\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} \begin{thebibliography}{99} \bibitem{a1} H. Amann, M. G. Crandall; \emph{On some existence theorems for semi-linear elliptic equations}, M.R.C. Tech. Report 1772, Madison Wisconsin. 1977. \bibitem{a2} M. Amara, D. Capatina-Papaghiuc, E. Chacon-Vera, D. Trijullo; \emph{Vorticity velocity pressure formulation for Navier- Stokes equations}, Comput. Vis. Sci. 6(2004), 47-52. \bibitem{a3} P. Auscher, S. Dubois, P. Tchamitchian; \emph{On the stability of global solutions to Navier-Stokes equations in the space}, Journal de mathematiques pures et appliquees 83, 673-697 (2004) \bibitem{b1} C. Bernardi, B. Metivet, R. Verfurth; \emph{Analyse numerique d'indicateurs d'erreur}, Chap. 8 in Maillage et adaptation, P.-L. George ed., Hermes (2001), 251-278. \bibitem{b2} C. Bernardi, Y. Mady, F. Rapetti; \emph{Discretisations variationnelles de problèmes aux limites elliptiques}, Springer-Verlag Berlin Heidelberg (2004) . \bibitem{c1} P. Constantin, C. Foias; \emph{Navier Stokes Equations}, The University of Chicago Press, Chicago, (1988). \bibitem{d1} F. Dubois, M. Salaun, S. Salmon; \emph{orticity–velocity–pressure and stream function-vorticity formulations for the Stokes problem}, J. Math. Pures Appl. 82 (2003), 1395–1451. \bibitem{g1} D. Gilbarg , N. S. Trudinger; \emph{Elliptic Partial Differential equation of second order}, Springer-Verlag Heidelberg (1983). \bibitem{g2} V. Girault, P.-A. Raviart; \emph{Finite Element Methods for Navier-Stokes Equations}, Springer, 1986. \bibitem{g3} M. Gunzburger, S. Manservisi; \emph{Analysis and approximation of the velocity tracking problem for Navier Stokes flows with distributed controls}, SIAM J. Numer. Anal. 37 (5) (2000) 1481 to 1512. \bibitem{h1} E. Hopf; \emph{On Non-Linear Partial Differential Equations}. Lecture Series of the Symp. on partial Diff. Equations Berkeley. (1955). \bibitem{r1} Jean-Emile Rakotoson, Jean-Michel Rakotoson; \emph{Analyse fonctionnelle appliquee aux equations aux derivees partielles}, Presses universitaires de France Paris (1999) \bibitem{r2} J. P. Raymond; \emph{Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions}, Ann. Inst. H. Poincare Anal. Non Lineaire, 24(6):921–951, 2007. \bibitem{r3} J. C. de los Reyes, R. Griesse; \emph{State-constrained optimal control of the three-dimensional stationary Navier-Stokes equations}, J. Math. Anal. Appl. 343 (2008) 257-272. \bibitem{w1} J. Wang, X. Wang, X. Ye; \emph{Finite element methods for the Navier–Stokes equations by H(div) elements}, J. Comput. Math. 26 (2008) 1–28. \bibitem{t1} R. Temam; \emph{Navier-Stokes Equations, Theory and Numerical Analysis}, North-Holland, Amsterdam, 1984. \bibitem{y1} A. Younes, S. Abidi; \emph{The Dirichlet Navier-Stokes Problem in the Velocity-Pressure Formulation}, International Journal of Applied Mathematics, Volume 24 No. 3 2011, 469-477. \end{thebibliography} \end{document}