# Energy equality in the isentropic compressible Navier-Stokes equations
allowing vacuum

Yulin Ye111School of Mathematics and Statistics, Henan University, Kaifeng,
475004, P. R. China. Email<EMAIL_ADDRESS>Yanqing Wang222Corresponding
author. College of Mathematics and Information Science, Zhengzhou University
of Light Industry, Zhengzhou, Henan 450002, P. R. China Email:
<EMAIL_ADDRESS>and Wei Wei333School of Mathematics and Center for
Nonlinear Studies, Northwest University, Xi’an, Shaanxi 710127, P. R. China
Email<EMAIL_ADDRESS>
###### Abstract

It is well-known that a Leray-Hopf weak solution in $L^{4}(0,T;L^{4}(\Omega))$
for the incompressible Navier-Stokes system is persistence of energy due to
Lions [19]. In this paper, it is shown that Lions’s condition for energy
balance is also valid for the weak solutions of the isentropic compressible
Navier-Stokes equations allowing vacuum under suitable integrability
conditions on the density and its derivative. This allows us to establish
various sufficient conditions implying energy equality for the compressible
flow as well as the non-homogenous incompressible Navier-Stokes equations.
This is an improvement of corresponding results obtained by Yu in [32, Arch.
Ration. Mech. Anal., 225 (2017)], and our criterion via the gradient of the
velocity partially answers a question posed by Liang in [18, Proc. Roy. Soc.
Edinburgh Sect. A (2020)].

MSC(2000): 35Q30, 35Q35, 76D03, 76D05
Keywords: compressible Navier-Stokes equations; energy equality; vacuum

##  1 Introduction

In this paper, we are concerned with the classical isentropic compressible
Navier-Stokes equations in a periodic domain $\Omega=\mathbb{T}^{d}$ with
$d=2,3$

$\left\\{\begin{aligned} &\rho_{t}+\nabla\cdot(\rho v)=0,\\\ &(\rho
v)_{t}+\mathrm{div}\,(\rho v\otimes v)+\nabla P(\rho)-\mu\Delta
v-(\mu+\lambda)\nabla\text{div\,}v=0,\\\ \end{aligned}\right.$ (1.1)

where $\rho\geq 0$ stands for the density of the flow, $v$ represent the fluid
velocity field and $P(\rho)=\rho^{\gamma}$ with $\gamma>1$ is the scalar
pressure. The viscosity coefficients $\mu$ and $\lambda$ satisfy $\mu\geq 0$
and $2\mu+d\lambda>0$. This model usually describes the motion of a
compressible viscous barotropic fluid. We complement equations (LABEL:INS)
with initial data

$\rho(0,x)=\rho_{0}(x),\ (\rho v)(0,x)=(\rho_{0}v_{0})(x),\ x\in\Omega,$ (1.2)

where we define $v_{0}=0$ on the sets $\\{x\in\Omega:\ \rho_{0}=0\\}.$

Recently, originated from Yu’s work [32], the energy conservation of weak
solutions in the compressible fluid equations attracts a lot of attention (see
e.g.[32, 18, 22, 24, 23, 6, 29, 5, 12, 1]). Towards critical regularity for a
weak solution keeping energy equality, the main result in [32] is that if a
weak solution $(\rho,v)$ in the sense of Definition 2.1 satisfies

$\displaystyle 0\leq\rho\leq c<\infty,\ \ \nabla\sqrt{\rho}\in
L^{\infty}(0,T;L^{2}(\mathbb{T}^{d})),$ (1.3) $\displaystyle v\in
L^{p}(0,T;L^{q}(\mathbb{T}^{d}))~{}~{}p\geq 4\ \text{ and}\ q\geq 6,\ and\
v_{0}\in L^{q_{0}},\ q_{0}\geq 3,$

the following energy equality is valid

$\displaystyle\int_{\mathbb{T}^{d}}\left(\frac{1}{2}\rho|v|^{2}+\frac{\rho^{\gamma}}{\gamma-1}\right)dx+\int_{0}^{T}\int_{\mathbb{T}^{d}}$
$\displaystyle\left[\mu|\nabla
v|^{2}+(\mu+\lambda)|\mathrm{div}\,v|^{2}\right]dxdt$ (1.4) $\displaystyle=$
$\displaystyle\int_{\mathbb{T}^{d}}\left(\frac{1}{2}\rho_{0}|v_{0}|^{2}+\frac{\rho_{0}^{\gamma}}{\gamma-1}\right)dx.$

The famous Onsager’s conjecture [25] is what is the critical regularity for a
weak solution conserving energy in the incompressible Euler equations. In this
direction, a celebrated result due to Constantin-E-Titi [9] states that a weak
solution of the Euler system conserves the energy if $v$ is in
$L^{3}(0,T;B^{\alpha}_{3,\infty})$ with $\alpha>1/3.$ The negative results of
Onsager’s conjecture can be found in [4, 14]. For the 3D incompressible
Navier-Stokes, a kind of well-known positive energy conservation class is
Lions-Shinbrot type criterion in terms of integrability of the velocity $v$.
Particularly, in [19], Lions showed that a weak solution $v$ satisfies the
energy equality in the incompressible Navier-Stokes in any spatial dimension
if the velocity $v$ meets

$v\in L^{4}(0,T;L^{4}(\mathbb{T}^{d})).$ (1.5)

Lions’s energy conservation criterion was generalized by Shinbrot in [26] to

$v\in
L^{p}(0,T;L^{q}(\mathbb{T}^{d})),~{}\text{with}~{}\frac{2}{p}+\frac{2}{q}=1~{}\text{and}~{}q\geq
4.$ (1.6)

For $q<4$, Lions-Shinbrot type criterion in tri-dimensional space states

$v\in
L^{p}(0,T;L^{q}(\mathbb{T}^{3})),~{}\text{with}~{}\frac{1}{p}+\frac{3}{q}=1~{}\text{and}~{}3<q<4,$
(1.7)

which was studied by Taniuchi [28] and Beirao da Veiga-Yang [2]. Recent
progress about Lions-Shinbrot type criterion can be found in [7, 8, 3, 34, 29,
31, 33, 30].

Now, we turn attention back to compressible fluid. For equations (LABEL:INS)
posed an open bounded domain $\Omega$ with Dirichlet boundary condition, Chen-
Liang-Wang-Xu [6] showed that (1.3) yields the energy balance law (1.4). For
the general compressible models including (LABEL:INS) in the absence of
vacuum, it is shown that the following conditions

$\displaystyle 0<c_{1}\leq\rho\leq c_{2}<\infty,v\in
L^{\infty}(0,T;L^{2}(\mathbb{T}^{d})),\nabla v\in
L^{2}(0,T;L^{2}(\mathbb{T}^{d})),$ (1.8)
$\displaystyle\sup_{t\in(0,T)}\sup_{|h|<\varepsilon}|h|^{-\frac{1}{2}}\|\rho(\cdot+h,t)-\rho(\cdot,t)\|_{L^{2}(\mathbb{T}^{d})}<\infty,$
$\displaystyle v\in L^{4}(0,T;L^{4}(\mathbb{T}^{d})).$

ensure that a weak solution meets the energy equality by Nguye-Nguye-Tang in
[22]. The energy conservation criterion $\eqref{NNT}$ is still valid if both
$\eqref{NNT}_{2}$ and $\eqref{NNT}_{3}$ are replaced by (1.6) or (1.7), which
was recently proved in [29]. We also refer the readers to [18, 29] for energy
equality class via the gradient of the velocity. Roughly speaking, from the
aforementioned works, the Lions-Shinbrot type energy conservation class for
the isentropic compressible Navier-Stokes equations in the case away from
vacuum is the same as the incompressible Naiver-Stokes equations. A natural
question is weather this type energy conservation class is valid when the
initial data of system (LABEL:INS) contain vacuums. The objective of this
paper will be devoted to this issue.

We state our first result as follows.

######  Theorem 1.1.

The energy equality (1.4) of weak solutions $(\rho,v)$ to the isentropic
compressible Navier-Stokes equations (LABEL:INS)-(1.2) holds if one of the
following four conditions is satisfied

  1. (1)

$0\leq\rho\leq c_{2}<\infty$, $\nabla\sqrt{\rho}\in
L^{4}(0,T;L^{4}(\mathbb{T}^{d})),$ $\nabla v\in
L^{2}(0,T;L^{2}(\mathbb{T}^{d}))$, $v_{0}\in L^{2}(\mathbb{T}^{d})$,

$v\in L^{4}(0,T;L^{4}(\mathbb{T}^{d}));$ (1.9)

  2. (2)

$0\leq\rho\leq c_{2}<\infty$, $\nabla\sqrt{\rho}\in
L^{4}(0,T;L^{4}(\mathbb{T}^{d})),$ $\nabla v\in
L^{2}(0,T;L^{2}(\mathbb{T}^{d}))$, $v_{0}\in L^{2}(\mathbb{T}^{d})$,

$v\in
L^{p}(0,T;L^{q}(\mathbb{T}^{d}))~{}\text{with}~{}\frac{1}{p}+\frac{3}{q}=1,3<q\leq
4;$ (1.10)

  3. (3)

$0\leq\rho\leq c_{2}<\infty$, $\nabla\sqrt{\rho}\in
L^{4}(0,T;L^{2}(\mathbb{T}^{d}))$, $\nabla v\in
L^{2}(0,T;L^{2}(\mathbb{T}^{d}))$, $v_{0}\in L^{3}(\mathbb{T}^{d})$

$v\in L^{4}(0,T;L^{6}(\mathbb{T}^{d}));$ (1.11)

  4. (4)

$0\leq\rho\leq c_{2}<\infty$, $\nabla\sqrt{\rho}\in
L^{4}(0,T;L^{4}(\mathbb{T}^{d})),$ and $\nabla v\in
L^{2}(0,T;L^{2}(\mathbb{T}^{d}))$, $v_{0}\in L^{2}(\mathbb{T}^{d})$

$\nabla v\in
L^{p}(0,T;L^{r}(\mathbb{T}^{d}))~{}\text{with}~{}\frac{1}{p}+\frac{3}{r}=1+\frac{3}{d},\frac{3d}{d+3}<r\leq\frac{4d}{d+4}.$
(1.12)

######  Remark 1.1.

This theorem extends the well-known Lions’s energy equality criterion to the
isentropic compressible equations (LABEL:INS) with vacuum. Sufficient
conditions (1.7) implying energy equality are also generalized from the
incompressible fluid to the compressible flow allowing vacuum.

######  Remark 1.2.

Compared with Yu’ class (1.3), the condition (1.11) relaxes the integrability
of the density.

######  Remark 1.3.

In [18, Remark 1.7, Proc. Roy. Soc. Sect. A (2020)], Liang mentioned a
question that one considers the energy equality sufficient condition via the
gradient of the velocity when the vacuum state is allowed. From (1.12), we see
that this result gives an affirmative answer to this question.

######  Remark 1.4.

By means of a global mollification combined with an independent boundary cut-
off used in [6], one may generalize Theorem 1.1 from periodic boundary to
Dirichlet boundary.

We give some comments on the proof of this theorem. In contrast with the works
of [29, 22] without vacuum, it seems that the test function and the
Constantin-E-Titi type commutators on mollifying kernel break down in the
presence of vacuum. The test function as [32], a Lions’s type commutator
involving space-time mollifier and the strategy for energy equality condition
in terms of both the velocity and its gradient in [29] help us prove Theorem
1.1 and deal with the vacuum, (see also the Theorem 1.3). It is remarking that
a new ingredient is that we present a generalized Lions’s type commutators
involving the integral norms with different exponents in space and time, and
outline its proof, see Lemma 2.1 for the details. Moreover, since the initial
vacuum is allowed, one just obtains the energy $\sqrt{\rho}v\in
L^{\infty}(0,T;L^{2}(\Omega))$ rather than $v\in
L^{\infty}(0,T;L^{2}(\Omega))$. It is the main difficulty caused by this to
derive the blow up criterion of a strong solution of system with vacuum in
terms of the gradient of the velocity. We refer the readers to [15] for the
detail. In [21, Remark 5.1, p4], Lions mentioned that one can get further
regularity $v\in L^{2}(0,T;L^{2}(\Omega))$ in equations (LABEL:INS) for the
periodic case via careful Poincaré type inequality. However, it seems that it
is difficult to use $v\in L^{2}(0,T;L^{2}(\Omega))$ in the derivation of
(1.12). To prove (1.12) and deal with the vacuum, in the spirit of [21, Remark
5.1, p4], we make full use of $\nabla v\in L^{p}(0,T;L^{q}(\mathbb{T}^{3}))$
and apply the generalized Lions type Poincaré inequality to overcome the
difficulty. Based on this, roughly speaking, we observe that $\nabla v\in
L^{p}(0,T;L^{q}(\mathbb{T}^{3}))$ means $\int_{\Omega}vdx\in L^{p}(0,T)$. This
is the critical point to get (1.12).

As a by-product of above theorem, one further derives various energy
conservation class for the three-dimensional isentropic compressible Navier-
Stokes equations (LABEL:INS).

######  Corollary 1.2.

Let $(\rho,v)$ be a weak solution of isentropic compressible Navier-Stokes
equations (LABEL:INS)-(1.2) in the sense of Definition 2.1. Assume the density
$\rho$ satisfies one of the following condition

  1. (1)

$0\leq\rho\leq c_{2}<\infty,\nabla\sqrt{\rho}\in
L^{4}(0,T;L^{4}(\mathbb{T}^{3}));$

  2. (2)

$\nabla\sqrt{\rho}\in L^{\infty}(0,T;L^{4}(\mathbb{T}^{3}))$;

  3. (3)

$0\leq\rho\leq c<\infty,$ $\nabla\sqrt{\rho}\in
L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))$ and $\nabla\sqrt{\rho}\in
L^{p_{1}}(0,T;L^{q_{1}}(\mathbb{T}^{3})),with\frac{1}{p_{1}}+\frac{1}{q_{1}}=\frac{1}{2},q_{1}\geq
4$;

and the velocity meet one of the following

  1. (i)

$\nabla v\in L^{2}(0,T;L^{2}(\mathbb{T}^{3}))$ $v\in
L^{p}(0,T;L^{q}(\mathbb{T}^{3}))~{}\text{with}~{}\frac{1}{p}+\frac{3}{q}=1,3<q\leq
4,\ v_{0}\in L^{2}(\mathbb{T}^{3});$

  2. (ii)

$\nabla v\in L^{2}(0,T;L^{2}(\mathbb{T}^{3}))$ $\nabla v\in
L^{p}(0,T;L^{r}(\mathbb{T}^{3}))~{}\text{with}~{}\frac{1}{p}+\frac{3}{r}=2,\
\frac{3}{2}<r\leq\frac{12}{7},\ v_{0}\in L^{2}(\mathbb{T}^{3}).$

As mentioned above, the key point for the proof of Theorem 1.1 is the
following energy equality condition in terms of the velocity and its gradient.
Indeed, Theorem 1.1 is an immediate consequence of the following theorem.

######  Theorem 1.3.

For any $p,q\geq 4$ and $dp<2q+3d$ with $d\geq 2$, the energy equality (1.4)
of weak solutions $(\rho,v)$ to the isentropic compressible Navier-Stokes
equation (LABEL:INS)-(1.2) is valid provided

$\displaystyle 0\leq\rho<c<\infty,\nabla\sqrt{\rho}\in
L^{\frac{p}{p-3}}(0,T;L^{\frac{q}{q-3}}(\mathbb{T}^{d})),$ (1.13)
$\displaystyle v\in L^{p}(0,T;L^{q}(\mathbb{T}^{d})),\nabla v\in
L^{\frac{p}{p-2}}(0,T;L^{\frac{q}{q-2}}(\mathbb{T}^{d})),\ v_{0}\in
L^{\frac{q}{2}}(\mathbb{T}^{d}).$

Finally, following the path of the above theorem, one can consider the non-
homogenous incompressible Navier-Stokes equations below

$\left\\{\begin{aligned} &\rho_{t}+\mathrm{div}\,(\rho v)=0,\\\ &(\rho
v)_{t}+\mathrm{div}\,(\rho v\otimes v)-\mu\Delta v+\nabla P=0,\\\
&\mathrm{div}\,v=0,\\\ \end{aligned}\right.$ (1.14)

and derive a similar result as Theorem 1.3. Since the velocity is divergence-
free, the pressure part in the proof becomes trivial. Hence, we just formulate
the corresponding result here.

######  Theorem 1.4.

Let $(\rho,v)$ be a weak solution to the Navier-Stokes equation (LABEL:nhins).
Assume that the pair $(\rho,v)$ obeys

$\displaystyle 0\leq\rho\leq c_{2}<\infty,\nabla\sqrt{\rho}\in
L^{4}(0,T;L^{4}(\mathbb{T}^{3})),\nabla v\in
L^{2}(0,T;L^{2}(\mathbb{T}^{3})),$ (1.15) $\displaystyle v\in
L^{p}(0,T;L^{q}(\mathbb{T}^{3}))~{}\text{with}~{}\frac{1}{p}+\frac{3}{q}=1,3<q\leq
4,~{}\text{and}~{}v_{0}\in L^{2}(\mathbb{T}^{3});$

then, the energy equality below holds

$\int_{\mathbb{T}^{3}}\frac{1}{2}\rho|v|^{2}dx+\int_{0}^{T}\int_{\mathbb{T}^{3}}\mu|\nabla
v|^{2}dxdt=\int_{\mathbb{T}^{3}}\frac{1}{2}\rho_{0}|v_{0}|^{2}dx.$ (1.16)

######  Remark 1.5.

When the density $\rho$ becomes a constant, Theorem 1.4 reduces to the Lions’s
classical result (1.5).

######  Remark 1.6.

Previous results on energy balance proved in [17, 22, 29] are all without
vacuum. It seems that this theorem will be the first criterion on energy
equality of weak solutions to non-homogenous incompressible Navier-Stokes
equations in the presence of vacuum.

The remainder of this paper is organized as follows. Section 2 is devoted to
the auxiliary lemmas involving mollifier and the key inequality. In Section 3,
we first present the proof of Theorem 1.3. Then, based on Theorem 1.3, we
complete the proof of Theorem 1.1 and 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 $\|f(t,\cdot)\|_{X}$ belonging
to $L^{p}(0,\,T)$. The classical Sobolev space $W^{k,p}(\Omega)$ is equipped
with the norm
$\|f\|_{W^{k,p}(\Omega)}=\sum\limits_{|\alpha|=0}^{k}\|D^{\alpha}f\|_{L^{p}(\Omega)}$.
For simplicity, we denote by

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

Let $\eta$ be non-negative smooth function supported in the space-time ball of
radius 1 and its integral equals to 1. We define the rescaled space-time
mollifier
$\eta_{\varepsilon}(t,x)=\frac{1}{\varepsilon^{d+1}}\eta(\frac{t}{\varepsilon},\frac{x}{\varepsilon})$

$f^{\varepsilon}(t,x)=\int_{0}^{T}\int_{\Omega}f(s,y)\eta_{\varepsilon}(t-s,x-y)dyds.$

######  Definition 2.1.

A pair ($\rho,v$) is called a weak solution to (LABEL:INS) with initial data
(1.2) if ($\rho,v$) satisfies

  1. (i)

equation (LABEL:INS) holds in $D^{\prime}(0,T;\Omega)$ and

$P(\rho),\rho|v|^{2}\in L^{\infty}(0,T;L^{1}(\Omega)),\ \ \ \nabla v\in
L^{2}(0,T;L^{2}(\Omega)),$ (2.1)

  2. (ii)

the density $\rho$ is a renormalized solution of (LABEL:INS) in the sense of
[10].

  3. (iii)

the energy inequality holds

$\displaystyle E(t)+\int_{0}^{T}\int_{\Omega}\left(\mu|\nabla
v|^{2}+(\mu+\lambda)|\mathrm{div}\,v|^{2}\right)dxdt\leq E(0),$ (2.2)

where
$E(t)=\int_{\Omega}\left(\frac{1}{2}\rho|v|^{2}+\frac{\rho^{\gamma}}{\gamma-1}\right)dx$.

To the knowledge of the authors, the following statement about Lions’s type
commutators involving space-time mollifier with $p=q,p_{i}=q_{i}$, $(i=1,2)$
can be found in [16], which plays an important role in recent works [32, 5, 6]
and whose complete rigorous proof was not presented any where (see [20] for
original version). To make our work more self-contained and more readable, we
shall outline its proof.

######  Lemma 2.1.

Let $1\leq p,q,p_{1},q_{1},p_{2},q_{2}\leq\infty$, with
$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$ and
$\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$. Let $\partial$ be a partial
derivative in space or time, in addition, let $\partial_{t}f,\ \nabla f\in
L^{p_{1}}(0,T;L^{q_{1}}(\Omega))$, $g\in L^{p_{2}}(0,T;L^{q_{2}}(\Omega))$.
Then, there holds

$\|{\partial(fg)^{\varepsilon}}-\partial(f\,{g}^{\varepsilon})\|_{L^{p}(0,T;L^{q}(\Omega))}\leq
C\left(\|\partial_{t}f\|_{L^{p_{1}}(0,T;L^{q_{1}}(\Omega))}+\|\nabla
f\|_{L^{p_{1}}(0,T;L^{q_{1}}(\Omega))}\right)\|g\|_{L^{p_{2}}(0,T;L^{q_{2}}(\Omega))},$

for some constant $C>0$ independent of $\varepsilon$, $f$ and $g$. Moreover,

${\partial{(fg)^{\varepsilon}}}-\partial{(f\,{g^{\varepsilon}})}\to
0\quad\text{ in }{L^{p}(0,T;L^{q}(\Omega))},$

as $\varepsilon\to 0$ if $p_{2},q_{2}<\infty.$

###### Proof.

First, we let

$\displaystyle\partial(fg)^{\varepsilon}-\partial(fg^{\varepsilon})=\partial(fg)^{\varepsilon}-f\partial
g^{\varepsilon}-\partial fg^{\varepsilon}=G^{\varepsilon}-\partial
fg^{\varepsilon},$ (2.3)

where $G^{\varepsilon}=\partial(fg)^{\varepsilon}-f\partial g^{\varepsilon}$.
Due to the properties of the Mollifier and the integration by parts, we have

$\displaystyle|G^{\varepsilon}(t,x)|=$
$\displaystyle\left|\int\int\left(f(s,y)-f(t,x)\right)g(s,y)\frac{1}{\varepsilon^{d+1}}\frac{1}{\varepsilon}\partial\eta(\frac{t-s}{\varepsilon},\frac{x-y}{\varepsilon})dyds\right|$
(2.4) $\displaystyle=$
$\displaystyle\left|\int\int\left[f(s,y)-f(t,y)+f(t,y)-f(t,x)\right]g(s,y)\frac{1}{\varepsilon^{d+1}}\frac{1}{\varepsilon}\partial\eta(\frac{t-s}{\varepsilon},\frac{x-y}{\varepsilon})dyds\right|$
$\displaystyle\leq$ $\displaystyle
C\int_{t-\varepsilon}^{t+\varepsilon}\int_{B(x,\varepsilon)}\frac{1}{\varepsilon^{d+1}}\int_{0}^{1}|\partial_{t}f(t+\tau(s-t),y)|d\tau|g(s,y)|dyds$
$\displaystyle\ \ \ \ \ \
+\int_{t-\varepsilon}^{t+\varepsilon}\int_{B(x,\varepsilon)}\frac{1}{\varepsilon^{d+1}}\int_{0}^{1}|\nabla
f(t,x+\tau(y-x))|d\tau|g(s,y)|dyds$ $\displaystyle=$ $\displaystyle
I_{1}+I_{2},$

here, $B(x,\varepsilon)=\\{y\in\Omega;|y-x|<\varepsilon\\}$ is an open ball
centered at $x$. To control $I_{1}$ and $I_{2}$, by the Hölder’s inequality,
we know that

$\displaystyle I_{1}\leq$ $\displaystyle
C\int_{t-\varepsilon}^{t+\varepsilon}\frac{1}{\varepsilon}\left(\int_{B(x,\varepsilon)}\frac{1}{\varepsilon^{d}}\int_{0}^{1}|\partial_{t}f(t+\tau(s-t),y)|^{s_{1}}d\tau
dy\right)^{\frac{1}{s_{1}}}\left(\int_{B(x,\varepsilon)}\frac{1}{\varepsilon^{d}}|g(s,y)|^{s_{2}}dy\right)^{\frac{1}{s_{2}}}ds$
(2.5) $\displaystyle\leq$ $\displaystyle
C\int_{t-\varepsilon}^{t+\varepsilon}\frac{1}{\varepsilon}\left(\int_{B(0,\varepsilon)}\frac{1}{\varepsilon^{d}}\int_{0}^{1}|\partial_{t}f(t+\tau(s-t),x-z)|^{s_{1}}d\tau
dz\right)^{\frac{1}{s_{1}}}\left(\int_{B(0,\varepsilon)}\frac{1}{\varepsilon^{d}}|g(s,x-z)|^{s_{2}}dz\right)^{\frac{1}{s_{2}}}ds,$

where $\frac{1}{s_{1}}+\frac{1}{s_{2}}=1,1\leq s_{1}\leq q_{1}$ and $1\leq
s_{2}\leq q_{2}$. Moreover, notice that

$\displaystyle\int_{B(0,\varepsilon)}\frac{1}{\varepsilon^{d}}\int_{0}^{1}|\partial_{t}f(t+\tau(s-t),x-z)|^{s_{1}}d\tau
dz$ (2.6) $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\int_{0}^{1}|\partial_{t}f(t+\tau(s-t),x-z)|^{s_{1}}d\tau\frac{{\bf
1}_{B(0,\varepsilon)}(z)}{\varepsilon^{d}}dz$ $\displaystyle=$
$\displaystyle\left(\int_{0}^{1}|\partial_{t}f(t+\tau(s-t))|^{s_{1}}d\tau*J_{\varepsilon}\right)(x),$

and

$\displaystyle\int_{B(0,\varepsilon)}\frac{1}{\varepsilon^{d}}|g(s,x-z)|^{s_{2}}dz$
(2.7) $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}|g(s,x-z)|^{s_{2}}\frac{{\bf
1}_{B(0,\varepsilon)}(z)}{\varepsilon^{d}}dz$ $\displaystyle=$
$\displaystyle\left(|g(s)|^{s_{2}}*J_{\varepsilon}\right)(x),$

where $J_{\varepsilon}(z)=\frac{{\bf
1}_{B(0,\varepsilon)}(z)}{\varepsilon^{d}}\geq 0$ and
$\|J_{\varepsilon}\|_{L^{1}{(\mathbb{R}^{d})}}=$measure of $B(0,1)$. Then
substituting (2.6) and (2.7) into (2.5), we infer

$I_{1}\leq
C\int_{t-\varepsilon}^{t+\varepsilon}\frac{1}{\varepsilon}\left(\int_{0}^{1}|\partial_{t}f(t+\tau(s-t))|^{s_{1}}d\tau*J_{\varepsilon}(x)\right)^{\frac{1}{s_{1}}}\Big{(}|g(s)|^{s_{2}}*J_{\varepsilon}(x)\Big{)}^{\frac{1}{s_{2}}}ds.$
(2.8)

Then using the Minkowski inequality and Hölder’s inequality, one can derive
that

$\displaystyle\|I_{1}\|_{L^{q}(\Omega)}$ (2.9) $\displaystyle\leq$
$\displaystyle
C\Big{\|}\int_{t-\varepsilon}^{t+\varepsilon}\frac{1}{\varepsilon}\left(\int_{0}^{1}|\partial_{t}f(t+\tau(s-t))|^{s_{1}}d\tau*J_{\varepsilon}(x)\right)^{\frac{1}{s_{1}}}\Big{(}|g(s)|^{s_{2}}*J_{\varepsilon}(x)\Big{)}^{\frac{1}{s_{2}}}ds\Big{\|}_{L^{q}(\Omega)}$
$\displaystyle\leq$ $\displaystyle
C\int_{t-\varepsilon}^{t+\varepsilon}\frac{1}{\varepsilon}\Big{\|}\left(\int_{0}^{1}|\partial_{t}f(t+\tau(s-t))|^{s_{1}}d\tau*J_{\varepsilon}(x)\right)^{\frac{1}{s_{1}}}\Big{(}|g(s)|^{s_{2}}*J_{\varepsilon}(x)\Big{)}^{\frac{1}{s_{2}}}\Big{\|}_{L^{q}(\Omega)}ds$
$\displaystyle\leq$ $\displaystyle
C\int_{t-\varepsilon}^{t+\varepsilon}\frac{1}{\varepsilon}\Big{\|}\left(\int_{0}^{1}|\partial_{t}f(t+\tau(s-t))|^{s_{1}}d\tau*J_{\varepsilon}(x)\right)^{\frac{1}{s_{1}}}\Big{\|}_{L^{q_{1}}(\Omega)}\Big{\|}\Big{(}|g(s)|^{s_{2}}*J_{\varepsilon}(x)\Big{)}^{\frac{1}{s_{2}}}\Big{\|}_{L^{q_{2}}(\Omega)}ds$
$\displaystyle\leq$ $\displaystyle
C\int_{t-\varepsilon}^{t+\varepsilon}\frac{1}{\varepsilon}\left(\int_{0}^{1}\|\partial_{t}f(t+\tau(s-t))\Big{\|}_{L^{q_{1}}(\Omega)}^{s_{1}}d\tau\right)^{\frac{1}{s_{1}}}\|g(s)\|_{L^{q_{2}}(\Omega)}ds$
$\displaystyle\leq$ $\displaystyle
C\left(\int_{t-\varepsilon}^{t+\varepsilon}\frac{1}{\varepsilon}\int_{0}^{1}\|\partial_{t}f(t+\tau(s-t))\|_{L^{q_{1}}(\Omega)}^{s_{1}}d\tau
ds\right)^{\frac{1}{s_{1}}}\left(\int_{t-\varepsilon}^{t+\varepsilon}\frac{1}{\varepsilon}\|g(s)\|_{L^{q_{2}}(\Omega)}^{{s_{2}}}ds\right)^{\frac{1}{s_{2}}}$
$\displaystyle\leq$ $\displaystyle
C\left(\int_{0}^{1}\frac{1}{\tau\varepsilon}\int_{-\tau\varepsilon}^{\tau\varepsilon}\|\partial_{t}f(t-\varsigma)\|_{L^{q_{1}}(\Omega)}^{s_{1}}d\varsigma
d\tau\right)^{\frac{1}{s_{1}}}\left(\int_{-\varepsilon}^{\varepsilon}\frac{1}{\varepsilon}\|g(t-\varsigma)\|_{L^{q_{2}}(\Omega)}^{s_{2}}d\varsigma\right)^{\frac{1}{s_{2}}}$
$\displaystyle\leq$ $\displaystyle
C\Big{(}\|\partial_{t}f(t-\varsigma)\|_{L^{q_{1}}(\Omega)}^{s_{1}}*J_{1\varepsilon}(\varsigma)\Big{)}^{\frac{1}{s_{1}}}\Big{(}\|g(t-\varsigma)\|_{L^{q_{2}}(\Omega)}^{s_{2}}*J_{2\varepsilon}(\varsigma)\Big{)}^{\frac{1}{s_{2}}},$
(2.10)

where $J_{1\varepsilon}=\int_{0}^{1}\frac{{\bf
1}_{(-\tau\varepsilon,\tau\varepsilon)}}{\tau\varepsilon}(\varsigma)d\tau\geq
0,\ J_{2\varepsilon}=\frac{{\bf
1}_{(-\varepsilon,\varepsilon)}}{\varepsilon}(\varsigma)\geq 0$ and
$\|J_{1\varepsilon}\|_{L^{1}{(\mathbb{R})}}=\|J_{2\varepsilon}\|_{L^{1}{(\mathbb{R})}}=2$.

Using the Hölder’s inequality again, $1\leq s_{1}\leq p_{1}$, and $1\leq
s_{2}\leq p_{2}$, this immediately gives

$\displaystyle\|I_{1}\|_{L^{p}(L^{q})}$ (2.11) $\displaystyle\leq$
$\displaystyle
C\left\|\Big{(}\|\partial_{t}f(t-\varsigma)\|_{L^{q_{1}}(\mathbb{T}^{d})}^{s_{1}}*J_{1\varepsilon}(\varsigma)\Big{)}^{\frac{1}{s_{1}}}\right\|_{L^{p_{1}}(0,T)}\left\|\Big{(}\|g(t-\varsigma)\|_{L^{q_{2}}(\mathbb{T}^{d})}^{s_{2}}*J_{2\varepsilon}(\varsigma)\Big{)}^{\frac{1}{s_{2}}}\right\|_{L^{p_{2}}(0,T)}$
$\displaystyle\leq$ $\displaystyle
C\|\partial_{t}f\|_{L^{p_{1}}(L^{q_{1}})}\|g\|_{L^{p_{2}}(L^{q_{2}})}.$

As the same manner of the derivation above, we see that

$\displaystyle I_{2}$ $\displaystyle\leq
C\int_{t-\varepsilon}^{t+\varepsilon}\frac{1}{\varepsilon}\left(\int_{B(x,\varepsilon)}\frac{1}{\varepsilon^{d}}\int_{0}^{1}|\nabla
f(t,x+\tau(y-x))|^{s_{1}}d\tau
dy\right)^{\frac{1}{s_{1}}}\left(\int_{B(x,\varepsilon)}\frac{1}{\varepsilon^{d}}|g(s,y)|^{s_{2}}dy\right)^{\frac{1}{s_{2}}}ds$
(2.12) $\displaystyle\leq
C\int_{t-\varepsilon}^{t+\varepsilon}\frac{1}{\varepsilon}\left(|\nabla
f(t,x-z)|^{s_{1}}*\overline{J}_{\varepsilon}(z)\right)^{\frac{1}{s_{1}}}\left(|g(s)|^{s_{2}}*J_{\varepsilon}(z)\right)^{\frac{1}{s_{2}}}ds$

where $\overline{J}_{\varepsilon}(z)=\frac{{\bf
1}_{B(0,\tau\varepsilon)}}{(\tau\varepsilon)^{d}}\geq 0$ and
$\|\overline{J}_{\varepsilon}(z)\|_{L^{1}(\mathbb{R}^{d})}=$ measure of ball
$B(0,1)$.

Hence, we further get

$\displaystyle\|I_{2}\|_{L^{q}(\Omega)}\leq$ $\displaystyle
C\Big{\|}\int_{t-\varepsilon}^{t+\varepsilon}\frac{1}{\varepsilon}\left(|\nabla
f(t,x-z)|^{s_{1}}*\overline{J}_{\varepsilon}(z)\right)^{\frac{1}{s_{1}}}\left(|g(s,x-z)|^{s_{2}}*J_{\varepsilon}(z)\right)^{\frac{1}{s_{2}}}ds\Big{\|}_{L^{q}(\Omega)}$
$\displaystyle\leq$ $\displaystyle
C\int_{t-\varepsilon}^{t+\varepsilon}\frac{1}{\varepsilon}\|\left(|\nabla
f(t,x-z)|^{s_{1}}*\overline{J}_{\varepsilon}(z)\right)^{\frac{1}{s_{1}}}\left(|g(s,x-z)|^{s_{2}}*J_{\varepsilon}(z)\right)^{\frac{1}{s_{2}}}\|_{L^{q}(\Omega)}ds$
$\displaystyle\leq$ $\displaystyle
C\int_{t-\varepsilon}^{t+\varepsilon}\frac{1}{\varepsilon}\|\nabla
f(t)\|_{L^{q_{1}}(\Omega)}\|g(s)\|_{L^{q_{2}}(\Omega)}ds$ $\displaystyle\leq$
$\displaystyle C\|\nabla
f(t)\|_{L^{q_{1}}(\Omega)}\int_{t-\varepsilon}^{t+\varepsilon}\frac{1}{\varepsilon}\|g(s)\|_{L^{q_{2}}(\Omega)}ds$
$\displaystyle\leq$ $\displaystyle C\|\nabla
f(t)\|_{L^{q_{1}}(\Omega)}\left(\|g(t-\varsigma)\|_{L^{q_{2}}(\Omega)}*J_{2\varepsilon}(\varsigma)\right),$
(2.13)

which implies that

$\displaystyle\|I_{2}\|_{L^{p}(L^{q})}$ $\displaystyle\leq C\Big{\|}\|\nabla
f(t)\|_{L^{q_{1}}(\Omega)}\left(\|g(t-\varsigma)\|_{L^{q_{2}}(\Omega)}*J_{2\varepsilon}(\varsigma)\right)\Big{\|}_{L^{p}(0,T)}$
(2.14) $\displaystyle\leq C\|\nabla
f\|_{L^{p_{1}}(L^{q_{1}})}\left\|\left(\|g(t-\varsigma)\|_{L^{q_{2}}(\Omega)}*J_{2\varepsilon}(\varsigma)\right)\right\|_{L^{p_{2}}}$
$\displaystyle\leq C\|\nabla
f\|_{L^{p_{1}}(L^{q_{1}})}\|g\|_{L^{p_{2}}(L^{q_{2}})}.$

Then together with (2.3), (2.11) and (2.14), we obtain that

$\displaystyle\|\partial(fg)^{\varepsilon}-\partial(fg^{\varepsilon})\|_{L^{p}(L^{q})}$
$\displaystyle\leq C\left(\|G^{\varepsilon}\|_{L^{p}(L^{q})}+\|\partial
fg^{\varepsilon}\|_{L^{p}(L^{q})}\right)$ (2.15) $\displaystyle\leq
C\left(\|\partial_{t}f\|_{L^{p_{1}}(L^{q_{1}})}+\|\nabla
f\|_{L^{p_{1}}(L^{q_{1}})}\right)\|g\|_{L^{p_{2}}(L^{q_{2}})}.$

Furthermore, if $1\leq p_{2},q_{2}<\infty$, let $\\{g_{n}\\}\in
C_{0}^{\infty}(\Omega)$ with $g_{n}\rightarrow g$ strongly in
$L^{p_{2}}(L^{q_{2}})$. Thus, by the density arguments and properties of the
standard mollification, we arrive at

$\displaystyle\|\partial(fg)^{\varepsilon}-\partial(fg^{\varepsilon})\|_{L^{p}(L^{q})}$
(2.16) $\displaystyle\leq$ $\displaystyle
C\|\partial\left(f\left(g-g_{n}\right)\right)^{\varepsilon}-\partial\left(f\left(g-g_{n}\right)^{\varepsilon}\right)+\left(\partial\left(fg_{n}\right)^{\varepsilon}-\partial\left(fg_{n}^{\varepsilon}\right)\right)\|_{L^{p}(L^{q})}$
$\displaystyle\leq$ $\displaystyle
C\left(\|\partial_{t}f\|_{L^{p_{1}}(L^{q_{1}})}+\|\nabla
f\|_{L^{p_{1}}(L^{q_{1}})}\right)\|g-g_{n}\|_{L^{p_{2}}(L^{q_{2}})}+C\|\partial(fg_{n})^{\varepsilon}-\partial(fg_{n})\|_{L^{p}(L^{q})}$
$\displaystyle\ \ \ \
+C\|\partial(f(g_{n}-g_{n}^{\varepsilon}))\|_{L^{p}(L^{q})}\rightarrow 0,\ \
as\ \varepsilon\rightarrow 0.$

This concludes the proof. ∎

######  Lemma 2.2.

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

$\|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\Omega))}\rightarrow
0,\ \ \ as\ \varepsilon\rightarrow 0.$ (2.17)

###### Proof.

By the triangle inequality, one have

$\displaystyle\|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
C\left(\|(fg)^{\varepsilon}-(fg)\|_{L^{p}(0,T;L^{q}(\Omega))}+\|fg-f^{\varepsilon}g\|_{L^{p}(0,T;L^{q}(\Omega))}+\|f^{\varepsilon}g-f^{\varepsilon}g^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\Omega))}\right)$
$\displaystyle\leq$ $\displaystyle
C\Big{(}\|(fg)^{\varepsilon}-fg\|_{L^{p}(0,T;L^{q}(\Omega))}+\|f-f^{\varepsilon}\|_{L^{p_{1}}(0,T;L^{q_{1}}(\Omega))}\|g\|_{L^{p_{2}}(0,T;L^{q_{2}}(\Omega))}$
$\displaystyle\ \ \ \ \
+\|f^{\varepsilon}\|_{L^{p_{1}}(0,T;L^{q_{1}}(\Omega))}\|g-g^{\varepsilon}\|_{L^{p_{2}}(0,T;L^{q_{2}}(\Omega))}\Big{)},$

then, together with the properties of the standard mollification, we can
obtain (2.17). ∎

Finally, we recall the generalized Aubin-Lions Lemma to extend the energy
equality up to the initial time.

######  Lemma 2.3 ([27]).

Let $X\hookrightarrow B\hookrightarrow Y$ be three Banach spaces with compact
imbedding $X\hookrightarrow\hookrightarrow Y$. Further, let there exist
$0<\theta<1$ and $M>0$ such that

$\|v\|_{B}\leq M\|v\|_{X}^{1-\theta}\|v\|_{Y}^{\theta}\ \ for\ all\ v\in X\cap
Y.$ (2.18)

Denote for $T>0$,

$W(0,T):=W^{s_{0},r_{0}}((0,T),X)\cap W^{s_{1},r_{1}}((0,T),Y)$ (2.19)

with

$\displaystyle s_{0},s_{1}\in\mathbb{R};\ r_{0},r_{1}\in[1,\infty],$ (2.20)
$\displaystyle s_{\theta}:=(1-\theta)s_{0}$ $\displaystyle+\theta s_{1},\
\frac{1}{r_{\theta}}:=\frac{1-\theta}{r_{0}}+\frac{\theta}{r_{1}},\
s^{*}:=s_{\theta}-\frac{1}{r_{\theta}}.$

Assume that $s_{\theta}>0$ and $F$ is a bounded set in $W(0,T)$. Then, we have

If $s_{*}\leq 0$, then $F$ is relatively compact in $L^{p}((0,T),B)$ for all
$1\leq p<p^{*}:=-\frac{1}{s^{*}}$.

If $s_{*}>0$, then $F$ is relatively compact in $C((0,T),B)$.

##  3 Proof of Theorem 1.1, Theorem 1.3 and Corollary 1.2

In this section, we first present the proof of Theorem 1.3. Then, making the
use of interpolation inequality and the Poincaré inequality, we prove the
Theorem 1.1 and Corollary 1.2 by the results of Theorem 1.3.

###### Proof of Theorem 1.3.

Let $\phi(t)$ be a smooth function compactly supported in $(0,+\infty)$.
Multiplying $\eqref{INS}_{2}$ by $(\phi v^{\varepsilon})^{\varepsilon}$, then
integrating over $(0,T)\times\Omega$ , we infer that

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

To pass the limit of $\varepsilon$, we reformulate every term of the last
equation. A straightforward computation leads to

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

It follows from integration by parts and the mass equation $\eqref{INS}_{1}$
that

$\displaystyle\int_{0}^{T}\int\phi(t)v^{\varepsilon}\mathrm{div}\,(\rho
v\otimes v)^{\varepsilon}$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\int\phi(t)v^{\varepsilon}\mathrm{div}\,[(\rho
v\otimes v)^{\varepsilon}-(\rho v)\otimes
v^{\varepsilon}]+\int_{0}^{T}\int\phi(t)v^{\varepsilon}\mathrm{div}\,(\rho
v\otimes v^{\varepsilon})$ $\displaystyle=$
$\displaystyle-\int_{0}^{T}\int\phi(t)\nabla v^{\varepsilon}[(\rho v\otimes
v)^{\varepsilon}-(\rho v)\otimes
v^{\varepsilon}]+\int_{0}^{T}\int\phi(t)\left(\mathrm{div}\,(\rho
v)|v^{\varepsilon}|^{2}+\frac{1}{2}\rho v\nabla|v^{\varepsilon}|^{2}\right)$
$\displaystyle=$ $\displaystyle-\int_{0}^{T}\int\phi(t)\nabla
v^{\varepsilon}[(\rho v\otimes v)^{\varepsilon}-(\rho v)\otimes
v^{\varepsilon}]+\frac{1}{2}\int_{0}^{T}\int\phi(t)\mathrm{div}\,(\rho
v)|v^{\varepsilon}|^{2}$ $\displaystyle=$
$\displaystyle-\int_{0}^{T}\int\phi(t)\nabla v^{\varepsilon}[(\rho v\otimes
v)^{\varepsilon}-(\rho v)\otimes
v^{\varepsilon}]-\frac{1}{2}\int_{0}^{T}\int\phi(t)\partial_{t}\rho|v^{\varepsilon}|^{2}.$
(3.3)

We rewrite the pressure term as

$\displaystyle\int_{0}^{T}\int\phi(t)v^{\varepsilon}\nabla(\rho^{\gamma})^{\varepsilon}=\int_{0}^{T}\int\phi(t)[v^{\varepsilon}\nabla(\rho^{\gamma})^{\varepsilon}-v\nabla(\rho^{\gamma})]+\int_{0}^{T}\int\phi(t)v\nabla(\rho^{\gamma}).$
(3.4)

Then using the integration by parts and mass equation $\eqref{INS}_{1}$ again,
we find

$\displaystyle\int_{0}^{T}\int\phi(t)v\cdot\nabla(\rho^{\gamma})=$
$\displaystyle-\int_{0}^{T}\int\phi(t)\rho^{\gamma-1}\rho\text{div\,}v$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\int\phi(t)\rho^{\gamma-1}(\partial_{t}\rho+v\cdot\nabla\rho)$
$\displaystyle=$
$\displaystyle\frac{1}{\gamma}\int_{0}^{T}\int\phi(t)\partial_{t}\rho^{\gamma}+\frac{1}{\gamma}\int_{0}^{T}\int\phi(t)v\cdot\nabla\rho^{\gamma},$

which in turn means that

$\displaystyle\int_{0}^{T}\int\phi(t)v\cdot\nabla(\rho^{\gamma})=\frac{1}{\gamma-1}\int_{0}^{T}\int\phi(t)\partial_{t}\rho^{\gamma}.$
(3.5)

Thanks to integration by parts, we arrive at

$\displaystyle-\mu\int_{0}^{T}\int\phi(t)v^{\varepsilon}\Delta
v^{\varepsilon}=\mu\int_{0}^{T}\int\phi(t)|\nabla v^{\varepsilon}|^{2},$ (3.6)
$\displaystyle-(\mu+\lambda)\int_{0}^{T}\int\phi(t)v^{\varepsilon}\nabla\text{div\,}v^{\varepsilon}=(\mu+\lambda)\int_{0}^{T}\int\phi(t)|\mathrm{div}\,{v^{\varepsilon}}|^{2}.$

Plugging (3.2)-(3.6) into (3.1) and using the integration by parts, we
conclude that

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

It is enough to prove that the terms on the right hand-side of (3.7) tend to
zero as $\varepsilon\rightarrow 0$. In view of Hölder’s inequality and Lemma
(2.1), we know that

$\displaystyle\int_{s}^{t}\int\phi(t)v^{\varepsilon}\Big{[}\partial_{t}(\rho
v)^{\varepsilon}-\partial_{t}(\rho v^{\varepsilon})\Big{]}$ $\displaystyle\leq
C\|v^{\varepsilon}\|_{L^{p}(L^{q})}\|\partial_{t}(\rho
v)^{\varepsilon}-\partial_{t}(\rho
v^{\varepsilon})\|_{L^{\frac{p}{p-1}}(L^{\frac{q}{q-1}})}$ (3.8)
$\displaystyle\leq
C\|v\|^{2}_{L^{p}(L^{q})}\left(\|\rho_{t}\|_{L^{\frac{p}{p-2}}(L^{\frac{q}{q-2}})}+\|\nabla\rho\|_{L^{\frac{p}{p-2}}(L^{\frac{q}{q-2}})}\right).$

To bound $\rho_{t}$ and $\nabla\rho$, we employ mass equation to obtain

$\rho_{t}=-2\sqrt{\rho}v\cdot\nabla\sqrt{\rho}-{\rho}\text{div}v,\ and\
\nabla\rho=2\sqrt{\rho}\nabla\sqrt{\rho}.$

As a consequent, the triangle inequality and Hölder’s inequality guarantee
that

$\displaystyle\|\rho_{t}\|_{L^{\frac{p}{p-2}}(L^{\frac{q}{q-2}})}$
$\displaystyle\leq
C\left(\|-2\sqrt{\rho}v\cdot\nabla\sqrt{\rho}\|_{L^{\frac{p}{p-2}}(L^{\frac{q}{q-2}})}+\|{\rho}\text{div}v\|_{L^{\frac{p}{p-2}}(L^{\frac{q}{q-2}})}\right)$
(3.9) $\displaystyle\leq
C\left(\|v\|_{L^{p}(L^{q})}\|\nabla\sqrt{\rho}\|_{L^{\frac{p}{p-3}}(L^{\frac{q}{q-3}})}+\|\nabla
v\|_{L^{\frac{p}{p-2}}(L^{\frac{q}{q-2}})}\right),$

and

$\displaystyle\|\nabla\rho\|_{L^{\frac{p}{p-2}}(L^{\frac{q}{q-2}})}\leq$
$\displaystyle
C\|\sqrt{\rho}\nabla\sqrt{\rho}\|_{L^{\frac{p}{p-2}}(L^{\frac{q}{q-2}})}\leq
C\|\nabla\sqrt{\rho}\|_{L^{\frac{p}{p-3}}(L^{\frac{q}{q-3}})}.$ (3.10)

Plugging (3.9) and (3.10) into (3.8), we get

$\displaystyle\int_{0}^{T}\int\phi(t)v^{\varepsilon}\Big{[}\partial_{t}(\rho
v)^{\varepsilon}-\partial_{t}(\rho v^{\varepsilon})\Big{]}$ (3.11)
$\displaystyle\leq$ $\displaystyle
C\|v\|^{2}_{L^{p}(L^{q})}\left(\left(\|v\|_{L^{p}(L^{q})}+1\right)\|\nabla\sqrt{\rho}\|_{L^{\frac{p}{p-3}}(L^{\frac{q}{q-3}})}+\|\nabla
v\|_{L^{\frac{p}{p-2}}(L^{\frac{q}{q-2}})}\right)\leq C.$

From Lemma (2.1), we end up with, as $\varepsilon\rightarrow 0$,

$\int_{0}^{T}\int\phi(t)v^{\varepsilon}\Big{[}\partial_{t}(\rho
v)^{\varepsilon}-\partial_{t}(\rho v^{\varepsilon})\Big{]}\rightarrow 0.$

Using the integration by parts, we observe that

$\displaystyle\left|\int_{0}^{T}\int\phi(t)\nabla v^{\varepsilon}[(\rho
v\otimes v)^{\varepsilon}-(\rho v)\otimes v^{\varepsilon}]\right|$ (3.12)
$\displaystyle\leq$ $\displaystyle C\|\nabla
v^{\varepsilon}\|_{L^{\frac{p}{p-2}}(L^{\frac{q}{q-2}})}\|(\rho v\otimes
v)^{\varepsilon}-(\rho v)\otimes
v^{\varepsilon}\|_{L^{\frac{p}{2}}(L^{\frac{q}{2}})}$ $\displaystyle\leq$
$\displaystyle C\|\nabla
v\|_{L^{\frac{p}{p-2}}(L^{\frac{q}{q-2}})}\left(\|(\rho v\otimes
v)^{\varepsilon}-\rho v\otimes v\|_{L^{\frac{p}{2}}(L^{\frac{q}{2}})}+\|\rho
v\otimes v-\rho v\otimes
v^{\varepsilon}\|_{L^{\frac{p}{2}}(L^{\frac{q}{2}})}\right)$
$\displaystyle\leq$ $\displaystyle C\|\nabla
v\|_{L^{\frac{p}{p-2}}(L^{\frac{q}{q-2}})}\left(\|(\rho v\otimes
v)^{\varepsilon}-\rho v\otimes v\|_{L^{\frac{p}{2}}(L^{\frac{q}{2}})}+\|\rho
v\|_{L^{p}(L^{q})}\|v-v^{\varepsilon}\|_{L^{p}(L^{q})}\right).$

Hence, by the standard properties of the mollification, we have

$\int_{0}^{T}\int\phi(t)\nabla v^{\varepsilon}[(\rho v\otimes
v)^{\varepsilon}-(\rho v)\otimes v^{\varepsilon}]\rightarrow 0\text{ as
}\varepsilon\rightarrow 0.$

According to the upper bound of the density, Hölder’s inequality on bound
domain, we observe that

$\|\nabla(\rho^{\gamma})\|_{L^{\frac{p}{p-1}}(L^{\frac{q}{q-1}})}\leq
C\|\nabla\sqrt{\rho}\|_{L^{\frac{p}{p-1}}(L^{\frac{q}{q-1}})}\leq
C\|\nabla\sqrt{\rho}\|_{L^{\frac{p}{p-3}}(L^{\frac{q}{q-3}})},$

which in turn implies that

$\displaystyle\int_{0}^{T}\int\phi(t)[v^{\varepsilon}\nabla(\rho^{\gamma})^{\varepsilon}-v\nabla(\rho^{\gamma})]\rightarrow
0,$ (3.13)

where we have used lemma 2.2.

Then together with (3.11)-(3.13), passing to the limits as
$\varepsilon\rightarrow 0$, we know that

$\displaystyle-\int_{0}^{T}\int\phi_{t}\left(\frac{1}{2}\rho|v|^{2}+\frac{\rho^{\gamma}}{\gamma-1}\right)+\int_{0}^{T}\int\phi(t)\left(\mu|\nabla
v|^{2}+(\mu+\lambda)|\mathrm{div}\,v|^{2}\right)=0.$ (3.14)

The next objective is to get the energy equality up to the initial time $t=0$
by the similar method in [6] and [32], for the convenience of the reader and
the integrity of the paper, we give the details. First, we derive from the
mass equation $\eqref{INS}_{1}$ that

$\displaystyle\partial_{t}(\rho^{\gamma})=-\gamma\rho^{\gamma}\mathrm{div}\,v-2\gamma\rho^{\gamma-\frac{1}{2}}v\cdot\nabla\sqrt{\rho},$
(3.15)

and

$\displaystyle\partial_{t}(\sqrt{\rho})=\frac{\sqrt{\rho}}{2}\mathrm{div}\,v+v\cdot\nabla\sqrt{\rho},$
(3.16)

which together with (1.13) gives

$\partial_{t}(\rho^{\gamma},\sqrt{\rho})\in
L^{\frac{p}{p-2}}(0,T;L^{\frac{q}{q-2}}(\Omega)),\ \ \
\nabla(\rho^{\gamma},\sqrt{\rho})\in
L^{\frac{p}{p-3}}(0,T;L^{\frac{q}{q-3}}(\Omega)).$

Hence, using the Aubin-Lions Lemma 2.3, we can obtain

$(\rho^{\gamma},\ \sqrt{\rho})\in C([0,T];L^{\frac{q}{q-3}}(\Omega)),\ for\
dp<2q+3d,\ p\geq 4\ and\ q\geq 4.$ (3.17)

Furthermore, by the moumentum equality $\eqref{INS}_{2}$, we know that

$\rho v\in L^{\infty}(0,T;L^{2}(\Omega))\cap H^{1}(0,T;H^{-1}(\Omega)),$

then, in light of the Aubin-Lions Lemma 2.3, we conclude that

$\rho v\in C([0,T];L^{2}_{weak}(\Omega)).$ (3.18)

Meanwhile, using the natural energy (2.2), (3.17) and (3.18), we have

$\displaystyle 0$ $\displaystyle\leq\overline{\lim_{t\rightarrow
0}}\int|\sqrt{\rho}v-\sqrt{\rho_{0}}v_{0}|^{2}dx$ (3.19)
$\displaystyle=2\overline{\lim_{t\rightarrow
0}}\left(\int\left(\frac{1}{2}\rho|v|^{2}+\frac{1}{\gamma-1}\rho^{\gamma}\right)dx-\int\left(\frac{1}{2}\rho_{0}|v_{0}|^{2}+\frac{1}{\gamma-1}\rho_{0}^{\gamma}\right)dx\right)$
$\displaystyle\ \ \ +2\overline{\lim_{t\rightarrow
0}}\left(\int\sqrt{\rho_{0}}v_{0}\left(\sqrt{\rho_{0}}v_{0}-\sqrt{\rho}v\right)dx+\frac{1}{\gamma-1}\int\left(\rho_{0}^{\gamma}-\rho^{\gamma}\right)dx\right)$
$\displaystyle\leq 2\overline{\lim_{t\rightarrow
0}}\int\sqrt{\rho_{0}}v_{0}\left(\sqrt{\rho_{0}}v_{0}-\sqrt{\rho}v\right)dx$
$\displaystyle=2\overline{\lim_{t\rightarrow 0}}\int
v_{0}\left(\rho_{0}v_{0}-\rho v\right)dx+2\overline{\lim_{t\rightarrow 0}}\int
v_{0}\sqrt{\rho}v\left(\sqrt{\rho}-\sqrt{\rho_{0}}\right)dx=0,$

from which it follows

$\sqrt{\rho}v(t)\rightarrow\sqrt{\rho}v(0)\ \ strongly\ in\ L^{2}(\Omega)\ as\
t\rightarrow 0^{+}.$ (3.20)

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

$\sqrt{\rho}v(t)\rightarrow\sqrt{\rho}v(t_{0})\ \ strongly\ in\ L^{2}(\Omega)\
as\ t\rightarrow t_{0}^{+}.$ (3.21)

Before we go any further, it should be noted that (3.14) 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

$\phi_{\tau}(t)=\left\\{\begin{array}[]{lll}0,&0\leq t\leq\tau,\\\
\frac{t-\tau}{\alpha},&\tau\leq t\leq\tau+\alpha,\\\ 1,&\tau+\alpha\leq t\leq
t_{0},\\\ \frac{t_{0}-t}{\alpha},&t_{0}\leq t\leq t_{0}+\alpha,\\\
0,&t_{0}+\alpha\leq t.\end{array}\right.$ (3.22)

Then substituting this test function into (3.14), we arrive at

$\displaystyle-\int_{\tau}^{\tau+\alpha}\int$
$\displaystyle\frac{1}{\alpha}\left(\frac{1}{2}\rho
v^{2}+\frac{1}{\gamma-1}\rho^{\gamma}\right)+\frac{1}{\alpha}\int_{t_{0}}^{t_{0}+\alpha}\int\left(\frac{1}{2}\rho
v^{2}+\frac{1}{\gamma-1}\rho^{\gamma}\right)$ (3.23)
$\displaystyle+\int_{\tau}^{t_{0}+\alpha}\int\phi_{\tau}\left(\mu|\nabla
v|^{2}+\left(\mu+\lambda\right)|\mathrm{div}\,v|^{2}\right)=0.$

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

$\displaystyle-\int$ $\displaystyle\left(\frac{1}{2}\rho
v^{2}+\frac{1}{\gamma-1}\rho^{\gamma}\right)(\tau)dx+\int\left(\frac{1}{2}\rho
v^{2}+\frac{1}{\gamma-1}\rho^{\gamma}\right)(t_{0})dx$ (3.24)
$\displaystyle+\int_{\tau}^{t_{0}}\int\left(\mu|\nabla
v|^{2}+\left(\mu+\lambda\right)|\mathrm{div}\,v|^{2}\right)dxds=0.$

Finally, letting $\tau\rightarrow 0$, using the continuity of
$\int_{0}^{t}\int\left(\mu|\nabla
v|^{2}+\left(\mu+\lambda\right)|\mathrm{div}\,v|^{2}\right)$, (3.17) and
(3.20), we can obtain

$\displaystyle\int\left(\frac{1}{2}\rho
v^{2}+\frac{1}{\gamma-1}\rho^{\gamma}\right)(t_{0})dx+\int_{0}^{t_{0}}\int$
$\displaystyle\left(\mu|\nabla
v|^{2}+\left(\mu+\lambda\right)|\mathrm{div}\,v|^{2}\right)dxds$ (3.25)
$\displaystyle=$
$\displaystyle\int\left(\frac{1}{2}\rho_{0}v_{0}^{2}+\frac{1}{\gamma-1}\rho_{0}^{\gamma}\right)dx.$

Then we complete the proof of Theorem 1.3. ∎

Next, with the help of Theorem 1.3, we are in a position to prove Theorem 1.1.

###### Proof of Theorem 1.1.

(1) Choose $p=q=4$ in (1.13), we obtain Lions type energy conservation
condition.

$\displaystyle 0\leq\rho<c<\infty,\nabla\sqrt{\rho}\in L^{4}(L^{4}),$ (3.26)
$\displaystyle v\in L^{4}(L^{4}),\nabla v\in L^{2}(L^{2})\ and\ v_{0}\in
L^{2}(\mathbb{T}^{d}).$

(2) With the help of interpolation, we show (1.10) can be reduced to (1.9).
Indeed, taking advantage of the Gagliardo-Nirenberg inequality on bounded
domains, Hölder’s inequality and Young’s inequality, we know that

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

From the the result just proved, we obtain energy equality via (1.9). We
finish the proof of this case.

(3) Taking $p=4,q=6$ in (1.13), we immediately get the desired result.

(4) We move to the proof of the last case. Firstly, we temporarily assume that
the following fact is valid,

$\int_{\Omega}vdx\in L^{p}(0,T).$ (3.28)

It follows from the triangle inequality and Poincaré-Sobolev inequality that,
for $q<d$,

$\displaystyle\|v\|_{L^{p}(L^{\frac{dq}{d-q}})}\leq$
$\displaystyle\Big{\|}v-\frac{1}{|\Omega|}\int_{\Omega}vdy\Big{\|}_{L^{p}(L^{\frac{dq}{d-q}})}+\Big{\|}\frac{1}{|\Omega|}\int_{\Omega}vdy\Big{\|}_{L^{p}(L^{\frac{dq}{d-q}})}$
$\displaystyle\leq$ $\displaystyle C\|\nabla
v\|_{L^{p}(L^{q})}+C\Big{\|}\int_{\Omega}vdy\Big{\|}_{L^{p}(0,T)}.$

From the result case (2) in just proved, it is enough to prove the quality
(3.28) we have assumed. The Hölder inequality and classical Poincaré
inequality ensure that

$\displaystyle\Big{|}\int_{\Omega}\rho\left(v-\frac{1}{|\Omega|}\int_{\Omega}vdy\right)dx\Big{|}\leq$
$\displaystyle\|\rho\|_{L^{\frac{q}{q-1}}}\Big{\|}v-\frac{1}{|\Omega|}\int_{\Omega}vdy\Big{\|}_{L^{q}}$
(3.29) $\displaystyle\leq$ $\displaystyle
C\|\rho\|_{L^{\frac{q}{q-1}}}\|\nabla v\|_{L^{q}}.$

Using the Hölder’s inequality once again and the upper bound of the density,
we find

$\int\rho vdx\leq C\|\sqrt{\rho}v\|_{L^{2}}.$ (3.30)

In view of the triangle inequality, (3.29) and (3.30), we infer that

$\displaystyle\Big{|}\int_{\Omega}vdy\Big{|}=$
$\displaystyle\frac{|\Omega|}{\int_{\Omega}\rho_{0}dx}\Big{|}\int_{\Omega}\left(\frac{1}{|\Omega|}\int_{\Omega}vdy\right)\rho
dx\Big{|}$ $\displaystyle\leq$
$\displaystyle\frac{C}{\int_{\Omega}\rho_{0}dx}\Big{|}\int_{\Omega}\Big{[}\left(\frac{1}{|\Omega|}\int_{\Omega}vdy\right)-v\Big{]}\rho
dx\Big{|}+\frac{C}{\int_{\Omega}\rho_{0}dx}\Big{|}\int_{\Omega}\rho
vdx\Big{|}$ $\displaystyle\leq$ $\displaystyle C\|\nabla
v\|_{L^{q}}+C\|\sqrt{\rho}v\|_{L^{2}}.$

which means

$\displaystyle\Big{\|}\int_{\Omega}vdy\Big{\|}_{L^{p}(0,T)}\leq$
$\displaystyle C\|\nabla
v\|_{L^{p}(L^{q})}+C\|\sqrt{\rho}v\|_{L^{\infty}(L^{2})}.$

Thus, it follows from the triangle inequality and Poincaré-Sobolev inequality
that

$\displaystyle\|v\|_{L^{p}(L^{q})}\leq$
$\displaystyle\Big{\|}v-\frac{1}{|\Omega|}\int_{\Omega}vdy\Big{\|}_{L^{p}(L^{q})}+\Big{\|}\frac{1}{|\Omega|}\int_{\Omega}vdy\Big{\|}_{L^{p}(L^{q})}$
$\displaystyle\leq$ $\displaystyle C\|\nabla
v\|_{L^{p}(L^{q})}+C\Big{\|}\int_{\Omega}vdy\Big{\|}_{L^{p}(0,T)}$
$\displaystyle\leq$ $\displaystyle C\left(\|\nabla
v\|_{L^{p}(L^{q})}+\|\sqrt{\rho}v\|_{L^{\infty}(L^{2})}\right),$

which means $v\in L^{p}(D^{1,q})$ implies $v\in L^{p}(W^{1,q})$. which implies

$v\in L^{p}(0,T;W^{1,q}(\mathbb{T}^{d})),\ q<d.$

In summary, we have shown

$\nabla v\in
L^{p}(0,T;L^{r}(\mathbb{T}^{d}))~{}\text{with}~{}\frac{1}{p}+\frac{3}{r}=1+\frac{3}{d},\
\frac{3d}{d+3}<r\leq\frac{4d}{d+4},$

turns out

$v\in
L^{p}(0,T;L^{q}(\mathbb{T}^{d}))~{}\text{with}~{}\frac{1}{p}+\frac{3}{q}=1,3<q\leq
4.$

At this stage, we complete the proof of Theorem 1.1. ∎

######  Remark 3.1.

To prove (1.12), one may directly employ the following Poincaré inequality
which can be proved by slightly modifying the proof of [11, Lemma 3.2, page
47]:
Let $v\in W^{1,q}(\Omega)$ with $q\geq 2$, and let $\rho$ be a non-negative
function such that

$0<M\leq\int_{\Omega}\rho dx,\ \int_{\Omega}\rho^{\gamma}dx\leq E_{0}<\infty,$
(3.31)

where $\Omega\subset\mathbb{R}^{d}$ is a bounded domain and $\gamma>1$. Then
there exists a constant $c$ depending solely on $M$ and $E_{0}$ such that

$\|v\|_{L^{q}(\Omega)}^{q}\leq c(E_{0},M)\left(\|\nabla
v\|_{L^{q}(\Omega)}^{q}+\left(\int_{\Omega}\rho|v|dx\right)^{q}\right).$
(3.32)

Next, with the help of Theorem 1.1, we prove Corollary 1.2.

###### Proof of Corollary 1.2.

According to Theorem 1.1, it suffices to show that all the density hypotheses
yield

$0\leq\rho\leq c_{2}<\infty,\nabla\sqrt{\rho}\in
L^{4}(0,T;L^{4}(\mathbb{T}^{3})).$

On the one hand, for the case (2), it is enough to show $0\leq\rho\leq
c_{2}<\infty.$ The Sobolev embedding theorem ensures that
$\nabla\sqrt{\rho}\in L^{\infty}(0,T;L^{4}(\mathbb{T}^{3}))$ and ${\rho}\in
L^{\infty}(0,T;L^{\gamma}(\mathbb{T}^{3}))$ with $\gamma>\frac{3}{2}$ (due to
the theory of the global existence of weak solutions to (LABEL:INS)-(1.2)
obtained by Feireisl- Novotný-Petzeltová in [13]) leads to

$0\leq\rho\leq c_{2}<\infty.$

For the case (3), it follows from the well-know interpolation inequality and
Hölder’s inequality that, for $\frac{1}{p_{1}}+\frac{1}{q_{1}}=\frac{1}{2}$,
with $q_{1}\geq 4$,

$\displaystyle\|\nabla\sqrt{\rho}\|_{L^{4}(0,T;L^{4}(\mathbb{T}^{3})))}\leq$
$\displaystyle
C\|\nabla\sqrt{\rho}\|_{L^{\infty}\left(0,T;L^{2}(\mathbb{T}^{3})\right)}^{\frac{(q_{1}-4)}{2(q_{1}-2)}}\|\nabla\sqrt{\rho}\|_{L^{p_{1}}(0,T;L^{{q_{1}}}(\mathbb{T}^{3}))}^{\frac{q_{1}}{2(q_{1}-2)}}<\infty.$

This completes the proof of this corollary. ∎

## Acknowledgement

Ye was partially supported by the National Natural Science Foundation of China
under grant (No.11701145) and China Postdoctoral Science Foundation (No.
2020M672196). Wang was partially supported by the National Natural Science
Foundation of China under grant (No. 11971446, No. 12071113 and No. 11601492).
Wei was partially supported by the National Natural Science Foundation of
China under grant (No. 11601423, No. 11771352, No. 11871057).

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