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Unique solvability for the non-Newtonian magneto-micropolar fluid

Changjia Wang

Author Affiliations

School of Science, Changchun University of Science and Technology, Changchun, 130022, P.R. China

Boundary Value Problems 2013, 2013:182  doi:10.1186/1687-2770-2013-182


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/182


Received:8 January 2013
Accepted:24 July 2013
Published:8 August 2013

© 2013 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, a motion of an incompressible non-Newtonian magneto-micropolar fluid is considered. We assume that the stress tensor has a p-structure, and we establish the global in time existence and uniqueness of the weak solutions with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M1">View MathML</a> in three dimensions.

1 Introduction and main results

This paper is concerned about the existence and uniqueness of the weak solutions to the non-Newtonian magneto-micropolar fluid equations in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M2">View MathML</a>, which are described by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M3">View MathML</a>

(1.1)

here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M4">View MathML</a> is an open-bounded domain with Lipschitz boundaries, and the unknowns u, ω, b, π denote the velocity of the fluid, the micro-rotational velocity, magnetic field and hydrostatic pressure, respectively. χ, μ, λ are positive numbers associated with properties of the material: χ is the vortex viscosity, μ is spin viscosity and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M5">View MathML</a> is the magnetic Reynold. In (1.1), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M6">View MathML</a> is the symmetric part of the velocity gradient, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M7">View MathML</a>

To (1.1) we append the following initial and boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M8">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M9">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M10">View MathML</a>.

The theory of micropolar fluid was first proposed by Eringen [1] in 1966, which enabled us to consider some physical phenomena that cannot be treated by the classical Navier-Stokes equations for the viscous incompressible fluid, for example, the motion of animal blood, liquid crystals and dilute aqueous polymer solutions, etc.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M11">View MathML</a>, system (1.1) reduces to the classical magneto-micropolar fluid equations, and there are many earlier results concerning the weak and strong solvability in a bounded domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M4">View MathML</a>. For strong solutions, Galdi and Rionero [2] stated, without proof, the results of existence and uniqueness of strong solutions. Rojas-Medar [3] studied it and established the local in time existence and uniqueness of strong solutions by the spectral Galerkin method. In 1999, Ortega-Torres and Rojas-Medar [4] proved global existence of strong solutions with the small initial values. For weak solutions, Rojas-Medar and Boldrini [5] proved the existence of weak solutions, and in the 2D case, also proved the uniqueness of the weak solutions.

On the other hand, there are few existence results about the non-Newtonian magneto-micropolar fluid, i.e., the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M13">View MathML</a> case. In a recent work, Gunzburger et al.[6] studied the reduced problem (with both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M15">View MathML</a>), and gave the global unique solvability of the first initial-boundary value problem in a bounded two or three-dimensional domain. Improved results are proved for the periodic boundary condition case.

In this paper, we will prove the global existence and uniqueness of the weak solutions for the full system (1.1)-(1.3) under the condition that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M16">View MathML</a>. These results are based on the Galerkin method and a series of uniform estimates, which do not depend on the parameters.

Throughout this work, we use a standard notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M17">View MathML</a> (normed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M18">View MathML</a>) for Lebesgue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M19">View MathML</a>-spaces, as well as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M20">View MathML</a> (normed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M21">View MathML</a>) for the usual Sobolev spaces. As usual, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M22">View MathML</a> denotes the set of all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M23">View MathML</a>-functions with the compact support in Ω. Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M24">View MathML</a> and a Banach space X, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M25">View MathML</a> Bochner spaces, which are equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M26">View MathML</a>

We also introduce the following functional vector spaces:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M27">View MathML</a>

We next introduce the definition of a weak solution for problems (1.1)-(1.3).

Definition 1.1 We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M28">View MathML</a> is a weak solution to problems (1.1)-(1.3) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M29">View MathML</a>

(1.4)

satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M30">View MathML</a>

(1.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M31">View MathML</a>

(1.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M32">View MathML</a>

(1.7)

where the symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M33">View MathML</a> denotes a generic duality pairing.

The following theorem gives the main results of this paper.

Theorem 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M4">View MathML</a>be an open-bounded domain with a Lipschitz boundaryΩ. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M37">View MathML</a>. Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M38">View MathML</a>, there exists a unique weak solution to problem (1.1)-(1.3) in the sense of Definition 1.1.

Remark 1.1 If (1.4)-(1.5) hold, it could be easy to introduce the pressure <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M40">View MathML</a>. This will be done at the end of Section 3.

For latter use, let us state some useful inequalities.

Lemma 1.1 (See [7]) (Korn’s inequality)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M41">View MathML</a>. Then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M42">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M43">View MathML</a>

(1.8)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M44">View MathML</a>is open and bounded with a Lipschitz boundary.

Lemma 1.2 (See [8]) (On negative norm)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M41">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M46">View MathML</a>. Then there exists a constantCsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M47">View MathML</a>

Lemma 1.3 (See [9])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M48">View MathML</a>. For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M49">View MathML</a>, there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M50">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M51">View MathML</a>

By using Hölder’s inequality and the imbedding inequality, we could arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M52">View MathML</a>

(1.9)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M53">View MathML</a>

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M54">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M55">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M56">View MathML</a>. We will also apply the so-called multiplicative inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M57">View MathML</a>

(1.10)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M58">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M55">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M60">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M61">View MathML</a>.

Finally, the paper is organized as follows. In Section 2, we focus on the derivation of the priori estimates for the smooth solutions. On the bases of these estimates, in Section 3, we get the existence result with the help of the Galerkin method. The aim of Section 4 is to give the uniqueness criterion.

2 The priori estimates

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M28">View MathML</a> be a smooth solution to system (1.1)-(1.3). The goal of this section is to derive some priori estimates about it. In all the following sections, we always assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M35">View MathML</a> holds.

Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M64">View MathML</a> in (1.5), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M65">View MathML</a> in (1.6), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M66">View MathML</a> in (1.7), and observing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M67">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M68">View MathML</a>

Adding the identities above, noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M69">View MathML</a>, and Korn’s inequality (1.8), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M70">View MathML</a>

After choosing ε properly small, integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M72">View MathML</a>, the Gronwall’s inequality yields that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M73">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M74">View MathML</a> is a constant depending on the time T and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M77">View MathML</a>.

Next, we derive the higher order estimates for ω and b. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M78">View MathML</a> in (1.6), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M79">View MathML</a>

(2.2)

for the first term on the right hand side, we compute by the divergence free conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M80">View MathML</a>

(2.3)

where Hölder’s, Young’s inequality and (1.9), (1.10) have been used.

Inserting (2.3) into (2.2), choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M81">View MathML</a> and integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M83">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M84">View MathML</a>

(2.4)

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M35">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M86">View MathML</a>

(2.5)

Gronwall’s inequality and estimate (2.1) now provide the bound

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M87">View MathML</a>

(2.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M88">View MathML</a> is a constant depending on the time T, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M74">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M90">View MathML</a>.

Next, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M91">View MathML</a> in (1.7) to discover

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M92">View MathML</a>

(2.7)

Reasoning similar to (2.3), we could find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M93">View MathML</a>

(2.8)

For the second term on the right hand side of (2.7), we compute

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M94">View MathML</a>

(2.9)

where we have used Hölder’s, Young’s inequality and (1.9), (1.10). Choosing ε and δ properly small, inserting (2.8)-(2.9) into (2.7) and integrating it over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M83">View MathML</a>, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M97">View MathML</a>

(2.10)

Observing (2.5) and estimate (2.1), then Gronwall’s inequality yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M98">View MathML</a>

(2.11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M99">View MathML</a> is a constant depending on the time T, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M74">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M101">View MathML</a>.

Reasoning analogously to (2.6) and (2.11), it is easy to see that identity (1.6) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M102">View MathML</a>, (1.7) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M103">View MathML</a>, with the help of (2.6) and (2.11), guarantee the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M104">View MathML</a>

(2.12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M106">View MathML</a> are both constants depending only on the time T and some norm of the initial values.

In fact (we here only take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M107">View MathML</a> as an example), set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M103">View MathML</a> in (1.7), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M109">View MathML</a>

(2.13)

Now, we compute, by using Hölder’s, Young’s inequality and (1.9), (1.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M110">View MathML</a>

(2.14)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M111">View MathML</a>

(2.15)

Combining (2.13)-(2.15), by choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M112">View MathML</a>, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M113">View MathML</a>

noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M35">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M115">View MathML</a>, and now estimate (2.1), (2.11) and Gronwall’s inequality imply the estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M107">View MathML</a> in (2.12).

In the following, we will derive the bound for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M117">View MathML</a>. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M118">View MathML</a> in (1.5), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M119">View MathML</a>

Integrating it over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M83">View MathML</a>, by choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M112">View MathML</a> and Korn’s inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M123">View MathML</a>

(2.16)

Now, we compute, by (2.11)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M124">View MathML</a>

(2.17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M125">View MathML</a>

(2.18)

Inserting (2.17)-(2.18) into (2.16), by appealing to Korn’s inequality, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M126">View MathML</a>

(2.19)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M127">View MathML</a> depends on T and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M99">View MathML</a>. Now, Gronwall’s inequality and (2.1) yield that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M129">View MathML</a>

(2.20)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M130">View MathML</a> depends on the time T, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M99">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M133">View MathML</a>.

3 Approximate solutions and existence result

In this section, we show the existence of a weak solution to the system (1.1)-(1.3) via the Galerkin approximations. For this purpose, we take the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M134">View MathML</a> formed by the eigenvectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M136">View MathML</a> , of the Stokes operator and the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M137">View MathML</a> formed by the eigenvectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M136">View MathML</a> , of the Laplace operator. According to the Appendix of [7], the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M134">View MathML</a> form a basis in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M141">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M142">View MathML</a>. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M143">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M144">View MathML</a>, we construct the Galerkin approximations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M145">View MathML</a> being of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M146">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M149">View MathML</a> solve the system of ordinary equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M150">View MathML</a>

(3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M151">View MathML</a>

(3.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M152">View MathML</a>

(3.3)

Moreover, we require that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M155">View MathML</a> satisfy the following initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M156">View MathML</a>

(3.4)

The local solvability is guaranteed by the Carathéodory theorem, and the global unique solvability follows from the fact that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M157">View MathML</a>

with upper bounds C that do not depend on k. Moreover, we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M154">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M155">View MathML</a> the same estimates for all norms we have obtained in Section 2. More precisely, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M161">View MathML</a>

(3.5)

with a constant C that does not depend on k.

Uniform estimates (3.5) imply that there exists a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M163">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M164">View MathML</a> (not relabeled) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M165">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M166">View MathML</a>. Therefore, by making use of the Aubin-Lions lemma (see Lions [10], Theorem 1.5.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M167">View MathML</a>

With the convergence above, it is easy to pass to the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M168">View MathML</a> in (3.1)-(3.3) to find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M169">View MathML</a>

(3.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M170">View MathML</a>

(3.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M171">View MathML</a>

(3.8)

Next, to complete the existence proof, we need to verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M172">View MathML</a>

(3.9)

By Lemma 1.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M173">View MathML</a>

Considering <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M174">View MathML</a> of this identity together with (3.6) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M175">View MathML</a>

and thus (3.9) follows.

Having the estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M176">View MathML</a>

(3.10)

we can now introduce the pressure from (1.5). For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M177">View MathML</a>, define the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M178">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M179">View MathML</a>

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M180">View MathML</a>

By using De Rahm’s theorem (see [11], Lemma 2.7), we obtain a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M181">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M182">View MathML</a>

Moreover, due to estimates (3.10),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M183">View MathML</a>

Then, by Lemma 1.2, there is a generic constant C, depending only on the data such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M184">View MathML</a>

Now, we complete the proof of the existence part of Theorem 1.1.

4 Uniqueness criterion

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M185">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M186">View MathML</a> be both solutions of the problem. Then, for their difference <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M188">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M189">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M190">View MathML</a>

(4.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M191">View MathML</a>

(4.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M192">View MathML</a>

(4.3)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M193">View MathML</a> in (4.1), by Lemma 1.3 and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M194">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M195">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M196">View MathML</a>

for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M197">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M198">View MathML</a>, it follows from Hölder’s, Young’s inequality and (1.9), (1.10) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M199">View MathML</a>

so, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M200">View MathML</a>

(4.4)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M201">View MathML</a> in (4.2) and noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M202">View MathML</a>, it follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M203">View MathML</a>

and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M204">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M205">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M206">View MathML</a>

thus, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M207">View MathML</a>

(4.5)

Similarly, by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M208">View MathML</a> in (4.3), reasoning analogous as above, we could get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M209">View MathML</a>

(4.6)

Adding (4.4)-(4.6) and observing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M210">View MathML</a>, after choosing ε properly small, we finally get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M211">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M212">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M213">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M35">View MathML</a>, then Gronwall’s inequality and the estimates obtained in Section 2 yield that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/182/mathml/M215">View MathML</a>

This completes the proof of the theorem.

Competing interests

The author declares that they have no competing interests.

Authors’ contributions

The author completed the paper. The author read and approved the final manuscript.

Acknowledgements

The author would like to thank the referees for valuable comments and suggestions for improving this paper. This work was partially supported by the National NSF (Grant No. 10971080) of China.

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