A regularity criterion for liquid crystal flows in terms of the component of velocity and the horizontal derivative components of orientation field

<abstract><p>In this paper, we establish a regularity criterion for the 3D nematic liquid crystal flows. More precisely, we prove that the local smooth solution $ (u, d) $ is regular provided that velocity component $ u_{3} $, vorticity component $ \omega_{3} $ and the horizontal derivative components of the orientation field $ \nabla_{h}d $ satisfy</p>

<p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \int_{0}^{T}|| u_{3}||_{L^{p}}^{\frac{2p}{p-3}}+||\omega_{3}||_{L^{q}}^{\frac{2q}{2q-3}}+||\nabla_{h} d||_{L^{a}}^{\frac{2a}{a-3}} \mbox{d} t<\infty,\nonumber \\ with\ \ 3< p\leq\infty,\ \frac{3}{2}< q\leq\infty,\ 3< a\leq\infty. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p>

</abstract>


Introduction
In this paper, we will consider the following three-dimensional (3D) nematic liquid crystal flows: where u = (u 1 , u 2 , u 3 ) ∈ R 3 is the velocity field, d = (d 1 , d 2 , d 3 ) ∈ R 3 is the macroscopic average of molecular orientation field and p represents the scalar pressure. The notation ∇d ∇d represents the 3 × 3 matrix of which the (i, j) entry can be denoted by u 0 is the initial velocity with ∇ · u 0 = 0, d 0 is initial orientation vector with |d 0 | ≤ 1. Here, µ, λ, γ, η are all positive constants. And to simplify the presentation, we shall assume that µ = λ = γ = η = 1 in this paper.
The hydrodynamic theory of liquid crystals was established by Ericksen and Leslie during 1960s (see [4,10]). And the system (1.1) is a simplified version of the Ericksen-Leslie model which still retains most of the essential features of the hydrodynamic equations for nematic liquid crystal (see [8]). One of the most significant studies in this area was made by Lin and Liu [9], where they established the existence of global-in-time weak solutions and local-in-time classical solutions. When the orientation field d equals a constant, the above equations reduce to the incompressible Navier-Stokes equations. For well-known Prodi-Serrin type regularity criterion, people paid much focus on decomposing the integral term about u · ∇u and got some improving results based on the components of velocity field u and the gradient of the velocity field ∇u, readers can refer to [1-3, 7, 14, 20, 21, 23, 24]. Naturally, these related results were extended to the liquid crystal flows, see [5,6,11,12,[16][17][18][19]22], and references therein. Moreover, these Prodi-Serrin type regularity criteria based on velocity field indicate that the velocity field u plays a more dominate role than the orientation field d does on the regularity of solutions to the system (1.1).
In [13], Qian established the regularity criterion for system (1.1). That is, if where u h = (u 1 , u 2 ), ω 3 = ∂ 1 u 2 − ∂ 2 u 1 , then the solution is regular. Later, Qian [15] proved the following regularity criterion: Inspired by the above results, we establish the following regularity criterion: then (u, d) can be extended beyond T .
Remark 1.1. In [20], Zhang has decomposed the integral R 3 (u · ∇)u · ∆udx into the several integrals containing u 3 and ω 3 for the Navier-Stokes equation, and the corresponding criterion is So the condition on ∂ 3 u h in (1.2) can be removed and the condition on ∂ 3 u h in (1.3) can be replaced. And, the regularity condition of orientation field d is needed to control the term ∇ · (∇d ∇d) in view of (1.3 Throughout this paper, the letter C means a generic constant which may vary from line to line, and the directional derivatives of a function ϕ are denoted by ∂ i ϕ = ∂ϕ ∂x i (i = 1, 2, 3).

Proof of Theorem 1.1
According to the local well-posedness of smooth solution established by Lin and Liu [9] , we only need to establish the priori estimates. And we have the following standard L 2 estimate (for example, see [17, p.2-3] ) By an argument similar to [17, Eq (2.7)], we have In the following part, we estimate the terms above one by one. For I 1 referring to [20, (2.1)-(2.7)], (or see [11]), I 1 can be decomposed as follows: where α mni jkl , 1 ≤ m, n ≤ 2, 1 ≤ i, j, k, l ≤ 3, are suitable integers. And the purpose is to rewrite ∂ m u n by u 3 and ω 3 , 1 ≤ m, n ≤ 2.

by (2.3) and integration by parts.
Because the Riesz transformation is bounded from L p (R 2 ) to L p (R 2 ) for 1 < p < ∞, we have where p > 3, q > 3 2 .

Conclusions
In this paper, we prove a regular criterion of solution for the 3D nematic liquid crystal flows via velocity component u 3 , vorticity component ω 3 and the horizontal derivative components of the orientation field ∇ h d, and we hope that the condition on ∇ h d will be removed in future study.