Regularity Criterion for Weak Solution to the 3D Micropolar Fluid Equations
Regularity criterion for the 3D micropolar fluid equations is investigated. We prove that, for some , if , where and , then the solution can be extended smoothly beyond . The derivative can be substituted with any directional derivative of .
In the paper, we investigate the initial value problem for the micropolar fluid equations in : with the initial value where , , and stand for the divergence free velocity field, nondivergence free microrotation field (angular velocity of the rotation of the particles of the fluid), the scalar pressure, respectively is the Newtonian kinetic viscosity, is the dynamics microrotation viscosity, and are the angular viscosity (see, e.g., Lukaszewicz ).
The micropolar fluid equations was first proposed 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 that consists 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 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 is important to the scientists working with the hydrodynamic fluid problems and phenomena. For more background, we refer to  and references therein. Besides their physical applications, the micropolar fluid equations are also mathematically significant. The existences of weak and strong solutions for micropolar fluid equations were treated by Galdi and Rionero  and Yamaguchi , respectively. 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. Fundamental mathematical issues such as the global regularity of their solutions have generated extensive research, and many interesting results have been obtained (see [6–8]). A Beale-Kato-Madja criterion (see ) of smooth solutions to a related model with (1.1) was established in .
If and , then (1.1) reduces to be the Navier-Stokes equations. Besides its physical applications, the Navier-Stokes equations are also mathematically significant. In the last century, Leray  and Hopf  constructed weak solutions to the Navier-Stokes equations. The solution is called the Leray-Hopf weak solution. Later on, much effort has been devoted to establish the global existence and uniqueness of smooth solutions to the Navier-Stokes equations. Different criteria for regularity of the weak solutions have been proposed, and many interesting results are established (see [13–31]).
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 then the solution can be extended smoothly beyond .
The paper is organized as follows. We first state some important inequalities in Section 2, which play an important roles in the proof of our main result. Then, we give definition of weak solution and state main results in Section 3 and then prove main result in Section 4.
Lemma 2.1. Assume that and satisfy Assume that , , and . Then, there exists a positive constant such that Especially, when , there exists a positive constant such that which holds for any and with .
Proof. It is not difficult to find Then, we obtain Integrating with respect to and using Hölder inequality, we have Integrating with respect to and using Hölder inequality, we obtain It follows from Hölder inequality that By the above inequality, we get (2.2).
Lemma 2.2. Let and assume that . Then, there exists a positive constant such that
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 weak solution to (1.1), (1.2) (see ).
Definition 3.1 (Weak solutions). Let , , and . A measurable -valued triple is said to be a weak solution to (1.1), (1.2) on if the following conditions hold the following.(1)(2)Equations (1.1), (1.2) are satisfied in the sense of distributions; that is, for every and with , hold (3)The energy inequality, that is,
4. Proof of Theorem 3.2
Proof. Multiplying the first equation of (1.1) by and integrating with respect to on , using integration by parts, we obtain
Similarly, we get
Summing up (4.1)-(4.2), we deduce that
By integration by parts and Cauchy inequality, we obtain
Combining (4.3)-(4.4) yields
Integrating with respect to , we have
Differentiating (1.1) with respect to , we obtain
Taking the inner product of with the first equation of (4.7) and using integration by parts yields
Similarly, we get
Combining (4.8)–(4.9) yields
Using integration by parts and Cauchy inequality, we obtain
Combining (4.10)–(4.11) yields
In what follows, we estimate . By integration by parts and Hölder inequality, we obtain where It follows from the interpolating inequality that From (2.3), we get where When , we have and application of Young inequality yields where From Hölder inequality, we obtain where Combining (4.12)–(4.20) yields From Gronwall inequality, we get
Multiplying the first equation of (1.1) by and integrating with respect to on , then using integration by parts, we obtain Similarly, we get Collecting (4.24) and (4.25) yields Thanks to integration by parts and Cauchy inequality, we get It follows from (4.26)-(4.27) and integration by parts that
In what follows, we estimate .
By (2.9) and Young inequality, we deduce that where .
By (2.9) and Young inequality, we have where .
Combining (4.28)–(4.30) yields From (4.31), Gronwall inequality, (4.6), and (4.23), we know that . Thus, can be extended smoothly beyond . We have completed the proof of Theorem 3.2.
This work was supported in part by the NNSF of China (Grant no. 10971190) and the Research Initiation Project for High-level Talents (201031) of the North China University of Water Resources and Electric Power.
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