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On the well-posedness of the incompressible porous media equation in Triebel-Lizorkin spaces

Wenxin Yu1* and Yigang He2

Author Affiliations

1 College of Electrical and Information Engineering, Hunan University, Changsha, 410000, P.R. China

2 School of Electrical and Automation Engineering, Hefei University of Technology, Anhui Pro., Hefei, 230009, P.R. China

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Boundary Value Problems 2014, 2014:95  doi:10.1186/1687-2770-2014-95

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/95


Received:22 January 2014
Accepted:24 March 2014
Published:6 May 2014

© 2014 Yu and He; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

In this paper, we prove the local well-posedness for the incompressible porous media equation in Triebel-Lizorkin spaces and obtain blow-up criterion of smooth solutions. The main tools we use are the Fourier localization technique and Bony’s paraproduct decomposition.

MSC: 76S05, 76D03.

Keywords:
well-posedness; incompressible porous media equation; blow-up criterion; Fourier localization; Bony’s paraproduct decomposition; Triebel-Lizorkin space

1 Introduction

In this paper, we are concerned with the incompressible porous media equation (IPM) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M1">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M2">View MathML</a> or 3):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M3">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M5">View MathML</a>, θ is the liquid temperature, u is the liquid discharge, p is the scalar pressure, k is the matrix of position-independent medium permeabilities in the different directions, respectively, divided by the viscosity, g is the acceleration due to gravity, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M6">View MathML</a> is the last canonical vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M7">View MathML</a>. For simplicity, we only consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M8">View MathML</a>.

By rewriting Darcy’s law we obtain the expression of velocity u only in terms of temperature θ[1,2]. In the 2D case, thanks to the incompressibility, taking the curl operator first and the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M9">View MathML</a> operator second on both sides of Darcy’ law, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M10">View MathML</a>

thus the velocity u can be recovered as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M11">View MathML</a>

Through integration by parts we finally get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M12">View MathML</a>

(1.2)

where the kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M13">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M14">View MathML</a>

Similarly, in 3D case, applying the curl operator twice to Darcy’s law, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M15">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M16">View MathML</a>, thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M17">View MathML</a>

(1.3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M18">View MathML</a>

We observe that, in general, each coefficient of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M19">View MathML</a> (with t as parameter) is only the linear combination of the Calderoń-Zygmund singular integral (with the definition see the sequel) of θ and θ itself. We write the general version as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M20">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M24">View MathML</a> are all operators mapping scalar functions to vector-valued functions and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M25">View MathML</a> equals a constant multiplication operator whereas <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M26">View MathML</a> means a Calderón-Zygmund singular integral operator. Especially the corresponding specific forms in 2D or 3D are shown as (1.2) or (1.3).

We observe that the system (IPM) is not more than a transport equation with non-local divergence-free velocity field (the specific relationship between velocity and temperature as (1.4) shows). It shares many similarities with another flow model - the 2D dissipative quasi-geostrophic (QG) equation, which has been intensively studied by many authors [3-8]. From a mathematical point of view, the system (IPM) is somewhat a generalization of the (QG) equation. Very recently, the system (IPM) was introduced and investigated by Córdoba et al. In [2], they treated the (IMP) in 2D case and obtained the local existence and uniqueness in Hölder space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M27">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M28">View MathML</a> by the particle-trajectory method and gave some blow-up criteria of smooth solutions. Recently, they proved non-uniqueness for weak solutions of (IPM) in [9]. For the dissipative system related (IPM), in [1], the authors obtained some results on strong solutions, weak solutions and attractors. For finite energy they obtained global existence and uniqueness in the subcritical and critical cases. In the supercritical case, they obtained local results in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M30">View MathML</a> and extended to be global under a small condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M31">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M32">View MathML</a>, where c is a small fixed constant.

Recently, Chae studied the local well-posedness and blow-up criterion for the incompressible Euler equations [10,11], and quasi-geostrophic equations [12] in Triebel-Lizorkin spaces. As is well known, Triebel-Lizorkin spaces are the unification of several classical function spaces such as Lebegue spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M33">View MathML</a>, Sobolev spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M34">View MathML</a>, Lipschitz spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M35">View MathML</a>, and so on. In [10], the author first used the Littlewood-Paley operator to localize the Euler equation to the frequency annulus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M36">View MathML</a>, then obtained an integral representation of the frequency-localized solution on the Lagrangian coordinates by introducing a family of particle-trajectory mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M37">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M38">View MathML</a>

(1.5)

where v is a divergence-free velocity field and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M39">View MathML</a> is a frequency projection to the ball <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M40">View MathML</a> (see Section 2). He also used the following equivalent relation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M41">View MathML</a>

(1.6)

to estimate the frequency-localized solutions of the Euler equations or quasi-geostrophic equations in Triebel-Lizorkin spaces. However, it seems difficult to give a strict proof for the above equivalent relation (1.6) and its related counterpart due to the lack of a uniform change of the coordinates independent of j. To avoid this trouble, Chen et al.[13] introduced a particle trajectory mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M42">View MathML</a> independent of j defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M43">View MathML</a>

and then established a new commutator estimate to obtain the local well-posedness of the ideal MHD equations in the Triebel-Lizorkin spaces.

In this paper, we will adapt the method of Chen et al.[13] to establish the local well-posedness for the incompressible porous media equation (1.1) and to obtain a blow-up criterion of smooth solutions in the framework of Triebel-Lizorkin spaces.

Now we state our result as follows.

Theorem 1.1 (i) Local-in-time existence. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M45">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M46">View MathML</a>, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M47">View MathML</a>such that (1.1) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M48">View MathML</a>.

(ii) Blow-up criterion. The local-in-time solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M49">View MathML</a>constructed in (i) blows up at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M50">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M51">View MathML</a>, i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M52">View MathML</a>

if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M53">View MathML</a>

2 Preliminaries

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M54">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M55">View MathML</a>. Choose two nonnegative smooth radial functions χ, φ supported, respectively, in ℬ and such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M57">View MathML</a>

We denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M59">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M60">View MathML</a>. Then the dyadic blocks <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M61">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M62">View MathML</a> can be defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M63">View MathML</a>

Formally, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M64">View MathML</a> is a frequency projection to the annulus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M36">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M62">View MathML</a> is a frequency projection to the ball <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M40">View MathML</a>. One easily verifies that with our choice of φ

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M68">View MathML</a>

(2.1)

With the introduction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M61">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M62">View MathML</a>, let us recall the definition of the Triebel-Lizorkin space. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M72">View MathML</a>, the homogeneous Triebel-Lizorkin space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M73">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M74">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M75">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M76">View MathML</a> denotes the dual space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M77">View MathML</a> and can be identified by the quotient space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M78">View MathML</a> with the polynomials space .

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M72">View MathML</a>, we define the inhomogeneous Triebel-Lizorkin space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M51">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M83">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M84">View MathML</a>

We refer to [14] for more details.

Lemma 2.1 (Bernstein’s inequality) [15]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M85">View MathML</a>. There exists a constantCindependent offandjsuch that, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M86">View MathML</a>, the following inequalities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M87">View MathML</a>

Lemma 2.2[14]

For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M85">View MathML</a>, there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M89">View MathML</a>such that the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M90">View MathML</a>

Proposition 2.1[10]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M92">View MathML</a>, or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M93">View MathML</a>, then there exists a constantCsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M94">View MathML</a>

Proposition 2.2[10]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M95">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M96">View MathML</a>. Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M97">View MathML</a>, then there exists a constantCsuch that the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M98">View MathML</a>

Proposition 2.3[13]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M92">View MathML</a>, or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M93">View MathML</a>, andfbe a solenoidal vector field. Then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M80">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M102">View MathML</a>

(2.2)

or for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M103">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M104">View MathML</a>

(2.3)

The classical Calderón-Zygmund singular integrals are operators of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M105">View MathML</a>

where Ω is defined on the unit sphere of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M107">View MathML</a>, and is integrable with zero average and where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M108">View MathML</a>. Clearly, the definition is meaningful for Schwartz functions. Moreover if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M110">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M111">View MathML</a> bounded, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M112">View MathML</a>.

The general version (1.4) of the relationship between u and θ is in fact ensured by the following result (see e.g.[16]).

Lemma 2.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M113">View MathML</a>be a homogeneous function of degree 0, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M114">View MathML</a>be the corresponding multiplier operator defined by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M115">View MathML</a>, then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M116">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M117">View MathML</a>with zero average such that for any Schwartz functionf,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M118">View MathML</a>

Remark 2.1 Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M119">View MathML</a>, the Fourier multiplier of the operator is rather clear. In fact, each component of its multiplier is the linear combination of the term like <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M122">View MathML</a>, which of course belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M123">View MathML</a> and is homogeneous of degree 0.

3 Proof of Theorem 1.1

We divide the proof of Theorem 1.1 into several steps.

Step 1. A priori estimates.

Taking the operation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M124">View MathML</a> on both sides of the first equation of (1.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M125">View MathML</a>

(3.1)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M42">View MathML</a> be the solution of the following ordinary differential equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M127">View MathML</a>

(3.2)

Then it follows from (3.1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M128">View MathML</a>

(3.3)

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M129">View MathML</a>

(3.4)

Multiplying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M130">View MathML</a>, taking the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M131">View MathML</a> norm on both sides of (3.4), we get by using the Minkowski inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M132">View MathML</a>

(3.5)

Next, taking the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M111">View MathML</a> norm with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M134">View MathML</a> on both sides of (3.5), we get by using the Minkowski inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M135">View MathML</a>

(3.6)

Using the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M136">View MathML</a> is a volume-preserving diffeomorphism due to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M137">View MathML</a>, we get from (3.6) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M138">View MathML</a>

(3.7)

Thanks to Proposition 2.3, the last term on the right side of (3.7) is dominated by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M139">View MathML</a>

(3.8)

and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M140">View MathML</a>

(3.9)

where we used (1.4) and the boundedness of the Calderón-Zygmund singular integral operator on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M73">View MathML</a>.

Now from (1.1) we have immediately

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M142">View MathML</a>

(3.10)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M143">View MathML</a>, since div <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M144">View MathML</a>. Summing up (3.9) and (3.10) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M145">View MathML</a>

(3.11)

which together with the Gronwall inequality gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M146">View MathML</a>

(3.12)

Step 2. Approximate solutions and uniform estimates.

We construct the approximate solutions of (1.1). Define the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M147">View MathML</a> by solving the following systems:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M148">View MathML</a>

(3.13)

We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M149">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M150">View MathML</a> be the solutions of the following ordinary differential equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M151">View MathML</a>

(3.14)

for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M152">View MathML</a>. Then, following the same procedure of estimate leading to (3.11), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M153">View MathML</a>

(3.15)

where we used the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M154">View MathML</a>, Sobolev embedding theorem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M155">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M156">View MathML</a>, (1.4) and the boundedness of the Calderón-Zygmund singular integral operator on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M51">View MathML</a>. Equation (3.15) together with the Gronwall inequality implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M158">View MathML</a>

(3.16)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M159">View MathML</a> independent of n. Thus, if we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M160">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M161">View MathML</a>

we have, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M162">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M163">View MathML</a>

(3.17)

by the standard induction arguments. Then, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M164">View MathML</a>. Moreover, it follows from Proposition 2.1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M165">View MathML</a>

by Sobolev embedding and the boundedness of the Calderón-Zygmund singular integral operator on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M51">View MathML</a>, and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M167">View MathML</a>

(3.18)

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M168">View MathML</a>. This together with (3.17) gives the uniform estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M169">View MathML</a> in n.

Step 3. Existence.

We will show that there exists a positive time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M170">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M171">View MathML</a>) independent of n such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M172">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M173">View MathML</a> are Cauchy sequences in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M174">View MathML</a>. For this purpose, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M175">View MathML</a>

Then, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M176">View MathML</a> satisfies the equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M177">View MathML</a>

(3.19)

Applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M124">View MathML</a> to the first equation of (3.19), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M179">View MathML</a>

(3.20)

Exactly as in the proof of (3.7), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M180">View MathML</a>

(3.21)

where we used Proposition 2.1, Proposition 2.3, the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M155">View MathML</a>, and the boundedness of the Calderón-Zygmund singular integral operator on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M51">View MathML</a>. Thanks to the Fourier support of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M183">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M184">View MathML</a>

(3.22)

Now, we estimate the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M111">View MathML</a> norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M186">View MathML</a>. Multiplying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M187">View MathML</a> on both sides of the first equation of (3.19), and integrating the resulting equations over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M1">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M189">View MathML</a>

which together with (3.21) and (3.22) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M190">View MathML</a>

(3.23)

Equation (3.23) together with (3.17) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M191">View MathML</a>

(3.24)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M192">View MathML</a>. Thus, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M193">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M194">View MathML</a>

This implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M195">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M196">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M197">View MathML</a>. By the standard argument, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M198">View MathML</a>, the limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M199">View MathML</a> solves (1.1) with the initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M200">View MathML</a>. The fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M201">View MathML</a> follows from the uniform estimate (3.18).

Step 4. Uniqueness.

Consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M202">View MathML</a> is another solution to (1.1) with the same initial data. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M203">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M204">View MathML</a>. Then δθ satisfies the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M205">View MathML</a>

In the same way as the derivation in (3.24), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M206">View MathML</a>

for sufficiently small T. This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M207">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M208">View MathML</a>.

Blow-up criterion.

For the a priori estimate (3.12), we only need to dominate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M209">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M210">View MathML</a>. From Proposition 2.2 and the boundedness of the Calderón-Zygmund operator from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M211">View MathML</a> into itself, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M212">View MathML</a>

Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M213">View MathML</a>

Thus, the a priori estimate (3.12) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M214">View MathML</a>

By the Gronwall inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M215">View MathML</a>

Therefore, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M216">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M217">View MathML</a>.

On the other hand, it follows from the Sobolev embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M218">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M219">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M220">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M217">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/95/mathml/M216">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgements

This work was supported by the National Natural Science Funds of China for Distinguished Young Scholar under Grant No. 50925727, The National Defense Advanced Research Project Grant Nos. C1120110004, 9140A27020211DZ5102, the Key Grant Project of Chinese Ministry of Education under Grant No. 313018, and the Fundamental Research Funds for the Central Universities (2012HGCX0003).

References

  1. Castro, A, Córdoba, D, Gancedo, F, Orive, R: Incompressible flow in porous media with fractional diffusion. Nonlinearity. 22, 1791–1815 (2009). Publisher Full Text OpenURL

  2. Córdoba, D, Gancedo, F, Orive, R: Analytical behaviour of the two-dimensional incompressible flow in porous media. J. Math. Phys.. 48, 1–19 (2007)

  3. Abidi, H, Hmidi, T: On the global well posedness of the critical quasi-geostrophic equation. SIAM J. Math. Anal.. 40, 167–185 (2008). Publisher Full Text OpenURL

  4. Caffarelli, L, Vasseur, V: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equations. Ann. Math.. 171, 1903–1930 (2010). Publisher Full Text OpenURL

  5. Chen, Q, Miao, C, Zhang, Z: A new Bernstein’s inequality and the 2D dissipative quasigeostrophic equation. Commun. Math. Phys.. 271, 821–838 (2007). Publisher Full Text OpenURL

  6. Córdoba, A, Córdoba, D: A maximum principle applied to the quasi-geostrophic equations. Commun. Math. Phys.. 249, 511–528 (2004)

  7. Kislev, A, Nazarov, F, Volberg, A: Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Invent. Math.. 167, 445–453 (2007). Publisher Full Text OpenURL

  8. Wu, J: Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces. SIAM J. Math. Anal.. 36, 1014–1030 (2004)

  9. Córdoba, D, Faraco, D, Gancedo, F: Lack of uniqueness for weak solutions of the incompressible porous media equation. Arch. Ration. Mech. Anal.. 200, 725–746 (2011). Publisher Full Text OpenURL

  10. Chae, D: On the well-posedness of the Euler equations in the Triebel-Lizorkin spaces. Commun. Pure Appl. Math.. 55, 654–678 (2002). Publisher Full Text OpenURL

  11. Chae, D: On the Euler equations in the critical Triebel-Lizorkin spaces. Arch. Ration. Mech. Anal.. 170, 185–210 (2003). Publisher Full Text OpenURL

  12. Chae, D: The quasi-geostrophic equations in the Triebel-Lizorkin spaces. Nonlinearity. 16, 479–495 (2003). Publisher Full Text OpenURL

  13. Chen, Q, Miao, C, Zhang, Z: On the well-posedness of the ideal MHD equations in the Triebel-Lizorkin spaces. Arch. Ration. Mech. Anal.. 195, 561–578 (2010). Publisher Full Text OpenURL

  14. Triebel, H: Theory of Function Spaces, Birkhäuser Verlag, Basel (1983)

  15. Chemin, J-Y: Perfect Incompressible Fluids, Oxford University Press, New York (1998)

  16. Duoandikoetxea, J: Fourier Analysis, Am. Math. Soc., Providence (2001) (Translated and revised by D. Cruz-Uribe)