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.