The continuous dependence for the Hal-MHD equations with fractional magnetic diffusion

TL;DR

This paper proves that solutions to the incompressible Hall-MHD equations with fractional magnetic diffusion vary continuously with initial conditions in a specific Sobolev space, ensuring stability of solutions.

Contribution

It establishes the continuous dependence of solutions on initial data for the Hall-MHD system with fractional magnetic diffusion in certain Sobolev spaces.

Findings

Solutions depend continuously on initial data in $H^s$ for $s > 1 + d/2$

Provides mathematical validation for stability of solutions

Extends previous results to fractional magnetic diffusion cases

Abstract

In this paper we show that the solutions to the incompressible Hall-MHD system with fractional magnetic diffusion depend continuously on the initial data in $H^s(\mathbb{R}^d)$, $s>1+\frac{d}{2}$.

Equations

$$\{\begin{array}{l}{\partial }_{t}u+u\cdot \nabla u+\nabla P=b\cdot \nabla b,\\ {\partial }_{t}b+u\cdot \nabla b+\nabla \times ((\nabla \times b)\times b)+(-\Delta {)}^{\alpha }b=b\cdot \nabla u,\\ \mathrm{div}u=\mathrm{div}b=0,(u,b){|}_{t=0}=(u_0,b_0),\end{array}$$

$$\{\begin{array}{l}{\partial }_t{u}^n+u^n\cdot \nabla u^n+\nabla P_n=b^n\cdot \nabla b^n,\\ {\partial }_t{b}^n+u^n\cdot \nabla b^n+\nabla \times ((\nabla \times b^n)\times b^n)+(-\Delta {)}^{\alpha }{b}^n=b^n\cdot \nabla u^n,\\ \mathrm{div}{u}^n=\mathrm{div}{b}^n=0,(u^n,b^n){|}_{t=0}=(u_0^n,b_0^n),\end{array}$$

$$\displaystyle\lim_{n\rightarrow\infty}\big{(}||u^{n}-u||_{L^{\infty}_{T}(H^{s}(\mathbb{R}^{d}))}+||b^{n}-b||_{L^{\infty}_{T}(H^{s}(\mathbb{R}^{d}))}\big{)}=0.$$

$$\|fg\|_{H^{s}}\leq C\big{(}\|f\|_{L^{\infty}}||g||_{H^{s}}+||f||_{H^{s}}\|g\|_{L^{\infty}}\big{)}.$$

$$\big{|}\big{|}(2^{j\sigma}||R_{j}||_{L^{2}})_{j\geq-1}\big{|}\big{|}_{\ell^{r}}\leq\begin{cases}C||\nabla v||_{H^{\frac{d}{2}}\cap L^{\infty}}||f||_{H^{\sigma}},\ \mathrm{if}\quad\sigma<1+\frac{d}{2},\\ C||\nabla v||_{H^{\frac{d}{2}+1}}||f||_{H^{\sigma}},\quad\quad\mathrm{if}\quad\sigma=1+\frac{d}{2},\\ C||\nabla v||_{H^{\sigma-1}}||f||_{H^{\sigma}},\quad\mathrm{if}\quad\sigma>1+\frac{d}{2}.\end{cases}$$

$$\big{|}\big{|}(2^{j\sigma}||R_{j}||_{L^{2}})_{j\geq-1}\big{|}\big{|}_{\ell^{r}}\leq C(||\nabla v||_{L^{\infty}}||f||_{H^{\sigma}}+||\nabla f||_{L^{\infty}}||\nabla v||_{H^{\sigma-1}}).$$

$$||u(t)|{|}_{H^s}^2+||b(t)|{|}_{H^s}^2\le (||u_0|{|}_{H^s}^2+||b_0|{|}_{H^s}^2)e^{C{\int }_0^T(||u(t)|{|}_{C^{0,1}}+||b(t)|{|}_{C^{0,1}}+||b(t)|{|}_{C^{0,1}}^2)\mathrm{d}t}.$$

$$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}(||{\Delta }_{j}u|{|}_{L^2}^2+||{\Delta }_{j}b|{|}_{L^2}^2)+C_0{2}^{2j\alpha }||{\Delta }_{j}b|{|}_{L^2}^2\le \sum _{i=1}^5{I}_i,$$

$$I_1=-{\int }_{{\mathbb{R}}^d}[{\Delta }_j,u\cdot \nabla ]u\cdot {\Delta }_{j}u\mathrm{d}x,I_2=-{\int }_{{\mathbb{R}}^d}[{\Delta }_j,u\cdot \nabla ]b\cdot {\Delta }_{j}b\mathrm{d}x,$$

$$I_3={\int }_{{\mathbb{R}}^d}[{\Delta }_j,b\cdot \nabla ]b\cdot {\Delta }_{j}u\mathrm{d}x,I_4={\int }_{{\mathbb{R}}^d}[{\Delta }_j,b\cdot \nabla ]u\cdot {\Delta }_{j}b\mathrm{d}x,$$

$$I_5={\int }_{{\mathbb{R}}^d}[{\Delta }_j,b\times ](\nabla \times b)\cdot (\nabla \times {\Delta }_{j}b)\mathrm{d}x.$$

$$|I_1|\lesssim ||[{\Delta }_j,u\cdot \nabla ]u|{|}_{L^2}||{\Delta }_{j}u|{|}_{L^2}\lesssim 2^{-2js}{c}_j^2||\nabla u|{|}_{L^{\infty }}||u|{|}_{H^s}^2,$$

$$\displaystyle|I_{2}|\lesssim||[\Delta_{j},u\cdot\nabla]b||_{L^{2}}||\Delta_{j}b||_{L^{2}}\lesssim 2^{-2js}c^{2}_{j}\big{(}||\nabla u||_{L^{\infty}}||b||^{2}_{H^{s}}+||\nabla b||_{L^{\infty}}||u||_{H^{s}}||b||_{H^{s}}\big{)},$$

$$|I_3|\lesssim ||[{\Delta }_j,b\cdot \nabla ]b|{|}_{L^2}||{\Delta }_{j}u|{|}_{L^2}\lesssim 2^{-2js}{c}_j^2||\nabla b|{|}_{L^{\infty }}||u|{|}_{H^s}||b|{|}_{H^s},$$

$$\displaystyle|I_{4}|\lesssim||[\Delta_{j},b\cdot\nabla]u||_{L^{2}}||\Delta_{j}b||_{L^{2}}\lesssim 2^{-2js}c^{2}_{j}\big{(}||\nabla b||_{L^{\infty}}||u||_{H^{s}}||b||_{H^{s}}+||\nabla u||_{L^{\infty}}||b||^{2}_{H^{s}}\big{)}.$$

$$|I_5|$$

$$\le C||[{\Delta }_j,b\times ](\nabla \times b)|{|}_{L^2}^2+\frac{C_0}{8}{2}^{2j\alpha }||{\Delta }_{j}b|{|}_{L^2}^2$$

$$\le C{2}^{-2j(s-1)}{c}_j^2||\nabla b|{|}_{L^{\infty }}^2||b|{|}_{H^s}^2+\frac{C_0}{8}{2}^{2j\alpha }||{\Delta }_{j}b|{|}_{L^2}^2.$$

$$|I_5|\lesssim ||b|{|}_{C^{0,1}}||b|{|}_{H^s}^2.$$

$$||u(t)|{|}_{H^s}^2+||b(t)|{|}_{H^s}^2\le ||u_0|{|}_{H^s}^2+||b_0|{|}_{H^s}^2$$

$$+{\int }_0^t(||u|{|}_{H^s}^2+||b|{|}_{H^s}^2)(||u(\tau )|{|}_{C^{0,1}}+||b(\tau )|{|}_{C^{0,1}}+||b(\tau )|{|}_{C^{0,1}}^2)\mathrm{d}\tau .$$

$$||\delta u(t)|{|}_{H^{s-1}}^2+||\delta b(t)|{|}_{H^{s-1}}^2\le (||\delta u_0|{|}_{H^{s-1}}^2+||\delta b_0|{|}_{H^{s-1}}^2)e^{\mathrm{A}(t)},$$

$$||\delta u(t)|{|}_{H^s}^2$$

$$\displaystyle\quad+C\int^{t}_{0}(||u^{2}||^{2}_{H^{s+1}}+||b^{2}||^{2}_{H^{s+1}})(||\delta u||^{2}_{H^{s-1}}+||\delta b||^{2}_{H^{s-1}})\mathrm{d}\tau\Big{)}e^{\mathrm{A}(t)},$$

$$\mathrm{A}(t)=C{\int }_0^t(1+||u^1|{|}_{H^s}+||u^2|{|}_{H^s}+||b^1|{|}_{H^s}+||b^2|{|}_{H^s}+||b^1|{|}_{H^s}^2+||b^2|{|}_{H^s}^2)\mathrm{d}\tau .$$

$$\{\begin{array}{l}{\partial }_t\delta u+u^1\cdot \nabla \delta u+\delta u\cdot \nabla u^2+\nabla \bar{P}=b^1\cdot \nabla \delta b+\delta b\cdot \nabla b^2,\\ {\partial }_t\delta b+u^1\cdot \nabla \delta b+\delta u\cdot \nabla b^2+(-\Delta {)}^{\alpha }\delta b+\nabla \times ((\nabla \times \delta b)\times b^1)\\ +\nabla \times ((\nabla \times b^2)\times \delta b)=b^1\cdot \nabla \delta u+\delta b\cdot \nabla u^2,\\ \mathrm{div}\delta u=\mathrm{div}\delta b=0,(\delta u,\delta b){|}_{t=0}=(\delta u_0,\delta b_0).\end{array}$$

$$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}(||{\Delta }_j\delta u|{|}_{L^2}^2+||{\Delta }_j\delta b|{|}_{L^2}^2)+C_0{2}^{2j\alpha }||{\Delta }_j\delta b|{|}_{L^2}^2\le \sum _{i=1}^8{J}_i,$$

$$J_1=-{\int }_{{\mathbb{R}}^d}[{\Delta }_j,u^1\cdot \nabla ]\delta u\cdot {\Delta }_j\delta u\mathrm{d}x,J_2=-{\int }_{{\mathbb{R}}^d}[{\Delta }_j,u^1\cdot \nabla ]\delta b\cdot {\Delta }_j\delta b\mathrm{d}x,$$

$$J_3={\int }_{{\mathbb{R}}^d}[{\Delta }_j,b^1\cdot \nabla ]\delta b\cdot {\Delta }_j\delta u\mathrm{d}x,J_4={\int }_{{\mathbb{R}}^d}[{\Delta }_j,b^1\cdot \nabla ]\delta u\cdot {\Delta }_j\delta b\mathrm{d}x,$$

$$J_5={\int }_{{\mathbb{R}}^d}{\Delta }_j(\delta b\cdot \nabla b^2-\delta u\cdot \nabla u^2){\Delta }_j\delta u\mathrm{d}x,$$

$$J_6={\int }_{{\mathbb{R}}^d}{\Delta }_j(\delta b\cdot \nabla u^2-\delta u\cdot \nabla b^2){\Delta }_j\delta b\mathrm{d}x,$$

$$J_7={\int }_{{\mathbb{R}}^d}[{\Delta }_j,b^1\times ](\nabla \times \delta b)\cdot (\nabla \times {\Delta }_j\delta b)\mathrm{d}x,$$

$$J_8=-{\int }_{{\mathbb{R}}^d}{\Delta }_j((\nabla \times b^2)\times \delta b)\cdot (\nabla \times {\Delta }_j\delta b)\mathrm{d}x.$$

$$\begin{array}{rl}& |J_1|\lesssim ||[{\Delta }_j,u^1\cdot \nabla ]\delta u|{|}_{L^2}||{\Delta }_j\delta u|{|}_{L^2}\lesssim 2^{-2j(s-1)}{c}_j^2||\nabla u^1|{|}_{H^{s-1}}||\delta u|{|}_{H^{s-1}}^2,\\ & |J_2|\lesssim ||[{\Delta }_j,u^1\cdot \nabla ]\delta b|{|}_{L^2}||{\Delta }_j\delta b|{|}_{L^2}\lesssim 2^{-2j(s-1)}{c}_j^2||\nabla u^1|{|}_{H^{s-1}}||\delta b|{|}_{H^{s-1}}^2,\\ & |J_3|\lesssim ||[{\Delta }_j,b^1\cdot \nabla ]\delta b|{|}_{L^2}||{\Delta }_j\delta u|{|}_{L^2}\lesssim 2^{-2j(s-1)}{c}_j^2||\nabla b^1|{|}_{H^{s-1}}||\delta u|{|}_{H^{s-1}}||\delta b|{|}_{H^{s-1}},\\ & |J_4|\lesssim ||[{\Delta }_j,b^1\cdot \nabla ]\delta u|{|}_{L^2}||{\Delta }_j\delta b|{|}_{L^2}\lesssim 2^{-2j(s-1)}{c}_j^2||\nabla b^1|{|}_{H^{s-1}}||\delta u|{|}_{H^{s-1}}||\delta b|{|}_{H^{s-1}},\\ & |J_5|\lesssim 2^{-2j(s-1)}{c}_j^2(||b^2|{|}_{H^s}||\delta u|{|}_{H^{s-1}}||\delta b|{|}_{H^{s-1}}+||u^2|{|}_{H^s}||\delta u|{|}_{H^{s-1}}^2),\\ & |J_6|\lesssim 2^{-2j(s-1)}{c}_j^2(||b^2|{|}_{H^s}||\delta u|{|}_{H^{s-1}}||\delta b|{|}_{H^{s-1}}+||u^2|{|}_{H^s}||\delta b|{|}_{H^{s-1}}^2).\end{array}$$

$$\begin{array}{rl}|J_7|& \lesssim ||[{\Delta }_j,b^1\times ](\nabla \times \delta b)|{|}_{L^2}\cdot (2^j||{\Delta }_j\delta b|{|}_{L^2})\\ & \le C||[{\Delta }_j,b^1\times ](\nabla \times \delta b)|{|}_{L^2}^2+\frac{C_0}{8}{2}^{2j\alpha }||{\Delta }_j\delta b|{|}_{L^2}^2\\ & \le C{2}^{-2j(s-1)}{c}_j^2||b^1|{|}_{H^s}^2||\delta b|{|}_{H^{s-1}}^2+\frac{C_0}{8}{2}^{2j\alpha }||{\Delta }_j\delta b|{|}_{L^2}^2.\end{array}$$

$$\begin{array}{rl}|J_8|& \lesssim ||{\Delta }_j((\nabla \times b^2)\times \delta b)|{|}_{L^2}\cdot (2^j||{\Delta }_j\delta b|{|}_{L^2})\\ & \le C||{\Delta }_j((\nabla \times b^2)\times \delta b)|{|}_{L^2}^2+\frac{C_0}{8}{2}^{2j\alpha }||{\Delta }_j\delta b|{|}_{L^2}^2\\ & \le C{2}^{-2j(s-1)}{c}_j^2||b^2|{|}_{H^s}^2||\delta b|{|}_{H^{s-1}}^2+\frac{C_0}{8}{2}^{2j\alpha }||{\Delta }_j\delta b|{|}_{L^2}^2.\end{array}$$

$$|J_7|+|J_8|\lesssim (||b^1|{|}_{H^s}+||b^2|{|}_{H^s})||\delta b|{|}_{H^{s-1}}^2.$$

Full text

1. Introduction and main result

In this paper, we are concerned with the study on the Cauchy problem for the incompressible Hall-MHD system with fractional magnetic diffusion:

[TABLE]

where $u$ and $b$ represent the flow velocity vector and the magnetic field vector respectively, $P$ is a scalar pressure, while $u_{0}(x)$ and $b_{0}(x)$ are the given initial velocity and initial magnetic field with $\nabla\cdot u_{0}=\nabla\cdot b_{0}=0$. The Hall term $\nabla\times((\nabla\times b)\times b)$ plays an important role in magnetic reconnection which is happening in the case of large magnetic shear. The fractional Laplacian operator $(-\Delta)^{\alpha}$ is defined in terms of the Fourier transform by $\widehat{(-\Delta)^{\alpha}}f(\xi)=|\xi|^{2\alpha}\hat{f}(\xi)$.

The applications of the Hall-MHD system cover a very wide range of physical objects, such as, in the magnetic reconnection in space plasmas, the star formation, neutron stars and geo-dynamos.

Recently, there are many researches on the standard Hall-MHD equations with $-\Delta u$ and $-\Delta b$ , concerning global weak solutions [1], local and global (small) strong solutions [2, 3], and the large time behavior of weak and strong solutions [4, 5, 6, 7]. For the system (1.1), Chae, Wan and Wu [8] proved the local existence and uniqueness of the solution to the Hall-MHD equations with only a fractional Laplacian magnetic diffusion $(-\Delta)^{\alpha}b$ in the space $H^{s}(\mathbb{R}^{d})$ for $s>1+\frac{d}{2}$ and $\alpha>\frac{1}{2}$. To the knowledge of the author, there are few results on continuous dependence upon the initial data for the Hall-MHD system. Therefore, the purpose of our paper is to discuss the continuous dependence of solutions on the initial data in (1.1) and give a precise estimate.

Our argument can state as follows:

Theorem 1.1**.**

Let $d\geq 2$, $\alpha\geq 1$ and $s>1+\frac{d}{2}$. Suppose that $u^{n}_{0}\in H^{s}(\mathbb{R}^{d})$ goes to $u_{0}\in H^{s}(\mathbb{R}^{d})$ in $H^{s}(\mathbb{R}^{d})$ and $b^{n}_{0}\in H^{s}(\mathbb{R}^{d})$ goes to $b_{0}\in H^{s}(\mathbb{R}^{d})$ in $H^{s}(\mathbb{R}^{d})$ when $n$ goes to infinity. Then there exists a positive $T>0$ independent of $n$ such that $(u^{n},b^{n})\in\mathcal{C}([0,T];H^{s}(\mathbb{R}^{d}))$ be the solution of

[TABLE]

and $(u,b)\in\mathcal{C}([0,T];H^{s}(\mathbb{R}^{d}))$ be the solution of (1.1) with initial data $(u_{0},b_{0})$. Moreover, there holds

[TABLE]

Notations. Given a Banach space $X$, we denote its norm by $\|\cdot\|_{X}$. Since all spaces of functions are over $\mathbb{R}^{d}$, for simplicity, we drop $\mathbb{R}^{d}$ in our notations of function spaces if there is no ambiguity. The symbol $A\lesssim B$ denotes that there exists a constant $\bar{c}_{0}>0$ independent of $A$ and $B$, such that $A\leq\bar{c}_{0}B$. We denote by $\{c_{j}\}_{j\geq-1}$ a sequence such that $||c_{j}||_{\ell^{2}}\leq 1$.

2. Preliminaries

In this section we collect some preliminary definitions and lemmas. For more details we refer the readers to [9].

Let $\chi:{\mathbb{R}}^{d}\to[0,1]$ be a radial, non-negative, smooth and radially decreasing function which is supported in $\mathcal{B}\triangleq\{\xi:|\xi|\leq\frac{4}{3}\}$ and $\chi\equiv 1$ for $|\xi|\leq\frac{5}{4}$. Let $\varphi(\xi)=\chi(\frac{\xi}{2})-\chi(\xi)$. Then $\varphi$ is supported in the ring $\mathcal{C}\triangleq\{\xi\in\mathbb{R}^{d}:\frac{3}{4}\leq|\xi|\leq\frac{8}{3}\}$. For $u\in\mathcal{S}^{\prime}$, $q\in{\mathbb{Z}}$, we define the Littlewood-Paley operators: $\dot{\Delta}_{q}{u}=\mathcal{F}^{-1}(\varphi(2^{-q}\cdot)\mathcal{F}u)$, ${\Delta}_{q}{u}=\dot{\Delta}_{q}{u}$ for $q\geq 0$, ${\Delta}_{q}{u}=0$ for $q\leq-2$ and $\Delta_{-1}u=\mathcal{F}^{-1}(\chi\mathcal{F}u)$, and S_{q}{u}=\mathcal{F}^{-1}\big{(}\chi(2^{-q}\xi)\mathcal{F}u\big{)}. Here we use ${\mathcal{F}}(f)$ or $\widehat{f}$ to denote the Fourier transform of $f$.

Lemma 2.1**.**

[9]** Let $s>0$. Then there exists a constant $C$, depending only on $d,s$, such that for all $f,g\in H^{s}$,

[TABLE]

Lemma 2.2**.**

[10*]** Let $\sigma\in\mathbb{R}$ and $v$ be a vector field over $\mathbb{R}^{d}$. Assume that $\sigma>-d\min\{1-\frac{1}{p},\frac{1}{p}\}$. Define $R_{j}=[v\cdot\nabla,\Delta_{j}]f$. There a constant $C=C(p,\sigma,d)$ such that

[TABLE]

Moreover, if $\sigma>0$, we have

[TABLE]

3. Some useful estimates

In this section, motivated by [11], we will establish some useful estimates for smooth solutions of (1.1), which is the key component in the proof of Theorems 1.1. The estimates can be stated as follows.

Lemma 3.1**.**

Let $d\geq 2$ and $s>1+\frac{d}{2}$. Suppose that $(u,b)\in\mathcal{C}([0,T];H^{s})$ is the solution of (1.1) with initial data $(u_{0},b_{0})$. Then, we have for all $t\in[0,T]$,

[TABLE]

Proof.

Now, we apply $\Delta_{j}$ to (1.1), and take the inner product with $(\Delta_{j}u,\Delta_{j}b)$ and integrate by parts to have

[TABLE]

where

[TABLE]

According to Lemmas 2.1-2.2, it is easy to estimate

[TABLE]

By Lemma 2.2, we can gain that for $j\geq 0$,

[TABLE]

If $j=-1$, it is easy to deduce that

[TABLE]

Integrating (3.1) over $[0,t]$, multiplying the inequality above by $2^{2j{s}}$ and summing over $j\geq-1$, we have

[TABLE]

This along with the Gronwall inequality yields the result of this lemma. ∎

Lemma 3.2**.**

Let $d\geq 2$ and $s>1+\frac{d}{2}$. Suppose that $(u^{1},b^{1})$ and $(u^{2},b^{2})$ are two smooth solutions of (1.1) with initial data $(u^{1}_{0},b^{1}_{0})$ and $(u^{2}_{0},b^{2}_{0})$ respectively. Denote $\delta u=u^{1}-u^{2}$ and $\delta b=b^{1}-b^{2}$. Then, we have for all $t\in[0,T]$,

[TABLE]

and

[TABLE]

with

[TABLE]

Proof.

It is easy to show that

[TABLE]

Now, we apply $\Delta_{j}$ to (3.2), and take the inner product with $(\Delta_{j}\delta u,\Delta_{j}\delta b)$ and integrate by parts to have

[TABLE]

where

[TABLE]

On the one hand, according to Lemmas 2.1-2.2, it is easy to estimate

[TABLE]

By Lemma 2.2, we obtain for $j\geq 0$,

[TABLE]

By Lemma 2.1, we have for $j\geq 0$,

[TABLE]

If $j=-1$, it is easy to deduce that

[TABLE]

Integrating (3.3) over $[0,t]$, multiplying the inequality above by $2^{2j(s-1)}$ and summing over $j\geq-1$, we infer from (3.4)-(3.7) that

[TABLE]

This along with the Gronwall inequality yields the first part of the lemma. On the other hand, according to Lemmas 2.1-2.2, it is easy to estimate

[TABLE]

By Lemma 2.2, we can gain for $j\geq 0$,

[TABLE]

By Lemma 2.1, we have for $j\geq 0$,

[TABLE]

If $j=-1$, by (3.7), we have

[TABLE]

Integrating (3.3) over $[0,t]$, multiplying the inequality above by $2^{2j{s}}$ and summing over $j\geq-1$, we infer from (3.8)-(3.11) that

[TABLE]

This along with the Gronwall inequality yields the second part of the lemma. Therefore, we complete the proof of this lemma. ∎

4. Proof of Theorem 1.1

In this section, we will give the details of the proof for the Theorem 1.1.

Proof of Theorem 1.1. First, according to classical results, there exist a positive $T_{n}>0$ such that (1.2) have a solution $(u^{n},b^{n})\in\mathcal{C}([0,T_{n});H^{s})$. Indeed, by Lemma 3.1, we have

[TABLE]

Denote $R=\sup\limits_{n\geq 0}(||u^{n}_{0}||_{H^{s}}+||b^{n}_{0}||_{H^{s}})$. Therefore, by continuity arguments, there exists a positive $T=T(s,d,R)$ satisfying $T_{n}>T$ such that

[TABLE]

Moreover, for all $\gamma>s$, we have for all $t\in[0,T]$,

[TABLE]

Let $(u^{n}_{j},b^{n}_{j})\in\mathcal{C}([0,T];H^{s})$ be the solution of

[TABLE]

Then, according to Lemma 3.2, we have

[TABLE]

which along with the fact that $||u^{n}_{j}||_{L^{\infty}_{T}(H^{s+1})}+||b^{n}_{j}||_{L^{\infty}_{T}(H^{s+1})}\leq C2^{j}$ leads to

[TABLE]

Using Lemma 3.2, we can obtain

[TABLE]

Therefore, combing (4.2)-(4.3), we obtain

[TABLE]

This implies the result of Theorem 1.1.

Acknowledgements. This work was partially supported by NSFC (No. 11361004).