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\roman{•}
\LTR
\title{error estimate of the Linear Cahn-Hilliard-Cook equation in the semidiscrete case }




\begin{document}

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		\hyperref[sec:Intro]{Introduction and definition  of  functional analys} 
	    \vskip 1cm
		\hyperref[sec:section2]{semigroup approach and Hilbert schmidt operator }
\vskip 1cm
	\hyperref[sec:section3]{stochastic calculus}
	\vskip 1cm
	\hyperref[sec:section4]{Cahn-Hilliard-Cook equation}
 \vskip 3cm
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\end{staticcontents*}
\begin{plainslide}
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\centering% \LARGE
\color[rgb]{0,0.6,0}{\huge\titr{\makeatletter\@title\makeatother}}
 \vskip 1cm
\color{black}{By: Bahareh Naseri}
\vskip 1cm
supervisor: Dr. Fardin Seadpanah
\vskip 1cm
Advisor: Dr. Morad Ahmadnasab
\vskip 1cm
Department of Mathematices
\vskip 1cm
University of Kurdestan
\vskip 1cm
2015
\distance{2}
%	\tableofcontents
\vskip 1cm
\end{plainslide}
1. Introduction and definition  of  functional analys
\begin{enumerate}
\item[-]
$\mathcal{C}^k $ spaces
\item[-]
$H^k$ spaces
\item[-]
Laplacian operator
\item[-]
$\dot{H}^k$ spaces
\item[-]
discrete Laplacian operator
\item[-]
orthogonal projection
\item[-]
Rits projection
\end{enumerate}
2. semigroup approach and  Hilbert schmidt operator\

3. stochastic calculus\
\begin{enumerate}
\item[-]
Notation and definition
\item[-]
Brownian motion
\item[-]
$-Q$ Winer process
\item[-]
stochastic Integral
\end{enumerate}
4. Cahn-Hilliard-Cook equation
\begin{enumerate}
\item[-]
Notation and definition
\item[-]
Linearized Cahn-Hilliard-Cook equation
\item[-]
Weak form 
\item[-]
Spatial semidiscrete form
\item[-]
error estimate for the Linearized Cahn-Hilliard-Cook equation in the semidiscrete case
\end{enumerate}
\end{plainslide}
\roman{•}
\LTR
\section{Introduction and definition  of  functional analys} \label{sec:Intro}

\begin{plainslide}
\mytarif{\begin{definition} 
\roman{•}
\LTR
)$\mathcal{C}^k $ spaces):\\

for
 $M \in \mathbb{R}^d$ 
 we denoted by 
 $\mathcal{C}(M)$ the linear space of continuous functions on $M$.\\
 
$\mathcal{C}_b(M) \subset \mathcal{C}(M) $
 the space of all bounded functions of 
 $\mathcal{C}(M) $ 
 made a normed linear space by \\
 \begin{center}
 $ \|v\|_{\mathcal{C}(M) } = sup_{x \in M} |v(x)| $
 \end{center}
\end{definition}}

\mytarif{\begin{definition}
\roman{•}
\LTR
)$H^k$ spaces):\\

$H^{k}=H^{k}(\Omega)=\lbrace v\in L_{2}\vert\,D^{\alpha}v \in L_{2},\quad |\alpha| \leq k\rbrace.$\\

$\alpha=(\alpha_{1},...,\alpha_{d})\,$\\ 

$|\alpha|=\sum_{i=1}^{d}\alpha_{i}$\\


$D^{\alpha}v=\dfrac{\partial^{|\alpha|} v}{\partial x_{1}^{\alpha_{1}}...\partial x_{d}^{\alpha_{d}}},$\\


$(v,w)_{k}=(v,w)_{H^{k}}=\sum_{|\alpha|\leq k}\int_{\Omega} D^{\alpha}vD^{\alpha}w dx=\sum_{|\alpha|\leq k}\Big(D^{\alpha}v,D^{\alpha}w\Big).$\\
 

$\|v\|_{k}=\|v\|_{H^{k}}=(v,v)^{\frac{1}{2}}_{{H^{k}}}=\Big(\sum_{|\alpha|\leq k}\int_{\Omega}(D^{\alpha}v)^{2}dx\Big)^{\frac{1}{2}}.$\\ 


$H_{0}^{1}(\Omega)=\lbrace v \in H^{1}(\Omega)\ , v|_\Gamma =0 \rbrace,$\\ 

\end{definition}}

\end{plainslide}

\begin{plainslide}
\mytarif{\begin{definition}
\roman{•}
\LTR
) Laplacian operator):\\

$\Lambda =-\Delta$\\

$\Lambda$ is symetric, positive definite and self adjoint with domain $D(\Lambda)=H^(\Omega) \cap H_0^1(\Omega)$\\

$(\Lambda u,v)=(\nabla u,\nabla v),\forall u,v \in H_0^1.$  (green theorem)\\
\begin{equation*}
\Lambda \varphi_j =\lambda_j \varphi_j, \j=1,\cdots,\infty : \left\{
\begin{array}{ll}
\lbrace \lambda_j \rbrace^\infty_{i=1} \quad  & \quad  are\ eignvalus \\
 with & \quad  0=\lambda_0 \leq \lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_j \leq \cdots \rightarrow \infty \\
\lbrace \varphi_j \rbrace^\infty_{i=1} &\quad  are\ eignfunctions\ (ONB)\ in $H_{0}^{1}$ and $\varphi_{0}=|D|^{\frac{1}{2}} $
\end{array}\right.
\end{equation*}

$\Lambda^\alpha v=\sum^\infty_{j=1} \lambda_j^\alpha (v,\varphi_j)\phi_j$\\

\end{definition}}
\begin{plainslide}
\mytarif{\begin{definition}
\roman{•}
\LTR
)$\dot{H}^k$ spaces) :\\

$\dot{H}^{\alpha}=D(\Lambda^ \frac{\alpha}{2})=\lbrace v \in H^\alpha \vert \Delta^j v|_\Gamma =0, j<\frac{\alpha}{2} \rbrace \quad  \forall  \alpha \in \mathbb{R} $\\


$\Vert v \Vert_{\dot{H}^{\alpha}}=\vert v\vert_{\alpha}=\Vert \Lambda^ \frac{\alpha}{2} v \Vert =(\Lambda^ \alpha v,v)^\frac{1}{2}=(\Big\sum^\infty_{j=1} \lambda_j^\alpha (v,\phi_j)^2)^\frac{1}{2}, \quad \forall v \in \dot{H}^{\alpha}$\\



$|V|_{\alpha} =\| \Lambda^\frac{\alpha}{2} v\|$
\end{definition}}
\end{plainslide}
\mytarif{\begin{definition}
\roman{•}
\LTR
(discrete Laplacian operator):\\

$S_h= \lbrace v \in \mathcal{C}(\overline{\Omega}), v linear in K for each K \in \tau_h, v=0 on \Gamma \rbrace$  $N_h=\dim (S_h)$\\

$ A_h: S_h \rightarrow S_h $ be the discrete Laplacian operator such that\\

$(A_h v_h,\chi)=(\nabla v_h,\nabla \chi) \quad \forall v_h,\chi \in S_h$\\

$A_h \varphi_{h,j}=\lambda_{h,j} \varphi_{h,j};   \lbrace \lambda_{h,j}, \varphi_{h,j}\rbrace_{j=1}^{N_h}$ are eignpairs and $\varphi_{h,0}=\varphi_{0}$ \\

\end{definition}}
\mytarif{\begin{definition}
\roman{•}
\LTR
(orthogonal projection):\\

$(\mathcal{P}_{h}:\dot{H}^{0}\rightarrow S_h)$\\ 


$\left(\mathcal{P}_{h}v,\chi\right) =\left( v,\chi\right),\hspace{0.4cm} \forall \chi \in S_{h},v\in \dot{H^{0}},$\\ 


stability property of $ \mathcal{P}_{h}$ on $S_h$ :\\

$$\|\mathcal{P}_h g\|_{L_p(\mathcal{D})}\hspace{0.1cm}  \leq \hspace{0.1cm}C \|g\|_{L_p( \mathcal{D})},
\hspace{0.5cm}g \in L_{p\left( \mathcal{D}\right)},$$
\end{definition}}
\end{plainslide}
\begin{plainslide}
\mytarif{\begin{definition}
\roman{•}
\LTR
(Rits projection):\\ 

$ \mathcal{R}_{h}: \dot{H^{1}} \rightarrow S_{h}$\\ 


$\left(\nabla \mathcal{R}_{h}v,\nabla \chi\right) =\left( \nabla v,\nabla \chi\right),\hspace{0.4cm} \forall \chi \in S_{h}$\\ 


\begin{theorem}
$we\ have\ the \following\ error\ estimates:$
\begin{quotation}
& \|\left(\mathcal{ R}_{h}v-v\right)\|\leq C h^{\beta} \|v\|_{\beta},\quad \forall v \in \dot{H}^{1}
\end{quotation}
\begin{quotation}
&\|\left(\mathcal{ P}_{h}v-v\right) \| \leq C h^{\beta} \|v\|_{\beta},\quad \forall v \in \dot{H}^{1}
\end{quotation}\\
\end{theorem}
\end{definition}}
 \end{plainslide}
 \roman{•}
\LTR
 \section{semigroup approach and Hilbert schmidt operator} \label{sec:Intro}
 \begin{plainslide}
 \roman{•}
\LTR
semigroup approach:\\
let:
\begin{enumerate}
\item[*]
$X$ be a banach space.\\

\item[*]
$\left E\left( t\right) \right ,0\leq t <\infty, $  a family of bounded linear operator on $X$ then
\end{enumerate}

$\left( E\left( t\right):X \rightarrow X\right)$ is a semigroup of bounded linear operator on $X$ if:
\begin{enumerate}
\item[1.]
$\,E\left( 0\right)=I,$              )$I$ is identity operator on $X$(\\
\item[2.]
$\,E\left( t+s\right)=E\left( t\right)E\left( s\right),\hspace{0.4cm}\forall t,s\geq0.$       (semigroup property)
\end{enumerate}
$A$ is the generator of semigroup defined by\\
\begin{center}
$Ax=\lim_{t \downarrow 0}\dfrac{E\left(t\right) x-x}{t}=\dfrac{d^{+}E\left( t\right) x}{dt},
\hspace{0.4cm}\forall x\in D\left( A\right),$\\ 
\end{center} 
with domain
\begin{center}
 $ D\left( A\right)=\left\lbrace   x \in X: \lim_{t \downarrow 0}\dfrac{E\left(t\right) x-x}{t} وجود داشته باشد\right\rbrace $\\ 
\end{center} 
the  $C_{0}-$ semigroup defined by:
$\lim_{t\downarrow 0} E\left( t\right) x=x,\hspace{0.4cm}  \forall x \in X.$ \\

\mymesal{\begin{example}
 \roman{•}
\LTR
$E(t)= e^{-t \Lambda^2}$ is an example of Analytic semigroup .
\end{example}}

for $v \in H$ we define :
\begin{center}
$e^{-t \Lambda^2} v= \sum_{j=0}^{\infty} e^{-t \lambda_{j}^2} (v,\varphi_{j})\varphi_{j} $
\end{center}

\begin{theorem}
let $-A^2$ is an infenitisimal generator of $-C_{0}$ semigroup $E(t)= e^{-tA^2}$ then:
$$\|A_{h}^{2\beta} E_{h}(t) P_{h} Pv\| + \| A^{2\beta} E(t) Pv\| \leq c t^{-\beta} e^{-ct} \|v\|$$
$$ \int_{0}^{t} \|A_{h} E_{h}(s) P_{h} Pv\|^2 ds +  \int_{0}^{t} \|A E(s) P v\|^2 ds \leq c  \|v\|^2 $$
\end{theorem}
 \end{plainslide}
 \begin{plainslide}
 \roman{•}
\LTR
Hilbert schmidt operator:\\

Let $U,H$ be separable Hilbert spaces:\\

$T:U \rightarrow H$ , bounded linear operator\\

the space of these shown by $\mathcal{L} (U,H)$\\ 

$T \in \mathcal{L} (U,H)$ is Hilbert schmidt operator if $\sum_{k=1}^{\infty} \|Te_k\|_H^2 <\infty,$\\

where $\left\lbrace e_k \right\rbrace _{k=1}^{\infty}$ are arbitrary ONB of $U$ \\ 

thay are denoted by $\mathcal{L}_2(U,H)$\\

with scalar product and corresponding norm : \\ 

 $(T,S)_{\mathcal{L}_2(U,H)} = \sum_{k=1}^{\infty} (T e_k,Se_k)_H,\hspace{1.2cm}$\\
 
 $\|T\|_{\mathcal{L}_2(U,H)} =\|T\|_{HS}= (\sum_{k=1}^{\infty} \|Te_k\|_H^2)^\frac{1}{2}.$\\

 \end{plainslide}
 \roman{•}
\LTR