On Global Well-Posedness and Temporal Decay for 3D Magnetic Induction Equations with Hall Effect

Abstract

The main purpose of this paper is to study the global existence and uniqueness of solutions for three-dimensional incompressible magnetic induction equations with Hall effect provided that ${\parallel u_0\parallel }_{H^{\frac{3}{2}+\epsilon }}+{\parallel b_0\parallel }_{H^2}$$(0<\epsilon <1)$ is sufficiently small. Moreover, using the Fourier splitting method and the properties of decay character $r^{*}$, one also shows the algebraic decay rate of a higher order derivative of solutions to magnetic induction equations with the Hall effect.

1. Introduction

Supposing that $\rho$ denotes the density, u describes the velocity field of the fluid, b means the magnetic field and $\pi$ is the pressure, a high-resolution, non-oscillatory, central scheme for the Hall–MHD model [1,2] can be introduced in the following:

$$\begin{array}{ccc}& & {\partial }_t\rho =-\nabla ·\left(\rho u\right),\hfill \end{array}$$

(1)

$$\begin{array}{ccc}& & {\partial }_t\left(\rho u\right)=-\nabla ·\left\{\rho u{u}^t+\left(\pi +\frac{b^2}{2}\right)I_{3\times 3}-b{b}^t\right\},\hfill \end{array}$$

(2)

$$\begin{array}{ccc}& & {\partial }_{t}b=-\nabla \times E,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}& & {\partial }_{t}U=-\nabla ·\left\{\left(U+\pi -\frac{b^2}{2}\right)u+E\times B\right\},\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& & \nabla ·u=\nabla ·b=0.\hfill \end{array}$$

(5)

Systems (1)–(5) follow from the MHD equations after normalizing as the Geospace Environment Modeling (GEM) challenge. The total energy, U, momentum, $\rho u$ and magnetic field, b, can be coupled through the following state equation:

$$U=\frac{\pi }{\gamma -1}+\frac{\rho u^2}{2}+\frac{b^2}{2}.$$

(6)

Moreover, one expresses the electric field in the generalized Ohm’s law [1,2]:

$$\begin{array}{ccc}& & E=-u\times b+\eta j+\frac{{\delta }_i}{L_0}\frac{j\times b}{\rho }-\frac{{\delta }_i}{L_0}\frac{\nabla \pi }{\rho }+{\left(\frac{{\delta }_e}{L_0}\right)}^2\frac{1}{\rho }\left[{\partial }_{t}j+(u·\nabla )j\right],\hfill \end{array}$$

(7)

$$\begin{array}{ccc}& & j=\nabla \times b,\hfill \end{array}$$

(8)

where $L_0$, ${\delta }_e$ and ${\delta }_i$ denotes the normalizing length limit, electron inertia and ion inertia, respectively. For the simulations considered in the work, the electron pressure tensor $-\frac{{\delta }_i}{L_0}\frac{\nabla \pi }{\rho }$ will be ignored [2].

Considering the incompressible case, denoting $\rho \equiv 1$, combining (1)–(8) together yields the following three-dimensional system [1,3,4,5]:

$$\begin{array}{ccc}& & \mathrm{div}u=\mathrm{div}b=0,\hfill \end{array}$$

(9)

$$\begin{array}{ccc}& & {\partial }_{t}u+u·\nabla u+\nabla \left(\pi +\frac{1}{2}{|b|}^2\right)-\Delta u=b·\nabla b,\hfill \end{array}$$

(10)

$$\begin{array}{ccc}& & {\partial }_{t}b-{\left(\frac{{\delta }_e}{L_0}\right)}^2{\partial }_t\Delta b+u·\nabla b-b·\nabla u-\Delta b=\frac{{\delta }_i}{L_0}\mathrm{rot}(b\times \mathrm{rot}b)-{\left(\frac{{\delta }_e}{L_0}\right)}^2\mathrm{rot}\left((u·\nabla )\mathrm{rot}b\right),\hfill \end{array}$$

(11)

$$\begin{array}{ccc}& & (u,b)(·,0)=(u_0,b_0)(·)\mathrm{in}{\mathbb{R}}^3.\hfill \end{array}$$

(12)

For simplicity, ${\delta }_i=1$ and ${\left(\frac{{\delta }_e}{L_0}\right)}^2=1$ in this paper.

If ${\delta }_e=0$, systems (9)–(12) reduce to the three-dimensional incompressible Hall–MHD system, whose applications cover a very wide range of physical objects, for example, magnetic reconnection in space plasmas, star formation, neutron stars and geo-dynamo. The global well-posedness, regularity criterion and decay characterization of solutions to 3D incompressible Hall–MHD system were studied by many authors [6,7,8,9,10,11,12,13,14]. It is worth pointing out that Wan et al. [15] assumed that the initial data $(u_0,b_0)\in H^m\left({\mathbb{R}}^3\right)$ with $m>\frac{5}{2}$, $\nabla ·u_0=\nabla ·b_0=0$, and $\parallel u_0{\parallel }_{{\dot{B}}_{2,\infty }^{\frac{1}{2}+\epsilon }}+{\parallel b_0\parallel }_{{\dot{B}}_{2,\infty }^{\frac{3}{2}}}$ are sufficiently small, proving that the 3D Hall–MHD system admits a unique global solution $(u,b)\in C(0,\infty ;H^m\left({\mathbb{R}}^3\right))$, which may be the latest result on the small initial data global well-posedness for the Hall–MHD system.

For systems (9)–(12), Fan et al. [16] established the existence of global weak solutions, existence of local strong solutions and some blow-up criteria. They pointed out that if $u_0\in L^2\left({\mathbb{R}}^3\right)$, $b_0\in H^1\left({\mathbb{R}}^3\right)$ and $\nabla ·u_0=\nabla ·b_0=0$, then there exists a weak solution $(u,b)$ for systems (9)–(12), which satisfies the energy inequality

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^3}{(|u|}^2+{|b|}^2+{|\nabla b|}^2)dx+2{\int }_0^T{\int }_{{\mathbb{R}}^3}{(|\nabla u|}^2+{|\nabla b|}^2)dxdt\le {\int }_{{\mathbb{R}}^3}(|u_0{|}^2+|b_0{|}^2+|\nabla b_0{|}^2)dx.\end{array}$$

Latterly, Ma et al. [17] proved the global existence of strong solutions to 3D two-fluid MHD equations provided that $\parallel u_0{\parallel }_{{\dot{H}}^{\frac{1}{2}}}+\parallel b_0{\parallel }_{{\dot{H}}^{\frac{1}{2}}}+{\parallel b_0\parallel }_{{\dot{H}}^{\frac{3}{2}}}$ is sufficiently small. The main difference between systems (9)–(12), the Hall–MHD system and the two-fluid MHD system is the nonlinear term $\mathrm{rot}\left(\right(u·\nabla \left)\mathrm{rot}b\right)$. Because of the existence of this nonlinear term, it is difficult to obtain the global well-posedness of systems (9)–(12) under the same assumption as Wan et al. [15] and Ma et al. [17].

The first purpose of this paper is to to prove the following theorem on the global well-posedness of systems (9)–(12).

Theorem 1.

Let $\epsilon \in (0,1)$ and $m,p,K\in \mathbb{N}$, $K\ge max\{m,3\}$. Assume that the initial data $(u_0,b_0)\in H^K\left({\mathbb{R}}^3\right)\times H^{K+1}\left({\mathbb{R}}^3\right)$ with $\nabla ·u_0=\nabla ·b_0=0$ such that

$$\parallel u_0{\parallel }_{H^{\frac{3}{2}+\epsilon }}+{\parallel b_0\parallel }_{H^2}<\eta \left(\alpha \right),$$

(13)

for some small enough constant $\eta \left(\alpha \right)>0$. Then, there exists a unique global solution $(u,b)$ for systems (9)–(12), such that

$$\parallel {\partial }_t^p{\mathsf{\Lambda }}^m{u\parallel }_{L^2}^2+\parallel {\partial }_t^p{\mathsf{\Lambda }}^m{b\parallel }_{L^2}^2+\parallel {\partial }_t^p{\mathsf{\Lambda }}^{m+1}{b\parallel }_{L^2}^2\le C(\parallel u_0{\parallel }_{H^K}^2+\parallel b_0{\parallel }_{H^K}^2),$$

(14)

for all $m+2p\le K$.

Remark 1.

In the above and the following, ${\mathsf{\Lambda }}^m$ is defined by

$${\mathsf{\Lambda }}^{m}f\left(x\right)={(-\Delta )}^{\frac{m}{2}}f\left(x\right)={\int }_{{\mathbb{R}}^3}{|\xi |}^m\widehat{f}\left(\xi \right)e^{2\pi ix·\xi }d\xi .$$

(15)

The temporal decay rate of solutions is also an interesting topic in the study of dissipative equations. One of the main tools to study the temporal decay rate is the Fourier splitting method, which was introduced by Schonbek in [18,19]. Laterly, this method was well extended to investigate the decay for the solutions of PDE from mathematical physics, see, e.g., Schonbek et al. [20] for the MHD sysem, Brandolese et al. [21] for the viscous Boussinesq system, Dai et al. [22] for liquid crystal systems, Weng [14] and Chae et al. [8] for the Hall–MHD system, Niche [23] for the Navier–Stokes–Voigt equations, Ferreira et al. [24] for quasi-geostrophic equations, Zhao et al. [25] for third grade fluids, etc.

Recently, in order to characterize the decay rate of dissipative equations more profoundly, Bjorland et al. [26] and Niche et al. [27] introduced the idea of decay indicator $P_r$ and decay character $r^{*}$. Latterly, Brandolese [28] improved the definition of the decay indicator and the decay character by taking advantage of the insight provided by the Littlewood–Paley analysis and the use of Besov spaces. For more details on $P_r$ and $r^{*}$, we refer to Section 2.

In consequence, it is desirable to understand the asymptotic behavior of the magnetic induction equations with the Hall effect. With the aid of the classical Fourier splitting method and the properties of decay character $r^{*}$, the decay rate of solutions to systems (9)–(12) has been characterized:

Lemma 1

([29]). Assume that $(u_0,b_0)\in L^2\left({\mathbb{R}}^3\right)\times H^1\left({\mathbb{R}}^3\right)$, $\nabla ·u_0=\nabla ·b_0=0$ and $r^{*}=r^{*}\left(u_0\right)=r^{*}\left(b_0\right)\in (-\frac{3}{2},+\infty )$ is the decay character. Let $(u,b)$ be the solution of systems (9)–(12) with initial value $(u_0,b_0)$. Then

$${\parallel u\left(t\right)\parallel }_{L^2}^2+{\parallel b\left(t\right)\parallel }_{L^2}^2+{\parallel \nabla b\left(t\right)\parallel }_{L^2}^2\le C{(1+t)}^{-min\{\frac{3}{2}+r^{*},\frac{5}{2}\}},\forall t>0,$$

where the constant C depends essentially on $\parallel u_0{\parallel }_{L^2}$, $\parallel b_0{\parallel }_{L^2}$ and $\parallel \nabla b_0{\parallel }_{L^2}$.

Lemma 2

([30]). Suppose that $m\in \mathbb{N}$, $K_0\ge max\{3,m\}$, $(u_0,b_0)\in H^{K_0}\left({\mathbb{R}}^3\right)\times H^{K_0+1}\left({\mathbb{R}}^3\right)$ and $\nabla ·u_0=\nabla ·b_0=0$. Let $r^{*}=r^{*}\left(u_0\right)=r^{*}\left(b_0\right)\in (-\frac{3}{2},+\infty )$ be the decay character. Then, for the small global-in-time solution $(u,b)$, there exists a positive constant $C=C(,\parallel u_0{\parallel }_{H^{K_0}},\parallel b_0{\parallel }_{H^{K_0+1}})$, such that

$$\parallel {\mathsf{\Lambda }}^m{u\parallel }_{L^2}^2+\parallel {\mathsf{\Lambda }}^m{b\parallel }_{L^2}^2+{\parallel {\mathsf{\Lambda }}^{m+1}b\parallel }_{L^2}^2\le C{(1+t)}^{-min\{\frac{3}{2}+r^{*}+m,\frac{5}{2}+m\}},\mathit{for}\mathit{large}t.$$

On the basis of Lemmas 1 and 2, using the properties of decay character $r^{*}$ and Fourier splitting method, one can continue to study the decay characterization of solutions to systems (9)–(12), establish the decay rate of higher-order derivative of solutions on both time and space. Note that the global in-time existence and uniqueness can be guaranteed for sufficiently small initial data. The result can be described as follows:

Theorem 2.

Let $m,p\in \mathbb{N}$, $K\ge max\{3+2p,m+2p\}$ and $r^{*}=r^{*}\left(u_0\right)=r^{*}\left(b_0\right)\in (-\frac{3}{2},+\infty )$ be the decay character. Suppose that $(u_0,b_0)\in H^K\left({\mathbb{R}}^3\right)\times H^{K+1}\left({\mathbb{R}}^3\right)$ with $\nabla ·u_0=\nabla ·b_0=0$. Then, there exists a positive constant $C=C(,\parallel u_0{\parallel }_{H^K},\parallel b_0{\parallel }_{H^{K+1}})$, such that

$$\parallel {\partial }_t^p{\mathsf{\Lambda }}^m{u\parallel }_{L^2}^2+\parallel {\partial }_t^p{\mathsf{\Lambda }}^m{b\parallel }_{L^2}^2+{\parallel {\partial }_t^p{\mathsf{\Lambda }}^{m+1}b\parallel }_{L^2}^2\le C{(1+t)}^{-min\{\frac{3}{2}+r^{*}+m+2p,\frac{5}{2}+m+2p\}},$$

(16)

for large t.

The rest of this paper is organized as follows. In Section 2, we give some preliminary results on the properties of decay character $r^{*}$. Section 3 is devoted to the proofs of Theorem 1. The proof of Theorem 2 is given in Section 4. Conclusions are outlined in Section 5.

2. Properties of Decay Character

2.1. Definition and Properties of Decay Character

The definitions of decay indicator $P_r\left(u_0\right)$ and decay character $r^{*}$ was first introduced in [26], Brandolese [28] redefined them, which seems more precise.

Definition 1

([28]). Suppose that $v_0\in L^2\left({\mathbb{R}}^n\right)$, $B_{\rho }=\{\xi \in {\mathbb{R}}^n:|\xi |\le \rho \}$ and $\mathsf{\Lambda }={(-\Delta )}^{\frac{1}{2}}$. If the following two lower and upper limits exist, they are the lower and upper decay indicators of $v_0$:

$$P_r\left(v_0\right){)}_{-}=\underset{\rho \to 0^{-}}{lim}{\rho }^{-2r-n}{\int }_{B\left(\rho \right)}{|\widehat{v_0}\left(\xi \right)|}^{2}d\xi ,$$

$$P_r\left(v_0\right){)}_{+}=\underset{\rho \to 0^{+}}{lim}{\rho }^{-2r-n}{\int }_{B\left(\rho \right)}{|\widehat{v_0}\left(\xi \right)|}^{2}d\xi .$$

When $P_r{\left(v_0\right)}_{-}=P_r{\left(v_0\right)}_{+}$, then $P_r\left(v_0\right)=P_r{\left(v_0\right)}_{-}=P_r{\left(v_0\right)}_{+}$ can be defined as the decay indicator corresponding to $v_0$.

Definition 2

([28]). The upper and lower decay characters of $v_0\in L^2\left({\mathbb{R}}^n\right)$ are defined as

$$r{\left(v_0\right)}_{+}=sup\{r\in \mathbb{R}:P_r{\left(v_0\right)}_{+}<\infty \},$$

$$r{\left(v_0\right)}_{-}=sup\{r\in \mathbb{R}:P_r{\left(v_0\right)}_{-}<\infty \}.$$

Definition 3

([28]). If $v_0\in L^2\left({\mathbb{R}}^n\right)$ is such that there exists $r^{*}\in (-\frac{n}{2},\infty )$ such that

$$r^{*}\left(v_0\right)=max\{r\in \mathbb{R}:P_r{\left(v_0\right)}_{+}<\infty \}=min\{r\in \mathbb{R}:P_r{\left(v_0\right)}_{-}>0\}.$$

then this number $r^{*}=r^{*}\left(v_0\right)$ can be called the decay character of $v_0$. The decay character of $v_0$ in the two limit situations is defined as follows:

$$r^{*}\left(v_0\right)=+\infty ,\mathrm{if}r{\left(v_0\right)}_{+}=r{\left(v_0\right)}_{-}=+\infty ,$$

$$r^{*}\left(v_0\right)=-\frac{n}{2},\mathrm{if}r{\left(v_0\right)}_{+}=r{\left(v_0\right)}_{-}=-\frac{n}{2}.$$

Lemma 3

([27]). Let $v_0\in H^s\left({\mathbb{R}}^n\right)$, with $s>0$. Then

 (1) 

if $-\frac{n}{2}<r^{*}\left(v_0\right)<\infty$, then $-\frac{n}{2}+s<r_s^{*}\left(v_0\right)<\infty$ and $r_s^{*}\left(v_0\right)=s+r^{*}\left(v_0\right)$;

 (2) 

$r_s^{*}\left(v_0\right)=\infty$ if and only if $r^{*}\left(v_0\right)=\infty$;

 (3) 

$r^{*}\left(v_0\right)=-\frac{n}{2}$ if and only if $r_s^{*}\left(v_0\right)=-\frac{n}{2}+s$.

Remark 2.

The decay character $r^{*}=r^{*}\left(v_0\right)$ measures the “order" of ${\widehat{v}}_0\left(\xi \right)$ at $\xi =0$ in frequency space. The theory of [26,27] allows defining the decay character only in the following three situations:

 (1) 

Either, $\exists r\in (-\frac{n}{2}+s,+\infty )$ such that $0<P_r^s\left(u_0\right)<+\infty$, and in this case r is unique,

 (2) 

Or $\forall r\in (-\frac{n}{2}+s,+\infty )$, one has $P_r^s\left(u_0\right)=0$,

 (3) 

Or $\forall r\in (-\frac{n}{2}+s,+\infty )$, one has $P_r^s\left(u_0\right)=+\infty$.

However, not in the other cases (e.g., it can happen that

$$\exists r,r^{\prime }\in (-\frac{n}{2}+s,+\infty )\mathrm{such}\mathrm{that}{P}_r^s\left(u_0\right)=0\mathrm{and}{P}_{r^{\prime }}^s\left(u_0\right)=+\infty .$$

In addition, it can also happen that the limit-defining $P_r^s\left(u_0\right)$ does not exist.

2.2. Decay Characterization of a Linear Equation

Consider the linear part of (11):

$$\left\{\begin{array}{c}{v}_t-\Delta v_t-\Delta v=0,\hfill \\ v(x,0)=v_0\left(x\right),\hfill \end{array}\right.x\in {\mathbb{R}}^n,n\in {\mathbb{N}}^{+}.$$

(17)

Define the space $H^1\left({\mathbb{R}}^3\right)$, such that

$${\parallel v\parallel }_{H^1\left({\mathbb{R}}^3\right)}^2={\parallel v\parallel }_{L^2\left({\mathbb{R}}^3\right)}^2+{\parallel \nabla v\parallel }_{L^2}^2,s\ge 0.$$

Hence,

$$\frac{d}{dt}{(\parallel v\parallel }_{L^2}^2+{\parallel \nabla v\parallel }_{L^2}^2{)+2\parallel \nabla v\parallel }_{L^2}=0.$$

In the Fourier space, the solution to (17) is

$$\widehat{v}(\xi ,t)=e^{-\frac{{|\xi |}^2}{1+{|\xi |}^2}t}{\widehat{v}}_0\left(\xi \right),$$

and

$${\partial }_t^p\widehat{v}(\xi ,t)={(-1)}^p{\left(\frac{{|\xi |}^2}{1+{|\xi |}^2}\right)}^p{e}^{-\frac{{|\xi |}^2}{1+{|\xi |}^2}t}{\widehat{v}}_0\left(\xi \right).$$

The $L^2$-decay characterization of solutions to system (17) was established by Niche [23].

Lemma 4

([23]). Assume that $v_0\in H^1\left({\mathbb{R}}^n\right)$, which has decay character $r^{*}\left(v_0\right)=r^{*}$, is a solution to system (17). Then

 (1) 

If $-\frac{n}{2}<r^{*}<+\infty$, there exist two positive constants $C_1,C_2>0$, such that

$$C_1{(1+t)}^{-(\frac{n}{2}+r^{*})}\le {\parallel v\left(t\right)\parallel }_{L^2}^2+{\parallel \nabla v\left(t\right)\parallel }_{L^2}^2\le C_2{(1+t)}^{-(\frac{n}{2}+r^{*})};$$

 (2) 

if $r^{*}=-\frac{n}{2}$, there exists $C=C\left(\epsilon \right)>0$, such that

$${\parallel v\left(t\right)\parallel }_{L^2}^2+{\parallel \nabla v\left(t\right)\parallel }_{L^2}^2\ge C{(1+t)}^{-\epsilon },\forall \epsilon >0,$$

which means the decay of ${\parallel v\parallel }_{L^2}^2+{\parallel \nabla v\left(t\right)\parallel }_{L^2}^2$ is slower than any uniform algebraic rate;

 (3) 

if $r^{*}=+\infty$, there exists a $C>0$ such that

$${\parallel v\left(t\right)\parallel }_{L^2}^2+{\parallel \nabla v\left(t\right)\parallel }_{L^2}^2\le C{(1+t)}^{-m},\forall m>0,$$

that is, the decay of ${\parallel v\parallel }_{L^2}^2+{\parallel \nabla v\left(t\right)\parallel }_{L^2}^2$ is faster than any algebraic rate.

Therei s also a lemma on the ${\dot{H}}^s$-decay rate of solutions to system (17).

Lemma 5

([31,32]). Suppose that ${\mathsf{\Lambda }}^s{v}_0\in H^1\left({\mathbb{R}}^n\right)$ ($s>0$) has decay character $r_s^{*}=r_s^{*}\left(v_0\right)$. Then

 (1) 

If $-\frac{n}{2}\le r^{*}<+\infty$, there exist two positive constants $C_1,C_2>0$, such that

$$C_1{(1+t)}^{-(\frac{n}{2}+r^{*}+s)}\le \parallel {\mathsf{\Lambda }}^s{v\left(t\right)\parallel }_{L^2}^2+{\parallel {\mathsf{\Lambda }}^{s+1}v\left(t\right)\parallel }_{L^2}^2\le C_2{(1+t)}^{-(\frac{n}{2}+r^{*}+s)};$$

 (2) 

if $r^{*}=+\infty$, there exists a $C>0$ such that

$${\parallel v\left(t\right)\parallel }_{L^2}^2+{\parallel {\mathsf{\Lambda }}^{s+1}v\left(t\right)\parallel }_{L^2}^2\le C{(1+t)}^{-m},\forall m>0,$$

that is, the decay of $\parallel {\mathsf{\Lambda }}^s{v\parallel }_{L^2}^2+{\parallel {\mathsf{\Lambda }}^{s+1}v\parallel }_{L^2}^2$ is faster than any algebraic rate.

The following result regarding the decay characterization of solutions to (17) can be found in [32].

Lemma 6.

Let $v_0\in H^{K+1}\left({\mathbb{R}}^n\right)$ ($s>0$) have decay character $r_s^{*}=r_s^{*}\left(v_0\right)$. Then, for all $0<m+2p\le K$, the following decay estimates hold:

 (1) 

If $-\frac{n}{2}\le r^{*}<\infty$, then there exists a positive constant $C_1$ such that

$$\parallel {\partial }_t^p{\mathsf{\Lambda }}^m{v\left(t\right)\parallel }_{L^2}^2+{\parallel {\partial }_t^p{\mathsf{\Lambda }}^{m+1}v\left(t\right)\parallel }_{L^2}^2\le C_1{(1+t)}^{-(r^{*}+m+2p+\frac{n}{2})};$$

 (2) 

if $r^{*}=\infty$, given any $s>0$, there exists a positive constant $C_2=C_2\left(s\right)$ such that

$$\parallel {\partial }_t^p{\mathsf{\Lambda }}^m{v\left(t\right)\parallel }_{L^2}^2+{\parallel {\partial }_t^p{\mathsf{\Lambda }}^{m+1}v\left(t\right)\parallel }_{L^2}^2\le C_2{(1+t)}^{-s},$$

which means the decay is faster than any algebraic rate.

2.3. Decay Characterization of the Linear Part for Systems (9)–(12)

For the linear part of systems (9)–(12):

$$\left\{\begin{array}{c}{\overline{u}}_t-\Delta \overline{u}=0,\hfill \\ {\overline{b}}_t-\Delta {\overline{b}}_t-\Delta \overline{b}=0,\hfill \\ \overline{u}(x,0)={\overline{u}}_0\left(x\right),\hfill \\ \overline{b}(x,0)={\overline{b}}_0\left(x\right),\hfill \end{array}\right.x\in {\mathbb{R}}^3.$$

(18)

Combining the results of Niche [23], Niche et al. [27], Anh et al. [33], Zhao [30], we obtain the following three lemmas:

Lemma 7

([32]). Assume that $({\overline{u}}_0,{\overline{b}}_0)\in L^2\left({\mathbb{R}}^3\right)\times H^1\left({\mathbb{R}}^3\right)$, which has decay character $r^{*}$. Then

 (1) 

If $-\frac{3}{2}<r^{*}<+\infty$, there exist two positive constants $C_1,C_2>0$, such that

$$C_1{(1+t)}^{-(\frac{3}{2}+r^{*})}\le \parallel \overline{u}{\left(t\right)\parallel }_{L^2}^2+\parallel \overline{b}{\left(t\right)\parallel }_{L^2}^2+{\parallel \nabla \overline{b}\left(t\right)\parallel }_{L^2}^2\le C_2{(1+t)}^{-(\frac{3}{2}+r^{*})};$$

 (2) 

if $r^{*}=-\frac{3}{2}$, there exists $C=C\left(\epsilon \right)>0$, such that

$$\parallel \overline{u}{\left(t\right)\parallel }_{L^2}^2+\parallel \overline{b}{\left(t\right)\parallel }_{L^2}^2+{\parallel \nabla \overline{b}\left(t\right)\parallel }_{L^2}^2\ge C{(1+t)}^{-\epsilon },\forall \epsilon >0,$$

which means the decay of $\parallel \overline{u}{\left(t\right)\parallel }_{L^2}^2+\parallel \overline{b}{\left(t\right)\parallel }_{L^2}^2+{\parallel \nabla \overline{b}\left(t\right)\parallel }_{L^2}^2$ is slower than any uniform algebraic rate;

 (3) 

if $r^{*}=+\infty$, there exists a $C>0$ such that

$$\parallel \overline{u}{\left(t\right)\parallel }_{L^2}^2+\parallel \overline{b}{\left(t\right)\parallel }_{L^2}^2+{\parallel \nabla \overline{b}\left(t\right)\parallel }_{L^2}^2\le C{(1+t)}^{-m},\forall m>0,$$

that is, the decay of $\parallel \overline{u}{\left(t\right)\parallel }_{L^2}^2+\parallel \overline{b}{\left(t\right)\parallel }_{L^2}^2+{\parallel \nabla \overline{b}\left(t\right)\parallel }_{L^2}^2$ is faster than any algebraic rate.

Lemma 8

([32]). Suppose that $({\mathsf{\Lambda }}^s{\overline{u}}_0,{\mathsf{\Lambda }}^s{\overline{b}}_0)\in L^2\left({\mathbb{R}}^3\right)\times H^1\left({\mathbb{R}}^3\right)$ ($s>0$) has decay character $r_s^{*}$. Then

 (1) 

If $-\frac{3}{2}\le r^{*}<+\infty$, there exist two positive constants $C_1,C_2>0$, such that

$$C_1{(1+t)}^{-(\frac{3}{2}+r^{*}+s)}\le \parallel {\mathsf{\Lambda }}^s\overline{u}{\left(t\right)\parallel }_{L^2}^2+\parallel {\mathsf{\Lambda }}^s\overline{b}{\left(t\right)\parallel }_{L^2}^2+{\parallel {\mathsf{\Lambda }}^s\nabla \overline{b}\left(t\right)\parallel }_{L^2}^2\le C_2{(1+t)}^{-(\frac{3}{2}+r^{*}+s)};$$

 (2) 

if $r^{*}=+\infty$, there exists a $C>0$ such that

$$\parallel {\mathsf{\Lambda }}^s\overline{u}{\left(t\right)\parallel }_{L^2}^2+\parallel {\mathsf{\Lambda }}^s\overline{b}{\left(t\right)\parallel }_{L^2}^2+{\parallel {\mathsf{\Lambda }}^s\nabla \overline{b}\left(t\right)\parallel }_{L^2}^2\le C{(1+t)}^{-m},\forall m>0,$$

that is, the decay of $\parallel {\mathsf{\Lambda }}^s\overline{u}{\left(t\right)\parallel }_{L^2}^2+\parallel {\mathsf{\Lambda }}^s\overline{b}{\left(t\right)\parallel }_{L^2}^2+{\parallel {\mathsf{\Lambda }}^s\nabla \overline{b}\left(t\right)\parallel }_{L^2}^2$ is faster than any algebraic rate.

Lemma 9

([32]). Suppose that $K\in {\mathbb{Z}}^{+}$. Let $({\overline{u}}_0,{\overline{b}}_0)\in H^K\left({\mathbb{R}}^3\right)\times H^{K+1}\left({\mathbb{R}}^3\right)$ have decay character $r*$. Then, for all $0<m+2p\le K$, the following decay estimates hold:

 (1) 

If $-\frac{3}{2}\le r^{*}<\infty$, then there exists a positive constant $C_1$ such that

$$\parallel {\partial }_t^p{\mathsf{\Lambda }}^m\overline{u}{\left(t\right)\parallel }_{L^2}^2+\parallel {\partial }_t^p{\mathsf{\Lambda }}^m\overline{b}{\left(t\right)\parallel }_{L^2}^2+{\parallel {\partial }_t^p{\mathsf{\Lambda }}^{m+1}\overline{b}\left(t\right)\parallel }_{L^2}^2\le C_1{(1+t)}^{-(r^{*}+m+2p+\frac{3}{2})};$$

 (2) 

if $r^{*}=\infty$, given any $s>0$, there exists a positive constant $C_2=C_2\left(s\right)$ such that

$$\parallel {\partial }_t^p{\mathsf{\Lambda }}^m\overline{u}{\left(t\right)\parallel }_{L^2}^2+\parallel {\partial }_t^p{\mathsf{\Lambda }}^m\overline{b}{\left(t\right)\parallel }_{L^2}^2+{\parallel {\partial }_t^p{\mathsf{\Lambda }}^{m+1}\overline{b}\left(t\right)\parallel }_{L^2}^2\le C_2{(1+t)}^{-s},$$

that is, the decay is faster than any algebraic rate.

3. Proof of Theorems 1

One first proves that (14) holds for $p=0$.

Testing by u and b, respectively, adding them together gives

$$\begin{array}{c}\hfill \frac{1}{2}\frac{d}{dt}\left({\parallel u\parallel }_{L^2}^2+{\parallel b\parallel }_{L^2}^2+{\parallel \nabla b\parallel }_{L^2}^2\right)+{\parallel \nabla u\parallel }_{L^2}^2+{\parallel \nabla b\parallel }_{L^2}^2=0.\end{array}$$

(19)

Taking $\mathsf{\Lambda }$ to (11), testing by $\mathsf{\Lambda }b$, respectively, it yields that

$$\begin{array}{cc}& \frac{1}{2}\frac{d}{dt}\left({\parallel \mathsf{\Lambda }b\parallel }_{L^2}^2+{\parallel {\mathsf{\Lambda }}^{2}b\parallel }_{L^2}^2\right)+{\parallel {\mathsf{\Lambda }}^{2}b\parallel }_{L^2}^2\hfill \\ \hfill =& -{\int }_{{\mathbb{R}}^3}\mathsf{\Lambda }(u·\nabla b)\mathsf{\Lambda }bdx+{\int }_{{\mathbb{R}}^3}\mathsf{\Lambda }(b·\nabla u)\mathsf{\Lambda }bdx\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^3}\mathsf{\Lambda }\left(\mathrm{rot}(b\times \mathrm{rot}b)\right)\mathsf{\Lambda }bdx-{\int }_{{\mathbb{R}}^3}\mathsf{\Lambda }\left(\mathrm{rot}\left((u·\nabla )\mathrm{rot}b\right)\right)\mathsf{\Lambda }bdx\hfill \\ \hfill \le & {C\parallel u\parallel }_{L^3}{\parallel \nabla b\parallel }_{L^6}\parallel {\mathsf{\Lambda }}^2{b\parallel }_{L^2}+{C\parallel b\parallel }_{L^6}{\parallel \nabla u\parallel }_{L^3}{\parallel {\mathsf{\Lambda }}^{2}b\parallel }_{L^2}\hfill \\ \hfill & +{C(\parallel \nabla b\parallel }_{L^3}{\parallel \nabla b\parallel }_{L^6}+{\parallel b\parallel }_{L^{\infty }}\parallel {\mathsf{\Lambda }}^2{b\parallel }_{L^2})\parallel {\mathsf{\Lambda }}^2{b\parallel }_{L^2}+{C\parallel \nabla u\parallel }_{L^{\infty }}{\parallel {\mathsf{\Lambda }}^{2}b\parallel }_{L^2}\hfill \\ \hfill \le & {C(\parallel u\parallel }_{H^{\frac{3}{2}+\epsilon }}+{\parallel b\parallel }_{H^2}{\left)\right(\parallel \nabla b\parallel }_{H^1}^2+{\parallel \nabla u\parallel }_{H^{\frac{3}{2}+\epsilon }}^2),\hfill \end{array}$$

(20)

where one has used

$$\begin{array}{cc}\hfill & \left|{\int }_{{\mathbb{R}}^3}\mathsf{\Lambda }\left(\mathrm{rot}\left((u·\nabla )\mathrm{rot}b\right)\right)\mathsf{\Lambda }bdx\right|\hfill \\ \hfill \le & \left|{\int }_{{\mathbb{R}}^3}\mathsf{\Lambda }\left((u·\nabla )\mathrm{rot}b\right)\mathsf{\Lambda }\mathrm{rot}bdx-{\int }_{{\mathbb{R}}^3}\left((u·\nabla )\mathsf{\Lambda }\mathrm{rot}b\right)\mathsf{\Lambda }\mathrm{rot}bdx\right|\hfill \\ \hfill \le & {C\parallel \nabla u\parallel }_{L^{\infty }}{\parallel {\mathsf{\Lambda }}^{2}b\parallel }_{L^2}.\hfill \end{array}$$

Taking ${\mathsf{\Lambda }}^{\frac{3}{2}+\epsilon }$ to (10), testing by ${\mathsf{\Lambda }}^{\frac{3}{2}+\epsilon }u$, it yields that

$$\begin{array}{cc}& \frac{1}{2}\frac{d}{dt}\parallel {\mathsf{\Lambda }}^{\frac{3}{2}+\epsilon }{u\parallel }_{L^2}^2+{\parallel {\mathsf{\Lambda }}^{\frac{5}{2}+\epsilon }u\parallel }_{L^2}^2\hfill \\ \hfill =& -{\int }_{{\mathbb{R}}^3}{\mathsf{\Lambda }}^{\frac{3}{2}+\epsilon }(u·\nabla u){\mathsf{\Lambda }}^{\frac{3}{2}+\epsilon }udx+{\int }_{{\mathbb{R}}^3}{\mathsf{\Lambda }}^{\frac{3}{2}+\epsilon }(b·\nabla b){\mathsf{\Lambda }}^{\frac{3}{2}+\epsilon }udx\hfill \\ \hfill \le & {C\parallel u\parallel }_{L^3}\parallel {\mathsf{\Lambda }}^{\frac{3}{2}+\epsilon }{u\parallel }_{L^6}\parallel {\mathsf{\Lambda }}^{\frac{5}{2}+\epsilon }{u\parallel }_{L^2}+{C\parallel b\parallel }_{L^{\infty }}\parallel {\mathsf{\Lambda }}^{\frac{3}{2}+\epsilon }{b\parallel }_{L^2}{\parallel {\mathsf{\Lambda }}^{\frac{5}{2}+\epsilon }u\parallel }_{L^2}\hfill \\ \hfill \le & C\parallel {\mathsf{\Lambda }}^{\frac{1}{2}}{u\parallel }_{L^2}\parallel {\mathsf{\Lambda }}^{\frac{5}{2}+\epsilon }{u\parallel }_{L^2}^2+{C\parallel b\parallel }_{H^2}{\parallel \nabla b\parallel }_{H^1}{\parallel {\mathsf{\Lambda }}^{\frac{5}{2}+\epsilon }u\parallel }_{L^2}\hfill \\ \hfill \le & C\parallel {\mathsf{\Lambda }}^{\frac{1}{2}}{u\parallel }_{L^2}\parallel {\mathsf{\Lambda }}^{\frac{5}{2}+\epsilon }{u\parallel }_{L^2}^2+{C\parallel b\parallel }_{H^2}{(\parallel \nabla b\parallel }_{H^1}^2+\parallel {\mathsf{\Lambda }}^{\frac{5}{2}+\epsilon }u{\parallel }_{L^2}^2),\hfill \end{array}$$

(21)

Combining (19)–(21) together gives

$$\begin{array}{c}\hfill \frac{d}{dt}\left({\parallel u\parallel }_{H^{\frac{3}{2}+\epsilon }}^2+{\parallel b\parallel }_{H^2}^2\right)+{\parallel \nabla u\parallel }_{H^{\frac{3}{2}+\epsilon }}^2+{\parallel \nabla b\parallel }_{H^1}^2\le {C(\parallel u\parallel }_{H^{\frac{3}{2}+\epsilon }}+{\parallel b\parallel }_{H^2}{\left)\right(\parallel \nabla b\parallel }_{H^1}^2+{\parallel \nabla u\parallel }_{H^{\frac{3}{2}+\epsilon }}^2).\end{array}$$

Combining condition (13) with proof by contradiction, the global bound as follows can be obtained:

$$\begin{array}{c}\hfill {\parallel u\parallel }_{H^{\frac{3}{2}+\epsilon }}^2+{\parallel b\parallel }_{H^2}^2+{\int }_0^t{(\parallel \nabla u\parallel }_{H^{\frac{3}{2}+\epsilon }}^2+{\parallel \nabla b\parallel }_{H^1}^2)d\tau \le C,\forall t>0.\end{array}$$

(22)

From the local well-posedness result (see [16]), (19) and (22), one easily proves that Theorem 1 holds for $p=0$, i.e.,

$$\parallel {\mathsf{\Lambda }}^m{u\parallel }_{L^2}^2+\parallel {\mathsf{\Lambda }}^m{b\parallel }_{L^2}^2+\parallel {\mathsf{\Lambda }}^{m+1}{b\parallel }_{L^2}^2\le C(\parallel u_0{\parallel }_{H^K}^2+\parallel b_0{\parallel }_{H^K}^2),$$

(23)

provided that $K\ge max\{m,3\}$.

In the following, the time derivatives of the solution in terms of the space derivatives will be bounded. Let $\mathcal{A}=(I-\Delta )$, then ${\mathcal{A}}^{-1}={(I-\Delta )}^{-1}$. Applying ${\partial }_t^p{\mathsf{\Lambda }}^m$ to the solution of systems (9)–(12), it yields that

$$\parallel {\partial }_t^{p+1}{\mathsf{\Lambda }}^m{u\parallel }_{L^2}^2\le C(\parallel {\partial }_t^p{\mathsf{\Lambda }}^{m+2}{u\parallel }_{L^2}^2+\parallel {\partial }_t^p{\mathsf{\Lambda }}^m{(u·\nabla u)\parallel }_{L^2}^2+\parallel {\partial }_t^p{\mathsf{\Lambda }}^m(b·\nabla b){\parallel }_{L^2}^2),$$

(24)

$$\begin{array}{cc}\hfill & \parallel {\partial }_t^{p+1}{\mathsf{\Lambda }}^m{(b-\Delta b)\parallel }_{L^2}^2\hfill \\ \hfill \le & C(\parallel {\partial }_t^p{\mathsf{\Lambda }}^{m+2}b{\parallel }_{L^2}^2+\parallel {\partial }_t^p{\mathsf{\Lambda }}^m(u·\nabla b){\parallel }_{L^2}^2+\parallel {\partial }_t^p{\mathsf{\Lambda }}^m(b·\nabla u){\parallel }_{L^2}^2\hfill \\ & +\parallel {\partial }_t^p{\mathsf{\Lambda }}^m{\nabla \times [b\times (\nabla \times b)]\parallel }_{L^2}^2+\parallel {\partial }_t^p{\mathsf{\Lambda }}^m\nabla \times \left[(u·\nabla )(\nabla \times b)\right]{\parallel }_{L^2}^2).\hfill \end{array}$$

(25)

and

$$\begin{array}{cc}\hfill & \parallel {\partial }_t^{p+1}{\mathsf{\Lambda }}^m{b\parallel }_{L^2}^2\hfill \\ \hfill \le & C(\parallel {\partial }_t^p{\mathcal{A}}^{-1}{\mathsf{\Lambda }}^{m+2}b{\parallel }_{L^2}^2+\parallel {\partial }_t^p{\mathcal{A}}^{-1}{\mathsf{\Lambda }}^m(u·\nabla b){\parallel }_{L^2}^2+\parallel {\partial }_t^p{\mathcal{A}}^{-1}{\mathsf{\Lambda }}^m(b·\nabla u){\parallel }_{L^2}^2\hfill \\ & +\parallel {\partial }_t^p{\mathcal{A}}^{-1}{\mathsf{\Lambda }}^m{\nabla \times [b\times (\nabla \times b)]\parallel }_{L^2}^2+\parallel {\partial }_t^p{\mathcal{A}}^{-1}{\mathsf{\Lambda }}^m\nabla \times \left[(u·\nabla )(\nabla \times b)\right]{\parallel }_{L^2}^2).\hfill \end{array}$$

(26)

Using Gagliardo–Nirenberg inequality, the second term on the right hand side of (24) can be bounded as

$$\begin{array}{cc}\hfill \parallel {\partial }_t^p{\mathsf{\Lambda }}^m{(u·\nabla u)\parallel }_{L^2}^2\le & \sum _{\mathfrak{p}=0}^p\sum _{\mathfrak{m}=0}^m{C}_p^{\mathfrak{p}}{C}_m^{\mathfrak{m}}\parallel {\partial }_t^{\mathfrak{p}}{\mathsf{\Lambda }}^{\mathfrak{m}}{u\parallel }_{L^3}^2{\parallel {\partial }_t^{p-\mathfrak{p}}{\mathsf{\Lambda }}^{m-\mathfrak{m}+1}u\parallel }_{L^6}^2\hfill \\ \hfill \le & C\sum _{\mathfrak{p}=0}^p\sum _{\mathfrak{m}=0}^m\parallel {\partial }_t^{\mathfrak{p}}{\mathsf{\Lambda }}^{\mathfrak{m}+\frac{1}{2}}{u\parallel }_{L^2}^2{\parallel {\partial }_t^{p-\mathfrak{p}}{\mathsf{\Lambda }}^{m+2-\mathfrak{m}}u\parallel }_{L^2}^2.\hfill \end{array}$$

(27)

Similarly,

$$\begin{array}{c}\hfill \parallel {\partial }_t^p{\mathsf{\Lambda }}^m{(b·\nabla b)\parallel }_{L^2}^2\le C\sum _{\mathfrak{p}=0}^P\sum _{\mathfrak{m}=0}^M\parallel {\partial }_t^{\mathfrak{p}}{\mathsf{\Lambda }}^{\mathfrak{m}+\frac{1}{2}}{b\parallel }_{L^2}^2{\parallel {\partial }_t^{p-\mathfrak{p}}{\mathsf{\Lambda }}^{m+2-\mathfrak{m}}b\parallel }_{L^2}^2.\end{array}$$

(28)

$$\begin{array}{cc}\hfill & \parallel {\partial }_t^p{\mathcal{A}}^{-1}{\mathsf{\Lambda }}^m{(u·\nabla b)\parallel }_{L^2}^2+{\parallel {\partial }_t^p{\mathcal{A}}^{-1}{\mathsf{\Lambda }}^m(b·\nabla u)\parallel }_{L^2}^2\hfill \\ \hfill \le & C\parallel {\partial }_t^p{\mathsf{\Lambda }}^m{(u·\nabla b)\parallel }_{L^2}^2+{\parallel {\partial }_t^p{\mathsf{\Lambda }}^m(b·\nabla u)\parallel }_{L^2}^2\hfill \\ \hfill \le & C\sum _{\mathfrak{p}=0}^p\sum _{\mathfrak{m}=0}^m\left(\parallel {\partial }_t^{\mathfrak{p}}{\mathsf{\Lambda }}^{\mathfrak{m}+\frac{1}{2}}{u\parallel }_{L^2}^2\parallel {\partial }_t^{p-\mathfrak{p}}{\mathsf{\Lambda }}^{m+2-\mathfrak{m}}{u\parallel }_{L^2}^2+\parallel {\partial }_t^{\mathfrak{p}}{\mathsf{\Lambda }}^{\mathfrak{m}+\frac{1}{2}}{b\parallel }_{L^2}^2{\parallel {\partial }_t^{p-\mathfrak{p}}{\mathsf{\Lambda }}^{m+2-\mathfrak{m}}b\parallel }_{L^2}^2\right).\hfill \end{array}$$

(29)

In addition,

$$\begin{array}{cc}\hfill & \parallel {\partial }_t^p{\mathcal{A}}^{-1}{\mathsf{\Lambda }}^m{\nabla \times [b\times (\nabla \times b)]\parallel }_{L^2}^2\hfill \\ \hfill \le & C\parallel {\partial }_t^p{\mathsf{\Lambda }}^m{\nabla \times [b\times (\nabla \times b)]\parallel }_{L^2}^2\hfill \\ \hfill \le & C\sum _{\mathfrak{p}=0}^p\sum _{\mathfrak{m}=0}^{m+1}{C}_p^{\mathfrak{p}}{C}_{m+1}^{\mathfrak{m}}\parallel {\partial }_t^{\mathfrak{p}}{\mathsf{\Lambda }}^{\mathfrak{m}}{b\parallel }_{L^{\infty }}^2{\parallel {\partial }_t^{p-\mathfrak{p}}{\mathsf{\Lambda }}^{m+2-\mathfrak{m}}b\parallel }_{L^2}^2\hfill \\ \hfill \le & C\sum _{\mathfrak{p}=0}^p\sum _{\mathfrak{m}=0}^{m+1}\parallel {\partial }_t^{\mathfrak{p}}{\mathsf{\Lambda }}^{\mathfrak{m}+1}{b\parallel }_{L^2}\parallel {\partial }_t^{\mathfrak{p}}{\mathsf{\Lambda }}^{\mathfrak{m}+2}{b\parallel }_{L^2}{\parallel {\partial }_t^{p-\mathfrak{p}}{\mathsf{\Lambda }}^{m+2-\mathfrak{m}}b\parallel }_{L^2}^2,\hfill \end{array}$$

(30)

and

$$\begin{array}{cc}\hfill & \parallel {\partial }_t^p{\mathcal{A}}^{-1}{\mathsf{\Lambda }}^m{\nabla \times \left[(u·\nabla )(\nabla \times b)\right]\parallel }_{L^2}^2\hfill \\ \hfill \le & C\parallel {\partial }_t^p{\mathsf{\Lambda }}^m{\nabla \times \left[(u·\nabla )(\nabla \times b)\right]\parallel }_{L^2}^2\hfill \\ \hfill \le & C\sum _{\mathfrak{p}=0}^p\sum _{\mathfrak{m}=0}^{m+1}{C}_p^{\mathfrak{p}}{C}_{m+1}^{\mathfrak{m}}\parallel {\partial }_t^{\mathfrak{p}}{\mathsf{\Lambda }}^{\mathfrak{m}}{u\parallel }_{L^{\infty }}^2{\parallel {\partial }_t^{p-\mathfrak{p}}{\mathsf{\Lambda }}^{m+3-\mathfrak{m}}b\parallel }_{L^2}^2\hfill \\ \hfill \le & C\sum _{\mathfrak{p}=0}^p\sum _{\mathfrak{m}=0}^{m+1}\parallel {\partial }_t^{\mathfrak{p}}{\mathsf{\Lambda }}^{\mathfrak{m}+1}{u\parallel }_{L^2}\parallel {\partial }_t^{\mathfrak{p}}{\mathsf{\Lambda }}^{\mathfrak{m}+2}{u\parallel }_{L^2}{\parallel {\partial }_t^{p-\mathfrak{p}}{\mathsf{\Lambda }}^{m+3-\mathfrak{m}}b\parallel }_{L^2}^2.\hfill \end{array}$$

(31)

Moreover,

$$\begin{array}{cc}\hfill \parallel {\partial }_t^{p+1}{\mathsf{\Lambda }}^{m+1}{b\parallel }_{L^2}^2\le & C(\parallel {\partial }_t^{p+1}{\mathsf{\Lambda }}^{m}b{\parallel }_{L^2}^2+\parallel {\partial }_t^{p+1}{\mathsf{\Lambda }}^{m+2}b{\parallel }_{L^2}^2)\hfill \\ \hfill \le & C(\parallel {\partial }_t^{p+1}{\mathsf{\Lambda }}^{m}b{\parallel }_{L^2}^2+\parallel {\partial }_t^{p+1}{\mathsf{\Lambda }}^m(b-\Delta b){\parallel }_{L^2}^2).\hfill \end{array}$$

(32)

Putting (24)–(32) together gives

$$\begin{array}{cc}& \parallel {\partial }_t^{p+1}{\mathsf{\Lambda }}^m{u\parallel }_{L^2}^2+\parallel {\partial }_t^{p+1}{\mathsf{\Lambda }}^m{b\parallel }_{L^2}^2+{\parallel {\partial }_t^{p+1}{\mathsf{\Lambda }}^{m+1}bu\parallel }_{L^2}^2\hfill \\ \hfill \le & C(\parallel {\partial }_t^{p}u{\parallel }_{H^{m+2}}^2+\parallel {\partial }_t^{p}b{\parallel }_{H^{m+2}}^2+\parallel {\partial }_t^p\mathsf{\Lambda }b{\parallel }_{H^{m+2}}^2),\hfill \end{array}$$

(33)

which means, for all $m,p,K\in \mathbb{N}$ such that $K\ge max\{m,3\}+2p$,

$$\begin{array}{c}\hfill \parallel {\partial }_t^p{\mathsf{\Lambda }}^m{u\parallel }_{L^2}^2+\parallel {\partial }_t^p{\mathsf{\Lambda }}^m{b\parallel }_{L^2}^2+\parallel {\partial }_t^p{\mathsf{\Lambda }}^{m+1}{b\parallel }_{L^2}^2\le C(\parallel u_0{\parallel }_{H^K}^2+\parallel b_0{\parallel }_{H^K}^2+\parallel \mathsf{\Lambda }{b}_0{\parallel }_{H^K}^2).\end{array}$$

This complete the proof.

4. Proof of Theorem 2

4.1. Auxiliary Lemmas

In [29], the author established the following result:

Lemma 10

(see [29]). Let $(u_0,b_0)\in L^2\left({\mathbb{R}}^3\right)\times H^1\left({\mathbb{R}}^3\right)$ with $\mathit{div}{u}_0=\mathit{div}{b}_0=0$. Suppose that $(u,b)$ is the solution of systems (9)–(12) with initial value $(u_0,b_0)$. Then

$$|\widehat{u}{(\xi ,t)|}^2\le C\left[e^{-{2|\xi |}^{2}t}|\widehat{u_0}{\left(\xi \right)|}^2+{|\xi |}^2{\left({\int }_0^t{(\parallel u\left(t\right)\parallel }_{L^2}^2+{\parallel b\left(t\right)\parallel }_{L^2}^2)ds\right)}^2\right],$$

(34)

and

$$\begin{array}{c}\hfill |\widehat{b}{(\xi ,t)|}^2\le C\left[e^{-\frac{{2|\xi |}^{2}t}{1+{|\xi |}^2}}|\widehat{b_0}{\left(\xi \right)|}^2+{(|\xi |}^2+{|\xi |}^4+{|\xi |}^6){\left({\int }_0^t{(\parallel u\left(t\right)\parallel }_{L^2}^2+{\parallel b\left(t\right)\parallel }_{L^2}^2)ds\right)}^2\right].\end{array}$$

(35)

In order to characterize the decay estimates of systems (9)–(12), the following lemma is introduced.

Lemma 11.

Suppose that the assumptions listed in Lemma 10 are satisfied. Then, for $p>0$,

$$\begin{array}{cc}\hfill |{\partial }_t^p\widehat{u}{(\xi ,t)|}^2\le & {C|\xi |}^{4p}{e}^{-{2|\xi |}^{2}t}|{\widehat{u}}_0{(\xi ,t)|}^2+C{|\xi |}^{4p+2}{\left[{\int }_0^t{(\parallel u\parallel }_{L^2}^2+{\parallel b\parallel }_{L^2}^2)ds\right]}^2\hfill \\ & +C\sum _{\mathfrak{p}=0}^{p-1}\sum _{l=0}^{\mathfrak{p}}{|\xi |}^{4p-4\mathfrak{p}-2}(\parallel {\partial }_t^l{u\parallel }_{L^2}^2{\parallel {\partial }_t^{\mathfrak{p}-l}u\parallel }_{L^2}^2,\hfill \end{array}$$

(36)

and

$$\begin{array}{cc}\hfill |{\partial }_t^p\widehat{b}{(\xi ,t)|}^2\le & {C|\xi |}^{4p}{e}^{-2\frac{{|\xi |}^2}{1+{|\xi |}^2}t}|{\widehat{b}}_0{(\xi ,t)|}^2+C{|\xi |}^{4p+2}{\left[{\int }_0^t{(\parallel u\parallel }_{L^2}^2+{\parallel b\parallel }_{L^2}^2)ds\right]}^2\hfill \\ & +C\sum _{\mathfrak{p}=0}^{p-1}\sum _{l=0}^{\mathfrak{p}}{|\xi |}^{4p-4\mathfrak{p}-2}{(1+|\xi |}^2+{|\xi |}^4)\hfill \\ & \times (\parallel {\partial }_t^{l}u{\parallel }_{L^2}^2\parallel {\partial }_t^{\mathfrak{p}-l}b{\parallel }_{L^2}^2+\parallel {\partial }_t^{l}b{\parallel }_{L^2}^2\parallel {\partial }_t^{\mathfrak{p}-l}u{\parallel }_{L^2}^2+\parallel {\partial }_t^{l}b{\parallel }_{L^2}^2\parallel {\partial }_t^{\mathfrak{p}-l}b{\parallel }_{L^2}^2).\hfill \end{array}$$

(37)

 Proof. 

Note that

$$\begin{array}{cc}\hfill {\partial }_t^p\widehat{u}(\xi ,t)=& {(-1)}^p{|\xi |}^{2p}{e}^{-{|\xi |}^{2}t}{\widehat{u}}_0\left(\xi \right)+\sum _{\mathfrak{p}=0}^{p-1}{(-1)}^{p-\mathfrak{p}-1}{|\xi |}^{2p-2\mathfrak{p}-2}{\partial }_t^{\mathfrak{p}}(\widehat{u·\nabla u}-\widehat{b·\nabla b})(\xi ,t)\hfill \\ & +{\int }_0^t{(-1)}^p{|\xi |}^{2p}{e}^{-{|\xi |}^2(t-s)}(\widehat{u·\nabla u}-\widehat{b·\nabla b})(\xi ,s)ds.\hfill \end{array}$$

(38)

In addition,

$$\begin{array}{cc}\hfill |{\partial }_t^{\mathfrak{p}}(\widehat{u·\nabla u}-\widehat{b·\nabla b})(\xi ,t)|\le & |{\partial }_t^{\mathfrak{p}}\sum _j{\xi }_j\widehat{u_{j}u}(\xi ,t)|+|{\partial }_t^{\mathfrak{p}}\sum _j{\xi }_j\widehat{b_{j}b}(\xi ,t)|\hfill \\ \hfill \le & C\sum _{l=0}^{\mathfrak{p}}|\xi |(\parallel {\partial }_t^l{u\parallel }_{L^2}\parallel {\partial }_t^{\mathfrak{p}-l}{u\parallel }_{L^2}+\parallel {\partial }_t^l{b\parallel }_{L^2}\parallel {\partial }_t^{\mathfrak{p}-l}b{\parallel }_{L^2}).\hfill \end{array}$$

(39)

Adding (38) and (39) together, by using (34), it yields (36). On the other hand, the following equality holds:

$$\begin{array}{cc}\hfill {\partial }_t^p\widehat{b}(\xi ,t)=& {(-1)}^p{|\xi |}^{2p}{e}^{-\frac{{|\xi |}^2}{1+{|\xi |}^2}t}{\widehat{b}}_0\left(\xi \right)+\sum _{\mathfrak{p}=0}^{p-1}{(-1)}^{p-\mathfrak{p}-1}{|\xi |}^{2(p-\mathfrak{p}-1)}\left[\widehat{u·\nabla b}-\widehat{b·\nabla u}\right.\hfill \\ & \left.+\widehat{\nabla \times \left[\right(\nabla \times b)\times b]}+\widehat{\nabla \times [u·\nabla (\nabla \times b\left)\right]}\right](\xi ,t)\hfill \\ & +{\int }_0^t{(-1)}^p{|\xi |}^{2p}{e}^{-\frac{{|\xi |}^2}{1+{|\xi |}^2}(t-s)}\left[\widehat{u·\nabla b}-\widehat{b·\nabla u}+\widehat{\nabla \times \left[\right(\nabla \times b)\times b]}\right.\hfill \\ & \left.+\widehat{\nabla \times [u·\nabla (\nabla \times b\left)\right]}\right](\xi ,s)ds.\hfill \end{array}$$

(40)

Moreover,

$$|{\partial }_t^{\mathfrak{p}}(\widehat{u·\nabla b}-\widehat{b·\nabla u})(\xi ,t)|\le C\sum _{l=0}^{\mathfrak{p}}|\xi |(\parallel {\partial }_t^l{u\parallel }_{L^2}\parallel {\partial }_t^{\mathfrak{p}-l}{b\parallel }_{L^2}+\parallel {\partial }_t^l{b\parallel }_{L^2}\parallel {\partial }_t^{\mathfrak{p}-l}u{\parallel }_{L^2}),$$

(41)

$$|{\partial }_t^{\mathfrak{p}}\widehat{\nabla \times \left(\right(\nabla \times b)\times b)}(\xi ,t)|\le C\sum _{l=0}^{\mathfrak{p}}{|\xi |}^2\parallel {\partial }_t^l{b\parallel }_{L^2}{\parallel {\partial }_t^{\mathfrak{p}-l}b\parallel }_{L^2},$$

(42)

and

$$|{\partial }_t^{\mathfrak{p}}\widehat{\nabla \times (u·\nabla (\nabla \times b\left)\right)}(\xi ,t)|\le C\sum _{l=0}^{\mathfrak{p}}{|\xi |}^3(\parallel {\partial }_t^l{u\parallel }_{L^2}\parallel {\partial }_t^{\mathfrak{p}-l}{b\parallel }_{L^2}+\parallel {\partial }_t^l{b\parallel }_{L^2}\parallel {\partial }_t^{\mathfrak{p}-l}u{\parallel }_{L^2}),$$

(43)

Combining (40)–(43) together, applying (35), the estimate (38) is obtained, and the proof is completed.

□

4.2. Proof of Theorem 2

Theorem 2 is proven using the mathematical induction in this subsection.

First of all, the fact that Theorem 2 holds for the case $p=1$ is proven:

Lemma 12.

Let $m\in \mathbb{N}$, $K\ge max\{5,m+2\}$ and $r^{*}=r^{*}\left(u_0\right)=r^{*}\left(b_0\right)\in (-\frac{3}{2},+\infty )$ be the decay character. Suppose that $(u_0,b_0)\in H^K\left({\mathbb{R}}^3\right)\times H^{K+1}\left({\mathbb{R}}^3\right)$ with $\nabla ·u_0=\nabla ·b_0=0$. Then, there exists a positive constant $C=C(\parallel u_0{\parallel }_{H^K},\parallel b_0{\parallel }_{H^{K+1}})$, such that

$$\parallel {\partial }_t{\mathsf{\Lambda }}^m{u\parallel }_{L^2}^2+\parallel {\partial }_t{\mathsf{\Lambda }}^m{b\parallel }_{L^2}^2+{\parallel {\partial }_t{\mathsf{\Lambda }}^{m+1}b\parallel }_{L^2}^2\le C{(1+t)}^{-min\{\frac{7}{2}+r^{*}+m,\frac{9}{2}+m\}},\mathit{for}\mathit{large}t.$$

(44)

 Proof. 

In order to prove Lemma 12, one first proves the case $m=0$. Applying ${\partial }_t$ to (10) and (11), multiplying both side by ${\partial }_{t}u$ and ${\partial }_{t}b$ respectively, integrating over ${\mathbb{R}}^3$, gives that

$$\begin{array}{cc}& \frac{1}{2}\frac{d}{dt}(\parallel {\partial }_t{u\parallel }_{L^2}^2+\parallel {\partial }_t{b\parallel }_{L^2}^2+\parallel {\partial }_t{\mathsf{\Lambda }b\parallel }_{L^2}^2)+\parallel {\partial }_t{\mathsf{\Lambda }u\parallel }_{L^2}^2+{\parallel {\partial }_t\mathsf{\Lambda }b\parallel }_{L^2}^2\hfill \\ \hfill \le & |(u·{\partial }_t\mathsf{\Lambda }u,{\partial }_{t}u)|+|(b·{\partial }_t\mathsf{\Lambda }u,{\partial }_{t}b)|+|(u·{\partial }_t\mathsf{\Lambda }b,{\partial }_{t}b)|+|(b·{\partial }_t\mathsf{\Lambda }b,{\partial }_{t}u)|+|(b·{\partial }_t\mathsf{\Lambda }b,{\partial }_t\mathsf{\Lambda }b)|\hfill \\ & +|({\partial }_{t}u·\nabla (\nabla \times b),\nabla \times b)|+|({\partial }_{t}u·{\partial }_t\mathsf{\Lambda }u,u)|+|({\partial }_{t}b·{\partial }_t\mathsf{\Lambda }u,b)|+|({\partial }_{t}u·{\partial }_t\mathsf{\Lambda }b,b)|\hfill \\ & +|({\partial }_{t}b·{\partial }_t\mathsf{\Lambda }b,u)|+|({\partial }_{t}b·{\partial }_t\mathsf{\Lambda }b,\mathsf{\Lambda }b)|+|(u·\nabla (\nabla \times b_t),\nabla \times b_t)|\hfill \\ \hfill \le & {\parallel u\parallel }_{L^3}\parallel {\partial }_t{\mathsf{\Lambda }u\parallel }_{L^2}\parallel {\partial }_t{u\parallel }_{L^6}+{\parallel b\parallel }_{L^3}\parallel {\partial }_t{\mathsf{\Lambda }u\parallel }_{L^2}\parallel {\partial }_t{b\parallel }_{L^6}+{\parallel u\parallel }_{L^3}\parallel {\partial }_t{\mathsf{\Lambda }b\parallel }_{L^2}{\parallel {\partial }_{t}b\parallel }_{L^6}\hfill \\ & +{\parallel b\parallel }_{L^3}\parallel {\partial }_t{\mathsf{\Lambda }b\parallel }_{L^2}\parallel {\partial }_t{u\parallel }_{L^6}+{\parallel b\parallel }_{L^{\infty }}\parallel {\partial }_t{\mathsf{\Lambda }b\parallel }_{L^2}^2+\parallel {\mathsf{\Lambda }}^2{b\parallel }_{L^3}\parallel {\partial }_t{\mathsf{\Lambda }b\parallel }_{L^2}{\parallel {\partial }_{t}u\parallel }_{L^6}\hfill \\ & +{\parallel u\parallel }_{L^3}\parallel {\partial }_t{u\parallel }_{L^6}\parallel {\partial }_t{\mathsf{\Lambda }u\parallel }_{L^2}+{\parallel b\parallel }_{L^3}\parallel {\partial }_t{b\parallel }_{L^6}\parallel {\partial }_t{\mathsf{\Lambda }u\parallel }_{L^2}+{\parallel u\parallel }_{L^3}\parallel {\partial }_t{b\parallel }_{L^6}{\parallel {\partial }_t\mathsf{\Lambda }b\parallel }_{L^2}\hfill \\ & +{\parallel b\parallel }_{L^3}\parallel {\partial }_t{u\parallel }_{L^6}\parallel {\partial }_t{\mathsf{\Lambda }b\parallel }_{L^2}+{\parallel \mathsf{\Lambda }b\parallel }_{L^3}\parallel {\partial }_t{b\parallel }_{L^6}\parallel {\partial }_t{\mathsf{\Lambda }b\parallel }_{L^2}+{\parallel \mathsf{\Lambda }u\parallel }_{L^{\infty }}{\parallel {\partial }_t\mathsf{\Lambda }b\parallel }_{L^2}^2\hfill \\ \hfill \le & (\parallel {\mathsf{\Lambda }}^{\frac{1}{2}}u{\parallel }_{L^2}+\parallel {\mathsf{\Lambda }}^{\frac{1}{2}}b{\parallel }_{L^2}+\parallel {\mathsf{\Lambda }}^{\frac{1}{2}}u{\parallel }_{L^2}+\parallel \mathsf{\Lambda }b{\parallel }_{L^2}^{\frac{1}{2}}\parallel {\mathsf{\Lambda }}^{2}b{\parallel }_{L^2}^{\frac{1}{2}}+\parallel {\mathsf{\Lambda }}^{\frac{5}{2}}b{\parallel }_{L^2}+\parallel {\mathsf{\Lambda }}^{\frac{3}{2}}b{\parallel }_{L^2}\hfill \\ & +{\parallel u\parallel }_{H^K}\left)\right(\parallel {\partial }_t{\mathsf{\Lambda }u\parallel }_{L^2}^2+\parallel {\partial }_t\mathsf{\Lambda }b{\parallel }_{L^2}^2)\hfill \\ \hfill \le & \frac{1}{2}(\parallel {\partial }_t{\mathsf{\Lambda }u\parallel }_{L^2}^2+\parallel {\partial }_t\mathsf{\Lambda }b{\parallel }_{L^2}^2),\mathrm{for}\mathrm{large}t,\hfill \end{array}$$

where

$$(u·\nabla (\nabla \times b_t),\nabla \times b_t)=0.$$

has been used. Then,

$$\frac{d}{dt}(\parallel {\partial }_t{u\parallel }_{L^2}^2+\parallel {\partial }_t{b\parallel }_{L^2}^2+\parallel {\partial }_t{\mathsf{\Lambda }b\parallel }_{L^2}^2)+\parallel {\partial }_t{\mathsf{\Lambda }u\parallel }_{L^2}^2+{\parallel {\partial }_t\mathsf{\Lambda }b\parallel }_{L^2}^2\le 0,\mathrm{for}\mathrm{large}t.$$

(45)

Applying Plancherel’s theorem to (45) gives

$$\begin{array}{cc}& \frac{d}{dt}{\int }_{{\mathbb{R}}^3}\left[|\widehat{{\partial }_{t}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_{t}b}{(\xi ,t)|}^2\right]d\xi \hfill \\ & +{\int }_{{\mathbb{R}}^3}\frac{{|\xi |}^2}{1+{|\xi |}^2}\left[|\widehat{{\partial }_{t}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_{t}b}{(\xi ,t)|}^2\right]d\xi \le 0,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(46)

Set

$$B\left(t\right):=\left\{\xi \in {\mathbb{R}}^3{||\xi |}^2\le \frac{g^{\prime }\left(t\right)}{g\left(t\right)-g^{\prime }\left(t\right)}\right\},B^c\left(t\right):={\mathbb{R}}^3\backslash B\left(t\right),$$

where $g\left(t\right)$ is a differentiable function of t satisfying

$$g\left(0\right)=1,g^{\prime }\left(t\right)>0\mathrm{and}2g\left(t\right)>g^{\prime }\left(t\right),\forall t>0.$$

Multiplying (46) by $g\left(t\right)$ gives

$$\begin{array}{cc}& \frac{d}{dt}\left\{g\left(t\right){\int }_{{\mathbb{R}}^3}\left[|\widehat{{\partial }_{t}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_{t}b}{(\xi ,t)|}^2\right]d\xi \right\}\hfill \\ \hfill \le & g^{\prime }\left(t\right){\int }_{B\left(t\right)}\left[|\widehat{{\partial }_{t}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_{t}b}{(\xi ,t)|}^2\right]d\xi ,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

It then follows from Lemma 11 that

$$\begin{array}{cc}\hfill & g\left(t\right){\int }_{{\mathbb{R}}^3}\left[|\widehat{{\partial }_{t}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_{t}b}{(\xi ,t)|}^2\right]d\xi \hfill \\ \hfill \le & C+{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^4\left[e^{-{2|\xi |}^{2}t}|{\widehat{u}}_0{\left(\xi \right)|}^2{+(1+|\xi |}^2)e^{-2\frac{{|\xi |}^2}{1+{|\xi |}^2}t}{|{\widehat{b}}_0\left(\xi \right)|}^2\right]d\xi ds\hfill \\ \hfill & +C{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^2\left({\parallel u\left(s\right)\parallel }_{L^2}^4+{\parallel b\left(s\right)\parallel }_{L^2}^4+{\parallel u\left(s\right)\parallel }_{L^2}^2{\parallel b\left(s\right)\parallel }_{L^2}^2\right)d\xi ds\hfill \\ \hfill & +C{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^6{(1+|\xi |}^2+{|\xi |}^4){\left[{\int }_0^t{(\parallel u\left(\tau \right)\parallel }_{L^2}^2+{\parallel b\left(\tau \right)\parallel }_{L^2}^2)d\tau \right]}^{2}d\xi ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(47)

The right hand side of (47) is estimated in the following. For the first term, by using the estimates from Lemma 6, it yields that

$$\begin{array}{cc}\hfill & {\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^4\left[e^{-{2|\xi |}^{2}t}|{\widehat{u}}_0{\left(\xi \right)|}^2{+(1+|\xi |}^2)e^{-2\frac{{|\xi |}^2}{1+{|\xi |}^2}t}{|{\widehat{b}}_0\left(\xi \right)|}^2\right]d\xi ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right)(\parallel {\partial }_t\overline{u}{\left(s\right)\parallel }_{L^2}^2+\parallel {\partial }_t\overline{b}{\left(s\right)\parallel }_{L^2}^2+\parallel {\partial }_t\mathsf{\Lambda }\overline{b}\left(s\right){\parallel }_{L^2}^2)ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-min\{r^{*}+\frac{7}{2},\frac{9}{2}\}}ds,\hfill \end{array}$$

(48)

where $(\overline{u},\overline{b})$ is the solution to the linear system (18). For the second term, after integrating in polar coordinates in $B\left(t\right)$, one can deduce that

$$\begin{array}{cc}\hfill & C{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^2\left({\parallel u\left(s\right)\parallel }_{L^2}^4+{\parallel b\left(s\right)\parallel }_{L^2}^4+{\parallel u\left(s\right)\parallel }_{L^2}^2{\parallel b\left(s\right)\parallel }_{L^2}^2\right)d\xi ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-\frac{5}{2}}{(1+s)}^{-2min\{r^{*}+\frac{3}{2},\frac{5}{2}\}}ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-min\{2r^{*}+\frac{11}{2},\frac{15}{2}\}}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(49)

In addition, if $r^{*}+\frac{3}{2}<\frac{5}{2}$, the following estimate holds:

$${\left[{\int }_0^t{(\parallel u\left(\tau \right)\parallel }_{L^2}^2+{\parallel b\left(\tau \right)\parallel }_{L^2}^2)d\tau \right]}^2\le C{(1+t)}^{-2r^{*}-1},\mathrm{for}\mathrm{large}t,$$

then the third term of the right hand side of (47) can be estimated as

$$\begin{array}{cc}\hfill & C{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^6{(1+|\xi |}^2+{|\xi |}^4){\left[{\int }_0^t{(\parallel u\left(\tau \right)\parallel }_{L^2}^2+{\parallel b\left(\tau \right)\parallel }_{L^2}^2)d\tau \right]}^{2}d\xi ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-\frac{9}{2}}{(1+s)}^{-2r^{*}-1}ds\le C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-2r^{*}-\frac{11}{2}}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(50)

If $r^{*}+\frac{3}{2}\ge \frac{5}{2}$,

$${\left[{\int }_0^t{(\parallel u\left(\tau \right)\parallel }_{L^2}^2+{\parallel b\left(\tau \right)\parallel }_{L^2}^2)d\tau \right]}^2\le C,\mathrm{for}\mathrm{large}t,$$

the third term of the right hand side of (47) satisfies

$$\begin{array}{cc}\hfill & C{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^6{(1+|\xi |}^2+{|\xi |}^4){\left[{\int }_0^t{(\parallel u\left(\tau \right)\parallel }_{L^2}^2+{\parallel b\left(\tau \right)\parallel }_{L^2}^2)d\tau \right]}^{2}d\xi ds\hfill \\ & \le C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-\frac{9}{2}}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(51)

For a fixed $r^{*}$, choose $g\left(t\right)={(1+t)}^m$, for some $m>max\{r^{*}+\frac{7}{2},\frac{9}{2}\}$. Then $\rho \left(t\right)=C{(1+t)}^{-\frac{1}{2}}$. It follows from (47)–(51) that

$$\begin{array}{cc}\hfill & \parallel {\partial }_t{u\parallel }_{L^2}^2+\parallel {\partial }_t{b\parallel }_{L^2}^2+{\parallel {\partial }_t\mathsf{\Lambda }b\parallel }_{L^2}^2\hfill \\ \hfill \le & C{(1+t)}^{-m}+C{(1+t)}^{-min\{r^{*}+\frac{7}{2},\frac{9}{2}\}}+C{(1+t)}^{-min\{2r^{*}+\frac{11}{2},\frac{15}{2}\}}\hfill \\ & +C{(1+t)}^{-min\{2r^{*}+\frac{11}{2},\frac{9}{2}\}}\hfill \\ \hfill \le & C{(1+t)}^{-min\{r^{*}+\frac{7}{2},\frac{9}{2}\}},\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(52)

Suppose that Lemma 12 holds for $m\le N\in {\mathbb{N}}^{+}$, then one can prove that it also holds for $m=N+1$. Applying ${\partial }_t{\mathsf{\Lambda }}^{N+1}$ to (10) and (11), multiplying both side by ${\partial }_t{\mathsf{\Lambda }}^{N+1}u$ and ${\partial }_t{\mathsf{\Lambda }}^{N+1}b$ respectively, integrating over ${\mathbb{R}}^3$, gives

$$\begin{array}{cc}\hfill & \frac{1}{2}\frac{d}{dt}\left(\parallel {\partial }_t{\mathsf{\Lambda }}^{N+1}{u\parallel }_{L^2}^2+\parallel {\partial }_t{\mathsf{\Lambda }}^{N+1}{b\parallel }_{L^2}^2+{\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}b\parallel }_{L^2}^2\right)\hfill \\ & +\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}^2+{\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}b\parallel }_{L^2}^2\hfill \\ \hfill =& \sum _{p=0}^1\sum _{m=0}^{N+1}{C}_1^p{C}_{N+1}^m\left[-({\partial }_t^p{\mathsf{\Lambda }}^{m}u·{\partial }_t{\mathsf{\Lambda }}^{N+2}u,{\partial }_t^{1-p}{\mathsf{\Lambda }}^{N+1-m}u)\right.\hfill \\ & +({\partial }_t^p{\mathsf{\Lambda }}^{m}b·{\partial }_t{\mathsf{\Lambda }}^{N+2}u,{\partial }_t^{1-p}{\mathsf{\Lambda }}^{N+1-m}b)-({\partial }_t^p{\mathsf{\Lambda }}^{m}u·{\partial }_t{\mathsf{\Lambda }}^{N+2}b,{\partial }_t^{1-p}{\mathsf{\Lambda }}^{N+1-m}b)\hfill \\ \hfill & +({\partial }_t^p{\mathsf{\Lambda }}^{m}b·{\partial }_t{\mathsf{\Lambda }}^{N+2}b,{\partial }_t^{1-p}{\mathsf{\Lambda }}^{N+1-m}u)+({\partial }_t^p{\mathsf{\Lambda }}^{m}b·{\partial }_t{\mathsf{\Lambda }}^{N+2}b,{\partial }_t^{1-p}{\mathsf{\Lambda }}^{N+2-m}b)\hfill \\ \hfill & \left.+({\partial }_t^p{\mathsf{\Lambda }}^{m}u·{\partial }_t{\mathsf{\Lambda }}^{N+2}b,{\partial }_t^{1-p}{\mathsf{\Lambda }}^{N+3-m}b)\right]\hfill \\ \hfill =& I_1+I_2+I_3+I_4+I_5+I_6.\hfill \end{array}$$

Note that

$$\begin{array}{cc}\hfill I_1\le & \sum _{m=0}^{N+1}{C}_{N+1}^m\left[|({\mathsf{\Lambda }}^{m}u·{\partial }_t{\mathsf{\Lambda }}^{N+2}u,{\partial }_t{\mathsf{\Lambda }}^{N+1-m}u)|+|({\partial }_t{\mathsf{\Lambda }}^{m}u·{\partial }_t{\mathsf{\Lambda }}^{N+2}u,{\mathsf{\Lambda }}^{N+1-m}u)|\right]\hfill \\ \hfill \le & {C\parallel u\parallel }_{L^3}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+1}{u\parallel }_{L^6}+{C\parallel \mathsf{\Lambda }u\parallel }_{L^3}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}{\parallel {\partial }_t{\mathsf{\Lambda }}^{N}u\parallel }_{L^6}\hfill \\ & +C\sum _{m=2}^{N+1}\parallel {\mathsf{\Lambda }}^m{u\parallel }_{L^3}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}{\parallel {\partial }_t{\mathsf{\Lambda }}^{N+1-m}u\parallel }_{L^6}\hfill \\ & +C\sum _{m=0}^{N-1}\parallel {\partial }_t{\mathsf{\Lambda }}^m{u\parallel }_{L^6}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}{\parallel {\mathsf{\Lambda }}^{N+1-m}u\parallel }_{L^3}\hfill \\ & +C\parallel {\partial }_t{\mathsf{\Lambda }}^N{u\parallel }_{L^6}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}{\parallel \mathsf{\Lambda }u\parallel }_{L^3}+C\parallel {\partial }_t{\mathsf{\Lambda }}^{N+1}{u\parallel }_{L^6}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}{\parallel u\parallel }_{L^3}\hfill \\ \hfill \le & C\parallel {\mathsf{\Lambda }}^{\frac{1}{2}}{u\parallel }_{L^2}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}^2+C\sum _{m=2}^{N+1}\parallel {\mathsf{\Lambda }}^{m+\frac{1}{2}}{u\parallel }_{L^2}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}{\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2-m}u\parallel }_{L^2}\hfill \\ & +C\parallel {\mathsf{\Lambda }}^{\frac{3}{2}}{u\parallel }_{L^2}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}^{\frac{3}{2}}\parallel {\partial }_t{\mathsf{\Lambda }}^N{u\parallel }_{L^2}^{\frac{1}{2}}+C\sum _{m=0}^{N-1}\parallel {\partial }_t{\mathsf{\Lambda }}^{m+1}{u\parallel }_{L^2}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}{\parallel {\mathsf{\Lambda }}^{N+\frac{3}{2}-m}u\parallel }_{L^2}\hfill \\ \hfill \le & \frac{1}{8}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}^2+C\parallel {\mathsf{\Lambda }}^{\frac{3}{2}}{u\parallel }_{L^2}^2\parallel {\partial }_t{\mathsf{\Lambda }}^N{u\parallel }_{L^2}^2+C\sum _{m=2}^{N+1}\parallel {\mathsf{\Lambda }}^{m+\frac{1}{2}}{u\parallel }_{L^2}^2{\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2-m}u\parallel }_{L^2}^2\hfill \\ & +C\sum _{m=0}^{N-1}\parallel {\partial }_t{\mathsf{\Lambda }}^{m+1}{u\parallel }_{L^2}^2{\parallel {\mathsf{\Lambda }}^{N+\frac{3}{2}-m}u\parallel }_{L^2}^2,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}\hfill I_2\le & \frac{1}{8}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}^2+C\parallel {\mathsf{\Lambda }}^{\frac{3}{2}}{b\parallel }_{L^2}^2\parallel {\partial }_t{\mathsf{\Lambda }}^N{b\parallel }_{L^2}^2+C\sum _{m=2}^{N+1}\parallel {\mathsf{\Lambda }}^{m+\frac{1}{2}}{b\parallel }_{L^2}^2{\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2-m}b\parallel }_{L^2}^2\hfill \\ & +C\sum _{m=0}^{N-1}\parallel {\partial }_t{\mathsf{\Lambda }}^{m+1}{b\parallel }_{L^2}^2{\parallel {\mathsf{\Lambda }}^{N+\frac{3}{2}-m}b\parallel }_{L^2}^2,\mathrm{for}\mathrm{large}t,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & I_3+I_4\hfill \\ \hfill \le & \frac{1}{8}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}^2+C(\parallel {\mathsf{\Lambda }}^{\frac{3}{2}}{u\parallel }_{L^2}^2+\parallel {\mathsf{\Lambda }}^{\frac{3}{2}}{b\parallel }_{L^2}^2\left)\right(\parallel {\partial }_t{\mathsf{\Lambda }}^N{u\parallel }_{L^2}^2+\parallel {\partial }_t{\mathsf{\Lambda }}^{N}b{\parallel }_{L^2}^2)\hfill \\ & +C\sum _{m=2}^{N+1}(\parallel {\mathsf{\Lambda }}^{m+\frac{1}{2}}{u\parallel }_{L^2}^2+\parallel {\mathsf{\Lambda }}^{m+\frac{1}{2}}{b\parallel }_{L^2}^2\left)\right(\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2-m}{u\parallel }_{L^2}^2+\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2-m}b{\parallel }_{L^2}^2)\hfill \\ & +C\sum _{m=0}^{N-1}(\parallel {\partial }_t{\mathsf{\Lambda }}^{m+1}{u\parallel }_{L^2}^2+\parallel {\partial }_t{\mathsf{\Lambda }}^{m+1}{b\parallel }_{L^2}^2\left)\right(\parallel {\mathsf{\Lambda }}^{N+\frac{3}{2}-m}{u\parallel }_{L^2}^2+\parallel {\mathsf{\Lambda }}^{N+\frac{3}{2}-m}b{\parallel }_{L^2}^2),\mathrm{for}\mathrm{large}t,\hfill \end{array}$$

The following estimate also holds:

$$\begin{array}{cc}\hfill I_5\le & \sum _{m=0}^{N+1}{C}_{N+1}^m\left[|({\mathsf{\Lambda }}^{m}b·{\partial }_t{\mathsf{\Lambda }}^{N+2}b,{\partial }_t{\mathsf{\Lambda }}^{N+2-m}b)|+|({\partial }_t{\mathsf{\Lambda }}^{m}b·{\partial }_t{\mathsf{\Lambda }}^{N+2}b,{\mathsf{\Lambda }}^{N+2-m}b)|\right]\hfill \\ \hfill \le & {\parallel b\parallel }_{L^{\infty }}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}^2+{\parallel \mathsf{\Lambda }b\parallel }_{L^3}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+1}{b\parallel }_{L^6}+\parallel {\mathsf{\Lambda }}^2{b\parallel }_{L^2}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}{\parallel {\partial }_t{\mathsf{\Lambda }}^{N}b\parallel }_{L^6}\hfill \\ \hfill & +\sum _{m=3}^{N+1}\parallel {\mathsf{\Lambda }}^m{b\parallel }_{L^3}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2-m}{b\parallel }_{L^6}+\sum _{m=0}^{N-1}\parallel {\partial }_t{\mathsf{\Lambda }}^m{b\parallel }_{L^6}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}{\parallel {\mathsf{\Lambda }}^{N+2-m}b\parallel }_{L^3}\hfill \\ & +\parallel {\partial }_t{\mathsf{\Lambda }}^N{b\parallel }_{L^6}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}\parallel {\mathsf{\Lambda }}^2{b\parallel }_{L^3}+\parallel {\partial }_t{\mathsf{\Lambda }}^{N+1}{b\parallel }_{L^6}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}{\parallel \mathsf{\Lambda }b\parallel }_{L^3}\hfill \\ \hfill \le & \frac{1}{16}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}^2+\parallel {\mathsf{\Lambda }}^{\frac{5}{2}}{b\parallel }_{L^2}^2\parallel {\partial }_t{\mathsf{\Lambda }}^N{b\parallel }_{L^2}^2+\sum _{m=3}^{N+1}\parallel {\mathsf{\Lambda }}^{m+\frac{1}{2}}{b\parallel }_{L^2}^2{\parallel {\partial }_t{\mathsf{\Lambda }}^{N+3-m}b\parallel }_{L^2}^2\hfill \\ & +\sum _{m=0}^{N-1}\parallel {\partial }_t{\mathsf{\Lambda }}^{m+1}{b\parallel }_{L^2}^2{\parallel {\mathsf{\Lambda }}^{N+\frac{5}{2}-m}b\parallel }_{L^2}^2,\mathrm{for}\mathrm{large}t,\hfill \end{array}$$

Moreover, $I_6$ satisfies

$$\begin{array}{cc}\hfill I_6=& \sum _{m=0}^{N+1}{C}_{N+1}^m\left[|({\mathsf{\Lambda }}^{m}u·{\partial }_t{\mathsf{\Lambda }}^{N+2}b,{\partial }_t{\mathsf{\Lambda }}^{N+3-m}b)|+|({\partial }_t{\mathsf{\Lambda }}^{m}u·{\partial }_t{\mathsf{\Lambda }}^{N+2}b,{\mathsf{\Lambda }}^{N+3-m}b)|\right]\hfill \\ \hfill \le & C[(u·\nabla {\mathsf{\Lambda }}^{N+2}{b}_t,{\mathsf{\Lambda }}^{N+2}{b}_t)+\parallel \mathsf{\Lambda }u{\parallel }_{L^{\infty }}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}b{\parallel }_{L^2}^2+\parallel {\mathsf{\Lambda }}^{2}u{\parallel }_{L^3}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}b{\parallel }_{L^2}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+1}b{\parallel }_{L^6}\hfill \\ \hfill & +\parallel {\mathsf{\Lambda }}^3{u\parallel }_{L^2}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}\parallel {\partial }_t{\mathsf{\Lambda }}^N{b\parallel }_{L^{\infty }}+\sum _{m=4}^{N+1}\parallel {\mathsf{\Lambda }}^m{u\parallel }_{L^3}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}{\parallel {\partial }_t{\mathsf{\Lambda }}^{N+3-m}b\parallel }_{L^6}\hfill \\ \hfill & +\parallel {\partial }_t{\mathsf{\Lambda }}^{N+1}{u\parallel }_{L^6}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}\parallel {\mathsf{\Lambda }}^2{b\parallel }_{L^3}+\parallel {\partial }_t{\mathsf{\Lambda }}^N{u\parallel }_{L^6}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}{\parallel {\mathsf{\Lambda }}^{3}b\parallel }_{L^3}\hfill \\ \hfill & +\sum _{m=0}^{N-1}\parallel {\partial }_t{\mathsf{\Lambda }}^m{u\parallel }_{L^6}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}\parallel {\mathsf{\Lambda }}^{N+2-m}b{\parallel }_{L^3}]\hfill \\ \hfill \le & \frac{1}{16}\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}^2+\parallel {\mathsf{\Lambda }}^{\frac{7}{2}}{b\parallel }_{L^2}^2\parallel {\partial }_t{\mathsf{\Lambda }}^N{b\parallel }_{L^2}^2+\parallel {\mathsf{\Lambda }}^3{u\parallel }_{L^2}^2{\parallel {\partial }_t{\mathsf{\Lambda }}^{N+1}b\parallel }_{L^2}^2\hfill \\ \hfill & +\sum _{m=4}^{N+1}\parallel {\mathsf{\Lambda }}^{m+\frac{1}{2}}{u\parallel }_{L^2}^2\parallel {\partial }_t{\mathsf{\Lambda }}^{N+4-m}{b\parallel }_{L^2}^2+\sum _{m=0}^{N-1}\parallel {\partial }_t{\mathsf{\Lambda }}^{m+1}{u\parallel }_{L^2}^2{\parallel {\mathsf{\Lambda }}^{N+\frac{5}{2}-m}b\parallel }_{L^2}^2,\mathrm{for}\mathrm{large}t,\hfill \end{array}$$

where

$$\parallel {\partial }_t{\mathsf{\Lambda }}^{N+1}{v\parallel }_{L^2}^2\le \epsilon \parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{v\parallel }_{L^2}^2+C_{\epsilon }{\parallel {\partial }_t{\mathsf{\Lambda }}^{N}v\parallel }_{L^2}^2.$$

has been used. Note that ${\parallel v\parallel }_{L^{\infty }}\le {C\parallel \mathsf{\Lambda }v\parallel }_{L^2}^{\frac{1}{2}}{\parallel {\mathsf{\Lambda }}^{2}v\parallel }_{L^2}^{\frac{1}{2}}$. By using the previous decay results, it yields that

$$\begin{array}{cc}\hfill & \frac{d}{dt}(\parallel {\partial }_t{\mathsf{\Lambda }}^{N+1}{u\parallel }_{L^2}^2+\parallel {\partial }_t{\mathsf{\Lambda }}^{N+1}{b\parallel }_{L^2}^2+\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}^2)+\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}^2+{\parallel {\partial }_t{\mathsf{\Lambda }}^{N+2}b\parallel }_{L^2}^2\hfill \\ \hfill & \le C{(1+t)}^{-min\{2r^{*}+\frac{15}{2}+N,\frac{19}{2}+N\}},\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(53)

Applying the Plancherel’s theorem to (53) gives

$$\begin{array}{cc}& \frac{d}{dt}{\int }_{{\mathbb{R}}^3}\left[|\widehat{{\partial }_t{\mathsf{\Lambda }}^{N+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t{\mathsf{\Lambda }}^{N+1}b}{(\xi ,t)|}^2\right]d\xi \hfill \\ & +{\int }_{{\mathbb{R}}^3}\frac{{|\xi |}^2}{1+{|\xi |}^2}\left[|\widehat{{\partial }_t{\mathsf{\Lambda }}^{N+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t{\mathsf{\Lambda }}^{N+1}b}{(\xi ,t)|}^2\right]d\xi \hfill \\ \hfill \le & C{(1+t)}^{-min\{2r^{*}+\frac{15}{2}+N,\frac{19}{2}+N\}},\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(54)

Multiplying (54) by $g\left(t\right)$, it yields

$$\begin{array}{cc}& \frac{d}{dt}\left\{g\left(t\right){\int }_{{\mathbb{R}}^3}\left[|\widehat{{\partial }_t{\mathsf{\Lambda }}^{N+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t{\mathsf{\Lambda }}^{N+1}b}{(\xi ,t)|}^2\right]d\xi \right\}\hfill \\ \hfill \le & g^{\prime }\left(t\right){\int }_{B\left(t\right)}\left[|\widehat{{\partial }_t{\mathsf{\Lambda }}^{N+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t{\mathsf{\Lambda }}^{N+1}b}{(\xi ,t)|}^2\right]d\xi \hfill \\ \hfill & +Cg\left(t\right){(1+t)}^{-min\{2r^{*}+\frac{15}{2}+N,\frac{19}{2}+N\}},\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

Hence

$$\begin{array}{cc}\hfill & g\left(t\right){\int }_{{\mathbb{R}}^3}\left[|\widehat{{\partial }_t{\mathsf{\Lambda }}^{N+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t{\mathsf{\Lambda }}^{N+1}b}{(\xi ,t)|}^2\right]d\xi \hfill \\ \hfill \le & C+{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}\left[|\widehat{{\partial }_t{\mathsf{\Lambda }}^{N+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t{\mathsf{\Lambda }}^{N+1}b}{(\xi ,t)|}^2\right]d\xi ds\hfill \\ \hfill & +C{\int }_0^{t}g\left(s\right){(1+s)}^{-min\{2r^{*}+\frac{15}{2}+N,\frac{19}{2}+N\}}ds\hfill \\ \hfill \le & C+{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^{2N+2}\left[|\widehat{{\partial }_{t}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_{t}b}{(\xi ,t)|}^2\right]d\xi ds\hfill \\ \hfill & +C{\int }_0^{t}g\left(s\right){(1+s)}^{-min\{2r^{*}+\frac{15}{2}+N,\frac{19}{2}+N\}}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(55)

It then follows from (36), (38) and (55) that

$$\begin{array}{cc}\hfill & g\left(t\right){\int }_{{\mathbb{R}}^3}\left[|\widehat{{\partial }_t{\mathsf{\Lambda }}^{N+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t{\mathsf{\Lambda }}^{N+1}b}{(\xi ,t)|}^2\right]d\xi \hfill \\ \hfill \le & C+{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^{2N+6}\left[e^{-{2|\xi |}^{2}t}|{\widehat{u}}_0{\left(\xi \right)|}^2{+(1+|\xi |}^2)e^{-2\frac{{|\xi |}^2}{1+{|\xi |}^2}t}{|{\widehat{b}}_0\left(\xi \right)|}^2\right]d\xi ds\hfill \\ \hfill & +C{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^{2N+4}\left({\parallel u\left(s\right)\parallel }_{L^2}^4+{\parallel b\left(s\right)\parallel }_{L^2}^4+{\parallel u\left(s\right)\parallel }_{L^2}^2{\parallel b\left(s\right)\parallel }_{L^2}^2\right)d\xi ds\hfill \\ \hfill & +C{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^{2N+8}{(1+|\xi |}^2+{|\xi |}^4){\left[{\int }_0^t{(\parallel u\left(\tau \right)\parallel }_{L^2}^2+{\parallel b\left(\tau \right)\parallel }_{L^2}^2)d\tau \right]}^{2}d\xi ds\hfill \\ \hfill & +C{\int }_0^{t}g\left(s\right){(1+s)}^{-min\{2r^{*}+\frac{15}{2}+N,\frac{19}{2}+N\}}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(56)

Consider the first term of the right hand side of (56):

$$\begin{array}{cc}\hfill & {\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^{2N+6}\left[e^{-{2|\xi |}^{2}t}|{\widehat{u}}_0{\left(\xi \right)|}^2{+(1+|\xi |}^2)e^{-2\frac{{|\xi |}^2}{1+{|\xi |}^2}t}{|{\widehat{b}}_0\left(\xi \right)|}^2\right]d\xi ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-min\{r^{*}+\frac{9}{2}+N,\frac{11}{2}+N\}}ds.\hfill \end{array}$$

(57)

For the second term, after integrating in polar coordinates in $B\left(t\right)$,

$$\begin{array}{cc}\hfill & C{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^{2N+4}\left({\parallel u\left(s\right)\parallel }_{L^2}^4+{\parallel b\left(s\right)\parallel }_{L^2}^4+{\parallel u\left(s\right)\parallel }_{L^2}^2{\parallel b\left(s\right)\parallel }_{L^2}^2\right)d\xi ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-\frac{7}{2}-N}{(1+s)}^{-2min\{r^{*}+\frac{3}{2},\frac{5}{2}\}}ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-min\{2r^{*}+\frac{13}{2}+N,\frac{17}{2}+N\}}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(58)

In addition, if $r^{*}+\frac{3}{2}<\frac{5}{2}$, the third term of the right hand side of (56) can be estimated as

$$\begin{array}{cc}\hfill & C{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^{2N+8}{(1+|\xi |}^2+{|\xi |}^4){\left[{\int }_0^t{(\parallel u\left(\tau \right)\parallel }_{L^2}^2+{\parallel b\left(\tau \right)\parallel }_{L^2}^2)d\tau \right]}^{2}d\xi ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-\frac{11}{2}-N}{(1+s)}^{-2r^{*}-1}ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-(2r^{*}+\frac{13}{2}+N)}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(59)

If $r^{*}+\frac{3}{2}\ge \frac{5}{2}$, then, the third term of the right hand side of (56) satisfies

$$\begin{array}{cc}\hfill & C{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^{2N+8}{(1+|\xi |}^2+{|\xi |}^4){\left[{\int }_0^t{(\parallel u\left(\tau \right)\parallel }_{L^2}^2+{\parallel b\left(\tau \right)\parallel }_{L^2}^2)d\tau \right]}^{2}d\xi ds\hfill \\ & \le C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-(N+\frac{11}{2})}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(60)

The last term satisfies

$$\begin{array}{cc}& C{\int }_0^{t}g\left(s\right){(1+s)}^{-min\{2r^{*}+\frac{15}{2}+N,\frac{19}{2}+N\}}ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-min\{2r^{*}+\frac{13}{2}+N,\frac{17}{2}+N\}}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(61)

For a fixed $r^{*}$, we choose $g\left(t\right)={(1+t)}^m$, for some $m>max\{r^{*}+\frac{9}{2}+N,\frac{11}{2}+N\}$. Then $\rho \left(t\right)=C{(1+t)}^{-\frac{1}{2}}$ and from (56)–(61), it yields that

$$\begin{array}{cc}& \parallel {\partial }_t{\mathsf{\Lambda }}^{N+1}{u\parallel }_{L^2}^2+\parallel {\partial }_t{\mathsf{\Lambda }}^{N+1}{b\parallel }_{L^2}^2+{\parallel {\partial }_t{\mathsf{\Lambda }}^{N+1}b\parallel }_{L^2}^2\hfill \\ \hfill & \le C{(1+t)}^{-min\{r^{*}+\frac{9}{2}+N,\frac{11}{2}+N\}},\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(62)

Through mathematical induction, one concludes that for $P=1$ and $0<M+\frac{5}{2}P<K$,

$$\begin{array}{c}\hfill \parallel {\partial }_t{\mathsf{\Lambda }}^M{u\parallel }_{L^2}^2+\parallel {\partial }_t{\mathsf{\Lambda }}^M{b\parallel }_{L^2}^2+{\parallel {\partial }_t{\mathsf{\Lambda }}^{M}b\parallel }_{L^2}^2\le C{(1+t)}^{-min\{r^{*}+\frac{7}{2}+M,\frac{9}{2}+M\}},\mathrm{for}\mathrm{large}t.\end{array}$$

(63)

Hence, the proof is complete. □

Now, suppose that Theorem 2 holds for $p\le P_0\in {\mathbb{N}}^{+}$, and prove it also holds for $p=P_0+1$.

Lemma 13.

Let $m\in \mathbb{N}$, $K=max\{5+2P_0,m+2P_0+2\}$ and $r^{*}=r^{*}\left(u_0\right)=r^{*}\left(b_0\right)\in (-\frac{3}{2},+\infty )$ be the decay character. Suppose that $(u_0,b_0)\in H^K\left({\mathbb{R}}^3\right)\times H^{K+1}\left({\mathbb{R}}^3\right)$ with $\nabla ·u_0=\nabla ·b_0=0$. Then, there exists a positive constant $C=C(,\parallel u_0{\parallel }_{H^K},\parallel b_0{\parallel }_{H^{K+1}})$, such that

$$\begin{array}{cc}& \parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^m{u\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^m{b\parallel }_{L^2}^2+{\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{m+1}b\parallel }_{L^2}^2\hfill \\ & \le C{(1+t)}^{-min\{\frac{3}{2}+r^{*}+m+2\left(P_0+1\right),\frac{5}{2}+m+2\left(P_0+1\right)\}},\mathit{for}\mathit{large}t.\hfill \end{array}$$

(64)

 Proof. 

First, assume that $m=0$. Applying ${\partial }_t^{P_0+1}$ to (10) and (11), multiplying both side by ${\partial }_t^{P_0+1}u$ and ${\partial }_t^{P_0+1}b$, respectively, integrating over ${\mathbb{R}}^3$, it yields that

$$\begin{array}{cc}\hfill & \frac{1}{2}\frac{d}{dt}(\parallel {\partial }_t^{P_0+1}{u\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1}{b\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }b\parallel }_{L^2}^2)+\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }u\parallel }_{L^2}^2+{\parallel {\partial }_t^{P_0+1}\mathsf{\Lambda }b\parallel }_{L^2}^2\hfill \\ \hfill \le & \sum _{p=0}^{P_0+1}\left[|({\partial }_t^{p}u·{\partial }_t^{P_0+1}\mathsf{\Lambda }u,{\partial }_t^{P_0+1-p}u)|+|({\partial }_t^{p}b·{\partial }_t^{P_0+1}\mathsf{\Lambda }u,{\partial }_t^{P_0+1-p}b)|\right.\hfill \\ & \left.+|({\partial }_t^{p}u·{\partial }_t^{P_0+1}\mathsf{\Lambda }b,{\partial }_t^{P_0+1-p}b)|+|({\partial }_t^{p}b·{\partial }_t^{P_0+1}\mathsf{\Lambda }b,{\partial }_t^{P_0+1-p}u)|+|({\partial }_t^{p}b·{\partial }_t^{P_0+1}\mathsf{\Lambda }b,{\partial }_t^{P_0+1-p}\mathsf{\Lambda }b)|\right]\hfill \\ \hfill & +|(u·{\partial }_t^{P_0+1}\mathsf{\Lambda }b,{\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{2}b)|+\sum _{p=1}^{P_0+1}|({\partial }_t^{p}u·{\partial }_t^{P_0+1}\mathsf{\Lambda }b,{\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{2}b)|.\hfill \end{array}$$

A simple calculation shows that

$$\begin{array}{cc}\hfill & \frac{1}{2}\frac{d}{dt}(\parallel {\partial }_t^{P_0+1}{u\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1}{b\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }b\parallel }_{L^2}^2)+\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }u\parallel }_{L^2}^2+{\parallel {\partial }_t^{P_0+1}\mathsf{\Lambda }b\parallel }_{L^2}^2\hfill \\ \hfill \le & {2\parallel u\parallel }_{L^3}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }u\parallel }_{L^2}\parallel {\partial }_t^{P_0+1}{u\parallel }_{L^6}+\sum _{p=1}^{P_0}\parallel {\partial }_t^p{u\parallel }_{L^6}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }u\parallel }_{L^2}{\parallel {\partial }_t^{P_0+1-p}u\parallel }_{L^3}\hfill \\ \hfill & +{2\parallel b\parallel }_{L^3}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }u\parallel }_{L^2}\parallel {\partial }_t^{P_0+1}{b\parallel }_{L^6}+\sum _{p=1}^{P_0}\parallel {\partial }_t^p{b\parallel }_{L^6}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }u\parallel }_{L^2}{\parallel {\partial }_t^{P_0+1-p}b\parallel }_{L^3}\hfill \\ \hfill & +{2\parallel u\parallel }_{L^3}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }b\parallel }_{L^2}\parallel {\partial }_t^{P_0+1}{b\parallel }_{L^6}+\sum _{p=1}^{P_0}\parallel {\partial }_t^p{u\parallel }_{L^6}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }b\parallel }_{L^2}{\parallel {\partial }_t^{P_0+1-p}b\parallel }_{L^3}\hfill \\ & +{2\parallel b\parallel }_{L^3}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }b\parallel }_{L^2}\parallel {\partial }_t^{P_0+1}{u\parallel }_{L^6}+\sum _{p=1}^{P_0}\parallel {\partial }_t^p{b\parallel }_{L^6}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }b\parallel }_{L^2}{\parallel {\partial }_t^{P_0+1-p}u\parallel }_{L^3}\hfill \\ \hfill & +{\parallel b\parallel }_{L^{\infty }}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }b\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1}{b\parallel }_{L^6}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }b\parallel }_{L^2}{\parallel \mathsf{\Lambda }b\parallel }_{L^3}\hfill \\ \hfill & +\sum _{p=1}^{P_0}\parallel {\partial }_t^p{b\parallel }_{L^6}\parallel {\partial }_t^{P_0+1}{b\parallel }_{L^2}\parallel {\partial }_t^{P_0+1-p}{\mathsf{\Lambda }b\parallel }_{L^3}+{\parallel \mathsf{\Lambda }u\parallel }_{L^{\infty }}{\parallel {\partial }_t^{P_0+1}\mathsf{\Lambda }b\parallel }_{L^2}^2\hfill \\ & +\parallel {\mathsf{\Lambda }}^2{b\parallel }_{L^3}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }b\parallel }_{L^2}\parallel {\partial }_t^{P_0+1}{u\parallel }_{L^6}+\sum _{p=1}^{P_0}\parallel {\partial }_t^p{u\parallel }_{L^6}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^b{\parallel }_{L^2}{\parallel {\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{2}b\parallel }_{L^3}\hfill \\ \hfill \le & \frac{1}{2}(\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }u\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }b\parallel }_{L^2}^2)+C\sum _{p=1}^{P_0}(\parallel {\partial }_t^p{\mathsf{\Lambda }u\parallel }_{L^2}^2+\parallel {\partial }_t^p{\mathsf{\Lambda }b\parallel }_{L^2}^2\left)\right(\parallel {\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{\frac{1}{2}}{u\parallel }_{L^2}^2\hfill \\ \hfill & +\parallel {\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{\frac{1}{2}}{b\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{\frac{3}{2}}{b\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{\frac{5}{2}}b{\parallel }_{L^2}^2),\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

Applying the previous decay estimates, it yields that

$$\begin{array}{cc}\hfill & \frac{d}{dt}(\parallel {\partial }_t^{P_0+1}{u\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1}{b\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }b\parallel }_{L^2}^2)+\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }u\parallel }_{L^2}^2+{\parallel {\partial }_t^{P_0+1}\mathsf{\Lambda }b\parallel }_{L^2}^2\hfill \\ \hfill \le & C\sum _{p=1}^{P_0}(\parallel {\partial }_t^p{\mathsf{\Lambda }u\parallel }_{L^2}^2+\parallel {\partial }_t^p{\mathsf{\Lambda }b\parallel }_{L^2}^2\left)\right(\parallel {\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{\frac{1}{2}}{u\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{\frac{1}{2}}{b\parallel }_{L^2}^2+{\parallel {\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{\frac{3}{2}}b\parallel }_{L^2}^2\hfill \\ \hfill & +\parallel {\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{\frac{5}{2}}b{\parallel }_{L^2}^2)\hfill \\ \hfill \le & C{(1+t)}^{-min\{2r^{*}+\frac{13}{2}+2P_0,\frac{17}{2}+2P_0\}},\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(65)

Using Plancherel’s theorem to (65) gives

$$\begin{array}{cc}& \frac{d}{dt}{\int }_{{\mathbb{R}}^3}\left[|\widehat{{\partial }_t^{P_0+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t^{P_0+1}b}{(\xi ,t)|}^2\right]d\xi \hfill \\ & +{\int }_{{\mathbb{R}}^3}\frac{{|\xi |}^2}{1+{|\xi |}^2}\left[|\widehat{{\partial }_t^{P_0+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t^{P_0+1}b}{(\xi ,t)|}^2\right]d\xi \hfill \\ \hfill \le & C{(1+t)}^{-min\{2r^{*}+\frac{13}{2}+2P_0,\frac{17}{2}+2P_0\}},\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(66)

Multiplying (66) by $g\left(t\right)$ gives

$$\begin{array}{cc}& \frac{d}{dt}\left\{g\left(t\right){\int }_{{\mathbb{R}}^3}\left[|\widehat{{\partial }_t^{P_0+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t^{P_0+1}b}{(\xi ,t)|}^2\right]d\xi \right\}\hfill \\ \hfill \le & g^{\prime }\left(t\right){\int }_{B\left(t\right)}\left[|\widehat{{\partial }_t^{P_0+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t^{P_0+1}b}{(\xi ,t)|}^2\right]d\xi \hfill \\ \hfill & +Cg\left(t\right){(1+t)}^{-min\{2r^{*}+\frac{13}{2}+2P_0,\frac{17}{2}+2P_0\}},\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

Hence

$$\begin{array}{cc}\hfill & g\left(t\right){\int }_{{\mathbb{R}}^3}\left[|\widehat{{\partial }_t^{P_0+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t^{P_0+1}b}{(\xi ,t)|}^2\right]d\xi \hfill \\ \hfill \le & C+{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}\left[|\widehat{{\partial }_t^{P_0+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t^{P_0+1}b}{(\xi ,t)|}^2\right]d\xi ds\hfill \\ \hfill & +C{\int }_0^{t}g\left(s\right){(1+s)}^{-min\{2r^{*}+\frac{13}{2}+2P_0,\frac{17}{2}+2P_0\}}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(67)

It then follows from (36), (38) and (67) that

$$\begin{array}{cc}\hfill & g\left(t\right){\int }_{{\mathbb{R}}^3}\left[|\widehat{{\partial }_t^{P_0+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t^{P_0+1}b}{(\xi ,t)|}^2\right]d\xi \hfill \\ \hfill \le & C+{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^{4P_0+4}\left[e^{-{2|\xi |}^{2}t}|{\widehat{u}}_0{\left(\xi \right)|}^2{+(1+|\xi |}^2)e^{-2\frac{{|\xi |}^2}{1+{|\xi |}^2}t}{|{\widehat{b}}_0\left(\xi \right)|}^2\right]d\xi ds\hfill \\ \hfill & +C{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}\sum _{p=0}^{P_0}\sum _{l=0}^p{|\xi |}^{4P_0+2-4p}(\parallel {\partial }_t^l{u\parallel }_{L^2}^2+\parallel {\partial }_t^l{b\parallel }_{L^2}^2\left)\right(\parallel {\partial }_t^{p-l}{u\parallel }_{L^2}^2+\parallel {\partial }_t^{p-l}b{\parallel }_{L^2}^2)d\xi ds\hfill \\ \hfill & +C{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^{4P_0+6}{(1+|\xi |}^2+{|\xi |}^4){\left[{\int }_0^t{(\parallel u\left(\tau \right)\parallel }_{L^2}^2+{\parallel b\left(\tau \right)\parallel }_{L^2}^2)d\tau \right]}^{2}d\xi ds\hfill \\ \hfill & +C{\int }_0^{t}g\left(s\right){(1+s)}^{-min\{2r^{*}+\frac{13}{2}+2P_0,\frac{17}{2}+2P_0\}}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(68)

For the first term of the right hand side of (68),

$$\begin{array}{cc}\hfill & {\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^{4P_0+4}\left[e^{-{2|\xi |}^{2}t}|{\widehat{u}}_0{\left(\xi \right)|}^2{+(1+|\xi |}^2)e^{-2\frac{{|\xi |}^2}{1+{|\xi |}^2}t}{|{\widehat{b}}_0\left(\xi \right)|}^2\right]d\xi d\xi ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-(r^{*}+\frac{7}{2}+2P_0)}ds.\hfill \end{array}$$

(69)

For the second term, after integrating in polar coordinates in $B\left(t\right)$,

$$\begin{array}{cc}\hfill & C{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}\sum _{p=0}^{P_0}\sum _{l=0}^p{|\xi |}^{4P_0+2-4p}(\parallel {\partial }_t^l{u\parallel }_{L^2}^2+\parallel {\partial }_t^l{b\parallel }_{L^2}^2\left)\right(\parallel {\partial }_t^{p-l}{u\parallel }_{L^2}^2+\parallel {\partial }_t^{p-l}b{\parallel }_{L^2}^2)d\xi ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-(2P_0-2p+\frac{5}{2})}{(1+s)}^{-min\{r^{*}+\frac{3}{2}+2l,\frac{5}{2}+2l\}}{(1+s)}^{-min\{r^{*}+\frac{3}{2}+2p-2l,\frac{5}{2}+2p-2l\}}ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-min\{2r^{*}+\frac{11}{2}+2P_0,\frac{15}{2}+2P_0\}}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(70)

In addition, if $r^{*}+\frac{3}{2}<\frac{5}{2}$, the third term of the right hand side of (68) can be estimated as

$$\begin{array}{cc}\hfill & C{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^{4P_0+6}{(1+|\xi |}^2+{|\xi |}^4){\left[{\int }_0^t{(\parallel u\left(\tau \right)\parallel }_{L^2}^2+{\parallel b\left(\tau \right)\parallel }_{L^2}^2)d\tau \right]}^{2}d\xi ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-\frac{9}{2}-2P_0}{(1+s)}^{-2r^{*}-1}ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-(2r^{*}+\frac{11}{2}+2P_0)}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(71)

If $r^{*}+\frac{3}{2}\ge \frac{5}{2}$,

$$\begin{array}{cc}\hfill & C{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^{4P_0+6}{(1+|\xi |}^2+{|\xi |}^4){\left[{\int }_0^t{(\parallel u\left(\tau \right)\parallel }_{L^2}^2+{\parallel b\left(\tau \right)\parallel }_{L^2}^2)d\tau \right]}^{2}d\xi ds\hfill \\ & \le C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-(2P_0+\frac{9}{2})}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(72)

The last term satisfies

$$\begin{array}{cc}& C{\int }_0^{t}g\left(s\right){(1+s)}^{-min\{2r^{*}+\frac{13}{2}+2P_0,\frac{17}{2}+2P_0\}}ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-min\{2r^{*}+\frac{11}{2}+2P_0,\frac{15}{2}+2P_0\}}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(73)

For a fixed $r^{*}$, choose $g\left(t\right)={(1+t)}^m$, for some $m>max\{r^{*}+\frac{7}{2}+2P_0,\frac{9}{2}+2P_0\}$. Then $\rho \left(t\right)=C{(1+t)}^{-\frac{1}{2}}$ and from (68)–(73),

$$\begin{array}{c}\hfill \parallel {\partial }_t^{P_0+1}{u\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1}{b\parallel }_{L^2}^2+{\parallel {\partial }_t^{P_0+1}b\parallel }_{L^2}^2\le C{(1+t)}^{-min\{r^{*}+\frac{7}{2}+2P_0,\frac{9}{2}+2P_0\}},\mathrm{for}\mathrm{large}t.\end{array}$$

(74)

Suppose that Lemma 13 holds for $m\le N\in {\mathbb{N}}^{+}$, then one can prove it also holds for $m=N+1$. Applying ${\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}$ to (10) and (11), multiplying both sides by ${\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}u$ and ${\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}b$, respectively, integrating over ${\mathbb{R}}^3$, it yields that

$$\begin{array}{cc}\hfill & \frac{1}{2}\frac{d}{dt}(\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}{u\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}{b\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}b{\parallel }_{L^2}^2)\hfill \\ \hfill & +\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}^2+{\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}b\parallel }_{L^2}^2\hfill \\ \hfill \le & \sum _{p=0}^{P_0+1}\sum _{m=0}^{N+1}{C}_{P_0+1}^p{C}_{N+1}^m\left[|({\partial }_t^p{\mathsf{\Lambda }}^{m}u·{\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}u,{\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{N+1-m}u)|\right.\hfill \\ \hfill & +|({\partial }_t^p{\mathsf{\Lambda }}^{m}b·{\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}u,{\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{N+1-m}b)|+|({\partial }_t^p{\mathsf{\Lambda }}^{m}u·{\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}b,{\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{N+1-m}b)|\hfill \\ & +|({\partial }_t^p{\mathsf{\Lambda }}^{m}b·{\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}u,{\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{N+1-m}b)|+|({\partial }_t^p{\mathsf{\Lambda }}^{m}b·{\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}b,{\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{N+2-m}b)|\hfill \\ & \left.+|({\partial }_t^p{\mathsf{\Lambda }}^{m}u·{\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}b,{\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{N+3-m}b)|\right]\hfill \\ \hfill :=& I_1+I_2+I_3+I_4+I_5+I_6.\hfill \end{array}$$

First of all, $I_1$ satisfies

$$\begin{array}{cc}\hfill I_1\le & \sum _{m=0}^{N+1}{C}_{N+1}^m\parallel {\mathsf{\Lambda }}^m{u\parallel }_{L^3}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}{\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1-m}\parallel }_{L^6}\hfill \\ \hfill & +\sum _{m=0}^{N+1}{C}_{N+1}^m\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^m{u\parallel }_{L^3}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}{\parallel {\mathsf{\Lambda }}^{N+1-m}\parallel }_{L^6}\hfill \\ \hfill & +\sum _{p=1}^{P_0}\sum _{m=0}^{N+1}{C}_{P_0+1}^p{C}_{N+1}^m\parallel {\partial }_t^p{\mathsf{\Lambda }}^m{u\parallel }_{L^6}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}{\parallel {\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{N+1-m}\parallel }_{L^3}\hfill \\ \hfill \le & \frac{1}{16}{\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}u\parallel }_{L^2}^2\hfill \\ \hfill & +C\left[\sum _{m=0}^{N+1}\left(\parallel {\mathsf{\Lambda }}^{m+\frac{1}{2}}{u\parallel }_{L^2}^2\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2-m}{u\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{m+1}{u\parallel }_{L^2}^2{\parallel {\mathsf{\Lambda }}^{N+\frac{3}{2}-m}u\parallel }_{L^2}^2\right)\right]\hfill \\ \hfill & +C\left[\sum _{p=1}^{P_0}\sum _{m=0}^{N+1}\parallel {\partial }_t^p{\mathsf{\Lambda }}^{m+\frac{1}{2}}{u\parallel }_{L^2}^2{\parallel {\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{N+2-m}u\parallel }_{L^2}^2\right]\hfill \\ \hfill \le & \frac{1}{8}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}^2+C\sum _{m=1}^{N+1}\parallel {\mathsf{\Lambda }}^{m+\frac{1}{2}}{u\parallel }_{L^2}^2{\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2-m}u\parallel }_{L^2}^2\hfill \\ \hfill & +C\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^N{u\parallel }_{L^2}^2\parallel {\mathsf{\Lambda }}^{\frac{3}{2}}{u\parallel }_{L^2}^2+C\sum _{m=0}^{N-1}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{m+1}{u\parallel }_{L^2}^2{\parallel {\mathsf{\Lambda }}^{N+\frac{3}{2}-m}u\parallel }_{L^2}^2\hfill \\ \hfill & +C\left[\sum _{p=1}^{P_0}\sum _{m=0}^{N+1}\parallel {\partial }_t^p{\mathsf{\Lambda }}^{m+\frac{1}{2}}{u\parallel }_{L^2}^2{\parallel {\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{N+2-m}u\parallel }_{L^2}^2\right],\mathrm{for}\mathrm{large}t,\hfill \end{array}$$

where the following fact

$$\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}{u\parallel }_{L^2}^2\le \frac{1}{2}(\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N}u{\parallel }_{L^2}^2).$$

has been used. Similarly,

$$\begin{array}{cc}\hfill I_2\le & \frac{1}{8}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}^2+C\sum _{m=1}^{N+1}\parallel {\mathsf{\Lambda }}^{m+\frac{1}{2}}{b\parallel }_{L^2}^2{\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2-m}b\parallel }_{L^2}^2\hfill \\ \hfill & +C\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^N{b\parallel }_{L^2}^2\parallel {\mathsf{\Lambda }}^{\frac{3}{2}}{b\parallel }_{L^2}^2+C\sum _{m=0}^{N-1}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{m+1}{b\parallel }_{L^2}^2{\parallel {\mathsf{\Lambda }}^{N+\frac{3}{2}-m}b\parallel }_{L^2}^2\hfill \\ \hfill & +C\left[\sum _{p=1}^{P_0}\sum _{m=0}^{N+1}\parallel {\partial }_t^p{\mathsf{\Lambda }}^{m+\frac{1}{2}}{b\parallel }_{L^2}^2{\parallel {\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{N+2-m}b\parallel }_{L^2}^2\right],\mathrm{for}\mathrm{large}t,\hfill \end{array}$$

$$\begin{array}{cc}\hfill I_3\le & \frac{1}{8}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}^2+C\sum _{m=1}^{N+1}\parallel {\mathsf{\Lambda }}^{m+\frac{1}{2}}{u\parallel }_{L^2}^2{\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2-m}b\parallel }_{L^2}^2\hfill \\ \hfill & +C\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^N{u\parallel }_{L^2}^2\parallel {\mathsf{\Lambda }}^{\frac{3}{2}}{b\parallel }_{L^2}^2+C\sum _{m=0}^{N-1}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{m+1}{u\parallel }_{L^2}^2{\parallel {\mathsf{\Lambda }}^{N+\frac{3}{2}-m}b\parallel }_{L^2}^2\hfill \\ \hfill & +C\left[\sum _{p=1}^{P_0}\sum _{m=0}^{N+1}\parallel {\partial }_t^p{\mathsf{\Lambda }}^{m+\frac{1}{2}}{u\parallel }_{L^2}^2{\parallel {\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{N+2-m}b\parallel }_{L^2}^2\right],\mathrm{for}\mathrm{large}t,\hfill \end{array}$$

$$\begin{array}{cc}\hfill I_4\le & \frac{1}{8}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}^2+C\sum _{m=1}^{N+1}\parallel {\mathsf{\Lambda }}^{m+\frac{1}{2}}{b\parallel }_{L^2}^2{\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2-m}u\parallel }_{L^2}^2\hfill \\ \hfill & +C\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^N{b\parallel }_{L^2}^2\parallel {\mathsf{\Lambda }}^{\frac{3}{2}}{u\parallel }_{L^2}^2+C\sum _{m=0}^{N-1}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{m+1}{b\parallel }_{L^2}^2{\parallel {\mathsf{\Lambda }}^{N+\frac{3}{2}-m}u\parallel }_{L^2}^2\hfill \\ \hfill & +C\left[\sum _{p=1}^{P_0}\sum _{m=0}^{N+1}\parallel {\partial }_t^p{\mathsf{\Lambda }}^{m+\frac{1}{2}}{b\parallel }_{L^2}^2{\parallel {\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{N+2-m}u\parallel }_{L^2}^2\right],\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

For $I_5$,

$$\begin{array}{cc}\hfill I_5\le & {\parallel b\parallel }_{L^{\infty }}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}^2+\sum _{m=1}^{N+1}{C}_{N+1}^m\parallel {\mathsf{\Lambda }}^m{b\parallel }_{L^3}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}{\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1-m}b\parallel }_{L^6}\hfill \\ \hfill & +\parallel {\partial }_t^{P_0+1}{b\parallel }_{L^6}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}{\parallel \mathsf{\Lambda }b\parallel }_{L^3}+\sum _{m=0}^N\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^m{b\parallel }_{L^6}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}{\parallel {\mathsf{\Lambda }}^{N+2-m}b\parallel }_{L^3}\hfill \\ \hfill & +\sum _{p=1}^{P_0}\sum _{m=0}^{N+1}\parallel {\partial }_t^p{\mathsf{\Lambda }}^m{b\parallel }_{L^6}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}{\parallel {\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{N+2-m}b\parallel }_{L^3}\hfill \\ \hfill \le & \frac{1}{8}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}^2+C\sum _{m=2}^{N+1}\parallel {\mathsf{\Lambda }}^{m+\frac{1}{2}}{b\parallel }_{L^2}^2{\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}b\parallel }_{L^2}^2\hfill \\ \hfill & +\sum _{m=0}^{N-1}\parallel {\mathsf{\Lambda }}^{N+\frac{5}{2}-m}{b\parallel }_{L^2}^2\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{m+1}{b\parallel }_{L^2}^2+(\parallel {\mathsf{\Lambda }}^{\frac{3}{2}}{b\parallel }_{L^2}^2+\parallel {\mathsf{\Lambda }}^{\frac{5}{2}}{b\parallel }_{L^2}^2)\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^N{b\parallel }_{L^2}^2\hfill \\ \hfill & +\sum _{p=1}^{P_0}\sum _{m=0}^{N+1}\parallel {\partial }_t^p{\mathsf{\Lambda }}^{m+1}{b\parallel }_{L^2}^2{\parallel {\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{N+\frac{5}{2}-m}b\parallel }_{L^2}^2,\mathrm{for}\mathrm{large}t,\hfill \end{array}$$

where

$$\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}{b\parallel }_{L^2}^2\le \frac{1}{2}(\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N}b{\parallel }_{L^2}^2).$$

has been used. In addition,

$$\begin{array}{cc}\hfill I_6\le & C[\parallel \mathsf{\Lambda }u{\parallel }_{L^{\infty }}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}b{\parallel }_{L^2}^2+\sum _{m=2}^{N+1}\parallel {\mathsf{\Lambda }}^{m}u{\parallel }_{L^3}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}b{\parallel }_{L^2}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+3-m}b{\parallel }_{L^6}\hfill \\ \hfill & +\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}{u\parallel }_{L^6}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}\parallel {\mathsf{\Lambda }}^2{b\parallel }_{L^3}+\sum _{m=0}^N\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^m{u\parallel }_{L^6}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}{\parallel {\mathsf{\Lambda }}^{N+3-m}b\parallel }_{L^2}\hfill \\ \hfill & +\sum _{p=1}^{P_0}\sum _{m=0}^{N+1}\parallel {\partial }_t{\mathsf{\Lambda }}^m{u\parallel }_{L^6}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}\parallel {\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{N+3-m}b{\parallel }_{L^3}]\hfill \\ \hfill \le & \frac{1}{8}(\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{u\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}{b\parallel }_{L^2}^2)+C[\parallel {\mathsf{\Lambda }}^{\frac{7}{2}}{u\parallel }_{L^2}^2{\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N}b\parallel }_{L^2}^2\hfill \\ \hfill & +\sum _{m=4}^N\parallel {\mathsf{\Lambda }}^{\frac{1}{2}+m}{u\parallel }_{L^2}^2\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+4-m}{b\parallel }_{L^2}^2+\sum _{m=0}^{N-1}\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{m+1}{u\parallel }_{L^2}^2{\parallel {\mathsf{\Lambda }}^{N+\frac{7}{2}-m}b\parallel }_{L^2}^2\hfill \\ \hfill & +\parallel {\mathsf{\Lambda }}^{\frac{7}{2}}{b\parallel }_{L^2}^2\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^N{u\parallel }_{L^2}^2+\sum _{p=1}^{P_0}\sum _{m=0}^{N+1}\parallel {\partial }_t^p{\mathsf{\Lambda }}^{m+1}{u\parallel }_{L^2}^2\parallel {\partial }_t^{P_0+1-p}{\mathsf{\Lambda }}^{N+\frac{7}{2}-m}b{\parallel }_{L^2}^2],\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

Summing up, using the previous decay results gives

$$\begin{array}{cc}\hfill & \frac{d}{dt}(\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}{u\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}{b\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+2}b{\parallel }_{L^2}^2)\hfill \\ \hfill \le & C{(1+t)}^{-min\{2r^{*}+\frac{15}{2}+N+2P_0,\frac{19}{2}+N+2P_0\}},\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(75)

Applying Plancherel’s theorem to (75), it yields that

$$\begin{array}{cc}& \frac{d}{dt}{\int }_{{\mathbb{R}}^3}\left[|\widehat{{\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}b}{(\xi ,t)|}^2\right]d\xi \hfill \\ & +{\int }_{{\mathbb{R}}^3}\frac{{|\xi |}^2}{1+{|\xi |}^2}\left[|\widehat{{\partial }_t^{P_0+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t^{P_0+1}b}{(\xi ,t)|}^2\right]d\xi \hfill \\ \hfill \le & C{(1+t)}^{-min\{2r^{*}+\frac{15}{2}+N+2P_0,\frac{19}{2}+N+2P_0\}},\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(76)

Multiplying (76) by $g\left(t\right)$ gives

$$\begin{array}{cc}& \frac{d}{dt}\left\{g\left(t\right){\int }_{{\mathbb{R}}^3}\left[|\widehat{{\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}b}{(\xi ,t)|}^2\right]d\xi \right\}\hfill \\ \hfill \le & g^{\prime }\left(t\right){\int }_{B\left(t\right)}\left[|\widehat{{\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}b}{(\xi ,t)|}^2\right]d\xi \hfill \\ \hfill & +Cg\left(t\right){(1+t)}^{-min\{2r^{*}+\frac{15}{2}+N+2P_0,\frac{19}{2}+N+2P_0\}},\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

Hence

$$\begin{array}{cc}\hfill & g\left(t\right){\int }_{{\mathbb{R}}^3}\left[|\widehat{{\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}b}{(\xi ,t)|}^2\right]d\xi \hfill \\ \hfill \le & C+{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}\left[|\widehat{{\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}b}{(\xi ,t)|}^2\right]d\xi ds\hfill \\ \hfill & +C{\int }_0^{t}g\left(s\right){(1+s)}^{-min\{2r^{*}+\frac{15}{2}+N+2P_0,\frac{19}{2}+N+2P_0\}}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(77)

It then follows from (36), (38) and (67) that

$$\begin{array}{cc}\hfill & g\left(t\right){\int }_{{\mathbb{R}}^3}\left[|\widehat{{\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}u}{(\xi ,t)|}^2{+(1+|\xi |}^2)|\widehat{{\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}b}{(\xi ,t)|}^2\right]d\xi \hfill \\ \hfill \le & C+{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^{4P_0+2N+6}\left[e^{-{2|\xi |}^{2}t}|{\widehat{u}}_0{\left(\xi \right)|}^2{+(1+|\xi |}^2)e^{-2\frac{{|\xi |}^2}{1+{|\xi |}^2}t}{|{\widehat{b}}_0\left(\xi \right)|}^2\right]d\xi ds\hfill \\ \hfill & +C{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}\sum _{p=0}^{P_0}\sum _{l=0}^p{|\xi |}^{4P_0+4+2N-4p}(\parallel {\partial }_t^l{u\parallel }_{L^2}^2+\parallel {\partial }_t^l{b\parallel }_{L^2}^2\left)\right(\parallel {\partial }_t^{p-l}{u\parallel }_{L^2}^2+\parallel {\partial }_t^{p-l}b{\parallel }_{L^2}^2)d\xi ds\hfill \\ \hfill & +C{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^{4P_0+2N+8}{(1+|\xi |}^2+{|\xi |}^4){\left[{\int }_0^t{(\parallel u\left(\tau \right)\parallel }_{L^2}^2+{\parallel b\left(\tau \right)\parallel }_{L^2}^2)d\tau \right]}^{2}d\xi ds\hfill \\ \hfill & +C{\int }_0^{t}g\left(s\right){(1+s)}^{-min\{2r^{*}+\frac{13}{2}+2P_0,\frac{17}{2}+2P_0\}}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(78)

Consider the first term of the right hand side of (78),

$$\begin{array}{cc}\hfill & {\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^{4P_0+4}\left[e^{-{2|\xi |}^{2}t}|{\widehat{u}}_0{\left(\xi \right)|}^2{+(1+|\xi |}^2)e^{-2\frac{{|\xi |}^2}{1+{|\xi |}^2}t}{|{\widehat{b}}_0\left(\xi \right)|}^2\right]d\xi d\xi ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-(r^{*}+\frac{9}{2}+2P_0+N)}ds.\hfill \end{array}$$

(79)

For the second term, after integrating in polar coordinates in $B\left(t\right)$,

$$\begin{array}{cc}\hfill & C{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}\sum _{p=0}^{P_0}\sum _{l=0}^p{|\xi |}^{4P_0+4+2N-4p}(\parallel {\partial }_t^l{u\parallel }_{L^2}^2+\parallel {\partial }_t^l{b\parallel }_{L^2}^2\left)\right(\parallel {\partial }_t^{p-l}{u\parallel }_{L^2}^2+\parallel {\partial }_t^{p-l}b{\parallel }_{L^2}^2)d\xi ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-min\{2r^{*}+N+\frac{13}{2}+2P_0,\frac{17}{2}+N+2P_0\}}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(80)

Moreover, if $r^{*}+\frac{3}{2}<\frac{5}{2}$, the third term of the right hand side of (78) can be estimated as

$$\begin{array}{cc}\hfill & C{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^{4P_0+2N+8}{(1+|\xi |}^2+{|\xi |}^4){\left[{\int }_0^t{(\parallel u\left(\tau \right)\parallel }_{L^2}^2+{\parallel b\left(\tau \right)\parallel }_{L^2}^2)d\tau \right]}^{2}d\xi ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-(2r^{*}+N+\frac{13}{2}+2P_0)}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(81)

If $r^{*}+\frac{3}{2}\ge \frac{5}{2}$,

$$\begin{array}{cc}\hfill & C{\int }_0^t{g}^{\prime }\left(s\right){\int }_{B\left(s\right)}{|\xi |}^{4P_0+2N+8}{(1+|\xi |}^2+{|\xi |}^4){\left[{\int }_0^t{(\parallel u\left(\tau \right)\parallel }_{L^2}^2+{\parallel b\left(\tau \right)\parallel }_{L^2}^2)d\tau \right]}^{2}d\xi ds\hfill \\ & \le C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-(2P_0+\frac{11}{2}+N)}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(82)

The last term can be estimated as

$$\begin{array}{cc}& C{\int }_0^{t}g\left(s\right){(1+s)}^{-min\{2r^{*}+\frac{15}{2}+N+2P_0,\frac{19}{2}+N+2P_0\}}ds\hfill \\ \hfill \le & C{\int }_0^t{g}^{\prime }\left(s\right){(1+s)}^{-min\{2r^{*}+\frac{13}{2}+N+2P_0,\frac{17}{2}+N+2P_0\}}ds,\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(83)

For a fixed $r^{*}$, choose $g\left(t\right)={(1+t)}^m$, for some $m>max\{r^{*}+\frac{9}{2}+N+2P_0,\frac{11}{2}+N+2P_0\}$. Then $\rho \left(t\right)=C{(1+t)}^{-\frac{1}{2}}$ and from (78)–(83), one can obtain

$$\begin{array}{cc}& \parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}{u\parallel }_{L^2}^2+\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}{b\parallel }_{L^2}^2+{\parallel {\partial }_t^{P_0+1}{\mathsf{\Lambda }}^{N+1}b\parallel }_{L^2}^2\hfill \\ \hfill \le & C{(1+t)}^{-min\{r^{*}+\frac{9}{2}+N+2P_0,\frac{11}{2}+N+2P_0\}},\mathrm{for}\mathrm{large}t.\hfill \end{array}$$

(84)

This completes the proof of Lemma 13. □

The proof of Theorem 2 is given in the following.

Proof of Theorem 2.

Lemma 12 implies Theorem 2 holds for the case $p=1$. Then, supposing that Theorem 2 holds for $p\le P_0\in {\mathbb{N}}^{+}$, one can also obtain that it holds for $p=P_0+1$ (Lemma 13). Hence, through mathematical induction, the proof of Theorem 2 is complete. □

5. Conclusions

The magnetic induction equation with Hall effect is a typical Hall–MHD equation. This model can be used to describe the reconnection phenomenon by simulating flows with differential typical scales $L_0$. From a mathematical point of view, the local well-posedness, global well-posedness and large time behavior of solutions are very interesting. In the previous works of Fan et al. [11], the authors studied the local well-posedness of strong solutions and gave the preliminary result on the small initial data global well-posedness; Zhao [29,30] considered the large time behavior of solutions, established the decay estimates for the weak solution (see also Lemma 1) and the strong solution (see Lemma 2). In this paper, one only assumes that $\parallel u_0{\parallel }_{H^{\frac{3}{2}+\epsilon }}+{\parallel b_0\parallel }_{H^2}$ is sufficiently small, obtains the global well-posedness of strong solution and establishes the a priori estimates on higher order time and spatial derivatives of solutions. Moreover, by using the properties of decay character and the Fourier splitting method, one also shows the optimal decay rates for higher order time and spatial derivatives of solutions. In a sense, the results of this paper can be seen as an improvement of the previous results in [11,29,30].