Time decay of solutions for compressible isentropic non-Newtonian fluids

Abstract

In this paper, we consider the Cauchy problem of a compressible Navier–Stokes system of Eills-type non-Newtonian fluids. We investigate the time decay properties of classical solutions for the compressible non-Newtonian fluid equations. More specifically, we construct a new linearized system in terms of a combination of the solutions, and then we investigate the long-time behavior of the Cauchy problem for the three-dimensional isentropic compressible Eills-type non-Newtonian fluids with an initial perturbation.

1 Introduction

Motivated by the well-posedness result of non-Newtonian fluids in [1] and the long-time behavior result of the Navier–Stokes system in [2], we investigate the time decay properties of solutions for compressible non-Newtonian fluid equations. More specifically, we investigate the optimal decay rate of the highest-order derivative of solutions to the equations of compressible non-Newtonian fluids, which defined in a bounded domain $\Omega\subset\mathbb{R}^{3}$ is governed by the following equations [3]:

$$ \textstyle\begin{cases} \rho _{t}+\operatorname{div}(\rho \boldsymbol{v}) =0, \\ (\rho v)_{t}+\operatorname{div}(\rho \boldsymbol{v}\otimes \boldsymbol{v})+\nabla p(\rho )= \operatorname{div}\mathbb{S}, \end{cases} $$

(1.1)

where the unknowns $\rho :=\rho (x, t)$, $\boldsymbol{v}:=\boldsymbol{v}(x, t)$ denote the density and the velocity of the non-Newtonian fluids, respectively, $p:=p(\rho )$ is the fluid pressure, which is a smooth function depending on ρ, and $\mathbb{S}$ represents the viscous stress tensor, which depends on the rate of strain $D_{ij}(\nabla \boldsymbol{v})$, where $D_{ij}(\nabla \boldsymbol{v})$ is given as $D_{ij}(\nabla \boldsymbol{v})=\frac{\partial v_{i}}{\partial x_{j}}+ \frac{\partial v_{j}}{\partial x_{i}}$. We mention that (1.1)₁ is the continuity equation and (1.1)₂ describes the balance law of momentum.

If the relation between the stress and the rate of strain is linear, i.e., $S_{ij}=\mu (\frac{\partial v_{i}}{\partial x_{j}}+ \frac{\partial v_{j}}{\partial x_{i}} )$, then the fluid is called Newtonian. The coefficient μ is called the viscosity coefficient, which depends on temperature, density, and pressure. For example, water, alcohols, and simple hydrocarbon compounds turn out to be Newtonian fluids. The governing equations of motion can be written by the Navier–Stokes equations.

If the relation between the stress and the rate of strain is nonlinear, the fluid is called non-Newtonian. Examples of non-Newtonian fluids include molten plastics, greases, paper pulp, and biological fluids like blood. The simplest stress–strain relation in non-Newtonian fluids is given by $S_{ij}=\mu (\frac{\partial v_{i}}{\partial x_{j}}+ \frac{\partial v_{j}}{\partial x_{i}} )^{q}$ for $0< q<1$ [4].

Recently, the following stress–strain relation has been widely investigated [3]:

$$ S_{ij}= \bigl(\mu _{0}+ \mu _{1} \bigl\vert D(\nabla \boldsymbol{v}) \bigr\vert ^{p-2} \bigr)D_{ij}(\nabla \boldsymbol{v}), $$

where

$$ \textstyle\begin{cases} \text{Newtonian}, & \text{for } \mu _{0}>0,\mu _{1}=0; \\ \text{Rabinowitsch} , & \text{for } \mu _{0},\mu _{1}>0,p=4; \\ \text{Eills}, & \text{for } \mu _{0},\mu _{1}>0,p>2; \\ \text{Ostwald--de Waele}, & \text{for } \mu _{0}=0,\mu _{1}>0,p>1; \\ \text{Bingham}, & \text{for } \mu _{0},\mu _{1}>0,p=1. \end{cases} $$

The above five types of stress–strain relation can be found in non-Newtonian fluids. For example, for $\mu _{0}=0$, it is a pseudo-plastic fluid in the case of $p<2$ and it is a dilatant fluid in the case of $p>2$. From a physical point of view, the stress–strain relation describes a shear thickening fluid if $p>2$ and a shear thinning fluid if $1< p<2$. The values of the parameters p and $\mu _{1}$ of the pseudo-plastic Ostwald–de Waele models are presented in [3].

On account of the physical importance of non-Newtonian fluids, they have attracted attention from many engineers, mathematicians, physicists, and so on. However, both the problems of well-posedness and dynamical behaviors of motion equations of non-Newtonian fluids are very difficult to investigate because of the singularity.

Even so, important progress has been made in the theoretical analysis of non-Newtonian fluid systems: Bothe–Pruss studied a class of non-Newtonian fluids based on $L_{p}$-theory [5]; Feireisl–Kwon studied the long-time behavior of dissipative solutions to models of non-Newtonian compressible fluids [6]; Moscariello–Porzio investigated the behavior in time of solutions to motion of Non-Newtonian fluids [7]. More results of non-Newtonian fluids can be found in [1, 3, 8–11] and the references cited therein.

In this paper, we investigate the corresponding Eills-type non-Newtonian fluids, i.e.,

$$\begin{aligned} &\mathbb{S}= \bigl(\mu _{0}+ \mu _{1} \bigl\vert D( \nabla \boldsymbol{v}) \bigr\vert ^{p-2} \bigr)D(\nabla \boldsymbol{v}), \\ &\operatorname{div}\mathbb{S}=\mu _{0}(\Delta \boldsymbol{v}+\nabla \operatorname{div}\boldsymbol{v}) +\mu _{1}\operatorname{div} \bigl( \bigl\vert D(\nabla \boldsymbol{v}) \bigr\vert ^{p-2}D(\nabla \boldsymbol{v}) \bigr). \end{aligned}$$

Thus, equation (1.1) reduces to

$$ \textstyle\begin{cases} \rho _{t}+\operatorname{div}(\rho \boldsymbol{v})=0, \\ \rho \boldsymbol{v}_{t}+\operatorname{div}(\rho \boldsymbol{v}\otimes \boldsymbol{v})+\nabla p( \rho ) -\mu _{0}(\Delta \boldsymbol{v}+\nabla \operatorname{div}\boldsymbol{v}) =\mu _{1} \operatorname{div} ( \vert D(\nabla \boldsymbol{v}) \vert ^{p-2}D(\nabla \boldsymbol{v}) ). \end{cases} $$

(1.2)

We can reformulate the system (1.2). Without loss of generality, we take $P'(1) = 1$ and thus derive the following equations from (1.2) by applying the formulas above:

$$ \textstyle\begin{cases} \rho _{t}+\operatorname{div}(\rho \boldsymbol{v})=0, \\ \boldsymbol{v}_{t}+\boldsymbol{v}\cdot \nabla \boldsymbol{v}+p'(\rho )\nabla \rho /\rho -\mu _{0}( \Delta \boldsymbol{v}+\nabla \operatorname{div}\boldsymbol{v})/\rho \\ \quad =\mu _{1}\operatorname{div} ( \vert D(\nabla \boldsymbol{v}) \vert ^{p-2}D(\nabla \boldsymbol{v}) )/\rho . \end{cases} $$

(1.3)

Much important progress has been made in the investigation of long-time behaviors of global smooth solutions to compressible Navier–Stokes systems. For instance, the global existence of strong solutions to compressible Navier–Stokes equations in multidimensional whole space was obtained first by Matsumura–Nishida [12, 13], who also showed that the global solution tends to its equilibrium state in large time. The optimal $L^{p} (p\geqslant 2)$ decay rates were established later by Ponce [14]. To conclude, the optimal $L^{2}$ time decay rate for isentropic compressible Navier–Stokes equations in three dimensions is

$$\begin{aligned} \bigl\Vert (\rho -\bar{\rho}, \boldsymbol{v}) (t) \bigr\Vert _{L^{2}(\mathbb{R}^{3})}\leqslant C(1+t)^{- \frac{3}{4}}, \end{aligned}$$

where $(\bar{\rho}, \boldsymbol{0})$ is the constant state.

Then Liu–Wang [15] investigated the properties of the Green’s function for isentropic Navier–Stokes systems and showed an interesting pointwise convergence of global solutions to the diffusive waves with the optimal time decay rate in odd dimension where the important phenomenon of the weaker Huygen principle is also justified due to the dispersion effects of compressible viscous fluids in multidimensional odd space. This was generalized to the full system later in [16], where the wave motions of other types are also introduced. Therein, the optimal $L^{\infty}$ time decay rate in three dimensions is

$$\begin{aligned} \bigl\Vert (\rho -\bar{\rho}, \boldsymbol{v}) (t) \bigr\Vert _{L^{\infty}(\mathbb{R}^{3})} \leqslant C(1+t)^{-\frac{3}{2}}. \end{aligned}$$

The same decay property also appears in the exterior domain problem [17] and an infinite layer [18]. Li–Matsumura–Zhang [19] obtained the optimal time decay of the Navier–Stokes–Poisson system, which is different from the pure Navier–Stokes equations.

Next, Li–Zhang investigated the long-time behavior and optimal decay rates of global strong solutions to three-dimensional isentropic compressible Navier–Stokes systems; when the regular initial data also belong to some Sobolev space $H^{l}(\mathbb{R}^{3})\cap \dot{B}^{-s}_{1,\infty}(\mathbb{R}^{3})$ with $l\geqslant 4$ and $s\in [0, 1]$, they show that the global solution to this system converges to the equilibrium state at a faster decay rate in time.

Later, Tan–Wang derived the optimal time decay rates for the higher-order spatial derivatives of solutions to magnetohydrodynamic equations [20]. Besides, Gao–Chen–Yao [21] further deduced higher time decay rates for the higher-order spatial derivatives of solutions, which improve the result of Tan–Wang [20]. Recently, exploiting the technique of decomposition of solutions into low and high frequencies in [22], Huang–Lin–Wang proved that this result also holds for $k=N$ [23]. At present, the above isentropic results have been further extended to non-isentropic cases; see [21, 24] for examples.

There are many other important results on time decay estimation. Abdallah–Jiang–Tan studied the decay estimates for isentropic compressible magnetohydrodynamic equations in a bounded domain [25], Fan–Jiang studied the long-time behavior of liquid crystal flows with a trigonometric condition in two dimensions [26], Guo–Tan studied the long-time behavior of solutions to a class of non-Newtonian compressible fluids [27], Chen–Tan–Wu have given the time decay rates for the equations of compressible heat-conductive flow through porous media [28], and Tan–Wu studied the long-time behavior of solutions for compressible Euler equations with damping in $\mathbb{R}^{3}$ [29]. Zhang studied the decay of the three-dimensional inviscid liquid–gas two-phase flow model [30] and the decay of the three-dimensional viscous liquid–gas two-phase flow model with damping [31]. Zhang–Wu studied the global well-posedness and long-time behavior of the viscous liquid–gas two-phase flow model [32] and the global existence and asymptotic behavior for the three-dimensional compressible non-isentropic Euler equations with damping [33]. Zhang–Tan studied the existence [34] and asymptotic behavior of global smooth solutions for p-systems with damping and boundary effects and the asymptotic behavior of solutions to the Navier–Stokes equations of a two-dimensional compressible flow [35]. Zhang–Tan–Ming studied the global existence and asymptotic behavior of smooth solutions to a coupled hyperbolic-parabolic system [36]. Jiang–Zhang studied the existence and asymptotic behavior of global smooth solutions for p-systems with nonlinear damping and fixed boundary effects [37]. Qiu–Zhang studied the decay of the three-dimensional quasilinear hyperbolic equations with nonlinear damping [38]. Then Hu–Qiu–Wang–Yang studied the incompressible limit for compressible viscoelastic flows with large velocity [39], Zhao–Li–Yan studied the global Sobolev regular solution for Boussinesq systems [40], and Panasenko–Pileckas studied the non-stationary Poiseuille flow of a non-Newtonian fluid with the shear rate-dependent viscosity [41].

In this paper, we are further interested in non-Newtonian fluids. The well-posedness problem of non-Newtonian fluids has been widely investigated; see [42] for the existence results. Motivated by the optimal time decay rates of solutions of Newtonian fluids and the result of the long-time behavior of solutions to non-Newtonian compressible fluids, using the methods in [43], we further investigate the global well-posedness and optimal time decay rates of the solutions for the system (1.2). To this purpose, we provide the following initial condition for the system (1.2):

$$\begin{aligned} (\rho , \boldsymbol{v})|_{t=0}=\bigl(\rho ^{0}, \boldsymbol{v}^{0}\bigr). \end{aligned}$$

(1.4)

We will consider the existence problem and the optimal time decay rates of small perturbation solutions around the rest state $(\bar{\rho}, \boldsymbol{0})$ of the system (1.2), where ρ̄ is always taken to be equal to one for the sake of simplicity.

By taking the new change of variables

$$\begin{aligned} \varrho =\rho -1,\qquad \boldsymbol{u}=\boldsymbol{v}-\boldsymbol{0}, \end{aligned}$$

we can further rewrite the system (1.3)–(1.4) into the perturbation forms:

$$ \textstyle\begin{cases} \varrho _{t}+\operatorname{div}\boldsymbol{u}=-\operatorname{div}(\varrho \boldsymbol{u}), \\ \boldsymbol{u}_{t}-\mu _{0}(\Delta \boldsymbol{u}+\nabla \operatorname{div}\boldsymbol{u})+ \nabla \varrho =\mathcal{N}, \\ (\varrho , \boldsymbol{u})|_{t=0}=(\varrho ^{0}, \boldsymbol{u}^{0})=(\rho ^{0}-1, \boldsymbol{u}^{0}), \end{cases} $$

(1.5)

where the nonlinear terms $\mathcal{N}$ are defined by

$$\begin{aligned} \mathcal{N}& = -\boldsymbol{u}\cdot \nabla \boldsymbol{u}-h_{3}( \varrho )\nabla \varrho -h_{2}(\varrho ) (\Delta \boldsymbol{u} + \nabla \operatorname{div} \boldsymbol{u}) \\ &\quad{} + h_{1}(\varrho )\operatorname{div} \bigl( \bigl\vert D( \nabla \boldsymbol{u}) \bigr\vert ^{p-2}D( \nabla \boldsymbol{u}) \bigr), \end{aligned}$$

where the nonlinear functions are defined as follows:

$$\begin{aligned} h_{1}(\varrho )=\frac{\mu _{1}}{\varrho +1},\qquad h_{2}(\varrho )= \frac{\mu _{0}\varrho}{\varrho +1},\qquad h_{3}(\varrho )= \frac{p'(\varrho +1)}{\varrho +1}-1. \end{aligned}$$

Before presenting our main result, we introduce the following notations, which are used frequently throughout the paper.

1.1 Notations

(1) Basic notations:

The notation $\langle \cdot , \cdot \rangle$ stands for the inner product in $L^{2} (\mathbb{R}^{3})$; $a \lesssim b$ means that $a\leqslant Cb$ for some constant $C>0$. For simplicity, we also denote $a\approx b$ if $a\lesssim b$ and $a\gtrsim b$. The symbol $\nabla ^{l}$ with an integer $l\geqslant 1$ represents as usual any spatial derivatives of order l; $C_{i}>0$ represents a generic constant that may vary from line to line for $i\in \mathbb{Z}^{+}$; and the integral symbol $\int :=\int _{\mathbb{R}^{3}}$.

(2) Notations of function spaces:

We employ $L^{r} (\mathbb{R}^{3})$ to denote the usual $L^{r}$ spaces and $H^{s} (\mathbb{R}^{3})$ to denote the Sobolev spaces with norm $\|\cdot \|_{L^{r}}$ and $\|\cdot \|_{H^{s}}$, respectively, where $1\leqslant r\leqslant \infty , s\in \mathbb{R}$.

In addition, Λ is a pseudo-differential operator, which is defined by

$$\begin{aligned} \Lambda ^{s}f=\mathcal{F}^{-1}\bigl( \vert \xi \vert ^{s} \hat{f}\bigr) \quad \text{for } s\in \mathbb{R}, \end{aligned}$$

where f̂ and $\mathcal{F}^{-1}(f)$ denote the Fourier transform and the inverse Fourier transform, respectively.

Let $\varphi (\xi )$ be a smooth cut-off function, which satisfies $0 \leqslant \varphi (\xi ) \leqslant 1 (\xi \in \mathbb{R}^{3})$ and $\varphi (\xi )=1, |\xi |\leqslant 1$, $\varphi (\xi )=0$, $|\xi |>1$. Then we can define a frequency decomposition for the function $f(x)\in L^{2}(\mathbb{R}^{3})$ as follows:

$$\begin{aligned} f^{L}(x)=\varphi (D_{x})f(x),\qquad f^{H}(x)= \bigl(I-\varphi (D_{x}) \bigr)f(x), \end{aligned}$$

where $D_{x}:= \frac{1}{\sqrt{-1}}(\partial _{x_{1}}, \partial _{x_{2}}, \partial _{x_{3}})$ and $\varphi (D_{x})$ is a pseudo-differential operator with respect to $\varphi (\xi )$. Moreover, $f(x)$ can be expressed as

$$\begin{aligned} f(x)=f^{L}(x)+f^{H}(x). \end{aligned}$$

(1.6)

1.2 Main result

Now, we are in a position to present our main result.

Theorem 1.1

Suppose $(\rho ^{0}-1, \boldsymbol{u}^{0})\in H^{3} (\mathbb{R}^{3})$ and

$$\begin{aligned} \bigl\Vert \bigl(\rho ^{0}-1, \boldsymbol{u}^{0} \bigr) (t) \bigr\Vert _{H^{3} (\mathbb{R}^{3})}\leqslant \epsilon , \end{aligned}$$

where ϵ is a sufficiently small constant. Then the Cauchy problem (1.5) admits a unique global solution $(\rho , \boldsymbol{u})$ satisfying

$$\begin{aligned} &\rho -1\in C^{0} \bigl([0, \infty ); H^{3}\bigl( \mathbb{R}^{3}\bigr) \bigr) \cap C^{1} \bigl([0, \infty ); H^{2}\bigl(\mathbb{R}^{3}\bigr) \bigr), \\ &\boldsymbol{u}\in C^{0} \bigl([0, \infty ); H^{3} \bigl(\mathbb{R}^{3}\bigr) \bigr) \cap C^{1} \bigl([0, \infty ); H^{1}\bigl(\mathbb{R}^{3}\bigr) \bigr). \end{aligned}$$

Moreover, if the initial data $(\rho ^{0}-1, \boldsymbol{u}^{0})$ are bounded in $L^{1}(\mathbb{R}^{3})$ space, then, for any $t \geqslant 0$, the classical solution $(\rho , \boldsymbol{u})$ enjoys

$$\begin{aligned} \bigl\Vert \nabla ^{k}(\rho -1, \boldsymbol{u}) (t) \bigr\Vert _{H^{3}(\mathbb{R}^{3})}\leqslant C(1+t)^{- \frac{3}{4}-\frac{k}{2}},\quad k=0, 1, 2, 3. \end{aligned}$$

The first step involves constructing a new linearized system by combining the solutions. Subsequently, our focus is on establishing the global existence and uniqueness of the solution for the Cauchy problem (1.5). The a priori estimates will be provided in Proposition 3.1, where our main focus lies on estimating the nonlinear terms of a specific class of compressible Eills-type non-Newtonian fluids with $\mathbb{S}= (\mu _{0}+ \mu _{1} \vert D(\nabla \boldsymbol{u}) \vert ^{p-2} )D(\nabla \boldsymbol{u})$. We employ an energy estimation method to address the challenges posed by nonlinear structures, thereby enabling us to obtain the energy estimation under the $H^{3}$ norm. Subsequently, we present a technique for eliminating the low frequency component and provide a decay estimate for this particular part. Finally, we establish decay rates for the nonlinear system.

The rest of this paper is organized as follows. First, in Sect. 2, we list some well-known mathematical results, which will be used in Sects. 3 and 4. In Sect. 3, we establish a priori estimates of solutions and then prove the existence of the global-in-time solution based on the local existence of unique solutions. In Sect. 4, we will obtain the optimal time decay rates of the non-homogeneous system by constructing some decay estimates of the linearized system based on the technique of decomposition of solutions into low and high frequencies [22].

2 Basic analysis tools

This section is devoted to providing some important mathematical results, which will be used in the next sections.

Lemma 2.1

([22]). For any given integers i, j, k, we have

$$\begin{aligned} &\bigl\Vert \nabla ^{j}f^{L} \bigr\Vert _{L^{2}}\leqslant r^{j-i}_{0} \bigl\Vert \nabla ^{i}f^{L} \bigr\Vert _{L^{2}},\qquad \bigl\Vert \nabla ^{j}f^{H} \bigr\Vert _{L^{2}} \leqslant \frac{1}{R^{k-i}_{0}} \bigl\Vert \nabla ^{k}f^{L} \bigr\Vert _{L^{2}}, \\ &\bigl\Vert \nabla ^{j}f^{L} \bigr\Vert _{L^{2}}\leqslant \bigl\Vert \nabla ^{k}f \bigr\Vert _{L^{2}} \quad \textit{and}\quad \bigl\Vert \nabla ^{j}f^{H} \bigr\Vert _{L^{2}}\leqslant \bigl\Vert \nabla ^{k}f \bigr\Vert _{L^{2}}, \end{aligned}$$

where $f\in H^{n}(\mathbb{R}^{3})$ and $i\leqslant j\leqslant k\leqslant n$. Moreover, we have

$$\begin{aligned} r^{j}_{0} \bigl\Vert f^{n} \bigr\Vert _{L^{2}}\leqslant \bigl\Vert \nabla ^{j}f^{n} \bigr\Vert _{L^{2}} \leqslant R^{j}_{0} \bigl\Vert f^{n} \bigr\Vert _{L^{2}}, \end{aligned}$$

for some constant $r_{0}>0$ and $R_{0}>0$.

Lemma 2.2

([44]). Let $f \in H^{2}(\mathbb{R}^{3})$. Then we can have

$$\begin{aligned}& \Vert f \Vert _{L^{p}}\leqslant \Vert f \Vert _{H^{1}} \quad \textit{for }2\leqslant p \leqslant 6, \\& \Vert f \Vert _{L^{\infty}}\leqslant \Vert \nabla f \Vert ^{\frac{1}{2}}_{L^{2}} \Vert \nabla f \Vert ^{\frac{1}{2}}_{H^{1}} \leqslant \Vert \nabla f \Vert _{H^{1}}, \\& \Vert f \Vert _{L^{6}}\leqslant \Vert \nabla f \Vert _{L^{2}}. \end{aligned}$$

Lemma 2.3

([45]). We have

$$\begin{aligned} \bigl\Vert \nabla ^{l}(fg) \bigr\Vert _{L^{p}} \leqslant \Vert f \Vert _{L^{q_{1}}} \bigl\Vert \nabla ^{l}g \bigr\Vert _{L^{q_{2}}} + \bigl\Vert \nabla ^{l}f \bigr\Vert _{L^{q_{3}}} \Vert g \Vert _{L^{q_{4}}}, \end{aligned}$$

where $l\geqslant 1$, $1\leqslant q_{i}\leqslant +\infty$, and

$$\begin{aligned} \frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}= \frac{1}{q_{3}}+ \frac{1}{q_{4}}. \end{aligned}$$

Lemma 2.4

(Gagliardo–Nirenberg inequality). Suppose $0\leqslant i, j\leqslant k$. Then we have

$$\begin{aligned} \bigl\Vert \nabla ^{i}g \bigr\Vert _{L^{p}}\leqslant \bigl\Vert \nabla ^{j}g \bigr\Vert ^{1-\sigma}_{L^{p_{1}}} \bigl\Vert \nabla ^{k}g \bigr\Vert ^{\sigma}_{L^{p_{2}}}, \end{aligned}$$

where $0\leqslant \sigma \leqslant 1$ and

$$\begin{aligned} \frac{i}{3}-\frac{1}{p}= \biggl(\frac{j}{3}- \frac{1}{p_{1}} \biggr) (1- \sigma ) + \biggl(\frac{k}{3}+ \frac{1}{p_{2}} \biggr)\sigma . \end{aligned}$$

In particular, if $p=\infty$, then $0<\sigma <1$ is required.

Lemma 2.5

([46]). Let $\psi (\omega )$ be a smooth function of ω with bounded derivatives of any order. If $\|\omega \|_{L^{\infty}(\mathbb{R}^{3})}\leqslant 1$, then for any integer $i\geqslant 1$, we have

$$\begin{aligned} \bigl\Vert \nabla ^{i}\psi (\omega ) \bigr\Vert _{L^{p}(\mathbb{R}^{3})}\leqslant \bigl\Vert \nabla ^{i}\omega \bigr\Vert _{L^{p}(\mathbb{R}^{3})}, \end{aligned}$$

for $1\leqslant p\leqslant \infty$.

For the decay estimates of solutions, we further introduce the following basic inequalities.

Lemma 2.6

([47]). Suppose $c_{1}, c_{2}, c_{3}\in \mathbb{R}^{3}$ and $0\leqslant c_{1}\leqslant c_{2}, c_{3}>0$. We have, for $t\in \mathbb{R}_{+}$,

$$\begin{aligned} \int ^{t}_{0}(1+t-\tau )^{-c_{1}}(1+ \tau )^{-c_{2}}\,\mathrm{d} {\tau} \leqslant C(c_{1}, c_{2}) (1+t)^{-c_{1}} \end{aligned}$$

and

$$\begin{aligned} \int ^{t}_{0}(1+\tau )^{-c_{1}}e^{-c_{3}(t-\tau )} \,\mathrm{d} {\tau} \leqslant C(c_{1}, c_{3}) (1+t)^{-c_{1}}, \end{aligned}$$

where constants $C(c_{1}, c_{2})>0$, $C(c_{1}, c_{3})>0$ only depend on $c_{1}$, $c_{2}$, $c_{3}$.

3 Global existence and uniqueness for the nonlinear system

This section is devoted to establishing the global existence and uniqueness of the solution for the Cauchy problem (1.5). More precisely, with the help of a priori estimates, we extend the local classical solution to the global one by the standard continuity method.

3.1 Global existence of solutions

First, we provide the relevant space for the system (1.5) by

$$\begin{aligned} \Omega (0, T)={}& \bigl\{ (\varrho , \boldsymbol{u})|\varrho \in C^{0} \bigl(0, T; H^{3}\bigl( \mathbb{R}^{3}\bigr)\bigr)\cap C^{1}\bigl(0, T; H^{2}\bigl(\mathbb{R}^{3} \bigr)\bigr), \\ &\boldsymbol{u}\in C^{0}\bigl(0, T; H^{3}\bigl( \mathbb{R}^{3}\bigr)\bigr)\cap C^{1}\bigl(0, T; H^{1}\bigl( \mathbb{R}^{3}\bigr)\bigr), \\ &\nabla \varrho \in L^{2}\bigl(0, T; H^{2}\bigl( \mathbb{R}^{3}\bigr)\bigr);\nabla \boldsymbol{u} \in L^{2} \bigl(0, T; H^{3}\bigl(\mathbb{R}^{3}\bigr)\bigr) \bigr\} \end{aligned}$$

for any $0 \leqslant T\leqslant \infty$.

By a method similar to the one in [13, 48], we can get the local existence of unique solutions to (1.5).

Proposition 3.1

(Local existence). Suppose $(\varrho ^{0}, \boldsymbol{u}^{0})\in H^{3}(\mathbb{R}^{3})$ and $\inf \{\varrho ^{0}+1\}>0$. Then there exists a constant $T_{0}>0$ depending on $\|\varrho ^{0}, \boldsymbol{u}^{0})\|_{H^{3}(\mathbb{R}^{3})}$ such that the system (1.5) has a unique solution $(\varrho , \boldsymbol{u})\in \Omega (0, T_{0})$, which satisfies

$$\begin{aligned} \inf_{x\in \mathbb{R}^{3}, 0\leqslant t\leqslant T_{0}}\{\varrho +1 \}>0 \end{aligned}$$

and

$$\begin{aligned} \bigl\Vert (\varrho , \boldsymbol{u}) (t) \bigr\Vert _{H^{3}}+ \biggl( \int ^{t}_{0} \bigl\Vert \nabla ( \varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{H^{3}}\,\mathrm{d} {\tau} \biggr)^{\frac{1}{2}} \leqslant \sqrt{C_{1}} \bigl\Vert \bigl( \varrho ^{0}, \boldsymbol{u}^{0}\bigr) \bigr\Vert _{H^{3}}, \end{aligned}$$

where $C_{1}>0$ is a constant.

Proof

With the iteration technique and the fixed point theorem in hand, the conclusion is obvious; please refer to [13, 48] for the details. □

Proposition 3.2

(A priori estimates). Assume that the Cauchy problem (1.5) has a solution $(\varrho , \boldsymbol{u})\in \Omega (0, T)$ with a constant $T >0$. Then there exists a sufficiently small constant $\epsilon _{0}>0$ such that if

$$\begin{aligned} \sup_{0\leqslant t\leqslant T} \bigl\Vert (\varrho , \boldsymbol{u}) (t) \bigr\Vert _{H^{3}} \leqslant \epsilon _{0}, \end{aligned}$$

(3.1)

then for any $t\in [0, T]$ we have

$$\begin{aligned} \bigl\Vert (\varrho , \boldsymbol{u}) (t) \bigr\Vert ^{2}_{H^{3}}+ \int ^{t}_{0} \bigl( \bigl\Vert \nabla \varrho (\tau ) \bigr\Vert ^{2}_{H^{2}} + \bigl\Vert \nabla \boldsymbol{u}(\tau ) \bigr\Vert ^{2}_{H^{3}} \bigr) \,\mathrm{d} {\tau}\leqslant C_{2} \bigl\Vert \bigl(\varrho ^{0}, \boldsymbol{u}^{0}\bigr) \bigr\Vert ^{2}_{H^{3}}, \end{aligned}$$

(3.2)

where the constant $C_{2}>0$ is independent of T.

The details of the proof of Proposition 3.2 will be given in Sect. 3.2.

Theorem 3.1

(Global existence). Suppose $(\varrho ^{0}, \boldsymbol{u}^{0}) \in H^{3}(\mathbb{R}^{3})$. Then there exists a constant $\epsilon >0$ such that when

$$\begin{aligned} \mathcal{C}_{0}< \min \biggl\{ \frac{\epsilon}{\sqrt{C_{1}}}, \frac{\epsilon}{\sqrt{C_{1} C_{2}}} \biggr\} < \infty , \end{aligned}$$

the Cauchy problem of (1.5) admits a unique solution $(\varrho , \boldsymbol{u})$, which satisfies for any $t>0$

$$\begin{aligned} \bigl\Vert (\varrho , \boldsymbol{u}) (t) \bigr\Vert ^{2}_{H^{3}}+ \int ^{t}_{0} \bigl( \bigl\Vert \nabla \varrho (\tau ) \bigr\Vert ^{2}_{H^{2}} + \bigl\Vert \nabla \boldsymbol{u}(\tau ) \bigr\Vert ^{2}_{H^{3}} \bigr) \,\mathrm{d} {\tau}\leqslant C_{2}\mathcal{C}^{2}_{0}, \end{aligned}$$

where $\mathcal{C}_{0}:=\|(\varrho ^{0}, \boldsymbol{u}^{0})\|_{H^{3}}$ and $C_{1}, C_{2}>0$ are constants.

Proof

With Propositions 3.1 and 3.2 in hand, we can easily derive Theorem 3.1 by a classical method. We omit it here due to space constraints; please refer to [13, 48] for the details. □

Remark 3.1

With the Sobolev imbedding inequality in hand, it is easy to get $\frac{1}{2}\leqslant \rho +1\leqslant \frac{3}{2}$. Then under the assumptions in Proposition 3.2, we can have

$$\begin{aligned} \bigl\vert h_{1}(\varrho ) \bigr\vert \leqslant C_{3}, \bigl\vert (h_{2}, h_{3}) (\varrho ) \bigr\vert \leqslant C_{3} \vert \varrho \vert ,\qquad \bigl\vert \bigl(h^{(l)}_{1}, h^{(l)}_{2}, h^{(l)}_{3}\bigr) ( \varrho ) \bigr\vert \leqslant C_{3}\quad \text{for } l \geqslant 1, \end{aligned}$$

where $C_{3}>0$ is a constant.

3.2 Proof of Proposition 3.2

In this subsection, we aim to complete the proof of Proposition 3.2. The key step in the proof is to derive the energy estimates of the solution $(\varrho , \boldsymbol{u})$ for the transformed Cauchy problem (1.5) by the energy method.

Lemma 3.1

We have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}{\mathcal{F}_{1}}+\frac{\gamma _{1}}{4} \Vert \nabla \varrho \Vert ^{2}_{L^{2}} + \frac{\mu _{0}}{2} \Vert \nabla \boldsymbol{u} \Vert ^{2}_{H^{1}}+ \frac{\mu _{0}}{2} \Vert \operatorname{div}\boldsymbol{u} \Vert ^{2}_{H^{1}}\leqslant 0, \end{aligned}$$

where

$$\begin{aligned} \mathcal{F}_{1}=\frac{1}{2} \bigl( \Vert \varrho \Vert ^{2}_{H^{1}}+ \Vert \boldsymbol{u} \Vert ^{2}_{H^{1}} \bigr) +\gamma _{1} \int{\nabla \varrho \cdot \boldsymbol{u}\,\mathrm{d}x}, \end{aligned}$$

where $0<\gamma _{1}<\{\frac{1}{4}, \frac{1}{8\mu _{0}}\}$ is a given constant.

Proof

Multiplying $\nabla ^{k}\text{(1.5)}_{1}$, $\nabla ^{k}\text{(1.5)}_{2}$ by $\nabla ^{k}\varrho$, $\nabla ^{k} \boldsymbol{u}$, respectively, and integrating over $\mathbb{R}^{3}$ by parts, we have

$$\begin{aligned} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \bigl( \bigl\Vert \nabla ^{k}\varrho \bigr\Vert ^{2}_{L^{2}} + \bigl\Vert \nabla ^{k}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \bigr) +\mu _{0} \bigl\Vert \nabla ^{k}\nabla \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+ \mu _{0} \bigl\Vert \nabla ^{k} \operatorname{div} \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \\ & \quad = \bigl\langle \nabla ^{k}\varrho , -\nabla ^{k} \operatorname{div}(\varrho \boldsymbol{u})\bigr\rangle +\bigl\langle \nabla ^{k} \boldsymbol{u}, \nabla ^{k}\mathcal{N} \bigr\rangle . \end{aligned}$$

(3.3)

By $\langle \nabla \text{(1.5)}_{1}, \boldsymbol{u}\rangle + \langle \text{(1.5)}_{2}, \nabla \varrho \rangle$, we find that

$$\begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}t} \int \nabla \varrho \cdot \boldsymbol{u} \,\mathrm{d}x + \Vert \nabla \varrho \Vert ^{2}_{L^{2}} \\ & \quad = \Vert \operatorname{div}\boldsymbol{u} \Vert ^{2}_{L^{2}} +\mu _{0} \int \nabla \varrho \cdot \Delta \boldsymbol{u} \,\mathrm{d}x+\mu _{0} \int \nabla \varrho \cdot \nabla \operatorname{div}\boldsymbol{u}\, \mathrm{d}x \\ &\quad \quad{} - \int \nabla \operatorname{div}(\varrho \boldsymbol{u})\cdot \boldsymbol{u}\,\mathrm{d}x + \int \mathcal{N}\cdot \nabla \varrho \,\mathrm{d}x. \end{aligned}$$

(3.4)

Then using Young’s inequality, we can estimate that for some fixed constant $\gamma _{1}$,

$$\begin{aligned} &\gamma _{1}\mu _{0} \int \nabla \varrho \cdot \Delta \boldsymbol{u}\,\mathrm{d}x \leqslant \frac{\gamma _{1}}{4} \Vert \nabla \varrho \Vert ^{2}_{L^{2}} + \gamma _{1}\mu ^{2}_{0} \Vert \Delta \boldsymbol{u} \Vert ^{2}_{L^{2}}, \\ &\gamma _{1}\mu _{0} \int \nabla \varrho \cdot \nabla \operatorname{div}\boldsymbol{u}\, \mathrm{d}x \leqslant \frac{\gamma _{1}}{4} \Vert \nabla \varrho \Vert ^{2}_{L^{2}}+\gamma _{1}\mu ^{2}_{0} \Vert \nabla \operatorname{div}\boldsymbol{u} \Vert ^{2}_{L^{2}}. \end{aligned}$$

(3.5)

Adding up the two identities $\gamma _{1}\cdot \text{(3.4)}$ and $\sum_{0\leqslant k\leqslant 1}\text{(3.3)}$, then using (3.5), we have

$$\begin{aligned} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \biggl( \Vert \varrho \Vert ^{2}_{H^{1}}+ \Vert \boldsymbol{u} \Vert ^{2}_{H^{1}} +2\gamma _{1} \int \nabla \varrho \cdot \boldsymbol{u} \,\mathrm{d}x \biggr) \\ &\quad \quad{} + \frac{\gamma _{1}}{2} \Vert \nabla \varrho \Vert ^{2}_{L^{2}}+\mu _{0} \Vert \nabla \boldsymbol{u} \Vert ^{2}_{H^{1}}+\mu _{0} \Vert \operatorname{div}\boldsymbol{u} \Vert ^{2}_{H^{1}} \\ &\quad \leqslant \Vert \operatorname{div}\boldsymbol{u} \Vert ^{2}_{L^{2}}+\gamma _{1}\mu ^{2}_{0} \Vert \Delta \boldsymbol{u} \Vert ^{2}_{L^{2}}+\gamma _{1}\mu ^{2}_{0} \Vert \nabla \operatorname{div}\boldsymbol{u} \Vert ^{2}_{L^{2}} - \int \varrho \operatorname{div}( \varrho \boldsymbol{u})\,\mathrm{d}x \\ &\quad \quad{} - \int \nabla \varrho \cdot \nabla \operatorname{div}(\varrho \boldsymbol{u}) \,\mathrm{d}x+ \int \boldsymbol{u}\cdot \mathcal{N}\,\mathrm{d}x+ \int \nabla \boldsymbol{u}\cdot \nabla \mathcal{N}\,\mathrm{d}x \\ &\quad \quad{} - \gamma _{1} \int \boldsymbol{u}\cdot \nabla \operatorname{div}(\varrho \boldsymbol{u}) \,\mathrm{d}x +\gamma _{1} \int \nabla \varrho \cdot \mathcal{N} \,\mathrm{d}x. \end{aligned}$$

(3.6)

The nonlinear terms on the right-hand side of (3.6) can be bounded as follows. With Young’s inequality and Hölder’s inequality in hand, integrating by parts and using Lemmas 2.2 and 2.3 and (3.1), we obtain

$$\begin{aligned}& - \int \varrho \operatorname{div}(\varrho \boldsymbol{u})\,\mathrm{d}x \leqslant C \epsilon _{0} \bigl\Vert \nabla (\varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}}, \end{aligned}$$

(3.7)

$$\begin{aligned}& - \int \nabla \varrho \cdot \nabla \operatorname{div}(\varrho \boldsymbol{u}) \,\mathrm{d}x \leqslant C\epsilon _{0}\bigl( \bigl\Vert \nabla ^{2}\varrho \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \nabla (\varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}} \bigr), \end{aligned}$$

(3.8)

$$\begin{aligned}& - \int \boldsymbol{u}\cdot \nabla \operatorname{div}(\varrho \boldsymbol{u})\,\mathrm{d}x = \int \operatorname{div}\boldsymbol{u}\operatorname{div}(\varrho \boldsymbol{u})\,\mathrm{d}x \leqslant C\epsilon _{0} \bigl\Vert \nabla (\varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}} \end{aligned}$$

(3.9)

and

$$\begin{aligned}& \int \boldsymbol{u}\cdot \mathcal{N}\,\mathrm{d}x \\& \quad = \int \boldsymbol{u}\cdot \bigl(-\boldsymbol{u}\cdot \nabla \boldsymbol{u}-h_{3}(\varrho ) \nabla \varrho -h_{2}( \varrho ) (\Delta \boldsymbol{u}+\nabla \operatorname{div}\boldsymbol{u}) \bigr)\, \mathrm{d}x \\& \quad \quad{} + \int \boldsymbol{u}\cdot h_{1}(\varrho )\operatorname{div} \bigl( \bigl\vert D(\nabla \boldsymbol{u}) \bigr\vert ^{p-2}D(\nabla \boldsymbol{u}) \bigr)\,\mathrm{d}x \\& \quad \leqslant C \Vert \boldsymbol{u} \Vert _{L^{6}}\bigl( \Vert \boldsymbol{u} \Vert _{L^{3}} \Vert \nabla \boldsymbol{u} \Vert _{L^{2}} + \bigl\Vert h_{3}(\varrho ) \bigr\Vert _{L^{3}} \Vert \nabla \varrho \Vert _{L^{2}}\bigr) \\& \quad \quad{} + C \Vert \boldsymbol{u} \Vert _{L^{6}} \bigl\Vert h_{2}(\varrho ) \bigr\Vert _{L^{3}} \bigl\Vert \nabla ^{2}\boldsymbol{u} \bigr\Vert _{L^{2}} \\& \quad \quad{} + C \Vert \boldsymbol{u} \Vert _{L^{6}} \bigl\Vert h_{1}(\varrho ) \bigr\Vert _{L^{\infty}} \bigl\Vert \operatorname{div} \bigl\vert D(\nabla \boldsymbol{u}) \bigr\vert ^{p-2} \bigr\Vert _{L^{2}} \bigl\Vert D(\nabla \boldsymbol{u}) \bigr\Vert _{L^{3}} \\& \quad \quad{} + C \Vert \boldsymbol{u} \Vert _{L^{6}} \bigl\Vert h_{1}(\varrho ) \bigr\Vert _{L^{\infty}}\||D(\nabla \boldsymbol{u})|^{p-2}\|_{L^{3}} \bigl\Vert \operatorname{div}D(\nabla \boldsymbol{u}) \bigr\Vert _{L^{2}} \\& \quad \leqslant C\epsilon _{0} \bigl( \bigl\Vert \nabla ^{2}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} + \bigl\Vert \nabla (\varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}}+ \Vert \nabla \boldsymbol{u} \Vert ^{2}_{H^{2}} \bigr), \end{aligned}$$

(3.10)

$$\begin{aligned}& \begin{aligned}[b] \int \nabla \boldsymbol{u}\cdot \nabla \mathcal{N}\,\mathrm{d}x & = \int \nabla \boldsymbol{u}\cdot \nabla \bigl(-\boldsymbol{u}\cdot \nabla \boldsymbol{u}-h_{3}(\varrho )\nabla \varrho \bigr)\,\mathrm{d}x \\ &\quad{} - \int \nabla \boldsymbol{u}\cdot \nabla \bigl(h_{2}(\varrho ) (\Delta \boldsymbol{u}+ \nabla \operatorname{div}\boldsymbol{u})\bigr)\, \mathrm{d}x \\ &\quad{} + \int \nabla \boldsymbol{u}\cdot \nabla \bigl(h_{1}(\varrho ) \operatorname{div} \bigl( \bigl\vert D(\nabla \boldsymbol{u}) \bigr\vert ^{p-2}D(\nabla \boldsymbol{u}) \bigr) \bigr)\,\mathrm{d}x \\ & \leqslant C \bigl\Vert \nabla ^{2}\boldsymbol{u} \bigr\Vert _{L^{2}}\bigl( \Vert \nabla \boldsymbol{u} \Vert _{L^{2}} \Vert \boldsymbol{u} \Vert _{L^{\infty}} + \bigl\Vert h_{3}(\varrho ) \bigr\Vert _{L^{\infty}} \Vert \nabla \varrho \Vert _{L^{2}}\bigr) \\ &\quad{} + C \bigl\Vert \nabla ^{2}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \bigl\Vert h_{2}(\varrho ) \bigr\Vert _{L^{\infty}} \\ &\quad{} + C \bigl\Vert \nabla ^{2}\boldsymbol{u} \bigr\Vert _{L^{2}} \bigl\Vert h_{1}(\varrho ) \bigr\Vert _{L^{\infty}} \bigl\Vert \operatorname{div} \bigl\vert D(\nabla \boldsymbol{u}) \bigr\vert ^{p-2} \bigr\Vert _{L^{2}} \bigl\Vert D(\nabla \boldsymbol{u}) \bigr\Vert _{L^{\infty}} \\ &\quad{} + C \bigl\Vert \nabla ^{2}\boldsymbol{u} \bigr\Vert _{L^{2}} \bigl\Vert h_{1}(\varrho ) \bigr\Vert _{L^{\infty}} \||D( \nabla \boldsymbol{u})|^{p-2} \|_{L^{\infty}} \bigl\Vert \operatorname{div}D(\nabla \boldsymbol{u}) \bigr\Vert _{L^{2}} \\ & \leqslant C\epsilon _{0} \bigl( \bigl\Vert \nabla ^{2}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} + \bigl\Vert \nabla (\varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}}+ \Vert \nabla \boldsymbol{u} \Vert ^{2}_{H^{2}} \bigr), \end{aligned} \end{aligned}$$

(3.11)

$$\begin{aligned}& \begin{aligned}[b] \gamma _{1} \int \nabla \varrho \cdot \mathcal{N}\,\mathrm{d}x & = \gamma _{1} \Vert \nabla \varrho \Vert _{L^{2}} \Vert \mathcal{N} \Vert _{L^{2}} \\ & \leqslant C\gamma _{1} \Vert \nabla \varrho \Vert _{L^{2}}\bigl( \Vert \nabla \boldsymbol{u} \Vert _{L^{2}} \Vert \boldsymbol{u} \Vert _{L^{\infty}} + \bigl\Vert h_{3}(\varrho ) \bigr\Vert _{L^{\infty}} \Vert \nabla \varrho \Vert _{L^{2}}\bigr) \\ &\quad{} + C \Vert \nabla \varrho \Vert _{L^{2}} \bigl\Vert h_{2}(\varrho ) \bigr\Vert _{L^{\infty}} \bigl\Vert \nabla ^{2}\boldsymbol{u} \bigr\Vert _{L^{2}} \\ &\quad{} + C \Vert \nabla \varrho \Vert _{L^{2}} \bigl\Vert h_{1}(\varrho ) \bigr\Vert _{L^{\infty}} \bigl\Vert \operatorname{div} \bigl\vert D(\nabla \boldsymbol{u}) \bigr\vert ^{p-2} \bigr\Vert _{L^{2}} \bigl\Vert D(\nabla \boldsymbol{u}) \bigr\Vert _{L^{\infty}} \\ &\quad{} + C \Vert \nabla \varrho \Vert _{L^{2}} \bigl\Vert h_{1}(\varrho ) \bigr\Vert _{L^{\infty}} \||D( \nabla \boldsymbol{u})|^{p-2}\|_{L^{\infty}} \bigl\Vert \operatorname{div}D(\nabla \boldsymbol{u}) \bigr\Vert _{L^{2}} \\ & \leqslant C\epsilon _{0}\bigl( \bigl\Vert \nabla ^{2}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} + \bigl\Vert \nabla (\varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}}+ \Vert \nabla \boldsymbol{u} \Vert ^{2}_{H^{2}}\bigr). \end{aligned} \end{aligned}$$

(3.12)

Putting the estimates (3.7)–(3.12) into (3.6), we have

$$\begin{aligned} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \biggl( \Vert \varrho \Vert ^{2}_{H^{1}}+ \Vert \boldsymbol{u} \Vert ^{2}_{H^{1}} +2\gamma _{1} \int \nabla \varrho \cdot \boldsymbol{u} \,\mathrm{d}x \biggr) \\ &\quad \quad{} + \frac{\gamma _{1}}{4} \Vert \nabla \varrho \Vert ^{2}_{L^{2}}+ \frac{\mu _{0}}{2} \Vert \nabla \boldsymbol{u} \Vert ^{2}_{H^{1}}+\frac{\mu _{0}}{2} \Vert \operatorname{div}\boldsymbol{u} \Vert ^{2}_{H^{1}} \\ &\quad \leqslant C(1+\gamma _{1})\epsilon _{0} \bigl( \bigl\Vert \nabla ^{2}(\varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}} + \bigl\Vert \nabla (\varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}}+ \Vert \nabla \boldsymbol{u} \Vert ^{2}_{H^{2}} \bigr), \end{aligned}$$

(3.13)

where $0<\gamma _{1}<\{\frac{1}{4}, \frac{1}{8\mu _{0}}\}$ is a fixed constant. Then the desired estimate follows from (3.13). Thus, the proof of Lemma 3.1 is complete. □

Now we establish the energy estimate on the highest-order derivatives of the solution $(\varrho , \boldsymbol{u})$ for the Cauchy problem (1.5).

Lemma 3.2

We have

$$\begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}t}{\mathcal{F}_{2}} + \frac{\gamma _{2}}{4} \bigl\Vert \nabla ^{2}\varrho \bigr\Vert ^{2}_{L^{2}} + \frac{\mu _{0}}{2} \bigl\Vert \nabla ^{2}\operatorname{div}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+ \frac{\mu _{0}}{2} \bigl\Vert \nabla ^{3} \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \\ & \quad \leqslant \frac{1}{4} \Vert \nabla \operatorname{div} \boldsymbol{u} \Vert ^{2}_{L^{2}}+C \epsilon _{0} \bigl( \bigl\Vert \nabla ^{2}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+ \Vert \nabla \boldsymbol{u} \Vert ^{2}_{H^{2}} \bigr), \end{aligned}$$

(3.14)

where

$$\begin{aligned} \mathcal{F}_{2} =\frac{1}{2} \bigl( \bigl\Vert \nabla ^{2}\varrho \bigr\Vert ^{2}_{L^{2}} + \bigl\Vert \nabla ^{2}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \bigr) +\gamma _{2} \int{\nabla ^{2} \varrho \cdot \nabla \boldsymbol{u}\, \mathrm{d}x}, \end{aligned}$$

where $0<\gamma _{2}<\{\frac{1}{4}, \frac{1}{8\mu _{0}}\}$ is a given constant.

Proof

Multiplying $\nabla ^{2}\text{(1.5)}_{1}$, $\nabla ^{2}\text{(1.5)}_{2}$ by $\nabla ^{2}\varrho$, $\nabla ^{2} \boldsymbol{u}$, respectively, and integrating over $\mathbb{R}^{3}$ by parts, we have

$$\begin{aligned} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \bigl( \bigl\Vert \nabla ^{2}\varrho \bigr\Vert ^{2}_{L^{2}} + \bigl\Vert \nabla ^{2}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \bigr) +\mu _{0} \bigl\Vert \nabla ^{2}\nabla \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+ \mu _{0} \bigl\Vert \nabla ^{2} \operatorname{div} \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \\ &\quad = \bigl\langle \nabla ^{2}\varrho , -\nabla ^{2} \operatorname{div}(\varrho \boldsymbol{u})\bigr\rangle +\bigl\langle \nabla ^{2} \boldsymbol{u}, \nabla ^{2}\mathcal{N} \bigr\rangle . \end{aligned}$$

(3.15)

Multiplying $\nabla ^{2}\text{(1.5)}_{1}$ by ∇u and then exploiting $\nabla \text{(1.5)}_{2}\cdot \nabla ^{2}\varrho$ and Young’s inequality, we can estimate that

$$\begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}t} \int \nabla ^{2}\varrho \cdot \nabla \boldsymbol{u}\, \mathrm{d}x + \int \bigl\vert \nabla ^{2}\varrho \bigr\vert ^{2} \,\mathrm{d}x \\ &\quad = \Vert \nabla \operatorname{div}\boldsymbol{u} \Vert ^{2}_{L^{2}}+ \mu _{0} \int \nabla ^{2}\varrho \cdot \nabla \Delta \boldsymbol{u}\, \mathrm{d}x+\mu _{0} \int \nabla ^{2}\varrho \cdot \nabla ^{2} \operatorname{div}\boldsymbol{u} \,\mathrm{d}x \\ &\quad \quad{} - \int \nabla ^{2}\operatorname{div}(\varrho \boldsymbol{u}) \cdot \nabla \boldsymbol{u} \,\mathrm{d}x + \int \nabla \mathcal{N}\cdot \nabla ^{2}\varrho \,\mathrm{d}x \\ &\quad \leqslant \frac{1}{2} \bigl\Vert \nabla ^{2}\varrho \bigr\Vert ^{2}_{L^{2}}+\mu _{0}^{2} \Vert \nabla \Delta \boldsymbol{u} \Vert ^{2}_{L^{2}} + \mu _{0}^{2} \bigl\Vert \nabla ^{2} \operatorname{div}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+ \Vert \nabla \operatorname{div} \boldsymbol{u} \Vert ^{2}_{L^{2}} \\ &\quad \quad{} - \int \nabla ^{2}\operatorname{div}(\varrho \boldsymbol{u}) \cdot \nabla \boldsymbol{u} \,\mathrm{d}x + \int \nabla \mathcal{N}\cdot \nabla ^{2}\varrho \, \mathrm{d}x. \end{aligned}$$

(3.16)

Thus, summing up (3.15) and $\gamma _{2}\cdot \text{(3.16)}$, we have

$$\begin{aligned} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \biggl( \bigl\Vert \nabla ^{2} \varrho \bigr\Vert ^{2}_{L^{2}} + \bigl\Vert \nabla ^{2}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} +2\gamma _{2} \int \nabla ^{2}\varrho \cdot \nabla \boldsymbol{u}\, \mathrm{d}x \biggr) \\ &\quad \quad{} + \frac{\gamma _{2}}{2} \bigl\Vert \nabla ^{2}\varrho \bigr\Vert ^{2}_{L^{2}}+\mu _{0} \bigl\Vert \nabla ^{2}\nabla \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+ \mu _{0} \bigl\Vert \nabla ^{2} \operatorname{div} \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \\ &\quad \leqslant \gamma _{2}\mu _{0}^{2} \Vert \nabla \Delta \boldsymbol{u} \Vert ^{2}_{L^{2}} +\gamma _{2}\mu _{0}^{2} \bigl\Vert \nabla ^{2}\operatorname{div}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} +\gamma _{2} \Vert \nabla \operatorname{div}\boldsymbol{u} \Vert ^{2}_{L^{2}} \\ &\quad \quad{} - \int _{R^{3}}\nabla ^{2}\varrho \cdot \nabla ^{2}\operatorname{div}( \varrho \boldsymbol{u})\,\mathrm{d}x+ \int \nabla ^{2}\boldsymbol{u}\cdot \nabla ^{2} \mathcal{N}\,\mathrm{d}x \\ &\quad \quad{} - \gamma _{2} \int \nabla \boldsymbol{u}\cdot \nabla ^{2} \operatorname{div}(\varrho \boldsymbol{u})\,\mathrm{d}x+ \gamma _{2} \int \nabla ^{2}\varrho \cdot \nabla \mathcal{N}\, \mathrm{d}x. \end{aligned}$$

(3.17)

Next we estimate for the nonlinear terms on the right-hand side of (3.17). Thanks to Lemmas 2.2–2.4, Hölder’s inequality, and Young’s inequality, after integrating by parts, we can have

$$\begin{aligned}& - \int \nabla ^{2}\varrho \cdot \nabla ^{2} \operatorname{div}(\varrho \boldsymbol{u})\,\mathrm{d}x\leqslant C\epsilon _{0} \bigl( \bigl\Vert \nabla ^{3}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \nabla ^{2}( \varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}} \bigr), \end{aligned}$$

(3.18)

$$\begin{aligned}& - \int \nabla \boldsymbol{u}\cdot \nabla ^{2} \operatorname{div}(\varrho \boldsymbol{u}) \,\mathrm{d}x \leqslant C\epsilon _{0} \bigl\Vert \nabla ^{2}(\varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}} \end{aligned}$$

(3.19)

and

$$\begin{aligned} & \int \nabla ^{2}\boldsymbol{u}\cdot \nabla ^{2} \mathcal{N}\,\mathrm{d}x \\ &\quad = \int \nabla ^{2}\boldsymbol{u}\cdot \nabla ^{2} \bigl(-\boldsymbol{u}\cdot \nabla \boldsymbol{u} -h_{3}(\varrho ) \nabla \varrho -h_{2}(\varrho ) (\Delta \boldsymbol{u}+ \nabla \operatorname{div}\boldsymbol{u}) \bigr)\,\mathrm{d}x \\ &\quad \quad{} + \int \nabla ^{2}\boldsymbol{u}\cdot \nabla ^{2} \bigl(h_{1}(\varrho ) \operatorname{div} \bigl( \bigl\vert D(\nabla \boldsymbol{u}) \bigr\vert ^{p-2}D(\nabla \boldsymbol{u}) \bigr) \bigr)\,\mathrm{d}x \\ &\quad \leqslant C \bigl( \bigl\vert \bigl\langle \nabla ^{3} \boldsymbol{u}, \nabla (\boldsymbol{u}\cdot \nabla \boldsymbol{u})\bigr\rangle \bigr\vert + \bigl\vert \bigl\langle \nabla ^{3}\boldsymbol{u}, \nabla \bigl(h_{3}( \varrho )\nabla \varrho \bigr)\bigr\rangle \bigr\vert \bigr) \\ &\quad \quad{} + C \bigl( \bigl\vert \bigl\langle \nabla ^{3} \boldsymbol{u}, \nabla \bigl(h_{2}( \varrho )\Delta \boldsymbol{u} \bigr)\bigr\rangle \bigr\vert + \bigl\vert \bigl\langle \nabla ^{3}\boldsymbol{u}, \nabla \bigl(h_{2}( \varrho )\nabla \operatorname{div}\boldsymbol{u}\bigr) \bigr\rangle \bigr\vert \bigr) \\ &\quad \quad{} + C \bigl( \bigl\vert \bigl\langle \nabla ^{3} \boldsymbol{u}, \nabla \bigl(h_{1}(\varrho ) \operatorname{div} \bigl( \bigl\vert D(\nabla \boldsymbol{u}) \bigr\vert ^{p-2}D(\nabla \boldsymbol{u}) \bigr) \bigr)\bigr\rangle \bigr\vert \bigr) \\ &\quad \leqslant C \bigl\Vert \nabla ^{3}\boldsymbol{u} \bigr\Vert _{L^{2}}\bigl( \Vert \nabla \boldsymbol{u} \Vert _{L^{6}} \Vert \nabla \boldsymbol{u} \Vert _{L^{3}}+ \Vert \boldsymbol{u} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{2} \boldsymbol{u} \bigr\Vert _{L^{2}}\bigr) \\ &\quad \quad{} + C \bigl\Vert \nabla ^{3}\boldsymbol{u} \bigr\Vert _{L^{2}}\bigl( \bigl\Vert \nabla h_{3}(\varrho ) \bigr\Vert _{L^{ \infty}} \Vert \nabla \varrho \Vert _{L^{2}}+ \bigl\Vert h_{3}(\varrho ) \bigr\Vert _{L^{\infty}} \bigl\Vert \nabla ^{2} \varrho \bigr\Vert _{L^{2}}\bigr) \\ &\quad \quad{} + C \bigl\Vert \nabla ^{3}\boldsymbol{u} \bigr\Vert _{L^{2}}\bigl( \bigl\Vert \nabla h_{2}(\varrho ) \bigr\Vert _{L^{ \infty}} \bigl\Vert \nabla ^{2} \boldsymbol{u} \bigr\Vert _{L^{2}}+ \bigl\Vert h_{2}(\varrho ) \bigr\Vert _{L^{\infty}} \bigl\Vert \nabla ^{3} \boldsymbol{u} \bigr\Vert _{L^{2}}\bigr) \\ &\quad \quad{} + C \bigl\Vert \nabla ^{3}\boldsymbol{u} \bigr\Vert _{L^{2}}\bigl( \bigl\Vert h_{1}(\varrho ) \bigr\Vert _{L^{\infty}}+ \bigl\Vert \nabla h_{1}(\varrho ) \bigr\Vert _{L^{\infty}}\bigr) \Vert \boldsymbol{u} \Vert ^{p-1}_{H^{2}} \\ &\quad \leqslant C\epsilon _{0} \bigl( \Vert \nabla \boldsymbol{u} \Vert ^{2}_{H^{2}}+ \bigl\Vert \nabla ^{2}( \varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}} \bigr), \end{aligned}$$

(3.20)

$$\begin{aligned} & \int \nabla ^{2}\varrho \cdot \nabla \mathcal{N}\,\mathrm{d}x \\ &\quad = \int \nabla ^{2}\varrho \cdot \nabla \bigl(-\boldsymbol{u} \cdot \nabla \boldsymbol{u}-h_{3}(\varrho )\nabla \varrho -h_{2}(\varrho ) (\Delta \boldsymbol{u}+ \nabla \operatorname{div} \boldsymbol{u}) \bigr)\,\mathrm{d}x \\ &\quad \quad{} + \int \nabla ^{2}\varrho \cdot \nabla \bigl(h_{1}( \varrho ) \operatorname{div} \bigl( \bigl\vert D(\nabla \boldsymbol{u}) \bigr\vert ^{p-2}D(\nabla \boldsymbol{u}) \bigr) \bigr)\,\mathrm{d}x \\ &\quad \leqslant C\gamma _{2} \bigl\Vert \nabla ^{2}\varrho \bigr\Vert _{L^{2}} \bigl( \Vert \nabla \boldsymbol{u} \Vert _{L^{6}} \Vert \nabla \boldsymbol{u} \Vert _{L^{3}}+ \Vert \boldsymbol{u} \Vert _{L^{ \infty}} \bigl\Vert \nabla ^{2} \boldsymbol{u} \bigr\Vert _{L^{2}} \bigr) \\ &\quad \quad{} + C \bigl\Vert \nabla ^{2}\varrho \bigr\Vert _{L^{2}} \bigl( \bigl\Vert \nabla h_{3}(\varrho ) \bigr\Vert _{L^{ \infty}} \Vert \nabla \varrho \Vert _{L^{2}}+ \bigl\Vert h_{3}(\varrho ) \bigr\Vert _{L^{\infty}} \bigl\Vert \nabla ^{2} \varrho \bigr\Vert _{L^{3}} \bigr) \\ &\quad \quad{} + C \bigl\Vert \nabla ^{2}\varrho \bigr\Vert _{L^{2}} \bigl( \bigl\Vert \nabla h_{2}(\varrho ) \bigr\Vert _{L^{ \infty}} \bigl\Vert \nabla ^{2} \boldsymbol{u} \bigr\Vert _{L^{2}}+ \bigl\Vert h_{2}(\varrho ) \bigr\Vert _{L^{\infty}} \bigl\Vert \nabla ^{3} \boldsymbol{u} \bigr\Vert _{L^{2}} \bigr) \\ &\quad \quad{} + C \bigl\Vert \nabla ^{2}\varrho \bigr\Vert _{L^{2}}\bigl( \bigl\Vert h_{1}(\varrho ) \bigr\Vert _{L^{\infty}}+ \bigl\Vert \nabla h_{1}(\varrho ) \bigr\Vert _{L^{\infty}}\bigr) \Vert \boldsymbol{u} \Vert ^{p-1}_{H^{3}} \\ &\quad \leqslant C\epsilon _{0} \bigl( \Vert \nabla \boldsymbol{u} \Vert ^{2}_{H^{2}}+ \bigl\Vert \nabla ^{2}( \varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}} \bigr). \end{aligned}$$

(3.21)

Plugging (3.18)–(3.21) into (3.17), we can deduce that

$$\begin{aligned} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \biggl( \bigl\Vert \nabla ^{2} \varrho \bigr\Vert ^{2}_{L^{2}} + \bigl\Vert \nabla ^{2}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} +2\gamma _{2} \int \nabla ^{2}\varrho \cdot \nabla \boldsymbol{u} \, \mathrm{d}x \biggr) \\ &\quad \quad{} + \frac{\gamma _{2}}{4} \bigl\Vert \nabla ^{2}\varrho \bigr\Vert ^{2}_{L^{2}} + \frac{\mu _{0}}{2} \bigl\Vert \nabla ^{2}\nabla \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+ \frac{\mu _{0}}{2} \bigl\Vert \nabla ^{2}\operatorname{div}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \\ &\quad \leqslant \frac{1}{4} \Vert \nabla \operatorname{div} \boldsymbol{u} \Vert ^{2}_{L^{2}}+C \epsilon _{0}\bigl( \bigl\Vert \nabla ^{2}(\varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}}+ \Vert \nabla \boldsymbol{u} \Vert ^{2}_{H^{2}}\bigr), \end{aligned}$$

where $0<\gamma _{2}<\{\frac{1}{4}, \frac{1}{8\mu _{0}}\}$ is a fixed constant. Consequently, we complete the proof of (3.14). □

Lemma 3.3

We have

$$\begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}t}{\mathcal{F}_{3}} + \frac{\gamma _{3}}{4} \bigl\Vert \nabla ^{3}\varrho \bigr\Vert ^{2}_{L^{2}} + \frac{\mu _{0}}{2} \bigl\Vert \nabla ^{3}\operatorname{div}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+ \frac{\mu _{0}}{2} \bigl\Vert \nabla ^{4} \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \\ &\quad \leqslant \frac{1}{4} \bigl\Vert \nabla ^{2} \operatorname{div}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+C \epsilon _{0} \bigl( \Vert \nabla \boldsymbol{u} \Vert ^{2}_{H^{3}} + \bigl\Vert \nabla ^{3} \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \bigr), \end{aligned}$$

(3.22)

where

$$\begin{aligned} \mathcal{F}_{3}=\frac{1}{2} \bigl( \bigl\Vert \nabla ^{3}\varrho \bigr\Vert ^{2}_{L^{2}} + \bigl\Vert \nabla ^{3}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \bigr) +\gamma _{3} \int{\nabla ^{3} \varrho \cdot \nabla ^{2} \boldsymbol{u}\,\mathrm{d}x}, \end{aligned}$$

where $0<\gamma _{3}<\{\frac{1}{4}, \frac{1}{8\mu _{0}}\}$ is a given constant.

Proof

Multiplying $\nabla ^{3}\text{(1.5)}_{1}$, $\nabla ^{3}\text{(1.5)}_{2}$ by $\nabla ^{3}\varrho$, $\nabla ^{3} \boldsymbol{u}$, respectively, and integrating over $\mathbb{R}^{3}$ by parts, we have

$$\begin{aligned} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \bigl( \bigl\Vert \nabla ^{3}\varrho \bigr\Vert ^{2}_{L^{2}} + \bigl\Vert \nabla ^{3}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \bigr) +\mu _{0} \bigl\Vert \nabla ^{3}\nabla \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+ \mu _{0} \bigl\Vert \nabla ^{3} \operatorname{div} \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \\ & \quad = \bigl\langle \nabla ^{3}\varrho , -\nabla ^{3} \operatorname{div}(\varrho \boldsymbol{u})\bigr\rangle +\bigl\langle \nabla ^{3} \boldsymbol{u}, \nabla ^{3}\mathcal{N} \bigr\rangle . \end{aligned}$$

(3.23)

Multiplying $\nabla ^{3}\text{(1.5)}_{1}$ by $\nabla ^{2}\boldsymbol{u}$ and then exploiting $\nabla ^{2}\text{(1.5)}_{2}\cdot \nabla ^{3}\varrho$ and Young’s inequality, we can estimate that

$$\begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}t} \int \nabla ^{3}\varrho \cdot \nabla ^{2} \boldsymbol{u}\,\mathrm{d}x + \int \bigl\vert \nabla ^{3}\varrho \bigr\vert ^{2} \,\mathrm{d}x \\ &\quad = \bigl\Vert \nabla ^{2}\operatorname{div}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+\mu _{0} \int \nabla ^{3}\varrho \cdot \nabla ^{2}\Delta \boldsymbol{u}\,\mathrm{d}x +\mu _{0} \int \nabla ^{3}\varrho \cdot \nabla ^{3} \operatorname{div}\boldsymbol{u} \,\mathrm{d}x \\ &\quad \quad{} - \int \nabla ^{3}\operatorname{div}(\varrho \boldsymbol{u}) \cdot \nabla ^{2} \boldsymbol{u}\,\mathrm{d}x + \int \nabla ^{2}\mathcal{N}\cdot \nabla ^{3} \varrho \,\mathrm{d}x \\ &\quad \leqslant \frac{1}{2} \bigl\Vert \nabla ^{3}\varrho \bigr\Vert ^{2}_{L^{2}}+\mu _{0}^{2} \bigl\Vert \nabla ^{2}\Delta \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} +\mu _{0}^{2} \bigl\Vert \nabla ^{3} \operatorname{div}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \nabla ^{2} \operatorname{div}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \\ &\quad \quad{} - \int \nabla ^{3}\operatorname{div}(\varrho \boldsymbol{u}) \cdot \nabla ^{2}\boldsymbol{u} \,\mathrm{d}x + \int \nabla ^{2}\mathcal{N}\cdot \nabla ^{3} \varrho \,\mathrm{d}x. \end{aligned}$$

(3.24)

Summing up (3.23) and $\gamma _{3}\cdot \text{(3.24)}$, we have

$$\begin{aligned} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \biggl( \bigl\Vert \nabla ^{3} \varrho \bigr\Vert ^{2}_{L^{2}} + \bigl\Vert \nabla ^{3}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} +2\gamma _{3} \int \nabla ^{3}\varrho \cdot \nabla ^{2} \boldsymbol{u} \,\mathrm{d}x \biggr) \\ &\quad \quad{} + \frac{\gamma _{3}}{2} \bigl\Vert \nabla ^{3}\varrho \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \nabla ^{3} \nabla \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \nabla ^{3}\operatorname{div}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \\ &\quad \leqslant \gamma _{3}\mu _{0}^{2} \bigl\Vert \nabla ^{2}\Delta \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} +\gamma _{3}\mu _{0}^{2} \bigl\Vert \nabla ^{3} \operatorname{div}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} + \gamma _{3} \bigl\Vert \nabla ^{2}\operatorname{div} \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \\ &\quad \quad{} - \int \nabla ^{3}\varrho \cdot \nabla ^{3} \operatorname{div}(\varrho \boldsymbol{u})\,\mathrm{d}x+ \int \nabla ^{3}\boldsymbol{u}\cdot \nabla ^{3} \mathcal{N} \,\mathrm{d}x \\ &\quad \quad{} - \gamma _{3} \int \nabla ^{2} \boldsymbol{u}\cdot \nabla ^{3} \operatorname{div}( \varrho \boldsymbol{u})\,\mathrm{d}x+ \gamma _{3} \int \nabla ^{3}\varrho \cdot \nabla ^{2} \mathcal{N}\,\mathrm{d}x. \end{aligned}$$

(3.25)

Next we estimate for the nonlinear terms on the right-hand side of (3.25). With Lemmas 2.2–2.4, Hölder’s inequality, and Young’s inequality in hand, after integrating by parts, we get

$$\begin{aligned}& - \int \nabla ^{3}\varrho \cdot \nabla ^{3} \operatorname{div}(\varrho \boldsymbol{u})\,\mathrm{d}x\leqslant C\epsilon _{0}\bigl( \bigl\Vert \nabla ^{3}(\varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}} + \bigl\Vert \nabla ^{4} \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \bigr), \end{aligned}$$

(3.26)

$$\begin{aligned}& - \int \nabla ^{2}\boldsymbol{u}\cdot \nabla ^{3} \operatorname{div}(\varrho \boldsymbol{u}) \,\mathrm{d}x \leqslant C\epsilon _{0} \bigl\Vert \nabla ^{3}(\varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}} \end{aligned}$$

(3.27)

and

$$\begin{aligned} & \int \nabla ^{3}\boldsymbol{u}\cdot \nabla ^{3} \mathcal{N}\,\mathrm{d}x \\ &\quad = \int \nabla ^{3}\boldsymbol{u}\cdot \nabla ^{3} \bigl(-\boldsymbol{u}\cdot \nabla \boldsymbol{u}-h_{3}(\varrho )\nabla \varrho -h_{2}(\varrho ) (\Delta \boldsymbol{u}+ \nabla \operatorname{div}\boldsymbol{u}) \bigr)\,\mathrm{d}x \\ &\quad \quad{} + \int \nabla ^{3}\boldsymbol{u}\cdot \nabla ^{3} \bigl(h_{1}(\varrho ) \operatorname{div} \bigl( \bigl\vert D(\nabla \boldsymbol{u}) \bigr\vert ^{p-2}D(\nabla \boldsymbol{u}) \bigr) \bigr)\,\mathrm{d}x \\ &\quad \leqslant C\bigl( \bigl\vert \bigl\langle \nabla ^{4} \boldsymbol{u}, \nabla ^{2}(\boldsymbol{u}\cdot \nabla \boldsymbol{u}) \bigr\rangle \bigr\vert + \bigl\vert \bigl\langle \nabla ^{4} \boldsymbol{u}, \nabla ^{2}\bigl(h_{3}( \varrho )\nabla \varrho \bigr)\bigr\rangle \bigr\vert \bigr) \\ &\quad \quad{} + C\bigl( \bigl\vert \bigl\langle \nabla ^{4} \boldsymbol{u}, \nabla ^{2}\bigl(h_{2}(\varrho )\Delta \boldsymbol{u}\bigr)\bigr\rangle \bigr\vert + \bigl\vert \bigl\langle \nabla ^{4}\boldsymbol{u}, \nabla ^{2}\bigl(h_{2}( \varrho )\nabla \operatorname{div}\boldsymbol{u}\bigr)\bigr\rangle \bigr\vert \bigr) \\ &\quad \quad{} + C ( \bigl\vert \bigl\langle \nabla ^{4}\boldsymbol{u}, \nabla ^{2} \bigl(h_{1}( \varrho )\operatorname{div} \bigl( \bigl\vert D(\nabla \boldsymbol{u}) \bigr\vert ^{p-2}D(\nabla \boldsymbol{u}) \bigr) \bigr) \bigr\rangle \bigr\vert \\ &\quad \leqslant C \bigl\Vert \nabla ^{4}\boldsymbol{u} \bigr\Vert _{L^{2}}\bigl( \Vert \nabla \boldsymbol{u} \Vert _{L^{6}} \bigl\Vert \nabla ^{2} \boldsymbol{u} \bigr\Vert _{L^{3}}+ \Vert \boldsymbol{u} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{3} \boldsymbol{u} \bigr\Vert _{L^{2}} \bigr) \\ &\quad \quad{} + C \bigl\Vert \nabla ^{4}\boldsymbol{u} \bigr\Vert _{L^{2}}\bigl( \bigl\Vert \nabla ^{2} h_{3}( \varrho ) \bigr\Vert _{L^{ \infty}} \Vert \nabla \varrho \Vert _{L^{2}}+ \bigl\Vert \nabla h_{3}(\varrho ) \bigr\Vert _{L^{ \infty}} \bigl\Vert \nabla ^{2} \varrho \bigr\Vert _{L^{2}}\bigr) \\ &\quad \quad{} + C \bigl\Vert \nabla ^{4}\boldsymbol{u} \bigr\Vert _{L^{2}} \bigl\Vert h_{3}(\varrho ) \bigr\Vert _{L^{\infty}} \bigl\Vert \nabla ^{3} \varrho \bigr\Vert _{L^{2}} \\ &\quad \quad{} + C \bigl\Vert \nabla ^{4}\boldsymbol{u} \bigr\Vert _{L^{2}}\bigl( \bigl\Vert \nabla ^{2} h_{2}( \varrho ) \bigr\Vert _{L^{ \infty}} \bigl\Vert \nabla ^{2} \boldsymbol{u} \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla h_{2}(\varrho ) \bigr\Vert _{L^{ \infty}} \bigl\Vert \nabla ^{3} \boldsymbol{u} \bigr\Vert _{L^{2}}\bigr) \\ &\quad \quad{} + C \bigl\Vert \nabla ^{4}\boldsymbol{u} \bigr\Vert _{L^{2}} \bigl\Vert h_{2}(\varrho ) \bigr\Vert _{L^{\infty}} \bigl\Vert \nabla ^{4} \boldsymbol{u} \bigr\Vert _{L^{2}} \\ &\quad \quad{} + C \bigl\Vert \nabla ^{4}\boldsymbol{u} \bigr\Vert _{L^{2}}\bigl( \bigl\Vert h_{1}(\varrho ) \bigr\Vert _{L^{\infty}}+ \bigl\Vert \nabla h_{1}(\varrho ) \bigr\Vert _{L^{\infty}}+ \bigl\Vert \nabla ^{2} h_{1}( \varrho ) \bigr\Vert _{L^{ \infty}}\bigr) \Vert \boldsymbol{u} \Vert ^{p-2}_{H^{3}} \Vert \boldsymbol{u} \Vert _{H^{4}} \\ &\quad \leqslant C\epsilon _{0} \bigl( \Vert \nabla \boldsymbol{u} \Vert ^{2}_{H^{3}}+ \bigl\Vert \nabla ^{3}( \varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}} \bigr), \end{aligned}$$

(3.28)

$$\begin{aligned} & \int \nabla ^{3}\varrho \cdot \nabla ^{2} \mathcal{N}\,\mathrm{d}x \\ &\quad = \int \nabla ^{3}\varrho \cdot \nabla ^{2} \bigl(- \boldsymbol{u}\cdot \nabla \boldsymbol{u}-h_{3}(\varrho )\nabla \varrho -h_{2}(\varrho ) (\Delta \boldsymbol{u}+ \nabla \operatorname{div}\boldsymbol{u}) \bigr)\,\mathrm{d}x \\ &\quad \quad{} + \int \nabla ^{3}\varrho \cdot \nabla ^{2} \bigl(h_{1}(\varrho ) \operatorname{div} \bigl( \bigl\vert D(\nabla \boldsymbol{u}) \bigr\vert ^{p-2}D(\nabla \boldsymbol{u}) \bigr) \bigr)\,\mathrm{d}x \\ &\quad \leqslant C \bigl\Vert \nabla ^{3}\varrho \bigr\Vert _{L^{2}}\bigl( \Vert \nabla \boldsymbol{u} \Vert _{L^{6}} \bigl\Vert \nabla ^{2} \boldsymbol{u} \bigr\Vert _{L^{3}}+ \Vert \boldsymbol{u} \Vert _{L^{\infty}} \bigl\Vert \nabla ^{3} \boldsymbol{u} \bigr\Vert _{L^{2}} \bigr) \\ &\quad \quad{} + C \bigl\Vert \nabla ^{3}\varrho \bigr\Vert _{L^{2}}\bigl( \bigl\Vert \nabla ^{2} h_{3}( \varrho ) \bigr\Vert _{L^{ \infty}} \Vert \nabla \varrho \Vert _{L^{2}}+ \bigl\Vert \nabla h_{3}(\varrho ) \bigr\Vert _{L^{ \infty}} \bigl\Vert \nabla ^{2} \varrho \bigr\Vert _{L^{2}}\bigr) \\ &\quad \quad{} + C \bigl\Vert \nabla ^{3}\varrho \bigr\Vert _{L^{2}} \bigl\Vert h_{3}(\varrho ) \bigr\Vert _{L^{\infty}} \bigl\Vert \nabla ^{3} \varrho \bigr\Vert _{L^{2}} \\ &\quad \quad{} + C \bigl\Vert \nabla ^{3}\varrho \bigr\Vert _{L^{2}}\bigl( \bigl\Vert \nabla ^{2} h_{2}( \varrho ) \bigr\Vert _{L^{ \infty}} \bigl\Vert \nabla ^{2} \boldsymbol{u} \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla h_{2}(\varrho ) \bigr\Vert _{L^{ \infty}} \bigl\Vert \nabla ^{3} \boldsymbol{u} \bigr\Vert _{L^{2}}\bigr) \\ &\quad \quad{} + C \bigl\Vert \nabla ^{3}\varrho \bigr\Vert _{L^{2}} \bigl\Vert h_{2}(\varrho ) \bigr\Vert _{L^{\infty}} \bigl\Vert \nabla ^{4} \boldsymbol{u} \bigr\Vert _{L^{2}} \\ &\quad \quad{} + C \bigl\Vert \nabla ^{3}\varrho \bigr\Vert _{L^{2}}\bigl( \bigl\Vert h_{1}(\varrho ) \bigr\Vert _{L^{\infty}}+ \bigl\Vert \nabla h_{1}(\varrho ) \bigr\Vert _{L^{\infty}} + \bigl\Vert \nabla ^{2} h_{1}( \varrho ) \bigr\Vert _{L^{\infty}}\bigr) \Vert \boldsymbol{u} \Vert ^{p-2}_{H^{3}} \Vert \boldsymbol{u} \Vert _{H^{4}} \\ &\quad \leqslant C\epsilon _{0} \bigl( \Vert \nabla \boldsymbol{u} \Vert ^{2}_{H^{3}}+ \bigl\Vert \nabla ^{3}( \varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}} \bigr). \end{aligned}$$

(3.29)

Plugging (3.26)–(3.29) into (3.25), we can deduce that

$$\begin{aligned} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \biggl( \bigl\Vert \nabla ^{3} \varrho \bigr\Vert ^{2}_{L^{2}} + \bigl\Vert \nabla ^{3}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} +2\gamma _{3} \int \nabla ^{3}\varrho \cdot \nabla ^{2} \boldsymbol{u} \,\mathrm{d}x \biggr) \\ &\quad \quad{} + \frac{\gamma _{3}}{4} \bigl\Vert \nabla ^{3}\varrho \bigr\Vert ^{2}_{L^{2}} + \frac{\mu _{0}}{2} \bigl\Vert \nabla ^{3}\nabla \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} + \frac{\mu _{0}}{2} \bigl\Vert \nabla ^{3}\operatorname{div}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \\ &\quad \leqslant \frac{1}{4} \bigl\Vert \nabla ^{2} \operatorname{div}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} +C \epsilon _{0} \bigl( \Vert \nabla \boldsymbol{u} \Vert ^{2}_{H^{3}} + \bigl\Vert \nabla ^{3}( \varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}} \bigr), \end{aligned}$$

where $0<\gamma _{3}<\{\frac{1}{4}, \frac{1}{8\mu _{0}}\}$ is a fixed constant. Consequently, we completed the proof of (3.22). □

With Lemmas 3.1–3.2 in hand, we easily further obtain Proposition 3.2. In fact, keeping in mind the Young’s inequality and the definitions of $\mathcal{F}_{1}$, $\mathcal{F}_{2}$, $\mathcal{F}_{3}$, we have

$$\begin{aligned} \frac{1}{C_{4}} \bigl\Vert (\varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{H^{3}}\leqslant \mathcal{F}_{1} + \mathcal{F}_{2}+\mathcal{F}_{3}\leqslant C_{4} \bigl\Vert ( \varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{H^{3}}, \end{aligned}$$

where $C_{4}>0$ is a constant, which yields

$$\begin{aligned} \mathcal{F}_{1} +\mathcal{F}_{2}+\mathcal{F}_{3} \approx \bigl\Vert (\varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{H^{3}}. \end{aligned}$$

Thanks to the three lemmas above, integrating the resulting inequality over $(0, t)$, (3.2) holds for the small enough $\epsilon _{0}$. This completes the proof of Proposition 3.2.

4 Time decay rates of the solution

In this section we shall show the time decay rates of the Cauchy problem (1.5). The proof will be broken up into two subsections.

4.1 Cancellation of the low frequency part

Inspired by the observation of canceling the low frequency part of the solution, we draw the following conclusion.

Lemma 4.1

We have

$$\begin{aligned} \bigl\Vert \nabla ^{3}(\varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}} \leqslant Ce^{C_{5}t} \bigl\Vert \nabla ^{3}\bigl(\varrho ^{0}, \boldsymbol{u}^{0}\bigr) \bigr\Vert ^{2}_{L^{2}} +C \int ^{t}_{0}e^{C_{5}(t- \tau )} \bigl\Vert \nabla ^{3}\bigl(\varrho ^{L}, \boldsymbol{u}^{L} \bigr) (\tau ) \bigr\Vert ^{2}_{L^{2}} \,\mathrm{d} {\tau}, \end{aligned}$$

(4.1)

where $C, C_{5}> 0$ are constants.

Proof

Multiplying $\nabla ^{3}\text{(1.5)}_{1}$ by $\nabla ^{2}\boldsymbol{u}$, exploiting $\nabla ^{2}\text{(1.5)}_{2}\cdot \nabla ^{3}\varrho ^{L}$ in $L^{2}$, and integrating by parts, we can estimate that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \int \nabla ^{3}\varrho ^{L}\cdot \nabla ^{2}\boldsymbol{u} \,\mathrm{d}x & = \mu _{0} \int \nabla ^{3}\varrho ^{L} \cdot \nabla ^{2}\Delta \boldsymbol{u}\,\mathrm{d}x +\mu _{0} \int \nabla ^{3} \varrho ^{L}\cdot \nabla ^{3}\operatorname{div}\boldsymbol{u}\,\mathrm{d}x \\ &\quad{} + \int \nabla ^{2}\operatorname{div}\boldsymbol{u}\cdot \nabla ^{2} \operatorname{div}\boldsymbol{u}^{L}\,\mathrm{d}x - \int \nabla ^{3}\varrho ^{L} \cdot \nabla ^{3}\varrho \,\mathrm{d}x \\ &\quad{} + \int \nabla ^{2}\mathcal{N}\cdot \nabla ^{3} \varrho ^{L} \,\mathrm{d}x + \int \nabla ^{2}\operatorname{div}(\varrho \boldsymbol{u})^{L}\cdot \nabla ^{2} \operatorname{div} \boldsymbol{u}\,\mathrm{d}x. \end{aligned}$$

Similarly to (3.5), using the Young’s inequality, we have

$$\begin{aligned} &{-} \frac{\mathrm{d}}{\mathrm{d}t} \int \nabla ^{3}\varrho ^{L}\cdot \nabla ^{2}\boldsymbol{u}\,\mathrm{d}x \\ &\quad \leqslant \frac{\mu _{0}}{2} \bigl\Vert \nabla ^{2}\Delta \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+ \frac{\mu _{0}}{2} \bigl\Vert \nabla ^{3}\operatorname{div} \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \nabla ^{2}\operatorname{div}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \\ &\quad \quad{} + \frac{1}{2} \bigl\Vert \nabla ^{2} \operatorname{div}\boldsymbol{u}^{L} \bigr\Vert ^{2}_{L^{2}}+ \frac{5+2\mu _{0}}{2} \bigl\Vert \nabla ^{3}\varrho ^{L} \bigr\Vert ^{2}_{L^{2}} + \frac{1}{8} \bigl\Vert \nabla ^{3}\varrho \bigr\Vert ^{2}_{L^{2}} \\ &\quad \quad{} + \frac{1}{2} \bigl\Vert \nabla ^{2} \operatorname{div}(\varrho \boldsymbol{u})^{L} \bigr\Vert ^{2}_{L^{2}} +\frac{1}{2} \bigl\Vert \nabla ^{2}\mathcal{N} \bigr\Vert ^{2}_{L^{2}} . \end{aligned}$$

(4.2)

By virtue of the Plancherel theorem and Lemma 3.3, we estimate that

$$\begin{aligned} &\frac{1}{2} \bigl\Vert \nabla ^{2} \operatorname{div}(\varrho \boldsymbol{u})^{L} \bigr\Vert ^{2}_{L^{2}} +\frac{1}{2} \bigl\Vert \nabla ^{2}\mathcal{N} \bigr\Vert ^{2}_{L^{2}} \\ &\quad \leqslant C\epsilon _{0} \bigl( \bigl\Vert \nabla ^{4}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \nabla ^{3}(\varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}} \bigr) . \end{aligned}$$

(4.3)

Adding up (4.1) and $\gamma _{2}\cdot \text{(4.2)}$ with some positive constants, we obtain

$$\begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}t} \biggl({\mathcal{F}_{3}} -\gamma _{3} \int \nabla ^{3}\varrho ^{L}\cdot \nabla ^{2}\boldsymbol{u}\,\mathrm{d}x \biggr) \\ &\quad \quad{} + \frac{\gamma _{3}}{8} \bigl\Vert \nabla ^{3}\varrho \bigr\Vert ^{2}_{L^{2}} + \frac{\mu _{0}}{4} \bigl\Vert \nabla ^{3}\boldsymbol{u}^{H} \bigr\Vert ^{2}_{L^{2}} + \frac{\mu _{0}}{4} \bigl\Vert \nabla ^{4} \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+ \frac{\mu _{0}}{2} \bigl\Vert \nabla ^{3}\operatorname{div} \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \\ &\quad \leqslant \biggl(\frac{1}{4}+\gamma _{3} \biggr) \bigl\Vert \nabla ^{2} \operatorname{div}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+\frac{\gamma _{3}\mu _{0}}{2} \bigl\Vert \nabla ^{2}\Delta \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} + \frac{\gamma _{3}\mu _{0}}{2} \bigl\Vert \nabla ^{3}\operatorname{div} \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \\ &\quad \quad{} + C\gamma _{3} \bigl( \bigl\Vert \nabla ^{3} \varrho ^{L} \bigr\Vert ^{2}_{L^{2}} + \bigl\Vert \nabla ^{2}\operatorname{div}\boldsymbol{u}^{L} \bigr\Vert ^{2}_{L^{2}} \bigr) \\ &\quad \quad{} + C\epsilon _{0}(1+\gamma _{3}) \bigl\Vert \nabla ^{3}(\varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}} +C\epsilon _{0} \bigl\Vert \nabla ^{4} \boldsymbol{u} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{3} \boldsymbol{u} \bigr\Vert ^{p-2}_{L^{2}}. \end{aligned}$$

In addition, by the frequency decomposition (1.6), we further put $\frac{\mu _{0}}{4}\|\nabla ^{2}\boldsymbol{u}^{L}\|^{2}_{L^{2}}$ on both sides of (4.1) to get

$$\begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}t} \biggl({\mathcal{F}_{3}} -\gamma _{3} \int \nabla ^{3}\varrho ^{L}\cdot \nabla ^{2}\boldsymbol{u}\,\mathrm{d}x \biggr) \\ &\quad \quad{} + \frac{\gamma _{3}}{8} \bigl\Vert \nabla ^{3}\varrho \bigr\Vert ^{2}_{L^{2}} + \frac{\mu _{0}}{8} \bigl\Vert \nabla ^{3}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} + \frac{\mu _{0}}{4} \bigl\Vert \nabla ^{4} \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+ \frac{\mu _{0}}{2} \bigl\Vert \nabla ^{3}\operatorname{div}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \\ &\quad \leqslant \biggl(\frac{1}{4}+\gamma _{3} \biggr) \bigl\Vert \nabla ^{2} \operatorname{div}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} +\frac{\gamma _{3}\mu _{0}}{2} \bigl\Vert \nabla ^{2}\Delta \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} + \frac{\gamma _{3}\mu _{0}}{2} \bigl\Vert \nabla ^{3}\operatorname{div} \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \\ &\quad \quad{} + C\gamma _{3} \bigl\Vert \nabla ^{3}\varrho ^{L} \bigr\Vert ^{2}_{L^{2}}+ \biggl( \frac{1}{4}+C\gamma _{3} \biggr) \bigl\Vert \nabla ^{3} \boldsymbol{u}^{L} \bigr\Vert ^{2}_{L^{2}} \\ &\quad \quad{} + C\epsilon _{0}(1+\gamma _{3}) \bigl\Vert \nabla ^{3}(\varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}} +C\epsilon _{0} \bigl\Vert \nabla ^{4} \boldsymbol{u} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{3} \boldsymbol{u} \bigr\Vert ^{p-2}_{L^{2}}. \end{aligned}$$

Furthermore, noting the smallness of $\epsilon _{0}$, we obviously have

$$\begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}t} \biggl({\mathcal{F}_{3}}- \gamma _{3} \int \nabla ^{3}\varrho ^{L}\cdot \nabla ^{2}\boldsymbol{u} \,\mathrm{d}x \biggr) \\ &\quad \quad{} + \frac{\gamma _{3}}{16} \bigl\Vert \nabla ^{3}\varrho \bigr\Vert ^{2}_{L^{2}} + \frac{\mu _{0}}{16} \bigl\Vert \nabla ^{3}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} + \frac{\mu _{0}}{8} \bigl\Vert \nabla ^{4} \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}+ \frac{\mu _{0}}{8} \bigl\Vert \nabla ^{3}\operatorname{div}\boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \\ &\quad \leqslant C \bigl\Vert \nabla ^{3}\bigl(\varrho ^{L}, \boldsymbol{u}^{L}\bigr) \bigr\Vert ^{2}_{L^{2}}. \end{aligned}$$

(4.4)

In view of the frequency decomposition (1.6), we get

$$\begin{aligned} &{\mathcal{F}_{3}}-\gamma _{3} \int \nabla ^{3}\varrho ^{L}\cdot \nabla ^{2}\boldsymbol{u} \,\mathrm{d}x \\ &\quad = \frac{1}{2} \bigl( \bigl\Vert \nabla ^{3}\varrho \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \nabla ^{3} \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \bigr) +\gamma _{3} \int \nabla ^{3}\varrho ^{H} \cdot \nabla ^{2}\boldsymbol{u}\,\mathrm{d}x \\ &\quad \leqslant \frac{1}{2} \bigl( \bigl\Vert \nabla ^{3} \varrho \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \nabla ^{3} \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}} \bigr) +\frac{\gamma _{3}}{2} \bigl\Vert \nabla ^{2}\varrho \bigr\Vert ^{2}_{L^{2}} +\frac{\gamma _{3}}{2} \bigl\Vert \nabla ^{3} \boldsymbol{u} \bigr\Vert ^{2}_{L^{2}}. \end{aligned}$$

(4.5)

Next, recombining (4.5), we have

$$\begin{aligned} &{\mathcal{F}_{3}} -\gamma _{3} \int \nabla ^{3}\varrho ^{L} \cdot \nabla ^{2}\boldsymbol{u}\,\mathrm{d}x \approx \bigl\Vert \nabla ^{3}(\varrho , \boldsymbol{u}) \bigr\Vert ^{2}_{L^{2}}, \end{aligned}$$

(4.6)

and with the help of (4.4) and (4.6), we can deduce that for a suitable constant $C_{5}$,

$$\begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}t} \biggl({\mathcal{F}_{3}} -\gamma _{3} \int \nabla ^{3}\varrho ^{L}\cdot \nabla ^{2}\boldsymbol{u} \,\mathrm{d}x \biggr) +C_{5} \biggl({\mathcal{F}_{3}} -\gamma _{3} \int \nabla ^{3} \varrho ^{L}\cdot \nabla ^{2}\boldsymbol{u}\,\mathrm{d}x \biggr) \\ &\quad \leqslant C \bigl\Vert \nabla ^{3}\bigl(\varrho ^{L}, \boldsymbol{u}^{L}\bigr) \bigr\Vert ^{2}_{L^{2}}. \end{aligned}$$

Consequently, thanks to the Gronwall inequality, we conclude that

$$\begin{aligned} &{\mathcal{F}_{3}}(t)-\gamma _{3} \int \nabla ^{3}\varrho ^{L}\cdot \nabla ^{2} \boldsymbol{u} \,\mathrm{d}x \\ &\quad \leqslant C_{5}e^{-C_{5}t} \biggl({\mathcal{F}_{3}}(0) -\gamma _{3} \int \nabla ^{3}\varrho ^{L}_{0} \cdot \nabla ^{2}\boldsymbol{u}_{0} \,\mathrm{d}x \biggr) \\ &\quad \quad{} + C \int ^{t}_{0}e^{-C_{5}(t-\tau )} \bigl\Vert \nabla ^{3}\bigl(\varrho ^{L}, \boldsymbol{u}^{L} \bigr) \bigr\Vert ^{2}_{L^{2}}\,\mathrm{d} {\tau}. \end{aligned}$$

This completes the proof of Lemma 4.1. □

4.2 Decay estimate of the low frequency part

We will present the estimate of the low frequency part of the constructed solution by analyzing the structure of the semigroup of the system (1.5). To this end, by Hausdorff decomposition [49], we first decompose the velocity u into $m=\Lambda ^{-1}\operatorname{div}\boldsymbol{u}$ and $M=\Lambda ^{-1}\operatorname{curl}\boldsymbol{u}$, where $\operatorname{curl}_{ij}=\partial _{j}u_{i}-\partial _{i}u_{j}$ and $\Lambda =\sqrt{-\Delta}$. Then, the system (1.5) can be decoupled into the following systems:

$$ \textstyle\begin{cases} \varrho _{t}+\Lambda m=-\operatorname{div}(\varrho \boldsymbol{u}), \\ m_{t}-2\mu _{0}\Delta m-\Lambda \varrho =\Lambda ^{-1}\operatorname{div} \mathcal{N}, \\ (\varrho , m)|_{t=0}=(\varrho ^{0}, m^{0})(x) \end{cases} $$

(4.7)

and

$$ \textstyle\begin{cases} M_{t}-\mu _{0}\Delta M=\Lambda ^{-1}\operatorname{curl}\mathcal{N}, \\ M(0, x)=M^{0}(x), \end{cases} $$

(4.8)

where $m^{0}:=\Lambda ^{-1}\operatorname{div}\boldsymbol{u}^{0}$ and $M^{0}=: \Lambda ^{-1}\operatorname{curl}\boldsymbol{u}^{0}$. Then a direct calculation yields the following lemma; see [22] for the proof.

Lemma 4.2

Let $M(t, x)$ be the solution to the linearized system of (4.8). Then, for all $|\xi |^{2}\geqslant 0$, we have

$$\begin{aligned} \bigl\vert \hat{M}(t, \xi ) \bigr\vert ^{2}\leqslant Ce^{- \vert \xi \vert ^{2}t} \bigl\vert \hat{M}(0, \xi ) \bigr\vert ^{2}, \end{aligned}$$

where $C>0$ is a constant and M̂ denotes the Fourier transform of M.

Now we turn to considering the linearized system of (4.7). The following system can be obtained by applying the Fourier transform:

$$ \textstyle\begin{cases} \hat{\varrho _{t}}=- \vert \xi \vert \hat{m}, \\ \hat{m_{t}}=- \vert \xi \vert \hat{\varrho}-2\mu _{0} \vert \xi \vert ^{2}\hat{m}, \end{cases} $$

(4.9)

which is also rewritten by

$$\begin{aligned} \hat{U_{t}} =\hat{\mathcal{A}}\bigl( \vert \xi \vert \bigr)\hat{U}, \end{aligned}$$

(4.10)

where $\hat{U}=(\hat{\varrho}, \hat{m})$ and

$$ \hat{\mathcal{A}}\bigl( \vert \xi \vert \bigr)= \begin{pmatrix} 0, & - \vert \xi \vert \\ \vert \xi \vert , & -2\mu _{0} \vert \xi \vert ^{2} \end{pmatrix}. $$

According to the standard theory of ordinary differential equations, the system (4.10) admits a solution which can be expressed by

$$\begin{aligned} \hat{U} =e^{t\hat{\mathcal{A}}(|\xi |)}\hat{U}(0). \end{aligned}$$

(4.11)

By taking the inverse Fourier transform on both sides of (4.11), we can obtain

$$\begin{aligned} U =A(t)U(0), \end{aligned}$$

where

$$\begin{aligned} A(t)U=\mathcal{F}^{-1} \bigl(e^{t\hat{\mathcal{A}}(|\xi |)}\hat{U}( \xi ) \bigr), \end{aligned}$$

which implies the solution to the linearized system of (4.7). Additionally, we further work out the eigenvalues $\lambda _{i}(\xi )$ ($i=1, 2$) of matrix $\hat{\mathcal{A}}(|\xi |)$ and express them by

$$\begin{aligned} \det \bigl(\hat{\mathcal{A}}\bigl( \vert \xi \vert \bigr)- \lambda I \bigr) =\lambda ^{2}+2 \mu _{0} \vert \xi \vert \lambda + \vert \xi \vert ^{2}=0. \end{aligned}$$

(4.12)

Then the eigenvalues $\lambda _{i}(\xi )$ ($i=1, 2$) of $\hat{\mathcal{A}}(|\xi |)$ can be calculated by (4.12) as follows:

$$ \textstyle\begin{cases} \lambda _{1}(\xi )=-\mu _{0} \vert \xi \vert ^{2}+ \vert \xi \vert \sqrt{\mu _{0}^{2} \vert \xi \vert ^{2}-1}, \\ \lambda _{2}(\xi )=-\mu _{0} \vert \xi \vert ^{2}- \vert \xi \vert \sqrt{\mu _{0}^{2} \vert \xi \vert ^{2}-1}. \end{cases} $$

Based on the semigroup decomposition theory given in [2], we have

$$\begin{aligned} e^{t\hat{\mathcal{A}}(|\xi |)} =e^{\lambda _{1}t}P_{1}(\xi )+e^{ \lambda _{2}t}P_{2}(\xi ), \end{aligned}$$

where

$$\begin{aligned} P_{i}(\xi )=\prod_{j\neq i} \frac{A( \vert \xi \vert )-\lambda _{i} I}{\lambda _{i}-\lambda _{j}}\quad (i, j=1, 2) \end{aligned}$$

is a projection operator.

Then by tedious and careful calculations, we can present the asymptotic expansions of $\lambda _{i}(\xi ), P_{i}(\xi )$ ($i=1, 2$) and $e^{t\hat{\mathcal{A}}(|\xi |)}$ in the low and high frequency situations, which give rise to the following lemma.

Lemma 4.3

For any $|\xi |\leqslant 1$, $\lambda _{i}(\xi )$ ($i=1, 2$) has the Taylor series expansion

$$ \textstyle\begin{cases} \lambda _{1}(\xi )=- \vert \xi \vert ^{2}+i ( \vert \xi \vert +O( \vert \xi \vert ^{3}) ), \\ \lambda _{2}(\xi )=- \vert \xi \vert ^{2}-i ( \vert \xi \vert +O( \vert \xi \vert ^{3}) ). \end{cases} $$

We omit the proof of this lemma here; please refer to [17]. With the help of Lemmas 4.2 and 4.3, one can obtain the time decay estimates of the low frequency part of the solution to the linear system (4.9).

Proposition 4.1

Let $1\leqslant q\leqslant 2$. Then for any integer $k\geqslant 0$, the solution to the linearized system of (1.5) satisfies

$$\begin{aligned} \bigl\Vert \nabla ^{k}\bigl(\varrho ^{L}, m^{L}, \boldsymbol{u}^{L}\bigr) (t) \bigr\Vert _{L^{2}}\leqslant C(1+t)^{- \frac{3}{2} (\frac{1}{q}-\frac{1}{2})-\frac{k}{2}} \bigl\Vert \bigl(\varrho ^{0}, m^{0}, \boldsymbol{u}^{0}\bigr) (t) \bigr\Vert _{L^{q}}, \end{aligned}$$

where $C>0$ is a constant independent of T.

Proof

Following the arguments in [43], we can deduce that

$$\begin{aligned} \bigl\Vert \nabla ^{k}\bigl(\varrho ^{L}, m^{L}, M^{L}\bigr) (t) \bigr\Vert _{L^{2}}\leqslant C(1+t)^{- \frac{3}{2} (\frac{1}{q}-\frac{1}{2})-\frac{k}{2}} \bigl\Vert \bigl(\varrho ^{0}, m^{0}, M^{0}\bigr) (t) \bigr\Vert _{L^{q}}. \end{aligned}$$

(4.13)

Note that

$$\begin{aligned} \boldsymbol{u}=\Delta ^{-1}(\nabla \operatorname{div}\boldsymbol{u}-\operatorname{curl} \operatorname{curl} \boldsymbol{u}) =-\Lambda ^{-1}\nabla m+\Lambda ^{-1} \operatorname{curl}M, \end{aligned}$$

(4.14)

which together with (4.13) implies that

$$\begin{aligned} \bigl\Vert \nabla ^{k}\boldsymbol{u}^{L}(t) \bigr\Vert _{L^{2}}= \bigl\Vert \nabla ^{k} \bigl(m^{L}, M^{L}\bigr) (t) \bigr\Vert _{L^{2}} \leqslant C(1+t)^{-\frac{3}{2} (\frac{1}{q}-\frac{1}{2}) -\frac{k}{2}} \bigl\Vert \boldsymbol{u}^{0}(t) \bigr\Vert _{L^{q}}. \end{aligned}$$

(4.15)

Combining (4.13) and (4.15), we complete the proof of Proposition 4.1. □

4.3 Decay rates for the nonlinear system

This subsection is devoted to investigating the optimal time decay rates of the solutions for the nonlinear system (1.5). For convenience, we define

$$\begin{aligned} W(t)= \bigl(\varrho (t), \boldsymbol{u}(t) \bigr)^{T} \end{aligned}$$

and

$$ Q= \begin{pmatrix} 0, &\operatorname{div} \\ \nabla , & -\mu _{0}\Delta -\mu _{0} \nabla \operatorname{div} \end{pmatrix}. $$

Then it follows from the nonlinear system (1.5) that

$$\begin{aligned} W_{t}+QW=\mathcal{N}_{1}(W) \end{aligned}$$

with the initial data $W|_{t=0}=W(0)$, where $\mathcal{N}_{1}(W)$ has been defined by

$$\begin{aligned} \mathcal{N}_{1}(W)= \bigl(-\operatorname{div}(\varrho \boldsymbol{u}), \mathcal{N} \bigr)^{T}. \end{aligned}$$

Thanks to Duhamel’s principle, the solution of the nonlinear system can be presented as

$$\begin{aligned} W(t)=Q(0)W(0)+ \int ^{t}_{0}Q(t-\tau )\mathcal{N}_{1}(W) (\tau ) \, \mathrm{d} {\tau}, \end{aligned}$$

where $Q(0)W(0)$ is the initial data of the solution to the linearized system of (1.5). In addition, with Proposition 4.1 in hand, we have the following lemma.

Lemma 4.4

For any integer $k\geqslant 0$, we have

$$\begin{aligned} \bigl\Vert \nabla ^{k}W^{L}(t) \bigr\Vert _{L^{2}}& \leqslant C_{6}(1+t)^{-(\frac{3}{4}+ \frac{k}{2})} \bigl\Vert W(0) \bigr\Vert _{L^{1}} \\ &\quad{} + C_{6} \int ^{\frac{t}{2}}_{0}(1+t-\tau )^{-(\frac{3}{4} +\frac{k}{2})} \bigl\Vert \mathcal{N}_{1}(W) (\tau ) \bigr\Vert _{L^{1}}\, \mathrm{d} {\tau} \\ &\quad{} + C_{6} \int ^{t}_{\frac{t}{2}} (1+t-\tau )^{\frac{k}{2}} \bigl\Vert \mathcal{N}_{1}(W) ( \tau ) \bigr\Vert _{L^{2}}\,\mathrm{d} {\tau}, \end{aligned}$$

(4.16)

where $C_{6}>0$ is a constant.

Based on Lemmas 4.1 and 4.4, we are in a position to establish the optimal time decay rates of solutions.

Lemma 4.5

(Optimal time decay rates). With the assumptions in Theorem 1.1, we have, for any $t\in [0,\infty )$,

$$\begin{aligned} \bigl\Vert \nabla ^{l}(\varrho , \boldsymbol{u}) (t) \bigr\Vert _{L^{2}}\leqslant C(1+t)^{-( \frac{3}{4}+\frac{l}{2})},\quad l=0, 1, 2, 3, \end{aligned}$$

where $C>0$ is a constant.

Proof

We first define a non-decreasing Lyapunov function $\mathcal{G}(\tau )$ as

$$\begin{aligned} \mathcal{G}(\tau ):=\sup_{0\leqslant \tau \leqslant t}\sum ^{3}_{l=0}(1+ \tau )^{\frac{3}{4}+\frac{l}{2}} \bigl\Vert \nabla ^{l}(\varrho , \boldsymbol{u}) (\tau ) \bigr\Vert _{L^{2}}, \end{aligned}$$

which implies that for $0\leqslant \tau \leqslant t$ and $0\leqslant l\leqslant 3$,

$$\begin{aligned} \bigl\Vert \nabla ^{l}(\varrho , \boldsymbol{u}) (\tau ) \bigr\Vert _{L^{2}}\leqslant C_{7}(1+ \tau )^{-(\frac{3}{4}+\frac{l}{2})}\mathcal{G}(\tau ), \end{aligned}$$

(4.17)

where the constant $C_{7}>0$ is independent of $\epsilon _{0}$.

Due to (4.17) and Hölder’s inequality, we have

$$\begin{aligned} \bigl\Vert \mathcal{N}_{1}(W) (\tau ) \bigr\Vert _{L^{1}}& \leqslant \bigl\Vert (\varrho , \boldsymbol{u}) \bigr\Vert _{L^{2}} \bigl\Vert \nabla (\varrho , \boldsymbol{u}) \bigr\Vert _{L^{2}}+ \Vert \varrho \Vert _{L^{2}} \bigl\Vert \nabla ^{2} \boldsymbol{u} \bigr\Vert _{L^{2}} \\ &\quad{} + \Vert \nabla \boldsymbol{u} \Vert ^{2}_{L^{2}}+ \Vert \nabla \boldsymbol{u} \Vert ^{p-2}_{L^{2(q-2)}} \Vert \nabla \boldsymbol{u} \Vert _{H^{1}} \\ & \leqslant \epsilon _{0}\mathcal{G}(t) (1+\tau )^{-\frac{5}{4}} \end{aligned}$$

and

$$\begin{aligned} \bigl\Vert \mathcal{N}_{1}(W) (\tau ) \bigr\Vert _{L^{2}}& \leqslant \bigl\Vert (\varrho , \boldsymbol{u}) \bigr\Vert _{L^{3}} \bigl\Vert \nabla (\varrho , \boldsymbol{u}) \bigr\Vert _{L^{6}}+ \Vert \varrho \Vert _{L^{ \infty}} \bigl\Vert \nabla ^{2} \boldsymbol{u} \bigr\Vert _{L^{2}} \\ &\quad{} + \Vert \nabla \boldsymbol{u} \Vert ^{2}_{L^{3}} + \Vert \nabla \boldsymbol{u} \Vert ^{2}_{L^{6}}+ \Vert \nabla \boldsymbol{u} \Vert ^{p-2}_{L^{\infty}} \Vert \nabla \boldsymbol{u} \Vert _{H^{1}} \\ & \leqslant \epsilon _{0}^{1-\vartheta}\mathcal{G}^{1+\vartheta}(t) (1+ \tau )^{-(\frac{7}{4}+\frac{3\vartheta}{4})}, \end{aligned}$$

where $\vartheta \in (0, \frac{1}{2})$ is a given constant. Combining with Lemmas 2.6 and 4.4, we have

$$\begin{aligned} \bigl\Vert \nabla ^{l}W^{L}(t) \bigr\Vert _{L^{2}}& \leqslant C(1+t)^{-(\frac{3}{4}+ \frac{l}{2})} \bigl\Vert W(0) \bigr\Vert _{L^{1}} \\ &\quad{} + C_{6} \int ^{\frac{t}{2}}_{0}(1+t-\tau )^{-(\frac{3}{4}+\frac{l}{2})} \epsilon _{0}\mathcal{G}(\tau ) (1+\tau )^{-\frac{5}{4}}\, \mathrm{d} { \tau} \\ &\quad{} + C_{6} \int ^{t}_{\frac{t}{2}}(1+t-\tau )^{\frac{l}{2}} \epsilon _{0}^{1- \vartheta}\mathcal{G}^{1+\vartheta}(\tau ) (1+ \tau )^{-(\frac{7}{4}+ \frac{3\vartheta}{4})}\,\mathrm{d} {\tau} \\ & \leqslant C(1+t)^{-(\frac{3}{4}+\frac{l}{2})} \bigl( \bigl\Vert W(0) \bigr\Vert _{L^{1}} + \epsilon _{0}\mathcal{G}(\tau ) +\epsilon _{0}^{1-\vartheta} \mathcal{G}^{1+\vartheta}(\tau ) \bigr), \end{aligned}$$

(4.18)

where $0\leqslant l\leqslant 3$. Putting (4.18) into (4.1) and using Lemma 2.6, we can deduce that

$$\begin{aligned} \bigl\Vert \nabla ^{3}W(t) \bigr\Vert ^{2}_{L^{2}}& \leqslant Ce^{-C_{3}t} \bigl\Vert \nabla ^{3}W(0) \bigr\Vert ^{2}_{L^{2}} \\ &\quad{} + C \bigl( \bigl\Vert W(0) \bigr\Vert ^{2}_{L^{1}}+ \epsilon ^{2}_{0}\mathcal{G}^{2}(t) \bigr) \int ^{t}_{0}e^{-C_{3}(t-\tau )} (1+\tau )^{-\frac{7}{2}} \,\mathrm{d} {\tau} \\ &\quad{} + C\epsilon _{0}^{2-2\vartheta}\mathcal{G}^{2+2\vartheta}(t) \int ^{t}_{0}e^{-C_{3}(t- \tau )} (1+t)^{-\frac{7}{2}}\,\mathrm{d} {\tau} \\ &\leqslant C(1+t)^{-\frac{7}{2}} \bigl( \bigl\Vert W(0) \bigr\Vert ^{2}_{H^{2}\cap L^{1}} + \epsilon ^{2}_{0} \mathcal{G}^{2}(t) +\epsilon _{0}^{2-2\vartheta} \mathcal{G}^{2+2\vartheta}(t) \bigr). \end{aligned}$$

(4.19)

Making use of (1.6) and Lemma 2.1, we have

$$\begin{aligned} \bigl\Vert \nabla ^{l}W(t) \bigr\Vert ^{2}_{L^{2}}&\leqslant C \bigl\Vert \nabla ^{l}W^{L}(t) \bigr\Vert ^{2}_{L^{2}} +C \bigl\Vert \nabla ^{l}W^{H}(t) \bigr\Vert ^{2}_{L^{2}} \\ &\leqslant C \bigl\Vert \nabla ^{l}W^{L} \bigr\Vert ^{2}_{L^{2}}+C \bigl\Vert \nabla ^{3}W \bigr\Vert ^{2}_{L^{2}}. \end{aligned}$$

(4.20)

Therefore, by putting (4.18)–(4.19) into (4.20), we deduce that for $0\leqslant l\leqslant 3$,

$$\begin{aligned} \bigl\Vert \nabla ^{l}W(t) \bigr\Vert ^{2}_{L^{2}} \leqslant C(1+t)^{-(\frac{3}{2}+l)} \bigl( \bigl\Vert W(0) \bigr\Vert ^{2}_{H^{2}\cap L^{1}}+\epsilon ^{2}_{0} \mathcal{G}^{2}(t) +\epsilon _{0}^{2-2\vartheta} \mathcal{G}^{2+2\vartheta}(t) \bigr). \end{aligned}$$

Recalling the definition of $\mathcal{G}(t)$, we can derive for sufficiently small $\epsilon _{0}$ that

$$\begin{aligned} \mathcal{G}^{2}(t)\leqslant \frac{C_{8}}{2} \bigl( \bigl\Vert (\varrho , u) (0) \bigr\Vert ^{2}_{H^{2} \cap L^{1}}+ \epsilon ^{2}_{0}\mathcal{G}^{2}(t) +\epsilon _{0}^{2-2 \vartheta}\mathcal{G}^{2+2\vartheta}(t) \bigr), \end{aligned}$$

(4.21)

where $C_{8}$ is independent of $\epsilon _{0}$.

For the last term on the right-hand side of (4.21), by Young’s inequality, we obtain

$$\begin{aligned} C_{8}\epsilon _{0}^{2-2\vartheta} \mathcal{G}^{2+2\vartheta}(t) \leqslant \frac{1-\vartheta}{2} C^{\frac{2}{1-\vartheta}}_{8} + \frac{1+\vartheta}{2} \epsilon _{0}^{\frac{1-\vartheta}{1+\vartheta}} \mathcal{G}^{4}(t). \end{aligned}$$

(4.22)

From (4.21)–(4.22), we have

$$\begin{aligned} \mathcal{G}^{2}(t)\leqslant \mathcal{I}_{0}+C_{\epsilon _{0}} \mathcal{G}^{4}(t), \end{aligned}$$

where

$$\begin{aligned} C_{\epsilon _{0}} :=\frac{1+\vartheta}{2}\epsilon _{0}^{ \frac{4(1-\vartheta )}{1+\vartheta}} \end{aligned}$$

and

$$\begin{aligned} \mathcal{I}_{0} :=C_{8} \bigl\Vert (\varrho , u) (0) \bigr\Vert ^{2}_{H^{2}\cap L^{1}}+ \frac{1-\vartheta}{2} C_{8}^{\frac{2}{1-\vartheta}}. \end{aligned}$$

Now we prove that $\mathcal{G}(t)$ is a bounded function by contradiction. Suppose $\mathcal{G}^{2}(t)>2\mathcal{I}_{0}$ for any $t \geqslant t_{1}$ with a constant $t_{1}>0$. Noting that $\mathcal{G}(t)\in C^{0}[0, +\infty )$ and $\mathcal{G}^{2}(0)$ is small, we have

$$\begin{aligned} \mathcal{G}^{2}(t_{0})=2 \mathcal{I}_{0} \end{aligned}$$

(4.23)

with some $t_{0}\in (0, t_{1})$. Moreover, from (4.23), we have

$$\begin{aligned} \mathcal{G}^{2}(t_{0})\leqslant \mathcal{I}_{0}+C_{\epsilon _{0}} \mathcal{G}^{4}(t_{0}), \end{aligned}$$

which implies

$$\begin{aligned} \mathcal{G}^{2}(t_{0}) \leqslant \frac{\mathcal{I}_{0}}{1-C_{\epsilon _{0}}\mathcal{G}^{2}(t_{0})}. \end{aligned}$$

(4.24)

Assume $\epsilon _{0}$ is a small constant such that $C_{\epsilon _{0}}< \frac{1}{4\mathcal{I}_{0}}$, which leads to $C_{\epsilon _{0}}\mathcal{G}^{2}(t_{0})<\frac{1}{2}$. This fact together with (4.24) implies

$$\begin{aligned} \mathcal{G}^{2}(t_{0})< 2 \mathcal{I}_{0}. \end{aligned}$$

(4.25)

Clearly, (4.25) contradicts (4.23). Therefore, one always gets $\mathcal{G}^{2}(t)\leqslant 2\mathcal{I}_{0}$ for any $t\geqslant t_{1}$. Keeping in mind that $\mathcal{G}(t)$ is non-decreasing, we can deduce $\mathcal{G}(t)\leqslant C$ for any $t\in [0, +\infty )$. This completes the proof. □

5 Conclusion

In this paper, we investigated the time decay properties of solutions for compressible non-Newtonian fluid equations. More specifically, we investigated the long-time behavior of the Cauchy problem for Eills-type three-dimensional isentropic compressible fluids by the well-posedness result for the non-Newtonian fluid equations in [1], as well as the long-time behavior result for the Navier–Stokes system in [2]. Li–Zhang investigated the long-time behavior of Newtonian fluids; we further investigated the long-time behavior of Eills-type non-Newtonian fluids.