|\rho^{\varepsilon}\right\|_{L^{\infty}(\mathbb{T}^{2})}+1\right)+\frac{1}{\varepsilon}\sqrt{\left|\log\left(l\right)\right|}\left(\left\|\rho^{\varepsilon}\right\|_{L^{2}(\mathbb{T}^{2})}+1\right).$

Proof.

$\nabla\Phi^{\varepsilon}=\frac{1}{\varepsilon}\int\frac{x-y}{\left|x-y\right|^{2}}\left(\rho^{\varepsilon}-1\right)(t,y)dy+\frac{1}{\varepsilon}\nabla
V_{0}\star\left(\rho^{\varepsilon}-1\right).$

By 2.1, (b) we have $\nabla V_{0}\in C^{\infty}(\mathbb{T}^{2})$ and therefore
we can estimate

$\displaystyle\left\|\nabla
V_{0}\star\left(\rho^{\varepsilon}-1\right)\right\|_{L^{\infty}(\mathbb{T}^{2})}\leq\left\|\nabla
V_{0}\right\|_{L^{\infty}(\mathbb{T}^{2})}\left\|\rho^{\varepsilon}-1\right\|_{{}_{L^{1}(\mathbb{T}^{2})}}\leq
2\left\|\nabla V_{0}\right\|_{L^{\infty}(\mathbb{T}^{2})}.$

As for the first integral we split it as

$=\frac{1}{\varepsilon}\int_{\left|x-y\right|\leq
l}\frac{x-y}{\left|x-y\right|^{2}}\left(\rho^{\varepsilon}-1\right)(t,y)dy+\frac{1}{\varepsilon}\int_{\left|x-y\right|\geq
l}\frac{x-y}{\left|x-y\right|^{2}}\left(\rho^{\varepsilon}-1\right)(t,y)dy.$
(4.3)

For the first integral in (4.3)), note that

$\left|\int_{\left|x-y\right|\leq
l}\frac{x-y}{\left|x-y\right|^{2}}\left(\rho^{\varepsilon}-1\right)(t,y)dy\right|\leq\left\|\frac{\mathbf{1}_{\left|\cdot\right|\leq
l}}{\left|\cdot\right|}\star(\rho^{\varepsilon}-1)\right\|_{L^{\infty}(\mathbb{T}^{2})}\leq
l\left\|\rho^{\varepsilon}-1\right\|_{L^{\infty}(\mathbb{T}^{2})}\leq
l\left(\left\|\rho^{\varepsilon}\right\|_{L^{\infty}(\mathbb{T}^{2})}+1\right).$
(4.4)

For the second integral, observe that

$\displaystyle\left\|\frac{\mathbf{1}_{\left|\cdot\right|\geq
l}}{\left|\cdot\right|}\star\left(\rho^{\varepsilon}-1\right)\right\|_{L^{\infty}(\mathbb{T}^{2})}$
$\displaystyle\leq\left\|\frac{\mathbf{1}_{\left|\cdot\right|\geq
l}}{\left|\cdot\right|}\right\|_{L^{2}(\mathbb{T}^{2})}\left\|\rho^{\varepsilon}\right\|_{L^{2}(\mathbb{T}^{2})}$
$\displaystyle\leq\sqrt{\left|\log\left(l\right)\right|}\left(\left\|\rho^{\varepsilon}\right\|_{L^{2}(\mathbb{T}^{2})}+1\right).$

Gathering the inequalities gives

$\left\|\nabla\Phi^{\varepsilon}\right\|_{L^{\infty}(\mathbb{T}^{2})}\leq\frac{2}{\varepsilon}\left\|\nabla
V_{0}\right\|_{L^{\infty}(\mathbb{T}^{2})}+\frac{1}{\varepsilon}l\left(\left\|\rho^{\varepsilon}\right\|_{L^{\infty}(\mathbb{T}^{2})}+1\right)+\frac{1}{\varepsilon}\sqrt{\left|\log\left(l\right)\right|}\left(\left\|\rho^{\varepsilon}\right\|_{L^{2}(\mathbb{T}^{2})}+1\right).$

$\square$

We are now in a good position to study quantitatively the propagation in time
of $\left\|\rho^{\varepsilon}(t,\cdot)\right\|_{\infty}.$

######  Lemma 4.6.

Let assumptions $\mathbf{H1}$-$\mathbf{H2}$ hold. Let
$f^{\varepsilon}(t,x,\xi)$ be the solution to the Cauchy problem (VP) with
initial data $f_{0}^{\varepsilon}$. Then, for all $t\in[0,\infty)$ it holds
that

$\left\|\rho^{\varepsilon}(t,\cdot)\right\|_{\infty}\lesssim\mathbf{M}_{\varepsilon}+\mathbf{M}_{\varepsilon}^{3}t^{2}\left(1+\log\left(1+\mathbf{M}_{\varepsilon}t\right)\right)\coloneqq\Gamma_{\varepsilon}(t),$

where
$\mathbf{M}_{\varepsilon}\coloneqq\frac{1+(\overline{M_{\varepsilon}})^{\frac{1}{2}}}{\varepsilon}$.

###### Proof.

We can write the solution $f^{\varepsilon}(t,x,\xi)$ in a Lagrangian manner,
namely

$f^{\varepsilon}(t,x,\xi)=f_{0}^{\varepsilon}(X_{\left(t,x,\xi\right)}(0),\Xi_{\left(t,x,\xi\right)}(0))$

for a.e. $(x,\xi)\in\mathbb{T}^{2}\times\mathbb{R}^{2}$ where
$(X_{\left(t,x,\xi\right)},\Xi_{\left(t,x,\xi\right)})$ are given by (4.1). By
4.5 applied with
$l=\frac{1}{1+\left\|\rho^{\varepsilon}(t,\cdot)\right\|_{\infty}}$, 4.3, and
4.4 we have

$\left\|\nabla\Phi_{\varepsilon}(t,\cdot)\right\|_{L^{\infty}(\mathbb{T}^{2})}\lesssim\frac{1}{\varepsilon}+\frac{1+(\overline{M_{\varepsilon}})^{\frac{1}{2}}}{\varepsilon}\sqrt{\left|\log\left(\overline{M}_{\varepsilon}(1+Q_{\ast}^{2}(t))\right)\right|}.$
(4.5)

Setting
$\mathbf{M}_{\varepsilon}\coloneqq\frac{1+(\overline{M_{\varepsilon}})^{\frac{1}{2}}}{\varepsilon}$
we see that

$\left|\Xi_{\left(s,x,\xi\right)}(0)-\xi\right|\leq\left|\int_{0}^{s}\nabla\Phi_{\left(t,x,\xi\right)}^{\varepsilon}(\tau,X(\tau))d\tau\right|$

which shows that

$\leq\mathbf{M}_{\varepsilon}t\underset{s\in[0,t]}{\sup}\left(1+\sqrt{\left|\log\left(\mathbf{M}_{\varepsilon}(1+Q_{\ast}^{2}(s))\right)\right|}\right),$

so that

$\displaystyle\underset{s\in[0,t]}{\sup}Q_{\ast}(s)\leq\underset{s\in[0,t]}{\sup}\int_{0}^{s}\mathbf{M}_{\varepsilon}\left(1+\sqrt{\log(\mathbf{M}_{\varepsilon}(1+Q_{\ast}^{2}(\tau)))}\right)d\tau$
$\displaystyle\leq
t\mathbf{M}_{\varepsilon}\left(1+\sqrt{\log\left(\mathbf{M}_{\varepsilon}\left(1+\underset{s\in[0,t]}{\sup}Q_{\ast}^{2}(s)\right)\right)}\right).$

Solving the above inequality (see Lemma A.1 in [17] for more details) yields

$\underset{s\in[0,t]}{\sup}Q_{\ast}(s)\lesssim\mathbf{M}_{\varepsilon}t\sqrt{1+\log\left(1+\mathbf{M}_{\varepsilon}t\right)}.$

Thanks to 4.4

$\left\|\rho^{\varepsilon}(t,\cdot)\right\|_{\infty}\lesssim\mathbf{M}_{\varepsilon}+\mathbf{M}_{\varepsilon}^{3}t^{2}\left(1+\log\left(1+\mathbf{M}_{\varepsilon}t\right)\right)\coloneqq\Gamma_{\varepsilon}(t).$

∎

###  4.2 The Grönwall estimate for the modulated energy

We consider the modulated energy, which we recall is given by

$E_{\varepsilon}(t)\coloneqq\frac{1}{2}\int\left|\xi-u(t,x)\right|^{2}f^{\varepsilon}(t,x,\xi)dxd\xi+\frac{\varepsilon}{2}\int\left|\nabla\Phi^{\varepsilon}\right|^{2}(t,x)dx.$
(4.6)

Here $f^{\varepsilon}(t,x,\xi)$ is a solution of the Vlasov–Poisson system

$\left\\{\begin{array}[]{c}\partial_{t}f^{\varepsilon}+\xi\cdot\nabla_{x}f^{\varepsilon}-\nabla\Phi^{\varepsilon}\cdot\nabla_{\xi}f^{\varepsilon}=0,\
f^{\varepsilon}(0,x,\xi)=f_{0}^{\varepsilon}\\\
\rho^{\varepsilon}-1=-\varepsilon\Delta\Phi^{\varepsilon},\
\rho^{\varepsilon}(t,x)=\int_{\mathbb{R}^{2}}f^{\varepsilon}(t,x,\xi)d\xi,\end{array}\right.$
(4.7)

and $(\omega,u)$ is a solution to the 2D incompressible Euler equation

$\partial_{t}\omega+\mathrm{div}(\omega u)=0,\ \omega=\Delta\psi,\
u=-\nabla^{\bot}\psi.$ (4.8)

We introduce the notation

$d(u)\coloneqq\frac{1}{2}\left(\nabla u+\nabla^{T}u\right)$

and

$A(u)\coloneqq\partial_{t}u+u\cdot\nabla u.$

The following calculation can be found in Appendix 7 of [4].

######  Theorem 4.7.

Let assumptions H1-H4 and let the assumption of 2.3 hold. Then, it holds that

$\displaystyle\dot{E_{\varepsilon}}(t)=-$
$\displaystyle\int_{\mathbb{T}^{2}}\int_{\mathbb{R}^{2}}d(u)(t,x):(\xi-u(t,x))\otimes(\xi-u(t,x))f^{\varepsilon}(t,x,\xi)dxd\xi$
$\displaystyle+\varepsilon\int_{\mathbb{T}^{2}}d(u)(t,x):\nabla\Phi^{\varepsilon}\otimes\nabla\Phi^{\varepsilon}dx+\int_{\mathbb{T}^{2}}A(u)(t,x)\cdot
u(t,x)\left(\rho^{\varepsilon}(t,x)-1\right)dx\coloneqq\stackrel{{\scriptstyle[}}{{k}}=1]{3}{\sum}I_{k}.$

In order to conclude the Grönwall estimate, we now aim to estimate each of the
summands $I_{k}$ by means of the modulated energy $E_{\varepsilon}(t)$. The
main theorem of this section, from which our main result 1.1 follows directly,
is

######  Theorem 4.8.

Let assumptions H1-H4 hold. Let $f^{\varepsilon}(t,x,\xi)$ and
$(\omega(t,x),u(t,x))$ be the solutions to (4.7) and (4.8) respectively
(ensured by A.1 and 2.3 respectively). Let $E_{\varepsilon}(t)$ be given by
formula (4.6). Then, there is some
$T_{\ast}=T_{\ast}(\left\|\omega_{0}\right\|_{L^{\infty}(\mathbb{T}^{2})})$
such that
$\underset{t\in[0,T_{\ast}]}{\sup}E_{\varepsilon}(t)\underset{\varepsilon\rightarrow
0}{\rightarrow}0$.

Proof. Step 1. Estimate on $I_{1}$. Let
$F^{\varepsilon}:\mathbb{T}^{2}\rightarrow\mathbb{R}$ be defined as follows

$F^{\varepsilon}(t,x)\coloneqq\int_{\mathbb{R}^{2}}(\xi-u(t,x))^{i}(\xi-u(t,x))^{j}f^{\varepsilon}(t,x,\xi)d\xi.$

First, using Einstein’s summation

$\displaystyle\left|I_{1}\right|=$
$\displaystyle\left|\int_{\mathbb{T}^{2}\times\mathbb{R}^{2}}\partial_{x_{i}}u^{j}(t,\cdot)(\xi-u(t,\cdot))^{i}(\xi-u(t,\cdot))^{j}f^{\varepsilon}(t,x,\xi)dxd\xi\right|$
$\displaystyle\leq\underset{t\in[0,T]}{\sup}\left\|\nabla
u(t,\cdot)\right\|_{\mathrm{BMO(\mathbb{T}^{2})}}\left\|\mathcal{M}F^{\varepsilon}\right\|_{L^{1}(\mathbb{T}^{2})}\lesssim_{\left\|\omega_{0}\right\|_{L^{\infty}(\mathbb{T}^{2})}}\left\|\mathcal{M}F^{\varepsilon}\right\|_{L^{1}(\mathbb{T}^{2})},$

where we obtain the first inequality thanks to 3.6, and the second inequality
is by 2.6. Now, by 3.7 we get:

$\displaystyle\left\|\mathcal{M}F^{\varepsilon}(t,\cdot)\right\|_{L^{1}(\mathbb{T}^{2})}\leq\eta+2\int_{\mathbb{T}^{2}}\left|F^{\varepsilon}(t,x)\right|\log_{+}\left|F^{\varepsilon}(t,x)\right|dx+2\log\left(\frac{1}{\eta}\right)\left\|F^{\varepsilon}(t,\cdot)\right\|_{L^{1}(\mathbb{T}^{2})}.$

In order to bound
$\int_{\mathbb{R}^{2}}\left|F^{\varepsilon}(t,x)\right|\log_{+}\left|F^{\varepsilon}(t,x)\right|dx$
by means of $\left\|F^{\varepsilon}(t,\cdot)\right\|_{L^{1}(\mathbb{T}^{2})}$,
we need an estimate on
$\left\|F^{\varepsilon}(t,\cdot)\right\|_{L^{\infty}(\mathbb{T}^{2})}$. By 2.7
and 4.2 we have

$\displaystyle\left|F^{\varepsilon}(t,x)\right|\leq\int_{\mathbb{R}^{2}}\left|\xi\right|^{2}f^{\varepsilon}(t,x,\xi)d\xi+\left\|\omega_{0}\right\|_{L^{\infty}(\mathbb{T}^{2})}\int_{\mathbb{R}^{2}}f^{\varepsilon}(t,x,\xi)d\xi$
$\lesssim_{\left\|\omega_{0}\right\|_{L^{\infty}(\mathbb{T}^{2})}}\mathcal{E}_{\varepsilon}^{0}(0)+\left\|\rho_{f}^{\varepsilon}(t,\cdot)\right\|_{L^{\infty}(\mathbb{T}^{2})}.$
(4.9)

Hence, 4.2 and 4.6 entails

$\displaystyle\left\|\mathcal{M}F^{\varepsilon}(t,\cdot)\right\|_{L^{1}(\mathbb{T}^{2})}\lesssim$
${}_{\left\|\omega_{0}\right\|_{L^{\infty}(\mathbb{T}^{2})}}\eta+\log\left(\mathcal{E}_{\varepsilon}^{0}(0)+\left\|\rho_{f}^{\varepsilon}(t,\cdot)\right\|_{L^{\infty}(\mathbb{T}^{2})}\right)\left\|F^{\varepsilon}(t,\cdot)\right\|_{L^{1}(\mathbb{T}^{2})}+\log\left(\frac{1}{\eta}\right)\left\|F^{\varepsilon}(t,\cdot)\right\|_{L^{1}(\mathbb{T}^{2})}$
$\displaystyle\lesssim$
$\displaystyle\eta+\log\left(\mathcal{E}_{\varepsilon}^{0}(0)+\Gamma_{\varepsilon}(t)\right)E_{\varepsilon}(t)+\log\left(\frac{1}{\eta}\right)E_{\varepsilon}(t).$

To conclude

$|I_{1}(t)|\lesssim_{\left\|\omega_{0}\right\|_{L^{\infty}(\mathbb{T}^{2})}}\left(\eta+\log\left(\mathcal{E}_{\varepsilon}^{0}(0)+\Gamma_{\varepsilon}(t)\right)E_{\varepsilon}(t)+\log\left(\frac{1}{\eta}\right)E_{\varepsilon}(t)\right).$
(4.10)

Step 2. Estimate on $I_{2}$. The estimate for $I_{2}$ is obtained via
considerations similar to the ones in Step 1. We have

$I_{2}(t)=\frac{1}{\varepsilon}\int_{\mathbb{T}^{2}}\mathrm{div}(u)\left|\nabla
V\star(\rho_{f}^{\varepsilon}-1)(t,x)\right|^{2}dx.$

Thanks to 2.6 and 3.6 we have:

$\displaystyle\left|\int_{\mathbb{T}^{2}}\mathrm{div}(u)\left|\nabla
V\star(\rho_{f}^{\varepsilon}-1)(t,x)\right|^{2}dx\right|$
$\displaystyle\leq\left\|\mathrm{div}(u(t,\cdot))\right\|_{\mathrm{BMO(\mathbb{T}^{2})}}\left\|\mathcal{M}\left|\nabla
V\star(\rho_{f}^{\varepsilon}-1)\right|^{2}(t,\cdot)\right\|_{{L}^{1}(\mathbb{T}^{2})}$
$\displaystyle\lesssim_{\left\|\omega_{0}\right\|_{L^{\infty}(\mathbb{T}^{2})}}\left\|\mathcal{M}\left|\nabla
V\star(\rho_{f}^{\varepsilon}-1)\right|^{2}(t,\cdot)\right\|_{L^{1}(\mathbb{T}^{2})}.$

Notice that, using almost the same analysis as in the proof of 4.5, we obtain
the bound

$\left\|\left|\nabla
V\star(\rho_{f}^{\varepsilon}-1)(t,x)\right|^{2}\right\|_{L^{\infty}(\mathbb{T}^{2})}\lesssim
1+\left\|\rho_{f}^{\varepsilon}(t,\cdot)\right\|_{L^{1}(\mathbb{T}^{2})}^{2}+\left\|\rho_{f}^{\varepsilon}(t,\cdot)\right\|_{L^{\infty}(\mathbb{T}^{2})}^{2}\lesssim
1+\Gamma_{\varepsilon}^{2}(t).$

Thanks to 3.7 and proceeding as in Step 1 we find

$|I_{2}(t)|\lesssim_{\left\|\omega_{0}\right\|_{L^{\infty}(\mathbb{T}^{2})}}\eta+\log\left(1+\Gamma_{\varepsilon}^{2}(t)\right)E_{\varepsilon}(t)+\log\left(\frac{1}{\eta}\right)E_{\varepsilon}(t).$
(4.11)

Step 3. Estimate on $I_{3}$. Let $\varphi(t,x)=A(t,x)\cdot u(t,x)$. We have

$\displaystyle\left|\int_{\mathbb{T}^{2}}\varphi(t,x)\left(\rho^{\varepsilon}(t,x)-1\right)dx\right|$
$\displaystyle\leq\underset{t\in[0,T]}{\sup}\left\|\nabla\varphi(t,\cdot)\right\|_{2}\left\|\rho^{\varepsilon}(t,\cdot)-1\right\|_{H^{-1}(\mathbb{T}^{d})}$
$\displaystyle\leq\varepsilon\underset{t\in[0,T]}{\sup}\left\|\nabla\varphi(t,\cdot)\right\|_{2}^{2}+\varepsilon\left\|\nabla\Phi^{\varepsilon}(t)\right\|_{L^{2}(\mathbb{T}^{2})}^{2}$
$\displaystyle\leq\varepsilon\underset{t\in[0,T]}{\sup}\left\|\nabla\varphi(t,\cdot)\right\|_{2}^{2}+E_{\varepsilon}(t)\lesssim_{\left\|\omega_{0}\right\|_{L^{\infty}(\mathbb{T}^{2})}}\varepsilon+E_{\varepsilon}(t).$

Step 4. Conclusion. Altogether, we obtain

$\dot{E_{\varepsilon}}(t)\lesssim_{\left\|\omega_{0}\right\|_{L^{\infty}(\mathbb{T}^{2})}}\varepsilon+\eta+\left(\log\left(\mathcal{E}_{\varepsilon}^{0}(0)+\Gamma_{\varepsilon}(t)\right)+\log\left(1+\Gamma_{\varepsilon}^{2}(t)\right)+\log\left(\frac{1}{\eta}\right)\right)E_{\varepsilon}(t).$

By Grönwall’s lemma we get

$E_{\varepsilon}(t)\leq\frac{\left(\mathcal{E}_{\varepsilon}^{0}(0)+\Gamma_{\varepsilon}(t)\right)^{CT}\left(1+\Gamma_{\varepsilon}^{2}(t)\right)^{CT}}{\eta^{CT}}\left(\varepsilon+\eta+E_{\varepsilon}(0)\right)\leq\frac{1}{\varepsilon^{\gamma
CT}\eta^{CT}}\left(\varepsilon+\eta+E_{\varepsilon}(0)\right),$

where $C=C(\left\|\omega_{0}\right\|_{L^{\infty}(\mathbb{T}^{2})})$ and
$\gamma=\gamma(\alpha)>0$. Choosing $\eta\sim\varepsilon^{\delta}$ for some
arbitrary $\delta>0$ and
$T_{\ast}=T_{\ast}(\left\|\omega_{0}\right\|_{L^{\infty}(\mathbb{T}^{2})},\beta,\alpha,\delta)>0$
sufficiently small, and using H4, we finally get
$\underset{t\in[0,T_{\ast}]}{\sup}E_{\varepsilon}(t)\underset{\varepsilon\rightarrow
0}{\rightarrow}0$.

$\square$

######  Remark 4.9.

The convergence
$\underset{t\in[0,T]}{\sup}E_{\varepsilon}(t)\underset{\varepsilon\rightarrow
0}{\rightarrow}0$ implies
$\rho^{\varepsilon}(t,\cdot)\underset{\varepsilon\rightarrow
0}{\rightharpoonup}\omega(t,\cdot)$ weakly in the sense of measures uniformly
on $[0,T]$.

##  5  Construction of well prepared initial data

Take

$f_{0}^{\varepsilon}(x,\xi)\coloneqq\frac{1}{\varepsilon^{\beta}}\exp\left(-\frac{\left|\xi-u^{0}(x)\right|^{2}}{2\varepsilon^{\beta}}\right).$

By variable change we see that

$\int_{\mathbb{R}^{2}}f_{0}^{\varepsilon}(x,\xi)d\xi=1$

hence

$\iint_{\mathbb{T}^{2}\times\mathbb{R}^{2}}f_{0}^{\varepsilon}(x,\xi)d\xi
dx=1,$

so that H1 holds. This also shows that for this choice the interaction part
vanishes identicaly

$E_{\varepsilon}^{2}(0)=0.$

As for the kinetic part, we compute

$E_{\varepsilon}^{1}(0)=\frac{1}{2}\iint_{\mathbb{T}^{2}\times\mathbb{R}^{2}}\left|\xi-
u_{0}(x)\right|^{2}f_{0}^{\varepsilon}(x,\xi)dxd\xi=\frac{1}{\varepsilon^{\beta}}\iint_{\mathbb{T}^{2}\times\mathbb{R}^{2}}\left|\xi\right|^{2}\exp\left(-\frac{\left|\xi\right|^{2}}{2\varepsilon^{\beta}}\right)dxd\xi$

$=\varepsilon^{\beta}\iint_{\mathbb{T}^{2}\times\mathbb{R}^{2}}\left|\xi\right|^{2}\exp\left(-\frac{\left|\xi\right|^{2}}{2}\right)d\xi
dx=O(\varepsilon^{\beta}).$ (5.1)

This shows that $E_{\varepsilon}(0)=O(\varepsilon^{\beta})$, which is
requirement H2. To verify H3, note that

$\left\|f_{0}^{\varepsilon}\right\|_{L^{\infty}\cap
L^{1}}\leq\frac{1}{\varepsilon^{\beta}}+1$

and

$\left\|\left|\xi\right|^{k_{0}}f_{0}^{\varepsilon}\right\|_{L^{\infty}\cap
L^{1}}\leq\frac{1}{\varepsilon^{\beta}}+\frac{1}{\varepsilon^{\beta}}\iint_{\mathbb{T}^{2}\times\mathbb{R}^{2}}\left|\xi\right|^{k_{0}}\exp\left(-\frac{\left|\xi-u^{0}(x)\right|^{2}}{2\varepsilon^{\beta}}\right)d\xi
dx.$ (5.2)

After changing variables the integral on the right hand side of (5.2) writes

$\displaystyle\frac{1}{\varepsilon^{\beta}}\iint_{\mathbb{T}^{2}\times\mathbb{R}^{2}}\left|\xi+u^{0}\right|^{k_{0}}\exp\left(-\frac{\left|\xi\right|^{2}}{2\varepsilon^{\beta}}\right)d\xi
dx\lesssim_{k_{0}}$
$\displaystyle\frac{1}{\varepsilon^{\beta}}\iint_{\mathbb{T}^{2}\times\mathbb{R}^{2}}\left|\xi\right|^{k_{0}}\exp\left(-\frac{\left|\xi\right|^{2}}{2\varepsilon^{\beta}}\right)d\xi
dx$
$\displaystyle+\left\|u^{0}\right\|_{\infty}^{k}\frac{1}{\varepsilon^{\beta}}\iint_{\mathbb{T}^{2}\times\mathbb{R}^{2}}\exp\left(-\frac{\left|\xi\right|^{2}}{2\varepsilon^{\beta}}\right)d\xi
dx$
$\displaystyle=\left\|u^{0}\right\|_{\infty}^{k_{0}}+\varepsilon^{\frac{\beta
k_{0}}{2}}\iint_{\mathbb{T}^{2}\times\mathbb{R}^{2}}\left|\xi\right|^{k_{0}}\exp\left(-\frac{\left|\xi\right|^{2}}{2}\right)d\xi
dx$ $\displaystyle\lesssim_{k_{0},\left\|u^{0}\right\|_{\infty}}1.$

So to conclude this shows that

$\left\|(1+\left|\xi\right|^{k_{0}})f_{0}^{\varepsilon}\right\|_{L^{\infty}\cap
L^{1}}\lesssim_{k_{0},\left\|u^{0}\right\|_{\infty}}\varepsilon^{-\beta}.$

##  Appendix A Well-posedness of the Vlasov–Poisson system on the two-
dimensional torus

For completeness, we recall the well-posedness theory of the Vlasov–Poisson
system on $\mathbb{T}^{2}\times\mathbb{R}^{2}$. Here the Debye length doesn’t
play any role, thus we set $\varepsilon=1$. We follow the presentation of the
well-posedness theory for (VP) in [15].

######  Theorem A.1.

(Well-posedness for Vlasov–Poisson on the $2$ dimensional torus) Let $f_{0}\in
L^{\infty}\cap L^{1}(\mathbb{T}^{2}\times\mathbb{R}^{2})$ be a probability
density. Assume that

  1. 1.

There are some $k_{0}>2$ and $\kappa_{0}>0$ such that for a.e.
$(x,\xi)\in\mathbb{T}^{2}\times\mathbb{R}^{2}$ it holds that

$f_{0}(x,\xi)\leq\frac{\kappa_{0}}{1+\left|\xi\right|^{k_{0}}}.$

  2. 2.

There are some $m_{0}>2$ and $\mu_{0}>0$ such that

$\iint_{\mathbb{T}^{2}\times\mathbb{R}^{2}}\left|\xi\right|^{m_{0}}f_{0}(x,\xi)dxd\xi\leq\mu_{0}.$

Then, there exists a unique solution $f\in
C([0,\infty);\mathcal{P}(\mathbb{T}^{2}\times\mathbb{R}^{2}))$ to (VP).

The assumptions on $f_{0}$ in the theorem above correspond to H1-H2 in 1.1.
These assumptions are sufficient to define a unique weak solution to the
Vlasov–Poisson system [2, 20], which also satisfies propagation of velocity
moments. This is an important property which also implies propagation of
regularity. Since we couldn’t find a detailed proof of propagation of velocity
moments for solutions to (VP) on the two-dimensional torus, we write it below.

#### Propagation of velocity moments

We consider the velocity moment of a weak solution $f$ to (VP) of order $k\geq
0$ defined by

$M_{k}(t)\coloneqq\iint_{\mathbb{T}^{2}\times\mathbb{R}^{2}}\left\lvert\xi\right\rvert^{k}f(t,x,v)dxd\xi.$
(A.1)

We also define the following partial moment for $k\geq 0$,

$m_{k}(f)(x):=\int_{\mathbb{R}^{2}}\left\lvert\xi\right\rvert^{k}fd\xi.$

We will show that $M_{k}(t)$ is bounded for all time by showing it satisfies a
Grönwall inequality. Before showing this, we recall a classical inequality for
velocity moments, which is a generalization of 4.1.

######  Lemma A.2.

Let $0\leq k^{\prime}\leq k<\infty$ and $r=\frac{k+2}{k^{\prime}+2}$. If $f\in
L^{\infty}(\mathbb{T}^{2}\times\mathbb{R}^{2})$ with $M_{k}(f)<\infty$ then
$m_{k^{\prime}}(f)\in L^{r}(\mathbb{T}^{2})$ and

$\left\|m_{k^{\prime}}(f)\right\|_{r}\leq
c\left\|f\right\|_{\infty}^{\frac{k-k^{\prime}}{k+2}}M_{k}(f)^{\frac{k^{\prime}+2}{k+2}}$
(A.2)

where $c=c(k,k^{\prime},p)>0$.

First we notice that $M_{k}$ verifies the following differential inequality

$\displaystyle\left\lvert\frac{d}{dt}M_{k}(t)\right\rvert$
$\displaystyle=\left\lvert\iint\left\lvert\xi\right\rvert^{k}(-v\cdot\nabla_{x}f-\nabla_{x}\Phi\cdot\nabla_{v}f)d\xi
dx\right\rvert$
$\displaystyle=\left\lvert\iint\left\lvert\xi\right\rvert^{k}\mathop{\rm
div_{v}}\left(\nabla_{x}\Phi f\right)d\xi dx\right\rvert=\left\lvert\iint
k\left\lvert\xi\right\rvert^{k-2}v\cdot\nabla_{x}\Phi fd\xi dx\right\rvert$
$\displaystyle\leq\iint
k\left\lvert\xi\right\rvert^{k-1}fd\xi\left\lvert\nabla_{x}\Phi\right\rvert
dx\leq
k\left\|\nabla_{x}\Phi(t)\right\|_{k+2}\left\|\int_{\mathbb{R}^{2}}f(t,\cdot,\xi)d\xi\right\|_{\frac{k+2}{k+1}}.$

By A.2 with $k^{\prime}=k-1$, we have

$\left\|\int_{\mathbb{T}^{2}}\left\lvert\xi\right\rvert^{k-1}f(t,\cdot,\xi)d\xi\right\|_{\frac{k+2}{k+1}}\leq
CM_{k}(t)^{\frac{k+1}{k+2}}.$

Therefore we have

$\dot{M}_{k}(t)\leq
kC\left\|\nabla_{x}\Phi(t)\right\|_{k+2}M_{k}(t)^{\frac{k+1}{k+2}}.$

To close the Grönwall inequality, we aim to obtain
$\left\|\nabla_{x}\Phi(t)\right\|_{k+2}\leq M_{k}(t)^{\frac{1}{k+2}}$. Hence
we need to study the electric field $\nabla_{x}\Phi$ more carefully. We recall
that $\nabla_{x}\Phi=-\nabla_{x}G\ast\rho$ where $G$ is the Green function for
the negative Laplacian on the torus. Indeed $G$ is the solution to $-\Delta
G=\delta_{0}-1$ and is given by

$G(x)=-\frac{1}{2\pi}\log\left\lvert x\right\rvert+G_{0}(x)$

where $G_{0}\in C^{\infty}(\overline{B_{1/4}(0)})$.

Therefore $G$ has the same singularity as the Green function on the full
space. Thus, $\nabla G$ belongs to the Lorentz space $\in
L^{2,\infty}(\mathbb{T}^{2})$. Hence, using the weak Young inequality we
obtain

$\left\|\nabla_{x}\Phi(t)\right\|_{k+2}\leq
C\left\|\rho(t)\right\|_{\frac{2k+4}{k+4}}.$

Now applying A.2 with $k^{\prime}=0$ we have

$\left\|\rho(t)\right\|_{\frac{k+2}{2}}\leq CM_{k}(t)^{\frac{2}{k+2}}.$

Since $\frac{2k+4}{k+4}\leq\frac{k+2}{2}$ we have the following Hölder’s
inequality

$\left\|\rho(t)\right\|_{\frac{2k+4}{k+4}}\leq\left\|\rho(t)\right\|_{1}^{1-\theta}\left\|\rho(t)\right\|_{\frac{k+2}{2}}^{\theta},$

and after carefully computing the exponents in the inequality we finally
obtain $1-\theta=\theta=\frac{1}{2}$. Therefore

$\left\|\nabla_{x}\Phi(t)\right\|_{k+2}\leq
C\left\|\rho(t)\right\|_{\frac{2k+4}{k+4}}\leq
C\left\|\rho(t)\right\|_{\frac{k+2}{2}}^{\frac{1}{2}}\leq
CM_{k}(t)^{\frac{1}{k+2}}.$

This inequality allows us to close the Grönwall estimate and so $M_{k}(t)$ is
bounded for all time. We notice that, once we obtain the propagation of
velocity moments, we have a bound on the density that implies that the
electric field is bounded. Once the electric field is bounded, Assumption 1 is
verified for all times. For more details see [20, 15, 16, 17].

#### Uniqueness

Thanks to Assumption 1 in A.1, and to the propagation of moments, we can show
that $\rho\in L^{\infty}_{loc}(\mathbb{R}^{+};L^{\infty}(\mathbb{T}^{2}))$ for
all times. Therefore, the uniqueness criterion established in Loeper’s work
[22] is verified, which concludes the proof of A.1.

#### Acknowledgments

IBP was supported by the EPSRC grant number EP/V051121/1. This work was also
supported by the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex
Particle Dynamics: Phase Transitions, Patterns and Synchronization) of the
European Research Council Executive Agency (ERC) under the European Union’s
Horizon 2020 research and innovation programme (grant agreement No. 883363).

AR was supported by the NCCR SwissMAP which was funded by the Swiss National
Science Foundation grant number 205607. AR would like to thank the Swiss
National Science Foundation for its financial support.

We would like to thank Pierre-Emmanuel Jabin for many stimulating discussions
which have greatly contributed to this work, and to Matthew Rosenzweig for his
interest in this work.

## References

  * [1] H. Bahouri, J.Y. Chemin and R. Danchin. Fourier analysis and nonlinear partial differential equations. Grundlehren der mathematischen Wissenschaften. Vol. 343. (2011).
  * [2] J. Batt and G. Rein. Global classical solutions of the periodic Vlasov–Poisson system in three dimensions. C. R. Acad. Sci. Paris Sér. I Math., 313(6):411–416, (1991).
  * [3] I. Ben-Porat. Derivation of Euler’s equations of perfect fluids from von Neumann’s equation with magnetic field. Journal of Statistical Physics (2023) Vol. 190:121.
  * [4] Y. Brenier. Convergence of the Vlasov–Poisson system to the incompressible Euler equations. Comm. Partial Differential equations Vol. 25 (3–4): 737–754 (2000).
  * [5] Y. Brenier and E. Grenier. Limite singulière du système de Vlasov–Poisson dans le régime de quasi neutralité : le cas indépendant du temps. C. R. Acad. Sci. Paris Sér. I Math., 318(2):121–124, (1994).
  * [6] H. Brézis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer (2010).
  * [7] J.A. Carrillo and Y.P. Choi. Mean-Field Limits: From Particle Descriptions to Macroscopic Equations. Archive for Rational Mechanics and Analysis. Volume 241, pages 1529–1573, (2021).
  * [8] G. Crippa and G. Stefani. An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces. arXiv:2110.15648.
  * [9] J.M. Delort. Existence de nappes de tourbillon en dimension deux. Journal of the American Mathematical Society, 4 (1991), pp. 553–586.
  * [10] M. Duerinckx. Mean-field limits for some Riesz interaction gradient flows. SIAM Journal on Mathematical Analysis, 48 (2016), pp. 2269–2300.
  * [11] J. Duoandikoetxea. Fourier analysis. American Mathematical Society. 29 (2000).
  * [12] F. Golse and L. Saint-Raymond. The Vlasov–Poisson system with strong magnetic field. J. Math. Pures Appl. (9), 78(8):791–817 (1999).
  * [13] E. Grenier. Defect measures of the Vlasov–Poisson system in the quasineutral regime. Comm. Partial Differential Equations, 20(7-8):1189–1215, (1995).
  * [14] E. Grenier. Oscillations in quasineutral plasmas. Comm. Partial Differential Equations, 21(3-4):363–394, (1996).
  * [15] M. Griffin-Pickering and M. Iacobelli. Recent developments on the well-posedness theory for Vlasov-type equations. In From particle systems to partial differential equations, volume 352 of Springer Proc. Math. Stat., pages 301–319. Springer, Cham, 2021.
  * [16] M. Griffin-Pickering and M. Iacobelli. Global well-posedness for the Vlasov–Poisson system with massless electrons in the 3-dimensional torus. Comm. Partial Differential Equations, 46(10):1892–1939, 2021.
  * [17] M. Griffin-Pickering and M. Iacobelli. Stability in quasineutral plasmas with thermalized electrons. Preprint (2023).
  * [18] D. Han-Kwan and M. Iacobelli. From Newton’s second law to Euler’s equations of perfect fluids. Proc. Amer. Math. Soc., Vol. 149 (7): 3045–3061 (2021).
  * [19] A. Jonsson, P. Sjögren and H. Wallin.  Hardy and Lipschitz spaces on subsets of $\mathbb{R}^{n}$. Studia Mathematica Volume: 80, Issue: 2, page 141-166, (1984).
  * [20] P. L. Lions and B. Perthame. Propagation of moments and regularity for the 3-dimensional Vlasov–Poisson system. Inventiones mathematicae Vol. 105, pp 415–430 (1991).
  * [21] P.-L. Lions. Mathematical topics in fluid mechanics. Vol. 1. Incompressible models, Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, New York, 1996.
  * [22] G. Loeper. Uniqueness of the solution to the Vlasov–Poisson system with bounded density. J. Math. Pures Appl. (9), 86(1):68–79, (2006).
  * [23] A. J. Majda and A. L. Bertozzi. Vorticity and Incompressible Flow. Vol. 27 of Cambridge Texts in Applied Mathematics. Cambridge University Press (2002).
  * [24] N. Masmoudi. From Vlasov–Poisson system to the incompressible Euler system. Comm. Partial Differential Equations, 26(9-10), (2001).
  * [25] M. Ménard Mean-Field Limit Derivation of a Monokinetic Spray Model with Gyroscopic Effects. SIAM Journal on Mathematical Analysis, 56 (2022), pp. 1068–1113.
  * [26] E. Miot. On the gyrokinetic limit for the two-dimensional Vlasov–Poisson system.  arXiv:1603.04502v1.
  * [27] K. Pfaffelmoser. Global classical solutions of the Vlasov–Poisson system in three dimensions for general initial data. J. Differential Equations, 95(2):281–303, (1992).
  * [28] M. Rosenzweig. Mean-field convergence of point vortices to the incompressible Euler equation with vorticity in $L^{\infty}$. Arch. Ration. Mech. Anal. Vol. 243 (2022), no. 3, 1361-1431.
  * [29] M. Rosenzweig. From Quantum Many-Body Systems to Ideal Fluids. arXiv:2110.04195 (2021).
  * [30] W. Rudin. Real and Complex Analysis. Mc Graw Hill, Singapore, 1986.
  * [31] L. Saint-Raymond. Control of large velocities in the two-dimensional gyrokinetic approximation. J. Math. Pures Appl. 81 (2002) 379–399.
  * [32] S. Serfaty. Mean field limit for Coulomb-type flows. (Appendix in collaboration with Duerinckx, M.) Duke Math. J. 169, 2887–2935 (2020).
  * [33] E. M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993.
  * [34] V. Yudovich. Non-stationary flow of an ideal incompressible liquid. USSR Computational Mathematics and Mathematical Physics, 3 (1963), pp. 1407 – 1456.
  * [35] A. Zygmund. Trigonometric Series. Cambridge University Press; 3rd edition.