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Blow-up criterion for 3D compressible viscous magneto-micropolar fluids with initial vacuum

Peixin Zhang

Author Affiliations

School of Mathematical Sciences, Huaqiao University, Quanzhou, 362021, P.R. China

Boundary Value Problems 2013, 2013:160  doi:10.1186/1687-2770-2013-160


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/160


Received:18 February 2013
Accepted:16 June 2013
Published:1 July 2013

© 2013 Zhang; 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, the author establishes a blow-up criterion of strong solutions to 3D compressible viscous magneto-micropolar fluids. It is shown that if the density and the velocity satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M1">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M3">View MathML</a>, then the strong solutions to the Cauchy problem can exist globally over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M4">View MathML</a>. The initial density may vanish on open sets, that is, the initial vacuum is allowed.

MSC: 76N10, 35B44, 35B45.

Keywords:
compressible magneto-micropolar fluids; blow-up criterion; strong solution; vacuum

1 Introduction

In this paper, we consider the following 3D compressible viscous magneto-micropolar fluids:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M5">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M6">View MathML</a> is the spacial coordinate and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M7">View MathML</a> is the time. The unknown functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M12">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M14">View MathML</a>) are the fluid density, velocity, micro-rotational velocity, magnetic field and pressure, respectively. The constants μ, λ, ξ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M16">View MathML</a> and σ are the viscosity coefficients of the fluid satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M17">View MathML</a>

(1.2)

System (1.1)-(1.2) describing the motion of aggregates of small solid ferromagnetic particles relative to viscous magnetic fluids, such as water, hydrocarbon, ester, fluorocarbon, etc., in which they are immersed, covers a wide range of heat and mass transfer phenomena, under the action of magnetic fields, and is of great importance in practical and mathematics applications (see [1]). Indeed, (1.1) is composed of the balance laws of mass, momentum, moment of momentum and magnetohydrodynamic, respectively. Due to its importance in mathematics and physics, there is a lot of literature devoted to the mathematical theory of the compressible viscous magneto-micropolar system (see [2-4]).

For the incompressible magneto-micropolar fluid models where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M18">View MathML</a>, Rojas-Medar [5] established local existence and uniqueness of strong solutions by the Galerkin method. Ortega-Torres and Rojas-Medar [6] proved global existence of strong solutions for small initial data. A BKM type blow-up criterion for smooth solution that relies on the vorticity of velocity only was obtained by Yuan [7]. For regularity results, refer to Yuan [8] and Gala [9].

In particular, if the effect of angular velocity field of the particle’s rotation is omitted, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M19">View MathML</a>, then (1.1) reduces to compressible magnetohydrodynamic equations (MHD). There are numerous important progress on compressible MHD (see [10-12] and the references therein). The local strong solutions to the compressible MHD with large initial data were respectively obtained by Vol’pert-Khudiaev [10] and Fan-Yu [11] in cases that the initial density is strictly positive and the initial density may vanish. Xu-Zhang [12] proved a blow-up criterion that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M20">View MathML</a> is the maximal time of existence of a strong solution, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M21">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M22">View MathML</a> is the weak <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M23">View MathML</a> space and r, s satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M24">View MathML</a>

(1.3)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M25">View MathML</a>, (1.1) reduces to compressible micropolar fluid equations. Mujakovic [13,14] considered the one-dimensional motion of compressible viscous micropolar fluids and studied the local/global existence. The global existence of strong solutions to the 1D model with initial vacuum was also obtained in [15]. For multi-dimensional compressible magneto-micropolar equations, Amirat and Hamdache [16] proved the global existence of weak solutions with finite energy and the adiabatic constant for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M26">View MathML</a>, which generalized Lions’ pioneering work [17] and the work by Feireisl et al.[18]. Chen [19] established the local existence and uniqueness of strong solutions under the assumption that the initial density may vanish, and in [20] Chen et al. proved a blow-up criterion that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M27">View MathML</a>

where r, s satisfy (1.3).

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M19">View MathML</a>, (1.1) reduces to isentropic compressible Navier-Stokes equations. In [21], the authors established a Serrin-type blow-up criterion that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M30">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M31">View MathML</a>

where r, s satisfy (1.3).

In this paper, our main purpose is to establish a blow-up criterion of strong solutions for system (1.1) with the following conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M32">View MathML</a>

(1.4)

To proceed, we introduce the following notations. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M33">View MathML</a>, we denote the standard homogeneous and inhomogeneous Sobolev spaces as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M34">View MathML</a>

To present the main result, we first give the following local existence and uniqueness of strong solutions to the Cauchy problem (1.1), (1.2) and (1.4) with initial vacuum (without proof), which can be obtained by the same method developed by Choe-Kim in [22] (see also Fan-Yu [11] and Chen [19] for MHD and compressible micropolar fluids, respectively).

Theorem 1.1Assume that for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M35">View MathML</a>, the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M36">View MathML</a>satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M37">View MathML</a>

(1.5)

and the compatibility conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M38">View MathML</a>

(1.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M39">View MathML</a>

(1.7)

with some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M40">View MathML</a>. Then there exists a positive time<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M41">View MathML</a>such that the problem (1.1), (1.2) and (1.4) has a unique strong solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M42">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M43">View MathML</a>satisfying, for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M44">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M45">View MathML</a>

(1.8)

Motivated by [20,21] and [12], we have the main purpose in this paper to prove a blow-up criterion for the problem (1.1), (1.2) and (1.4). More precisely, the main result in this paper reads as follows.

Theorem 1.2Assume that the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M46">View MathML</a>satisfies (1.5)-(1.7). Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M47">View MathML</a>be a strong solution of the Cauchy problem (1.1), (1.2) and (1.4) with the regularities (1.8). If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M48">View MathML</a>is the maximal time of existence, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M49">View MathML</a>

(1.9)

for anyrandssatisfying (1.3).

Remark 1.3 Theorem 1.1 proves that the strong solutions of (1.1), (1.2) and (1.4) can exist only in a small time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M50">View MathML</a>, which means that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M50">View MathML</a> is the maximal time of existence, then there must be some component of the fluid mechanics blow-ups. Theorem 1.2 points out one kind of blow-up mechanics.

Remark 1.4 There is no any additional growth condition on the micro-rotational velocity w and magnetic field H. This reveals that the density and the linear velocity play a more important role compared to the angular velocity of rotation of particles and the magnetic field in the regularity theory of solutions to 3D compressible magneto-micropolar fluid flows.

The rest of the paper is devoted to completing the proof of Theorem 1.2.

2 Proof of Theorem 1.2

First, we give the following well-known Gagliardo-Nirenberg inequality that will be used frequently.

Lemma 2.1For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M53">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M54">View MathML</a>, there exists some generic constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M55">View MathML</a>, which may depend onp, qandr, such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M56">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M57">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M58">View MathML</a>

(2.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M59">View MathML</a>

(2.2)

The following BKM’s type inequality which will be used to estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M60">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M61">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M62">View MathML</a> can be found in [12].

Lemma 2.2For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M63">View MathML</a>, there is a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M64">View MathML</a>, depending only onq, such that the following estimate holds for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M65">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M66">View MathML</a>

(2.3)

The proof of Theorem 1.2 is based on the contradiction arguments. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M42">View MathML</a> be a strong solution of the problem (1.1), (1.2) and (1.4) as described in Theorem 1.1. Suppose that (1.9) is false, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M68">View MathML</a>

(2.4)

where r, s satisfy (1.3) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M69">View MathML</a> is a constant.

One can easily deduce from the following energy estimate (1.1), (1.2) and (1.4).

Lemma 2.3It holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M70">View MathML</a>

(2.5)

Here and hereafter, C denotes a generic positive constant which may depend onμ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M15">View MathML</a>, λ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M16">View MathML</a>, ξ, σ, A, γ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M78">View MathML</a>, Tand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M69">View MathML</a>.

We denote the material derivative of f by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M80">View MathML</a> and set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M81">View MathML</a>

(2.6)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M82">View MathML</a> due to (1.1)5, we have from (1.1)2 and (1.1)3 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M83">View MathML</a>

(2.7)

Thus, from the standard <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M84">View MathML</a>-estimate of an elliptic system, we have the following lemma.

Lemma 2.4Under the condition (2.4), it holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M85">View MathML</a>

(2.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M86">View MathML</a>

(2.9)

Proof In view of standard <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M87">View MathML</a>-estimates of elliptic system (2.7), one immediately obtains (2.8). By (2.1) and (2.4), we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M88">View MathML</a>

which, combined with (2.8), yields (2.9) immediately. □

The next lemma is concerned with the higher integrability of H under the assumption (2.4).

Lemma 2.5Under the condition (2.4), it holds for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M89">View MathML</a>that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M90">View MathML</a>

(2.10)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M91">View MathML</a>is a positive constant depending onq.

The proof is similar to Lemma 3.3 in [12] and is omitted here.

With the help of (2.4) and Lemmas 2.3-2.5, we can prove the following key lemma.

Lemma 2.6Under the condition (2.4), it holds that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M92">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M93">View MathML</a>

(2.11)

Proof Multiplying (1.1)2, (1.1)3 and (1.1)4 by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M94">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M95">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M96">View MathML</a>, respectively, and integrating the resulting equations by parts, one obtains after summing up that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M97">View MathML</a>

(2.12)

To estimate the first term on the right-hand side of (2.12), we observe that P satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M98">View MathML</a>

Hence, using (2.4), (2.5) and (2.10) yields that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M99">View MathML</a>

(2.13)

where we have used Young’s inequality and (2.1).

For the second term, we have, after integration by parts, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M100">View MathML</a>

(2.14)

and by the Cauchy-Schwarz inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M101">View MathML</a>

(2.15)

Similarly, integrating by parts and using the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M102">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M103">View MathML</a>

(2.16)

For the last three terms on the right-hand side of (2.12), one has from (2.4) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M104">View MathML</a>

(2.17)

Thus, putting (2.13)-(2.17) into (2.12) and choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M105">View MathML</a> suitably small, we infer from (2.8) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M106">View MathML</a>

(2.18)

For any r, s satisfying (1.3), we have by the Hölder and Sobolev inequalities that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M107">View MathML</a>

(2.19)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M108">View MathML</a>.

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M109">View MathML</a>, H, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M110">View MathML</a>, ∇w, ∇H into (2.19) and using (2.10), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M111">View MathML</a>

(2.20)

By the standard <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M84">View MathML</a>-estimate, one can deduce from (2.1), (2.4), (2.8) and (2.10) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M113">View MathML</a>

(2.21)

Furthermore, it follows from (1.1)4 and Sobolev’s embedding inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M114">View MathML</a>

(2.22)

putting (2.21) and (2.22) into (2.20), such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M115">View MathML</a>

(2.23)

which, together (2.21) and (2.18), choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M116">View MathML</a> suitably small, gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M117">View MathML</a>

(2.24)

It is easily seen that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M118">View MathML</a>

Taking this into account, we conclude from (2.4), (2.24) and Gronwall’s inequality that part of (2.11) holds for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M92">View MathML</a>. Note that the estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M120">View MathML</a> is a consequence of (2.4), (2.22) and (2.23). The proof of this lemma is completed. □

Next we prove the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M123">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M124">View MathML</a> by the compatibility conditions (1.6) and (1.7).

Lemma 2.7Under the condition (2.4), it holds that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M92">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M126">View MathML</a>

(2.25)

Proof Applying the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M127">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M127">View MathML</a> to both sides of (1.1)2 and (1.1)3, respectively, and using (1.1)1, one can obtain, after a straightforward calculation, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M129">View MathML</a>

(2.26)

We get after integration by parts that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M130">View MathML</a>

(2.27)

Similarly, we also have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M131">View MathML</a>

(2.28)

After integration by parts, using (1.1)1 and (2.11), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M132">View MathML</a>

(2.29)

Using the definition of the material derivation and integrating by parts, we deduce from (2.1), (2.5) and (2.11) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M133">View MathML</a>

(2.30)

and, similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M134">View MathML</a>

(2.31)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M135">View MathML</a>

(2.32)

The ninth term on the right-hand side of (2.26) can be estimated as follows, integrating by parts, using (2.1), (2.5), (2.10), (2.11) and Hölder’s inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M136">View MathML</a>

(2.33)

In a similar manner, one also has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M137">View MathML</a>

(2.34)

Putting (2.27)-(2.34) into (2.26), using the Cauchy-Schwarz inequality and choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M116">View MathML</a> suitably small, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M139">View MathML</a>

(2.35)

To estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M123">View MathML</a>, one can differentiate (1.1)4 with respect to t, multiply the resulting equations by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M96">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M87">View MathML</a>, and integrate by parts over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M143">View MathML</a> to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M144">View MathML</a>

(2.36)

Integrating by parts and using (1.1)5, (2.1), (2.10) and (2.11), then we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M145">View MathML</a>

for some positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M146">View MathML</a>. For the second term on the right-hand side of (2.36), integrating by parts and using (2.1) give

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M147">View MathML</a>

Putting the estimates of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M149">View MathML</a> into (2.36) and choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M150">View MathML</a> small enough, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M151">View MathML</a>

(2.37)

Then, combining (2.35) and (2.37), using Young’s inequality, and choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M152">View MathML</a> suitably small yield that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M153">View MathML</a>

(2.38)

Firstly, we use (2.4)-(2.6), (2.9), (2.10), (2.11), (2.1) and (2.2) to infer from the standard <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M84">View MathML</a>-estimate that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M155">View MathML</a>

(2.39)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M156">View MathML</a>

(2.40)

Moreover, by the standard <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M87">View MathML</a>-estimate of an elliptic system, we infer from (1.1)4, (2.1), (2.2) and (2.11) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M158">View MathML</a>

and hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M159">View MathML</a>

(2.41)

Combining (2.39)-(2.41), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M160">View MathML</a>

(2.42)

Now, putting (2.41) and (2.42) into (2.38), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M161">View MathML</a>

from which and (2.11), we immediately obtain (2.25) by Gronwall’s inequality, (1.6) and (1.7). As a result of (2.41), we can also deduce the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M124">View MathML</a>. □

The next lemma is used to bound the density gradient and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M163">View MathML</a>.

Lemma 2.8Under the condition (2.4), it holds that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M92">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M165">View MathML</a>

(2.43)

for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M35">View MathML</a>.

Proof Differentiating (1.1)1 with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M167">View MathML</a> and multiplying it by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M168">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M169">View MathML</a>) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M87">View MathML</a>, we obtain, after integrating by parts and summing up, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M171">View MathML</a>

(2.44)

It follows from (2.1), (2.4), (2.6)-(2.9), (2.25) and the interpolation inequality that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M35">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M173">View MathML</a>

where (2.2) and (2.25) were also used to get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M174">View MathML</a>. So, putting this into (2.44) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M175">View MathML</a>

(2.45)

We now estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M60">View MathML</a>. To do this, we first observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M177">View MathML</a>

Hence, using the standard <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M84">View MathML</a>-estimate of an elliptic system leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M179">View MathML</a>

From (1.1)3 and the standard <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M87">View MathML</a>-estimate of the elliptic system, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M181">View MathML</a>

(2.46)

and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M182">View MathML</a>

(2.47)

This, together with Lemmas 2.2 and 2.6, gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M183">View MathML</a>

(2.48)

Now, if we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M184">View MathML</a> and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M185">View MathML</a>

then it is seen from (2.45) and (2.48) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M186">View MathML</a>

due to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M187">View MathML</a>. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M188">View MathML</a>

(2.49)

On the other hand, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M189">View MathML</a>

we thus deduce from (2.11), (2.25), (2.4), (2.8), (2.9) and (2.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M190">View MathML</a>

(2.50)

As a result, it follows from (2.49) and Gronwall’s inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M191">View MathML</a>

and consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M192">View MathML</a>

(2.51)

From this and (2.25), (2.48), (2.50), one obtains

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M193">View MathML</a>

(2.52)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M194">View MathML</a> in (2.45), we get, by using (2.52) and (2.25) and Gronwall’s inequality, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M195">View MathML</a>

(2.53)

Moreover, the standard <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M87">View MathML</a>-estimate of an elliptic system and (1.1)2, together with (2.4), (2.11) and (2.25), implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M197">View MathML</a>

(2.54)

Similar to the proof of (2.47), there are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M198">View MathML</a>

where we have used (2.1), (2.11), (2.46), (2.51) and (2.54). From this, together with (2.25), (2.46) and (2.51)-(2.54), we can deduce (2.43). □

As a consequence of Lemmas 2.6-2.8, we have the following lemma.

Lemma 2.9Under the condition (2.4), it holds that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M92">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M200">View MathML</a>

(2.55)

The proof is the same as that of Lemma 3.6 in [20] and is omitted here.

With the help of Lemmas 2.3, 2.6-2.9 and the local existence theorem, we can complete the proof of Theorem 1.2 by the contradiction arguments. In fact, in view of Lemmas 2.3, 2.6-2.9, it is easy to see that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M201">View MathML</a> have the same regularities imposed on the initial data (1.5) at the time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M202">View MathML</a>. This implies that the compatibility conditions (1.6) and (1.7) are satisfied at the time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M203">View MathML</a>. Thus, we can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M204">View MathML</a> as the initial data and apply the local existence theorem to extend the local strong solutions beyond <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M203">View MathML</a>. This contradicts the assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/160/mathml/M203">View MathML</a> is the maximal time of existence.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

This work is partially supported by the Fundamental Research Funds for the Central Universities (Grant No. 11QZR16), the National Natural Science Foundation of China (Grant No. 11001090).

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