Abstract
This paper establishes the global existence and uniqueness of classical solutions to the 2D micropolar fluid flows with mixed partial dissipation and angular viscosity.
1. Introduction
In this paper, we investigate the Cauchy problem for the viscous incompressible micropolar fluid flows. In three-dimensional case it can be expressed as Here, is the divergence-free fluid velocity field, is a scalar pressure, is the microrotation field (angular velocity of the rotation of the particles of the fluid), and the constant is the Newtonian kinetic viscosity, is the dynamics microrotation viscosity, and are the angular viscosities (see, e.g., [1, 2]).
The micropolar fluid equations (1) enable us to consider some physical phenomena that cannot be treated by the classical Navier-Stokes equations ( in (1)), such as the motion of animal blood, liquid crystals, and dilute aqueous polymer solutions. Physically, (1)1 represents the conservation of linear momentum, (1)2 reflects the conservation of angular momentum, and (1)3 is the incompressibility of the fluid, specifying the conservation of mass.
Besides their physical applications, the micropolar fluid equations (1) are also mathematically important. The existence of weak and strong solutions was established by Galdi and Rionero [3] and Yamaguchi [4], respectively.
In this paper, we study the global regularity problem of the D micropolar fluid equations. Assuming that the velocity component in the -direction is zero and the axes of rotation of particles are parallel to the -axis, that is, we obtain by gathering (2) into (1) where is a vector and is a scalar. Here and in what follows, we use the notations The global regularity of (3) with full viscosity has been established by Łukaszewicz [2] (see also [5] for more explicit result). The purpose of this paper is to investigate the global regularity of the D micropolar fluid flows with mixed partial dissipation and angular viscosity. To be precise, we will consider the following system:
Our study is partially motivated by the global well-posedness of the D MHD equations with partial viscosities (see [6, 7], for instance), that of the D Boussinesq equations with partial viscosity (see, e.g., [8, 9]), and that of the D micropolar fluid equations with zero angular viscosity [10].
The main result of this paper now reads.
Theorem 1. Suppose , , with . Then (5) with initial data possesses a unique global classical solution . In addition, for any , satisfies where is the vorticity.
Remark 2. Using the same method in this paper, we may also establish the global regularity for the following system:
The rest of this paper is organized as follows. In Section 2, we recall an elementary lemma from [7]. Section 3 is devoted to establishing the a priori bounds for and , while the bounds for and are provided in Section 4. With the a priori estimates in Sections 3 and 4, we may conclude the proof of Theorem 1 as in [7]. Throughout this paper, the -norm of a function is denoted by .
2. An Elementary Lemma
We recall in this section the following elementary lemma from [7].
Lemma 3. Assume that , , , , and all belong to . Then,
Proof. We provide a proof of (8) simpler than that of [7].
Applying Hölder inequality,
Thus,
Consequently,
3. A Priori Bounds for and
In this section, we establish the a priori bounds for and . First, we have the following energy estimates.
Proposition 4. Assume solves (5) on . Then, Here is a constant depending only on , , , and .
Proof. Taking the inner product of (5)1 with and (5)2 with in , respectively, we deduce where we use the following facts (the first one being well-known in the mathematical theory of fluid dynamics, and its proof is provided in the appendix): Now, can be dominated as Substituting (15) into (13), we obtain (12) by invoking Gronwall inequality.
Remark 5. Due to the partial dissipation and angular viscosity, we are not able to establish the uniform boundedness of on but rather the exponential growth:
Now, we are in a position to derive the bounds for and .
Proposition 6. Assume as in Proposition 4. Then the vorticity and satisfy
Proof. Taking the curl of (5)1, we find Then, taking the inner product of (18) with and (5)2 with in , respectively, we obtain Applying Gronwall inequality, we may complete the proof of Proposition 6.
4. A Priori Bounds for and
This section is devoted to deriving the a priori bounds for and .
Proposition 7. Assume as in Proposition 6. Then, Here is a constant depending only on , , , and .
Proof. Taking the inner product of (18) with and (5)2 with in , respectively, we find Gathering the above equations together, noticing that we see Expanding the right-hand side of (23) gives For , applying Hölder inequality yields For , integrating by parts gives For , we apply Lemma 3 to deduce Similarly, we have Now, for , , we use Lemma 3 and Young inequality to see Gathering (25)–(30) into (24) yields According to Proposition 6, we may invoke Gronwall inequality to deduce (20).
Appendix
In this appendix, we provide the proof of (14) for reader’s convenience.
Lemma A.1. Let be divergence-free; that is, . Then
Proof. Integration by parts formula gives Noticing that , we have Consequently, we have as desired.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
Zujin Zhang was partially supported by the Youth Natural Science Foundation of Jiangxi Province (20132BAB211007) and the National Natural Science Foundation of China (11326138).