\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 98, pp. 1--21.\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-2017/98\hfil Navier-Stokes equations] {Navier-Stokes equations in the half-space in variable exponent spaces of Clifford-valued functions} \author[R. Niu, H. Zheng, B. Zhang \hfil EJDE-2017/98\hfilneg] {Rui Niu, Hongtao Zheng, Binlin Zhang} \address{Rui Niu \newline College of Power and Energy Engineering, Harbin Engineering University, 150001 Harbin, China. \newline Department of Mathematics, Heilongjiang Institute of Technology, 150050 Harbin, China} \email{ruiniu1981@gmail.com} \address{Hongtao Zheng (corresponding author)\newline College of Power and Energy Engineering, Harbin Engineering University, 150001 Harbin, China} \email{zht-304@163.com} \address{Binlin Zhang \newline Department of Mathematics, Heilongjiang Institute of Technology, 150050 Harbin, China} \email{zhangbinlin2012@163.com} \dedicatory{Communicated by Vicentiu Radulescu} \thanks{Submitted January 28, 2017. Published April 5, 2017.} \subjclass[2010]{30G35, 35J60, 35Q30, 46E30, 76D03} \keywords{Clifford analysis; variable exponent; Navier-Stokes equations; \hfill\break\indent half-space} \begin{abstract} In this article, we study the steady generalized Navier-Stokes equations in a half-space in the setting of variable exponent spaces. We first establish variable exponent spaces of Clifford-valued functions in a half-space. Then, using this operator theory together with the contraction mapping principle, we obtain the existence and uniqueness of solutions to the stationary Navier-Stokes equations and Navier-Stokes equations with heat conduction in a half-space under suitable hypotheses. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} Since Kov\'{a}\v{c}ik and R\'{a}kosn\'{i}k \cite{25} first studied the spaces $L^{p(x)}$ and $W^{k, p(x)}$, more and more attention are paid to Lebesgue and Sobolev variable exponent spaces and their applications to differential equations. See \cite{9, 10} for basic properties of variable exponent spaces and \cite{23, VR} for recent overviews of differential equations with variable growth. It is well-known that 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 \cite{28, 29}, which can be described by the boundary-value problem for the generalized Navier-Stokes equations. In the setting of variable exponent spaces, Diening et al.\ \cite{4} proved the existence and uniqueness of strong and weak solutions of the Stokes system and Poisson equations for bounded domains, the whole-space and the half-space, respectively. In the previous decades, the study of these spaces has been stimulated by problems in elastic mechanics, calculus of variations and differential equations with variable growth conditions, see \cite{12, FS, RZ, RR, DR, 32, 33} 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 begin 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 \cite{21}. After that, Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. Clifford analysis as an active branch of mathematics concerned with 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 the monograph of Brackx et al.\ \cite{BDS}. It is worthy mentioning that G\"urlebeck and Spr\"o{\ss}ig \cite{16, 17} 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 \cite{2, 18}. As Galdi \cite{22} pointed out, the study of the Stokes problem in the half-space possesses an independent interest and it will be fundamental for the treatment of other linear and nonlinear problems when the region of flow is either an exterior domain or a domain with a suitable unbounded boundary. On the one hand, Diening et al.\ \cite{DMS} studied the following model introduced in \cite{28, 29} to describe motions of electrorheological fluids: \begin{equation} \begin{gathered} -\operatorname{div}\mathcal{M}(\mathrm{D}u)+(u\cdot \nabla)u + \nabla \pi = f\quad x\in\Omega\\ \operatorname{div}u=0\quad x\in\Omega\\ u=0\quad x\in\partial\Omega, \end{gathered}\label{e1.1} \end{equation} where $\Omega$ is a bounded domain with Lipschitz boundary $\partial\Omega$, $f\in (W^{1,p(x)}_0(\Omega))^{*}=W^{-1,p'(x)}(\Omega)$, $2n/(n+2)0\}$ and $\Sigma=\partial \mathbb{R}^n_{+}$. A Clifford-valued function $u:\mathbb{R}^n_{+}\to\mathrm{C}\ell _n$ can be written as $u=\Sigma_{I}u_{I}\mathrm{e}_{I}$, where the coefficients $u_{I}:\mathbb{R}^n_{+}\to \mathbb{R}$ are real-valued functions. The Dirac operator on the Euclidean space used here is introduced by $$ D=\sum_{j=1}^ne_{j}\frac{\partial}{\partial x_{j}} :=\sum_{j=1}^ne_{j}\partial_{j}. $$ If $u$ is a real-valued function defined on $\mathbb{R}^n_{+}$, then $Du=\nabla u=(\partial_1u,\partial_2u,\ldots,\partial_nu)$. 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 $x\in\mathbb{R}^n_{+}$. 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 \cite{14, 19} and references therein. \subsection{Variable exponent spaces of Clifford-valued functions} Next we recall some basic properties of variable exponent spaces of Clifford-valued functions. In what follows, we use the short notation $L^{p(x)}(\mathbb{R}^N_{+})$, $W^{1,p(x)}(\mathbb{R}^N_{+})$, etc., instead of $L^{p(x)}(\mathbb{R}^N_{+}, \mathbb{R})$, $W^{1,p(x)}(\mathbb{R}^N_{+},\mathbb{R})$, etc. Throughout this paper we always assume (unless declared specially) \begin{equation} p \in \mathcal {P}^{\rm log}(\mathbb{R}^n_{+}) \mathrm{ and } 1 < p_{-}:= \inf_{x \in \overline{\mathbb{R}^n_{+}}}p(x) \leq p(x) \leq \sup_{x \in \overline{\mathbb{R}^n_{+}}}p(x) =:p_{+} < \infty. \label{e2.1} \end{equation} where $\mathcal{P}^{\rm log}(\mathbb{R}^n_{+})$ is the set of exponent $p$ satisfying the so-called log-H\"older continuity, i.e., $$ |p(x) - p(y)| \leq \frac{C}{\log (e + |x-y|^{-1})},\quad |p(x) - p(\infty)| \leq \frac{C}{\log (e + |x|^{-1})} $$ hold for all $x,y \in \mathbb{R}^N_{+}$, where $p(\infty)=\lim_{|x|\to \infty}p(x)$, see \cite{3, 4}. Let $\mathcal{P}(\mathbb{R}^n_{+})$ be the set of all Lebesgue measurable functions $p: \mathbb{R}^n_{+} \to (1,\infty)$. Given $p \in \mathcal{P}(\mathbb{R}^n_{+})$ we define the conjugate function $p'(x)\in \mathcal{P}(\mathbb{R}^n_{+})$ by $$ p'(x)=\frac{p(x)}{p(x)-1}\quad\text{for each } x\in\mathbb{R}^N_{+}. $$ The variable exponent Lebesgue space $L^{p(x)}(\mathbb{R}^n_{+})$ is defined by $$ L^{p(x)}(\mathbb{R}^n_{+})=\big\{ u\in \mathcal{P}(\mathbb{R}^n_{+}): \int_{\mathbb{R}^n_{+}}|u|^{p(x)}dx <\infty \big\}, $$ with the norm $$ \|u\|_{L^{p(x)}(\mathbb{R}^n_{+})} =\inf \big\{t>0:\int_{\mathbb{R}^n_{+}}|\frac{u}{t}|^{p(x)}dx \leq 1\big\}. $$ The variable exponent Sobolev space $W^{1,p(x)}(\mathbb{R}^n_{+})$ in a half-space is defined by $$ W^{1,p(x)}(\mathbb{R}^n_{+})=\big\{u\in L^{p(x)}(\mathbb{R}^n_{+}) : |\nabla u| \in L^{p(x)}(\mathbb{R}^n_{+})\big\}, $$ with the norm \begin{equation} \|u\|_{W^{1,p(x)}(\mathbb{R}^n_{+})} =\|\nabla u\|_{L^{p(x)}(\mathbb{R}^n_{+})} + \|u\|_{L^{p(x)}(\mathbb{R}^n_{+})}.\label{e2.2} \end{equation} Denote $W_0^{1,p(x)}(\mathbb{R}^n_{+})$ by the completion of $C_0^{\infty}(\mathbb{R}^n_{+})$ in $W^{1,p(x)}(\mathbb{R}^n_{+})$ with respect to the norm \eqref{e2.2}. The space $W^{-1,p(x)}(\mathbb{R}^n_{+})$ is defined as the dual of the space $W_0^{1,p'(x)}(\mathbb{R}^n_{+})$. For more details we refer to \cite{3, 9, 10} and reference therein. In the following, we say that $u\in L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell_n)$ can be understood coordinate wise. For example, $u\in L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ means that $\{u_{I}\} \subset L^{p(x)}(\mathbb{R}^n_{+})$ for $u=\Sigma_{I}u_{I}e_{I}\in \mathrm{C}\ell _n$ with the norm $\|u\|_{L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)} = \sum_{I}\|u_{I}\|_{L^{p(x)}(\mathbb{R}^n_{+})}$. In the same way, spaces $W^{1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$, $W_0^{1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$, $C_0^{\infty}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$, etc., can be understood similarly. In particular, the space $L^2(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ can be converted into a right Hilbert $\mathrm{C}\ell _n$-module by defining the following Clifford-valued inner product (see \cite[Definition 3.74]{16}) \begin{equation} \big(f,g\big)_{\mathrm{C}\ell _n} =\int_{\mathbb{R}^n_{+}}{\overline{f(x)}g(x)}dx. \label{e2.3} \end{equation} \begin{remark} \label{re11} \rm Following the same arguments as in \cite{13, 34}, we can calculate easily that $\|u\|_{L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)}$ is equivalent to the norm $\||u|\|_{L^{p(x)}(\mathbb{R}^n_{+})}$. Furthermore, we also prove that for every $u \in W^{1,p(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$, $\|Du\|_{L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)}$ is an equivalent norm of $\|u\|_{W^{1,p(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)}$. \end{remark} \begin{lemma}[\cite{13}] \label{le111} Assume that $p(x) \in \mathcal{P}(\mathbb{R}^n_{+})$. Then $$ \int_{\mathbb{R}^n_{+}}|uv|dx \leq C(n,p)\|u\|_{L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)}\|v\|_{L^{p'(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)} $$ for every $u \in L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ and $ v \in L^{p'(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$. \end{lemma} \begin{lemma}[\cite{13, 14}] \label{le11} If $p(x) \in \mathcal{P}(\mathbb{R}^n_{+})$, then $L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ and $W^{1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ are reflexive Banach spaces. \end{lemma} Based on the Cauchy kernel $G(x)$ we can introduce the Teodorescu operator. There exist a number of applications and methods based on the properties of this Teodorescu operator. But in our case of considering the domain $\Omega$ as an unbounded domain, the main problem in applying this operator is that the Cauchy kernel does not have good enough behaviour near infinity. For example, the Teodorescu operator is an unbounded operator over the usual function spaces on $\Omega$. In this paper, we will follow the idea from \cite{2, GKRS} by using add-on terms to the Cauchy kernel. More precisely, we choose a fixed point $z$ lying in the complement of $\mathbb{R}^N_{+}$. Then we consider the following operators. \begin{definition}[\cite{2, 16, 17}] \rm (i) Let $u \in C(\mathbb{R}^n_{+}, \mathrm{C}\ell_n)$. The Teodorescu operator is defined by $$ Tu(x) = \int_{\mathbb{R}^n_{+}}K_{z}(x, y)u(y)dy, $$ where $K_z(x,y)=G(x-y)-G(y-z)$, $G(x)$ is the above-mentioned generalized Cauchy kernel. (ii) Let $u \in C^{1}(\mathbb{R}^n_{+}, \mathrm{C}\ell_n) \cap C(\overline{\mathbb{R}^n_{+}},\mathrm{C}\ell_n)$. The {\it boundary operator} is defined by $$ Fu(x) = \int_{\Sigma}K_{z}(x, y)\alpha(y)u(y)dS_{y}, $$ where $\alpha(y)$ denotes the outward normal unit vector at $y$. (iii) Let $u \in L^{1}_{\rm loc}(\mathbb{R}^n)$. Then the Hardy-Littlewood maximal operator is defined by $$ Mu\big(x\big) = \sup_{x\in Q}\frac{1}{|Q|} \int_{Q}|u(y)|dy. $$ for all $x\in \mathbb{R}^n$, where the supremum is taken over all cubes (or ball) $Q\subset \mathbb{R}^n$ which contain $x$. \end{definition} The Teodorescu operator was first introduced in \cite{GKRS} and the operator properties in the scale of $W^{k,2}$-spaces were given in \cite{UK}, see also \cite{2} for the corresponding operator properties in the $W^{k,q}$-spaces over unbounded domains. Its main advantage is a faster decay to infinity of the kernel. \begin{lemma}\label{le1} {\rm (see \cite{3})} Let $x \in \mathbb{R}^n$, $\delta>0$ and $u \in L^{1}_{\rm loc}(\mathbb{R}^n)$. Then $$ \int_{B(x, \delta)}\frac{1}{|x-y|^{n-1}}|u(y)|dy \leq C(\delta) Mu(x). $$ where $C(\delta)>0$ is a positive constant. Moreover, if $u\in L^{p(x)}(\mathbb{R}^n)$ with $\|u\|_{p(x)}\leq 1$, then $$ \int_{\mathbb{R}^n\setminus B(x, \delta)}\frac{1}{|x-y|^{n-1}}|u(y)|dy \leq C(n,p,\delta, |B|). $$ where $C(n,p,\delta, |B|)$ is a positive constant. \end{lemma} \begin{lemma}[\cite{3}] \label{le2} Let $p(x)$ satisfy \eqref{e2.1}. Then $M$ is bounded in $L^{p(x)}(\mathbb{R}^n)$. \end{lemma} \begin{lemma}[\cite{GKRS}] \label{le3} Let $u \in C^{1}(\mathbb{R}^n_{+}, \mathrm{C}\ell_n)$. Then $$ \partial_kTu(x) = \frac{1}{\omega_n}\int_{\mathbb{R}^n_{+}} \frac{\partial}{\partial x_k}G(x-y)u(y)dy + \frac{u(x)}{n}\overline{e}_k. $$ \end{lemma} \begin{lemma}[\cite{3}] \label{le4} Let $\Phi$ be a Calder\'on-Zygmund operator with Calder\'on-Zygmund kernel $K$ on $\mathbb{R}^n \times\mathbb{R}^n$. Then $\Phi$ is bounded on $L^{p(x)}(\mathbb{R}^n)$. \end{lemma} \begin{lemma}\label{le6} The following operators are continuous linear operators: \begin{itemize} \item[(i)] $T: L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n) \to W^{1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$. \item[(ii)] $\widetilde{T}: W^{-1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n) \to L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$. \end{itemize} \end{lemma} \begin{proof} (i) We divide the proof into two parts: \smallskip \noindent\textbf{Step 1:} The operator $\partial_k\mathrm{T}: L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell_n) \to L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell_n)$ is continuous. By Lemma \ref{le3} we have for $u \in C^{\infty}_0(\mathbb{R}^n_{+}, \mathrm{C}\ell_n)$ $$ \partial_k\mathrm{T}u(x) = \frac{1}{\omega_n}\int_{\mathbb{R}^n_{+}} \frac{\partial}{\partial x_k}K_{z}(x, y)u(y)dy + \frac{u(x)}{n}\overline{e}_k. $$ Let $\mathcal{K}(x,y) = \frac{1}{\omega_n}\frac{\partial}{\partial x_k}K_{z}(x, y)$. Since $\frac{1}{\omega_n}\frac{\partial}{\partial x_k}K_{z}(x, y) =\frac{1}{\omega_n}\frac{\partial}{\partial x_k}G(x-y)$ and $$ \frac{\partial}{\partial x_k}G(x-y) =\frac{1}{|x-y|^n}\Big(\overline{e}_k - n\sum^n_{i=1}\frac{(x_k - y_k)(x_{i} - y_{i})}{|x-y|^2}\overline{e}_{i}\Big), $$ we obtain $$ \big|\frac{\partial}{\partial x_k}G(x-y)\big| \leq \frac{n^2 + 1}{|x-y|^n},\quad (k=1,\dots,n). $$ Notice that $$ \int_{S_1}\Big(\overline{e}_k - n\sum^n_{i=1}\frac{(x_k - y_k)(x_{i} - y_{i})}{|x-y|^2}\overline{e}_{i}\Big)dS=0, $$ where $S_1 = \{y \in \mathbb{R}^n_{+}:|x-y|= 1\}$. Then it is easy to verify that $\mathcal{K}(x,y)$ satisfies the following properties: \begin{itemize} \item[(a)] $|\mathcal{K}(x,y) | \leq C|x-y|^{-n}$; \item[(b)] $\mathcal{K}\big(t(x,y)\big) = t^{-n}K(x,y),t>0$; \item[(c)] $\int_{S_1}\mathcal{K}(x,y)dS = 0$. \end{itemize} Now we define $u(x) = 0$ for $x \in \mathbb{R}^n \setminus \mathbb{R}^n_{+}$. Then $\mathcal{K}(x,y)$ satisfies the conditions of Calder\'on-Zygmund kernal on $\mathbb{R}^n \times \mathbb{R}^n$. By Theorem \ref{th211}, we know the inequality can be extended to $L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell_n)$. Therefore, we obtain by Lemma \ref{le1} and Lemma \ref{le2} \begin{equation} \|\frac{1}{\omega_n}\int_{\Omega}\partial_{k,x}G(x-y)u(y)dy\|_{L^{p(x)} (\mathbb{R}^n_{+}, \mathrm{C}\ell_n)} \leq C(n,p)\|u\|_{L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell_n)}\label{e2.3b} \end{equation} On the other hand, \begin{equation} \|\frac{u(x)}{n}\overline{e}_k\|_{L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell_n)} \leq \frac{1}{n}\|u\|_{L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell_n)}\label{e2.4} \end{equation} Combining \eqref{e2.3} with \eqref{e2.4}, we obtain $$ \|\partial_k\mathrm{T}u\|_{L^{p(x)}(\mathbb{R}^n, Cl_n)} \leq C(n,p)\|u\|_{L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell_n)}. $$ \noindent\textbf{Step 2:} The operator $\mathrm{T}: L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell_n) \to L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell_n)$ is continuous. We define $u(x) = 0$ for $x \in \mathbb{R}^n \setminus \mathbb{R}^n_{+}$. Since $$ |G(x-y)+G(y-z)|\leq \frac{1}{\omega_n}\Big(\frac{1}{|x-y|^{n-1}} +\frac{1}{|y-z|^{n-1}}\Big), $$ we have \begin{align*} |Tu(x)| & \leq C(n)\int_{\mathbb{R}^n_{+}}\left(|G(x-y)|+|G(y-z)|\right)|u(y)|dy \\ &\leq C\Big(\int_{\mathbb{R}^n_{+}}\frac{1}{|x-y|^{n-1}}|u(y)|dy +\int_{\mathbb{R}^n_{+}}\frac{1}{|y-z|^{n-1}}|u(y)|dy\Big). \end{align*} Then by Lemmas \ref{le1} and \ref{le2}, we obtain $$ \|Tu\|_{L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)} \leq C(n, p)\|u\|_{L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)}. $$ Finally, combining Step 1 with Step 2, we have \begin{align*} \|Tu\|_{W^{1,p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)} &= \|Tu\|_{L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)}+\sum^n_{k=1}\|\partial_k\mathrm{T}u\|_{L^{p(x)} (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}\\ &\leq C(n,p)\|u\|_{L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)}. \end{align*} Then we obtain the desired conclusion (i). (ii) In view of \cite[Proposition 12.3.2]{3}, we know that for each $f \in W^{-1,p(x)}(\mathbb{R}^n_{+})$, there exists $f_k \in L^{p(x)}(\mathbb{R}^n_{+}),k=0,1,\ldots,n$, such that \begin{equation} \langle f,\varphi\rangle = \sum^n_{k=0}\int_{\mathbb{R}^n_{+}} f_k\frac{\partial \varphi}{\partial x_k} dx,\label{e2.4b} \end{equation} for all $\varphi \in W^{1,p'(x)}_0(\mathbb{R}^n_{+})$. Moreover, $\|f\|_{W^{-1,p(x)}(\mathbb{R}^n_{+})}$ is equivalent to $\sum^n_{k=0}\|f_k\|_{L^{p(x)}(\mathbb{R}^n_{+})}$. Obviously, for every $f \in W^{-1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ the equality \eqref{e2.4} still holds for $f_k \in L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$, $k=0,1,\ldots,n$. Moreover, $\|f\|_{W^{-1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)}$ is equivalent to $\sum^n_{k=0}\|f_k\|_{L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)}$. On the other hand, by \cite[Proposition 12.3.4]{3}, it is easy to show that the space $C^{\infty}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ is dense in $W^{-1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$. Thus we may choose $$ u^j = u_0^j + \sum^n_{k=1}\frac{\partial u_k^j}{\partial x_k}, $$ where $u_0^j, u_k^j \in C^{\infty}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$, such that $\|u^j-f\|_{W^{-1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)}\to 0$ and $\|u_k^j-f_k\|_{L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)}\to 0$ as $j\to\infty$, where $k=0,1,\ldots,n$. Here, we are using the fact that $C^{\infty}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ is dense in $L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$(see \cite{3}). Then we consider $$ Tu^j = \int_{ \mathbb{R}^n_{+}}K_z(x,y)u^j(y)dy. $$ Then we have \begin{align*} Tu^j &= \int_{\mathbb{R}^n_{+}}K_z(x,y)\Big(u_0^j(y) + \sum^n_{k=1}\frac{\partial}{\partial y_k}u_k^j(y)\Big)dy\\ &=\int_{\mathbb{R}^n_{+}}K_z(x,y)u_0^j(y)dy -\sum^n_{k=1} \int_{ \mathbb{R}^n_{+}}\frac{\partial}{\partial y_k}K_z(x,y)u_k^j(y)dy. \end{align*} Since $$ \big|\int_{ \mathbb{R}^n_{+}}K_z(x,y)u_0^j(y)dy\big| \leq \int_{ \mathbb{R}^n_{+}}\frac{1}{|x-y|^{n-1}}\big|u_0^j(y)\big|dy +\int_{\mathbb{R}^n_{+}}\frac{1}{|y-z|^{n-1}}\big|u_0^j(y)\big|dy. $$ By Remark \ref{re11}, Lemma \ref{le1} and Lemma \ref{le2}, there exists a constant $C_0> 0$ such that \begin{equation} \big\|\int_{\mathbb{R}^n_{+}}K_z(x,y)u_0^j(y)dy \big\|_{L^{p(x)} (\mathbb{R}^n_{+},\mathrm{C}\ell _n)} \leq C_0\|u_0^j\|_{L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)}.\label{e2.5} \end{equation} Now let us extend $u_k^j(x)$ by zero to $\mathbb{R}^n\setminus \mathbb{R}^n_{+}$. Note that the position of $z$ which is outside of a half space $\overline{\mathbb{R}^n_{+}}$ leads to the fact that $G(y-z)$ has no singularities for any $y\in \mathbb{R}^n_{+}$. Thus it is easy to show that $\frac{\partial}{\partial y_k}K_z(x,y)$ satisfies the conditions of Calder\'{o}n-Zygmund kernel on $\mathbb{R}^n \times \mathbb{R}^n$. In view of Lemma \ref{le4}, there exist positive constant $C_k(k=1,\ldots,n)$ such that \begin{equation} \big\|\int_{\mathbb{R}^n_{+}}\frac{\partial}{\partial y_k}K_z(x,y)u_k^j(y) \big\|_{L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)} \leq C_k\|u_k^j\|_{L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)}. \label{e2.6} \end{equation} Combining \eqref{e2.5} with \eqref{e2.6}, we have \begin{align*} \|Tu^j\|_{L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)} \leq C_0\|u_0^j\|_{L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)} + \sum^n_{k=1} C_k\|u_k^j\|_{L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)}. \end{align*} Letting $j\to \infty$, 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 $f \in W^{-1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$, there exists a constant $\widetilde{C}>0$ such that \begin{align*} \|\widetilde{T}f\|_{L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)} &\leq C\Big(\|f_0\|_{L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)} + \sum^n_{k=1}\|f_k\|_{L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)}\Big)\\ &\leq \widetilde{C}\|f\|_{W^{-1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)}. \end{align*} Hence claim (ii) follows. \end{proof} \begin{lemma}\label{le7} The following operators are continuous linear operators: \begin{itemize} \item[(i)] $D: W^{1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n) \to L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$. \item[(ii)] $\widetilde{D}:L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n) \to W^{-1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$. \end{itemize} \end{lemma} \begin{proof} (i) The proof is similar to that of \cite[Lemma 2.6]{14}, so we omit it. (ii) We consider the following Dirichlet problem of the Poisson equation with homogeneous boundary data \begin{equation} \begin{gathered} -\Delta u = f, \quad \text{in }\mathbb{R}^n_{+} \\ u = 0, \quad \text{on } \Sigma \end{gathered}\label{e2.7} \end{equation} It is easy to see that for all $f \in W^{-1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ problem \eqref{e2.7} still has a unique weak solution $u\in W^{1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$, see Diening, Lengeler and Ru\v{z}i\v{c}ka \cite{4}. We denote by $\Delta^{-1}_0$ the solution operator. On the other hand, it is clear that the operator $$ \Delta : W^{1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n) \to W^{-1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n) $$ is continuous, so we obtain from Lemma \ref{le6} that the operator $\widetilde{D}=-\Delta T:L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n) \to W^{-1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ is continuous, where the operator $\widetilde{D}$ can be considered as a unique continuous linear extension of the operator $D$. \end{proof} \begin{lemma}\label{le8} Let $p(x) \in \mathcal{P}(\mathbb{R}^n_+)$. \begin{itemize} \item[(i)] If $u \in W^{1,p(x)}(\mathbb{R}^n_+,\mathrm{C}\ell _n)$, then the Borel-Pompeiu formula $Fu(x)+TDu(x) = u(x)$ holds for all $x \in \mathbb{R}^n_+$. \item[(ii)] If $u \in L^{p(x)}(\mathbb{R}^n_+,\mathrm{C}\ell _n)$, then the equation $DTu(x) = u(x)$ holds for all $x \in \mathbb{R}^n_+$. \end{itemize} \end{lemma} \begin{proof} Let us denote by $C_0^{\infty}(\mathbb{R}^n_+)$ the space of all restrictions of functions from $C_0^{\infty}(\mathbb{R}^n)$ to $\mathbb{R}^n_+$. Furthermore, suppose $\varphi\in C_0^{\infty}(\overline{\mathbb{R}^n_+})$. Now, let us consider a point $y\in \Omega$ and the open ball $B(0,r)$ with origin $0$, radius $r$, and boundary $S(0, r)$. If $r$ is sufficiently large such that $y$ lies in the domain $\Omega(r)=B(0,r)\cap\mathbb{R}^n_+$. For this domain, we have $$ F_{S(0, r)}\varphi (y)=\varphi(y)-T_{\Omega(r)}D\varphi (y). $$ see \cite{MRZ} for more details. This can be written in the form \begin{align*} &\lim_{r\to \infty}\Big(\int_{\Sigma\cap B(0,r)}+\int_{S(0, r)\cap \mathbb{R}^n_+} \Big)K_z(x,y)\alpha(y)u(y)dS_{y}\\ &=\varphi(y)-\lim_{r\to \infty}T_{\Omega(r)}D\varphi (y) \end{align*} Since \[ \lim_{r\to \infty}\int_{\Sigma\cap B(0,r)}K_z(x,y)\alpha(y)u(y)dS_{y} =\int_{\Sigma}K_z(x,y)\alpha(y)u(y)dS_{y} \] and \[ \lim_{r\to \infty}T_{\Omega(r)}D\varphi (y)=T_{\mathbb{R}^n_+}D\varphi (y),\quad \lim_{r\to \infty}\int_{S(0, r)\cap \mathbb{R}^n_+}K_z(x,y)\alpha(y)u(y)dS_{y}=0, \] we obtain the Borel-Pompeiu formula in case of $\varphi\in C_0^{\infty}(\overline{\mathbb{R}^n_+})$. Finally, the desired result $(i)$ follows immediately from the density document. (ii) Using the same idea with (i), we can get directly the desired result from \cite[Lemma 2.6]{MRZ}. \end{proof} \begin{lemma}\label{le9} Let $p(x)$ satisfy \eqref{e2.1}. \begin{itemize} \item[(i)] If $u \in L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$, then $\widetilde{T}\widetilde{D}u(x) = u(x)$ for all $x \in \mathbb{R}^n_{+}$. \item[(ii)] If $u \in W^{-1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$, then $\widetilde{D}\widetilde{T}u(x) = u(x)$ for all $x \in \mathbb{R}^n_{+}$. \end{itemize} \end{lemma} \begin{proof} (i) follows from Lemma \ref{le8} (i) and the denseness of $W^{1,p(x)}_0(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)$ in $L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)$. (ii) follows from Lemma \ref{le8} (ii) and the denseness of $C^{\infty}_0(\overline{\mathbb{R}^n_{+}},\mathrm{C}\ell _n)$ in the space $W^{-1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$, see \cite[Proposition 12.3.4]{3} for the details. \end{proof} G\"urlebeck and Spr\"o{\ss}ig \cite{16, 17} showed that the orthogonal decomposition of the space $L^2(\Omega)$ holds in the hyper-complex function theory: \begin{equation} L^2(\Omega,\mathrm{C}\ell _n) = (\ker D \cap L^2(\Omega,\mathrm{C}\ell _n)) \oplus DW^{1,2}_0(\Omega,\mathrm{C}\ell _n)\label{e2.8} \end{equation} with respect to the Clifford-valued product \eqref{e2.3}. Note that $\ker D$ 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 \cite{6} for the complex case and \cite{16} for the hyper-complex case. K\"{a}hler \cite{24} extended the orthogonal decomposition \eqref{e2.8} to the spaces $L^{p}(\Omega)$ in form of a direct decomposition in the case of Clifford analysis. In \cite{9}, Fu et al.\ extended the direct decomposition to the case of variable exponent Lebesgue spaces in bounded smooth domains. \begin{theorem}\label{th211} The space $L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ allows the Hodge-type decomposition \begin{equation} L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n) =(\ker \widetilde{D} \cap L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)) \oplus DW^{1,p(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)\label{e3.1} \end{equation} with respect to the Clifford-valued product \eqref{e2.3}. \end{theorem} \begin{proof} Similar to the proof of \cite[Theorem 6]{24}, we first show that the intersection of $(\ker \widetilde{D} \cap L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n))$ and $DW^{1,p(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ is empty. Suppose $f$ belongs to both $\ker \widetilde{D}\cap L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ and $DW^{1,p(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$, then $\widetilde{D}f = 0$ . Because $f$ belongs to $DW^{1,p(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$, there exists a function $v \in W^{1,p(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ such that $Dv =f$. Hence, we obtain that $-\Delta v = 0$ and $v = 0$ on $\Sigma$. From the uniqueness of $\Delta^{-1}_0$ we obtain $v = 0$. Consequently, $f= 0$. Therefore, the sum of the two subspaces is a direct one. Now let $u \in L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$. Then $u_2 = D\Delta^{-1}_0\widetilde{D}u \in DW^{1,p(x)}_0(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)$. Let $u_1 = u - u_2$. Then $u_1 \in L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$. Furthermore, we take $u_k\in W^{1,p(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ such that $u_k \to u $ in $L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)$, then by the denseness of $W^{1,p(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ in $L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)$ and Lemma \ref{le111}, we have for any $\varphi \in W^{1,p'(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ \begin{align*} \big(u_1,D\varphi \big)_{\mathrm{C}\ell _n} &=\big(u - u_2,D\varphi \big)_{\mathrm{C}\ell _n} \\ &=\lim_{k \to \infty}\big(Du_k - DD\Delta^{-1}_0Du_k, \varphi \big)_{\mathrm{C}\ell _n} \\ &=\lim_{k \to \infty}\big(Du_k - Du_k,\varphi \big)_{\mathrm{C}\ell _n}=0, \end{align*} which implies that $u_1 \in \ker \widetilde{D}$. Since $u \in L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ is arbitrary, the desired result follows. \end{proof} From this decomposition we can get the following two projections \begin{gather*} P: L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n) \to \ker \widetilde{D}\cap L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n), \\ Q: L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n) \to DW^{1,p(x)}_0(\mathbb{R}^n_{+}, \mathrm{C}\ell _n). \end{gather*} Moreover, we have $$ Q = D\Delta^{-1}_0\widetilde{D},\ \ P = I - Q. $$ \begin{corollary}\label{le10} The space $L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n) \cap \operatorname{im}Q$ is a closed subspace of $L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$. \end{corollary} The proof can be easily done by combining Theorem \ref{th211}, Lemma \ref{le11} with Lemma \ref{le7}. We refer the reader to \cite[Lemma 2.6]{ZFR} for a similar argument. \begin{corollary}\label{le101} $\big(L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n) \cap \operatorname{im}Q\big)^{\ast} = L^{p'(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n) \cap \operatorname{im}Q$. Namely, the linear operator $$ \Phi : DW^{1,p'(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n) \to \big(DW^{1,p(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)\big)^{\ast} $$ given by $$ \Phi(Du)(D\varphi) = (D\varphi , Du)_{Sc}:= \int_{\mathbb{R}^n_{+}} \big[\overline{D\varphi}Du\big]_0dx $$ is a Banach space isomorphism. \end{corollary} \begin{proof} In terms of Lemma \ref{le10}, $DW^{1,p(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ and $DW^{1,p'(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ are reflexive Banach spaces since they are closed subspaces in $L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ and $L^{p'(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ respectively. The linearity of $\Phi$ is clear. For injectivity, suppose \begin{equation} \Phi(Du)(D\varphi) = (D\varphi , Du)_{Sc} = 0\label{e2.10} \end{equation} for all $\varphi \in W^{1,p(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ and some $u \in W^{1,p'(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$. For any $\omega \in L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$, according to \eqref{e3.1}, we may write $\omega = \alpha + \beta$ with $\alpha \in \ker \widetilde{D} \cap L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ and $\beta \in DW^{1,p(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$. Thus we obtain $$ (\omega , Du)_{Sc} = (\alpha + \beta , Du)_{Sc} = (\beta, Du)_{Sc}. $$ This and \eqref{e2.10} gives $(\omega , Du)_{Sc} = 0$. This means that $Du = 0$. It follows that $\Phi$ is injective. To get subjectivity, let $f \in \big(DW^{1,p(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)\big)^{\ast}$. By the Hahn-Banach Theorem, there is $F \in \big(L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)\big)^{\ast}$ with $\|F\| = \|f\|$ and $F|_{DW^{1,p(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)} = f$. Moreover, there exists $\varphi \in L^{p'(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ such that $F(u) = (u, \varphi)_{Sc}$ for any $u \in L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$. According to \eqref{e3.1}, we can write $\varphi =\eta+ D\alpha$, where $\eta \in \ker \widetilde{D} \cap L^{p'(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$, $D\alpha \in DW^{1,p'(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$. For any $Du \in DW^{1,p(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$, we have \begin{align*} f(Du)= (Du, \varphi)_{Sc} = (Du, D\alpha)_{Sc} = \Phi (D\alpha)(Du). \end{align*} Consequently, $\Phi (D\alpha) = f$. It follows that $\Phi$ is surjective. By \cite[Theorem 3.1]{13} we have $$ |\Phi (Du)(D\varphi)| \leq C\|D\varphi\|_{L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)}\|Du\|_{L^{p'(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)}. $$ This means that $\Phi$ is continuous. Furthermore, it is immediate that $\Phi^{-1}$ is continuous from the Inverse Function Theorem. This ends the proof of Lemma \ref{le11}. \end{proof} \section{Stokes equations in the half-space} In the section, we consider the Stokes system which consists in finding a solution $(u,\pi)$ for \begin{gather} -\Delta u + \frac{1}{\mu}\nabla\pi = \frac{\rho}{\mu}f\quad \text{in } \mathbb{R}^n_{+}, \label{e3.11}\\ \operatorname{div} u = f_0\quad \text{in }\mathbb{R}^n_{+}, \label{e3.12} \\ u = v_0\quad \text{on }\Sigma. \label{e3.13} \end{gather} With $\int_{\Omega}f_0dx=\int_{\partial\Omega}n\cdot v_0dx$ 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 \eqref{e3.13} 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 \cite{2, 16, 17, 18, 22}. In this paper, for $f=\sum_{i=1}^nf_{i}\mathrm{e}_{i}$ and $u=\sum_{i=1}^nu_{i}\mathrm{e}_{i}$, we consider the following Stokes system in the hyper-complex formulation (see [16, 17]): \begin{gather} \widetilde{D}Du + \frac{1}{\mu}D\pi = \frac{\rho}{\mu}f\quad \text{in } \mathbb{R}^n_{+}, \label{e3.14} \\ [Du]_0 = 0\quad \text{in }\mathbb{R}^n_{+}, \label{e3.15} \\ u = 0\quad \text{on }\Sigma. \label{e3.16} \end{gather} \begin{definition}\label{def21} \rm We say that $(u,\pi) \in W^{1,p(x)}_0(\mathbb{R}^n_{+}, \mathrm{C}\ell _n) \times L^{p(x)}(\mathbb{R}^n_{+})$ is a solution of \eqref{e3.14}--\eqref{e3.16} provided that it satisfies the system \eqref{e3.14}--\eqref{e3.16} for all $f\in W^{-1,p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)$. \end{definition} \begin{definition}\label{def22} \rm The operator $\widetilde{\nabla}:L^{p(x)}(\mathbb{R}^n_{+})\to (W^{-1,p(x)}(\mathbb{R}^n_{+}))^n$ is defined by $$ \langle \widetilde{\nabla} f,\varphi\rangle =-\langle f,\operatorname{div} \varphi\rangle :=-\int_{\mathbb{R}^n_{+}}{f\operatorname{div} \varphi}dx $$ for all $f \in L^{p(x)}(\mathbb{R}^n_{+})$ and $\varphi\in (C^{\infty}_0(\mathbb{R}^n_{+}))^n$. \end{definition} \begin{theorem} \label{th31} Suppose $f \in W^{-1,p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)$. Then the Stokes system \eqref{e3.14}--\eqref{e3.16} has a unique solution $(u,\pi) \in W^{1,p(x)}_0(\mathbb{R}^n_{+}, \mathrm{C}\ell _n) \times L^{p(x)}(\mathbb{R}^n_{+})$ of the form $$ u + \frac{1}{\mu}TQ\pi = \frac{\rho}{\mu}TQ\widetilde{T}f, $$ with respect to the estimate $$ \|Du\|_{L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)} + \frac{1}{\mu}\|Q\pi\|_{L^{p(x)}(\mathbb{R}^n_{+})} \leq C\frac{\rho}{\mu}\|Q\widetilde{T}f\|_{L^{p(x)} (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}. $$ Here, $C\geq1$ is a constant and the hydrostatic pressure $\pi$ is unique up to a constant. \end{theorem} \begin{proof} We first prove that if $f \in W^{-1,p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)$, then we have the representation $$ TQ\widetilde{T}f= u + TQ\omega. $$ Indeed, let $\varphi_n \in W^{1,p(x)}_0(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)$ with $\varphi_n \to \varphi$ in $L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)$. By Lemma \ref{le8}, we have $$ TQT(D\varphi_n) = TQ\varphi_n. $$ Since $W^{1,p(x)}_0(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)$ is dense in $L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)$, it follows that $TQ\widetilde{T}\widetilde{D}\varphi =TQ\varphi$. Thus, for $u \in W^{1,p(x)}_0(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)$ and $\pi \in L^{p(x)}(\mathbb{R}^n_{+})$ 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 \eqref{e3.14}--\eqref{e3.15} is equivalent to the system \begin{gather} u + \frac{1}{\mu}QTQ\pi = \frac{\rho}{\mu}TQ\widetilde{T}f, \label{e3.19}\\ [Q\pi]_0 = [Q\widetilde{T}f]_0. \label{e3.20} \end{gather} Obviously, the equality \eqref{e3.14} is equivalent to the equality \begin{equation} Du + \frac{1}{\mu}Q\pi = \frac{\rho}{\mu}Q\widetilde{T}f. \label{e3.21} \end{equation} Now we need to show that for each $f\in W^{-1,p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n^{1})$, the function $QTf$ can be decomposed into two functions $Du$ and $Q\pi$. Suppose $Du + Q\pi=0$ for $u \in W^{1,p(x)}_0(\mathbb{R}^n_{+}, \mathrm{C}\ell _n^{1}) \cap \ker \operatorname{div}$ and $\pi \in L^{p(x)}(\mathbb{R}^n_{+})$. Then \eqref{e3.15} 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 $\operatorname{im}Q$. Next we have to ask about the existence of a functional $\mathcal{F}\in (L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n^{1}) \cap \operatorname{im}Q)^{\ast}$ with $\mathcal {F}(Du)=0$ and $\mathcal {F}(Q\pi)=0$ but $\mathcal{F}(Q\widetilde{T}f)\neq 0$. This is equivalent to ask if there exists $g \in W^{-1,p'(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n^{1})$, such that for all $u \in W^{1,p(x)}_0(\mathbb{R}^n_{+}, \mathrm{C}\ell _n^{1}) \cap \mathrm{ker \ div}$ and $\omega \in L^{p(x)}(\mathbb{R}^n_{+})$, \begin{gather} (Du, Q\widetilde{T}g)_{Sc} = 0, \label{e3.22} \\ (Q\pi, Q\widetilde{T}g)_{Sc} = 0, \label{e3.23} \end{gather} but $(Q\widetilde{T}f, Q\widetilde{T}g)_{Sc} \neq 0$. Here, Lemmas \ref{le6} and Corollary \ref{le101} are employed. Thus, let us consider the system \eqref{e3.22} and \eqref{e3.23} with $g \in W^{-1,p'(x)}(\Omega, \mathrm{C}\ell _n^{1})$ for all open cubes $\Omega\subset \mathbb{R}^n_+$. Notice that, with the help of Lemma \ref{le7}, \eqref{e3.22} 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 $h \in L^{p'(x)}_{\rm loc}(\mathbb{R}^n_{+})$ because of \cite[Lemma 2.8]{ZFR}. Furthermore, by Definition \ref{def22}, it is easy to see that if $g \in W^{-1,p'(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n^{1})$, then $h \in L^{p'(x)}(\mathbb{R}^n_{+})$. Thus it follows from \eqref{e3.23} and Lemma \ref{le11}, $$ (Q\pi, Q\widetilde{T}g)_{Sc} = (Q\pi, Q\widetilde{T}\widetilde{D}h)_{Sc} = (Q\pi, Qh)_{Sc} = 0 $$ holds for each $\pi \in L^{p(x)}(\mathbb{R}^n_{+})$. 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 obtain $$ (Q\widetilde{T}f, Q\widetilde{T}g)_{Sc} = 0,\quad \text{for all } f \in W^{-1,p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n^{1}). $$ Finally, \eqref{e3.21} yields $$ \|Du\|_{L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)} + \frac{1}{\mu}\|Q\pi\|_{L^{p(x)}(\mathbb{R}^n_{+})} \geq \frac{\rho}{\mu}\|Q\widetilde{T}f\|_{L^{p(x)} (\mathbb{R}^n_{+}, \mathrm{C}\ell _n)}. $$ By the Norm Equivalence Theorem, we obtain $$ \|Du\|_{L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)} + \frac{1}{\mu}\|Q\pi\|_{L^{p(x)}(\mathbb{R}^n_{+})} \leq C\frac{\rho}{\mu}\|Q\widetilde{T}f\|_{L^{p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)}. $$ By Remark \ref{re11}, Lemma \ref{le6} and the boundedness of the operator $Q$, we obtain \begin{equation} \|u\|_{W^{1,p(x)}_0(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)} + \frac{1}{\mu}\|Q\pi\|_{L^{p(x)}(\mathbb{R}^n_{+})} \leq C\frac{\rho}{\mu}\|f\|_{W^{-1,p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)}, \label{e3.24} \end{equation} which implies the uniqueness of solution. Note that $Q\pi = 0$ implies $\pi \in \ker \widetilde{D}$. Therefore, $\pi$ is unique up to a constant. The proof is complete. \end{proof} \section{N-S equations in the half-space} In this section, we consider the time-independent Navier-Stokes equations in variable exponent spaces of Clifford-valued functions in a half-space: \begin{gather} -\Delta u +\frac{\rho}{\mu}(u\cdot \nabla)u + \frac{1}{\mu}\nabla\pi = \frac{\rho}{\mu}f \quad \text{in }\mathbb{R}^n_{+}, \label{e4.1}\\ \operatorname{div}u=f_0 \quad \text{in }\mathbb{R}^n_{+},\label{e4.2} \\ u=v_0\quad \text{on }\Sigma.\label{e4.3} \end{gather} 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 \cite{15} 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\"urlebeck and Spr\"o{\ss}ig \cite{16, 17, 19} 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 $6/52$. Evidently, Lemma \ref{le41} is a direct generalization of \cite[Lemma 4.1]{2} to the variable exponent context in a half-space. \end{remark} Now we are in a position to prove our main result. \begin{theorem}\label{th41} Let $p(x)$ satisfy \eqref{e2.1} and $n/2\leq p_{-}\leq p(x)\leq p_{+}<\infty$. Then the system \eqref{e4.4}--\eqref{e4.6} has a unique solution $(u,\pi)\in W^{1,p(x)}_0(\mathbb{R}^n_{+},\mathrm{C}\ell _n) \times L^{p(x)}(\mathbb{R}^n_{+},\mathbb{R})$ ($\pi$ is unique up to a real constant) if the right-hand side $f$ satisfies the condition \begin{equation} \|f\|_{W^{-1,p(x)} (\mathbb{R}^n_{+},\mathrm{C}\ell _n)} < \frac{\nu^2}{4 C_1 C_4^2} , \label{e4.7} \end{equation} with $\nu=\mu/\rho$, $C_4=C_2(1+C_3)$, where $C_3 \geq 1$ indicated in \eqref{e4.11} and $$ C_2=\|T\|_{[L^{p(x)}\cap \operatorname{im}Q,W^{1,p(x)}_0]}\|Q\|_{[L^{p(x)}, L^{p(x)}\cap\operatorname{im}Q]}\|\widetilde{T}\|_{[W^{-1,p(x)}, L^{p(x)}\cap \operatorname{im}Q]}. $$ For any function $u_0 \in W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ with \begin{equation} \|u_0\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)} \leq\frac{\nu}{2C_1 C_2}-\mathcal{M}, \label{e4.8} \end{equation} here, $\mathcal{M}=\sqrt{\frac{\nu^2}{4C_1 ^2C_4 ^2}-\frac{1}{C_1}\|f\|_{W^{-1,p(x)} (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}}$, the iteration process \begin{gather} u_k + \frac{1}{\mu}TQ\pi_k=\frac{\rho}{\mu}TQ\widetilde{T}F(u_{k-1}),\quad k=1,2,\ldots \label{e4.9} \\ \frac{1}{\mu}\big[Q\pi_k\big]_0= \frac{\rho}{\mu}\big[Q\widetilde{T}F(u_{k-1}) \big]_0, \label{e4.10} \end{gather} converges in $W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)\times L^{p(x)}(\mathbb{R}^n_{+},\mathbb{R})$. \end{theorem} \begin{proof} The proof is similar to one of \cite[theorem 4.6.8]{17}. For the reader's convenience, we would like to give some details. Replacing $f$ by $F(u_{k-1})$ in the proof of Theorem \ref{th31} we obtain the unique solvability of the Stokes equations \eqref{e4.9}--\eqref{e4.10} which we have to solve in each step. Moreover, we have the following estimate: \begin{equation} \|Du_k\|_{L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)} + \frac{1}{\mu}\|Q\pi_k\|_{L^{p(x)}(\mathbb{R}^n_{+})} \leq C_3\frac{\rho}{\mu}\|Q\widetilde{T}F(u_{k-1})\|_{L^{p(x)} (\mathbb{R}^n_{+},\mathrm{C}\ell _n)} \label{e4.11} \end{equation} where $C_3 \geq1$ is a constant. The only remaining problem is the convergence of our iteration procedure. From Theorem \ref{th31} we know $$ Du_k +\frac{1}{\mu} Q\pi_k =\frac{\rho}{\mu} Q\widetilde{T}F(u_{k-1}). $$ Then it follows from \eqref{e4.11} that $$ \frac{1}{\mu}\|Q(\pi_k -\pi_{k-1})\|_{L^{p(x)}(\mathbb{R}^n_{+})} \leq\frac{C_3}{\nu}\|Q\widetilde{T}(F(u_{k-1})-F(u_{k-2})) \|_{L^{p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)}. $$ Hence \begin{align*} \|u_k -u_{k-1}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)} &\leq \frac{1}{\mu}\|TQ(\pi_k -\pi_{k-1})\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)} \\ &\quad +\frac{\rho}{\mu}\|TQ\widetilde{T}(F(u_{k-1})-F(u_{k-2}))\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}\\ &\leq\frac{C_2 (1+C_3)}{\nu}\|F(u_{k-1})-F(u_{k-2})\|_{W^{-1,p(x)} (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}. \end{align*} In terms of Lemma \ref{le41}, one obtain \begin{align*} &\|F(u_{k-1})-F(u_{k-2})\|_{W^{-1,p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)}\\ &\leq C_1 \|u_{k-1} -u_{k-2}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}(\|u_{k-1}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}+\|u_{k-2}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}). \end{align*} Let $L_k =\frac{C_1 C_4}{\nu}(\|u_{k-1}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}+\|u_{k-1}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)})$ with $C_4=C_2(1+C_3)$. Then we obtain \begin{equation} \|u_k -u_{k-1}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)} \leq L_k \|u_{k-1} -u_{k-2}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}. \label{e4.12} \end{equation} On the other hand, by \eqref{e3.11} and Lemma \ref{le41}, we have \begin{align*} \|u_k\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)} &\leq\frac{1}{\mu}\|TQ\pi_k\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+}, \mathrm{C}\ell _n)}+\frac{\rho}{\mu}\|TQ\widetilde{T}F(u_{k-1}) \|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}\\ &\leq\frac{C_1 C_4}{\nu}\|u_{k-1}\|^2_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}+\frac{C_4}{\nu}\|f\|_{W^{-1,p(x)} (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}. \end{align*} Now we have to ensure that $$ \|u_k\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}\leq\|u_{k-1}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}. $$ For this we notice that $$ \frac{C_1 C_4}{\nu}\|u_{k-1}\|^2_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}+\frac{C_4}{\nu} \|f\|_{W^{-1,p(x)} (\mathbb{R}^n_{+},\mathrm{C}\ell _n)} \leq\|u_{k-1}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+}, \mathrm{C}\ell _n)}, $$ which is equivalent to $$ \|u_{k-1}\|^2_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)} -\frac{\nu}{C_1 C_4}\|u_{k-1}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},Cl_n)} +\frac{1}{C_1}\|f\|_{W^{-1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)}\leq0, $$ which is equivalent to $$ \Big(\|u_{k-1}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}-\frac{\nu}{2C_1 C_4}\Big)^2 \leq\frac{\nu^2}{(2C_1 C_4)^2}-\frac{1}{C_1}\|f\|_{W^{-1,p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell_n)}. $$ According to the assumption \eqref{e4.7}, we have $$ \Big| \|u_{k-1}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)} -\frac{\nu}{2C_1 C_4} \Big|\leq \mathcal{W} $$ with \[ \mathcal{M}=\Big(\frac{\nu^2}{4C_1 ^2C_4 ^2}-\frac{1}{C_1}\|f\|_{W^{-1,p(x)} (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}\Big)^{1/2}. \] This leads to the following condition for $\|u_{k-1}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}$, $$ \frac{\nu}{2C_1 C_4}-\mathcal{M}\leq \|u_{k-1}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+}, \mathrm{C}\ell _n)}\leq \frac{\nu}{2C_1 C_4}+\mathcal{M}. $$ Now assume that $\|u_{k-1}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}\leq \frac{\nu}{2C_1 C_4}-\mathcal{M}$. Then it follows that \begin{align*} &\|u_k\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)} \\ &\leq\frac{1}{\mu}\|TQ\pi_k\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)} +\frac{\rho}{\mu}\|TQ\widetilde{T}F(u_{k-1})\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+}, \mathrm{C}\ell _n)}\\ &\leq\frac{C_1 C_4}{\nu}\|u_{k-1}\|^2_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}+\frac{C_4}{\nu}\|f\|_{W^{-1,p(x)} (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}\\ &\leq\frac{C_1 C_4}{\nu}\Big(\frac{\nu}{2C_1 C_4}-\mathcal{M}\Big)^2 +\frac{C_4}{\nu}\|f\|_{W^{-1,p(x)} (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}\\ &\leq\frac{\nu}{2C_1 C_4}-\mathcal{M}. \end{align*} Consequently, using the inequality $\|u_{k-2}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}\leq \frac{\nu}{2C_1 C_4}-\mathcal{M}$ and \eqref{e4.12} we have \begin{align*} &\|u_k -u_{k-1}\|_{W^{1,p(x)}_0 \left(\mathbb{R}^n_{+},\mathrm{C}\ell _n\right)} \\ &\leq\frac{2C_1 C_4}{\nu}\Big(\frac{\nu}{2C_1 C_4}-\mathcal{M}\Big)\|u_{k-1} -u_{k-2}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}\\ &\leq\Big(1-\frac{2C_1 C_4}{\nu}\mathcal{M}\Big)\|u_k -u_{k-1}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}. \end{align*} And $$ L_k\leq 1-\frac{2C_1 C_4}{\nu}\mathcal{M}:=\mu<1. $$ In this case one has \begin{equation} \|u_k -u_{k-1}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)} \leq \mu\|u_{k-1} -u_{k-2}\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+}, \mathrm{C}\ell _n)}\label{e4.13} \end{equation} with $0<\mu<1$ and fixed. The convergence of the sequence $\{u_k\}$ is therefore obtained by Banach's contraction mapping principle, and hence the convergence of the sequence $\{\pi_k\}$ immediately follows from \eqref{e4.9}. \end{proof} \begin{remark}\rm Here, $\nu$ is the kinematic viscosity of the fluid. Our result states that under certain smallness condition of the external force, there exists a unique solution to the stationary Navier-Stokes equations. \end{remark} \begin{remark} \rm Here we would like to point out that the obtained solutions in Theorem \ref{th31} and Theorem \ref{th41} are weak solutions, see \cite[ Theorem 4.2 and Theorem 5.1]{NZZ} for the similar proofs in the case of bounded domains, so we omit all the details. \end{remark} Now it is straightforward to obtain the following results based on Theorem \ref{th41}. \begin{corollary}\label{co41} Under the assumptions in Theorem \ref{th41}, we have the a-priori estimate \begin{equation} \|u\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)} \leq \frac{\nu}{2C_1 C_4}-\mathcal{M}.\label{e4.14} \end{equation} An a-priori estimate for the term $\|Q\pi\|_{L^{p(x)}(\mathbb{R}^n_{+})}$ is easy to obtain. \end{corollary} \begin{corollary}\label{co42} There exists the error estimate $$ \|u_k -u\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)} \leq \frac{L^k}{1-L} \|u_0 -u\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}. $$ In the case of $u_0=0$ we have $$ \|u_k -u\|_{W^{1,p(x)}_0 (\mathbb{R}^n_{+},Cl_n)} \leq \frac{L^k}{1-L}\Big(\frac{\nu}{2C_1 C_4}-\mathcal{M}\Big). $$ \end{corollary} \section{N-S equations with heat conduction in the half-space} In this section we will study the flow of a viscous fluid under the influence of temperature. Similar to \cite{17} the above method for treating the stationary Navier-Stokes equations can be applied to more complicated problems. More specifically, we consider the following problem: \begin{gather} -\Delta u +\frac{\rho}{\mu}(u\cdot \nabla)u + \frac{1}{\mu}\nabla \pi +\frac{\gamma}{\mu}gw= -f\quad \text{in }\mathbb{R}^n_{+}, \label{e5.1} \\ -\Delta w +\frac{m}{\kappa}(u\cdot \nabla)w = \frac{1}{\kappa}h\quad \text{in }\mathbb{R}^n_{+}, \label{e5.2} \\ \operatorname{div} u=0\quad \text{in }\mathbb{R}^n_{+}, \label{e5.3}\\ u=0, \quad w=0\quad \text{on }\Sigma. \label{e5.4} \end{gather} 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. For the detailed account about the Grasshof number, the Prandtl number and the Reynolds number, we refer to \cite{18}. \begin{remark} \rm In the case of $\Omega$ a bounded domain and space $W_0 ^{k,2}(\Omega,\mathbb{H})$, the problem \eqref{e5.1}--\eqref{e5.4} was already studied by G\"urlebeck and Spr\"o{\ss}ig \cite{18}. In the case of $\Omega$ a unbounded domain and space $W_0 ^{1,p}(\Omega,\mathrm{C}\ell _n)$, the problem \eqref{e5.1}--\eqref{e5.4} was already investigated by Cerejeiras and K\"{a}hler \cite{2}, see also \cite{ZFR} for the corresponding results in the setting of variable exponents in bounded smooth domains. \end{remark} In analogy to the case of the Navier-Stokes equations, we consider the following equivalent hyper-complex problem: \begin{gather} u+\frac{1}{\mu}TQ\pi=-TQ\widetilde{T}(F(u)-\frac{\gamma}{\mu}e_n w) \quad \text{in }\mathbb{R}^n_{+},\label{e5.5} \\ \frac{1}{\mu}\big[Q\pi\big]_0=\big[Q\widetilde{T}(F(u) -\frac{\gamma}{\mu}e_n w)\big]_0 \quad \text{in }\mathbb{R}^n_{+},\label{e5.6}\\ w=-\frac{m}{\kappa}TQ\widetilde{T}\big[uD\big]_0 w+\frac{1}{\kappa}TQ\widetilde{T}h \quad \text{in }\mathbb{R}^n_{+},\label{e5.7} \end{gather} with $F(u):=f+\frac{\rho}{\mu} \big[uD\big]_0 u$. Then the problem can be solved by the following iteration process: \begin{gather} u_k+\frac{1}{\mu}TQ\pi_k=-TQ\widetilde{T}(F(u_{k-1})-\frac{\gamma}{\mu}e_n w_{k-1}) \quad \text{in }\mathbb{R}^n_{+},\label{e5.8} \\ \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 \quad \text{in }\mathbb{R}^n_{+},\label{e5.9}\\ w_k=-\frac{m}{\kappa}TQ\widetilde{T}\big[u_k D\big]_0w_k +\frac{1}{\kappa}TQ\widetilde{T}h \quad \text{in }\mathbb{R}^n_{+}.\label{e5.10} \end{gather} Equations \eqref{e5.8} and \eqref{e5.9} represent an iteration similar to the case of the Navier-Stokes equations. Hence we have to study the solvability of equation \eqref{e5.10}. To this end, in analogy to \cite{17}, we give the following ``inner'' iteration: \begin{equation} w_k ^i=-\frac{m}{\kappa}TQ\widetilde{T}(u_k\cdot \nabla)w^{i-1} _k +\frac{1}{\kappa}TQ\widetilde{T}h. \label{e5.11} \end{equation} Similar to the proof of \cite[Theorem 4.1]{ZFR}, one can obtain the following result. \begin{theorem}\label{th51} Let $u_k \in W_0 ^{1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$, where $p(x)$ satisfies \eqref{e2.1} and $n/2\leq p_{-} \leq p(x) \leq p_{+}<\infty$. Furthermore, suppose \begin{itemize} \item[(i)] $\|u_k\|_{W_0 ^{1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)} <\kappa/mC_1 C_2 $; \item[(ii)] $m\nu<2\kappa(1+C_3)$. \end{itemize} Then the iteration procedure $\eqref{e5.11}$ converges in $W_0^{1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)$ to a unique solution of \eqref{e5.10} and we have a-priori estimate $$ \|w_k\|_{W_0 ^{1,p(x)}(\mathbb{R}^n_{+},\mathrm{C}\ell _n)} \leq\frac{2(1+C_3)C_2}{2\kappa(1+C_3)-m\nu} \|h\|_{W^{-1,p(x)} (\mathbb{R}^n_{+},\mathrm{C}\ell _n)}. $$ \end{theorem} Combining theorem \ref{th51} with our considerations in the case of the Navier-Stokes equations leads to the following theorem. See a similar proof in \cite[Theorem 4.2]{ZFR}. \begin{theorem}\label{th52} Let $f \in W^{-1,p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)$, $h \in W^{-1,p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)$, where $p(x)$ satisfies \eqref{e2.1} and $n/2 \leq p_{-}\leq p(x) \leq p_{+} < \infty$. Furthermore, assume \begin{itemize} \item[(a)]$\nu\parallel f\parallel_{W^{-1,p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)} + C_5\parallel h\parallel_{W^{-1,p(x)} (\mathbb{R}^n_{+}, \mathrm{C}\ell _n)} < C_6$; \item[(b)] $\parallel h\parallel_{W^{-1,p(x)}(\mathbb{R}^n_{+}, \mathrm{C}\ell _n)} < C_7$; \item[(c)] $m\nu < 2\kappa (1+C_3)$, \end{itemize} where $$ C_5=\frac{2\gamma (1+C_3)C_2}{\mu(2\kappa (1+C_3)-m\nu)},\quad C_6=\frac{3\nu^2}{16C_1C^2_4},\gamma C_7=\frac{\mu(2\kappa (1+C_3)-m\nu)^2}{8\gamma mC_1C^3_2(1+C_3)^3}. $$ Then the problem \eqref{e5.5}--\eqref{e5.7} has a unique solution $(u,w,\pi)$ in $ W^{1,p(x)}_0(\mathbb{R}^n_{+}, \mathrm{C}\ell _n) \times W^{1,p(x)}_0(\mathbb{R}^n_{+}, \mathrm{C}\ell _n) \times L^{p(x)}(\mathbb{R}^n_{+},\mathbb{R})$, where $u$ and $w$ are uniquely defined, and $\pi$ uniquely up to a constant. Our iteration procedure \eqref{e5.8}--\eqref{e5.10} converges to the solution of \eqref{e5.5}--\eqref{e5.7}. \end{theorem} \subsection*{Acknowledgements} B. Zhang was supported by the Natural Science Foundation of Heilongjiang Province of China (No. A201306), by the Research Foundation of Heilongjiang Educational Committee (No. 12541667), and by the Doctoral Research Foundation of Heilongjiang Institute of Technology (No. 2013BJ15). \begin{thebibliography}{99} \bibitem{BDS} F. Brackx, R. Delanghe, F. Sommen; \emph{Clifford analysis}, London: Pitman Research Notes in Mathematics, 1982. \bibitem{2} P. Cerejeiras, U. K\"{a}hler; \emph{Elliptic boundary value problems of fluid dynamics over unbounded domains}, Mathematical Methods in the Applied Sciences, 23 (2000), 81--101. \bibitem{3} L. Diening, P. Harjulehto, P. H\"{a}st\"o, M. R$\mathring{\mathrm{u}}$\v{z}i\v{c}ka; \emph{Lebesgue and Sobolev Spaces with VariableExponents}, Springe-Verlag, Berlin, 2011. \bibitem{DMS} L. Diening, J. M\'alek, M. Steinhauer; \emph{On Lipschitz trunctions of Sobolev function (with variable exponent) and their selected applications}, ESAIM: Control, Optimisation and Calculus of Variations, 14 (2008), 211--232. \bibitem{4} L. Diening, D. Lengeler, M. R$\mathring{\mathrm{u}}$\v{z}i\v{c}ka; \emph{The Stokes and Poisson problem in variable exponent spaces}, Complex Varibles and Elliptic Equations, 56 (2011), 789--811. \bibitem{6} J. A. Dubinski; \emph{On a nonlinear analytic problem}, Doklady Mathematics, 48 (1998), 370--375. \bibitem{9} X. L. Fan, D. Zhao; \emph{On the spaces $L^{p(x)}$ and $W^{m,p(x)}$}, Journal of Mathematical Analysis and Applications, 263 (2001), 424--446. \bibitem{10} X. L. Fan, J. S. Shen, D. Zhao; \emph{Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega)$}, Journal of Mathematical Analysis and Applications, 262 (2001), 749--760. \bibitem{12} Y. Q. Fu, V. R\u{a}dulescu, B. L. Zhang; \emph{Hodge decomposition of variable exponent spaces of Clifford-valued functions and applications to Dirac and Stokes equations}, Computers and Mathematics with Applications, 70 (2015), 691--704. \bibitem{13} Y. Q. Fu, B. L. Zhang; \emph{Clifford valued weighted variable exponent spaces with an application to obstacle problems}, Adv. Appl. Clifford Algebras 23 (2013), 363--376. \bibitem{14} Y. Q. Fu, B. L. Zhang; \emph{Weak solutions for elliptic systems with variable growth in Clifford analysis}, Czechoslovak Mathematical Journal, 63 (2013), 643--670. \bibitem {FS} Y. Q. Fu, Y. Y. Shan; \emph{On the removability of isolated singular points for elliptic equations involving variable exponent}, Advances in Nonlinear Analysis, 5 (2016), 121--132. \bibitem{15} R. Finn; \emph{On the exterior stationary problem for the Navier-Stokes equations and associated pertubation problems}, Arch. Rat. Mech. Anal. 19 (1965), 363--406. \bibitem{16} K. G\"urlebeck, W. Spr\"o{\ss}ig, \begin{itshape}Quaternionic and Clifford Calculus for Physicists and Engineers\end{itshape}, John Wiley \& Sons, New York, 1997. \bibitem{17} K. G\"urlebeck, W. Spr\"o{\ss}ig; \emph{Quaternionic Analysis and Elliptic Boundary Value Problems}, Birkh\"{a}user, Boston, 1990. \bibitem{18} K. G\"urlebeck, W. Spr\"o{\ss}ig; \emph{Fluid flow problems with quaternionic analysis-an alternative conception}, Geometric Algebra Computing 2010, 345--380. \bibitem{19} K. G\"urlebeck; \emph{Approximate solution of the stationary Navier-Stokes equations}, Math. Nachr., 145 (1990), 297--308. \bibitem{GKRS} K. G\"urlebeck, U. K\"{a}hler, J. Ryan, W. Spr\"o{\ss}ig; \emph{Clifford analysis over unbounded domains}, Adv. Appl. Math., 19 (1997), 216--239. \bibitem{21} J. Gilbert, M. A. M. Murray; \emph{Clifford Algebra and Dirac Oprators in Harmonic Analysis}, Oxford University Press, Oxford, 1993. \bibitem{22} G. P. Galdi; \emph{An Introduction to the Mathematical Theory of the Navier-Stokes Equations}, Springer, New York, 2011. \bibitem{23} P. Harjulehto, P. H\"{a}st\"o, U. V. L\^{e}, M. Nuortio; \emph{Overview of differential equations with non-standard growth}, Nonlinear Analysis, 72 (2010), 4551--4574. \bibitem{24} U. K\"{a}hler; \emph{On a direct decomposition in the space $L^{p}(\Omega)$}, Zeitschrift f\"ur Analysis und ihre Anwendungen, 4 (1999), 839--848. \bibitem{UK} U. K\"{a}hler; \emph{Clifford Analysis and the Navier--Stokes equations over unbounded domains}, Adv. Appl. Clifford Algebras 11 (2001), 305--318. \bibitem{25} O. Kov\'{a}\v{c}ik, J. R\'{a}kosn\'{i}k; \emph{On spaces $L^{p(x)}$ and $W^{m,p(x)}$}, Czechoslovak Mathematical Journal 41 (1991), 592--618. \bibitem{MRZ} G. Molica Bisci, V. R\u{a}dulescu, B. L. Zhang; \emph{Existence of stationary states for $A$-Dirac equations with variable growth}, Adv. Appl. Clifford Algebras 25 (2015), 385--402. \bibitem{26} C. A. Nolder; \emph{$A$-harmonic equations and the Dirac operator}, Journal of Inequalities and Applications, 2010, Article ID 124018. \bibitem{27} C. A. Nolder; \emph{Nonlinear $A$-Dirac equations} Adv. Appl. Clifford Algebras, 21 (2011), 429--440. \bibitem{NZZ} R. Niu, H. T. Zhang, B. L. Zhang; \emph{Navier-Stokes equations with variable viscosity in variable exponent spaces of Clifford-valued functions}, Boundary value Problems 2015 (2015), 1--17. \bibitem{28} K. R. Rajagopa, M. R$\mathring{\mathrm{u}}$\v{z}i\v{c}ka; \emph{On the modeling of electrorheological materials}, Mech. Res. Commun. 23 (1996), 401--407. \bibitem{29} M. R$\mathring{\mathrm{u}}$\v{z}i\v{c}ka; \emph{Electrorheological Fluids: Modeling and Mathematical Theory}, Springer-Verlag, Berlin, 2000. \bibitem{RZ} V. R\u{a}dulescu, B. L. Zhang; \emph{Morse theory and local linking for a nonlinear degenerate problem arising in the theory of electrorheological fluids}, Nonlinear Analysis: Real World Applications, 17 (2014), 311--321. \bibitem{RR} V. R\v{a}dulescu, D. Repov\v{s}; \emph{Partial differential equations with variable exponents: variational methods and qualitative analysis}, CRC Press, Taylor \& Francis Group, Boca Raton FL, 2015. \bibitem{VR} V. R\v{a}dulescu; \emph{Nonlinear elliptic equations with variable exponent: old and new}, Nonlinear Anal., 121 (2015), 336--369. \bibitem {DR} D. Repov\v{s}; \emph{Stationary waves of Schr\"odinger-type equations with variable exponent}, Anal. Appl., 13 (2015), 645--661. \bibitem{32} V. V. Zhikov; \emph{Averaging of functionals of the calculus variations and elasticity theory}, Izv. Akad. Nauk SSSR Ser. Mat. 877 (1986), 675--710. \bibitem{33} V. V. Zhikov; \emph{Solvability of the three-dimensional thermistor problem}, Tr. Mat. Inst. Steklova, 261 (2008), 101--114. \bibitem{34} B. L. Zhang, Y. Q. Fu; \emph{Weak solutions for $A$-Dirac equations with variable growth in Clifford analysis}, Electronic Journal of Differential Equations, 227 (2012), 1--10. \bibitem{ZFR} B. L. Zhang, Y. Q. Fu, V. R\u{a}dulescu; \emph{The stationary Navier-Stokes equations in variable exponent spaces of Clifford-valued functions}, Adv. Appl. Clifford Algebras 24 (2014), 231--252. \end{thebibliography} \end{document}