Global Existence and Large-Time Behavior for 3D Full Compressible Magneto-Micropolar System Without Heat Conductivity

Abstract

The system of full compressible magneto-micropolar flows is discussed in 3D bounded domains with slip boundary conditions. Based on the energy method, after establishing some key a priori exponential decay-in-times rates of the strong solutions, we obtain both the global existence and exponential stability of strong solutions. In particular, it should be pointed out that the estimates of ${\parallel (\mathrm{curl}\mathbf{u},\mathrm{curl}\mathbf{w})\parallel }_{L^2}$ and ${\parallel (\mathrm{div}\mathbf{u},\mathrm{div}\mathbf{w})\parallel }_{L^2}$ are established separately, which implies that the growth rate of $(\mathrm{div}\mathbf{u},\mathrm{div}\mathbf{w})$ in $L^2$ are faster than that of $(\mathrm{curl}\mathbf{u},\mathrm{curl}\mathbf{w})$ under the condition that the diameter of the domain is suitably large. Compared with previous works, we no longer consider the pressure P as $\rho \theta$, but as variable in $(x,t)$, and directly deal with ${\parallel \nabla P\parallel }_{L^2}$. Based on slip boundary conditions, we established the $L^p$-norm for the gradient of effective viscous flux, and the term ${\parallel \nabla P\parallel }_{L^2}$ can be controlled by $\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}$. Through precise calculations, we found that $\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}$ is dependent on ${\parallel \nabla P\parallel }_{L^2}$. Therefore, the smallness condition we propose does not depend on the $L^r$-norm of the density gradient, which means that density can contain large oscillations.

1. Introduction

The full compressible magneto-micropolar system reads as [1]:

$$\left\{\begin{array}{c}{\rho }_t+\mathrm{div}\left(\rho \mathbf{u}\right)=0,\hfill \\ {\left(\rho \mathbf{u}\right)}_t+\mathrm{div}(\rho \mathbf{u}\otimes \mathbf{u})+\nabla P+(-\mu -\zeta )\Delta \mathbf{u}+(-\mu -\lambda +\zeta )\nabla \mathrm{div}\mathbf{u}\hfill \\ =2\zeta \nabla \times \mathbf{w}+\left(\mathrm{curl}\mathbf{b}\right)\times \mathbf{b},\hfill \\ {\left(\rho \mathbf{w}\right)}_t+\mathrm{div}(\rho \mathbf{u}\otimes \mathbf{w})+4\zeta \mathbf{w}-(c_a+c_d)\Delta \mathbf{w}-(c_0+c_d-c_a)\nabla \mathrm{div}\mathbf{w}=2\zeta \nabla \times \mathbf{u},\hfill \\ {\left(\rho e\right)}_t+\mathrm{div}\left(\rho e\mathbf{u}\right)+P(\rho ,e)\mathrm{div}\mathbf{u}-\kappa \Delta \theta ={2\mu \left|\mathbb{D}\left(\mathbf{u}\right)\right|}^2+\lambda {\left(\mathrm{div}\mathbf{u}\right)}^2+\zeta {|\nabla \times \mathbf{u}-2\mathbf{w}|}^2\hfill \\ +c_0{\left(\mathrm{div}\mathbf{w}\right)}^2+(c_a+c_d){|\nabla \mathbf{w}|}^2+(c_d-c_a)\nabla \mathbf{w}:\nabla {\mathbf{w}}^{\top }+\nu {\left|\mathrm{curl}\mathbf{b}\right|}^2,\hfill \\ {\mathbf{b}}_t-\nabla \times (\mathbf{u}\times \mathbf{b})=-\nu \nabla \times \left(\mathrm{curl}\mathbf{b}\right),\hfill \\ \mathrm{div}\mathbf{b}=0,\hfill \end{array}\right.$$

(1)

with $t⩾0$ and $x=(x_1,x_2,x_3)\in \Omega \subset {\mathbb{R}}^3$. The unknown functions $\rho ,\mathbf{u}={(u^1,u^2,u^3)}^{\top }$, $\mathbf{w}={(w^1,w^2,w^3)}^{\top }$, $\theta$, $\mathbf{b}={(b^1,b^2,b^3)}^{\top }$, and P stand for the density, velocity, micro-rotational velocity, temperature, magnetic field, and pressure, respectively. The deformation tensor $\mathbb{D}\left(\mathbf{u}\right)$ is represented as follows:

$$\mathbb{D}\left(\mathbf{u}\right)\triangleq {\displaystyle \frac{\nabla \mathbf{u}+{(\nabla \mathbf{u})}^{\top }}{2}}.$$

(2)

The physical constants $\mu$, $\lambda$ and $\zeta$ are the shear viscosity, the bulk viscosity, and the dynamics micro-rotation viscosity, respectively, which satisfy $\mu >0,2\mu +3\lambda ⩾0$ and $\mu +\lambda -\zeta >0$. The positive constants $c_0$, $c_a$ and $c_d$ are the angular viscosities, which satisfy $c_0+c_d-c_a>0$. The non-negative constant $\kappa$ represents the heat conductivity coefficient, and the constant $\nu >0$ represents the magnetic resistivity coefficient. The pressure $P=P(\rho ,e)$ is determined through the equations of state for ideal polytropic fluids

$$P(\rho ,e)\triangleq (\gamma -1)\rho e=R\rho \theta ,e\triangleq {\displaystyle \frac{R\theta }{\gamma -1}}=c_v\theta ,$$

(3)

where e is the internal energy. The adiabatic exponent is represented by $\gamma$ which satisfies $\gamma >1$, the perfect gas constant is represented by $R>0$, and the heat capacity is represented by $c_v>0$.

To make the paper more readable, we provide an optional figure as follows which shows system variables and their interactions.

The complex interplay between fluid dynamics and magnetic fields in the full compressible magneto-micropolar system (1) has motivated considerable headway in the analysis of its solutions, particularly regarding well-posedness and dynamic behavior. (see, e.g., [1,2,3,4,5]). Recent mathematical analysis of $\kappa$ has seen considerable progress. Since the main purpose of this paper is to study the situation without thermal diffusional, we only recall some results concerning the thermal diffusional coefficient $\kappa$.

When $\kappa >0$, we refer to [6,7,8,9] for one-dimensional case and [10,11,12] for multi-dimensional case under the condition of $\mathbf{w}=0$. A series of studies have advanced the understanding of the magneto-micropolar system. Following Chen-Wang’s [13] result on global existence and uniqueness for large $H^1$ data, Hu-Wang [14] considered the Cauchy problem, proving weak solution compactness in ${\mathbb{R}}^3$ and periodic domains even with density-dependent, vacuum-vanishing thermal and magnetic coefficients. Building on this, Huang-Li [15] obtained a Serrin-type blow-up criterion. The well-posedness for the case $\mathbf{w}\ne 0$ has been studied in [3,4,5]. Regarding the incompressible case with vacuum, local existence was obtained by Tang-Sun [16] and local well-posedness by Fan-Zhang-Zhou [17]. For the compressible system, Fan-Ozawa [2] proved uniform-in-$\epsilon$ existence via the Banach fixed point theorem. Numerical aspects have also been investigated in [18,19,20].

The absence of heat conductivity ($\kappa =0$) introduces considerable complexity in establishing the global well-posedness of smooth solutions and often demands more stringent conditions. This is evidenced by the work of Lu-Huang [21], who secured local strong solutions in two dimensions under a specific decay condition on the initial density and magnetic field at infinity. Yu’s contributions further illuminate this context: in [22], global strong solutions were obtained for the 3D Cauchy problem under a small initial energy assumption, a result complemented by [23], which provides global existence and decay rates for solutions to an initial-boundary value problem.

The primary objective of this paper, inspired by [2,22,23], is the analysis of global well-posedness and asymptotic behavior for the 3D full compressible magneto-micropolar system in the presence of large oscillations in the initial density. Our study pursues this by examining the scenario with zero heat conductivity ($\kappa =0$). We set $c_v=R=1$ and $\gamma =2$ for simplicity, and let $\Omega \subset {\mathbb{R}}^3$ be a simply connected bounded domain. Consequently, the system (1) can be reformulated as:

$$\left\{\begin{array}{c}{\rho }_t+\mathrm{div}\left(\rho \mathbf{u}\right)=0,\hfill \\ {\left(\rho \mathbf{u}\right)}_t+\mathrm{div}(\rho \mathbf{u}\otimes \mathbf{u})+\nabla P-(\mu +\zeta )\Delta \mathbf{u}-(\mu +\lambda -\zeta )\nabla \mathrm{div}\mathbf{u}\hfill \\ =2\zeta \nabla \times \mathbf{w}+(\nabla \times \mathbf{b})\times \mathbf{b},\hfill \\ {\left(\rho \mathbf{w}\right)}_t+\mathrm{div}(\rho \mathbf{u}\otimes \mathbf{w})+4\zeta \mathbf{w}-(c_a+c_d)\Delta \mathbf{w}-(c_0+c_d-c_a)\nabla \mathrm{div}\mathbf{w}=2\zeta \nabla \times \mathbf{u},\hfill \\ P_t+\mathrm{div}\left(P\mathbf{u}\right)+P\mathrm{div}\mathbf{u}={2\mu \left|\mathbb{D}\left(\mathbf{u}\right)\right|}^2+\lambda {\left(\mathrm{div}\mathbf{u}\right)}^2+\zeta {|\nabla \times \mathbf{u}-2\mathbf{w}|}^2\hfill \\ +c_0{\left(\mathrm{div}\mathbf{w}\right)}^2+(c_a+c_d){|\nabla \mathbf{w}|}^2+(c_d-c_a)\nabla \mathbf{w}:\nabla {\mathbf{w}}^T+\nu {|\nabla \times \mathbf{b}|}^2,\hfill \\ {\mathbf{b}}_t+\mathbf{u}·\nabla \mathbf{b}-\mathbf{b}·\nabla \mathbf{u}+\mathbf{b}\mathrm{div}\mathbf{u}=-\nu \nabla \times (\nabla \times \mathbf{b}),\hfill \\ \mathrm{div}\mathbf{b}=0.\hfill \end{array}\right.$$

(4)

In 3D full compressible magneto-micropolar fluid model with large oscillations of the initial density, setting the thermal conductivity coefficient to zero is a mathematical idealization rather than a strict physical requirement. Physically, thermal conductivity represents the fluid’s ability to conduct heat, and a zero value implies a perfectly adiabatic process with no heat diffusion. The primary rationale for this assumption is $mathematicaltractability$. When dealing with large initial density oscillations, the system becomes extremely complex. The energy equation, coupled with the mass, momentum, and angular momentum equations, creates significant challenges for proving the existence of solutions. The term involving thermal conductivity (typically a second-order Laplacian in the temperature or internal energy) introduces strong dissipative effects that are difficult to handle analytically in this high-dimensional, nonlinear, and coupled system, especially with low regularity in the initial data. By setting thermal conductivity to zero, the energy equation simplifies, often reducing to an adiabatic relation (like $DP/Dt=c^{2}D\rho /Dt$ for an ideal gas) or a transport equation for the temperature. This simplification is crucial for closing the estimates needed in a rigorous mathematical analysis, such as using weak convergence methods or constructing approximate solutions. It allows mathematicians to isolate and understand the effects of compressibility, magnetic fields, and micropolarity without the additional complication of heat diffusion. Therefore, this assumption is a necessary $mathematicalsimplification$ to make the initial-boundary value problem analytically accessible, even though real fluids always have some, however small, thermal conductivity.

This paper addresses the initial-boundary value problem for system (4), subject to the following initial conditions:

$$(\rho ,P,\mathbf{u},\mathbf{w},\mathbf{b})(x,0)=({\rho }_0,P_0,{\mathbf{u}}_0,{\mathbf{w}}_0,{\mathbf{b}}_0)\left(x\right)\mathrm{with}x\in \Omega ,$$

(5)

and slip boundary conditions

$$\left\{\begin{array}{c}\mathbf{u}·\mathbf{n}=0,\mathrm{curl}\mathbf{u}\times \mathbf{n}=0,\mathrm{on}\partial \Omega ,\hfill \\ \mathbf{w}·\mathbf{n}=0,\mathrm{curl}\mathbf{w}\times \mathbf{n}=0,\mathrm{on}\partial \Omega ,\hfill \\ \mathbf{b}·\mathbf{n}=0,\mathrm{curl}\mathbf{b}\times \mathbf{n}=0,\mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

(6)

where the unit outward normal vector on the domain boundary $\partial \Omega$ is denoted by $\mathbf{n}={(n^1,n^2,n^3)}^{\top }$.

The principal mathematical obstacle to a deeper understanding of the 3D fully compressible magneto-micropolar model stems from the non-trivial superposition and nonlinear coupling of its constituent physics, which creates a multi-scale competitive environment absent in classical theories. Unlike standard magnetohydrodynamics (MHD), this model incorporates an additional transport equation for the angular velocity field of micro-elements, introducing new dissipative mechanisms. The core difficulty is that the system’s energy is partitioned and exchanged between three distinct dynamical realms: the hyperbolic transport of the magnetic field (governed by Maxwell’s equations), the acoustic pressure waves of the compressible fluid (introducing a second hyperbolic subsystem), and the dissipative, parabolic nature of the micro-rotational viscosity. This triad creates a formidable analytical challenge. The coupling is not merely additive, it is multiplicative through nonlinear terms like the Lorentz force, the Coriolis-type coupling between linear and angular velocity, and the advection terms. This intricate interplay resists the standard energy methods effective for simpler models. Specifically, it disrupts the delicate balances needed for a priori estimates, complicates the identification of a dissipative structure capable of controlling all solution components uniformly, and severely limits the applicability of standard compactness arguments. Consequently, establishing global well-posedness or even prolonged existence for large data in three dimensions remains a profound and open problem, as the system’s multi-scale nature–where parabolic, hyperbolic, and dispersive phenomena compete–defies a unified analytical treatment.

As a preliminary step, we establish the following conventions.

$$\int gdx\triangleq {\int }_{\Omega }gdx,\overline{g}\triangleq {\displaystyle \frac{1}{|\Omega |}}{\int }_{\Omega }{gdx,\parallel (f,g)\parallel }_X\triangleq {\parallel f\parallel }_X+{\parallel g\parallel }_X.$$

The initial total energy for system (4) is denoted by

$$E_0\triangleq \int \left({\displaystyle \frac{1}{2}}{\rho }_0|{\mathbf{u}}_0{|}^2+{\displaystyle \frac{1}{2}}{\rho }_0|{\mathbf{w}}_0{|}^2+P_0+{\displaystyle \frac{1}{2}}{|{\mathbf{b}}_0|}^2\right)dx.$$

The principal result of this paper is formulated below.

Theorem 1.

Let the initial data $({\rho }_0,P_0,{\mathbf{u}}_0,{\mathbf{w}}_0,{\mathbf{b}}_0)$, for $r\in (3,p]$ with $p\in (3,6)$ and $0<\underline{\rho }<\overline{\rho }<\infty$, be such that it satisfies

$${\rho }_0\in W^{1,r},P_0\in W^{1,p},\underline{\rho }⩽{\rho }_0⩽\overline{\rho },({\mathbf{u}}_0,{\mathbf{w}}_0,{\mathbf{b}}_0)\in H^2,$$

(7)

Then there exists a positive constant $\epsilon >0$, depending on $\mu ,\lambda ,\nu$, $\zeta ,c_0,c_a,c_d$, $\underline{\rho },\overline{\rho }$, $|\Omega |$ and initial data, such that if

$$C_0\triangleq \parallel ({\mathbf{u}}_0,{\mathbf{w}}_0,{\mathbf{b}}_0){\parallel }_{H^2}^2+{\parallel \nabla P_0\parallel }_{L^2}^2⩽\epsilon ,$$

a unique global strong solution $(\rho ,P,\mathbf{u},\mathbf{w},\mathbf{b})$ to the problem (4)–(6) exists on $\Omega \times (0,\infty )$ and possesses the following properties:

$${\displaystyle \frac{1}{2}}\underline{\rho }⩽\rho (x,t)⩽2\overline{\rho },{\displaystyle \frac{E_0}{2|\Omega |}}⩽P(x,t)⩽{\displaystyle \frac{3E_0}{2|\Omega |}},\forall (x,t)\in \Omega \times [0,\infty ),$$

and

$$\left\{\begin{array}{c}\rho \in C([0,\infty );W^{1,r}),{\rho }_t\in C([0,\infty );L^r),\hfill \\ P\in C([0,\infty );W^{1,p}),P_t\in C([0,\infty );L^p),\hfill \\ (\mathbf{u},\mathbf{w},\mathbf{b})\in C([0,\infty );H^2)\cap L^2(0,T;W^{2,p}),\hfill \\ ({\mathbf{u}}_t,{\mathbf{w}}_t,{\mathbf{b}}_t)\in L^{\infty }(0,\infty ;L^2)\cap L^2(0,\infty ;H^1).\hfill \end{array}\right.$$

Additionally, for a domain Ω with a suitably large diameter, positive constants C, σ, and $\eta (\eta <\sigma )$ exist that are functions only of $\mu ,\lambda ,\nu ,\zeta ,c_0,c_a,c_d$, and $|\Omega |$, and which for any $t>1$ yield the following:

$$\begin{array}{cc}\hfill & \parallel ({\rho }^{1/2}\mathbf{u},{\rho }^{1/2}{\mathbf{w}}_t,\mathbf{b})\left(t\right){\parallel }_{L^2}^2+{\parallel P^{1/2}\mathrm{div}\mathbf{u}\left(t\right)\parallel }_{L^2}^2\hfill \\ \hfill & +\parallel ({\rho }^{1/2}{\mathbf{u}}_t,{\rho }^{1/2}{\mathbf{w}}_t,{\mathbf{b}}_t)\left(t\right){\parallel }_{L^2}^2+{\parallel \nabla P\left(t\right)\parallel }_{L^2}^2⩽C{e}^{-\sigma t},\hfill \end{array}$$

and

$${\parallel (\mathrm{curl}\mathbf{u},\mathrm{curl}\mathbf{w},\mathrm{curl}\mathbf{b})\left(t\right)\parallel }_{L^2}^2⩽C{e}^{-\frac{\sigma t}{2}},{\parallel \nabla P\left(t\right)\parallel }_{L^p}^2⩽C{e}^{-\eta t}.$$

Next, we will give the remark on Physical Implications. Firstly, the proof of exponential decay-in-time for the strong solutions indicates that the system is highly effective at dissipating disturbances. Physically, this means that any initial perturbation in the fluid velocity, microrotation, or magnetic field will be smoothed out at a rapid, exponential rate. The system is not only stable but returns to its equilibrium state predictably and quickly. For the magnetic field, this implies a robust mechanism for suppressing magnetic fluctuations, which is crucial for the stability of plasma configurations in confinement devices. For the microrotation, it signifies the rapid attenuation of local vortex-like motions of the fluid’s micro-structures, leading to a uniform state.

Secondly, the separate analysis of the curl (related to rotational motion and vorticity) and divergence (related to compressibility and expansion) is physically significant. Our finding that the $L^2$-norm of $(\mathrm{div}\mathbf{u},\mathrm{div}\mathbf{w})$ can grow faster than that of $(\mathrm{curl}\mathbf{u},\mathrm{curl}\mathbf{w})$ in large domains reveals an important property: compressibility effects (acoustic waves, expansion/contraction) can become more dynamically active or persistent at larger scales compared to rotational effects (vortices, micro-rotations). This suggests that in large-scale systems, the dominant mechanisms for energy transfer or instability may be driven more by acoustic phenomena than by vortical dynamics. Furthermore, the slip boundary condition $\mathbf{u}·\mathbf{n}=0$ inherently suppresses the generation of strong vorticity at the boundary, which can indirectly amplify the relative influence of the compressible (divergence) components within the fluid bulk.

This paper is structured as follows. Section 2 presents fundamental preliminary results and essential inequalities. Section 3 is dedicated to deriving the necessary a priori estimates, which culminate in the proof of Theorem 1.

2. Preliminaries

This preliminary section presents lemmas that will be utilized in the forthcoming sections. To commence, we state the local existence result for strong solutions, which can be proven by employing arguments similar to those in [24]. In order to ensure the completeness and reproducibility of the paper, we briefly mention the key ideas.

Lemma 1.

Assume that $({\rho }_0,P_0,{\mathbf{u}}_0,{\mathbf{w}}_0,{\mathbf{b}}_0)$ satisfies (7). Then, there exists a positive time $T_{\ast }>0$ such that the problem (4)–(6) has a strong solution $(\rho ,P,\mathbf{u},\mathbf{w},\mathbf{b})$ in $\Omega \times [0,T_{\ast }]$.

In order to obtain the local existence and uniqueness of strong solution of the problem (4)–(6), we firstly using the Galerkin method or linearization/iteration to construct a suitable approximate system whose solutions are easier to handle. This scheme is designed to satisfy the necessary physical constraints, such as the divergence-free condition for the magnetic field. Secondly, Deriving a series of energy estimates for the approximate solutions in high-order Sobolev spaces to obtain the uniform A Priori Estimates.mThe core of this step is to show that for a small time $T_{\ast }>0$, which depends only on the initial data, the norms of the approximate solutions, e.g., ${\parallel (\rho ,P,\mathbf{u},\mathbf{w},\mathbf{b})\parallel }_{H^s}$, remain uniformly bounded. This relies critically on the initial data satisfying the compatibility conditions (7) and being bounded away from vacuum ${\rho }_0>0$. Thirdly, Using the uniform bounds from Step 2 to prove that the sequence of approximate solutions converges strongly to a limit function, and then, one must verify that this limit function satisfies the original nonlinear system (4)–(6). Finally, Proving the uniqueness by assuming two strong solutions exist for the same initial data. The key is to estimate the difference between these two solutions in a lower-order norm. By subtracting the equations they satisfy and performing energy estimates, one can typically show that the norm of their difference is zero, implying the solutions are identical. This often relies on the Ladyzhenskaya-type inequalities to handle the nonlinear terms.

Lemma 2

([25]). Consider a simply connected bounded domain $\Omega \subset {\mathbb{R}}^3$ with smooth boundary $\partial \Omega$. For parameters $p\in [2,6]$, $q\in (1,\infty )$, and $r\in (3,\infty )$, suppose the functions $\mathbf{f}\in H^1$ and $\mathbf{g}\in L^q\cap W^{1,r}$ are given. Then one can find generic constants $C,C_1,C_2>0$, potentially depending on $p,q,r$, and Ω, satisfying

$${\parallel \mathbf{f}\parallel }_{L^p}⩽{C\parallel \mathbf{f}\parallel }_{L^2}^{\frac{6-p}{2p}}{\parallel \nabla \mathbf{f}\parallel }_{L^2}^{\frac{3p-6}{2p}}+C_1{\parallel \mathbf{f}\parallel }_{L^2},$$

(8)

and

$${\parallel \mathbf{g}\parallel }_{L^{\infty }}⩽{C\parallel \mathbf{g}\parallel }_{L^q}^{\frac{q(r-3)}{3r+q(r-3)}}{\parallel \nabla \mathbf{g}\parallel }_{L^r}^{\frac{3r}{3r+q(r-3)}}+C_2{\parallel \mathbf{g}\parallel }_{L^2}.$$

(9)

In addition, if $\mathbf{f}·\mathbf{n}=0$ on $\partial \Omega$ (resp. $\mathbf{g}·\mathbf{n}=0$ on $\partial \Omega$), then one can take $C_1=0$ (resp. $C_2=0$).

Lemma 2 is a classical inequality, and its standard proof can be found in [25].

We examine the Lamé system represented by

$$\left\{\begin{array}{c}-\mu \Delta \mathbf{u}-(\mu +\lambda )\nabla \mathrm{div}\mathbf{u}=\mathbf{f},\mathrm{in}\Omega ,\hfill \\ \mathbf{u}·\mathbf{n}=0,\mathrm{curl}\mathbf{u}\times \mathbf{n}=0,\mathrm{on}\partial \Omega .\hfill \end{array}\right.$$

(10)

This system admits the following estimate:

Lemma 3

(see [26]). Under the assumptions that $p>1$, $k⩾0$, $\mathbf{f}\in W^{k,p}$, and that $\mathbf{u}\in W^{k+2,p}$ solves (10), one can find a positive constant C whose dependence is restricted to p and Ω, for which

$${\parallel \mathbf{u}\parallel }_{W^{k+2,p}}⩽C\left({\parallel \mathbf{f}\parallel }_{W^{k,p}}+{\parallel \mathbf{u}\parallel }_{L^p}\right).$$

(11)

The proof of the estimate (11) relies on the foundational theory of linear elliptic systems developed by Agmon, Douglis, and Nirenberg (AND). The core logic proceeds as follows: The first and most critical step is to verify that the Lamé system, combined with the specific boundary conditions $\mathbf{u}·\mathbf{n}=0$ and $\left(\mathrm{curl}\mathbf{u}\right)\times \mathbf{n}=0$, constitutes an elliptic boundary value problem in the sense of ADN. Once the system is established as an elliptic boundary value problem satisfying the complementing condition, the general ADN theory can be applied. A central result of this theory states that for such a system, an a priori estimate holds: there exists a constant $C>0$ such that

$${\parallel \mathbf{u}\parallel }_{W^{k+2,p}}⩽C\left({\parallel \mathbf{f}\parallel }_{W^{k,p}}+{\parallel \mathbf{u}\parallel }_{L^p}+{\parallel \mathrm{Boundary}\mathrm{Data}\parallel }_{W^{k+2-1/p,p}(\partial \Omega )}\right).$$

In this specific case, the boundary data for our system is homogeneous (i.e., zero), so the boundary term vanishes, yielding:

$${\parallel \mathbf{u}\parallel }_{W^{k+2,p}}⩽C\left({\parallel \mathbf{f}\parallel }_{W^{k,p}}+{\parallel \mathbf{u}\parallel }_{L^p}\right).$$

Lemma 4.

Let $\Omega \subset {\mathbb{R}}^3$ be a simply connected bounded domain with $C^{1,1}$ boundary $\partial \Omega$. If $\mathbf{v}\in W^{1,2}$ with $\mathbf{v}·\mathbf{n}=0$ on $\partial \Omega$, then

(i) (See [27], Proposition 2.4) A positive constant $\underline{\mu }=\underline{\mu }(\mu ,\lambda ,\Omega )$ exists yielding

$$\int \left({\mu \left|\mathrm{curl}\mathbf{v}\right|}^2+(2\mu +\lambda ){\left(\mathrm{div}\mathbf{v}\right)}^2\right)dx⩾\underline{\mu }\int {|\nabla \mathbf{v}|}^{2}dx.$$

(12)

(ii) (See [28], Exercise II. 5.6) It holds that

$${\parallel \mathbf{v}\parallel }_{L^2}⩽C{\parallel \nabla \mathbf{v}\parallel }_{L^2},$$

(13)

for some $C=C(\Omega )>0$.

(iii) (See [29], Lemma 2.9) If in addition $\mathrm{div}\mathbf{v}=0$, then

$${\parallel \nabla \mathbf{v}\parallel }_{L^2}⩽C{\parallel \mathrm{curl}\mathbf{v}\parallel }_{L^2},$$

(14)

with a constant $C=C(\Omega )>0$.

The inequality in (i) is a specific form of Korn’s inequality for a bounded domain with slip boundary conditions $(\mathbf{v}·\mathbf{n}=0)$. The inequality in (ii) is a version of the Poincaré inequality. The inequality in (iii) is a fundamental result in vector analysis for domains with nice boundaries, often called a Poincaré inequality for solenoidal fields.

3. Proof of Theorem 1

The objective of this subsection is to derive the necessary a priori estimates for the local strong solutions $(\rho ,P,\mathbf{u},\mathbf{w},\mathbf{b})$—whose existence is postulated in Lemma 1—to the initial-boundary value problem (4)–(6). Accordingly, consider a fixed time $T>0$ and let $(\rho ,P,\mathbf{u},\mathbf{w},\mathbf{b})$ be the smooth solution on $\Omega \times [0,T]$ evolving from smooth initial data $({\rho }_0,P_0,{\mathbf{u}}_0,{\mathbf{w}}_0,{\mathbf{b}}_0)$ adhering to (7). Set

$$\begin{array}{cc}\hfill \mathcal{H}\left(t\right)& \triangleq \parallel ({\rho }^{1/2}\mathbf{u},{\rho }^{1/2}\mathbf{w},\mathbf{b})\left(t\right){\parallel }_{L^2}^2+{\parallel P^{1/2}\mathrm{div}\mathbf{u}\left(t\right)\parallel }_{L^2}^2\hfill \\ \hfill & +\parallel ({\rho }^{1/2}{\mathbf{u}}_t,{\rho }^{1/2}{\mathbf{w}}_t,{\mathbf{b}}_t)\left(t\right){\parallel }_{L^2}^2+{\parallel \nabla P\left(t\right)\parallel }_{L^2}^2,\hfill \end{array}$$

and assume

$${\displaystyle \frac{1}{3}}\underline{\rho }⩽\rho (x,t)⩽3\overline{\rho },$$

(15)

$${\displaystyle \frac{E_0}{3|\Omega |}}⩽P(x,t)⩽{\displaystyle \frac{2E_0}{|\Omega |}},$$

(16)

$$\underset{t\in [0,T]}{sup}\mathcal{H}\left(t\right)⩽2C_0^{1/2},$$

(17)

$$\underset{t\in [0,T]}{sup}{\parallel (\mathrm{curl}\mathbf{u},\mathrm{curl}\mathbf{w},\mathrm{curl}\mathbf{b})\left(t\right)\parallel }_{L^2}^2⩽2C_0^{1/4},$$

(18)

$$\underset{t\in [0,T]}{sup}{\parallel \nabla P\left(t\right)\parallel }_{L^p}^2⩽3{\parallel \nabla P_0\parallel }_{L^p}^{2}p\in (3,6).$$

(19)

Using (12) and (16)–(18) gives

$$\begin{array}{cc}\hfill & \underset{t\in [0,T]}{sup}{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\left(t\right)\parallel }_{L^2}^2\hfill \\ \hfill & ⩽C\underset{t\in [0,T]}{sup}\left({\parallel (\mathrm{curl}\mathbf{u},\mathrm{curl}\mathbf{w},\mathrm{curl}\mathbf{b})\left(t\right)\parallel }_{L^2}^2+{\parallel (\mathrm{div}\mathbf{u},\mathrm{div}\mathbf{w})\left(t\right)\parallel }_{L^2}^2\right)\hfill \\ \hfill & ⩽C{C}_0^{1/4}.\hfill \end{array}$$

(20)

Hereafter, generic positive constants independent of time $T>0$ and $C_0$ are denoted by C or $C_i$ (with $i=1,2,\cdots$), for brevity. With the help of (15)–(18), we now present the following essential a priori estimates.

Lemma 5.

Consider a smooth solution $(\rho ,P,\mathbf{u},\mathbf{w},\mathbf{b})$ of the system (4)–(6), defined on $\Omega \times (0,T]$, which also adheres to the conditions (15)–(18), then

$$\underset{t\in [0,T]}{sup}{\parallel ({\nabla }^2\mathbf{u},{\nabla }^2\mathbf{w},{\nabla }^2\mathbf{b})\left(t\right)\parallel }_{L^2}^2⩽C{C}_0^{1/4}.$$

(21)

Proof. 

It follows from (6)₂, (8), (11), (13), (15) and (20) that

$$\begin{array}{cc}\hfill \parallel {\nabla }^2{\mathbf{u}\parallel }_{L^2}& ⩽C\left(\parallel ({\mathbf{u}}_t,\nabla P,\nabla \mathbf{w},\mathbf{u}){\parallel }_{L^2}+{\parallel \nabla \mathbf{u}\parallel }_{L^3}{\parallel \mathbf{u}\parallel }_{L^6}+{\parallel \left|\mathbf{b}\right||\nabla \mathbf{b}|\parallel }_{L^2}\right)\hfill \\ \hfill & ⩽C\left(\parallel ({\mathbf{u}}_t,\nabla \mathbf{u},\nabla \mathbf{w},\nabla P){\parallel }_{L^2}+{\parallel \nabla \mathbf{u}\parallel }_{L^2}^{3/2}\parallel {\nabla }^2{\mathbf{u}\parallel }_{L^2}^{1/2}+{\parallel \nabla \mathbf{b}\parallel }_{L^2}^{3/2}{\parallel {\nabla }^2\mathbf{b}\parallel }_{L^2}^{1/2}\right)\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{2}}\parallel {\nabla }^2{\mathbf{u}\parallel }_{L^2}+{\displaystyle \frac{1}{4}}\parallel {\nabla }^2{\mathbf{b}\parallel }_{L^2}+C{\parallel ({\mathbf{u}}_t,\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b},\nabla P)\parallel }_{L^2},\hfill \end{array}$$

which, implies

$$\parallel {\nabla }^2{\mathbf{u}\parallel }_{L^2}⩽{\displaystyle \frac{1}{2}}\parallel {\nabla }^2{\mathbf{b}\parallel }_{L^2}+C{\parallel ({\mathbf{u}}_t,\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b},\nabla P)\parallel }_{L^2}.$$

(22)

Based on (4)₃, one can deduce from (8), (13) and (15) that

$$\begin{array}{cc}\hfill \parallel {\nabla }^2{\mathbf{w}\parallel }_{L^2}& ⩽C\parallel {\mathbf{w}}_t{\parallel }_{L^2}+{C\parallel \nabla \mathbf{w}\parallel }_{L^3}{\parallel \mathbf{u}\parallel }_{L^6}+{C\parallel \mathbf{w}\parallel }_{L^2}+C{\parallel \nabla \mathbf{u}\parallel }_{L^2}\hfill \\ \hfill & ⩽C\left(\parallel {\mathbf{w}}_t{\parallel }_{L^2}+{\parallel \nabla \mathbf{w}\parallel }_{L^2}^{3/2}\parallel {\nabla }^2{\mathbf{w}\parallel }_{L^2}^{1/2}+{\parallel (\nabla \mathbf{u},\nabla \mathbf{w})\parallel }_{L^2}\right)\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{2}}{\parallel {\nabla }^2\mathbf{w}\parallel }_{L^2}+C\left(\parallel {\mathbf{w}}_t{\parallel }_{L^2}+{\parallel (\nabla \mathbf{u},\nabla \mathbf{w})\parallel }_{L^2}\right),\hfill \end{array}$$

so that

$$\begin{array}{cc}\hfill \parallel {\nabla }^2{\mathbf{w}\parallel }_{L^2}& ⩽C\parallel ({\mathbf{w}}_t,\nabla \mathbf{u},\nabla \mathbf{w}){\parallel }_{L^2}.\hfill \end{array}$$

(23)

Similarly, due to (4)₅ and (20), one also has

$$\begin{array}{cc}\hfill \parallel {\nabla }^2{\mathbf{b}\parallel }_{L^2}& ⩽C\left(\parallel {\mathbf{b}}_t{\parallel }_{L^2}+{\parallel \nabla \mathbf{b}\parallel }_{L^3}{\parallel \mathbf{u}\parallel }_{L^6}+{\parallel \mathbf{b}\parallel }_{L^6}{\parallel \nabla \mathbf{u}\parallel }_{L^3}\right)\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{2}}\parallel {\nabla }^2{\mathbf{b}\parallel }_{L^2}+{\displaystyle \frac{1}{4}}\parallel {\nabla }^2{\mathbf{u}\parallel }_{L^2}+C{\parallel ({\mathbf{b}}_t,\nabla \mathbf{u},\nabla \mathbf{b})\parallel }_{L^2},\hfill \end{array}$$

which gives

$$\parallel {\nabla }^2{\mathbf{b}\parallel }_{L^2}⩽{\displaystyle \frac{1}{2}}\parallel {\nabla }^2{\mathbf{u}\parallel }_{L^2}+C{\parallel ({\mathbf{b}}_t,\nabla \mathbf{u},\nabla \mathbf{b})\parallel }_{L^2}.$$

(24)

The inequality (24), together with (22) and (23), implies

$$\parallel ({\nabla }^2\mathbf{u},{\nabla }^2\mathbf{w},{\nabla }^2\mathbf{b}){\parallel }_{L^2}⩽C{\parallel ({\mathbf{u}}_t,{\mathbf{w}}_t,{\mathbf{b}}_t,\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b},\nabla P)\parallel }_{L^2},$$

(25)

which, together with (17) and (20), implies (21). □

The following lemma plays a crucial role in the subsequent attenuation estimation. We first estimate the $L^2$ of the gradient of pressure, which is closely related to the estimation of velocity, rather than giving the energy inequality first. The application of this method further enhances that the initial density can contain large oscillations, which is different from previous methods.

Lemma 6.

Given a smooth solution $(\rho ,P,\mathbf{u},\mathbf{w},\mathbf{b})$ of (4)–(6) in $\Omega \times (0,T]$ which satisfies (15)–(18), a positive constant $C_1$ can be established, with $C_1=C(\mu ,\lambda ,\nu ,\zeta ,c_0,c_a,c_d,\Omega )$, such that

$${\displaystyle \frac{d}{dt}}\parallel ({\rho }^{1/2}\mathbf{u},{\rho }^{1/2}\mathbf{w},\mathbf{b}){\parallel }_{L^2}^2+{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^2⩽C_1{\parallel \nabla P\parallel }_{L^2}^2.$$

(26)

Proof. 

Multiplying (4)₂, (4)₃ and (4)₅ by $\mathbf{u}$, $\mathbf{w}$ and $\mathbf{b}$ in $L^2$, respectively, and using the integration by parts over $\Omega$, one has from we (6) and (12)–(14) that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}\int \left({\displaystyle \frac{1}{2}}{\rho \left|\mathbf{u}\right|}^2+{\displaystyle \frac{1}{2}}{\rho \left|\mathbf{w}\right|}^2+{\displaystyle \frac{1}{2}}{\left|\mathbf{b}\right|}^2\right)dx+{C\parallel \nabla \mathbf{b}\parallel }_{L^2}^2+\underline{\mu }{\parallel \nabla \mathbf{u}\parallel }_{L^2}^2+C{\parallel \nabla \mathbf{w}\parallel }_{L^2}^2\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel ({\rho }^{1/2}\mathbf{u},\mathbf{b}){\parallel }_{L^2}^2+{\parallel ({\nu }^{1/2}\mathrm{curl}\mathbf{b},{\mu }^{1/2}\mathrm{curl}\mathbf{u},{(2\mu +\lambda )}^{1/2}\mathrm{div}\mathbf{u})\parallel }_{L^2}^2\hfill \\ \hfill & {+\zeta \parallel \mathrm{curl}\mathbf{u}-2\mathbf{w}\parallel }_{L^2}^2+(c_a+c_d){\parallel \mathrm{curl}\mathbf{w}\parallel }_{L^2}^2+(c_0+2c_d){\parallel \mathrm{div}\mathbf{w}\parallel }_{L^2}^2\hfill \\ \hfill & =-\int \mathbf{u}·\nabla Pdx⩽{C\parallel \mathbf{u}\parallel }_{L^2}{\parallel \nabla P\parallel }_{L^2}\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{2}}\underline{\mu }{\parallel \nabla \mathbf{u}\parallel }_{L^2}^2+C{\parallel \nabla P\parallel }_{L^2}^2,\hfill \end{array}$$

which, together with (12), implies (26). Therefore, the proof of Lemma 6 is completed. □

Next, we will deal with the $L^2$-norm of the gradient of pressure. Based on the assumption in (16), i.e., there is a lower bound for pressure. By using the effective viscous flux, we can construct the desired estimate.

Lemma 7.

Let $(\rho ,P,\mathbf{u},\mathbf{w},\mathbf{b})$ be a smooth solution of (4)–(6) on $\Omega \times (0,T]$ satisfying (15)–(19). Then, there exist some positive constants ${\epsilon }_1$ and $C_2$, depending only on $\mu ,\lambda ,\zeta ,\nu ,\overline{\rho },\underline{\rho }$, $c_0,c_a,c_d$ and Ω, such that for $p\in (3,6)$,

$${\displaystyle \frac{d}{dt}}{\parallel \nabla P\parallel }_{L^2}^2+{\parallel \nabla P\parallel }_{L^2}^2⩽C_2\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}^2+C{C}_0^{\frac{p-3}{20p-24}}{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^2,$$

(27)

provided $C_0⩽{\epsilon }_1$.

Proof. 

Define

$$F\triangleq (2\mu +\lambda )\mathrm{div}\mathbf{u}-(P-\overline{P})-{\displaystyle \frac{1}{2}}{\left|\mathbf{b}\right|}^2.$$

We first present the equation satisfied by the pressure P, then multiply both sides of this equation by ${|\nabla P|}^{q-2}\nabla P$ with $q\in [2,p],p\in (3,6)$ to derive the final equality satisfied by $\nabla P$. For each term on the right-hand side of this equality, we apply classical inequalities (e.g., Sobolev inequality, Hölder inequality, Cauchy-Schwarz inequality and Gronwall inequality, etc.) to obtain the desired estimates. First, Operating ∇ to both sides of (4)₄ yields

$$\begin{array}{cc}\hfill & \nabla P_t+(\mathbf{u}·\nabla )\nabla P+\nabla \mathbf{u}\nabla P+2\mathrm{div}\mathbf{u}\nabla P-\nabla \left({2\mu \left|\mathbb{D}\left(\mathbf{u}\right)\right|}^2+\lambda {\left(\mathrm{div}\mathbf{u}\right)}^2+\nu {\left|\mathrm{curl}\mathbf{b}\right|}^2\right)\hfill \\ \hfill & -\nabla \left({\zeta |\nabla \times \mathbf{u}-2\mathbf{w}|}^2+c_0{\left(\mathrm{div}\mathbf{w}\right)}^2+(c_a+c_d){|\nabla \mathbf{w}|}^2+(c_d-c_a)\nabla \mathbf{w}:\nabla {\mathbf{w}}^{\top }\right)\hfill \\ \hfill & =-2P\nabla \mathrm{div}\mathbf{u}=-{\displaystyle \frac{2P}{2\mu +\lambda }}\left(\nabla F+\nabla P+\nabla \mathbf{b}·\mathbf{b}\right).\hfill \end{array}$$

Multiplying the above equation by ${|\nabla P|}^{q-2}\nabla P$ with $q\in [2,p],p\in (3,6)$, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{p}}{(|\nabla P|}^q{)}_t+{\displaystyle \frac{1}{q}}{\mathrm{div}\left(\right|\nabla P|}^q\mathbf{u})+{\displaystyle \frac{2q-1}{q}}{|\nabla P|}^q\mathrm{div}\mathbf{u}+{|\nabla P|}^{q-2}{(\nabla P)}^{tr}\nabla \mathbf{u}\nabla P\hfill \\ \hfill & -{|\nabla P|}^{q-2}\nabla P·\nabla \left({2\mu \left|\mathbb{D}\left(\mathbf{u}\right)\right|}^2+\mu {\left(\mathrm{div}\mathbf{u}\right)}^2{+\nu |\nabla \times \mathbf{b}|}^2+\zeta {|\nabla \times \mathbf{u}-2\mathbf{w}|}^2\right)\hfill \\ \hfill & -{|\nabla P|}^{q-2}\nabla P·\nabla (c_0{\left(\mathrm{div}\mathbf{w}\right)}^2+(c_a+c_d){|\nabla \mathbf{w}|}^2+(c_d-c_a)\nabla \mathbf{w}:\nabla {\mathbf{w}}^{\top })\hfill \\ \hfill & +{\displaystyle \frac{2P}{2\mu +\lambda }}{\left(\right|\nabla P|}^{q-2}\nabla P)·\left(\nabla F+\nabla P+\nabla \mathbf{b}·\mathbf{b}\right)=0,\hfill \end{array}$$

which, together with (16), yields

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}{\parallel \nabla P\parallel }_{L^q}^q+{\displaystyle \frac{q}{2\mu +\lambda }}{\parallel \nabla P\parallel }_{L^q}^q\hfill \\ \hfill & ⩽{C\parallel \nabla \mathbf{u}\parallel }_{L^{\infty }}{\parallel \nabla P\parallel }_{L^q}^q+C\left({\parallel \nabla \mathbf{w}\parallel }_{L^{\infty }}{\parallel \nabla \mathbf{u}\parallel }_{L^q}+{\parallel \mathbf{w}\parallel }_{L^{\infty }}{\parallel ({\nabla }^2\mathbf{u},\nabla \mathbf{w})\parallel }_{L^q}\right){\parallel \nabla P\parallel }_{L^q}^{q-1}\hfill \\ \hfill & +C\left({\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^{\infty }}\parallel ({\nabla }^2\mathbf{u},{\nabla }^2\mathbf{w},{\nabla }^2\mathbf{b}){\parallel }_{L^q}+{\parallel (\nabla F,\nabla \mathbf{b}·\mathbf{b})\parallel }_{L^q}\right){\parallel \nabla P\parallel }_{L^q}^{q-1}.\hfill \end{array}$$

(28)

By virtue of (4)₂ and (4)₅, one has from (8), (9), (11), (13), (15) and (21) that for $p\in (3,6)$,

$$\begin{array}{cc}\hfill {\parallel \nabla \mathbf{u}\parallel }_{W^{1,p}}& ⩽C\parallel \rho {\mathbf{u}}_t{+\rho \mathbf{u}·\nabla \mathbf{u}+\nabla P-\nabla \mathbf{b}·\mathbf{b}-2\zeta \nabla \times \mathbf{w}\parallel }_{L^p}+C{\parallel \mathbf{u}\parallel }_{L^p}\hfill \\ \hfill & ⩽C\left(\parallel {\mathbf{u}}_t{\parallel }_{L^p}+{\parallel (\mathbf{u},\mathbf{b})\parallel }_{L^{\infty }}{\parallel (\nabla \mathbf{u},\nabla \mathbf{b})\parallel }_{L^p}+{\parallel \nabla P\parallel }_{L^p}+{\parallel \nabla \mathbf{w}\parallel }_{L^p}+{\parallel \mathbf{u}\parallel }_{H^1}\right)\hfill \\ \hfill & ⩽C\parallel {\mathbf{u}}_t{\parallel }_{H^1}+C\left({\parallel (\nabla \mathbf{u},\nabla \mathbf{b})\parallel }_{L^2}^{1/2}\parallel ({\nabla }^2\mathbf{u},{\nabla }^2\mathbf{b}){\parallel }_{L^2}^{1/2}+{\parallel (\nabla \mathbf{u},\nabla \mathbf{b})\parallel }_{L^2}\right)\hfill \\ \hfill & \times \left[{\parallel (\nabla \mathbf{u},\nabla \mathbf{b})\parallel }_{L^2}^{\frac{6-p}{2p}}\parallel ({\nabla }^2\mathbf{u},{\nabla }^2\mathbf{b}){\parallel }_{L^2}^{\frac{3p-6}{2p}}+{\parallel (\nabla \mathbf{u},\nabla \mathbf{b})\parallel }_{L^2}\right]\hfill \\ \hfill & +{\parallel \nabla P\parallel }_{L^p}+C\left({\parallel \nabla \mathbf{w}\parallel }_{L^2}^{\frac{6-p}{2p}}\parallel {\nabla }^2{\mathbf{w}\parallel }_{L^2}^{\frac{3p-6}{2p}}+{\parallel \nabla \mathbf{w}\parallel }_{L^2}\right)+C{\parallel \nabla \mathbf{u}\parallel }_{L^2}\hfill \\ \hfill & ⩽C\parallel \nabla {\mathbf{u}}_t{\parallel }_{L^2}+{C\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}+{\parallel \nabla P\parallel }_{L^p},\hfill \end{array}$$

(29)

and

$$\begin{array}{cc}\hfill {\parallel \nabla \mathbf{b}\parallel }_{W^{1,p}}& ⩽C\parallel {\mathbf{b}}_t{+\mathbf{u}·\nabla \mathbf{b}\parallel }_{L^p}{+C\parallel \mathbf{b}·\nabla \mathbf{u}\parallel }_{L^p}+{C\parallel \mathbf{b}\mathrm{div}\mathbf{u}\parallel }_{L^p}+C{\parallel \mathbf{b}\parallel }_{L^p}\hfill \\ \hfill & ⩽C\left(\parallel ({\mathbf{b}}_t,\mathbf{b}){\parallel }_{L^p}+\parallel (\mathbf{u},\mathbf{b}){\parallel }_{L^{\infty }}{\parallel (\nabla \mathbf{b},\nabla \mathbf{u})\parallel }_{L^p}\right)\hfill \\ \hfill & ⩽C\parallel \nabla {\mathbf{b}}_t{\parallel }_{L^2}+C{\parallel (\nabla \mathbf{u},\nabla \mathbf{b})\parallel }_{L^2},\hfill \end{array}$$

(30)

similar to (29), one also has

$$\begin{array}{cc}\hfill {\parallel \nabla \mathbf{w}\parallel }_{W^{1,p}}& ⩽C\left(\parallel {\mathbf{w}}_t{\parallel }_{L^p}+{\parallel \mathbf{u}\parallel }_{L^{\infty }}{\parallel \nabla \mathbf{w}\parallel }_{L^p}+{\parallel \nabla \mathbf{u}\parallel }_{L^p}+{\parallel \mathbf{w}\parallel }_{H^1}\right)\hfill \\ \hfill & ⩽C\parallel {\mathbf{w}}_t{\parallel }_{H^1}+C\left({\parallel \nabla \mathbf{u}\parallel }_{L^2}^{\frac{6-p}{2p}}\parallel {\nabla }^2{\mathbf{u}\parallel }_{L^2}^{\frac{3p-6}{2p}}+{\parallel \nabla \mathbf{u}\parallel }_{L^2}\right)+C{\parallel \nabla \mathbf{w}\parallel }_{L^2}\hfill \\ \hfill & +C\left({\parallel \nabla \mathbf{u}\parallel }_{L^2}^{1/2}\parallel {\nabla }^2{\mathbf{u}\parallel }_{L^2}^{1/2}+{\parallel \nabla \mathbf{u}\parallel }_{L^2}\right)\left({\parallel \nabla \mathbf{w}\parallel }_{L^2}^{\frac{6-p}{2p}}\parallel {\nabla }^2{\mathbf{w}\parallel }_{L^2}^{\frac{3p-6}{2p}}+{\parallel \nabla \mathbf{w}\parallel }_{L^2}\right)\hfill \\ \hfill & ⩽C\parallel \nabla {\mathbf{w}}_t{\parallel }_{L^2}+C{\parallel (\nabla \mathbf{u},\nabla \mathbf{w})\parallel }_{L^2},\hfill \end{array}$$

(31)

Collecting (9), (17), (19), (20) and (29)–(31), gives

$$\begin{array}{cc}\hfill & {\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^{\infty }}\hfill \\ \hfill & ⩽C\left({\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^{\frac{2p-6}{5p-6}}\parallel ({\nabla }^2\mathbf{u},{\nabla }^2\mathbf{w},{\nabla }^2\mathbf{b}){\parallel }_{L^2}^{\frac{3p}{5p-6}}+{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}\right)\hfill \\ \hfill & ⩽{C\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^{\frac{p-3}{5p-6}}\left({\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^{1/2}+{\parallel ({\nabla }^2\mathbf{u},{\nabla }^2\mathbf{w},{\nabla }^2\mathbf{b})\parallel }_{L^p}\right)\hfill \\ \hfill & +{C\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}\hfill \\ \hfill & ⩽C{C}_0^{\frac{p-3}{40p-48}}\left(\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}+1\right)+C{C}_0^{1/8}\hfill \\ \hfill & ⩽C{C}_0^{\frac{p-3}{40p-48}}{\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t)\parallel }_{L^2}+C{C}_0^{\frac{p-3}{40p-48}}.\hfill \end{array}$$

(32)

For ${\parallel \nabla F\parallel }_{L^q}$, we get from (4)₂ and the definition of F that

$$\rho {\mathbf{u}}_t+\rho \mathbf{u}·\nabla \mathbf{u}=\nabla F-\mu \nabla \times \left(\mathrm{curl}\mathbf{u}\right)-\zeta \nabla \times \left(\mathrm{curl}\mathbf{u}\right)+2\zeta \nabla \times \mathbf{w}+\mathrm{div}(\mathbf{b}\otimes \mathbf{b}),$$

(33)

thus

$$\Delta F=\mathrm{div}(\rho {\mathbf{u}}_t+\rho \mathbf{u}·\nabla \mathbf{u})-\mathrm{div}\mathrm{div}(\mathbf{b}\otimes \mathbf{b}).$$

(34)

Therefore, for any $\phi \in C^{\infty }(\Omega )$, using (33) and (34) and the integration by parts gives

$$\begin{array}{cc}\hfill & \int \mathrm{div}(\rho {\mathbf{u}}_t+\rho \mathbf{u}·\nabla \mathbf{u})\phi dx-\int \mathrm{div}\mathrm{div}(\mathbf{b}\otimes \mathbf{b})\phi dx=\int \Delta F\phi dx\hfill \\ \hfill & ={\int }_{\partial \Omega }{\displaystyle \frac{\partial F}{\partial \mathbf{n}}}\phi dS-\int \nabla F·\nabla \phi dx={\int }_{\partial \Omega }{\displaystyle \frac{\partial F}{\partial \mathbf{n}}}\phi dS\hfill \\ \hfill & -\int \left(\rho {\mathbf{u}}_t+\rho \mathbf{u}·\nabla \mathbf{u}-\mathrm{div}(\mathbf{b}\otimes \mathbf{b})-2\zeta \nabla \times \mathbf{w}+(\mu +\zeta )\nabla \times \left(\mathrm{curl}\mathbf{u}\right)\right)·\nabla \phi dx\hfill \\ \hfill & ={\int }_{\partial \Omega }\left[{\displaystyle \frac{\partial F}{\partial \mathbf{n}}}-\left[\rho {\mathbf{u}}_t+\rho \mathbf{u}·\nabla \mathbf{u}-\mathrm{div}(\mathbf{b}\otimes \mathbf{b})-2\zeta \nabla \times \mathbf{w}+(\mu +\zeta )\nabla \times \left(\mathrm{curl}\mathbf{u}\right)\right]·\mathbf{n}\right]\phi dS\hfill \\ \hfill & +\int \left(\mathrm{div}(\rho {\mathbf{u}}_t+\rho \mathbf{u}·\nabla \mathbf{u})-\mathrm{div}\mathrm{div}(\mathbf{b}\otimes \mathbf{b})\right)\phi dx,\hfill \end{array}$$

which, gives

$${\displaystyle \frac{\partial F}{\partial \mathbf{n}}}=(\rho {\mathbf{u}}_t+\rho \mathbf{u}·\nabla \mathbf{u}-\mathrm{div}(\mathbf{b}\otimes \mathbf{b})-2\zeta \nabla \times \mathbf{w}+(\mu +\zeta )\nabla \times \left(\mathrm{curl}\mathbf{u}\right))·\mathbf{n}\mathrm{a}.\mathrm{e}.\mathrm{on}\partial \Omega .$$

(35)

Hence, using (13), (21), (25), (34) and (35), we get from [30] that

$$\begin{array}{cc}\hfill {\parallel \nabla F\parallel }_{L^q}& ⩽C\parallel \rho {\mathbf{u}}_t{+\rho \mathbf{u}·\nabla \mathbf{u}-\nabla \mathbf{b}·\mathbf{b}\parallel }_{L^q}\hfill \\ \hfill & ⩽C\left(\parallel {\mathbf{u}}_t{\parallel }_{L^q}+{\parallel \mathbf{u}\parallel }_{L^{\infty }}{\parallel \nabla \mathbf{u}\parallel }_{L^q}+{\parallel \mathbf{b}\parallel }_{L^{\infty }}{\parallel \nabla \mathbf{b}\parallel }_{L^q}\right)\hfill \\ \hfill & ⩽C\parallel {\mathbf{u}}_t{\parallel }_{H^1}+C{\parallel (\nabla \mathbf{u},\nabla \mathbf{b})\parallel }_{H^1}^2\hfill \\ \hfill & ⩽C\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}+C{C}_0^{1/8}\left({\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}+{\parallel \nabla P\parallel }_{L^2}\right).\hfill \end{array}$$

(36)

Choosing $q=2$ in (28) and collecting (8), (9), (15)–(20), (32) and (36) yield

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}{\parallel \nabla P\parallel }_{L^2}^2+{\parallel \nabla P\parallel }_{L^2}^2\hfill \\ \hfill & ⩽C\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}{\parallel \nabla P\parallel }_{L^2}+C{C}_0^{\frac{p-3}{40p-48}}\parallel ({\nabla }^2\mathbf{u},{\nabla }^2\mathbf{w},{\nabla }^2\mathbf{b}){\parallel }_{L^2}{\parallel \nabla P\parallel }_{L^2}\hfill \\ \hfill & +C{C}_0^{\frac{p-3}{40p-48}}{\parallel \nabla P\parallel }_{L^2}^2+C{C}_0^{1/8}{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}{\parallel \nabla P\parallel }_{L^2}+C{C}_0^{1/8}{\parallel \nabla P\parallel }_{L^2}^2\hfill \\ \hfill & ⩽C\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}{\parallel \nabla P\parallel }_{L^2}+C{C}_0^{\frac{p-3}{40p-48}}{\parallel \nabla P\parallel }_{L^2}^2\hfill \\ \hfill & +C{C}_0^{\frac{p-3}{40p-48}}{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}{\parallel \nabla P\parallel }_{L^2}\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{2}}{\parallel \nabla P\parallel }_{L^2}^2+C\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}^2+C_3{C}_0^{\frac{p-3}{40p-48}}{\parallel \nabla P\parallel }_{L^2}^2\hfill \\ \hfill & +C{C}_0^{\frac{p-3}{40p-48}}{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^2,\hfill \end{array}$$

provided that

$$C_0⩽{\epsilon }_1\triangleq min\left\{{\displaystyle \frac{1}{4}},{\left({\displaystyle \frac{1}{4C_3}}\right)}^{\frac{40p-48}{p-3}}\right\}.$$

Thus, the proof of Lemma 7 is completed. □

According to Lemma 5, the quantity ${\parallel \nabla P\parallel }_{L^2}$ depends on $\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}$. We will therefore shift our focus to establishing estimates for $\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}$.

Lemma 8.

Suppose $(\rho ,P,\mathbf{u},\mathbf{w},\mathbf{b})$ is a smooth solution of (4)–(6) in $\Omega \times (0,T]$, fulfilling the estimates (15)–(19). Then one can establish a positive constant ${\epsilon }_2$, which is a function of $\mu ,\lambda ,\nu ,\overline{\rho },\underline{\rho }$, $\zeta ,c_0,c_a,c_d$ and Ω alone, for which

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}\parallel ({\rho }^{1/2}{\mathbf{u}}_t,{\rho }^{1/2}{\mathbf{w}}_t,P^{1/2}\mathrm{div}\mathbf{u},{\mathbf{b}}_t){\parallel }_{L^2}^2+{\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t,{\mathbf{w}}_t)\parallel }_{L^2}^2\hfill \\ \hfill & ⩽C{C}_0^{1/8}{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^2+C{C}_0^{1/8}{\parallel \nabla P\parallel }_{L^2}^2,\hfill \end{array}$$

(37)

provided $C_0⩽{\epsilon }_2$.

Proof. 

We begin by presenting the equation satisfied by ${\mathbf{u}}_t$. Then, we multiply both sides of the equation by ${\mathbf{u}}_t$ and integrate over the domain $\Omega$ to derive an estimate for $\parallel {\rho }^{1/2}{\mathbf{u}}_t{\parallel }_{L^2}^2$. Subsequently, we perform careful estimates for each term on the right-hand side of the resulting equation. Following the same approach, we establish inequalities for $\parallel {\rho }^{1/2}{\mathbf{w}}_t{\parallel }_{L^2}^2$, $\parallel P^{1/2}{\mathrm{div}\mathbf{u}\parallel }_{L^2}^2$ and ${\mathbf{b}}_t$. Finally, by summing these three inequalities and integrating over the time interval $[0,T]$, we obtain the desired estimate by applying the smallness condition in Theorem 1. First, Operating ${\partial }_t$ to (4)₂, yields

$$\begin{array}{cc}\hfill & \rho {\mathbf{u}}_{tt}+\rho \mathbf{u}·\nabla {\mathbf{u}}_t-(\mu +\zeta )\Delta {\mathbf{u}}_t-(\mu +\lambda -\zeta )\nabla \mathrm{div}{\mathbf{u}}_t\hfill \\ \hfill & =\mathrm{div}\left(\rho \mathbf{u}\right)({\mathbf{u}}_t+\mathbf{u}·\nabla \mathbf{u})-\rho {\mathbf{u}}_t·\nabla \mathbf{u}-\nabla P_t+2\zeta \nabla \times {\mathbf{w}}_t\hfill \\ \hfill & +\mathbf{b}·\nabla {\mathbf{b}}_t+{\mathbf{b}}_t·\nabla \mathbf{b}-\nabla {\mathbf{b}}_t·\mathbf{b}-\nabla \mathbf{b}·{\mathbf{b}}_t.\hfill \end{array}$$

(38)

We multiply (38) by ${\mathbf{u}}_t$ in $L^2$ and use the integration by parts that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel {\rho }^{1/2}{\mathbf{u}}_t{\parallel }_{L^2}^2+{\parallel ({\mu }^{1/2}\nabla {\mathbf{u}}_t,{(\mu +\lambda )}^{1/2}\mathrm{div}{\mathbf{u}}_t)\parallel }_{L^2}^2\hfill \\ \hfill & =\int \mathrm{div}\left(\rho \mathbf{u}\right){\mathbf{u}}_t·{\mathbf{u}}_{t}dx+\int \mathrm{div}\left(\rho \mathbf{u}\right)\mathbf{u}·\nabla \mathbf{u}·{\mathbf{u}}_{t}dx-\int \rho {\mathbf{u}}_t·\nabla \mathbf{u}·{\mathbf{u}}_{t}dx\hfill \\ \hfill & -\int P_t\mathrm{div}{\mathbf{u}}_{t}dx+2\zeta \int \nabla \times {\mathbf{w}}_t·{\mathbf{u}}_{t}dx+\zeta \int (\Delta {\mathbf{u}}_t-\nabla \mathrm{div}{\mathbf{u}}_t)·{\mathbf{u}}_{t}dx\hfill \\ \hfill & +\int \left(\mathbf{b}·\nabla {\mathbf{b}}_t+{\mathbf{b}}_t·\nabla \mathbf{b}-\nabla {\mathbf{b}}_t·\mathbf{b}-\nabla \mathbf{b}·{\mathbf{b}}_t\right)·{\mathbf{u}}_{t}dx\triangleq \sum _{i=1}^7{N}_i.\hfill \end{array}$$

(39)

Based on (8), (13), (15) and (20) that

$$\begin{array}{cc}\hfill N_1& =-2\int \rho \mathbf{u}·\nabla {\mathbf{u}}_t·{\mathbf{u}}_{t}dx⩽{C\parallel \mathbf{u}\parallel }_{L^6}\parallel \nabla {\mathbf{u}}_t{\parallel }_{L^2}{\parallel {\mathbf{u}}_t\parallel }_{L^3}\hfill \\ \hfill & ⩽{C\parallel \nabla \mathbf{u}\parallel }_{L^2}{\parallel \nabla {\mathbf{u}}_t\parallel }_{L^2}\left(\parallel {\mathbf{u}}_t{\parallel }_{L^2}^{1/2}\parallel \nabla {\mathbf{u}}_t{\parallel }_{L^2}^{1/2}+{\parallel {\mathbf{u}}_t\parallel }_{L^2}\right)\hfill \\ \hfill & ⩽C_4{C}_0^{1/8}\parallel \nabla {\mathbf{u}}_t{\parallel }_{L^2}^2⩽{\displaystyle \frac{1}{10}}\underline{\mu }{\parallel \nabla {\mathbf{u}}_t\parallel }_{L^2}^2,\hfill \end{array}$$

provided that $C_0⩽{\epsilon }_{2,1}\triangleq min\{{\epsilon }_1,{\left(\frac{\underline{\mu }}{10C_4}\right)}^8\}$. Similarly,

$$\begin{array}{cc}\hfill N_3& ⩽C\parallel {\mathbf{u}}_t{\parallel }_{L^6}{\parallel \nabla \mathbf{u}\parallel }_{L^2}\parallel {\mathbf{u}}_t{\parallel }_{L^3}⩽C\parallel \nabla {\mathbf{u}}_t{\parallel }_{L^2}^2{\parallel \nabla \mathbf{u}\parallel }_{L^2}\hfill \\ \hfill & ⩽C_5{C}_0^{1/8}\parallel \nabla {\mathbf{u}}_t{\parallel }_{L^2}^2⩽{\displaystyle \frac{1}{10}}\underline{\mu }{\parallel \nabla {\mathbf{u}}_t\parallel }_{L^2}^2,\hfill \end{array}$$

provided that $C_0⩽{\epsilon }_{2,2}\triangleq min\{{\epsilon }_1,{\left(\frac{\underline{\mu }}{10C_5}\right)}^8\}$. It follows from (13) and (25) that

$$\begin{array}{cc}\hfill N_2& =-\int \rho \mathbf{u}·\nabla (\mathbf{u}·\nabla \mathbf{u}·{\mathbf{u}}_t)dx\hfill \\ \hfill & ⩽{C\parallel \nabla \mathbf{u}\parallel }_{L^2}{\parallel \nabla \mathbf{u}\parallel }_{H^1}{\parallel \nabla {\mathbf{u}}_t\parallel }_{L^2}\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{20}}\underline{\mu }\parallel \nabla {\mathbf{u}}_t{\parallel }_{L^2}^2+{C\parallel \nabla \mathbf{u}\parallel }_{L^2}^2{\parallel \nabla \mathbf{u}\parallel }_{H^1}^2\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{20}}\underline{\mu }\parallel \nabla {\mathbf{u}}_t{\parallel }_{L^2}^2+C_6{C}_0^{1/4}{\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t,\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^2\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{10}}\underline{\mu }\parallel \nabla {\mathbf{u}}_t{\parallel }_{L^2}^2+C{C}_0^{1/4}{\parallel (\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t)\parallel }_{L^2}^2+C{C}_0^{1/4}\left({\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^2+{\parallel \nabla P\parallel }_{L^2}^2\right),\hfill \end{array}$$

provided that $C_0⩽{\epsilon }_{2,3}\triangleq min\{{\epsilon }_1,{\left(\frac{\underline{\mu }}{20C_6}\right)}^8\}$. Using (6)₄, (13), (15)–(20) and (25), gives

$$\begin{array}{cc}\hfill N_4& ⩽-{\displaystyle \frac{d}{dt}}\int P{\left(\mathrm{div}\mathbf{u}\right)}^{2}dx+{C\parallel \mathbf{u}\parallel }_{L^6}{\parallel \nabla P\parallel }_{L^2}{\parallel \nabla \mathbf{u}\parallel }_{L^6}^2+{C\parallel \nabla \mathbf{u}\parallel }_{L^3}^3+C{\parallel \nabla \mathbf{u}\parallel }_{L^2}^4\hfill \\ \hfill & +{C\parallel (\nabla \mathbf{w},\nabla \mathbf{b},\mathbf{w})\parallel }_{L^4}^2{\parallel \nabla \mathbf{u}\parallel }_{L^4}^2+{C\parallel \mathbf{u}\parallel }_{L^6}{\parallel \mathbf{w}\parallel }_{L^2}{\parallel \nabla \mathbf{u}\parallel }_{L^6}^2\hfill \\ \hfill & +C\parallel \nabla {\mathbf{u}}_t{\parallel }_{L^2}{\parallel \nabla P\parallel }_{L^2}{\parallel \mathbf{u}\parallel }_{H^2}+{C\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b},\mathbf{w})\parallel }_{L^4}^2{\parallel \nabla {\mathbf{u}}_t\parallel }_{L^2}\hfill \\ \hfill & ⩽-{\displaystyle \frac{d}{dt}}\parallel P^{1/2}{\mathrm{div}\mathbf{u}\parallel }_{L^2}^2+{\displaystyle \frac{1}{20}}\underline{\mu }\parallel \nabla {\mathbf{u}}_t{\parallel }_{L^2}^2+C{C}_0^{1/4}{\parallel \nabla \mathbf{u}\parallel }_{H^1}^2\hfill \\ \hfill & +{C\parallel \nabla \mathbf{u}\parallel }_{H^1}^3+C{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{H^1}^4\hfill \\ \hfill & ⩽-{\displaystyle \frac{d}{dt}}\parallel P^{1/2}{\mathrm{div}\mathbf{u}\parallel }_{L^2}^2+{\displaystyle \frac{1}{20}}\underline{\mu }{\parallel \nabla {\mathbf{u}}_t\parallel }_{L^2}^2\hfill \\ \hfill & +C_7{C}_0^{1/8}\left(\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}^2+{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b},\nabla P)\parallel }_{L^2}^2\right)\hfill \\ \hfill & ⩽-{\displaystyle \frac{d}{dt}}\parallel P^{1/2}{\mathrm{div}\mathbf{u}\parallel }_{L^2}^2+{\displaystyle \frac{1}{10}}\underline{\mu }\parallel \nabla {\mathbf{u}}_t{\parallel }_{L^2}^2+C{C}_0^{1/8}{\parallel (\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t)\parallel }_{L^2}^2\hfill \\ \hfill & +C{C}_0^{1/8}{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b},\nabla P)\parallel }_{L^2}^2,\hfill \end{array}$$

provided that $C_0⩽{\epsilon }_{2,4}\triangleq min\{{\epsilon }_1,{\left(\frac{\underline{\mu }}{20C_7}\right)}^8\}$. Using integration by parts gives

$$\begin{array}{cc}\hfill N_5+N_6& =2\zeta \int \nabla \times {\mathbf{u}}_t·{\mathbf{w}}_{t}dx-\zeta \int \nabla \times (\nabla \times {\mathbf{u}}_t)·{\mathbf{u}}_{t}dx\hfill \\ \hfill & =2\zeta \int \nabla \times {\mathbf{u}}_t·{\mathbf{w}}_{t}dx-\zeta \int {|\nabla \times {\mathbf{u}}_t|}^{2}dx,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill N_7& =\int \left(\mathbf{b}·\nabla {\mathbf{b}}_t·{\mathbf{u}}_t-{\mathbf{b}}_t·\nabla {\mathbf{u}}_t·\mathbf{b}+\mathrm{div}{\mathbf{u}}_t{\mathbf{b}}_t·\mathbf{b}\right)dx\hfill \\ \hfill & ⩽C\left({\parallel \mathbf{b}\parallel }_{L^2}^{1/2}{\parallel \nabla \mathbf{b}\parallel }_{L^2}^{1/2}+{\parallel \mathbf{b}\parallel }_{L^2}\right){\parallel (\nabla {\mathbf{b}}_t,\nabla {\mathbf{u}}_t)\parallel }_{L^2}\left(\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}+{\parallel ({\mathbf{u}}_t,{\mathbf{b}}_t)\parallel }_{L^2}\right)\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{10}}\underline{\mu }\parallel \nabla {\mathbf{u}}_t{\parallel }_{L^2}^2+C{C}_0^{1/4}{\parallel \nabla {\mathbf{b}}_t\parallel }_{L^2}^2.\hfill \end{array}$$

Collecting $N_j(j=1,2,·,7)$ into (39), gives

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}\parallel ({\rho }^{1/2}{\mathbf{u}}_t,P^{1/2}\mathrm{div}\mathbf{u}){\parallel }_{L^2}^2+{\parallel \nabla {\mathbf{u}}_t\parallel }_{L^2}^2+\zeta \int \left(|\nabla \times {\mathbf{u}}_t{|}^2-2\zeta \nabla \times {\mathbf{u}}_t·{\mathbf{w}}_t\right)dx\hfill \\ \hfill & ⩽C_8{C}_0^{1/8}{\parallel (\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t)\parallel }_{L^2}^2+C{C}_0^{1/8}\left({\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^2+{\parallel \nabla P\parallel }_{L^2}^2\right).\hfill \end{array}$$

(40)

Operating ${\partial }_t$ to (4)₃, we get from (4)₁ that

$$\begin{array}{cc}\hfill & \rho {\mathbf{w}}_{tt}+\rho \mathbf{u}·\nabla {\mathbf{w}}_t+4\zeta {\mathbf{w}}_t-(c_a+c_d)\Delta {\mathbf{w}}_t-(c_0+c_d-c_a)\nabla \mathrm{div}{\mathbf{w}}_t\hfill \\ \hfill & =\mathrm{div}\left(\rho \mathbf{u}\right)({\mathbf{w}}_t+\mathbf{u}·\nabla \mathbf{w})-\rho {\mathbf{u}}_t·\nabla \mathbf{w}+2\zeta \nabla \times {\mathbf{u}}_t.\hfill \end{array}$$

(41)

Multiplying (41) by ${\mathbf{w}}_t$ in $L^2$ and using the integration by parts, leads to

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel {\rho }^{1/2}{\mathbf{w}}_t{\parallel }_{L^2}^2+(c_a+c_d)\parallel \nabla {\mathbf{w}}_t{\parallel }_{L^2}^2+(c_0+c_d-c_a){\parallel \mathrm{div}{\mathbf{w}}_t\parallel }_{L^2}^2\hfill \\ \hfill & =\int \mathrm{div}\left(\rho \mathbf{u}\right){\mathbf{w}}_t·{\mathbf{w}}_{t}dx+\int \mathrm{div}\left(\rho \mathbf{u}\right)\mathbf{u}·\nabla \mathbf{w}·{\mathbf{w}}_{t}dx-\int \rho {\mathbf{u}}_t·\nabla \mathbf{w}·{\mathbf{w}}_{t}dx\hfill \\ \hfill & +2\zeta \int \nabla \times {\mathbf{u}}_t·{\mathbf{w}}_{t}dx-4\zeta {\parallel {\mathbf{w}}_t\parallel }_{L^2}^2\hfill \\ \hfill & \triangleq Q_1+Q_2+Q_3+2\zeta \int \nabla \times {\mathbf{u}}_t·{\mathbf{w}}_{t}dx-4\zeta \int {\left|{\mathbf{w}}_t\right|}^{2}dx.\hfill \end{array}$$

(42)

By virtue of (8), (13), (15), (25) and (29) that

$$\begin{array}{cc}\hfill Q_1& ⩽C\int \rho |\mathbf{u}||\nabla {\mathbf{w}}_t\left|\right|{\mathbf{w}}_t{|dx⩽C\parallel \mathbf{u}\parallel }_{L^6}\parallel \nabla {\mathbf{w}}_t{\parallel }_{L^2}{\parallel {\mathbf{w}}_t\parallel }_{L^3}\hfill \\ \hfill & ⩽{C\parallel \nabla \mathbf{u}\parallel }_{L^2}{\parallel \nabla {\mathbf{w}}_t\parallel }_{L^2}\left(\parallel {\mathbf{w}}_t{\parallel }_{L^2}^{1/2}\parallel \nabla {\mathbf{w}}_t{\parallel }_{L^2}^{1/2}+{\parallel {\mathbf{w}}_t\parallel }_{L^2}\right)\hfill \\ \hfill & ⩽{\tilde{C}}_4{C}_0^{1/8}\parallel \nabla {\mathbf{w}}_t{\parallel }_{L^2}^2⩽{\displaystyle \frac{c_a+c_d}{6}}{\parallel \nabla {\mathbf{w}}_t\parallel }_{L^2}^2,\hfill \end{array}$$

provided that $C_0⩽{\epsilon }_{2,5}\triangleq min\{{\epsilon }_1,{\left(\frac{c_a+c_d}{6{\tilde{C}}_4}\right)}^8\}$. Similarly, due to (13), one has

$$\begin{array}{cc}\hfill Q_3& ⩽C\parallel {\mathbf{u}}_t{\parallel }_{L^6}{\parallel \nabla \mathbf{w}\parallel }_{L^2}{\parallel {\mathbf{w}}_t\parallel }_{L^3}\hfill \\ \hfill & ⩽C\left(\parallel \nabla {\mathbf{u}}_t{\parallel }_{L^2}+{\parallel {\mathbf{u}}_t\parallel }_{L^2}\right){\parallel \nabla \mathbf{w}\parallel }_{L^2}\left(\parallel {\mathbf{w}}_t{\parallel }_{L^2}^{1/2}\parallel \nabla {\mathbf{w}}_t{\parallel }_{L^2}^{1/2}+{\parallel {\mathbf{w}}_t\parallel }_{L^2}\right)\hfill \\ \hfill & ⩽C\parallel \nabla {\mathbf{u}}_t{\parallel }_{L^2}\parallel \nabla {\mathbf{w}}_t{\parallel }_{L^2}{\parallel \nabla \mathbf{w}\parallel }_{L^2}\hfill \\ \hfill & ⩽C{C}_0^{1/8}\parallel \nabla {\mathbf{u}}_t{\parallel }_{L^2}^2+{\tilde{C}}_5{C}_0^{1/8}{\parallel \nabla {\mathbf{w}}_t\parallel }_{L^2}^2\hfill \\ \hfill & ⩽C{C}_0^{1/4}\parallel \nabla {\mathbf{u}}_t{\parallel }_{L^2}^2+{\displaystyle \frac{c_a+c_d}{6}}{\parallel \nabla {\mathbf{w}}_t\parallel }_{L^2}^2,\hfill \end{array}$$

provided that $C_0⩽{\epsilon }_{2,6}\triangleq min\{{\epsilon }_1,\frac{1}{2},{\left(\frac{c_a+c_d}{6{\tilde{C}}_5}\right)}^8\}$. It follows from (13) and (25) that

$$\begin{array}{cc}\hfill Q_2& =-\int \rho \mathbf{u}·\nabla (\mathbf{u}·\nabla \mathbf{w}·{\mathbf{w}}_t)dx\hfill \\ \hfill & ⩽{C\parallel \mathbf{u}\parallel }_{L^6}{\parallel \nabla \mathbf{u}\parallel }_{L^2}{\parallel \nabla \mathbf{w}\parallel }_{L^6}\parallel {\mathbf{w}}_t{\parallel }_{L^6}+{C\parallel \mathbf{u}\parallel }_{L^6}^2\parallel {\nabla }^2{\mathbf{w}\parallel }_{L^2}{\parallel {\mathbf{w}}_t\parallel }_{L^6}\hfill \\ \hfill & +{C\parallel \mathbf{u}\parallel }_{L^6}^2{\parallel \nabla \mathbf{w}\parallel }_{L^6}\parallel \nabla {\mathbf{w}}_t{\parallel }_{L^2}⩽{C\parallel \nabla \mathbf{u}\parallel }_{L^2}{\parallel \nabla \mathbf{w}\parallel }_{H^1}{\parallel \nabla {\mathbf{w}}_t\parallel }_{L^2}\hfill \\ \hfill & ⩽{\displaystyle \frac{c_a+c_d}{12}}\parallel \nabla {\mathbf{w}}_t{\parallel }_{L^2}^2+{C\parallel \nabla \mathbf{u}\parallel }_{L^2}^2{\parallel \nabla \mathbf{w}\parallel }_{H^1}^2\hfill \\ \hfill & ⩽{\displaystyle \frac{c_a+c_d}{12}}\parallel \nabla {\mathbf{w}}_t{\parallel }_{L^2}^2+{\tilde{C}}_6{C}_0^{1/4}({\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t)\parallel }_{L^2}^2\hfill \\ \hfill & +{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^2+{\parallel \nabla P\parallel }_{L^2}^2)\hfill \\ \hfill & ⩽{\displaystyle \frac{c_a+c_d}{6}}\parallel \nabla {\mathbf{w}}_t{\parallel }_{L^2}^2+C{C}_0^{1/4}{\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{b}}_t)\parallel }_{L^2}^2\hfill \\ \hfill & +C{C}_0^{1/4}\left({\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^2+{\parallel \nabla P\parallel }_{L^2}^2\right),\hfill \end{array}$$

provided that $C_0⩽{\epsilon }_{2,7}\triangleq min\{{\epsilon }_1,{\left(\frac{c_a+c_d}{12{\tilde{C}}_6}\right)}^8\}$. Substituting $Q_1$, $Q_2$ and $Q_3$ into (42), one has

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}\parallel {\rho }^{1/2}{\mathbf{w}}_t{\parallel }_{L^2}^2+{\parallel \nabla {\mathbf{w}}_t\parallel }_{L^2}^2+\zeta \int \left({\left|2\mathbf{w}\right|}^2-2\nabla \times {\mathbf{u}}_t·{\mathbf{w}}_t\right)dx\hfill \\ \hfill & ⩽C{C}_0^{1/4}{\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{b}}_t)\parallel }_{L^2}^2+C{C}_0^{1/4}\left({\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^2+{\parallel \nabla P\parallel }_{L^2}^2\right)\hfill \end{array}$$

(43)

According to the observation, one has

$$\parallel {\mathbf{w}}_t{\parallel }_{L^2}^2⩽C\int {|\nabla \times {\mathbf{u}}_t-2{\mathbf{w}}_t|}^{2}dx.$$

This together with (40) and (43), yields

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}\parallel ({\rho }^{1/2}{\mathbf{u}}_t,P^{1/2}\mathrm{div}\mathbf{u},{\rho }^{1/2}{\mathbf{w}}_t){\parallel }_{L^2}^2+{\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,{\mathbf{w}}_t)\parallel }_{L^2}^2\hfill \\ \hfill & ⩽C{C}_0^{1/4}{\parallel \nabla {\mathbf{b}}_t\parallel }_{L^2}^2+C{C}_0^{1/4}\left({\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^2+{\parallel \nabla P\parallel }_{L^2}^2\right)\hfill \end{array}$$

(44)

Taking the time derivative of (6)₄ yields

$$\mathbf{b}tt-\nu \Delta {\mathbf{b}}_t=-{\mathbf{u}}_t·\nabla \mathbf{b}-\mathbf{u}·\nabla {\mathbf{b}}_t+{\mathbf{b}}_t·\nabla \mathbf{u}+\mathbf{b}·\nabla {\mathbf{u}}_t-{\mathbf{b}}_t\mathrm{div}\mathbf{u}-\mathbf{b}\mathrm{div}{\mathbf{u}}_t.$$

(45)

Then, forming the $L^2$ inner product of (45) with ${\mathbf{b}}_t$ and applying integration by parts, we derive from (8), (14) and (20) that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel {\mathbf{b}}_t{\parallel }_{L^2}^2+C_9\nu \parallel \nabla {\mathbf{b}}_t{\parallel }_{L^2}^2⩽{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel {\mathbf{b}}_t{\parallel }_{L^2}^2+\nu {\parallel \mathrm{curl}{\mathbf{b}}_t\parallel }_{L^2}^2\hfill \\ \hfill & ⩽C\left(\parallel \nabla {\mathbf{u}}_t{\parallel }_{L^2}{\parallel \mathbf{b}\parallel }_{L^3}\parallel {\mathbf{b}}_t{\parallel }_{L^6}+\parallel {\mathbf{u}}_t{\parallel }_{L^6}{\parallel \mathbf{b}\parallel }_{L^3}{\parallel \nabla {\mathbf{b}}_t\parallel }_{L^2}\right)\hfill \\ \hfill & +{C\parallel \mathbf{u}\parallel }_{L^6}\parallel \nabla {\mathbf{b}}_t{\parallel }_{L^2}\parallel {\mathbf{b}}_t{\parallel }_{L^3}+C\parallel {\mathbf{b}}_t{\parallel }_{L^6}{\parallel \nabla \mathbf{u}\parallel }_{L^2}{\parallel {\mathbf{b}}_t\parallel }_{L^3}\hfill \\ \hfill & ⩽{\displaystyle \frac{C_9\nu }{4}}\parallel \nabla {\mathbf{b}}_t{\parallel }_{L^2}^2+{C\parallel \nabla \mathbf{b}\parallel }_{L^2}^2\parallel \nabla {\mathbf{u}}_t{\parallel }_{L^2}^2+C_{10}{C}_0^{1/8}{\parallel \nabla {\mathbf{b}}_t\parallel }_{L^2}^2\hfill \\ \hfill & ⩽{\displaystyle \frac{C_9\nu }{2}}\parallel \nabla {\mathbf{b}}_t{\parallel }_{L^2}^2+C{C}_0^{1/4}{\parallel \nabla {\mathbf{u}}_t\parallel }_{L^2}^2,\hfill \end{array}$$

provided that $C_0⩽{\epsilon }_{2,8}\triangleq min\{{\epsilon }_1,{\left(\frac{C_9\nu }{4C_{10}}\right)}^8\}$. Hence

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}\parallel {\mathbf{b}}_t{\parallel }_{L^2}^2+\parallel \nabla {\mathbf{b}}_t{\parallel }_{L^2}^2⩽C_{11}{C}_0^{1/4}{\parallel \nabla {\mathbf{u}}_t\parallel }_{L^2}^2,\end{array}$$

which, together with (44), implies (37), provided that

$$C_0⩽{\epsilon }_2\triangleq min\left\{{\epsilon }_{2,i},{\left({\displaystyle \frac{1}{2C_{11}}}\right)}^4,{\left({\displaystyle \frac{1}{2C_8}}\right)}^8\right\},\mathrm{for}i=1,2,\cdots ,7.$$

With this, the proof of Lemma 8 is finished. □

Building upon the results we have established in Lemmas 6–8, we now proceed to estimate $\mathcal{H}\left(t\right)$.

Lemma 9.

Consider a smooth solution $(\rho ,P,\mathbf{u},\mathbf{w},\mathbf{b})$ to the system (4)–(6) on $\Omega \times (0,T]$ that also satisfies (15)–(19). Then there exist positive constants ${\epsilon }_3$ and σ, whose dependence is restricted to $\mu ,\lambda ,\nu ,\overline{\rho },\underline{\rho }$, $\zeta ,c_0,c_a,c_d$, and Ω, such that

$$\underset{t\in [0,T]}{sup}{e}^{\sigma t}\mathcal{H}\left(t\right)⩽C_0^{1/2},$$

(46)

and

$${\int }_0^T{e}^{\frac{\sigma t}{2}}{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b},\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t,\nabla P)\parallel }_{L^2}^{2}dt⩽C{C}_0^{1/2},$$

(47)

provided $C_0⩽{\epsilon }_3$.

Proof. 

Multiplying (26) and (37) by $\frac{1}{2C_1}$ and $2C_2$, respectively, then, adding the resulting inequalities to (28), implies

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}{\parallel ({\rho }^{1/2}\mathbf{u},{\rho }^{1/2}\mathbf{w},\mathbf{b},\nabla P,{\rho }^{1/2}{\mathbf{u}}_t,{\rho }^{1/2}{\mathbf{w}}_t,P^{1/2}\mathrm{div}\mathbf{u},{\mathbf{b}}_t)\parallel }_{L^2}^2\hfill \\ \hfill & +\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b},\nabla P,\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}^2\hfill \\ \hfill & ⩽C_{12}{C}_0^{1/8}{\parallel \nabla P\parallel }_{L^2}^2+C_{13}{C}_0^{\frac{p-3}{20p-24}}{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^2\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{2}}{\parallel \nabla P\parallel }_{L^2}^2+{\displaystyle \frac{1}{2}}{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^2,\hfill \end{array}$$

provided that $C_0⩽{\epsilon }_{3,1}\triangleq min\{{\epsilon }_2,{\left(\frac{1}{2C_{12}}\right)}^8,{\left(\frac{1}{2C_{13}}\right)}^{\frac{20p-24}{p-3}}\}$. Therefore,

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}\mathcal{G}\left(t\right)+{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b},\nabla P,\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t)\parallel }_{L^2}^2⩽0,\end{array}$$

(48)

with

$$\begin{array}{c}\hfill \mathcal{G}\left(t\right)\triangleq \parallel ({\rho }^{1/2}\mathbf{u},{\rho }^{1/2}\mathbf{w},\mathbf{b},\nabla P,{\rho }^{1/2}{\mathbf{u}}_t,{\rho }^{1/2}{\mathbf{w}}_t,P^{1/2}\mathrm{div}\mathbf{u},{\mathbf{b}}_t){\parallel }_{L^2}^2.\end{array}$$

With the help of (13) and (15), there exists a positive constant $\sigma$ such that

$$\begin{array}{cc}\hfill & \sigma \parallel ({\rho }^{1/2}\mathbf{u},{\rho }^{1/2}\mathbf{w},\mathbf{b},{\rho }^{1/2}{\mathbf{u}}_t,{\rho }^{1/2}{\mathbf{w}}_t,P^{1/2}\mathrm{div}\mathbf{u},{\mathbf{b}}_t){\parallel }_{L^2}^2\hfill \\ \hfill & ⩽\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b},\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}^2,\hfill \end{array}$$

which gives

$$\begin{array}{c}\hfill \sigma \mathcal{G}\left(t\right)⩽\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b},\nabla P,\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}^2.\end{array}$$

(49)

The combination of (48) and (49) leads to

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}\mathcal{G}\left(t\right)+\sigma \mathcal{G}\left(t\right)⩽0,\end{array}$$

which imples

$$\underset{t\in [0,T]}{sup}{e}^{\sigma t}\mathcal{H}\left(t\right)⩽C_{14}{C}_0⩽C_0^{1/2},$$

(50)

provided that

$$C_0⩽{\epsilon }_3\triangleq min\left\{{\epsilon }_{3,1},{\left({\displaystyle \frac{1}{C_{14}}}\right)}^2\right\},$$

which yields (46). Multiplying (48) by $e^{\frac{\sigma t}{2}}$ and integrating the results over $[0,T]$, we have from (50) that

$$\begin{array}{cc}\hfill & \underset{t\in [0,T]}{sup}{e}^{\frac{\sigma t}{2}}\mathcal{H}\left(t\right)+{\int }_0^T{e}^{\frac{\sigma t}{2}}{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b},\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t,\nabla P)\parallel }_{L^2}^{2}dt\hfill \\ \hfill & ⩽\underset{t\in [0,T]}{sup}{e}^{\frac{\sigma t}{2}}\mathcal{H}\left(t\right)+C{C}_0⩽C{C}_0^{1/2}{\int }_0^T{e}^{-\frac{\sigma t}{2}}dt+C{C}_0⩽C{C}_0^{1/2},\hfill \end{array}$$

which leads to (47). The assertion of Lemma 9 is now verified. □

According to Lemma 9, ${\parallel \mathrm{div}\mathbf{u}\parallel }_{L^2}$ is bounded. Since ${\parallel \nabla \mathbf{u}\parallel }_{L^2}\le {C(\parallel \mathrm{div}\mathbf{u}\parallel }_{L^2}+{\parallel \mathrm{curl}\mathbf{u}\parallel }_{L^2})$, we therefore focus on estimating ${\parallel \mathrm{curl}\mathbf{u}\parallel }_{L^2}$.

Lemma 10.

Consider a smooth solution $(\rho ,P,\mathbf{u},\mathbf{w},\mathbf{b})$ to the system (4)–(6) on $\Omega \times (0,T]$ that also satisfies (15)–(19). Then there exists a positive constant ${\epsilon }_4$, whose dependence is confined to $\mu ,\lambda ,\nu ,\overline{\rho },\underline{\rho }$, $\zeta ,c_0,c_a,c_d$ and Ω, such that

$$\underset{t\in [0,T]}{sup}{e}^{\frac{\sigma t}{2}}{\parallel (\mathrm{curl}\mathbf{u},\mathrm{curl}\mathbf{w},\mathrm{curl}\mathbf{b})\left(t\right)\parallel }_{L^2}^2+{\int }_0^T{e}^{\frac{\sigma t}{2}}{\parallel ({\rho }^{1/2}{\mathbf{u}}_t,{\rho }^{1/2}{\mathbf{w}}_t,{\mathbf{b}}_t)\parallel }_{L^2}^2⩽C_0^{1/4},$$

(51)

provided $C_0⩽{\epsilon }_4$.

Proof. 

An $L^2$-multiplication of (4)₂ with ${\mathbf{u}}_t$ yields, upon application of (9), (13)–(15) and (20), the following result:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\left[{\mu \parallel \mathrm{curl}\mathbf{u}\parallel }_{L^2}^2+(2\mu +\lambda ){\parallel \mathrm{div}\mathbf{u}\parallel }_{L^2}^2\right]+\parallel {\rho }^{1/2}{\mathbf{u}}_t{\parallel }_{L^2}^2+{\displaystyle \frac{\zeta }{2}}{\int [(|\nabla \times \mathbf{u}|}^2{)}_t-4\nabla \times {\mathbf{u}}_t·\mathbf{w}]dx\hfill \\ \hfill & =-\int \rho \mathbf{u}·\nabla \mathbf{u}·{\mathbf{u}}_{t}dx-\int {\mathbf{u}}_t·\nabla Pdx+\int \left(\mathbf{b}·\nabla \mathbf{b}-{\displaystyle \frac{1}{2}}\nabla {\left|\mathbf{b}\right|}^2\right)·{\mathbf{u}}_{t}dx\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{2}}\parallel {\rho }^{1/2}{\mathbf{u}}_t{\parallel }_{L^2}^2+{C\parallel \mathbf{u}\parallel }_{L^{\infty }}^2{\parallel \nabla \mathbf{u}\parallel }_{L^2}^2+{C\parallel \nabla P\parallel }_{L^2}^2+{C\parallel \mathbf{b}\parallel }_{L^{\infty }}^2{\parallel \nabla \mathbf{b}\parallel }_{L^2}^2\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{2}}\parallel {\rho }^{1/2}{\mathbf{u}}_t{\parallel }_{L^2}^2+{C\parallel (\mathbf{u},\mathbf{b})\parallel }_{H^2}^2{\parallel (\nabla \mathbf{u},\nabla \mathbf{b})\parallel }_{L^2}^2+C{\parallel \nabla P\parallel }_{L^2}^2\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{2}}\parallel {\rho }^{1/2}{\mathbf{u}}_t{\parallel }_{L^2}+C{C}_0^{1/4}\parallel ({\rho }^{1/2}{\mathbf{w}}_t,{\mathbf{b}}_t){\parallel }_{L^2}^2+C{C}_0^{1/4}\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b}){\parallel }_{L^2}^2+C{\parallel \nabla P\parallel }_{L^2}^2,\hfill \end{array}$$

which, leads to

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}\left[{\parallel \mathrm{curl}\mathbf{u}\parallel }_{L^2}^2+{\parallel \mathrm{div}\mathbf{u}\parallel }_{L^2}^2\right]+\parallel {\rho }^{1/2}{\mathbf{u}}_t{\parallel }_{L^2}^2+{\displaystyle \frac{\zeta }{2}}{\int [(|\nabla \times \mathbf{u}|}^2{)}_t-4\nabla \times {\mathbf{u}}_t·\mathbf{w}]dx\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{4}}\parallel ({\rho }^{1/2}{\mathbf{w}}_t,{\mathbf{b}}_t){\parallel }_{L^2}^2+C{C}_0^{1/4}\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b}){\parallel }_{L^2}^2+C{\parallel \nabla P\parallel }_{L^2}^2.\hfill \end{array}$$

(52)

Similar to (42), one has from (4)₃ that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}\left[{\parallel \mathrm{curl}\mathbf{w}\parallel }_{L^2}^2+{\parallel \mathrm{div}\mathbf{w}\parallel }_{L^2}^2\right]+\parallel {\rho }^{1/2}{\mathbf{w}}_t{\parallel }_{L^2}^2+{\displaystyle \frac{\zeta }{2}}{\int [(|\mathbf{w}|}^2{)}_t-4\nabla \times \mathbf{u}·{\mathbf{w}}_t]dx\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{4}}\parallel ({\rho }^{1/2}{\mathbf{u}}_t,{\mathbf{b}}_t){\parallel }_{L^2}^2+C{C}_0^{1/4}\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b}){\parallel }_{L^2}^2+C{\parallel \nabla P\parallel }_{L^2}^2.\hfill \end{array}$$

(53)

Using (4)₅, (8), (9), (13) and (20) yields

$$\begin{array}{cc}\hfill & {\nu (\parallel \mathrm{curl}\mathbf{b}\parallel }_{L^2}^2{)}_t+{\nu }^2{\parallel \nabla \times \mathrm{curl}\mathbf{b}\parallel }_{L^2}^2+{\parallel {\mathbf{b}}_t\parallel }_{L^2}^2\hfill \\ \hfill & =\int {({\mathbf{b}}_t+\nu \nabla \times \mathrm{curl}\mathbf{b})}^{2}dx=\int {|\mathbf{b}·\nabla \mathbf{u}-\mathbf{u}·\nabla \mathbf{b}-\mathbf{b}\mathrm{div}\mathbf{u}|}^{2}dx\hfill \\ \hfill & ⩽C(\parallel \mathbf{b}{\parallel }_{L^6}^2\parallel \nabla \mathbf{u}{\parallel }_{L^3}^2+\parallel \mathbf{u}{\parallel }_{L^6}^2\parallel \nabla \mathbf{b}{\parallel }_{L^2}\parallel {\nabla }^2\mathbf{b}{\parallel }_{L^2})\hfill \\ \hfill & ⩽{C\parallel \nabla \mathbf{b}\parallel }_{L^2}^2{\parallel \nabla \mathbf{u}\parallel }_{H^1}^2+{C\parallel \nabla \mathbf{u}\parallel }_{L^2}^2{\parallel \nabla \mathbf{b}\parallel }_{H^1}^2\hfill \\ \hfill & ⩽C{C}_0^{1/4}{\parallel (\nabla \mathbf{u},\nabla \mathbf{b})\parallel }_{L^2}^2.\hfill \end{array}$$

(54)

Collecting (52)–(54), one has

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}\left({\parallel (\mathrm{curl}\mathbf{u},\mathrm{div}\mathbf{u},\mathrm{curl}\mathbf{w},\mathrm{div}\mathbf{w},\mathrm{curl}\mathbf{b})\parallel }_{L^2}^2+{\parallel \nabla \times \mathbf{u}-2\mathbf{w}\parallel }_{L^2}^2\right)\hfill \\ \hfill & +\parallel ({\rho }^{1/2}{\mathbf{u}}_t,{\rho }^{1/2}{\mathbf{w}}_t,{\mathbf{b}}_t){\parallel }_{L^2}^2+{\parallel \nabla \times \mathrm{curl}\mathbf{b}\parallel }_{L^2}^2\hfill \\ \hfill & ⩽C{C}_0^{1/4}{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^2+C_{15}{\parallel \nabla P\parallel }_{L^2}^2.\hfill \end{array}$$

(55)

Multiplying (55) by $e^{\frac{\sigma t}{2}}$ and integrating the results over $[0,T]$, we obtain from (47) that

$$\begin{array}{cc}\hfill & \underset{t\in [0,T]}{sup}{e}^{\frac{\sigma t}{2}}{\parallel (\mathrm{curl}\mathbf{u},\mathrm{curl}\mathbf{w},\mathrm{curl}\mathbf{b})\left(t\right)\parallel }_{L^2}^2+{\int }_0^T{e}^{\frac{\sigma t}{2}}{\parallel ({\rho }^{1/2}{\mathbf{u}}_t,{\rho }^{1/2}{\mathbf{w}}_t,{\mathbf{b}}_t)\parallel }_{L^2}^2\hfill \\ \hfill & ⩽C{C}_0+C{\int }_0^T{e}^{\frac{\sigma t}{2}}{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b},\nabla P)\parallel }_{L^2}^{2}dt\hfill \\ \hfill & ⩽C_{16}{C}_0^{1/2},\hfill \end{array}$$

provided that

$$C_0⩽{\epsilon }_4\triangleq min\left\{{\epsilon }_3,{\left({\displaystyle \frac{1}{C_{16}}}\right)}^4\right\}.$$

Therefore, the proof of Lemma 10 is completed. □

Remark 1.

We now explain why we do not estimate $\mathrm{curl}\mathbf{u}$ in Lemma 19 directly. Operating (26) $\times \frac{1}{2C_1}$ + (27) + (37) $\times 2C_2$ + (55) $\times \frac{1}{8C_{15}}$, yields

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}\parallel \left({\rho }^{1/2}\mathbf{u},{\rho }^{1/2}\mathbf{w},\mathbf{b},P^{1/2}\mathrm{div}\mathbf{u},\nabla P,{\rho }^{1/2}{\mathbf{u}}_t,{\rho }^{1/2}{\mathbf{w}}_t,{\mathbf{b}}_t,\mathrm{curl}\mathbf{u},\mathrm{curl}\mathbf{w},\mathrm{div}\mathbf{u},\mathrm{div}\mathbf{w},\mathrm{curl}\mathbf{b}\right){\parallel }_{L^2}^2\hfill \\ \hfill & +\parallel \left(\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b},\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t,\nabla P,{\rho }^{1/2}{\mathbf{u}}_t,{\rho }^{1/2}{\mathbf{w}}_t,{\mathbf{b}}_t\right){\parallel }_{L^2}^2⩽0,\hfill \end{array}$$

provided that $C_0$ is suitably small. Thus, there exists a positive constant δ such that

$$\begin{array}{cc}\hfill & \delta \mathcal{H}\left(t\right)+{\delta \parallel (\mathrm{curl}\mathbf{u},\mathrm{div}\mathbf{u},\mathrm{curl}\mathbf{w},\mathrm{div}\mathbf{w},\mathrm{curl}\mathbf{b})\parallel }_{L^2}^2\hfill \\ \hfill & ⩽\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b},\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t,\nabla P,{\rho }^{1/2}{\mathbf{u}}_t,{\rho }^{1/2}{\mathbf{w}}_t,{\mathbf{b}}_t){\parallel }_{L^2}^2,\hfill \end{array}$$

which implies

$$\begin{array}{cc}\hfill & \delta \parallel ({\rho }^{1/2}\mathbf{u},{\rho }^{1/2}\mathbf{w},\mathbf{b},P^{1/2}\mathrm{div}\mathbf{u},\mathrm{curl}\mathbf{u},\mathrm{div}\mathbf{u},\mathrm{curl}\mathbf{w},\mathrm{div}\mathbf{w},\mathrm{curl}\mathbf{b}){\parallel }_{L^2}^2\hfill \\ \hfill & ⩽{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^2.\hfill \end{array}$$

Compared to (49), it has $\delta <\sigma$. This implies that ${\parallel \mathrm{curl}\mathbf{u}\parallel }_{L^2}$ and ${\parallel \mathrm{curl}\mathbf{w}\parallel }_{L^2}$ decay slower that $\mathcal{H}\left(t\right)$.

Lemma 11.

Let $(\rho ,P,\mathbf{u},\mathbf{w},\mathbf{b})$ be a smooth solution of (4)–(6) on $\Omega \times (0,T]$ satisfying (15)–(19). Then, there exist some positive constants ${\epsilon }_5$ and $\varsigma (\varsigma <\sigma )$, depending only on $\mu ,\lambda ,\nu ,\overline{\rho },\underline{\rho }$, $\zeta ,c_a,c_d,c_0$, Ω and $\parallel \nabla P_0{\parallel }_{L^p}$ with $p\in (3,6)$ such that

$$\underset{t\in [0,T]}{sup}{e}^{\varsigma t}{\parallel \nabla P\left(t\right)\parallel }_{L^p}^2⩽2{\parallel \nabla P_0\parallel }_{L^p}^2,$$

(56)

and

$${\int }_0^T{e}^{\frac{\varsigma t}{2}}\left(\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b},{\rho }^{1/2}{\mathbf{u}}_t,{\rho }^{1/2}{\mathbf{w}}_t,{\mathbf{b}}_t,\nabla P){\parallel }_{L^2}^2+{\parallel \nabla P\parallel }_{L^p}^2\right)⩽C,$$

(57)

provided $C_0⩽{\epsilon }_5$.

Proof. 

Choosing $q=p$ in (28) and dividing the resulting inequality by ${\parallel \nabla P\parallel }_{L^p}^{p-2}$, one has from (29)–(32) and (36) that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}{\parallel \nabla P\parallel }_{L^p}^2+{\displaystyle \frac{q}{2\mu +\lambda }}{\parallel \nabla P\parallel }_{L^p}^2\hfill \\ \hfill & ⩽{C\parallel \nabla \mathbf{u}\parallel }_{L^{\infty }}{\parallel \nabla P\parallel }_{L^p}^2+C\left({\parallel \nabla \mathbf{w}\parallel }_{L^{\infty }}{\parallel \nabla \mathbf{u}\parallel }_{L^p}+{\parallel \mathbf{w}\parallel }_{L^{\infty }}{\parallel ({\nabla }^2\mathbf{u},\nabla \mathbf{w})\parallel }_{L^p}\right){\parallel \nabla P\parallel }_{L^p}\hfill \\ \hfill & +C\left({\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^{\infty }}\parallel ({\nabla }^2\mathbf{u},{\nabla }^2\mathbf{w},{\nabla }^2\mathbf{b}){\parallel }_{L^p}+{\parallel (\nabla F,\nabla \mathbf{b}·\mathbf{b})\parallel }_{L^p}\right){\parallel \nabla P\parallel }_{L^p}\hfill \\ \hfill & ⩽C\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}{\parallel \nabla P\parallel }_{L^p}+C{C}_0^{\frac{p-3}{40p-48}}{\parallel \nabla P\parallel }_{L^p}^2\hfill \\ \hfill & +C{C}_0^{\frac{p-3}{40p-48}}\parallel ({\nabla }^2\mathbf{u},{\nabla }^2\mathbf{w},{\nabla }^2\mathbf{b}){\parallel }_{L^p}{\parallel \nabla P\parallel }_{L^p}+C\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b}){\parallel }_{L^2}{\parallel \nabla P\parallel }_{L^p}\hfill \\ \hfill & +{C\parallel \nabla P\parallel }_{L^2}{\parallel \nabla P\parallel }_{L^p}+C\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}{\parallel ({\nabla }^2\mathbf{u},{\nabla }^2\mathbf{w},{\nabla }^2\mathbf{b})\parallel }_{L^p}\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{4}}{\parallel \nabla P\parallel }_{L^p}^2+C\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}^2+C_{17}{C}_0^{\frac{p-3}{40p-48}}{\parallel \nabla P\parallel }_{L^p}^2\hfill \\ \hfill & +{C\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^2+C{\parallel \nabla P\parallel }_{L^2}^2,\hfill \end{array}$$

provided that $C_0⩽{\epsilon }_{5,1}\triangleq min\{{\epsilon }_4,{\left(\frac{1}{4C_{17}}\right)}^{\frac{40p-48}{p-3}}\}$, so that

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}{\parallel \nabla P\parallel }_{L^p}^2+{\parallel \nabla P\parallel }_{L^p}^2⩽C_{18}{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b},\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t,\nabla P)\parallel }_{L^2}^2.\end{array}$$

(58)

Multiplying (58) by $\frac{\frac{1}{4C_1}+C_2+\frac{1}{4}}{3C_{18}}$, one obtains from (48) that

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}\left[{\parallel \nabla P\parallel }_{L^p}^2+\mathcal{G}\left(t\right)\right]+{\parallel \nabla P\parallel }_{L^p}^2+{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b},\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t,\nabla P)\parallel }_{L^2}^2⩽0.\end{array}$$

(59)

It follows that one may choose a positive constant $\varsigma$, strictly less than $\sigma$, such that

$$\begin{array}{cc}\hfill \underset{t\in [0,T]}{sup}{e}^{\varsigma t}{\parallel \nabla P\left(t\right)\parallel }_{L^p}^2& ⩽\parallel \nabla P_0{\parallel }_{L^p}^2+C\underset{t\in [0,T]}{sup}{e}^{\varsigma t}\mathcal{G}\left(t\right)\hfill \\ \hfill & ⩽\parallel \nabla P_0{\parallel }_{L^p}^2+C_{19}{C}_0^{1/2}\hfill \\ \hfill & ⩽2\parallel \nabla P_0{\parallel }_{L^p}^2,\hfill \end{array}$$

provided that

$$C_0⩽{\epsilon }_5\triangleq min\left\{{\epsilon }_{5,1},{\left({\displaystyle \frac{\parallel \nabla P_0{\parallel }_{L^p}^2}{C_{19}}}\right)}^2\right\}.$$

Alternatively, (59) implies that

$$\begin{array}{cc}\hfill & \underset{t\in [0,T]}{sup}{e}^{\frac{\varsigma t}{2}}\left[{\parallel \nabla P\parallel }_{L^p}^2+\mathcal{G}\left(t\right)\right]+{\int }_0^T{e}^{\frac{\varsigma t}{2}}{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b},\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t,\nabla P)\parallel }_{L^2}^{2}dt\hfill \\ \hfill & +{\int }_0^T{e}^{\frac{\varsigma t}{2}}{\parallel \nabla P\parallel }_{L^p}^{2}dt\hfill \\ \hfill & ⩽C{\int }_0^T{e}^{\frac{\varsigma t}{2}}\left[{\parallel \nabla P\parallel }_{L^p}^2+\mathcal{G}\left(t\right)\right]dt+C\hfill \\ \hfill & ⩽C{\int }_0^T{e}^{-\frac{\varsigma t}{2}}dt+C⩽C.\hfill \end{array}$$

The assertion of Lemma 11 is now verified. □

Lemma 12.

Consider a smooth solution $(\rho ,P,\mathbf{u},\mathbf{w},\mathbf{b})$ to the system (4)–(6) on $\Omega \times (0,T]$ that also satisfies (15)–(19). Then there exists a positive constant ${\epsilon }_6$, whose dependence is restricted to $\mu ,\lambda ,\nu ,\overline{\rho },\underline{\rho }$, $\zeta ,c_a,c_0,c_d$, $\Omega ,E_0$ and $|\nabla P_0{|}_{L^p}$ for $p\in (3,6)$, such that

$${\displaystyle \frac{E_0}{2|\Omega |}}⩽P(x,t)⩽{\displaystyle \frac{3E_0}{2|\Omega |}},$$

(60)

provided $C_0⩽{\epsilon }_6$.

Proof. 

It follows from (13), (14), (25) and (47) that

$${\int }_0^T{\parallel ({\nabla }^2\mathbf{u},{\nabla }^2\mathbf{w},{\nabla }^2\mathbf{b})\parallel }_{L^2}^{2}dt⩽C{C}_0^{1/2}.$$

(61)

Integrating (4)₄ with $\kappa =0$ over $\Omega \times [0,t]$ leads to

$$\begin{array}{cc}\hfill \int Pdx& =E_0-{\displaystyle \frac{1}{2}}\int ({\rho }_0{\mathbf{u}}_0^2+{\rho }_0{\mathbf{w}}_0^2+{\mathbf{b}}_0^2)dx-{\int }_0^t\int P\mathrm{div}\mathbf{u}dxdt\hfill \\ \hfill & +{\int }_0^t{\int \left(2\mu \right|\mathbb{D}\left(\mathbf{u}\right)|}^2+\lambda {\left(\mathrm{div}\mathbf{u}\right)}^2+\zeta {|\nabla \times \mathbf{u}-2\mathbf{w}|}^2+c_0{\left(\mathrm{div}\mathbf{w}\right)}^2\hfill \\ \hfill & +(c_a+c_d){|\nabla \mathbf{w}|}^2+(c_d-c_a)\nabla \mathbf{w}:\nabla {\mathbf{w}}^T+\nu {|\nabla \times \mathbf{b}|}^2)dxdt.\hfill \end{array}$$

(62)

Using (13), (14), (17), (25)–(27) and (61) yields

$$\begin{array}{cc}\hfill -{\int }_0^t\int P\mathrm{div}\mathbf{u}dxdt& ={\int }_0^t\int \mathbf{u}·\nabla Pdxdt\hfill \\ \hfill & ⩽C{\int }_0^t\left({\parallel \nabla \mathbf{u}\parallel }_{L^2}^2+{\parallel \nabla P\parallel }_{L^2}^2\right)dt\hfill \\ \hfill & ⩽C{C}_0^{1/2},\hfill \end{array}$$

(63)

and

$$\begin{array}{cc}\hfill & |{\int }_0^t{\int (2\mu |\mathbb{D}\left(\mathbf{u}\right)|}^2+\lambda {\left(\mathrm{div}\mathbf{u}\right)}^2+\zeta {|\nabla \times \mathbf{u}-2\mathbf{w}|}^2+c_0{\left(\mathrm{div}\mathbf{w}\right)}^2\hfill \\ \hfill & +(c_a+c_d){|\nabla \mathbf{w}|}^2+(c_d-c_a)\nabla \mathbf{w}:\nabla {\mathbf{w}}^T+\nu {|\nabla \times \mathbf{b}|}^2\left)dxdt\right|\hfill \\ \hfill & ⩽C{\int }_0^t\left({\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^2+{\parallel \mathbf{w}\parallel }_{L^2}^2\right)dt\hfill \\ \hfill & ⩽C_{20}{C}_0^{1/4},\hfill \end{array}$$

(64)

thus, the combination of (62)–(64) implies

$${\displaystyle \frac{3}{4|\Omega |}}{E}_0⩽\overline{P}={\displaystyle \frac{1}{|\Omega |}}\int Pdx⩽{\displaystyle \frac{5}{4|\Omega |}}{E}_0,$$

provided that $C_0⩽{\epsilon }_{6,1}\triangleq min\{{\epsilon }_5,{\left(\frac{E_0}{4C_{20}}\right)}^4\}$. Using (9) and (46), one has

$$\parallel P-\overline{P}{\parallel }_{L^{\infty }}⩽C\parallel P-\overline{P}{\parallel }_{L^2}^{\frac{2p-6}{5p-6}}{\parallel \nabla P\parallel }_{L^p}^{\frac{3p}{5p-6}}⩽C{\parallel \nabla P\parallel }_{L^2}^{\frac{2p-6}{5p-6}}⩽C_{21}{C}_0^{\frac{p-3}{10p-12}},$$

taking

$$C_0⩽{\epsilon }_6\triangleq min\left\{{\epsilon }_{6,1},{\left({\displaystyle \frac{E_0}{4|\Omega |C_{21}}}\right)}^{\frac{10p-12}{p-3}}\right\},$$

The assertion of Lemma 12 is now fully established. □

Lemma 13.

Let $(\rho ,P,\mathbf{u},\mathbf{w},\mathbf{b})$ be a smooth solution of (4)–(6) on $\Omega \times (0,T]$ satisfying (15)–(19). Then, there exists a positive constant ε as described in Theorem 1 such that

$${\displaystyle \frac{1}{2}}\underline{\rho }⩽\rho (x,t)⩽2\overline{\rho },$$

(65)

provided that $C_0⩽\epsilon$.

Proof. 

We rewrite (6)₁ as follows:

$$D_{t}ln\rho =-\mathrm{div}\mathbf{u},$$

(66)

and

$$D_{t}ln{\rho }^{-1}=\mathrm{div}\mathbf{u}.$$

(67)

Here, $D_t\rho ={\rho }_t+\mathbf{u}·\nabla \rho$. With the help of (47) and (57), one has

$$\begin{array}{c}\hfill {\int }_0^T{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}dt⩽{\left({\int }_0^T{e}^{\frac{\sigma t}{2}}{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^{2}dt\right)}^{1/2}{\left({\int }_0^T{e}^{-\frac{\sigma t}{2}}dt\right)}^{1/2}⩽C{C}_0^{1/4},\end{array}$$

and

$$\begin{array}{cc}\hfill & {\int }_0^T\left(\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b},\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t,\nabla P){\parallel }_{L^2}+{\parallel \nabla P\parallel }_{L^p}\right)dt\hfill \\ \hfill & ⩽{\left({\int }_0^T{e}^{\frac{\varsigma t}{2}}\left(\parallel (\nabla \mathbf{u},\nabla \mathbf{b},\nabla {\mathbf{u}}_t,\nabla {\mathbf{b}}_t,\nabla P){\parallel }_{L^2}^2+{\parallel \nabla P\parallel }_{L^p}^2\right)dt\right)}^{1/2}{\left({\int }_0^T{e}^{-\frac{\varsigma t}{2}}dt\right)}^{1/2}\hfill \\ \hfill & ⩽C.\hfill \end{array}$$

(68)

Thanks to (6)₄, we deduce

$$\begin{array}{cc}\hfill \mathrm{div}\mathbf{u}& ={\displaystyle \frac{{2\mu \left|\mathbb{D}\left(\mathbf{u}\right)\right|}^2+\lambda {\left(\mathrm{div}\mathbf{u}\right)}^2+\nu {|\nabla \times \mathbf{b}|}^2}{2P}}-{\displaystyle \frac{D_{t}lnP}{2}}\hfill \\ \hfill & +{\displaystyle \frac{{\zeta |\nabla \times \mathbf{u}-2\mathbf{w}|}^2+c_0{\left(\mathrm{div}\mathbf{w}\right)}^2+(c_a+c_d){|\nabla \mathbf{w}|}^2+(c_d-c_a)\nabla \mathbf{w}:\nabla {\mathbf{w}}^T}{2P}}.\hfill \end{array}$$

(69)

Integrating (69) over $[0,T]$ and using (29), (30) and (68) lead to

$$\begin{array}{cc}\hfill & \left|{\int }_0^T\mathrm{div}\mathbf{u}dt\right|\hfill \\ \hfill & ⩽\left|{\int }_0^T{\displaystyle \frac{{2\mu \left|\mathbb{D}\left(\mathbf{u}\right)\right|}^2+\lambda {\left(\mathrm{div}\mathbf{u}\right)}^2+\nu {|\nabla \times \mathbf{b}|}^2}{2P}}dt\right|+\left|{\int }_0^T{\displaystyle \frac{D_{t}lnP}{2}}dt\right|\hfill \\ \hfill & +\left|{\int }_0^T{\displaystyle \frac{{\zeta |\nabla \times \mathbf{u}-2\mathbf{w}|}^2+c_0{\left(\mathrm{div}\mathbf{w}\right)}^2+(c_a+c_d){|\nabla \mathbf{w}|}^2+(c_d-c_a)\nabla \mathbf{w}:\nabla {\mathbf{w}}^T}{2P}}dt\right|\hfill \\ \hfill & ⩽C{\int }_0^T{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\mathbf{w},\nabla \mathbf{b})\parallel }_{L^{\infty }}^{2}dt+C\underset{t\in [0,T]}{sup}|lnP\left(t\right)|\hfill \\ \hfill & ⩽C{\int }_0^T\left({\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}^{\frac{2p-6}{5p-6}}\parallel ({\nabla }^2\mathbf{u},{\nabla }^2\mathbf{w},{\nabla }^2\mathbf{b}){\parallel }_{L^p}^{\frac{3p}{5p-6}}+{\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})\parallel }_{L^2}\right)dt\hfill \\ \hfill & +C\underset{t\in [0,T]}{sup}{\parallel P-\overline{P}\parallel }_{L^{\infty }}\hfill \\ \hfill & ⩽C_{21}{C}_0^{\frac{p-3}{10p-12}}.\hfill \end{array}$$

(70)

This utilizes the $L^{\infty }$ comparability $|lnP|\sim |P-\overline{P}|$. Consequently, performing a time integration of (66) and (67) from 0 to T and invoking estimate (70), we derive

$$\begin{array}{cc}\hfill \underset{t\in [0,T]}{sup}{\parallel \rho \parallel }_{L^{\infty }}& ⩽\parallel {\rho }_0{\parallel }_{L^{\infty }}exp\left\{-\left|{\int }_0^T\mathrm{div}\mathbf{u}dt\right|\right\}\hfill \\ \hfill & ⩽\overline{\rho }exp\left\{C_{21}{C}_0^{\frac{p-3}{10p-12}}\right\}\hfill \\ \hfill & ⩽2\overline{\rho },\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \underset{t\in [0,T]}{sup}{\parallel {\rho }^{-1}\parallel }_{L^{\infty }}& ⩽\parallel {\rho }_0^{-1}{\parallel }_{L^{\infty }}exp\left\{\left|{\int }_0^T\mathrm{div}\mathbf{u}dt\right|\right\}\hfill \\ \hfill & ⩽{\underline{\rho }}^{-1}exp\left\{C_{21}{C}_0^{\frac{p-3}{10p-12}}\right\}⩽2{\underline{\rho }}^{-1},\hfill \end{array}$$

provided that

$$C_0⩽\epsilon \triangleq min\left\{{\epsilon }_6,{\left({\displaystyle \frac{ln2}{C_{21}}}\right)}^{\frac{10p-12}{p-3}}\right\},$$

which, leads to (65). The assertion of Lemma 13 is now fully established. □

Lemma 14.

For a given $p\in (3,6)$ and any $r\in (3,p]$, let $(\rho ,P,\mathbf{u},\mathbf{w},\mathbf{b})$ be a smooth solution of (4)–(6) on $\Omega \times (0,T]$ that obeys (15)–(19). Then

$$\underset{t\in [0,T]}{sup}{\parallel \nabla \rho \parallel }_{L^r}⩽C.$$

(71)

Proof. 

Differentiating (6)₁ with respect to $x_i$, and multiplying it by $r|{\partial }_i{\rho |}^{r-2}{\partial }_i\rho$ yields

$$\begin{array}{cc}\hfill & {\left({|\nabla \rho |}^r\right)}_t+\mathrm{div}\left({|\nabla \rho |}^r\mathbf{u}\right)+(r-1){|\nabla \rho |}^r\mathrm{div}\mathbf{u}\hfill \\ \hfill & +{r|\nabla \rho |}^{r-2}{(\nabla \rho )}^{tr}\nabla \mathbf{u}(\nabla \rho )+r\rho {|\nabla \rho |}^{r-2}\nabla \rho ·\nabla \mathrm{div}\mathbf{u}=0.\hfill \end{array}$$

(72)

Integrating (72) on $\Omega$ and using (9), (32) and (36), gives

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{\parallel \nabla \rho \parallel }_{L^r}& ⩽{C\parallel \nabla \mathbf{u}\parallel }_{L^{\infty }}{\parallel \nabla \rho \parallel }_{L^r}+{C\parallel \nabla F\parallel }_{L^r}+C{\parallel \nabla P\parallel }_{L^r}\hfill \\ \hfill & ⩽{C(\parallel \nabla \rho \parallel }_{L^r}+1)\left(\parallel ({\mathbf{u}}_t,{\mathbf{w}}_t,{\mathbf{b}}_t){\parallel }_{L^r}^2+\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b}){\parallel }_{L^r}^2+{\parallel \nabla P\parallel }_{L^r}^2\right),\hfill \end{array}$$

which, together with (68) and Gronwall’s inequality, gives rise to

$$\begin{array}{cc}\hfill & \underset{t\in [0,T]}{sup}{\parallel \nabla \rho \parallel }_{L^r}\hfill \\ \hfill & ⩽(\parallel \nabla {\rho }_0{\parallel }_{L^r}+1)exp\left\{C{\int }_0^T\left(\parallel ({\mathbf{u}}_t,{\mathbf{w}}_t,{\mathbf{b}}_t){\parallel }_{L^r}^2+\parallel (\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b}){\parallel }_{L^r}^2+{\parallel \nabla P\parallel }_{L^r}^2\right)dt\right\}\hfill \\ \hfill & ⩽C(\parallel \nabla {\rho }_0{\parallel }_{L^r}+1),\hfill \end{array}$$

which, implies (71). The demonstration of Lemma 14 is now finished. □

Proof of Theorem 1.

The primary objective of this work is to establish Theorem 1, building upon the fundamental uniform-in-time estimates derived in Section 3. By applying Lemma 1, we deduce the existence of a positive time $T_{\ast }>0$ for which the system (4)–(6) possesses a strong solution $(\rho ,P,\mathbf{u},\mathbf{w},\mathbf{b})$ in $\Omega \times (0,T_{\ast }]$. Equipped with the full set of a priori estimates, we now proceed to extend this local strong solution globally in time. Define

$$T^{\ast }\triangleq sup\left\{T\right|(\rho ,P,\mathbf{u},\mathbf{w},\mathbf{b})\mathrm{is}\mathrm{a}\mathrm{strong}\mathrm{solution}\mathrm{on}[0,T]\}.$$

(73)

We assert that

$$T^{\ast }=\infty .$$

(74)

Suppose, for contradiction, that $T^{\ast }<\infty$. Then, by virtue of Lemmas 5–14, the solution $(\rho ,P,\mathbf{u},\mathbf{w},\mathbf{b})$ evaluated at $T=T^{\ast }$ is shown to satisfy the regularity conditions (7). Consequently, Lemma 1 implies the existence of a time $T^{\ast \ast }>T^{\ast }$ such that $(\rho ,P,\mathbf{u},\mathbf{w},\mathbf{b})$ can be extended to a strong solution of (4)–(6) on $\Omega \times [0,T)$, thereby contradicting the maximality of $T^{\ast }$ in (73). We therefore conclude that (74) must hold, which completes the proof of Theorem 1. □

4. Conclusions

This work establishes the global existence and exponential stability of strong solutions for the three-dimensional full compressible magneto-micropolar fluid system within a general bounded domain under slip boundary conditions. By employing the energy method, we derived key a priori estimates demonstrating exponential decay in time. A pivotal finding is the separate treatment of the curl and divergence estimates for the velocity and micro-rotation fields, which revealed that the growth rate of the divergence components in the $L^2$-norm can outpace that of the curl components in sufficiently large domains. A significant methodological advancement lies in our treatment of the pressure P not as a function of density and temperature $\rho \theta$ but as an independent variable in $(x,t)$, allowing us to directly control its gradient ${\parallel \nabla P\parallel }_{L^2}$. Leveraging the slip boundary conditions, we established an $L^p$-estimate for the gradient of the effective viscous flux, creating a control loop where ${\parallel \nabla P\parallel }_{L^2}$ is bounded by $\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}$ and vice versa. This self-contained argument enables our smallness assumption to be independent of the density gradient’s norm, thereby permitting initial data with large density oscillations.

Future Research Directions. Several promising avenues extend from this research. Firstly, incorporating heat conductivity with a positive lower bound would enhance the physical realism of the model. Secondly, analyzing the system’s behavior under different boundary conditions, such as no-slip or mixed Dirichlet–Neumann conditions, would be a valuable generalization. Finally, while this study focuses on a three-dimensional system, investigating its dynamics in more complex geometries or under the influence of external forces presents a natural next step.

Implications and Applications. Our findings have direct implications for both physical modeling and numerical simulation. The demonstrated stability under slip boundary conditions suggests that such configurations can effectively model confined magneto-micropolar flows without triggering small-wavelength instabilities at the boundary, a crucial insight for designing lab-on-a-chip devices or studying lubrication with microstructures. The identified faster growth of divergence components highlights a potential mechanism for compressibility-driven instabilities in large domains, which must be accounted for in predictive models. Furthermore, the developed energy method framework provides a robust analytical foundation for validating numerical schemes, such as the spectral Galerkin method. While Galerkin methods are powerful for constructing approximate solutions, the energy method employed here offers a more direct path for establishing crucial, solution-norm-based a priori estimates essential for proving global existence and stability, thereby complementing numerical approaches.