A regularity criterion via horizontal components of velocity and molecular orientations for the 3D nematic liquid crystal flows

Qiang Li; Mianlu Zou; Qiang Li; Mianlu Zou

doi:10.3934/math.2022514

AIMS Mathematics

2022, Volume 7, Issue 5: 9278-9287. doi: 10.3934/math.2022514

Previous Article Next Article

Research article

A regularity criterion via horizontal components of velocity and molecular orientations for the 3D nematic liquid crystal flows

Qiang Li ^{1
,
,},
Mianlu Zou ²

1.
School of Mathematics and Statistics, Xinyang College, Henan 464000, China
2.
School of Big Data and Artificial Intelligence, Xinyang College, Henan 464000, China

Received: 07 January 2022 Revised: 19 February 2022 Accepted: 04 March 2022 Published: 09 March 2022
MSC : 35B65, 35Q35, 76A15

In this paper, we obtain a regularity criterion via horizontal components of velocity and molecular orientations for the 3D nematic liquid crystal flows. That is the smooth solution $(u, d)$ can be extended beyond T, provided that $\int_{0}^{T}(||u_{h}||_{\dot{B}_{\infty, \infty}^{0}}^{2}+||\nabla d||_{\dot{B}_{\infty, \infty}^{0}}^{2}) \mbox{d} t < \infty$ .

Keywords:

Citation: Qiang Li, Mianlu Zou. A regularity criterion via horizontal components of velocity and molecular orientations for the 3D nematic liquid crystal flows[J]. AIMS Mathematics, 2022, 7(5): 9278-9287. doi: 10.3934/math.2022514

Related Papers:

[1]	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. AIMS Mathematics, 2022, 7(3): 4168-4175. doi: 10.3934/math.2022231
[2]	Junling Sun, Xuefeng Han . Existence of Sobolev regular solutions for the incompressible flow of liquid crystals in three dimensions. AIMS Mathematics, 2022, 7(9): 15759-15794. doi: 10.3934/math.2022863
[3]	Qiang Li, Baoquan Yuan . Blow-up criterion for the 3D nematic liquid crystal flows via one velocity and vorticity components and molecular orientations. AIMS Mathematics, 2020, 5(1): 619-628. doi: 10.3934/math.2020041
[4]	Tariq Mahmood, Zhaoyang Shang . Blow-up criterion for incompressible nematic type liquid crystal equations in three-dimensional space. AIMS Mathematics, 2020, 5(2): 746-765. doi: 10.3934/math.2020051
[5]	Zhongying Liu, Yang Liu, Yiqi Jiang . Global solvability of 3D non-isothermal incompressible nematic liquid crystal flows. AIMS Mathematics, 2022, 7(7): 12536-12565. doi: 10.3934/math.2022695
[6]	Xiufang Zhao . Decay estimates for three-dimensional nematic liquid crystal system. AIMS Mathematics, 2022, 7(9): 16249-16260. doi: 10.3934/math.2022887
[7]	Danxia Wang, Ni Miao, Jing Liu . A second-order numerical scheme for the Ericksen-Leslie equation. AIMS Mathematics, 2022, 7(9): 15834-15853. doi: 10.3934/math.2022867
[8]	Zhengyan Luo, Lintao Ma, Yinghui Zhang . Optimal decay rates of higher–order derivatives of solutions for the compressible nematic liquid crystal flows in $ \mathbb R^3 $. AIMS Mathematics, 2022, 7(4): 6234-6258. doi: 10.3934/math.2022347
[9]	Abdelkader Moumen, Ramsha Shafqat, Azmat Ullah Khan Niazi, Nuttapol Pakkaranang, Mdi Begum Jeelani, Kiran Saleem . A study of the time fractional Navier-Stokes equations for vertical flow. AIMS Mathematics, 2023, 8(4): 8702-8730. doi: 10.3934/math.2023437
[10]	Sadek Gala, Maria Alessandra Ragusa . A logarithmically improved regularity criterion for the 3D MHD equations in Morrey-Campanato space. AIMS Mathematics, 2017, 2(1): 16-23. doi: 10.3934/Math.2017.1.16

Abstract

1. Introduction

In this paper, we consider the following nematic liquid crystal flows in 3-dimensions:

$\begin{eqnarray} \begin{cases} \partial_t u + u\cdot \nabla u -\mu\Delta u+\nabla p = -\lambda \nabla\cdot (\nabla d\odot \nabla d),\\ \partial_t d + u\cdot \nabla d = \gamma(\Delta d+|\nabla d|^{2}d), \\ \nabla \cdot u = 0, |d| = 1,\\ u(x, 0) = u_0(x), d(x, 0) = d_0(x), \end{cases} \end{eqnarray}$

(1.1)

here, $u = u(x, t) = (u_{1}(x, t), u_{2}(x, t), u_{3}(x, t))$ denotes the velocity field, $d = d(x, t) = (d_{1}(x, t), d_{2}(x, t), d_{3}(x, t))$ denotes the macroscopic average of molecular orientation field and $p = p(x, t)$ is the scalar pressure. And $\mu$ , $\lambda$ , $\gamma$ are positive constants, which will be assumed to be 1 because their specific values play no roles in our arguments. The notation $\nabla d\odot \nabla d$ represents the $3\times 3$ matrix whose the $(i, j)th$ component is given by $\partial_{i}d_{k} \partial_{j}d_{k}(i, j\leq 3)$ .

The above system (1.1) is a simplified version of the Ericksen-Leslie equations, which was first introduced by Lin ^[9] to describe the nematic liquid crystals flows. It is well-known that the system (1.1) has a unique local smooth solution $(u, d)$ provided that initial data $u_{0}\in H^{s}(\mathbb{R}^{n}, \mathbb{R}^{n})$ with $\nabla\cdot u_{0} = 0$ and $d_{0}\in H^{s+1}(\mathbb{R}^{n}, \mathbb{S}^{2})$ for $s\geq n$ . For the regularity criteria readers may refer to ^{[7,8,10,12,14,15,16]}.

Obviously, the above system (1.1) reduce to the incompressible Navier-Stokes equations when the orientation field $d$ equals a constant. For Navier-Stokes equations, in ^[2], Dong and Zhang established the following regularity criterion:

$\begin{eqnarray} \int_{0}^{T}\|\nabla_{h}u_{h}\|_{\dot{B}_{\infty,\infty}^{0}} \mbox{d} t < \infty, \end{eqnarray}$

(1.2)

where $\nabla_{h} = (\partial_{1}, \partial_{2}), u_{h} = (u_{1}, u_{2})$ . And they left a question that had been solved by Gala and Ragusa in ^[3], which is that whether (1.2) can be replaced by the following condition:

$\begin{eqnarray} \int_{0}^{T}\|u_{h}\|_{\dot{B}_{\infty,\infty}^{0}}^{2} \mbox{d} t < \infty. \end{eqnarray}$

(1.3)

In this paper, we are aimed to extend the criterion (1.3) to the system (1.1). Our main results are stated as follows:

Theorem 1.1. Let initial data $u_{0}\in H^{3}(\mathbb{R}^{3})$ , $d_{0}\in H^{4}(\mathbb{R}^{3}, \mathbb{S}^{2})$ , $\nabla\cdot u_{0} = 0$ . Suppose $(u, d)$ is a local smooth solution to the equations of (1.1) on $[0, T)$ for some $0 < T < \infty$ . If $(u, d)$ satisfies

$\begin{eqnarray} \int_{0}^{T}(\|u_{h}\|_{\dot{B}_{\infty,\infty}^{0}}^{2}+\|\nabla d\|_{\dot{B}_{\infty,\infty}^{0}}^{2}) \mathit{\mbox{d}} t < \infty, \end{eqnarray}$

(1.4)

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

Remark 1.1. In ^[14], Yuan and Wei obtained a regularity condition that is

$\int_{0}^{T}(\|\omega\|_{\dot{B}_{\infty,\infty}^{-r}}^{\frac{2}{2-r}}+||\nabla d||_{\dot{B}_{\infty,\infty}^{0}}^{2}) \mbox{d} t < \infty,\ 0 < r < 2.$

Later, Li and Yuan ^[7] improved the above regularity condition by

$\int_{0}^{T}(||\omega_{3}||_{L^{p}}^{\frac{2p}{2p-3}}+||u_{3}||_{L^{q}}^{\frac{2q}{q-3}} +||\nabla d||_{\dot{B}_{\infty,\infty}^{0}}^{2}) \mbox{d} t < \infty,\ with\ \frac{3}{2} < p\leq\infty,\ 3 < q\leq\infty.$

Compared to the two regularity conditons, (1.4) only contains two components of the velocity field $u$ in Besov spaces, which is an improved result.

2. Preliminaries

In this section, we will give some useful inequalities which play an important role in our proof.

Lemma 2.1. (Page 82 in ^[1]) Let $1 < q < p < \infty$ and $\alpha$ be a positive real number. Then there exists a constantC such that

$\begin{eqnarray*} \|f\|_{L^{p}}\leq C\|f\|_{\dot{B}_{\infty,\infty}^{-\alpha}}^{1-\theta}\| f\|_{\dot{B}_{q,q}^{\beta}}^{\theta}, with\ \beta = \alpha(\frac{p}{q}-1), \theta = \frac{q}{p}. \end{eqnarray*}$

In particular, when $\beta = 1$ , $q = 2$ and $p = 4$ , we have $\alpha = 1$ and

$\begin{eqnarray} \|f\|_{L^{4}}\leq C\|f\|_{\dot{B}_{\infty,\infty}^{-1}}^{\frac{1}{2}}\|\nabla f\|_{L^{2}}^{\frac{1}{2}}. \end{eqnarray}$

(2.1)

Lemma 2.2. (Product and commutator estimate ^[6,11]) Let $s > 0$ , $1 < p < \infty$ , and $\frac{1}{p} = \frac{1}{p_1}+\frac{1}{p_2} = \frac{1}{p_3}+\frac{1}{p_4}$ with $p_2$ , $p_3\in(1, +\infty)$ and $p_1$ , $p_4\in[1, +\infty]$ . Then,

$\begin{eqnarray} \|\Lambda^{s}(fg)\|_{L^{p}}\leq C(\|g\|_{L^{p_{1}}}\|\Lambda^{s}f\|_{L^{p_{2}}}+\|\Lambda^{s}g\|_{L^{p_{3}}} \|f\|_{L^{p_{4}}}), \end{eqnarray}$

(2.2)

$\begin{eqnarray} \|[\Lambda^{s},f\cdot \nabla]g\|_{L^{p}}\leq C(\|\nabla f\|_{L^{p_{1}}}\|\Lambda^{s}g\|_{L^{p_{2}}}+\|\Lambda^{s}f\|_{L^{p_{3}}}\|\nabla g\|_{L^{p_{4}}}), \end{eqnarray}$

(2.3)

where $[\Lambda^{s}, f]g = \Lambda^{s}(fg)-f\Lambda^{s}g$ .

Lemma 2.3. (^[5], Theorem 2.1) Let $s > \frac{3}{2}$ , then there exists a constant C such that

$\begin{eqnarray} \|f\|_{\dot{B}_{\infty,2}^{0}}\leq C(1+\|f\|_{\dot{B}_{\infty,\infty}^{0}}\log^{\frac{1}{2}}(1+\|f\|_{\dot{H}^{s}})), \end{eqnarray}$

(2.4)

for any $f\in \dot{H}^{s}$ .

Lemma 2.4. (^[13], Lemma 2.3 or page 21 in ^[17]) Let $f\in BMO$ and $g, h\in {H}^{1}(\mathbb{R}^{3})$ , we have

$\begin{eqnarray} \int_{R^{3}}f\nabla(gh) \mathit{\mbox{d}} x\leq C\|f\|_{BMO}(\|\nabla g\|_{L^{2}}\|h\|_{L^{2}}+\|g\|_{L^{2}}\|\nabla h\|_{L^{2}}). \end{eqnarray}$

(2.5)

3. Proof of Theorem 1.1

Let's recall a result that will be a bridge to prove our conclusion. In ^[4], Huang and Wang established a BKM type blow-up criterion for the system (1.1). If $T$ is the maximal time, $0 < T < \infty$ , one has

$\begin{eqnarray} \int_{0}^{T}(\|\omega\|_{L^{\infty}}+||\nabla d||_{L^{\infty}}^{2}) \mbox{d} t = \infty, \end{eqnarray}$

(3.1)

where $\omega = \nabla\times u$ . Under the conditions (1.4) and (3.1), if we can show

$\begin{eqnarray} \int_{0}^{T}(||\nabla\Delta u||_{L^{2}}^{2}+||\Delta^{2} d||_{L^{2}}^{2}) \mbox{d} t < C, \end{eqnarray}$

(3.2)

then Theorem 1.1 is valid by using the Sobolev embedding $H^{2}(\mathbb{R}^{3})\hookrightarrow L^{\infty}(\mathbb{R}^{3})$ .

Noticing that $\nabla\cdot u = 0$ , we obtain

$\begin{eqnarray*} \int_{ \mathbb{R}^{3}}(u\cdot\nabla u)\cdot u \mbox{d} x = 0,\int_{ \mathbb{R}^{3}}\nabla p\cdot u \mbox{d} x = 0. \end{eqnarray*}$

Mutiplying $u$ to Eq $(1.1)_{1}$ and integrating over $\mathbb{R}^{3}$ yields

$\begin{eqnarray*} \frac{1}{2}\frac{ \mbox{d}}{ \mbox{d} t}\|u\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{2} = -\int_{ \mathbb{R}^{3}}\nabla d\cdot \Delta d \cdot u \mbox{d} x. \end{eqnarray*}$

Similarly, multiplying $(1.1)_{2}$ by $-\Delta d$ and integrating over $\mathbb{R}^{3}$ one has

$\begin{eqnarray*} \frac{1}{2}\frac{ \mbox{d}}{ \mbox{d} t}\|\nabla d\|_{L^{2}}^{2}+\|\Delta d\|_{L^{2}}^{2} = \int_{ \mathbb{R}^{3}}u\cdot\nabla d\cdot \Delta d-|\nabla d|^{2}d\Delta d \mbox{d} x\nonumber. \end{eqnarray*}$

By adding the above equalities and using the facts $|d| = 1, \ \Delta(|d|^{2}) = 0\Rightarrow |\nabla d|^{2} = -d\cdot\Delta d$ , we have

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

(3.3)

Integrating (3.3) in time gives

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

Similarly going on the above process, multiplying $(1.1)_{1}$ by $-\Delta u$ and integrating over $\mathbb{R}^{3}$ , then applying $\Delta$ to Eq $(1.1)_{2}$ , and taking the inner product with $\Delta d$ , after that adding the resulting equalities, we achieve that

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

(3.4)

For $I_{1}$ , by the incompressibility condition and integration by parts several times, we conclude that

$\begin{align*} I_{1} = &\int_{ \mathbb{R}^{3}}(u\cdot\nabla)u\cdot\Delta u \mbox{d} x\nonumber \\ = & \int_{ \mathbb{R}^{3}}(\sum\limits_{k,j = 1}^{3}\sum\limits_{i = 1}^{2}u_{j} \partial_{j}u_{i} \partial_{k} \partial_{k} u_{i}+\sum\limits_{k,j = 1}^{3}u_{j} \partial_{j}u_{3} \partial_{k} \partial_{k} u_{3}) \mbox{d} x\nonumber \\ = & \int_{ \mathbb{R}^{3}}(\sum\limits_{k,j = 1}^{3}\sum\limits_{i = 1}^{2}u_{j} \partial_{j}u_{i} \partial_{k} \partial_{k} u_{i}+\sum\limits_{k = 1}^{3}\sum\limits_{j = 1}^{2}u_{j} \partial_{j}u_{3} \partial_{k} \partial_{k} u_{3}-\frac{1}{2}\sum\limits_{k = 1}^{3} \partial_{3}u_{3} \partial_{k}u_{3} \partial_{k} u_{3}) \mbox{d} x\nonumber \\ = & \int_{ \mathbb{R}^{3}}[(\sum\limits_{k,j = 1}^{3}\sum\limits_{i = 1}^{2}u_{j} \partial_{j}u_{i} \partial_{k} \partial_{k} u_{i}+\sum\limits_{k = 1}^{3}\sum\limits_{j = 1}^{2}u_{j} \partial_{j}u_{3} \partial_{k} \partial_{k} u_{3}+\frac{1}{2}\sum\limits_{k = 1}^{3} \partial_{k}u_{3}( \partial_{1}u_{1}+ \partial_{2}u_{2}) \partial_{k} u_{3}] \mbox{d} x\nonumber \\ = &\int_{ \mathbb{R}^{3}}\{\sum\limits_{k,j = 1}^{3}\sum\limits_{i = 1}^{2}u_{i} \partial_{k}( \partial_{k}u_{j} \partial_{j} u_{i})+\sum\limits_{k = 1}^{3}\sum\limits_{j = 1}^{2}[-u_{j} \partial_{j}(\frac{ \partial_{k}u_{3} \partial_{k} u_{3}}{2})+u_{j} \partial_{k}( \partial_{j}u_{3} \partial_{k}u_{3})]\nonumber \\ &-\frac{1}{2}\sum\limits_{k = 1}^{3}[u_{1} \partial_{1}( \partial_{k}u_{3} \partial_{3} u_{3})+u_{2} \partial_{2}( \partial_{k}u_{3} \partial_{3} u_{3})]\} \mbox{d} x \nonumber \\ \leq&C\int_{ \mathbb{R}^{3}}|u_{h}||\nabla (\nabla u\nabla u)| \mbox{d} x.\nonumber \end{align*}$

With the inequality (2.5), the Sobolev embedding ${\dot{B}_{\infty, 2}^{0}}\hookrightarrow BMO$ and the inequality (2.4), we have

$\begin{align} I_{1}&\leq C\| u_{h}\|_{BMO}\|\nabla u\|_{L^{2}}\|\Delta u\|_{L^{2}} \\ &\leq C \| u_{h}\|_{BMO}^{2}\|\nabla u\|_{L^{2}}^{2}+\frac{1}{4}\|\Delta u\|_{L^{2}}^{2} \\ &\leq C\| u_{h}\|_{\dot{B}_{\infty,2}^{0}}^{2}\|\nabla u\|_{L^{2}}^{2}+\frac{1}{4}\|\Delta u\|_{L^{2}}^{2} \\ &\leq C[1+||u_{h}||_{\dot{B}_{\infty,\infty}^{0}}^{2}\log(1+\|\Delta u\|_{L^{2}})]\|\nabla u\|_{L^{2}}^{2}+\frac{1}{4}\|\Delta u\|_{L^{2}}^{2}. \end{align}$

(3.5)

Adding $I_{2}$ and $I_{3}$ together, it follows by the divergence free condition $\nabla\cdot u = 0$

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

Hence, it can be deduced from inequality (2.1) that

$\begin{eqnarray} I_{2}+I_{3}&\leq& C||\nabla u||_{L^{2}}||\Delta d||_{L^{4}}^{2} \\ &\leq& C||\nabla u||_{L^{2}}||\nabla d||_{\dot{B}_{\infty,\infty}^{0}}||\nabla\Delta d||_{L^{2}} \\ &\leq& C||\nabla d||_{\dot{B}_{\infty,\infty}^{0}}^{2}||\nabla u||_{L^{2}}^{2}+\frac{1}{4}||\nabla\Delta d||_{L^{2}}^{2}. \end{eqnarray}$

(3.6)

For $I_{4}$ , by the product estimate (2.2) and inequality (2.1), we obtain

$\begin{eqnarray} I_{4}& = &\int_{ \mathbb{R}^{3}}\Delta (|\nabla d|^{2}d)\cdot\Delta d \mbox{d} x \\&\leq& \|\Delta(|\nabla d|^{2}d)\|_{L^{\frac{4}{3}}}\|\Delta d\|_{L^{4}} \\&\leq& C(\|\nabla\Delta d\|_{L^{2}}\|\nabla d\|_{L^{4}}\|d\|_{L^{\infty}}+\|\Delta d\|_{L^{4}}\|\nabla d\|_{L^{4}}^{2}) \|\Delta d\|_{L^{4}} \\&\leq& C\|\Delta d\|_{L^{4}}^{2}\|\nabla d\|_{L^{4}}^{2}+\frac{1}{8}\|\nabla\Delta d\|_{L^{2}}^{2} \\&\leq& C\|\nabla d\|_{\dot{B}_{\infty,\infty}^{0}} \|\nabla\Delta d\|_{L^{2}}\|d\|_{L^{\infty}}\|\Delta d\|_{L^{2}}+\frac{1}{8}\|\nabla\Delta d\|_{L^{2}}^{2} \\&\leq& C\|\nabla d\|_{\dot{B}_{\infty,\infty}^{0}}^{2}\|\Delta d\|_{L^{2}}^{2} +\frac{1}{4}\|\nabla\Delta d\|_{L^{2}}^{2}. \end{eqnarray}$

(3.7)

Combining (3.4)–(3.7), one has

$\begin{eqnarray} &&\frac{ \mbox{d}}{ \mbox{d} t}(||\nabla u||_{L^{2}}^{2}+||\Delta d||_{L^{2}}^{2})+||\Delta u||_{L^{2}}^{2}+||\nabla\Delta d||_{L^{2}}^{2} \\ &\leq& C(1+||u_{h}||_{\dot{B}_{\infty,\infty}^{0}}^{2}\log(1+\|\Delta u\|_{L^{2}})+||\nabla d||_{\dot{B}_{\infty,\infty}^{0}}^{2})(||\nabla u||_{L^{2}}^{2}+||\Delta d||_{L^{2}}^{2}). \end{eqnarray}$

(3.8)

Noting (1.4), one concludes that for any small constant $\epsilon > 0$ , there exists $T_{0} < T$ such that

$\begin{align} \int_{T_{0}}^{T}||u_{h}||_{\dot{B}_{\infty,\infty}^{0}}^{2} \mbox{d} t < \epsilon. \end{align}$

(3.9)

For any $T_{0}\leq t < T$ , we set

$\begin{align} M(t) = \sup\limits_{T_{0}\leq s\leq t} (||\Delta u(s)||_{L^{2}}^{2}+||\nabla\Delta d(s)||_{L^{2}}^{2}). \end{align}$

(3.10)

Employing Gronwall inequality for (3.8) in the interval $[T_{0}, t]$ and using (3.9), (3.10) yields

$\begin{align} & ||\nabla u(t)||_{L^{2}}^{2}+||\Delta d(t)||_{L^{2}}^{2}+\int_{T_{0}}^{t}(||\Delta u(s)||_{L^{2}}^{2}+||\nabla\Delta d(s)||_{L^{2}}^{2}) \mbox{d} s \\ \leq & (||\nabla u(T_{0})||_{L^{2}}^{2}+||\Delta d(T_{0})||_{L^{2}}^{2})\exp\left\{C\int_{T_{0}}^{t}(1+||u_{h}||_{\dot{B}_{\infty,\infty}^{0}}^{2}\log(1+\|\Delta u\|_{L^{2}})+||\nabla d||_{\dot{B}_{\infty,\infty}^{0}}^{2}) \mbox{d} s\right\} \\ \leq & C_{0}C_{1}\exp\left\{C\int_{T_{0}}^{t}||u_{h}||_{\dot{B}_{\infty,\infty}^{0}}^{2} \log(1+\|\Delta u\|_{L^{2}}^{2}+||\nabla\Delta d||_{L^{2}}^{2}) \mbox{d} s\right\} \\ \leq & C_{0}C_{1}\exp\left\{C\epsilon\log(1+M(t))\right\} \\ \leq & C_{0}C_{1}(1+M(t))^{C\epsilon}, \end{align}$

(3.11)

where the letter $C_{0}$ means a constant depending on $(||\nabla u(T_{0})||_{L^{2}}^{2}+||\Delta d(T_{0})||_{L^{2}}^{2})$ , $C_{1}$ depends on $\exp\left\{C\int_{T_{0}}^{t}(1+||\nabla d||_{\dot{B}_{\infty, \infty}^{0}}^{2}) \mbox{d} s\right\}$ , and $C$ is a generic constant which may be different from line to line.

Then we need to bound the norm $\|\Delta u\|_{L^{2}}$ and $\|\nabla\Delta d\|_{L^{2}}$ so as to confirm the validness of inequality (3.2). Applying $\Delta$ and $\nabla\Delta$ to Eqs $(1.1)_{1}$ and $(1.1)_{2}$ respectively, and taking the $L^{2}$ inner product with $(\Delta u, \nabla\Delta d)$ , we obtain that

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

(3.12)

According to $\nabla\cdot u = 0$ , H $\rm\ddot{o}$ lder inequality, the commutator estimate (2.3), interpolation inequality and Young inequality, $J_{1}$ can be estimated by

$\begin{eqnarray} J_{1} & = &-\int_{ \mathbb{R}^{3}}[\Delta,u\cdot\nabla]u\cdot\Delta u \mbox{d} x \\ &\leq& \|[\Delta,u\cdot\nabla]u\|_{L^{\frac{4}{3}}}\|\Delta u\|_{L^{4}} \\ &\leq& C(\|\nabla u\|_{L^{2}}\|\Delta u\|_{L^{4}}+\|\Delta u\|_{L^{4}}\|\nabla u\|_{L^{2}})\|\Delta u\|_{L^{4}} \\ &\leq& C\|\nabla u\|_{L^{2}}\|\Delta u\|_{L^{4}}^{2} \leq C\|\nabla u\|_{L^{2}}\|\nabla u\|_{L^{2}}^{\frac{1}{4}}\|\nabla\Delta u\|_{L^{2}}^{\frac{7}{4}} \\ &\leq& C\|\nabla u\|_{L^{2}}^{10}+\frac{1}{6}\|\nabla\Delta u\|_{L^{2}}^{2}. \end{eqnarray}$

(3.13)

By the product estimate (2.2), we have

$\begin{eqnarray} J_{2} & = &\int_{ \mathbb{R}^{3}} \partial_{i}(\nabla d_{j}\Delta d_{j})\cdot \partial_{i}\Delta u \mbox{d} x \\ &\leq& \|\nabla(\nabla d\Delta d)\|_{L^{2}}\|\nabla\Delta u\|_{L^{2}} \\ &\leq& (\|\nabla d\|_{L^{4}}\|\nabla\Delta d\|_{L^{4}}+ \|\Delta d\|_{L^{4}}\|\Delta d\|_{L^{4}})\|\nabla\Delta u\|_{L^{2}} \\ &\leq& C\|\nabla d\|_{L^{4}}^{2}\|\nabla\Delta d\|_{L^{4}}^{2}+C\|\Delta d\|_{L^{4}}^{4}+\frac{1}{6} \|\nabla\Delta u\|_{L^{2}}^{2} \\ &\leq& C\| d\|_{L^{\infty}}\|\Delta d\|_{L^{2}}\|\Delta d\|_{L^{2}}^{\frac{1}{4}}\|\Delta\Delta d\|_{L^{2}}^{\frac{7}{4}}+C\|\Delta d\|_{L^{2}}^{\frac{5}{2}}\|\Delta\Delta d\|_{L^{2}}^{\frac{3}{2}}+\frac{1}{6}\|\nabla\Delta u\|_{L^{2}}^{2} \\ &\leq& C\|\Delta d\|_{L^{2}}^{\frac{5}{4}}\|\Delta^{2} d\|_{L^{2}}^{\frac{7}{4}} +C\|\Delta d\|_{L^{2}}^{\frac{5}{2}}\|\Delta^{2} d\|_{L^{2}}^{\frac{3}{2}}+\frac{1}{6}\|\nabla\Delta u\|_{L^{2}}^{2} \\ &\leq& C\|\Delta d\|_{L^{2}}^{10}+\frac{1}{6}\|\Delta^{2} d\|_{L^{2}}^{2}+\frac{1}{6}\|\nabla\Delta u\|_{L^{2}}^{2}. \end{eqnarray}$

(3.14)

Similar as (3.13), one may conclude

$\begin{eqnarray} J_{3}& = &-\int_{ \mathbb{R}^{3}}[\nabla\Delta,u\cdot\nabla]d\cdot\nabla\Delta d \mbox{d} x \\&\leq& \|[\nabla\Delta,u\cdot\nabla]d\|_{L^{\frac{4}{3}}}\|\nabla\Delta d\|_{L^{4}} \\ &\leq& (\|\nabla u\|_{L^{2}}\|\nabla\Delta d\|_{L^{4}}+ \|\nabla d\|_{L^{4}}\|\nabla\Delta u\|_{L^{2}})\|\nabla\Delta d\|_{L^{4}} \\ &\leq& C\|\nabla u\|_{L^{2}}\|\nabla\Delta d\|_{L^{4}}^{2}+C\|\nabla d\|_{L^{4}}^{2}\|\nabla\Delta d\|_{L^{4}}^{2}+\frac{1}{6}\|\nabla\Delta u\|_{L^{2}}^{2} \\ &\leq& C\|\nabla u\|_{L^{2}}\|\Delta d\|_{L^{2}}^{\frac{1}{4}}\|\Delta\Delta d\|_{L^{2}}^{\frac{7}{4}}+C\|d\|_{L^{\infty}}\|\Delta d\|_{L^{2}}\|\Delta d\|_{L^{2}}^{\frac{1}{4}}\|\Delta\Delta d\|_{L^{2}}^{\frac{7}{4}} +\frac{1}{6}\|\nabla\Delta u\|_{L^{2}}^{2} \\ &\leq& C\|\nabla u\|_{L^{2}}^{8}\|\Delta d\|_{L^{2}}^{2}+C\|\Delta d\|_{L^{2}}^{10}+\frac{1}{6}\|\Delta^{2} d\|_{L^{2}}^{2}+\frac{1}{6}\|\nabla\Delta u\|_{L^{2}}^{2}. \end{eqnarray}$

(3.15)

By the product estimate (2.2) and the fact $\ |\nabla d|^{2} = -d\cdot\Delta d$ , we infer that

$\begin{eqnarray} J_{4}& = &-\int_{ \mathbb{R}^{3}}\nabla\Delta (|\nabla d|^{2}d)\cdot\nabla\Delta d \mbox{d} x \end{eqnarray}$

(3.16)

$\begin{eqnarray} & = &\int_{ \mathbb{R}^{3}}\Delta (|\nabla d|^{2}d)\cdot(\nabla\nabla\Delta d ) \mbox{d} x \\&\leq& \|\Delta (|\nabla d|^{2}d)\|_{L^{2}}\|\nabla\nabla\Delta d\|_{L^{2}} \\&\leq& C(\|\Delta (|\nabla d|^{2})d\|_{L^{2}}+\||\nabla d|^{2}\Delta d\|_{L^{2}})\|\Delta^{2} d\|_{L^{2}} \\&\leq& C(\|\nabla d\|_{L^{4}}\|\nabla\Delta d\|_{L^{4}}\|d\|_{L^{\infty}}+\|d\cdot\Delta d\Delta d\|_{L^{2}})\|\Delta^{2} d\|_{L^{2}} \end{eqnarray}$

(3.17)

$\begin{eqnarray} &\leq& C(\|\nabla d\|_{L^{4}}\|\nabla\Delta d\|_{L^{4}}+\|\Delta d\|_{L^{4}}^{2})\|\Delta^{2} d\|_{L^{2}} \\&\leq& C\|d\|_{L^{\infty}}^{\frac{1}{2}}\|\Delta d\|_{L^{2}}^{\frac{1}{2}}||\Delta d||_{L^{2}}^{\frac{1}{8}}\|\Delta^{2} d\|_{L^{2}}^{\frac{7}{8}}\|\Delta^{2} d\|_{L^{2}}+C\|\Delta d\|_{L^{2}}^{\frac{5}{4}}\|\Delta^{2} d\|_{L^{2}}^{\frac{3}{4}}\|\Delta^{2} d\|_{L^{2}} \\&\leq& C\|\Delta d\|_{L^{2}}^{10}+\frac{1}{6}\|\Delta^{2} d\|_{L^{2}}^{2}.\nonumber \end{eqnarray}$

Inserting the above estimates (3.13)–(3.16) to (3.12), and combining (3.11), we obtain

$\begin{eqnarray*} &&\frac{ \mbox{d}}{ \mbox{d} t}(1+\|\Delta u\|_{L^{2}}^{2}+\|\nabla\Delta d\|_{L^{2}}^{2})+\|\nabla\Delta u\|_{L^{2}}^{2}+\|\Delta^{2} d\|_{L^{2}}^{2} \\ &\leq& C(\|\nabla u\|_{L^{2}}^{10}+\|\Delta d\|_{L^{2}}^{10}+\|\nabla u\|_{L^{2}}^{8}\|\Delta d\|_{L^{2}}^{2}) \\ &\leq& C C_{0} C_{1}(1+M(t))^{5C\epsilon}. \end{eqnarray*}$

Integrating the above inequality with respect to time from $T_{0}$ to $t$ , $T_{0}\leq t < T$ , it follows that

$\begin{eqnarray*} &&(1+\|\Delta u(t)\|_{L^{2}}^{2}+\|\nabla\Delta d(t)\|_{L^{2}}^{2})+\int_{T_{0}}^{t} (\|\nabla\Delta u\|_{L^{2}}^{2}+\|\Delta^{2} d\|_{L^{2}}^{2}) \mbox{d} \tau\nonumber \\ &\leq& 1+\|\Delta u(T_{0})\|_{L^{2}}^{2}+\|\nabla\Delta d(T_{0})\|_{L^{2}}^{2}+ \int_{T_{0}}^{t}C C_{0}C_{1}(1+M(\tau))^{5C\epsilon} \mbox{d} \tau\nonumber \\ & = & 1+\|\Delta u(T_{0})\|_{L^{2}}^{2}+\|\nabla\Delta d(T_{0})\|_{L^{2}}^{2}+ \int_{T_{0}}^{t}C C_{0}C_{1}(1+M(\tau)) \mbox{d} \tau \end{eqnarray*}$

by choosing $\epsilon = \frac{1}{5C}$ . And we can derive from the above inequality and (3.10) that

$\begin{eqnarray*} &&(1+M(t))+\int_{T_{0}}^{t} (\|\nabla\Delta u\|_{L^{2}}^{2}+\|\Delta^{2} d\|_{L^{2}}^{2}) \mbox{d} \tau \\ &\leq& 1+\|\Delta u(T_{0})\|_{L^{2}}^{2}+\|\nabla\Delta d(T_{0})\|_{L^{2}}^{2}+ \int_{T_{0}}^{t}C C_{0}C_{1}(1+M(\tau)) \mbox{d} \tau. \end{eqnarray*}$

Gronwall's inequality implies

$\begin{eqnarray*} &&(1+M(t))+\int_{T_{0}}^{t} (\|\nabla\Delta u\|_{L^{2}}^{2}+\|\Delta^{2} d\|_{L^{2}}^{2}) \mbox{d} \tau \\ &\leq& (1+\|\Delta u(T_{0})\|_{L^{2}}^{2}+\|\nabla\Delta d(T_{0})\|_{L^{2}}^{2}) \exp\left\{CC_{0}C_{1}(T-T_{0})\right\}. \end{eqnarray*}$

The proof of Theorem 1.1. is thus completed.

4. Conclusions

In this paper, we establish a regularity criterion for the 3D nematic liquid crystal flows via velocity component $u_{h}$ and orientation field $\nabla d$ in Besov space. Furthermore, there is a lack of references about regularity criterion via component of orientation field $d$ for the system (1.1) and we hope to weaken the condition (1.4) by the components of both $u$ and $d$ in more general spaces in future study.

Conflict of interest

All authors declare no conflicts of interest in this paper.

References

[1]	H. Bahouri, J. Y. Chemin, R. Danchin, Fourier analysis and nonlinear partial differential equations, Berlin, Heidelberg: Springer, 2011. https://doi.org/10.1007/978-3-642-16830-7
[2]	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
[3]	S. Gala, M. A. Ragusa, A new regularity criterion for the Navier-Stokes equations in terms of the two components of the velocity, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), 1–9. https://doi.org/10.14232/ejqtde.2016.1.26 doi: 10.14232/ejqtde.2016.1.26
[4]	T. Huang, C. Y. Wang, Blow up criterion for nematic liquid crystal flows, Commmun. Part. Differ. Equ., 37 (2012), 875–884. https://doi.org/10.1080/03605302.2012.659366 doi: 10.1080/03605302.2012.659366
[5]	H. Kozono, T. Ogawa, Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z., 242 (2002), 251–278. https://doi.org/10.1007/s002090100332 doi: 10.1007/s002090100332
[6]	T. Kato, G. Ponce, Commutator estimates and the Euler and the Navier-Stokes equations, Commun. Pure Appl. Math., 41 (1988), 891–907. https://doi.org/10.1002/cpa.3160410704 doi: 10.1002/cpa.3160410704
[7]	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
[8]	Q. Li, B. Q. Yuan, A regularity criterion for liquid crystal flows in terms of the component of velocity and the horizontal derivative components of orientation field, AIMS Math., 7 (2022), 4168–4175. https://doi.org/10.3934/math.2022231 doi: 10.3934/math.2022231
[9]	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
[10]	Q. Liu, P. Wang, The 3D nematic liquid crystal equations with blow-up criteria in terms of pressure, Nonlinear Anal. Real World Appl., 40 (2018), 290–306. https://doi.org/10.1016/j.nonrwa.2017.08.008 doi: 10.1016/j.nonrwa.2017.08.008
[11]	A. J. Majda, A. L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, Cambridge: Cambridge University Press, 2002. https: //doi.org/10.1115/1.1483363
[12]	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
[13]	B. Q. Yuan, X. Li, Regularity of weak solutions to the 3D magneto-micropolar equations in Besov spaces, Acta Appl. Math., 163 (2019), 207–223. https://doi.org/10.1007/s10440-018-0220-z doi: 10.1007/s10440-018-0220-z
[14]	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.
[15]	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
[16]	J. H. Zhao, BKM's criterion for the 3D nematic liquid crystal flows via two velocity components and molecular orientations, Math. Method. Appl. Sci., 40 (2016), 871–882. https://doi.org/10.1002/mma.4014 doi: 10.1002/mma.4014
[17]	X. X. Zheng, A regularity criterion for the tridimensional Navier-Stokes equations in term of one velocity component, J. Differ. Equ., 256 (2014), 283–309. https://doi.org/10.1016/j.jde.2013.09.002 doi: 10.1016/j.jde.2013.09.002

This article has been cited by:

1.	Ruirui Wang, Qiang Mu, Yulei Wang, Qiang Li, Yongqiang Fu, Two Logarithmically Improved Regularity Criteria for the 3D Nematic Liquid Crystal Flows, 2022, 2022, 2314-4785, 1, 10.1155/2022/4454497
2.	Jae-Myoung Kim, Blow-up criteria for the full compressible Navier-Stokes equations involving temperature in Vishik Spaces, 2022, 7, 2473-6988, 15693, 10.3934/math.2022859
3.	Ines Ben Omrane, Mourad Ben Slimane, Sadek Gala, Maria Alessandra Ragusa, A new regularity criterion for the 3D nematic liquid crystal flows, 2025, 23, 0219-5305, 287, 10.1142/S0219530524500313

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(1555) PDF downloads(74) Cited by(3)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Mathematics

A regularity criterion via horizontal components of velocity and molecular orientations for the 3D nematic liquid crystal flows

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. Proof of Theorem 1.1

4. Conclusions

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Catalog

AIMS Mathematics

A regularity criterion via horizontal components of velocity and molecular orientations for the 3D nematic liquid crystal flows

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. Proof of Theorem 1.1

4. Conclusions

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog