Regularity Criterion for the Nematic Liquid Crystal Flows in Terms of Velocity

and Applied Analysis 3 In the following, we establish the bounds of I 1 –I 3 , for the first term I 1 ; thanks to Lemma 5 and using Young’s inequality, we have I 1 ≤ ∫ |u| |∇u| |Δu| dx ≤ 󵄩 󵄩 󵄩 󵄩 󵄩 ‖u‖ L γ 3 󵄩 󵄩 󵄩 󵄩 󵄩L α 1,2 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ‖∇u‖ L 2γ/(γ−2) 3 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩L 2α/(α−2)


Introduction
This paper is devoted to the regularity criterion for the threedimensional nematic liquid crystal flows: with initial data where ( , ) is the velocity field, ( , ) represents the macroscopic average of the nematic liquid crystal orientation field, and ( , ) is the scalar pressure. The symbol ∇ ⊙ ∇ denotes a matrix whose ( , )th entry is given by ⋅ for 1 ≤ , ≤ 3; here ( ) = (1/ 2 )(| | 2 −1) . Since the sizes of the viscosity constants ], , , do not play important roles in our proof, for simplicity, we assume all these positive constants to be one.
The hydrodynamic theory of liquid crystals was established by Ericksen and Leslie [1][2][3][4]; the model (1) is a simplified system of Ericksen-Leslie model which was first introduced by Lin in [5], and one of the most significant works is given by Lin and Liu [6]; more precisely, they established global existence for weak solutions and classical solutions. Recently, Liu et al. in [7] established the regularity criterion for (1) as follows: One may refer to some interesting and important regularity criteria of nematic liquid crystal flows studied by many authors (see, e.g., [8][9][10][11][12][13] and the references therein). When is constant, the system (1) becomes the well-known Navier-Stokes equations. The regularity of solutions to the 3D NS equations has been widely investigated during the past fifty years; see, for example, [14][15][16][17][18][19][20][21][22] and so on. The aim of this paper is to establish a new regularity criterion by providing sufficient condition in terms of velocity or the fractional derivative of velocity in one direction in the framework of anisotropic Lebesgue space. Throughout the paper, the norm of the Lebesgue spaces belongs to the permutation group = span{1, 2, 3}. Denote with the initial data div 0 = 0, and let the pair ( , ) be the weak solution to the liquid crystal flows then ( , ) can be extended beyond .
with the initial data div 0 = 0, and let the pair ( , ) be the weak solution to the liquid crystal flows then ( , ) can be extended beyond .

Corollary 3.
Under the assumption of Theorem 2, if we fix = 1, then the sufficient condition is that Remark 4. Comparing with the corresponding results in [7], it is obvious that the conclusion of Corollary 3 is an improvement version of Theorem 1.1 in [7] in some sense.

The Proof of Theorems 1 and 2
In this section, we will prove Theorems 1 and 2. For convenience, we first recall the following three-dimensional Sobolev and Ladyzhenskaya inequalities in the whole space (see, e.g., [23][24][25]).
Besides, we multiply (1) 2 by | | 4 and integrate over 3 and get which implies Multiplying the first equation of (1) by −Δ and integrating over 3 . Similarly, by taking Δ on both sides of the second equation of (1), by multiplying the resulting equation by Δ , by integrating over 3 , and then by adding two results above and taking the divergence-free condition div = 0 into account, we obtain Abstract and Applied Analysis 3 In the following, we establish the bounds of 1 -3 , for the first term 1 ; thanks to Lemma 5 and using Young's inequality, we have For the second term 2 , similar to estimate of 1 , we have For the term 3 , using Hö lder's inequality, Young's inequality, and (13), one has Substituting the above estimates (15)- (17) into (14), we obtain Integrating (18) from 0 to , using Hö lder's inequality and Young's inequality, one has Finally, applying Gronwall's inequality and using condition (5), then ( , ) can be extended beyond . This completes the proof of Theorem 1.
Proof of Theorem 2. When ∈ [0, 1/ ), combining Theorem 1 and using the following imbedding theorem, one can get the conclusion that When ∈ (1/ , 1), our strategy is to show that is a sufficient condition. We can verify that integral term , satisfies the conditions of Theorem 1 with ∈ (2, ∞). Applying Lemma 5, Hö lder's inequality, and the interpolation theorem, one can conclude that, for ∈ [ (2 + 1)/ , ], This together with Theorem 1 gives the desired result of Theorem 2.