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

Qiang Li; Baoquan Yuan; Qiang Li; Baoquan Yuan

doi:10.3934/math.2022231

AIMS Mathematics

2022, Volume 7, Issue 3: 4168-4175. doi: 10.3934/math.2022231

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Research article

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

Qiang Li ¹,
Baoquan Yuan ^{2
,
,}

1.
School of Mathematics and Statistics, Xinyang College, Henan 464000, China
2.
School of Mathematics and Information Science, Henan Polytechnic University, Henan 454000, China

Received: 14 October 2021 Revised: 06 December 2021 Accepted: 07 December 2021 Published: 17 December 2021
MSC : 35B65, 35Q35, 76A15

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

$\begin{eqnarray*} \int_{0}^{T}|| u_{3}||_{L^{p}}^{\frac{2p}{p-3}}+||\omega_{3}||_{L^{q}}^{\frac{2q}{2q-3}}+||\nabla_{h} d||_{L^{a}}^{\frac{2a}{a-3}} \mbox{d} t<\infty,\nonumber \\ with\ \ 3< p\leq\infty,\ \frac{3}{2}< q\leq\infty,\ 3< a\leq\infty. \end{eqnarray*}$

Keywords:

Citation: Qiang Li, Baoquan Yuan. A regularity criterion for liquid crystal flows in terms of the component of velocity and the horizontal derivative components of orientation field[J]. AIMS Mathematics, 2022, 7(3): 4168-4175. doi: 10.3934/math.2022231

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Abstract

1. Introduction

In this paper, we will consider the following three-dimensional (3D) nematic liquid crystal flows:

$\begin{eqnarray} \begin{cases} \partial_t u + u\cdot \nabla u - \mu \Delta u+\nabla p = -\lambda \nabla\cdot (\nabla d\odot \nabla d),\\ \partial_t d + u\cdot \nabla d = \gamma(\Delta d-f(d)), \\ \nabla \cdot u = 0,\\ u(x, 0) = u_0(x), d(x, 0) = d_0(x), \end{cases} \end{eqnarray}$

(1.1)

where $u = (u_{1}, u_{2}, u_{3})\in\mathbb{R}^{3}$ is the velocity field, $d = (d_{1}, d_{2}, d_{3})\in\mathbb{R}^{3}$ is the macroscopic average of molecular orientation field and $p$ represents the scalar pressure. The notation $\nabla d\odot \nabla d$ represents the $3\times 3$ matrix of which the $(i, j)$ entry can be denoted by

$\sum\limits_{k = 1}^{3} \partial_{i}d_{k} \partial_{j}d_{k}(1\leq i,j\leq 3),$

and

$f(d) = \frac{1}{|\eta|^{2}}(|d|^{2}-1)d.$

$u_{0}$ is the initial velocity with $\nabla\cdot u_{0} = 0$ , $d_{0}$ is initial orientation vector with $|d_{0}|\leq 1$ . Here, $\mu$ , $\lambda$ , $\gamma$ , $\eta$ are all positive constants. And to simplify the presentation, we shall assume that $\mu = \lambda = \gamma = \eta = 1$ 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 $u\cdot\nabla u$ and got some improving results based on the components of velocity field $u$ and the gradient of the velocity field $\nabla u$ , readers can refer to ^{[1,2,3,7,14,20,21,23,24]}. 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

$\begin{eqnarray} \int_{0}^{T}|| u_{3}||_{L^{p}}^{q}+||\omega_{3}||_{L^{a}}^{b}+|| \partial_{3} u_{h}||_{L^{a}}^{b} \mbox{d} t < M,\ for\ some\ \ M > 0 \\ and\ \ \frac{3}{p}+\frac{2}{q} = 1,\ \frac{3}{a}+\frac{2}{b} = 2,\ 3 < p\leq\infty, \frac{3}{2} < a\leq\infty \end{eqnarray}$

(1.2)

where $u_{h} = (u_{1}, u_{2})$ , $\omega_{3} = \partial_{1}u_{2}- \partial_{2}u_{1}$ , then the solution is regular. Later, Qian ^[15] proved the following regularity criterion:

$\begin{eqnarray} \int_{0}^{T}|| u_{3}||_{L^{p}}^{q}+|| \partial_{3} u_{h}||_{L^{a}}^{b}+||\nabla_{h}\nabla d||_{L^{a}}^{b} \mbox{d} t < M,\ for\ some\ \ M > 0 \\ and\ \ \frac{3}{p}+\frac{2}{q} = 1,\ \frac{3}{a}+\frac{2}{b} = 2,\ 3 < p\leq\infty, \frac{3}{2} < a\leq\infty. \end{eqnarray}$

(1.3)

Inspired by the above results, we establish the following regularity criterion:

Theorem 1.1. Suppose the initial data $u_{0}\in H^{1}(\mathbb{R}^{3})$ with $\nabla\cdot u_{0} = 0$ , $d_{0}\in H^{2}(\mathbb{R}^{3})$ , and let $(u, d)$ be a smooth solution to the system (1.1) on $[0, T)$ for some $0 < T < \infty$ . If $(u, d)$ satisfies the following condition

$\begin{eqnarray} \int_{0}^{T}|| u_{3}||_{L^{p}}^{\frac{2p}{p-3}}+||\omega_{3}||_{L^{q}}^{\frac{2q}{2q-3}}+||\nabla_{h} d||_{L^{a}}^{\frac{2a}{a-3}} \mathit{\mbox{d}} t < \infty, with\ \ 3 < p\leq\infty,\ \frac{3}{2} < q\leq\infty,\ 3 < a\leq\infty, \end{eqnarray}$

(1.4)

then $(u, d)$ can be extended beyond $T$ .

Remark 1.1. In ^[20], Zhang has decomposed the integral $\int_{ \mathbb{R}^{3}}(u\cdot\nabla)u\cdot\Delta u \mathit{\mbox{d}} x$ into the several integrals containing $u_{3}$ and $\omega_{3}$ for the Navier-Stokes equation, and the corresponding criterion is

$\begin{eqnarray} \int_{0}^{T}||u_{3}||_{L^{p}}^{\frac{2p}{p-3}}+||\omega_{3}||_{L^{q}}^{\frac{2q}{2q-3}} \mathit{\mbox{d}} t < \infty,\ with\ 3 < p\leq\infty,\ \frac{3}{2} < q\leq\infty. \end{eqnarray}$

(1.5)

So the condition on $\partial_{3} u_{h}$ in (1.2) can be removed and the condition on $\partial_{3} u_{h}$ in (1.3) can be replaced. And, the regularity condition of orientation field $d$ is needed to control the term $\nabla\cdot(\nabla d\odot\nabla d)$ in view of (1.3).

Remark 1.2. Compared with the corresponding results (1.2), we replace the conditions on $\partial_{3}u_{h}$ with $\nabla_{h}d$ because we can not control the 2-order higher derivatives term $\nabla\cdot(\nabla d\odot\nabla d)$ by only $u_{3}$ and $\omega_{3}$ . Compared with (1.3), we reduce 1-order derivative on orientation field $d$ , which improves the result 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 $\varphi$ are denoted by $\partial_{i}\varphi = \frac{ \partial \varphi}{ \partial x_{i}}\ (i = 1, 2, 3)$ .

2. 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])

$\begin{eqnarray} (||u||_{L^{2}}^{2}+||\nabla d||_{L^{2}}^{2})+2\int_{0}^{T}(||\nabla u||_{L^{2}}^{2}+||\Delta d||_{L^{2}}^{2}+||d|\nabla d|||_{L^{2}}^{2}+\frac{1}{2}||\nabla|d|^{2}||_{L^{2}}^{2}) \mbox{d} t \\ \leq C(||u_{0}||_{L^{2}}^{2}+||\nabla d_{0}||_{L^{2}}^{2}). \end{eqnarray}$

(2.1)

By an argument similar to [17,Eq (2.7)], we have

$\begin{align} &\; \frac{1}{2}\frac{ \mbox{d}}{ \mbox{d} t}(||\nabla u||_{L^{2}}^{2}+||\Delta d||_{L^{2}}^{2})+||\Delta u||_{L^{2}}^{2}+||\nabla\Delta d||_{L^{2}}^{2} \\ = & \int_{ \mathbb{R}^{3}}(u\cdot\nabla)u\cdot\Delta u \mbox{d} x+\int_{ \mathbb{R}^{3}} \nabla\cdot(\nabla d \odot\nabla d )\cdot\Delta u \mbox{d} x \\&- \int_{ \mathbb{R}^{3}}\Delta(u\cdot\nabla d)\cdot\Delta d \mbox{d} x-\int_{ \mathbb{R}^{3}}\Delta (| d|^{2}d-d)\cdot\Delta d \mbox{d} x \\ : = & I_{1}+I_{2}+I_{3}+I_{4}. \end{align}$

(2.2)

In the following part, we estimate the terms above one by one. For $I_{1}$ referring to [,(2.1)–(2.7)], (or see ^[11]), $I_1$ can be decomposed as follows:

$\begin{align*} I_{1} = &\sum\limits_{i,j,k,l = 1}^{3}\int_{ \mathbb{R}^{3}}\alpha_{11ijkl} \partial_{1}u_{1} \partial_{i}u_{j} \partial_{k} u_{l}\\&+\sum\limits_{i,j,k,l = 1}^{3}\int_{ \mathbb{R}^{3}}\alpha_{12ijkl} \partial_{1}u_{2} \partial_{i}u_{j} \partial_{k}u_{l} \\& + \sum\limits_{i,j,k,l = 1}^{3}\int_{ \mathbb{R}^{3}}\alpha_{21ijkl} \partial_{2}u_{1} \partial_{i}u_{j} \partial_{k} \\&+\sum\limits_{i,j,k,l = 1}^{3}\int_{ \mathbb{R}^{3}}\alpha_{22ijkl} \partial_{2}u_{2} \partial_{i}u_{j} \partial_{k} \\ = & I_{11}+I_{12}+I_{13}+I_{14}, \end{align*}$

where $\alpha_{mnijkl}, \ 1\leq m, n\leq 2, \ 1\leq i, j, k, l\leq 3$ , are suitable integers. And the purpose is to rewrite $\partial_{m}u_{n}$ by $u_{3}$ and $\omega_{3}$ , $1\leq m, n\leq 2$ .

Denoting by $\Delta_{h} = \partial_{1} \partial_{1}+ \partial_{2} \partial_{2}$ the horizontal Laplacian, and $\Re_{m} = \frac{ \partial_{m}}{\sqrt{-\Delta_{h}}}$ the two-dimension Riesz transformation, it was shown in [20,(2.2)–(2.4)], that

$\begin{eqnarray} \Delta_{h}u_{1} = - \partial_{2}\omega_{3}- \partial_{1} \partial_{3}u_{3},\ \Delta_{h}u_{2} = \partial_{1}\omega_{3}- \partial_{2} \partial_{3}u_{3}. \end{eqnarray}$

$\begin{eqnarray} \partial_{m}u_{1} = \frac{ \partial_{2}}{\sqrt{-\Delta_{h}}}\frac{ \partial_{m}}{\sqrt{-\Delta_{h}}}\omega_{3}+ \frac{ \partial_{1}}{\sqrt{-\Delta_{h}}}\frac{ \partial_{m}}{\sqrt{-\Delta_{h}}} \partial_{3}u_{3} = \Re_{2}\Re_{m}\omega_{3}+\Re_{1}\Re_{m} \partial_{3}u_{3}, \end{eqnarray}$

(2.3)

$\begin{eqnarray} \partial_{m}u_{2} = \Re_{1}\Re_{m}\omega_{3}+\Re_{2}\Re_{m} \partial_{3}u_{3},\ 1\leq m,n\leq 2. \end{eqnarray}$

(2.4)

The term $I_{11}$ can be expressed as

$\begin{align*} I_{11} = &\sum\limits_{i,j,k,l = 1}^{3}\int_{ \mathbb{R}^{3}}\alpha_{11ijkl} \partial_{1}u_{1} \partial_{i}u_{j} \partial_{k} u_{l} \mbox{d} x \\ = & \sum\limits_{i,j,k,l = 1}^{3}\int_{ \mathbb{R}^{3}}\alpha_{11ijkl}(\Re_{2}\Re_{1}\omega_{3}+ \Re_{1}\Re_{1} \partial_{3}u_{3}) \partial_{i}u_{j} \partial_{k} u_{l} \mbox{d} x \\ = & \sum\limits_{i,j,k,l = 1}^{3}\int_{ \mathbb{R}^{3}}\alpha_{11ijkl}\Re_{2}\Re_{1}\omega_{3} \partial_{i}u_{j} \partial_{k} u_{l} \mbox{d} x \\ -& \sum\limits_{i,j,k,l = 1}^{3}\int_{ \mathbb{R}^{3}}\alpha_{11ijkl}\Re_{1}\Re_{1}u_{3} ( \partial_{3} \partial_{i}u_{j} \partial_{k} u_{l}+ \partial_{i}u_{j} \partial_{3} \partial_{k} u_{l}) \mbox{d} x, \end{align*}$

by (2.3) and integration by parts. Because the Riesz transformation is bounded from $L^{p}(\mathbb{R}^{2})$ to $L^{p}(\mathbb{R}^{2})$ for $1 < p < \infty$ , we have

$\begin{align*} I_{11} \leq&C||u_{3}||_{L^{p}}||\nabla u||_{L^{\frac{2p}{p-2}}}||\nabla^{2} u||_{L^{2}}+C||\omega_{3}||_{L^{q}}||\nabla u||_{L^{\frac{2q}{q-1}}}^{2} \\ \leq& C||u_{3}||_{L^{p}}||\nabla u||_{L^{2}}^{\frac{p-3}{p}}||\nabla^{2} u||_{L^{2}}^{\frac{p+3}{p}}+C||\omega_{3}||_{L^{q}}||\nabla u||_{L^{2}}^{\frac{2q-3}{q}}||\nabla^{2} u||_{L^{2}}^{\frac{3}{q}} \\ \leq& C(||u_{3}||_{L^{p}}^{\frac{2p}{p-3}}+||\omega_{3}||_{L^{q}}^{\frac{2q}{2q-3}})||\nabla u||_{L^{2}}^{2}+\frac{1}{16}||\Delta u||_{L^{2}}^{2}, \end{align*}$

where $p > 3, q > \frac{3}{2}$ .

The similar argument as $I_{11}$ can be used to terms $I_{12}, I_{13}, I_{14}$ , therefore it can be deduced that

$\begin{eqnarray} I_{1}\leq&C(||u_{3}||_{L^{p}}^{\frac{2p}{p-3}}+||\omega_{3}||_{L^{q}}^{\frac{2q}{2q-3}})||\nabla u||_{L^{2}}^{2}+\frac{1}{4}||\Delta u||_{L^{2}}^{2}. \end{eqnarray}$

(2.5)

For $I_{2}$ and $I_{3}$ , by using the fact $\nabla\cdot u = 0$ and integrating by parts several times, we can rewrite it as follows

$\begin{align*} I_{2}+I_{3} = &\int_{ \mathbb{R}^{3}}\sum\limits_{i,j,k = 1}^{3}[( \partial_{i} \partial_{j}d_{k} \partial_{j}d_{k}+ \partial_{i}d_{k} \partial_{j} \partial_{j}d_{k})\Delta u_{i}\\ &-(\Delta u_{i} \partial_{i}d_{k}\Delta d_{k} + 2\nabla u_{i} \partial_{i}\nabla d_{k}\Delta d_{k}+u_{i} \partial_{i} \Delta d_{k}\Delta d_{k})] \mbox{d} x\nonumber \\ = & \int_{ \mathbb{R}^{3}}\sum\limits_{i,j,k = 1}^{3}-2\nabla u_{i} \partial_{i}\nabla d_{k}\Delta d_{k} \mbox{d} x\nonumber \\ = & \int_{ \mathbb{R}^{3}}-2\sum\limits_{j,k = 1}^{3}\sum\limits_{i = 1}^{2} \partial_{j} u_{i} \partial_{i} \partial_{j} d_{k} \Delta d_{k} \mbox{d} x-\int_{ \mathbb{R}^{3}}2\sum\limits_{j,k = 1}^{3} \partial_{j} u_{3} \partial_{3} \partial_{j} d_{k} \Delta d_{k} \mbox{d} x\nonumber \\ = & I_{21}+I_{22}.\nonumber \end{align*}$

$\begin{align*} \label{2.6} I_{21} = \int_{ \mathbb{R}^{3}}2\sum\limits_{j,k = 1}^{3}\sum\limits_{i = 1}^{2} \partial_{j} u_{i} \partial_{i} d_{k} \partial_{j}\Delta d_{k} \mbox{d} x +\int_{ \mathbb{R}^{3}}2\sum\limits_{j,k = 1}^{3}\sum\limits_{i = 1}^{2} \partial_{j} \partial_{j} u_{i} \partial_{i} d_{k} \Delta d_{k} \mbox{d} x\nonumber = I_{211}+I_{212}.\nonumber \end{align*}$

Next, employing the H $\rm\ddot{o}$ lder inequality, interpolation inequality and Young's inequality, we have

$\begin{align} I_{211} \leq& C\|\nabla_{h} d\|_{L^{a}}\|\nabla u\|_{L^{\frac{2a}{a-2}}}\|\nabla\Delta d\|_{L^{2}} \\ \leq& C\|\nabla_{h} d\|_{L^{a}}\|\nabla u\|_{L^{2}}^{\frac{a-3}{a}}\|\Delta u\|_{L^{2}}^{\frac{3}{a}}\|\nabla\Delta d\|_{L^{2}} \\ \leq& C\|\nabla_{h} d\|_{L^{a}}^{\frac{2a}{a-3}}\|\nabla u\|_{L^{2}}^{2}+\frac{1}{8}\|\Delta u\|_{L^{2}}^{2}+\frac{1}{8}\|\nabla\Delta d\|_{L^{2}}^{2}, \end{align}$

(2.6)

$\begin{align} I_{212} \leq& C\|\nabla_{h} d\|_{L^{a}}\|\Delta d\|_{L^{\frac{2a}{a-2}}}\|\Delta u\|_{L^{2}} \\ \leq& C\|\nabla_{h} d\|_{L^{a}}\|\Delta d\|_{L^{2}}^{\frac{a-3}{a}}\|\nabla\Delta d\|_{L^{2}}^{\frac{3}{a}}\|\Delta u\|_{L^{2}} \\ \leq& C\|\nabla_{h} d\|_{L^{a}}^{\frac{2a}{a-3}}\|\Delta d\|_{L^{2}}^{2}+\frac{1}{8}\|\Delta u\|_{L^{2}}^{2}+\frac{1}{8}\|\nabla\Delta d\|_{L^{2}}^{2}. \end{align}$

(2.7)

In the same way, the term $I_{22}$ can be bounded as follows

$\begin{align} I_{22} = &\int_{ \mathbb{R}^{3}}2\sum\limits_{j,k = 1}^{3}( u_{3} \partial_{3} \partial_{j} \partial_{j} d_{k} \Delta d_{k}+ u_{3} \partial_{3} \partial_{j} d_{k} \partial_{j}\Delta d_{k} \mbox{d} x) \mbox{d} x \\ \leq& C\|u_{3}\|_{L^{p}}\|\Delta d\|_{L^{\frac{2p}{p-2}}}\|\nabla\Delta d\|_{L^{2}} \\ \leq& C\|u_{3}\|_{L^{p}}\|\Delta d\|_{L^{2}}^{\frac{p-3}{p}}\|\nabla\Delta d\|_{L^{2}}^{\frac{3}{p}}\|\nabla\Delta d\|_{L^{2}} \\ \leq& C\|u_{3}\|_{L^{p}}^{\frac{2p}{p-3}}\|\Delta d\|_{L^{2}}^{2}+\frac{1}{8}\|\nabla\Delta d\|_{L^{2}}^{2}. \end{align}$

(2.8)

Adding the above inequalities (2.6)–(2.8) together, one obtains

$\begin{align} I_{2}+I_{3} \leq& C(\| u_{3}\|_{L^{p}}^{{\frac{2p}{p-3}}} +\| \nabla_{h}d\|_{L^{a}}^{{\frac{2a}{a-3}}})(\|\nabla u\|_{L^{2}}^{2}+ \|\Delta d\|_{L^{2}}^{2}) + \frac{1}{4}\|\Delta u\|_{L^{2}}^{2}+\frac{3}{8}\|\nabla\Delta d\|_{L^{2}}^{2}. \end{align}$

(2.9)

For $I_{4}$ , we have

$\begin{align} I_{4} \leq& \int_{ \mathbb{R}^{3}} |\Delta d|^{2}+\Delta(|d|^{2}d)\cdot\Delta d \mbox{d} x \\ \leq& ||\Delta d||_{L^{2}}^{2}+C(||\Delta |d|^{2}||_{L^{2}}||d||_{L^{6}}||\Delta d||_{L^{3}}+||\Delta d||_{L^{3}}||d||_{L^{6}}^{2}||\Delta d||_{L^{3}}) \\ \leq& ||\Delta d||_{L^{2}}^{2}+C||\Delta d||_{L^{3}}||d||_{L^{6}}^{2}||\Delta d||_{L^{3}} \\ \leq& ||\Delta d||_{L^{2}}^{2}+C||\Delta d||_{L^{2}}||\nabla\Delta d||_{L^{2}} \\ \leq& C||\Delta d||_{L^{2}}^{2}+{\frac{1}{8}}||\nabla\Delta d||_{L^{2}}^{2}. \end{align}$

(2.10)

Hence, inserting (2.5), (2.9) and (2.10) into (2.2) yields

$\begin{align*} &\frac{ \mbox{d}}{ \mbox{d} t}(||\nabla u||_{L^{2}}^{2}+||\Delta d||_{L^{2}}^{2})+||\Delta u||_{L^{2}}^{2}+||\nabla\Delta d||_{L^{2}}^{2} \\ \leq & C(1+\| u_{3}\|_{L^{p}}^{{\frac{2p}{p-3}}}+||\omega_{3}||_{L^{q}}^{\frac{2q}{2q-3}} +\| \nabla_{h}d\|_{L^{a}}^{{\frac{2a}{a-3}}})(||\nabla u||_{L^{2}}^{2}+||\Delta d||_{L^{2}}^{2}), \end{align*}$

and it could be derived by Gronwall inequality that

$\begin{align*} &||\nabla u||_{L^{2}}^{2}+||\Delta d||_{L^{2}}^{2}+\int_{0}^{T}(||\Delta u||_{L^{2}}^{2}+||\nabla\Delta d||_{L^{2}}^{2}) \mbox{d} t \\ \leq& (||\nabla u_{0}||_{L^{2}}^{2}+||\Delta d_{0}||_{L^{2}}^{2})\exp\left\{\int_{0}^{T}C(1+\| u_{3}\|_{L^{p}}^{{\frac{2p}{p-3}}}+||\omega_{3}||_{L^{q}}^{\frac{2q}{2q-3}} +\| \nabla_{h}d\|_{L^{a}}^{{\frac{2a}{a-3}}}) \mbox{d} t\right\} . \end{align*}$

Then the proof of Theorem 1.1 is completed.

3. 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 $\omega_{3}$ and the horizontal derivative components of the orientation field $\nabla_{h}d$ , and we hope that the condition on $\nabla_{h}d$ will be removed in future study.

Acknowledgments

The authors are appreciated for the helpful suggestions of referees.

Conflict of interest

All authors declare no conflict of interest in this paper.

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AIMS Mathematics

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

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Abstract

1. Introduction

2. Proof of Theorem 1.1

3. Conclusions

Acknowledgments

Conflict of interest

References

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AIMS Mathematics

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

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Abstract

1. Introduction

2. Proof of Theorem 1.1

3. Conclusions

Acknowledgments

Conflict of interest

References

This article has been cited by:

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