Existence of Sobolev regular solutions for the incompressible flow of liquid crystals in three dimensions

1.   Introduction and main results

In this paper, we consider the simplified Ericksen-Leslie system, it models the hydrodynamics of nematic liquid crystals in dimension three. It takes the following form:

where $(t, x)\in \mathbb R^+\times\Omega$ with $\Omega = \mathbb R^+\times \mathbb R^2$ (or $\Omega$ being a bounded smooth domain in $\mathbb R^3$), $u(t, x): \mathbb R^+\times\Omega\rightarrow \mathbb R^3$ is the fluid velocity, $P(t, x): \mathbb R^+\times\Omega\rightarrow \mathbb R$ stands for the pressure in the fluid, $d(t, x): \mathbb R^+\times\Omega\rightarrow \mathbb R^3$ represents the director field for the averaged macro-scopic molecular orientations. The constants $\nu$, $\kappa$ and $\gamma$ are positive constants representing the viscosity of the fluid, the competition between kinetic and potential energy, and the microscopic elastic relaxation time for the molecular orientation field, respectively. The symmetric $3\times3$ matrix $\nabla d\odot\nabla d$ denotes the Ericksen stress tensor whose $(i, j)$-th entry is given by $\nabla_id\cdot\nabla_jd$ for $1\leq i, \; j\leq 3$. The divergence free condition in the second equations of problem (1.1) guarantees the incompressibility of the fluid. The vector valued nonlinear function $f(d)$ is the gradient of certain smooth scalar potential function $\mathcal F(d): \mathbb R^3\rightarrow \mathbb R$ such that $f(d) = \nabla_d \mathcal F(d)$, we take $\mathcal F$ to be the Ginzburg-Landau approximation

It is simply the Ginzburg-Landau approximation of the constraint $|d|$ for small ${\varepsilon}$ as given in ^[1,2].

The pressure takes the form

Liquid crystals are substances that exhibit a phase of matter that has properties between those of a conventional liquid, and those of a solid crystal. It may flow like a liquid, but its molecules may be oriented in a crystal-like way. In the 1960s, Ericksen ^[3] and Leslie ^[4] established the hydrodynamic theory of liquid crystals. The Ericksen-Lislie theory describes the liquid crystal flow, including the velocity vector $u$ and direction vector $d$ of the fluid. Since the general Ericksen-Leslie system is very complicated, a simplified model of the Ericksen-Leslie system ^[1,2,5] is meanfully to be considered, and there are large diffculties even in the well-posedness theory. One of difficulties is from the three dimension Navier-Stokes type equations in the first equations in (1.1). Another difficulties is from the harmonic maps in the second equations in (1.1) with the constraint condition $|d| = 1$.

When the direction vector $d\equiv0$, Eq (1.1) is reduced into the famous three dimension incompressible Navier-Stokes equations

The question of whether a solution of the $3$D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data with finite energy is one of the Millennium Prize problems ^[6]. In 1934, Leray ^[7] showed that the $3$D incompressible Navier-Stokes Eq (1.3) admit global-forward-in-time weak solutions of the initial value problem. Caffarelli et al. ^[8] established a ${\varepsilon}$-regularity criterion for Eq (1.3). After that, Lin ^[9] gave a new and simpler proof for the result of Caffarelli, Kohn and Nirenberg. Buckmaster and Vicol ^[10] proved that the non-uniqueness of solutions for Eq (1.3) in $H^{s}$-space for some small $s > 0$. One can see ^{[11,12,13,14,15,16,17,18]} for more results on this equations.

When the direction vector $d\not\equiv0$ and $f(d) = |\nabla d|^2d$, there is a constraint condition $|d| = 1$. This makes the Ericksen-Leslie system more complicated. For this case, Lin ea al. ^[19] and Hong ^[20] established the existence of global weak solutions for this system in $\mathbb R^2$. Lin and Wang ^[21] obtained the global existence of a weak solution for the case of three dimension. After that, Huang ea al. ^[22] showed that two examples of non-trivial solutions to this $3$D system with finite time singularity in the unit ball centered at $0$.

In order to relax the constraint condition $|d| = 1$ for the Dirichlet energy

Lin and Liu ^[1,2] used the Ginzburg-Landau energy

to replace the Dirichlet energy, then they ^[1] obtained the global existence of the solution of (1.1) in dimension two and three, after that, they ^[2] got partial regularity of weak solutions to this system. Furthermore, they ^[23] proved existence of solutions for system (1.1) and also analyzed the limits of weak solutions of it as ${\varepsilon}\rightarrow0$. Specially, by setting the term $f(d) = 0$, Hu and Wang ^[24] proved that all the global weak solutions constructed in ^[1] be equal to the unique strong solution. Cavaterra et al. ^[25] established the existence of optimal boundary controls for this system in two dimension. We point out that any solution convergence to the equilibrium state (constants or functions with infinite energy in the unbounded domain) admits the infinite energy. To the author's knowledgement, there is few result on the global finite energy solution with regularity for this system in the unbounded domain or the smooth bounded domain.

In this paper, we will prove the global existence of Sobolev regular solutions for the three-dimensional incompressible flow of liquid crystals in the unbounded domain and the smooth bounded domain. Meanwhile, we give the explicit asymptotic expansion formula of Sobolev regular solutions for system (1.1).

1.1.   The case of unbounded domain $\Omega= \mathbb R^+\times \mathbb R^2$

We supplement the three-dimensional incompressible nematic liquid crystals (1.1) with the initial data

and the boundary condition

We assume that the initial data (1.4) satisfies the following conditions

We now state one of main results in this paper.

Theorem 1.1. Let the parameter $\nu, \gamma, \kappa, {\varepsilon} > 0$ in (1.1). For any fixed constant $s\geq1$, if the initial data (1.4) of the three dimension nematic liquid crystal flow (1.1) satisfies the condition (1.6), and there exists a small constant $\delta\in(0, 1)$ such that

then the three dimension nematic liquid crystal flow (1.1) with the boundary conditions (1.5) admit a global Sobolev regular solution with finite energy

Moreover, it holds

for any $(t, x)\in(0, \infty)\times\Omega$. Here, the pressure is given by (1.2).

As an application, we have the following asymptotic expansion of solutions.

Corollary 1.1. Let the parameter $\nu, \gamma, \kappa, {\varepsilon} > 0$ in (1.1). Assume that the integers $p > 0, 2 < q < 2(p+1)$ and the parameter $0 < \delta\ll1$. Assume that the small initial data (1.4) satisfies the condition (1.6). Then the three dimension nematic liquid crystal flow (1.1) in the unbounded domain $\mathbb R^+\times \mathbb R^2$ admit an explicit expansion of global Sobolev regular solution with finite energy as follows

for all $x = (x_1, x_2, x_3)\in \mathbb R^+\times \mathbb R^2$, where

and the remainder term $\mathcal R_k(t, x)\in H^s(\Omega)$ with $s\geq1$ and $k = 1, 2$ satisfies

Furthermore, one can see that

Moreover, the pressure is determined by

Remark 1.1. Corollary 1.1 is derived directly from the proof of Theorem 1.1. From Corollary 1.1, we can observe that Eq (1.1) admit a global Sobolev regular solution with finite energy by chosing different constants $p$ and $q$. Moreover, if the vector functions

and

satisfy the conditions

and

Moreover, for any fixed $s\geq1$ and $(t, x)\in\Omega$ and $i, j = 1, 2, 3$, it also needs the conditions

and

then we can use the proof of Theorem 1.1 to construct more global Sobolev regular solutions with finite energy of (1.1) in the unbounded domain $\mathbb R^+\times \mathbb R^2$.

1.2.   The case of smooth bounded domain $\Omega\subset \mathbb R^3$

We have the following results.

Theorem 1.2. Let the parameter $\nu, \gamma, \kappa, {\varepsilon} > 0$ in (1.1) and ${\gamma\over{\varepsilon}^2} < 1$. For any fixed constant $s\geq1$, if the initial data (1.4) of the three dimension nematic liquid crystal flow (1.1) satisfies the conditions (1.6), and there exists a small constant $0 < \delta\ll\min\{1, \nu, \gamma\}$ such that

then the three dimension nematic liquid crystal flow (1.1) with the boundary conditions (1.5) admit a global Sobolev regular solution with finite energy

Moreover, it holds

for any $(t, x)\in(0, \infty)\times\Omega$. Here, the pressure is given by (1.2).

In particularly, if the domain $\Omega: = ([0, T])^3$ (a regular polyhedron) with the finite constant $T > 0$, we have the following asymptotic expansion of solutions.

Corollary 1.2. Let the parameter $\nu, \gamma, \kappa, {\varepsilon} > 0$ in (1.1) and ${\gamma\over{\varepsilon}^2} < 1$. Assume that the integers $p > 1$ and $q > 2$, and the parameter $0 < \delta\ll1$. The three dimension nematic liquid crystal flow (1.1) in the smooth bounded domain $\Omega$ admits an explicit expansion of Sobolev regular solutions with finite energy as follows

where $\forall x = (x_1, x_2, x_3)\in \Omega: = ([0, T])^3$, and

with

and the remainder term $\mathcal R_k(t, x)\in H^s(\Omega)$ with $s\geq1$ and $k = 1, 2$ satisfies

Furthermore, one can see that

Moreover, the pressure is determined by

The organization of this paper is as follows. In Section 2, we show how to choose a suitable initial approximation functions, which lead to the dissipative structure of linearized system. After that, we give the existence of global time-decay Sobolev solution for the linearized equations of the first approximation step. In Section 3, we establish the general approximation step for the construction of the Nash-Moser iteration scheme. This method has been used in ^{[26,27,28,29,30,31]}. For the general Nash-Moser implicit function theorem, we refer to the seminal papers of Nash ^[32], Moser ^[33] and Hörmander ^[34]. This last section will show how to construct a global Sobolev solution for the $3$D nematic liquid crystal flow (1.1) by using the proof of convergence for the Nash-Moser iteration scheme.

2.   The first approximation step

Throughout this paper, let $\Omega: = \mathbb R^+\times \mathbb R^2$ or a smooth bounded domain. We denote the usual norms of $\mathbb L^2(\Omega)$ and the Sobolev space $\mathbb H^s(\Omega)$ by $\|\cdot\|_{ \mathbb L^2}$ and $\|\cdot\|_{ \mathbb H^s}$, respectively. The norm of the Sobolev space $H^s(\Omega): = (\mathbb H^s(\Omega))^3$ is denoted by $\|\cdot\|_{H^s}$. The symbol $a\lesssim b$ means that there exists a positive constant $C$ such that $a\leq Cb$. We denote by $(a, b, c)^T$ the column vector in $\mathbb R^3$. The letter $C$ with subscripts is used to denote dependencies stands for a positive constant that might change its value at each occurrence.

The proof of Theorem 1.2 is the same with Theorem 1.1 by a small modification of relationship of parameters $\nu, \gamma, {\varepsilon}$. Thus we give a proof process including both Theorems 1.1 and 1.2.

We introduce a family of smooth operators possessing the following properties.

Lemma 2.1. ^[34,35] There is a family $\{\Pi_{\theta}\}_{\theta\geq1}$ of smoothing operators in the space $H^s(\Omega)$ acting on the class of functions such that

where $C$ is a positive constant and $(s_1-s_2)_+: = \max(0, s_1-s_2)$.

In our iteration scheme, we set

where $N_0$ is a fixed positive constant, then by (2.1), it holds

We consider the approximation problem of the $3$D nematic liquid crystal Eq (1.1) as follows

with the initial data (1.4), the boundary condition (1.5) and the incompressible condition

2.1.   The initial approximation function

Let $s\geq1$ be a fixed finite constant and $0 < \delta_0 < \delta^2\ll1$. For any $x\in\Omega$, we choose the initial approximation functions

where we require

and

Moreover, for any fixed $s\geq1$, and $\forall t > 0$ and $x\in\Omega$ and $i, j = 1, 2, 3$, it also needs the conditions

and

and the initial error term

where $E^{(0)}$ and $\overline{E}^{(0)}$ denote the error term taking the form

with the vector form

There are two family of explicit examples. In fact, many vector functions can be chosen to satisfy conditions (2.4)–(2.8). We now give two families of exact examples of the initial approximation function satisfying (2.4)–(2.8) for the unbounded domain and the smooth bounded domain.

For the unbounded domain $\Omega = \mathbb R^+\times \mathbb R^2$, let the integers $p > 1$ and $2 < q < 2(p+1)$. We choose the initial approximation functions of the following form

where

For the smooth bounded domain $\Omega: = ([0, T])^3$ (a regular polyhedron) with the finite constant $T > 0$, we set the integers $p > 1$ and $q > 2$, and

with

By straightforward computations, we obtain

and

Moreover, for every fixed integers $p$ and $q$, the assumptions (2.4)–(2.8) holds.

Here, the initial approximation pressure satisfies

2.2.   The time-decay of first approximation step

We now construct the first approximation solution denoted by $\Big(u^{(1)}(t, \lambda x), d^{(1)}(t, \lambda x)\Big)$ of system (2.3). There should be two cases on the parameter $\lambda$ according to the domain.

Case 2.1. If the domain is the unbounded domain $\mathbb R^+\times \mathbb R^2$, then we set the parameter $\lambda > 1$.

Case 2.2. If the domain is a bounded domain $\Omega$, we set the parameter $\lambda = 1$, meanwhile, we should require ${\lambda\over{\varepsilon}^2} < 1$.

The first approximation step between the initial approximation function and first approximation solution is denoted by

Then we linearize the nonlinear system (2.3) around $(u^{(0)}, d^{(0)})$ to get the linearized operators as follows

and

where $\mathcal D_{u^{(0)}}$ denotes the Fréchet derivatives on $u^{(0)}$, and by (1.2), it takes the form

We now consider the linear system

and the boundary condition

from which, the solution of it gives the first approximation step of $3$D nematic liquid crystal flow (1.1).

Before we carry out some a priori estimates, for $j = 1, 2, 3$, we rewrite equations of (2.12) into a coupled system as follows

coupled with

with the initial data

and the boundary condition (2.13).

We will use the weighted function estimate to get the decay in time of solutions. It is like the Carleman estimate. Let $\psi(x_1)$ be a function defined in $(0, +\infty)$ such that

and $e^{-\psi(x_1)}$ is bounded in $(0, +\infty)$. The condition (2.17) implies $\psi''(x_1)\geq\kappa > 1$. In fact, there are many functions can satisfy above conditions. For a simple example, we take the function as the form

We now derive $\mathbb L^2$-estimate of solution for the linear systems (2.14) and (2.15).

Lemma 2.2. If the domain is the unbounded domain $\mathbb R^+\times \mathbb R^2$, let the parameter $\lambda > 1$. If the domain is a bounded domain $\Omega$, let the parameter $\lambda = 1$ and ${\gamma\over{\varepsilon}^2} < 1$. Assume that the initial approximation function $(u^{(0)}, d^{(0)})$ satisfies conditions (2.4)–(2.8). Then the solution $(\textbf{h}^{(1)}(t, \lambda x), \textbf{w}^{(1)}(t, \lambda x))$ of the linear systems (2.14) and (2.15) satisfies

where $C_{\nu, \gamma, \lambda, {\varepsilon}, \delta}$ denotes a positive constant depending on $\nu, \gamma, \lambda, {\varepsilon}, \delta$.

Proof. Multiplying both sides of equations in (2.14) and (2.15) by $e^{-\psi(x_1)}h_j^{(1)}$ and $e^{-\psi(x_1)}w_j^{(1)}$, respectively, then integrating over $\Omega$ by noticing the boundary condition (2.13), for $j = 1, 2, 3$, it follows that

and

We sum up (2.19) and (2.20) from $j = 1$ to $j = 3$, hence

On the one hand, note that we have chosen the initial approximation function $(u^{(0)}, d^{(0)})$ satisfying (2.4)–(2.8). We integrate by parts and we get

Since the initial approximation function $u^{(0)}$ satisfies $\nabla\cdot u^{(0)}$, inequality (2.23) reduces to

By direct computation we find

Next, noticing the incompressible condition

we obtain

then, we integrate by part to derive

furthermore, from (2.11), using the standard Calderon-Zygmund theory and Young's inequality, it holds

thus, it holds

where

Note that $\psi'(x_1)$ is bounded in $\Omega$. It follows from inquality (2.27) that

On the other hand, we integrate by part to derive

from which, by Young's inequality, we obtain

and

and

and by $\psi''(x_1) > {1\over4}$, it holds

If the domain is the unbounded domain $\mathbb R^+\times \mathbb R^2$, we choose the parameter $\lambda > 1$. If the domain is a bounded domain $\Omega$, we choose the parameter $\lambda = 1$ and ${\gamma\over{\varepsilon}^2} < 1$. Next, for a suitable constant $\lambda > 1$, we substitute (2.23) and (2.34) into (2.21) to get

where $C_{\lambda}$ denotes a positive constant depending on $\lambda$.

Since $\psi(x_1)$ satisfies the ODE inequality (2.17), for suitable positive constants $\lambda$ and $\delta$, there exist two positive constants $C_{\nu, \gamma, \lambda}$ and $C_{\nu, \gamma, \lambda, {\varepsilon}, \delta}$ such that

thus, integrating (2.35) over $(0, t)$ we find

which gives the following inequality

Therefore, we apply Gronwall's inequality to inequality (2.36) to obtain

which yields

Furthermore, since $e^{-\psi(x_1)}$ is a bounded smooth function in $\Omega$, it follows that

The proof is now complete.

Next, we derive the higher order derivatives estimates. For a fixed constant $s\geq1$, we apply $D_i^s: = \partial _{x_i}^s$ $(\forall i = 1, 2, 3)$ to both sides of (2.14). Therefore

coupled with

with the boundary condition

where the constant $1\leq l\leq s$, and

Next, we derive higher derivative estimate of solution for (2.14).

Lemma 2.3. If the domain is the unbounded domain $\mathbb R^+\times \mathbb R^2$, let the parameter $\lambda > 1$. If the domain is a bounded domain $\Omega$, let the parameter $\lambda = 1$ and ${\gamma\over{\varepsilon}^2} < 1$. Assume that the initial approximation function $(u^{(0)}, d^{(0)})$ satisfies conditions (2.4)–(2.8). Then the solution $(\textbf{h}^{(1)}(t, \lambda x), \textbf{w}^{(1)}(t, \lambda x))$ of the linear system (2.14) and (2.15) satisfies

where $C_{\nu, \gamma, \lambda, {\varepsilon}, \delta}$ denotes a positive constant depending on $\nu, \gamma, \lambda, {\varepsilon}, \delta$.

Proof. We use an induction argument. Let $s = 1$. By (2.38), we obtain

coupled with

with the boundary condition

We also choose the weighted function satisfies (2.17). Multiplying both sides of (2.42) and (2.43) by $D_i^1h_j^{(1)}e^{-\psi(x_1)}$ and $D_i^1w_j^{(1)}e^{-\psi(x_1)}$, respectively, then integrating over $\Omega$ by noticing (2.44), and summing up those equalities from $j = 1$ to $j = 3$, it follows that

and

where

We now estimate each term in (2.45) and (2.46). On the one hand, since we have chosen the initial approximation function $(u^{(0)}, d^{(0)})$ satisfying (2.4)–(2.8), we derive

By the incompressibility condition $\nabla\cdot\textbf{h}^{(1)} = 0$ and integration by parts we find

from which, similar to (2.28), we use the standard Calderon-Zygmund theory and Young's inequality to derive

and

and

On the other hand, by Young's inequality, it holds

and we integrate by parts to estimate

Thus, summing up (2.45) and (2.46) from $i = 1$ to $i = 3$, we use (2.47)–(2.52) to derive

If the domain is the unbounded domain $\mathbb R^+\times \mathbb R^2$, we choose the parameter $\lambda > 1$. If the domain is a bounded domain $\Omega$, we choose the parameter $\lambda = 1$ and ${\gamma\over{\varepsilon}^2} < 1$. Furthermore, note that $\psi(x_1)$ satisfies the ODE inequality (2.17), for suitable positive constants $\lambda$ and $\delta$, there exist two positive constants $C_{\nu, \gamma, \lambda}$ and $C_{\nu, \gamma, \lambda, {\varepsilon}, \delta}$ such that we integrate inequality (2.53) over $(0, t)$ to get

We observe that the last term in the right-hand side of (2.54) can be controlled by using (2.37). Therefore

Hence, by (2.37), we apply Gronwall's inequality to (2.54) to derive

Assume that the $2\leq l\leq s-1$ derivative case holds, that is,

and

We now prove the $s$th derivative case holds. Multiplying both sides of Eqs (2.38) and (2.39) by $D_i^sh^{(1)}_je^{-\psi(x_1)}$ and $D_i^sw^{(1)}_je^{-\psi(x_1)}$, then integrating over $\Omega$ by using the boundary condition (2.40), and summing up those equalities from $i, j = 1$ to $i, j = 3$, with similar arguments as for getting (2.53), we can obtain

where we intergrate by parts to estimate the term $(D_i^{s_1}\partial _{x_{k_1}}d^{(0)}_{k_2})(D_i^{s_2}\partial _{x_{k_1}}\partial _{x_j}w^{(1)}_{k_2})$ in $F_j$. Thus, by (2.57) and (2.58), in a similar way as for getting (2.55) and (2.56), applying Gronwall's inequality to (2.59), we obtain

and

Noticing that $e^{-\psi(x_1)}$ is be a bounded function in $(0, +\infty)$, it follows from inequality (2.60) that

The proof is now complete.

2.3.   Existence of the first approximation step

Based on above a priori estimates, we are ready to prove the existence of the first approximation step by using semigroup theory arguments, see ^[36].

Proposition 2.1. If the domain is the unbounded domain $\mathbb R^+\times \mathbb R^2$, let the parameter $\lambda > 1$. If the domain is a bounded domain $\Omega$, let the parameter $\lambda = 1$ and ${\gamma\over{\varepsilon}^2} < 1$. Assume the initial approximation function $(u^{(0)}, d^{(0)})$ satisfying (2.4)–(2.8). Then the linearized system

and the boundary condition

admit a global solution

Moreover, the global solution satisfies

Proof. The idea of proof is based on the semigroup theory. We first rewrite it as an evolution equation, then by the semigroup generator given in ^[36], we know the evolution operator generate a semigroup, a local existence can been shown. It combine with the dissipative energy estimate given in Lemma 2.3, the global existence of linear evolution is obtained. More precisely, let $\mathbb P$ be the Leray projector onto the space of divergence free functions. We apply the Leray projector to Eq (2.12). Therefore

where

and

For convenience, we rewrite (2.62) as

We notice that the term $\left(\begin{array}{cccc} \mathbb N_1(\textbf{h}^{(1)}, \textbf{w}^{(1)})\\ \mathbb N_2(\textbf{h}^{(1)}, \textbf{w}^{(1)}) \end{array} \right)$ can be seen as a bounded perturbation of the linear operator $ML: = \left(\begin{array}{cccc} -\nu\lambda^2 \mathbb P\triangle & 0\\ 0 & -\gamma\lambda^2 \mathbb P\triangle \end{array} \right)$. We follow the idea of ^[36,37] to show that the linear operator

can generate a strongly continuous semigroup $e^{ \mathbb Z \tau}$ in Sobolev space $H^s(\Omega)\times H^s(\Omega)$ by the Lumer-Phillips theorem ^[38]. Hence linear Eq (2.60) has a solution in $\mathbb C((0, +\infty); H^s(\Omega)\times H^s(\Omega))$. Furthermore, it follows from Lemmas 2.2 and 2.3 that the estimate (2.61) holds.

3.   The $m$th approximation step

Let ${\varepsilon}\in(0, 1)$ be a fixed constant. We define

with the integer $2\leq k\leq m-1$ and the constant $s\geq1$.

Assume that the $m$-th approximation solutions of (2.3) is denoted by $(\textbf{h}^{(m)}(t, \lambda x), \textbf{w}^{(m)}(t, \lambda x))$ with $m = 2, 3, \ldots$. Let

it holds

We linearize the nonlinear Eq (2.3) around $(u^{(m-1)}(t, \lambda x), d^{(m-1)}(t, \lambda x))$ to get the following initial value problem

with the boundary conditions

where the error term is

and

and

This is also the nonlinear term in the approximation problem (2.3) at $(u^{(m-1)}(t, \lambda x), d^{(m-1)}(t, \lambda x))$. Here the approximation pressure satisfies

The following result describes how to construct the $m$-th approximation solution.

Proposition 3.1. If the domain is the unbounded domain $\mathbb R^+\times \mathbb R^2$, let the parameter $\lambda > 1$. If the domain is a bounded domain $\Omega$, let the parameter $\lambda = 1$ and ${\gamma\over{\varepsilon}^2} < 1$. Assume the initial approximation function $(u^{(0)}, d^{(0)})$ satisfying (2.4)–(2.8). $(u^{(m-1)}, d^{(m-1)})\in \mathcal B_{{\varepsilon}}$ and $\sum_{i = 1}^{m-1}\Big(\|\textbf{h}^{(i)}\|^2_{H^s}+\|\textbf{w}^{(i)}\|^2_{H^s}\Big)\lesssim\delta^2$. Then the linearized problem (3.2) with the boundary condition (3.3) admits a global solution

which satisfies

where the error term verifies

Proof. By direct computation we find

By the assumption

for a fixed constant $\lambda > 1$, it follows that

Thus, noticing that $(u^{(0)}(t, x), d^{(0)}(t, x))$ satisfies (2.4)–(2.8), by small modification of $\partial _{x_i}u^{(0)}_j(t, x)$ and $\partial _{x_i}d^{(0)}_j(t, x))$, it follows that

and

Moreover, we notice that the $(m-1)$-th approximation solution is

and

We obtain

and

Then we will find the $m$-th $(m\geq2)$ approximation solution $(u^{(m)}(t, \lambda x), d^{(m)}(t, \lambda x))$, which is equivalent to find $(\textbf{h}^{(m)}, \textbf{w}^{(m)})$ such that

Substituting (3.12) into (2.3), we have

Set

which we supplement with the boundary conditions (3.3).

Since we assume $(u^{(m-1)}(t, \lambda x), d^{(m-1)}(t, \lambda x))\in \mathcal B_{{\varepsilon}}$, there is the same structure between the linear system (2.12) and the linear system of $m$th approximation solutions. Thus, by the same arguments as in the proof of Proposition 2.1, we can show that above problem admits a solution $(\textbf{h}^{(m)}, \textbf{w}^{(m)})\in H^s(\Omega)\times H^s(\Omega)$. Here, we should use (2.2). Furthermore, similar to (2.61), we can use (3.9) and (3.10) to derive

where one can see the $(m-1)$-th error term $(E^{(m-1)}, \overline{E}^{(m-1)})$ such that

Moreover, by (3.5) and (3.6) and the standard Calderon-Zygmund theory, it follows that

The proof is now complete.

4.   Convergence of the approximation scheme

Our target is to prove that the nonlinear Eq (1.1) admits global solutions

This is equivalent to show that the series

For a fixed constant $s > 1$, let $1 < s = \bar{k} < k_0\leq k$ and

which yields

Proposition 4.1. If the domain is the unbounded domain $\mathbb R^+\times \mathbb R^2$, let the parameter $\lambda > 1$. If the domain is a bounded domain $\Omega$, let the parameter $\lambda = 1$ and ${\gamma\over{\varepsilon}^2} < 1$. Assume the initial approximation function $(u^{(0)}, d^{(0)})$ satisfying (2.4)–(2.8). Then the three dimension nematic liquid crystal flow (1.1) with the small initial data (1.4) and the boundary condition (1.5) admit a global Sobolev solution

Moreover, it holds the estimate

Proof. The proof is based on the induction. For convenience, we first deal with the case of zero initial data, that is, $u_0(x) = \Big(0, 0, 0\Big)^T$ and $d_0(x) = \Big(0, 0, 0\Big)^T$. After that, we discuss the case of non-zero initial data. Note that $N_m = N_0^m$ with $N_0 > 1$. For all $m = 1, 2, \ldots$, we claim that there exists a sufficient small positive constant $\delta$ such that

For the case of $m = 1$, we recall the assumptions (2.4)–(2.8) on the initial approximation function $(u^{(0)}(t, x), d^{(0)}(t, x))$. By (2.61), letting $0 < \delta_0 < N_0^{-(8+k-\bar{k})}\delta^2\ll1$, we obtain

Moreover, by (3.8) and the above estimate,

and

which means that $(u^{(1)}, d^{(1)})\in \mathcal B_{\delta}$.

Assume that the case of $m-1$ holds, that is,

Then we prove that the case of $m$ holds. Using (2.2), (3.7) and the second inequality of (4.3), we derive

which combined with (2.2), (3.8) and (4.1) yields

We choose a sufficiently small positive constant $\delta_0$ such that

Thus, by (4.5) we have

and

So, the error term goes to $0$ as $m\rightarrow \infty$, that is,

On the other hand, note that $N_m = N_0^m$, by (4.3) and (4.4). It follows that

This means that $(u^{(m)}, d^{(m)})\in \mathcal B_{\delta}$. Hence we conclude that (4.2) holds.

Therefore, the three dimension nematic liquid crystal flow (1.1) with the zero initial data admit global solutions

Next, we discuss the case of small non-zero initial data.

We introduce the auxiliary function

Thus, the initial data reduces to

and the boundary condition (1.5) is changed into the Dirichlet boundary condition

and Eq (1.1) are transformed into equations of $(\overline{u}, \overline{d})$.

Thus, we can follow the above iteration scheme to construct a global Sobolev solution $(\overline{u}, \overline{d})$. Furthermore, the global Sobolev solution of Eq (1.1) with a small non-zero initial data takes the form $(\overline{u}(t, x)+u_0, \overline{d}(t, x)+d_0)$. We notice that by the auxiliary function, the new first equation contain the non-autonomous force:

and the new second equation contain the non-autonomous force:

Thus, we have to require the initial data $(u_0, d_0)\in H^{s+2}(\Omega)\times H^{s+2}(\Omega)$ to be small and $\nabla\cdot u_0 = 0$.

Finally, using (1.2), by the standard Calderon-Zygmund theory for the Riesz operator $\mathcal R$, we have $\| \mathcal R u\|_{ \mathbb L^{s_0}}\leq\|u\|_{ \mathbb L^{s_0}}$ with $1 < s_0 < \infty$, hence

This completes the proof.

Conflict of interest

The authors declare that they have no competing interests.