# A new perspective on Wasserstein distances
for kinetic problems

Mikaela Iacobelli

###### Abstract.

We introduce a new class of Wasserstein-type distances specifically designed
to tackle questions concerning stability and convergence to equilibria for
kinetic equations. Thanks to these new distances, we improve some classical
estimates by Loeper [49] and Dobrushin [16] on Vlasov-type equations, and we
present an application to quasi-neutral limits.

ETH Zürich, Department of Mathematics, Rämistrasse 101, 8092 Zürich,
Switzerland.
Email<EMAIL_ADDRESS>
##  1\. Introduction

###  1.1. General overview

Monge-Kantorovich distances, also known as Wasserstein distances, play a
central role in statistical mechanics, especially in the theory of propagation
of chaos and studying large particle systems’ mean behavior. From the late
1970s, there have been many applications of Wasserstein distances in kinetic
theory, as is beautifully described in the bibliographical notes of [60,
Chapter 6]. In particular, these distances are frequently used to prove the
uniqueness and stability of solutions to kinetic equations, study singular
limits, and measure convergence to equilibrium.

The first celebrated result relying on Monge-Kantorovich-Wasserstein distances
in non-collisional kinetic theory is the proof by Dobrushin [16] on the well-
posedness for Vlasov equations with $C^{1,1}$ potentials, where existence,
uniqueness, and stability are proved via a fixed point argument in the
bounded-Lipschitz or the $1$-Wasserstein distance. As a consequence of this
argument, one also obtains the validity of the mean-field limit for Vlasov
equations with smooth potentials. The interested reader may refer to [20,
Chapter 1.4] and [40, Chapter 3.3] for a detailed explanation of Dobrushin’s
stability estimate, its consequences on the mean-field limit for the Vlasov
equation, and of the role of Monge-Kantorovich-Wasserstein distances.
Dobrushin’s estimate is at the core of several kinetic theory arguments; see
for example [10, 11, 12, 13, 15, 18, 22, 29] for some applications.

In recent times, Golse and Paul in [23] introduced a quantum analog of the
$2$-Wasserstein distance to measure the approximation of the $N$-body quantum
dynamics by its mean-field limit. In [21] the authors prove quantitative
stability estimates that are reminiscent of Dobrushin’s, and they show that,
in the case of $C^{1,1}$ potentials, the mean-field limit of the quantum
mechanics of $N$ identical particles is uniform in the classical limit.

Another fundamental stability estimate was proved by Loeper [49], who
established uniqueness and stability of solutions with bounded density for the
Vlasov-Poisson equation. Loeper’s argument relies on the fact that the Coulomb
kernel is generated by a potential solving Poisson’s equation and exploits the
strong connection between the $2$-Wasserstein distance and the $H^{-1}$-norm.
Besides providing the best-known uniqueness criterion for Vlasov-Poisson, this
approach also gives a new proof of uniqueness à la Yudovich for $2D$ Euler.
Loeper’s result has been generalized to less singular kernels [37], and it is
the cornerstone for several other stability arguments [7, 14, 36, 42, 44, 48,
58]. Also, Loeper’s uniqueness criterion for Vlasov-Poisson has been extended
to solutions whose associated density belongs to some suitable Orlicz spaces
[51, 38]. In the following, we will focus our attention on some applications
of Loeper’s stability estimate related to quasi-neutral limit for the Vlasov-
Poisson equation [28, 30, 34, 35].

In general, extending Dobrushin’s and Loeper’s estimates is a delicate matter.
A possible idea is to introduce an anisotropic metric that weights spatial and
momentum coordinates differently. For example, in [43], the author considers a
variant of the $2$-Wasserstein distance where the cost for moving points in
the $x$-variable is higher than for the $v$-variable. By suitably selecting
the parameters, this allows the author to extend the validity ranges for the
mean-field limit for the Vlasov-Poisson system. Also, as shown in [28, 30], an
analogous method provides better convergence estimates when considering
combined mean-field and quasi-neutral limits in Vlasov-Poisson-type systems.
At the same time as this paper was written, another variant of this idea was
introduced in [56], where the author improves the trend to equilibrium for
$1$-D kinetic Fokker-Planck equations via estimates measured in an analog of
the $2$-Wasserstein metric.

This work aims to push further the idea that, when applied to kinetic
problems, Wasserstein distances should be modified to reflect the natural
anisotropy between position and momentum variables. Moreover, since these
metrics are used to measure the distance between PDEs’ solutions, we will
introduce time-dependent counterparts that can vary along with the
characteristic flow. Still, it is worth noticing that our method could be
applied, beyond the kinetic framework, to equations where the evolution in one
of the variables enjoys better regularity properties than the others.

Before stating our main results, let us emphasize that the idea of finding
appropriate generalised Wasserstein distances has been used successfully in
other contexts in the optimal transport and evolution PDE community, see for
instance [17, 19, 45, 46, 54, 55] and references therein.

###  1.2. Definitions and main results

Let us recall the definition of Wasserstein distances (see for instance [1,
60]). In what follows, $\mathcal{X}$ will be either the $d$-dimensional torus
$\mathbb{T}^{d}$ or the Euclidean space $\mathbb{R}^{d}$.

######  Definition 1.1.

Let $\mu,\nu$ be two probability measures on
$\mathcal{X}\times\mathbb{R}^{d}$. We denote with $\Pi(\mu,\nu)$ the set of
all probability measures on $(\mathcal{X}\times\mathbb{R}^{d})^{2}$ with
_marginals_ $\mu$ and $\nu.$ More precisely, $\pi\in\Pi(\mu,\nu)$ if

$\pi[A\times(\mathcal{X}\times\mathbb{R}^{d})]=\mu[A],\ \ \
\pi[(\mathcal{X}\times\mathbb{R}^{d})\times B]=\nu[B],\qquad\text{for all
$A,B\subset\mathcal{X}\times\mathbb{R}^{d}$ Borel.}$

We shall call _coupling_ (between $\mu$ and $\nu$) an element in
$\Pi(\mu,\nu).$

For $p\geq 1$, the $p$-Wasserstein distance between $\mu$ and $\nu$ is defined
as

$W_{p}(\mu,\nu):=\left(\inf_{\gamma\in\Pi(\mu,\nu)}\int_{(\mathcal{X}\times\mathbb{R}^{d})^{2}}\left(|x-y|^{p}+|v-w|^{p}\right)\mathrm{d}\gamma(x,v,y,w)\right)^{1/p}.$

(1) A free-flow $W_{1}$-type distance for the Vlasov equations with $C^{1,1}$
potential. Consider two solutions $f_{1},f_{2}$ of the Vlasov equation on
$\mathcal{X}$, namely

$\partial_{t}f+v\cdot\nabla_{x}f+F[f]\cdot\nabla_{v}f=0,\qquad F[f]:=\nabla
K\ast\rho_{f},\qquad\rho_{f}:=\int fdv,$

where $\|D^{2}K\|_{\infty}=:B<\infty$. The classical Dobrushin’s argument
shows that

$W_{1}(f_{1}(t),f_{2}(t))\leq e^{(1+2B)t}W_{1}(f_{1}(0),f_{2}(0)).$

In particular, when the potential $K$ is identically zero, this bound provides
an exponential stability for $W_{1}$ that is far from optimal. Indeed, since
the solution is simply given by $f(t,x,v)=f(0,x-tv,v)$, it is clear that in
this case $W_{1}(f_{1}(t),f_{2}(t))\sim t$ for $t\gg 1$.

By introducing a $W_{1}$-type distance adapted to the free flow, we can prove
that

$W_{1}(f_{1}(t),f_{2}(t))\leq\min\left\\{(1+t)e^{\frac{2}{3}B\left((1+t)^{3}-1\right)},e^{(1+2B)t}\right\\}W_{1}(f_{1}(0),f_{2}(0))$

(see Theorem 2.1 below).

This estimate gives the optimal bound when $K\equiv 0$. Moreover, for $B\leq
1$, this provides a better estimate compared to the usual Dobrushin’s bound
when $t\in[0,T_{B}]$ with $T_{B}\simeq B^{-1/2}$.

(2) An improved $W_{2}$-stability estimate for Vlasov-Poisson with bounded
density. For this second application, we focus on the case of the torus for
simplicity, but a completely similar analysis works on the whole space.

Consider two solutions $f_{1},f_{2}$ of the Vlasov-Poisson equation on
$\mathbb{T}^{d}$, namely

$\partial_{t}f+v\cdot\nabla_{x}f+\nabla U\cdot\nabla_{v}f=0,\qquad-\Delta
U:=\rho_{f}-1,\qquad\rho_{f}:=\int fdv.$

As shown in [49], Loeper’s proof provides the following stability estimate
whenever $W_{2}(f_{1}(0),f_{2}(0))$ is sufficiently small (which is the
interesting case):

$W_{2}(f_{1}(t),f_{2}(t))\leq
c_{d}e^{\log\left(\frac{W_{2}(f_{1}(0),f_{2}(0))}{c_{d}}\right)e^{-Ct}},$

where $c_{d}>0$ is a dimensional constant, while $C$ depends on the
$L^{\infty}$ norm of $\rho_{f_{1}}$ and $\rho_{f_{2}}.$ This estimate can then
be applied to prove the validity of the quasi-neutral limit for Vlasov-Poisson
for initial data that are double-exponential perturbation of analytic
functions [34, 35] (see Remark 3.4).

To improve this result, given $(X_{i},V_{i})$ the characteristics associated
to $f_{i}$, we consider a nonlinear $W_{2}$-type quantity of the form

$Q(t):=\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left[\lambda(t)|X_{1}(t,x,v)-X_{2}(t,y,w)|^{2}+|V_{1}(t,x,v)-V_{2}(t,y,w)|^{2}\right]d\pi_{0}(x,v,y,w)$

where $\pi_{0}$ is an optimal coupling, and $\lambda(t)=|\log(Q(t))|.$ We then
prove that $Q(t)$ is well-defined whenever $Q(t)\ll 1,$ and finally, comparing
$Q(t)$ to $W_{2},$ we show that

$W_{2}(f_{1}(t),f_{2}(t))^{2}\leq
2e^{-\left(\sqrt{\left|\log\left\\{W_{2}(f_{1}(0),f_{2}(0))^{2}\left|\log\left(\frac{1}{2}W_{2}(f_{1}(0),f_{2}(0))^{2}\right)\right|\right\\}\right|}-Ct\right)^{2}}.$

(see Theorem 3.1). To better understand the improvement of our estimate with
respect to Loeper’s, one can think as follows: if
$W_{2}(f_{1}(0),f_{2}(0)=\theta\ll 1$, then Loeper’s estimate implies that
$W_{2}(f_{1}(t),f_{2}(t))\lesssim 1$ for $t\in[0,\log|\log\theta|]$. Instead,
our bound gives $W_{2}(f_{1}(t),f_{2}(t))\lesssim 1$ for
$t\in\bigl{[}0,|\log\theta|^{1/2}\bigr{]}$, so on a much longer time-interval.

######  Remark 1.2.

  * •

Note that a standard Gronwall estimate of the form
$W_{2}(f_{1}(t),f_{2}(t))\leq e^{Ct}W_{2}(f_{1}(0),f_{2}(0))$ would imply that
$W_{2}(f_{1}(t),f_{2}(t))\lesssim 1$ for $t\in[0,|\log\theta|]$. So, while
Loeper’s bound loses an extra logarithm in terms of time-scale, our bound only
loses a square root. Since the electric field for a solution with bounded
density is at most log-Lipschitz, an estimate of the form
$W_{2}(f_{1}(t),f_{2}(t))\leq e^{Ct}W_{2}(f_{1}(0),f_{2}(0))$ is not expected
to hold in this setting, and we believe our bound to be essentially sharp.

  * •

Our improvement from $\log|\log\theta|$ to $|\log\theta|^{1/2}$ is similar to
the one obtained for the $W_{1}$ distance, see [38, Remark 1.7]. In that
paper, the authors rely crucially on the second-order structure of the Vlasov
equation, namely $\ddot{X}=\nabla U(t,X)$. Our proof, instead, relies only on
the fact that $\dot{X}=a(t,X,V)$, where $a(t,\cdot,\cdot)$ is Lipschitz, and
it can be generalized to other contexts where the second-order structure
fails.

Our new stability estimate has interesting applications for what concerns some
singular limits for Vlasov-type equations. In particular, by considering the
Vlasov-Poisson system in appropriate dimensionless variables that take the
Debye length into account, we prove the validity of the quasi-neutral limit
for Vlasov-Poisson for initial data that are an exponential perturbation of
analytic functions, see also Remark 3.4.

The paper is structured as follows: in the next two sections, we will present
our two main results, and then in the final section of the paper, we will
discuss more generally our approach and how it leads to the introduction of a
new family of Wasserstein-type distances.

##  2\. Dobrushin’s estimate revisited

###  2.1. The Vlasov equation

The Vlasov equation is a non-linear partial differential equation providing a
statistical description for the collective behavior of large numbers of
charged particles in mutual, long-range interaction. This model was first
introduced by Jeans in the context of Newtonian stellar dynamics [41], and
later by Vlasov in his work on plasma physics [61, 62]. The unknown of the
Vlasov equation $f(t,x,v)$ is the distribution function of the system at time,
that is the number density of particles that are located at the position $x$
and have instantaneous velocity $v$ at time $t.$ The Vlasov equation for the
distribution function $f$ reads as follows:

$\partial_{t}f(t,x,v)+v\cdot\nabla_{x}f(t,x,v)+F[f](t,x)\cdot\nabla_{v}f(t,x,v)=0$
(2.1)

where

$F[f](x)=\iint\nabla K(x-y)\,f(dy,dw)=\nabla K\ast_{x,v}f.$

In other words, the Vlasov equation for particle systems is a kinetic model
where each particle is subject to the acceleration field $F[f]$ created by all
the other particles in the system.

The Vlasov equation is a transport equation and, for a sufficiently regular
force field, it can be described by the method of characteristics. The initial
distribution $f_{0}$ is transported by a characteristic flow $(X,V)$ generated
by the mean-field force $F[f]$: if we denote

$\left\\{\begin{array}[]{l}\dot{X}(t,x,v)=V(t,x,v),\\\
\dot{V}(t,x,v)=F[f](t,X(t,x,v)),\\\
X(0,x,v)=x,\,V(0,x,v)=v,\end{array}\right.$

then $f(t,X(t,x,v),V(t,x,v))=f(0,x,v)$. Also, since the vector field
$(v,F[f])$ is divergence free, one has conservation of mass and of all
$L^{p}$-norms. For an introduction to this topic we refer to the lecture notes
[20].

###  2.2. An improved Dobrushin’s estimate

Consider the Vlasov equation with smooth kernel. More precisely,

$\partial_{t}f+v\cdot\nabla_{x}f+F[f]\cdot\nabla_{v}f=0,\qquad F[f]:=\nabla
K\ast\rho_{f},\qquad\rho_{f}:=\int fdv,$ (2.2)

where $\|D^{2}K\|_{\infty}=:B<\infty$. As explained in the introduction, our
goal is to provide a stability estimate for solutions that is optimal in the
regime as $B$ tends to zero. Here is our result:

######  Theorem 2.1.

Let $f_{1},f_{2}$ be two solution of (2.2). Then

$W_{1}(f_{1}(t),f_{2}(t))\leq\min\left\\{(1+t)e^{\frac{2}{3}B\left((1+t)^{3}-1\right)},e^{(1+2B)t}\right\\}W_{1}(f_{1}(0),f_{2}(0)).$

###### Proof.

Let $(X_{i},V_{i})$ denote the characteristic flow associated to $f_{i}$, that
is

$\left\\{\begin{array}[]{l}\dot{X}_{i}(t,x,v)=V_{i}(t,x,v),\\\
\dot{V}_{i}(t,x,v)=\nabla\bigl{(}K\ast\rho_{f_{i}}\bigr{)}(t,X_{i}(t,x,v)),\\\
X_{i}(0,x,v)=x,\,V_{i}(0,x,v)=v.\end{array}\right.$

Note that, since $\nabla K$ is Lipschitz, the characteristic flow is well-
defined thanks to Cauchy-Lipschitz theory (see [20, Chapter 2]). To prove
Theorem 2.1, we consider $\pi_{0}$ an optimal $W_{1}$-coupling between
$f_{1}(0)$ and $f_{2}(0)$, and we define the quantity

$Q(t):=\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\big{[}|X_{1}(t,x,v)-tV_{1}(t,x,v)-(X_{2}(t,y,w)-tV_{2}(t,x,v))|+|V_{1}(t,x,v)-V_{2}(t,y,w)|\big{]}d\pi_{0}(x,v,y,w).$

Note that

$Q(0)=\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left[|x-y|+|v-w|\right]d\pi_{0}(x,v,y,w)=W_{1}(f_{1}(0),f_{2}(0)).$
(2.3)

Also

$\begin{split}&\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\big{[}|X_{1}(t,x,v)-X_{2}(t,y,w)|+|V_{1}(t,x,v)-V_{2}(t,y,w)|\big{]}d\pi_{0}(x,v,y,w)\\\
&\leq\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\big{[}|X_{1}(t,x,v)-tV_{1}(t,x,v)-(X_{2}(t,y,w)-tV_{2}(t,x,v))|+|V_{1}(t,x,v)-V_{2}(t,y,w)|\big{]}d\pi_{0}(x,v,y,w)\\\
&\qquad+t\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}|V_{1}(t,x,v)-V_{2}(t,y,w)|d\pi_{0}(x,v,y,w)\\\
&\leq(1+t)Q(t).\end{split}$ (2.4)

Since $\frac{d}{dt}(X_{i}-tV_{i})=-t\dot{V}_{i}$, one has

$\displaystyle\frac{d}{dt}Q(t)$
$\displaystyle\leq(1+t)\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left[|\dot{V}_{1}(t,x,v)-\dot{V}_{2}(t,y,w)|\right]d\pi_{0}(x,v,y,w)$
(2.5)
$\displaystyle=(1+t)\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left|\nabla\bigl{(}K\ast\rho_{f_{1}}\bigr{)}(t,X_{1}(t,x,v))-\nabla\bigl{(}K\ast\rho_{f_{2}}\bigr{)}(t,X_{2}(t,y,w))\right|d\pi_{0}(x,v,y,w)$
(2.6)
$\displaystyle\leq(1+t)\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left|\nabla\bigl{(}K\ast\rho_{f_{1}}\bigr{)}(t,X_{1}(t,x,v))-\nabla\bigl{(}K\ast\rho_{f_{2}}\bigr{)}(t,X_{1}(t,x,v))\right|d\pi_{0}(x,v,y,w)$
(2.7)
$\displaystyle\qquad+(1+t)\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left|\nabla\bigl{(}K\ast\rho_{f_{2}}\bigr{)}(t,X_{1}(t,x,v))-\nabla\bigl{(}K\ast\rho_{f_{2}}\bigr{)}(t,X_{2}(t,y,w))\right|d\pi_{0}(x,v,y,w)$
(2.8) $\displaystyle=:(1+t)\bigl{[}T_{1}+T_{2}\bigr{]}.$ (2.9)

We now observe that, since $\nabla K$ is $B$-Lipschitz, we can bound

$T_{2}\leq
B\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\big{[}|X_{1}(t,x,v)-X_{2}(t,y,w)|\big{]}d\pi_{0}(x,v,y,w)\leq
B(1+t)Q(t),$

where the second inequality follows from (2.4). For $T_{1}$, we note that

$\displaystyle\left|\nabla\bigl{(}K\ast\rho_{f_{1}}\bigr{)}(t,X_{1}(t,x,v))-\nabla\bigl{(}K\ast\rho_{f_{2}}\bigr{)}(t,X_{1}(t,x,v))\right|=\left|\int_{\mathbb{T}^{d}}\nabla
K(X_{1}(t,x,v)-z)d\,\bigl{(}\rho_{f_{1}(t)}(z)-\rho_{f_{2}(t)}(z)\bigr{)}\right|.$
(2.10)

Here, similarly to Dobrushin’s argument, we use that $W_{1}$ admits the
following dual formulation:

$W_{1}(\mu,\nu)=\sup_{\psi\,\text{1-Lip}}\int\psi\,d(\mu-\nu).$ (2.11)

Thanks to this fact, since $z\mapsto\nabla K(X_{1}(t,x,v)-z)$ is
$B$-Lipschitz, we deduce that

$\left|\int_{\mathbb{T}^{d}}\nabla
K(X_{1}(t,x,v)-z)\mathop{}\\!\mathrm{d}\bigl{(}\rho_{f_{1}(t)}(z)-\rho_{f_{2}(t)}(z)\bigr{)}\right|\leq
B\,W_{1}(\rho_{f_{1}(t)},\rho_{f_{2}(t)}),$

and therefore

$T_{1}\leq
B\,W_{1}(\rho_{f_{1}(t)},\rho_{f_{2}(t)})\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}d\pi_{0}(x,v,y,w)=B\,W_{1}(\rho_{f_{1}(t)},\rho_{f_{2}(t)}).$

Let
$\gamma_{t}=(X_{1}(t,\cdot,\cdot),X_{2}(t,\cdot,\cdot))_{\\#}\pi_{0}\in\Pi(\rho_{f_{1}},\rho_{f_{2}}).$
Then, by the definition of $W_{1}$ (see Definition 1.1),

$W_{1}(\rho_{f_{1}(t)},\rho_{f_{2}(t)})\leq\int_{(\mathbb{T}^{d})^{2}}|x-y|d\gamma_{t}(x,y)=\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}|X_{1}(t,x,v)-X_{2}(t,y,w)|d\pi_{0}(x,v,y,w),$

so using (2.4) we conclude that $T_{1}\leq B(1+t)Q(t)$.

In conclusion, we proved that

$Q^{\prime}(t)\leq 2B(1+t)^{2}Q(t),$

therefore

$Q(t)\leq e^{\frac{2}{3}B\left((1+t)^{3}-1\right)}Q(0).$

Recalling (2.3) and (2.4), this yields

$W_{1}(f_{1}(t),f_{2}(t))\leq(1+t)e^{\frac{2}{3}B\left((1+t)^{3}-1\right)}W_{1}(f_{1}(0),f_{2}(0)).$
(2.12)

As noted in the introduction, this estimate is more powerful than the usual
Dobrushin’s estimate 111Dobrushin’s argument is performed considering the so-
called bounded-Lipschitz distance on probability measures, which is defined by
duality against bounded Lipschitz functions. However, the same proof where one
replaces the bounded-Lipschitz distance with the $W_{1}$ distance (which can
be defined by duality against Lipschitz functions, as shown in (2.11)),
provides this bound.

$W_{1}(f_{1}(t),f_{2}(t))\leq e^{(1+2B)t}W_{1}(f_{1}(0),f_{2}(0))$ (2.13)

when $B$ is small. On the other hand, for large times, the term $(1+t)^{3}$ in
our estimate provides a worse bound (2.13). Hence, both bounds are helpful
depending on the mutual sizes of $B$ and $t$, and one can choose to apply
whichever gives the stronger bound. In conclusion, one has

$W_{1}(f_{1}(t),f_{2}(t))\leq\min\left\\{(1+t)e^{\frac{2}{3}B\left((1+t)^{3}-1\right)},e^{(1+2B)t}\right\\}W_{1}(f_{1}(0),f_{2}(0)),$
(2.14)

as desired. ∎

##  3\. Stability estimates for Vlasov-Poisson and quasi-neutral limits

###  3.1. The Vlasov-Poisson system

The Vlasov-Poisson system is the classical kinetic model describing dilute,
totally ionised, unmagnetized plasma. In its most common form, $f$ is the
distribution function of the electrons moving in a self-induced electrostatic
field, while the ions are assumed to act as a fixed background. In this
section, we consider the phase space to be
${\mathbb{T}}^{d}\times{\mathbb{R}}^{d},$ for reasons that will be explained
later.

$(VP):=\left\\{\begin{array}[]{ccc}\partial_{t}f+v\cdot\nabla_{x}f+E\cdot\nabla_{v}f=0,\\\
E=-\nabla U,\\\ \Delta U=1-\int_{{\mathbb{R}}^{d}}f\,dv=1-\rho_{f},\\\
f|_{t=0}=f_{0}\geq 0,\ \
\int_{{\mathbb{T}}^{d}\times{\mathbb{R}}^{d}}f_{0}\,dx\,dv=1.\end{array}\right.$
(3.1)

The well-posedness theory of this system has been extensively studied, see,
for example, the survey paper [32]. Global-in-time classical solutions have
been constructed under various conditions on the initial data (see for example
[4, 6, 47, 53, 57, 59]), while global-in-time weak solutions were presented in
[2] and [39] for $L^{p}$ initial data (see also [3, 5]). In this section, we
will focus on an important contribution to the uniqueness theory made by
Loeper [49], who proved uniqueness for solutions of (3.1) with bounded density
by means of a strong-strong stability estimate in Wasserstein.

###  3.2. Quasi-neutral limits

Since plasmas are excellent conductors of electricity, and any charges that
develop are readily neutralized, they can be treated as being quasi-neutral.
On the other hand, at small spatial and time scales, the quasi-neutrality is
no longer verified. The distance over which quasi-neutrality may break down
can be described in terms of the Debye length $\lambda_{D}$, and varies
according to the physical characteristics of the plasma. The Debye length is
usually much shorter than the typical observation scale. Therefore, we can
define the parameter $\varepsilon:=\lambda_{D}/L$ and consider the limit as
$\varepsilon$ tends to zero. This procedure is known as quasi-neutral limit.

When we take the Debye length into account, in appropriate dimensionless
variables, the Vlasov-Poisson system becomes:

$(VP)_{\varepsilon}:=\left\\{\begin{array}[]{ccc}\partial_{t}f_{\varepsilon}+v\cdot\nabla_{x}f_{\varepsilon}+E_{\varepsilon}\cdot\nabla_{v}f_{\varepsilon}=0,\\\
E_{\varepsilon}=-\nabla_{x}U_{\varepsilon},\\\
-\varepsilon^{2}\Delta_{x}U_{\varepsilon}=\rho_{f_{\varepsilon}}-1,\\\
f_{\varepsilon}|_{t=0}=f_{0,\varepsilon}\geq 0,\ \
\int_{\mathbb{T}^{d}\times\mathbb{R}^{d}}f_{0,\varepsilon}\,dx\,dv=1,\end{array}\right.$
(3.2)

and the energy of the rescaled system is the following:

$\mathcal{E}(f_{\varepsilon}(t)):=\frac{1}{2}\int_{\mathbb{T}^{d}\times\mathbb{R}^{d}}f_{\varepsilon}|v|^{2}dxdv+\frac{\varepsilon^{2}}{2}\int_{\mathbb{T}^{d}}|\nabla_{x}U_{\varepsilon}|^{2}dx.$
(3.3)

The quasi-neutral limit corresponds to a singular limit for the rescaled
system (3.2), in which the formal limiting system is the _Kinetic Isothermal
Euler_ system:

$(KIE):=\left\\{\begin{array}[]{ccc}\partial_{t}f+v\cdot\nabla_{x}f+E\cdot\nabla_{v}f=0,\\\
E=-\nabla_{x}U,\\\ \rho=1,\\\ f|_{t=0}=f_{0}\geq 0,\ \
\int_{\mathbb{T}^{d}\times\mathbb{R}^{d}}f_{0}\,dx\,dv=1.\end{array}\right.$
(3.4)

The force $E=-\nabla_{x}U$ is defined implicitly through the incompressibility
constraint $\rho=1$, and may be thought of as a Lagrange multiplier associated
to this constraint. In other words, electrons move under the effect of a
gradient in such a way that their density remains equal to $1$ everywhere.
Thus (KIE) is a “kinetic” version of the incompressible Euler equations. As
shown in [8], the potential $U$ formally satisfies the Laplace equation

$-\Delta_{x}U=\sum_{i,j}\partial_{x_{i}}\partial_{x_{j}}\int_{{\mathbb{R}}^{d}}v_{i}v_{j}f\mathop{}\\!\mathrm{d}\,v.$

As discussed in [31], the justification of this limit is very delicate. In
particular, it can fail even for smooth initial data. Still, a series of
positive results are available. In particular, as shown in [34, 35], a way to
get the validity of the quasi-neutral limit for a large class of data can be
achieved if one can prove some quantitative strong-strong stability at the
level of the $(VP)_{\varepsilon}$ system. Also, the stronger the stability
estimate, the larger the class of initial data for which the quasi-neutral
limit hold. In [34, 35] the authors prove that the quasi-neutral limit holds
for initial data that are an extremely small perturbation of an analytic
function. Here, by introducing a suitable non-linear version of the
Wasserstein distance, we can considerably improve that results.

Here is our main theorem, which provides us with a new $W_{2}$ stability
estimate. We prove the result with a general parameter $\varepsilon\leq 1$ as
this is necessary for the study of the quasi-neutral limit. The reader
interested in the Vlasov-Poisson case can simply apply our estimate with
$\varepsilon=1$.

######  Theorem 3.1.

Let $\varepsilon\leq 1$, and let $f_{1},f_{2}$ be two weak solutions of the
$(VP)_{\varepsilon}$ system (3.2), and set

$\rho_{1}:=\int_{\mathbb{R}^{d}}f_{1}\,dv,\quad\rho_{2}=\int_{\mathbb{R}^{d}}f_{2}\,dv.$

Define the function

$A(t):=\|\rho_{1}(t)\|_{L^{\infty}(\mathbb{T}^{d})}+\|\rho_{2}(t)\|_{L^{\infty}(\mathbb{T}^{d})},$
(3.5)

and assume that $A(t)\in L^{1}([0,T])$ for some $T>0$. There exist a
dimensional constant $C_{d}>0$ and a universal constant $c_{0}>0$ such that
the following holds: if $W_{2}(f_{1}(0),f_{2}(0))$ is sufficiently small so
that $W_{2}(f_{1}(0),f_{2}(0))\leq c_{0}\varepsilon$ and

$\sqrt{\left|\log\left(\varepsilon^{-2}W_{2}(f_{1}(0),f_{2}(0))^{2}\left|\log\left(\frac{1}{2}\varepsilon^{-2}W_{2}(f_{1}(0),f_{2}(0))^{2}\right)\right|\right)\right|}\geq\frac{C_{d}}{\varepsilon}\int_{0}^{T}A(s)\,ds+\sqrt{\left|\log\left(\frac{\varepsilon}{e}\right)\right|},$
(3.6)

then

$W_{2}(f_{1}(t),f_{2}(t))^{2}\leq
2e^{-\left(\sqrt{\left|\log\left\\{\varepsilon^{-2}W_{2}(f_{1}(0),f_{2}(0))^{2}\left|\log\left(\frac{1}{2}\varepsilon^{-2}W_{2}(f_{1}(0),f_{2}(0))^{2}\right)\right|\right\\}\right|}-\frac{C_{d}}{\varepsilon}\int_{0}^{t}A(s)\,ds\right)^{2}}\qquad\text{for
all }t\in[0,T].$

######  Remark 3.2.

The assumption (3.6) depends on the time interval $[0,T].$ If $T$ is very
small so that

$\frac{C_{d}}{\varepsilon}\int_{0}^{T}A(s)\,ds\leq\sqrt{\left|\log\left(\frac{\varepsilon}{e}\right)\right|}$

then (3.6) corresponds to $W_{2}(f_{1}(0),f_{2}(0))\leq\varepsilon^{3}.$ Of
course this is not the relevant regime since the time interval is usually at
least of size $1$. In particular, since $A(t)\geq 2,$ 222We recall that
$\int_{\mathbb{T}^{d}}\rho_{i}(x,t)dx=1,$ that implies
$\|\rho_{i}(\cdot,t)\|_{L^{\infty}(\mathbb{T}^{d})}\geq 1,$ $i=1,2.$ Therefore
$A(t)\geq 2.$

$\frac{C_{d}}{\varepsilon}\int_{0}^{T}A(s)\,ds=\frac{C_{T}}{\varepsilon}\qquad\text{for
some constant $C_{T}\gtrsim 1$}.$

Therefore (3.6) corresponds to asking $W_{2}(f_{1}(0),f_{2}(0))$ being bounded
by $e^{-C\varepsilon^{-2}}.$ This requirement is very natural in this context,
as also discussed in Remark 3.4.

As in [34, 35], Theorem 3.1 yields the validity of the quasi-neutral limit for
$W_{2}$-perturbations of analytic data. However, our estimate is stronger with
respect to the previous results and provides an almost optimal rate in the
quasi-neutral limit. More broadly, we believe that our approach for proving
Theorem 3.1 has its own interest and could be used in other settings.

To state our application to the quasi-neutral limit, we need to recall some
notation introduced by Grenier [26] in one of the first mathematical works on
this topic. In [26] the author relies on an interpretation of the plasma as a
superposition of a -possibly uncountable- collection of fluids and he shows
that the quasi-neutral limit holds when the sequence of initial data
$f_{0,\varepsilon}$ enjoys uniform analytic regularity with respect to the
space variable. As explained in [34] (see the discussion after Definition
$1.4$), this decomposition is purely a technical tool and it does not impose
any restriction on the initial datum. This result has been improved by Brenier
[9], who gives a rigorous justification of the quasi-neutral limit in the so
called “cold electron” case, i.e. when the initial distribution
$f_{0,\varepsilon}$ converges to a monokinetic profile

$f_{0}(x,v)=\rho_{0}(x)\delta_{v=v_{0}(x)}$

where $\delta_{v}$ denotes the Dirac measure in velocity, see also [9, 50,
25].

Let us define a suitable analytic norm, as in [26]: given $\delta>0$ and a
function $g:\mathbb{T}^{d}\to\mathbb{R}$, we define

$\|g\|_{B_{\delta}}:=\sum_{k\in\mathbb{Z}^{d}}|\widehat{g}(k)|\delta^{|k|},$

where $\widehat{g}(k)$ is the $k$-th Fourier coefficient of $g$. We define
$B_{\delta}$ as the space of functions $g$ such that
$\|g\|_{B_{\delta}}<+\infty$.

######  Corollary 3.3.

Let $d=2,3$, and let $\gamma$, $\delta_{0}$, and $C_{0}$ be positive
constants. Consider a sequence $(f_{0,\varepsilon})$ of non-negative initial
data in $L^{1}$ for (3.2) such that for all $\varepsilon\in(0,1)$, and all
$x\in\mathbb{T}^{d}$,

  * •

(uniform estimates)

$\|f_{0,\varepsilon}\|_{\infty}\leq
C_{0},\quad\mathcal{E}(f_{0,\varepsilon})\leq C_{0},$

  * •

(compact support in velocity)

$f_{0,\varepsilon}(x,v)=0\quad\text{if }|v|>\frac{1}{\varepsilon^{\gamma}},$

  * •

(analytic + perturbation) Assume the following decomposition:

$f_{0,\varepsilon}=g_{0,\varepsilon}+h_{0,\varepsilon},$

where $(g_{0,\varepsilon})$ is a sequence of continuous functions satisfying

$\sup_{\varepsilon\in(0,1)}\sup_{v\in\mathbb{R}^{d}}\,(1+|v|^{2})\|g_{0,\varepsilon}(\cdot,v)\|_{B_{\delta_{0}}}\leq
C_{0},$

admitting a limit $g_{0}$ in the sense of distributions. Furthemore,
$(h_{0,\varepsilon})$ is a sequence of functions satisfying for all
$\varepsilon>0$

$W_{2}(f_{0,\varepsilon},g_{0,\varepsilon})\leq
e^{-K\varepsilon^{-2\zeta}}\qquad\text{with
}\zeta=\left\\{\begin{array}[]{ll}1+2\max\\{\beta,\gamma\\})&\text{if }d=2,\\\
1+\max\\{38,3\gamma\\})&\text{if }d=3,\\\ \end{array}\right.$ (3.7)

for some constants $K>0$ and $\beta>2$.

For all $\varepsilon\in(0,1)$, consider $f_{\varepsilon}(t)$ a global weak
solution of (3.2) with initial condition $f_{0,\varepsilon}$, and define the
filtered distribution function

$\widetilde{f}_{\varepsilon}(t,x,v):=f_{\varepsilon}\Big{(}t,x,v-\frac{1}{i}(d_{+}(t,x)e^{\frac{it}{\sqrt{\varepsilon}}}-d_{-}(t,x)e^{-\frac{it}{\sqrt{\varepsilon}}})\Big{)}$
(3.8)

where $(d_{\pm})$ are the correctors are defined as the solution of

${\rm curl}\ d_{\pm}=0,\qquad{\rm
div}\bigg{(}\partial_{t}d_{\pm}+\left(\int\rho_{\theta}v_{\theta}\mu(d\theta)\cdot\nabla\right)d_{\pm}\bigg{)}=0,$
(3.9) ${\rm div}d_{\pm}(0)=\underset{\varepsilon\to 0}{\lim}{\rm
div}\frac{\sqrt{\varepsilon}E^{\varepsilon}(0)\pm
ij^{\varepsilon}(0)}{2},\qquad
j^{\varepsilon}:=\int\rho^{\varepsilon}_{\theta}v^{\varepsilon}_{\theta}\mu(d\theta).$
(3.10)

Then there exist $T>0$, and $g(t)$ a weak solution on $[0,T]$ of (3.4) with
initial condition $g_{0},$ such that

$\lim_{\varepsilon\to
0}\sup_{t\in[0,T]}W_{1}(\widetilde{f}_{\varepsilon}(t),g(t))=0.$

######  Remark 3.4.

Already in the one dimensional case, there is a negative result stating that
an initial rate of convergence of the form
$W_{2}(f_{0,\varepsilon},g_{0,\varepsilon})\leq\varepsilon^{k}$ for some $k>0$
is not sufficient to ensure the validity of the quasi-neutral limit for
positive times. This is the consequence of _instability mechanisms_ described
in [27] and [33]. Hence, our assumption on the size of
$W_{2}(f_{0,\varepsilon},g_{0,\varepsilon})$ considerably improves the results
in [34, 35], where a double exponential
$\exp\big{(}-\exp({K\varepsilon^{-\zeta}})\big{)}$ was required.

######  Remark 3.5.

In Corollary 3.3 we consider sequences of initial conditions with compact
support in velocity (yet, we allow the support to grow polynomially as
$\varepsilon$ goes to zero). The reason is that we need $L^{\infty}$ bounds on
the density $\rho_{f_{\varepsilon}}(t)=\int f_{\varepsilon}(t)\,dv$, so a
control on the support in velocity is needed. We have decided to put these
assumptions because they are the same as in [6] and so we can rely on some
estimates proved in that paper. However, using the argument in [52] (see also
[29]) one could relax the assumptions and require only a moment condition on
$f_{0,\varepsilon}$. Providing this extension is not difficult, but it would
require some work that would go beyond the main goal of this paper.

Before proving Proof of Theorem 3.1, we first show how it implies Corollary
3.3.

###### Proof of Corollary 3.3.

Let $g_{\varepsilon}(t)$ denote the solution of (3.2) starting from
$g_{0,\varepsilon}$. As shown in [34, Section 4], under the assumptions in the
statement, the following bounds hold:

$\|\rho_{g_{\varepsilon}}\|_{L^{\infty}([0,T]\times\mathbb{T}^{d})}\leq\bar{C},\qquad\|\rho_{f_{\varepsilon}}\|_{L^{\infty}([0,T]\times\mathbb{T}^{d})}\leq\bar{C}\varepsilon^{-(\zeta-1)},$

where $\zeta=\zeta(d)$ is as in the statement. Hence Theorem 3.1 and (3.7)
yield

$\sup_{[0,T]}W_{2}(f_{\varepsilon}(t),g_{\varepsilon}(t))\leq
2e^{-\left(\sqrt{\left|\log\left\\{\varepsilon^{-2}W_{2}(f_{0,\varepsilon},g_{0,\varepsilon})^{2}\left|\log\left(\frac{1}{2}\varepsilon^{-2}W_{2}(f_{0,\varepsilon},g_{0,\varepsilon})^{2}\right)\right|\right\\}\right|}-C_{d}\bar{C}T\varepsilon^{-\zeta}\right)^{2}}$

provided $C_{d}^{2}\bar{C}^{2}T^{2}<K$ (which can be guaranteed by taking $T$
smaller if necessary). This implies that
$\sup_{[0,T]}W_{2}(f_{\varepsilon}(t),g_{\varepsilon}(t))\to 0$ as
$\varepsilon\to 0,$ and we can now conclude as in [34, Proof of Theorem 1.7].
∎

###  3.3. Proof of Theorem 3.1

Before starting the proof we recall [34, Lemma 3.2], see also [29, Lemma 3.3].

######  Lemma 3.6.

Let $\Psi_{i}:\mathbb{T}^{d}\to\mathbb{R}$ solve

$-\varepsilon^{2}\Delta\Psi_{i}=\rho_{i}-1,\qquad i=1,2.$

Then

$\varepsilon^{2}\|\nabla\Psi_{1}-\nabla\Psi_{2}\|_{L^{2}(\mathbb{T}^{d})}\leq\Bigl{[}\max\bigl{\\{}\|\rho_{1}\|_{L^{\infty}(\mathbb{T}^{d})},\|\rho_{2}\|_{L^{\infty}(\mathbb{T}^{d})}\bigr{\\}}\Bigr{]}^{1/2}\,W_{2}(\rho_{1},\rho_{2}),$
$\varepsilon^{2}|\nabla\Psi_{i}(x)-\nabla\Psi_{i}(y)|\leq
C\,|x-y|\,\log\biggl{(}\frac{4\sqrt{d}}{|x-y|}\biggr{)}\,\|\rho_{i}-1\|_{L^{\infty}(\mathbb{T}^{d})}\qquad\forall\,x,y\in\mathbb{T}^{d},\,i=1,2.$

Let $(X_{i},V_{i})$ denote the characteristic flow associated to $f_{i}$, that
is

$\left\\{\begin{array}[]{l}\dot{X}_{i}(t,x,v)=V_{i}(t,x,v),\\\
\dot{V}_{i}(t,x,v)=E_{i}(t,X_{i}(t,x,v)),\\\
X_{i}(0,x,v)=x,\,\,V_{i}(0,x,v)=v,\end{array}\right.\qquad E_{i}=\nabla
U_{i},\qquad\varepsilon^{2}\Delta U_{i}=\rho_{f_{i}}-1.$

To prove Theorem 3.1, we consider $\pi_{0}$ an optimal $W_{2}$-coupling
between $f_{1}(0)$ and $f_{2}(0)$, and we define the quantity $Q(t)$ defined
as the unique constant (assuming it exists) such that

$Q(t)=\frac{1}{2}\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left[\varepsilon^{-2}|\log
Q(t)|\,|X_{1}(t,x,v)-X_{2}(t,y,w)|^{2}+|V_{1}(t,x,v)-V_{2}(t,y,w)|^{2}\right]d\pi_{0}(x,v,y,w).$

In other words, we are considering a quantity of the form

$Q(t)=\frac{1}{2}\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left[\lambda(t)|X_{1}(t,x,v)-X_{2}(t,y,w)|^{2}+|V_{1}(t,x,v)-V_{2}(t,y,w)|^{2}\right]d\pi_{0}(x,v,y,w),$

with $\lambda(t)$ depending on time, and we are assuming that actually
$\lambda(t)$ is a function of $Q(t)$ itself. The particular choice
$\lambda(t)=\varepsilon^{-2}|\log Q(t)|$ is specific to this problem: the
logarithm will help to compensate for the log-Lipschitz regularity of the
electric fields, while $\varepsilon^{-2}$ is the natural scaling in the
current setting.

Note that a priori is not clear that $Q(t)$ is well-defined. This will be
proved in Lemma 3.7 below. However, assuming for now that $Q(t)$ is well-
defined, we show how this quantity allows us to prove the result. We have

$\displaystyle Q^{\prime}(t)$
$\displaystyle=\frac{1}{2}\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\lambda^{\prime}(t)|X_{1}(t,x,v)-X_{2}(t,y,w)|^{2}\,d\pi_{0}(x,v,y,w)$
(3.11)
$\displaystyle\qquad+\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left[\lambda(t)(X_{1}(t,x,v)-X_{2}(t,y,w))\cdot(V_{1}(t,x,v)-V_{2}(t,y,w)\right]d\pi_{0}(x,v,y,w)$
(3.12)
$\displaystyle\qquad-\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left[(V_{1}(t,x,v)-V_{2}(t,y,w)\cdot(E_{1}(t,X_{1}(t,x,v))-E_{2}(t,X_{2}(t,y,w)))\right]d\pi_{0}(x,v,y,w)$
(3.13)

By Cauchy-Schwartz inequality and recalling the definition of $Q(t)$ we have:

$\displaystyle Q^{\prime}(t)$
$\displaystyle\leq\frac{1}{2}\lambda^{\prime}(t)\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}|X_{1}(t,x,v)-X_{2}(t,y,w)|^{2}\,d\pi_{0}(x,v,y,w)$
(3.15)
$\displaystyle\qquad+\lambda(t)\left(\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}|X_{1}(t,x,v)-X_{2}(t,y,w)|^{2}\,d\pi_{0}(x,v,y,w)\right)^{\frac{1}{2}}\cdot$
(3.16)
$\displaystyle\qquad\qquad\qquad\cdot\left(\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}|V_{1}(t,x,v)-V_{2}(t,y,w)|^{2}\,d\pi_{0}(x,v,y,w)\right)^{\frac{1}{2}}$
(3.17)
$\displaystyle\qquad+\left(\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}|V_{1}(t,x,v)-V_{2}(t,y,w)|^{2}\,d\pi_{0}(x,v,y,w)\right)^{\frac{1}{2}}\cdot$
(3.18)
$\displaystyle\qquad\qquad\qquad\cdot\left(\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}|E_{1}(t,X_{1}(t,x,v))-E_{2}(t,X_{2}(t,y,w))|^{2}\,d\pi_{0}(x,v,y,w)\right)^{\frac{1}{2}}$
(3.19)
$\displaystyle\leq\frac{1}{2}\lambda^{\prime}(t)\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}|X_{1}(t,x,v)-X_{2}(t,y,w)|^{2}\,d\pi_{0}(x,v,y,w)$
(3.20)
$\displaystyle+2\sqrt{\lambda(t)}Q(t)+\sqrt{Q(t)}\|E_{1}(t,X_{1}(t,x,v))-E_{2}(t,X_{2}(t,y,w))\|_{L^{2}(d\pi_{0}(x,v,y,w))}.$
(3.21)

Adding and subtracting $-E_{2}(t,X_{1})$ we obtain:

$\displaystyle Q^{\prime}(t)$
$\displaystyle\leq\frac{1}{2}\lambda^{\prime}(t)\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}|X_{1}(t,x,v)-X_{2}(t,y,w)|^{2}\,d\pi_{0}(x,v,y,w)$
(3.22)
$\displaystyle+2\sqrt{\lambda(t)}Q(t)+\sqrt{Q(t)}\left(T_{1}+T_{2}\right)$
(3.23)

where

$T_{1}=\|E_{2}(t,X_{1}(t,x,v))-E_{2}(t,X_{2}(t,y,w))\|_{L^{2}(d\pi_{0}(x,v,y,w))},$
$T_{2}=\|E_{1}(t,X_{1}(t,x,v))-E_{2}(t,X_{1}(t,x,v))\|_{L^{2}(d\pi_{0}(x,v,y,w))}.$

Thanks to Lemma 3.6 and by the very same argument in [34] we can bound $T_{1}$
and $T_{2}$ as follows: 333Note that, since $\rho_{i}\geq 0$ and
$\|\rho_{i}(\cdot,t)\|_{L^{\infty}(\mathbb{T}^{d})}\geq 1,$ then
$\|\rho_{i}(\cdot,t)-1\|_{L^{\infty}(\mathbb{T}^{d})}\leq\|\rho_{i}(\cdot,t)\|_{L^{\infty}(\mathbb{T}^{d})}\leq
A(t)$ for $i=1,2.$

$T_{2}\leq\frac{C}{\varepsilon^{2}}A(t)\sqrt{\frac{Q(t)}{\lambda(t)}},\qquad\text{and}\qquad
T_{1}\leq\frac{C}{\varepsilon^{2}}A(t)\sqrt{\phi\left(\frac{Q(t)}{\lambda(t)}\right)}$

where we have

$\phi(s)=\left\\{\begin{array}[]{ll}s\log^{2}(s)&\mbox{for}\,s\in(0,1/e]\\\
s&\mbox{for}\,s>1/e\end{array}\right.$ (3.24)

We now recall that $\lambda(t)=\varepsilon^{-2}|\log(Q(t))|$ and we substitute
this expression in the derivative of $Q(t).$ Notice that in this estimate we
are interested in small values of $Q(t)$ and in particular, as we will show
below, we will always be in the regime
$\varepsilon^{2}Q(t)/|\log(Q(t))|\in(0,1/e).$ Therefore we have

$T_{1}\leq\frac{C}{\varepsilon^{2}}A(t)\sqrt{\frac{\varepsilon^{2}\,Q(t)}{|\log(Q(t))|}\log^{2}\left(\frac{\varepsilon^{2}\,Q(t)}{|\log(Q(t))|}\right)},$

so by equation (3.22) we have

$Q^{\prime}(t)\leq\left(-\frac{1}{2}\frac{Q^{\prime}(t)}{|\log(Q(t))|}\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}|X_{1}(t,x,v)-X_{2}(t,y,w)|^{2}\,d\pi_{0}(x,v,y,w)\right)\\\
+\left(2\frac{\sqrt{|\log(Q(t))|}}{\varepsilon}+\frac{C\,A(t)}{\varepsilon\sqrt{|\log(Q(t))|}}\right)Q(t)+C\,A(t)\frac{\sqrt{Q(t)}}{\varepsilon}\sqrt{\frac{Q(t)}{|\log(Q(t))|}\log^{2}\left(\frac{\varepsilon^{2}\,Q(t)}{|\log(Q(t))|}\right)}.$

We now consider two cases, depending on the sign of $Q^{\prime}(t)$. If
$Q^{\prime}(t)\leq 0$, then we do not do anything. If instead
$Q^{\prime}(t)>0$, then the first term in the right-hand side above is
negative, and therefore

$\displaystyle
Q^{\prime}(t)\leq\left(2\frac{\sqrt{|\log(Q(t))|}}{\varepsilon}+\frac{C\,A(t)}{\varepsilon\sqrt{|\log(Q(t))|}}\right)Q(t)+C\,A(t)\frac{\sqrt{Q(t)}}{\varepsilon}\sqrt{\frac{Q(t)}{|\log(Q(t))|}\log^{2}\left(\frac{\varepsilon^{2}\,Q(t)}{|\log(Q(t))|}\right)}.$
(3.25)

Since the right-hand side above is nonnegative, independently of the sign of
$Q^{\prime}(t)$ we know that the bound above holds. We now observe that as
long as $Q(t)\leq\varepsilon$ then

$\log^{2}\left(\frac{\varepsilon^{2}\,Q(t)}{|\log(Q(t))|}\right)\leq
C\log^{2}(Q(t)).$

Thus,

$Q^{\prime}(t)\leq\left(2\frac{\sqrt{|\log(Q(t))|}}{\varepsilon}+\frac{C\,A(t)}{\varepsilon\sqrt{|\log(Q(t))|}}+\frac{C\,A(t)\sqrt{|\log(Q(t))|}}{\varepsilon}\right)Q(t)\qquad\text{provided
$Q(t)\leq\varepsilon$}.$

Since $A(t)\geq 1$ 444We recall that $\int_{\mathbb{T}^{d}}\rho(x,t)dx=1,$
that implies $\|\rho(\cdot,t)\|_{L^{\infty}(\mathbb{T}^{d})}\geq 1.$ Therefore
$A(t)\geq 1.$, provided $|\log(Q(t))|\geq 1$, the above bound reduces to

$Q^{\prime}(t)\leq\frac{2C_{d}\,A(t)}{\varepsilon}Q(t)\sqrt{|\log(Q(t))|}$

where $C_{d}$ is a dimensional constant. Note that the two conditions
$Q(t)\leq\varepsilon$ and $|\log(Q(t))|\geq 1$ are guaranteed if
$Q(t)\leq\frac{\varepsilon}{e}$ (recall that $\varepsilon\leq 1$ by
assumption).

Hence, provided that we are in the regime $Q(s)\leq\frac{\varepsilon}{e}$ on
$[0,t]$, this implies

$Q(t)\leq
R(t):=e^{-\left(\sqrt{|\log(Q(0))|}-\frac{C_{d}}{\varepsilon}\int_{0}^{t}A(s)\,ds\right)^{2}}.$
(3.26)

We observe that the bound (3.26) guarantees that

$\sup_{s\in[0,t]}Q(s)\leq\frac{\varepsilon}{e}\quad\text{ holds if
}\quad\sup_{s\in[0,t]}R(s)\leq\frac{\varepsilon}{e}.$

In particular, (3.26) holds if

$\sqrt{|\log(Q(0))|}\geq\frac{C_{d}}{\varepsilon}\int_{0}^{t}A(s)\,ds+\sqrt{\left|\log\left(\frac{\varepsilon}{e}\right)\right|}$
(3.27)

We now compare the quantity $Q$ to the Wasserstein distance. First of all,
since $Q(t)\leq\frac{\varepsilon}{e}$ then $\varepsilon^{-2}|\log(Q(t))|\geq
1$, therefore

$\displaystyle\frac{1}{2}W_{2}(f_{1}(t),f_{2}(t))^{2}$
$\displaystyle\leq\frac{1}{2}\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left[|X_{1}(t,x,v)-X_{2}(t,y,w)|^{2}+|V_{1}(t,x,v)-V_{2}(t,y,w)|^{2}\right]d\pi_{0}(x,v,y,w)$
(3.28) $\displaystyle\leq Q(t).$ (3.29)

On the other hand, since $\varepsilon^{-2}|\log(Q(0))|\geq 1$ and $\pi_{0}$ is
an optimal plan,

$Q(0)\leq\frac{1}{2}\varepsilon^{-2}|\log(Q(0))|\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left[|x-y|^{2}+|v-w|^{2}\right]d\pi_{0}(x,v,y,w)=\frac{1}{2}\varepsilon^{-2}|\log(Q(0))|W_{2}(f_{1}(0),f_{2}(0))^{2},$

or equivalently

$\frac{Q(0)}{|\log(Q(0))|}\leq\frac{1}{2}\varepsilon^{-2}W_{2}(f_{1}(0),f_{2}(0))^{2},$

We now observe that, near the origin, the inverse of the function
$s\mapsto\frac{s}{|\log s|}$ behaves like $\tau\mapsto\tau|\log\tau|$. In
particular, there exists a universal small constant $c_{0}>0$ such that

$\frac{s}{|\log s|}\leq\tau\qquad\text{for some }0\leq\tau\leq
c_{0}\qquad\Rightarrow\qquad s\leq 2\tau|\log\tau|.$

Hence, if $\frac{1}{2}\varepsilon^{-2}W_{2}(f_{1}(0),f_{2}(0))^{2}\leq c_{0}$,
we deduce that

$Q(0)\leq\varepsilon^{-2}W_{2}(f_{1}(0),f_{2}(0))^{2}\left|\log\left(\frac{1}{2}\varepsilon^{-2}W_{2}(f_{1}(0),f_{2}(0))^{2}\right)\right|.$

Combining these bounds with (3.26), and recalling (3.27), this implies

$W_{2}(f_{1}(t),f_{2}(t))^{2}\leq
2e^{-\left(\sqrt{\left|\log\left(\varepsilon^{-2}W_{2}(f_{1}(0),f_{2}(0))^{2}\left|\log\left(\frac{1}{2}\varepsilon^{-2}W_{2}(f_{1}(0),f_{2}(0))^{2}\right)\right|\right)\right|}-\frac{C_{d}}{\varepsilon}\int_{0}^{t}A(s)\,ds\right)^{2}}$

provided $\frac{1}{2}\varepsilon^{-2}W_{2}(f_{1}(0),f_{2}(0))^{2}\leq c_{0}$
and

$\sqrt{\left|\log\left(\varepsilon^{-2}W_{2}(f_{1}(0),f_{2}(0))^{2}\left|\log\left(\frac{1}{2}\varepsilon^{-2}W_{2}(f_{1}(0),f_{2}(0))^{2}\right)\right|\right)\right|}\geq\frac{C_{d}}{\varepsilon}\int_{0}^{t}A(s)\,ds+\sqrt{\left|\log\left(\frac{\varepsilon}{e}\right)\right|}.$

Finally, to complete the proof, we show the following:

######  Lemma 3.7.

With the notation and assumptions of the theorem, the quantity $Q(t)$ is well
defined and it is locally Lipschitz continuous where $Q(t)>0$. In particular
it is differentiable a.e.

###### Proof.

Set

$D(t):=\frac{1}{2}\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}|X_{1}(t,x,v)-X_{2}(t,y,w)|^{2}d\pi_{0}(x,v,y,w),$
$E(t):=\frac{1}{2}\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}|V_{1}(t,x,v)-V_{2}(t,y,w)|^{2}d\pi_{0}(x,v,y,w).$

We can assume that $D(t)$ and $E(t)$ are nonzero, otherwise we are in the
“degenerate” situation where $f_{1}\equiv f_{2}$, in which case $Q(t)$ is
trivially 0. Also, since $D(t)$ and $E(t)$ are written in terms of the
characteristic flow, it is standard to check that they are differentiable.
555There is no novelty here, as these are the quantities that appear also in
[49], where Loeper computes their derivatives and show that they can be
controlled in terms of $D(t)$ and $E(t)$ themselves. In particular, the
quantities $D(t)$ and $E(t)$ are also uniformly Lipschitz.

We note that the quantity $Q(t)$ is implicitly defined via the relation

$Q(t)=\varepsilon^{-2}|\log Q(t)|D(t)+E(t),$ (3.30)

or equivalently, for each fixed time $t$, $Q(t)$ is the solution of the
equation

$F(q,D(t),E(t))=0\qquad\text{with }F(q,r,s):=q+\varepsilon^{-2}\log
q\,r-s\qquad\text{for }q\in(0,1).$

Since the function $q\mapsto q+\varepsilon^{-2}\log q\,D(t)$ is strictly
increasing on $(0,1)$ and its image covers the interval $(0,1)$, we deduce
that the equation above has a unique solution provided $E(t)<1$. Hence, this
proves that $Q(t)\in(0,1)$ is well defined provided $E(t)<1$. In addition,
thanks to the implicit function theorem applied to the function $F\in
C^{1}_{\rm loc}((0,1)\times{\mathbb{R}}\times{\mathbb{R}})$, we deduce the
existence of a $C^{1}_{\rm loc}$ function $G$ such that $Q(t)=G(D(t),E(t))$.
Now, differentiating the relation (3.30) with respect to $t$ we obtain

$Q^{\prime}(t)\biggl{(}1+\varepsilon^{-2}\frac{D(t)}{Q(t)}\biggr{)}=\varepsilon^{-2}|\log
Q(t)|D^{\prime}(t)+E^{\prime}(t).$

Hence, since $D$ and $E$ are uniformly bounded and Lipschitz, for any
$\delta>0$ we deduce that

$|Q^{\prime}(t)|\leq\frac{\varepsilon^{-2}|\log
Q(t)|\,|D^{\prime}(t)|+|E^{\prime}(t)|}{1+\varepsilon^{-2}\frac{D(t)}{Q(t)}}\leq
C_{\delta}\qquad\text{where }Q(t)>\delta.$

This proves that, for any $\delta>0$, the function $t\mapsto Q(t)$ is
uniformly Lipschitz continuous inside the set $\\{Q(t)>\delta\\}$. This proves
that $Q(t)$ is locally Lipschitz continuous inside the region $\\{Q(t)>0\\}$.

So, to conclude the proof, we need to ensure that $E(t)<1$. Note that, since
by assumption $E(0)\leq\frac{1}{2}W_{2}(f_{1}(0),f_{2}(0))^{2}\ll 1$, by
continuity we have that $E(t)<1$ for $t>0$ small. So $Q(t)$ is well defined
for $t>0$ small. Also, as long as $Q(t)$ is well defined, we have that
$Q(t)\geq E(t)$. Hence, as long as $Q(t)$ is well defined, we have that

$E(t)\leq Q(t)\leq
e^{-\left(\sqrt{|\log(Q(0))|}-\frac{C_{d}}{\varepsilon}\int_{0}^{t}A(s)\,ds\right)^{2}}.$

Since, by our smallness assumption on $W_{2}(f_{1}(0),f_{2}(0))$, the right
hand side above remain small on $[0,T]$, the bound above guarantees that
$E(t)\ll 1$ for all $t\in[0,T]$. This proves that $Q(t)$ is well-defined on
$[0,T]$, which concludes the proof. ∎

######  Remark 3.8.

In the previous proof we considered $\lambda(t)=\varepsilon^{-2}|\log(Q(t))|$
and in Lemma 3.7 we proved that $Q(t)$ is well-defined provided it is small
enough. This restriction is due to the fact that the function
${\mathbb{R}}^{+}\ni s\mapsto\varepsilon^{-2}|\log s|$ is decreasing only for
$s\in(0,1).$ An alternative choice could have been to define

$\Phi(s)=\left\\{\begin{array}[]{ll}|\log s|&\mbox{for}\,s\in(0,1/e]\\\
e^{-1}s^{-1}&\mbox{for}\,s>1/e,\end{array}\right.$

and $\lambda(t):=\varepsilon^{-2}\Phi(Q(t)).$ With this choice, since
${\mathbb{R}}^{+}\ni s\mapsto\Phi(s)$ is decreasing and of class $C^{1}$, one
can define $Q(t)$ as the unique solution of

$F(q,D(t),E(t))=0\qquad\text{with
}F(q,r,s):=q+\varepsilon^{-2}\Phi(q)\,r-s\qquad\text{for }q\in(0,\infty).$

With this definition, the proof of Lemma 3.7 shows that $Q(t)$ is always well
defined (without any restriction on the size of $E(t)$), and it is locally
Lipschitz continuous where $Q(t)>0$.

Since in our setting we are interested in the case $Q(t)\ll 1,$ there is no
advantage in using this latter definition of $\lambda.$ However this
observation could be useful in other situations, see also Section 4 below.

##  4\. Summary, generalizations, and perspectives

As we have seen in the last two sections, suitably modifying Wasserstein
distances can be particularly useful in a kinetic setting to take advantage of
the asymmetry between $x$ and $v$. More precisely, let
$\mathcal{X}=\mathbb{T}^{d}$ or $\mathcal{X}=\mathbb{R}^{d}$, and let $\mu$
and $\nu$ be two probability measures on $\mathcal{X}\times\mathbb{R}^{d}$.
Also, let $\Pi(\mu,\nu)$ denote the collection of all measures on
$(\mathcal{X}\times\mathbb{R}^{d})^{2}$ with marginals $\mu$ and $\nu$ on the
first and second factors respectively.

The first natural generalization, given $p\geq 1$ and
$\lambda\in\mathbb{R}^{+}$, is to consider

$W_{\lambda,\,p}(\mu,\nu):=\left(\inf_{\pi\in\Pi(\mu,\nu)}\int_{(\mathcal{X}\times\mathbb{R}^{d})^{2}}\left(\lambda|x-y|^{p}+|v-w|^{p}\right)\mathrm{d}\pi(x,v,y,w)\right)^{1/p},$

as done in [28, 30, 43].

An alternative way, introduced in [56] for $p=2$, would be to consider three
parameters $a,b,c>0$ such that $\sqrt{ac}>b$ and define

$W_{a,b,c,\,p}(\mu,\nu):=\left(\inf_{\pi\in\Pi(\mu,\nu)}\int_{(\mathcal{X}\times\mathbb{R}^{d})^{2}}\left(a|x-y|^{2}+2b(x-y)\cdot(v-w)+c|v-w|^{2}\right)^{p/2}\mathrm{d}\pi(x,v,y,w)\right)^{1/p},$

In this paper, we have introduced two different generalizations.

  * (i)

First, we considered the nonlinear version of the $W_{\lambda,\,p}(\mu,\nu)$
by choosing $\lambda$ depending on the distance itself. We defined this along
a flow, but that can be also be defined in a general setting as follows:
given $p\geq 1$ and a decreasing function
$\Phi:{\mathbb{R}}^{+}\to{\mathbb{R}}^{+}$, for every $\pi\in\Pi(\mu,\nu)$ and
$\lambda$ we define $D_{p}(\pi,\Phi)$ as the unique number $s$ such that

$s-\Phi(s)\int_{(\mathcal{X}\times\mathbb{R}^{d})^{2}}|x-y|^{p}\mathrm{d}\pi(x,v,y,w)=\int_{(\mathcal{X}\times\mathbb{R}^{d})^{2}}|v-w|^{p}\mathrm{d}\pi(x,v,y,w)$

(arguing as in the proof of Lemma 3.7 it is easy to check that $D(\pi,\Phi)$
is well defined, see also Remark 3.8). Then, we set

$W_{\Phi,p}(\mu,\nu):=\left(\inf_{\pi\in\Pi(\mu,\nu)}D_{p}(\pi,\Phi)\right)^{1/p}.$

This definition with $\Phi(s)=\varepsilon^{-2}|\log s|$ for $s\in(0,1/e)$ and
$p=2$ essentially corresponds to the quantity used in the proof of Theorem
3.1, although there we considered the quantity $D(t)$ where we did not take
the infimum over couplings $\pi\in\Pi(\mu,\nu),$ since it was not needed for
our purpose.

  * (ii)

In a different direction, we modified the $W_{1}$ distance by introducing a
shift in position. Note that this second quantity cannot be defined as a
“static” distance since the shift $x-tv$ depends on the time $t$. Hence, one
can generalize it only as a time dependent quantity as follows:

$\widetilde{W}_{t,\,p}(\mu,\nu):=\left(\inf_{\pi\in\Pi(\mu,\nu)}\int_{(\mathcal{X}\times\mathbb{R}^{d})^{2}}\big{(}|(x-tv)-(y-tw)|^{p}+|v-w|^{p}\big{)}\mathrm{d}\pi(x,v,y,w)\right)^{1/p}.$

Of course, these approaches can be further combined by mixing the different
quantities defined above. Note that there is no universal “best” choice, and
each problem requires its adaptation. Still, we believe, as this paper shows,
that this approach can lead to an improvement to several existing results, as
well as to prove new estimates. In addition, the approach is very general and
can be useful in any situation where there is an asymmetry between the
variables involved.

To mention some concrete applications, our ideas could also be applied in the
setting of quantum systems by suitably modifying the quantum Wasserstein
distances introduced in [21, 24]. Also, our new Loeper-type estimate may be
helpful to obtain stability estimates in $W_{2}$ when the density belongs to
some suitable Orlicz spaces, in analogy to [38] where stability estimates have
been proved for $W_{1}$.

## Acknowledgments

We are grateful to Megan Griffin-Pickering and Evelyne Miot for their valuable
comments on a preliminary version of this paper. We also thank the anonymous
referees for their useful comments and observations.

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