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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 147, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/147\hfil Existence and regularity of solutions]
{Existence and regularity of solutions for Stokes systems with non-smooth boundary
data in a polyhedron}

\author[H. Adjal, M. Moussaoui, A. Zine \hfil EJDE-2017/147\hfilneg]
{Houria Adjal, Mohand Moussaoui, Abdelmalek Zine}

\address{Houria Adjal \newline
Laboratoire d'Analyse Non Lin\'eaire et Histoire des Maths,
E.N.S, B.P. 92 Vieux Kouba
16050 Algiers, Algeria}
\email{houria.adjal@yahoo.com}

\address{Mohand Moussaoui \newline
Laboratoire d'Analyse Non Lin\'eaire et Histoire des Maths,
E.N.S, B.P. 92 Vieux Kouba
16050 Algiers, Algeria}
\email{mmohand47@gmail.com}

\address{Abdelmalek Zine  (corresponding author)\newline
Univ. Lyon, Institut Camille Jordan,
 Ecole centrale de Lyon,
36 avenue Guy de Collonge, 69134 Ecully, France}
\email{abdel-malek.zine@ec-lyon.fr}

\dedicatory{Communicated by Mokhtar Kirane}

\thanks{Submitted October 24, 2016. Published June 22, 2017.}
\subjclass[2010]{35Q35, 76B03, 76N10}
\keywords{Stokes system; polyhedron; non smooth boundary data;
\hfill\break\indent existence and regularity}

\begin{abstract}
 The Stokes problem is fundamental in the study of fluid flows.
 In the case of smooth domains and data, this problem is extensively studied
 in the literature. But there are only a few
 results for  non-smooth boundary data.
 In \cite{MousZin98}, there are  some promising results in the
 2 dimensional case.  The aim of this work is to extend those
 results to a polyhedron domain with non-regular data.
\end{abstract}

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

\section{Introduction}

Let $\Omega$ be a convex polyhedron of $\mathbb{R}^3$ with boundary $\Gamma$. The steady,
creeping flow of an incompressible fluid is governed by Stokes system
\begin{equation} \label{eS}
\begin{gathered}
- \operatorname{div}( 2 \eta \mathbf{d}(\mathbf{u}) - p {\boldsymbol{\delta}} )  
=  \mathbf{f} \quad \text{in } \Omega, \\
\operatorname{div}\mathbf{u}  =  0 \quad \text{in } \Omega, \\
\mathbf{u}  =  \mathbf{g} \quad \text{on } \Gamma.
\end{gathered}
\end{equation}

Where $ \mathbf{u} $ the velocity field and $ p $ the pressure are the unknowns of 
the problem.
$\mathbf{f} $ and $ \mathbf{g} $ are given functions respectively defined on $\Omega$ 
and its boundary $\Gamma$, and
respectively representing the inertia forces and boundary data.
Finally, $\mathbf{d}(\mathbf{u}) = (\nabla \mathbf{u} + \nabla^t \mathbf{u} )/2 $ is the strain rate tensor, 
$ \boldsymbol{\delta} $ the
identity tensor and $ \eta $ the viscosity of the fluid, supposed to be constant
(Newtonian fluid).

In the bi-dimensional framework, the authors obtained in previous 
work \cite{MousZin98}, some promising results on the existence and regularity 
of the solution to the system \eqref{eS}. 
More precisely, the boundary $\Gamma$ is supposed to be a set of segments 
$\Gamma_i=]S_i, S_{i+1}[$, $i=1, \dots, N$ and the data 
$\mathbf{g}_{|\Gamma_i} \in (H^s(\Gamma_i))^2$
with $-1/2 < s < +1/2$.

This work represents a generalization of the results obtained in \cite{MousZin98}.
It concerns the existence and regularity results of solutions to non-homogeneous 
Stokes system in a polyhedron with non enough regular data $ \mathbf{g} $ on the boundary. 
More precisely, we assume that $\Omega$ is a convex polyhedron and it is supposed that
its
boundary $\Gamma$ is composed of surfaces $F_i, i=1, \dots, N$:
\[
\Gamma = \cup_{i=1}^N F_{i}, \quad
\Gamma_i = \partial F_i = \cup_{j=1}^{N_i} \Gamma_{ij}.
\]
For $j=1, \dots, N_i $,
$\Gamma_{ij}$ stands for the edges of the $i^{\text{th}}$
face $F_i$ see Figure \ref{fig1}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig1.pdf}
\end{center}
\caption{Boundary of the domain }
\label{fig1}
\end{figure}

For each face $F_i$, we define a local direct basis 
$ ( \mathbf{n}^i, \boldsymbol{\tau}_1^i, \boldsymbol{\tau}_2^i ) $,
$ \mathbf{n}^i $ being the outward unit normal to $F_i$ and 
$ ( \boldsymbol{\tau}_1^i, \boldsymbol{\tau}_2^i) $ a basis of the tangent plan containing the face $ F_i $.

Note that for a Newtonian fluid, the viscosity is constant. 
Then, dividing by $ \eta > 0$,
the first equation of \eqref{eS} becomes
\[
-\Delta \mathbf{u} + \nabla p = \mathbf{f}, \quad \text{in } \Omega.
\]

To simplify, it is assumed that $\mathbf{f}=0$ in $ \Omega$. Given a family of 
distributions
$ \mathbf{g}_i = ( g_1^i,g_2^i,g_3^i ) $ defined on $F_i$. Our purpose is to show
existence, uniqueness and regularity results for $(\mathbf{u},p)$ solution to the 
Stokes problem
\begin{gather}
- \Delta \mathbf{u} + \nabla p  =  0, \quad \text{in  } \Omega, \label{eq:0.1} \\
\operatorname{div}\mathbf{u}  =  0, \quad \text{in } \Omega, \label{eq:0.2} \\
\mathbf{u}  =  \mathbf{g}_i, \quad \text{on } F_{i},\; i=1,\dots,N,\label{eq:0.3}
\end{gather}
where \eqref{eq:0.1} and \eqref{eq:0.2} are verified in the distribution sense and
\eqref{eq:0.3} is verified in generalized sense of traces. 

Let $ \mathbb{H}^s(\cdot)$ denote the product space $(H^s(\cdot))^3$, where 
$H^s(\cdot)$ denotes the usual Sobolev space (See e.g. \cite{Grisvard85}).

\begin{remark} \label{rmk1.1} \rm
Let $ \mathbf{g} = (\mathbf{g}_1, \mathbf{g}_2, \dots, \mathbf{g}_N )$. We then have the 
classical result:

If $ \mathbf{g} \in \prod_{i=1,N} \mathbb{H}^{1/2}(F_i)$ and $ \mathbf{g} $ is the
trace of a function of
$\mathbb{H}^1(\Omega)$
then the previous Stokes problem has a unique solution in
$ \mathbb{H}^{1} (\Omega) \times L_{0}^{2} (\Omega )$.
Here $ L_{0}^{2} (\Omega ) $ is the sub-space of $L^{2}$-functions with zero
mean.
\end{remark}

\section{Existence and uniqueness result}

We consider the case where the data satisfy
\[
\mathbf{g}_i \in \mathbb{H}^s ( F_{i}), \quad \text{for $i=1, \dots, N$  with 
$| s| <\frac{1}{2}$}.
\]
Let $H=( H_i )_{i=1}^{N}$ such that $H_i=\mathbf{g}_i \cdot \mathbf{n}_i$ on the face
$F_i$. We will assume that the $H_i$ functions satisfy
\begin{equation} \label{Eq:H}
\sum_{i=1}^N \langle H_i ,  1 \rangle =0.
\end{equation}
Note that \eqref{Eq:H} is a compatibility condition which is necessary for the
existence of solutions to the Stokes problem \eqref{eq:0.1}--\eqref{eq:0.3}.

Let us also define the tangential part of $\mathbf{g}_i$ on $\Gamma_i$ by
\begin{equation} \label{Eq:G}
\mathbf{G}=( \mathbf{G}_i )_{i=1}^{N} \quad \text{where }  \mathbf{G}_i = (\mathbf{g}_i)_{t} = \mathbf{g}_i-(\mathbf{g}_i \cdot
\mathbf{n}_i)\mathbf{n}_i. 
\end{equation}
In this case, the solution of the system \eqref{eq:0.1}--\eqref{eq:0.3} is not 
regular enough and then it can not be variational. We then use the 
transposition technique as in the search for very weak solutions.
This contains, in particular, the interesting case with regular data on each 
face $F_i$ of the boundary but without connection or compatibility at the edges 
$\Gamma_{ij}$ of $\Gamma$.

The approach developed here may be extended, in natural way, to the case where 
$ \Omega $ is a non convex polyhedron of $\mathbb{R}^3$ and also to the case where
the data belong to $W^{s,p}$ spaces. This could be done by using the results 
of Dauge \cite{Dauge89}, Maz'ya-Plamenevski\v{i} \cite{MazPla84} and 
Grisvard \cite{Grisvard67}. 
The main result of this work is the following theorem.

\begin{theorem} \label{Theo:1}
Let $ \mathbf{g}_i = (g^1_i,g^2_i,g^3_i )$ given in $\mathbb{H}^s(F_{i})$ for $ i=1,\dots,N $ and
$ -1/2 <s <1/2 $.
Then, there exists a unique $ \mathbf{u} \in \cap_{\sigma <s + \frac{1}{2} }
\mathbb{H}^{\sigma}(\Omega )$ and a unique (up to an additive constant distribution)
$ p \in H^{s-\frac{1}{2}}(\Omega )$, solution to
\eqref{eq:0.1}--\eqref{eq:0.3}.
\end{theorem}

The proof of the Theorem \ref{Theo:1} is done in two steps. 
First, we recall some results on the traces of functions in $H^{s}(\Omega)$ 
and establish some useful lemma for the rest of the presentation. 
We will then use the result of Dauge \cite {Dauge89} and
transposition techniques to show that the problem \eqref{eq:0.1}--\eqref{eq:0.3} 
has a solution $ (\mathbf{u}, p) \in ( L^2(\Omega) )^3 \times H^{-1}(\Omega)$. 
And then, in section 3, we will use interpolation techniques to increase 
the regularity of the solution, and establish the proof of Theorem \ref{Theo:1}.

\begin{remark} \label{rmk2.2} \rm
We did not get the optimal regularity that one could hope, namely,
$ \mathbf{u} \in \mathbb{H}^{s + \frac{1}{2}} (\Omega )$. This is because of the following
fact: 
Noting $H^{1/2} (\Gamma )$ the space of traces of
functions $H^{1}(\Omega )$, it is difficult to characterize its
dual $ (H^{1/2} (\Gamma ) )'$.
$H^{1/2}(\Gamma )$ is actually a dense, but non closed, sub-space
of $ \prod_{j=1}^{N} H^{1/2} (F_{j} )$.
\end{remark}

\subsection{Theorem of traces}

Let us first recall some brief results on vector fields. 
For more details, we refer to
Girault-Raviart \cite{GiRav}. For any
$\mathbf{v} = (v_1,v_2,v_3) \in ( \mathcal{D}' (\Omega) )^3 $,
$ \operatorname{curl} \mathbf{v}$ is defined by
\[
\operatorname{curl} \mathbf{v} = \Big(
\frac{\partial v_3}{\partial x_2}-\frac{\partial v_2}{\partial x_3},
\frac{\partial v_1}{\partial x_3}-\frac{\partial v_3}{\partial x_1},
\frac{\partial v_2}{\partial x_1}-\frac{\partial v_1}{\partial x_2}
\Big). 
\]
It is then easy to see that
\begin{equation} \label{Eq:04}
\operatorname{curl}( \operatorname{curl} \mathbf{v} ) 
= - \Delta \mathbf{v} + \nabla(\operatorname{div}\mathbf{v}).
\end{equation}

As on Figure \ref{fig1}, $\{\mathbf{n},\boldsymbol{\tau}_1,\boldsymbol{\tau}_2\}$ denote the local
coordinates and $\mathbf{v}_{\mathbf{t}} = \mathbf{v} - (\mathbf{v} \cdot \mathbf{n} ) \mathbf{n} $ the tangential
part of the vector
$\mathbf{v}$. 
In the sequel, $ (\cdot  ,  \cdot  ) $ denotes the usual inner product on
$ ( L^2(\Omega) )^3 $ and  $ \langle \cdot  , \cdot \rangle $ the duality
pairing.
Let
\[
\mathbb{H}(\operatorname{curl} ,\Omega) = \{ \mathbf{v} \in ( L^2(\Omega) )^3 ;
\operatorname{curl}  \mathbf{v} \in  ( L^2(\Omega) )^3 \}.
\]
We recall in the following theorem the tangential trace and a Green's 
formula that we use later

\begin{theorem}[{Girault-Raviart \cite[ Prop. 1.2]{GiRav}}]
The mapping $ \gamma_{\mathbf{t}} : \mathbf{v} \to \gamma_{\mathbf{t}} \mathbf{v} =
( \mathbf{v} \times \mathbf{n} )_{\Gamma} $ defined on
$[ \mathcal{D}(\bar \Omega)]^3$ extends by continuity to a linear continuous
mapping, still denoted $\gamma_{\mathbf{t}}$,
from $ \mathbb{H}(\operatorname{curl},\Omega) $ to $ \mathbb{H}^{-1/2} ( \Gamma ) $. Moreover,
\begin{equation} \label{Eq:GreenRT}
\int_\Omega \operatorname{curl}\mathbf{v} \cdot \boldsymbol{\varphi}  =  \int_\Omega \mathbf{v} \cdot
\operatorname{curl}\boldsymbol{\varphi} +
\langle \gamma_{\mathbf{t}} \mathbf{v} , \boldsymbol{\varphi} \rangle_{\Gamma}, \;\;
\forall \mathbf{v} \in \mathbb{H}(\operatorname{curl},\Omega) \text{ and } \forall \boldsymbol{\varphi} \in
\mathbb{H}^1(\Omega).
\end{equation}
\end{theorem}

Now, by definition of the domain $\Omega$, see Figure \ref{fig1}, each face $F_i$ of
$\Gamma = \partial \Omega$ is a convex polygon, the boundary
 $ \Gamma_i = \partial F_i$ is Lipschitz-continuous.
For $ x \in F_i$, $ \rho_i(x)$ denotes the distance of $x$ to $\Gamma_i$. We then may
construct the following sub-space (see Grisvard \cite{Grisvard67}):
\[
\mathbb{H}^{1/2}_{0,0} (F_i) = \{ \mathbf{v} \in \mathbb{H}^{1/2}(F_i):
                       \frac{\mathbf{v}}{\sqrt{\rho_i}} \in \mathbb{L}^2(F_i) \}.
\]
Equipped with the norm
\[
 \|| \mathbf{v} \|| = \big\{ \| \mathbf{v} \|^2_{1/2,F_i} +
              \| \frac{ \mathbf{v} }{ \sqrt{\rho_i} } \|^2_{0,F_i} \big\}^{1/2} ,
\]
where  $ \mathbb{H}^{1/2}_{0,0} (F_i) $ is a Hilbert space.

Introducing the space
\[
\mathbb{W} = \mathbb{H}^3(\Omega) \cap \mathbb{H}^2_0 (\Omega),
\]
we first give a result on the tangential trace of the Laplacian operator.


\begin{theorem} \label{Theo:2}
The mapping $T_1 : \boldsymbol{\varphi} \to  ( \gamma_{\mathbf{t}} \Delta \boldsymbol{\varphi}_{|_{F_i} }
)_{i=1}^{N} $ from $\mathbb{W}$ on $ \prod_{j=1}^{N} ( H^{1/2}_{0,0} (F_j))^2 $
is linear, continuous, onto and has a linear continuous right inverse.
\end{theorem}

\begin{proof}
 let us first show that for any $\boldsymbol{\varphi} \in \mathbb{W}$,
\[
 \Big( \gamma_{\mathbf{t}} \Delta \boldsymbol{\varphi}_{|_{F_i}} \Big)_{i=1}^N \in
\prod_{j=1}^{N} ( H^{1/2}_{0,0} (F_{j}) )^2.
\]
Let $\boldsymbol{\varphi} \in \mathbb{W}$, by definition of the tangential trace $\gamma_{\mathbf{t}}$,
we have
$ \gamma_{\mathbf{t}} \Delta \boldsymbol{\varphi}  
= \Delta \boldsymbol{\varphi} \times \mathbf{n}$ and then
$ \gamma_{\mathbf{t}} \Delta \boldsymbol{\varphi}$ belongs to the tangential plan. Therefore, it is
sufficient to verify that
$ \Delta \boldsymbol{\varphi}_{|_{F_j}} \in ( H^{1/2}_{0,0}(F_j) )^2, \forall
j=1,\dots,N$.
To do so, we will first show that for any
$ \psi \in (\mathcal{C}^2(\overline{\Omega}
))^3 \cap \mathbb{H}^{2}_{0}(\Omega )$, we have $\Delta \psi = 0$ on
$\Gamma_{ij}$,
where $ \Gamma_{ij} $ is an edge of the boundary $\Gamma$. 

Without loss of generality, one may suppose that the boundary $\Gamma$
admits a face $ F_1 $ belonging to the plan $ xoy$ and another face $F_2$
adjacent to $F_1$ having an angle $ \omega < \pi $ with the plan $xoy$, see
\ref{fig1}. 

As, $ \psi \in \mathbb{H}^2_0(\Omega) $, we have  $ \nabla \psi = 0 $ on 
$ F_1 $ and $ F_2 $.
Then:

On $ F_1 $, the basis of the tangent plan is 
$(\boldsymbol{\tau}_1,\boldsymbol{\tau}_2)$ with $\boldsymbol{\tau}_1 = \mathbf{i} 
$ and $\boldsymbol{\tau}_2 = \mathbf{j}$, then
\begin{gather*}
 \frac{\partial }{\partial \boldsymbol{\tau}_1 }( \nabla \psi ) 
=  \frac{\partial }{\partial \mathbf{i}} ( \nabla \psi ) 
= \nabla (\nabla \psi) \cdot \mathbf{i} 
=  \frac{\partial }{\partial x} ( \nabla \psi ) = 0,
\\
\frac{\partial }{\partial \boldsymbol{\tau}_2 }( \nabla \psi ) 
=  \frac{\partial }{\partial \mathbf{j}} ( \nabla \psi )
= \nabla (\nabla \psi) \cdot \mathbf{j} 
=  \frac{\partial }{\partial y} ( \nabla \psi ) = 0.
\end{gather*}
Hence,
\begin{gather*}
\frac{\partial^2 \psi}{\partial x^2} =
\frac{\partial^2 \psi}{\partial x \partial y} =
\frac{\partial^2 \psi}{\partial x \partial z} = 0 \quad \text{and} \quad
\frac{\partial^2 \psi}{\partial y^2} =
\frac{\partial^2 \psi}{\partial y \partial x} =
\frac{\partial^2 \psi}{\partial y \partial z} = 0.
\end{gather*}


On $ F_2 $, $ \boldsymbol{\tau} = \alpha \mathbf{i} + \beta \mathbf{k} $, 
with $ \beta \ne 0$. $ \frac{\partial }{\partial \boldsymbol{\tau} } 
( \nabla \psi )
 = \nabla ( \nabla \psi ) \boldsymbol{\tau} = 0$, then
\[
\alpha \frac{\partial^2 \psi}{\partial x \partial z} +
\beta  \frac{\partial^2 \psi}{\partial z^2} = 0.
\]
We get $ \Delta \psi = 0 $ on $ \Gamma_{12} $ the common edge to both faces 
$ F_1 $ and $ F_2$. 

 To show that $T_1$ is onto, we use the trace theorem by Grisvard \cite{Grisvard67}.
\end{proof}

Let us now consider the space
\[
\mathbf{E} = \{ \mathbf{v} \in \mathbb{H}^{2}(\Omega) 
\cap \mathbb{H}^{1}_{0}(\Omega): \operatorname{div}\mathbf{v} \in H^{1}_{0}(\Omega) \}.
\]
We get, as a corollary of Theorem \ref{Theo:2}, the following result.

\begin{corollary} \label{Coro:1}
The mapping $ T_2 : \mathbf{v} \to ( \gamma_{\mathbf{t}}
\operatorname{curl}\mathbf{v}_{|_{F_i}})_{i=1}^{N} $ from $\mathbf{E}$ on $
\prod_{j=1}^{N} ( H^{1/2}_{0,0} (F_j) )^2 $
is linear, continuous and onto.
\end{corollary}

\begin{proof}
Let $ \mathbf{v} \in \mathbf{E} $, to show that
$ T_2(\mathbf{v}) = ( \gamma_{\mathbf{t}} \operatorname{curl}\mathbf{v}_{|_{F_j}}
)_{j=1}^{N}
\in \prod_{j=1}^{N} ( H^{1/2}_{0,0} (F_{j}) )^2$, we use the
same procedure as for the previous Theorem \ref{Theo:2}. 

 First we show that $T_2$ is an onto mapping:
let $\mathbf{h} \in \prod_{j=1}^{N} ( H^{1/2}_{0,0} (F_j) )^2$.
By Theorem \ref{Theo:2}, there exists $ \boldsymbol{\varphi} \in \mathbb{W} $ such that
$ \mathbf{h} = \gamma_{\mathbf{t}} \Delta \boldsymbol{\varphi} $.
We set $ \mathbf{v} = - \operatorname{curl}\boldsymbol{\varphi} $, it is easy to see that $ \mathbf{v} \in
\mathbf{E} $.
On  the other hand,
\[
\gamma_{\mathbf{t}} \operatorname{curl}\mathbf{v} =-\gamma_{\mathbf{t}} (
\operatorname{curl}(\operatorname{curl}\boldsymbol{\varphi}
))
 =\gamma_{\mathbf{t}} ( \Delta \boldsymbol{\varphi} ) -
 \gamma_{\mathbf{t}} ( \nabla (\operatorname{div}\boldsymbol{\varphi} ) ).
\]
As, $ \boldsymbol{\varphi} \in \mathbb{H}_0^2( \Omega )$, we obtain $ \operatorname{div}\boldsymbol{\varphi} = 0 $ on $ \Gamma $.
Therefore $\gamma_{\mathbf{t}} (\nabla (\operatorname{div}\boldsymbol{\varphi} ))=0$.
Then, there exists $ \mathbf{v} $ in $ \mathbf{E} $ such that 
$ \mathbf{h} = \gamma_{\mathbf{t}} \operatorname{curl}{\bf v} $.
Hence we obtain the expected result.
\end{proof}

In the sequel, we will need the Hilbert space
\[
\mathbb{V} =  \{ \boldsymbol{\psi} \in \mathbb{H}^1 (\Omega ): \Delta^2 \boldsymbol{\psi} \in \mathbb{L}^2( \Omega )\}.
\]
Equipped with the inner product
$$ (\!(\mathbf{u},\mathbf{v})\!)_{\mathbb{V}} = (\!(\mathbf{u},\mathbf{v})\!)_{\mathbb{H}^1} + (\!(\Delta^2 \mathbf{u},\Delta^2
\mathbf{v})\!)_{\mathbb{L}^2}\,. 
$$
Note that thanks to formula \eqref{Eq:GreenRT}, one also obtains the 
Green's formula:
for all $\boldsymbol{\varphi} \in \mathbb{W}$  and all $\boldsymbol{\psi} \in \mathbb{V}$, 
\[
\int_{\Omega}  \operatorname{curl}\boldsymbol{\psi} \cdot \operatorname{curl}\Delta \boldsymbol{\varphi}
=- \int_{\Omega} \Delta \boldsymbol{\psi} \cdot \Delta \boldsymbol{\varphi}  + 
\langle \gamma_{\mathbf{t}} \Delta \boldsymbol{\varphi} , \operatorname{curl}\boldsymbol{\psi} \rangle.
\]

\begin{proposition} \label{Prop:1} 
\begin{itemize}
\item
$( \mathcal{D} (\overline{ \Omega }) )^3 $ is dense in $\mathbb{V} $.
\item
The mapping  $ \gamma_{\mathbf{t}} $ defined on 
$( \mathcal{D}(\overline{\Omega}))^3$
by $ \gamma_{\mathbf{t}} \boldsymbol{\psi} = \boldsymbol{\psi} \times \mathbf{n}_{|_{\Gamma}} $ may be 
extended in a linear, continuous mapping from $ \mathbb{V} $ on 
$  \prod_{j=1}^{N} \big( H^{-1/2}(F_j) \big)^2$.
\end{itemize}
\end{proposition}


\begin{proof} 
 To show that $( \mathcal{D} (\overline{ \Omega }) )^3 $ is dense in
$\mathbb{V}$, one just need to show that if $\mathcal{L}$ is a linear continuous 
form on $\mathbb{V}$ vanishing on $( \mathcal{D} (\overline{ \Omega }) )^3 $ 
then it is identically equal to zero. 

Let $\mathcal{L} (\cdot) $ be a linear continuous form on $\mathbb{V}$. 
Then, there exists a unique $ \mathbf{u} \in \mathbb{H}^1(\Omega)$, 
and  $ \mathbf{f} \in \mathbb{L}^2(\Omega) $ such that
\[
\mathcal{L}( \mathbf{v} ) = \int_{\Omega}( \nabla \mathbf{u} \cdot \nabla \mathbf{v} 
+ \mathbf{u} \cdot \mathbf{v}) \,dx 
+ \int_{\Omega} \mathbf{f} \cdot \Delta^2 \mathbf{v} \,dx \quad \forall \mathbf{v} \in \mathbb{V}.
\]
Suppose that $\mathcal{L}(\boldsymbol{\varphi})=0$ for all 
$\boldsymbol{\varphi} \in (\mathcal{D} (\overline{\Omega}))^3$.
Noting $\tilde{\mathbf{u}},  \widetilde{\nabla \mathbf{u}} $, and  $ \tilde{\mathbf{f}} $
the extensions of $ \mathbf{u} $,  $ \nabla \mathbf{v} $, and  $ \mathbf{f} $ by $ 0 $ to  $ \mathbb{R}^3 $.
Then
\[
\int_{\mathbb{R}^3} ( \tilde{ \mathbf{u} } \cdot \boldsymbol{\varphi}  +
\widetilde{\nabla \mathbf{u} } \cdot \nabla \boldsymbol{\varphi} +
\tilde{\mathbf{f}} \cdot \Delta^2 \boldsymbol{\varphi} )\,dx = 0,\quad
\forall \boldsymbol{\varphi} \in (\mathcal{D}(\mathbb{R}^{3}))^3;
\]
using a Green formula we obtain
\[
\Delta^2 \tilde{\mathbf{f}} - \operatorname{div}(\widetilde{\nabla \mathbf{u}}) + \tilde{\mathbf{u}}=0.
\]
We therefore deduce that $ \tilde{ \mathbf{f} } \in \mathbb{H}^{3}( \mathbb{R}^3 )$.
Since $\mathbf{f}$ vanishes outside of the domain $ \Omega $, we conclude that 
$ \mathbf{f} $ is in $ \mathbb{H}^{3}_{0}( \Omega ) $. 
Let, $ (\mathbf{f}_{n} )_{n \in \mathbb{N} } $, be a sequence
of $ (\mathcal{D}(\Omega))^3 $ functions which converge, in $\mathbb{H}^{3}(\Omega) $, to
$\mathbf{f}$.
Using the sequence $\mathbf{f}_n$, we obtain
\begin{align*}
\mathcal{L}( \mathbf{v}) 
& = \int_{\Omega} ( \mathbf{u} \cdot \mathbf{v} + \nabla \mathbf{u} \cdot \nabla \mathbf{v} ) \,dx 
+\lim_{n \to \infty } \int_{\Omega} \mathbf{f}_{n} \cdot \Delta^2 \mathbf{v} \,dx,  \\
& =  \int_{\Omega} ( \mathbf{u} \cdot \mathbf{v} + \nabla \mathbf{u} \cdot \nabla \mathbf{v}
) \,dx 
 - \lim_{n \to \infty }\int_{\Omega} \nabla( \Delta \mathbf{f}_{n})  
 \cdot \nabla \mathbf{v} \,dx,        \\
 & =  \int_{\Omega} ( \mathbf{u} \cdot \mathbf{v} + \nabla \mathbf{u} \cdot \nabla \mathbf{v} ) \,dx 
 - \int_{\Omega}  \nabla( \Delta \mathbf{f}) \cdot \nabla \mathbf{v} \,dx. 
\end{align*}

$\mathcal{L} $ is then a linear continuous form on $ \mathbb{V} $ for the 
$\mathbb{H}^1(\Omega)$-norm.
Finally, as $\mathcal{L} $ vanishes on 
$ (\mathcal{D}(\overline{\Omega}))^3 $ which is dense in
$ \mathbb{H}^1(\Omega) $, it vanishes everywhere. This ends the proof of the first point. 


The second statement in the  proposition is a consequence of the denseness
of $\mathcal{D} (\overline{\Omega}))^3 $ in $ \mathbb{V} $ and the trace theorem 
\ref{Theo:2}.
\end{proof}

\section{Proof of theorem \ref{Theo:1}}

We first introduce some additional functional spaces and some auxiliary problems. 
Let $\mathbf{Z}$ be a subspace of $H^1$ with a zero-mean functions given by
\begin{equation} \label{Eq:Z}
\mathbf{Z} = \{ z \in H^1(\Omega): \int_\Omega z = 0 \}.
\end{equation}
Let $\mathbf{g}_i$ be the boundary data of our initial problem 
\eqref{eq:0.1}--\eqref{eq:0.3}, and 
consider the  variational problem
\begin{equation} \label{ePa}
\begin{gathered}
z \in \mathbf{Z}, \\
\int_\Omega \nabla z \cdot \nabla v =
\sum_{i=1}^N \langle \mathbf{g}_i \cdot \mathbf{n}_i, \gamma_0 v \rangle, \quad \forall
v \in \mathbf{Z}.
\end{gathered}
\end{equation}
The  auxiliary problem \eqref{ePa} admits a unique solution $z \in \mathbf{Z}$.
Moreover, $z$ is the solution of the boundary-value problem
\begin{gather*}
\Delta z = 0, \\
\nabla z \cdot \mathbf{n} = H, \\
\int_\Omega z = 0.
\end{gather*}
Let us now define the vector
\[
\mathbf{u}_1 = \nabla z.
\]
It is easy to check that $\mathbf{u}_1$ satisfy the following properties
\[
\Delta \mathbf{u}_1 = 0, \quad \operatorname{curl} \mathbf{u}_1 = 0, \quad
\operatorname{div}(\mathbf{u}_1) = 0.
\]
Therefore,
\[
\mathbf{u}_1 \in \mathbb{H}(\operatorname{div}, \operatorname{curl}, \Omega) \; \Longrightarrow  \;
(\mathbf{u}_1 \cdot \mathbf{n})_{|\Gamma} \in H^{-1/2}(\Gamma).
\]
Where
\[
\mathbb{H}(\operatorname{div}, \operatorname{curl}, \Omega) = \big\{ \mathbf{v} \in
L^2(\Omega)^3; \operatorname{div}(\mathbf{v}) \in L^2(\Omega)  \text{ and }
 \operatorname{curl}( \mathbf{v}) \in L^2(\Omega)^3
\big\}.
\]
Now, using the result in Jerison-Kenig \cite[Theorem 2]{JerKen81}
and the interpolation result, one gets
\[
 (\mathbf{u}_1 \cdot \mathbf{n})_{|\Gamma} \in H^{-s'}(\Gamma), \quad s < s' <\frac{1}{2}.
\]

Let $p_1 =0$, then it is easy to see that the couple $(\mathbf{u}_1,p_1)$ 
is a solution to the Stokes problem
\begin{equation}\label{Eq:u1}
\begin{gathered}
- \Delta \mathbf{u}_1 + \nabla p_1  =  0, \; \text{in  } \Omega, \\
\operatorname{div}\mathbf{u}_1  =  0, \quad \text{in } \Omega, \\
\mathbf{u}_1 \cdot \mathbf{n}  =  H, \quad \text{on } \Gamma.
\end{gathered}
\end{equation}
We also consider the  auxiliary Stokes problem
\begin{equation}\label{Eq:w}
\begin{gathered}
- \Delta \mathbf{w} + \nabla p_0  =  0, \quad \text{in  } \Omega, \\
\operatorname{div}\mathbf{w}  =  0, \quad \text{in } \Omega, \\
\mathbf{w} \cdot \mathbf{n}   =  0, \quad \text{on } \Gamma, \\
\mathbf{w} \times \mathbf{n}  =  \mathbf{G} - \mathbf{u}_1 \times \mathbf{n} = \mathbf{G}', \quad \text{on } \Gamma.
\end{gathered}
\end{equation}

To show the main result of this paper, given in Theorem \ref{Theo:1},
in addition to the space $\mathbf{Z}$ previously introduced \eqref{Eq:Z},
we will need also the use of the  space
\[
\mathbf{Z}_0 =  \big\{ g \in H^1_{0}(\Omega); \;\;\int_{\Omega} g=0 \big\}. 
\]
Following Dauge \cite{Dauge89}, the Stokes operator 
\[
\mathbf{S}(\mathbf{v},q) = ( -\Delta \mathbf{v} + \nabla q , \operatorname{div}\mathbf{v} ) 
\]
is an isomorphism from $ \mathbf{E} \times  \mathbf{Z} $ to 
$ L^2(\Omega) \times \mathbf{Z}_0 $.
Indeed this gives, by transposition, the following result.

\begin{proposition} \label{Prop:4}
For any linear continuous form $ \mathcal{L}(\cdot,\cdot)$ on
$ \mathbf{E} \times \mathbf{Z} $, there exists a unique $(\mathbf{u}  , p)$, element of
$ L^2(\Omega) \times H^{-1}(\Omega) / \mathbb{R} $ satisfying
\[
\int_\Omega \mathbf{u} \cdot \big( -\Delta \mathbf{v} + \nabla q \big) dx
- \big< p , \operatorname{div}\mathbf{v} \big>_{ H^{-1}(\Omega), H^1_0(\Omega)}
= \mathcal{L}( \mathbf{v},q), \quad \forall (\mathbf{v},q) \in \mathbf{E} \times \mathbf{Z}.
\]
\end{proposition}

\begin{remark} \label{Rema:2}\rm
Note that any $ p $ in $\mathbf{Z}_0' $, the dual space of $\mathbf{Z}_0$,  may be written
as $ p = p_{0} + c $ where $ p_0 \in H^{-1} (\Omega) $ and $ c $ is some constant.
\end{remark}

Taking into account the previous remark, concerning $\mathbf{Z}_0'$ the dual space of
$\mathbf{Z}_0$, one may give an equivalent formulation of the proposition \ref{Prop:4} 
as follows:
If $ \mathcal{L} $ is a linear continuous form on
$\mathbf{E} \times \mathbf{Z}$, then there exists a unique  $\mathbf{u}$ in $\mathbb{L}^2(\Omega)$
and unique (up to a constant) $p_0 \in H^{-1} (\Omega) $ such that 
\begin{equation}
\int_{\Omega} \mathbf{u} \cdot ( -\Delta \mathbf{v} + \nabla q )
- \langle p_{0}  ,  \operatorname{div}\mathbf{v} \rangle_{ H^{-1}(\Omega), H^1_0(\Omega)}
 =  \mathcal{L}( \mathbf{v},q), \quad  \forall (\mathbf{v},q) \in \mathbf{E} \times \mathbf{Z}.
\label{eq:11}
\end{equation}
We now have all the ingredients to prove  theorem \ref{Theo:1}.

\subsection*{Proof of Theorem \ref{Theo:1}}
Let $ s \in \mathbb{R} $ satisfying $ -1/2 < s <1/2$. Then, there exists
$\varepsilon  \in  ]0,1[$, such that $s = - \frac{1}{2} + \varepsilon $.
Consider now $ \mathbf{g} $ belonging to 
$\prod_{j=1}^{N} \mathbb{H}^{-\frac{1}{2}+\varepsilon}(F_j)$.
For all $i=1,\dots,N$, we set  $\mathbf{g}_i = \mathbf{g}_{|_{F_i}}$ and recall the
tangential part
$\mathbf{G}_i$ of functions of $\mathbf{g}_i$ on $F_i$, previously defined in \eqref{Eq:G}. 
Since $\mathbf{g}_i \in \mathbb{H}^{-\frac{1}{2}+\varepsilon}(F_i ) $, we obtain
\[
\mathbf{G}_i \in
H^{-\frac{1}{2} + \varepsilon} ( F_{i} ). 
\]
Let us now define the linear form $\mathcal{L}$ on the space
$\mathbf{E} \times \mathbf{Z}$:
\begin{equation}
\mathcal{L}(\mathbf{v},q) = \sum_{i=1}^N \langle \mathbf{G}_i'  , 
 \gamma_{\mathbf{t}} \operatorname{curl} \mathbf{v}_{|_{F_i}} \rangle,
\label{Eq:Form_Lin}
\end{equation}
where $\mathbf{G}'$ is given by $\mathbf{G}' = \mathbf{G} - \mathbf{u}_1 \times \mathbf{n}$, see the last
equation of \eqref{Eq:w}. 

\begin{proposition}\label{Prop:3}
Let $ \mathcal{L} $ be the linear form given in \eqref{Eq:Form_Lin}. 
There exists
$ \mathbf{w} \in \mathbb{L}^2 (\Omega) $ unique,
and
$ p_0 \in H^{-1} (\Omega) $ unique up to a constant,
satisfying 
\begin{equation} \label{eq:1}
 \int_{\Omega} \mathbf{w} \cdot (-\Delta \mathbf{v} + \nabla q)-
\langle  p_0,\operatorname{div}\mathbf{v} \rangle _{H^{-1}(\Omega),H^{1}_0(\Omega)}
= \mathcal{L} (\mathbf{v},q), \quad \forall (\mathbf{v},q) \in \mathbf{E} \times \mathbf{Z}.
\end{equation}
Moreover, $(\mathbf{w},p_0)$ is a solution to the Stokes
 problem \eqref{Eq:w}. 
\end{proposition}

\begin{proof}
On one hand, it is quite easy to see that the form $\mathcal{L}$ given by
\eqref{Eq:Form_Lin} is linear and continuous on $\mathbf{E} \times
\mathbf{Z}$. On the other hand, following the remark \ref{Rema:2}  and the Proposition
\ref{Prop:4}, there exists 
$ ( \mathbf{w} , p_0 ) \in \mathbb{L}^2 (\Omega) \times H^{-1} (\Omega ) $
satisfying the equality \eqref{eq:1}. 

Let us show now that the above pair $(\mathbf{w}, p_0)$, obtained in the previous 
proposition, verify the Stokes equations \eqref{Eq:w}. Let $w_j$, $j=1,2,3$ denote
the components of $\mathbf{w}$.

(1) Choosing  $\mathbf{v} = (v_1,0,0) \in (\mathcal{D} (\Omega))^3$ and $q=0$, we obtain the
following equation
\[
-\Delta w_1 + \frac{\partial p_0}{\partial x_1} = 0, \quad
 \text{in the distribution sense}. 
\]
and, in a same way, for $ v_j$, $j=2,3, $ in $ \mathcal{D} (\Omega) $,
$ v_k = 0$, $k=1,2,3$ with $k \neq j$ and $ q = 0 $, one gets
\[
-\Delta w_j + \frac{\partial p_0}{\partial x_j} = 0, 
\quad j=1,2,  \text{ in the distribution sense}.
\]
Then $-\Delta \mathbf{w} + \nabla p_0 = 0$, which is precisely the first equation of
\eqref{Eq:w}.

(2) Now, let $ q_0$ be any element of $\mathcal{D}(\Omega )$, choosing $\mathbf{v} = 0$
 and $ q = q_0 - \frac{1}{\mid \Omega \mid } \int_{\Omega} q_0 $. Using
\eqref{eq:1} and \eqref{Eq:H}, one gets the incompressibility condition 
$ \operatorname{div} \mathbf{w} = 0 $. 

Finally,
\begin{gather*}
\mathbf{w}  \in \mathbb{L}^2(\Omega)  \\
\operatorname{div}\mathbf{w} = 0  
\end{gather*}
implies that $\mathbf{w} \cdot \mathbf{n}$ is well defined as an element of the dual
$[H^{1/2} (\Gamma) ]'$. 
Moreover, since $ q $ is an element of $ H^1(\Omega) $, we have
\[
\int_{\Omega} q \operatorname{div}\mathbf{w} + \int_{\Omega} \nabla q \cdot \mathbf{w}
= \langle \mathbf{w} \cdot \mathbf{n}  ,  \gamma_0 q \rangle. 
\]
Thus, noting that $ \gamma_0 ( H^1(\Omega) ) = \gamma_0 \mathbf{Z} $,
it follows that
\begin{equation}
\label{Eq:01}
\mathbf{w} \cdot \mathbf{n} = 0 \quad \text{and then }  \mathbf{w} \cdot \mathbf{n}_i = 0.
\end{equation}

(3) It remains to show that
\[
 \mathbf{w}_t = \mathbf{G}_i' \quad \text{on }  F_i, \text{for } i=1,\dots,N.
\]
On the one hand, as $\mathbf{w} \in \mathbb{L}^2(\Omega)$ satisfying 
$ \operatorname{div}\mathbf{w} = 0$, there exists
$\boldsymbol{\psi} \in \mathbb{H}^1(\Omega)$ such that
\[
\mathbf{w}=\operatorname{curl} \boldsymbol{\psi} \quad \text{and} \quad 
 \operatorname{div}\boldsymbol{\psi} = 0 \quad \text{since }  \mathbf{w} \cdot \mathbf{n} =0.
\]
Consequently, $\boldsymbol{\psi} \in \mathbb{V} $. 

 On the other hand, $\mathbf{w}$ satisfies the first equation of the Stokes equation
$-\Delta \mathbf{w}+\nabla p_0=0$. Thus, applying the $\operatorname{curl}$ operator to the Stokes
equation and using \eqref{Eq:04}, we obtain
\[
- \Delta( \operatorname{curl}( \operatorname{curl}\boldsymbol{\psi}) )+\operatorname{curl}( \nabla p_0)
=\Delta^2 \boldsymbol{\psi} - \nabla ( \operatorname{div}\boldsymbol{\psi} ) = \Delta^2 \boldsymbol{\psi} = 0.
\]
Therefore,
\[
\boldsymbol{\psi} \in \mathbb{H}^1(\Omega) \quad\text{and} \quad  \Delta^2 \boldsymbol{\psi} = 0.
\]
Then $ \boldsymbol{\psi} \in \mathbb{V} $. 
Furthermore, we already saw that
\[
\boldsymbol{\varphi} \in \mathbb{W} \Longrightarrow \mathbf{v} = -\operatorname{curl}\boldsymbol{\varphi} \in \mathbf{E}.
\]
Consequently, let $\mathbf{v} \in \mathbf{E}$ such that $\operatorname{div}\mathbf{v} =0$;
thanks to equality \eqref{eq:1} one obtains
\[
\int_{\Omega} \mathbf{w} \cdot \Delta \mathbf{v} = \int_\Omega \mathbf{w} \cdot \nabla q
+ \sum_{i} \langle \mathbf{G}_i', \gamma_{\mathbf{t}} \operatorname{curl}\mathbf{v}_{|_{F_i}}
\rangle. 
\]
And then, thanks to \eqref{Eq:01}, we have
\[
\int_{\Omega} \mathbf{w} \cdot \Delta \mathbf{v} =  \sum_{i=1}^N \langle \mathbf{G}_i'  , 
\gamma_{\mathbf{t}} \operatorname{curl} \mathbf{v}_{|_{F_i}} \rangle 
 =  \langle \mathbf{w}_t , \gamma_{\mathbf{t}} \operatorname{curl}\mathbf{v} \rangle.
\]
Therefore, using the surjectivity obtained at Corollary \ref{Coro:1}, we obtain
\[
{\mathbf{w}_t} = \mathbf{G}_i'  \text{ on } 
F_i, \; \forall i=1,\dots,N 
\]
 and then $\mathbf{w}_t = \mathbf{G}'$  on $\Gamma$.
It is therefore deduced that $(\mathbf{u},p_0) = (\mathbf{u}_1 + \mathbf{w}, p_0)$ is a
 solution of our initial
problem. This completes the proof. 
\end{proof}

In summary, we have the following results:
\begin{itemize}
\item[(i)]
considering the proposition \ref{Prop:3}:
Let $ \varepsilon > 0 $, for all
$ (g_1,g_2,g_3) \in \mathbb{H}^{-\frac{1}{2}+\varepsilon} ( \Gamma ) $, there exists
$ \mathbf{u} \in \mathbb{L}^2(\Omega) $ unique and $ p $ in $ H^{-1}(\Omega) $ unique 
(up to an additive constant) such that
\begin{gather*}
-\Delta \mathbf{u} + \nabla p  =  0 \\
 \operatorname{div}\mathbf{u}  =  0 \\
\mathbf{u} =  (g_1,g_2,g_3) \quad \text{on } \Gamma.
\end{gather*}

 \item[(ii)]
For $ (g_1,g_2,g_3) \in \mathbb{H}^{1/2}(\Gamma) $, there exists
(see Temam \cite{Temam79}) $( \mathbf{u},p) $ unique in
 $ \mathbb{H}^{1}(\Omega) \times L^{2}_{0}(\Omega) $
solution of the Stokes system \eqref{eq:1}--\eqref{Eq:H}.
\end{itemize}

Then Theorem \ref{Theo:1} follows then by interpolation.


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\end{document}