Global Existence of Strong Solutions for Beris–Edwards’s Liquid Crystal System in Dimension Three

Abstract

We consider a system, established by Beris and Edwards in the Q-tensor framework, modeling the incompressible flow of nematic liquid crystals. The coupling system consists of the Navier–Stokes equation and the evolution equation for the Q-tensor. We prove the global existence of strong solutions in a three-dimensional bounded domain with homogeneous Dirichlet boundary conditions, under the assumption that the viscosity is sufficiently large.

1. Introduction

Liquid crystals are mesomorphic states with physical properties intermediate between those of crystalline solids and isotropic fluids. The nematic is one of the most common phases, in which the rod-like molecules possess orientational order but no positional order. The anisotropic behavior of liquid crystals are described by the different order parameters. In the context of continuum mechanics, there are three widely accepted models to describe nematic liquid crystal flows: the Beris–Edwards model [1], the Qian–Sheng model [2] and the Ericksen–Leslie model [3,4]. The first two are derived by a variational method based on the Landau–de Gennes theory. This model introduces a symmetric and traceless order parameter $Q(\mathbf{x})$ to describe the alignment behaviour of molecule orientations. In physics, $Q(\mathbf{x})$ can be understood as the renormalized second-order moment of f:

$$\begin{array}{c}\hfill Q(\mathbf{x})={\int }_{{\mathbb{S}}^2}(\mathbf{m}\mathbf{m}-\frac{1}{3}\mathbf{I})f(\mathbf{x},\mathbf{m})\mathrm{d}\mathbf{m},\end{array}$$

(1)

where $f(\mathbf{x},\mathbf{m})$ corresponds to the orientational distribution function of molecules along the direction $\mathbf{m}$ at material point $\mathbf{x}$. For $Q(\mathbf{x})\in {\mathbb{S}}_0^3$, one can find $s(\mathbf{x}),r(\mathbf{x})\in \mathbb{R},\mathbf{n},\mathbf{m}\in {\mathbb{S}}^2$ with $\mathbf{n}·\mathbf{m}=0$ such that

$$\begin{array}{c}\hfill Q(\mathbf{x})=s(\mathbf{x})\left(\mathbf{n}\mathbf{n}-\frac{1}{3}\mathbf{I}\right)+r(\mathbf{x})\left(\mathbf{m}\mathbf{m}-\frac{1}{3}\mathbf{I}\right),\end{array}$$

(2)

where $\mathbf{I}$ is a $3\times 3$ identity matrix. Liquid crystals are said to be isotropic when $s=r=0$, uniaxial when $s\ne 0,r=0$, and biaxial when $s\ne 0,r\ne 0$.

Taking no account of external fields, the general Landau–de Gennes (LdG) energy functional is given by

$$\begin{array}{cc}\hfill \mathcal{F}(& Q,\nabla Q)\hfill \\ \hfill =& {\int }_{\mathsf{\Omega }}\{\underset{f_B:\mathrm{bulk}\mathrm{energy}}{\underbrace{\frac{a}{2}\mathrm{tr}(Q^2)-\frac{b}{3}\mathrm{tr}(Q^3)+\frac{c}{4}(\mathrm{tr}{(Q^2)}^2}}\hfill \\ \hfill & +\underset{f_E:\mathrm{elastic}\mathrm{energy}}{\underbrace{\frac{1}{2}\left(L_1{|\nabla Q|}^2+L_2{Q}_{ij,j}{Q}_{ik,k}+L_3{Q}_{ij,k}{Q}_{ik,j}+L_4{Q}_{ij}{Q}_{kl,i}{Q}_{kl,j}\right)}}\}\mathrm{d}\mathbf{x}.\hfill \end{array}$$

(3)

The above free energy consists of the bulk energy $f_B(Q)$ and the elastic energy $f_E(Q,\nabla Q)$. In Label (3), $a,b,c$ are non-zero constants that depend on the material and temperature, in particular, $a=\alpha ({\tilde{T}}^{*}-\tilde{T})$, where the constant $\alpha >0$ depends on the material, $\tilde{T}$ is the absolute temperature and ${\tilde{T}}^{*}$ a characteristic liquid crystal temperature. In addition, $L_i(i=1,2,3,4)$ are material dependent elastic constants. For brevity, we only consider the case of $L_1=L>0$ and $L_2=L_3=L_4=0$. As for the LdG theory, the interested reader can refer to [5,6] for more details. Concerning investigations of the equilibrium solutions for the LdG functional, one can refer to [7,8] and the references therein.

We consider the Beris–Edwards model from [1], which takes the following form:

$$\begin{array}{cc}\hfill \frac{\partial Q}{\partial t}+\mathbf{v}·\nabla Q& =\Gamma \mathbf{H}+\mathbf{W}·Q-Q·\mathbf{W},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}·\nabla \mathbf{v}& =-\nabla p+\mu \Delta \mathbf{v}+\nabla ·\left({\sigma }^d+\sigma \right),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \nabla ·\mathbf{v}& =0,\hfill \end{array}$$

(6)

where $\mathbf{v}$ is the fluid velocity, p the pressure, $\Gamma >0$ a collective rotational diffusion constant, and $\mu >0$ the kinematic viscosity. The molecular field $\mathbf{H}$ is given by

$$\begin{array}{c}\hfill \mathbf{H}\stackrel{\mathrm{def}}{=}-\frac{\delta \mathcal{F}}{\delta Q}=L\Delta Q-aQ+b(Q^2-\frac{1}{3}{\left|Q\right|}^2{\mathbf{I})-c|Q|}^{2}Q.\end{array}$$

The notation $\mathbf{W}=\frac{1}{2}(\nabla \mathbf{v}-{(\nabla \mathbf{v})}^T)$ represents the skew-symmetric part of the strain rate, and $\mathbf{W}·Q-Q·\mathbf{W}$ are induced by rotation part of the velocity gradient. In addition, the distortion stress ${\sigma }^d$ and the orientational-induced stress $\sigma$ are respectively defined by

$$\begin{array}{c}\hfill {\sigma }^d=-L\nabla Q\odot \nabla Q,\sigma =Q·\mathbf{H}-\mathbf{H}·Q,\end{array}$$

(7)

in which we use the notation ${(\nabla Q\odot \nabla Q)}_{ij}={\partial }_i{Q}_{kl}{\partial }_j{Q}_{kl}$. We define

$$\begin{array}{c}\hfill f(Q)\stackrel{\mathrm{def}}{=}aQ-b(Q^2-\frac{1}{3}{\left|Q\right|}^2{\mathbf{I})+c|Q|}^{2}Q.\end{array}$$

Then, we can see that

$$\begin{array}{c}\hfill \mathbf{H}=L\Delta Q-f(Q),f(Q)={\nabla }_Q{f}_B(Q)+\frac{b}{3}{\left|Q\right|}^2\mathbf{I}.\end{array}$$

In this paper, we consider the systems (4)–(6) in the bounded domain $\mathsf{\Omega }\times {\mathbb{R}}^{+}$ with the following conditions (cf. [9]):

$$\begin{array}{cc}\hfill & \mathbf{v}(\mathbf{x},0)={\mathbf{v}}_0(\mathbf{x}),\mathrm{with}\nabla ·{\mathbf{v}}_0=0,Q(\mathbf{x},0)=Q_0(\mathbf{x}),\mathrm{for}\mathbf{x}\in \mathsf{\Omega },\hfill \\ \hfill & \mathbf{v}(\mathbf{x},t)=0,Q(\mathbf{x},t)=0,\mathrm{for}(\mathbf{x},t)\in \partial \mathsf{\Omega }\times {\mathbb{R}}^{+}.\hfill \end{array}$$

We remark that the above initial and boundary conditions, which can be also verified as in [10], will ensure the well-posedness of the system.

It is worth pointing out that the constitutive relations (7) are special cases of more general expressions [11] for the corresponding tensors, so Label (7) do not include alignment effects due to the flow. In other words, the coupling parameter $\xi$ between the order tensor Q and the stretch tensor $\mathbf{D}=\frac{1}{2}(\nabla \mathbf{v}+{(\nabla \mathbf{v})}^T)$ is zero.

Up to now, there have been some analytic results on the Beris–Edwards system. The well-posedness study of the Beris–Edwards system was initiated in the seminal work of Paicu and Zarnescu [12,13]. For the case of the whole space, when $\xi =0$, the authors [12] investigated the global well-posedness for weak solutions, and obtained the higher global regularity in dimension two. Large time behavior of the solution to the Cauchy problem in ${\mathbb{R}}^3$ with $\xi =0$ was studied in [14]. When the full system with coupling parameter $\xi \ne 0$ but is sufficiently small, the existence of global weak solutions in ${\mathbb{R}}^d$ with $d=2,3$ was showed in [13], while the uniqueness of weak solutions in ${\mathbb{R}}^2$ was established by De Anna and Zarnescu [15,16]. Concerning the Beris–Edwards system with anisotriopic elastic energy ($L_2+L_3>0$ and $L_4=0$ in (3)), Huang–Ding [17] proved the existence of global weak solutions, and the global well-posedness of strong solutions in ${\mathbb{R}}^3$ if the fluid viscosity is sufficiently large. For the case of the bounded domain, Abels–Dolzmann–Liu [18] presented the global and local well-posedness with higher time-regularity for the initial boundary value problem with inhomogeneous boundary data. In [9], the existence and uniqueness of local strong solution was also showed in the case of homogeneous Dirichlet boundary conditions. In addition, the well-posedness of weak solutions to initial boundary value problems subject to various boundary conditions were also studied in [19,20] when $\xi =0$. Recently, Liu-Wang [21] studied the well-posedness of strong solutions to the Beris–Edwards model with an anisotropic elastic energy ($L_1>0,L_1+L_2+L_3>0$ and $L_4=0$ in (3)).

On the other hand, under the periodic boundary condition, Wilkinson [22] established the well-posedness of weak solutions for $\xi \ne 0$ and higher regularity in dimension two for $\xi =0$, in which the bulk energy $f_B$ is replaced by a singular potential that ensures the order tensor parameter Q possesses physically reasonable values. The global well-posedness and long-time behavior of system with $\xi \ne 0$ in the two-dimensional periodic case was investigated in [23].

Some numerical works have been developed to simulate liquid crystal flow systems governed by the Beris–Edwards model. For example, Denniston et al. [11] used the Lattice Boltzmann method to solve the Beris–Edwards model to study structures in nematic liquid crystals. Zhao–Wang [24] proposed several semi-discrete energy stable schemes for the the Beris–Edwards model, to study flowing behavior of nematic liquid crystals in various geometries. For the same model, Zhao et al. [25] developed a novel linear second order unconditionally energy stable scheme to simulate dynamics in flows of liquid crystals including the interesting defect behaviour of flows. We emphasize that since the Beris–Edwards model, capable of describing many interesting physical phenomena, is a rather complicated system coupling the Navier–Stokes equation with the evolution equation for the Q-tensor, the related numerical study is an important and challenging issue.

The major objective of this paper is to investigate the global existence of strong solutions in a bounded domain $\mathsf{\Omega }\in {\mathbb{R}}^3$ with homogeneous Dirichlet boundary conditions, under the assumption that the viscosity coefficient $\mu$ is sufficiently large.

This paper is organized as follows. In Section 2, we first introduce several lemmas that will be used throughout the paper, and then present the main result of this paper. In Section 3, we are devoted to the proof of Theorem 1 by showing that the higher-order energy we construct is uniformly bounded under the condition of the large viscosity.

2. Preliminaries and the Main Result

Notations and Conventions. The Einstein summation convention is utilized throughout this paper. The space of symmetric traceless tensors is defined as

$$\begin{array}{c}\hfill {\mathbb{S}}_0^3\stackrel{\mathrm{def}}{=}\left\{Q\in {\mathbb{R}}^{3\times 3}:Q_{ij}=Q_{ji},Q_{ii}=0\right\},\end{array}$$

(8)

which is endowed with the inner product $Q_1:Q_2=Q_{1ij}{Q}_{2ij}.$ The set ${\mathbb{S}}_0^3$ is a five-dimensional linear subspace of ${\mathbb{R}}^{3\times 3}.$ The matrix norm on ${\mathbb{S}}_0^3$ is defined as $\left|Q\right|\stackrel{\mathrm{def}}{=}\sqrt{\mathrm{Tr}{Q}^2}=\sqrt{Q_{ij}{Q}_{ij}}$. According to this norm, the Sobolev space will be defined as

$$\begin{array}{c}\hfill H^k(\mathsf{\Omega },{\mathbb{S}}_0^3)\stackrel{\mathrm{def}}{=}\left\{f:{\int }_{\mathsf{\Omega }}{\displaystyle \sum _{|{\alpha }^{\prime }|\le k}}{\left|{\partial }^{{\alpha }^{\prime }}f(\mathbf{x})\right|}^2\mathrm{d}\mathbf{x}<\infty \right\},\end{array}$$

with k being a non-negative integer and ${\alpha }^{\prime }$ being a multi-index. For two tensors $A,B\in {\mathbb{S}}_0^3$, we denote ${(A·B)}_{ij}=A_{ik}{B}_{kj}$ and $A:B=A_{ij}{B}_{ij}$. For any $Q_1,Q_2\in L^2{({\mathbb{R}}^3)}^{3\times 3}$, the corresponding inner product is defined by

$$\begin{array}{c}\hfill \langle Q_1,Q_2\rangle \stackrel{\mathrm{def}}{=}{\int }_{\mathsf{\Omega }}{Q}_{1ij}(\mathbf{x}):Q_{2ij}(\mathbf{x})\mathrm{d}\mathbf{x}.\end{array}$$

We denote by ${\mathbf{n}}_1\otimes {\mathbf{n}}_2$ the tensor product of two vectors ${\mathbf{n}}_1$ and ${\mathbf{n}}_2$, and we omit the symbol ⊗ for simplicity. We use $f_{,i}$ to denote ${\partial }_{i}f$ and $\mathbf{I}$ to denote the $3\times 3$ order identity tensor. In addition, the space V is defined by $V=\{\mathbf{v}\in H_0^1(\mathsf{\Omega }),\nabla ·\mathbf{v}=0\}$.

The following Lemmas 1 and 2 are well known and they will be used frequently in this paper. The arguments for the two lemmas can be referred to [10,26,27]. The more general form for Lemma 2 can be found in [28].

Lemma 1.

If Ω is a smooth and bounded domain in ${\mathbb{R}}^3$, then, for all $u\in H_0^1(\mathsf{\Omega })$,

$$\begin{array}{c}\hfill {\parallel u\parallel }_{L^4}\le {C\parallel u\parallel }_{L^2}^{\frac{1}{4}}{\parallel \nabla u\parallel }_{L^2}^{\frac{3}{4}}{,\parallel u\parallel }_{L^3}\le {C\parallel u\parallel }_{L^2}^{\frac{1}{2}}{\parallel \nabla u\parallel }_{L^2}^{\frac{1}{2}}.\end{array}$$

In addition, if $u\in H^2(\mathsf{\Omega })$, then

$$\begin{array}{c}\hfill {\parallel u\parallel }_{L^{\infty }}\le {C\parallel u\parallel }_{H^1}^{\frac{1}{2}}{\parallel u\parallel }_{H^2}^{\frac{1}{2}}{,\parallel u\parallel }_{L^{\infty }}\le {C\parallel u\parallel }_{L^2}^{\frac{1}{4}}{\parallel u\parallel }_{H^2}^{\frac{3}{4}}.\end{array}$$

Lemma 2 

(Gagliardo–Nirenberg inequalities). Under the assumption of Lemma 1, then there holds

$$\begin{array}{c}\hfill \parallel D^i{u\parallel }_{L^r}\le {C\parallel u\parallel }_{L^q}^{1-\frac{i}{m}}{\parallel D^{m}u\parallel }_{L^p}^{\frac{i}{m}},\end{array}$$

where $0\le i\le m$ and $\frac{1}{r}=\frac{i}{m}\frac{1}{p}+(1-\frac{i}{m})\frac{1}{q}$.

Lemma 3.

Suppose that $(Q,\mathbf{v})$ is a strong solution to systems (4)–(6). Then, it follows that

$$\begin{array}{c}\hfill {\parallel \mathbf{v}(t)\parallel }_{L^2}+{\parallel Q(t)\parallel }_{H^1}\le C,\forall t>0,\end{array}$$

where the constant $C>0$ depends on $\parallel {\mathbf{v}}_0{\parallel }_{L^2},{\parallel Q_0\parallel }_{H^1},L,a,b,c$ and Ω. Furthermore, there holds that

$$\begin{array}{c}\hfill {\int }_0^T{\int }_{\mathsf{\Omega }}\left({\mu |\nabla \mathbf{v}|}^2+\Gamma {\left|\mathbf{H}(Q)\right|}^2\right)\mathrm{d}\mathbf{x}\mathrm{d}t\le C_T,\forall T>0,\end{array}$$

where $C_T>0$ may moreover depend on $\mu ,\Gamma$ and T.

Since we are working with the bounded domain with homogeneous Dirichlet boundary conditions, the proof Lemma 3 can be easily obtained from the works [12,23].

The main result of this paper is stated as follows.

Theorem 1.

For any $({\mathbf{v}}_0,Q_0)\in V\times H^2$, under the large viscosity assumption

$$\begin{array}{c}\hfill \mu \ge {\mu }_0(\Gamma ,a,b,c,L,\parallel {\mathbf{v}}_0{\parallel }_{H^1},\parallel Q_0{\parallel }_{H^2}),\end{array}$$

systems (4)–(6) have a solution $(\mathbf{v},Q)$ that satisfies

$$\begin{array}{c}\hfill \mathbf{v}\in L^{\infty }(0,\infty ;V)\cap L^2(0,\infty ;H^2),Q\in L^{\infty }(0,\infty ;H^2)\cap L^2(0,\infty ;H^3).\end{array}$$

We remark that the main idea in the proof of Theorem 1 comes from the works in [29,30].

3. Global Existence of Strong Solutions

In this section, we first derive a higher-order energy inequality, and then prove the global existence of strong solutions when the viscosity $\mu$ is large enough.

Proposition 1.

Assume that, for arbitrary ${\mu }_0>0$,

$$\begin{array}{c}\hfill \mathcal{A}(t)={\parallel \nabla \mathbf{v}(t)\parallel }_{L^2}^2+{\parallel \mathbf{H}(t)\parallel }_{L^2}^2.\end{array}$$

(9)

If $\mu \ge {\mu }_0>0$, and $(Q,\mathbf{v})$ is the strong solution of systems (4)–(6), then there holds:

$$\begin{array}{cc}\hfill \frac{\mathrm{d}}{\mathrm{d}t}\tilde{\mathcal{A}}(t)\le & -\left(\mu -C_1{\mu }^{\frac{1}{2}}\tilde{\mathcal{A}}(t)\right){\parallel \Delta \mathbf{v}\parallel }_{L^2}^2\hfill \\ \hfill & -\left(\frac{\Gamma L}{2}-C_2{\mu }^{-\frac{1}{4}}\tilde{\mathcal{A}}(t)\right){\parallel \nabla \mathbf{H}\parallel }_{L^2}^2+C_3\tilde{\mathcal{A}}(t),\hfill \end{array}$$

(10)

where $\tilde{\mathcal{A}}=\mathcal{A}+1$, and positive constants $C_i(i=1,2,3)$ depend on $\mathsf{\Omega },f,\parallel {\mathbf{v}}_0{\parallel }_{L^2},{\parallel Q_0\parallel }_{H^1},L,a$, $b,c,{\mu }_0$ and Γ.

Proof. 

Using systems (4)–(6) and integration by parts, we obtain

$$\begin{array}{cc}\hfill \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{A}(t)=& -\langle \Delta \mathbf{v},{\mathbf{v}}_t\rangle +〈\mathbf{H},L\Delta Q_t-f^{\prime }(Q)Q_t〉\hfill \\ \hfill =& \langle \Delta \mathbf{v},\mathbf{v}·\nabla \mathbf{v}\rangle -{\mu \parallel \Delta \mathbf{v}\parallel }_{L^2}^2-〈\Delta \mathbf{v},\nabla ·({\sigma }^d+\sigma )〉\hfill \\ \hfill & -L〈\mathbf{H},\Delta (\mathbf{v}·\nabla Q)〉-\Gamma L{\parallel \nabla \mathbf{H}\parallel }_{L^2}^2+L〈\mathbf{H},\Delta (\mathbf{W}·Q-Q·\mathbf{W})〉\hfill \\ \hfill & -〈\mathbf{H},f^{\prime }(Q)\left(-\mathbf{v}·\nabla Q+\Gamma \mathbf{H}+\mathbf{W}·Q-Q·\mathbf{W}\right)〉.\hfill \end{array}$$

(11)

By the incompressibility condition $\nabla ·\mathbf{v}=0$ and the boundary conditions ${(Q,\mathbf{v})|}_{\partial \mathsf{\Omega }}=0$, we see from Equation (4) that

$$\begin{array}{c}\hfill {\mathbf{H}|}_{\partial \mathsf{\Omega }}=\frac{1}{\Gamma }\left(\frac{\partial Q}{\partial t}+\mathbf{v}·\nabla Q-\mathbf{W}·Q+Q·\mathbf{W}\right){|}_{\partial \mathbf{W}}=0.\end{array}$$

Note that $\Delta \mathbf{v}$ is also divergence free, i.e., ${\partial }_k\Delta v_k=0$. Then, using the fact that $f_B{(Q)|}_{\partial \mathsf{\Omega }}=0$ and ${\partial }_k{f}_B(Q)={\nabla }_Q{f}_B(Q){\partial }_{k}Q$, we have the following:

$$\begin{array}{c}\hfill {\int }_{\mathsf{\Omega }}f(Q_{ij})\Delta v_k{\partial }_k{Q}_{ij}\mathrm{d}\mathbf{x}={\int }_{\mathsf{\Omega }}\Delta v_k{\partial }_k{f}_B(Q_{ij})\mathrm{d}\mathbf{x}+\frac{b}{3}{\int }_{\mathsf{\Omega }}{\left|Q\right|}^2{\delta }_{ij}\Delta v_k{\partial }_k{Q}_{ij}\mathrm{d}\mathbf{x}=0.\end{array}$$

(12)

By utilizing integration by parts and (12), we can derive that

$$\begin{array}{cc}\hfill & -L〈\mathbf{H},\Delta (\mathbf{v}·\nabla Q)〉\hfill \\ \hfill & =-L{\int }_{\mathsf{\Omega }}{H}_{ij}\Delta v_k{\partial }_k{Q}_{ij}\mathrm{d}\mathbf{x}-2L{\int }_{\mathsf{\Omega }}{H}_{ij}{\partial }_l{v}_k{\partial }_l{\partial }_k{Q}_{ij}\mathrm{d}\mathbf{x}-L\langle \mathbf{H},\mathbf{v}·\nabla \Delta Q\rangle \hfill \\ \hfill & =-L{\int }_{\mathsf{\Omega }}(L\Delta Q_{ij}-f(Q_{ij}))\Delta v_k{\partial }_k{Q}_{ij}\mathrm{d}\mathbf{x}-2L{\int }_{\mathsf{\Omega }}{H}_{ij}{\partial }_l{v}_k{\partial }_l{\partial }_k{Q}_{ij}\mathrm{d}\mathbf{x}-\langle \mathbf{H},\mathbf{v}·\nabla (\mathbf{H}+f)\rangle \hfill \\ \hfill & =-L^2{\int }_{\mathsf{\Omega }}\Delta Q_{ij}\Delta v_k{\partial }_k{Q}_{ij}\mathrm{d}\mathbf{x}+2L{\int }_{\mathsf{\Omega }}{\partial }_k{H}_{ij}{\partial }_l{v}_k{\partial }_l{Q}_{ij}\mathrm{d}\mathbf{x}-\langle \mathbf{H},\mathbf{v}·\nabla f\rangle .\hfill \end{array}$$

(13)

Similarly, using the divergence free ${\partial }_k\Delta v_k=0$ and integration by parts yields

$$\begin{array}{cc}\hfill & -\langle \Delta \mathbf{v},\nabla ·({\sigma }^d+\sigma )\rangle \hfill \\ \hfill & =L〈\Delta \mathbf{v},\nabla ·(\nabla Q\odot \nabla Q)〉-〈\Delta \mathbf{v},\nabla ·(Q·\mathbf{H}-\mathbf{H}·Q)〉\hfill \\ \hfill & =L{\int }_{\mathsf{\Omega }}\Delta v_i{\partial }_i{Q}_{kl}\Delta Q_{kl}\mathrm{d}\mathbf{x}+L〈\Delta \mathbf{v},\nabla \left(\frac{{|\nabla Q|}^2}{2}\right)〉-〈\Delta \mathbf{v},\nabla ·(Q·\mathbf{H}-\mathbf{H}·Q)〉\hfill \\ \hfill & =L{\int }_{\mathsf{\Omega }}\Delta v_i{\partial }_i{Q}_{kl}\Delta Q_{kl}\mathrm{d}\mathbf{x}-〈\Delta \mathbf{v},\nabla ·(Q·\mathbf{H}-\mathbf{H}·Q)〉.\hfill \end{array}$$

(14)

Notice that Q and $\mathbf{H}(Q)$ are symmetric. Using the following basic algebra for any $A,B,C\in {\mathbb{R}}^{3\times 3}$,

$$\begin{array}{c}\hfill (AB):C=B:(A^{T}C)=A:(C{B}^T),\end{array}$$

we can deduce from integration by parts that

$$\begin{array}{cc}\hfill & 〈\mathbf{H},\Delta (\mathbf{W}·Q-Q·\mathbf{W})〉\hfill \\ \hfill & ={\int }_{\mathsf{\Omega }}\Delta \mathbf{H}:(\mathbf{W}·Q-Q·\mathbf{W})\mathrm{d}\mathbf{x}\hfill \\ \hfill & ={\int }_{\mathsf{\Omega }}\nabla \mathbf{v}:(\Delta \mathbf{H}·Q-Q·\Delta \mathbf{H})\mathrm{d}\mathbf{x}\hfill \\ \hfill & =〈\Delta \mathbf{v},\nabla ·(Q·\mathbf{H}-\mathbf{H}·Q)〉-2{\int }_{\mathsf{\Omega }}{\partial }_j{v}_i({\partial }_j{Q}_{kj}{\partial }_l{H}_{ik}-{\partial }_l{Q}_{ik}{\partial }_l{H}_{kj})\mathrm{d}\mathbf{x}\hfill \\ \hfill & -{\int }_{\mathsf{\Omega }}{\partial }_j{v}_i(H_{ik}\Delta Q_{kj}-\Delta Q_{ik}{H}_{kj})\mathrm{d}\mathbf{x}.\hfill \end{array}$$

(15)

Therefore, substituting (13)–(15) into (11) and discarding some cancellations we obtain

$$\begin{array}{cc}\hfill & \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{A}(t)+{\mu \parallel \Delta \mathbf{v}\parallel }_{L^2}^2+\Gamma L{\parallel \nabla \mathbf{H}\parallel }_{L^2}^2\hfill \\ \hfill & =\langle \Delta \mathbf{v},\mathbf{v}·\nabla \mathbf{v}\rangle +2L{\int }_{\mathsf{\Omega }}{\partial }_k{H}_{ij}{\partial }_l{v}_k{\partial }_l{Q}_{ij}\mathrm{d}\mathbf{x}\hfill \\ \hfill & +L(1-L){\int }_{\mathsf{\Omega }}\Delta v_i{\partial }_i{Q}_{kl}\Delta Q_{kl}\mathrm{d}\mathbf{x}-\langle \mathbf{H},\mathbf{v}·\nabla f\rangle \hfill \\ \hfill & -2{\int }_{\mathsf{\Omega }}{\partial }_j{v}_i({\partial }_j{Q}_{kj}{\partial }_l{H}_{ik}-{\partial }_l{Q}_{ik}{\partial }_l{H}_{kj})\mathrm{d}\mathbf{x}\hfill \\ \hfill & -{\int }_{\mathsf{\Omega }}{\partial }_j{v}_i(H_{ik}\Delta Q_{kj}-\Delta Q_{ik}{H}_{kj})\mathrm{d}\mathbf{x}\hfill \\ \hfill & -\Gamma {\int }_{\mathsf{\Omega }}{f}^{\prime }(Q){\left|\mathbf{H}\right|}^2\mathrm{d}\mathbf{x}+{\int }_{\mathsf{\Omega }}{f}^{\prime }(Q)(\mathbf{v}·\nabla Q):\mathbf{H}\mathrm{d}\mathbf{x}\hfill \\ \hfill & -〈\mathbf{H},f^{\prime }(Q)(\mathsf{\Omega }·Q-Q·\mathbf{W})〉\hfill \\ \hfill & \stackrel{\mathrm{def}}{=}{J}_1+\cdots +J_9.\hfill \end{array}$$

(16)

First, we present some useful estimates. By Lemmas 1–3 and the Sobolev embedding relation $H^1\hookrightarrow L^6$ for the domain $\mathsf{\Omega }$ in ${\mathbb{R}}^3$, we have

$$\begin{array}{cc}\hfill {\parallel \Delta Q\parallel }_{L^2}\le & \frac{1}{L}{\parallel \mathbf{H}\parallel }_{L^2}+\frac{1}{L}\parallel aQ-b(Q^2-\frac{1}{3}{\left|Q\right|}^2\mathbf{I})+c|Q^2|Q{\parallel }_{L^2}\hfill \\ \hfill \le & \frac{1}{L}{\parallel \mathbf{H}\parallel }_{L^2}+C{\parallel Q\parallel }_{H^1}\hfill \\ \hfill \le & C(\parallel \mathbf{H}{\parallel }_{L^2}+1),\hfill \\ \hfill {\parallel \nabla Q\parallel }_{L^6}\le & {C\parallel \nabla Q\parallel }_{H^1}\le {C(1+\parallel \Delta Q\parallel }_{L^2}{)\le C(\parallel \mathbf{H}\parallel }_{L^2}+1),\hfill \\ \hfill {\parallel Q\parallel }_{L^{\infty }}\le & {C\parallel Q\parallel }_{H^1}^{\frac{1}{2}}{\parallel Q\parallel }_{H^2}^{\frac{1}{2}}\le {C(1+\parallel \Delta Q\parallel }_{L^2}^{\frac{1}{2}}{)\le C(1+\parallel \mathbf{H}\parallel }_{L^2}^{\frac{1}{2}}).\hfill \end{array}$$

By using a similar argument, from Lemmas 1–3, we find that

$$\begin{array}{cc}\hfill {\parallel \nabla \Delta Q\parallel }_{L^2}\le & \frac{1}{L}{\parallel \nabla \mathbf{H}\parallel }_{L^2}+\frac{1}{L}{\parallel \nabla f(Q)\parallel }_{L^2}\hfill \\ \hfill \le & \frac{1}{L}{\parallel \nabla \mathbf{H}\parallel }_{L^2}+\frac{1}{L}{(1+\parallel Q\parallel }_{L^{\infty }}^2{)\parallel \nabla Q\parallel }_{L^2}\hfill \\ \hfill \le & \frac{1}{L}{\parallel \nabla \mathbf{H}\parallel }_{L^2}+\frac{1}{L}{(1+\parallel \Delta Q\parallel }_{L^2})\hfill \\ \hfill \le & \frac{1}{L}{\parallel \nabla \mathbf{H}\parallel }_{L^2}+\frac{1}{L}\left(1+{\parallel \nabla Q\parallel }_{L^2}^{\frac{1}{2}}{\parallel \nabla \Delta Q\parallel }_{L^2}^{\frac{1}{2}}+{\parallel \nabla Q\parallel }_{L^2}\right)\hfill \\ \hfill \le & \frac{1}{L}{\parallel \nabla \mathbf{H}\parallel }_{L^2}+\frac{1}{2}{\parallel \nabla \Delta Q\parallel }_{L^2}+C,\hfill \end{array}$$

which leads to

$$\begin{array}{c}\hfill {\parallel \nabla \Delta Q\parallel }_{L^2}\le {C(\parallel \nabla \mathbf{H}\parallel }_{L^2}+1).\end{array}$$

In addition, by Lemma 1, we have

$$\begin{array}{cc}\hfill {\parallel \nabla Q\parallel }_{L^{\infty }}\le & {C\parallel \nabla Q\parallel }_{L^2}^{\frac{1}{4}}{\parallel \nabla Q\parallel }_{H^2}^{\frac{3}{4}}\le {C(\parallel \nabla \Delta Q\parallel }_{L^2}^{\frac{3}{4}}+1)\hfill \\ \hfill \le & C(\parallel \nabla \mathbf{H}{\parallel }_{L^2}^{\frac{3}{4}}+1).\hfill \end{array}$$

(17)

We next estimate (16). For the first term $J_1$, using H$\ddot{o}$lder inequality and Lemma 1 lead to

$$\begin{array}{cc}\hfill J_1\le & {\parallel \Delta \mathbf{v}\parallel }_{L^2}{\parallel \mathbf{v}\parallel }_{L^4}{\parallel \nabla \mathbf{v}\parallel }_{L^4}\hfill \\ \hfill \le & {C\parallel \Delta \mathbf{v}\parallel }_{L^2}{\parallel \mathbf{v}\parallel }_{L^2}^{\frac{1}{4}}{\parallel \nabla \mathbf{v}\parallel }_{L^2}^{\frac{3}{4}}{\parallel \nabla \mathbf{v}\parallel }_{L^2}^{\frac{1}{4}}{\parallel \Delta \mathbf{v}\parallel }_{L^2}^{\frac{3}{4}}\hfill \\ \hfill \le & {\mu }^{\frac{1}{2}}{\parallel \Delta \mathbf{v}\parallel }_{L^2}^2+{\mu }^{\frac{1}{2}}{\parallel \nabla \mathbf{v}\parallel }_{L^2}^2{\parallel \Delta \mathbf{v}\parallel }_{L^2}^2+C{\mu }^{-\frac{7}{2}}{\parallel \nabla \mathbf{v}\parallel }_{L^2}^2\hfill \\ \hfill \le & {\mu }^{\frac{1}{2}}\tilde{\mathcal{A}}{\parallel \Delta \mathbf{v}\parallel }_{L^2}^2+C{\mu }^{-\frac{7}{2}}\mathcal{A}.\hfill \end{array}$$

Similarly, we can deduce that

$$\begin{array}{cc}\hfill J_2\le & {C\parallel \nabla \mathbf{H}\parallel }_{L^2}{\parallel \nabla \mathbf{v}\parallel }_{L^3}{\parallel \nabla Q\parallel }_{L^6}\hfill \\ \hfill \le & {C\parallel \nabla \mathbf{H}\parallel }_{L^2}{\parallel \nabla \mathbf{v}\parallel }_{L^2}^{\frac{1}{2}}{\parallel \Delta \mathbf{v}\parallel }_{L^2}^{\frac{1}{2}}{(\parallel \mathbf{H}\parallel }_{L^2}+1)\hfill \\ \hfill \le & {\mu }^{\frac{1}{2}}{\parallel \nabla \mathbf{v}\parallel }_{L^2}{\parallel \Delta \mathbf{v}\parallel }_{L^2}+C{\mu }^{-\frac{1}{4}}\tilde{\mathcal{A}}{\parallel \nabla \mathbf{H}\parallel }_{L^2}^2\hfill \\ \hfill \le & {\mu }^{\frac{1}{2}}{\parallel \Delta \mathbf{v}\parallel }_{L^2}^2+C{\mu }^{-\frac{1}{4}}\tilde{\mathcal{A}}{\parallel \nabla \mathbf{H}\parallel }_{L^2}^2+C\mathcal{A}.\hfill \end{array}$$

By the H$\ddot{o}$lder inequality and (17), the term $J_3$ can be estimated as follows:

$$\begin{array}{cc}\hfill J_3\le & {C\parallel \Delta \mathbf{v}\parallel }_{L^2}{\parallel \Delta Q\parallel }_{L^2}{\parallel \nabla Q\parallel }_{L^{\infty }}\hfill \\ \hfill \le & {C\parallel \Delta \mathbf{v}\parallel }_{L^2}{(\parallel \mathbf{H}\parallel }_{L^2}+{1)(\parallel \nabla \mathbf{H}\parallel }_{L^2}^{\frac{3}{4}}+1)\hfill \\ \hfill \le & {\mu }^{\frac{1}{2}}{\parallel \Delta \mathbf{v}\parallel }_{L^2}^2+C{\mu }^{-\frac{2}{3}}{(\parallel \mathbf{H}\parallel }_{L^2}^2+{1)\parallel \nabla \mathbf{H}\parallel }_{L^2}^2+{C(\parallel \mathbf{H}\parallel }_{L^2}^2+1)\hfill \\ \hfill \le & {\mu }^{\frac{1}{2}}{\parallel \Delta \mathbf{v}\parallel }_{L^2}^2+C{\mu }^{-\frac{2}{3}}\tilde{\mathcal{A}}{\parallel \nabla \mathbf{H}\parallel }_{L^2}^2+C\tilde{\mathcal{A}}.\hfill \end{array}$$

For the term $J_4$, we see that

$$\begin{array}{cc}\hfill J_4\le & {C\parallel \mathbf{H}\parallel }_{L^3}{\parallel \mathbf{v}\parallel }_{L^6}{\parallel \nabla f(Q)\parallel }_{L^2}\hfill \\ \hfill \le & {C\parallel \mathbf{H}\parallel }_{L^2}^{\frac{1}{2}}{\parallel \nabla \mathbf{H}\parallel }_{L^2}^{\frac{1}{2}}{\parallel \nabla \mathbf{v}\parallel }_{L^2}{(1+\parallel Q\parallel }_{L^{\infty }}^2{)\parallel \nabla Q\parallel }_{L^2}\hfill \\ \hfill \le & {C(1+\parallel \mathbf{H}\parallel }_{L^2}{)(\parallel \nabla \mathbf{H}\parallel }_{L^2}+{\parallel \mathbf{H}\parallel }_{L^2}{)\parallel \nabla \mathbf{v}\parallel }_{L^2}\hfill \\ \hfill \le & {\mu }^{\frac{1}{2}}{(1+\parallel \mathbf{H}\parallel }_{L^2}^2{)\parallel \Delta \mathbf{v}\parallel }_{L^2}^2+{\mu }^{-\frac{1}{4}}{(1+\parallel \mathbf{H}\parallel }_{L^2}^2{)\parallel \nabla \mathbf{H}\parallel }_{L^2}^2\hfill \\ \hfill & +C(1+{\mu }^{-\frac{1}{2}}+{\mu }^{-\frac{3}{4}}){\parallel \mathbf{H}\parallel }_{L^2}^2\hfill \\ \hfill \le & {\mu }^{\frac{1}{2}}\tilde{\mathcal{A}}{\parallel \Delta \mathbf{v}\parallel }_{L^2}^2+C{\mu }^{-\frac{1}{4}}\tilde{\mathcal{A}}{\parallel \nabla \mathbf{H}\parallel }_{L^2}^2+C\mathcal{A}.\hfill \end{array}$$

Recalling the Sobolev embedding relation $H^1\hookrightarrow L^6$ in $\mathsf{\Omega }\in {\mathbb{R}}^3$, by using H$\ddot{o}$lder inequality and Lemma 1, we can infer that

$$\begin{array}{cc}\hfill J_5\le & {C\parallel \nabla Q\parallel }_{L^3}{\parallel \nabla \mathbf{v}\parallel }_{L^6}{\parallel \nabla \mathbf{H}\parallel }_{L^2}\hfill \\ \hfill \le & {C\parallel \nabla Q\parallel }_{L^2}^{\frac{1}{2}}{\parallel \nabla Q\parallel }_{H^1}^{\frac{1}{2}}{\parallel \nabla \mathbf{H}\parallel }_{L^2}{\parallel \nabla \mathbf{v}\parallel }_{H^1}\hfill \\ \hfill \le & {C(1+\parallel \mathbf{H}\parallel }_{L^2}{)\parallel \nabla \mathbf{H}\parallel }_{L^2}{(\parallel \Delta \mathbf{v}\parallel }_{L^2}+1)\hfill \\ \hfill \le & \frac{1}{2}{\mu }^{\frac{1}{2}}{\parallel \Delta \mathbf{v}\parallel }_{L^2}^2+C{\mu }^{-\frac{1}{2}}\tilde{\mathcal{A}}{\parallel \nabla \mathbf{H}\parallel }_{L^2}^2.\hfill \end{array}$$

Similarly, the term $J_6$ can be estimated as

$$\begin{array}{cc}\hfill J_6\le & {C\parallel \nabla \mathbf{v}\parallel }_{L^6}{\parallel \mathbf{H}\parallel }_{L^3}{\parallel \Delta Q\parallel }_{L^2}\hfill \\ \hfill \le & {C\parallel \Delta \mathbf{v}\parallel }_{L^2}{\parallel \mathbf{H}\parallel }_{L^2}^{\frac{1}{2}}{\parallel \nabla \mathbf{H}\parallel }_{L^2}^{\frac{1}{2}}{(1+\parallel \mathbf{H}\parallel }_{L^2})\hfill \\ \hfill \le & \frac{1}{2}{\mu }^{\frac{1}{2}}{\parallel \Delta \mathbf{v}\parallel }_{L^2}^2+C{\mu }^{-\frac{1}{2}}{\parallel \mathbf{H}\parallel }_{L^2}{\parallel \nabla \mathbf{H}\parallel }_{L^2}{(1+\parallel \mathbf{H}\parallel }_{L^2}{)}^2\hfill \\ \hfill \le & \frac{1}{2}{\mu }^{\frac{1}{2}}{\parallel \Delta \mathbf{v}\parallel }_{L^2}^2+C{\mu }^{-\frac{1}{4}}\tilde{\mathcal{A}}{\parallel \nabla \mathbf{H}\parallel }_{L^2}^2+C\tilde{\mathcal{A}}.\hfill \end{array}$$

As for the estimate of $J_7$, we infer that

$$\begin{array}{cc}\hfill J_7\le & C\parallel f^{\prime }{(Q)\parallel }_{L^3}{\parallel \mathbf{H}\parallel }_{L^3}^2\le {C(1+\parallel Q\parallel }_{L^6}^2{)\parallel \mathbf{H}\parallel }_{L^3}^2\hfill \\ \hfill \le & {C(\parallel \nabla \mathbf{H}\parallel }_{L^2}+{\parallel \mathbf{H}\parallel }_{L^2}{)\parallel \mathbf{H}\parallel }_{L^2}\hfill \\ \hfill \le & \frac{\Gamma L}{2}{\parallel \nabla \mathbf{H}\parallel }_{L^2}^2+C{\parallel \mathbf{H}\parallel }_{L^2}^2.\hfill \end{array}$$

For the term $J_8$, we have

$$\begin{array}{cc}\hfill J_8\le & C\parallel f^{\prime }{(Q)\parallel }_{L^3}{\parallel \mathbf{v}\parallel }_{L^6}{\parallel \mathbf{H}\parallel }_{L^3}{\parallel \nabla Q\parallel }_{L^6}\hfill \\ \hfill \le & {C(1+\parallel Q\parallel }_{L^6}^2{)\parallel \nabla \mathbf{v}\parallel }_{L^2}{\parallel \mathbf{H}\parallel }^{\frac{1}{2}}{\parallel \nabla \mathbf{H}\parallel }_{L^2}^{\frac{1}{2}}{(\parallel \mathbf{H}\parallel }_{L^2}+1)\hfill \\ \hfill \le & {C\parallel \nabla \mathbf{v}\parallel }_{L^2}{\parallel \nabla \mathbf{H}\parallel }_{L^2}{(\parallel \mathbf{H}\parallel }_{L^2}+1)\hfill \\ \hfill \le & {\mu }^{\frac{1}{2}}{\parallel \Delta \mathbf{v}\parallel }_{L^2}^2+C{\mu }^{-\frac{1}{4}}\tilde{\mathcal{A}}{\parallel \nabla \mathbf{H}\parallel }_{L^2}^2.\hfill \end{array}$$

We now turn to control the last term $J_9$, and it is easy to deduce that

$$\begin{array}{cc}\hfill J_9\le & {C(1+\parallel Q\parallel }_{L^6}^2{)\parallel \mathbf{H}\parallel }_{L^3}{\parallel Q\parallel }_{L^6}{\parallel \nabla \mathbf{v}\parallel }_{L^6}\hfill \\ \hfill \le & {C(\parallel \mathbf{H}\parallel }_{L^2}+{\parallel \nabla \mathbf{H}\parallel }_{L^2}{)\parallel \nabla \mathbf{v}\parallel }_{L^2}^{\frac{1}{2}}{\parallel \Delta \mathbf{v}\parallel }_{L^2}^{\frac{1}{2}}\hfill \\ \hfill \le & {\mu }^{\frac{1}{2}}{\parallel \Delta \mathbf{v}\parallel }_{L^2}^2+C{\mu }^{-\frac{1}{4}}{\parallel \nabla \mathbf{H}\parallel }_{L^2}^2+C\tilde{\mathcal{A}}.\hfill \end{array}$$

Therefore, summarizing all the previous estimates implies that the inequality (10) holds, and this completes the proof Proposition 1. □

Proof of Theorem 1.

We can infer from (10) that

$$\begin{array}{cc}\hfill & \frac{\mathrm{d}}{\mathrm{d}t}\tilde{\mathcal{A}}(t)+\left(\mu -C_1{\mu }^{\frac{1}{2}}\tilde{\mathcal{A}}(t)\right){\parallel \Delta \mathbf{v}\parallel }_{L^2}^2\hfill \\ \hfill & +\left(\frac{\Gamma L}{2}-C_2{\mu }^{-\frac{1}{4}}\tilde{\mathcal{A}}(t)\right){\parallel \nabla \mathbf{H}\parallel }_{L^2}^2\hfill \\ \hfill & \le C_3\tilde{\mathcal{A}}(t).\hfill \end{array}$$

Moreover, we derive from Lemma 3 that

$$\begin{array}{c}\hfill {\int }_t^{t+1}\tilde{\mathcal{A}}(s)ds\le {\int }_t^{t+1}\mathcal{A}(s)ds+1\le M,\forall t\in [0,T-1],\end{array}$$

where the positive constant M depends on $\Gamma ,a,b,c,L,\mu$ and the initial data. Assuming that $\mu$ is large enough and

$$\begin{array}{c}\hfill {\mu }^{\frac{1}{2}}\ge {\mu }_0^{\frac{1}{2}}\stackrel{\mathrm{def}}{=}{C}_1(\tilde{\mathcal{A}}(0)+2M+C_{3}M)+\frac{4C_2^2}{{\Gamma }^2{L}^2}{(\tilde{\mathcal{A}}(0)+2M+C_{3}M)}^2+1,\end{array}$$

then initially there exists some $T_0>0$ such that

$$\begin{array}{c}\hfill \mu -C_1{\mu }^{\frac{1}{2}}\tilde{\mathcal{A}}(t)\ge 0,\frac{\Gamma L}{2}-C_2{\mu }^{-\frac{1}{4}}\mathcal{A}(t)\ge 0,\forall t\in [0,T_0].\end{array}$$

(18)

Hence, in this time interval, we obtain that

$$\begin{array}{c}\hfill \frac{\mathrm{d}}{\mathrm{d}t}\tilde{\mathcal{A}}(t)\le C_3\tilde{\mathcal{A}}(t).\end{array}$$

Suppose that $T_{*}$ is the largest one among such $T_0$. Applying a similar argument in ([30], Theorem 4.3), the argument by contradiction yields $T_{*}=T$. Thus, we see that $\mathcal{A}(t)$ is uniformly bounded for all $t\ge 0$ and fulfills (18).

The uniform estimate enables us to pass to the limit for the approximate solution $(Q_m,{\mathbf{v}}_m)$ corresponding to systems (4)–(6) when $m\to \infty$. Based on a similar argument to [29], it can be proved that there exist global strong solutions of systems (4)–(6) such that $(Q,\mathbf{v})$ satisfies

$$\begin{array}{c}\hfill \mathbf{v}\in L^{\infty }(0,\infty ;V)\cap L^2(0,\infty ;H^2),Q\in L^{\infty }(0,\infty ;H^2)\cap L^2(0,\infty ;H^3).\end{array}$$

Therefore, the proof of Theorem 1 is complete. □

4. Conclusions

This paper is mainly concerned with the coupled Beris–Edwards system descrbing nematic liquid crystal flows when material coefficient $\xi =0$. In a bounded domain $\mathsf{\Omega }\in {\mathbb{R}}^3$ with homogeneous Dirichlet boundary conditions, we first establish the uniform estimate of higher-order energy functionals, and then prove global existence of strong solutions to the Beris–Edwards system under the assumption that the viscosity $\mu$ is large enough. From a physical point of view, the domain of the liquid crystal model is usually bounded, which will lead to impartant interesting physical properties such as defects. At physical boundary, liquid crystals are usually taken as the uniaxial phase but not as the isotropic phase; hence, the well-posedness study with inhomogeneous boundary conditions and $\xi \ne 0$ is a more meaningful work.