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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 169, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/169\hfil 
incompressible Navier-Stokes equation]
{Vanishing viscosity limit for the 3D nonhomogeneous incompressible
Navier-Stokes equation with special slip boundary condition}

\author[P. Chen, Y. Xiao, H. Zhang \hfil EJDE-2017/169\hfilneg]
{Pengfei Chen, Yuelong Xiao, Hui Zhang}

\address{Pengfei Chen \newline
School of Mathematical Sciences,
Xiangtan University,
Hunan 411105, China}
\email{cpfxtu@163.com}

\address{Yuelong Xiao (corresponding author)\newline
School of Mathematical Sciences,
Xiangtan University,
Hunan 411105, China}
\email{xyl@xtu.edu.cn}

\address{Hui Zhang \newline
 School of Mathematics and computation Sciences,
Anqing Normal University,
AnHui,  246133 China}
\email{zhangaqtc@126.com}

\thanks{Submitted April 3, 2017. Published July 6, 2017.}
\subjclass[2010]{35Q35, 35B40, 35B65, 76B03}
\keywords{Navier-Stokes equation; slip boundary conditions;
\hfill\break\indent vanishing viscosity limit}

\begin{abstract}
 In this article we consider the three-dimensional nonhomogeneous
 incompressible Navier-Stokes equation with special slip boundary conditions
 in a bounded domain. We discuss the problem of the vanishing viscosity limit
 and provide a rate of convergence estimates for the strong solution.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks


\section{Introduction}

Let $\Omega\subset R^3$ be a bounded smooth domain,
the initial boundary value problem of the
nonhomogeneous incompressible Navier-Stokes equation is given by
\begin{gather}
\label{1.1} \rho\partial_{t}u-\nu\Delta u+\rho u\cdot\nabla u+\nabla p=0,
 \quad\text{in }\Omega,\\
\label{1.2} \partial_{t}\rho+u\cdot\nabla \rho=0,\quad\text{in }\Omega,\\
\label{1.3} \nabla\cdot u=0,\quad\text{in }\Omega,\\
\label{1.4} u(0,x)=u_0,\rho(0,x)=\rho_0,\quad\text{in }\Omega,
\end{gather}
equipped with the  vorticity boundary conditions
\begin{equation}\label{1.5}
u\cdot n=0, \quad\omega\cdot n=0, \quad n\times(\Delta u)=0\quad\text{on }
\partial\Omega.
\end{equation}
Here the constant $\nu>0$, n, $\rho, u, p$
represent the viscosity coefficient, the outward unit normal vector,
the mass density, the velocity field and the pressure of the fluids,
respectively.
The initial density $\rho_0(x)$ is assumed to satisfy the condition
$m\leq\rho_0(x)\leq M$
with $m$ and $M$ are given positive constants.

The vanishing viscosity limit for the nonhomogeneous incompressible 
Navier-Stokes equation with the cauchy problem and the periodic boundary conditions
has been investigated by Itoh \cite{Shigeharu},
Itoh and Tani \cite{ShigeharuItoh} and Danchin \cite{R. Danchin2}, respectlvely.
In the presence of a physical boundary,
the vanishing viscosity limit problems become more challenging and 
significance because of the emergence of the boundary layer.
Formally, when the viscous term is vanishing,
system \eqref{1.1}-\eqref{1.4} degenerates into the nonhomogeneous 
incompressible Euler equation
\begin{gather}
\label{1.6} \rho^{0}\partial_{t}u^{0}+\rho^{0} u^{0}\cdot\nabla u^{0}
+\nabla p^{0}=0,\quad\text{in }\Omega,\\
\label{1.7} \partial_{t}\rho^{0}+u^{0}\cdot\nabla \rho^{0}=0,\quad\text{in }\Omega,\\
\label{1.8} \nabla\cdot u^{0}=0,\quad\text{in }\Omega,\\
\label{1.9} u^{0}(0,x)=u_0,\rho^{0}(0,x)=\rho_0,\quad\text{in }\Omega,
\end{gather}
with the slip boundary conditions
\begin{equation}\label{1.10}
u^{0}\cdot n=0,\quad\text{on }\partial\Omega.
\end{equation}
The initial boundary value problem of the equation
\eqref{1.6}-\eqref{1.10} has a smooth solution at least local in time,
it has been addressed
by several authors, see, e.g. \cite{H0,ShigeharuItoh,A. VALLI}.
Concerning the nonhomogeneous incompressible Navier-Stokes equation,
one of the most common physical boundary conditions is the classical 
no-slip boundary conditions
\begin{equation}\label{1.11}
u=0,\quad\text{on }\partial\Omega,
\end{equation}
which means that fluid particles are adherent to the boundary because of the
 positive viscosity, it was proposed by Stokes in \cite{stokes}.
This Dirichlet type problem has been addressed
in \cite{R. Danchin1,R.Salvi} and references therein.
However, the asymptotic convergence of the solution
is one of the major open problem except some special cases,
the main challenging is a discrepancy between the no-slip boundary conditions
for the nonhomogeneous incompressible Navier-Stokes equation and the tangential
boundary conditions for the nonhomogeneous incompressible Euler equation.

Another class of familiar boundary conditions
is the Navier-slip boundary conditions,
which can be shown as follows
\begin{equation}
u\cdot n=0, \quad 2(S(u)n)_\tau=-\gamma u_\tau,\quad\text{on } \partial\Omega,
\end{equation}
it was first introduced in \cite{N},
where $2S(u)n=(\nabla u+(\nabla u)^\top)$ is the viscous stress tensor,
$\gamma$ is a given smooth function on the boundary.
We can also write the equivalently form as the following vorticity-slip condition
\begin{equation}\label{1.13}
u\cdot n=0, n\times \omega=\beta u,\quad\text{on }\partial\Omega.
\end{equation}
The result of weak convergence have been considered by Ferreira and 
Planas \cite{LCF}.
As $\beta=0$, the special vorticity-slip conditions have initially been applied
to three-dimensional incompressible Navier-Stokes equation in \cite{xiao1}.
Based on the above works,
the author and coauthor found an additional condition for the density
to obtain the strong convergence rate for the nonhomogeneous Navier-Stokes equation
on the flat domain in \cite{pfc}.
However, to our best knowledge,
it is still unknown if the similar strong convergence results
can be established in a general bounde domain.
There are many references on inviscid limit for Navier-Stokes equation
with Navier-slip boundary conditions,
the readers can be referred in
\cite{HF,H,LB,W. Craig, Iftimie,D Iftimie,Kelliher,Lions,Zhang Jianwen}.

Our main goal in this paper is to show the vanishing viscosity limit problem
with the vorticity boundary condition \eqref{1.5}.
This type of boundary condition,
which was initially established in \cite{xiao4}
for the homogeneous incompressible Navier-Stokes equation,
where the author established the mathematical result on rate of convergence 
for strong solution.
Our approach here is motivated by the ideas \cite{xiao4} to study the problem for
the nonhomogeneous incompressible Navier-Stokes equation
and is based on the following observations:
First, we need to add the some additional boundary conditions
for the density, which is described by
\begin{equation}\label{1.14}
\nabla \rho=0,\quad\text{on }\partial\Omega.
\end{equation}
The boundary condition \eqref{1.14} can
balance well the momentum equation \eqref{3.2} with boundary conditions \eqref{3.4},
we can obtain the strong solutions local in time.
Second, we need to construct a new system \eqref{3.1}-\eqref{3.7},
which can be regarded as a relaxed vorticity system of
nonhomogeneous incompressible Navier-Stokes equation.
The fact shows that the pressure vanishes in the new system,
yet the new system is indeed
the vorticity system of the equations \eqref{1.1}-\eqref{1.5}.
Our first main result is concerned with the local well-posedness of
the initial boundary value problem for the equations \eqref{1.1}-\eqref{1.5}.

\begin{theorem}\label{t1.1}
Let $\Omega$ be the bounded smooth domain,
denote H by the space $\{u\in L^2(\Omega);\nabla\cdot u=0,\text{ in }
 \Omega, u\cdot n=0\text{ on }\partial\Omega\}$, 
$u_0\in H^1(\Omega)\cap H$, $\rho_0\in H^2(\Omega)$,
$\omega_0\in H^1(\Omega)\cap H$. Then there exists $T^\nu=T^\nu(\omega_0)>0$,
such that the initial boundary value problem \eqref{1.1}-\eqref{1.2} 
has a unique solution $(\rho,u,p)$ satisfying
\begin{gather*}
u\in L^2(0,T;H^3(\Omega))\cap C([0,T^\nu);H^2(\Omega)),\\
\rho\in C([0,T^\nu);H^2(\Omega)),\ u'\in L^2(0,T;V),
\end{gather*}
for any $T\in(0,T^\nu)$, and
\begin{gather*}
  -\Delta p =\rho\partial_iu_j\partial_ju_i, \\
 \partial_np=(\Delta u-\rho u\cdot\nabla u)\cdot n,\\
 \int_\Omega p=0,
\end{gather*}
 for $t\in[0,T^\nu)$.
\end{theorem}

\begin{remark} \label{rmk1.2} \rm
To obtain the results above,
we need to construct a new initial boundary value problem \eqref{3.1}-\eqref{3.7}.
Since there is one more condition in \eqref{1.5} than that normally Navier-slip 
boundary conditions, thus it is non-trivial to show the consistency of 
the boundary conditions to get the well-posedness.
\end{remark}

As the viscosity coefficient $\nu$ tends to be zero,
we show the following convergence of rate.

\begin{theorem}\label{t1.2}
Let $\rho_0\in H^4(\Omega),u_0\in H\cap H^4(\Omega)$ satisfy
$\nabla\rho_0\cdot n=0,\nabla\times u_0\in H$,
$\rho^0(t),u^0(t)$ be the solution to the Euler equations for nonhomogeneous fluids
on $[0,T]$ with initial data $\rho_0,u_0$,
$\rho(t),u(t)$ be the solution in Theorem \ref{t1.1}. Then, we have the following
\begin{equation}\label{1.15}
\|\rho-\rho^0\|_2^2+\|u-u^0\|_2^2+\nu\int_0^t\|u-u^0\|_3^2dt\leq c\nu^{1-s}
\end{equation}
on the interval $[0,T]$ with $T=T(\sigma,s)>0$ independent of $\nu\in(0,\sigma)$
for $s>0$ and $\nu\in(0,\sigma)$.
\end{theorem}

\begin{remark} \label{rmk1.4} \rm
Under the vorticity boundary conditions,
we can get a result mathematically of strong convergence estimate to the solutions.
The rate of convergence \eqref{1.15} is better than those
for the Navier-slip boundary conditions cases in \cite{LCF}.
Compared with the case of co-normal uniform estimate as in \cite{Masmoudi,Y. wang},
our problem here does not so tedious and complicated,
it can be proved only by standard energy estimates.
\end{remark}

The rest of this article is organized as follows: 
Section 2, we recall some notations, definitions, and
preliminary facts. 
Section 3, we give the local well-posedness to the  initial boundary 
value problem for the nonhomogeneous Navier-Stokes equations \eqref{1.1}-\eqref{1.5}.
Section 4, we establish the rate of convergence to the solutions.

\section{Preliminaries}

 Let us start by recalling the standard notation of some function spaces
and operators which are familiar in the mathematical theory of fluids 
modelled by Navier-Stokes system, see \cite{xiao1,xiao4}.
For convenience, note the inner product by $(\cdot,\cdot)$
and the norm of the standard Hilbert space
$L^2(\Omega)$, $H^s(\Omega)$ by $\|\cdot\|$, $\|\cdot\|_s$, respectively.
We also denote $[A,B]=AB-BA$, the commutator between
two operators $A$ and $B$. Set
\begin{gather*}
H = \{u\in L^{2}(\Omega); \nabla\cdot u = 0,\text{ in } \Omega, \;
u\cdot n = 0\text{ on }\Omega\},\\
V = H^1 (\Omega)\cap H, \\
W = \{u\in H^{2}(\Omega); n\times(\nabla\times u)=0\text{ on }\Omega\}.
\end{gather*}
Let $\psi, \phi$ be two vector function,
the following formula is shown by direct calculations:
\begin{gather}
\label{2.1} \nabla\times (\psi\times \phi)
=\phi\cdot\nabla \psi-\psi\cdot\nabla \phi+\psi\nabla\cdot \phi
 -\phi\nabla\cdot \psi, \\
\label{2.2} \nabla\times(\psi\cdot\nabla\phi)
=\psi\cdot\nabla(\nabla\times\phi)+\nabla\psi^\perp\cdot\nabla\phi,
\end{gather}
where $\nabla\psi^\perp$ is expressed in components by
\[  
(\nabla\psi^\perp\cdot\nabla\phi)_j=(-1)^{j+1}\partial_{j+1}\psi\cdot\nabla\phi_{j+1}
+(-1)^{j+2}\partial_{j+2}\psi\cdot\nabla\phi_{j+2}
\]
with the index modulated by $3$.
We denote by $A= -\Delta$ the Stokes operator with 
$D(A) = W\subset V$ is the self-adjoint extension of the positive closed
with its inverse being compact, and there is a countable eigenvalues
$\{\lambda_{j}\}$ such that
\[ 
0 < \lambda_{1}\leq\lambda_{2}\cdots \to \infty,
\]
the corresponding eigenvector $\{e_{j}\}\subset W\cap
C^{\infty}(\Omega)$ makes an orthogonal complete basis of $H$.
We first show the following estimate.

\begin{lemma}[\cite{xiao1}]
Let $s\geq 0$ be an integer. Let $u\in H^s$ be a vector-valued function, then
\begin{gather*}
\|u\|_s \leq C(\|\nabla\times u\|_{s-1}+\|\nabla\cdot u\|_{s-1}
 +|n\cdot u|_{s-\frac{1}{2}}),\\
\|u\|_s \leq C(\|\nabla\times u\|_{s-1}+\|\nabla\cdot u\|_{s-1}
 +|n\times u|_{s-\frac{1}{2}}+\| u\|_{s-1}).
\end{gather*}
\end{lemma}

Assuming that $\phi(t),\psi(t),f(t)$
are smooth non-negative functions defined for all $t\geq 0$,
we show the following differential inequality.

\begin{lemma}[\cite{R.Salvi}]\label{l2.2}
Suppose $\phi(0)=\phi_0$ and $\frac{d\phi(t)}{dt}+\psi(t)\leq g(\phi(t))+f(t)$ 
for $t\geq 0$, where $g$ is a non-negative Lipschitz continuous function 
defined for $\phi\geq 0$.
Then $\phi(t)\leq F(t;\phi_0)$ for $t\in [0,T(\phi_0))$ where $F(\cdot;\phi_0)$ 
is the solution of the initial value problem $\frac{dF(t)}{dt}=g(F(t))+f(t)$; 
$F(0)=\phi_0$ and $[0,T(\phi_0))$
is the largest interval to which it can be continued.
Also, if $g$ is nondecreasing, then
\[
\int_0^t\psi(\tau)d\tau\leq \widetilde{F}(t;\phi_0)
\]
with
\[
\widetilde{F}(t;\phi_0)=\phi_0+\int_0^t[g(F(\tau;\phi_0))+f(\tau)]d\tau.
\]
\end{lemma}

\section{Local well-posedness results}
Our main purpose in this section is to solve
 the initial boundary value problem \eqref{1.1}-\eqref{1.5}.
Firstly, we give the following additional boundary condition for density:

\begin{lemma}\label{lem1}
Let the initial density satisfy the condition $\nabla\rho_0=0$ on the boundary,
then the density have the persistence property that $\nabla\rho(t,\cdot)=0$
on the boundary.
\end{lemma}

\begin{proof}
Applying the gradient operator $\nabla$ to the transport equation \eqref{1.2},
it follows that
$$
\frac{D}{dt}(\nabla\rho)+\nabla u\cdot\nabla\rho=0,
$$
the ordinary differential equations is linear and
the initial data satisfies $\nabla\rho_0=0$, we can prove the lemma.
\end{proof}

On the other hand,  to obtain the strong solution,
we need to construct the following system,
which is called a relaxed vorticity equation of \eqref{1.1}-\eqref{1.5}:
\begin{gather}
\label{3.1} \rho_t+u\cdot\nabla\rho=0,\quad\text{in }\Omega,\\
\label{3.2} \rho(\partial_t\omega+u\cdot\nabla \omega
 -\omega\cdot\nabla u)+\nabla\rho\times(\partial_tu+u\cdot\nabla u)
 -\nu \Delta\omega+\nabla q=0,\quad\text{in }\Omega,\\
\label{3.3} \nabla\cdot \omega=0,\quad\text{in }\Omega,\\
\label{3.4} \omega\cdot n=0, n\times(\nabla\times\omega)=0,\quad\text{on }
 \partial\Omega,
\end{gather}
with $u=T\omega$ given by
\begin{gather}
 \label{3.5} \nabla\times u=\omega,\quad\text{in } \Omega,\\
\label{3.6} \nabla\cdot u=0,\quad\text{in }\Omega,\\
\label{3.7} u\cdot n=0, \quad\text{on }\partial\Omega,
\end{gather}
Where the linear operator satisfy $T: H\to V$ with $u=T\omega$,
which is the unique solution of  equations \eqref{3.5}-\eqref{3.7},
is continuous.
We claim that the initial boundary value problem \eqref{3.1}-\eqref{3.7}
possesses exactly one strong solution in a maximal time interval.
Let $P_k$ the orthogonal project of H onto the space $H_k$
spanned by the k first eigenfunctions $e_1,\cdots e_k$ of A.
Then the solutions of  system \eqref{3.1}-\eqref{3.7}
can be obtained by using a Semi-Galerkin approximations method
determined by the spaces $H_k$ and the operators $P_k$.
For each fixed k, we consider the following finite dimensional problem:
Find $T_k\in(0,T]$ such that
\begin{gather*}
\begin{aligned}
&P_m(\rho^{(m)}\partial_{t}\omega^{(m)}+\rho^{(m)} T\omega^{(m)}
 \cdot\nabla \omega^{(m)}-\rho^{(m)} \omega^{(m)}\cdot\nabla T\omega^{(m)}) \\
&+P_m(\nabla\rho^{(m)}\times(\partial_{t}T\omega^{(m)}
 +T\omega^{(m)}\cdot\nabla T\omega^{(m)}))
 -\nu\Delta P_m\omega^{(m)}=0,
\end{aligned}\\
\rho^{(m)}_t+T\omega^{(m)}\cdot\nabla\rho^{(m)}=0,\\
\omega^{(m)}(0,x)=P_m\omega_0(x),\ \rho^{(m)}(0,x)=\rho_0(x),\\
e_m\cdot n=0, n\times(\nabla\times e_m)=0.
\end{gather*}
We have an initial boundary value problem for
a system of ordinary differential equations
coupled to a transport equation.
By using the characteristics method, it can prove the system possesses 
exactly one solution $(\rho^{(m)},\omega^{(m)})$ defined in a time 
interval $[0,T_k)$.
The kth approximated problem can also be written in the form
\begin{gather*}
\begin{aligned}
&(\rho^{(m)}\partial_{t}\omega^{(m)}+\rho^{(m)} 
 T\omega^{(m)}\cdot\nabla \omega^{(m)}-\rho^{(m)} 
 \omega^{(m)}\cdot\nabla T\omega^{(m)},v) \\
&+(\nabla\rho^{(m)}\times(\partial_{t}T\omega^{(m)}+T\omega^{(m)}\cdot\nabla 
 T\omega^{(m)}),v) -\nu(\Delta \omega^{(m)},v)=0,
\end{aligned}\\
\rho^{(m)}_t+T\omega^{(m)}\cdot\nabla\rho^{(m)}=0,\\
\omega^{(m)}(0,x)=P_m\omega_0(x),\ \rho^{(m)}(0,x)=\rho_0(x),\\
e_m\cdot n=0, n\times(\nabla\times e_m)=0.
\end{gather*}
Through the Semi-Galerkin approximation method,
the rest of the process to estimate the solutions of \eqref{3.1}-\eqref{3.7}
is rather standard.
We do not give the detailed proof,
the reader can be referred to Chapter 3 in \cite{SN}.
The main theorem in this section is the following.

\begin{theorem}\label{t3.1}
Let $\rho_0\in H^2(\Omega)$ and $\omega_0\in V$, then there exists 
$T^\nu=T^\nu(\rho_0,\omega_0)>0$,
such that  problem \eqref{3.1}-\eqref{3.7} has a unique solution 
$(\rho,\omega,q)$ on the interval $[0,T^\nu)$ satisfying
\begin{gather*}
\rho\in C([0,T^\nu);W),\\
\omega\in L^2(0,T^\nu;W)\cap C([0,T^\nu);V), \omega'\in L^2(0,T^\nu;H),
\end{gather*}
and the energy equation
\begin{equation}
\|\rho(t)\|_2^2+\|\nabla\times\omega(t)\|^2
+\nu\int_0^t\|\partial_t\omega\|^2dx+\nu\int_0^t\|\omega(s)\|_2^2ds \leq c
\end{equation}
hold on $[0,t]$ for any $t\in(0,T^\nu$), and q is given uniquely by
\begin{gather}
\label{3.21} \Delta q =0,\\
\label{3.22} \partial_nq =-\rho(u\cdot\nabla\omega-\omega\cdot\nabla u)\cdot n,\\
\label{3.23} \int_{\partial\Omega}q =0,
\end{gather}
 for a.e. $t\in(0,T^\nu)$.
\end{theorem}

\begin{lemma}\label{l3.3}
Let $\omega\in V$, $\nabla\rho=0$ on boundary. Then
\[
\rho(T\omega\cdot\nabla \omega-\omega\cdot\nabla(T\omega))\in H.
\]
\end{lemma}

\begin{proof}
Since $\omega\in V$, it follows that $T\omega\in H^2(\Omega)\cap V$.
Then $\omega\times T\omega\in H^1(\Omega)$. 
The boundary condition $T\omega\cdot n=0$ and $\omega\cdot n=0$ implies
\[
n\times(\omega\times T\omega)=0,\text{ on } \partial\Omega.
\]
This completes the proof.
\end{proof}

From Lemma \ref{l3.3} we have the following corollary.

\begin{corollary} \label{coro3.4}
The solution $q$ in theorem \ref{t3.1} satisfies $q=0$, for a.e. $t\in(0,T^\nu)$.
\end{corollary}

From the analysis above, it follows that \eqref{3.2} is the curl of the
equation \eqref{1.1}. Thus Theorem \ref{t1.1} is proved.

\begin{remark} \label{rmk3.5} \rm
It should be noted that constructing system \eqref{3.1}-\eqref{3.6} is necessary.
If the boundary condition is replaced by the non slip boundary $\omega=0$, 
then $(\Delta\omega)\cdot n$ may not be zero,
from  equations \eqref{3.21}-\eqref{3.23},
hence $\nabla q$ may not be zero.
Then the momentum equation should be of the form
\[
\rho\partial_{t}u-\nu\Delta u+\rho u\cdot\nabla u+F(q)+\nabla p=0,\quad
\text{in } \Omega,
\]
for some vector function $F$ of $q$.
\end{remark}

\section{Convergence of solutions}


In this section we prove Theorem \ref{t1.2}.
Let us show the following lemma before giving the convergence estimate.

\begin{lemma} \label{lem4.1}
Let $\rho,u$ be a smooth solution to the nonhomogeneous incompressible 
Euler equations on the interval $[0,T]$ with initial $\rho_0\in H^3(\Omega)$,  
$u_0\in H^3(\Omega)\cap H$ and
$\nabla\rho_0=0$, $\nabla\times u_0\in H$.
Then  $(\nabla\times u^0)\cdot n=0,\quad\text{on }\partial\Omega$ for all
$t\in [0,T]$.
\end{lemma}

\begin{proof}
Note that the particle path forms a diffeomorphism on the boundary.
The vorticity equations of the nonhomogeneous incompressible Navier-Stokes 
equation is 
\begin{gather}
\rho^0_t+u^0\cdot\nabla\rho^0=0,\quad\text{in }\Omega,\\
\label{4.2}\rho^0(\partial_t\omega^0+u^0\cdot\nabla \omega^0-\omega^0\cdot\nabla u^0)
+\nabla\rho^0\times (\partial_tu^0+u^0\cdot\nabla u^0)
=0,\quad\text{in }\Omega,
\end{gather}
From Lemma \ref{lem1}, it follows that 
$\nabla\rho^0\times (\partial_tu^0+u^0\cdot\nabla u^0)$ vanishes on the boundary. 
Multiplying \eqref{4.2} by the unit outward norm vector yields
\[
\frac{D(\omega^0\cdot n)}{dt}=(\omega^0\cdot\nabla)u^0\cdot n
+\omega^0\cdot(u^0\cdot\nabla)n.
\]
From  \cite[Lemma 3.1]{xiao4}, there exist $\alpha,\beta$ such that
\[
\frac{D(\omega^0\cdot n)}{dt}=(\alpha+\beta)(\omega^0\cdot n).
\]
Since $\omega_0\cdot n=0\quad\text{on }\partial\Omega$, one has
$\omega^0(x,t)\cdot n=0$ on $\partial\Omega$.
This complete the proof.
\end{proof}

\begin{remark} \label{rmk4.2} \rm
To obtain the asymptotic convergence of the solutions,
we need some additional conditions for nonhomogeneous Euler equation
to overcome the boundary layer.
If the nonhomogeneous Euler equation
match the boundary conditions $\omega^0\cdot n=0$
in mathematical structure,
it can coincide with that of
nonhomogeneous Navier-Stokes equation
in the tangential directions.
Hence, we restrict the initial data condition
of the density satisfy $\nabla\rho_0=0$.
\end{remark}

\begin{proof}[Proof of Theorem \ref{t1.2}]
First, we denote  $a=\rho-\rho^0$, $v=u-u^0$, $w=\omega-\omega^0$.
From  the transport equations, it follows that
\begin{equation}\label{4.3}
\frac{d}{dt}a+u^0\cdot\nabla a=-v\cdot\nabla\rho.
\end{equation}
Applying the operate $D^2$ and taking the inner product
of \eqref{4.3} with $D^2a$, we have
\[
\frac{d}{dt}\|a(t)\|_2^2+(u^0\cdot\nabla D^2a,D^2a)+([D^2,u^0\cdot\nabla]a,D^2a)
=-(D^2(v\cdot\nabla\rho),D^2a).
\]
Hence, by  Young's inequality, it is easy to obtain
\begin{equation}\label{4.4}
\frac{d}{dt}\|a(t)\|_2^2\leq c\delta\nu\|\Delta w\|^2+\nu^{-1}\|a\|_2^4
+\|a\|_2^2+\|v\|_2^2+c\nu.
\end{equation}
Secondly, we estimate the difference system
between the vorticity equation of \eqref{1.1}
and the vorticity equation of \eqref{1.6}:
\begin{equation}\label{4.5}
a w_t+(\rho v+au^0)\cdot\nabla w+\rho^0 w_t+\rho^0u^0\cdot\nabla w
+\Phi-\nu\Delta w=\nu\Delta\omega^0,
\end{equation}
with the boundary conditions
\begin{equation}
u\cdot n=0, w \cdot n=0\quad\text{on } \partial\Omega,
\end{equation}
where $\Phi=A+B$,
\begin{align*}
A&=a w^0_t+a v\cdot\nabla\omega^0+\rho^0 v\cdot\nabla\omega^0
 +au^0\cdot\nabla\omega^0+aw\cdot\nabla v \\
&\quad +aw\cdot\nabla u^0+a\omega^0\cdot\nabla v+a\omega^0\cdot\nabla u^0
 +\rho^0w\cdot\nabla v+\rho^0w\cdot\nabla u^0
 +\rho^0\omega^0\cdot\nabla v \\
&\quad +\nabla a\times(\partial_tv+\partial_tu^0+v\cdot\nabla v
 +v\cdot\nabla u^0+u^0\cdot\nabla v+u^0\cdot\nabla u^0) \\
&\quad +\nabla \rho^0\times(\partial_tv+v\cdot\nabla v+v\cdot\nabla u^0
 +u^0\cdot\nabla v),
\end{align*}
and
\begin{align*}
B&=\nabla a\times(\partial_tv+\partial_tu^0+v\cdot\nabla v+v\cdot\nabla u^0
 +u^0\cdot\nabla v+u^0\cdot\nabla u^0) \\
&\quad+\nabla \rho^0\times(\partial_tv+v\cdot\nabla v+v\cdot\nabla u^0
 +u^0\cdot\nabla v).
\end{align*}
Taking the inner product of \eqref{4.5} with $-\Delta w$, it follows that
\begin{equation}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}(\|\sqrt{a}\nabla\times w\|^2
 +\|\sqrt{\rho^0}\nabla\times w\|^2)ds+\nu\|\Delta w\|^2-(\Phi,\Delta w) \\
&=\int_{\partial\Omega}((a w)_t+(\rho_0 w)_t)\cdot(n\times(\nabla\times w))ds
+\nu(\Delta\omega^0,\Delta w).
\end{aligned}
\end{equation}
Integrating by part yields
\begin{equation}\label{4.8}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}(\|\sqrt{a}\nabla\times w
 +\|\sqrt{\rho^0}\nabla\times w\|^2 \\
&+2\int_{\partial\Omega}(a w+\rho_0w)\cdot(n\times(\nabla\times \omega^0)))ds
 +\nu\|\Delta w\|^2 \\
&=(\Phi,\Delta w)+\int_{\partial\Omega}(a w+\rho_0w)
 \cdot(n\times\partial_t(\nabla\times \omega^0))ds
+\nu(\Delta\omega^0,\Delta w).
\end{aligned}
\end{equation}
Here we use the property that $n\times(\nabla\times\omega)=0, v\cdot n=0, w\cdot n=0$
  on $\partial\Omega$, it follows that
\begin{equation}\label{4.9}
\begin{aligned}
(\Phi,-\Delta w)
&=\int_{\partial\Omega}\Phi\cdot n\times(\nabla\times\omega^0)-
(\nabla\times\Phi,\nabla\times w) \\
&=(\Phi,-\Delta\omega^0)-(\nabla\times\Phi,\nabla\times\omega^0)
-(\nabla\times\Phi,\nabla\times w).
\end{aligned}
\end{equation}

Next, we list some basic facts to be used later.
The unit out normal vector $n$ has been extended as follows:
\[
 n(x)=\frac{\nabla\varphi(r(x))}{|\nabla\varphi(r(x))|},\ x\in\Omega
\]
and
\[ 
r(x)=\min_{y\in\partial\Omega}d(x,y)=d(x,y_0),\quad  y_0\in\partial\Omega,
\]
which is unique when $r(x)\leq \sigma$ for some $\sigma>0$,
and the function is smooth and compact supported in $[0,\sigma)$ such that
$$
\varphi(0)=1,\quad  \varphi'(0)=1.
$$
First we estimate on $(\nabla\times\Phi,\nabla\times w)$ from \eqref{4.9}, 
recall that
\[
(\nabla\times\Phi,\nabla\times w)=(\nabla\times(A+B),\nabla\times w).
\]
It follows from the definition of $A$ that
\begin{align*}
|(\nabla\times (a w^0_t),\nabla\times w)|
&=|(\nabla a\times w^0_t+a\nabla\times w^0_t),\nabla\times w)|\\
&\leq c\|a\|_2\|\nabla\times w^0_t\|\|\nabla\times w\|\\
&\leq c(\|\nabla\times w\|^2+\|a\|_2^2),
\end{align*}
and
\begin{align*}
&|(\nabla\times(aw\cdot\nabla v+\rho^0w\cdot\nabla v),\nabla\times w)|\\
&=|(\nabla(aw)^\perp\cdot\nabla v+aw\cdot\nabla w+\nabla(\rho^0w)^\perp\cdot\nabla v+\rho^0w\cdot\nabla w,\nabla\times w)|\\
&\leq c(\|a\|_2\|\nabla\times w\|_1^{1/2}\|\nabla\times w\|^{\frac{5}{2}}+\|\nabla\times w\|_1^{1/2}\|\nabla\times w\|^{\frac{5}{2}})\\
&\leq c\delta\nu\|\Delta w\|^2+c (\nu^{-1/3}\|\nabla\times w\|^{\frac{10}{3}}+\|a\|_2^2
+\nu^{-1}\|\nabla\times w\|^{10})+c\nu.
\end{align*}
similarly, it obtains that
\begin{align*}
&|(\nabla\times(a v\cdot\nabla\omega^0+\rho^0 v\cdot\nabla\omega^0+au^0\cdot\nabla\omega^0+aw\cdot\nabla u^0 \\
&+a\omega^0\cdot\nabla v+a\omega^0\cdot\nabla u^0+\rho^0w\cdot\nabla u^0
+\rho^0\omega^0\cdot\nabla v,\nabla\times w)|\\
&\leq c(\|a\|_2^2+\|\nabla\times w\|^2+\|\nabla\times w\|^4).
\end{align*}
Next, we calculate the term B, note that
\begin{align*}
&|(\nabla\times B,\nabla\times w)|=|(\nabla\times(\nabla a\times(\partial_tv+\partial_tu^0+v\cdot\nabla v+v\cdot\nabla u^0\\
&+u^0\cdot\nabla v+u^0\cdot\nabla u^0)
+\nabla \rho^0\times(\partial_tv+v\cdot\nabla v+v\cdot\nabla u^0+u^0\cdot\nabla v)),\nabla\times w)|.
\end{align*}
it follows that
\begin{align*}
 &|(\nabla\times(\nabla a\times(\partial_tv+v\cdot\nabla v)),\nabla\times w)|
 =|(\nabla a\cdot\nabla(\partial_tv+v\cdot\nabla v)\\
 &-(\partial_tv+v\cdot\nabla v)\cdot\nabla(\nabla a)+\nabla a\nabla\cdot(v\cdot\nabla v)-
 (\partial_tv+v\cdot\nabla v)\Delta a,\nabla\times w)|\\
&\leq c(\|a\|_2\|\nabla\times w\|_1^{1/2}\|\nabla\times w\|^{\frac{5}{2}}
+\|a\|_2\|\partial_tw\|\|\nabla\times w\|_1^{1/2}\|\nabla\times w\|^{1/2})\\
&\leq c\delta(\nu\|\Delta w\|^2+\epsilon\|\partial_tw\|^2+\|a\|_2^2
+\nu^{-3/2}\|a\|_2^8
+\nu^{-1}\|\nabla\times w\|^{10} \\
&\quad +\nu^{-1/2}\|\nabla\times w\|^4+\nu).
\end{align*}
Similarly, we can get
\begin{align*}
&|(\nabla\times(\nabla a\times(\partial_tu^0+v\cdot\nabla u^0
+u^0\cdot\nabla v+u^0\cdot\nabla u^0)\\
&+\nabla \rho^0\times(\partial_tv+v\cdot\nabla v+v\cdot\nabla u^0
 +u^0\cdot\nabla v)),\nabla\times w)|\\
&\leq c(\|a\|_2\|\nabla\times w\|+\|\nabla\times w\|_2^2+c\nu),
\end{align*}
and
\begin{align*}
&|(\nabla \rho^0\times(\partial_tv+v\cdot\nabla v+v\cdot\nabla u^0
 +u^0\cdot\nabla v)),\nabla\times w)|\\
&\leq c(\|\partial_tv\|^2+\|\nabla\times w\|^2+\|\nabla\times w\|^4).
\end{align*}
Hence, it follows that
\begin{equation}\label{4.10}
\begin{aligned}
(\nabla\times\Phi,\nabla\times w)
&\leq c(\delta\nu\|\Delta w\|^2+\epsilon\|\partial_tw\|^2+\nu^{-1}\|a\|_2^8
 \\
&\quad +\nu^{-1/3}\|\nabla\times w\|^{\frac{10}{3}}
 +\nu^{-1}\|\nabla\times w\|^{10}+\nu^{-1}\|\nabla\times w\|^4
 \\
&\quad +\|a\|_2^2+\|\nabla\times w\|^2+\|\nabla\times w\|^4)+
\epsilon\|\partial_t w\|^2+c\nu.
\end{aligned}
\end{equation}
Second, we estimate on the term $(\nabla\times\Phi,\nabla\times\omega^0)$:
\[
(\nabla\times\Phi,\nabla\times \omega^0)=(\nabla\times(A+B),
\nabla\times \omega^0).
\]
Recall that
\begin{align*}
&(\nabla\times(\rho^0 v\cdot\nabla\omega^0+au^0\cdot\nabla\omega^0
 +a\omega^0\cdot\nabla u^0+\rho^0w\cdot\nabla u^0
 +\rho^0\omega^0\cdot\nabla v),\nabla\times \omega^0)\\
&=\int_{\partial\Omega}n\times\Big(\rho^0 v\cdot\nabla\omega^0
 +au^0\cdot\nabla\omega^0+a\omega^0\cdot\nabla u^0
 +\rho^0w\cdot\nabla u^0\\
&\quad +\rho^0\omega^0\cdot\nabla v\Big)\nabla\times \omega^0ds
 -\Big(\rho^0 v\cdot\nabla\omega^0+au^0\cdot\nabla\omega^0
+a\omega^0\cdot\nabla u^0+\rho^0w\cdot\nabla u^0\\
&\quad +\rho^0\omega^0\cdot\nabla v, 
 \nabla\times \omega^0,-\Delta\omega^0\Big),
\end{align*}
then, it follows from the trace theorem that
\begin{align*}
&\nu\int_{\partial\Omega}n\times(a w^0_t
 +\rho^0 v\cdot\nabla\omega^0+au^0\cdot\nabla\omega^0
 +a\omega^0\cdot\nabla u^0\\
&+\rho^0w\cdot\nabla u^0+\rho^0\omega^0\cdot\nabla v),\nabla\times
  \omega^0)\nabla\times \omega^0ds\\
&\leq c\nu(\|\nabla v\|_s+\|w\|_s+\|a\|_s)\|\nabla\times \omega^0\|_1\\
&\leq c\nu(\|\omega\|^{1-s}\|\nabla\times \omega\|^s+\|a\|^{1-s}\|\nabla a\|^s)\\
&\leq c\nu(\|\nabla\times \omega\|^2+\|\nabla a\|^2_1+\nu^{2-s}).
\end{align*}
At the same time, the remaining term of $A$ is estimated as
\begin{align*}
&(\nabla\times(a v\cdot\nabla\omega^0
+aw\cdot\nabla v+aw\cdot\nabla u^0+a\omega^0\cdot\nabla v
 +\rho^0w\cdot\nabla v,\nabla\times \omega^0)  \\
&\leq c(\|a\|^2_2+\|\nabla\times \omega\|^2+\|\nabla\times \omega\|^4),
\end{align*}
By the definition of $B$, it follows that
\begin{align*}
(\nabla\times B,\nabla\times \omega^0)
&=\int_{\Omega}n\times B \nabla\times \omega^0ds-(B,\Delta\omega^0)\\
&\leq c(\|\nabla\times \omega\|^2+\|a\|^2_2+\epsilon\|\partial_tw\|^2+\nu^{1-s}).
\end{align*}
Therefore, we can deduce that
\begin{equation}
|(\nabla\times\Phi,\nabla\times\omega^0)|
\leq c(\|\nabla\times \omega\|^2+\|a\|^2_2+\epsilon\|\partial_tw\|^2+\nu^{1-s}).
\end{equation}
Finally, we estimate on $(\Phi,-\Delta\omega^0)$:
\begin{align*}
&|(a w^0_t+\rho^0 v\cdot\nabla\omega^0+au^0\cdot\nabla\omega^0
 +a\omega^0\cdot\nabla u^0+\rho^0\omega^0\cdot\nabla v
 +\rho^0w\cdot\nabla u^0,-\Delta\omega^0)|\\
&\leq c(\|\nabla\times \omega\|^2+\|a\|^2_2+\nu^{1-s}),
\end{align*}
\begin{align*}
&|(a v\cdot\nabla\omega^0+aw\cdot\nabla v+aw\cdot\nabla u^0+a\omega^0\cdot\nabla v+\rho^0w\cdot\nabla  v,-\Delta\omega^0)|\\
&\leq c(\|a\|^2_2+\|\nabla\times \omega\|^4+\nu),
\end{align*}
and
\[
|(B,-\Delta\omega^0)|
\leq c(\|\nabla\times \omega\|^2+\|a\|^2_2+\epsilon\|\partial_tw\|^2+\nu^{1-s}).
\]
So
\begin{equation}
|(\Phi,-\Delta\omega^0)|\leq c(\|\nabla\times \omega\|^2
+\|\nabla\times \omega\|^4+\|a\|^2_2+\epsilon\|\partial_tw\|^2+\nu^{1-s}).
\end{equation}
The remaining terms in\eqref{4.8} can be estimated as follows:
\begin{equation}
|\int_{\partial\Omega}(a w+\rho_0w)\cdot(n\times\partial_t(\nabla\times \omega^0))ds|
\leq \|\nabla\times \omega\|^2+\|a\|^2_2+c\nu^{1-s},
\end{equation}
\begin{equation}
\nu|(\Delta\omega^0,\Delta w)|\leq c\delta\nu\|\Delta w\|^2+c\nu^{1-s},
\end{equation}
\begin{equation}\label{4.15}
|\int_{\partial\Omega}(a w+\rho_0w)\cdot(n\times(\nabla\times \omega^0)))ds|
\leq \frac{1}{4}\|\nabla\times \omega\|^2+\|a\|^2_2+c\nu^{1-s}.
\end{equation}

In order to estimate $\|\partial_tw\|^2$,
taking the inner product \eqref{4.5} with $\partial_tw$, it follows that
\begin{equation}\label{4.16}
\begin{aligned}
&\int_{\Omega}a |w_t|^2+\rho^0 |w_t|^2+\nu\frac{d}{dt}\|\nabla\times w\|^2 \\
&=\int_{\Omega}((\rho v+au^0)\cdot\nabla w+\rho^0u^0\cdot\nabla w
+\Phi+\nu\Delta\omega^0)\partial_tw \\
&\quad +\int_{\partial\Omega} n\times(\nabla\times w)w_t.
\end{aligned}
\end{equation}
From the boundary condition $n\times(\nabla\times \omega)=0$,
we have
\begin{equation}\label{4.17}
\int_{\partial\Omega} n\times(\nabla\times w)w_tds=-\frac{d}{dt}\int_{\partial\Omega} n\times(\nabla\times \omega^0)wds+\int_{\partial\Omega} n\times(\nabla\times \omega_t^0)wds.
\end{equation}
It follows from the formula \eqref{4.16} and \eqref{4.17} that
\begin{align*}
&\int_{\Omega}\rho |w_t|^2dx+\nu\frac{d}{dt}(\|\nabla\times w\|^2+
\int_{\partial\Omega} n\times(\nabla\times \omega^0)wds)\\
&=\int_{\Omega}((\rho v+au^0)\cdot\nabla w+\rho^0u^0\cdot\nabla w)\partial_tw
 +\int_{\Omega}\Phi\partial_tw
 +\int_{\partial\Omega} n\times(\nabla\times \omega_t^0)w\\
&=I+II+III.
\end{align*}
Hence,
\[
I\leq c(\|a\|_2^2+\|\nabla\times w\|^2+\|\nabla\times w\|^4
+\|a\|_2^4)+\frac{m}{4}\|\partial_tw\|^2,
\]
and
\[
II\leq c\|\Phi\|^2+\frac{m}{4}\|\partial_tw\|^2
\leq c(\|a\|_2^2+\|\nabla\times w\|^2
+\|\partial_tv\|^2)^3+\frac{m}{4}\|\partial_tw\|^2
\]
It follows from the trace theorem that
\[
III\leq c\|\omega\|^{1-s}\|\nabla\times \omega\|^s
\leq \frac{1}{4}\|\nabla\times \omega\|^2+c\nu^{1-s}.
\]
It follows that
\begin{equation}\label{4.18}
\begin{aligned}
&m\|w_t\|^2+\nu\frac{d}{dt}(\|\nabla\times w\|^2+
\int_{\partial\Omega} n\times(\nabla\times \omega^0)wds) \\
&\leq c(\|a\|_2^2+\|\nabla\times w\|^2+\|\partial_tv\|^2)^3
 +\frac{m}{2}\|\partial_tw\|^2+c\nu^{1-s}.
\end{aligned}
\end{equation}
Through the estimates \eqref{4.4}, \eqref{4.10}-\eqref{4.15}, \eqref{4.18}
 we obtain
\begin{equation}
\begin{aligned}
&\frac{d}{dt}(\|\sqrt{a}\nabla\times w\|^2+\|\nabla\times w\|^2
 +\nu\|\nabla\times w\|^2+\|a\|_2^2)+\nu\|\Delta w\|^2+m\|w_t\|^2 \\
&=c((\|a\|_2^2+\|\nabla\times w\|^2+\|\partial_tv\|^2)^3
 +\nu^{-1}\|a\|_2^4+\nu^{-3/2}\|a\|_2^8 \\
&+\nu^{-1/3}\|\nabla\times w\|^{\frac{10}{3}}
 +\nu^{-1}\|\nabla\times w\|^{10}+c\nu^{-1/2}\|\nabla\times w\|^4+\nu+\nu^{1-s}).
\end{aligned}
\end{equation}

If $s\in(0,1/2)$ and
\[
\|a\|_2^2\leq c\nu^{1-s},\quad    \|\nabla\times \omega\|^2\leq c\nu^{1-s}.
\]
So we  deduce that
\[
\nu^{-3/2}\|a\|_2^4+\nu^{-1/3}\|\nabla\times w\|^{\frac{10}{3}}
+\nu^{-1}\|\nabla\times w\|^{10}+c\nu^{-1/2}\|\nabla\times w\|^4=o(\nu^{1-s}),
\]
and there exists some constant $c$ such that
\[
\nu^{-1}\|a\|_2^4\leq c\nu^{1-s}.
\]
Using the initial data $a(0)=0, w(0)=0$, by the lemma \ref{l2.2}, we obtain
\[
\|a\|_2^2+\|\sqrt{a}\nabla\times w(t)\|^2
+\|\nabla\times w(t)\|^2+\int_{\Omega}\|w_t\|^2dx
+\nu\int_\Omega\|\Delta w(s)\|^2dx\leq c\nu^{1-s}.
\]
on the interval $[0,T_1]$ for $s\in(0,\frac{1}{2})$ and 
$\nu\in(0,\nu_1)\subset(0,\nu_0)$,
where $T_1=T_1(\nu_1,s)>0$ is independent of $\nu\in(0,\nu_0)$.
If $s\geq \frac{1}{2}$, we can chose a $s'\in (0,1/3)$ such that
$\nu^{s'}\leq c\nu^{s}$.
 The proof is complete.
\end{proof}

\subsection*{Acknowledgments}
The authors want to express their sincere
thanks to  anonymous referee for his or her useful comments and suggestions
 which helped improve the paper greatly.
P. F. Chen was partially supported by the NSFC (No. 11371300) and 
the Hunan Provincial Innovation Foundation For Postgraduate(CX2015B205).
Y. L. Xiao was supported by the NSFC (No. 11371300) and the Research Fund 
for the Doctoral Program of Higher Education of China(20134301110008).
H. Zhang was partially supported by the Research Fund of SMS at Anhui 
University and Anhui Education Bureau(AQKJ2014B009).


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\end{document}