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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 250, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/250\hfil Navier-Stokes problem]
{Navier-Stokes problem in velocity-pressure formulation:
 Newton linearization and convergence}

\author[A. Younes,  A. Jarray,  M. Bouchiba \hfil EJDE-2014/250\hfilneg]
{Anis Younes, Abdennaceur Jarray, Mohamed Bouchiba}  % in alphabetical order

\address{Anis Younes \newline
 Tunis El Manar University, Faculty of Sciences of Tunis, Tunisia}
\email{younesanis@yahoo.fr}

\address{Abdennaceur Jarray \newline
Tunis El Manar University, Faculty of Sciences of Tunis, Tunisia}
\email{abdennaceur.jarray@gmail.com}

\address{Mohamed Bouchiba \newline
 Carthage University, National Institute of Applied Sciences and Technology,
Tunisia}
\email{mohamed.bouchiba@yahoo.fr}

\thanks{Submitted January 3, 2013. Published December 1, 2014.}
\subjclass[2000]{35J20, 49J96}
\keywords{Navier-Stokes equations; Newton's algorithm; convergence}

\begin{abstract}
 In this article we study the nonlinear Navier-Stokes problem in
 velocity-pressure formulation. We construct a sequence of a Newton-linearized
 problems and we show that the sequence of weak solutions converges towards
 the solution of the nonlinear one in a quadratic way.
\end{abstract}

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


\section{Introduction}

The stationary Navier-Stokes problem may be written in the form
\begin{equation} \label{eQ}
 \begin{gathered}
- \nu \Delta u + (u \cdot \nabla) u  + \nabla p  = f \quad\text{ in } \Omega\\
\operatorname{div}u = 0 \quad\text{in } \Omega\\
u =0 \quad\text{on } \Gamma =\partial \Omega
\end{gathered}
\end{equation}
This equation describes the motion of an incompressible fluid contained
in $\Omega$ and subjected to an outside forces $f$, $u$ is the velocity of fluid
flow, $p$ is the pressure and $\nu$ its viscosity.

The variational formulation of the Navier Stokes equations in the classic
form is well studied in \cite{g1,g2,t1}. In most publications they
uses a trilinear form in the variational formulation for studying the
nonlinear term presented in the equation of momentum.

This paper is devoted to give another idea:  we construct a sequence of
a Newton-linearized problems and we show, using Lax-Milgramm theorem,
that the variational formulation of each one has an unique solution.
We show then that the sequence of weak solutions converges towards the
solution of the nonlinear one in a quadratic way.

The outline of the paper is as follows:
 In Section 2 we start by a Newton-linearisation of the  Navier Stokes
equations. We obtain a sequence of linear problems and we show the existence
of a weak solution.
In Section 3 we show  the quadratic  convergence of the sequence of the
solutions in Theorem \ref{thm3}.
In section 4  the  nonhomogeneous problem is treated.

\section{Linearized problems}

\subsection*{Linearization}
Let $\Omega$  a bounded domain of $\mathbb{R}^{2}$ with Lipschitz-continuous
boundary $\Gamma$, and let
\[
V = \{ v \in  (H_0^{1}(\Omega))^{2} , \operatorname{div} v = 0 \} 
\]
with norm  $\|u\|_V = \max \{\|u_1\|_{H_0^{1}}, \|u_2\|_{H_0^{1}}\} $.
We set $ L_0^{2} = ({L}_0^{2}(\Omega))^{2}$, and
$ H_0^{1}(\Omega) = (H_0^{1}(\Omega))^{2}$ with norm
$\|u\|_{H_0^{1}} = \max \{\|u_1\|_{H_0^{1}}, \|u_2\|_{H_0^{1}}\} $
and $W =  H_0^{1}(\Omega) \times  L_0^{2} $.

The nonlinear term
$$
(u\cdot\nabla) u = \begin{pmatrix}
  u_{1} \frac{\partial u_{1}}{\partial x} + u_{2} \frac{\partial u_{1}}{\partial y }
 \\
  u_{1} \frac{\partial u_{2}}{\partial x} + u_{2} \frac{\partial u_{2}}{\partial y }
\end{pmatrix}
$$
can be written as
\[
(u \cdot \nabla) u = \frac{1}{2} \nabla |u|^{2} + \operatorname{rot} u \wedge u.
\]
To solve \eqref{eQ} we construct a sequence of Newton-linearized problems.
Starting from an arbitrary $u_0 \in H_0^{1}(\Omega)$ and
$p_0 \in L_0^{2}$ we consider
the  iterative scheme:
\begin{equation} \label{eQn+1}
 \begin{gathered}
- \nu \Delta u_{n+1} + (u_{n+1} \cdot \nabla) u_{n}  + (u_{n} \cdot \nabla) u_{n+1}
+ \nabla p_{n+1}  = f_n \quad \text{in } \Omega\\
\operatorname{div} u_{n+1} = 0 \quad \text{in } \Omega\\
 u_{n+1} =0 \quad\text{on } \Gamma =\partial \Omega
\end{gathered}
\end{equation}
where $ f_n = f + ( u_n \cdot\nabla) u_n $. Problem \eqref{eQn+1} is linear.

\subsection*{Variational formulation}
The variational formulation of \eqref{eQn+1} is
\begin{equation} \label{eQVn+1}
 \begin{gathered}
\text{Find   $(u_{n+1},p_{n+1}) \in W $ such  that}\\
a_0(u_{n+1} , v) + a_{n}(u_{n+1} ,v) + a^{n}(u_{n+1} ,v) + b(p_{n+1},v)
= L_n(v)\quad \forall v \in H_0^{1}(\Omega) \\
b(q,u_{n+1}) =0 \quad \forall q \in L_0^{2}
\end{gathered}
\end{equation}
where the bilinear forms $a_0$, $a_{n}$ , $a^{n}$  are given for
$v ,u \in H_0^{1}(\Omega)$ and $ p \in L_0^{2}$  by
\begin{gather*}
 a_0(u,v) = \nu \int_{\Omega} \nabla u  \nabla v \,dx ,\quad
 a^{n}(u,v) =  \int_{\Omega} ( u \cdot\nabla u_{n}) v \,dx ,\\
 a_{n}(u,v) =  \int_{\Omega} ( u_n \cdot\nabla u)\; v \,dx ,\quad
  b(p,v) = \int_{\Omega} \nabla p  v \,dx = -\int_{\Omega} p \operatorname{div}
 v \,dx
\end{gather*}
 and $L_n(v)= \langle f_n, v \rangle$ \
Using Green formula and $\operatorname{div} v = 0$ we have
 $b(p, v) = 0$.

Then we associate  to  \eqref{eQVn+1} the  problem
\begin{equation} \label{ePVn+1}
 \begin{gathered}
\text{Find $u_{n+1} \in V$ such  that}\\
a_0(u_{n+1} , v) + a_{n}(u_{n+1} ,v) + a^{n}(u_{n+1} ,v) = L_n(v)
\quad \forall v \in V\,.
\end{gathered}
\end{equation}

\begin{lemma}\label{lem1}
Problem  \eqref{eQVn+1} is equivalent to  problem \eqref{ePVn+1}.
\end{lemma}

\begin{proof}
Indeed, if $(u_{n+1},p_{n+1})$ is  a solution of problem \eqref{eQVn+1}
 then $u_{n+1}$ is a solution of \eqref{ePVn+1}.
 Reciprocally, if $u_{n+1}$ is a solution of  the problem
 \eqref{ePVn+1} then we  apply de Rham's theorem:
Let $\Omega$ a bounded regular domain of $\mathbb{R}^{2}$ and
$\mathcal{K}$ a continuous linear form on $(H_0^{1}(\Omega))^{2}$.
Then the linear form  $\mathcal{K}$ vanishes on $V$ if and only if
there exists a unique function $p_{n+1} \in L^{2}(\Omega)/\mathbb{R}$
 such that for all $v \in H_0^{1}(\Omega)$,
\[
\mathcal{K}(v) = \int_{\Omega} p_{n+1}  \operatorname{div} v  \,dx \,.
\]


Let the linear form satisfies
\[
 \mathcal{K}(v) = a_0(u_{n+1} , v) + a_{n}(u_{n+1} , v)+ a^{n}(u_{n+1} , v)
 - L_{n} (v) .
\]
 Therefore we have $ \mathcal{K}(v)= 0 $ for all $v \in V  $,
then de Rham's theorem implies that there exists a unique function
$p_{n+1} \in \mathrm{L}^{2}(\Omega)/\mathbb{R}$
 such that
 \[ %\label{e34}
 a_0(u_{n+1} , v) + a_{n}(u_{n+1} , v)+ a^{n}(u_{n+1} , v)
 - L_{n} (v) = \int_{\Omega} p_{n+1}   \operatorname{div} v \,dx \quad
 \forall v \in H_0^{1}(\Omega);
\]
therefore,
\begin{equation*}\label{e35}
a_0(u_{n+1} , v) + a_{n}(u_{n+1} , v)+ a^{n}(u_{n+1} , v) -
 \int_{\Omega} p_{n+1}  div  v  dx  =  L_{n} (v)  \quad \forall
 v \in H_0^{1}(\Omega)\,.
\end{equation*}
Which gives the desired result.
\end{proof}

Let us now show that problem \eqref{ePVn+1} has an unique solution for
each $n$. For this, we need the following lemma.

\begin{lemma}\label{lem1b}
For fixed $u_{n} \in V$ the form $(u, v) \to  a_{n}(u, v)$ and
$(u, v) \to  a^{n}(u, v)$ are  continuous on $H_0^{1}(\Omega)$.
\end{lemma}

\begin{proof}
We have
\begin{gather*}
a_n(u,v)=\sum^2_{i,j=1} \int_{\Omega}u_{n,j}\frac{\partial u_i}{\partial x_j}v_i\,dx,\\
a^n(u,v)=\sum^2_{i,j=1} \int_{\Omega}u_{j}\frac{\partial u_{n,i}}{\partial x_j}v_i
\,dx
\end{gather*}
by Holder's inequality we have
\begin{equation}\label{e1}
\big|  \int_{\Omega}u_{n,j}\frac{\partial u_i}{\partial x_j}v_i\,dx\big|
\leq \|u_{n,j}\|_{L^4}  \|v_i\|_{L^4}  \|\frac{\partial u_i}{\partial x_j}\|_{L^2}
\end{equation}
According to the Sobolev Imbedding Theorem, the space $H^{1}(\Omega)$
is continuously embedded in $L^{4}(\Omega)$. Then  there exists  $C_1>0 $ such that
\begin{equation}\label{e1b}
|a_{n}(u, v)| \leq  C_1  \|u\|_{H_0^{1}(\Omega)}  \|v\|_{H_0^{1}(\Omega)}
\|u_{n}\|_{H_0^{1}(\Omega)}.
\end{equation}
The same result holds with the term $ a^n$,
\begin{equation}\label{e2}
|a^{n}(u, v)| \leq  C_2  \|u\|_{H_0^{1}(\Omega)}  \|v\|_{H_0^{1}(\Omega)}
\|u_{n}\|_{H_0^{1}(\Omega)}.
\end{equation}
\end{proof}

To show the coercivity of the form $a=a_0 + a_{n} + a^n$ we have
the following lemma.

\begin{lemma}\label{lem2}
We have $a_{n}(u, u) = 0$ for all $u \in V$.
\end{lemma}

\begin{proof}
Note that
\begin{equation}\label{e3}
a_{n}(u,u)= \int_{\Omega} (u_n . \nabla) u \; u \,dx
= \frac{1}{2}  \int_{\Omega}  u_{n}  \nabla (|u|^{2}) \,dx
\end{equation}
where
$$
\nabla (|u|^{2})  =  \begin{pmatrix}
    \frac{\partial( u_{1}^{2} +  u_{2}^{2})}{\partial x }  \\
    \frac{\partial (u_{1}^{2}  + u_{2}^{2})} {\partial y}
\end{pmatrix}
$$
 Using Green's formula and $\operatorname{div}u_{n} = 0$ and boundary
conditions we have
\begin{equation}\label{e4}
2 a_{n}(u,u)= \int_{\Omega}  \nabla |u|^{2} u_{n} \,dx
=  -\int_{\Omega} \operatorname{div} u_{n} |u|^{2} \,dx = 0 .
\end{equation}

\end{proof}
 For $ \alpha > 0$, let
$B_{\alpha}= \{ v \in V : \|v\|_{H_0^{1}(\Omega)} \leq \alpha \}$.

\begin{lemma}\label{lem3}
We have
\begin{equation}\label{e5}
  a(u,u) \geq (\nu C_3 - \alpha C_2 ) \|u\|^2_{H_0^{1}(\Omega)} \quad
 \forall u \in V ,\; \forall u_n \in B_{\alpha}
\end{equation}
with  $C_3 = \min[\frac{1}{ 2(C_{p}(\Omega))^{2}} ,\frac{1}{ 2}]$.
\end{lemma}

\begin{proof}
Using \eqref{e4}  we obtain $ a(u,u)= a_0(u,u) + a^n(u,u) $.
By the Poincare inequality,
\begin{equation}\label{e6}
\|u\|_{L^2(\Omega)} \leq C_{p}(\Omega)\|\nabla u\|_{L^2(\Omega)},
\end{equation}
we obtain
\[
 a_0(u,u) = \nu \|\nabla u \|^2_{L^2(\Omega)}\geq
\nu \min[\frac{1}{ 2(C_{p}(\Omega))^{2}} , \frac{1}{2}] \|u\|^2_{H_0^{1}(\Omega)}\,.
\]
 Then
\begin{equation}\label{e7}
 a_0(u,u)   \geq  \nu  C_3 \|u\|_{H_0^{1}(\Omega)}^2\,.
\end{equation}
Using \eqref{e2} we have
\begin{equation}\label{e8}
 a^n(u,u) \leq C_2 \alpha \|u\|^2_{H_0^{1}(\Omega)} \quad \forall u \in V,\;
u_n \in B_{\alpha}
\end{equation}
which gives
 \begin{equation}\label{e9}
  a^n(u,u) \geq - C_2 \alpha \|u\|^2_{H_0^{1}(\Omega)} \quad
\forall u \in V ,\; u_n \in B_{\alpha}
 \end{equation}
 with \eqref{e7} we have the result.
\end{proof}

\begin{lemma}\label{lem4}
For  $\|f\|_{(L^2(\Omega))^2} $ small enough or $\nu $  large enough there is
$ \alpha^{*} >0 $  independent of $n$  such that
$\|u_n\|_{H_0^{1}(\Omega)} \leq  \alpha^{*}$ for all
$n \in \mathbb{N}$ where $ u_n $ is solution of
\eqref{ePVn+1} with $n$ instead of $n+1$.
\end{lemma}

\begin{proof}
 We must have   $(\nu C_3- C_2 \alpha^{*})>0$.
 So we choose $\alpha^{*}<\frac{\nu C_3}{ C_2}$.

Remains to show by induction that if $u_n$ is solution of \eqref{ePVn+1} with $n$
instead of $n+1$,
then $\|  u_n  \|_{H_0^{1}(\Omega)} \leq \alpha^{*}$  for all $n \in \mathbf{N}$.
 Let  $u_0 \in B_{\alpha^{*}}$ and assume that  $u_n  \in B_{\alpha^{*}} $.
We note $ u= u_{n+1}$ is a solution of \eqref{ePVn+1} and
$ \|f\|_2 = \|f\|_{(L^2(\Omega))^2}$.
 We have
\begin{equation} \label{e10}
a(u,u)=L_n(u)=\int_{\Omega}(f +(u_n\nabla) u_n) u\,dx\,.
\end{equation}
Then $a(u,u)\leq(\|f\|_2+C {\alpha^{*}}^2)\|u\|_{H_0^{1}(\Omega)}$.

From \eqref{e5} we obtain 
$(\nu C_3- C_2\alpha^{*})\|u\|_{H_0^{1}}(\Omega) \leq(\|f\|_2+C {\alpha^{*}}^2)$
which gives
\[
{\|u\|}_{H_0^{1}(\Omega)} \leq \frac{\|f\|_2+
C {\alpha^{*}}^2}{(\nu C_3- C_2\alpha^{*})}\,.
\]
So to deduce the result we must have
 $$
 \frac{\|f\|_2+C {\alpha^{*}}^2}{(\nu C_3- C_2\alpha^{*})}\leq\alpha^{*}\,.
 $$
We put
\[
P(\alpha^{*})=(C+C_2){\alpha^{*}}^2-\nu C_3\alpha^{*}+\|f\|_2 \leq 0,\quad \alpha^{*} <\frac{\nu C_3}{ C_2} %(*)
\]
Therefore, the discriminant of the  polynomial $P(\alpha^{*})$  must verify
 \begin{equation}
\Delta=\nu^2 {C_3}^2-4 (C+C_2)\|f\|_2> 0\,.
 \end{equation}
Then
\begin{equation}\label{e11}
\|f\|_2< \frac{\nu^2 {C_3}^2}{4(C+C_2)}
 \end{equation}
 and hence $ P(\alpha^{*})$ has two roots
\[
\alpha_1=\frac{\nu C_3-  \sqrt{\Delta}}{2(C+C_2)},\quad
\alpha_2=\frac{\nu C_3+  \sqrt{\Delta}}{2(C+C_2)}
\]
Since $ \alpha_2 > 0 $ we can choose
$0  < \alpha^{*}  <  \min(\frac{\nu C_3}{ C_2},\alpha_2)$.
\end{proof}

\begin{theorem} \label{thm1}
(1) For $f \in (L^{2}(\Omega))^{2}$ satisfying (\ref{e10}),
 problem \eqref{ePVn+1} has a unique solution $u_{n+1} \in V \cap B_{\alpha^{*}} $.

(2) If $ u_0 \in B_{\alpha^{*}} \cap H^2(\Omega) $, then
 $ u_{n+1} \in H^2(\Omega)$.
\end{theorem}

\begin{proof}
(1) Since $ u_n \in B_{\alpha^{*}}$, we have
$$
|L_n(v)| \leq ( \|f\|_2 + C {\alpha^{*}}^2) \|v\|_{{H_0^{1}(\Omega)}}
$$
 which gives the continuity of  $L_n $ and using Lemma \ref{lem1},
Lemma \ref{lem2} and Lemma \ref{lem3} with Lax-Milgram Theorem we obtain the result.

(2) We assume that $ u_{n} \in H^2(\Omega)$ then
$(u_n\nabla) u_n \in (L^2(\Omega)^2 $, which implies that
 $ f_n= f +(u_n\nabla) u_n \in (L^2(\Omega)^2 $ for $ f \in (L^2(\Omega)^2 $,
and by classical regularity Theorem we have
$ u_{n+1} \in H^2(\Omega)$.
\end{proof}

\section{Convergence}

The sequence $(u_{n})_{n \in \mathbb{N}}$, solutions of \eqref{ePVn+1} with $n$ instead of $n+1$,
satisfy
\begin{equation}\label{e12}
 \|u_{n}\|_{{H_0^{1}(\Omega)}} \leq  \alpha^{*}  \quad  \forall n \geq 0,
\end{equation}
which implies that the sequence $(u_{n})_{n \in \mathbb{N}}$ is bounded in
$H_0^{1}(\Omega)$.
Then there exist a subsequence that converges weakly to $\phi$ in
$H_0^{1}(\Omega)$.
Since the injection of $H_0^{1}(\Omega)$ in $(L^{2}(\Omega))^{2}$ is compact,
there exists a subsequence still noted $u_{n}$ which converges strongly to $\phi$
in  $(L^{2}(\Omega))^{2}$.

We need the following  result.

\begin{lemma}\label{lem5}
For $ v\in V $, we have:
\begin{itemize}
\item[(1)]  $\lim_{n \to \infty} a_0 ( u_{n+1} , v) = a_0 (\phi , v)$;

\item[(2)]  $\lim_{n \to \infty} a_{n} ( u_{n+1} , v) = a_{\infty}
(\phi , v) = \int_{\Omega} (\phi\dot\nabla) \phi v dx$;

\item[(3)] $\lim_{n \to \infty} a^{n} ( u_{n+1} , v) = a^{\infty}
(\phi , v) = \int_{\Omega} (\phi\dot\nabla) \phi v dx$;

\item[(4)] We have $\lim_{n \to \infty} L_{n}(v)
= L_{\infty}(v)= \int_{\Omega} [f + (\phi\dot\nabla) \phi] v dx$.
\end{itemize}
\end{lemma}

\begin{proof}
 (1) Since $ u_n \rightharpoonup \phi $, and by linearity of $ u \to a_0(u,v)$
 we have $ a_0(u_{n+1} , v ) \to a_0(\phi , v)$ for all $v \in V $.

(2) Let
 \begin{equation}\label{e16}
 E=|a^{n}(u_{n+1} , v ) - a^{\infty}(\phi , v)|
= \big| \int_{\Omega} \{ (u_{n+1}\dot\nabla)u_{n} -
 (\phi \dot\nabla) \phi \} v \,dx \big|
\end{equation}
We can write
 \begin{equation}\label{e17}
(u_{n+1}\cdot \nabla)u_{n} - (\phi . \nabla)\phi
 = ((u_{n+1} - \phi ) \cdot \nabla)u_{n} + (\phi \cdot \nabla)(u_{n} -\phi)
\end{equation}
which gives with $ u_n \in H^{2}(\Omega)$ and using Green's theorem,
 \begin{equation}
E  \leq  C  [\|u_{n+1} - \phi \|_{2}  \|u_{n}\|_{H^{1}}\|v\|_{H^{1}}
+ \| u_{n} - \phi \|_{2}( \|\nabla v\|_{H^{1}}\|\phi\|_{H^{1}}
+ \|\nabla \phi\|_{H^{1}}\|v\|_{H^{1}} )].
\end{equation}
Since $u_{n}$  converges strongly to $\phi$  in  $(L^{2}(\Omega))^{2}$,
it follows that $ E \to 0 $.

\item[(3)] Let
\[
F= |a_{n}(u_{n+1} , v ) - a_{\infty}(\phi , v)|
=  \int_{\Omega} \{ (u_{n}\cdot\nabla)u_{n+1} -
 (\phi \cdot\nabla) \phi \} v \,dx\,.
\]
Then
 \begin{equation}
F \leq  C  [\|u_{n} - \phi \|_{2}   \|u_{n+1}\|_{H^{1}}\|v\|_{H^{1}} +
  \| u_{n+1} - \phi \|_{2}( \|v\|_{H^{1}}\|\nabla \phi\|_{H^{1}}
+\|\phi\|_{H^{1}}\|\nabla v\|_{H^{1}})];
\end{equation}
thus $ F \to 0 $.

\item[(4)] Let
\[
G=|L_n( v ) - L_{\infty}( v )| \leq  \int_{\Omega} |(u_n \nabla u_{n})
- (\phi \nabla \phi )| |\nabla v |\,dx.
\]
Then
\begin{equation}\label{e18}
G \leq C [ \| u_{n} -  \phi\|_{2} (\|u_n\|_{H^{1}}\|v\|_{H^{1}}
  + \|\phi\|_{H^{1}}  \|\nabla v\|_{H^{1}}+
  \|\nabla \phi\|_{H^{1}}  \|v\|_{H^{1}})].
  \end{equation}
Then Lemma \ref{lem5} gives the desired result.
\end{proof}

For using de Rham's Theorem, let  $\mathcal{L}$ a continuous linear form
on $(H_0^{1}(\Omega))^{2}$
which vanishes on $V$ if and only if there exists a unique function
$\varphi \in L^{2}(\Omega)/\mathbb{R}$
 such that for all $v \in H_0^{1}(\Omega)$,
\[
\mathcal{L}(v) = \int_{\Omega} \varphi  \operatorname{div} v \,dx .
\]

\begin{theorem} \label{thm2}
 We have $\lim_{n \to \infty} u_{n} = \phi$ in $V$ then $ \phi$
is a solution of \eqref{eQ}.
\end{theorem}

\begin{proof}
It follows from Lemma \ref{lem5} that
 $$
\lim_{n \to \infty} a_0 ( u_{n+1} , v ) + a_{n} (u_{n+1} , v)
+ a^{n} (u_{n+1} , v)
 = a_0(\phi , v) + 2 a_{\infty}(\phi , v) = L_{\infty}(v)\,.
$$
Let the linear form
$\mathcal{L}(v) = a_0(\phi , v) + a_{\infty}(\phi , v)+ a^{\infty}(\phi , v)
- L_{\infty} (v) $.
Therefore  $ \mathcal{L}(v)= 0 $ for all $v \in V  $,
then de Rham's theorem implies that there exists a unique function
$p \in \mathrm{L}^{2}(\Omega)/\mathbb{R}$  such that
 \begin{equation}\label{e26}
 a_0(\phi , v) + 2 a_{\infty}(\phi , v) - L_{\infty} (v)
=\int_{\Omega} p   \operatorname{div} v \,dx \quad \forall  v \in H_0^{1}(\Omega)
\end{equation}
which gives
\begin{gather}\label{e27}
\nu \int_{\Omega} \nabla \phi \;\nabla v \,dx +  \int_{\Omega}
(\phi \dot\nabla) \phi  v \,dx -
 \int_{\Omega} p  \operatorname{div} v \,dx
 =  \int_{\Omega} f   v \,dx  \quad \forall  v \in H_0^{1}(\Omega),\\
\label{e28}
\int_{\Omega} ( - \nu \Delta \phi  +   (\phi \dot\nabla) \phi   + \nabla p
 - f )  v \,dx  = 0 \quad   \forall   v   \in   H_0^{1}(\Omega)\,.
\end{gather}
Then in $\mathcal{D}'(\Omega)$,
\begin{equation}\label{e29}
- \nu \Delta \phi  +   (\phi \dot\nabla) \phi   + \nabla p  - f   = 0\,.
\end{equation}
Since $\phi \in V$ we  conclude that $\phi$ is the solution of \eqref{eQ}.
\end{proof}

\begin{theorem}\label{thm3}
Let $ u_{n+1}$ be the solution of \eqref{eQVn+1}, and  $\phi$ be the
 solution of \eqref{eQ}. Then convergence of the sequence
$(u_{n+1})_{n \in \mathbb{N}}$ towards $ \phi $ is quadratic; i.e.,
\begin{equation}\label{e30}
 \|u_{n+1}- \phi\|_{H_0^{1}(\Omega)} \leq C_{2} \|u_{n}- \phi\|_{H_0^{1}(\Omega)}^2
\end{equation}
\end{theorem}

\begin{proof}
Let $ \omega_n = u_n - \phi $ and $ \chi_n = p_n -p $.
 Subtracting problem \eqref{eQn+1} from  \eqref{eQ} we obtain
\begin{equation} \label{eDn+1}
\begin{gathered}
- \nu \Delta \omega_{n+1} + (\omega_{n+1}\nabla) u_n
+ ( u_n \nabla ) \omega_{n+1} + \nabla \chi_{n+1}
 = (\omega_{n}\nabla)\omega_{n} \quad \text{in } \Omega\\
  \operatorname{div}\omega_{n+1} = 0 \quad \text{in } \Omega\\
 \omega_{n+1}  = 0\quad \text{on } \Gamma
\end{gathered}
\end{equation}
The variational formulation of \eqref{eDn+1} is
\begin{equation} \label{eDVn+1}
\begin{gathered}
\text{Find $(\omega_{n+1}, \chi_{n+1}) \in W $ such  that} \\
a(\omega_{n+1} , v) + b(\chi_{n+1},v) = F_n(v)\quad \forall v \in H_0^{1}(\Omega) \\
b(q,\omega_{n+1}) =0 \quad \forall q \in L_0^{2},
\end{gathered}
\end{equation}
where $ a=a_0+a^n+a_n$ and
\[
 b(q,\omega_{n+1}) =  -\int_{\Omega} q \operatorname{div} \omega_{n+1} \,dx,\quad
F_n (v) = \int_\Omega (\omega_{n}\nabla)\omega_{n}  v \,dx .
\]
Since $\operatorname{div} \omega_{n+1} = 0$, using  Lemma\ref{lem1}
and Lemma \ref{lem4}, for $ u_n \in B_{\alpha^{*}}$ and $ v=\omega_{n+1}$,
we obtain
\begin{equation}\label{e31}
\begin{aligned}
 ( \nu C_1  - C \alpha^{*} ) \|\omega_{n+1}\|_{H_0^{1}(\Omega)}^2
&\leq  a ( \omega_{n+1} ,\omega_{n+1}) = F_(\omega_{n+1}) \\
&\leq  C \|\omega_{n}\|_{H_0^{1}(\Omega)}^2 \|\omega_{n+1}\|_{H_0^{1}(\Omega)}\,.
\end{aligned}
\end{equation}
This gives \ref{e29}, with $C_{2} = \frac{C}{( \nu C_1  - C \alpha^{*} )} $
 and the convergence is quadratic.
\end{proof}

\section{Nonhomogeneous problem}

We are concerned now with the   nonhomogeneous problem
\begin{equation} \label{eP}
 \begin{gathered}
- \nu \Delta u + (u\cdot\nabla) u  + \nabla p  = f \quad\text{in } \Omega\\
 \operatorname{div}u = 0 \quad \text{in } \Omega\\
 u =g \quad \text{on } \Gamma
\end{gathered}
\end{equation}
Where the state $ u $ is sought in the space $ (H^1(\Omega))^2 \cap V $.

Throughout this section $ \Omega $ denotes a bounded domain in
$ \mathbb{R}^2$, with Lipschitz-continuous boundary
 $ \Gamma= \cap \Gamma_i$ $i=1,\dots,4 $.
We assume in this section that
\begin{equation}\label{e32}
 \int_{\Gamma_i} g.n_i \; d\sigma =0  \quad \text{with }
 g \in H= (H^{1/2}(\Gamma))^2  \text{ and }f \in K=(H^{-1}(\Omega))^2.
\end{equation}
We assume also that for a given $ g \in H $ satisfying \ref{e30},
for any $c>0$ there exists  a function
$ w_0 \in (H^1(\Omega))^2$ such that
\begin{gather}\label{e33}
\operatorname{div} w_0 = 0, \quad w_0{|\Gamma} = g ,\\
\label{e34}
  |a_n(w_0,u_n)| \leq c \|u_n\|^2_{H_0^{1}(\Omega)}  \quad \forall  u_n   \in V .
\end{gather}

The existence of $ w_0 $ satisfying \ref{e30}, \ref{e31}
is a technical result due to Hopf \cite{h1}.

\begin{theorem}\label{thm4}
Given $ (g,f) \in K \times H $ satisfying \ref{e31}, there exists
a pair $ (u, p) \in (H^1(\Omega))^2 \times L^2_0(\Omega) $ which is a
solution of \eqref{eP}.
\end{theorem}

\begin{proof}
 Let $ \xi_0=u_0 - w_0 $ where  $ w_0 $ verify \ref{e30}, \ref{e31}
and an arbitrary $ u_0 \in V$.
We consider  the sequence of linear problems
\begin{equation} \label{eFn+1}
\begin{gathered}
- \nu \Delta \xi_{n+1} + (\xi_{n+1}. \nabla) \xi_{n}
+  (\xi_{n}\cdot \nabla) \xi_{n+1}+\nabla p_{n+1}
 = \mathfrak{f_n} \quad \text{in } \Omega\\
\operatorname{div} \xi_{n+1} = 0 \quad  \text{in } \Omega\\
 \xi_{n+1} =0 \quad \text{on } \Gamma
\end{gathered}
\end{equation}
with $ \xi_{n+1}=  u_{n+1} - w_0 $,
$u_{n+1} \in H_0^{1}(\Omega) $ and
$ \mathfrak{f_n}= f + (\xi_n \nabla) \xi_n
 + \nu \Delta w_0 - (w_0\cdot \nabla) w_0 $.

Then $ \xi_{n+1} $ is a solution of the variational problem
\begin{equation} \label{eFVn+1}
\begin{gathered}
\text{Find $ \xi_{n+1} \in V$ such  that}\\
a(\xi_{n+1},v) =  L_n( v )\quad  \forall v \in V\,,
\end{gathered}
\end{equation}
where $a(\xi,v)=a_0(\xi , v) + a_{n}(\xi ,v) + a^{n}(\xi ,v) +a_{\star}(\xi,v)$
with
\[
a_{\star}(\xi,v)=  \int_{\Omega} ( \xi \dot\nabla ) w_0  v \,dx
+  \int_{\Omega} ( w_0 \dot\nabla ) \xi  v \,dx
\]
and  $ L_n(v) = \langle \mathfrak{f_n},v \rangle $.

Taking  $ c >  \nu $ and using \ref{e5} we obtain
\begin{equation}\label{e35b}
  |a(\xi_{n+1},\xi_{n+1})| \geq (\nu - c) \|\xi_{n+1}\|^2_{H_0^{1}(\Omega)}
\end{equation}
Thus we have the coercivity and $ L $ is obviously continuous on $ V$.
We observe that problem
\eqref{eFVn+1} fits into the framework of section 1 and therefore
the sequence $ \xi_{n}$ converges towards a solution of \eqref{eP}.
\end{proof}

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\end{document}