# Regularity of a gradient flow generated by the anisotropic Landau-de Gennes
energy with a singular potential

Yuning Liu111NYU Shanghai, 1555 Century Avenue, Shanghai 200122, China, and
NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan
Road North, Shanghai, 200062, China<EMAIL_ADDRESS>Xin Yang Lu222Department of
Mathematical Sciences, Lakehead University, 955 Oliver Rd., Thunder Bay, ON
P7B 5E1 Canada<EMAIL_ADDRESS>333 Department of Mathematics and
Statistics, McGill University, 805 Sherbrooke St. W., Montreal, QC H3A0B9,
Canada<EMAIL_ADDRESS>Xiang Xu444 Department of Mathematics and
Statistics, Old Dominion University, Norfolk, VA 23529, USA<EMAIL_ADDRESS>
###### Abstract

In this paper we study a gradient flow generated by the Landau-de Gennes free
energy that describes nematic liquid crystal configurations in the space of
$Q$-tensors. This free energy density functional is composed of three
quadratic terms as the elastic energy density part, and a singular potential
in the bulk part that is considered as a natural enforcement of a physical
constraint on the eigenvalues of $Q$. The system is a non-diagonal parabolic
system with a singular potential which trends to infinity logarithmically when
the eigenvalues of $Q$ approaches the physical boundary. We give a rigorous
proof that for rather general initial data with possibly infinite free energy,
the system has a unique strong solution after any positive time $t_{0}$.
Furthermore, this unique strong solution detaches from the physical boundary
after a sufficiently large time $T_{0}$. We also give estimate of the
Hausdorff measure of the set where the solution touches the physical boundary
and thus prove a partial regularity result of the solution in the intermediate
stage $(0,T_{0})$.

##  1 Introduction

The Landau-de Gennes theory is a continuum theory of nematic liquid crystals
[10]. When formulating static or dynamic continuum theories a crucial step is
to select an appropriate order parameter that captures the microscopic
structure of the rod-like molecule systems. In our framework the order
parameter is a matrix-valued function that takes values in the following so
called $Q$-tensor space

$\mathcal{Q}:=\Big{\\{}M\in\mathbb{R}^{3\times
3}\big{|}\operatorname{tr}M=0;\,M=M^{T}\Big{\\}}.$ (1.1)

It is considered as a suitably normalized second order moment of the
probability distribution function that dictates locally preferred orientations
of nematic molecular directors (cf. [2, 24, 23]).

To formulate the problem, let $\mathbb{T}^{n}$ be the unit box/square in
$\mathbb{R}^{n}$ with $n=2$ or $3$. For each order parameter
$Q:\mathbb{T}^{n}\to\mathcal{Q}$, the associated free energy functional
$\mathcal{E}(Q)$ consists of the elastic and the bulk parts, which reads

$\mathcal{E}(Q):=\mathcal{G}(Q)+\mathcal{B}(Q)-\alpha\|Q\|^{2}_{L^{2}(\mathbb{T}^{n})}.$
(1.2)

Here $\mathcal{G}$ stands for the anisotropic elastic energy that contains
three quadratic terms of $\nabla{Q}$:

$\mathcal{G}(Q):=\begin{cases}\displaystyle\int_{\mathbb{T}^{3}}\big{(}L_{1}{\partial}_{k}Q_{ij}{\partial}_{k}Q_{ij}+L_{2}{\partial}_{j}Q_{ik}{\partial}_{k}Q_{ij}+L_{3}{\partial}_{j}Q_{ij}{\partial}_{k}Q_{ik}\big{)}\,\mathrm{d}x,\quad\mbox{if
}Q\in H^{1}(\mathbb{T}^{n}),\\\
+\infty,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\mbox{otherwise},\end{cases}$
(1.3)

where $L_{1},L_{2},L_{3}$ are material dependent constants. Here and in the
sequel $\partial_{k}Q_{ij}$ denotes the $k$-th spatial partial derivative of
the $ij$-th component of $Q$, and we adopt Einstein summation convention by
summing over repeated Latin letters. Following [21], we assume

$L_{1}>3|L_{2}+L_{3}|,$ (1.4)

which ensures that (1.3) fulfills the strong Legendre condition.

Further, $\mathcal{B}(Q)$ denotes the bulk energy

$\mathcal{B}(Q):=\int_{\mathbb{T}^{3}}\psi(Q)\,\mathrm{d}{x},$

where the integrand $\psi(Q)$ is the singular potential introduced in [4]:

$\psi(Q):=\begin{cases}\displaystyle\inf_{\rho\in\mathcal{A}_{Q}}\int_{\mathbb{S}^{2}}\rho(p)\ln{\rho(p)}\,\mathrm{d}p,\quad\mbox{if
}-\frac{1}{3}<\lambda_{i}(Q)<\frac{2}{3},\;1\leq i\leq 3,\\\
+\infty,\qquad\qquad\qquad\qquad\quad\mbox{otherwise}.\end{cases}$ (1.5)

Here $\lambda_{i}(Q)$ denotes the $i$-th eigenvalue of the matrix $Q$ and
$\mathcal{A}_{Q}$ is the admissible class defined by

$\mathcal{A}_{Q}=\left\\{\rho(p):\mathbb{S}^{2}\rightarrow\overline{\mathbb{R}^{+}}~{}\big{|}~{}\,\|\rho\|_{L^{1}(\mathbb{S}^{2})}=1;\;\int_{\mathbb{S}^{2}}\big{(}p\otimes{p}-\frac{1}{3}\mathbb{I}\big{)}\rho(p)\,\mathrm{d}{p}=Q\right\\}.$

It is noted that the singular potential (1.5) imposes physical constraints on
the eigenvalues of $Q$. Futher, $\alpha>0$ in (1.2) is a temperature dependent
constant which characterizes the relative intensity of the molecular Brownian
motion and the molecular interaction [4]. We refer interested readers to [2,
4, 5, 3, 16, 22] for detailed discussions of basic analytic properties of
$\psi$, such as convexity, smoothness in its effective domain, blow-up rates
near the physical boundary, etc. Meanwhile, various problems in static and
dynamic configurations concerning $\psi$ can be found in [7, 12, 14, 15, 16,
17, 29]. Specifically, the free energy in related dynamic problems considered
so far in the existing literature [12, 15, 16, 29] only involves the $L_{1}$
isotropic term. Therefore, we are motivated to study the dynamic problem whose
free energy contains anisotropic $L_{2},L_{3}$ terms. It is worth pointing out
that the presence of such terms is more than a mere technical challenge, since
they make it impossible to recover any kind of maximum principle, which was
crucial in [29].

This paper is concerned with a rigorous study of the gradient flow generated
by $\mathcal{E}(Q)$ in the Hilbert space $L^{2}(\mathbb{T}^{n};\mathcal{Q})$:

$\left\\{\begin{array}[]{rlr}\partial_{t}Q(t,\cdot)&=-\partial\mathcal{E}(Q(t,\cdot)),&t>0,\\\
Q(0,x)&=Q_{0}(x),&x\in\mathbb{T}^{n}\end{array}\right.$ (1.6)

subject to periodic boundary condition

$Q(t,x+e_{i})=Q(t,x),\quad\mbox{for
}(t,x)\in\mathbb{R}^{+}\times\partial\mathbb{T}^{n}.$ (1.7)

Here in (1.6), $\partial\mathcal{E}(Q)$ is formally the variation of the free
energy (1.2). However, due to the singular feature of $\psi(Q)$, it should be
understood as sub-differential (see Lemma 3.3 for more details.)

Parallel to the $Q$-tensor space (1.1), we introduce the physical $Q$-tensor
space by

$\mathcal{Q}_{phy}:=\Big{\\{}M\in\mathcal{Q}\big{|}\,-\frac{1}{3}<\lambda_{1}(M)\leq\lambda_{2}(M)\leq\lambda_{3}(M)<\frac{2}{3}\Big{\\}},$
(1.8)

where $\lambda_{i}(M)$ denotes the $i$-th eigenvalue of the matrix $M$,
ordered non-decreasingly. Any element in $\mathcal{Q}_{phy}$ is called a
physical $Q$-tensor.

Our first main result ensures the existence and uniqueness of solutions to the
gradient flow (1.6) with rather general initial data.

######  Theorem 1.1.

Let $n=2$ or $3$. For any initial data

$Q_{0}\in\overline{\\{Q\in
L^{2}(\mathbb{T}^{n};\mathcal{Q}_{phy})\mid\mathcal{E}(Q)<\infty\\}}^{L^{2}(\mathbb{T}^{n})},$
(1.9)

there exists a unique global solution
$Q(t,x):\mathbb{R}^{+}\times\mathbb{T}^{n}\to\mathcal{Q}_{phy}$ of (1.6) such
that

$\displaystyle\partial_{t}Q_{ij}$
$\displaystyle=2L_{1}\Delta{Q}_{ij}+2(L_{2}+L_{3}){\partial}_{j}{\partial}_{k}Q_{ik}-\frac{2}{3}(L_{2}+L_{3}){\partial}_{k}{\partial}_{\ell}Q_{\ell
k}\delta_{ij}$
$\displaystyle\qquad-\frac{\partial\psi}{\partial{Q}_{ij}}+\frac{1}{3}\operatorname{tr}\Big{(}\frac{\partial\psi}{\partial{Q}}\Big{)}\delta_{ij}+2\alpha
Q_{ij}$ (1.10)

holds almost everywhere in $(0,\infty)\times\mathbb{T}^{n}$. And for any fixed
$t_{0}>0$, the solution satisfies

$Q\in L^{\infty}(t_{0},\infty;H^{1}(\mathbb{T}^{n})),\qquad\partial_{t}Q\in
L^{2}_{loc}(t_{0},\infty;L^{2}(\mathbb{T}^{n})),$ (1.11)

and the energy dissipative equality

$\int_{t_{0}}^{T}\big{(}\|\partial_{t}Q(t,\cdot)\|_{L^{2}(\mathbb{T}^{3})}^{2}+\|\partial\mathcal{E}(Q(t,\cdot))\|^{2}_{L^{2}(\mathbb{T}^{3})}\big{)}\,\mathrm{d}{t}=2\mathcal{E}(Q(t_{0}))-2\mathcal{E}(Q(T))$
(1.12)

for all $0<t_{0}<T<+\infty$. Further, $Q(t,\cdot)$ is physical in the sense
that

$Q(t,x)\in\mathcal{Q}_{phy},\quad\forall t>0,\;a.e.\;x\in\mathbb{T}^{n}.$
(1.13)

It is worthy to point out that due to the energy dissipative property of the
gradient flow as well as the convexity of the singular potential, for any
$T>0$ one can formally establish the a priori estimate of $Q$ in
$L^{\infty}_{loc}(0,T;H^{1}(\mathbb{T}^{n}))\cap
L^{2}_{loc}(0,T;H^{2}(\mathbb{T}^{n}))$. As a consequence, existence of weak
solutions to (1.6) can be achieved by using two level approximation schemes as
in [29], i.e., regularizing the initial data and the singular free energy.
However, such arguments involve fairly complicated approximation procedures.
Fortunately Ambrosio–Gigli–Savaré [1] provides a powerful framework to obtain
the solution under very general assumptions of the initial data.

To establish higher regularity, namely a uniform-in-time $H^{2}$ bound of the
solution $Q$, essential difficulties arise from the anisotropic terms. Without
the $L_{2}+L_{3}$ terms, the convexity of $\psi$ as well as the classical
$L^{1}-L^{\infty}$ estimate of heat equation ensure the strict physicality at
any positive time (see section $8$ in [29] for details) and henceforth
conventional energy method applies. Concerning the gradient flow (1.6),
unfortunately the anisotropic terms make such maximum principle argument
invalid. As a consequence, the proof of higher regularity of the solution
becomes quite subtle in the sense that $Q$ might not stay inside any compact
subset of $\mathcal{Q}_{phy}$. To overcome such a difficulty, we need to make
a careful exploitation of its gradient flow structure, as well as to combine
several results on the gradient flow theory given in [1] and Gamma-convergence
of gradient flows discussed in [28, 27]. These lead to the next theorem which
improves the regularity by establishing the uniform-in-time $H^{2}$ bound of
the unique solution to the gradient flow (1.6). Further, it can be shown that
this unique solution detaches from its physical boundary after a sufficiently
large time $T_{0}$.

######  Theorem 1.2.

For any $t_{0}>0$ the solution established in Theorem 1.1 enjoys the improved
regularity $Q\in L^{\infty}(t_{0},+\infty;H^{2}(\mathbb{T}^{n}))$, and for
almost every $t\geq t_{0}$ there holds

$\|\Delta Q(t,\cdot)\|_{L^{2}(\mathbb{T}^{n})}\leq
C_{L}\left(e^{4\alpha}\sqrt{\mathcal{E}(Q(t_{0}))-\inf\mathcal{E}+1}+2\alpha\|Q(t,\cdot)\|_{L^{2}(\mathbb{T}^{n})}\right),$
(1.14)

where $C_{L}$ is expressed by

$C_{L}:=\frac{1}{2(L_{1}-|L_{2}+L_{3}|)}\sqrt{\frac{L_{1}+|L_{2}+L_{3}|}{L_{1}+|L_{2}+L_{3}|-2\sqrt{L_{1}|L_{2}+L_{3}|}}}$
(1.15)

Furthermore, under the stronger assumption

$L_{1}-3|L_{2}+L_{3}|-\alpha{C}_{\mathbb{T}^{n}}^{2}>0,$ (1.16)

where $C_{\mathbb{T}^{n}}=(2\pi)^{n}$ is the Poincaré constant in
$\mathbb{T}^{n}$, there exists $T_{0}>0$ such that the unique solution is
strictly physical for all $t\geq T_{0}$ in the sense that

$-\dfrac{1}{3}+\kappa\leq\lambda_{i}(Q(t,x))\leq\dfrac{2}{3}-\kappa,\quad\forall
x\in\mathbb{T}^{n}$ (1.17)

for some constant $\kappa\in(0,1/6)$.

During the period $(0,T_{0})$, a partial regularity result of the unique
solution can be established, i.e. Hausdorff dimension of the set where the
solution touches the physical boundary $\partial\mathcal{Q}_{phy}$:

######  Theorem 1.3.

Let $Q(t,x)$ be the unique strong solution of (1.6) established in Theorem
1.2. Then for a.e. $t\in(0,T_{0})$, the contact set

$\Sigma_{t}:=\\{x\in\mathbb{T}^{n}\mid
Q(t,x)\in{\partial}\mathcal{Q}_{phy}\\}$ (1.18)

has the following estimate:

  * •

$\mbox{dim}_{\mathcal{H}}(\Sigma_{t})\leq 2$ for $n=3$.

  * •

$\mbox{dim}_{\mathcal{H}}(\Sigma_{t})=0$ for $n=2$.

The rest of the paper is organized as follows. Some notations and
preliminaries are provided in Section 2. The proofs of the three main results,
namely Theorems 1.1, 1.2, and 1.3, are given in Sections 3, 4, 5,
respectively.

##  2 Preliminaries

We start with a few basic notations in $Q$-tensor theory. For any
$Q\in\mathcal{Q}$, $|Q|:=\sqrt{\operatorname{tr}(Q^{t}Q)}$ represents the
Frobenius norm of $Q$. The gradient of the function $\psi(Q)$ will be
abbreviated by $\psi^{\prime}(Q)$, and its components are denoted by
$\psi^{\prime}_{ij}(Q):=\frac{{\partial}\psi(Q)}{{\partial}Q_{ij}}$. Moreover,
we denote $L^{2}(\mathbb{T}^{n};\mathcal{Q})$ the Hilbert space endowed with
the $L^{2}$ metric

$\|Q\|_{L^{2}(\mathbb{T}^{n})}=\sqrt{\int_{\mathbb{T}^{n}}\mbox{tr}(Q^{t}Q)}=\sqrt{\int_{\mathbb{T}^{n}}\mbox{tr}(Q^{2})},\quad\mbox{for
}Q:\mathbb{T}^{n}\rightarrow\mathcal{Q}.$

Here and after, for brevity, $\|\cdot\|_{L^{2}(\mathbb{T}^{n})}$ will often be
written as $\|\cdot\|_{L^{2}}$, or simply $\|\cdot\|$.

Next we provide some preliminaries of gradient flow theory in a Hilbert space.
We start with some basic definitions in a Hilbert space $H$, with inner
product $\langle\cdot,\cdot\rangle$ and norm $\|\cdot\|$ (cf. [9, 13]).

######  Definition 2.1.

A function $f:H\rightarrow\mathbb{R}\cup\\{+\infty\\}$ is called proper if $f$
is not identically equal to $+\infty$. The effective domain of $f$ is defined
by

$D(f)=\big{\\{}u\in H|\,f(u)<+\infty\big{\\}}.$

By [4], the effective domain $D(\psi)$ is equivalent to (1.5).

######  Definition 2.2.

Let $\lambda\in\mathbb{R}$, a $\lambda$-convex function
$F:H\rightarrow(-\infty,+\infty]$ is a function satisfying

$F((1-t)u+tv)\leq(1-t)F(u)+tF(v)-\frac{\lambda}{2}t(1-t)\|u-v\|^{2},\qquad\forall
u,v\in H.$

For each $u\in H$, $\partial{F}[u]$ is defined as the set of $w\in H$ such
that

$F(u)+\langle w,v-u\rangle+\frac{\lambda}{2}\|u-v\|^{2}\leq F(v),\quad\forall
v\in H.$

The mapping $\partial{F}:H\rightarrow 2^{H}$ is called the subdifferential of
$F$. Further, We say $u\in D(\partial F)$, the domain of $\partial F$,
provided $\partial{F}[u]$ is not empty.

######  Definition 2.3.

We say $u(t)$ is a gradient flow of $F$ starting from $u_{0}\in H$ if it is a
locally absolutely continuous curve in $(0,+\infty)$ such that

$\begin{cases}\partial_{t}u(t)=-\partial F(u(t)),\quad\mbox{a.e.}\;t>0\\\
\displaystyle\lim_{t\rightarrow 0^{+}}u(t)=u_{0}.\end{cases}$ (2.1)

The next result is due to [1, Theorem 4.0.4] which was originally stated under
metric space setting. For the purpose of proving Theorem 1.1, it suffices to
rewrite it in the Hilbert space setting:

######  Proposition 2.1.

Let $\lambda\in\mathbb{R}$ and $F:H\rightarrow(-\infty,+\infty]$ be a proper,
bounded from below, and lower semicontinuous functional. Suppose for each
$\tau\in(0,\frac{1}{\lambda^{-}})$ with $\lambda^{-}:=\max\\{0,-\lambda\\}$
and each fixed $w\in D(F)$, the functional

$\Phi(\tau,w;v)=\frac{1}{2\tau}\|v-w\|^{2}+F(v),~{}\forall v\in D(F)$ (2.2)

satisfies the following inequality for every $v_{0},v_{1}\in D(F)$:

$\displaystyle\Phi(\tau,w;t)\leq(1-t)\Phi(\tau,w;v_{0})+t\Phi(\tau,w;v_{1})-\frac{1+\lambda\tau}{2\tau}t(1-t)\|v_{0}-v_{1}\|^{2}.$

Then for each $u_{0}\in\overline{D(F)}$,
$u(t)=\displaystyle\lim_{k\rightarrow+\infty}J_{t/k}^{k}(u_{0})$ with
$J_{\tau}$ being the resolvent

$X\in J_{\tau}(Y)\Longleftrightarrow
X\in\mbox{argmin}\Big{\\{}F(\cdot)+\frac{1}{2\tau}\|Y-\cdot\|^{2}\Big{\\}},$
(2.3)

satisfies

  1. 1.

Variational inequality: $u$ is the unique solution to the evolution
variational inequality

$\frac{1}{2}\dfrac{d}{dt}\|u(t)-v\|^{2}-\lambda\|u(t)-v\|^{2}+F(u(t))\leq
F(v),\qquad\mbox{for a.e. }t>0\mbox{ and }v\in D(F),$ (2.4)

among all the locally absolutely continuous curves such that $u(t)\rightarrow
u_{0}$ as $t\downarrow 0^{+}$.

  2. 2.

Regularizing effect: $u$ is locally Lipschitz regular, and $u(t,\cdot)\in
D(\mathcal{E})$ for all $t>0$.

######  Remark 2.1.

It is well known that for a $\lambda$-convex function
$F:H\rightarrow(-\infty,+\infty]$, a locally absolutely continuous curve
$u(t)$ in $(0,+\infty)$ satisfies (2.1) if and only if it satisfies the
evolution variational inequality (2.4).

Now we turn to the $\Gamma$-convergence of gradient flows in a Hilbert space,
a theory developed in [27] and [28]. Let $\\{u_{n}\\}$ be the solution to the
gradient flow

$\partial_{t}u_{n}=-\nabla E_{n}(u_{n})$ (2.5)

of a $C^{1}$ functional sequence $\\{E_{n}\\}$. Assume $E_{n}$
$\Gamma$-converges to a functional $F$, and there is a general sense of
convergence $u_{n}\overset{S}{\rightarrow}u$, relative to which the
$\Gamma$-convergence of $E_{n}$ to $F$ holds. We introduce the “energy-excess”
along a family of curves $u_{n}(t)$ with
$u_{n}(t)\overset{S}{\rightarrow}u(t)$ by setting

$\tilde{D}(t)=\limsup_{n\rightarrow\infty}E_{n}(u_{n}(t))-F(u(t)).$

The main result of [28] is the following:

######  Proposition 2.2.

Assume $E_{n}$ and $F$ satisfy a $\Gamma-\liminf$ relation: if
$u_{n}\overset{S}{\rightarrow}u$ as $n\rightarrow\infty$ then

$\displaystyle\liminf_{n\rightarrow\infty}E_{n}(u_{n})\geq F(u).$

Assume that the following two additional conditions hold:

  1. 1.

(Lower bound on the velocities) If $u_{n}(t)\overset{S}{\rightarrow}u(t)$ for
all $t\in[0,T)$ then there exists $f\in L^{1}(0,T)$ such that for every
$s\in[0,T)$

$\liminf_{n\rightarrow\infty}\int_{0}^{s}\|\partial_{t}u_{n}(t)\|_{H}^{2}\,dt\geq\int_{0}^{s}\big{[}\|\partial_{t}u(t)\|_{H}^{2}-f(t)\tilde{D}(t)\big{]}\,dt.$
(2.6)

  2. 2.

(Lower bound for the slopes) If $u_{n}\overset{S}{\rightarrow}u$ then

$\liminf_{n\rightarrow\infty}\|\nabla E_{n}(u_{n})\|_{H}^{2}\,dt\geq\|\nabla
F(u)\|_{H}^{2}-C\tilde{D},$ (2.7)

where $C$ is a universal constant, and $\|\nabla F(u)\|$ denotes the minimal
norm of the elements in $\partial F(u)$.

Assume $u_{n}(t)$ is a family of solutions to (2.5) on $[0,T)$ with
$u_{n}(t)\overset{S}{\rightarrow}u(t)$ for all $t\in[0,T)$, such that

$E_{n}(u_{n}(0))-E_{n}(u_{n}(t))=\int_{0}^{t}\|\partial_{t}u_{n}(s)\|_{H}^{2}\,ds,\quad\forall
t\in[0,T).$

Assume also that

$\displaystyle\lim_{n\rightarrow\infty}E_{n}(u_{n}(0))=F(u(0)),$

then $u\in H^{1}(0,T;H)$ and is a solution to $\partial_{t}u=-\partial F(u)$
on $[0,T)$. Moreover, $\tilde{D}(t)=0$ for all $t$ (that is the solutions
“remain well-prepared”) and

$\|\partial_{t}u_{n}\|_{H}\xrightarrow{n\to\infty}\|\partial_{t}u\|_{H},\quad\|\nabla
E_{n}(u_{n})\|_{H}\xrightarrow{n\to\infty}\|\nabla F(u)\|_{H}\quad\mbox{in
}\;L^{2}(0,T).$

##  3 Proof of Theorem 1.1: Existence of solutions

This section is devoted to the proof of Theorem 1.1. First of all, with the
choices $H=L^{2}(\mathbb{T}^{n};\mathcal{Q})$ and $F=\mathcal{E}$ (defined in
(1.2)), we show that the assumptions in Proposition 2.1 are satisfied, so that
there exists a unique solution $Q(t,\cdot)$ in variational inequality setting
(2.4) (see Proposition 3.1 below). Moreover, since the free energy is
$-2\alpha$ convex, the solution achieved in Proposition 3.1 is equivalent to
the solution of the gradient flow (1.6) in sub-differential setting. As a
consequence, we compute explicitly the sub-differential of $\mathcal{E}$, and
obtain a unique strong solution to equation (1.1). Finally, we apply two
theorems in [1] to show further regularity properties of $Q$ in Theorem 1.1.
Since all the following arguments are valid for both $\mathbb{T}^{3}$ and
$\mathbb{T}^{2}$ with minor modifications, for brevity we discuss the case of
$\mathbb{T}^{3}$ only.

In this subsection we consider the following settings:

$(H,\|\cdot\|)=L^{2}(\mathbb{T}^{3};\mathcal{Q}),\qquad F=\mathcal{E}(Q).$

To begin with, we need to verify all assumptions in Proposition 2.1 are valid,
which is given in the following two lemmas.

######  Lemma 3.1.

The free energy functional $\mathcal{E}$ is proper, bounded from below,
$-2\alpha$ convex and lower semicontinuous in
$L^{2}(\mathbb{T}^{3};\mathcal{Q})$.

###### Proof.

First we show that the elastic energy $\mathcal{G}$ is nonnegative, convex,
and lower semicontinuous in $L^{2}(\mathbb{T}^{3};\mathcal{Q})$. It is proved
in [21] that when $L_{1}>0,L_{1}+L_{2}+L_{3}>0$, $\mathcal{G}$ satisfies the
strong Legendre condition, which implies the convexity of $\mathcal{G}$. It
suffices to show that $\mathcal{G}$ is nonnegative when $Q\in
H^{1}(\mathbb{T}^{3})$, which follows from the coefficient assumption (1.4),
integration by parts and the Cauchy-Schwarz inequality:

$\displaystyle\mathcal{G}(Q)$
$\displaystyle\geq\int_{\mathbb{T}^{3}}\Big{(}L_{1}\partial_{k}Q_{ij}\partial_{k}Q_{ij}+L_{2}\partial_{k}Q_{ik}\partial_{k}Q_{ij}+L_{3}\partial_{j}Q_{ij}\partial_{k}Q_{ik}\Big{)}\,dx$
$\displaystyle=\int_{\mathbb{T}^{3}}\Big{[}L_{1}\partial_{k}Q_{ij}\partial_{k}Q_{ij}+(L_{2}+L_{3})\partial_{j}Q_{ik}\partial_{k}Q_{ij}\Big{]}\,dx$
$\displaystyle\geq\int_{\mathbb{T}^{3}}\big{(}L_{1}-3|L_{2}+L_{3}|\big{)}|\nabla{Q}|^{2}\,dx\geq
0.$ (3.1)

Besides, since $\mathcal{G}$ is convex and quadratic, it is lower
semicontinuous [18, Theorem 8.1].

Next we show that the functional $\mathcal{B}$ is convex, bounded from below,
and lower semicontinuous in $L^{2}(\mathbb{T}^{3};\mathcal{Q})$. The convexity
of $\mathcal{B}$ follows from [4, 16]. A lower bounded can be derived from the
inequality $x\ln{x}\geq-1/e$ for any $x\geq 0$:

$\mathcal{B}=\int_{\mathbb{T}^{3}}\psi(Q)\,\mathrm{d}{x}\geq-4\pi^{2}|\mathbb{T}^{3}|/e.$
(3.2)

To show the lower semicontinuity of $\mathcal{B}$, let $Q_{n}\rightarrow Q$
strongly in $L^{2}(\mathbb{T}^{3})$. If
$\displaystyle\liminf_{n\rightarrow\infty}\psi(Q_{n})=+\infty$ on a set of
positive measure, then the proof is done. Thus upon subsequence we assume

$\displaystyle\liminf_{n\rightarrow\infty}\mathcal{B}(Q_{n})=\displaystyle\lim_{n\rightarrow\infty}\mathcal{B}(Q_{n})<+\infty,\;\;\mbox{and}\;\;Q_{n}(x)\xrightarrow{n\to\infty}Q(x)\;\;\mbox{for
a.e. }x\in\mathbb{T}^{3}.$ (3.3)

Consequently, for all $n\in\mathbb{N}$ sufficiently large and a.e.
$x\in\mathbb{T}^{3}$, all eigenvalues of $Q_{n}(x)$ are in $(-1/3,2/3)$.
Moreover, the eigenvalues of $Q(x)$ are in $[-1/3,2/3]$ since convergence of
eigenvalues follows from convergence of the matrices (cf. [26]).

We claim that for a.e. $x\in\mathbb{T}^{3}$ the eigenvalues of $Q(x)$ are in
$(-1/3,2/3)$. To this aim, we argue by contradiction. Assume the opposite,
i.e. $E=\big{\\{}x\in\mathbb{T}^{3},\lambda_{1}(Q(x))=-1/3\big{\\}}$ has
positive measure. Then it follows from [4] that
$\psi(Q_{n}(x))\xrightarrow{n\to\infty}+\infty$ in $E$, and henceforth Fatou’s
lemma implies

$\displaystyle\displaystyle\liminf_{n\rightarrow\infty}\mathcal{B}(Q_{n})$
$\displaystyle\geq\displaystyle\liminf_{n\rightarrow\infty}\int_{E}\psi(Q_{n})\,\mathrm{d}{x}+\displaystyle\liminf_{n\rightarrow\infty}\int_{\mathbb{T}^{3}\setminus
E}\psi(Q_{n})\,\mathrm{d}{x}$
$\displaystyle\geq\int_{E}\displaystyle\liminf_{n\rightarrow\infty}\psi(Q_{n})\,\mathrm{d}{x}-\dfrac{4\pi^{2}}{e}|\mathbb{T}^{3}\setminus
E|=+\infty,$

which is in contradiction with (3.3). Thus the claim is proved. Since $\psi$
is smooth in $D(\psi)=\mathcal{Q}_{phy}$ (see [16]), we have
$\psi(Q_{n}(x))\xrightarrow{n\to\infty}\psi(Q(x))$ for a.e.
$x\in\mathbb{T}^{3}$. Thus Fatou’s lemma implies

$\displaystyle\displaystyle\liminf_{n\rightarrow\infty}\mathcal{B}(Q_{n})\geq\int_{\mathbb{T}^{3}}\displaystyle\liminf_{n\rightarrow\infty}\psi(Q_{n})\,\mathrm{d}{x}=\int_{\mathbb{T}^{3}}\psi(Q(x))\,\mathrm{d}{x}=\mathcal{B}(Q).$

To sum up, $\mathcal{E}$ is $-2\alpha$ convex, and lower semicontinuous in
$L^{2}(\mathbb{T}^{3};\mathcal{Q})$. It remains to show $\mathcal{E}$ is
proper and bounded from below. Clearly, $\mathcal{E}(Q)<+\infty$ provided
$Q\in H^{1}(\mathbb{T}^{3})$ and $Q(x)\in D(\psi)$ a.e. $x\in\mathbb{T}^{3}$,
hence $\mathcal{E}$ is proper. Further, if $Q$ is not physical then
$\mathcal{E}(Q)=+\infty$ , while if $Q$ is physical then
$\|Q\|_{L^{2}(\mathbb{T}^{3})}$ is bounded, hence $\mathcal{E}$ is bounded
from below since both $\mathcal{G}$ and $\psi$ are bounded from below.

∎

######  Remark 3.1.

It is noted that the coefficient assumption (1.4) is different from the one in
[11, 20] which ensures the elastic energy $\mathcal{G}$ is coercive only.

######  Lemma 3.2.

For any $R,P_{0},P_{1}\in D(\mathcal{E})$, denote
$\gamma_{t}=(1-t)P_{0}+tP_{1}$, $t\in[0,1]$, then for each $0<\tau<1/2\alpha$
the functional

$\Phi(\tau,R;\gamma_{t}):=\dfrac{\|\gamma_{t}-R\|^{2}}{2\tau}+\mathcal{E}(\gamma_{t})$
(3.4)

is $(1/\tau-2\alpha)$-convex on $\gamma_{t}$, in the sense that

$\displaystyle\Phi$ $\displaystyle(\tau,R;\gamma_{t})$
$\displaystyle\leq(1-t)\Phi(\tau,R;P_{0})+t\Phi(\tau,R;P_{1})-\dfrac{(1-2\alpha\tau)}{2\tau}t(1-t)\|P_{1}-P_{0}\|^{2},\quad\forall
t\in[0,1].$ (3.5)

###### Proof.

We infer from the convexity of $\mathcal{G}$ and $\mathcal{B}$ that

$\displaystyle\Phi(\tau,R;\gamma_{t})$
$\displaystyle=\dfrac{\big{\|}(1-t)P_{0}+tP_{1}-R\big{\|}^{2}}{2\tau}+(\mathcal{G}+\mathcal{B})((1-t)P_{0}+tP_{1})-\alpha\big{\|}(1-t)P_{0}+tP_{1}\big{\|}^{2}$
$\displaystyle\leq\dfrac{(1-t)\|P_{0}-R\|^{2}+t\|P_{1}-R\|^{2}-t(1-t)\|P_{0}-P_{1}\|^{2}}{2\tau}+(1-t)(\mathcal{G}+\mathcal{B})(P_{0})$
$\displaystyle\qquad+t(\mathcal{G}+\mathcal{B})(P_{1})-\alpha\big{[}(1-t)\|P_{0}\|^{2}+t\|P_{1}\|^{2}-t(1-t)\|P_{0}-P_{1}\|^{2}\big{]}$
$\displaystyle=(1-t)\Phi(\tau,R;P_{0})+t\Phi(\tau,R;P_{1})-\dfrac{(1-2\alpha\tau)}{2\tau}t(1-t)\|P_{1}-P_{0}\|^{2}$

∎

To sum up, we manage to verify that all assumptions of Proposition 2.1 are
satisfied, which leads to the following theorem.

######  Proposition 3.1.

Let $n=3$. For any initial data

$Q_{0}\in\overline{D(\mathcal{E})}:=\overline{\\{Q\in
L^{2}(\mathbb{T}^{3};\mathcal{Q}_{phy})\mid\mathcal{E}(Q)<\infty\\}}^{L^{2}(\mathbb{T}^{3})},$
(3.6)

Let $Q(t)=\displaystyle\lim_{k\rightarrow+\infty}J_{t/k}^{k}(Q_{0})$ with $J$
being the resolvent

$X\in J_{\tau}(Y)\Longleftrightarrow
X\in\mbox{argmin}\Big{\\{}F(\cdot)+\frac{1}{2\tau}\|Y-\cdot\|^{2}\Big{\\}}.$

Then we have

  1. 1.

Variational inequality: $Q$ is the unique solution to the evolution
variational inequality

$\begin{split}\dfrac{d}{dt}\|Q(t,\cdot)-P\|^{2}_{L^{2}(\mathbb{T}^{3})}&-\alpha\|Q(t,\cdot)-P\|^{2}_{L^{2}(\mathbb{T}^{3})}+\mathcal{E}(Q(t,\cdot))\leq\mathcal{E}(P),\\\
&\quad\mbox{for a.e. }t>0\mbox{ and }P\in D(\mathcal{E}),\end{split}$ (3.7)

among all locally absolutely continuous curves such that
$Q(t,\cdot)\rightarrow Q_{0}$ as $t\downarrow 0^{+}$.

  2. 2.

Regularizing effect: $Q$ is locally Lipschitz, and $Q(t,\cdot)\in
D(\mathcal{E})$ for all $t>0$. In particular, $Q$ is physical in the sense
that

$-\dfrac{1}{3}<\lambda_{i}(Q(t,x))<+\dfrac{2}{3},\quad\forall
t>0,\;a.e.\;x\in\mathbb{T}^{3}$ (3.8)

To proceed, note that $\mathcal{E}$ is $-2\alpha$ convex in
$L^{2}(\mathbb{T}^{3};\mathcal{Q})$, hence by Remark 2.1 we know that
$Q(t,\cdot)$ constructed in Proposition 3.1 is the unique solution to the
gradient flow (1.6). The following lemma computes explicitly the sub-
differential of the free energy $\mathcal{E}$.

######  Lemma 3.3.

For any $Q\in D(\partial\mathcal{E})$ and $1\leq i,j\leq 3$, we have

$\displaystyle-\partial\mathcal{E}(Q)_{ij}$
$\displaystyle=2L_{1}\Delta{Q}_{ij}+2(L_{2}+L_{3}){\partial}_{kj}Q_{ik}-\dfrac{2(L_{2}+L_{3})}{3}{\partial}_{k\ell}Q_{k\ell}\delta_{ij}$
$\displaystyle\qquad-\psi^{\prime}(Q)_{ij}+\dfrac{\mathrm{tr}(\psi^{\prime}(Q))}{3}\delta_{ij}+2\alpha{Q}_{ij}.$

###### Proof.

To begin with, it is immediate to derive

$\partial\mathcal{G}(Q)_{ij}=-2L_{1}\Delta
Q_{ij}-2(L_{2}+L_{3})\partial_{k}\partial_{j}Q_{ik}+\frac{2}{3}(L_{2}+L_{3})\partial_{\ell}\partial_{k}Q_{k\ell}\delta_{ij}.$
(3.9)

Next we need to verify that

$\partial\mathcal{B}(Q)=\Big{\\{}\psi^{\prime}(Q)-\frac{1}{3}\mathrm{tr}(\psi^{\prime}(Q))\mathbb{I}_{3}\Big{\\}}.$

Case 1: $Q$ is strictly physical.

Let $R\in C_{c}^{\infty}(\mathbb{T}^{3},\mathcal{Q})$. Then by convexity and
smoothness of $\psi$, any element $\xi\in\partial\mathcal{B}(Q)$ satisfies

$\displaystyle\int_{\mathbb{T}^{3}}\big{[}\psi^{\prime}(Q):R\big{]}\,d{x}$
$\displaystyle=\lim_{{\varepsilon}\rightarrow
0^{+}}\frac{\mathcal{B}(Q+{\varepsilon}R)-\mathcal{B}(Q)}{{\varepsilon}}\geq\langle\xi,R\rangle_{L^{2}(\mathbb{T}^{3})},$
$\displaystyle-\int_{\mathbb{T}^{3}}\big{[}\psi^{\prime}(Q):R\big{]}\,d{x}$
$\displaystyle=\lim_{{\varepsilon}\rightarrow
0^{+}}\frac{\mathcal{B}(Q-{\varepsilon}R)-\mathcal{B}(Q)}{{\varepsilon}}\geq-\langle\xi,R\rangle_{L^{2}(\mathbb{T}^{3})},$

which indicates
$\langle\xi,R\rangle_{L^{2}(\mathbb{T}^{3})}=\langle\psi^{\prime}(Q),R\rangle_{L^{2}(\mathbb{T}^{3})}$.
By density,

$\xi=\psi^{\prime}(Q)-\mathrm{tr}(\psi^{\prime}(Q))\mathbb{I}_{3}/3$ (3.10)

as an element in the Hilbert space $L^{2}(\mathbb{T}^{n};\mathcal{Q})$. Hence
$\partial\mathcal{B}(Q)$
$=\\{\psi^{\prime}(Q)-\mathrm{tr}(\psi^{\prime}(Q))\mathbb{I}_{3}/3\\}$ for
any uniformly physical $Q$.

Case 2: $Q$ is not strictly physical.

We define

$\rho(P):=\min_{1\leq i\leq
3}\Big{\\{}\lambda_{i}(P)+\frac{1}{3},\,\frac{2}{3}-\lambda_{i}(P)\Big{\\}},\qquad\forall\,P\in\mathcal{Q},$
(3.11)

and $A_{\eta}:=\\{x\in\mathbb{T}^{3}:\rho(Q(x))<\eta\\}$ for arbitrarily small
$\eta>0$. Since $Q\in D(\partial\mathcal{E})\subset D(\mathcal{E})$, we have
$\psi(Q)<+\infty$, thus $|A_{\eta}|\rightarrow 0$ as $\eta\rightarrow 0^{+}$.
Let

$T_{\eta}(Q):=\\{R\in L^{\infty}(\mathbb{T}^{3};\mathcal{Q}):\ R\equiv 0\mbox{
on }A_{\eta}\\}.$

Let us consider any fixed $\eta>0$, and $R\in T_{\eta}(Q)$. Then it is easy to
check that $\rho(Q)\geq\eta$ outside $A_{\eta}$, and for all sufficiently
small ${\varepsilon}$ we have $\rho(Q\pm{\varepsilon}R)\geq\eta/2$ outside
$A_{\eta}$, hence

$\mathcal{B}(Q+{\varepsilon}R)=\int_{\mathbb{T}^{3}\setminus
A_{\eta}}\psi(Q+{\varepsilon}R)\,\mathrm{d}{x}+\int_{A_{\eta}}\psi(Q)\,\mathrm{d}{x}<+\infty.$

This together with the covexity of $\psi$ implies that for any sufficiently
small ${\varepsilon}>0$

$\displaystyle{\varepsilon}\int_{\mathbb{T}^{3}\setminus
A_{\eta}}\big{[}\psi^{\prime}(Q+{\varepsilon}R):R\big{]}\,\mathrm{d}{x}\geq\mathcal{B}(Q+{\varepsilon}R)-\mathcal{B}(Q)$
$\displaystyle=$ $\displaystyle\int_{\mathbb{T}^{3}\setminus
A_{\eta}}\big{[}\psi(Q+{\varepsilon}R)-\psi(Q)\big{]}\,\mathrm{d}{x}\geq{\varepsilon}\int_{\mathbb{T}^{3}\setminus
A_{\eta}}\big{[}\psi^{\prime}(Q):R\big{]}\,\mathrm{d}{x},$

and dividing by ${\varepsilon}$ gives

$\displaystyle\liminf_{{\varepsilon}\rightarrow
0^{+}}\frac{\mathcal{B}(Q+{\varepsilon}R)-\mathcal{B}(Q)}{{\varepsilon}}$
$\displaystyle\geq\int_{\mathbb{T}^{3}\setminus
A_{\eta}}\big{[}\psi^{\prime}(Q):R\big{]}\,\mathrm{d}{x},$ (3.12)
$\displaystyle\limsup_{{\varepsilon}\rightarrow
0^{+}}\frac{\mathcal{B}(Q+{\varepsilon}R)-\mathcal{B}(Q)}{{\varepsilon}}$
$\displaystyle\leq\limsup_{{\varepsilon}\rightarrow
0^{+}}\int_{\mathbb{T}^{3}\setminus
A_{\eta}}\big{[}\psi^{\prime}(Q+{\varepsilon}R):R\big{]}\,\mathrm{d}{x}.$
(3.13)

Meanwhile, note that

$\psi^{\prime}\in L^{\infty}(\mathbb{T}^{3}\setminus
A_{\eta}),\;\;\lim_{{\varepsilon}\rightarrow{0^{+}}}\int_{\mathbb{T}^{3}\setminus
A_{\eta}}\big{[}\psi^{\prime}(Q+{\varepsilon}R):R\big{]}\,\mathrm{d}{x}=\int_{\mathbb{T}^{3}\setminus
A_{\eta}}\big{[}\psi^{\prime}(Q):R\big{]}\,\mathrm{d}{x}=\int_{\mathbb{T}^{3}}\big{[}\psi^{\prime}(Q):R\big{]}\,\mathrm{d}{x}$

due to the fact that $R\equiv 0$ on $A_{\eta}$. This together with (3.12),
(3.13) yields

$\lim_{{\varepsilon}\rightarrow
0^{+}}\frac{\mathcal{B}(Q+{\varepsilon}R)-\mathcal{B}(Q)}{{\varepsilon}}=\int_{\mathbb{T}^{3}}\big{[}\psi^{\prime}(Q):R\big{]}\,\mathrm{d}{x}.$

Further, as discussed in case 1, for any $R\in\bigcup_{\eta>0}T_{\eta}(Q)$ we
have

$\langle\xi,R\rangle_{L^{2}(\mathbb{T}^{3})}=\langle\psi^{\prime}(Q),R\rangle_{L^{2}(\mathbb{T}^{3})}.$

By density, it follows
$\xi=\psi^{\prime}(Q)-\mathrm{tr}(\psi^{\prime}(Q))\mathbb{I}_{3}/3$ as
elements of the Hilbert space $L^{2}(\mathbb{T}^{3};\mathcal{Q})$. Hence
$\partial\mathcal{B}(Q)=\\{\psi^{\prime}(Q)-\mathrm{tr}(\psi^{\prime}(Q))\mathbb{I}_{3}/3\\}$
even if $Q\subset D(\partial\mathcal{E})$ is not strictly physical.

By Lemma 3.1, $\mathcal{G}$ and $\mathcal{B}$ are proper, convex and lower
semicontinuous, and the intersection
$D(\mathcal{G})\cap\text{int}D(\mathcal{B})$ is non empty, we get
$\partial(\mathcal{G}+\mathcal{B})(Q)=\partial\mathcal{G}(Q)+\partial\mathcal{B}(Q)$
by [6, Theorem 2.10]. Since the last term
$-\alpha\|Q\|_{L^{2}(\mathbb{T}^{3})}^{2}$ is a $C^{1}$ perturbation of the
energy, we infer

$\partial\mathcal{E}(Q)=\partial(\mathcal{G}+\mathcal{B})(Q)-2\alpha{Q}=\partial\mathcal{G}(Q)+\partial\mathcal{B}(Q)-2\alpha{Q},$
(3.14)

which concludes the proof. ∎

Proposition 3.1 leads to higher regularity of the solution $Q$. Since the
energy $\mathcal{E}$ is $-2\alpha$-convex, and since the solution $Q(t,\cdot)$
satisfies $Q(t,\cdot)\in D(\mathcal{E})$ for all time $t>0$, we conclude that
for any $t\geq t_{0}>0$ the function $Q(t,\cdot)$ is the gradient flow of
$\mathcal{E}$ in $L^{2}(\mathbb{T}^{n};\mathcal{Q})$ with initial datum
$Q(t_{0},\cdot)\in D(\mathcal{E})$. Thus we can apply [1, Theorem 2.4.15] to
obtain

######  Proposition 3.2.

Let $Q(t,\cdot)$ be the solution given by Theorem 1.1. Then the map

$t\mapsto e^{-2\alpha
t}\|\partial\mathcal{E}(Q(t,\cdot))\|_{L^{2}(\mathbb{T}^{3})}$

is nonincreasing, right continuous on $[t_{0},+\infty)$ for all $t_{0}>0$.

Finally, combining [1, Corollary 2.4.11] and [1, Theorem 2.3.3] we obtain the
energy identity (1.12)

######  Proposition 3.3.

The solution $Q(t,\cdot)$ given by Theorem 1.1 satisfies the energy equality

$\int_{t_{0}}^{T}\frac{1}{2}\big{(}\|\partial_{t}Q(t,\cdot)\|_{L^{2}(\mathbb{T}^{3})}^{2}+\frac{1}{2}\|\partial\mathcal{E}(Q(t,\cdot))\|^{2}_{L^{2}(\mathbb{T}^{3})}\big{)}\,\mathrm{d}{t}=\mathcal{E}(Q(t_{0}))-\mathcal{E}(Q(T))$

for all $0<t_{0}<T<+\infty$.

In conclusion, the proof of Theorem 1.1 is complete.

######  Remark 3.2.

The nonincreasing property of the energy term $e^{-2\alpha
t}\|\partial\mathcal{E}(Q(t,\cdot))\|_{L^{2}(\mathbb{T}^{3})}$ in Proposition
3.2 will play an essential role in the proof of Theorem 1.2, which is the main
reason that the Ambrosio-Gigli-Savare’s gradient flow theory in [1] is adopted
in this section other than the classical Brezis-Pazy’s theory.

##  4 Proof of Theorem 1.2: higher regularity of solutions

This section is devoted to the proof of Theorem 1.2. Since there is only minor
modification of arguments between $\mathbb{T}^{2}$ and $\mathbb{T}^{3}$, we
only discuss the case in $\mathbb{T}^{3}$.

It is noted that the maximum principle argument utilized in [29] fails due to
the presence of the anisotropic terms, hence the solution $Q(t,\cdot)$ is not
ensured to stay detached from the physical boundary ${\partial}\mathcal{Q}$ at
any positive time $t$. To achieve the proof, we have to put together several
results in [1, 27, 28], and to make a full exploitation of the gradient flow
structure in (1.6).

Our main strategy is as follows. First of all, to avoid the singular feature
of $\partial\mathcal{E}(Q)$ in (1.6), we shall consider a sequence of smooth
gradient flows (4.19) that are generated by an approximation sequence
$\\{\mathcal{E}_{n}\\}$ defined in (4.6) of the free energy $\mathcal{E}$.
Secondly, we will prove $\Gamma-$ convergence of $\\{\mathcal{E}_{n}\\}$ to
$\mathcal{E}$ in Proposition 4.1, which together with energy dissipative
equality achieved in Proposition 3.3 we can show the “convergence” of the
gradient flow sequence (4.19) to (1.6). Next, we will show in Proposition 4.2
that the solution sequence $\\{Q_{n}\\}$ to the gradient flow sequence (4.19)
is in $H^{2}(\mathbb{T}^{3};\mathcal{Q})$ space, and give a corresponding
estimate of the $H^{2}$ bound of $Q$ in terms of $\|\partial\mathcal{E}(Q)\|$,
which together with Proposition 3.2 leads to the uniform-in-time bound for
$\|\partial\mathcal{E}(Q)\|$. Finally, we make use of the convexity of $\psi$
and Sobolev interpolation inequalities to derive strict physicality of the
solution at all large times.

Here and after, we denote $\\{\psi_{n}\\}$ the sequence of functions that is
used in [29] to approximate the Ball-Majumdar bulk potential $\psi$: first, we
introduce the Moreau-Yosida approximations

$\tilde{\psi}_{n}(Q):=\inf_{A\in\mathcal{Q}}\\{n|A-Q|^{2}+\psi(A)\\},\qquad
Q\in\mathcal{Q}.$ (4.1)

Then using a smooth regularization we define

$\psi_{n}(Q)=n^{5}\int_{\mathcal{Q}}\tilde{\psi}_{n}\big{(}n(Q-R)\big{)}\phi(R)\,dR,\qquad
Q\in\mathcal{Q}.$ (4.2)

Here $\phi\in C_{c}^{\infty}(\mathcal{Q},\overline{R^{+}})$ is of unit mass.
Let us recall [29, Proposition 3.1] for each $n\geq 1$, we have

$(\textbf{M0})$ $\psi_{n}$ is an isotropic function of $Q$.

$(\textbf{M1})$ $\psi_{n}$ is both smooth and convex in $\mathcal{Q}$.

$(\textbf{M2})$ $\psi_{n}$ is bounded from below, i.e.,
$-4\pi^{2}|\mathbb{T}^{3}|/e\leq\psi_{n}(R)$, $\forall R\in\mathcal{Q}$,
$\forall n\geq 1$.

$(\textbf{M3})$ $\psi_{n}\leq\psi_{n+1}\leq\psi$ on $\mathcal{Q}$ for $n\geq
1$.

$(\textbf{M4})$ $\psi_{n}\rightarrow\psi$ in
$L_{loc}^{\infty}\big{(}D(\psi)\big{)}$ as $n\rightarrow\infty$, and
$\psi_{n}$ is uniformly divergent on $\mathcal{Q}\setminus D(\psi)$

$(\textbf{M5})$
$\frac{\partial\psi_{n}}{\partial{Q}}\rightarrow\frac{\partial\psi}{\partial{Q}}$
in $L_{loc}^{\infty}\big{(}D(\psi)\big{)}$ as $n\rightarrow\infty$.

$(\textbf{M6})$ There exist constants $\lambda_{n},\Lambda_{n}>0$ that may
depend on $n$, such that

$\lambda_{n}|R|-\Lambda_{n}\leq\Big{|}\psi_{n}^{\prime}(R)-\frac{1}{3}\mathrm{tr}(\psi_{n}^{\prime}(R))\mathbb{I}_{3}\Big{|}\leq\lambda_{n}|R|+\Lambda_{n},\quad\forall
R\in\mathcal{Q}$

Besides the aforementioned properties $(\textbf{M0})$ to $(\textbf{M6})$, we
need to further derive the following finer estimate of the sequence
$\\{\psi_{n}\\}$, in order to prove the later Proposition 4.1 and Lemma 4.2.

######  Lemma 4.1.

For any $n\in\mathbb{N}$, there exists a generic constant $C_{n}>0$ such that

$\displaystyle\psi_{n}(Q)$ $\displaystyle\geq C_{n}|Q|^{2},\qquad\mbox{outside
a fixed compact subset in }\mathcal{Q}.$ (4.3)

Moreover, $C_{n}\rightarrow+\infty$ as $n\rightarrow+\infty$.

###### Proof.

Since the null matrix $0\in D(\psi)$, taking $A=0$ in (4.1) we get the upper
bound

$\tilde{\psi}_{n}(Q)\leq n|Q|^{2}+\psi(0).$

On the other hand, since such infimum in (4.1) is finite, there exists a
minimizing sequence $A_{m}\subset D(\psi)$ such that

$\displaystyle\tilde{\psi}_{n}(Q)=\lim_{m\rightarrow+\infty}(n\,|A_{m}-Q|^{2}+\psi(A_{m}))$
$\displaystyle\geq$
$\displaystyle\lim_{m\rightarrow+\infty}n|A_{m}-Q|^{2}+\inf\psi\geq
n\,\mbox{dist}(Q,D(\psi))^{2}+\inf\psi.$ (4.4)

By the triangle inequality, we have

$|Q|\leq\mbox{dist}(Q,D(\psi))+\mbox{dist}(0,D(\psi)),$

and since $0\in D(\psi)$, we can further get

$\mbox{dist}(Q,D(\psi))\geq|Q|-\mbox{diam}(D(\psi)).$

As a consequence, we see

$\tilde{\psi}_{n}(Q)\geq
n|Q|^{2}-2n\,\mbox{diam}(D(\psi))|Q|+\mbox{diam}(D(\psi))^{2}+\inf\psi\geq
n|Q|\Big{(}|Q|-2\mbox{diam}(D(\psi))\Big{)}.$

Hence we arrive at the uniform quadratic estimate

$\frac{n}{2}|Q|^{2}\leq\tilde{\psi}_{n}(Q)$ (4.5)

in $\\{|Q|>4\mbox{diam}(D(\psi))\\}$. Then following the mollification of
$\tilde{\psi}_{n}$ in (4.2) one can get (4.3). ∎

We introduce the energy sequence
$\mathcal{E}_{n}:\mathcal{Q}\rightarrow\mathbb{R}\cup\\{+\infty\\}$

$\mathcal{E}_{n}(Q)=\begin{cases}\mathcal{G}(Q)+\int_{\mathbb{T}^{3}}\psi_{n}(Q)\,\mathrm{d}{x}-\alpha\|Q\|_{L^{2}(\mathbb{T}^{3})}^{2},\quad\mbox{if
}Q\in H^{1}(\mathbb{T}^{3})\\\
+\infty,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\mbox{otherwise},\end{cases}$
(4.6)

and establish a $\Gamma$-convergence result.

######  Proposition 4.1.

The sequence of energies $\\{\mathcal{E}_{n}\\}$ $\Gamma$-converges to
$\mathcal{E}$.

###### Proof.

We first show compactness. Let us assume
$\displaystyle\liminf_{n\rightarrow+\infty}\mathcal{E}_{n}(Q_{n})<+\infty$.
Upon subsequence, we may assume

$\liminf_{n\rightarrow+\infty}\mathcal{E}_{n}(Q_{n})=\lim_{n\rightarrow+\infty}\mathcal{E}_{n}(Q_{n})<+\infty,\qquad\displaystyle\sup_{n\in\mathbb{N}}\mathcal{E}_{n}(Q_{n})<+\infty.$
(4.7)

We need first to ensure the existence of a strong limit $Q$. We claim that
$Q_{n}$ is uniformly bounded in $L^{2}(\mathbb{T}^{3})$. Otherwise there
exists a subsequence $\\{Q_{n_{k}}\\}$, such that
$\|Q_{n_{k}}\|_{L^{2}(\mathbb{T}^{3})}\rightarrow+\infty$, it then follows
directly from Lemma 3.1 and (4.3) that
$\mathcal{E}_{n_{k}}(Q_{n_{k}})\rightarrow+\infty$, which is in contradiction
with the assumption (4.7).

Further, note that

$\displaystyle(L_{1}-3|L_{2}+L_{3}|)\sup_{n}\|\nabla{Q}_{n}\|_{L^{2}(\mathbb{T}^{3})}^{2}$
$\displaystyle\leq\sup_{n}\mathcal{G}(Q_{n})\leq\sup_{n}\mathcal{E}_{n}(Q_{n})+\alpha\sup_{n}\|Q_{n}\|_{L^{2}(\mathbb{T}^{3})}^{2}+\inf_{n,P}\int_{\mathbb{T}^{3}}|\psi_{n}(P)|\,\mathrm{d}{x}<+\infty.$

Thus $Q_{n}$ is uniformly bounded in $H^{1}(\mathbb{T}^{3})$, and
$Q_{n}\rightarrow Q$ (up to a subsequence) strongly in
$L^{2}(\mathbb{T}^{3})$.

Next we show $\Gamma-\limsup$ inequality. That is, for any $Q\in\mathcal{Q}$
there exists a recovery sequence $Q_{n}$ such that

$\limsup_{n\rightarrow+\infty}\mathcal{E}_{n}(Q_{n})\leq\mathcal{E}(Q).$ (4.8)

Without loss of generality we assume $\mathcal{E}(Q)<+\infty$. Taking
$Q_{n}=Q$, for every $n\geq 1$ we get from $(\textbf{M3})$ that

$\mathcal{G}(Q_{n})-\alpha\|Q_{n}\|_{L^{2}(\mathbb{T}^{3})}^{2}\equiv\mathcal{G}(Q)-\alpha\|Q\|_{L^{2}(\mathbb{T}^{3})}^{2},\quad\int_{\mathbb{T}^{3}}\psi_{n}(Q_{n})\,\mathrm{d}x\leq\int_{\mathbb{T}^{3}}\psi(Q)\,\mathrm{d}x,\quad\forall
n\geq 1.$

In all, (4.8) is verified.

We proceed to show $\Gamma-\liminf$ inequality. That is, for any sequence
$Q_{n}\rightarrow Q$ strongly in $L^{2}(\mathbb{T}^{3})$, it holds

$\liminf_{n\rightarrow+\infty}\mathcal{E}_{n}(Q_{n})\geq\mathcal{E}(Q).$ (4.9)

Without loss of generality we assume
$\displaystyle\liminf_{n\rightarrow+\infty}\mathcal{E}_{n}(Q_{n})<+\infty$.
Again upon subsequence, we may assume

$\liminf_{n\rightarrow+\infty}\mathcal{E}_{n}(Q_{n})=\lim_{n\rightarrow+\infty}\mathcal{E}_{n}(Q_{n})<+\infty,\qquad\displaystyle\sup_{n\in\mathbb{N}}\mathcal{E}_{n}(Q_{n})<+\infty.$

As discussed earlier we have $Q_{n}\rightarrow Q$ strongly in
$L^{2}(\mathbb{T}^{3})$, and upon further extracting a subsequence we may
assume $Q_{n}\rightarrow Q$ a.e. in $\mathbb{T}^{3}$. Hence together with the
lower semicontinuity of $\mathcal{G}$ achieved in Lemma 3.1 it yields

$\lim_{n\rightarrow+\infty}\|Q_{n}(\cdot)\|_{L^{2}(\mathbb{T}^{3})}^{2}=\|Q(\cdot)\|_{L^{2}(\mathbb{T}^{3})}^{2},\qquad\liminf_{n\rightarrow+\infty}\mathcal{G}(Q_{n})\geq\mathcal{G}(Q).$
(4.10)

It remains to prove

$\liminf_{n\rightarrow+\infty}\int_{\mathbb{T}^{3}}\psi_{n}(Q_{n})\,\mathrm{d}x\geq\int_{\mathbb{T}^{3}}\psi(Q)\,\mathrm{d}x.$
(4.11)

To proceed, we denote

$D:=\big{\\{}x\in\mathbb{T}^{3}:Q(x)\in D(\psi)\big{\\}},$

and we distinguish between two cases.

Case 1: $\mathcal{B}(Q)<+\infty$. For any sufficiently small ${\varepsilon}>0$
let us define

$\mathbb{T}^{3}_{\varepsilon}=D\cap\left\\{x\in\mathbb{T}^{3}:-\frac{1}{3}+{\varepsilon}\leq\lambda_{i}(Q(x))\leq\frac{2}{3}-{\varepsilon},\;1\leq
i\leq 3\right\\}.$ (4.12)

Since $\big{|}\mathbb{T}^{3}\setminus{D}\big{|}=0$ we get

$\displaystyle\int_{\mathbb{T}^{3}}\big{[}\psi_{n}(Q_{n})-\psi(Q)\big{]}\,\mathrm{d}{x}$
$\displaystyle=\int_{D}\big{[}\psi_{n}(Q_{n})-\psi(Q)\big{]}\,\mathrm{d}{x}$
$\displaystyle=\int_{D\setminus\mathbb{T}^{3}_{\varepsilon}}\big{[}\psi_{n}(Q_{n})-\psi(Q)]\,\mathrm{d}{x}+\int_{\mathbb{T}^{3}_{\varepsilon}}[\psi_{n}(Q_{n})-\psi(Q)\big{]}\,\mathrm{d}{x}.$
(4.13)

On the one hand, for any $n\geq 1$,

$\displaystyle\int_{D\setminus\mathbb{T}^{3}_{\varepsilon}}\psi_{n}(Q_{n})\,\mathrm{d}{x}$
$\displaystyle\geq\lim_{{\varepsilon}\to
0}(\inf_{n,P}\psi_{n}(P))\big{|}D\setminus\mathbb{T}^{3}_{\varepsilon}\big{|}=0,$

and since $\psi\in L^{1}(\mathbb{T}^{3})$, as
$|D\setminus\mathbb{T}^{3}_{\varepsilon}|\rightarrow 0$, we have

$\displaystyle\lim_{{\varepsilon}\to
0}\int_{D\setminus\mathbb{T}^{3}_{\varepsilon}}\psi(Q)\,\mathrm{d}{x}=0.$

Hence for all $\delta>0$, there exists
${\varepsilon}_{0}={\varepsilon}_{0}(\delta)>0$, such that

$\int_{D\setminus\mathbb{T}^{3}_{\varepsilon}}\big{[}\psi_{n}(Q_{n})-\psi(Q)\big{]}\,\mathrm{d}{x}>-\dfrac{\delta}{2},\quad\forall
n\geq 1\quad\mbox{whenever }\,{\varepsilon}\leq{\varepsilon}_{0}.$ (4.14)

On the other hand, it follows from $(\textbf{M5})$ that
$\psi_{n}(Q(x))\rightarrow\psi(Q(x))$,
$\psi_{n}^{\prime}(Q(x))\rightarrow\psi^{\prime}(Q(x))$ as
$n\rightarrow+\infty$ in $L^{\infty}(\mathbb{T}^{3}_{\varepsilon})$. Thus it
implies for any fixed ${\varepsilon}\leq{\varepsilon}_{0}$ it holds

$\lim_{n\rightarrow\infty}\int_{\mathbb{T}^{3}_{\varepsilon}}[\psi_{n}(Q_{n})-\psi(Q)\big{]}\,\mathrm{d}{x}=0.$

Specifically, for ${\varepsilon}={\varepsilon}_{0}$ there exists
$N_{0}\in\mathbb{N}$, such that

$\int_{\mathbb{T}^{3}_{{\varepsilon}_{0}}}[\psi_{n}(Q_{n})-\psi(Q)\big{]}\,\mathrm{d}{x}>-\dfrac{\delta}{2},\quad\forall
n\geq N_{0}$ (4.15)

Combining (4.14) and (4.15) we finish the proof of (4.11) by the arbitrariness
of $\delta$.

Case 2: $\mathcal{B}(Q)=+\infty$. In this case it suffices to check

$\liminf_{n\rightarrow\infty}\int_{\mathbb{T}^{3}}\psi_{n}(Q_{n})\,\mathrm{d}{x}=+\infty.$
(4.16)

If $|\mathbb{T}^{3}\setminus{D}|>0$, since $Q_{n}\rightarrow Q$ a.e. it
follows from Egorov’s theorem that there exists a set
$F\subset(\mathbb{T}^{3}\setminus{D})$, $|F|>0$, such that $Q_{n}\rightarrow
Q$ uniformly on $F$. Note that $Q(x)\in\mathcal{Q}\setminus D(\psi)$, $\forall
x\in F$. Hence the uniform convergence of $Q_{n}$ to $Q$ on $F$ implies there
exists a sequence ${\varepsilon}_{n}\searrow 0^{+}$, such that

$\lambda_{i}(Q_{n}(x))\leq-\frac{1}{3}+{\varepsilon}_{n}\;\mbox{or
}\;\lambda_{i}(Q_{n}(x))\geq\frac{2}{3}-{\varepsilon}_{n},\quad\forall\,1\leq
i\leq 3.$

Then Fatou’s lemma and $(\textbf{M4})$ yields

$\liminf_{n\rightarrow+\infty}\int_{F}\psi_{n}(Q_{n})\,\mathrm{d}{x}\geq\int_{F}\liminf_{n\rightarrow+\infty}\psi_{n}(Q_{n})\,\mathrm{d}{x}=+\infty.$

Therefore, (4.16) is verified in this case.

Alternatively if $|\mathbb{T}^{3}\setminus{D}|=0$, then using similar argument
as in _Case 1_ we have

$\int_{\mathbb{T}^{3}}\psi_{n}(Q_{n})\,\mathrm{d}{x}=\int_{D}\psi_{n}(Q_{n})\,\mathrm{d}{x}=\int_{D\setminus\mathbb{T}^{3}_{{\varepsilon}}}\psi_{n}(Q_{n})\,\mathrm{d}{x}+\int_{\mathbb{T}^{3}_{{\varepsilon}}}\psi_{n}(Q_{n})\,\mathrm{d}{x},$

where $\mathbb{T}^{3}_{{\varepsilon}}$ is given in (4.12). On one hand,

$\int_{D\setminus\mathbb{T}^{3}_{{\varepsilon}}}\psi_{n}(Q_{n})\,\mathrm{d}{x}\geq(\inf_{n,P}\psi_{n}(P))\big{|}\mathbb{T}^{3}\setminus\mathbb{T}^{3}_{{\varepsilon}}\big{|}\geq-\frac{4\pi|\mathbb{T}^{3}|^{2}}{e},\quad\forall{\varepsilon}>0.$
(4.17)

On the other hand, since $\mathcal{B}(Q)=+\infty$ and
$\mathbb{T}^{3}_{{\varepsilon}}\nearrow D$, we infer that $\forall M>0$, there
exists ${\varepsilon}_{0}>0$, such that

$\int_{\mathbb{T}^{3}_{{\varepsilon}}}\psi(Q)\,\mathrm{d}{x}>M+1+\frac{4\pi|\mathbb{T}^{3}|^{2}}{e},\quad\forall{\varepsilon}\leq{\varepsilon}_{0}.$

Meanwhile, $(\textbf{M5})$ indicates that
$\psi_{n}(Q(x))\rightarrow\psi(Q(x))$,
$\psi_{n}^{\prime}(Q(x))\rightarrow\psi^{\prime}(Q(x))$ in
$L^{\infty}(\mathbb{T}^{3}_{{\varepsilon}_{0}})$, which gives

$\int_{\mathbb{T}^{3}_{{\varepsilon}_{0}}}\psi_{n}(Q_{n})\,\mathrm{d}{x}\rightarrow\int_{\mathbb{T}^{3}_{{\varepsilon}_{0}}}\psi(Q)\,\mathrm{d}{x}.$

Thus there exists $N\in\mathbb{N}$, such that

$\int_{\mathbb{T}^{3}_{{\varepsilon}_{0}}}\psi_{n}(Q_{n})\,\mathrm{d}{x}>\int_{\mathbb{T}^{3}_{{\varepsilon}_{0}}}\psi(Q)\,\mathrm{d}{x}-1>M+\frac{4\pi|\mathbb{T}^{3}|^{2}}{e},\quad\forall
n\geq N.$ (4.18)

Putting together (4.17) and (4.18), we conclude that (4.16) is valid. Hence
the proof is complete. ∎

Since in Theorem 1.1 we allow the initial data
$Q_{0}\in\overline{D(\mathcal{E})}$, one may not expect any further regularity
at time $t=0$. Thus we only aim to prove the regularity of $Q(t,\cdot)$ for
positive times $t>0$.

Let $t_{0}>0$ be arbitrarily given. Consider the gradient flow sequence

$\displaystyle\begin{cases}\partial_{t}Q_{n}=-\partial\mathcal{E}_{n}(Q_{n}),\quad
t\geq t_{0},\\\ Q_{n}(t_{0},x)=Q(t_{0},x)\end{cases}$ (4.19)

subject to periodic boundary conditions (1.7). Here we let time start from
$t_{0}$ only as a matter of convenience, to avoid an (unnecessary) time
shifting. Note that the energies $\mathcal{E}_{n}$ are also $-2\alpha$-convex,
proper, lower semicontinuous, and uniformly bounded from below. Thus we can
apply [1, Theorem 4.0.4] to get the same conclusions as in Theorem 1.1,
Proposition 3.2 and Proposition 3.3.

As a consequence, we manage to prove

######  Lemma 4.2.

Let $Q_{n}$ be the solution of (4.19). Then for every $t_{0}\in(0,T)$

$\|\partial\mathcal{E}_{n}(Q_{n}(t,\cdot))\|_{L^{2}(\mathbb{T}^{3})}\rightarrow\|\partial\mathcal{E}(Q(t,\cdot))\|_{L^{2}(\mathbb{T}^{3})},\quad\|\partial_{t}Q_{n}(t,\cdot)\|_{L^{2}(\mathbb{T}^{3})}\rightarrow\|\partial_{t}Q(t,\cdot)\|_{L^{2}(\mathbb{T}^{3})}\;\text{
in }L^{2}(t_{0},T),$

up to a subsequence.

###### Proof.

By Proposition 4.1, we have
$\mathcal{E}_{n}\overset{\Gamma}{\rightarrow}\mathcal{E}$. We aim to show that
we are under the hypotheses of Proposition 2.2, where the general sense of
convergence is considered to be the strong convergence in
$L^{2}(\mathbb{T}^{3};\mathcal{Q})$. Let us first check that the conditions
(2.6), (2.7) are valid for the solutions $Q_{n}$ of (4.19).

By Proposition 3.3 and $(\textbf{M3})$ we have

$\mathcal{E}_{n}(Q_{n})(t)\leq\mathcal{E}_{n}(Q(t_{0}))\leq\mathcal{E}(Q(t_{0})),\quad\forall\,t>t_{0},n\in\mathbb{N}.$

Hence following a similar argument in the proof of compactness part in
Proposition 4.1 we get

$\\{Q_{n}\\}\;\;\mbox{is uniformly bounded in
}L^{\infty}(t_{0},T;H^{1}(\mathbb{T}^{3})).$ (4.20)

Next, by Proposition 3.3 and $(\textbf{M3})$, the energy equality

$\displaystyle\int_{t_{0}}^{T}\big{(}\|\partial_{t}Q_{n}(t,\cdot)\|^{2}+\|\partial\mathcal{E}_{n}(Q_{n}(t,\cdot))\|^{2}\big{)}\,\mathrm{d}{t}$
$\displaystyle=2\mathcal{E}_{n}(Q(t_{0}))-2\mathcal{E}_{n}(Q_{n}(T))$
$\displaystyle\leq 2\mathcal{E}(Q(t_{0}))-2\inf\mathcal{E}_{n}$ (4.21)

holds for all $t_{0}<T<+\infty$. Hence

$\\{\partial_{t}Q_{n}\\}\;\mbox{is uniformly bounded
in}\;L^{2}(t_{0},T;L^{2}(\mathbb{T}^{3}))$ (4.22)

and we can apply the Aubin-Lions lemma to yield

$Q_{n}\rightarrow\tilde{Q}\;\;\mbox{strongly in
}C\big{(}[t_{0},T];L^{2}(\mathbb{T}^{3})\big{)},$ (4.23)

and (4) implies that

$\partial_{t}Q_{n}\rightarrow\partial_{t}\tilde{Q}\;\;\mbox{weakly in
}L^{2}(0,T;L^{2}(\mathbb{T}^{3})).$ (4.24)

Hence the condition (2.6) is satisfied by taking $f=0$. Since each
$\mathcal{E}_{n}$ is $-2\alpha$ convex, by [25, Proposition 13] we know that
the condition (2.7) is also satisfied by taking $C=0$.

Finally, note that the initial data are “well-prepared” in the sense of
$\mathcal{E}_{n}(Q(t_{0}))\rightarrow\mathcal{E}(Q(t_{0}))$, thus all the
assumptions in Proposition 2.2 are satisfied, which in turn gives that
$\tilde{Q}$ is a solution of the gradient flow

$\begin{cases}\partial_{t}\tilde{Q}=-\partial\mathcal{E}(\tilde{Q}),\\\
\tilde{Q}(t_{0})=Q(t_{0}).\end{cases}$

Since $Q$ is already a solution, and by Theorem 1.1 we know that such a
solution is unique, we infer $\tilde{Q}=Q$, and the proof is complete.

∎

Now we turn to the $H^{2}$-regularity of the sequence $\\{Q_{n}\\}$. The
following technical implies the coercivity of $\partial\mathcal{G}$:

######  Lemma 4.3.

For any $P\in H^{2}(\mathbb{T}^{3};\mathcal{Q})$ the operator
$\partial\mathcal{G}(P)$ satisfies

$2(L_{1}-|L_{2}+L_{3}|)\|\Delta
P\|^{2}_{L^{2}(\mathbb{T}^{3})}\leq\langle-\partial\mathcal{G}(P),\Delta
P\rangle_{L^{2}}\leq
2(L_{1}+|L_{2}+L_{3}|)\|\Delta{P}\|_{L^{2}(\mathbb{T}^{3})}^{2}.$

###### Proof.

Direct computations yield

$\displaystyle\langle-\partial\mathcal{G}(P),\Delta P\rangle_{L^{2}}$
$\displaystyle=\int_{\mathbb{T}^{3}}\bigg{[}2L_{1}\Delta{P}_{ij}+2(L_{2}+L_{3})\partial_{k}\partial_{j}P_{ik}-\frac{2}{3}(L_{2}+L_{3})\partial_{\ell}\partial_{k}P_{k\ell}\delta_{ij}\bigg{]}\Delta{P}_{ij}\,\mathrm{d}{x}$
$\displaystyle=2L_{1}\|\Delta
P\|^{2}_{L^{2}(\mathbb{T}^{3})}+2(L_{2}+L_{3})\int_{\mathbb{T}^{3}}\partial_{j}\partial_{k}P_{ik}\Delta
P_{ij}\,\mathrm{d}{x},$

where the last term of R.H.S. can be treated by integration by parts and the
Hölder’s inequality:

$\displaystyle
2(L_{2}+L_{3})\int_{\mathbb{T}^{3}}\partial_{k}\partial_{j}P_{ik}\Delta
P_{ij}\,\mathrm{d}{x}$
$\displaystyle=2(L_{2}+L_{3})\int_{\mathbb{T}^{3}}\partial_{\ell}\partial_{j}P_{ik}\partial_{\ell}\partial_{k}P_{ij}\,\mathrm{d}{x}$
$\displaystyle\leq
2|L_{2}+L_{3}|\int_{\mathbb{T}^{3}}\sqrt{\displaystyle\sum_{i,j,k,\ell}(\partial_{\ell}\partial_{j}P_{ik})^{2}}\sqrt{\displaystyle\sum_{i,j,k,\ell}(\partial_{\ell}\partial_{k}P_{ij})^{2}}\,\mathrm{d}{x}$
$\displaystyle=2|L_{2}+L_{3}|\int_{\mathbb{T}^{3}}|\nabla^{2}P|^{2}\,\mathrm{d}{x}=2(|L_{2}+L_{3}|)\|\Delta
P\|^{2}_{L^{2}(\mathbb{T}^{3})}.$

∎

Next, we recall the notion of angles between two elements in a Hilbert space
$(H,\langle,\rangle)$. Given two nonzero elements $u,v\in H$, the angle
between $u$ and $v$ is the unique value

$\angle(u,v):=\arccos\frac{\langle
u,v\rangle}{\|u\|\|v\|}\in[0,\pi],\qquad\text{where}\quad\|u\|_{H}=\sqrt{\langle
u,u\rangle}.$

Then the following triangle inequality is valid.

######  Lemma 4.4.

For any three unit vectors $\vec{u},\vec{v},\vec{w}\in\mathbb{S}^{2}$, it
holds

$\angle(\vec{u},\vec{v})\leq\angle(\vec{u},\vec{w})+\angle(\vec{v},\vec{w}),$
(4.25)

where $\angle(\vec{u},\vec{v})\in[0,\pi]$ stands for the angle between the two
vectors $\vec{u}$ and $\vec{v}$.

###### Proof.

By rotation invariance, we assume the unit vectors $\vec{u}$, $\vec{v}$ lie on
the $xy$-plane. And in particular, w.l.o.g. we suppose $\vec{u}$ points in the
direction of $x$-axis:

$\vec{u}=\langle 1,0,0\rangle,\quad\vec{v}=\langle
a,\sqrt{1-a^{2}},0\rangle,\quad\vec{w}=\langle
c_{1},c_{2},c_{3}\rangle\in\mathbb{S}^{2},\quad-1\leq a\leq 1.$

Correspondingly, we have

$\displaystyle\cos\angle(\vec{u},\vec{v})$
$\displaystyle=a,\quad\cos\angle(\vec{u},\vec{w})=c_{1},\quad\cos\angle(\vec{v},\vec{w})=ac_{1}+\sqrt{1-a^{2}}c_{2},$
$\displaystyle\sin\angle(\vec{u},\vec{w})$
$\displaystyle=\sqrt{1-c_{1}^{2}},\quad\sin\angle(\vec{v},\vec{w})=\sqrt{1-2a\sqrt{1-a^{2}}c_{1}c_{2}-a^{2}c_{1}^{2}-(1-a^{2})c_{2}^{2}}.$

To prove (4.25), it is equivalent to show

$\cos\angle(\vec{u},\vec{v})\geq\cos\big{(}\angle(\vec{u},\vec{w})+\angle(\vec{v},\vec{w})\big{)}.$

That is,

$a\geq
c_{1}(ac_{1}+\sqrt{1-a^{2}}c_{2})-\sqrt{1-c_{1}^{2}}\sqrt{1-2a\sqrt{1-a^{2}}c_{1}c_{2}-a^{2}c_{1}^{2}-(1-a^{2})c_{2}^{2}},$

which is equivalent to

$\sqrt{1-c_{1}^{2}}\sqrt{1-2a\sqrt{1-a^{2}}c_{1}c_{2}-a^{2}c_{1}^{2}-(1-a^{2})c_{2}^{2}}\geq
a(c_{1}^{2}-1)+\sqrt{1-a^{2}}c_{1}c_{2}.$ (4.26)

If the R.H.S. of (4.26) is non-positive, the proof is complete. Otherwise, it
is equivalent to prove

$\displaystyle\qquad(1-c_{1}^{2})\big{[}1-2a\sqrt{1-a^{2}}c_{1}c_{2}-a^{2}c_{1}^{2}-(1-a^{2})c_{2}^{2}\big{]}$
$\displaystyle\qquad\qquad\geq
a^{2}(1-c_{1}^{2})^{2}+(1-a^{2})c_{1}^{2}c_{2}^{2}-2a\sqrt{1-a^{2}}(1-c_{1}^{2})c_{1}c_{2}$
$\displaystyle\Leftrightarrow\quad(1-c_{1}^{2})(1-a^{2}c_{1}^{2})-(1-a^{2})c_{2}^{2}\geq
a^{2}(1-c_{1}^{2})^{2}$
$\displaystyle\Leftrightarrow\quad(1-c_{1}^{2}-c_{2}^{2})(1-a^{2})\geq 0,$

which is obviously true. ∎

Based on Lemma 4.2 and Lemma 4.3, we show that the $H^{2}$-norm of $Q_{n}$ can
be estimated in terms of $\|\partial\mathcal{E}_{n}(Q_{n}(t))\|$ uniformly in
$n\in\mathbb{N}$.

######  Proposition 4.2.

For any $n\in\mathbb{N}$, a.e. $t\in(t_{0},T)$, it holds

$\displaystyle\big{\|}\Delta
Q_{n}(t,\cdot)\big{\|}_{L^{2}(\mathbb{T}^{3})}\leq
C_{L}\big{(}\big{\|}\partial\mathcal{E}_{n}(Q_{n}(t,\cdot))\big{\|}_{L^{2}(\mathbb{T}^{3})}+2\alpha\big{\|}Q_{n}(t,\cdot)\big{\|}_{L^{2}(\mathbb{T}^{3})}\big{)},$

where $C_{L}$ is defined in (1.15).

###### Proof.

By Lemma 4.2, we have that (up to a subsequence)
$\|\partial{\mathcal{E}}_{n}(Q_{n}(t,\cdot))\|$ is convergent to
$\|\partial\mathcal{E}(Q(t,\cdot))\|$ for almost every fixed $t\in[t_{0},T]$.
Hence for any fixed $t\in[t_{0},T]$ (after removing a set of measure zero),
for any $n\in\mathbb{N}$ it holds Recall (4.6), we have for every $n\geq 1$
and almost every $t\in(t_{0},T)$ that

$\displaystyle\Big{\|}\partial\mathcal{G}(Q_{n}(t,\cdot))+\psi_{n}^{\prime}(Q_{n}(t,\cdot))-\frac{1}{3}\operatorname{tr}(\psi_{n}^{\prime})(Q_{n}(t,\cdot))\mathbb{I}_{3}\Big{\|}_{L^{2}(\mathbb{T}^{3})}$
$\displaystyle\leq\big{\|}\partial\mathcal{E}_{n}(Q_{n}(t,\cdot))\big{\|}_{L^{2}(\mathbb{T}^{3})}+2\alpha\big{\|}Q_{n}(t,\cdot)\big{\|}_{L^{2}(\mathbb{T}^{3})}.$
(4.27)

On the other hand, by $(\textbf{M6})$, and $\\{Q_{n}\\}\in
L^{\infty}(0,T;H^{1}(\mathbb{T}^{3}))$, we know that

$\Big{\|}\psi_{n}^{\prime}(Q_{n}(t,\cdot))-\frac{1}{3}\operatorname{tr}(\psi_{n}^{\prime})(Q_{n}(t,\cdot))\mathbb{I}_{3}\Big{\|}_{L^{2}(\mathbb{T}^{3})}\leq\lambda_{n}\|Q_{n}(t,\cdot)\|_{L^{2}(\mathbb{T}^{3})}+\Lambda_{n}|\mathbb{T}^{3}|<+\infty$

which together with (4) implies that

$\big{\|}\partial\mathcal{G}(Q_{n}(t,\cdot))\big{\|}_{L^{2}(\mathbb{T}^{3})}<\infty.$

By lemma 4.3, we conclude that $Q_{n}(t,\cdot)\in H^{2}(\mathbb{T}^{3})$, a.e.
$t\geq 0$ for any $n\in\mathbb{N}$.

In the rest of this lemma, for simplicity $Q_{n}(\cdot,t)$ is abbreviated by
$Q_{n}$, and w.l.o.g. we assume $\|\Delta{Q}_{n}\|_{L^{2}(\mathbb{T}^{3})}>0$.
By (1.4) and Lemma 4.3, we have

$\displaystyle\frac{\int_{\mathbb{T}^{3}}\left(\partial\mathcal{G}(Q_{n}):-\Delta{Q}_{n}\right)\,dx}{\|\partial\mathcal{G}(Q_{n})\|_{L^{2}}\|\Delta{Q}_{n}\|_{L^{2}}}\geq\frac{2(L_{1}-|L_{2}+L_{3}|)\|\Delta{Q}_{n}\|_{L^{2}}^{2}}{2(L_{1}+|L_{2}+L_{3}|)\|\Delta{Q}_{n}\|_{L^{2}}^{2}}\geq\frac{L_{1}-|L_{2}+L_{3}|}{L_{1}+|L_{2}+L_{3}|}>0.$

Hence the angle between $\partial\mathcal{G}(Q_{n})$ and $-\Delta{Q}_{n}$ is
bounded by

$\angle\big{(}\partial\mathcal{G}(Q_{n}),-\Delta{Q}_{n}\big{)}\leq\arccos\frac{L_{1}-|L_{2}+L_{3}|}{L_{1}+|L_{2}+L_{3}|}.$

Meanwhile, by $(\textbf{M1})$ we know that

$\int_{\mathbb{T}^{3}}\left(\partial\psi_{n}(Q_{n}):-\Delta{Q}_{n}\right)\,dx\geq
0,$

which together with Lemma 4.4 gives

$\displaystyle\angle\big{(}\partial\mathcal{G}(Q_{n}),\partial\psi_{n}(Q_{n})\big{)}$
$\displaystyle\leq\angle\big{(}\partial\mathcal{G}(Q_{n}),-\Delta{Q}_{n}\big{)}+\angle\big{(}\partial\psi_{n}(Q_{n}),-\Delta{Q}_{n}\big{)}$
$\displaystyle\leq\frac{\pi}{2}+\arccos\frac{L_{1}-|L_{2}+L_{3}|}{L_{1}+|L_{2}+L_{3}|}.$

As a consequence, we obtain that

$\displaystyle\int_{\mathbb{T}^{3}}\left(\partial\mathcal{G}(Q_{n}):\partial\psi_{n}(Q_{n})\right)\,dx$
(4.28)
$\displaystyle\geq\cos\Big{(}\frac{\pi}{2}+\arccos\frac{L_{1}-|L_{2}+L_{3}|}{L_{1}+|L_{2}+L_{3}|}\Big{)}\|\partial\mathcal{G}(Q_{n})\|_{L^{2}(\mathbb{T}^{3})}\|\partial\psi_{n}(Q_{n})\|_{L^{2}(\mathbb{T}^{3})}$
$\displaystyle=-\frac{2\sqrt{L_{1}|L_{2}+L_{3}|}}{L_{1}+|L_{2}+L_{3}|}\|\partial\mathcal{G}(Q_{n})\|_{L^{2}(\mathbb{T}^{3})}\|\partial\psi_{n}(Q_{n})\|_{L^{2}(\mathbb{T}^{3})}.$
(4.29)

Note that $2\sqrt{L_{1}|L_{2}+L_{3}|}/(L_{1}+|L_{2}+L_{3}|)<1$ by (1.4).

In all, after combining Lemma 4.3, (4), (4.28) and the Cauchy-Schwarz
inequality we conclude that

$\displaystyle\big{(}\|\partial\mathcal{E}_{n}(Q_{n})\|_{L^{2}(\mathbb{T}^{3})}+2\alpha\|Q_{n}\|_{L^{2}(\mathbb{T}^{3})}\big{)}^{2}$
$\displaystyle\geq\|\partial\mathcal{G}(Q_{n})\|_{L^{2}(\mathbb{T}^{3})}^{2}+\|\partial\psi_{n}(Q_{n})\|_{L^{2}(\mathbb{T}^{3})}^{2}-\frac{4\sqrt{L_{1}|L_{2}+L_{3}|}}{L_{1}+|L_{2}+L_{3}|}\|\partial\mathcal{G}(Q_{n})\|_{L^{2}(\mathbb{T}^{3})}\|\partial\psi_{n}(Q_{n})\|_{L^{2}(\mathbb{T}^{3})}$
$\displaystyle\geq\frac{L_{1}+|L_{2}+L_{3}|-2\sqrt{L_{1}|L_{2}+L_{3}|}}{L_{1}+|L_{2}+L_{3}|}\|\partial\mathcal{G}(Q_{n})\|_{L^{2}(\mathbb{T}^{3})}^{2}$
$\displaystyle\geq\frac{L_{1}+|L_{2}+L_{3}|-2\sqrt{L_{1}|L_{2}+L_{3}|}}{L_{1}+|L_{2}+L_{3}|}4(L_{1}-|L_{2}+L_{3}|)^{2}\|\Delta{Q}_{n}\|_{L^{2}(\mathbb{T}^{3})}^{2}$
(4.30)

The proof is complete. ∎

Summing up the results established in this subsection, we are ready to
establish the improved regularity properties of the unique solution $Q$
obtained in Theorem 1.1.

###### Proof of Theorem 1.2.

Part 1: uniform-in-time bound (1.14) It is proved in (4.23) that (up to a
subsequence) $Q_{n}(t)\rightarrow Q(t,\cdot)$ in
$C\big{(}[t_{0},T];L^{2}(\mathbb{T}^{3})\big{)}$. Moreover, it follows from
Proposition 4.2 that for a.e. $t\in(t_{0},T)$, $\|\Delta
Q_{n}(t)\|_{L^{2}(\mathbb{T}^{3})}$ is (up to a subsequence) uniformly bounded
in $n\in\mathbb{N}$. Hence $\Delta Q(t,\cdot)\in L^{2}(\mathbb{T}^{3})$ a.e.,
and from Lemma 4.2 we can further make the following estimates

$\displaystyle\|\Delta Q(t,\cdot)\|_{L^{2}(\mathbb{T}^{3})}$
$\displaystyle\leq\liminf_{n\rightarrow\infty}\|\Delta
Q_{n}(t,\cdot)\|_{L^{2}(\mathbb{T}^{3})}$
$\displaystyle\leq\liminf_{n\rightarrow\infty}C_{L}\big{(}\|\partial\mathcal{E}_{n}(Q(t,\cdot))\|_{L^{2}(\mathbb{T}^{3})}+2\alpha\|Q_{n}(t,\cdot)\|_{L^{2}(\mathbb{T}^{3})}\big{)}$
$\displaystyle=C_{L}\big{(}\|\partial\mathcal{E}(Q(t,\cdot))\|_{L^{2}(\mathbb{T}^{3})}+2\alpha\|Q(t,\cdot)\|_{L^{2}(\mathbb{T}^{3})}\big{)},$
(4.31)

where $C_{L}$ is defined in (1.15).

W.l.o.g. we assume $T=t_{0}+100$. In view of Lemma (4.31), it suffices to
provide the $L^{\infty}$-bound for
$\|\partial\mathcal{E}(Q(t,\cdot))\|_{L^{2}(\mathbb{T}^{3})}^{2}$. By equation
(1.12) we have

$\int_{t_{0}}^{+\infty}\|\partial\mathcal{E}(Q(t,\cdot))\|_{L^{2}(\mathbb{T}^{3})}^{2}\,\mathrm{d}{t}\leq\mathcal{E}(Q(t_{0}))-\inf\mathcal{E}.$
(4.32)

Thus for any $n\in\mathbb{N}$ it holds

$\int_{t_{0}+n}^{t_{0}+n+1}\|\partial\mathcal{E}(Q(t,\cdot))\|_{L^{2}(\mathbb{T}^{3})}^{2}\,\mathrm{d}{t}\leq\mathcal{E}(Q(t_{0}))-\inf\mathcal{E},$
(4.33)

hence there exists a set of positive measure $A_{n}\subset[t_{0}+n,t_{0}+n+1]$
such that

$\|\partial\mathcal{E}(Q(t,\cdot))\|_{L^{2}(\mathbb{T}^{3})}^{2}\leq\mathcal{E}(Q(t_{0}))-\inf\mathcal{E}+1,\quad\forall\,t\in
A_{n}.$ (4.34)

For any $n\in\mathbb{N}$, $n\leq 99$, let us choose a time $s_{n}\in A_{n}$,
and for the sake of convenience we set $s_{0}=t_{0}$. Obviously
$\\{s_{n}\\}_{n\leq 99}$ is monotone increasing, and $s_{n+1}-s_{n}\leq 2$.

Let us then consider the gradient flow sequence

$\begin{cases}\partial_{t}P_{n}=-\partial\mathcal{E}(P_{n}),\\\
P_{n}(0,\cdot)=Q(s_{n},\cdot),\end{cases}$ (4.35)

subject to periodic boundary condition, whose solution $P_{n}$ is given by the
time shift $P_{n}(t,\cdot):=Q(s_{n}+t,\cdot)$, $\forall\,t\geq 0$. By
Proposition 3.2, the function $t\mapsto
e^{-2\alpha{t}}\|\partial\mathcal{E}(P_{n}(t))\|_{L^{2}(\mathbb{T}^{3})}$ is
nonincreasing, thus together with (4.34) we have

$\displaystyle\|\partial\mathcal{E}(Q(t+s_{n},\cdot))\|_{L^{2}(\mathbb{T}^{3})}^{2}$
$\displaystyle=\|\partial\mathcal{E}(P_{n}(t,\cdot))\|_{L^{2}(\mathbb{T}^{3})}^{2}\leq
e^{4\alpha(s_{n+1}-s_{n})}\|\partial\mathcal{E}(P_{n}(0,\cdot))\|_{L^{2}(\mathbb{T}^{3})}^{2}$
$\displaystyle\leq
e^{8\alpha}\|\partial\mathcal{E}(Q(s_{n},\cdot))\|_{L^{2}(\mathbb{T}^{3})}^{2}\leq
e^{8\alpha}(\mathcal{E}(Q(t_{0}))-\inf\mathcal{E}+1),$

for a.e. $t\in(0,s_{n+1}-s_{n})$. Repeating this argument for all $n$ finally
gives

$\|\partial\mathcal{E}(Q(t,\cdot))\|_{L^{2}(\mathbb{T}^{3})}\leq
e^{4\alpha}\sqrt{\mathcal{E}(Q(t_{0}))-\inf\mathcal{E}+1},\qquad\mbox{for a.e.
}t\in(t_{0},t_{0}+99).$

In addition, we recall that all eigenvalues of $Q(t,\cdot)$ are in
$(-1/3,2/3)$ for a.e. $t\in(t_{0},t_{0}+99)$, hence
$\|Q(t,\cdot)\|_{H^{2}(\mathbb{T}^{3})}$ is uniform-in-time bounded in
$(t_{0},t_{0}+99)$. Thus (1.14) could be proved by iteration.

Part 2: strict physicality (1.17). Now that $Q$ is a solution of

$\partial_{t}Q=-\partial\mathcal{G}(Q(t,\cdot))-\psi^{\prime}(Q(t,\cdot))+\frac{\mathrm{tr}(\psi^{\prime}(Q(t,\cdot)))}{3}\mathbb{I}_{3}+2\alpha{Q}(t,\cdot)\qquad\forall\,t>t_{0},$

with $Q\in L^{\infty}(t_{0},+\infty;H^{2})$, let us take the inner product
with $-\Delta Q(t,\cdot)$. Then it gives

$\displaystyle\dfrac{1}{2}\dfrac{d}{d{t}}\|\nabla{Q}(t,\cdot)\|^{2}_{L^{2}(\mathbb{T}^{3})}$
$\displaystyle=\langle\partial\mathcal{G}(Q(t,\cdot)),\Delta{Q}(t,\cdot)\rangle_{L^{2}(\mathbb{T}^{3})}+\langle\psi^{\prime}(Q(t,\cdot)),\Delta{Q}(t,\cdot)\rangle_{L^{2}(\mathbb{T}^{3})}$
$\displaystyle\qquad+2\alpha\|\nabla{Q}(t,\cdot)\|^{2}_{L^{2}(\mathbb{T}^{3})}.$
(4.36)

Note that by (1.16), Lemma 4.3, and the Poincaré inequality, we have

$\displaystyle\langle\partial\mathcal{G}(Q(t,\cdot)),\Delta{Q}(t)\rangle_{L^{2}(\mathbb{T}^{3})}$
$\displaystyle\leq-2(L_{1}-|L_{2}+L_{3}|)\|\Delta{Q}(t,\cdot)\|_{L^{2}(\mathbb{T}^{3})}^{2}$
$\displaystyle\leq-\dfrac{2(L_{1}-|L_{2}+L_{3}|)}{C_{\mathbb{T}^{3}}^{2}}\|\nabla{Q}(t,\cdot)\|_{L^{2}(\mathbb{T}^{3})}^{2}$
(4.37)

On the other hand,

$\displaystyle\langle\psi^{\prime}(Q(t,\cdot)),\Delta{Q}(t,\cdot)\rangle_{L^{2}(\mathbb{T}^{3})}$
$\displaystyle=\int_{\mathbb{T}^{3}}\psi^{\prime}(Q(t,x))\Delta{Q}(t,x)\,\mathrm{d}{x}$
$\displaystyle=-\int_{\mathbb{T}^{3}}\psi^{\prime\prime}(Q(t,x))|\nabla{Q}(t,x)|^{2}\,\mathrm{d}{x}\leq
0,$ (4.38)

due to the convexity of $\psi$. Inserting (4) and (4) into (4), we get

$\dfrac{d}{dt}\|\nabla{Q}(t,\cdot)\|^{2}_{L^{2}(\mathbb{T}^{3})}\leq
4\Big{(}-\frac{L_{1}-|L_{2}+L_{3}|}{C_{\mathbb{T}^{3}}^{2}}+\alpha\Big{)}\|\nabla{Q}(t,\cdot)\|_{L^{2}(\mathbb{T}^{3})}^{2},\quad\forall\,t\geq
t_{0}.$

Since $Q(t_{0},\cdot)\in H^{1}(\mathbb{T}^{3})$, it follows from Gronwall’s
inequality that

$\|\nabla{Q}(t,\cdot)\|^{2}_{L^{2}(\mathbb{T}^{3})}\leq\exp\bigg{(}4\Big{[}-\frac{L_{1}-|L_{2}+L_{3}|}{C_{\mathbb{T}^{3}}^{2}}+\alpha\Big{]}(t-t_{0})\bigg{)}\|\nabla{Q}(t_{0},\cdot)\|^{2}_{L^{2}(\mathbb{T}^{3})}\qquad\forall\,t\geq
t_{0}.$ (4.39)

Denote by
$\bar{Q}(t):=|\mathbb{T}^{3}|^{-1}\int_{\mathbb{T}^{3}}Q(t,\cdot)\,\mathrm{d}{x}$
the mean value of $Q(t,\cdot)$ over $\mathbb{T}^{3}$. Due to the convexity of
$\psi$ one can apply Jensen’s inequality to derive

$\displaystyle\psi(\bar{Q}(t))$
$\displaystyle=\psi(\bar{Q}(t))|\mathbb{T}^{3}|\leq\int_{\mathbb{T}^{3}}\psi(Q(t,x))\,\mathrm{d}{x}=\mathcal{E}(Q(t))-\mathcal{G}(Q(t))+\alpha\|Q(t,\cdot)\|_{L^{2}(\mathbb{T}^{3})}^{2}$
$\displaystyle\leq\mathcal{E}(Q(t_{0}))+\alpha\sup_{s\geq
t_{0}}\|Q(s,\cdot)\|_{L^{\infty}(\mathbb{T}^{3})}^{2}|\mathbb{T}^{3}|,$

since $\mathcal{G}\geq 0$ by Lemma 3.1. It is noted that the last bound above
is independent of $t\geq t_{0}$. Thus

$\rho_{0}:=\inf_{t\geq t_{0}}\rho(\\-Q(t,\cdot))>0,$ (4.40)

where $\rho$ is define in (3.11).

Finally, by (4.31) and the Gagliardo-Nirenberg inequality, we obtain

$\displaystyle\|Q(t,\cdot)$
$\displaystyle-\bar{Q}(t)\|_{L^{\infty}(\mathbb{T}^{3})}$ $\displaystyle\leq
C^{\prime}\|\nabla{Q}(t,\cdot)\|_{L^{2}(\mathbb{T}^{3})}^{1/2}\|\Delta{Q}(t,\cdot)\|_{L^{2}(\mathbb{T}^{3})}^{1/2}$
$\displaystyle\leq
C^{\prime}\exp\bigg{(}\Big{[}-\frac{L_{1}-|L_{2}+L_{3}|}{C_{\mathbb{T}^{3}}^{2}}+\alpha\Big{]}(t-t_{0})\bigg{)}\big{\|}\nabla{Q}(t_{0},\cdot)\big{\|}_{L^{2}(\mathbb{T}^{3})}^{1/2}\cdot
C(t_{0})^{1/2}$ (4.41)

for some geometric constant $C^{\prime}$, and a.e. $t\geq t_{0}$. This
together with (1.16) implies
$\|Q(t,\cdot)-\bar{Q}(t)\|_{L^{\infty}(\mathbb{T}^{3})}$ decays uniformly to
zero as $t\rightarrow\infty$. Since the convergence in the $L^{\infty}$ norm
implies the uniform convergence of all eigenvalues, due to (4.40) we conclude
that there exists some $T_{0}>0$, such that

$\rho(Q(t,\cdot))\geq\dfrac{\rho_{0}}{2},\quad\forall\,t\geq T_{0}.$

Therefore, the proof of Theorem 1.2 is finished. ∎

##  5 Proof of Theorem 1.3: Size of the contact set

In this section we shall estimate the Hausdorff dimension of the singular set
$\Sigma_{t}$ where the unique global solution $Q(t,x)$ to (1.6) touches the
physical boundary

$\Sigma_{t}:=\\{x\in\mathbb{T}^{n}\mid
Q(t,x)\in{\partial}\mathcal{Q}_{phy}\\}.$ (5.1)

Here ${\partial}\mathcal{Q}_{phy}$ is the boundary of $\mathcal{Q}_{phy}$
where the smallest eigenvalue of any element equals $-1/3$.

To begin with, we state [22, Theorem 1.2] which provides a lower bound of the
blowup rate of ${\partial}\psi(P)$ as $P\in\mathcal{Q}_{phy}$ approaches its
physical boundary.

######  Proposition 5.1.

For any $P\in\mathcal{Q}_{phy}$, as $\lambda_{1}(P)\rightarrow-1/3$ it holds

$|\partial\psi(P)|\geq\frac{C_{1}}{\lambda_{1}(P)+\frac{1}{3}}$ (5.2)

with the constant $C_{1}$ given by

$\displaystyle C_{1}=\frac{\sqrt{3}}{9\sqrt{2\pi}e}\cdot\inf_{\xi\geq
0}\frac{e^{-\xi}\mathrm{I}_{0}(\xi)}{e^{\frac{-\xi}{2}}\mathrm{I}_{0}(\frac{\xi}{2})}>0,$
(5.3)

where $\mathrm{I}_{0}(\cdot)$ is the zeroth order modified Bessel function of
first kind.

######  Remark 5.1.

As was pointed out in [17, Appendix C] that
$d(P,\partial\mathcal{Q}_{phy})=\frac{\sqrt{6}}{2}\big{[}\lambda_{1}(P)+\frac{1}{3}\big{]}$
for any $P\in\mathcal{Q}_{phy}$. Together with Proposition 5.1, it is
immediate to see that there exists a generic and suitably small constant
$\delta_{0}>0$, such that

$|\partial\psi(P)|\geq\frac{\sqrt{6}C_{1}}{2d(P,\partial\mathcal{Q}_{phy})},\qquad\mbox{whenever
}\;d(P,\partial\mathcal{Q}_{phy})<\delta_{0}.$ (5.4)

The following technical lemma is necessary.

######  Lemma 5.1.

For any $s>0$ there exists a sequence of coverings $V_{m}=\\{B_{i,m}\\}$ of
$\Sigma_{t}$, where $B_{i,m}=B(x_{i,m},r_{i,m})$, such that

$\displaystyle\lim_{m\rightarrow\infty}\sum_{i}r_{i,m}^{s}=\mathcal{H}^{s}(\Sigma_{t})\leq\liminf_{m\rightarrow\infty}\sum_{j}\big{|}5r_{j,m}^{\ast}\big{|}^{s}$

Here $B_{i,m}=B(x_{i,m},r_{i,m})$ such that
$B_{i,m}\subset\\{x\in\mathbb{T}^{n}|\,d(x,\Sigma_{t})\leq\delta\\}$ for some
$\delta>0$, and $r_{j,m}^{\ast}$ are the radii of the balls $B_{j,m}^{\ast}$,
i.e., the sub-covering given by the Vitali covering lemma.

###### Proof.

First, by recalling the definition of Hausdorff content

$\mathcal{H}_{\delta}^{s}(\Sigma_{t})\,:=\inf\Big{\\{}\sum_{i}r_{i}^{s}:\,\Sigma_{t}\subset\bigcup_{i}B_{i},\,\sup_{i}r_{i}<\delta\Big{\\}},$

and the Hausdorff measure

$\mathcal{H}^{s}(\Sigma_{t})=\lim_{\delta\rightarrow
0}\mathcal{H}_{\delta}^{s}(\Sigma_{t}),$

we know that for any $\delta>0$ we have a sequence of coverings
$V_{m}=\\{B_{i,m,\delta}\\}$ such that

$\lim_{m\rightarrow\infty}\sum_{i}r_{i,m,\delta}^{s}=\mathcal{H}_{\delta}^{s}(\Sigma_{t})$

Therefore, using the standard diagonal argument we can choose a sufficiently
large $m=m(\delta)$ such that

$0\leq\sum_{i}r_{i,m,\delta}^{s}-\mathcal{H}_{\delta}^{s}(\Sigma_{t}):=\varepsilon_{\delta}\ll
1,\qquad\mbox{where }\;\varepsilon_{\delta}\rightarrow 0\Leftrightarrow
m\rightarrow+\infty,$

which further gives

$\lim_{m\rightarrow\infty}\sum_{i}r_{i,m}^{s}=\mathcal{H}^{s}(\Sigma_{t}),\qquad\mbox{where
}\;r_{i,m}:=r_{i,m,\delta(m)}.$

W.l.o.g., we may choose $m$ to be bijective in $\delta$. Thus
$V_{m}=\\{B_{i,m}\\}$ is an admissible sequence to achieve the Hausdorff
measure.

Note that $\\{5B_{j,m}^{\ast}\\}$ is another covering with balls for
$\Sigma_{t}$, hence for each $m$, we have

$\sum_{j}|5r_{j,m}^{\ast}|^{s}\geq\mathcal{H}_{\delta(m)}^{s}(\Sigma_{t})=\sum_{i}r_{i,m,\delta(m)}^{s}-\varepsilon_{\delta},$

which implies

$\liminf_{m\rightarrow\infty}\sum_{j}|5r_{j,m}^{\ast}|^{s}\geq\liminf_{m\rightarrow\infty}\mathcal{H}_{\delta(m)}^{s}(\Sigma_{t})=\mathcal{H}^{s}(\Sigma_{t}).$

∎

Using Proposition 5.1 and lemma 5.1, now we are ready to finish the proof of
Theorem 1.3.

###### Proof of Theorem 1.3.

First of all, for any $t_{0}>0$, by Theorems 1.1 and 1.2, we know that the
unique strong solution $Q(t,x)$ to equation (1.1) satisfies

$Q\in L^{\infty}(t_{0},+\infty;H^{2}(\mathbb{T}^{n})),\qquad\partial_{t}Q\in
L^{\infty}(t_{0},T_{0};L^{2}(\mathbb{T}^{n})),$ (5.5)

which together with equation (1.1) gives

$\partial\psi(Q)\in L^{\infty}(t_{0},T_{0};L^{2}(\mathbb{T}^{n})).$

Here and after, we are only concerned with $t\in(0,T_{0})$ that satisfies

$\big{\|}\partial\psi(Q(t,\cdot))\big{\|}_{L^{2}(\mathbb{T}^{n})}<+\infty,$

which obviously has a full measure in $(0,T_{0})$.

Case 1: $n=3$

By Sobolev embedding $H^{2}(\mathbb{T}^{3})\hookrightarrow
C^{0,\frac{1}{2}}(\mathbb{T}^{3})$ and $Q\in L^{\infty}(t_{0},+\infty;H^{2})$,
there exists a generic constant $C_{H}>0$ that is independent of $t$, such
that

$|Q(t,x)-Q(t,y)|\leq C_{H}|x-y|^{\frac{1}{2}},\qquad\forall
x,y\in\mathbb{T}^{3}.$ (5.6)

In particular, for any given $x$, let $x^{\perp}\in\Sigma_{t}$ be a projection
such that

$|x-x^{\perp}|=d(x,\Sigma_{t}):=\mbox{dist}(x,\Sigma_{t}),$ (5.7)

and henceforth we get

$d(Q(t,x),\partial\mathcal{Q}_{phy})\leq|Q(t,x)-Q(t,x^{\perp})|\leq
C_{H}|x-x^{\perp}|^{\frac{1}{2}}=C_{H}\sqrt{\mbox{dist}(x,\Sigma_{t})}.$

This combined with Proposition 5.1 implies for any ball $B$ and $\delta\ll 1$
that

$\displaystyle\int_{\\{x\in
B|\,d(x,\Sigma_{t})\leq\delta\\}}\|\partial\psi(Q(t,x))\|_{F}^{2}\,dx$
$\displaystyle\geq C_{1}^{2}\int_{\\{x\in
B|\,d(x,\Sigma_{t})\leq\delta\\}}d(Q(t,x),\partial\mathcal{Q}_{phy})^{-2}dx$
$\displaystyle\geq\frac{C_{1}^{2}}{C_{H}^{2}}\int_{\\{x\in
B|\,d(x,\Sigma_{t})\leq\delta\\}}\frac{1}{\mbox{dist}(x,\Sigma_{t})}\,dx.$
(5.8)

To proceed, by applying Lemma 5.1 with $s=2$, we obtain the existence of a
sequence of covering $\\{B_{i,m}\\}$ of $\Sigma_{t}$ with balls
$B_{i,m}=B(x_{i,m},r_{i,m})$ such that
$B_{i,m}\subset\\{x\in\mathbb{T}^{3}|\,d(x,\Sigma_{T})\leq\delta\\}$ and

$\lim_{m\rightarrow\infty}\sum_{i}r_{i,m}^{s}=\mathcal{H}^{s}(\Sigma_{t}).$

We can assume that each such ball $B_{i,m}$ intersects $\Sigma_{t}$, for
otherwise we can just remove it from the covering. By Vitali’s covering lemma,
for each $m$ we can choose a sub-collection of mutually disjoint balls
$\\{B_{j,m}^{\ast}\\}$, with $B_{j,m}^{\ast}=B(x_{j,m}^{\ast},r_{j,m}^{\ast})$
such that

$\Sigma_{t}\subset\bigcup_{i}B_{i,m}\subset\bigcup_{j}5B_{j,m}^{\ast}.$

On each such ball $B_{j,m}^{\ast}$, it follows from (5) that

$\int_{B_{j,m}^{\ast}}|\partial\psi(Q(t,x))|^{2}\,dx\geq\frac{C_{1}^{2}}{C_{H}^{2}}\int_{B_{j,m}^{\ast}}\frac{1}{\mbox{dist}(x,\Sigma_{t})}\,dx.$
(5.9)

As each of such ball $B_{j,m}$ intersects $\Sigma_{t}$, we can choose an
arbitrary intersection point $y_{j,m}\in B_{j,m}^{\ast}\cap\Sigma_{t}$, and

$\int_{B_{j,m}^{\ast}}\frac{1}{\mbox{dist}(x,\Sigma_{t})}\,dx\geq\int_{B_{j,m}^{\ast}}\frac{1}{|x-y_{j,m}|}\,dx.$
(5.10)

We claim that the last integral in (5.10) is minimized when
$y_{j,m}\in\partial B_{j,m}^{\ast}$. To prove this claim, w.l.o.g. we assume
$B_{j,m}^{\ast}$ is the unit ball $B$ centered at the origin, and we denote
$y_{j,m}=y=(y_{1},0,0)$, $0\leq y_{1}\leq 1$. Hence we consider

$f(y):=\int_{B}\frac{1}{|x-y|}\,dx.$

Then

$\frac{\partial{f}}{\partial{y_{1}}}=\int_{B}\frac{x_{1}-y_{1}}{|x-y|^{3}}\,dx,\qquad\mbox{where
}\;x=(x_{1},x_{2},x_{3}).$

If $y_{1}=0$, by symmetry we see $\frac{\partial{f}}{\partial{y_{1}}}=0$. If
$0<y_{1}\leq 1$, we denote $A=\\{x\in\mathbb{R}^{3}|\,x_{1}\geq y_{1}\\}$, and
$A^{\prime}$ the reflection of $A$ across
$\\{x\in\mathbb{R}^{3}|\,x_{1}=y_{1}\\}$. Note that $A\cup A^{\prime}$ is
symmetric with respect to both $\\{x\in\mathbb{R}^{3}|\,x_{1}=y_{1}\\}$ and
the point $y=(y_{1},0,0)$. Thus we have

$\frac{\partial{f}}{\partial{y_{1}}}=\int_{B\setminus(A\cup
A^{\prime})}\frac{x_{1}-y_{1}}{|x-y|^{3}}\,dx+\underbrace{\int_{A\cup
A^{\prime}}\frac{x_{1}-y_{1}}{|x-y|^{3}}\,dx}_{=0}<0,$

due to the fact that $B\setminus(A\cup A^{\prime}$ is entirely left to
$\\{x\in\mathbb{R}^{3}|\,x_{1}=y_{1}\\}$. Hence the claim is proved.

As a consequence,

$\int_{B_{j,m}^{\ast}}\frac{1}{|x-y_{j,m}|}\,dx\geq\frac{4\pi}{3}|r_{j,m}^{\ast}|^{2}.$
(5.11)

which together with (5.9) and (5.10) gives

$\int_{B_{j,m}^{\ast}}|\partial\psi(Q(t,x))|^{2}\,dx\geq\tilde{C}|r_{j,m}^{\ast}|^{\frac{5}{2}},\qquad\mbox{where
}\,\tilde{C}:=\frac{4\pi}{3}\frac{C_{1}^{2}}{C_{H}^{2}}.$ (5.12)

Since $\\{B_{j,m}^{\ast}\\}$ are non-overlapping, after summing up the above
inequality over $j$ we obtain

$\|\partial\psi(Q(t,x))\|_{L^{2}(\mathbb{T}^{3})}^{2}\geq\sum_{j}\int_{B_{j,m}^{\ast}}|\partial\psi(Q(t,x))|^{2}\,dx\geq\tilde{C}\sum_{j}|r_{j,m}^{\ast}|^{2},$
(5.13)

which together with Lemma 5.1 yields

$\|\partial\psi(Q(t,x))\|_{L^{2}(\mathbb{T}^{3})}^{2}\geq\tilde{C}\liminf_{m\rightarrow\infty}\sum_{j}|r_{j,m}^{\ast}|^{2}\geq\dfrac{\tilde{C}}{5^{2}}\mathcal{H}^{2}(\Sigma_{t}).$
(5.14)

Thus we conclude that $\mbox{dim}_{\mathcal{H}}(\Sigma_{t})\leq 2$.

Case 2: $n=2$

In the $2D$ case the Sobolev embedding reads
$H^{2}(\mathbb{T}^{2})\hookrightarrow C^{\beta}(\mathbb{T}^{2})$ for all
$\beta\in(0,1)$. Correspondingly we have

$\|Q(t,x)-Q(t,y)\|\leq C_{\beta}|x-y|^{\beta},\qquad\forall
x,y\in\mathbb{T}^{2},$ (5.15)

and (5) is replaced by

$\displaystyle\int_{\\{x\in
B|\,d(x,\Sigma_{t})\leq\delta\\}}\|\partial\psi(Q(t,x))\|^{2}\,dx\geq\frac{C_{1}^{2}}{C_{\beta}^{2}}\int_{\\{x\in
B|\,d(x,\Sigma_{t})\leq\delta\\}}\frac{1}{\mbox{dist}^{2\beta}(x,\Sigma_{t})}\,dx.$
(5.16)

As a consequence, applying Lemma 5.1 with $s=2-2\beta$, and using Vitali’s
covering argument exactly as in Case 1, one may replace (5.14) by

$\|\partial\psi(Q(t,x))\|_{L^{2}(\mathbb{T}^{2})}^{2}\geq\tilde{C}^{\prime}\liminf_{m\rightarrow\infty}\sum_{j}|r_{j,m}^{\ast}|^{2-2\beta}\geq\dfrac{\tilde{C}^{\prime}}{5^{2-2\beta}}\mathcal{H}^{2-2\beta}(\Sigma_{t}),\quad\mbox{where
}\;\tilde{C}^{\prime}=\frac{4\pi C_{1}^{2}}{C_{\beta}^{2}}.$ (5.17)

In conclusion, we obtain $\mbox{dim}_{\mathcal{H}}(\Sigma_{t})\leq 2-2\beta$
for any $\beta\in(0,1)$. The proof is complete by the arbitrariness of
$\beta\in(0,1)$. ∎

Acknowledgements Y. N. Liu’s work is partially supported by NSF of China under
Grant 11971314. X. Y. Lu’s work is supported by his NSERC Discovery Grant
“Regularity of minimizers and pattern formation in geometric minimization
problems”. X. Xu’s work is supported by the NSF grant DMS-2007157 and the
Simons Foundation Grant No. 635288. We want to thank Professors Hongjie Dong,
Fanghua Lin and Chun Liu for their kind discussions.

## References

  * [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, second edition, 2008.
  * [2] J. Ball, Mathematics of liquid crystals, Cambridge Centre for Analysis short course, 13–17, 2012.
  * [3] J. Ball, Analysis of liquid crystals and their defects, Lecture notes of the Scuola Estiva GNFM, Ravello 17–22 Sep. 2018\.
  * [4] J. Ball and A. Majumdar, Nematic liquid crystals: from Maier-Saupe to a continuum theory, Mol. Cryst. Liq. Cryst., 525, 1–11, 2010\.
  * [5] J. Ball and A. Majumdar, Passage from the mean-field Maier-Saupe to the continuum Landau-de Gennes theory for nematic liquid crystals, work in progress.
  * [6] V. Barbu, Nonlinear Differential Equations of Montone Types in Banach Spaces, Springer Monographs in Mathematics, Springer, New York, 2010.
  * [7] P. Bauman and D. Phillips, Regularity and the behavior of eigenvalues for minimizers of a constrained Q-tensor energy for liquid crystals, Calc. Var. Partial Differential Equations, 55, 55–81, 2016,
  * [8] S. Boyd and L. Vandenberghe, _Convex Optimization_ , Cambridge University Press, Cambridge, 2004.
  * [9] H. Brezis, _Functional analysis, Sobolev spaces and partial differential equations_ , Universitext, Springer, New York, 2011.
  * [10] P.G. de Gennes and J. Prost, _The Physics of Liquid Crystals_ , Oxford Science Publications, Oxford, 1993.
  * [11] T. A. Davis and E. C. Gartland, Finite element analysis of the Landau-de Gennes minimization problem for liquid crystals, SIAM J. Numer. Anal., 35(1), 336–362, 1998.
  * [12] H. R. Du, X. P. Hu and C. Y. Wang, Suitable weak solutions for the co-rotational Beris-Edwards system in dimension three, Arch. Rational Mech. Anal., 238, 749-803, 2020.
  * [13] L. C. Evans, _Partial differential equations_ , Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.
  * [14] L. C. Evans, O. Kneuss and H. Tran, Partial regularity for minimizers of singular energy functionals, with application to liquid crystal models, Trans. Amer. Math. Soc., 368, 3389–3413, 2016.
  * [15] E. Feireisl, E. Rocca, G. Schimperna and A. Zarnescu, Evolution of non-isothermal Landau-de Gennes nematic liquid crystals flows with singular potential, Commun. Math. Sci., 12, 317–343, 2014.
  * [16] E. Feireisl, E. Rocca, G. Schimperna and A. Zarnescu, Nonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy, Annali di Mat. Pura ed App., 194(5), 1269–1299, 2015.
  * [17] Z. Y. Geng and J. J. Tong, Regularity of minimizers of a tensor-valued variational obstacle problem in three dimensions, Calc. Var., 59(57), doi.org/10.1007/s00526-020-1717-7, 2020.
  * [18] M. Giaquinta and L. Martinazzi, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, Second edition, Edizioni della Normale, Pisa, 2012
  * [19] G. Iyer, X. Xu and A. Zarnescu, Dynamic cubic instability in a 2D Q-tensor model for liquid crystals, Math. Models Methods Appl. Sci., 25(8), 1477–1517, 2015.
  * [20] G. Kitavtsev, J. M. Robbins, V. Slastikov and A. Zarnescu, Liquid crystal defects in the Landau-de Gennes theory in two dimensions-beyond the one-constant approximation, Math. Models Methods Appl. Sci., 26(14), 2769–2808, 2016.
  * [21] Y. N. Liu and W. Wang, On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals, Disc. Cont. Dyn. Sys.-B, 23(9), 3879-3899, 2018.
  * [22] X. Y. Lu, X. Xu and W. J. Zhang, Blowup rate estimates of the Ball-Majumdar potential and its gradient in the Landau-de Gennes theory, arXiv:2009.00537v2, 2020.
  * [23] N. J. Mottram and J. P. Newton, Introduction to Q-tensor theory, arXiv preprint, arXiv:1409.3542, 2014.
  * [24] A. Majumdar and A. Zarnescu, Landau-De Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond, Arch. Rational Mech. Anal., 196, 227–280, 2010.
  * [25] C. Ortner, Two Variational Techniques for the Approximation of Curves of Maximal Slope, Technical report NA05/10, Oxford University Computing Laboratory, Oxford, UK, 2005.
  * [26] P. J. Ryan, Homogeneity and some curvature conditions for hypersurfaces, Tohoku Math. J., 21, 363-388, 1969.
  * [27] E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to ginzburg-landau, Communications on Pure and Applied mathematics, 57, 1627-1672, 2004.
  * [28] S. Serfaty, Gamma-convergence of gradient flows on hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst, 31, 1427-1451, 2011.
  * [29] M. Wilkinson, Strict physicality of global weak solutions of a Navier-Stokes Q-tensor system with singular potential, Arch. Ration. Mech. Anal., 218, 487-526, 2015.