THE 3D LIQUID CRYSTAL SYSTEM WITH CANNONE TYPE INITIAL DATA AND LARGE VERTICAL VELOCITY

. The hydrodynamic theory of the nematic liquid crystals was established by Ericksen [4] and Leslie [8]. In this paper, based on a new technique, we obtain global well-posedness to a simpliﬁed model introduced by Lin [9] in the critical Besov space with Cannone type initial data and large vertical velocity, which improves the main result in [15]. In addition, the small condition on u 0 is independent of another small condition on d 0 − ¯ d 0 , which is quite diﬀerent from the previous works [15, 16].

We often see the liquid crystal state as an intermediate state between liquid and solid. Although there are three main types of liquid crystals: nematic, termed smectic and cholesteric, the nematic phase appears to be the most common one. The molecules in the nematic liquid crystals do not exhibit any potential order, however, they have long-range orientation order. The hydrodynamic theory of the nematic liquid crystals had been studied by Ericksen [4] and Leslie [8]. In both 2D and 3D, Lin-Liu [10] obtained the global weak solution to an approximate model, which can be given by replacing |∇d| 2 d by −∇ 1 (|d| 2 − 1) 2 . The initial condition was u 0 ∈ L 2 and d 0 ∈ H 1 in that paper. Provided the initial data satisfied u 0 ∈ H 1 and d 0 ∈ H 2 , they also obtained a unique global classical solution in 2D, and in 3D with additional large viscosity. We refer to [11] for the result concerning the partial regularity of suitable weak solution.

RENHUI WAN
For the system (1), the authors in [6,12] proved global existence of the Leray-Hopf type weak solutions in 2D. It was [14] that established the existence of global weak solutions when u 0 ∈ L 2 and d 0 ∈ H 1 with the initial director field d 0 maps to the upper hemisphere. We refer to [13] for the uniqueness of weak solution and [5,15,16,17] for the studies of small data solution.
Denote D = d −d 0 with D 0 = d 0 −d 0 , then we can rewrite system (1) as By using Littlewood-Paley Theory, [5] obtained local well-posedness for (2) with the initial data satisfying (u 0 , d 0 −d 0 ) ∈Ḃ 1 2 2,1 (R 3 ) ×Ḃ 3 2 2,1 (R 3 ) and global well-posedness with the assumption of the small initial data, i.e., By applying the idea dealing with 3D Navier-Stokes equations, Liu et al. [15] proved a new global well-posedness with large initial vertical velocity. Precisely, they assumed which was generalized to the L p framework in [16]. In fact, when , they obtained the local well-posedness with (r, q) satisfying Under the small initial condition they also showed the global well-posedness. In these works, [15,16,5], the viscosity parameters ν and µ occur in (2) and the small conditions are depend on these parameters. In the present work, we have set ν = µ = 1 for simplicity. Now, we state our first result.
(2) One can also get a generalization of Theorem 1.2 via using our idea. In fact, we can get global solution under where (r, q) meets (4), which improves the result in [16].
(3) It is worth pointing out that the index p in (7) and (8) is independent of q, which means the small condition on u 0 is independent of another small condition on d 0 −d 0 . This is very different from [15,16]. Furthermore, our results allow large Lip norm of the velocity. (4) Although global well-posedness for (1) with the Koch-Tartar type data (see [7]) was obtained in [17], we ensure that it is difficult to get large vertical velocity under this data.
Let us show the idea. To avoid the coupling between u and D, we shall split the Navier-Stokes equations from (1). Indeed, we suffice to obtain global well-posedness for the following two new systems: and where (9) is the well-known Navier-Stokes equations which admits global solution with Cannone type dataḂ 3 p −1 p,∞ (see [2]). By combining with some product estimates and blow-up criteria, we can get that the solution for (9) satisfies Then using (11) and continuous arguments, one can also get global well-posedness for (10). At last, combing these estimates, we can obtain the global bound of (2), see section 3 and section 4 for the details. The present paper is structured as follows: In section 2, we provide some definitions of spaces and several lemmas. The third section and the fourth section devote to the proof of Theorem 1.1 and Theorem 1.2, respectively. We provide some estimates in the Appendix.
Let us complete this section by describing the notations we shall use in this paper.
Notations. For A, B two operator, we denote by [A, B] = AB − BA the commutator between A and B. In some places of this paper, we may use L p andḂ s p,r to stand for L p (R 3 ) andḂ s p,r (R 3 ), respectively. The uniform constant C may be different on different lines. We use f p to denote the L p (R 3 ) norm of f . We denote L r t (X) the space L r ([0, t]; X). (c j ) j∈Z will be a generic element of l 1 (Z), i.e., j∈Z c j ≤ 1. Finally, we denote 2. Preliminaries. In this section, we give some necessary definitions, propositions and lemmas.
The Fourier transform is defined by Choose a nonnegative smooth radial function ϕ supported in C such that We denote ϕ j = ϕ(2 −j ξ) and h = F −1 ϕ, where F −1 stands for the inverse Fourier transform. Then the dyadic blocks ∆ j and S j can be defined as follows One easily verifies that with our choice of ϕ Let us recall the definition of the Besov space.
multi-index} and can be identified by the quotient space of S /P with the polynomials space P.
The norm of the spaceL r1 t (Ḃ s p,r ) is defined by , which is also used in our proof.
The following proposition and lemma provide Bernstein type inequalities and standard commutator estimate in d dimensions.
There exists a constant C such that for any Lipschitz function a with gradient in L p (R d ) and any function b in L q (R d ), we have for any positive λ, There exists a constant c depending on R2 R1 and such that Let us introduce the homogeneous Bony's decomposition. where 3. Proof of Theorem 1.1. In this section, we give the proof of Theorem 1.1. As the mention in section 1, we shall consider two new systems (9) and (10). Now, we begin the proof.
Step 1. Local well-posedness and regularity criterion for (9). One can easily get the local well-posedness of the solution for (9), that is, there exists a T 1 > 0 such that (9) admits a unique solution V satisfying ), see [5] for the details. Next, we will see that the condition where p ∈ [1, ∞] and K 1 is a constant, can yield global regularity.
Applying the operator ∆ j to the both sides of (9), taking the L 2 product with ∆ j V , and using the Bernstein's inequality, one gets where we have used Dividing by ∆ j V 2 , and integrating over (0, t), we can obtain By Bony's decomposition, we have For I 1 and I 2 . By commutator estimate (13), Bernstein's inequality and Young's inequality for series, For I 3 and I 4 . Similar arguments lead In a same way, one can also get Using (15), setting in (5) small enough such that CK 1 ε < 1 2 , then V L∞ (0,t;Ḃ Hence, we have proved global regularity under the condition (15).
Using (14) and Bernstein's inequality, then we have Integrating in time, multiplying by 2 j( 3 p −1) , we can obtain .
For L 3 , by Young's inequality for series and Bernstein's inequality, we have .
Combining with the above three estimates, one can obtain .
Setting = c 8C , we can get Thus we can get a contradiction with the previous assumption by continuous arguments. Hence,T = T 1 . Namely, we can get (18) holds for all t ∈ (0, T 1 ). Combining with interpolation inequality we can get (15) with K 1 = C 1 . It means that we have proved (9) has a unique global solution satisfying Step 3. Global solution to (10). Thanks to the global bound (19) of V , one can easily get there exists T 2 > 0 such that (10)  2,1 ). To get the global solution of (10), we suffice to prove for all t ∈ (0, T 2 ) there holds and As the previous steps, by Bernstein's inequality, we have By (12),Ḃ 3 2 2,1 → L ∞ and Bernstien's inequality, we have Combining with the above estimates and using interpolation inequality, we obtain )dτ where η 1 > 0 fixed later is a small constant. We assume thatT < T 2 . For all t ∈ (0,T ), we have )dτ Choosing η 1 = c 4C , and applying Gronwall's lemma yields Thanks to the estimate where we have use the result in step 2, then for all t ∈ (0,T ), By setting = η1 2 in (5), we can obtain a contradiction with the previous assumption by continuous arguments. Thus, we can getT = T 2 . So we have proved (20) and (21).
Step 4. Global solution to (2). Following the way in [5], one can get the local well-posedness of the solution (u, D) for (2). In addition, we can see (V + W, D) is also a solution to (2). Due to the uniqueness, there holds u = W + V . Thanks to the previous steps, we can obtain the global bound of (u, D), which completes the proof of Theorem 1.1.

4.
Proof of Theorem 1.2. In this section, we prove Theorem 1.2. The procedure is similar to the previous section, but we need a new regularity criterion to get global well-posedness for (9), which is given in the following lemma. 2,1 (R 3 ) with ∇ · u 0 = 0. There exists a unique solution V to (9) in the spaceC where p ∈ [2, 6) and 0 is a small enough constant, then T 1 = ∞.
Proof of Lemma 4.1. Repeating step 1 in section 3 gives Here we have split V · ∇V into three terms and applied divergence free condition. Thanks to ), (24) the proof of which will be given in the Appendix, we have So we have obtained global regularity under (23). Now, we begin the proof of Theorem 1.2. It suffices to prove that (23) holds when 2 ≤ p < 6. In fact, once (23) holds, (9) has a unique global solution, and then following the procedure in section 3 line by line can yield the Theorem 1.2.
Step 1. The estimate for V 3 . One can get V 3 satisfies As the procedure in section 3, we can obtain

RENHUI WAN
Let us split I 1 into two terms: and . Using Bony's decomposition, we have By Bernstein's inequality, . By Bernstein's inequality, p < 6 and Young's inequality for series , one gets .

Thus we have
.
For I 12 , using ∂ 3 V 3 = −∇ h · V h and similar procedure, we have .

THE 3D LIQUID CRYSTAL SYSTEM WITH CANNONE TYPE DATA 5533
Collecting the above estimates, one can obtain ).
For I 2 , let us recall that p can be given by and then Following the estimate in step 2 of the section 3, one can get ).
For L 23 , it follows from using the procedure as the estimate of I 11 and p < 6 that .
Proof. The lemma is a variant of Lemma 3.2 in [18]. We split it into two cases: p ∈ [2, 5) and p ∈ [5,6). We only show the details of the case p ∈ [2, 5), since the difference between the two cases have been given in [18]. By Bony's decomposition, For I 1 , by Bernstein's inequality, and interpolation inequality one gets here we have used Bernstein's inequality in two dimensions and Minkowski inequality. For I 2 , using Bernstein's inequality and (31), For I 3 , applying Bernstein's inequality and (31) again, Similarly, we also have For the third term on the left hand side of (30), we can get a similar bound by modifying the proof of the first two terms slightly. Hence, (30) with α = 1 p can be proved.