# A general sufficient criterion for energy conservation in the Navier-Stokes
system

Yanqing Wang111 College of Mathematics and Information Science, Zhengzhou
University of Light Industry, Zhengzhou, Henan 450002, P. R. China Email:
<EMAIL_ADDRESS>and Yulin Ye222Corresponding author. School of
Mathematics and Statistics, Henan University, Kaifeng, 475004, P. R. China.
Email<EMAIL_ADDRESS>
###### Abstract

In this paper, we derive an energy conservation criterion based on a
combination of velocity and its gradient for the weak solutions of both the
homogeneous incompressible Navier-Stokes equations and the general
compressible Navier-Stokes equations. For the incompressible case, this class
implies most known corresponding results on periodic domain via either the
velocity or its gradient including the famous Lions’ energy conservation
criterion obtained in [20]. For the compressible case, this helps us to extend
the previously known criteria for the energy conservation of weak solutions
from the incompressible fluid to compressible flow and improve the recent
results due to Nguyen-Nguyen-Tang in [21, Nonlinearity 32 (2019)] and Liang in
[17, Proc. Roy. Soc. Edinburgh Sect. A (2020)].

MSC(2000): 35Q30, 35Q35, 76N06, 76N10
Keywords: Navier-Stokes equations; Compressible Navier-Stokes equations;
Energy conservation

##  1 Introduction

The homogeneous Navier-Stokes equations describing the motion of
incompressible fluid in three-dimensional space read

$\left\\{\begin{aligned} &v_{t}-\Delta v+\mathrm{div}\,(v\otimes
v)+\nabla\pi=0,\quad(x,t)\in\Omega\times(0,+\infty),\\\ &\mathrm{div}\,v=0,\\\
&v_{0}=v(x,0).\end{aligned}\right.$ (1.1)

Here, $v$ stands for the velocity field of the flow and $\pi$ represents the
pressure of the fluid, respectively. The initial datum satisfies
$\mathrm{div}\,v_{0}=0$. Usually, one considers the Navier-Stokes equations on
the periodic domain $(\Omega=\mathbb{T}^{3})$, on smooth bounded domain
$\Omega$ with Dirichlet boundary condition or on the whole space
$(\Omega=\mathbb{R}^{3})$.

It is well known that Leray-Hopf weak solutions of the Navier-Stokes equations
(LABEL:NS) obeys the energy inequality

$\|v(T)\|_{L^{2}(\Omega)}^{2}+2\int_{0}^{T}\|\nabla
v\|_{L^{2}(\Omega)}^{2}ds\leq\|v_{0}\|_{L^{2}(\Omega)}^{2},$

rather than energy equality. The first attempts to determine sufficient
conditions implying energy conservation of Leray-Hopf weak solutions in the
homogeneous incompressible Navier-Stokes equations were given by Lions [20]
and Prodi [25] provided that the criterion

$v\in L^{4}(0,T;L^{4}(\Omega))$ (1.2)

is satisfied. Then Serrin [26] showed the energy conservation by giving a
criterion in a scaling invarant space, that is

$v\in L^{p}(0,T;L^{q}(\Omega)),\ \text{with}\ \frac{2}{p}+\frac{d}{q}\leq 1\
\text{and}\ q\geq d,$

where $d$ is the spatial dimension. However, the weak solution which satisfy
the given criterion will immediately become a classical one. Later, Shinbrot
extended Lions’ condition in [20] for energy conservation to

$v\in L^{p}(0,T;L^{q}(\Omega))~{}\text{
with}~{}\frac{2}{p}+\frac{2}{q}=1,q\geq 4.$ (1.3)

It is worth remarking that condition (1.3) is weaker than (1.2) when the
dimension $d\geq 4$ and more importantly, it is true regardless of the
dimension of the underlying space. On the other hand, the energy conservation
condition (1.3) can be replaced by

$v\in L^{p}(0,T;L^{q}(\Omega))\text{ with}\ \frac{1}{p}+\frac{3}{q}=1,3<q<4;$
(1.4)

which was recently obtained by Beirao da Veiga-Yang in [4]. Using the Fourier
methods, Cheskidov-Friedlander-Shvydkoy [10] gave the following sufficient
condition for energy conservation (here $A$ denotes the Stokes operator
associated to the Dirichlet boundary conditions) that

$A^{5/12}v\in L^{3}(0,T;L^{2}(\Omega)),$

in fact, this criterion is equivalent in terms of scaling to $v\in
L^{3}(0,T;L^{\frac{9}{2}}(\Omega)).$ Very recently, Berselli-Chiodaroli [5]
and Zhang [35] obtained energy equality via the following condition,

$\nabla v\in
L^{p}\left(0,T;L^{q}\left(\Omega\right)\right),\frac{1}{p}+\frac{3}{q}=2,\frac{3}{2}<q<\frac{9}{5}~{}~{}\text{
or}~{}~{}\frac{1}{p}+\frac{6}{5q}=1,\frac{9}{5}\leq q.$ (1.5)

It is worth remarking that the domain $\Omega$ in most the aforementioned
conditional results is the smooth bounded. The first objective of this paper
is to show the following sufficient criterion for the weak solutions keeping
the energy of the Navier-Stokes equations on the periodic domain
$\mathbb{T}^{3},$

$v\in
L^{\frac{2p}{p-1}}(0,T;L^{\frac{2q}{q-1}}(\mathbb{T}^{3}))~{}~{}\text{and}~{}~{}\nabla
v\in L^{p}(0,T;L^{q}(\mathbb{T}^{3})).$ (1.6)

Surprisingly, this result covers the corresponding results of (1.3), (1.4) and
(1.5) on periodic domain and further discussion will be found in Remark 1.2
and 1.3. Indeed, we will also prove this class for the following compressible
Navier-Stokes equations with degenerate viscosities and general pressure law
(GNS)

$\left\\{\begin{aligned} &\rho_{t}+\mathrm{div}\,(\rho v)=0,\\\ &(\rho
v)_{t}+\mathrm{div}\,(\rho v\otimes v)+\nabla
p(\rho)-\mathrm{div}\,(\nu(\rho)\mathbb{D}v)-\nabla(\mu(\rho)\mathrm{div}\,v)=0,\\\
\end{aligned}\right.$ (1.7)

with the initial data

$\displaystyle\rho(0,x)=\rho_{0}(x)\ \text{and}\ (\rho
v)(0,x)=\rho_{0}(x)v_{0}(x),\ \ x\in\Omega,$ (1.8)

where the unknown functions $\rho$ and $v$ denote the density of the fluid and
velocity of the fluid, respectively; $\mathbb{D}v=\frac{1}{2}(\nabla
v\otimes\nabla v^{T})$ stands for the stain tensor; The general pressure
$0\leq p(\rho)\in C^{1}(0,\infty)$ with $p^{{}^{\prime}}(\cdot)>0$ and the
viscosity coefficients $\nu(\rho),\mu(\rho):(0,\infty)\rightarrow[0,\infty)$
are continuous functions of density. We will consider the case of bounded
domain with periodic boundary conditions, namely $\Omega=\mathbb{T}^{d}$,
where $d\geq 2$ is the dimension of the domain. It should be noted that when
$\nu(\rho)\equiv\nu,\mu(\rho)\equiv\mu$ and $p(\rho)=\rho^{\gamma}$ with
$\gamma>1$, then (GNS) will reduce to the classical isentropic compressible
Navier-Stokes equations (ICNS):

$\left\\{\begin{aligned} &\rho_{t}+\mathrm{div}\,(\rho v)=0,\\\ &(\rho
v)_{t}+\mathrm{div}\,(\rho v\otimes v)+\nabla\rho^{\gamma}-\nu\Delta
v-\mu\nabla\text{div\,}v=0,\end{aligned}\right.$ (1.9)

and when $\nu(\rho)=\rho,\mu(\rho)\equiv 0$ and $p(\rho)=\rho^{\gamma}$ with
$\gamma>1$, then (GNS) reduces to the compressible Navier-Stokes equations
with degenerate viscosity and $\gamma$-pressure law (CNSD) as follows:

$\left\\{\begin{aligned} &\rho_{t}+\mathrm{div}\,(\rho v)=0,\\\ &(\rho
v)_{t}+\mathrm{div}\,(\rho v\otimes
v)+\nabla\rho^{\gamma}-\mathrm{div}\,(\rho\mathbb{D}v)=0.\end{aligned}\right.$
(1.10)

The global existence of weak solutions which satisfy the energy inequality has
already been known, see P. L. Lions [19] and Feireisl-Novotný-Petzeltová [14]
for (ICNS) with constant viscosity coefficients case, Vassure-Yu [30] for
(CNSD) and Li-Xin [16] for (GNS) with degenerate viscosity coefficients case,
but the regularity and even the uniqueness of weak solutions are still open
problems. Since the weak solutions satisfy the energy inequality rather than
equality due to the basic a priori estimates and the lack of regularity, this
anomalous dissipation of the energy opens a possibility for an energy sink
other than the natural viscous dissipation, however, such a property of real
fluids is not expected to exist physically, so it is therefore a famous
problem to consider the sufficient criterion for the energy conservation of
weak solutions. Roughly speaking, this addresses the question how much
regularities are needed for a weak solution to conserve energy, which is also
involving the uniqueness of the weak solutions. On the other hand, energy
conservation is also one aspect of the Onsager’s conjecture in the context of
homogeneous incompressible Euler equations in [24], in which Onsager
conjectured that the kinetic energy is globally conserved for Hölder
continuous solutions with the exponent greater than $\frac{1}{3}$, while an
energy dissipation phenomenon occurs for Hölder continuous solutions with the
exponent less than $\frac{1}{3}$. For the positive part, the milestone work is
due to Constantin-E-Titi [12], in which it was proved that the energy of 3D
incompressible Euler equations is conserved for every weak solution in
$L^{3}(0,T;B_{3,\infty}^{\alpha})$ with $\alpha>1/3$. On the other hand, Isett
resolved the “negative”part of Onsager’s conjecture for 3D incompressible
Euler equations in [15], where he proved that for any $\alpha<\frac{1}{3}$
there is a nonzero weak solution to the incompressible Euler equations in the
class $v\in C^{\alpha}_{t,x}$ and $p\in C^{2\alpha}_{t,x}$ such that $v$ is
identically $0$ outside a finite time interval. In particular, the solution
$v$ fails to conserve the energy. We refer the reader to [1, 2, 3, 7, 8, 9,
13, 23, 22, 6] for recent progress in this direction.

Compared with the incompressible Navier-Stokes equations, due to the stronger
nonlinearity, the energy conservation of the weak solutions is more
challenging and hence the results are few. When the density is strictly away
from vacuum, for the weak solutions to the general compressible models
(LABEL:GNS), Nguye-Nguye-Tang [21] established the Shinbrot-type criterion and
showed that if the weak solutions satisfied

$\displaystyle 0<c_{1}\leq\rho\leq c_{2}<\infty,v\in
L^{\infty}(0,T;L^{2}(\mathbb{T}^{3})),\nabla v\in
L^{2}(0,T;L^{2}(\mathbb{T}^{3})),$ (1.11)
$\displaystyle\sup_{t\in(0,T)}\sup_{|h|<\varepsilon}|h|^{-\frac{1}{2}}\|\rho(\cdot+h,t)-\rho(\cdot,t)\|_{L^{2}(\mathbb{T}^{3})}<\infty,$
$\displaystyle v\in L^{p}(0,T;L^{q}(\mathbb{T}^{3}))\text{
with}~{}~{}\left\\{\begin{aligned} &\frac{2}{p}+\frac{2}{q}=1,q\geq 4,\\\
&\frac{1}{p}+\frac{3}{q}=1,3<q<4,\end{aligned}\right.$

then the energy of weak solutions is globally conserved, which means the
energy equality holds for any $t\in[0,T]$. It is worth noting that though the
part $\frac{1}{p}+\frac{3}{q}=1,3<q<4$ was not mentioned in [21], it is a
direct consequence from interpolation and $v\in
L^{4}(0,T;L^{4}(\mathbb{T}^{3}))$ (see [4] and the corresponding proof in
Theorem 1.2).

Later, Liang [17] derived a energy conservation criterion via the gradient of
velocity for isentropic Navier-Stokes equations (ICNS) under the following
condition

$\displaystyle 0<c_{1}\leq\rho\leq c_{2}<\infty,v\in
L^{\infty}(0,T;L^{2}(\mathbb{T}^{3})),\nabla v\in
L^{2}(0,T;L^{2}(\mathbb{T}^{3})),$ (1.12) $\displaystyle\nabla v\in
L^{p}\left(0,T;L^{s}(\mathbb{T}^{3})\right)~{}\text{with}~{}\left\\{\begin{aligned}
\frac{1}{p}+\frac{3}{s}<2,\ \ \ \ \ \ &\frac{3}{2}<s<\frac{9}{5},\\\
\frac{5}{p}+\frac{6}{s}<5,\ \ \ \ \ \ &\frac{9}{5}\leq s\leq 3,\\\
\frac{1}{p}+\frac{2}{s+2}<1,\ \ &3<s<\infty,\end{aligned}\right.$

then the energy of weak solutions is locally conserved, which means the energy
equality holds in the sense of distribution in $(0,T)$.

When the density may contain vacuum, in the spirit of well-known Shinbrot’s
criterion in [27], Yu [34] showed that if a weak solution $(\rho,v)$ of
(LABEL:CNSD) or (LABEL:ICNS) satisfies

$\displaystyle\sqrt{\rho}v\in L^{\infty}(0,T;L^{2}(\Omega)),\sqrt{\rho}\nabla
v\in L^{2}(0,T;L^{2}(\Omega)),$ (1.13) $\displaystyle 0\leq\rho\leq c<\infty,\
\ \nabla\sqrt{\rho}\in L^{\infty}(0,T;L^{2}(\Omega))$ $\displaystyle v\in
L^{p}(0,T;L^{q}(\Omega))~{}~{}\frac{1}{p}+\frac{1}{q}\leq\frac{5}{12}\ \text{
and}\ q\geq 6,$

then the energy is globally conserved. Recently, for equations (LABEL:ICNS),
Chen-Liang-Wang-Xu [11] obtained the energy balance in a bounded domain with
physical boundaries under the following condition

$\displaystyle\sqrt{\rho}v\in L^{\infty}(0,T;L^{2}(\Omega)),\sqrt{\rho}\nabla
v\in L^{2}(0,T;L^{2}(\Omega)),$ (1.14) $\displaystyle 0\leq\rho\leq
c<\infty,\nabla\sqrt{\rho}\in L^{\infty}(0,T;L^{2}(\Omega)),$ $\displaystyle
v\in L^{p}(0,T;L^{q}(\Omega)),p\geq 4,q\geq 6.$

As mentioned above, the energy conservation for fluid equations addresses the
question how much regularities are needed for a weak solution to conserve
energy, which will help us further to consider the uniqueness and regularity
of the weak solutions. However, up to now, the related results on energy
conservation of weak solutions for compressible Navier-Stokes equations are
less satisfactory than incompressible ones. For example, compared with the
Shinbrot’s condition (1.3) for the incompressible Navier-Stokes equations, the
criterion (LABEL:NNT) obtained by Nguye-Nguye-Tang in [21] requires additional
constraint that
$\sup\limits_{t\in(0,T)}\sup\limits_{|h|<\varepsilon}|h|^{-\frac{1}{2}}\|\rho(\cdot+h,t)-\rho(\cdot,t)\|_{L^{2}(\mathbb{T}^{3})}<\infty$.
Note that, due to the blow-up criteria only via the density for strong
solutions to the Cauchy problem of compressible isentropic Navier-Stokes
equations in $\mathbb{R}^{3}$, under the assumptions on the coefficients of
viscosity established in [29, 32] it implies the bound of density to the 3-D
compressible Navier-Stokes equations yields strong solutions and the strong
solutions are expected to meet energy conservation. Based on this, the second
objective of this paper is to remove $\eqref{NNT}_{2}$ to obtain the
persistence of energy. Moreover, when we see the criterion obtained by Liang
in [17], on the one hand, the restrictions on the indexes $p$ and $s$ are
“subcritical”other than “critical”and are stronger when $s>3$ compared with
the result (1.5) for incompressible case. On the other hand, the criterion
obtained in [17] only implies the energy conserved “locally ”not “globally”.
Hence, our third objective of this paper is to improve the criterion via the
gradient of velocity and to show $\eqref{NNT}_{1}$ and (1.5) guarantee the
energy equality in system (LABEL:GNS) globally.

Before stating the main results, we introduce the definition of the weak
solutions.

######  Definition 1.1.

A pair ($\rho,v$) is called a weak solution to (LABEL:GNS) with initial data
($\rho_{0},v_{0}$) if ($\rho,v$) satisfy

  1. (i)

equations (LABEL:GNS) hold in $\mathcal{D}^{\prime}(0,T;\mathbb{T}^{d})$ and

$P(\rho),\rho|v|^{2}\in L^{\infty}(0,T;L^{1}(\mathbb{T}^{d})),\ \ \ \nabla
v\in L^{2}(0,T;L^{2}(\mathbb{T}^{d})),$ (1.15)

  2. (ii)

$\rho(\cdot,t)\rightharpoonup\rho_{0}$ in $\mathcal{D}^{\prime}(\Omega)$ as
$t\rightarrow 0$, i.e.

$\lim_{t\rightarrow
0}\int_{\mathbb{T}^{d}}\rho(x,t)\varphi(x)dx=\int_{\mathbb{T}^{d}}\rho_{0}(x)\varphi(x)dx,$
(1.16)

for every test function $\varphi\in C_{0}^{\infty}(\mathbb{T}^{d}).$

  3. (iii)

$(\rho v)(\cdot,t)\rightharpoonup\rho_{0}v_{0}$ in
$\mathcal{D}^{\prime}(\mathbb{T}^{d})$ as $t\rightarrow 0$ i.e.

$\lim_{t\rightarrow 0}\int_{\mathbb{T}^{d}}(\rho
v)(x,t)\psi(x)dx=\int_{\mathbb{T}^{d}}(\rho_{0}v_{0})(x)\psi(x)dx,$ (1.17)

for every test vector field $\psi\in C_{0}^{\infty}(\mathbb{T}^{d})^{d}$.

  4. (iv)

the energy inequality holds

$\displaystyle\mathcal{E}(t)+\int_{0}^{T}\int_{\mathbb{T}^{d}}$
$\displaystyle\left[\nu(\rho)|\mathbb{D}v|^{2}+\mu(\rho)|\mathrm{div}\,v|^{2}\right]dxdt\leq\mathcal{E}(0),$
(1.18)

where
$\mathcal{E}(t)=\int_{\mathbb{T}^{d}}\left[\frac{1}{2}\rho|v|^{2}+P(\rho)\right]dx$
and $P(\rho)=\rho\int_{1}^{\rho}\frac{p(z)}{z^{2}}dz.$

We formulate our first result as follows:

######  Theorem 1.1.

For any dimension $d\geq 2$, let $(\rho,v)$ be a weak solution to the general
compressible Navier-Stokes equations (LABEL:GNS). Assume that $1<p,q<\infty$
and

$\left\\{\begin{aligned} &0<c_{1}\leq\rho\leq c_{2}<\infty,v\in
L^{\infty}(0,T;L^{2}(\mathbb{T}^{d})),\nabla v\in
L^{2}(0,T;L^{2}(\mathbb{T}^{d})),\\\ &v\in
L^{\frac{2p}{p-1}}(0,T;L^{\frac{2q}{q-1}}(\mathbb{T}^{d})),\ \nabla v\in
L^{p}(0,T;L^{q}(\mathbb{T}^{d}))\ and\ \sqrt{\rho_{0}}v_{0}\in L^{2+\delta}\
for\ any\ \delta>0,\end{aligned}\right.$ (1.19)

then the energy of weak solutions is globally conserved, that is, for any
$t\in[0,T]$

$\displaystyle\mathcal{E}(t)+\int_{0}^{T}\int_{\mathbb{T}^{d}}$
$\displaystyle\left[\nu(\rho)|\mathbb{D}v|^{2}+\mu(\rho)|\mathrm{div}\,v|^{2}\right]dxdt=\mathcal{E}(0),$
(1.20)

where
$\mathcal{E}(t)=\int_{\Omega}\left[\frac{1}{2}\rho|v|^{2}+P(\rho)\right]dx$
and $P(\rho)=\rho\int_{1}^{\rho}\frac{p(z)}{z^{2}}dz.$

######  Remark 1.1.

We follow the path of [21] to prove Theorem 1.1. The improvement of their
condition (LABEL:NNT) are threefold. First, Theorem 1.1 removed the additional
restriction on the regularity of density. Second, Theorem 1.1 not only covers
their result (LABEL:NNT) but also allow us to derive new criterion ( see the
following corollary). Third, the regularity of pressure $p(\rho)$ is relaxed
from $C^{2}(0,\infty)$ in [21] to $C^{1}(0,\infty)$.

######  Remark 1.2.

At first glance, energy conservation criteria (LABEL:u) based on a combination
of velocity and its gradient are more complicated than (1.22) and (1.23),
however, (LABEL:u) together with natural energy $v\in
L^{\infty}(0,T;L^{2}(\mathbb{T}^{3})),\nabla v\in
L^{2}(0,T;L^{2}(\mathbb{T}^{3}))$ leads to (1.22) and (1.23) in the following
corollary.

######  Remark 1.3.

By small modification of proof in Theorem 1.1, the results in Theorem 1.1 also
hold for homogenous incompressible Navier-Stokes equations (LABEL:NS), that
is, $v\in L^{\frac{2p}{p-1}}(0,T;L^{\frac{2q}{q-1}}(\mathbb{T}^{3}))$ and
$\nabla v\in L^{p}(0,T;L^{q}(\mathbb{T}^{3}))$ means the energy equality in
the classical homogenous incompressible Navier-Stokes equations. The special
case $p=q=2$ reduces to the famous Lions’ energy conservation criterion (1.2).
As mentioned in latter remark, this result covers the (1.3)-(1.5), hence,
roughly speaking, this unifies the known energy conservation criteria via the
velocity and its gradient in incompressible Navier-Stokes equations. After we
finished this paper, we learnt that a special case that $p=3,q=9/5$ and away
from $1/2$-Hölder continuous curve in time for general energy equality in the
homogeneous Navier-Stokes equations (LABEL:NS) in $\mathbb{R}^{3}$ was
considered in [28].

######  Remark 1.4.

The new ingredient in the proof of this theorem is the application of the
following inequality

$\Big{\|}\nabla\Big{(}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{)}\Big{\|}_{L^{p}(\mathbb{T}^{d})}\leq
C\|\nabla v\|_{L^{p}(\mathbb{T}^{d})}.$ (1.21)

This help us to pass the limit of pressure term only with the positive bounded
density, which removes the additional restriction of the density
$\eqref{NNT}_{2}$ in [21]. For the proof of (1.21), we refer the readers to
Lemma 2.3 (see also [17, page 7]).

######  Remark 1.5.

One can consider Theorem 1.1 and Corollary 1.2 on smooth bounded domain.
Combining the framework for bounded domain in [21] and the proof here, one
only needs to deal with the boundary terms caused by integrations by parts.
Fortunately, these additional terms are the lower order terms.

######  Remark 1.6.

In dimension $d=2$, the Gagliardo-Nirenberg inequality guarantees that

$\|v\|_{L^{4}(0,T;L^{4}(\mathbb{T}^{2}))}\leq
C\|v\|_{L^{\infty}(0,T;L^{2}(\mathbb{T}^{2}))}^{\frac{1}{2}}\|\nabla
v\|_{L^{2}(0,T;L^{2}(\mathbb{T}^{2}))}^{\frac{1}{2}}\leq C.$

Therefore, according to Theorem 1.1, the bounded density with positive lower
bound and natural energy yield the energy conservation of the weak solutions.

Taking the natural energy of weak solutions into account, one immediately
derives the following corollary.

######  Corollary 1.2.

When the dimension $d=3$, if the weak solutions $(\rho,v)$ to the Navier-
Stokes equation (LABEL:GNS) satisfy one of the following two conditions

  1. (1)

$0<c_{1}\leq\rho\leq c_{2}<\infty$ , $v\in
L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))$, $\nabla v\in
L^{2}(0,T;L^{2}(\mathbb{T}^{3}))$ and $\sqrt{\rho_{0}}v_{0}\in
L^{2+\delta}(\mathbb{T}^{3})$ for any $\delta>0$,

$v\in L^{p}(0,T;L^{q}(\mathbb{T}^{3}))~{}\text{with}~{}\left\\{\begin{aligned}
&\frac{2}{p}+\frac{2}{q}=1,q\geq 4,\\\
&\frac{1}{p}+\frac{3}{q}=1,3<q<4;\end{aligned}\right.$ (1.22)

  2. (2)

$0<c_{1}\leq\rho\leq c_{2}<\infty,$ $v\in
L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))$ , $\nabla v\in
L^{2}(0,T;L^{2}(\mathbb{T}^{3}))$ and $\sqrt{\rho_{0}}v_{0}\in
L^{2+\delta}(\mathbb{T}^{3})$ for any $\delta>0$,

$\nabla v\in
L^{p}(0,T;L^{q}(\mathbb{T}^{3}))~{}\text{with}~{}\left\\{\begin{aligned}
&\frac{1}{p}+\frac{3}{q}=2,\ \frac{3}{2}<q<\frac{9}{5},\\\
&\frac{1}{p}+\frac{6}{5q}=1,\frac{9}{5}\leq q,\end{aligned}\right.$ (1.23)

then the energy is globally conserved, that is, for any $t\in[0,T]$,

$\displaystyle\mathcal{E}(t)+\int_{0}^{T}\int_{\mathbb{T}^{3}}$
$\displaystyle\left[\nu(\rho)|\mathbb{D}v|^{2}+\mu(\rho)|\mathrm{div}\,v|^{2}\right]dxdt=\mathcal{E}(0),$
(1.24)

where
$\mathcal{E}(t)=\int_{\mathbb{T}^{3}}\left[\frac{1}{2}\rho|v|^{2}+P(\rho)\right]dx$
and $P(\rho)=\rho\int_{1}^{\rho}\frac{p(z)}{z^{2}}dz.$

######  Remark 1.7.

Compared with result (LABEL:NNT) obtained by Nguye-Nguye-Tang [21], conditions
1.22 only required the density is bounded from below and above. Hence, result
(1.22) is an improvement of (LABEL:NNT) in [21].

######  Remark 1.8.

We extend the energy conservation criteria (1.3)-(1.5) from incompressible
Navier-Stokes equations to general compressible Navier-Stokes equations with
no vacuum.

######  Remark 1.9.

In contrast with (1.12), the generalization in (1.23) is threefold: first, to
improve the corresponding results in (1.23); second, to consider the more
general equations; third, we can get the energy conservation up to the initial
time $t=0$.

######  Remark 1.10.

It seems that a new strategy for studying the energy equality of fluid
equations is to firstly establish a conservation criterion based on a
combination of velocity and its gradient, which may be applied to other
incompressible and compressible fluid equations. A successful application can
be found in [31].

######  Remark 1.11.

In the forthcoming work [33], the energy conservation criterion for the weak
solutions of general compressible Navier-Stokes equations allowing vacuum will
be considered.

Finally, as [21], one can establish the results parallel to Theorem 1.1 and
Corollary 1.2 for the non-homogenous incompressible Navier-Stokes equations
below

$\left\\{\begin{aligned} &\rho_{t}+\mathrm{div}\,(\rho v)=0,\\\ &(\rho
v)_{t}+\mathrm{div}\,(\rho v\otimes
v)-\mathrm{div}\,(\nu(\rho)\mathbb{D}v)+\nabla\pi=0,\\\ &\mathrm{div}\,v=0,\\\
&(\rho,v)|_{t=0}=(\rho_{0},u_{0}),\end{aligned}\right.$ (1.25)

we leave this to the interested readers.

The remainder of this paper is organized as follows. Section 2 is devoted to
the auxiliary lemmas involving mollifier and the key inequality (1.21). In
section 3, we first present the proof of Theorem 1.1. Then, based on Theorem
1.1, we complete the proof of Corollary 1.2.

##  2 Notations and some auxiliary lemmas

First, we introduce some notations used in this paper. For $p\in[1,\,\infty]$,
the notation $L^{p}(0,\,T;X)$ stands for the set of measurable functions on
the interval $(0,\,T)$ with values in $X$ and $\|f(\cdot,t)\|_{X}$ belonging
to $L^{p}(0,\,T)$. The classical Sobolev space $W^{k,p}(\mathbb{T}^{d})$ is
equipped with the norm
$\|f\|_{W^{k,p}(\mathbb{T}^{d})}=\sum\limits_{\alpha=0}^{k}\|D^{\alpha}f\|_{L^{p}(\mathbb{T}^{d})}$.
The space $C^{\infty}_{b}(\mathbb{T}^{d})$ is the bounded smooth functions on
$\mathbb{T}^{d}$. $c_{1},c_{2}$ and $C$ are positive constants. For
simplicity, we denote by

$\int_{0}^{T}\int_{\mathbb{T}^{d}}f(x,t)dxdt=\int_{0}^{T}\int f\
~{}~{}\text{and}~{}~{}\|f\|_{L^{p}(0,T;X)}=\|f\|_{L^{p}(X)}.$

Let $\eta_{\varepsilon}:\mathbb{R}^{d}\rightarrow\mathbb{R}$ be a standard
mollifier.i.e. $\eta(x)=C_{0}e^{-\frac{1}{1-|x|^{2}}}$ for $|x|<1$ and
$\eta(x)=0$ for $|x|\geq 1$, where $C_{0}$ is a constant such that
$\int_{\mathbb{R}^{d}}\eta(x)dx=1$. For $\varepsilon>0$, we define the
rescaled mollifier
$\eta_{\varepsilon}(x)=\frac{1}{\varepsilon^{d}}\eta(\frac{x}{\varepsilon})$.
For any function $f\in L^{1}_{loc}(\Omega)$, its mollified version is defined
as

$f^{\varepsilon}(x)=(f*\eta_{\varepsilon})(x)=\int_{\mathbb{R}^{d}}f(x-y)\eta_{\varepsilon}(y)dy,\
\ x\in\Omega_{\varepsilon},$

where $\Omega_{\varepsilon}=\\{x\in\Omega:d(x,\partial\Omega)>\varepsilon\\}.$

We first recall the results involving the mollifier established in [21].

######  Lemma 2.1.

([21]) Suppose that $f\in L^{p}(0,T;L^{q}(\mathbb{T}^{d}))$. Then for any
$\varepsilon>0$, there holds

$\|\nabla f^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}\leq
C\varepsilon^{-1}\|f\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))},$ (2.1)

and, if $p,q<\infty$

$\limsup_{\varepsilon\rightarrow 0}\varepsilon\|\nabla
f^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}=0.$

Moreover, if $0<c_{1}\leq g\leq c_{2}<\infty$, then there holds, for any
$\varepsilon>0$,

$\displaystyle\Big{\|}\nabla\frac{f^{\varepsilon}}{g^{\varepsilon}}\Big{\|}_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}\leq
C\varepsilon^{-1}\|f\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))},$ (2.2)

and if $p,q<\infty$

$\limsup_{\varepsilon\rightarrow
0}\varepsilon\Big{\|}\nabla\frac{f^{\varepsilon}}{g^{\varepsilon}}\Big{\|}_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}=0.$
(2.3)

The next lemma with $p=q,p_{1}=q_{1},p_{2}=q_{2}$ was proved in [21]. We
generalize it by extending the integral norms with different exponents in
space and time.

######  Lemma 2.2.

Let $1\leq p,q,p_{1},p_{2},q_{1},q_{2}\leq\infty$ with
$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$ and
$\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$. Assume $f\in
L^{p_{1}}(0,T;W^{1,q_{1}}(\mathbb{T}^{d}))$ and $g\in
L^{p_{2}}(0,T;L^{q_{2}}(\mathbb{T}^{d}))$. Then for any $\varepsilon>0$, there
holds

$\displaystyle\|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}\leq
C\varepsilon\|f\|_{L^{p_{1}}(0,T;W^{1,q_{1}}(\mathbb{T}^{d}))}\|g\|_{L^{p_{2}}(0,T;L^{q_{2}}(\mathbb{T}^{d}))}.$
(2.4)

Moreover, if $p_{2},q_{2}<\infty$ then

$\displaystyle\limsup_{\varepsilon\to
0}\varepsilon^{-1}\|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}=0.$
(2.5)

###### Proof.

Thanks to the fact observed in [12] and the ideas in [18], we know that

$\displaystyle(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}=R^{\varepsilon}-(f^{\varepsilon}-f)(g^{\varepsilon}-g),$
(2.6)

where

$\displaystyle
R^{\varepsilon}(x,t):=\int_{\mathbb{R}^{d}}\big{(}f\left(y,t\right)-f(x,t)\big{)}\big{(}g(y,t)-g(x,t)\big{)}\eta_{\varepsilon}(x-y)dy.$

Using the triangle’s inequality, it yields that

$\displaystyle\|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}\|_{L^{q}(\mathbb{T}^{d})}\leq
C\left(\|R^{\varepsilon}\|_{L^{q}(\mathbb{T}^{d})}+\|(f-f^{\varepsilon})(g-g^{\varepsilon})\|_{L^{q}(\mathbb{T}^{d})}\right).$
(2.7)

Let $B(x,\varepsilon)=\\{y\in\mathbb{T}^{d};|x-y|<\varepsilon\\}$, then by
means of Hölder’s inequality and direct computation, we see that

$\displaystyle|R^{\varepsilon}|$
$\displaystyle\leq\int_{B(x,\varepsilon)}\frac{1}{\varepsilon^{d}}|f(y)-f(x)||g(y)-g(x)|dy$
(2.8) $\displaystyle\leq
C\left(\frac{1}{\varepsilon^{d}}\int_{B(x,\varepsilon)}|f(y)-f(x)|^{s_{1}}dy\right)^{\frac{1}{s_{1}}}\left(\frac{1}{\varepsilon^{d}}\int_{B(x,\varepsilon)}|g(y)-g(x)|^{s_{2}}dy\right)^{\frac{1}{s_{2}}}$
$\displaystyle\leq
C\varepsilon\left(\frac{1}{\varepsilon^{d}}\int_{B(x,\varepsilon)}\int_{0}^{1}|\nabla
f(x+(y-x)s)|^{s_{1}}dsdy\right)^{\frac{1}{s_{1}}}\left(\frac{1}{\varepsilon^{d}}\int_{B(x,\varepsilon)}|g(y)|^{s_{2}}dy+|g(x)|^{s_{2}}\right)^{\frac{1}{s_{2}}}$
$\displaystyle\leq C\varepsilon\left(\int_{B(0,1)}\int_{0}^{1}|\nabla
f(x+\omega\varepsilon
s)|^{s_{1}}dsd\omega\right)^{\frac{1}{s_{1}}}\left(\int_{B(0,1)}|g(x+\omega\varepsilon)|^{s_{2}}d\omega+|g(x)|^{s_{2}}\right)^{\frac{1}{s_{2}}}$
$\displaystyle\leq C\varepsilon\left(\int_{\mathbb{R}^{d}}|\nabla
f(x-z)|^{s_{1}}\int_{0}^{1}\frac{\bf{1}_{B(0,\varepsilon s)}(z)}{(\varepsilon
s)^{d}}dsdz\right)^{\frac{1}{s_{1}}}\left(\int_{\mathbb{R}^{d}}|g(x-z)|^{s_{2}}\frac{\bf{1}_{B(0,\varepsilon)(z)}}{\varepsilon^{d}}dz+|g(x)|^{s_{2}}\right)^{\frac{1}{s_{2}}}$
$\displaystyle\leq C\left(|\nabla
f|^{s_{1}}*J_{\varepsilon}\right)^{\frac{1}{s_{1}}}\left(|g|^{s_{2}}*J_{1\varepsilon}+|g(x)|^{s_{2}}\right)^{\frac{1}{s_{2}}},$

where $s_{1}\leq q_{1},s_{2}\leq q_{2}$ with
$\frac{1}{s_{1}}+\frac{1}{s_{2}}=1$,
$J_{\varepsilon}=\int_{0}^{1}\frac{\bf{1}_{B(0,\varepsilon s)}}{(\varepsilon
s)^{d}}ds\geq 0$,
$J_{1\varepsilon}=\frac{\bf{1}_{B(0,\varepsilon)}}{\varepsilon^{d}}\geq 0$ and
$\int_{\mathbb{R}^{d}}\int_{0}^{1}\frac{\bf{1}_{B(0,\varepsilon
s)}}{(\varepsilon
s)^{d}}dsdz=\int_{\mathbb{R}^{d}}\frac{\bf{1}_{B(0,\varepsilon)}}{\varepsilon^{d}}dz=measure(B(0,1)).$

Then in view of the Minkowski inequality, we conclude that

$\displaystyle\|R^{\varepsilon}\|_{L^{q}}$ $\displaystyle\leq
C\varepsilon\|\left(|\nabla
f|^{s_{1}}*J_{\varepsilon}\right)^{\frac{1}{s_{1}}}\left(|g|^{s_{2}}*J_{1\varepsilon}+|g(x)|^{s_{2}}\right)^{\frac{1}{s_{2}}}\|_{L^{q}}$
(2.9) $\displaystyle\leq C\varepsilon\left[\|(|\nabla
f|^{s_{1}}*J_{\varepsilon})^{\frac{1}{s_{1}}}\|_{L^{q_{1}}}\left(\|(|g|^{s_{2}}*J_{1\varepsilon})^{\frac{1}{s_{2}}}\|_{L^{q_{2}}}+\|g\|_{L^{q_{2}}}\right)\right]$
$\displaystyle\leq C\varepsilon\|\nabla f\|_{L^{q_{1}}}\|g\|_{L^{q_{2}}}.$

Furthermore, one has

$\displaystyle|(f^{\varepsilon}-f)(g^{\varepsilon}-g)|$ (2.10)
$\displaystyle\leq$
$\displaystyle\int|(f(y)-f(x))|\eta_{\varepsilon}(x-y)dy\int|(g(y)-g(x))|\eta_{\varepsilon}(x-y)dy$
$\displaystyle\leq$ $\displaystyle
C\varepsilon\left(\frac{1}{\varepsilon^{d}}\int_{B(x,\varepsilon)}\int_{0}^{1}|\nabla
f(x+(y-x)s)|dsdy\right)\left(\frac{1}{\varepsilon^{d}}\int_{B(x,\varepsilon)}|g(y)-g(x)|dy\right)$
$\displaystyle\leq$ $\displaystyle
C\varepsilon\left(\frac{1}{\varepsilon^{d}}\int_{B(x,\varepsilon)}\int_{0}^{1}|\nabla
f(x+(y-x)s)|^{s_{1}}dsdy\right)^{\frac{1}{s_{1}}}\left(\frac{1}{\varepsilon^{d}}\int_{B(x,\varepsilon)}|g(y)-g(x)|^{s_{2}}dy\right)^{\frac{1}{s_{2}}}.$

Along the same lines of derivation of (2.8) and (2.9), we arrive at

$\|(f^{\varepsilon}-f)(g^{\varepsilon}-g)\|_{L^{q}}\leq C\varepsilon\|\nabla
f\|_{L^{q_{1}}}\|g\|_{L^{q_{2}}}.$ (2.11)

In combination with (2.6), (2.9) and (2.11) and using the Hölder’s inequality
with respect to time, we can deduce the result (2.4).

Furthermore, if $q_{1},q_{2}<\infty$, let $\\{g_{n}\\}\in
C_{b}^{\infty}(\mathbb{T}^{d})$ with $g_{n}\rightarrow g$ strongly in
$L^{q_{2}}$. Thus, by density arguments, we find that

$\displaystyle\|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}\|_{L^{q}}$
$\displaystyle\leq
C\left\|(\left(f(g-g_{n})\right)^{\varepsilon}+(fg_{n})^{\varepsilon}-f^{\varepsilon}(g-g_{n})^{\varepsilon}-f^{\varepsilon}g_{n}^{\varepsilon}\|_{L^{q}}\right)$
(2.12) $\displaystyle\leq
C\left(\|\left(f(g-g_{n})\right)^{\varepsilon}-f^{\varepsilon}(g-g_{n})^{\varepsilon}\|_{L^{q}}+\|(fg_{n})^{\varepsilon}-f^{\varepsilon}g_{n}^{\varepsilon}\|_{L^{q}}\right)$
$\displaystyle\leq C\left(\varepsilon\|\nabla
f\|_{L^{q_{1}}}\|g-g_{n}\|_{L^{q_{2}}}+\varepsilon^{2}\|\nabla
f\|_{L^{q_{1}}}\|\nabla g_{n}\|_{L^{q_{2}}}\right),$

which means

$\displaystyle\varepsilon^{-1}\|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}\|_{L^{q}}\leq
C\left(\|\nabla f\|_{L^{q_{1}}}\|g-g_{n}\|_{L^{q_{2}}}+\varepsilon\|\nabla
f\|_{L^{q_{1}}}\|\nabla g_{n}\|_{L^{q_{2}}}\right),$ (2.13)

hence, as $\varepsilon\rightarrow 0$ and $n\rightarrow\infty$ , we can obtain
that

$\displaystyle\varepsilon^{-1}\|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}\|_{L^{p}(L^{q})}$
(2.14) $\displaystyle\leq$ $\displaystyle C\left(\int_{0}^{T}\left(\|\nabla
f\|_{L^{q_{1}}}\|g-g_{n}\|_{L^{q_{2}}}+\varepsilon\|\nabla
f\|_{L^{q_{1}}}\|\nabla g_{n}\|_{L^{q_{2}}}\right)^{p}dt\right)^{\frac{1}{p}}$
$\displaystyle\leq$ $\displaystyle C\left(\int_{0}^{T}(\|\nabla
f\|_{L^{q_{1}}}\|g-g_{n}\|_{L^{q_{2}}})^{p}dt\right)^{\frac{1}{p}}+C\varepsilon\left(\int_{0}^{T}(\|\nabla
f\|_{L^{q_{1}}}\|\nabla g_{n}\|_{L^{q_{2}}})^{p}dt\right)^{\frac{1}{p}}$
$\displaystyle\leq$ $\displaystyle C\|\nabla
f\|_{L^{p_{1}}(L^{q_{1}})}\|g-g_{n}\|_{L^{p_{2}}(L^{q_{2}})}+\varepsilon\|\nabla
f\|_{L^{p_{1}}(L^{q_{1}})}\|\nabla g_{n}\|_{L^{p_{2}}(L^{q_{2}})}\rightarrow
0.$

Then, we have completed the proof of Lemma 2.2. ∎

The next lemma is the key to remove $\eqref{NNT}_{2}$.

######  Lemma 2.3.

Assume that $0<\underline{\rho}\leq\rho(x,t)\leq\overline{\rho}<\infty$ and
$v\in W^{1,p}(\mathbb{T}^{d})$ with $1\leq p\leq\infty$. Then

$\Big{\|}\partial\left(\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\right)\Big{\|}_{L^{p}(\mathbb{T}^{d})}\leq
C\|\nabla v\|_{L^{p}(\mathbb{T}^{d})}.$ (2.15)

###### Proof.

By direct computation, one has

$\partial\left(\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\right)=\frac{\partial(\rho
v)^{\varepsilon}-v\partial\rho^{\varepsilon}}{\rho^{\varepsilon}}-\frac{\left((\rho
v)^{\varepsilon}-\rho^{\varepsilon}v\right)\partial\rho^{\varepsilon}}{(\rho^{\varepsilon})^{2}}:=I_{1}+I_{2}.$
(2.16)

Let $B(x,\varepsilon)=\\{y\in\mathbb{T}^{d};|x-y|<\varepsilon\\}$, then Using
the Hölder’s inequality, we have

$\displaystyle|I_{1}|$ $\displaystyle\leq
C|\int\rho(y)\left(v(y)-v(x)\right)\nabla_{x}\eta_{\varepsilon}(x-y)dy|$
(2.17) $\displaystyle\leq
C\|\rho\|_{L^{\infty}}|\int_{\mathbb{R}^{d}}|v(y)-v(x)|\frac{1}{\varepsilon^{d}}\nabla\eta(\frac{x-y}{{}^{\varepsilon}})\frac{1}{{}^{\varepsilon}}dy|$
$\displaystyle\leq
C\left(\frac{1}{\varepsilon^{d}}\int_{B(x,\varepsilon)}\frac{|v(y)-v(x)|^{p}}{\varepsilon^{p}}dy\right)^{\frac{1}{p}}.$

Then using the mean value theorem, one can obtain

$\displaystyle\frac{1}{\varepsilon^{d}}\int_{B(x,\varepsilon)}\frac{|v(y)-v(x)|^{p}}{\varepsilon^{p}}dy$
$\displaystyle\leq
C\frac{1}{\varepsilon^{d}}\int_{B(x,\varepsilon)}\int_{0}^{1}|\nabla
v(x+(y-x)s)|^{p}\frac{|y-x|^{p}}{\varepsilon^{p}}dsdy$ (2.18)
$\displaystyle\leq C\int_{0}^{1}\int_{B(0,1)}|\nabla
v(x+s\varepsilon\omega)|^{p}d\omega ds$ $\displaystyle\leq
C\int_{\mathbb{R}^{d}}|\nabla
v(x-z)|^{p}\int_{0}^{1}\frac{\bf{1}_{(B(0,\varepsilon s)}(z)}{(\varepsilon
s)^{d}}dsdz$ $\displaystyle=(|\nabla v|^{p}*J_{\varepsilon})(x),$

where $J_{\varepsilon}(z)=\int_{0}^{1}\frac{\bf{1}_{(B(0,\varepsilon
s)}(z)}{(\varepsilon s)^{d}}ds\geq 0$ and it’s easy to check that
$\int_{\mathbb{R}^{d}}J_{\varepsilon}dz={\text{\ }measure\ of\ }(B(0,1))$.
Next, to estimate $I_{2}$, due to the Hölder’s inequality, one deduces

$\displaystyle|I_{2}|$
$\displaystyle=|\int\rho(y)\left(v(y)-v(x)\right)\eta_{\varepsilon}(x-y)dy\frac{\int\rho(y)\nabla_{x}\eta_{\varepsilon}(x-y)dy}{\left(\int\rho(y)\eta_{\varepsilon}(x-y)dy\right)^{2}}|$
(2.19) $\displaystyle\leq
C\|\rho\|_{L^{\infty}}^{2}\int_{B(x,\varepsilon)}|v(y)-v(x)|\frac{1}{\varepsilon^{d}}dy\int_{B(x,\varepsilon)}\frac{1}{\varepsilon^{d}}|\nabla\eta(\frac{x-y}{\varepsilon})|\frac{1}{\varepsilon}dy$
$\displaystyle\leq
C\left(\frac{1}{\varepsilon^{d}}\int_{B(x,\varepsilon)}\frac{|v(y)-v(x)|^{p}}{\varepsilon^{p}}dy\right)^{\frac{1}{p}}.$

Therefore, by the same arguments as in (2.18), in combination with
(2.16)-(2.19), we have

$|I_{1}|+|I_{2}|\leq C\left(\nabla
v|^{p}*J_{\varepsilon}\right)^{\frac{1}{p}}.$ (2.20)

Then from the Minkowski’s inequality, we arrive at

$\displaystyle\Big{\|}\partial\left(\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\right)\Big{\|}_{L^{p}(\mathbb{T}^{d})}$
$\displaystyle\leq C\|\left(|\nabla
v|^{p}*J_{\varepsilon}\right)^{\frac{1}{p}}\|_{L^{p}}$ (2.21)
$\displaystyle\leq C\|\nabla
v\|_{L^{p}}\|J_{\varepsilon}\|_{L^{1}}^{\frac{1}{p}}\leq C\|\nabla
v\|_{L^{p}}.$

Then we have completed the proof of lemma (2.3). ∎

##  3 Proof of Theorem 1.1 and Corollary 1.2

In this section, we first present the proof of Theorem 1.1. Then, making use
of interpolation and the natural energy, we prove Corollary 1.2 by the results
of Theorem 1.1.

###### Proof of Theorem 1.1.

Let $\phi(t)$ be a smooth function compactly supported in $(0,+\infty)$.
Multiplying $\eqref{GNS}_{2}$ by $\left(\phi(t)\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\right)^{\varepsilon}$, then integrating
it over $(0,T)\times\mathbb{T}^{d}$, we have

$\displaystyle\int_{0}^{T}\int\phi(t)\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{[}\partial_{t}(\rho
v)^{\varepsilon}+\mathrm{div}\,(\rho v\otimes v)^{\varepsilon}+\nabla
p(\rho)^{\varepsilon}-\mathrm{div}\,(\nu(\rho)\mathbb{D}v)^{\varepsilon}-\nabla(\mu(\rho)\mathrm{div}\,v)^{\varepsilon}\Big{]}=0.$
(3.1)

We will rewrite every term of the last equality to pass the limit of
$\varepsilon$. For the first term in (3.1), a straightforward calculation and
$\eqref{GNS}_{1}$ yields that

$\displaystyle\int_{0}^{T}\int\phi(t)\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\partial_{t}\Big{(}\rho
v\Big{)}^{\varepsilon}$
$\displaystyle=\int_{0}^{T}\int\phi(t)\left[\frac{1}{2}\partial_{t}(\frac{|(\rho
v)^{\varepsilon}|^{2}}{\rho^{\varepsilon}})+\frac{1}{2}\partial_{t}\rho^{\varepsilon}\frac{|(\rho
v)^{\varepsilon}|^{2}}{(\rho^{\varepsilon})^{2}}\right]$ (3.2)
$\displaystyle=\int_{0}^{T}\int\phi(t)\left[\frac{1}{2}\partial_{t}\Big{(}\frac{|(\rho
v)^{\varepsilon}|^{2}}{\rho^{\varepsilon}}\Big{)}-\frac{1}{2}\mathrm{div}\,(\rho
v)^{\varepsilon}\frac{|(\rho
v)^{\varepsilon}|^{2}}{(\rho^{\varepsilon})^{2}}\right].$

For the second term in (3.1), by integration by parts, it gives that

$\displaystyle\int_{0}^{T}\int\phi(t)\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\mathrm{div}\,(\rho v\otimes
v)^{\varepsilon}$ (3.3) $\displaystyle=$
$\displaystyle-\int_{0}^{T}\int\phi(t)\nabla\Big{(}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{)}[(\rho v\otimes
v)^{\varepsilon}-(\rho v)^{\varepsilon}\otimes
v^{\varepsilon}]-\int_{0}^{T}\int\phi(t)\nabla\Big{(}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{)}(\rho v)^{\varepsilon}\otimes
v^{\varepsilon}.$

For the second term on the right hand side of above equality (3.3), it follows
from the integration by parts once again that

$\displaystyle-\int_{0}^{T}\int\phi(t)\nabla\Big{(}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{)}(\rho v)^{\varepsilon}\otimes
v^{\varepsilon}$ (3.4) $\displaystyle=$
$\displaystyle\int_{0}^{T}\int\phi(t)\left(\mathrm{div}\,v^{\varepsilon}\frac{|(\rho
v)^{\varepsilon}|^{2}}{\rho^{\varepsilon}}+\frac{1}{2}\frac{v^{\varepsilon}}{\rho^{\varepsilon}}\nabla|(\rho
v)^{\varepsilon}|^{2}\right)$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\int\phi(t)\left(\frac{1}{2}\mathrm{div}\,v^{\varepsilon}\frac{|(\rho
v)^{\varepsilon}|^{2}}{\rho^{\varepsilon}}-\frac{1}{2}v^{\varepsilon}\nabla({\frac{1}{\rho^{\varepsilon}}})|(\rho
v)^{\varepsilon}|^{2}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{T}\int\phi(t)\mathrm{div}\,(\rho^{\varepsilon}v^{\varepsilon})\frac{|(\rho
v)^{\varepsilon}|^{2}}{(\rho^{\varepsilon})^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{T}\int\phi(t)\mathrm{div}\,\Big{[}\rho^{\varepsilon}v^{\varepsilon}-(\rho
v)^{\varepsilon}\Big{]}\frac{|(\rho
v)^{\varepsilon}|^{2}}{(\rho^{\varepsilon})^{2}}+\frac{1}{2}\int_{0}^{T}\int\phi(t)\mathrm{div}\,(\rho
v)^{\varepsilon}\frac{|(\rho v)^{\varepsilon}|^{2}}{(\rho^{\varepsilon})^{2}}$
$\displaystyle=$
$\displaystyle-\int_{0}^{T}\int\phi(t)\Big{[}\rho^{\varepsilon}v^{\varepsilon}-(\rho
v)^{\varepsilon}\Big{]}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\nabla\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}+\frac{1}{2}\int_{0}^{T}\int\phi(t)\mathrm{div}\,(\rho
v)^{\varepsilon}\frac{|(\rho
v)^{\varepsilon}|^{2}}{(\rho^{\varepsilon})^{2}}.$

Then inserting (3.4) into (3.3), we have

$\displaystyle\int_{0}^{T}\int\phi(t)\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\mathrm{div}\,(\rho v\otimes
v)^{\varepsilon}$ (3.5) $\displaystyle=$
$\displaystyle-\int_{0}^{T}\int\phi(t)\nabla\Big{(}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{)}[(\rho v\otimes
v)^{\varepsilon}-(\rho v)^{\varepsilon}\otimes v^{\varepsilon}]$
$\displaystyle-\int_{0}^{T}\int\phi(t)\Big{[}\rho^{\varepsilon}v^{\varepsilon}-(\rho
v)^{\varepsilon}\Big{]}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\nabla\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}+\frac{1}{2}\int_{0}^{T}\int\phi(t)\mathrm{div}\,(\rho
v)^{\varepsilon}\frac{|(\rho
v)^{\varepsilon}|^{2}}{(\rho^{\varepsilon})^{2}}.$

For the pressure term in (3.1), together with the integration by parts, one
has

$\displaystyle\int_{0}^{T}\int\phi(t)\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\nabla(p(\rho))^{\varepsilon}$ (3.6)
$\displaystyle=$ $\displaystyle\int_{0}^{T}\int\phi(t)\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\nabla\left[(p(\rho))^{\varepsilon}-p(\rho^{\varepsilon})\right]+\int_{0}^{T}\int\phi(t)\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\nabla p(\rho^{\varepsilon})$
$\displaystyle=$
$\displaystyle-\int_{0}^{T}\int\phi(t)\mathrm{div}\,\Big{[}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{]}[(p(\rho))^{\varepsilon}-p(\rho^{\varepsilon})]+\int_{0}^{T}\int\phi(t)\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\nabla p(\rho^{\varepsilon}).$

Using the mass equation $\eqref{GNS}_{1}$, the second term on the right hand-
side of (3.6) can be rewritten as

$\displaystyle\int_{0}^{T}\int\phi(t)\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\nabla p(\rho^{\varepsilon})$
$\displaystyle=\int_{0}^{T}\int\phi(t)(\rho
v)^{\varepsilon}\nabla\int_{1}^{\rho^{\varepsilon}}\frac{p^{{}^{\prime}}(z)}{z}dzdxdt$
(3.7)
$\displaystyle=\int_{0}^{T}\int\phi(t)\partial_{t}\rho^{\varepsilon}\left[\frac{p(\rho^{\varepsilon})}{\rho^{\varepsilon}}+\int_{1}^{\rho^{\varepsilon}}\frac{p(z)}{z^{2}}dz\right]dxdt$
$\displaystyle=\int_{0}^{T}\int\phi(t)\partial_{t}P(\rho^{\varepsilon}),$

where
$P(\rho^{\varepsilon})=\rho^{\varepsilon}\int_{1}^{\rho^{\varepsilon}}\frac{p(z)}{z^{2}}dz.$
Finally, for the viscous terms in (3.1), using the integration by parts, we
have

$\displaystyle-\int_{0}^{T}\int\phi(t)\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\mathrm{div}\,(\nu(\rho)\mathbb{D}v)^{\varepsilon}$
(3.8) $\displaystyle=$
$\displaystyle\int_{0}^{T}\int\phi(t)\left(-\mathrm{div}\,(\nu(\rho)\mathbb{D}v)^{\varepsilon}v^{\varepsilon}-\mathrm{div}\,(\nu(\rho)\mathbb{D}v)^{\varepsilon}\frac{(\rho
v)^{\varepsilon}-\rho^{\varepsilon}v^{\varepsilon}}{\rho^{\varepsilon}}\right),$

and

$\displaystyle-\int_{0}^{T}\int\phi(t)\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\nabla(\mu(\rho)\mathrm{div}\,v)^{\varepsilon}$
(3.9) $\displaystyle=$
$\displaystyle\int_{0}^{T}\int\phi(t)\left(-\nabla(\mu(\rho)\mathrm{div}\,v)^{\varepsilon}v^{\varepsilon}-\nabla(\mu(\rho)\mathrm{div}\,v)^{\varepsilon}\frac{(\rho
v)^{\varepsilon}-\rho^{\varepsilon}v^{\varepsilon}}{\rho^{\varepsilon}}\right).$

Then substituting (3.2), (3.5)-(3.9) into (3.1), we see that

$\displaystyle\int_{0}^{T}\int\phi(t)\partial_{t}\left(\frac{1}{2}\frac{|(\rho
v)^{\varepsilon}|^{2}}{\rho^{\varepsilon}}+P(\rho^{\varepsilon})\right)-\int_{0}^{T}\int\phi(t)\left(\mathrm{div}\,\left(\nu(\rho)\mathbb{D}v\right)^{\varepsilon}v^{\varepsilon}+\nabla(\mu(\rho)\mathrm{div}\,v)^{\varepsilon}v^{\varepsilon}\right)$
(3.10) $\displaystyle=$
$\displaystyle\int_{0}^{T}\int\phi(t)\mathrm{div}\,(\nu(\rho)\mathbb{D}v)^{\varepsilon}\frac{(\rho
v)^{\varepsilon}-\rho^{\varepsilon}v^{\varepsilon}}{\rho^{\varepsilon}}+\int_{0}^{T}\int\phi(t)\nabla(\mu(\rho)\mathrm{div}\,v)^{\varepsilon}\frac{(\rho
v)^{\varepsilon}-\rho^{\varepsilon}v^{\varepsilon}}{\rho^{\varepsilon}}$
$\displaystyle\ \ \ +\int_{0}^{T}\int\phi(t)\mathrm{div}\,\Big{[}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{]}[(p(\rho))^{\varepsilon}-p(\rho^{\varepsilon})]$
$\displaystyle+$ $\displaystyle\int_{0}^{T}\int\phi(t)\nabla\Big{(}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{)}[(\rho v\otimes
v)^{\varepsilon}-(\rho v)^{\varepsilon}\otimes
v^{\varepsilon}]+\int_{0}^{T}\int\phi(t)\Big{[}\rho^{\varepsilon}v^{\varepsilon}-(\rho
v)^{\varepsilon}\Big{]}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\nabla\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}.$

Next, we need to prove that the terms on the right hand-side of (3.10) tend to
zero as $\varepsilon\rightarrow 0$.

Firstly, it follows from Lemma 2.1 and Lemma 2.2 that

$\displaystyle\|\mathrm{div}\,(\nu(\rho)\mathbb{D}v)^{\varepsilon}\|_{L^{2}(L^{2})}\leq
C\varepsilon^{-1}\|\nu(\rho)\mathbb{D}v\|_{L^{2}(L^{2})}\leq
C\varepsilon^{-1}\|\nabla v\|_{L^{2}(L^{2})},$ (3.11)
$\displaystyle\limsup_{\varepsilon\rightarrow
0}\varepsilon\|\mathrm{div}\,(\nu(\rho)\mathbb{D}v)^{\varepsilon}\|_{L^{2}(L^{2})}=0,$
$\displaystyle\|(\rho
v)^{\varepsilon}-\rho^{\varepsilon}v^{\varepsilon}\|_{L^{2}(L^{2})}\leq
C\varepsilon\|\rho\|_{L^{\infty}(L^{\infty})}\|v\|_{L^{2}(W^{1,2})}.$

Moreover, due to the Hölder’s inequality, we can obtain that

$\displaystyle\Big{|}\int_{0}^{T}\int\phi(t)\mathrm{div}\,(\nu(\rho)\mathbb{D}v)^{\varepsilon}\frac{(\rho
v)^{\varepsilon}-\rho^{\varepsilon}v^{\varepsilon}}{\rho^{\varepsilon}}\Big{|}$
(3.12) $\displaystyle\leq$ $\displaystyle
C\|\mathrm{div}\,(\nu(\rho)\mathbb{D}v)^{\varepsilon}\|_{L^{2}(L^{2})}\|\frac{(\rho
v)^{\varepsilon}-\rho^{\varepsilon}v^{\varepsilon}}{\rho^{\varepsilon}}\|_{L^{2}(L^{2})}$
$\displaystyle\leq$ $\displaystyle
C\varepsilon\|\mathrm{div}\,(\nu(\rho)\mathbb{D}v)^{\varepsilon}\|_{L^{2}(L^{2})}\|\rho\|_{L^{\infty}L^{\infty}}\|v\|_{L^{2}(W^{1,2})}.$

As a consequence, in combination with (3.11) and (3.12), we have

$\limsup_{\varepsilon\rightarrow
0}\Big{|}\int_{0}^{T}\int\phi(t)\mathrm{div}\,(\nu(\rho)\mathbb{D}v)^{\varepsilon}\frac{(\rho
v)^{\varepsilon}-\rho^{\varepsilon}v^{\varepsilon}}{\rho^{\varepsilon}}\Big{|}=0.$

Likewise, there also holds

$\displaystyle\limsup_{\varepsilon\rightarrow
0}\Big{|}\int_{0}^{T}\int\phi(t)\nabla(\mu(\rho)\mathrm{div}\,v)^{\varepsilon}\frac{(\rho
v)^{\varepsilon}-\rho^{\varepsilon}v^{\varepsilon}}{\rho^{\varepsilon}}\Big{|}=0.$
(3.13)

Next, by means of the triangle inequality, the Hölder inequality and Lemma
2.3, we obtain

$\displaystyle\int_{0}^{T}\int\phi(t)\mathrm{div}\,\Big{[}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{]}[(p(\rho))^{\varepsilon}-p(\rho^{\varepsilon})]$
(3.14) $\displaystyle\leq$
$\displaystyle\int_{0}^{T}\int\phi(t)\Big{|}\mathrm{div}\,\Big{[}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{]}\Big{|}|(p(\rho))^{\varepsilon}-p(\rho)|+\int_{0}^{T}\int\phi(t)\Big{|}\mathrm{div}\,\Big{[}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{]}\Big{|}|p(\rho)-p(\rho^{\varepsilon})|$
$\displaystyle\leq$ $\displaystyle C\|\mathrm{div}\,\Big{[}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{]}\|_{L^{2}(L^{2})}\Big{(}\|(p(\rho))^{\varepsilon}-p(\rho)\|_{L^{2}(L^{2})}+\|p(\rho)-p(\rho^{\varepsilon})\|_{L^{2}(L^{2})}\Big{)}$
$\displaystyle\leq$ $\displaystyle C\|\nabla
v\|_{L^{2}(L^{2})}\Big{(}\|(p(\rho))^{\varepsilon}-p(\rho)\|_{L^{2}(L^{2})}+\|p^{\prime}\|_{L^{\infty}(L^{\infty})}\|\rho-\rho^{\varepsilon}\|_{L^{2}(L^{2})}\Big{)},$

which implies that

$\limsup_{\varepsilon\rightarrow
0}\int_{0}^{T}\int\phi(t)\mathrm{div}\,\left(\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\right)\big{(}(p(\rho))^{\varepsilon}-p(\rho^{\varepsilon})\big{)}=0.$

At this stage, it is enough to show

$\displaystyle\limsup_{\varepsilon\rightarrow 0}\int_{0}^{T}\int$
$\displaystyle\phi(t)\nabla\Big{(}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{)}[(\rho v\otimes
v)^{\varepsilon}-(\rho v)^{\varepsilon}\otimes v^{\varepsilon}]$ (3.15)
$\displaystyle+\limsup_{\varepsilon\rightarrow
0}\int_{0}^{T}\int\phi(t)\Big{[}\rho^{\varepsilon}v^{\varepsilon}-(\rho
v)^{\varepsilon}\Big{]}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\nabla\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}=0,$

under the hypothesis

$v\in L^{\frac{2p}{p-1}}(L^{\frac{2q}{q-1}})~{}~{}\text{ and }~{}~{}\nabla
v\in L^{p}(L^{q}).$ (3.16)

To do this, applying Lemma 2.2, we obtain that

$\displaystyle\|(\rho v\otimes v)^{\varepsilon}-(\rho v)^{\varepsilon}\otimes
v^{\varepsilon}\|_{L^{\frac{2p}{p+1}}(L^{\frac{2q}{q+1}})}\leq
C\varepsilon\|v\|_{L^{p}(W^{1,q})}\|\rho
v\|_{L^{\frac{2p}{p-1}}(L^{\frac{2q}{q-1}})},$ (3.17)
$\displaystyle\Big{\|}\nabla\Big{(}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{)}\Big{\|}_{L^{\frac{2p}{p-1}}(L^{\frac{2q}{q-1}})}\leq
C\varepsilon^{-1}\|\rho v\|_{L^{\frac{2p}{p-1}}(L^{\frac{2q}{q-1}})},$
$\displaystyle\limsup_{\varepsilon\rightarrow
0}\varepsilon\Big{\|}\nabla\Big{(}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{)}\Big{\|}_{L^{\frac{2p}{p-1}}(L^{\frac{2q}{q-1}})}=0.$

Using the Hölder’s inequality and Lemma 2.1, we find

$\displaystyle\Big{|}\int_{0}^{T}\int\phi(t)\nabla\Big{(}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{)}[(\rho v\otimes
v)^{\varepsilon}-(\rho v)^{\varepsilon}\otimes v^{\varepsilon}]\Big{|}$ (3.18)
$\displaystyle\leq$ $\displaystyle C\Big{\|}\nabla\Big{(}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{)}\Big{\|}_{L^{\frac{2p}{p-1}}(L^{\frac{2q}{q-1}})}\|(\rho
v\otimes v)^{\varepsilon}-(\rho v)^{\varepsilon}\otimes
v^{\varepsilon}\|_{L^{\frac{2p}{p+1}}(L^{\frac{2q}{q+1}})}$
$\displaystyle\leq$ $\displaystyle
C\varepsilon\Big{\|}\nabla\Big{(}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{)}\Big{\|}_{L^{\frac{2p}{p-1}}(L^{\frac{2q}{q-1}})}\|v\|_{L^{p}(W^{1,q})}\|\rho
v\|_{L^{\frac{2p}{p-1}}(L^{\frac{2q}{q-1}})}$ $\displaystyle\leq$
$\displaystyle C\varepsilon\Big{\|}\nabla\Big{(}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{)}\Big{\|}_{L^{\frac{2p}{p-1}}(L^{\frac{2q}{q-1}})}\|v\|_{L^{p}(W^{1,q})}\|v\|_{L^{\frac{2p}{p-1}}(L^{\frac{2q}{q-1}})}$
$\displaystyle\leq$ $\displaystyle
C\varepsilon\Big{\|}\nabla\Big{(}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{)}\Big{\|}_{L^{\frac{2p}{p-1}}(L^{\frac{2q}{q-1}})},$

which in turn gives

$\limsup_{\varepsilon\rightarrow
0}\Big{|}\int_{0}^{T}\int\phi(t)\nabla\Big{(}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{)}[(\rho v\otimes
v)^{\varepsilon}-(\rho v)^{\varepsilon}\otimes v^{\varepsilon}]\Big{|}=0.$

Now, we turn our attentions to the term
$\int_{0}^{T}\int\phi(t)\Big{[}\rho^{\varepsilon}v^{\varepsilon}-(\rho
v)^{\varepsilon}\Big{]}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\nabla\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}$. Since $\rho v\in
L^{\frac{2p}{p-1}}(L^{\frac{2q}{q-1}})$, we derive from Lemma 2.1 that

$\limsup_{\varepsilon\rightarrow 0}\varepsilon\Big{\|}\nabla\left(\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\right)\Big{\|}_{L^{\frac{2p}{p-1}}(L^{\frac{2q}{q-1}})}=0.$
(3.19)

In addition, we conclude from Lemma 2.2 that

$\|\rho^{\varepsilon}v^{\varepsilon}-(\rho
v)^{\varepsilon}\|_{L^{p}(L^{q})}\leq
C\varepsilon\|v\|_{L^{p}(W^{1,q})}\|\rho\|_{L^{\infty}(L^{\infty})}.$ (3.20)

Then, together with the Hölder’s inequality and (3.20), we have,

$\displaystyle\Big{|}\int_{0}^{T}\int\phi(t)\Big{[}\rho^{\varepsilon}v^{\varepsilon}-(\rho
v)^{\varepsilon}\Big{]}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\nabla\left(\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\right)\Big{|}$ (3.21)
$\displaystyle\leq$ $\displaystyle C\|\rho^{\varepsilon}v^{\varepsilon}-(\rho
v)^{\varepsilon}\|_{L^{p}(L^{q})}\Big{\|}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{\|}_{L^{\frac{2p}{p-1}}(L^{\frac{2q}{q-1}})}\Big{\|}\nabla\left(\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\right)\Big{\|}_{L^{\frac{2p}{p-1}}(L^{\frac{2q}{q-1}})}$
$\displaystyle\leq$ $\displaystyle
C\|v\|_{L^{p}(W^{1,q})}\|\rho\|_{L^{\infty}(L^{\infty})}\|v\|_{L^{\frac{2p}{p-1}}(L^{\frac{2q}{q-1}})}\varepsilon\Big{\|}\nabla\left(\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\right)\Big{\|}_{L^{\frac{2p}{p-1}}(L^{\frac{2q}{q-1}})}$
$\displaystyle\leq$ $\displaystyle C\varepsilon\Big{\|}\nabla\left(\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\right)\Big{\|}_{L^{\frac{2p}{p-1}}(L^{\frac{2q}{q-1}})},$

which together with (3.19) yields that

$\limsup_{\varepsilon\rightarrow
0}\Big{|}\int_{0}^{T}\int\phi(t)\Big{[}\rho^{\varepsilon}v^{\varepsilon}-(\rho
v)^{\varepsilon}\Big{]}\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\nabla\frac{(\rho
v)^{\varepsilon}}{\rho^{\varepsilon}}\Big{|}=0.$

Collecting all the above estimates, using the integration by parts with
respect to $t$ and letting $\varepsilon\rightarrow 0$, for any
$\phi(t)\in\mathcal{D}(0,T)$, we have

$\displaystyle-\int_{0}^{T}\int\partial_{t}\phi(t)\left(\frac{1}{2}\rho
v^{2}+P(\rho)\right)+\int_{0}^{T}\int\phi(t)\left(\nu(\rho)|\mathbb{D}v|^{2}+\mu(\rho)|\mathrm{div}\,v|^{2}\right)=0.$
(3.22)

The next objective is to get the energy equality up to the initial time $t=0$.
First, we claim that for any $t_{0}\geq 0$,

$\lim\limits_{t\rightarrow
t_{0}^{+}}\|\sqrt{\rho}v(t)\|_{L^{2}(\mathbb{T}^{d})}=\|\sqrt{\rho}v(t_{0})\|_{L^{2}(\mathbb{T}^{d})}\
\ \text{and}\ \ \lim\limits_{t\rightarrow
t_{0}^{+}}\|P(\rho)(t)\|_{L^{1}(\mathbb{T}^{d})}=\|P(\rho)(t_{0})\|_{L^{1}(\mathbb{T}^{d})}.$

In fact, by the energy estimates (1.15) and the weak continuity of $\rho$ and
$\rho v$ in (1.16) and (1.17), we have

$\displaystyle\rho v\in L^{\infty}(0,T;L^{2}(\mathbb{T}^{d}))\cap
H^{1}(0,T;W^{-1,1}(\mathbb{T}^{d}))\hookrightarrow
C([0,T];L^{2}_{weak}(\mathbb{T}^{d})),$ (3.23) and
$\displaystyle\sqrt{\rho}\in L^{\infty}(0,T;L^{\infty}(\mathbb{T}^{d}))\cap
H^{1}(0,T;H^{-1}(\mathbb{T}^{d}))\hookrightarrow
C([0,T];L^{l}_{weak}(\mathbb{T}^{d})),\ \text{for\ any}\ l\in(1,+\infty)$

Due to the convexity of $\rho\mapsto P(\rho)$, we have

$\displaystyle\int_{\mathbb{T}^{d}}P(\rho)(t_{0})\leq\underline{\lim\limits_{t\rightarrow
t_{0}^{+}}}\int_{\mathbb{T}^{d}}P(\rho)(t),\ \ \text{for\ any}\ t_{0}\geq 0.$
(3.24)

Meanwhile, using the natural energy (1.15),(1.18), (3.23)and (3.24), we have

$\displaystyle 0$ $\displaystyle\leq\overline{\lim_{t\rightarrow
0^{+}}}\int|\sqrt{\rho}v-\sqrt{\rho_{0}}v_{0}|^{2}dx$ (3.25)
$\displaystyle=2\overline{\lim_{t\rightarrow
0^{+}}}\left(\int\left(\frac{1}{2}\rho|v|^{2}+P(\rho)\right)dx-\int\left(\frac{1}{2}\rho_{0}|v_{0}|^{2}+P(\rho_{0})\right)dx\right)$
$\displaystyle\ \ \ +2\overline{\lim_{t\rightarrow
0^{+}}}\left(\int\sqrt{\rho_{0}}v_{0}\left(\sqrt{\rho_{0}}v_{0}-\sqrt{\rho}v\right)dx+\int\left(P(\rho_{0})-P(\rho)\right)dx\right)$
$\displaystyle\leq 2\overline{\lim_{t\rightarrow
0^{+}}}\int\sqrt{\rho_{0}}v_{0}\left(\sqrt{\rho_{0}}v_{0}-\sqrt{\rho}v\right)dx$
$\displaystyle=0,$

where the last equality sign comes from

$\displaystyle 2\overline{\lim_{t\rightarrow
0^{+}}}\int\sqrt{\rho_{0}}v_{0}\left(\sqrt{\rho_{0}}v_{0}-\sqrt{\rho}v\right)dx$
(3.26) $\displaystyle=$ $\displaystyle 2\overline{\lim_{t\rightarrow
0^{+}}}\int\frac{\sqrt{\rho_{0}}v_{0}}{\sqrt{\rho}}\left(\sqrt{\rho}\sqrt{\rho_{0}}v_{0}-\rho
v\right)dx$ $\displaystyle\leq$ $\displaystyle 2\overline{\lim_{t\rightarrow
0^{+}}}\int\frac{\sqrt{\rho_{0}}v_{0}}{\sqrt{\rho}}\left(\sqrt{\rho}\sqrt{\rho_{0}}v_{0}-\rho_{0}v_{0}\right)dx+2\overline{\lim_{t\rightarrow
0^{+}}}\int\frac{\sqrt{\rho_{0}}v_{0}}{\sqrt{\rho}}\left({\rho_{0}}v_{0}-\rho
v\right)dx$ $\displaystyle\leq$ $\displaystyle 2\overline{\lim_{t\rightarrow
0^{+}}}\int(\sqrt{\rho_{0}}v_{0})^{2}\left(\sqrt{\rho}-\sqrt{\rho_{0}}\right)dx+2\overline{\lim_{t\rightarrow
0^{+}}}\int\sqrt{\rho_{0}}v_{0}\left({\rho_{0}}v_{0}-\rho v\right)dx$
$\displaystyle=$ $\displaystyle 0,$

where we used (1.15), (3.23) and $\sqrt{\rho_{0}}v_{0}\in L^{2+\delta}$ for
any $\delta>0$. Then we have

$\sqrt{\rho}v(t)\rightarrow\sqrt{\rho}v(0)\ \ strongly\ in\
L^{2}(\mathbb{T}^{d})\ as\ t\rightarrow 0^{+}.$ (3.27)

Similarly, one has the right temporal continuity of $\sqrt{\rho}v$ in
$L^{2}(\mathbb{T}^{d})$, hence, for any $t_{0}\geq 0$, we infer that

$\sqrt{\rho}v(t)\rightarrow\sqrt{\rho}v(t_{0})\ \ strongly\ in\
L^{2}(\mathbb{T}^{d})\ as\ t\rightarrow t_{0}^{+}.$ (3.28)

Next, it follows from (1.18) that

$\overline{\lim_{t\rightarrow
t_{0}^{+}}}\mathcal{E}(t)\leq\mathcal{E}(t_{0}).$ (3.29)

This and (3.28) imply

$\overline{\lim\limits_{t\rightarrow
t_{0}^{+}}}\|P(\rho)(t)\|_{L^{1}(\mathbb{T}^{d})}\leq\|P(\rho)(t_{0})\|_{L^{1}(\mathbb{T}^{d})}.$
(3.30)

Notice from (1.15) and the mass equation $\eqref{GNS}_{1}$ that

$P(\rho)\in L^{\infty}(0,T;L^{\infty}(\mathbb{T}^{d}))\cap
H^{1}(0,T;H^{-1}(\mathbb{T}^{d}))\hookrightarrow
C([0,T];L^{2}_{\text{w}eak}(\mathbb{T}^{d})).$ (3.31)

Hence, (3.24), (3.30) and (3.31) guarantee

$\lim\limits_{t\rightarrow
t_{0}^{+}}\|P(\rho)(t)\|_{L^{1}(\mathbb{T}^{d})}=\|P(\rho)(t_{0})\|_{L^{1}(\mathbb{T}^{d})}.$
(3.32)

Before we go any further, it should be noted that (3.22) remains valid for
function $\phi$ belonging to $W^{1,\infty}$ rather than $C^{1}$, then for any
$t_{0}>0$, we redefine the test function $\phi$ as $\phi_{\tau}$ for some
positive $\tau$ and $\alpha$ such that $\tau+\alpha<t_{0}$, that is

$\phi_{\tau}(t)=\left\\{\begin{array}[]{lll}0,&0\leq t\leq\tau,\\\
\frac{t-\tau}{\alpha},&\tau\leq t\leq\tau+\alpha,\\\ 1,&\tau+\alpha\leq t\leq
t_{0},\\\ \frac{t_{0}-t}{\alpha},&t_{0}\leq t\leq t_{0}+\alpha,\\\
0,&t_{0}+\alpha\leq t.\end{array}\right.$ (3.33)

Then substituting this test function into (3.22), we arrive at

$\displaystyle-\int_{\tau}^{\tau+\alpha}\int$
$\displaystyle\frac{1}{\alpha}\left(\frac{1}{2}\rho
v^{2}+P(\rho)\right)+\frac{1}{\alpha}\int_{t_{0}}^{t_{0}+\alpha}\int\left(\frac{1}{2}\rho
v^{2}+P(\rho)\right)$ (3.34)
$\displaystyle+\int_{\tau}^{t_{0}+\alpha}\int\phi_{\tau}\left(\nu(\rho)|\mathbb{D}v|^{2}+\mu(\rho)|\mathrm{div}\,v|^{2}\right)=0.$

Taking $\alpha\rightarrow 0$ and using the fact that
$\int_{0}^{t}\int\left(\nu(\rho)|\mathbb{D}v|^{2}+\mu(\rho)|\mathrm{div}\,v|^{2}\right)$
is continuous with respect to $t$ and the Lebesgue point Theorem, we deduce
that

$\displaystyle-\int$ $\displaystyle\left(\frac{1}{2}\rho
v^{2}+P(\rho)\right)(\tau)dx+\int\left(\frac{1}{2}\rho
v^{2}+P(\rho)\right)(t_{0})dx$ (3.35)
$\displaystyle+\int_{\tau}^{t_{0}}\int\left(\nu(\rho)|\mathbb{D}v|^{2}+\mu(\rho)|\mathrm{div}\,v|^{2}\right)=0.$

Finally, letting $\tau\rightarrow 0$, using the continuity of
$\int_{0}^{t}\int\left(\nu(\rho)|\mathbb{D}v|^{2}+\mu(\rho)|\mathrm{div}\,v|^{2}\right)$,
(3.28) and (3.32), we can obtain

$\displaystyle\int\left(\frac{1}{2}\rho
v^{2}+P(\rho)\right)(t_{0})dx+\int_{0}^{t_{0}}\int$
$\displaystyle\left(\nu(\rho)|\mathbb{D}v|^{2}+\mu(\rho)|\mathrm{div}\,v|^{2}\right)dxds$
(3.36) $\displaystyle=$
$\displaystyle\int\left(\frac{1}{2}\rho_{0}v_{0}^{2}+P(\rho_{0})\right)dx.$

Then we have completed the proof of Theorem 1.1.

∎

We are in a position to prove Corollary 1.2.

###### Proof of Corollary 1.2.

(1) The natural energy gives $v\in L^{2}(0,T;H^{1}(\mathbb{T}^{3}))$. Choosing
$p=q=2$ in (LABEL:u), we immediately prove that the condition $v\in
L^{4}(0,T;L^{4}(\mathbb{T}^{3}))$ yields energy equality.

It is worth remarking that the rest proof in (1.22) can be reduced to this
special case. Next, we first deal with the case (1.22) in Corollary 1.2 with
$q\geq 4$ and $\frac{2}{p}+\frac{2}{q}=1$. The Gagliardo-Nirenberg inequality
guarantees that

$\displaystyle\|v\|_{L^{4}(0,T;L^{4}(\mathbb{T}^{3}))}\leq$ $\displaystyle
C\|v\|_{L^{\infty}\left(0,T;L^{2}(\mathbb{T}^{3})\right)}^{\frac{(q-4)}{2q-4}}\|v\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{3}))}^{\frac{q}{2q-4}}\leq
C.$ (3.37)

From the result just proved, we obtain energy equality via (1.22) with $q\geq
4$.
Then we consider (1.22) with $3<q<4$ and $\frac{1}{p}+\frac{3}{q}=1$. Using
the Gagliardo-Nirenberg inequality again, we know that

$\displaystyle\|v\|_{L^{4}(0,T;L^{4}(\mathbb{T}^{3}))}$ $\displaystyle\leq
C\|v\|_{L^{2}(0,T;L^{6}(\mathbb{T}^{3}))}^{\frac{3(4-q)}{2(6-q)}}\|v\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{3}))}^{\frac{q}{2(6-q)}}$
(3.38) $\displaystyle\leq C\left(\|\nabla
v\|_{L^{2}(0,T;L^{2}(\mathbb{T}^{3}))}+\|v\|_{L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))}\right)^{\frac{3(4-q)}{2(6-q)}}\|v\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{3}))}^{\frac{q}{2(6-q)}}\leq
C.$

We finish the proof of (1.22).

(2) Now, we focus on the proof of (1.23). Indeed, note that $v\in
L^{p}(0,T;W^{1,q}(\mathbb{T}^{3}))$, therefore, according to Theorem 1.1, it
suffices to derive $v\in
L^{\frac{2p}{p-1}}(0,T;L^{\frac{2q}{q-1}}(\mathbb{T}^{3}))$ from (1.23). For
$q\geq\frac{9}{5}$, by the Gagliardo-Nirenberg inequality, we get

$\displaystyle\|v\|_{L^{\frac{2q}{q-1}}(\mathbb{T}^{3})}\leq
C\|v\|_{L^{2}(\mathbb{T}^{3})}^{\frac{5q-9}{5q-6}}\|\nabla
v\|_{L^{q}(\mathbb{T}^{3})}^{\frac{3}{5q-6}}.$ (3.39)

Thanks to $\frac{1}{p}+\frac{6}{5q}=1$, we further infer that

$\displaystyle\|v\|_{L^{\frac{2p}{p-1}}(0,T;L^{\frac{2q}{q-1}}(\mathbb{T}^{3}))}\leq
C\|v\|_{L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))}^{\frac{5q-9}{5q-6}}\|\nabla
v\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{3}))}^{\frac{3}{5q-6}}\leq C.$ (3.40)

In light of Theorem 1.1, we have proved (1.23) for $q\geq\frac{9}{5}$.

Finally, for $\frac{3}{2}<q<\frac{9}{5}$, it follows the Gagliardo-Nirenberg
inequality that

$\displaystyle\|v\|_{L^{\frac{2q}{q-1}}(\mathbb{T}^{3})}\leq
C\|v\|_{L^{6}(\mathbb{T}^{3})}^{\frac{9-5q}{6-3q}}\|\nabla
v\|_{L^{q}(\mathbb{T}^{3})}^{\frac{2q-3}{6-3q}},$ (3.41)

Thanks to $\frac{1}{p}+\frac{3}{q}=2$, we further have

$\displaystyle\|v\|_{L^{\frac{2p}{p-1}}(0,T;L^{\frac{2q}{q-1}}(\mathbb{T}^{3}))}$
$\displaystyle\leq
C\|v\|_{L^{2}(0,T;L^{6}(\mathbb{T}^{3}))}^{\frac{9-5q}{6-3q}}\|\nabla
v\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{3}))}^{\frac{2q-3}{6-3q}}$ (3.42)
$\displaystyle\leq\left(\|\nabla
v\|_{L^{2}(0,T;L^{2}(\mathbb{T}^{3}))}+\|v\|_{L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))}\right)^{\frac{9-5q}{6-3q}}\|\nabla
v\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{3}))}^{\frac{2q-3}{6-3q}},$

then we conclude the desired result from Theorem 1.1. The proof of this
Corollary is completed. ∎

## Acknowledgement

Wang was partially supported by the National Natural Science Foundation of
China under grant (No. 11971446, No. 12071113 and No. 11601492). Ye was
partially supported by the National Natural Science Foundation of China under
grant (No.11701145) and China Postdoctoral Science Foundation (No.
2020M672196).

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