Blow-up criterion for the 3D nematic liquid crystal flows via one velocity and vorticity components and molecular orientations

1.   Introduction

In this paper, we are interested in the following hydrodynamic system modeling the flow of the nematic liquid crystal materials in 3-dimensions:

where $u$ is the velocity field, $d$ is the macroscopic average of molecular orientation field and $p$ represents the scalar pressure. And $\mu$ is the kinematic viscosity, $\lambda$ is the competition between the kinetic and potential energies, and $\gamma$ is the microscopic elastic relation time for the molecular orientation field. The notation $\nabla d\odot \nabla d$ represents the $3\times 3$ matrix, of which the $(i, j)th$ component can be denoted by $\partial_{i}d_{k} \partial_{j}d_{k}(i, j\leq 3)$.

The model of the hydrodynamic theory for liquid crystals was established by Ericksen and Leslie ^[8,12,13], and the system (1.1) was first introduced by Lin ^[14] as a simplified version to the Ericksen-Leslie system describing the flow of nematic liquid crystals. Later, Lin and Liu had done many significant works such as ^[15,16].

When the orientation field $d$ equals a constant, the above equations become the incompressible Navier-Stokes equations. Many regularity results on the weak solutions to the three-dimensional Navier-Stokes equations have been well studied, for example see ^{[3,4,5,6,7,9,17,18,21,22,23,30,32,33]}, and references therein, where they have proved that the solution is a smooth one if the velocity, or vorticity, or the gradient of velocity, or their components are regular. In their famous work ^[2], J. Beale et al. proved that the smooth solution $u$ blows up at a finite time $t = T^{\ast}$ for the 3D Euler equations, if $\int_{0}^{T^{\ast}}\|\omega\|_{L^{\infty}} \mbox{d} t = \infty,$ which also holds for the Navier-Stokes equations. In ^[31], Zhang has investigated a regularity criterion via one velocity and one vorticity component. On the other hand, when the velocity field $u = 0$, the system (1.1) becomes to the heat flow of harmonic maps onto a sphere. Wang proved in ^[24] that, if $0 < T^{\ast} < \infty$ is the maximal time for the unique smooth solution $d\in C^{\infty}(\mathbb{R}^{n}; (0, T^{\ast}])$, then $\|\nabla d\|_{L^{n}}$ blows up as time $t$ tends to $T^{\ast}$. Motivated by these developments, the global smooth solution on the nematic liquid crystal model (1.1) are studied in a series papers ^{[10,19,20,26,27,28,29]}. Huang and Wang ^[10] established a BKM type blow-up criterion for the system (1.1). That is, if $T^{\ast}$ is the maximal time, $0 < T^{\ast} < \infty$, then

This result is improved by Zhao ^[29] via two velocity components and molecular orientations. More precisely, the smooth solution (u, d) of the system (1.1) blows up at time $t = T^{\ast} < \infty$, if and only if

Recently, Yuan and Wei ^[27] consider the blow-up criterion in terms of the vorticity in Besov space of negative index and the orientation field in the homogeneous Besov space. If

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

Inspired by ^[27] and ^[31], we are aimed to replace the gradient of velocity in (1.3) and the vorticity in (1.4) by one velocity and one vorticity component. Our main results are stated as follows:

Theorem 1.1. Assume the initial data $u_{0}\in H^{3}(\mathbb{R}^{3})$ with $\nabla\cdot u_{0} = 0$, and $d_{0}\in H^{4}(\mathbb{R}^{3}, \mathbb{S}^{2})$, $(u, d)$ is a smooth solution to the equations of (1.1) on $[0, T)$ for some $0 < T < \infty$. Then $(u, d)$ can be extended beyond $T$, provided that

Remark 1.2. As we know, if the initial data $u_{0}\in H^{s}(\mathbb{R}^{n})$ with $\nabla\cdot u_{0} = 0$ and $d_{0}\in H^{s+1}(\mathbb{R}^{n}, \mathbb{S}^{2})$ for $s\geq n$, then there exists a positive time $T$ depending only on the initial value such that system (1.1) has a unique smooth solution $(u, d)\in (\mathbb{R}^{n}\times[0, T))$ satisfying (see for example ^[25])

In the following part, we shall use simplified notations. we shall use the letter $C$ to denote a generic constant which may be different from line to line, and write $\partial_{t}u = \frac{ \partial u}{ \partial t}, \partial_{i} = \frac{ \partial}{ \partial x_{i}}$. Since the concrete values of the constants $\mu, \lambda, \gamma$ play no role in our discussion, to simplify the presentation, we shall assume that $\mu = \lambda = \gamma = 1$ in this paper.

2.   Preliminaries

In this section, we shall recall the interpolation inequality in ^[1] and the commutator estimate in ^[11], which will be used in the process of the proof of Theorem 1.1.

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

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

Lemma 2.2. (Commutator estimate ^[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,

3.   Proof of Theorem 1.1

In this section, we will prove Theorem 1.1 by energy methods. Under the condition (1.5), it suffices to show that, there exists a constant C such that

which is enough to guarantee the extension of smooth solution $(u, d)$ beyond the time $T$, for details refer to ^[10].

Firstly, taking the $L^{2}$ inner product with $u$ and $-\Delta d$ to the equations $(1.1)_{1}$ and $(1.1)_{2}$ respectively, and adding them together, it follows that

where we have used the facts $|d| = 1, \ |\nabla d|^{2} = -d\cdot\Delta d$, and the following equalities, due to $\nabla\cdot u = 0$,

Integrating (3.2) in time, we get

Next, we are devoted to obtaining the the $H^1$ estimate of $u$ and $\nabla d$. Applying $\Delta$ to the Eq. $(1.1)_{2}$, and taking the inner product with $\Delta d$, we obtain

Multiplying $(1.1)_{1}$ by $-\Delta u$, and integrating by parts, one has

Summing up (3.3) and (3.4), it could be derived that

For the term $I_{1}$ one may refer to ^[31], for the completeness, We here give the deduction as follows:

We classify the the terms $\partial_{k}u_{j} \partial_{j}u_{i} \partial_{k} u_{i}, 1\leq i, j, k\leq 3$ as

(1) If $k = j = 3$, or $j = i = 3$, or $k = i = 3$, we then invoke the divergence free condition to replace $\partial_{3}u_{3}$ by $- \partial_{1}u_{1}- \partial_{2}u_{2}$;

(2) Otherwise, at least two indices belong to $\{1, 2\}$. Thus $I_{1}$ will be

where $\alpha_{mnijkl}, \ 1\leq m, n\leq 2, \ 1\leq i, j, k, l\leq 3$, are suitable integers. Next, we want to represent $\partial_{m}u_{n}, 1\leq m, n\leq 2$ by $u_{3}$ and $\omega_{3}$. Denoting by $\Delta_{h} = \partial_{1} \partial_{1}+ \partial_{2} \partial_{2}$ the horizontal Laplacian, we have

Based on the computations above, we can use the two-dimension Riesz transformation $\Re_{m} = \frac{ \partial_{m}}{\sqrt{-\Delta_{h}}}$ to denote the term $\partial_{m}u_{n}, 1\leq m, n\leq 2$,

By (3.6), the term $I_{11}$ could be turned into

Because of the Riesz transformation being bounded in $L^{p}(\mathbb{R}^{2})$ to $L^{p}(\mathbb{R}^{2})$ for $1 < p < \infty$, and using H$\rm\ddot{o}$lder and Gagliardo-Nirenberg inequalities yields

The estimates of terms $I_{12}, I_{21}, I_{22}$ are similar to $I_{11}$, thus we can get

Next, we estimate the terms $I_{2}, I_{3}, I_{4}$.

We deduce from the Lemma 2.1 that

For $I_{4}$, it is easy to check that

Inserting (3.8), (3.9) and (3.10) into (3.5) yields

Applying the Gronwall inequality leads to

At last, under the $H^{1}$ estimates of $\nabla u$ and $\Delta d$, we will show

where C is a constant.

Applying $\Delta$ and $\nabla\Delta$ to the 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

Using the inequality (3.11) and commutator estimate, $J_{1}, J_{2}, J_{3}$ can be estimated by

To bound $J_{4}$, by the facts $|d| = 1, \ |\nabla d|^{2} = -d\cdot\Delta d$, it follows that

Putting the estimates (3.14)–(3.17) to (3.13), we get

which gives us the desired result (3.12) by the Gronwall inequality. Finally, by using the Sobolev embedding $H^{2}(\mathbb{R}^{3})\hookrightarrow L^{\infty}(\mathbb{R}^{3})$, (3.12) leads to the BKM's criterion (3.1) immediately, which completes the proof of Theorem 1.1.

Acknowledgments

The research of Baoquan Yuan was partially supported by the National Natural Science Foundation of China (No. 11471103).

Conflict of interest

All authors declare no conflicts of interest in this paper.