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.

References

[1]	C. S. Cao, Sufficient conditions for the regularity to the 3D Navier-Stokes equations, Discrete Contin. Dyn. Syst., 26 (2010), 1141–1151. http://dx.doi.org/10.3934/dcds.2010.26.1141 doi: 10.3934/dcds.2010.26.1141
[2]	C. S. Cao, E. S. Titi, Regularity criteria for the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J., 57 (2008), 2643–2661.
[3]	B. Q. Dong, Z. F. Zhang, The BKM criterion for the 3D Navier-Stokes equations via two velocity components, Nonlinear Anal.: Real World Appl., 11 (2010), 2415–2421. https://doi.org/10.1016/j.nonrwa.2009.07.013 doi: 10.1016/j.nonrwa.2009.07.013
[4]	J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Rational Mech. Anal., 9 (1962), 371–378. https://doi.org/10.1007/BF00253358 doi: 10.1007/BF00253358
[5]	J. S. Fan, B. L. Guo, Regularity criterion to some liquid crystal models and the Landau-Lifshitz equations in $\mathbb{R}^{3}$ , Sci. China Ser. A-Math., 51 (2008), 1787–1797. https://doi.org/10.1007/s11425-008-0013-3 doi: 10.1007/s11425-008-0013-3
[6]	W. J. Gu, B. Samet, Y. Zhou, A regularity criterion for a simplified non-isothermal model for nematic liquid crystals, Funkcial. Ekvac., 63 (2020), 247–258.
[7]	X. J. Jia, Y. Zhou, Remarks on regularity criteria for the Navier-Stokes equations via one velocity component, Nonlinear Anal.: Real World Appl., 15 (2014), 239–245. https://doi.org/10.1016/j.nonrwa.2013.08.002 doi: 10.1016/j.nonrwa.2013.08.002
[8]	F. H. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena, Commun. Pure Appl. Math., 42 (1989), 789–814. https://doi.org/10.1002/cpa.3160420605 doi: 10.1002/cpa.3160420605
[9]	F. H. Lin, C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Commun. Pure Appl. Math., 48 (1995), 501–537. https://doi.org/10.1002/cpa.3160480503 doi: 10.1002/cpa.3160480503
[10]	F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265–283. https://doi.org/10.1007/BF00251810 doi: 10.1007/BF00251810
[11]	Q. Li, B. Q. Yuan, Blow-up criterion for the 3D nematic liquid crystal flows via one velocity and vorticity components and molecular orientations, AIMS Math., 5 (2020), 619–628. https://doi.org/10.3934/math.2020041 doi: 10.3934/math.2020041
[12]	Q. Liu, J. H. Zhao, S. B. Cui, A regularity criterion for the three-dimensional nematic liquid crystal flow in terms of one directional derivative of the velocity, J. Math. Phys., 52 (2011), 033102. https://doi.org/10.1063/1.3567170 doi: 10.1063/1.3567170
[13]	C. Y. Qian, Remarks on the regularity criterion for the nematic liquid crystal flows in $\mathbb{R}^{3}$ , Appl. Math. Lett., 274 (2016), 679–689. https://doi.org/10.1016/j.amc.2015.11.007 doi: 10.1016/j.amc.2015.11.007
[14]	C. Y. Qian, A remark on the global regularity for the 3D Navier-Stokes equations, Appl. Math. Comput., 57 (2016), 126–131. https://doi.org/10.1016/j.aml.2016.01.016 doi: 10.1016/j.aml.2016.01.016
[15]	C. Y. Qian, A further note on the regularity criterion for the 3D nematic liquid crystal flows, Appl. Math. Comput., 290 (2016), 258–266. https://doi.org/10.1016/j.amc.2016.06.011 doi: 10.1016/j.amc.2016.06.011
[16]	R. Y. Wei, Y. Li, Z. A. Yao, Two new regularity criteria for nematic liquid crystal flows, J. Math. Anal. Appl., 424 (2015), 636–650. https://doi.org/10.1016/j.jmaa.2014.10.089 doi: 10.1016/j.jmaa.2014.10.089
[17]	B. Q. Yuan, Q. Li, Note on global regular solution to the 3D liquid crystal equations, Appl. Math. Lett., 109 (2020), 106491. https://doi.org/10.1016/j.aml.2020.106491 doi: 10.1016/j.aml.2020.106491
[18]	B. Q. Yuan, C. Z. Wei, BKM's criterion for the 3D nematic liquid crystal flows in Besov spaces of negative regular index, J. Nonlinear Sci. Appl., 10 (2017), 3030–3037. http://dx.doi.org/10.22436/jnsa.010.06.17 doi: 10.22436/jnsa.010.06.17
[19]	B. Q. Yuan, C. Z. Wei, Global regularity of the generalized liquid crystal model with fractional diffusion, J. Math. Anal. Appl., 467 (2018), 948–958. https://doi.org/10.1016/j.jmaa.2018.07.047 doi: 10.1016/j.jmaa.2018.07.047
[20]	Z. J. Zhang, Serrin-type regularity criterion for the Navier-Stokes equations involving one velocity and one vorticity component, Czech. Math. J., 68 (2018), 219–225. https://doi.org/10.21136/CMJ.2017.0419-16 doi: 10.21136/CMJ.2017.0419-16
[21]	Z. J. Zhang, F. Alzahrani, T. Hayat, Y. Zhou, Two new regularity criteria for the Navier-Stokes equations via two entries of the velocity Hessian tensor, Appl. Math. Lett., 37 (2014), 124–130. https://doi.org/10.1016/j.aml.2014.06.011 doi: 10.1016/j.aml.2014.06.011
[22]	L. L. Zhao, F. Q. Li, On the regularity criteria for 3-D liquid crystal flows in terms of the horizontal derivative components of the pressure, J. Math. Rese. Appl., 40 (2020), 165–168.
[23]	Y. Zhou, M. Pokorný, On the regularity to the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097–1107.
[24]	Z. J. Zhang, W. J. Yuan, Y. Zhou, Some remarks on the Navier-Stokes equations with regularity in one direction, Appl. Math., 64 (2019), 301–308. https://doi.org/10.21136/AM.2019.0264-18 doi: 10.21136/AM.2019.0264-18

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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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