Existence of Steady Solutions for a Model for Micropolar Electrorheological Fluid Flows with Not Globally log-Hölder Continuous Shear Exponent

In this paper, we study the existence of weak solutions to a steady system that describes the motion of a micropolar electrorheological fluid. The constitutive relations for the stress tensors belong to the class of generalized Newtonian fluids. The analysis of this particular problem leads naturally to weighted variable exponent Sobolev spaces. We establish the existence of solutions for a material function p^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{p}}$$\end{document} that is log\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log $$\end{document}-Hölder continuous and an electric field E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{E}$$\end{document} for that |E|2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert \textbf{E}\vert ^2$$\end{document} is bounded and smooth. Note that these conditions do not imply that the variable shear exponent p=p^∘|E|2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p={\hat{p}}\circ \vert \textbf{E}\vert ^2$$\end{document} is globally log\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log $$\end{document}-Hölder continuous.


Introduction
In this paper we establish the existence of solutions of the system 1 Here, Ω ⊆ R d , d ≥ 2, is a bounded domain.The three equations in (1.1) represent the balance of momentum, mass and angular momentum for an incompressible, micropolar electrorheological fluid.In it, v denotes the velocity, ω the microrotation, π the pressure, S the mechanical extra stress tensor, N the couple stress tensor, the electromagnetic couple force, f = f + χ E div(E ⊗ E) the body force, where f is the mechanical body force, χ E the dielectric susceptibility and E the electric field.The electric field E solves the quasi-static Maxwell's equations div E = 0 in Ω , where n is the outer normal vector field of ∂Ω and E 0 is a given electric field.The system (1.1), (1.2) is the steady version of a model derived in [9], which generalizes previous models of electrorheological fluids in [29], [31].The model in [9] contains a more realistic description of the dependence of the electrorheological effect on the direction of the electric field.Since Maxwell's equations (1.2) are separated from the balance laws (1.1) and due to the well developed mathematical theory for Maxwell's equations (cf.Section 3), we can view the electric field E with appropriate properties as a given quantity in (1.1).As a consequence, we concentrate in this paper on the investigation of the mechanical properties of the electrorheological fluid governed by (1.1).
Micropolar fluids have been introduced by Eringen in the sixties (cf.[10]).A model for electrorheological fluids was proposed in [30], [29], [31].While there exist many investigations of micropolar fluids or electrorheological fluids (cf.[23], [31]), there exist to our knowledge no mathematical investigations of steady motions of micropolar electrorheological fluids except the PhD thesis [11], the diploma thesis [33], the paper [12] and the recent result [21].Except for the latter contribution these investigations only treat the case of constant shear exponents.
For the existence theory of problems of similar type as (1.1), the Lipschitz truncation technique (cf.[14], [6]) has proven to be very powerful.This method is available in the setting of Sobolev spaces (cf.[13], [6], [8]), variable exponent Sobolev spaces (cf.[6], [8]), solenoidal Sobolev spaces (cf.[1]), Sobolev spaces with Muckenhoupt weights (cf.[12]) and functions of bounded variation (cf.[2]).Since, in general, |E| 2 does not belong to the correct Muckenhoupt class, the results in [12] are either sub-optimal with respect to the lower bound for the shear exponent p or require additional assumptions on the electric field E. These deficiencies are overcome in [21] by an thorough localization of the arguments.Moreover, [21] contains the first treatment of the full model for micropolar electrorheological fluids in weighted variable exponent spaces under the assumption that the shear exponent is globally log-Hölder continuous.The present paper relaxes this condition and shows existence of solutions under the only assumption that the electric field E is bounded and smooth.
This paper is organized as follows: In Section 2, we introduce the functional setting for the treatment of the variable exponent weighted case, and collect auxiliary results.Then, Section 3 is devoted to the analysis of the electric field, while Section 4 is devoted to the weak stability of the stress tensors.Eventually, in Section 5, we deploy the Lipschitz truncation technique to prove the existence of weak solutions of (1.1), (1.2).

Basic notation and standard function spaces
We employ the customary Lebesgue spaces L p (Ω), 1 ≤ p ≤ ∞, and Sobolev spaces W 1,p (Ω), 1 ≤ p ≤ ∞, where Ω ⊆ R d , d ∈ N, is a bounded domain.We denote by • p the norm in L p (Ω) and by • 1,p the norm in W 1,p (Ω).The space W 1,p 0 (Ω), 1 ≤ p < ∞, is defined as the completion of C ∞ 0 (Ω) with respect to the gradient norm ∇ • p , while the space having a vanishing trace, i.e., u| ∂G = 0. We use small boldface letters, e.g., v, to denote vector-valued functions and capital boldface letters, e.g., S, to denote tensor-valued functions 3 .However, we do not distinguish between scalar, vector-valued and tensor-valued function spaces in the notation.The standard scalar product between vectors is denoted by v • u, while the standard scalar product between tensors is denoted by A : B. For a normed linear vector space X, we denote its topological dual space by X * .Moreover, we employ the notation u, v := ´Ω uv dx, whenever the right-hand side is well-defined.We denote by |M | the d-dimensional Lebesgue measure of a measurable set M .The mean value of a locally integrable function u ∈ L 1 loc (Ω) over a measurable set M ⊆ Ω is denoted by ffl M u dx := 1 |M | ´M u dx.By L p 0 (Ω) and C ∞ 0,0 (Ω), resp., we denote the subspace of L p (Ω) and C ∞ 0 (Ω), resp., consisting of all functions u with vanishing mean value, i.e., ffl Ω u dx = 0.

Weighted variable exponent Lebesgue and Sobolev spaces
In this section, we will give a brief introduction into weighted variable exponent Lebesgue and Sobolev spaces.
A weight σ on R d is a locally integrable function satisfying 0 < σ < ∞ a.e. 4 .To each weight σ, we associate a Radon measure ν σ defined via ν σ (A) := ´A σ dx for every measurable set A ⊆ R d .
Let us now introduce variable exponent Sobolev spaces in the weighted and unweighted case.Let us start with the unweighted case.Due to L p(•) (Ω) → L 1 loc (Ω), we can define the variable exponent Sobolev space W 1,p(•) (Ω) as the subspace of L p(•) (Ω) consisting of all functions u ∈ L p(•) (Ω) whose distributional gradient satisfies ∇u ∈ L p(•) (Ω).The norm into a separable Banach space (cf.[5,Thm. 8.1.6]).Then, we define the space W (Ω) and W 1,p(•) 0,div (Ω) are reflexive (cf.[5,Thm. 8.1.6]).Note that the velocity field v : Ω → R d solving (1.1), in view of the properties of the extra stress tensor (cf.Assumption 4.1), necessarily satisfies Dv ∈ L p(•) (Ω).Even though we have that v = 0 on ∂Ω, we cannot resort to Korn's inequality in the setting of variable exponent Sobolev spaces (cf.[5,Thm. 14.3.21]),since we do not assume that p = p•|E| 2 ∈ P ∞ (Ω) is globally log-Hölder continuous.However, if we switch by means of Hölder's inequality from the variable exponent p ∈ P ∞ (Ω) to its lower bound p − , for which Korn's inequality is available, also using Poincaré's inequality, we can expect that a solution v of (1.1) satisfies v ∈ L p − (Ω).Thus, the natural energy space for the velocity possesses a different integrability for the function and its symmetric gradient.This motivates the introduction of the following variable exponent function spaces.
be open and q, p ∈ P ∞ (Ω).Then, the space X (Ω), by definition, is a Banach space.For the separability and reflexivity, we first observe that Π : d×d is an isometry and, thus, an isometric isomorphism into its range R(Π).Hence, R(Π) inherits the separability and reflexivity from L q(•) (Ω) d ×L p(•) (Ω) d×d , and by virtue of the isometric isomorphism be open and q, p ∈ P ∞ (Ω).Then, we define the spaces For the treatment of the micro-rotation ω, we also need weighted variable exponent Sobolev spaces.In analogy with [21, Ass.2.2], we make the following assumption on the weight σ.
shows that Assumption 2.7 is satisfied for every q, p ∈ P ∞ (Ω).Definition 2.9.Let Ω ⊆ R d , d ∈ N, be open and let σ satisfy Assumption 2.7 for q, p ∈ P ∞ (Ω).For u ∈ C ∞ (Ω), we define The weighted variable exponent Sobolev space X In other words, w ∈ X in Ω with ∇w = ∇w for all w ∈ W 1,p(•) (Ω).However, in general, ∇w and the usual distributional gradient ∇w do not coincide.
Definition 2.12.Let Ω ⊆ R d , d ∈ N, be open and let Assumption 2.7 be satisfied for q, p ∈ P ∞ (Ω).Then, we define the space In the particular case σ = 1 a.e. in Ω, we employ the abbreviations The following local embedding result will play a key role for the applicability of the Lipschitz truncation technique in the proof of the existence result in Theorem 5.1.
d+2 .Then, for each Ω ⊂⊂ Ω, there exists a constant c(p, Ω ) > 0 such that for every u ∈ X . (2.14) i.e., we have that The proof is postponed to the Appendix A.

log-Hölder continuity and related results
We say that a bounded exponent p ∈ P ∞ (G) is locally log-Hölder continuous, if there is a constant c 1 > 0 such that for all x, y ∈ G We say that p ∈ P ∞ (G) satisfies the log-Hölder decay condition, if there exist constants c 2 > 0 and We say that p is globally log-Hölder continuous on G, if it is locally log-Hölder continuous and satisfies the log-Hölder decay condition.Then, the maximum c log (p) := max{c 1 , c 2 } is just called the log-Hölder constant of p.Moreover, we denote by P log (G) the set of globally log-Hölder continuous functions on G. log-Hölder continuity is a special modulus of continuity for variable exponents that is sufficient for the validity of the following results.

The electric field E
We first note that the system (1.2) is separated from (1.1), in the sense that one can first solve the quasi-static Maxwell's equations yielding an electric field E, which then, in turn, enters into (1.1) as a parameter through the stress tensors.
It is proved in [27], [28], [31], that for bounded Lipschitz domains, there exists a solution 5 A more detailed analysis of the properties of the electric field E can be found in [11].Let us summarize these results here.Combining (1.2) 1 and (1.2) 2 , we obtain that i.e., the electric field is a harmonic function and, thus, real analytic.In particular, for a harmonic function, we can characterize its zero set as follows: Proof: See [11], [12,Lem. 3.1].
Finally, we observe that using the regularity theory for Maxwell's equations (cf.[32], [31]), one can give conditions on the boundary data E 0 ensuring that the electric field E is globally bounded, i.e., E ∞ ≤ c(E 0 ).Based on these two observations, we will make the following assumption on the electric field E: Note that there, indeed, exist solutions of the quasi-static Maxwell's equations that satisfy Assumption 3.3, but do not belong to any Hölder space.In particular, there exist a solution of the quasi-static Maxwell's equations such that for a standard choice of p ∈ P log (R), we have that p := p • |E| 2 / ∈ P log (Ω).
In the sequel, we do not use that E is the solution of the quasi-static Maxwell's equations (1.2), but we will only use Assumption 3.3.

A weak stability lemma
The weak stability of problems of p-Laplace type is well-known (cf.[6]).It also holds for our problem (1.1) if we make appropriate natural assumptions on the extra stress tensor S and on the couple stress tensor N, which are motivated by the canonical example in (1.3).We denote the symmetric and the skew-symmetric part, resp., of a tensor A ∈ R d×d by A sym := 1 2 (A + A ) and For the extra stress tensor S : R d×d sym × R d×d skew × R d → R d and some p ∈ P log (R) with p− > 1, there exist constants c, C > 0 such that: Assumption 4.2.For the couple stress tensor N : R d×d × R d → R d×d and some p ∈ P log (R) with p− > 1, there exist constants c, C > 0 such that:  (Ω).
Proof: Using all rational tuples contained in Ω 0 as centers, we find a countable family (B k ) k∈N of balls covering Ω 0 such that B k := 2B k ⊂⊂ Ω 0 for every k ∈ N.

Main theorem
Now we have everything at our disposal to prove our main result, namely the existence of solutions to the problem (1.1), (1.2) for p − > 2d d+2 even if the shear exponent p is not globally log-Hölder continuous.

Figure 1 :
Figure 1: Plots of the first (blue/left) and the second (red/right) component of the numerically determined electric field.