SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

A single exponential BKM type estimate for the 3D incompressible ideal MHD equations

Jianli Liu1, Fenglun Wei1 and Kejia Pan23*

Author Affiliations

1 Department of Mathematics, Shanghai University, Shanghai, 200444, P.R. China

2 School of Mathematics and Statistics, Central South University, Changsha, 410083, P.R. China

3 Key Laboratory of Metallogenic Prediction of Nonferrous Metals, Ministry of Education, Central South University, Changsha, 410083, P.R. China

For all author emails, please log on.

Boundary Value Problems 2014, 2014:96  doi:10.1186/1687-2770-2014-96

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/96


Received:27 February 2014
Accepted:15 April 2014
Published:6 May 2014

© 2014 Liu et al.; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

In this paper, we give a Beale-Kato-Majda type criterion of strong solutions to the incompressible ideal MHD equations. Instead of double exponential estimates, we get a single exponential bound on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M1">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M2">View MathML</a>). It can be applied to a system of an ideal viscoelastic flow.

MSC: 35B65, 76W05.

Keywords:
MHD equations; ideal viscoelastic flow; Beale-Kato-Majda criterion; single exponential bound

1 Introduction

In this paper, we will get the Beale-Kato-Majda type criterion for the breakdown of smooth solutions to the incompressible ideal MHD equations in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M3">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M4">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M6">View MathML</a>, u is the flow velocity, h is the magnetic field, p is the pressure, while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M8">View MathML</a> are, respectively, the given initial velocity and initial magnetic field satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M10">View MathML</a>.

Using the standard energy method [1], it is well known that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M12">View MathML</a>, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M13">View MathML</a> such that the Cauchy problem (1) has a unique smooth solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M14">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M15">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M16">View MathML</a>

(2)

Recently, Caflisch et al.[2] extended the well-known result of Beale et al.[3] to the 3D ideal MHD equations. More precisely, they showed that if the smooth solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M14">View MathML</a> satisfies the following condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M18">View MathML</a>

(3)

then the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M14">View MathML</a> can be extended beyond <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M20">View MathML</a>, namely, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M22">View MathML</a>. Many authors also considered the blow-up criterion of the ideal MHD equations in other spaces; see [4-6] and references therein. More recently, for the following incompressible Euler equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M23">View MathML</a>

(4)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M24">View MathML</a>, Chen and Pavlovic [7] showed that if the solution u to (4) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M25">View MathML</a>

(5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M28">View MathML</a>, then the solution u can be extended beyond <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M20">View MathML</a>. The quantity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M30">View MathML</a> was introduced by Constantin in [8] (see also the work of Constantin et al.[9]). For the blow-up criterion of incompressible Euler equations, we refer to [7,10] and references therein.

2 Main results

In this short note, we develop these ideas further and establish an analogous blow-up criterion for solutions of the 3D ideal MHD equations (1). More precisely, we can get the following theorem.

Theorem 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M14">View MathML</a>be a solution to (1) in the class (2) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M32">View MathML</a>. Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M33">View MathML</a>

(6)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M36">View MathML</a>and the definition of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M37">View MathML</a>as above. Then there exists a finite positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M38">View MathML</a>independent of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M14">View MathML</a>andtsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M40">View MathML</a>

(7)

holds for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M41">View MathML</a>.

Remark 2.1 Using a similar method, we also can get the blow-up criterion result about ideal viscoelastic flow

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M42">View MathML</a>

(8)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M44">View MathML</a>.

Theorem 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M45">View MathML</a>be a solution to (8) in the class (2) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M32">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M47">View MathML</a>is defined as above, and that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M48">View MathML</a>

(9)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M49">View MathML</a>. Then there exists a finite positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M38">View MathML</a>independent of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M45">View MathML</a>andtsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M52">View MathML</a>

(10)

holds for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M41">View MathML</a>.

This system arises in the Oldroyd model for an ideal viscoelastic flow, i.e. a viscoelastic fluid whose elastic properties dominate its behavior. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M54">View MathML</a> represents the local deformation gradient of the fluid. The blow-up criterion of the ideal viscoelastic system can be found in [11] and references therein.

3 Proof of Theorem 2.1

For the proof of our main result, firstly we give some properties about the gradient of velocity. Recall that the full gradient of the velocity, ∇u, can be decomposed into symmetric and antisymmetric parts,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M55">View MathML</a>

(11)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M56">View MathML</a>

(12)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M57">View MathML</a> is called the deformation tensor.

In the following lemmas, we recall some important properties of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M57">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M59">View MathML</a> without proof [7,8].

Lemma 3.1For both the symmetric and the antisymmetric parts<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M59">View MathML</a>ofu, the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M62">View MathML</a>bound

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M63">View MathML</a>

(13)

holds.

The antisymmetric part<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M59">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M65">View MathML</a>

(14)

for any vector<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M66">View MathML</a>. The vorticityωsatisfies the identity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M67">View MathML</a>

(15)

(‘<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M68">View MathML</a>denotes principal value) where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M69">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M70">View MathML</a>. Notably,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M71">View MathML</a>

(16)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M72">View MathML</a>denotes the standard measure on the sphere<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M73">View MathML</a>.

The matrix components of the symmetric part have the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M74">View MathML</a>

(17)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M75">View MathML</a>are the vector components ofω, and where the integral kernels<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M76">View MathML</a>have the properties

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M77">View MathML</a>

(18)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M78">View MathML</a>

(19)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M79">View MathML</a>

(20)

Thus, in particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M80">View MathML</a>is a Calderon-Zygmund operator, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M81">View MathML</a>.

We can also give the following useful lemma to provide an upper bound of singular integral operator for the incompressible Euler equations in [7].

Lemma 3.2For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M82">View MathML</a>fixed, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M83">View MathML</a>, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M30">View MathML</a>be defined as above. Moreover, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M85">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M86">View MathML</a>) denote the components of the vorticity vector<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M87">View MathML</a>. Then any singular integral operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M88">View MathML</a>

(21)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M89">View MathML</a>

(22)

satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M90">View MathML</a>

(23)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M91">View MathML</a>and the constantCindependent ofuandt.

Now we are ready to give a proof of Theorem 2.1, which is based on combining an energy estimate for ideal MHD equations with the estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M92">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M93">View MathML</a>, we recall the definitions of the homogeneous and inhomogeneous Besov norms for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M94">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M95">View MathML</a>

(24)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M96">View MathML</a>

(25)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M97">View MathML</a> is the Paley-Littlewood projection of f of scale j. We take the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M98">View MathML</a> Besov norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M99">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M100">View MathML</a>; then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M101">View MathML</a>

(26)

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M102">View MathML</a>

(27)

However, applying the results of Lemma 3.1 and Lemma 3.2 to u and h, and by the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M103">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M104">View MathML</a>

(28)

Therefore, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M105">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/96/mathml/M106">View MathML</a>. Thus we complete the proof of Theorem 2.1.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

The first author was supported by the Excellent Young Teachers Program of Shanghai, Doctoral Fund of Ministry of Education of China (No. 20133108120002) and The First-class Discipline of Universities in Shanghai. The research of the third author, who is the corresponding author, was supported by the Natural Science Foundation of China (No. 41204082), the Research Fund for the Doctoral Program of Higher Education of China (No. 20120162120036), Special Foundation of China Postdoctoral Science (No. 2013T60781) and Mathematics and Interdisciplinary Sciences Project of Central South University. Moreover, the authors are grateful to anonymous referees for their constructive comments and suggestions.

References

  1. Majda, AJ: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer, New York (1984)

  2. Caflisch, RE, Klapper, I, Steele, G: Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD. Commun. Math. Phys.. 184, 443–455 (1997). Publisher Full Text OpenURL

  3. Beale, JT, Kato, T, Majda, AJ: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys.. 94, 61–66 (1984). Publisher Full Text OpenURL

  4. Chen, QL, Miao, CX, Zhang, ZF: 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

  5. Du, Y, Liu, Y, Yao, ZA: Remarks on the blow-up criteria for three-dimensional ideal magnetohydrodynamics equations. J. Math. Phys.. 50, (2009) Article ID 023507

  6. Zhang, ZF, Liu, XF: On the blow-up criterion of smooth solutions to the 3D ideal MHD equations. Acta Math. Appl. Sinica (Engl. Ser.). 20, 695–700 (2004)

  7. Chen, T, Pavlovic, N: A lower bound on blowup rates for the 3D incompressible Euler equations and a single exponential Beale-Kato-Majda type estimate. Commun. Math. Phys.. 314, 265–280 (2012). Publisher Full Text OpenURL

  8. Constantin, P: Geometric statistics in turbulence. SIAM Rev.. 36, 73–98 (1994). Publisher Full Text OpenURL

  9. Constantin, P, Fefferman, C, Majda, AJ: Geometric constraints on potentially singular solutions for the 3-D Euler equations. Commun. Partial Differ. Equ.. 21, 559–571 (1996). Publisher Full Text OpenURL

  10. Deng, J, Hou, TY, Yu, X: Improved geometric conditions for non-blowup of the 3D incompressible Euler equation. Commun. Partial Differ. Equ.. 31, 293–306 (2006). Publisher Full Text OpenURL

  11. Hu, XP, Ryan, H: Blowup criterion for ideal viscoelastic flow. J. Math. Fluid Mech.. 15, 431–437 (2013). Publisher Full Text OpenURL