On the blow-up criterion for the Hall-MHD problem with partial dissipation in $\mathbb{R}^{3}$

Abstract

In this paper, we investigate the 3D incompressible Hall-magnetohydrodynamics with partial dissipation. Based on the results in (Du in Bound. Value Probl. 2022:6, 2022; Du and Liu in Acta Math. Sci. 42A:5, 2022; Fei and Xiang in J. Math. Phys. 56:051504, 2015), we establish an improved blow-up criterion for classical solutions. Furthermore, using the blow-up criterion, we also obtain the existence of the classical solutions only under the condition that the initial data $\|V_{0}\|_{H^{1}}+\|B_{0}\|_{H^{2}}$ are sufficiently small.

1 Introduction

We discuss the following Hall-magnetohydrodynamic problem in three-dimensions:

$$\begin{aligned} \textstyle\begin{cases} V_{t}+(V\cdot \nabla )V+\nabla (p+\pi )=\rho _{1}V_{x_{1}x_{1}}+\rho _{2}V_{x_{2}x_{2}} +\rho _{3}V_{x_{3}x_{3}}+(B\cdot \nabla )B, \\ B_{t}+(V\cdot \nabla )B=(B\cdot \nabla )V-\Delta B-\nabla \times (( \nabla \times B)\times B), \\ DivV=0, \qquad DivB=0, \\ V(0,x)=V_{0}, \qquad B(0,x)=B_{0}, \end{cases}\displaystyle \end{aligned}$$

(1)

where $(t,x)\in \mathbb{R}^{+}\times \mathbb{R}^{3}$, V and B denote the velocity and magnetic fields, respectively, and $\rho _{1}$, $\rho _{2}$, $\rho _{3}$ are the kinematic viscosities. The Hall-MHD equations include $\nabla \times ((\nabla \times B)\times B)$ (Hall term), which is different from the MHD equations. The Hall term is used to describe magnetic reconnection, and it appears when the magnetic shear is large.

Many mathematical results for MHD equations have been obtained [11, 12, 16, 19, 22, 23, 25–30, 32]. Recently, there are some contributions on the Hall-MHD system. The paper [1] presented derivations of Hall-MHD system. Chae and Lee [5] established two optimal blow-up criterions. For more research on the Hall-MHD system, we refer to [6–9, 13, 18, 20, 21, 24]. The authors of the papers [2, 3, 15] investigated the Boussinesq (or MHD) system with partial viscosity. Fei and Xiang [21] obtained a blow-up criterion for (1) with $\rho _{1}=\rho _{2}=1$ and $\rho _{3}=0$; they also proved the global small data solution. Du [13] obtained the large data existence to $2\frac{1}{2}$-dimensional Hall-MHD problem with partial dissipation, provided that the coefficients of dissipation and magnetic diffusion are sufficiently large. Paper [14] established an improved blow-up criterion in terms of BMO norm for 3D Hall-magnetohydrodynamics with partial dissipation. Du [15] obtained the global existence of the classical solutions, provided that $(\|u_{0}\|_{L^{2}}^{2}+\|B_{0}\|_{L^{2}}^{2})(\|\nabla u_{0}\|_{L^{2}}^{2}+ \|\nabla B_{0}\|_{L^{2}}^{2}+\|\nabla ^{2}u_{0}\|_{L^{2}}^{2}+\| \nabla ^{2}B_{0}\|_{L^{2}}^{2})/Q^{4}$ is sufficiently small.

In this paper, we study the incompressible Hall-magnetohydrodynamics (1) with partial dissipation in three dimensions. Inspired by [2–5, 13–15, 17, 21], we establish an improved blow-up criterion for classical solutions. Furthermore, we also get the small data global well-posedness.

Theorem 1.1

Let $\rho _{1},\rho _{2}>0$, $\rho _{3}=0$, and $(V_{0},B_{0})\in H^{3}(\mathbb{R}^{3})$ with $DivV_{0}=DivB_{0}=0$. Then the following two equalities are equivalent:

$$ \begin{aligned}&\textit{(a)} \quad \limsup_{t\nearrow \widetilde{\mathbf{T}}}\bigl( \bigl\Vert V(t) \bigr\Vert _{H^{3}}^{2}+ \bigl\Vert B(t) \bigr\Vert _{H^{3}}^{2}\bigr)=\infty , \\ &\textit{(b)} \quad \int _{0}^{\widetilde{\mathbf{T}}}\bigl( \Vert V \Vert _{L^{s}}^{k}+ \Vert \nabla B \Vert _{L^{\lambda}}^{\alpha} \bigr)\,dt=\infty , \end{aligned} $$

where $\widetilde{\mathbf{T}}<\infty$ is the first blow-up time to (1), and s, k, λ, α satisfy

$$ \begin{aligned}\frac{3}{s}+\frac{2}{k}\leq 1, \qquad \frac{3}{\lambda}+ \frac{2}{\alpha}\leq 1, \quad \textit{and}\quad s\in (12,\infty ], \lambda \in (3, \infty ]. \end{aligned} $$

Remark 1.1

Compared to [19], the blow-up criterion imposes the condition on $\int _{0}^{\widetilde{\mathbf{T}}}(\|V\|_{L^{s}}^{k}+\|\nabla B\|_{L^{ \lambda}}^{\alpha})\,dt<\infty$ instead of $\int _{0}^{\widetilde{\mathbf{T}}}(\|\nabla V\|_{L^{p}}^{q}+\| \Delta B\|_{L^{\beta}}^{\gamma})\,dt<\infty$, where $p,\beta \in (3, \infty ]$.

Based on the Theorem 1.1, we can get the following small data existence to system (1) with $\rho _{1},\rho _{2}>0$ and $\rho _{3}=0$.

Theorem 1.2

Suppose $\rho _{1},\rho _{2}>0$, $\rho _{3}=0$, $(V_{0},B_{0})\in H^{3}(\mathbb{R}^{3})$, $DivV_{0}=DivB_{0}=0$, and there exists a constant $L>0$, such that $\|V_{0}\|_{H^{1}}+\|B_{0}\|_{H^{2}}< L$. Then (1) has a unique classical solution $(V,B)\in L^{\infty}(0,\infty ;H^{3}(\mathbb{R}^{3}))$.

We can also obtain results similar to Theorems 1.1 and 1.2 for problem (1) with $\rho _{2}=0$, $\rho _{1},\rho _{3}>0$ and $\rho _{1}=0$, $\rho _{2},\rho _{3}>0$.

Remark 1.2

Compared to the previous results, the smallness conditions are given for $\|V_{0}\|_{H^{1}}+\|B_{0}\|_{H^{2}}$ instead of sufficiently small $\|V_{0}\|_{H^{3}}+\|B_{0}\|_{H^{3}}$ in [21] and $\|V_{0}\|_{H^{2}}+\|B_{0}\|_{H^{2}}$ in [14].

In the paper, $\partial _{j}$ and $V_{j}$ represent the jth components of ∇ and V, and C denotes a generic positive constant. We adopt the following simplified notation:

$$ \begin{aligned}&D:=\nabla =(\partial _{1},\partial _{2}, \partial _{3}); \qquad D_{p}:=( \partial _{1}, \partial _{2},0); \qquad V_{p}:=(V_{1},V_{2},0); \\ &\Vert \cdot \Vert _{s}\triangleq \Vert \cdot \Vert _{L^{s}}; \qquad \rho :=\min \{\rho _{1}, \rho _{2}\}; \qquad \rho _{0}:=\min \{\rho ,1\}. \end{aligned} $$

2 Some a priori estimates

To establish Theorems 1.1 and 1.2, we need the following lemmas.

Lemma 2.1

([10])

Suppose f, g, h, $D_{p}f$, $D_{p}g$, and $\partial _{3}h$ are all in $L^{2}(\mathbb{R}^{3})$. Then

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

Lemma 2.2

([31])

Let f, g, h, $\partial _{1}g$, $\partial _{2}g$, $\partial _{2}h$, and $\partial _{3}h$ be all in $L^{2}(\mathbb{R}^{3})$. Then

$$ \begin{aligned} \int _{\mathbb{R}^{3}} \vert fgh \vert \,dx\leq C \Vert f \Vert _{2} \Vert \partial _{1}g \Vert _{2}^{\frac{1}{2}} \Vert \partial _{3}h \Vert _{2}^{\frac{1}{2}} \Vert g \Vert _{2}^{ \frac{1}{4}} \Vert h \Vert _{2}^{\frac{1}{4}} \Vert \partial _{2}g \Vert _{2}^{ \frac{1}{4}} \Vert \partial _{2}h \Vert _{2}^{\frac{1}{4}}. \end{aligned} $$

Let $\rho _{1}>0$, $\rho _{2}>0$, and $\rho _{3}=0$, Taking the scalar products of (1)₁ and (1)₂ with V and B, we get

$$\begin{aligned} \begin{aligned}\frac{1}{2}\frac{d}{dt}\bigl( \bigl\Vert V(t) \bigr\Vert _{2}^{2}+ \bigl\Vert B(t) \bigr\Vert _{2}^{2}\bigr)+ \rho _{0}\bigl( \Vert D_{p}V \Vert _{2}^{2}+ \Vert DB \Vert _{2}^{2}\bigr)=0. \end{aligned} \end{aligned}$$

(2)

Integrating (2) over $(0,T)$, we get

$$\begin{aligned} \begin{aligned} \Vert V \Vert _{2}^{2}+ \Vert B \Vert _{2}^{2}+ 2\rho _{0} \int _{0}^{T}\bigl( \bigl\Vert D_{p}V(t) \bigr\Vert _{2}^{2}+ \bigl\Vert DB(t) \bigr\Vert _{2}^{2}\bigr)\,dt= \Vert V_{0} \Vert _{2}^{2}+ \Vert B_{0} \Vert _{2}^{2}. \end{aligned} \end{aligned}$$

(3)

Proposition 2.1

Assume that $\rho _{1}>0$, $\rho _{2}>0$, $\rho _{3}=0$, and $(V,B)$ is a solution to (1). Then

$$\begin{aligned} \begin{aligned}&\frac{d}{dt}\bigl( \Vert DV \Vert _{2}^{2}+ \Vert DB \Vert _{2}^{2}\bigr)+ \rho _{0}\bigl( \Vert DD_{p}V \Vert _{2}^{2}+ \bigl\Vert D^{2}B \bigr\Vert _{2}^{2}\bigr) \\ &\quad \leq C\bigl( \Vert DV \Vert _{2}^{2}+ \Vert DB \Vert _{2}^{2}\bigr) \bigl( \Vert V \Vert _{s}^{\frac{2s}{s-6}}+ \Vert DB \Vert _{\lambda}^{\frac{2\lambda}{\lambda -3}} \bigr). \end{aligned} \end{aligned}$$

(4)

Proof

We operate D to (1)₁ and (1)₂. Then taking the inner product of them with DV and DB, we get

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{d}{dt}\bigl( \bigl\Vert DV(t) \bigr\Vert _{2}^{2}+ \bigl\Vert DB(t) \bigr\Vert _{2}^{2}\bigr)+ \rho _{1} \Vert \partial _{1}DV \Vert _{2}^{2}+\rho _{2} \Vert \partial _{2}DV \Vert _{2}^{2}+ \bigl\Vert D^{2}B \bigr\Vert _{2}^{2} \\ &\quad=- \int _{\mathbb{R}^{3}}D(V\cdot DB)\cdot DB\,dx + \int _{\mathbb{R}^{3}}D(B \cdot DV)\cdot DB\,dx + \int _{\mathbb{R}^{3}}D(B\cdot DB)\cdot DV\,dx \\ &\quad\quad{}- \int _{\mathbb{R}^{3}}D(V\cdot DV)\cdot DV\,dx - \int _{\mathbb{R}^{3}}D\bigl[D \times \bigl((D\times B)\times B\bigr)\bigr] \cdot DB\,dx \\ &\quad=M_{1}+M_{2}+M_{3}+M_{4}+M_{5}. \end{aligned} \end{aligned}$$

(5)

Firstly, we use $DivV=0$, integration by parts, and interpolation to deduce that

$$\begin{aligned} \begin{aligned} \vert M_{1} \vert &= \biggl\vert \int _{\mathbb{R}^{3}}(V\cdot D B)\cdot D^{2}B\,dx \biggr\vert \\ &\leq C \Vert V \Vert _{s} \Vert D B \Vert _{\frac{2s}{s-2}} \bigl\Vert D^{2}B \bigr\Vert _{2} \\ &\leq C \Vert V \Vert _{s} \Vert D B \Vert _{2}^{\frac{s-3}{s}} \bigl\Vert D^{2}B \bigr\Vert _{2}^{ \frac{s+3}{s}} \\ &\leq C \Vert V \Vert _{s}^{\frac{2s}{s-3}} \Vert DB \Vert _{2}^{2}+\frac{1}{10} \bigl\Vert D^{2}B \bigr\Vert _{2}^{2}. \end{aligned} \end{aligned}$$

(6)

By cancelation property and integration by parts we rewrite $M_{2}+M_{3}$ as follows:

$$ \begin{aligned}M_{2}+M_{3}=-\sum _{j=1}^{3} \int _{\mathbb{R}^{3}}2V( \partial _{j}B\cdot D)\partial _{j}B +V\bigl(\partial _{j}^{2}B\cdot D\bigr)B+V( \partial _{j}B\cdot D)\partial _{j}B\,dx. \end{aligned} $$

Hence, similarly to (6), we get

$$\begin{aligned} \begin{aligned} \vert M_{2}+M_{3} \vert \leq C \Vert V \Vert _{s}^{\frac{2s}{s-3}} \Vert D B \Vert _{2}^{2}+ \frac{3}{10} \bigl\Vert D^{2}B \bigr\Vert _{2}^{2}. \end{aligned} \end{aligned}$$

(7)

We can decompose $M_{4}$ into three terms:

$$\begin{aligned} \begin{aligned}M_{4}&=- \int _{\mathbb{R}^{3}}(D_{p}V\cdot D)VD_{p}V\,dx- \int _{ \mathbb{R}^{3}}(\partial _{3}V_{p}\cdot D_{p})V\partial _{3}V\,dx+ \int _{ \mathbb{R}^{3}}(D_{p}\cdot V_{p})\partial _{3}V\partial _{3}V\,dx \\ &=M_{41}+M_{42}+M_{43}. \end{aligned} \end{aligned}$$

(8)

Integrating by parts $M_{41}$ and $M_{42}$, we have

$$ \begin{aligned} \vert M_{41} \vert &=2 \biggl\vert \int _{\mathbb{R}^{3}}VD_{p}VDD_{p}V\,dx \biggr\vert \\ &\leq C \Vert V \Vert _{s} \Vert D_{p}V \Vert _{\frac{2s}{s-2}} \Vert DD_{p}V \Vert _{2} \\ &\leq C \Vert V \Vert _{s} \Vert D V \Vert _{2}^{\frac{s-3}{s}} \Vert DD_{p}V \Vert _{2}^{ \frac{s+3}{s}} \\ &\leq C \Vert V \Vert _{s}^{\frac{2s}{s-3}} \Vert D V \Vert _{2}^{2}+\frac{\rho}{6} \Vert DD_{p}V \Vert _{2}^{2} \end{aligned} $$

and

$$ \begin{aligned} \vert M_{42} \vert &= \biggl\vert \int _{\mathbb{R}^{3}}\partial _{3}D_{p}V_{p}V \partial _{3}V+ \partial _{3}V_{p}V\partial _{3}D_{p}V\,dx \biggr\vert \\ &\leq C \Vert V \Vert _{s} \Vert D V \Vert _{\frac{2s}{s-2}} \Vert DD_{p}V \Vert _{2} \\ &\leq C \Vert V \Vert _{s}^{\frac{s}{s-6}} \Vert DV \Vert _{2}^{\frac{s-12}{s-6}} \Vert DD_{p}V \Vert _{2} \\ &\leq C \Vert V \Vert _{s}^{\frac{2s}{s-6}} \Vert DV \Vert _{2}^{2}+\frac{\rho}{6} \Vert DD_{p}V \Vert _{2}^{2}. \end{aligned} $$

Similarly, we have

$$ \begin{aligned} \vert M_{43} \vert \leq C \Vert V \Vert _{s}^{\frac{2s}{s-6}} \Vert DV \Vert _{2}^{2}+ \frac{\rho}{6} \Vert DD_{p}V \Vert _{2}^{2}. \end{aligned} $$

Collecting the above estimates, we get

$$\begin{aligned} \begin{aligned} \vert M_{4} \vert \leq C \Vert V \Vert _{s}^{\frac{2s}{s-6}} \Vert DV \Vert _{2}^{2}+ \frac{\rho}{2} \Vert DD_{p}V \Vert _{2}^{2}. \end{aligned} \end{aligned}$$

(9)

Applying cancelation property and interpolation inequality, we get

$$\begin{aligned} \begin{aligned} \vert M_{5} \vert &= \biggl\vert \int _{\mathbb{R}^{3}}\bigl[D\bigl((\nabla \times B)\times B\bigr)-D( \nabla \times B)\times B\bigr]\cdot D(\nabla \times B)\,dx \biggr\vert \\ &\leq C \Vert D B \Vert _{\lambda} \Vert D B \Vert _{\frac{2\lambda}{\lambda -2}} \bigl\Vert D^{2}B \bigr\Vert _{2} \\ &\leq C \Vert D B \Vert _{\lambda} \Vert D B \Vert _{2}^{\frac{\lambda -3}{\lambda}} \bigl\Vert D^{2} B \bigr\Vert _{2}^{\frac{\lambda +3}{\lambda}} \\ &\leq C \Vert D B \Vert _{\lambda}^{\frac{2\lambda}{\lambda -3}} \Vert D B \Vert _{2}^{2}+ \frac{1}{10} \bigl\Vert D^{2} B \bigr\Vert _{2}^{2}. \end{aligned} \end{aligned}$$

(10)

Combining (5)–(10) yields (4). □

Applying Gronwall’s inequality to (4), we get the following inequality:

$$\begin{aligned} \begin{aligned}&\sup_{0< t< T}\bigl( \bigl\Vert DV(t) \bigr\Vert _{2}^{2}+ \bigl\Vert D B(t) \bigr\Vert _{2}^{2}\bigr)+ \rho _{0} \int _{0}^{T}\bigl( \bigl\Vert DD_{p}V(t) \bigr\Vert _{2}^{2}+ \bigl\Vert D^{2}B(t) \bigr\Vert _{2}^{2}\bigr)\,dt \\ &\quad\leq \bigl( \Vert DV_{0} \Vert _{2}^{2}+ \Vert DB_{0} \Vert _{2}^{2}\bigr) (1+C \int _{0}^{T}\bigl( \bigl\Vert V(t) \bigr\Vert _{s}^{\frac{2s}{s-6}}+ \bigl\Vert D B(t) \bigr\Vert _{\lambda}^{ \frac{2\lambda}{\lambda -3}}\bigr)\,dt \\ &\quad\quad{}\times \exp \biggl(C \int _{0}^{T}\bigl( \bigl\Vert V(t) \bigr\Vert _{k}^{\frac{2s}{s-6}}+ \bigl\Vert DB(t) \bigr\Vert _{ \lambda}^{\frac{2\lambda}{\lambda -3}}\bigr)\,dt\biggr). \end{aligned} \end{aligned}$$

(11)

Proposition 2.2

Suppose the conditions in Proposition 2.1hold. Then

$$\begin{aligned} \begin{aligned}&\frac{d}{dt}\bigl( \bigl\Vert \Delta V(t) \bigr\Vert _{2}^{2}+ \bigl\Vert \Delta B(t) \bigr\Vert _{2}^{2}\bigr)+ \rho _{0}\bigl( \Vert \Delta D_{p}V \Vert _{2}^{2}+ \bigl\Vert D^{3}B \bigr\Vert _{2}^{2}\bigr) \\ &\quad \leq C\bigl( \Vert \Delta V \Vert _{2}^{2}+ \Vert \Delta B \Vert _{2}^{2}\bigr) \bigl( \Vert V \Vert _{s}^{ \frac{2s}{s-3}}+ \Vert D B \Vert _{\lambda}^{\frac{2\lambda}{\lambda -3}}+ \Vert DV \Vert _{2}^{6}\bigr). \end{aligned} \end{aligned}$$

(12)

Proof

Similarly to (5), we have

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{d}{dt}\bigl( \bigl\Vert D^{2}V(t) \bigr\Vert _{2}^{2}+ \bigl\Vert D^{2}B(t) \bigr\Vert _{2}^{2}\bigr)+ \rho _{1} \bigl\Vert \partial _{1}D^{2}V \bigr\Vert _{2}^{2}+\rho _{2} \bigl\Vert \partial _{2}D^{2}V \bigr\Vert _{2}^{2}+ \bigl\Vert D^{3}B \bigr\Vert _{2}^{2} \\ &\quad=- \int _{\mathbb{R}^{3}}D^{2}(V\cdot DB)\cdot D^{2}B\,dx + \int _{ \mathbb{R}^{3}}D^{2}(B\cdot DV)\cdot D^{2}B\,dx + \int _{\mathbb{R}^{3}}D^{2}(B \cdot DB)\cdot D^{2}V\,dx \\ &\quad\quad{}- \int _{\mathbb{R}^{3}}D^{2}(V\cdot DV)\cdot D^{2}V\,dx - \int _{ \mathbb{R}^{3}}D^{2}\bigl[D\times \bigl((D\times B)\times B\bigr)\bigr]\cdot D^{2}B\,dx \\ &\quad=R_{1}+R_{2}+R_{3}+R_{4}+R_{5}. \end{aligned} \end{aligned}$$

(13)

We decompose $R_{1}$ into two parts:

$$\begin{aligned} \begin{aligned}R_{1}&=- \int _{\mathbb{R}^{3}}\bigl(D^{2}V\cdot D\bigr)B\cdot D^{2}B\,dx -2 \int _{\mathbb{R}^{3}}(DV\cdot D)DB\cdot D^{2}B\,dx \\ &=R_{11}+R_{12}. \end{aligned} \end{aligned}$$

(14)

Further, we decompose $R_{11}$ into two parts:

$$ \begin{aligned}R_{11}&= \int _{\mathbb{R}^{3}}(D V\cdot D)DB\cdot D^{2}B\,dx + \int _{\mathbb{R}^{3}}(DV\cdot D)B\cdot D^{3}B\,dx =R_{111}+R_{112}. \end{aligned} $$

Applying integration by parts and interpolation, we have

$$ \begin{aligned} \vert R_{111} \vert &= \biggl\vert \int _{\mathbb{R}^{3}}(V\cdot D)D B\cdot D^{3}B+(V \cdot D)D^{2}B\cdot D^{2}B\,dx \biggr\vert \\ &\leq C \Vert V \Vert _{s} \Vert \Delta B \Vert _{\frac{2s}{s-2}} \bigl\Vert D^{3}B \bigr\Vert _{2} \\ &\leq C \Vert V \Vert _{s} \Vert \Delta B \Vert _{2}^{\frac{s-3}{s}} \bigl\Vert D^{3}B \bigr\Vert _{2}^{ \frac{s+3}{s}} \\ &\leq C \Vert V \Vert _{s}^{\frac{2s}{s-3}} \Vert \Delta B \Vert _{2}^{2}+\frac{1}{22} \bigl\Vert D^{3}B \bigr\Vert _{2}^{2} \end{aligned} $$

and

$$ \begin{aligned} \vert R_{12} \vert \leq C \Vert V \Vert _{s}^{\frac{2s}{s-3}} \Vert \Delta B \Vert _{2}^{2}+ \frac{1}{22} \bigl\Vert D^{3}B \bigr\Vert _{2}^{2}. \end{aligned} $$

We estimate $R_{112}$ as

$$ \begin{aligned} \vert R_{112} \vert &\leq C \Vert DV \Vert _{2} \Vert D B \Vert _{L^{\infty}} \bigl\Vert D^{3}B \bigr\Vert _{2} \\ &\leq C \Vert DV \Vert _{2} \Vert \Delta B \Vert _{2}^{\frac{1}{2}} \bigl\Vert D^{3}B \bigr\Vert _{2}^{ \frac{3}{2}} \\ &\leq C \Vert DV \Vert _{2}^{4} \Vert \Delta B \Vert _{2}^{2}+\frac{1}{22} \bigl\Vert D^{3}B \bigr\Vert _{2}^{2}. \end{aligned} $$

Hence we have

$$\begin{aligned} \begin{aligned} \vert R_{1} \vert \leq C\bigl( \Vert V \Vert _{s}^{\frac{2s}{s-3}}+ \Vert DV \Vert _{2}^{4} \bigr) \Vert \Delta B \Vert _{2}^{2} +\frac{3}{22} \bigl\Vert D^{3}B \bigr\Vert _{2}^{2}. \end{aligned} \end{aligned}$$

(15)

We can rewrite $R_{2}+R_{3}$ as

$$\begin{aligned} \begin{aligned}R_{2}+R_{3}&= \int _{\mathbb{R}^{3}}\bigl(D^{2}B\cdot D\bigr)B\cdot D^{2}V\,dx +2 \int _{\mathbb{R}^{3}}(DB\cdot D)D B\cdot D^{2}V\,dx \\ &\quad{}+ \int _{\mathbb{R}^{3}}\bigl(D^{2}B\cdot DV\bigr)\cdot D^{2}B\,dx +2 \int _{ \mathbb{R}^{3}}(D B\cdot D)DV\cdot D^{2}B\,dx \\ &=R_{231}+R_{232}+R_{233}+R_{234}. \end{aligned} \end{aligned}$$

(16)

The terms $R_{231}$, $R_{232}$, $R_{234}$, and $R_{233}$ can be estimated as $R_{11}$, $R_{12}$. Hence we have

$$\begin{aligned} \begin{aligned} \vert R_{2}+R_{3} \vert \leq C\bigl( \Vert V \Vert _{s}^{\frac{2s}{s-3}}+ \Vert DV \Vert _{2}^{4}\bigr) \Vert \Delta B \Vert _{2}^{2} +\frac{7}{22} \bigl\Vert D^{3}B \bigr\Vert _{2}^{2}. \end{aligned} \end{aligned}$$

(17)

The term $R_{4}$ can be written as

$$\begin{aligned} \begin{aligned}R_{4}=- \int _{\mathbb{R}^{3}}\bigl(D^{2}u\cdot D\bigr)u\cdot D^{2}u\,dx -2 \int _{\mathbb{R}^{3}}(Du\cdot D)D u\cdot D^{2}u\,dx=R_{41}+R_{42}. \end{aligned} \end{aligned}$$

(18)

We further decompose $R_{41}$ and $R_{42}$ into three parts:

$$\begin{aligned}& \begin{aligned}R_{41}&=- \int _{\mathbb{R}^{3}}(D_{p}DV\cdot D)V\cdot D_{p}DV\,dx- \int _{\mathbb{R}^{3}}\bigl(\partial _{3}^{2}V_{p} \cdot D_{p}\bigr)V\cdot \partial _{3}^{2}V\,dx\\ &\quad {}+ \int _{\mathbb{R}^{3}}(\partial _{3}D_{p}\cdot V_{p}) \partial _{3}V\cdot \partial _{3}^{2}V\,dx \\ &=R_{411}+R_{412}+R_{413}, \end{aligned} \end{aligned}$$

(19)

$$\begin{aligned}& \begin{aligned}R_{42}&=-2 \int _{\mathbb{R}^{3}}(D_{p}V\cdot D)DV\cdot D_{p}DV\,dx-2 \int _{\mathbb{R}^{3}}(\partial _{3}V_{p}\cdot D_{p})DV\cdot \partial _{3}^{2}V\,dx\\ &\quad {}+2 \int _{\mathbb{R}^{3}}(D_{p}\cdot V_{p}) \partial _{3}DV\cdot \partial _{3}^{2}V\,dx \\ &=R_{421}+R_{422}+R_{423}. \end{aligned} \end{aligned}$$

(20)

By integration by parts and interpolation we obtain

$$ \begin{aligned} \vert R_{411} \vert &=2 \biggl\vert \int _{\mathbb{R}^{3}}VDD_{p}VD^{2}D_{p}V\,dx \biggr\vert \\ &\leq C \Vert V \Vert _{s} \Vert DD_{p}V \Vert _{\frac{2s}{s-2}} \Vert \Delta D_{p}V \Vert _{2} \\ &\leq C \Vert V \Vert _{s} \Vert DD_{p}V \Vert _{2}^{\frac{s-3}{s}} \Vert \Delta D_{p}V \Vert _{2}^{ \frac{s+3}{s}} \\ &\leq +C \Vert V \Vert _{s}^{\frac{2s}{s-3}} \Vert DD_{p}V \Vert _{2}^{2}+\frac{\rho}{12} \Vert \Delta D_{p}V \Vert _{2}^{2} \end{aligned} $$

By integration by parts and Lemma 2.2 we get

$$ \begin{aligned} \vert R_{412} \vert &= \biggl\vert \int _{\mathbb{R}^{3}}\partial _{3}^{2}D_{p}V_{p}V \cdot \partial _{3}^{2}V+\partial _{3}^{2}V_{p} V\cdot \partial _{3}^{2}D_{p}V\,dx \biggr\vert \\ &\leq C \bigl\Vert \partial _{3}^{2}D_{p}V \bigr\Vert _{2} \Vert \partial _{3}V \Vert _{2}^{ \frac{1}{2}} \bigl\Vert \partial _{1}\partial _{3}^{2} V \bigr\Vert _{2}^{\frac{1}{2}} \Vert V \Vert _{2}^{\frac{1}{4}} \bigl\Vert \partial _{3}^{2}V \bigr\Vert _{2}^{\frac{1}{4}} \Vert \partial _{2}V \Vert _{2}^{\frac{1}{4}} \bigl\Vert \partial _{2}\partial _{3}^{2}V \bigr\Vert _{2}^{ \frac{1}{4}} \\ &\leq C \Vert \Delta D_{p}V \Vert _{2}^{\frac{7}{4}} \Vert DV \Vert _{2}^{\frac{3}{4}} \Vert \Delta V \Vert _{2}^{\frac{1}{4}} \\ &\leq C \Vert DV \Vert _{2}^{6} \Vert \Delta V \Vert _{2}^{2}+\frac{\rho}{12} \Vert \Delta D_{p}V \Vert _{2}^{2}. \end{aligned} $$

We can use Lemma 2.1 to estimate $R_{313}$ as follows:

$$ \begin{aligned} \vert R_{413} \vert &\leq C \Vert \partial _{3}D_{p}V \Vert _{2}^{\frac{1}{2}} \Vert \partial _{3}V \Vert _{2}^{\frac{1}{2}} \bigl\Vert \partial _{3}^{2}V \bigr\Vert _{2}^{ \frac{1}{2}} \bigl\Vert \partial _{3}^{2}D_{p}V \bigr\Vert _{2}^{\frac{1}{2}} \Vert \partial _{3}D_{p}V \Vert _{2}^{\frac{1}{2}} \bigl\Vert \partial _{3}^{2}D_{p}V \bigr\Vert _{2}^{\frac{1}{2}} \\ &\leq C \bigl\Vert D^{2}D_{p}V \bigr\Vert _{2} \Vert DD_{p}V \Vert _{2} \Vert DV \Vert _{L^{2}}^{\frac{1}{2}} \bigl\Vert D^{2}V \bigr\Vert _{2}^{\frac{1}{2}} \\ &\leq C \bigl\Vert D^{2}D_{p}V \bigr\Vert _{2}^{\frac{3}{2}} \Vert DV \Vert _{2} \bigl\Vert D^{2}V \bigr\Vert _{2}^{ \frac{1}{2}} \\ &\leq C \Vert DV \Vert _{2}^{4} \Vert \Delta V \Vert _{2}^{2}+\frac{\rho}{12} \bigl\Vert D^{2}D_{p}V \bigr\Vert _{2}^{2}. \end{aligned} $$

Therefore we have

$$ \begin{aligned} \vert R_{41} \vert \leq C\bigl( \Vert DV \Vert _{2}^{6}+ \Vert V \Vert _{s}^{\frac{2s}{s-3}} \bigr) \Vert \Delta V \Vert _{2}^{2}+\frac{3\rho}{12} \Vert \Delta D_{p}V \Vert _{2}^{2}. \end{aligned} $$

Clearly, $R_{421}$, $R_{422}$, $R_{423}$ can be estimated as $R_{411}$, $R_{413}$, $R_{412}$. Hence we have

$$ \begin{aligned} \vert R_{42} \vert \leq C\bigl( \Vert DV \Vert _{2}^{6}+ \Vert V \Vert _{s}^{\frac{2s}{s-3}} \bigr) \Vert \Delta V \Vert _{2}^{2}+\frac{3\rho}{12} \Vert \Delta D_{p}V \Vert _{2}^{2}. \end{aligned} $$

Therefore we get

$$\begin{aligned} \begin{aligned} \vert R_{4} \vert \leq C\bigl( \Vert DV \Vert _{2}^{6}+ \Vert V \Vert _{s}^{\frac{2s}{s-3}} \bigr) \Vert \Delta V \Vert _{2}^{2} +\frac{\rho}{2} \Vert \Delta D_{p}V \Vert _{2}^{2}. \end{aligned} \end{aligned}$$

(21)

Using the cancelation property, we obtain

$$\begin{aligned} \begin{aligned} \vert R_{5} \vert &= \biggl\vert \int _{\mathbb{R}^{3}}\bigl[D^{2}\bigl((\nabla \times B) \times B \bigr)-D^{2}(\nabla \times B)\times B\bigr]\cdot D^{2}(\nabla \times B)\,dx \biggr\vert \\ &\leq C \bigl\Vert D^{3}B \bigr\Vert _{2} \Vert DB \Vert _{\lambda} \bigl\Vert D^{2}B \bigr\Vert _{ \frac{2\lambda}{\lambda -2}} \\ &\leq C \bigl\Vert D^{3}B \bigr\Vert _{2}^{\frac{\lambda +3}{\lambda}} \Vert D B \Vert _{\lambda} \bigl\Vert D^{2}B \bigr\Vert _{2}^{\frac{\lambda -3}{\lambda}} \\ &\leq C \Vert DB \Vert _{\lambda}^{\frac{2\lambda}{\lambda -3}} \bigl\Vert D^{2}B \bigr\Vert _{2}^{2}+ \frac{1}{22} \bigl\Vert D^{3}B \bigr\Vert _{2}^{2}. \end{aligned} \end{aligned}$$

(22)

Combining (13)–(22), we get (12). □

Proposition 2.3

Let $(V,B)$ solve system (1) with $\rho _{1}>0$, $\rho _{2}>0$, and $\rho _{3}=0$. Then

$$\begin{aligned} \begin{aligned}&\frac{d}{dt}\bigl( \bigl\Vert D^{3}V(t) \bigr\Vert _{2}^{2}+ \bigl\Vert D^{3}B(t) \bigr\Vert _{2}^{2}\bigr)+ \rho _{0}\bigl( \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2}^{2}+ \bigl\Vert D^{4}B \bigr\Vert _{2}^{2}\bigr) \\ &\quad \leq C\bigl( \bigl\Vert D^{3}V \bigr\Vert _{2}^{2}+ \bigl\Vert D^{3}B \bigr\Vert _{2}^{2}\bigr) \bigl( \Vert V \Vert _{s}^{ \frac{2s}{s-3}}+ \Vert D B \Vert _{\lambda}^{\frac{2\lambda}{\lambda -3}}+ \Vert DV \Vert _{2}^{6} + \Vert \Delta B \Vert _{2}^{2}\bigr). \end{aligned} \end{aligned}$$

(23)

Proof

Similarly to the derivation of (5), we have

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{d}{dt}\bigl( \bigl\Vert D^{3}V(t) \bigr\Vert _{2}^{2}+ \bigl\Vert D^{3}B(t) \bigr\Vert _{2}^{2}\bigr)+ \rho _{1} \bigl\Vert \partial _{1}D^{3}V \bigr\Vert _{2}^{2}+\rho _{2} \bigl\Vert \partial _{2}D^{3}V \bigr\Vert _{2}^{2}+ \bigl\Vert D^{4}B \bigr\Vert _{2}^{2} \\ &\quad=- \int _{\mathbb{R}^{3}}D^{3}(V\cdot DB)\cdot D^{3}B\,dx + \int _{ \mathbb{R}^{3}}D^{3}(B\cdot DV)\cdot D^{3}B\,dx + \int _{\mathbb{R}^{3}}D^{3}(B \cdot DB)\cdot D^{3}V\,dx \\ &\quad\quad{}- \int _{\mathbb{R}^{3}}D^{3}(V\cdot DV)\cdot D^{3}V\,dx - \int _{ \mathbb{R}^{3}}D^{3}\bigl[D\times \bigl((D\times B)\times B\bigr)\bigr]\cdot D^{3}B\,dx \\ &\quad=T_{1}+T_{2}+T_{3}+T_{4}+T_{5}. \end{aligned} \end{aligned}$$

(24)

We decompose $T_{1}$ into three parts:

$$ \begin{aligned}T_{1}&=- \int _{\mathbb{R}^{3}}\bigl(D^{3}V\cdot D\bigr)B\cdot D^{3}B\,dx -3 \int _{\mathbb{R}^{3}}\bigl(D^{2}V\cdot D\bigr)DB\cdot D^{3}B\,dx \\ &\quad {}-3 \int _{ \mathbb{R}^{3}}(DV\cdot D)D^{2}B\cdot D^{3}B\,dx \\ &=T_{11}+T_{12}+T_{13}. \end{aligned} $$

We apply the Hölder inequality and interpolation to estimate $T_{11}$:

$$ \begin{aligned} \vert T_{11} \vert &\leq C \bigl\Vert D^{3}V \bigr\Vert _{2} \Vert D B \Vert _{3} \bigl\Vert D^{3}B \bigr\Vert _{6} \\ &\leq C \bigl\Vert D^{3}V \bigr\Vert _{2} \Vert \Delta B \Vert _{2}^{\frac{3}{4}} \Vert B \Vert _{2}^{ \frac{1}{4}} \bigl\Vert D^{4}B \bigr\Vert _{2} \\ &\leq C \Vert \Delta B \Vert _{2}^{2} \bigl\Vert D^{3}V \bigr\Vert _{2}^{2}+\frac{1}{20} \bigl\Vert D^{4}B \bigr\Vert _{2}^{2}. \end{aligned} $$

Using integration by parts and $DivV=0$, we get

$$ \begin{aligned} \vert T_{12} \vert &=3 \biggl\vert \int _{\mathbb{R}^{3}}D^{2}VDB\cdot D^{4}B\,dx \biggr\vert \\ &\leq C \bigl\Vert D^{4}B \bigr\Vert _{2} \Vert \Delta V \Vert _{6} \Vert D B \Vert _{3} \\ &\leq C \bigl\Vert D^{4}B \bigr\Vert _{2} \bigl\Vert D^{3}V \bigr\Vert _{2} \Vert \Delta B \Vert _{2}^{\frac{3}{4}} \Vert B \Vert _{2}^{\frac{1}{4}} \\ &\leq C \Vert \Delta B \Vert _{2}^{2} \bigl\Vert D^{3}V \bigr\Vert _{2}^{2}+\frac{1}{20} \bigl\Vert D^{4}B \bigr\Vert _{2}^{2}. \end{aligned} $$

We used the boundedness of $\|B\|_{L^{2}}$ in the above two estimates. Similarly, we estimate $T_{13}$ as follows:

$$ \begin{aligned} \vert T_{13} \vert &=3 \biggl\vert \int _{\mathbb{R}^{3}}(V\cdot D)D^{2}B\cdot D^{4}B+(V \cdot D)D^{3}B\cdot D^{3}B\,dx \biggr\vert \\ &\leq C \Vert V \Vert _{s} \bigl\Vert D^{3}B \bigr\Vert _{\frac{2s}{s-2}} \bigl\Vert D^{4}B \bigr\Vert _{2} \\ &\leq C \Vert V \Vert _{s} \bigl\Vert D^{3}B \bigr\Vert _{2}^{\frac{s-3}{s}} \bigl\Vert D^{4}B \bigr\Vert _{2}^{ \frac{s+3}{s}} \\ &\leq C \Vert V \Vert _{s}^{\frac{2s}{s-3}} \bigl\Vert D^{3}B \bigr\Vert _{2}^{2}+\frac{1}{20} \bigl\Vert D^{4}B \bigr\Vert _{2}^{2}. \end{aligned} $$

Hence we get

$$\begin{aligned} \begin{aligned} \vert T_{1} \vert \leq C\bigl( \Vert V \Vert _{s}^{\frac{2s}{s-3}}+ \Vert \Delta B \Vert _{2}^{2} \bigr) \bigl( \bigl\Vert D^{3}V \bigr\Vert _{2}^{2}+ \bigl\Vert D^{3}B \bigr\Vert _{2}^{2}\bigr)+ \frac{3}{20} \bigl\Vert D^{4}B \bigr\Vert _{2}^{2}. \end{aligned} \end{aligned}$$

(25)

We decompose $T_{2}+T_{3}$ into six terms:

$$ \begin{aligned}T_{2}+T_{3}&= \int _{\mathbb{R}^{3}}\bigl(D^{3}B\cdot D\bigr)B\cdot D^{3}V\,dx +3 \int _{\mathbb{R}^{3}}\bigl(D^{2}B\cdot D\bigr)DB\cdot D^{3}V\,dx\\ &\quad {} +3 \int _{ \mathbb{R}^{3}}(DB\cdot D)D^{2}B\cdot D^{3}V\,dx+ \int _{\mathbb{R}^{3}}\bigl(D^{3}B\cdot D\bigr)V\cdot D^{3}B\,dx \\ &\quad{} +3 \int _{ \mathbb{R}^{3}}\bigl(D^{2}B\cdot D\bigr)DV\cdot D^{3}B\,dx +3 \int _{\mathbb{R}^{3}}(DB \cdot D)D^{2}V\cdot D^{3}B\,dx \\ &=T_{231}+T_{232}+T_{233}+T_{234}+T_{235}+T_{236}. \end{aligned} $$

Integrating by parts $T_{232}$, we get

$$ \begin{aligned}T_{232}&=-3 \int _{\mathbb{R}^{3}}\bigl(D^{3}B\cdot D\bigr)DB\cdot D^{2}V\,dx -3 \int _{\mathbb{R}^{3}}\bigl(D^{2}B\cdot D\bigr)D^{2}B \cdot D^{2}V\,dx. \end{aligned} $$

Therefore $(T_{231},T_{233},T_{236})$, $(T_{232},T_{235})$, and $T_{234}$ can be estimated as $T_{11}$, $T_{12}$, $T_{13}$. Hence we get

$$\begin{aligned} \begin{aligned} \vert T_{2}+T_{3} \vert \leq C\bigl( \Vert V \Vert _{s}^{\frac{2s}{s-3}}+ \Vert \Delta B \Vert _{2}^{2}\bigr) \bigl( \bigl\Vert D^{3}V \bigr\Vert _{2}^{2}+ \bigl\Vert D^{3}B \bigr\Vert _{2}^{2}\bigr)+\frac{6}{20} \bigl\Vert D^{4}B \bigr\Vert _{2}^{2}. \end{aligned} \end{aligned}$$

(26)

We split $T_{4}$ into three terms:

$$ \begin{aligned}T_{4}&=- \int _{\mathbb{R}^{3}}\bigl(D^{3}V\cdot D\bigr)VD^{3}V\,dx -3 \int _{\mathbb{R}^{3}}\bigl(D^{2}V\cdot D\bigr)DVD^{3}V\,dx \\ &\quad {}-3 \int _{\mathbb{R}^{3}}(DV \cdot D)D^{2}VD^{3}V\,dx \\ &=T_{41}+T_{42}+T_{43}, \end{aligned} $$

and $T_{41}$ and $T_{42}$ can be further decomposed into three parts as follows:

$$\begin{aligned}& \begin{aligned}T_{41}&=- \int _{\mathbb{R}^{3}}\bigl(D^{2}D_{p}V\cdot D \bigr)VD^{2}D_{p}V\,dx - \int _{\mathbb{R}^{3}}\bigl(\partial _{3}^{3}V_{p} \cdot D_{p}\bigr)V\partial _{3}^{3}V\,dx \\ &\quad {}+ \int _{\mathbb{R}^{3}}\bigl(\partial _{3}^{2}D_{p} \cdot V_{p}\bigr)\partial _{3}V \partial _{3}^{3}V\,dx \\ &=T_{411}+T_{412}+T_{413}, \end{aligned} \\& \begin{aligned} T_{42}&=-3 \int _{\mathbb{R}^{3}}(DD_{p}V\cdot D)DV D^{2}D_{p}V\,dx -3 \int _{\mathbb{R}^{3}}\bigl(\partial _{3}^{2}V_{p} \cdot D_{p}\bigr)\partial _{3}V \partial _{3}^{3}V\,dx \\ &\quad {}+3 \int _{\mathbb{R}^{3}}(\partial _{3}D_{p} \cdot V_{p})\partial _{3}^{2}V \partial _{3}^{3}V\,dx \\ &=T_{421}+T_{422}+T_{423}. \end{aligned} \end{aligned}$$

Integrating by parts in $T_{411}$ and applying the Hölder inequality, we get

$$ \begin{aligned} \vert T_{411} \vert &=2 \biggl\vert \int _{\mathbb{R}^{3}}(VD^{2}D_{p}VD^{3}D_{p}V\,dx \biggr\vert \\ &\leq C \Vert V \Vert _{s} \bigl\Vert D^{2}D_{p}V \bigr\Vert _{\frac{2s}{s-2}} \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2} \\ &\leq C \Vert V \Vert _{s} \bigl\Vert D^{2}D_{p}V \bigr\Vert _{2}^{\frac{s-3}{s}} \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2}^{ \frac{s+3}{s}} \\ &\leq C \Vert V \Vert _{s}^{\frac{2s}{s-3}} \bigl\Vert D^{3}V \bigr\Vert _{2}^{2}+\frac{\rho}{22} \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2}^{2}. \end{aligned} $$

Integrating by parts $T_{412}$, we have

$$ \begin{aligned} \vert T_{412} \vert &= \biggl\vert \int _{\mathbb{R}^{3}}V\partial _{3}^{3}D_{p}V_{p} \partial _{3}^{3}V +V\partial _{3}^{3}V_{p} \partial _{3}^{3}D_{p}V\,dx \biggr\vert \\ &\leq C \bigl\Vert \partial _{3}^{3}D_{p}V \bigr\Vert _{2} \Vert \partial _{3}V \Vert _{2}^{ \frac{1}{2}} \bigl\Vert \partial _{1}\partial _{3}^{3}V \bigr\Vert _{2}^{\frac{1}{2}} \Vert V \Vert _{2}^{\frac{1}{4}} \bigl\Vert \partial _{3}^{3}V \bigr\Vert _{2}^{\frac{1}{4}} \Vert \partial _{2}V \Vert _{2}^{\frac{1}{4}} \bigl\Vert \partial _{2}\partial _{3}^{3}V \bigr\Vert _{2}^{\frac{1}{4}} \\ &\leq C \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2}^{\frac{7}{4}} \Vert DV \Vert _{2}^{\frac{3}{4}} \bigl\Vert D^{3}V \bigr\Vert _{2}^{\frac{1}{4}} \\ &\leq C \Vert DV \Vert _{2}^{6} \bigl\Vert D^{3}V \bigr\Vert _{2}^{2}+\frac{\rho}{22} \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2}^{2}. \end{aligned} $$

By Lemma 2.1 we have

$$\begin{aligned} \vert T_{413} \vert &\leq C \bigl\Vert \partial _{3}^{2}D_{p}V \bigr\Vert _{2}^{\frac{1}{2}} \Vert \partial _{3}V \Vert _{2}^{\frac{1}{2}} \bigl\Vert \partial _{3}^{3}V \bigr\Vert _{2}^{ \frac{1}{2}} \bigl\Vert \partial _{3}^{3}D_{p}V \bigr\Vert _{2}^{\frac{1}{2}} \Vert \partial _{3}D_{p}V \Vert _{2}^{\frac{1}{2}} \bigl\Vert \partial _{3}^{3}D_{p}V \bigr\Vert _{2}^{\frac{1}{2}} \\ &\leq C \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2} \bigl\Vert D^{2}D_{p}V \bigr\Vert _{L^{2}}^{\frac{1}{2}} \Vert DD_{p}V \Vert _{2}^{\frac{1}{2}} \Vert DV \Vert _{2}^{\frac{1}{2}} \bigl\Vert D^{3}V \bigr\Vert _{2}^{ \frac{1}{2}} \\ &\leq C \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2}^{\frac{3}{2}} \Vert DV \Vert _{2} \bigl\Vert D^{3}V \bigr\Vert _{2}^{ \frac{1}{2}} \\ &\leq C \Vert DV \Vert _{2}^{4} \bigl\Vert D^{3}V \bigr\Vert _{2}^{2}+\frac{\rho}{22} \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2}^{2}. \end{aligned}$$

Therefore

$$ \begin{aligned} \vert T_{41} \vert \leq C\bigl( \Vert DV \Vert _{2}^{6}+ \Vert V \Vert _{s}^{\frac{2s}{s-3}} \bigr) \bigl\Vert D^{3}V \bigr\Vert _{2}^{2} + \frac{3\rho}{22} \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2}^{2}. \end{aligned} $$

We can use Lemma 2.1 to estimate $T_{421}$ as follows:

$$ \begin{aligned} \vert T_{421} \vert &\leq C \Vert DD_{p}V \Vert _{2}^{\frac{1}{2}} \bigl\Vert D^{2}V \bigr\Vert _{2}^{ \frac{1}{2}} \bigl\Vert D^{2}D_{p}V \bigr\Vert _{2}^{\frac{1}{2}} \Vert \partial _{3}DD_{p}V \Vert _{2}^{\frac{1}{2}} \bigl\Vert D^{2}D_{p}V \bigr\Vert _{2}^{\frac{1}{2}} \bigl\Vert D^{2}D_{p}^{2}V \bigr\Vert _{2}^{\frac{1}{2}} \\ &\leq C \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2}^{\frac{1}{2}} \bigl\Vert D^{2}D_{p}V \bigr\Vert _{2}^{ \frac{3}{2}} \Vert DD_{p}V \Vert _{2}^{\frac{1}{2}} \bigl\Vert D^{2}V \bigr\Vert _{2}^{\frac{1}{2}} \\ &\leq C \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2}^{\frac{5}{3}} \Vert D_{p}V \Vert _{2}^{\frac{5}{6}} \bigl\Vert D^{3}V \bigr\Vert _{2}^{\frac{1}{3}} \Vert V \Vert _{2}^{\frac{1}{6}} \\ &\leq C \Vert DV \Vert _{2}^{5} \bigl\Vert D^{3}V \bigr\Vert _{2}^{2}+\frac{\rho}{22} \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2}^{2}, \end{aligned} $$

where we used the boundedness of $\|u\|_{2}$. We further divide $T_{422}$ into two terms:

$$ \begin{aligned}T_{422}&=-3 \int _{\mathbb{R}^{3}}\bigl(\partial _{3}^{2}V_{p} \cdot D_{p}\bigr)\partial _{3}V_{p}\partial _{3}^{3}V_{p}\,dx -3 \int _{ \mathbb{R}^{3}}\bigl(\partial _{3}^{2}V_{p} \cdot D_{p}\bigr)\partial _{3}V_{3} \partial _{3}^{3}V_{3}\,dx \\ &=\frac{3}{2} \int _{\mathbb{R}^{3}}\bigl(\partial _{3}^{2}V_{p} \cdot D_{p}\bigr) \partial _{3}^{2}V_{p} \partial _{3}^{2}V_{p}\,dx +3 \int _{\mathbb{R}^{3}}\bigl( \partial _{3}^{2}V\cdot D_{p}\bigr)\partial _{3}V_{3}\bigl(\partial _{3}^{2}D_{p} \cdot V_{p}\bigr)\,dx \\ &=T_{4221}+T_{4222}. \end{aligned} $$

We estimate $T_{4221}$ as

$$ \begin{aligned} \vert T_{4221} \vert &\leq C \bigl\Vert \partial _{3}^{2}V \bigr\Vert _{2}^{\frac{1}{2}} \bigl\Vert \partial _{3}^{2}D_{p}V \bigr\Vert _{2}^{\frac{1}{2}} \bigl\Vert \partial _{3}^{2}V \bigr\Vert _{2}^{ \frac{1}{2}} \bigl\Vert \partial _{3}^{2}D_{p}V \bigr\Vert _{2}^{\frac{1}{2}} \bigl\Vert \partial _{3}^{3}D_{p}V \bigr\Vert _{2}^{\frac{1}{2}} \bigl\Vert \partial _{3}^{2}D_{p}V \bigr\Vert _{2}^{\frac{1}{2}} \\ &\leq C \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2}^{\frac{1}{2}} \bigl\Vert D^{2}D_{p}V \bigr\Vert _{2}^{ \frac{3}{2}} \bigl\Vert D^{2}V \bigr\Vert _{2} \\ &\leq C \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2}^{\frac{3}{2}} \Vert DV \Vert _{2} \bigl\Vert D^{3}V \bigr\Vert _{2}^{ \frac{1}{2}} \\ &\leq C \Vert DV \Vert _{2}^{4} \bigl\Vert D^{3}V \bigr\Vert _{2}^{2}+\frac{\rho}{22} \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2}^{2}. \end{aligned} $$

We can estimate $T_{4222}$ as $T_{421}$:

$$ \begin{aligned} \vert T_{4222} \vert \leq C \Vert DV \Vert _{2}^{5} \bigl\Vert D^{3}V \bigr\Vert _{2}^{2} + \frac{\rho}{22} \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2}^{2}. \end{aligned} $$

Similarly,

$$ \begin{aligned} \vert T_{423} \vert \leq C \Vert DV \Vert _{2}^{5} \bigl\Vert D^{3}V \bigr\Vert _{2}^{2} + \frac{2\rho}{22} \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2}^{2}. \end{aligned} $$

Hence we get

$$ \begin{aligned} \vert T_{42} \vert \leq C \Vert DV \Vert _{2}^{5} \bigl\Vert D^{3}V \bigr\Vert _{2}^{2} + \frac{5\rho}{22} \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2}^{2}. \end{aligned} $$

We can estimate $T_{43}$ as $T_{41}$:

$$ \begin{aligned} \vert T_{43} \vert \leq C\bigl( \Vert DV \Vert _{2}^{6}+ \Vert V \Vert _{s}^{\frac{2s}{s-3}} \bigr) \bigl\Vert D^{3}V \bigr\Vert _{2}^{2} + \frac{3\rho}{22} \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2}^{2}. \end{aligned} $$

Putting the above estimates together, we obtain

$$\begin{aligned} \begin{aligned} \vert T_{4} \vert \leq C\bigl( \Vert DV \Vert _{2}^{6}+ \Vert V \Vert _{s}^{\frac{2s}{s-3}} \bigr) \bigl\Vert D^{3}V \bigr\Vert _{2}^{2} + \frac{\rho}{2} \bigl\Vert D^{3}D_{p}V \bigr\Vert _{2}^{2}. \end{aligned} \end{aligned}$$

(27)

Similarly to (16), we get

$$\begin{aligned} \begin{aligned} \vert T_{5} \vert &= \biggl\vert \int _{\mathbb{R}^{3}}\bigl[D^{3}\bigl((\nabla \times B) \times B \bigr)-D^{3}(\nabla \times B)\times B\bigr]\cdot D^{3}(\nabla \times B)\,dx \biggr\vert \\ &\leq C \Vert D B \Vert _{\lambda} \bigl\Vert D^{3}B \bigr\Vert _{\frac{2\lambda}{\lambda -2}} \bigl\Vert D^{4}B \bigr\Vert _{2} \\ &\leq C \Vert D B \Vert _{\lambda} \bigl\Vert D^{3}B \bigr\Vert _{2}^{\frac{\lambda -3}{\beta}} \bigl\Vert D^{4}B \bigr\Vert _{2}^{\frac{\lambda +3}{\lambda}} \\ &\leq C \Vert D B \Vert _{\lambda}^{\frac{2\lambda}{\lambda -3}} \bigl\Vert D^{3} B \bigr\Vert _{2}^{2} +\frac{1}{20} \bigl\Vert D^{4}B \bigr\Vert _{2}^{2}. \end{aligned} \end{aligned}$$

(28)

Combining (24)–(28) yields (23). □

3 Proof of Theorem 1.1

Putting (2), (4), (12), and (23) together, we get

$$ \begin{aligned}&\frac{d}{dt}\bigl( \bigl\Vert V(t) \bigr\Vert _{H^{3}}^{2}+ \bigl\Vert B(t) \bigr\Vert _{H^{3}}^{2}\bigr)+ \rho _{0}\bigl( \Vert D_{p}V \Vert _{H^{3}}^{2}+ \Vert DB \Vert _{H^{3}}^{2}\bigr) \\ &\quad \leq C\bigl( \Vert V \Vert _{s}^{\frac{2s}{s-3}}+ \Vert DB \Vert _{\lambda}^{ \frac{2\lambda}{\lambda -3}}+ \Vert D V \Vert _{2}^{6}+ \bigl\Vert D^{2}B \bigr\Vert _{2}^{2}\bigr) \bigl( \Vert V \Vert _{H^{3}}^{2}+ \Vert B \Vert _{H^{3}}^{2}\bigr). \end{aligned} $$

Applying Gronwall’s inequality to this inequality, we obtain

$$ \begin{aligned}&\sup_{0< t< T}\bigl( \bigl\Vert V(t) \bigr\Vert _{H^{3}}^{2}+ \bigl\Vert B(t) \bigr\Vert _{H^{3}}^{2}\bigr)+ \rho _{0} \int _{0}^{T}\bigl( \bigl\Vert D_{p}V(t) \bigr\Vert _{H^{3}}^{2}+ \bigl\Vert DB(t) \bigr\Vert _{H^{3}}^{2}\bigr)\,dt \\ &\quad\leq \bigl( \Vert V_{0} \Vert _{H^{3}}^{2}+ \Vert B_{0} \Vert _{H^{3}}^{2}\bigr)\\ &\qquad {}\times (1+C \int _{0}^{T}\bigl( \bigl\Vert V(t) \bigr\Vert _{s}^{\frac{2s}{s-6}}+ \bigl\Vert D B(t) \bigr\Vert _{\lambda}^{ \frac{2\lambda}{\lambda -3}}+ \bigl\Vert DV(t) \bigr\Vert _{2}^{6}+ \bigl\Vert D^{2}B(t) \bigr\Vert _{2}^{2}\bigr)\,dt) \\ &\quad\quad{}\times \exp \biggl[C \int _{0}^{T}\bigl( \bigl\Vert V(t) \bigr\Vert _{s}^{\frac{2s}{s-6}}+ \bigl\Vert DB(t) \bigr\Vert _{ \lambda}^{\frac{2\lambda}{\lambda -3}} + \bigl\Vert DV(t) \bigr\Vert _{2}^{6}+ \bigl\Vert D^{2}B(t) \bigr\Vert _{2}^{2}\bigr)\,dt\biggr]. \end{aligned} $$

which, together with (11), gives that if

$$ \begin{aligned} \int _{0}^{\widetilde{\mathbf{T}}}\bigl( \bigl\Vert V(t) \bigr\Vert _{s}^{ \frac{2s}{s-6}} + \bigl\Vert DB(t) \bigr\Vert _{\lambda}^{\frac{2\lambda}{\lambda -3}}\bigr)\,dt< \infty , \end{aligned} $$

then

$$ \begin{aligned}&\sup_{0< t< \widetilde{\mathbf{T}}}\bigl( \bigl\Vert V(t) \bigr\Vert _{H^{3}}^{2}+ \bigl\Vert B(t) \bigr\Vert _{H^{3}}^{2}\bigr)+ \rho _{0} \int _{0}^{\widetilde{\mathbf{T}}}\bigl( \bigl\Vert D_{p}V(t) \bigr\Vert _{H^{3}}^{2}+ \bigl\Vert DB(t) \bigr\Vert _{H^{3}}^{2}\bigr)\,dt< \infty . \end{aligned} $$

Notice that $k\geq \frac{2s}{s-3}$ and $\alpha \geq \frac{2\lambda}{\lambda -3}$. Hence Theorem 1.1 holds.

4 Proof of Theorem 1.2

We begin with estimating the terms $M_{1}$–$M_{5}$ in (5), First, we estimate $M_{1}$:

$$\begin{aligned} \begin{aligned} \vert M_{1} \vert &= \biggl\vert \int _{\mathbb{R}^{3}}(DV\cdot D)B\cdot D B\,dx \biggr\vert \\ &\leq C \Vert DV \Vert _{2} \Vert DB \Vert _{4}^{2} \\ &\leq C \Vert DV \Vert _{2} \Vert DB \Vert _{2}^{\frac{1}{2}} \bigl\Vert D^{2}B \bigr\Vert _{2}^{\frac{3}{2}} \\ &\leq C\bigl( \Vert DV \Vert _{2}^{2}+ \Vert DB \Vert _{2}\bigr) \bigl\Vert D^{2}B \bigr\Vert _{2}^{\frac{3}{2}}. \end{aligned} \end{aligned}$$

(29)

The sum $M_{2}+M_{3}$ can be rewritten as

$$ \begin{aligned}M_{2}+M_{3}=\sum _{j=1}^{3} \int _{\mathbb{R}^{3}}(\partial _{j}B \cdot D)V\cdot \partial _{j}B +(\partial _{j}B\cdot D)B\cdot \partial _{j}V\,dx, \end{aligned} $$

and hence we can estimate $M_{2}+M_{3}$ as $M_{1}$ to obtain

$$\begin{aligned} \begin{aligned} \vert M_{2}+M_{3} \vert \leq C\bigl( \Vert DV \Vert _{2}^{2}+ \Vert DB \Vert _{2} \bigr) \bigl\Vert D^{2}B \bigr\Vert _{2}^{ \frac{3}{2}}. \end{aligned} \end{aligned}$$

(30)

For $M_{4}$, we estimate each term $M_{41}$–$M_{43}$ in (8). By Lemma 2.1 and interpolation we have

$$ \begin{aligned} \vert M_{41} \vert &\leq C \Vert D_{p}V \Vert _{2}^{\frac{1}{2}} \Vert DV \Vert _{2}^{ \frac{1}{2}} \Vert D_{p}V \Vert _{2}^{\frac{1}{2}} \Vert \partial _{3}D_{p}V \Vert _{2}^{ \frac{1}{2}} \Vert DD_{p}V \Vert _{2}^{\frac{1}{2}} \bigl\Vert D_{p}^{2}V \bigr\Vert _{2}^{ \frac{1}{2}} \\ &\leq C \Vert DV \Vert _{2}^{\frac{3}{2}} \Vert DD_{p}V \Vert _{2}^{\frac{3}{2}} \end{aligned} $$

and

$$ \begin{aligned} \vert M_{42} \vert &\leq C \Vert \partial _{3}V \Vert _{2}^{\frac{1}{2}} \Vert D_{p}V \Vert _{2}^{\frac{1}{2}} \Vert \partial _{3}V \Vert _{2}^{\frac{1}{2}} \Vert \partial _{3}D_{p}V_{p} \Vert _{2}^{\frac{1}{2}} \Vert \partial _{3}D_{p}V \Vert _{2}^{\frac{1}{2}} \Vert \partial _{3}D_{p}V \Vert _{2}^{\frac{1}{2}} \\ &\leq C \Vert D V \Vert _{2}^{\frac{3}{2}} \Vert DD_{p}V \Vert _{2}^{\frac{3}{2}}. \end{aligned} $$

In a similar manner, we have

$$ \begin{aligned} \vert M_{43} \vert \leq C \Vert DV \Vert _{2}^{\frac{3}{2}} \Vert DD_{p}V \Vert _{2}^{ \frac{3}{2}}. \end{aligned} $$

Therefore

$$\begin{aligned} \begin{aligned} \vert M_{4} \vert \leq C \Vert DV \Vert _{2}^{\frac{3}{2}} \Vert DD_{p}V \Vert _{2}^{ \frac{3}{2}}. \end{aligned} \end{aligned}$$

(31)

By the boundedness of $\|B\|_{2}$ we get

$$\begin{aligned} \begin{aligned} \vert M_{5} \vert &= \biggl\vert \int _{\mathbb{R}^{3}}\bigl((\nabla \times B)\times B\bigr) \cdot D^{2}(\nabla \times B)\,dx \biggr\vert \\ &\leq C \Vert D B \Vert _{6} \Vert B \Vert _{3} \bigl\Vert D^{3}B \bigr\Vert _{2} \\ &\leq C \bigl\Vert D^{2}B \bigr\Vert _{2} \bigl\Vert D^{2}B \bigr\Vert _{2}^{\frac{1}{4}} \Vert B \Vert _{2}^{ \frac{3}{4}} \bigl\Vert D^{3}B \bigr\Vert _{2} \\ &\leq C \bigl\Vert D^{2}B \bigr\Vert _{2}^{\frac{5}{4}} \bigl\Vert D^{3}B \bigr\Vert _{2}. \end{aligned} \end{aligned}$$

(32)

Putting (5) and (29)–(32) together, we have

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{d}{dt}\bigl( \Vert DV \Vert _{2}^{2}+ \Vert DB \Vert _{2}^{2} \bigr)+ \rho _{0}\bigl( \Vert DD_{p}V \Vert _{2}^{2}+ \bigl\Vert D^{2}B \bigr\Vert _{2}^{2}\bigr) \\ &\quad \leq C\bigl( \Vert DV \Vert _{2}^{2}+ \Vert DB \Vert _{2}^{2}\bigr) \bigl( \Vert DD_{p}V \Vert _{2}^{2}+ \bigl\Vert D^{2}B \bigr\Vert _{2}^{2}\bigr) + \Vert \Delta B \Vert _{2}^{2} \bigl\Vert D^{3}B \bigr\Vert _{2}^{2}. \end{aligned} \end{aligned}$$

(33)

Next, we estimate each term $R_{1}$–$R_{5}$ in (13). To estimate $R_{1}$, we need to estimate $R_{11}$ and $R_{2}$ in (14). Applying the Hölder inequality and interpolation inequality, we have

$$ \begin{aligned} \vert R_{11} \vert &\leq C \bigl\Vert D^{2}V \bigr\Vert _{2} \Vert D B \Vert _{3} \bigl\Vert D^{2}B \bigr\Vert _{6} \\ &\leq C \bigl\Vert D^{2}V \bigr\Vert _{2} \bigl\Vert D^{2}B \bigr\Vert _{2}^{\frac{1}{2}} \Vert D B \Vert _{2}^{ \frac{1}{2}} \bigl\Vert D^{3} B \bigr\Vert _{2} \\ &\leq C \Vert \Delta B \Vert _{2} \bigl\Vert D^{3}B \bigr\Vert _{2}^{2} +C \Vert D B \Vert _{2} \Vert \Delta V \Vert _{2}^{2} \end{aligned} $$

and

$$ \begin{aligned} \vert R_{12} \vert &\leq C \bigl\Vert D^{2}B \bigr\Vert _{6} \bigl\Vert D^{2}B \bigr\Vert _{2} \Vert D V \Vert _{3} \\ &\leq C \bigl\Vert D^{3}B \bigr\Vert _{2} \bigl\Vert D^{2}B \bigr\Vert _{L^{2}} \bigl\Vert D^{2}V \bigr\Vert _{2}^{\frac{3}{4}} \Vert V \Vert _{2}^{\frac{1}{4}} \\ &\leq C \Vert \Delta B \Vert _{2}^{2} \bigl\Vert D^{3}B \bigr\Vert _{2}^{2} +C \Vert \Delta V \Vert _{2}^{ \frac{3}{2}}, \end{aligned} $$

where we used the boundedness of $\|u\|_{2}$. Hence we get

$$\begin{aligned} \begin{aligned} \vert R_{1} \vert \leq C \Vert \Delta B \Vert _{2}^{2} \bigl\Vert D^{3}B \bigr\Vert _{2}^{2} +C\bigl(1+ \Vert DB \Vert _{2}\bigr) \Vert \Delta V \Vert _{2}^{2}. \end{aligned} \end{aligned}$$

(34)

Based on (16), we can, similarly to $R_{11}$ and $R_{12}$, estimate $(R_{231},R_{232},R_{234})$, $R_{233}$. Hence, we have

$$\begin{aligned} \begin{aligned} \vert R_{2}+R_{3} \vert \leq C \Vert \Delta B \Vert _{2}^{2} \bigl\Vert D^{3}B \bigr\Vert _{2}^{2} +C\bigl(1+ \Vert DB \Vert _{2} \bigr) \Vert \Delta V \Vert _{2}^{2}. \end{aligned} \end{aligned}$$

(35)

For $R_{4}$, we estimate each term $R_{411}-R_{413}$ in (19). Applying the Hölder inequality and interpolation inequality, we have

$$ \begin{aligned} \vert R_{411} \vert &\leq C \Vert DV \Vert _{2} \Vert DD_{p}V \Vert _{4}^{2} \\ &\leq C \Vert DV \Vert _{2} \Vert DD_{p}V \Vert _{2}^{\frac{7}{4}} \Vert D_{p}V \Vert _{2}^{ \frac{1}{4}} \\ &\leq C \Vert DV \Vert _{2}^{\frac{5}{4}} \Vert DD_{p}V \Vert _{2}^{\frac{7}{4}}. \end{aligned} $$

We can further divide $R_{412}$ of (19) into two terms:

$$ \begin{aligned}R_{412}=- \int _{\mathbb{R}^{3}}\bigl(\partial _{3}^{2}V_{p} \cdot D_{p}\bigr)V_{p} \cdot \partial _{3}^{2}V_{p}\,dx + \int _{\mathbb{R}^{3}}\bigl(\partial _{3}^{2}V_{p} \cdot D_{p}\bigr)V_{3}(\partial _{3}D_{p} \cdot V_{p})\,dx=R_{4121}+R_{4122}. \end{aligned} $$

Applying integration by parts and Lemma 2.2, we have

$$ \begin{aligned} \vert R_{4121} \vert &=2 \biggl\vert \int _{\mathbb{R}^{3}}V_{p}\partial _{3}^{2}V_{p} \partial _{3}^{2}D_{p}V_{p}\,dx \biggr\vert \\ &\leq C \bigl\Vert \partial _{3}^{2}D_{p}V_{p} \bigr\Vert _{2} \Vert \partial _{3}V_{p} \Vert _{2}^{ \frac{1}{2}} \bigl\Vert \partial _{1}\partial _{3}^{2}V_{p} \bigr\Vert _{2}^{\frac{1}{2}} \Vert V_{p} \Vert _{2}^{\frac{1}{4}} \bigl\Vert \partial _{3}^{2}V_{p} \bigr\Vert _{2}^{ \frac{1}{4}} \Vert \partial _{2}V_{p} \Vert _{2}^{\frac{1}{4}} \bigl\Vert \partial _{2} \partial _{3}^{2}V_{p} \bigr\Vert _{2}^{\frac{1}{4}} \\ &\leq C \Vert \Delta D_{p}V \Vert _{2}^{\frac{7}{4}} \Vert DV \Vert _{2}^{\frac{1}{2}} \Vert \Delta V \Vert _{2}^{\frac{1}{4}} \Vert D_{p}V \Vert _{2}^{\frac{1}{4}} \\ &\leq C \Vert D V \Vert _{2}^{\frac{4}{7}} \Vert \Delta D_{p}V \Vert _{2}^{2} +C \Vert D_{p}V \Vert _{2}^{2} \Vert \Delta V \Vert _{2}^{2}. \end{aligned} $$

We estimate $R_{4122}$ as follows:

$$ \begin{aligned} \vert R_{4122} \vert &\leq C \bigl\Vert \partial _{3}^{2}V_{p} \bigr\Vert _{2}^{\frac{1}{2}} \Vert D_{p}V_{3} \Vert _{2}^{\frac{1}{2}} \Vert \partial _{3}D_{p}V_{p} \Vert _{2}^{ \frac{1}{2}} \bigl\Vert \partial _{3}^{2}D_{p}V_{p} \bigr\Vert _{2}^{\frac{1}{2}} \Vert \partial _{3}D_{p}V_{3} \Vert _{2}^{\frac{1}{2}} \bigl\Vert \partial _{3}D_{p}^{2}V_{p} \bigr\Vert _{2}^{\frac{1}{2}} \\ &\leq C \Vert \Delta D_{p}V \Vert _{2} \Vert DD_{p}V \Vert _{2} \Vert D_{p}V \Vert _{2}^{ \frac{1}{2}} \Vert \Delta V \Vert _{2}^{\frac{1}{2}} \\ &\leq C \Vert \Delta D_{p}V \Vert _{2}^{\frac{3}{2}} \Vert D_{p}V \Vert _{2}^{\frac{1}{2}} \Vert D_{p}V \Vert _{2}^{\frac{1}{2}} \Vert \Delta V \Vert _{2}^{\frac{1}{2}} \\ &\leq C \Vert DV \Vert _{2}^{\frac{2}{3}} \Vert \Delta D_{p}V \Vert _{2}^{2} +C \Vert D_{p}V \Vert _{2}^{2} \Vert \Delta V \Vert _{2}^{2}. \end{aligned} $$

In a similar manner, we obtain

$$ \begin{aligned} \vert R_{413} \vert \leq C \Vert DV \Vert _{2}^{\frac{2}{3}} \Vert \Delta D_{p}V \Vert _{2}^{2} +C \Vert D_{p}V \Vert _{2}^{2} \Vert \Delta V \Vert _{2}^{2}. \end{aligned} $$

Clearly, $R_{421}$, $R_{422}$, and $R_{423}$ in (20) can be estimated as $R_{411}$, $R_{413}$, and $R_{412}$. Hence we have

$$\begin{aligned} \begin{aligned} \vert R_{4} \vert \leq C \Vert DV \Vert _{2}^{\frac{5}{4}} \Vert \Delta D_{p}V \Vert _{2}^{2} +C \Vert D_{p}V \Vert _{2}^{2} \Vert \Delta V \Vert _{2}^{2}. \end{aligned} \end{aligned}$$

(36)

Similarly to (29), we have

$$\begin{aligned} \begin{aligned} \vert R_{5} \vert &= \biggl\vert \int _{\mathbb{R}^{3}}\bigl[D^{2}\bigl((\nabla \times B) \times B \bigr)-D^{2}(\nabla \times B)\times B\bigr]\cdot D^{2}(\nabla \times B)\,dx \biggr\vert \\ &\leq C \Vert D B \Vert _{\infty} \bigl\Vert D^{2}B \bigr\Vert _{2} \bigl\Vert D^{3}B \bigr\Vert _{2} \\ &\leq C \Vert \Delta B \Vert _{2}^{\frac{3}{2}} \bigl\Vert D^{3}B \bigr\Vert _{2}^{\frac{3}{2}}. \end{aligned} \end{aligned}$$

(37)

Combining (13) and (34)–(37), we get

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{d}{dt}\bigl( \Vert \Delta V \Vert _{2}^{2}+ \Vert \Delta B \Vert _{2}^{2}\bigr)+ \rho _{0}\bigl( \Vert \Delta D_{p}V \Vert _{2}^{2}+ \bigl\Vert D^{3}B \bigr\Vert _{2}^{2}\bigr) \\ &\quad \leq C\bigl( \Vert DV \Vert _{2}^{2}+ \Vert \Delta B \Vert _{2}^{2}\bigr) \bigl( \Vert \Delta D_{p}V \Vert _{2}^{2}+ \bigl\Vert D^{3}B \bigr\Vert _{2}^{2}\bigr) \\ &\qquad {}+C\bigl( \Vert D_{p}V \Vert _{2}^{2}+ \Vert DB \Vert _{2}^{2} \bigr) \bigl( \Vert \Delta V \Vert _{2}^{2}+ \Vert \Delta B \Vert _{2}^{2}\bigr). \end{aligned} \end{aligned}$$

(38)

Adding (2), (33), and (38) together, we have

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{d}{dt}\bigl( \Vert V \Vert _{H^{2}}^{2}+ \Vert B \Vert _{H^{2}}^{2} \bigr)+\bigl[ \rho _{0}-C\bigl( \Vert DV \Vert _{2}^{2}+ \Vert D B \Vert _{2}^{2}+ \Vert \Delta B \Vert _{2}^{2}\bigr)\bigr] \\ &\quad{}\times \bigl( \Vert D_{p}V \Vert _{H^{2}}^{2}+ \Vert DB \Vert _{H^{2}}^{2}\bigr)\leq C\bigl( \Vert D_{p}V \Vert _{2}^{2} + \Vert DB \Vert _{2}^{2}\bigr) \bigl( \Vert V \Vert _{H^{2}}^{2}+ \Vert B \Vert _{H^{2}}^{2}\bigr), \end{aligned} \end{aligned}$$

(39)

Choose L so small that

$$ \begin{aligned}C\bigl( \Vert DV_{0} \Vert _{2}^{2}+ \Vert DB_{0} \Vert _{2}^{2}+ \Vert \Delta B_{0} \Vert _{2}^{2}\bigr) \leq \frac{\rho _{0}}{2}. \end{aligned} $$

Substituting this inequality into (39), we get

$$\begin{aligned} \begin{aligned}&\frac{d}{dt}\bigl( \Vert V \Vert _{H^{2}}^{2}+ \Vert B \Vert _{H^{2}}^{2}\bigr)+\rho _{0}\bigl( \Vert D_{p}V \Vert _{H^{2}}^{2}+ \Vert DB \Vert _{H^{2}}^{2}\bigr) \\ &\quad \leq C\bigl( \Vert D_{p}V \Vert _{2}^{2}+ \Vert DB \Vert _{2}^{2}\bigr) \bigl( \Vert V \Vert _{H^{2}}^{2}+ \Vert B \Vert _{H^{2}}^{2} \bigr). \end{aligned} \end{aligned}$$

(40)

Applying Gronwall’s inequality to (40), we have

$$ \begin{aligned}&\sup_{0< t< T}\bigl( \bigl\Vert V(t) \bigr\Vert _{H^{2}}^{2}+ \bigl\Vert B(t) \bigr\Vert _{H^{2}}^{2}\bigr)+ \rho _{0} \int _{0}^{T}\bigl( \bigl\Vert D_{p}V(t) \bigr\Vert _{H^{2}}^{2}+ \bigl\Vert DB(t) \bigr\Vert _{H^{2}}^{2}\bigr)\,dt \\ &\quad\leq \bigl( \Vert V_{0} \Vert _{H^{2}}^{2}+ \Vert B_{0} \Vert _{H^{2}}^{2}\bigr)\biggl[1+C \int _{0}^{T}\bigl( \bigl\Vert D_{p}V(t) \bigr\Vert _{2}^{2}+ \bigl\Vert DB(t) \bigr\Vert _{2}^{2}\bigr)\,dt\biggr] \\ &\quad\quad{}\times \exp \biggl[C \int _{0}^{T}\bigl( \bigl\Vert D_{p}V(t) \bigr\Vert _{2}^{2}+ \bigl\Vert DB(t) \bigr\Vert _{2}^{2}\bigr)\,dt\biggr]. \end{aligned} $$

which, together with (3), yields that for any $T\in (0,\widetilde{\mathbf{T}})$,

$$ \begin{aligned}(V,B)\in L^{\infty}\bigl(0,T;H^{2}\bigr), \qquad (D_{p}V,D B)\in L^{2}\bigl(0,T;H^{2}\bigr). \end{aligned} $$

Noticing that

$$ \begin{aligned}H^{2}\bigl(\mathbb{R}^{3}\bigr) \hookrightarrow L^{\infty}\bigl(\mathbb{R}^{3}\bigr), \end{aligned} $$

we get

$$ \begin{aligned}(V,DB)\in L^{2}\bigl(0,T;L^{\infty}\bigl( \mathbb{R}^{3}\bigr)\bigr), \quad \forall T \in (0,\widetilde{ \mathbf{T}}). \end{aligned} $$

Based on Theorem 1.1 ($p=\beta =\infty$ and $q=\gamma =2$), we have $\widetilde{\mathbf{T}}=\infty$, which yields Theorem 1.2.