Open Access Research

Blow-up criteria for 3D nematic liquid crystal models in a bounded domain

Jishan Fan1, Gen Nakamura2 and Yong Zhou3*

Author Affiliations

1 Department of Applied Mathematics, Nanjing Forestry University, Nanjing, P.R. China

2 Department of Mathematics, Inha University, Incheon, 402-751, Republic of Korea

3 Department of Mathematics, Zhejiang Normal University, Jinhua, P.R. China

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Boundary Value Problems 2013, 2013:176  doi:10.1186/1687-2770-2013-176


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/176


Received:9 April 2013
Accepted:11 July 2013
Published:26 July 2013

© 2013 Fan 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 prove some blow-up criteria for two 3D density-dependent nematic liquid crystal models in a bounded domain.

MSC: 35Q30, 76D03, 76D09.

Keywords:
liquid crystal; blow-up criterion; bounded domain

1 Introduction

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M1">View MathML</a> be a bounded domain with smooth boundary Ω, and let ν be the unit outward normal vector on Ω. We consider the regularity criterion to the density-dependent incompressible nematic liquid crystal model as follows [1-4]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M2">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M3">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M4">View MathML</a>

(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M5">View MathML</a>

(1.4)

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M6">View MathML</a> with initial and boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M7">View MathML</a>

(1.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M8">View MathML</a>

(1.6)

where ρ denotes the density, u the velocity, π the pressure, and d represents the macroscopic molecular orientations, respectively. The symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M9">View MathML</a> denotes a matrix whose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M10">View MathML</a>th entry is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M11">View MathML</a>, and it is easy to find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M12">View MathML</a>.

When d is a given constant vector, then (1.1)-(1.3) represent the well-known density-dependent Navier-Stokes system, which has received many studies; see [5-7] and references therein.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M13">View MathML</a>, Guillén-González et al.[8] proved the blow-up criterion

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M14">View MathML</a>

(1.7)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M15">View MathML</a>.

It is easy to prove that the problem (1.1)-(1.6) has a unique local-in-time strong solution [6,9], and thus we omit the details here. The aim of this paper is to consider the regularity criterion; we will prove the following theorem.

Theorem 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M18">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M19">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M21">View MathML</a>in Ω and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M22">View MathML</a>onΩ. We also assume that the following compatibility condition holds true: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M23">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M24">View MathML</a>

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M25">View MathML</a>be a local strong solution to the problem (1.1)-(1.6). Ifusatisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M26">View MathML</a>

(1.8)

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M15">View MathML</a>, then the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M25">View MathML</a>can be extended beyond<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M29">View MathML</a>.

Remark 1.1 When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M30">View MathML</a>, our result improves (1.7) to (1.8).

Remark 1.2 By similar calculations as those in [6], we can replace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M31">View MathML</a>-norm in (1.8) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M32">View MathML</a>-norm, and thus we omit the details here.

Remark 1.3 When the space dimension <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M33">View MathML</a>, we can prove that the problem (1.1)-(1.6) has a unique global-in-time strong solution by the same method as that in [10], and thus we omit the details here.

Next we consider another liquid model: (1.1), (1.2), (1.3), (1.5), (1.6) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M34">View MathML</a>

(1.9)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M35">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M6">View MathML</a>. Li and Wang [9] proved that the problem has a unique local strong solution. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M37">View MathML</a>, Fan et al.[11] proved a regularity criterion. The aim of this paper is to study the regularity criterion of the problem in a bounded domain. We will prove the following theorem.

Theorem 1.2Let the initial data satisfy the same conditions in Theorem 1.1 and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M38">View MathML</a>in Ω. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M39">View MathML</a>be a local strong solution to the problem (1.1)-(1.3), (1.5), (1.6) and (1.9). Ifuanddsatisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M40">View MathML</a>

(1.10)

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M15">View MathML</a>, then the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M25">View MathML</a>can be extended beyond<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M29">View MathML</a>.

2 Proof of Theorem 1.1

We only need to establish a priori estimates.

Below we shall use the notation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M44">View MathML</a>

First, thanks to the maximum principle, it follows from (1.1) and (1.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M45">View MathML</a>

(2.1)

Testing (1.3) by u and using (1.1) and (1.2), we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M46">View MathML</a>

(2.2)

Testing (1.4) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M47">View MathML</a> and using (1.1), we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M48">View MathML</a>

(2.3)

Summing up (2.2) and (2.3), we have the well-known energy inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M49">View MathML</a>

(2.4)

Next, we prove the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M50">View MathML</a>

(2.5)

Without loss of generality, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M51">View MathML</a>. Multiplying (1.4) by 2d, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M52">View MathML</a>

(2.6)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M54">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M55">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M56">View MathML</a>. Then (2.5) follows from (2.6) by the maximum principle.

In the following calculations, we use the following Gauss-Green formula [12]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M57">View MathML</a>

(2.7)

and the following estimate [13,14]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M58">View MathML</a>

(2.8)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M59">View MathML</a>.

Taking ∇ to (1.4)i, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M60">View MathML</a>

Testing the above equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M61">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M62">View MathML</a>), using (1.1), (2.7), (2.8), (2.5) and summing over i, we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M63">View MathML</a>

which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M64">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M65">View MathML</a>

(2.9)

Testing (1.3) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M66">View MathML</a>, using (1.1), (1.2), (2.1) and (2.9), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M67">View MathML</a>

(2.10)

By the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M68">View MathML</a>-regularity theory of the Stokes system, it follows from (1.3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M69">View MathML</a>

which yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M70">View MathML</a>

(2.11)

Testing (1.4) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M71">View MathML</a>, using (2.5) and (2.9), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M72">View MathML</a>

(2.12)

On the other hand, by the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M73">View MathML</a>-regularity theory of the elliptic equation, from (1.4), (2.5) and (2.9) we infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M74">View MathML</a>

which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M75">View MathML</a>

(2.13)

Combining (2.11) and (2.13), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M76">View MathML</a>

(2.14)

Putting (2.14) into (2.10) and (2.12) and summing up, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M77">View MathML</a>

which leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M78">View MathML</a>

(2.15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M79">View MathML</a>

(2.16)

It follows from (2.14), (2.15) and (2.16) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M80">View MathML</a>

(2.17)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M81">View MathML</a> to (1.3), testing by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M66">View MathML</a>, using (1.1), (1.2) and (2.15), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M83">View MathML</a>

(2.18)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M81">View MathML</a> to (1.4), testing by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M71">View MathML</a>, using (2.5), (2.15), (2.16) and (2.17), we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M86">View MathML</a>

(2.19)

Combining (2.18) and (2.19), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M87">View MathML</a>

(2.20)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M88">View MathML</a>

(2.21)

It follows from (1.4), (2.21) and (2.16) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M89">View MathML</a>

(2.22)

It follows from (2.14), (2.15), (2.20) and (2.21) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M90">View MathML</a>

(2.23)

It follows from (1.3), (2.20) and (2.23) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M91">View MathML</a>

(2.24)

from which it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M92">View MathML</a>

(2.25)

Applying ∇ to (1.2), testing by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M93">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M94">View MathML</a>) and using (2.25), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M95">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M96">View MathML</a>

(2.26)

and therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M97">View MathML</a>

(2.27)

This completes the proof.

3 Proof of Theorem 1.2

This section is devoted to the proof of Theorem 1.2. We only need to establish a priori estimates.

First, we still have (2.1) and (2.2).

Next, we easily infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M98">View MathML</a>

(3.1)

Testing (1.9) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M99">View MathML</a> and using (1.1) and (3.1), we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M100">View MathML</a>

(3.2)

Summing up (2.2) and (3.2), we have the well-known energy inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M101">View MathML</a>

(3.3)

Taking ∇ to (1.9)i, testing by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M61">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M62">View MathML</a>), using (1.1), (2.7), (2.8) and (3.1), similarly to (2.9), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M104">View MathML</a>

which yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M105">View MathML</a>

(3.4)

We still have (2.10) and (2.11).

Similarly to (2.12), testing (1.9) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M71">View MathML</a>, using (3.1) and (3.4), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M107">View MathML</a>

(3.5)

Similarly to (2.13), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M108">View MathML</a>

(3.6)

Combining (2.11) and (3.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M109">View MathML</a>

(3.7)

Putting (3.7) into (3.5) and (2.10) and using the Gronwall inequality, we still have (2.15), (2.16), (2.17) and (2.18).

Similarly to (2.19), applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M81">View MathML</a> to (1.9), testing by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M71">View MathML</a> and using (3.4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/176/mathml/M112">View MathML</a>

(3.8)

Combining (2.18) and (3.8) and using the Gronwall inequality, we still obtain (2.20) and (2.21).

By similar calculations as those in (2.22)-(2.27), we still arrive at (2.22)-(2.27).

This completes the proof.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

This work is partially supported by the Zhejiang Innovation Project (Grant No. T200905), the ZJNSF (Grant No. R6090109) and the NSFC (Grant No. 11171154). The authors are indebted to the referee for some helpful suggestions.

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