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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 182, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/182\hfil Weak-strong uniqueness of hydrodynamic flow]
{Weak-strong uniqueness of hydrodynamic flow of
nematic liquid crystals}

\author[J. Zhao, Q. Liu\hfil EJDE-2012/182\hfilneg]
{Ji-hong Zhao, Qiao Liu}  

\address{Ji-hong Zhao \newline
College of Science, Northwest A\&F University, 
Yangling, Shaanxi 712100, China}
\email{zhaojihong2007@yahoo.com.cn}

\address{Qiao Liu \newline
 Department of Mathematics, Hunan Normal University, 
 Changsha, Hunan 410081, China}
\email{liuqao2005@163.com}

\thanks{Submitted July 12, 2012. Published October 19, 2012.}
\subjclass[2000]{35A02, 35B35, 76A15}
\keywords{Nematic liquid crystal flow; weak solutions;
stability; \hfill\break\indent weak-strong uniqueness}

\begin{abstract}
 This article concerns a simplified model for a hydrodynamic system
 of incompressible nematic liquid crystal materials. It is shown that
 the weak-strong uniqueness holds for the class of weak solutions
 provided that either
  $(\mathbf{u}, \nabla\mathbf{d})\in C([0,T),L^3(\mathbb{R}^3))$; or
  $(\mathbf{u}, \nabla\mathbf{d})\in L^q(0,T; \dot{B}^{-1+3/p+2/q}_{p,q}
 (\mathbb{R}^3))$ with
 $2\leq p<\infty$, $2<q<\infty$ and $\frac{3}{p}+\frac{2}{q}>1$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

In this article, we study uniqueness criteria for solutions of a
hydrodynamical system modeling the flow of nematic liquid crystals
in the whole space $\mathbb{R}^3$, namely the Cauchy problem
\begin{equation}\label{eq1.1}
\begin{gathered}
\partial_{t} \mathbf{u}-\nu\Delta \mathbf{u}
+\mathbf{u}\cdot\nabla\mathbf{u}+\nabla{\pi}
  =-\lambda\operatorname{div}(\nabla \mathbf{d} \odot\nabla \mathbf{d}),\\
\partial_{t}  \mathbf{d}+\mathbf{u}\cdot\nabla\mathbf{d}
=\gamma(\Delta \mathbf{d}-g(\mathbf{d})),\\
\operatorname{div} \mathbf{u}=0,\\
(\mathbf{u}, \mathbf{d})|_{t=0}=(\mathbf{u}_0, \mathbf{d}_0).
\end{gathered}
\end{equation}
This system describes the time evolution of nematic liquid crystal
materials (cf. \cite{L89}), where $\mathbf{u}\in\mathbb{R}^3$ and
$\pi\in\mathbb{R}$ denote, respectively, the velocity field and the
pressure of the fluid, and $\mathbf{d}\in\mathbb{R}^3$ denotes the
director field of the nematic liquid crystals;
$\nu,\lambda,\gamma$ are positive constants, and
$g(\mathbf{d})=\nabla G(\mathbf{d})$ with
$G(\mathbf{d})=\frac{|\mathbf{d}|^{4}}{4}-\frac{|\mathbf{d}|^2}{2}$
is a Ginzburg-Landau approximation function; the unusual term
$\nabla \mathbf{d}\odot\nabla \mathbf{d}=(\langle \partial_{x_{i}}
\mathbf{d}, \partial_{x_{j}} \mathbf{d}\rangle )_{1\leq i, j\leq 3}$ is the
stress tensor induced by the director field $\mathbf{d}$, and the
notation $\langle\cdot, \cdot\rangle$ denotes the inner product in
$\mathbb{R}^3$. Since the sizes of the viscosity constants $\nu$,
$\lambda$ and $\gamma$ do not play important roles in the proof of
our main result, we shall assume that $\nu=\lambda=\gamma=1$
throughout this paper.

As the authors pointed out in \cite{LL95}, although system
\eqref{eq1.1} is a simplified version of the liquid crystal model
proposed by Ericksen \cite{E61} and Leslie \cite{L68}, but it
still retains most of the interesting mathematical properties. We
refer the reader to see \cite{E87,HK87,L79,L89}
and the references therein for more discussions of the
physical background of this problem.  In \cite{LL95},  using the
modified Galerkin method and the compactness argument, Lin and Liu
proved global existence of weak solutions of \eqref{eq1.1} with
$g(\mathbf{d})=\nabla G(\mathbf{d})$ for some smooth and bounded
function $G: \mathbb{R}^3\to \mathbb{R}$. Moreover, when
$g(\mathbf{d})=0$, they established global existence of strong
solutions if the initial data is sufficiently small (or if the
viscosity $\nu$ is sufficiently large). The same as for the
Navier-Stokes equations (which are equations obtained by putting
$\mathbf{d}=\mathbf{0}$ in \eqref{eq1.1}), it is well known that
weak solution of \eqref{eq1.1} is unique and regular in
$\mathbb{R}^2$. However, the question of regularity and uniqueness
of weak solution is an outstanding open problem in $\mathbb{R}^3$.
Hence, it is meaningful to find sufficient conditions on a strong
solution of \eqref{eq1.1} such that all weak solutions sharing the
same initial data must coincide with the one which additionally
satisfies these sufficient conditions, and we say then weak-strong
uniqueness holds. For the three dimensional Navier-Stokes equations,
Prodi \cite{P59} and Serrin \cite{S63} proved that weak-strong
uniqueness holds in the class
$$
  \mathcal{P}=L^q(0, T; L^{p}(\mathbb{R}^3)) \quad  \text{with }
  \frac{3}{p}+\frac{2}{q}=1,\; 3< p\leq \infty.
$$
Von Wahl \cite{V85} and Giga \cite{G86} improved this result in the
class
$$
  \mathcal{P}=C([0, T], L^3(\mathbb{R}^3)).
$$
Moreover, this last result was extended in the limit case by Kozono
and Sohr \cite{KS96}, and Escauriaza, Seregin and \u{S}ver\'{a}k
\cite{ESS03}, who proved that weak strong uniqueness holds for
$$
  \mathcal{P}=L^{\infty}(0, T; L^3(\mathbb{R}^3)).
$$
For uniqueness criteria related to the Sobolev spaces, we refer the
reader to \cite{B95,R02}. Recently,  many
researches have refined the above results. Kozono and Taniuchi
\cite{KT00} proved that weak-strong uniqueness holds in the class
$$
  \mathcal{P}=L^2(0, T; BMO).
$$
Gallagher and Planchon \cite{GP02} proved that weak-strong
uniqueness holds for
$$
  \mathcal{P}=L^q(0,T; \dot{B}^{-1+3/p+2/q}_{p,q}(\mathbb{R}^3))
  \quad \text{with $2\leq p<\infty$, $2<q<\infty$ and $\frac{3}{p}+\frac{2}{q}>1$}.
$$
Lemari\'{e}-Rieusset \cite{L02} and Germain \cite{G06} proved that
weak-strong uniqueness holds for
$$
  \mathcal{P}=C([0,T], X_{1}^{(0)})\quad \text{or}\quad
 \mathcal{P}=L^{2/(1-r)}(0,T;  X_{r})\quad \text{with } r\in[-1,1).
$$
 Chen, Miao and Zhang \cite{CMZ09} improved the above
results by showing weak-strong uniqueness for
$$
  \mathcal{P}=L^q(0,T; \dot{B}^{r}_{p,\infty}(\mathbb{R}^3))
  \quad \text{with $\frac{3}{1+r}< p\leq\infty, r\in(0,1]$
  and $(p,r)\neq(\infty,1)$}.
$$
We refer the reader to see \cite{G06} and \cite{L02} for definitions
of these function spaces.

In this article, we are interested in finding  uniqueness criteria
for weak solutions of \eqref{eq1.1}.
For the two $n\times n$ matrixes
$A=(a_{ij})_{i,j=1}^{n}$ and $B=(b_{ij})_{i,j=1}^{n}$, we define
$A:B=\sum^{n}_{i,j=1}a_{ij}b_{ij}$, and denote  by $\otimes$ the
tensor product.  Let us recall the definition of weak solutions.

\begin{definition}\label{def1.1} \rm
The vector-valued function $(\mathbf{u}, \mathbf{d})$ is called a
weak solution of \eqref{eq1.1} on $\mathbb{R}^3\times(0,T)$ if it
satisfies the following conditions:
\begin{itemize}
\item [(1)] $(\mathbf{u},\nabla\mathbf{d})\in L^{\infty}(0,T;
L^2(\mathbb{R}^3))\cap
L^2(0,T;\dot{H}^1(\mathbb{R}^3)):=(\mathcal{LS})$, where
$\dot{H}^1(\mathbb{R}^3)$ is the usual homogeneous Sobolev
space; i.e., the space of functions whose gradient belongs to
$L^2(\mathbb{R}^3)$.


\item [(2)] $(\mathbf{u},\mathbf{d})$ satisfies
\eqref{eq1.1} in the sense of distributions; i.e.,
$\operatorname{div}\mathbf{u}=0$ in the distributional sense and for all
$\mathbf{v}\in C_0^{\infty}(\mathbb{R}^3\times(0,T))$ and
$\mathbf{e}\in C_0^{\infty}(\mathbb{R}^3\times(0,T))$ with
$\operatorname{div}\mathbf{v}=0$, we have
\begin{align*}
&\int^T_0\int_{\mathbb{R}^3} \mathbf{u}\cdot\partial_{t}\mathbf{v} \,dx\,dt
-\int^T_0\int_{\mathbb{R}^3}\nabla\mathbf{u}:\nabla\mathbf{v}\,dx\,dt
  +\int^T_0\int_{\mathbb{R}^3}\mathbf{u}\otimes\mathbf{u}:\nabla\mathbf{v}\,dx\,dt\\
&=-\int^T_0\int_{\mathbb{R}^3}\nabla\mathbf{d}\odot\nabla\mathbf{d}:\nabla\mathbf{v}\,dx\,dt
\end{align*}
and
\begin{align*}
&\int^T_0\int_{\mathbb{R}^3} \mathbf{d}\cdot\partial_{t}\mathbf{e} \,dx\,dt
-\int^T_0\int_{\mathbb{R}^3}\nabla \mathbf{d}:\nabla\mathbf{e}\,dx\,dt
  +\int^T_0\int_{\mathbb{R}^3}\mathbf{u}\otimes\mathbf{d}:
  \nabla  \mathbf{e}\,dx\,dt\\
&=\int^T_0\int_{\mathbb{R}^3}g(\mathbf{d})\cdot \mathbf{e}\,dx\,dt.
\end{align*}

\item [(3)] The following energy inequality holds
(see \eqref{eq3.6} in the appendix):
\begin{align*}
&\|\mathbf{u}(t)\|_{L^2}^2+\|\nabla\mathbf{d}(t)\|_{L^2}^2
  +2\int_0^t(\|\nabla\mathbf{u}(\tau)\|_{L^2}^2
  +\|\Delta\mathbf{d}(\tau)\|_{L^2}^2)d\tau\\
 & +6\int_0^t\|\mathbf{d}\cdot\nabla\mathbf{d}\|_{L^2}^2(\tau)d\tau\\
 &\leq
 \|\mathbf{u}_0\|_{L^2}^2+\|\nabla\mathbf{d}_0\|_{L^2}^2
 +\int_0^t\|\nabla\mathbf{d}(\tau)\|_{L^2}^2d\tau\quad
 \text{for all }t\geq 0.
\end{align*}
\end{itemize}
\end{definition}


Before presenting the exact statement of our result, let us first
recall the definition of the homogeneous Besov spaces. Let
$\mathcal{S}(\mathbb{R}^3)$ be the Schwartz space. We denote by
$\{\Delta_{j}, S_{j}\}_{j\in\mathbb{Z}}$ the Littlewood-Paley
decomposition. Let $\mathcal{Z}(\mathbb{R}^3)=\big\{f\in
\mathcal{S}(\mathbb{R}^3): \ \ \partial^{\alpha}\widehat{f}(0)=0,
\ \forall\alpha\in(\mathbb{N}\cup\{0\})^3\big\}$, and denote its
dual by $\mathcal{Z}'(\mathbb{R}^3)$. Recall that for
$s\in\mathbb{R}$ and $(p,q)\in[1, \infty]\times[1, \infty]$, the
homogeneous Besov space $\dot{B}^{s}_{p,q}(\mathbb{R}^3)$ is
defined by
\begin{equation*}
  \dot{B}^{s}_{p,q}(\mathbb{R}^3)=\big\{f\in \mathcal{Z}'(\mathbb{R}^3):
  \|f\|_{\dot{B}^{s}_{p,q}}<\infty\big\},
\end{equation*}
where
\begin{equation*}
  \|f\|_{\dot{B}^{s}_{p,q}}=
\begin{cases}
  \big(\sum_{j\in\mathbb{Z}}2^{jsq}\|\Delta_{j}f\|_{L^{p}}^q\big)^{1/q}\
  &\text{for } 1\leq q<\infty,\\
  \sup_{j\in\mathbb{Z}}2^{js}\|\Delta_{j}f\|_{L^{p}} &\text{for } q=\infty.
\end{cases}
\end{equation*}
It is well-known that if either $s<\frac{3}{p}$ or $s=\frac{3}{p}$
and $q=1$, then
$(\dot{B}^{s}_{p,q}(\mathbb{R}^3),\|\cdot\|_{\dot{B}^{s}_{p,q}})$
is a Banach space. For more details about the homogeneous Besov
spaces, we refer the reader to see \cite{L02}. Next we introduce
some notations. Given $0<T<\infty$ and a Banach space $X$, we denote
by $C([0,T],X)$ the Banach space of all bounded and continuous
mappings from $[0,T]$ to $X$, and for $p\geq1$, we denote by
$L^{p}(0,T; X)$ the set of Bochner measurable $X$-valued time
dependent functions $f$ such that $t\to\|f\|_{X}$ belongs to
$L^{p}(0,T)$. The product of Banach spaces
$\mathcal{X}\times\mathcal{Y}$ will be equipped with the usual norm
$\|(f,g)\|_{\mathcal{X}\times\mathcal{Y}}=\|f\|_{\mathcal{X}}+\|g\|_{\mathcal{Y}}$,
and if $\mathcal{X}=\mathcal{Y}$, we use $\|(f,g)\|_{\mathcal{X}}$
to denote $\|(f,g)\|_{\mathcal{X}\times\mathcal{X}}$.

The main result of this paper is as follows.

\begin{theorem}\label{th1.2}
Assume that $(\mathbf{u}, \mathbf{d})$ and $(\tilde{\mathbf{u}},
\tilde{\mathbf{d}})$ are two weak solutions of \eqref{eq1.1} for a
given initial data $(\mathbf{u}_0, \nabla\mathbf{d}_0)\in
L^2(\mathbb{R}^3)$. Assume furthermore that for some $T>0$,
either
\begin{equation}\label{eq1.2}
 (\mathbf{u},\nabla\mathbf{d})\in C([0,T], L^3(\mathbb{R}^3))
\end{equation}
or
\begin{equation}\label{eq1.3}
  (\mathbf{u}, \nabla\mathbf{d})\in L^q(0,T;\dot{B}^{-1+3/p+2/q}_{p,q}(\mathbb{R}^3))
\end{equation}
with $2\leq p<\infty$, $2<q<\infty$ and $\frac{3}{p}+\frac{2}{q}>1$.
Then $\mathbf{u}=\tilde{\mathbf{u}}$ and
$\mathbf{d}=\tilde{\mathbf{d}}$ on the time interval $[0,T]$.
\end{theorem}

\begin{remark} \label{rmk1.1} \rm
Theorem \ref{th1.2} holds
with $\frac{3}{p}+\frac{2}{q}=1$ in \eqref{eq1.3} as well, with the
space $L^q(0,T; \dot{B}^{-1+3/p+2/q}_{p,q}(\mathbb{R}^3))$
replaced by $L^q(0,T; L^{p}(\mathbb{R}^3))$, when $p>3$, namely,
if we assume that
\begin{equation*}
  (\mathbf{u},\nabla\mathbf{d})\in L^q(0,T;,
  L^{p}(\mathbb{R}^3)) \quad
   \text{with $3< p\leq\infty$, $2\leq q<\infty$ and
  $\frac{3}{p}+\frac{2}{q}=1$},
\end{equation*}
then $\mathbf{u}=\tilde{\mathbf{u}}$ and
$\mathbf{d}=\tilde{\mathbf{d}}$ on the time interval $[0,T]$. This
can be seen as a consequence of Prodi-Serrin's uniqueness criterion.
\end{remark}

\begin{remark} \label{rmk1.2} \rm
We extend, in Theorem \ref{th1.2}, the
uniqueness criteria of weak solutions of \cite{V85} and \cite{GP02}
for the system \eqref{eq1.1}.
\end{remark}

Let us sketch an idea leading to the proof of Theorem \ref{th1.2}.
We introduce the  function
$$
  F=\nabla\mathbf{d}.
$$
Let $F^T$ be the transpose of $F$. Then, taking the
gradient of second equation of \eqref{eq1.1}, noticing the facts
that $F\odot F=F^TF$ and
$$
  \frac{\partial}{\partial x_{k}}\Big(\sum_{j=1}^{n}\mathbf{u}_{j}
  \frac{\partial \mathbf{d}_{i}}{\partial
  x_{j}}\Big)=\sum_{j=1}^{n}\frac{\partial \mathbf{u}_{j}}{\partial x_{k}}
  \frac{\partial \mathbf{d}_{i}}{\partial
  x_{j}}+\sum_{j=1}^{n}\mathbf{u}_{j}\frac{\partial}{\partial x_{j}}
  \Big(\frac{\partial \mathbf{d}_{i}}{\partial
  x_{k}}\Big)=(F\nabla \mathbf{u}+\mathbf{u}\cdot\nabla F)_{ik}
$$
for all $i, k=1,2, \dots, n$,  system \eqref{eq1.1} reads
\begin{equation}\label{eq1.4}
\begin{gathered}
  \partial_{t} \mathbf{u}-\Delta \mathbf{u}
  =-\nabla\pi-\mathbf{u}\cdot\nabla\mathbf{u}-\operatorname{div}(F^TF),\\
  \partial_{t} F-\Delta F=-\mathbf{u}\cdot\nabla F-F\nabla
  \mathbf{u}-(3|\mathbf{d}|^2-1)F,\\
\operatorname{div}\mathbf{u}=0,\\
  (\mathbf{u},F)|_{t=0}=(\mathbf{u}_0,F_0),
\end{gathered}
\end{equation}
where $F_0=\nabla \mathbf{d}_0$. System \eqref{eq1.4} is
more related to the viscoelastic fluids, which had attracted much
attention recently; see for instance \cite{LLZ05}. Using the
technical matrixes analysis, the energy inequality and the similar
argument in the studying of the incompressible Navier-Stokes
equations in \cite{V85} and \cite{GP02}, we can obtain some
important estimates which yield the proof of Theorem \ref{th1.2}.



Before ending this section, we mention some well-posedness results
of the system \eqref{eq1.1}. Recently, when $g(\mathbf{d})=0$, by
using the maximal regularity of Stokes equations and the parabolic
equations, Hu and Wang \cite{HW10} proved global existence of strong
solutions to the system \eqref{eq1.1} for small initial data
belonging to Besov spaces of positive-order. They also proved that
when the strong solution exists, all global weak solutions
constructed by \cite{LL95} must be equal to the unique strong
solution.  In \cite{LLW10} and \cite{LW10}, the authors studied the
system \eqref{eq1.1} with
$g(\mathbf{d})=|\nabla\mathbf{d}|^2\mathbf{d}$ in two dimensions.
They established the global existence, uniqueness and partial
regularity of weak solutions and performed the blow-up analysis at
each singular time. Hong \cite{H11} proved independently the global
existence of weak solutions of the system \eqref{eq1.1} in two
dimensions. In \cite{W11}, Wang established global well-posedness of
\eqref{eq1.1} with $g(\mathbf{d})=|\nabla\mathbf{d}|^2\mathbf{d}$
for small initial data in $BMO^{-1}\times BMO$. Some regularity
criteria for weak solutions of the system \eqref{eq1.1} were also
established, see \cite{G10}, \cite{LC10} and \cite{LZC11}.

The rest of this paper is organized as follows. In Section 2,  we
present the proof of Theorem \ref{th1.2}. In appendix, we shall
establish the basic energy inequality of the system \eqref{eq1.1},
which gives global existence of weak solutions of \eqref{eq1.1}.

\section{The proof of Theorem \ref{th1.2}}

Throughout this section, we assume that
$(\mathbf{u}_0,\nabla\mathbf{d}_0)$,
$(\tilde{\mathbf{u}}_0,\nabla\tilde{\mathbf{d}}_0)\in
L^2(\mathbb{R}^3)$,  and denote by $(\mathbf{u}, \mathbf{d})$
and $(\tilde{\mathbf{u}}, \tilde{\mathbf{d}})$, respectively, be two
weak solutions associated with initial conditions
$(\mathbf{u}_0,\nabla\mathbf{d}_0)$ and
$(\tilde{\mathbf{u}}_0,\nabla\tilde{\mathbf{d}}_0)$,
respectively.

Let us define $F=\nabla\mathbf{d}$,
$\tilde{F}=\nabla\tilde{\mathbf{d}}$, $F_0=\nabla\mathbf{d}_0$
and $\tilde{F}_0=\nabla\tilde{\mathbf{d}}_0$. Obviously, by
Definition \ref{def1.1}, $(\mathbf{u}, F)$ and
$(\tilde{\mathbf{u}}, \tilde{F})$ verify equations \eqref{eq1.4} and satisfy
\begin{equation} \label{eq2.1}
\begin{aligned}
&\|\mathbf{u}(t)\|_{L^2}^2+\|F(t)\|_{L^2}^2
  +2\int_0^t(\|\nabla\mathbf{u}(\tau)\|_{L^2}^2+\|\nabla F(\tau)\|_{L^2}^2)d\tau\\
&+6\int_0^t\||\mathbf{d}| F\|_{L^2}^2(\tau)d\tau \\
&\leq
  \|\mathbf{u}_0\|_{L^2}^2+\|F_0\|_{L^2}^2
  +2\int_0^t\|F(\tau)\|_{L^2}^2d\tau,
\end{aligned}\
\end{equation}
\begin{equation}  \label{eq2.2}
\begin{aligned}
&\|\tilde{\mathbf{u}}(t)\|_{L^2}^2+\|\tilde{F}(t)\|_{L^2}^2
  +2\int_0^t(\|\nabla\tilde{\mathbf{u}}(\tau)\|_{L^2}^2
  +\|\nabla \tilde{F}(\tau)\|_{L^2}^2)d\tau\\
&+6\int_0^t\||\tilde{\mathbf{d}}| \tilde{F}\|_{L^2}^2(\tau)d\tau \\
&\leq   \|\tilde{\mathbf{u}}_0\|_{L^2}^2+\|\tilde{F}_0\|_{L^2}^2
+2\int_0^t\|\tilde{F}(\tau)\|_{L^2}^2d\tau.
\end{aligned}
\end{equation}
Setting $\mathbf{w}=\mathbf{u}-\tilde{\mathbf{u}}$, $E=F-\tilde{F}$,
$\mathbf{w}_0=\mathbf{u}_0-\tilde{\mathbf{u}}_0$ and
$E_0=F_0-\tilde{F}_0$, we divide the proof of Theorem
\ref{th1.2} into the following two cases.

\textbf{Case 1.} $(\mathbf{u}, \nabla\mathbf{d})\in
C([0,T],L^3(\mathbb{R}^3))$. We shall prove the following
stability result.

\begin{proposition}\label{pro2.1}
Assume that $(\mathbf{u}, \nabla\mathbf{d})\in
C([0,T],L^3(\mathbb{R}^3))$. Then
\begin{equation} \label{eq2.3}
\begin{aligned}
&\|(\mathbf{w}(t), E(t))\|_{L^2}^2
 +2\int^t_0\|(\nabla\mathbf{w}(\tau), \nabla E(\tau))\|_{L^2}^2d\tau
   \\
&\leq\|(\mathbf{w}_0, E_0)\|_{L^2}^2
  \exp\Big(Ct\big(\|(\mathbf{u}, F)\|_{C([0,T],L^3(\mathbb{R}^3))}^2+1\big)\Big),
\end{aligned}
\end{equation}
where $C$ is a constant depending on
$\|(\mathbf{d}, \tilde{\mathbf{d}})\|_{L^{\infty}(0,T;\dot{H}^1)}$ and
$\|(\mathbf{d}, \tilde{\mathbf{d}})\|_{L^2(0,T;\dot{H}^2)}$.
\end{proposition}

It is clear that, under the condition \eqref{eq1.2}, Theorem
\ref{th1.2} is an immediate consequence of Proposition \ref{pro2.1}.
Note that, by \eqref{eq2.1} and \eqref{eq2.2}, the left hand side of
\eqref{eq2.3} satisfies
\begin{equation} \label{eq2.4}
\begin{aligned}
&\|(\mathbf{w}(t), E(t))\|_{L^2}^2+2\int^t_0\|(\nabla\mathbf{w}(\tau),
 \nabla E(\tau))\|_{L^2}^2d\tau\\
&=\|(\mathbf{u}(t), F(t))\|_{L^2}^2+\|(\tilde{\mathbf{u}}(t),
  \tilde{F}(t))\|_{L^2}^2 
 +2\int^t_0\|(\nabla\mathbf{u}(\tau), \nabla  F(\tau))\|_{L^2}^2d\tau\\
&\quad +2\int^t_0\|(\nabla\tilde{\mathbf{u}}(\tau), \nabla
  \tilde{F}(\tau))\|_{L^2}^2d\tau-2\big(\mathbf{u}(t)|\tilde{\mathbf{u}}(t)\big) \\
&\quad -2\big(F(t)|
  \tilde{F}(t)\big)-4\int_0^t\big(\nabla\mathbf{u}(\tau)|
  \nabla\tilde{\mathbf{u}}(\tau)\big)d\tau-4\int_0^t\big(\nabla
  F(\tau)|
  \nabla\tilde{F}(\tau)\big)d\tau \\
&\leq\|(\mathbf{u}_0, F_0)\|_{L^2}^2
 +2\int_0^t\|F(\tau)\|_{L^2}^2d\tau-6\int_0^t\||\mathbf{d}| F\|_{L^2}^2(\tau)d\tau
  +\|(\tilde{\mathbf{u}}_0, \tilde{F}_0)\|_{L^2}^2 \\
&\quad +2\int_0^t\|\tilde{F}(\tau)\|_{L^2}^2d\tau
 -6\int_0^t\||\tilde{\mathbf{d}}|
  \tilde{F}\|_{L^2}^2(\tau)d\tau-2\big(\mathbf{u}(t)| \tilde{\mathbf{u}}(t)\big)-2\big(F(t)|\tilde{F}(t)\big) \\
&\quad -4\int_0^t\big(\nabla\mathbf{u}(\tau)|
  \nabla\tilde{\mathbf{u}}(\tau)\big)d\tau-4\int_0^t\big(\nabla
  F(\tau)|\nabla\tilde{F}(\tau)\big)d\tau,
\end{aligned}
\end{equation}
where we denote by $(\cdot|\cdot)$ the scalar product in
$L^2(\mathbb{R}^3)$. Hence, we aim at proving the following
lemma.

\begin{lemma}\label{le2.2}
Under the assumptions of Proposition \ref{pro2.1}, the following
equality holds for all $t\leq T$,
\begin{equation} \label{eq2.5}
\begin{aligned}
&\big(\mathbf{u}(t)| \tilde{\mathbf{u}}(t)\big)+\big(F(t)|
  \tilde{F}(t)\big)+2\int_0^t\big(\nabla\mathbf{u}(\tau)|
  \nabla\tilde{\mathbf{u}}(\tau)\big)d\tau+2\int_0^t\big(\nabla
  F(\tau)|
  \nabla\tilde{F}(\tau)\big)d\tau \\
&=(\mathbf{u}_0|\tilde{\mathbf{u}}_0)+(F_0|\tilde{F}_0)
  -\int_0^t\int_{\mathbb{R}^3}\mathbf{w}\cdot\nabla\mathbf{w}\cdot\mathbf{u}\,dx\,d\tau
  +\int_0^t\int_{\mathbb{R}^3}\tilde{F}^T\tilde{F}:\nabla\mathbf{u}\,dx\,d\tau \\
&\quad +\int_0^t\int_{\mathbb{R}^3}F^TF:\nabla\mathbf{u}\,dx\,d\tau
  -\int_0^t\int_{\mathbb{R}^3}\nabla\mathbf{u}:(\tilde{F}^TF+F^T\tilde{F})\,dx\,d\tau \\
&\quad -\int_0^t\int_{\mathbb{R}^3}\mathbf{w}\cdot\nabla F:\tilde{F}\,dx\,d\tau
  +\int_0^t\int_{\mathbb{R}^3}F:\tilde{F}\nabla\mathbf{w}\,dx\,d\tau \\
&\quad -\int_0^t\int_{\mathbb{R}^3}(3|\mathbf{d}|^2-1)F:\tilde{F}\,dx\,d\tau
  -\int_0^t\int_{\mathbb{R}^3}(3|\tilde{\mathbf{d}}|^2-1)\tilde{F}:F\,dx\,d\tau.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
Let us choose two smooth sequences 
$\{(\tilde{\mathbf{u}}_n,\tilde{F}_n)\}$
 ($\operatorname{div}\tilde{\mathbf{u}}_n=0$) and $\{(\mathbf{u}_n, F_n)\}$
($\operatorname{div}\mathbf{u}_n=0$) such that
\begin{equation}\label{eq2.6}
\begin{gathered}
  \lim_{n\to\infty}(\tilde{\mathbf{u}}_n,
  \tilde{F}_n)=(\tilde{\mathbf{u}},\tilde{F})\quad\text{in }
  L^2(0,T; \dot{H}^1(\mathbb{R}^3)),\\
  \lim_{n\to\infty}(\tilde{\mathbf{u}}_n,
  \tilde{F}_n)=(\tilde{\mathbf{u}},\tilde{F})\quad \text{weakly-star in }
  L^{\infty}(0,T; L^2(\mathbb{R}^3))
\end{gathered}
\end{equation}
and
\begin{equation}\label{eq2.7}
\begin{gathered}
  \lim_{n\to\infty}(\mathbf{u}_n,
  F_n)=(\mathbf{u},F)\quad\text{in }
  L^2(0,T; \dot{H}^1(\mathbb{R}^3))\cap
  C([0,T],L^3(\mathbb{R}^3)),\\
  \lim_{n\to\infty}(\mathbf{u}_n,
  F_n)=(\mathbf{u},F)\quad  \text{weakly-star in }
  L^{\infty}(0,T; L^2(\mathbb{R}^3)).
\end{gathered}
\end{equation}
We split the proof into the following two steps.


\textbf{Step 1.} Taking the scalar product with
$\tilde{\mathbf{u}}_n$ and $\mathbf{u}_n$ of the equation
\eqref{eq1.4} on $\mathbf{u}$ and $\tilde{\mathbf{u}}$ respectively,
after integration in time and integration by parts in the space
variables, we obtain
\begin{equation}\label{eq2.8}
  \int_0^t\Big((\partial_{\tau}\mathbf{u}| \tilde{\mathbf{u}}_n)+(\nabla\mathbf{u}| \nabla\tilde{\mathbf{u}}_n)
  +(\mathbf{u}\cdot\nabla\mathbf{u}|\tilde{\mathbf{u}}_n)+(\operatorname{div}(F^TF)|
  \tilde{\mathbf{u}}_n)\Big)d\tau=0
\end{equation}
and
\begin{equation}\label{eq2.9}
  \int_0^t\Big((\partial_{\tau}\tilde{\mathbf{u}}| \mathbf{u}_n)+(\nabla\tilde{\mathbf{u}}| \nabla\mathbf{u}_n)
  +(\tilde{\mathbf{u}}\cdot\nabla\tilde{\mathbf{u}}|\mathbf{u}_n)+(\operatorname{div}(\tilde{F}^T\tilde{F})|
  \mathbf{u}_n)\Big)d\tau=0.
\end{equation}
By \eqref{eq2.6} and \eqref{eq2.7}, it is obvious that
\begin{equation}\label{eq2.10}
  \lim_{n\to\infty}\Big(\int_0^t(\nabla\mathbf{u}| \nabla\tilde{\mathbf{u}}_n)d\tau+\int_0^t(\nabla\tilde{\mathbf{u}}| \nabla\mathbf{u}_n)
  d\tau\Big)=2\int_0^t(\nabla\mathbf{u}|
  \nabla\tilde{\mathbf{u}})d\tau.
\end{equation}
Applying the H\"{o}lder inequality and the Sobolev embedding
inequality, it follows that
\begin{equation} \label{eq2.11}
\begin{aligned}
  \int_0^t(\tilde{\mathbf{u}}\cdot\nabla\tilde{\mathbf{u}}|\mathbf{u}_n)d\tau
 &\leq
  C\int_0^t\|\tilde{\mathbf{u}}\|_{L^{6}}\|\nabla\tilde{\mathbf{u}}
  \|_{L^2}\|\mathbf{u}_n\|_{L^3}d\tau \\
 &\leq C\|\tilde{\mathbf{u}}\|_{L^2(0,T;\dot{H}^1)}^2\|\mathbf{u}_n
  \|_{C([0,T],L^3)}.
\end{aligned}
\end{equation}
Since $\mathbf{u}_n$ converges to $\mathbf{u}$ in
$C([0,T],L^3(\mathbb{R}^3))$, \eqref{eq2.11} implies that
\begin{equation}\label{eq2.12}
  \lim_{n\to\infty}\int_0^t(\tilde{\mathbf{u}}\cdot\nabla\tilde{\mathbf{u}}
|\mathbf{u}_n)d\tau
  =\int_0^t(\tilde{\mathbf{u}}\cdot\nabla\tilde{\mathbf{u}}|\mathbf{u})d\tau.
\end{equation}
Similarly, by applying \eqref{eq2.6}, \eqref{eq2.7} and
\eqref{eq2.11},  we obtain the following three equalities:
\begin{equation} \label{eq2.13}
\begin{aligned}
&\lim_{n\to\infty}\int_0^t(\mathbf{u}\cdot\nabla\mathbf{u}|
 \tilde{\mathbf{u}}_n)d\tau
=-\lim_{n\to\infty}\int_0^t(\mathbf{u}\cdot\nabla\tilde{\mathbf{u}}_n|
 \mathbf{u})d\tau \\
&=-\int_0^t(\mathbf{u}\cdot\nabla\tilde{\mathbf{u}}|\mathbf{u})d\tau
  =\int_0^t(\mathbf{u}\cdot\nabla\mathbf{u}|\tilde{\mathbf{u}})d\tau,
\end{aligned}
\end{equation}
\begin{equation}\label{eq2.14}
\begin{aligned}
&\lim_{n\to\infty}\int_0^t(\operatorname{div}(F^TF)|
  \tilde{\mathbf{u}}_n)d\tau=-\lim_{n\to\infty}\int_0^t(F^TF|
  \nabla\tilde{\mathbf{u}}_n)d\tau \\
&=-\int_0^t(F^TF|  \nabla\tilde{\mathbf{u}})d\tau
=\int_0^t(\operatorname{div}(F^TF)|\tilde{\mathbf{u}})d\tau,
\end{aligned}
\end{equation}
and
\begin{equation} \label{eq2.15}
\begin{aligned}
&\lim_{n\to\infty}\int_0^t(\operatorname{div}(\tilde{F}^T\tilde{F})\big|
  \mathbf{u}_n)d\tau
=\lim_{n\to\infty}\int_0^t\Big(\sum_{i=1}^3(\partial_{x_{i}}\tilde{F}^T\tilde{F}+\tilde{F}^T\partial_{x_{i}}\tilde{F})|
  \mathbf{u}_n\Big)d\tau \\
&=\int_0^t\Big(\sum_{i=1}^3(\partial_{x_{i}}\tilde{F}^T\tilde{F}+\tilde{F}^T\partial_{x_{i}}\tilde{F})\big|
  \mathbf{u}\Big)d\tau=\int_0^t(\operatorname{div}(\tilde{F}^T\tilde{F})|
  \mathbf{u})d\tau.
\end{aligned}
\end{equation}
Since $\partial_{t}\tilde{\mathbf{u}}=\Delta\tilde{\mathbf{u}}
-\tilde{\mathbf{u}}\cdot\nabla\tilde{\mathbf{u}}
-\operatorname{div}(\tilde{F}^T\tilde{F})-\nabla\pi$ holds in the sense of
distribution, the estimates \eqref{eq2.10},
\eqref{eq2.12}--\eqref{eq2.15} and $\operatorname{div}\mathbf{u}_n=0$
imply in particular that
\begin{align*}
  \lim_{n\to\infty}\int_0^t(\partial_{\tau}\tilde{\mathbf{u}}|
  \mathbf{u}_n)d\tau
&=-\lim_{n\to\infty}\int_0^t\Big((\nabla\tilde{\mathbf{u}}| \nabla\mathbf{u}_n)
  +(\tilde{\mathbf{u}}\cdot\nabla\tilde{\mathbf{u}}|\mathbf{u}_n)+(\operatorname{div}(\tilde{F}^T\tilde{F})|
  \mathbf{u}_n)\Big)d\tau \\
&=-\int_0^t\Big((\nabla\tilde{\mathbf{u}}|\nabla\mathbf{u})
  +(\tilde{\mathbf{u}}\cdot\nabla\tilde{\mathbf{u}}|\mathbf{u})+(\operatorname{div}(\tilde{F}^T\tilde{F})|
  \mathbf{u})\Big)d\tau \\
&=\int_0^t(\partial_{\tau}\tilde{\mathbf{u}}|
  \mathbf{u})d\tau.
\end{align*}
It can be proved analogously that
\[
  \lim_{n\to\infty}\int_0^t(\partial_{\tau}\mathbf{u}| \tilde{\mathbf{u}}_n)d\tau
  =\int_0^t(\partial_{\tau}\mathbf{u}| \tilde{\mathbf{u}})d\tau.
\]
Putting these estimates together, and noticing that
\begin{gather}\label{eq2.16}
 \int_0^t(\partial_{\tau}\tilde{\mathbf{u}}|
  \mathbf{u})+(\partial_{\tau}\mathbf{u}| \tilde{\mathbf{u}})d\tau=(\mathbf{u}(t)|
  \tilde{\mathbf{u}}(t)),
  %-(\mathbf{u}_0|  \tilde{\mathbf{u}}_0),
\\ \label{eq2.17}
 \int_0^t\Big((\tilde{\mathbf{u}}\cdot\nabla\tilde{\mathbf{u}}|\mathbf{u})-(\mathbf{u}\cdot\nabla\tilde{\mathbf{u}}|\mathbf{u})\Big)d\tau
 =\int_0^t(\mathbf{w}\cdot\nabla\mathbf{w}| \mathbf{u})d\tau,
\end{gather}
we obtain
\begin{equation} \label{eq2.18}
\begin{aligned}
&(\mathbf{u}(t)|
  \tilde{\mathbf{u}}(t))+2\int_0^t(\nabla\mathbf{u}|
  \nabla\tilde{\mathbf{u}})d\tau\\
&=(\mathbf{u}_0|
  \tilde{\mathbf{u}}_0)-\int_0^t(\mathbf{w}\cdot\nabla\mathbf{w}| \mathbf{u})d\tau 
+\int_0^t\int_{\mathbb{R}^3}\tilde{F}^T\tilde{F}:\nabla\mathbf{u}\,dx\,d\tau\\
&\quad+\int_0^t\int_{\mathbb{R}^3}F^TF:\nabla\mathbf{u}\,dx\,d\tau.
\end{aligned}
\end{equation}

\textbf{Step 2.}  Proceeding in the same way as \eqref{eq2.8} and
\eqref{eq2.9}, we obtain 
\begin{equation}\label{eq2.19}
\begin{split}
&\int_0^t\Big((\partial_{\tau}F| \tilde{F}_n)+(\nabla F|\nabla\tilde{F}_n)
  +(\mathbf{u}\cdot\nabla F|\tilde{F}_n)\\
& +(F\nabla\mathbf{u}|
  \tilde{F}_n)+((3|\mathbf{d}|^2-1)F|\tilde{F}_n)\Big)d\tau=0
\end{split}
\end{equation}
and
\begin{equation}\label{eq2.20}
\begin{split}
&\int_0^t\Big((\partial_{\tau}\tilde{F}| F_n)+(\nabla\tilde{F}| \nabla F_n)
  +(\tilde{\mathbf{u}}\cdot\nabla\tilde{F}|F_n)\\
&+(\tilde{F}\nabla\tilde{\mathbf{u}}|
  F_n)+((3|\tilde{\mathbf{d}}|^2-1)\tilde{F}|F_n)\Big)d\tau=0.
\end{split}
\end{equation}
By using assumptions \eqref{eq2.6}--\eqref{eq2.7} and similar
argument in the proof of \eqref{eq2.11},  we obtain
\begin{gather}\label{eq2.21}
  \lim_{n\to\infty}\Big(\int_0^t(\nabla F| \nabla\tilde{F}_n)d\tau
+\int_0^t(\nabla\tilde{F}| \nabla F_n)   d\tau\Big)
=2\int_0^t(\nabla F| \nabla\tilde{F})d\tau,
\\ \label{eq2.22}
  \lim_{n\to\infty}\int_0^t(\tilde{\mathbf{u}}\cdot\nabla\tilde{F}|F_n)d\tau
  =\int_0^t(\tilde{\mathbf{u}}\cdot\nabla\tilde{F}|F)d\tau,
\\ \label{eq2.23}
  \lim_{n\to\infty}\int_0^t(\mathbf{u}\cdot\nabla F|\tilde{F}_n)d\tau
  =\int_0^t(\mathbf{u}\cdot\nabla F|\tilde{F})d\tau,
\\ \label{eq2.24}
  \lim_{n\to\infty}\int_0^t(\tilde{F}\nabla\tilde{\mathbf{u}}|
  F_n)d\tau
  =\int_0^t(\tilde{F}\nabla\tilde{\mathbf{u}}|
  F)d\tau,
\\ \label{eq2.25}
  \lim_{n\to\infty}\int_0^t(F\nabla\mathbf{u}|
  \tilde{F}_n)d\tau
  =\int_0^t(F\nabla\mathbf{u}|
  \tilde{F})d\tau.
\end{gather}
To estimate the remaining terms, the H\"{o}lder inequality and the
Sobolev embedding theorem yield 
\begin{equation*}
  \int_0^t(F(\tau)|\tilde{F}_n(\tau))d\tau
  \leq \|F\|_{L^2(0,T;L^2)}\|\tilde{F}_n\|_{L^2(0,T;L^2)}
\end{equation*}
and
\begin{align*}
  \int_0^t(3|\mathbf{d}|^2F|\tilde{F}_n)(\tau)d\tau
&\leq   C\int_0^t\|\mathbf{d}(\tau)\|_{L^{6}}\|F(\tau)\|_{L^2}
 \|\tilde{F}_n(\tau)\|_{L^{6}}d\tau\\
&\leq   C\|\mathbf{d}\|_{L^{\infty}(0,T;\dot{H}^1)}\|F\|_{L^2(0,T;L^2)}
\|\tilde{F}_n\|_{L^2(0,T;\dot{H}^1)}.
\end{align*}
Hence, by \eqref{eq2.6}--\eqref{eq2.7}, we can easily see that
\begin{equation}\label{eq2.26}
  \lim_{n\to\infty}\int_0^t((3|\mathbf{d}|^2-1)F|\tilde{F}_n)d\tau
  =\int_0^t((3|\mathbf{d}|^2-1)F|\tilde{F})d\tau.
\end{equation}
Similarly,
\begin{equation}\label{eq2.27}
  \lim_{n\to\infty}\int_0^t((3|\tilde{\mathbf{d}}|^2-1)\tilde{F}|F_n)d\tau
  =\int_0^t((3|\tilde{\mathbf{d}}|^2-1)\tilde{F}|F)d\tau.
\end{equation}
As in the derivations of estimates \eqref{eq2.16} and
\eqref{eq2.17}, the above estimates \eqref{eq2.21}--\eqref{eq2.27}
imply 
\begin{gather*}
  \lim_{n\to\infty}\int_0^t((\partial_{\tau}F| \tilde{F}_n)d\tau
=\int_0^t(\partial_{\tau}F|  \tilde{F})d\tau,
\\
  \lim_{n\to\infty}\int_0^t(\partial_{\tau}\tilde{F}| F_n)d\tau
  =\int_0^t(\partial_{\tau}\tilde{F}| F)d\tau.
\end{gather*}
Since
\begin{gather*}
  \int_0^t(\partial_{\tau}F| \tilde{F})+(\partial_{\tau}\tilde{F}| F)d\tau=(F(t)|
  \tilde{F}(t))-(F_0|
  \tilde{F}_0),
\\
  \int_0^t\int_{\mathbb{R}^3}\Big(\tilde{\mathbf{u}}\cdot\nabla\tilde{F}:F+\tilde{\mathbf{u}}\cdot\nabla
  F:\tilde{F}\Big)\,dx\,d\tau=\int_0^t\int_{\mathbb{R}^3}
\tilde{\mathbf{u}}\cdot\nabla(\tilde{F}:F)\,dx\,d\tau=0,
\\
 \tilde{F}\nabla\mathbf{u}:F+\tilde{F}:F\nabla\mathbf{u}=\nabla\mathbf{u}
:(\tilde{F}^TF+F^T\tilde{F}),
\end{gather*}
we have
\begin{align*}
&\int_0^t\int_{\mathbb{R}^3}\Big(\mathbf{u}\cdot\nabla F:\tilde{F}
  +\tilde{\mathbf{u}}\cdot\nabla\tilde{F}:F
  +F\nabla\mathbf{u}:\tilde{F}
  +\tilde{F}\nabla\tilde{\mathbf{u}}:F\Big)\,dx\,d\tau\\
&=\int_0^t\int_{\mathbb{R}^3}\Big(\nabla\mathbf{u}:(\tilde{F}^TF+F^T\tilde{F})
 +(\mathbf{u}-\tilde{\mathbf{u}})\cdot\nabla
  F:\tilde{F}-F:\tilde{F}\nabla(\mathbf{u}-\tilde{\mathbf{u}})
  \Big)\,dx\,d\tau.
\end{align*}
Here we have used the facts $\operatorname{div}\tilde{\mathbf{u}}=0$ and
$AB:C=A:CB^T=B:A^TC$ for any three $n\times n$ matrixes $A$, $B$
and $C$. Finally, putting all above estimates together, we obtain
\begin{equation} \label{eq2.28}
\begin{aligned}
&(F(t)| \tilde{F}(t))+2\int_0^t(\nabla F|  \nabla\tilde{F})d\tau\\
&=(F_0|  \tilde{F}_0)-\int_0^t\int_{\mathbb{R}^3}\Big(\nabla\mathbf{u}
 :(\tilde{F}^TF+F^T\tilde{F})  +\mathbf{w}\cdot\nabla F:\tilde{F} 
-F:\tilde{F}\nabla\mathbf{w} \Big)\,dx\,d\tau\\
&\quad -\int_0^t\int_{\mathbb{R}^3}\Big(\big((3|\mathbf{d}
|^2-1)F:\tilde{F}\big)
  +\big((3|\tilde{\mathbf{d}}|^2-1)\tilde{F}:F\big)\Big)\,dx\,d\tau.
\end{aligned}
\end{equation}
Now it is easy see that \eqref{eq2.5} follows from \eqref{eq2.18}
and \eqref{eq2.28}. This proves Lemma \ref{le2.3}.
\end{proof}

The following result plays a very important role in the proof of
Proposition \ref{pro2.1}.

\begin{lemma}[\cite{V85}] \label{le2.3} 
Let $u$ be a  measurable function in $(\mathcal{LS})\cap C([0, T],
L^3(\mathbb{R}^3))$. Then for each $\varepsilon>0$ we can split
$u$ on $[0,T]$ in $u=m+l$ with $m\in
L^{\infty}([0,T]\times\mathbb{R}^3)$ and
$\|l\|_{L^{\infty}(0,T;L^3)}<\varepsilon$.
\end{lemma}

\begin{proof}
The proof of this lemma is due to \cite{V85}, but we 
give it for completeness. Since $u\in C([0, T],
L^3(\mathbb{R}^3))$, by the uniform continuity, we can choose
$N$ large enough such that
$$
  \big\|u(x,t)-\sum_{k=0}^{N-1}\chi_{[\frac{k}{N}T,\frac{k+1}{N}T]}(t)
u(x,\frac{k}{N}T)\big\|_{L^{\infty}(0,T;L^3)}<\frac{\varepsilon}{2},
$$
where $\chi_{[a,b]}$ denotes the characteristic function on the
interval $[a,b]$. Now we may approximate each
$u(\cdot,\frac{k}{N}T)$ by a function $m_{k,N}\in
L^{\infty}(\mathbb{R}^3)$ with an error controlled in $L^3$-norm
by
$\|u(\cdot,\frac{k}{N}T)-m_{k,N}(\cdot)\|_{L^3}<\varepsilon/2$.
Now we define $m$ as $m(x,t)=\sum_{0\leq k\leq
N-1}\chi_{[\frac{k}{N}T,\frac{k+1}{N}T]}(t)m_{k,N}(x)$, and $l=u-m$.
This proves Lemma.
\end{proof}

\subsection*{Proof of Proposition \ref{pro2.1}}
Since $\operatorname{div}\mathbf{w}=0$, we obtain
$\int_0^t\int_{\mathbb{R}^3}\mathbf{w}\cdot\nabla
F:F\,dx\,d\tau=0$.  By \eqref{eq2.4} and Lemma \ref{le2.2}, it follows
immediately that
\begin{equation} \label{eq2.29}
\begin{aligned}
&\|(\mathbf{w}(t), E(t))\|_{L^2}^2+2\int^t_0\|(\nabla\mathbf{w}(\tau), \nabla
  E(\tau))\|_{L^2}^2d\tau\\
&\leq\|(\mathbf{u}_0, F_0)\|_{L^2}^2+\|(\tilde{\mathbf{u}}_0,
  \tilde{F}_0)\|_{L^2}^2
  -2(\mathbf{u}_0|\tilde{\mathbf{u}}_0)-2(F_0|\tilde{F}_0)\\
&\quad  +2\int_0^t\int_{\mathbb{R}^3}\mathbf{w}\cdot\nabla\mathbf{w}\cdot\mathbf{u}\,dx\,d\tau
  -2\int_0^t\int_{\mathbb{R}^3}\tilde{F}^T\tilde{F}:\nabla\mathbf{u}\,dx\,d\tau
 \\
&\quad-2\int_0^t\int_{\mathbb{R}^3}F^TF:\nabla\tilde{\mathbf{u}}\,dx\,d\tau
  +2\int_0^t\int_{\mathbb{R}^3}\nabla\mathbf{u}:(\tilde{F}^TF+F^T\tilde{F})\,dx\,d\tau
\\
&\quad +2\int_0^t\int_{\mathbb{R}^3}\mathbf{w}\cdot\nabla F:\tilde{F}\,dx\,d\tau
  -2\int_0^t\int_{\mathbb{R}^3}F:\tilde{F}\nabla\mathbf{w}\,dx\,d\tau
 -2\int_0^t\|E\|_{L^2}^2d\tau
\\
&\quad -6\int_0^t\int_{\mathbb{R}^3}\big(|\mathbf{d}|^2E:F+|\tilde{\mathbf{d}}|^2E:
 \tilde{F}\big)\,dx\,d\tau
\\
&\leq\|(\mathbf{w}_0, E_0)\|_{L^2}^2+2\int_0^t\int_{\mathbb{R}^3}\mathbf{w}
\cdot\nabla\mathbf{w}\cdot\mathbf{u}\,dx\,d\tau
  -2\int_0^t\int_{\mathbb{R}^3}E^TE:\nabla\mathbf{u}\,dx\,d\tau
\\
&\quad -2\int_0^t\int_{\mathbb{R}^3}\mathbf{w}\cdot\nabla F:E\,dx\,d\tau
  +2\int_0^t\int_{\mathbb{R}^3}E^TF:\nabla\mathbf{w}\,dx\,d\tau
 -2\int_0^t\|E\|_{L^2}^2d\tau \\
&\quad -6\int_0^t\int_{\mathbb{R}^3}\big(|\mathbf{d}|^2E:F+|\tilde{\mathbf{d}}|^2E:
\tilde{F}\big)\,dx\,d\tau.
\end{aligned}
\end{equation}
Since we have assumed that $(\mathbf{u},F)\in
C([0,T],L^3(\mathbb{R}^3))$, by Lemma \ref{le2.3}, we can split
$\mathbf{u}=\mathbf{u}_{1}+\mathbf{u}_{2}$ and $F=F_{1}+F_{2}$ such
that $(\mathbf{u}_{1}, F_{1})\in
L^{\infty}([0,T]\times\mathbb{R}^3)$ and
$\|(\mathbf{u}_{2}, F_{2})\|_{L^{\infty}(0,T;L^3)}<\varepsilon$, respectively, where
$\varepsilon>0$ is a constant to be determined later. Then we see
that
\begin{equation} \label{eq2.30}
\begin{aligned}
&\big|\int_0^t\int_{\mathbb{R}^3}\mathbf{w}\cdot\nabla\mathbf{w}\cdot\mathbf{u}
 \,dx\,d\tau\big| \\
&\leq C\|\mathbf{u}_{2}\|_{C([0,T],L^3)}\int_0^t\|\nabla
  \mathbf{w}\|_{L^2}^2d\tau \\
&\quad +\|\mathbf{u}_{1}\|_{L^{\infty}([0,T]\times\mathbb{R}^3)}
 \Big(\int_0^t\|\nabla   \mathbf{w}\|_{L^2}^2d\tau\Big)^{1/2}
 \Big(\int_0^t\|\mathbf{w}\|_{L^2}^2d\tau\Big)^{1/2} \\
&\leq 2C\varepsilon\int^t_0\|\nabla\mathbf{w}\|_{L^2}^2d\tau
 +\frac{4}{C\varepsilon}\|\mathbf{u}_{1}\|_{L^{\infty}([0,T]
\times\mathbb{R}^3)}^2\int_0^t\|\mathbf{w}\|_{L^2}^2d\tau.
\end{aligned}
\end{equation}
Similarly, we obtain
\begin{equation} \label{eq2.31}
\begin{aligned}
&\Big|\int_0^t\int_{\mathbb{R}^3}E^TE:\nabla\mathbf{u}\,dx\,d\tau\Big|\\
&=\Big|-\int_0^t\int_{\mathbb{R}^3}\operatorname{div}(E^TE)\cdot\mathbf{u}
 \,dx\,d\tau\Big| \\
 &\leq 2C\varepsilon\int^t_0\|\nabla E\|_{L^2}^2d\tau
  +\frac{4}{C\varepsilon}\|\mathbf{u}_{1}\|_{L^{\infty}([0,T]
 \times\mathbb{R}^3)}^2\int_0^t\|E\|_{L^2}^2d\tau;
\end{aligned}
\end{equation}
\begin{equation} \label{eq2.32}
\begin{aligned}
&\Big|\int_0^t\int_{\mathbb{R}^3}\mathbf{w}\cdot\nabla F:E\,dx\,d\tau\Big|\\
&=\Big|\int_0^t\int_{\mathbb{R}^3}(\mathbf{w}\otimes F)\cdot\nabla
  E\,dx\,d\tau\Big| \\
&\leq 2C\varepsilon\int^t_0\|(\nabla\mathbf{w},\nabla E)\|_{L^2}^2d\tau
  +\frac{4}{C\varepsilon}\|F_{1}\|_{L^{\infty}([0,T]\times\mathbb{R}^3)}^2\int_0^t\|\mathbf{w}\|_{L^2}^2d\tau;
\end{aligned}
\end{equation}
\begin{equation} \label{eq2.33}
\begin{aligned}
 \Big|\int_0^t\int_{\mathbb{R}^3}E^TF:\nabla\mathbf{w}\,dx\,d\tau\Big|
&\leq 2C\varepsilon\int^t_0\|(\nabla\mathbf{w},\nabla E)\|_{L^2}^2d\tau \\
&\quad +\frac{4}{C\varepsilon}\|F_{1}\|_{L^{\infty}([0,T]\times\mathbb{R}^3)}^2
\int_0^t\|E\|_{L^2}^2d\tau;
\end{aligned}
\end{equation}
\begin{equation} \label{eq2.34}
\begin{aligned}
&\Big|\int_0^t\int_{\mathbb{R}^3}\big(|\mathbf{d}|^2E:F+|\tilde{\mathbf{d}}|^2
E:\tilde{F}\big)\,dx\,d\tau\Big|\\
& \leq C\int_0^t(\|\mathbf{d}\|_{L^{6}}^2\|F\|_{L^{6}}
 +\|\tilde{\mathbf{d}}\|_{L^{6}}^2\|\tilde{F}\|_{L^{6}})\|E\|_{L^2}d\tau \\
&\leq C\int_0^t(\|\mathbf{d}\|_{\dot{H}^1}^2\|F\|_{\dot{H}^1}
 +\|\tilde{\mathbf{d}}\|_{\dot{H}^1}^2\|\tilde{F}\|_{\dot{H}^1})\|E\|_{L^2}d\tau \\
&\leq C\int_0^t\|E\|_{L^2}^2d\tau,
\end{aligned}
\end{equation}
where $C$ is a constant depending on $\|(\mathbf{d},
\tilde{\mathbf{d}})\|_{L^{\infty}(0,T;\dot{H}^1)}$ and
$\|(\mathbf{d}, \tilde{\mathbf{d}})\|_{L^2(0,T;\dot{H}^2)}$.
Returning back to the estimate \eqref{eq2.29}, putting
\eqref{eq2.30}--\eqref{eq2.34} together, and choosing $\varepsilon$
sufficiently small such that $16C\varepsilon<1$, we obtain
\begin{equation} \label{eq2.35}
\begin{aligned}
&\|(\mathbf{w}(t),E(t))\|_{L^2}^2+\int^t_0\|(\nabla\mathbf{w}(\tau), \nabla
  E(\tau))\|_{L^2}^2d\tau \\
&\leq\|(\mathbf{w}_0, E_0)\|_{L^2}^2
  +C\Big(\|(\mathbf{u}_{2}, F_{2})\|_{L^{\infty}((0,T)\times\mathbb{R}^3)}^2+1\Big)\int_0^t(\|\mathbf{w}\|_{L^2}^2
  +\|E\|_{L^2}^2)d\tau.
\end{aligned}
\end{equation}
The estimate above together with the Gronwall inequality yield the
desired estimate \eqref{eq2.3} immediately.  We complete the proof
of Proposition \ref{pro2.1}.  \hfill\qed


\textbf{Case 2.} $(\mathbf{u}, \nabla\mathbf{d})\in
L^q(0,T;\dot{B}^{-1+3/p+2/q}_{p,q}(\mathbb{R}^3))$. It suffices
to establish the following stability result.

\begin{proposition}\label{pro2.4}
Assume that $(\mathbf{u}, \nabla\mathbf{d})\in
L^q(0,T;\dot{B}^{-1+3/p+2/q}_{p,q}(\mathbb{R}^3))$ with
$2\leq p<\infty$, $2<q<\infty$ and $\frac{3}{p}+\frac{2}{q}>1$.
Then
\begin{equation} \label{eq2.36}
\begin{aligned}
&\|(\mathbf{w}(t), E(t))\|_{L^2}^2+2\int^t_0\|(\nabla\mathbf{w}(\tau),
 \nabla E(\tau))\|_{L^2}^2d\tau   \\
&\leq\|(\mathbf{w}_0, E_0)\|_{L^2}^2
  \times\exp\Big(Ct+C\int^t_0\|(\mathbf{u}(\tau),
  F(\tau))\|_{\dot{B}^{-1+3/p+2/q}_{p,q}}^qd\tau\Big).
\end{aligned}
\end{equation}
where $C$ is a constant depending on $\|(\mathbf{d},
\tilde{\mathbf{d}})\|_{L^{\infty}(0,T;\dot{H}^1)}$ and
$\|(\mathbf{d}, \tilde{\mathbf{d}})\|_{L^2(0,T;\dot{H}^2)}$.
\end{proposition}

To prove Proposition \ref{pro2.4}, the key tool we shall use is the
following Lemma whose proof can be found in \cite{GP02}.

\begin{lemma}[\cite{GP02}] \label{le2.5}
Let  $2\leq p<\infty$ and $2<q<\infty$ such that
$\frac{2}{q}+\frac{3}{p}>1$. Then for every $T>0$, the trilinear
form
\begin{equation*}
  (\mathbf{u},\mathbf{v},\mathbf{w})\in(\mathcal{LS})\times(\mathcal{LS})\times
  L^q(0,T;\dot{B}^{-1+3/p+2/q}_{p,q}(\mathbb{R}^3))\mapsto\int_0^T\int_{\mathbb{R}^3}\mathbf{u}\cdot\nabla
  \mathbf{v}\cdot
  \mathbf{w}\,dx\,dt
\end{equation*}
is continuous. In particular, the following estimate holds:
\begin{equation} \label{eq2.37}
\begin{aligned}
  &\Big|\int_0^T\int_{\mathbb{R}^3}\mathbf{u}\cdot\nabla\mathbf{
  v}\cdot\mathbf{w}\,dx\,dt\Big| \\
&\leq
  C\|\mathbf{u}\|_{L^{\infty}(0,T;L^2)}^{2/q}\|\nabla
  \mathbf{u}\|_{L^2(0,T;L^2)}^{1-2/q}\|\nabla
  \mathbf{v}\|_{L^2(0,T;L^2)}\|\mathbf{w}\|_{L^q(0,T;\dot{B}^{-1+3/p+2/q}_{p,q})} \\
&\quad +\|\nabla
  \mathbf{u}\|_{L^2(0,T;L^2)}\|\mathbf{v}\|_{L^{\infty}(0,T;L^2)}^{2/q}\|\nabla
  \mathbf{v}\|_{L^2(0,T;L^2)}^{1-2/q}\|\mathbf{w}\|_{L^q(0,T;\dot{B}^{-1+3/p+2/q}_{p,q})} \\
&\quad +\|\mathbf{u}\|_{L^{\infty}(0,T;L^2)}^{1/q}\|\nabla
  \mathbf{u}\|_{L^2(0,T;L^2)}^{1-1/q}\|\mathbf{v}\|_{L^{\infty}(0,T;L^2)}^{1/q}\|\nabla
  \mathbf{v}\|_{L^2(0,T;L^2)}^{1-1/q}\\
&\quad\times \|\mathbf{w}\|_{L^q(0,T;\dot{B}^{-1+3/p+2/q}_{p,q})}.
\end{aligned}
\end{equation}
\end{lemma}

% \label{rmk2.1}
Note that \eqref{eq2.37} holds in both scalar and vector cases.

Note that for the Navier-Stokes equations, Gallagher and Planchon
\cite{GP02} proved that weak-strong uniqueness holds in the class
$$
  \mathcal{P}=L^q(0,T; \dot{B}^{-1+3/p+2/q}_{p,q}(\mathbb{R}^3))
  \quad \text{with $2\leq p<\infty$, $2<q<\infty$ and $\frac{3}{p}+\frac{2}{q}>1$}.
$$ 
Hence, we need only to deal with the remaining terms $\operatorname{div}(F^TF)$ 
and $F\nabla\mathbf{u}$ (the term $\mathbf{u}\cdot\nabla F$ can be treated 
as the term $\mathbf{u}\cdot\nabla\mathbf{u}$). Similarly as we have done
before, we choose two smooth sequences of
$\{(\tilde{\mathbf{u}}_n, \tilde{F}_n)\}$ ($\operatorname{div}\tilde{\mathbf{u}}_n=0$)
 and $\{(\mathbf{u}_n, F_n)\}$
($\operatorname{div}\mathbf{u}_n=0$) such that
\begin{gather*}
  \lim_{n\to\infty}(\tilde{\mathbf{u}}_n,
  \tilde{F}_n)=(\tilde{\mathbf{u}},\tilde{F})\quad\text{in }
  L^2(0,T; \dot{H}^1(\mathbb{R}^3)),\\
  \lim_{n\to\infty}(\tilde{\mathbf{u}}_n,
  \tilde{F}_n)=(\tilde{\mathbf{u}},\tilde{F})\quad \text{weakly-star in }
  L^{\infty}(0,T; L^2(\mathbb{R}^3))
\end{gather*}
and
\begin{gather*}
  \lim_{n\to\infty}(\mathbf{u}_n,
  F_n)=(\mathbf{u},F)\quad\text{in }
  L^2(0,T; \dot{H}^1(\mathbb{R}^{n}))\cap
  L^q(0,T;\dot{B}^{-1+n/p+2/q}_{p,q}(\mathbb{R}^3)),\\
  \lim_{n\to\infty}(\mathbf{u}_n,
  F_n)=(\mathbf{u},F)\quad \text{weakly-star in }  L^{\infty}(0,T; L^2(\mathbb{R}^3)).
\end{gather*}
Applying the above assumptions and Lemma \ref{le2.5}, we obtain
\begin{equation} \label{eq2.38}
\begin{aligned}
  &\lim_{n\to\infty}\int_0^t(\operatorname{div}(F^TF)|
  \tilde{\mathbf{u}}_n)d\tau=-\lim_{n\to\infty}\int_0^t(F^TF|
  \nabla\tilde{\mathbf{u}}_n)d\tau \\
  &=-\int_0^t(F^TF| \nabla\tilde{\mathbf{u}})d\tau=\int_0^t(\operatorname{div}(F^TF)|\tilde{\mathbf{u}})d\tau
\end{aligned}
\end{equation}
and
\begin{equation} \label{eq2.39}
\begin{aligned}
&\lim_{n\to\infty}\int_0^t(\operatorname{div}(\tilde{F}^T\tilde{F})\big|
  \mathbf{u}_n)d\tau
  =\lim_{n\to\infty}\int_0^t\Big(\sum_{i=1}^{n}(\partial_{x_{i}}\tilde{F}^T\tilde{F}+\tilde{F}^T\partial_{x_{i}}\tilde{F})|
  \mathbf{u}_n\Big)d\tau \\
&=\int_0^t\Big(\sum_{i=1}^{n}(\partial_{x_{i}}\tilde{F}^T\tilde{F}+\tilde{F}^T\partial_{x_{i}}\tilde{F})\big|
  \mathbf{u}\Big)d\tau=\int_0^t(\operatorname{div}(\tilde{F}^T\tilde{F})|
  \mathbf{u})d\tau.
\end{aligned}
\end{equation}
Hence, \eqref{eq2.18} still holds under the assumption of
Proposition \ref{pro2.4}.

It is clear that by Lemma \ref{le2.5},
\begin{equation}\label{eq2.40}
  \lim_{n\to\infty}\int_0^t(\tilde{F}\nabla\tilde{\mathbf{u}}|
  F_n)d\tau
  =\int_0^t(\tilde{F}\nabla\tilde{\mathbf{u}}|
  F)d\tau.
\end{equation}
Since $\nabla\tilde{F}_n$ converges to $\nabla\tilde{F}$ in
$L^2(0,T; L^2(\mathbb{R}^3))$, and $\{\tilde{F}_n\}$ is
bounded in he space $L^{\infty}(0,T; L^2(\mathbb{R}^3))$ which was
ensured by the Banach-Steinhaus theorem due to $\tilde{F}_n$
weakly-star converge to $\tilde{F}$ in $L^{\infty}(0,T;
L^2(\mathbb{R}^3))$, by Lemma \ref{le2.5}, we obtain
\begin{equation}\label{eq2.41}
  \lim_{n\to\infty}\int_0^t(F\nabla\mathbf{u}|
  \tilde{F}_n)d\tau
  =\int_0^t(F\nabla\mathbf{u}| \tilde{F})d\tau.
\end{equation}
The two estimates \eqref{eq2.40}--\eqref{eq2.41} imply that the
equality \eqref{eq2.28} still holds under the assumption of
Proposition \ref{pro2.4}.


Now we finish the proof of Proposition \ref{pro2.4}. Using the
similar argument as in the proof of Lemma \ref{le2.5} (see
\cite{GP02}), we obtain 
\begin{equation} \label{eq2.42}
\begin{aligned}
&\Big|\int_0^t\int_{\mathbb{R}^3}\mathbf{w}\cdot\nabla\mathbf{w}
 \cdot\mathbf{u}\,dx\,d\tau\Big|\\
&\leq C\int_0^t\|\mathbf{w}\|_{L^2}^{2/q}\|\nabla
  \mathbf{w}\|_{L^2}^{2-2/q}\|\mathbf{u}\|_{\dot{B}^{-1+3/p+2/q}_{p,q}}d\tau \\
&\leq\frac{1}{2}\int^t_0\|\nabla\mathbf{w}\|_{L^2}^2d\tau
  +C\int_0^t\|\mathbf{w}\|_{L^2}^2
  \|\mathbf{u}\|_{\dot{B}^{-1+3/p+2/q}_{p,q}}^qd\tau;
\end{aligned}
\end{equation}
\begin{equation} \label{eq2.43}
\begin{aligned}
&\Big|\int_0^t\int_{\mathbb{R}^3}E^TE:\nabla\mathbf{u}\,dx\,d\tau\Big|\\
&=\Big|-\int_0^t\int_{\mathbb{R}^3}\operatorname{div}(E^TE)\cdot\mathbf{u}
 \,dx\,d\tau\Big| \\
&\leq C\int_0^t\|E\|_{L^2}^{2/q}\|\nabla
  E\|_{L^2}^{2-2/q}\|\mathbf{u}\|_{\dot{B}^{-1+3/p+2/q}_{p,q}}d\tau \\
&\leq\frac{1}{2}\int^t_0\|\nabla E\|_{L^2}^2d\tau
  +C\int_0^t\|E\|_{L^2}^2\|\mathbf{u}\|_{\dot{B}^{-1+3/p+2/q}_{p,q}}^qd\tau;
\end{aligned}
\end{equation}
\begin{equation} \label{eq2.44}
\begin{aligned}
&\Big|\int_0^t\int_{\mathbb{R}^3}\mathbf{w}\cdot\nabla F:E\,dx\,d\tau\Big|\\
&=\Big|\int_0^t\int_{\mathbb{R}^3}(\mathbf{w}\otimes F)\cdot\nabla
  E\,dx\,d\tau\Big| \\
&\leq\frac{1}{2}\int^t_0\|(\nabla\mathbf{w},\nabla E)\|_{L^2}^2d\tau
  +C\int_0^t\|(\mathbf{w}, E)\|_{L^2}^2\|F\|_{\dot{B}^{-1+3/p+2/q}_{p,q}}^qd\tau
\end{aligned}
\end{equation}
and
\begin{equation} \label{eq2.45}
\begin{aligned}
  \Big|\int_0^t\int_{\mathbb{R}^3}E^TF:\nabla\mathbf{w}\,dx\,d\tau\Big|
 &\leq\frac{1}{2}\int^t_0\|(\nabla\mathbf{w},\nabla E)\|_{L^2}^2d\tau \\
 &\quad +C\int_0^t\|(\mathbf{w},
  E)\|_{L^2}^2\|F\|_{\dot{B}^{-1+3/p+2/q}_{p,q}}^qd\tau.
\end{aligned}
\end{equation}
Returning back to the estimate \eqref{eq2.29} and putting the above
estimates \eqref{eq2.42}--\eqref{eq2.45} and \eqref{eq2.34}
together, we obtain
\begin{equation} \label{eq2.46}
\begin{aligned}
&\|(\mathbf{w}(t),E(t))\|_{L^2}^2+2\int^t_0\|(\nabla\mathbf{w}(\tau), \nabla
  E(\tau))\|_{L^2}^2d\tau \\
&\leq\|(\mathbf{w}_0, E_0)\|_{L^2}^2 +C\int_0^t(\|\mathbf{w}\|_{L^2}^2
  +\|E\|_{L^2}^2)(1+\|\mathbf{u}\|_{\dot{B}^{-1+3/p+2/q}_{p,q}}^q\\
&\quad  +\|F\|_{\dot{B}^{-1+3/p+2/q}_{p,q}}^q)d\tau.
\end{aligned}
\end{equation}
Applying the Gronwall inequality, we obtain \eqref{eq2.36}
immediately. The proof of Proposition \ref{pro2.4} is complete.
\hfill\qed


\section{Appendix}

In this section we shall establish the basic energy inequality (see
Definition \ref{def1.1}) governing the system \eqref{eq1.1}. In
order to do so, let us consider a classical solution $(\mathbf{u},
\mathbf{d})$ of the problem \eqref{eq1.1}. We first multiply the
first equation of \eqref{eq1.1} by $\mathbf{u}$, integrate over
$\mathbb{R}^3$, and use the fact $\nabla\cdot(\nabla \mathbf{d}
\odot\nabla
\mathbf{d})=\nabla(\frac{|\nabla\mathbf{d}|^2}{2})
+\Delta\mathbf{d}\cdot\nabla\mathbf{d}$,
we see that
\begin{equation}\label{eq3.1}
  \frac{1}{2}\frac{d}{dt}\|\mathbf{u}\|_{L^2}^2+\|\nabla\mathbf{u}\|_{L^2}^2
+(\Delta\mathbf{d}\cdot\nabla\mathbf{d},
  \mathbf{u})=0.
\end{equation}
Next, we multiply the second equation of \eqref{eq1.1} by $-\Delta
\mathbf{d}+g(\mathbf{d})$, integrate over $\mathbb{R}^3$, and use
the fact that $(\mathbf{u}\cdot\nabla\mathbf{d},
g(\mathbf{d}))=(\mathbf{u}, \nabla G(\mathbf{d}))=0$, we see that
\begin{equation}\label{eq3.2}
  \frac{1}{2}\frac{d}{dt}\|\nabla\mathbf{d}\|_{L^2}^2
+\frac{d}{dt}\int_{\mathbb{R}^3}G(\mathbf{d})dx+\|\Delta\mathbf{d}
-g(\mathbf{d})\|_{L^2}^2
  -(\mathbf{u}\cdot\nabla\mathbf{d}, \Delta\mathbf{d})=0.
\end{equation}
Equations \eqref{eq3.1} and \eqref{eq3.2} together imply 
\begin{equation}\label{eq3.3}
  \frac{1}{2}\frac{d}{dt}\Big(\|\mathbf{u}\|_{L^2}^2
+\|\nabla\mathbf{d}\|_{L^2}^2+2\int_{\mathbb{R}^3}G(\mathbf{d})dx\Big)
  +\|\nabla\mathbf{u}\|_{L^2}^2+\|\Delta\mathbf{d}-g(\mathbf{d})\|_{L^2}^2=0.
\end{equation}
Note that
$\int_{\mathbb{R}^3}G(\mathbf{d})dx=\frac{1}{4}\|\mathbf{d}(t)\|_{L^4}^4-\frac{1}{2}\|\mathbf{d}(t)\|_{L^2}^2$.
Hence, in order to calculate the term
$\frac{d}{dt}\int_{\mathbb{R}^3}G(\mathbf{d})dx$, we multiply the
second equation of \eqref{eq1.1} by $\mathbf{d}$ to yield that
\begin{equation*}
  \frac{1}{2}\frac{d}{dt}\|\mathbf{d}\|_{L^2}^2+\|\nabla\mathbf{d}\|_{L^2}^2
  +\int_{\mathbb{R}^3}g(\mathbf{d})\cdot \mathbf{d}dx=0;
\end{equation*}
i.e.,
\begin{equation}\label{eq3.4}
  \frac{1}{2}\frac{d}{dt}\|\mathbf{d}\|_{L^2}^2+\|\nabla\mathbf{d}\|_{L^2}^2
  +\|\mathbf{d}\|_{L^4}^4=\|\mathbf{d}\|_{L^2}^2.
\end{equation}
Similarly, multiplying the second equation of \eqref{eq1.1} by
$|\mathbf{d}|^2\mathbf{d}$, we obtain
\begin{equation}\label{eq3.5}
  \frac{1}{4}\frac{d}{dt}\|\mathbf{d}\|_{L^{4}}^{4}+3\|\mathbf{d}\cdot\nabla\mathbf{d}\|_{L^2}^2
  +\|\mathbf{d}\|_{L^6}^6=\|\mathbf{d}\|_{L^4}^{4}.
\end{equation}
On the other hand, it is obvious that
\begin{align*}
&\|\Delta\mathbf{d}-g(\mathbf{d})\|_{L^2}^2=(\Delta\mathbf{d}-|\mathbf{d}|^2\mathbf{d}+\mathbf{d},
  \Delta\mathbf{d}-|\mathbf{d}|^2\mathbf{d}+\mathbf{d})\\
&=\|\Delta\mathbf{d}\|_{L^2}^2-2(\Delta\mathbf{d},
  |\mathbf{d}|^2\mathbf{d})+2(\Delta\mathbf{d},
  \mathbf{d})-2(|\mathbf{d}|^2\mathbf{d},
  \mathbf{d})+(|\mathbf{d}|^2\mathbf{d},|\mathbf{d}|^2\mathbf{d})+(\mathbf{d},\mathbf{d})\\
&=\|\Delta\mathbf{d}\|_{L^2}^2+6\||\mathbf{d}|\nabla\mathbf{d}\|_{L^2}^2-2\|\nabla\mathbf{d}\|_{L^2}^2-2\|\mathbf{d}\|_{L^4}^4
  +\|\mathbf{d}\|_{L^6}^6+\|\mathbf{d}\|_{L^2}^2.
\end{align*}
Putting the estimates \eqref{eq3.3}--\eqref{eq3.5} together, we
obtain
\begin{equation}\label{eq3.6}
  \frac{1}{2}\frac{d}{dt}\Big(\|\mathbf{u}\|_{L^2}^2+\|\nabla\mathbf{d}\|_{L^2}^2\Big)
  +\|\nabla\mathbf{u}\|_{L^2}^2+\|\Delta\mathbf{d}\|_{L^2}^2+3\||\mathbf{d}|\nabla\mathbf{d}\|_{L^2}^2
  =\|\nabla\mathbf{d}\|_{L^2}^2.
\end{equation}
This yields immediately the energy inequality in Definition
\ref{def1.1}. Finally, by applying the Gronwall inequality, we obtain
the following basic energy inequality:
\begin{equation} \label{eq3.7}
\begin{aligned}
&\|\mathbf{u}(t)\|_{L^2}^2+\|\nabla\mathbf{d}(t)\|_{L^2}^2
  +\int_0^t\big(\|\nabla\mathbf{u}(\tau)\|_{L^2}^2+\|\Delta\mathbf{d}(\tau)\|_{L^2}^2\big)d\tau \\
&\leq C(\|\mathbf{u}_0\|_{L^2}^2,
  \|\nabla\mathbf{d}_0\|_{L^2}^2)e^{2t}.
\end{aligned}
\end{equation}
Combining the above energy estimate,  the Galerkin approximate
procedure and the compactness argument give global existence of weak
solutions of \eqref{eq1.1}.

\subsection*{Acknowledgments}
This research was supported by the National Natural Science Foundation of
China (11171357) and by the Doctoral Fund of Northwest A\&F University
(Z109021118).


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\end{document}