Abstract
Keywords:
magnetomicropolar fluid equations; weak solution; regularity criterion1 Introduction
In the paper we investigate the initial value problem for magnetomicropolar fluid equations in
with the initial value
where , , and denote the velocity of the fluid, the microrotational velocity, magnetic field and hydrostatic pressure, respectively. μ is the kinematic viscosity, χ is the vortex viscosity, γ and κ are spin viscosities and is the magnetic Reynold.
The incompressible magnetomicropolar fluid equations (1.1) have been studied extensively (see [16] and [710]). The existence and uniqueness of local strong solutions is proved by the Galerkin method in [5]. In [4], the author proved global existence of a strong solution with the small initial data. The existence of weak solutions and the uniqueness of weak solutions in 2D case were established in [6]. Yuan [8] obtained a BealeKatoMajda type blowup criterion for a smooth solution to the Cauchy problem for (1.1) that relies on the vorticity of velocity only. Wang et al.[10] established a BealeKatoMajda blowup criterion of smooth solutions to the 3D magnetomicropolar fluid equation with partial viscosity. Fundamental mathematical issues such as the regularity of weak solutions have generated extensive research and many interesting results have been established (see [1,7] and [9]).
If , (1.1) reduces to micropolar fluid equations. The micropolar fluid equations were first proposed by Eringen [11] (see also [12]). The existence of weak and strong solutions for micropolar fluid equations was obtained by Galdi and Rionero [13] and Yamaguchi [14], respectively. Dong and Chen [15] established regularity criteria of weak solutions to the threedimensional micropolar fluid equations. In [3], the authors gave sufficient conditions on the kinematics pressure in order to obtain the regularity and uniqueness of weak solutions to the micropolar fluid equations. For more details on regularity criteria, see [16,17] and [18].
If both and , then equations (1.1) reduce to be magnetohydrodynamic(MHD) equations. Magnetohydrodynamics (MHD), the science of motion of an electrically conducting fluid in the presence of a magnetic field, consists essentially of the interaction between the fluid velocity and the magnetic field (see [19]). Besides their physical applications, the MHD equations are also mathematically significant. The local existence of solutions to the Cauchy problem (1.1), (1.2) in the usual Sobolev spaces was established in [20] for any given initial data , . But whether the local solution can be extended to a global solution is a challenging open problem in the mathematical fluid mechanics. There are numerous important progresses on the fundamental issue of the regularity for the weak solution to (1.1), (1.2) (see [2128] and [2932]).
The purpose of this paper is to establish the regularity criteria of weak solutions to (1.1), (1.2) via the derivative of the velocity in one direction. It is proved that if with and , then the solution can be extended smoothly beyond .
The paper is organized as follows. We first state some important inequalities in Section 2. Then we give the definition of a weak solution and state main results in Section 3, and then we prove the main result in Section 4.
2 Preliminaries
In order to prove our main result, we need the following lemma, which may be found in [33] (see also [21,34] and [35]).
Lemma 2.1Assume thatand satisfy
Assume that, and. Then there exists a positive constant such that
Especially, when, there exists a positive constantsuch that
Lemma 2.2Letand assume that. Then there exists a positive constantsuch that
Proof
It follows from the interpolating inequality that
Combining (2.4) and (2.5) immediately yields (2.3). □
3 Main results
Before stating our main results, we introduce some function spaces. Let
The subspace
is obtained as the closure of with respect to norm . is the closure of with respect to the norm
Before stating our main results, we give the definition of a weak solution to (1.1), (1.2) (see [1,7] and [9]).
Definition 3.1 (Weak solutions)
Let , , . A measurable valued triple is said to be a weak solution to (1.1), (1.2) on if the following conditions hold:
1.
and
2. (1.1), (1.2) is satisfied in the sense of distributions, i.e., for every and with , the following hold:
and
3. The energy inequality, that is,
Theorem 3.1Letwith. Assume thatis a weak solution to (1.1), (1.2) on some interval. If
where
4 Proof of Theorem 3.1
Proof Multiplying the first equation of (1.1) by u and integrating with respect to x on , using integration by parts, we obtain
Similarly, we get
and
Summing up (4.1)(4.3), we deduce that
By integration by parts and the Cauchy inequality, we obtain
Using integration by parts, we obtain
Combining (4.4)(4.6) yields
Integrating with respect to t, we have
Differentiating (1.1) with respect to , we obtain
Taking the inner product of with the first equation of (4.8) and using integration by parts yield
Similarly, we get
and
Combining (4.9)(4.11) yields
Using integration by parts and the Cauchy inequality, we obtain
Using integration by parts, we have
Combining (4.12)(4.14) yields
In what follows, we estimate (). By integration by parts and the Hölder inequality, we obtain
where
It follows from the interpolating inequality that
From (2.2), we get
where
When , we have , and the application of the Young inequality yields
where
From integration by parts and the Hölder inequality, we obtain
where
Similarly,
and
where
By integration by parts and the inequality, we have
where
When , we have , and the application of the Young inequality yields
where
Combining (4.15)(4.20) yields
From the Gronwall inequality, we get
Multiplying the first equation of (1.1) by and integrating with respect to x on , and then using integration by parts, we obtain
Similarly, we get
and
Collecting (4.22)(4.24) yields
Thanks to integration by parts and the Cauchy inequality, we get
It follows from (4.25)(4.26) and integration by parts that
In what follows, we estimate ().
By (2.3) and the Young inequality, we deduce that
By (2.3) and the Young inequality, we have
Similarly, we obtain
and
Combining (4.27)(4.32) yields
From (4.33), the Gronwall inequality, (4.7) and (4.21), we know that . Thus, can be extended smoothly beyond . We have completed the proof of Theorem 3.1. □
Competing interests
The author declares that she has no competing interests.
Author’s contributions
The author completed the paper herself. The author read and approved the final manuscript.
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