Navier-Stokes equations with variable viscosity in variable exponent spaces of Clifford-valued functions
- Rui Niu^{1, 2},
- Hongtao Zheng^{1} and
- Binlin Zhang^{2}Email author
https://doi.org/10.1186/s13661-015-0291-y
© Niu et al.; licensee Springer. 2015
Received: 17 October 2014
Accepted: 21 January 2015
Published: 11 February 2015
Abstract
In this paper we study the stationary generalized Navier-Stokes equations when the viscosity is not only a constant but also a function which depends on the position and the shear-velocity. For this we establish an improved decomposition of variable exponent Lebesgue spaces of Clifford-valued 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 Navier-Stokes equations with variable viscosity in the setting of variable exponent spaces of Clifford-valued functions. Furthermore, the equivalences of solutions and weak solutions for the aforementioned equations are justified.
Keywords
MSC
1 Introduction
In order to study the time-dependent 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 Navier-Stokes 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 Navier-Stokes 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 Navier-Stokes 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 Navier-Stokes equations in variable exponent spaces of Clifford-valued 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 A-Dirac 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 A-Dirac 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 A-Dirac equations with variable growth. For this purpose, the authors established a theory of variable exponent spaces of Clifford-valued functions with applications to homogeneous and nonhomogeneous A-Dirac equations; see also [25, 26]. Very recently, Fu et al. in [27, 28] established a Hodge-type decomposition of variable exponent Lebesgue spaces of Clifford-valued functions with applications to the Stokes equations, the Navier-Stokes equations and the A-Dirac equations \(DA(Du)=0\). By using the Hodge-type decomposition and variational methods, Molica Bisci et al. in [29] studied the properties of weak solutions to the homogeneous and nonhomogeneous A-Dirac equations with variable growth. For an overview for the nonlinear A-Dirac equations with variable growth, we refer the reader to [30].
This paper is organized as follows. In Section 2, we start with a brief summary of Clifford algebra and variable exponent spaces of Clifford-valued 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 above-mentioned equations are showed. In Section 5, using similar methods to [28], we prove the existence and uniqueness of solutions to the generalized Navier-Stokes 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 above-mentioned equations are presented.
2 Preliminaries
First we recall some related definitions and results concerning Clifford algebra and variable exponent spaces of Clifford-valued functions. For a detailed account we refer to [6, 34, 35].
A Clifford-valued 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 real-valued functions.
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])
Lemma 2.2
(see [28])
- (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])
- (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])
- (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])
- (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
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
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.
Corollary 3.1
Proof
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 time-independent Maxwell equations over a three-dimensional 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
Proof
4 Stokes equations with variable viscosity
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].
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
Lemma 4.1
(see [28])
Theorem 4.1
Proof
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 \(\operatorname{im}Q_{a}\).
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
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
5 Navier-Stokes equations with variable viscosity
Our aim in this section is to investigate the existence of solution of the time-independent generalized Navier-Stokes equations (1.7). Evidently, the main difference from the above-mentioned 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 fixed-pointed 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.
Lemma 5.1
(see [28])
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
Proof
Definition 5.1
In the following we would like to point out the equivalence of solutions and weak solutions for the generalized Navier-Stokes 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).
Declarations
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).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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