# On singularities of Ericksen-Leslie system in dimension three

Tao Huang111Department of Mathematics, Wayne State University, Detroit, MI
48202, USA. Peiyong Wang∗

###### Abstract

In this paper, we consider the initial and boundary value problem of Ericksen-
Leslie system modeling nematic liquid crystal flows in dimension three. Two
examples of singularity at finite time are constructed. The first example is
constructed in a special axisymmetric class with suitable axisymmetric initial
and boundary data, while the second example is constructed for an initial data
with small energy but nontrivial topology. A counter example of maximum
principle to the system is constructed by utilizing the Poiseuille flow in
dimension one.

##  1 Introduction

Nematic liquid crystals are composed of rod-like molecules characterized by
the average alignment of the long axes of neighboring molecules, which have
simplest structure among various types of liquid crystals. The dynamic theory
of nematic liquid crystals has been first proposed by Ericksen [5] and Leslie
[12] in the 1960’s, which is a macroscopic continuum description of the time
evolution of both the flow velocity field and the orientation order parameter
of rod-like liquid crystals. More precisely, we would consider the following
Ericksen-Leslie system in $\Omega\times(0,\infty)$, where
$\Omega\subset\mathbb{R}^{3}$ is a bounded domain with smooth boundary

$\begin{cases}u_{t}+u\cdot\nabla u+\nabla P=-\nabla\cdot\big{(}\nabla
d\odot\nabla d\big{)}+\nabla\cdot(\sigma^{L}(u,d)),\\\ \nabla\cdot u=0,\\\
\lambda_{1}(d_{t}+u\cdot\nabla d-\Lambda d)+\lambda_{2}Ad=\Delta d+|\nabla
d|^{2}d+\lambda_{2}(d^{T}Ad)d,\end{cases}$ (1.1)

where $u(x,t):\Omega\times(0,\infty)\rightarrow\mathbb{R}^{3}$ is the fluid
velocity field, $d(x,t):\Omega\times(0,\infty)\rightarrow\mathbb{S}^{2}$ is
the orientation order parameter of nematic material at $(x,t)$, and
$P(x,t):\Omega\times(0,\infty)\rightarrow\mathbb{R}$ is the pressure. Denote
$u=(u_{1},u_{2},u_{3})\in\mathbb{R}^{3}$ and
$d=(d_{1},d_{2},d_{3})\in\mathbb{S}^{2}$. Then

$\big{(}\nabla d\odot\nabla d\big{)}_{ij}=\nabla_{i}d\cdot\nabla_{j}d,$

and

$A_{ij}=\frac{1}{2}\left(\frac{\partial u_{j}}{\partial x_{i}}+\frac{\partial
u_{i}}{\partial
x_{j}}\right),\quad\Lambda_{ij}=\frac{1}{2}\left(\frac{\partial
u_{i}}{\partial x_{j}}-\frac{\partial u_{j}}{\partial x_{i}}\right),\quad
N_{i}=\partial_{t}d_{i}+u\cdot\nabla d_{i}-\Lambda_{ij}d_{j}.$

denote the rate of the strain tensor, the skew-symmetric part of the strain
rate, and the rigid rotation part of the direction changing rate by fluid
vorticity, respectively. The left side of the third equation in (1.1) is the
kinematic transport, which represents the effect of the macroscopic flow field
on the microscopic structure. The material coefficients $\lambda_{1}$ and
$\lambda_{2}$ reflect the molecular shape and the slippery part between the
fluid and the particles. The term with $\lambda_{1}$ represents the rigid
rotation of molecules, while the term with $\lambda_{2}$ stands for the
stretching of molecules by the flow. The viscous (Leslie) stress tensor
$\sigma^{L}$ has the following form (cf. [13])

$\sigma_{ij}^{L}(u,d)=\mu_{1}d_{k}d_{p}A_{kp}d_{i}d_{j}+\mu_{2}N_{i}d_{j}+\mu_{3}d_{i}N_{j}+\mu_{4}A_{ij}+\mu_{5}A_{ik}d_{k}d_{j}+\mu_{6}A_{jk}d_{k}d_{i}.$

The viscous coefficients $\mu_{i}$, $i=1,\cdots,6,$ are called the Leslie’s
coefficients and the following relations are often assumed in the literature

$\mu_{2}+\mu_{3}=\mu_{6}-\mu_{5}$ (1.2)
$\lambda_{1}=\mu_{3}-\mu_{2}>0,\quad\lambda_{2}=\mu_{6}-\mu_{5}$ (1.3)
$\mu_{4}>0,\quad 2\mu_{1}+3\mu_{4}+2\mu_{5}+2\mu_{6}>0,\quad
2\mu_{4}+\mu_{5}+\mu_{6}>\frac{\lambda_{2}^{2}}{\lambda_{1}}.$ (1.4)

The first relation is called the Parodi’s relation, which has been derived
from the Onsager reciprocal relations expressing the equality of certain
relations between flows and forces in thermodynamic systems out of equilibrium
(cf. [20]). The second set of relations are the compatibility conditions. The
third empirical relations are necessary to obtain the energy inequality (cf.
[13], [23]). Throughout the paper, we assume that (1.2), (1.3) and (1.4) are
valid for system (1.1).

We consider the initial data

$u(x,0)=u_{0},\quad d(x,0)=d_{0}$ (1.5)

with $\mbox{div\,}u_{0}=0$ and $|d_{0}|=1$, and the boundary data

$u(x,t)|_{\partial\Omega\times[0,\infty)}=0,\quad
d(x,t)|_{\partial\Omega\times[0,\infty)}=\mathbf{e}$ (1.6)

with compatible conditions

$u_{0}|_{\partial\Omega}=0,\quad d_{0}|_{\partial\Omega}=\mathbf{e}.$ (1.7)

Here $\mathbf{e}=(0,0,1)$ is a constant vector.

In dimension two, the existence of global weak solutions to the Cauchy problem
of (1.1) has been established in [9]. The weak solution has been proved to be
regular except for finitely many times (see also [8] for related results). The
uniqueness of such a weak solution has been proved in [14, 22]. We would like
to point out that the assumptions of Leslie coefficients in [9] is stronger
than ours, (1.4). However, their results are still valid with weaker
assumption (1.4) since we can prove the similar energy inequality (see Lemma
2.3 below for more details). In dimension three, the global well-posedness
combining with long time behaviors for the system (1.1) around equilibrium
under various assumptions on the Leslie coefficients has been studied in [23,
24, 7].

There is a simplified system that has been first proposed in [15] by
neglecting the Leslie stress. There have been many results on the existence
and partial regularity of this simplified system in [16, 18, 17, 10, 11] and
the references therein.

The paper is organized as follows. In Section 2, we will state our main
results. In Section 3, a special form of axisymmetric solution will be
derived. In Section 4, global existence and singularities at finite time will
be discussed. In Section 5, another example of finite time singularity will be
constructed with an initial data with small energy but with large topology. In
Section 6, we will construct a counter example to show that the maximum
principle is not necessarily valid for Ericksen-Leslie system via the
Poiseuille flow in dimension one.

##  2 Main results

Let $B_{1}^{n}\subset\mathbb{R}^{n}$ denote the unit ball centered at $0$.
Inspired by the results in [10], we first consider the domain
$\Omega=B_{1}^{2}\times[0,1]$ and an axisymmetric solution $(u,P,d)$ to the
Ericksen-Leslie system (1.1) in the following special form

$\begin{cases}u(r,\theta,z,t):=v(r,t)\mathbf{e}^{r}+w(z,t)\mathbf{e}^{3},\\\
d(r,\theta,z,t):=\sin\varphi(r,t)\mathbf{e}^{r}+\cos\varphi(r,t)\mathbf{e}^{3},\\\
P(r,\theta,z,t):=Q(r,t)+R(z,t).\end{cases}$ (2.1)

For any ${\bf x}=(x,y,z)\in\Omega$, we take the initial data as follows

$u_{0}({\bf x})=(x,y,-2z),$ (2.2)

and

$\displaystyle d_{0}({\bf
x})=\Big{(}\frac{x}{\sqrt{x^{2}+y^{2}}}\sin\varphi_{0}\big{(}\sqrt{x^{2}+y^{2}}\big{)},\frac{y}{\sqrt{x^{2}+y^{2}}}\sin\varphi_{0}\big{(}\sqrt{x^{2}+y^{2}}\big{)},\cos\varphi_{0}\big{(}\sqrt{x^{2}+y^{2}}\big{)}\Big{)},$
(2.3)

with $\varphi_{0}\in C^{\infty}([0,1])$ and $\varphi_{0}(0)=0$. To derive the
special axisymmetric form of system (1.1), we also need to consider the
following boundary conditions

$\begin{cases}\ u({\bf x},t)=u_{0}({\bf x})&\ {\bf x}\in\partial\Omega,\
t>0,\\\ \ d({\bf x},t)=d_{0}({\bf x})&\ {\bf x}\in\partial
B_{1}^{2}\times[0,1],\ t>0,\\\ \frac{\partial d}{\partial z}({\bf x},t)=0&\
{\bf x}\in B_{1}^{2}\times\\{0,1\\},\ t>0.\end{cases}$ (2.4)

Hence the Ericksen-Leslie system (1.1) becomes (see Section 3.1 for details)

$v(r,t)=r,\quad w(z,t)=-2z,$
$\lambda_{1}(\varphi_{t}+r\varphi_{r})=\varphi_{rr}+\frac{1}{r}\varphi_{r}-\frac{\sin(2\varphi)}{2r^{2}}-3\lambda_{2}\sin\varphi\cos\varphi$
(2.5)

Our first result concerns the global existence of the equation (2.5), which
implies the global existence of the Ericksen-Leslie system (1.1).

###### Theorem 2.1

Suppose $\phi_{0}\in C^{\infty}([0,1])$ satisfies $\varphi_{0}(0)=0$ and

$0\leq\varphi_{0}(r)<\pi,$

for all $r\in[0,1]$. Then there is no finite time blowup for the smooth
solution to (2.5).

We can also construction an example of singularity to the system (1.1) in this
axisymmetric class.

###### Theorem 2.2

There exists $\phi_{0}\in C^{\infty}([0,1])$, with $\phi_{0}(0)=0$ and
$\varphi_{0}(1)>\pi$, such that the short time smooth solution $\varphi$ to
(2.5), with initial and boundary conditions (2.2)-(2.4), must blow up at
$T_{0}$ for some $0<T_{0}=T_{0}(\phi_{0})<+\infty$. More precisely,
$\varphi_{r}(0,t)\rightarrow\infty$ as $t\rightarrow T_{0}^{-}$.

The main idea to prove the Theorems 2.1 and 2.2 is to construct suitable
supersolution and subsolution, which has been first utilized in the
breakthrough work [1] when the authors studied the singularity of harmonic
heat flows. Similar results have been proved for simplified Ericksen-Leslie
system in [10]. However, there are still two difficulties for the full
Ericksen-Leslie system (1.1). The first one is that the comparison principle
and maximum principle may not be valid for the system (1.1) because of the
terms related to the skew-symmetric part $\Lambda$ of the strain rate and the
symmetric part $A$. Fortunately, it holds $\Lambda=0$, and $A$ is a diagonal
matrix for our special axisymmetric solution (2.1) (cf. Section 3.1).
Therefore, we still have comparison and maximum principles for equation (2.5)
(cf. Section 3.2). The second difficulty comes from the nonlinear term
$-3\lambda_{2}\sin\varphi\cos\varphi$ in (2.5). It is not hard to see that
this term will not produce any singularity. However, since we don’t know the
sign of $\lambda_{2}$, it brings more technique difficulties to the
construction of a supersolution and a subsolution.

We also want to point out that the special class of axisymmetric solutions may
not satisfy the following energy inequality, which has been proved in [23] and
[3].

###### Lemma 2.3

Suppose that $(u,d)$ is a regular solution to the system (1.1) with the
initial and boundary conditions (1.5) and (1.6) under the assumptions (1.2),
(1.3) and (1.4). For any $t\in(0,\infty)$, the following energy equality holds

$\frac{d}{dt}\frac{1}{2}\int_{\Omega}\left(|u|^{2}+|\nabla
d|^{2}\right)\,\leq-\int_{\Omega}\left(\alpha_{0}|\nabla
u|^{2}+\frac{1}{\lambda_{1}}\left|\Delta d+|\nabla
d|^{2}d\right|^{2}\right)\,,$ (2.6)

for some constant $\alpha_{0}>0$.

Therefore, we would like to construct another example of finite time
singularity of (1.1) for a generic initial data, in which the solution
satisfies the energy dissipation inequality. For simplicity, we take
$\Omega=B_{1}^{3}$, the unit ball in dimension three, and denote

$C^{\infty}_{0,{\rm{div}}}(B^{3}_{1},\mathbb{R}^{3}):=\Big{\\{}v\in
C^{\infty}(B_{1}^{3},\mathbb{R}^{3})\ \big{|}\ \nabla\cdot v=0\Big{\\}},$
$C_{\bf e}^{\infty}(B_{1}^{3},\mathbb{S}^{2}):=\Big{\\{}d\in
C^{\infty}(B_{1}^{3},\mathbb{S}^{2})\ \big{|}\ d={\bf e}\ {\rm{on}}\ \partial
B_{1}^{3}\Big{\\}}.$

For continuous maps $f,g\in
C\big{(}\overline{B_{1}^{3}},\mathbb{S}^{2}\big{)}$, with $f=g$ on $\partial
B_{1}^{3}$, we say that $f$ is homotopic to $g$ relative to $\partial
B_{1}^{3}$ if there exists a continuous map $\Phi\in
C\big{(}\overline{B_{1}^{3}}\times[0,1],\mathbb{S}^{2}\big{)}$ such that
$\Phi(\cdot,t)=f(\cdot)=g(\cdot)$ on $\partial B_{1}^{3}$, for all $0\leq
t\leq 1$, and $\Phi(\cdot,0)=f(\cdot)$ and $\Phi(\cdot,1)=g(\cdot)$ in
$B_{1}^{3}$. Now we are ready to state our main result.

###### Theorem 2.4

There exists $\epsilon_{0}>0$ such that if $u_{0}\in
C^{\infty}_{0,{\rm{div}}}(B_{1}^{3},\mathbb{R}^{3})$ and $d_{0}\in C_{\bf
e}^{\infty}(B_{1}^{3},\mathbb{S}^{2})$ satisfy that $d_{0}$ is not homotopic
to the constant map ${\bf e}:B_{1}^{3}\to\mathbb{S}^{2}$ relative to $\partial
B_{1}^{3}$, and

$E(u_{0},d_{0}):=\frac{1}{2}\int_{B_{1}^{3}}(|u_{0}|^{2}+|\nabla
d_{0}|^{2})\leq\epsilon_{0}^{2}.$ (2.7)

Then the short time smooth solution
$(u,d,P):B_{1}^{3}\times[0,T)\to\mathbb{R}^{3}\times\mathbb{S}^{2}\times\mathbb{R}$
to the Ericksen-Leslie system (1.1), with the initial-boundary condition (1.5)
and (1.6) under the assumptions (1.2), (1.3) and (1.4) must blow up before
time $T=1$.

Due to the energy estimate (2.6), the proof of Theorem 2.4 is quite similar to
those in [10] for the simplified system. For completeness, we will briefly
sketch the main steps of the proof in Sec 5.

We would like to remark that there does exist $(u_{0},d_{0})\in
C^{\infty}_{0,{\rm{div}}}(B_{1}^{3},\mathbb{R}^{3})\times C^{\infty}_{\bf
e}(B_{1}^{3},\mathbb{S}^{2})$ satisfying the conditions of Theorem 2.4 which
has been previously used by [10] in construction of finite time singularity of
simplified liquid crystal flows (see also [19], [4] for other applications).
More precisely, let
$H(z,w)=(|z|^{2}-|w|^{2},2zw):\mathbb{S}^{3}\equiv\big{\\{}(z,w)\in\mathbb{C}\times\mathbb{C}:|z|^{2}+|w|^{2}=1\big{\\}}\to\mathbb{S}^{2}\subset\mathbb{R}\times\mathbb{C}$
be the Hopf map. Let $D_{\lambda}({\bf x})=\lambda{\bf
x}:\mathbb{R}^{3}\to\mathbb{R}^{3}$ be the dilation map for $\lambda>0$,
$\Pi:\mathbb{S}^{3}\to\overline{\mathbb{R}^{3}}$ be the stereographic
projection map from ${\bf e}$, and $\Psi_{\lambda}=\Pi^{-1}\circ
D_{\lambda}\circ\Pi:\mathbb{S}^{3}\to\mathbb{S}^{3}$. Then direct calculations
imply that the Dirichlet energy of
$H\circ\Psi_{\lambda}:\mathbb{S}^{3}\to\mathbb{S}^{2}$ satisfies

$\lim_{\lambda\rightarrow\infty}\int_{\mathbb{S}^{3}}\big{|}\nabla(H\circ\Psi_{\lambda})\big{|}^{2}\,d\sigma=0.$

Moreover, it is easy to see that $H\circ\Psi_{\lambda}$ is not homotopic to
the constant map ${\bf e}:\mathbb{S}^{3}\to\mathbb{S}^{2}$. Let $\Phi\in
C^{\infty}\big{(}\overline{B_{1}^{3}},\mathbb{S}^{3}\big{)}$ such that
$\Phi:B_{1}^{3}\to\mathbb{S}^{3}\setminus\\{\bf e\\}$ is a diffeomorphism and
$\Phi={\bf e}$ on $\partial B_{1}^{3}$. Now we can check that for any $u\in
C^{\infty}_{0,{\rm{div}}}(B_{1}^{3},\mathbb{R}^{3})$, since

$\lim_{\lambda\rightarrow\infty}E(\lambda^{-1}u,H\circ\Psi_{\lambda}\circ\Phi)=0,$

we can find a sufficiently large $\lambda_{0}>0$ depending on $u$, $H$, and
$\Phi$ such that

$(u_{0},d_{0}):=(\lambda_{0}^{-1}u,H\circ\Psi_{\lambda_{0}}\circ\Phi):B_{1}^{3}\to\mathbb{R}^{3}\times\mathbb{S}^{2}$

satisfies the condition (2.7) of Theorem 2.4, and $d_{0}$ is not homotopic to
the constant map ${\bf e}$ relative to $\partial B_{1}^{3}$.

It is also an interesting topic to investigate more on the homotopic condition
of $d_{0}$ in Theorem 2.4. If we assume that $d_{0}$ is homotopic to the
constant map ${\bf e}$ relative to $\partial B_{1}^{3}$, then the angle
$\varphi$ between $d_{0}$ and ${\bf e}$ should belong to $[0,\pi)$. Because
the maximum principle (cf. Lemma 3.4) may not be valid for the system (1.1),
it is natural to ask if $d_{0}$ is homotopic to the constant map ${\bf e}$
relative to $\partial B_{1}^{3}$ and the initial energy of the system is
small, whether the angle $\varphi$ between $d_{0}$ and ${\bf e}$ can go beyond
$\pi$ at some time $t$. If this happens, $d(\cdot,t)$ is not homotopic to the
constant map ${\bf e}$ relative to $\partial B_{1}^{3}$, which, combined with
the energy inequality (2.6) and Theorem 2.4, implies the finite time
singularity. Unfortunately, we can’t verify this argument in dimension three
due to technical difficulties. However, utilizing the Poiseuille flows in
dimension one, we can construct a counter example of the maximum principle
(cf. Lemma 3.4), which does imply the maximum principle may not be valid in
the general case of Ericksen-Leslie system (1.1) (see Section 6 for more
details).

##  3 Axisymmetric solutions

Let $(r,\theta,z)$ denote the cylindrical coordinates of $\mathbb{R}^{3}$, and
set

$\mathbf{e}^{r}=(\cos\theta,\sin\theta,0)^{T},\quad\mathbf{e}^{\theta}=(-\sin\theta,\cos\theta,0)^{T},\quad\mathbf{e}^{3}=(0,0,1)^{T}$

as the canonical orthonormal base of $\mathbb{R}^{3}$ in the cylindrical
coordinates. For $\alpha\in[0,2\pi]$, let $R_{\alpha}\in{\rm{SO}}(3)$ denote
the rotation map of angle $\alpha$ with respect to the $z$-axis. Recall that a
vector field $v:\mathbb{R}^{3}\to\mathbb{R}^{3}$ is axisymmetric if

$R_{\alpha}^{-1}\circ v\circ R_{\alpha}=v,\ \forall\ \alpha\in[0,2\pi].$

Hence any axisymmetric vector field $v$ can be written as

$v(r,\theta,z)=v^{r}(r,z)\mathbf{e}^{r}+v^{\theta}(r,z)\mathbf{e}^{\theta}+v^{3}(r,z)\mathbf{e}^{3}.$
(3.1)

If, in addition, $v^{\theta}\equiv 0$, we say $v$ is axisymmetric without
swirls. A solution $(u,d,P)$ to the Ericksen-Leslie system (1.1) is said to be
axisymmetric without swirls, if it holds that

$\begin{cases}u(r,\theta,z,t)=u^{r}(r,z,t)\mathbf{e}^{r}+u^{3}(r,z,t)\mathbf{e}^{3},\\\
d(r,\theta,z,t)=\sin\varphi(r,z,t)\mathbf{e}^{r}+\cos\varphi(r,z,t)\mathbf{e}^{3},\\\
P(r,\theta,z,t)=P(r,z,t).\end{cases}$

A domain $\Omega\subset\mathbb{R}^{3}$ is axisymmetric if it is invariant
under a rotation map $R_{\alpha}$ for any $\alpha\in[0,2\pi]$.

###  3.1 Derivation of a special axisymmetric system

It is not hard to see that the initial condition (2.2) for $(v,w,\phi)$
reduces to

$\begin{cases}v|_{t=0}=r,\ \ \ \ \ \ \ 0\leq r\leq 1,\\\ w|_{t=0}=-2z,\ \
0\leq z\leq 1,\\\ \varphi|_{t=0}=\varphi_{0}(r),\ 0\leq r\leq 1,\end{cases}$
(3.2)

for some $\varphi_{0}\in C^{\infty}([0,1])$, with $\varphi_{0}(0)=0$. The
boundary condition (2.4) becomes

$\begin{split}\begin{cases}v(0,t)=0\\\
v(1,t)=1,\end{cases}&\quad\begin{cases}w(0,t)=0\\\
w(1,t)=-2,\end{cases}\quad\begin{cases}\varphi(0,t)=0\\\
\varphi(1,t)=\varphi_{0}(1).\end{cases}\end{split}$ (3.3)

###### Lemma 3.1

For $0<T\leq+\infty$, suppose that $v,w\in C^{\infty}([0,1]\times[0,T))$
satisfies

$\displaystyle\frac{1}{r}(rv)_{r}+w_{z}=0,\ (r,z)\in[0,1]\times[0,1],$ (3.4)

and

$\begin{split}\begin{cases}v(0,t)=0\\\
v(1,t)=1,\end{cases}&\quad\begin{cases}w(0,t)=0\\\
w(1,t)=-2.\end{cases}\end{split}$ (3.5)

Then $v(r,t)=r$ for any $(r,t)\in[0,1]\times[0,T)$, and $w(z,t)=-2z$ for any
$(z,t)\in[0,1]\times[0,T)$.

Proof. Differentiating (3.4) with respect to $z$ yields

$\displaystyle w_{zz}(z,t)=0$

so that $w(z,t)=a_{1}(t)z+a_{2}(t)$ for some functions $a_{1}(t)$ and
$a_{2}(t)$. Since $w(0,t)=0$ and $w(1,t)=-2$, we see that $a_{2}(t)\equiv 0$
and $a_{1}(t)\equiv-2$. Thus $w(z,t)=-2z.$

Similarly, differentiating (3.4) with respect to $r$ yields

$\displaystyle\big{(}\frac{1}{r}(rv)_{r}\big{)}_{r}(r,t)=0,$

which implies that $rv(r,t)=b_{1}(t)r^{2}+b_{2}(t)$ for some functions
$b_{1}(t)$ and $b_{2}(t)$. Since $v(0,t)=0$ and $v(1,t)=1$, we see that
$b_{2}(t)\equiv 0$ and $b_{1}(t)\equiv 1$. Thus $v(r,t)=r.$ The proof is
complete. $\Box$

Hence we have

$u=(x,y,-2z),\quad A=\left[\begin{array}[]{rrr}1&0&0\\\ 0&1&0\\\
0&0&-2\end{array}\right],\qquad\Lambda={\bf 0}_{3\times 3}.$
$Ad=\big{(}\cos\theta\sin\varphi,\sin\theta\sin\varphi,-2\cos\varphi\big{)}^{T}$
$(d^{T}Ad)d=(\sin^{2}\varphi-2\cos^{2}\varphi)\big{(}\cos\theta\sin\varphi,\sin\theta\sin\varphi,\cos\varphi\big{)}^{T}.$

Then

$(d^{T}Ad)d-Ad=-3\cos\varphi\sin\varphi\big{(}\cos\theta\cos\varphi,\sin\theta\cos\varphi,-\sin\varphi\big{)}^{T},$

$\displaystyle\Delta d+|\nabla
d|^{2}d=\big{(}\varphi_{rr}+\frac{1}{r}\varphi_{r}-\frac{\sin(2\varphi)}{2r^{2}}\big{)}\big{(}\cos\theta\cos\varphi,\sin\theta\cos\varphi,-\sin\varphi\big{)}^{T},$
$\displaystyle d_{t}+u\cdot\nabla
d=\left(\varphi_{t}+r\varphi_{r}\right)(\cos\theta\cos\varphi,\sin\theta\cos\varphi,-\sin\varphi)^{T}.$

Combining all above, the equation of $d$ in system (1.1) becomes

$\lambda_{1}(\varphi_{t}+r\varphi_{r})=\varphi_{rr}+\frac{1}{r}\varphi_{r}-\frac{\sin(2\varphi)}{2r^{2}}-3\lambda_{2}\sin\varphi\cos\varphi$
(3.6)

If we assume $\lambda_{2}=0$, this is exactly the equation in [10] for the
simplified liquid crystal flows. In this paper, we would investigate the more
general case when $\lambda_{2}\neq 0$.

The proof of the existence of a local smooth solution to equation (3.6)
proceeds similarly as in [10] with minor modification to take care of the
extra (smooth) external force term $3\lambda_{2}\sin\varphi\cos\varphi$.
Therefore, we omit the proof here.

###  3.2 Apriori estimates

In general, we can not expect the energy inequality like (2.6) for the
equation (3.6) with the initial and boundary conditions (3.2) and (3.3).
However, the following apriori estimate is still valid for (3.6).

###### Lemma 3.2

For any smooth solution $\varphi(r,t):[0,1]\times[0,T)\rightarrow\mathbb{R}$
to the equation (3.6) with initial and boundary conditions (3.2) and (3.3), we
have the following estimate

$\int_{0}^{1}\left(|\varphi_{r}|^{2}+\frac{\sin^{2}\varphi}{r^{2}}\right)r\,dr+\int_{0}^{T}\int_{0}^{1}|\varphi_{t}|^{2}r\,dr\leq
C(\varphi_{0},T).$ (3.7)

Proof. Multiplying the equation (4.1) by $\varphi_{t}r$ and integrating with
respect to $r$ over $[0,1]$, we obtain

$\begin{split}&\frac{d}{dt}\frac{1}{2}\int_{0}^{1}\left(|\varphi_{r}|^{2}+\frac{\sin^{2}\varphi}{r^{2}}\right)r\,dr+\lambda_{1}\int_{0}^{1}|\varphi_{t}|^{2}r\,dr\\\
=&-\lambda_{1}\int_{0}^{1}\varphi_{r}\varphi_{t}r^{2}\,dr-3\lambda_{2}\int_{0}^{1}\sin\varphi\cos\varphi\varphi_{t}r\,dr\\\
\leq&\frac{\lambda_{1}}{2}\int_{0}^{1}|\varphi_{t}|^{2}r\,dr+C\int_{0}^{1}\left(|\varphi_{r}|^{2}+\frac{\sin^{2}\varphi}{r^{2}}\right)r\,dr.\end{split}$

Standard Gronwall arguments should give us the desired energy estimates.

$\Box$

We also need the following lemma.

######  Lemma 3.3 (comparison principle)

Suppose the functions $\varphi$, $f$ and $g$ are a smooth solution,
subsolution and supersolution to (3.6) on $[0,1]\times[0,T)$ respectively, and

$f(r,t)\leq\varphi(r,t)\leq g(r,t)\ {\rm{on}}\
([0,1]\times\\{0\\})\cup(\\{0,1\\}\times(0,T)).$

Then we have

$f(r,t)\leq\varphi(r,t)\leq g(r,t),\ \forall\ (r,t)\in[0,1]\times[0,T).$ (3.8)

Proof. Set $\bar{f}=f-\varphi$. Then $\bar{f}$ satisfies

$\displaystyle\bar{f}_{t}+r\bar{f}_{r}\geq~{}\bar{f}_{rr}+\frac{\bar{f}_{r}}{r}+p_{2}(r,t)\bar{f},$

where

$\displaystyle
p_{2}(r,t):=-\frac{\sin(2f(r,t))-\sin(2\varphi(r,t))}{2r^{2}(f(r,t)-\varphi(r,t))}-3\lambda_{2}\frac{\sin(2f(r,t))-\sin(2\varphi(r,t))}{2(f(r,t)-\varphi(r,t))},$

and

$\bar{f}(r,t)\leq 0\ {\rm{on}}\
([0,1]\times\\{0\\})\cup(\\{0,1\\}\times(0,T)).$

For any $t\in(0,T)$, there exists a small $r_{2}\in(0,1)$ such that
$p_{2}(r,\tau)<0$ on $(r,\tau)\in(0,r_{2})\times(0,t)$. Combining it with the
fact that $p_{2}(r,\tau)$ is bounded on $(r_{2},1)\times(0,t)$, we conclude
that $p_{2}(r,\tau)$ is bounded from above on $(r,\tau)\in(0,1)\times(0,t)$.
By the standard maximum principle (see [6] or [21]), we conclude that
$\bar{f}\leq 0$ or $\varphi\geq f$ on $[0,1]\times(0,T)$. Similarly, one can
prove $\varphi\leq g$.

$\Box$

A natural corollary of the comparison lemma is the following maximum
principle.

######  Lemma 3.4 (maximum principle)

Suppose that $\varphi(r,t):[0,1]\times[0,T)\rightarrow\mathbb{R}$ is a smooth
solution to equation (3.6). If

$\varphi_{0}(0)=0,\ 0\leq\varphi_{0}(r)\leq\pi,\ \forall\ r\in[0,1],$

then it holds

$0<\varphi(r,t)<\pi,\ \forall\ r\in(0,1)\ {\rm{and}}\ t\in(0,T).$ (3.9)

##  4 Existence and singularity of axisymmetric solutions

In this section, we mainly concentrate on the equation

$\lambda_{1}(\varphi_{t}+r\varphi_{r})=\varphi_{rr}+\frac{1}{r}\varphi_{r}-\frac{\sin(2\varphi)}{2r^{2}}-3\lambda_{2}\sin\varphi\cos\varphi$
(4.1)

and provide the proof of Theorem 2.1 and Theorem 2.2.

Proof of Theorem 2.1. We argue by contradiction. Suppose $\varphi$ has the
first singularity at finite time $0<T_{1}<\infty$. For any $r_{0}\in(0,1)$ and
$t_{1}\in(0,T_{1})$, by the standard regularity theory of parabolic equations,
we can prove

$\big{\|}\varphi\big{\|}_{C^{k}([r_{0},1]\times(t_{1},T_{1}))}\leq
Cr_{0}^{-k},\ \forall k\geq 1.$ (4.2)

This implies that the possible singularity of the solution $\varphi$ can only
happen at $r=0$. Combining this fact with the blowup criterion, Theorem 1.4 in
[9], it holds for any $0<R<1$

$\liminf\limits_{t\rightarrow
T_{1}^{-}}\int_{0}^{R}|\varphi_{r}(r,t)|^{2}\,rdr\geq 4.$ (4.3)

We can choose $r_{m}\rightarrow 0^{+}$ and $t_{1}<t_{m}\rightarrow T_{1}^{-}$
as $m\rightarrow\infty$ and consider the blowup sequence

$\varphi^{m}(r,t)=\varphi\left(r_{m}r,t_{m}+r_{m}^{2}t\right).$ (4.4)

The rescaling functions $\varphi^{m}(r,t)$ are smooth solutions to the
following equation

$\lambda_{1}\left(\varphi^{m}_{t}+r_{m}^{2}\varphi^{m}_{r}\right)=\varphi^{m}_{rr}+\frac{1}{r}\varphi^{m}_{r}-\frac{\sin(2\varphi^{m})}{2r^{2}}-3\lambda_{2}r_{m}^{2}\sin\varphi^{m}\cos\varphi^{m},$
(4.5)

on $[0,1/r_{m}]\times[(t_{1}-t_{m})/r_{m}^{2},0]$. By the blowup criterion
(4.3), we conclude that $\varphi^{m}$ is not constant. By the energy estimates
in Lemma 3.2, we have

$\int_{0}^{1/r_{m}}|\varphi^{m}_{r}|^{2}\,rdr=\int_{0}^{1}|\varphi_{r}(r,t_{m}+r_{m}^{2}t)|^{2}\,rdr<\infty,$
(4.6)

and for any fixed $T>0$

$\int_{-T}^{0}\int_{0}^{1/r_{m}}|\varphi^{m}_{t}|^{2}\,rdrdt=\int_{t_{m}-Tr_{m}^{2}}^{t_{m}}\int_{0}^{1}|\varphi_{r}(r,t)|^{2}\,rdrdt\rightarrow
0$ (4.7)

as $m\rightarrow\infty$. It is not hard to see that

$r_{m}^{2}|\varphi^{m}_{r}|+3r_{m}^{2}|\lambda_{2}||\sin\varphi^{m}\cos\varphi^{m}|\rightarrow
0,\quad\mbox{in }L^{2}_{loc}([0,\infty)\times(-\infty,0))$

as $m\rightarrow\infty$. Therefore, there is a nonconstant smooth function
$\omega(r):[0,\infty)\rightarrow\mathbb{R}$ with finite energy such that

$\varphi^{m}(r,t)\rightarrow\omega\quad\mbox{in
}C^{2}_{\mbox{loc}}([0,\infty)\times(-\infty,0]),$

and $\omega(r)$ is a solution to the axisymmetric harmonic maps in dimension
two

$\omega_{rr}+\frac{1}{r}\omega_{r}-\frac{\sin(2\omega)}{2r^{2}}=0.$

Combining the fact $0\leq\varphi_{0}<\pi$ with the lemma 3.4, we obtain

$0<\varphi(r,t)<\pi,\ r\in[0,1],\ t_{m}\leq t<T,$

which implies

$0\leq\omega(r)\leq\pi\quad\mbox{for any }r\in[0,\infty).$

Hence, we conclude that

$d_{\infty}=(\sin\omega\cos\theta,\sin\omega\sin\theta,\cos\omega):\mathbb{R}^{2}\rightarrow\mathbb{S}^{2}$

is a nontrivial harmonic map with finite energy.

To secure a contradiction, we need to construct suitable supersolutions and
subsolutions to (4.1). Denote

$\overline{\varphi}(r,t,c)=2\arctan\left(\frac{re^{bt}}{c}\right)\quad\mbox{and}\quad\underline{\varphi}(r,t,c)=2\arctan\left(-\frac{re^{bt}}{c}\right),$

for some positive constant $c$ and $b=3|\lambda_{2}|/\lambda_{1}$. It is easy
to see that $\overline{\varphi}(r,c)$ and $\underline{\varphi}(r,c)$ are
smooth functions. Direct calculation implies

$\lambda_{1}(\overline{\varphi}_{t}+r\overline{\varphi}_{r})-\overline{\varphi}_{rr}-\frac{\overline{\varphi}_{r}}{r}+\frac{\sin(2\overline{\varphi})}{2r^{2}}+\frac{3\lambda_{2}}{2}\sin(2\overline{\varphi})=\frac{2rce^{bt}}{c^{2}+r^{2}e^{2bt}}\left(\lambda_{1}(1+b)+3\lambda_{2}\cos\overline{\varphi}\right)\geq
0,$

where we have used the facts $\lambda_{1}>0$ and
$\left|\cos\overline{\varphi}\right|\leq 1$ in the last inequality. Thus
$\overline{\varphi}$ is a supersolution of (4.1). Denote

$\eta_{0}=\pi-\max\limits_{0\leq r\leq 1}|\varphi_{0}(r)|.$

By the assumption on $\varphi_{0}$, we have $\eta_{0}\geq 0$ and
$|\varphi_{0}|\leq\pi-\eta_{0}$. Since
$\overline{\varphi}_{r}(0,0,c)=\frac{2}{c}$ and
$\overline{\varphi}(0,t,c)=\varphi_{0}(0)=0$, we can find a sufficiently small
$c>0$ such that

$\overline{\varphi}(r,0,c)\geq\varphi_{0}(r)$

for any $r\in[0,1]$, with equality holds iff $r=0$. Similarly, we can prove
that $\underline{\varphi}(r,t,c)$ is a subsolution to (4.1) and

$\varphi_{0}(r)\geq\underline{\varphi}(r,0,c)\ {\rm{in}}[0,1]$

for a sufficiently small $c>0$. By lemma 3.3, we conclude that
$\overline{\varphi}(r,t,c)\geq\varphi(r,t)\geq\underline{\varphi}(r,t,c)$ for
$r\in[0,1]$ and $t>0$.

Now by the definition of
$\overline{\varphi}(r,t,c),\underline{\varphi}(r,t,c)$, it is not hard to see
that

$0<\varphi(r,t)<\pi-c_{0},\ r\in[0,1],\ t_{m}\leq t<T,$

for some fixed constant $c_{0}>0$. Thus we have

$0\leq\omega(r)<\pi\quad\mbox{for any }r\in[0,\infty),$

which implies that

$d_{\infty}=(\sin\omega\cos\theta,\sin\omega\sin\theta,\cos\omega):\mathbb{R}^{2}\rightarrow\mathbb{S}^{2}$

is a nontrivial harmonic map with finite energy and degree zero, which is
impossible. Therefore, there is no finite time singularity for this case.
$\Box$

Now, we will construct the suitable subsolution and initial data to show the
existence of finite time singularity of (4.1), by modifying the arguments in
[10] and [1] .

Proof of Theorem 2.2. We first consider the following potential subsolution to
the equation (4.1)

$\eta(r,t)=2\arctan\left(\frac{r}{\beta(t)}\right),$

where $\beta(t)$ satisfies the following ODE

$\frac{d\beta}{dt}=-\beta^{\frac{2}{3}},\quad\beta(0)=\beta_{0},$ (4.8)

where $\beta_{0}$ is a positive constant that will be determined later. It is
not hard to see that

$\beta^{\frac{1}{3}}(t)=\frac{1}{3}(3\beta_{0}^{\frac{1}{3}}-t)$ (4.9)

Setting $T_{0}=3\beta_{0}^{\frac{1}{3}}$, we see that $\beta\rightarrow 0$ as
$t\rightarrow T_{0}^{-}$. Direct computation implies

$\begin{split}&\lambda_{1}(\eta_{t}+r\eta_{r})-\eta_{rr}-\frac{1}{r}\eta_{r}+\frac{\sin(2\eta)}{2r^{2}}+3\lambda_{2}\sin\eta\cos\eta\\\
=&\frac{2r}{\beta^{2}+r^{2}}\left(\lambda_{1}\beta^{\prime}+\lambda_{1}\beta+3\lambda_{2}\beta\cos\eta\right)\\\
\leq&\frac{2r}{\beta^{2}+r^{2}}\left(-\lambda_{1}\beta^{\frac{2}{3}}+(\lambda_{1}+3|\lambda_{2}|)\beta\right)\\\
=&\frac{2r\beta^{\frac{2}{3}}}{\beta^{2}+r^{2}}\left(-\lambda_{1}+(\lambda_{1}+3|\lambda_{2}|)\beta^{\frac{1}{3}}\right).\end{split}$
(4.10)

If we choose
$\beta_{0}^{\frac{1}{3}}<\frac{\lambda_{1}}{\lambda_{1}+3|\lambda_{2}|}$, then
for any $(r,t)\in(0,1)\times(0,3\beta_{0}^{\frac{1}{3}})$ it holds

$\lambda_{1}(\eta_{t}+r\eta_{r})-\eta_{rr}-\frac{1}{r}\eta_{r}+\frac{\sin(2\eta)}{2r^{2}}+3\lambda_{2}\sin\eta\cos\eta\leq
0.$ (4.11)

Since $\eta(1,0)<\pi$, we can find initial data $\varphi_{0}(r)$ such that
$\varphi_{0}(0)=0$, $\varphi_{0}(1)>\pi$ and

$\eta(r,0)\leq\varphi_{0}(r)$ (4.12)

for any $r\in[0,1]$. By the definition of $\eta$, it is easy to see that for
any $t\in(0,3\beta_{0}^{\frac{1}{3}})$

$\eta(0,t)=\varphi_{0}(0)=0,\quad\eta(1,t)\leq\pi<\varphi_{0}(1).$ (4.13)

Combining the facts (4.11)-(4.13) with Lemma 3.3, we conclude that
$\eta(r,t)\leq\varphi(r,t)$ for any
$(r,t)\in[0,1]\times[0,3\beta_{0}^{\frac{1}{3}})$. Direct computation implies

$\eta_{r}(0,t)=\frac{2}{\beta(t)}\rightarrow\infty,\quad\mbox{as }t\rightarrow
T_{0}^{-}.$

This combined with the fact $\eta(0,t)=\varphi(0,t)=0$ implies the finite time
blowup of $\varphi(r,t)$, which completes the proof of Theorem 2.2. $\Box$

##  5 Second example of singularity

We first need the following result on the existence of local smooth solution
to (1.1) with the initial and boundary conditions (1.5) and (1.6) under the
assumptions (1.2), (1.3) and (1.4).

###### Theorem 5.1

For $(u_{0},d_{0})\in
C^{\infty}_{0,{\rm{div}}}(B_{1}^{3},\mathbb{R}^{3})\times C_{\bf
e}^{\infty}(B_{1}^{3},\mathbb{S}^{2})$, there exist
$T_{0}=T_{0}(u_{0},d_{0})>0$ and a unique smooth solution $(u,d)\in
C^{\infty}\big{(}\overline{B_{1}^{3}}\times[0,T_{0}),\mathbb{R}^{3}\times\mathbb{S}^{2}\big{)}$
to the system (1.1) along with the initial-boundary conditions (1.5) and
(1.6). Moreover, the energy dissipation inequality

$\frac{d}{dt}\frac{1}{2}\int_{B_{1}^{3}}\left(|u|^{2}+|\nabla
d|^{2}\right)\,\leq-\int_{B_{1}^{3}}\left(\alpha_{0}|\nabla
u|^{2}+\frac{1}{\lambda_{1}}\left|\Delta d+|\nabla
d|^{2}d\right|^{2}\right)\,,$ (5.1)

holds for $0\leq t<T_{0}$ and some $\alpha_{0}>0$.

Proof. The existence of local smooth solution has been proved in [7], and the
energy inequality has been given by Lemma 2.3.

$\Box$

Now we would like to proceed to the proof of Theorem 2.4. Due to the energy
estimate (2.6), the proof is quite similar to that in [10] for the simplified
system. For the sake of completeness, we include a brief sketch of the main
steps.

Proof of Theorem 2.4. Assume $T_{0}>0$ is the end of the maximal time interval
for the short time smooth solution $(u,d)$ from lemma 5.1. Our goal is to show
that If $\epsilon_{0}>0$ is sufficiently small, then $T_{0}<1$.

We argue by contradiction. Suppose for any $\epsilon>0$ we can find
$(u_{0},d_{0})\in C^{\infty}_{0,{\rm{div}}}(B_{1}^{3},\mathbb{R}^{3})\times
C^{\infty}_{\bf e}(B_{1}^{3},\mathbb{S}^{2})$ satisfying the assumption of
Theorem 2.4, and a smooth solution $(u,d)\in
C^{\infty}\big{(}\overline{B_{1}^{3}}\times[0,1],\mathbb{R}^{3}\times\mathbb{S}^{2}\big{)}$
to the Ericksen-Leslie system (1.1) with the initial-boundary conditions (1.5)
and (1.6) under the assumptions (1.2), (1.3) and (1.4). Integrating (5.1) over
$t$ yields that $(u,d)$ satisfies the energy inequality

$E(u(t),d(t))+\int_{0}^{t}\int_{B_{1}^{3}}\big{(}\alpha_{0}|\nabla
u|^{2}+\frac{1}{\lambda_{1}}|\Delta d+|\nabla d|^{2}d|^{2}\big{)}\leq
E(u_{0},d_{0})\leq\epsilon^{2},$ (5.2)

for all $0\leq t\leq 1$. Applying Fubini’s theorem to (5.2), we find that
there exists $t_{1}\in(\frac{1}{2},1)$ such that

$E(u(t_{1}),d(t_{1}))+\int_{B_{1}^{3}}\big{(}|\nabla u(t_{1})|^{2}+|\Delta
d(t_{1})+|\nabla d(t_{1})|^{2}d(t_{1})|^{2}\big{)}\leq 8\epsilon^{2}.$ (5.3)

From (5.3) and $\epsilon$-apriori estimate Lemma 6.2 in [10] on approximate
harmonic maps, we conclude that there exists a constant $C>0$ such that

$\big{[}d(t_{1})\big{]}_{C^{\frac{1}{2}}(B_{1}^{3})}\leq C\sqrt{\epsilon}.$
(5.4)

Thus $d(t_{1})(B_{1}^{3})\subset B^{3}_{C\sqrt{\epsilon}}({\bf
e})\cap\mathbb{S}^{2}$ and hence $d(t_{1})$ is homotopic to $\bf e$ relative
to $\partial B_{1}^{3}$, provided $\epsilon>0$ is chosen to be sufficiently
small. Since $d\in
C^{\infty}\big{(}\overline{B_{1}^{3}}\times[0,t_{1}],\mathbb{S}^{2}\big{)}$
and $d={\bf e}$ on $\partial B_{1}^{3}\times[0,t_{1}]$, we see that $d(t_{1})$
is homotopic to $d_{0}$ relative to $\partial B_{1}^{3}$ and hence $d_{0}$ is
homotopic to $\bf e$ relative to $\partial B_{1}^{3}$, which is a
contradiction. This completes the proof of Theorem 2.4. $\Box$

##  6 Counter example on maximum principle

In this section, we will construct a counter example to show that the maximum
principle is not necessarily valid for Ericksen-Leslie system. To this end, we
consider the following Poiseuille flows for Ericksen-Leslie system (1.1) in
dimension one, i.e., for $(x,t)\in\mathbb{R}\times(0,\infty)$

$\begin{cases}w_{t}+a=(g(\varphi)w_{x}+h(\varphi)\varphi_{t})_{x},\\\
\lambda_{1}\varphi_{t}=\varphi_{xx}-h(\varphi)w_{x},\end{cases}$ (6.1)

where $g(\varphi)$ and $h(\varphi)$ are defined as following

$\displaystyle\begin{split}g(\varphi):=&\mu_{1}\sin^{2}\varphi\cos^{2}\varphi+\frac{\mu_{5}-\mu_{2}}{2}\sin^{2}\varphi+\frac{\mu_{3}+\mu_{6}}{2}\cos^{2}\varphi+\frac{\mu_{4}}{2},\\\
h(\varphi):=&\mu_{3}\cos^{2}\varphi-\mu_{2}\sin^{2}\varphi=\frac{\lambda_{1}+\lambda_{2}\cos(2\varphi)}{2}.\end{split}$
(6.2)

The system (6.1) can be derived from the Ericksen-Leslie system (1.1) by using
the Poiseuille flows (see Appendix in [2] for details)

$u=(0,0,w(x,t)),\quad d=(\sin\varphi(x,t),0,\cos\varphi(x,t)).$

We consider a simplified case of (6.1) by letting

$a=0,\quad\quad\mu_{1}=\mu_{5}=\mu_{6}=0,\quad\mu_{2}=-1,\quad\mu_{3}=1,\quad\mu_{4}=3,\quad\lambda_{1}=\mu_{3}-\mu_{2}=2.$

By the Onsager-Parodi relation (1.2), one has

$\lambda_{2}=\mu_{6}-\mu_{5}=\mu_{2}+\mu_{3}=0,$

which implies

$g(\varphi)=2,\quad h(\varphi)=1.$

Then the system (6.1) becomes

$\begin{cases}w_{t}=2w_{xx}+\varphi_{tx},\\\
2\varphi_{t}=\varphi_{xx}-w_{x}.\end{cases}$ (6.3)

The energy inequality is as follows

$\frac{d}{dt}\frac{1}{2}\int_{\mathbb{R}}|w|^{2}+|\varphi_{x}|^{2}\,dx+\int_{\mathbb{R}}|w_{x}|^{2}+|\varphi_{t}|^{2}+|w_{x}+\varphi_{t}|^{2}\,dx=0.$

Denote $v(x,t)=\int_{-\infty}^{x}w(y,t)\,dy$. Then it holds by the first
equation of (6.3)

$v_{t}=2w_{x}+\varphi_{t}=2v_{xx}+\varphi_{t}.$

Therefore the system (6.3) can be written into a system of $v,\varphi$

$\begin{cases}v_{t}=2v_{xx}+\varphi_{t},\\\
2\varphi_{t}=\varphi_{xx}-v_{xx}.\end{cases}$ (6.4)

Adding these two equations, we should have

$(v+\varphi)_{t}=(v+\varphi)_{xx}.$

Thus

$(v+\varphi)(x,t)=H(x,t)*(v_{0}+\varphi_{0})(x)=\frac{1}{\sqrt{4\pi
t}}\int_{\mathbb{R}}\exp{\left(-\frac{|x-y|^{2}}{4t}\right)}(v_{0}+\varphi_{0})(y)\,dy$

where $H(x,t)$ is the fundamental solution of heat equation in dimension one,
and $(v_{0},\varphi_{0})$ is the initial data of the system (6.3). Plugging in
this back to the equation of $\varphi$, we should have

$2\varphi_{t}=2\varphi_{xx}-\left(H(x,t)*(v_{0}+\varphi_{0})(x)\right)_{xx}=2\varphi_{xx}-H(x,t)*(v_{0}+\varphi_{0})_{xx}(x).$

We may take the smooth initial data $v_{0}$ and $\varphi_{0}$ such that

$v_{0}(x)=-x^{2},\quad\varphi_{0}(x)=0.$

Then

$H(x,t)*(v_{0}+\varphi_{0})_{xx}=-2,$

and

$\varphi(x,t)=t,$

which does not satisfy the maximum principle in the form of Lemma 3.4.

## References

  * [1] K. C. Chang, W. Y. Ding, R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces. J. Diff. Geom., 36 (1992) 507-515.
  * [2] G. Chen, T. Huang, and W. S. Liu.  Poiseuille flow of nematic liquid crystals via the full Ericksen-Leslie model.  Archive for Rational Mechanics and Analysis, 236:839–891, 2020.
  * [3] F. de Anna and C. Liu, Non-isothermal General Ericksen–Leslie System: Derivation, Analysis and Thermodynamic Consistency. Arch. Ration. Mech. Anal., 231 (2019) 637-717.
  * [4] S. J. Ding, C. Y. Wang, Finite time singularity of the Landau-Lifshitz-Gilbert equation. Int. Math. Res. Not. IMRN 2007, no. 4, Art. ID rnm012, 25 pp.
  * [5] J. L. Ericksen, Hydrostatic theory of liquid crystals. Arch. Ration. Mech. Anal. 9 (1962) 371-378.
  * [6] A. Friedman, Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs, New Jersey, 1964.
  * [7] M. Hieber, J. Prüss, Dynamics of the Ericksen-Leslie equations with general Leslie stress I: the incompressible isotropic case. Math. Ann. 369(3-4) (2017) 977-996.
  * [8] M. C. Hong and Z. P. Xin, Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in $\mathbb{R}^{2}$. Adv. Math. 231 (2012) 1364-1400.
  * [9] J. R. Huang, F. H. Lin and C. Y. Wang, Regularity and existence of global solutions to the Ericksen-Leslie system in $R^{2}$. Comm. Math. Phys., 331(2) (2014) 805-850.
  * [10] T Huang, F. H. Lin, C. Liu, and C. Y. Wang, Finite time singularity of the nematic liquid crystal flow in dimension three. Arch. Ration. Mech. Anal., 221(3) (2016) 1223–1254.
  * [11] C. C. Lai, F. H. Lin, C. Y. Wang, J. C. Wei, and Y. F. Zhou, Finite time blow-up for the nematic liquid crystal flow in dimension two. Comm. Pure Appl. Math., (2021) https://doi.org/10.1002/cpa.21993.
  * [12] F. M. Leslie, Some thermal effects in cholesteric liquid crystals. Proc. Roy. Soc. A. 307 (1968) 359-372.
  * [13] F. M. Leslie, Theory of Flow Phenomena in Liquid Crystals. Advances in Liquid Crystals, Vol. 4, 1-81. Academic Press, New York, 1979.
  * [14] J. K. Li, E. Titi, and Z. P. Xin, On the uniqueness of weak solutions to the Ericksen-Leslie liquid crystal model in $\mathbb{R}^{2}$. Math. Models Methods Appl. Sci. 26(4) (2016) 803-822.
  * [15] F. H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena. Comm. Pure Appl. Math. 42 (1989) 789-814.
  * [16] F. H. Lin, J. Y. Lin, and C. Y. Wang, Liquid crystal flows in two dimensions. Arch. Ration. Mech. Anal. 197 (2010) 297-336.
  * [17] F. H. Lin and C. Y. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three. Comm. Pure Appl. Math. 69 (2016) 1532-1571.
  * [18] F. H. Lin and C. Y. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals. Chin. Ann. Math., Ser. B, 31 (2010) 921-938.
  * [19] F. H. Lin, C. Y. Wang, The Analysis of Harmonic Maps and Their Heat Flows. The World Scientific, 2008.
  * [20] O. Parodi, Stress tensor for a nematic liquid crystal. J. Phys. 31 (1970) 581-584.
  * [21] M. H. Protter, H. F. Weinberge, Maximum principle in differential equations. Pretice-Hall, Englewood Cliffs, New Jersey, 1967.
  * [22] M. Wang and W. D. Wang, Global existence of weak solution for the 2-D Ericksen-Leslie system. Calc. Var. Partial Differential Equations, 51(3-4) (2014) 915-962.
  * [23] W. Wang, P. W. Zhang, and Z. F. Zhang, Well-posedness of the Ericksen-Leslie system. Arch. Ration. Mech. Anal. 210(3) (2013) 837-855.
  * [24] H. Wu, X. Xu, C. Liu, On the general Ericksen-Leslie system: Parodi’s relation, well-posedness and stability. Arch. Ration. Mech. Anal. 208 (2013) 59-107.