# Exponential Stability of the Inhomogeneous Navier-Stokes-Vlasov System in
Vacuum

Hai-Liang Li  School of Mathematical Sciences, Capital Normal University,
Beijing 100048, P. R. China  Academy for Multidisciplinary Studies, Capital
Normal University, Beijing 100048, P. R. China  Ling-Yun Shou  Department of
Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing
211106, P. R. China  Yue Zhang Corresponding author School of Mathematical
Sciences, Capital Normal University, Beijing 100048, P. R. China  Academy for
Multidisciplinary Studies, Capital Normal University, Beijing 100048, P. R.
China

###### Abstract

The main purpose of the present paper is to study the influence of the vacuum
on the asymptotic behaviors of solutions to the inhomogeneous Navier-Stokes-
Vlasov system in $\mathbb{R}^{3}\times\mathbb{R}^{3}$. To this end, we
establish the uniform bound of the macroscopic density associated with the
distribution function and prove the global existence and uniqueness of strong
solutions to the Cauchy problem with vacuum for either small initial energy or
large viscosity coefficient. The uniform boundedness and the presence of
vacuum enable us to show that as the time evolves, the fluid velocity decays,
while the distribution function concentrates towards a Dirac measure in
velocity centred at $0$, with an exponential rate.

††footnotetext: E-mails<EMAIL_ADDRESS>(H.-L. Li),
<EMAIL_ADDRESS>(L.-Y. Shou<EMAIL_ADDRESS>(Y.
Zhang).††footnotetext: MSC2020: 35Q30, 35Q83, 35A01, 35B40.††footnotetext:
Keywords: Inhomogeneous Navier-Stokes-Vlasov system, Cauchy problem, Strong
solution, Global existence, Exponential stability.††footnotetext: Funding:
H.-L. Li and Y. Zhang are partially supported by the National Natural Science
Foundation of China grants 11931010, 12226326, 12226327, and the key research
project of Academy for Multidisciplinary Studies, Capital Normal University.
L.-Y. Shou is supported by the National Natural Science Foundation of China
grant 12301275.

##  1 Introduction

The transport of small particles dispersed in dense fluids can be described by
the fluid-particle models which have a wide range of applications including
chemical engineering, pollution settling processes, medical treatments and
Diesel engines, etc [1, 4, 7, 26, 36, 40].

We investigate the inhomogeneous Navier-Stokes-Vlasov system for the fluid-
particle motion:

$\left\\{\begin{aligned} &\rho_{t}+{\rm{div}}_{x}(\rho u)=0,\\\ &(\rho
u)_{t}+{\rm{div}}_{x}(\rho u\otimes
u)+\nabla_{x}P=\mu\Delta_{x}u-\int_{\mathbb{R}^{3}}\kappa(u-v)fdv,\\\
&{\rm{div}}_{x}u=0,\\\
&f_{t}+v\cdot\nabla_{x}f+{\rm{div}}_{v}(\kappa(u-v)f)=0,\end{aligned}\right.$
(1.1)

where $\rho=\rho(t,x)\geq 0$, $u=(u^{1},u^{2},u^{3})(t,x)$, and $P=P(t,x)$
denote the density, the velocity, and the pressure of the fluid, respectively,
at the time $t\in\mathbb{R}_{+}$ and the position
$x=(x_{1},x_{2},x_{2})\in\mathbb{R}^{3}$, and $f=f(t,x,v)$ is the distribution
function of particles moving with the velocity
$v=(v_{1},v_{2},v_{3})\in\mathbb{R}^{3}$. In addition, the constants $\mu>0$
and $\kappa>0$ stand for the viscosity coefficient and the drag force
coefficient, respectively. In this paper, we consider the Cauchy problem of
(1.1) subject to the initial data

$\rho(0,x)=\rho_{0}(x),~{}~{}u(0,x)=u_{0}(x),~{}~{}f(0,x,v)=f_{0}(x,v),\quad(x,v)\in\mathbb{R}^{3}\times\mathbb{R}^{3}.$
(1.2)

The Navier-Stokes-Vlasov systems have been intensively investigated with much
significant mathematical progress (refer to [6, 25, 10, 13, 41, 23, 20, 39, 9,
37] and the references therein). Concerning the homogenous incompressible
Navier-Stokes-Vlasov model, the global existence of weak solutions was proved
by Hamdache [21] without the convection term in bounded domains with specular
reflection boundary conditions and by Boudin et al [6] with the convection
term in spatial periodic domains. The asymptotic stability of stationary
states in a pipe with partially absorbing boundary conditions was investigated
by Glass et al [19]. Under the condition that the initial energy and the
$\dot{H}^{\frac{1}{2}}$-norm of the initial data are sufficiently small, Han-
Kwan et al [23] established the uniform-in-time boundedness of the macroscopic
density for the particles and proved the uniqueness and exponential-in-time
stability of global weak solutions to the initial value problem in spatial
periodic domains. The methods in [23] were then applied to the study of the
large-time behaviors of weak solutions to the initial boundary value problem
with exponential decay rates [18] and to the Cauchy problem with algebraic
decay rates [22].

For the inhomogeneous Navier-Stokes-Vlasov system (1.1), Wang and Yu [39]
showed the global existence of weak solutions to the initial boundary value
problem, Su et al [37] justified the hydrodynamic limit of system (1.1) to a
macroscopical two-fluid system, and Choi [16] studied the large-time
solvability of the initial value problem with small initial velocity and
distribution function in either the spatial periodic domain or whole space.
The exponential time-decay rate in the spatial periodic space was also
obtained in [16], provided that the solutions satisfy some uniform-in-time
estimates. Furthermore, there is much important progress has been made
recently on the well-posedness and asymptotic behaviors of solutions for other
related fluid-particle models, cf. [28, 29, 11, 10, 13, 12, 15, 34, 33, 17,
20, 27, 8, 30].

In the present paper, we aim to prove a new large-time asymptotic result for
the Cauchy problem of the three-dimensional inhomogeneous Navier-Stokes-Vlasov
system (1.1). Based on the structure of the fluid-particle interactions and
the appearance of the vacuum, we establish some uniform a-priori estimates
with respect to time, which enable us to not only obtain the global-in-time
existence of weak and strong solutions to the Cauchy problem (1.1)–(1.2) with
vacuum, but also justify the exponential decay-in-time properties.

Before stating the main results, we present some notations below.

Notations. For $p\in[1,\infty]$ and integer $k\geq 0$, the simplified Lebesgue
and Sobolev spaces are given by

$\left\\{\begin{aligned} &L^{p}:=L^{p}(\mathbb{R}^{3}),\quad
W^{k,p}:=W^{k,p}(\mathbb{R}^{3}),\quad H^{k}:=W^{k,2},\\\ &D^{1}=\\{u\in
L^{6}~{}|~{}\nabla u\in L^{2}\\},\quad C_{0,\sigma}^{\infty}:=\\{u\in
C_{0}^{\infty}(\mathbb{R}^{3})~{}|~{}\hbox{\rm div}\,u=0\\},\quad
D_{0,\sigma}^{1}:=\overline{C_{0,\infty}^{\infty}}^{\|\cdot\|_{D^{1}}},\\\
&L_{x,v}^{p}:=L^{p}(\mathbb{R}^{3}\times\mathbb{R}^{3}),\quad
W_{x,v}^{k,p}:=W^{k,p}(\mathbb{R}^{3}\times\mathbb{R}^{3}).\end{aligned}\right.$

We define the kinetic energy of the system (1.1) as

$E(t):=\frac{1}{2}\int_{\mathbb{R}^{3}}\rho|u|^{2}dx+\frac{1}{2}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|v|^{2}fdvdx,$
(1.3)

and the corresponding initial energy by

$E_{0}=\frac{1}{2}\int_{\mathbb{R}^{3}}\rho_{0}|u_{0}|^{2}dx+\frac{1}{2}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|v|^{2}f_{0}dvdx.$
(1.4)

For any strong solution $(\rho,u,f)$ to (1.1), one has the basic energy
inequality

${\rm{\frac{d}{dt}}}E(t)+\mu\int_{\mathbb{R}^{3}}|\nabla
u|^{2}dx+\kappa\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx\leq 0.$
(1.5)

In addition, we define the constant

$\displaystyle M:=\int_{\mathbb{R}^{3}}|\nabla
u_{0}|^{2}dx+\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u_{0}-v|^{2}f_{0}dvdx.$
(1.6)

Let $n_{f}$ and $j_{f}$ be the macroscopical density and momentum,
respectively, related to the moments of the distribution function $f$ in (1.1)
as follows

$\displaystyle n_{f}=n_{f}(t,x):=\int_{\mathbb{R}^{3}}f(t,x,v)dv,\quad
j_{f}=j_{f}(t,x):=\int_{\mathbb{R}^{3}}vf(t,x,v)dv.$ (1.7)

Throughout this paper, $C>0$ ($C_{T}>0$) denotes some positive constants
independent (dependent) of $E_{0}$, $\mu$ and time.

First, we have the global existence, uniqueness, and large-time behaviors of
strong solutions to the Cauchy problem (1.1)–(1.2) with vacuum.

######  Theorem 1.1.

Let $\mu\geq 1$, and $R_{0}>0$ be a given constant. Assume that the initial
data $(\rho_{0},u_{0},f_{0})$ satisfies

$\left\\{\begin{aligned} &0\leq\rho_{0}\in L^{\frac{3}{2}}\cap L^{\infty}\cap
H^{1},\\\ &u_{0}\in D^{1}_{0,\sigma},\quad\sqrt{\rho_{0}}u_{0}\in L^{2},\\\
&0\leq f_{0}\in L^{1}_{x,v}\cap L^{\infty}_{x,v}\cap
W^{1,\frac{3}{2}}_{x,v},\\\
&{\rm{Supp}}_{v}f_{0}(x,v)\subset\left\\{v\in\mathbb{R}^{3}~{}|~{}|v|\leq
R_{0}\right\\}.\end{aligned}\right.$ (1.8)

Then, there exists a constant $\varepsilon_{0}>0$ depending on $\kappa$, $M$,
$\|(\rho_{0},n_{f_{0}})\|_{L^{\frac{3}{2}}\cap L^{\infty}}$ and
$\|j_{f_{0}}\|_{L^{\infty}}$ such that if

$E_{0}\leq\varepsilon_{0}\mu,$ (1.9)

the Cauchy problem (1.1)–(1.2) admits a unique global solution $(\rho,u,f)$
satisfying for any $T>0$ that

$\left\\{\begin{aligned} &\rho\in C([0,T];L^{\frac{3}{2}}\cap H^{1})\cap
L^{\infty}(0,T;L^{\infty}),\quad\sqrt{\rho}u_{t}\in L^{2}(0,T;L^{2}),\\\ &u\in
L^{\infty}(0,T;D^{1}_{0,\sigma})\cap L^{1}(0,T;W^{1,\infty}),\quad P\in
L^{2}(0,T;L^{2}),\\\ &\sqrt{t}(\nabla u,P)\in C([0,T];L^{2})\cap
L^{\infty}(0,T;H^{1})\cap L^{2}(0,T;W^{1,6}),\quad\sqrt{t}\sqrt{\rho}u_{t}\in
L^{\infty}(0,T;L^{2}),\\\ &f\in C([0,T];L_{x,v}^{1}\cap
W^{1,\frac{3}{2}}_{x,v})\cap L^{\infty}(0,T;L^{\infty}_{x,v}),\\\
\end{aligned}\right.$ (1.10)

and for any $(t,x,v)\in(0,T]\times\mathbb{R}^{3}\times\mathbb{R}^{3}$ that

$\left\\{\begin{aligned}
&0\leq\rho(t,x)\leq\|\rho_{0}\|_{L^{\infty}},\quad\|\nabla\rho(t)\|_{L^{2}}\leq
2\|\nabla\rho_{0}\|_{L^{2}},\\\ &0\leq f(t,x,v)\leq e^{3\kappa
T}\|f_{0}\|_{L^{\infty}_{x,v}},\\\ &0\leq n_{f}(t,x)\leq
2\|n_{f_{0}}\|_{L^{\infty}}.\end{aligned}\right.$ (1.11)

Moreover, for any $t\in[1,T]$, there exists a asymptotic profile $n^{*}\in
L^{\infty}$ of $n_{f}$ such that it holds the following large-time behaviors

$\left\\{\begin{aligned} &E(t)+\|\nabla u(t)\|_{H^{1}}+\|P(t)\|_{H^{1}}\leq
C_{1}e^{-\lambda_{1}t},\\\
&W_{1}(f(t),n^{*}\otimes\delta_{v})+W_{1}(n_{f}(t),n^{*})+\|j_{f}(t)\|_{L^{1}\cap
L^{\infty}}\leq C_{1}e^{-\lambda_{1}t},\\\
&{\rm{Supp}}_{v}f(t,x,v)\subset\left\\{v\in\mathbb{R}^{3}~{}|~{}|v|\leq(R_{0}+C_{1})e^{-\lambda_{1}t}\right\\},\end{aligned}\right.$
(1.12)

where $C_{1}>0$ and $\lambda_{1}>0$ are two constants independent of $T$,
$W_{1}$ represents the $1$-Wasserstein distance (refer to Definition A.1), and
$\delta_{v}$ denotes the Dirac measure in velocity supported at $\\{v=0\\}$.

######  Remark 1.1.

Theorem 1.1 shows that the fluid velocity $u$ and the distribution function
$f$ asymptotically converge to their equilibrium state $0$ and
$n^{*}\otimes\delta_{v}$, respectively, at an exponential rate in time, which
is quite different from the algebraic decay rates of the homogeneous
incompressible Navier-Stokes-Vlasov equations in whole space [22]. This result
is also a new comparison of the spatial periodic case [16].

######  Remark 1.2.

The condition (1.9) is satisfied, provided that either the initial energy
$E_{0}$ is suitable small or the viscosity coefficient $\mu$ is large enough.

######  Remark 1.3.

For the global existence and uniqueness in Theorem 1.1, the compact support
condition $\eqref{incd}_{4}$ can be relaxed to $(1+|v|^{q})f_{0}\in
L^{\infty}_{x,v}$ for some positive integer $q$ which is sufficiently large.

######  Remark 1.4.

Under the condition (1.9), we are able to establish higher-order estimates of
classical solutions for regular initial data.

We further investigate the global existence and large-time behaviors of weak
solutions to the Cauchy problem (1.1)–(1.2) with only bounded densities.

######  Definition 1.1.

For any time $T>0$, $(\rho,u,f)$ is said to be a global weak solution to the
Cauchy problem (1.1)–(1.2) if the following properties hold

  * •

(1) $(\rho,u,f)$ satisfies

$\left\\{\begin{aligned} &0\leq\rho\in C([0,T];L^{\frac{3}{2}})\cap
L^{\infty}(0,T;L^{\infty}),\\\ &\rho u\in C([0,T];H^{-1})\cap
L^{\infty}(0,T;L^{2}),\quad u\in L^{2}(0,T;D_{0,\sigma}^{1}),\\\ &0\leq f\in
C([0,T];L^{1}_{x,v})\cap L^{\infty}(0,T;L^{\infty}_{x,v}),\quad|v|^{2}f\in
L^{\infty}(0,T;L^{1}_{x,v});\\\ \end{aligned}\right.$ (1.13)

  * •

(2) $(\rho,u,f)$ solves the Cauchy problem (1.1)–(1.2) in the sense of
distributions;

  * •

(3) for a.e. $t\in(0,T]$, the energy inequality (1.5) holds.

######  Theorem 1.2.

For any given $q>4$, assume that the initial data $(\rho_{0},u_{0},f_{0})$
satisfies

$\left\\{\begin{aligned} &0\leq\rho_{0}\in L^{\frac{3}{2}}\cap L^{\infty},\\\
&\sqrt{\rho_{0}}u_{0}\in L^{2},\quad u_{0}\in D^{1}_{0,\sigma},\\\ &0\leq
f_{0}\in L^{1}_{x,v}\cap L^{\infty}_{x,v},\quad(1+|v|^{q})f_{0}\in
L^{1}_{x,v}\cap L^{\infty}_{x,v}.\end{aligned}\right.$ (1.14)

Then, there exists a constant $\varepsilon_{0}^{*}>0$ depending on $\kappa$,
$M$, $\|(\rho_{0},n_{f_{0}})\|_{L^{\frac{3}{2}}\cap L^{\infty}}$ and
$\|j_{f_{0}}\|_{L^{\infty}}$ such that if

$E_{0}\leq\varepsilon_{0}^{*}\mu,$ (1.15)

the Cauchy problem (1.1)–(1.2) admits a global weak solution $(\rho,u,f)$ in
the sense of Definition 1.1.

Furthermore, it holds for any $t\in[1,T]$ that

$\left\\{\begin{aligned} &E(t)+\|\nabla u(t)\|_{H^{1}}+\|P(t)\|_{H^{1}}\leq
C_{2}e^{-\lambda_{2}t},\\\
&W_{1}(f(t),n^{*}\otimes\delta_{v})+W_{1}(n_{f}(t),n^{*})+\|j_{f}(t)\|_{L^{1}\cap
L^{\infty}}\leq C_{2}e^{-\lambda_{2}t},\end{aligned}\right.$ (1.16)

where $C_{2}>0$ and $\lambda_{2}>0$ are two constants independent of $T$,
$W_{1}$ represents the $1$-Wasserstein distance (refer to Definition A.1),
$n^{*}$ is a asymptotic profile of $n_{f}$, and $\delta_{v}$ denotes the Dirac
measure in velocity supported at $\\{v=0\\}$.

We explain the main strategies and ideas to prove the above Theorems 1.1–1.2
about the global existence and large-time behaviors of solutions to the Cauchy
problem (1.1)–(1.2). To begin with, we prove the coercive estimate

$\displaystyle\mu\int_{\mathbb{R}^{3}}|\nabla
u|^{2}dx+\kappa\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx\geq\lambda
E(t)\quad\text{with}~{}~{}\lambda\sim\frac{1}{1+\displaystyle\sup_{t>0}\|(\rho,n_{f})(t)\|_{L^{\frac{3}{2}}}}.$
(1.17)

By (1.17) and the energy inequality (1.5), we obtain the exponential stability
of the basic energy $E(t)$ (refer to Proposition 2.1). Since the transport
nature in $\eqref{VNS}_{1}$ with vacuum leads to the uniform evolution of the
$L^{\frac{3}{2}}$-norm for the fluid density $\rho$, our key ingredient is to
show the uniform bound

$\displaystyle\sup_{t>0}\|n_{f}(t)\|_{L^{\frac{3}{2}}}<\infty.$ (1.18)

To justify (1.18), we rewrite the Vlasov equation $\eqref{VNS}_{4}$ as

$f(t,x,v)=e^{3\kappa t}f_{0}(X(0;t,x,v),V(0;t,x,v)),$ (1.19)

where the characteristic curves $X(\tau;t,x,v)$ and $V(\tau;t,x,v)$ are solved
by

$\left\\{\begin{aligned}
&{\rm{\frac{d}{d\tau}}}X(\tau;t,x,v)=V(\tau;t,x,v),\\\
&{\rm{\frac{d}{d\tau}}}V(\tau;t,x,v)=\kappa\left(u(\tau,X(\tau;t,x,v))-V(\tau;t,x,v)\right),\\\
&X(t;t,x,v)=x,\quad V(t;t,x,v)=v.\end{aligned}\right.$ (1.20)

Note that the formula (1.19) implies the time growth of the distribution
function $f$, so one may not obtain (1.18) directly. As in [23, 18, 22], the
Jacobian matrix of the map $\Gamma^{-1}_{t,x}:V(0;t,x,v)\rightarrow v$ has the
upper bound $2e^{-3\kappa t}$ (refer to Lemma 3.2) such that

$n_{f}(t,x)=\int_{\mathbb{R}^{3}}e^{3\kappa
t}f_{0}(X(0;t,x,v),V(0;t,x,v))dv\leq 2n_{f_{0}}(x),\quad
t>0,~{}~{}x\in\mathbb{R}^{3},$ (1.21)

as long as the following uniform Lipschitz regularity of $u$ holds

$\displaystyle\int_{0}^{t}\|\nabla u(\tau)\|_{L^{\infty}}d\tau\ll 1.$ (1.22)

However, it is difficult to enclose the a-priori estimate (1.22) since the
maximal regularity estimates used in [23, 18, 22] can not be applied to the
momentum equation $\eqref{VNS}_{2}$ containing the degenerate term $\rho
u_{t}$ near the vacuum. To overcome this difficulty, we observe the
interactions between the fluid and particles via the drag force and employ the
time-weighted estimates for the higher-order derivatives of $u$, inspired by
the work [24] for the pure inhomogeneous Navier-Stokes equations. To be more
specific, we introduce the new energy

$\displaystyle E_{1}(t):=\frac{1}{2}\int_{\mathbb{R}^{3}}|\nabla
u|^{2}dx+\frac{\kappa}{2\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx,$
(1.23)

which comes essentially from the dissipation term in (1.5) and leads to the
uniform bounds on the gradient of $u$. In addition, we estimate another new
energy

$\displaystyle
E_{2}(t):=\frac{1}{2\mu}\int_{\mathbb{R}^{3}}\rho|u_{t}|^{2}dx.$ (1.24)

In view of (1.5), (1.23), (1.24) and the Stokes structure in
$\eqref{VNS}_{2}$, we are able to derive the higher-order estimates of
$(u,P)$. With the aid of careful initial layer analysis on (1.23) and (1.24),
we obtain the short-time integration part of (1.22). Meanwhile, the large-time
integration part relies on the improvement of the exponential time-decay rate
for the energy $E(t)$ (refer to Lemma 3.3) to the higher-order energy
$E_{1}(t)$ and $E_{2}(t)$ (refer to Lemmas 3.4 and 3.5). Based on these
uniform estimates and the bootstrap argument, we enclose (1.22) and thence
deduce the higher-order regularities of $(\rho,f)$ and the quantitive decay
estimates (1.12).

The rest of this paper is organized as follows. In Section 2, we devote to the
exponential time-decay rate of the energy $E(t)$ under the additional bound
(1.18) of $n_{f}$. In Section 3, we justify the uniform bound (1.18), enclose
the a-priori Lipschitz regularity (1.22) of $u$ and further obtain the higher-
order estimates and large-time behaviors of the solution. The main results,
Theorems 1.1–1.2, are proved in Section 4. Some definitions and lemmas are
presented in Appendix.

##  2 Conditional large-time behavior

In this section, we prove the exponential stability of solutions to the Cauchy
problem (1.1)–(1.2) under the additional condition that the
$L^{\frac{3}{2}}$-norm of the macroscopical density $n_{f}$ is uniformly
bounded in time. Compared with the use of the Poincaré inequality in the
spatial periodic case [16], our proof relies on the integrability of densities
with vacuum.

######  Proposition 2.1.

For given time $T>0$, let $(\rho,u,f)$ be a strong solution to the Cauchy
problem (1.1)–(1.2) for $t\in(0,T]$. Suppose that there exist some constant
$C_{*}>0$ independent of $T$ such that

$\displaystyle\sup_{t\in(0,T]}\|n_{f}(t)\|_{L^{\frac{3}{2}}}\leq C_{*}.$ (2.1)

Then it holds for $t\in(0,T]$ that

$\displaystyle E(t)\leq e^{-2\alpha t}E_{0},$ (2.2)

with the constant

$\alpha:=\frac{\min\\{\mu,\kappa\\}}{2+C_{S}(\|\rho_{0}\|_{L^{\frac{3}{2}}}+2C_{*})}>0,$
(2.3)

where $C_{S}>0$ denotes the Sobolev constant such that
$\|\cdot\|_{L^{6}}^{2}\leq C_{S}\|\nabla\cdot\|_{L^{2}}^{2}$ holds.

###### Proof.

In view of (1.1), one can obtain the energy inequality (1.5) for any
$t\in(0,T]$. We claim that

$\displaystyle\mu\int_{\mathbb{R}^{3}}|\nabla
u|^{2}dx+\kappa\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx\geq
2\alpha E(t),$ (2.4)

with $\alpha$ given in (2.3). It then follows from (1.5), (2.4) and the
Grönwall inequality to derive the exponential stability (2.2).

Indeed, by the incompressible condition $\eqref{VNS}_{3}$ and the transport
nature of $\eqref{VNS}_{1}$, a direct computation (cf. [31]) deduces for
$t\in(0,T]$ that

$\|\rho(t)\|_{L^{\frac{3}{2}}}=\|\rho_{0}\|_{L^{\frac{3}{2}}},$ (2.5)

which together with the Sobolev inequality yields

$\displaystyle\int_{\mathbb{R}^{3}}\rho|u|^{2}dx\leq\|\rho(t)\|_{L^{\frac{3}{2}}}\|u(t)\|_{L^{6}}^{2}\leq
C_{S}\|\rho_{0}\|_{L^{\frac{3}{2}}}\|\nabla u(t)\|_{L^{2}}^{2}.$ (2.6)

Similarly, one has

$\displaystyle\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u|^{2}fdvdx\leq
C_{S}\|n_{f}(t)\|_{L^{\frac{3}{2}}}\|\nabla u(t)\|_{L^{2}}^{2}.$ (2.7)

Therefore, by (2.6)–(2.5), we obtain

$\displaystyle\int_{\mathbb{R}^{3}}\rho|u|^{2}dx+\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|v|^{2}fdvdx$
(2.8)
$\displaystyle\leq\int_{\mathbb{R}^{3}}\rho|u|^{2}dx+2\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u|^{2}fdvdx+2\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx$
$\displaystyle\leq\alpha^{-1}\left(\mu\int_{\mathbb{R}^{3}}|\nabla
u|^{2}dx+\kappa\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx\right),$

which gives rise to (2.4), and thus the proof is completed. ∎

##  3 The uniform a-priori estimates

To begin with, we state the local existence and uniqueness of strong solutions
to the Cauchy problem (1.1)–(1.2) with vacuum.

######  Proposition 3.1.

Suppose that the initial data $(\rho_{0},u_{0},f_{0})$ satisfies (1.8), then
there exists a small time $T_{0}>0$ such that the Cauchy problem (1.1)–(1.2)
admits a unique strong solution $(\rho,u,f)$ on
$(0,T_{0}]\times\mathbb{R}^{3}\times\mathbb{R}^{3}$ fulfilling (1.10).

The construction of the local-in-time solution is based on the limit of
approximated solutions away from vacuum as in [2, 32, 14], and here we omit
the details for brevity. The proof of the uniqueness is postponed to Section
4.

###  3.1 The bootstrap

In order to extend the local solution globally in time, we establish the
uniform a-priori bounds below.

######  Proposition 3.2.

There exists a constant $\varepsilon_{0}\in(0,1)$ depending only on $\kappa$,
$M$, $\|(\rho_{0},n_{f_{0}})\|_{L^{\frac{3}{2}}\cap L^{\infty}}$ and
$\|j_{f_{0}}\|_{L^{\infty}}$ such that if $E_{0}\leq\varepsilon_{0}\mu$, and
$(\rho,u,f)$ is a smooth solution to (1.1) satisfying

$\int_{0}^{T}(\|\nabla u(t)\|_{L^{2}}^{4}+\|u(t)\|_{L^{\infty}})dt\leq
1,\quad\int_{0}^{T}\|\nabla u(t)\|_{L^{\infty}}dt\leq\delta_{0},$ (3.1)

with $\delta_{0}$ given by $\kappa
e^{\kappa\delta_{0}}\delta_{0}=\frac{1}{10}$, then it holds

$\int_{0}^{T}(\|\nabla
u(t)\|_{L^{2}}^{4}+\|u(t)\|_{L^{\infty}})dt\leq\frac{1}{2},\quad\int_{0}^{T}\|\nabla
u(t)\|_{L^{\infty}}dt\leq\frac{\delta_{0}}{2}.$ (3.2)

Before giving the proof of Proposition 3.2, let us show some necessary uniform
bounds on $(\rho,u,f)$. By standard maximal principles for the mass
conservation equation $\eqref{VNS}_{1}$ and the Vlasov equation
$\eqref{VNS}_{4}$, one has the following lemma.

######  Lemma 3.1.

Let $T>0$ and $(\rho,u,f)$ be the solution to (1.1) for
$(t,x,v)\in(0,T]\times\mathbb{R}^{3}\times\mathbb{R}^{3}$ with the initial
data $(\rho_{0},u_{0},f_{0})$ satisfying (1.8). Then under the condition
(3.1), it holds

$\displaystyle 0\leq\rho(t,x)\leq\|\rho_{0}\|_{L^{\infty}},\quad 0\leq
f(t,x,v)\leq e^{3\kappa T}\|f_{0}\|_{L^{\infty}_{x,v}}.$ (3.3)

Note that one can not estimate the moments $n_{f}$ and $j_{f}$ by (3.3)
directly due to the time growth bound of $f$. In order to justify the key
condition (2.1), inspired by [23, 18, 22], we prove the following estimates
based on the representation formula solution to the Vlasov equation
$\eqref{VNS}_{4}$ via characteristic curves.

######  Lemma 3.2.

Let $T>0$ and $(\rho,u,f)$ be the solution to (1.1) for $t\in(0,T]$ with the
initial data $(\rho_{0},u_{0},f_{0})$ satisfying (1.8). Then under the
condition (3.1), it holds for any $1\leq p\leq\infty$ that

$\displaystyle\sup_{t\in(0,T]}\|n_{f}(t)\|_{L^{p}}\leq
2^{1-\frac{1}{p}}\|n_{f_{0}}\|_{L^{p}}.$ (3.4)

In addition, we have

$\left\\{\begin{aligned} &\sup_{t\in(0,T]}\|j_{f}(t)\|_{L^{2}}\leq
2\|n_{f_{0}}\|_{L^{\infty}}^{\frac{1}{2}}E_{0}^{\frac{1}{2}},\\\
&\sup_{t\in(0,T]}\|j_{f}(t)\|_{L^{6}}\leq
2\|n_{f_{0}}\|_{L^{\infty}}^{\frac{1}{6}}(\|j_{f_{0}}\|_{L^{\infty}}+\kappa\|n_{f_{0}}\|_{L^{\infty}})^{\frac{2}{3}}E_{0}^{\frac{1}{6}},\\\
&\sup_{t\in(0,T]}\|j_{f}(t)\|_{L^{\infty}}\leq
2(\|j_{f_{0}}\|_{L^{\infty}}+\kappa\|n_{f_{0}}\|_{L^{\infty}}).\end{aligned}\right.$
(3.5)

###### Proof.

Let the characteristic curves $X(\tau;t,x,v)$ and $V(\tau;t,x,v)$ be given by
(1.20). Integrating $\eqref{VNS}_{4}$ over
$[0,t]\times\mathbb{R}^{3}\times\mathbb{R}^{3}$ for $t\in(0,T]$, we get

$\displaystyle\|n_{f}(t)\|_{L^{1}}=\|n_{f_{0}}\|_{L^{1}}.$ (3.6)

According to the definition (1.7) of $n_{f}$ and the formula (1.19) of $f$, it
holds

$\displaystyle n_{f}(t,x)$ $\displaystyle=e^{3\kappa
t}\int_{\mathbb{R}^{3}}f_{0}(X(0;t,x,v),V(0;t,x,v))dv$ (3.7)
$\displaystyle=e^{3\kappa
t}\int_{\mathbb{R}^{3}}f_{0}(X(0;t,x,\Gamma_{t,x}^{-1}(w)),w)\left|\hbox{\rm
det}\,D_{v}\Gamma_{t,x}(\Gamma_{t,x}^{-1}(w))\right|^{-1}dw,$

with the mapping $\Gamma_{t,x}:v\mapsto w=V(0;t,x,v)$. To control (3.7), one
needs to justify that $\Gamma_{t,x}$ is $C^{1}$-diffeomorphism. Based on
(1.20), one deduces

$e^{\kappa
t}v-V(0;t,x,v)=\kappa\int_{0}^{t}e^{\kappa\tau}u(\tau,X(\tau;t,x,v))d\tau.$
(3.8)

Thence taking the derivative of (3.8) with respect to $v$, we have

$e^{\kappa
t}{\rm{Id}}-D_{v}\Gamma_{t,x}(v)=\kappa\int_{0}^{t}e^{\kappa\tau}\nabla
u(\tau,X(\tau;t,x,v)D_{v}X(\tau;t,x,v)d\tau,$ (3.9)

which gives rise to

$\displaystyle\|e^{-\kappa
t}D_{v}\Gamma_{t,x}(v)-{\rm{Id}}\|_{L^{\infty}_{x,v}}$
$\displaystyle\leq\kappa\int_{0}^{t}e^{-\kappa(t-\tau)}\|\nabla
u(\tau)\|_{L^{\infty}}\|D_{v}X(\tau;t,x,v)\|_{L^{\infty}_{x,v}}d\tau$ (3.10)
$\displaystyle\leq\kappa\int_{0}^{t}e^{-\kappa(t-\tau)}\|\nabla
u(\tau)\|_{L^{\infty}}e^{\kappa(t-\tau)}\exp{\left(\kappa\int_{\tau}^{t}\|\nabla
u(s)\|_{L^{\infty}}ds\right)}d\tau$
$\displaystyle\leq\kappa\int_{0}^{t}\|\nabla
u(\tau)\|_{L^{\infty}}d\tau\exp{\left(\kappa\int_{0}^{t}\|\nabla
u(\tau)\|_{L^{\infty}}d\tau\right)}.$

It then follows from (3.1) by choosing $\delta_{0}>0$ such that $\kappa
e^{\kappa\delta_{0}}\delta_{0}=\frac{1}{10}$, and Lemma A.3 with
$\phi=e^{-\kappa t}\Gamma_{t,x}$ to obtain

$|\hbox{\rm det}\,D_{v}\Gamma_{t,x}(v)|=e^{3\kappa t}|\hbox{\rm
det}\,\nabla\phi|\geq\frac{1}{2}e^{3\kappa t},$ (3.11)

which combined with (3.7) yields

$\displaystyle n_{f}(t,x)\leq$ $\displaystyle e^{3\kappa
t}\int_{\mathbb{R}^{3}}f_{0}(X(0;t,x,\Gamma_{t,x}^{-1}(w)),w)\cdot
2e^{-3\kappa t}dw$ (3.12) $\displaystyle\leq$ $\displaystyle
2\int_{\mathbb{R}^{3}}f_{0}(X(0;t,x,\Gamma_{t,x}^{-1}(w)),w)dw\leq
2\|n_{f_{0}}\|_{L^{\infty}}.$

Hence, the Riesz-Thorin interpolation theorem (refer to [3]) between (3.6) and
(3.12) concludes (3.4).

As for $j_{f}$, with the help of (1.5) and (3.12), it is easy to deduce

$\|j_{f}(t)\|_{L^{2}}\leq\|n_{f}(t)\|_{L^{\infty}}^{\frac{1}{2}}\left(\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|v|^{2}fdvdx\right)^{\frac{1}{2}}\leq
2\|n_{f_{0}}\|_{L^{\infty}}^{\frac{1}{2}}E_{0}^{\frac{1}{2}}.$ (3.13)

In addition, similarly to (3.7), $j_{f}$ fulfills the following description

$\displaystyle j_{f}(t,x)$ $\displaystyle=e^{3\kappa
t}\int_{\mathbb{R}^{3}}\Gamma_{t,x}^{-1}(w)f_{0}(X(0;t,x,\Gamma_{t,x}^{-1}(w)),w)\left|\hbox{\rm
det}\,D_{v}\Gamma_{t,x}(\Gamma_{t,x}^{-1}(w))\right|^{-1}dw$ (3.14)
$\displaystyle\leq e^{3\kappa
t}\int_{\mathbb{R}^{3}}\left|\Gamma_{t,x}^{-1}(w)\right|f_{0}(X(0;t,x,\Gamma_{t,x}^{-1}(w)),w)2e^{-3\kappa
t}dw$ $\displaystyle\leq 2e^{-\kappa
t}\int_{\mathbb{R}^{3}}V(0;t,x,\Gamma_{t,x}^{-1}(w))f_{0}(X(0;t,x,\Gamma_{t,x}^{-1}(w)),w)dw$
$\displaystyle\quad+2\kappa\int_{0}^{t}\int_{\mathbb{R}^{3}}e^{-\kappa(t-\tau)}\|u(\tau)\|_{L^{\infty}}f_{0}(X(0;t,x,\Gamma_{t,x}^{-1}(w)),w)dwd\tau,$

and this as well as (3.1) leads to

$\|j_{f}(t)\|_{L^{\infty}}\leq
2\|j_{f_{0}}\|_{L^{\infty}}+2\kappa\|n_{f_{0}}\|_{L^{\infty}}.$ (3.15)

By (3.13), (3.15), and the Hölder inequality, we conclude (3.5) and complete
the proof. ∎

Next, combining Lemma 3.2 with Proposition 2.1, we have the following
exponential stability.

######  Lemma 3.3.

Let $T>0$ and $(\rho,u,f)$ be the solution to (1.1) for $t\in(0,T]$ with the
initial data $(\rho_{0},u_{0},f_{0})$ satisfying (1.8). Then under the
condition (3.1), it holds

$\displaystyle\sup_{t\in(0,T]}e^{\alpha_{1}t}E(t)+\frac{1}{2}\int_{0}^{T}e^{\alpha_{1}t}\left(\mu\int_{\mathbb{R}^{3}}|\nabla
u|^{2}dx+\kappa\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx\right)dt\leq
E_{0},$ (3.16)

with the constant

$\displaystyle\alpha_{1}:=\frac{\min\\{1,\kappa\\}}{2+3C_{S}\|(\rho_{0},n_{f_{0}})\|_{L^{\frac{3}{2}}}}>0,$
(3.17)

where $E(t)$ and $E_{0}$ are given by (1.3) and (1.4), respectively, and
$C_{S}$ is the Sobolev constant.

###### Proof.

Due to (1.5), (2.4), (3.4) and the fact $\mu\geq 1$, it is easy to verify

${\rm{\frac{d}{dt}}}E(t)+\alpha_{1}E(t)+\frac{1}{2}D(t)\leq 0,$ (3.18)

with $\alpha_{1}$ given in (3.17). Multiplying two sides of (3.18) by
$e^{\alpha_{1}t}$ and then integrating the resulting inequality over $[0,t]$
for all $t\in(0,T]$, we conclude (3.16) and complete the proof. ∎

Then, we provide the following key Lemma concerning the higher-order uniform
estimates of $u$.

######  Lemma 3.4.

Let $T>0$ and $(\rho,u,f)$ be the solution to (1.1) for $t\in(0,T]$ with the
initial data $(\rho_{0},u_{0},f_{0})$ satisfying (1.8). Then under the
condition (3.1), it holds

$\displaystyle\sup_{t\in(0,T]}\left(\int_{\mathbb{R}^{3}}|\nabla
u|^{2}dx+\frac{1}{\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx\right)$
(3.19)
$\displaystyle\qquad+\int_{0}^{T}\int_{\mathbb{R}^{3}}\left(\frac{1}{\mu}\rho|u_{t}|^{2}+\mu|\nabla^{2}u|^{2}+\frac{1}{\mu}|\nabla
P|^{2}\right)dxdt+\frac{1}{\mu}\int_{0}^{T}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdxdt\leq
C_{3}M.$

Moreover, for any $T>1$, we have the short-time time-weighted estimates

$\displaystyle\sup_{t\in(0,1]}t^{\beta_{1}}\left(\int_{\mathbb{R}^{3}}|\nabla
u|^{2}dx+\frac{1}{\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx\right)$
(3.20)
$\displaystyle\qquad+\int_{0}^{1}t^{\beta_{1}}\int_{\mathbb{R}^{3}}\left(\frac{1}{\mu}\rho|u_{t}|^{2}+\mu|\nabla^{2}u|^{2}+\frac{1}{\mu}|\nabla
P|^{2}\right)dxdt+\frac{1}{\mu}\int_{0}^{1}t^{\beta_{1}}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdxdt$
$\displaystyle\leq\frac{C_{3}}{\mu^{\beta_{1}}}E_{0}^{\beta_{1}}M^{1-\beta_{1}},\quad\beta_{1}\in[0,1],$

and the large-time exponential stability

$\displaystyle\sup_{t\in[1,T]}e^{\alpha_{1}t}\left(\int_{\mathbb{R}^{3}}|\nabla
u|^{2}dx+\frac{1}{\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx\right)$
(3.21)
$\displaystyle\quad+\int_{1}^{T}e^{\alpha_{1}t}\int_{\mathbb{R}^{3}}\left(\frac{1}{\mu}\rho|u_{t}|^{2}+\mu|\nabla^{2}u|^{2}+\frac{1}{\mu}|\nabla
P|^{2}\right)dxdt\leq\frac{C_{3}}{\mu}E_{0}.$

Here $M$ and $\alpha_{1}$ are given by (1.6) and (3.17), respectively, and
$C_{3}>0$ is a constant depending only on $\kappa$ and
$\|(\rho_{0},n_{f_{0}})\|_{L^{\frac{3}{2}}\cap L^{\infty}}$.

###### Proof.

We first show the uniform estimate (3.19). Multiplying $\eqref{VNS}_{2}$ by
$u_{t}$ and then integrating by parts, one has

$\displaystyle\frac{\mu}{2}{\rm{\frac{d}{dt}}}\int_{\mathbb{R}^{3}}|\nabla
u|^{2}dx+\int_{\mathbb{R}^{3}}\rho|u_{t}|^{2}dx=-\int_{\mathbb{R}^{3}}\rho
u\cdot\nabla u\cdot
u_{t}dx-\kappa\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}u_{t}\cdot(u-v)fdvdx.$
(3.22)

It follows from $\eqref{VNS}_{3}$ that

$\displaystyle-\kappa\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}u_{t}\cdot(u-v)fdvdx$
(3.23)
$\displaystyle=-\frac{\kappa}{2}{\rm{\frac{d}{dt}}}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}(|u|^{2}-2u\cdot
v)fdvdx+\frac{\kappa}{2}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}(|u|^{2}-2u\cdot
v)f_{t}dvdx$
$\displaystyle=-\frac{\kappa}{2}{\rm{\frac{d}{dt}}}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}(|u|^{2}-2u\cdot
v)fdvdx+\kappa\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}(v\cdot\nabla_{x}u)\cdot(u-v)fdvdx$
$\displaystyle\quad-\kappa^{2}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}u\cdot(u-v)fdvdx$
$\displaystyle=-\frac{\kappa}{2}{\rm{\frac{d}{dt}}}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx+\kappa\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}(v\cdot\nabla_{x}u)\cdot(u-v)fdvdx-\kappa^{2}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx,$

where we noticed that the energy of the Vlasov equation $\eqref{VNS}_{3}$
satisfies

$\displaystyle\frac{1}{2}{\rm{\frac{d}{dt}}}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|v|^{2}fdvdx=\kappa\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}v\cdot(u-v)fdvdx.$
(3.24)

The combination of (3.22) and (3.23) yields

$\displaystyle{\rm{\frac{d}{dt}}}E_{1}(t)+\frac{1}{\mu}\int_{\mathbb{R}^{3}}\rho|u_{t}|^{2}dx+\frac{\kappa^{2}}{\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx$
(3.25) $\displaystyle=-\frac{1}{\mu}\int_{\mathbb{R}^{3}}\rho u\cdot\nabla
u\cdot
u_{t}dx+\frac{\kappa}{\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}(v\cdot\nabla_{x}u)\cdot(u-v)fdvdx,$

where the new energy $E_{1}(t)$ is defined by (1.23). To estimate the terms on
the right-hand side of (3.25), we make use of (3.3) and the Gagliardo-
Nirenberg inequality in Lemma A.1 to obtain

$\displaystyle\left|\frac{1}{\mu}\int_{\mathbb{R}^{3}}\rho u\cdot\nabla u\cdot
u_{t}dx\right|$
$\displaystyle\leq\frac{1}{\mu}\|\rho_{0}\|_{L^{\infty}}^{\frac{1}{2}}\|\sqrt{\rho}u_{t}(t)\|_{L^{2}}\|u(t)\|_{L^{6}}\|\nabla
u(t)\|_{L^{3}}$ (3.26)
$\displaystyle\leq\frac{C}{\mu}\|\sqrt{\rho}u_{t}(t)\|_{L^{2}}\|\nabla
u(t)\|_{L^{2}}^{\frac{3}{2}}\|\nabla^{2}u(t)\|_{L^{2}}^{\frac{1}{2}}$
$\displaystyle\leq\frac{\eta}{\mu}\|\sqrt{\rho}u_{t}(t)\|_{L^{2}}^{2}+\eta\mu\|\nabla^{2}u(t)\|_{L^{2}}^{2}+\frac{C}{\eta^{3}\mu^{3}}\|\nabla
u(t)\|_{L^{2}}^{6},$

and by (3.4), it holds

$\displaystyle\left|\frac{\kappa}{\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}(v\cdot\nabla_{x}u)\cdot(u-v)fdvdx\right|$
(3.27)
$\displaystyle\leq\frac{\kappa}{\mu}\left|\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}((v-u)\cdot\nabla_{x}u)\cdot(u-v)fdvdx\right|+\frac{\kappa}{\mu}\left|\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}(u\cdot\nabla_{x}u)\cdot(u-v)fdvdx\right|$
$\displaystyle\leq\frac{\kappa}{\mu}\|\nabla
u(t)\|_{L^{\infty}}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx+\frac{C\kappa}{\mu}\left(\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx\right)^{\frac{1}{2}}\|\nabla
u(t)\|_{L^{2}}^{\frac{3}{2}}\|\nabla^{2}u(t)\|_{L^{2}}^{\frac{1}{2}}$
$\displaystyle\leq\left(\frac{\kappa}{\mu}\|\nabla
u(t)\|_{L^{\infty}}+\frac{\eta\kappa^{2}}{\mu}\right)\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx+\eta\mu\|\nabla^{2}u(t)\|_{L^{2}}^{2}+\frac{C}{\eta^{3}\mu^{3}}\|\nabla
u(t)\|_{L^{2}}^{6}.$

where $\eta>0$ is a small constant to be determined later. To handle the
$L^{2}$-norm of $\nabla^{2}u$ involved in (3.26) and (3.27), we apply Lemma
A.2 to $\eqref{VNS}_{2}$ as well as (3.3), (3.12) and the Gagliardo-Nirenberg
inequality to deduce

$\displaystyle\mu\|\nabla^{2}u(t)\|_{L^{2}}^{2}+\frac{1}{\mu}\|\nabla
P(t)\|_{L^{2}}^{2}$ $\displaystyle\leq\frac{C}{\mu}\|\rho u_{t}+\rho
u\cdot\nabla u+\kappa\int_{\mathbb{R}^{3}}(u-v)fdv\|_{L^{2}}^{2}$
$\displaystyle\leq\frac{C}{\mu}\|\rho_{0}\|_{L^{\infty}}\|\sqrt{\rho}u_{t}(t)\|_{L^{2}}^{2}+\frac{C}{\mu}\|\rho_{0}\|_{L^{\infty}}^{2}\|u(t)\|_{L^{6}}^{2}\|\nabla
u(t)\|_{L^{3}}^{2}+\frac{2C\kappa^{2}}{\mu}\|n_{f_{0}}\|_{L^{\infty}}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx$
$\displaystyle\leq\frac{C}{\mu}\|\sqrt{\rho}u_{t}(t)\|_{L^{2}}^{2}+\frac{C}{\mu}\|\nabla
u(t)\|_{L^{2}}^{3}\|\nabla^{2}u(t)\|_{L^{2}}+\frac{C\kappa^{2}}{\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx$
$\displaystyle\leq\frac{\mu}{4}\|\nabla^{2}u(t)\|_{L^{2}}^{2}+\frac{C}{\mu}\|\sqrt{\rho}u_{t}(t)\|_{L^{2}}^{2}+\frac{C}{\mu^{3}}\|\nabla
u(t)\|_{L^{2}}^{6}+\frac{C\kappa^{2}}{\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx,$

which implies

$\displaystyle\mu\|\nabla^{2}u(t)\|_{L^{2}}^{2}+\frac{1}{\mu}\|\nabla
P(t)\|_{L^{2}}^{2}\leq\frac{C}{\mu}\|\sqrt{\rho}u_{t}(t)\|_{L^{2}}^{2}+\frac{C}{\mu^{3}}\|\nabla
u(t)\|_{L^{2}}^{6}+\frac{C\kappa^{2}}{\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx.$
(3.28)

Combining (LABEL:mmmmm3) with (3.25)–(3.27) and then taking $\eta>0$
sufficiently small, we conclude

$\displaystyle{\rm{\frac{d}{dt}}}E_{1}(t)+\frac{1}{2\mu}\int_{\mathbb{R}^{3}}\rho|u_{t}|^{2}dx+\frac{\kappa^{2}}{2\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx+\frac{1}{2C}\int_{\mathbb{R}^{3}}(\mu|\nabla^{2}u|^{2}+\frac{1}{\mu}|\nabla
P|^{2})dx$ (3.29) $\displaystyle\leq C\left(\|\nabla
u(t)\|_{L^{\infty}}+\frac{1}{\mu^{3}}\|\nabla
u(t)\|_{L^{2}}^{4}\right)E_{1}(t),$

which together with the Grönwall inequality, (3.1) and the fact $\mu\geq 1$
gives rise to (3.19).

Next, we multiply (3.29) by the time weight $t$ to have

$\displaystyle{\rm{\frac{d}{dt}}}\Big{(}tE_{1}(t)\Big{)}+\frac{t}{2\mu}\int_{\mathbb{R}^{3}}\rho|u_{t}|^{2}dx+\frac{\kappa^{2}t}{2\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx+\frac{t}{2C}\int_{\mathbb{R}^{3}}(\mu|\nabla^{2}u|^{2}+\frac{1}{\mu}|\nabla
P|^{2})dx$ (3.30) $\displaystyle\leq C\left(\|\nabla
u(t)\|_{L^{\infty}}+\frac{1}{\mu^{3}}\|\nabla
u(t)\|_{L^{2}}^{4}\right)tE_{1}(t)+\frac{1}{2}\int_{\mathbb{R}^{3}}|\nabla
u|^{2}dx+\frac{\kappa}{2\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx.$

Integrating (3.30) over $[0,t]$ for all $t\in(0,1]$ and employing (3.1),
(3.16) and the Grönwall inequality, we obtain the time-weighted estimate

$\displaystyle\sup_{t\in(0,1]}t\left(\int_{\mathbb{R}^{3}}|\nabla
u|^{2}dx+\frac{1}{\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx\right)$
(3.31)
$\displaystyle\qquad+\int_{0}^{1}t\int_{\mathbb{R}^{3}}\left(\frac{1}{\mu}\rho|u_{t}|^{2}+\mu|\nabla^{2}u|^{2}+\frac{1}{\mu}|\nabla
P|^{2}\right)dxdt+\frac{1}{\mu}\int_{0}^{1}t\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdxdt\leq\frac{C}{\mu}E_{0}.$

Hence, a interpolation argument between (3.19) and (3.31) leads to (3.20).

Finally, multiplying (3.29) by the exponential weight $e^{\alpha_{1}t}$, one
has

$\displaystyle{\rm{\frac{d}{dt}}}\Big{(}e^{\alpha_{1}t}E_{1}(t)\Big{)}+\frac{1}{2}e^{\alpha_{1}t}\int_{\mathbb{R}^{3}}\left(\frac{1}{\mu}\rho|u_{t}|^{2}+\mu|\nabla^{2}u|^{2}+\frac{1}{\mu}|\nabla
P|^{2}\right)dx+\frac{\kappa^{2}}{2\mu}e^{\alpha_{1}t}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx$
(3.32) $\displaystyle\leq C\left(\|\nabla
u(t)\|_{L^{\infty}}+\frac{1}{\mu^{3}}\|\nabla
u(t)\|_{L^{2}}^{4}\right)e^{\alpha_{1}t}E_{1}(t)+\frac{1}{2}\alpha_{1}e^{\alpha_{1}t}\int_{\mathbb{R}^{3}}|\nabla
u|^{2}dx$
$\displaystyle\quad+\frac{\kappa}{2\mu}\alpha_{1}e^{\alpha_{1}t}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx,$

which, combined with the Grönwall inequality on $[1,T]$ with respect to $t$,
(3.1), (3.16) and (3.31), yields (3.21). Thus, we complete the proof of Lemma
3.4. ∎

######  Lemma 3.5.

Let $T>1$ and $(\rho,u,f)$ be the solution to (1.1) for $t\in(0,T]$ with the
initial data $(\rho_{0},u_{0},f_{0})$ satisfying (1.8). Then under the
condition (3.1), one has the short-time estimates

$\displaystyle\sup_{t\in(0,1]}\frac{t^{1+\beta_{2}}}{\mu}\int_{\mathbb{R}^{3}}\rho|u_{t}|^{2}dx+\int_{0}^{1}t^{1+\beta_{2}}\left(\int_{\mathbb{R}^{3}}|\nabla
u_{t}|^{2}dx+\frac{1}{\mu}\int_{\mathbb{R}^{3}}n_{f}|u_{t}|^{2}dx\right)dt$
(3.33)
$\displaystyle\leq\frac{C_{4}(1+M^{2})}{\mu^{\beta_{2}}}E_{0}^{\beta_{2}}M^{1-\beta_{2}},\quad\beta_{2}\in[0,1],$

and the large-time exponential stability

$\displaystyle\sup_{t\in[1,T]}\frac{e^{\alpha_{1}t}}{\mu}\int_{\mathbb{R}^{3}}\rho|u_{t}|^{2}dx+\int_{1}^{T}e^{\alpha_{1}t}\left(\int_{\mathbb{R}^{3}}|\nabla
u_{t}|^{2}dx+\frac{1}{\mu}\int_{\mathbb{R}^{3}}n_{f}|u_{t}|^{2}dx\right)dt\leq\frac{C_{4}(1+M^{2})}{\mu}E_{0}.$
(3.34)

Here $M$ and $\alpha_{1}$ are given by (1.6) and (3.17), respectively, and
$C_{4}>0$ is a constant depending only on $\kappa$,
$\|(\rho_{0},n_{f_{0}})\|_{L^{\frac{3}{2}}\cap L^{\infty}}$ and
$\|j_{f_{0}}\|_{L^{\infty}}$.

###### Proof.

Applying $\partial_{t}$ to $\eqref{VNS}_{2}$, we have

$\rho(u_{tt}+u\cdot\nabla u_{t})+\nabla P_{t}-\mu\Delta u_{t}+\kappa
n_{f}u_{t}=-\rho_{t}u_{t}-(\rho u)_{t}\cdot\nabla
u-\kappa\int_{\mathbb{R}^{3}}(u-v)f_{t}dv.$ (3.35)

Multiplying (3.35) by $u_{t}$ and integrating by parts, we deduce

$\displaystyle\frac{1}{2\mu}{\rm{\frac{d}{dt}}}\int_{\mathbb{R}^{3}}\rho|u_{t}|^{2}dx+\int_{\mathbb{R}^{3}}|\nabla
u_{t}|^{2}dx+\frac{\kappa}{\mu}\int_{\mathbb{R}^{3}}n_{f}|u_{t}|^{2}dx$ (3.36)
$\displaystyle=-\frac{1}{\mu}\int_{\mathbb{R}^{3}}(\rho_{t}u_{t}+(\rho
u)_{t}\cdot\nabla u)\cdot
u_{t}dx-\frac{\kappa}{\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}u_{t}\cdot(u-v)f_{t}dvdx.$

It follows from $\eqref{VNS}_{1}$, (3.3), the Gagliardo-Nirenberg and Sobolev
inequalities that

$\displaystyle\left|\frac{1}{\mu}\int_{\mathbb{R}^{3}}(\rho_{t}u_{t}+(\rho
u)_{t}\cdot\nabla u)\cdot u_{t}dx\right|$ (3.37)
$\displaystyle\leq\frac{2}{\mu}\left|\int_{\mathbb{R}^{3}}\rho u\cdot\nabla
u_{t}\cdot u_{t}dx\right|+\frac{1}{\mu}\left|\int_{\mathbb{R}^{3}}\rho
u_{t}\cdot\nabla u\cdot
u_{t}dx\right|+\frac{1}{\mu}\left|\int_{\mathbb{R}^{3}}\rho
u\cdot\nabla(u\cdot\nabla u\cdot u_{t})dx\right|$
$\displaystyle\leq\frac{2}{\mu}\|\rho u_{t}\|_{L^{3}}\|u\|_{L^{6}}\|\nabla
u_{t}\|_{L^{2}}+\frac{1}{\mu}\|\rho u_{t}\|_{L^{3}}\|\nabla
u\|_{L^{2}}\|u_{t}\|_{L^{6}}$
$\displaystyle\quad+\frac{1}{\mu}\|\rho_{0}\|_{L^{\infty}}\|u\|_{L^{6}}\left(\|\nabla
u\|_{L^{2}}\|\nabla
u\|_{L^{6}}\|u_{t}\|_{L^{6}}+\|u\|_{L^{6}}\|\nabla^{2}u\|_{L^{2}}\|u_{t}\|_{L^{6}}+\|u\|_{L^{6}}\|\nabla
u\|_{L^{6}}\|\nabla u_{t}\|_{L^{2}}\right)$
$\displaystyle\leq\frac{C}{\mu}\|\sqrt{\rho}u_{t}\|_{L^{2}}^{\frac{1}{2}}\|\nabla
u_{t}\|_{L^{2}}^{\frac{3}{2}}\|\nabla u\|_{L^{2}}+\frac{C}{\mu}\|\nabla
u\|_{L^{2}}^{2}\|\nabla^{2}u\|_{L^{2}}\|\nabla u_{t}\|_{L^{2}}$
$\displaystyle\leq\frac{1}{4}\|\nabla
u_{t}(t)\|_{L^{2}}^{2}+\frac{C}{\mu^{3}}\|\nabla
u(t)\|_{L^{2}}^{4}\left(\frac{1}{\mu}\|\sqrt{\rho}u_{t}(t)\|_{L^{2}}^{2}+\mu\|\nabla^{2}u(t)\|_{L^{2}}^{2}\right).$

With the help of $\eqref{VNS}_{4}$, (3.4), and (3.5), it holds

$\displaystyle\left|\frac{\kappa}{\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}(u-v)\cdot
u_{t}f_{t}dvdx\right|$ (3.38)
$\displaystyle\leq\frac{\kappa}{\mu}\left|\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}(v\cdot\nabla
u)\cdot
u_{t}fdvdx\right|+\frac{\kappa}{\mu}\left|\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}(v\cdot\nabla
u_{t})\cdot(u-v)fdvdx\right|$
$\displaystyle\quad+\frac{\kappa^{2}}{\mu}\left|\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}u_{t}\cdot(u-v)fdvdx\right|$
$\displaystyle\leq\frac{C}{\mu}\|j_{f}\|_{L^{2}}\|\nabla
u\|_{L^{2}}^{\frac{1}{2}}\|\nabla^{2}u\|_{L^{2}}^{\frac{1}{2}}\|\nabla
u_{t}\|_{L^{2}}+\frac{C}{\mu}\|j_{f}\|_{L^{\infty}}^{\frac{1}{2}}\left(\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx\right)^{\frac{1}{2}}\|\nabla
u_{t}\|_{L^{2}}$
$\displaystyle\quad+\frac{\kappa^{2}}{\mu}\left(\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx\right)^{\frac{1}{2}}\|\sqrt{n_{f}}u\|_{L^{2}}$
$\displaystyle\leq\frac{1}{4}\|\nabla
u_{t}(t)\|_{L^{2}}^{2}+\frac{\kappa}{2\mu}\|\sqrt{n_{f}}u(t)\|_{L^{2}}^{2}+C\|\nabla
u(t)\|_{L^{2}}^{2}+C\mu\|\nabla^{2}u(t)\|_{L^{2}}^{2}+\frac{C}{\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx,$

where one has used $\mu\geq 1$ and $\|j_{f}(t)\|_{L^{2}}\leq
CE^{\frac{1}{2}}_{0}\leq C\mu^{\frac{1}{2}}$ due to
$E_{0}\leq\varepsilon_{0}\mu$ with $\varepsilon_{0}\in(0,1)$. In addition, it
is derived from (LABEL:mmmmm3) that

$\displaystyle\mu\|\nabla^{2}u(t)\|_{L^{2}}^{2}\leq\frac{C}{\mu}\|\sqrt{\rho}u_{t}(t)\|_{L^{2}}^{2}+\frac{C}{\mu^{3}}\|\nabla
u(t)\|_{L^{2}}^{6}+\frac{C}{\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx,$
(3.39)

which together with (LABEL:312), (3.38) and (LABEL:311) gives rise to

$\displaystyle{\rm{\frac{d}{dt}}}E_{2}(t)+\int_{\mathbb{R}^{3}}|\nabla
u_{t}|^{2}dx+\frac{\kappa}{\mu}\int_{\mathbb{R}^{3}}n_{f}|u_{t}|^{2}dx$ (3.40)
$\displaystyle\leq\frac{C}{\mu^{3}}\|\nabla
u(t)\|_{L^{2}}^{4}E_{2}(t)+\frac{C}{\mu^{6}}\|\nabla
u(t)\|_{L^{2}}^{10}+\frac{C}{\mu^{4}}\|\nabla
u\|_{L^{2}}^{4}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx$
$\displaystyle\quad+C\left(\mu\|\nabla^{2}u\|_{L^{2}}^{2}+\|\nabla
u(t)\|_{L^{2}}^{2}+\frac{1}{\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx\right),$

where the new energy $E_{2}(t)$ is defined by (1.24). Multiplying
(LABEL:ddtut) by $t^{1+\beta_{2}}$ with $\beta_{2}\in[0,1]$, we obtain

$\displaystyle{\rm{\frac{d}{dt}}}\Big{(}t^{1+\beta_{2}}E_{2}(t)\Big{)}+t^{1+\beta_{2}}\int_{\mathbb{R}^{3}}|\nabla
u_{t}|^{2}dx+\frac{\kappa
t^{1+\beta_{2}}}{\mu}\int_{\mathbb{R}^{3}}n_{f}|u_{t}|^{2}dx$ (3.41)
$\displaystyle\leq\frac{C}{\mu^{3}}\|\nabla
u(t)\|_{L^{2}}^{4}t^{1+\beta_{2}}E_{2}(t)+\frac{Ct^{1+\beta_{2}}}{\mu^{6}}\|\nabla
u(t)\|_{L^{2}}^{10}+\frac{Ct^{1+\beta_{2}}}{\mu^{4}}\|\nabla
u\|_{L^{2}}^{4}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx$
$\displaystyle\quad+Ct^{1+\beta_{2}}\left(\mu\|\nabla^{2}u\|_{L^{2}}^{2}+C\|\nabla
u(t)\|_{L^{2}}^{2}+\frac{1}{\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx\right)+\frac{(1+\beta_{2})t^{\beta_{2}}}{\mu}\int_{\mathbb{R}^{3}}\rho|u_{t}|^{2}dx.$

Thanks to (3.1), (3.19), and (3.20) with
$\beta_{1}=\frac{\beta_{2}}{3}\in[0,1]$, it holds

$\displaystyle\int_{0}^{1}t^{1+\beta_{2}}\|\nabla u(t)\|_{L^{2}}^{10}dt$
$\displaystyle\leq\sup_{t\in(0,1]}t^{\beta_{2}}\|\nabla
u(t)\|_{L^{2}}^{6}\int_{0}^{1}\|\nabla
u(t)\|_{L^{2}}^{4}dt\leq\frac{CM^{2}}{\mu^{\beta_{2}}}E_{0}^{\beta_{2}}M^{1-\beta_{2}}.$
(3.42)

Hence we are able to conclude (3.33) from the Grönwall inequality, (3.41),
(3.1), (3.20) and (3.42).

Furthermore, multiplying (LABEL:ddtut) by the exponential weight
$e^{\alpha_{1}t}$, we have

$\displaystyle{\rm{\frac{d}{dt}}}\Big{(}e^{\alpha_{1}t}E_{2}(t)\Big{)}+e^{\alpha_{1}t}\int_{\mathbb{R}^{3}}|\nabla
u_{t}|^{2}dx+\frac{\kappa
e^{\alpha_{1}t}}{\mu}\int_{\mathbb{R}^{3}}n_{f}|u_{t}|^{2}dx$ (3.43)
$\displaystyle\leq\frac{C}{\mu^{3}}\|\nabla
u(t)\|_{L^{2}}^{4}e^{\alpha_{1}t}E_{2}(t)+\frac{Ce^{\alpha_{1}t}}{\mu^{6}}\|\nabla
u(t)\|_{L^{2}}^{10}+\frac{Ce^{\alpha_{1}t}}{\mu^{4}}\|\nabla
u\|_{L^{2}}^{4}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx$
$\displaystyle\quad+Ce^{\alpha_{1}t}\left(\mu\|\nabla^{2}u\|_{L^{2}}^{2}+C\|\nabla
u(t)\|_{L^{2}}^{2}+\frac{1}{\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx\right)+\frac{\alpha_{1}e^{\alpha_{1}t}}{\mu}\int_{\mathbb{R}^{3}}\rho|u_{t}|^{2}dx,$

which together with the Grönwall inequality on $[1,T]$ with respect to $t$,
(3.1), (3.33) with $\beta_{2}=1$, (3.19), (3.16) and (3.21) leads to (3.34).
Thus, we complete the proof of Lemma 3.5. ∎

######  Lemma 3.6.

Let $T>1$ and $(\rho,u,f)$ be the solution to (1.1) for $t\in(0,T]$ with the
initial data $(\rho_{0},u_{0},f_{0})$ satisfying (1.8). Then under the
condition (3.1), it holds

$\displaystyle\frac{1}{\mu}\int_{0}^{T}\|P(t)\|_{L^{2}}^{2}dt\leq
C_{5}(1+M^{2}).$ (3.44)

Moreover, we have the short-time estimate

$\displaystyle\sup_{t\in(0,1]}t^{1+\beta_{3}}\left(\mu\|\nabla
u(t)\|_{H^{1}}^{2}+\frac{1}{\mu}\|P(t)\|_{H^{1}}^{2}\right)\leq\frac{C_{5}(1+M^{2})}{\mu^{\beta_{3}}}E_{0}^{\beta_{3}}M^{1-\beta_{3}},\quad\beta_{3}\in[0,1],$
(3.45)

and the large-time exponential stability

$\displaystyle\sup_{t\in[1,T]}e^{\alpha_{1}t}\left(\mu\|\nabla
u(t)\|_{H^{1}}^{2}+\frac{1}{\mu}\|P(t)\|_{H^{1}}^{2}\right)\leq\frac{C_{5}(1+M^{2})}{\mu}E_{0}.$
(3.46)

Here $M$ and $\alpha_{1}$ are given by (1.6) and (3.17), respectively, and
$C_{5}>0$ is a constant depending only on $\kappa$,
$\|(\rho_{0},n_{f_{0}})\|_{L^{\frac{3}{2}}\cap L^{\infty}}$ and
$\|j_{f_{0}}\|_{L^{\infty}}$.

###### Proof.

In view of (A.4) in Lemma A.2 to $\eqref{VNS}_{2}$, it follows from (2.5),
(3.3), (3.4) and the Gagliardo-Nirenberg inequality that

$\displaystyle\mu\|\nabla
u(t)\|_{L^{2}}^{2}+\frac{1}{\mu}\|P(t)\|_{L^{2}}^{2}$ (3.47)
$\displaystyle\leq\frac{C}{\mu}\|(\rho u_{t}+\rho u\cdot\nabla
u+\kappa\int_{\mathbb{R}^{3}}(u-v)fdv)(t)\|_{L^{\frac{6}{5}}}^{2}$
$\displaystyle\leq\frac{C}{\mu}\|\rho_{0}\|_{L^{\frac{3}{2}}}\|\sqrt{\rho}u_{t}(t)\|_{L^{2}}^{2}+\frac{C}{\mu}\|\rho_{0}\|_{L^{\infty}}\|\rho_{0}\|_{L^{\frac{3}{2}}}\|u(t)\|_{L^{6}}^{2}\|\nabla
u(t)\|_{L^{3}}^{2}$
$\displaystyle\quad+\frac{C\kappa^{2}}{\mu}\|n_{f}(t)\|_{L^{\frac{3}{2}}}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx$
$\displaystyle\leq\frac{C}{\mu}\|\sqrt{\rho}u_{t}(t)\|_{L^{2}}^{2}+C\mu\|\nabla^{2}u(t)\|_{L^{2}}^{2}+\frac{C}{\mu^{3}}\|\nabla
u(t)\|_{L^{2}}^{6}+\frac{C\kappa^{2}}{\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx,$

which together with (3.1) and (3.19) leads to (3.44).

In addition, combined (3.47) with (LABEL:mmmmm3), one has

$\displaystyle\mu\|\nabla
u(t)\|_{H^{1}}^{2}+\frac{1}{\mu}\|P(t)\|_{H^{1}}^{2}$ (3.48)
$\displaystyle\leq\frac{C}{\mu}\|\sqrt{\rho}u_{t}(t)\|_{L^{2}}^{2}+\frac{C}{\mu^{3}}\|\nabla
u(t)\|_{L^{2}}^{6}+\frac{C\kappa^{2}}{\mu}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|u-v|^{2}fdvdx.$

It is derived from Lemmas 3.3–3.5 and (LABEL:mmmd1) that (3.45) and (3.46)
hold, and the proof is completed. ∎

With the help of Lemmas 3.3–3.6, we are ready to prove Proposition 3.2.

###### Proof of Proposition 3.2.

Assume that the initial data $(\rho_{0},u_{0},f_{0})$ satisfies (1.8). For any
given time $T>0$, let $(\rho,u,f)$ be the solution to the Cauchy problem
(1.1)–(1.2) satisfying (3.1). Our goal is to justify the a-priori estimates
(3.2). First, by Lemma 3.3 and (3.19), it is easy to verify that

$\displaystyle\int_{0}^{T}\|\nabla u(t)\|_{L^{2}}^{4}dt$
$\displaystyle\leq\sup_{t\in(0,T]}\|\nabla
u(t)\|_{L^{2}}^{2}\int_{0}^{T}\|\nabla u(t)\|_{L^{2}}^{2}dt\leq
CM\mu^{-1}E_{0},$ (3.49)

which gives rise to

$\displaystyle\int_{0}^{T}\|\nabla
u(t)\|_{L^{2}}^{4}dt<\frac{1}{4},\quad\text{if}~{}~{}\mu^{-1}E_{0}<\varepsilon_{1}:=\frac{1}{4C(1+M)}.$
(3.50)

Next, we establish the $L^{1}(0,T;L^{\infty})$-norm of $u$. It follows from
the Gagliardo-Nirenberg inequality that

$\displaystyle\int_{0}^{T}\|u(t)\|_{L^{\infty}}dt$ $\displaystyle\leq
C\int_{0}^{T}\|\nabla
u(t)\|_{L^{2}}^{\frac{1}{2}}\|\nabla^{2}u(t)\|_{L^{2}}^{\frac{1}{2}}dt.$
(3.51)

For short time $T\leq 1$, we deduce from Lemma 3.3 and (3.20) with
$\beta_{1}=1$ that

$\displaystyle\int_{0}^{T}\|\nabla
u(t)\|_{L^{2}}^{\frac{1}{2}}\|\nabla^{2}u(t)\|_{L^{2}}^{\frac{1}{2}}dt$
$\displaystyle\leq\mu^{-\frac{1}{2}}\left(\int_{0}^{T}\mu\|\nabla
u(t)\|_{L^{2}}^{2}dt\right)^{\frac{1}{4}}\left(\int_{0}^{1}t\mu\|\nabla^{2}u(t)\|_{L^{2}}^{2}dt\right)^{\frac{1}{4}}\left(\int_{0}^{1}t^{-\frac{1}{2}}dt\right)^{\frac{1}{2}}$
$\displaystyle\leq C\mu^{-\frac{3}{4}}E_{0}^{\frac{1}{2}}\leq
C\left(\mu^{-1}E_{0}\right)^{\frac{1}{2}}.$

As for large time $T\geq 1$, the time-integral on $(0,1]$ can be handled
similarly as above, and on the rest part $[1,T]$ one has

$\displaystyle\int_{1}^{T}\|\nabla
u(t)\|_{L^{2}}^{\frac{1}{2}}\|\nabla^{2}u(t)\|_{L^{2}}^{\frac{1}{2}}dt$
$\displaystyle\leq\mu^{-\frac{1}{2}}\left(\int_{1}^{T}e^{\alpha_{1}t}\mu\|\nabla
u(t)\|_{L^{2}}^{2}dt\right)^{\frac{1}{4}}\left(\int_{1}^{T}e^{\alpha_{1}t}\mu\|\nabla^{2}u(t)\|_{L^{2}}^{2}dt\right)^{\frac{1}{4}}\left(\int_{1}^{T}e^{-\alpha_{1}t}dt\right)^{\frac{1}{2}}$
$\displaystyle\leq C\left(\mu^{-1}E_{0}\right)^{\frac{1}{2}},$

due to (3.16) and (3.21). It thus holds

$\displaystyle\int_{0}^{T}\|u(t)\|_{L^{\infty}}dt\leq\widetilde{C}_{1}\left(\mu^{-1}E_{0}\right)^{\frac{1}{2}}<\frac{1}{4},\quad\text{if}~{}~{}\mu^{-1}E_{0}<\varepsilon_{2}:=\frac{1}{(4\widetilde{C}_{1})^{2}}.$
(3.52)

for some generic constant $\widetilde{C}_{1}>0$.

In addition, we shall control the $L^{1}(0,T;L^{\infty})$-norm of $\nabla u$.
It can be derived from the Gagliardo-Nirenberg inequality that

$\|\nabla u(t)\|_{L^{\infty}}\leq C\|\nabla
u(t)\|_{L^{2}}^{\frac{1}{4}}\|\nabla^{2}u(t)\|_{L^{6}}^{\frac{3}{4}},$ (3.53)

which requires the estimates of $\|\nabla^{2}u(t)\|_{L^{6}}$. To that matter,
we apply (A.5) to $\eqref{VNS}_{2}$, which together with the Sobolev
inequality, (3.3), (3.4) and (3.5) yields

$\displaystyle\mu\|\nabla^{2}u(t)\|_{L^{6}}+\|\nabla P(t)\|_{L^{6}}$ (3.54)
$\displaystyle\leq\|(\rho u_{t}+\rho u\cdot\nabla
u+\kappa\int_{\mathbb{R}^{3}}(u-v)fdv)(t)\|_{L^{6}}$
$\displaystyle\leq\|\rho_{0}\|_{L^{\infty}}\|u_{t}(t)\|_{L^{6}}+\|\rho_{0}\|_{L^{\infty}}\|u(t)\|_{L^{6}}\|\nabla
u(t)\|_{L^{\infty}}+\kappa\left(\|n_{f}(t)\|_{L^{\infty}}\|u(t)\|_{L^{6}}+\|j_{f}(t)\|_{L^{6}}\right)$
$\displaystyle\leq C\|\nabla u_{t}(t)\|_{L^{2}}+C\|\nabla
u(t)\|_{L^{2}}\|\nabla u(t)\|_{L^{\infty}}+C\|\nabla
u(t)\|_{L^{2}}+C\mu^{\frac{1}{6}},$

where one has used $\|j_{f}(t)\|_{L^{6}}\leq CE_{0}^{\frac{1}{6}}\leq
C\mu^{\frac{1}{6}}$. Then, it follows from (3.53), (LABEL:dfgg) and the Young
inequality that

$\displaystyle\|\nabla u(t)\|_{L^{\infty}}$ $\displaystyle\leq
C\mu^{-\frac{3}{4}}\|\nabla u(t)\|_{L^{2}}^{\frac{1}{4}}\|\nabla
u_{t}(t)\|_{L^{2}}^{\frac{3}{4}}+C\mu^{-3}\|\nabla u(t)\|_{L^{2}}^{4}$ (3.55)
$\displaystyle\quad+C\mu^{-\frac{3}{4}}\|\nabla
u(t)\|_{L^{2}}+C\mu^{-\frac{5}{8}}\|\nabla u(t)\|_{L^{2}}^{\frac{1}{4}}.$

By virtue of (3.55), (3.49) and the short-time estimates (3.31), (3.20) with
$\beta_{1}=\frac{1}{4}$ and (3.33) with $\beta_{2}=\frac{1}{4}$, there exists
a positive constant
$\widetilde{C}_{2}=\widetilde{C}_{2}(M,\kappa,\delta_{0},\|(\rho_{0},n_{f_{0}})\|_{L^{\frac{3}{2}}\cap
L^{\infty}},\|j_{f_{0}}\|_{L^{\infty}})$ such that

$\displaystyle\int_{0}^{1}\|\nabla u(t)\|_{L^{\infty}}dt$ $\displaystyle\leq
C\mu^{-\frac{3}{4}}\left(\sup_{t\in(0,1]}t^{\frac{1}{4}}\|\nabla
u(t)\|_{L^{2}}^{2}\right)^{\frac{1}{8}}\left(\int_{0}^{1}t^{\frac{5}{4}}\|\nabla
u_{t}(t)\|_{L^{2}}^{2}dt\right)^{\frac{3}{8}}\left(\int_{0}^{1}t^{-\frac{4}{5}}dt\right)^{\frac{5}{8}}$
$\displaystyle\quad+C\mu^{-3}\int_{0}^{1}\|\nabla
u(t)\|_{L^{2}}^{4}dt+C\mu^{-\frac{3}{4}}\left(\sup_{t\in(0,1]}t\|\nabla
u(t)\|_{L^{2}}^{2}\right)^{\frac{1}{2}}\int_{0}^{1}t^{-\frac{1}{2}}dt$
$\displaystyle\quad+C\mu^{-\frac{5}{8}}\left(\sup_{t\in(0,1]}t\|\nabla
u(t)\|_{L^{2}}^{2}\right)^{\frac{1}{8}}\int_{0}^{1}t^{-\frac{1}{8}}dt$
$\displaystyle\leq
C(\mu^{-1}E_{0})^{\frac{1}{8}}+C\mu^{-1}E_{0}+C(\mu^{-1}E_{0})^{\frac{1}{2}}+C(\mu^{-1}E_{0})^{\frac{1}{8}}\leq\widetilde{C}_{2}(\mu^{-1}E_{0})^{\frac{1}{8}}.$

On the other hand, due to (3.55), (3.49), and the large-time estimates (3.21)
and (3.34), it holds for some positive constant
$\widetilde{C}_{3}=\widetilde{C}_{2}(M,\kappa,\delta_{0},\|(\rho_{0},n_{f_{0}})\|_{L^{\frac{3}{2}}\cap
L^{\infty}},\|j_{f_{0}}\|_{L^{\infty}})$ that

$\displaystyle\int_{1}^{T}\|\nabla u(t)\|_{L^{\infty}}dt$ $\displaystyle\leq
C\mu^{-\frac{3}{4}}\left(\sup_{t\in[1,T]}e^{\alpha_{1}t}\|\nabla
u(t)\|_{L^{2}}^{2}\right)^{\frac{1}{8}}\left(\int_{1}^{T}e^{\alpha_{1}t}\|\nabla
u_{t}(t)\|_{L^{2}}^{2}dt\right)^{\frac{3}{8}}\left(\int_{1}^{T}e^{-\frac{4}{5}\alpha_{1}t}dt\right)^{\frac{5}{8}}$
$\displaystyle\quad+C\mu^{-3}\int_{1}^{T}\|\nabla
u(t)\|_{L^{2}}^{4}dt+C\mu^{-\frac{3}{4}}\left(\sup_{t\in[1,T]}e^{\alpha_{1}t}\|\nabla
u(t)\|_{L^{2}}^{2}\right)^{\frac{1}{2}}\int_{1}^{T}e^{-\frac{1}{2}\alpha_{1}t}dt$
$\displaystyle\quad+C\mu^{-\frac{5}{8}}\left(\sup_{t\in[1,T]}e^{\alpha_{1}t}\|\nabla
u(t)\|_{L^{2}}^{2}\right)^{\frac{1}{8}}\int_{1}^{T}e^{-\frac{1}{8}\alpha_{1}t}dt$
$\displaystyle\leq
C(\mu^{-1}E_{0})^{\frac{1}{2}}+C\mu^{-1}E_{0}+C(\mu^{-1}E_{0})^{\frac{1}{2}}+C(\mu^{-1}E_{0})^{\frac{1}{8}}\leq\widetilde{C}_{3}(\mu^{-1}E_{0})^{\frac{1}{8}}.$

To sum up, one has

$\displaystyle\int_{0}^{T}\|\nabla
u(t)\|_{L^{\infty}}dt<\frac{\delta_{0}}{2},\quad\text{if}~{}~{}\mu^{-1}E_{0}<\varepsilon_{3}:=\min\Bigg{\\{}1,\frac{\delta_{0}^{8}}{(2(\widetilde{C}_{2}+\widetilde{C}_{3}))^{8}}\Bigg{\\}}.$
(3.56)

Therefore, we end up with (3.2) by taking
$\mu^{-1}E_{0}<\varepsilon_{0}:=\min\\{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}\\}$
with $\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}$ given in (3.50),
(3.52), (3.56), respectively, and thus complete the proof of Proposition 3.2.
∎

###  3.2 Large-time behavior of the distribution function

We further capture the concentration phenomena of the distribution function
$f$.

######  Lemma 3.7.

Let $T>0$ and $(\rho,u,f)$ be the solution to (1.1) for $t\in(0,T]$ with the
initial data $(\rho_{0},u_{0},f_{0})$ satisfying (1.8). Then under the
condition (3.1), it holds for
$\alpha_{2}=\min\left\\{\frac{\kappa}{2},\frac{\alpha_{1}}{4}\right\\}$ that

$\left\\{\begin{aligned} &\|j_{f}(t)\|_{L^{1}}\leq
C_{6}e^{-\alpha_{2}t}E^{\frac{1}{2}}_{0},\quad\quad\|j_{f}(t)\|_{L^{\infty}}\leq
C_{6}e^{-\alpha_{2}t}\left(\|j_{f_{0}}\|_{L^{\infty}}+E_{0}^{\frac{1}{2}}\right),\\\
&{\rm{Supp}}_{v}f(t,x,v)\subset\left\\{v\in\mathbb{R}^{3}~{}\Big{|}~{}|v|\leq
C_{6}e^{-\alpha_{2}t}(R_{0}+E_{0}^{\frac{1}{2}})\right\\},\quad(t,x)\in(0,T]\times\mathbb{R}^{3},\end{aligned}\right.$
(3.57)

and as $T\rightarrow+\infty$, there exists a profile $n^{*}\in L^{\infty}$
such that

$W_{1}(f(t),n^{*}\otimes\delta_{v})+W_{1}(n_{f}(t),n^{*})\leq
C_{6}e^{-\alpha_{1}t}E^{\frac{1}{2}}_{0}.$ (3.58)

Here $E_{0}$ and $\alpha_{1}$ are given by (1.4) and (3.17), respectively, and
$C_{6}>0$ is a constant depending only on $\kappa$, $M$,
$\|(\rho_{0},n_{f_{0}})\|_{L^{\frac{3}{2}}\cap L^{\infty}}$ and
$\|j_{f_{0}}\|_{L^{\infty}}$.

###### Proof.

For any $t\in(0,T]$, one deduces by (3.6) and (3.16) that

$\displaystyle\|j_{f}(t)\|_{L^{1}}\leq\left(\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}fdvdx\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|v|^{2}fdvdx\right)^{\frac{1}{2}}\leq
Ce^{-\frac{\alpha_{1}}{2}t}E_{0}^{\frac{1}{2}}.$ (3.59)

In terms of (3.14), (3.20) and (3.46), it holds for
$\alpha_{2}=\min\left\\{\frac{\kappa}{2},\frac{\alpha_{1}}{4}\right\\}$ that

$\displaystyle\|j_{f}(t)\|_{L^{\infty}}$ (3.60) $\displaystyle\leq 2e^{-\kappa
t}\|j_{f_{0}}\|_{L^{\infty}}+C\left\\{\int_{0}^{1}+\int_{1}^{t}\right\\}e^{-\kappa(t-\tau)}\|\nabla
u(\tau)\|_{L^{2}}^{\frac{1}{2}}\|\nabla^{2}u(\tau)\|_{L^{2}}^{\frac{1}{2}}d\tau$
$\displaystyle\leq 2e^{-\kappa t}\|j_{f_{0}}\|_{L^{\infty}}+Ce^{-\kappa
t}\left(\sup_{\tau\in(0,1]}\tau\|\nabla
u(\tau)\|_{L^{2}}^{2}\right)^{\frac{1}{4}}\left(\int_{0}^{1}\tau\mu\|\nabla^{2}u(\tau)\|_{L^{2}}^{2}d\tau\right)^{\frac{1}{4}}\left(\int_{0}^{1}\tau^{-\frac{2}{3}}d\tau\right)^{\frac{3}{4}}$
$\displaystyle\quad+C\left(\sup_{\tau\in[1,t]}e^{\alpha_{1}t}\mu\|\nabla
u(\tau)\|_{H^{1}}^{2}\right)^{\frac{1}{2}}\int_{1}^{t}e^{-\kappa(t-\tau)}e^{-\frac{\alpha_{1}}{2}\tau}d\tau$
$\displaystyle\leq
Ce^{-\alpha_{2}t}\left(\|j_{f_{0}}\|_{L^{\infty}}+E_{0}^{\frac{1}{2}}\right).$

Then, we deal with the large-time behavior of the compact support of $f$ with
respect to $v$. Due to the characteristics (1.20), it is straightforward to
obtain

$v=e^{-\kappa
t}V(0;t,x,v)+\int_{0}^{t}e^{-\kappa(t-\tau)}u(\tau,X(\tau;t,x,v))d\tau.$
(3.61)

Recalling (1.19) that

$f(t,x,v)=e^{3\kappa t}f_{0}(X(0;t,x,v),V(0;t,x,v)),$

we deduce

$f(t,x,v)\neq 0\Rightarrow
V(0;t,x,v)\in{\rm{Supp}}_{v}f_{0}(X(0;t,x,v),V(0;t,x,v))\Rightarrow|V(0;t,x,v)|\leq
R_{0}.$ (3.62)

In a similar argument as (3.60), for any $(t,x)\in(0,T]\times\mathbb{R}^{3}$
and $v\in\text{Supp}_{v}f(t,x,v)$, one has

$\displaystyle|v|$ $\displaystyle\leq e^{-\kappa
t}R_{0}+\int_{0}^{t}e^{-\kappa(t-\tau)}\|u(\tau)\|_{L^{\infty}}d\tau\leq
Ce^{-\alpha_{2}t}\left(R_{0}+E_{0}^{\frac{1}{2}}\right).$ (3.63)

Furthermore, we establish the large-time behavior of the distribution function
$f$ and the moment $n_{f}$. By the Monge-Kantorovitch duality (refer to Lemma
A.4), we have

$\displaystyle W_{1}(f(t),n_{f}(t)\otimes\delta_{v})$
$\displaystyle=\sup_{\|\nabla_{x,v}\psi\|_{L^{\infty}}\leq
1}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}f(t,x,v)\left(\psi(x,v)-\psi(x,0)\right)dvdx$
(3.64) $\displaystyle\leq\sup_{\|\nabla_{x,v}\psi\|_{L^{\infty}}\leq
1}\|\nabla_{v}\psi\|_{L^{\infty}}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|v|fdvdx$
$\displaystyle\leq\left(\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}fdvdx\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|v|^{2}fdvdx\right)^{\frac{1}{2}}$
$\displaystyle\leq\sqrt{2}\|n_{f}\|_{L^{1}}^{\frac{1}{2}}E(t)^{\frac{1}{2}},$

which together with (3.4) and (3.16) leads to

$W_{1}(f(t),n_{f}(t)\otimes\delta_{v})\leq
Ce^{-\frac{\alpha_{1}}{2}t}E_{0}^{\frac{1}{2}}.$ (3.65)

In addition, the integration of $\eqref{VNS}_{4}$ with respect to
$v\in\mathbb{R}^{3}$ yields

$(n_{f})_{t}+{\rm{div}}_{x}j_{f}=0.$ (3.66)

Thus it holds for any smooth function $\varphi(x)\in
C^{\infty}(\mathbb{R}^{3})$ that

${\rm{\frac{d}{dt}}}\int_{\mathbb{R}^{3}}\varphi
n_{f}dx-\int_{\mathbb{R}^{3}}\nabla_{x}\varphi j_{f}dx=0,$ (3.67)

which implies for any $t\in(0,T]$ that

$\displaystyle\left|\int_{\mathbb{R}^{3}}\varphi
n_{f}(T)dx-\int_{\mathbb{R}^{3}}\varphi n_{f}(t)dx\right|$
$\displaystyle\leq\int_{t}^{T}\left(\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|v|^{2}fdvdx\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^{3}}n_{f}|\nabla_{x}\varphi|^{2}dx\right)^{\frac{1}{2}}d\tau$
(3.68)
$\displaystyle\leq\|\nabla_{x}\varphi\|_{L^{\infty}}\int_{t}^{T}\|n_{f}(\tau)\|_{L^{1}}^{\frac{1}{2}}E(\tau)^{\frac{1}{2}}d\tau.$

More precisely, if $\|\nabla\varphi\|_{L^{\infty}}\leq 1$, in terms of (3.4)
and (3.16), we have

$\left|\int_{\mathbb{R}^{3}}\varphi n_{f}(T)dx-\int_{\mathbb{R}^{3}}\varphi
n_{f}(t)dx\right|\leq\|n_{f_{0}}\|_{L^{1}}^{\frac{1}{2}}E_{0}^{\frac{1}{2}}\int_{t}^{T}e^{-\frac{\alpha_{1}}{2}\tau}d\tau,$
(3.69)

and this extends to Lipschitz functions $\varphi$ by a standard approximation
argument so that the Monge-Kantorovitch duality formula leads to

$W_{1}(n_{f}(t),n_{f}(T))\leq
CE_{0}^{\frac{1}{2}}\frac{2}{\alpha_{1}}(e^{-\frac{\alpha_{1}}{2}t}-e^{-\frac{\alpha_{1}}{2}T}).$
(3.70)

Therefore the Cauchy criterion for $n_{f}(t)$ is verified as
$T\rightarrow\infty$, which means there exist some measure $n^{*}$ such that
$n_{f}(t)\rightarrow n^{*}$ as $t\rightarrow\infty$ in the sense of
$1$-Wasserstein distance. Since $n_{f}(t)$ satisfies the uniform estimate
(3.4), $n^{*}\in L^{\infty}(\mathbb{R}^{3})$ holds. By letting
$T\rightarrow\infty$ in (3.69), we conclude

$\displaystyle W_{1}(n_{f}(t),n^{*})\leq
Ce^{-\frac{\alpha_{1}}{2}t}E_{0}^{\frac{1}{2}},$ (3.71)

which together with (3.65) leads to

$W_{1}(f(t),n^{*}\otimes\delta_{v})\leq
Ce^{-\frac{\alpha_{1}}{2}t}E_{0}^{\frac{1}{2}}.$ (3.72)

Thus, we complete the proof of Lemma 3.7. ∎

###  3.3 The higher-order estimates

The following Lemmas are essential for the further proof on the uniqueness of
the strong solution.

######  Lemma 3.8.

Let $T>0$ and $(\rho,u,f)$ be the solution to (1.1) for $t\in(0,T]$ with the
initial data $(\rho_{0},u_{0},f_{0})$ satisfying (1.8). Then under the
condition (3.1), it holds

$\displaystyle\sup_{t\in(0,T]}\|\nabla\rho(t)\|_{L^{2}}\leq
2\|\nabla\rho_{0}\|_{L^{2}}.$ (3.73)

###### Proof.

Applying $\nabla$ to $\eqref{VNS}_{1}$ and noticing the incompressible
condition $\eqref{VNS}_{3}$, we have

$\nabla\rho_{t}+u\cdot\nabla(\nabla\rho)+\nabla u\cdot\nabla\rho=0,$ (3.74)

which gives rise to

$\frac{1}{2}{\rm{\frac{d}{dt}}}\|\nabla\rho(t)\|_{L^{2}}^{2}\leq\frac{3}{2}\|\nabla
u(t)\|_{L^{\infty}}\|\nabla\rho(t)\|_{L^{2}}^{2}.$ (3.75)

Dividing (3.75) by $\sqrt{\|\nabla\rho(t)\|_{L^{2}}^{2}+\eta_{0}}$ with
$\eta_{0}>0$, integrating the resulting inequality over $[0,t]$ for
$t\in(0,T]$, and then taking the limit $\eta_{0}\rightarrow 0$, we obtain

$\|\nabla\rho(t)\|_{L^{2}}\leq\|\nabla\rho_{0}\|_{L^{2}}\exp{\left(\frac{3}{2}\int_{0}^{T}\|\nabla
u(t)\|_{L^{\infty}}dt\right)}\leq 2\|\nabla\rho_{0}\|_{L^{2}},$ (3.76)

where one has used (3.2). Thus, the proof of Lemma 3.8 is completed. ∎

######  Lemma 3.9.

Let $T>0$ and $(\rho,u,f)$ be the solution to (1.1) for $t\in(0,T]$ with the
initial data $(\rho_{0},u_{0},f_{0})$ satisfying (1.8). Then under the
condition (3.1), it holds

$\sup_{t\in(0,T]}\|f(t)\|_{W^{1,\frac{3}{2}}_{x,v}}\leq
C_{T}\|f_{0}\|_{W^{1,\frac{3}{2}}_{x,v}}.$ (3.77)

###### Proof.

Multiplying $\eqref{VNS}_{4}$ by $f^{\frac{1}{2}}$ and integrating the
resulting equality by parts over $\mathbb{R}^{3}\times\mathbb{R}^{3}$, we get

$\displaystyle\frac{2}{3}{\rm{\frac{d}{dt}}}\|f(t)\|_{L^{\frac{3}{2}}_{x,v}}^{\frac{3}{2}}\leq\kappa\|f(t)\|_{L^{\frac{3}{2}}_{x,v}}^{\frac{3}{2}},$
(3.78)

which implies

$\displaystyle\sup_{t\in(0,T]}\|f(t)\|_{L^{\frac{3}{2}}_{x,v}}\leq e^{\kappa
t}\|f_{0}\|_{L^{\frac{3}{2}}}.$ (3.79)

Moreover, it can be derived from $\eqref{VNS}_{4}$ that

$(\partial_{x}^{i}f)_{t}+v\cdot\nabla_{x}\partial_{x}^{i}f+\kappa\partial_{x}^{i}u\cdot\nabla_{v}f+\kappa(u-v)\cdot\nabla_{v}\partial_{x}^{i}f-3\kappa\partial_{x}^{i}f=0,~{}~{}i=1,2,3.$
(3.80)

Multiplying (3.80) by $|\partial_{x}^{i}f|^{\frac{1}{2}}\hbox{\rm
sign}\,\partial_{x}^{i}f$, summing up for $i=1,2,3$, and integrating the
resulting equation over $\mathbb{R}^{3}\times\mathbb{R}^{3}$ , we obtain

$\displaystyle\frac{2}{3}{\rm{\frac{d}{dt}}}\|\nabla_{x}f(t)\|_{L^{\frac{3}{2}}_{x,v}}^{\frac{3}{2}}$
$\displaystyle\leq\kappa\|\nabla
u(t)\|_{L^{\infty}}\|\nabla_{v}f(t)\|_{L^{\frac{3}{2}}_{x,v}}^{\frac{1}{2}}\|\nabla_{x}f(t)\|_{L^{\frac{3}{2}}}+\kappa\|\nabla_{x}f(t)\|_{L^{\frac{3}{2}}_{x,v}}^{\frac{3}{2}}$
(3.81) $\displaystyle\leq C(\|\nabla
u(t)\|_{L^{\infty}}+1)\|(\nabla_{x}f,\nabla_{v}f)(t)\|_{L^{\frac{3}{2}}_{x,v}}^{\frac{3}{2}}.$

Similarly, it yields

$\displaystyle\frac{2}{3}{\rm{\frac{d}{dt}}}\|\nabla_{v}f(t)\|_{L^{\frac{3}{2}}_{x,v}}^{\frac{3}{2}}$
$\displaystyle\leq
C\|(\nabla_{x}f,\nabla_{v}f)(t)\|_{L^{\frac{3}{2}}_{x,v}}^{\frac{3}{2}}.$
(3.82)

Hence, the combination of (3.2), (3.79), (3.81), (3.82) and the Grönwall
inequality gives rise to (3.77), and the proof of Lemma 3.9 is completed. ∎

##  4 Proofs of main results

With the help of all the estimates obtained in Section 3, we are ready to
prove Theorems 1.1 and 1.2.

Proof of Theorem 1.1: Let $(\rho_{0},u_{0},f_{0})$ be the initial data
satisfying (1.8) and (1.9). According to Proposition 3.1, there exists a
maximal time $T_{*}>0$ such that the Cauchy problem (1.1)–(1.2) has a unique
strong solution $(\rho,u,f)$ fulfilling (1.10) on
$(0,T_{*}]\times\mathbb{R}^{3}\times\mathbb{R}^{3}$. Now we define

$T_{0}:=\sup\left\\{T\in(0,T_{*}]~{}\Big{|}~{}\int_{0}^{T}(\|\nabla
u(t)\|_{L^{2}}^{4}+\|u(t)\|_{L^{\infty}})dt\leq 1,\quad\int_{0}^{T}\|\nabla
u(t)\|_{L^{\infty}}dt\leq\delta_{0}\right\\}.$ (4.1)

By (1.10), there exists a small time $T_{1}\in(0,T_{*})$ such that $T_{0}\geq
T_{1}>0$.

We claim that

$T_{*}=T_{0}=\infty,$

as long as (1.9) holds. Indeed, if $T_{0}<T_{*}$, then one deduces from
Proposition 3.2 that

$\displaystyle\int_{0}^{T_{0}}(\|\nabla
u(t)\|_{L^{2}}^{4}+\|u(t)\|_{L^{\infty}})dt\leq\frac{1}{2},\quad\int_{0}^{T_{0}}\|\nabla
u(t)\|_{L^{\infty}}dt\leq\frac{\delta_{0}}{2},$ (4.2)

for all $T\in(0,T_{0}]$. This contradicts the definition of $T_{0}$ in (4.1),
and hence we have $T_{0}=T_{*}$. In order to show $T_{*}=\infty$, we assume
$T_{*}<\infty$ toward a contradiction. Since the uniform estimates (4.2) hold
for all $T\in(0,T_{*}]$, by virtue of Lemmas 3.1–3.6, one can take
$(\rho,u,f)(t)$ for a time $t$ sufficiently close to $T_{*}$ as a new initial
data and obtain the existence of solutions beyond the time $T_{*}$ due to
Proposition 3.1, which contradicts the maximality of $T_{*}$. Therefore,
$(\rho,u,f)$ is indeed a global strong solution to the Cauchy problem
(1.1)–(1.2) with vacuum. In addition, according to (4.1), (LABEL:dfgg) and
Lemmas 3.1–3.7 , one can show that the global solution $(\rho,u,f)$ satisfies
the property (1.10).

To complete the proof of Theorem 1.1, we finally prove the uniqueness issue.
For given time $T>0$, let $(\rho_{1},u_{1},P_{1},f_{1})$ and
$(\rho_{2},u_{2},P_{2},f_{2})$ be two solutions satisfying (1.10) subject to
the same initial data $(\rho_{0},u_{0},f_{0})$. We set

$(\tilde{\rho},\tilde{u},\tilde{P},\tilde{f})=(\rho_{1}-\rho_{2},u_{1}-u_{2},P_{1}-P_{2},f_{1}-f_{2}),$
(4.3)

which satisfies

$\left\\{\begin{aligned}
&\tilde{\rho}_{t}+u_{1}\cdot\nabla\tilde{\rho}=-\tilde{u}\cdot\nabla\rho_{2},\\\
&\rho_{1}(\tilde{u}_{t}+u_{1}\cdot\nabla\tilde{u})+\nabla\tilde{P}-\mu\Delta\tilde{u}+\kappa\int_{\mathbb{R}^{3}}\tilde{u}f_{1}dv=-\tilde{\rho}(u_{2})_{t}-\rho_{1}\tilde{u}\cdot\nabla
u_{2}-\kappa\int_{\mathbb{R}^{3}}(u_{2}-v)\tilde{f}dv,\\\ &\hbox{\rm
div}\,\tilde{u}=0,\\\
&\tilde{f}_{t}+v\cdot\nabla_{x}\tilde{f}+\kappa{\rm{div}}_{v}((u_{1}-v)\tilde{f})=-\kappa\tilde{u}\cdot\nabla_{v}f_{2}.\end{aligned}\right.$
(4.4)

Multiplying $\eqref{diso}_{1}$ by $|\tilde{\rho}|^{\frac{1}{2}}\hbox{\rm
sign}\,\tilde{\rho}$ and integrating over $\mathbb{R}^{3}$, we have

$\displaystyle\frac{2}{3}{\rm{\frac{d}{dt}}}\|\tilde{\rho}(t)\|_{L^{\frac{3}{2}}}^{\frac{3}{2}}$
$\displaystyle\leq\|\nabla
u_{1}(t)\|_{L^{\infty}}\|\tilde{\rho}(t)\|_{L^{\frac{3}{2}}}^{\frac{3}{2}}+\|\nabla\rho_{2}(t)\|_{L^{2}}\|\tilde{u}(t)\|_{L^{6}}\|\tilde{\rho}(t)\|_{L^{\frac{3}{2}}}^{\frac{1}{2}}$
(4.5) $\displaystyle\leq\|\nabla
u_{1}(t)\|_{L^{\infty}}\|\tilde{\rho}(t)\|_{L^{\frac{3}{2}}}^{\frac{3}{2}}+C\|\nabla\tilde{u}(t)\|_{L^{2}}\|\tilde{\rho}(t)\|_{L^{\frac{3}{2}}}^{\frac{1}{2}},$

which leads to

$\|\tilde{\rho}(t)\|_{L^{\frac{3}{2}}}\leq C\exp{\left(\int_{0}^{t}\|\nabla
u_{1}(\tau)\|_{L^{\infty}}d\tau\right)}\int_{0}^{t}\|\nabla\tilde{u}(\tau)\|_{L^{2}}d\tau\leq
Ct^{\frac{1}{2}}\left(\int_{0}^{t}\|\nabla\tilde{u}(\tau)\|_{L^{2}}^{2}d\tau\right)^{\frac{1}{2}}.$
(4.6)

Next, multiplying $\eqref{diso}_{4}$ by $|\tilde{f}|^{\frac{1}{5}}\hbox{\rm
sign}\,\tilde{f}$ and then integrating over
$\mathbb{R}^{3}\times\mathbb{R}^{3}$, we get

$\displaystyle\frac{6}{5}{\rm{\frac{d}{dt}}}\|\tilde{f}(t)\|_{L^{\frac{6}{5}}_{x,v}}^{\frac{6}{5}}$
$\displaystyle\leq\frac{\kappa}{2}\|\tilde{f}(t)\|_{L^{\frac{6}{5}}_{x,v}}^{\frac{6}{5}}+\|\tilde{u}(t)\|_{L^{6}}\|\nabla_{v}f_{2}(t)\|_{L^{\frac{3}{2}}_{x}(L^{\frac{6}{5}}_{v})}\|\tilde{f}(t)\|_{L^{\frac{6}{5}}_{x,v}}^{\frac{1}{5}}$
(4.7) $\displaystyle\leq
C\|\tilde{f}(t)\|_{L^{\frac{6}{5}}_{x,v}}^{\frac{6}{5}}+C\|\nabla\tilde{u}(t)\|_{L^{2}}\|\tilde{f}(t)\|_{L^{\frac{6}{5}}_{x,v}}^{\frac{1}{5}},$

where we have used
$\|\nabla_{v}f_{2}(t)\|_{L^{\frac{3}{2}}_{x}(L^{\frac{6}{5}}_{v})}\leq
C\|\nabla_{v}f(t)\|_{L^{\frac{3}{2}}_{x,v}}$ due to the compact support of $f$
with respect to $v$. Thus, the Grönwall inequality to (4.7) gives

$\displaystyle\|\tilde{f}(t)\|_{L^{\frac{6}{5}}_{x,v}}\leq
C_{T}\int_{0}^{t}\|\nabla\tilde{u}(\tau)\|_{L^{2}}d\tau\leq
C_{T}t^{\frac{1}{2}}\left(\int_{0}^{t}\|\nabla\tilde{u}(\tau)\|_{L^{2}}^{2}d\tau\right)^{\frac{1}{2}}.$
(4.8)

Since $f$ is compactly supported in $v$, the estimate (4.8) implies

$\displaystyle\|(n_{\tilde{f}},j_{\tilde{f}})(t)\|_{L^{\frac{6}{5}}}\leq
C_{T}t^{\frac{1}{2}}\left(\int_{0}^{t}\|\nabla\tilde{u}(\tau)\|_{L^{2}}^{2}d\tau\right)^{\frac{1}{2}}.$
(4.9)

Furthermore, multiplying $\eqref{diso}_{2}$ by $\tilde{u}$, integrating the
resulting equation by parts and using (4.6) and (4.9), we deduce that

$\displaystyle\frac{1}{2}{\rm{\frac{d}{dt}}}\int_{\mathbb{R}^{3}}\rho_{1}|\tilde{u}|^{2}dx+\mu\int_{\mathbb{R}^{3}}|\nabla\tilde{u}|^{2}dx+\kappa\int_{\mathbb{R}^{3}}n_{f_{1}}|\tilde{u}|^{2}dx$
$\displaystyle\leq\|(u_{2})_{t}(t)\|_{L^{6}}\|\tilde{\rho}(t)\|_{L^{\frac{3}{2}}}\|\tilde{u}(t)\|_{L^{6}}+\|\nabla
u_{2}(t)\|_{L^{\infty}}\int_{\mathbb{R}^{3}}\rho_{1}|\tilde{u}|^{2}dx$
$\displaystyle\quad+\|u_{2}(t)\|_{L^{6}}\|\nabla
u_{2}(t)\|_{L^{\infty}}\|\tilde{\rho}(t)\|_{L^{\frac{3}{2}}}\|\tilde{u}(t)\|_{L^{6}}$
$\displaystyle\quad+\|u_{2}(t)\|_{L^{\infty}}\|\tilde{u}(t)\|_{L^{6}}\|n_{\tilde{f}}(t)\|_{L^{\frac{6}{5}}}+\|\tilde{u}(t)\|_{L^{6}}\|j_{\tilde{f}}(t)\|_{L^{\frac{6}{5}}}$
$\displaystyle\leq\frac{\mu}{2}\int_{\mathbb{R}^{3}}|\nabla\tilde{u}|^{2}dx+C_{T}\Big{(}\|\nabla
u(t)\|_{L^{\infty}}+t\|\nabla(u_{2})_{t}(t)\|_{L^{2}}^{2}+t\|\nabla
u_{2}(t)\|_{H^{1}\cap
W^{1,6}}^{2}\Big{)}\int_{0}^{t}\|\nabla\tilde{u}(\tau)\|_{L^{2}}^{2}d\tau,$

from which we obtain for $t\in(0,T]$ that

$\displaystyle{\rm{\frac{d}{dt}}}\widetilde{G}(t)\leq C_{T}\Big{(}\|\nabla
u(t)\|_{L^{\infty}}+t\|\nabla(u_{2})_{t}(t)\|_{L^{2}}^{2}+t\|\nabla
u_{2}(t)\|_{H^{1}\cap W^{1,6}}^{2}\Big{)}\widetilde{G}(t),$ (4.10)

where $\widetilde{G}(t)$ is defined by

$\widetilde{G}(t):=\int_{\mathbb{R}^{3}}\rho_{1}|\tilde{u}|^{2}dx+\mu\int_{0}^{t}\int_{\mathbb{R}^{3}}|\nabla\tilde{u}|^{2}dx+\kappa\int_{\mathbb{R}^{3}}n_{f_{1}}|\tilde{u}|^{2}dx.$
(4.11)

Due to $t^{\frac{1}{2}}\nabla(u_{2})_{t}\in L^{2}(0,T;L^{2})$ and
$t^{\frac{1}{2}}\nabla u\in L^{2}(0,T;H^{1}\cap W^{1,6})$, making use of the
Grönwall inequality on (4.10), one has $\widetilde{G}(t)\equiv 0$ for
$t\in(0,T]$. This together with (4.6) and (4.8) implies that
$(\rho_{1},u_{1},f_{1})\equiv(\rho_{2},u_{2},f_{2})$. Finally, it follows from
the Stokes estimate on $\eqref{diso}_{2}$ that $P_{1}\equiv P_{2}$, and thus
the proof of Theorem 1.1 is completed.

Proof of Theorem 1.2: Let $(\rho_{0},u_{0},f_{0})$ be the initial data
satisfying (1.14) and (1.15). For any $\varepsilon\in(0,1)$, we regularize the
initial data as follows:

$(\rho_{0}^{\varepsilon}(x),u_{0}^{\varepsilon}(x),f_{0}^{\varepsilon}(x,v)):=(J_{1}^{\varepsilon}\ast\rho_{0}(x)+\varepsilon,J_{1}^{\varepsilon}\ast
u_{0}(x),J_{1}^{\varepsilon}\ast
J_{2}^{\varepsilon}\ast(f_{0}\mathbf{1}_{|v|\leq\varepsilon^{-1}})(x,v)),$
(4.12)

where $J_{1}^{\varepsilon}(x)$ and $J_{2}^{\varepsilon}(v)$ are the Friedrichs
mollifier with respect to the variables $x$ and $v$, respectively, and
$\mathbf{1}_{|v|\leq\varepsilon^{-1}}\in C_{0}^{\infty}(\mathbb{R})$ is the
cut-off function such that $\mathbf{1}_{|v|\leq\varepsilon^{-1}}=1$ for
$|v|\leq\varepsilon^{-1}$, and $\mathbf{1}_{|v|\leq\varepsilon^{-1}}=0$ for
$|v|\geq 2\varepsilon^{-1}$. It is easy to verify that
$(\rho_{0}^{\varepsilon},u_{0}^{\varepsilon},f_{0}^{\varepsilon})$ satisfies
the assumptions (1.14)–(1.15) uniformly in $\varepsilon\in(0,1)$ and as
$\varepsilon\rightarrow 0$, one has

$\left\\{\begin{aligned} &\begin{aligned}
&\rho^{\varepsilon}_{0}\rightarrow\rho_{0}&&\text{in}~{}~{}L^{p}\times
L^{p},\quad\frac{3}{2}\leq p<\infty,\\\ &\nabla
u_{0}^{\varepsilon}\rightarrow\nabla u_{0}&&\text{in}~{}~{}L^{2},\\\
&f_{0}^{\varepsilon}\rightarrow f_{0}&&\text{in}~{}~{}L^{p},\quad 1\leq
p<\infty,\end{aligned}\\\
&\frac{1}{2}\int_{\mathbb{R}^{3}}\rho^{\varepsilon}_{0}|u^{\varepsilon}_{0}|^{2}dx+\frac{1}{2}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}|v|^{2}f^{\varepsilon}_{0}dvdx\rightarrow
E_{0}.\end{aligned}\right.$

In addition, it holds for $q>4$ that

$\displaystyle\|(n_{f_{0}^{\varepsilon}},j_{f_{0}^{\varepsilon}})\|_{L^{\infty}}\leq\|(1+|v|^{q})f^{\varepsilon}_{0}\|_{L^{\infty}_{x,v}}\int_{\mathbb{R}^{3}}\frac{1+|v|}{1+|v|^{q}}dv\leq
C\|(1+|v|^{q})f_{0}\|_{L^{\infty}_{x,v}}.$

Owing to Theorem 1.1, there exists a unique global strong solution
$(\rho^{\varepsilon},u^{\varepsilon},P^{\varepsilon},f^{\varepsilon})$ for any
$\varepsilon\in(0,1)$ to the system (1.1) with the initial data
$(\rho_{0}^{\varepsilon},u_{0}^{\varepsilon},f_{0}^{\varepsilon})$. For any
given time $T>0$, it follows from Proposition 3.1 and Lemmas 3.1–3.6 that
$(\rho^{\varepsilon},u^{\varepsilon},P^{\varepsilon},f^{\varepsilon})$ has the
uniform bounds (3.2), (3.3), (3.16), (3.20)–(3.21), (3.33)–(3.34) and
(3.45)–(3.46). Thus, by virtue of the Aubin-Lions lemma and the Cantor
diagonal argument, there exists a limit $(\rho,u,P,f)$ such that up to a
subsequence, it holds as $\varepsilon\rightarrow 0$ that

$\left\\{\begin{aligned}
&\rho^{\varepsilon}\overset{\ast}{\rightharpoonup}\rho&&\text{in}~{}~{}L^{\infty}(0,T;L^{\frac{3}{2}}\cap
L^{\infty}),\\\ &u^{\varepsilon}\rightarrow
u&&\text{in}~{}~{}L^{2}(0,T;H^{1}_{loc}),\\\
&P^{\varepsilon}\overset{\ast}{\rightharpoonup}P&&\text{in}~{}~{}L^{\infty}(0,T;L^{\frac{3}{2}}\cap
L^{\infty}),\\\
&f^{\varepsilon}\overset{\ast}{\rightharpoonup}f&&\text{in}~{}~{}L^{\infty}(0,T;L^{\infty}_{x,v}).\end{aligned}\right.$

Therefore, it is easy to check that $(\rho,u,f)$ solves the system (1.1) in
the sense of distributions. Due to Fatou properties, $(\rho,u,f)$ satisfies
the uniform estimates in Lemmas 3.1–3.6, the large-time behavior
$\eqref{r2}_{1}$ and the energy inequality (1.5) for a.e. $t\in(0,T]$. From
the renormalized arguments for $\eqref{VNS}_{1}$ and $\eqref{VNS}_{4}$ (refer
to [31, 5]), one has $\rho\in C([0,T];L^{\frac{3}{2}})$ and $f\in
C([0,T];L^{1}_{x,v})$. It follows from $\rho u\in L^{\infty}(0,T;L^{2})$ and
$(\rho u)_{t}\in L^{\infty}(0,T;H^{-1})$ that $\rho u\in C([0,T];H^{-1})$, and
therefore $(\rho,u,f)$ is indeed a global weak solution to the Cauchy problem
(1.1)–(1.2) in the sense of Definition 1.1. Repeating the same argument as in
Lemma 3.7, we have the asymptotic behavior $\eqref{r2}_{2}$. Thus, the proof
of Theorem 1.2 is completed.

## Appendix

In this section, we provide some auxiliary lemmas that are used frequently
throughout this paper. We first show the Gagliardo-Nirenberg inequality (e.g.,
refer to [35]).

######  Lemma A.1.

Let $1\leq p,q,r\leq\infty$. For $0\leq k,l\leq s$, there exists a constant
$C>0$ such that

$\|\partial_{x}^{k}f\|_{L^{p}}\leq
C\|\partial_{x}^{l}f\|_{L^{q}}^{1-\theta}\|\partial_{x}^{s}f\|_{L^{r}}^{\theta},$
(A.1)

where $\theta\in[0,1]$ satisfies

$\frac{k}{3}-\frac{1}{p}=\left(\frac{l}{3}-\frac{1}{q}\right)(1-\theta)+\left(\frac{s}{3}-\frac{1}{r}\right)\theta.$
(A.2)

Next, we consider the Cauchy problem for the Stokes equations below

$\left\\{\begin{aligned} &-\mu\Delta u+\nabla P=g,\quad x\in\mathbb{R}^{3},\\\
&\hbox{\rm div}\,u=0,\\\ &u(x)\rightarrow 0~{}~{}\text{as
}|x|\rightarrow{\infty},\end{aligned}\right.$ (A.3)

and we have the following regularity estimates.

######  Lemma A.2.

If $g\in L^{\frac{6}{5}}\cap L^{p}$ with $p\in(\frac{6}{5},\infty)$, there
exists some positive constant $C$ depending only on $p$ such that the unique
weak solution $(u,P)\in D_{0,\sigma}^{1}\times L^{2}$ to the Cauchy problem
(A.3) satisfies

$\mu\|\nabla u\|_{L^{2}}+\|P\|_{L^{2}}\leq C\|g\|_{L^{\frac{6}{5}}},$ (A.4)

and

$\mu\|\nabla^{2}u\|_{L^{p}}+\|\nabla P\|_{L^{p}}\leq C\|g\|_{L^{p}}.$ (A.5)

###### Proof.

Multiplying $\eqref{steq}_{1}$ by $u$ and then integrating by parts together
with $\eqref{steq}_{2}$, we obtain

$\mu\|\nabla u\|_{L^{2}}^{2}=\int_{\mathbb{R}^{3}}g\cdot
udx\leq\|g\|_{L^{\frac{6}{5}}}\|u\|_{L^{6}}\leq
C\|g\|_{L^{\frac{6}{5}}}\|\nabla u\|_{L^{2}},$

which implies

$\mu\|\nabla u\|_{L^{2}}\leq C\|g\|_{L^{\frac{6}{5}}}.$ (A.6)

In addition, we can derive

$P=-\mu(-\Delta)^{-1}\hbox{\rm div}\,\Delta u-(-\Delta)^{-1}\hbox{\rm
div}\,g,$

which combined with the Sobolev inequality leads to

$\|P\|_{L^{2}}\leq\|(-\Delta)^{-1}\hbox{\rm div}\,g\|_{L^{2}}\leq
C\|g\|_{L^{\frac{6}{5}}}.$ (A.7)

Thus, one can conclude (A.4) from (A.6) and (A.7), and a similar argument
leads to (A.5) by $\eqref{steq}$. ∎

We also need the following version of the inverse function theorem, which can
be shown by direct computations.

######  Lemma A.3.

For $\Omega=\mathbb{R}^{3}$, if $\phi:\Omega\rightarrow\Omega$ is $C^{1}$ and
satisfies $\|\nabla\phi\|_{L^{\infty}}<1$, then $f:={\rm{Id}}+\phi$ is a
$C^{1}$-diffeomorphism of $\Omega$ onto itself satisfying $\|\nabla
f\|_{L^{\infty}}\leq\left(1-\|\nabla\phi\|_{L^{\infty}}\right)^{-1}$. If
furthermore $\|\nabla\phi\|_{L^{\infty}}\leq\frac{1}{9}$, then $\hbox{\rm
det}\,\nabla f\geq\frac{1}{2}$.

Finally, we recall the definition of Wasserstein distances and the Monge-
Kantorovitch duality. The interested readers can refer to the book [38].

######  Definition A.1.

Denote either $\mathbb{R}^{3}$ or $\mathbb{R}^{3}\times\mathbb{R}^{3}$ by $X$
and let $\mu_{1},\mu_{2}$ be the Borel measures on $X$. The $p$-Wasserstein
distance between $\mu_{1}$ and $\mu_{2}$ is defined by

$W_{p}(\mu_{1},\mu_{2}):=\left(\inf_{\omega\in\Gamma(\mu_{1},\mu_{2})}\int_{X\times
X}|z-z^{\prime}|^{p}d\omega(z,z^{\prime})\right)^{\frac{1}{p}},$ (A.8)

where $\Gamma(\mu_{1},\mu_{2})$ represents the collection of all measures on
$X\times X$ that the first and second marginal equal to $\mu_{1}$ and
$\mu_{2}$, respectively.

######  Lemma A.4.

Fix $p=1$ and $\mu_{1},\mu_{2}$ be the Borel measures on $X$, then it holds

$W_{1}(\mu_{1},\mu_{2})=\sup\left\\{\int_{X}\psi(z)d\mu_{1}(z)-\int_{X}\psi(z)d\mu_{2}(z)\Big{|}~{}\forall\psi\in{\rm{Lip}}(X),~{}\|\nabla\psi\|_{L^{\infty}(X)}\leq
1\right\\}.$ (A.9)

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