# Stability estimates for the Vlasov-Poisson system in $p$-kinetic Wasserstein
distances

Mikaela Iacobelli  and  Jonathan Junné

###### Abstract.

We extend Loeper’s $L^{2}$-estimate [12, Theorem 2.9] relating the electric
fields to the densities for the Vlasov-Poisson system to $L^{p}$, with
$1<p<+\infty$, based on the Helmholtz-Weyl decomposition. This allows us to
generalize both the classical Loeper’s $2$-Wasserstein stability estimate [12,
Theorem 1.2] and the recent stability estimate by the first author relying on
the newly introduced kinetic Wasserstein distance [10, Theorem 3.1] to kinetic
Wasserstein distances of order $1<p<+\infty$.

ETH Zürich, Department of Mathematics, Rämistrasse 101, 8092 Zürich,
Switzerland.
Email<EMAIL_ADDRESS>
TU Delft, Delft Institute of Applied Mathematics, Mekelweg 4, 2628 CD Delft,
Netherlands.
Email<EMAIL_ADDRESS>
##  1\. Introduction

###  1.1. General overview

Monge-Kantorovich distances, also known as Wasserstein distances, are used
extensively in kinetic theory, in particularly in the context of stability,
convergence to equilibrium and mean-field limits. A first celebrated result
for the $1$-Monge-Kantorovich distance is due to Dobrushin [2, Theorem 1], who
proved the well-posedness for Vlasov equations with $C^{1,1}$ potentials. An
explanation of Dobrushin’s stability estimate and its consequences on the
mean-field limit for the Vlasov equation can be found in [5, Chapter 1] and
[11, Chapter 3], and we refer to [7, Section 3] for a survey on well-posedness
for the Vlasov-Poisson system.

Regarding the $2$-Wasserstein distance, Loeper proved [12, Theorem 1.2] a
uniqueness criterion for solutions with bounded density based on a
$2$-Wasserstein distance stability estimate using both a link between the
$\dot{H}^{-1}$-seminorm and the $2$-Wasserstein distance, and the fact that
the Coulomb kernel is generated by a potential solving the Poisson equation.
In addition to the Vlasov-Poisson system, this criterion gives a new proof of
uniqueness à la Yudovich for $2D$ Euler. Beyond bounded density, Loeper’s
uniqueness criterion has been extended for some suitable Orlicz spaces using
the 1-Monge-Kantorovich distance by Miot [15, Theorem 1.1] and Miot, Holding
[9, Theorem 1.1].

On the Torus, Loeper’s criterion was improved by Han-Kwan, Iacobelli [8,
Theorem 3.1] for the Vlasov-Poisson system, and more recently for the Vlasov-
Poisson system with massless electron by Griffin-Pickering, Iacobelli [6,
Theorem 4.1].

The aim of this work is twofold. The first goal is to generalize Loeper’s
$2$-Wasserstein distance stability estimate to $p$-Wasserstein distances for
$1<p<+\infty$. The second goal is to extend the recent stability estimate [10,
Theorem 3.1] by the first author relying on the newly introduced kinetic
Wasserstein distance [10, Theorem 3.1] to kinetic Wasserstein distances of
order $1<p<+\infty$.

###  1.2. Definitions and main results

We first recall the classical Wasserstein distance (see [19, Chapter 6]) on
the product space $\mathcal{X}\times\mathbb{R}^{d}$, with $\mathcal{X}$
denoting in the sequel 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
$\mathcal{X}\times\mathbb{R}^{d}$. The _Wasserstein distance_ of order $p$,
with $p\geq 1$, between $\mu$ and $\nu$ is defined as

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

where $\Pi(\mu,\nu)$ is the set of _couplings_ ; that is, the set of
probability measures with _marginals_ $\mu$ and $\nu$. A coupling is said to
be _optimal_ if it minimizes the Wasserstein distance.

We consider two solutions $f_{1},f_{2}$ of the Vlasov-Poisson system on
$\mathcal{X}$, namely,

$\partial_{t}f+v\cdot\partial_{x}f+\nabla U\cdot\nabla_{v}f=0,\quad-\Delta
U:=\rho_{f}-1,\quad\rho_{f}:=\int_{\mathbb{R}^{d}}f\>dv$ (1.1)

on the torus, and

$\partial_{t}f+v\cdot\partial_{x}f+\nabla U\cdot\nabla_{v}f=0,\quad-\Delta
U:=\rho_{f},\quad\rho_{f}:=\int_{\mathbb{R}^{d}}f\>dv$ (1.2)

on the whole space, with initial profiles $f_{1}(0),f_{2}(0)$, and denote by
$Z_{1}:=(X_{1},V_{1})$ and $Z_{2}:=(X_{2},V_{2})$ their respective flows.

(1) A new $L^{p}$-estimate for the difference of electric fields. Loeper
estimates the $L^{2}$-norm [12, Theorem 2.9] of the difference of electric
fields with the Wasserstein distance between the densities. We extend the
$L^{2}$-estimate to $L^{p}$ for $1<p<+\infty$ using the Helmholtz-Weyl
decomposition of $L^{p}(\mathcal{X})=G_{p}(\mathcal{X})\oplus
H_{p}(\mathcal{X})$ into its hydrodynamic space that we recall (see [3,
Chapter III]):

######  Definition 1.2.

The _hydrodynamic spaces_ are the closed subspaces of $L^{p}(\mathcal{X})$
defined as

$G_{p}(\mathcal{X}):=\left\\{u\in L^{p}(\mathcal{X});\quad u=\nabla w\text{
for some }w\in W^{1,p}_{\operatorname{loc}}(\mathcal{X})\right\\}$

and

$H_{p}(\mathcal{X}):=\overline{\left\\{u\in
C_{c}^{\infty}(\mathcal{X});\quad\operatorname{div}u=0\text{ on
}\mathcal{X}\right\\}}.$

######  Remark 1.3.

Note that this decomposition breaks down for $p=1$ or $p=+\infty$ and does not
hold for general domains in $L^{p}$. It is equivalent to the solvability of an
Neumann problem (see [3, Lemma III.1.2]), while an orthogonal decomposition in
$L^{2}$ is always possible, whatever the domain is.

Using optimal transport techniques, Loeper manages to link the strong dual
homogeneous Sobolev norm and Wasserstein distances, and we recall those
notions:

######  Definition 1.4.

Let $1<p<+\infty$. The _homogeneous Sobolev space_ is the space

$\dot{W}^{1,p}(\mathcal{X}):=\left\\{[g];\quad g\in
W^{1,p}_{\operatorname{loc}}(\mathcal{X}),\quad\nabla g\in
L^{p}(\mathcal{X})\right\\},$

where $[\cdot]:=\\{\cdot+c;\>c\in\mathbb{R}\\}$ denotes the equivalence class
of functions up to a constant, together with the norm

$\left\lVert[g]\right\rVert_{\dot{W}^{1,p}(\mathcal{X})}:=\left\lVert\nabla
g\right\rVert_{L^{p}(\mathcal{X})}.$

This is a Banach space for which the equivalence classes of test functions

$\dot{\mathcal{D}}(\mathcal{X}):=\\{[\phi];\quad\phi\in
C^{\infty}_{c}(\mathcal{X})\\}$

are dense in it (see [16, Theorem 2.1]).

######  Definition 1.5.

We define the _dual homogeneous Sobolev space_ $\dot{W}^{-1,p}(\mathcal{X})$
to be the topological dual of $\dot{W}^{1,p^{\prime}}(\mathcal{X})$ equipped
with the _strong dual homogeneous Sobolev norm_. For a function $h$ with $\int
h=0$, by density,

$\displaystyle\left\lVert h\right\rVert_{\dot{W}^{-1,p}(\mathcal{X})}$
$\displaystyle:=\sup\left\\{\int_{\mathcal{X}}h\,[g]\>dx;\quad
g\in\dot{W}^{1,p^{\prime}}(\mathcal{X}),\quad\left\lVert[g]\right\rVert_{\dot{W}^{1,p^{\prime}}(\mathcal{X})}\leq
1\right\\}$
$\displaystyle=\sup\left\\{\int_{\mathcal{X}}h\,[\phi]\>dx;\quad\phi\in\dot{\mathcal{D}}(\mathcal{X}),\quad\left\lVert[\phi]\right\rVert_{\dot{W}^{1,p^{\prime}}(\mathcal{X})\leq
1}\right\\}.$

In our setting; that is either on the torus or on the Euclidean space, the
Helmholtz-Weyl decomposition holds [3, Theorem III.1.1 & Theorem III.1.2];

######  Theorem 1.6 (Helmholtz-Weyl decomposition).

The Helmholtz-Weyl decomposition holds for $L^{p}(\mathcal{X})$, for any
$1<p<+\infty$; that is,

$L^{p}(\mathcal{X})=G_{p}(\mathcal{X})\oplus H_{p}(\mathcal{X}).$

Moreover, when $p=2$, this decomposition is orthogonal.

The validity of the Helmholtz-Weyl decomposition implies the existence of an
Helmholtz-Weyl bounded linear projection operator (see [3, Remark III.1.1])

$\displaystyle P_{p}:L^{p}(\mathcal{X})\to H_{p}(\mathcal{X})$

with range $H_{p}(\mathcal{X})$ and with $G_{p}(\mathcal{X})$ as null space.
More precisely, there is a constant $C_{P_{p}}>0$ that only depends on $p$ and
$\mathcal{X}$ such that for all $u\in L^{p}(\mathcal{X})$, it holds

$\displaystyle\left\lVert P_{p}(u)\right\rVert_{L^{p}(\mathcal{X})}\leq
C_{P_{p}}\left\lVert u\right\rVert_{L^{p}(\mathcal{X})}.$ (1.3)

First, we extend this connection for densities to $L^{p}$. Using the machinery
of Helmholtz-Weyl decomposition, we generalize [12, Lemma 2.10] into the
following:

######  Lemma 1.7.

Let $\rho_{1},\rho_{2}\ L^{\infty}(\mathcal{X})$ be two probability measures,
and let $U_{i}$ satisfy $-\Delta U_{i}=\rho_{i}$ for
$\mathcal{X}=\mathbb{R}^{d}$, or $-\Delta U_{i}=\rho_{i}-1$ for
$\mathcal{X}=\mathbb{T}^{d}$, with $i=1,2$, in the distributional sense. Let
$1<p<+\infty$. Then there is a constant $C_{\mathrm{HW}}>0$ that only depends
on $p$ and $\mathcal{X}$ such that

$\left\lVert\nabla U_{1}-\nabla U_{2}\right\rVert_{L^{p}(\mathcal{X})}\leq
C_{\mathrm{HW}}\left\lVert\rho_{1}-\rho_{2}\right\rVert_{\dot{W}^{-1,p}(\mathcal{X})}.$
(1.4)

Second, we adapt Loeper’s argument of the $L^{2}$-estimate [12, Theorem 2.9]
with this new link to get the new $L^{p}$-estimate allowing us to generalize
stability estimates;

######  Proposition 1.8.

Let $\rho_{1},\rho_{2}\in L^{\infty}(\mathcal{X})$ be two probability
measures, and let $U_{i}$ satisfy $-\Delta U_{i}=\rho_{i}$ for
$\mathcal{X}=\mathbb{R}^{d}$, or $-\Delta U_{i}=\rho_{i}-1$ for
$\mathcal{X}=\mathbb{T}^{d}$, with $i=1,2$, in the distributional sense. Let
$1<p<+\infty$. Then there is a constant $C_{\mathrm{HW}}>0$ that only depends
on $p$ and $\mathcal{X}$ such that

$\left\lVert\nabla U_{1}-\nabla U_{2}\right\rVert_{L^{p}(\mathcal{X})}\leq
C_{\mathrm{HW}}\max\left\\{\left\lVert\rho_{1}\right\rVert_{L^{\infty}(\mathcal{X})},\left\lVert\rho_{2}\right\rVert_{L^{\infty}(\mathcal{X})}\right\\}^{1/p^{\prime}}W_{p}(\rho_{1},\rho_{2}).$
(1.5)

(2) Loeper’s stability estimate in $W_{p}$. Loeper noted [12, Lemma 3.6] that
both the Wasserstein distance of order two of the solutions and of the
associated densities are bounded by a flow quantity $Q$ given by

$Q(t):=\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\left\lvert
X_{1}(t,x,v)-X_{2}(t,x,v)\right\rvert^{2}+\left\lvert
V_{1}(t,x,v)-V_{2}(t,x,v)\right\rvert^{2}\>df^{0}(x,v),$

and the bounds read as

$W_{2}^{2}(f_{1}(t),f_{2}(t))\leq Q(t),\quad
W_{2}^{2}(\rho_{f_{1}}(t),\rho_{f_{2}}(t))\leq Q(t).$ (1.6)

Loeper uses the quantity $Q(t)$ together with the $L^{2}$-estimate on the
electric fields to prove the stability estimate [12, Theorem 1.2] leading to
the uniqueness of weak solutions. By modifying the quantity $Q(t)$ to

$\displaystyle Q_{p}(t)$
$\displaystyle:=\int_{(\mathcal{X}\times\mathbb{R}^{d})^{2}}\left\lvert
X_{1}(t,x,v)-X_{2}(t,y,w)\right\rvert^{p}+\left\lvert
V_{1}(t,x,v)-V_{2}(t,y,w)\right\rvert^{p}\>d\pi_{0}(x,v,y,w)$
$\displaystyle=\int_{(\mathcal{X}\times\mathbb{R}^{d})^{2}}\left\lvert
x-v\right\rvert^{p}+\left\lvert y-w\right\rvert^{p}\>d\pi_{t}(x,v,y,w),$

where $\pi_{t}\in\Pi(f_{1}(t),f_{2}(t))$ and $\pi_{0}$ is an optimal $W_{p}$
coupling (see [6, Section 4] for a construction of $\pi_{t}$), we are able to
generalize Loeper’s stability estimate [12, Theorem 1.2], and [8, Theorem 3.1]
in the torus case, to any Wasserstein distance of order $p$, with
$1<p<+\infty$;

######  Theorem 1.9.

Let $f_{1},f_{2}$ be two weak solutions to the Vlasov-Poisson system on
$\mathbb{T}^{d}$ (1.1) with respective densities

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

Let $1<p<+\infty$, and set

$A(t):=\left\lVert\rho_{f_{2}}(t)-1\right\rVert_{L^{\infty}(\mathbb{T}^{d})}+\left\lVert\rho_{f_{1}}(t)\right\rVert^{1/p}\max\left\\{\left\lVert\rho_{f_{1}}(t)\right\rVert_{L^{\infty}(\mathbb{T}^{d})},\left\lVert\rho_{f_{2}}(t)\right\rVert_{L^{\infty}(\mathbb{T}^{d})}\right\\}^{1/p^{\prime}},$
(1.7)

which is assumed to be in $L^{1}([0,T))$ for some $T>0$. Then there is a
constant $C_{\mathrm{L}}>0$ that only depends on $p$ and $d$ such that if
$W_{p}^{p}(f_{1}(0),f_{2}(0))$ is sufficiently small so that
$W_{p}^{p}(f_{1}(0),f_{2}(0))\leq(4\sqrt{d}/e)^{p}$ and

$\left\lvert\log\left(\frac{W_{p}^{p}(f_{1}(0),f_{2}(0))}{\left(4\sqrt{d}\right)^{p}}\right)\right\rvert\geq
p\exp\left(-C_{\mathrm{L}}\int_{0}^{T}A(s)\>ds\right)$ (1.8)

then

$W_{p}^{p}(f_{1}(t),f_{2}(t))\leq\left(4\sqrt{d}\right)^{p}\exp\left\\{\log\left(\frac{W_{p}^{p}(f_{1}(0),f_{2}(0))}{\left(4\sqrt{d}\right)^{p}}\right)\exp\left(C_{\mathrm{L}}\int_{0}^{t}A(s)\>ds\right)\right\\}$
(1.9)

(3) An improved $W_{p}$ stability estimate via kinetic Wasserstein distance.
Due to the anisotropy between position and momentum variables, we use an
adapted Wasserstein distance designed for kinetic problems taking this into
account as introduced in [10, Section 4]:

######  Definition 1.10.

Let $\mu,\nu$ be two probability measures on
$\mathcal{X}\times\mathbb{R}^{d}$. The _kinetic Wasserstein distance_ of order
$p$, with $p\geq 1$, between $\mu$ and $\nu$ is defined as

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

where $D_{p}(\pi,\lambda)$ is the unique number $s$ such that

$s-\lambda(s)\int_{(\mathcal{X}\times\mathbb{R}^{d})^{2}}\left\lvert
x-y\right\rvert^{p}\>d\pi(x,y,v,w)=\int_{(\mathcal{X}\times\mathbb{R}^{d})^{2}}\left\lvert
v-w\right\rvert^{p}\>d\pi(x,y,v,w),$

with $\lambda:\mathbb{R}^{+}\to\mathbb{R}^{+}$ a decreasing function.

We consider the quantity $D_{p}(t)$ for $\pi_{t}$ and
$\lambda(t)=\left\lvert\log D_{p}(t)\right\rvert$ (see [10, Lemma 3.7] for the
proof of existence) given by

$\displaystyle D_{p}(t)$
$\displaystyle:=\frac{1}{p}\int_{(\mathcal{X}\times\mathbb{R}^{d})^{2}}\lambda(t)\left\lvert
X_{1}(t,x,v)-X_{2}(t,y,w)\right\rvert^{p}+\left\lvert
V_{1}(t,x,v)-V_{2}(t,y,w)\right\rvert^{p}\>d\pi_{0}(x,v,y,w)$
$\displaystyle=\frac{1}{p}\int_{(\mathcal{X}\times\mathbb{R}^{d})^{2}}\lambda(t)\left\lvert
x-v\right\rvert^{p}+\left\lvert y-w\right\rvert^{p}\>d\pi_{t}(x,v,y,w).$

This quantity also compares to the usual Wasserstein distance $W_{p}$ as does
$Q_{p}(t)$, and this allows us to generalize the recent Iacobelli’s stability
estimate [10, Theorem 3.1] to the following:

######  Theorem 1.11.

Let $f_{1},f_{2}$ be two weak solutions to the Vlasov-Poisson system on
$\mathbb{T}^{d}$ (1.1) with respective densities

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

Let $1<p<+\infty$, and set

$A(t):=\left\lVert\rho_{f_{2}}(t)-1\right\rVert_{L^{\infty}(\mathbb{T}^{d})}+\left\lVert\rho_{f_{1}}(t)\right\rVert^{1/p}\max\left\\{\left\lVert\rho_{f_{1}}(t)\right\rVert_{L^{\infty}(\mathbb{T}^{d})},\left\lVert\rho_{f_{2}}(t)\right\rVert_{L^{\infty}(\mathbb{T}^{d})}\right\\}^{1/p^{\prime}},$
(1.10)

which is assumed to be in $L^{1}([0,T))$ for some $T>0$. Then there is a
universal constant $c_{0}>0$ and a constant $C_{\mathrm{KW}}>0$ that depends
only on $p$ and $d$ such that if $W_{p}^{p}(f_{1}(0),f_{2}(0))$ is
sufficiently small so that $W_{p}^{p}(f_{1}(0),f_{2}(0))\leq pc_{0}$ and

$\sqrt{\left\lvert\log\Bigg{\\{}W_{p}^{p}(f_{1}(0),f_{2}(0))\left\lvert\log\left(\frac{1}{p}W_{p}^{p}(f_{1}(0),f_{2}(0))\right)\right\rvert\Bigg{\\}}\right\rvert}\geq
C_{\mathrm{KW}}\int_{0}^{T}A(s)\>ds+1,$ (1.11)

then

$W_{p}^{p}(f_{1}(t),f_{2}(t))\\\ \leq
p\exp\Bigg{\\{}-\left(\sqrt{\left\lvert\log\Bigg{\\{}W_{p}^{p}(f_{1}(0),f_{2}(0))\left\lvert\log\left(\frac{1}{p}W_{p}^{p}(f_{1}(0),f_{2}(0))\right)\right\rvert\Bigg{\\}}\right\rvert}-C_{\mathrm{KW}}\int_{0}^{t}A(s)\>ds\right)^{2}\Bigg{\\}}.$
(1.12)

##  2\. A new $L^{p}$-estimate via the Helmholtz-Weyl decomposition for
$1<p<+\infty$

###  2.1. Proof of the $L^{p}$-estimate

###### Proof of Lemma 1.7.

Let $[\phi]\in\dot{\mathcal{D}}(\mathcal{X})$ be a quotient test function.
Note that $\int\rho_{1}-\rho_{2}=0$ as both $\rho_{1}$ and $\rho_{2}$ are
probability measures. Then integrating by parts,

$\int_{\mathcal{X}}[\phi]\left(\rho_{1}-\rho_{2}\right)\>dx=\int_{\mathcal{X}}\phi\left(\rho_{1}-\rho_{2}\right)\>dx=\int_{\mathcal{X}}\nabla\phi\cdot(\nabla
U_{1}-\nabla U_{2})\>dx.$

We use the Helmholtz-Weyl decomposition given by Theorem 1.6 to write any
$\mathbb{R}^{d}$-valued test function $\Phi\in C_{c}^{\infty}(\mathcal{X})$ as
$\Phi=\nabla\phi+g$, where $\nabla\phi\in G_{p^{\prime}}(\mathcal{X})$ and
$g\in H_{p^{\prime}}(\mathcal{X})$ with $1/p+1/p^{\prime}=1$. By definition,
there is a divergence-free sequence of test functions
$(g_{k})_{k\in\mathbb{N}}$, with compact support lying in some sequence of
open balls $(B_{r_{k}}(\mathcal{X}))_{k\in\mathbb{N}}$, whose
$L^{p^{\prime}}$-limit is $g$. An integration by parts yields

$\displaystyle\left\lVert\nabla U_{1}-\nabla
U_{2}\right\rVert_{L^{p}(\mathcal{X})}$
$\displaystyle=\sup_{\left\lVert\Phi\right\rVert_{L^{p^{\prime}}(\mathcal{X})}\leq
1}\left\\{\int_{\mathcal{X}}\Phi\cdot\left(\nabla U_{1}-\nabla
U_{2}\right)\>dx\right\\}$
$\displaystyle=\sup_{\left\lVert\Phi\right\rVert_{L^{p^{\prime}}(\mathcal{X})}\leq
1}\left\\{\int_{\mathcal{X}}\nabla\phi\cdot\left(\nabla U_{1}-\nabla
U_{2}\right)\>dx-\lim_{k\to\infty}\int_{B_{r_{k}}(\mathcal{X})}\operatorname{div}g_{k}\left(U_{1}-U_{2}\right)\>dx\right\\}$
$\displaystyle=\sup_{\phi;\>\left\lVert\Phi\right\rVert_{L^{p^{\prime}}(\mathcal{X})}\leq
1}\left\\{\int_{\mathcal{X}}\nabla\phi\cdot\left(\nabla U_{1}-\nabla
U_{2}\right)\>dx\right\\}.$

Since the projection operator $P_{p^{\prime}}:\Phi\to g$ is bounded from
$L^{p^{\prime}}(\mathcal{X})$ to $H_{p^{\prime}}(\mathcal{X})$, we have that

$\left\lVert\nabla\phi\right\rVert_{L^{p^{\prime}}(\mathcal{X})}=\left\lVert
g-\Phi\right\rVert_{L^{p^{\prime}}(\mathcal{X})}\leq\left(1+C_{P_{p^{\prime}}}\right)\left\lVert\Phi\right\rVert_{L^{p^{\prime}}(\mathcal{X})},$

where $C_{P_{p^{\prime}}}$ is the constant from (1.3), and we set
$C_{\mathrm{HW}}:=1+C_{P_{p^{\prime}}}$. Consider the larger set

$\left\\{\left\lVert\nabla\phi\right\rVert_{L^{p^{\prime}}(\mathcal{X})}/C_{\mathrm{HW}}\leq
1\right\\}\supset\left\\{\phi;\>\left\lVert\nabla\phi\right\rVert_{L^{p^{\prime}}(\mathcal{X})}/C_{\mathrm{HW}}\leq\left\lVert\Phi\right\rVert_{L^{p^{\prime}}(\mathcal{X})}\leq
1\right\\}$

that does not depend on $\Phi$ anymore. By replacing the supremum over this
set, we obtain

$\displaystyle\left\lVert\nabla U_{1}-\nabla
U_{2}\right\rVert_{L^{p}(\mathcal{X})}$ $\displaystyle\leq
C_{\mathrm{HW}}\sup_{\left\lVert\nabla\phi\right\rVert_{L^{p^{\prime}}(\mathcal{X})}\leq
1}\left\\{\int_{\mathcal{X}}\nabla\phi\cdot\left(\nabla U_{1}-\nabla
U_{2}\right)\>dx\right\\}$
$\displaystyle=C_{\mathrm{HW}}\sup_{\left\lVert\nabla[\phi]\right\rVert_{L^{p^{\prime}}(\mathcal{X})}\leq
1}\left\\{\int_{\mathcal{X}}[\phi]\left(\rho_{1}-\rho_{2}\right)\>dx\right\\}.$

We conclude by density of quotient test functions
$\dot{\mathcal{D}}(\mathcal{X})$ in $\dot{W}^{1,p}(\mathcal{X})$ and by the
definition of the strong dual homogeneous Sobolev norm. ∎

Before proving our new $L^{p}$-estimate, we first state the existence of an
optimal transport map adapted to our context;

######  Theorem 2.1 (Gangbo-McCann [4, Theorem 1.2]).

Let $\rho_{1},\rho_{2}$ be two probability measures on $\mathcal{X}$ that are
absolutely continuous with respect to the Lebesgue measure. Then

$\displaystyle
W_{p}(\rho_{1},\rho_{2})=\left(\inf_{T_{\\#}\rho_{1}=\rho_{2}}\bigg{\\{}\int_{\mathcal{X}}\left\lvert
x-T(x)\right\rvert^{p}\>d\rho_{1}(x)\bigg{\\}}\right)^{1/p},$

where the infimum runs over all measurable mappings
$T:\mathcal{X}\to\mathcal{X}$ that push forward $\rho_{1}$ onto $\rho_{2}$.
Moreover, the infimum is reached by a $\rho_{1}(dx)$-almost surely unique
mapping $T$, and there is a $\left\lvert\cdot\right\rvert^{p}$-convex function
$\psi$ such that
$T=\operatorname{Id}_{\mathcal{X}}-\left(\nabla\left\lvert\cdot\right\rvert^{p}\right)^{-1}\circ\nabla\psi$,
where we denote $\left(\nabla h^{*}\right)$ by $\left(\nabla h\right)^{-1}$
for a function $h$ with $h^{*}$ its Legendre transform.

###### Proof of Proposition 1.8.

Let us denote by

$\rho_{\theta}:=\left[\left(\theta-1\right)T+\left(2-\theta\right)\operatorname{Id}_{\mathcal{X}}\right]_{\\#}\rho_{1}$

the interpolant measure between $\rho_{1}$ and $\rho_{2}$, where $T$ is the
optimal transport map of Theorem 2.1. Let $\phi\in
C_{c}^{\infty}(\mathcal{X})$ be a test function. By the properties of
pushforwards of measures, it follows immediately that

$\int_{\mathcal{X}}\phi(x)\>d\rho_{\theta}(x)=\int_{\mathcal{X}}\phi\left((\theta-1)T(x)+(2-\theta)x\right)\>d\rho_{1}(x).$

Lebesgue’s dominated convergence theorem yields

$\frac{d}{d\theta}\int_{\mathcal{X}}\phi(x)\>d\rho_{\theta}(x)=\int_{\mathcal{X}}\nabla\phi\left((\theta-1)T(x)+(2-\theta)x\right)\cdot\left(T(x)-x\right)\>d\rho_{1}(x).$

Now, by using Hölder inequality with respect to the measure $\rho_{1}$, we get

$\displaystyle\frac{d}{d\theta}\int_{\mathcal{X}}\phi(x)\rho_{\theta}(x)\>dx$
$\displaystyle\leq\left(\int_{\mathcal{X}}\left\lvert\nabla\phi\left((\theta-1)T(x)+(2-\theta)x\right)\right\rvert^{p^{\prime}}\>d\rho_{1}(x)\right)^{1/p^{\prime}}\left(\int_{\mathcal{X}}\left\lvert
x-T(x)\right\rvert^{p}\>d\rho_{1}(x)\right)^{1/p}$
$\displaystyle=\left(\int_{\mathcal{X}}\left\lvert\nabla\phi(x)\right\rvert^{p^{\prime}}\>d\rho_{\theta}(x)\right)^{1/p^{\prime}}\left(\int_{\mathcal{X}}\left\lvert
x-T(x)\right\rvert^{p}\>d\rho_{1}(x)\right)^{1/p}.$

The second term in the product is exactly $W_{p}(\rho_{1},\rho_{2})$ by
Theorem 2.1. For the first one, thanks to [17, Remark 9], the
$L^{\infty}$-norm of the interpolant is controlled by the one of the two
measures;

$\left\lVert\rho_{\theta}\right\rVert_{L^{\infty}(\mathcal{X})}\leq\max\left\\{\left\lVert\rho_{1}\right\rVert_{L^{\infty}(\mathcal{X})},\left\lVert\rho_{2}\right\rVert_{L^{\infty}(\mathcal{X})}\right\\}.$

Therefore,

$\frac{d}{d\theta}\int_{\mathcal{X}}\phi(x)\rho_{\theta}(x)\>dx\leq\max\left\\{\left\lVert\rho_{1}\right\rVert_{L^{\infty}(\mathcal{X})},\left\lVert\rho_{2}\right\rVert_{L^{\infty}(\mathcal{X})}\right\\}^{1/p^{\prime}}\left(\int_{\mathcal{X}}\left\lvert\nabla\phi(x)\right\rvert^{p^{\prime}}\>dx\right)^{1/p^{\prime}}W_{p}(\rho_{1},\rho_{2}).$

Combining the above estimate with the fact that $\int\rho_{2}-\rho_{1}=0$ and
Fubini’s theorem yields

$\displaystyle\int_{\mathcal{X}}[\phi](x)\left(\rho_{2}(x)-\rho_{1}(x)\right)\>dx$
$\displaystyle=\int_{\mathcal{X}}\phi(x)\left(\rho_{2}(x)-\rho_{1}(x)\right)\>dx$
$\displaystyle=\int_{1}^{2}\left(\frac{d}{d\theta}\int_{\mathcal{X}}\phi(x)\rho_{\theta}(x)\>dx\right)d\theta$
$\displaystyle\leq\max\left\\{\left\lVert\rho_{1}\right\rVert_{L^{\infty}(\mathcal{X})},\left\lVert\rho_{2}\right\rVert_{L^{\infty}(\mathcal{X})}\right\\}^{1/p^{\prime}}\left(\int_{\mathcal{X}}\left\lvert\nabla[\phi](x)\right\rvert^{p^{\prime}}\>dx\right)^{1/p^{\prime}}W_{p}(\rho_{1},\rho_{2}).$

By restricting to quotient test functions $[\phi]$ such that
$\left\lVert\nabla[\phi]\right\rVert_{L^{p^{\prime}}(\mathcal{X})}\leq 1$, we
get the strong dual homogeneous norm so that

$\left\lVert\rho_{1}-\rho_{2}\right\rVert_{\dot{W}^{-1,p}(\mathcal{X})}\leq\max\left\\{\left\lVert\rho_{1}\right\rVert_{L^{\infty}(\mathcal{X})},\left\lVert\rho_{2}\right\rVert_{L^{\infty}(\mathcal{X})}\right\\}^{1/p^{\prime}}W_{p}(\rho_{1},\rho_{2}),$

and we conclude by Lemma 1.7. ∎

######  Remark 2.2.

Loeper uses extensively that the optimal transport map $T$ is convex to rely
on the gas internal energy theory developed by McCann (see [14, Section 2]) to
estimate the $L^{\infty}$-norm of the interpolant. Here, we only have
$|\cdot|^{p}$-convexity instead, while still the $L^{\infty}$-estimate on the
interpolant is valid as showed, for instance, by Santambrogio [17, Remark 9].
Loeper gives also an alternative proof [13, Proposition 3.1] using the
Benamou-Brenier formula [1, Proposition 1.1]. The interpolant measure
satisfies the continuity equation

$\partial_{\theta}\rho_{\theta}+\operatorname{div}_{x}(\rho_{\theta}v_{\theta})=0$

for a vector field $v_{\theta}$ related to the Wasserstein distance as through

$\int_{\mathcal{X}}\left\lvert
v_{\theta}(x)\right\rvert^{2}\>d\rho_{\theta}(x)=W_{2}^{2}(\rho_{1},\rho_{2}).$

Differentiating both sides of Poisson’s equation gives

$\Delta\partial_{\theta}U_{\theta}=-\partial_{\theta}\rho_{\theta}=\operatorname{div}_{x}(\rho_{\theta}v_{\theta}),$

and integrating by parts against $\partial_{\theta}U_{\theta}$ itself as test
function yields

$\int_{\mathcal{X}}\left\lvert\partial_{\theta}\nabla
U_{\theta}\right\rvert^{2}\>dx=\int_{\mathcal{X}}\rho_{\theta}v_{\theta}\cdot\partial_{\theta}\nabla
U_{\theta}\>dx$

so that

$\left\lVert\partial_{\theta}\nabla
U_{\theta}\right\rVert_{L^{2}(\mathcal{X})}\leq\left\lVert\rho_{\theta}\right\rVert_{L^{\infty}(\mathcal{X})}^{1/2}W_{2}(\rho_{1},\rho_{2})\leq\max\left\\{\left\lVert\rho_{1}\right\rVert_{L^{\infty}(\mathcal{X})},\left\lVert\rho_{2}\right\rVert_{L^{\infty}(\mathcal{X})}\right\\}^{1/2}W_{2}(\rho_{1},\rho_{2}),$

and the conclusion follows after integrating over $\theta\in[1,2]$.

Even though there is a $W_{p}$ version of Benamou-Brenier formula [18, Theorem
5.28], there is no analog test function that allows to mimic this proof for
$L^{p}$.

##  3\. Stability estimates revisited for Wasserstein-like distances

###  3.1. Loeper’s estimate revisited

The proof of Loeper’s stability estimate in $W_{p}$ on the torus
$\mathbb{T}^{d}$ is similar to [8, Theorem 3.1] using the new $L^{p}$-estimate
(1.5) from Proposition 1.8. It relies on the modified quantity (see [6,
Section 4])

$\displaystyle Q_{p}(t)$
$\displaystyle:=\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left\lvert
X_{1}(t,x,v)-X_{2}(t,y,w)\right\rvert^{p}+\left\lvert
V_{1}(t,x,v)-V_{2}(t,y,w)\right\rvert^{p}\>d\pi_{0}(x,v,y,w)$
$\displaystyle=\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left\lvert
x-v\right\rvert^{p}+\left\lvert y-w\right\rvert^{p}\>d\pi_{t}(x,v,y,w),$

where $\pi_{t}\in\Pi(f_{1}(t),f_{2}(t))$ and $\pi_{0}$ is an optimal coupling
that satisfies the marginal property;

$\int_{\left(\mathbb{T}^{d}\times\mathbb{R}^{d}\right)^{2}}\phi(x,v,y,w)\>d\pi_{t}(x,v,y,w)\\\
=\int_{\left(\mathbb{T}^{d}\times\mathbb{R}^{d}\right)^{2}}\phi\left(Z_{1}(t,x,v),Z_{2}(t,y,w)\right)\>d\pi_{0}(x,v,y,w),\quad\forall\phi\in
C_{b}((\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}).$ (3.1)

The last ingredients are the following estimates analog to (1.6) relying on
the definition of Wasserstein distance (see [12, Lemma 3.6]):

$W_{p}^{p}(f_{1}(t),f_{2}(t))\leq Q_{p}(t),\quad
W_{p}^{p}(\rho_{f_{1}}(t),\rho_{f_{2}}(t))\leq Q_{p}(t).$ (3.2)

###  3.2. Kinetic Wasserstein distance revisited

We proof the recent Iacobelli’s stability estimate in $W_{p}$ on the torus
$\mathbb{T}^{d}$ adapting the proof of [10, Theorem 3.1].

The proof relies on the modified quantity from the kinetic Wasserstein
distance (see [10, Section 4])

$\displaystyle D_{p}(t)$
$\displaystyle:=\frac{1}{p}\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\lambda(t)\left\lvert
X_{1}(t,x,v)-X_{2}(t,y,w)\right\rvert^{p}+\left\lvert
V_{1}(t,x,v)-V_{2}(t,y,w)\right\rvert^{p}\>d\pi_{0}(x,v,y,w)$
$\displaystyle=\frac{1}{p}\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\lambda(t)\left\lvert
x-y\right\rvert^{p}+\left\lvert y-w\right\rvert^{p}\>d\pi_{t}(x,v,y,w),$

where $\lambda(t)=\left\lvert\log D_{p}(t)\right\rvert^{p/2}$. We first state
the Log-Lipschitz estimate on the force fields [8, Lemma 3.2], see also [6,
Lemma 3.3]:

######  Lemma 3.1.

Let $U_{i}$ satisfy $-\Delta U_{i}=\rho_{f_{i}}-1,\>i=1,2$, on
$\mathbb{T}^{d}$ in the distributional sense. Then there is a constant $C>0$
such that for all $x,y\in\mathbb{T}^{d},\>i=1,2$, it holds

$\left\lvert\nabla U_{i}(x)-\nabla U_{i}(y)\right\rvert\leq C\left\lvert
x-y\right\rvert\log\left(\frac{4\sqrt{d}}{\left\lvert
x-y\right\rvert}\right)\left\lVert\rho_{f_{i}}-1\right\rVert_{L^{\infty}(\mathbb{T}^{d})}.$
(3.3)

###### Proof of Theorem 1.11.

$\displaystyle\dot{D}_{p}(t)=$
$\displaystyle\>\frac{1}{p}\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\dot{\lambda}(t)\left\lvert
X_{1}-X_{2}\right\rvert^{p}\>d\pi_{0}$
$\displaystyle+\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\lambda(t)\left\lvert
X_{1}-X_{2}\right\rvert^{p-2}(X_{1}-X_{2})\cdot(V_{1}-V_{2})\>d\pi_{0}$
$\displaystyle+\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left\lvert
V_{1}-V_{2}\right\rvert^{p-2}(V_{1}-V_{2})\cdot(\nabla_{x}U_{2}(t,X_{2})-\nabla_{x}U_{1}(t,X_{1}))\>d\pi_{0}.$

The last two terms are estimated using Hölder inequality with respect to the
measure $\pi_{0}$, and we have

$\dot{D}_{p}(t)\leq\frac{1}{p}\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\dot{\lambda}(t)\left\lvert
X_{1}-X_{2}\right\rvert^{p}\>d\pi_{0}\\\
+\lambda^{1/p}(t)\left(pD_{p}(t)\right)+\left(pD_{p}(t)\right)^{1/p^{\prime}}\left(\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left\lvert\nabla_{x}U_{2}(t,X_{2})-\nabla_{x}U_{1}(t,X_{1})\right\rvert^{p}\>d\pi_{0}\right)^{1/p}.$
(3.4)

Recall the separation of the difference force fields;

$\left\lvert\nabla_{x}U_{2}(t,X_{2})-\nabla_{x}U_{1}(t,X_{1})\right\rvert\leq\left\lvert\nabla_{x}U_{2}(t,X_{2})-\nabla_{x}U_{2}(t,X_{1})\right\rvert+\left\lvert\nabla_{x}U_{1}(t,X_{1})-\nabla_{x}U_{2}(t,X_{2})\right\rvert,$

whence

$\left(\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left\lvert\nabla_{x}U_{2}(t,X_{2})-\nabla_{x}U_{1}(t,X_{1})\right\rvert^{p}\>d\pi_{0}\right)^{1/p}\leq
T_{1}(t)+T_{2}(t),$ (3.5)

where

$T_{1}(t):=\left(\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left\lvert\nabla_{x}U_{2}(t,X_{2})-\nabla_{x}U_{2}(t,X_{1})\right\rvert^{p}\>d\pi_{0}\right)^{1/p},\\\
T_{2}(t):=\left(\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left\lvert\nabla_{x}U_{1}(t,X_{1})-\nabla_{x}U_{2}(t,X_{1})\right\rvert^{p}\>d\pi_{0}\right)^{1/p}.$
(3.6)

We estimate $T_{1}$ (3.6) using the non-decreasing concave function on
$[0,+\infty)$ given by

$\Phi_{p}(s):=\begin{cases}s\log^{p}\left(\frac{(4\sqrt{d})^{p}}{s}\right)&\text{if
}s\leq(4\sqrt{d}/e)^{p},\\\ \left(\frac{4p\sqrt{d}}{e}\right)^{p}&\text{if
}s>(4\sqrt{d}/e)^{p},\end{cases}$

together with the Log-Lipschitz estimate (3.3) from Lemma 3.1 and (1.10) to
get

$T_{1}(t)\leq\frac{C}{p}A(t)\left(\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\Phi_{p}\left(\left\lvert
X_{1}-X_{2}\right\rvert^{p}\right)\>d\pi_{0}\right)$

provided that $\left\lvert X_{1}-X_{2}\right\rvert^{p}\leq(4\sqrt{d}/e)^{p}$,
but this is always the case since the distance between points in the torus
cannot exceed $\sqrt{d}$. Thus by Jensen’s inequality we have

$\displaystyle T_{1}(t)$
$\displaystyle\leq\frac{CA(t)}{p}\Bigg{[}\Phi_{p}\left(\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left\lvert
X_{1}-X_{2}\right\rvert^{p}\>d\pi_{0}\right)\Bigg{]}^{1/p}\leq\frac{CA(t)}{p}\Bigg{[}\Phi_{p}\left(\frac{pD_{p}(t)}{\lambda(t)}\right)\Bigg{]}^{1/p}.$

Now, the considered regime becomes

$\frac{pD_{p}(t)}{\lambda(t)}\leq\left(4\sqrt{d}/e\right)^{p},$ (3.7)

so that

$T_{1}(t)\leq\frac{CA(t)}{p}\left(\frac{pD_{p}(t)}{\lambda(t)}\right)^{1/p}\Bigg{[}\left\lvert\log\left(\frac{pD_{p}(t)}{\lambda(t)}\right)\right\rvert+p\log\left(4\sqrt{d}\right)\Bigg{]}.$
(3.8)

We replace $\lambda(t)=\left\lvert\log D_{p}(t)\right\rvert^{p/2}$, and
consider yet another regime, now dictated by

$D_{p}(t)\leq\frac{1}{e},$ (3.9)

so that $\left\lvert\log D_{p}(t)\right\rvert\geq 1$. Note that this regime is
compatible with the regime (3.7) needed for the function $\Phi_{p}$ in the
sense that if $pD_{p}(t)\leq(4\sqrt{d}/e)^{p}$ holds, then
$pD_{p}(t)/\left\lvert\log D_{p}(t)\right\rvert^{p/2}\leq
pD_{p}(t)\leq(4\sqrt{d}/e)^{p}$, and since $p/e\leq(4\sqrt{d}/e)^{p}$, we can
only consider the regime (3.9). We estimate the logarithms in (3.8) in this
new regime (3.9) using an elementary inequality valid within this regime;

$\left\lvert\log\left(\frac{pD_{p}(t)}{\left\lvert\log
D_{p}(t)\right\rvert^{p/2}}\right)\right\rvert\leq 2\left(1+\log
p+\frac{p}{2}\right)\left\lvert\log D_{p}(t)\right\rvert,$

and set $C_{p}=2(1+\log p+p/2)$. Hence (3.8) becomes

$T_{1}(t)\leq\frac{CA(t)}{p}\left(\frac{pD_{p}(t)}{\lambda(t)}\right)^{1/p}\Big{[}C_{p}\left\lvert\log
D_{p}(t)\right\rvert+p\log\left(4\sqrt{d}\right)\Big{]}.$ (3.10)

We move to the estimation of $T_{2}$ (3.6). The $L^{p}$-estimate (1.5) from
Proposition 1.8 yields

$T_{2}(t)\leq C_{\mathrm{HW}}A(t)W_{p}(\rho_{f_{1}}(t),\rho_{f_{2}}(t))$
(3.11)

recalling (1.10). Since $(X_{1}(t),X_{2}(t))_{\\#}\pi_{0}$ has marginals
$\rho_{f_{1}}(t)$ and $\rho_{f_{2}}(t)$, we can estimate the Wasserstein
distance between the densities by $D_{p}$ (see [12, Lemma 3.6]). More
precisely, by the definition of the Wasserstein distance, we have

$\displaystyle W_{p}^{p}(\rho_{f_{1}}(t),\rho_{f_{2}}(t))$
$\displaystyle=\inf_{\gamma\in\Pi(\rho_{f_{1}}(t),\rho_{f_{2}}(t))}\int_{\mathbb{T}^{d}\times\mathbb{T}^{d}}\left\lvert
x-y\right\rvert^{p}\>d\gamma(x,y)$
$\displaystyle\leq\int_{\mathbb{T}^{d}\times\mathbb{T}^{d}}\left\lvert
x-y\right\rvert^{p}\>d\Big{[}\left(X_{1}(t),X_{2}(t)\right)_{\\#}\pi_{0}\Big{]}(x,y)$
$\displaystyle=\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left\lvert
X_{1}(t,x,v)-X_{2}(t,y,w)\right\rvert^{p}\>d\pi_{0}(x,v,y,w)\leq\frac{pD_{p}(t)}{\lambda(t)}.$

We replace $\lambda(t)=\left\lvert\log D_{p}(t)\right\rvert^{p/2}$ and the
above estimate in (3.11) to get

$T_{2}(t)\leq C_{\mathrm{HW}}A(t)\left(\frac{pD_{p}(t)}{\left\lvert\log
D_{p}(t)\right\rvert^{p/2}}\right)^{1/p}.$ (3.12)

Putting altogether estimates (3.4, 3.5, 3.10, 3.12) gives in the considered
regime (3.9) that

$\dot{D}_{p}(t)\leq\frac{1}{p}\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\frac{-\dot{D}_{p}(t)}{D_{p}(t)}\left\lvert\log
D_{p}(t)\right\rvert^{p/2-1}\left\lvert
X_{1}-X_{2}\right\rvert^{p}\>d\pi_{0}\\\
+\left(pD_{p}(t)\right)\Bigg{[}\sqrt{\left\lvert\log
D_{p}(t)\right\rvert}+\frac{CA(t)}{p}\left(C_{p}\sqrt{\left\lvert\log
D_{p}(t)\right\rvert}(t)+\frac{p\log\left(4\sqrt{d}\right)}{\sqrt{\left\lvert\log
D_{p}(t)\right\rvert}}\right)+\frac{C_{\mathrm{HW}}A(t)}{\sqrt{\left\lvert\log
D_{p}(t)\right\rvert}}\Bigg{]}.$

If $\dot{D}_{p}\leq 0$, then we de not do anything. Otherwise, then the first
term in the above estimate is negative, and therefore

$\dot{D}_{p}(t)\leq\left(pD_{p}(t)\right)\Bigg{[}\sqrt{\left\lvert\log
D_{p}(t)\right\rvert}+\frac{CA(t)}{p}\left(C_{p}\sqrt{\left\lvert\log
D_{p}(t)\right\rvert}(t)+\frac{p\log\left(4\sqrt{d}\right)}{\sqrt{\left\lvert\log
D_{p}(t)\right\rvert}}\right)+\frac{C_{\mathrm{HW}}A(t)}{\sqrt{\left\lvert\log
D_{p}(t)\right\rvert}}\Bigg{]}.$

Since the right-hand side is non-negative, independently of the sign of
$\dot{D}_{p}$, this bound is always valid in the regime, and using that
$\left\lvert\log D_{p}(t)\right\rvert\geq 1$, together with $A(t)\geq 1$, we
get

$\dot{D}_{p}(t)\leq\tilde{C}_{\mathrm{KW}}A(t)D_{p}(t)\sqrt{\left\lvert\log
D_{p}(t)\right\rvert},$

where
$\tilde{C}_{\mathrm{KW}}:=1+C/p\times\left(C_{p}+p\log\left(4\sqrt{d}\right)\right)+C_{\mathrm{HW}}$.
Therefore,

$D_{p}(t)\leq\exp\left\\{-\left(\sqrt{\left\lvert\log
D_{p}(0)\right\rvert}-C_{\mathrm{KW}}\int_{0}^{t}A(s)\>ds\right)^{2}\right\\},$
(3.13)

where $C_{\mathrm{KW}}:=\tilde{C}_{\mathrm{KW}}/2$ depends only on $p$ and
$d$. This implies in particular that (3.9) holds if

$\sqrt{\left\lvert\log D_{p}(0)\right\rvert}\geq
C_{\mathrm{KW}}\int_{0}^{T}A(s)\>ds+1.$ (3.14)

It remains to compare $D_{p}$ to the Wasserstein distance between the
solutions in the regime (3.9). By the definition of the Wasserstein distance,
since $(Z_{1}(t),Z_{2}(t))_{\\#}\pi_{0}$ has marginals $f_{1}(t)$ and
$f_{2}(t)$, we have that

$\displaystyle W_{p}^{p}(f_{1}(t),f_{2}(t))$
$\displaystyle=\inf_{\gamma\in\Pi(f_{1}(t),f_{2}(t))}\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left\lvert
x-y\right\rvert^{p}\>d\gamma(x,v,y,w)$
$\displaystyle\leq\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left\lvert
x-y\right\rvert^{p}+\left\lvert
v-w\right\rvert^{p}\>d\Big{[}\left(Z_{1}(t),Z_{2}(t)\right)_{\\#}\pi_{0}\Big{]}(x,v,y,w)$
$\displaystyle=\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left\lvert
X_{1}-X_{2}\right\rvert^{p}+\left\lvert
V_{1}-V_{2}\right\rvert^{p}\>d\pi_{0}\leq pD_{p}(t).$

For the initial Wasserstein distance, since $\pi_{0}$ is optimal, we get

$D_{p}(0)\leq\frac{1}{p}\left\lvert\log
D_{p}(t)\right\rvert\int_{(\mathbb{T}^{d}\times\mathbb{R}^{d})^{2}}\left\lvert
x-y\right\rvert^{p}+\left\lvert
v-w\right\rvert^{p}\>d\pi_{0}=\frac{1}{p}\left\lvert\log D_{p}(t)\right\rvert
W_{p}^{p}(f_{1}(0),f_{2}(0)),$

which we rewrite as

$\frac{D_{p}(0)}{\left\lvert\log
D_{p}(0)\right\rvert}\leq\frac{1}{p}W_{p}^{p}(f_{1}(0),f_{2}(0)).$

Note that, near the origin, the inverse of the function $s\mapsto
s/\left\lvert\log s\right\rvert$ behaves like
$\tau\mapsto\tau\left\lvert\log\tau\right\rvert$. In particular, there is a
universal constant $c_{0}>0$ such that

$\frac{s}{\left\lvert\log s\right\rvert}\leq\tau\quad\text{for some
$0\leq\tau\leq c_{0}$}\quad\implies\quad s\leq
p\tau\left\lvert\log\tau\right\rvert.$

Hence for sufficiently small initial distance such that
$W_{p}^{p}(f_{1}(0),f_{2}(0))\leq pc_{0}$, then

$D_{p}(0)\leq
W_{p}^{p}(f_{1}(0),f_{2}(0))\left\lvert\log\left(\frac{1}{p}W_{p}^{p}(f_{1}(0),f_{2}(0))\right)\right\rvert.$

Combining these bounds with (3.13), and recalling (3.14), this implies

$W_{p}^{p}(f_{1}(t),f_{2}(t))\\\ \leq
p\exp\Bigg{\\{}-\left(\sqrt{\left\lvert\log\Bigg{\\{}W_{p}^{p}(f_{1}(0),f_{2}(0))\left\lvert\log\left(\frac{1}{p}W_{p}^{p}(f_{1}(0),f_{2}(0))\right)\right\rvert\Bigg{\\}}\right\rvert}-C_{\mathrm{KW}}\int_{0}^{t}A(s)\>ds\right)^{2}\Bigg{\\}}$

provided $W_{p}^{p}(f_{1}(0),f_{2}(0))\leq pc_{0}$ and

$\sqrt{\left\lvert\log\Bigg{\\{}W_{p}^{p}(f_{1}(0),f_{2}(0))\left\lvert\log\left(\frac{1}{p}W_{p}^{p}(f_{1}(0),f_{2}(0))\right)\right\rvert\Bigg{\\}}\right\rvert}\geq
C_{\mathrm{KW}}\int_{0}^{T}A(s)\>ds+1.$

We conclude the proof by [10, Lemma 3.7 & Remark 3.8]. ∎

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