\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
Variational and Topological Methods:
Theory, Applications, Numerical Simulations, and Open Problems (2012).
{\em Electronic Journal of Differential Equations},
Conference 21 (2014),  pp. 149--172.
ISSN: 1072-6691.  http://ejde.math.txstate.edu,
http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document} \setcounter{page}{149}
\title[\hfilneg EJDE-2014/Conf/21 \hfil
 Hydrodynamic flow of nematic liquid crystals]
{A survey of results and open problems for the hydrodynamic flow
of nematic liquid crystals}

\author[J. L. Hineman \hfil EJDE-2014/Conf/21\hfilneg]
{Jay Lawrence Hineman}  % in alphabetical order

\address{Jay Lawrence Hineman \newline
Department of Mathematics\\
University of Kentucky \\
Lexington, KY 40506, USA}
\email{jay.hineman@uky.edu}

\thanks{Published February 10, 2014.}
\subjclass[2000]{76A15, 58E20}
\keywords{Liquid crystals; Ericksen-Leslie; harmonic maps}

\begin{abstract}
 The mathematical study of the hydrodynamic flow of nematic liquid
 crystals is a mix of material science, Navier-Stokes theory, and the
 study of harmonic maps. In this article, the basic model for the
 hydrodynamic flow of nematic liquid crystals is presented -- that is,
 the Ericksen-Leslie equations. There are a number of simplifications
 of the Ericksen-Leslie equations exist in the literature.
 We give a short survey of the known mathematical results for the
 Ericksen-Leslie equations and its simplifications. Along the way,
 open problems in analysis and computation for the Ericksen-Leslie
 system are discussed.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{claim}[theorem]{Claim}
\allowdisplaybreaks


\section{Introduction}

Liquid crystals and their mathematical description have a long and rich
history dating back over 100 years.
Liquid crystals can be thought of as materials that exhibit an intermediate
phase between liquid and solid. That is, while liquid
crystals may flow like liquids, they exhibit additional structural properties.

Many chemical compounds have liquid crystal phases. For example concentrated
solutions of rigid polymers in suitable solvents, DNA, and
certain viruses all exhibit liquid crystal phases. Since there are many
possible microscopic structures, there are accordingly many liquid
crystal phases, for example,  nematic, smectic, and cholesteric.

The simplest example of a liquid crystal phase occurring in nature is
the nematic phase for a single chemical species.
Physically, in the single-species case, liquid crystals that exhibit a
nematic phase, or nematic liquid crystals for short, are liquids
that are uniformly composed of rod-like molecules whose structure induces
a preferred average directional order.
A historical example is the compound MBBA,
\emph{N-(p-methoxybenzylidene)-p-butylaniline}, which is in the nematic
phase for approximately the
range temperatures $20^\circ C$ to $47^\circ C$ and whose length is on
the scale of Angstroms.

For the modeling of a single species of a nematic at a fixed temperature
one can consider a continuum theory which disregards
the individual molecular structure. Such a continuum assumption is valid
since the distance over which directional order occurs is
much larger than the molecular dimensions (that is proportional to $\mu m$
versus proportional to Angstroms).

For a more in depth and complete introduction to liquid crystals, their history,
and their mathematical study one should consult the texts of
Virga \cite{virga_variational_1995}, DeGennes and
Prost \cite{gennes_physics_1995}, and or Stewart \cite{stewart_static_2004}.

This survey we will focus the mathematical theory for the hydrodynamic
flow of nematic liquid crystals. Specifically we will examine the
Ericksen-Leslie equations for the hydrodynamic flow of nematic liquid
crystals and simplifications. We begin in the next section with an outline
of the derivation of the Ericksen-Leslie equations from balance laws.
 After this we discuss results for modifications and simplifications
 of the equations. The Ericksen-Leslie equations have three major
 facts that make their analysis difficult.

\begin{enumerate}
\item They are a strongly coupled system of nonlinear equations --
they contain a Navier-Stokes like system and a nonlinear system for
the transported heat flow of harmonic maps.

\item They contain physically relevant terms beyond those in the
standard Navier-Stokes equations and transported heat flow of harmonic maps.
\item There is a natural, point-wise nonlinear constraint on the system.
\end{enumerate}
The first fact cannot be avoided and it is noted that in general that
 mathematical results about the well-posedness of the Ericksen-Leslie
equations cannot surpass those for Navier-Stokes. Simplification of
the Ericksen-Leslie equations have been employed to deal with the
second and third facts. Specifically, the difficulties introduced by
the nonlinear constraint can be eliminated by analyzing an associated
penalized system. This is system is analyzed in detail in
Section \ref{penalizedSection}.

Recently, the author and others have examined simplifications of the
Ericksen-Leslie equations that retain the original constraint.
These works have still simplified equations and primarily examine a
strongly coupled system of the Navier-Stokes equations and the transported
heat flow of harmonic maps. This system, and its differences from the
penalized system are discussed the Sections \ref{nonPenalizedSection}
and \ref{newFindings}. The latter provides an outline of the recent work
by the author and C.Y. Wang on local well-posedness
\cite{hineman_well-posedness_2012}.

We record open problems for both the penalized and non-penalized
simplifications of the Ericksen-Leslie equations in both analysis
and numerical analysis. Each of these problem should be viewed as a
part of the over-arching problem of constructing a well-posedness
theory for the Ericksen-Leslie equations. For other directions and problems,
the interested reader is encourage to consult the surveys by Lin and
Liu \cite{lin_static_2001} and Lin \cite{lin_analytical_2012}.

Throughout this survey we will consider functions
$\mathbf{u}: \Omega \to \mathbb{R}^3$ (usually fluid velocity field)
and $\mathbf{d}: \Omega \to \mathbb{S}^2$ (usually the director or orientation
field) for $\Omega \subset \mathbb{R}^3$ and where $\mathbb{S}^2$
is the unit sphere in $\mathbb{R}^3$. We use the convention that upper
indices represent components and lower indices represent derivatives.
We also use the Einstein summation convention throughout -- that is,
we sum implicitly over any repeated index. Components are often cumbersome
and hence we will often use the common operators from tensor calculus
to provide concise expressions -- namely, we use $\cdot,:$ for the vector
and tensor inner-products respectively, and $\wedge$ for the outer or
cross product of vectors or tensors.

\section{Ericksen-Leslie equations}

The Ericksen-Leslie equations for the hydrodynamic flow of nematic
liquid crystals can be derived from the following conservation and
balance laws in the way that the Navier-Stokes equations is often derived
(for a derivation of the incompressible Navier-Stokes see
\cite{lions_mathematical_1996}, \cite{chorin_mathematical_1993}).
The Ericksen-Leslie equations were derived as result of extending the
earlier hydrostatic theory for liquid crystals given by Oseen, Zocher,
and Frank. In 1961, Ericksen \cite{ericksen_conservation_1961} provide
conservation laws for the hydrodynamic system for nematic liquid crystals.
The system was completed by Leslie \cite{leslie_constitutive_1968} in 1968 with
the addition of constitutive relations. For our development we assume the
following conservation and balance laws:
\begin{enumerate}
\item \emph{conservation of mass}
\begin{equation}
  \frac{D}{Dt} \int_\Omega \rho dV  = 0;  \label{nMass}
\end{equation}
\item \emph{balance of linear momentum}
\begin{equation}
\frac{D}{Dt} \int_\Omega \rho \mathbf{u} dV
= \int_{\Omega} \rho \mathbf{f}_{\rm body} dV + \int_{\partial \Omega}
\mathbf{f}_{\rm surf} dS; \label{nLinMoment}
\end{equation}
\item \emph{balance of angular momentum}
\begin{equation}
\frac{D}{Dt} \int_\Omega \rho (\mathbf{x} \wedge \mathbf{u})dV
= \int_\Omega \rho (\mathbf{x}\wedge \mathbf{f}_{\rm body}
+ \mathbf{k}) dV + \int_{\partial \Omega} (\mathbf{x} \wedge
\mathbf{f}_{\rm surf} + \mathbf{l}) dS. \label{nAngMoment}
\end{equation}
\end{enumerate}
Where, $\Omega \subset \mathbb{R}^3$, $\frac{D}{Dt}$ is material
or convective time derivative, and
\begin{gather*}
    \rho = \text{ density}, \quad
    \mathbf{x}  = \text{ position}, \\
    \mathbf{u}  = \text{ fluid velocity}, \\
    \mathbf{f}_{\rm body} = \text{ external body force}, \\
    \mathbf{f}_{\rm surf} = \text{ surface force (stress)}, \\
    \mathbf{k} = \text{ external body moment, and} \\
    \mathbf{l} = \text{ surface moment}.
\end{gather*}
If $\rho$ is constant, the Reynolds transport theorem implies that
\begin{equation}
  \label{pw:incompress}
    \nabla \cdot \mathbf{u} = 0 \quad\text{(incompressibility).}
\end{equation}
The application of the Reynolds transport theorem in the remaining
integral balance laws yields:
\begin{gather}
  \rho \frac{Du^i}{Dt} = \rho f_{\rm body}^{i} + \nabla_j F^{ij}
\quad \text{(point-wise balance of linear momentum)} \label{pw:linMomentBal} \\
  \rho K^i + \epsilon^{ijk} F^{kj} + \nabla_j L^{ij} = 0 \quad
\text{(point-wise balance of angular momentum)} \label{pw:angMomentBal}
\end{gather}
where
\begin{equation*}
  f^i_{\rm surf} = F^{ij} \nu^j \text{ and } l^i = L^{ij}\nu^j, \quad
 \nu= \text{normal to } \partial \Omega
\end{equation*}
and $\mathbf{F} = F^{ij}$ is the \emph{stress tensor} and
$\mathbf{L} = L^{ij}$ is the \emph{couple stress tensor}.

The equations giving mass conservation (incompressiblity), balance of
linear momentum, and balance of angular momentum can be
thought of as kinematic equations
(equations \eqref{pw:incompress}-\eqref{pw:angMomentBal} respectively).
That is, they are equations that describe motion, but they do not explain
the origin of the forces generating the motion.
Dynamic equations would also include the constitutive relations for the
material that describe the origins
of forces. One constitutive hypothesis that can be made for the derivation
of the Ericksen-Leslie equations is a
\emph{rate of work} assumption. Namely,
\begin{equation}  \label{RateOfWork:1}
  \begin{aligned}
    &\int_\Omega \rho (\mathbf{f}_{\rm body} \cdot \mathbf{u} + \mathbf{k} \cdot \hat{\mathbf{w}}) dV
    + \int_{\partial \Omega} (\mathbf{f}_{\rm surf} \cdot \mathbf{u} + \mathbf{l}\cdot \hat{\mathbf{w}}) dS \\
    &= \frac{D}{Dt} \int_\Omega \Big(\frac{1}{2} \rho \mathbf{u} \cdot \mathbf{u}
 + \sigma_F \Big) dV + \int_\Omega \mathcal{D} dV
  \end{aligned}
\end{equation}
where $\mathcal{D}$ is the rate of viscous dissipation per unit volume,
$\sigma_F$ is the Oseen-Zocher-Frank elastic energy density, and
$\hat{\mathbf{w}}$ is the axial vector
(representing \emph{local angular velocity}). The Oseen-Zocher-Frank elastic
energy density is assumed to be \emph{frame-indifferent},
\emph{material-symmetric}, \emph{even}, and \emph{positive definite}
 (see \cite{virga_variational_1995} or \cite{stewart_static_2004}).
These assumptions lead to \emph{Frank's Formula}
(see \cite{virga_variational_1995}) for $\sigma_F$ namely:
\begin{equation}   \label{franksFormula}
  \begin{aligned}
    \sigma_F(\mathbf{d}, \nabla \mathbf{d})
    &= k_1(\operatorname{div} \mathbf{d})^2 + k_2(\mathbf{d}
 \cdot \operatorname{curl} \mathbf{d})^2 + k_3|\mathbf{d}
 \wedge \operatorname{curl} \mathbf{d}|^2  \\
    &\quad + (k_2 + k_4)(\operatorname{tr}(\nabla \mathbf{d})^2
 - (\operatorname{div} \mathbf{d})^2).
  \end{aligned}
\end{equation}
Using the point-wise balance of linear momentum \eqref{pw:linMomentBal}
and point-wise balance of angular momentum \eqref{pw:angMomentBal} one may
simplify the rate of work assumption \eqref{RateOfWork:1} through the
application of the Reynolds transport theorem to find the point-wise relation:
\begin{equation}  \label{ptRateOfWork}
  F^{ij}\nabla_ju^j + L^{ij}\nabla_jw^i - w^i \epsilon^{ijk} F^{kj}_s
= \frac{D\sigma_F}{Dt} + \mathcal{D}.
\end{equation}
Here, $\mathbf{w} = \hat{\mathbf{w}} - \frac12 \nabla \wedge \mathbf{u}$
is the \emph{relative angular velocity}.

What remains is to find specific forms of the stress tensor $F_s^{ij}$
and the couple stress tensor $L^{ij}$. This is
done using Ericksen's identity \cite{ericksen_conservation_1961}:
\begin{equation}  \label{EricksenId}
  \epsilon^{ijk} \Big(d^j \frac{\partial \sigma_F}{\partial d^k}
 + \nabla_p d^j \frac{\partial \sigma_F}{\partial \nabla_p d^k}
    +\nabla_p d^j \frac{\partial \sigma_F}{\partial \nabla_k d^p} \Big) = 0.
\end{equation}
One finds that the stress tensor $F^{ij}$ and the couple stress tensor
$L^{ij}$ may take the forms
\begin{equation}
  \label{dynStressAndCouple}
  \begin{gathered}
    F^{ij} = -p \delta^{ij} - \frac{\partial \sigma_F}{\partial \nabla_j d^p}
\nabla_i d^p + \tilde{F}^{ij} \\
    L^{ij} = \epsilon^{iqp} d^q \frac{\partial \sigma_F}{\partial \nabla_j d^p}
+ \tilde{L}^{ij}.
  \end{gathered}
\end{equation}
Here, $p$ is the pressure arising from incompressibility and $\tilde{F}^{ij}$
and $\tilde{L}^{ij}$ are the
possible dynamic contributions. Thus, one has the following expression for
$\mathcal{D}$:
\begin{equation}  \label{dissipationIneq}
  \tilde{F}^{ij} \nabla_j u^i + \tilde{L}^{ij} \nabla_j \hat{w}^i
+ \hat{w}^i \epsilon^{ijk} \tilde{F}^{kj} = \mathcal{D}.
\end{equation}

To find constitutive relations for the dynamic terms $\tilde{F}^{ij}$,
$\tilde{L}^{ij}$ it is assumed that the
dissipation $\mathcal{D}$ is positive.  That is,
\begin{equation*}
  \tilde{F}^{ij} \nabla_j u^i + \tilde{L}^{ij} \nabla_j \hat{w}^i
+ \hat{w}^i \epsilon^{ijk} \tilde{F}^{kj} = \mathcal{D} \geq 0.
\end{equation*}
This assumption along with material-frame indifference, nematic symmetry,
and linear dependence on $d^i$, $N^i$ and $S^{ij}$ leads to
\begin{equation}  \label{leslieStress}
  \begin{gathered}
    \tilde{L}^{ij} = 0 \\
    \tilde{F}^{ij}
    = \alpha_1 d^k S^{kp}d^p d^i d^j + \alpha_2 N^id^j + \alpha_3 d^iN^j
    + \alpha_4 S^{ij}
    + \alpha_5 d^j S^{ik} d^k + \alpha_6 d^i S^{jk} d^k.
  \end{gathered}
\end{equation}
Where $\alpha_1, \ldots, \alpha_6$ are the Leslie viscosities and
\begin{gather*}
    S^{ij} = \frac{1}{2}(u_j^i + u_i^j) =  \text{ rate of strain tensor,} \\
    W^{ij} = \frac{1}{2}(u_j^i - u_i^j) =  \text{ vorticity tensor,} \\
    N^i = \frac{Dd^i}{Dt} - W^{ij}d^j = \text{ co-rotational time flux of
the director } \mathbf{d}.
\end{gather*}
Further manipulations, including a second application of Ericksen's identity,
and the fact that $d^id_j^i = 0$ lead to the full (isothermal)
Ericksen-Leslie equations that we summarize now.
\medskip

\noindent {\bf Summary of the Ericksen-Leslie equations:}
 We may summarize the Ericksen-Leslie equations in component form as
\begin{equation}  \label{el:pw:summary}
  \begin{gathered}
 \rho \frac{Du^i}{Dt} = \rho F^i - (p + \sigma_F)_i + \tilde{g}^jd_i^j
 + G_j d_i^j + \tilde{F}_j^{ij} \\
  u_i^i = 0 \\
   \Big( \frac{\partial \sigma_F}{\partial d^i_j} \Big)_j
- \frac{\partial \sigma_F}{\partial d^i} + \tilde{g}^i + G^i = \lambda d^i \\
    d^id^i = 1.
  \end{gathered}
\end{equation}
with constitutive relations for the viscous/dynamic stress tensor
$\tilde{F}^{ij}$ and vector $\tilde{g}^i$ given by
\begin{equation}  \label{dynamicStressAndVector}
  \begin{gathered}
    \tilde{t}^{ij}
    = \alpha_1 d^k S^{kp}d^p d^i d^j + \alpha_2 N^id^j + \alpha_3 d^iN^j
    + \alpha_4 S^{ij}
    + \alpha_5 d^j S^{ik} d^k + \alpha_6 d^i S^{jk} d^k \\
    \tilde{g}^i = -\gamma_1 N^i - \gamma_2 A^{ip} d^p.
  \end{gathered}
\end{equation}
and
\begin{equation}
  \begin{gathered}
    f^i_{\rm body} = \text{external body for per unit mass}, \\
    G^i = \text{generalized body force}, \\
    \rho = \text{density}, \quad
    p = \text{pressure},\\
    \sigma_F = \text{Oseen-Zocher-Frank elastic energy density for nematics,}\\
    \lambda = \text{Lagrange multiplier corresponding to the constraint } d^id^i = 1, \\
    \gamma_1  = \alpha_3 - \alpha_2 \geq 0 \\
    \gamma_2  = \alpha_3 + \alpha_2 = \alpha_6 -\alpha_5 = \text{Parodi's relation,} \\
    S^{ij} = \frac{1}{2}(u_j^i + u_i^j) = \text{ rate of strain tensor,} \\
    W^{ij} = \frac{1}{2}(u_j^i - u_i^j) =  \text{ vorticity tensor}, \\
    N^i = \frac{Dd^i}{Dt} - W^{ij}d^j = \text{ co-rotational time flux of the director } \mathbf{d}.
  \end{gathered}
\end{equation}

\section{Penalized simplifications of the Ericksen-Leslie equations}
\label{penalizedSection}

Intuitively, based upon the similarities between isotropic fluid flow modeled
by the Navier-Stokes equation and the Ericksen-Leslie equations, early
simplifications considered the Ericksen-Leslie equations as a system where
the microscopic alignment is governed by
the transported heat flow of harmonic maps into the sphere and the macroscopic
fluid behavior is governed by the Navier-Stokes equations.
This intuitive picture first appeared in the survey by
 Lin \cite{lin_nonlinear_1989} and was in explored in detail by Lin and Liu in
\cite{lin_nonparabolic_1995}. Lin and Liu in \cite{lin_nonparabolic_1995}
gave a particular simplification of the Ericksen-Leslie
equations inspired by the then recent work of Chen and Struwe
\cite{chen_weak_1989}, \cite{chen_existence_1989} on the heat flow harmonic maps.
They  provided a formulation of the Ericksen-Leslie equations that avoided the
constraint that the director must have unit length. Specifically they studied
the system
\begin{gather}
 \mathbf{u}_t + (\mathbf{u} \cdot \nabla) \mathbf{u}
 = \nu \Delta \mathbf{u} - \nabla \cdot (p\mathbb{I}
+ \lambda \nabla \mathbf{d} \odot \nabla \mathbf{d}) \label{LL:momentum} \\
 \nabla \cdot \mathbf{u} = 0 \label{LL:divfree} \\
 \mathbf{d}_t + (\mathbf{u} \cdot \nabla) \mathbf{d}
 = \gamma(\Delta \mathbf{d} - f(\mathbf{d})) \label{LL:director}.
\end{gather}
in $\Omega \times (0,\infty)$ for $\Omega \subset \mathbb{R}^m$, $m =2,3$.
Here $\nabla \mathbf{d} \odot \nabla \mathbf{d} = d_j^id_k^i$.
To arrive at this system:
\begin{enumerate}
\item The elastic constants in Oseen-Zocher-Frank elastic energy density
are set as $k_1 = k_2 = k_3 = 1$ and $k_4 = 0$. This reduces the
  Oseen-Zocher-Frank elastic energy to the simpler Dirichlet energy
$\frac12 \int_\Omega |\nabla \mathbf{d}|^2$
\item The sphere constraint $d^id^i = 1$ is handled via the penalization
$F(\mathbf{d}) = \frac{1}{4\epsilon^2}(|\mathbf{d}|^2 - 1)^2$. That is,
  the Oseen-Zocher-Frank elastic energy is replaced by the functional
$\int_\Omega \frac12|\nabla \mathbf{d}|^2
+ \frac{1}{4\epsilon^2}(|\mathbf{d}|^2 - 1)^2$.
\item It is assumed for Leslie viscosities $\alpha_1,\ldots,\alpha_6$
in~\eqref{dynamicStressAndVector} that $\alpha_1 = \alpha_5 = \alpha_6 = 0$.
  Hence the Ericksen-Leslie equations are reduced to equations having only
three constants $\nu$, $\gamma$, and $\lambda$
\item Body force terms $\mathbf{f}_{\rm body}$, $G^i$ and other terms are
removed.
\end{enumerate}

For the system \eqref{LL:momentum}-\eqref{LL:director}, Lin and Liu
demonstrated in \cite{lin_nonparabolic_1995} the following:

\begin{theorem}[Existence global weak solutions]
  Under the assumptions that \sloppy{$\mathbf{u}_0(x) \in L^2(\Omega)$}
and ${\mathbf{d}_0(x) \in H^1(\Omega)}$ with
  ${\mathbf{d}_0|_{\partial \Omega} \in H^{3/2}(\partial \Omega)}$,
the system \eqref{LL:momentum}-\eqref{LL:director} has a global weak
solution $(\mathbf{u},\mathbf{d})$ such that
  \begin{gather*}
      \mathbf{u} \in L^2(0,T, H^1(\Omega)) \cap L^\infty(0,T,L^2(\Omega)) \\
      \mathbf{d} \in L^2(0,T, H^2(\Omega)) \cap L^\infty(0,T,H^1(\Omega))
  \end{gather*}
for all $T \in (0,\infty)$.
\end{theorem}

\begin{theorem}[Wellposedness for large viscosity]
  Problem \eqref{LL:momentum}-\eqref{LL:director} has a unique global
classical solution $(\mathbf{u},\mathbf{d})$ provided that
$\mathbf{u}_0(x) \in H^1(\Omega)$,
  $\mathbf{d}_0(x) \in H^2(\Omega)$, $\dim(\Omega)=2,3$, and
$\nu \geq \nu_0(\lambda, \gamma, \mathbf{u}_0, \mathbf{d}_0)$.
\end{theorem}

Many others have studied the system \eqref{LL:momentum}-~\eqref{LL:director}.
Some of the major contributions have been the following:

(1) {\bf Partial Regularity.}
In \cite{lin_partial_1996}, Lin and Liu also demonstrated partial regularity
of suitable weak solutions to
  \eqref{LL:momentum}-\eqref{LL:director} in $\Omega \subset \mathbb{R}^3$.
This extended the work of Caffarelli, Kohn, and Nirenberg on the
  partial regularity of suitable weak solutions for the Navier-Stokes equations
(see \cite{caffarelli_partial_1982}). In particular, Lin and
  Liu were able to show that for sufficiently smooth initial and boundary data
that there exists a suitable weak solution such that
  the singular set of this solution has one-dimensional parabolic Hausdorff
measure zero. Specifically, a \emph{suitable weak solution}
  $(\mathbf{u}, \mathbf{d})$ to \eqref{LL:momentum}-~\eqref{LL:director} in
an open set $D \subset \mathbb{R}^3 \times \mathbb{R}$ i.
  satisfies \eqref{LL:momentum}-~\eqref{LL:director} weakly, ii.
$(\mathbf{u},\mathbf{d})$ has the bounds
  \begin{gather*}
      \int_{D \cap (\mathbb{R}^3 \times t)} (|\mathbf{u}|^2 + |\nabla \mathbf{d}|^2 + F(\mathbf{d})) dx < E_1 = \text{constant} \\
      \int \int_{D} (|\mathbf{u}|^2 + |\Delta \mathbf{d} - f(\mathbf{d})|) dxdt < E_2 = \text{constant}
  \end{gather*}
  and, iii. $(\mathbf{u}, \mathbf{d})$ satisfies the
\emph{generalized energy inequality}
  \begin{equation}
    \label{linLiuGeneralizedEnergy1996}
    \begin{aligned}
      &2 \int_0^T\int_\Omega (|\nabla \mathbf{u}|^2 + |\nabla^2 \mathbf{d}|^2) \phi  dx dt  \\
      &\leq \int_0^T \int_\Omega (|\mathbf{u}|^2 + |\nabla u|^2)(\phi_t + \Delta\phi) dx dt \\
      &\quad + \int_0^T \int_\Omega  (|\mathbf{u}|^2 + |\nabla u|^2 + 2 P)\mathbf{u}\cdot \nabla \phi dx dt \\
      &\quad + \int_0^T \int_\Omega ((\mathbf{u} \cdot \nabla) \mathbf{d}) \cdot \nabla \phi dx dt  + \int_0^T \int_\Omega \nabla f(\mathbf{d}) \nabla \mathbf{d} \phi dx dt.
    \end{aligned}
  \end{equation}

(2) {\bf Well-posedness of penalized Ericksen-Leslie equations.}
Lin and Liu also examined the existence of global weak
  solutions to the Ericksen-Leslie equations in the late 1990s in
\cite{lin_existence_2000}.
  Their main result was that under assumption $\mathbf{u}_0 \in L^2(\Omega)$
and $\mathbf{d}_0 \in H^1(\Omega)$ with
  $\mathbf{d}_0|_{\partial\Omega} \in H^{3/2}(\partial \Omega)$, the initial
and boundary-value problem consisting
  of a penalized version of \eqref{el:pw:summary}; that is,
  \begin{equation}    \label{LL:EL}
    \begin{gathered}
      \rho \frac{Du^i}{Dt} = -(p - \sigma_{LL})_i + \tilde{F}^{ij}_j \\
      u_i^i = 0 \\
     \Big( \frac{\partial \sigma_F}{\partial d^i_j} \Big)_j
 - \frac{\partial \sigma_F}{\partial d^i} + \tilde{g}^i = 0
    \end{gathered}
  \end{equation}
Here $d^id^i = 1$ has been excluded, and the energy density must be modified
is modified. For example,
  $\sigma_F = \frac12|\nabla d|^2+\frac{1}{4\epsilon^2}(1-|d|^2)^2$.
The system \eqref{LL:EL} retains many of the structures present in
  \eqref{el:pw:summary}. Lin and Liu showed under certain assumptions that
the system \eqref{LL:EL} has a global weak solution $(\mathbf{u}, \mathbf{d})$
such that
  \begin{gather*}
      \mathbf{u} \in L^2(0,T;H^1(\Omega)) \cap L^\infty(0,T;L^2(\Omega)) \\
      \mathbf{d} \in L^2(0,T;H^2(\Omega)) \cap L^\infty(0,T;H^1(\Omega))
  \end{gather*}
  for all $T \in (0,\infty)$. To achieve this result Lin and Liu required
the following assumptions on the Leslie viscosities to insure control
  of the norm of $\mathbf{d}$:
  \begin{equation}
    \label{LL:EL2000assump}
    \alpha_1,\alpha_4, \alpha_5 + \alpha_6 > 0, \ \gamma_1 < 0,  \text{ and }
\gamma_2 = 0.
  \end{equation}
  This simplification, as indicated by the authors, may have no physical
meaning. Similar equations for the hydrodynamic flow of nematic liquid
  crystals were studied using different techniques by Coutand and Shkoller
\cite{coutand_well-posedness_2001}.

  Very recently, Wu, Xu, and Liu in \cite{wu_general_2011} continued the
investigation of the well-posedness of the penalized
  Ericksen-Leslie system. They were able to examine the general penalized
Ericksen-Leslie system with out un-natural assumption on the
  Leslie viscosities, namely, $\gamma_2=0$. Their main result
\cite[Theorem 4.1]{wu_general_2011} states that the system
  \eqref{LL:EL} admits a unique global solution for larger viscosity
and initial data $(\mathbf{u}_0,\mathbf{d}_0) \in V \times H^2$.

  Other simplified models of the hydrodynamical flow of nematic liquid
crystals along the lines of the Lin--Liu equations
  \eqref{LL:momentum}-~\eqref{LL:director} have been proposed and analyzed.
  In the works \cite{sun_energetic_2008,wu_asymptotic_2009,cavaterra_3d_2011},
the authors considered
  \begin{equation}    \label{EVNematicModel}
    \begin{gathered}
      \nabla \cdot \mathbf{u} = 0 \\
      \mathbf{u}_t + \mathbf{u} \cdot \nabla \mathbf{u} = \nabla \cdot \mathcal{T} + \mathcal{F} \\
      \mathbf{d}_t + \mathbf{u} \cdot \nabla \mathbf{d} - \alpha \mathbf{d} \cdot \nabla \mathbf{u} + (1-\alpha)\mathbf{d} \cdot \nabla^T \mathbf{u}  = \gamma (\Delta \mathbf{d} - \nabla_{\mathbf{d}} W(\mathbf{d}))
    \end{gathered}
  \end{equation}
  with
  \begin{gather*}
     \mathcal{T} = \nu \mathcal{S} - \lambda \big [ \nabla \mathbf{d}
 \odot \nabla \mathbf{d} + \alpha(\Delta \mathbf{d}
 - \nabla_{\mathbf{d}} W(\mathbf{d})) \otimes \mathbf{d}
     - (1-\alpha) \mathbf{d} \otimes (\Delta \mathbf{d}
 - \nabla_{\mathbf{d}} W(\mathbf{d})) \big ] \\
    \mathcal{S} = \nu (\nabla \mathbf{u} + \nabla^T \mathbf{u})
  \end{gather*}
  for $(\mathbf{u},\mathbf{d},p): \Omega \times (0,T)
\to \mathbb{R}^3\times \mathbb{R}^3 \times \mathbb{R}$ where
$\mathbf{u}$ represents the velocity field, $\mathbf{d}$ represents the
  averaged macroscopic/continuum orientation, $p$ is the hydrodynamic pressure.
The constants $\nu$, $\lambda$, and $\gamma$ represent
  the viscosity, ratio of kinetic to potential energy, and the macroscopic
relaxation time or Deborah number. The parameter $\alpha  \in [0,1]$
  is related to the shape of the liquid crystal molecule---$\alpha=1$ yields
disc-like molecules, $\alpha =1/2$ yields spherical molecules, and
  $\alpha = 1$ yields rod like molecules.

  For this model, it is shown in \cite{wu_general_2011} (Theorem 2.1)
that under appropriate assumptions on initial data there exists global
  weak solutions with periodic boundary conditions. The authors of
\cite{cavaterra_3d_2011} have a similar result, but with appropriate
  Dirichlet and Neumann boundary data for the velocity field and the
orientation field.

(3) {\bf Limits of sequences of solutions to the penalized equations.}
  In \cite{lin_existence_2000} (Theorem 7.1) it was shown, up to a subsequence,
solutions $(\mathbf{u}_{\epsilon},\mathbf{d}_{\epsilon})$
  of the penalized Ericksen-Leslie equations converge strongly in
$L^2(0,T;L^2(\Omega)) \times L^2(0,T;H^1(\Omega))$ to
  the sharp system \eqref{el:pw:summary} with the addition of matrix defect
measure appearing with the other stress terms.
  Whether this defect measure is identically zero is an outstanding and
difficult open problem.


\section{Non-penalized simplifications of the Ericksen-Leslie equations}
\label{nonPenalizedSection}

Beginning in the late 1980s Hardt, Kinderlehrer and Lin started the
mathematical analysis of static solutions to
the Ericksen-Leslie equations or minimizers of the Oseen-Zocher-Frank elastic
energy functional \cite{hardt_existence_1986}. The equations for static
solutions follow naturally from the Ericksen-Leslie equations by setting
the velocity field $\mathbf{u}$ to zero. Such equations are the Euler-Lagrange
equations of the Oseen-Zocher-Frank elastic energy functional
($\int_\Omega \sigma_F$). Hardt, Kinderlehrer, and Lin used
direct methods to study the Oseen-Zocher-Frank elastic energy functional.
They established existence and partial regularity of minimizers of the
Oseen-Zocher-Frank functional. To be precise, the major results of their
 work are summarized in the following two theorems.

\begin{theorem}[Existence of minimizers]
For $\mathbf{d}_0: \partial \Omega \to \mathbb{S}^2$ a Lipshitz map,
the admissible class of minimizers
\begin{equation*}
  \mathcal{A}(\mathbf{d}_0) := \{\mathbf{d} \in H^1(\Omega, \mathbb{S}^2):
 \mathbf{d}_0 = \text{ trace of } \mathbf{d} \text{ on } \partial \Omega \}
\end{equation*}
is non-empty. Furthermore, for any
$\mathbf{d}_0: \partial \Omega \to \mathbb{S}^2$ there exists
 $\mathbf{d} \in \mathcal{A}(\mathbf{d}_0)$ such that
\begin{equation*}
\mathcal{F}[\mathbf{d}] = \inf_{\mathbf{d} \in \mathcal{A}(\mathbf{d}_0)}
 \int_\Omega \sigma_F(\mathbf{d},\nabla \mathbf{d}).
\end{equation*}
\end{theorem}

\begin{theorem}[Interior partial regularity]
  If $\mathbf{d} \in H^1(\Omega,\mathbb{S}^2)$ is a minimizer of $\mathcal{F}$,
then $\mathbf{d}$ is analytic on $\Omega \setminus Z$ for some relatively
closed subset $Z$  of $\Omega$ which has one-dimensional Hausdorff measure zero.
\end{theorem}

The work of Hardt, Kinderlehrer and Lin \cite{hardt_existence_1986}
inspired many other mathematical works on the static theory
of liquid crystals. Hardt and Lin expanded the study of defects of
nematics in \cite{hardt_stability_1989}.
The authors Cohen \cite{robert_cohen_weak_1990}, Kinderlehrer, Ou
\cite{kinderlehrer_second_1992}, and  H\'elein \cite{helein_minima_1987}
 examined the stability of static solutions.  Lin, Luskin, Alouges, Ghidaglia,
 and others explored static solutions of nematic liquid crystals by
numerically optimizing the Oseen-Zocher-Frank elastic energy functional
(see \cite{lin_relaxation_1989}, \cite{alouges_new_1997},
\cite{alouges_minimizing_1997}).
There remain open problems for the static theory of liquid crystals,
many of which revolve around better understanding of the nature of the
Oseen-Zocher-Frank energy functional. Though, not the focus of this survey
it worth mentioning a few very interesting problems for the static theory
of liquid crystals.


(1) {\bf Monotonicity of Oseen-Zocher-Frank functional.} To the best of the
knowledge of the author there is no mathematical
  proof of monotonicity or example demonstrating the lack of monotonicity.
That is, it is unknown that if $\mathbf{d}$ is a static solution
  (critical point of the Oseen-Zocher-Frank functional) whether one has
  \begin{equation*}
    \int_{B_r(x_0)} \sigma_F(\mathbf{d}, \nabla \mathbf{d}) \leq \int_{B_R(x_0)} \sigma_F(\mathbf{d}, \nabla \mathbf{d}), \ 0 < r \leq R.
  \end{equation*}
  As mentioned in the survey \cite{lin_static_2001} by F.H. Lin and C. Liu (without reference),
  there exist reliable computations that indicate that the Oseen-Zocher-Frank
 energy is monotone.

(2) {\bf Numerical exploration of static solutions of nematic liquid crystals.}
Beginning with the work Lin and Luskin \cite{lin_relaxation_1989} in the
late 1980s many mathematicians  have explored techniques for numerical
solution of the harmonic map problem.
Namely, Lin and Luskin   explored harmonic maps into spheres via relaxation.
Starting in the mid 1990s Alouges \cite{alouges_new_1997}
presented a creative and efficient algorithm to find harmonic maps by
minimization of the   Dirichlet energy functional into spheres or convex targets.
The algorithm transforms the numerical   difficulty of constraining the range
to a manifold (a nonlinear constraint) present in numerical solution
of harmonic maps into a more tractable constraint on tangent space
(a linear constraint). One of the
key properties that allows the Alouges algorithm to work is the fact that
projection onto the target (say
the sphere) decreases the Dirichlet energy. The same is not true for
the Oseen-Zocher-Frank functional.
Alouges and Ghidaglia were able to deal with the more general
Oseen-Zocher-Frank functional by constructing
auxilliary functionals that matched the Oseen-Zocher-Frank
functional when properly restricted. Alouges and
Ghidaglia in \cite{alouges_minimizing_1997} were unable to provide a proof
of the convergence of the Alouges algorithm
for the Oseen-Zocher-Frank functional. To the best of the knowledge of
the author the program  of Alouges and Ghidaglia remains unfinished.

  More recently,  Bartels re-imagined the Alouges algorithm in a
finite element context \cite{bartels_stability_2005}. Bartels
found that there must be a restriction upon the finite element
discretization in order to achieve the energy  decreasing property.
Bartels demonstrated that the use of \emph{acute} finite elements yield
an energy   decreasing algorithm. The work \cite{bartels_stability_2005}
only explores the Dirichlet energy. A similar analysis of
the more general Oseen-Zocher-Frank functional may be possible if
suitable constraint on the discretization   can be found.


Although, it is a natural transition to consider the dynamic non-penalized
Er\-icksen-Leslie equations \eqref{el:pw:summary} such an analysis
has only recently begun and is as result still incomplete. Here we discuss
known results a non-penalized Lin--Liu type system
\eqref{LL:momentum}-\eqref{LL:director}.
For this system there are fairly complete results for two dimensional
domains and preliminary results in three dimensions. For the two-dimensional
work, we discuss
the work of Lin, Lin and Wang \cite{lin_liquid_2009} and for three dimensions,
 we discuss the work of Hineman and Wang \cite{hineman_well-posedness_2012}.
One should also see Hong \cite{hong_global_2011} and
Xu-Zhang \cite{xu_global_2012} and Hong--Xin \cite{hong_global_2010}.

Lin, Lin, and Wang \cite{lin_liquid_2009} analyzed the simplified
 Ericksen-Leslie system
\begin{gather}
  \mathbf{u}_t + \mathbf{u} \cdot \nabla \mathbf{u} - \nu \Delta \mathbf{u}
+ \nabla P = -\lambda \nabla \cdot (\nabla \mathbf{d} \odot \mathbf{d}) \label{LLW:NS}\\
  \nabla \cdot \mathbf{u}  = 0  \label{LLW:divfree}\\
  \mathbf{d}_t + \mathbf{u} \cdot \nabla \mathbf{d}
= \gamma (\Delta \mathbf{d} + |\nabla \mathbf{d}|^2 \mathbf{d}) \label{LLW:director}
\end{gather}
for $\mathbf{u}: \Omega \to \mathbb{R}^2$ and
$\mathbf{d}: \Omega \to \mathbb{S}^2$ where $\Omega \subset \mathbb{R}^2$
and subject to
\begin{equation}
  \label{linlinwang:data}
  \mathbf{u}_0 \in \mathbf{H}, \mathbf{d}_0 \in H^1(\Omega,\mathbb{S}^2),
\text{ and } d_0 \in C^{2,\beta}(\partial \Omega, \mathbb{S}^2)
  \text{ for some } \beta \in (0,1).
\end{equation}
Where,
\begin{gather*}
    \mathbf{H} = \text{closure of } C_0^{\infty}(\Omega, \mathbb{R}^2)
 \cap \{\mathbf{u} : \nabla \cdot \mathbf{u} = 0 \} \text{ in }
 L^2(\Omega, \mathbb{R}^2),\\
    \mathbf{J} = \text{closure of } C_0^{\infty}(\Omega, \mathbb{R}^2)
 \cap \{\mathbf{u} : \nabla \cdot \mathbf{u} = 0 \} \text{ in }
 H^1(\Omega, \mathbb{R}^2), \text{ and}\\
    H^1(\Omega, \mathbb{S}^2) = \{\mathbf{d} \in H^1(\Omega, \mathbb{R}^3):
 \mathbf{d}(x) \in \mathbb{S}^2 \text{ a.e. } x \in \Omega \}.
\end{gather*}

They proved the following result.

\begin{theorem}[Regularity]  \label{linlinwang:thm1}
  For $0 < T < + \infty$ assume 
$$
\mathbf{u} \in L^\infty([0,T],\mathbf{H}) \cap L^2([0,T],\mathbf{J})
$$ 
and $\mathbf{d} \in
L^2([0,T],H^1(\Omega,\mathbb{S}^2))$ is a
  suitable weak solution of \eqref{LLW:NS}-\eqref{LLW:director} with
\eqref{linlinwang:data}. If in addition, $\mathbf{d} \in L^2([0,T],H^2(\Omega))$,
  then $(\mathbf{u},\mathbf{d}) \in C^\infty(\Omega \times (0,T])
\cap C_\beta^{2,1}(\bar{\Omega}) \times (0,T]$.
\end{theorem}

\begin{theorem}[Global Weak Solutions of Partial Regularity]
\label{linlinwang:thm2}
  For data satisfying \eqref{linlinwang:data}, there exist global
suitably weak solutions
  $\mathbf{u} \in L^\infty([0,\infty),\mathbf{H})
\cap L^2([0,\infty),\mathbf{J})$ and $\mathbf{d} \in L^\infty([0,\infty),
H^1(\Omega, \mathbb{S}^2))$
  of the equations \eqref{LLW:NS}-\eqref{LLW:director}, which has the
following properties:

(1) There exists $L \in \mathbb{N}$ depending only on
$(\mathbf{u}_0,\mathbf{d}_0)$ and
    $0<T_1< \cdots < T_L$, $1\leq i \leq L$, such that
    \begin{equation*}
      (\mathbf{u},\mathbf{d}) \in C^{\infty}(\Omega \times ((0,\infty) \setminus \{T_i\}_{i=1}^L)) \cap
      C_{\beta}^{2,1}(\bar{\Omega}\times((0,\infty) \setminus \{T_i\}_{i=1}^L)).
    \end{equation*}

(2) Each singular time $T_i$, $1\leq i \leq L$, can be characterized by
    \begin{equation*}
      \liminf_{t \uparrow T_i}\max_{x \in \bar{\Omega}} \int_{\Omega \cap B_r(x)}(|\mathbf{u}|^2 + |\nabla \mathbf{d}|^2)(y,t)dy \geq 8\pi, \ \forall r > 0.
    \end{equation*}
    Moreover, there exist $x_m^i \to x_0^i \in \Omega$, $t_i \uparrow T_i$, $r_m^i \downarrow 0$ and a
    non-constant smooth harmonic map $\omega_i: \mathbb{R}^2 \to \mathbb{S}^2$ with finite energy such that as $m \to \infty$,
    \begin{equation*}
      (\mathbf{u}_m^i, \mathbf{d}_m^i) \to (0,\omega_i) \text{ in } C_{\rm loc}^2(\mathbb{R}^2 \times [\infty, 0]),
    \end{equation*}
    where
    \begin{equation*}
      \mathbf{u}_m^i(x,t) = r_m^i\mathbf{u}(x_m^i+r_m^ix, t_m^i + (r_m^i)^2t), \ \mathbf{d}_m^i(x,t) = \mathbf{d}(x_m^i+r_m^ix, t_m^i + (r_m^i)^2t).
    \end{equation*}

(3) Set $T_0 = 0$. Then, for $0 \leq i \leq L-1$,
    \begin{equation*}
      |\mathbf{d}_t| + |\nabla^2 \mathbf{d}| \in L^2(\Omega\times [T_i,T_{i+1}- \epsilon]), \
      |\mathbf{u}_t| + |\nabla^2 \mathbf{u}| \in L^{4/3}(\Omega\times [T_i,T_{i+1}- \epsilon])
    \end{equation*}
    for any $\epsilon > 0$, and for any $0 < T_L < T < \infty$,
    \begin{equation*}
      |\mathbf{d}_t| + |\nabla^2 \mathbf{d}| \in L^2(\Omega\times [T_L,T]), \
      |\mathbf{u}_t| + |\nabla^2 \mathbf{u}| \in L^{4/3}(\Omega\times [T_L,T]).
    \end{equation*}

(4) There exist $t_k \uparrow \infty$ and a harmonic map
    $\mathbf{d}_\infty \in C^\infty(\Omega, \mathbb{S}^2) \cap C^{2,\beta}(\bar{\Omega},\mathbb{S}^2)$
    with $\mathbf{d}_\infty = \mathbf{d}_0$ on $\partial \Omega$ such that $\mathbf{u}(\cdot,t_k) \to 0$
    in $H^1(\Omega)$, $\mathbf{d}(\cdot,t_k) \to \mathbf{d}_\infty$ weakly in $H^1(\Omega)$, and there exist
    $l \in \mathbb{N}$, points $\{x_i\}_{i=1}^l \in \Omega$, and $\{m_i \}_{i=1}^l \subset \mathbb{N}$ such that
    \begin{equation*}
      |\nabla \mathbf{d}(\cdot,t_k)|^2dx \to |\nabla \mathbf{d}_\infty|^2dx + \sum_{i=1}^l 8\pi m_i \delta_{x_i}.
    \end{equation*}

(5)  If $(\mathbf{u}_0,\mathbf{d}_0)$ satisfies
    \begin{equation}
      \int_{\Omega}(|\mathbf{u}_0|^2 + |\nabla \mathbf{d}_0|^2) \leq 8 \pi,
    \end{equation}
    then
    $(\mathbf{u},\mathbf{d}) \in C^\infty(\Omega\times(0,\infty))
 \cap C_{\beta}^{2,1}(\bar{\Omega} \times (0,\infty))$.
    Moreover, there exist $t_k \uparrow \infty$ and
    $\mathbf{d}_\infty \in C^\infty(\Omega,\mathbb{S}^2)
 \cap C^{2,\beta}(\bar{\Omega}, \mathbb{S}^2)$ with $\mathbf{d}_\infty = d_0$ on
    $\partial \Omega$ such that
$(\mathbf{u}(\cdot, t_k),\mathbf{d}(\cdot, t_k)) \to (0,\mathbf{d}_\infty)$
in $C^2(\Omega)$.
\end{theorem}

Though the system treated Lin, Lin and Wang in \cite{lin_liquid_2009} is
a greatly simplified version of the Ericksen-Leslie equations
it preserves the sphere constraint on the director. This nonlinear constraint
seems to make it impossible apply a Galerkin-type technique to the problem.
To prove their main theorem on the local existence
(Theorem \ref{linlinwang:thm2}) the authors of \cite{lin_liquid_2009}
approximated weak solutions
with strong solutions that they obtained through fixed point iteration.
The proof is completed by using energy inequalities and the regularity result
Theorem \ref{linlinwang:thm1} to determine the maximal singular time.
This method was originally used by Leray in \cite{leray_sur_1934} to prove
local existence (and uniqueness) of weak solutions to the Navier-Stokes
equation. Many others have explored and extended the techniques of
Leray \cite{serrin_initial_1963,fujita_navier-stokes_1964,kato_strongl_1984,
escauriaza_l_2003,germain_multipliers_2006,lions_mathematical_1996,
lemarie-rieusset_recent_2002}.

The work Lin, Lin, and Wang in \cite{lin_liquid_2009} did not include
results on the uniqueness of weak solutions of
\eqref{LLW:NS}-\eqref{LLW:director} but indicated that they believed
the solution to be unique. A proof the uniqueness of solution to
\eqref{LLW:NS}-\eqref{LLW:director} was given by Lin and Wang
\cite{lin_uniqueness_2010}, specifically, they proved the following result.

\begin{theorem}
For $0 < T \leq +\infty$, $\mathbf{u}_0 \in L^2(\Omega, \mathbb{R}^2)$ with
$\nabla \cdot \mathbf{u}_0 = 0$ and $\mathbf{d}_0: \Omega \to \mathbb{S}^2$
with $\nabla \mathbf{d}_0 \in L^2(\Omega, \mathbb{R}^6)$,
suppose that for $i = 1,2$
$\mathbf{u}_i \in L_t^\infty L_x^2 \cap L_t^2 H_x^1(\Omega \times [0,T],
\mathbb{R}^2)$, $\nabla P_i \in L_t^{4/3}L_x^{4/3}(\Omega \times [0,T],
\mathbb{R}^2)$,
and $\mathbf{d}_i \in L_t^\infty \dot{H}_x^1 \cap L_t^2
\dot{H}_x^2(\Omega \times [0,T], \mathbb{S}^2)$
are a pair of weak solution to \eqref{LLW:NS}-\eqref{LLW:director}
under either
\begin{enumerate}
\item when $\Omega = \mathbb{R}^2$, the same initial condition:
  \begin{equation*}
    (\mathbf{u}_i, \mathbf{d}_i) |_{t=0} = (\mathbf{u}_0, \mathbf{d}_0), \; i=1,2,
  \end{equation*}
\item or, when $\Omega \subset \mathbb{R}^2$ is a bounded domain,
the same initial and boundary conditions:
  \begin{equation*}
    (\mathbf{u}_i, \mathbf{d}_i) = (u_0, d_0), \text{ on }
\Omega \times \{0\}, \ (\mathbf{u}_i,\mathbf{d}_i)
= (0,\mathbf{d}_0), \text{ on } \partial\Omega \times (0,T), \; i=1,2
  \end{equation*}
  with $\mathbf{d}_0 \in C^{2,\beta}(\partial \Omega, \mathbb{S}^2)$
for some $\beta \in (0,1)$.
\end{enumerate}
Then $(\mathbf{u}_1, \mathbf{d}_1) \equiv (\mathbf{u}_2, \mathbf{d}_2)$
in $\Omega \times [0,T)$.
\end{theorem}

An alternative method for uniqueness of solutions in Serrin's class
\cite{serrin_initial_1963} to \eqref{LLW:NS}-\eqref{LLW:director}
will be discussed in the forthcoming article by Huang and
Wang \cite{huang_regularity_2012}. This work builds upon the work of
 Huang and Wang on the uniqueness of heat flow of harmonic maps investigated
in \cite{huang_uniqueness_2012}.

As this work is still very recent, there remain many open problems.
In the two-dimensional case it would it would still benefit our understanding
of the full Ericksen-Leslie system to re-introduce various terms into the
simplified equations \eqref{LLW:NS}-\eqref{LLW:director}. The problems facing
closer approximations of the non-penalized Erick\-sen-Leslie will require
different tools than were employed for the penalized Ericksen\-Leslie
equations \eqref{LL:EL} described above.

\section{New findings in three dimensions}
\label{newFindings}

Recently, the author and  Wang considered the Cauchy problem for a
simplified hydrodynamic system modeling the
flow of nematic liquid crystal materials in $\mathbb R^3$
\cite{hineman_hydrodynamic_2012}. Specifically, they considered the
system for
$(\mathbf{u},P,\mathbf{d}):\mathbb{R}^3\times[0,T)
\to \mathbb{R}^3\times\mathbb{R}\times \mathbb{S}^2$, $0<T\le\infty$ given by
\begin{equation}\label{LLF}
\begin{gathered}
\mathbf{u}_t + \mathbf{u} \cdot \nabla \mathbf{u} - \nu\Delta \mathbf{u}
+ \nabla P = -\lambda\nabla\cdot(\nabla \mathbf{d} \odot \nabla \mathbf{d}),
\quad \text{in } \mathbb R^3\times (0,T),\\
\nabla \cdot \mathbf{u} = 0, \quad \text{in } \mathbb R^3\times (0,T),\\
\mathbf{d}_t + \mathbf{u} \cdot \nabla \mathbf{d}
=\gamma(\Delta \mathbf{d}+ |\nabla \mathbf{d}|^2 \mathbf{d}), \quad \text{in }
\mathbb R^3\times (0,T),\\
(\mathbf{u}, \mathbf{d})=(\mathbf{u}_0, \mathbf{d}_0), \quad \text{on }
\mathbb R^3\times \{0\}
\end{gathered}
\end{equation}
for a given initial data $(\mathbf{u}_0,\mathbf{d}_0):\mathbb R^3\to\mathbb R^3
\times \mathbb{S}^2$ with $\nabla\cdot \mathbf{u}_0=0$.
Here again $\mathbf{u}:\mathbb R^3\to\mathbb R^3$ represents the velocity
field of the fluid, $\mathbf{d}:\mathbb R^3\to \mathbb{S}^2$ is a unit
vector field representing the macroscopic molecular orientation of the
nematic liquid crystal material,
$P:\mathbb R^3\to\mathbb{R}$ represents the pressure function.
The constants $\nu,\lambda,$ and $\gamma$ are positive
constants that represent the viscosity of the fluid, the competition
between kinetic and potential energy, and the microscopic
elastic relaxation time for the molecular orientation field.

Because of the super-critical nonlinear term
$\nabla\cdot(\nabla \mathbf{d}\odot\nabla \mathbf{d})$ in \eqref{LLF}$_1$,
it has been an outstanding open problem whether there exists a global
Leray-Hopf type weak solution to
\eqref{LLF} in $\mathbb R^3$ for any initial data
$(\mathbf{u}_0, \mathbf{d}_0)\in L^2(\mathbb{R}^3,\mathbb{R}^3)
\times \dot{W}^{1,2}(\mathbb{R}^3, S^2)$ with $\nabla\cdot \mathbf{u}_0=0$.
That is, a solution to \eqref{LLF} for which one has:
\begin{enumerate}
\item $(\mathbf{u},P,\mathbf{d})$ is a unique weak solution to \eqref{LLF}.
\item $(\mathbf{u},P,\mathbf{d})$ satisfies the natural energy inequality.
\end{enumerate}

Preceding this work, it has been shown that in $\mathbb R^3$ there exists
a local, unique, strong solution to \eqref{LLF} for any initial data
$\mathbf{u}_0\in W^{s,2}(\mathbb R^3)$ and
$\mathbf{d}_0\in W^{s+1,2}(\mathbb R^3, \mathbb{S}^2)$ for $s>3$
with $\nabla\cdot \mathbf{u}_0=0$, see for example \cite{wen_solutions_2011}.
Huang--Wang in\cite{huang_blow_2012} established a blow up criterion
for local strong solutions to \eqref{LLF}, similar to the Beale-Kato-Majda
criterion for the Navier-Stokes equation (see \cite{beale_remarks_1984}).
Li-Wang \cite{li_global_2011} obtained the global existence of strong
solutions for to \eqref{LLF} with small initial data in certain Besov spaces.
In \cite{wang_well-posedness_2010}, Wang obtained the global (or local)
well-posedness of \eqref{LLF} for initial data $(\mathbf{u}_0, \mathbf{d}_0)$
belonging to the space ${\text{BMO}}^{-1}\times {\text{BMO}}$ with
$\nabla\cdot \mathbf{u}_0=0$, which is a invariant space under parabolic
scaling associated with \eqref{LLF}, with small norms.

The main contribution of this work is the local well-posedness of \eqref{LLF}
for any initial data $(\mathbf{u}_0, \mathbf{d}_0)$ such that
$(\mathbf{u}_0,\nabla \mathbf{d}_0)\in L^3_{\rm uloc}(\mathbb{R}^3)$.
Henceforth $L^3_{\rm uloc}(\mathbb{R}^3)$ denotes the space of uniformly
locally $L^3$-integrable functions. It turns out that
$L^3_{\rm uloc}(\mathbb R^3)$ is also invariant under parabolic
scaling associated with \eqref{LLF}. Similar scaling symmetry for
the Navier-Stokes equation has inspired exploration of the Cauchy
problem with initial data in various $L^p$ spaces (see for example
\cite{serrin_initial_1963,caffarelli_partial_1982,kato_strongl_1984,
escauriaza_l_2003}). Uniform local spaces space have also been employed
in a number of natural ways in the analysis
of the Navier-Stokes equation
(see \cite{basson_uniformly_2006,lemarie-rieusset_recent_2002}).

Now we give the definition of $L^3_{\rm uloc}(\mathbb R^3)$.
The readers can consult the monograph by Lemari\'e-Rieusset
\cite{lemarie-rieusset_recent_2002}  for applications of the
space $L^3_{\rm uloc}(\mathbb R^3)$ to the Navier-Stokes equation.

\begin{definition}\label{l3space} \rm
A function $f\in L^3_{\rm loc}(\mathbb R^3)$ belongs to the space
$L^3_{\rm uloc}(\mathbb R^3)$ consisting of uniformly locally
$L^3$-integrable functions, if there exists $0<R<+\infty$ such that
\begin{equation}\label{l3space_norm}
\|f\|_{L^3_R(\mathbb R^3)}
:=\sup_{x\in\mathbb R^3}\Big (\int_{B_R(x)}|f|^3\Big)^{1/3}<+\infty.
\end{equation}
For an open set $U\subset\mathbb R^3$, $f\in L^3_{\rm uloc}(U)$ if $f\chi_U\in L^3_{\rm uloc}(\mathbb R^3)$,
here $\chi_U$ is the characteristic function of $U$.
\end{definition}
\noindent It is clear that
\begin{itemize}
\item $L^3(\mathbb R^3)\subset L^3_{\rm uloc}(\mathbb R^3)$.
\item If $f\in L^3_{\rm uloc}(\mathbb R^3)$,
then $ \|f\|_{L^3_R(\mathbb R^3)}$
is finite for any $0<R<+\infty$. For any two $0<R_1\le R_2<\infty$,
it holds
\begin{equation}\label{equiv_norm}
\|f\|_{L^3_{R_1}(\mathbb R^3)}
\le \|f\|_{L^3_{R_2}(\mathbb R^3)}
\lesssim \Big(\frac{R_2}{R_1}\Big)\|f\|_{L^3_{R_1}(\mathbb R^3)},\quad
 \forall  f\in L^3_{\rm uloc}(\mathbb R^3).
\end{equation}

\item $ L^3_{\rm uloc}(\mathbb R^3)\subset
\cap_{0<R<\infty} {\text{BMO}}^{-1}_R(\mathbb R^3)$
(see \cite{koch_well-posedness_2001} or \cite{wang_well-posedness_2010}).
Moreover, for \newline any $0<R<\infty$, it holds
\begin{equation}\label{inclusion}
[f]_{\text{BMO}^{-1}_R(\mathbb R^3)}\lesssim \|f\|_{L^3_{R}(\mathbb R^3)},
\quad \forall \ f\in L^3_{\rm uloc}(\mathbb R^3).
\end{equation}
\end{itemize}

\begin{theorem}  \label{wellposedness}
  There exist $\epsilon_0 > 0$ and $\tau_0 > 0$ such that if
$\mathbf{u}_0:\mathbb{R}^3 \to \mathbb{R}^3$, with $\nabla\cdot \mathbf{u}_0=0$,
and $\mathbf{d}_0:\mathbb{R}^3 \to \mathbb{S}^2$
  satisfies, for some $0<R<+\infty$ and $e_0\in S^2$,
  \begin{equation}\label{small_condition}
    \begin{gathered}
    ||| (\mathbf{u}_0, \nabla \mathbf{d}_0)|||_{L^3_R(R^3)}
      :=\sup_{x\in\mathbb{R}^3} \Big(\int_{B_R(x)} |\mathbf{u}_0|^3
+|\nabla \mathbf{d}_0|^3\Big)^{1/3} \leq \epsilon_0 , \\
    \lim_{K\uparrow\infty}\|\mathbf{d}_0-e_0\|_{L^3_R(\mathbb R^3\setminus B_K)}
\le \epsilon_0,
    \end{gathered}
  \end{equation}
  then there exist $T_0\ge \tau_0 R^2$ and a unique solution
$(\mathbf{u}, \mathbf{d}):\mathbb{R}^3 \times [0,T_0) \to \mathbb{R}^3
 \times\mathbb{R}\times \mathbb{S}^2$
  of \eqref{LLF} that enjoys the following properties:
\\
(i) For $t\downarrow 0$,
  $ (\mathbf{u}(t), \mathbf{d}(t))\to
 (\mathbf{u}_0, \mathbf{d}_0)$ and $\nabla \mathbf{d}(t)\to \nabla \mathbf{d}_0$
  in $L^3_{\rm loc}(\mathbb{R}^3)$.
\\
(ii)
  \begin{gather*}
(\mathbf{u},\mathbf{d}) \in \cap_{0<\delta<T_0} C^\infty_b(\mathbb{R}^3
\times [\delta,T_0-\delta], \mathbb R^3\times \mathbb{S}^2), \\
   (\mathbf{u},\nabla \mathbf{d})\in\cap_{0<T'<T_0} C_*^0([0, T'],
L^3_{\rm uloc}(\mathbb R^3)).
  \end{gather*}
(iii) \begin{equation}\label{small_condition0}
    |||(\mathbf{u}(t),\nabla \mathbf{d}(t))|||_{L^\infty([0,\tau_0 R^2],
L^3_R(\mathbb{R}^3))}\le C\epsilon_0.
  \end{equation}
(iv) If $T_0 < +\infty$ is the maximum time interval then it must hold
  \begin{equation}\label{blowup_criterion}
    \limsup_{t \uparrow T_0} |||(\mathbf{u}(t), \nabla \mathbf{d}(t))
|||_{L_{r}^3(\mathbb{R}^3)}> \epsilon_0,
    \quad \forall \ 0<r<\infty.
  \end{equation}
\end{theorem}

\begin{remark}\label{outline:wellposedness} \rm
 The ideas to prove Theorem \ref{wellposedness} are motivated by those
employed by \cite{lin_liquid_2009}.
There are six main ingredients:
\begin{enumerate}
\item Approximate the initial data $(\mathbf{u}_0,\mathbf{d}_0)$ strongly in $L^p_{\rm loc}\cap W^{1,p}_{\rm loc}(\mathbb{R}^3)$
  by a sequence of smooth initial data $ \left\{(\mathbf{u}_0^k, \mathbf{d}_0^k)\right\}
  \subset C^\infty(\mathbb{R}^3,\mathbb{R}^3\times \mathbb{S}^2)\cap (L^p(\mathbb{R}^3,\mathbb{R}^3)\times \dot{W}^{1,p}(\mathbb{R}^3,\mathbb{S}^2))$ for $p=2,3$, with
  $\nabla\cdot \mathbf{u}_0^k=0$ and $\|(\mathbf{u}_0^k, \nabla \mathbf{d}_0^k)\|_{L^3_1(\mathbb{R}^3)}$
  uniformly small for all $k$ (see Lemma \ref{approx} below). We would like to point out that the existence of
  such an sequence is highly nontrivial.

\item Apply the fixed-point argument, similar to that by Lin-Lin-Wang \cite{lin_liquid_2009}, to
  obtain a sequence of smooth solutions $(\mathbf{u}^k, P^k, \mathbf{d}^k)$ of \eqref{LLF} in $\mathbb R^3\times [0, T_k]$ for $0<T_k<+\infty$,
  with the initial data $(\mathbf{u}_0^k, \mathbf{d}_0^k)$, such that $(\mathbf{u}^k, \mathbf{d}^k)\in C^\infty(\mathbb{R}^3\times [0,T_k],\mathbb{R}^3\times \mathbb{S}^2)
  \cap C([0,T_k], L^2(\mathbb{R}^3)\times \dot{W}^{1,2}(\mathbb{R}^3))$.

\item Utilize the local $L^3$-energy inequality \eqref{localEnergyInequality0} for $(\mathbf{u}^k, P^k, \mathbf{d}^k)$ to show that
  there is a uniform lower bound $\tau_0>0$ for $T_k$ such that 
 $\sup_{0\le t\le \tau_0} \|(\mathbf{u}^k(t),\\
    \nabla \mathbf{d}^k(t))\|_{L^3_1(\mathbb{R}^3)}$  is uniformly small for all $k$. It turns out that
  $(\mathbf{u}^k,\nabla \mathbf{d}^k)\in C([0, T_k], L^2(\mathbb{R}^3)\times L^2(\mathbb{R}^3))$ plays an important role in
  the estimate of $P^k$ in $L^{3/2}_{\rm loc}(\mathbb{R}^3)$.

\item Apply the $\epsilon_0$-regularity Theorem \ref{smooth:solutions}, 
which is analogous to that by Caffa\-relli-Kohn-Nirenberg
  \cite{caffarelli_partial_1982} on the Navier-Stokes equation, to obtain a priori  derivative estimates of $(\mathbf{u}^k, \mathbf{d}^k)$ in
  $\mathbb{R}^3\times [0,\tau_0]$. In order to handle the nonlinearity $\nabla\cdot(\nabla \mathbf{d}^k\odot\nabla \mathbf{d}^k)$
  in \eqref{LLF}$_1$, we do assume
  an additional smallness condition of $\|\nabla \mathbf{d}^k\|_{L^\infty_tL^3_x(P_1)}$ in Theorem \ref{smooth:solutions}. Then take
  a limit of $(\mathbf{u}^k, P^k, \mathbf{d}^k)$ to obtain the local existence of $L^3_{\rm uloc}$-solution $(\mathbf{u},P,\mathbf{d})$ of \eqref{LLF}
  in $\mathbb{R}^3\times [0,\tau_0]$. 

\item Apply Theorem \ref{smooth:solutions} again to characterize the 
finite maximal time interval $T_0$ of $(\mathbf{u}, P, \mathbf{d})$. 

\item Adapt the proof of \cite{wang_well-posedness_2010} to show that the uniqueness holds for $L^3_{\rm uloc}$-solutions that satisfy the
  same properties as the solution $(\mathbf{u},P,\mathbf{d})$ constructed as above.
\end{enumerate}
\end{remark}

\begin{remark} \rm
  Since the exact values of $\nu, \lambda, \gamma$ do not play a role
 in the proof  of Theorem \ref{wellposedness}, we may henceforth assume
$\nu=\lambda=\gamma=1$.
\end{remark}

 For a solution $(\mathbf{u}, P, \mathbf{d})$ to \eqref{LLF}, denote its
$L^3$-energy by
$$
E_3(u,\nabla \mathbf{d})(t)=\int_{\mathbb R^3}(|\mathbf{u}(t)|^3
+|\nabla \mathbf{d}(t)|^3), \quad t\ge 0.
$$
Concerning the global well-posedness of \eqref{LLF}, we have the following
result.

\begin{theorem}\label{gwp}
  There exists an $\epsilon_0 > 0$ such that if
$$
(\mathbf{u}_0,\mathbf{d}_0) \in L^3(\mathbb{R}^3,\mathbb{R}^3)
\times \dot{W}^{1,3}(\mathbb{R}^3,\mathbb{S}^2),
$$
with $\nabla\cdot \mathbf{u}_0=0$,  satisfies
  \begin{equation}\label{small_l3}
    E_3(\mathbf{u}_0,\nabla \mathbf{d}_0)  \leq \epsilon_0^3,
  \end{equation}
  then there exists a unique global solution
$(\mathbf{u},\mathbf{d}):\mathbb R^3\times [0,\infty)
\to \mathbb R^3\times\mathbb R\times \mathbb{S}^2$  of \eqref{LLF} such that
$(\mathbf{u}, \mathbf{d}) \in C^\infty(\mathbb R^3\times (0,+\infty))
\cap C([0,\infty), L^3(\mathbb{R}^3)\times \dot{W}^{1,3}(\mathbb{R}^3))$,
$E_3(\mathbf{u},\nabla \mathbf{d})(t)$ is monotone decreasing for $t\ge 0$, and
\begin{equation}\label{uniform_gradient_estimate}
\|\nabla^m \mathbf{u}(t)\|_{L^\infty(\mathbb{R}^3)}
+\|\nabla^{m+1} \mathbf{d}(t)\|_{L^\infty(\mathbb{R}^3)}
\le \frac{C\epsilon_0}{t^{\frac{m}2}},
\quad \forall\ t>0, \; m\ge 0.
\end{equation}
\end{theorem}

\begin{remark} \rm
The first conclusion of Theorem \ref{gwp} has already been proven by
Lin \cite{lin_well-posedness_2012}, based on refinement of the argument by
 Wang in \cite{wang_well-posedness_2010}.
\end{remark}

In the remainder of this section we discuss some of the details of the
proof of Theorem \ref{wellposedness} outlined in
Remark \ref{outline:wellposedness}. For complete details the reader
is encouraged to consult the work of Hineman and Wang
\cite{hineman_well-posedness_2012}.
\medskip

\noindent {\bf Step 1: Approximation.}
The first two ingredients listed in the Remark \ref{outline:wellposedness}
involve the approximation of rough initial data
by smooth initial data. This is precisely

\begin{lemma}\label{approx}
 For a sufficiently small $\epsilon_0>0$,
let $(\mathbf{u}_0, \mathbf{d}_0):\mathbb{R}^3\to \mathbb{R}^3\times
\mathbb{S}^2$, with divergence free
$\mathbf{u}_0\in L^3_{\rm uloc}(\mathbb{R}^3)$,
satisfy,  for some $e_0\in \mathbb{S}^2$,
\begin{equation}\label{small_condition1}
 |||(\mathbf{u}_0,\nabla \mathbf{d}_0)|||_{L^3_{1}(\mathbb{R}^3)}\leq \epsilon_0
\ {\text{and}}\ \lim_{K\to\infty}\|\mathbf{d}_0-e_0\|_{L^3_1(\mathbb R^3\setminus B_K)}
\le\epsilon_0,
\end{equation}
then there exist
$$\{(\mathbf{u}_0^k, \mathbf{d}_0^k)\}\subset
C^\infty(\mathbb{R}^3,\mathbb{R}^3\times \mathbb{S}^2)\cap \cap_{p=2}^3
(L^p(\mathbb{R}^3,\mathbb{R}^3)\times
\dot{W}^{1,p}(\mathbb{R}^3,\mathbb{S}^2))
$$
such that the following properties hold:\\
(i) $\nabla\cdot \mathbf{u}_0^k=0$ in $\mathbb{R}^3$ for all $k\ge 1$.\\
(ii) As $k\to \infty$,
\begin{equation}\label{local_convergence}
(\mathbf{u}_0^k,\mathbf{d}_0^k)\to (\mathbf{u}_0, \mathbf{d}_0) \ {\text{and}}\ \nabla \mathbf{d}_0^k\to \nabla \mathbf{d}_0
\ {\text{in}}\ L^p_{\rm loc}(\mathbb{R}^3) \ {\text{for}}\ p=2, 3.
\end{equation}
(iii) There exist $C_0>0$ and  $k_0>1$ such that for any $k\ge k_0$,
\begin{equation} \label{uniform_l3loc}
|||(\mathbf{u}_0^k,\nabla \mathbf{d}_0^k)|||_{L^3_{1}(\mathbb{R}^3)}\leq C_0\epsilon_0.
\end{equation}
\end{lemma}

\begin{remark} \rm
  The proof of Lemma \ref{approx} is non-trivial and technical.
It naturally requires separate approximations for velocity initial
data $\mathbf{u}_0$ and the   director initial data $\mathbf{d}_0$.
The approximation of $\mathbf{u}_0$ is similar to
\cite[Theorem 1.4]{basson_uniformly_2006}. Both approximations
  make critical use of $L^3_{\rm uloc}$.
\end{remark}

The fixed point argument for local existence of strong solutions of
 Lin-Lin-Wang \cite{lin_liquid_2009} is not dimension dependent. Thus
modifying the solution space $X$ of Lin-Lin-Wang in
\cite[Theorem 3.1]{lin_liquid_2009} one can consider for
$K > 0$, $0 < \alpha < 1$
the space
\begin{align*}
    X_T=\Big\{&(\mathbf{u},\mathbf{d}): \mathbb{R}^3\times [0,T]
 \to\mathbb{R}^3\times\mathbb{R}^3: \nabla\cdot \mathbf{u}=0, \\
    &\nabla^2f, \partial_t f\in C_b(\mathbb{R}^3\times [0, T])\cap
    C^\alpha(\mathbb{R}^3\times [0,T]), \\
    & (\mathbf{u},\mathbf{d})|_{t=0}=(\mathbf{u}_0^k, \mathbf{d}_0^k),
 \ \|(\mathbf{u}-\mathbf{u}_0^k, \mathbf{d}
-\mathbf{d}_0^k)\|_{C^{2,1}_\alpha(\mathbb{R}^3\times [0,1])}\le K\Big\},
  \end{align*}
Such considerations allow us to find solutions
$(\mathbf{u}^k, P^k, \mathbf{d}^k)$ of \eqref{LLF} in
$\mathbb R^3\times [0, T_k]$ for $0<T_k<+\infty$,
and the initial data $(\mathbf{u}_0^k, \mathbf{d}_0^k)$, such that
$(\mathbf{u}^k, \mathbf{d}^k)\in C^\infty(\mathbb{R}^3\times
[0,T_k],\mathbb{R}^3\times \mathbb{S}^2)
\cap C([0,T_k], L^2(\mathbb{R}^3)\times \dot{W}^{1,2}(\mathbb{R}^3))$.
These smooth solution will serve as approximate solutions to \eqref{LLF} with
$(\mathbf{u}_0, \nabla \mathbf{d}_0) \in L_{\rm uloc}^3(\mathbb{R}^3)$.
\medskip

\noindent{\bf Step 2: Uniform lower bounds.} The next to step
in the proof of Theorem \ref{wellposedness} is to find uniform
lower bounds on the
intervals of existence for the approximating smooth solutions.
That is we need to verify the claim:

\begin{claim} \label{lowerBoundClaim}
  There exists $\tau_0 > 0$ such that if $T_k$ is the maximal time interval
for the smooth solutions $(\mathbf{u}^k, \mathbf{d}^k)$ obtained
  by approximation process described in {\bf Step 1} (that is, from fixed-point iteration), then $T_k \geq \tau_0$, and one has that
  \begin{equation} \label{hinemanWang5.5}
    \sup_{0 \leq t \leq \tau_0} |||(\mathbf{u}^k(t),\nabla
\mathbf{d}^k(t)|||^3_{L^{1/2}(\mathbb{R}^3)} \leq 2 C_0^3 \epsilon_0^3.
  \end{equation}
\end{claim}

To verify Claim \ref{lowerBoundClaim}, we employ a local $L^3$ energy inequality,
specifically:

\begin{lemma}\label{local_ineq}
There exists $C>0$ such that  for $0<T\le \infty$, if
  $ (\mathbf{u}, \mathbf{d})\in C^\infty(\mathbb R^3\times [0, T),
  \mathbb R^3\times \mathbb{S}^2)\cap C([0,T), L^2(\mathbb{R}^3)
\times \dot{W}^{1,2}(\mathbb{R}^3))$  is a smooth solution of the
system \eqref{LLF}, then
  \begin{equation}
    \begin{aligned}
&\frac{d}{dt} \int_{\mathbb{R}^3} (|\mathbf{u}|^3
 + |\nabla \mathbf{d}|^3)\phi^2 + \int_{\mathbb{R}^3}
 \Big(|\nabla(|\mathbf{u}|^{3/2}\phi)|^2
 + |\nabla(|\nabla \mathbf{d}|^{3/2}\phi)|^2\Big) \\
&\leq C \int_{\mathbb{R}^3} (|\mathbf{u}|^3
 + |\nabla \mathbf{d}|^3)|\nabla \phi|^2
 + CR^{-2}\sup_{y\in\mathbb R^3}\Big(\int_{B_R(y)}|\mathbf{u}|^3
 +|\nabla \mathbf{d}|^3\Big)^{5/3} \\
&\quad+ C \Big(\int_{\text{spt}\phi} |\mathbf{u}|^3 +|\nabla \mathbf{d}|^3
 \Big)^{2/3}
  \int_{\mathbb{R}^3}\Big(|\nabla(|\mathbf{u}|^{3/2}\phi)|^2
 + |\nabla(|\nabla \mathbf{d}|^{3/2}\phi)|^2\Big),
    \end{aligned}
    \label{localEnergyInequality0}
  \end{equation}
holds for any $\phi \in C_0^\infty(\mathbb{R}^3)$,
with $0\le \phi\le 1$, the support of $\phi$ is
$\text{spt} \phi=B_R(x_0)$  for some $R>0$ and $x_0\in\mathbb R^3$, and
$|\nabla\phi|\le 4R^{-1}$.
\end{lemma}

\begin{proof}[Outline of proof.]
  Proof has three basic steps:
  \begin{enumerate}
  \item To obtain the $\nabla \mathbf{d}$ estimates, start by
 multiplying \eqref{LLF}$_3$ by the test function
$\phi^2|\nabla \mathbf{d}|\mathbf{d}$
    and integrating by parts. Using standard techniques one may
estimate the local $L^3$ energy of $\nabla \mathbf{d}$.
  \item Similarly, multiplying \eqref{LLF}$_1$ by the
test function $\phi^2|\mathbf{u}|\mathbf{u}$ and integrating by parts
one may estimate the local $L^3$ energy of $\mathbf{u}$.
  \item Estimate the pressure $P$. This step requires significant
 work which is recorded in Lemma \ref{L3:pressure:lemma}.
  \end{enumerate}
\end{proof}

\begin{lemma}  \label{L3:pressure:lemma}
Under the same assumptions as in Lemma \ref{local_ineq}, assume
that $\phi \in C_0^\infty(\mathbb{R}^3)$ satisfies $0\le\phi\le 1$,
$\text{spt}\ \phi =B_R(x_0)$   for some $x_0\in\mathbb R^3$,
and $|\nabla\phi|\le 2R^{-1}$. Then there exists $C>0$
such that for any $t\in (0, T)$ there is $c(t)\in \mathbb{R}$
so that the following estimate holds
  \begin{equation}
    \label{p-estimate}
    \begin{aligned}
      \Big( \int_{\mathbb{R}^3} |P(t)- c(t)|^3 \phi^3 \Big)^{1/3}
&\leq C \Big(\int_{\text{spt}\phi}|\mathbf{u}(t)|^3+|\nabla \mathbf{d}(t)|^3
 \Big)^{1/6}\\
&\Big(\int_{\mathbb R^3}(|\nabla(|\mathbf{u}(t)|^{3/2}\phi)|^2
 +|\nabla(|\nabla \mathbf{d}(t)|^{3/2}\phi)|^2\Big)^{1/2}\\
&\quad+CR^{-1}      \sup_{y \in \mathbb{R}^3}
 \Big(\int_{B_R(y)} |\mathbf{u}(t)|^3 + |\nabla \mathbf{d}(t)|^3\Big)^{2/3}.
\end{aligned}
  \end{equation}
\end{lemma}

\begin{proof}[Outline of proof]
  The proof of the pressure estimate relies on writing $P$ in terms
of Riesz transforms and making a commutator estimate.
  That is, one has
  \begin{equation*}
    -\Delta P  = \nabla_{jk}^2 (g^{jk}), \quad
 g^{jk} := u^ju^k + \nabla_j\mathbf{d} \cdot \nabla_k\mathbf{d},
  \end{equation*}
and so
  \begin{equation*}
    P = - {\mathbf R}_j{\mathbf R}_k(g^{jk})
  \end{equation*}
where ${\mathbf R}_j$ is the $j$-th Riesz transform on $\mathbb R^3$.

  One proceeds by writing
  \begin{equation}
    \begin{aligned}
      (P-c)\phi
      &= -{\bf R}_j {\bf R}_k(g^{jk})\phi  - c\phi\\
      &= -{\bf R}_j {\bf R}_k(g^{jk}\phi) - [\phi, {\bf R}_j{\bf R}_k](g^{jk}) -c\phi
    \end{aligned}
    \label{RRcommutator}
  \end{equation}
  and estimating each part separately.
\end{proof}


\noindent {\bf Step 3: Uniform estimation of $(\mathbf{u}_k, \mathbf{d}_k)$.}
For the approximation sequence $(\mathbf{u}^k,P^k,\mathbf{d}^k)$ one has
the pressure Poisson equation
\begin{equation*}
  -\Delta P^k  = \operatorname{div}^2(\mathbf{u} \otimes \mathbf{u}
+ \nabla \mathbf{d}^k \odot \nabla \mathbf{d}^k) \text{ in } \mathbb{R}^3.
\end{equation*}
Using the generalized energy inequality for suitable weak solutions
(see below Theorem \ref{local_energy_inequality}), the uniform lower bound
found in Claim \ref{lowerBoundClaim} and the pressure estimate used
in the local $L^3$ energy inequality (Lemma \ref{L3:pressure:lemma}) one has
\begin{equation}\label{pressure-estimate10}
\sup_{0\le t\le\tau_0}\sup_{x\in\mathbb{R}^3}\|P^k(t)-c_x^k(t)\|_{L^3(B_1(x))}
\le C\epsilon_0,
\end{equation}
where $c_x^k(t)\in\mathbb{R}$ depends on both $x\in\mathbb{R}^3$ and
$t\in [0,\tau_0]$.
From Claim \ref{lowerBoundClaim} and \eqref{pressure-estimate10}, we see that
for any $x_0\in\mathbb{R}^3$, $(\mathbf{u}^k, P^k-c_{x_0}^k, \mathbf{d}^k)$
satisfies the conditions
of Theorem \ref{smooth:solutions} (below) in
$ P_{\sqrt{\tau_0}}(x_0,\tau_0):=B_{\sqrt{\tau_0}}(x_0)\times [0,\tau_0]$.
Hence by Theorem \ref{smooth:solutions} we obtain that
$(\mathbf{u}^k, \mathbf{d}^k)\in C^\infty(\mathbb{R}^3\times (0, \tau_0),
\mathbb{R}^3\times \mathbb{S}^2)$, and
\begin{equation}\label{uniform_estimate10}
\sup_k \|(\mathbf{u}^k, \nabla \mathbf{d}^k)\|_{C^m(\mathbb{R}^3
\times [\delta,\tau_0])}\le C(m, \delta, \epsilon_0)
\end{equation}
holds for any $0<\delta<\tau_0/2$ and $m\ge 0$.

Here Lemma \ref{local_energy_inequality} and Theorem \ref{smooth:solutions}
are concerned with the regularity of suitable weak solutions. Specifically,
Lemma \ref{local_energy_inequality} is generalized energy inequality in the
spirit of Caffarelli-Kohn-Nirenberg \cite{caffarelli_partial_1982}.
Here we adapt notion suitable weak solution to \eqref{LLF}, a similar
definition is given by Lin \cite{lin_new_1998} on the Navier-Stokes equation.

Let $0<T\le\infty$ and  $\Omega\subset\mathbb R^3$ be a bounded smooth domain.

\begin{definition} \rm
  A triplete of functions
$(\mathbf{u},P,\mathbf{d}): \Omega \times (0,T) \to \mathbb{R}^3
\times \mathbb{R} \times \mathbb{S}^2$ is called a
\emph{suitable  weak solution} to the system \eqref{LLF} in
 $\Omega \times (0,T)$ if the following properties hold:
  \begin{enumerate}
    \item $\mathbf{u} \in L_t^\infty L_x^2 \cap L_t^2H_x^1(\Omega \times (0,T))$, $P \in L^{\frac{3}{2}}(\Omega \times (0,T))$
      and $\mathbf{d} \in L_t^2H_x^2(\Omega \times (0,T))$;
    \item $(\mathbf{u},P,\mathbf{d})$ satisfies the system \eqref{LLF} in the sense of distributions; and
    \item $(\mathbf{u},P,\mathbf{d})$ satisfies the local energy inequality \eqref{locEnergy}.
  \end{enumerate}
\end{definition}

\begin{lemma} \label{local_energy_inequality}
Suppose that $(\mathbf{u},\mathbf{d})\in C^\infty(\Omega\times(0,T),
\mathbb R^3\times \mathbb R\times \mathbb{S}^2)$ is a solution of
\eqref{LLF} in $\Omega \times (0,T)$.
 Then for any nonnegative  $\phi \in C_0^\infty(\Omega \times (0,T))$,
it holds that
  \begin{equation}
    \begin{aligned}
& 2 \int_{\Omega \times (0,T)} \left ( |\nabla \mathbf{u} |^2
 + | \Delta \mathbf{d} + |\nabla \mathbf{d}|^2 \mathbf{d}|^2 \right )\phi\\
&\leq \int_{\Omega \times (0,T)} \left ( |\mathbf{u}|^2
 + |\nabla \mathbf{d}|^2 \right )(\phi_t + \Delta \phi)
 + \int_{\Omega \times (0,T)} (|\mathbf{u}|^2
 + |\nabla \mathbf{d}|^2 + 2P)\mathbf{u}\cdot \nabla \phi \\
&\quad + 2 \int_{\Omega \times (0,T)} \left (\nabla \mathbf{d} \odot
 \nabla \mathbf{d} - |\nabla \mathbf{d}|^2 \mathbb I_3 \right):\nabla^2 \phi \\
&\quad + 2 \int_{\Omega \times (0,T)} \nabla \mathbf{d} \odot \nabla \mathbf{d}:
 \mathbf{u} \otimes \nabla \phi.
    \end{aligned}
    \label{locEnergy}
  \end{equation}
  \label{locEnergyLemma}
\end{lemma}

With this framework in place it is verified by Hineman-Wang
\cite{hineman_well-posedness_2012} suitable weak solutions of
\eqref{LLF} with small renormalized energy are smooth. That is:

\begin{theorem}  \label{smooth:solutions}
For any $\delta>0$, there exists $\epsilon_0>0$ such that
  $(\mathbf{u},P,\mathbf{d}):\Omega\times (0, T)\to\mathbb R^3\times\mathbb
R\times \mathbb{S}^2$ is a suitable
  weak solution to \eqref{LLF}, and satisfies, for
$z_0=(x_0,t_0)\in\Omega\times (0,T)$ and $P_{r_0}(z_0)\subset\Omega\times (0, T)$,
  \begin{equation}\label{eps0}
    \Big( r_0^{-2} \int_{P_{r_0}(z_0)} |\mathbf{u}|^3 \Big)^{1/3}
    + \Big( r_0^{-2} \int_{P_{r_0}(z_0)} |P|^{\frac{3}{2}} \Big)^{2/3}
    + \Big( r_0^{-2} \int_{P_{r_0}(z_0)} |\nabla \mathbf{d}|^3 \Big)^{1/3}
    \leq \epsilon_0,
  \end{equation}
  and
  \begin{equation}\label{l3infinity}
\|\nabla \mathbf{d}\|_{L^\infty_tL^3_x(P_{r_0}(z_0))}
< \frac{1-\delta}{\mathcal{C}(3)\footnotemark},
  \end{equation}
then $(\mathbf{u},\mathbf{d}) \in C^{\infty}(P_{\frac{r_0}4}(z_0),
\mathbb R^3\times \mathbb{S}^2)$, and the following estimate holds:
  \begin{equation} \label{cm:estimate}
\|(\mathbf{u},\mathbf{d})\|_{C^m(P_{\frac{r_0}4}(z_0))}
\leq C(m,r_0, \epsilon_0), \quad \forall\ m\ge 0.
  \end{equation}
\end{theorem}

\footnotetext{For the embedding $W^{1,2}(\Omega) \hookrightarrow L^6(\Omega)$,
where $\Omega \subset \mathbb{R}^3$, define \emph{the optimal Sobolev
embedding constant} $\mathcal{C}(3)$ by
\begin{equation*}
  \mathcal{C}(3) = \inf\big\{ \frac{\|\nabla f\|_{L^2(\Omega)}}
{\|f\|_{L^6(\Omega)}}: f \ne 0, f \in C_0^\infty(\mathbb{R}^3) \big\}.
\end{equation*}
It is noted that given $\Omega$ one may explicitly calculate $\mathcal{C}(3)$.}

\begin{remark} \rm
The proof of Theorem~\ref{smooth:solutions} employs a decay lemma,
Lemma \ref{decayThm} and potential estimates between Morrey spaces
similar to those developed for harmonic maps by Huang and
Wang \cite{huang_notes_2010}. For the complete details, please consult
the recent article \cite{hineman_well-posedness_2012}.
\end{remark}

\begin{lemma}  \label{decayThm}
For any $\delta>0$, there exist $\epsilon_0 > 0$ and
$\theta_0 \in (0,\frac{1}{2})$ such that
if $(\mathbf{u},P,\mathbf{d}): \Omega\times (0,T)
\to\mathbb R^3\times\mathbb R\times \mathbb{S}^2$ is a suitable weak
solution of \eqref{LLF},  and satisfies,
for $z_0=(x_0,t_0)\in\Omega\times (0,T)$ and
$P_{r_0}(z_0)\subset\Omega\times (0, T)$,
both \eqref{eps0} and \eqref{l3infinity}, then it holds that
  \begin{equation}    \label{decay:ineq}
    \begin{aligned}
&\Big( (\theta_0 r_0)^{-2} \int_{P_{\theta_0r_0}(z_0)} |\mathbf{u}|^3 \Big)^{1/3}
 +\Big( (\theta_0 r_0)^{-2} \int_{P_{\theta_0r_0}(z_0)} |P|^{\frac{3}{2}}
 \Big)^{2/3} \\
&+\Big( (\theta_0 r_0)^{-2} \int_{P_{\theta_0r_0}(z_0)} |\nabla \mathbf{d}|^3
  \Big)^{1/3}\\
&\leq \frac{1}{2} \Big[
        \Big( r_0^{-2} \int_{P_{r_0}(z_0)} |\mathbf{u}|^3 \Big)^{1/3}
        +\Big( r_0^{-2} \int_{P_{r_0}(z_0)} |P|^{\frac{3}{2}} \Big)^{2/3}
        +\Big( r_0^{-2} \int_{P_{r_0}(z_0)} |\nabla \mathbf{d}|^3 \Big)^{1/3}
      \Big].
    \end{aligned}
  \end{equation}
\end{lemma}

\begin{proof}[Outline of Proof]
Lemma \ref{decayThm} is proven by contradiction. Specifically, a sequence
of suitable weak solutions satisfying the decay inequality~\eqref{decay:ineq}
is constructed and it is shown that its $L^3_tL^3_x$-strong limit does
not satisfy the decay inequality \eqref{decay:ineq}. To achieve this goal, one
uses

(1) The generalized energy inequality described in Lemma
\ref{local_energy_inequality} to select an $L^2$-weakly convergent
subsequence of suitable weak solutions. The limit of such a
subsequence satisfies
\begin{gather*}
        \partial_t \mathbf{v} - \Delta \mathbf{v} + \nabla Q = 0, \\
        \nabla \cdot \mathbf{v} = 0, \\
        \partial_t \mathbf{e}- \Delta \mathbf{e} = 0.
\end{gather*}
    and hence through the Sobolev inequality and standard estimates on the
linear Stokes and heat equations one finds the limit
    $(\mathbf{v}, Q, \mathbf{e})$ satisfies
    \begin{equation}      \label{cacciopoli:est}
      \begin{gathered}
        \theta^{-2} \int_{P_\theta} (|\mathbf{v}|^3
+ |\nabla \mathbf{e}|^3) \leq C  \theta^{3} \int_{P_{3/4}}(|\mathbf{v}|^3
+|\nabla \mathbf{e}|^3)
 \le C\theta^3, \quad \theta \in (0,\frac12)
\\
\theta^{-2}\int_{P_\theta}|Q|^{3/2}
\leq C\theta\int_{P_{3/4}}|Q|^{3/2}\le C\theta \quad\theta \in (0,\frac12).
      \end{gathered}
    \end{equation}

(2)  Aubin-Lions Lemma \cite{temam_navier-stokes_2001}:

    \begin{lemma}\label{aubin}
      Let $X_0 \subset X \subset X_1$ be Banach spaces such that $X_0$
is compactly embedded in $X$, $X$ is continuously embedded
      in $X_1$, and $X_0$, $X_1$ are reflexive.  Then for
$1 < \alpha_0, \alpha_1 < \infty$,
      \begin{equation*}
\big\{u \in L^{\alpha_0}(0,T;X_0) :  \partial_t u \in L^{\alpha_1}(0,T;X_1)
\big\} \text{ is compactly embedded in }
L^{\alpha_0}(0,T;X).
      \end{equation*}
    \end{lemma}
allows the weak convergence of $v_i \to v$ and $\nabla e_i \to e_i$ to
be upgraded to strong convergence in $L^2_tL^2_x$ and after interpolation
    strong convergence in $L^3_tL^3_x$. To apply the Aubin-Lions lemma
one must employ the appropriate $W^{k,p}_\alpha$ estimates for auxiliary Stokes
    and Heat equations.

(3) Separate pressure estimate:

   \begin{lemma} \label{pressure:est}
 Suppose that $(\mathbf{u},P,\mathbf{d})$ is a suitable weak solution
of \eqref{LLF} on $P_1$.
      Then for any $0<r\le 1$ and $\tau \in (0,r/2)$, it holds that
      \begin{equation}     \label{pressure:est:1}
\frac{1}{\tau^2} \int_{P_\tau} |P|^{3/2}
        \leq C\Big[
          \big( \frac{r}{\tau} \big)^2 \frac{1}{r^2}
          \int_{P_r} (|\mathbf{u} - \mathbf{u}_{r}(t)|^3 + |\nabla \mathbf{d}|^3)
          + \big( \frac{\tau}{r}\big) \frac{1}{r^2}
          \int_{P_r} |P|^{3/2}
        \Big],
      \end{equation}
where $ \mathbf{u}_{r}(t) = \frac{1}{|B_r|} \int_{B_r} \mathbf{u}(x,t) $
for $- r^2 \leq t \leq 0$.
      In particular, it holds that
      \begin{equation}        \label{pressure:est:2}
        \begin{aligned}
          \frac{1}{\tau^2} \int_{P_\tau} |P|^{3/2}
          &\leq C  \big(\frac{r}{\tau}\big)^2
          \Big( \sup_{- r^2\leq t \leq 0} \frac{1}{r}
\int_{B_r} |\mathbf{u}|^2 \Big)^{3/4}
          \Big( \frac{1}{r} \int_{P_r} |\nabla u|^2 \Big)^{3/4} \\
          &+C\Big[
         \big(\frac{r}{\tau}\big)^2 \frac{1}{r^2}
         \int_{P_r} |\nabla \mathbf{d}|^3
         +\big(\frac{\tau}{r}\big) \frac{1}{r^2}
         \int_{P_r} |P|^{3/2}
       \Big].
     \end{aligned}
   \end{equation}
 \end{lemma}
\end{proof}


\noindent {\bf Step 4: Passage to the limit.}
Based on the estimates of $(\mathbf{u}^k, \mathbf{d}^k)$, we may assume,
after taking subsequences, that
$ (\mathbf{u},\mathbf{d})\in \cap_{0<\delta<\tau_0}
C^\infty_b(\mathbb{R}^3\times [\delta, \tau_0], \mathbb{R}^3\times \mathbb{S}^2)$,
 with $ (\mathbf{u},\nabla \mathbf{d})\in
 L^\infty([0, \tau_0], L^3_{\rm uloc}(\mathbb{R}^3)$, such that
\begin{gather*}
(\mathbf{u}^k, \nabla \mathbf{d}^k)\to (\mathbf{u},\nabla \mathbf{d})
\quad \text{weakly in } L^3(\mathbb{R}^3\times [0,\tau_0]), \\
(\mathbf{u}^k,\mathbf{d}^k)\to (\mathbf{u}, \mathbf{d}) \quad \text{in }
 C^m(B_R\times [\delta, \tau_0]), \; \forall\, m\ge 0, R>0, \delta<\tau_0.
\end{gather*}
Sending $k\to\infty$ one may find that
$$
\sup_{0\le t\le \tau_0}
|||(\mathbf{u},\nabla \mathbf{d}) |||_{L^3_1(\mathbb{R}^3)}\le C\epsilon_0.
$$
We can check from \eqref{LLF} and that for any $R>0$,
$$
\|(\partial_t \mathbf{u}^k, \partial_t \mathbf{d}^k)\|_{L^{3/2}
([0,\tau_0], W^{-1,\frac32}(B_R))}\le C(R)<+\infty.
$$
This implies that
\begin{equation}\label{trace}
 (\mathbf{u}(t),\nabla \mathbf{d}(t))\to (\mathbf{u}_0,\nabla \mathbf{d}_0)
\quad {\text{strongly\ in}}\ L^3_{\rm loc}(\mathbb{R}^3)  {\text{ as }}
 t\downarrow 0.
\end{equation}
In particular, we have that
$(\mathbf{u}_0,\nabla \mathbf{d}_0)\in C^0_*([0,\tau_0], L^3_{\rm uloc}
(\mathbb{R}^3))$.
\medskip

\noindent {\bf Step 5: Characterization of the maximal time interval $T_0$.}
Let $T_0>\tau_0$ be the maximal time interval for
the solution $(\mathbf{u},\mathbf{d})$ constructed in step 4.
Suppose that $T_0<+\infty$ and \eqref{blowup_criterion} were false. Then there exists
$r_0>0$ so that
$$\limsup_{t \uparrow T_0} |||(\mathbf{u}(t), \nabla \mathbf{d}(t))
|||_{L_{r_0}^3(\mathbb{R}^3)}\le \epsilon_0.$$
In particular, there exists $r_1\in (0, r_0]$ such that
$$
\sup_{T_0-r_1^2\le t\le T_0} |||(\mathbf{u}(t),
\nabla \mathbf{d}(t)) |||_{L_{r_1}^3(\mathbb{R}^3)}\le \epsilon_0.
$$
Hence by Theorem \ref{smooth:solutions},
$(\mathbf{u},\mathbf{d})\in C^\infty_b(\mathbb{R}^3\times [0, T_0])
\cap L^\infty([0, T_0], L^3_{\rm uloc}(\mathbb{R}^3))$.
This contradicts the maximality of $T_0$. Hence \eqref{blowup_criterion} holds.
\medskip

\noindent {\bf Step 6: Uniqueness.}
Let $(\mathbf{u}_1, \mathbf{d}_1), (\mathbf{u}_2,\mathbf{d}_2)
:\mathbb{R}^3\times [0, T_0]\to\mathbb{R}^3\times \mathbb{S}^2$ be two solutions
of \eqref{LLF}, under the same initial condition $(\mathbf{u}_0,\mathbf{d}_0)$,
that satisfy the properties of Theorem \ref{wellposedness}.
We first show $(\mathbf{u}_1,\mathbf{d}_1)\equiv (\mathbf{u}_2,\mathbf{d}_2)$
in $\mathbb{R}^3\times [0,\tau_0]$. This can be done
by the argument of \cite{wang_well-posedness_2010} pages 15-16 and is recorded
in \cite{hineman_well-posedness_2012}. We repeat it here for completeness.

Set $\mathbf{u}=\mathbf{u}_1-\mathbf{u}_2$, $\mathbf{d}=\mathbf{d}_1-\mathbf{d}_2$.  Then $(\mathbf{u},\mathbf{d})$ satisfies
\begin{gather*}
\partial_t \mathbf{u}-\Delta \mathbf{u}
 = -\mathbb P\nabla\cdot[\mathbf{u}_1\otimes \mathbf{u}_1-\mathbf{u}_2\otimes
 \mathbf{u}_2 +\nabla \mathbf{d}_1\odot\nabla
 \mathbf{d}_1-\nabla \mathbf{d}_2\odot\nabla \mathbf{d}_2]\\
\partial_t \mathbf{d}-\Delta \mathbf{d}
 = -(\mathbf{u}_1\cdot\nabla \mathbf{d}_1-\mathbf{u}_2\cdot\nabla \mathbf{d}_2)
 +|\nabla \mathbf{d}_1|^2\mathbf{d}_1-|\nabla \mathbf{d}_2|^2 \mathbf{d}_2\\
 (\mathbf{u},\mathbf{d})|_{t=0}= (0,0).
\end{gather*}
By the Duhamel formula, we have
\begin{gather*}
 \mathbf{u}(t)=-\mathbb V[\mathbf{u}_1\otimes \mathbf{u}_1-\mathbf{u}_2
 \otimes \mathbf{u}_2 +\nabla \mathbf{d}_1\odot\nabla \mathbf{d}_1
 -\nabla \mathbf{d}_2\odot\nabla \mathbf{d}_2]\\
 \mathbf{d}(t)=-\mathbb S[(\mathbf{u}_1\cdot\nabla \mathbf{d}_1
 -\mathbf{u}_2\cdot\nabla \mathbf{d}_2)-(|\nabla \mathbf{d}_1|^2\mathbf{d}_1
 -|\nabla \mathbf{d}_2|^2 \mathbf{d}_2)],
\end{gather*}
where
\begin{gather*}
\mathbb Sf(t)=\int_0^t e^{-(t-s)\Delta} f(s)\,ds, \\
\mathbb Vf(t)=\int_0^t e^{-(t-s)\Delta}\mathbb P\nabla\cdot f(s)\,ds, \quad
 \forall f:\mathbb R^3\times [0,+\infty)\to\mathbb R^3.
\end{gather*}
Recall the three function spaces used in \cite{wang_well-posedness_2011}.
Let $\mathbf X_{\tau_0}$ denote the space of functions $f$ on
$\mathbb{R}^3\times [0,\tau_0]$ such that
$$
|||f|||_{\mathbf X_{\tau_0}}:=\sup_{0<t\le\tau_0}
\|f(t)\|_{L^\infty(\mathbb{R}^3)}+\|f\|_{X_{\tau_0}}<+\infty,
$$
where
$$
\|f\|_{\mathbf X_{\tau_0}}:=\sup_{0<t\le\tau_0}\sqrt{t}
\|\nabla f(t)\|_{L^\infty(\mathbb{R}^3)}
+\sup_{x\in\mathbb{R}^3, 0<r\le\sqrt{\tau_0}}(r^{-3}\int_{P_r(x, r^2)}
|\nabla f|^2)^{1/2},
$$
 $\mathbf Y_{\tau_0}$ denote the space of functions $g$ on
$\mathbb{R}^3\times [0,\tau_0]$ such that
$$
||g||_{\mathbf Y_{\tau_0}}:=\sup_{0<t\le\tau_0}t
\|g(t)\|_{L^\infty(\mathbb{R}^3)}
+\sup_{x\in\mathbb{R}^3, 0<r\le\sqrt{\tau_0}}r^{-3}\int_{P_r(x, r^2)}|g|
<+\infty,
$$
and $\mathbf Z_{\tau_0}$ the space of functions $h$ on
$\mathbb{R}^3\times [0,\tau_0]$ such that
$$
\|h\|_{\mathbf Z_{\tau_0}}:=\sup_{0<t\le\tau_0}
\sqrt{t}\|h(t)\|_{L^\infty(\mathbb{R}^3)}
+\sup_{x\in\mathbb{R}^3, 0<r\le\sqrt{\tau_0}}(r^{-3}
\int_{P_r(x, r^2)}|h|^2)^{1/2}<+\infty.
$$
Since $(\mathbf{u}_i,\mathbf{d}_i)\in L^\infty([0,\tau_0],
L^2(\mathbb{R}^3)\times \dot{W}^{1,2}(\mathbb{R}^3))$ satisfies
\eqref{small_condition0} for $i=1,2$,
Theorem \ref{smooth:solutions} and the H\"older inequality imply that
$\mathbf{u}_i\in \mathbf Z_{\tau_0}, \mathbf{d}_i\in \mathbf X_{\tau_0}$
for $i=1,2$, and
$$
\sum_{i=1}^2 (\|\mathbf{u}_i\|_{\mathbf Z_{\tau_0}}
+\|\mathbf{d}_i\|_{\mathbf X_{\tau_0}})\le C\epsilon_0.
$$
It follows from Lemma 3.1 and Lemma 4.1 of \cite{wang_well-posedness_2011} that
\begin{align*}
 &\|\mathbf{u}\|_{\mathbf Z_{\tau_0}}+|||\mathbf{d}|||_{\mathbf X_{\tau_0}}\\
&\lesssim \Big\|(|\mathbf{u}_1|+|\mathbf{u}_2|)|\mathbf{u}|+(|\nabla \mathbf{d}_1|+|\nabla \mathbf{d}_2|)|\nabla \mathbf{d}|\Big\|_{\mathbf Y_{\tau_0}}\\
&\quad +\Big\| |\mathbf{u}||\nabla \mathbf{d}_2|
  +|\mathbf{u}_1||\nabla \mathbf{d}|+(|\nabla \mathbf{d}_1|
  +|\nabla \mathbf{d}_2|)|\nabla \mathbf{d}|
  +|\nabla \mathbf{d}_2|^2|\mathbf{d}|\Big\|_{\mathbf Y_{\tau_0}}\\
&\lesssim \Big[\sum_{i=1}^2(\|\mathbf{d}_i\|_{\mathbf X_{\tau_0}}
 +\|\mathbf{u}_i\|_{\mathbf Z_{\tau_0}})\Big]\|\mathbf{u}\|_{\mathbf Z_{\tau_0}}
  +\Big[\sum_{i=1}^2(\|\mathbf{u}_i\|_{\mathbf Z_{\tau_0}}+\|\mathbf{d}_i\|_{\mathbf X_{\tau_0}})\Big]
 |||\mathbf{d}|||_{\mathbf X_{\tau_0}}\\
&\lesssim \epsilon_0 [\|\mathbf{u}\|_{\mathbf Z_{\tau_0}}
 +|||\mathbf{d}|||_{\mathbf X_{\tau_0}}].
\end{align*}
This clearly implies that
$(\mathbf{u}_1,\mathbf{d}_1)\equiv (\mathbf{u}_2,\mathbf{d}_2)$ in
$\mathbb{R}^3\times [0,\tau_0]$. Since $(\mathbf{u}_1, \mathbf{d}_1)$ and
$(\mathbf{u}_2,\mathbf{d}_2)$ are classical solutions of \eqref{LLF} in
$\mathbb{R}^3\times [\tau_0, T_0)$, and
$(\mathbf{u}_1,\mathbf{d}_1)=(\mathbf{u}_2,\mathbf{d}_2)$ at $t=\tau_0$, it is standard that $(\mathbf{u}_1,\mathbf{d}_1)\equiv  (\mathbf{u}_2,\mathbf{d}_2)$ in
$\mathbb{R}^3\times [\tau_0, T_0)$. The proof is complete.

\subsection{Open problems}

Again, as this analysis is preliminary, there are many open problems
surrounding the hydrodynamic flow of nematic liquid crystals in $\mathbb{R}^3$.
It seems that the ultimate goal of future investigations should be to
re-introduce terms removed from \eqref{el:pw:summary} to arrive at \eqref{LLF}.
 There do remain a number of open problems in the analysis of the system
\eqref{LLF}, we mention a particular one of interest:

(1) {\bf Partial regularity of suitable weak solutions in three dimensions.}
In the proof of Theorem \ref{smooth:solutions} it was required to prove a
quantitative   decay lemma, Lemma \ref{decayThm}.
To control certain terms one assumed \eqref{l3infinity}
(for further details, please consult
  \cite{hineman_well-posedness_2012}). It is an obvious question to ask if the
  assumption \eqref{l3infinity} can be eliminated. If one can overcome
the difficulties associated with the elimination of this condition and,
  following \cite{caffarelli_partial_1982}, proves a theorem concerning
the rate at which singularities develop for $|\nabla \mathbf{u}|$
  and $|\nabla^2 \mathbf{d}|$ one would have the following
\emph{partial regularity} theorem.

  \begin{conjecture}[Partial Regularity]
    For any suitable weak solution of the system \eqref{LLF}
on an open set in space-time, the set
    \begin{equation*}
      \mathcal{S} = \text{``the singular set"} = \{(x,t):
\|(\mathbf{u},\nabla \mathbf{d})\|_{L_{\rm loc}^\infty(P_r(x,t))} = +\infty \}
    \end{equation*}
has $1$-dimensional parabolic Hausdorff measure 0; that is,
    \begin{equation*}
      \mathcal{P}^1(\mathcal{S}) = 0
      \text{ where }
      \mathcal{P}^k(X)
      := \lim_{\delta \to 0^+} \inf \Big\{ \sum_{i=1}^\infty r_i^k: X
\subset \bigcup_i P_{r_i}, r_i < \delta \Big\}.
    \end{equation*}
  \end{conjecture}

  Such a result would be analogous to the celebrated work of Caffarelli, Kohn,
and Nirenberg in \cite{caffarelli_partial_1982}
  and would extend the work of Lin-Lin-Wang \cite{lin_liquid_2009} and
Hardt-Kinderleher-Lin \cite{hardt_existence_1986}.

\subsection*{Conclusions}
Liquid crystals, and in general materials with fine structure will
require new tools in mathematics, modeling, and computation. This survey
has primarily focused on the mathematical difficulties and open problems
surrounding the Ericksen-Leslie equations for the hydrodynamic flow of
nematic liquid crystals. We have made the case that there many open problems
in the analysis and numerical of these equations that require new mathematics.

It is worth mentioning in closing that there are alternative models of
liquid crystals and their flow. For example, the $Q$-tensor theory of
De Gennes \cite{gennes_physics_1995}, \cite{majumdar_landau-gennes_2010}
and the micropolar theory of Eringen \cite{eringen_unified_1997}.
The analysis and numerical analysis of these models in also preliminary
 and offers additional directions for research.


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\end{thebibliography}

\end{document}