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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 128, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/128\hfil Blow-up criteria of smooth solutions]
{Blow-up criteria of smooth solutions to a 3D model of electro-kinetic fluids
 in a bounded domain}

\author[M. Chen, Q. Liu \hfil EJDE-2016/128\hfilneg]
{Miaochao Chen, Qilin Liu}

\address{Miaochao Chen (corresponding author) \newline
School of Applied Mathematics, Chaohu University,
Hefei 238000,  China}
\email{chenmiaochao@chu.edu.cn}

\address{Qilin Liu \newline
Department of Mathematics,
Southeast University, Nanjing 211189, China}
\email{Liuqlseu@126.com}

\thanks{Submitted August 8, 2015. Published May 19, 2016.}
\subjclass[2010]{35Q30, 76D03, 76D05, 76D07}
\keywords{Euler system; regularity criterion; bounded domain; bmo}

\begin{abstract}
 We prove that a smooth solution of a 3D model for electro-kinetic fluids
 in a bounded domain breaks down blows up at  the same time as  certain norm
 of vorticity. This norm is weaker than bmo-norm.
\end{abstract}

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

\section{Introduction}

Let $\Omega\subset\mathbb{R}^3$ be a bounded, simply connected domain 
with smooth boundary $\partial\Omega$, and $\nu$ is the unit outward normal 
vector to $\partial\Omega$. We consider the following model of 
electro-hydrodynamics in $\Omega\times(0,\infty)$ \cite{B-S2004, Pro1994}:
\begin{gather}
\partial_t u+(u\cdot\nabla)u+\nabla \pi=\Delta \phi\nabla\phi,\label{11} \\
\operatorname{div} u= 0, \label{12} \\
\partial_t n+u\cdot\nabla n= \nabla\cdot(\nabla n-n\nabla\phi),\label{13} \\
\partial_t p+u\cdot\nabla p= \nabla\cdot(\nabla p+p\nabla\phi),\label{14} \\
-\Delta \phi=p-n,\quad \int_\Omega \phi dx=0,\label{15} \\
u\cdot\nu=0,\quad\frac{\partial n}{\partial\nu}
=\frac{\partial p}{\partial\nu}=\frac{\partial \phi}{\partial\nu}=0\quad \text{on }
\partial\Omega\times(0,\infty),\label{16} \\
(u,n,p)(x,0)=(u_0,n_0,p_0)(x),\quad x\in\Omega\subset\mathbb{R}^3.\label{17}
\end{gather}
The unknowns $u$, $\pi$, $\phi$, $n$ and $p$ denote the velocity, pressure, 
electric potential, anion concentration and cation concentration, respectively.

Equations \eqref{13}--\eqref{15} are known as the electro-chemical equations 
\cite{C-B2005} or semiconductor equations \cite{K-O2008,B-H-N1994,D-G-J2001}, 
and electro-rheological systems \cite{Pro1994, T-C2002} when formally setting $u=0$.

Equations \eqref{11} and \eqref{12} are the Euler equations with the Lorentz 
force $(n-p)\nabla\phi=\Delta\phi\nabla\phi$. 
Ogawa-Taniuchi \cite{O-T2003} proved that a smooth solution breaks down 
if a certain norm of vorticity blows up at the same time. Here this norm 
is weaker than bmo-norm.
Zhang and Yin \cite{Z-Y2015} proved the global well-posedness of 
 problem \eqref{11}--\eqref{17} when $\Omega:=\mathbb{R}^2$.

Before presenting our results, we introduce some function spaces, 
and notation.

Let $\eta$, $\phi_j$, $j=0,\pm 1,\pm2, \pm3, \dots$ be the Littlewood-Paley 
dyadic decomposition of unity that satisfies 
\begin{gather*}
\eta\in C_0^\infty(B(0,1)),\quad
 \phi\in C_0^\infty (B(0,2)\setminus B(0,\frac{1}{2})),\\
 \phi_j(\xi)=\phi(2^{-j}\xi),\quad
\eta(\xi)+\sum_{j=0}^\infty \phi_j(\xi)=1
\end{gather*}
for all $\xi\in\mathbb{R}^3$, where $B(x,r)$ denotes the ball centered at 
$x$ of radius $r$.
We first recall the space of Besov type introduced by Vishik \cite{Vis2000}.

\begin{definition}[\cite{Vis2000}] \label{def11} \rm
Let $\Theta(\alpha)(\geq 1)$ be a nondecreasing function on $[1,\infty)$. 
$V_{\Theta}:=\{ f\in\mathscr{S}':\|f\|_{V_{\Theta}}<\infty \}$ 
with the norm
$$ 
\|f\|_{V_{\Theta}}:=\sup_{N=1,2,\dots}
\frac{\|(n\hat f)^\vee\|_{L^\infty}+\sum_{j=0}^N\|
(\phi_j\hat f)^\vee\|_{L^\infty}}{\Theta(N)},
$$ 
where $\hat f$ and $\check{f}$ denote the Fourier and inverse 
Fourier transforms.
\end{definition}

We note that
$$
\|f\|_{V_{\Theta}}\leq C \|f\|_{B^o_{\infty,\infty}}
\leq C\|f\|_{bmo} \leq C \|f\|_{L^\infty}, \quad \text{if } \Theta(N)\geq N.
$$
Now let us introduce the space of bmo type used in \cite{O-T2003}.

\begin{definition}\label{def12} \rm
Let $\beta(r)$ be a positive function on $(0,1]$ and $\Omega\subset \mathbb{R}^3$ 
be a domain with $\partial\Omega\in C^\infty$.

(1) $bmo_{\beta}(\mathbb{R}^3)$ is defined as the set of functions $f$
in $L^1_{\rm loc}(\mathbb{R}^3)$  such that
\begin{align*}
\|f\|_{bmo_\beta (\mathbb{R}^3)}
&:=\sup_{0<r<1,x\in\mathbb{R}^3}\frac{1}{|B(x,r)|\beta(r)}
 \int_{B(x,r)}|f(y)-\bar f_{B(x,r)}|dy\\
&\quad +\sup_{x\in\mathbb{R}^3}\frac{1}{|B(x,1)|}\int_{B(x,1)}|f(y)|dy\leq \infty,
\end{align*}
where $\bar f_B:=\frac{1}{|B|}\int_B f(y)dy$.

(2) On $\Omega\subset\mathbb{R}^3$ we define $bmo_\beta$ as restrictions
 of the above space $bmo_\beta(\mathbb{R}^3)$:
$$ 
bmo_\beta(\Omega):= \{ f|_\Omega;\,f\in bmo_\beta(\mathbb{R}^3)\}, 
$$
where $f|_\Omega$ is the restriction of $f$ on $\Omega$. The norm of this space 
is defined by
$$ 
\|f\|_{bmo_\beta(\Omega)}:=\inf \big\{ \|\tilde f\|_{bmo_\beta (\mathbb{R}^3)};
\tilde f\in bmo_\beta (\mathbb{R}^3) \text{ with }\tilde f=f\text{ in }\Omega 
\big\}. 
$$
\end{definition}

In particular if $\beta(r)=1$, we write 
$bmo_\beta (\mathbb{R}^3)=bmo(\mathbb{R}^3)$ and 
$bmo_\beta (\Omega)=bmo(\Omega)$. Obviously, $bmo\subset bmo_\beta$ 
if $\beta\geq 1$.

\begin{definition}\label{def13} \rm
Let $\Theta(\alpha)(\geq 1)$ be a nondecreasing function on $[1,\infty)$.
$$
Y_{\Theta}(\Omega):=\{f\in L^1(\Omega):\|f\|_{Y_{\Theta}(\Omega)}<\infty\},
$$ 
where
$$
\|f\|_{Y_{\Theta}(\Omega)}:=\sup_{p\geq 1}\frac{\|f\|_{L^p}}{\Theta(p)}.
$$
$$
M_{\Theta}(\Omega):=\{f\in L^1(\Omega): \|f\|_{M_{\Theta}(\Omega)}<\infty\},
$$ 
where
$$\|f\|_{M_{\Theta}(\Omega)}:=\sup_{p\geq 1}\frac{1}{\Theta(p)} 
\sup_{0<r<1,x\in\mathbb{R}^3}
\Big( r^{-3+\frac{3}{p}}\int_{B(x,r)\cap\Omega}|f(y)|dy \Big).
$$
\end{definition}

We note that these spaces have the following relations.
\begin{equation}\label{18}
\|f\|_{M_{\Theta}(\Omega)}
\leq C \|f\|_{Y_{\Theta}(\Omega)}
\leq C \|f\|_{bmo(\Omega)}.
\end{equation}

Let 
\[
\beta(r):=\frac{\Theta(\log(e+\frac{1}{r}))}{\log(e+\frac{1}{r})}.
\]
In this article we use  the following assumptions:
\begin{itemize}
\item[(H1)] $\Theta(\alpha)$ is a positive and nondecreasing function on
 $[1,\infty)$ satisfying
\begin{equation}\label{19}
\int^{+\infty} \frac{d\alpha}{\Theta(\alpha)}=\infty,\quad
\Theta(\alpha)\geq\alpha.
\end{equation}

\item[(H2)] For all $s\geq 1$ there exists $C(s)$ such that
$$
\Theta(s\alpha)\leq C(s)\Theta(\alpha)\quad \text{for all } \alpha\geq 1.
$$


\item[(H3)] $\beta(r)$ is a non-increasing function on $(0,1]$.
\end{itemize}

Ogawa-Taniuchi \cite{O-T2003} proved the following blowup criterion
\begin{equation}\label{110}
\int_0^T \|\omega(t)\|_{bmo_\beta(\Omega)} 
+ \|\omega(t)\|_{M_{\Theta}(\Omega_\epsilon)}dt=\infty,
\end{equation}
where $\omega:=\operatorname{curl} u$ and for all $\epsilon>0$ and 
$\Omega_\epsilon:=\{ x\in\Omega;
\operatorname{dist}(x,\partial\Omega)<\epsilon \}$ or
\begin{equation}\label{111}
\int_0^T \|\omega(t)\|_{bmo_\beta(\Omega_{3\epsilon})} 
+ \|\omega(t)\|_{M_{\Theta}(\Omega_{3\epsilon})}
+\|\rho\omega(t)\|_{V_{\Theta}}dt=\infty,
\end{equation}
for all $0<\epsilon<\epsilon_0$ and all $\rho\in C^\infty(\mathbb{R}^3)$ with 
$\rho\equiv 1$ in $\Omega\setminus\Omega_\epsilon$ and $\rho\equiv 0$ in 
$\mathbb{R}^3\setminus\Omega$. $\epsilon_0$ is a small positive constant 
depending only on $\Omega$.

Since $\beta(r)\geq 1$, we have
 $$
\|f\|_{bmo_\beta(\Omega)}\leq \|f\|_{bmo(\Omega)}.
$$ 
By this inequality and \eqref{18}, \eqref{110} implies
\begin{equation}\label{112}
\int_0^T \|\omega(t)\|_{bmo(\Omega)}dt=\infty.
\end{equation}

The aim of this article is to prove a similar result for problem \eqref{11}--\eqref{17}.
 It is easy to show that  \eqref{11}--\eqref{17} has a unique local smooth 
solution with $u_0\in H^3$ and $(n_0,p_0)\in H^2$. Thus we omit the details here.
 However, the global regularity is still open, which this paper aims to study.
 We will prove the following result.

\begin{theorem}\label{thm11}
Let $u_0\in H^3, (n_0,p_0)\in H^2$, $n_0,p_0\geq 0, \operatorname{div} u_0=0$ in
$\Omega$, 
$u_0\cdot\nu=\frac{\partial n_0}{\partial\nu}=\frac{\partial p_0}{\partial\nu}$ on 
$\partial\Omega$ and $\int_\Omega n_0 dx=\int_\Omega p_0dx$. Suppose that 
$(u,n,p)$ is a local smooth solution to \eqref{11}--\eqref{17} on $[0,T)$.
 If $T$ is maximal, then \eqref{110} and \eqref{111} hold.
\end{theorem}

In Section 2, we will give some preliminaries. 
Section 3 is devoted to the proof of Theorem \ref{thm11}.

\section{Preliminaries}

\begin{lemma}[\cite{B-B1974}] \label{lem21}
For any $u\in W^{s,p}$ with $\operatorname{div} u=0$ in $\Omega$ and $u\cdot\nu=0$ on $\partial\Omega$, there holds
$$ \|u\|_{W^{s,p}}\leq C (\|u\|_{L^p}+\|\operatorname{curl} u\|_{W^{s-1,p}}) $$
for any $s\geq 1$ and $p\in(1,\infty).$
\end{lemma}

\begin{lemma}[\cite{Fer1993}] \label{lem22}
Let $s\geq 1$.

(1) If $f,g\in H^s(\Omega)\cap C(\Omega)$, then
$$
\|fg\|_{H^s(\Omega)}\leq C\left( \|f\|_{H^s(\Omega)}
\|g\|_{L^\infty(\Omega)}+\|f\|_{L^\infty(\Omega)} \|g\|_{H^s(\Omega)} \right).
$$

(2) If $f\in H^s(\Omega)\cap C^1(\Omega)$ and $g\in H^{s-1}(\Omega)\cap C(\Omega)$, 
then for $|\alpha|\leq s$,
$$
\|D^\alpha(fg)-fD^\alpha g\|_{L^2(\Omega)}
\leq C \left( \|f\|_{H^s(\Omega)}\|g\|_{L^\infty(\Omega)}
+\|f\|_{W^{1,\infty}(\Omega)} \|g\|_{H^{s-1}(\Omega)} \right).
$$
\end{lemma}

\begin{lemma}[\cite{O-T2003}] \label{lem23}
For all $\epsilon>0$, we have
\begin{align*}
&\|\nabla u\|_{L^\infty(\Omega)} \\
&\leq  C \left( 1+\|u\|_{L^2(\Omega)} 
+ \|\operatorname{curl}u\|_{bmo_\beta(\Omega)} 
+ \|\operatorname{curl}u\|_{M_{\Theta}(\Omega\epsilon)} \right)\\
 \Theta (\log(e+\|u\|_{H^3(\Omega)}))
\end{align*}
for all $u\in H^3(\Omega)$ with $\operatorname{div} u=0$ in $\Omega$ and
$u\cdot\nu=0$ on $\partial \Omega$.
\end{lemma}

\begin{lemma}[\cite{O-T2003}] \label{lem24}
There exists a constant $\epsilon_0$ depending only on $\Omega$ such 
that:
For all $0<\epsilon<\epsilon_0$ and for all
 $\rho\in C^\infty(\mathbb{R}^3)$ with $\rho\equiv 1$ in 
$\Omega\setminus\Omega_\epsilon$ and $\rho\equiv 0$ in
 $\mathbb{R}^3\setminus\Omega$ there exists constant $C$ depending only on 
$\epsilon,\rho,\Omega$ and $\Theta$ such that
\begin{align*}
\|\nabla u\|_{L^\infty(\Omega)} 
&\leq  C \Big( 1+\|u\|_{L^2(\Omega)} 
+ \|\operatorname{curl}u\|_{bmo_\beta(\Omega_{3\epsilon})} 
+ \|\operatorname{curl}u\|_{M_{\Theta}(\Omega_{3\epsilon})} \\
&\quad+\|\rho\operatorname{curl} u\|_{V_{\Theta}} \Big)
\Theta (\log(e+\|u\|_{H^3(\Omega)}))
\end{align*}
for all $u\in H^3(\Omega)$ with $\operatorname{div} u=0$ in $\Omega$ and 
$u\cdot\nu=0$ on $\partial\Omega$.
\end{lemma}

\begin{lemma}[\cite{Har1982}] \label{lem25}
Let $\psi$ be nonnegative function on $(0,T)$ with $\int_0^T\psi(t)dt<\infty$, 
let $\Theta(\alpha)$ be a positive and  nondecreasing for $\alpha\geq 1$ 
and $\int^{+\infty}\frac{d\alpha}{\Theta(\alpha)}=\infty$.
 Assume that $v\in C([0,T))$ and 
$$
0\leq v(t)\leq v(0)+\int_0^t\psi(s)\Theta(v(s))ds\quad \text{for all }
 0\leq t<T.
$$ 
Then $\sup_{0\leq t\leq T}v(t)<\infty$.
\end{lemma}

\section{Proof of Theorem \ref{thm11}}

Since the proof of \eqref{111} is similar to that of \eqref{110}, we only 
need to prove \eqref{110}. By the standard argument of continuation of 
local solutions, it suffices to prove that if
\begin{equation}\label{31}
\int_0^T \|\omega(t)\|_{bmo_\beta(\Omega)}
+\|\omega(t)\|_{M_{\Theta}(\Omega\epsilon)}dt<\infty\quad\text{for some }
 \epsilon>0,
\end{equation}
then
\begin{equation}\label{32}
u\in L^\infty(0,T;H^3),\quad (n,p)\in L^\infty (0,T;H^2)\cap L^2(0,T;H^3).
\end{equation}

First, by the maximum principle, it is easy to prove that $n,p\geq 0$ 
in $\Omega\times (0,\infty)$.

Testing \eqref{13} by $n$ and testing \eqref{14} by $p$, using 
\eqref{15}, \eqref{12} and summing up the resulting inequality, we easily get
$$
\frac{1}{2}\int n^2+p^2 dx +\int_0^T\int |\nabla n|^2 +|\nabla p|^2 
+\frac{1}{2}(p-n)^2(n+p)dxdt\leq \frac{1}{2}\int u_0^2+p_0^2dx,
$$
whence
\begin{equation}\label{33}
\|(n,p)\|_{L^\infty(0,T;L^2)} + \|(n,p)\|_{L^2(0,T;H^1)}\leq C.
\end{equation}

Testing \eqref{13} by $n^{k-1}$ and testing \eqref{14} by $p^{k-1}$, 
using \eqref{12}, \eqref{15} and $n,p\geq 0$, we find that
$$
\int n^k+p^k dx\leq \int n_0^k+p_0^k \leq \int (n_0+p_0)^kdx,
$$
which gives
$$
\|n\|_{L^k}\leq \|n_0+p_0\|_{L^k},\quad
\|p\|_{L^k}\leq \|n_0+p_0\|_{L^k}.
$$
Taking $k\to \infty$, we obtain
\begin{equation}\label{34}
\|(n,p)\|_{L^\infty(0,T;L^\infty)} \leq C.
\end{equation}
Testing \eqref{11} by $u$, using \eqref{12}-\eqref{15}, we infer that
\begin{equation}\label{35}
\frac{1}{2}\frac{d}{dt} \int u^2 +|\nabla \phi|^2dx
+\int |\Delta\phi|^2+(n+p)|\nabla \phi|^2dx=0,
\end{equation}
which leads to
\begin{equation}\label{36}
\|u\|_{L^\infty(0,T;L^2)}\leq C.
\end{equation}

It follows from \eqref{35}, \eqref{34}, \eqref{33} and \eqref{15} that
\begin{equation}\label{37}
\nabla\phi\in L^\infty(0,T; H^1\cap L^\infty)\cap L^2(0,T;H^2).
\end{equation}

Testing \eqref{13} by $-\Delta n$, using \eqref{12}, \eqref{15}, \eqref{16}, 
\eqref{34} and \eqref{37}, we have
\begin{align*}
&\frac{1}{2}\frac{d}{dt}\int |\nabla n|^2dx +\int |\Delta n|^2dx\\
&= \int (u\cdot \nabla)n\cdot\Delta n dx + \int (n\Delta \phi 
 +\nabla n\cdot\nabla\phi)\Delta n dx\\
&= \sum_{i,j}\int u_i\partial_i n \partial_j^2 n dx+\int (n\Delta\phi 
 +\nabla n\cdot\nabla \phi)\Delta ndx\\
&= -\sum_{i,j}\int \partial_j u_i\partial_i n \partial_j n dx
 +\int (n(n-p) +\nabla n\cdot\nabla \phi)\Delta ndx\\
&\leq  C \|\nabla u\|_{L^\infty}\|\nabla n\|_{L^2}^2+C\|\Delta n\|_{L^2}
 +C \|\nabla n\|_{L^2} \|\nabla \phi\|_{L^\infty} \|\Delta n\|_{L^2}\\
&\leq  \frac{1}{2}\|\Delta n\|_{L^2}^2 
 + C \|\nabla u\|_{L^\infty}\|\nabla n\|_{L^2}^2 +C +C \|\nabla n\|_{L^2}^2,
\end{align*}
which implies
\begin{equation}\label{38}
\frac{d}{dt}\int |\nabla n|^2dx +\int |\Delta n|^2dx 
\leq C+C\|\nabla n\|_{L^2}^2+C \|\nabla u\|_{L^\infty}\|\nabla n\|_{L^2}^2.
\end{equation}
Similarly for the $p$-equation, we have
\begin{equation}\label{39}
\frac{d}{dt}\int |\nabla p|^2dx +\int |\Delta p|^2dx 
\leq C+C\|\nabla p\|_{L^2}^2+C \|\nabla u\|_{L^\infty}\|\nabla p\|_{L^2}^2.
\end{equation}

Equations \eqref{13} and \eqref{16} can be rewritten as
\begin{gather*}
\Delta n=f:=\partial_t n+u\cdot\nabla n +\nabla\cdot(n\nabla\phi), 
\quad \text{in }\Omega\times(0,\infty)\\
\frac{\partial n}{\partial\nu}=0,\quad \text{on } \partial \Omega\times(0,\infty).
\end{gather*}
By the classical regularity theory of elliptic equation, using \eqref{36}, 
\eqref{34} and \eqref{37}, we deduce that
\begin{equation} \label{310}
\begin{aligned}
\|n\|_{H^3}& \leq  C\|f\|_{H^1} \\
&\leq  C \|\partial_t n\|_{H^1} + C\|u\cdot\nabla n\|_{H^1}
 + C\|\nabla\cdot(n\nabla\phi)\|_{H^1} \\
&\leq  C \|\partial_t n\|_{H^1} + C \|u\|_{L^2}\|\nabla n\|_{L^\infty}
 + C \|u\|_{L^6} \|\Delta n\|_{L^3} \\
&\quad +C\|\nabla u\|_{L^\infty}\|\nabla n\|_{L^2}+C \|n\Delta\phi\|_{L^2}
 +  C \|\nabla n\cdot\nabla \phi\|_{L^2} \\
&\quad +C\|n\|_{L^\infty}\|\nabla\Delta\phi\|_{L^2}
 +C \|\nabla n\|_{L^\infty} \|\nabla^2 \phi\|_{L^2}
 + C \|\nabla\phi\|_{L^6}\|\Delta n\|_{L^3} \\
&\leq   C \|\partial_t n\|_{H^1} + C \|\nabla n\|_{L^\infty}
 + C \|u\|_{L^6}\|\Delta n\|_{L^3} \\
&\quad +C\|\nabla u\|_{L^\infty}\|\nabla n\|_{L^2} + C + C\|\nabla n\|_{L^2} \\
&\quad +C\|\nabla(n-p)\|_{L^2}+C\| \Delta n \|_{L^3}.
\end{aligned}
\end{equation}

Now we use the following Gagliardo-Nirenberg inequalities:
\begin{gather}
\|\nabla n\|_{L^\infty}  
 \leq   C \|n\|^{1/3}_{L^\infty} \|n\|^{2/3}_{H^3},\label{311} \\
\|\nabla n\|_{L^3}  \leq   C \|n\|^{1/3}_{L^\infty} \|n\|^{2/3}_{H^3},\label{312} \\
\|u\|_{L^6}^3  \leq   C \|u\|^{2}_{L^2} \|u\|_{H^3}. \label{313}
\end{gather}
It follows from \eqref{310}, \eqref{311}, \eqref{312}, \eqref{313}, \eqref{36}, 
\eqref{34} and the Young inequality that
\begin{equation}\label{314}
\begin{aligned}
\|n\|_{H^3}
&\leq  C \|\partial_t n\|_{H^1}+C+C\|u\|_{H^3}
 +C\|\nabla u\|_{L^\infty} \|\nabla n\|_{L^2} \\
&\quad+C\|\nabla n\|_{L^2} + C\|\nabla p\|_{L^2}.
\end{aligned}
\end{equation}
Similarly to the $p$- equation, we have
\begin{equation}\label{315}
\begin{aligned}
\|p\|_{H^3}&\leq  C \|\partial_t p\|_{H^1}+C+C\|u\|_{H^3}
+C\|\nabla u\|_{L^\infty} \|\nabla p\|_{L^2} \\
&\quad +C\|\nabla n\|_{L^2} + C\|\nabla p\|_{L^2}.
\end{aligned}
\end{equation}

Applying the curl to \eqref{11}, using \eqref{12}, we obtain
\begin{equation}\label{316}
\partial_t \omega + u\cdot \nabla \omega = \omega\cdot\nabla u 
+ \operatorname{curl}(\Delta\phi \nabla \phi).
\end{equation}

Applying $\Delta$ to \eqref{316}, testing by $\Delta \omega$, using \eqref{12}, 
we find that
\begin{equation}\label{317}
\begin{aligned}
\frac{1}{2}\frac{d}{dt} \int |\Delta \omega|^2dx 
&= -\int (\Delta (u\cdot\nabla\omega)-u\nabla\Delta\omega)\Delta \omega dx \\
&\quad +\int \Delta(\omega\cdot\nabla u)\cdot \Delta\omega dx 
 +\int \Delta \operatorname{curl} (\Delta\phi \nabla \phi)\cdot\Delta\omega dx \\
&\leq  \big(  \| \Delta (u\cdot\nabla\omega)-u\nabla\Delta\omega \|_{L^2} 
 +\|\Delta(\omega\cdot\nabla u)\|_{L^2} \\
&\quad + \|\Delta \operatorname{curl} (\Delta\phi \nabla \phi)  \|_{L^2} \big)
 \|\Delta \omega\|_{L^2} \\
 &=: (I_1+I_2+I_3)\|\Delta \omega\|_{L^2}.
\end{aligned}
\end{equation}
Using \eqref{12} and Lemma \ref{lem22}, $I_1$ and $I_2$ can be bounded as follows.
\begin{align*}
I_1&= \sum_i \| \Delta \partial_i (u_i\omega)-u_i\partial_i\Delta\omega \|_{L^2}\\
&\leq C \| \nabla u \|_{L^\infty} \|\Delta \omega\|_{L^2} + C \| \omega \|_{L^\infty} \|\nabla^3 u\|_{L^2}\\
&\leq C \| \nabla u \|_{L^\infty} \|u\|_{H^3},
\end{align*}
\[
I_2\leq  C \| \omega \|_{L^\infty} \|u\|_{H^3} 
+  C \| \nabla u \|_{L^\infty} \|\omega\|_{H^2}
\leq  C \| \nabla u \|_{L^\infty} \|u\|_{H^3}.
\]
Noting that
$$ 
\Delta\phi\cdot\nabla\phi =\sum_{i,j}\partial_j(\partial_j\phi \partial_i \phi)
-\frac{1}{2} \sum_{i,j} \partial_i(\partial_j\phi)^2,
$$
using Lemma \ref{lem22} and \eqref{37}, we have
$$ 
I_3\leq C \| \nabla\phi \|_{L^\infty} \|\nabla\phi\|_{H^4}
\leq C \|\nabla\phi\|_{H^4} \leq C \|\phi\|_{H^5}\leq C \|n-p\|_{H^3}.
$$
Inserting the above estimates into \eqref{317}, we obtain
\begin{equation}\label{318}
\frac{1}{2} \frac{d}{dt} \int |\Delta \omega|^2 dx 
\leq C (\|\nabla u\|_{L^\infty} \|u\|_{H^3} + \|n-p\|_{H^3}  )
 \|\Delta \omega\|_{L^2}.
\end{equation}
Testing \eqref{11} by $\partial_t u$, using \eqref{12}, \eqref{36}, \eqref{37} 
and \eqref{313}, we infer that
\begin{equation}\label{319}
\begin{aligned}
\|\partial_t u\|_{L^2} 
&\leq  \| \Delta\phi \nabla\phi \|_{L^2} + \| u\cdot  \nabla u \|_{L^2} \\
&\leq  \| \nabla \phi \|_{L^\infty}  \| \Delta \phi \|_{L^2} 
 +  \| u \|_{L^6} \| \nabla u \|_{L^3} \\
&\leq  C + C   \| u \|_{L^2}^{2/3}  \| u \|_{H^3}^{1/3}  
 \| u \|_{L^2}^{1/2}  \| u \|_{H^3}^{1/2} \\
&\leq  C+ C \| u \|_{H^3}^{5/6}.
\end{aligned}
\end{equation}
Here we have used the Gagliardo-Nirenberg inequality
$$ 
\| \nabla u \|^2_{L^3} \leq C \| u \|_{L^2} \| u \|_{H^3}. 
$$
Applying $\partial_t$ to \eqref{13}, we see that
$$ 
\partial_t^2 n + u\cdot \nabla \partial_t n -\Delta \partial_t n 
= -\partial_t u\cdot \nabla n - \nabla\cdot \partial_t (n\nabla\phi). 
$$
Testing the above equation by $\partial_t n$, using \eqref{12}, \eqref{16},
 \eqref{34}, \eqref{37}, \eqref{319} and \eqref{15}, we derive
\begin{align*}
&\frac{1}{2}\frac{d}{dt} \int (\partial_t n)^2dx 
 + \int |\nabla \partial_t n|^2dx\\
&= -\int (\partial_t u\cdot \nabla) n\cdot \partial_t n dx 
 + \int \partial_t (n\nabla\phi)\cdot \nabla \partial_t n dx\\
&= \int \partial_t u\cdot n \nabla \partial_t n dx 
 + \int \partial_t (n\nabla\phi)\cdot\partial_t n dx\\
&\leq  \left(  \| n \|_{L^\infty}  \| \partial_t u\|_{L^2}
  +  \| \nabla\phi \|_{L^\infty}  \| \partial_t n\|_{L^2} 
 +  \| n \|_{L^\infty}  \| \nabla \partial_t \phi\|_{L^2} \right) \| \nabla \partial_t n \|_{L^2}\\
&\leq  C ( \|\partial_t u\|_{L^2} + \|\partial_t n\|_{L^2} 
 + \|\partial_t (n-p)\|_{L^2} ) \|\nabla \partial_t n\|_{L^2}\\
&\leq  \frac{1}{2} \| \nabla \partial_t n \|^2_{L^2} + C + C \|u\|^2_{H^3} 
 + C\| \partial_t n\|^2_{L^2} +C\| \partial_t p\|^2_{L^2},
\end{align*}
whence
\begin{equation}\label{320}
\frac{d}{dt} \int |\partial_t n|^2dx + \int |\nabla \partial_t n|^2dx
\leq C + C \|u\|^2_{H^3} + C \|\partial_t (n,p)\|_{L^2}^2.
\end{equation}
Similarly, for the $p$-equation, we have
\begin{equation}\label{321}
\frac{d}{dt} \int (\partial_t p)^2dx + \int |\nabla \partial_t p|^2dx
\leq C + C \|u\|^2_{H^3} +  C \|\partial_t (n,p)\|_{L^2}^2.
\end{equation}
Combining \eqref{38}, \eqref{39}, \eqref{314}, \eqref{315}, \eqref{318}, \eqref{320} 
and \eqref{321}, using \eqref{36}, Lemma \ref{lem21}, Lemma \ref{lem23},
 and Lemma \ref{lem25}, we conclude that \eqref{32} holds. 
This completes the proof. 


\subsection*{Acknowledgements}
The author is indebted to the referees for their valuable suggestions.
This work is supported by the Natural Science Foundation of Chaohu University
(No. XLY-201503), the University Natural Science Foundation of Anhui
(No. KJ2015A270).


\begin{thebibliography}{99}

\bibitem{B-S2004} M. Bazant, T. Squires;
\emph{Induced-charge electro-kinetic phenomena, theory and microfluidic applications}. 
Phys. Rev. Lett. 92 (2004), 066101.

\bibitem{Pro1994} R. E. Problestein;
\emph{Physicochemical Hydrodynamics, An introduction}. 
John Wiley and Sons, INC., 1994.

\bibitem{C-B2005} K. Chu, M. Bazant;
\emph{Electrochemical thin films at and above the classical limiting current}. 
SIAM J. Appl. Math. 65 (2005), 1485-1505.

\bibitem{K-O2008} M. Kurokiba, T. Ogawa;
\emph{$L^p$ well-posedness for the drift-diffusion system arising from the 
semiconductor device simulation}. J. Math. Anal. Appl. 342 (2008), 1052-1067.

\bibitem{B-H-N1994} P. Biler, W. Hebisch, T. Nadzieja;
\emph{The Debye system: existence and large time behavior of solutions}. 
Nonlinear Anal. TMA 23 (1994), 1189-1209.

\bibitem{D-G-J2001} J. I. D\'{i}az, G. Galiano, A. J\"{u}ngel;
\emph{On a quasilinear degenerate system arising in semiconductor theory. 
Part I: Existence and uniqueness of solutions}. Nonlinear Analysis: 
Real World Applications 2(2001), 305-336.

\bibitem{T-C2002} S. Thamida, H. C. Chang;
\emph{Nonlinear electro-kinetic ejection and entrainment due to polarization 
at nearly insulated wedges}. Phys. Fluids 14 (2002), 4315-4328.

\bibitem{O-T2003} T. Ogawa, Y. Taniuchi;
\emph{On blow-up criteria of smooth solutions to the 3D Euler equations 
in a bounded domain}. J. Differential Equations 190 (2003), 39-63.

\bibitem{Z-Y2015} Z. Zhang, Z. Yin;
\emph{Global well-posedness for the Euler-Nernst-Planck-Poisson system 
in dimension two}, Nonlinear Anal. 125(2015), 30-53.

\bibitem{Vis2000} M. Vishik;
\emph{Incompressible flows of an ideal fluid with unbounded vorticity}. 
Comm. Math. Phys. 213 (2000), 697-731.

\bibitem{B-B1974} J. P. Bourguignon, H. Brezis;
\emph{Remarks on the Euler equation}. J. Funct. Anal. 15 (1974), 341-363.

\bibitem{Fer1993} A. B. Ferrari;
\emph{On the blow-up of solutions of 3-D Euler equations in a bounded domain}. 
Comm. Math. Phys. 155 (1993), 277-294.

\bibitem{Har1982} Ph. Hartman;
\emph{Ordinary Differential Equations}, 2nd Edition. Birkh\"auser, 
Boston, Basel, Stuttgart, 1982.

\end{thebibliography}

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