On the regularity criteria for the 3D magneto-micropolar fluids in terms of one directional derivative
© Xiang and Yang; licensee Springer. 2012
Received: 1 August 2012
Accepted: 12 November 2012
Published: 27 November 2012
In this paper, we establish two new regularity criteria for the 3D magneto-micropolar fluids in terms of one directional derivative of the velocity or of the pressure and the magnetic field.
MSC:35Q35, 76W05, 35B65.
where u is the fluid velocity, w is the micro-rotational velocity, b is the magnetic field and π is the pressure. Equations (1.1) describe the motion of a micropolar fluid which is moving in the presence of a magnetic field (see ). The positive parameters μ, χ, γ, κ and ν in (1.1) are associated with the properties of the materials: μ is the kinematic viscosity, χ is the vortex viscosity, ν and κ are the spin viscosities and is the magnetic Reynolds number.
Recently, Yuan  investigated the local existence and uniqueness of the strong solutions to the magneto-micropolar fluid equations (1.1) (see also [3–6] for the bounded domain cases). Thus, the further problem at the center of the mathematical theory concerning equations (1.1) is whether or not it has a global in time smooth solution for any prescribed smooth initial data, which is still a challenging open problem. In the absence of a global well-posedness theory, the development of regularity criteria is of major importance for both theoretical and practical purposes. We would like to recall some related results in this direction.
In this paper, we establish two new regularity criteria for the 3D magneto-micropolar fluid equations (1.1) in terms of one directional derivative of the velocity u or of the pressure π and the magnetic field b by adapting the method of . Without loss of generality, we set the viscous coefficients .
We now state our main results as follows.
then can be extended beyond T.
Note that when , and thus the corresponding assumption in (1.4) should be understood as .
Remark 1.1 Theorem 1.1 improves the regularity criterion in  (see (1.3)) in the sense that it depends only on one directional derivative of the velocity u.
then can be extended beyond T.
Remark 1.2 When , we also obtain a new regularity criterion for the micropolar equations determined by one direction derivative of the pressure π alone.
We shall prove our results in the next section. For simplicity, we denote by the norm and by the inner product throughout the paper. The letter C denotes an inessential constant which might vary from line to line, but does not depend on particular solutions or functions.
2 Proof of the main results
for any φ satisfying and .
which implies that T is not the maximum existence time and thus the solution can be extended beyond T by the standard arguments of continuation of local solutions.
Now we split the proof of the estimates (2.1) into two steps.
Step 1: Estimates for .
which is the desired estimates.
Step 2: Estimates for .
for any , which implies that the desired estimates (2.1) hold and thus the solution can be extended beyond T. □
Now we turn our attention to proving Theorem 1.2. We will first transform equations (1.1) into a symmetric form.
Firstly, taking the inner product of , w and with the above equations, respectively, and integrating by parts, we can obtain the energy estimates similar to (2.2).
The case can be similarly dealt with.
Thus, Gronwall’s inequality together with the assumption (1.5) and the energy estimates gives the desired estimates (2.12) and thus the solution can be extended beyond T. □
The authors would like to thank the referees for their valuable comments and remarks. This work was partially supported by the NNSF of China (No. 11101068), the Sichuan Youth Science & Technology Foundation (No. 2011JQ0003), the SRF for ROCS, SEM, and the Fundamental Research Funds for the Central Universities (ZYGX2009X019).
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