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Remarks on the regularity criteria for the 3D MHD equations in the multiplier spaces

Zujin Zhang1*, Xiqin Ouyang2, Dingxing Zhong1 and Shulin Qiu1

Author Affiliations

1 School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou, Jiangxi, 341000, P.R. China

2 College of Mathematics, Gannan Institute of Education, Ganzhou, Jiangxi, 341000, P.R. China

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Boundary Value Problems 2013, 2013:270  doi:10.1186/1687-2770-2013-270

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/270


Received:27 September 2013
Accepted:22 November 2013
Published:12 December 2013

© 2013 Zhang et al.; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we consider the regularity criteria for the 3D MHD equations. It is proved that if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M1">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M2">View MathML</a>

then the solution actually is smooth. This extends the previous results given by Guo and Gala (Anal. Appl. 10:373-380, 2013), Gala (Math. Methods Appl. Sci. 33:1496-1503, 2010).

MSC: 35B65, 35Q35, 76D03.

Keywords:
MHD equations; regularity criteria; regularity of solutions

1 Introduction

In this paper, we consider the following three-dimensional (3D) magnetohydrodynamic (MHD) equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M3">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M4">View MathML</a> is the fluid velocity field, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M5">View MathML</a> is the magnetic field, π is a scalar pressure, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M6">View MathML</a> are the prescribed initial data satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M7">View MathML</a> in the distributional sense. Physically, (1) governs the dynamics of the velocity and magnetic fields in electrically conducting fluids such as plasmas, liquid metals, and salt water. Moreover, (1)1 reflects the conservation of momentum, (1)2 is the induction equation, and (1)3 specifies the conservation of mass.

Besides its physical applications, MHD system (1) is also mathematically significant. Duvaut and Lions [1] constructed a global weak solution to (1) for initial data with finite energy. However, the issue of regularity and uniqueness of such a given weak solution remains a challenging open problem. Many sufficient conditions (see, e.g., [2-16] and the references therein) were derived to guarantee the regularity of the weak solution. Among these results, we are interested in regularity criteria involving only partial components of the velocity u, the magnetic field b, the pressure gradient ∇π, etc.

In [8], Jia and Zhou used an intricate decomposition technique and delicate inequalities to obtain the following regularity criterion:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M8">View MathML</a>

(2)

that is, if (2) holds, then the solution of (1) is smooth. Applying a more subtle decomposition technique (see [[13], Remark 3]), Zhang, Li, and Yu [13] could be able to prove smoothness condition (2) in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M9">View MathML</a>. Noticeably, Zhang [17] treated the range <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M10">View MathML</a> in a unified approach.

As far as one directional derivative of the velocity field is concerned, Cao and Wu [2] proved the following regularity criterion:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M11">View MathML</a>

Jia and Zhou [10] showed that if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M12">View MathML</a>

then the solution is regular. These results were improved by Zhang [18] to be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M13">View MathML</a>

We would like to give another contribution in this direction and prove that if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M14">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M15">View MathML</a>

then the solution actually is smooth. Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M16">View MathML</a> is the multiplier spaces, which is strictly larger than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M17">View MathML</a> (see Section 2 for details).

Notice that our result extends

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M18">View MathML</a>

which is in [19] for the Navier-Stokes equations. At this moment, for MHD equations (1), we could not be able to add the regularity condition on one directional derivative of the velocity field only, since we need to convert (1) into a symmetric form. We also improve the result

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M19">View MathML</a>

of [20] in the sense that we need only one directional derivative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M20">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M21">View MathML</a> to ensure the smoothness of the solution.

Before stating the precise result, let us recall the weak formulation of MHD equations (1).

Definition 1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M22">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M7">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M24">View MathML</a>. A measurable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M25">View MathML</a>-valued pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M26">View MathML</a> is called a weak solution to (1) with initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M6">View MathML</a> provided that the following three conditions hold:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M29">View MathML</a>;

(2) (1)1,2,3,4 are satisfied in the distributional sense;

(3) the energy inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M30">View MathML</a>

Now, our main results read as follows.

Theorem 2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M31">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M7">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M24">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M26">View MathML</a>is a given weak solution pair of MHD system (1) with initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M6">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M36">View MathML</a>. If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M37">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M38">View MathML</a>

then the solution is smooth on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M36">View MathML</a>.

The proof of Theorem 2 will be given in Section 3. In Section 2, we introduce the multiplier spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M16">View MathML</a> and recall their fine properties.

2 Preliminaries

In this section, we recall the definition and fine properties of the multiplier spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M16">View MathML</a> (see [21,22] for example).

Definition 3 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M42">View MathML</a>, the homogeneous space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M16">View MathML</a> is defined as the space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M44">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M45">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M46">View MathML</a> is the space of distributions u such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M47">View MathML</a>

We have the following scaling properties:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M48">View MathML</a>

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M49">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M50">View MathML</a>

where BMO is the homogeneous space of bounded mean oscillations associated with the semi-norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M51">View MathML</a>

Furthermore, we have the following strict imbeddings:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M52">View MathML</a>

which could be justified simply as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M53">View MathML</a>

3 Proof of Theorem 2

In this section, we shall prove Theorem 2. As we will see in the proof below, we need only to consider the case that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M54">View MathML</a>

(3)

First, let us convert (1) into a symmetric form. Writing

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M55">View MathML</a>

we find by adding and subtracting (1)1 with (1)2,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M56">View MathML</a>

(4)

The rest of the proof is organized as follows. In the first step, we dominate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M57">View MathML</a> and the time integral of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M58">View MathML</a>, while the second step is devoted to controlling <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M59">View MathML</a>.

Step 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M60">View MathML</a> estimates.

Taking the inner product of (4)1 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M61">View MathML</a>, (4)2 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M62">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M63">View MathML</a> respectively, adding them together, and noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M64">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M65">View MathML</a>

(5)

We now estimate I as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M66">View MathML</a>

(6)

Plugging (6) into (5), invoking the Gronwall inequality, and noticing (3), we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M67">View MathML</a>

(7)

Step 2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M68">View MathML</a> estimates.

Taking the inner product of (4)1 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M69">View MathML</a>, (4)2 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M70">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M63">View MathML</a> respectively, adding them together, and noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M64">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M73">View MathML</a>

(8)

Utilizing interpolation inequalities and the following multiplicative Gagliardo-Nireberg inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M74">View MathML</a>

(9)

it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M75">View MathML</a>

(10)

Putting (10) into (8), utilizing the Gronwall inequality, and noticing (7), we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M76">View MathML</a>

and consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/270/mathml/M77">View MathML</a>

The classical Serrin-type regularity criterion [6] then concludes the proof of Theorem 2.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

We declare that all authors collaborated and dedicated the same amount of time in order to perform this article.

Acknowledgements

This work was partially supported by the Youth Natural Science Foundation of Jiangxi Province (20132BAB211007), the Science Foundation of Jiangxi Provincial Department of Education (GJJ13658, GJJ13659) and the National Natural Science Foundation of China (11326138, 11361004). The authors thank the referees for their careful reading and suggestions which led to an improvement of the work.

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