1 Introduction

Micropolar fluid equations, which were suggested and introduced by Eringen in the 1960s (see [3]), are a significant step toward the generalization of the Navier–Stokes equations. It is a type of fluids that exhibits micro-rotational effects and micro-rotational inertia and can be viewed as a non-Newtonian fluid. Physically, micropolar fluid may represent fluids that consist of rigid, randomly oriented (or spherical particles) suspended in a viscous medium, where the deformation of fluid particles is ignored. It can describe many phenomena that appear in a large number of complex fluids such as the suspensions, animal blood, and liquid crystals which cannot be characterized appropriately by the Navier–Stokes system, and that it is important to the scientists working with the hydrodynamic-fluid problems and phenomena. We refer the reader to the monograph [14], which provides a detailed derivation of the micropolar fluid equations from the general constitutive laws, together with an extensive review of the mathematical theory and the applications of this particular model. For more background on micropolar fluids, we refer to [8, 9] and references therein.

Let \(\Omega \subseteq \mathbb {R}^3\) be a bounded smooth domain, we are concerned with an initial-boundary-value problem of three-dimensional (3D for short) nonhomogeneous micropolar fluid equations with density-dependent viscosity in \(\Omega \times (0,T)\):

$$\begin{aligned} {\left\{ \begin{array}{ll} \rho _t+{{\,\textrm{div}\,}}(\rho \textbf{u}) = 0,\\ (\rho \textbf{u})_t+{{\,\textrm{div}\,}}(\rho \textbf{u}\otimes \textbf{u}) -{{\,\textrm{div}\,}}((\mu (\rho )+\xi )\nabla \textbf{u})+\nabla P =2\xi {{\,\textrm{curl}\,}}\textbf{w},\\ (\rho \textbf{w})_t+{{\,\textrm{div}\,}}(\rho \textbf{u}\otimes \textbf{w}) +4\xi \textbf{w}-\eta \Delta \textbf{w} -(\eta +\lambda )\nabla {{\,\textrm{div}\,}}\textbf{w} =2\xi {{\,\textrm{curl}\,}}\textbf{u},\\ {{\,\textrm{div}\,}}\textbf{u}=0, \end{array}\right. } \end{aligned}$$
(1.1)

with the initial condition

$$\begin{aligned} (\rho ,\rho \textbf{u},\rho \textbf{w})(x,0)=(\rho _0,\rho _0\textbf{u}_0,\rho _0\textbf{w}_0)(x),\ \ x\in \Omega , \end{aligned}$$
(1.2)

and the Dirichlet boundary condition

$$\begin{aligned} (\textbf{u},\textbf{w})(x,t)=(\textbf{0},\textbf{0}),\ x\in \partial \Omega ,\ t\ge 0. \end{aligned}$$
(1.3)

Here \(\rho \), \(\textbf{u}\), \(\textbf{w}\), and P denote the density, velocity, micro-rotational velocity, and pressure of the fluid, respectively. The viscosity coefficient \(\mu (\rho )\) is a function of the density \(\rho \) satisfying

$$\begin{aligned} \mu \in C^1[0,\infty ),\ \mu \ge \underline{\mu }>0, \end{aligned}$$
(1.4)

for some positive constant \(\underline{\mu }\), while the constants \(\lambda ,\xi \), and \(\eta \) are all positive.

When there is no microstructure (\(\xi =0\) and \(\textbf{w}=\textbf{0}\)), the system (1.1) reduces to nonhomogeneous Navier–Stokes equations with density-dependent viscosity. In 1996, Lions [10, Chapter 2] showed the global existence of weak solutions in any space dimensions for the initial density allowing vacuum states. Under the compatibility condition

$$\begin{aligned} -{{\,\textrm{div}\,}}(\mu (\rho _0)\nabla \textbf{u}_0)+\nabla P_0=\sqrt{\rho _0}\textbf{g},\ \ \text {for some}\ (P_0,\textbf{g})\in H^1\times L^2, \end{aligned}$$
(1.5)

Zhang [19], and independently by Huang and Wang [7], established the global existence and uniqueness of strong solutions in 3D bounded domains provided that \(\Vert \nabla \textbf{u}_0\Vert _{L^2}\) is suitably small. Applying Desjardins’ interpolation inequality, Zhong [21] derived global strong solutions for 2D problem provided that \(\Vert \nabla \mu (\rho _0)\Vert _{L^q}\ (q>2)\) is suitably small. Very recently, He–Li–Lü [6] obtained the global existence and exponential stability of strong solutions in \(\mathbb {R}^3\) if \(\Vert \textbf{u}_0\Vert _{\dot{H}^\beta }\) with \(\beta \in (\frac{1}{2},1]\) is small enough, while Liu [12] proved a similar result under the assumption that the initial density is suitably small. It should be mentioned that there is no need to impose some compatibility condition on the initial data in [6, 12, 21] via time-weighted estimates.

Recently, there have been some studies on the global strong solutions to the system (1.1) with constant viscosity \(\mu \). On the one hand, when the initial density is strictly away from vacuum (i.e., \(\rho _0\) is strictly positive), Braz e Silva et al. [1] investigated global existence and uniqueness of solutions for 3D Cauchy problem through a Lagrangian approach. The authors [2] studied global strong solutions in 3D thin domains \(\mathbb {R}^2\times (0,\varepsilon )\) provided that \(\varepsilon (\Vert \nabla \textbf{u}_0\Vert _{L^2}^2+\Vert \nabla \textbf{w}_0\Vert _{L^2}^2)\) is suitably small. Moreover, they showed that the linear and angular velocities tend to vanish away from the initial time when \(\varepsilon \) approaches to zero. On the other hand, for the initial density allowing vacuum states, by imposing the following compatibility condition

$$\begin{aligned} {\left\{ \begin{array}{ll} -(\mu +\xi )\Delta \textbf{u}_0+\nabla P_0 -2\xi {{\,\textrm{curl}\,}}\textbf{w}_0=\sqrt{\rho _0}\textbf{g}_1,\\ -\eta \Delta \textbf{w}_0-(\eta +\lambda )\nabla {{\,\textrm{div}\,}}\textbf{w}_0 +4\xi \textbf{w}_0-2\xi {{\,\textrm{curl}\,}}\textbf{u}_0=\sqrt{\rho _0}\textbf{g}_2, \end{array}\right. } \end{aligned}$$
(1.6)

for some \((\nabla P_0,\textbf{g}_1,\textbf{g}_2)\in L^2(\mathbb {R}^3)\), Zhang and Zhu [20] derived the global existence of strong solution under the assumption that the initial density or the initial energy is small enough. Later on, Ye [18] improved their result by removing (1.6) and furthermore obtained exponential decay estimates of solutions. Meanwhile, Wu and Zhong [17] obtained global well-posedness and exponential decay estimates of strong solutions on 3D bounded domains provided that the initial energy is suitably small. Moreover, with the help of spatial-weighted energy estimates, Zhong [22] established global strong solutions to the 2D Cauchy problem with large initial data and vacuum at infinity.

In the present paper, we shall establish the global existence of strong solutions to the problem (1.1)–(1.3) under small initial energy. Moreover, we will investigate the large-time behavior of such global solutions. It should be remarked that the strong interaction between viscosity and velocity will bring some serious difficulties in the mathematical study of global theory for the case of density-dependent viscosity, thus the method [17] used for the constant viscosity case cannot be applied directly. The main novelty of the present paper consists in the absence of the positive lower bound for the initial density as well as the absence of compatibility conditions for the initial data.

For \(1\le r\le \infty \) and integer \(k\ge 1\), we denote the standard homogeneous and inhomogeneous Sobolev spaces as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} L^r=L^r(\Omega ), \ D^{k,r}=D^{k,r}(\Omega )=\{v\in L^1_{{{\,\textrm{loc}\,}}}(\Omega )|\nabla ^kv\in L^r(\Omega )\}, \\ D^{k,r}_{0,\sigma }=\overline{\{\textbf{v}\in C^\infty _{0}(\Omega )|{{\,\textrm{div}\,}}\textbf{v}=0\}}\ \text {closure in the norm of}\ D^{k,r}, \\ D^k=D^{k,2},\ D^k_{0,\sigma }=D^{k,2}_{0,\sigma },\ W^{k,r}= W^{k,r}(\Omega ),\ H^k= W^{k,2}. \end{array}\right. } \end{aligned}$$

Our main result can be stated as follows.

Theorem 1.1

For constants \(q\in (3,6)\) and \(0<\bar{\rho }<\infty \), assume that the initial data \((\rho _0,\textbf{u}_0,\textbf{w}_0)\) satisfies

$$\begin{aligned} 0\le \rho _0\le \bar{\rho },\ \rho _0\in H^1,\ \textbf{u}_0\in D_{0,\sigma }^1,\ \textbf{w}_0\in H_{0}^1,\ \nabla \mu (\rho _0)\in L^q. \end{aligned}$$
(1.7)

Then there exists some small positive constant \(\varepsilon _0\) depending on \(\Omega ,q,\xi ,\eta ,\underline{\mu }, \bar{\mu }\triangleq \sup \limits _{[0,\bar{\rho }]}\mu (\rho )\), \(\bar{\rho }\), \(\Vert \nabla \mu (\rho _0)\Vert _{L^q}\), \(\Vert \nabla \textbf{u}_0\Vert _{L^2}\), and \(\Vert \nabla \textbf{w}_0\Vert _{L^2}\) such that if

$$\begin{aligned} C_0\triangleq \Vert \sqrt{\rho _0}\textbf{u}_0\Vert _{L^2}^2 +\Vert \sqrt{\rho _0}\textbf{w}_0\Vert _{L^2}^2\le \varepsilon _0, \end{aligned}$$
(1.8)

the problem (1.1)–(1.3) has a unique global strong solution \((\rho ,\textbf{u},\textbf{w})\) satisfying that, for any \(0<\tau <\infty \) and \(r\in (2,q)\),

$$\begin{aligned} {\left\{ \begin{array}{ll} 0\le \rho \le \bar{\rho },\ \rho \in C([0,\infty );H^1),\ \nabla \mu (\rho )\in C([0,\infty );L^q), \\ \sqrt{\rho }\textbf{u},\ \nabla \textbf{u},\ t\sqrt{\rho }\textbf{u}_t,\ \sqrt{t}\nabla P,\ \sqrt{t}\nabla ^2\textbf{u}\in L^\infty (0,\infty ; L^2), \\ \sqrt{\rho }\textbf{w},\ \nabla \textbf{w},\ t\sqrt{\rho }\textbf{w}_t,\ \sqrt{t}\nabla ^2\textbf{w}\in L^\infty (0,\infty ; L^2), \\ \textbf{u}, \textbf{w}\in L^2(0,\infty ;H^2)\cap L^2(\tau ,\infty ; W^{2,r}), \\ \nabla P\in L^2(0,\infty ;L^2)\cap L^2(\tau ,\infty ;L^r), \\ \sqrt{\rho }\textbf{u}_t,\ \sqrt{\rho }\textbf{w}_t,\ t\nabla \textbf{u}_t, \ t\nabla \textbf{w}_t\in L^2(\Omega \times (0,\infty )). \end{array}\right. } \end{aligned}$$
(1.9)

Moreover, it holds that

$$\begin{aligned} \sup \limits _{0\le t<\infty }\Vert \nabla \rho \Vert _{L^2} \le 2\Vert \nabla \rho _0\Vert _{L^2},\ \sup \limits _{0\le t<\infty }\Vert \nabla \mu (\rho )\Vert _{L^q} \le 2\Vert \nabla \mu (\rho _0)\Vert _{L^q}, \end{aligned}$$
(1.10)

and there exists a positive constant C depending only on \(\Omega ,q,\xi ,\eta ,\underline{\mu },\bar{\mu }\), \(\bar{\rho }\), and the initial data such that, for \(t\ge 1\),

$$\begin{aligned} \Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^2 +\Vert \sqrt{\rho }\textbf{w}_t\Vert _{L^2}^2 +\Vert \textbf{u}(\cdot ,t)\Vert _{H^2}^2 +\Vert \nabla P(\cdot ,t)\Vert _{L^2}^2 +\Vert \textbf{w}(\cdot ,t)\Vert _{H^2}^2 \le Ce^{-\sigma t}, \end{aligned}$$
(1.11)

where \(\sigma \triangleq \min \left\{ \frac{\underline{\mu }}{\bar{\rho }d^2}, \frac{\eta }{\bar{\rho }d^2}\right\} \) with d being the diameter of \(\Omega \).

Remark 1.1

Our Theorem 1.1 extends the recent work [17] obtained by Wu and Zhong to the density-dependent viscosity case. However, it is not a simple exercise due to the strong coupling between viscosity coefficient and velocity.

Remark 1.2

Very recently, Liu and Zhong [11] proved the global existence and exponential decay estimates of strong solutions for 2D nonhomogeneous micropolar fluids with density-dependent viscosity and vacuum in bounded domains provided that \(\Vert \nabla \mu (\rho _0)\Vert _{L^q}\) is suitably small. Here we should point out that it seems impossible to obtain similar results under such smallness condition for 3D case. Otherwise, we can establish a global large the strong solution for the 3D problem with constant viscosity. As is well known, this is the biggest open problem in the studies of fluid equations.

We now make some comments on the key ingredient for the proof of Theorem 1.1. To extend the local solution to be a global one, one needs to obtain global a priori estimates on solutions in suitable higher norms. We will adapt some basic ideas used in [7, 19], where the authors investigated the global existence of strong solutions to the 3D nonhomogeneous Navier–Stokes equations with density-dependent viscosity and vacuum. However, compared with [7, 19], the proof of Theorem 1.1 is much more involved due to the strong coupling between the velocity and the micro-rotational velocity and the absence of the compatibility condition. Consequently, some new ideas are needed to overcome these difficulties.

First, applying the upper bounds on the density (see (3.1)) and the Poincaré’s inequality, we derive that \(\Vert \sqrt{\rho }\textbf{u}\Vert ^2_{L^2} +\Vert \sqrt{\rho }\textbf{w}\Vert ^2_{L^2}\) decays at the rate of \(e^{-\sigma t}\) for some \(\sigma >0\) depending only on \(\underline{\mu },\eta ,\bar{\rho }\), and the diameter of the \(\Omega \) (see (3.3)). Next, we need to obtain time-weighted estimates of \(\Vert \nabla \textbf{u}\Vert _{L^2}^{2}+\Vert \nabla \textbf{w}\Vert _{L^2}^{2}\). To overcome the difficulties caused by the density-dependent viscosity, motivated by [7, 19], we assume that \(\Vert \nabla \mu (\rho )(t)\Vert _{L^q} \le 4\Vert \nabla \mu (\rho _0)\Vert _{L^q}\) (see (3.7)). Moreover, owing to the strong coupling between the velocity and the micro-rotational velocity, we assume the condition (3.10). Then, with the help of regularity properties of Stokes equations (see Lemma 2.4) and elliptic equations, we can obtain the desired bound of \(\Vert \nabla \textbf{u}\Vert _{L^2}^{2}+\Vert \nabla \textbf{w}\Vert _{L^2}^{2}\) under the assumption that the initial energy is suitably small (see Lemma 3.2). These bounds are crucial in deriving the time-weighted estimates on the \(L^\infty (0,T;L^2)\)-norm of \(\sqrt{\rho }\textbf{u}_t\) and \(\sqrt{\rho }\textbf{w}_t\). The next key step is to show that the quantity \(\Vert \nabla \mu (\rho )(t)\Vert _{L^q}\) is in fact less than \(2\Vert \nabla \mu (\rho _0)\Vert _{L^q}\). To this end, it needs to tackle \(\Vert \nabla \textbf{u}\Vert _{L^1(0,T;L^\infty )}\). Indeed, based on t-weighted estimates (see Lemmas 3.1–3.3), we find that the uniform bound (with respect to time) on the \(L^1(0,T;L^\infty )\)-norm of \(\nabla \textbf{u}\) is bounded by the initial energy (see (3.38)). This in particular completes the proof of (3.8) as long as the assumption (1.8) stated in Theorem 1.1 holds true (see (3.46)). Finally, the higher-order estimates on solutions are obtained (see Lemma 3.6) by considering the time-weighted type due to the lacking of the compatibility conditions.

When there is no microstructure (\(\xi =0\) and \(\textbf{w}=\textbf{0}\)), the system (1.1) reduces to nonhomogeneous Navier–Stokes equations with density-dependent viscosity. Our method can be applied to this case and obtain the following global existence and exponential decay estimates of strong solutions.

Theorem 1.2

For constants \(q\in (3,6)\) and \(0<\bar{\rho }<\infty \), assume that the initial data \((\rho _0,\textbf{u}_0)\) satisfies

$$\begin{aligned} 0\le \rho _0\le \bar{\rho },\ \rho _0\in H^1,\ \textbf{u}_0\in D_{0,\sigma }^1,\ \nabla \mu (\rho _0)\in L^q. \end{aligned}$$

Then there exists some small positive constant \(\varepsilon _0\) depending on \(\Omega ,q,\underline{\mu }, \bar{\mu }\triangleq \sup \limits _{[0,\bar{\rho }]}\mu (\rho )\), \(\bar{\rho }\), \(\Vert \nabla \mu (\rho _0)\Vert _{L^q}\), and \(\Vert \nabla \textbf{u}_0\Vert _{L^2}\) such that if

$$\begin{aligned} \Vert \sqrt{\rho _0}\textbf{u}_0\Vert _{L^2}^2\le \varepsilon _0, \end{aligned}$$

the 3D nonhomogeneous Navier–Stokes equations with density-dependent viscosity, that is (1.1)–(1.3) with \(\xi =0\) and \(\textbf{w}=\textbf{0}\), has a unique global strong solution \((\rho ,\textbf{u})\) satisfying that, for any \(0<\tau <\infty \) and \(r\in (2,q)\),

$$\begin{aligned} {\left\{ \begin{array}{ll} 0\le \rho \le \bar{\rho },\ \rho \in C([0,\infty );H^1),\ \nabla \mu (\rho )\in C([0,\infty );L^q), \\ \sqrt{\rho }\textbf{u},\ \nabla \textbf{u},\ t\sqrt{\rho }\textbf{u}_t,\ \sqrt{t}\nabla P,\ \sqrt{t}\nabla ^2\textbf{u}\in L^\infty (0,\infty ; L^2), \\ \textbf{u}\in L^2(0,\infty ;H^2)\cap L^2(\tau ,\infty ; W^{2,r}), \\ \nabla P\in L^2(0,\infty ;L^2)\cap L^2(\tau ,\infty ;L^r), \\ \sqrt{\rho }\textbf{u}_t,\ t\nabla \textbf{u}_t\in L^2(\Omega \times (0,\infty )). \end{array}\right. } \end{aligned}$$

Moreover, it holds that

$$\begin{aligned} \sup \limits _{0\le t<\infty }\Vert \nabla \rho \Vert _{L^2} \le 2\Vert \nabla \rho _0\Vert _{L^2},\ \sup \limits _{0\le t<\infty }\Vert \nabla \mu (\rho )\Vert _{L^q} \le 2\Vert \nabla \mu (\rho _0)\Vert _{L^q}, \end{aligned}$$

and there exists a positive constant C depending only on \(\Omega ,q,\underline{\mu },\bar{\mu }\), \(\bar{\rho }\), and the initial data such that, for \(t\ge 1\),

$$\begin{aligned} \Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^2 +\Vert \textbf{u}(\cdot ,t)\Vert _{H^2}^2 +\Vert \nabla P(\cdot ,t)\Vert _{L^2}^2 \le Ce^{-\sigma t}, \end{aligned}$$

where \(\sigma \triangleq \frac{\underline{\mu }}{\bar{\rho }d^2}\) with d being the diameter of \(\Omega \).

Remark 1.3

Theorem 1.2 generalizes previous results for the 3D Navier–Stokes equations in [7, 19], which need some compatibility conditions on the initial data and \(\Vert \nabla \textbf{u}_0\Vert _{L^2}\) to be small. Moreover, we get exponential decay rates of the solution rather than algebraic decay rates as those in [7, 19].

The rest of the paper is organized as follows. In Sect. 2, we collect some elementary facts and inequalities which will be needed in later analysis. Section 3 is devoted to the proof of Theorem 1.1.

2 Preliminaries

In this section, we will recall some known facts and elementary inequalities which will be used frequently later.

We start with the local existence of strong solutions to the problem (1.1)–(1.3), whose proof can be performed by using similar strategies as those in [13].

Lemma 2.1

Assume that \((\rho _0,\textbf{u}_0,\textbf{w}_0)\) satisfies (1.7), then there exist a small time \(T>0\) and a unique strong solution \((\rho ,\textbf{u}, \textbf{w})\) to the problem (1.1)–(1.3) in \(\Omega \times (0,T]\).

Next, the following Gronwall’s inequality (see [16, pp. 12–13]) will play a central role in showing a priori estimates on strong solutions \((\rho ,\textbf{u},\textbf{w})\).

Lemma 2.2

Suppose that h and r are integrable on (a, b) and nonnegative a.e. in (a, b). Further assume that \(y\in C[a, b], y'\in L^1(a, b)\), and

$$\begin{aligned} y'(t)\le h(t)+r(t)y(t)\ \ \text {for}\ a.e\ t\in (a,b). \end{aligned}$$

Then

$$\begin{aligned} y(t)\le \left[ y(a)+\int _{a}^{t}h(s)\textrm{d}s\right] \exp \left( \int _{a}^{t}r(s)\textrm{d}s\right) ,\ \ t\in [a,b]. \end{aligned}$$

Next, the following Gagliardo–Nirenberg inequality (see [4, Theorem 10.1, p. 27]) will be useful in the next section.

Lemma 2.3

(1) Let \(p\in [2,\frac{3r}{3-r}]\) for \(r\in [2,3)\), or \(p\in [2,\infty )\) for \(r=3\), then there exists some generic positive constant C which may depend on r such that for any \(f\in L^2\cap D^{1,r}_0\),

$$\begin{aligned} \Vert f\Vert _{L^p}^p\le C\Vert f\Vert _{L^2}^{p-\frac{3r(p-2)}{5r-6}}\Vert \nabla f\Vert _{L^r}^{\frac{3r(p-2)}{5r-6}}. \end{aligned}$$
(2.1)

(2) Let \(q\in (1,\infty )\) and \(r\in (3,\infty )\), then there exists some generic positive constant C which may depend on q, r such that for any \(g\in L^q\cap D^{1,r}_0\),

$$\begin{aligned} \Vert g\Vert _{L^\infty }\le C\Vert g\Vert _{L^q}^{\frac{q(r-3)}{3r+q(r-3)}}\Vert \nabla g\Vert _{L^r}^{\frac{3r}{3r+q(r-3)}}. \end{aligned}$$
(2.2)

Next, we state the following regularity properties on the Stokes system, please refer to [7, Lemma 2.1] for the proof.

Lemma 2.4

For constants \(q\in (3,6)\), \(\underline{\mu }, \bar{\mu }>0\), let the function \(\mu \) satisfy

$$\begin{aligned} \nabla \mu \in L^q,\ \underline{\mu }\le \mu \le \bar{\mu }. \end{aligned}$$
(2.3)

Assume that \((\textbf{u},P)\in D^1_{0,\sigma }\times L^2\) is the unique weak solution to the following problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -{{\,\textrm{div}\,}}(\mu \nabla \textbf{u})+\nabla P=\textbf{F},\ {} &{} x\in \Omega ,\\ {{\,\textrm{div}\,}}\textbf{u}=0,\ {} &{} x\in \Omega ,\\ \textbf{u}=\textbf{0},\ {} &{} x\in \partial \Omega . \end{array}\right. } \end{aligned}$$
(2.4)

Then there exists some positive constant C depending only on \(\underline{\mu },\bar{\mu },q\), and \(\Omega \) such that the followings hold.

  • If \(\textbf{F}\in L^2\), then \((\textbf{u}, \nabla P)\in H^2\times L^2\) and

    $$\begin{aligned} \Vert \textbf{u}\Vert _{H^2}+\Vert \nabla P\Vert _{L^2} \le C\Vert \textbf{F}\Vert _{L^2} \left( 1+\Vert \nabla \mu \Vert _{L^q}^{\frac{q}{q-3}}\right) . \end{aligned}$$
    (2.5)

  • If \(\textbf{F}\in L^r\) for some \(r\in (2,q)\), then \((\textbf{u},\nabla P)\in W^{2,r}\times L^{r}\) and

    $$\begin{aligned} \Vert \textbf{u}\Vert _{W^{2,r}}+ \Vert \nabla P\Vert _{L^r} \le C \Vert \textbf{F}\Vert _{L^r} \left( 1+\Vert \nabla \mu \Vert _{L^q}^{\frac{q(5r-6)}{2r(q-3)}}\right) . \end{aligned}$$
    (2.6)

3 Proof of Theorem 1.1

3.1 A Priori Estimates

In this subsection, we will establish some necessary a priori bounds of local strong solutions \((\rho ,\textbf{u},\textbf{w})\) to the problem (1.1)–(1.3) whose existence is guaranteed by Lemma 2.1. Thus, let \(T>0\) be a fixed time and \((\rho ,\textbf{u},\textbf{w})\) be the smooth solution (1.1)–(1.3) on \(\Omega \times (0,T]\) with smooth initial data \((\rho _0,\textbf{u}_0,\textbf{w}_0)\) satisfying (1.7). In what follows, we write

$$\begin{aligned} \int \cdot \textrm{d}x=\int _{\Omega }\cdot \textrm{d}x. \end{aligned}$$

For simplicity, we will use the letter \(C_i\ (i=1,2,\cdots )\) to denote the generic positive constant which may depend on \(\Omega ,q,\xi ,\eta ,\underline{\mu },\bar{\mu },\bar{\rho },\) and the initial data, but independent of T and \(C_0\).

We start with the following standard energy estimate and the boundedness of the density.

Lemma 3.1

It holds that

$$\begin{aligned}&0\le \rho (x,t)\le \bar{\rho }\ \ \text {for}\ \ (x,t)\in \Omega \times [0,T], \end{aligned}$$
(3.1)
$$\begin{aligned}&\sup _{0\le t\le T} \left( \Vert \sqrt{\rho }\textbf{u}\Vert ^2_{L^2} +\Vert \sqrt{\rho }\textbf{w}\Vert _{L^2}^2\right) +\int _{0}^{T}\left( \underline{\mu }\Vert \nabla \textbf{u}\Vert _{L^2}^2 +\eta \Vert \nabla \textbf{w}\Vert _{L^2}^2\right) \textrm{d}t \le C_0, \end{aligned}$$
(3.2)

and

$$\begin{aligned}&\sup _{0\le t\le T}\left[ e^{\sigma t}\left( \Vert \sqrt{\rho }\textbf{u}\Vert ^2_{L^2} +\Vert \sqrt{\rho }\textbf{w}\Vert _{L^2}^2\right) \right] +\int _{0}^{T}e^{\sigma t}\left( \underline{\mu }\Vert \nabla \textbf{u}\Vert _{L^2}^2 +\eta \Vert \nabla \textbf{w}\Vert _{L^2}^2\right) \textrm{d}t \le C_0, \end{aligned}$$
(3.3)

where \(\sigma \triangleq \min \left\{ \frac{\underline{\mu }}{\bar{\rho }d^2}, \frac{\eta }{\bar{\rho }d^2}\right\} \) with d being the diameter of \(\Omega \).

Proof

1. (1.1)\(_1\) and (1.1)\(_4\) show that \(\rho \) satisfies a transport equation, thus we obtain immediately (3.1).

2. Multiplying (1.1)\(_2\) by \(\textbf{u}\) and (1.1)\(_3\) by \(\textbf{w}\), respectively, we get after integration by parts that

$$\begin{aligned}&\frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\left( \Vert \sqrt{\rho }\textbf{u}\Vert _{L^2}^2 +\Vert \sqrt{\rho }\textbf{w}\Vert _{L^2}^2\right) +\int (\mu (\rho )+\xi )|\nabla \textbf{u}|^2\textrm{d}x +\eta \Vert \nabla \textbf{w}\Vert _{L^2}^2\nonumber \\&\qquad +4\xi \Vert \textbf{w}\Vert _{L^2}^2+(\eta +\lambda )\Vert {{\,\textrm{div}\,}}\textbf{w}\Vert _{L^2}^2 \nonumber \\&\quad =4\xi \int {{\,\textrm{curl}\,}}\textbf{u}\cdot \textbf{w}\textrm{d}x \le \xi \Vert \nabla \textbf{u}\Vert _{L^2}^2+4\xi \Vert \textbf{w}\Vert _{L^2}^2, \end{aligned}$$

which together with (1.4) gives that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\left( \Vert \sqrt{\rho }\textbf{u}\Vert _{L^2}^2 +\Vert \sqrt{\rho }\textbf{w}\Vert _{L^2}^2\right) +2\underline{\mu }\Vert \nabla \textbf{u}\Vert _{L^2}^2+2\eta \Vert \nabla \textbf{w}\Vert _{L^2}^2 \le 0. \end{aligned}$$
(3.4)

Integrating (3.4) over [0, T] leads to (3.2).

3. It follows from (3.1) and Poincaré’s inequality (see [15, (A.3), p. 266]) that

$$\begin{aligned} \Vert \sqrt{\rho }\textbf{u}\Vert _{L^2}^2+\Vert \sqrt{\rho }\textbf{w}\Vert _{L^2}^2 \le \Vert \rho \Vert _{L^\infty }\left( \Vert \textbf{u}\Vert _{L^2}^2 +\Vert \textbf{w}\Vert _{L^2}^2\right) \le \bar{\rho }d^2\left( \Vert \nabla \textbf{u}\Vert _{L^2}^2 +\Vert \nabla \textbf{w}\Vert _{L^2}^2\right) , \end{aligned}$$
(3.5)

where d is the diameter of \(\Omega \). Hence, for \(\sigma =\min \big \{\frac{\underline{\mu }}{\bar{\rho }d^2}, \frac{\eta }{\bar{\rho }d^2}\big \}\), we derive from (3.5) and (3.4) that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}&\left( \Vert \sqrt{\rho }\textbf{u}\Vert _{L^2}^2 +\Vert \sqrt{\rho }\textbf{w}\Vert _{L^2}^2\right) +\sigma \left( \Vert \sqrt{\rho }\textbf{u}\Vert _{L^2}^2+\Vert \sqrt{\rho }\textbf{w}\Vert _{L^2}^2\right) \\ {}&\quad +\underline{\mu }\Vert \nabla \textbf{u}\Vert _{L^2}^2+\eta \Vert \nabla \textbf{w}\Vert _{L^2}^2 \le 0. \end{aligned}$$

This implies that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\left[ e^{\sigma t}\left( \Vert \sqrt{\rho }\textbf{u}\Vert _{L^2}^2 +\Vert \sqrt{\rho }\textbf{w}\Vert _{L^2}^2\right) \right] +e^{\sigma t}\left( \underline{\mu }\Vert \nabla \textbf{u}\Vert _{L^2}^2 +\eta \Vert \nabla \textbf{w}\Vert _{L^2}^2\right) \le 0. \end{aligned}$$

Integrating the above inequality over [0, T] yields (3.3) and completes the proof of Lemma 3.1. \(\square \)

Remark 3.1

Since \(\mu (\rho )\) is a continuously differentiable function, we deduce from (3.1) and (1.4) that

$$\begin{aligned} 0<\underline{\mu }\le \mu (\rho )\le \bar{\mu } \triangleq \max _{0\le \rho \le \bar{\rho }}\mu (\rho )<\infty . \end{aligned}$$
(3.6)

Next, we have the following key estimates on \((\rho ,\textbf{u},\textbf{w})\).

Proposition 3.1

There exists some positive constant \(\varepsilon _0\) depending only on \(\Omega ,q,\xi ,\eta ,\underline{\mu }\), \(\bar{\mu }\), \(\bar{\rho }\), \(\Vert \nabla \mu (\rho _0)\Vert _{L^q}\), \(\Vert \nabla \textbf{u}_0\Vert _{L^2}\), and \(\Vert \nabla \textbf{w}_0\Vert _{L^2}\) such that if \((\rho ,\textbf{u},\textbf{w})\) is a smooth solution of (1.1)–(1.3) on \(\Omega \times (0,T]\) satisfying

$$\begin{aligned} \sup \limits _{0\le t\le T}\Vert \nabla \mu (\rho )\Vert _{L^q} \le 4\Vert \nabla \mu (\rho _0)\Vert _{L^q}, \end{aligned}$$
(3.7)

then the following estimate holds

$$\begin{aligned} \sup \limits _{0\le t\le T}\Vert \nabla \mu (\rho )\Vert _{L^q} \le 2\Vert \nabla \mu (\rho _0)\Vert _{L^q}, \end{aligned}$$
(3.8)

provided that

$$\begin{aligned} C_0\le \varepsilon _0. \end{aligned}$$
(3.9)

Before proving Proposition 3.1, we establish some necessary a priori estimates. We have the following time-weighted estimates on the \(L^\infty (0,T;L^2)\)-norm of the gradients of velocity and micro-rotational velocity.

Lemma 3.2

Let \((\rho , \textbf{u}, \textbf{w})\) be a smooth solution to (1.1)–(1.3) satisfying (3.7). Then there exists some positive constant \(\varepsilon _1\) depending only on \(\Omega ,\bar{\rho },q,\underline{\mu },\bar{\mu },\xi ,\eta ,\) \(\Vert \nabla \mu (\rho _0)\Vert _{L^q}\), \(\Vert \nabla \textbf{u}_0\Vert _{L^2}^2\), and \(\Vert \nabla \textbf{w}_0\Vert _{L^2}^2\) such that if

$$\begin{aligned}&\sup _{0\le t\le T}\left( \underline{\mu }\Vert \nabla \textbf{u}\Vert _{L^2}^2 +\eta \Vert \nabla \textbf{w}\Vert _{L^2}^2+(\eta +\lambda )\Vert {{\,\textrm{div}\,}}\textbf{w}\Vert _{L^2}^2 +\xi \Vert 2\textbf{w}-{{\,\textrm{curl}\,}}\textbf{u}\Vert _{L^2}^2\right) \nonumber \\&\quad \le 4\left( \bar{\mu }\Vert \nabla \textbf{u}_0\Vert _{L^2}^2 +\eta \Vert \nabla \textbf{w}_0\Vert _{L^2}^2+(\eta +\lambda )\Vert {{\,\textrm{div}\,}}\textbf{w}_0\Vert _{L^2}^2 +\xi \Vert 2\textbf{w}_0-{{\,\textrm{curl}\,}}\textbf{u}_0\Vert _{L^2}^2\right) , \end{aligned}$$
(3.10)

then

$$\begin{aligned}&\sup _{0\le t\le T}\left( \underline{\mu }\Vert \nabla \textbf{u}\Vert _{L^2}^2 +\eta \Vert \nabla \textbf{w}\Vert _{L^2}^2+(\eta +\lambda )\Vert {{\,\textrm{div}\,}}\textbf{w}\Vert _{L^2}^2 +\xi \Vert 2\textbf{w}-{{\,\textrm{curl}\,}}\textbf{u}\Vert _{L^2}^2\right) \nonumber \\&\qquad +\int _{0}^{T}\left( \Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^2 +\Vert \sqrt{\rho }\textbf{w}_t\Vert _{L^2}^2\right) \textrm{d}t \nonumber \\&\quad \le 2\left( \bar{\mu }\Vert \nabla \textbf{u}_0\Vert _{L^2}^2 +\eta \Vert \nabla \textbf{w}_0\Vert _{L^2}^2+(\eta +\lambda )\Vert {{\,\textrm{div}\,}}\textbf{w}_0\Vert _{L^2}^2 +\xi \Vert 2\textbf{w}_0-{{\,\textrm{curl}\,}}\textbf{u}_0\Vert _{L^2}^2\right) , \end{aligned}$$
(3.11)

provided that

$$\begin{aligned} C_0\le \varepsilon _1. \end{aligned}$$
(3.12)

Moreover, for \(t\in \{1,2\}\), one has that

$$\begin{aligned}&\sup _{0\le t\le T}\left[ t^i\left( \Vert \nabla \textbf{u}\Vert _{L^2}^2+\Vert \nabla \textbf{w}\Vert _{L^2}^2\right) \right] +\int _{0}^{T}t^i\left( \Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^2 +\Vert \sqrt{\rho }\textbf{w}_t\Vert _{L^2}^2\right) \textrm{d}t \le CC_0, \end{aligned}$$
(3.13)

with some positive constant C depending only on \(\Omega ,\bar{\rho },q,\underline{\mu },\bar{\mu },\xi ,\eta ,\) \(\Vert \nabla \mu (\rho _0)\Vert _{L^q}\), \(\Vert \nabla \textbf{u}_0\Vert _{L^2}\), and \(\Vert \nabla \textbf{w}_0\Vert _{L^2}\).

Proof

1. Multiplying (1.1)\(_{2}\) by \(\textbf{u}_t\) and noting that

$$\begin{aligned} {[}\mu (\rho )]_t+\textbf{u}\cdot \nabla \mu (\rho )=0, \end{aligned}$$
(3.14)

we obtain after using \(\Delta \textbf{u}=\nabla {{\,\textrm{div}\,}}\textbf{u}-{{\,\textrm{curl}\,}}({{\,\textrm{curl}\,}}\textbf{u}) =-{{\,\textrm{curl}\,}}({{\,\textrm{curl}\,}}\textbf{u})\) and integration by parts that

$$\begin{aligned}&\frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\int \mu (\rho )|\nabla \textbf{u}|^2\textrm{d}x +\frac{\xi }{2}\frac{\textrm{d}}{\textrm{d}t}\int |{{\,\textrm{curl}\,}}\textbf{u}|^2\textrm{d}x +\int \rho |\textbf{u}_t|^2\textrm{d}x \nonumber \\&\quad = -\int \rho \textbf{u}\cdot \nabla \textbf{u}\cdot \textbf{u}_t\textrm{d}x -\frac{1}{2}\int \textbf{u}\cdot \nabla \mu (\rho )|\nabla \textbf{u}|^2\textrm{d}x +2\xi \int {{\,\textrm{curl}\,}}\textbf{w}\cdot \textbf{u}_t\textrm{d}x \nonumber \\&\quad = -\int \rho \textbf{u}\cdot \nabla \textbf{u}\cdot \textbf{u}_t\textrm{d}x -\frac{1}{2}\int \textbf{u}\cdot \nabla \mu (\rho )|\nabla \textbf{u}|^2\textrm{d}x +2\xi \int {{\,\textrm{curl}\,}}\textbf{u}_t\cdot \textbf{w}\textrm{d}x \nonumber \\&\quad = -\int \rho \textbf{u}\cdot \nabla \textbf{u}\cdot \textbf{u}_t\textrm{d}x -\frac{1}{2}\int \textbf{u}\cdot \nabla \mu (\rho )|\nabla \textbf{u}|^2\textrm{d}x +2\xi \frac{\textrm{d}}{\textrm{d}t}\int {{\,\textrm{curl}\,}}\textbf{u}\cdot \textbf{w}\textrm{d}x\nonumber \\&\qquad -2\xi \int {{\,\textrm{curl}\,}}\textbf{u}\cdot \textbf{w}_t\textrm{d}x. \end{aligned}$$
(3.15)

Multiplying (1.1)\(_{3}\) by \(\textbf{w}_t\) and integration by parts, we get that

$$\begin{aligned}&\frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\int \left( \eta |\nabla \textbf{w}|^2 +(\eta +\lambda )({{\,\textrm{div}\,}}\textbf{w})^2 +4\xi |\textbf{w}|^2\right) \textrm{d}x +\int \rho |\textbf{w}_t|^2\textrm{d}x \nonumber \\&\quad = -\int \rho \textbf{u}\cdot \nabla \textbf{w}\cdot \textbf{w}_t\textrm{d}x +2\xi \int {{\,\textrm{curl}\,}}\textbf{u}\cdot \textbf{w}_t\textrm{d}x, \end{aligned}$$
(3.16)

which combined with (3.15) leads to

$$\begin{aligned}&\frac{1}{2}B'(t)+\int \rho |\textbf{u}_t|^2\textrm{d}x+\int \rho |\textbf{w}_t|^2\textrm{d}x \nonumber \\&\quad = -\int \rho \textbf{u}\cdot \nabla \textbf{u}\cdot \textbf{u}_t\textrm{d}x -\int \rho \textbf{u}\cdot \nabla \textbf{w}\cdot \textbf{w}_t\textrm{d}x -\frac{1}{2}\int \textbf{u}\cdot \nabla \mu (\rho )|\nabla \textbf{u}|^2\textrm{d}x \triangleq \sum _{i=1}^3J_i, \end{aligned}$$
(3.17)

where

$$\begin{aligned} B(t)\triangleq \int \left( \mu (\rho )|\nabla \textbf{u}|^2 +\eta |\nabla \textbf{w}|^2+(\eta +\lambda )({{\,\textrm{div}\,}}\textbf{w})^2 +\xi |2\textbf{w}-{{\,\textrm{curl}\,}}\textbf{u}|^2\right) \textrm{d}x \end{aligned}$$

satisfies

$$\begin{aligned}&\underline{\mu }\Vert \nabla \textbf{u}\Vert _{L^2}^2 +\eta \Vert \nabla \textbf{w}\Vert _{L^2}^2+(\eta +\lambda )\Vert {{\,\textrm{div}\,}}\textbf{w}\Vert _{L^2}^2 +\xi \Vert 2\textbf{w}-{{\,\textrm{curl}\,}}\textbf{u}\Vert _{L^2}^2 \le B(t)\nonumber \\&\quad \le C\Vert \nabla \textbf{u}\Vert _{L^2}^2+C\Vert \nabla \textbf{w}\Vert _{L^2}^2 \end{aligned}$$
(3.18)

due to (1.4), Poincaré’s inequality, and the following

$$\begin{aligned} \left| -4\xi \int {{\,\textrm{curl}\,}}\textbf{u}\cdot \textbf{w}\textrm{d}x\right| \le \xi \int |\nabla \textbf{u}|^2\textrm{d}x+4\xi \int |\textbf{w}|^2\textrm{d}x. \end{aligned}$$

From (3.1), Hölder’s inequality, Sobolev’s inequality, and (3.7), we have that

$$\begin{aligned} |J_1|&\le \bar{\rho }^{\frac{1}{2}}\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2} \Vert \textbf{u}\Vert _{L^6}\Vert \nabla \textbf{u}\Vert _{L^3} \nonumber \\&\le C(\Omega ,\bar{\rho })\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2} \Vert \nabla \textbf{u}\Vert _{L^2}^{\frac{3}{2}} \Vert \nabla \textbf{u}\Vert _{L^6}^{\frac{1}{2}} \nonumber \\&\le \frac{1}{2}\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^2 +C\Vert \nabla \textbf{u}\Vert _{L^2}^3\Vert \nabla \textbf{u}\Vert _{H^1}, \end{aligned}$$
(3.19)
$$\begin{aligned} |J_2|&\le \bar{\rho }^{\frac{1}{2}}\Vert \sqrt{\rho }\textbf{w}_t\Vert _{L^2} \Vert \textbf{u}\Vert _{L^6}\Vert \nabla \textbf{w}\Vert _{L^3} \nonumber \\&\le C(\Omega ,\bar{\rho })\Vert \sqrt{\rho }\textbf{w}_t\Vert _{L^2} \Vert \nabla \textbf{u}\Vert _{L^2}\Vert \nabla \textbf{w}\Vert _{L^2}^{\frac{1}{2}} \Vert \nabla \textbf{w}\Vert _{L^6}^{\frac{1}{2}} \nonumber \\&\le \frac{1}{2}\Vert \sqrt{\rho }\textbf{w}_t\Vert _{L^2}^2 +C\Vert \nabla \textbf{u}\Vert _{L^2}^2 \Vert \nabla \textbf{w}\Vert _{L^2}\Vert \nabla \textbf{w}\Vert _{H^1}, \end{aligned}$$
(3.20)
$$\begin{aligned} |J_3|&\le \frac{1}{2}\Vert \textbf{u}\Vert _{L^6}\Vert \nabla \mu \Vert _{L^3} \Vert \nabla \textbf{u}\Vert _{L^4}^2\nonumber \\&\le C(\Omega ,q)\Vert \nabla \textbf{u}\Vert _{L^2}\Vert \nabla \mu \Vert _{L^q} \Vert \nabla \textbf{u}\Vert _{L^2}^{\frac{1}{2}}\Vert \nabla \textbf{u}\Vert _{L^6}^{\frac{3}{2}} \nonumber \\&\le C\Vert \nabla \textbf{u}\Vert _{L^2}^{\frac{3}{2}} \Vert \nabla \textbf{u}\Vert _{H^1}^{\frac{3}{2}}. \end{aligned}$$
(3.21)

Thus, inserting (3.19)–(3.21) into (3.17) gives that

$$\begin{aligned}&B'(t)+\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^2+\Vert \sqrt{\rho }\textbf{w}_t\Vert _{L^2}^2 \nonumber \\&\quad \le C\Vert \nabla \textbf{u}\Vert _{L^2}^3\Vert \nabla \textbf{u}\Vert _{H^1} +C\Vert \nabla \textbf{u}\Vert _{L^2}^2\Vert \nabla \textbf{w}\Vert _{L^2}\Vert \nabla \textbf{w}\Vert _{H^1} +C\Vert \nabla \textbf{u}\Vert _{L^2}^{\frac{3}{2}} \Vert \nabla \textbf{u}\Vert _{H^1}^{\frac{3}{2}}. \end{aligned}$$
(3.22)

2. Since \((\textbf{u},P)\) satisfies the following Stokes system

$$\begin{aligned} {\left\{ \begin{array}{ll} -{{\,\textrm{div}\,}}((\mu (\rho )+\xi )\nabla \textbf{u})+\nabla P = -\rho \textbf{u}_t-\rho \textbf{u}\cdot \nabla \textbf{u}+2\xi {{\,\textrm{curl}\,}}\textbf{w},\,\,\,\,&{}x\in \Omega ,\\ {{\,\textrm{div}\,}}\textbf{u}=0, \,\,\,&{}x\in \Omega ,\\ \textbf{u}(x)=\textbf{0},\,\,\,\,&{}x\in \partial \Omega , \end{array}\right. } \end{aligned}$$
(3.23)

applying Lemma 2.4 with \(\textbf{F}=-\rho \textbf{u}_t-\rho \textbf{u}\cdot \nabla \textbf{u} +2\xi {{\,\textrm{curl}\,}}\textbf{w}\), we then infer from (3.7) and (3.1) that

$$\begin{aligned} \Vert \textbf{u}\Vert _{H^2}+\Vert \nabla P\Vert _{L^2}&\le C \Vert \textbf{F}\Vert _{L^2}\left( 1+\Vert \nabla \mu \Vert _{L^q}^{\frac{q}{q-3}}\right) \nonumber \\&\le C\Vert \rho \textbf{u}_t\Vert _{L^2}+C\Vert \rho \textbf{u}\cdot \nabla \textbf{u}\Vert _{L^2} + C\Vert \nabla \textbf{w}\Vert _{L^2} \nonumber \\&\le C\bar{\rho }^{\frac{1}{2}}\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2} +C\bar{\rho }\Vert \textbf{u}\Vert _{L^6}\Vert \nabla \textbf{u}\Vert _{L^3} +C\Vert \nabla \textbf{w}\Vert _{L^2} \nonumber \\&\le C\Vert \sqrt{\rho } \textbf{u}_t\Vert _{L^2}+C\Vert \nabla \textbf{u}\Vert _{L^2} ^{\frac{3}{2}} \Vert \nabla \textbf{u}\Vert _{H^1}^{\frac{1}{2}}+C\Vert \nabla \textbf{w}\Vert _{L^2} \nonumber \\&\le \frac{1}{2}\Vert \nabla \textbf{u}\Vert _{H^1} +C\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}+C\Vert \nabla \textbf{u}\Vert _{L^2}^3 +C\Vert \nabla \textbf{w}\Vert _{L^2}, \end{aligned}$$

which implies that

$$\begin{aligned} \Vert \textbf{u}\Vert _{H^2}+\Vert \nabla P\Vert _{L^2} \le C\Vert \sqrt{\rho } \textbf{u}_t\Vert _{L^2} +C\Vert \nabla \textbf{w}\Vert _{L^2}+C\Vert \nabla \textbf{u}\Vert _{L^2}^3. \end{aligned}$$
(3.24)

Multiplying (1.1)\(_3\) by \(\textbf{w}\) and integration by parts yield that

$$\begin{aligned}&4\xi \Vert \textbf{w}\Vert _{L^2}^2+\eta \Vert \nabla \textbf{w}\Vert _{L^2}^2 +(\eta +\lambda )\Vert {{\,\textrm{div}\,}}\textbf{w}\Vert _{L^2} \le \Vert 2\xi {{\,\textrm{curl}\,}}\textbf{u}-\rho \textbf{w}_t-\rho \textbf{u}\cdot \nabla \textbf{w}\Vert _{L^2}\Vert \textbf{w}\Vert _{L^2} \\&\qquad \le \xi \Vert \textbf{w}\Vert _{L^2}^2+C\left( \Vert \rho \textbf{w}_t\Vert _{L^2}^2 +\Vert \rho \textbf{u}\cdot \nabla \textbf{w}\Vert _{L^2}^2 +\Vert \nabla \textbf{u}\Vert _{L^2}^2\right) , \end{aligned}$$

which implies that

$$\begin{aligned} \Vert \textbf{w}\Vert _{H^1} \le C\left( \Vert \rho \textbf{w}_t\Vert _{L^2} +\Vert \rho \textbf{u}\cdot \nabla \textbf{w}\Vert _{L^2} +\Vert \nabla \textbf{u}\Vert _{L^2}\right) . \end{aligned}$$
(3.25)

Employing the \(L^2\)-theory of elliptic equations (see [5, Chapter 9]), we get from (1.1)\(_3\) that

$$\begin{aligned} \Vert \nabla ^2\textbf{w}\Vert _{L^2}&\le C\Vert 2\xi {{\,\textrm{curl}\,}}\textbf{u}-\rho \textbf{w}_t-\rho \textbf{u}\cdot \nabla \textbf{w} -4\xi \textbf{w}\Vert _{L^2} \\&\le C\Vert \textbf{w}\Vert _{L^2}+C\left( \Vert \rho \textbf{w}_t\Vert _{L^2} +\Vert \rho \textbf{u}\cdot \nabla \textbf{w}\Vert _{L^2} +\Vert \nabla \textbf{u}\Vert _{L^2}\right) . \end{aligned}$$

This combined with (3.25) and (3.1) leads to

$$\begin{aligned} \Vert \textbf{w}\Vert _{H^2}&\le C\left( \Vert \rho \textbf{w}_t\Vert _{L^2}+\Vert \rho \textbf{u}\cdot \nabla \textbf{w}\Vert _{L^2} +\Vert \nabla \textbf{u}\Vert _{L^2}\right) \nonumber \\&\le C\bar{\rho }^{\frac{1}{2}}\Vert \sqrt{\rho }\textbf{w}_t\Vert _{L^2} +C\bar{\rho }\Vert \textbf{u}\Vert _{L^6}\Vert \nabla \textbf{w}\Vert _{L^3} + C\Vert \nabla \textbf{u}\Vert _{L^2} \nonumber \\&\le C\Vert \sqrt{\rho }\textbf{w}_t\Vert _{L^2} +C\Vert \nabla \textbf{u}\Vert _{L^2}\Vert \nabla \textbf{w}\Vert _{L^2}^{\frac{1}{2}} \Vert \nabla \textbf{w}\Vert _{H^1}^{\frac{1}{2}}+C\Vert \nabla \textbf{u}\Vert _{L^2} \nonumber \\&\le \frac{1}{2}\Vert \nabla \textbf{w}\Vert _{H^1} +C\Vert \sqrt{\rho } \textbf{w}_t\Vert _{L^2}+C\Vert \nabla \textbf{u}\Vert _{L^2} +C\Vert \nabla \textbf{u}\Vert _{L^2}^2\Vert \nabla \textbf{w}\Vert _{L^2}, \end{aligned}$$

which yields that

$$\begin{aligned} \Vert \textbf{w}\Vert _{H^2} \le C\Vert \sqrt{\rho }\textbf{w}_t\Vert _{L^2} +C\Vert \nabla \textbf{u}\Vert _{L^2} +C\Vert \nabla \textbf{u}\Vert _{L^2}^2\Vert \nabla \textbf{w}\Vert _{L^2}. \end{aligned}$$
(3.26)

Substituting (3.24) and (3.26) into (3.22), one gets after using Young’s inequality and (3.10) that

$$\begin{aligned} B'(t)+\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^2 +\Vert \sqrt{\rho }\textbf{w}_t\Vert _{L^2}^2 \le C\left( \Vert \nabla \textbf{u}\Vert _{L^2}^2+\Vert \nabla \textbf{w}\Vert _{L^2}^2\right) . \end{aligned}$$
(3.27)

Integrating (3.27) over [0, T], we obtain from (3.2) that

$$\begin{aligned}&\sup _{0\le t\le T}\left( \underline{\mu }\Vert \nabla \textbf{u}\Vert _{L^2}^2 +\eta \Vert \nabla \textbf{w}\Vert _{L^2}^2+(\eta +\lambda )\Vert {{\,\textrm{div}\,}}\textbf{w}\Vert _{L^2}^2 +\xi \Vert 2\textbf{w}-{{\,\textrm{curl}\,}}\textbf{u}\Vert _{L^2}^2\right) \nonumber \\&\qquad +\int _{0}^{T}\left( \Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^2 +\Vert \sqrt{\rho }\textbf{w}_t\Vert _{L^2}^2\right) \textrm{d}t \nonumber \\&\quad \le \bar{\mu }\Vert \nabla \textbf{u}_0\Vert _{L^2}^2 +\eta \Vert \nabla \textbf{w}_0\Vert _{L^2}^2+(\eta +\lambda )\Vert {{\,\textrm{div}\,}}\textbf{w}_0\Vert _{L^2}^2 +\xi \Vert 2\textbf{w}_0-{{\,\textrm{curl}\,}}\textbf{u}_0\Vert _{L^2}^2+C_1C_0, \end{aligned}$$
(3.28)

which implies (3.11) provided that

$$\begin{aligned} C_0\le \varepsilon _1\triangleq \frac{\bar{\mu }\Vert \nabla \textbf{u}_0\Vert _{L^2}^2 +\eta \Vert \nabla \textbf{w}_0\Vert _{L^2}^2+(\eta +\lambda )\Vert {{\,\textrm{div}\,}}\textbf{w}_0\Vert _{L^2}^2 +\xi \Vert 2\textbf{w}_0-{{\,\textrm{curl}\,}}\textbf{u}_0\Vert _{L^2}^2}{C_1}. \end{aligned}$$

3. Multiplying (3.27) by \(t^i\ (i=1,2)\), we get from (3.18) that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}(t^iB(t))+t^i\left( \Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^2 +\Vert \sqrt{\rho }\textbf{w}_t\Vert _{L^2}^2\right) \le C\left( t^i+t^{i-1}\right) \left( \Vert \nabla \textbf{u}\Vert _{L^2}^2+\Vert \nabla \textbf{w}\Vert _{L^2}^2\right) . \end{aligned}$$

This along with Gronwall’s inequality, (3.18), and (3.3) leads to (3.13). \(\square \)

As an application of Lemmas 3.1 and 3.2, we get the following time-weighted estimates on \(\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^2\) and \(\Vert \sqrt{\rho }\textbf{w}_t\Vert _{L^2}^2\), which play crucial roles in dealing with time-independent bound on the \(L^1(0,T;L^\infty )\)-norm of \(\nabla \textbf{u}\).

Lemma 3.3

Let \((\rho , \textbf{u}, \textbf{w})\) be a smooth solution to (1.1)–(1.3) satisfying (3.7) and (3.10). Then there exists a positive constant C depending only on \(\Omega ,q,\bar{\rho },\xi ,\eta ,\underline{\mu },\overline{\mu },\) \(\Vert \nabla \mu (\rho _0)\Vert _{L^q}\), \(\Vert \nabla \textbf{u}_0\Vert _{L^2}\), and \(\Vert \nabla \textbf{w}_0\Vert _{L^2}\) such that, for \(t\in \{1,2\}\),

$$\begin{aligned}&\sup _{0\le t\le T}\left[ t^i\left( \Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^2 +\Vert \sqrt{\rho }\textbf{w}_t\Vert _{L^2}^2\right) \right] +\int _{0}^{T}t^i\left( \Vert \nabla \textbf{u}_t\Vert _{L^2}^2 +\Vert \nabla \textbf{w}_t\Vert _{L^2}^2\right) \textrm{d}t \le CC_0, \end{aligned}$$
(3.29)

provided that (3.12) holds true.

Proof

Differentiating (1.1)\(_2\) and (1.1)\(_3\) with respect to t and using (1.1)\(_1\) give rise to

$$\begin{aligned}&\rho \textbf{u}_{tt}+\rho \textbf{u}\cdot \nabla \textbf{u}_t -{{\,\textrm{div}\,}}((\mu (\rho )+\xi )\nabla \textbf{u})_t +\nabla P_t =(\textbf{u}\cdot \nabla \rho )(\textbf{u}_t+\textbf{u}\cdot \nabla \textbf{u})\nonumber \\&\qquad -\rho \textbf{u}_t\cdot \nabla \textbf{u}+2\xi {{\,\textrm{curl}\,}}\textbf{w}_t, \end{aligned}$$
(3.30)
$$\begin{aligned}&\rho \textbf{w}_{tt}+\rho \textbf{u}\cdot \nabla \textbf{w}_t+4\xi \textbf{w}_t -\eta \Delta \textbf{w}_t -(\eta +\lambda )\nabla {{\,\textrm{div}\,}}\textbf{w}_t \nonumber \\&\quad =(\textbf{u}\cdot \nabla \rho )(\textbf{w}_t +\textbf{u}\cdot \nabla \textbf{w}) -\rho \textbf{u}_t\cdot \nabla \textbf{w}+2\xi {{\,\textrm{curl}\,}}\textbf{u}_t. \end{aligned}$$
(3.31)

Multiplying (3.30) by \(\textbf{u}_t\), (3.31) by \(\textbf{w}_t\), and integrating the resulting equalities by parts over \(\Omega \) and summing them, we obtain from (3.14) that

$$\begin{aligned}&\frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\int (\rho |\textbf{u}_t|^2+\rho |\textbf{w}_t|^2)\textrm{d}x +\int (\mu (\rho )+\xi )|\nabla \textbf{u}_t|^2 \textrm{d}x+4\xi \int |\textbf{w}_t|^2\textrm{d}x \nonumber \\&\qquad +\eta \int |\nabla \textbf{w}_t|^2\textrm{d}x +(\eta +\lambda )\int ({{\,\textrm{div}\,}}\textbf{w}_t)^2\textrm{d}x \nonumber \\&\quad =\int ((\textbf{u}\cdot \nabla \rho )(\textbf{u}_t+\textbf{u}\cdot \nabla \textbf{u}) \cdot \textbf{u}_{t}-\rho \textbf{u}_t\cdot \nabla \textbf{u}\cdot \textbf{u}_{t})\textrm{d}x \nonumber \\&\qquad +\int ((\textbf{u}\cdot \nabla \rho )(\textbf{w}_t+\textbf{u}\cdot \nabla \textbf{w}) \cdot \textbf{w}_{t}-\rho \textbf{u}_t\cdot \nabla \textbf{w}\cdot \textbf{w}_{t})\textrm{d}x \nonumber \\&\qquad +2\xi \int ({{\,\textrm{curl}\,}}\textbf{w}_t\cdot \textbf{u}_t +{{\,\textrm{curl}\,}}\textbf{u}_t\cdot \textbf{w}_t)\textrm{d}x +\int \textbf{u}\cdot \nabla \mu (\rho )\nabla \textbf{u}\cdot \nabla \textbf{u}_t\textrm{d}x \triangleq \sum _{i=1}^4 \hat{I}_i. \end{aligned}$$
(3.32)

After integration by parts, we infer from Hölder’s inequality, Sobolev’s inequality, (3.1), (3.10), and Young’s inequality that

$$\begin{aligned} \hat{I}_1&\le C\int \rho |\textbf{u}|\left( |\textbf{u}_t||\nabla \textbf{u}_t| +|\textbf{u}||\nabla ^2\textbf{u}||\textbf{u}_t| +|\textbf{u}||\nabla \textbf{u}||\nabla \textbf{u}_t| +|\nabla \textbf{u}|^2|\textbf{u}_t|\right) \textrm{d}x\nonumber \\ {}&\qquad +\int \rho |\textbf{u}_t|^2|\nabla \textbf{u}|\textrm{d}x \nonumber \\&\le C\bar{\rho }^{\frac{1}{2}}\Vert \textbf{u}\Vert _{L^6} \Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^3}\Vert \nabla \textbf{u}_{t}\Vert _{L^2} +C\bar{\rho }\Vert \textbf{u}\Vert _{L^6}^2 \Vert \nabla ^2\textbf{u}\Vert _{L^2}\Vert \textbf{u}_t\Vert _{L^6}\nonumber \\ {}&\qquad +C\bar{\rho }\Vert \textbf{u}\Vert _{L^6}^2 \Vert \nabla \textbf{u}\Vert _{L^6}\Vert \nabla \textbf{u}_t\Vert _{L^2} \nonumber \\&\quad +C\bar{\rho }\Vert \textbf{u}\Vert _{L^6}\Vert \nabla \textbf{u}\Vert _{L^6} \Vert \nabla \textbf{u}\Vert _{L^2}\Vert \textbf{u}_{t}\Vert _{L^6} +\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^3}\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^6} \Vert \nabla \textbf{u}\Vert _{L^2} \nonumber \\&\le C\bar{\rho }^{\frac{1}{2}}\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^{\frac{1}{2}} \Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^6}^{\frac{1}{2}}\Vert \nabla \textbf{u}_t\Vert _{L^2} +C\bar{\rho }\Vert \nabla \textbf{u}\Vert _{L^2}^2 \Vert \nabla \textbf{u}\Vert _{H^1}\Vert \nabla \textbf{u}_t\Vert _{L^2} \nonumber \\&\quad +C\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^{\frac{1}{2}} \Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^6}^{\frac{3}{2}}\Vert \nabla \textbf{u}\Vert _{L^2} \nonumber \\&\le C\bar{\rho }^{\frac{3}{4}}\Vert \sqrt{\rho }\textbf{u}_{t}\Vert _{L^2}^{\frac{1}{2}} \Vert \nabla \textbf{u}_{t}\Vert _{L^2}^{\frac{3}{2}} +C\Vert \nabla \textbf{u}\Vert _{H^1}\Vert \nabla \textbf{u}_{t}\Vert _{L^2} \nonumber \\&\le \delta \Vert \nabla \textbf{u}_{t}\Vert _{L^{2}}^{2}+C\Vert \sqrt{\rho } \textbf{u}_{t}\Vert _{L^2}^{2}+C\Vert \nabla \textbf{u}\Vert _{H^1}^2. \end{aligned}$$
(3.33)

Similarly, one has that

$$\begin{aligned} \hat{I}_2 \le&C\int \rho |\textbf{w}|\left( |\textbf{w}_t||\nabla \textbf{w}_t| +|\textbf{u}||\nabla ^2\textbf{w}||\textbf{w}_t| +|\textbf{u}||\nabla \textbf{w}||\nabla \textbf{w}_t| +|\nabla \textbf{u}||\nabla \textbf{w}||\textbf{w}_t|\right) \textrm{d}x\nonumber \\&+\int \rho |\textbf{u}_t||\textbf{w}_t||\nabla \textbf{w}|\textrm{d}x \nonumber \\ \le&\delta \Vert \nabla \textbf{w}_{t}\Vert _{L^{2}}^{2} +C\Vert \sqrt{\rho }\textbf{w}_{t}\Vert _{L^2}^{2} +C\Vert \nabla \textbf{w}\Vert _{H^1}^2. \end{aligned}$$
(3.34)

We obtain from integration by parts and Cauchy–Schwarz inequality that

$$\begin{aligned} \hat{I}_3 =4\xi \int {{\,\textrm{curl}\,}}\textbf{u}_t\cdot \textbf{w}_t\textrm{d}x \le 4\xi \Vert \textbf{w}_t\Vert _{L^2}^2+\xi \Vert \nabla \textbf{u}_t\Vert _{L^2}^2. \end{aligned}$$
(3.35)

By virtue of Hölder’s inequality, Sobolev’s inequality, and (3.7), we derive that

$$\begin{aligned} \hat{I}_4 \le C\Vert \nabla \mu (\rho )\Vert _{L^q}\Vert \textbf{u}\Vert _{L^\infty } \Vert \nabla \textbf{u}\Vert _{L^{\frac{2q}{q-2}}}\Vert \nabla \textbf{u}_t\Vert _{L^2}&\le C\Vert \nabla \textbf{u}\Vert _{H^1}^2 \Vert \nabla \textbf{u}_t\Vert _{L^2}\nonumber \\ {}&\le \delta \Vert \nabla \textbf{u}_t\Vert _{L^2}^2 +C\Vert \nabla \textbf{u}\Vert _{H^1}^4. \end{aligned}$$
(3.36)

Substituting (3.33)–(3.36) into (3.32), we obtain after choosing \(\delta \) suitably small that

$$\begin{aligned}&\frac{\textrm{d}}{\textrm{d}t}\left( \Vert \sqrt{\rho }\textbf{u}_{t}\Vert _{L^2}^{2} +\Vert \sqrt{\rho }\textbf{w}_{t}\Vert _{L^2}^{2}\right) +\Vert \nabla \textbf{u}_{t}\Vert _{L^{2}}^{2} +\Vert \nabla \textbf{w}_{t}\Vert _{L^{2}}^{2} \nonumber \\&\quad \le C\left( \Vert \sqrt{\rho }\textbf{u}_{t}\Vert _{L^2}^{2} +\Vert \sqrt{\rho }\textbf{w}_{t}\Vert _{L^2}^2\right) +C\Vert \nabla \textbf{u}\Vert _{H^1}^2+C\Vert \nabla \textbf{w}\Vert _{H^1}^2+C\Vert \nabla \textbf{u}\Vert _{H^1}^4. \end{aligned}$$

This combined with (3.24), (3.26), and (3.10) yields that

$$\begin{aligned}&\frac{\textrm{d}}{\textrm{d}t}\left( \Vert \sqrt{\rho }\textbf{u}_{t}\Vert _{L^2}^{2} +\Vert \sqrt{\rho }\textbf{w}_{t}\Vert _{L^2}^{2}\right) +\Vert \nabla \textbf{u}_{t}\Vert _{L^{2}}^{2} +\Vert \nabla \textbf{w}_{t}\Vert _{L^{2}}^{2} \nonumber \\&\quad \le C\Vert \sqrt{\rho }\textbf{w}_{t}\Vert _{L^2}^2\Vert \sqrt{\rho }\textbf{w}_{t}\Vert _{L^2}^2+C\left( \Vert \sqrt{\rho }\textbf{u}_{t}\Vert _{L^2}^{2} +\Vert \sqrt{\rho }\textbf{w}_{t}\Vert _{L^2}^{2}\right) +C\left( \Vert \nabla \textbf{u}\Vert _{L^2}^2+\Vert \nabla \textbf{w}\Vert _{L^2}^2\right) , \end{aligned}$$
(3.37)

which multiplied by \(t^i\ (i\in \{1,2\})\) together with Gronwall’s inequality, (3.3), (3.11), and (3.13) leads to (3.29). \(\square \)

Now we can use Lemmas 3.2 and 3.3 to obtain the following time-in-uniform bound on the \(L^1(0,T;L^\infty )\)-norm of \(\nabla \textbf{u}\) which is very important for proving Proposition 3.1.

Lemma 3.4

Let \((\rho , \textbf{u}, \textbf{w})\) be a smooth solution to (1.1)–(1.3) satisfying (3.7) and (3.10). Then there exists a generic positive constant C depending only on \(\Omega ,q,\bar{\rho },\xi ,\eta ,\underline{\mu },\overline{\mu },\) \(\Vert \nabla \mu (\rho _0)\Vert _{L^q}\), \(\Vert \nabla \textbf{u}_0\Vert _{L^2}\), and \(\Vert \nabla \textbf{w}_0\Vert _{L^2}\) such that

$$\begin{aligned} \int _{0}^{T}\Vert \nabla \textbf{u}\Vert _{L^\infty }\textrm{d}t \le CC_0+CC_0^{\frac{1}{2}}, \end{aligned}$$
(3.38)

provided that (3.12) holds true.

Proof

It follows from Lemma 2.4, Hölder’s inequality, (3.7), (3.1), and Sobolev’s inequality that, for \(r\in (3,q)\),

$$\begin{aligned}&\Vert \textbf{u}\Vert _{W^{2,r}}+\Vert \nabla P\Vert _{L^r} \nonumber \\&\quad \le C \left( \Vert \rho \textbf{u}_t\Vert _{L^r}+\Vert \rho \textbf{u}\cdot \nabla \textbf{u}\Vert _{L^r}+\Vert \nabla \textbf{w}\Vert _{L^r}\right) \left( 1+\Vert \nabla \mu \Vert _{L^q}^{\frac{q(5r-6)}{2r(q-3)}}\right) \nonumber \\&\quad \le C\bar{\rho }^{\frac{1}{2}}\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^{\frac{6-r}{2r}} \Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^6}^{\frac{3r-6}{2r}} +C\bar{\rho }\Vert \textbf{u}\Vert _{L^6}\Vert \nabla \textbf{u}\Vert _{L^{\frac{6r}{6-r}}} +C\Vert \nabla \textbf{w}\Vert _{L^2}^{\frac{6-r}{2r}}\Vert \nabla \textbf{w}\Vert _{L^6}^{\frac{3r-6}{2r}} \nonumber \\&\quad \le C\bar{\rho }^{\frac{5r-6}{4r}}\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^{\frac{6-r}{2r}} \Vert \nabla \textbf{u}_t\Vert _{L^2}^{\frac{3r-6}{2r}} +C\Vert \nabla \textbf{u}\Vert _{L^2}^{\frac{6r-6}{5r-6}} \Vert \nabla \textbf{u}\Vert _{W^{1,r}}^{\frac{4r-6}{5r-6}} +C\Vert \nabla \textbf{w}\Vert _{L^2}^{\frac{6-r}{2r}}\Vert \nabla \textbf{w}\Vert _{H^1}^{\frac{3r-6}{2r}} \nonumber \\&\quad \le C\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^{\frac{6-r}{2r}} \Vert \nabla \textbf{u}_t\Vert _{L^2}^{\frac{3r-6}{2r}} +C\Vert \nabla \textbf{u}\Vert _{L^2}^{\frac{6r-6}{r}} +\frac{1}{2}\Vert \nabla \textbf{u}\Vert _{W^{1,r}} +C\Vert \nabla \textbf{w}\Vert _{L^2}^{\frac{6-r}{2r}}\Vert \nabla \textbf{w}\Vert _{H^1}^{\frac{3r-6}{2r}}. \end{aligned}$$

This implies that

$$\begin{aligned} \Vert \textbf{u}\Vert _{W^{2,r}} \le C\Vert \nabla \textbf{u}\Vert _{L^2}^{\frac{6r-6}{r}} +C\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^{\frac{6-r}{2r}} \Vert \nabla \textbf{u}_t\Vert _{L^2}^{\frac{3r-6}{2r}} +C\Vert \nabla \textbf{w}\Vert _{L^2}^{\frac{6-r}{2r}}\Vert \nabla \textbf{w}\Vert _{H^1}^{\frac{3r-6}{2r}}. \end{aligned}$$
(3.39)

We derive from (3.13) and (3.2) that

$$\begin{aligned} \int _{0}^{T}\Vert \nabla \textbf{u}\Vert _{L^2}^{\frac{6r-6}{r}}\textrm{d}t \le \left( \sup _{0\le t\le T}\Vert \nabla \textbf{u}\Vert _{L^2}^2\right) ^{\frac{2r-3}{r}} \int _{0}^{T}\Vert \nabla \textbf{u}\Vert _{L^2}^2\textrm{d}t\le CC_0. \end{aligned}$$
(3.40)

From (3.26), (3.11), (3.13), and (3.3), one has that, for \(i\in \{1,2\}\),

$$\begin{aligned} \int _{0}^{T}t^i\Vert \textbf{w}\Vert _{H^2}^2\textrm{d}t\le CC_0. \end{aligned}$$
(3.41)

If \(0\le T\le 1\), we deduce from Hölder’s inequality, (3.29), (3.13), and (3.41) that

$$\begin{aligned}&\int _{0}^{T}\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^{\frac{6-r}{2r}} \Vert \nabla \textbf{u}_t\Vert _{L^2}^{\frac{3r-6}{2r}}\textrm{d}t +\int _{0}^{T}\Vert \nabla \textbf{w}\Vert _{L^2}^{\frac{6-r}{2r}} \Vert \nabla \textbf{w}\Vert _{H^1}^{\frac{3r-6}{2r}}\textrm{d}t \nonumber \\&\quad = \int _{0}^{T}\left( t\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^2\right) ^{\frac{6-r}{4r}} \left( t\Vert \nabla \textbf{u}_t\Vert _{L^2}^2\right) ^{\frac{3r-6}{4r}}t^{-\frac{1}{2}}\textrm{d}t\nonumber \\ {}&\qquad +\int _{0}^{T}\left( t\Vert \nabla \textbf{w}\Vert _{L^2}^2\right) ^{\frac{6-r}{4r}} \left( t\Vert \nabla \textbf{w}\Vert _{H^1}^2\right) ^{\frac{3r-6}{4r}}t^{-\frac{1}{2}}\textrm{d}t \nonumber \\&\quad \le \sup _{0\le t\le T}\left( t\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^2\right) ^{\frac{6-r}{4r}} \left( \int _{0}^{T}t\Vert \nabla \textbf{u}_t\Vert _{L^2}^2\textrm{d}t\right) ^{\frac{3r-6}{4r}} \left( \int _{0}^{T}t^{-\frac{2r}{r+6}}\textrm{d}t\right) ^{\frac{r+6}{4r}} \nonumber \\&\quad \quad +\sup _{0\le t\le T} \left( t\Vert \nabla \textbf{w}\Vert _{L^2}^2\right) ^{\frac{6-r}{4r}} \left( \int _{0}^{T}t\Vert \nabla \textbf{w}\Vert _{H^1}^2\textrm{d}t\right) ^{\frac{3r-6}{4r}} \left( \int _{0}^{T}t^{-\frac{2r}{r+6}}\textrm{d}t\right) ^{\frac{r+6}{4r}} \nonumber \\&\quad \le CC_0^{\frac{1}{2}}. \end{aligned}$$
(3.42)

If \(T>1\), we get from Hölder’s inequality, (3.29), (3.13), and (3.41) that

$$\begin{aligned}&\int _{1}^{T}\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^{\frac{6-r}{2r}} \Vert \nabla \textbf{u}_t\Vert _{L^2}^{\frac{3r-6}{2r}}\textrm{d}t +\int _{1}^{T}\Vert \nabla \textbf{w}\Vert _{L^2}^{\frac{6-r}{2r}} \Vert \nabla \textbf{w}\Vert _{H^1}^{\frac{3r-6}{2r}}\textrm{d}t \nonumber \\&\quad = \int _{1}^{T}\left( t^2\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^2\right) ^{\frac{6-r}{4r}} \left( t^2\Vert \nabla \textbf{u}_t\Vert _{L^2}^2\right) ^{\frac{3r-6}{4r}}t^{-\frac{1}{2}}\textrm{d}t\nonumber \\ {}&\qquad +\int _{1}^{T}\left( t^2\Vert \nabla \textbf{w}\Vert _{L^2}^2\right) ^{\frac{6-r}{4r}} \left( t^2\Vert \nabla \textbf{w}\Vert _{H^1}^2\right) ^{\frac{3r-6}{4r}}t^{-\frac{1}{2}}\textrm{d}t \nonumber \\&\quad \le \sup _{1\le t\le T}\left( t^2\Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^2\right) ^{\frac{6-r}{4r}} \left( \int _{1}^{T}t^2\Vert \nabla \textbf{u}_t\Vert _{L^2}^2\textrm{d}t\right) ^{\frac{3r-6}{4r}} \left( \int _{1}^{T}t^{-\frac{4r}{r+6}}\textrm{d}t\right) ^{\frac{r+6}{4r}} \nonumber \\&\quad \quad +\sup _{1\le t\le T} \left( t^2\Vert \nabla \textbf{w}\Vert _{L^2}^2\right) ^{\frac{6-r}{4r}} \left( \int _{1}^{T}t^2\Vert \nabla \textbf{w}\Vert _{H^1}^2\textrm{d}t\right) ^{\frac{3r-6}{4r}} \left( \int _{1}^{T}t^{-\frac{4r}{r+6}}\textrm{d}t\right) ^{\frac{r+6}{4r}} \nonumber \\&\quad \le CC_0^{\frac{1}{2}}. \end{aligned}$$
(3.43)

Consequently, (3.38) follows from (3.39), Sobolev’s inequality, (3.40), (3.42), and (3.43). \(\square \)

Now, Proposition 3.1 is a direct consequence of Lemmas 3.1–3.4.

Proof of Proposition 3.1

Taking spatial derivative \(\nabla \) on the transport equation (3.14) leads to

$$\begin{aligned} (\nabla \mu (\rho ))_{t}+\textbf{u}\cdot \nabla ^2\mu (\rho ) +\nabla \textbf{u}\cdot \nabla \mu (\rho )=\textbf{0}. \end{aligned}$$
(3.44)

Multiplying (3.44) by \(q|\nabla \mu (\rho )|^{q-2}\nabla \mu (\rho )\) and integrating the resulting equation over \(\Omega \) give rise to

$$\begin{aligned}&\frac{\textrm{d}}{\textrm{d}t}\int |\nabla \mu (\rho )|^q\textrm{d}x +q\int \textbf{u}\cdot \nabla ^2\mu (\rho )\cdot |\nabla \mu (\rho )|^{q-2}\nabla \mu (\rho )\textrm{d}x\\&\qquad =-q\int \nabla \textbf{u}\cdot \nabla \mu (\rho )\cdot |\nabla \mu (\rho )|^{q-2}\nabla \mu (\rho )\textrm{d}x. \end{aligned}$$

Integration by parts together with \({{\,\textrm{div}\,}}\textbf{u}=0\) yields

$$\begin{aligned}&q\int \textbf{u}\cdot \nabla ^2\mu (\rho )\cdot |\nabla \mu (\rho )|^{q-2}\nabla \mu (\rho )\textrm{d}x =\int \textbf{u}\cdot \nabla (|\nabla \mu (\rho )|^q)\textrm{d}x\\&\qquad =-\int |\nabla \mu (\rho )|^q{{\,\textrm{div}\,}}\textbf{u}\textrm{d}x=0. \end{aligned}$$

Thus, we get that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\Vert \nabla \mu (\rho )\Vert _{L^q}^q \le q\int |\nabla \textbf{u}||\nabla \mu (\rho )|^q\textrm{d}x \le q\Vert \nabla \textbf{u}\Vert _{L^\infty }\Vert \nabla \mu (\rho )\Vert _{L^q}^q. \end{aligned}$$

This implies that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\Vert \nabla \mu (\rho )\Vert _{L^q} \le \Vert \nabla \textbf{u}\Vert _{L^\infty }\Vert \nabla \mu (\rho )\Vert _{L^q}, \end{aligned}$$
(3.45)

which together with Gronwall’s inequality and (3.38) yields that

$$\begin{aligned} \sup _{0\le t\le T}\Vert \nabla \mu (\rho )\Vert _{L^q}&\le \Vert \nabla \mu (\rho _0)\Vert _{L^q} \exp \left\{ \int _0^T\Vert \nabla \textbf{u}\Vert _{L^\infty }\textrm{d}t\right\} \nonumber \\&\le e^{C_2\left( C_0+C_0^{\frac{1}{2}}\right) }\Vert \nabla \mu (\rho _0)\Vert _{L^q} \le 2\Vert \nabla \mu (\rho _0)\Vert _{L^q}, \end{aligned}$$
(3.46)

provided that

$$\begin{aligned} C_0\le \varepsilon _0 \triangleq \min \left\{ \varepsilon _1,\frac{\left( \sqrt{1+\frac{4\ln 2}{C_2}}-1\right) ^2}{4}\right\} . \end{aligned}$$
(3.47)

This completes the proof of Proposition 3.1. \(\square \)

Next, we will show the following exponential decay-in-time estimates on the solution \((\rho ,\textbf{u},\textbf{w})\) for the large time.

Lemma 3.5

Let \((\rho , \textbf{u}, \textbf{w})\) be a smooth solution to (1.1)–(1.3) satisfying (3.7) and (3.10). Then there exists a generic positive constant C depending only on \(\Omega ,q,\bar{\rho },\xi ,\eta ,\underline{\mu },\overline{\mu },\) \(\Vert \nabla \mu (\rho _0)\Vert _{L^q}\), \(C_0\), \(\Vert \nabla \textbf{u}_0\Vert _{L^2}\), and \(\Vert \nabla \textbf{w}_0\Vert _{L^2}\) such that

$$\begin{aligned}&\sup _{0\le t\le T}\left[ e^{\sigma t}\left( \Vert \nabla \textbf{u}\Vert _{L^2}^2+\Vert \nabla \textbf{w}\Vert _{L^2}^2\right) \right] +\int _{0}^{T}e^{\sigma t}\left( \Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^2 +\Vert \sqrt{\rho }\textbf{w}_t\Vert _{L^2}^2\right) \textrm{d}t\le C, \end{aligned}$$
(3.48)

provided that (3.12) holds true. Moreover, for \(\sigma \) as that in Lemma 3.1, one has that

$$\begin{aligned}&\sup _{\zeta (T)\le t\le T}\left[ e^{\sigma t}\left( \Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^2 +\Vert \sqrt{\rho }\textbf{w}_t\Vert _{L^2}^2\right) \right] +\int _{\zeta (T)}^{T}e^{\sigma t}\left( \Vert \nabla \textbf{u}_t\Vert _{L^2}^2 +\Vert \nabla \textbf{w}_t\Vert _{L^2}^2\right) \textrm{d}t\le C, \end{aligned}$$
(3.49)

with \(\zeta (T)\triangleq \min \{1,T\}\).

Proof

Multiplying (3.27) by \(e^{\sigma t}\) and applying (3.18), we obtain that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}(e^{\sigma t}B(t))+e^{\sigma t}\left( \Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^2 +\Vert \sqrt{\rho }\textbf{w}_t\Vert _{L^2}^2\right) \le Ce^{\sigma t}(\Vert \nabla \textbf{u}\Vert _{L^2}^2+\Vert \nabla \textbf{w}\Vert _{L^2}^2), \end{aligned}$$

which along with Gronwall’s inequality, (3.3), and (3.18) yields (3.48).

Multiplying (3.37) by \(e^{\sigma t}\) gives rise to

$$\begin{aligned}&\frac{\textrm{d}}{\textrm{d}t}\left[ e^{\sigma t}\left( \Vert \sqrt{\rho }\textbf{u}_{t}\Vert _{L^2}^{2} +\Vert \sqrt{\rho }\textbf{w}_{t}\Vert _{L^2}^{2}\right) \right] +e^{\sigma t}\left( \Vert \nabla \textbf{u}_{t}\Vert _{L^{2}}^{2} +\Vert \nabla \textbf{w}_{t}\Vert _{L^{2}}^{2}\right) \nonumber \\&\quad \le C \Vert \sqrt{\rho }\textbf{w}_{t}\Vert _{L^2}^{2}\left( e^{\sigma t}\Vert \sqrt{\rho }\textbf{w}_{t}\Vert _{L^2}^{2}\right) +Ce^{\sigma t}\left( \Vert \sqrt{\rho }\textbf{u}_{t}\Vert _{L^2}^{2} +\Vert \sqrt{\rho }\textbf{w}_{t}\Vert _{L^2}^{2}\right) \\&\qquad +Ce^{\sigma t}\left( \Vert \nabla \textbf{u}\Vert _{L^2}^2 +\Vert \nabla \textbf{w}\Vert _{L^2}^2\right) , \end{aligned}$$

which combined with Gronwall’s inequality, (3.11), (3.48), and (3.3) leads to (3.49). \(\square \)

Finally, we will prove the estimates on the higher-order derivatives of the solution \((\rho ,\textbf{u},\textbf{w})\).

Lemma 3.6

Let \((\rho , \textbf{u}, \textbf{w})\) be a smooth solution to (1.1)–(1.3) satisfying (3.7) and (3.10). Then there exists a generic positive constant C depending only on \(\Omega ,q,\bar{\rho },\xi ,\eta ,\underline{\mu },\overline{\mu },\) \(\Vert \nabla \mu (\rho _0)\Vert _{L^q}\), and the initial data such that, for \(r\in (2,q)\),

$$\begin{aligned}&\sup _{0\le t\le T}\left[ \Vert \rho \Vert _{H^1}+t\left( \Vert \textbf{u}\Vert _{H^2}^2 +\Vert \nabla P\Vert _{L^2}^2+\Vert \textbf{w}\Vert _{H^2}^2\right) \right] \nonumber \\&\qquad +\int _{0}^{T}t\left( \Vert \textbf{u}\Vert _{W^{2,r}}^2 +\Vert \nabla P\Vert _{L^r}^2+\Vert \textbf{w}\Vert _{W^{2,r}}^2\right) \textrm{d}t\le C, \end{aligned}$$
(3.50)

provided that (3.9) holds true. Moreover, for \(\sigma \) as that in Lemma 3.1 and \(\zeta (T)\) as that in Lemma 3.5, one has that

$$\begin{aligned} \sup _{\zeta (T)\le t\le T}\left[ e^{\sigma t}\left( \Vert \textbf{u}\Vert _{H^2}^2 +\Vert \nabla P\Vert _{L^2}^2+\Vert \textbf{w}\Vert _{H^2}^2\right) \right] \le C. \end{aligned}$$
(3.51)

Proof

1. Similarly to (3.46), we get that

$$\begin{aligned} \sup _{0\le t\le T}\Vert \nabla \rho \Vert _{L^2}\le 2\Vert \nabla \rho _0\Vert _{L^2}, \end{aligned}$$
(3.52)

provided that (3.47) holds true. This together with Gronwall’s inequality, (3.38), and (3.1) implies that

$$\begin{aligned} \sup _{0\le t\le T}\Vert \rho \Vert _{H^1}\le C. \end{aligned}$$

It deduces from (3.24), (3.26), and (3.11) that

$$\begin{aligned} \Vert \textbf{u}\Vert _{H^2}^2+\Vert \nabla P\Vert _{L^2}^2+\Vert \textbf{w}\Vert _{H^2}^2 \le C\left( \Vert \sqrt{\rho }\textbf{u}_t\Vert _{L^2}^2+\Vert \sqrt{\rho }\textbf{w}_t\Vert _{L^2}^2\right) +C\left( \Vert \nabla \textbf{u}\Vert _{L^2}^2+\Vert \nabla \textbf{w}\Vert _{L^2}^2\right) , \end{aligned}$$
(3.53)

which combined with (3.13) and (3.29) yields that

$$\begin{aligned} \sup _{0\le t\le T}\left[ t\left( \Vert \textbf{u}\Vert _{H^2}^2 +\Vert \nabla P\Vert _{L^2}^2+\Vert \textbf{w}\Vert _{H^2}^2\right) \right] \le C, \end{aligned}$$
(3.54)

while (3.51) follows from (3.53), (3.48), and (3.49).

2. We obtain from (3.53), (3.11), (3.13), and (3.3) that

$$\begin{aligned} \int _0^T(1+t)\left( \Vert \textbf{u}\Vert _{H^2}^2+\Vert \textbf{w}\Vert _{H^2}^2\right) \textrm{d}t\le C. \end{aligned}$$
(3.55)

From Lemma 2.4, (3.7), (3.1), and Sobolev’s inequality, we have that, for \(r\in (2,q)\),

$$\begin{aligned} \Vert \textbf{u}\Vert _{W^{2,r}}^2+\Vert \nabla P\Vert _{L^r}^2&\le C\left( \Vert \rho \textbf{u}_t\Vert _{L^r}^2 +\Vert \rho \textbf{u}\cdot \nabla \textbf{u}\Vert _{L^r}^2 +\Vert \nabla \textbf{w}\Vert _{L^r}^2\right) \nonumber \\&\le C\Vert \rho \Vert _{L^\infty }^2\Vert \nabla \textbf{u}_t\Vert _{L^2}^2 +C\Vert \rho \Vert _{L^\infty }^2\Vert \textbf{u}\Vert _{L^\infty }^2\Vert \nabla \textbf{u}\Vert _{L^r}^2 +C\Vert \nabla \textbf{w}\Vert _{H^1}^2 \nonumber \\&\le C\Vert \nabla \textbf{u}_t\Vert _{L^2}^2 +C\Vert \textbf{u}\Vert _{H^2}^2\Vert \textbf{u}\Vert _{H^2}^2 +C\Vert \textbf{w}\Vert _{H^2}^2, \end{aligned}$$

which along with (3.29), (3.54), and (3.55) indicates that

$$\begin{aligned} \int _0^Tt\left( \Vert \textbf{u}\Vert _{W^{2,r}}^2+\Vert \nabla P\Vert _{L^r}^2\right) \textrm{d}t\le C. \end{aligned}$$

3. We derive from (1.1)\(_3\), regularity theories of elliptic equations, (3.1), and Sobolev’s inequality that

$$\begin{aligned} \Vert \textbf{w}\Vert _{W^{2,r}}^2&\le C\Vert \rho \textbf{w}_t\Vert _{L^r}^2+C\Vert \rho \textbf{u}\cdot \nabla \textbf{w}\Vert _{L^r}^2 + C\Vert \nabla \textbf{u}\Vert _{L^r}^2+ C\Vert \textbf{w}\Vert _{H^2}^2 \nonumber \\&\le C\Vert \rho \Vert _{L^\infty }^2\Vert \nabla \textbf{w}_t\Vert _{L^2}^2 +C\Vert \rho \Vert _{L^\infty }^2\Vert \textbf{u}\Vert _{L^\infty }^2\Vert \nabla \textbf{w}\Vert _{L^r}^2 + C\Vert \nabla \textbf{u}\Vert _{H^1}^2 + C\Vert \textbf{w}\Vert _{H^2}^2 \nonumber \\&\le C\Vert \nabla \textbf{w}_t\Vert _{L^2}^2 +C\Vert \textbf{u}\Vert _{H^2}^2\Vert \textbf{w}\Vert _{H^2}^2 +C\Vert \textbf{u}\Vert _{H^2}^2+C\Vert \textbf{w}\Vert _{H^2}^2, \end{aligned}$$

which combined with (3.29), (3.54), and (3.55) yields that

$$\begin{aligned} \int _0^Tt\Vert \textbf{w}\Vert _{W^{2,r}}^2\textrm{d}t\le C. \end{aligned}$$

This finishes the proof of Lemma 3.6. \(\square \)

3.2 Proof of Theorem 1.1

With all the a priori estimates established in Lemmas 3.1–3.6, we can immediately obtain the existence result of Theorem 1.1 by standard arguments as those in [6]. Here we omit the details for simplicity. Moreover, (1.10) follows from (3.52) and (3.8), while the decay estimate (1.11) follows from (3.49) and (3.51). \(\square \)