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NavierStokes equations with variable viscosity in variable exponent spaces of Cliffordvalued functions
Boundary Value Problems volume 2015, Article number: 30 (2015)
Abstract
In this paper we study the stationary generalized NavierStokes equations when the viscosity is not only a constant but also a function which depends on the position and the shearvelocity. For this we establish an improved decomposition of variable exponent Lebesgue spaces of Cliffordvalued functions. Using this decomposition together with Clifford operator calculus, we obtain the existence, uniqueness and representation of solutions for the generalized Stokes equations and the generalized NavierStokes equations with variable viscosity in the setting of variable exponent spaces of Cliffordvalued functions. Furthermore, the equivalences of solutions and weak solutions for the aforementioned equations are justified.
1 Introduction
In this paper we are concerned with the stationary generalized NavierStokes equations:
where the operator A is defined by \(Au=au\) with \(a(x, u): \overline {\Omega} \times\mathbb{R}^{n}\rightarrow\mathbb{R}^{+}\) and \(a\in C^{\infty}(\overline{\Omega} \times\mathbb{R}^{n})\) and \(\Omega\subset\mathbb{R}^{n}\) (\(n\geq2\)) is a bounded domain with sufficiently smooth boundary ∂Ω, u is the velocity, q the hydrostatic pressure, ρ the density, f the vector of the external forces. Notice that (1.1) may be written
where \(\pi:=q/\rho\). If the relation \(A(x, u):=\mu(\alpha+\mathrm {D}u^{2})^{\frac{p2}{2}}\) with \(\mu, \alpha>0\), where μ is the shear viscosity, \(\mathrm{D}u:=(1/2)(\nabla u+(\nabla u)^{\top})\) is the symmetric gradient, then the fluid is called nonNewtonian fluid with pstructure; see, for example, Acerbi and Mingione [1] for related discussions in this direction. Clearly, the fluid in the case \(p=2\) is a Newtonian fluid, and then (1.4) becomes
where \(\nu:=\mu/\rho\) is the kinematic viscosity. Here (1.5) is the famous NavierStokes equation, for the detailed account about the NavierStokes equations we refer to [2, 3]. In this paper we will consider the NavierStokes equations in a Clifford language under the assumption that the viscosity depends on the position and velocity, i.e., \(\nu=\nu(x, u)\in\mathbb {R}^{+}\). It is easy to see that the viscous term \(\nu\Delta u\) can be replaced by the Clifford expression \(D\nu Du\), where \(D=\sum_{j=1}^{n}\mathrm{e}_{j}\partial_{j}\) denotes the Dirac operator of a massless field, u is a Cliffordvalued function; for the details as regards the Clifford algebra we refer to the next section.
In order to study the timedependent motion of a viscous, incompressible fluid we need velocity fields u of the particles of the fluid along with their first spatial derivatives to be summable to the \(p(k)\)th power (\(p_{}\geq1\), \(k\in \mathbb{N}\)) at each time and each position. Additionally, u has to be solenoidal and vanishes at the boundary of the domain where the motion happens. Thus this gives rise to the introduction of spaces of divergence free functions in a generalized sense. Furthermore, the models of electrorheological fluids, which were introduced by Rajagopal and Růžička [4, 5], can be described by the boundary value problems for the generalized NavierStokes equations. This leads naturally to the establishment of function spaces with variable exponents. On the other hand, Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English geometer William Kingdon Clifford. We have a generalization of the complex numbers, the quaternions, and the exterior algebras; see [6]. As an active branch of mathematics over the past 40 years, Clifford analysis usually studies the solutions of the Dirac equation for functions defined on domains in Euclidean space and taking value in Clifford algebras; see, for example, [7]. In [8] the authors gave an overview of applications of Clifford analysis in mathematical physics. Hence, it makes sense to study the stationary NavierStokes equations in the setting of Clifford algebras.
It is worth pointing out that our attempt is to give a unified approach to deal with physical problems modeled by the generalized NavierStokes equations, which is quite different from the approaches of some authors, for example, we refer the reader to the monograph [9]. Based on the above consideration, we should study the generalized NavierStokes equations in variable exponent spaces of Cliffordvalued functions. Of course, the study of variable exponent spaces has been driven by various problems in elastic mechanics, calculus of variations and differential equations with variable growth; see [10–20] and the references therein.
Evidently, we should primarily be concerned with the study of the first term \(DA(Du)\) in (1.1). In [21, 22], Nolder first introduced the general nonlinear ADirac equations \(DA(x, Du)=0\) which arise in the study of many phenomena in the physical sciences. In particular, he developed some tools for the study of weak solutions to nonlinear ADirac equations in the space \(W^{1,p}_{0}(\Omega,\mathrm{C}\ell_{n})\). Inspired by his works, Fu and Zhang in [23, 24] were interested in the existence of weak solutions for the general nonlinear ADirac equations with variable growth. For this purpose, the authors established a theory of variable exponent spaces of Cliffordvalued functions with applications to homogeneous and nonhomogeneous ADirac equations; see also [25, 26]. Very recently, Fu et al. in [27, 28] established a Hodgetype decomposition of variable exponent Lebesgue spaces of Cliffordvalued functions with applications to the Stokes equations, the NavierStokes equations and the ADirac equations \(DA(Du)=0\). By using the Hodgetype decomposition and variational methods, Molica Bisci et al. in [29] studied the properties of weak solutions to the homogeneous and nonhomogeneous ADirac equations with variable growth. For an overview for the nonlinear ADirac equations with variable growth, we refer the reader to [30].
It is worth mentioning that Diening et al. in [31] studied the following model introduced in [4, 5] to describe the motions of electrorheological fluids:
where \(f\in(W^{1,p(x)}_{0}(\Omega))^{*}=W^{1,p'(x)}(\Omega)\), \(2n/(n+2)< p_{}\leq p_{+}<\infty\) and the operator A satisfies certain natural variable growth conditions. The authors obtained the existence of weak solutions in \((W^{1,p(x)}_{0}(\Omega))^{n}\times L^{s}_{0}(\Omega)\), here \(s:=\min \{(p_{+})', np_{}/2(np_{}) \}\) if \(p_{}< n\) and \(s:=(p_{+})'\) otherwise, \(L^{s}_{0}(\Omega):=\{\pi\in L^{s}(\Omega): \int_{\Omega}\pi\, dx=0\} \). From a practical point of view, we have to investigate the representation of solutions to system (1.6) besides existence and uniqueness. Based on the method developed by Sprößig in [2], it is possible to obtain the desired results if we consider system (1.6) as \(\mathcal {M}(\mathrm{D}u)=a(x,u)\mathrm{D}u\) under the assumption \(f\in W^{1,p(x)}(\Omega, \mathbb{R}^{n})\), here \(a(x,u)\) is a positive function.
Motivated by their works, our goal in this article is to give a generalization of related results in Sprößig [2] to the variable exponent setting. More precisely, we investigate properties of solutions for the following NavierStokes equations in variable exponent spaces of Cliffordvalued functions:
Throughout the paper, the operator A is defined by \(Au=au\) with \(a(x, u): \overline{\Omega} \times\mathbb{R}^{n}\rightarrow\mathbb{R}^{+}\) and \(a\in C^{\infty}(\overline{\Omega} \times\mathbb{R}^{n})\). The outline for this study is to first establish a modified decomposition of variable exponent Lebesgue spaces. Then the classic results about the Stokes equations and the NavierStokes equations obtained by Gürlebeck and Sprößig [32, 33] are extended to the variable exponent setting. In particular, we would like to point out the equivalences of solutions and weak solutions for the abovementioned equations, which has not been clearly stated by the previous works in [28].
This paper is organized as follows. In Section 2, we start with a brief summary of Clifford algebra and variable exponent spaces of Cliffordvalued functions. In Section 3, we establish a modified decomposition of variable exponent Lebesgue spaces, and then discuss its some applications, which will be needed later. In Section 4, we obtain the existence and uniqueness of the generalized Stokes equations in the context of variable exponent spaces. Moreover, the equivalence of solutions and weak solutions for the abovementioned equations are showed. In Section 5, using similar methods to [28], we prove the existence and uniqueness of solutions to the generalized NavierStokes equations in \(W^{1,p(x)}_{0}(\Omega, \mathrm {C}\ell _{n})\times L^{p(x)}(\Omega, \mathbb{R})\) under certain hypotheses. Moreover, the equivalence of solutions and weak solutions for the abovementioned equations are presented.
2 Preliminaries
First we recall some related definitions and results concerning Clifford algebra and variable exponent spaces of Cliffordvalued functions. For a detailed account we refer to [6, 34, 35].
Let \(\mathrm{C}\ell_{n}\) be the real universal Clifford algebra over \(\mathbb{R}^{n}\). Denote \(\mathrm{C}\ell_{n}\) by
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
where \(I\) denotes cardinal number of the set I. The Clifford algebra \(\mathrm{C}\ell_{n}\) is a graded algebra as
Any element \(a\in\mathrm{C}\ell_{n}\) may thus be written in a unique way as
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 quaternions. It is customary to identify 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
For \(u\in\mathrm{C}\ell_{n}\), we denote 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:
We denote
Then an inner product is thus obtained, giving rise to the norm \(\cdot\) on \(\mathrm{C}\ell_{n}\) given by
A Cliffordvalued function \(u:\Omega\rightarrow\mathrm{C}\ell_{n}\) can be written as \(u=\sum_{I}u_{I}\mathrm{e}_{I}\), where the coefficients \(u_{I}:\Omega\rightarrow\mathbb{R}\) are realvalued functions.
The Dirac operator on Euclidean space used here is introduced by
This is a special case of the AtiyahSingerDirac operator acting on sections of a spinor bundle. Note that the most famous Dirac operator describes the propagation of a free fermion in three dimensions.
If u is a realvalued function defined on a domain Ω in \(\mathbb{R}^{n}\), then \(Du=\nabla u=(\partial_{1}u,\partial_{2}u, \ldots,\partial_{n}u)\). Moreover, \(D^{2}=DD=\Delta\), where Δ 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
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.
Next we recall some basic properties of variable exponent spaces of Cliffordvalued 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)
where \(\mathcal{P}^{\mathrm{log}}(\Omega)\) is the set of exponent p satisfying the socalled logHölder continuity, i.e.,
holds for all \(x,y \in\Omega\); see [36]. 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
The variable exponent Lebesgue space \(L^{p(x)}(\Omega)\) is defined by
with the norm
The variable exponent Sobolev space \(W^{1,p(x)}(\Omega)\) is defined by
with the norm
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 [9] and references therein.
In what follows, 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 \(u=\sum_{I}u_{I}e_{I}\in\mathrm{C}\ell_{n}\) 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 Cliffordvalued inner product (see [32, Definition 3.74]):
Remark 2.1
A simple calculation leads to the claim that the norm \(\u\_{L^{p(x)}(\Omega, \mathrm{C}\ell_{n})}\) is equivalent to the norm \(\ u \_{L^{p(x)}(\Omega)}\). Furthermore, \(\Du\_{L^{p(x)}(\Omega,\mathrm{C}\ell_{n})}\) is an equivalent norm of \(\u\_{W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell_{n})}\) for every \(u \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})\); for the details we refer to [24, 25].
Lemma 2.1
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.
Definition 2.1
(see [32])
Let \(u \in C(\Omega, \mathrm{C}\ell _{n})\). The Teodorescu operator is defined by
where \(G(x)\) is the generalized Cauchy kernel mentioned above.
Lemma 2.2
(see [28])
The following operators are bounded linear operators:

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

(ii)
\(\widetilde{T}: W^{1,p(x)}(\Omega,\mathrm{C}\ell_{n}) \rightarrow L^{p(x)}(\Omega,\mathrm{C}\ell_{n})\), where the operator \(\widetilde {T}\) can be considered as a unique bounded linear extension of the Teodorescu operator.
Lemma 2.3
(see [28])
The following operators are bounded linear operators:

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

(ii)
\(\widetilde{D}:L^{p(x)}(\Omega,\mathrm{C}\ell_{n}) \rightarrow W^{1,p(x)}(\Omega,\mathrm{C}\ell_{n})\), where the operator \(\widetilde{D}\) can be considered as a unique bounded linear extension of the Dirac operator.
Lemma 2.4
(see [28])
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 \(x \in\Omega\).
Lemma 2.5
(see [28])
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 \(x \in\Omega\).
3 A modified decomposition of spaces
Gürlebeck and Sprößig [32, 33] showed that the orthogonal decomposition of the space \(L^{2}(\Omega)\) holds in the hypercomplex function theory:
with respect to the Cliffordvalued product (2.3). Kähler [37] extended the orthogonal decomposition (3.1) to the spaces \(L^{p}(\Omega, \mathrm{C}\ell_{n})\) in form of a direct decomposition in a bounded domain. In [38], Cerejeiras and Kähler investigated a direct decomposition of \(L^{p}(\Omega, \mathrm{C}\ell_{n})\) in an unbounded domain. In [28], Zhang et al. generalized (3.1) to the setting of \(L^{p(x)}(\Omega, \mathrm{C}\ell_{n})\) in a bounded domain.
In [36], Diening et al. showed that the Dirichlet problem of the Poisson equation with homogeneous boundary data
has a unique weak solution \(u\in W^{1,p(x)}(\Omega)\) for each \(f \in W^{1,p(x)}(\Omega)\). Moreover, the following estimate holds:
Here u is called a weak solution of problem (3.2) provided that
Then it is easy to see that for all \(f \in W^{1,p(x)}(\Omega,\mathrm{C}\ell_{n})\) the problem (3.2) 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.
Now we are in a position to prove a decomposition of the variable exponent Lebesgue spaces equipped with the modified inner product.
Theorem 3.1
The space \(L^{p(x)}(\Omega,\mathrm{C}\ell_{n})\) allows the Hodgetype decomposition
with respect to the inner product
where \(AB=BA=I\) and \(Bu:=b(x, u)u\) with \(ab=1\).
Proof
Similar to the proof of Theorem 6 in [37], we first show the intersection of spaces \(B\operatorname{ker}\widetilde{D} \cap L^{p(x)}(\Omega,\mathrm {C}\ell_{n})\) and spaces \(DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell_{n})\) contains one and only one zero element. Indeed, suppose \(f \in (B\operatorname{ker}\widetilde{D} \cap L^{p(x)}(\Omega,\mathrm{C}\ell_{n}) ) \cap DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell_{n})\), then \(\widetilde{D}Af = 0\). Moreover, \(f \in L^{p(x)}(\Omega,\mathrm{C}\ell_{n})\) shows that there exists a function \(w=TAf \in W^{1,p(x)}(\Omega,\mathrm{C}\ell_{n})\) because of Lemma 2.2; then \(f=BDw \in DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell_{n})\) due to Lemma 2.4. It follows that \(w \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell_{n})\). Therefore, we have \(\widetilde{D}D w = 0\) in Ω and \(w = 0\) on ∂Ω. From the uniqueness of \(\Delta^{1}_{0}\) we obtain \(w = 0\). Consequently, \(f = 0\). Therefore, the sum of the two subspaces is a direct one.
Now let \(u \in L^{p(x)}(\Omega,\mathrm{C}\ell_{n})\). If we take \(u_{2} = BD\Delta^{1}_{0}\widetilde{D}Au\), then using the same arguments as the first part, we deduce that \(u_{2} \in DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell_{n})\). Let \(u_{1} = u  u_{2}\). Then \(u_{1} \in L^{p(x)}(\Omega,\mathrm {C}\ell_{n})\). Furthermore, we take \(u_{k}\in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})\) such that \(u_{k} \rightarrow u \) in \(L^{p(x)}(\Omega, \mathrm{C}\ell _{n})\), then by the density of \(W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell_{n})\) in \(L^{p(x)}(\Omega, \mathrm{C}\ell_{n})\) and Lemma 2.3, we have for any \(\varphi\in W^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell_{n})\),
Thus, we obtain \(u_{1} \in B\operatorname{ker}\widetilde{D}\). Since \(u \in L^{p(x)}(\Omega,\mathrm{C}\ell_{n})\) is arbitrary, the desired result follows. □
From this decomposition we can get the following projections:
Moreover, we have
Corollary 3.1
Let \(f \in L^{p(x)}(\Omega,\mathrm{C}\ell_{n})\). Then the following equations with homogeneous boundary data:
has a unique solution \(u \in W^{2,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})\) which may be represented by the formula \(u=TQ_{a}BTf\).
Proof
For existence of solution, Theorem 3.1 implies that there exists a function \(u \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell_{n})\) such that \(Q_{a}BTf=Du\). Lemma 2.2 gives \(Tf \in W^{1,p(x)}(\Omega ,\mathrm{C}\ell_{n})\) and \(Q_{a}BTf \in W^{2,p(x)}_{0}(\Omega ,\mathrm{C}\ell_{n})\). Then from Lemma 2.4 it follows that \(TQ_{a}BTf=TDu=u\). Hence \(u \in W^{2,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})\). Further, we have
which implies \(DAD u = f\).
For the uniqueness of solution, consider the boundary value problems: \(DAD u=0\) in Ω and \(u=0\) on ∂Ω. It is easy to see that \(Du \in\operatorname{im}P_{a} \cap\operatorname{im}Q_{a}=\{0\}\). Then \(Du=0\). Hence \(u=TDu=0\) because of Lemma 2.4. □
Remark 3.1
Gürlebeck and Sprößig [33] pointed out that the equations of linear elasticity and timeindependent Maxwell equations over a threedimensional domain with zero boundary condition could be transformed into problem (3.5) in a quaternionic language; see Section 4.3 and Section 4.4 in [33] for more details.
Corollary 3.2
The space \(L^{p(x)}(\Omega,\mathrm{C}\ell_{n}) \cap\operatorname{im}Q_{a}\) is a closed subspace of \(L^{p(x)}(\Omega,\mathrm{C}\ell_{n})\).
Proof
The proof follows that of Lemma 6 in [28] almost word by word, we thus omit the details. □
Corollary 3.3
\((L^{p(x)}(\Omega,\mathrm{C}\ell_{n}) \cap\operatorname{im}Q_{a} )^{\ast} = L^{p'(x)}(\Omega,\mathrm {C}\ell_{n}) \cap \operatorname{im}Q_{a}\). In other words, the linear operator
given by
is a Banach space isomorphism.
Proof
Similar to the proof of Lemma 2.7 in [28], for the reader’s convenience, we will give a detailed treatment of the proof. In view of Corollary 3.2, \(DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell_{n})\) 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 Φ is clear. For injectivity, suppose
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 (3.4), we may write \(\omega= \alpha+ \beta\) with \(\alpha\in B\operatorname{ker}\widetilde{D} \cap L^{p(x)}(\Omega,\mathrm{C}\ell_{n})\) and \(\beta\in DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell_{n})\). Thus we obtain
Taking a similar argument to the first part of the proof in Theorem 3.1, we may write \(\alpha=B\zeta\) with \(\zeta\in\operatorname{ker}\widetilde{D} \cap L^{p(x)}(\Omega,\mathrm{C}\ell_{n})\) and \(Du=BD\eta\) with \(\eta\in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell_{n})\), then it is easy to see that \((\alpha, Du)_{\mathrm{Sc}}=0 \). This together with (3.6) gives \((\omega, Du)_{\mathrm{Sc}} = 0\). This leads to \(Du = 0\). It follows that Φ is injective. To get surjectivity, let \(f \in (DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell_{n}) )^{\ast}\). By the HahnBanach theorem, there is \(F \in (L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) )^{\ast}\) 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 \(F(u) = (u, \varphi )_{\mathrm{Sc}}\) for any \(u \in L^{p(x)}(\Omega,\mathrm{C}\ell_{n})\). According to (3.4), we can write \(\varphi=\xi+ D\alpha\), where \(\xi\in B\operatorname{ker}\widetilde{D} \cap L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})\), \(D\alpha \in DW^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell_{n})\). For any \(Du \in DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell_{n})\), we have
Consequently, \(\Phi(D\alpha) = f\). It follows that Φ is surjective. By Theorem 3.1 in [23] we have
This means that Φ is continuous. Furthermore, it is immediate that \(\Phi^{1}\) is continuous from the inverse function theorem. The proof of Corollary 3.3 is thus finished. □
4 Stokes equations with variable viscosity
In the study of the stationary NavierStokes equations, the corresponding Stokes equations play an important role. It can be said that any open question about the NavierStokes equations, such as global existence of strong solutions, uniqueness and regularity of weak solutions, and asymptotic behavior, is closely related with the qualitative and quantitative properties of the solutions of Stokes equations; see, for example, [38] for related discussions. To be precise, it is crucial to investigate the properties of solutions \((u,\pi)\) to the following Stokes system:
With \(\int_{\Omega}f_{0}\,dx=\int_{\partial\Omega}n\cdot v_{0}\,dx\), the necessary condition for the solvability is given. The scalar function \(f_{0}\) is a measure of the compressibility of fluid. The boundary condition (4.3) describes the adhesion at the boundary of the domain Ω 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 [3, 33].
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 hypercomplex formulation:
Definition 4.1
We say that \((u,\pi) \in W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell_{n}) \times L^{p(x)}(\Omega)\) a solution of (4.4)(4.6) provided that it satisfies the system (4.4)(4.6) for every \(f\in W^{1,p(x)}(\Omega, \mathrm{C}\ell_{n})\).
Definition 4.2
The operator \(\widetilde{\nabla}:L^{p(x)}(\Omega)\rightarrow (W^{1,p(x)}(\Omega))^{n}\) is defined by
for all \(f \in L^{p(x)}(\Omega)\) and \(\varphi\in(C^{\infty}_{0}(\Omega))^{n}\).
Lemma 4.1
(see [28])
Let Ω be a bounded Lipschitz domain of \(\mathbb{R}^{n}\). Let \(f\in(W^{1,p(x)}(\Omega))^{n}\) satisfy
for any \phi \in \mathcal{M}(\mathrm{\Omega}):=\{v\in {({W}_{0}^{1,{p}^{\prime}(x)}(\mathrm{\Omega}))}^{n}:divv=0\}. Then there exists \(q\in L^{p(x)}(\Omega)\) such that \(f=\widetilde{\nabla} q\).
Theorem 4.1
Let \(f \in W^{1,p(x)}(\Omega, \mathrm{C}\ell_{n})\). Then the Stokes system (4.4)(4.6) 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
with respect to the estimate
Here, \(C\geq1\) is a constant and the hydrostatic pressure π is unique up to a constant.
Proof
We first show that the following representation is valid for each \(f \in W^{1,p(x)}(\Omega, \mathrm{C}\ell_{n})\):
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
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 \(TQ_{a}B\widetilde {T}\widetilde{D}\varphi= TQ_{a}B\varphi\). Thus, for \(u \in W^{1,p(x)}_{0}(\Omega, \mathrm {C}\ell_{n})\) and \(\pi\in L^{p(x)}(\Omega)\), Lemma 2.4 yields
This implies that our system (4.4)(4.6) is equivalent to the system
Clearly, the equality (4.7) is equivalent to the following equality:
Now we need to show that for each \(f\in W^{1,p(x)}(\Omega, \mathrm{C}\ell_{n}^{1})\), the function \(Q_{a}BTf\) can be decomposed into two functions Du and \(Q_{a}B\pi\). Suppose \(Du + Q_{a}B\pi=0\) for \(u \in W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n}^{1}) \cap \operatorname{ker} \operatorname{div}\) and \(\pi\in L^{p(x)}(\Omega)\). Then (4.5) gives \([Q_{a}B\pi]_{0} = 0\). Thus, \(Q_{a}B\pi= 0\). Hence, \(Du = Q_{a}B\pi= 0\). This means that \(Du + Q_{a}B\pi\) is a direct sum, which is a subset of \(\operatorname{im}Q_{a}\).
Next we have to consider the existence of a functional \(\mathcal {F}\in(L^{p(x)}(\Omega, \mathrm{C}\ell_{n}^{1}) \cap\operatorname {im}Q_{a})^{\ast}\) with \(\mathcal{F}(Du)=0\) and \(\mathcal{F}(Q_{a}B\pi)=0\) but \(\mathcal {F}(Q_{a}B\widetilde{T}f)\neq0\). This is equivalent to asking if there exists \(g \in W^{1,p'(x)}(\Omega, \mathrm{C}\ell_{n}^{1})\), such that for all \(u \in W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell_{n}^{1}) \cap\operatorname{ker} \operatorname{div}\) and \(\pi\in L^{p(x)}(\Omega)\),
but \([Q_{a}B\widetilde{T}f, Q_{a}B\widetilde{T}g]_{A} \neq0\). Here, Corollary 3.3 is applied.
Thus, let us consider the system (4.10) and (4.11) with \(g \in W^{1,p'(x)}(\Omega, \mathrm{C}\ell_{n}^{1})\). Taking the arguments as the first part of the proof in Theorem 3.1, we may write \(Du=BDw\) with \(w \in W^{1,p(x)}_{0}(\Omega, \mathrm {C}\ell_{n}^{1})\). Then \(\operatorname{div}u=\operatorname{div}w=0\) because of \(b(x)>0\). According to Lemma 2.5, (4.10) yields
which implies \(g = \widetilde{\nabla} h = \widetilde{D}h\) with \(h \in L^{p'(x)}(\Omega)\) because of Lemma 4.1. Thus we obtain from (4.11) and Lemma 2.4
holds for each \(\pi\in L^{p(x)}(\Omega)\). Hence, \(Q_{a}B\pi=Q_{a}Bh^{p'(x)2}Q_{a}Bh\) gives \(Q_{a}Bh = 0\). Then we obtain
Furthermore, we get
Finally, (4.9) yields
By the norm equivalence theorem, we obtain
By Remark 2.1, Lemma 2.2, and the boundedness of the operator \(Q_{a}\), we get
From (4.12) the uniqueness of the solution follows. Note that \(Q_{a}B(\pi_{1} \pi_{2})= 0\) implies \(B(\pi_{1} \pi_{2}) \in B\operatorname{ker}\widetilde{D}\). Then \(\pi_{1}=\pi_{2} +c\) with \(c \in \mathbb{R}\). Therefore, π is unique up to a constant. The proof of Theorem 4.1 is now complete. □
In [28], the authors did not point out the relation of solutions and weak solutions for the Stokes equations. Next we should prove that solutions and weak solutions for the Stokes equations are equivalent.
Definition 4.3
The Cliffordvalued function u is called a weak solution of the system (4.3)(4.6), if for each \(v \in W^{1,p'(x)}_{0}(\Omega, \mathrm{C}\ell_{n}^{1}) \cap\operatorname {ker} \operatorname{div}\), the equality
holds, where \((u, v)_{\mathrm{Sc}}\) denotes the classical scalar product \((u, v)_{\mathrm{Sc}}=\int_{\Omega}[\overline{u}v]_{0}\,dx=\int_{\Omega}\sum_{i=1}^{n} u_{i} v_{i}\,dx\).
Theorem 4.2
Let \(u \in W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n}^{1})\) and \(\pi\in L^{p(x)}(\Omega)\) be a solution of system (4.7)(4.8). Then u is a weak solution of system (4.4)(4.6). If u is a weak solution of the system (4.4)(4.6), then there exists a function \(\pi\in L^{p(x)}(\Omega)\) such that the pair \((u, \pi)\) solves the system (4.7)(4.8).
Proof
We follow the idea of Theorem 4.5.2 in [33]. First we need to show that for all functions \(v \in W^{1,p'(x)}_{0}(\Omega, \mathrm{C}\ell_{n}^{1}) \cap\operatorname{ker} \operatorname{div}\), the following equality holds:
For this purpose, from (4.7) it follows that
for each \(v \in W^{1,p'(x)}_{0}(\Omega, \mathrm{C}\ell_{n}^{1}) \cap \operatorname{ker} \operatorname{div}\). The first term of the right side in (4.14) yields
thanks to Lemma 2.4, Lemma 2.5 and (3.4). For the second term of the right side in (4.14), we choose a sequence \(\{\pi_{k}\}_{k=1}^{\infty} \subset W^{1,p(x)}_{0}(\Omega)\) with \(\pi_{k}\rightarrow\pi\) in \(L^{p(x)}(\Omega)\), then
because \((\nabla\pi_{k}, v)_{\mathrm{Sc}}=(\pi_{k}, \operatorname{div} v)_{\mathrm{Sc}}=0\) for all \(v \in W^{1,p'(x)}_{0}(\Omega, \mathrm{C}\ell_{n}^{1}) \cap \operatorname{ker} \operatorname{div}\). Then (4.14), together with (4.15) and (4.16), gives (4.13).
Conversely, let \(u \in W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n}^{1}) \cap\operatorname{ker} \operatorname{div}\) be a weak solution for the system (4.4)(4.6). That means that (4.13) holds. It easily follows from Lemma 2.4 and Theorem 3.1 that
for all \(q \in L^{p(x)}(\Omega)\) and \(v \in W^{1,p'(x)}_{0}(\Omega, \mathrm{C}\ell_{n}^{1}) \cap\operatorname{ker} \operatorname{div}\). Then we have
On one hand, we have shown in Theorem 4.1, which is independent of the above considerations, that the system
has a solution \((\psi, \pi) \in W^{1,p(x)}_{0}(\Omega, \mathrm {C}\ell_{n}^{1}) \times L^{p(x)}(\Omega)\). Then substituting q in (4.17) by π, we obtain
which implies
On the other hand, it follows from (4.17) that
for all \(v \in W^{1,p'(x)}_{0}(\Omega, \mathrm{C}\ell_{n}^{1}) \cap \operatorname{ker} \operatorname{div}\). Obviously, \(W^{1,p'(x)}_{0}(\Omega, \mathrm{C}\ell_{n}^{1}) \cap \operatorname{ker} \operatorname{div}\) is a closed subspace in \(W^{1,p'(x)}_{0}(\Omega, \mathrm{C}\ell_{n}^{1})\), and hence \(W^{1,p'(x)}_{0}(\Omega, \mathrm {C}\ell_{n}^{1}) \cap\operatorname{ker} \operatorname{div}\) is a reflexive Banach space due to Lemma 2.1. Then (4.18) yields
Finally, by using the density argument and Corollary 3.1 we obtain
This ends the proof of Theorem 4.2. □
Remark 4.1
Actually, the last part of the proof in Theorem 4.2 is quite different from that of Theorem 4.5.2 in [33]. The reason lies in the difference of \(W^{1,2}_{0}(\Omega, \mathrm {C}\ell_{n})\) and \(W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell_{n})\).
5 NavierStokes equations with variable viscosity
Our aim in this section is to investigate the existence of solution of the timeindependent generalized NavierStokes equations (1.7). Evidently, the main difference from the abovementioned Stokes equations is the appearance of the nonlinear 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 [39] 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 fixedpointed theorem. Gürlebeck and Sprößig [32, 33] 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 Ω and \(6/5< p<3/2\). Cerejeiras and Kähler [38] obtained similar results provided the external force f belongs to \(W^{1,p}(\Omega,\mathrm{C}\ell_{n})\) for an unbounded domain Ω and \(n/2\leq p<\infty\). Zhang et al. [28] investigated similar results provided that the external force f belongs to \(W^{1,p(x)}(\Omega,\mathrm{C}\ell_{n})\) for a bounded domain Ω and \(n/2\leq p<\infty\). Our intention in this section is to extend these results to the more general case in variable exponent spaces.
For \(f=\sum_{i=1}^{n}f_{i} \mathrm{e}_{i}\), \(u=\sum_{i=1}^{n}u_{i} \mathrm{e}_{i}\), let us consider the following steady generalized NavierStokes equations in the hypercomplex notation:
with the nonlinear part \(F(u)=f [uD ]_{0} u\), where the operator A is mentioned above.
Lemma 5.1
(see [28])
Let \(p(x)\) satisfies (2.1) and \(n/2\leq p_{}\leq p(x) \leq p_{+}<\infty\). 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
where \(C_{1}=C_{1}(n,p,\Omega)\) is a positive constant.
Using Lemma 5.1, which is crucial to the convergence of the iteration method, we are able to give the main result as follows.
Theorem 5.1
Let \(p(x)\) satisfy (2.1) and \(n/2\leq p_{}\leq p(x)\leq p_{+}<\infty\). Then the system (5.1)(5.3) has a unique solution \((u,\pi)\in W^{1,p(x)}_{0} (\Omega,\mathrm{C}\ell_{n})\times L^{p(x)}(\Omega,\mathbb{R})\) (π is unique up to a real constant) if the righthand side f satisfies the condition
with \(\nu=\mu/\rho\), \(C_{4}=C_{2}(1+C_{3})\), where \(C_{3} \geq1\) indicated in (5.8) and
For any function \(u_{0} \in W^{1,p(x)}_{0} (\Omega,\mathrm{C}\ell_{n})\) with
here \(\mathcal{F}=\sqrt{\frac{\nu^{2}}{4C_{1} ^{2}C_{4} ^{2}}\frac{1}{C_{1}}\f\_{W^{1,p(x)} (\Omega,\mathrm{C}\ell_{n})}}\), the iteration procedure
converges in \(W^{1,p(x)}_{0} (\Omega,\mathrm{C}\ell_{n})\times L^{p(x)}(\Omega)\), where \(B_{k}h:=b(x, u_{k})h\).
Proof
The proof is similar to that of Theorem 4.6.8 in [33]. For the reader’s convenience, we will give the key details of the proof. Replacing f by \(F(u_{k1})\) in the proof of Theorem 4.1, we obtain the unique solvability of the Stokes equations (4.4)(4.6) which we have to solve in each step. Moreover, we have the following estimate:
where \(C_{3} \geq1\) is a constant. In the following, using the Banach fixed point theorem, we could finish the remaining proof by following the proof of Theorem 3.1 in [28] word by word; we thus omit the details. □
Definition 5.1
The Cliffordvalued function u is called a weak solution of the system (5.6)(5.7), if for all \(v \in W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell_{n}^{1}) \cap\operatorname{ker} \operatorname{div}\), the following equality holds:
that is to say, the following equality is valid:
for every \(v \in W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell_{n}^{1}) \cap \operatorname{ker} \operatorname{div}\).
In the following we would like to point out the equivalence of solutions and weak solutions for the generalized NavierStokes equations; this point is similar to the case of the Stokes equations.
Theorem 5.2
Let \(u \in W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n}^{1})\) and \(\pi\in L^{p(x)}(\Omega)\) be a solution of system (5.6)(5.7). Then u is a weak solution of system (5.1)(5.3). If u is a weak solution of the system (5.6)(5.7), then there exists a function \(\pi\in L^{p(x)}(\Omega)\) such that the pair \((u, \pi)\) solves the system (5.1)(5.3).
Proof
The proof is quite similar to the proof of Theorem 4.2, so we omit it. We also refer the reader to a similar proof of Theorem 4.6.2 in [33] by replacing Q with \(Q_{a}B\). □
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Acknowledgements
The authors would like to thank the referee for constructive suggestion which helps us to improve the quality of the paper. B Zhang was supported by the Research Foundation of Heilongjiang Educational Committee (Grant No. 12541667).
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Niu, R., Zheng, H. & Zhang, B. NavierStokes equations with variable viscosity in variable exponent spaces of Cliffordvalued functions. Bound Value Probl 2015, 30 (2015). https://doi.org/10.1186/s136610150291y
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DOI: https://doi.org/10.1186/s136610150291y