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Global existence of 2D nonhomogeneous incompressible magnetohydrodynamics with vacuum

Menglong Su1*, Xiaohui Qian2 and Jian Wang3

Author Affiliations

1 School of Mathematical Sciences, Luoyang Normal University, Luoyang, 471022, P.R. China

2 College of Science, Zhongyuan University of Technology, Zhengzhou, 450007, P.R. China

3 School of Mathematics, Ocean University of China, Qingdao, 266100, P.R. China

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Boundary Value Problems 2014, 2014:94  doi:10.1186/1687-2770-2014-94


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/94


Received:4 January 2014
Accepted:31 March 2014
Published:6 May 2014

© 2014 Su 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 credited.

Abstract

In this paper, we prove the existence of global strong solutions to the Cauchy problem of 2D incompressible magnetohydrodynamics (MHD) flows. Here, we emphasize that the initial density <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M1">View MathML</a> is permitted to contain vacuum states, and the initial velocity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M2">View MathML</a> and magnetic fields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M3">View MathML</a> can be arbitrarily large.

Keywords:
global strong solutions; 2D incompressible magnetohydrodynamics flows; vacuum states

1 Introduction

The mathematical model of magnetohydrodynamics (MHD) is used to simulate the motion of a conducting fluid under the effect of the electromagnetic field and has a very wide range of applications in astrophysics, plasma, and so on. The governing equations of nonhomogeneous MHD can be stated as follows [1,2]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M4">View MathML</a>

(1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M5">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M6">View MathML</a>

(3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M7">View MathML</a>

(4)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M9">View MathML</a>. The unknown functions ρ, u, P, and B denote the fluid density, velocity, pressure, and magnetic field, respectively. The constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M10">View MathML</a> is the viscosity coefficient. The constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M11">View MathML</a> is the resistivity coefficient, which is inversely proportional to the electrical conductivity constant and acts as the magnetic diffusivity of magnetic fields. Without loss of generality, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M12">View MathML</a> throughout the paper. In this paper, we assume the state equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M13">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M15">View MathML</a>) and study the Cauchy problem. Without loss of generality, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M16">View MathML</a>. In this paper, we consider the Cauchy problem for (1)-(4) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M17">View MathML</a> with given initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M19">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M2">View MathML</a>, as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M21">View MathML</a>

(5)

and far-field behavior

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M22">View MathML</a>

(6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M23">View MathML</a> is some fixed positive constant.

The global well-posedness and dynamical behaviors of MHD system are rather difficult to investigate because of the strong coupling and interplay interaction between the fluid motion and the magnetic fields. Recently, there is much more important progress on the mathematical analysis of these topics for the (nonhomogeneous or homogeneous) MHD system (see, for example, [3-20]). Here, we only mention some of them. Kawashima [14] obtained the global existence of smooth solutions in the two-dimensional case when the initial data are a small perturbation of some given constant state. Li-Xu-Zhang showed in [15] the global well-posedness and large-time behavior of classical solutions to the Cauchy problem of compressible MHD for regular initial data with small energy but possibly large oscillations. In [9,18], Hoff and Tsyganov obtained the global existence and uniqueness of weak solutions with small initial energy. Umeda-Kawashima-Shizuta [17] studied the global existence and time decay rate of smooth solutions to the linearized two-dimensional compressible MHD equations. The optimal decay estimates of classical solutions to the compressible MHD system were obtained by Zhang-Zhao [20] when the initial data are close to a nonvacuum equilibrium. Hu-Wang [10,11] and Fan-Yu [8] proved the global existence of renormalized solutions to the compressible MHD equations for general large initial data. When the viscosity and resistivity go to zero, Zhang [19] showed that the solution of the Cauchy problem for the nonhomogeneous incompressible MHD system converges to the solution of the ideal MHD system and the convergence rate was also obtained. Craig-Huang-Wang [7] obtained the global existence and uniqueness of strong solutions for initial data with small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M24">View MathML</a>-norm in the bounded or unbounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M25">View MathML</a>.

In [12], Huang-Wang considered the global strong solutions to (1)-(4) in the bounded domain with suitable boundary conditions on u and B. Their arguments actually depend on the size of the domain, and so they cannot be applied to the Cauchy problem directly. Then one natural question may be raised: whether the global strong solutions exist in the whole space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M26">View MathML</a>. Here, we want to answer the question. Our main result is stated as follows.

Theorem 1.1Assume that the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M2">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M19">View MathML</a>satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M30">View MathML</a>

(7)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M31">View MathML</a>. Then for any given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M32">View MathML</a>, there exists a unique global strong solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M33">View MathML</a>of (1)-(6) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M34">View MathML</a>

(8)

The proof of Theorem 1.1 is mainly based on a critical Sobolev inequality of logarithmic type which was recently proved by Huang-Wang [12] and is originally due to Brezis-Wainger [21]. The main difficulty compared with [12] is that we should bound all the desired estimates without the restriction on the size of the domain, especially that the Poincaré inequality is not the same from the bounded domain to the whole spaces.

For convenience, we explain the notions used throughout this paper. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M35">View MathML</a>

The standard homogeneous and inhomogeneous Sobolev spaces are defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M36">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M37">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M38">View MathML</a>.

The paper is organized as follows. In Section 2, we state some well-known inequalities and basic facts which will be used frequently later. The proof of Theorem 1.1 will be cast in Section 3.

2 Preliminaries

In this section, we list some useful lemmas which will be frequently used in the next sections. We start from the local existence of strong solutions, which is similar to [4] or [14].

Lemma 2.1Assume that the conditions of Theorem 1.1 hold. Then there exists a positive time<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M39">View MathML</a>such that the Cauchy problem (1)-(6) admits a unique strong solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M40">View MathML</a>.

Next is the well-known Gagliardo-Nirenberg inequality (see [22]).

Lemma 2.2For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M41">View MathML</a>, we have for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M42">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M43">View MathML</a>

(9)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M44">View MathML</a>is a positive constant depending only onp. In addition, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M45">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M46">View MathML</a>, then there exists a universal positive constantCsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M47">View MathML</a>

(10)

Next, we list the Poincaré type inequality, which yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M48">View MathML</a> even when the vacuum states appear.

Lemma 2.3Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M49">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M51">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M52">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M53">View MathML</a>

(11)

whereCdepends only on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M55">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M56">View MathML</a>.

Proof The proof of this lemma can easily be deduced by (9), Hölder’s inequality and the following equality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M57">View MathML</a>

so the details are omitted here. □

In the following, in order to improve the regularity of the velocity, we need to use the estimates of the Stokes equations. We refer the reader to [23,24] for details.

Lemma 2.4Consider the following stationary Stokes equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M58">View MathML</a>

Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M59">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M60">View MathML</a>), there exists a positive constantC, depending only onmandp, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M61">View MathML</a>

(12)

To improve the regularity of the magnetic fields, we need the following result on the elliptic system.

Lemma 2.5Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M62">View MathML</a>is a weak solution of the Poisson equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M63">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M64">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M65">View MathML</a>). Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M66">View MathML</a>

(13)

with some constantCdepending only onq.

To bound the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M67">View MathML</a>-norm of the gradient of the velocity, we will apply a critical Sobolev inequality of logarithmic type which was proved by Huang-Wang [12]. This is the key tool for the proof of Theorem 1.1.

Lemma 2.6For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M68">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M69">View MathML</a>, assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M70">View MathML</a>. Then there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M71">View MathML</a>, independent ofsandt, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M72">View MathML</a>

(14)

3 Proof of Theorem 1.1

This section is devoted to obtaining the proof of Theorem 1.1. According to Lemma 2.1, a local strong solution of the Cauchy problem (1)-(6) exists. Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M73">View MathML</a> is the first blowup time of the strong solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M74">View MathML</a> to the Cauchy problem, it suffices to prove there actually exists a generic positive constant M (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M75">View MathML</a>), depending only on the initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M73">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M78">View MathML</a>

(15)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M79">View MathML</a>. Then due to the local existence theorem (Lemma 2.1), it can easily be shown that the strong solution can be extended beyond <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M73">View MathML</a>. This conclusion contradicts the assumption on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M73">View MathML</a>. Thus, the strong solution exists globally on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M82">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M32">View MathML</a>. Hence the proof of Theorem 1.1 is therefore completed.

The proof of (15) is based on a series of lemmas. For simplicity, throughout the remainder of this paper, we denote by C a generic constant which depends only on the initial data and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M73">View MathML</a> and may change from line to line.

First, the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M85">View MathML</a>-norm of the density can be obtained easily by using the method of characteristics, we list the following lemma without proof.

Lemma 3.1For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M79">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M87">View MathML</a>

(16)

Next, the basic energy inequalities are used.

Lemma 3.2For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M79">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M89">View MathML</a>

(17)

The following estimates are the key estimates in the proof of Theorem 1.1, which depends on the critical Sobolev inequality of logarithmic type (see Lemma 2.6).

Lemma 3.3For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M79">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M91">View MathML</a>

(18)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M92">View MathML</a>is the material derivative off.

Proof First, multiplying (2) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M93">View MathML</a> and integrating the resultant equation by parts over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M26">View MathML</a> on x, one deduces that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M95">View MathML</a>

(19)

For the first term on the right-hand side of (19), using Young’s inequality and (16), one shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M96">View MathML</a>

Next, the second term can be deduced as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M97">View MathML</a>

Then, substituting the above two estimates into (19), one obtains

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M98">View MathML</a>

(20)

Multiplying (3) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M99">View MathML</a> and integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M26">View MathML</a> by parts, one deduces that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M101">View MathML</a>

(21)

The term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M102">View MathML</a> on the left-hand side of (20) cannot be determined positive or negative, thus we have to control it by some appropriate positive terms. Note that it follows from Gagliardo-Nirenberg inequality that we may deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M103">View MathML</a>

Then multiplying (21) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M104">View MathML</a>, adding it to (20), and integrating the resulting equation over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M105">View MathML</a> on time, we finally deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M106">View MathML</a>

(22)

To proceed, we have to estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M107">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M108">View MathML</a>. First, due to (11), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M109">View MathML</a>

(23)

For convenience, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M110">View MathML</a>

Then, combining (17), (22), and (23), we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M111">View MathML</a>

(24)

To proceed, we have to get the appropriate bound on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M107">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M108">View MathML</a>. Thus, due to (13), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M114">View MathML</a>

(25)

where we have used (11) and (24). Similarly, we conclude from (12), (11), and (24) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M115">View MathML</a>

(26)

Hence, combining (25) and (26), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M116">View MathML</a>

(27)

Thus, keeping the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M117">View MathML</a> in mind, we conclude from (27) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M118">View MathML</a>

Substituting the above estimate into (25), we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M119">View MathML</a>

(28)

It follows from the basic energy estimate that one can choose the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M120">View MathML</a> small enough, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M121">View MathML</a>

Substituting the above estimate into (25), we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M122">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M123">View MathML</a>

from which we complete the proof of this lemma. □

Remark 3.1 Due to (18) and the definition of the material derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M124">View MathML</a>, we show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M125">View MathML</a>

(29)

by the following simple fact, i.e.:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M126">View MathML</a>

where we have used (9), (16), (17), and (18).

The following lemma is devoted to improving the time regularity of u and B.

Lemma 3.4For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M79">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M128">View MathML</a>

(30)

Proof Differentiating (2) with respect to t, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M129">View MathML</a>

Multiplying the above equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M93">View MathML</a>, then integrating the resulting equation over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M131">View MathML</a> on x, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M132">View MathML</a>

(31)

Now, we estimate each term on the right-hand side of (31). First, due to (1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M133">View MathML</a>

Next, it follows from (1), (18), Hölder’s inequality, and Young’s inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M134">View MathML</a>

Then one obtains

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M135">View MathML</a>

where we have used (9), (11), and (24). Finally, as for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M136">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M137">View MathML</a>, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M138">View MathML</a>

Hence, substituting all the above estimates into (31), we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M139">View MathML</a>

(32)

From now on, we focus on the estimate for B. Differentiating equation (3) with respect to t, multiplying the resulting equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M99">View MathML</a>, and then integrating by parts over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M26">View MathML</a> on x, we finally obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M142">View MathML</a>

(33)

We estimate each term on the right-hand side of (33). First, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M143">View MathML</a>, ones deduce from (11) and (9) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M144">View MathML</a>

Similarly, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M145">View MathML</a>, we show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M146">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M147">View MathML</a>, one deduces

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M148">View MathML</a>

Then, substituting the above estimates on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M147">View MathML</a>, one deduces

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M152">View MathML</a>

(34)

Thus, combining (32) and (34), together with Gronwall’s inequality, one easily completes the proof of (30). This completes the proof of Lemma 3.4. □

Next, we will apply (12) and (13) to improve the higher regularity on the velocity u and magnetic fields B, respectively.

Lemma 3.5For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M79">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M154">View MathML</a>

(35)

Proof Let us rewrite (2) in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M155">View MathML</a>

Then, using Lemma 2.4, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M156">View MathML</a>

(36)

Similarly, due to Lemma 2.5, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M157">View MathML</a>

(37)

Thus, combining the above two inequalities and Young’s inequality, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M158">View MathML</a>

(38)

Then, by Lemmas 2.4 and 2.5, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M159">View MathML</a>

(39)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M160">View MathML</a>

(40)

Then, combining all the above estimates (36)-(40) together we show that (35). This completes the proof of Lemma 3.5. □

Lemma 3.6For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M79">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M162">View MathML</a>

(41)

Proof Differentiating (1) with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M163">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M164">View MathML</a>), multiplying the resultant equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M165">View MathML</a>, then integrating the resulting equation by parts over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M26">View MathML</a> with respect to x, we finally deduce after summing them up that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M167">View MathML</a>

which, combined with (35) and Gronwall’s inequality, yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M168">View MathML</a>

(42)

Similarly, we can also obtain from (1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M169">View MathML</a>

which combined with (42), together with (35) and Gronwall’s inequality, yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M170">View MathML</a>

It follows from (12) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M171">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M172">View MathML</a>. Similarly, we can obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/94/mathml/M173">View MathML</a>. Thus, we obtain (41), and thus complete the proof of Lemma 3.6. □

The proof of Theorem 1.1 is based on all the estimates that we deduced in Lemmas 3.1-3.6. From all the estimates obtained, we arrive at (15), and, finally, the proof of Theorem 1.1 is therefore completed.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

MS carried out the main work and drafted the manuscript. XQ participated in completing the proof of Lemma 3.5. JW participated in completing the proof of Lemma 3.6. All authors read and approved the final manuscript.

Acknowledgements

This work was supported by NSFC-Union Science Foundation of Henan (No. U1304103) and Natural Science Foundation of Henan Province (No. 122300410261).

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