1. Introduction

The motion of a general viscous compressible, isentropic fluid in a domain $\Omega \subset \mathbb{R}^3$ is governed by the following compressible Navier–Stokes equations

(1.1) \begin{equation} \begin{cases} \partial _t\rho +{\mathrm {div}}\,(\rho u)=0, \\ \partial _t(\rho u)+{\mathrm {div}}\,(\rho u\otimes u)-\mu \Delta u-(\mu +\lambda )\nabla {\mathrm {div}}\, u+\nabla P=0, \end{cases} \end{equation}

where $\rho$ , $u$ , $P=a\rho ^{\gamma }$ ( $a\gt 0$ ) denote the density, velocity and pressure, respectively, $\gamma \gt 1$ is the adiabatic exponent and $\mu$ and $\lambda$ represent the shear viscosity and bulk viscosity coefficients satisfying the physical restrictions:

(1.2) \begin{equation} \mu \gt 0,\quad \lambda +\frac {2}{3}\mu \geq 0. \end{equation}

The equations (1.1) will be equipped with initial data

(1.3) \begin{equation} (\rho , u)(x,0)=(\rho _0, u_0)(x), \quad x\in \Omega . \end{equation}

In this paper, we assume that $\Omega \subset \mathbb{R}^3$ is a simply connected bounded domain with smooth boundary $\partial \Omega$ . The system (1.1) is studied under the Navier-slip boundary conditions:

(1.4) \begin{equation} u\cdot n=0,\, {\mathrm {curl}}\, u\times n=-Au\ \mathrm { on }\ \partial \Omega , \end{equation}

where $n$ is the unit outer normal to $\partial \Omega$ , and $A=A(x)$ is a $3\times 3$ symmetric matrix defined on $\partial \Omega$ . There exist some different forms of slip boundary conditions related to (1.4), where the detailed discussions can be found in [Reference Cai and Li4].

There are large amounts of literature concerning the well-posedness theory for the compressible Navier–Stokes equations (1.1). The one-dimensional problem has been investigated extensively by many people, see, for instance, [Reference Hoff16, Reference Kazhikhov28, Reference Kazhikhov and Shelukhin29, Reference Serre41, Reference Serre42] and references therein. The local well-posedness of multidimensional problem was studied by Nash [Reference Nash36] and Serrin [Reference Serrin43] in the absence of vacuum, while for the initial density with vacuum, the existence and uniqueness of local strong solution were proved in [Reference Cho, Choe and Kim6–Reference Choe and Kim9, Reference Salvi and Straškraba40]. Matsumura and Nishida [Reference Matsumura and Nishida35] first proved the global existence of strong solutions with initial data close to the equilibrium and later was extended to the discontinuous initial data by Hoff [Reference Hoff17, Reference Hoff and Serre20]. For the existence of solutions for arbitrary large initial data in 3D, the major breakthrough was made by Lions [Reference Lions32], in which he showed global existence of weak solutions for the whole space, periodic domains or bounded domains with Dirichlet boundary conditions with $\gamma \geq \frac {9}{5}$ and later was extended to the case $\gamma \gt \frac {3}{2}$ by Feireisl [Reference Feireisl, Novotný and Petzeltová12–Reference Feireisl14]. When the initial data are assumed to have some spherically symmetric or axisymmetric properties, Jiang and Zhang [Reference Jiang and Zhang25, Reference Jiang and Zhang26] proved the existence of global weak solutions for any $\gamma \gt 1$ . Shortly thereafter, Hoff [Reference Hoff18] gave a new type of global weak solutions with small energy, which have extra regularity compared with the ones constructed by Lions-Feireisl in [Reference Feireisl, Novotný and Petzeltová12, Reference Lions32] under an additional condition on viscosity coefficients and the far-field density $\tilde {\rho }\gt 0$ .

Up to now, the uniqueness and regularity for weak solutions in [Reference Feireisl, Novotný and Petzeltová12, Reference Lions32] (with arbitrarily large initial data) still remain open. Recently, many important progress on global existence and uniqueness of classical solutions with large oscillations and vacuum to viscous compressible fluids in a barotropic regime has been made. Huang, Li and Xin [Reference Huang, Li and Xin24] first established the global existence of classical solutions to 3D Cauchy problem of the isentropic compressible Navier–Stokes equations with small initial total energy but possibly large oscillations. Later, Li and Xin [Reference Li and Xin30] studied the 2D Cauchy problem and the large time asymptotic behaviour of solutions with small initial total energy. Very recently, Hong-Hou-Peng-Zhu [Reference Hong, Hou, Peng and Zhu22] provided a positive result under the condition that the adiabatic exponent $\gamma$ is close to 1, which says that classical solutions to Cauchy problem of the isentropic compressible Navier–Stokes equations exist globally with allowing the large initial energy and the presence of vacuum. This type of solution can be viewed as the Nishida–Smoller type large solution, which is originally studied for the conservation laws with BV initial data in [Reference Nishida and Smoller38], where Nishida and Smoller showed the global existence of solutions to the Cauchy problem of 1D isentropic Euler equations under the condition that $(\gamma -1)\,\text{total var.}\{u_0,\rho _0\}$ Footnote ¹ is sufficiently small. In particular, this result implies that the initial energy could be large as $\gamma$ is sufficiently close to 1. For some generalizations of the Nishida–Smoller type results on inviscid or viscous flow, one can see, for instance, [Reference Hong, Hou, Peng and Zhu21, Reference Kawashima and Nishida27, Reference Liu, Yang, Zhao and Zou33, Reference Liu34, Reference Tan, Yang, Zhao and Zou44, Reference Temple45]. For compressible Navier–Stokes equations (1.1) in half space $\mathbb{R}^3_+$ with slip boundary conditions, Hoff [Reference Hoff18] established the global existence of weak solutions under the assumption that the initial energy is suitably small. For compressible Navier–Stokes equations (1.1) in a non axis-symmetric domain, Novotný and Străskraba [Reference Novotný and Străskraba39] proved global existence of weak solutions. For compressible Navier–Stokes equations (1.1) in a general bounded smooth domain, the global existence of strong (or classical) solutions has been investigated for the 3D case with slip boundary condition in [Reference Cai and Li4], and the 2D case with similar boundary condition in [Reference Fan, Li and Li11], both of which are equipped with small initial total energy and vacuum. Even for the 3D bounded domain with non-slip boundary condition, Fan and Li [Reference Fan and Li10] proved the global existence of classical solutions to the barotropic compressible Navier–Stokes system with small initial energy. Also, one can refer to [Reference Cai, Li and Lü5] for the exterior domain case.

In conclusion, all the works [Reference Cai and Li4, Reference Cai, Li and Lü5, Reference Hoff18, Reference Hong, Hou, Peng and Zhu22, Reference Huang, Li and Xin24, Reference Li and Xin30, Reference Nishida and Smoller38] depend essentially on small initial energy or the advantage of the whole space. Therefore, a natural and important problem is to study what will happen if both large initial energy and boundary effects are involved. That is to say, we aim to investigate the global well-posedness and long-time behaviour of Nishida–Smoller type large solutions to the compressible Navier–Stokes equations (1.1) with slip boundary conditions (1.4) and vacuum. However, to the best of our knowledge, up to now, this problem still remains open. The main purpose of this work is to resolve this problem. More precisely, we prove the global existence and uniqueness of classical solutions to the initial-boundary-value problem (1.1)–(1.4) with large initial energy and vacuum provided that the adiabatic exponent is sufficiently close to 1. Moreover, we also prove that the classical solutions have an exponential decay rate, which is decreasing with respect to the adiabatic exponent. Finally, we prove that the gradient of the density will grow unboundedly with an exponential rate if the vacuum appears (even at a point) initially. This generalizes the previous related works in [Reference Cai and Li4, Reference Cai, Li and Lü5, Reference Hoff18, Reference Hong, Hou, Peng and Zhu22, Reference Huang, Li and Xin24, Reference Li and Xin30, Reference Nishida and Smoller38], where either small initial energy is required or boundary effects are absent.

Before stating our result, let us introduce the following notations and conventions used throughout this paper. We set

\begin{equation*} \int f=\int _{\Omega }fdx,\quad \int _0^Tg=\int _0^Tgdt \end{equation*}

and

\begin{equation*} \bar {f}\,:\!=\frac {1}{|\Omega |}\int _{\Omega }fdx \end{equation*}

which is the average of $f$ on $\Omega$ .

For $1\leq r\leq \infty$ , and integer $k\geq 1$ , we denote the standard Sobolev spaces as follows:

(1.5) \begin{equation} \begin{cases} L^r=L^r(\Omega ),\,D^{k,r}=\{u\in L_{loc}^1(\Omega )\ :\ \|\nabla ^ku\|_{L^r}\lt \infty \},\\ W^{k,r}=L^r\cap D^{k,r},\,H^k=W^{k,2},\, D^k=D^{k,2},\\ D_0^1=\{u\in L^6\ :\ \|\nabla u\|_{L^2}\lt \infty ,\ \mathrm { and\ (1.4)\ holds}\},\\ H_0^1=L^2\cap D_0^1,\,\|u\|_{D^{k,r}}=\|\nabla ^ku\|_{L^r}. \end{cases} \end{equation}

For some $s\in (0,1)$ , the fractional Sobolev space $H^s(\Omega )$ is defined by

\begin{equation*} H^s(\Omega )\,:\!=\left \{u\in L^2(\Omega ): \int _{\Omega \times \Omega }\frac {|u(x)-u(y)|^2}{|x-y|^{n+2s}}dxdy\lt \infty \right \} \end{equation*}

with the norm:

\begin{equation*} \|u\|_{H^s(\Omega )}\,:\!=\|u\|_{L^2(\Omega )}+\left (\int _{\Omega \times \Omega }\frac {|u(x)-u(y)|^2}{|x-y|^{n+2s}}dxdy\right )^{\frac {1}{2}}. \end{equation*}

Additionally, the Einstein summation convention will be frequently used in the later sections:

\begin{equation*} A_iB_i=\sum _{i=1}^3A_iB_i. \end{equation*}

The initial total energy of (1.1) is defined as

(1.6) \begin{equation} E_0\,:\!=\int \left(\frac {1}{2}\rho _0|u_0|^2+\frac {1}{\gamma -1}P(\rho _0)\right)\!, \end{equation}

and the modified initial energy involving $\gamma -1$ is denoted as

(1.7) \begin{equation} \mathcal{E}_0\,:\!=\int \frac {1}{2}\rho _0|u_0|^2+(\gamma -1)E_0. \end{equation}

In the following, we denote by $C\gt 0$ a generic constant depending on $\mu , \lambda , a, \tilde {\rho }, \Omega , M$ and the matrix $A$ , but independent of $\gamma -1, E_0, \mathcal{E}_0$ and $t$ . And we write $C(\alpha )$ to emphasize the dependence of $C$ on the parameter $\alpha$ .

Now, we are in a position to state our main results.

Theorem 1.1. Let $\Omega$ be a simply connected bounded domain in $\mathbb{R}^3$ and its smooth boundary $\partial \Omega$ has a finite number of 2-dimensional connected components. For given positive constants $M$ and $\tilde {\rho }$ , suppose that the $3\times 3$ symmetric matrix $A$ in (1.4) is smooth and positive semi-definite, and the initial data $(\rho _0,u_0)$ satisfy for some $q\in (3,6)$ ,

(1.8) \begin{equation} (\rho _0,P(\rho _0))\in W^{2,q},\quad u_0\in \{\,f\in H^2\ :\ \ f\cdot n=0,\,{\mathrm {curl}}\ f\times n=-A\,f\ \mathrm { on }\ \partial \Omega \}, \end{equation}

(1.9) \begin{equation} 0\leq \rho _0\leq \tilde {\rho },\quad \|\nabla u_0\|_{L^2}\leq M, \end{equation}

and the compatibility condition

(1.10) \begin{equation} -\mu \Delta u_0-(\mu +\lambda )\nabla {\mathrm {div}}\, u_0+\nabla P(\rho _0)=\rho _0^{\frac {1}{2}}g, \end{equation}

for some $g\in L^2$ . Then, the initial-boundary value problem (1.1)–(1.4) admits a unique classical solution $(\rho ,u)$ in $\Omega \times (0,\infty )$ satisfying that

(1.11) \begin{equation} 0\leq \rho (x,t)\leq 2\tilde {\rho },\,(x,t)\in \Omega \times (0,\infty ), \end{equation}

and for any $0\lt \tau \lt T\lt \infty$ ,

(1.12) \begin{equation} \begin{cases} (\rho ,P)\in C([0,T];\ W^{2,q}),\\ \nabla u\in C([0,T];\ H^1)\cap L^{\infty }(\tau ,T;\ W^{2,q}),\\ u_t\in L^{\infty }(\tau ,T;\ H^2)\cap H^1(\tau ,T;\ H^1),\\ \sqrt {\rho }u_t\in L^{\infty }(0,\infty ;\ L^2), \end{cases} \end{equation}

provided that

(1.13) \begin{equation} \mathcal{E}_0\leq \epsilon , \quad \|A\|_{W^{1,6}}\leq \hat {\epsilon }. \end{equation}

Here, $\epsilon \gt 0$ is a small constant depending only on $\mu , \lambda , \gamma , a, \tilde {\rho }, \Omega , M, E_0$ , and the matrix $A$ , but independent of $\gamma -1$ and $t$ , $\hat {\epsilon }\gt 0$ is a small constant depending only on $\mu , \lambda$ and $\Omega$ . According to (3.45), (3.51), (3.56), (3.79) and (3.57), $\epsilon$ and $\hat {\epsilon }$ can be, respectively, precisely characterized in the following form:

\begin{equation*} \begin{aligned} \epsilon &=\min \left \{ \vphantom{\left (\frac {\tilde {\rho }}{4C(\tilde {\rho })(1+E_0)}\right )^3} 1,(4C(\tilde {\rho }))^{-12},(C(\tilde {\rho },M))^{-2}, (2C(\tilde {\rho }))^{-16},(2C(\tilde {\rho },M)(E_0+1))^{-2},\big(2C(\tilde {\rho })E_0^{\frac {1}{2}}\big)^{-32},\right .\\ &\qquad \quad \left . (2C(\tilde {\rho },M))^{-\frac {16}{9}},\left (\frac {\tilde {\rho }}{2C(\tilde {\rho },M)}\right )^{-12},(C(\tilde {\rho }))^{-1},\left (\frac {\tilde {\rho }}{4C(\tilde {\rho })(1+E_0)}\right )^3\right \} \end{aligned} \end{equation*}

and

\begin{equation*} \hat {\epsilon }=\min \big\{(2C)^{-\frac {1}{3}}, (27C(\Omega ))^{-\frac {1}{3}}, (2CC(\Omega ))^{-\frac {1}{3}}\big\} \end{equation*}

with $C$ here depending only on $\mu ,\lambda$ and $\Omega$ and $C(\Omega )$ only depending on $\Omega$ and $\mu$ .

Moreover, if $\bar {\rho }\leq 1$ and $\frac {\tilde {\rho }}{\bar {\rho }}\geq 3$ , then for any $\gamma \in (1,\frac {3}{2}]$ , $r\in [1,\infty )$ and $p\in [1,6]$ , there exists positive constants $\tilde {C}$ and $\eta _0$ , with $\tilde {C}$ depending only on $\mu ,\lambda ,\gamma ,a,\tilde {\rho },\bar {\rho },M,\Omega ,r,p$ and the matrix $A$ , and $\eta _0$ depending only on $\mu ,\lambda ,a,\Omega ,r,p$ , $\tilde {\rho }$ and $\frac {\tilde {\rho }}{\bar {\rho }}$ , but independent of $\gamma -1$ , such that for any $t\geq 1$ , it holds that

(1.14) \begin{equation} \|\rho -\bar {\rho }\|_{L^r}+\|u\|_{W^{1,p}}+\|\sqrt {\rho }\dot {u}\|_{L^2}\leq \tilde {C}e^{-\eta _0\bar {\rho }^{\gamma }t}. \end{equation}

On the other hand, if $\gamma \gt \frac {3}{2}$ , there exists similar exponent decay result as follows:

(1.15) \begin{equation} \|\rho -\bar {\rho }\|_{L^r}+\|u\|_{W^{1,p}}+\|\sqrt {\rho }\dot {u}\|_{L^2}\leq \tilde {C}_1e^{-\eta _1t} \end{equation}

where positive constants $\tilde {C}_1$ and $\eta _1$ depend only on $\mu ,\lambda ,\gamma ,a,\tilde {\rho },\bar {\rho },\Omega ,r,p$ and the matrix $A$ (with $\tilde {C}_1$ depending on $M$ as well).

With Theorem1.1 in hand, we will give a corollary to state the large-time behaviour of $\nabla \rho$ when vacuum appears initially. This corollary is a direct consequence from the exponent decay of classical solutions as in (1.14) and (1.15), and the proof can be referred to [Reference Cai and Li4] for details and is omitted here.

Corollary 1.2. Under the conditions of Theorem 1.1 , assume further that there exists certain point $x_0\in \Omega$ such that $\rho (x_0)=0$ . Then, the unique classical solution $(\rho , u)$ obtained in Theorem 1.1 satisfies that for any $r_1\gt 3$ , there exist positive constants $\tilde {C}_2$ and $\eta _2$ , with $\tilde {C}_2$ depending only on $\mu ,\lambda ,\gamma ,a,\tilde {\rho },\bar {\rho },\Omega ,r_1$ , and $\eta _2$ depending only on $\mu ,\lambda ,a,\Omega ,r_1$ , $\tilde {\rho }$ and $\frac {\tilde {\rho }}{\bar {\rho }}$ , but independent of $\gamma -1$ , such that for any $\gamma \in (1,\frac {3}{2}]$ and $t\geq 1$ ,

(1.16) \begin{equation} \|\nabla \rho (t)\|_{L^{r_1}}\geq \tilde {C}_2e^{\eta _2\bar {\rho }^{\gamma }t}, \end{equation}

and positive constants $\tilde {C}_3$ and $\eta _3$ depending only on $\mu ,\lambda ,\gamma ,a,\tilde {\rho },\bar {\rho },\Omega ,r_1$ such that for any $\gamma \in \left(\frac {3}{2},\infty \right)$ and $t\geq 1$ ,

(1.17) \begin{equation} \|\nabla \rho (t)\|_{L^{r_1}}\geq \tilde {C}_3e^{\eta _3t}. \end{equation}

Here we list some remarks as follows.

Now, we make some comments on the analysis of this paper. Similar to the arguments in [Reference Cai and Li4] and [Reference Hong, Hou, Peng and Zhu22], the key issue in our proof is to derive the time-independent upper bound on the density $\rho$ (see Lemma 3.9). Compared to [Reference Cai and Li4] where the analysis relies heavily on the smallness of the initial energy $E_0$ , the new difficulty here lies in the fact that in our case, the initial energy could be large when the adiabatic exponent $\gamma$ is sufficiently close to 1. Indeed, with the help of the smallness of the initial energy $E_0$ , Cai-Li [Reference Cai and Li4] derives the smallness of $\displaystyle \int _0^T\|\nabla u\|_{L^2}^2$ and $\displaystyle \|P\|_{L^1}$ by the elementary energy estimate directly (see Lemma 3.2). However, due to the absence of smallness on the initial energy, we cannot extract the smallness of $\displaystyle \int _0^T\|\nabla u\|_{L^2}^2$ from the elementary energy estimate. Therefore, we need to develop some new ingredients to overcome this difficulty. On the other hand, as compared to Hong-Hou-Peng-Zhu [Reference Hong, Hou, Peng and Zhu22] where the Cauchy problem is considered, we need to employ some new observations and ideas to overcome the difficulties from the Navier-slip boundary conditions (1.4). We now highlight the main differences and ingredients as follows:

• • As mentioned before, to derive the time-independent upper bound of the density $\rho$ , due to the lack of smallness of $E_0$ , the smallness of $\displaystyle \int _0^T\|\nabla u\|_{L^2}^2$ cannot be extracted from the basic energy estimate directly. However, we can obtain the smallness of $\|P\|_{L^1}$ by the basic energy estimate directly. This is very different from [Reference Hong, Hou, Peng and Zhu22] where the proof depends heavily on the relation between $\|P(\rho )-P(\tilde {\rho })\|_{L^2}^2$ and $\displaystyle \int G(\rho )$ . It should be mentioned that we give an explicit relation of these two terms in Remark A.2 by employing a careful analysis.

• • Since the initial energy $E_0$ could be large in our analysis, we can only get the smallness of $\displaystyle \int _0^{\sigma (T)}\|\nabla u\|_{L^2}^2$ rather than $\displaystyle \int _0^T\|\nabla u\|_{L^2}^2$ (see Lemma 3.3). By making delicate energy estimates, we can get the estimates of $A_1(T)$ and $A_2(T)$ stated in Lemma 3.4:

  (1.18) \begin{align} A_2(T)&\leq C A_1(\sigma (T))+CA_1^{\frac {3}{2}}(T)+C(\tilde {\rho })A_1^3(E_0+1)\nonumber\\ &\quad +\cdots +\int _0^T\sigma ^3\big(\|\nabla u\|_{L^4}^4+\|P-\bar {P}\|_{L^4}^4\big), \end{align}
  (1.19) \begin{align} A_1(T)&\leq A_1(\sigma (T))+C\int _{\sigma (T)}^T\int \sigma (|P-\bar {P}||\nabla u|^2+|\nabla u|^3)+C(\tilde {\rho })\int _{\sigma (T)}^T\sigma \|\nabla u\|_{L^2}^4\nonumber\\ &\leq A_1(\sigma (T))+C(\tilde {\rho })\left (\int _{\sigma (T)}^T\|\nabla u\|_{L^2}^2\right )^{\frac {1}{2}}A_1^{\frac {3}{4}}(T)A_2^{\frac {1}{4}}(T)\nonumber\\ &\leq A_1(\sigma (T))+C(\tilde {\rho })E_0^{\frac {1}{2}}A_1^{\frac {3}{4}}(T)A_2^{\frac {1}{4}}(T),\quad (\mathrm {see\ (3.75){-}(3.77)}) \end{align}

  which is new and very different from those in [Reference Cai and Li4, Reference Hong, Hou, Peng and Zhu22]. Indeed, by virtue of the Navier-slip boundary conditions (1.4), the term $A_1^{\frac {3}{2}}(T)$ in (1.18) is caused by boundary term $\displaystyle \int _{\partial \Omega }\sigma ^mu\cdot \nabla n\cdot uG$ (see (3.38)), which is inevitable. This term combined with the second estimate (1.19) implies that the order of $A_2(T)$ should be higher than that of $A_1(T)$ , but lower than that of $A_1^{\frac {3}{2}}(T)$ . Based on this key observation, we specifically choose $A_2(T)\sim A_1(T)^{\frac {4}{3}}$ in Proposition 3.1. It is worth mentioning that this is different from [Reference Hong, Hou, Peng and Zhu22] where $A_1(T)\sim A_2(T)^2$ is chosen. In addition, due to lack of the smallness of $E_0$ , we have to derive new estimates on almost all boundary terms and intermediate terms appearing in the control of $A_2(T)$ , such as $\displaystyle \int _{\partial \Omega }G(u\cdot \nabla u)\cdot \nabla n\cdot u$ in (3.26) and $\|\nabla u\|_{L^2}^2\|\nabla G\|_{L^3}^2$ in (3.32), which is very different from [Reference Cai and Li4].

• • The control of $\displaystyle \int _0^T\sigma ^3\big(\|\nabla u\|_{L^4}^4+\|P-\bar {P}\|_{L^4}^4\big)$ appearing in (1.18) is the most difficult part of this paper. Applying Lemma 2.7 or using the similar procedure to deal with $\displaystyle \int _0^T\sigma ^3\|P-\bar {P}\|_{L^4}^4$ as in Lemma 3.7 will yield the same trouble that

  (1.20) \begin{equation} \int _0^T\sigma ^3\|P-\bar {P}\|_{L^4}^4\leq C\int _0^T\sigma ^3\|\nabla u\|_{L^2}^4+\hbox{good terms}, \end{equation}
  (1.21) \begin{equation} \int _0^T\sigma ^3\|\nabla u\|_{L^4}^4\leq C\int _0^T\sigma ^3(\|\nabla u\|_{L^2}^4+\|P-\bar {P}\|_{L^4}^4)+\hbox{good terms}, \end{equation}
  where the trouble term $\displaystyle \int _0^T\sigma ^3\|\nabla u\|_{L^2}^4$ in (1.20) and (1.21) is out of control due to lack of smallness on $\displaystyle \int _{\sigma (T)}^T\sigma ^3\|\nabla u\|_{L^2}^4$ (smaller than $A_2(T)$ ). Notice that, for the Cauchy problem in [Reference Hong, Hou, Peng and Zhu22], the troubling term $\displaystyle \int _0^T\sigma ^3\|\nabla u\|_{L^2}^4$ in (1.20) and (1.21) exactly disappears. Then, by combining the relations (1.20) and (1.21), the authors in [Reference Hong, Hou, Peng and Zhu22] succeed in controlling $\displaystyle \int _0^T\sigma ^3\|P-\bar {P}\|_{L^4}^4$ and hence $\displaystyle \int _0^T\sigma ^3\|\nabla u\|_{L^4}^4$ . Therefore, we need to employ some new thoughts to overcome this difficulty. Fortunately, we observe that the term $\displaystyle \int _{\sigma (T)}^T\sigma ^3\|\nabla u\|_{L^2}^4$ is actually from ${\mathrm {curl}}\, u\times n|_{\partial \Omega }=-Au$ of Navier-slip boundary conditions (1.4). If some smallness condition is imposed on the matrix $A$ , then we can close the estimate of $\displaystyle \int _{\sigma (T)}^T\sigma ^3\|\nabla u\|_{L^4}^4$ and hence that of $\displaystyle \int _0^T\sigma ^3\|P-\bar {P}\|_{L^4}^4$ . This part is discussed in Remark 2.8, (3.61), (3.64) and (3.65). It is worth mentioning that the smallness condition in (1.13) imposed on the matrix $A$ is different from [Reference Hong, Hou, Peng and Zhu22], but it can be seen as a similar constraint on boundary as compared to the smallness assumption on far-field density in [Reference Hong, Hou, Peng and Zhu22]. In particular, our results hold for the usual case that the matrix $A=0$ . Additionally, we should remark that the smallness of $A$ only depends on $\mu ,\lambda$ and $\Omega$ but is independent of the density $\rho$ , velocity $u$ and pressure $P$ .

• • The next important issue is to give the exponent decay of the classical solution obtained in Theorem1.1. Since we aim to describe the monotonicity of exponent decay rate with respect to the adiabatic exponent $\gamma$ , the rough relation among $(\rho -\bar {\rho })^2,G(\rho )$ and $(P(\rho )-P(\bar {\rho }))(\rho -\bar {\rho })$ in [Reference Cai and Li4] is not applicable for our case. To overcome this difficulty, by employing several key observations, we give an explicit relation among three terms above in Lemma A.1. With the crucial Lemma A.1 in hand, we succeed in deriving the exponent decay rate for any $\gamma \in (1,\frac {3}{2}]$ in Proposition 3.10.

The rest of the paper is organized as follows: In the next section, we introduce some elementary lemmas that will be needed later. In Section 3, we give the proof of Theorem1.1.