# Long-time behavior to the 3D isentropic compressible Navier–Stokes equations
††thanks: Wu’s research was partially supported by Natural Science Foundation
of Fujian Province and Fujian Alliance of Mathematics (Nos. 2022J01304 and
2023SXLMMS08). Zhong’s research was partially supported by Fundamental
Research Funds for the Central Universities (No. SWU–KU24001) and National
Natural Science Foundation of China (No. 12371227).

Guochun Wu$\,{}^{\rm 1}\,$, Xin Zhong$\,{}^{\rm 2}\,$
${}^{\rm 1}\,$ School of Mathematical Sciences, Huaqiao University, Quanzhou
362021, China
${}^{\rm 2}\,$ School of Mathematics and Statistics, Southwest University,
Chongqing 400715, China E-mail addresses<EMAIL_ADDRESS>(G. Wu),
<EMAIL_ADDRESS>(X. Zhong).

###### Abstract

We are concerned with the long-time behavior of classical solutions to the
isentropic compressible Navier–Stokes equations in $\mathbb{R}^{3}$. Our main
results and innovations can be stated as follows:

  * •

Under the assumption that the density $\rho({\bf{x}},t)$ verifies
$\rho({\bf{x}},0)\geq c>0$ and $\sup_{t\geq
0}\|\rho(\cdot,t)\|_{L^{\infty}}\leq M$, we establish the optimal decay rates
of the solutions. This greatly improves the previous result (Arch. Ration.
Mech. Anal. 234 (2019), 1167–1222), where the authors require an extra
hypothesis $\sup_{t\geq 0}\|\rho(\cdot,t)\|_{C^{\alpha}}\leq M$ with $\alpha$
arbitrarily small.

  * •

We prove that the vacuum state will persist for any time provided that the
initial density contains vacuum and the far-field density is away from vacuum,
which extends the torus case obtained in (SIAM J. Math. Anal. 55 (2023),
882–899) to the whole space.

  * •

We derive the decay properties of the solutions with vacuum as far-field
density. This in particular gives the first result concerning the
$L^{\infty}$-decay with a rate $(1+t)^{-1}$ for the pressure to the 3D
compressible Navier–Stokes equations in the presence of vacuum.

The main ingredient of the proof relies on the techniques involving blow-up
criterion, a key time-independent positive upper and lower bounds of the
density, and a regularity interpolation trick.

Key words and phrases. Compressible Navier–Stokes equations; long-time
behavior; vacuum.

2020 Mathematics Subject Classification. 76N06; 76N10.

###### Contents

  1. 1 Introduction
    1. 1.1 Background and motivation
    2. 1.2 Main results
    3. 1.3 Strategy of the proofs
    4. 1.4 Structure of the paper
  2. 2 Preliminaries
  3. 3 A priori estimates
  4. 4 Proofs of Theorems 1.1 and 1.2
  5. 5 Proof of Theorem 1.3

##  1 Introduction

###  1.1 Background and motivation

We study the Cauchy problem of the isentropic compressible Navier–Stokes
equations in $\mathbb{R}^{3}$:

$\displaystyle\begin{cases}\rho_{t}+\operatorname{div}(\rho\mathbf{u})=0,\\\
(\rho\mathbf{u})_{t}+\operatorname{div}(\rho\mathbf{u}\otimes\mathbf{u})-\mu\Delta\mathbf{u}-(\mu+\lambda)\nabla\text{div}\mathbf{u}+\nabla
P(\rho)=0,\end{cases}$ (1.1)

with the initial condition

$\displaystyle(\rho,\rho\mathbf{u})(\mathbf{x},0)=(\rho_{0},\rho_{0}\mathbf{u}_{0})({\bf
x}),\ \ {\bf x}\in\mathbb{R}^{3},$ (1.2)

and the far-field behavior

$\displaystyle(\rho(\mathbf{x},t),\mathbf{u}(\mathbf{x},t))\rightarrow(\tilde{\rho},\mathbf{0}),\
\ \text{as}\ |\mathbf{x}|\rightarrow\infty,\ t>0,$ (1.3)

for a non-negative constant $\tilde{\rho}$. Here $\rho$,
$\mathbf{u}=(u^{1},u^{2},u^{3})$, and $P=\rho^{\gamma}\ (\gamma>1)$ denote the
density, velocity, and pressure of the fluid, respectively. The constants
$\mu$ and $\lambda$ are the shear viscosity and the bulk viscosity of the
fluid satisfying the physical restrictions

$\displaystyle\mu>0\ \ \text{and}\ \ 2\mu+3\lambda\geq 0.$

There is a huge literature on the global existence (well-posedness) of
solutions to the multi-dimensional compressible isentropic Navier–Stokes
equations. When the initial density is away from a vacuum, the global well-
posedness of strong solutions in three-dimensional (3D) domains was first
obtained by Matsumura and Nishida [35, 36]. With the help of the effective
viscous flux, Hoff [19, 20] proved the global existence of non-vacuum
solutions with discontinuous initial data, while Danchin [6] investigated the
global existence and uniqueness of strong solutions in the critical Besov
spaces (see also [3, 4, 17] for the $L^{p}$ setting). It should be pointed out
that the results stated above require smallness of the initial oscillation on
the perturbation of initial data near the non-vacuum equilibrium state. This
assumption concerning the small oscillations on the initial perturbation of a
non-vacuum state is recently removed by Huang–Li–Xin [26], where they proved
the global existence and uniqueness of classical solutions with smooth initial
data that are of small energy in $\mathbb{R}^{3}$ (see also [2] for the case
of 3D bounded domains). The major breakthrough on the solutions with large
initial data and vacuum is due to Lions [33], where he used the
renormalization skills introduced by DiPerna and Lions [11] to establish
global weak solutions in $\mathbb{R}^{n}\ (n=2,3)$ provided that
$\gamma\geq\frac{3n}{n+2}$. Later on, Feireisl–Novotný–Petzeltová [14]
improved Lions’ result to the case $\gamma>\frac{n}{2}$. At the same time,
Jiang and Zhang [27, 28] derived global weak solutions for any $\gamma>1$ when
the initial data are assumed to have some spherically symmetric or
axisymmetric properties. Nonetheless, due to the possible concentration of
finite kinetic energy in very small domains, it still seems to be a challenge
to show the global existence of weak solutions with general 3D data for
$\gamma\in(1,\frac{n}{2}]$. There are also very interesting investigations
about global large strong solutions in some sense for multi-dimensional
compressible isentropic Navier–Stokes equations, please refer to [7, 8, 22,
13, 45] and references therein.

It is also of great interest to study the large-time behavior of solutions.
The readers can refer to [9, 12, 21, 29, 30, 34, 35, 36] and references
therein for large-time behavior of global smooth solutions to the compressible
Navier–Stokes system under the smallness assumption on the initial
perturbation. Some important progress has been made about long time behavior
of large solutions for multi-dimensional compressible Navier–Stokes equations
by many authors. Feireisl and Petzeltová [15] first showed that any weak
solution converges to a fixed stationary state as time goes to infinity via
the weak convergence method. Under the assumptions that the density is
essentially bounded and has uniform-in-time positive lower bound, Padula [39]
proved that weak solutions decay exponentially to the equilibrium state in
$L^{2}$-norm. By using the Bogovskii operator and constructing a suitable
Lyapunov functional, Peng–Shi–Wu [40] improved the result in [39] to the case
that they didn’t need the time-independent upper and lower bounds of density.
He–Huang–Wang [18] investigated the global stability of large strong solutions
to the 3D Cauchy problem. More precisely, under the hypothesis that the
density $\rho({\bf x},t)$ verifies $\inf_{{\bf
x}\in\mathbb{R}^{3}}\rho_{0}({\bf x})\geq c_{0}>0$ and $\sup_{t\geq
0}\|\rho(t)\|_{C^{\alpha}}\leq M$ with arbitrarily small $\alpha$, they
established the convergence of the solutions to its associated equilibrium
states with an explicit decay rate which is the same as that of the heat
equation. It should be emphasized that the assumption $\sup_{t\geq
0}\|\rho(t)\|_{C^{\alpha}}\leq M$ plays an essential role in their analysis
(see the proof of Proposition 2.3 in [18] for details).

As mentioned in many papers (refer to [26, 2, 10, 43, 23, 24, 37, 38] for
instance), one of the main difficulties for the stability of solutions to the
compressible isentropic Navier–Stokes equations is the concentration of the
density. In particular, Desjardins [10] studied the regularity of weak
solutions for small time under periodic boundary conditions, and particularly
showed that weak solutions in $\mathbb{T}^{2}$ turn out to be smooth as long
as the density remains bounded in $L^{\infty}(\mathbb{T}^{2})$. Sun–Wang–Zhang
[43] proved that the concentration of the density must be responsible for the
loss of the regularity of the strong solution in finite time. Hu [23, 24]
investigated the concentration phenomenon of weak solutions with finite
energy, and showed that the concentration set has a parabolic Hausdorff
dimension related to $\gamma$ by a provided formula. More recently,
Merle–Raphaël–Rodnianski–Szeftel [37, 38] constructed a set of finite energy
smooth initial data for which the corresponding solutions to the compressible
3D Navier–Stokes and Euler equations implode (with infinite density) at a
later time at a point, and completely described the associated formation of
singularity. Then a natural question, but of importance and interest, is that
can we show the long-time behavior of global classical solutions to the system
(1.1) with bounded density? The main goal of the present paper is to give an
affirmative response to this problem.

###  1.2 Main results

Before stating our main result, we first explain the notations and conventions
used throughout this paper. We denote by $C$ a positive generic constant which
may vary at different places. The norms in Sobolev spaces
$H^{\ell}(\mathbb{R}^{3})$ and $W^{\ell,r}(\mathbb{R}^{3})$ are denoted
respectively by $\|\cdot\|_{H^{\ell}}$ and $\|\cdot\|_{W^{\ell,r}}$ for
$\ell\geq 0$ and $r\geq 1$. Particularly, for $\ell=0$, we will simply use
$\|\cdot\|_{L^{r}}$. For the sake of conciseness, we do not precise in
functional space names when they are concerned with scalar–valued or
vector–valued functions. $\|(f,g)\|_{X}$ denotes $\|f\|_{X}+\|g\|_{X}$. We
denote $\nabla=\partial_{x}=(\partial_{1},\partial_{2},\partial_{3})$, where
$\partial_{i}=\partial_{x_{i}}$, $\nabla_{i}=\partial_{i}$ and put
$\partial_{x}^{\ell}f=\nabla^{\ell}f=\nabla(\nabla^{\ell-1}f)$. We denote
$\|u\|_{D^{k,r}}\overset{\triangle}{=}\|\nabla^{k}u\|_{L^{r}}$,
$D^{k}(\mathbb{R}^{3})=D^{k,2}(\mathbb{R}^{3})$, and
$D^{1}(\mathbb{R}^{3})=\\{u\in L^{6}\big{|}\|\nabla u\|_{L^{2}}<\infty\\}$.
Let $\Lambda^{s}$ be the pseudo differential operator defined by

$\Lambda^{s}f=\mathfrak{F}^{-1}(|{\bf\xi}|^{s}\hat{f}),~{}\hbox{for}~{}s\in\mathbb{R},$

where both $\hat{f}$ and $\mathfrak{F}(f)$ denote the Fourier transform of
$f$. The homogenous Sobolev space $\dot{H}^{s}(\mathbb{R}^{3})$ is endowed
with the norm $\|f\|_{\dot{H}^{s}}:=\|\Lambda^{s}f\|_{L^{2}}$. We also drop
the domain $\mathbb{R}^{3}$ in integrands over $\mathbb{R}^{3}$, that is,
$\int\cdot\mathrm{d}{\bf x}=\int_{\mathbb{R}^{3}}\cdot\mathrm{d}{\bf x}$.
Introducing the Leray projector
$\mathcal{P}\triangleq\text{Id}+\nabla(-\Delta)^{-1}\operatorname{div}$ onto
divergence-free vector fields, and
$\mathcal{Q}\triangleq\text{Id}-\mathcal{P}$. In particular, because
$\mathcal{P}$ and $\mathcal{Q}$ are smooth homogeneous of degree $0$ Fourier
multipliers, they map $L^{p}$ into itself for any $1<p<\infty$.

We also recall the Littlewood–Paley decomposition. Choose a radial function
$\varphi\in S(\mathbb{R}^{3})$ supported in
$\mathcal{C}=\\{\xi\in\mathbb{R}^{N},\frac{3}{4}\leq|\xi|\leq\frac{8}{3}\\}$
such that

$\sum\limits_{q\in\mathbb{Z}}\varphi(2^{-q}\xi)=1,\hbox{\ \ for all\ \
}\xi\neq 0.$

For $q\in\mathbb{Z}$, we define the following dyadic blocks:

$\Delta_{q}f=\mathfrak{F}^{-1}(\varphi(2^{-q}\xi)\mathfrak{F}f).$

We denote the space $\mathcal{D}^{{}^{\prime}}(\mathbb{R}^{N})$ by the dual
space of $\mathcal{D}(\mathbb{R}^{N})=\\{f\in
S(\mathbb{R}^{3});D^{\alpha}\hat{f}(0)=0;\forall\alpha\in\mathbb{N}^{3}\hbox{multi-
index}\\}$, which also can be identified by the quotient space of
$S^{{}^{\prime}}(\mathbb{R}^{3})/\mathcal{P}$ with polynomials space
$\mathcal{P}$. The formal equality

$f=\sum\limits_{q\in\mathbb{Z}}\Delta_{q}f$

holds true for $f\in\mathcal{D}^{{}^{\prime}}(\mathbb{R}^{3})$ and is called
the homogeneous Littlewood–Paley decomposition.

######  Definition 1.1.

Let $s\in\mathbb{R},1\leq p,r\leq+\infty$. The homogeneous Besov space
$\dot{B}_{p,r}^{s}$ is defined by

$\dot{B}_{p,r}^{s}(\mathbb{R}^{3})=\\{f\in
D^{{}^{\prime}}(\mathbb{R}^{3}):\|f\|_{\dot{B}_{p,r}^{s}}<+\infty\\},$

where

$\|f\|_{\dot{B}_{p,r}^{s}}\triangleq\|2^{qs}\|\Delta_{q}f\|_{L^{p}}\|_{l^{r}}.$

As in [19], the effective viscous flux $F$ and vorticity ${\bf w}$ are defined
by

$\displaystyle F\overset{\text{def}}{=}(2\mu+\lambda)\operatorname{div}{\bf
u}-(P(\rho)-P(\tilde{\rho}))\ \ \text{and}\ \ {\bf
w}\overset{\text{def}}{=}\nabla\times{\bf u}.$ (1.4)

By (1.1)2, one has that

$\displaystyle\Delta F=\operatorname{div}(\rho\dot{{\bf u}})\ \ \text{and}\ \
\mu\Delta{\bf w}=\nabla\times(\rho\dot{\bf u}),$ (1.5)

where $``\ \dot{\ }\ "$ denotes the material derivative which is defined by

$\dot{f}\overset{\text{def}}{=}f_{t}+{\bf u}\cdot\nabla f.$

The potential energy $G(\rho)$ is defined by

$\displaystyle
G(\rho)=\rho\int^{\rho}_{{\tilde{\rho}}}\frac{P(s)-P(\tilde{\rho})}{s^{2}}\mathrm{d}s,$
(1.6)

it is not hard to deduce that there exists a positive constant $C$ depending
only on $\tilde{\rho}$ and $\bar{\rho}$ such that

$\displaystyle\begin{cases}G(\rho)=\frac{1}{\gamma-1}P,\ &\text{if}\
\tilde{\rho}=0,\\\
\frac{1}{C(\tilde{\rho},\bar{\rho})}(\rho-\tilde{\rho})^{2}\leq G(\rho)\leq
C(\tilde{\rho},\bar{\rho})(\rho-\tilde{\rho})^{2},\ &\text{if}\
\tilde{\rho}>0\ \text{and}\ \rho\leq\bar{\rho}.\end{cases}$ (1.7)

Our first main novelty of this paper is to establish the optimal convergence
rates of the solutions to the Cauchy problem (1.1)–(1.3).

######  Theorem 1.1.

Let $\tilde{\rho}=1$. Assume that the initial data $(\rho_{0},{\bf u}_{0})$
satisfies

$\displaystyle\inf\rho_{0}\geq c_{0}>0,\ \ K\triangleq\|(\rho_{0}-1,{\bf
u}_{0})\|_{H^{3}}<\infty.$

If $2\mu>\lambda$ and $(\rho,{\bf u})$ is a global classical solution to the
Cauchy problem (1.1)–(1.3) verifying that

$\displaystyle\sup\limits_{t\geq 0}\|\rho(\cdot,t)\|_{L^{\infty}}\leq M$ (1.8)

for some positive constant $M$, then the following properties hold.

(1) (Lower bound for the density) There exists a positive constant
$\underline{\rho}=\underline{\rho}(c_{0},M,K)$ such that

$\displaystyle\rho(t)\geq\underline{\rho},\ \ \text{for all}\ \ t\geq 0.$
(1.9)

(2) (Uniform-in-time bound for the solution) For all $t\geq 0$, it holds that

$\displaystyle\|(\rho-1,{\bf
u})(t)\|_{H^{3}}^{2}+\int_{0}^{t}\big{(}\|\nabla\rho(\tau)\|_{H^{2}}^{2}+\|\nabla{\bf
u}(\tau)\|_{H^{3}}^{2}\big{)}\mathrm{d}\tau\leq C(c_{0},M,K).$ (1.10)

(3) (Decay estimate for the solution) If additionally
$N_{0}\triangleq\|(\rho_{0}-1,{\bf u}_{0})\|_{\dot{B}^{-s}_{2,\infty}}<\infty$
with $0<s\leq\frac{3}{2}$, we have

$\displaystyle\|\nabla^{k}(\rho-1,{\bf u})(t)\|_{H^{3-k}}\leq
C(c_{0},M,K,N_{0})(1+t)^{-\frac{k+s}{2}t},\ \ k\in\\{0,1,2\\}.$ (1.11)

######  Remark 1.1.

Theorem 1.1 gives the global dynamics of the system (1.1) when the initial
data is far away from the equilibrium in the whole space. This is in sharp
contrast to the results of [9, 29, 30, 35, 36] where the small initial
perturbation near the non-vacuum equilibrium state is used to get the long-
time behavior of the solution.

######  Remark 1.2.

It should be noticed that in Theorem 1.1, the density is only assumed to be
bounded from above, while the global dynamics of large solutions in [18,
Theorem 1.1] requires an extra hypothesis $\sup_{t\geq
0}\|\rho(t)\|_{C^{\alpha}}\leq M$ with arbitrarily small $\alpha$, which plays
an essential role in establishing the uniform positive lower bound of the
density (see the proof of Proposition 2.3 in [18] for details). Thus, we
greatly improve the main result of [18, Theorem 1.1].

Next, we shall prove that the vacuum states will not vanish for any time
provided that the vacuum states are present initially and the far-field
density is away from a vacuum.

######  Theorem 1.2.

Let $\tilde{\rho}=1$. Suppose that the initial data $(\rho_{0},{\bf u}_{0})$
satisfies

$\displaystyle(\rho_{0}|{\bf u}_{0}|^{2}+G(\rho_{0}))\in
L^{1}(\mathbb{R}^{3}),\ \ {\bf u}_{0}\in D^{1}(\mathbb{R}^{3})\cap
D^{3}(\mathbb{R}^{3}),$ (1.12) $\displaystyle(\rho_{0}-1,P(\rho_{0})-P(1))\in
H^{3},\ \ 0=\inf\rho_{0}\leq\sup\rho_{0}\leq\bar{\rho},$ (1.13)

and the compatibility condition

$\displaystyle-\mu\Delta{\bf u}_{0}-(\mu+\lambda)\nabla\operatorname{div}{\bf
u}_{0}+\nabla P(\rho_{0})=\rho_{0}g$ (1.14)

for some $g\in D^{1}(\mathbb{R}^{3})$ with $\sqrt{\rho_{0}}g\in
L^{2}(\mathbb{R}^{3})$. Let $2\mu>\lambda$ and $(\rho,{\bf u})$ be a global
classical solution to the Cauchy problem (1.1)–(1.3) verifying (1.8), then it
holds that

$\displaystyle\lim\limits_{t\rightarrow\infty}\|\rho-1\|_{L^{q}}=0,\ \
\text{for any}\ \ q>2,$ (1.15)

and

$\displaystyle\inf\limits_{{\bf x}\in\mathbb{R}^{3}}\rho({\bf x},t)=0,\ \
\text{for any}\ \ t\geq 0.$ (1.16)

######  Remark 1.3.

(1.16) shows that the vacuum state will not vanish for any time, which
together with the large-time behavior (1.15) leads to the gradient of the
density has to blow up as time goes to infinity, in the sense that for any
$r>3$,

$\displaystyle\lim\limits_{t\rightarrow\infty}\|\nabla\rho(\cdot,t)\|_{L^{r}}=\infty.$

This is in sharp contrast to that in [6, 35] where the gradient of the density
is suitably small uniformly for all time.

######  Remark 1.4.

Theorem 1.2 extends the torus case [44] to the whole space. However, this is a
non-trivial exercise since the Poincaré inequality is crucial in the arguments
in [44] which does not hold in the whole space.

Finally, we have the following results concerning the decay properties of the
global classical solutions when the far-field density is vacuum (i.e.,
$\tilde{\rho}=0$).

######  Theorem 1.3.

Assume that $\tilde{\rho}=0$, $\rho_{0}\in L^{1}(\mathbb{R}^{3})$, and the
initial data $(\rho_{0},{\bf u}_{0})$ satisfies (1.12)–(1.14). Let
$7\mu>\lambda$ and $(\rho,{\bf u})$ be a global classical solution to the
Cauchy problem (1.1)–(1.3) verifying (1.8), then it holds that, for any $t\geq
0$,

$\displaystyle\left\|P(\cdot,t)\right\|_{L^{p}}\leq
C(1+t)^{-1+\frac{1}{p\gamma}},\ \ \text{for any}\ 1\leq p\leq\infty,$ (1.17)

and

$\displaystyle\int\rho|{\bf u}|^{2}\mathrm{d}{\bf x}\leq
C(1+t)^{-1-\frac{1}{3\gamma}}.$ (1.18)

######  Remark 1.5.

To the best of our knowledge, the $L^{\infty}$-decay with a rate $(1+t)^{-1}$
for the pressure in (1.17) is completely new to the 3D compressible
Navier–Stokes equations (1.1) in the presence of vacuum.

###  1.3 Strategy of the proofs

We shall now explain the major difficulties and techniques in the proof of our
main results. It should be pointed out that the methods used in [40, 44]
cannot be adopted here to prove Theorems 1.1 and 1.2 since their arguments
rely more or less on the fact that domain is bounded. For instance, the
Poincaré inequality and the $L^{p}$ bound of Bogovskii operator are used which
do not hold in the whole space. Moreover, as stated above, the analysis in
[18] relies heavily on the condition that the density is bounded uniformly in
time in the Hölder space $C^{\alpha}$ with $\alpha$ arbitrarily small.
Consequently, we cannot simply employ the frameworks of [40, 44, 18], but need
some new observations and ideas to establish the desired a priori estimates
under the assumption (1.8).

We first show the time-independent positive lower bound (1.9) of the density.
Motivated by [10], we rewrite the mass equation (1.1)1 in terms of $\ln\rho$,
which immediately yields the time-dependent positive lower bound of $\rho$
(see (3.7)). Then, arguing by contradiction, and making fully use of (1.1)1
and Lagrangian coordinates, we derive a crucial bound on $\rho-1$ for large
time (see (3)). This together with the dissipative estimate on the effective
viscous flux (see (3.1)) leads to the time-independent positive lower bound of
$\rho$ immediately. Such bound is important to show the large-time behavior of
$\|\rho(t)-1\|_{L^{\infty}}$ (see (3.15)). Next, we attempt to prove the
uniform-in-time bound (1.10) of the solution. It turns out that one of the key
issues is to derive the time-independent estimate on $L^{6}$-norm of the
gradient of $\rho$. Multiplying the equation of $\nabla\rho$ (see (3.18)) by
$|\nabla\rho|^{4}\nabla\rho$ yields that the time-dependent estimates on both
the $L^{\infty}_{t}L^{6}_{x}$-norm of the density gradient and the
$L^{2}_{t}L^{\infty}_{x}$-norm of the velocity gradient can be obtained by
solving a logarithm Gronwall inequality based on Lemma 2.5. Then, after some
careful analysis based on contradiction arguments, we succeed in obtaining
that, for large time, the $L^{6}$-norm of the gradient of the density can be
controlled with the help of an interpolation trick and Proposition 3.1. This
combined with (3.15) leads to the large-time behavior of
$\|\nabla\rho(t)\|_{L^{6}}$ and the time-independent estimate on the
$L^{2}_{t}L^{\infty}_{x}$-norm of the velocity gradient. With this key bound
for $L^{\infty}(0,\infty;W^{1,6})$-norm of the density at hand, we can deduce
(1.10) (see the detailed proof of Lemma 3.3). Then, using these a priori
estimates, one derives the decay estimate (1.11) by applying the arguments as
in [16]. Next, we show Theorem 1.2. To do so, the key step is to establish
that $\|\rho-1\|_{L^{6}}$ converges to zero as time goes to infinity (see
(4.3)), which combined with (3.1), the upper bound of the density, and the
interpolation inequality yields the desired results (1.15) and (1.16).

To prove Theorem 1.3, the key ingredient is to establish an algebraic decay
rate of the total energy (see (5.1)). By the basic energy estimate and making
fully use of the momentum equation (1.1)2, one can obtain an energy-
dissipation inequality of the form

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{M}_{1}(t)+C(\mathcal{M}_{1}(t))^{\frac{2\gamma-1}{\gamma-1}}\leq
0,$

where $\mathcal{M}_{1}(t)$ is equivalent to the total energy. This implies the
algebraic decay estimate in Lemma 5.1 immediately. Then, based on such
estimate and the large-time asymptotic behavior of solutions in [32] (see
(5.12) and (5.13)), we are able to derive the desired decay estimates (1.17)
and (1.18).

###  1.4 Structure of the paper

In Section 2, we collect some elementary facts and inequalities that will be
used later. Section 3 is devoted to the a priori estimates, which play
important roles in the proof of Theorems 1.1 and 1.2 in the next section.
Finally, we give the proof of Theorem 1.3 in Section 5.

##  2 Preliminaries

In this section, we collect some known facts and elementary inequalities that
will be used frequently later.

We start with the local existence and uniqueness of classical solutions when
the initial density may vanish in an open set, whose proof can be performed by
using standard procedures as in [5].

######  Lemma 2.1.

For $\tilde{\rho}\geq 0$, assume that the initial data satisfy (1.12)–(1.14).
Then there exist a small time $T_{*}$ and a unique classical solution
$(\rho,{\bf u})$ to the Cauchy problem (1.1)–(1.3) on
$\mathbb{R}^{3}\times(0,T_{*}]$ such that

$\displaystyle\begin{cases}(\rho-\tilde{\rho},P(\rho)-P(\tilde{\rho}))\in
C([0,T_{*}];H^{3}(\mathbb{R}^{3})),\\\ {\bf u}\in
C([0,T_{*}];D^{1}(\mathbb{R}^{3})\cap D^{3}(\mathbb{R}^{3}))\cap
L^{2}((0,T_{*});D^{4}(\mathbb{R}^{3})),\\\ {\bf u}_{t}\in
L^{\infty}((0,T_{*});D^{1}(\mathbb{R}^{3}))\cap
L^{2}((0,T_{*});D^{2}(\mathbb{R}^{3})),\\\ \sqrt{\rho}{\bf u}_{tt}\in
L^{2}((0,T_{*});L^{2}(\mathbb{R}^{3})),\ \sqrt{t}{\bf u}\in
L^{\infty}((0,T_{*});D^{4}(\mathbb{R}^{3})),\\\ \sqrt{t}{\bf u}_{t}\in
L^{\infty}((0,T_{*});D^{2}(\mathbb{R}^{3})),\ \sqrt{t}{\bf u}_{tt}\in
L^{2}((0,T_{*});D^{1}(\mathbb{R}^{3})),\\\ t^{\frac{1}{2}}\sqrt{\rho}{\bf
u}_{tt}\in L^{\infty}((0,T_{*});L^{2}(\mathbb{R}^{3})),\ t{\bf u}_{t}\in
L^{\infty}((0,T_{*});D^{3}(\mathbb{R}^{3})),\\\ t{\bf u}_{tt}\in
L^{\infty}((0,T_{*});D^{1}(\mathbb{R}^{3}))\cap
L^{2}((0,T_{*});D^{2}(\mathbb{R}^{3})),\\\ t\sqrt{\rho}{\bf u}_{ttt}\in
L^{2}((0,T_{*});L^{2}(\mathbb{R}^{3})),\ t^{\frac{3}{2}}{\bf u}_{tt}\in
L^{\infty}((0,T_{*});D^{2}(\mathbb{R}^{3})),\\\ t^{\frac{3}{2}}{\bf
u}_{ttt}\in L^{2}((0,T_{*});D^{1}(\mathbb{R}^{3})),\
t^{\frac{3}{2}}\sqrt{\rho}{\bf u}_{ttt}\in
L^{\infty}((0,T_{*});L^{2}(\mathbb{R}^{3})).\end{cases}$ (2.1)

Next, the following estimates follow from (1.4) and the standard
$L^{p}$-estimate of elliptic system, cf. [26].

######  Lemma 2.2.

Let $(\rho,{\bf u})$ be as in Theorems 1.1–1.3, there exists a generic
positive constant $C$ such that, for $p\in[2,6]$,

$\displaystyle\|\nabla F\|_{L^{p}}+\|\nabla{\bf w}\|_{L^{p}}$
$\displaystyle\leq C\|\rho\dot{\bf u}\|_{L^{p}},$ (2.2)
$\displaystyle\|{F}\|_{L^{p}}+\|{\bf w}\|_{L^{p}}$ $\displaystyle\leq
C\|\rho\dot{\bf u}\|_{L^{2}}^{\frac{3p-6}{2p}}\big{(}\|\nabla{\bf
u}\|_{L^{2}}+\|P-P(\tilde{\rho})\|_{L^{2}}\big{)}^{\frac{6-p}{2p}},$ (2.3)
$\displaystyle\|\nabla{\bf u}\|_{L^{p}}$ $\displaystyle\leq
C\big{(}\|F\|_{L^{p}}+\|{\bf
w}\|_{L^{p}}+\|P-P(\tilde{\rho})\|_{L^{p}}\big{)},$ (2.4)
$\displaystyle\|\nabla{\bf u}\|_{L^{p}}$ $\displaystyle\leq C\|\nabla{\bf
u}\|_{L^{2}}^{\frac{6-p}{2p}}\big{(}\|\rho\dot{\bf
u}\|_{L^{2}}+\|P-P(\tilde{\rho})\|_{L^{6}}\big{)}^{\frac{3p-6}{2p}}.$ (2.5)

We recall the well-known Gagliardo–Nirenberg inequality (see [31, Theorem
12.87]).

######  Lemma 2.3.

(Gagliardo–Nirenberg inequality, general case). Let $1\leq$
$p,q\leq\infty,m\in\mathbb{N},k\in\mathbb{N}_{0}$, with $0\leq k<m$, and let
$a$, $r$ be such that

$\displaystyle 0\leq a\leq 1-\frac{k}{m}$

and

$\displaystyle(1-a)\Big{(}\frac{1}{p}-\frac{m-k}{3}\Big{)}+a\Big{(}\frac{1}{q}+\frac{k}{3}\Big{)}=\frac{1}{r}\in[0,1].$

Then there exists a constant $C=C(m,p,q,a,k)>0$ such that

$\displaystyle\left\|\nabla^{k}f\right\|_{L^{r}}\leq
C\|f\|_{L^{q}}^{a}\left\|\nabla^{m}f\right\|_{L^{p}}^{1-a}$ (2.6)

for every $f\in
L^{q}\left(\mathbb{R}^{3}\right)\cap\dot{W}^{m,p}\left(\mathbb{R}^{3}\right)$(where
the homogeneous Sobolev space $\dot{W}^{m,p}\left(\mathbb{R}^{3}\right)$ is
the space of all functions $f\in L_{\text{loc}}^{1}(\mathbb{R}^{3})$ whose
$\alpha$-th weak derivative $\partial^{\alpha}f$ belongs to
$L^{p}(\mathbb{R}^{3})$ with $|\alpha|=m$), with the following exceptional
cases:

(i) If $k=0,mp<3$, and $q=\infty$, we assume that $f$ vanishes at infinity.

(ii) If $1<p<\infty$ and $m-k-\frac{3}{p}$ is a nonnegative integer, then
(2.6) only holds for $0<a\leq 1-\frac{k}{m}$.

We also need the following embedding and interpolation estimates on Besov
spaces, cf. [41].

######  Lemma 2.4.

(1) Let $0<s\leq\frac{3}{2}$ and $1\leq p<2$ satisfying
$\frac{1}{2}+\frac{s}{3}=\frac{1}{p}$, then there exists a constant
$C=C(s,p)>0$ such that

$\displaystyle\|f\|_{\dot{B}_{2,\infty}^{-s}}\leq C\|f\|_{L^{p}}.$ (2.7)

(2) Let $s>0$ and $\ell\geq 0$, then there exists a constant $C=C(s,\ell)>0$
such that

$\displaystyle\|\nabla^{\ell}f\|_{L^{2}}\leq
C\|\nabla^{\ell+1}f\|_{L^{2}}^{1-\alpha}\|f\|_{\dot{B}_{2,\infty}^{-s}}^{\alpha},\
\ \text{with}\ \alpha=\frac{1}{\ell+1+s}.$ (2.8)

Finally, the following Beale–Kato–Majda type inequality will be used to
estimate $\|\nabla\varrho\|_{L^{6}}$ and $\|\nabla{\bf u}\|_{L^{\infty}}$,
whose proof can be found in [25, Lemma 2.3].

######  Lemma 2.5.

Suppose that $3<q<\infty$, then there is a positive constant $C$ depending
only on $q$ such that, for any $\nabla{\bf u}\in L^{2}(\mathbb{R}^{3})\cap
D^{1,q}(\mathbb{R}^{3})$,

$\displaystyle\|\nabla{\bf u}\|_{L^{\infty}}\leq
C\big{(}\|\operatorname{div}{\bf u}\|_{L^{\infty}}+\|{\bf
w}\|_{L^{\infty}}\big{)}\ln\big{(}\text{e}+\|\nabla^{2}{\bf
u}\|_{L^{q}}\big{)}+C\|\nabla{\bf u}\|_{L^{2}}+C.$ (2.9)

##  3 A priori estimates

In this section, we will establish some necessary a priori bounds for
classical solutions $(\rho,\mathbf{u})$ to the Cauchy problem (1.1)–(1.3) with
$\tilde{\rho}=1$. In what follows, let $\varrho\overset{\text{def}}{=}\rho-1$
and $\sigma\overset{\text{def}}{=}P(\rho)-P(\tilde{\rho})=\rho^{\gamma}-1$.
Noting that

$\displaystyle\sigma=\varrho\int_{0}^{1}\gamma(\theta\rho+(1-\theta))^{\gamma-1}\mathrm{d}\theta,$

hence, if $\rho\leq M$, one has that

$\displaystyle|\sigma|\sim|\varrho|.$ (3.1)

Setting

$\displaystyle
H(\rho)=\frac{1}{2\mu+\lambda}\Big{[}\frac{\gamma^{2}}{2(2\gamma-1)}(\rho-1)^{2}-\Big{(}\frac{\gamma-1}{2(2\gamma-1)}\rho^{\gamma}+\frac{\gamma(\gamma-1)}{2(2\gamma-1)}\rho-\frac{\gamma^{2}+2\gamma-1}{2(2\gamma-1)}\Big{)}G(\rho)\Big{]},$

due to (1.7) and $\rho\leq M$, we get that

$\displaystyle|H(\rho)|\leq CG(\rho).$ (3.2)

The following Proposition 3.1 deals with the uniform-in-time bounds and the
dissipation inequality, whose proof can be found in [18, Proposition 2.2].

######  Proposition 3.1.

Let $2\mu>\lambda$, $\tilde{\rho}=1$, and $(\rho,{\bf u})$ be a classical
solution of the Cauchy problem (1.1)–(1.3) with $0\leq\rho\leq\bar{\rho}$,
then there exist constants $A_{i}\ (i=1,\cdots,6)$ such that

$\displaystyle A_{1}$ $\displaystyle\|(\rho^{\frac{1}{4}}{\bf
u})(t)\|^{4}_{L^{4}}+A_{2}\Big{(}\mu\|\nabla{\bf
u}(t)\|_{L^{2}}^{2}+(\mu+\lambda)\|\operatorname{div}{\bf
u}(t)\|_{L^{2}}^{2}-\int(\sigma\operatorname{div}{\bf u})(t)\mathrm{d}{\bf
x}+\int H(\rho)(t)\mathrm{d}{\bf x}\Big{)}$
$\displaystyle+A_{3}\|\sigma(t)\|_{L^{6}}^{2}+A_{4}\Big{(}\int
G(\rho)(t)\mathrm{d}{\bf x}+\|\sqrt{\rho}{\bf
u}(t)\|_{L^{2}}^{2}\Big{)}+A_{5}\|(\sqrt{\rho}\dot{\bf u})(t)\|_{L^{2}}^{2}$
$\displaystyle+A_{6}\int_{0}^{t}\big{(}\||{\bf u}||\nabla{\bf
u}|\|_{L^{2}}^{2}+\|\nabla{\bf u}\|_{L^{2}}^{2}+\|\sqrt{\rho}{\bf
u}_{t}\|_{L^{2}}^{2}+\|\nabla{\bf w}\|_{L^{2}}^{2}+\|\nabla
F\|_{L^{2}}^{2}+\|\sigma\|_{L^{6}}^{2}$ $\displaystyle+\|\nabla{\bf
u}\|^{2}_{L^{6}}+\|\nabla\dot{\bf u}\|^{2}_{L^{2}}+\|F\|_{W^{1,6}}^{2}+\|{\bf
w}\|_{W^{1,6}}^{2}\big{)}\mathrm{d}\tau\leq C(M,K),$ (3.3)

where

$\displaystyle A_{1}$ $\displaystyle\|\rho^{\frac{1}{4}}{\bf
u}\|^{4}_{L^{4}}+A_{2}\Big{(}\mu\|\nabla{\bf
u}\|_{L^{2}}^{2}+(\mu+\lambda)\|\operatorname{div}{\bf
u}\|_{L^{2}}^{2}-\int\sigma\operatorname{div}{\bf u}\mathrm{d}{\bf x}+\int
H(\rho)\mathrm{d}{\bf x}\Big{)}$
$\displaystyle+A_{3}\|\sigma\|_{L^{6}}^{2}+A_{4}\Big{(}\int
G(\rho)\mathrm{d}{\bf x}+\|\sqrt{\rho}{\bf
u}\|_{L^{2}}^{2}\Big{)}+A_{5}\|\sqrt{\rho}\dot{\bf u}\|_{L^{2}}^{2}$
$\displaystyle\sim$ $\displaystyle\ \|\rho-1\|_{L^{2}}+\|\sqrt{\rho}{\bf
u}\|_{L^{2}}^{2}+\|\rho^{\frac{1}{4}}{\bf u}\|^{4}_{L^{4}}+\|\nabla{\bf
u}\|_{L^{2}}^{2}+\|\rho-1\|_{L^{6}}+\|\sqrt{\rho}\dot{\bf u}\|_{L^{2}}^{2}.$
(3.4)

We have the following uniform positive lower bound of $\rho$.

######  Lemma 3.1.

Under the assumptions of Theorem 1.1, there exists a positive constant $c_{1}$
independent of $t$ such that

$\displaystyle\inf\limits_{{\bf x}\in\mathbb{R}^{3}}\rho({\bf x},t)\geq
c_{1},$ (3.5)

and

$\displaystyle\lim\limits_{t\rightarrow\infty}\|\rho(\cdot,t)-1\|_{L^{\infty}}=0.$
(3.6)

###### Proof.

Let ${\bf y}\in\mathbb{R}^{3}$ and define the corresponding particle path
${\bf x}(t,{\bf y})$ by

$\displaystyle\begin{cases}\frac{\mathrm{d}{\bf x}(t,{\bf
y})}{\mathrm{d}t}={\bf u}({\bf x}(t,{\bf y}),t),\\\ {\bf x}(t_{0},{\bf
y})={\bf y},\end{cases}$

then we rewrite the mass equation (1.1)1 as

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\ln\rho=-\operatorname{div}{\bf
u},$

which together with the definition of $F$ in (1.4), (1.8), and (3.1) implies
that

$\displaystyle\rho(t,{\bf
x})\geq\rho_{0}\text{e}^{{-\int_{0}^{t}}\|\operatorname{div}{\bf
u}\|_{L^{\infty}}\mathrm{d}\tau}\geq
c_{0}\text{e}^{{-C\int_{0}^{t}}\|(\sigma,F)\|_{L^{\infty}}\mathrm{d}\tau}\geq
c_{0}\text{e}^{-C(t+1)}.$ (3.7)

Moreover, due to (3.1), for any $\delta>0$, there exists a constant $T_{1}>0$
such that, for any $t\geq T_{1}$,

$\displaystyle\int_{t}^{\infty}\big{(}\|\nabla{\bf
u}\|_{L^{2}}^{2}+\|\sqrt{\rho}{\bf u}_{t}\|_{L^{2}}^{2}+\|\nabla{\bf
w}\|_{L^{2}}^{2}+\|\nabla F\|_{L^{2}}^{2}+\|\sigma\|_{L^{6}}^{2}+\|\nabla{\bf
u}\|^{2}_{L^{6}}+\|\nabla\dot{\bf u}\|^{2}_{L^{2}}$ $\displaystyle\ \ \
+\|F\|_{W^{1,6}}^{2}+\|{\bf
w}\|_{W^{1,6}}^{2}\big{)}\mathrm{d}\tau\leq\delta.$ (3.8)

Next, we argue by contradiction to show (3.5). Indeed, if (3.5) is false, then
there exists a time $T_{2}\geq T_{1}$ such that

$\displaystyle 0<\tilde{c}\triangleq\rho({\bf
x}(T_{2}),T_{2})\leq\Big{(}\frac{1}{2}\Big{)}^{\frac{1}{\gamma}}.$ (3.9)

Choosing a minimal value of $T_{3}>T_{2}$ such that $\rho({\bf
x}(T_{3}),T_{3})=\frac{\tilde{c}}{2}$, so one has that

$\displaystyle\rho({\bf x}(t),t)\in[\tilde{c}/2,\tilde{c}]\ \ \text{for}\ \
t\in[T_{2},T_{3}].$ (3.10)

By the definition of $F$ in (1.4), we rewrite (1.1)1 as

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}(\rho-1)+\frac{1}{2\mu+\lambda}\rho(\rho^{\gamma}-1)=-\frac{1}{2\mu+\lambda}\rho
F.$ (3.11)

Integrating (3.11) along particle trajectories from $T_{2}$ to $T_{3}$,
abbreviating $\rho({\bf x}(t),t)$ by $\rho(t)$ for convenience, we get that

$\displaystyle(2\mu+\lambda)(\rho-1)\Big{|}^{T_{3}}_{T_{2}}+\int^{T_{3}}_{T_{2}}\rho(\rho^{\gamma}-1)\mathrm{d}\tau=-\int^{T_{3}}_{T_{2}}\rho
F({\bf x}(\tau),\tau)\mathrm{d}\tau.$ (3.12)

By (3.9) and (3.10), one has that

$\displaystyle(2\mu+\lambda)(\rho-1)\Big{|}^{T_{3}}_{T_{2}}+\int^{T_{3}}_{T_{2}}\rho(\rho^{\gamma}-1)\mathrm{d}\tau$
$\displaystyle\leq-\frac{\tilde{c}(2\mu+\lambda)}{2}+\int^{T_{3}}_{T_{2}}\frac{\tilde{c}}{2}\Big{(}\frac{1}{2}-1\Big{)}\mathrm{d}\tau$
$\displaystyle=-\frac{\tilde{c}(2\mu+\lambda)}{2}-\frac{\tilde{c}}{4}(T_{3}-T_{2}).$
(3.13)

In virtue of (3) and (3.10), we deduce that

$\displaystyle-\int^{T_{3}}_{T_{2}}\rho F({\bf
x}(\tau),\tau)\mathrm{d}\tau\geq-\tilde{c}\int^{T_{3}}_{T_{2}}\|F\|_{L^{\infty}}\mathrm{d}\tau\geq-C\tilde{c}\int^{T_{3}}_{T_{2}}\|F\|_{W^{1,6}}\mathrm{d}\tau\geq-C\tilde{c}\delta^{\frac{1}{2}}(T_{3}-T_{2})^{\frac{1}{2}}.$
(3.14)

As a result, substituting (3) and (3.14) into (3.12), we derive that

$\displaystyle-\frac{\tilde{c}(2\mu+\lambda)}{2}-\frac{\tilde{c}}{4}(T_{3}-T_{2})\geq-C\tilde{c}\delta^{\frac{1}{2}}(T_{3}-T_{2})^{\frac{1}{2}},$

which is impossible if $\delta$ is small enough. Thus, the desired (3.5)
holds.

Now we turn to prove (3.6). Multiplying (3.11) by $\rho-1$, we arrive at

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}(\rho-1)^{2}+\frac{1}{2\mu+\lambda}\rho(\rho^{\gamma}-1)(\rho-1)=-\frac{1}{2\mu+\lambda}\rho(\rho-1)F.$

By (3.1) and (3.5), there exists a positive constant $c_{2}$ such that

$\displaystyle\frac{1}{2\mu+\lambda}\rho(\rho^{\gamma}-1)(\rho-1)\geq
2c_{2}(\rho-1)^{2}.$

Then, by Cauchy–Schwarz inequality, we conclude that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}(\rho-1)^{2}+c_{2}(\rho-1)^{2}\leq
C\|F\|_{L^{\infty}}^{2}\leq C\|F\|_{W^{1,6}}^{2},$ (3.15)

which together with Gronwall inequality shows that

$\displaystyle(\rho(t)-1)^{2}$
$\displaystyle\leq\text{e}^{-c_{2}t}(\rho(0)-1)^{2}+C\int_{0}^{t}\text{e}^{-c_{2}(t-\tau)}\|F(\tau)\|_{W^{1,6}}^{2}\mathrm{d}\tau$
$\displaystyle\leq\text{e}^{-c_{2}t}(\rho(0)-1)^{2}+C\int_{0}^{\frac{t}{2}}\text{e}^{-c_{2}(t-\tau)}\|F(\tau)\|_{W^{1,6}}^{2}\mathrm{d}\tau+C\int_{\frac{t}{2}}^{t}\text{e}^{-c_{2}(t-\tau)}\|F(\tau)\|_{W^{1,6}}^{2}\mathrm{d}\tau$
$\displaystyle\leq\text{e}^{-c_{2}t}(\rho(0)-1)^{2}+C\text{e}^{-\frac{c_{2}t}{2}}\int_{0}^{\frac{t}{2}}\|F(\tau)\|_{W^{1,6}}^{2}\mathrm{d}\tau+C\int_{\frac{t}{2}}^{t}\|F(\tau)\|_{W^{1,6}}^{2}\mathrm{d}\tau.$

Letting $t\rightarrow\infty$ and using (3.1), we obtain the desired (3.6). ∎

Next, we will prove the following key a priori estimates on $\nabla\varrho$,
which plays a crucial role in our analysis.

######  Lemma 3.2.

Under the assumptions of Theorem 1.1, there exists a positive constant $C$
depending on the initial data $(\rho_{0},{\bf u}_{0})$, $c_{1}$, and $M$ such
that

$\displaystyle\|\varrho\|_{L^{\infty}((0,+\infty);W^{1,6})\cap
L^{2}((0,+\infty);W^{1,6})}+\|\nabla{\bf
u}\|_{L^{2}((0,+\infty);L^{\infty})}\leq C,$ (3.16)

and

$\displaystyle\lim\limits_{t\rightarrow\infty}\|\nabla\varrho(\cdot,t)\|_{L^{6}}=0.$
(3.17)

###### Proof.

By (1.1)1 and the definition of $F$ in (1.4), it is not difficult to derive
that

$\displaystyle\partial_{t}\nabla\varrho+\frac{\gamma}{2\mu+\lambda}\rho^{\gamma}\nabla\varrho=-\frac{\rho^{\gamma}-1}{2\mu+\lambda}\nabla\varrho-\frac{1}{2\mu+\lambda}\nabla\varrho
F-\frac{1}{2\mu+\lambda}\rho\nabla F-\nabla{\bf u}\cdot\nabla\varrho-{\bf
u}\cdot\nabla\nabla\varrho.$ (3.18)

Multiplying (3.18) by $6|\nabla\varrho|^{4}\nabla\varrho$ and integrating the
resulting equation on $\mathbb{R}^{3}$, one can deduce that

$\displaystyle\partial_{t}\|\nabla\varrho\|_{L^{6}}^{6}+\frac{6\gamma}{2\mu+\lambda}\int\rho^{\gamma}|\nabla\varrho|^{6}\mathrm{d}{\bf
x}$
$\displaystyle=-\int\frac{6(\rho^{\gamma}-1)}{2\mu+\lambda}|\nabla\varrho|^{6}\mathrm{d}{\bf
x}-\int\frac{6}{2\mu+\lambda}F|\nabla\varrho|^{6}\mathrm{d}{\bf
x}-\int\frac{6}{2\mu+\lambda}\rho\nabla
F|\nabla\varrho|^{4}\nabla\varrho\mathrm{d}{\bf x}$ $\displaystyle\ \ \
-6\int\nabla{\bf
u}\cdot\nabla\varrho|\nabla\varrho|^{4}\nabla\varrho\mathrm{d}{\bf x}-\int{\bf
u}\cdot\nabla|\nabla\varrho|^{6}\mathrm{d}{\bf x}$ $\displaystyle\leq
C\|\varrho\|_{L^{\infty}}\|\nabla\varrho\|_{L^{6}}^{6}+C\|F\|_{L^{\infty}}\|\nabla\varrho\|_{L^{6}}^{6}+C\|\nabla
F\|_{L^{6}}\|\nabla\varrho\|_{L^{6}}^{5}$ $\displaystyle\ \ \ +C\|\nabla{\bf
u}\|_{L^{\infty}}\|\nabla\varrho\|_{L^{6}}^{6}+C\|\operatorname{div}{\bf
u}\|_{L^{\infty}}\|\nabla\varrho\|_{L^{6}}^{6}$ $\displaystyle\leq
C\|\varrho\|_{L^{\infty}}\|\nabla\varrho\|_{L^{6}}^{6}+C\|F\|_{W^{1,6}}\|\nabla\varrho\|_{L^{6}}^{5}+C\|\nabla{\bf
u}\|_{L^{\infty}}\|\nabla\varrho\|_{L^{6}}^{6}.$ (3.19)

By (1.1)2, we see that

$\mu\Delta{\bf u}+(\mu+\lambda)\nabla\operatorname{div}{\bf u}=\rho\dot{\bf
u}+\nabla P(\rho).$

Thus, for any $1<p<\infty$, it follows from the standard $L^{p}$-estimate for
the elliptic system that

$\displaystyle\|\nabla^{2}{\bf u}\|_{L^{p}}\leq C(\|\rho\dot{\bf
u}\|_{L^{p}}+\|\nabla P\|_{L^{p}}),$ (3.20)

which together with (2.9) yields that

$\displaystyle\|\nabla{\bf u}\|_{L^{\infty}}$ $\displaystyle\leq
C\big{(}\|\operatorname{div}{\bf u}\|_{L^{\infty}}+\|{\bf
w}\|_{L^{\infty}}\big{)}\ln(\text{e}+\|\nabla^{2}{\bf
u}\|_{L^{6}})+C\|\nabla{\bf u}\|_{L^{2}}+C$ $\displaystyle\leq
C\big{(}\|\operatorname{div}{\bf u}\|_{L^{\infty}}+\|{\bf
w}\|_{L^{\infty}}\big{)}\ln(\text{e}+\|\dot{\bf
u}\|_{L^{6}}+\|\nabla\varrho\|_{L^{6}})+C\|\nabla{\bf u}\|_{L^{2}}+C$
$\displaystyle\leq C\big{(}\|\operatorname{div}{\bf u}\|_{L^{\infty}}+\|{\bf
w}\|_{L^{\infty}}\big{)}\ln(\text{e}+\|\nabla\dot{\bf u}\|_{L^{2}})$
$\displaystyle\ \ \ +C\big{(}\|\operatorname{div}{\bf u}\|_{L^{\infty}}+\|{\bf
w}\|_{L^{\infty}}\big{)}\ln(\text{e}+\|\nabla\varrho\|_{L^{6}})+C\|\nabla{\bf
u}\|_{L^{2}}+C.$ (3.21)

Setting

$\displaystyle f(t)\triangleq\text{e}+\|\nabla\varrho\|_{L^{6}}^{6},\
g(t)\triangleq 1+\big{(}\|\operatorname{div}{\bf u}\|_{L^{\infty}}+\|{\bf
w}\|_{L^{\infty}}\big{)}\ln(\text{e}+\|\nabla\dot{\bf
u}\|_{L^{2}})+\|\nabla{\bf
u}\|_{L^{2}}+\|\varrho\|_{L^{\infty}}+\|F\|_{W^{1,6}},$

then one obtains from (3) and (3) that

$\displaystyle f^{\prime}(t)\leq Cg(t)f(t)+Cg(t)f(t)\ln f(t),$

which leads to

$\displaystyle(\ln f(t))^{\prime}\leq Cg(t)+Cg(t)\ln f(t).$ (3.22)

It follows from (1.4), (1.8), and (3.1) that

$\displaystyle\int_{0}^{T}g(t)\mathrm{d}t\leq$ $\displaystyle
C\int_{0}^{T}\big{(}1+\|\operatorname{div}{\bf u}\|^{2}_{L^{\infty}}+\|{\bf
w}\|^{2}_{L^{\infty}}+\|\nabla\dot{\bf u}\|_{L^{2}}^{2}+\|\nabla{\bf
u}\|_{L^{2}}+\|\varrho\|_{L^{\infty}}+\|F\|_{W^{1,6}}\big{)}\mathrm{d}t$
$\displaystyle\leq$ $\displaystyle
C(T)+C\int_{0}^{T}\big{(}\|F\|_{L^{\infty}}^{2}+\|{\bf
w}\|_{L^{\infty}}^{2}+\|\sigma\|_{L^{\infty}}^{2}\big{)}\mathrm{d}t$
$\displaystyle\leq$ $\displaystyle
C(T)+C\int_{0}^{T}\big{(}\|F\|_{W^{1,6}}^{2}+\|{\bf
w}\|_{W^{1,6}}^{2}\big{)}\mathrm{d}t$ $\displaystyle\leq$ $\displaystyle
C(T),$

which together with (3.22) and Gronwall inequality shows that

$\displaystyle\sup\limits_{0\leq t\leq T}f(t)\leq C(T).$

Consequently, one has that

$\displaystyle\sup\limits_{0\leq t\leq T}\|\nabla\rho\|_{L^{6}}\leq C(T).$
(3.23)

Combining (3.1) and (3.6), we arrive at

$\displaystyle\lim\limits_{t\rightarrow\infty}\big{(}\|\rho-1\|_{L^{p}}+\|\rho-1\|_{L^{\infty}}\big{)}=0,\
\ \text{for any}\ p>2.$

Therefore, for any $\delta>0$, there exists a constant $T_{4}>T_{1}$ such
that, for any $t\geq T_{4}$ and $p>2$,

$\displaystyle\|(\rho-1)(t)\|_{L^{p}}+\|(\rho-1)(t)\|_{L^{\infty}}\leq\delta^{2}.$
(3.24)

Now we claim that there exists a positive constant $C$ independent of $t$ such
that, for any $t\geq 0$,

$\displaystyle\|\nabla\varrho(t)\|_{L^{6}}\leq C.$ (3.25)

Indeed, if (3.25) is false, there exists a time $T_{5}>T_{4}$ such that
$\|\nabla\varrho(T_{5})\|_{L^{6}}=\frac{1}{\delta^{\kappa}}$ for a positive
constant $\kappa$. We choose a minimal value of $T_{7}>T_{5}$ such that
$\|\nabla\varrho(T_{7})\|_{L^{6}}=\frac{2}{\delta^{\kappa}}$, then choose a
maximal value of $T_{6}<T_{7}$ such that
$\|\nabla\varrho(T_{6})\|_{L^{6}}=\frac{1}{\delta^{\kappa}}$. Thus, one has
that

$\displaystyle\|\nabla\varrho(t)\|_{L^{6}}\in\Big{[}\frac{1}{\delta^{\kappa}},\frac{2}{\delta^{\kappa}}\Big{]}\
\ \text{for}\ \ t\in[T_{6},T_{7}].$ (3.26)

Therefore, for any $t\in[T_{6},T_{7}]$, one obtains from Bernstein inequality
and the interpolation inequality that

$\displaystyle\|\nabla^{2}(-\Delta)^{-1}\sigma\|_{L^{\infty}}\leq$
$\displaystyle\Big{\|}\nabla^{2}(-\Delta)^{-1}\sum\limits_{j\leq
N}\Delta_{j}\sigma\Big{\|}_{L^{\infty}}+\Big{\|}\nabla^{2}(-\Delta)^{-1}\sum\limits_{j\geq
N}\Delta_{j}\sigma\Big{\|}_{L^{\infty}}$ $\displaystyle\leq$ $\displaystyle
C2^{\frac{3N}{p}}\|\nabla^{2}(-\Delta)^{-1}\sigma\|_{L^{p}}+C\sum\limits_{j\geq
N}2^{-\frac{j}{2}}\|\nabla\sigma\|_{L^{6}}$ $\displaystyle\leq$ $\displaystyle
C\big{(}2^{\frac{3N}{p}}\|\varrho\|_{L^{p}}+2^{-\frac{N}{2}}\|\nabla\varrho\|_{L^{6}}\big{)}$
$\displaystyle\leq$ $\displaystyle C\delta,$ (3.27)

provided that $N\geq\log_{2}\frac{1}{\delta^{2(\kappa+1)}}$ and $p=3N$. This
implies that

$\displaystyle\|\nabla{\bf u}\|_{L^{\infty}}\leq$ $\displaystyle
C\big{(}\|\nabla\mathcal{P}{\bf u}\|_{L^{\infty}}+\|\nabla\mathcal{Q}{\bf
u}\|_{L^{\infty}}\big{)}$ $\displaystyle\leq$ $\displaystyle
C\big{(}\|\nabla\mathcal{P}{\bf
u}\|_{W^{1,6}}+\|\nabla^{2}(-\Delta)^{-1}F\|_{L^{\infty}}+\|\nabla^{2}(-\Delta)^{-1}\sigma\|_{L^{\infty}}\big{)}$
$\displaystyle\leq$ $\displaystyle C\big{(}\|{\bf
w}\|_{W^{1,6}}+\|F\|_{W^{1,6}}+\delta\big{)},$ (3.28)

where we have used

$\displaystyle\mathcal{Q}{\bf
u}=-\frac{\nabla(-\Delta)^{-1}(F+\sigma)}{2\mu+\lambda}.$

Inserting (3) into (3), we see that

$\displaystyle\partial_{t}\|\nabla\varrho\|_{L^{6}}^{6}+\frac{6\gamma}{2\mu+\lambda}\int\rho^{\gamma}|\nabla\varrho|^{6}\mathrm{d}{\bf
x}$ $\displaystyle\leq
C\delta\|\nabla\varrho\|_{L^{6}}^{6}+C\|F\|_{W^{1,6}}\|\nabla\varrho\|_{L^{6}}^{5}+C\big{(}\|{\bf
w}\|_{W^{1,6}}+\|F\|_{W^{1,6}}\big{)}\|\nabla\varrho\|_{L^{6}}^{6}.$

Thus, if $\delta$ is suitably small, we arrive at

$\displaystyle\partial_{t}\|\nabla\varrho\|_{L^{6}}^{6}+\frac{3\gamma}{2\mu+\lambda}\int\rho^{\gamma}|\nabla\varrho|^{6}\mathrm{d}{\bf
x}\leq C\|F\|_{W^{1,6}}\|\nabla\varrho\|_{L^{6}}^{5}+C\big{(}\|{\bf
w}\|_{W^{1,6}}+\|F\|_{W^{1,6}}\big{)}\|\nabla\varrho\|_{L^{6}}^{6}.$

Integrating the above inequality with respect to $t$ from $T_{6}$ to $T_{7}$
and using (3), one finds that

$\displaystyle\frac{1}{\delta^{6\kappa}}+\frac{3\gamma}{(2\mu+\lambda)2^{\gamma}}\frac{1}{\delta^{6\kappa}}(T_{7}-T_{6})$
$\displaystyle\leq C\left(\int_{T_{6}}^{T_{7}}(\|{\bf
w}\|_{W^{1,6}}^{2}+\|F\|_{W^{1,6}}^{2})\mathrm{d}t\right)^{\frac{1}{2}}(T_{7}-T_{6})^{\frac{1}{2}}\sup\limits_{T_{6}\leq
t\leq
T_{7}}\big{(}\|\nabla\varrho\|_{L^{6}}^{5}+\|\nabla\varrho\|_{L^{6}}^{6}\big{)}$
$\displaystyle\leq
C\frac{\delta^{\frac{1}{2}}}{\delta^{6\kappa}}(T_{7}-T_{6})^{\frac{1}{2}},$

which is impossible when $\delta$ is small enough and (3.25) follows
immediately.

For any $t\geq T_{4}$, substituting (3.24) and (3.25) into (3) gives that

$\displaystyle\|\nabla^{2}(-\Delta)^{-1}\sigma\|_{L^{\infty}}\leq
C\left(2^{\frac{3N}{p}}\|\varrho\|_{L^{p}}+2^{-\frac{N}{2}}\|\nabla\varrho\|_{L^{6}}\right)\leq
C_{1}2^{\frac{3N}{p}}\delta^{2}+C_{2}2^{-\frac{N}{2}}\leq\delta,$

provided that $\delta<\frac{1}{2C_{1}}$, $N\geq
2\log_{2}\frac{C_{2}}{\delta}$, and $p=3N$. Plugging the above estimate into
(3) yields that

$\displaystyle\|\nabla{\bf u}\|_{L^{\infty}}\leq C\big{(}\|{\bf
w}\|_{W^{1,6}}+\|F\|_{W^{1,6}}+\delta\big{)}.$ (3.29)

Inserting (3.29) into (3), we derive that

$\displaystyle\partial_{t}\|\nabla\varrho\|_{L^{6}}^{6}+\frac{6\gamma}{2\mu+\lambda}\int\rho^{\gamma}|\nabla\varrho|^{6}\mathrm{d}{\bf
x}$ $\displaystyle\leq
C\delta\|\nabla\varrho\|_{L^{6}}^{6}+C\|F\|_{W^{1,6}}\|\nabla\varrho\|_{L^{6}}^{5}+C\|{\bf
w}\|_{W^{1,6}}\|\nabla\varrho\|_{L^{6}}^{6}+C\|F\|_{W^{1,6}}\|\nabla\varrho\|_{L^{6}}^{6},$

which together with (3.25) implies that there exists a positive constant
$c_{3}$ such that

$\displaystyle\partial_{t}\|\nabla\varrho\|_{L^{6}}+c_{3}\|\nabla\varrho\|_{L^{6}}$
$\displaystyle\leq C\|F\|_{W^{1,6}}+C\|{\bf
w}\|_{W^{1,6}}\|\nabla\varrho\|_{L^{6}}+C\|F\|_{W^{1,6}}\|\nabla\varrho\|_{L^{6}}$
$\displaystyle\leq C\|F\|_{W^{1,6}}+C\|{\bf w}\|_{W^{1,6}}.$ (3.30)

By Gronwall inequality, we have

$\displaystyle\|\nabla\varrho(t)\|_{L^{6}}\leq$
$\displaystyle\text{e}^{-c_{3}(t-T_{4})}\|\nabla\varrho(T_{4})\|_{L^{6}}+C\int_{T_{4}}^{t}\text{e}^{-c_{3}(t-\tau)}\big{(}\|F\|_{W^{1,6}}+\|{\bf
w}\|_{W^{1,6}}\big{)}\mathrm{d}\tau$ $\displaystyle\leq$
$\displaystyle\text{e}^{-c_{3}(t-T_{4})}\|\nabla\varrho(T_{4})\|_{L^{6}}+C\int_{T_{4}}^{\frac{t+T_{4}}{2}}\text{e}^{-c_{3}(t-\tau)}\big{(}\|F\|_{W^{1,6}}+\|{\bf
w}\|_{W^{1,6}}\big{)}\mathrm{d}\tau$
$\displaystyle+C\int^{t}_{\frac{t+T_{4}}{2}}\text{e}^{-c_{3}(t-\tau)}(\|F\|_{W^{1,6}}+\|{\bf
w}\|_{W^{1,6}})\mathrm{d}\tau$ $\displaystyle\leq$
$\displaystyle\text{e}^{-c_{3}(t-T_{4})}\|\nabla\varrho(T_{4})\|_{L^{6}}+C\frac{t-T_{4}}{2}\text{e}^{-\frac{c_{3}(t+T_{4})}{2}}\left(\int_{T_{4}}^{\frac{t+T_{4}}{2}}\big{(}\|F\|_{W^{1,6}}^{2}+\|{\bf
w}\|_{W^{1,6}}^{2}\big{)}\mathrm{d}\tau\right)^{\frac{1}{2}}$
$\displaystyle+C\left(\int^{t}_{\frac{t+T_{4}}{2}}\text{e}^{-2c_{3}(t-\tau)}\mathrm{d}\tau\right)^{\frac{1}{2}}\left(\int^{t}_{\frac{t+T_{4}}{2}}\big{(}\|F\|_{W^{1,6}}^{2}+\|{\bf
w}\|^{2}_{W^{1,6}}\big{)}\mathrm{d}\tau\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\text{e}^{-c_{3}(t-T_{4})}\|\nabla\varrho(T_{4})\|_{L^{6}}+C\text{e}^{-\frac{c_{3}(t+T_{4})}{4}}\delta^{\frac{1}{2}}+C\left(\int^{t}_{\frac{t+T_{4}}{2}}\big{(}\|F\|_{W^{1,6}}^{2}+\|{\bf
w}\|^{2}_{W^{1,6}}\big{)}\mathrm{d}\tau\right)^{\frac{1}{2}},$

which leads to (3.17).

Finally, multiplying (3.30) by $2\|\nabla\varrho\|_{L^{6}}$ and using
Cauchy–Schwarz inequality, we see that

$\displaystyle\partial_{t}$
$\displaystyle\|\nabla\varrho\|_{L^{6}}^{2}+2c_{3}\|\nabla\varrho\|_{L^{6}}^{2}\leq
C\|F\|_{W^{1,6}}^{2}+C\|{\bf w}\|_{W^{1,6}}^{2}.$ (3.31)

Integrating (3.31) with respect to $t$ from $0$ to $\infty$ and using (3.1),
one has that

$\displaystyle\int_{0}^{\infty}\|\nabla\varrho\|_{L^{6}}^{2}\mathrm{d}t\leq
C\int_{0}^{\infty}\left(\|F\|_{W^{1,6}}^{2}+\|{\bf
w}\|_{W^{1,6}}^{2}\right)\mathrm{d}t\leq C,$

and

$\displaystyle\int_{0}^{\infty}\|\nabla{\bf
u}\|_{L^{\infty}}^{2}\mathrm{d}t\leq C\int_{0}^{\infty}\|\nabla{\bf
u}\|_{W^{1,6}}^{2}\mathrm{d}t\leq
C\int_{0}^{\infty}\left(\|F\|_{W^{1,6}}^{2}+\|\sigma\|_{W^{1,6}}^{2}+\|{\bf
w}\|_{W^{1,6}}^{2}\right)\mathrm{d}t\leq C,$

these together with (3.25) imply (3.16). ∎

Next, we will prove the following uniform-in-time bounds on the solution.

######  Lemma 3.3.

Under the assumptions of Theorem 1.1, it holds that

$\sup\limits_{0\leq t<\infty}\left(\|\varrho(t)\|_{H^{3}}^{2}+\|{\bf
u}(t)\|_{H^{3}}^{2}\right)+\int_{0}^{\infty}\left(\|\nabla\varrho(t)\|_{H^{2}}^{2}+\|\nabla{\bf
u}(t)\|_{H^{3}}^{2}\right)\mathrm{d}t\leq C.$ (3.32)

###### Proof.

Multiplying (3.18) by $2\nabla\varrho$ and integrating the resulting equation
over $\mathbb{R}^{3}$, one derives that

$\displaystyle\partial_{t}\|\nabla\varrho\|_{L^{2}}^{2}+\frac{2\gamma}{2\mu+\lambda}\int\rho^{\gamma}|\nabla\varrho|^{2}\mathrm{d}{\bf
x}$
$\displaystyle=-\int\frac{2(\rho^{\gamma}-1)}{2\mu+\lambda}|\nabla\varrho|^{2}\mathrm{d}{\bf
x}-\int\frac{2}{2\mu+\lambda}F|\nabla\varrho|^{2}\mathrm{d}{\bf
x}-\int\frac{2}{2\mu+\lambda}\rho\nabla F\cdot\nabla\varrho\mathrm{d}{\bf x}$
$\displaystyle\ \ \ -6\int\nabla{\bf
u}\cdot\nabla\varrho\cdot\nabla\varrho\mathrm{d}{\bf x}-\int{\bf
u}\cdot\nabla|\nabla\varrho|^{2}\mathrm{d}{\bf x}$ $\displaystyle\leq
C\|\varrho\|_{L^{\infty}}\|\nabla\varrho\|_{L^{2}}^{2}+C\|F\|_{L^{\infty}}\|\nabla\varrho\|_{L^{2}}^{2}+C\|\nabla
F\|_{L^{2}}\|\nabla\varrho\|_{L^{2}}$ $\displaystyle\ \ \ +C\|\nabla{\bf
u}\|_{L^{\infty}}\|\nabla\varrho\|_{L^{2}}^{2}+C\|\operatorname{div}{\bf
u}\|_{L^{\infty}}\|\nabla\varrho\|_{L^{2}}^{2}$ $\displaystyle\leq
C\|\varrho\|_{L^{\infty}}\|\nabla\varrho\|_{L^{2}}^{2}+C\|F\|_{W^{1,6}}\|\nabla\varrho\|_{L^{2}}^{2}+C\|\nabla
F\|_{L^{2}}\|\nabla\varrho\|_{L^{2}}+C\|\nabla{\bf
u}\|_{L^{\infty}}\|\nabla\varrho\|_{L^{2}}^{2},$

which combined with (3.5) and Cauchy–Schwarz inequality implies that

$\displaystyle\partial_{t}\|\nabla\varrho\|_{L^{2}}^{2}+\frac{\gamma}{2\mu+\lambda}\int\rho^{\gamma}|\nabla\varrho|^{2}\mathrm{d}{\bf
x}$ $\displaystyle\leq
C\|\varrho\|_{L^{\infty}}\|\nabla\varrho\|_{L^{2}}^{2}+C\|F\|_{W^{1,6}}^{2}\|\nabla\varrho\|_{L^{2}}^{2}+C\|\nabla
F\|_{L^{2}}^{2}+C\|\nabla{\bf
u}\|_{L^{\infty}}^{2}\|\nabla\varrho\|_{L^{2}}^{2}.$ (3.33)

This along with Gronwall inequality, (3.1), and (3.16) leads to

$\displaystyle\|\nabla\varrho(T)\|_{L^{2}}\leq C(T).$ (3.34)

For any $\delta>0$, it infers from (3.6) and (3.17) that there exists a
constant $T_{8}>T_{1}$ such that

$\displaystyle\|\varrho(\cdot,t)\|_{L^{\infty}}+\|\nabla\varrho(\cdot,t)\|_{L^{6}}\leq\delta,\
\ \text{for any}\ t\geq T_{8}.$ (3.35)

Therefore, by (3), it holds that, for any $t\geq T_{8}$,

$\displaystyle\partial_{t}\|\nabla\varrho\|_{L^{2}}^{2}+\frac{\gamma}{2(2\mu+\lambda)}\int\rho^{\gamma}|\nabla\varrho|^{2}\mathrm{d}{\bf
x}\leq C\|F\|_{W^{1,6}}^{2}\|\nabla\varrho\|_{L^{2}}^{2}+C\|\nabla
F\|_{L^{2}}^{2}+C\|\nabla{\bf
u}\|_{L^{\infty}}^{2}\|\nabla\varrho\|_{L^{2}}^{2},$ (3.36)

which together with Gronwall inequality, (3.1), (3.16), and (3.34) yields that

$\displaystyle\|\nabla\varrho(t)\|_{L^{2}}\leq C,\ \ \text{for any}\ t\geq 0.$

This combined with (3.1), (3.5) and (3.36) implies that

$\displaystyle\sup\limits_{0\leq
t<\infty}\|\nabla\varrho(t)\|_{L^{2}}^{2}+\int_{0}^{\infty}\|\nabla\varrho(t)\|_{L^{2}}^{2}\mathrm{d}t\leq
C,$ (3.37) $\int_{0}^{\infty}\|\nabla^{2}{\bf u}\|_{L^{2}}^{2}\mathrm{d}t\leq
C\int_{0}^{\infty}\left(\|\nabla F\|_{L^{2}}^{2}+\|\nabla{\bf
w}\|_{L^{2}}^{2}+\|\nabla\varrho\|_{L^{2}}^{2}\right)\mathrm{d}t\leq C,$
(3.38)

and

$\displaystyle\sup\limits_{0\leq t<\infty}\|\nabla^{2}{\bf
u}(t)\|_{L^{2}}^{2}\leq C\sup\limits_{0\leq
t<\infty}\left(\|\sqrt{\rho}\dot{\bf
u}\|_{L^{2}}^{2}+\|\nabla\varrho\|_{L^{2}}^{2}\right)\leq C.$ (3.39)

Applying the operator $\nabla$ to (3.18), one gets that

$\displaystyle\partial_{t}\nabla^{2}\varrho+\frac{\gamma}{2\mu+\lambda}\rho^{\gamma}\nabla^{2}\varrho=$
$\displaystyle-\frac{2\gamma^{2}}{2\mu+\lambda}\rho^{\gamma-1}\nabla\varrho\nabla\varrho-\frac{\rho^{\gamma}-1}{2\mu+\lambda}\nabla^{2}\varrho-\frac{1}{2\mu+\lambda}\nabla^{2}\varrho
F-\frac{2}{2\mu+\lambda}\nabla\varrho\nabla F$
$\displaystyle-\frac{1}{2\mu+\lambda}\rho\nabla^{2}F-\nabla^{2}{\bf
u}\cdot\nabla\varrho-2\nabla{\bf u}\cdot\nabla\nabla\varrho-{\bf
u}\cdot\nabla\nabla^{2}\varrho.$ (3.40)

Multiplying (3) by $2\nabla^{2}\varrho$ and integrating the resulting equation
over $\mathbb{R}^{3}$, we then infer from (3.5) and Cauchy–Schwarz inequality
that

$\displaystyle\partial_{t}\|\nabla^{2}\varrho\|_{L^{2}}^{2}+\frac{\gamma}{2\mu+\lambda}\int\rho^{\gamma}|\nabla^{2}\varrho|^{2}\mathrm{d}{\bf
x}$ $\displaystyle\leq
C\|\varrho\|_{L^{\infty}}\|\nabla^{2}\varrho\|_{L^{2}}^{2}+C\|\nabla\varrho\|_{L^{3}}^{2}\|\nabla\varrho\|_{L^{6}}^{2}+C\|F\|_{L^{\infty}}^{2}\|\nabla^{2}\varrho\|_{L^{2}}^{2}+C\|\nabla
F\|_{L^{6}}^{2}\|\nabla\varrho\|_{L^{3}}^{2}$
$\displaystyle\quad+C\|\nabla^{2}F\|_{L^{2}}^{2}+C\|\nabla^{2}{\bf
u}\|_{L^{6}}^{2}\|\nabla\varrho\|_{L^{3}}^{2}+C\|\nabla{\bf
u}\|_{L^{\infty}}^{2}\|\nabla^{2}\varrho\|_{L^{2}}^{2}.$ (3.41)

Noting that

$\displaystyle\|\nabla^{2}{\bf u}\|_{L^{6}}^{2}\leq C\big{(}\|\rho\dot{\bf
u}\|_{L^{6}}^{2}+\|\nabla\sigma\|_{L^{6}}^{2}\big{)}\leq
C\big{(}\|\nabla\dot{\bf
u}\|_{L^{2}}^{2}+\|\nabla\varrho\|_{L^{6}}^{2}\big{)},$ (3.42)
$\displaystyle\|\nabla^{2}F\|_{L^{2}}^{2}\leq C\|\nabla(\rho\dot{\bf
u})\|_{L^{2}}^{2}\leq C\|\nabla\dot{\bf
u}\|_{L^{2}}^{2}+C\|\nabla\varrho\|_{L^{3}}^{2}\|\dot{\bf u}\|_{L^{6}}^{2}\leq
C\|\nabla\dot{\bf u}\|_{L^{2}}^{2},$ (3.43)

one thus derives from (3) that

$\displaystyle\partial_{t}\|$
$\displaystyle\nabla^{2}\varrho\|_{L^{2}}^{2}+\frac{\gamma}{2\mu+\lambda}\int\rho^{\gamma}|\nabla^{2}\varrho|^{2}\mathrm{d}{\bf
x}$ $\displaystyle\leq$ $\displaystyle
C\|\varrho\|_{L^{\infty}}\|\nabla^{2}\varrho\|_{L^{2}}^{2}+C\|\nabla\varrho\|_{L^{3}}^{2}\|\nabla\varrho\|_{L^{6}}^{2}+C\|F\|_{W^{1,6}}^{2}\|\nabla^{2}\varrho\|_{L^{2}}^{2}+C\|\nabla
F\|_{L^{6}}^{2}\|\nabla\varrho\|_{L^{3}}^{2}$ $\displaystyle+C\|\nabla\dot{\bf
u}\|_{L^{2}}^{2}+C\|\nabla\dot{\bf
u}\|_{L^{2}}^{2}\|\nabla\varrho\|_{L^{3}}^{2}+C\|\nabla{\bf
u}\|_{L^{\infty}}^{2}\|\nabla^{2}\varrho\|_{L^{2}}^{2}.$ (3.44)

This together with Gronwall inequality, (3.1), (3.5), (3.16), and (3.37) shows
that

$\displaystyle\|\nabla^{2}\varrho(T)\|_{L^{2}}\leq C(T).$ (3.45)

By (3.16), (3.35), (3.37), and (3), it holds that, for any $t\geq T_{8}$,

$\displaystyle\partial_{t}\|\nabla^{2}\varrho\|_{L^{2}}^{2}+\frac{\gamma}{2(2\mu+\lambda)}\int|\nabla^{2}\varrho|^{2}\mathrm{d}{\bf
x}$ $\displaystyle\leq
C\|\nabla\varrho\|_{L^{3}}^{2}\|\nabla\varrho\|_{L^{6}}^{2}+C\|F\|_{W^{1,6}}^{2}\|\nabla^{2}\varrho\|_{L^{2}}^{2}+C\|\nabla
F\|_{L^{6}}^{2}$
$\displaystyle\quad+C\|\nabla^{2}F\|_{L^{2}}^{2}+C\|\nabla\dot{\bf
u}\|_{L^{2}}^{2}+C\|\nabla{\bf
u}\|_{L^{\infty}}^{2}\|\nabla^{2}\varrho\|_{L^{2}}^{2}.$ (3.46)

This along with Gronwall inequality, (3.1), (3.16), (3.37), and (3.45)
indicates that

$\displaystyle\|\nabla^{2}\varrho(t)\|_{L^{2}}\leq C,\ \ \text{for any}\ t\geq
0,$

which combined with (3) implies that

$\displaystyle\sup\limits_{0\leq
t<\infty}\|\nabla^{2}\varrho(t)\|_{L^{2}}^{2}+\int_{0}^{\infty}\|\nabla^{2}\varrho(t)\|_{L^{2}}^{2}\mathrm{d}t\leq
C,$ (3.47)

and

$\displaystyle\int_{0}^{\infty}\|\nabla^{3}{\bf
u}\|_{L^{2}}^{2}\mathrm{d}t\leq$ $\displaystyle
C\int_{0}^{\infty}\left(\|\nabla(\rho\dot{\bf
u})\|_{L^{2}}^{2}+\|\nabla^{2}\sigma\|_{L^{2}}^{2}\right)\mathrm{d}t$
$\displaystyle\leq$ $\displaystyle C\int_{0}^{\infty}\left(\|\nabla\dot{\bf
u}\|_{L^{2}}^{2}+\|\nabla\varrho\|_{L^{3}}^{2}\|\dot{\bf
u}\|_{L^{6}}^{2}+\|\nabla^{2}\varrho\|_{L^{2}}^{2}+\|\nabla\varrho\|_{L^{4}}^{4}\right)\mathrm{d}t$
$\displaystyle\leq$ $\displaystyle C.$ (3.48)

The system (1.1) can be equivalently reformulated in the linearized form as
follows

$\displaystyle\begin{cases}\varrho_{t}+\operatorname{div}\mathbf{u}=-\varrho\operatorname{div}\mathbf{u}-\mathbf{u}\cdot\nabla\varrho,\\\
\mathbf{u}_{t}-\mu\Delta\mathbf{u}-(\mu+\lambda)\nabla\operatorname{div}\mathbf{u}+\gamma\nabla\varrho=-\mathbf{u}\cdot\nabla\mathbf{u}-\frac{\mu\varrho}{\rho}\Delta\mathbf{u}-\frac{(\mu+\lambda)\varrho}{\rho}\nabla\operatorname{div}\mathbf{u}-\gamma(\rho^{\gamma-1}-1)\nabla\varrho.\end{cases}$
(3.49)

Applying $\nabla^{3}$ to (3.49)1 and (3.49)2, then multiplying them by
$\gamma\nabla^{3}\varrho$ and $\nabla^{3}{\bf u}$, respectively, and
integrating the resultant equations over $\mathbb{R}^{3}$ and summing up, it
is not difficult to deduce that

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}$
$\displaystyle\left(\gamma\|\nabla^{3}\varrho\|_{L^{2}}^{2}+\|\nabla^{3}{\bf
u}\|_{L^{2}}^{2}\right)+\mu\|\nabla^{4}{\bf
u}\|_{L^{2}}^{2}+(\mu+\lambda)\|\nabla^{3}\operatorname{div}{\bf
u}\|_{L^{2}}^{2}$ $\displaystyle\leq$ $\displaystyle C\|\nabla{\bf
u}\|_{H^{2}}\|\nabla^{3}\varrho\|_{L^{2}}^{2}+C\|\nabla\varrho\|_{H^{1}}\|\nabla^{4}{\bf
u}\|_{L^{2}}\|\nabla^{3}\varrho\|_{L^{2}}+C\|\varrho\|_{L^{\infty}}\|\nabla^{4}{\bf
u}\|_{L^{2}}\|\nabla^{3}\varrho\|_{L^{2}}$ $\displaystyle+C\|\nabla{\bf
u}\|_{H^{2}}^{2}\|\nabla^{4}{\bf
u}\|_{L^{2}}+C\|\varrho\|_{L^{\infty}}\|\nabla^{4}{\bf
u}\|_{L^{2}}^{2}+C\|\nabla^{2}\varrho\|_{L^{2}}\|\nabla^{3}{\bf
u}\|_{L^{2}}^{\frac{1}{2}}\|\nabla^{4}{\bf u}\|_{L^{2}}^{\frac{3}{2}}$
$\displaystyle+C\|\nabla^{3}\varrho\|_{L^{2}}\|\nabla^{2}{\bf
u}\|_{H^{1}}\|\nabla^{4}{\bf
u}\|_{L^{2}}+C\|\nabla\varrho\|_{H^{1}}^{2}\|\nabla^{2}{\bf
u}\|_{H^{1}}\|\nabla^{4}{\bf u}\|_{L^{2}}$
$\displaystyle+C\|\nabla\varrho\|_{H^{1}}\|\nabla^{3}\varrho\|_{L^{2}}^{2}+C\|\nabla\varrho\|_{H^{1}}^{3}\|\nabla^{3}\varrho\|_{L^{2}}.$
(3.50)

Applying $\nabla^{2}$ to (3.49)2 and multiplying the resulting equations by
$\nabla^{3}\varrho$, and then integrating over $\mathbb{R}^{3}$, one has that

$\displaystyle\gamma\|\nabla^{3}\varrho\|_{L^{2}}^{2}=$
$\displaystyle-\int\nabla^{2}{\bf u}_{t}\nabla^{3}\varrho\mathrm{d}{\bf
x}+\int\nabla^{2}\left(\frac{\mu\Delta\mathbf{u}+(\mu+\lambda)\operatorname{div}\mathbf{u}}{\rho}-{\bf
u}\cdot\nabla{\bf u}\right)\nabla^{3}\varrho\mathrm{d}{\bf x}$
$\displaystyle-\gamma\int\nabla^{2}\left((\rho^{\gamma-1}-1)\varrho\right)\nabla^{3}\varrho\mathrm{d}{\bf
x}$ $\displaystyle\leq$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int\nabla^{2}\operatorname{div}{\bf
u}\nabla^{2}\varrho\mathrm{d}{\bf x}+C\|\nabla^{3}{\bf
u}\|_{L^{2}}^{2}+C\|\nabla\varrho\|_{H^{1}}\|\nabla{\bf
u}\|_{H^{1}}\|\nabla^{4}{\bf
u}\|_{L^{2}}+C\|\nabla^{3}\varrho\|_{L^{2}}\|\nabla^{4}{\bf u}\|_{L^{2}}$
$\displaystyle+C\|\nabla\varrho\|_{H^{1}}\|\nabla^{3}\varrho\|_{L^{2}}\|\nabla^{4}{\bf
u}\|_{L^{2}}+C\|\nabla^{3}\varrho\|_{L^{2}}\|\nabla{\bf
u}\|_{H^{2}}^{2}+C\|\nabla^{3}\varrho\|_{L^{2}}\|\nabla\varrho\|_{H^{1}}^{2},$

which implies that

$\displaystyle-\frac{\mathrm{d}}{\mathrm{d}t}\int\nabla^{2}\operatorname{div}{\bf
u}\nabla^{2}\varrho\mathrm{d}{\bf
x}+\frac{\gamma}{2}\|\nabla^{3}\varrho\|_{L^{2}}^{2}\leq$ $\displaystyle
C\|\nabla^{3}{\bf u}\|_{L^{2}}^{2}+C\|\nabla\varrho\|_{H^{1}}^{2}\|\nabla{\bf
u}\|_{H^{1}}^{2}+C\|\nabla^{4}{\bf u}\|_{L^{2}}^{2}$
$\displaystyle+C\|\nabla\varrho\|_{H^{1}}^{2}\|\nabla^{3}\varrho\|_{L^{2}}^{2}+C\|\nabla{\bf
u}\|_{H^{2}}^{4}+C\|\nabla\varrho\|_{H^{1}}^{4}.$ (3.51)

Let $D>0$ be a large fixed constant. Summing up
$D\times\eqref{3.47}+\eqref{3.48}+\eqref{3.41}$ and applying Cauchy–Schwarz
inequality, then there exists an energy functional
$H(\|\nabla^{2}\varrho\|_{H^{1}},\|\nabla^{3}{\bf u}\|_{L^{2}})$ which is
equivalent to $\|\nabla^{2}\varrho\|_{H^{1}}^{2}+\|\nabla^{3}{\bf
u}\|_{L^{2}}^{2}$ such that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}H(t)+C_{1}\left(\|\nabla^{2}\varrho\|_{H^{1}}^{2}+\|\nabla^{4}{\bf
u}\|_{L^{2}}^{2}\right)\leq$ $\displaystyle C(\|\nabla{\bf
u}\|^{2}_{H^{2}}+\|\nabla\varrho\|^{2}_{H^{1}})H(t)+C\|\nabla\varrho\|_{H^{1}}^{2}+\|\nabla{\bf
u}\|_{H^{1}}^{2}$
$\displaystyle+C\|F\|_{W^{1,6}}^{2}\|\nabla^{2}\varrho\|_{L^{2}}^{2}+C\|\nabla
F\|_{L^{6}}^{2}+C\|\nabla\dot{\bf u}\|_{L^{2}}^{2},$ (3.52)

where we have used (3.1), (3.16), (3.37), (3.39), and (3.47). Applying
Gronwall inequality to (3.52), we see that

$\displaystyle H(t)\leq C,$

which combined with (3.52) leads to

$\displaystyle\sup\limits_{0\leq
t<\infty}\left(\|\nabla^{2}\varrho(t)\|_{H^{1}}^{2}+\|\nabla^{3}{\bf
u}(t)\|_{L^{2}}^{2}\right)+\int_{0}^{\infty}\left(\|\nabla^{2}\varrho(t)\|_{H^{1}}^{2}+\|\nabla^{4}{\bf
u}(t)\|_{L^{2}}^{2}\right)\mathrm{d}t\leq C.$

This along with (3.1), (3.37), and Sobolev inequality yields (3.32). ∎

The following Lemma 3.4 is necessary for decay estimates on the higher-order
derivatives of $(\varrho,{\bf u})$.

######  Lemma 3.4.

Under the assumptions of Theorem 1.1, it holds that

$\displaystyle\sup\limits_{0\leq
t<\infty}\Big{(}\|\varrho(t)\|_{\dot{B}^{-s}_{2,\infty}}+\|{\bf
u}(t)\|_{\dot{B}^{-s}_{2,\infty}}\Big{)}\leq C,\ \text{for}\ s\in(0,1/2],$
(3.53) $\displaystyle\|\varrho(T)\|_{\dot{B}^{-s}_{2,\infty}}+\|{\bf
u}(T)\|_{\dot{B}^{-s}_{2,\infty}}\leq C(T),\ \text{for}\ s\in(1/2,3/2].$
(3.54)

###### Proof.

Applying $2^{-2sq}\Delta_{q}\eqref{3.46}_{1}$,
$2^{-2sq}\Delta_{q}\eqref{3.46}_{2}$ by $\gamma 2^{-2sq}\Delta_{q}\varrho$,
$2^{-2sq}\Delta_{q}{\bf u}$, respectively, summing up, integrating the
resultant equation over $\mathbb{R}^{3}$, and then using (2.7), we obtain that

$\displaystyle\frac{1}{2}$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(\gamma\left\|2^{-2sq}\Delta_{q}\varrho\right\|_{L^{2}}^{2}+\left\|2^{-2sq}\Delta_{q}{\bf
u}\right\|_{L^{2}}^{2}\right)+\mu\left\|2^{-2sq}\Delta_{q}(\nabla{\bf
u})\right\|_{L^{2}}^{2}+(\mu+\lambda)\left\|2^{-2sq}\Delta_{q}(\operatorname{div}{\bf
u})\right\|_{L^{2}}^{2}$ $\displaystyle=$ $\displaystyle\gamma\int
2^{-2sq}\Delta_{q}\operatorname{div}(\varrho{\bf
u})2^{-2sq}\Delta_{q}\varrho\mathrm{d}{\bf x}+\int
2^{-2sq}\Delta_{q}\left(-\mathbf{u}\cdot\nabla\mathbf{u}-\gamma(\rho^{\gamma-1}-1)\nabla\varrho\right)2^{-2sq}\Delta_{q}{\bf
u}\mathrm{d}{\bf x}$ $\displaystyle+\int
2^{-2sq}\Delta_{q}\left(-\frac{\mu\varrho}{\rho}\Delta\mathbf{u}-\frac{(\mu+\lambda)\varrho}{\rho}\nabla\operatorname{div}\mathbf{u}\right)2^{-2sq}\Delta_{q}{\bf
u}\mathrm{d}{\bf x}$ $\displaystyle\leq$ $\displaystyle
C\left\|\left(\operatorname{div}(\varrho{\bf
u}),-\mathbf{u}\cdot\nabla\mathbf{u}-\gamma(\rho^{\gamma-1}-1)\nabla\varrho,-\frac{\mu\varrho}{\rho}\Delta\mathbf{u}-\frac{(\mu+\lambda)\varrho}{\rho}\nabla\operatorname{div}\mathbf{u}\right)\right\|_{\dot{B}^{-s}_{2,1}}$
$\displaystyle\times\left(\gamma\left\|2^{-2sq}\Delta_{q}\varrho\right\|_{L^{2}}^{2}+\left\|2^{-2sq}\Delta_{q}{\bf
u}\right\|_{L^{2}}^{2}\right)$ $\displaystyle\leq$ $\displaystyle
C\left\|\left(\operatorname{div}(\varrho{\bf
u}),-\mathbf{u}\cdot\nabla\mathbf{u}-\gamma(\rho^{\gamma-1}-1)\nabla\varrho,-\frac{\mu\varrho}{\rho}\Delta\mathbf{u}-\frac{(\mu+\lambda)\varrho}{\rho}\nabla\operatorname{div}\mathbf{u}\right)\right\|_{L^{\frac{6}{3+2s}}}$
$\displaystyle\times\left(\gamma\left\|2^{-2sq}\Delta_{q}\varrho\right\|_{L^{2}}^{2}+\left\|2^{-2sq}\Delta_{q}{\bf
u}\right\|_{L^{2}}^{2}\right)$ $\displaystyle\leq$ $\displaystyle
C\|(\varrho,{\bf u})\|_{L^{\frac{3}{s}}}\|\nabla(\varrho,{\bf
u})\|_{H^{1}}\left(\gamma\left\|2^{-2sq}\Delta_{q}\varrho\right\|_{L^{2}}^{2}+\left\|2^{-2sq}\Delta_{q}{\bf
u}\right\|_{L^{2}}^{2}\right).$ (3.55)

If $s\in(0,1/2]$, that is $\frac{3}{s}\geq 6$, then (3.53) follows from (3),
Sobolev inequality, and Gronwall inequality. When $s\in(1/2,3/2]$, then
$\frac{3}{s}\geq 2$, so (3.54) follows from (3), Gronwall inequality, and
(3.32). ∎

##  4 Proofs of Theorems 1.1 and 1.2

With all the a priori estimates in Section 3 at hand, we are ready to prove
Theorems 1.1 and 1.2.

Proof of Theorem 1.1. The desired (1.9) and (1.10) follow from (3.5) and
(3.32), respectively.

To show (1.11), in virtue of (3.32), there exists a positive constant $T_{9}$
such that

$\displaystyle\|\nabla\varrho(T_{9})\|_{H^{2}}+\|\nabla{\bf
u}(T_{9})\|_{H^{2}}\leq\delta,$

which together with (2.8) and (3.53) gives that

$\displaystyle\|\varrho(T_{9})\|_{L^{2}}+\|{\bf u}(T_{9})\|_{L^{2}}\leq
C\|\nabla(\varrho,{\bf u})\|_{L^{2}}^{\frac{s}{1+s}}\|\nabla(\varrho,{\bf
u})\|_{\dot{B}^{-s}_{2,\infty}}^{\frac{1}{1+s}}\leq C\delta^{\frac{s}{1+s}},\
\text{if}\ s\in(0,1/2],$ $\displaystyle\|\varrho(T_{9})\|_{L^{2}}+\|{\bf
u}(T_{9})\|_{L^{2}}\leq C\|\nabla(\varrho,{\bf
u})\|_{L^{2}}^{\frac{1}{3}}\|\nabla(\varrho,{\bf
u})\|_{\dot{B}^{-\frac{1}{2}}_{2,\infty}}^{\frac{2}{3}}\leq
C\delta^{\frac{1}{3}},\ \text{if}\ s\in(1/2,3/2].$

Therefore, we have

$\displaystyle\|\varrho(T_{9})\|_{L^{2}}+\|{\bf
u}(T_{9})\|_{L^{2}}\leq\delta^{\frac{1}{6}},$ (4.1)

provided that $\delta$ is properly small. Combining (4.1) with (3.53) and
(3.54), then using a family of scaled energy estimates with minimum derivative
counts and interpolations among them developed by Guo and Wang [16], we deduce
(1.11). The proof of Theorem 1.1 is completed. $\Box$

Proof of Theorem 1.2. It follows from (1.1)1 and the definition of $\sigma$
that

$\displaystyle\sigma_{t}+{\bf
u}\cdot\nabla\sigma+\gamma\sigma\operatorname{div}{\bf u}=0.$ (4.2)

Multiplying (4.2) by $6\sigma^{5}$ and integrating the resulting equality over
$\mathbb{R}^{3}$, one has that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left\|\sigma\right\|_{L^{6}}^{6}=-(4\gamma-1)\int\sigma^{6}\operatorname{div}{\bf
u}\mathrm{d}{\bf x}-\gamma\int\sigma^{5}\operatorname{div}{\bf
u}\mathrm{d}{\bf x},$

which combined with (1.4), (1.8), (3.1), and Young inequality yields that

$\displaystyle\int_{0}^{\infty}\left|\frac{\mathrm{d}}{\mathrm{d}t}\left\|\sigma\right\|_{L^{6}}^{6}\right|\mathrm{d}t\leq
C\int_{0}^{\infty}\big{(}\left\|\sigma\right\|_{L^{6}}^{6}+\left\|\operatorname{div}{\bf
u}\right\|_{L^{2}}^{2}\big{)}\mathrm{d}t\leq C,$

As a result, we have

$\displaystyle\lim\limits_{t\rightarrow\infty}\left\|\sigma\right\|_{L^{6}}=0.$

This along with (3.1) implies that

$\displaystyle\lim\limits_{t\rightarrow\infty}\|\rho-1\|_{L^{6}}=0,$ (4.3)

which together with (1.7), (1.8), (3.1), and the interpolation inequality
indicates (1.15).

On the other hand, thanks to (3.1), one gets that

$\displaystyle\inf\limits_{{\bf x}\in\mathbb{R}^{3}}\rho({\bf
x},t)\leq\inf\limits_{{\bf x}\in\mathbb{R}^{3}}\rho_{0}({\bf
x})\text{e}^{\int_{0}^{t}\|\operatorname{div}{\bf
u}\|_{L^{\infty}}\mathrm{d}\tau}\leq\inf\limits_{{\bf
x}\in\mathbb{R}^{3}}\rho_{0}({\bf
x})\text{e}^{C\int_{0}^{t}(\|F\|_{L^{\infty}}+\|\sigma\|_{L^{\infty}})\mathrm{d}\tau}\leq\inf\limits_{{\bf
x}\in\mathbb{R}^{3}}\rho_{0}({\bf x})\text{e}^{C(1+t)}=0,$

as desired (1.16). We complete the proof of Theorem 1.2. $\Box$

##  5 Proof of Theorem 1.3

This section is devoted to proving Theorem 1.3. To do so, we start with an
algebraic decay rate of the total energy.

######  Lemma 5.1.

Under the assumptions of Theorem 1.3, there exists a positive constant $C$
independent of $t$ such that

$\displaystyle\int\Big{(}\frac{1}{2}\rho|{\bf
u}|^{2}+G(\rho)\Big{)}\mathrm{d}{\bf x}\leq
C(1+t)^{-\frac{\gamma-1}{\gamma}}.$ (5.1)

###### Proof.

Multiplying (1.1)1 and (1.1)2 by $G^{\prime}(\rho)$ and ${\bf u}$,
respectively, summing up and integrating the resultant over $\mathbb{R}^{3}$,
we obtain that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int\Big{(}\frac{1}{2}\rho|{\bf
u}|^{2}+G(\rho)\Big{)}\mathrm{d}{\bf x}+\mu\left\|\nabla{\bf
u}\right\|_{L^{2}}^{2}+(\mu+\lambda)\left\|\nabla{\bf
u}\right\|_{L^{2}}^{2}=0,$ (5.2)

which indicates that

$\displaystyle\sup\limits_{0\leq t<\infty}\int\Big{(}\frac{1}{2}\rho|{\bf
u}|^{2}+G(\rho)\Big{)}\mathrm{d}{\bf
x}+\int_{0}^{\infty}\big{[}\mu\left\|\nabla{\bf
u}\right\|_{L^{2}}^{2}+(\mu+\lambda)\left\|\nabla{\bf
u}\right\|_{L^{2}}^{2}\big{]}\mathrm{d}t\leq\int\Big{(}\rho_{0}|{\bf
u}_{0}|^{2}+2G(\rho_{0})\Big{)}\mathrm{d}{\bf x}.$ (5.3)

Applying the operator $\Delta^{-1}\operatorname{div}$ to (1.1)2, we arrive at

$\displaystyle P(\rho)=\partial_{t}(-\Delta)^{-1}\operatorname{div}(\rho{\bf
u})+(2\mu+\lambda)\operatorname{div}{\bf
u}-{\mathcal{R}}_{i}{\mathcal{R}}_{j}(\rho u^{i}u^{j}),$ (5.4)

where ${\mathcal{R}}_{i}$ is the usual Riesz transform on $\mathbb{R}^{3}$:
${\mathcal{R}}_{i}=-(-\Delta)^{-\frac{1}{2}}\partial_{x_{i}}$. Taking the
$L^{2}$ product of (5.4) with $P(\rho)$, we infer that

$\displaystyle\int P^{2}(\rho)\mathrm{d}{\bf x}=$
$\displaystyle\int\partial_{t}(-\Delta)^{-1}\operatorname{div}(\rho{\bf
u})P(\rho)\mathrm{d}{\bf x}+\int(2\mu+\lambda)\operatorname{div}{\bf
u}P(\rho)\mathrm{d}{\bf x}-\int{\mathcal{R}}_{i}{\mathcal{R}}_{j}(\rho
u^{i}u^{j})P(\rho)\mathrm{d}{\bf x}$ $\displaystyle\triangleq$
$\displaystyle~{}I_{1}+I_{2}+I_{3}.$ (5.5)

We need to estimate each term $I_{1i}\ (i=1,2,3)$. It follows from (1.1)1 that

$\displaystyle(P(\rho))_{t}+\operatorname{div}[P(\rho){\bf
u}]+(\gamma-1)P(\rho)\operatorname{div}{\bf u}=0,$

which combined with (1.14), (5.3), the Marcinkiewicz multiplier theorem (see
[42, p. 96]), (2.6), Hölder inequality, and Cauchy–Schwarz inequality yields
that

$\displaystyle I_{1}=$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int(-\Delta)^{-1}\operatorname{div}(\rho{\bf
u})P(\rho)\mathrm{d}{\bf x}-\int(-\Delta)^{-1}\operatorname{div}(\rho{\bf
u})(P(\rho))_{t}\mathrm{d}{\bf x}$ $\displaystyle=$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int(-\Delta)^{-1}\operatorname{div}(\rho{\bf
u})P(\rho)\mathrm{d}{\bf
x}-\int\nabla(-\Delta)^{-1}\operatorname{div}(\rho{\bf u})P(\rho){\bf
u}\mathrm{d}{\bf x}$
$\displaystyle+\int(-\Delta)^{-1}\operatorname{div}(\rho{\bf
u})(\gamma-1)P(\rho)\operatorname{div}{\bf u}\mathrm{d}{\bf x}$
$\displaystyle\leq$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int(-\Delta)^{-1}\operatorname{div}(\rho{\bf
u})P(\rho)\mathrm{d}{\bf x}+C\big{(}\|\rho{\bf
u}\|_{L^{6}}\|P(\rho)\|_{L^{\frac{3}{2}}}\|{\bf
u}\|_{L^{6}}+\|\Delta^{-1}\operatorname{div}(\rho{\bf
u})\|_{L^{6}}\|P(\rho)\|_{L^{3}}\|\operatorname{div}{\bf u}\|_{L^{2}}\big{)}$
$\displaystyle\leq$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int(-\Delta)^{-1}\operatorname{div}(\rho{\bf
u})P(\rho)\mathrm{d}{\bf
x}+C\big{(}\|\rho\|_{L^{\infty}}\|P(\rho)\|_{L^{\frac{3}{2}}}\|{\bf
u}\|_{L^{6}}^{2}+\|\rho{\bf
u}\|_{L^{2}}\|P(\rho)\|_{L^{3}}\|\operatorname{div}{\bf u}\|_{L^{2}}\big{)}$
$\displaystyle\leq$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int(-\Delta)^{-1}\operatorname{div}(\rho{\bf
u})P(\rho)\mathrm{d}{\bf
x}+C\big{(}\|\rho\|_{L^{\infty}}\|P(\rho)\|_{L^{\frac{3}{2}}}\|{\bf
u}\|_{L^{6}}^{2}+\|\rho\|_{L^{3}}\|{\bf
u}\|_{L^{6}}\|\|P(\rho)\|_{L^{3}}\|\operatorname{div}{\bf u}\|_{L^{2}}\big{)}$
$\displaystyle\leq$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{R}^{3}}(-\Delta)^{-1}\operatorname{div}(\rho{\bf
u})P(\rho)\mathrm{d}{\bf x}+C\|\nabla{\bf u}\|^{2}_{L^{2}}.$ (5.6)

Similarly, $I_{12}$ and $I_{13}$ can be handled as

$\displaystyle I_{2}\leq\frac{1}{2}\|P(\rho)\|^{2}_{L^{2}}+C\|\nabla{\bf
u}\|^{2}_{L^{2}},$ (5.7) $\displaystyle I_{3}\leq C\|\rho
u^{i}u^{j}\|_{L^{3}}\|P(\rho)\|_{L^{\frac{3}{2}}}\leq
C\|\rho\|_{L^{\infty}}\|{\bf u}\|_{L^{6}}^{2}\|P(\rho)\|_{L^{\frac{3}{2}}}\leq
C\|\nabla{\bf u}\|^{2}_{L^{2}}.$ (5.8)

Hence, substituting (5)–(5.8) into (5) gives that

$\displaystyle-\frac{\mathrm{d}}{\mathrm{d}t}\int(-\Delta)^{-1}\operatorname{div}(\rho{\bf
u})P(\rho)\mathrm{d}{\bf x}+\frac{1}{2}\|P(\rho)\|_{L^{2}}^{2}\leq
C\|\nabla{\bf u}\|^{2}_{L^{2}}.$ (5.9)

For $t\geq 0$, choose a positive constant $D_{1}$ and define the temporal
energy functional

$\displaystyle\mathcal{M}_{1}(t)=D_{1}\int\Big{(}\frac{1}{2}\rho|{\bf
u}|^{2}\mathrm{d}{\bf x}+G(\rho)\Big{)}\mathrm{d}{\bf
x}-\int(-\Delta)^{-1}\operatorname{div}(\rho{\bf u})P(\rho)\mathrm{d}{\bf x}.$

By (1.8), (5.3), and $\|\rho\|_{L^{1}}=\|\rho_{0}\|_{L^{1}}$, we observe that

$\displaystyle\int(-\Delta)^{-1}\operatorname{div}(\rho{\bf
u})P(\rho)\mathrm{d}{\bf x}\leq$
$\displaystyle~{}C\|(-\Delta)^{-1}\operatorname{div}(\rho{\bf
u})\|_{L^{6}}\|P(\rho)\|_{L^{\frac{6}{5}}}$ $\displaystyle\leq$
$\displaystyle~{}C\|\rho{\bf u}\|_{L^{2}}\|P(\rho)\|_{L^{\frac{6}{5}}}$
$\displaystyle\leq$
$\displaystyle~{}C\left(\|\sqrt{\rho}\|_{L^{\infty}}^{6}\|\sqrt{\rho}{\bf
u}\|^{6}_{L^{2}}+\|P(\rho)\|_{L^{1}}\|P(\rho)\|_{L^{\infty}}^{\frac{1}{5}}\right)$
$\displaystyle\leq$ $\displaystyle~{}C\int\Big{(}\frac{1}{2}\rho|{\bf
u}|^{2}+G(\rho)\Big{)}\mathrm{d}{\bf x},$

and

$\displaystyle\int\Big{(}\frac{1}{2}\rho|{\bf
u}|^{2}+G(\rho)\Big{)}\mathrm{d}{\bf x}$
$\displaystyle=\frac{1}{2}\int\rho^{\frac{\gamma-1}{2\gamma-1}}(\sqrt{\rho}|{\bf
u}|)^{\frac{2\gamma}{2\gamma-1}}|{\bf
u}|^{\frac{2(\gamma-1)}{2\gamma-1}}\mathrm{d}{\bf x}+\int
G(\rho)\mathrm{d}{\bf x}$ $\displaystyle\leq
C\|\rho\|_{L^{\frac{3}{2}}}^{\frac{\gamma-1}{2\gamma-1}}\|\sqrt{\rho}{\bf
u}\|_{L^{2}}^{\frac{2\gamma}{2\gamma-1}}\|{\bf
u}\|_{L^{6}}^{\frac{2(\gamma-1)}{2\gamma-1}}+C\|\rho\|_{L^{1}}^{\frac{\gamma}{2\gamma-1}}\|\rho\|_{L^{2\gamma}}^{\frac{2\gamma(\gamma-1)}{2\gamma-1}}$
$\displaystyle\leq C\|(P(\rho),\nabla{\bf
u})\|^{\frac{2(\gamma-1)}{2\gamma-1}}_{L^{2}}.$ (5.10)

Thus, if $D_{1}$ is chosen large enough, one has that

$\displaystyle\mathcal{M}_{1}(t)\sim\int\Big{(}\frac{1}{2}\rho|{\bf
u}|^{2}+G(\rho)\Big{)}\mathrm{d}{\bf x}.$

Taking a linear combination of (5.2) and (5.9), and using (5), we see that
there exists a positive constant $C$ independent of $t$ such that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{M}_{1}(t)+C(\mathcal{M}_{1}(t))^{\frac{2\gamma-1}{\gamma-1}}\leq
0.$

Solving the above inequality directly yields that

$\displaystyle\mathcal{M}_{1}(t)\leq C(1+t)^{-\frac{\gamma-1}{\gamma}},$

as desired (5.1). ∎

For $\delta>0$, it infers from (5.1) and (5.3) that there exists a positive
constant $T_{10}$ such that

$\displaystyle\int\Big{(}\frac{1}{2}\rho|{\bf
u}|^{2}+G(\rho)\Big{)}(T_{10})\mathrm{d}{\bf x}+\left\|\nabla{\bf
u}(T_{10})\right\|_{L^{2}}\leq\delta.$ (5.11)

The following Proposition 5.1 (see [32, Theorem 1.3]) deals with the large-
time asymptotic behavior of global classical solutions.

######  Proposition 5.1.

Under the assumptions of Theorem 1.3, it holds that

$\displaystyle\sup\limits_{T_{10}\leq
t<\infty}\big{[}t\big{(}\left\|\nabla{\bf
u}\right\|_{L^{2}}^{2}+\left\|P\right\|_{L^{2}}^{2}\big{)}\big{]}+\int_{T_{10}}^{\infty}t\big{(}\|\sqrt{\rho}\dot{\bf
u}\|_{L^{2}}^{2}+\|P\|_{L^{3}}^{3}\big{)}\mathrm{d}t\leq C,$ (5.12)
$\displaystyle\sup\limits_{T_{10}\leq
t<\infty}\big{[}t^{2}\big{(}\|\sqrt{\rho}\dot{\bf
u}\|_{L^{2}}^{2}+\|P\|_{L^{3}}^{3}\big{)}\big{]}+\int_{T_{10}}^{\infty}t^{2}\big{(}\left\|\nabla{\dot{\bf
u}}\right\|_{L^{2}}^{2}+\left\|P\right\|^{4}_{L^{4}}\big{)}\mathrm{d}t\leq C.$
(5.13)

Now we are in a position to prove Theorem 1.3.

Proof of Theorem 1.3. By (1.1)1 and the definition of $F$ in (1.4), one gets
that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\rho+\frac{1}{2\mu+\lambda}\rho^{\gamma+1}=-\frac{1}{2\mu+\lambda}\rho
F,$ (5.14)

Multiplying (5.14) by $\gamma t^{2}\rho^{\gamma-1}$ indicates that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}(t^{2}\rho^{\gamma})+\frac{\gamma}{2\mu+\lambda}t^{2}\rho^{2\gamma}=2t\rho^{\gamma}-\frac{\gamma}{2\mu+\lambda}t^{2}\rho^{\gamma}F,$

which implies that

$\displaystyle
t^{2}\rho^{\gamma}(t)+\frac{\gamma}{2\mu+\lambda}\int_{T_{10}}^{t}\tau^{2}\rho^{2\gamma}\mathrm{d}\tau=T_{10}^{2}\rho^{\gamma}(T_{10})+2\int_{T_{10}}^{t}\tau\rho^{\gamma}\mathrm{d}\tau-\frac{\gamma}{2\mu+\lambda}\int_{T_{10}}^{t}\tau^{2}\rho^{\gamma}F\mathrm{d}\tau.$
(5.15)

By Cauchy–Schwarz inequality, we have

$\displaystyle\int_{T_{10}}^{t}\tau\rho^{\gamma}\mathrm{d}\tau\leq\frac{\gamma}{2(2\mu+\lambda)}\int_{T_{10}}^{t}\tau^{2}\rho^{2\gamma}\mathrm{d}\tau+C\int_{T_{10}}^{t}\mathrm{d}\tau\leq\frac{\gamma}{2(2\mu+\lambda)}\int_{T_{10}}^{t}\tau^{2}\rho^{2\gamma}\mathrm{d}\tau+Ct.$
(5.16)

It follows from (2.2), (2.3), (2.6), (5.12), and (5.13) that

$\displaystyle\int_{T_{10}}^{t}\tau^{2}\rho^{\gamma}F\mathrm{d}\tau$
$\displaystyle\leq\frac{\gamma}{2(2\mu+\lambda)}\int_{T_{10}}^{t}\tau^{2}\rho^{2\gamma}\mathrm{d}\tau+C\int_{T_{10}}^{t}\tau^{2}\left\|F\right\|_{L^{\infty}}^{2}\mathrm{d}t$
$\displaystyle\leq\frac{\gamma}{2(2\mu+\lambda)}\int_{T_{10}}^{t}\tau^{2}\rho^{2\gamma}\mathrm{d}\tau+\int_{T_{10}}^{t}\tau^{2}\left\|F\right\|_{L^{6}}\left\|\nabla
F\right\|_{L^{6}}\mathrm{d}t$
$\displaystyle\leq\frac{\gamma}{2(2\mu+\lambda)}\int_{T_{10}}^{t}\tau^{2}\rho^{2\gamma}\mathrm{d}\tau+C\int_{T_{10}}^{t}\tau^{2}\left\|\rho\dot{\bf
u}\right\|_{L^{2}}\left\|\rho\dot{\bf u}\right\|_{L^{6}}\mathrm{d}t$
$\displaystyle\leq\frac{\gamma}{2(2\mu+\lambda)}\int_{T_{10}}^{t}\tau^{2}\rho^{2\gamma}\mathrm{d}\tau+C\int_{T_{10}}^{t}\tau\left\|\nabla\dot{\bf
u}\right\|_{L^{2}}\mathrm{d}t$
$\displaystyle\leq\frac{\gamma}{2(2\mu+\lambda)}\int_{T_{10}}^{t}\tau^{2}\rho^{2\gamma}\mathrm{d}\tau+Ct.$
(5.17)

Substituting (5.16) and (5.17) into (5.15), we obtain that

$\displaystyle\left\|P(\cdot,t)\right\|_{L^{\infty}}\leq C(1+t)^{-1},$

which together with (5.1) and Hölder inequality leads to

$\displaystyle\left\|P(\cdot,t)\right\|_{L^{p}}\leq\left\|P(\cdot,t)\right\|_{L^{1}}^{\frac{1}{p}}\left\|P(\cdot,t)\right\|_{L^{\infty}}^{1-\frac{1}{p}}\leq
C(1+t)^{-1+\frac{1}{p\gamma}},\ \ \text{for any}\ 1\leq p\leq\infty,$

as desired (1.17).

Next, one infers from (1.17), (5.12), and Hölder inequality that

$\displaystyle\int\rho|{\bf u}|^{2}\mathrm{d}{\bf
x}\leq\left\|\rho\right\|_{L^{\frac{3}{2}}}\left\|\bf
u\right\|_{L^{6}}^{2}\leq
C(1+t)^{-1}\left\|\rho\right\|_{L^{1}}^{\frac{2}{3}}\left\|\rho\right\|_{L^{\infty}}^{\frac{1}{3}}\leq
C(1+t)^{-1-\frac{1}{3\gamma}},$

as desired (1.18).

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