On the Cauchy problem of compressible Micropolar fluids subjected to Hall current

1.   Introduction

In this paper, we are interested in the mathematical analysis of the equations in a viscous, electrically conducting, micropolar fluids in the presence of a magnetic field, taking into account the effect of Hall current. When the strength of the magnetic field is sufficiently large, Ohm's law $\mathbf{J} = \sigma(\mathbf{E}+ {\bf{u}} \times {\mathit{\boldsymbol{b}}})$ needs to be modified to include Hall current so that the electric current density $\mathbf{J}$ satisfies the relation

where $\mathbf{E}, {\mathit{\boldsymbol{b}}}$, and ${\bf{u}}$ stand for the electric field, magnetic induction, and the fluid velocity, respectively. $\omega_e, \tau_e, \sigma, e$, and $p_e$ are the cyclotron frequency of electrons, electron collision time, electric conductivity, electron charge, and electron pressure, respectively (see [7, p.101]). Then the electric field $\mathbf{E}$ can be written as

Faraday's and Ampère's laws yield

where $\mu_e$ is the magnetic permeability constant. Thus, the generalized magnetic induction equation with the Hall effect has the following form:

The theory of micropolar fluids was developed by Eringen ^[10,11,12] and a generalization including the effects of magnetic fields has been developed by Ahmadi and Shahinpoor ^[2]. It has important engineering applications, such as in the extraction of oils/gas from oil fields, fluid flow in chemical engineering and magnetohydrodynamic (MHD) generators with neutral fluid seeding in the form of rigid microinclusions (see ^[12] and the references therein).

This paper is concerned with the mathematical analysis of the equations describing the flow in viscous, electrically conducting, micropolar fluids in the presence of a magnetic field, taking into account the effect of Hall current. The three-dimensional (3D) compressible, viscous micropolar fluids subject to Hall current occupying a domain $\Omega \subset \mathbb{R}^3$ are given by

Here, $\rho\geqslant 0, {\bf{u}} = (u^1, u^3, u^3), {\bf{w}} = (w^1, w^2, w^3)$ and ${\mathit{\boldsymbol{b}}} = (b^1, b^2, b^3)$ stand for the density, velocity, micro-rotational velocity and magnetic field, respectively. $p(\rho) = a\rho^\gamma\; (a > 0, \; \gamma > 1)$ is the pressure. The constants $\mu_1$ and $\lambda_1$ denote the shear and bulk viscosity coefficients, $\zeta$ is the dynamics micro-rotation viscosity, $\mu_2,$ and $\lambda_2$ are the angular viscosities, $\nu$ is the magnetic diffusivity of the magnetic field and $\beta$ is the Hall coefficient, and they satisfy

Let $\Omega = \mathbb{R}^3$. We consider the Cauchy problem of (1.1) with the far-field behavior

and the initial data

Micropolar fluids are fluids with microstructures belonging to a class of fluids with a nonsymmetric stress tensor. They exhibit micro-rotation effects and micro-rotational inertia. Liquids, fluids with additives, some polymeric fluids, colloidal fluids, and animal blood are a few examples of micropolar fluids. The theory of micropolar fluids was introduced by Eringen ^[10] and subsequently extended widely to the case of electrically conducting fluids in the magnetic field and to polarized fluids in an electric field. With the deepening of the research of micropolar fluid theory, the fluids considering the Hall effect attracted more and more attention. It has important engineering applications, such as in the extraction of oils/gas from oil fields, fluid flow in chemical engineering, and magnetohydrodynamic generators with neutral fluids seeding in the form of microinclusions.

There is much literature on the Cauchy problems of the micropolar system. If the magnetic field $\mathbf{b} = 0$, then the system (1.1) reduces to the classical micropolar fluid system, which has been successfully applied for modeling rheologically complex liquids such as blood and suspensions (see, e.g., ^[10,11,12]). Physically it may represent the fluids consisting of bar-like elements. Mujakovi$\acute{c}$ ^[24,25] studied the one-dimensional problem. Amirat, Hamdache ^[3] proved the existence of a global weak solution in a bounded domain in $\mathbb{R}^3$. Later, Chen et al. ^[5] obtained the global weak solutions of 3D compressible micropolar fluids with discontinuous initial data and vacuum.

From the mathematical viewpoint, the system (1.1) becomes the classical compressible magneto-micropolar fluids provided $\beta = 0$. Compared with the classical magneto-micropolar fluids, the system (1.1) has the Hall term $\tfrac{(\nabla \times {\mathit{\boldsymbol{b}}})\times {\mathit{\boldsymbol{b}}}}{\rho}$ in (1.1)$_4$, which is very important in describing many phenomena such as magnetic reconnection in space plasmas, star formation, neutron stars and geo-dynamo (see ^{[16,23,27,29]} and references therein). When the Hall term is neglected, the system (1.1) reduces to the well-known compressible magneto-micropolar fluid system, which has received many studies (see ^{[17,28,30,32,33,35,36]}). Wei et al. ^[30] established the existence of global-in-time smooth solutions under the condition of the small perturbations of initial data in $H^3$-norm and also obtained the long-time behavior of magneto-micropolar fluids. Based on the time-weighted energy estimate, Zhang in ^[35] proved the asymptotic stability of steady state with the strictly positive constant density, vanishing velocity, micro-rotational velocity, and magnetic field. Their results were later improved by Tong and Tan ^[28], where they showed that the solution of the magneto-micropolar fluids converges to its constant equilibrium state at the exact same $L^2$-decay rate as the linearized equations, which shows that the convergence rate is optimal (see ^[6,8,35] for related results).

If we consider the effects of the Hall term in the MHD system, Xiang ^[31] investigated the large-time behavior of solutions to the 3D compressible Hall magneto-hydrodynamics equations in addition, he also obtained that the smooth solution of the compressible Hall-magneto-hydrodynamics system converges globally in time to the smooth solution of the compressible magneto-hydro-dynamics system as $\beta \to 0$. Later, Lai et al. ^[18] considered the 3D compressible full Hall-MHD equations, and they obtained the global existence and optimal decay rates when the initial data are appropriately small. Lai and Xu ^[19] established the global existence of strong solutions for planar compressible, viscous, heat-conductive Hall-MHD equations with large initial data (see also ^[13,15,34] for related results).

For the magneto-micropolar fluids with Hall term, it has attracted the attention of many physicists and mathematicians due to their important background, rich phenomena, mathematical complexity, and challenges (see ^[4,21,26]). Mekheimer and El Kot ^[21] investigated the influence of magnetic field and Hall current on blood flow through a stenotic artery. Rani and Tomar ^[26] investigated the thermal instability of a micropolar fluid layer heated from below in the presence of the Hall current, and showed that the Hall current parameter has a destabilizing effect on the system. Amirat and Hamdache ^[4] studied the system, which is a combination of the generalized magnetic induction, the equations of micropolar fluids, and the temperature equation, and obtained the global existence of weak solutions in a bounded domain of $\mathbb{R}^3$.

Motivated by the results as in ^[4,18,31], the aim of this paper is to study the large-time behavior of solutions to the 3D compressible micropolar fluids subjected to Hall current. Before stating the main results, we explain the notation and conventions used throughout this paper. We denote

For $1\leqslant r \leqslant \infty, k\in\mathbb{Z}$ and $\alpha > 0$, we denote the standard homogeneous and inhomogeneous Sobolev spaces:

We use $\langle \cdot, \cdot \rangle$ to denote the inner product over the Hilbert space $L^2(\mathbb{R}^3)$, i.e.,

and set

We now state the definition of strong solutions of (1.1)–(1.3) as follows.

Definition 1.1. A pair of functions $(\rho, \mathbf{u}, \mathbf{w}, \mathbf{b})$ is called a strong solution to the problem (1.1)–(1.3) in $\mathbb{R}^3\times (0, \infty)$, if and only if $(\rho, \mathbf{u}, \mathbf{w}, \mathbf{b})$ satisfies (1.1) almost everywhere in $\mathbb{R}^3\times (0, \infty)$, and belongs to the following class of functions in which the uniqueness can be shown to hold

for any $0 < T < \infty$.

Now, we state our main results as follows. First, we will give the global existence and decay rates of strong solutions of the system (1.1) as follows.

Theorem 1.1. I. Global existence. Suppose that $(\rho_0-1, \mathbf{u}_0, \mathbf{w}_0, \mathbf{b}_0)\in H^3$. Then there exists a positive constant $\varepsilon_0$, depending only on $\|\nabla^2(\rho_0-1, \mathbf{u}_0, \mathbf{w}_0, \mathbf{b}_0)\|_{H^1}$, $\mu_1, \lambda_1, \zeta, \mu_1, \lambda_2, \nu, \gamma$, and $a$, such that the system (1.1)–(1.3) possesses a unique global strong solution $(\rho, \mathbf{u}, \mathbf{w}, \mathbf{b})$ on $\mathbb{R}^3 \times (0, \infty)$ satisfying

for all $t \geqslant 0$, provided

where the positive constant $C$ is independent of $\beta$ and $t$.

II. Decay rates. Assume further that $(\rho_0-1, \mathbf{u}_0, \mathbf{w}_0, \mathbf{b}_0)\in L^1$. Then there exists a positive constant $\varepsilon_1\in (0, \varepsilon_0]$, depending on $\|\nabla^2(\rho_0-1, \mathbf{u}_0, \mathbf{w}_0, \mathbf{b}_0)\|_{H^1}$ and $\|(\rho_0-1, \mathbf{u}_0, \mathbf{w}_0, \mathbf{b}_0)\|_{L^1}$, such that for any $t\geqslant 0$

and that for any $t\geqslant \tilde{T}$ with $\tilde{T} > 0$ being large enough and depending on $\|\nabla^2(\rho_0-1, \mathbf{u}_0, \mathbf{w}_0, \mathbf{b}_0)\|_{H^1}$ and $\|(\rho_0-1, \mathbf{u}_0, \mathbf{w}_0, \mathbf{b}_0)\|_{L^1}$,

provided $\|(\rho_0-1, \mathbf{u}_0, \mathbf{w}_0, \mathbf{b}_0)\|_{H^1} \leqslant \varepsilon_1$.

Remark 1.1. It was worth noting that all of our estimates are uniform in the Hall coefficient $\beta$ in Theorem 1.1.

Remark 1.2. Compared with the decay estimates of the linear system (cf. Lemma 2.1), the decay rates stated in the second part of Theorem 1.1 are optimal, except for the one of $\|\nabla^3(\rho-1, \mathbf{u}, \mathbf{w}, \mathbf{b})\|_{L^2}$. Indeed, if $(\rho_0-1, \mathbf{u}_0, \mathbf{w}_0, \mathbf{b}_0)\in H^m$ with $m\geqslant 4$, then the optimal decay rates of the solutions can be obtained up to the $(m-1)$-th order derivatives of $(\rho-1, \mathbf{u}, \mathbf{w}, \mathbf{b})$ and the $m$-th order derivatives of $\mathbf{b}$. The lack of the optimal decay estimates of $\|\nabla^m(\rho-1, \mathbf{u}, \mathbf{w}, \mathbf{b})\|_{L^2}$ is mainly due to the insufficient dissipation of density and the strong coupling of fluid quantities.

Next, we will show that the unique smooth solution of the 3D compressible micropolar fluids subjected to Hall current converges globally in time to a smooth solution of the 3D compressible magneto-micropolar system as the Hall coefficient $\beta \to 0$.

Theorem 1.2 (Vanishing Hall limit). Suppose that $(\rho^\beta, \mathbf{u}^{\beta}, \mathbf{w}^{\beta}, \mathbf{b}^{\beta})$ and $(\rho^0, \mathbf{u}^{0}, \mathbf{w}^{0}, \mathbf{b}^0)$ are two smooth solutions to Eq (1.1) obtained in Theorem 1.1 corresponding to $\beta > 0$ and $\beta = 0$, respectively. Then for any $T\in (0, \infty)$, it holds that

as $\beta \to 0$. Moreover, there exists a positive constant $c$ depending on $T$ such that

The proofs of Theorems 1.1 and 1.2 are similar to the ones in ^[18,31,37], based on the standard $L^2$-method and the origin ideas developed by Matsumura and Nishida ^[22]. It is worth noting that though the $H^1$-perturbation of initial data is small, the higher-order derivatives could be of large oscillations. Compared with the results in ^[37], where the authors only obtained the optimal decay estimates for the $L^p$-norm $(p\in [2, 6])$ of the solution and the $L^2$-norm of its first-order derivative, the decay rates of both $\|(\nabla^2 \rho, \nabla^2 \mathbf{u}, \nabla^2 \mathbf{w}, \nabla^2 \mathbf{b})(t)\|_{L^2}$ and $\|\nabla^3 \mathbf{b}\|_{L^2}$ for large $t > 0$ are also optimal in the present paper. This will be achieved by making full use of the $H^1$-decay estimates and the Sobolev interpolation inequalities (see (3.81) and (3.82)). The key point here is that all the estimates are uniform in the Hall coefficient $\beta$.

2.   Reformulation

In this section, the global strong solutions near the state $(1, 0, 0, 0)$ to the Cauchy problem (1.1)–(1.3) will be constructed. Define

Then, the quantities $(\varrho, {\mathit{\boldsymbol{v}}}, \omega, {\mathit{\boldsymbol{h}}})$ satisfy

where the functions $R_1, R_2, R_3$, and $R_4$ are defined as

and $f(\varrho), g(\varrho)$, and $h(\varrho)$ given by

The left-hand side of (2.1) is indeed the linearized magneto-micropolar fluid equations. Thus, the $L^p-L^p$ time decay property of the linearized magneto-micropolar fluid equations of (2.1) can be obtained in similar arguments as used in [35, Theorem 2.1]. Thus, we can rewrite the solution of (2.1) as

where $V\triangleq (\varrho, {\mathit{\boldsymbol{v}}}, \omega, {\mathit{\boldsymbol{h}}}), V_0\triangleq(\varrho_0, {\mathit{\boldsymbol{v}}}_0, \omega_0, {\mathit{\boldsymbol{h}}}_0)$, and $\mathcal{L}$ is a matrix-valued differential operator given by

Therefore, due to ^[35], we have the following lemma.

Lemma 2.1. Suppose that $V_0 \in L^1\cap H^3$. Let $V\triangleq V(x, t)$, be the smooth solution of $V_t+\mathcal{L}V = 0$. Then for any $m\in \{0, 1, 2, 3, 4\}$,

where $C > 0$ is a generic constant depending only on $\mu_1, \lambda_1, \zeta, \mu_2, \lambda_2$ and $\nu$.

Lemma 2.1, together with the Duhamel principle, gives rise to the following lemma.

Lemma 2.2. Assume that a quadruple of $(\varrho, {\mathit{\boldsymbol{v}}}, \omega, {\mathit{\boldsymbol{h}}})$ is the smooth solution of (2.1) with the initial data $(\varrho_0, {\mathit{\boldsymbol{v}}}_0, \omega_0, {\mathit{\boldsymbol{h}}}_0)\in L^1 \cap H^3$. Then for any $m\in {0, 1, 2, 3}$,

where $C > 0$ is a generic constant depending only on $\mu_1, \lambda_1, \zeta, \mu_2, \lambda_2$ and $\nu$.

Next, we will give the product and commutator estimates, which can be found in ^[20].

Lemma 2.3. Suppose that $f$ and $g$ are the smooth functions in the Schwartz class. Then, for any $s > 0$ and $1 < p < +\infty$, there exists a generic positive constant $C$ such that

and

where $p_1, p_2 > 1$ satisfying

The following lemma, which can be found in ^[37], is essential for deriving the decay rates.

Lemma 2.4. Suppose that $a, b, c \in \mathbb{R}$ satisfy $a \in [0, b], b\in (1, \infty)$ and $c\in (0, \infty)$. Then, there exists a positive constant $C$, depending only on $a, b$, and $c$, such that for any $t > 0$,

Finally, we recall the local existence theorem of (1.1)–(1.3) (also cf. Eqs (2.1)–(2.3)), which can be proved in a similar way to ^[13].

Lemma 2.5. Suppose that the initial data satisfies

Then there exists a small positive time $T_*$ such that the problem (1.1)–(1.3) possesses a unique classical solution $(\rho, \mathbf{u}, \mathbf{w}, \mathbf{b})$ on $\mathbb{R}^3 \times [0, T_*]$ satisfying

Proof. In order to prove Lemma 2.5, we denote the Banach space

with the form

Let $\tilde{ {\bf{u}}}, \tilde{ {\bf{w}}}, \tilde{ {\mathit{\boldsymbol{b}}}}$ be given, the linear problem of (1.1)–(1.3) can be written as

Let $({\bf{u}}, {\bf{w}}, {\mathit{\boldsymbol{b}}})$ be the unique strong solution to the problem (2.14). We define the fixed point map:

with

We will prove the map $\mathcal{F}$ mapping $\mathcal{B}\times \mathcal{B}\times \mathcal{B}$ into $\mathcal{B}\times \mathcal{B}\times \mathcal{B}$ for suitable constant $K$ and small $T$, and $\mathcal{F}$ is a contraction mapping on $\mathcal{B}\times \mathcal{B}\times \mathcal{B}$, and thus $\mathcal{F}$ has a unique fixed point in $\mathcal{B}\times \mathcal{B}\times \mathcal{B}$. In order to do this, we will divide the proof into five steps.

Step I. For given $\tilde{ {\bf{u}}}\in \mathcal{B}$, we will prove that for some small $0 < T\leqslant 1$, the problems (2.14)$_1$ and (2.14)$_2$ has a unique solution $\rho$ satisfying

here and later on, $C$ will denote a constant independent of $K$.

Since (2.14)$_1$ is linear with regular $\tilde{ {\bf{u}}}$, the existence and uniqueness are well-known. We only need to establish (2.15). In order to prove (2.15), we know from (2.14)$_1$ that

which yields

thus

provided that $KT^{1/2}\leqslant 1$ and $T\leqslant 1$.

In a similar way, we use (2.16) to denote the expression of $\nabla \rho, \Delta \rho$ and $\nabla \Delta \rho$, and then give the estimation of $\nabla \rho$, $\Delta \rho$, and $\nabla \Delta \rho$ as follows:

and

provided that $T^{1/2}K\leqslant 1$ and $T\leqslant1$.

Similarly,

provided that $KT^{1/2}\leqslant 1$ and $T\leqslant1$, and thus (2.15) hold.

Step II. For given $\tilde{ {\bf{u}}}, \tilde{ {\mathit{\boldsymbol{b}}}}\in \mathcal{B}$, we will prove that for some small $0 < T\leqslant 1$, the problem (2.14)$_7$ and (2.14)$_8$ have a unique solution ${\mathit{\boldsymbol{b}}}$ satisfying

Since (2.14)$_7$ is linear with regular $\rho, \tilde{ {\bf{u}}}$, and $\tilde{ {\mathit{\boldsymbol{b}}}}$, the existence and uniqueness are well-known we only need to establish (2.17). In order to prove (2.17), we multiply (2.14)$_7$ by ${\mathit{\boldsymbol{b}}}$ and integrate over $\mathbb{R}^3$, after integration by parts, we have from the Gronwall inequality that

provided that $K^2 T\leqslant 1$.

Applying $\Delta$ to (2.14)$_7$, and then multiplying it by $\Delta {\mathit{\boldsymbol{b}}}$, after integration by parts, we get

We bound $J_1$ as follows:

Using (2.15), we bound $J_2$ as follows:

Due to the Gronwall inequality, one has

provided that $K^4 T+K^2 T\leqslant 1$.

Applying $\nabla \Delta$ to (2.14)$_7$, then multiplying it by $\nabla \Delta {\mathit{\boldsymbol{b}}}$, after integration by parts, we get from Gronwall inequality that

provided that $K^6T+K^4 T+K^2 T\leqslant 1$.

Applying $\partial_t$ to (2.14)$_7$, then multiplying it by $\partial_t {\mathit{\boldsymbol{b}}}$, after integration by parts, we infer that

Using Gronwall's inequality, one has

provided that $K^4 T\leqslant 1$. In the same way, applying $\Delta \partial_t$ to (2.14)$_7$, then multiplying it by $\Delta {\mathit{\boldsymbol{b}}}_t$, after integration by parts, we get from Gronwall's inequality that

provided that $K^6 T\leqslant 1$, which yields (2.17).

Step III. For given $\rho, \tilde{ {\bf{u}}}\in \mathcal{B}$, we will prove that for some small $0 < T\leqslant 1$, the problems (2.14)$_5$ and (2.14)$_6$ have a unique solution ${\bf{w}}$ satisfying

Since (2.14)$_5$ in linear with regular $\rho$ and $\tilde{ {\bf{u}}}$, the existence and uniqueness are well-known, we only need to establish (2.18). In order to prove (2.18), we multiply (2.14)$_5$ by ${\bf{w}}$ and integrate over $\mathbb{R}^3$. After integration by parts, we have

which, yields

Multiplying (2.14)$_5$ by ${\bf{w}}_t$ and integrating over $\mathbb{R}^3$, after integrating by parts, one deduces from (2.15) that

which gives

provided $K^2T\leqslant 1$.

Applying $\partial_t$ to (2.14)$_5$, then multiplying it by $\partial_t {\bf{w}}$ and integrating over $\mathbb{R}^3$, after integrating by parts, one deduce

which gives

provided $K^2T\leqslant 1$.

In a similar way, we deduce from the $\dot{H}^3$-theory of the elliptic system that

and

provided $K^2T\leqslant 1$.

Step IV. For given $\rho, \tilde{ {\bf{u}}}, \tilde{ {\bf{w}}}, \tilde{ {\mathit{\boldsymbol{b}}}}\in \mathcal{B}$, we will prove that for some small $0 < T\leqslant 1$, the problem (2.14)$_3$ and (2.14)$_4$ have a unique solution ${\bf{u}}$ satisfying

Since (2.14)$_3$ is linear with regular $\rho, \tilde{ {\bf{u}}}, \tilde{ {\mathit{\boldsymbol{b}}}}$ and $\tilde{ {\bf{w}}}$, the existence and uniqueness are well-known, we only need to establish (2.19). In order to prove (2.19), we multiply (2.14)$_3$ by ${\bf{u}}$ and integrate over $\mathbb{R}^3$. After integration by parts and taking (2.16)–(2.18) into consideration, we obtain

which, yields

Multiplying (2.14)$_3$ by ${\bf{u}}_t$ and integrating over $\mathbb{R}^3$, after integrating by parts, one deduces

which gives

provided $K^2T\leqslant 1$.

Applying $\partial_t$ to (2.14)$_3$, then multiplying it by $\partial_t {\bf{u}}$ and integrating over $\mathbb{R}^3$, after integrating by parts, one deduces

which gives

provided $K^2T\leqslant 1$.

In a similar way, we deduce from the $\dot{H}^3$-theory of the elliptic system that

and

provided $K^2T\leqslant 1$.

Step V. Due to the above analysis, we can take $K = \max\{C_1, C_2, C_3\}$, and thus $\mathcal{F}$ maps $\mathcal{B}\times \mathcal{B}\times \mathcal{B}$ into $\mathcal{B}\times \mathcal{B}\times \mathcal{B}$. Therefore, in this step, we will prove that $\mathcal{F}$ is contracted in the sense of a weaker norm, that is, there is a constant $\alpha \in (0, 1)$ such that for any $(\tilde{ {\bf{u}}}_i, \tilde{ {\mathit{\boldsymbol{b}}}}_i, \tilde{ {\bf{w}}}_i)\; (i = 1, 2)$ and some small $0 < T\leqslant 1$, the following estimate holds:

In order to obtain (2.20), we suppose that $(\rho_i, {\bf{u}}_i, {\bf{w}}_i, {\mathit{\boldsymbol{b}}}_i)\; (i = 1, 2)$ are the solutions to the problem (2.14) corresponding to $(\tilde{ {\bf{u}}}_i, \tilde{ {\bf{w}}}_i, \tilde{ {\mathit{\boldsymbol{b}}}}_i)$. Denote

Then, we obtain that

Texting (2.21)$_1$ by $\rho$, we obtain that for any $\varepsilon \in (0, 1)$,

Multiplying (2.21)$_4$ by ${\mathit{\boldsymbol{b}}}$, after integration by parts, we see that for any $\varepsilon \in (0, 1)$,

Multiplying (2.21)$_3$ by ${\bf{w}}$ and integrating over $\mathbb{R}^3$, we obtain that for any $\varepsilon \in (0, 1)$ and $\epsilon \in (0, 1)$,

Multiplying (2.21)$_2$ by ${\bf{u}}$ and integrating over $\mathbb{R}^3$, we obtain that for any $\varepsilon \in (0, 1)$ and $\epsilon \in (0, 1)$,

Combining the above inequalities and using the Gronwall inequality, and taking $\varepsilon$ and $\epsilon$ suitably small, we conclude that (2.20) holds true.

Due to the above steps and the Banach fixed point theorem, we finish the proof of Lemma 2.5. □

Lemma 2.6. Let $p\in[2, \tfrac{3s}{3-s}]$ for $s\in[2, 3)$, or $p\in[2, \infty]$ for $s = 3$, and let $q\in(1, \infty), r\in(3, \infty)$. There exists some generic constant $C > 0$ which may depend on $s$ and $r$ such that for $f\in L^2\cap D^{1, s}_0$ and $g\in L^q \cap D^{1, r}_0$, we have

and

3.   Proof of Theorem 1.1

3.1.   Global existence of classical solutions

In this section, we will establish some necessary a priori bounds for smooth solution to the system (2.1)–(2.3) to extend the local classical solution guaranteed by Lemma 2.5. Thus, let $T > 0$ be a fixed time, and $(\varrho, {\mathit{\boldsymbol{v}}}, \omega, {\mathit{\boldsymbol{h}}})$ be the smooth solution to (2.1)–(2.3) on $\mathbb{R}^3 \times [0, T]$ in the class (2.12) with smooth initial data $(\varrho_0, {\mathit{\boldsymbol{v}}}_0, \omega_0, {\mathit{\boldsymbol{h}}}_0)$ satisfying (2.11). To estimate this solution, we make the following a priori assumptions. For any given $L > 1$, (not necessarily small), suppose that

and

where $\alpha$ is a positive constant, depending on $L$, and satisfies

Here $C$ is the Sobolev embedding constant of (3.8).

We have the following key a priori estimates on $(\varrho, {\mathit{\boldsymbol{v}}}, \omega, {\mathit{\boldsymbol{h}}})$.

Proposition 3.1. For any given positive constant $M > 0$ (not necessarily small), suppose that

Then there exists a positive constant $\varepsilon$, depending only on $\mu_1, \lambda_1, \zeta, \mu_2, \lambda_2, \nu$ and $M$, such that for any $t > 0$, the system (2.1) possesses a unique global classical solution $(\varrho, {\mathit{\boldsymbol{v}}}, \omega, {\mathit{\boldsymbol{h}}})\in \mathbb{R}^3 \times [0, \infty)$ satisfying

and

provided

In order to prove Theorem 1.1, it was suffices to prove Proposition 3.1. However, the assumptions of (3.1) and (3.2) are crucial to prove Proposition 3.1. Therefore, our main aim in the next is to close the a priori assumptions (3.1) and (3.2). Obviously, we can infer from (3.1)–(3.3) and the Sobolev embedding inequality (cf. ^[1]) that

which implies that

moreover

and

for any $m\geqslant 1$. We now begin to derive a series of a priori estimates.

Lemma 3.1. Let the assumptions (3.1) and (3.2) be in force. Then

Proof. Multiplying (2.1)$_1$–(2.1)$_4$ by $a\gamma \varrho, {\mathit{\boldsymbol{v}}}, \omega$, and ${\mathit{\boldsymbol{h}}}$ in $L^2$, respectively, and integrating by parts, we obtain after adding them together that

To deal with the right-hand side of (3.13), we notice from (2.2) and (3.2) that

By using (3.1), (3.2), (3.9)–(3.11), and the Sobolev inequality (2.22), we deduce from (2.2) and the integration by parts that

where we have used the fact that $\alpha \in (0, 1)$ and $L > 1$. According to the observation of (3.13), one has

thus, by using a similar manner to (3.15) and choosing $\delta_1 = \min\{\tfrac{\zeta}{C}, \tfrac{\mu_1}{4C}\}$ in (3.16), we can deduce from the above inequality that

and

Substituting (3.14)–(3.17) into (3.13) immediately yields (3.12). □

Lemma 3.2. Let the assumptions (3.1) and (3.2) be in force. Then

Proof. Operating $\nabla$ to (2.1)$_1$–(2.1)$_4$, multiplying them by $a\gamma \nabla \varrho$, $\nabla {\mathit{\boldsymbol{v}}}$, $\nabla \omega$, and $\nabla {\mathit{\boldsymbol{h}}}$ in $L^2$, respectively, and integrating by parts, we obtain that

It follows from (3.1), (3.2), and the integration by parts that

Due to the Sobolev inequalities (2.22), (2.23), (3.10), and (3.11), it can be obtained by direct calculation that

thus, using (3.21), one has

Putting (3.20) and (3.22) into (3.19), we obtain (3.18). □

Remark 3.1. Obviously, we can infer from (3.20) that

Similarly,

which, combined with (3.19) and (3.23), yields

which will be used later to close the estimate of $\|\nabla (\varrho, {\mathit{\boldsymbol{v}}}, \omega, {\mathit{\boldsymbol{h}}})\|_{L^2}^2$.

Lemma 3.3. Let the assumptions (3.1) and (3.2) be in force. Then

Proof. Multiplying (2.1)$_2$ by $\nabla \varrho$ in $L^2$, one has

Thanks to (2.1)$_1$, we deduce from integration by parts that

and

The Cauchy-Schwarz inequality, together with (3.21), gives

Substituting (3.27)–(3.29) into (3.26) yields (3.25). □

Lemma 3.4. Let the assumptions (3.1) and (3.2) be in force. Then

Proof. Similar to the proof of Lemma 3.2, we infer from (2.1) that

Keeping in mind that $L > 1$, thus, the inequalities (2.7) and (2.8), together with Gagliardo-Nirenberg inequality (cf. ^[1]) show that

Due to the Sobolev inequalities (2.22) and (2.23), one has

Similar to the derivation of (3.33), we can obtain that

The combination of (3.8)–(3.11) and the interpolation inequality shows that

and

where $\delta_2$ is an undetermined positive constant. According to the observation of (3.31), we know that

thus, putting (3.37) into (3.36) and choosing $\delta_2 = \{\tfrac{\zeta}{C}, \tfrac{\mu_1}{4C}\}$, one has

Similar to the derivation of (3.35) and (3.38), one has

and

The combination of (3.33)–(3.35) and (3.38)–(3.41), gives

Therefore, substituting (3.32) and (3.42) into (3.31), we can obtain (3.30) by using Cauchy-Schwarz inequality. □

Lemma 3.5. Let the assumptions (3.1) and (3.2) be in force. Then

Proof. Operating ${\rm{div}}$ to (2.1)$_2$ and multiplying the resulting equation by $\Delta \varrho$ in $L^2$, after integrating by parts, one has from $\nabla \cdot (\nabla \times \omega) = 0$ that

Due to (2.1)$_1$, one has

On the other hand, it follows from (3.42) that

which, together with (3.44) and (3.45), gives (3.43). □

Lemma 3.6. Let the assumptions (3.1) and (3.2) be in force. Then

Proof. Similar to the proof of Lemma 3.4, we infer from (2.1) that

Keeping in mind that $L > 1$, thus, Lemma 2.3, together with Gagliardo-Nirenberg inequality (cf. ^[1]) shows that

Due to the Sobolev interpolation inequalities (2.22) and (2.23) (cf. ^[9,14]), one has from (2.7) that

Similar to the derivation of (3.49), we can get

The combination of (3.8)–(3.11) and the interpolation inequality shows that

and

where $\delta_3$ is an undetermined positive constant. According to the observation of (3.47), we know that

thus, putting (3.53) into (3.52) and choosing $\delta_3 = \{\tfrac{\zeta}{C}, \tfrac{\mu_1}{4C}\}$, one has

Similar to the derivation of (3.51) and (3.54), one has from (2.7) that

and

and

The combination of (3.49)–(3.51) and (3.54)–(3.57) yields

which, together with (3.47) and (3.48), gives (3.46). □

Lemma 3.7. Let the assumptions (3.1) and (3.2) be in force. Then

Proof. Operating $\nabla{\rm{div}}$ to (2.1)$_2$ and multiplying the resulting equation by $\nabla \Delta \varrho$ in $L^2$, after integrating by parts, one has from $\nabla \cdot (\nabla \times \omega) = 0$ that

Due to (2.1)$_1$, one has

On the other hand, it follows from (3.58) that

which, together with (3.60) and (3.61), gives (3.59). □

With all the a priori estimates obtained in Lemmas 3.1–3.7 at hand, we are ready to prove Proposition 3.1 next.

Proof of Proposition 3.1. Due to the inequalities of (3.12), (3.18), and (3.25), one has

and

provided $\alpha > 0$ is chosen to be small enough such that

Therefore, we infer from (3.62) that

which, multiply $M_1\triangleq \max\{4, 8C_2\}$ and added to (3.63), gives

where $C_3(M_1)$ is a positive number depending on $M_1$. Then, if $\alpha > 0$ is chosen to be small such that

then, we have

Integrating the above inequality over $[0, T]$, one has

moreover,

It follows from (3.24), (3.30), and (3.46) that

thus

provided $\alpha > 0$ is chosen to be small enough such that

Next, we infer from (3.43) and (3.59) that

then

provided $\alpha > 0$ is chosen to be small enough such that

Multiplying (3.65) by $M_2\triangleq\{8, 8C_5\}$, and adding the resulting inequality to (3.66), after integrating the resulting inequality over $[0, T]$, one has

Taking $L^2\triangleq 4\hat{C} \| \nabla (\varrho_0, {\mathit{\boldsymbol{v}}}_0, \omega_0, {\mathit{\boldsymbol{h}}}_0)\|_{H^2}^2$ in (3.67), and choosing $\varepsilon > 0$ sufficiently small such that $4\check{C} \| (\varrho_0, {\mathit{\boldsymbol{v}}}_0, \omega_0, {\mathit{\boldsymbol{h}}}_0)\|_{H^1}^2\leqslant \alpha^2$ in (3.64), then, we can close the a priori assumptions (3.1) and (3.2) by bootstrap arguments. This, together with the local existence result (cf. Lemma 2.5), finishes the proof of Proposition 3.1, and thus, the proof of the first part of Theorem 1.1 is complete. □

3.2.   Decay rates

In this subsection, our main aim is to derive the decay rates of the solutions $(\varrho, {\mathit{\boldsymbol{v}}}, \omega, {\mathit{\boldsymbol{h}}})$ obtained in the first part of Theorem 1.1.

Lemma 3.8. Let the conditions of Proposition 3.1 be in force. Assume that $\varepsilon > 0$ is small enough and $\|(\varrho_0, {\mathit{\boldsymbol{v}}}_0, \omega_0, {\mathit{\boldsymbol{h}}}_0)\|_{L^1}$ is bounded. Then for any $t\geqslant 0$,

and

Proof. Similar to the derivation of (3.64), we can infer from (3.24), (3.30), (3.43), (3.46), and (3.59) that there exist some positive constants $\tilde{M}$ and $\tilde{c}$ such that if $\varepsilon > 0$ is small enough, then

where

Therefore

Due to Cauchy-Schwarz inequality, it holds that

thus

For suitable large number $\tilde{M}\geqslant C_6+1 > 0$, then

then, we can infer from (3.70) that there exists a positive constant $c$ such that

thus

Thanks to (2.6), we have

The Cauchy-Schwarz inequality, together with (3.4)–(3.7), and (3.8)–(3.11), shows that

and similarly,

Substituting (3.73) and (3.74) into (3.72), one has from (3.4) that

Set

So that, the inequality (2.10), together with (3.75), gives

which, combined with (3.71), shows that

Thus, if $\varepsilon > 0$ is small enough, then we obtain that $\mathcal{F}(t)\leqslant C$. This, together with $\mathcal{A}(t)\sim \|\nabla(\varrho, {\mathit{\boldsymbol{v}}}, \omega, {\mathit{\boldsymbol{h}}})\|_{H^2}^2$, yields (3.68).

In order to prove (3.69), we use (2.6), (2.10), (3.68), (3.73), and (3.74) to obtain that

which implies the desired estimate (3.69). Therefore, we complete the proof of Lemma 3.8. □

Compared with (2.5) in Lemma 2.1, the decay rates of the $H^1$-norm of solutions stated in Lemma 3.8 are optimal. Next, our main aim is to show the decay estimates of higher derivatives.

Lemma 3.9. Let the conditions of Proposition 3.1 be in force. Then there exists a positive time $T_1$ such that if $\varepsilon > 0$ is small enough, the following estimate holds.

for any $t\geqslant T_1$.

Proof. In terms of (2.7)–(2.9) in Lemma 2.3, one has from (2.2) and (3.6) that

Due to Proposition 3.1 and Lemma 3.8, one has

Taking (3.78) and (3.79) into (3.31), one has from integration by parts that

which, together with (3.46) and (3.59), gives that there exist some positive constants $\bar{M}$ (suitably large) and $\bar{c}$ (suitably small) such that if $\varepsilon > 0$ is small enough, then

where

The combination of the Sobolev interpolation inequalities (2.22) and (2.23) and Cauchy-Schwarz inequality shows

where $\delta > 0$ is a positive number to be chosen later. Then

Similarly,

Putting (3.81) and (3.82) into (3.80), one has

If $t\geqslant \delta > 0$, then $\|\nabla^3 \varrho\|_{L^2}^2\geqslant \delta (1+t)^{-1}\|\nabla^3 \varrho\|_{L^2}^2$, so that, we infer from (3.83) that

Furthermore, due to $\mathcal{A}_1^{\prime}(t)\sim \|\nabla^2(\varrho, {\mathit{\boldsymbol{v}}}, \omega, {\mathit{\boldsymbol{h}}})\|_{H^1}^2$ for suitable large $\bar{M}$, then there exists a positive constant $c_1$ depending only on $\bar{M}$ and $\bar{c}$, such that

If $\delta = 4c_1^{-1}$, then it follows from (3.85) that

thus

which, integrated over $(0, t)$, gives

provided $t > 0$ is large enough such that

Dur to $\mathcal{A}_1^{\prime}(t)\sim \|\nabla^2(\varrho, {\mathit{\boldsymbol{v}}}, \omega, {\mathit{\boldsymbol{h}}})\|_{H^1}^2$, thus, we can obtain (3.77) from (3.86). Therefore, we complete the proof of Lemma 3.9. □

Lemma 3.10. Let the conditions of Proposition 3.1 be in force. Then there exists a positive time $T_2$ such that if $\varepsilon > 0$ is small enough, the following estimate holds

for any $t\geqslant T_2$.

Proof. Operating $\nabla^2$ to (2.1)$_4$, multiplying it by $\nabla \Delta {\mathit{\boldsymbol{h}}}$ in $L^2$, and integrating by parts, we infer from Cauchy-Schwarz inequality that

The right-hand side terms of (3.88) can be estimated as follows. Due to (2.7)–(2.9), (3.48), we infer from (3.68), (3.69), and (3.77) that

and

Substituting (3.89) and (3.90) into (3.88), we obtain that

Similar to the derivation of (3.81), one has from (3.77) that

which, together with (3.91), yields

Therefore, we deduce from (3.92) that

which, implies

provided

Thus, we can obtain (3.87) from (3.93). Therefore, we complete the proof of Lemma 3.10. □

Proof of decay rates. Collecting Lemmas 3.8–3.10 together, one immediately obtains the desired decay rates stated in the section part of Theorem 1.1. □

4.   Proof of Theorem 1.2

In order to prove Theorem 1.2, we first consider the standard MHD equations without Hall effects (i.e., $\beta = 0$) as follows:

with far-field boundary conditions and initial conditions:

where $p^0 = p(\rho^0) = a (\rho^0)^\gamma$.

According to the observation, we know that all of the global estimates and decay rates established in Section 3 hold for the system (4.1)-(4.2). Therefore, we give the following global existence result for the problem (4.1)-(4.2).

Proposition 4.1. Assume that the initial data $(\rho_0, \mathbf{u}_0, \mathbf{w}_0, \mathbf{b}_0)$ satisfy

for any given constants $L_0$ (not necessary small). Then, there exists a positive constant $\bar{\varepsilon}_0$ depending on $L_0$, $\mu_1, \lambda_1, \zeta, \mu_2, \lambda_2, \nu, a$, and $\gamma$ such that if

then the Cauchy problem (4.1)-(4.2) has a unique global classical solution $(\rho^0, \mathit{{\bf{u}}}^0, {\bf{w}}^0, {\mathit{\boldsymbol{b}}}^0)$ on $\mathbb{R}^3 \times (0, \infty)$ satisfying

for all $t \geqslant 0$, where $C$ is positive constant independent of $t$.

In order to prove the convergence rates stated in Theorem 1.2, we define

where $(\rho, {\bf{u}}, {\bf{w}}, {\mathit{\boldsymbol{b}}})$ is the solution of the problem (1.1)–(1.3), and $(\rho^0, {\bf{u}}^0, {\bf{w}}^0, {\mathit{\boldsymbol{b}}}^0)$ is the solution of problem (4.1)-(4.2). Then, the quadruple $(\pi, {\mathit{\boldsymbol{v}}}, \varpi, {\mathit{\boldsymbol{h}}})$ satisfies

Lemma 4.1. Let $(\pi, {\mathit{\boldsymbol{v}}}, \varpi, {\mathit{\boldsymbol{h}}})$ be the solution of system (4.6). There exists a positive constant $C = C(T)$, independent of $\beta$, such that for any $T\in (0, \infty)$, then

Proof. Multiplying (4.6)$_1$ by $\pi$ in $L^2$ and integrating by parts, we infer from (1.4), (4.5), and Cauchy-Schwarz inequality that

With the aid of (1.4) and (4.5), we can easily deduce from (4.6) that

Since it holds that

thus, multiplying (4.6)$_2$ by ${\mathit{\boldsymbol{v}}}$ in $L^2$ and integrating by parts, we infer from (1.4), (4.5), (4.9), and (4.10) that

Similarly, multiplying (4.6)$_3$ by $\varrho$ in $L^2$ and integrating by parts, one can deduce from (1.4), (4.5) and (4.9) that

Multiplying (4.6)$_4$ by ${\mathit{\boldsymbol{h}}}$ and integrating by parts, we can obtain from (1.4), (4.5) and (4.9) that

Multiplying (4.11) by a suitably large constant and adding the result to (4.8), we obtain from (4.12) and (4.13) that

which, together with the Gronwall inequality and $\rho > 0$, gives rise to (4.7). □

Lemma 4.2. Let $(\pi, {\mathit{\boldsymbol{v}}}, \varpi, {\mathit{\boldsymbol{h}}})$ be the solution of system (4.6). There exists a positive constant $C = C(T)$, independent of $\beta$, such that for any $T\in (0, \infty)$, then

Proof. Operating $\nabla$ on both sides of (4.6)$_1$, and multiplying it by $\nabla \pi$ in $L^2$, after integrating by parts, we deduce from (4.7) and the Cauchy-Schwarz inequality that

where $\sigma$ is an undetermined positive constant.

Multiplying (4.6)$_2$ by ${\mathit{\boldsymbol{v}}}_t$ in $L^2$ and integrating by parts, then, by virtue of (1.4), (4.5), (4.7), and (4.9) that

Multiplying (4.6)$_3$ by $\varpi_t$ in $L^2$ and integrating by parts, then, by virtue of (1.4), (4.5), (4.7), and (4.9) that

Similarly,

Due to (3.9), we know that $\rho$ is strictly lower-bounded. Thus, it follows from (4.15)–(4.18) that

The combination of (1.4), (4.5)–(4.7), and (4.9) gives

Multiplying (4.19) by $(C_9+1)$ and choosing $\sigma = (C_9+1)/2$ in (4.19), adding the resulting inequality to (4.20), one has

which, together with the Gronwall inequality, yields (4.14). □

Lemma 4.3. Let $(\pi, {\mathit{\boldsymbol{v}}}, \varpi, {\mathit{\boldsymbol{h}}})$ be the solution of system (4.6). There exists a positive constant $C = C(T)$, independent of $\beta$, such that for any $T\in (0, \infty)$, then

Proof. Due to (1.4), (2.7)–(2.9), (4.7), and (4.14), we deduce from (4.6)$_1$, Cauchy-Schwarz inequality, and the Sobolev inequality (2.22) that

Thanks to (4.6)$_2$, we have

Operating $\nabla^2$ on both sides of (4.23), multiplying the resulting equation by $\nabla^2 {\mathit{\boldsymbol{v}}}$ in $L^2$, and integrating by parts, we get from (2.7)–(2.9) and (4.7)–(4.14) that

Due to (4.6)$_3$, one has

Operating $\nabla^2$ on both sides of (4.25), multiplying the resulting equation by $\nabla^2 \varpi$ in $L^2$, and integrating by parts, we get from (2.7)–(2.9) and (4.7)–(4.14) that

Operating $\nabla^2$ on both sides of (4.6)$_4$, multiplying the resulting equation by $\nabla^2 {\mathit{\boldsymbol{h}}}$ in $L^2$, and integrating by parts, we get from (2.7)–(2.9) and (4.7)–(4.14) that

The combination of (4.22), (4.24), (4.26), and (4.27) gives that

which, together with the Gronwall inequality, yields (4.21). □

Proof of Theorem 1.2. Now, the convergence rates of the vanishing limit of the Hall coefficient stated in Theorem 1.2 readily follows from Lemmas 4.1–4.3. □

5.   Conclusions

This paper is concerned with the Cauchy problem of the compressible magneto-micropolar fluids subjected to Hall current in three-dimensional whole space. Both the global existence and optimal decay rates of strong solutions are obtained when the smooth initial data are sufficiently close to the non-vacuum equilibrium in $H^1$. As a by-product of the uniform estimates, the vanishing limit of the Hall coefficient is also justified. We refer to Theorems 1.1 and 1.2 for details.

Acknowledgments

This project is partially supported by Shandong Provincial Natural Science Foundation (Grant No. ZR2024MA033) and the Science and Technology Project of Weifang (2022GX006).

The author is very grateful to the experts who gave some guidance and suggestions during the writing process of this paper, and also to the anonymous reviewers for their comments on this paper, which greatly improved the quality of this paper.

Conflict of interest

The author declares that she has no conflict of interest.