We study the incompressible magneto-micropolar fluid equations with partial viscosity in . A blow-up criterion of smooth solutions is obtained. The result is analogous to the celebrated Beale-Kato-Majda type criterion for the inviscid Euler equations of incompressible fluids.
The incompressible magneto-micropolar 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 magneto-micropolar fluid equation (1.1) has been studied extensively (see [1–7]). In , 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  established local existence and uniqueness of strong solutions by the Galerkin method. Rojas-Medar and Boldrini  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  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 . For regularity results, refer to Yuan  and Gala .
If , (1.1) reduces to micropolar fluid equations. The micropolar fluid equations was first developed by Eringen . 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  and references therein. The existences of weak and strong solutions for micropolar fluid equations were proved by Galdi and Rionero  and Yamaguchi , respectively. Regularity criteria of weak solutions to the micropolar fluid equations are investigated in . In , 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 ). 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  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 . Zhou and Gala  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 ). In , 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 . Logarithmically improved regularity criteria for two related models to MHD equations were established in  and , respectively. Lei and Zhou  studied the magneto-hydrodynamic equations with and . Caflisch et al.  and Zhang and Liu  obtained blow-up criteria of smooth solutions to 3-D ideal MHD equations, respectively. Cannone et al.  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 this paper, we consider the magneto-micropolar fluid equations (1.1) with partial viscosity, that is, . Without loss of generality, we take . The corresponding magneto-micropolar fluid equations thus reads
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  says that any solution is smooth up to time under the assumption that . Beale-Kato-Majdas criterion is slightly improved by Kozono and Taniuchi  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.
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.
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 blow-up criterion of smooth solutions to the magneto-micropolar fluid equations (1.2) in Section 3.
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 Littlewood-Paley decomposition. Choose a nonnegative radial functions , supported in such that
The frequency localization operator is defined by
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 .
For any and , then
hold, where and are positive constants independent of and .
The following inequality is well-known Gagliardo-Nirenberg inequality.
There exists a uniform positive constant such that
holds for all .
The following lemma comes from .
The following calculus inequality holds:
There is a uniform positive constant , such that
holds for all vectors with .
The proof can be found in . For completeness, the proof will be also sketched here. It follows from Littlewood-Paley decomposition that
Using (2.7) and (2.11), we obtain
By the Biot-Savard 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.
In three space dimensions, the following inequalities
hold, and in two space dimensions, the following inequalities
(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
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
From Lemmas 2.2 and 2.5, Young inequality, and (3.8), we have
in 3D and
Consequently, we get
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
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.
This work was supported by the NNSF of China (Grant no. 10971190).
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