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\begin{document}
\title{The Navier-Stokes equations in variable exponent spaces
of Clifford-valued functions}

\author{Binlin Zhang$^{\mathrm{a},}$\footnote{Corresponding author.
E-mail address:\,{\tt zhangbinlin2012@yahoo.cn}}, Yongqiang Fu$^{\mathrm{a}}$, Vicen\c{t}iu D. R\u{a}dulescu$^{\mathrm{b},\mathrm{c}}$\\
\small $^{\mathrm{a}}$ Department of Mathematics, Harbin Institute of Technology,
Harbin 150001, China \\
\small $^{\mathrm{b}}$ Institute of Mathematics ``Simion Stoilow" of the Romanian Academy, 014700 Bucharest, Romania\\
\small $^{\mathrm{c}}$ Department of Mathematics, University of Craiova,  200585 Craiova, Romania
} \date{ }
 \maketitle

\begin{abstract}
{In the frame of variable exponent spaces of Clifford-valued
functions and using the Banach fixed-point theorem, we obtain the
existence and uniqueness of solutions to the stationary
Navier-Stokes equations and Navier-Stokes equations with heat
conduction under certain natural assumptions. In a sense, we extend some
results of P. Cerejeiras and U. K\"{a}hler [P. Cerejeiras and U.
K\"{a}hler, Elliptic boundary value problems of fluid dynamics over
unbounded domains, Mathematical Methods in the Applied Sciences,
23 (2000), pp. 81-101]. }
\end{abstract}
\emph{Keywords:} Clifford analysis; variable exponent; Navier-Stokes equations; heat conduction.\\
\emph{2010 Mathematics Subject Classification:} 30G35; 35J60; 35Q30; 46E30; 76D03.

{\section*{{\bf 1. Introduction }}} \setcounter{section}{1}
 \setcounter{equation}{0}

   Since Kov\'{a}\v{c}ik and R\'{a}kosn\'{i}k first thoroughly studied the spaces $L^{p(x)}$ and $W^{k, p(x)}$ in [1],
Lebes\-gue and Sobolev variable exponent spaces have attracted more and more
   attention, see for instance [2-5] for recent properties of variable exponent spaces and [6] for an overview of differential
   equations with variable growth.
   In particular, one of the reasons that forced the rapid
   expansion of the theory of variable exponent function spaces has
   been the models of electrorheological fluids introduced by
   Rajagopal and R$\mathring{\mathrm{u}}$\v{z}i\v{c}ka [7, 8], which can be described by the boundary-value problem for
   the generalized Navier-Stokes equations. Diening, Lengeler and R$\mathring{\mathrm{u}}$\v{z}i\v{c}ka [9] proved the
existence and uniqueness of strong and weak solutions of the Stokes
system and Poisson equations for bounded domains in the setting of
variable exponent spaces. Of course, the study of these spaces has
been stimulated by problems in elastic mechanics, calculus of
variations and differential equations with variable growth
conditions, see [10-13] and references therein.

   As a powerful tool for solving elliptic boundary value problems in the plane, the methods of complex function theory play an important role.
   One way to extend these ideas to higher dimension is to start with a generalization of algebraic and geometrical properties of the complex numbers.
   In this way, Hamilton studied the algebra of quaternion in 1843. Further generalizations were introduced by Clifford in 1878.
   He initiated the so-called geometric algebras or Clifford algebras, which are generalizations of the complex numbers, the quaternions,
   and the exterior algebras, see [14]. Clifford analysis is usually the study of Dirac equation or of a generalized Cauchy-Riemann
   system, in which solutions are defined on domains
   in the Euclidean space and take values in Clifford algebras, see [15,
   16]. In particular, G\"{u}rlebeck and Spr\"{o}{\ss}ig [17, 18]
   developed an operator calculus, which is analogous to the known complex analytic approach in the plane and based on three
   operators: a Cauchy-Riemann-type operator, a Teodorescu
   transform, and a generalized Cauchy-type integral operator, to
   investigate elliptic boundary value problems of fluid dynamics over bounded and unbounded domains,
   especially the Navier-Stokes equations and related equations. Of
   course, there are a number of unsolved basic problems involving the Navier-Stokes
   equations. This is mainly due to the problem concerning the
   solvability of the corresponding linear Stokes equations over
   domains, see [19, 20].

Our goal in this article
is to investigate the existence and uniqueness of solutions for
Navier-Stokes equations in the variable exponent context. For this
purpose, an iteration method introduced by G\"{u}rlebeck and
Spr\"{o}{\ss}ig [17, 18, 21], which requires the solution of a
Stokes-problem in every step of iteration, is used to treat this
kind of problems. Besides being of interest in its own as a
generalization of classical results, the results are of importance
in the study of fluid dynamics. Of course, the whole treatment
applies to a much larger class of elliptic problems.

This paper is organized as follows. In Section 2, we begin with a
brief summary of basic knowledge of Clifford algebras and variable
exponent spaces of Clifford-valued functions. Especially, we require
the existence and uniqueness of the Stokes equations in the context
of variable exponent spaces, which will be needed later. In Section
3, we investigate an iterative method for the solution of the
stationary Navier-Stokes equations. Using the Banach fixed-pointed
theorem, we prove the existence and uniqueness of a solution to the
Navier-Stokes equations in $W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell
_{n})\times L^{p(x)}(\Omega, \mathbb{R})$ under certain hypotheses.
In Section 4, with the help of the Banach fixed-pointed theorem, we
prove the existence and uniqueness of a solution in
$W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})\times
W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})\times L^{p(x)}(\Omega,
\mathbb{R})$ of the Navier-Stokes problem with heat conduction under some conditions.


{\section*{{\bf 2. Preliminaries}}}

\textbf{2.1.\begin{itshape} Clifford algebra \end{itshape}}

 We first recall some related notions and results concerning Clifford
algebras. For a detailed account we refer to [15, 16, 22-25].

Let $\mathrm{C}\ell _{n}$ be the real universal Clifford algebras
over $\mathbb{R}^{n}$. Denote $\mathrm{C}\ell _{n}$ by
$$\mathrm{C}\ell _{n} =\mathrm{ span} \{\mathrm{e}_{0},\mathrm{e}_{1},\mathrm{e}_{2},\ldots ,\mathrm{e}_{n},e_{1}e_{2},\ldots ,\mathrm{e}_{n-1}\mathrm{e}_{n},\ldots ,\mathrm{e}_{1}e_{2}\cdots \mathrm{e}_{n}\}$$
where $\mathrm{e}_{0}=1$(the identity element in $\mathbb{R}^{n}$),
$\{\mathrm{e}_{1},\mathrm{e}_{2},\ldots ,\mathrm{e}_{n}\}$ is an
orthonormal basis of $\mathbb{R}^{n}$ with the relation
$\mathrm{e}_{i}\mathrm{e}_{j}+\mathrm{e}_{j}\mathrm{e}_{i}=-2\delta_{ij}\mathrm{e}_{0}$.
Thus the dimension of $\mathrm{C}\ell _{n}$ is $2^{n}$. For
$I=\{i_{1},\ldots,i_{r}\}\subset \{1,\ldots ,n\}$
    with $1\leq i_{1}<i_{2}<\cdots <i_{n}\leq n$, put $\mathrm{e}_{I}=\mathrm{e}_{i_{1}}\mathrm{e}_{i_{2}}\cdots \mathrm{e}_{i_{r}}$, while for $I=\emptyset$, $\mathrm{e}_{\emptyset }=\mathrm{e}_{0}$.
     For $0\leq r\leq n$ fixed, the space $\mathrm{C}\ell _{n}^{r}$ is defined by
$$\mathrm{C}\ell _{n}^{r}=\mathrm{span}\{\mathrm{e}_{I}:|I|:=\mathrm{card}(I)=r\}.$$
The Clifford algebras $\mathrm{C}\ell _{n}$ is a graded algebra as
$$\mathrm{C}\ell _{n}=\bigoplus_{r}\mathrm{C}\ell _{n}^{r}.$$
    Any element $a\in \mathrm{C}\ell _{n}$ may thus be written in a unique way as
$$a=[a]_{0}+[a]_{1}+\cdots +[a]_{n}$$
    where $[\ ]_{r}:\mathrm{C}\ell _{n}\rightarrow \mathrm{C}\ell _{n}^{r}$ denotes the projection of $\mathrm{C}\ell _{n}$ onto $\mathrm{C}\ell _{n}^{r}$.
    In particular, by $\mathrm{C}\ell _{n}^{2}=\mathbb{H}$ we denote the
    algebra of real quaternion.
It is customary to identify $\mathbb{R}$ with $\mathrm{C}\ell
_{n}^{0}$ and identify $\mathbb{R}^{n}$ with $\mathrm{C}\ell
_{n}^{1}$ respectively.
     This means that each element $x$ of $\mathbb{R}^{n}$ may be represented by $$x=\sum_{i=1}^{n}x_{i}\mathrm{e}_{i}.$$ For $u\in \mathrm{C}\ell _{n}$,
     we denotes by $[u]_{0}$ the scalar part of $u$, that is the coefficient of the element $\mathrm{e}_{0}$. We define the Clifford conjugation as follows:
$$\overline{\mathrm{e}_{i_{1}}\mathrm{e}_{i_{2}}\cdots \mathrm{e}_{i_{r}}}=(-1)^{\frac{r(r+1)}{2}}\mathrm{e}_{i_{1}}\mathrm{e}_{i_{2}}\cdots \mathrm{e}_{i_{r}}.$$
 We denote
$$(A,B)=\big[\overline{A}B\big]_{0}.$$
Then an inner product is thus obtained, give rising to the norm
$|\cdot|$ on $\mathrm{C}\ell _{n}$ given by
$$|A|^{2}=\big[\overline{A}A\big]_{0}.$$



For all what follows let $\Omega \subset \mathbb{R}^{n}(n\geq2)$ be
a bounded domain with a sufficiently smooth boundary
$\partial\Omega$. A Clifford-valued function
$u:\Omega\rightarrow\mathrm{C}\ell _{n}$ can be written as
$u=\Sigma_{I}u_{I}\mathrm{e}_{I}$, where the coefficients
$u_{I}:\Omega\rightarrow \mathbb{R}$ are real-valued functions.


    The Dirac operator on Euclidean space used here is introduced by
$$D=\sum_{j=1}^{n}e_{j}\frac{\partial}{\partial x_{j}}:=\sum_{j=1}^{n}e_{j}\partial_{j}.$$


    If $u$ is a real-valued function defined on a domain $\Omega$ in $\mathbb{R}^{n}$,
    then $Du=\nabla u=(\partial_{1}u,\partial_{2}u,\ldots,\partial_{n}u)$.
    Moreover, $D^{2}=-\Delta$, where $\Delta$ is the
Laplace operator which operates only on coefficients. A function is
left monogenic if it satisfies the equation $Du(x)=0$ for each
$x\in\Omega$. A similar definition can be given for right monogenic
function. An important example of a left monogenic function is the
generalized Cauchy kernel
$$G(x)=\frac{1}{\omega_{n}}\frac{\overline{x}}{|x|^{n}},$$
where $\omega_{n}$ denotes the surface area of the unit ball in $\mathbb{R}^{n}$.
This function is a fundamental solution of the Dirac operator.
Basic properties of left monogenic functions one can refer to [16, 21, 29] and references therein.\\
\\
\textbf{ 2.2. \begin{itshape}Variable exponent spaces of
Clifford-valued
functions\end{itshape}}\\

Next we recall some basic properties of variable exponent spaces of
Clifford-valued functions. Note that in what follows, we use the
short notation $L^{p(x)}(\Omega)$, $W^{1,p(x)}(\Omega)$, etc.,
instead of $L^{p(x)}(\Omega, \mathbb{R})$,
$W^{1,p(x)}(\Omega,\mathbb{R})$, etc.

 Throughout this paper we
always assume (unless declare specially)
$$p \in \mathrm{\mathcal {P}^{log}}(\Omega) \ \mathrm{and} \ 1 < p_{-}:= \inf_{x \in \overline{\Omega}}p(x) \leq p(x)
\leq \sup_{x \in \overline{\Omega}}p(x) =:p_{+} < \infty.\eqno(2.1)$$
where  $\mathrm{\mathcal{P}^{log}}(\Omega)$ is the set of exponent
$p$ satisfying the so-called log-$\mathrm{H\ddot{o}lder}$
continuity, i.e.,
$$|p(x) - p(y)| \leq \frac{C}{\mathrm{log}(e + 1/|x-y|)}$$
holds for all $x,y \in \Omega$, see [9, 27]. Let $\mathcal{P}(\Omega)$  be the
set of all Lebesgue measurable functions $p: \Omega \rightarrow
(1,\infty)$. Given $p \in \mathcal{P}(\Omega)$ we define the conjugate
function $p'(x)\in \mathcal{P}(\Omega)$ by
$$p'(x)=\frac{p(x)}{p(x)-1}\ \  \mathrm{for}\ \mathrm{each} \
x\in\Omega.$$

The variable exponent Lebesgue space $L^{p(x)}(\Omega)$ is defined
by
$$L^{p(x)}(\Omega)=\{ u\in \mathcal{P}(\Omega):\int_{\Omega}|u|^{p(x)}dx <\infty \},$$
with the norm
$$||u||_{L^{p(x)}(\Omega)}=\mathrm{inf} \{t>0:\int_{\Omega}|\frac{u}{t}|^{p(x)}dx \leq 1\}.$$
The variable exponent Sobolev space $W^{1,p(x)}(\Omega)$ is defined
by
$$W^{1,p(x)}(\Omega)=\{u\in L^{p(x)}(\Omega): |\nabla  u| \in L^{p(x)}(\Omega)\},$$
with the norm
$$||u||_{W^{1,p(x)}(\Omega)}=||\nabla  u||_{L^{p(x)}(\Omega)} + ||u||_{L^{p(x)}(\Omega)}.\eqno(2.2)$$
Denote $W_{0}^{1,p(x)}(\Omega)$ by the completion of
$C_{0}^{\infty}(\Omega)$ in $W^{1,p(x)}(\Omega)$ with respect to the
norm (2.2). The space $W^{-1,p(x)}(\Omega)$ is defined as the dual
of the space $W_{0}^{1,p'(x)}(\Omega)$.
For more details we refer to [2-5, 26] and reference therein.\\

In the following, we say that $u\in L^{p(x)}(\Omega,\mathrm{C}\ell
_{n})$  can be understood coordinate wisely.
 For example, $u\in L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ means that $\{u_{I}\} \subset L^{p(x)}(\Omega)$
 for $u=\Sigma_{I}u_{I}e_{I}\in \mathrm{C}\ell _{n}$ with the norm $||u||_{L^{p(x)}(\Omega, \mathrm{C}\ell _{n})}= \sum_{I}||u_{I}||_{L^{p(x)}(\Omega)}$.
 In this way, spaces $W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$, $W_{0}^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$, $C_{0}^{\infty}(\Omega,\mathrm{C}\ell _{n})$, etc., can be understood similarly.
 In particular, the space $L^{2}(\Omega,\mathrm{C}\ell _{n})$ can be converted into a right Hilbert $\mathrm{C}\ell _{n}$-module by defining
 the following Clifford-valued inner product (see [17, Definition 3.74])
 $$\big(f,g\big)_{\mathrm{C}\ell _{n}}=\int_{\Omega}{\overline{f(x)}g(x)}dx.  \eqno(2.3)$$

\noindent\begin{itshape}Remark\end{itshape} 2.1. A simple
calculation yields that the norm $||u||_{L^{p(x)}(\Omega,
\mathrm{C}\ell _{n})}$ is equivalent to the norm
$||\ |u|\ ||_{L^{p(x)}(\Omega)}$. Furthermore, we also
have that for every $u \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell
_{n})$, $||Du||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}$ is an
equivalent norm of
$||u||_{W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})}$ (see [27, 28]).\\
\\
\textbf{Lemma 2.1.} (see [27,29]) \begin{itshape}If $p(x) \in \mathcal{P}(\Omega)$,
then $L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ and
$W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$ are reflexive Banach
spaces.\end{itshape}\\
\\
\textbf{Definition 2.1.}(see [17]) Let $u \in C(\Omega, \mathrm{C}\ell
_{n})$. The Teodorescu operator is defined by
$$Tu(x) = \int_{\Omega}G(x-y)u(y)dy,$$
where $G(x)$ is the generalized Cauchy kernel mentioned above.\\
\\
\textbf{Lemma 2.2.}\begin{itshape} The following operators are continuous linear operators:

 (i) $T: L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow
W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$.

(ii) $\widetilde{T}: W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})
\rightarrow
L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$.  \end{itshape}

\begin{proof}(i) see Theorem 2.7 in [29].

(ii) In view of Proposition 12.3.2 in [26], we know that
 for each $f \in W^{-1,p(x)}(\Omega)$, there exists $f_{k} \in L^{p(x)}(\Omega),k=0,1,\ldots,n,$ such that
$$<f,\varphi> = \sum^{n}_{k=0}\int_{\Omega}
f_{k}\frac{\partial \varphi}{\partial x_{k}} dx,\eqno(2.4)$$ for all
$\varphi \in W^{1,p'(x)}_{0}(\Omega)$. Moreover,
$\|f\|_{W^{-1,p(x)}(\Omega)}$ is equivalent to
$\sum^{n}_{k=0}\|f_{k}\|_{L^{p(x)}(\Omega)}$. Obviously, for every
$f \in W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$ the equality (2.4) still holds
for $f_{k} \in L^{p(x)}(\Omega,\mathrm{C}\ell _{n}),$ $k=0,1,\ldots,n$. Moreover,
$\|f\|_{W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})}$ is equivalent to
$\sum^{n}_{k=0}\|f_{k}\|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}$. On the other
hand, by Proposition 12.3.4 in [26], it is easy to show that the
space $C^{\infty}_{0}(\Omega,Cl_{n})$ is dense in
$W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$. Thus we may choose
$$u^{j} = u_{0}^{j} + \sum^{n}_{k=1}\frac{\partial u_{k}^{j}}{\partial x_{k}},$$
where $u_{0}^{j}, u_{k}^{j} \in C^{\infty}_{0}(\Omega,\mathrm{C}\ell _{n})$, such
that $\|u^{j}-f\|_{W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})}\rightarrow 0$ and
$\|u_{k}^{j}-f_{k}\|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}$ $\rightarrow 0$ as
$j\rightarrow\infty$, where $k=0,1,\ldots,n.$ Here, we are using the
fact that $C^{\infty}_{0}(\Omega,\mathrm{C}\ell _{n})$ is dense in
$L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$(see [29]). Then we consider
$$Tu^{j} = \int_{ \Omega}G(x-y)u^{j}(y)dy,$$
where $G(x)$ is the above-mentioned generalized Cauchy kernel. Further, we have
$$\aligned
Tu^{j} &= \int_{\Omega}G(x-y)\big(u_{0}^{j}(y) + \sum^{n}_{k=1}\frac{\partial}{\partial y_{k}}u_{k}^{j}(y)\big)dy\\
&=\int_{ \Omega}G(x-y)u_{0}^{j}(y)dy + \sum^{n}_{k=1} \int_{
\Omega}\frac{\partial}{\partial x_{k}}G(x-y)u_{k}^{j}(y)dy.
\endaligned$$
Since
$$\big|\int_{ \Omega}G(x-y)u_{0}^{j}(y)dy\big|\leq \int_{ \Omega}\frac{1}{|x-y|^{n-1}}\big|u_{0}^{j}(y)\big|dy.$$
By Remark 2.1, Lemma 2.7 and Lemma 2.8, there exists a constant
$C_{0}
> 0$ such that
$$\Big|\Big|\int_{ \Omega}G(x-y)u_{0}^{j}(y)dy \Big|\Big|_{L^{p(x)}(\Omega,Cl_{n})} \leq C_{0}||u_{0}^{j}||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}.\eqno(2.5)$$
Now let us extend $u_{k}^{j}(x)$ by zero to $\mathbb{R}^{n}\setminus
\Omega$. Then it is easy to show that $\frac{\partial}{\partial
x_{k}}G(x-y)$ satisfies the conditions of Calder\'{o}n-Zygmund
kernel on $\mathbb{R}^{n} \times \mathbb{R}^{n}$ (see [29]). In view
of Lemma 2.9, there exist positive constant $C_{k}(k=1,\ldots,n)$
such that
$$\Big|\Big|\int_{ \Omega}\frac{\partial}{\partial x_{k}}G(x-y)u_{k}^{j}(y) \Big|\Big|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})} \leq C_{k}\big\|u_{k}^{j}\big\|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}.\eqno(2.6)$$
Combining (2.5) with (2.6), we have
$$\aligned
\big|\big|&Tu^{j}\big|\big|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}& \\
\leq &\Big|\Big|\int_{ \Omega}G(x-y)u_{0}^{j}(y)dy \Big|\Big|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})} + \sum^{n}_{k=1}\Big|\Big|\int_{ \Omega}\frac{\partial}{\partial x_{k}}G(x-y)u_{k}^{j}(y) \Big|\Big|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}\\
\leq & C_{0}\big|\big|u_{0}^{j}\big|\big|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}
+ \sum^{n}_{k=1}
C_{k}\big|\big|u_{k}^{j}\big|\big|_{L^{p(x)}(\Omega,Cl_{n})}.
\endaligned$$
Letting $j\rightarrow \infty$, by means of the Continuous Linear
Extension Theorem, the operator $T$ can be uniquely extended to a
bounded linear operator $\widetilde{T}$ such that for all $f \in
W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$, there exists a constant
$\widetilde{C}>0$ such that
$$\big\|\widetilde{T}f\big\|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})} \leq C\Big(\big\|f_{0}\big\|_{L^{p(x)}(\Omega,Cl_{n})} + \sum^{n}_{k=1}\big\|f_{k}\big\|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}\Big) \leq
\widetilde{C}\big\|f\big\|_{W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})}.$$
Now we complete the proof of the lemma 2.2.
\end{proof}

\noindent\textbf{Lemma 2.3.}\begin{itshape} The following operators are
continuous linear operators:

 (i) $D: W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow
L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$.

 (ii) $\widetilde{D}:L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$.
\end{itshape}

\begin{proof} (i) see Lemma 2.6 in [29].

(ii) We consider the following Dirichlet problem of the Poisson equation with homogeneous
boundary data
$$\left\{
  \begin{array}{ll}
    -\Delta u = f, &\mathrm{in} \ \Omega \\
    u = 0, &\mathrm{on} \ \partial \Omega
  \end{array}
\right.\eqno(2.7)$$

L. Diening, D. Lengeler and M. Ru\v{z}i\v{c}ka [9] proved that the problem (2.7) possesses a unique weak solution $u\in
W^{1,p(x)}(\Omega)$ for each $f \in W^{-1,p(x)}(\Omega)$ with the
estimate
 $$||u||_{W^{1,p(x)}(\Omega)} \leq C(n,p,\Omega)||f||_{W^{-1,p(x)}(\Omega)}.$$
where we call $u$ a weak solution of (2.7) provided that
$$<f,\varphi>=\int_{\Omega}\nabla u\cdot \nabla \varphi dx ,\ \ \forall\  \varphi \in W^{1,p'(x)}_{0}(\Omega).$$ Then it is easy to see
that for all $f \in W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$ the problem (2.7)
still has a unique weak solution $u\in W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$.
We denote by $\Delta^{-1}_{0}$ the solution operator. On the other
hand, it is clear that the operator
$$\Delta : W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$$
is continuous, so we obtain that the operator $\widetilde{D}=-\Delta
T:L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$ is
continuous from Lemma 2.2, where the operator $\widetilde{D}$ can
be considered as a unique continuous linear extension of the
operator $D$.\end{proof}

\noindent\textbf{Lemma 2.4.}\begin{itshape} Let $p(x) \in \mathcal{P}(\Omega)$.

(i) If $u \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$, then the
equation
 $TDu(x) = u(x)$ holds for all $x \in \Omega$.

(ii) If $u \in L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$, then the
equation $DTu(x) =
u(x)$ holds for all $x \in \Omega$.\end{itshape}

\begin{proof}
By Remark 4.21 in [17], (i) and (ii) can be
deduced from $W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$ $\hookrightarrow
W^{1,p_{-}}(\Omega,\mathrm{C}\ell _{n})$ and
$L^{p(x)}(\Omega,\mathrm{C}\ell _{n})\hookrightarrow
L^{p_{-}}(\Omega,\mathrm{C}\ell _{n})$.
\end{proof}

\noindent\textbf{Lemma 2.5.}\begin{itshape} Let $p(x)$ satisfies (2.1).

(i) If $u \in L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$, then the
equation $\widetilde{T}\widetilde{D}u(x) = u(x)$ holds for all $x
\in
\Omega$.

(ii) If $u \in W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$, then the
equation $\widetilde{D}\widetilde{T}u(x) = u(x)$ holds
for all $x \in \Omega$.\end{itshape}

\begin{proof}
(i) follows from Lemma 2.4 (i) and the density of
$W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})$ in $L^{p(x)}(\Omega, \mathrm{C}\ell _{n})$.

(ii) follows from Lemma 2.4 (ii) and the density of
$C^{\infty}_{0}(\Omega,\mathrm{C}\ell _{n})$ in $W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$.
\end{proof}



G\"{u}rlebeck and Spr\"{o}{\ss}ig [17, 18] showed that the
orthogonal decomposition of the space $L^{2}(\Omega)$ holds in the
hyper-complex function theory:
 $$L^{2}(\Omega,\mathrm{C}\ell _{n}) = (\mathrm{ker}D \cap L^{2}(\Omega,\mathrm{C}\ell _{n})) \oplus DW^{1,2}_{0}(\Omega,\mathrm{C}\ell _{n})\eqno(2.8)$$
with respect to the Clifford-valued product (2.3). Note that
$\mathrm{ker}D$ denotes the set of all monogenic functions on
$\Omega$. This decomposition has a number of applications,
especially to the theory of partial differential equations, see [30]
for the complex case and [17] for the hyper-complex case. K\"{a}hler
[31] extended the orthogonal decomposition (2.8) to the spaces
$L^{p}(\Omega)$ in form of a direct decomposition in the case of
Clifford analysis. Similar to the proof of Theorem 6 in [31], we can
generalize the direct decomposition into the case of variable
exponent Lebesgue spaces as follows:
$$L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) =(\mathrm{ker}\widetilde{D} \cap L^{p(x)}(\Omega,\mathrm{C}\ell _{n})) \oplus DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})\eqno(2.9)$$
with respect to the Clifford-valued product (2.3).

>From this decomposition we can get the following projections
$$P: L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow \mathrm{ker}\widetilde{D}\cap L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$$
$$Q: L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n}).$$
Moreover, we have
$$Q = D\Delta^{-1}_{0}\widetilde{D},\ \  P = I - Q.$$


\noindent\textbf{Lemma 2.6.}\begin{itshape} The space
$L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) \cap \mathrm{im}Q$ is a closed
subspace of
$L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$.\end{itshape}

\begin{proof} Let $u \in \overline{L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) \cap \mathrm{im}Q} =
\overline{DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})}$. Then there exists $u_{k}
\in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ such that
 $||Du_{k} - u||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})} \rightarrow 0$ as $k \rightarrow \infty$. In view of Lemma 2.1,
we can extract a subsequence of $\{u_{k}\}$( still labeled by
$\{u_{k}\}$ ), such that $u_{k} \rightharpoonup v$ weakly in
$W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$. Since the norm in a Banach space is
weakly lower semicontinuous, we obtain from Lemma 2.3
$$||Dv - u||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})} \leq \liminf_{n \rightarrow \infty}||Du_{n} - u||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})} = 0.$$
Thus $u = Dv$, which completes the proof of Lemma 2.6. \end{proof}

\noindent\textbf{Lemma 2.7.}  \begin{itshape}$\big(L^{p(x)}(\Omega,\mathrm{C}\ell _{n})
\cap \mathrm{im}Q\big)^{\ast} = L^{p'(x)}(\Omega,\mathrm{C}\ell _{n}) \cap
\mathrm{im}Q$. Namely, the linear operator
$$\Phi : DW^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell _{n}) \rightarrow \big(DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})\big)^{\ast}$$
given by
$$\Phi(Du)(D\varphi) = (D\varphi , Du)_{Sc}:= \int_{\Omega}\Big[\overline{D\varphi}Du\Big]_{0}dx$$
is a Banach space isomorphism.\end{itshape}


\begin{proof} In terms of Lemma 2.6,
$DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ and
$DW^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ are reflexive Banach spaces since
they are closed in $L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ and
$L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})$ respectively. Linearity of $\Phi$ is
clear. For injectivity, suppose
$$\Phi(Du)(D\varphi) = (D\varphi , Du)_{Sc} = 0\eqno(2.10)$$
for all $\varphi \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ and some $u \in
W^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$. For any $\omega \in
L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$, according to (2.9), we may write
$\omega = \alpha + \beta$ with $\alpha \in \mathrm{ker}\widetilde{D}
\cap L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ and $\beta \in
DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$. Thus we obtain
$$(\omega , Du)_{Sc} = (\alpha + \beta , Du)_{Sc} = (\alpha , Du)_{Sc} + (\beta, Du)_{Sc} = (\beta, Du)_{Sc}.$$
This together with (2.10) gives $(\omega , Du)_{Sc} = 0$. This
results in $Du = 0$. It follows that $\Phi$ is injective. To get
surjectivity, let $f \in
\big(DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})\big)^{\ast}$. By the Hahn-Banach
Theorem, there is $F \in \big(L^{p(x)}(\Omega,\mathrm{C}\ell _{n})\big)^{\ast}$
with $||F|| = ||f||$ and $F|_{DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})} = f$.
Moreover, there exists $\varphi \in
L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})$ such that $F(u) = (u, \varphi)_{Sc}$ for
any $u \in L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$. According to (2.9), we
can write $\varphi =\eta+ D\alpha$, where $\eta \in
\mathrm{ker}\widetilde{D} \cap L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})$, $D\alpha
\in DW^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell _{n}).$ For any $Du \in
DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$, we have
\begin{align*}
f(Du)= (Du, \varphi)_{Sc} = (Du, \eta)_{Sc} + (Du, D\alpha)_{Sc}= (Du, D\alpha)_{Sc} = \Phi (D\alpha)(Du).
\end{align*}
Consequently, $\Phi (D\alpha) = f$. It follows that $\Phi$ is
surjective. By Theorem 3.1 in [27] we have
$$|\Phi (Du)(D\varphi)| = |(D\varphi, Du)_{Sc}| \leq C||D\varphi||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}||Du||_{L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})}.$$
This means that $\Phi$ is continuous. Furthermore, it is immediate that $\Phi^{-1}$ is continuous from the Inverse Function Theorem.
This finishes the proof of Lemma 2.7. \end{proof}


In the following, we consider the Stokes system which
consists in finding a solution $(u,\pi)$ :
$$-\Delta u + \frac{1}{\mu}\nabla\pi = \frac{\rho}{\mu}f\ \ \ \mathrm{in}\ \Omega, \eqno(2.11)$$
$$\mathrm{div}\  u = f_{0}\ \ \ \mathrm{in}\ \Omega, \eqno(2.12)$$
$$u = v_{0}\ \ \ \ \ \ \mathrm{on}\ \partial \Omega. \eqno(2.13)$$


With $\int_{\Omega}f_{0}dx=\int_{\partial\Omega}n\cdot v_{0}dx$ the necessary condition for the solvability is given.
Here, $u$ is the velocity, $\pi$ the hydrostatic pressure, $\rho$
the density, $\mu$ the viscosity, $f$ the vector of the external
forces and the scalar function $f_{0}$ a measure of the
compressibility of fluid.
 The boundary condition (2.13) describes the adhesion at the boundary of the domain
 $\Omega$
 for $v_{0} = 0$.
 This system describes the stationary flow of a homogeneous viscous fluid for small Reynold's numbers.
 For more details, we refer to [17-19, 33].


In this paper, for $f=\sum_{i=1}^{n}f_{i}\mathrm{e}_{i}$ and
$u=\sum_{i=1}^{n}u_{i}\mathrm{e}_{i}$, let us consider the following
Stokes system in the hyper-complex formulation (see [17, 18]):
$$\widetilde{D}Du + \frac{1}{\mu}D\pi = \frac{\rho}{\mu}f\ \ \ \mathrm{in}\  \Omega, \eqno(2.14)$$
$$[Du]_{0} = 0\ \ \ \mathrm{in}\  \Omega, \eqno(2.15)$$
$$u = 0\ \ \ \ \ \ \mathrm{on}\ \partial \Omega. \eqno(2.16)$$
\\
\textbf{Definition 2.2.} We call $(u,\pi) \in W^{1,p(x)}_{0}(\Omega,
\mathrm{C}\ell _{n}) \times L^{p(x)}(\Omega)$ a solution of
(2.14)-(2.16) provided that it satisfies the system (2.14)-(2.16) for
every $f\in W^{-1,p(x)}(\Omega, \mathrm{C}\ell _{n})$.\\

\noindent\textbf{Definition 2.3.} The operator
 $\widetilde{\nabla}:L^{p(x)}(\Omega)\rightarrow
 (W^{-1,p(x)}(\Omega))^{n}$ is defined by $$<\widetilde{\nabla} f,\varphi>=-<f,\mathrm{div} \varphi>:=-\int_{\Omega}{f\
\mathrm{div} \varphi}dx$$ for all $f \in L^{p(x)}(\Omega)$ and
$\varphi\in (C^{\infty}_{0}(\Omega))^{n}$.\\

\noindent \textbf{Proposition 4.1.}(see [32, Lemma 11.1]) \begin{itshape}Let $ E_{1}$ be a Banach space , $ E_{2}, E_{3}$
 normed spaces, $A$ an operator in
 $\mathscr{L}(E_{1},E_{2})$ and $B$ a compact operator in
 $\mathscr{L}(E_{1},E_{3})$, where we denote by $\mathscr{L}(E,F)$ the set of all bounded linear operators on $E$ to $F$. If $\|u\|_{E_{1}}$ is equivalent to
 $\|Au\|_{E_{2}}+\|Bu\|_{E_{3}}$ for each $u\in E_{1}$, then the
 range space $R(A)$ of the operator $A$ is a closed subspace of
 $E_{2}$.\end{itshape}\\

\noindent\textbf{Lemma 2.8.}\begin{itshape} Let $\Omega$ be a bounded Lipschitz
 domain of $\mathbb{R}^{n}$. Let $f\in (W^{-1,p(x)}(\Omega))^{n}$
 satisfies:$$<f,\varphi>=\int_{\Omega}{f\cdot \varphi}dx=0,$$ for any
 $\varphi\in \mathscr{W}(\Omega):=\big\{v\in
 (W^{1,p'(x)}_{0}(\Omega))^{n}:\mathrm{div}\ v=0\big\}$. Then there exists
 $q\in L^{p(x)}(\Omega)$ such that $f=\widetilde{\nabla} q$.\end{itshape}

\begin{proof} We first show that the range space $R(\widetilde{\nabla} )$ is a
 closed subspace of $(W^{-1,p(x)}(\Omega))^{n}$. Let
$E_{1}=L^{p(x)}(\Omega),E_{2}=(W^{-1,p(x)}(\Omega))^{n},E_{3}=W^{-1,p(x)}(\Omega)$
$A=\widetilde{\nabla}$, $B=I$ (the identity operator). Since the
domain is bounded, the canonical imbedding $B$ of $E_{1}$ into
$E_{3}$ is compact (see [9]). On the other hand, by means of
$\mathrm{H\ddot{o}lder}$ inequality, we have for every $f\in
L^{p(x)}(\Omega)$
$$\aligned
\big\|\widetilde{\nabla}
f\big\|_{W^{-1,p(x)}(\Omega)}&=\sup_{||g||_{W^{1,p'(x)}_{0}(\Omega)}\leq1}\big|<\widetilde{\nabla}
f,g>\big|\\
&=\sup_{||g||_{W^{1,p'(x)}_{0}(\Omega)}\leq1}\Big|\int_{\Omega}{f \
\mathrm{div} g}dx\Big|\leq C\big\|f\big\|_{L^{p(x)}(\Omega)}.
\endaligned$$
Then it is easy to show that
$$\big\|\widetilde{\nabla} f\big\|_{W^{-1,p(x)}(\Omega)}+\big\|f\big\|_{W^{-1,p(x)}(\Omega)} \leq
C\big\|f\big\|_{L^{p(x)}(\Omega)} \ \ for\ all\ f\in
L^{p(x)}(\Omega).\eqno(2.17)$$

On the other hand, by Theorem 14.3.18 in [26] we have
$$\big\|f\big\|_{L^{p(x)}(\Omega)}\leq C\big(\big\|\widetilde{\nabla} f\big\|_{W^{-1,p(x)}(\Omega)}+\big\|f\big\|_{W^{-1,p(x)}(\Omega)}\big), \eqno(2.18)$$
 holds for every $f\in L^{p(x)}(\Omega)$. Then (2.17) and (2.18) yield the
equivalence of the norms of both sides above. Thus, the
desired conclusion follows from Proposition 4.1.


Since $<-\widetilde{\nabla} u,g>=<u,\mathrm{div}\ g>$ for every $u\in
L^{p(x)}(\Omega)$ and $g\in(W^{1,p'(x)}_{0}(\Omega))^{n}$, it
follows that the operator $-\widetilde{\nabla} \in
\mathscr{L}({L^{p(x)}(\Omega)},(W^{-1,p(x)}(\Omega))^{n})$ is
 the adjoint of the operator $\mathrm{div} \in
 \mathscr{L}((W^{1,p'(x)}_{0}(\Omega))^{n},{L^{p'(x)}(\Omega)})$.
 Then the Closed
Range Theorem of Banach implies that
 $$R(\widetilde{\nabla})=(\mathrm{ker(div)})^{\perp}:=\{y\in (W^{-1,p(x)}(\Omega))^{n}:\
 <y,v>=0,\forall\
 v\in \mathscr{W}(\Omega)\}.$$ This implies the desired conclusion.\end{proof}

\noindent\textbf{Theorem 2.1.}\begin{itshape} Suppose $f \in
W^{-1,p(x)}(\Omega, \mathrm{C}\ell _{n})$. Then the Stokes system
(2.14)-(2.16) has a unique solution $(u,\pi) \in
W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n}) \times L^{p(x)}(\Omega)$
in the form
 $$u + \frac{1}{\mu}TQ\pi = \frac{\rho}{\mu}TQ\widetilde{T}f,$$
with respect to the estimate
$$||Du||_{L^{p(x)}(\Omega, \mathrm{C}\ell _{n})} + \frac{1}{\mu}||Q\pi||_{L^{p(x)}(\Omega)} \leq C\frac{\rho}{\mu}||Q\widetilde{T}f||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}.$$
Here, $C\geq1$ is a constant and the hydrostatic pressure $\pi$ is unique up to a constant.
\end{itshape}

\begin{proof} We first prove that if $f \in W^{-1,p(x)}(\Omega,
\mathrm{C}\ell _{n})$, then we have the following representation:
$$TQ\widetilde{T}f= u + TQ\omega.$$
Indeed, let $\varphi_{n} \in W^{1,p(x)}_{0}(\Omega,
\mathrm{C}\ell _{n})$ with $\varphi_{n} \rightarrow \varphi$ in $L^{p(x)}(\Omega,
\mathrm{C}\ell _{n})$. By Lemma 2.4, we have
$$TQT(D\varphi_{n}) = TQ\varphi_{n}.$$
Since $W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})$ is dense in $L^{p(x)}(\Omega,
\mathrm{C}\ell _{n})$, it follows that $TQ\widetilde{T}\widetilde{D}\varphi =
TQ\varphi$. Thus, for $u \in W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})$ and
$\pi \in L^{p(x)}(\Omega)$ we obtain
$$TQ\widetilde{T}(\frac{\rho}{\mu}f)= TQ\widetilde{T}(\widetilde{D}Du + \frac{1}{\mu}\widetilde{D}\pi) = u + \frac{1}{\mu}TQ\pi.$$

This implies that our system (2.14)-(2.15) is
equivalent to the system
\begin{align}
&u + \frac{1}{\mu}QTQ\pi = \frac{\rho}{\mu}TQ\widetilde{T}f, \tag{2.19}\\
&[Q\pi]_{0} = [Q\widetilde{T}f]_{0}. \tag{2.20}
\end{align}
Obviously, the equality (2.14) is equivalent to the following
equality
$$Du + \frac{1}{\mu}Q\pi = \frac{\rho}{\mu}Q\widetilde{T}f. \eqno(2.21)$$

Now we need to show that for each $f\in W^{-1,p(x)}(\Omega,
\mathrm{C}\ell _{n}^{1})$, the function $QTf$ can be decomposed into two
functions $Du$ and $Q\pi$. Suppose
$Du + Q\pi=0$ for $u \in W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n}^{1}) \cap
\mathrm{ker \ div}$ and $\pi \in L^{p(x)}(\Omega)$. Then (2.15)
gives $[Q\pi]_{0} = 0$. Thus, $Q\pi = 0$. Hence, $Du = Q\pi = 0$.
This means that $Du + Q\pi$ is a direct sum, which is a subset of $\mathrm{im}Q$.

Next we have to ask about the existence of a functional $\mathcal
{F}\in (L^{p(x)}(\Omega, \mathrm{C}\ell _{n}^{1}) \cap \mathrm{im}Q)^{\ast}$ with
$\mathcal {F}(Du)=0$ and $\mathcal {F}(Q\pi)=0$ but $\mathcal
{F}(Q\widetilde{T}f)\neq 0$. This amounts to asking if there exists
$g \in W^{-1,p'(x)}(\Omega, \mathrm{C}\ell _{n}^{1})$, such that for all $u \in
W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n}^{1}) \cap \mathrm{ker \ div}$ and
$\omega \in L^{p(x)}(\Omega)$
$$(Du, Q\widetilde{T}g)_{Sc} = 0, \eqno(2.22)$$
$$(Q\pi, Q\widetilde{T}g)_{Sc} = 0, \eqno(2.23)$$
but $(Q\widetilde{T}f, Q\widetilde{T}g)_{Sc} \neq 0$. Here, Lemma
2.7 is used.

Thus, let us consider the system (2.22) and (2.23) with $g \in
W^{-1,p'(x)}(\Omega, \mathrm{C}\ell _{n}^{1})$. Notice that, with the help of
Lemma 2.5, (2.22) yields
$$(Du, Q\widetilde{T}g)_{Sc} = (u, \widetilde{D}Q\widetilde{T}g)_{Sc} = (u, \widetilde{D}\widetilde{T}g - \widetilde{D}P\widetilde{T}g)_{Sc} = (u, g)_{Sc} = 0,$$
which implies $g = \widetilde{\nabla} h = \widetilde{D}h$ with $h
\in L^{p'(x)}(\Omega)$ because of Lemma 2.8. Thus we obtain from (2.23) and Lemma 2.7
$$(Q\pi, Q\widetilde{T}g)_{Sc} = (Q\pi, Q\widetilde{T}\widetilde{D}h)_{Sc} = (Q\pi, Qh)_{Sc} = 0$$
holds for each $\pi \in L^{p(x)}(\Omega)$. Hence,
$Q\pi=|Qh|^{p'(x)-2}Qh$ gives $Qh = 0$. Then we obtain
$$g =\widetilde{D}h= \widetilde{D}Qh + \widetilde{D}Ph= 0.$$ Furthermore, we
get
$$(Q\widetilde{T}f, Q\widetilde{T}g)_{Sc} = 0,\ \ \forall f \in W^{-1,p(x)}(\Omega, \mathrm{C}\ell _{n}^{1}).$$
Finally, (2.21) yields
$$||Du||_{L^{p(x)}(\Omega, \mathrm{C}\ell _{n})} + \frac{1}{\mu}||Q\pi||_{L^{p(x)}(\Omega)} \geq \frac{\rho}{\mu}||Q\widetilde{T}f||_{L^{p(x)}(\Omega, \mathrm{C}\ell _{n})}.$$
By the Norm Equivalence Theorem, we obtain
$$||Du||_{L^{p(x)}(\Omega, \mathrm{C}\ell _{n})} + \frac{1}{\mu}||Q\pi||_{L^{p(x)}(\Omega)} \leq C\frac{\rho}{\mu}||Q\widetilde{T}f||_{L^{p(x)}(\Omega, \mathrm{C}\ell _{n})}.$$
By Remark 2.1, Lemma 2.2 and the boundedness of the operator $Q$,
we get
$$||u||_{W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})} + \frac{1}{\mu}||Q\pi||_{L^{p(x)}(\Omega)} \leq C\frac{\rho}{\mu}||f||_{W^{-1,p(x)}(\Omega, \mathrm{C}\ell _{n})}.\eqno(2.24)$$
>From (2.24) the uniqueness of the solution follows. Notice that
$Q\pi = 0$ implies $\pi \in \mathrm{ker}\widetilde{D}$.
Therefore, $\pi$ is unique up to a constant. Now we complete the proof of Theorem 2.1.\end{proof}



{\section*{{\bf 3. Steady Navier-Stokes equations}}}

Our aim in this section is to seek after an iterative method for the
solution of the time-independent Navier-Stokes equations:
$$-\Delta u +\frac{\rho}{\mu}(u\cdot \nabla)u + \frac{1}{\mu}\nabla\pi = \frac{\rho}{\mu}f \ \ \ \mathrm{in}\ \Omega, \eqno(3.1)$$
$$\mathrm{div}u=f_0 \ \ \ \mathrm{in}\ \Omega,\eqno(3.2)$$
$$u=v_0\ \ \ \mathrm{on}\ \partial\Omega.\eqno(3.3)$$

In addition to the case of the Stokes system, the main difference
from the above-mentioned Stokes equations is the appearance of the
non-linear convection term $(u\cdot \nabla)u$. In 1928, Oseen showed
that one can get relatively good results if the convection term
$(u\cdot \nabla)u$ is replaced by $(v\cdot \nabla)u$ , where $v$ is
a solution of the corresponding Stokes equations. In 1965, Finn [34]
proved the existence of solutions for small external forces with a
spatial decreasing to infinity of order $|x|^{-1}$ for the case of
$n=3$, and used the Banach fixed-pointed theorem. G\"{u}rlebeck and
Spr\"{o}{\ss}ig [17, 18, 21] solved this system by a reduction to a
sequence of Stokes problems provided the external force $f$ belongs
to $L^p (\Omega,\mathbb{H})$ for a bounded domain $\Omega$ and
$\frac{6}{5}<p<\frac{3}{2}$. Cerejeiras and K\"{a}hler [2] obtained
the similar results provided the external force $f$ belongs to
$W^{-1,p}(\Omega,\mathrm{C}\ell _{n})$ for an unbounded domain
$\Omega$ and $ \frac{n}{2}\leq p<\infty$. Our interest in this
section is to extend these results to the setting of variable
exponent spaces.

In this paper, for $f=\sum_{i=1}^{n}f_i e_i$ , $u=\sum_{i=1}^{n}u_i
e_i$, let us consider the following steady Navier-Stokes equations
in the hyper-complex notation:
$$\widetilde{D}Du+\frac{1}{\mu}D\pi= \frac{\rho}{\mu}F(u)\ \  \  \mathrm{in}\ \Omega, \eqno(3.4)$$
$$\big[Du\big]_0=0\ \      \  \mathrm{in}\ \Omega,  \eqno(3.5)$$
$$u=0\ \      \  \mathrm{on}\  \partial\Omega,   \eqno(3.6)$$
with the non-linear part $F(u)=f-\big[uD\big]_0 u$.\\

We first give the following lemma, which is crucial to the convergence of the iteration method.\\
\\
\textbf{Lemma 3.1.} \begin{itshape}Let $p(x)$ satisfies (2.1) and
$\frac{n}{2}\leq p_{-}\leq p(x) \leq p_{+}<\infty$. Then the
operator $F:W_0 ^{1,p(x)}(\Omega,\mathrm{C}\ell _{n}^1)\rightarrow W
^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n}^1)$  is a continuous operator
and we have
$$\big\|\big[uD\big]_0 u\big\|_{W ^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})}\leq C_1 \big\|u\big\|^2 _{W_0 ^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})},$$\end{itshape}
where $C_1=C_1(n,p,\Omega)$ is a positive constant.\\
\begin{proof} Let $u=\sum_{i=1}^{n}u_i e_i \in W_0
^{1,p(x)}(\Omega,\mathrm{C}\ell _{n} ^1)$. Then
$$\aligned
\Big\|\big[uD\big]_0 u\Big\|_{W ^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})}&=\Big\|\sum_{j=1}^{n}(\sum_{i=1}^{n}u_i\partial_{i} u_j) e_j \Big\|_{W ^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})}\\
&\leq\sum_{i,j=1}^{n}\Big\|u_i\partial_{i} u_j \Big\|_{W
^{-1,p(x)}(\Omega)}.
\endaligned$$
In view of the continuous embedding $L^{s(x)}(\Omega)\hookrightarrow
W ^{-1,p(x)}(\Omega)$ for $s(x)=\frac{np(x)}{n+p(x)}$ (see [26]), we
have
$$\big\|u_i\partial_{i} u_j \big\|_{W ^{-1,p(x)}(\Omega)}\leq C(n,p,\Omega) \big\|u_i\partial_{i} u_j \big\|_{L
^{s(x)}(\Omega)}.$$ According to Theorem 2.3 in [1] and H\"{o}lder's
inequality, we obtain
$$\aligned
\big\|u_i\partial_{i} u_j \big\|_{L ^{s(x)}(\Omega)} &\leq
C\sup_{||\varphi_j||_{L^{s'(x)}(\Omega)}\leq1}\int_{\Omega}\big|u_i\partial_i u_j\big| \big|\varphi_j\big| dx\\
&\leq C \big\|u_i \big\|_{L ^{n}(\Omega)} \big\|\partial_{i} u_j
\big\|_{L ^{p(x)}(\Omega)} \big\|\varphi_j\big\|_{L
^{s'(x)}(\Omega)}\\
 &\leq C \big\|u_i \big\|_{L ^{n}(\Omega)}
\big\| u_{j} \big\|_{W^{1,p(x)}_{0}(\Omega)}.
\endaligned$$
In terms of the continuous embedding $W^{1,p(x)}_0
(\Omega)\hookrightarrow L^{n}(\Omega) $ for $\frac{n}{2}\leq
p_{-}\leq p(x)\leq p_{+}<\infty$ (see [26]), we get
$$\Big\|u_i\partial_{i} u_j \Big\|_{L ^{s(x)}(\Omega)}\leq
C(n,p,\Omega) \big\|u_i \big\|_{W^{1,p(x)}_{0}(\Omega)} \big\| u_{j}
\big\|_{W^{1,p(x)}_{0}(\Omega)}.$$ Finally, it is immediate to get
the desired estimate from inequalities mentioned above. Hence, it is
easy to get the continuity of the operator $F$ from $F(u)=f-[uD]_0
u$.\end{proof}

\noindent\begin{itshape}Remark\end{itshape} 3.1. Actually,
$\frac{n}{2}\leq p_{-}$ means $p_{-}\in (1,+\infty)$  for $n=2$
while $p_{-}\in[\frac{n}{2},+\infty)$ for $n>2$. Obviously, Lemma
3.1 is a direct generalization of  Lemma 4.1 in [19] to the variable exponent context.\\

 We are now in position to show our main result in this section as follows.\\
\\
\textbf{Theorem 3.1.} \begin{itshape}Let $p(x)$ satisfies (2.1) and
$\frac{n}{2}\leq p_{-}\leq p(x)\leq p_{+}<\infty$. Then the system
(3.4)-(3.6) has a unique solution $(u,\pi)\in W^{1,p(x)}_0
(\Omega,\mathrm{C}\ell _{n})\times L^{p(x)}(\Omega,\mathbb{R})$
($\pi$ is unique up to a real constant) if the right-hand side $f$
satisfies the condition
$$  \|f\|_{W^{-1,p(x)} (\Omega,\mathrm{C}\ell _{n})} < \frac{\nu^2}{4 C_1 C_4^2} ,                                   \eqno (3.7)$$
with $\nu=\frac{\mu}{\rho}$ , $C_4=C_2(1+C_3)$, where $C_3 \geq 1$
indicated in (3.11) and
$$C_2=\|T\|_{[L^{p(x)}\cap
\mathrm{im}Q,W^{1,p(x)}_0]}\|Q\|_{[L^{p(x)},
L^{p(x)}\cap\mathrm{im}Q]}\|\widetilde{T}\|_{[W^{-1,p(x)},
L^{p(x)}\cap \mathrm{im}Q]}.$$

For any function $u_0 \in W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})$
with
$$||u_0||_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}\leq\frac{\nu}{2C_1 C_2}-W,                                       \eqno(3.8)$$
here, $W=\sqrt{\frac{\nu^2}{4C_1 ^2C_4
^2}-\frac{1}{C_1}||f||_{W^{-1,p(x)} (\Omega,\mathrm{C}\ell _{n})}}$
, the iteration procedure
$$   u_k + \frac{1}{\mu}TQ\pi_k=\frac{\rho}{\mu}TQ\widetilde{T}F(u_{k-1}),\ k=1,2,\ldots    \eqno(3.9)$$
$$   \frac{1}{\mu}\big[Q\pi_k\big]_0= \frac{\rho}{\mu}\big[Q\widetilde{T}F(u_{k-1})\big]_0,  \eqno(3.10)$$
converges in $W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})\times
L^{p(x)}(\Omega,\mathbb{R})$.\end{itshape}\\

\begin{proof} We follow the idea similar to the one in [18], and for the reader's convenience, hear we will give the full details of the proof.
Replacing $f$  by $F(u_{k-1})$ in the proof of Theorem 2.1 we obtain the unique solvability of the
Stokes equations (3.9)-(3.10) which we have to solve in each step.
Moreover, we have the following estimate:
$$ \big\|Du_k\big\|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})} + \frac{1}{\mu}\big\|Q\pi_k\big\|_{L^{p(x)}(\Omega)}\leq C_3\frac{\rho}{\mu}\big\|Q\widetilde{T}F(u_{k-1})\big\|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}              \eqno(3.11)$$
where $C_3 \geq1$  is a constant. The only remaining problem is the
convergence of our iteration procedure. From Theorem 2.1 we know
$$Du_k +\frac{1}{\mu} Q\pi_k =\frac{\rho}{\mu} Q\widetilde{T}F(u_{k-1}). $$
Then we deduce from (3.11)
$$ \frac{1}{\mu}\big\|Q(\pi_k -\pi_{k-1})\big\|_{L^{p(x)}(\Omega)}
\leq\frac{C_3}{\nu}\big\|Q\widetilde{T}(F(u_{k-1})-F(u_{k-2}))\big\|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}.$$
Hence we have
$$\aligned
&\big\|u_k -u_{k-1}\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}\\
\leq& \frac{1}{\mu}\big\|TQ(\pi_k -\pi_{k-1})\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}+\frac{\rho}{\mu}\big\|TQ\widetilde{T}(F(u_{k-1})-F(u_{k-2}))||_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}\\
\leq&\frac{C_2
(1+C_3)}{\nu}\big\|F(u_{k-1})-F(u_{k-2})\big\|_{W^{-1,p(x)}
(\Omega,\mathrm{C}\ell _{n})}.
\endaligned$$
By means of Lemma 3.1 we deduce
$$\aligned
&\big\|F(u_{k-1})-F(u_{k-2})\big\|_{W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})}\\
=&\big\|[u_{k-1}D]_0 u_{k-1}-[u_{k-2}D]_0u_{k-2}\big\|_{W^{-1,p(x)} (\Omega,\mathrm{C}\ell _{n})}\\
=&\big\|((u_{k-1} -u_{k-2})\cdot \nabla)u_{k-1}\big\|_{W^{-1,p(x)} (\Omega,\mathrm{C}\ell _{n})}+\big\|(u_{k-2}\cdot \nabla)(u_{k-1} -u_{k-2})\big\|_{W^{-1,p(x)} (\Omega,\mathrm{C}\ell _{n})}\\
\leq&C_1 \big\|u_{k-1} -u_{k-2}\big\|_{W^{1,p(x)}_0
(\Omega,\mathrm{C}\ell _{n})}(\big\|u_{k-1}\big\|_{W^{1,p(x)}_0
(\Omega,\mathrm{C}\ell _{n})}+\big\|u_{k-2}\big\|_{W^{1,p(x)}_0
(\Omega,\mathrm{C}\ell _{n})}).
\endaligned$$
Let $L_k =\frac{C_1 C_4}{\nu}(\big\|u_{k-1}\big\|_{W^{1,p(x)}_0
(\Omega,\mathrm{C}\ell _{n})}+\big\|u_{k-1}\big\|_{W^{1,p(x)}_0
(\Omega,\mathrm{C}\ell _{n})})$ with $C_4=C_2(1+C_3)$. Then we
obtain
$$\big\|u_k -u_{k-1}\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}\leq L_k \big\|u_{k-1} -u_{k-2}\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}. \eqno(3.12)$$
On the other hand, by (3.11) and Lemma 3.1, we have
$$\aligned
\big\|u_k\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}&\leq\frac{1}{\mu}\big\|TQ\pi_k\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}+\frac{\rho}{\mu}\big\|TQ\widetilde{T}F(u_{k-1})\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}\\
&\leq\frac{C_1 C_4}{\nu}\big\|u_{k-1}\big\|^2_{W^{1,p(x)}_0
(\Omega,\mathrm{C}\ell
_{n})}+\frac{C_4}{\nu}\big\|f\big\|_{W^{-1,p(x)}
(\Omega,\mathrm{C}\ell _{n})}.
\endaligned$$
Now we have to ensure that $\big\|u_k\big\|_{W^{1,p(x)}_0
(\Omega,\mathrm{C}\ell _{n})}\leq\big\|u_{k-1}\big\|_{W^{1,p(x)}_0
(\Omega,\mathrm{C}\ell _{n})}$. For this we notice that
$$\aligned
&\frac{C_1 C_4}{\nu}\big\|u_{k-1}\big\|^2_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}+\frac{C_4}{\nu}\big\|f\big\|_{W^{-1,p(x)} (\Omega,\mathrm{C}\ell _{n})}\leq\big\|u_{k-1}\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}\\
&\Leftrightarrow\big\|u_{k-1}\big\|^2_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}-\frac{\nu}{C_1 C_4}\big\|u_{k-1}\big\|_{W^{1,p(x)}_0 (\Omega,Cl_n)}+\frac{1}{C_1}\big\|f\big\|_{W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})}\leq0\\
&\Leftrightarrow(\big\|u_{k-1}\big\|_{W^{1,p(x)}_0
(\Omega,\mathrm{C}\ell _{n})}-\frac{\nu}{2C_1 C_4})^2
\leq\frac{\nu^2}{(2C_1
C_4)^2}-\frac{1}{C_1}\big\|f\big\|_{W^{-1,p(x)}(\Omega,\mathrm{C}\ell
_{n})}.
\endaligned$$
According to the assumption (3.7), we have
$$\Big| \big\|u_{k-1}\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}-\frac{\nu}{2C_1 C_4} \Big|\leq W$$
with $W=\sqrt{\frac{\nu^2}{4C_1 ^2C_4
^2}-\frac{1}{C_1}\big\|f\big\|_{W^{-1,p(x)} (\Omega,\mathrm{C}\ell
_{n})}}$. This leads to the following condition for
$\big\|u_{k-1}\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}$ :
$$\frac{\nu}{2C_1 C_4}-W\leq \big\|u_{k-1}\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}\leq \frac{\nu}{2C_1 C_4}+W.$$
Now assume $\big\|u_{k-1}\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell
_{n})}\leq \frac{\nu}{2C_1 C_4}-W.$ Then it follows
$$\aligned
\big\|u_k\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}&\leq\frac{1}{\mu}\big\|TQ\pi_k\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}+\frac{\rho}{\mu}\big\|TQ\widetilde{T}F(u_{k-1})\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}\\
&\leq\frac{C_1 C_4}{\nu}\big\|u_{k-1}\big\|^2_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}+\frac{C_4}{\nu}\big\|f\big\|_{W^{-1,p(x)} (\Omega,\mathrm{C}\ell _{n})}\\
&\leq\frac{C_1 C_4}{\nu}(\frac{\nu}{2C_1 C_4}-W)^2+\frac{C_4}{\nu}\big\|f\big\|_{W^{-1,p(x)} (\Omega,\mathrm{C}\ell _{n})}\\
&\leq\frac{\nu}{2C_1 C_4}-W.
\endaligned$$
Consequently, using the inequality
$\big\|u_{k-2}\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}\leq
\frac{\nu}{2C_1 C_4}-W$  and (3.12) we have
$$\aligned
\big\|u_k -u_{k-1}\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}&\leq\frac{2C_1 C_4}{\nu}(\frac{\nu}{2C_1 C_4}-W)\big\|u_{k-1} -u_{k-2}\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}\\
&\leq(1-\frac{2C_1 C_4}{\nu}W)\big\|u_k -u_{k-1}\big\|_{W^{1,p(x)}_0
(\Omega,\mathrm{C}\ell _{n})}.
\endaligned$$
And
$$ L_k\leq 1-\frac{2C_1 C_4}{\nu}W:=L<1.$$
In the case one has
$$\big\|u_k -u_{k-1}\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}\leq L\big\|u_{k-1} -u_{k-2}\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}\eqno(3.13)$$
with $0<L<1$  and fixed. The convergence of the sequence $\{u_{k}\}$
is therefore obtained by making use of the Banach fixed-point
theorem, and hence the convergence of the sequence $\{\pi_{k}\}$
immediately follows from (3.9).\end{proof}
\noindent\begin{itshape}Remark\end{itshape} 3.2. Here, $\nu$ is the
kinematic viscosity of the fluid. Theorem 3.1 states that under
certain smallness condition of the external force, there exists a
unique solution to the stationary
Navier-Stokes equations.
\\
\textbf{Corollary 3.1.} \begin{itshape}Under the same hypotheses as
in Theorem 3.1, we have the a-priori estimate
$$ \big\|u\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}\leq \frac{\nu}{2C_1 C_4}-W.\eqno(3.14)$$
An a-priori estimate for the term
$\big\|Q\pi\big\|_{L^{p(x)}(\Omega)}$ is easy to
obtain.\end{itshape}\\
\begin{proof} (3.14) immediately follows from
$\big\|u_{k-1}\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}\leq
\frac{\nu}{2C_1 C_4}-W$. It is easy to obtain the estimate for
$\big\|Q\pi\big\|_{L^{p(x)}(\Omega)}$ from (3.11), (3.14) and Lemma
3.1. \end{proof}

Now it is immediate to obtain the following result from (3.13).\\
\\
\textbf{Corollary 3.2.}
\begin{itshape}There exists the error estimate
$$\big\|u_k -u\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}\leq \frac{L^k}{1-L} \big\|u_0 -u\big\|_{W^{1,p(x)}_0 (\Omega,\mathrm{C}\ell _{n})}.$$
In the case of $u_0=0$  we have
$$\big\|u_k -u\big\|_{W^{1,p(x)}_0 (\Omega,Cl_n)}\leq \frac{L^k}{1-L}\Big(\frac{\nu}{2C_1
C_4}-W\Big).$$\end{itshape}

{\section*{{\bf 4. Steady Navier-Stokes equations with
heat conduction}}}

In this section we will consider the flow of a viscous fluid under
the influence of temperature. Similar to [18] the above used method
for treating the steady Navier-Stokes equations can be applied to
more complicated problems. More precisely, we consider the following
problem:
$$ -\Delta u +\frac{\rho}{\mu}(u\cdot \nabla)u + \frac{1}{\mu}\nabla \pi +\frac{\gamma}{\mu}gw= -f\ \   \ \mathrm{in}\ \Omega, \eqno(4.1)$$
$$ -\Delta w +\frac{m}{\kappa}(u\cdot \nabla)w = \frac{1}{\kappa}h\ \   \ \mathrm{in}\ \Omega, \eqno(4.2)$$
$$ \mathrm{div} u=0\ \  \ \mathrm{in}\ \Omega, \eqno(4.3)$$
$$u,w=0\ \   \   \mathrm{on}\ \partial\Omega.   \eqno(4.4)$$

In addition to the case of Navier-Stokes equations,  $w$ denotes the temperature,  $\gamma$ the Grasshof number,
$m$  the Prandtl number,  $\kappa$ the number of temperature conductivity and  $g$ the vector $(0,0,\ldots,-1)^{T}$ ,
where only the $n$th component is different from zero.
\\
\noindent\begin{itshape}Remark\end{itshape} 4.1. The detailed account for the Grasshof number, the Prandtl number and the Reynolds number we refer to [20].\\
\\
\noindent\begin{itshape}Remark\end{itshape} 4.2. In the case of
$\Omega$  a bounded domain and space $W_0 ^{k,2}(\Omega,\mathbb{H})$
the problem (4.1)-(4.4) was already studied by G\"{u}rlebeck and
Spr\"{o}{\ss}ig [18]. In the case of $\Omega$ a unbounded domain and
space  $W_0 ^{1,p}(\Omega,\mathrm{C}\ell _{n})$ the problem
(4.1)-(4.4) was already
investigated by Cerejeiras and K\"{a}hler [19].

In analogy to the case of the Navier-Stokes equations, we consider
the following equivalent hyper-complex problem:

$$u+\frac{1}{\mu}TQ\pi=-TQ\widetilde{T}(F(u)-\frac{\gamma}{\mu}e_n w) \ \  \mathrm{in}\  \Omega,\eqno(4.5)$$
$$\frac{1}{\mu}\big[Q\pi\big]_0=\big[Q\widetilde{T}(F(u)-\frac{\gamma}{\mu}e_n w)\big]_0 \ \  \mathrm{in}\  \Omega,\eqno(4.6)$$
$$w=-\frac{m}{\kappa}TQ\widetilde{T}\big[uD\big]_0 w+\frac{1}{\kappa}TQ\widetilde{T}h \ \  \mathrm{in}\ \Omega,\eqno(4.7)$$
with $F(u):=f+\frac{\rho}{\mu} \big[uD\big]_0 u$. Then the problem
can be solved by the following iteration procedure:
$$u_k+\frac{1}{\mu}TQ\pi_k=-TQ\widetilde{T}(F(u_{k-1})-\frac{\gamma}{\mu}e_n w_{k-1}) \ \ \mathrm{in}\  \Omega,\eqno(4.8)$$
$$\frac{1}{\mu}\big[Q\pi_k\big]_0=\big[Q\widetilde{T}(F(u_{k-1})-\frac{\gamma}{\mu}e_n w_{k-1})\big]_0 \ \  \mathrm{in}\  \Omega,\eqno(4.9)$$
$$w_k=-\frac{m}{\kappa}TQ\widetilde{T}\big[u_k D\big]_0w_k+\frac{1}{\kappa}TQ\widetilde{T}h \ \ \mathrm{in}\  \Omega.\eqno(4.10)$$

The equations (4.8) and (4.9) represent an iteration similar to the
case of the Navier-Stokes equations. Therefore we have to study the
solvability of equation (4.10). For this purpose, in analogy to
[18], we propose the following "inner" iteration:
$$w_k ^i=-\frac{m}{\kappa}TQ\widetilde{T}(u_k\cdot \nabla)w^{i-1} _k+\frac{1}{\kappa}TQ\widetilde{T}h. \eqno(4.11)$$
\textbf{Theorem 4.1.} \begin{itshape}Let $u_k \in W_0 ^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$, where $p(x)$ satisfies (2.1) and $\frac{n}{2}\leq p_{-} \leq p(x) \leq p_{+}<\infty$. Furthermore, suppose\\
\\
(i) $\big\|u_k\big\|_{W_0 ^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})}<\kappa/mC_1 C_2 $;\\
\\
(ii) $m\nu<2\kappa(1+C_3).$ \\
Then the iteration procedure (4.11) converges in $W_0
^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$  to a unique solution of
(4.10) and we have a-priori estimate
$$ \big\|w_k\big\|_{W_0 ^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})}\leq\frac{2(1+C_3)C_2}{2\kappa(1+C_3)-m\nu}  \big\|h\big\|_{W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})}.          \eqno(4.12)$$\end{itshape}
\begin{proof} From the assumption (i), (4.10) and Lemma 3.1 it
follows
$$\big\|w_k ^i\big\|_{W_0 ^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})}\leq \frac{C_2\big\|h\big\|_{W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})}}{\kappa-mC_1 C_2 \big\|u_k\big\|_{W_0 ^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})}}$$
for each $i\in \mathbb{N}$. Consequently, the sequence  $\{w^i
_k\}_{i\in \mathbb{N}}$ is bounded in $W_0 ^{1,p(x)}(\Omega,\mathrm{C}\ell
_{n})$. In view of Lemma 3.1 we get
$$\aligned
\big\|w_k ^i-w_k ^{i-1}\big\|_{W_0 ^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})}&=\big\|\frac{m}{\kappa}TQ\widetilde{T}(u_k \cdot \nabla)(w_k ^{i-1}-w_k ^{i-2})\big\|_{W_0 ^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})}\\
&\leq \frac{m}{\kappa}C_1 C_2 \big\|u_k\big\|_{W_0
^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})}\big\|w_k ^i-w_k
^{i-1}\big\|_{W_0 ^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})}.
\endaligned$$
Therefore, if $\frac{m}{\kappa}C_1 C_2 \big\|u_k\big\|_{W_0
^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})}<1$, then we obtain the
convergence of the sequence $\{w^i _k\}_{i\in \mathbb{N}}$  in $W_0
^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$. Similar to the proof of
Theorem 3.1, we can obtain from (4.8)
\begin{align*}
&\big\|u_k\big\|_{W_0 ^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})}\\
\leq& \frac{\nu}{2C_{1}C_{4}} -\sqrt{\frac{\nu^{2}}{4C^{2}_{1}C^{2}_{4}} - \frac{\nu}{C_{1}}\parallel f\parallel_{W^{-1,p(x)}(\Omega, \mathrm{C}\ell _{n})} - \frac{\gamma }{\mu C_{1}}\parallel w_{k-1}\parallel_{W_0 ^{1,p(x)}(\Omega, \mathrm{C}\ell _{n})}}\\
\leq& \frac{\nu}{2C_{1}C_{4}}.\tag{4.13}
\end{align*}
This implies that the condition (ii) is a sufficient convergence
condition.\end{proof}

\noindent\textbf{Theorem 4.2.} \begin{itshape}Let $f \in W^{-1,p(x)}(\Omega, \mathrm{C}\ell _{n})$, $h \in W^{-1,p(x)}(\Omega, \mathrm{C}\ell _{n})$,
where $p(x)$ satisfies (2.1) and $\frac{n}{2} \leq p_{-}\leq p(x) \leq p_{+} < \infty$ . Furthermore, assume\\
\\
(a) $\nu\parallel f\parallel_{W^{-1,p(x)}(\Omega, \mathrm{C}\ell _{n})} + C_5\parallel h\parallel_{W^{-1,p(x)}(\Omega, \mathrm{C}\ell _{n})} < C_6$;\\
\\
(b) $\parallel h\parallel_{W^{-1,p(x)}(\Omega, \mathrm{C}\ell _{n})} < C_7$;\\
\\
(c) $m\nu <  2\kappa (1+C_{3}),$\\
where $$C_5=\frac{2\gamma (1+C_{3})C_{2}}{\mu(2\kappa
(1+C_{3})-m\nu)},\ \ C_6=\frac{3\nu^{2}}{16C_{1}C^{2}_{4}},\ \
C_7=\frac{\mu(2\kappa (1+C_{3})-m\nu)^{2}}{8\gamma
mC_{1}C^{3}_{2}(1+C_{3})^{3}}.$$ Then the problem (4.5)-(4.7) has a
unique solution $(u,w,\pi)$ in space $ W^{1,p(x)}_{0}(\Omega,
\mathrm{C}\ell _{n}) \times W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell
_{n}) \times L^{p(x)}(\Omega,\mathbb{R}),$ where  $u$ and  $w$ are
uniquely defined and  $\pi$ uniquely up to a constant. Our iteration
procedure (4.8)-(4.10) converges to the solution of
(4.5)-(4.7).\end{itshape}
\begin{proof} The proof is similar to the one in [18], here we sketch the key steps of the proof briefly for the reader's convenience.
It is immediate to get the following estimate from the
assumption (a)
$$\frac{\nu}{C_{1}}\parallel f\parallel_{W^{-1,p(x)}(\Omega, \mathrm{C}\ell _{n})}
+ \frac{C_{5}}{C_{1}}\parallel h\parallel_{W^{-1,p(x)}(\Omega,
\mathrm{C}\ell _{n})} < \frac{\nu^{2}}{4C^{2}_{1}C^{2}_{4}}.$$ In
virtue of the a-priori estimate (4.12), Similar to the proof of
Theorem 3.1, we obtain the following a-priori estimate
$$
\parallel u_{k}\parallel_{W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})} \leq
\frac{\nu}{2C_{1}C_{4}} -M \eqno(4.14)$$ with
$$ M=\sqrt{\frac{\nu^{2}}{4C^{2}_{1}C^{2}_{4}} -
\frac{\nu}{C_{1}}\parallel f\parallel_{W^{-1,p(x)}(\Omega,
\mathrm{C}\ell _{n})} - \frac{C_{5}}{C_{1}}\parallel
h\parallel_{W^{-1,p(x)}(\Omega, \mathrm{C}\ell _{n})}}.$$ So the
sequence $\{u_{k}\}_{k\in \mathbb{N}}$ and $\{w_{k}\}_{k\in
\mathbb{N}}$ are bounded in $W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell
_{n})$. Now (4.10) yields
$$\aligned
&\parallel w_{k} - w_{k-1}\parallel_{W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})} \\
\leq& \frac{m}{\kappa}C_{2}\parallel (u_{k} \cdot \nabla)(w_{k} -
w_{k-1}) + ((u_{k} - u_{k-1})\cdot
\nabla)w_{k-1}\parallel_{W^{-1,p(x)}(\Omega, \mathrm{C}\ell _{n})}.
\endaligned$$
Then Lemma 3.1, (4.12) and (4.13) give
$$\aligned
&\parallel w_{k} - w_{k-1}\parallel_{W^{1,p(x)}_{0}(\Omega,
\mathrm{C}\ell _{n})}\\
\leq& \frac{\mu}{2\gamma C_{4}C_{7}}\parallel h\parallel_{W^{-1,p(x)}(\Omega, \mathrm{C}\ell _{n})}
 \parallel u_{k} -
u_{k-1}\parallel_{W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})}.
\endaligned$$
Consequently, we obtain from (4.8)
$$\parallel u_{k} - u_{k-1}\parallel_{W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})} \leq L_{k}\| u_{k-1} - u_{k-2}\|_{W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})},$$
with
$$\aligned
L_{k} = &\frac{C_{1}C_{4}}{\nu}\big(\parallel u_{k-1}\parallel_{W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})} + \parallel u_{k-2}\parallel_{W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})}\big)
+ \frac{1}{2C_7}\parallel h\parallel_{W^{-1,p(x)}(\Omega,
\mathrm{C}\ell _{n})}.
\endaligned$$
Obviously, we can get from the conditions (a), (b) and (4.14)
$$L_{k} \leq L < 1,$$
where $0<L<1$ is a constant independent of $k$. Now we can apply the
Banach fixed-pointed theorem.\end{proof}

\noindent\textbf{Acknowledgements.}
The first author wishes to
express his appreciation to Professor Uwe K\"{a}hler for his
assistance during the preparation of the work.

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