On quasi-monoidal comonads and their corepresentations

Dingguo Wang; Xiaohui Zhang; Dingguo Wang; Xiaohui Zhang

doi:10.3934/era.2022160

Electronic Research Archive

2022, Volume 30, Issue 8: 3153-3171. doi: 10.3934/era.2022160

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Research article

On quasi-monoidal comonads and their corepresentations

Dingguo Wang ,
Xiaohui Zhang ^,

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

Received: 20 March 2022 Revised: 12 May 2022 Accepted: 17 May 2022 Published: 17 June 2022

In this paper, we define and study quasi-monoidal comonads on a monoidal category. It generalize the (Hom type) coquasi-bialgebras to a non-braided setting. We investigate their corepresentations and their coquasitriangular structures. We also discuss their gauge equivalence relations.

Keywords:

Citation: Dingguo Wang, Xiaohui Zhang. On quasi-monoidal comonads and their corepresentations[J]. Electronic Research Archive, 2022, 30(8): 3153-3171. doi: 10.3934/era.2022160

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Abstract

1. Introduction

In this paper, we consider the three-dimensional magneto-micropolar fluid equations with fractional dissipation

$\begin{equation} \left\{ \begin{array}{l} { \partial_{t}u+\mu (-\Delta)^{\alpha} u-\chi\Delta u+u\cdot\nabla u-b\cdot\nabla b+\nabla p-2\chi\nabla\times v = 0, }\\ { \partial_{t}v+\eta (-\Delta)^{\beta} v-\kappa\nabla\nabla\cdot v+4\chi v+u\cdot\nabla v-2\chi\nabla\times u = 0, }\\ { \partial_{t}b+\lambda (-\Delta)^{\gamma} b+u\cdot\nabla b-b\cdot\nabla u = 0, }\\ { \nabla\cdot u = 0, \nabla\cdot b = 0, }\\ \end{array} \right. \end{equation}$

(1.1)

with an initial value

$\begin{equation} t = 0: u = u_0(x), \; \; v = v_0(x), \; \; b = b_0(x), \; \; x\in \mathbb{R}^{3}. \end{equation}$

(1.2)

Here $u = u(x, t)$ , $v = v(x, t)$ , $b = b(x, t)\in \mathbb{R}^{3}$ , and $p = p(x, t)\in \mathbb{R}$ are the velocity, micro-rotational velocity, magnetic fields, and scalar pressure, respectively. $\mu$ , $\chi$ , and $\frac{1}{\lambda}$ represent the kinematic viscosity, vortex viscosity, and magnetic Reynolds number, respectively. $\eta$ and $\kappa$ are angular viscosities. $\alpha$ , $\beta$ and $\gamma$ are the parameters of the fractional dissipations corresponding to the velocity, micro-rotational velocity and magnetic field, respectively. The fractional Laplace operator $(-\Delta)^\alpha$ is defined through the Fourier transform as

$\begin{equation*} \widehat{(-\Delta)^\alpha f}(\xi) = \widehat{\Lambda ^{2\alpha }f} = |\xi|^{2\alpha}\hat f(\xi). \end{equation*}$

The incompressible magneto-micropolar fluid equations have made analytic studies a great challenge but offer new opportunities due to their distinctive mathematical features. Regularity criteria for weak solutions are established by Fan and Zhong ^[1] in pointwise multipliers for $1\leq \alpha = \beta = \gamma\leq \frac{5}{4}$ . Local and global well-posedness have been established in ^[2,3,4], respectively. For $\alpha = \beta = \gamma = 1$ , we refer to ^[5,6,7] for the existence of strong solutions and weak solutions, respectively. In the study field of the magneto-micropolar fluid equations, regularity criteria for weak solutions and blow-up criteria for smooth solutions are very important topics. The readers may refer to regularity criteria of weak solutions in Morrey-Campanato space ^[8], in Lorentz space ^[9], Besov space ^[10], Triebel-Lizorkin space ^[11] and other regularity criteria for weak solutions ^{[12,13,14,15]}, and ^[16,17] for blow-up criteria of smooth solutions in different function spaces, respectively. Serrin-type regularity criteria for weak solutions via the velocity fields and the gradient of the velocity field were established in Yuan ^[13], respectively. We may refer to ^[18,19,20] for global well-posedness. On the other hand, the global regularity of weak solutions to (1.1) with partial viscosities becomes more complex. In the case of 2D, we may refer to ^{[22,23,24,25]}, and in the case of 3D, we may refer to ^[26,27].

If $v = 0$ and $\chi = 0$ , then (1.1) reduces to MHD equations with fractional dissipation. The MHD equations govern the dynamics of the velocity and magnetic fields in electrically conducting fluids such as plasmas, liquid metals, and salt water. We only recall regularity criteria for our purpose. If $\alpha, \beta > \frac{5}{4}$ , some regularity criteria have been established by Wu ^[28,29], which are given in terms of the velocity $u$ . If $1\leq \alpha = \beta\leq \frac{3}{2}$ , Zhou ^[30] obtained the Serrin-type criteria $u\in L^p_TL^q_x$ with $\frac{2\alpha}{p}+\frac{3}{q}\leq 2\alpha-1$ and $\frac{3}{2\alpha-1} < q\leq\infty$ . Later, Yuan ^[14] extended the above function space $L^q$ to $B^s_{q, \infty}$ . Recently, the regularity criterion involving $u_3, b \in L^\omega_TL^q_x$ is given in ^[31]. We also refer to ^[32,33] for well-posedness and ^[34] for blow up criterion of smooth solutions.

Motivated by the Serrin-type regularity criterion of weak solutions to Navier-Stokes equations ^[35,36] and MHD equations ^[30,31]. The main purpose is to investigate the regularity criterion of weak solutions to the systems (1.1) and (1.2) in this paper and establish the Serrin-type regularity criterion of weak solutions involving partial components. We state our main result as follows:

Theorem 1.1. Let $1\le\alpha = \beta = \gamma \le \frac{3}{2}$ and $\chi, \kappa\geq 0$ . Assume that $(u_0, v_0, b_0)\in H^1(\mathbb{R}^3)$ and $\nabla\cdot u_0 = \nabla\cdot b_0 = 0$ . Furthermore, if

$\begin{equation*} u_3, v, b \in L^{\varrho} (0, T;L^{q}(\mathbb{R}^3)), \end{equation*}$

with

$\begin{equation} \frac{2\alpha }{\varrho} + \frac{3}{q} \le \frac{3}{4} (2\alpha-1 )+\frac{3(1-\epsilon )}{4q}, \ \ \frac{3+\epsilon }{2\alpha -1} < q\le \infty , \ \ 0 < \epsilon\le \frac{1}{3}, \end{equation}$

(1.3)

then the solution $(u, v, b)$ to the systems (1.1) and (1.2) remains smooth on $\big [0, T \big]$ .

Remark 1.2. Since the concrete values of the constants $\mu$ , $\eta$ , and $\lambda$ play no role in our proof, for this reason, we shall assume them to be all equal to one throughout this paper. For convenience of description, we define horizontal derivatives $\nabla_h: = (\partial_{1}, \partial_{2})$ .

Remark 1.3. When $v = 0$ and $\chi = 0$ , the conclusion in Theorem 1.1 is reduced to the one in ^[31].

Remark 1.4. Compared with ^[31], the main difficulty in this paper comes from the nonlinear term $u\cdot\nabla v$ . In order to overcome the difficulty caused by the nonlinear term, owing to the energy functional (see (2.2)), we first use integrating by parts and $\nabla\cdot u = 0$ to transform it into a control of the horizontal derivative, and then use Hölder's inequality, multiplicative Sobolev inequality, the Gagliardo-Nirenberg inequality, and Young's inequality to control the nonlinear term.

2. Proof of main result

In this section, our main purpose is to complete the proof of Theorem 1.1. To this end, we introduce the following lemma:

Lemma 2.1. (^[37]) The multiplicative Sobolev inequality

$\begin{equation} \|\nabla u\|_{L^{3q}}\le C\|\partial_1\nabla u\|_{L^2}^{\frac{1}{3}} \|\partial_2\nabla u\|_{L^2}^{\frac{1}{3}} \|\partial_3\nabla u\|_{L^q}^{\frac{1}{3}}, \ \ 1\le q < \infty, \end{equation}$

(2.1)

holds.

In what follows, we prove Theorem 1.1.

Proof. Let

$\begin{equation} \begin{split} E(t) : = & \|\nabla _hu(t)\|_{L^2}^2+\|\nabla _hv(t)\|_{L^2}^2+\|\nabla _hb(t)\|_{L^2}^2+ \int_0^t(\|\nabla _h\Lambda ^{\alpha } u(\tau)\|_{L^2}^2+\\ & \|\nabla _h\Lambda ^{\alpha } v(\tau)\|_{L^2}^2+\|\nabla _h\Lambda ^{\alpha } b(\tau)\|_{L^2}^2)d\tau+ \kappa\int_0^t\|\nabla _h \nabla\cdot v(\tau)\|_{L^2}^2d\tau. \end{split} \end{equation}$

(2.2)

The proof is divided into two cases: $\frac{3+\epsilon }{2\alpha -1} < q < \infty$ and $q = \infty$ . We first consider the case $\frac{3+\epsilon }{2\alpha -1} < q < \infty$ .

Taking the inner product of the first three equations of (1.1) with $(u, v, b)$ , and adding them up, using integrating by parts, the divergence-free condition, and Cauchy inequality, we obtain

$\begin{equation*} \label{5} \begin{split} & \frac{1}{2}\frac{d}{dt}(\|u(t)\|_{L^2}^2+\|v(t)\|_{L^2}^2+\|b(t)\|_{L^2}^2) +\|\Lambda ^{\alpha } u(t)\|_{L^2}^2+\|\Lambda ^{\alpha } v(t)\|_{L^2}^2 \\ & +\|\Lambda ^{\alpha } b(t)\|_{L^2}^2+\kappa\|\nabla\cdot v(t)\|_{L^2}^2 \leq 0. \end{split} \end{equation*}$

Integrating the above inequality with respect to $t$ and then obtaining

$\begin{equation*} \begin{split} & \|u(t)\|_{L^2}^2+\|v(t)\|_{L^2}^2+\|b(t)\|_{L^2}^2 +2\int^t_0(\|\Lambda ^{\alpha } u(\tau)\|_{L^2}^2+\|\Lambda ^{\alpha } v(\tau)\|_{L^2}^2 \\ & +\|\Lambda ^{\alpha } b(\tau)\|_{L^2}^2+\kappa\|\nabla\cdot v(\tau)\|_{L^2}^2)d\tau \leq \|u_0\|_{L^2}^2+\|v_0\|_{L^2}^2+\|b_0\|_{L^2}^2. \end{split} \end{equation*}$

By multiplying the first three equations of (1.1) by $\Delta_hu$ , $\Delta_hv$ , and $\Delta_hb$ , respectively, and adding them up, using integrating by parts and the divergence-free condition, we have

$\begin{eqnarray} &&\frac{1}{2}\frac{d}{dt}(\|\nabla _hu(t)\|_{L^2}^2+\|\nabla _hv(t)\|_{L^2}^2+\|\nabla _hb(t)\|_{L^2}^2) +\|\nabla _h\Lambda ^{\alpha } u(t)\|_{L^2}^2+\|\nabla _h\Lambda ^{\alpha } v(t)\|_{L^2}^2 \\ && +\|\nabla _h\Lambda ^{\alpha } b(t)\|_{L^2}^2+\kappa\|\nabla _h \nabla\cdot v(t)\|_{L^2}^2+\chi\|\nabla _h\nabla u(t)\|_{L^2}^2+ 4\chi\|\nabla _hv\|_{L^2}^2 \\ & : = & \sum\limits_{i = 1}^6I_i, \end{eqnarray}$

(2.3)

where

$\begin{eqnarray} \label{} I_1 & = &\int_{\mathbb{R}^3}(u\cdot\nabla u)\cdot\Delta_hu\;dx, \\ I_2& = &-\int_{\mathbb{R}^3}(b\cdot\nabla b)\cdot\Delta_hu\;dx, \\ I_3& = &\int_{\mathbb{R}^3}(u\cdot\nabla b)\cdot\Delta_hb\;dx, \\ I_4& = &-\int_{\mathbb{R}^3}(b\cdot\nabla u)\cdot\Delta_hb\;dx, \\ I_5& = &\int_{\mathbb{R}^3}(u\cdot\nabla v)\cdot\Delta_hv\;dx, \\ I_6& = &-2\chi \int_{\mathbb{R}^3}[(\nabla \times v)\cdot\Delta_h u + (\nabla \times u) \cdot\Delta_h v]\;dx. \end{eqnarray}$

Thanks to integration by parts and Cauchy's inequality, we arrive at

$\begin{eqnarray} I_6 & = & 4\chi\int_{\mathbb{R}^3} \nabla_h(\nabla \times u) \cdot \nabla_h v\;dx \\ &\leq & \chi\|\nabla_h (\nabla \times u)\|^2_{L^2}+4\chi\|\nabla_hv\|^2_{L^2} \\ & = & \chi\|\nabla_h \nabla u\|^2_{L^2}+4\chi\|\nabla_hv\|^2_{L^2}. \end{eqnarray}$

(2.4)

For $I_1$ , we divide it into the following three items: $I_{1i}(i = 1, 2, 3)$ as

$\begin{eqnarray} I_1 & = & \sum\limits_{j, k = 1}^2\int_{\mathbb{R}^3}u_j\partial_j u_k\Delta_hu_k\;dx +\sum\limits_{j = 1}^3\int_{\mathbb{R}^3}u_j\partial_j u_3\Delta_hu_3\;dx+ \\ && \sum\limits_{k = 1}^2\int_{\mathbb{R}^3}u_3\partial_3 u_k\Delta_hu_k\;dx \\ &: = & I_{11}+ I_{12}+ I_{13}. \end{eqnarray}$

(2.5)

The divergence-free condition and integration by parts entail that

$\begin{eqnarray} I_{11}& = &\sum\limits_{i, j, k = 1}^2\int_{\mathbb{R}^3}u_j\partial_j u_k\partial^2_{ii}u_k\;dx\\& = & -\sum\limits_{i, j, k = 1}^2\int_{\mathbb{R}^3}\partial_{i}u_j\partial_j u_k\partial_{i}u_k\;dx+\frac{1}{2}\sum\limits_{i, j, k = 1}^2\int_{\mathbb{R}^3}\partial_{j}u_j|\partial_i u_k|^2\;dx\\& = & -\sum\limits_{i, j, k = 1}^2\int_{\mathbb{R}^3}\partial_{i}u_j\partial_j u_k\partial_{i}u_k\;dx-\frac{1}{2}\sum\limits_{i, k = 1}^2\int_{\mathbb{R}^3}\partial_{3}u_3|\partial_i u_k|^2\;dx\\& = & -\int_{\mathbb{R}^3}\partial_{1}u_1\partial_1 u_1\partial_{1}u_1\;dx-\int_{\mathbb{R}^3}\partial_{1}u_1\partial_1 u_2\partial_{1}u_2\;dx-\int_{\mathbb{R}^3}\partial_{1}u_2\partial_2 u_1\partial_{1}u_1\;dx\\&&-\int_{\mathbb{R}^3}\partial_{1}u_2\partial_2 u_2\partial_{1}u_2\;dx-\int_{\mathbb{R}^3}\partial_{2}u_1\partial_1 u_1\partial_{2}u_1\;dx-\int_{\mathbb{R}^3}\partial_{2}u_1\partial_1 u_2\partial_{2}u_2\;dx\\&&-\int_{\mathbb{R}^3}\partial_{2}u_2\partial_2 u_1\partial_{2}u_1\;dx-\int_{\mathbb{R}^3}\partial_{2}u_2\partial_2 u_2\partial_{2}u_2\;dx-\frac{1}{2}\sum\limits_{i, k = 1}^2\int_{\mathbb{R}^3}\partial_{3}u_3|\partial_i u_k|^2\;dx\\& = & -\int_{\mathbb{R}^3}\partial_{1}u_1\partial_1 u_1\partial_{1}u_1\;dx-\int_{\mathbb{R}^3}\partial_{2}u_2\partial_2 u_2\partial_{2}u_2\;dx+\int_{\mathbb{R}^3}\partial_{3}u_3\partial_2 u_1\partial_{2}u_1\;dx\\&&+\int_{\mathbb{R}^3}\partial_{3}u_3\partial_1 u_2\partial_{1}u_2\;dx+\int_{\mathbb{R}^3}\partial_{3}u_3\partial_2 u_1\partial_{1}u_2\;dx-\frac{1}{2}\sum\limits_{i, k = 1}^2\int_{\mathbb{R}^3}\partial_{3}u_3|\partial_i u_k|^2\;dx\\& = & \frac{1}{2} \sum\limits_{j, k = 1}^2\int_{\mathbb{R}^3}\partial_3 u_3\partial_k u_j\partial_k u_j\;dx -\int_{\mathbb{R}^3}\partial_3 u_3\partial_1 u_1\partial_2 u_2\;dx +\int_{\mathbb{R}^3}\partial_3 u_3\partial_2 u_1\partial_1 u_2\;dx \\& = & -\sum\limits_{j, k = 1}^2\int_{\mathbb{R}^3} u_3\partial_{3k}^{2} u_j\partial_k u_j\;dx +\int_{\mathbb{R}^3} u_3(\partial_{32}^{2} u_2\partial_1 u_1+\partial_{31}^{2} u_1 \partial_2 u_2)\;dx \\&& -\int_{\mathbb{R}^3} u_3(\partial_{32}^{2} u_1\partial_1 u_2+\partial_{31}^{2} u_2 \partial_2 u_1)\;dx, \end{eqnarray}$

(2.6)

and

$\begin{eqnarray} I_{12} = - \sum\limits_{j = 1}^3\sum\limits_{l = 1}^2\int_{\mathbb{R}^3}\partial_l u_j\partial_j u_3\partial_l u_3\;dx = \sum\limits_{j = 1}^3\sum\limits_{l = 1}^2\int_{\mathbb{R}^3}\partial_l u_j u_3\partial_{jl}^{2} u_3\;dx. \end{eqnarray}$

(2.7)

Therefore, we obtain

$\begin{equation} |I_1|\le C\int_{\mathbb{R}^3} |u_3||\nabla u||\nabla_h\nabla u|\;dx. \end{equation}$

(2.8)

From Hölder's inequality, Lemma 2.1, the Gagliardo-Nirenberg inequality, and Young's inequality, it follows that

$\begin{eqnarray} |I_1| &\leq & C\int_{\mathbb{R}^3} |u_3||\nabla u||\nabla_h\nabla u|\;dx \\ &\leq & C\|u_3\|_{L^{q}}\| \nabla u\|_{L^{\theta _1}} \|\nabla_h\nabla u\|_{L^{\theta _2}} \\ &\leq & C\|u_3\|_{L^{q}}\| \nabla_h\nabla u\|_{L^{2} }^{\frac{2}{3} } \| \Delta u\|_{L^{\frac{\theta _1}{3} } }^{\frac{1}{3} } \|\nabla_h\nabla u\|_{L^{\theta _2}} \\ &\leq & C\|u_3\|_{L^{q}}\| \nabla_h u\|_{L^{2} }^{\frac{2s_1}{3} } \| \nabla _h\Lambda ^{\alpha } u\|_{L^{2} }^{\frac{2(1-s_1)}{3} }\| \nabla u\|_{L^{2} }^{\frac{s_2}{3} } \| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{\frac{1-s_2}{3} }\| \nabla_h u\|_{L^{2} }^{s_3 } \| \nabla _h\Lambda ^{\alpha } u\|_{L^{2} }^{1-s_3 }\\ &\leq & C\|u_3\|_{L^{q}}\| \nabla u\|_{L^{2} }^{\frac{2s_1}{3} } \| \nabla _h\Lambda ^{\alpha } u\|_{L^{2} }^{\frac{2(1-s_1)}{3} }\| \nabla u\|_{L^{2} }^{\frac{s_2}{3} } \| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{\frac{1-s_2}{3} }\| \nabla u\|_{L^{2} }^{s_3 } \| \nabla _h\Lambda ^{\alpha } u\|_{L^{2} }^{1-s_3 }\\ &\leq& C\|u_3\|_{L^{q}}\| \nabla u\|_{L^{2} }^{\frac{2s_1}{3}+\frac{s_2}{3}+s_3}\| \Lambda ^{\alpha+1} u\|_{L^{2}}^{\frac{1-s_2}{3} }\| \nabla _h\Lambda ^{\alpha } u\|_{L^{2} }^{\frac{2(1-s_1)}{3}+1-s_3 } \\ &\leq & C\big[ \|u_3\|_{L^{q}}\| \nabla u\|_{L^{2} }^{\frac{2s_1}{3}+\frac{s_2}{3}+s_3 } \| \Lambda ^{\alpha+1} u\|_{L^{2}}^{\frac{1-s_2}{3} } \big]^{m^{\prime }} +\frac{1}{6}\| \nabla _h\Lambda ^{\alpha } u\|_{L^{2} }^{(\frac{2(1-s_1)}{3}+1-s_3 )m}, \end{eqnarray}$

(2.9)

where the constants $1 < \theta _1, \theta _2, m, m^{\prime } < \infty$ and $0\le s_1, s_2, s_3\le1$ satisfy

$\begin{equation} \left\{ \begin{array}{l} { \frac{1}{\theta _1} + \frac{1}{\theta _2}+\frac{1}{q} = 1, }\\ { 2-\frac{3}{2} = \big(1-\frac{3}{2}\big) s_1+\big(1+\alpha-\frac{3}{2}\big) \big(1-s_1\big), }\\ { 2-\frac{3}{\theta_1/3} = \big(1-\frac{3}{2}\big) s_2+\big(1+\alpha-\frac{3}{2}\big)\big(1-s_2\big), }\\ { 2-\frac{3}{\theta_2} = \big(1-\frac{3}{2}\big) s_3+\big(1+\alpha-\frac{3}{2}\big)\big(1-s_3\big), }\\ { \frac{1}{m} + \frac{1}{m^{\prime }} = 1, }\\ { \big(\frac{2(1-s_1)}{3}+1-s_3\big)m = 2.} \end{array} \right. \end{equation}$

(2.10)

Noting that $1\le\alpha\le\dfrac{3}{2}$ and $\frac{3+\epsilon }{2\alpha -1} < q\le \infty$ , one solution to (2.10) can be written as

$\begin{equation} \left\{ \begin{array}{l} { \theta _1 = \frac{18q}{5q-18\epsilon}, }\\ { \theta _2 = \frac{18q}{13q-18(1-\epsilon)}, }\\ { s _1 = 1-\frac{1}{\alpha}, }\\ { s _2 = 1-\frac{9\epsilon}{\alpha q}, }\\ { s _3 = 1-\frac{1}{3\alpha}-\frac{3(1-\epsilon)}{\alpha q}, }\\ { m = \frac{2\alpha q}{q+3(1-\epsilon)}, }\\ { m^{\prime } = \frac{2\alpha q}{(2\alpha-1)q-3(1-\epsilon)}.} \end{array} \right. \end{equation}$

(2.11)

To bound $I_3$ , we decompose it into three pieces as

$\begin{eqnarray} I_3& = &\sum\limits_{j, k = 1}^2\int_{\mathbb{R}^3}u_j\partial_j b_k\Delta_hb_k\;dx +\sum\limits_{j = 1}^2\int_{\mathbb{R}^3}u_j\partial_j b_3\Delta_hb_3\;dx+\sum\limits_{k = 1}^3\int_{\mathbb{R}^3}u_3\partial_3 b_k\Delta_h b_k\;dx \\&: = & I_{31}+ I_{32}+ I_{33}. \end{eqnarray}$

(2.12)

By using integrating by parts (see^[31]), we have

$\begin{eqnarray} I_{31}& = & \sum\limits_{j, k, l = 1}^2\int_{\mathbb{R}^3} \big[\partial_{ll}^{2} u_j\partial_j b_k b_k +\partial_l u_j\partial_{lj}^{2} b_k b_k\big]\;dx - \\ && \frac{1}{2} \sum\limits_{j, k, l = 1}^2\int_{\mathbb{R}^3} \big[\partial_{lj}^{2} u_j\partial_l b_k b_k +\partial_j u_j\partial_{ll}^{2} b_k b_k\big]\;dx. \end{eqnarray}$

(2.13)

Similarly, we have

$\begin{eqnarray} I_{32} & = & \sum\limits_{j, l = 1}^2\int_{\mathbb{R}^3} \big[\partial_{ll}^{2} u_j\partial_j b_3 b_3 +\partial_l u_j\partial_{lj}^{2} b_3 b_3\big]\;dx -\\ && \frac{1}{2} \sum\limits_{j, k, l = 1}^2\int_{\mathbb{R}^3} \big[\partial_{lj}^{2} u_j\partial_l b_3 b_3 +\partial_j u_j\partial_{ll}^{2} b_3 b_3\big]\;dx, \end{eqnarray}$

(2.14)

and

$\begin{eqnarray} I_{33} & = & \sum\limits_{k = 1}^3 \sum\limits_{l = 1}^2\int_{\mathbb{R}^3} \big[\partial_{3l}^{2} u_3\partial_l b_k b_k +\partial_l u_3\partial_{3l}^{2} b_k b_k\big]\;dx +\\ && \frac{1}{2} \sum\limits_{k = 1}^3\sum\limits_{j, l = 1}^2\int_{\mathbb{R}^3} \big[\partial_{lj}^{2} u_j\partial_l b_k b_k +\partial_j u_j\partial_{ll}^{2} b_k b_k\big]\;dx. \end{eqnarray}$

(2.15)

Collecting (2.13)–(2.15), it is easy to derive that

$\begin{equation} |I_3|\le C\int_{\mathbb{R}^3} |b|(|\nabla u|+|\nabla b|)(|\nabla_h\nabla u|+|\nabla_h\nabla b|)\;dx. \end{equation}$

(2.16)

Furthermore, we have

$\begin{equation} |I_2+I_3+I_4|\le C\int_{\mathbb{R}^3} |b|(|\nabla u|+|\nabla b|)(|\nabla_h\nabla u|+|\nabla_h\nabla b|)\;dx. \end{equation}$

(2.17)

Similar to (2.13), it follows from Hölder's inequality, Lemma 2.1, Gagliardo-Nirenberg inequality, and Young's inequality that

$\begin{eqnarray} && |I_2+I_3+I_4| \\ &\le& C\int_{\mathbb{R}^3} |b|(|\nabla u|+|\nabla b|)(|\nabla_h\nabla u|+|\nabla_h\nabla b|)\;dx \\ &\le& C\|b\|_{L^{q}}\| |\nabla u|+|\nabla b|\|_{L^{\theta _1}} \||\nabla_h\nabla u|+|\nabla_h\nabla b|\|_{L^{\theta _2}} \\ &\le& C\|b\|_{L^{q}}\big(\| \nabla_h\nabla u\|_{L^{2} }^{\frac{2}{3} } \| \Delta u\|_{L^{\frac{\theta _1}{3} } }^{\frac{1}{3} } +\| \nabla_h\nabla b\|_{L^{2} }^{\frac{2}{3} } \| \Delta b\|_{L^{\frac{\theta _1}{3} } }^{\frac{1}{3} }\big)\cdot\\ && \quad \big(\|\nabla_h\nabla u\|_{L^{\theta _2}}+\|\nabla_h\nabla b\|_{L^{\theta _2}} \big) \\ &\le& C\|b\|_{L^{q}}\Big(\| \nabla u\|_{L^{2} }^{\frac{2s_1}{3} } \| \nabla _h\Lambda ^{\alpha } u\|_{L^{2} }^{\frac{2(1-s_1)}{3} } \| \nabla u\|_{L^{2} }^{\frac{s_2}{3} } \| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{\frac{1-s_2}{3} } + \\ && \quad\| \nabla b\|_{L^{2} }^{\frac{2s_1}{3} } \| \nabla _h\Lambda ^{\alpha } b\|_{L^{2} }^{\frac{2(1-s_1)}{3} } \| \nabla b\|_{L^{2} }^{\frac{s_2}{3} } \| \Lambda ^{\alpha+1 } b\|_{L^{2} }^{\frac{1-s_2}{3} }\Big)\cdot \\ && \quad \big(\| \nabla u\|_{L^{2} }^{s_3 } \| \nabla _h\Lambda ^{\alpha } u\|_{L^{2} }^{1-s_3 }+\| \nabla b\|_{L^{2} }^{s_3 } \| \nabla _h\Lambda ^{\alpha } b\|_{L^{2} }^{1-s_3 }\big) \\ &\le& C\|b\|_{L^{q}} \big(\| \nabla u\|_{L^{2} }^{\frac{2s_1}{3} } +\| \nabla b\|_{L^{2} }^{\frac{2s_1}{3} }\big) \big(\| \nabla _h\Lambda ^{\alpha } u\|_{L^{2} }^{\frac{2(1-s_1)}{3} }+\| \nabla _h\Lambda ^{\alpha } b\|_{L^{2} }^{\frac{2(1-s_1)}{3} }\big)\cdot \\ &&\quad \big(\| \nabla u\|_{L^{2} }^{\frac{s_2}{3} }+\| \nabla b\|_{L^{2} }^{\frac{s_2}{3} }\big) \big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{\frac{1-s_2}{3} }+\| \Lambda ^{\alpha+1 } b\|_{L^{2} }^{\frac{1-s_2}{3} }\big)\cdot \\ &&\quad \big(\| \nabla u\|_{L^{2} }^{s_3 }+\| \nabla b\|_{L^{2} }^{s_3 }\big) \big(\| \nabla _h\Lambda ^{\alpha } u\|_{L^{2} }^{1-s_3 }+\| \nabla _h\Lambda ^{\alpha } b\|_{L^{2} }^{1-s_3 }\big) \\ &\le& C\|b\|_{L^{q}} \big(\| \nabla u\|_{L^{2} }+\| \nabla b\|_{L^{2} }\big)^{\frac{2s_1}{3}+\frac{s_2}{3}+s_3 } \big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }+\| \Lambda ^{\alpha+1 } b\|_{L^{2} }\big)^{\frac{1-s_2}{3}}\cdot \\ &&\quad \big(\| \nabla _h\Lambda ^{\alpha } u\|_{L^{2} }+\| \nabla _h\Lambda ^{\alpha } b\|_{L^{2} }\big)^{\frac{2(1-s_1)}{3}+1-s_3}\\ &\le& C\big [ \|b\|_{L^{q}}\big(\| \nabla u\|_{L^{2} }+\| \nabla b\|_{L^{2} }\big)^{\frac{2s_1}{3}+\frac{s_2}{3}+s_3 } \big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }+\| \Lambda ^{\alpha+1 } b\|_{L^{2} }\big)^{\frac{1-s_2}{3} } \big ]^{m^{\prime}}+\\ &&\quad \frac{1}{6}\big(\| \nabla _h\Lambda ^{\alpha } u\|_{L^{2} }+\| \nabla _h\Lambda ^{\alpha } b\|_{L^{2}}\big)^{\big(\frac{2(1-s_1)}{3}+1-s_3 \big)m}, \end{eqnarray}$

(2.18)

where the constants $1 < \theta _1, \theta _2, m, m^{\prime } < \infty$ and $0\le s_1, s_2, s_3\le1$ satisfy (2.10).

Similar to $I_3$ , we bound $I_5$ as

$\begin{equation} |I_5|\le C\int_{\mathbb{R}^3} |v|(|\nabla u|+|\nabla v|)(|\nabla_h\nabla u|+|\nabla_h\nabla v|)\;dx. \end{equation}$

(2.19)

Using the same steps as (2.18), we obtain

$\begin{eqnarray} \label{22} |I_5| &\le& C\int_{\mathbb{R}^3} |v|(|\nabla u|+|\nabla v|)(|\nabla_h\nabla u|+|\nabla_h\nabla v|)\;dx \\ &\le& C\big [ \|v\|_{L^{q}}\big(\| \nabla u\|_{L^{2} }+\| \nabla v\|_{L^{2} }\big)^{\frac{2s_1}{3}+\frac{s_2}{3}+s_3 } \big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }+\| \Lambda ^{\alpha+1 } v\|_{L^{2} }\big)^{\frac{1-s_2}{3} } \big ]^{m^{\prime}}+\\ &&\quad \frac{1}{6}\big(\| \nabla _h\Lambda ^{\alpha } u\|_{L^{2} }+\| \nabla _h\Lambda ^{\alpha } v\|_{L^{2}}\big)^{\big(\frac{2(1-s_1)}{3}+1-s_3 \big)m}, \end{eqnarray}$

where the constants $1 < \theta _1, \theta _2, m, m^{\prime } < \infty$ and $0\le s_1, s_2, s_3\le1$ satisfy (2.10).

Combining (2.3), (2.4), (2.9), (2.18), and (2.20), we arrive at

$\begin{eqnarray} &&\frac{d}{dt}(\|\nabla _hu(t)\|_{L^2}^2+\|\nabla _hv(t)\|_{L^2}^2+\|\nabla _hb(t)\|_{L^2}^2) +\|\nabla _h\Lambda ^{\alpha } u(t)\|_{L^2}^2+\\ && \|\nabla _h\Lambda ^{\alpha } v(t)\|_{L^2}^2+\|\nabla _h\Lambda ^{\alpha } b(t)\|_{L^2}^2+\kappa\|\nabla _h \nabla\cdot v(t)\|_{L^2}^2\\ &\le& C\|u_3\|_{L^{q}}^{\frac{2\alpha q}{(2\alpha-1)q-3(1-\epsilon)}} \| \nabla u\|_{L^{2} }^{\frac{2\big((2\alpha-1) q-3\big)}{(2\alpha-1)q-3(1-\epsilon)}} \| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{\frac{6\epsilon }{(2\alpha-1)q-3(1-\epsilon)} } +\\ && \|b\|_{L^{q}}^{\frac{2\alpha q}{(2\alpha-1)q-3(1-\epsilon)}}\big(\| \nabla u\|_{L^{2} }+\| \nabla b\|_{L^{2} }\big)^{\frac{2\big((2\alpha-1) q-3\big)}{(2\alpha-1)q-3(1-\epsilon)} } \big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }+\| \Lambda ^{\alpha+1 } b\|_{L^{2} }\big)^{\frac{6\epsilon }{(2\alpha-1)q-3(1-\epsilon)} }+\\ && \|v\|_{L^{q}}^{\frac{2\alpha q}{(2\alpha-1)q-3(1-\epsilon)}}\big(\| \nabla u\|_{L^{2} }+\| \nabla v\|_{L^{2} }\big)^{\frac{2\big((2\alpha-1) q-3\big)}{(2\alpha-1)q-3(1-\epsilon)} } \big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }+\| \Lambda ^{\alpha+1 } v\|_{L^{2} }\big)^{\frac{6\epsilon }{(2\alpha-1)q-3(1-\epsilon)} }\\ &\le& C\big(\|u_3\|_{L^{q}}+\|b\|_{L^{q}}+\|v\|_{L^{q}}\big)^{\frac{2\alpha q}{(2\alpha-1)q-3(1-\epsilon)}}\big(\| \nabla u\|_{L^{2} }+\| \nabla b\|_{L^{2}}+\\ && \| \nabla v\|_{L^{2} }\big)^{\frac{2\big((2\alpha-1) q-3\big)}{(2\alpha-1)q-3(1-\epsilon)}}\big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }+\| \Lambda ^{\alpha+1 } b\|_{L^{2} }+\| \Lambda ^{\alpha+1 } v\|_{L^{2} }\big)^{\frac{6\epsilon }{(2\alpha-1)q-3(1-\epsilon)}}. \end{eqnarray}$

(2.20)

Set

$\begin{equation} \begin{array}{lll} { \Theta_1 = \frac{2\alpha q}{(2\alpha-1)q-3(1-\epsilon)}, }\\ { \Theta_2 = \frac{2\big((2\alpha-1) q-3\big)}{(2\alpha-1)q-3(1-\epsilon)}, }\\ { \Theta_3 = \frac{6\epsilon }{(2\alpha-1)q-3(1-\epsilon)}.} \end{array} \end{equation}$

(2.21)

Integrating (2.20) with respect to $t$ , we obtain

$\begin{equation} \begin{split} E(t) &\le CJ_0+C \int_0^t\big(\|u_3\|_{L^{q}}+\|b\|_{L^{q}}+\|v\|_{L^{q}}\big)^{\Theta_1} \big(\| \nabla u\|_{L^{2} }+\| \nabla b\|_{L^{2}}+\\ & \quad \| \nabla v\|_{L^{2} }\big)^{\Theta_2}\big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }+\| \Lambda ^{\alpha+1 } b\|_{L^{2} }+\| \Lambda ^{\alpha+1 } v\|_{L^{2} }\big)^{\Theta_3}d\tau, \end{split} \end{equation}$

(2.22)

where $J_0 = \|\nabla u(0)\|_{L^2}^2+\|\nabla v(0)\|_{L^2}^2+\|\nabla b(0)\|_{L^2}^2$ .

By taking the inner product of the first three equations of (1.1) with $(-\Delta u, -\Delta v, -\Delta b)$ and integrating by parts, the divergence-free condition, we have

$\begin{eqnarray} && \frac{1}{2}\frac{d}{dt}(\|\nabla u(t)\|_{L^2}^2+\|\nabla v(t)\|_{L^2}^2+\|\nabla b(t)\|_{L^2}^2) +\|\Lambda ^{\alpha+1 }u(t)\|_{L^2}^2+\|\Lambda ^{\alpha+1 }v(t)\|_{L^2}^2 +\\ && \|\Lambda ^{\alpha+1 } b(t)\|_{L^2}^2+\kappa\|\nabla \nabla\cdot v(t)\|_{L^2}^2+\chi\|\nabla \nabla u(t)\|_{L^2}^2+ 4\chi\|\nabla v(t)\|_{L^2}^2\\ &: = & \sum\limits_{i = 1}^6J_i, \end{eqnarray}$

(2.23)

where

$\begin{eqnarray} \label{} J_1& = &\int_{\mathbb{R}^3}(u\cdot\nabla u)\cdot\Delta u\;dx, \\ J_2& = &-\int_{\mathbb{R}^3}(b\cdot\nabla b)\cdot\Delta u\;dx, \\ J_3& = &\int_{\mathbb{R}^3}(u\cdot\nabla b)\cdot\Delta b\;dx, \\ J_4& = &-\int_{\mathbb{R}^3}(b\cdot\nabla u)\cdot\Delta b\;dx, \\ J_5& = &\int_{\mathbb{R}^3}(u\cdot\nabla v)\cdot\Delta v\;dx, \\ J_6& = &-2\chi \int_{\mathbb{R}^3}[(\nabla \times v)\cdot\Delta u + (\nabla \times u) \cdot\Delta v]\;dx. \end{eqnarray}$

By integration by parts and Cauchy's inequality, we arrive at

$\begin{eqnarray} J_6 & = & 4\chi\int_{\mathbb{R}^3} \nabla(\nabla \times u) \cdot \nabla v\;dx \\ &\leq& \chi\|\nabla (\nabla \times u)\|^2_{L^2}+4\chi\|\nabla v\|^2_{L^2} \\ & = & \chi\|\nabla \nabla u\|^2_{L^2}+4\chi\|\nabla v\|^2_{L^2}. \end{eqnarray}$

(2.24)

For $J_1$ , we divide it into the following three items: $J_{1i}(i = 1, 2, 3)$

$\begin{eqnarray} J_1 & = &\int_{\mathbb{R}^3}u_3\partial_3 u\cdot\Delta_hu\;dx +\sum\limits_{j = 1}^2\int_{\mathbb{R}^3}u_j\partial_j u\cdot\Delta u\;dx+\int_{\mathbb{R}^3}u_3\partial_3 u\cdot \partial_{33}^{2} u\;dx \\ &: = & J_{11}+ J_{12}+ J_{13}. \end{eqnarray}$

(2.25)

Integrating by parts and using the divergence-free condition yields

$\begin{equation} J_{11} = \frac{1}{2} \sum\limits_{k = 1}^3\sum\limits_{l = 1}^2\int_{\mathbb{R}^3}\partial_3 u_3\partial_l u_k\partial_l u_k\;dx -\sum\limits_{k = 1}^3\sum\limits_{l = 1}^2\int_{\mathbb{R}^3}\partial_l u_3\partial_3 u_k\partial_l u_k\;dx, \end{equation}$

(2.26)

$\begin{equation} J_{12} = \frac{1}{2} \sum\limits_{j = 1}^3\sum\limits_{k, l = 1}^3\int_{\mathbb{R}^3}\partial_j u_j\partial_l u_k\partial_l u_k\;dx -\sum\limits_{j = 1}^2\sum\limits_{k, l = 1}^3\int_{\mathbb{R}^3}\partial_l u_j\partial_j u_k\partial_l u_k\;dx, \end{equation}$

(2.27)

and

$\begin{equation} J_{13} = \frac{1}{2} \sum\limits_{k = 1}^3\int_{\mathbb{R}^3}(\partial_1 u_1+\partial_2 u_2)\partial_3 u_k\partial_3 u_k\;dx. \end{equation}$

(2.28)

Therefore, we have

$\begin{equation} |J_1|\le C\int_{\mathbb{R}^3} |\nabla_h u||\nabla u|^{2}\;dx. \end{equation}$

(2.29)

From Hölder's inequality and Lemma 2.1, it follows that

$\begin{eqnarray} |J_1| &\le & C\|\nabla_h u\|_{L^{2}}\| \nabla u\|_{L^4}^{2} \\ &\le& C\|\nabla_h u\|_{L^{2}}\| \nabla u\|_{L^2}^{2-\frac{3}{2\alpha}} \| \Lambda ^{\alpha } u\|_{L^{6} }^{\frac{3}{2\alpha} } \\ &\le& C\|\nabla_h u\|_{L^{2}}\| \nabla u\|_{L^2}^{2-\frac{3}{2\alpha}} \| \nabla_h\Lambda ^{\alpha } u\|_{L^{2} }^{\frac{1}{\alpha} }\| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{\frac{1}{2\alpha} }. \end{eqnarray}$

(2.30)

By using integrating by parts and the divergence-free condition, we have

$\begin{eqnarray} J_{3} & = & -\sum\limits_{j, k, l = 1}^3\int_{\mathbb{R}^3}\partial_l (u_j\partial_j b_k)\partial_l b_k\;dx \\ & = & -\sum\limits_{j, k, l = 1}^3\int_{\mathbb{R}^3} (\partial_l u_j\partial_j b_k\partial_l b_k+u_j\partial_{lj}^{2} b_k\partial_l b_k)\;dx \\ & = & \sum\limits_{j, k, l = 1}^3\int_{\mathbb{R}^3}b_k\partial_l (\partial_lu_j\partial_j b_k)\;dx\\ & = & \sum\limits_{j, k, l = 1}^3\int_{\mathbb{R}^3} (b_k\partial_{ll}^{2} u_j\partial_j b_k+b_k\partial_l u_j\partial_{jl}^{2} b_k)\;dx. \end{eqnarray}$

(2.31)

Then we arrive at

$\begin{equation} |J_3|\le C\int_{\mathbb{R}^3} |b|(|\nabla u|+|\nabla b|)(|\Delta u|+|\Delta b|)\;dx. \end{equation}$

(2.32)

Furthermore, we have

$\begin{equation} |J_2+J_3+J_4|\le C\int_{\mathbb{R}^3} |b|(|\nabla u|+|\nabla b|)(\Delta u|+|\Delta b|)\;dx. \end{equation}$

(2.33)

It follows from the same procedure (2.18) that

$\begin{eqnarray} && |J_2+J_3+J_4| \\ &\le& C\int_{\mathbb{R}^3} |b|(|\nabla u|+|\nabla b|)(|\Delta u|+|\Delta b|)\;dx \\ &\le& C\|b\|_{L^{q}}\| |\nabla u|+|\nabla b|\|_{L^{\theta _1}} \||\Delta u|+|\Delta b|\|_{L^{\theta _2}} \\ &\le& C\|b\|_{L^{q}}\big(\| \Delta u\|_{L^{2} }^{\frac{2}{3} } \| \Delta u\|_{L^{\frac{\theta _1}{3}}}^{\frac{1}{3} } +\| \Delta b\|_{L^{2} }^{\frac{2}{3} } \| \Delta b\|_{L^{\frac{\theta _1}{3} } }^{\frac{1}{3}}\big) \big(\|\Delta u\|_{L^{\theta _2}}+\|\Delta b\|_{L^{\theta _2}} \big) \\ &\le& C\|b\|_{L^{q}}\Big(\| \nabla u\|_{L^{2} }^{\frac{2s_1}{3}} \| \Lambda ^{\alpha+1 } u\|_{L^{2}}^{\frac{2(1-s_1)}{3} } \| \nabla u\|_{L^{2} }^{\frac{s_2}{3} } \| \Lambda ^{\alpha+1 } u\|_{L^{2}}^{\frac{1-s_2}{3}} +\\ && \| \nabla b\|_{L^{2} }^{\frac{2s_1}{3} } \| \Lambda ^{\alpha+1 } b\|_{L^{2} }^{\frac{2(1-s_1)}{3} } \| \nabla b\|_{L^{2} }^{\frac{s_2}{3} } \| \Lambda ^{\alpha+1 } b\|_{L^{2} }^{\frac{1-s_2}{3} }\Big)\times \\ && \big(\| \nabla u\|_{L^{2} }^{s_3 } \| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{1-s_3 }+\| \nabla b\|_{L^{2} }^{s_3 } \| \Lambda ^{\alpha+1 } b\|_{L^{2} }^{1-s_3 }\big) \\ &\le & C\|b\|_{L^{q}} \big(\| \nabla u\|_{L^{2} }+\| \nabla b\|_{L^{2} }\big)^{\frac{2s_1}{3}+\frac{s_2}{3}+s_3 } \big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }+\| \Lambda ^{\alpha+1 } b\|_{L^{2} }\big)^{\frac{2(1-s_1)}{3}+\frac{1-s_2}{3}+1-s_3 }\\ &\le& C\|b\|_{L^{q}}^{\frac{2\alpha q}{(2\alpha-1)q-3}} \big(\| \nabla u\|_{L^{2} }^{2}+\| \nabla b\|_{L^{2} }^{2}\big) +\frac{1}{8}\big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{2}+\| \Lambda ^{\alpha+1 } b\|_{L^{2} }^{2}\big), \end{eqnarray}$

(2.34)

where the constants $1 < \theta _1, \theta _2, m, m^{\prime } < \infty$ and $0\le s_1, s_2, s_3\le1$ satisfy (2.10).

Similar to $J_3$ , we bound $J_5$ as

$\begin{equation} |J_5|\le C\int_{\mathbb{R}^3} |v|(|\nabla u|+|\nabla v|)(|\Delta u|+|\Delta v|)\;dx. \end{equation}$

(2.35)

The same procedure leads to (2.34) yields

$\begin{eqnarray} \label{38} |J_5| &\le& C\int_{\mathbb{R}^3} |v|(|\nabla u|+|\nabla v|)(|\Delta u|+|\Delta v|)\;dx \\ &\le& C\|v\|_{L^{q}}^{\frac{2\alpha q}{(2\alpha-1)q-3}} \big(\| \nabla u\|_{L^{2} }^{2}+\| \nabla v\|_{L^{2} }^{2}\big) +\frac{1}{8}\big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{2}+\| \Lambda ^{\alpha+1} v\|_{L^{2}}^{2}\big). \end{eqnarray}$

Combining (2.23), (2.24), (2.30), (2.34), and (2.36), we have

$\begin{eqnarray} && \frac{1}{2}\frac{d}{dt}(\|\nabla u(t)\|_{L^2}^2+\|\nabla v(t)\|_{L^2}^2+\|\nabla b(t)\|_{L^2}^2) +\frac{3}{4}(\|\Lambda ^{\alpha+1 } u(t)\|_{L^2}^2+\|\Lambda ^{\alpha+1 } v(t)\|_{L^2}^2)\\ && +\frac{3}{4}\|\Lambda ^{\alpha+1}b(t)\|_{L^2}^2+\kappa\|\nabla \nabla\cdot v(t)\|_{L^2}^2\\ &\le& C(\|b\|_{L^{q}}^{\frac{2\alpha q}{(2\alpha-1)q-3}}+\|v\|_{L^{q}}^{\frac{2\alpha q}{(2\alpha-1)q-3}}) \big(\| \nabla u\|_{L^{2} }^{2}+\| \nabla b\|_{L^{2} }^{2}+\| \nabla v\|_{L^{2} }^{2}\big)+ \\ && C\|\nabla_h u\|_{L^{2}}\| \nabla u\|_{L^2}^{2-\frac{3}{2\alpha}} \| \nabla_h\Lambda ^{\alpha } u\|_{L^{2} }^{\frac{1}{\alpha} }\| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{\frac{1}{2\alpha}}. \end{eqnarray}$

(2.36)

Integrating (2.36) over the interval $(0, t)$ and using Hölder's inequality, it was deduced that

$\begin{eqnarray} && \frac{1}{2}(\|\nabla u(t)\|_{L^2}^2+\|\nabla v(t)\|_{L^2}^2+\|\nabla b(t)\|_{L^2}^2)+ \frac{3}{4}\int_0^t(\|\Lambda ^{\alpha+1 } u(\tau)\|_{L^2}^2+\\ && \|\Lambda ^{\alpha+1 } v(\tau)\|_{L^2}^2+\|\Lambda ^{\alpha+1 } b(\tau)\|_{L^2}^2)d\tau +\int_0^t\kappa\|\nabla \nabla\cdot v(\tau)\|_{L^2}^2d\tau \\ &\le& C+ C\int_0^t(\|b\|_{L^{q}}^{\frac{2\alpha q}{(2\alpha-1)q-3}}+\|v\|_{L^{q}}^{\frac{2\alpha q}{(2\alpha-1)q-3}}) \big(\| \nabla u\|_{L^{2} }^{2}+\| \nabla b\|_{L^{2} }^{2}+\| \nabla v\|_{L^{2} }^{2}\big) d\tau +\\ && C \int_0^t\|\nabla_h u\|_{L^{2}}\| \nabla u\|_{L^2}^{2-\frac{3}{2\alpha}} \| \nabla_h\Lambda ^{\alpha } u\|_{L^{2} }^{\frac{1}{\alpha} }\| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{\frac{1}{2\alpha} } d\tau \\ &\le& C+ C\int_0^t(\|b\|_{L^{q}}^{\frac{2\alpha q}{(2\alpha-1)q-3}}+\|v\|_{L^{q}}^{\frac{2\alpha q}{(2\alpha-1)q-3}}) \big(\| \nabla u\|_{L^{2} }^{2}+\| \nabla b\|_{L^{2} }^{2}+\| \nabla v\|_{L^{2} }^{2}\big) d\tau+\\ && C\sup\limits_{0\leq\tau\leq t}\|\nabla_h u\|_{L^{2}} \int_0^t\| \nabla u\|_{L^2}^{2-\frac{3}{2\alpha}} \| \nabla_h\Lambda ^{\alpha } u\|_{L^{2} }^{\frac{1}{\alpha} }\| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{\frac{1}{2\alpha} } d\tau. \end{eqnarray}$

(2.37)

From Young's inequality, it follows that

$\begin{eqnarray} && C\sup\limits_{0\leq\tau\leq t}\|\nabla_h u\|_{L^{2}} \int_0^t\| \nabla u\|_{L^2}^{2-\frac{3}{2\alpha}} \| \nabla_h\Lambda ^{\alpha } u\|_{L^{2} }^{\frac{1}{\alpha} }\| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{\frac{1}{2\alpha} } d\tau \\ &\le& C\sup\limits_{0\leq\tau\leq t}\|\nabla_h u\|_{L^{2}} \big[ \int_0^t \| \nabla u\|_{L^{2} }^{2} d\tau\big]^{1-\frac{3}{4\alpha}} \big[ \int_0^t \| \nabla_h\Lambda ^{\alpha } u\|_{L^{2} }^{2} d\tau\big]^{\frac{1}{2\alpha}}\big[ \int_0^t \| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{2} d\tau\big]^{\frac{1}{4\alpha}} \\ &\le& C\sup\limits_{0\leq\tau\leq t}\|\nabla_h u\|_{L^{2}} \big[ \int_0^t \| u\|_{L^{2} }^{\frac{2\alpha}{1+\alpha}} \| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{\frac{2}{1+\alpha}} d\tau\big]^{1-\frac{3}{4\alpha}} \big[ \int_0^t \| \nabla_h\Lambda ^{\alpha } u\|_{L^{2} }^{2} d\tau\big]^{\frac{1}{2\alpha}}\big[ \int_0^t \| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{2} d\tau\big]^{\frac{1}{4\alpha}} \\ &\le & C\sup\limits_{0\leq\tau\leq t}\|\nabla_h u\|_{L^{2}} \big[ \int_0^t \| \nabla_h\Lambda ^{\alpha } u\|_{L^{2} }^{2} d\tau\big]^{\frac{1}{2\alpha}}\big[ \int_0^t \| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{2} d\tau\big]^{\frac{1}{4\alpha}+\frac{4\alpha-3}{4\alpha(1+\alpha)}} \\ &\le& C\sup\limits_{0\leq\tau\leq t}\|\nabla_h u\|_{L^{2}} \big[ \big( \int_0^t \| \nabla_h\Lambda ^{\alpha } u\|_{L^{2}}^{2} d\tau\big)^{\frac{1}{2}} +1\big]\big[\big( \int_0^t \| \Lambda ^{\alpha+1 } u\|_{L^{2}}^{2} d\tau\big)^{\frac{1}{4}}+1 \big] \\ &\le& CE(t)\big[ \int_0^t \| \Lambda ^{\alpha+1 } u\|_{L^{2}}^{2} d\tau \big]^{\frac{1}{4}}+C\sup\limits_{0\leq\tau\leq t}\|\nabla_h u\|_{L^{2}} \big[ \int_0^t \| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{2} d\tau \big]^{\frac{1}{4}}+\\ && CE(t)+C\sup\limits_{0\leq\tau\leq t}\|\nabla_h u\|_{L^{2}}\\ &\le & CE(t)\big[ \int_0^t \| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{2} d\tau \big]^{\frac{1}{4}}+C\big( \sup\limits_{0\leq\tau\leq t}\|\nabla_h u\|^{2}_{L^{2}}+1 \big) \big[ \int_0^t \| \Lambda ^{\alpha+1 } u\|_{L^{2}}^{2} d\tau \big]^{\frac{1}{4}} +\\ && CE(t)+C\sup\limits_{0\leq\tau\leq t}\|\nabla_h u\|^{2}_{L^{2}}+C \\ &\le& CE(t)\big[ \int_0^t \| \Lambda ^{\alpha+1 } u\|_{L^{2}}^{2} d\tau \big]^{\frac{1}{4}}+C \big[ \int_0^t \| \Lambda ^{\alpha+1} u\|_{L^{2}}^{2} d\tau \big]^{\frac{1}{4}}+CE(t)+C. \end{eqnarray}$

(2.38)

Then, we have

(2.39)

By using Hölder's inequality, Young's inequality, and (2.22), we deduce that

$\begin{eqnarray} CE(t)&\le& C+C \int_0^t\big(\|u_3\|_{L^{q}}+\|b\|_{L^{q}}+\|v\|_{L^{q}}\big)^{\Theta_1}\big(\| \nabla u\|_{L^{2} }+\| \nabla b\|_{L^{2}}+\\ && \| \nabla v\|_{L^{2} }\big)^{\Theta_2}\big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }+\| \Lambda ^{\alpha+1 } b\|_{L^{2} }+\| \Lambda ^{\alpha+1} v\|_{L^{2} }\big)^{\Theta_3} d\tau \\ &\le& C+C\big[ \int_0^t\big(\|u_3\|_{L^{q}}+\|b\|_{L^{q}}+\|v\|_{L^{q}}\big)^{\frac{2\alpha q}{(2\alpha-1)q-3}}\big(\| \nabla u\|_{L^{2} }+\| \nabla b\|_{L^{2}}+\\ && \| \nabla v\|_{L^{2} }\big)^{2} d\tau \big]^{\Theta_2} \big[ \int_0^t\big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }+\| \Lambda ^{\alpha+1 } b\|_{L^{2} }+\| \Lambda ^{\alpha+1 } v\|_{L^{2} }\big)^{2 }d\tau\big]^{\frac{1}{2}\Theta_3} \\ &\le& C+C \int_0^t\big(\|u_3\|_{L^{q}}+\|b\|_{L^{q}}+\|v\|_{L^{q}}\big)^{\frac{2\alpha q}{(2\alpha-1)q-3}}\big(\| \nabla u\|_{L^{2} }+\| \nabla b\|_{L^{2}}+\| \nabla v\|_{L^{2} }\big)^{2} d\tau + \\ && \frac{1}{16} \int_0^t\big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }+\| \Lambda ^{\alpha+1 } b\|_{L^{2} }+\| \Lambda ^{\alpha+1 } v\|_{L^{2} }\big)^{2 }d\tau. \end{eqnarray}$

(2.40)

Similarly, it follows from (2.22) and Hölder's inequality and Young's inequality that

$\begin{eqnarray} && CE(t)\big[ \int_0^t \| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{2} d\tau \big]^{\frac{1}{4}} \\ &\le& C\big[ \int_0^t \| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{2} d\tau \big]^{\frac{1}{4}}+C\big[ \int_0^t \| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{2} d\tau \big]^{\frac{1}{4}}\int_0^t\big(\|u_3\|_{L^{q}}+\|b\|_{L^{q}}+\\ && \|v\|_{L^{q}}\big)^{\Theta_1}\big(\| \nabla u\|_{L^{2} }+\| \nabla b\|_{L^{2}}+\| \nabla v\|_{L^{2} }\big)^{\frac{\Theta_2}{2}}\big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }+\| \Lambda ^{\alpha+1 } b\|_{L^{2} }+\| \Lambda ^{\alpha+1 } v\|_{L^{2} }\big)^{\Theta_3}d\tau\\ &\le & C\big[ \int_0^t \| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{2} d\tau \big]^{\frac{1}{4}}+C\big[ \int_0^t \| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{2} d\tau \big]^{\frac{1}{4}}\big[ \int_0^t\big(\|u_3\|_{L^{q}}+\|b\|_{L^{q}}+\\ && \|v\|_{L^{q}}\big)^{\frac{2\alpha q}{(2\alpha-1)q-3}}\big(\| \nabla u\|_{L^{2} }+\| \nabla b\|_{L^{2}}+\| \nabla v\|_{L^{2} }\big)^{2} d\tau \big] ^{\frac{\Theta_2}{2}}\big[ \int_0^t\big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }+\\ && \| \Lambda ^{\alpha+1 } b\|_{L^{2} }+\| \Lambda ^{\alpha+1 } v\|_{L^{2} }\big)^{2 }d\tau\big]^{\frac{\Theta_3}{2}} \\ &\le& C\big[ \int_0^t \| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{2} d\tau \big]^{\frac{1}{4}}+C\big[ \int_0^t\big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }+\| \Lambda ^{\alpha+1 } b\|_{L^{2} }+\| \Lambda ^{\alpha+1 } v\|_{L^{2} }\big)^{2 }d\tau\big]^{\frac{2\Theta_3+1}{4}}\cdot \\ &&\big[ \int_0^t\big(\|u_3\|_{L^{q}}+\|b\|_{L^{q}}+\|v\|_{L^{q}}\big)^{\frac{2\alpha q}{(2\alpha-1)q-3}}\big(\| \nabla u\|_{L^{2} }+\| \nabla b\|_{L^{2}}+\| \nabla v\|_{L^{2} }\big)^{2} d\tau \big] ^{\Theta_2}\\ &\le& C\big[ \int_0^t \| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{2} d\tau \big]^{\frac{1}{4}}+C\big[ \int_0^t\big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }+\| \Lambda ^{\alpha+1 } b\|_{L^{2} }+\| \Lambda ^{\alpha+1 } v\|_{L^{2} }\big)^{2 }d\tau\big]^{\frac{2\Theta_3+1}{4}}\cdot\\ &&\big[ \int_0^t\big(\|u_3\|_{L^{q}}+\|b\|_{L^{q}}+\|v\|_{L^{q}}\big)^{\Theta_4}\big(\| \nabla u\|_{L^{2} }+\| \nabla b\|_{L^{2}}+\| \nabla v\|_{L^{2} }\big)^{2} d\tau \big] ^{\frac{3(2\alpha-1)q+3(1-\epsilon)-12}{4\big[ (2\alpha-1)q-3(1-\epsilon)\big] } } \\ &\le& C+C \int_0^t\big(\|u_3\|_{L^{q}}+\|b\|_{L^{q}}+\|v\|_{L^{q}}\big)^{\Theta_4}\big(\| \nabla u\|_{L^{2} }+\| \nabla b\|_{L^{2}}+\\ && \| \nabla v\|_{L^{2} }\big)^{2} d\tau + \frac{1}{16} \int_0^t\big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }+\| \Lambda ^{\alpha+1 } b\|_{L^{2} }+\| \Lambda ^{\alpha+1 } v\|_{L^{2} }\big)^{2 }d\tau, \end{eqnarray}$

(2.41)

where $\Theta_4 = \frac{8\alpha q}{3(2\alpha-1)q+3(1-\epsilon)-12}.$

We substitute (2.40) and (2.41) into (2.39) and then use Young's inequality to obtain

$\begin{eqnarray} && \frac{1}{2}(\|\nabla u(t)\|_{L^2}^2+\|\nabla v(t)\|_{L^2}^2+\|\nabla b(t)\|_{L^2}^2)+\frac{3}{4}\int_0^t(\|\Lambda ^{\alpha+1 } u(\tau)\|_{L^2}^2+\\ &&\|\Lambda ^{\alpha+1 } v(\tau)\|_{L^2}^2+\|\Lambda ^{\alpha+1 } b(\tau)\|_{L^2}^2)d\tau+\int_0^t\kappa\|\nabla \nabla\cdot v(\tau)\|_{L^2}^2d\tau \\ &\le& C+ C\int_0^t(\|b\|_{L^{q}}^{\frac{2\alpha q}{(2\alpha-1)q-3}}+\|v\|_{L^{q}}^{\frac{2\alpha q}{(2\alpha-1)q-3}}) \big(\| \nabla u\|_{L^{2} }^{2}+\| \nabla b\|_{L^{2} }^{2}+\| \nabla v\|_{L^{2} }^{2}\big) d\tau+\\ && C \int_0^t\big(\|u_3\|_{L^{q}}+\|b\|_{L^{q}}+\|v\|_{L^{q}}\big)^{\Theta_4}\big(\| \nabla u\|_{L^{2} }+\| \nabla b\|_{L^{2}}+\| \nabla v\|_{L^{2} }\big)^{2} d\tau +\\ && C \int_0^t\big(\|u_3\|_{L^{q}}+\|b\|_{L^{q}}+\|v\|_{L^{q}}\big)^{\frac{2\alpha q}{(2\alpha-1)q-3}}\big(\| \nabla u\|_{L^{2} }+\| \nabla b\|_{L^{2}}+\| \nabla v\|_{L^{2} }\big)^{2} d\tau +\\ && \frac{1}{8}\big[ \int_0^t\big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }+\| \Lambda ^{\alpha+1 } b\|_{L^{2} }+\| \Lambda ^{\alpha+1 } v\|_{L^{2} }\big)^{2}d\tau\big]\\ &\le& C+C \int_0^t\big(\|u_3\|_{L^{q}}^{\Theta_4}+\|b\|_{L^{q}}^{\Theta_4}+\|v\|_{L^{q}}^{\Theta_4}\big)\big(\| \nabla u\|_{L^{2} }+\| \nabla b\|_{L^{2}}+\| \nabla v\|_{L^{2} }\big)^{2} d\tau +\\ && \frac{1}{4} \int_0^t\big(\| \Lambda ^{\alpha+1 } u\|_{L^{2} }^{2}+\| \Lambda ^{\alpha+1 } b\|_{L^{2} }^{2}+\| \Lambda ^{\alpha+1 } v\|_{L^{2} }^{2}\big) d\tau. \end{eqnarray}$

(2.42)

Then we have

$\begin{eqnarray} && \|\nabla u(t)\|_{L^2}^2+\|\nabla v(t)\|_{L^2}^2+\|\nabla b(t)\|_{L^2}^2+\int_0^t(\|\Lambda ^{\alpha+1 } u(\tau)\|_{L^2}^2+\\ && \|\Lambda ^{\alpha+1 } v(\tau)\|_{L^2}^2+\|\Lambda ^{\alpha+1 } b(\tau)\|_{L^2}^2)d\tau+\int_0^t\kappa\|\nabla \nabla\cdot v(\tau)\|_{L^2}^2d\tau \\ &\le& C+C \int_0^t\big(\|u_3\|_{L^{q}}^{\Theta_4}+\|b\|_{L^{q}}^{\Theta_4}+\|v\|_{L^{q}}^{\Theta_4}\big)\big(\| \nabla u\|_{L^{2} }+\| \nabla b\|_{L^{2}}+\| \nabla v\|_{L^{2} }\big)^{2} d\tau. \end{eqnarray}$

(2.43)

Thanks to Gronwall's inequality and condition (1.3), we obtain

$\begin{eqnarray} && \|\nabla u(t)\|_{L^2}^2+\|\nabla v(t)\|_{L^2}^2+\|\nabla b(t)\|_{L^2}^2+\int_0^t(\|\Lambda ^{\alpha+1 } u(\tau)\|_{L^2}^2+\\ && \|\Lambda ^{\alpha+1 } v(\tau)\|_{L^2}^2+\|\Lambda ^{\alpha+1 } b(\tau)\|_{L^2}^2)d\tau+\int_0^t\kappa\|\nabla \nabla\cdot v(\tau)\|_{L^2}^2d\tau \\ &\le& C\exp \big[ C \int_0^t\big(\|u_3\|_{L^{q}}^{\Theta_4}+\|b\|_{L^{q}}^{\Theta_4}+\|v\|_{L^{q}}^{\Theta_4}\big)d\tau \big] < \infty. \end{eqnarray}$

(2.44)

Finally, we consider the case $q = \infty$ . By repeating the above procedure, we derive that

$\begin{equation*} \label{48} E(t) \le CJ_0+C \int_0^t\big(\|u_3\|_{L^{\infty}}+\|b\|_{L^{\infty}}+\|v\|_{L^{\infty}}\big)^{\frac{2\alpha }{2\alpha-1}}\big(\| \nabla u\|_{L^{2} }+\| \nabla b\|_{L^{2}}+\| \nabla v\|_{L^{2} }\big)^{2 }d\tau. \end{equation*}$

Thanks to Gronwall's inequality and condition (1.3), we obtain

$\begin{eqnarray} && \|\nabla u(t)\|_{L^2}^2+\|\nabla v(t)\|_{L^2}^2+\|\nabla b(t)\|_{L^2}^2 +\int_0^t(\|\Lambda ^{\alpha+1 } u(\tau)\|_{L^2}^2+\\ && \|\Lambda ^{\alpha+1 } v(\tau)\|_{L^2}^2+\|\Lambda ^{\alpha+1 } b(\tau)\|_{L^2}^2)d\tau +\int_0^t\kappa\|\nabla \nabla\cdot v(\tau)\|_{L^2}^2d\tau \\ &\le& C\exp \big[ C \int_0^t\big(\|u_3\|_{L^{\infty}}^{\frac{8\alpha }{3(2\alpha-1)}}+\|b\|_{L^{\infty}}^{\frac{8\alpha }{3(2\alpha-1)}}+\|v\|_{L^{\infty}}^{\frac{8\alpha }{3(2\alpha-1)}}\big)d\tau\big ] < \infty. \end{eqnarray}$

(2.45)

By the above steps, we establish a higher-order a priori estimate of the solutions, and then we obtain that the higher-order norm of the solutions is bounded, thus proving the smoothness of the solutions. This completes the proof of Theorem 1.1.

3. Conclusions

In this paper, the regularity criterion of the weak solution of the three-dimensional magnetic micropolar fluid equation when $1\le\alpha = \beta = \gamma \le \frac{3}{2}$ is studied. However, the regularity of the weak solution of the magnetic micropolar fluid equation when $1\le\alpha, \beta, \gamma \le \frac{3}{2}$ on $\mathbb{R}^3$ is still an open problem, and it is hoped that the method in this paper can provide inspiration for the solution of this problem.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was supported by [the Basic Research Project of Key Scientific Research Project Plan of Universities in Henan Province (Grant No. 20ZX002)].

Conflict of interest

The authors declare there is no conflict of interest.

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