Abstract
In this paper, we establish two new regularity criteria for the 3D magnetomicropolar fluids in terms of one directional derivative of the velocity or of the pressure and the magnetic field.
MSC: 35Q35, 76W05, 35B65.
Keywords:
magnetomicropolar fluid equations; regularity criteria1 Introduction
In this paper, we consider the Cauchy problem of the 3D incompressible magnetomicropolar fluid equations
where u is the fluid velocity, w is the microrotational 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 [1]). 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 [2] investigated the local existence and uniqueness of the strong solutions to the magnetomicropolar fluid equations (1.1) (see also [36] 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 wellposedness 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.
Note that if the microrotation effects and the magnetic filed are not taken into account, i.e., , equations (1.1) reduce to the classical NavierStokes equations. The global regularity issue has been thoroughly investigated for the 3D NavierStokes equations and many important regularity criteria have been established (see [716] and the references therein). In particular, the first wellknown regularity criterion is due to Serrin [14]: if the LerayHopf weak solution u of the 3D NavierStokes equations satisfies
then u is regular on . Beirao da Veiga [8] and Penel and Pokorny [13] established another regularity criteria by replacing the above conditions with the following ones:
or
More recently, Cao and Titi [17] established a regularity criterion in terms of only one of the nine components of the gradient of a velocity field, that is, the solution u is regular on if
This result on is stronger than a similar result of Zhou and Pokorny [18] in the sense of allowing for much smaller values of p. These regularity criteria are of physical relevance since experimental measurements are usually obtained for quantities of the form . The regularity criterion by imposing the growth conditions on the pressure field are also examined by, for example, Berselli and Galdi [9], Chae and Lee [10] and Zhou [15,16], i.e., if
or
then the solution u is regular on (see also [14,17] for the Besov spaces cases). For the 3D NavierStokes equations with boundary conditions, Cao and Titi first introduced a regularity criterion in terms of only one component of the pressure gradient based on the breakthrough of the global regularity of the 3D primitive equations [19]. Recently, Cao and Titi [20] established a similar regularity criterion for the Cauchy problem of the 3D NavierStokes equations, that is, the solution u is regular on if
When the microrotation effects are neglected, i.e., , equations (1.1) become the usual magnetohydrodynamic (MHD) equations. Some of the regularity criteria established for the NavierStokes equations can be extended to the 3D MHD equations by making assumptions on both u and b (see [21,22]). Moreover, He and Xin [23,24] showed that the velocity field u plays a dominant role in the regularity issue and derived a criterion in terms of the velocity field u alone (see also [25,26] for the Besov spaces cases). Recently, Cao and Wu [27] further proved that if
or
then is regular on . More recently, Liu, Zhao and Cui [28] have adapted the method of [27] to establish a similar regularity criterion for the 3D nematic liquid crystal flow.
If we ignore the magnetic filed, i.e., , equations (1.1) reduce to the micropolar fluid equations. The theory of micropolar fluid has attracted more and more scholars’ attention in recent years. In particular, Dong, Jia and Chen [29] recently established a regularity criterion via the pressure field, which says that if
then is regular on (see also [30,31] for the Lorentz spaces cases).
For the full magnetomicropolar fluid equations (1.1), Yuan [32] recently showed that the solution is regular on if
or
For other regularity criteria of equations (1.1), we refer to Gala [33], Geng, Chen and Gala [34], Wang, Hu and Wang [35], Yuan [2] and Zhang, Yao and Wang [36].
In this paper, we establish two new regularity criteria for the 3D magnetomicropolar 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 [27]. Without loss of generality, we set the viscous coefficients .
We now state our main results as follows.
Theorem 1.1Assume thatwith. Letbe the corresponding local smooth solution to the magnetomicropolar fluid equations (1.1) onfor some. If the velocityusatisfies
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 [32] (see (1.3)) in the sense that it depends only on one directional derivative of the velocity u.
Theorem 1.2Assume thatwith. Letbe the corresponding local smooth solution to the magnetomicropolar fluid equations (1.1) onfor some. If the pressureπand the magnetic fieldbsatisfy
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
In this section, we give the proof of Theorem 1.1 and Theorem 1.2. The following lemma plays an important role in our arguments. Its proof can be found in [37] or [27].
Lemma 2.1Let the parameters, , andrsatisfy
and suppose that (). Then there exists a constantsuch that
In particular, whenand, there exists a constantsuch that
Proof of Theorem 1.1 Observe that for any with , there exists a unique local smooth solution to equations (1.1) (see [2]). Let be the maximum existence time. To prove Theorem 1.1, it is sufficient to show that the assumption (1.4) implies . Indeed, we shall prove that under the condition (1.4), there exists a constant such that
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.
Firstly, we derive the energy inequality. For this purpose, we take the inner product of u, w and b with equations (1.1), respectively, sum the resulting equations and then integrate by parts to obtain
where we used in the first equality and Hölder’s inequality in the last inequality. Thus,
It follows from Gronwall’s inequality that
Now we split the proof of the estimates (2.1) into two steps.
To this end, differentiating the first three equations in (1.1) with respect to , taking the inner product of , and with the resulting equations, respectively, and then performing a space integration by parts, we get
where we used the facts
by , we can sum the above equations to obtain
We now estimate the above terms one by one. To bound , we first integrate by parts and then apply Hölder’s inequality to obtain
It follows from the GagliardoNirenberg inequality that
and from Lemma 2.1 that
Substituting these two estimates into (2.3) and then using Young’s inequality, we see that for
For , by Hölder’s inequality, the GagliardoNirenberg inequality and Young’s inequality, we have for
Applying similar procedure to and , we have for
and
For the term , by using Hölder’s inequality and Young’s inequality, it can be bounded as follows:
Finally, we can follow the steps as in the bound of to estimate . Precisely, by integrations by parts and Hölder’s inequality, we have
Then the GagliardoNirenberg inequality, Lemma 2.1 and Young’s inequality yield that for
Combining the estimates (2.4)(2.12), we see that for
Thus, Gronwall’s inequality together with the energy inequality (2.2) and the assumption (1.4) implies that for
Then
which is the desired estimates.
For this purpose, taking the inner product of Δu, Δw and Δb with the first three equations in (1.1), respectively, and then performing a space integration by parts, we have
Noticing , we sum the above equations and integrate by parts to obtain
By using the interpolation inequality and taking in Lemma 2.1, we have
where . Then Young’s inequality yields
Similarly,
Substituting the above two estimates into (2.15), we have
By using Gronwall’s inequality, the energy inequality (2.2) and the estimate (2.14), we conclude that
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.
Proof of Theorem 1.2 Following from Serrin type criteria (1.2) with and on the 3D magnetomicropolar fluid equations (1.1), it is sufficient to prove that
To do this, we set
and then equations (1.1) are converted to the following 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).
Next we take the inner product of , and with the first three equations in (2.17), respectively, and then integrate by parts to obtain
We now bound the above terms one by one. For , we have
It follows from the integration by parts, we see
Similarly, we have
and
The process for estimating is more subtle. It follows from Hölder’s inequality and Lemma 2.1 that
To estimate the term involving , we take the divergence of the first equation of (2.17) and find
by . Then the CalderónZygmund inequality, Hölder’s inequality and the interpolation inequality imply that
Similarly, we have
If , combining the above two estimates, we see
The case can be similarly dealt with.
Summarily, we conclude that
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. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
ZX wrote the first draft and HY corrected and improved it. Both authors read and approved the final draft.
Acknowledgements
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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