Global Well-Posedness for 2D Nonhomogeneous Magneto-Micropolar Equations with Density-Dependent Viscosity

Abstract

We study an initial boundary value problem of 2D nonhomogeneous magneto-micropolar equations with density-dependent viscosity in smooth bounded domains. When the initial density can contain vacuum states, we prove that there is a unique global strong solution for the system under the assumption that initial velocity is suitably small. In particular, the initial data can be arbitrarily large except the gradient of velocity. Finally, we obtain the exponential decay rates of strong solutions by using the energy method.

1 Introduction

In the paper, we first introduce the following standard 3D nonhomogeneous incompressible magneto-micropolar equations

$$\begin{aligned} \left\{ \begin{aligned}&\rho _t +\text {div}(\rho u)=0,\\&(\rho u)_t+\text {div}(\rho u\otimes u)-\text {div}((\mu (\rho )+\kappa )\nabla u)+\nabla P=2\kappa \nabla \times w+b\cdot \nabla b,\\&(\rho w)_t+\text {div}(\rho u\otimes w)+4\kappa w-\gamma \triangle w-\lambda \nabla \text {div}w=2\kappa \nabla \times u,\\&b_t+\nu \triangle b+u\cdot \nabla b-b\cdot \nabla u=0,\\&\text {div} u=\text {div} b=0, \end{aligned} \right. \end{aligned}$$

(1.1)

where $\rho$, u, w, b and P are the density, velocity field, micro-rotational, magnetic field and pressure of the fluid, respectively. The positive constant $\gamma$ and $\lambda$ are the angular viscosities and $\kappa$ is the micro-rotation viscosity, while $\nu >0$ is the magnetic diffusive coefficient. The kinematic viscosity $\mu (\rho )$ satisfies the following hypothesis

$$\begin{aligned} \mu (\cdot )\in C^1[0, \infty ),\quad and\quad 0<{\underline{\mu }}\le \mu (\phi )\le {\bar{\mu }}<\infty ,\quad for\quad \forall \phi \in [0, \infty ). \end{aligned}$$

(1.2)

where ${\underline{\mu }}$ and ${\bar{\mu }}$ are some positive constant. In the special case when

$$\begin{aligned} \left\{ \begin{aligned}&\rho =\rho (x_1, x_2, t),\quad u=(u_1(x_1, x_2,t), u_2(x_1, x_2,t), 0),\\&b=(b_1(x_1, x_2,t), b_2(x_1, x_2,t), 0),\quad w=(0, 0, w(x_1, x_2,t)), \end{aligned} \right. \end{aligned}$$

(1.3)

the 3D micropolar equations reduce to the 2D micropolar equations

$$\begin{aligned} \left\{ \begin{aligned}&\rho _t +u\cdot \nabla \rho =0,\\&\rho u_t+\rho u\cdot \nabla u-\text {div}((\mu (\rho )+\kappa )\nabla u)+\nabla P=-2\kappa \nabla ^\bot w+b\cdot \nabla b,\\&\rho w_t+\rho u\cdot \nabla w+4\kappa w-\gamma \triangle w=2\kappa \nabla ^\bot \cdot u,\\&b_t+\nu \triangle b+u\cdot \nabla b-b\cdot \nabla u=0,\\&\text {div} u=\text {div} b=0. \end{aligned} \right. \end{aligned}$$

(1.4)

Here $u=(u_1, u_2)$ is a vector with the corresponding scalar vorticity, and w is a scalar function in what follows,

$$\begin{aligned} \nabla ^\bot \cdot u=\partial _1u_2-\partial _2u_1,\quad \nabla ^\bot w=(-\partial _2w,\partial _1w). \end{aligned}$$

(1.5)

Let $\Omega \in {\mathbb {R}}^2$ be a bounded smooth domain, and we consider the initial boundary value problem of (1.4) with the initial condition and the Dirichlet boundary condition:

$$\begin{aligned} \left\{ \begin{aligned}&(\rho , \rho u, \rho w, b)(x, 0)=(\rho _0, \rho _0 u_0, \rho _0 w_0, b_0)(x),\quad x\in \Omega ,\\&(u, w, b)(x, t)=(0, 0, 0),\quad x\in \partial \Omega ,\quad t>0. \end{aligned} \right. \end{aligned}$$

(1.6)

The system (1.1) describes the motion of electrically conducting micropolar fluids in the presence of a magnetic field. The magneto-micropolar fluid model was first proposed by Ahmadi–Shahinpoor [1] in the 1970s, which extends the valid domain of MHD equations and accounts for microrotation effect. There are some literatures focused on the mathematical theory of the incompressible viscous magneto-micropolar system, in particular, studying well-posedness of solutions to the magneto-micropolar fluid equations, also refer to [2] for relevant background. However, if the initial density includes the vacuum state, it has a few results to the existence of solutions for this system [3,4,5]. When the fluid is homogeneous $(\mathrm {i.e.}\, \rho =const)$, the local existence and uniqueness of strong solutions were firstly established by Rojas-Medar [6] with using the Galerkin method. Yamazaki [7] studied the global regularity of the two-dimensional magneto-micropolar fluid system, and they showed that with zero angular viscosity the solution triple remains smooth for all time. Next, Shang–Zhao [8] proved the global regularity of classical solutions to 2D magneto-micropolar fluid equations with only micro-rotational velocity dissipation and magnetic diffusion. Recently, appealing to a refined pure energy method, Tan-Wu-Zhou [9] investigated the global existence and decay estimate of solutions to magneto-micropolar fluid equations by assuming that the $H^3$-norm of the initial data is small, but the higher order derivatives can be arbitrary large. Lin and Xiang [10] considered the global well-posedness for the 2D incompressible magneto-micropolar fluid system with partial viscosity. Yet for the density-dependent viscosity and the initial density allowing vacuum states, it seems to need solve much more difficult problems to the system (1.1).

When the system do not consider the magnetic field $(\mathrm {i.e.}\,b=0)$, the system (1.1) reduces to the nonhomogeneous micropolar fluid equations. Eringen [11] first introduced the micropolar fluids, which accounts for micro-rotation effects and micro-rotation in a fluid motion system, and can be viewed as non-Newtonian fluids with nonsymmetric stress tensor. When the connected open set $\Omega \in {\mathbb {R}}^3$ replaces the whole space ${\mathbb {R}}^3$, and the solution vanishes on $\partial \Omega \times [0, T]$, Galdi–Rionero [12] showed existence and uniqueness of weak solutions to the initial boundary value problem for the micropolar system. Dong–Zhang [13] and Liu–Wang [14] proved the global regularity of smooth solutions to the 2D micropolar fluid with the micro-rotation viscosity $\gamma =0$. Zhang–Zhu [15] studied the global strong and classical solution for the 3D micropolar equations with vacuum, which assumed $\Vert \rho _0\Vert _{L^\infty }$ or $\Vert \sqrt{\rho _0}u_0\Vert ^2_{L^2}+\Vert \sqrt{\rho _0}w_0\Vert ^2_{L^2}$ is small enough. Recently, Song [16] concerned the global well-posedness for the 3D compressible micropolar system in the critical Besov space, and proposed the linear system for the compressible micropolar equations could be decomposed into a compressible Navier–Stokes equation and an incompressible micropolar system. However, if the influence of magnetic field on the motion is considered, it will also bring some difficulties on a priori estimates of the system (1.1).

For the incompressible magneto-micropolar fluid model, many scholars has been attracted by its research significance in physics and mathematics. When the initial density allowing vacuum states and the density-dependent viscosity, Zhang–Zhu [5] established the global strong solutions for the 3D nonhomogeneous incompressible magneto-micropolar equations under the condition that the initial energy is small enough. It is worth noting that these results are valid under the following compatibility conditions

$$\begin{aligned} \left\{ \begin{aligned}&-(\mu _1+\xi )\triangle u_0+\nabla P_0-2\xi \textrm{curl}w_0-b_0\cdot \nabla b_0=\sqrt{\rho _0}g_1,\\&-\mu _2\triangle w_0-(\mu _2+\lambda )\nabla \textrm{div}w_0+4\xi w_0-2\xi \textrm{curl}u_0=\sqrt{\rho _0}g_2, \end{aligned} \right. \end{aligned}$$

(1.7)

for some $(\nabla P_0, g_1, g_2)\in L^2$. Meanwhile, they also obtain the algebraic decay rates of the solutions provided that the initial energy is small enough. Later on, Zhong [17] extended this result to the entire two-dimensional space, and showed the local existence of strong solutions to 2D nonhomogeneous magneto-micropolar fluid with vacuum as far field density. Then, he established the global existence and exponential decay of strong solutions of nonhomogeneous magneto-micropolar fluid equations with large initial data and vacuum in paper [18]. Recently, With the help of weighted function and the duality principle of BMO space and Hardy space, Zhong [19] investigated the global well-posedness to nonhomogeneous magneto-micropolar fluid equations with zero density at infinity in ${\mathbb {R}}^2$. Furthermore, for the homogeneous Dirichlet boundary conditions of the velocity and micro-rotational velocity and Navier-slip boundary condition of the magnetic field, he proved the initial boundary value problem of 3D nonhomogeneous magneto-micropolar fluid equations in [20]. The above results are all the density-independent viscosity. It will bring much more difficulties to estimate the $L^\infty (0, T; L^2)$-norm for the gradients of velocity because of the density-dependent viscosity. However, the paper [5] proved the global regularity of this system (1.1) in 3D space and only obtained the algebraic decay rate. It is worth noting that they require must be satisfied some small energy conditions. As a result of the standard Sobolev embedding theorem, a prior estimate for the 3D case cannot be applied to the 2D case. The purpose of this paper is to establish the global well-posedness of solutions for (1.4)–(1.6) with vacuum in smooth bounded domains. Especially, there is no need other small initial energy but only need the initial velocity is suitably small, and we also yielded the exponential decay rate of strong solutions.

Now, we go back to (1.1). Before stating the main results, we first explain the notations and conventions used throughout this paper. For $1\le r\le \infty$ and $k\ge 0$, the standard Lebesgue and Sobolve spaces are defined as $L^r=L^r(\Omega )$, $W^{k, r}=W^{k, r}(\Omega )$, and $H^k(\Omega )=W^{k, 2}(\Omega )\nonumber$, $r=2$. The space $H^1_{0, \sigma }$ represent the closure in $H^1$ of the space $C^\infty _{0, \sigma }\triangleq \{f\in C^\infty _0(\Omega )|\textrm{div}f=0\}$.

The following is our main result of the paper:

Theorem 1.1

Let $q\in (2,\infty )$ be a fixed constant, assume that the initial data $(\rho _0, u_0, w_0, b_0)$ satisfy

$$\begin{aligned} \begin{aligned} 0\le \rho _0\in W^{1,q}, u_0\in H_{0,\sigma }^1, w_0\in H_0^1, b_0\in H_{0,\sigma }^1. \end{aligned} \end{aligned}$$

(1.8)

Then there exist some small positive constant $\varepsilon _0$ depending only on q, $\kappa$, $\nu$, $\Omega$, ${\underline{\mu }}$, ${\bar{\mu }}\triangleq \sup \limits _{[0, {\bar{\rho }}]}\mu (\rho )$, ${\bar{\rho }}$, $\Vert \nabla u_0\Vert _{L^2}$, $\Vert \nabla w_0\Vert _{L^2}$ and $\Vert \nabla b_0\Vert _{L^2}$, such that if

$$\begin{aligned} \Vert \nabla u_0\Vert ^2_{L^2}\le \varepsilon _0, \end{aligned}$$

(1.9)

there is a unique strong solution $(\rho , u, w, b, P)$ satisfying that for any $0<\tau<T<\infty$ and $2<r<min\{q,3\}$,

$$\begin{aligned} \left\{ \begin{aligned}&0\le \rho \in C\left( [0, T]; W^{1,q}\right) , \nabla \mu (\rho )\in C\left( [0,T]; L^q\right) ,\\&\nabla u\in L^\infty \left( 0, T; L^2\right) \cap L^\infty \left( \tau , T; H^1\right) \cap L^2\left( \tau , T; W^{1,r}\right) ,\\&\nabla w\in L^\infty \left( 0, T; L^2\right) \cap L^\infty \left( \tau , T; H^1\right) \cap L^2\left( \tau , T; H^2\right) ,\\&\nabla b\in L^\infty \left( 0, T; L^2\right) \cap L^\infty \left( 0, T; H^1\right) \cap L^2\left( 0, T; W^{1,r}\right) ,\\&P\in L^\infty (0, T; L^2)\cap L^\infty \left( \tau , T; H^1\right) \cap L^2\left( \tau , T; W^{1,r}\right) ,\\&t\sqrt{\rho }u_t, t\sqrt{\rho }w_t, tb_t\in L^\infty \left( 0, T; L^2\right) , \\&t\nabla u_t, t\nabla w_t, t\nabla b_t\in L^2\left( 0, T; L^2\right) ,\\&e^{\frac{\sigma t}{2}}\nabla u, e^{\frac{\sigma t}{2}}\nabla w, e^{\frac{\sigma t}{2}}\nabla b, e^{\frac{\sigma t}{2}}\sqrt{\rho }u_t, e^{\frac{\sigma t}{2}}\sqrt{\rho }w_t, e^{\frac{\sigma t}{2}}\triangle b\in L^2\left( 0, T; L^2\right) , \end{aligned} \right. \end{aligned}$$

(1.10)

where $\sigma \triangleq \min \{\frac{{\underline{\mu }}}{C_p{\bar{\rho }}}, \frac{\kappa }{C_p{\bar{\rho }}}, \frac{\nu }{C_p{\bar{\rho }}}\}$ with $C_p$ being the constant of Poincar$\acute{e}$’s inequality. Moreover, there exists some positive constant C depending only on $\Omega$, q, $\kappa$, $\nu$, ${\underline{\mu }}$, ${\bar{\mu }}$, ${\bar{\rho }}$, $\Vert \nabla u_0\Vert _{L^2}$, $\Vert \nabla w_0\Vert _{L^2}$ and $\Vert \nabla b_0\Vert _{L^2}$ such that for all $t \ge 1$,

$$\begin{aligned} \left\{ \begin{aligned}&\Vert \nabla u(\cdot , t)\Vert _{H^1}+\Vert \nabla w(\cdot , t)\Vert _{H^1}+\Vert \nabla b(\cdot , t)\Vert _{H^1}+\Vert \nabla P(\cdot , t)\Vert _{L^2}\le Ce^{-\sigma t},\\&\Vert \sqrt{\rho }u_t(\cdot , t)\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t(\cdot , t)\Vert ^2_{L^2}+\Vert b_t(\cdot , t)\Vert ^2_{L^2}\le Ce^{-\sigma t}. \end{aligned} \right. \end{aligned}$$

(1.11)

Remark 1.1

For the Theorem 1.1, it holds for any function $\mu (\rho )$ satisfying (1.2) and for arbitrarily large initial density, which can contain vacuum condition. The initial data is no need satisfy any compatibility conditions [21]

$-div((\mu (\rho _0)+\kappa )\nabla u_0)+\nabla P_0=\rho _0^{\frac{1}{2}}g$,

for some $(P_0, g)\in H^1\times L^2$.

Remark 1.2

It should be noted that we improved the results of [5], we consider the density-dependent viscosity and the initial data can be arbitrarily large except $\Vert \nabla u_0\Vert ^2_{L^2}$ ((1.9)). Finally, we also obtain time-independent estimates and exponential decay rates for the solutions.

Now we simply present the main idea of the proof and give the main difficulty in this paper. The local existence and uniqueness of strong solutions to the systems (1.4)–(1.6) follow from the paper [17] (see Lemma 2.1). We need to deduce some global a priori estimates on strong solution to (1.4)–(1.6) in proper higher regularity, and then extend the local solution to the global solution. Due to we consider the initial density can contain vacuum states and even have compact support, in particular, the viscosity coefficient is also affected by the density. Hence it not only increases the difficulty of estimating the the $L^\infty (0, T; L^2(\Omega ))$-norm of $\Vert \nabla u\Vert ^2_{L^2}$, but also solves the strong coupling between $u\cdot \nabla u$, $u\cdot \nabla b$ and $b\cdot \nabla u$. Firstly, the key ingredient here is to get the time-independent $L^1(0, T; L^\infty (\Omega ))$-norm of $\nabla u$. we derive that the bound on $L^2(0, T; L^2(\Omega ))$-norm of $e^{\frac{\sigma t}{2}}\nabla u$ by applying the upper bounds on the density (3.3) and the Poincar$\acute{e}$ inequality. And then the most important thing is to estimate $L^\infty (0, T; L^2(\Omega ))$-norm of $\nabla u$ by the Lemma 2.3 and (3.1). Next, we will obtain a key estimate (3.28) by multiplying (1.4)$_4$ by $b|b|^2$ and Gronwall’s inequality, which is used to deal with the strong coupling between $b\cdot \nabla u$. In addition, we need to define a function $\zeta (t)\triangleq \min \{1, t\}$ to get the estimates on $L^\infty (0, T; L^2(\Omega ))$-norm of $t^{\frac{1}{2}}\sqrt{\rho }u_t$ and $L^\infty (\zeta (T), T; L^2(\Omega ))$-norm of $e^{\frac{1}{2}\sigma t}\sqrt{\rho }u_t$, which avoids the singularity of $\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}$ at $t=0$. Because of the coupling between magnetic field and the gradient of velocity and magnetic field, it is important to obtain the estimates on $H^2(\Omega )$-norm of w and $L^2(\Omega )$-norm of $b_t$ by using the standard $L^2(\Omega )$-estimates of the elliptic system, the Eq. (1.4)$_4$ and Sobolev’s inequality, which is used to control the $L^\infty (\zeta (T), T; L^2(\Omega ))$-norm of $e^{\frac{1}{2}\sigma t}\sqrt{\rho }w_t$ and $L^\infty (\zeta (T), T; L^2(\Omega ))$-norm of $e^{\frac{1}{2}\sigma t}b_t$, further in order to get the $L^1(0, T; L^\infty (\Omega ))$-norm of $\nabla u$. Finally, with a priori estimates stated above, we are in a position to prove Proposition 3.1. Meanwhile, it can bound the estimation of the time derivatives for the solutions $(\rho , u, w, b, P)$ to extend the local solution to all time, and thus claims the proof of Theorem 1.1.

The rest of this paper is organized as follows: we introduce some elementary facts and inequalities in Sect. 2. The Sect. 3 is devoted to a priori estimates. Finally, we will give the proof of Theorem 1.1 in Sect. 4.

2 Preliminaries

We will recall some known facts and elementary inequalities which will be used frequently later. The following local existence of strong solutions whose proof is similar to [17].

Lemma 2.1

Assume that $(\rho _0, u_0, w_0, b_0)$ satisfies (1.8). Then there exist a small time $T >0$ and a unique strong solution $(\rho , u, w, b, P)$ to the problem (1.4)–(1.6) in $\Omega \times (0, T)$ satisfies (1.10)–(1.11).

Lemma 2.2

(See [22]) (Gagliardo–Nirenberg) Let v belongs to $L^q(\Omega )$, and its derivatives of order m, $\nabla ^mv$, belong to $L^r(\Omega )$, $1\le s, r\le \infty$. Then for the derivatives $\nabla ^jv$, $0\le j<m$, we have

$$\begin{aligned} \Vert \nabla ^jv\Vert _{L^s(\Omega )}\le {\tilde{C}}\Vert \nabla ^mv\Vert _{L^r(\Omega )}^\alpha \Vert v\Vert _{L^s(\Omega )}^{1-\alpha }, \end{aligned}$$

(2.1)

where

$\frac{1}{s}=\frac{j}{n}+\alpha \left( \frac{1}{r}-\frac{m}{n}\right) +\left( 1-\alpha \right) \frac{1}{s}$,

for all a in the interval

$\frac{j}{m}\le \alpha \le 1,$

and the constant ${\tilde{C}}$ depends only on n, m, j, s, r, $\alpha$.

Next, we need the following regularity on the Stokes equations to derive the estimates of the derivatives of the solutions, whose proof can be found in [23].

Lemma 2.3

Assume that $\rho \in W^{1,q}$, $q\in (2,\infty )$, $0\le \rho \le {\bar{\rho }}$ and $\mu (\rho )$ satisfies (1.2) on $[0, {\bar{\rho }}]$. Let $(u, P)\in H^1_{1, \sigma }\times L^2$ be the unique weak solution to the following boundary value problem

$$\begin{aligned} -div(\mu (\rho )\nabla u)+\nabla P=F,\quad divu=0\quad in\quad \Omega ,\quad \int Pdx=0. \end{aligned}$$

(2.2)

Then we have the following regularity results:

(i) If $F\in L^2$, then $(u, P)\in H^2\times H^1$ and

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

(2.3)

(ii) If $F\in L^r$ for some $r\in (2, q)$, then $(u, P)\in W^{2, r}\times W^{1, r}$ and

$$\begin{aligned}&\Vert u\Vert _{W^{2, r}}\le C\Vert F\Vert _{L^r}(1+\Vert \nabla \mu \left( \rho )\Vert _{L^q}\right) ^{\frac{qr}{2(q-r)}},\nonumber \\&\Vert P\Vert _{W^{1, r}}\le C\Vert F\Vert _{L^r}(1+\Vert \nabla \mu \left( \rho )\Vert _{L^q}\right) ^{1+\frac{qr}{2(q-r)}}. \end{aligned}$$

(2.4)

Here the constant C depends on $\Omega$, q, r, ${\underline{\mu }}$, ${\bar{\mu }}$.

3 A Priori Estimates

In this section, we first let $T>0$ be a fixed time and $(\rho , u, w, b, P)$ be the smooth solution to (1.4)–(1.6) on $\Omega \times (0, T]$ with smooth initial data $(\rho _0, u_0, w_0)$ satisfying (1.8). Because of the Lemma 2.1, we will establish some necessary a priori bounds for strong solutions $(\rho , u, w, b, P)$ to the initial boundary value problem (1.4)–(1.6) to extend the local strong solution. Firstly, we assume that the following a priori hypothesis holds:

Proposition 3.1

There exists a small positive constant $\varepsilon _0$ depending only on $\Omega$, q, $\kappa$, $\nu$, ${\underline{\mu }}$, ${\bar{\mu }}$, ${\bar{\rho }}$, $\Vert \nabla u_0\Vert _{L^2}$, $\Vert \nabla w_0\Vert _{L^2}$ and $\Vert \nabla b_0\Vert _{L^2}$ such that if $(\rho , u, w, b, P)$ is a smooth solution of (1.4)–(1.6) on $\Omega \times (0, T]$ satisfying

$$\begin{aligned} \sup \limits _{t\in [0, T]}\Vert \nabla \mu (\rho )\Vert _{L^q}\le 2M,\quad \sup \limits _{t\in [0, T]}e^{\sigma t}\Vert \nabla u\Vert ^2_{L^2}\le 4\Vert \nabla u_0\Vert ^2_{L^2}, \end{aligned}$$

(3.1)

and the following estimate holds:

$$\begin{aligned} \sup \limits _{t\in [0, T]}\Vert \nabla \mu (\rho )\Vert _{L^q}\le M,\quad \sup \limits _{t\in [0, T]}e^{\sigma t}\Vert \nabla u\Vert ^2_{L^2}\le 2\Vert \nabla u_0\Vert ^2_{L^2}, \end{aligned}$$

(3.2)

provided that $\Vert \nabla u_0\Vert ^2_{L^2}\le \varepsilon _0$.

We begin with the following boundedness of density and elementary estimates.

Lemma 3.1

It holds that

$$\begin{aligned} 0\le \rho (x, t)\le \sup \limits _{t\in [0, T]}\Vert \rho \Vert _{L^\infty }={\bar{\rho }}, \end{aligned}$$

(3.3)

$$\begin{aligned}&\sup \limits _{t\in [0, T]}\left( \Vert \sqrt{\rho }u\Vert ^2_{L^2}+\Vert \sqrt{\rho }w\Vert ^2_{L^2}+\Vert b\Vert ^2_{L^2}\right) \nonumber \\&\qquad +\int _0^T\left( {\underline{\mu }}\Vert \nabla u\Vert _{L^2}^2+\gamma \Vert \nabla w\Vert _{L^2}^2+\nu \Vert \nabla b\Vert _{L^2}^2\right) dt\nonumber \\&\quad \le \left( \Vert \sqrt{\rho _0}u_0\Vert ^2_{L^2}+\Vert \sqrt{\rho }_0w_0\Vert ^2_{L^2}+\Vert b_0\Vert ^2_{L^2}\right) , \end{aligned}$$

(3.4)

$$\begin{aligned}&\sup \limits _{t\in [0, T]}e^{\sigma t}\left( \Vert \sqrt{\rho }u\Vert ^2_{L^2}+\Vert \sqrt{\rho }w\Vert ^2_{L^2}+\Vert b\Vert ^2_{L^2}\right) \nonumber \\&\qquad +\int _0^Te^{\sigma t}\left( {\underline{\mu }}\Vert \nabla u\Vert _{L^2}^2+\gamma \Vert \nabla w\Vert _{L^2}^2+\nu \Vert \nabla b\Vert _{L^2}^2\right) dt\nonumber \\&\quad \le \left( \Vert \sqrt{\rho _0}u_0\Vert ^2_{L^2}+\Vert \sqrt{\rho }_0w_0\Vert ^2_{L^2}+\Vert b_0\Vert ^2_{L^2}\right) , \end{aligned}$$

(3.5)

where $\sigma :=min\{\frac{{\underline{\mu }}}{C_p{\bar{\rho }}}, \frac{\gamma }{C_p{\bar{\rho }}}, \frac{\nu }{C_p{\bar{\rho }}}\}\}$ with $C_p$ being the constant of Poincar$\acute{e}$’s inequality.

Proof

It follows from the transport equation (1.4)$_1$ to get the (3.3) (see Lious [17]). Next, we prove the (3.4), adding (1.4)$_2\times u$ to (1.4)$_2\times w$ and integrating by parts leads to

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\left( \Vert \sqrt{\rho }u\Vert ^2_{L^2}+\Vert \sqrt{\rho }w\Vert ^2_{L^2}\right) + \left( {\underline{\mu }}+\kappa \right) \Vert \nabla u\Vert _{L^2}^2+ \gamma \Vert \nabla w\Vert _{L^2}^2+4\kappa \Vert w\Vert _{L^2}^2\nonumber \\&\quad \le 4\kappa \int w\nabla ^\bot \cdot udx \le 4\kappa \Vert w\Vert _{L^2}^2+\kappa \Vert \nabla u\Vert _{L^2}^2, \end{aligned}$$

(3.6)

which gives

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

(3.7)

Multiplying (1.4)$_4$ by b and integration by parts over $\Omega$, we derive that

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\Vert b\Vert ^2_{L^2}+\nu \Vert \nabla b\Vert ^2_{L^2}=0, \end{aligned}$$

(3.8)

this combining with (3.7) and integrating the inequality in t gives (3.4). It follows from the Poincar$\acute{e}$’s inequality and (3.3) that

$$\begin{aligned}&\Vert \sqrt{\rho }u\Vert ^2_{L^2}\le \Vert \rho \Vert _{L^\infty }\Vert u\Vert ^2_{L^2}\le \frac{C_p{\bar{\rho }}}{{\underline{\mu }}}({\underline{\mu }}\Vert \nabla u\Vert ^2_{L^2})\le \sigma ^{-1}({\underline{\mu }}\Vert \nabla u\Vert ^2_{L^2}),\nonumber \\&\Vert \sqrt{\rho }w\Vert ^2_{L^2}\le \Vert \rho \Vert _{L^\infty }\Vert w\Vert ^2_{L^2}\le \frac{C_p{\bar{\rho }}}{\gamma }(\gamma \Vert \nabla w\Vert ^2_{L^2})\le \sigma ^{-1}(\gamma \Vert \nabla w\Vert ^2_{L^2}),\nonumber \\&\Vert b\Vert ^2_{L^2}\le \frac{C_p}{\nu }(\nu \Vert \nabla b\Vert ^2_{L^2})\le \sigma ^{-1}(\nu \Vert \nabla b\Vert ^2_{L^2}), \end{aligned}$$

(3.9)

where $\sigma \triangleq \min \{\frac{{\underline{\mu }}}{C_p{\bar{\rho }}}, \frac{\gamma }{C_p{\bar{\rho }}}, \frac{\nu }{C_p}\}$ with $C_p$ being the constant of Poincar$\acute{e}$’s inequality. Form (3.9), (3.9) and (3.7) yields that

$$\begin{aligned}&\frac{d}{dt}\left( \Vert \sqrt{\rho }u\Vert ^2_{L^2}+\Vert \sqrt{\rho }w\Vert ^2_{L^2}+\Vert b\Vert ^2_{L^2}\right) +\sigma \Vert \sqrt{\rho }u\Vert ^2_{L^2}+\sigma \Vert \sqrt{\rho }w\Vert ^2_{L^2}+\sigma \Vert b\Vert ^2_{L^2}\nonumber \\&\quad +{\underline{\mu }}\Vert \nabla u\Vert ^2_{L^2}+\gamma \Vert \nabla w\Vert ^2_{L^2}+\nu \Vert \nabla b\Vert ^2_{L^2}\le 0, \end{aligned}$$

(3.10)

then this multiplies by $e^{\sigma t}$ we have

$$\begin{aligned}&\frac{d}{dt}[e^{\sigma t}\left( \Vert \sqrt{\rho }u\Vert ^2_{L^2}+\Vert \sqrt{\rho }w\Vert ^2_{L^2}+\Vert b\Vert ^2_{L^2}\right) ]\nonumber \\&\quad +e^{\sigma t}\left( {\underline{\mu }}\Vert \nabla u\Vert _{L^2}^2+\gamma \Vert \nabla w\Vert _{L^2}^2+\nu \Vert \nabla b\Vert ^2_{L^2}\right) \le 0. \end{aligned}$$

(3.11)

Integrating the above inequality in t leads to (3.6) and completes the Proof of Lemma 3.1.

Lemma 3.2

Let $(\rho , u, w, b, P)$ be a smooth solution to (1.4)–(1.6) satisfying (3.1). Then there exists some positive constant C depending only on $\Omega$, q, $\kappa$, $\nu$, ${\underline{\mu }}$, ${\bar{\mu }}$, ${\bar{\rho }}$, $\Vert \nabla u_0\Vert _{L^2}$, $\Vert \nabla w_0\Vert _{L^2}$ and $\Vert \nabla b_0\Vert _{L^2}$ such that

$$\begin{aligned}&\sup \limits _{t\in [0, T]}\left( \Vert \nabla u\Vert ^2_{L^2}+\Vert \nabla w\Vert ^2_{L^2}+\Vert \nabla b\Vert ^2_{L^2}\right) +\int _0^T\left( \Vert \sqrt{\rho }u_t\Vert _{L^2}^2+\Vert \sqrt{\rho }w_t\Vert _{L^2}^2\right. \nonumber \\&\quad \left. +\Vert \triangle b\Vert ^2_{L^2}+\Vert |b||\nabla b|\Vert ^2_{L^2}\right) dt\le C\Vert \nabla u_0\Vert ^2_{L^2}, \end{aligned}$$

(3.12)

$$\begin{aligned}&\sup \limits _{t\in [0, T]}e^{\sigma t}\left( \Vert \nabla u\Vert ^2_{L^2}+\Vert \nabla w\Vert ^2_{L^2} +\Vert \nabla b\Vert ^2_{L^2}\right) +\int _0^T e^{\sigma t}\left( \Vert \sqrt{\rho }u_t\Vert _{L^2}^2+\Vert \sqrt{\rho }w_t\Vert _{L^2}^2\right. \nonumber \\&\quad \left. +\Vert \triangle b\Vert ^2_{L^2}+\Vert |b||\nabla b|\Vert ^2_{L^2}\right) dt\le C\Vert \nabla u_0\Vert ^2_{L^2}. \end{aligned}$$

(3.13)

Proof

We can get the following equation from (1.4)$_1$

$$\begin{aligned} \mu (\rho )_t+u\cdot \nabla \mu (\rho )=0. \end{aligned}$$

(3.14)

Next, multiplying (1.4)$_2$ and (1.4)$_3$ by $u_t$ and $w_t$ respectively and integrating by parts, we have

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int \left( \left( \mu (\rho )+\kappa \right) |\nabla u|^2+\gamma |\nabla w|^2+4\kappa w^2\right) dx+\int \rho |u_t|^2dx+\int \rho |w_t|^2dx\nonumber \\&\quad =-\int \rho u\cdot \nabla u\cdot u_tdx-\int \rho u\cdot \nabla w\cdot w_tdx- \frac{1}{2}\int u\cdot \nabla \mu (\rho )|\nabla u|^2dx\nonumber \\&\qquad -2\kappa \int \nabla ^\bot wu_tdx+2\kappa \int \nabla ^\bot \cdot uw_tdx+\int b\cdot \nabla u_tdx \triangleq \sum \limits _{i=1}^{6}I_i. \end{aligned}$$

(3.15)

Now, we will estimate each term on the right hand of (3.15) as following. Firstly, Applying to the Hölder’s, Sobolev’s and Gagliardo–Nirenberg inequality along with (3.1) that

$$\begin{aligned}&|I_1|+|I_2|\le |\int \rho u\cdot \nabla u\cdot u_tdx|+|\int \rho u\cdot \nabla w\cdot w_tdx|\nonumber \\&\quad \le C{\bar{\rho }}^{\frac{1}{2}}\Vert u\Vert _{L^\infty }(\Vert \sqrt{\rho }u_t\Vert _{L^2}\Vert \nabla u\Vert _{L^2}+\Vert \sqrt{\rho }w_t\Vert _{L^2}\Vert \nabla w\Vert _{L^2})\nonumber \\&\quad \le C{\bar{\rho }}^{\frac{1}{2}}\Vert u\Vert ^{\frac{1}{2}}_{L^2}\Vert \nabla ^2u\Vert ^{\frac{1}{2}}_{L^2}(\Vert \sqrt{\rho }u_t\Vert _{L^2}\Vert \nabla u\Vert _{L^2}+\Vert \sqrt{\rho }w_t\Vert _{L^2}\Vert \nabla w\Vert _{L^2})\nonumber \\&\quad \le \frac{1}{2}(\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t\Vert ^2_{L^2})+C(\Vert \nabla u\Vert ^3_{L^2}+\Vert \nabla w\Vert ^2_{L^2})\Vert \nabla u\Vert _{H^1}. \end{aligned}$$

(3.16)

According to H$\ddot{o}$lder’s inequality and (3.1), we have

$$\begin{aligned} |I_3|&\le \frac{1}{2}|\int u\cdot \mu (\rho )|\nabla u|^2dx|\nonumber \\&\le C\Vert \nabla \mu (\rho )\Vert _{L^q}\Vert u\Vert _{L^{\frac{2q}{q-2}}}\Vert \nabla u\Vert ^{2}_{L^4}\nonumber \\&\le C\Vert \nabla u\Vert ^2_{L^2}\Vert \nabla u\Vert _{H^1}. \end{aligned}$$

(3.17)

Using integration by parts, we get

$$\begin{aligned} I_4+I_5&=-2\kappa \int \nabla ^{\bot }wu_tdx+2\kappa \int \nabla ^{\bot }\cdot uw_tdx\nonumber \\&=2\kappa \int \nabla ^{\bot }\cdot u_twdx+2\kappa \int \nabla ^{\bot }\cdot uw_tdx\nonumber \\&=2\kappa \frac{d}{dt}\int \nabla ^{\bot }\cdot uwdx. \end{aligned}$$

(3.18)

It follows from integration by parts, Sobolev’s inequality, (1.4)$_4$ and (1.4)$_5$ together with $b|_{\partial \Omega }$=0 that

$$\begin{aligned} I_6&=-\frac{d}{dt}\int b\cdot \nabla u\cdot bdx+\int b_t\cdot \nabla u\cdot bdx+\int b\cdot \nabla u\cdot b_t dx\nonumber \\&=-\frac{d}{dt}\int b\cdot \nabla u\cdot bdx+\int (\nu \triangle b-u\cdot \nabla b+b\cdot \nabla u)\cdot \nabla u\cdot bdx\nonumber \\&\quad +\int b\cdot \nabla u\cdot (\nu \triangle b-u\cdot \nabla b+b\cdot \nabla u)dx\nonumber \\&\le -\frac{d}{dt}\int b\cdot \nabla u\cdot bdx+2\nu \int |\nabla u||b||\triangle b|dx+2\int |b|^2|\nabla u|^2dx\nonumber \\&\quad +2\int |u||\nabla u||b||\nabla b|dx\nonumber \\&\le -\frac{d}{dt}\int b\cdot \nabla u\cdot bdx+\frac{\nu }{4}\Vert \triangle b\Vert ^2_{L^2}+C\Vert b\Vert ^6_{L^6}+C\Vert \nabla u\Vert ^3_{L^3}\nonumber \\&\quad +C\Vert u\Vert _{L^\infty }\Vert \nabla b\Vert _{L^2}\Vert |b||\nabla u|\Vert _{L^2}\nonumber \\&\le -\frac{d}{dt}\int b\cdot \nabla u\cdot bdx+\frac{\nu }{4}\Vert \triangle b\Vert ^2_{L^2}+C\Vert b\Vert ^2_{L^2}\Vert \nabla b\Vert ^4_{L^2}+C\Vert \nabla u\Vert ^2_{L^2}\Vert \nabla ^2\Vert _{L^2}\nonumber \\&\quad +C\Vert u\Vert ^{\frac{1}{2}}_{L^4}\Vert \nabla u\Vert ^{\frac{1}{2}}_{L^4}\Vert b\Vert _{L^4}\Vert \nabla b\Vert _{L^2}\Vert \nabla u\Vert _{L^4}\nonumber \\&\le -\frac{d}{dt}\int b\cdot \nabla u\cdot bdx+\frac{\nu }{4}\Vert \triangle b\Vert ^2_{L^2}+C\Vert b\Vert ^2_{L^2}\Vert \nabla b\Vert ^4_{L^2}+C\Vert \nabla u\Vert ^2_{L^2}\Vert \nabla ^2\Vert _{L^2}\nonumber \\&\quad +C\Vert b\Vert _{L^4}\Vert \nabla b\Vert _{L^2}\Vert \nabla u\Vert _{L^2}\Vert \nabla u\Vert _{H^1}. \end{aligned}$$

(3.19)

Therefore, substituting (3.16)–(3.19) into (3.15), we have

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int \left( \left( \mu (\rho )+\kappa \right) |\nabla u|^2+\gamma |\nabla w|^2+4\kappa w^2- \nabla ^\bot \cdot uw+b\cdot \nabla u\cdot b\right) dx\nonumber \\&\qquad +\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t\Vert ^2_{L^2}\nonumber \\&\quad \le \frac{1}{2}(\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t\Vert ^2_{L^2})+C(\Vert \nabla u\Vert ^3_{L^2}+\Vert \nabla w\Vert ^2_{L^2})\Vert \nabla u\Vert _{H^1}\nonumber \\&\qquad +\frac{\nu }{4}\Vert \triangle b\Vert ^2_{L^2}+C\Vert b\Vert ^2_{L^2}\Vert \nabla b\Vert ^4_{L^2}+C\Vert b\Vert _{L^4}\Vert \nabla b\Vert _{L^2}\Vert \nabla u\Vert _{L^2}\Vert \nabla u\Vert _{H^1}. \end{aligned}$$

(3.20)

Multiplying (1.4)$_4$ by $\triangle b$ and integrating the resulting equality over $\Omega$ along with H$\ddot{o}$lder’s and Gagliardo-Nirenberg inequalities that

$$\begin{aligned}&\frac{d}{dt}\Vert \nabla b\Vert ^2_{L^2}+2\nu \Vert \triangle b\Vert ^2_{L^2}\nonumber \\&\quad \le \int |\nabla u||\nabla b|^2dx+\int |\nabla u||b||\triangle b|dx\nonumber \\&\quad \le C\Vert \nabla u\Vert ^2_{L^2}\Vert \nabla ^2u\Vert _{L^2}+C(1+\Vert b\Vert ^2_{L^2})\Vert \nabla b\Vert ^4_{L^2} +\frac{\nu }{4}\Vert \triangle b\Vert ^2_{L^2}, \end{aligned}$$

(3.21)

which combining with (3.20), we can directly yield that

$$\begin{aligned}&A'(t)+\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t\Vert ^2_{L^2}+\nu \Vert \triangle b\Vert ^2_{L^2}\nonumber \\&\quad \le C(\Vert \nabla u\Vert ^3_{L^2}+\Vert \nabla w\Vert ^2_{L^2})\Vert \nabla u\Vert _{H^1}+C\Vert \nabla b\Vert ^4_{L^2}\nonumber \\&\qquad +C\Vert b\Vert _{L^4}\Vert \nabla b\Vert _{L^2}\Vert \nabla u\Vert _{L^2}\Vert \nabla u\Vert _{H^1}, \end{aligned}$$

(3.22)

where

$$\begin{aligned} A(t)\triangleq \int ((\mu (\rho )+\kappa )|\nabla u|^2+\gamma |\nabla w|^2+|\nabla b|^2+4\kappa w^2-\nabla ^\bot \cdot uw+b\cdot \nabla u\cdot b)dx, \end{aligned}$$

(3.23)

and satisfies

$$\begin{aligned}&\frac{{\underline{\mu }}}{2}\Vert \nabla u\Vert ^2_{L^2}+\gamma \Vert \nabla w\Vert ^2_{L^2}+\Vert \nabla b\Vert ^2_{L^2}-C_1\Vert b\Vert ^4_{L^4}\nonumber \\&\quad \le A(t) \le C\Vert \nabla u\Vert ^2_{L^2}+C\Vert \nabla w\Vert ^2_{L^2}+C\Vert \nabla b\Vert ^2_{L^2}. \end{aligned}$$

(3.24)

According to Lemma 2.3 with $F = \rho u_t -\rho u\cdot \nabla u-2\kappa \nabla ^\bot w+b\cdot \nabla b$ and combining with (3.1) and (3.3), we obtain

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

(3.25)

which directly yields that

$$\begin{aligned} \Vert u\Vert _{H^2}+\Vert P\Vert _{H^1}\le C\Vert \sqrt{\rho }u_t\Vert _{L^2}+C\Vert \nabla u\Vert _{L^2}\Vert \nabla u\Vert ^2_{L^2}+C\Vert \nabla w\Vert _{L^2}+C\Vert |b||\nabla b|\Vert _{L^2}. \end{aligned}$$

(3.26)

Next, multiplying (1.4)$_4$ by $b|b|^2$ and integrating the resulting equality over $\Omega$ and together with Gagliardo–Nirenberg inequalities, we derive

$$\begin{aligned}&\frac{1}{4}\frac{d}{dt}\Vert b\Vert ^4_{L^4}+\nu \Vert |b||\nabla b|\Vert ^2_{L^2}+\frac{\nu }{2}\Vert \nabla |b|^2\Vert ^2_{L^2}\nonumber \\&\quad \le C\Vert \nabla u\Vert _{L^2}\Vert |b|^2\Vert ^2_{L^4}\nonumber \\&\quad \le C\Vert \nabla u\Vert _{L^2}\Vert |b|^2\Vert _{L^2}\Vert \nabla |b|^2\Vert _{L^2}\nonumber \\&\quad \le \frac{\nu }{4}\Vert \nabla |b|^2\Vert ^2_{L^2}+C\Vert \nabla u\Vert ^2_{L^2}\Vert b\Vert ^4_{L^4}\nonumber \\&\quad \le \frac{\nu }{4}\Vert \nabla |b|^2\Vert ^2_{L^2}+C\Vert \nabla u\Vert ^4_{L^2}+C\Vert \nabla b\Vert ^4_{L^2}, \end{aligned}$$

(3.27)

this together with Gronwall’s inequality and (3.4) implies

$$\begin{aligned} \sup \limits _{t\in [0, T]}\Vert b\Vert ^4_{L^4}+\int ^T_0\Vert |b||\nabla b|\Vert ^2_{L^2}dt\le C. \end{aligned}$$

(3.28)

Putting (3.26) into (3.22) and along with the above inequality, Young’s inequality and (3.1), we show that

$$\begin{aligned}&A'(t)+\frac{1}{2}\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t\Vert ^2_{L^2}+\nu \Vert \triangle b\Vert ^2_{L^2}\nonumber \\&\quad \le C\Vert \nabla u\Vert ^6_{L^2}+C\Vert \nabla w\Vert ^4_{L^2}+C\Vert \nabla b\Vert ^4_{L^2}+C\Vert \nabla w\Vert ^2_{L^2}\Vert \nabla u\Vert ^3_{L^2}\nonumber \\&\qquad +C\Vert \nabla b\Vert _{L^2}\Vert \nabla u\Vert _{L^2}\Vert \nabla u\Vert ^3_{L^2}\nonumber \\&\quad \le C\Vert \nabla u\Vert ^6_{L^2}+C\Vert \nabla w\Vert ^4_{L^2}+C\Vert \nabla b\Vert ^4_{L^2}+\epsilon \Vert |b||\nabla b|\Vert _{L^2}. \end{aligned}$$

(3.29)

Then, adding (3.27) multiplied by $4(C_1+1)$ to (3.29) and choosing $\epsilon$ suitably small, it follows from (3.1)

$$\begin{aligned}&\frac{d}{dt}(A(t)+4(C_1+1)\Vert b\Vert ^4_{L^4})+\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t\Vert ^2_{L^2}+\nu \Vert \triangle b\Vert ^2_{L^2}+\Vert |b||\nabla b|\Vert _{L^2}\nonumber \\&\quad \le C\Vert \nabla u\Vert ^6_{L^2}+C\Vert \nabla w\Vert ^4_{L^2}+C\Vert \nabla b\Vert ^4_{L^2}\nonumber \\&\quad \le C(\Vert \nabla u\Vert ^4_{L^2}+\Vert \nabla w\Vert ^2_{L^2}+\Vert \nabla b\Vert ^2_{L^2})(A(t)+4(C_1+1)\Vert b\Vert ^4_{L^4}). \end{aligned}$$

(3.30)

By (3.1) and (3.4), noting that

$$\begin{aligned} \int _0^T\Vert \nabla u\Vert ^4_{L^2}dt\le \frac{1}{e^{0\cdot \sigma }}\sup \limits _{t\in [0, T]}e^{\sigma t}\Vert \nabla u\Vert ^2_{L^2}\int _0^T\Vert \nabla u\Vert ^2_{L^2}dt\le C\Vert \nabla u_0\Vert ^2_{L^2}. \end{aligned}$$

(3.31)

Hence, integrating (3.30) over [0, T] and together with G$\ddot{o}$nwall’s inequality and (3.4) leads to (3.12). Then, multiplying (3.21) by $e^{\sigma t}$ and combining with (3.24) and (3.12) yields

$$\begin{aligned} \frac{d}{dt}&e^{\sigma t}(A(t)+4(C_1+1)\Vert b\Vert ^4_{L^4})+e^{\sigma t}(\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}\nonumber \\&\qquad +\Vert \sqrt{\rho }w_t\Vert ^2_{L^2}+\nu \Vert \triangle b\Vert ^2_{L^2}+\Vert |b||\nabla b|\Vert _{L^2})\nonumber \\&\quad \le Ce^{\sigma t}(A(t)+4(C_1+1)\Vert b\Vert ^4_{L^4})(\Vert \nabla u\Vert ^2_{L^2}+\Vert \nabla w\Vert ^2_{L^2}+\Vert \nabla b\Vert ^2_{L^2})+\sigma e^{\sigma t}A(t)\nonumber \\&\qquad +4\sigma e^{\sigma t}(C_1+1)\Vert b\Vert ^4_{L^4}, \end{aligned}$$

(3.32)

which applying to the Gönwall’s inequality, we can deduce (3.13) from (3.1), (3.5) and (3.12). It finishes the proof of Lemma 3.2.$\square$

Lemma 3.3

Let $(\rho , u, w, b, P)$ be a smooth solution to (1.4)–(1.6) satisfying (3.1). Then there exists some positive constant C depending only on $\Omega$, q, $\kappa$, $\nu$, ${\underline{\mu }}$, ${\bar{\mu }}$, ${\bar{\rho }}$, $\Vert \nabla u_0\Vert _{L^2}$, $\Vert \nabla w_0\Vert _{L^2}$ and $\Vert \nabla b_0\Vert _{L^2}$ such that

$$\begin{aligned}&\sup \limits _{t\in [0, T]}t(\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t\Vert ^2_{L^2}+\Vert b_t\Vert ^2_{L^2})+\int ^{\zeta (T)}_0 t(\Vert \nabla u_t\Vert ^2_{L^2}\nonumber \\&\quad +\Vert \nabla w_t\Vert ^2_{L^2}+\Vert \nabla b_t\Vert ^2_{L^2})dt\le C\Vert \nabla u_0\Vert ^2_{L^2}, \end{aligned}$$

(3.33)

$$\begin{aligned}&\sup \limits _{t\in [\zeta (T), T]}e^{\sigma t}(\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t\Vert ^2_{L^2}+\Vert b_t\Vert _{L^2})+\int ^T_{\zeta (T)}e^{\sigma t}(\Vert \nabla u_t\Vert ^2_{L^2}\nonumber \\&\quad +\Vert \nabla w_t\Vert ^2_{L^2}+\Vert \nabla b_t\Vert ^2_{L^2})dt\le C\Vert \nabla u_0\Vert ^2_{L^2}. \end{aligned}$$

(3.34)

Here, $\zeta (T)$ is defined by $\zeta (t)\triangleq \min \{1, t\}$.

Proof

Differentiating (1.4)${}_2$ and (1.4)$_3$ with respect to t respectively yields that

$$\begin{aligned}&\rho u_{tt}+\rho u\cdot \nabla u_t-div((\mu (\rho )+\kappa )\nabla u_t)+\nabla P_t+div(\mu (\rho )_t\nabla u)\nonumber \\&\quad =(u\cdot \nabla \rho )(u_t+u\cdot \nabla u)-\rho u_t\cdot \nabla u-2\kappa \nabla ^\bot w_t+b_t\cdot \nabla b+b\cdot \nabla b_t, \end{aligned}$$

(3.35)

$$\begin{aligned}&\rho w_{tt}+\rho u\cdot \nabla w_t+4\kappa w_t-\gamma \triangle w_t=(u\cdot \nabla \rho )(w_t+u\cdot \nabla w)-\rho u_t\cdot \nabla w\nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad +2\kappa \nabla ^\bot \cdot u_t. \end{aligned}$$

(3.36)

Next, multiplying (3.35) by $u_t$ and (3.36) by $w_t$, together with integration by parts and (1.4)$_1$, we get

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int (\rho |u_t|^2+\rho |w_t|^2)dx+\int ((\mu (\rho )+\kappa )|\nabla u_t|^2+\gamma |\nabla w_t|^2+4\kappa w_t^2)dx\nonumber \\&\quad =-\int ((u\cdot \nabla \rho )(u_t+u\cdot \nabla u)+\rho u_t\cdot \nabla u)\cdot u_tdx\nonumber \\&\qquad -\int ((u\cdot \nabla \rho )(w_t+u\cdot \nabla w)+\rho u_t\cdot \nabla w)w_tdx\nonumber \\&\qquad -2\kappa \int (\nabla ^\bot w_t\cdot u_t+\nabla ^\bot \cdot u_tw_t)dx+\int u\cdot \nabla \mu (\rho )\nabla u\cdot \nabla u_tdx\nonumber \\&\qquad +\int b_t\cdot \nabla b\cdot u_tdx+\int b\cdot \nabla b_t\cdot u_tdx \triangleq \sum \limits _{i=1}^{6}J_i. \end{aligned}$$

(3.37)

It follows from integration by parts, H$\ddot{o}$lder’s, Gagliardo–Nirenberg and Sobolev’s inequality together with (3.1) and (3.12) that

$$\begin{aligned} J_1&\le \int (\rho |u|(|u||u_t||\nabla ^2u|+|u||\nabla u_t||\nabla u|+|u_t||\nabla u|^2+|u_t||\nabla u_t|)+\rho |u_t|^2|\nabla u|)dx\nonumber \\&\le {\bar{\rho }}\Vert u\Vert ^2_{L^6}\Vert u_t\Vert _{L^6}\Vert \nabla ^2u\Vert _{L^2}+{\bar{\rho }}\Vert u\Vert ^2_{L^6}\Vert \nabla u\Vert _{L^6}\Vert \nabla u_t\Vert _{L^2}\nonumber \\&\quad +{\bar{\rho }}\Vert u_t\Vert _{L^6}\Vert u\Vert _{L^6}\Vert \nabla u\Vert _{L^2}\Vert \nabla u\Vert _{L^6}\nonumber \\&\quad +2{\bar{\rho }}^{\frac{1}{2}}\Vert u\Vert _{L^6}\Vert \sqrt{\rho }u_t\Vert _{L^3}\Vert \nabla u_t\Vert _{L^2}+\Vert \sqrt{\rho }u_t\Vert ^2_{L^4}\Vert \nabla u\Vert _{L^2}\nonumber \\&\le C{\bar{\rho }}\Vert \nabla u_t\Vert _{L^2}\Vert \nabla u\Vert ^2_{L^2}\Vert u\Vert _{H^2}+C{\bar{\rho }}^{\frac{1}{2}}\Vert \nabla u\Vert _{L^2}\Vert \sqrt{\rho }u_t\Vert ^{\frac{1}{2}}_{L^2}\Vert \sqrt{\rho }u_t\Vert ^{\frac{1}{2}}_{L^6}\Vert \nabla u_t\Vert _{L^2}\nonumber \\&\quad +C\Vert \nabla u\Vert _{L^2}\Vert \sqrt{\rho }u_t\Vert ^{\frac{1}{2}}_{L^2}\Vert \sqrt{\rho }u_t\Vert ^{\frac{3}{2}}_{L^6}\nonumber \\&\le C{\bar{\rho }}\Vert \nabla u_t\Vert _{L^2}\Vert \nabla u\Vert ^2_{L^2}\Vert u\Vert _{H^2}+C{\bar{\rho }}^{\frac{3}{4}}\Vert \nabla u\Vert _{L^2}\Vert \sqrt{\rho }u_t\Vert ^{\frac{1}{2}}_{L^2}\Vert \nabla u_t\Vert ^{\frac{3}{2}}_{L^2}\nonumber \\&\le \frac{{\underline{\mu }}}{8}\Vert \nabla u_t\Vert ^2_{L^2}+C\Vert u\Vert ^2_{H^2}+C\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}. \end{aligned}$$

(3.38)

Similarly, we can directly yield that

$$\begin{aligned} J_2&\le C\int (\rho |u|(|u||w_t||\nabla ^2w|+|u||\nabla w_t||\nabla w|+|w_t||\nabla u||\nabla w|\nonumber \\&\quad +2|w_t||\nabla w_t|)+\rho |u_t||w_t||\nabla w|)dx\nonumber \\&\le {\bar{\rho }}\Vert u\Vert ^2_{L^6}\Vert w_t\Vert _{L^6}\Vert \nabla ^2w\Vert _{L^2}+{\bar{\rho }}\Vert u\Vert ^2_{L^6}\Vert \nabla w\Vert _{L^6}\Vert \nabla w_t\Vert _{L^2}\nonumber \\&\quad +{\bar{\rho }}\Vert u\Vert _{L^6}\Vert \nabla u\Vert _{L^2}\Vert \nabla w\Vert _{L^6}\Vert w_t\Vert _{L^6}\nonumber \\&\quad +2{\bar{\rho }}^{\frac{1}{2}}\Vert u\Vert _{L^6}\Vert \sqrt{\rho }w_t\Vert _{L^3}\Vert \nabla w_t\Vert _{L^2}+{\bar{\rho }}^{\frac{1}{2}}\Vert \sqrt{\rho }w_t\Vert _{L^3}\Vert u_t\Vert _{L^6}\Vert \nabla w\Vert _{L^2}\nonumber \\&\le C{\bar{\rho }}\Vert \nabla w_t\Vert _{L^2}\Vert \nabla u\Vert ^2_{L^2}\Vert w\Vert _{H^2}+C{\bar{\rho }}^{\frac{1}{2}}\Vert \nabla u\Vert _{L^2}\Vert \sqrt{\rho }w_t\Vert ^{\frac{1}{2}}_{L^2}\Vert \sqrt{\rho }w_t\Vert ^{\frac{1}{2}}_{L^6}\Vert \nabla w_t\Vert _{L^2}\nonumber \\&\quad +C{\bar{\rho }}^{\frac{1}{2}}\Vert \nabla u_t\Vert _{L^2}\Vert \sqrt{\rho }w_t\Vert ^{\frac{1}{2}}_{L^2}\Vert \sqrt{\rho }w_t\Vert ^{\frac{1}{2}}_{L^6}\Vert \nabla w\Vert _{L^2}\nonumber \\&\le C{\bar{\rho }}\Vert \nabla w_t\Vert _{L^2}\Vert \nabla u\Vert ^2_{L^2}\Vert w\Vert _{H^2}+C{\bar{\rho }}^{\frac{1}{2}}\Vert \nabla u\Vert _{L^2}\Vert \sqrt{\rho }w_t\Vert ^{\frac{1}{2}}_{L^2}\Vert \nabla w_t\Vert ^{\frac{3}{2}}_{L^2}\nonumber \\&\quad +C{\bar{\rho }}^{\frac{1}{2}}\Vert \nabla u_t\Vert _{L^2}\Vert \sqrt{\rho }w_t\Vert ^{\frac{1}{2}}_{L^2}\Vert \nabla w_t\Vert ^{\frac{1}{2}}_{L^2}\Vert \nabla w\Vert _{L^2}\nonumber \\&\le \frac{{\underline{\mu }}}{8}\Vert \nabla u_t\Vert ^2_{L^2}+\frac{\gamma }{2}\Vert \nabla w_t\Vert ^2_{L^2}+C(\Vert \sqrt{\rho }w_t\Vert ^2_{L^2}+\Vert w\Vert ^2_{H^2}). \end{aligned}$$

(3.39)

It deduces from the integration by parts and Cauchy-Schwarz inequality that

$$\begin{aligned} I_3=4\kappa \int \nabla ^\bot \cdot u_tw_tdx\le 4\kappa \Vert w_t\Vert ^2_{L^2}+\kappa \Vert \nabla u_t\Vert ^2_{L^2}. \end{aligned}$$

(3.40)

With the help of (3.1) and Sobolev’s inequality, we get

$$\begin{aligned} J_4&\le C\Vert u\Vert _{L^{\frac{4q}{q-2}}}\Vert \nabla \mu (\rho )\Vert _{L^q}\Vert \nabla u\Vert _{L^{\frac{4q}{q-2}}}\Vert \nabla u_t\Vert _{L^2}\nonumber \\&\le C\Vert \nabla u\Vert _{L^2}\Vert \nabla u\Vert _{H^1}\Vert \nabla u_t\Vert _{L^2}\nonumber \\&\le \frac{{\underline{\mu }}}{8}\Vert \nabla u_t\Vert ^2_{L^2}+C\Vert u\Vert ^2_{H^2}. \end{aligned}$$

(3.41)

By Sobolev’s inequality and (3.28), we have

$$\begin{aligned} J_5+J_6&\le \Vert b_t\Vert _{L^4}\Vert \nabla u_t\Vert _{L^2}\Vert b\Vert _{L^4}\le C\Vert b_t\Vert ^{\frac{1}{2}}_{L^2}\Vert \nabla b_t\Vert ^{\frac{1}{2}}_{L^2}\Vert \nabla u_t\Vert _{L^2}\nonumber \\&\le \frac{{\underline{\mu }}}{8}\Vert \nabla u_t\Vert ^2_{L^2}+C(\varepsilon )\Vert b_t\Vert ^2_{L^2}+\varepsilon \Vert \nabla b_t\Vert ^2_{L^2}. \end{aligned}$$

(3.42)

Substituting (3.38) and (3.42) into (3.37), it shows

$$\begin{aligned}&\frac{d}{dt}(\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t\Vert ^2_{L^2})+{\underline{\mu }}\Vert \nabla u_t\Vert ^2_{L^2}+\gamma \Vert \nabla w_t\Vert ^2_{L^2}\nonumber \\&\quad \le C(\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t\Vert ^2_{L^2}+\Vert u\Vert ^2_{H^2}+\Vert w\Vert ^2_{H^2})+C(\varepsilon )\Vert b_t\Vert ^2_{L^2}+\varepsilon \Vert \nabla b_t\Vert ^2_{L^2}. \end{aligned}$$

(3.43)

Multiplying (1.4)$_3$ by w and integrating by parts yield

$$\begin{aligned} 4\kappa \Vert w\Vert ^2_{L^2}+\gamma \Vert \nabla w\Vert ^2_{L^2}&\le \Vert -\rho w_t-\rho u\cdot \nabla w-2\kappa \nabla ^\bot \cdot u\Vert _{L^2}\Vert w\Vert _{L^2}\nonumber \\&\le \kappa \Vert w\Vert ^2_{L^2}+C(\Vert \rho w_t\Vert ^2_{L^2}+\Vert \rho u\cdot \nabla w\Vert ^2_{L^2}+\Vert \nabla u\Vert ^2_{L^2}), \end{aligned}$$

(3.44)

which implies that

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

(3.45)

According to the standard $L^2$-estimates of the elliptic system (see [24]) and together with (3.1), (3.3), (3.18) and (3.45), we obtain

$$\begin{aligned} \Vert w\Vert _{H^2}&=\Vert w\Vert _{H^1}+\Vert \nabla ^2w\Vert _{L^2}\nonumber \\&\le \Vert w\Vert _{H^1}+C\Vert \rho w_t -\rho u\cdot \nabla w-4\kappa w-2\kappa \nabla ^\bot \cdot u\Vert _{L^2}\nonumber \\&\le C(\Vert \rho w_t\Vert _{L^2}+\Vert \rho u\cdot \nabla w\Vert _{L^2}+\Vert \nabla u\Vert _{L^2})\nonumber \\&\le C{\bar{\rho }}^{\frac{1}{2}}\Vert \sqrt{\rho } w_t\Vert _{L^2}+C{\bar{\rho }}\Vert u\Vert _{L^6}\Vert \nabla w\Vert _{L^3}+C\Vert \nabla u\Vert _{L^2}\nonumber \\&\le C\Vert \sqrt{\rho } w_t\Vert _{L^2}+C{\bar{\rho }}\Vert \nabla u\Vert _{L^2}\Vert \nabla w\Vert ^{\frac{1}{2}}_{L^2}\Vert \nabla ^2w\Vert ^{\frac{1}{2}}_{L^2}+C\Vert \nabla u\Vert _{L^2}\nonumber \\&\le C\Vert \sqrt{\rho } w_t\Vert _{L^2}+\frac{1}{2}\Vert w\Vert _{H^2}+C\Vert \nabla u\Vert ^2_{L^2}\Vert \nabla w\Vert _{L^2}+C\Vert \nabla u\Vert _{L^2}, \end{aligned}$$

(3.46)

which gives

$$\begin{aligned} \Vert w\Vert _{H^2}\le C\Vert \sqrt{\rho } w_t\Vert _{L^2}+C\Vert \nabla w\Vert _{L^2}+C\Vert \nabla u\Vert _{L^2}. \end{aligned}$$

(3.47)

Combining with (1.4)$_4$, (3.12), Gagliardo-Nirenberg and Sobolev’s inequality leads to

$$\begin{aligned} \Vert b_t\Vert ^2_{L^2}&\le C\Vert \triangle b\Vert ^2_{L^2}+C\Vert u\Vert ^2_{L^\infty }\Vert \nabla b\Vert ^2_{L^2}+C\Vert b\Vert ^2_{L^4}\Vert \nabla u\Vert ^2_{L^4}\nonumber \\&\le C\Vert \triangle b\Vert ^2_{L^2}+C\Vert u\Vert _{L^4}\Vert \nabla u\Vert _{L^4}\Vert \nabla b\Vert _{L^2}+C\Vert \nabla b\Vert ^2_{L^2}\Vert \nabla u\Vert _{L^2}\Vert \nabla u\Vert _{H^1}\nonumber \\&\le C\Vert \triangle b\Vert ^2_{L^2}+\Vert \nabla u\Vert _{L^2}\Vert \nabla u\Vert ^{\frac{1}{2}}_{L^2}\Vert \nabla u\Vert ^{\frac{1}{2}}_{H^1}\Vert \nabla b\Vert _{L^2}+C\Vert \nabla b\Vert ^2_{L^2}\Vert \nabla u\Vert _{L^2}\Vert \nabla u\Vert _{H^1}\nonumber \\&\le C\Vert \triangle b\Vert ^2_{L^2}+C\Vert \nabla u\Vert ^2_{H^1}+C\Vert \nabla u\Vert ^2_{L^2}+C\Vert \nabla b\Vert ^2_{L^2}. \end{aligned}$$

(3.48)

Then, we deduce from (3.22), (3.26), (3.47) and (3.27) that

$$\begin{aligned}&\frac{d}{dt}(\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t\Vert ^2_{L^2})+{\underline{\mu }}\Vert \nabla u_t\Vert ^2_{L^2}+\gamma \Vert \nabla w_t\Vert ^2_{L^2}\nonumber \\&\quad \le C(\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t\Vert ^2_{L^2}+\Vert \triangle b\Vert ^2_{L^2}+\Vert |b||\nabla b|\Vert ^2_{L^2})\nonumber \\&\qquad +C(\Vert \nabla u\Vert ^2_{L^2}+\Vert \nabla w\Vert ^2_{L^2}+\Vert \nabla b\Vert ^2_{L^2})+\varepsilon \Vert \nabla b_t\Vert ^2_{L^2}. \end{aligned}$$

(3.49)

Differentiating (1.4)$_4$ with respect to t, we get

$$\begin{aligned} b_{tt}-\nu \triangle b_t+u_t\cdot \nabla b+u\cdot \nabla b_t+b_t\cdot \nabla u+b\cdot \nabla u_t=0, \end{aligned}$$

(3.50)

which multiplying by $b_t$ and along with integration by parts, Sobolev’s inequality and (3.12), we have

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\Vert b_t\Vert ^2_{L^2}+\nu \Vert \nabla b_t\Vert ^2_{L^2}&\le C(\Vert |u_t||b|\Vert _{L^2}+\Vert |u|b_t\Vert _{L^2})\Vert \nabla b_t\Vert _{L^2}\nonumber \\&\le C(\Vert u_t\Vert _{L^4}\Vert b\Vert _{L^4}+\Vert u\Vert _{L^4}\Vert b_t\Vert _{L^4})\Vert \nabla b_t\Vert _{L^2}\nonumber \\&\le C(\Vert \nabla u_t\Vert _{L^2}\Vert \nabla b\Vert _{L^2}+\Vert \nabla u\Vert _{L^2}\Vert b_t\Vert ^\frac{1}{2}_{L^2}\Vert \nabla b_t\Vert ^\frac{1}{2}_{L^2})\Vert \nabla b_t\Vert _{L^2}\nonumber \\&\le \frac{\nu }{2}\Vert \nabla b_t\Vert ^2_{L^2}+C_2\Vert \nabla u_t\Vert ^2_{L^2}+C\Vert b_t\Vert ^2_{L^2}, \end{aligned}$$

(3.51)

where $C_2$ is a positive constant. Next, adding (3.49)$\times \frac{2C_2}{{\underline{\mu }}}$ to (3.51) and choosing $\varepsilon =\frac{\nu }{4C_2}$, it yields

$$\begin{aligned}&\frac{d}{dt}\frac{2C_2}{{\underline{\mu }}}(\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t\Vert ^2_{L^2}+\Vert b_t\Vert ^2_{L^2})+C_2\Vert \nabla u_t\Vert ^2_{L^2}+\gamma \Vert \nabla w_t\Vert ^2_{L^2}+\frac{\nu }{2}\Vert \nabla b_t\Vert ^2_{L^2}\nonumber \\&\quad \le C(\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t\Vert ^2_{L^2}+\Vert \triangle b\Vert ^2_{L^2}+\Vert |b||\nabla b|\Vert ^2_{L^2})\nonumber \\&\qquad +C(\Vert \nabla u\Vert ^2_{L^2}+\Vert \nabla w\Vert ^2_{L^2}+\Vert \nabla b\Vert ^2_{L^2}). \end{aligned}$$

(3.52)

Multiplying (3.52) by t and integrating it over [0, T], then together with (3.5) and (3.13) leads to

$$\begin{aligned}&\sup \limits _{t\in [0, T]}t(\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t\Vert ^2_{L^2}+\Vert b_t\Vert ^2_{L^2})\nonumber \\&\qquad +\int ^T_0 t(\Vert \nabla u_t\Vert ^2_{L^2}+\Vert \nabla w_t\Vert ^2_{L^2}+\Vert \nabla b_t\Vert _{L^2})dt\nonumber \\&\quad \le C\sup \limits _{t\in [0, T]}(te^{-\sigma t})\int _0^Te^{\sigma t}(\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t\Vert ^2_{L^2}+\Vert \triangle b\Vert ^2_{L^2}+\Vert |b||\nabla b|\Vert ^2_{L^2})dx\nonumber \\&\qquad +C\sup \limits _{t\in [0, T]}(te^{-\sigma t})\int _0^Te^{\sigma t}(\Vert \nabla u\Vert ^2_{L^2}+\Vert \nabla w\Vert ^2_{L^2}+\Vert \nabla b\Vert ^2_{L^2})dt\nonumber \\&\quad \le C\Vert \nabla u_0\Vert ^2_{L^2}. \end{aligned}$$

(3.53)

Multiplying (3.52) by $e^{\sigma t}$ and together with (3.48), we derive

$$\begin{aligned}&\frac{d}{dt}\frac{2C_2}{{\underline{\mu }}}e^{\sigma t}(\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t\Vert ^2_{L^2}+\Vert b_t\Vert ^2_{L^2})\nonumber \\&\qquad +e^{\sigma t}(\Vert \nabla u_t\Vert ^2_{L^2}+\Vert \nabla w_t\Vert ^2_{L^2}+\Vert \nabla b_t\Vert ^2_{L^2})\nonumber \\&\quad \le Ce^{\sigma t}(\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t\Vert ^2_{L^2}+\Vert \triangle b\Vert ^2_{L^2}+\Vert |b||\nabla b|\Vert ^2_{L^2})+\sigma Ce^{\sigma t}\Vert b_t\Vert ^2_{L^2}\nonumber \\&\qquad +Ce^{\sigma t}(\Vert \nabla u\Vert ^2_{L^2}+\Vert \nabla w\Vert ^2_{L^2}+\Vert \nabla b\Vert ^2_{L^2})+\sigma Ce^{\sigma t}(\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t\Vert ^2_{L^2})\nonumber \\&\quad \le Ce^{\sigma t}(\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }w_t\Vert ^2_{L^2}+\Vert \triangle b\Vert ^2_{L^2}+\Vert |b||\nabla b|\Vert ^2_{L^2})\nonumber \\&\qquad +Ce^{\sigma t}(\Vert \nabla u\Vert ^2_{L^2}+\Vert \nabla w\Vert ^2_{L^2}+\Vert \nabla b\Vert ^2_{L^2}). \end{aligned}$$

(3.54)

Finally, integrating (3.54) by t over $[\zeta (T), T]$, and it deduces from (3.5) and (3.13) to lead to (3.34). The proof of Lemma 3.3 is finished.

Lemma 3.4

Let $(\rho , u, w, b, P)$ be a smooth solution to (1.4)–(1.6) satisfying (3.1). Then there exists some positive constant C depending only on $\Omega$, q, $\kappa$, $\nu$, ${\underline{\mu }}$, ${\bar{\mu }}$, ${\bar{\rho }}$, $\Vert \nabla u_0\Vert _{L^2}$, $\Vert \nabla w_0\Vert _{L^2}$ and $\Vert \nabla b_0\Vert _{L^2}$ such that

$$\begin{aligned} \int ^T_0\Vert \nabla u\Vert _{L^\infty }dt\le C\Vert \nabla u_0\Vert ^2_{L^2}. \end{aligned}$$

(3.55)

Proof

First, it follows from Lemma 2.2, (3.3), (3.1), H$\ddot{o}$lder’s, Sobolev’s and Gagliardo-Nirenberg inequalities that for any $r\in (2, min(q, 3))$,

$$\begin{aligned} \Vert \nabla u\Vert _{L^\infty }&\le \Vert u\Vert _{W^{2, r}}\le C(\Vert \rho u_t\Vert _{L^r}+\Vert \rho u\cdot \nabla u\Vert _{L^r}+\Vert \nabla w\Vert _{L^r}+\Vert b\cdot \nabla b\Vert _{L^r})\nonumber \\&\quad \times (1+\Vert \nabla \mu \left( \rho )\Vert _{L^q}\right) ^{\frac{qr}{2(q-r)}}\nonumber \\&\le C\Vert \rho u_t\Vert _{L^3}+C\Vert |u||\nabla u|\Vert _{L^3}+C\Vert \nabla w\Vert _{L^4}+C\Vert b\Vert _{L^\infty }\Vert \nabla b\Vert _{L^4}\nonumber \\&\le C{\bar{\rho }}^{\frac{3}{4}}\Vert \sqrt{\rho } u_t\Vert _{L^2}^{\frac{1}{2}}\Vert u_t\Vert _{L^6}^{\frac{1}{2}}+C\Vert u\Vert _{L^6}\Vert \nabla u\Vert _{L^6}\nonumber \\&\quad +C\Vert \nabla ^2w\Vert _{L^2}+C\Vert b\Vert ^{\frac{1}{2}}_{L^4}\Vert \nabla b\Vert ^{\frac{1}{2}}_{L^4}\nonumber \\&\le C\Vert \sqrt{\rho }u_t\Vert _{L^2}^{\frac{1}{2}}\Vert \nabla u_t\Vert _{L^2}^{\frac{1}{2}}+C\Vert u\Vert _{H^1}+C\Vert w\Vert _{H^1}+C\Vert \nabla b\Vert ^2_{L^2}+C\Vert \triangle b\Vert ^2_{L^2}, \end{aligned}$$

(3.56)

which together with (3.26) and (3.47), it shows

$$\begin{aligned} \Vert \nabla u\Vert _{L^\infty }&\le C\Vert \sqrt{\rho }u_t\Vert _{L^2}^{\frac{1}{2}}\Vert \nabla u_t\Vert _{L^2}^{\frac{1}{2}}+C(\Vert \sqrt{\rho }u_t\Vert _{L^2}+\Vert \sqrt{\rho }w_t\Vert _{L^2})+C\Vert \nabla b\Vert ^2_{L^2}\nonumber \\&\quad +C(\Vert \nabla u\Vert _{L^2}+\Vert \nabla w\Vert _{L^2})+C\Vert |b||\nabla b|\Vert _{L^2}. \end{aligned}$$

(3.57)

Hence, it follows from (3.1), (3.4), (3.33) and (3.34), that for $t\in [0, \zeta (T)]$,

$$\begin{aligned}&\int _0^{\zeta (T)}\Vert \nabla u\Vert _{L^\infty }dt\nonumber \\&\quad \le C(\int _0^{\zeta (T)}t^{-{\frac{1}{2}}}\Vert \sqrt{\rho }u_t\Vert _{L^2}^{\frac{2}{3}}dt)^{\frac{3}{4}}(\int _0^{\zeta (T)}t^{{\frac{3}{2}}}\Vert \nabla u_t\Vert _{L^2}^2dt)^{\frac{1}{4}}\nonumber \\&\qquad +C\int _0^{\zeta (T)}t^{-{\frac{1}{2}}}(t^{{\frac{1}{2}}}\Vert \sqrt{\rho }u_t\Vert _{L^2}+t^{{\frac{1}{2}}}\Vert \sqrt{\rho }w_t\Vert _{L^2})dt+\int _0^{\zeta (T)}\Vert \nabla b\Vert ^2_{L^2}dt\nonumber \\&\qquad +C\int _0^{\zeta (T)}(\Vert \nabla u\Vert _{L^2}+\Vert \nabla w\Vert _{L^2})dt+C\int _0^{\zeta (T)}(e^{\frac{\sigma t}{2}}\Vert |b||\nabla b|\Vert _{L^2})e^{-\frac{\sigma t}{2}}dt\nonumber \\&\quad \le C\sup \limits _{t\in [0, \zeta (T)]}(t\Vert \sqrt{\rho }u_t\Vert ^2_{L^2})^{\frac{1}{4}}(\int _0^{\zeta (T)}t^{-{\frac{1}{2}}}\cdot t^{-{\frac{1}{3}}}dt)^\frac{3}{4}(\int _0^{\zeta (T)}t^{{\frac{3}{2}}}\Vert \nabla u_t\Vert _{L^2})^{\frac{1}{4}}\nonumber \\&\qquad +C\int _0^{\zeta (T)}e^{\sigma t}\Vert |b||\nabla b|\Vert _{L^2}dt\int _0^{\zeta (T)}e^{-\sigma t}dt \le C\Vert \nabla u_0\Vert ^2_{L^2}. \end{aligned}$$

(3.58)

Similarly, it follows from (3.1), (3.5) and (3.34), that for $t\in [\zeta (T), T]$,

$$\begin{aligned} \int _{\zeta (T)}^T\Vert \nabla u\Vert _{L^\infty }dt&\le C\Vert \nabla u_0\Vert _{L^2}+C(\int _{\zeta (T)}^Te^{-\sigma t}dt)^{\frac{1}{2}}(\int _{\zeta (T)}^Te^{\sigma t}\Vert \nabla u_t\Vert ^2_{L^2}dt)^{\frac{1}{2}}\nonumber \\&\le C\Vert \nabla u_0\Vert ^2_{L^2}, \end{aligned}$$

(3.59)

this together with (3.58) yields (3.55). This completes the proof of Lemma 3.4.$\square$

With Lemmas 3.1–3.4 at hand, we are in a position to prove Proposition 3.1.

Proof of Proposition 3.1

First, it follows from (1.4)$_1$, multiplying (3.14) by $|\nabla \mu (\rho )|^{q-2}\partial _j\mu (\rho )$ and integrating the resulting equality by parts, we obtain that

$$\begin{aligned} \frac{d}{dt}\Vert \nabla \mu (\rho )\Vert _{L^q}\le C\Vert \nabla u\Vert _{L^\infty }\Vert \nabla \mu (\rho )\Vert _{L^q}, \end{aligned}$$

(3.60)

which together with Gronwall’s inequality and (3.55) yields

$$\begin{aligned} \sup \limits _{t\in [0, T]}\Vert \nabla \mu (\rho )\Vert _{L^q}&\le \Vert \nabla \mu (\rho _0)\Vert _{L^q}\exp \{q\int ^T_0\Vert \nabla u\Vert _{L^\infty }dt\}\nonumber \\&\le \Vert \nabla \mu (\rho _0)\Vert _{L^q}\exp \{C_3\Vert \nabla u_0\Vert ^2_{L^2}\}\nonumber \\&\le 2\Vert \nabla \mu (\rho _0)\Vert _{L^q}, \end{aligned}$$

(3.61)

where $\Vert \nabla u_0\Vert ^2_{L^2}\le \varepsilon _1\triangleq \min \{1, \frac{ln2}{C_3}\}$. Next, it deduces from (3.30), (3.5) and (3.12) gives

$$\begin{aligned}&\sup \limits _{t\in [0, T]}e^{\sigma t}\Vert \nabla u\Vert ^2_{L^2}\nonumber \\&\quad \le \sup \limits _{t\in [0, T]}(\Vert \nabla u\Vert ^2_{L^2})^2\int _0^Te^{\sigma t}\Vert \nabla u\Vert ^2_{L^2}dt+\sup \limits _{t\in [0, T]}\Vert \nabla w\Vert ^2_{L^2}\int _0^Te^{\sigma t}\Vert \nabla w\Vert ^2_{L^2}dt\nonumber \\&\qquad +\sup \limits _{t\in [0, T]}\Vert \nabla b\Vert ^2_{L^2}\int _0^Te^{\sigma t}\Vert \nabla b\Vert ^2_{L^2}dt\nonumber \\&\quad \le C(\Vert \nabla u_0\Vert ^2_{L^2})^3\le \Vert \nabla u_0\Vert ^2_{L^2}, \end{aligned}$$

(3.62)

where $\Vert \nabla u_0\Vert ^2_{L^2}\le \varepsilon _0\triangleq \min \{\varepsilon _1, C^{-\frac{1}{4}}\}$. Thus, we gain the (3.1) from (3.61) and (3.62). It completes the proof of the Proposition 3.1.$\square$

Lemma 3.5

Let $(\rho , u, w, b, P)$ be a smooth solution to (1.4)–(1.6) satisfying (3.1). Then there exists some positive constant C depending only on $\Omega$, q, $\kappa$, $\nu$, ${\underline{\mu }}$, ${\bar{\mu }}$, ${\bar{\rho }}$, $\Vert \nabla u_0\Vert _{L^2}$, $\Vert \nabla w_0\Vert _{L^2}$ and $\Vert \nabla b_0\Vert _{L^2}$ such that

$$\begin{aligned}&\sup \limits _{t\in [0, T]}\Vert \rho \Vert _{H^1\cap W^{1, q}}+\sup \limits _{t\in [0, T]}t(\Vert u\Vert ^2_{H^2}+\Vert P\Vert ^2_{H^1}+\Vert w\Vert ^2_{H^2}+\Vert b\Vert ^2_{H^2})\nonumber \\&\quad +\int _0^T\zeta e^{\sigma t}(\Vert u\Vert ^2_{H^2}+\Vert P\Vert ^2_{H^1}+\Vert w\Vert ^2_{H^2}+\Vert b\Vert ^2_{H^2}\nonumber \\&\quad +\Vert u\Vert ^2_{W^{2, r}}+\Vert P\Vert ^2_{W^{1, r}}+\Vert w\Vert ^2_{W^{2, r}}+\Vert b\Vert ^2_{W^{2, r}})dt\le C. \end{aligned}$$

(3.63)

Proof

It is easy to deduce from (3.60) and (3.61) that

$$\begin{aligned} \Vert \nabla \rho \Vert _{L^2\cap L^q}\le 2\Vert \nabla \rho _0\Vert _{L^2\cap L^q}. \end{aligned}$$

(3.64)

We notice that (1.4)$_4$ combining with (3.4), (3.12), Gagliardo-Nirenberg and Sobolev’s inequality that

$$\begin{aligned} \Vert b\Vert ^2_{H^2}&\le C(\Vert b_t\Vert ^2_{L^2}+\Vert u\cdot \nabla b\Vert ^2_{L^2}+\Vert b\cdot \nabla u\Vert ^2_{L^2}+\Vert b\Vert ^2_{H^1})\nonumber \\&\le C\Vert b_t\Vert ^2_{L^2}+C\Vert u\Vert ^2_{L^6}\Vert \nabla b\Vert ^2_{L^3}+C\Vert b\Vert ^2_{L^\infty }\Vert \nabla u\Vert ^2_{L^2}+C\Vert \nabla b\Vert ^2_{L^2}\nonumber \\&\le C\Vert b_t\Vert ^2_{L^2}+C\Vert \nabla u\Vert ^2_{L^2}\Vert \nabla b\Vert _{L^2}\Vert \nabla ^2b\Vert _{L^2}+C\Vert b\Vert _{L^2}\Vert b\Vert _{H^2}\Vert \nabla u\Vert ^2_{L^2}+C\Vert \nabla b\Vert ^2_{L^2}\nonumber \\&\le C\Vert b_t\Vert ^2_{L^2}+C\Vert \nabla b\Vert ^2_{L^2}+\frac{1}{2}\Vert b\Vert ^2_{H^2}, \end{aligned}$$

(3.65)

which combining with (3.25) and (3.47) shows

$$\begin{aligned}&\Vert u\Vert ^2_{H^2}+\Vert P\Vert ^2_{H^1}+\Vert w\Vert ^2_{H^2}+\Vert b\Vert ^2_{H^2}\nonumber \\&\quad \le C(\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho } w_t\Vert _{L^2}+\Vert b_t\Vert ^2_{L^2})\nonumber \\&\qquad +C(\Vert \nabla u\Vert ^2_{L^2}+\Vert \nabla w\Vert ^2_{L^2}+\Vert \nabla b\Vert ^2_{L^2})+C\Vert |b||\nabla b|\Vert ^2_{L^2}\nonumber \\&\quad \le C(\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho } w_t\Vert _{L^2}+\Vert b_t\Vert ^2_{L^2})+C(\Vert \nabla u\Vert ^2_{L^2}+\Vert \nabla w\Vert ^2_{L^2}+\Vert \nabla b\Vert ^2_{L^2}), \end{aligned}$$

(3.66)

where

$$\begin{aligned} \Vert |b||\nabla b|\Vert ^2_{L^2}\le C\Vert b\Vert ^2_{H^2}\Vert \nabla b\Vert ^2_{L^2}\le C\Vert b_t\Vert ^2_{L^2}+C\Vert \nabla b\Vert ^2_{L^2}. \end{aligned}$$

(3.67)

Then, it follows from (3.5), (3.13), and (3.34) that

$$\begin{aligned}&\sup \limits _{t\in [0, T]}t(\Vert u\Vert ^2_{H^2}+\Vert P\Vert ^2_{H^1}+\Vert w\Vert ^2_{H^2}+\Vert b\Vert ^2_{H^2})\nonumber \\&\quad +\int _0^T\zeta e^{\sigma t}(\Vert u\Vert ^2_{H^2}+\Vert P\Vert ^2_{H^1}+\Vert w\Vert ^2_{H^2}+\Vert b\Vert ^2_{H^2})dt\le C. \end{aligned}$$

(3.68)

It deduces from Lemma 2.3, Sobolev’s inequality, (3.3), (3.12), (3.13), (3.66) and (3.68) that for any $r\in (2, q)$,

$$\begin{aligned}&\Vert u\Vert ^2_{W^{2, r}}+\Vert P\Vert ^2_{W^{1, r}}\nonumber \\&\quad \le C\left( \Vert \rho u_t\Vert ^2_{L^r}+\Vert \rho u\cdot \nabla u\Vert ^2_{L^r}+\Vert \nabla w\Vert ^2_{L^r}+\Vert b\cdot \nabla b\Vert ^2_{L^r}\right) \nonumber \\&\qquad \times (1+\Vert \nabla \mu \left( \rho )\Vert _{L^q}\right) ^{1+\frac{qr}{2(q-r)}}\nonumber \\&\quad \le C\left( {\bar{\rho }}^2\Vert u_t\Vert ^2_{L^r}+{\bar{\rho }}^2\Vert u\Vert ^2_{L^{\frac{qr}{q-r}}}\Vert \nabla u\Vert ^2_{L^q}+\Vert \nabla ^2w\Vert ^2_{L^2}+\Vert b\Vert ^2_{L^{\frac{qr}{q-r}}}\Vert \nabla b\Vert ^2_{L^q}\right) \nonumber \\&\quad \le C\left( \Vert \nabla u_t\Vert ^2_{L^2}+\Vert \nabla u\Vert ^2_{L^2}\Vert \nabla ^2u\Vert ^2_{L^2}+\Vert \nabla ^2w\Vert ^2_{L^2}+\Vert \nabla b\Vert ^2_{L^2}\Vert \nabla ^2b\Vert ^2_{L^2}\right) \nonumber \\&\quad \le C\left( \Vert \nabla u_t\Vert ^2_{L^2}+\Vert \triangle b\Vert ^2_{L^2}\right) +C\left( \Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \sqrt{\rho } w_t\Vert ^2_{L^2}+\Vert b_t\Vert ^2_{L^2}\right) \nonumber \\&\qquad +C\left( \Vert \nabla u\Vert ^2_{L^2}+\Vert \nabla w\Vert ^2_{L^2}+\Vert \nabla b\Vert ^2_{L^2}\right) +C\Vert |b||\nabla b|\Vert ^2_{L^2}. \end{aligned}$$

(3.69)

We infer from (1.4)$_3$, regularity theory of elliptic equations, (3.3), (3.12), (3.13), (3.66) and (3.68) that for any $r\in (2, q)$,

$$\begin{aligned} \Vert w\Vert ^2_{W^{2, r}}&\le C\left( \Vert \rho w_t\Vert ^2_{L^r}+\Vert \rho u\cdot \nabla w\Vert ^2_{L^r}+\Vert w\Vert ^2_{L^r}+\Vert \nabla u\Vert ^2_{L^r}\right) \nonumber \\&\le C\left( {\bar{\rho }}^2\Vert w_t\Vert ^2_{L^r}+{\bar{\rho }}^2\Vert u\Vert ^2_{L^{\frac{qr}{q-r}}}\Vert \nabla w\Vert ^2_{L^q}+\Vert \nabla w\Vert ^2_{L^2}+\Vert \nabla ^2w\Vert ^2_{L^2}\right) \nonumber \\&\le C\left( \Vert \nabla w_t\Vert ^2_{L^2}+\Vert \nabla u\Vert ^2_{L^2}\Vert \nabla ^2w\Vert ^2_{L^2}+\Vert \nabla w\Vert ^2_{L^2}+\Vert \nabla ^2w\Vert ^2_{L^2}\right) \nonumber \\&\le C\left( \Vert \nabla w_t\Vert ^2_{L^2}+\Vert \sqrt{\rho } w_t\Vert _{L^2}\right) +C\left( \Vert \nabla u\Vert ^2_{L^2}+\Vert \nabla w\Vert ^2_{L^2}\right) . \end{aligned}$$

(3.70)

Similarly, we can obtain from (1.4)$_4$ and Sobolev’s inequality that

$$\begin{aligned} \Vert b\Vert ^2_{W^{2, r}}&\le C\left( \Vert b_t\Vert ^2_{L^r}+\Vert u\cdot \nabla b\Vert ^2_{L^r}+\Vert b\cdot \nabla u\Vert ^2_{L^r}\right) \nonumber \\&\le C\Vert \nabla b_t\Vert ^2_{L^2}+\Vert u\Vert ^2_{L^{\frac{qr}{q-r}}}\Vert \nabla b\Vert ^2_{L^q}+\Vert b\Vert ^2_{L^{\frac{qr}{q-r}}}\Vert \nabla u\Vert ^2_{L^q})\nonumber \\&\le C\left( \Vert \nabla b_t\Vert ^2_{L^2}+\Vert \nabla u\Vert ^2_{L^2}\Vert \nabla ^2b\Vert ^2_{L^2}+\Vert \nabla b\Vert ^2_{L^2}\Vert \nabla ^2u\Vert ^2_{L^2}\right) \nonumber \\&\le C(\Vert \nabla b_t\Vert ^2_{L^2}+\Vert \sqrt{\rho }u_t\Vert ^2_{L^2}+\Vert \triangle b\Vert ^2_{L^2}+C\Vert |b||\nabla b|\Vert ^2_{L^2}\nonumber \\&\quad +C\left( \Vert \nabla u\Vert ^2_{L^2}+\Vert \nabla w\Vert ^2_{L^2}\right) , \end{aligned}$$

(3.71)

and this along with (3.69) and (3.70) by (3.5) (3.3), (3.12), (3.13) and (3.68) gives

$$\begin{aligned} \int _0^T\zeta e^{\sigma t}\left( \Vert u\Vert ^2_{W^{2, r}}+\Vert P\Vert ^2_{W^{1, r}}+\Vert w\Vert ^2_{W^{2, r}}+\Vert b\Vert ^2_{W^{2, r}}\right) dt\le C. \end{aligned}$$

(3.72)

This combines (3.64) and (3.68) by indicates (3.63) and the proof of Lemma 3.5 is finished.$\square$

4 Proof of Theorem 1.1

With all the a priori estimates obtained in Sect. 3 at hand, we are now in a position to prove Theorem 1.1.

By Lemma 2.1, there exists a $T_{*}>0$ such that the problem (1.4)–(1.6) has a unique local strong solution $(\rho , u, w, b)$ on $\Omega \times (0, T_{*})$. We plan to extend the local solution to all time.

Set

$$\begin{aligned} T^{*}=\sup \{\,T\,|\,\left( \rho , u, w, b\right) \, is\, a\, strong\, solution\, on \times (0, T]\}. \end{aligned}$$

(4.1)

First, for any $0<\tau<T_{*}< T\le T^{*}$ with T finite, one deduces from (3.12), (3.33) and (3.63) that for any $q > 2$,

$$\begin{aligned} \nabla u, \nabla w, \nabla b\in C([\zeta (t), T]; L^q), \end{aligned}$$

(4.2)

where one has used the standard embedding

$L^\infty (\zeta (t), T; H^1)\cap H^1(\zeta (t), T; H^{-1})\hookrightarrow C(\zeta (t), T; L^q) \,for\, any\, q\in (2, \infty )$.

Moreover, it deduces from (3.64), (3.3), (3.12), (3.13) and ( [25], Lemma 2.3) that

$$\begin{aligned} \rho \in C([0, T]; W^{1, q}),\, \rho u\in C([0, T]; L^2),\, \rho w\in C([0, T]; L^2). \end{aligned}$$

(4.3)

Finally, if $T_{*}<\infty$, it follows from (4.2), (4.3), (3.4) and (3.12) that

$(\rho , u, w, b)(x, T^{*})=\lim \limits _{t\rightarrow T^{*}}(\rho , u, w, b)(x, t)$

satisfies the initial condition (1.8) at $t=T^{*}$. Thus, taking $(\rho , u, w, b)(x, T^{*})$ as the initial data, due to Lemma 2.1, it can extend the strong solutions beyond $T^{*}$. This contradicts the assumption of $T^{*}$ in (4.1). The proof of Theorem 1.1 is completed.