# On three-dimensional Hall-magnetohydrodynamic equations with partial dissipation

## Abstract

In this paper, we address the Hall-MHD equations with partial dissipation. Applying some important inequalities (such as the logarithmic Sobolev inequality using BMO space, bilinear estimates in BMO space, Young’s inequality, cancellation property, interpolation inequality) and delicate energy estimates, we establish an improved blow-up criterion for the strong solution. Moreover, we also obtain the existence of the strong solution for small initial data, the smallness conditions of which are given by the suitable Sobolev norms.

## 1 Introduction

The incompressible Hall-magnetohydrodynamic equations with full dissipation in three dimensions read as:

\begin{aligned} \textstyle\begin{cases} u_{t}+(u\cdot \nabla )u+\nabla (p+\pi )=\kappa _{1}u_{x_{1}x_{1}}+ \kappa _{2}u_{x_{2}x_{2}} +\kappa _{3}u_{x_{3}x_{3}}+(B\cdot \nabla )B, \\ \operatorname{div}u=0, \\ B_{t}+(u\cdot \nabla )B=(B\cdot \nabla )u+\Delta B-\nabla \times (( \nabla \times B)\times B), \\ \operatorname{div}B=0, \\ u(0,x)=u_{0},\qquad B(0,x)=B_{0}. \end{cases}\displaystyle \end{aligned}
(1)

Here $$u(t,x)$$, $$B(t,x)$$ denote velocity field and magnetic field, respectively; $$\kappa _{1}$$, $$\kappa _{2}$$, $$\kappa _{3}$$ are the kinematic viscosity, $$(t,x) \in \mathbb{R}^{+}\times \mathbb{R}^{3}$$.

Compared to usual MHD system and the Boussinesq equations, Hall-MHD equations involve $$\nabla \times ((\nabla \times B)\times B)$$, it is Hall term and plays a crucial position in magnetic reconnection due to Ohm’s law. Magnetic reconnection corresponds to changes in the topology of magnetic field lines, which are ubiquitously observed in space. The Hall term becomes important when large magnetic shear appears because it has second-order derivatives, and it restores the influence of the electric current in the Lorentz force occurring in Ohm’s law, which was neglected in usual MHD. Therefore, Hall-MHD is very important for such problems as magnetic reconnection in neutron stars, geo-dynamo, space plasmas, and star formation. The paper  introduces the physical background to Hall-magnetohydrodynamics, and papers [7, 8, 13, 1518, 24] present the recent progress of the Hall-MHD system.

The nonlinear Jordan–Moore–Gibson–Thompson equation with memory read as

\begin{aligned} \tau u_{ttt}+u_{tt}-c^{2} \Delta u-\beta \Delta u_{t}- \int _{0}^{t}h(t-s) \Delta u(s)\,ds= \frac{\partial }{\partial t}\biggl(\frac{1}{c^{2}}\frac{B}{2A}(u_{t})^{2}+ \vert \nabla u \vert ^{2}\biggr), \end{aligned}

where $$u = u(x,t)$$ denotes the scalar acoustic velocity. The Jordan–Moore–Gibson–Thompson equation is one of the nonlinear sound equations that describe the propagation of sound waves in gases and liquids. Recent works on the Jordan–Moore–Gibson–Thompson equation can be found in [4, 12]. The Hall-MHD Eqs. (1) describe the magnetic properties for a conductive fluid moving in a magnetic field, in which magnetic reconnection happens in the case of large magnetic shear. In the Hall-MHD Eqs. (1), $$u= u(x,t)$$, $$B = B(x,t)$$ are non-dimensional quantities corresponding to the fluid velocity field, the magnetic field.

Many results on usual MHD system have been obtained in [10, 11, 14, 17, 2123, 2628, 3033]. However, the Hall-MHD system had few results until recently. The paper  got the local existence and global small solutions for the Hall-magnetohydrodynamics. Some results on the Boussinesq and MHD equations with partial viscosity were obtained in [5, 6, 15, 24]. Two new blow-up criteria for the system (1) with $$\kappa _{1}=\kappa _{2}=\kappa _{3}=1$$ were obtained by Chae and Lee in . Fei and Xiang  got a blow-up criterion and small existence to (1) with $$\kappa _{1}=\kappa _{2}=1$$, $$\kappa _{3}=0$$.

The paper  established regularity criterion for the Hall-MHD equations without viscosity and full dissipation, the papers [2, 3] obtained regularity criterion for the Hall-MHD equations with full viscosity and full dissipation in different spaces. In this paper, we investigate the Hall-magnetohydrodynamic system with full viscosity and partial dissipation. Inspired by [8, 13, 19, 33], we find a new blow-up criterion for strong solution, which imposes the condition is $$(u,\nabla B)\in L^{2}(0,T;\mathrm{BMO})$$. Additionally, we also get the existence of the strong solution for small initial data.

The first aim of this paper is to get blow-up criterion for the strong solution to (1) with $$\kappa _{1},\kappa _{2}>0$$, $$\kappa _{3}=0$$.

### Theorem 1.1

Assume that $$\kappa _{1}>0$$, $$\kappa _{2}>0$$, $$\kappa _{3}=0$$, $$(u_{0},B_{0})\in H^{3}(\mathbb{R}^{3})$$, and $$\operatorname{div}u_{0}=\operatorname{div}B_{0}=0$$, let $$T_{0}<\infty$$ be the first blow-up time to the problem (1), then

\begin{aligned} \limsup_{t\nearrow T_{0}}\bigl( \bigl\Vert u(t) \bigr\Vert _{H^{3}}^{2}+ \bigl\Vert B(t) \bigr\Vert _{H^{3}}^{2}\bigr)= \infty , \end{aligned}

is equivalent to

\begin{aligned} \int _{0}^{T_{0}}\bigl( \bigl\Vert u(t) \bigr\Vert _{\mathrm{BMO}}^{2}+ \bigl\Vert \nabla B(t) \bigr\Vert _{\mathrm{BMO}}^{2}\bigr)\,dt= \infty . \end{aligned}

### Remark 1.1

Compared to previous results, the blow-up condition $$\int _{0}^{T_{0}}(\|u\|_{\mathrm{BMO}}^{2}+ \|\nabla B\|_{\mathrm{BMO}}^{2})\,dt<\infty$$ instead of $$\int _{0}^{T^{*}}(\|\nabla u\|_{L^{p}}^{q}+\|\Delta B\|_{L^{\beta }}^{ \gamma })\,dt<\infty$$ with $$p, \beta \in (3,\infty ]$$ in . Noticing the fact $$W^{1,p}(\mathbb{R}^{3})\hookrightarrow L^{\infty }(\mathbb{R}^{3}) \hookrightarrow \mathrm{BMO}(\mathbb{R}^{3})$$, $$p>3$$, thus the above blow-up criterion is meaningful.

### Remark 1.2

The similar blow-up criterion can also be established for the system (1) with cases when $$\kappa _{1}=0,\kappa _{2}$$, $$\kappa _{3}>0$$ and $$\kappa _{1}>0$$, $$\kappa _{2}=0$$, $$\kappa _{3}>0$$.

Based on the Theorem 1.1, we can obtain the small initial data solutions to (1) with $$\kappa _{1},\kappa _{2}>0$$, $$\kappa _{3}=0$$.

### Theorem 1.2

Suppose the conditions in Theorem 1.1hold, there exists a universal positive constant $$\varepsilon ^{*}$$, then (1) has a solution $$(u,B)\in L^{\infty }(0,\infty ;~H^{3}(\mathbb{R}^{3}))$$, provided that $$\|u_{0}\|_{H^{2}}+\|B_{0}\|_{H^{2}}<\varepsilon ^{*}$$.

### Remark 1.3

Compared to , the smallness condition $$\|(u_{0},B_{0})\|_{H^{2}}$$ instead of $$\|(u_{0},B_{0})\|_{H^{3}}$$ in  is sufficiently small.

## 2 Notations and preliminaries

Through the paper, $$\partial _{k}$$ and $$u_{k}$$ represent the k th components of and u, and the following simplified notation will be adopted throughout the paper:

\begin{aligned}& \int _{\mathbb{R}^{3}}f\,dx:= \int \int \int _{\mathbb{R}^{3}}f(t,x)\,dx_{1}\,dx_{2}\,dx_{3};\qquad \Vert \cdot \Vert _{k}:= \Vert \cdot \Vert _{L^{k}}; \\& f_{0}:=f(0,x); \qquad \nabla _{p}:=(\partial _{1},\partial _{2},0);\qquad u_{p}:=(u_{1},u_{2},0). \end{aligned}

Next, some lemmas are given.

### Lemma 2.1

(See )

Let $$f,g,h,\nabla _{p}f,\nabla _{p}g,\partial _{3}h\in L^{2}(\mathbb{R}^{3})$$, then

\begin{aligned} \int _{\mathbb{R}^{3}}fgh\,dx\leq C \Vert f \Vert _{2}^{\frac{1}{2}} \Vert \nabla _{p}f \Vert _{2}^{\frac{1}{2}} \Vert g \Vert _{2}^{\frac{1}{2}} \Vert \nabla _{p}g \Vert _{2}^{ \frac{1}{2}} \Vert h \Vert _{2}^{\frac{1}{2}} \Vert \partial _{3}h \Vert _{2}^{ \frac{1}{2}}. \end{aligned}

### Lemma 2.2

(See )

Suppose $$\nabla g\in W^{1,q}(\mathbb{R}^{3})\cap L^{2}(\mathbb{R}^{3})$$, then

\begin{aligned} \Vert \nabla g \Vert _{L^{\infty }}\leq C \bigl[ \Vert \nabla g \Vert _{\mathrm{BMO}}\ln ^{\frac{1}{2}}\bigl(e+ \Vert \nabla g \Vert _{W^{1,q}}+ \Vert g \Vert _{L^{\infty }} \bigr)+1\bigr], \end{aligned}

here $$q>3$$.

### Lemma 2.3

(See [25, Lemma 1])

The bilinear estimates in BMO space, let $$h_{1},h_{2}\in \mathrm{BMO}\cap H^{|\zeta |+|\eta |}$$. Then

\begin{aligned} \bigl\Vert \partial ^{\zeta }h_{1} \cdot \partial ^{\eta }h_{2} \bigr\Vert _{2} \leq C\bigl( \Vert h_{1} \Vert _{\mathrm{BMO}} \bigl\Vert (- \Delta )^{\frac{ \vert \zeta \vert + \vert \eta \vert }{2}}h_{2} \bigr\Vert _{2} + \Vert h_{2} \Vert _{\mathrm{BMO}} \bigl\Vert (-\Delta )^{\frac{ \vert \zeta \vert + \vert \eta \vert }{2}}h_{1} \bigr\Vert _{2}\bigr), \end{aligned}

where $$\zeta =(\zeta _{1},\zeta _{2},\zeta _{3}), \eta =(\eta _{1},\eta _{2}, \eta _{3})$$, and $$|\zeta |,|\eta |\geq 1$$.

## 3 Proof of Theorem 1.1

We adopt the following notations: $$\nabla ^{\varrho }:=\partial ^{|\varrho |}/\partial _{1}^{\varrho _{1}} \partial _{2}^{\varrho _{2}}\partial _{3}^{\varrho _{3}}$$, where $$\varrho =(\varrho _{1},\varrho _{2},\varrho _{3})\in (\mathbb{N} \cup \{0\})^{3}$$ with $$|\varrho |=\varrho _{1}+\varrho _{2}+\varrho _{3}\leq 3$$, $$\kappa :=\min \{\kappa _{1},\kappa _{2}\}$$ and $$\kappa _{0}:=\min \{\kappa ,1\}$$.

Operating $$\nabla ^{\varrho }$$ on (1)1 and (1)3, and multiplying them by $$\nabla ^{\varrho }u$$ and $$\nabla ^{\varrho }B$$, respectively, and then integrating by parts, one gets

\begin{aligned} \begin{aligned}[b] &\frac{1}{2}\,\frac{d}{dt}\bigl( \bigl\Vert u(t) \bigr\Vert _{H^{3}}^{2}+ \bigl\Vert B(t) \bigr\Vert _{H^{3}}^{2}\bigr)+ \kappa _{1} \Vert \partial _{1}u \Vert _{H^{3}}^{2}+\kappa _{2} \Vert \partial _{2}u \Vert _{H^{3}}^{2}+ \Vert \nabla B \Vert _{H^{3}}^{2} \\ &\quad =-\sum_{0\leq \vert \varrho \vert \leq 3} \int _{\mathbb{R}^{3}}\nabla ^{ \varrho }\bigl[\nabla \times \bigl(( \nabla \times B)\times B\bigr)\bigr]\cdot \nabla ^{ \varrho }B\,dx \\ &\qquad {}-\sum_{0\leq \vert \varrho \vert \leq 3} \int _{\mathbb{R}^{3}}\nabla ^{ \varrho }(u\cdot \nabla B)\cdot \nabla ^{\varrho }B\,dx -\sum_{0\leq \vert \varrho \vert \leq 3} \int _{\mathbb{R}^{3}}\nabla ^{\varrho }(u\cdot \nabla u)\cdot \nabla ^{\varrho }u\,dx \\ &\qquad {}+\sum_{0\leq \vert \varrho \vert \leq 3} \int _{\mathbb{R}^{3}}\nabla ^{ \varrho }(B\cdot \nabla u)\cdot \nabla ^{\varrho }B\,dx +\sum_{0\leq \vert \varrho \vert \leq 3} \int _{\mathbb{R}^{3}}\nabla ^{\varrho }(B\cdot \nabla B)\cdot \nabla ^{\varrho }u\,dx \\ &\quad :=H_{1}+H_{2}+H_{3}+H_{4}+H_{5}. \end{aligned} \end{aligned}
(2)

Using Lemma 2.2, we have

\begin{aligned} \Vert \nabla B \Vert _{L^{\infty }}\leq C \bigl[ \Vert \nabla B \Vert _{\mathrm{BMO}}\ln ^{\frac{1}{2}}\bigl(e+ \Vert \nabla B \Vert _{W^{1,\frac{7}{2}}}+ \Vert B \Vert _{L^{\infty }} \bigr)+1\bigr] .\end{aligned}

Noticing the fact that $$H^{2}(\mathbb{R}^{3})\hookrightarrow W^{1,\frac{7}{2}}(\mathbb{R}^{3})$$, and $$H^{3}(\mathbb{R}^{3})\hookrightarrow L^{\infty }(\mathbb{R}^{3})$$, we get

\begin{aligned} \Vert \nabla B \Vert _{L^{\infty }}\leq C \bigl[ \Vert \nabla B \Vert _{\mathrm{BMO}}\ln ^{\frac{1}{2}}\bigl(e+ \Vert B \Vert _{H^{3}}\bigr)+1\bigr] .\end{aligned}

By the above inequality, cancellation property and Young’s inequality, one obtains

\begin{aligned} \begin{aligned}[b] \vert H_{1} \vert &\leq C \Vert B \Vert _{H^{3}} \Vert \nabla B \Vert _{\infty } \Vert \nabla B \Vert _{H^{3}} \\ &\leq \frac{1}{8} \Vert \nabla B \Vert _{H^{3}}^{2}+C \Vert \nabla B \Vert _{\infty }^{2} \Vert B \Vert _{H^{3}}^{2} \\ &\leq \frac{1}{8} \Vert \nabla B \Vert _{H^{3}}^{2}+C \bigl(\ln \bigl(e+ \Vert B \Vert _{H^{3}}\bigr) \Vert \nabla B \Vert _{\mathrm{BMO}}^{2}+1\bigr) \Vert B \Vert _{H^{3}}^{2}. \end{aligned} \end{aligned}
(3)

We apply cancellation property and Lemma 2.3 to deduce that

\begin{aligned} \begin{aligned}[b] \vert H_{2} \vert &= \biggl\vert \sum_{0\leq \vert \varrho \vert \leq 3} \int _{\mathbb{R}^{3}}\bigl[ \nabla ^{\varrho }(u\cdot \nabla B)-(u \cdot \nabla )\nabla ^{\varrho }B\bigr] \cdot \nabla ^{\varrho }B\,dx \biggr\vert \\ &\leq C\bigl( \Vert B \Vert _{H^{3}} \Vert \nabla B \Vert _{H^{3}} \Vert u \Vert _{\mathrm{BMO}}+ \Vert B \Vert _{H^{3}} \Vert u \Vert _{H^{3}} \Vert \nabla B \Vert _{\mathrm{BMO}}\bigr) \\ &\leq \frac{1}{8} \Vert \nabla B \Vert _{H^{3}}^{2}+C \Vert u \Vert _{\mathrm{BMO}}^{2} \Vert B \Vert _{H^{3}}^{2} +C \Vert \nabla B \Vert _{\mathrm{BMO}} \Vert u \Vert _{H^{3}} \Vert B \Vert _{H^{3}}. \end{aligned} \end{aligned}
(4)

For $$H_{3}$$, when $$|\varrho |=0$$, the $$H_{3}$$ have cancelled. When $$|\varrho |=1$$, by $$\operatorname{div} u=0$$, $$H_{3}$$ can be rewritten as follows

\begin{aligned} H_{31}&=- \int _{\mathbb{R}^{3}}(\nabla u\cdot \nabla )u\cdot \nabla u\,dx \\ &=- \int _{\mathbb{R}^{3}}(\nabla _{p}u\cdot \nabla )u \nabla _{p}u\,dx- \int _{\mathbb{R}^{3}}(\partial _{3} u_{p}\cdot \nabla _{p})u \partial _{3}u\,dx+ \int _{\mathbb{R}^{3}}(\nabla _{p}\cdot u_{p}) \partial _{3}u\partial _{3}u\,dx. \end{aligned}

Thus using the Höledr inequality and Lemma 2.3, one obtains

\begin{aligned} \begin{aligned}[b] \vert H_{31} \vert &\leq C \Vert \nabla u\nabla _{p}u \Vert _{2} \Vert \nabla u \Vert _{2} \\ &\leq C \Vert \nabla _{p}\nabla u \Vert _{2} \Vert u \Vert _{\mathrm{BMO}} \Vert \nabla u \Vert _{2} \\ &\leq C \Vert u \Vert _{\mathrm{BMO}} \Vert \nabla _{p}u \Vert _{H^{1}} \Vert u \Vert _{H^{1}} \\ &\leq \frac{3\kappa }{36} \Vert \nabla _{p}u \Vert _{H^{1}}^{2}+C \Vert u \Vert _{\mathrm{BMO}}^{2} \Vert u \Vert _{H^{1}}^{2}. \end{aligned} \end{aligned}
(5)

When $$|\varrho |=2$$, one can write $$H_{3}$$ as

\begin{aligned} H_{32}&=- \int _{\mathbb{R}^{3}}\bigl(\nabla ^{2}u\cdot \nabla \bigr)u \nabla ^{2}u\,dx-2 \int _{\mathbb{R}^{3}}(\nabla u\cdot \nabla )\nabla u \nabla ^{2}u\,dx \\ &=H_{321}+H_{322} \end{aligned}

$$H_{321}$$, $$H_{322}$$ can be further decomposed into three parts, respectively.

\begin{aligned}& \begin{aligned} H_{321}={}&{-} \int _{\mathbb{R}^{3}}(\nabla \nabla _{p}u\cdot \nabla )u \nabla \nabla _{p}u\,dx - \int _{\mathbb{R}^{3}}\bigl(\partial _{3}^{2}u_{p} \cdot \nabla _{p}\bigr)u\partial _{3}^{2}u\,dx\\ &{} + \int _{\mathbb{R}^{3}}( \partial _{3}\nabla _{p} \cdot u_{p})\partial _{3}u\partial _{3}^{2}u\,dx \\ ={}&H_{3211}+H_{3212}+H_{3213}. \end{aligned} \\& \begin{aligned} H_{322}={}&{-}2 \int _{\mathbb{R}^{3}}(\nabla _{p}u\cdot \nabla )\nabla u \nabla \nabla _{p}u\,dx -2 \int _{\mathbb{R}^{3}}(\partial _{3}u_{p} \cdot \nabla _{p})\nabla u\partial _{3}^{2}u\,dx\\ &{} +2 \int _{\mathbb{R}^{3}}( \nabla _{p}\cdot u_{p}) \partial _{3}\nabla u\partial _{3}^{2}u\,dx \\ ={}&H_{3221}+H_{3222}+H_{3223}. \end{aligned} \end{aligned}

By the Höledr inequality and Lemma 2.3, we have

\begin{aligned} \begin{aligned}[b] \vert H_{3211} \vert &\leq C \Vert \nabla _{p}\nabla u\nabla u \Vert _{2} \bigl\Vert \nabla ^{2}u \bigr\Vert _{2} \\ &\leq C \Vert u \Vert _{\mathrm{BMO}} \bigl\Vert \nabla _{p} \nabla ^{2} u \bigr\Vert _{2} \bigl\Vert \nabla ^{2}u \bigr\Vert _{2} \\ &\leq C \Vert u \Vert _{\mathrm{BMO}} \Vert \nabla _{p}u \Vert _{H^{2}} \Vert u \Vert _{H^{2}} \\ &\leq \frac{\kappa }{36} \Vert \nabla _{p}u \Vert _{H^{2}}^{2}+C \Vert u \Vert _{\mathrm{BMO}}^{2} \Vert u \Vert _{H^{2}}^{2}, \end{aligned} \end{aligned}
(6)

and

\begin{aligned} \begin{aligned}[b] \vert H_{3212} \vert &\leq C \bigl\Vert \nabla _{p}u\partial _{3}^{2}u \bigr\Vert _{2} \bigl\Vert \partial _{3}^{2}u \bigr\Vert _{2} \\ &\leq C \Vert u \Vert _{\mathrm{BMO}} \bigl\Vert \nabla _{p} \nabla ^{2} u \bigr\Vert _{2} \bigl\Vert \nabla ^{2}u \bigr\Vert _{2} \\ &\leq C \Vert u \Vert _{\mathrm{BMO}} \Vert \nabla _{p}u \Vert _{H^{2}} \Vert u \Vert _{H^{2}} \\ &\leq \frac{\kappa }{36} \Vert \nabla _{p}u \Vert _{H^{2}}^{2}+C \Vert u \Vert _{\mathrm{BMO}}^{2} \Vert u \Vert _{H^{2}}^{2}. \end{aligned} \end{aligned}
(7)

Similarly to $$H_{3211}$$ and $$H_{3212}$$, we have

\begin{aligned} \begin{aligned} \vert H_{3213},H_{3221},H_{3222},H_{3223} \vert \leq \frac{\kappa }{36} \Vert \nabla _{p}u \Vert _{H^{2}}^{2}+C \Vert u \Vert _{\mathrm{BMO}}^{2} \Vert u \Vert _{H^{2}}^{2}. \end{aligned} \end{aligned}
(8)

Collecting (7), (8), and (9), we have

\begin{aligned} \begin{aligned} \vert H_{32} \vert \leq \frac{6\kappa }{36} \Vert \nabla _{h}u \Vert _{H^{2}}^{2}+C \Vert u \Vert _{\mathrm{BMO}}^{2} \Vert u \Vert _{H^{2}}^{2}. \end{aligned} \end{aligned}
(9)

When $$|\varrho |=3$$, we rewrite $$H_{3}$$ as follows

\begin{aligned} H_{33}={}&{-} \int _{\mathbb{R}^{3}}\bigl(\nabla ^{3}u\cdot \nabla \bigr)u \cdot \nabla ^{3}u\,dx -3 \int _{\mathbb{R}^{3}}\bigl(\nabla ^{2}u\cdot \nabla \bigr) \nabla u\cdot \nabla ^{3}u\,dx\\ &{} -3 \int _{\mathbb{R}^{3}}(\nabla u\cdot \nabla )\nabla ^{2}u \cdot \nabla ^{3}u\,dx \\ ={}&H_{331}+H_{332}+H_{3233}. \end{aligned}

Since

\begin{aligned} H_{331}={}&{-} \int _{\mathbb{R}^{3}}\bigl(\nabla ^{2}\nabla _{p}u\cdot \nabla \bigr)u \nabla ^{2}\nabla _{p}u\,dx - \int _{\mathbb{R}^{3}}\bigl(\partial _{3}^{3}u_{p} \cdot \nabla _{p}\bigr)u\partial _{3}^{3}u\,dx \\ &{}+ \int _{\mathbb{R}^{3}}\bigl( \partial _{3}^{2} \nabla _{p}\cdot u_{p}\bigr)\partial _{3}u \partial _{3}^{3}u\,dx \\ ={}&H_{3311}+H_{3312}+H_{3313}, \end{aligned}

and

\begin{aligned} H_{332}={}&{-}3 \int _{\mathbb{R}^{3}}\bigl(\nabla ^{2}u\cdot \nabla \bigr) \nabla _{p}u \nabla ^{2}\nabla _{p}u\,dx -3 \int _{\mathbb{R}^{3}}\bigl(\partial _{3}^{2}u_{p} \cdot \nabla _{p}\bigr)\partial _{3}u\partial _{3}^{3}u\,dx \\ &{}+3 \int _{ \mathbb{R}^{3}}(\partial _{3}\nabla _{p} \cdot u_{p})\partial _{3}^{2}u \partial _{3}^{3}u\,dx \\ ={}&H_{3321}+H_{3322}+H_{3323}. \end{aligned}

Applying the Höledr inequality, Lemma 2.3, and Young’s inequality, one has

\begin{aligned} \begin{aligned}[b] \vert H_{3311} \vert &\leq C \bigl\Vert \nabla _{p}\nabla ^{2}u\nabla u \bigr\Vert _{2} \bigl\Vert \nabla _{p} \nabla ^{2}u \bigr\Vert _{2} \\ &\leq C \Vert u \Vert _{\mathrm{BMO}} \bigl\Vert \nabla ^{3} \nabla _{p}u \bigr\Vert _{2} \bigl\Vert \nabla _{p}\nabla ^{2}u \bigr\Vert _{2} \\ &\leq C \Vert u \Vert _{\mathrm{BMO}} \Vert \nabla _{p}u \Vert _{H^{3}} \Vert u \Vert _{H^{3}} \\ &\leq \frac{\kappa }{36} \Vert \nabla _{p}u \Vert _{H^{3}}^{2}+C \Vert u \Vert _{\mathrm{BMO}}^{2} \Vert u \Vert _{H^{3}}^{2}, \end{aligned} \end{aligned}
(10)

and

\begin{aligned} \begin{aligned}[b] \vert H_{3312} \vert &\leq C \bigl\Vert \partial _{3}^{3}u_{p}\nabla _{p}u \bigr\Vert _{2} \bigl\Vert \partial _{3}^{3}u \bigr\Vert _{2} \\ &\leq C \Vert u \Vert _{\mathrm{BMO}} \bigl\Vert \nabla _{p} \nabla ^{3}u \bigr\Vert _{2} \bigl\Vert \nabla ^{3}u \bigr\Vert _{2} \\ &\leq C \Vert u \Vert _{\mathrm{BMO}} \Vert \nabla _{p}u \Vert _{H^{3}} \Vert u \Vert _{H^{3}} \\ &\leq \frac{\kappa }{36} \Vert \nabla _{p}u \Vert _{H^{3}}^{2}+C \Vert u \Vert _{\mathrm{BMO}}^{2} \Vert u \Vert _{H^{3}}^{2}. \end{aligned} \end{aligned}
(11)

Similarly to (11) and (12), one obtains

\begin{aligned} \begin{aligned}[b] \vert H_{3321} \vert &\leq C \bigl\Vert \nabla _{p}\nabla ^{2}u \bigr\Vert _{2} \bigl\Vert \nabla ^{2}u\nabla _{p} \nabla u \bigr\Vert _{2} \\ &\leq C \bigl\Vert \nabla ^{3}\nabla _{p}u \bigr\Vert _{2} \bigl\Vert \nabla ^{2}\nabla _{p}u \bigr\Vert _{2} \Vert u \Vert _{\mathrm{BMO}} \\ &\leq \frac{\kappa }{36} \Vert \nabla _{p}u \Vert _{H^{3}}^{2}+C \Vert u \Vert _{\mathrm{BMO}}^{2} \Vert u \Vert _{H^{3}}^{2}, \end{aligned} \end{aligned}
(12)

and

\begin{aligned} \begin{aligned}[b] \vert H_{3322} \vert &\leq C \bigl\Vert \partial _{3}^{2}u_{p}\nabla _{p}\partial _{3}u \bigr\Vert _{2} \bigl\Vert \partial _{3}^{3}u \bigr\Vert _{2} \\ &\leq C \Vert u \Vert _{\mathrm{BMO}} \bigl\Vert \nabla _{p} \nabla ^{3} u \bigr\Vert _{2} \bigl\Vert \nabla ^{3}u \bigr\Vert _{2} \\ &\leq \frac{\kappa }{36} \Vert \nabla _{p}u \Vert _{H^{3}}^{2}+C \Vert u \Vert _{\mathrm{BMO}}^{2} \Vert u \Vert _{H^{3}}^{2}. \end{aligned} \end{aligned}
(13)

One can estimate $$H_{3313}$$, $$H_{3323}$$ as $$H_{3312}$$, $$H_{3322}$$ to get

\begin{aligned} \begin{aligned} \vert H_{3313},H_{3323} \vert \leq \frac{\kappa }{36} \Vert \nabla _{p}u \Vert _{H^{3}}^{2}+C \Vert u \Vert _{\mathrm{BMO}}^{2} \Vert u \Vert _{H^{3}}^{2}. \end{aligned} \end{aligned}
(14)

Clearly, $$H_{333}$$ can be similarly estimated as $$H_{331}$$, so we have

\begin{aligned} \begin{aligned} \vert H_{333} \vert \leq \frac{3\kappa }{36} \Vert \nabla _{p}u \Vert _{H^{3}}^{2}+C \Vert u \Vert _{\mathrm{BMO}}^{2} \Vert u \Vert _{H^{3}}^{2}. \end{aligned} \end{aligned}
(15)

Putting (10)–(15) together, we obtain

\begin{aligned} \begin{aligned} \vert H_{33} \vert \leq \frac{9\kappa }{36} \Vert \nabla _{p}u \Vert _{H^{3}}^{2}+C \Vert u \Vert _{\mathrm{BMO}}^{2} \Vert u \Vert _{H^{3}}^{2}. \end{aligned} \end{aligned}
(16)

Combining (5), (9), and (16), we get

\begin{aligned} \begin{aligned} \vert H_{3} \vert \leq \frac{18\kappa }{36} \Vert \nabla _{p}u \Vert _{H^{3}}^{2}+C \Vert u \Vert _{\mathrm{BMO}}^{2} \Vert u \Vert _{H^{3}}^{2}. \end{aligned} \end{aligned}
(17)

Applying cancellation property and integration by parts, one can deduce that

\begin{aligned} H_{4}+H_{5}&=\sum _{0\leq |\varrho |\leq 3} \int _{\mathbb{R}^{3}}\bigl[ \nabla ^{\varrho }(B u)- B\nabla ^{\varrho }u\bigr]\cdot \nabla \nabla ^{ \varrho }B+\bigl[\nabla ^{\varrho }(B\cdot D B)- (B\cdot D)\nabla ^{\varrho }B\bigr] \cdot \nabla ^{\varrho }u\,dx \\ &=H_{41}+H_{42}. \end{aligned}

By the Höledr inequality, Lemma 2.3, and Young’s inequality, we get

\begin{aligned} \begin{aligned}[b] \vert H_{41} \vert &\leq C \Vert \nabla B \Vert _{H^{3}}\bigl( \Vert u \Vert _{H^{3}} \Vert B \Vert _{\mathrm{BMO}}+ \Vert B \Vert _{H^{3}} \Vert u \Vert _{\mathrm{BMO}}\bigr) \\ &\leq \frac{1}{8} \Vert \nabla B \Vert _{H^{3}}^{2}+C \bigl( \Vert B \Vert _{H^{3}}^{2}+ \Vert u \Vert _{H^{3}}^{2}\bigr) \bigl( \Vert u \Vert _{\mathrm{BMO}}^{2}+ \Vert B \Vert _{\mathrm{BMO}}^{2} \bigr), \end{aligned} \end{aligned}
(18)

and

\begin{aligned} \begin{aligned}[b] \vert H_{42} \vert &\leq C \Vert B \Vert _{\mathrm{BMO}} \Vert \nabla B \Vert _{H^{3}} \Vert u \Vert _{H^{3}} \\ &\leq \frac{1}{8} \Vert \nabla B \Vert _{H^{3}}^{2}+C \Vert B \Vert _{\mathrm{BMO}}^{2} \Vert u \Vert _{H^{3}}^{2}. \end{aligned} \end{aligned}
(19)

Collecting (18) and (19), we have

\begin{aligned} \vert H_{4}+H_{5} \vert \leq \frac{2}{8} \Vert \nabla B \Vert _{H^{3}}^{2}+C \bigl( \Vert B \Vert _{\mathrm{BMO}}^{2}+ \Vert u \Vert _{\mathrm{BMO}}^{2}\bigr) \bigl( \Vert u \Vert _{H^{3}}^{2}+ \Vert B \Vert _{H^{3}}^{2} \bigr). \end{aligned}
(20)

Combining (2)–(4), (17), and (20), we get

\begin{aligned} \begin{aligned}[b] &\frac{d}{dt}\bigl( \Vert u \Vert _{H^{3}}+ \Vert B \Vert _{H^{3}}\bigr)+\kappa \Vert \nabla _{p}u \Vert _{H^{3}}+ \Vert \nabla B \Vert _{H^{3}} \\ &\quad \leq C\bigl(1+ \Vert B \Vert _{\mathrm{BMO}}^{2}+ \Vert u \Vert _{\mathrm{BMO}}^{2}+ \Vert \nabla B \Vert _{\mathrm{BMO}}^{2}\ln \bigl(e+ \Vert B \Vert _{H^{3}} \bigr)\bigr) \bigl( \Vert u \Vert _{H^{3}}^{2}+ \Vert B \Vert _{H^{3}}^{2}\bigr). \end{aligned} \end{aligned}
(21)

Setting $$R(t):=e+\|u\|_{H^{3}}+\|B\|_{H^{3}}$$, from (21), one obtains

$$\frac{d}{dt}R(t)\leq C\bigl( \Vert \nabla B \Vert _{\mathrm{BMO}}^{2}+ \Vert u \Vert _{\mathrm{BMO}}^{2}+C\bigr)R(t) \ln R(t).$$

Applying the Gronwall inequality, one gets

$$\sup_{0\leq t\leq T}R(t)\leq \bigl( \Vert u_{0} \Vert _{H^{3}}^{2}+ \Vert B_{0} \Vert _{H^{3}}^{2}+e\bigr) \exp \biggl(C\exp \biggl( \int _{0}^{T} \Vert u \Vert _{\mathrm{BMO}}^{2}+ \Vert \nabla B \Vert _{\mathrm{BMO}}^{2}\,dt \biggr)\biggr),$$

which implies the blow-up criterion in Theorem 1.1 holds.

## 4 Proof of Theorem 1.2

Operating to (1)1, (1)3, taking the scalar product of them with u, B, one gets

\begin{aligned} \begin{aligned}[b] &\frac{1}{2}\,\frac{d}{dt}\bigl( \Vert \nabla u \Vert _{2}^{2}+ \Vert \nabla B \Vert _{2}^{2}\bigr)+ \kappa _{1} \Vert \partial _{1}\nabla u \Vert _{2}^{2}+\kappa _{2} \Vert \partial _{2} \nabla u \Vert _{2}^{2}+ \Vert \Delta B \Vert _{2}^{2} \\ &\quad =- \int _{\mathbb{R}^{3}}\nabla \bigl[\nabla \times \bigl((\nabla \times B) \times B\bigr)\bigr]\cdot \nabla B\,dx - \int _{\mathbb{R}^{3}}\nabla (u\cdot \nabla B)\cdot \nabla B\,dx \\ &\qquad {}- \int _{\mathbb{R}^{3}}\nabla (u\cdot \nabla u)\cdot \nabla u\,dx + \int _{\mathbb{R}^{3}}\nabla (B\cdot \nabla B)\cdot \nabla u\,dx + \int _{ \mathbb{R}^{3}}\nabla (B\cdot \nabla u)\cdot \nabla B\,dx \\ &\quad :=K_{1}+K_{2}+K_{3}+K_{4}+K_{5}. \end{aligned} \end{aligned}
(22)

Firstly, applying the Hölder inequality, commutator estimate and interpolation, one gets

\begin{aligned} \begin{aligned}[b] \vert K_{1} \vert &\leq C \bigl\Vert \nabla \bigl((\nabla \times B)\times B\bigr)-\nabla (\nabla \times B) \times B \bigr\Vert _{\frac{6}{5}} \bigl\Vert \nabla ^{2}B \bigr\Vert _{6} \\ &\leq C \Vert \nabla B \Vert _{2} \Vert \nabla B \Vert _{3} \bigl\Vert \nabla ^{2}B \bigr\Vert _{6} \\ &\leq C \Vert \nabla B \Vert _{2} \bigl\Vert \nabla ^{3} B \bigr\Vert _{2}^{2}, \end{aligned} \end{aligned}
(23)

here we use the fact that $$\|\nabla B\|_{3}\leq C\| B\|_{2}^{\frac{1}{2}}\|\nabla ^{3}B\|_{2}^{ \frac{1}{2}}$$, $$\|\nabla ^{2}B\|_{6}\leq C\|\nabla ^{3}B\|_{2}$$ due to the Gagliardo-Nirenberg-Sobolev inequality. By the Hölder inequality, one obtains

\begin{aligned} \begin{aligned}[b] \vert K_{2} \vert &\leq C \Vert \nabla u \Vert _{2} \Vert \nabla B \Vert _{4}^{2} \\ &\leq C \Vert \nabla u \Vert _{2} \bigl\Vert \nabla ^{2}B \bigr\Vert _{2}^{2}. \end{aligned} \end{aligned}
(24)

Reviewing $$H_{31}$$ in Sect. 3, we know $$K_{3}=H_{31}$$. Hence, applying Lemma 2.1, one obtains

\begin{aligned} \begin{aligned}[b] \vert K_{3} \vert &\leq C \Vert \nabla u \Vert _{2}^{\frac{1}{2}} \Vert \nabla u \Vert _{2}^{ \frac{1}{2}} \Vert \nabla _{p} u \Vert _{2}^{\frac{1}{2}} \Vert \nabla \nabla _{p}u \Vert _{2}^{\frac{1}{2}} \Vert \nabla \nabla _{p}u \Vert _{2}^{\frac{1}{2}} \Vert \nabla \nabla _{p} u \Vert _{2}^{\frac{1}{2}} \\ &\leq C \Vert \nabla _{p}\nabla u \Vert _{2}^{2} \Vert \nabla u \Vert _{2}. \end{aligned} \end{aligned}
(25)

$$K_{4}+K_{5}$$ can be written into two parts:

$$K_{4}+K_{5}= \int _{\mathbb{R}^{3}}(\nabla B\cdot \nabla u)\cdot \nabla B\,dx + \int _{\mathbb{R}^{3}}(\nabla B\cdot \nabla B)\cdot \nabla u\,dx.$$

By the Höledr inequality, we obtain

\begin{aligned} \begin{aligned}[b] \vert K_{4}+K_{5} \vert &\leq C \Vert \nabla u \Vert _{2} \Vert \nabla B \Vert _{4}^{2} \\ &\leq C \Vert \nabla u \Vert _{2} \bigl\Vert \nabla ^{2}B \bigr\Vert _{2}^{2}. \end{aligned} \end{aligned}
(26)

Combining (22)–(26), we get

\begin{aligned} \begin{aligned}[b] &\frac{1}{2}\,\frac{d}{dt}\bigl( \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2}+ \bigl\Vert \nabla B(t) \bigr\Vert _{2}^{2}\bigr) +\kappa \Vert \nabla _{p}\nabla u \Vert _{2}^{2}+ \Vert \Delta B \Vert _{2}^{2} \\ &\quad \leq C\bigl( \Vert \nabla B \Vert _{2}+ \Vert \nabla u \Vert _{2}\bigr) \bigl( \Vert \Delta B \Vert _{2}^{2}+ \Vert \nabla \nabla _{p}u \Vert _{2}^{2}\bigr) +C \Vert \nabla B \Vert _{2} \bigl\Vert \nabla ^{3} B \bigr\Vert _{2}^{2}. \end{aligned} \end{aligned}
(27)

Similarly to derivation of (22), one gets

\begin{aligned} \begin{aligned}[b] &\frac{1}{2}\,\frac{d}{dt}\bigl( \bigl\Vert \Delta u(t) \bigr\Vert _{2}^{2}+ \bigl\Vert \Delta B(t) \bigr\Vert _{2}^{2}\bigr)+ \kappa _{1} \Vert \partial _{1}\Delta u \Vert _{2}^{2}+\kappa _{2} \Vert \partial _{2} \Delta u \Vert _{2}^{2}+ \bigl\Vert \nabla ^{3} B \bigr\Vert _{2}^{2} \\ &\quad =- \int _{\mathbb{R}^{3}}D^{2}\bigl[\nabla \times \bigl((\nabla \times B) \times B\bigr)\bigr]\cdot D^{2}B\,dx - \int _{\mathbb{R}^{3}}D^{2}(u\cdot \nabla B) \cdot D^{2}B\,dx \\ &\qquad {}- \int _{\mathbb{R}^{3}}D^{2}(u\cdot \nabla u)\cdot D^{2}u\,dx + \int _{ \mathbb{R}^{3}}D^{2}(B\cdot \nabla u)\cdot D^{2}B\,dx + \int _{ \mathbb{R}^{3}}D^{2}(B\cdot \nabla B)\cdot D^{2}u\,dx \\ &\quad :=E_{1}+E_{2}+E_{3}+E_{4}+E_{5}. \end{aligned} \end{aligned}
(28)

We apply cancellation property, the Hölder inequality, commutator estimate to estimate $$E_{1}$$ as follows

\begin{aligned} \begin{aligned}[b] \vert E_{1} \vert &\leq C \bigl\Vert D^{2}\bigl[(\nabla \times B)\times B\bigr]-D^{2}( \nabla \times B) \times B \bigr\Vert _{2} \bigl\Vert \nabla ^{3}B \bigr\Vert _{2} \\ &\leq C \Vert \nabla B \Vert _{3} \Vert \Delta B \Vert _{6} \bigl\Vert \nabla ^{3}B \bigr\Vert _{2} \\ &\leq C \Vert \Delta B \Vert _{2} \bigl\Vert \nabla ^{3}B \bigr\Vert _{2}^{2}. \end{aligned} \end{aligned}
(29)

$$E_{2}$$ can be split into two terms:

\begin{aligned} E_{2}&=- \int _{\mathbb{R}^{3}}\bigl(D^{2}u\cdot \nabla \bigr)B\cdot D^{2}B\,dx -2 \int _{\mathbb{R}^{3}}(D u\cdot \nabla )\nabla B\cdot D^{2}B\,dx \\ &=E_{21}+E_{22}. \end{aligned}

Noticing the fact that $$\|\nabla B\|_{3}\leq C\| B\|_{2}^{\frac{1}{2}}\|\nabla ^{3}B\|_{2}^{ \frac{1}{2}}$$, $$\|\nabla ^{2}B\|_{6}\leq C\|\nabla ^{3}B\|_{2}$$ due to the Gagliardo-Nirenberg-Sobolev inequality, we have

\begin{aligned} \vert E_{21} \vert \leq& C \bigl\Vert \nabla ^{2}u \bigr\Vert _{2} \Vert \nabla B \Vert _{3} \bigl\Vert \nabla ^{2}B \bigr\Vert _{6} \leq C \bigl\Vert \nabla ^{2}u \bigr\Vert _{2} \bigl\Vert \nabla ^{3}B \bigr\Vert _{2}^{2}, \end{aligned}
(30)

and

\begin{aligned} \vert E_{22} \vert \leq &C \Vert \nabla u \Vert _{3} \bigl\Vert \nabla ^{2}B \bigr\Vert _{3}^{2} \leq C \bigl\Vert \nabla ^{2}u \bigr\Vert _{2}^{\frac{3}{4}} \bigl\Vert \nabla ^{3}B \bigr\Vert _{2}^{ \frac{5}{3}} \leq C \bigl\Vert \nabla ^{2}u \bigr\Vert _{2} \bigl\Vert \nabla ^{3}B \bigr\Vert _{2}^{2}. \end{aligned}
(31)

Collecting (30) and (31), we have

\begin{aligned} \vert E_{2} \vert \leq C \bigl\Vert \nabla ^{2}u \bigr\Vert _{2} \bigl\Vert \nabla ^{3}B \bigr\Vert _{2}^{2}. \end{aligned}
(32)

Obviously, $$E_{3}=H_{32}$$, hence we get

\begin{aligned} E_{3}&=H_{32} \\ &=H_{321}+H_{322} \\ &=H_{3211}+H_{3212}+H_{3213}+H_{3221}+H_{3222}+H_{3223}. \end{aligned}

One can use the Hölder inequality to deduce that

\begin{aligned}& \begin{aligned}[b] \vert H_{3211} \vert &= \biggl\vert \int _{\mathbb{R}^{3}}(\nabla _{p}\nabla u\cdot \nabla )u \cdot \nabla _{p}\nabla u dx \biggr\vert \\ &\leq C \Vert \nabla _{p}\nabla u \Vert _{2}^{\frac{1}{2}} \Vert \nabla u \Vert _{2}^{ \frac{1}{2}} \Vert \nabla _{p}\nabla u \Vert _{2}^{\frac{1}{2}} \bigl\Vert \nabla _{p}^{2} \nabla u \bigr\Vert _{2}^{\frac{1}{2}} \Vert \nabla _{p}\nabla u \Vert _{2}^{\frac{1}{2}} \Vert \partial _{3}\nabla _{p}\nabla u \Vert _{L^{2}}^{\frac{1}{2}} \\ &\leq C \bigl\Vert \nabla _{p}\nabla ^{2}u \bigr\Vert _{2} \Vert \nabla _{p}\nabla u \Vert _{2}^{ \frac{3}{2}} \Vert \nabla u \Vert _{2}^{\frac{1}{2}} \\ &\leq C \bigl\Vert \nabla _{p}\nabla ^{2}u \bigr\Vert _{2} \Vert \nabla _{p}\nabla u \Vert _{2} \Vert \nabla _{p}\nabla u \Vert _{2}^{\frac{1}{2}} \Vert \nabla u \Vert _{2}^{\frac{1}{2}} \\ &\leq C \bigl\Vert \nabla _{p}\nabla ^{2}u \bigr\Vert _{2}^{2} \bigl\Vert \nabla ^{2}u \bigr\Vert _{2}. \end{aligned} \end{aligned}
(33)
\begin{aligned}& \begin{aligned}[b] \vert H_{3212} \vert &= \biggl\vert \int _{\mathbb{R}^{3}}\bigl(\partial _{3}^{2}u_{p} \cdot \nabla _{p}\bigr)u\cdot \partial _{3}^{2}u\,dx \biggr\vert \\ &\leq C \bigl\Vert \partial _{3}^{2}u \bigr\Vert _{2}^{\frac{1}{2}} \Vert \nabla _{p}u \Vert _{2}^{ \frac{1}{2}} \bigl\Vert \partial _{3}^{2} u \bigr\Vert _{2}^{\frac{1}{2}} \bigl\Vert \nabla _{p} \partial _{3}^{2}u \bigr\Vert _{2}^{\frac{1}{2}} \Vert \nabla _{p}\partial _{3} u \Vert _{2}^{ \frac{1}{2}} \bigl\Vert \nabla _{p}\partial _{3}^{2}u \bigr\Vert _{2}^{\frac{1}{2}} \\ &\leq C \bigl\Vert \nabla _{p}\nabla ^{2}u \bigr\Vert _{2} \bigl\Vert \nabla ^{2}u \bigr\Vert _{2} \Vert \nabla _{p} \nabla u \Vert _{2}^{\frac{1}{2}} \Vert \nabla _{p} u \Vert _{2}^{\frac{1}{2}} \\ &\leq C \bigl\Vert \nabla _{p}\nabla ^{2}u \bigr\Vert _{2}^{2} \bigl\Vert \nabla ^{2}u \bigr\Vert _{2}. \end{aligned} \end{aligned}
(34)
\begin{aligned}& \begin{aligned}[b] \vert H_{3213} \vert &= \biggl\vert \int _{\mathbb{R}^{3}}(\partial _{3}\nabla _{p} \cdot u_{p}) \partial _{3}u\cdot \partial _{3}^{2}u\,dx \biggr\vert \\ &\leq C \Vert \partial _{3}\nabla _{p}\cdot u_{p} \Vert _{2}^{\frac{1}{2}} \Vert \partial _{3}u \Vert _{2}^{\frac{1}{2}} \bigl\Vert \partial _{3}^{2}u \bigr\Vert _{2}^{ \frac{1}{2}} \bigl\Vert \nabla _{p}\partial _{3}^{2} u \bigr\Vert _{2}^{\frac{1}{2}} \Vert \nabla _{p} \partial _{3}u \Vert _{2}^{\frac{1}{2}} \bigl\Vert \nabla _{p}\partial _{3}^{2}u \bigr\Vert _{2}^{\frac{1}{2}} \\ &\leq C \bigl\Vert \nabla _{p}\nabla ^{2}u \bigr\Vert _{2} \bigl\Vert \nabla ^{2}u \bigr\Vert _{2}^{ \frac{1}{2}} \Vert \nabla _{p}\nabla u \Vert _{2} \Vert \nabla u \Vert _{2}^{\frac{1}{2}} \\ &\leq C \bigl\Vert \nabla _{p}\nabla ^{2}u \bigr\Vert _{2}^{\frac{3}{2}} \bigl\Vert \nabla ^{2}u \bigr\Vert _{2}^{ \frac{3}{4}} \\ &\leq C \bigl\Vert \nabla _{p}\nabla ^{2}u \bigr\Vert _{2}^{2} \bigl\Vert \nabla ^{2}u \bigr\Vert _{2}. \end{aligned} \end{aligned}
(35)

Similarly to the above calculation, one gets

\begin{aligned} \begin{aligned}[b] \vert H_{3221} \vert &= \biggl\vert \int _{\mathbb{R}^{3}}(\nabla _{p}u\cdot \nabla )\nabla u \cdot \nabla _{p}\nabla u\,dx \biggr\vert \\ &\leq C \Vert \nabla _{p}u \Vert _{2}^{\frac{1}{2}} \bigl\Vert \nabla ^{2}u \bigr\Vert _{2}^{ \frac{1}{2}} \Vert \nabla _{p}\nabla u \Vert _{2}^{\frac{1}{2}} \bigl\Vert \nabla _{p}^{2}u \bigr\Vert _{2}^{\frac{1}{2}} \bigl\Vert \nabla _{p}\nabla ^{2}u \bigr\Vert _{2}^{\frac{1}{2}} \Vert \nabla _{p}\partial _{3}\nabla u \Vert _{2}^{\frac{1}{2}} \\ &\leq C \bigl\Vert \nabla _{p}\nabla ^{2}u \bigr\Vert _{2} \bigl\Vert \nabla ^{2}u \bigr\Vert _{2} \Vert \nabla _{p}u \Vert _{2}^{\frac{1}{2}} \Vert \nabla _{h}\nabla u \Vert _{2}^{\frac{1}{2}} \Vert \nabla u \Vert _{2}^{\frac{1}{2}} \\ &\leq C \bigl\Vert \nabla _{p}\nabla ^{2}u \bigr\Vert _{2}^{2} \bigl\Vert \nabla ^{2}u \bigr\Vert _{2}. \end{aligned} \end{aligned}
(36)

In similar manner as $$H_{3213}$$ and $$H_{3212}$$, one gets

\begin{aligned} \vert H_{3222},H_{3223} \vert \leq C \bigl\Vert \nabla _{p}\nabla ^{2}u \bigr\Vert _{2}^{2} \bigl\Vert \nabla ^{2}u \bigr\Vert _{2}. \end{aligned}
(37)

Combining (33)–(37), we have

\begin{aligned} \vert E_{3} \vert \leq C \bigl\Vert \nabla _{p}\nabla ^{2}u \bigr\Vert _{2}^{2} \bigl\Vert \nabla ^{2}u \bigr\Vert _{2}. \end{aligned}
(38)

One can split $$E_{4}+E_{5}$$ into four terms:

\begin{aligned} E_{4}+E_{5}={}& \int _{\mathbb{R}^{3}}\bigl(\nabla ^{2}B\cdot \nabla \bigr)B \cdot \nabla ^{2}u\,dx +2 \int _{\mathbb{R}^{3}}(\nabla B\cdot \nabla )\nabla B \cdot \nabla ^{2}u\,dx \\ &{}+ \int _{\mathbb{R}^{3}}\bigl(\nabla ^{2}B\cdot \nabla u\bigr) \cdot \nabla ^{2}B\,dx +2 \int _{\mathbb{R}^{3}}(\nabla B\cdot \nabla )\nabla u\cdot \nabla ^{2}B\,dx \\ ={}&E_{41}+E_{42}+E_{43}+E_{44}. \end{aligned}

Similarly to $$E_{21}$$ and $$E_{22}$$, one has

\begin{aligned} \vert E_{41},E_{42},E_{43},E_{44} \vert \leq C \bigl\Vert \nabla ^{2}u \bigr\Vert _{2} \bigl\Vert \nabla ^{3}B \bigr\Vert _{2}^{2}. \end{aligned}

Hence, one gets

\begin{aligned} \begin{aligned} \vert E_{4}+E_{5} \vert \leq C \bigl\Vert \nabla ^{2}u \bigr\Vert _{2} \bigl\Vert \nabla ^{3}B \bigr\Vert _{2}^{2}. \end{aligned} \end{aligned}
(39)

Combining (28), (29), (32), (38), and (39), we have

\begin{aligned} \begin{aligned}[b] &\frac{1}{2}\,\frac{d}{dt}\bigl( \bigl\Vert \Delta u(t) \bigr\Vert _{2}^{2}+ \bigl\Vert \Delta B(t) \bigr\Vert _{2}^{2}\bigr)+ \kappa \Vert \nabla _{p}\Delta \ u \Vert _{2}^{2}+ \bigl\Vert \nabla ^{3}B \bigr\Vert _{2}^{2} \\ &\quad \leq C\bigl( \Vert \Delta u \Vert _{2}+ \Vert \Delta B \Vert _{2}\bigr) \bigl( \Vert \nabla _{p}\Delta u \Vert _{2}^{2}+ \bigl\Vert \nabla ^{3}B \bigr\Vert _{2}^{2}\bigr). \end{aligned} \end{aligned}
(40)

Adding (27) to (40), we get

\begin{aligned} &\frac{1}{2}\,\frac{d}{dt}\bigl( \Vert \nabla B \Vert _{2}^{2}+ \Vert \nabla u \Vert _{2}^{2}+ \Vert \Delta B \Vert _{2}^{2}+ \Vert \Delta u \Vert _{2}^{2} \bigr) \\ &\qquad {}+\kappa \Vert \nabla _{p}\nabla u \Vert _{2}^{2}+ \Vert \Delta B \Vert _{2}^{2}+\kappa \Vert \nabla _{p}\Delta u \Vert _{2}^{2}+ \bigl\Vert \nabla ^{3} B \bigr\Vert _{2}^{2} \\ &\quad \leq C\bigl( \Vert \nabla u \Vert _{2}+ \Vert \nabla B \Vert _{2}+ \bigl\Vert \nabla ^{2}u \bigr\Vert _{2}+ \bigl\Vert \nabla ^{2}B \bigr\Vert _{2}\bigr)\\ &\qquad {}\times \bigl( \Vert \nabla _{p}\nabla u \Vert _{2}^{2}+ \Vert \Delta B \Vert _{2}^{2}+ \Vert \nabla _{p}\Delta u \Vert _{2}^{2}+ \bigl\Vert \nabla ^{3}B \bigr\Vert _{2}^{2} \bigr). \end{aligned}

Therefore, one gets

\begin{aligned} &\frac{1}{2}\,\frac{d}{dt}\bigl( \Vert \nabla B \Vert _{2}^{2}+ \Vert \nabla u \Vert _{2}^{2}+ \Vert \Delta B \Vert _{2}^{2}+ \Vert \Delta u \Vert _{2}^{2} \bigr) \\ &\quad {}+\bigl[\kappa _{0}-C\bigl( \Vert \nabla u \Vert _{2}+ \Vert \nabla B \Vert _{2}+ \bigl\Vert \nabla ^{2}u \bigr\Vert _{2}+ \bigl\Vert \nabla ^{2}B \bigr\Vert _{2}\bigr)\bigr] \\ &\quad {}\times \bigl( \Vert \nabla _{p}\nabla u \Vert _{2}^{2}+ \Vert \Delta B \Vert _{2}^{2}+ \Vert \nabla _{p}\Delta u \Vert _{2}^{2}+ \bigl\Vert \nabla ^{3}B \bigr\Vert _{2}^{2} \bigr)\leq 0. \end{aligned}

where $$\kappa _{0}=\min \{\kappa ,1\}$$. Choose $$\varepsilon ^{*}$$ sufficiently small such that

\begin{aligned} C\bigl( \Vert \nabla u_{0} \Vert _{2}+ \Vert \nabla B_{0} \Vert _{2}+ \Vert \Delta u_{0} \Vert _{2}+ \Vert \Delta B_{0} \Vert _{2}\bigr)\leq \frac{\kappa _{0}}{2}. \end{aligned}

Then one obtains:

$$(B,u)\in L^{\infty }\bigl(0,T;H^{2}\bigr), (\nabla _{p}u,\nabla B)\in L^{2}\bigl(0,T;H^{2} \bigr), \quad \forall T\in (0,T_{0}),$$

noticing

\begin{aligned} H^{2}\bigl(\mathbb{R}^{3} \bigr)\hookrightarrow \mathrm{BMO}\bigl(\mathbb{R}^{3}\bigr), \end{aligned}

yields for any $$T\in (0,T_{0})$$

\begin{aligned} (\nabla B,u)\in \bigl(0,T;\mathrm{BMO}\bigl( \mathbb{R}^{3}\bigr)\bigr). \end{aligned}

By Theorem 1.1, applying continuation argument, we obtain the result of Theorem 1.2.

Not applicable.

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## Acknowledgements

The author is indebted to the referee and the associate editor for their detailed comments and valuable suggestions, which greatly improved the manuscript. The author is also grateful to Prof. Lili Du for useful direction on this paper. This research was supported by High-level Talent Sailing Project of Yibin University (2021QH07).

## Funding

This research was supported by High-level Talent Sailing Project of Yibin University (2021QH07).

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### Contributions

BD prepared the manuscript initially and performed all the steps of the proofs in this research. The main idea of this paper was proposed by BD. All authors read and approved the final manuscript.

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Correspondence to Baoying Du.

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