\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 173, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/173\hfil Regularity of solutions]
{Regularity of solutions to 3-D nematic liquid crystal flows}

\author[Q. Liu, S. Cui \hfil EJDE-2010/173\hfilneg]
{Qiao Liu, Shangbin Cui} 

\address{Qiao Liu \newline
Department of Mathematics, Sun Yat-sen University, Guangzhou,
Guangdong 510275, China} 
\email{liuqao2005@lzu.cn, liuqao2005@163.com}

\address{Shangbin Cui \newline
Department of Mathematics, Sun Yat-sen
University, Guangzhou, Guangdong 510275, China}
\email{cuisb3@yahoo.com.cn}

\thanks{Submitted November 4, 2010. Published December 6, 2010.}
\thanks{Supported by grant 10771223 from  the National Natural
 Science Foundation of China}
\subjclass[2000]{76A15, 35B65, 35Q35}
\keywords{Liquid crystal flow; initial value problem;
 regularity of solutions}

\begin{abstract}
 In this note we consider the regularity of solutions to 3-D
 nematic liquid crystal flows, we prove that if either
 $u\in L^{q}(0,T;L^p(\mathbb{R}^3))$,
 $\frac{2}{q}+\frac{3}{p}\leq1$,  $3<p\leq\infty$;
 or $u\in L^{\alpha}(0,T;L^{\beta}(\mathbb{R}^3))$,
 $\frac{2}{\alpha}+\frac{3}{\beta}\leq 2$,
 $\frac{3}{2}< \beta\leq\infty$, then the solution $(u,d)$
 is regular on $(0,T]$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]

\section{Introduction}\label{intro}

In this note we study the following hydrodynamical systems
modelling the flow of nematic liquid crystal material
\cite{LL,LL2}:
\begin{gather}
u_{t}-\nu\Delta u +(u\cdot\nabla)u+\nabla{P}
=-\lambda\nabla\cdot(\nabla d \odot\nabla d),\quad
\text{in }\mathbb{R}^3\times (0,\infty), \label{e1.1}\\
d_{t}+(u\cdot\nabla)d=\gamma(\Delta d-f(d)),\quad
\text{in }\mathbb{R}^3\times (0,\infty), \label{e1.2}\\
\operatorname{div}u=0,\quad\text{in }\mathbb{R}^3\times (0,\infty),
\label{e1.3}\\
(u,d)|_{t=0}=(u_0,d_0)\quad\text{in }\mathbb{R}^3. \label{e1.4}
\end{gather}
Here $u=u(x,t)=(u_1(x,t),u_2(x,t),u_3(x,t))$ is the velocity
field of the flow, $d=d(x,t)=(d_1(x,t),d_2(x,t),d_3(x,t))$ is
the (averaged) macroscopic/continuum molecule direction, $P=P(x,t)$
is a scalar function representing the pressure, $\nu,\lambda,\gamma$
are positive constants, and
$f(d)=\frac{1}{\epsilon^2}(|d|^2-1)d$.
The term $\nabla d\odot\nabla d$ denotes the $3\times 3$ matrix
whose $(i,j)$-th entry is equal to $\partial_{i}d\cdot
\partial_{j}d$ (for $1\leq i,j\leq 3$). For simplicity, we assume
that $\nu=\lambda=\gamma=\epsilon=1$ throughout this paper.

The above system is a simplified version of the Ericksen-Leslie
model (see \cite{LL}) which retains many essential features of the
hydrodynamic equations for nematic liquid crystal. The existence of
global-in-time weak solutions and local-in-time classical solutions
for this system have been established by Lin and Liu \cite{LL}.
Later, in \cite{LL2}, they also proved that the one dimensional
spacetime Hausdorff measure of the singular set of the ``suitable''
weak solutions is zero. Recently, Zhou and Fan in \cite{ZF} proved a
regularity criterion for another system of partial differential
equations modelling nematic liquid crystal flows, which is considered
by Sun and Liu \cite{SL} and is similar to \eqref{e1.1}--\eqref{e1.4}; their
result says that if the local solution $(u,b)$ satisfies
\[
\int_0^{T}\frac{\|\nabla u\|_{p}^{r}}{1+ln(e+\|\nabla
u\|_{p})}\,d t<\infty\quad \text{with}\quad
\frac{2}{r}+\frac{3}{p}=2,\; 2\leq p\leq 3,
\]
then $(u,d)$ is regular on $(0,T]$.

We notice that if $d\equiv 0$, then the system \eqref{e1.1}--\eqref{e1.4} becomes
to the Navier-Stokes equations. There have been a lot of works on
regularity criteria of the solution to the 3-D Navier-Stokes
equations. The following results in this direction are well-known:
If one of the following two conditions holds
\begin{itemize}
\item[(1)] $u\in L^{q}(0,T;L^p)$ for
$\frac{2}{q}+\frac{3}{p}\leq 1$  and $3<p\leq\infty$;
\item[(2)] $\nabla u\in L^{\alpha}(0,T;L^{\beta})$
for $\frac{2}{\alpha}+\frac{3}{\beta}\leq 2$  and
$\frac{3}{2}<\beta\leq\infty$,
\end{itemize}
then the solution to the 3-D  Navier-Stokes equations is regular
\cite{BV,YG,HX,JS}. In this note we
want to show that the above regularity criteria still hold for the
nematic liquid crystal flow \eqref{e1.1}--\eqref{e1.4}. More precisely, we have
the following results:


\begin{theorem}\label{thm1.1}
Let $(u_0,d_0)\in H^{1}(\mathbb{R}^3)\times H^2(\mathbb{R}^3)$
with $\operatorname{div}u_0=0$. Suppose that $(u,d)$ is a local
smooth solution of the liquid crystal flow \eqref{e1.1}--\eqref{e1.4} on the
time interval $[0,T)$ associate with the initial value
$(u_0,d_0)$. Assume that one of the following two conditions holds
\begin{itemize}
\item[(a)] $u\in L^{q}(0,T;L^p(\mathbb{R}^3))$, for
$\frac{2}{q}+\frac{3}{p}\leq1$  with $3< p\leq\infty$;

\item[(b)] $\nabla u\in L^{\alpha}(0,T;L^{\beta}(\mathbb{R}^3))$,
 for $\frac{2}{\alpha}+\frac{3}{\beta}\leq 2$  with
 $\frac{3}{2}<\beta\leq\infty$.
\end{itemize}
Then $(u,d)$ can be extended beyond $T$.
\end{theorem}

We shall give the proof of this result in the following section.
As usual, we use the notation $C$ to denote a ``generic''
constant which may change from line to line, and use
$\|\cdot\|_{p}$ to denote the norm of the Lebesgue space $L^p$.

\section{Proof of Theorem \ref{thm1.1}}

Assume that $[0,T_{max})$ is the maximal interval of the existence
of local smooth solution. To conclude our proof, we only need to
show that $T<T_{max}$. Arguing by contradiction, we assume that
$T_{max}\leq T$, and either (a) or (b) holds. If we can establish
the estimate
\begin{equation} \label{e2.1}
\lim_{t\to T^{-}}(\|\nabla u\|_2+\|\Delta d\|_2)<\infty,
\end{equation}
then $[0,T)$ is not a maximal interval of the existence of solution,
which leads to an contradiction.

We multiply \eqref{e1.1} by $u$ and integrate over $\mathbb{R}^3$, and
multiply \eqref{e1.2} by $-\Delta d+f(d)$ and integrate over
$\mathbb{R}^3$. By adding the two results above, we obtain
\begin{align}\label{e2.2}
\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^3}(|u|^2+|\nabla
d|^2+2F(d))\,d x+\int_{\mathbb{R}^3}(|\nabla u|^2+|\Delta
d-f(d)|^2)\,d x=0,
\end{align}
where $F(d)$ is the  primitive function of $f(d)$; i.e.,
$F(d)=\frac{|d|^{4}}{4}-\frac{|d|^2}{2}$. Here we used the
condition $\operatorname{div} u=0$ and the fact that
\[
((u\cdot\nabla) u,u)=(u,\nabla P)=((u\cdot \nabla) d,f(d))=(u,\nabla
\frac{|\nabla d|^2}{2})=0.
\]
Hence
\begin{equation} \label{e2.3}
\|u\|_{L^{\infty}(0,T;L^2)}+\|u\|_{L^2(0,T;H^{1})}\leq C.
\end{equation}
Multiply \eqref{e1.2} by $|d|^2d$ and integrate by parts yields
\[
\frac{1}{4}\frac{d}{dt}\int_{\mathbb{R}^3}|d|^{4}(t,x)
\,d x+\int_{\mathbb{R}^3}(3d^2|\nabla
d|^2+|d|^{6})(t,x)\,d x
=\int_{\mathbb{R}^3}|d|^{4}(t,x)\,d x,
\]
which implies
\begin{equation} \label{e2.4}
\|d (t,\cdot)\|_{L^{\infty}(0,T;L^{4})}
+\int_0^{t}\int_{\mathbb{R}^3}(3d^2|\nabla
d|^2+|d|^{6})(\tau,x)\,d x\,d \tau\leq C.
\end{equation}
Multiply \eqref{e1.2} by $f(d)$ and integrate by parts,
we obtain
\begin{equation} \label{e2.5}
\frac{d}{dt}\int_{\mathbb{R}^3}F(d)(t,x)\,d x
=\int_{\mathbb{R}^3}(\Delta d  f(d)-|f(d)|^2)(t,x)\,d x.
\end{equation}
By \eqref{e2.2}--\eqref{e2.5}, the Gronwall's inequality and
the fact $f(d)=(|d|^2-1)d$, we obtain
\begin{equation} \label{e2.6}
\|d\|_{L^{\infty}(0,T;H^{1})}+\|d\|_{L^2(0,T;H^2)}\leq C.
\end{equation}


Noticing that the $i$-th ($i$=1,2,3) component of $u$ satisfies
\begin{equation} \label{e2.7}
\partial_{t}u_{i}+(u\cdot\nabla)u_{i}-\Delta
u_{i}+\partial_{i}P=-\sum_{j=1}^3\partial_{j}
\Big(\sum_{k=1}^3\partial_{i}d_{k}\partial_{j}d_{k}\Big).
\end{equation}
Multiplying \eqref{e2.7} by $-\Delta u_{i}$, summing over $i$, using
integration by parts, and noting that $\operatorname{div} u=0$,
 we obtain
\begin{equation} \label{e2.8}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^3}|\nabla u|^2\,d x
+\int_{\mathbb{R}^3}|\Delta u|^2\,d x \\
&=\sum_{i=1}^3\int_{\mathbb{R}^3}(u\cdot\nabla) u_{i} \Delta
u_{i}\,d x-\sum_{i,k=1}^3\int_{\mathbb{R}^3}\partial_{i}
d_{k}\Delta d_{k}\Delta u_{i}\,d x.
\end{aligned}
\end{equation}
Applying $\Delta$ to both sides of \eqref{e1.2}, multiplying
them with $\Delta d$, and using \eqref{e1.3}, we obtain
\begin{equation} \label{e2.9}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^3}|\Delta
d|^2\,d x+\int_{\mathbb{R}^3}\left(|\nabla \Delta
d|^2+\Delta f(d) \Delta d\right)\,d x \\
&=\sum_{i,k}^3\int_{\mathbb{R}^3}\Delta
u_{i}\partial_{i}d_{k}\Delta
d_{k}\,d x-2\sum_{i,k=1}^3\int_{\mathbb{R}^3}\nabla
u_{i}\partial_{i}\nabla d_{k}\Delta d_{k}\,d x,
\end{aligned}
\end{equation}
where we used the condition $\operatorname{div} u=0$.
Putting \eqref{e2.8} and \eqref{e2.9} together, we obtain
\begin{equation} \label{e2.10}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^3}\left(|\nabla
u|^2+|\Delta d|^2\right)\,d x+\int_{\mathbb{R}^3}(|\Delta
u|^2+|\nabla \Delta d|^2)\,d x \\
&= \sum_{i=1}^3\int_{\mathbb{R}^3} (u\cdot \nabla)u_{i}\Delta
u_{i}\,d x-2\sum_{i,k=1}^3\int_{\mathbb{R}^3}\nabla
u_{i}\partial_{i}\nabla d_{k}\Delta
d_{k}\,d x-\int_{\mathbb{R}^3}\Delta
f(d)\Delta d\,d x \\
&=: I_1+I_2+I_3.
\end{aligned}
\end{equation}

Now, we first consider the case that the smooth solution $(u,d)$
satisfies the condition (a). For $I_1$, we can do estimates for it
as
\begin{equation} \label{e2.11}
\begin{aligned}
I_1&\leq  C\|u\|_{{p}}\|\nabla u\|_{{\frac{2p}{p-2}}}\|\Delta
u\|_2\quad\text{(H\"{o}lder's inequality)} \\
&\leq  C\| u\|_{{p}}\|\nabla u\|_{{2}}^{\frac{p-3}{p}}\|\Delta
u\|_{{2}}^{1+\frac{3}{p}}\quad\text{(Gagliardo-Nirenberg
inequality)} \\
&\leq  \frac{1}{2}\|\Delta u\|_2^2+C\|
u\|_{p}^{\frac{2p}{p-3}}\|\nabla u\|_2^2\quad \text{(Young
inequality)}.
\end{aligned}
\end{equation}
Similarly, we can estimate $I_2$ and $I_3$ as
\begin{equation} \label{e2.12}
\begin{aligned}
I_2&= 2\int_{\mathbb{R}^3}u_{i}\partial_{i}\nabla d_{k}\nabla
\Delta d_{k}\,d x \\
&\leq  C\|u\|_{p}\|\nabla^2d\|_{\frac{2p}{p-2}}\|\Delta d\|_2\quad\text{ (H\"{o}lder's Inequality)} \\
&\leq  C\| u\|_{{p}}\|\nabla^2
d\|_{{2}}^{\frac{p-3}{p}}\|\nabla\Delta
d\|_{{2}}^{1+\frac{3}{p}}\quad\text{(Gagliardo-Nirenberg
inequality)} \\
&\leq  \frac{1}{4}\|\nabla\Delta d\|_2^2+C\|
u\|_{p}^{\frac{2p}{p-3}}\|\Delta d\|_2^2\quad \text{(Young
inequality)},
\end{aligned}
\end{equation}
\begin{equation} \label{e2.13}
\begin{aligned}
I_3&= \sum_{i=1}^3\int_{\mathbb{R}^3}\partial_{i}[(|d|^2-1)d]\partial_{i}\Delta
d\,d x \\
&= \sum_{i=1}^33\int_{\mathbb{R}^3}\partial_{i}d\partial_{i}\Delta
d|d|^2\,d x-\sum_{i=1}^3\int_{\mathbb{R}^3}\partial_{i}d\partial_{i}\Delta
d\,d x \\
&\leq  C \|\nabla d\|_{6}\|\nabla \Delta
d\|_2\|d\|_{6}^2+C\|\nabla d\|_2\|\nabla \Delta
d\|_2\quad\text{ (H\"{o}lder's inequality)} \\
&\leq  C\|\Delta d\|_2\|\nabla \Delta d\|_2\|\nabla
d\|_2^2+C\|\nabla d\|_2\|\nabla \Delta d\|_2\quad
\text{(Sobolev embedding)} \\
&\leq  \frac{1}{4}\|\nabla \Delta d\|_2^2+C(\|\nabla
d\|_2^2+\|\nabla d\|_2^2\|\Delta
d\|_2^2)\quad\text{(Young inequality)} \\
&\leq  \frac{1}{4}\|\nabla \Delta d\|_2^2+C\|\Delta
d\|_2^2+C.
\end{aligned}
\end{equation}
Substituting the above estimates \eqref{e2.11}--\eqref{e2.13}
into \eqref{e2.10}, we obtain
\begin{equation} \label{e2.14}
\begin{aligned}
&\frac{d}{dt}\int_{\mathbb{R}^3}\left(|\nabla u|^2+|\Delta
d|^2\right)\,d x+\int_{\mathbb{R}^3}(|\Delta
u|^2+|\nabla \Delta d|^2)\,d x \\
&\leq  C\| u\|_{p}^{\frac{2p}{p-3}}(\|\nabla u\|_2^2+\|\Delta
d\|_2^2)+C\|\Delta
d\|_2^2+C \\
&\leq C(1+\|u\|_{p}^{\frac{2p}{p-3}})(\|\nabla u\|_2^2+\|\Delta
d\|_2^2)+C.
\end{aligned}
\end{equation}
Hence, the Gronwall's inequality yields
\begin{equation} \label{e2.15}
\sup_{0\leq t\leq T}\{\|\nabla u\|_2^2+\|\Delta d\|_2^2\}
\leq C e^{CT}e^{\int_0^{T}\|u\|_{p}^{2p/(p-3)}\,d t}<
\infty.
\end{equation}

Next we consider the case that the smooth solution $(u,d)$ satisfies
the condition (b). We estimate $I_1$ as follows:
\begin{equation} \label{e2.16}
\begin{aligned}
I_1&= -\sum_{i,j=1}^3\int_{\mathbb{R}^3}
(\partial_{i}u\cdot \nabla)u_{i}\partial_{j}u_{i}\,d x \\
&\leq  C\|\nabla u\|_{{\beta}}\|\nabla u\|_{{\frac{2\beta}{\beta-1}}}^2\quad\text{ (H\"{o}lder's inequality)} \\
&\leq  C\|\nabla u\|_{{\beta}}\|\nabla
u\|_{{2}}^{\frac{2\beta-3}{\beta}}\|\Delta
u\|_{{2}}^{\frac{3}{\beta}}\quad\text{(Gagliardo-Nirenberg
inequality)} \\
&\leq  \frac{1}{2}\|\Delta u\|_2^2+C\|\nabla
u\|_{\beta}^{\frac{2\beta}{2\beta-3}}\|\nabla u\|_2^2\quad
\text{(Young inequality)}.
\end{aligned}
\end{equation}
Similarly, we can do estimates for $I_2$ as
\begin{equation} \label{e2.17}
\begin{aligned}
I_2
&\leq  C\|\nabla u\|_{\beta}\|\Delta
d\|_{\frac{2\beta}{\beta-1}}^2\quad\text{ (H\"{o}lder's inequality)} \\
&\leq  C\|\nabla u\|_{{\beta}}\|\Delta
d\|_{{2}}^{\frac{2\beta-3}{\beta}}\|\nabla\Delta
d\|_{{2}}^{\frac{3}{\beta}}\quad\text{(Gagliardo-Nirenberg
inequality)} \\
&\leq  \frac{1}{4}\|\nabla\Delta d\|_2^2+C\|\nabla
u\|_{\beta}^{\frac{2\beta}{2\beta-3}}\|\Delta d\|_2^2\quad
\text{(Young inequality)},
\end{aligned}
\end{equation}
and for $I_3$ as
\begin{equation} \label{e2.18}
I_3 \leq  \frac{1}{4}\|\nabla \Delta d\|_2^2+C\|\Delta d\|_2^2+C.
\end{equation}
Putting the above estimates for \eqref{e2.15}--\eqref{e2.18}
into \eqref{e2.10}, we
obtain
\begin{align*}
&\frac{d}{dt}\int_{\mathbb{R}^3}\left(|\nabla u|^2+|\Delta
d|^2\right)\,d x+\int_{\mathbb{R}^3}(|\Delta
u|^2+|\nabla \Delta d|^2)\,d x \\
&\leq C(1+\|\nabla u\|_{\beta}^{\frac{2\beta}{2\beta-3}})(\|\nabla
u\|_2^2+\|\Delta d\|_2^2)+C.
\end{align*}
Hence, the Gronwall's inequality yields
\begin{equation} \label{e2.19}
\sup_{0\leq t\leq T}\{\|\nabla u\|_2^2+\|\Delta d\|_2^2\}
\leq C e^{CT}e^{\int_0^{T}\|\nabla
u\|_{\beta}^{\frac{2\beta}{2\beta-3}}\,d t}< \infty.
\end{equation}
By \eqref{e2.15} and \eqref{e2.19}, we see that \eqref{e2.1} follows.
This proves Theorem \ref{thm1.1}.

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\end{document}