We study the incompressible magnetomicropolar fluid equations with partial viscosity in . A blowup criterion of smooth solutions is obtained. The result is analogous to the celebrated BealeKatoMajda type criterion for the inviscid Euler equations of incompressible fluids.
1. Introduction
The incompressible magnetomicropolar fluid equations in take the following form:
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 magnetomicropolar 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 twodimensional case. In the threedimensional case, a uniqueness result similar to the one for NavierStokes equations is given and the same result concerning fractional derivatives is obtained, but only for a more regular weak solution. RojasMedar [4] established local existence and uniqueness of strong solutions by the Galerkin method. RojasMedar 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. OrtegaTorres and RojasMedar [3] proved global existence of strong solutions for small initial data. A BealeKatoMajda type blowup 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 nonNewtonian 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 NavierStokes equations, and that it is important to the scientists working with the hydrodynamicfluid 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 Eulerlike system was found.
If both and , then (1.1) reduces to be the magnetohydrodynamic (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 Serrintype 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 MorreyCampanato 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 magnetohydrodynamic equations with and . Caflisch et al. [28] and Zhang and Liu [29] obtained blowup criteria of smooth solutions to 3D ideal MHD equations, respectively. Cannone et al. [30] showed a losing estimate for the ideal MHD equations and applied it to establish an improved blowup criterion of smooth solutions to ideal MHD equations.
In this paper, we consider the magnetomicropolar fluid equations (1.1) with partial viscosity, that is, . Without loss of generality, we take . The corresponding magnetomicropolar fluid equations thus reads
In the absence of global wellposedness, the development of blowup/non blowup theory is of major importance for both theoretical and practical purposes. For incompressible Euler and NavierStokes equations, the wellknown BealeKatoMajda's criterion [31] says that any solution is smooth up to time under the assumption that . BealeKatoMajdas criterion is slightly improved by Kozono and Taniuchi [32] under the assumption . In this paper, we obtain a BealeKatoMajda type blowup criterion of smooth solutions to the magnetomicropolar fluid equations (1.2).
Now we state our results as follows.
Theorem 1.1.
Let , with , . Assume that is a smooth solution to (1.2) with initial data , , for . If satisfies
then the solution can be extended beyond .
We have the following corollary immediately.
Corollary 1.2.
Let , with , . Assume that is a smooth solution to (1.2) with initial data , , for . Suppose that is the maximal existence time, then
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 blowup criterion of smooth solutions to the magnetomicropolar fluid equations (1.2) in Section 3.
2. Preliminaries
Let be the Schwartz class of rapidly decreasing functions. Given , its Fourier transform is defined by
and for any given , its inverse Fourier transform is defined by
Next, let us recall the LittlewoodPaley decomposition. Choose a nonnegative radial functions , supported in such that
The frequency localization operator is defined by
Let us now define homogeneous function spaces (see e.g., [33, 34]). For and , the homogeneous TriebelLizorkin space as the set of tempered distributions such that
BMO denotes the homogenous space of bounded mean oscillations associated with the norm
Thereafter, we will use the fact .
In what follows, we will make continuous use of Bernstein inequalities, which comes from [35].
Lemma 2.1.
For any and , then
hold, where and are positive constants independent of and .
The following inequality is wellknown GagliardoNirenberg inequality.
Lemma 2.2.
There exists a uniform positive constant such that
holds for all .
The following lemma comes from [36].
Lemma 2.3.
The following calculus inequality holds:
Lemma 2.4.
There is a uniform positive constant , such that
holds for all vectors with .
Proof.
The proof can be found in [37]. For completeness, the proof will be also sketched here. It follows from LittlewoodPaley decomposition that
Using (2.7) and (2.11), we obtain
By the BiotSavard law, we have a representation of in terms of as
where , denote the Riesz transforms. Since is a bounded operator in BMO, this yields
with . Taking
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.
In three space dimensions, the following inequalities
hold, and in two space dimensions, the following inequalities
hold.
Proof.
(2.16) and (2.17) are of course well known. In fact, we can obtain them by Sobolev embedding and the scaling techniques. In what follows, we only prove the last inequality in (2.16) and (2.17). Sobolev embedding implies that for . Consequently, we get
For any given and , let
By (2.18) and (2.19), we obtain
which is equivalent to
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.
Multiplying (1.2) by , respectively, then integrating the resulting equation with respect to on and using integration by parts, we get
where we have used and .
Integrating with respect to , we obtain
Applying to (1.2) and taking the inner product of the resulting equation with , with help of integration by parts, we have
It follows from (3.3) and , that
By Gronwall inequality, we get
Thanks to (1.3), we know that for any small constant , there exists such that
Let
It follows from (3.5), (3.6), (3.7), and Lemma 2.4 that
where depends on , while is an absolute positive constant.
Applying to the first equation of (1.2), then taking inner product of the resulting equation with , using integration by parts, we get
Similarly, we obtain
Using (3.9), (3.10), , and integration by parts, we have
In what follows, for simplicity, we will set .
From Hölder inequality and Lemma 2.3, we get
Using integration by parts and Hölder inequality, we obtain
By Lemma 2.5, Young inequality, and (3.8), we deduce that
in 3D and
in 2D.
From Lemmas 2.2 and 2.5, Young inequality, and (3.8), we have
in 3D and
in 2D.
Consequently, we get
provided that
It follows from (3.13) and (3.18) that
Similarly, we obtain
Combining (3.11), (3.12), (3.20), and (3.21) yields
for all .
Integrating (3.22) with respect to from to and using Lemma 2.4, we have
which implies
For all , from Gronwall inequality and (3.24), we obtain
where depends on .
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.
Acknowledgment
This work was supported by the NNSF of China (Grant no. 10971190).
References

Gala, S: Regularity criteria for the 3D magnetomicropolar fluid equations in the MorreyCampanato space. Nonlinear Differential Equations and Applications. 17(2), 181–194 (2010). Publisher Full Text

OrtegaTorres, EE, RojasMedar, MA: On the uniqueness and regularity of the weak solution for magnetomicropolar fluid equations. Revista de Matemáticas Aplicadas. 17(2), 75–90 (1996)

OrtegaTorres, EE, RojasMedar, MA: Magnetomicropolar fluid motion: global existence of strong solutions. Abstract and Applied Analysis. 4(2), 109–125 (1999). Publisher Full Text

RojasMedar, MA: Magnetomicropolar fluid motion: existence and uniqueness of strong solution. Mathematische Nachrichten. 188, 301–319 (1997). Publisher Full Text

RojasMedar, MA, Boldrini, JL: Magnetomicropolar fluid motion: existence of weak solutions. Revista Matemática Complutense. 11(2), 443–460 (1998)

Yuan, BQ: Regularity of weak solutions to magnetomicropolar fluid equations. Acta Mathematica Scientia. 30(5), 1469–1480 (2010). Publisher Full Text

Yuan, J: Existence theorem and blowup criterion of the strong solutions to the magnetomicropolar fluid equations. Mathematical Methods in the Applied Sciences. 31(9), 1113–1130 (2008). Publisher Full Text

Eringen, AC: Theory of micropolar fluids. Journal of Mathematics and Mechanics. 16, 1–18 (1966)

Łukaszewicz, G: Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology,p. xvi+253. Birkhäuser, Boston, Mass, USA (1999)

Galdi, GP, Rionero, S: A note on the existence and uniqueness of solutions of the micropolar fluid equations. International Journal of Engineering Science. 15(2), 105–108 (1977). Publisher Full Text

Yamaguchi, N: Existence of global strong solution to the micropolar fluid system in a bounded domain. Mathematical Methods in the Applied Sciences. 28(13), 1507–1526 (2005). Publisher Full Text

Dong, BQ, Chen, ZM: Regularity criteria of weak solutions to the threedimensional micropolar flows. Journal of Mathematical Physics. 50(10, article 103525), 13 (2009)

OrtegaTorres, E, RojasMedar, M: On the regularity for solutions of the micropolar fluid equations. Rendiconti del Seminario Matematico della Università di Padova. 122, 27–37 (2009)

OrtegaTorres, E, VillamizarRoa, EJ, RojasMedar, MA: Micropolar fluids with vanishing viscosity. Abstract and Applied Analysis. 2010, (2010)

Cao, C, Wu, J: Two regularity criteria for the 3D MHD equations. Journal of Differential Equations. 248(9), 2263–2274 (2010). Publisher Full Text

Fan, J, Jiang, S, Nakamura, G, Zhou, Y: Logarithmically improved regularity criteria for the NavierStokes 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. 227(1), 113–152 (2005). Publisher Full Text

Zhou, Y: Remarks on regularities for the 3D MHD equations. Discrete and Continuous Dynamical Systems. Series A. 12(5), 881–886 (2005)

Zhou, Y: Regularity criteria for the 3D MHD equations in terms of the pressure. International Journal of NonLinear Mechanics. 41(10), 1174–1180 (2006). Publisher Full Text

Zhou, Y, Gala, S: Regularity criteria for the solutions to the 3D MHD equations in the multiplier space. Zeitschrift für Angewandte Mathematik und Physik. 61(2), 193–199 (2010). PubMed Abstract  Publisher Full Text

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. 72(910), 3643–3648 (2010). PubMed Abstract  Publisher Full Text

Zhou, Y, Fan, J: A regularity criterion for the 2D MHD system with zero magnetic diffusivity. Journal of Mathematical Analysis and Applications. 378(1), 169–172 (2011). Publisher Full Text

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. 24(3), 491–505 (2007)

Zhou, Y, Fan, J: Regularity criteria of strong solutions to a problem of magnetoelastic interactions. Communications on Pure and Applied Analysis. 9(6), 1697–1704 (2010)

Zhou, Y, Fan, J: A regularity criterion for the nematic liquid crystal flows. journal of Inequalities and Applications. 2010, (2010)

Lei, Z, Zhou, Y: BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity. Discrete and Continuous Dynamical Systems. Series A. 25(2), 575–583 (2009)

Caflisch, RE, Klapper, I, Steele, G: Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD. Communications in Mathematical Physics. 184(2), 443–455 (1997). Publisher Full Text

Zhang, ZF, Liu, XF: On the blowup criterion of smooth solutions to the 3D ideal MHD equations. Acta Mathematicae Applicatae Sinica. 20(4), 695–700 (2004). Publisher Full Text

Cannone, M, Chen, Q, Miao, C: A losing estimate for the ideal MHD equations with application to blowup criterion. SIAM Journal on Mathematical Analysis. 38(6), 1847–1859 (2007). Publisher Full Text

Beale, JT, Kato, T, Majda, A: Remarks on the breakdown of smooth solutions for the 3D Euler equations. Communications in Mathematical Physics. 94(1), 61–66 (1984). Publisher Full Text

Kozono, H, Taniuchi, Y: Bilinear estimates in BMO and the NavierStokes equations. Mathematische Zeitschrift. 235(1), 173–194 (2000). Publisher Full Text

Bergh, J, Löfström, J: Interpolation Spaces, Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany (1976)

Triebel, H: Theory of Function Spaces, Monographs in Mathematics,p. 284. Birkhäuser, Basel, Switzerland (1983)

Chemin, JY: Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and Its Applications,p. x+187. The Clarendon Press Oxford University Press, New York, NY, USA (1998)

Majda, AJ, Bertozzi, AL: Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics,p. xii+545. Cambridge University Press, Cambridge, UK (2002)

Zhou, Y, Lei, Z: Logarithmically improved criterion for Euler and NavierStokes equations. preprint