Global stability solution of the 2D incompressible anisotropic magneto-micropolar fluid equations

1.   Introduction

The incompressible magneto-micropolar fluid equations describes the motion of an incompressible conducting micropolar fluid in an arbitrary magnetic field. In this paper, we consider the 2D incompressible anisotropic magneto-micropolar fluid equations,

where $u = (u_1, u_2)$, $b = (b_1, b_2)$, $\nabla^{\perp} m = (- \partial_2m, \partial_1m)$. Also $m$ and $P$ are the scalars. The nonnegative parameters $\mu > 0$, $\nu > 0$ and $\chi > 0$ denote the kinematic viscosity, magnetic diffusion coefficient and the dynamic micro-rotation viscosity. Besides, $\gamma$ and $\kappa$ are the angular viscosities. The operators $\partial_1$, $\partial_2$ represent the horizontal and vertical direction respectively.

Our goal is to investigate the stability problem on the perturbation $(u, b, m)$ near the steady solution $\big(u^{0}, B^{0}, m^{0}\big)$ with $b = B-B^0$. Here

Without loss of generality, set $\mu = \chi = \frac12$ and $\nu = \kappa = 1$. It is easy to verify that $(u, b, m)$ satisfies

The standard magneto-micropolar fluid equations with full velocity field dissipation, magnetic diffusion and angular viscosities can be written as

Because of the mathematically significant, the magneto-micropolar fluid equations and closely related equations have attracted considerable attentions for mathematical scholars and many important results have been achieved. Major works mainly concentrated on the global well-posedness and global regularity of the solution. Let's recall some of these results.

For the 2D incompressible magneto-micropolar equations, Yuan and Qiao ^[1] established the global smooth solution for the equations with zero angular viscosity and zero magnetic diffusion or with only angular viscosity and magnetic diffusion. In a two dimensional bounded domain with Navier type boundary condition for the velocity, Fan and Zhou ^[2] proved the existence and uniqueness of global strong solutions to the incompressible magneto-micropolar system. Ma in ^[3] obtained the global existence and regularity of classical solutions to the equations with mixed partial dissipation, magnetic diffusion and angular viscosity. In addition, some conditional regularity of strong solutions also be obtained. Guo and Shang in ^[4] showed the global regularity of solutions to the 2D incompressible magneto-micropolar equations with partial dissipation. For $2{\frac {1} {2}}$ dimensional system, the results of the global well-posedness for the incompressible magneto-micropolar fluid equations with mixed partial dissipation have been obtained (see e.g., ^[5,6]). Besides, for 3D case, the global existence results for the Cauchy problem in ${\mathbb{R}}^3$ are obtained by Tan and Wu in ^[7]. The global well-posedness and global regularity of the incompressible magneto-micropolar system have been studied in ^[8,9]. For more results, we refer to ^{[7,10,11,12,13,14]} and references therein.

However, there are few results to our knowledge on the large-time behavior of the magneto-micropolar fluids. Shang and Gu ^[15] obtained the $L^2$-decay estimates of solutions for the two-dimensional incompressible magneto-micropolar fluid equations, which is $\|u(t)\|_{L^2}+\|w(t)\|_{L^2}\leq C(1+t)^{-\frac {4} {3}}$ and $\|b\|_{L^2}\leq C(1+t)^{-\frac {1} {2}}$. Moreover, by proving the optimal decay for $\|b(t)\|_{L^\infty}$, the authors optimized the decay rates to $\|u(t)\|_{L^2}+\|w(t)\|_{L^2}\leq Ct^{-2}$ in ^[15]. In this paper, we will show decay rates for the linear system (1.2) in the next section.

If the magnetic field $B = 0$, the equations (1.3) become the micropolar fluid equations. For the 2D incompressible micropolar equations, Liu in ^[16] studied the global well-posedness to the Cauchy problem of 2D micropolar equations with large initial data and vacuum, and showed that the problem admits a unique global strong solution. Ye ^[17] studied the global regularity for the system of the 2D incompressible micropolar equations with vertical dissipation in the horizontal velocity equation, horizontal dissipation in the vertical velocity equation. Dong and Li ^[18] studied the global regularity in time and large time behavior of solutions to the 2D micropolar equations with only angular viscosity dissipation. The more results of the well-posedness, regularity and large time decay problems on the micropolar fluid equations can be shown in ^{[19,20,21,22]}. On the other hand, if $\chi = 0$ and $m = 0$, the equations in (1.3) reduce to the magneto-hydrodynamic equations (MHD). The case of full dissipation and magnetic diffusion, the classical solution is global (see e.g., ^[23]). There are numerous works on the global regularity and the stability. One of the significant works on the global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion is completed by Wu and Cao in ^[24]. One can also refer to ^{[25,26,27,28,29]} and so on.

Motivated by the results of the magneto-micropolar fluid equations and closely related equations, this paper investigates the stability of the solution of the system (1.2). We attempt to achieve two main goals. The first is to give the linear asymptotic stability, which is equivalent to assessing the small data global well-posedness. We need to consider the corresponding linear system of (1.2) to illustrate it,

With some assumptions on the initial data, we establish explicit decay rates of the solutions for the linear system (1.4). To give decay rates, we define the fractional operator $\Lambda^\alpha f$ via the Fourier transform,

Our first result is as follows.

Theorem 1.1. For any $s\ge0$, let the initial data $(u_0, b_0, m_0)\in {H}^s(\mathbb R^2)$ with $\nabla\cdot u_0 = \nabla\cdot b_0 = 0$. Suppose that $(u, b, m)$ is the solution of the linear system (1.4).

1. Assume that $({\nabla} u_0, {\nabla} b_0, {\nabla} m_0) \in {H}^s(\mathbb R^2)$. Then the decay rates holds

2. Suppose $(\Lambda_1^{-\sigma}u_0, \Lambda_1^{-\sigma}b_0, \Lambda_1^{-\sigma}m_0), (\Lambda_2^{-\sigma}u_0, \Lambda_2^{-\sigma}b_0, \Lambda_2^{-\sigma}m_0) \in H^s(\mathbb{R}^2)$, where $\sigma > 0$ is a real number. Then $(u, b, m)$ satisfies

The second goal is to prove the stability of the nonlinear system in (1.2).

Theorem 1.2. Suppose that $(u_0, b_0, m_0)\in H^2(\mathbb{R}^2)$ with $\nabla\cdot u_0 = \nabla\cdot b_0 = 0$. Let $(u, b, m)$ be the solution of the nonlinear system (1.2). Then there exists $\delta > 0$, such that if

then (1.2) possesses a unique global solution $(u, b, m)\in C(0, \infty; H^2({\mathbb{R}}^2))$ satisfying

for all $t\ge0$, where $C$ is a pure constant.

By the technology of bootstrapping argument (see [30,p.21]) and the energy method, we are able to obtain the nonlinear stability. To prove Theorem 1.1, we first introduce the energy $E(t)$ as follows

for any $t\geq0$. Our efforts concentrate on establishing the a priori estimate of $E(t)$ in Section 3,

Then the bootstrapping argument implies the global bound and also the stability.

This paper is organized as follows. In section 2, we give the proof of decay rates in Theorem 1.1. In section 3, by employing the energy method and using the bootstrapping argument, we establish the $H^2$-estimate and then complete the proof of Theorem 1.2.

2.   Proof of Theorem 1.1

In this section, we will show the decay rates in $H^s$ for the solutions based on the linearized system (1.2). Under the different assumptions on the initial data, we establish the asymptotic stability for the linear system. Before stating our results in (1.5), we first give a tool which will be used in the proof of (1.5).

Lemma 2.1. Let $f = f(t)$ be a nonnegative continuous function satisfying, for two constants $a_0 > 0$ and $a_1 > 0$,

Then, for any $t > 0$, for $a_2 = \max\{2a_0 f(0), 2a_1 a_0\}$,

The tool of Lemma 2.1 (see^[25]) will be used to establish the decay rate (1.5). It indicates that generalized monotone nonnegative integrable functions have a precise decay rate.

2.1.   Proof of (1.5)

Proof of (1.5). We show the proof of $s = 0$, then by the iterate, we can derive the case of $s > 0$. First of all, we consider the first condition of monotonous. Taking the $L^2$-inner product of (1.4) with $(\Delta u, \Delta b, \Delta m)$, we have

where we used the fact that

We denote the $f(t)$ as

Thus (2.3) implies that

for any $s < t$.

As a result, we prove the first condition. Next we verify the second condition that is $\int_0^\infty f(t)dt \leq C$. First, we have the $H^1$-estimates,

By integration by parts and Hölder's inequality, we infer

thus, it implies

Combining (2.6), we can deduce

Besides, from (2.6), we can infer

Now it suffices to prove the integrability for $\|\partial_{2}b\|_{L^{2}}^{2}$ on the time. Dotting $\partial_2b$ to the velocity equation of (1.4), integrating over ${\mathbb{R}}^2$, then replacing $\partial_tb$ by the other terms of the magnetic equation, we obtain

The four integral terms above can be estimated as

where we used the Hölder's inequality and Young's inequality. Inserting (2.12) into (2.11) and integrating it on $[0, t]$, we have

Then, adding $\lambda\times$ (2.13) into (2.6), where $\lambda > 0$ is a small number, we have

From (2.14), we infer

Collecting (2.9), (2.10) and (2.15), it suffices to verify $f(t)$ satisfying the second condition which is nonnegative integrable. As a consequence, by Lemma 2.1 we conclude,

where $C$ is a constant and (2.16) gives the desired result (2.2). Thus, we conclude the proof of (1.5).

2.2.   Proof of (1.6)

Proof of (1.6). Using the consideration of the itera again, we only prove the case of $s = 0$ similarly to the (1.5). We have the $H^1$-estimate as follows,

where

and

Applying $\Lambda_1^{-\sigma}$ and $\Lambda_2^{-\sigma}$ to the linear system (1.4) respectively, and dotting the correspending equations with $(\Lambda_1^{-\sigma}u_1, \Lambda_1^{-\sigma}u_2, \Lambda_1^{-\sigma}b, \Lambda_1^{-\sigma}m)$ and $(\Lambda_2^{-\sigma}u_1, \Lambda_2^{-\sigma}u_2, \Lambda_2^{-\sigma}b, \Lambda_2^{-\sigma}m)$, then integrating over ${\mathbb{R}}^2$, we have

and,

Combining (2.17) and (2.18), then integrating by parts, it infers that

which indicates that

Then the estimate of $\|u_1\|_{L^2}$ follows from the Plancherel's identity and Hölder's inequality, which can be written as

Similarly,

and

Collecting (2.20), (2.22), (2.23) and (2.19), we have

Then we infer that

Which immediately leads to

Thus, we finish the proof of Theorem 1.1.

3.   Proof of Theorem 1.2

This section proves the stability of the nonlinear system (1.2). By exploiting the methods of bootstrapping and energy method, we establish the $H^2$-estimate. Before our proof, we give two useful tools. The first provides an anisotropic inequality for the integral of triple product and the proof can be found in ^[24]. The second shows a basic fact.

Lemma 3.1. Suppose that $f$, $g$, $\partial_2 g$, $h$ and $\partial_1 h$ are all in $L^2(\mathbb{R}^2)$. Then, for some constant $C > 0$,

Lemma 3.2. Due to $\nabla\cdot u = 0$, we have the fact that

The key step in the proof is to deal with the nonlinear and coupled terms. Therefore, we will take full use of the Lemma 3.1. Combining Sobolev inequality, Hölder's inequality and Young's inequality, the closed priori estimate of the energy $E(t)$ can be established.

Proof. This section aims to obtain the $H^2$-estimate. Since the equivalent norms,

it suffices to make the estimate on $\parallel(u_{1}, u_{2}, b, m) \parallel^{2}_{L^{2}}$ and $\parallel(\partial_{i}^{2}u_{1}, \partial_{i}^{2}u_{2}, \partial_{i}^{2}b, \partial_{i}^{2}m) \parallel^{2}_{L^{2}}$ respectively.

First of all, by taking $L^2$-inner product of (1.2) with $(u, b, m)$ and using integration by parts, it easily infers

Next, it suffices to estimate $\|(u_{1}, u_{2}, b, m) \|^{2}_{\dot H^{2}}$. Applying $\partial_{1}^{2}$ and $\partial_{2}^{2}$ to every equation in (1.2) respectively, then taking the $L^2$-inner product with $(\partial_{i}^{2}u_{1}, \partial_{i}^{2}u_{2}, \partial_{i}^{2}b, \partial_{i}^{2}m)$, and integrating them on $[0, t]$, we have

They can be written as follows respectively,

where we have used the facts that

and

Then we estimate $I_{1}+I_{2}\cdots+I_{7}$ one by one. In the following calculations, the Hölder's inequality, Young's inequality and Sobolev embedding inequality will be applied frequently. By using the Lemma (3.2), $I_{1}$ and $I_{3}$ can be estimated together,

The term $I_2$ can be transformed into four terms,

For $I_{21}$ and $I_{22}$, we have

When we estimate $I_{23}$ and $I_{24}$, the incompressible condition $\nabla\cdot b = 0$ will be used,

Collecting the bounds for $I_2$, we have

For $I_{4}$, due to $\nabla\cdot u = \nabla\cdot b = 0$, it can be estimated as follows,

Now we think about $I_{5}$, which can be rewritten as following four parts,

$I_{51}$ and $I_{52}$ can be estimated easily. By Sobolev embedding inequality, we have

To deal with $I_{53}$ and $I_{54}$, writing them into four terms,

Using the Hölder's inequality and Young's inequality, $J_{1}$ and $J_{3}$ can be estimated as follows,

Integrating by parts and using Lemma 3.1, we obtain

Inserting (3.20) and (3.21) into (3.19),

Collecting (3.18) and (3.22), we infer

Similarly to $I_{5}$, $I_{6}$ can be written as

For $I_{61}$ and $I_{62}$, by the Sobolev embedding inequality, we obtain

$I_{63}$ and $I_{64}$ can be divided into six terms,

Estimating $K_{2}$, $K_{3}$, $K_{5}$ and $K_{6}$ together,

Considering $K_{1}$, integrating it by parts,

Similarly to $K_{1}$, and using the Lemma 3.1 again,

Hence,

Consequently, combining (3.25) and (3.30), we have

Now we concerning the last term $I_7$, which can be rewritten as follows,

First, by combining the Hölder's inequality and Young's inequality, $I_{71}$ becomes

We can infer $I_{72}$,

Therefore, $I_7$ is estimated as

Collecting all the estimate $I_{1}+I_{2}\cdots+I_{7}$ and inserting them into (3.5), we deduce

Then we finished the estimate of $\parallel(\partial_{i}^{2}u_{1}, \partial_{i}^{2}u_{2}, \partial_{i}^{2}b, \partial_{i}^{2}m) \parallel^{2}_{L^{2}}$. Combining (3.4) and (3.36), then integrating the resulted equation on $[0, t]$, we conclude

which indicates the desired estimate

Thus this completes the proof of (1.8).

Proof of Theorem 1.2. We have the energy inequality, namely

where $C$ is a pure constant. Due to the assumption that $\|(u_0, b_0, m_0)\|_{H^2({\mathbb{R}}^2)}\le\delta$ is sufficiently small, such that

To initiate the bootstrapping argument to the energy inequality, we make the ansatz

It then implies that

Substituting (3.43) into (3.40) and combining with (3.41), we obtain

Then we have obtained that $E(t)$ actually admits an smaller upper bound, which is

By the bootstrapping argument, this completes the proof of Theorem 1.2.

4.   Conclusions

In this paper, the stability of the 2D incompressible anisotropic magneto-micropolar fluid equations near a background magnetic field with partial mixed velocity dissipations, magnetic diffusion and horizontal vortex viscosity is considered. We obtained the explicit decay rates for the solution of the linear system in $H^s({\mathbb{R}}^2)$ Sobolev space and the stability of nonlinear system. And the results reveal that the background magnetic field can stabilize the electrically conducting fluids.

Acknowledgments

We would like to thank the reviewers for their careful reading of our paper and for their insightful comments and suggestions. Also we would like to express sincere gratitude to Professor Hongxia Lin.

Conflict of interest

The authors declare that there are no conflicts of interest.