# A Beale-Kato-Madja Criterion for Magneto-Micropolar Fluid Equations with Partial Viscosity

- Yu-Zhu Wang
^{1}Email author, - Liping Hu
^{2}and - Yin-Xia Wang
^{1}

**2011**:128614

**DOI: **10.1155/2011/128614

© Yu-Zhu Wang et al. 2011

**Received: **18 February 2011

**Accepted: **7 March 2011

**Published: **15 March 2011

## Abstract

## 1. Introduction

where , , and denote the velocity of the fluid, the microrotational velocity, magnetic field, and hydrostatic pressure, respectively. is the kinematic viscosity, is the vortex viscosity, and are spin viscosities, and is the magnetic Reynold.

The incompressible magneto-micropolar fluid equation (1.1) has been studied extensively (see [1–7]). In [2], the authors have proven that a weak solution to (1.1) has fractional time derivatives of any order less than in the two-dimensional case. In the three-dimensional case, a uniqueness result similar to the one for Navier-Stokes equations is given and the same result concerning fractional derivatives is obtained, but only for a more regular weak solution. Rojas-Medar [4] established local existence and uniqueness of strong solutions by the Galerkin method. Rojas-Medar and Boldrini [5] also proved the existence of weak solutions by the Galerkin method, and in 2D case, also proved the uniqueness of the weak solutions. Ortega-Torres and Rojas-Medar [3] proved global existence of strong solutions for small initial data. A Beale-Kato-Majda type blow-up criterion for smooth solution to (1.1) that relies on the vorticity of velocity only is obtained by Yuan [7]. For regularity results, refer to Yuan [6] and Gala [1].

If , (1.1) reduces to micropolar fluid equations. The micropolar fluid equations was first developed by Eringen [8]. It is a type of fluids which exhibits the microrotational effects and microrotational inertia, and can be viewed as a non-Newtonian fluid. Physically, micropolar fluid may represent fluids consisting of rigid, randomly oriented (or spherical particles) suspended in a viscous medium, where the deformation of fluid particles is ignored. It can describe many phenomena that appeared in a large number of complex fluids such as the suspensions, animal blood, and liquid crystals which cannot be characterized appropriately by the Navier-Stokes equations, and that it is important to the scientists working with the hydrodynamic-fluid problems and phenomena. For more background, we refer to [9] and references therein. The existences of weak and strong solutions for micropolar fluid equations were proved by Galdi and Rionero [10] and Yamaguchi [11], respectively. Regularity criteria of weak solutions to the micropolar fluid equations are investigated in [12]. In [13], the authors gave sufficient conditions on the kinematics pressure in order to obtain regularity and uniqueness of the weak solutions to the micropolar fluid equations. The convergence of weak solutions of the micropolar fluids in bounded domains of was investigated (see [14]). When the viscosities tend to zero, in the limit, a fluid governed by an Euler-like system was found.

If both and , then (1.1) reduces to be the magneto-hydrodynamic (MHD) equations. There are numerous important progresses on the fundamental issue of the regularity for the weak solution to MHD systems (see [15–23]). Zhou [18] established Serrin-type regularity criteria in term of the velocity only. Logarithmically improved regularity criteria for MHD equations were established in [16, 23]. Regularity criteria for the 3D MHD equations in term of the pressure were obtained [19]. Zhou and Gala [20] obtained regularity criteria of solutions in term of and in the multiplier spaces. A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field in Morrey-Campanato spaces was established (see [21]). In [22], a regularity criterion for the 2D MHD system with zero magnetic diffusivity was obtained.

Regularity criteria for the generalized viscous MHD equations were also obtained in [24]. Logarithmically improved regularity criteria for two related models to MHD equations were established in [25] and [26], respectively. Lei and Zhou [27] studied the magneto-hydrodynamic equations with and . Caflisch et al. [28] and Zhang and Liu [29] obtained blow-up criteria of smooth solutions to 3-D ideal MHD equations, respectively. Cannone et al. [30] showed a losing estimate for the ideal MHD equations and applied it to establish an improved blow-up criterion of smooth solutions to ideal MHD equations.

In the absence of global well-posedness, the development of blow-up/non blow-up theory is of major importance for both theoretical and practical purposes. For incompressible Euler and Navier-Stokes equations, the well-known Beale-Kato-Majda's criterion [31] says that any solution is smooth up to time under the assumption that . Beale-Kato-Majdas criterion is slightly improved by Kozono and Taniuchi [32] under the assumption . In this paper, we obtain a Beale-Kato-Majda type blow-up criterion of smooth solutions to the magneto-micropolar fluid equations (1.2).

Now we state our results as follows.

Theorem 1.1.

then the solution can be extended beyond .

We have the following corollary immediately.

Corollary 1.2.

The paper is organized as follows. We first state some preliminaries on functional settings and some important inequalities in Section 2 and then prove the blow-up criterion of smooth solutions to the magneto-micropolar fluid equations (1.2) in Section 3.

## 2. Preliminaries

Thereafter, we will use the fact .

In what follows, we will make continuous use of Bernstein inequalities, which comes from [35].

Lemma 2.1.

hold, where and are positive constants independent of and .

The following inequality is well-known Gagliardo-Nirenberg inequality.

Lemma 2.2.

The following lemma comes from [36].

Lemma 2.3.

Lemma 2.4.

Proof.

It follows from (2.12), (2.14), and (2.15) that (2.10) holds. Thus, the lemma is proved.

In order to prove Theorem 1.1, we need the following interpolation inequalities in two and three space dimensions.

Lemma 2.5.

hold.

Proof.

Taking and and , respectively. From (2.21), we immediately get the last inequality in (2.16) and (2.17). Thus, we have completed the proof of Lemma 2.5.

## 3. Proof of Main Results

Proof of Theorem 1.1.

where depends on , while is an absolute positive constant.

In what follows, for simplicity, we will set .

in 2D.

in 2D.

Noting that (3.2) and the right hand side of (3.25) is independent of for , we know that . Thus, Theorem 1.1 is proved.

## Declarations

### Acknowledgment

This work was supported by the NNSF of China (Grant no. 10971190).

## Authors’ Affiliations

## References

- Gala S: Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey-Campanato space.
*Nonlinear Differential Equations and Applications*2010, 17(2):181-194. 10.1007/s00030-009-0047-4View ArticleMathSciNetGoogle Scholar - Ortega-Torres EE, Rojas-Medar MA: On the uniqueness and regularity of the weak solution for magneto-micropolar fluid equations.
*Revista de Matemáticas Aplicadas*1996, 17(2):75-90.MathSciNetGoogle Scholar - Ortega-Torres EE, Rojas-Medar MA: Magneto-micropolar fluid motion: global existence of strong solutions.
*Abstract and Applied Analysis*1999, 4(2):109-125. 10.1155/S1085337599000287View ArticleMathSciNetGoogle Scholar - Rojas-Medar MA: Magneto-micropolar fluid motion: existence and uniqueness of strong solution.
*Mathematische Nachrichten*1997, 188: 301-319. 10.1002/mana.19971880116View ArticleMathSciNetGoogle Scholar - Rojas-Medar MA, Boldrini JL: Magneto-micropolar fluid motion: existence of weak solutions.
*Revista Matemática Complutense*1998, 11(2):443-460.View ArticleMathSciNetGoogle Scholar - Yuan BQ: Regularity of weak solutions to magneto-micropolar fluid equations.
*Acta Mathematica Scientia*2010, 30(5):1469-1480. 10.1016/S0252-9602(10)60139-7View ArticleMathSciNetGoogle Scholar - Yuan J: Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations.
*Mathematical Methods in the Applied Sciences*2008, 31(9):1113-1130. 10.1002/mma.967View ArticleMathSciNetGoogle Scholar - Eringen AC: Theory of micropolar fluids.
*Journal of Mathematics and Mechanics*1966, 16: 1-18.MathSciNetGoogle Scholar - Łukaszewicz G:
*Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology*. Birkhäuser, Boston, Mass, USA; 1999:xvi+253.Google Scholar - Galdi GP, Rionero S: A note on the existence and uniqueness of solutions of the micropolar fluid equations.
*International Journal of Engineering Science*1977, 15(2):105-108. 10.1016/0020-7225(77)90025-8View ArticleMathSciNetGoogle Scholar - Yamaguchi N: Existence of global strong solution to the micropolar fluid system in a bounded domain.
*Mathematical Methods in the Applied Sciences*2005, 28(13):1507-1526. 10.1002/mma.617View ArticleMathSciNetGoogle Scholar - Dong B-Q, Chen Z-M: Regularity criteria of weak solutions to the three-dimensional micropolar flows.
*Journal of Mathematical Physics*2009, 50(10, article 103525):13.View ArticleMathSciNetGoogle Scholar - Ortega-Torres E, Rojas-Medar M: On the regularity for solutions of the micropolar fluid equations.
*Rendiconti del Seminario Matematico della Università di Padova*2009, 122: 27-37.View ArticleMathSciNetGoogle Scholar - Ortega-Torres E, Villamizar-Roa EJ, Rojas-Medar MA: Micropolar fluids with vanishing viscosity.
*Abstract and Applied Analysis*2010, 2010:-18.Google Scholar - Cao C, Wu J: Two regularity criteria for the 3D MHD equations.
*Journal of Differential Equations*2010, 248(9):2263-2274. 10.1016/j.jde.2009.09.020View ArticleMathSciNetGoogle Scholar - Fan J, Jiang S, Nakamura G, Zhou Y: Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations.
*Journal of Mathematical Fluid Mechanics*. In press - He C, Xin Z: Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations.
*Journal of Functional Analysis*2005, 227(1):113-152. 10.1016/j.jfa.2005.06.009View ArticleMathSciNetGoogle Scholar - Zhou Y: Remarks on regularities for the 3D MHD equations.
*Discrete and Continuous Dynamical Systems. Series A*2005, 12(5):881-886.View ArticleMathSciNetGoogle Scholar - Zhou Y: Regularity criteria for the 3D MHD equations in terms of the pressure.
*International Journal of Non-Linear Mechanics*2006, 41(10):1174-1180. 10.1016/j.ijnonlinmec.2006.12.001View ArticleMathSciNetGoogle Scholar - Zhou Y, Gala S: Regularity criteria for the solutions to the 3D MHD equations in the multiplier space.
*Zeitschrift für Angewandte Mathematik und Physik*2010, 61(2):193-199. 10.1007/s00033-009-0023-1View ArticleMathSciNetGoogle Scholar - Zhou Y, Gala S: A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field.
*Nonlinear Analysis. Theory, Methods & Applications*2010, 72(9-10):3643-3648. 10.1016/j.na.2009.12.045View ArticleMathSciNetGoogle Scholar - Zhou Y, Fan J: A regularity criterion for the 2D MHD system with zero magnetic diffusivity.
*Journal of Mathematical Analysis and Applications*2011, 378(1):169-172. 10.1016/j.jmaa.2011.01.014View ArticleMathSciNetGoogle Scholar - Zhou Y, Fan J: Logarithmically improved regularity criteria for the 3D viscous MHD equations.
*Forum Math*. In press - Zhou Y: Regularity criteria for the generalized viscous MHD equations.
*Annales de l'Institut Henri Poincaré. Analyse Non Linéaire*2007, 24(3):491-505.View ArticleGoogle Scholar - Zhou Y, Fan J: Regularity criteria of strong solutions to a problem of magneto-elastic interactions.
*Communications on Pure and Applied Analysis*2010, 9(6):1697-1704.View ArticleMathSciNetGoogle Scholar - Zhou Y, Fan J: A regularity criterion for the nematic liquid crystal flows.
*journal of Inequalities and Applications*2010, 2010:-9.Google Scholar - Lei Z, Zhou Y: BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity.
*Discrete and Continuous Dynamical Systems. Series A*2009, 25(2):575-583.View ArticleMathSciNetGoogle Scholar - Caflisch RE, Klapper I, Steele G: Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD.
*Communications in Mathematical Physics*1997, 184(2):443-455. 10.1007/s002200050067View ArticleMathSciNetGoogle Scholar - Zhang Z-F, Liu X-F: On the blow-up criterion of smooth solutions to the 3D ideal MHD equations.
*Acta Mathematicae Applicatae Sinica*2004, 20(4):695-700. 10.1007/s10255-004-0207-6View ArticleMathSciNetGoogle Scholar - Cannone M, Chen Q, Miao C: A losing estimate for the ideal MHD equations with application to blow-up criterion.
*SIAM Journal on Mathematical Analysis*2007, 38(6):1847-1859. 10.1137/060652002View ArticleMathSciNetGoogle Scholar - Beale JT, Kato T, Majda A: Remarks on the breakdown of smooth solutions for the 3-D Euler equations.
*Communications in Mathematical Physics*1984, 94(1):61-66. 10.1007/BF01212349View ArticleMathSciNetGoogle Scholar - Kozono H, Taniuchi Y: Bilinear estimates in BMO and the Navier-Stokes equations.
*Mathematische Zeitschrift*2000, 235(1):173-194. 10.1007/s002090000130View ArticleMathSciNetGoogle Scholar - Bergh J, Löfström J:
*Interpolation Spaces, Grundlehren der Mathematischen Wissenschaften*. Springer, Berlin, Germany; 1976.Google Scholar - Triebel H:
*Theory of Function Spaces, Monographs in Mathematics*.*Volume 78*. Birkhäuser, Basel, Switzerland; 1983:284.View ArticleGoogle Scholar - Chemin J-Y:
*Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and Its Applications*.*Volume 14*. The Clarendon Press Oxford University Press, New York, NY, USA; 1998:x+187.Google Scholar - Majda AJ, Bertozzi AL:
*Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics*.*Volume 27*. Cambridge University Press, Cambridge, UK; 2002:xii+545.Google Scholar - Zhou Y, Lei Z: Logarithmically improved criterion for Euler and Navier-Stokes equations. preprint

## Copyright

This article is published under license to BioMed Central Ltd. 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.