On the regularity criteria for the 3D magneto-micropolar fluids in terms of one directional derivative
Boundary Value Problemsvolume 2012, Article number: 139 (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.
In this paper, we consider the Cauchy problem of the 3D incompressible magneto-micropolar fluid equations
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.
Note that if the micro-rotation effects and the magnetic filed are not taken into account, i.e., , equations (1.1) reduce to the classical Navier-Stokes equations. The global regularity issue has been thoroughly investigated for the 3D Navier-Stokes equations and many important regularity criteria have been established (see [7–16] and the references therein). In particular, the first well-known regularity criterion is due to Serrin : if the Leray-Hopf weak solution u of the 3D Navier-Stokes equations satisfies
More recently, Cao and Titi  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
where and , or
This result on is stronger than a similar result of Zhou and Pokorny  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 , Chae and Lee  and Zhou [15, 16], i.e., if
then the solution u is regular on (see also [14, 17] for the Besov spaces cases). For the 3D Navier-Stokes 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 . Recently, Cao and Titi  established a similar regularity criterion for the Cauchy problem of the 3D Navier-Stokes equations, that is, the solution u is regular on if
When the micro-rotation effects are neglected, i.e., , equations (1.1) become the usual magnetohydrodynamic (MHD) equations. Some of the regularity criteria established for the Navier-Stokes 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  further proved that if
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  recently established a regularity criterion via the pressure field, which says that if
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.
Theorem 1.1 Assume that with . Let be the corresponding local smooth solution to the magneto-micropolar fluid equations (1.1) on for some . If the velocity u satisfies
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.
Theorem 1.2 Assume that with . Let be the corresponding local smooth solution to the magneto-micropolar fluid equations (1.1) on for some . If the pressure π and the magnetic field b satisfy
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
Lemma 2.1 Let the parameters , , and r satisfy
and suppose that (). Then there exists a constant such that
In particular, when and , there exists a constant such that
for any φ satisfying and .
Proof of Theorem 1.1 Observe that for any with , there exists a unique local smooth solution to equations (1.1) (see ). 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.
Step 1: Estimates for .
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 . Noticing that
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 Gagliardo-Nirenberg 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
and that for
For , by Hölder’s inequality, the Gagliardo-Nirenberg inequality and Young’s inequality, we have for
Applying similar procedure to and , we have for
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 Gagliardo-Nirenberg inequality, Lemma 2.1 and Young’s inequality yield that for
Combining the estimates (2.4)-(2.12), we see that for
and that for
Thus, Gronwall’s inequality together with the energy inequality (2.2) and the assumption (1.4) implies that for
with , and
which is the desired estimates.
Step 2: Estimates for .
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
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 magneto-micropolar 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
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 Calderón-Zygmund 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. □
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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).
The authors declare that they have no competing interests.
ZX wrote the first draft and HY corrected and improved it. Both authors read and approved the final draft.