A REMARK ON BLOW UP CRITERION OF THREE-DIMENSIONAL NEMATIC LIQUID CRYSTAL FLOWS

. In this paper, we study the initial value problem for the three-dimensional nematic liquid crystal ﬂows. Blow up criterion of smooth solutions is established by the energy method, which reﬁnes the previous result.


Introduction. We investigate the nematic liquid crystal flows in three space
with the initial data t = 0 : u = u 0 (x), d = d 0 (x), (2) where u(t, x) : R 3 −→ R 3 represents the velocity field d : R 3 −→ S 2 denotes the macroscopic average of the nematic liquid crystal orientation filed, p(t, x) denotes the pressure and ν > 0 is the kinematic viscosity.
The materials called liquid crystals have attracted lots of scientists attention since the end of the last century. Briefly, liquid crystals are states of matter which are capable of flow, and in which the molecular arrangement gives rise to a preferred direction. Many of their properties in an extensive review were discussed in [1]. A number of attempts have been made to formulate continuum theories to describe properties of these peculiar liquids(see [22], [7]). Liquid crystals are divided into different phases according to the behavior of the molecules. The most important are the nematic phase, the cholesteric phase and the smetic phase. The hydrodynamic theory of liquid crystals in the nematic case has been established by Ericksen [4] and Leslie [15] (see also [6], [8] and [16]), which has been widely used for theoretical and experimental research(see [5]). (1) is a macroscopic continuum description of the time evolution of the materials under the influence of both the flow field u(x, t), and the macroscopic description of the microscopic orientation configurations d(x, t) of rod-like liquid crystals. Recall that the Ericksen-Leslie theory reduces to the Ossen-Frank theory in the static case, see Hardt-Lin-Kinderlehrer [10] and references therein. The mathematical theory is presently still under a wide development and the study of the full Ericksen-Leslie model presents relevant mathematical difficulties. Therefore, a simplified model (1) of the general Ericksen-Leslie equations (see [17], [18]) that keeps many of the mathematical difficulties of the original equations that was introduced in [17].
In [19], global weak solutions to the initial boundary value for (1) on bounded domains in two space dimensions was established ( see also [11]). The uniqueness of such weak solutions is obtained by Lin and Wang [20]. Wen and Ding [27] obtained local existence and uniqueness of solution. Moreover, they also established that the global existence and uniqueness of the solution with small initial data.
Local existence of smooth solutions has been announced in [12]: for any 0 < T < T 0 . (1.1) has distinct physical background, rich mathematical connotation and important theoretical value. The problem (1), (2) has great challenge and attracts many Mathematician's interest. Some interesting results have been established, we may refer to [2], [9], [12] , [24], [25], [28] and [29]. The existence of strong solution to liquid crystals system in critical Besov space and a criterion which is similar to Serrins criterion on regularity of weak solution to Navier-Stokes equations have been established by Hao and Liu [9]. Ladyzhenskaya-Prodi-Serrin type criterion for the breakdown of classical solutions has been established by Chen, Tan and Wu [2]. Huang and Wang [12] established a blow up criterion for classical solutions to (1.1) in three space dimensions. More precisely, they proved that 0 < T < +∞ is the maximal time interval iff T 0 ( ∇ × u L ∞ + ∇d 2 L ∞ )dt = ∞. This blow up criterion was refined in [29]. Very recently, Zhang, Tan and Wu [28] obtained a blow up criterion of smooth solutions. Their result says (u, d) is smooth up to time T provided that The condition on d is only a special case of the Serrin-type regularity criteria. In [25], the author studied the problem (1), (2) with ν = 0 and obtained a logarithmically improved blow up criterion of smooth solution.
In the paper, our main purpose is to refine and improve the result in [28]. More precisely, we extend the blow up criterion obtained in [28] toḂ 0 ∞,∞ and L p spaces. The blow up criterion obtained in this paper is different from blow-up criteria of smooth solutions to three-dimensional magneto-micropolar fluid equations in terms of the vorticity in a homogenous negative exponent Besov space in [26]. We overcome the difficulty which has been caused by the fact that theḂ 0 ∞,∞ and L p spaces replace the BM O and L 4 spaces. The proof is based on more sophisticated energy estimate. Now we state our results as follows.
then the solution (u, d) can be extended beyond t = T .
We have the following corollary immediately.
The plan of the paper is arranged as follows. We first state some preliminary on functional settings and some important inequalities in Section 2 and then prove the blow up criterion of smooth solutions to (1), (2) in Section 3.

2.
Preliminaries. Let S(R n ) be the Schwartz class of rapidly decreasing functions. Given f ∈ S(R n ), its Fourier transform Ff =f is defined bŷ and for any given g ∈ S(R n ), its inverse Fourier transform F −1 g =ǧ is defined by Firstly, we recall the littlewood-Paley decomposition. Choose a non-negative radial functions φ ∈ S(R n ), supported in C = {ξ ∈ R n : 3 The frequency localization operator is defined by Next we recall the definition of homogeneous function spaces (see [23]). For (p, q) ∈ [1, ∞] 2 and s ∈ R, the homogeneous Besov spaceḂ s p,q is defined as the set of f up to polynomials such that The following inequality is well-known Gagliardo-Nirenberg inequality.
Lemma 2.1. Let j, m be any integers satisfying 0 ≤ j < m, and let 1 ≤ q, r ≤ ∞, Then for all f ∈ L q (R n ) W m,r (R n ), there is a positive constant C depending only on n, m, j, q, r, θ such that the following inequality holds: with the following exception: if 1 < r < ∞ and m − j − n r is a nonnegative integer, then (6) holds only for a satisfying j m ≤ θ < 1. The following Lemma comes from [13] and [21].
where 1 ≤ α ≤ m and 1 In what follows, we shall make use of Bernstein inequalities, which comes from [3].
hold, where c and C are positive constants independent of f and k.
Lemma 2.4. There exists a uniform positive constant C, such that holds for all vectors f ∈ H 3 (R n )(n = 2, 3) with ∇ · f = 0.
Proof. The proof can be founded in [14]. For the convenience of the readers, the proof will be also sketched here. It follows from Littlewood -Paley decomposition that Using (8), (9) and (11), we obtain By the Biot-Savard law, we have a representation of ∇f in terms of ∇ × f as where R = (R 1 , · · · , R n ), R j = ∂ ∂xj (−∆) − 1 2 denote the Riesz transforms. Since R is a bounded operator inḂ 0 ∞,∞ , this yields ∇f Ḃ0 with C = C(n). Taking It follows from (12), (13) and (14) that (10) holds. Thus, the lemma is proved.

Proof of main results.
Proof. For the classical solutions (u, d), we have Applying ∇ to the first equation of (1), multiplying the resulting equation by ∇u and integrating with respect to x over R 3 , with help of integration by parts, we have (16) Similarly, we obtain By (16), (17) and ∇ · u = 0, we deduce that It follows from Lemma 2.2 that By integration by parts, Hölder inequality, Gagliardo-Nirenberg inequality and Young inequality, we obtain

YINXIA WANG
Similarly, we have For the term I 4 , we apply integration by parts and Hölder inequality. This yields For the term I 41 , using Gagliardo-Nirenberg inequality and Young inequality, we get For the term I 42 , Gagliardo-Nirenberg inequality and Young inequality give Combining (18) We differentiate the second equation of (1) with respect to x. This yields Multiplying (26) by |∇d| p−2 ∇d and and integrating with respect to x over R 3 , with help of integration by parts and Hölder inequality, we have 1 p For the term ∇d p+2 L p+2 , we apply Gagliardo-Nirenberg inequality. This yields For the term C ∇ 2 d L 2 ∇d 4 L 8 , applying Gagliardo-Nirenberg inequality, we obtain Collecting these estimates yielding Adding (27) and (28) yielding Owing to (3), we know that for any small constant ε > 0, there exists T < T such that Let It follows from (15), (29), (30), (31) and Lemma 2.4 that where C 1 depends on ∇u(T ) 2 Applying ∇ m to the first equation of (1), then taking L 2 inner product with ∇ m u and using integration by parts, we have Likewise, we obtain It follows (33), (34) and integration by parts that In what follows, for simplicity, we will set m = 3. With help of integration by parts and Hölder inequality, we derive that It follows from Gagliardo-Nirenberg inequality and Young inequality Substituting the above two estimates into (36), we obtain Using integration by parts and Hölder inequality, we get