2.1 Proof of Theorem 1.3
First, we give the following equality implying conservation of the energy:
$$ \frac{1}{2}\frac {d}{dt}\bigl( \Vert u \Vert _{L^{2}}^{2}+ \Vert B \Vert _{L^{2}}^{2}+ \Vert J \Vert _{L^{2}}^{2}\bigr)+ \mu \bigl( \Vert \nabla u \Vert _{L^{2}}^{2}+ \Vert \nabla B \Vert _{L^{2}}^{2}\bigr)=0. $$
(2.1)
In addition, for \(k\geq 0\), Bae and Shin [1] established the following inequality:
$$ \begin{aligned} &\frac{1}{2} \frac {d}{dt}\bigl( \bigl\Vert \Lambda ^{k}u \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{k}B \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{k}J \bigr\Vert _{L^{2}}^{2}\bigr)+\mu \bigl( \bigl\Vert \Lambda ^{k+1}u \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{k+1}B \bigr\Vert _{L^{2}}^{2} \bigr) \\ &\quad \leq C\bigl( \Vert \nabla u \Vert _{L^{\infty}}+ \Vert \nabla B \Vert _{L^{\infty}}+ \Vert \nabla J \Vert _{L^{\infty}}\bigr) \bigl( \bigl\Vert \Lambda ^{k}u \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{k}B \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{k}J \bigr\Vert _{L^{2}}^{2} \bigr). \end{aligned} $$
(2.2)
By using the embedding \(H^{2}(\mathbb{R}^{3})\rightarrow L^{\infty}(\mathbb{R}^{3})\) and Lemma 1.1, one has
$$ \begin{aligned} &\frac{1}{2} \frac {d}{dt}\bigl( \bigl\Vert \Lambda ^{k}u \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{k}B \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{k}J \bigr\Vert _{L^{2}}^{2}\bigr)+\mu \bigl( \bigl\Vert \Lambda ^{k+1}u \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{k+1}B \bigr\Vert _{L^{2}}^{2} \bigr) \\ &\quad \leq C\bigl( \Vert \nabla u \Vert _{H^{2}}+ \Vert \nabla B \Vert _{H^{2}}+ \Vert \nabla J \Vert _{H^{2}}\bigr) \bigl( \bigl\Vert \Lambda ^{k}u \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{k}B \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{k}J \bigr\Vert _{L^{2}}^{2} \bigr) \\ &\quad \leq C\mathcal{E}_{0}\bigl( \bigl\Vert \Lambda ^{k}u \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{k}B \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{k}J \bigr\Vert _{L^{2}}^{2}\bigr). \end{aligned} $$
(2.3)
Summing (2.3) over \(k=1,2,3,\ldots ,N\), and adding (2.1) to the resulting inequality, yields that
$$ \begin{aligned} & \frac {d}{dt}\bigl( \Vert u \Vert _{H^{N}}^{2}+ \Vert B \Vert _{H^{N}}^{2}+ \Vert J \Vert _{H^{N}}^{2} \bigr)+2 \mu \bigl( \Vert \nabla u \Vert _{H^{N}}^{2}+ \Vert \nabla B \Vert _{H^{N}}^{2}\bigr) \\ &\quad \leq C_{0} \mathcal{E}_{0}\bigl( \Vert \nabla u \Vert _{H^{N}}^{2}+ \Vert \nabla B \Vert _{H^{N}}^{2} \bigr). \end{aligned} $$
(2.4)
Thus, \((u,B,J)\) exists globally-in-time when \(C_{0}\mathcal{E}_{0}<2\mu \), and the proof of Theorem 1.3 complete.
2.2 Proof of Theorem 1.4
We will derive the evolution of the negative Sobolev norms of the solution to system (1.1). In order to estimate the nonlinear terms, we need to restrict ourselves to that \(s\in [0,\frac{1}{2}]\) and \(s\in (\frac{1}{2},\frac{3}{2})\), respectively.
First, taking \(\Lambda ^{-s }\) to (1.1)1, by taking the inner product of them with \(\Lambda ^{-s }u\) and \(\Lambda ^{-s}B\), respectively, we deduce that
$$ \begin{aligned} &\frac{1}{2} \frac {d}{dt}\bigl( \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{-s}J \bigr\Vert _{L^{2}}^{2}\bigr)+\mu \bigl( \bigl\Vert \Lambda ^{-s} \nabla u \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{-s}\nabla B \bigr\Vert _{L^{2}}^{2} \bigr) \\ &\quad =- \int _{\mathbb{R}^{3}}\Lambda ^{-s}(u\cdot \nabla u)\cdot \Lambda ^{-s}u\,dx+ \int _{\mathbb{R}^{3}}\Lambda ^{-s}(B\cdot \nabla B)\cdot \Lambda ^{-s}u\,dx \\ &\qquad{} - \int _{\mathbb{R}^{3}}\Lambda ^{-s}(J\cdot \nabla J)\cdot \Lambda ^{-s}u\,dx - \int _{\mathbb{R}^{3}}\Lambda ^{-s}(u\cdot \nabla B)\cdot \Lambda ^{-s}B\,dx \\ &\qquad{} + \int _{\mathbb{R}^{3}}\Lambda ^{-s}(B\cdot \nabla u)\cdot \Lambda ^{-s}B\,dx - \int _{\mathbb{R}^{3}}\Lambda ^{-s}(J\times B)\cdot \Lambda ^{-s}B\,dx \\ &\qquad {}- \int _{\mathbb{R}^{3}}\Lambda ^{-s}(u\cdot \nabla J)\cdot \Lambda ^{-s}B\,dx- \int _{\mathbb{R}^{3}}\Lambda ^{-s}(J\cdot \nabla u)\cdot \Lambda ^{-s}B\,dx \\ &\qquad {}+ \int _{\mathbb{R}^{3}}\Lambda ^{-s}(J\cdot \nabla J)\cdot \Lambda ^{-s}B\,dx \\ &\quad =:I_{1}+I_{2}+\cdots +I_{9}. \end{aligned} $$
(2.5)
The main tool to estimate the nonlinear terms on the right-hand side of (2.5) is the Sobolev interpolation inequality. This forces us to require that \(s\in (0,\frac{3}{2})\). If \(s\in (0,\frac{1}{2}]\), we have \(\frac{1}{2}+\frac {s}{3}<1\) and \(\frac {3}{s}\geq 6\). Hence, applying the Kato–Ponce inequality [3], the Sobolev embedding theorem, Hölder’s inequality, and Young’s inequality, we obtain
$$\begin{aligned}& \begin{aligned} I_{1} \leq{}& \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}} \bigl\Vert \Lambda ^{-s}(u\cdot \nabla u ) \bigr\Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}} \Vert u \cdot \nabla u \Vert _{L^{ \frac{1}{\frac{1}{2}+\frac {s}{3}}}} \\ \leq{}&C \bigl\Vert \Lambda ^{-s} u \bigr\Vert _{L^{2}} \Vert u \Vert _{L^{\frac {3}{s}}} \Vert \nabla u \Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}} \Vert \nabla u \Vert _{L^{2}}^{ \frac{1}{2}+s} \Vert \Delta u \Vert _{L^{2}}^{\frac{1}{2}-s} \Vert \nabla u \Vert _{L^{2}} \\ \leq{}&C \Vert \nabla u \Vert _{H^{1}}^{2} \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}}, \end{aligned} \end{aligned}$$
(2.6)
$$\begin{aligned}& \begin{aligned} I_{2}\leq{}& \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}} \bigl\Vert \Lambda ^{-s}(B\cdot \nabla B ) \bigr\Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}} \Vert B\cdot \nabla B \Vert _{L^{ \frac{1}{\frac{1}{2}+\frac {s}{3}}}} \\ \leq{}&C \bigl\Vert \Lambda ^{-s} u \bigr\Vert _{L^{2}} \Vert B \Vert _{L^{\frac {3}{s}}} \Vert \nabla B \Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}} \Vert \nabla B \Vert _{L^{2}}^{ \frac{1}{2}+s} \Vert \Delta u \Vert _{L^{2}}^{\frac{1}{2}-s} \Vert \nabla B \Vert _{L^{2}} \\ \leq{}&C \Vert \nabla J \Vert _{H^{1}}^{2} \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}}, \end{aligned} \end{aligned}$$
(2.7)
$$\begin{aligned}& \begin{aligned} I_{3}\leq{}& \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}} \bigl\Vert \Lambda ^{-s}(J\cdot \nabla J ) \bigr\Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}} \Vert J\cdot \nabla J \Vert _{L^{ \frac{1}{\frac{1}{2}+\frac {s}{3}}}} \\ \leq{}&C \bigl\Vert \Lambda ^{-s} u \bigr\Vert _{L^{2}} \Vert J \Vert _{L^{\frac {3}{s}}} \Vert \nabla J \Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}} \Vert \nabla J \Vert _{L^{2}}^{ \frac{1}{2}+s} \Vert \Delta J \Vert _{L^{2}}^{\frac{1}{2}-s} \Vert \nabla J \Vert _{L^{2}} \\ \leq{}&C \Vert \nabla J \Vert _{H^{1}}^{2} \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}}, \end{aligned} \end{aligned}$$
(2.8)
$$\begin{aligned}& \begin{aligned} I_{4}\leq{}& \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \bigl\Vert \Lambda ^{-s}(u\cdot \nabla B ) \bigr\Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \Vert u \cdot \nabla B \Vert _{L^{ \frac{1}{\frac{1}{2}+\frac {s}{3}}}} \\ \leq{}&C \bigl\Vert \Lambda ^{-s} B \bigr\Vert _{L^{2}} \Vert u \Vert _{L^{\frac {3}{s}}} \Vert \nabla B \Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \Vert \nabla u \Vert _{L^{2}}^{ \frac{1}{2}+s} \Vert \Delta u \Vert _{L^{2}}^{\frac{1}{2}-s} \Vert \nabla B \Vert _{L^{2}} \\ \leq{}&C\bigl( \Vert \nabla u \Vert _{H^{1}}^{2} + \Vert \nabla B \Vert _{L^{2}}^{2}\bigr) \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}}, \end{aligned} \end{aligned}$$
(2.9)
$$\begin{aligned}& \begin{aligned} I_{5}\leq{}& \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \bigl\Vert \Lambda ^{-s}(B\cdot \nabla u ) \bigr\Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \Vert B \cdot \nabla u \Vert _{L^{ \frac{1}{\frac{1}{2}+\frac {s}{3}}}} \\ \leq{}&C \bigl\Vert \Lambda ^{-s} B \bigr\Vert _{L^{2}} \Vert B \Vert _{L^{\frac {3}{s}}} \Vert \nabla u \Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \Vert \nabla B \Vert _{L^{2}}^{ \frac{1}{2}+s} \Vert \Delta B \Vert _{L^{2}}^{\frac{1}{2}-s} \Vert \nabla u \Vert _{L^{2}} \\ \leq{}&C\bigl( \Vert \nabla B \Vert _{H^{1}}^{2} + \Vert \nabla u \Vert _{L^{2}}^{2}\bigr) \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}}, \end{aligned} \end{aligned}$$
(2.10)
$$\begin{aligned}& \begin{aligned} I_{6}\leq{}& \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \bigl\Vert \Lambda ^{-s}(J\times B ) \bigr\Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \Vert J\times B \Vert _{L^{ \frac{1}{\frac{1}{2}+\frac {s}{3}}}} \\ \leq{}&C \bigl\Vert \Lambda ^{-s} B \bigr\Vert _{L^{2}} \Vert B \Vert _{L^{\frac {3}{s}}} \Vert J \Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \Vert \nabla B \Vert _{L^{2}}^{\frac{1}{2}+s} \Vert \Delta B \Vert _{L^{2}}^{\frac{1}{2}-s} \Vert J \Vert _{L^{2}} \\ \leq{}&C \Vert \nabla B \Vert _{H^{1}}^{2} \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}}, \end{aligned} \end{aligned}$$
(2.11)
$$\begin{aligned}& \begin{aligned} I_{7}\leq{}& \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \bigl\Vert \Lambda ^{-s}(u\cdot \nabla J) \bigr\Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \Vert u \cdot \nabla J \Vert _{L^{ \frac{1}{\frac{1}{2}+\frac {s}{3}}}} \\ \leq{}&C \bigl\Vert \Lambda ^{-s} B \bigr\Vert _{L^{2}} \Vert u \Vert _{L^{\frac {3}{s}}} \Vert \nabla J \Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \Vert \nabla u \Vert _{L^{2}}^{ \frac{1}{2}+s} \Vert \Delta u \Vert _{L^{2}}^{\frac{1}{2}-s} \Vert \nabla J \Vert _{L^{2}} \\ \leq{}&C \bigl( \Vert \nabla B \Vert _{H^{1}}^{2}+ \Vert \nabla u \Vert _{H^{1}}^{2}\bigr) \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}}, \end{aligned} \end{aligned}$$
(2.12)
$$\begin{aligned}& \begin{aligned} I_{8}\leq{}& \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \bigl\Vert \Lambda ^{-s}(J\cdot \nabla u) \bigr\Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \Vert J \cdot \nabla u \Vert _{L^{ \frac{1}{\frac{1}{2}+\frac {s}{3}}}} \\ \leq{}&C \bigl\Vert \Lambda ^{-s} B \bigr\Vert _{L^{2}} \Vert \nabla u \Vert _{L^{\frac {3}{s}}} \Vert J \Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \Vert \Delta u \Vert _{L^{2}}^{ \frac{1}{2}+s} \Vert \nabla \Delta u \Vert _{L^{2}}^{\frac{1}{2}-s} \Vert J \Vert _{L^{2}} \\ \leq{}&C \bigl( \Vert \nabla B \Vert _{L^{2}}^{2}+ \Vert \Delta u \Vert _{H^{1}}^{2}\bigr) \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \end{aligned} \end{aligned}$$
(2.13)
and
$$ \begin{aligned} I_{9}\leq{}& \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \bigl\Vert \Lambda ^{-s}(J\cdot \nabla J) \bigr\Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \Vert J \cdot \nabla J \Vert _{L^{ \frac{1}{\frac{1}{2}+\frac {s}{3}}}} \\ \leq{}&C \bigl\Vert \Lambda ^{-s} B \bigr\Vert _{L^{2}} \Vert J \Vert _{L^{\frac {3}{s}}} \Vert \nabla J \Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \Vert \nabla J \Vert _{L^{2}}^{ \frac{1}{2}+s} \Vert \Delta J \Vert _{L^{2}}^{\frac{1}{2}-s} \Vert \nabla J \Vert _{L^{2}} \\ \leq{}&C \Vert \nabla B \Vert _{H^{2}}^{2} \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}}. \end{aligned} $$
(2.14)
Summing (2.5)–(2.14) gives
$$ \begin{aligned} &\frac{1}{2} \frac {d}{dt}\bigl( \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{-s}J \bigr\Vert _{L^{2}}^{2}\bigr)+\mu \bigl( \bigl\Vert \Lambda ^{-s} \nabla u \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{-s}\nabla B \bigr\Vert _{L^{2}}^{2} \bigr) \\ &\quad \leq C\bigl( \Vert \nabla B \Vert _{H^{2}}^{2}+ \Vert \nabla u \Vert _{H^{2}}^{2}\bigr) \bigl( \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}} + \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \bigr). \end{aligned} $$
(2.15)
If \(s\in (\frac{1}{2},\frac{3}{2})\), we will estimate the right-hand sides of (2.5) and obtain the negative Sobolev norm estimates in another way. Since \(s\in (\frac{1}{2},\frac{3}{2})\), we easily obtain \(\frac{1}{2}+\frac {s}{3}<1\) and \(\frac {3}{s}\in (2,6)\). Therefore, using the Kato–Ponce inequality and Sobolev’s embedding theorem, we arrive at
$$\begin{aligned}& \begin{aligned} I_{1} \leq C \bigl\Vert \Lambda ^{-s} u \bigr\Vert _{L^{2}} \Vert u \Vert _{L^{\frac {3}{s}}} \Vert \nabla u \Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}} \Vert u \Vert _{L^{2}}^{s- \frac{1}{2} } \Vert \nabla u \Vert _{L^{2}}^{\frac{3}{2}-s} \Vert \nabla u \Vert _{L^{2}} , \end{aligned} \end{aligned}$$
(2.16)
$$\begin{aligned}& \begin{aligned} I_{2} \leq C \bigl\Vert \Lambda ^{-s} u \bigr\Vert _{L^{2}} \Vert B \Vert _{L^{\frac {3}{s}}} \Vert \nabla B \Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}} \Vert B \Vert _{L^{2}}^{s- \frac{1}{2} } \Vert \nabla B \Vert _{L^{2}}^{\frac{3}{2}-s} \Vert \nabla B \Vert _{L^{2}} , \end{aligned} \end{aligned}$$
(2.17)
$$\begin{aligned}& \begin{aligned} I_{3} \leq{}& C \bigl\Vert \Lambda ^{-s} u \bigr\Vert _{L^{2}} \Vert J \Vert _{L^{\frac {3}{s}}} \Vert \nabla J \Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}} \Vert J \Vert _{L^{2}}^{s- \frac{1}{2} } \Vert \nabla J \Vert _{L^{2}}^{\frac{3}{2}-s} \Vert \nabla J \Vert _{L^{2}} , \\ \leq{}&C \Vert \nabla B \Vert _{H^{2}}^{2} \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}}, \end{aligned} \end{aligned}$$
(2.18)
$$\begin{aligned}& \begin{aligned} I_{4} \leq C \bigl\Vert \Lambda ^{-s} B \bigr\Vert _{L^{2}} \Vert u \Vert _{L^{\frac {3}{s}}} \Vert \nabla B \Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \Vert u \Vert _{L^{2}}^{s- \frac{1}{2} } \Vert \nabla u \Vert _{L^{2}}^{\frac{3}{2}-s} \Vert \nabla B \Vert _{L^{2}} , \end{aligned} \end{aligned}$$
(2.19)
$$\begin{aligned}& \begin{aligned} I_{5} \leq C \bigl\Vert \Lambda ^{-s} B \bigr\Vert _{L^{2}} \Vert B \Vert _{L^{\frac {3}{s}}} \Vert \nabla u \Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \Vert B \Vert _{L^{2}}^{s- \frac{1}{2} } \Vert \nabla B \Vert _{L^{2}}^{\frac{3}{2}-s} \Vert \nabla u \Vert _{L^{2}} , \end{aligned} \end{aligned}$$
(2.20)
$$\begin{aligned}& \begin{aligned} I_{6} \leq C \bigl\Vert \Lambda ^{-s} B \bigr\Vert _{L^{2}} \Vert B \Vert _{L^{\frac {3}{s}}} \Vert J \Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \Vert B \Vert _{L^{2}}^{s-\frac{1}{2} } \Vert \nabla B \Vert _{L^{2}}^{\frac{3}{2}-s} \Vert \nabla B \Vert _{L^{2}} , \end{aligned} \end{aligned}$$
(2.21)
$$\begin{aligned}& \begin{aligned} I_{7} \leq C \bigl\Vert \Lambda ^{-s} B \bigr\Vert _{L^{2}} \Vert u \Vert _{L^{\frac {3}{s}}} \Vert \nabla J \Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \Vert u \Vert _{L^{2}}^{s- \frac{1}{2} } \Vert \nabla u \Vert _{L^{2}}^{\frac{3}{2}-s} \Vert \nabla J \Vert _{L^{2}} , \end{aligned} \end{aligned}$$
(2.22)
$$\begin{aligned}& \begin{aligned} I_{8} \leq{}&C \bigl\Vert \Lambda ^{-s} B \bigr\Vert _{L^{2}} \Vert \nabla u \Vert _{L^{ \frac {3}{s}}} \Vert J \Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \Vert \nabla u \Vert _{L^{2}}^{s-\frac{1}{2}} \Vert \Delta u \Vert _{L^{2}}^{\frac{3}{2}-s} \Vert J \Vert _{L^{2}} \\ \leq{}&C \bigl( \Vert \nabla u \Vert _{H^{1}}^{2}+ \Vert \nabla B \Vert _{L^{2}}^{2}\bigr) \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \end{aligned} \end{aligned}$$
(2.23)
and
$$ \begin{aligned} I_{9}\leq{}&C \bigl\Vert \Lambda ^{-s} u \bigr\Vert _{L^{2}} \Vert J \Vert _{L^{\frac {3}{s}}} \Vert \nabla J \Vert _{L^{2}} \leq C \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \Vert J \Vert _{L^{2}}^{s- \frac{1}{2} } \Vert \nabla J \Vert _{L^{2}}^{\frac{3}{2}-s} \Vert \nabla J \Vert _{L^{2}} \\ \leq{}&C \Vert \nabla B \Vert _{H^{1}}^{2} \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}}. \end{aligned} $$
(2.24)
Summing (2.5) and (2.16)–(2.24) gives
$$ \begin{aligned} &\frac{1}{2} \frac {d}{dt}\bigl( \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{-s}J \bigr\Vert _{L^{2}}^{2}\bigr)+\mu \bigl( \bigl\Vert \Lambda ^{-s} \nabla u \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{-s}\nabla B \bigr\Vert _{L^{2}}^{2} \bigr) \\ &\quad \leq C\bigl( \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}} + \bigl\Vert \Lambda ^{-s}B \bigr\Vert _{L^{2}} \bigr) \\ &\qquad{} \times \bigl[ \Vert \nabla B \Vert _{H^{2}}^{2}+ \Vert \nabla u \Vert _{H^{1}}^{2}+\bigl( \Vert u \Vert _{L^{2}}+ \Vert B \Vert _{L^{2}} \bigr)^{s-\frac{1}{2} } \\ &\qquad{}\times \bigl( \Vert \nabla u \Vert _{L^{2}}+ \Vert \nabla B \Vert _{L^{2}}\bigr)^{ \frac{3}{2}-s}( \Vert \nabla u \Vert _{L^{2}}+ \Vert \nabla B \Vert _{H^{1}} \bigr].\end{aligned} $$
(2.25)
Next, by using the negative Sobolev norm estimates (2.15) and (2.25), we establish the decay estimates for the solution of system (1.1).
First, one considers the case \(s\in (0,\frac{1}{2}]\). Define
$$ \mathcal{E}_{l} (t)= \bigl\Vert \nabla ^{l}u \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{l}B \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{l }J \bigr\Vert _{L^{2}}^{2} $$
and
$$ \mathcal{E}_{-s}(t)= \bigl\Vert \Lambda ^{-s}u \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Lambda ^{-s}B \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \Lambda ^{-s}J \bigr\Vert _{L^{2}}^{2} . $$
Integrating in time (2.15), and applying (1.3), we obtain
$$ \begin{aligned} \mathcal{E}_{-s}(t)\leq{}& \mathcal{E}_{-s}(0)+C \int _{0}^{t}\bigl( \Vert \nabla B \Vert _{H^{2}}^{2}+ \Vert \nabla u \Vert _{H^{2}}^{2} \bigr)\sqrt{\mathcal{E}_{-s}(\tau )}d \tau \\ \leq{}&C_{0} \Bigl(1+\sup_{0\leq \tau \leq t}\sqrt{ \mathcal{E}_{-s}( \tau )}\,d\tau \Bigr), \end{aligned} $$
which implies (1.4) for \(s\in [0,\frac{1}{2}]\), that is
$$ \bigl\Vert \Lambda ^{-s} u(t) \bigr\Vert _{L^{2}} + \bigl\Vert \Lambda ^{-s}B(t) \bigr\Vert _{L^{2}} + \bigl\Vert \Lambda ^{-s} J(t) \bigr\Vert _{L^{2}} \leq C_{0}. $$
(2.26)
In addition, if \(l=1,2,\ldots ,N\), by the Sobolev interpolation inequality, we deduce that
$$ \bigl\Vert \nabla ^{l+1}v \bigr\Vert _{L^{2}}\geq C \bigl\Vert \nabla ^{l}v \bigr\Vert _{L^{2}}^{1+ \frac{1}{l+s}} \bigl\Vert \Lambda ^{-s}v \bigr\Vert _{L^{2}}^{-\frac{1}{l+s}}. $$
By this fact and (2.26), we derive that
$$ \bigl\Vert \nabla ^{l+1}(u,B,J) \bigr\Vert _{L^{2}}^{2} \geq C_{0} \bigl\Vert \nabla ^{l}(u,B,J ) \bigr\Vert _{L^{2}}^{2})^{1+\frac{1}{l+s}}. $$
Hence, by using (2.4), one has
$$ \frac {d}{dt}\mathcal{E}_{l} +C_{0} ( \mathcal{E}_{l} )^{1+ \frac{1}{l+s}}\leq 0,\quad \text{for } l=1,2,\ldots ,N, $$
that is
$$ \mathcal{E}_{l} (t)\leq C_{0}(1+ t)^{-l-s}, \quad \text{for } l=1,2, \ldots ,N , $$
which implies that (1.5) holds for the case \(s\in [0,\frac{1}{2}]\).
In addition, the arguments for \(s\in [0,\frac{1}{2}]\) cannot be applied to \(s\in (\frac{1}{2},\frac{3}{2})\). However, observing that \(u_{0}, B_{0}, J_{0}\in \dot{H}^{-\frac{1}{2}}\) hold since \(\dot{H}^{-s}\cap L^{2}\subset \dot{H}^{-s'}\) for any \(s'\in [0,s]\), we can deduce from what we have proved for (1.4) and (1.5) with \(s=\frac{1}{2}\) that
$$ \bigl\Vert \nabla ^{l}u \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{l} B \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{l} J \bigr\Vert _{L^{2}}^{2} \leq C_{0}(1+t)^{-\frac{1}{2}-l}, \quad \text{for } l=0,1, \ldots ,N . $$
(2.27)
By (2.25), we have
$$ \begin{aligned} \mathcal{E}_{-s}(t)\leq{}& \mathcal{E}_{-s}(0)+C \int _{0}^{t} \Vert u \Vert _{L^{2}}^{s- \frac{1}{2}} \Vert \nabla u \Vert _{L^{2}}^{\frac{3}{2}-s} \sqrt{\mathcal{E}_{-s}( \tau )}\,d\tau \\ &{}+C \int _{0}^{t}\bigl( \Vert \nabla u \Vert ^{2}_{H^{1}}+ \Vert \varrho \Vert _{H^{3}}^{2} \bigr) \sqrt{\mathcal{E}_{-s}(\tau )}\,d\tau \\ \leq{}&C+C \int _{0}^{t}(1+\tau )^{-\frac{7}{4}-\frac {s}{2}}\,d\tau \sup_{\tau \in [0,t]} \sqrt{\mathcal{E}_{-s}(\tau )}+C \sup _{\tau \in [0,t]} \sqrt{\mathcal{E}_{-s}(\tau )} \\ \leq{}&C+C\sup_{\tau \in [0,t]} \sqrt{\mathcal{E}_{-s}(\tau )},\quad \text{for } s\in \biggl(\frac{1}{2},\frac{3}{2} \biggr), \end{aligned} $$
which means (1.4) holds for \(s\in (\frac{1}{2},\frac{3}{2})\). Moreover, we can repeat the arguments leading to (1.5) for \(s\in [0,\frac{1}{2}]\) to prove that they also hold for \(s\in (\frac{1}{2},\frac{3}{2})\). Hence, we complete the proof.