\def\papertitle{VANISHING VISCOSITY FOR THE  NAVIER-STOKES BOUSSINESQ SYSTEM}
\def\paperauthorA{\underline{ Youssouf Maafa}$^1$, Mohamed Zerguine$^1$}
\def\paperauthorB{Mohamed Zerguine}
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\newtheorem{Theo}{Theorem}[section]
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\newtheorem{Rema}[Theo]{Remark}

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{$1$-University of Batna -2- ,LEDPA}

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\begin{abstract}
This paper deals with the local well-posedness in time for the Navier-Stokes Boussinesq equations in two dimensions in the framework of a smooth vortex patch. Furthermore, we  provide the inviscid limit for the velocity and the density.
\end{abstract}

\section{Introduction}
The system of the  Navier-Stokes  Boussinseq  with viscosity $\mu >0 $ given by the coupled equation, 
\begin{equation}\label{Eq-3}
\left\{ \begin{array}{ll} 
\partial_{t}v_\mu+v_\mu\cdot\nabla v_\mu-\mu \Delta v_\mu+\nabla p=\theta_\mu\vec e_2 & \textrm{ if } (t,x)\in \RR_+\times\RR^2,\\
 \partial_{t}\theta_\mu+v_\mu\cdot\nabla \theta_\mu =0 & \textrm{if $(t,x)\in \RR_+\times\RR^2$,}\\ 
\Div v_\mu =0, &\\ 
({v_\mu},{\theta_\mu})_{| t=0}=({v}_\mu^0,{\theta_\mu}^0).\tag{NSB$_\mu$} 
\end{array} \right .
\end{equation}
where $v=(v^1,v^2)\RR^2$ refers to the velocity vector filed located in position $x\in\RR^2$ at a time $t$ which assumed to be incompressible, the scalar function $\theta(t,x) \in\RR_{+}$ denotes  the temperature or the density, $p(t, x)\in\RR$ is the pressure which relates $v$ and $\theta$ through an elliptic equation. $\theta_\mu\vec e_2 $ is the buoyancy force in the direction  $\vec{e}_2=(0,1)$.

\hspace{0.5cm}Taking the  $\Curl$ operator to the momentum equation in \eqref{Eq-3} we get
\begin{equation}\label{Eq-5}
\left\{ \begin{array}{ll}
\partial_{t}\omega_{\mu}+v_{\mu}\cdot\nabla\omega_{\mu}-\mu\Delta\omega_{\mu}=\partial_1\theta_{\mu}, & \\
\partial_{t}\theta_\mu+v_\mu\cdot\nabla \theta_\mu =0, & \\

(\theta_{\mu},\omega_{\mu})_{| t=0}=(\theta_{\mu}^0,\omega_{\mu}^0).
\end{array} \right.\tag{VD$_{\mu} $}
\end{equation}
\hspace{0.5cm}Our goal is to prove that the system \eqref{Eq-3} is locally well-posed whenever the initial vorticity is a smooth vortex patch, that is $\omega^0_\mu=1_{\Omega^0}$, with the boundary $\partial\Omega_0 $ is a Jordan curve with $C^{\varepsilon+1}$ regularity, $0<\varepsilon<1$. In addition, we prove the local persistence of geometric structures as follows, equivalently the image $\Omega_t=\Psi_\mu(t,\Omega^0)$ keeps its initial regularity, with  $\Psi_\mu$ is the flow generated by the velocity $v_\mu$,
\begin{equation*}\label{Eq-7}
\left\{ \begin{array}{ll} 
\partial_{t}\Psi_\mu(t,x)=v(t,\Psi_\mu(t,x)), & \\
\Psi_\mu(0,x)=x.
\end{array} \right.
\end{equation*} 
%We assume that   $\omega^0 \in L^1 \cap L^\infty $ then  we prove the local existence of solutions using the classical methods .\\   
%Moreover the question we ask : is the regularity of the domain $\Omega_0 $ preserved by  the flow map? To answer this question , we study the persistence regularity of the vortex patch for \eqref{Eq-3}.\\
\hspace{0.5cm}Our second task is to study the inviscid limi of the system \eqref{Eq-3}  towards the system \eqref{Eq-1} given by the so-called  Euler  Boussinseq   
\begin{equation}\label{Eq-1}
\left\{ \begin{array}{ll} 
\partial_{t}v+v\cdot\nabla v+\nabla p= \theta\vec e_2& \textrm{if $(t,x)\in \RR_+\times\RR^2$,}\\
 \partial_{t}\theta+v\cdot\nabla \theta  =0 & \textrm{if $(t,x)\in \RR_+\times\RR^2$,}\\ 
\Div v =0, &\\ 
({v},{\theta})_{| t=0}=({v}^0,{\theta}^0).\tag{EB} 
\end{array} \right .
\end{equation} 
and we evaluate the rate of convergence between velocities, densities and vortices when the viscosity.

Let us briefly mention some results related to the classical Euler equation. J.Y. Chemin \cite{Chemin2} showed that if the initial boundary $\partial\Omega_0$ is $C^{1+\EE},$ for $\EE\in(0,1)$ then $\partial\Omega_t$ is still of class $C^{1+\EE}$. In this context, Hmidi and Zerguine \cite{Hmidi-Zerguine} extended the result of \cite{Chemin2} to the stratified Euler equation. For other connected subjects in different situations we refer the reader to \cite{Abidi-Danchin,Danchin0,Danchin-Zhang,Danchin-Paicu,Fanelli,HH,Hmidi-1,Meddour} and the references therein. On the other hand Meddour and Zerguine \cite{Meddour} explored the inviscid limit of the Navier-stokes Boussinesq system to the Stratifed Euler equation in the vortex patch setting. Inspired by the works \cite{Chemin2,Hmidi-Zerguine,Meddour}, we are mainly interested by studying the Navier-Stokes Boussinesq equations. The first result deals with the local existence and the local persistence of geometric structures of the system \ref{Eq-3}.

In particular, we have the following Theorem. 
\begin{Theo}\label{Th:1.1}
Let $0<\EE<1, a \in (1,\infty) $ and  $X_0$ be a family of admissible vector fields and $v_{\mu}^0$ be a free-divergence vector field in the sense that $\omega^0_{\mu}\in L^a\cap C^{\EE}(X_0)$. Let $  \theta _\mu^0 \in {L^{2}}\cap C^{\EE+1}(X_0)$ with $\nabla \theta^0_\mu \in L^a$ , then for $\mu\in]0,1[$ the system \eqref{Eq-3} admits a unique global solution 
\begin{equation*}
(v_\mu,\theta_\mu)\in L^\infty\big([0,T];\Lip\big)\times L^\infty\big([0,T];\Lip\cap L^2\big). 
\end{equation*}
and 
\begin{equation*}
\omega_\mu \in L^\infty\big([0,T];L^a \cap L^\infty \big). 
\end{equation*}
More precisely,
\begin{equation*}
\| \nabla v_\mu\|_{L^\infty_t L^{\infty}}\leq  C_0e^{C_0t}   . 
\end{equation*}
Furthermore,
\begin{equation*}
\| \omega_\mu\|_{L^\infty_tC^{\epsilon}(X_t)}+{\widetilde{\|}} X_{\lambda }\|_{L^\infty_t C^{\epsilon}(X_t)} +\|  \Psi_\mu \|_{{L^\infty_tC^\epsilon}(X_t)} \leq   C_0e^{\exp C_0t}.
\end{equation*}
\end{Theo} 

\hspace{0.5cm}The second result of this paper deals with the inviscid limit for the systm \eqref{Eq-3} to the stratified-Euler system. More precisely, we have  
\begin{Theo}\label{Theo1.2}
Let $(v_{\mu}, \theta_{\mu})$, $(v, \theta)$,  be the solution of the \eqref{Eq-3}, \eqref{Eq-1} respectively with the same initial data  satisfies the condition of Theorem \ref{Th:1.1} such that  $\omega_\mu^0=\omega^0= \mathbf{1}_{\Omega_0}$ where $ \Omega_0 $ is simply connected bounded domain. Then for all $t\ge0, \mu\in]0,1[,$  we have 
$$ \Vert v_\mu(t) -v (t)\Vert_{L^2} + \Vert \theta_\mu (t)-\theta(t) \Vert_{L^2 } \le C_0 (\mu t)^{\frac{1}{2}} .$$ 
\end{Theo}

\section{Tool Box}
\subsection{Function spaces} Let$(\chi,\varphi)\in\mathscr{D}(\RR^2)\times \mathscr{D}(\RR^2)$
be a radial cut-off functions be such that $\supp\chi\subset\{\xi\in\RR^2: \|\xi\|\le1\}$ and $\supp\varphi(\xi)\subset\{\xi\in\RR^2: 1/2\le\|\xi\|\leq 2\}$, so that
\begin{equation*}
\chi(\xi)+\sum_{q\ge0}\varphi(2^{-q}\xi)=1.
\end{equation*}
Through $\chi$ and $\varphi$, the Littlewood-Paley or frequency cut-off operators $(\Delta_q)_{q\ge-1}$ and $(\dot\Delta_q)_{q\ge-1}$ are defined for $u\in \mathscr{S}'(\RR^2)$   
\begin{equation*}
\Delta_{-1} u=\chi(\DD)u,\; \Delta_qu=\varphi(2^{-q}\DD)u\;\;\mbox{for}\;\;q\in\NN,\quad \dot\Delta_qu=\varphi(2^{-q}\DD)u\;\; \mbox{for}\;\; q\in\ZZ. 
\end{equation*}
where in general case $f(\DD)$ stands the pseudo-differential operator $u\mapsto\mathscr{F}^{-1}(f \mathscr{F}u)$ with constant symbol.  The lower frequencies sequence $(S_{q})_{q\ge0}$ is defined for $q\ge0$, 
\begin{equation*}
S_{q}u\triangleq\sum_{j\le q-1}\Delta_{j}u.
\end{equation*}     
In accordance of the previous properties we derive the well-known decomposition of unity
$$
u=\sum_{q\ge-1}\Delta_q u,\quad u=\sum_{q\in\ZZ}\dot\Delta_q u.
$$
The results  currently available allow us to define the inohomogeneous Besov denoted $B_{p,r}^s $ (resp. $\dot B_{p,r}^s$) and defined in the following way.
\begin{defi} For $(p,r,s)\in[1,  +\infty]^2\times\RR $, the inhomogeneous Besov spaces $B_{p,  r}^s$ (resp. homogeneous Besov spaces $\dot B_{p,  r}^s$) are defined by 
$$
B_{p,  r}^s=\{u\in \mathcal{S}'(\RR^2):\|u\|_{B_{p,  r}^s} <+\infty\}, \quad\dot B_{p,  r}^s=\{u\in  \mathcal{S}'(\RR^2)_{|\mathbb{P}}:\|u\|_{\dot B_{p,  r}^s} <+\infty\},
$$
where $\mathbb{P}$ refers to the set of polynomial functions in $\RR^2$ so that
$$
\|u\|_{B_{p,r}^s}\triangleq\left\{\begin{array}{ll}
\Big(\sum_{q\ge-1}2^{rqs}\|\Delta_q u\|_{L^{p}}^r\Big)^{1/r} & \textrm{if $r\in[1, +\infty[,$}\\
\sup_{q\ge-1}2^{qs}\Vert\Delta_q u\Vert_{L^p} & \textrm{if $r=+\infty$.}
\end{array}
\right.
$$
and 
$$
 \|u\|_{\dot B_{p,r}^s}\triangleq\left\{\begin{array}{ll}
\Big(\sum_{q\in\ZZ}2^{rqs}\|\dot\Delta_q u\|_{L^{p}}^r\Big)^{1/r} & \textrm{if $r\in[1, +\infty[,$}\\
\sup_{q\in\ZZ}2^{qs}\Vert\dot\Delta_q u\Vert_{L^p} & \textrm{if $r=+\infty$.}
\end{array}
\right.
$$
\end{defi}
\hspace{0.5cm}The Bernstein's inequalities are listed in the following lemma. 
\begin{lem}\label{Bernstein} There exists a constant $C>0$ such that for $1\le a\le b\le\infty$, for every function $u$ and every $q\in\NN\cup\{-1\}$, we have
\begin{enumerate}
\item[{\bf(i)}] $\sup_{\vert\alpha\vert=k}\Vert\partial^{\alpha}S_{q}u\Vert_{L^{b}}\le C^{k}2^{q(k+2({1}/{a}-{1}/{b}))}\Vert S_{q}u\Vert_{L^{a}}$.
\item[{\bf(ii)}] $C^{-k}2^{qk}\Vert\Delta_{q}u\Vert_{L^{a}}\le\sup_{\vert\alpha\vert=k}\Vert\partial^{\alpha}\Delta_{q}u\Vert_{L^{a}}\le C^{k}2^{qk}\Vert\Delta_{q}u\Vert_{L^{a}}$.
\end{enumerate}
\end{lem}
As a consequence of Bernstein inequality, we have
\begin{prop}\label{Emb} For $(s,\widetilde{s},p,p_1,p_2,r_1,r_2)\in\RR^2\times]1,\infty[\times[1,\infty]^4$ with $\widetilde{s} \le s, p_1\le p_2$ and $r_1\le r_2$, then we have
\begin{enumerate}
\item[{\bf(i)}] $B_{p,r}^{s} \hookrightarrow B_{p,r}^{\widetilde{s}} $.  
\item[{\bf(ii)}] $B_{p_{1},r_1}^{s}(\RR^2)  \hookrightarrow B_{p_2,r_2}^{s+2(1/p_2 -1/p_1)}(\RR^2)$. 
\end{enumerate}
\end{prop}
 %For $s\in\NN$, the obtained spaces are well-known Zugmund-H\"older spaces and noted sometimes by $C^s_{\star}$ equipped with the following norm
%\begin{equation*}
%\|u\|_{C^s_{\star}}=\sup_{q\ge-1}2^{qs}\|\Delta_q u\|_{L^\infty}.
%\end{equation*}
\hspace{0.5cm}Now, we state Bony's decomposition \cite{Bony} which allows us to split formally the product of two tempered distributions $u$ and $v$ into three pieces. More precisely, we have.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{defi}\label{def:1} For a given $u, v\in\mathcal{S}'$ we have
 $$
uv=T_u v+T_v u+\mathscr{R}(u,v),
$$
with
$$T_u v=\sum_{q}S_{q-1}u\Delta_q v,\quad  \mathscr{R}(u,v)=\sum_{q}\Delta_qu\widetilde\Delta_{q}v  \quad\hbox{and}\quad \widetilde\Delta_{q}=\Delta_{q-1}+\Delta_{q}+\Delta_{q+1}.
$$
\end{defi}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The mixed space-time spaces are stated as follows. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{defi} Let $T>0$ and $(s,\beta,p,r)\in\RR\times[1, \infty]^3$.  We define the spaces $L^{\beta}_{T}B_{p,r}^s$ and $\widetilde L^{\beta}_{T}B_{p,r}^s$ respectively by: 
$$
L^\beta_{T}B_{p,r}^s\triangleq\Big\{u: [0,T]\to\mathcal{S}^{'}; \Vert u\Vert_{L_{T}^{\beta}B_{p, r}^{s}}=\big\Vert\big(2^{qs}\Vert \Delta_{q}u\Vert_{L^{p}}\big)_{\ell^{r}}\big\Vert_{L_{T}^{\beta}}<\infty\Big\},
$$
$$
\widetilde L^{\beta}_{T}B_{p,r}^s\triangleq\Big\{u:[0,T]\to\mathcal{S}^{'}; \Vert u\Vert_{\widetilde L_{T}^{\beta}{B}_{p, r}^{s}}=\big(2^{qs}\Vert \Delta_{q}u\Vert_{L_{T}^{\beta}L^{p}}\big)_{\ell^{r}}<\infty\Big\}.
$$
\end{defi}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The following result is a useful result in our approach. For the proof see \cite[Corollary 1 ]{HH}  
\begin{cor}\label{Coro:1} Given $\EE\in]0,1[$ and $X$ be a vector field be such that $X, \Div X\in C^\EE$. Then for $f$ be a Lipschitz scalar function $k\in\{1,2\}$ the following statement holds.
\begin{equation*}
\|(\partial_k X)\cdot\nabla f\|_{C^{\EE-1}}\le C\|\nabla f\|_{L^\infty}\big(\|\Div X\|_{C^\EE}+\|X\|_{C^\EE}\big).
\end{equation*}
\end{cor}
   






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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%At the end of this subsection, we furnish a Theorem which treats the action of composition law with a smooth function in the Besov spaces and its proof can be found in \cite{Alinhac-Gerard}.
%\begin{Theo}\label{thm2.8}
%Let $G \in C^{[s]+2}$, with $G(0)=0$ and $s\in[0,\infty]$. Assume that $\theta \in B^{s}_{p,r} \cap L^{\infty} ,$ with $(p,r) \in [1,+\infty]^2$, then $G\circ \theta \in B^{s}_{p,r}$ and satisfying
%$$\|G \circ \theta\|_{B^{s}_{p,r}} \leq C(s)\sup_{|y|\leq C\|\theta\|_{L^\infty}}\|G^{[s]+2}(y)\|_{L^\infty} \|\theta\|_{B^{s}_{p,r}}.$$
%\end{Theo}

\subsection{particle results} In this subsection, we give some preparatory results freely used throughout our analysis. 

\begin{equation}\label{PC1.0}
\left\{ \begin{array}{ll}
\partial_{t}a + v\cdot\nabla a -\mu\Delta a=g,&\\ 
a_{\mid t=0}=a^{0}. 
\end{array} 
\right.
\end{equation}  
We start with the persistence of Besov regularity for \eqref{PC1.0} whose proof may be found in  \cite{Bahouri-Chemin-Danchin}
\begin{prop}\label{prop2.9}
Let $(s,r,p) \in ]-1, 1[ \times[1, \infty]^{2}$ and $v$ be a smooth vector field in free-divergence. Assume that $(a^{0},g)\in B_{p,r}^{s}\times L_{\loc}^{1}(\mathbb{R}_+; B_{p,r}^{s})$. Then for every smooth solution $a$ of $\eqref{PC1.0}$ and $t\geq 0$ we have
\begin{align*}
\Vert a(t) \Vert_{B_{p,r}^{s}}\leq Ce^{CV(t)}\left( \Vert a^{0}\Vert_{B_{p,r}^{s}}+ \int_0^{t}e^{-CV(\tau)}\Vert g(\tau)\Vert_{B_{p,r}^{s}}d\tau\right), 
\end{align*}
with the notation 
\begin{align*}
V(t)=\int_0^{t}\Vert\nabla v(\tau) \Vert_{L^{\infty}}d\tau,
\end{align*}
where $C=C(s)$ being a positive constant.
\end{prop}

The statement of maximal regularity for \eqref{PC1.0} in mixed space-time Besov space is given by the following result. For the proof see \cite{Bahouri-Chemin-Danchin}.
\begin{prop}\label{prop2.10}
Let $(s,p_1,p_2,r)\in ]-1, 1[\times[1,\infty]^{3}$ and $v$ be a free-divergence vector field belongs to $L_{loc}^{1}(\mathbb{R}_+;\Lip)$ then there exists a constant $C\geq0$, so that for every smooth solution $a$ of \eqref{PC1.0} we have for all $ t\geq 0 $
\begin{equation*}
\mu^{\frac{1}{r}}\Vert a\Vert_{\widetilde{L}_t^{r}B_{p_{1},p_{2}}^{{s+\frac{2}{r}}}}\leq Ce^{CV(t)}(1 + \mu t)^{\frac{1}{r}}\Big(\Vert a^{0}\Vert_{B_{p_{1},p_{2}}^{{s}}} + \Vert g\Vert_{L_t^{1}B_{p_1,p_2}^{{s}}}\Big).
\end{equation*}
\end{prop}
 
Next, we have the classical C\'alderon Zygmund inequality.


\begin{prop}\label{C.Z}
Let $p\in ]1, +\infty[$ and $v$ be a free-divergence vector field whose vorticity $\omega \in L^{p}$. Then $\nabla v \in L^{p}$ and  
\begin{align*}
\Vert \nabla v\Vert_{L^{p}}\leq C \frac{p^{2}}{p-1}\Vert \omega\Vert_{L^{p}}.
\end{align*}
with $ C $ being a universal constant.
\end{prop}

%To estimate $\theta-$variable in transport-diffusion regime, we require some standard properties of heat semigroup $(\Ss(t))_{t>0}$, in particular the estimate $L^p L^q $ with $\Ss(t)=e^{t\Delta}$. Especially we have.

%\begin{prop}\label{P,S} The family $(\Ss)_{t\ge0}$  is a strongly continuous semigroup of bounded linear operators from $L^p(\RR^2)$ for any $p\in[1,\infty]$. Furtheremore, for $1\le p\le q\le\infty$ the following assertions are hold.
%\begin{enumerate}
%\item[{\bf(i)}] For $f\in L^p(\RR^2)$, then for every $t>0$ we have 
%\begin{equation}\label{S(t)}
%\|\Ss(t) f \|_{L^q(\RR^2)} \leq \frac{C}{t^{\frac{1}{p}-\frac{1}{q}}}\|f\|_{L^p(\RR^2)}.
%\end{equation}
%\item[{\bf(ii)}] For $f\in L^p(\RR^2)$ and $\alpha\in\NN^N$, then for every $t>0$ we have
%\begin{equation}\label{S(t)-alpha}
%\|\Ss(t)\partial^{|\alpha|}f\|_{L^q(\RR^2)}\le\frac{C}{t^{\frac{|\alpha|}{2}\frac{1}{2}+\frac{1}{p}-\frac{1}{q}}}\|f\|_{L^p(\RR^2)}.
%\end{equation}
%\end{enumerate}
%\end{prop} 

At this stage, we define the anisotropic H\"older spaces as follows
\begin{defi}\label{3.1}
Let $\epsilon \in ]0, 1[$. A family of vector fields $X = (X_\lambda)_{\lambda \in \Lambda}$ is said to be admissible if and only if   the following assertions are hold.\\
\begin{enumerate}
\item[{\bf(i)}]Regularity: $\forall \lambda \in \Lambda \quad X_\lambda, \Div X_\lambda   \in C^{\epsilon}.$
\item[{\bf(ii)}] Non-degeneray: $ I(X)\triangleq \inf_{x\in \mathbb{R}^{N}}\sup_{\lambda \in \Lambda }\mid X_\lambda (x)\mid>0.$
\end{enumerate}
Setting
\begin{equation*}
\widetilde{\|}X_\lambda\|{C^{\epsilon}}\triangleq \| X_\lambda\|_{C^{\epsilon}} + \| \Div X_\lambda\|_{C^{\epsilon}}.
\end{equation*}
\end{defi}
\begin{defi}\label{3.2}
Let $X  = (X_\lambda)_{\lambda \in \Lambda }$ be an admissible family. The action of each factor $X_\lambda$ on $u \in L^{\infty}$ is defined as the directional derivative of $ u$ along $X_\lambda$ by the formula,
$$\partial_{X_{\lambda}} u = \Div(uX_\lambda) - u \Div X_\lambda .$$
The concept of anisotropic H\"{o}lder space, will be noted by $C^{\epsilon}(X)$ is defined below.
\end{defi}
\begin{defi}\label{def:3.2.1}
Let $\epsilon \in ]0, 1[$ and $X $ be an admissible family of vector fields. We say that $u \in C^{\epsilon}(X)$ if and only if:\\
\begin{enumerate}
\item[{\bf(i)}] $ u \in L^{\infty}$ and satisfies
\begin{equation*}
\forall \lambda \in \Lambda, \partial_{X_{\lambda}} u \in C^{\epsilon-1}, \quad \sup_{\lambda \in \Lambda }\Vert\partial_{X_{\lambda}} u\Vert_{C^{\epsilon-1}}< +\infty .
\end{equation*}   
\item[{\bf(ii)}]  $ C^{\epsilon}(X)$ is a normed space with
\begin{equation*}    
\Vert u\Vert_{C^{\epsilon}(X)}\triangleq \frac{1}{I(X)}\left( \Vert u\Vert_{L^{\infty}}\sup_{\lambda \in \Lambda}\widetilde{\Vert}X_\lambda\Vert_{C^{\epsilon}}+ \sup_{\lambda \in \Lambda }\Vert \partial_{X_{\lambda}} u\Vert_{C^{\epsilon-1}}\right) .
\end{equation*} 
\end{enumerate}
The next result play a major role in the proof of our main results. We refer the reader to \cite{Chemin2}. 
\begin{Theo}\label{The estimate log}
Let $\epsilon \in ]0, 1[$ and $X = (X_{t,\lambda})_{\lambda \in \Lambda}$ be a family of vector fields as in Definition \ref{def:3.2.1}. Let $v$ be a free-divergence vector field such that its vorticity $\omega$ belongs to $ L^{2}\cap C^{\epsilon}(X)$. Then there exists a constant C depending only on $\epsilon $, such that
\begin{align}
\Vert \nabla v\Vert_{L^{\infty}}\leq C\left( \Vert \omega \Vert_{L^{2}} + \Vert \omega\Vert_{L^{\infty}} \log \left( e+ \frac{\Vert \omega \Vert_{C^{\epsilon}(X)}}{\Vert \omega \Vert_{L^{\infty}}} \right) \right) .
\end{align}
\end{Theo}
\end{defi}
According Danchin's result \cite{Danchin0}, the class $C^{\EE}_{\Sigma}$ doesn't covers only the vortex patch of the type $\omega_0={\bf 1}_{\Omega_0}$, but also encompass the so-called general vortex. Specifically, we have.
\begin{prop}\label{G-V} Let $\Omega_0$ be a $C^{1+\EE}-$bounded domain, with $0<\EE<1$. Then for every function $f\in C^\EE$, we have
\begin{equation*}
f{\bf 1}_{\Omega_0}\in C^{\EE}_{\Sigma}.
\end{equation*}
\end{prop}

%The proof of the previous theorem involves some a priori estimates, respectively for the density, vorticity, and the velocity in different functional spaces listed below.  
%\section{proof f theorem \eqref{the3.8} }
%The existence and uniqueness of the solution the system \eqref{Eq:1} was proved in  \cite{M.Paicu-N.Zhu}  , we only need to give some necessary a priori estimates for the proof of Theorem \eqref{the3.8}

\subsection{A priori estimates} In this part we shall give some a priori estimates for the velocity and the vorticity.
  
\begin{prop}\label{Prop:1}Let $v_\mu$ be a smooth divergence-free vector field and $\theta_\mu$ be a smooth solution of the equation \eqref{Eq:3.1}. Then the following assertions are hold.
\begin{enumerate}

\item[{\bf(i)}]For  $p\in[1,\infty]$ and  $t\ge0$ we have 
$$\|\theta_\mu(t)\|_{L^p}\le \|\theta^0_\mu\|_{L^p}.$$
\item[{\bf(ii)}] For  $p\in[1,\infty]$ and  $t\ge0$ we have 
$$\|\nabla\theta_\mu(t)\|_{L^p}\le \|\nabla\theta_\mu^0\|_{L^p}e^{CV_\mu(t)}.$$ with
$V_\mu(t)=\int_0^t\|\nabla v_\mu(\tau)\|_{L^\infty}d\tau.$
\item[{\bf(iii)}] For  $p\in[1,\infty]$ and  $t\ge0$ we have 
$$\|\omega_\mu (t)\|_{L^p}\le C_0e^{CV_\mu(t)}.$$ with
$V(t)=\int_0^t\|\nabla v_\mu(\tau)\|_{L^\infty}d\tau.$
\end{enumerate}
\end{prop}
\begin{proof}{\bf(i)}According to \eqref{Eq-7} we can  express the density $\theta_\mu(t)$ by the initial value $ \theta^0_\mu $ and the flow $\Psi$ as follows 
\begin{equation*}
\theta_\mu (t,x)= \theta^0_\mu (\Psi^{-1}(t,x)).
\end{equation*}
Taking the $L^p$-norm to this equation and thanks to incompressible condition we infer that 
\begin{equation}\label{the<th-o}
\| \theta_\mu (t) {\|}_{L^p} \leq \| \theta_\mu ^0 {\|}_{L^p}. 
\end{equation}.\\
{\bf(ii)}  Taking  the partial derivative $\partial_j$ to $\theta_\mu -$ equation to obtain 
\begin{equation*}
\partial_t\partial_j\theta+v\cdot\nabla\theta=-\partial_j\theta\cdot\nabla v,
\end{equation*}
The $L^p-$estimate for the above   gives
\begin{equation*}
\|\nabla\theta(t)\|_{L^p}\lesssim\|\nabla\theta_0\|_{L^p}+\int_0^t\|\nabla\theta(\tau)\|_{L^p}\|\nabla v(\tau)\|_{L^\infty}d\tau.
\end{equation*}
The Gronwall's inequality implies that 
$$\|\nabla\theta(t)\|_{L^p}\le \|\nabla\theta_0\|_{L^p}e^{CV(t)}.$$
{\bf(iii)} 
The $L^p-$estimate for the $\omega_\mu$ equation  gives
\begin{equation*}
\|\omega_\mu(t)\|_{L^p}\lesssim\|\omega_\mu^0\|_{L^p}+\int_0^t\|\nabla\theta(\tau)\|_{L^p}\|\nabla v(\tau)\|_{L^\infty}d\tau.
\end{equation*}
Combining the last two estimates we find the desired result .
\end{proof}

{\bf{Proof of Theorem \ref{Th:1.1}}}
The existence part of the theorem is classical and can be done for example by using a standard recursive method, see, e.g. \cite{HH}. We will control the quantities  $\|\nabla v (t) \|_{L^\infty}$ and   $ \|\omega_{\mu}(t)\|_{C^\EE (X)}$ for every $t \ge 0$ . For this aim, appalling the operator $\partial_{X_t,\lambda}$ to  $\omega_\mu$ equation, we have 
\begin{equation*}\label{Eq8}
(\partial_t+v\cdot\nabla-\mu\Delta)\partial_{X_t,\lambda}\omega_\mu={X_{t,\lambda}} \cdot \nabla \partial_1\theta_\mu-\mu[\Delta,{X_{t,\lambda}}]\omega_\mu.
\end{equation*}
For commutator $\mu[\Delta,{X_{t,\lambda}}]\omega_\mu$. From Bony's decomposition, we write
\begin{equation*}
\mu[\Delta,X_{t,\lambda}]\omega_\mu=\mathfrak{A}+\mu\mathfrak{B},
\end{equation*}
with
\begin{equation*}
\mathfrak{A}\triangleq2\mu T_{\nabla X^{i}_{t,\lambda}}\partial_i\nabla\omega_\mu+2\mu T_{\partial_i\nabla\omega_\mu} \nabla  X^{i}_{t,\lambda}+ \mu T_{\Delta  X^{i}_{t,\lambda}} \partial_i\omega_\mu+\mu T_{\partial_i\omega_\mu}\Delta  X{^{i}_{t,\lambda}}.
\end{equation*}
and
\begin{equation*}
\mathfrak{B} \triangleq 2\mathcal{R}(\nabla X^i_{t,\lambda},\partial_i\nabla \omega_\mu)+\mathcal{R}(\Delta X^i_{t,\lambda},\partial_i\omega_\mu).
\end{equation*}
According to Theorem 3.38 page 162 in \cite{Bahouri-Chemin-Danchin} , we have
\begin{equation}\label{X omega}
\Vert\partial_{X_{\lambda}} \omega_\mu \Vert_{L_t^{\infty}C^{\epsilon-1}}\le C e^{CV_\mu(t)}\Big(\Vert\partial_{X_0,\lambda}\omega^0_\mu\Vert_{C^{\epsilon-1}}+(1+\mu t)\Vert \mathfrak{A} \Vert_{{L_t^\infty}C^{\epsilon-3}}+ \mu \Vert \mathfrak{B} \Vert_{\widetilde{L}_t^1C^{\epsilon-1}}+\Vert\partial_{X_{\lambda}}\partial_1\theta_\mu \Vert_{L^1_tC^{\epsilon-1}}\Big )\\.
\end{equation}
From \cite{Bahouri-Chemin-Danchin,Hmidi-1} we have the following estimate 
\begin{equation*}
\| \mathfrak{A} \|_{{L_t^\infty}C^{\epsilon-3}} \leq C\| \omega \|_{L_t^{\infty}L^{\infty}}\| X_\lambda \|_{{L_t^\infty}C^{\epsilon}}.
\end{equation*}
Thanks to Proposition \ref{Prop:1}, we get 
\begin{equation}\label{A}
\| \mathfrak{A} \|_{{L_t^\infty}C^{\epsilon-3}} \le C_{0}e^{CV_{\mu}(t)} \| X_\lambda \|_{{L_t^\infty}C^{\epsilon}} .
\end{equation}
Again from \cite{Bahouri-Chemin-Danchin,Hmidi-1} we have the following estimate
\begin{equation}\label{B}
\| \mathfrak{B} \Vert _{{\tilde{L}^1_t}C^{\epsilon-1}}\leq C   \| \omega \Vert _{{\tilde{L}^1_t}B^2_{\infty,\infty}}\| X_\lambda \|_{{L_t^\infty}C^{\epsilon}} .
\end{equation}
For  the term $\Vert\omega_\mu\Vert _{\widetilde{L}^1_tB^2_{\infty,\infty}}$, we use the Proposition \ref{prop2.10} for $a=\omega_\mu, g=\partial_1\theta_\mu, r=1, s=0$, and $p_1=p_2=\infty$, we obtain 
\begin{equation*} 
\mu \Vert\omega_\mu\Vert _{\widetilde{L}^1_tB^2_{\infty,\infty}}   \leq Ce^{CV_\mu(t)}(1+\mu t) \Big( \Vert\omega^0_\mu\Vert _{B^0_{\infty,\infty}}+ \int_{0}^{t} \Vert \partial_1\theta_\mu(\tau) \Vert _{B^0_{\infty,\infty}} d\tau \Big ).
\end{equation*}
 The embedding $L^\infty \hookrightarrow B^0_{{\infty,\infty}}$ implies that
\begin{equation*} 
\mu \Vert\omega_\mu\Vert _{\widetilde{L}^1_tB^2_{\infty,\infty}}  \leq Ce^{CV_\mu(t)}(1+\mu t) \big(\Vert\omega^0 _\mu \Vert _{L^{\infty}}+  \Vert \nabla \theta_\mu \Vert _{L^1_t L^{\infty}}\big).
\end{equation*}
Using Proposition \ref{prop2.10} we obtain 
\begin{equation*} 
\mu \Vert\omega_\mu\Vert _{\widetilde{L}^1_tB^2_{\infty,\infty}}  \leq C_{0}e^{CV_\mu(t)}(1+\mu t) (1+t) 
\end{equation*}
Plugging the last estimate into \eqref{B}, we infer that 
 \begin{equation}\label{B}
\| \mathfrak{B} \Vert _{{\tilde{L}^1_t}C^{\epsilon-1}}\le  C_{0}e^{CV_\mu(t)}(1+\mu t) (1+t)  \| X_\lambda \|_{{L_t^\infty}C^{\epsilon}} .
\end{equation}
To treat the quantity $\| \partial_{X_{\lambda}}  \partial_1 \theta_\mu \|_{L^{1}_tC^{\epsilon-1}} $  we note that 
\begin{equation}
 \partial_{X_{\tau,\lambda}} \partial_1 \theta_\mu  =\partial_1(\partial_{X_{\tau,\lambda}}\theta_\mu )-\partial_{\partial_1X_{\tau,\lambda}}\theta_\mu .
\end{equation}
It follows , from taking the $C^{\epsilon-1}-$ norm  to this equation 
$$\|\partial_{X_{\tau,\lambda}} \partial_1 \theta_\mu (\tau)\|_{C^{\epsilon-1}} \lesssim \|\partial_1(\partial_{X_{\tau,\lambda}}\theta_\mu )(\tau) \|_{C^{\epsilon-1}}+\|{(\partial_1  X_{\tau,\lambda}}) \cdot \nabla \theta_\mu (\tau) \|_{C^{\epsilon-1}}. $$ 
Moreover  using  the fact $\partial_1 : C^\epsilon \longrightarrow  C^{\epsilon-1} $ is a continuous map  and Corollary \ref{Coro:1},  we get 
\begin{equation*}
\| \partial_{X_{\tau,\lambda}} \partial_1 \theta_\mu  (\tau)\|_{C^{\epsilon-1}} \lesssim \|\partial_{X_{\tau,\lambda}}\theta_\mu (\tau) \|_{C^{\epsilon}}+\|\nabla\theta_\mu (\tau) \|_{L^\infty}  \tilde{\|} X_{\tau,\lambda}\|_{C^{\epsilon}}  e^{CV_\mu(t)}   .
\end{equation*} 
From Proposition \ref{prop2.9} and Proposition \ref{Prop:1}, we find
\begin{equation}\label{RHO}
\|\partial_{X_{\tau,\lambda}} \partial_1 \theta_\mu (\tau)\|_{L^1_t C^{\epsilon-1}} \le \|\partial_{X(0)_{\lambda}}\theta^0_\mu\|_{C^{\epsilon}} e^{CV_\mu(t)} t+C_{0}\tilde{\|} X_{\tau,\lambda}\|_{C^{\epsilon}}  e^{CV_\mu(t)} 
\end{equation}

Summing \eqref{A},\eqref{B},\eqref{RHO} and punting them in \eqref{X omega},such that $\mu \in ]0,1[ $ we infer that
\begin{equation}\label{Est11} 
\Vert\partial_{X_{t,\lambda}} \omega_\mu\Vert_{L_t^{\infty}C^{\epsilon-1}} \leq C_0e^{CV_\mu(t)}(1+t )^{2} \widetilde{\Vert} X_{\lambda,t} \Vert_{L^{\infty}C^{\epsilon}}.
\end{equation}
On other hand bound  by using Proposition \ref{prop2.9}, we find 
\begin{equation}\label{X C epsilon -1}
\Vert X_{t,\lambda }\Vert_{C^{\epsilon}}\leq Ce^{CV_\mu(t)}\Big(\Vert X_{0,\lambda}\Vert_{C^{\epsilon}}+ \int_0^t e^{- CV_{\mu}(\tau)} \Vert  \partial_{X_{\tau,\lambda}} v_\mu(\tau)\Vert_{C^{\epsilon}} d\tau\Big).             
\end{equation}
We  use the following result which its proof can be found in \cite{Bahouri-Chemin-Danchin,Chemin2}
\begin{equation*} 
\Vert  \partial_{X_{t,\lambda}} v_\mu(t)\Vert_{C^{\epsilon}} \leq  C\big( \Vert \nabla v_\mu(t) \Vert_{L^\infty} \widetilde{\Vert}  X_{t,\lambda }\Vert_{C^{\epsilon}}+\Vert \omega_\mu(t) \Vert_{C^{\epsilon-1}}\big).
\end{equation*}
Thanks to \eqref{Est11}, we find
\begin{equation}\label{X v}
\Vert  \partial_{X_{t,\lambda}} v_\mu(t)\Vert_{C^{\epsilon}} \leq  C \widetilde{\Vert}  X_{t,\lambda }\Vert_{C^{\epsilon}}\Big(\Vert \nabla v_\mu(t) \Vert_{L^\infty} + C_0e^{CV_\mu(t)}(1+t )^{2}\Big).
\end{equation}
Plunging \eqref{X v} in\eqref{X C epsilon -1}, we get
\begin{equation*}
{\Vert} X_{t,\lambda }\Vert_{C^{\epsilon}}\leq {Ce^{CV_\mu(t)}\bigg(\Vert X_{0,\lambda}\Vert_{C^{\epsilon}} + C_0\int_0^t e^{-{CV_{\mu}(\tau)}} \widetilde{\Vert}}  X_{\tau,\lambda }\Vert_{C^\epsilon}\Big( \Vert \nabla v_\mu(\tau) \Vert_{L^\infty} +(1+\tau )^{2}\Big)d\tau \bigg).  
\end{equation*}
For the term ${\Vert} \Div X_{t,\lambda }\Vert_{C^{\epsilon}}$ by applying Proposition \ref{prop2.9} to  $(\partial_t+v\cdot\nabla) \Div {X_{t,\lambda}}=0$, we get  
\begin{equation}\label{Div-X}
\|\Div {X_{t,\lambda}}\|_{C^\EE}\le Ce^{CV_\mu(t)}\|\Div X_{0,\lambda}\|_{C^\EE}.
\end{equation}
Combining the last two estimates we have
\begin{equation}\label{3.52}
e^{-CV_\mu(t)}\widetilde{\Vert} X_{t,\lambda }\Vert_{C^{\epsilon}}\le C\bigg(\widetilde{\Vert} X_{0,\lambda}\Vert_{C^{\epsilon}} + C_0\int_0^t e^{-{CV_{\mu}(\tau)}} \tilde{\Vert}  X_{\tau,\lambda }\Vert_{C^\epsilon}\Big( \Vert \nabla v_\mu(\tau) \Vert_{L^\infty}+(1+\tau)^{2}\Big)d\tau\bigg).
\end{equation}   
The Gronwall's inequality gives
\begin{equation}\label{Est-X}
{\widetilde{\Vert}} X_{t,\lambda }\Vert_{C^{\epsilon}} \leq C_0 e^{ C_0  V_\mu(t)}e^{ C_0 t^{3}}.
\end{equation}
Gathering \eqref{Est11} and \eqref{Est-X}, one has
\begin{equation*}\label{3.26.}
\Vert\partial_{X_{t,\lambda}} \omega_\mu(t)\Vert_{C^{\epsilon-1}} \leq C_0e^{ C_0  V_\mu(t)}e^{ C_0 t^{3}}.
\end{equation*}
Moreover, from the last two estimates and Propitiation \ref{Prop:1}, we get   
\begin{equation}\label{3.27}
\Vert\partial_{X_{t,\lambda}} \omega_\mu(t)\Vert_{C^{\epsilon-1}}+\Vert \omega_\mu(t)\Vert_{L^\infty} {\widetilde{\Vert}} X_{t,\lambda }\Vert_{C^{\epsilon}} \leq C_0 e^{ C_0  V_\mu(t)} e^{ C_0 t^{3}}. 
\end{equation}
Now, we recall that 
\begin{equation}\label{3.56}
\Vert \omega \Vert_{C^{\epsilon}(X)}\triangleq \frac{1}{I(X_t)}\bigg( \Vert \omega\Vert_{L^{\infty}}\sup_{\lambda \in \Lambda}\widetilde{\Vert}X_\lambda\Vert_{C^{\epsilon}}+ \sup_{\lambda \in \Lambda }\Vert \partial_{X_{\lambda}} \omega \Vert_{C^{\epsilon-1}}\bigg).
\end{equation}
To control the term  $I(X_t)$ we apply the derivative in time to the quantitity $\partial_{X_{0,\lambda}}\Psi$, it follows  
\begin{equation*}
\left\{ \begin{array}{ll}
\partial_t \partial_{X_0,\lambda} \Psi(t,x) = \nabla v(t,\Psi(t,x)) \partial_{X_0,\lambda} \psi(t,x) &\\
\partial_{X_{0,\lambda}} \Psi(0,x) ={X_{0,\lambda}}.
\end{array} 
\right.
\end{equation*}
The time reversibilty of the previous equation and Gronwall's inequality ensure that 
\begin{equation*}
| X_{0,\lambda}(x)| \leq |\partial_{X_{0,\lambda}}\Psi(t,x)|e^{V_{\mu}(t)}.
\end{equation*}   
From {\bf(ii)} in  Definition \ref{3.1} we get 
\begin{equation}\label{I-X-t}
 I(X_t)\geq I(X_0)e^{-V_{\mu}(t)} > 0.
\end{equation}  
Thanks to \eqref{3.27}, \eqref{3.56} and \eqref{I-X-t}, we have
\begin{equation}\label{Est15}
\Vert \omega_\mu(t)\Vert_{C^{\epsilon}(X_t)} \leq C_0 e^{ C_0 t^{3}}e^{ C_0  V_\mu(t)} .
\end{equation}
According to Theorem \ref{The estimate log} and   Proposition \ref{Prop:1} , we obtain
$$ \Vert \nabla v_\mu(t)\Vert_{L^{\infty}}\leq C\left( C_0+ C_0  \log \left( e+ \frac{\Vert \omega_\mu(t) \Vert_{C^{\epsilon}(X)}}{\Vert \omega_\mu(t) \Vert_{L^{\infty}}} \right) \right).  $$
The  monotonicity of function $ x\longmapsto \log (e+\frac{a}{x})$ gives
$$ \Vert \nabla v_\mu(t)\Vert_{L^{\infty}}\leq  C_0 \left( C_{0}+ C_0 \log \left( e+ \frac{\Vert \omega_\mu(t) \Vert_{C^{\epsilon}(X)}}{\Vert \omega^0_\mu \Vert_{L^{\infty}}} \right) \right).  $$ 
It follows from \eqref{Est15} that
$$ \Vert \nabla v_\mu(t)\Vert_{L^{\infty}}\leq C_0 \Big( (1+t)^{3}+\int_0^t  \Vert \nabla v_\mu(\tau)\Vert_{L^{\infty}}d\tau\Big). $$
Again, Gronwall's inequality gives 
\begin{equation}\label{Lip v}
\Vert \nabla v_\mu(t)\Vert_{L^{\infty}}\leq C_0e^{C_0t }.
\end{equation}
Together with \eqref{Est15}, we have   
\begin{equation}\label{Est16}
\Vert \omega_\mu(t)\Vert_{C^{\epsilon}(X_t)} \leq C_0 e^{ \exp C_0t^{t}} .
\end{equation} 
To control he term $\Psi_\mu$ in $C^\epsilon(X_t)$.  we recall that $\partial_{X_{0,\lambda}} \Psi_\mu(t) = {X_{t,\lambda}}\circ\Psi_\mu(t) $. Thus, we get
\begin{equation*}
 \Vert  {X_{t,\lambda}}\circ\Psi_\mu(t) \Vert_{C^\epsilon} \leq  \Vert  {X_{t,\lambda}}\Vert _{C^\epsilon}{\Vert \nabla \Psi_\mu(t) \Vert}^{\EE}_{L^{\infty}} \leq  \Vert  {X_{t,\lambda}}\Vert _{C^\epsilon} e^{CV_{\mu}(t)},
\end{equation*}
where, we have used ${\Vert \nabla \Psi_\mu(t) \Vert}_{L^{\infty}} \leq e^{CV_{\mu}(t)}$. Consequently,  
\begin{equation}\label{3.31}
\Vert  \Psi_\mu(t) \Vert_{{C^\epsilon}(X_{t})} \leq C_0 e^{ \exp C_0t}.
\end{equation}
The proof of Theorem \ref{Th:1.1} is finished. 


\section{Inviscid limit}
{\bf{Proof of Theorem \ref{Theo1.2}}}      

Taking the difference between \eqref{Eq-3} and \eqref{Eq-1}, by setting $ U=v_\mu -v, \Theta =\theta_\mu -\theta $ and $P =p_\mu-p$ we find out that the triplet $(U,\Theta,P)$ gouverns the following evolution system. 
\begin{displaymath}\label{D-mu}
\left\{ \begin{array}{ll} 
\partial_{t} U+v_\mu \cdot\nabla U-\mu\Delta U =\Delta v-\nabla P+\Theta \vec{e}_2 -U \cdot\nabla  v,  & \\
\partial_{t}\Theta +v_\mu\cdot\nabla \Theta=-U \cdot\nabla  \theta ,  & \\ 
\nabla\cdot U =0, &\\ 
({U},{\Theta})_{| t=0}=({U}_0,{\Theta}_0).\tag{D$_\mu$} 
\end{array} \right .
\end{displaymath}
Multiplying the first equation in the system \eqref{D-mu} by $ U $ and integrating by part over $\RR^2$, such that $\Div v_{\mu}=\Div v=0 $ and H\"older's inequality ensure that
\begin{equation*} 
\frac{1}{2} \frac{d}{dt} \Vert  U(t) {\Vert}^2_{L^2} +\mu \Vert \nabla U(t) {\Vert}^2_{L^2}  \le  \mu \Vert \nabla v (t) \Vert_{L^2}\Vert \nabla U(t) {\Vert}_{L^2} +\Vert \nabla v (t) \Vert_{L^\infty}\Vert U(t) {\Vert}_{L^2}  + \Vert \Theta(t) \Vert_{L^2}\Vert U(t) {\Vert}_{L^2}.
\end{equation*}
Young inequality implies that
\begin{equation*} 
\frac{1}{2} \frac{d}{dt} \Vert  U(t) {\Vert}^2_{L^2} +\mu \Vert \nabla U(t) {\Vert}^2_{L^2}  \le  \frac{\mu}{2} \Vert \nabla v (t) {\Vert}^2_{L^2} + \frac{\mu}{2} \Vert \nabla U(t) {\Vert}^2_{L^2} +\Big(\Vert U(t) {\Vert}^2_{L^2}  + \Vert \Theta(t) {\Vert}^2_{L^2}\Big)\big(\Vert \nabla v (t) \Vert_{L^\infty} +1\Big).
\end{equation*}
Integrating in time this  inequality ,we find 
\begin{equation}\label{U}   
 \Vert  U(t) {\Vert}^2_{L^2} +\frac{\mu}{2} \Vert \nabla U(t) {\Vert}^2_{L^2}  \le \Vert  U^0 {\Vert}^2_{L^2}+ \frac{\mu}{2} \Vert \nabla v  {\Vert}_{L^2_tL^2}  +\int_0^t \Big(\Vert U(\tau) {\Vert}^2_{L^2}  + \Vert \Theta(\tau) {\Vert}^2_{L^2}\Big)\big(\Vert \nabla v (\tau) \Vert_{L^\infty} +1\Big)d\tau.
\end{equation}
 
Similarly for $\Theta-$equation, we have

\begin{equation*}
\frac{1}{2} \frac{d}{dt} \Vert \Theta(t) {\Vert}^2_{L^2} \le  \Vert \nabla \theta (t) {\Vert}_{L^\infty}\Vert  U(t) {\Vert}^2_{L^2}.
\end{equation*}
Integrating in time over $[0,t]$, it follows 
\begin{equation}\label{T}
\Vert \Theta(t) {\Vert}^2_{L^2} \le \Vert \Theta^0{\Vert}^2_{L^2} +\int_0^t \Vert \nabla \theta (\tau) {\Vert}_{L^\infty}\Vert  U(\tau) {\Vert}^2_{L^2} d\tau .
 \end{equation} 
Gathering \eqref{U} and \eqref{T}, one has 
\begin{eqnarray}
\Pi(t)  &\lesssim & \Pi(0) + \int_0^t  \Pi(\tau) \Big(1 +\Vert \nabla \theta (\tau) {\Vert}_{L^\infty}  +\Vert \nabla v(\tau) \Vert_{L^\infty} \Big)d\tau  \nonumber\\  &+& \mu \Vert \nabla v  {\Vert}_{L^2_tL^2},
\end{eqnarray}
with $\Pi(t)=\Vert v_\mu(t) -v (t){\Vert}^2_{L^2} + \Vert \theta_\mu (t)-\theta(t) {\Vert}^2_{L^2 } $, so Gronwall's inequality gives 
\begin{equation*}
\Pi(t) \leq Ce^{t+ V_\mu (t)+V(t)+\Vert \nabla \theta  {\Vert}_{ L^1_tL^\infty}} \Big(\Pi(0) + \mu \Vert \Delta v_\mu \Vert_{ L^2_tL^2}\Big).
 \end{equation*}
Again, H\"older's inequality in time variable and using Proposition \ref{C.Z}, we get
\begin{equation*}
\Pi(t) \leq Ce^{t+ V_\mu (t)+V(t)+\Vert \nabla \theta  {\Vert}_{ L^1_tL^\infty}} \Big(\Pi(0) + (\mu t) \Vert \omega_ \mu \Vert_{ L^\infty_t L^2}\Big).
 \end{equation*}
Thanks to Theorem and Proposition \ref{Prop:1}, we find that
\begin{equation*}
\Pi(t) \leq C_{0}e^{C_{0}t} (\mu t).
 \end{equation*}
Consequently,
$$\Vert v_\mu(t) -v (t){\Vert}_{L^2} + \Vert \theta_\mu (t)-\theta(t) {\Vert}_{L^2 } \leq C_{0} (\mu t)^{\frac{1}{2}}.$$
%\begin{proof} 

This completes the proof of  Theorem \ref{Theo1.2}.

%\end{proof}
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