Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 925629, 6 pages

http://dx.doi.org/10.1155/2013/925629

Research Article

Global Regularity Criterion for the Magneto-Micropolar Fluid Equations

Fanhui Meng¹ and Gang Wang²

¹Changchun Finance College, Changchun, Jilin 130028, China

²Shandong Transport Vocational College, Weifang, Shandong 261206, China

Received 24 December 2012; Accepted 22 February 2013

Academic Editor: Hua Su

Copyright © 2013 Fanhui Meng and Gang Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We are concerned with the magneto-micropolar fluid equations in . Using Littlewood-Paley decomposition, we obtain an Osgood type global regularity criterion for the system.

1. Introduction

In this paper, we consider the following magneto-micropolar fluid equations in :

where denotes the velocity of the fluid at a point , , and denote, respectively, the microrotational velocity, the magnetic field, and the hydrostatic pressure. are positive numbers associated with properties of the material: is the kinematic viscosity, is the vortex viscosity, and are spin viscosities, and is the magnetic Reynold. are initial data for the velocity, the angular velocity, and the magnetic field with properties and .

It is well known that the question of global existence or finite time blowup of smooth solutions for the 3D incompressible Euler or Navier-Stokes equations has been one of the most outstanding open problems in applied analysis, as well as that for the 3D incompressible magneto-micropolar fluid equations. This challenging problem has attracted significant attention. Therefore, it is interesting to study the global regularity criterion of the solutions for system (1). But there are few theories about regularity and blow-up criteria of magneto-micropolar fluid equations. Some blow-up criterion are obtained by Yuan [1] in 2010. His paper implies that most classical blow-up criteria of smooth solutions to Navier-Stokes or magneto-hydrodynamic equations also hold for magneto-micropolar fluid equations. In particular, using Fourier frequency localization, Yuan proved the Beale-Kato-Majda criterion only relying on ; that is, if

then the solution can be extended past time . In 2008, Yuan [2] obtain the following blow-up criteria: if

then the solution can be extended . Recently,  Xu [3] also studied the regularity of weak solutions to magneto-micropolar fluid equations in Besov spaces.

In this paper, we establish a refined global regularity criterion by means of Osgood norm which improves the results (2) and (3). As we know, Osgood condition plays an important role in solving uniqueness of solutions to the incompressible fluids equations. This induces us to apply it to global regularity criterion problems of smooth solution. To achieve this goal, taking full advantage of Fourier frequency localization method and using the low-high decomposition technique, we show the following main results.

Theorem 1. Suppose that for , is the smooth solution to (1). If the following Osgood type condition:

then the solution can be extended past time . Here one denotes that .

Remark 2. The Osgood type condition (4) is weaker than (2) and (3). Note that, for , we have

2. Preliminaries

The proof of the results presented in this paper is based on a dyadic partition of unity in Fourier variables, the so-called homogeneous Littlewood-Paley decomposition. So, we first introduce the Littlewood-Paley decomposition and review the so-called Beinstein estimate and commutator estimate, which are to be used in the proof of our theorem.

Let be the Schwartz class of rapidly decreasing functions. Given , the Fourier transform of is defined by

We consider that , respectively, support in and , such that

Setting , then if and if . Let and ; the dyadic blocks are defined as follows:

Informally, is a frequency projection to the annulus , while is frequency projection to the ball . The details of Littlewood-Paley decomposition can be found in Triebel [4] and Chemin [5]. Now Besov spaces in can be defined as follows:

where denotes the dual space of .

Now we introduce well-known Bernstetin's Lemma and commutator estimate, the proof is omitted here, and we can find the details in Chemin [5], Chemin and Lerner [6], and Kato and Ponce [7].

Lemma 3 (Bernstein's lemma). Let . Assume that , then there exist constants independent of , such that

Remark 4. From the above Beinstein estimate, we easily know that in , for the Reisz transform , it has for

If suppose vector valued funtion be divergence free, by Biot Savard law with and the boundedness of Reisz transform on , we have, there exist constants independent such that

If the frequency of is restricted to annulus , then (11) implies that

Now we denote that , which satisfies

can be defined in the same way as follows:

Using the perivious notation, we define the norm of Sobolev space

especially by Fourier transform, and can be defined as

where

Lemma 5 (Commutator estimate). Let ; assume that , then there exists a constant independent of , such that

with , such that

Here .

3. Proof of the Theorem 1

First we go on with the estimates of the solution under the condition (3). Denote that ,  and we take curl on both sides of (1); we get the following equation:

with

which uses the fact .

Multiplying the three equations with , respectively, integrating by parts over about the variable , then adding the resulting equations yields that

where we use the following fact due to the divergence free condition of :

Let us begin with estimating and . Using Littlewood-Paley decomposition to , we have

For the terms and , using Hölder's inequality, Beinstein's inequality, and (12), (13), we obtain

for , we get

where we use the interpolation inequality

Summing up (26)-(27), we have

can be treated in the same way, and we decompose it as

then we obtain

Now we study , we decompose by using Littlewood-Paley theory; that is,

then

Similarly for , and , we have

Using Young's inequality, the term can be written as

Summing up (29)–(35) and taking the sum into (23), by Young's inequality, we get

If we let , that is, if we choose

where stands for the integral parts of , , then we have

where .

Using Gronwall's inequality, we have

On the other hand, by multiplying , it can be easily derived from magneto-micropolar fluid equation (1) that

Equation (39) along with (40) implies that the estimate of solution .

Next, we will show how to deduce estimates based on the estimates. We apply operator on the two sides of (1), multiply by the resulting equations and integrate the final form over , and ge

where we use the fact

Furthermore, the divergence free conditions of imply that

then

By Lemma 5, Hölder's inequality, Gagliardo-Nirenberg's inequality

and Young's inequality

we deduce that

Similarly, we estimate as follows:

Finally, we estimate the last term

Summing up (47)–(49) with (44), we obtain

Using Gronwall's inequality we obtain

Hence by (39) and (51), we can get the regularity at time ; that is, the smooth solution can be extended past time .

References

B. Yuan, “Regularity of weak solutions to magneto-micropolar fluid equations,” Acta Mathematica Scientia B, vol. 30, no. 5, pp. 1469–1480, 2010.

J. Yuan, “Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations,” Mathematical Methods in the Applied Sciences, vol. 31, no. 9, pp. 1113–1130, 2008.

F. Xu, “Regularity criterion of weak solution for the 3D magneto-micropolar fluid equations in Besov spaces,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 6, pp. 2426–2433, 2012.

H. Triebel, Theory of Function Spaces, vol. 78 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 1983.

J.-Y. Chemin, Perfect Incompressible Fluids, vol. 14 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, New York, NY, USA, 1998.

J.-Y. Chemin and N. Lerner, “Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes,” Journal of Differential Equations, vol. 121, no. 2, pp. 314–328, 1995.

T. Kato and G. Ponce, “Commutator estimates and the Euler and Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 891–907, 1988.