Space–Time Decay Rate of the Compressible Adiabatic Flow Through Porous Media in ${\mathbb {R}}^3$

Abstract

We are concerned with the space–time decay rate of the compressible adiabatic flow through porous media in ${\mathbb {R}}^3$. Based on the previous temporal decay results for this system, it is shown that the space–time decay rate of the pressure and velocity (p, u) in the weighted Lebesgue space $L_\gamma ^2$ is $t^{-\frac{3}{4}+\frac{\gamma }{2}}$, and the space–time decay rate of $k(1\le k \le 3)$th-order spatial derivative of the pressure and velocity $\nabla ^k(p,u)$ in the weighted Lebesgue space $L_\gamma ^2$ is $t^{-\frac{5}{4}+\frac{\gamma }{2}}$. Our methods mainly involve delicate weighted energy estimates and interpolation trick.

1 Introduction and Main Results

In this paper, we investigate the space–time decay rate of the 3D compressible adiabatic flow through porous media. The system takes the following form

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t \rho +{\text {div}}(\rho u)=0, \\ \partial _t(\rho u)+{\text {div}}(\rho u \otimes u)+\nabla p=-\alpha \rho u, \\ \partial _t\left( \rho \left( e+\frac{1}{2} u^2\right) \right) +{\text {div}}\left( \rho u\left( e+\frac{1}{2} u^2\right) +u p\right) =-\alpha \rho u^2, \end{array}\right. \end{aligned}$$

(1.1)

where $t \ge 0$, $x \in {\mathbb {R}}^3$. The functions $\rho (t, x)$ denote density, $u(t, x)=\left( u_1, u_2, u_3\right)$ represents velocity, p is pressure, and e is specific internal energy per unit mass. The constant $\alpha >0$ models friction coefficient. For simplicity, as in [34], we only consider polytropic fluids. This means that the equations of state are $p=R \rho \theta$ and $e=C_v \theta$, where $R>0, C_v>0$ are the universal gas constant and the specific heat at constant volume, respectively.

1.1 History of the Problem

When the entropy $s=$ constant, the system (1.1) reduces into the following compressible Euler equations with damping

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t \rho +{\text {div}}(\rho u)=0, \\ \partial _t(\rho u)+{\text {div}}(\rho u \otimes u)+\nabla p=-\alpha \rho u. \end{array}\right. \end{aligned}$$

(1.2)

We review some closely related results of the system. Nishida [24, 25] proved the global existence of a smooth solution with small initial data in 1-D case, Wang–Yang [32] established the global existence and the pointwise estimates of the solution with small data in 3-D case, and Fang–Xu [5] obtained the existence and asymptotic behavior of $C^1$ solutions on the framework of Besov space. Jang et al. studied the well-posedness for compressible Euler equations with physical vacuum singularity in [15, 16]. The convergence rates for Cauchy problems in both 1-D and multi-D were deeply studied, and we refer to [6, 7, 12,13,14, 20, 26, 27, 37, 42] and the references therein. If the initial data are small in $L^1$, Sideris–Thomases–Wang [29] showed that the $L^2$-norm of the solution decays at the rate $(1+t)^{-3 / 4}$. However, Tan and Wu [31] improve the $L^2$-decay rate into $(1+t)^{-3 / 4-s / 2}$ provided that the initial data are small in $\dot{B}_{1, \infty }^{-s}$ with $s\in [0,1]$. Later, Chen and Tan [1] removed the smallness of those low frequency assumption of the initial data and showed the optimal decay rates of the higher-order spatial derivatives. Recently, based on the temporal decay results of [1], for any integer $\ell \ge 3$, Kim [18] obtained that the space–time decay rate of $k(0 \le k \le \ell -2)$th-order spatial derivative of solution to the 3D compressible Euler equations with damping (1.2) in weighted Lebesgue space $L_\gamma ^2$ is $t^{-\frac{2k+3}{4}+\frac{\gamma }{2}}$. And we also refer the interested readers to [4, 10, 17, 28, 32, 36] for the initial boundary value problem and to [2, 3, 21] for the half space problem.

When $s \ne$ constant in (1.1), it is the adiabatic flow, which is more complex due to a non-decaying component in linear level which causes great troubles in nonlinear couplings. As in [34], we consider the variables (p, u, s); that is, the equation of state for the gas is given by

$$\begin{aligned} \rho =\rho (p, s)=\kappa p \frac{c_v}{\frac{c}{c_v+R}} e^{-\frac{s}{c_v+R}}, \end{aligned}$$

(1.3)

where $\kappa$ is a constant. Under the aforementioned assumptions, the system (1.1) can be rewritten as

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t p+\frac{R+c_v}{c_v} p {\text {div}} u+u \cdot \nabla p=0, \\ \partial _t u+(u \cdot \nabla ) u+\frac{\nabla p}{\rho }=-\alpha u, \\ \partial _t s+(u \cdot \nabla ) s=0, \end{array}\right. \end{aligned}$$

(1.4)

where $\rho = \rho (p, s)$ is defined by (1.3). Notice that (1.4) is a hyperbolic system, where the dissipation comes from damping. We consider the initial value problem to (1.4) in the whole space with the initial data

$$\begin{aligned} \left( p(t, x), u(t, x), s(t, x)\right) |_{t=0}=\left( p_0(x), u_0(x), s_0(x)\right) \rightarrow \left( p_{\infty }, 0, s_{\infty }\right) \text{ as } |x|\rightarrow \infty ,\nonumber \\ \end{aligned}$$

(1.5)

where $p_{\infty }>0$ and $s_{\infty }$ are given constants. For one-dimensional version of the system (1.4), Hsiao et al. established the global existence of smooth solution to Cauchy problem with small data in [11, 41], studied the large-time behavior of these solutions later in [8, 22], and investigated the initial boundary value problem in [9]. For multi-dimensional problem of the system (1.4), under the assumptions that the initial perturbation $\left( p_0-p_{\infty }, u_0, s_0\right)$ is small in $H^3\left( {\mathbb {R}}^3\right)$ and $\left( p_0-p_{\infty }, u_0\right)$ is bounded in $L^1\left( {\mathbb {R}}^3\right)$, Wu–Tan–Huang [34] obtained the global existence and the decay estimates of solution as follows,

$$\begin{aligned} \begin{aligned}&\left\| \left( p-p_{\infty }\right) (t)\right\| _{L^2} \leqslant C_1(1+t)^{-\frac{3}{4}},\\&\left\| \nabla \left( p-p_{\infty }\right) (t)\right\| _{H^2}+\Vert u\Vert _{H^3} \leqslant C_1(1+t)^{-\frac{5}{4}}, \\&\left\| \partial _t(p, u, s)(t)\right\| _{L^2} \leqslant C_1(1+t)^{-\frac{5}{4}}. \end{aligned} \end{aligned}$$

(1.6)

Recently, Wu and Miao [35] removed the assumption that $\left( p_0-p_{\infty }, u_0\right)$ is bounded in $L^1\left( {\mathbb {R}}^3\right)$ and then obtained the $L^2-L^2$-norm decay rate of the solution.

However, to the best of our knowledge, up to now, there is no result on the space–time decay rate of the compressible adiabatic flow through porous media (1.4). The main motivation of this paper is to give a definite answer to this issue. More precisely, we will establish the space–time decay rates of the $k(0\le k\le 3)$-order derivative of strong solution to the Cauchy problem (1.4)–(1.5) in the Lebesgue space $L_\gamma ^2$.

1.2 Notations

C, $C^*$ or $C_i$ are time independent constants, which may vary at different places. $L^{p}$ and $H^{\ell }$ denote the usual Lebesgue space $L^{p}\left( {\mathbb {R}}^{3}\right)$ and Sobolev spaces $H^{\ell }\left( {\mathbb {R}}^{3}\right) =W^{\ell , 2}\left( {\mathbb {R}}^{3}\right)$ with norms $\Vert \cdot \Vert _{L^{p}}$ and $\Vert \cdot \Vert _{H^{\ell }}$, respectively. We denote $\Vert (f, g)\Vert _{X}:=\Vert f\Vert _{X}+\Vert g\Vert _{X}$ for simplicity. The notation $f \lesssim g$ means that $f \le C g$. We often drop x-dependence of differential operators; that is, $\nabla f=\nabla _{x} f=\left( \partial _{x_{1}} f, \partial _{x_{2}} f, \partial _{x_{3}} f\right)$ and $\nabla ^{k}$ denotes any partial derivative $\partial ^{\alpha }$ with multi-index $\alpha$, $|\alpha |=k$. For any $\gamma \in {\mathbb {R}}$, denote the weighted Lebesgue space by $L_{\gamma }^{p}\left( {\mathbb {R}}^{3}\right) (2 \le p<+\infty )$ with respect to the spatial variables as follows,

$$\begin{aligned} L_{\gamma }^{p}\left( {\mathbb {R}}^{3}\right) :=\left\{ f(x): {\mathbb {R}}^{3} \rightarrow {\mathbb {R}}, \Vert f\Vert _{L_{\gamma }^{p}\left( {\mathbb {R}}^{3}\right) }^{p}:=\int _{{\mathbb {R}}^{3}}|x|^{p \gamma }|f(x)|^{p} d x<+\infty \right\} . \end{aligned}$$

Then, we can define the weighted Sobolev space as follows,

$$\begin{aligned} H_{\gamma }^{s}\left( {\mathbb {R}}^{3}\right) :=\left\{ f \in L_{\gamma }^{2}\left( {\mathbb {R}}^{3}\right) \mid \Vert f\Vert _{H_{\gamma }^{s}\left( {\mathbb {R}}^{3}\right) }^{2}:=\sum _{k \le s}\left\| \nabla ^{k} u\right\| _{L_{\gamma }^{2}\left( {\mathbb {R}}^{3}\right) }^{2}<+\infty \right\} . \end{aligned}$$

1.3 Main Results

Based on the temporal decay results (1.6), we investigate the space–time decay rates of strong solution in the weighted Lebesgue space $L_\gamma ^2$ as follows.

Theorem 1.1

Let (p, u, s) be the strong solution to the system (1.4)–(1.5) with initial data $\left( p_{0}-p_{\infty }, u_{0}, s_{0}-s_{\infty }\right) \in H^{3}\left( {\mathbb {R}}^{3}\right)$, where $\left\| \left( p_0-p_{\infty }, u_0, s_0-s_{\infty }\right) \right\| _{H^3}$ is sufficiently small, and $\left( p_{0}-p_{\infty }, u_{0}\right) \in L^{1}\left( {\mathbb {R}}^{3}\right) \bigcap H_\gamma ^{3}\left( {\mathbb {R}}^{3}\right)$, then there exists a large enough T such that

$$\begin{aligned}&\left\| (p,u)(t)\right\| _{L^{2}_{\gamma }} \lesssim t^{-\frac{3}{4}+\frac{\gamma }{2}},\end{aligned}$$

(1.7)

$$\begin{aligned}&\left\| \nabla ^k(p,u)(t)\right\| _{L^{2}_{\gamma }} \lesssim t^{-\frac{5}{4}+\frac{\gamma }{2}}, \end{aligned}$$

(1.8)

for all $1\le k \le 3$, $t>T$ and $\gamma \ge 0$.

Remark 1.2

Applying the Gagliardo–Nirenberg–Sobolev inequality and the weighted interpolation inequality, we can obtain the space–time decay rates of smooth solution in weighted Lebesgue space $L_\gamma ^q(2 \le q \le \infty , \gamma \ge 0)$ as follows. For $\gamma \ge 2$, applying the Gagliardo–Nirenberg–Sobolev inequality, (1.7) and (1.8) we have

$$\begin{aligned} \Vert |x|^\gamma (p,u) \Vert _{L^{\infty }}&\lesssim \Vert |x|^\gamma (p,u) \Vert _{L^{2}}^\frac{1}{4}\Vert \nabla ^2(|x|^\gamma (p,u)) \Vert _{L^{2}}^\frac{3}{4}\\&\lesssim \Vert (p,u) \Vert _{L_\gamma ^2}^\frac{1}{4} \left( \left\| \nabla ^2 (p,u)\right\| _{L_\gamma ^2}+\left\| \nabla (p,u)\right\| _{L_{\gamma -1}^2}+\Vert (p,u)\Vert _{L_{\gamma -2}^2}\right) ^\frac{3}{4}\\&\lesssim t^{-\frac{9}{8}+\frac{\gamma }{2}}. \end{aligned}$$

Using the weighted interpolation inequality, one has

$$\begin{aligned} \Vert (p,u) \Vert _{L_\gamma ^q}&\lesssim \Vert (p,u) \Vert _{L_\gamma ^2}^{\theta }\Vert |x|^\gamma (p,u) \Vert _{L^\infty }^{1-\theta }\\&\lesssim t^{-\frac{3}{4}(\frac{3}{2}-\frac{1}{q})+\frac{\gamma }{2}}. \end{aligned}$$

And the conclusion for the case of $\gamma \in [0,2)$ is proved by weighted interpolation inequality. Using the weighted interpolation inequality with $\Vert \nabla (p,u) \Vert _{L_\gamma ^2} \lesssim t^{-\frac{5}{4}+\frac{\gamma }{2}}$ and $\Vert |x|^\gamma \nabla (p,u) \Vert _{L^\infty } \lesssim t^{-\frac{5}{4}+\frac{\gamma }{2}}$, we have

$$\begin{aligned} \Vert \nabla (p,u) \Vert _{L_\gamma ^q} \lesssim t^{-\frac{5}{4}+\frac{\gamma }{2}}. \end{aligned}$$

Due to the non-dissipation property of the entropy s, we can only obtain $\left\| \nabla ^k\left( p,u\right) \right\| _{L^2}$$\leqslant C(1+t)^{-\frac{5}{4}}$ $(k=1,2,3)$, please refer to [34] for more details. Thus, we can only establish $\Vert |x|^\gamma \nabla ^k(p,u) \Vert _{L^\infty } \lesssim t^{-\frac{5}{4}+\frac{\gamma }{2}}(k=0,1)$. Then, we can only obtain $\Vert \nabla ^k(p,u) \Vert _{L_\gamma ^q}(k=0,1)$ by the weighted interpolation inequality.

Remark 1.3

The space–time decay rates of $L_{\gamma }^{2}$-norm of the 0th-order and 1th-order spatial derivatives for the solution are $(1 + t)^{-\frac{3}{4}+\frac{\gamma }{2}}$ and $(1 + t)^{-\frac{5}{4}+\frac{\gamma }{2}}$, respectively, which coincide with those of the heat equation. Thus, the space–time decay rates of $L_{\gamma }^{2}$-norm of the 0th-order and 1th-order spatial derivatives are optimal in this sense. However, due to the non-dissipation property of the entropy s, we can only show that the space–time decay rates of the 2th-order and 3th-order spatial derivatives of the solution are $(1 + t)^{-\frac{5}{4}+\frac{\gamma }{2}}$, which are slower than those of the heat equation.

Remark 1.4

It is interesting to compare our methods with ones of incompressible flows [19, 33, 38,39,40]. Using the parabolic interpolation inequality and some weighted estimates, Kukavica [19] obtained the sharp decay rate for the strong solution to the incompressible Navier–Stokes equation, and Weng [33] and Zhao [38,39,40] addressed the space–time decay properties for higher-order spatial derivatives of strong solutions to the incompressible viscous resistive Hall-MHD equations, viscous Boussinesq system, and nematic liquid crystal system, respectively. However, for the 3D compressible adiabatic flow through porous media (1.4), the new difficulty comes from the fact that there is no dissipation of the entropy s, and the pressure p depends on s. To overcome the difficulty, we make full use of the dissipation structure of the system (2.1) and employ the interpolation trick and weighted energy methods. We refer to (1.11) and its estimates of $K_1-K_3$ for more details.

Remark 1.5

Compared to Kim [18] where the $H^\ell (\ell \ge 3)$-norm space–time estimate of $k(0\le k \le \ell -2)$-order spatial derivative of solution to the compressible Euler equations with damping in ${\mathbb {R}}^3$ is established, we need to develop new ingredients in the proof to overcome the difficulty arising from the appearance of the non-isentropic term. We refer the readers to the proofs of (3.9), (3.64)–(3.68) and (3.69) for details.

Now, let us outline the strategies of proving Theorem 1.1 and explain the main difficulties in the process. We use the delicate weighted energy estimates to obtain that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t} \textrm{E}(t) \le C_{0} t^{-\frac{5}{4}} \textrm{E}(t)+C_{1} t^{-\frac{3}{2 \gamma }} \textrm{E}(t)^{\frac{\gamma -1}{\gamma }}, \end{aligned}$$

(1.9)

where $\textrm{E}(t)=\left\| (p,u) \right\| _{L_\gamma ^2}^2$, and

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t}\widehat{ \textrm{E}}(t) \le C_{0} t^{-\frac{5}{4}}\widehat{\textrm{E}}(t)+C_{1} t^{-\frac{5}{2 \gamma }} \widehat{\textrm{E}}(t)^{\frac{\gamma -1}{\gamma }}+C_3 t^{-\frac{7}{2}+\gamma }, \end{aligned}$$

(1.10)

where $\widehat{\textrm{E}}(t)=\left\| \nabla ^{k}(p,u) \right\| _{L_\gamma ^2}^2(k=1,2,3)$, $C_i>0(i=0,1,3)$. Applying Lemma 2.5 to (1.9)–(1.10) and interpolation trick, we can prove Theorem 1.1. The two main difficulties in the process can be outlined as follows. The first difficulty is that Lemma 2.2 does not work in Weighted Lebesgue space $L_\gamma ^2$, and the second difficulty is that there is no dissipative estimate for p in (3.4) and (3.19), where the relationship between p and the entropy s can be seen in (1.3). To overcome these difficulties, we fully use the structure of (2.1) to reduce the order of spatial derivative of solution and make delicate weighted energy estimates. For the sake of simplicity, we only take the trouble term $\left\langle |x|^{2 \gamma }\nabla ^{3} p, \nabla ^{3}(p{\text {div}} u)\right\rangle$ as an example. First of all, by fully using the structure of (2.1)$_1$ to obtain an equation of ${\text {div}} u$ given in (3.21), and substituting (3.21) into this term, one has

$$\begin{aligned} -\left\langle |x|^{2 \gamma }\nabla ^{3} p, \nabla ^{3}(p{\text {div}} u)\right\rangle&=\left\langle |x|^{2 \gamma }\nabla ^{3} p, \nabla ^{3}\left( p\frac{p_t+\kappa _1 u \cdot \nabla p}{\kappa _2-\kappa _3 p}\right) \right\rangle \nonumber \\&=\left\langle |x|^{2 \gamma }\nabla ^{3} p,\left( \frac{p}{\kappa _2-\kappa _3 p}\right) u\nabla ^{4}p\right\rangle \nonumber \\&\quad +\left\langle |x|^{2 \gamma }\nabla ^3 p, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \nabla ^3 p_t\right\rangle \nonumber \\&\quad +\left\langle |x|^{2 \gamma }\nabla ^3 p, \nabla ^3\left( \frac{p}{\kappa _2-\kappa _3 p}\right) \nabla u \right\rangle +\text {good terms} \nonumber \\&:=K_1+K_2+K_3+\text {good terms}. \end{aligned}$$

(1.11)

Next, we will focus on the main trouble terms $K_{j}(j=1,2,3)$. For the term $K_1$, by employing integration by parts, it holds that

$$\begin{aligned} \left\langle |x|^{2 \gamma }\nabla ^{3} p, u\nabla ^{4}p\right\rangle =-\frac{1}{2}\left\langle \nabla (|x|^{2 \gamma } u), |\nabla ^{3}p|^2\right\rangle . \end{aligned}$$

(1.12)

For the term $K_2$, we have

$$\begin{aligned}&\left\langle |x|^{2 \gamma }\nabla ^3 p, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \nabla ^3 p_t\right\rangle \nonumber \\&\quad =\frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\left\langle |x|^{2 \gamma }|\nabla ^3 p|^2, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\rangle -\frac{1}{2}\left\langle |x|^{2 \gamma }|\nabla ^3 p|^2, \frac{\textrm{d}}{\textrm{d}t}\left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\rangle , \end{aligned}$$

(1.13)

where $\left\| \nabla ^3p\right\| _{L_{\gamma }^{2}}^{2}-\left\langle |x|^{2 \gamma }|\nabla ^3 p|^2, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\rangle$ is equivalent to $\left\| \nabla ^3p\right\| _{L_{\gamma }^{2}}^{2}$, since $\left\| p_0 \right.$$\left. -p_{\infty }\right\| _{H^3}$ is sufficiently small, and the fact that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\left( \frac{p}{\kappa _2-\kappa _3 p}\right) =-\frac{\kappa _2\left[ \left( \kappa _2-\kappa _3 p\right) \nabla u+ \kappa _1 u\cdot \nabla p\right] }{(\kappa _2-\kappa _3 p)^2}. \end{aligned}$$

Noticing that Lemma 2.2 does not work in $L_\gamma ^2$, for the term $K_3$, we need employ some new idea. The key observation here is to use the Hölder’s inequality skillfully to get

$$\begin{aligned} \left\| |x|^{2\gamma }\nabla ^3 p\nabla ^3\left( \frac{p }{\kappa _2-\kappa _3 p}\right) \nabla u \right\| _{L^1}&\lesssim \left\| \nabla ^3 p\right\| _{L_\gamma ^2}\left\| \nabla ^3\left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\| _{L^2}\left\| |x|^{\gamma }\nabla u\right\| _{L^\infty }\\ {}&\lesssim \left\| \nabla ^3 p\right\| _{L_\gamma ^2}\left\| \nabla ^3 p \right\| _{L^2}\left\| |x|^{\gamma }\nabla u\right\| _{L^\infty },\nonumber \end{aligned}$$

(1.14)

where

$$\begin{aligned}&\left\| |x|^{\gamma }\nabla u\right\| _{L^{\infty }}\nonumber \\&\quad \lesssim \left( \left\| \nabla ^3u\right\| _{L_\gamma ^2}+\left\| \nabla ^2 u\right\| _{L_{\gamma -1}^2}+\Vert \nabla u\Vert _{L_{\gamma -2}^2}+\left\| \nabla ^2 u\right\| _{L_\gamma ^2}+\Vert \nabla u\Vert _{L_{\gamma -1}^2}\right) \text {(see (3.63))}. \nonumber \end{aligned}$$

(2.1)

With (1.12)–(1.14) in hand, we can bound the trouble terms $K_1$-$K_3$ properly.

The paper will be organized as follows. In Sect. 2, we rewrite the Cauchy problem (1.4)–(1.5) and present some lemmas, which are used frequently throughout this paper. In Sect. 3, using the weighted energy methods, Lemma 2.5 and interpolation trick, we prove Theorem 1.1.

2 Reformulation and Preliminaries

2.1 Reformulation

In this section, we reformulate the Cauchy problem (1.4)–(1.5) as follows. Set

$$\begin{aligned} \kappa _1=\sqrt{\frac{c_v}{\left( R+c_v\right) \rho _{\infty } p_{\infty }}}, \quad \kappa _2=\sqrt{\frac{\left( R+c_v\right) p_{\infty }}{c_v \rho _{\infty }}}, \end{aligned}$$

where $\rho _{\infty }=\rho \left( p_{\infty }, s_{\infty }\right)$. Taking change of variables by $(p, u, s) \rightarrow \left( p+p_{\infty }, \kappa _1 u, \right.$$\left. s+s_{\infty }\right)$, the Cauchy problem(1.4)–(1.5) can be rewritten as

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t p+\kappa _2 {\text {div}} u=-\frac{\left( R+c_v\right) \kappa _1}{c_v} p {\text {div}} u-\kappa _1 u \cdot \nabla p:=\kappa _3 p {\text {div}} u-\kappa _1 u \cdot \nabla p, \\ \partial _t u+\kappa _2 \nabla p+\alpha u=-\kappa _1(u \cdot \nabla ) u-\frac{1}{\kappa _1}\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) \nabla p, \\ \partial _t s+\kappa _1(u \cdot \nabla ) s=0, \\ \left( p, u, s\right) |_{t=0}=\left( p_0, u_0, s_0\right) \rightarrow (0,0,0) \text{ as } |x|\rightarrow \infty . \end{array}\right. \end{aligned}$$

(2.2)

Here and after, we still denote the reformulated variables by (p, u, s) without confusion.

Following are several useful tools, which will be frequently used in the whole article.

Lemma 2.1

(Gagliardo–Nirenberg inequality) Let $0 \le i, j \le k$, then

$$\begin{aligned} \left\| \nabla ^{i} f\right\| _{L^{p}} \lesssim \left\| \nabla ^{j} f\right\| _{L^{q}}^{1-a}\left\| \nabla ^{k} f\right\| _{L^{r}}^{a}, \end{aligned}$$

where a satisfies

$$\begin{aligned} \frac{i}{3}-\frac{1}{p}=\left( \frac{j}{3}-\frac{1}{q}\right) (1-a)+\left( \frac{k}{3}-\frac{1}{r}\right) a. \end{aligned}$$

Particularly, when $p=3,q=r=2,i=j=0,k=1$, combining Cauchy’s inequality, we have

$$\begin{aligned} \left\| f\right\| _{L^{3}} \lesssim \left\| f\right\| _{L^{2}}^{\frac{1}{2}}\left\| \nabla f\right\| _{L^{2}}^{\frac{1}{2}} \lesssim \left\| f\right\| _{H^{1}}, \end{aligned}$$

(2.3)

and when $p=\infty ,q=r=2,i=0,j=1,k=2$, combining Cauchy’s inequality, we have

$$\begin{aligned} \left\| f\right\| _{L^{\infty }} \lesssim \left\| \nabla f\right\| _{L^{2}}^{\frac{1}{2}}\left\| \nabla ^2 f\right\| _{L^{2}}^{\frac{1}{2}} \lesssim \left\| \nabla f\right\| _{H^{1}}, \end{aligned}$$

(2.4)

while $i=j=0,k=1,a=1,p=q=r=2$ and using Minkowski’s inequality, we have

$$\begin{aligned} \Vert |x|^{\gamma }f\Vert _{L^{6}} \lesssim \left\| \nabla \left( |x|^{\gamma } f\right) \right\| _{L^{2}} \lesssim \left( \Vert \nabla f\Vert _{L_{\gamma }^{2}}+\Vert f\Vert _{L_{\gamma -1}^{2}}\right) . \end{aligned}$$

(2.5)

Proof

This is a special case of [23] and some inequalities based on our needs.

Lemma 2.2

Assume that the function f(m) satisfies

$$\begin{aligned} f(m) \sim m \text{ and } \left\| f^{(k)}(m)\right\| \le C_k \text{ for } \text{ any } k \ge 1, \end{aligned}$$

then for any integer $k \ge 0$ and $p \ge 2$, we have

$$\begin{aligned} \left\| \nabla ^k f(m)\right\| _{L^p} \le C_k\left\| \nabla ^k m\right\| _{L^p}, \end{aligned}$$

where $C_k$ is a constant independent of t. Particularly, in this paper, we have the fact that $\frac{1}{\rho }-\frac{1}{\rho _{\infty }} \sim {\mathcal {O}}(1)(p+s)$ from (1.3), and $\frac{p}{\kappa _2-\kappa _3 p}\sim {\mathcal {O}}(1)(p)$.

Proof

Refer to Lemma A.4 of [1] for $p = 2$ and Lemma 2.2 of [30] for $p \ge 2$.

Lemma 2.3

The vector function $f\in C_{0}^{\infty }\left( {\mathbb {R}}^{\textrm{3}}\right)$ and bounded scalar function g such that

$$\begin{aligned} \left| \int _{{\mathbb {R}}^{3}}\left( \nabla \left| x\right| ^{2\gamma }\right) \cdot fg \mathrm {~d} x\right| \lesssim \Vert g\Vert _{L_{\gamma }^{2}}\Vert f\Vert _{L_{\gamma -1}^{2}}. \end{aligned}$$

Proof

The left side of the above inequality can be rewritten as

$$\begin{aligned} \left| 2 \gamma \int _{{\mathbb {R}}^{3}}|x|^{2 \gamma -2} x_{j} \partial _{i} x_{j}g f_{i} d x\right| . \end{aligned}$$

Using Hölder’s inequality, we have

$$\begin{aligned} \left| \int _{{\mathbb {R}}^{3}}\left( \nabla \left| x\right| ^{2\gamma }\right) \cdot f g\mathrm {~d} x\right| \lesssim \Vert g\Vert _{L_{\gamma }^{2}}\Vert f\Vert _{L_{\gamma -1}^{2}}. \end{aligned}$$

Lemma 2.4

(Weighted interpolation inequality) If $p, r \geqslant 1,s+n / r,\alpha +n /p, \beta +n / q>0,$ and $0 \leqslant \theta \leqslant 1$, then

$$\begin{aligned} \left\| f\right\| _{L_{s}^{r}} \le \left\| f\right\| _{L_{\alpha }^{p}}^{\theta }\left\| f\right\| _{L_{\beta }^{q}}^{1-\theta }, \end{aligned}$$

for $f \in C_{0}^{\infty }\left( {\mathbb {R}}^{\textrm{n}}\right)$ provided that

$$\begin{aligned} \frac{1}{r}=\frac{\theta }{p}+\frac{1-\theta }{q}, \end{aligned}$$

and

$$\begin{aligned} s=\theta \alpha +(1-\theta ) \beta . \end{aligned}$$

Particularly, while $s=p=q=2,\theta =\frac{\gamma -1}{\gamma },s=\gamma -1,\alpha =\gamma ,\beta =0$, we have

$$\begin{aligned} \Vert f\Vert _{L_{\gamma -1}^{2}} \le \Vert f\Vert _{L_{\gamma }^{2}}^{\frac{\gamma -1}{\gamma }}\Vert f\Vert _{L^{2}}^{\frac{1}{\gamma }}. \end{aligned}$$

(3.1)

Proof

We compute

$$\begin{aligned} \int _{U}|x|^{sr}|f|^{r} \textrm{d} x= & {} \int _{U}|x|^{\alpha \theta r}|f|^{\theta r}|x|^{\beta (1-\theta ) r}|f|^{(1-\theta ) r} \textrm{d} x \\\le & {} \left( \int _{U}\left( |x|^{\alpha \theta r}|f|^{\theta r }\right) ^{\frac{p}{\theta r}} \textrm{d} x\right) ^{\frac{\theta r}{p}}\\{} & {} \quad \left( \int _{U}\left( |x|^{\beta (1-\theta )r}|f|^{(1-\theta )r }\right) ^{\frac{q}{(1-\theta ) r}} \textrm{d} x\right) ^{\frac{(1-\theta )r}{q}}. \end{aligned}$$

Thus, we complete the proof of Lemma 2.4. $\square$

Lemma 2.5

(Gronwall-type Lemma) Let $\alpha _{0}>1, \alpha _{1}<1, \alpha _{2}<1$, and $\beta _{1}<1, \beta _{2}<1$. Assume that a continuously differential function $F:[1, \infty ) \rightarrow [0, \infty )$ satisfies

$$\begin{aligned} \begin{aligned}&\frac{\textrm{d}}{\textrm{d} t} F(t) \le C_{0} t^{-\alpha _{0}} F(t)+C_{1} t^{-\alpha _{1}} F(t)^{\beta _{1}}+C_{2} t^{-\alpha _{2}} F(t)^{\beta _{2}}+C_{3} t^{\gamma _{1}-1}, t \ge 1 \\&F(1) \le K_{0}, \end{aligned} \end{aligned}$$

where $C_{0}, C_{1}, C_{2}, C_{3}, K_{0}\ge 0$ and $\gamma _{i}=\frac{1-\alpha _{i}}{1-\beta _{i}}>0$ for $i=1,2$. Assume that $\gamma _{1} \ge \gamma _{2}$, then there exists a constant $C^{*}$ depending on $\alpha _{0}, \alpha _{1}, \beta _{1}, \alpha _{2}, \beta _{2}, K_{0}, C_{i}, i=1,2,3$, such that

$$\begin{aligned} F(t) \le C^{*}t^{\gamma _{1}}, \end{aligned}$$

for all $t \ge 1$.

Proof

We can refer to Lemma 2.1 of [33]. $\square$

3 The proof of Theorem 1.1

In this section, we are going to derive the space–time decay rate of the solution (p, u) under the priori assumption that

$$\begin{aligned} \sup _{0 \le \tau \le t}\Vert (p, u, s)\Vert _{H^3} \le \epsilon , \text{ for } \epsilon >0 \text{ is } \text{ sufficiently } \text{ small. } \end{aligned}$$

(3.2)

According to (1.6), it is clear that there exists a large enough T, such that

$$\begin{aligned} \begin{aligned} \left\| p(t)\right\| _{L^{2}} \lesssim t^{-\frac{3}{4}},\quad \left\| u(t)\right\| _{L^{2}} \lesssim t^{-\frac{5}{4}},\quad \left\| \nabla ^{k}(p, u)(t)\right\| _{H^{1}} \lesssim t^{-\frac{5}{4}}, \end{aligned} \end{aligned}$$

(3.3)

for all $1\le k\le 2,t> T$.

Lemma 3.1

Under the assumptions of Theorem 1.1, then there exists a large enough T such that the solution (p, u) of system (2.1) has the estimate

$$\begin{aligned} \left\| (p,u)(t)\right\| _{{L}_\gamma ^2}&\lesssim t^{-\frac{3}{4}+\frac{\gamma }{2}}, \end{aligned}$$

(3.4)

for all $t> T$ and $\gamma \ge 0$, where C is a positive constant independent of t.

Proof

Multiplying the equations (2.1$_{1}$) and (2.1$_{2}$) by $|x|^{2 \gamma }p,|x|^{2 \gamma }u$, respectively, summing them up, integrating over ${\mathbb {R}}^{3}$, and then using integration by parts to simplify, one has

$$\begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d} t}\left\| (p,u)\right\| _{L_{\gamma }^{2}}^{2} +\alpha \left\| u\right\| _{L_{\gamma }^{2}}^{2} \nonumber \\&\quad = \kappa _2\int _{{\mathbb {R}}^{3}}\nabla \left( |x|^{2 \gamma }\right) \cdot up \mathrm {~d} x+\kappa _3 \int _{{\mathbb {R}}^{3}}|x|^{2 \gamma }|p|^2 {\text {div}}u \mathrm {~d} x-\kappa _1 \int _{{\mathbb {R}}^{3}}|x|^{2 \gamma }pu\cdot \nabla p \mathrm {~d}x \nonumber \\&\quad -\kappa _1 \int _{{\mathbb {R}}^{3}}|x|^{2 \gamma }u(u\cdot \nabla )u \mathrm {~d} x-\frac{1}{\kappa _1}\int _{{\mathbb {R}}^{3}}|x|^{2 \gamma }u \left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) \nabla p\mathrm {~d} x \nonumber \\&\quad :=\sum _{j=1}^{4}I_{j}+\frac{1}{\kappa _1}I_{5}. \end{aligned}$$

(3.5)

Applying Lemma 2.3 and Cauchy’s inequality, we have

$$\begin{aligned} I_{1}\lesssim & {} \left\| \nabla (|x|^{2\gamma } )p u\right\| _{L^{1}}\nonumber \\\lesssim & {} \Vert p\Vert _{L_{\gamma }^{2}}\Vert u\Vert _{L_{\gamma -1}^{2}}\nonumber \\\le & {} \epsilon \Vert u\Vert _{L_{\gamma }^{2}}^2+C(\epsilon )\Vert p\Vert _{L_{\gamma -1}^{2}}^2. \end{aligned}$$

(3.6)

Using Hölder’s inequality, Lemma 2.1 (Gagliardo–Nirenberg inequality) and (3.2), we have

$$\begin{aligned} I_{2}\lesssim & {} \Vert \nabla u\Vert _{L^{\infty }}\Vert p\Vert _{L_{\gamma }^{2}}^2\nonumber \\\lesssim & {} \left\| \nabla ^{2} u\right\| _{H^{1}}\Vert p\Vert _{L_{\gamma }^{2}}^2\nonumber \\\lesssim & {} t^{-\frac{5}{4}}\Vert p\Vert _{L_{\gamma }^{2}}^2. \end{aligned}$$

(3.7)

Applying the same arguments as $I_{2}$ (3.6) and Cauchy’s inequality, we obtain

$$\begin{aligned} I_{3}+I_{4}\lesssim t^{-\frac{5}{4}}\Vert (p,u)\Vert _{L_{\gamma }^{2}}^2. \end{aligned}$$

(3.8)

For $I_{5}$, we need to use the following formula for $\nabla p$ derived from Eq. (2.1)$_2$,

$$\begin{aligned} \nabla p=-\frac{1}{\kappa _2-c_v\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) }\left[ \kappa _1(u\cdot \nabla )u+u_t+au\right] . \end{aligned}$$

(3.9)

Inserting (3.8) into $I_5$, one has

$$\begin{aligned} I_{5}= & {} \int _{{\mathbb {R}}^{3}}|x|^{2 \gamma }u\frac{\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) } \left[ \kappa _1(u\cdot \nabla )u+u_t+au\right] \mathrm {~d} x\nonumber \\= & {} \int _{{\mathbb {R}}^{3}}|x|^{2 \gamma }u \frac{\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }u_t\mathrm {~d} x\nonumber \\{} & {} +\int _{{\mathbb {R}}^{3}}|x|^{2 \gamma }u \frac{\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\left[ \kappa _1(u\cdot \nabla )u+au\right] \mathrm {~d} x\nonumber \\= & {} \frac{1}{2}\frac{d}{dt}\int _{{\mathbb {R}}^{3}}|x|^{2 \gamma }|u|^2 \frac{\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\mathrm {~d} x\nonumber \\{} & {} -\frac{1}{2}\int _{{\mathbb {R}}^{3}}|x|^{2 \gamma }|u|^2 \frac{d}{dt}\left( \frac{\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \mathrm {~d} x\nonumber \\{} & {} +\int _{{\mathbb {R}}^{3}}|x|^{2 \gamma }u \frac{\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\left[ \kappa _1(u\cdot \nabla )u+au\right] \mathrm {~d} x\nonumber \\= & {} \frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\int _{{\mathbb {R}}^{3}}|x|^{2 \gamma }|u|^2 \frac{\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\mathrm {~d} x+\sum _{j=1}^{2 }I{_{5,j}}. \end{aligned}$$

(3.10)

Applying $\rho =\rho (p,s)$ derived from (1.3), $p_t$ derived from (2.1)$_1$, $s_t$ derived from (2.1)$_3$, and the priori estimate assumption (3.1), we have

$$\begin{aligned} I_{5,1}\le \epsilon \left\| u\right\| _{L_{\gamma }^{2}}^2. \end{aligned}$$

(3.11)

Applying Minkowski’s inequality, Hölder’s inequality, Lemmas 2.1, 2.2 (Gagliardo–Nirenberg inequality), Cauchy’s inequality, (3.2), and the priori estimates (3.1), we have

$$\begin{aligned} I_{5,2}&\lesssim \left\| |x|^{2\gamma } u \frac{ \left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\left[ \kappa _1(u\cdot \nabla )u+au\right] \right\| _{L^{1}}\nonumber \\&\lesssim \left\| \frac{ \left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) } \right\| _{L^{\infty }}\left[ \left\| \nabla u\right\| _{L^{\infty }} \left\| u\right\| _{L_{\gamma }^{2}}^2+\left\| u\right\| _{L_{\gamma }^{2}}^2\right] \nonumber \\&\lesssim \left\| \left( p,s\right) \right\| _{L^{\infty }}\left[ \left\| \nabla u\right\| _{L^{\infty }} \left\| u\right\| _{L_{\gamma }^{2}}^2+\left\| u\right\| _{L_{\gamma }^{2}}^2\right] \nonumber \\&\lesssim \Vert \nabla (p,s)\Vert _{H^{1}}\Vert \nabla ^2 u\Vert _{H^{1}}\Vert u\Vert _{L_{\gamma }^{2}}^{2}+\Vert \nabla (p,s)\Vert _{H^{1}}\left\| u\right\| _{L_{\gamma }^{2}}^2\nonumber \\&\le \epsilon t^{-\frac{5}{4}}\left\| u\right\| _{L_{\gamma }^{2}}^2+\epsilon \left\| u\right\| _{L_{\gamma }^{2}}^2\nonumber \\&\le 2\epsilon \left\| u\right\| _{L_{\gamma }^{2}}^2. \end{aligned}$$

(3.12)

Substituting (3.10) and (3.11) into (3.9), it yields

$$\begin{aligned} \begin{aligned} I_{5} \le&\frac{1}{2}\frac{d}{dt}\int _{{\mathbb {R}}^{3}}|x|^{2 \gamma }|u|^2 \frac{\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\mathrm {~d} x +3\epsilon \left\| u\right\| _{L_{\gamma }^{2}}^2. \end{aligned} \end{aligned}$$

(3.13)

Substituting (3.5)–(3.7) and (3.12) into (3.4), and noting that $\epsilon$ is small enough, we have

$$\begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d} t}\left[ \left\| (p,u)\right\| _{L_{\gamma }^{2}}^{2}-\frac{1}{\kappa _1}\int _{{\mathbb {R}}^{3}}|x|^{2 \gamma }|u|^2 \frac{\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\mathrm {~d} x\right] +\frac{\alpha }{2}\left\| u\right\| _{L_{\gamma }^{2}}^{2} \nonumber \\ \lesssim&t^{-\frac{5}{4}}\Vert (p,u)\Vert _{L_{\gamma }^{2}}^2+\Vert p\Vert _{L_{\gamma -1}^{2}}^2. \end{aligned}$$

(3.14)

Defining $H(t)= \left\| (p,u)\right\| _{L_{\gamma }^{2}}^{2}-\frac{1}{\kappa _1}\int _{{\mathbb {R}}^{3}}|x|^{2 \gamma }|u|^2 \frac{\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\mathrm {~d} x$, it is obvious that there exist two positive constants $\overline{C }$ and $\underline{C}$ such that $\underline{C }\left\| (p,u)\right\| _{L_{\gamma }^{2}}^{2}\le H(t) \le \overline{C}\left\| (p,u)\right\| _{L_{\gamma }^{2}}^{2}$. Thus, H(t) is equivalent to $\left\| (p,u)\right\| _{L_{\gamma }^{2}}^{2}$, and (3.13) can be rewritten as

$$\begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d} t}\left\| (p,u)\right\| _{L_{\gamma }^{2}}^{2} +\frac{\alpha }{2} \left\| u\right\| _{L_{\gamma }^{2}}^{2} \nonumber \\ \lesssim&t^{-\frac{5}{4}}\Vert (p,u)\Vert _{L_{\gamma }^{2}}^2+\Vert p\Vert _{L_{\gamma -1}^{2}}^2.\end{aligned}$$

(3.15)

Substituting (2.5) and (3.2) into (3.14), we have

$$\begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d} t}\left\| (p,u)\right\| _{L_{\gamma }^{2}}^{2} +\frac{\alpha }{2}\left\| u\right\| _{L_{\gamma }^{2}}^{2} \nonumber \\ \lesssim&t^{-\frac{5}{4}}\Vert (p,u)\Vert _{L_{\gamma }^{2}}^2 +\Vert (p,u)\Vert _{L_{\gamma }^{2}}^{\frac{2(\gamma -1)}{\gamma }}\Vert (p,u)\Vert _{L^{2}}^{\frac{2}{\gamma }}\nonumber \\ \lesssim&t^{-\frac{5}{4}}\Vert (p,u)\Vert _{L_{\gamma }^{2}}^2 + t^{-\frac{3}{2 \gamma }}\Vert (p,u)\Vert _{L_{\gamma }^{2}}^{\frac{2(\gamma -1)}{\gamma }}. \end{aligned}$$

(3.16)

Denoting $\textrm{E}(t):=\left\| (p,u)\right\| _{L_{\gamma }^{2}}^{2}$, we can obtain

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t} \textrm{E}(t) \le C_{0} t^{-\frac{5}{4}} \textrm{E}(t)+C_{1} t^{-\frac{3}{2 \gamma }} \textrm{E}(t)^{\frac{\gamma -1}{\gamma }}. \end{aligned}$$

If $\gamma >\frac{3}{2}$, then we can apply Lemma 2.5 with $\alpha _{0}=\frac{5}{4}>1,\alpha _{1}=\frac{3}{2\gamma }<1, \beta _{1}=\frac{\gamma -1}{\gamma }<1$, $\gamma _{1}=\frac{1-\alpha _{1}}{1-\beta _{1}}=-\frac{3}{2}+\gamma$. Thus,

$$\begin{aligned} \textrm{E}(t) \le C t^{-\frac{3}{2}+\gamma }, \end{aligned}$$

(3.17)

for all $t > T$. Lemma 3.1 is proved for all $\gamma >\frac{3}{2}$, and the conclusion for the case of $[0,\frac{3}{2}]$ is proved by Lemma 2.4 (weighted interpolation inequality). More precisely, combining (3.2) and (3.16), we have

$$\begin{aligned} \Vert (p,u)(t)\Vert _{L_{\gamma _{0}}^{2}}&\lesssim \Vert (p,u)(t)\Vert _{L^{2}}^{1-\frac{\gamma _{0}}{\gamma }}\Vert (p,u)(t)\Vert _{L_{\gamma }^{2}}^{\frac{\gamma _{0}}{\gamma }} \lesssim t^{-\frac{3}{4}+\frac{\gamma _{0}}{2}}, \end{aligned}$$

(3.18)

for all $t > T$ and $\gamma _{0} \in [0, \gamma ]$, where $[0,\frac{3}{2}] \subset [0, \gamma ](\gamma >\frac{3}{2})$. Thus, we have proved Lemma 3.1, that is the (1.7). $\square$

We prove the space–time decay rate of high-order spatial derivative of solution as follows.

Lemma 3.2

Under the assumptions of Theorem 1.1, then there exists a large enough T such that the solution (p, u) of system (2.1) has the estimate

$$\begin{aligned} \left\| \nabla ^k\left( p, u\right) (t)\right\| _{{L}_\gamma ^2}\lesssim t^{-\frac{5}{4}+\frac{\gamma }{2}}, \end{aligned}$$

(3.19)

for all $1\le k \le 3$, $t> T$ and $\gamma \ge 0$, where C is a positive constant independent of t.

Proof

Applying $\nabla ^{k}$ to each equation of (2.1$_{1}$) and (2.1$_{2}$), then multiplying Eqs. (2.1$_{1}$) and (2.1$_{2}$) by $|x|^{2 \gamma }\nabla ^{k} p$, $|x|^{2 \gamma }\nabla ^{k}u$, respectively, $k=1,2,3$, summing them up, integrating over ${\mathbb {R}}^{3}$, using integration by parts to simplify, one has

$$\begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d} t}\left\| \nabla ^{k}(p,u)\right\| _{L_{\gamma }^{2}}^{2} +\alpha \left\| \nabla ^{k} u\right\| _{L_{\gamma }^{2}}^{2} \nonumber \\&\quad =\kappa _2\int _{{\mathbb {R}}^{3}}\nabla \left( |x|^{2 \gamma }\right) \cdot \nabla ^{k} u \nabla ^{k} p \mathrm {~d} x +\kappa _3\left\langle |x|^{2 \gamma }\nabla ^{k} p, \nabla ^{k}(p{\text {div}} u)\right\rangle \nonumber \\&\qquad -\kappa _1 \left\langle |x|^{2 \gamma } \nabla ^{k} p,\nabla ^{k}( u\cdot \nabla p)\right\rangle -\kappa _1 \left\langle |x|^{2 \gamma }\nabla ^{k} u,\nabla ^{k}\left( u\cdot \nabla u\right) \right\rangle \nonumber \\&\qquad -\frac{1}{\kappa _1}\left\langle |x|^{2 \gamma }\nabla ^{k} u,\nabla ^{k}\left( \left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) \nabla p\right) \right\rangle . \end{aligned}$$

(3.20)

Applying Lemma 2.3 and Cauchy’s inequality to the first term of above (3.19), we have

$$\begin{aligned} \begin{aligned} \kappa _2\int _{{\mathbb {R}}^{3}}\nabla \left( |x|^{2 \gamma }\right) \cdot \nabla ^{k} u \nabla ^{k} p \mathrm {~d} x&\le \kappa _2 \left\| \nabla (|x|^{2\gamma } )\nabla ^k p \nabla ^k u\right\| _{L^{1}}\\&\le \kappa _2\Vert \nabla ^k u\Vert _{L_{\gamma }^{2}}\Vert \nabla ^k p\Vert _{L_{\gamma -1}^{2}}\\&\le \epsilon \kappa _2 \Vert \nabla ^k u\Vert _{L_{\gamma }^{2}}^2+C(\epsilon )\kappa _2 \Vert \nabla ^k p\Vert _{L_{\gamma -1}^{2}}^2. \end{aligned} \end{aligned}$$

(3.21)

To reduce the order of $\nabla ^{k+1}(p,u)$ in (3.19) by integration by parts, we need to use (3.8) and the following formula for ${\text {div}} u$ derived from Eq. (2.1)$_1$,

$$\begin{aligned}&{\text {div}} u=-\frac{p_t+\kappa _1 u \cdot \nabla p}{\kappa _2-\kappa _3 p}. \end{aligned}$$

(3.22)

Inserting (3.20), (3.21) and (3.8) into (3.19), we have

$$\begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d} t}\left\| \nabla ^{k}(p,u)\right\| _{L_{\gamma }^{2}}^{2} +\alpha \left\| \nabla ^{k} u\right\| _{L_{\gamma }^{2}}^{2} \nonumber \\&\quad \le \epsilon \kappa _2 \Vert \nabla ^k u\Vert _{L_{\gamma }^{2}}^2+C(\epsilon )\kappa _2 \Vert \nabla ^k p\Vert _{L_{\gamma -1}^{2}}^2 -\kappa _3\left\langle |x|^{2 \gamma }\nabla ^{k} p, \nabla ^{k}\left( p\frac{p_t+\kappa _1 u \cdot \nabla p}{\kappa _2-\kappa _3 p}\right) \right\rangle \nonumber \\&\qquad -\kappa _1 \left\langle |x|^{2 \gamma } \nabla ^{k} p,\nabla ^{k}( u\cdot \nabla p)\right\rangle -\kappa _1 \left\langle |x|^{2 \gamma }\nabla ^{k} u,\nabla ^{k}\left( u\cdot \nabla u\right) \right\rangle \nonumber \\&\qquad +\frac{1}{\kappa _1}\left\langle |x|^{2 \gamma }\nabla ^{k} u,\nabla ^{k}\left( \frac{ \left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\left[ \kappa _1(u\cdot \nabla )u+u_t+au\right] \right) \right\rangle \nonumber \\&\quad =\epsilon \kappa _2 \Vert \nabla ^k u\Vert _{L_{\gamma }^{2}}^2+C(\epsilon )\kappa _2 \Vert \nabla ^k p\Vert _{L_{\gamma -1}^{2}}^2 -\kappa _3\left\langle |x|^{2 \gamma }\nabla ^{k} p, \nabla ^{k}\left( \frac{pp_t}{\kappa _2-\kappa _3 p}\right) \right\rangle \nonumber \\&\qquad -\kappa _1\kappa _3\left\langle |x|^{2 \gamma }\nabla ^{k} p, \nabla ^{k}\left( \frac{pu \cdot \nabla p}{\kappa _2-\kappa _3 p}\right) \right\rangle -\kappa _1 \left\langle |x|^{2 \gamma } \nabla ^{k} p,\nabla ^{k}( u\cdot \nabla p)\right\rangle \nonumber \\&\qquad -\kappa _1\left\langle |x|^{2 \gamma }\nabla ^{k} u,\nabla ^{k}\left( u\cdot \nabla u\right) \right\rangle +\left\langle |x|^{2 \gamma }\nabla ^{k} u,\nabla ^{k}\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }u\cdot \nabla u\right) \right\rangle \nonumber \\&\qquad +\frac{1}{\kappa _1}\left\langle |x|^{2 \gamma }\nabla ^{k} u,\nabla ^{k}\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }u_t\right) \right\rangle \nonumber \\&\qquad +\frac{a}{\kappa _1}\left\langle |x|^{2 \gamma }\nabla ^{k} u,\nabla ^{k}\left( \frac{\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) }u\right) \right\rangle \nonumber \\&\quad :=\epsilon \kappa _2 \Vert \nabla ^k u\Vert _{L_{\gamma }^{2}}^2+C(\epsilon )\kappa _2 \Vert \nabla ^k p\Vert _{L_{\gamma -1}^{2}}^2+\sum _{j=1}^{7}J_{j}^k, \end{aligned}$$

(3.23)

where $k=1,2,3$, $p_t$ and $u_t$ derived from (3.21) and (3.8), respectively, such that

$$\begin{aligned} p_t&=-\left[ \left( \kappa _2-\kappa _3 p\right) {\text {div}} u+ \kappa _1 u\cdot \nabla p\right] ,\end{aligned}$$

(3.24)

$$\begin{aligned} u_t&=-\left[ \kappa _2\nabla p-c_v\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) \nabla p+\kappa _1(u\cdot \nabla )u+au \right] . \end{aligned}$$

(3.25)

We discuss $J_{j}^k(j=1,2...7)$ by three steps, which correspond to the case of $k=1,2,3$, respectively, as follows.

Step 1 ($L_\gamma ^2$ estimate of $\varvec{\nabla (p,u)}$): Inserting (3.23) into $J_1^1$, we have

$$\begin{aligned} J_1^1= & {} -\kappa _3\left\langle |x|^{2 \gamma }\nabla p, \nabla \left( \frac{pp_t}{\kappa _2-\kappa _3 p}\right) \right\rangle \nonumber \\= & {} -\kappa _3\left\langle |x|^{2 \gamma }\nabla p, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \nabla p_t\right\rangle -\kappa _3\left\langle |x|^{2 \gamma }\nabla p, \nabla \left( \frac{p}{\kappa _2-\kappa _3 p}\right) p_t\right\rangle \nonumber \\= & {} -\frac{\kappa _3}{2}\frac{\textrm{d}}{\textrm{d}t}\left\langle |x|^{2 \gamma }|\nabla p|^2, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\rangle +\frac{\kappa _3}{2}\left\langle |x|^{2 \gamma }|\nabla p|^2, \frac{\textrm{d}}{\textrm{d}t}\left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\rangle \nonumber \\{} & {} -\kappa _3\left\langle |x|^{2 \gamma }\nabla p, \nabla \left( \frac{p}{\kappa _2-\kappa _3 p}\right) p_t\right\rangle \nonumber \\= & {} -\frac{\kappa _3}{2}\frac{\textrm{d}}{\textrm{d}t}\left\langle |x|^{2 \gamma }|\nabla p|^2, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\rangle \nonumber \\{} & {} -\frac{\kappa _2\kappa _3}{2}\left\langle |x|^{2 \gamma }|\nabla p|^2, \left( \frac{ \left( \kappa _2-\kappa _3 p\right) \nabla u+ \kappa _1 u\cdot \nabla p}{(\kappa _2-\kappa _3 p)^2}\right) \right\rangle \nonumber \\{} & {} +\kappa _3\left\langle |x|^{2 \gamma }\nabla p, \nabla \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \left[ \left( \kappa _2-\kappa _3 p\right) \nabla u+ \kappa _1 u\cdot \nabla p\right] \right\rangle \nonumber \\= & {} -\frac{\kappa _3}{2}\frac{\textrm{d}}{\textrm{d}t}\left\langle |x|^{2 \gamma }|\nabla p|^2, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\rangle +\sum _{j=1}^{2}J_{1,j}^1. \end{aligned}$$

(3.26)

Using Minkowski’s inequality, Hölder’s inequality, Lemmas 2.1, 2.2 (Gagliardo–Nirenberg inequality), Cauchy’s inequality and (3.2), we have

$$\begin{aligned} J_{1,1}^1&\lesssim \left\| |x|^{2\gamma }|\nabla p|^2\left( \frac{ \left( \kappa _2-\kappa _3 p\right) \nabla u+ \kappa _1 u\cdot \nabla p}{(\kappa _2-\kappa _3 p)^2}\right) \right\| _{L^1}\nonumber \\&\lesssim \left\| \nabla p\right\| _{L_\gamma ^2}^2\left( \left\| \frac{ 1 }{\kappa _2-\kappa _3 p} \right\| _{L^\infty } \left\| \nabla u \right\| _{L^\infty }+\left\| \frac{ 1}{(\kappa _2-\kappa _3 p)^2} \right\| _{L^\infty }\left\| u \right\| _{L^\infty }\left\| \nabla p \right\| _{L^\infty } \right) \nonumber \\&\lesssim \left\| \nabla p\right\| _{L_\gamma ^2}^2\left( \left\| \nabla p \right\| _{H^1}\left\| \nabla ^2u \right\| _{H^1}+\left\| \nabla p \right\| _{H^1}\left\| \nabla u \right\| _{H^1}\left\| \nabla ^2p \right\| _{H^1} \right) \nonumber \\&\lesssim t^{-\frac{5}{2}}\left\| \nabla p\right\| _{L_\gamma ^2}^2,\end{aligned}$$

(3.27)

$$\begin{aligned} J_{1,2}^1&\lesssim \left\| |x|^{2\gamma }\nabla p\nabla \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \left[ \left( \kappa _2-\kappa _3 p\right) \nabla u+ \kappa _1 u\cdot \nabla p\right] \right\| _{L^1}\nonumber \\&\lesssim \left\| \nabla p\right\| _{L_\gamma ^2}\left\| \nabla \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\| _{L^\infty }\left[ \left\| \left( \kappa _2-\kappa _3 p\right) \right\| _{L^\infty }\left\| \nabla u\right\| _{L_\gamma ^2}+\left\| u \right\| _{L^\infty }\left\| \nabla p\right\| _{L_\gamma ^2}\right] \nonumber \\&\lesssim \left\| \nabla ^2p \right\| _{H^1} \left\| \nabla p \right\| _{H^1}\left\| \nabla (p,u)\right\| _{L_\gamma ^2}^2+\left\| \nabla ^2p \right\| _{H^1} \left\| \nabla u \right\| _{H^1}\left\| \nabla p\right\| _{L_\gamma ^2}^2\nonumber \\&\lesssim t^{-\frac{5}{2}}\left\| \nabla (p,u)\right\| _{L_\gamma ^2}^2. \end{aligned}$$

(3.28)

Substituting (3.26) and (3.27) into (3.25), we have

$$\begin{aligned} J_1^1\le -\frac{\kappa _3}{2}\frac{\textrm{d}}{\textrm{d}t}\left\langle |x|^{2 \gamma }|\nabla p|^2, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\rangle +Ct^{-\frac{5}{2}}\left\| \nabla (p,u)\right\| _{L_\gamma ^2}^2. \end{aligned}$$

(3.29)

Using integration by parts, Minkowski’s inequality, Hölder’s inequality, Lemmas 2.1, 2.2, 2.3 (Gagliardo–Nirenberg inequality), (3.2), and Cauchy’s inequality, we have

$$\begin{aligned} J_{2}^1&\lesssim \left\| |x|^{2\gamma }\nabla p\nabla \left( \frac{pu\cdot \nabla p}{\kappa _2-\kappa _3 p}\right) \right\| _{L^1}\nonumber \\&\lesssim \left\| \frac{p}{\kappa _2-\kappa _3 p} \right\| _{L^\infty }\left\| \nabla (|x|^{2\gamma }u)|\nabla p|^2\right. \nonumber \\&\left. +|x|^{2\gamma }\nabla u|\nabla p|^2 \right\| _{L^1}+\left\| |x|^{2\gamma }\nabla \left( \frac{p}{\kappa _2-\kappa _3 p}\right) u|\nabla p|^2 \right\| _{L^1}\nonumber \\&\lesssim \left\| p \right\| _{L^\infty }\left[ \left\| u\right\| _{L^\infty }\left\| \nabla p\right\| _{L_\gamma ^2}\left\| \nabla p\right\| _{L_{\gamma -1}^2}+\left\| \nabla u\right\| _{L^\infty }\left\| \nabla p\right\| _{L_\gamma ^2}^2\right] \nonumber \\&+\left\| \nabla p \right\| _{L^\infty } \left\| u\right\| _{L^\infty }\left\| \nabla p\right\| _{L_\gamma ^2}^2\nonumber \\&\lesssim \left\| \nabla p \right\| _{H^1}\left\| \nabla u \right\| _{H^1}\left\| \nabla p\right\| _{L_\gamma ^2}\left\| \nabla p\right\| _{L_{\gamma -1}^2}+\left\| \nabla p \right\| _{H^1}\left\| \nabla ^2 u \right\| _{H^1}\left\| \nabla p\right\| _{L_\gamma ^2}^2\nonumber \\&\quad +\left\| \nabla ^2 p \right\| _{H^1}\left\| \nabla u \right\| _{H^1}\left\| \nabla p\right\| _{L_\gamma ^2}^2\nonumber \\&\lesssim t^{-\frac{5}{2}}\left\| \nabla p\right\| _{L_\gamma ^2}^2+t^{-\frac{5}{2}}\left\| \nabla p\right\| _{L_{\gamma -1}^2}^2. \end{aligned}$$

(3.30)

Using the same arguments as (3.29), we can obtain

$$\begin{aligned} J_{3}^1+J_{4}^1&\lesssim t^{-\frac{5}{2}}\left\| \nabla (p,u)\right\| _{L_\gamma ^2}^2+t^{-\frac{5}{2}}\left\| \nabla (p,u)\right\| _{L_{\gamma -1}^2}^2. \end{aligned}$$

(3.31)

Applying integration by parts, Minkowski’s inequality, Hölder’s inequality, Lemmas 2.1, 2.2, 2.3 (Gagliardo–Nirenberg inequality), the priori estimates (3.1), (3.2), and Cauchy’s inequality, we have

$$\begin{aligned} J_{5}^1&\lesssim \left\| |x|^{2\gamma }\nabla u\nabla \left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }u\cdot \nabla u\right) \right\| _{L^1}\nonumber \\&\lesssim \left\| \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) } \right\| _{L^\infty }\left\| \nabla (|x|^{2\gamma }u)|\nabla u|^2+|x|^{2\gamma }\nabla u|\nabla u|^2 \right\| _{L^1}\nonumber \\&\quad +\left\| \nabla \left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) } \right) \right\| _{L^\infty } \left\| |x|^{2\gamma }u|\nabla u|^2\right\| _{L^1}\nonumber \\&\lesssim \left\| (p,s) \right\| _{L^\infty }\left[ \left\| u\right\| _{L^\infty }\left\| \nabla u\right\| _{L_\gamma ^2}\left\| \nabla u\right\| _{L_{\gamma -1}^2}+\left\| \nabla u\right\| _{L^\infty }\left\| \nabla u\right\| _{L_\gamma ^2}^2\right] \nonumber \\&\quad +\left\| \nabla (p,s) \right\| _{L^\infty } \left\| u\right\| _{L^\infty }\left\| \nabla u\right\| _{L_\gamma ^2}^2\nonumber \\&\lesssim \left\| \nabla (p,s) \right\| _{H^1}\left\| \nabla u \right\| _{H^1}\left\| \nabla u\right\| _{L_\gamma ^2}\left\| \nabla u\right\| _{L_{\gamma -1}^2}+\left\| \nabla (p,s) \right\| _{H^1}\left\| \nabla ^2 u \right\| _{H^1}\left\| \nabla u\right\| _{L_\gamma ^2}^2\nonumber \\&\quad +\left\| \nabla ^2 (p,s) \right\| _{H^1}\left\| \nabla u \right\| _{H^1}\left\| \nabla u\right\| _{L_\gamma ^2}^2\nonumber \\&\le \epsilon t^{-\frac{5}{4}}\left\| \nabla u\right\| _{L_\gamma ^2}^2+\epsilon t^{-\frac{5}{4}}\left\| \nabla u\right\| _{L_{\gamma -1}^2}^2. \end{aligned}$$

(3.32)

Applying (3.24), $J_6^1$ can be rewritten as

$$\begin{aligned} J_6^1&=\frac{1}{\kappa _1}\left\langle |x|^{2 \gamma }\nabla u,\nabla \left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }u_t\right) \right\rangle \nonumber \\&=\frac{1}{\kappa _1}\left\langle |x|^{2 \gamma }\nabla u,\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \nabla u_t\right\rangle \nonumber \\&+\frac{1}{\kappa _1}\left\langle |x|^{2 \gamma }\nabla u,\nabla \left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) u_t\right\rangle \nonumber \\&=\frac{1}{2\kappa _1}\frac{\textrm{d}}{\textrm{d}t}\left\langle |x|^{2 \gamma }|\nabla u|^2,\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\rangle \nonumber \\&\quad -\frac{1}{2\kappa _1}\left\langle |x|^{2 \gamma }|\nabla u|^2,\frac{\textrm{d}}{\textrm{d}t}\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\rangle \nonumber \\&\quad -\frac{1}{\kappa _1}\left\langle |x|^{2 \gamma }\nabla u,\nabla \left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right. \nonumber \\&\qquad \left. \left[ \kappa _2\nabla p-c_v\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) \nabla p+\kappa _1(u\cdot \nabla )u+au \right] \right\rangle \nonumber \\&:=\frac{1}{2\kappa _1}\frac{\textrm{d}}{\textrm{d}t}\left\langle |x|^{2 \gamma }|\nabla u|^2,\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\rangle +\sum _{j=1}^{2}J_{6,j}^1. \end{aligned}$$

(3.33)

Using $\rho =\rho (p,s)$ derived from (1.3), $p_t$ derived from (2.1)$_1$, $s_t$ derived from (2.1)$_3$, and the priori estimate assumption (3.1), we have

$$\begin{aligned} J_{6,1}^1\le \epsilon \left\| \nabla u\right\| _{L_{\gamma }^{2}}^2. \end{aligned}$$

(3.34)

Applying Minkowski’s inequality, Hölder’s inequality, Lemmas 2.1, 2.2 (Gagliardo–Nirenberg inequality), Cauchy’s inequality, the priori estimates (3.1), (3.2), and (3.3), we have

$$\begin{aligned} I_{6,2}^1&\lesssim \left\| |x|^{2\gamma }\nabla u \frac{ \left( \frac{1}{{\rho }}{-}\frac{1}{\rho _{\infty }}\right) }{\kappa _2{-}c_v\left( \frac{1}{\rho } {-}\frac{1}{\rho _{\infty }}\right) }\left[ \kappa _2\nabla p{-}c_v\left( \frac{1}{\rho }{-}\frac{1}{\rho _{\infty }}\right) \nabla p{+}\kappa _1(u\cdot \nabla )u{+}au \right] \right\| _{L^{1}}\nonumber \\&\lesssim \left\| \nabla u\right\| _{L_{\gamma }^{2}}\left\| \nabla \left( \frac{\left( \frac{1}{\rho } {-}\frac{1}{\rho _{\infty }}\right) }{\kappa _2{-}c_v\left( \frac{1}{\rho } {-}\frac{1}{\rho _{\infty }}\right) }\right) \right\| _{L^{\infty }}\left[ \left\| \nabla p\right\| _{L_{\gamma }^{2}}{+}\left\| \left( \frac{1}{\rho } {-}\frac{1}{\rho _{\infty }}\right) \right\| _{L^{\infty }}\left\| \nabla p\right\| _{L_{\gamma }^{2}}\right. \nonumber \\&\quad \left. +\left\| u \right\| _{L^{\infty }}\left\| \nabla u\right\| _{L_{\gamma }^{2}}+\left\| u\right\| _{L_{\gamma }^{2}}\right] \nonumber \\&\lesssim \left\| \nabla u\right\| _{L_{\gamma }^{2}}\left\| \nabla \left( p,s\right) \right\| _{L^{\infty }}\left[ \left\| \nabla p\right\| _{L_{\gamma }^{2}}+\left\| \left( p,s\right) \right\| _{L^{\infty }}\left\| \nabla p\right\| _{L_{\gamma }^{2}} \right. \nonumber \\&\left. +\left\| u \right\| _{L^{\infty }}\left\| \nabla u\right\| _{L_{\gamma }^{2}}+\left\| u\right\| _{L_{\gamma }^{2}} \right] \nonumber \\&\lesssim \Vert \nabla ^2 (p,s)\Vert _{H^{1}}\Vert \nabla (p,u)\Vert _{L_{\gamma }^{2}}^{2}{+}\Vert \nabla ^2 (p,s)\Vert _{H^{1}}\Vert \nabla (p,s)\Vert _{H^{1}}\Vert \nabla (p,u)\Vert _{L_{\gamma }^{2}}^{2}\nonumber \\&\quad {+}\Vert \nabla ^2 (p,s)\Vert _{H^{1}}\Vert \nabla u\Vert _{H^{1}}\Vert \nabla u\Vert _{L_{\gamma }^{2}}^{2} {+}\Vert \nabla ^2 (p,s)\Vert _{H^{1}}\Vert \nabla u\Vert _{L_{\gamma }^{2}}^{2}{+}\Vert \nabla ^2 (p,s)\Vert _{H^{1}}\Vert u\Vert _{L_{\gamma }^{2}}^{2}\nonumber \\&\le \epsilon t^{-\frac{3}{2}+\gamma }+\epsilon \left\| \nabla (p,u)\right\| _{L_{\gamma }^{2}}^2. \end{aligned}$$

(3.35)

Combining (3.33) and (3.34), we have

$$\begin{aligned} J_6^1\le \frac{1}{2\kappa _1}\frac{\textrm{d}}{\textrm{d}t}\left\langle |x|^{2 \gamma }|\nabla u|^2,\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\rangle +\epsilon t^{-\frac{3}{2}+\gamma }+\epsilon \left\| \nabla (p,u)\right\| _{L_{\gamma }^{2}}^2. \end{aligned}$$

(3.36)

Using Hölder’s inequality, Lemmas 2.1, 2.2 (Gagliardo–Nirenberg inequality), the priori estimates (3.1), (3.2), (3.3), and Cauchy’s inequality, we have

$$\begin{aligned} J_{7}^1&\lesssim \left\| |x|^{2\gamma }\nabla u\nabla \left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }u\right) \right\| _{L^1}\nonumber \\&\lesssim \left\| \nabla u\right\| _{L_\gamma ^2}\left\| \nabla \left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\| _{L^\infty }\left\| u\right\| _{L_\gamma ^2}\nonumber \\&+\left\| \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) } \right\| _{L^\infty }\left\| \nabla u\right\| _{L_\gamma ^2}^2\nonumber \\&\lesssim \left\| \nabla ^2(p,s) \right\| _{H^1}\left\| \nabla u\right\| _{L_\gamma ^2}\left\| u\right\| _{L_\gamma ^2}+\left\| \nabla (p,s) \right\| _{H^1}\left\| \nabla u\right\| _{L_\gamma ^2}^2\nonumber \nonumber \\&\le \epsilon t^{-\frac{3}{4}+\frac{\gamma }{2}}\left\| \nabla u\right\| _{L_\gamma ^2}+ \epsilon \left\| \nabla u\right\| _{L_\gamma ^2}^2\nonumber \\&\le \epsilon t^{-\frac{3}{2}+\gamma }+\epsilon \left\| \nabla u\right\| _{L_\gamma ^2}^2. \end{aligned}$$

(3.37)

Substituting the estimates (3.28)–(3.31), (3.35) and (3.36) into (3.22), note that $\epsilon$ is small enough and there exists a large enough T such that

$$\begin{aligned} \epsilon t^{-\frac{3}{2}+\gamma }\lesssim t^{-\frac{7}{2}+\gamma },\quad \epsilon<t^{-\frac{5}{2}},\quad t^{-\frac{5}{2}}<1,\quad \epsilon t^{-\frac{5}{4}}<1, \end{aligned}$$

(3.38)

for all $t>T$. Thus,

$$\begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d} t}\left[ \left\| \nabla (p,u)\right\| _{L_{\gamma }^{2}}^{2} +\kappa _3\left\langle |x|^{2 \gamma }|\nabla p|^2, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\rangle \nonumber \right. \\&\qquad \left. -\frac{1}{\kappa _1}\left\langle |x|^{2 \gamma }|\nabla u|^2,\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\rangle \right] +\alpha \left\| \nabla u\right\| _{L_{\gamma }^{2}}^{2} \nonumber \\&\quad \lesssim t^{-\frac{7}{2}+\gamma }+t^{-\frac{5}{2}}\left\| \nabla (p,u)\right\| _{L_{\gamma }^{2}}^{2} +\Vert \nabla (p,u)\Vert _{L_{\gamma -1}^{2}}^2. \end{aligned}$$

(3.39)

Defining

$$\begin{aligned}&H_1(t)= \left\| \nabla (p,u)\right\| _{L_{\gamma }^{2}}^{2} +\kappa _3\left\langle |x|^{2 \gamma }|\nabla p|^2, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\rangle \\&\quad -\frac{1}{\kappa _1}\left\langle |x|^{2 \gamma }|\nabla u|^2,\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\rangle , \end{aligned}$$

it is obvious that there exist two positive constants $\overline{C }$ and $\underline{C}$ such that $\underline{C }\left\| \nabla (p,u)\right\| _{L_{\gamma }^{2}}^{2}\le H_1(t) \le \overline{C }\left\| \nabla (p,u)\right\| _{L_{\gamma }^{2}}^{2}$. Thus, $H_1(t)$ is equivalent to $\left\| \nabla (p,u)\right\| _{L_{\gamma }^{2}}^{2}$, and (3.38) can be rewritten as

$$\begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d} t}\left\| \nabla (p,u)\right\| _{L_{\gamma }^{2}}^{2}+\alpha \left\| \nabla u\right\| _{L_{\gamma }^{2}}^{2} \nonumber \\&\quad \lesssim t^{-\frac{7}{2}+\gamma }+t^{-\frac{5}{2}}\left\| \nabla (p,u)\right\| _{L_{\gamma }^{2}}^{2} +\Vert \nabla (p,u)\Vert _{L_{\gamma -1}^{2}}^2. \end{aligned}$$

(3.40)

Substituting (2.5) and (3.2) into (3.39), we have

$$\begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d} t}\left\| \nabla (p,u)\right\| _{L_{\gamma }^{2}}^{2} +\alpha \left\| \nabla u\right\| _{L_{\gamma }^{2}}^{2} \nonumber \\&\quad \lesssim t^{-\frac{7}{2}+\gamma }+\left\| \nabla (p,u)\right\| _{L_{\gamma }^{2}}^{2} +t^{-\frac{5}{2}}\Vert \nabla (p,u)\Vert _{L_{\gamma }^{2}}^{\frac{2(\gamma -1)}{\gamma }}\Vert \nabla (p,u)\Vert _{L^{2}}^{\frac{2}{\gamma }} \nonumber \\&\quad \lesssim t^{-\frac{5}{2}}\Vert \nabla (p,u)\Vert _{L_{\gamma }^{2}}^2+t^{-\frac{5}{2\gamma }}\Vert \nabla (p,u)\Vert _{L_{\gamma }^{2}}^{\frac{2(\gamma -1)}{\gamma }}+t^{-\frac{7}{2}+\gamma }. \end{aligned}$$

(3.41)

Denoting $\widehat{\textrm{E}}(t):=\left\| \nabla (p,u)\right\| _{L_{\gamma }^{2}}^{2}$, we can obtain

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t} \widehat{\textrm{E}}(t)\le C_{0} t^{-\frac{5}{2}} \widehat{\textrm{E}}(t)+C_{1} t^{-\frac{5}{2 \gamma }} \widehat{\textrm{E}}(t)^{\frac{\gamma -1}{\gamma }}+C_3t^{-\frac{7}{2}+\gamma }. \end{aligned}$$

If $\gamma >\frac{5}{2}$, then we can apply Lemma 2.5 with $\alpha _{0}=\frac{5}{2}>1,\alpha _{1}=\frac{5}{2\gamma }<1, \beta _{1}=\frac{\gamma -1}{\gamma }<1$, $\gamma _{1}=\frac{1-\alpha _{1}}{1-\beta _{1}}=-\frac{5}{2}+\gamma$. Thus,

$$\begin{aligned} \widehat{\textrm{E}}(t) \le C t^{-\frac{5}{2}+ \gamma }, \end{aligned}$$

(3.42)

for all $t > T$. Lemma 3.2 is proved for all $\gamma >\frac{5}{2}$, and the conclusion for $\gamma \in [0,\frac{5}{2}]$ is proved by Lemma 2.4 (weighted interpolation inequality). Thus, for the case of $k=1$, we have

$$\begin{aligned} \left\| \nabla \left( p, u\right) (t)\right\| _{{L}_\gamma ^2}\lesssim t^{-\frac{5}{4}+\frac{\gamma }{2}}, \end{aligned}$$

(3.43)

for all $t> T$ and $\gamma \ge 0$.

Step 2 ($L_\gamma ^2$ estimate of $\varvec{\nabla ^2(p,u)}$): Using the same arguments as $J_1^1$ (3.25), we have

$$\begin{aligned} J_1^2=&-\kappa _3\left\langle |x|^{2 \gamma }\nabla ^2 p, \nabla ^2\left( \frac{pp_t}{\kappa _2-\kappa _3 p}\right) \right\rangle \nonumber \\ =&-\frac{\kappa _3}{2}\frac{\textrm{d}}{\textrm{d}t}\left\langle |x|^{2 \gamma }|\nabla ^2 p|^2, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\rangle \nonumber \\&-\frac{\kappa _2\kappa _3}{2}\left\langle |x|^{2 \gamma }|\nabla ^2 p|^2, \left( \frac{ \left( \kappa _2-\kappa _3 p\right) \nabla u+ \kappa _1 u\cdot \nabla p}{(\kappa _2-\kappa _3 p)^2}\right) \right\rangle \nonumber \\&+\kappa _3\left\langle |x|^{2 \gamma }\nabla ^2 p, \nabla \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \nabla \left[ \left( \kappa _2-\kappa _3 p\right) \nabla u+ \kappa _1 u\cdot \nabla p\right] \right\rangle \nonumber \\&+\kappa _3\left\langle |x|^{2 \gamma }\nabla ^2 p, \nabla ^2\left( \frac{p}{\kappa _2-\kappa _3 p}\right) \left[ \left( \kappa _2-\kappa _3 p\right) \nabla u+ \kappa _1 u\cdot \nabla p\right] \right\rangle \nonumber \\ :=&-\frac{\kappa _3}{2}\frac{\textrm{d}}{\textrm{d}t}\left\langle |x|^{2 \gamma }|\nabla ^2 p|^2, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\rangle +\sum _{j=1}^{3}J_{1,j}^2. \end{aligned}$$

(3.44)

Applying a similar method to the proof of (3.26), we have

$$\begin{aligned} J_{1,1}^2\lesssim t^{-\frac{5}{2}}\left\| \nabla ^2 p\right\| _{L_\gamma ^2}^2. \end{aligned}$$

(3.45)

Using Minkowski’s inequality, Hölder’s inequality, Lemmas 2.1, 2.2 (Gagliardo–Nirenberg inequality), and (3.2), we have

$$\begin{aligned} J_{1,2}^2\lesssim&\left\| |x|^{2\gamma }\nabla ^2 p\nabla \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \nabla \left[ \left( \kappa _2-\kappa _3 p\right) \nabla u+ \kappa _1 u\cdot \nabla p\right] \right\| _{L^1}\nonumber \\ \lesssim&\left\| \nabla \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\| _{L^\infty }\left\| \nabla ^2 p\right\| _{L_\gamma ^2}\nonumber \\&\times \left[ \left\| \nabla \left( \kappa _2-\kappa _3 p\right) \right\| _{L^\infty }\left\| \nabla u\right\| _{L_\gamma ^2}+\left\| \left( \kappa _2-\kappa _3 p\right) \right\| _{L^\infty }\left\| \nabla ^2 u\right\| _{L_\gamma ^2} \nonumber \right. \\&\left. +\left\| \nabla ^2 p\right\| _{L_\gamma ^2}\left\| u \right\| _{L^\infty }+\left\| \nabla u \right\| _{L^\infty }\left\| \nabla p\right\| _{L_\gamma ^2}\right] \nonumber \\ \lesssim&\left\| \nabla ^2p \right\| _{H^1} \left\| \nabla (p,u) \right\| _{H^1}\left\| \nabla ^2 (p,u)\right\| _{L_\gamma ^2}^2+\left\| \nabla ^2p \right\| _{H^1} \left\| \nabla (p,u) \right\| _{H^1}\left\| \nabla (p,u) \right\| _{L_\gamma ^2}^2\nonumber \\ \lesssim&t^{-\frac{7}{2}+\gamma }+t^{-\frac{5}{2}}\left\| \nabla (p,u)\right\| _{L_\gamma ^2}^2,\\ J_{1,3}^2\lesssim&\left\| |x|^{2\gamma }\nabla ^2 p\nabla ^2\left( \frac{p}{\kappa _2-\kappa _3 p}\right) \left[ \left( \kappa _2-\kappa _3 p\right) \nabla u+ \kappa _1 u\cdot \nabla p\right] \right\| _{L^1}\nonumber \\ \lesssim&\left\| \nabla ^2 p\right\| _{L_\gamma ^2}\left\| \nabla ^2\left( \frac{p}{\kappa _2{-}\kappa _3 p}\right) \right\| _{L^3}\left[ \left\| \left( \kappa _2{-}\kappa _3 p\right) \right\| _{L^\infty }\left\| \nabla u\right\| _{L_\gamma ^6}{+}\left\| u \right\| _{L^\infty }\left\| \nabla p\right\| _{L_\gamma ^6}\right] \nonumber \\ \lesssim&\left\| \nabla ^2p \right\| _{H^1}\left\| \nabla ^2 p\right\| _{L_\gamma ^2}\left\| \nabla (p,u) \right\| _{H^1}\left( \left\| \nabla ^2 (p,u)\right\| _{L_\gamma ^2}+\left\| \nabla (p,u)\right\| _{L_{\gamma -1}^2}\right) \nonumber \\ \lesssim&t^{-\frac{5}{2}} \left\| \nabla ^2 (p,u)\right\| _{L_\gamma ^2}^2+t^{-\frac{5}{2}}\times t^{-\frac{5}{2}+\gamma -1} \nonumber \\ \lesssim&t^{-\frac{7}{2}+\gamma }+ t^{-\frac{5}{2}}\left\| \nabla ^2 (p,u)\right\| _{L_\gamma ^2}^2. \end{aligned}$$

(3.46)

Combining (3.43)–(3.46), we have

$$\begin{aligned} J_{1}^2\le&-\frac{\kappa _3}{2}\frac{\textrm{d}}{\textrm{d}t}\left\langle |x|^{2 \gamma }|\nabla ^2 p|^2, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\rangle +C t^{-\frac{7}{2}+\gamma }+Ct^{-\frac{5}{2}}\left\| \nabla ^2 (p,u)\right\| _{L_\gamma ^2}^2. \end{aligned}$$

(3.47)

Using integration by parts, Minkowski’s inequality, Hölder’s inequality, Lemmas 2.1, 2.2, 2.3 (Gagliardo–Nirenberg inequality), (3.2), (3.42) and Cauchy’s inequality, we have

$$\begin{aligned} J_{2}^2\lesssim&\left\| |x|^{2\gamma }\nabla ^2 p\nabla ^2\left( \frac{pu\cdot \nabla p}{\kappa _2-\kappa _3 p}\right) \right\| _{L^1}\nonumber \\ \lesssim&\left\| \frac{p}{\kappa _2-\kappa _3 p} \right\| _{L^\infty }\left\| \nabla (|x|^{2\gamma }u)|\nabla ^2 p|^2+|x|^{2\gamma }\nabla u|\nabla ^2 p|^2+|x|^{2\gamma }\nabla ^2 p\nabla ^2 u\nabla p \right\| _{L^1}\nonumber \\&+\left\| \nabla \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\| _{L^\infty }\left\| |x|^{2\gamma }u|\nabla ^2 p|^2+|x|^{2\gamma }\nabla ^2 p\nabla u\nabla p \right\| _{L^1}\nonumber \\&+\left\| |x|^{2\gamma }\nabla ^2 p\nabla ^2\left( \frac{p}{\kappa _2-\kappa _3 p}\right) u\nabla p \right\| _{L^1}\nonumber \\ \lesssim&\left\| \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\| _{L^\infty }\left[ \left\| u\right\| _{L^\infty }\left\| \nabla ^2 p\right\| _{L_\gamma ^2}\left\| \nabla ^2 p\right\| _{L_{\gamma -1}^2}+\left\| \nabla u\right\| _{L^\infty }\left\| \nabla ^2 p\right\| _{L_\gamma ^2}^2\right] \nonumber \\&+\left\| \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\| _{L^\infty }\left\| \nabla ^2 u\right\| _{L^3}\left\| \nabla ^2 p\right\| _{L_\gamma ^2}\left\| \nabla p\right\| _{L_\gamma ^6}\nonumber \\&+\left\| \nabla \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\| _{L^\infty }\left[ \left\| u\right\| _{L^\infty }\left\| \nabla ^2p\right\| _{L_\gamma ^2}^2+ \left\| \nabla u\right\| _{L^\infty }\left\| \nabla p\right\| _{L_\gamma ^2}\left\| \nabla ^2 p\right\| _{L_\gamma ^2}\right] \nonumber \\&+\left\| u\right\| _{L^\infty }\left\| \nabla ^2 \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\| _{L^3}\left\| \nabla ^2 p\right\| _{L_\gamma ^2}\left\| \nabla p\right\| _{L_\gamma ^6}\nonumber \\ \lesssim&\left\| \nabla p \right\| _{H^1}\left( \left\| \nabla u\right\| _{H^1}\left\| \nabla ^2 p\right\| _{L_\gamma ^2}\left\| \nabla ^2 p\right\| _{L_{\gamma -1}^2}+\left\| \nabla ^2 u\right\| _{H^1}\left\| \nabla ^2 p\right\| _{L_\gamma ^2}^2\right) \nonumber \\&+\left\| \nabla p \right\| _{H^1}\left\| \nabla ^2 u\right\| _{H^1}\left\| \nabla ^2 p\right\| _{L_\gamma ^2}(\left\| \nabla ^2 p\right\| _{L_\gamma ^2}+\left\| \nabla p\right\| _{L_{\gamma -1}^2})\nonumber \\&+\left\| \nabla ^2 p \right\| _{H^1}\left( \left\| \nabla u\right\| _{H^1}\left\| \nabla ^2p\right\| _{L_\gamma ^2}^2+ \left\| \nabla ^2 u\right\| _{H^1}\left\| \nabla p\right\| _{L_\gamma ^2}\left\| \nabla ^2 p\right\| _{L_\gamma ^2}\right) \nonumber \\&+\left\| \nabla u\right\| _{H^1}\left\| \nabla ^2 p\right\| _{H^1}\left\| \nabla ^2 p\right\| _{L_\gamma ^2}(\left\| \nabla ^2 p\right\| _{L_\gamma ^2}+\left\| \nabla p\right\| _{L_{\gamma -1}^2})\nonumber \\ \lesssim&t^{-\frac{5}{2}}\left\| \nabla ^2 p\right\| _{L_\gamma ^2}\left\| \nabla ^2 p\right\| _{L_{\gamma -1}^2}+t^{-\frac{5}{2}}\left\| \nabla ^2 p\right\| _{L_\gamma ^2}^2+ t^{-\frac{15}{4}+\frac{\gamma -1}{2}}\left\| \nabla ^2 p\right\| _{L_\gamma ^2}\nonumber \\&+ t^{-\frac{15}{4}+\frac{\gamma }{2}}\left\| \nabla ^2 p\right\| _{L_\gamma ^2}\nonumber \\ \lesssim&t^{-\frac{7}{2}+\gamma }+ t^{-\frac{5}{2}}\left\| \nabla ^2 p\right\| _{L_\gamma ^2}^2+t^{-\frac{5}{2}}\left\| \nabla p\right\| _{L_{\gamma -1}^2}^2. \end{aligned}$$

(3.48)

Using the same arguments as (3.48), we can obtain

$$\begin{aligned} J_{3}^2+J_{4}^2 \lesssim&t^{-\frac{7}{2}+\gamma }+t^{-\frac{5}{2}}\left\| \nabla ^2 (p,u)\right\| _{L_\gamma ^2}^2+t^{-\frac{5}{2}}\left\| \nabla ^2 (p,u)\right\| _{L_{\gamma -1}^2}^2. \end{aligned}$$

(3.49)

Applying the priori estimates (3.1) additionally, we have

$$\begin{aligned} J_{5}^2\lesssim&t^{-\frac{7}{2}+\gamma }+\epsilon t^{-\frac{5}{4}}\left\| \nabla ^2 u\right\| _{L_\gamma ^2}^2+\epsilon t^{-\frac{5}{4}}\left\| \nabla ^2 u\right\| _{L_{\gamma -1}^2}^2. \end{aligned}$$

(3.50)

Applying a similar method to the proof of (3.32), we have

$$\begin{aligned} J_6^2&=\frac{1}{\kappa _1}\left\langle |x|^{2 \gamma }\nabla ^2 u,\nabla ^2\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }u_t\right) \right\rangle \nonumber \\&=\frac{1}{2\kappa _1}\left[ \frac{\textrm{d}}{\textrm{d}t}\left\langle |x|^{2 \gamma }|\nabla ^2 u|^2,\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\rangle \right. \nonumber \\&\left. \quad -\left\langle |x|^{2 \gamma }|\nabla ^2 u|^2,\frac{\textrm{d}}{\textrm{d}t}\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\rangle \right] \nonumber \\&\quad -\frac{1}{\kappa _1}\left\langle |x|^{2 \gamma }\nabla ^2 u,\nabla \left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \nabla \right. \nonumber \\&\left. \quad \left[ \kappa _2\nabla p-c_v\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) \nabla p+\kappa _1(u\cdot \nabla )u+au \right] \right\rangle \nonumber \\&\quad -\frac{1}{\kappa _1}\left\langle |x|^{2 \gamma }\nabla ^2 u,\nabla ^2\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right. \nonumber \\&\left. \quad \left[ \kappa _2\nabla p-c_v\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) \nabla p+\kappa _1(u\cdot \nabla )u+au \right] \right\rangle \nonumber \\&:=\frac{1}{2\kappa _1}\frac{\textrm{d}}{\textrm{d}t}\left\langle |x|^{2 \gamma }|\nabla ^2 u|^2,\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\rangle +\sum _{j=1}^{3}J_{6,j}^2. \end{aligned}$$

(3.51)

Using the same method used in the proof of (3.33), one has

$$\begin{aligned} J_{6,1}^2\lesssim \epsilon \Vert \nabla ^2u\Vert _{L_\gamma ^2}^2. \end{aligned}$$

(3.52)

Applying same arguments used in (3.34), we have

$$\begin{aligned} J_{6,2}^2\le \epsilon t^{-\frac{5}{2}+\gamma }\left\| u\right\| _{L_{\gamma }^{2}}^2+\epsilon \left\| \nabla ^2 (p,u)\right\| _{L_{\gamma }^{2}}^2. \end{aligned}$$

(3.53)

Using Minkowski’s inequality, Hölder’s inequality, Lemmas 2.1, 2.2 (Gagliardo–Nirenberg inequality), the priori estimates (3.1), (3.2), (3.3), (3.42), and Cauchy’s inequality, we have

$$\begin{aligned} |J_{6,3}^2|\lesssim&\left\| |x|^{2 \gamma }\nabla ^2 u\nabla ^2\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right. \nonumber \\&\left. \quad \left[ \kappa _2\nabla p-c_v\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) \nabla p+\kappa _1(u\cdot \nabla )u+au \right] \right\| _{L^1}\nonumber \\ \lesssim&\left\| \nabla ^2 u\right\| _{L_\gamma ^2}\left\| \nabla ^2\left( \frac{\left( \frac{1}{\rho } {-}\frac{1}{\rho _{\infty }}\right) }{\kappa _2{-}c_v\left( \frac{1}{\rho } {-}\frac{1}{\rho _{\infty }}\right) }\right) \right\| _{L^3}\left[ \left\| \nabla p\right\| _{L_\gamma ^6}{+}\left\| \left( \kappa _2{-}\kappa _3 p\right) \right\| _{L^\infty }\left\| \nabla p\right\| _{L_\gamma ^6}\right] \nonumber \\&+\left\| \nabla ^2 u\right\| _{L_\gamma ^2}\left\| \nabla ^2\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\| _{L^3}\left[ \left\| u \right\| _{L^\infty }\left\| \nabla u\right\| _{L_\gamma ^6}+\left\| u\right\| _{L_\gamma ^6}\right] \nonumber \\ \lesssim&\left\| \nabla ^2(p,s) \right\| _{H^1}\left\| \nabla ^2 u\right\| _{L_\gamma ^2}\left( \left\| \nabla ^2 p\right\| _{L_\gamma ^2}+\left\| \nabla p\right\| _{L_{\gamma -1}^2}\right) (1+\left\| \nabla p \right\| _{H^1}) \nonumber \\&+\left\| \nabla ^2(p,s) \right\| _{H^1}\left\| \nabla u \right\| _{H^1}\left\| \nabla ^2 u\right\| _{L_\gamma ^2}\left( \left\| \nabla ^2 u\right\| _{L_\gamma ^2}+\left\| \nabla u\right\| _{L_{\gamma -1}^2}\right) \nonumber \\&+\left\| \nabla ^2(p,s) \right\| _{H^1}\left\| \nabla ^2 u\right\| _{L_\gamma ^2}\left( \left\| \nabla u\right\| _{L_\gamma ^2}+\left\| u\right\| _{L_{\gamma -1}^2}\right) \nonumber \\ \le&\epsilon t^{-\frac{5}{2}+\gamma }+ \epsilon \left\| \nabla ^2 (p,u)\right\| _{L_\gamma ^2}^2. \end{aligned}$$

(3.54)

Substituting (3.52)–(3.54) into (3.51), we obtain

$$\begin{aligned} J_6^2\le \frac{1}{2\kappa _1}\frac{\textrm{d}}{\textrm{d}t}\left\langle |x|^{2 \gamma }|\nabla ^2 u|^2,\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\rangle +\epsilon t^{-\frac{5}{2}+\gamma }+ \epsilon \left\| \nabla ^2 (p,u)\right\| _{L_\gamma ^2}^2.\nonumber \\ \end{aligned}$$

(3.55)

Using the same arguments as (3.48), we have

$$\begin{aligned} J_{7}^2\lesssim&\epsilon t^{-\frac{5}{2}+\gamma }+\epsilon \left\| \nabla ^2 u\right\| _{L_\gamma ^2}^2. \end{aligned}$$

(3.56)

Substituting the estimates (3.47)–(3.50), (3.55) and (3.56) into (3.22), note that $\epsilon$ is small enough and there exists a large enough T such that

$$\begin{aligned} \epsilon t^{-\frac{5}{2}+\gamma }\lesssim t^{-\frac{7}{2}+\gamma },\quad \epsilon<t^{-\frac{5}{2}},\quad t^{-\frac{5}{2}}<1,\quad \epsilon t^{-\frac{5}{4}}<1, \end{aligned}$$

(3.57)

for all $t>T$. Thus,

$$\begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d} t}\left[ \left\| \nabla ^2(p,u)\right\| _{L_{\gamma }^{2}}^{2} +\kappa _3\left\langle |x|^{2 \gamma }|\nabla ^2 p|^2, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\rangle \nonumber \right. \\&\left. -\frac{1}{\kappa _1}\left\langle |x|^{2 \gamma }|\nabla ^2 u|^2,\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _1\kappa _2+\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\rangle \right] +\alpha \left\| \nabla ^2 u\right\| _{L_{\gamma }^{2}}^{2} \nonumber \\ \lesssim&t^{-\frac{7}{2}+\gamma }+t^{-\frac{5}{2}}\left\| \nabla ^2(p,u)\right\| _{L_{\gamma }^{2}}^{2} +\Vert \nabla ^2(p,u)\Vert _{L_{\gamma -1}^{2}}^2. \end{aligned}$$

(3.58)

Defining

$$\begin{aligned} H_2(t)=&\left\| \nabla ^2(p,u)\right\| _{L_{\gamma }^{2}}^{2} +\kappa _3\left\langle |x|^{2 \gamma }|\nabla ^2 p|^2, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\rangle \nonumber \\&-\frac{1}{\kappa _1}\left\langle |x|^{2 \gamma }|\nabla ^2 u|^2,\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _1\kappa _2+\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\rangle , \end{aligned}$$

it is obvious that there exist two positive constants $\overline{C }$ and $\underline{C}$ such that $\underline{C }\left\| \nabla ^2(p,u)\right\| _{L_{\gamma }^{2}}^{2}\le H_2(t) \le \overline{C }\left\| \nabla ^2(p,u)\right\| _{L_{\gamma }^{2}}^{2}$. Thus, $H_2(t)$ is equivalent to $\left\| \nabla ^2(p,u)\right\| _{L_{\gamma }^{2}}^{2}$, and (3.58) can be rewritten as

$$\begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d} t}\left\| \nabla ^2(p,u)\right\| _{L_{\gamma }^{2}}^{2}+\alpha \left\| \nabla ^2 u\right\| _{L_{\gamma }^{2}}^{2}\nonumber \\ \lesssim&t^{-\frac{7}{2}+\gamma }+t^{-\frac{5}{2}}\left\| \nabla ^2(p,u)\right\| _{L_{\gamma }^{2}}^{2} +\Vert \nabla ^2(p,u)\Vert _{L_{\gamma -1}^{2}}^2. \end{aligned}$$

(3.59)

Substituting (2.5) and (3.2) into (3.59), we have

$$\begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d} t}\left\| \nabla ^2(p,u)\right\| _{L_{\gamma }^{2}}^{2} +\alpha \left\| \nabla ^2 u\right\| _{L_{\gamma }^{2}}^{2} \nonumber \\ \lesssim&t^{-\frac{7}{2}+\gamma }+t^{-\frac{5}{2}}\left\| \nabla ^2(p,u)\right\| _{L_{\gamma }^{2}}^{2} +\Vert \nabla ^2(p,u)\Vert _{L_{\gamma }^{2}}^{\frac{2(\gamma -1)}{\gamma }}\Vert \nabla ^2(p,u)\Vert _{L^{2}}^{\frac{2}{\gamma }} \nonumber \\ \lesssim&t^{-\frac{5}{2}}\Vert \nabla ^2(p,u)\Vert _{L_{\gamma }^{2}}^2+t^{-\frac{5}{2\gamma }}\Vert \nabla ^2 (p,u)\Vert _{L_{\gamma }^{2}}^{\frac{2(\gamma -1)}{\gamma }}+t^{-\frac{7}{2}+\gamma }. \end{aligned}$$

(3.60)

Denoting $\widehat{\textrm{E}}(t):=\left\| \nabla ^2(p,u)\right\| _{L_{\gamma }^{2}}^{2}$, we can obtain

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t} \widehat{\textrm{E}}(t) \le C_{0} t^{-\frac{5}{2}} \widehat{\textrm{E}}(t)+C_{1} t^{-\frac{5}{2 \gamma }} \widehat{\textrm{E}}(t)^{\frac{\gamma -1}{\gamma }}+C_3t^{-\frac{7}{2}+\gamma }. \end{aligned}$$

If $\gamma >\frac{5}{2}$, then we can apply Lemma 2.5 with $\alpha _{0}=\frac{5}{2}>1,\alpha _{1}=\frac{5}{2\gamma }<1, \beta _{1}=\frac{\gamma -1}{\gamma }<1$, $\gamma _{1}=\frac{1-\alpha _{1}}{1-\beta _{1}}=-\frac{5}{2}+\gamma$. Thus,

$$\begin{aligned} \widehat{\textrm{E}}(t) \le C t^{-\frac{5}{2}+\gamma }, \end{aligned}$$

(3.61)

for all $t > T$. Lemma 3.2 is proved for all $\gamma >\frac{5}{2}$, and the conclusion for $\gamma \in [0,\frac{5}{2}]$ is proved by Lemma 2.4 (weighted interpolation inequality). Thus, for the case of $k=2$, we have

$$\begin{aligned} \left\| \nabla ^2\left( p, u\right) (t)\right\| _{{L}_\gamma ^2}\lesssim t^{-\frac{5}{4}+\frac{\gamma }{2}}, \end{aligned}$$

(3.62)

for all $t> T$ and $\gamma \ge 0$.

Step 3 ($L_\gamma ^2$ estimate of $\varvec{\nabla ^3(p,u)}$): For the case of $k=3$, noticing that Lemma 2.2 does not work in $L_\gamma ^2$, for the terms like $\left\| |x|^{2\gamma }\nabla ^3 p\nabla ^3\left( \frac{p}{\kappa _2-\kappa _3 p}\right) u\nabla p \right\| _{L^1}$, we need employ some new idea. The key observation here is to use Hölder’s inequality skillfully to get

$$\begin{aligned}{} & {} \left\| |x|^{2\gamma }\nabla ^3 p\nabla ^3\left( \frac{p}{\kappa _2-\kappa _3 p}\right) u\nabla p \right\| _{L^1}\\{} & {} \qquad \lesssim \left\| \nabla p\right\| _{L^\infty }\left\| \nabla ^3 \left( \frac{p}{\kappa _2-\kappa _3 p} \right) \right\| _{L^2}\left\| \nabla ^3 p\right\| _{L_\gamma ^2}\left\| |x|^\gamma u\right\| _{L^\infty }. \end{aligned}$$

Thus, before discussing the case of $J_3^k$, for any smooth function f, we need to discuss the term $\left\| |x|^{\gamma }f\right\| _{L^{\infty }}$ with $\left\| f\right\| _{L_\gamma ^{2}}\lesssim t^{-\frac{3}{4}+\frac{\gamma }{2}}$ and $\left\| \nabla ^kf\right\| _{L_\gamma ^{2}}\lesssim t^{-\frac{5}{4}+\frac{\gamma }{2}}(k=1,2)$. Applying Lemma 2.1 (Gagliardo–Nirenberg inequality) and Cauchy’s inequality as follows, one has

$$\begin{aligned}&\left\| |x|^{\gamma }f\right\| _{L^{\infty }}\lesssim \left( \left\| \nabla \left( |x|^\gamma f\right) \right\| _{L^2}^{\frac{1}{2}}\left\| \nabla ^2\left( |x|^\gamma f\right) \right\| _{L^2}^{\frac{1}{2}}\right) \nonumber \\&\quad \lesssim \left( \left\| \nabla ^2\left( |x|^\gamma f\right) \right\| _{L^2}+\left\| \nabla \left( |x|^\gamma f\right) \right\| _{L^2}\right) \nonumber \\&\quad \lesssim \left( \left\| \nabla ^2 f\right\| _{L_\gamma ^2}+\left\| \nabla f\right\| _{L_{\gamma -1}^2}+\Vert f\Vert _{L_{\gamma -2}^2}+\left\| \nabla f\right\| _{L_\gamma ^2}+\Vert f\Vert _{L_{\gamma -1}^2}\right) \nonumber \\&\quad \lesssim t^{-\frac{5}{4}+\frac{\gamma }{2}}. \end{aligned}$$

(3.63)

Similar to the methods used in (3.25), $J_1^3$ can be rewritten as

$$\begin{aligned} J_1^3=&-\frac{\kappa _3}{2}\frac{\textrm{d}}{\textrm{d}t}\left\langle |x|^{2 \gamma }|\nabla ^3 p|^2, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\rangle \nonumber \\&-\frac{\kappa _2\kappa _3}{2}\left\langle |x|^{2 \gamma }|\nabla ^3 p|^2, \left( \frac{ \left( \kappa _2-\kappa _3 p\right) \nabla u+ \kappa _1 u\cdot \nabla p}{(\kappa _2-\kappa _3 p)^2}\right) \right\rangle \nonumber \\&+\kappa _3\left\langle |x|^{2 \gamma }\nabla ^3 p, \nabla \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \nabla ^2\left[ \left( \kappa _2-\kappa _3 p\right) \nabla u+ \kappa _1 u\cdot \nabla p\right] \right\rangle \nonumber \\&+\kappa _3\left\langle |x|^{2 \gamma }\nabla ^3 p, \nabla ^2\left( \frac{p}{\kappa _2-\kappa _3 p}\right) \nabla \left[ \left( \kappa _2-\kappa _3 p\right) \nabla u+ \kappa _1 u\cdot \nabla p\right] \right\rangle \nonumber \\&+\kappa _3\left\langle |x|^{2 \gamma }\nabla ^3 p, \nabla ^3\left( \frac{p}{\kappa _2-\kappa _3 p}\right) \left[ \left( \kappa _2-\kappa _3 p\right) \nabla u+ \kappa _1 u\cdot \nabla p\right] \right\rangle \nonumber \\ :=&-\frac{\kappa _3}{2}\frac{\textrm{d}}{\textrm{d}t}\left\langle |x|^{2 \gamma }|\nabla ^3 p|^2, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\rangle +\sum _{j=1}^{4}J_{1,j}^3. \end{aligned}$$

(3.64)

Similar to the arguments used in (3.26),

$$\begin{aligned} J_{1,1}^3&\lesssim t^{-\frac{5}{2}}\left\| \nabla ^3 p\right\| _{L_\gamma ^2}^2. \end{aligned}$$

(3.65)

Applying the similar methods used in (3.45) and (3.46), one has

$$\begin{aligned} J_{1,2}^3+J_{1,3}^3&\lesssim t^{-\frac{7}{2}+\gamma }+ t^{-\frac{5}{2}}\left\| \nabla ^3 (p,u)\right\| _{L_\gamma ^2}^2. \end{aligned}$$

(3.66)

Using Minkowski’s inequality, Hölder’s inequality, Lemmas 2.1, 2.2 (Gagliardo–Nirenberg inequality), (3.63), (3.2), (3.3), (3.42), (3.62), and Cauchy’s inequality, we have

$$\begin{aligned} J_{1,4}^3&\lesssim \left\| |x|^{2\gamma }\nabla ^3 p\nabla ^3\left( \frac{p }{\kappa _2-\kappa _3 p}\right) \left[ \left( \kappa _2-\kappa _3 p\right) \nabla u+ \kappa _1 u\cdot \nabla p\right] \right\| _{L^1}\nonumber \\&\lesssim \left\| \nabla ^3 p\right\| _{L_\gamma ^2}\left\| \nabla ^3\left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\| _{L^2}\left[ \left\| |x|^{\gamma }\nabla u\right\| _{L^\infty }+\left\| \nabla u\right\| _{L^\infty }\left\| |x|^{\gamma }p\right\| _{L^\infty }\right. \nonumber \\&\left. +\left\| \nabla p\right\| _{L^\infty }\left\| |x|^{\gamma } u\right\| _{L^\infty }\right] \nonumber \\&\lesssim \left\| \nabla ^3p \right\| _{L^2}\left\| \nabla ^3 p\right\| _{L_\gamma ^2} \left( \left\| \nabla ^3 u\right\| _{L_\gamma ^2}+t^{-\frac{5}{4}+\frac{\gamma }{2}}+t^{-\frac{5}{4}+\frac{\gamma }{2}}\left\| \nabla ^2(p,u)\right\| _{H^1}\right) \nonumber \\&\lesssim t^{-\frac{5}{4}}\left\| \nabla ^3 p\right\| _{L_\gamma ^2} \left( \left\| \nabla ^3 u\right\| _{L_\gamma ^2}+t^{-\frac{5}{4}+\frac{\gamma }{2}}+t^{-\frac{5}{2}+\frac{\gamma }{2}}\right) \nonumber \\&\lesssim t^{-\frac{7}{2}+\gamma }+ t^{-\frac{5}{4}}\left\| \nabla ^3 (p,u)\right\| _{L_\gamma ^2}^2. \end{aligned}$$

(3.67)

Substituting (3.65)–(3.67) into (3.64), we have

$$\begin{aligned} J_1^3\le -\frac{\kappa _3}{2}\frac{\textrm{d}}{\textrm{d}t}\left\langle |x|^{2 \gamma }|\nabla ^3 p|^2, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\rangle +C t^{-\frac{5}{4}}\left\| \nabla ^3 (p,u)\right\| _{L_\gamma ^2}^2+Ct^{-\frac{7}{2}+\gamma }. \end{aligned}$$

(3.68)

Using integration by parts, Minkowski’s inequality, Hölder’s inequality, Lemmas 2.1, 2.2, 2.3 (Gagliardo–Nirenberg inequality), (3.63), (3.2), (3.42), (3.62), and Cauchy’s inequality, we have

$$\begin{aligned} J_{2}^3\lesssim&\left\| |x|^{2\gamma }\nabla ^3 p\nabla ^3\left( \frac{pu\cdot \nabla p}{\kappa _2-\kappa _3 p}\right) \right\| _{L^1}\nonumber \\ \lesssim&\left\| \frac{p}{\kappa _2{-}\kappa _3 p} \right\| _{L^\infty } \left\| \nabla (|x|^{2\gamma }u)|\nabla ^3 p|^2{+}|x|^{2\gamma } \nabla ^3 p\left( \nabla u\nabla ^3 p{+} \nabla ^2 u\nabla ^2 p {+}\nabla ^3 u\nabla p \right) \right\| _{L^1} \nonumber \\&+\left\| \nabla \left( \frac{p}{\kappa _2-\kappa _3 p} \right) \right\| _{L^\infty }\left\| |x|^{2\gamma }\nabla ^3 p\left( u\nabla ^3 p+\nabla u\nabla ^2 p+\nabla ^2 u\nabla p\right) \right\| _{L^1}\nonumber \\&+\left\| |x|^{2\gamma }\nabla ^3 p\nabla ^2\left( \frac{p}{\kappa _2-\kappa _3 p}\right) \left( \nabla u\nabla p+ u\nabla ^2 p \right) \right\| _{L^1}\nonumber \\&+\left\| |x|^{2\gamma }\nabla ^3 p\nabla ^3\left( \frac{p}{\kappa _2-\kappa _3 p}\right) u\nabla p \right\| _{L^1}\nonumber \\ \lesssim&\left\| \left( \frac{p}{\kappa _2-\kappa _3 p} \right) \right\| _{L^\infty }\left[ \left\| u\right\| _{L^\infty }\left\| \nabla ^3 p\right\| _{L_\gamma ^2}\left\| \nabla ^3 p\right\| _{L_{\gamma -1}^2}+\left\| \nabla u\right\| _{L^\infty }\left\| \nabla ^3 p\right\| _{L_\gamma ^2}^2 \nonumber \right. \\&\left. \left\| \nabla ^2 u\right\| _{L^3}\left\| \nabla ^3 p\right\| _{L_\gamma ^2}\left\| \nabla ^2 p\right\| _{L_\gamma ^6}+\left\| \nabla p \right\| _{L^\infty } \left\| \nabla ^3p\right\| _{L_\gamma ^2}\left\| \nabla ^3 u\right\| _{L_\gamma ^2}\right] \nonumber \\&+\left\| \nabla \left( \frac{p}{\kappa _2-\kappa _3 p} \right) \right\| _{L^\infty }\left\| \nabla ^3 p\right\| _{L_\gamma ^2}\left[ \left\| u \right\| _{L^\infty }\left\| \nabla ^3p\right\| _{L_\gamma ^2}+\left\| \nabla (p,u) \right\| _{L^\infty }\left\| \nabla ^2 (p,u)\right\| _{L_\gamma ^2}\right] \nonumber \\&{+}\left\| \nabla ^2 \left( \frac{p}{\kappa _2{-}\kappa _3 p} \right) \right\| _{L^3}\left[ \left\| \nabla u \right\| _{L^\infty }\left\| \nabla ^3 p\right\| _{L_\gamma ^2}\left\| \nabla p\right\| _{L_\gamma ^6}{+} \left\| u \right\| _{L^\infty }\left\| \nabla ^3 p\right\| _{L_\gamma ^2}\left\| \nabla ^2 p\right\| _{L_\gamma ^6}\right] \nonumber \\&{+}\left\| \nabla p\right\| _{L^\infty }\left\| \nabla ^3 \left( \frac{p}{\kappa _2{-}\kappa _3 p} \right) \right\| _{L^2}\left\| \nabla ^3 p\right\| _{L_\gamma ^2}\left\| |x|^\gamma u\right\| _{L^\infty }\nonumber \\ \lesssim&\left\| \nabla p\right\| _{H^1}\left[ \left\| \nabla u\right\| _{H^1}\left\| \nabla ^3 p\right\| _{L_\gamma ^2}\left\| \nabla ^3 p\right\| _{L_{\gamma -1}^2}+\left\| \nabla ^2 u\right\| _{H^1}\left\| \nabla ^3 (p,u)\right\| _{L_\gamma ^2}^2\right] \nonumber \\&+\left\| \nabla p\right\| _{H^1}\left[ \left\| \nabla ^2 u\right\| _{H^1}\left\| \nabla ^3 p\right\| _{L_\gamma ^2}\left( \left\| \nabla ^3 p\right\| _{L_\gamma ^2}+\left\| \nabla ^2 p\right\| _{L_{\gamma -1}^2}\right) \right. \nonumber \\&\left. +\left\| \nabla ^2 p\right\| _{H^1}\left\| \nabla ^3 p\right\| _{L_\gamma ^2}\left\| \nabla ^3 u\right\| _{L_\gamma ^2}\right] \nonumber \\&+\left\| \nabla ^2p\right\| _{H^1}\left\| \nabla ^3 p\right\| _{L_\gamma ^2}\left[ \left\| \nabla u\right\| _{H^1}\left\| \nabla ^3 p\right\| _{L_\gamma ^2}+ \left\| \nabla ^2 (p,u)\right\| _{H^1}\left\| \nabla ^2 (p,u)\right\| _{L_\gamma ^2}\right] \nonumber \\&+\left\| \nabla ^2 p\right\| _{H^1}\left\| \nabla ^3 p\right\| _{L_\gamma ^2}\left[ \left\| (\nabla u,\nabla ^2 u)\right\| _{H^1}(\left\| \nabla ^3 p\right\| _{L_\gamma ^2}+\left\| \nabla ^2 p\right\| _{L_{\gamma -1}^2})\right. \nonumber \\&\left. +\left\| \nabla ^3p \right\| _{L^2}\times t^{-\frac{5}{4}+\frac{\gamma }{2}}\right] \nonumber \\ \lesssim&t^{-\frac{5}{2}}\left\| \nabla ^3 p\right\| _{L_\gamma ^2}\left\| \nabla ^3 p\right\| _{L_{\gamma -1}^2}+t^{-\frac{5}{2}}\left\| \nabla ^3 (p,u)\right\| _{L_\gamma ^2}^2\nonumber \\&+t^{-\frac{5}{2}}\times t^{-\frac{9}{4}+\frac{\gamma }{2}}\left\| \nabla ^3 p\right\| _{L_\gamma ^2}+t^{-\frac{5}{2}}\times t^{-\frac{5}{4}+\frac{\gamma }{2}}\left\| \nabla ^3 p\right\| _{L_\gamma ^2} \nonumber \\ \lesssim&t^{-\frac{7}{2}+\gamma }+ t^{-\frac{5}{2}}\left\| \nabla ^3 (p,u)\right\| _{L_\gamma ^2}^2+t^{-\frac{5}{2}}\left\| \nabla ^3 p\right\| _{L_{\gamma -1}^2}^2. \end{aligned}$$

(3.69)

Using the same arguments as (3.69), we obtain

$$\begin{aligned} J_{3}^3+J_{4}^3 \lesssim&t^{-\frac{7}{2}+\gamma }+t^{-\frac{5}{2}}\left\| \nabla ^3 (p,u)\right\| _{L_\gamma ^2}^2+t^{-\frac{5}{2}}\left\| \nabla ^3 (p,u)\right\| _{L_{\gamma -1}^2}^2. \end{aligned}$$

(3.70)

Using the priori estimates (3.1) additionally, one has

$$\begin{aligned} J_{5}^3\lesssim&t^{-\frac{7}{2}+\gamma }+\epsilon t^{-\frac{5}{4}}\left\| \nabla ^3 u\right\| _{L_\gamma ^2}^2+\epsilon t^{-\frac{5}{4}}\left\| \nabla ^3 u\right\| _{L_{\gamma -1}^2}^2. \end{aligned}$$

(3.71)

Applying the similar arguments used in (3.32), we have

$$\begin{aligned} J_6^3&=\frac{1}{\kappa _1}\left\langle |x|^{2 \gamma }\nabla ^3 u,\nabla ^3\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }u_t\right) \right\rangle \nonumber \\&=\frac{1}{2\kappa _1}\left[ \frac{\textrm{d}}{\textrm{d}t}\left\langle |x|^{2 \gamma }|\nabla ^3 u|^2,\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\rangle \right. \nonumber \\&\quad \left. -\left\langle |x|^{2 \gamma }|\nabla ^3 u|^2,\frac{\textrm{d}}{\textrm{d}t}\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\rangle \right] \nonumber \\&\quad -\frac{1}{\kappa _1}\left\langle |x|^{2 \gamma }\nabla ^3 u,\nabla \left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \nabla ^2 \right. \nonumber \\&\quad \left. \left[ \kappa _2\nabla p-c_v\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) \nabla p+\kappa _1(u\cdot \nabla )u+au \right] \right\rangle \nonumber \\&\quad -\frac{1}{\kappa _1}\left\langle |x|^{2 \gamma }\nabla ^2 u,\nabla ^2\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \nabla \right. \nonumber \\&\left. \left[ \kappa _2\nabla p-c_v\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) \nabla p+\kappa _1(u\cdot \nabla )u+au \right] \right\rangle \nonumber \\&\quad -\frac{1}{\kappa _1}\left\langle |x|^{2 \gamma }\nabla ^3 u,\nabla ^3\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right. \nonumber \\&\left. \left[ \kappa _2\nabla p-c_v\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) \nabla p+\kappa _1(u\cdot \nabla )u+au \right] \right\rangle \nonumber \\&:=\frac{1}{2\kappa _1}\frac{\textrm{d}}{\textrm{d}t}\left\langle |x|^{2 \gamma }|\nabla ^3 u|^2,\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\rangle +\sum _{j=1}^{4}J_{6,j}^3. \end{aligned}$$

(3.72)

Using the same method as in (3.33), we have

$$\begin{aligned} J_{6,1}^3\lesssim \epsilon \Vert \nabla ^3u\Vert _{L_\gamma ^2}^2. \end{aligned}$$

(3.73)

Applying the same arguments used in (3.34) and (3.54) for $J_{6,2}^3$ and $J_{6,3}^3$, respectively, one has

$$\begin{aligned} J_{6,2}^3+J_{6,3}^3\le \epsilon \Vert \nabla ^3(p,u)\Vert _{L_\gamma ^2}^2+\epsilon t^{-\frac{5}{2}+\gamma }. \end{aligned}$$

(3.74)

Applying Minkowski’s inequality, Hölder’s inequality, Lemmas 2.1, 2.2 (Gagliardo–Nirenberg inequality), (3.63), the priori estimates (3.1), (3.2), (3.3), (3.42), (3.62), and Cauchy’s inequality, we have

$$\begin{aligned} |J_{6,4}^3|\lesssim&\left\| |x|^{2 \gamma }\nabla ^3 u,\nabla ^3\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right. \nonumber \\&\left. \left[ \kappa _2\nabla p-c_v\left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) \nabla p+\kappa _1(u\cdot \nabla )u+au \right] \right\| _{L^1}\nonumber \\ \lesssim&\left\| \nabla ^3 u\right\| _{L_\gamma ^2}\left\| \nabla ^3\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\| _{L^2}\nonumber \\&\left[ \left\| |x|^{\gamma }\nabla p\right\| _{L^{\infty }} \left( 1+ \left\| \left( \frac{1}{\rho }-\frac{1}{\rho _{\infty }}\right) \right\| _{L^{\infty }}\right) \right] \nonumber \\&+\left\| \nabla ^3 u\right\| _{L_\gamma ^2}\left\| \nabla ^3\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _2-c_v\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\| _{L^2}\left[ \left\| |x|^{\gamma }u\right\| _{L^{\infty }} \left( 1+ \left\| \nabla u\right\| _{L^{\infty }}\right) \right] \nonumber \\ \lesssim&\left\| \nabla ^3(p,s) \right\| _{L^2}\left\| \nabla ^3 u\right\| _{L_\gamma ^2}\left[ (\left\| \nabla ^3 p\right\| _{L_\gamma ^2}+t^{-\frac{5}{4}+\frac{\gamma }{2}})(1+\left\| \nabla (p,s) \right\| _{H^1})\right. \nonumber \\&\left. +t^{-\frac{5}{4}+\frac{\gamma }{2}}(1+\left\| \nabla ^2 u \right\| _{H^1})\right] \nonumber \\ \le&\epsilon \left\| \nabla ^3 u\right\| _{L_\gamma ^2}\left[ (\left\| \nabla ^3 p\right\| _{L_\gamma ^2}+t^{-\frac{5}{4}+\frac{\gamma }{2}})(1+\epsilon )+t^{-\frac{5}{4}+\frac{\gamma }{2}}(1+t^{-\frac{5}{4}})\right] \nonumber \\ \le&\epsilon t^{-\frac{5}{2}+\gamma }+C\epsilon \left\| \nabla ^3 (p,u)\right\| _{L_\gamma ^2}^2. \end{aligned}$$

(3.75)

Substituting (3.73)–(3.75) into (3.72), we have

$$\begin{aligned} J_6^3\le \frac{1}{2\kappa _1}\frac{\textrm{d}}{\textrm{d}t}\left\langle |x|^{2 \gamma }|\nabla ^3 u|^2,\left( \frac{\left( \frac{1}{\rho } {-}\frac{1}{\rho _{\infty }}\right) }{\kappa _2{-}c_v\left( \frac{1}{\rho } {-}\frac{1}{\rho _{\infty }}\right) }\right) \right\rangle +C\epsilon \left\| \nabla ^3 (p,u)\right\| _{L_\gamma ^2}^2+C\epsilon t^{-\frac{5}{2}+\gamma }. \end{aligned}$$

(3.76)

Using the same methods used in (3.69), we have

$$\begin{aligned} J_{7}^3 \le&\epsilon t^{-\frac{5}{2}+\gamma }+\epsilon \left\| \nabla ^3 u\right\| _{L_\gamma ^2}^2. \end{aligned}$$

(3.77)

Substituting the (3.68)–(3.71), (3.76) and (3.77) into (3.22), note that $\epsilon$ is small enough and then there exists a large enough T such that

$$\begin{aligned} \epsilon t^{-\frac{5}{2}+\gamma }\lesssim t^{-\frac{7}{2}+\gamma },\quad \epsilon<t^{-\frac{5}{2}},\quad t^{-\frac{5}{2}}<1,\quad \epsilon t^{-\frac{5}{4}}<1, \end{aligned}$$

(3.78)

for all $t>T$. Thus,

$$\begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d} t}\left[ \left\| \nabla ^3(p,u)\right\| _{L_{\gamma }^{2}}^{2} +\kappa _3\left\langle |x|^{2 \gamma }|\nabla ^3 p|^2, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\rangle \nonumber \right. \\&\qquad \left. -\frac{1}{\kappa _1}\left\langle |x|^{2 \gamma }|\nabla ^3 u|^2,\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _1\kappa _2+\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\rangle \right] +\alpha \left\| \nabla ^3 u\right\| _{L_{\gamma }^{2}}^{2} \nonumber \\&\quad \lesssim t^{-\frac{7}{2}+\gamma }+t^{-\frac{5}{2}}\left\| \nabla ^3(p,u)\right\| _{L_{\gamma }^{2}}^{2} +\Vert \nabla ^3(p,u)\Vert _{L_{\gamma -1}^{2}}^2. \end{aligned}$$

(3.79)

Defining

$$\begin{aligned} H_3(t)=&\left\| \nabla ^3(p,u)\right\| _{L_{\gamma }^{2}}^{2} +\kappa _3\left\langle |x|^{2 \gamma }|\nabla ^3 p|^2, \left( \frac{p}{\kappa _2-\kappa _3 p}\right) \right\rangle \nonumber \\&-\frac{1}{\kappa _1}\left\langle |x|^{2 \gamma }|\nabla ^3 u|^2,\left( \frac{\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }{\kappa _1\kappa _2+\left( \frac{1}{\rho } -\frac{1}{\rho _{\infty }}\right) }\right) \right\rangle , \end{aligned}$$

it is obvious that there exist two positive constants $\overline{C }$ and $\underline{C}$ such that $\underline{C }\left\| \nabla ^3(p,u)\right\| _{L_{\gamma }^{2}}^{2}\le H_3(t) \le \overline{C }\left\| \nabla ^3(p,u)\right\| _{L_{\gamma }^{2}}^{2}$. Thus, $H_3(t)$ is equivalent to $\left\| \nabla ^3(p,u)\right\| _{L_{\gamma }^{2}}^{2}$, and (3.79) can be rewritten as

$$\begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d} t}\left\| \nabla ^3(p,u)\right\| _{L_{\gamma }^{2}}^{2} +\alpha \left\| \nabla ^3 u\right\| _{L_{\gamma }^{2}}^{2} \nonumber \\&\quad \lesssim t^{-\frac{7}{2}+\gamma }+t^{-\frac{5}{4}}\left\| \nabla ^3(p,u)\right\| _{L_{\gamma }^{2}}^{2} +\Vert \nabla ^3(p,u)\Vert _{L_{\gamma -1}^{2}}^2. \end{aligned}$$

(3.80)

Substituting (2.5) and (3.2) into (3.80), we have

$$\begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d} t}\left\| \nabla ^3(p,u)\right\| _{L_{\gamma }^{2}}^{2} +\alpha \left\| \nabla ^3 u\right\| _{L_{\gamma }^{2}}^{2} \nonumber \\&\quad \lesssim t^{-\frac{7}{2}+\gamma }+t^{-\frac{5}{4}}\left\| \nabla ^3(p,u)\right\| _{L_{\gamma }^{2}}^{2} +\Vert \nabla ^3(p,u)\Vert _{L_{\gamma }^{2}}^{\frac{2(\gamma -1)}{\gamma }}\Vert \nabla ^3(p,u)\Vert _{L^{2}}^{\frac{2}{\gamma }} \nonumber \\&\quad \lesssim t^{-\frac{5}{4}}\Vert \nabla ^3(p,u)\Vert _{L_{\gamma }^{2}}^2+t^{-\frac{5}{2\gamma }}\Vert \nabla ^3 (p,u)\Vert _{L_{\gamma }^{2}}^{\frac{2(\gamma -1)}{\gamma }}+t^{-\frac{7}{2}+\gamma }. \end{aligned}$$

(3.81)

Denoting $\widehat{\textrm{E}}(t):=\left\| \nabla ^3(p,u)\right\| _{L_{\gamma }^{2}}^{2}$, we can obtain

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t} \widehat{\textrm{E}}(t) \le C_{0} t^{-\frac{5}{4}} \widehat{\textrm{E}}(t)+C_{1} t^{-\frac{5}{2 \gamma }} \widehat{\textrm{E}}(t)^{\frac{\gamma -1}{\gamma }}+C_3t^{-\frac{7}{2}+\gamma }. \end{aligned}$$

If $\gamma >\frac{5}{2}$, then we can apply Lemma 2.5 with $\alpha _{0}=\frac{5}{4}>1,\alpha _{1}=\frac{5}{2\gamma }<1, \beta _{1}=\frac{\gamma -1}{\gamma }<1$, $\gamma _{1}=\frac{1-\alpha _{1}}{1-\beta _{1}}=-\frac{5}{2}+\gamma$. Thus,

$$\begin{aligned} \widehat{\textrm{E}}(t) \le C t^{-\frac{5}{2}+\gamma }, \end{aligned}$$

(3.82)

for all $t > T$. Lemma 3.2 is proved for all $\gamma >\frac{5}{2}$, and the conclusion for $\gamma \in [0,\frac{5}{2}]$ is proved by Lemma 2.4 (weighted interpolation inequality). Thus, for the case of $k=3$, we have

$$\begin{aligned} \left\| \nabla ^3\left( p, u\right) (t)\right\| _{{L}_\gamma ^2}\lesssim t^{-\frac{5}{4}+\frac{\gamma }{2}}, \end{aligned}$$

(3.83)

for all $t> T$ and $\gamma \ge 0$.

Combining (3.42), (3.62), and (3.83), we have completed the proof of Lemma 3.2, that is (1.8). And combining Lemmas 3.1 and 3.2, we have completed the proof of Theorem 1.1. $\square$