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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

Blow-up criteria for smooth solutions to the generalized 3D MHD equations

Liping Hu1* and Yinxia Wang2

Author Affiliations

1 College of Information and Management Sciences, Henan Agricultural University, Zhengzhou, 450011, China

2 School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou, 450011, China

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Boundary Value Problems 2013, 2013:187  doi:10.1186/1687-2770-2013-187


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/187


Received:22 April 2013
Accepted:5 August 2013
Published:21 August 2013

© 2013 Hu and Wang; 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 focus on the generalized 3D magnetohydrodynamic equations. Two logarithmically blow-up criteria of smooth solutions are established.

MSC: 76D03, 76W05.

Keywords:
generalized MHD equations; blow-up criteria

1 Introduction

We study blow up criteria of smooth solutions to the incompressible generalized magnetohydrodynamics (GMHD) equations in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M1">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M2">View MathML</a>

(1.1)

with the initial condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M3">View MathML</a>

(1.2)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M6">View MathML</a> are non-dimensional quantities corresponding to the flow velocity, the magnetic field and the total kinetic pressure at the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M7">View MathML</a>, while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M9">View MathML</a> are the given initial velocity and initial magnetic field with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M11">View MathML</a>, respectively.

The GMHD equations is a generalized model of MHD equations. It has important physical background. Therefore, the GMHD equations are also mathematically significant. For 3D Navier-Stokes equations, whether there exists a global smooth solution to 3D impressible GMHD equations is still an open problem. In the absence of global well-posedness, the development of blow-up/ non blow-up theory is of major importance for both theoretical and practical purposes. Fundamental mathematical issues such as the global regularity of their solutions have generated extensive research and many interesting results have been established (see [1-5]).

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M12">View MathML</a>, (1.1) reduces to MHD equations. There are numerous important progresses on the fundamental issue of the regularity for the weak solution to (1.1), (1.2) (see [6-18]). A criterion for the breakdown of classical solutions to (1.1) with zero viscosity and positive resistivity in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M1">View MathML</a> was derived in [9]. Some sufficient integrability conditions on two components or the gradient of two components of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M15">View MathML</a> in Morrey-Campanato spaces were obtained in [10]. A logarithmal improved blow-up criterion of smooth solutions in an appropriate homogeneous Besov space was obtained by Wang et al.[11]. Zhou and Fan [15] established various logarithmically improved regularity criteria for the 3D MHD equations in terms of the velocity field and pressure, respectively. These regularity criteria can be regarded as log in time improvements of the standard Serrin criteria established before. Two new regularity criteria for the 3D incompressible MHD equations involving partial components of the velocity and magnetic fields were obtained by Jia and Zhou [17].

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M17">View MathML</a>, (1.1) reduces to Navier-Stokes equations. Leray [19] and Hopf [20] constructed weak solutions to the Navier-Stokes equations, respectively. The solution is called the Leray-Hopf weak solution. Later on, much effort has been devoted to establish the global existence and uniqueness of smooth solutions to the Navier-Stokes equations. Different criteria for regularity of the weak solutions have been proposed and many interesting results have been obtained [21-25].

In the paper, we obtain two logarithmically blow-up criteria of smooth solutions to (1.1), (1.2) in Morrey-Campanato spaces. We hope that the study of equations (1.1) can improve the understanding of the problem of Navier-Stokes equations and MHD equations.

Now we state our results as follows.

Theorem 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M19">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M11">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M22">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M23">View MathML</a>is a smooth solution to (1.1), (1.2) on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M24">View MathML</a>. Ifusatisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M25">View MathML</a>

(1.3)

then the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M26">View MathML</a>can be extended beyond<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M27">View MathML</a>.

We have the following corollary immediately.

Corollary 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M19">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M11">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M32">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M23">View MathML</a>is a smooth solution to (1.1), (1.2) on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M24">View MathML</a>. Suppose thatTis the maximal existence time, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M35">View MathML</a>

(1.4)

Theorem 1.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M19">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M11">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M32">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M23">View MathML</a>is a smooth solution to (1.1), (1.2) on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M24">View MathML</a>. If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M43">View MathML</a>

(1.5)

then the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M26">View MathML</a>can be extended beyond<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M27">View MathML</a>.

We have the following corollary immediately.

Corollary 1.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M19">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M11">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M32">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M23">View MathML</a>is a smooth solution to (1.1), (1.2) on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M24">View MathML</a>. Suppose thatTis the maximal existence time, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M53">View MathML</a>

(1.6)

The paper is organized as follows. We first state some preliminaries on function spaces and some important inequalities in Section 2. Then we prove main results in Section 3 and Section 4, respectively.

2 Preliminaries

Before stating our main results, we recall the definition and some properties of the homogeneous Morrey-Campanato space.

Definition 2.1 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M54">View MathML</a>, the Morrey-Campanato space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M55">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M56">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M57">View MathML</a> denotes the ball of center x with radius R.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M58">View MathML</a>, we define the homogeneous space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M59">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M60">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M61">View MathML</a> is the space of all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M62">View MathML</a> functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M63">View MathML</a> with compact support. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M64">View MathML</a> is a Banach space when it is equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M65">View MathML</a>

where the infimum is taken over all possible decompositions.

Lemma 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M66">View MathML</a>andp, qsatisfy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M67">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M68">View MathML</a>is the dual space of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M64">View MathML</a>.

Lemma 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M71">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M72">View MathML</a>. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M73">View MathML</a>

Then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M74">View MathML</a>such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M75">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M76">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M77">View MathML</a>

The following lemma comes from [21].

Lemma 2.3Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M78">View MathML</a>. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M79">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M81">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M82">View MathML</a>

(2.1)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M83">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M84">View MathML</a>.

The following inequality is the well-known Gagliardo-Nirenberg inequality.

Lemma 2.4Letj, mbe any integers satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M85">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M86">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M88">View MathML</a>be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M89">View MathML</a>

Then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M90">View MathML</a>, there is a positive constantCdepending only onn, m, j, q, r, θsuch that the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M91">View MathML</a>

(2.2)

with the following exception: if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M81">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M93">View MathML</a>is a nonnegative integer, then (2.2) holds only for a satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M94">View MathML</a>.

3 Proof of Theorem 1.1

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M95">View MathML</a>. We multiply the first equation of (1.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M96">View MathML</a> and use integration by parts. This yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M97">View MathML</a>

(3.1)

Similarly, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M98">View MathML</a>

(3.2)

Summing up (3.1) and (3.2), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M99">View MathML</a>

(3.3)

By using Lemmas 2.1, 2.2 and (2.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M100">View MathML</a>

(3.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M101">View MathML</a>.

We apply integration by parts, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M102">View MathML</a>, Lemmas 2.1, 2.2 and (2.2). This gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M103">View MathML</a>

(3.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M101">View MathML</a>.

Similarly, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M105">View MathML</a>

(3.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M101">View MathML</a>.

Substituting (3.4)-(3.6) into (3.3) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M107">View MathML</a>

(3.7)

Owing to (1.3), we know that for any small constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M108">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M109">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M110">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M111">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M112">View MathML</a>

By (3.7), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M113">View MathML</a>

(3.8)

It follows from (3.8) and Gronwall’s inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M114">View MathML</a>

(3.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M115">View MathML</a>.

Applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M116">View MathML</a> to the first equation of (1.1), then taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M117">View MathML</a> inner product of the resulting equation with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M118">View MathML</a> and using integration by parts, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M119">View MathML</a>

(3.10)

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M120">View MathML</a>

(3.11)

Combining (3.10)-(3.11), using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M102">View MathML</a> and integration by parts yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M123">View MathML</a>

(3.12)

In what follows, for simplicity, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M124">View MathML</a>.

Using Hölder’s inequality and (2.1), (2.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M125">View MathML</a>

(3.13)

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M126">View MathML</a>

(3.14)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M127">View MathML</a>

(3.15)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M128">View MathML</a>

(3.16)

Inserting (3.13)-(3.16) into (3.12) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M129">View MathML</a>

Gronwall’s inequality implies the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M130">View MathML</a>-norm of u and B provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M131">View MathML</a>, which can be achieved by the absolute continuous property of integral (1.3). We have completed the proof of Theorem 1.1. □

4 Proof of Theorem 1.2

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M95">View MathML</a>. Multiplying the first equation of (1.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M96">View MathML</a> and using integration by parts, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M134">View MathML</a>

(4.1)

Similarly, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M135">View MathML</a>

(4.2)

Summing up (4.1) and (4.2), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M136">View MathML</a>

(4.3)

Using Lemmas 2.1, 2.2 and (2.2), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M137">View MathML</a>

(4.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M101">View MathML</a>.

Similarly, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M139">View MathML</a>

(4.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M101">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M141">View MathML</a>

(4.6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M142">View MathML</a>

(4.7)

Combining (4.3)-(4.7) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M143">View MathML</a>

(4.8)

Thanks to (1.5), we know that for any small constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M108">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M109">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M146">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M111">View MathML</a>, set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M148">View MathML</a>

(4.9)

By (4.8) and (4.9), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M149">View MathML</a>

(4.10)

Equation (4.10) and Gronwall’s inequality give the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M150">View MathML</a>

(4.11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M115">View MathML</a>.

From (4.11), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/187/mathml/M130">View MathML</a> estimate for this case is the same as that for Theorem 1.1. Thus, Theorem 1.2 is proved. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

LH and YW carried out the proof of the main part of this article. All authors have read and approved the final manuscript.

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