Electrorheological Fluids - Modeling and Mathematical Theory

Michael Rfi~i6ka

Electrorheological Fluids: Modeling and Mathematical Theor~J

Springer

Author Michael Rfl~i6ka Institute of Applied Mathematics University of Freiburg Eckerstr. 1 79104 Freiburg, Germany E-mail: rose @mathematik.uni-freiburg.de

Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme R~i6ka, Michael: Electrorheological fluids : modeling and mathRematical theory 1 Michael R0~iEka. - Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Singapore ; Tokyo : Springer, 2000 (Lecture notes in mathematics ; 1748) ISBN 3-540-41385-5

Mathematics Subject Classification (2000): 76Axx, 35Q35, 76D03, 35Dxx, 35K55, 35J60, 3502 ISSN 0075- 8434 ISBN 3-540-41385-5 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH © Springer-Verlag Berlin Heidelberg 2000 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 10724266 41/3142-543210 - Printed on acid-free paper

MEINEN ELTERN

Introduction Electrorheological fluids are special viscous liquids, that are characterized by their ability to undergo significant changes in their mechanical properties when an electric field is applied. This property can be exploited in technological applications, e.g. actuators, clutches, shock absorbers, and rehabilitation equipment to name a few. Winslow [131] is credited for the first observation of the behaviour of electrorheological fluids in 1949. Since then great strides have been made to overcome the impediments of early electrorheological fluids, as the abrasive nature and the instability of the suspension and the enormous voltage requirements that are necessary for a significant change in the material properties. Nowadays existing electrorheological fluids, which make the above mentioned devices possible, are the result of intensive efforts to manufacture materials without these impediments. In this book we present a theoretical investigation of electrorheological fluids. Firstly, we develop a model for these liquids within the framework of Rational Mechanics, which takes into account the complex interactions between the electro-magnetic fields and the moving liquid. Secondly, we carry out a mathematical analysis of the resulting system of partial differential equations which possesses so-called non-standard growth conditions. We discuss the functional setting, namely generalized Lebesgue and generalized Sobolev spaces, and show existence and uniqueness of weak and strong solutions respectively. In this introduction we want to give a general overview of the issues which are discussed in the following chapters in detail. Electrorheological'fluids can be modeled in many ways. One possibility consists in the investigation of the underlying microstructure to obtain a macroscopic description of the material (cf. Klingenberg, van Swol, Zukoski [54], Halsey, Toot [43], Bonnecaze, Brady [16], Parthasarathy, Klingenberg [97]). Another approach uses the framework of continuum mechanics and treats the electrorheological fluid as a homogenized single continuum (cf. Atkin, Shi, Bullogh [6], Rajagopal, Wineman [107], Wineman, Rajagopal [130]) or models it using the theory of mixture (cf. Rajagopal, Yalamanchili, Wineman [108]). A completely different perspective is provided by modeling based on direct numerical simulations taking into account the dynamics and interaction of particles (cf. Whittle [133], Bailey, Gillies, Heyes, Sutcliffe [7]). In all these models the electric field is treated as a constant parameter. However, there is strong evidence in the literature (cf. Katsikopoulos, Zukoski [52], Abu-Jdayil, Brunn [1], [2l, [3], Wunderlich, Brunn [1341) that the behaviour of electrorheological fluids is

viii strongly influenced by non-homogeneous electric fields. In many experiments one tries to modify the geometry in such a way that the so-called ER-effect (i.e. the reduction in volumetric flow rate at constant pressure drop) increases at the same field strength. In order to capture such effects we develop a model that takes into account the interaction of the electro-magnetic fields and the moving liquid and thus, the electric field is treated as a variable, that has to be determined. The interaction of the electro-magnetic fields and the moving material is described on the basis of the "dipole current-loop" model (cf. Grot [41], Pao [96]). In the first part of Chapter 1 we carry out the modeling, which builds on ideas from Rajagopal, Rfi~iSka [104], [105]. More precisely, we start with the general balance laws for mass, linear momentum, angular momentum, energy, the second law of thermodynamics in the form of the Clausius-Duhem inequality and Maxwell's equations in their Minkowskian formulation. We then choose dependent and independent variables, reflecting the nature of the processes we are interested in, and impose the requirement of Galilean invariance of both the constitutive relations and the balance laws. In the next step we simplify the system by incorporating the physical properties of an electrorheological fluid, namely that it can be considered as a non-conducting dielectric and that the mechanical response does not change if a magnetic field is applied. In the last step we carry out a dimensional analysis and a subsequent approximation, restricting the validity of the resulting system to certain but typical situations. The main assumptions are that the magnetic field is of secondary importance and that the oscillations of the electro-magnetic fields are not larger than the reaction time of the fluid. The final result of the whole procedure is the system (1.2.32)-(1.2.36) and (1.2.43)-(1.2.46), which governs the motion of an electrorheological fluid. For example, if the fluid is assumed to be incompressible it turns out that the relevant equations are div (E + P ) = 0, curl E = 0, 0v P0-~

div S + p0[Vv]v + V¢ = p 0 f + [VE]P,

(0.1)

(0.2)

div v = 0, where E is the electric field, P the polarization, P0 the density, v the velocity, S the extra stress, ¢ the pressure, and f the mechanical force. In the remaining part of Chapter 1 we discuss various special forms of the constitutive relation for the extra stress S. In Section 1.3 we consider models in which the extra stress S depends linearly on the symmetric velocity gradient D, when the fluid is either compressible or incompressible or mechanically incompressible but electrically compressible. In particular, the additional restrictions on the material response function imposed by the Clausius-Duhem inequality are examined. Experimental evidence for concrete electrorheological fluids however suggests that a linear dependence is in general too

ix simple. In the last section we therefore propose a model capable of explaining many of the observed phenomena. In this model the extra stress has the form S = aul ((1 + IDlU)~

- 1)E ® E + ((~31+ aa3[E[2)( 1 + [D[2) a~-22D

(0.3)

+ a51(1 + [D[2) 2a~(DE ® E + E @ D E ) , where oqj are material constants and where the material function p depends on the strength of the electric field IEI 2 and satisfies 1 < p ~ < v(IEI ~) < p0 < ~ .

(0.4)

To illustrate the features of (0.3) we solve the boundary value problem for a flow driven by a pressure drop between infinite parallel planes. Finally, we discuss the consequences of the Clausius-Duhem inequality for the model (0.3), which from the mathematical point of view is related to the coercivity of the elliptic operator, and conditions which ensure the uniform monotonicity of the operator induced by - div S (cf. Lemma 1.4.46, Lemma 1.4.64). Let us now have a closer look at the system (0.1)-(0.4) under the assumption that the polarization depends linearly on the electric field, i.e. P = xEE. Fortunately, the system is separated into the quasi-static Maxwell's equations (0.1) and the equation of motion and the conservation of mass (0.2), where E can be viewed as a parameter. Maxwell's equations are widely studied in the literature and well understood. For details, we refer the reader to the overview article Milani, Picard [83] within the context of Hilbert space methods and to Schwarz [115], where questions concerning the regularity of solutions to (0.1) are discussed. In Section 2.3 we have collected and indicated the proofs of the results for the system (0.1) that we need in the sequel. The situation for the system (0.2)-(0.4) is quite different and hence the remaining part of this book is devoted to the study of it. Since the material function p, which essentially determines S, depends on the magnitude of the electric field [El 2, we have to deal with an elliptic or parabolic system of partial differential equations with socalled non-standard growth conditions, i.e. the elliptic operator S satisfies

S(D,E). D > co(l +

lE[2)(1+

[DI2) 2

ID[2, (0.5)

IS(D,E)[ _< c1(1 + [D[2) ~2a~AIEI2. The problem of regularity of solutions to equations, variational integrals and systems with non-standard growth conditions has been extensively studied in the last ten years, starting with counterexamples by Giaquinta {37] and Marcellini [76]. At the beginning of Chapter 2 we give a short overview of some of the results which have been achieved in the last years. Since the solution E of Maxwell's equations is in general not constant it is clear from the form of the extra stress S that the canonical functional setting are the spaces /y(x)(~) and WI'P(x)(~), so-called generalized Lebesgue and generalized

Sobolev spaces, respectively. These spaces have been studied e.g. by Hudzik [44] and Kov~ik, R~kosn~k [55]. Since they are not so well known and because we also need weighted versions of them, we will discuss their properties in some detail in Section 2.2. We would like to point out that generalized Lebesgue and generalized Sobolev spaces have a lot in common with the classical Lebesgue and classical Sobolev spaces, however there are also many open fundamental questions, which are well understood in the theory of classical Lebesgue and classical Sobolev spaces. For example it is not known, even for very "nice" functions p(x), whether smooth functions are dense in the space WI,P(x)(~) or whether the maximal operator is continuous from/2(x)(f~) into In Chapter 3 we study steady flows of incompressible shear dependent electrorheological fluids. Their motion is governed by the system div E = 0

in~,

curl E = 0 E. n = E0.n

on Off,

- div S(D, E) + [Vv]v + V¢ = f + xE[VE]E, div v = 0 v = 0

(0.6)

inf',

(0.7)

on 0 ~ ,

where ~ is a bounded domain and the data f and E0 are given. The extra stress S is given by (0.3), (0.4) and the coefficients aij are such that the operator is coercive and uniformly monotone. In Section 3.2 we show the existence of weak solutions to the system (0.6), (0.7) whenever the lower bound Poo of the material function p is larger than 9/5 (cf. Theorem 3.2.4). The solution is unique if the data are small enough (cf. Proposition 3.2.38). The existence result is accomplished by adapting the theory of monotone operators and a compactness argument to our situation. This is possible since the energy estimate ensures that, within the context of generalized weighted Sobolev spaces, the operator induced by - d i v S(D, E) maps the natural energy space 1 Ep(~),, into its dual. In the remaining part of Chapter 3 we present a different approach to the steady problem, which ensures the existence of strong solutions to the system (0.6), (0.7), (0.3), (0.4), but does not use the theory of monotone operators. The importance of this method will become clear when unsteady problems are treated. The main problem of course consists in the identification of the limiting element for the sequence S(D(v~), E), where v n is some approximate solution to (0.7). This problem can be solved by using Vitali's theorem if we derive apriori estimates that ensure the almost everywhere convergence of D(vn). To this end we can build on ideas initiated by Ne~as [93] and developed by M~lek, Ne~as, Rfi~i~ka [71], [73], Bellout, Bloom, Ne~as [9], M~lek, Ne~as, Rokyta, Rfi~i~ka [70] in their study of unsteady flows of generalized 1see (2.2.47) for the definition of Ep(z},~

xi Newtonian fluids 2. The desired estimate will be obtained using essentially - A v n ~ 2 as a test function in the weak formulation. Since this test function is not divergence free also terms containing the pressure ¢ have to be estimated. The regularity properties of ¢ have to be determined from (0.7), viewed as an operator equation in some negative Sobolev space. It turns out that due to the growth conditions of the extra stress S it is not possible to handle the pressure terms. Therefore, we approximate S(D, E) by some sequence SA(D, E), for A -+ co, which is constructed explicitly. The approximation SA(D, E) is chosen such that on the one hand the growth properties allow us to estimate (in dependence on A) the pressure terms and that on the other hand we do not lose too much in the coercivity and monotonicity properties of SA(D, E) compared with the coercivity and monotonicity properties of S ( D , E ) . What this vague statement means precisely is formulated and proven in Section 3.3.1. The final approximation of the system (0.7), which we will denote by (0.7)~,A, is obtained by a mollification of the convective term. It is easy to see that the system (0.7)~,A possesses weak solutions (cf. Proposition 3.4.2). Using the difference quotient method one can show that locally second order derivatives belong to an appropriate Lebesgue space. This property in turn enables us to use --AvE'A~2a, ~ > 1, as a test function and to derive local estimates for the second order derivatives V2v ~'A, the gradient of the extra stress VSA(D(v~'A), E) and the pressure ce,A, which are independent of A (cf. Proposition 3.5.7). We would like to point out that these local estimates are possible because we have at our disposal some global information from the energy estimate, which is sufficient to compensate the non-local nature of the pressure. These estimates enable the limiting process A -~ oo and we arrive at an approximation of the system (0.7), where only the convective term is mollified. In a final step, it remains to estimate the convective term, which stands on the right-hand side of our estimates (cf. Proposition 3.5.42), by the information we have on the left-hand sides 3. It turns out that this procedure works under certain restrictions on Po in terms of poo that can be found in the formulation of Theorem 3.3.7, our main result concerning strong solutions of the steady system (0.6), (0.7), (0.3), (0.4). Let us point out that the lower bound for p~, which is governed by the convective term, is the same as in Theorem 3.2.4, where monotonicity methods have been used. However, in contrast to Theorem 3.2.4 upper bounds for P0 appear, which are in the case poo _ 2 due to the chosen approximation, and in the case p ~ < 2 due to the non-standard growth conditions of the elliptic operator S. Finally, we would like to mention that Theorem 3.2.4 and Theorem 3.3.7 are to my knowledge the only existence result for elliptic systems with non-standard growth conditions with a nonlinear right-hand side b(u, Vu). In the case of elliptic equations with non-standard growth conditions there 2Generalized Newtonian fluids can by considered as a special case of electrorheological fluids if we set E -- 0 and thus p = const. The aim in these papers was to establish the existence of weak solutions to the equations of motion when the theory of monotone operators fails because p~ is too small (cf. Ladyzhenskaya [58], Lions [68]). 3In this and the previous step various technical assertions are needed, which are proved in the Appendix.

xii are existence results by Marcellini [78] and Boccardo, Gallouet, Marcellini [12] under the assumption that the right-hand side b(x) belongs to the space Llo ~ and L 1, respectively. Of course, in both the scalar case and the vectorial case, the existence of minimizers of variational integrals is clear (using lower semicontinuity arguments) under certain conditions on the integrand. In the last chapter we turn our attention to the study of unsteady flows of shear dependent electrorheological fluids. We treat the system (0.1), (0.2) together with appropriate boundary and initial conditions for two different models for the extra stress tensor S. The main part of the chapter is devoted to the case when we set in (0.3) a21 = a51 = 0 and c~31= a33, i.e. S has the form S ( D ( v ) , E ) = a31(1 + IEI2)(1 + {D(v)l 2) 2 D ( v ) ,

(0.8)

where ~31 > 0 and p = p(IEI 2) satisfies (0.4) with Poo _> 2. In the second model that we consider we assume that the extra stress tensor S depends linearly on D, i.e. in (0.3) we set ¢~m = 0 and p = 2. In both cases we again assume that the polarization is given by P = x E E . The second model behaves very similar to the unsteady Navier-Stokes system and we prove global in time existence of weak solutions for large data (cf. Theorem 4.1.20). The proof of this result is not carried out since only minor changes in the standard proofs of the corresponding result in the Navier-Stokes theory are necessary. On the other hand, the system (0.1), (0.2) with S given by (0.8) behaves quite differently. First of all and in contrast to the steady system (0.6), (0.7) we cannot use monotonicity methods to prove the existence of solutions to the system (0.1), (0.2), (0.8), (0.4) due to the non-standard growth of the operator. More precisely, the canonical functional setting would be the space L p(~'=)(QT), since the energy method provides the estimate T

f//(l+[D(v)l f 2) =

]D(v)l2 dxdt <_ c,

(0.9)

.J

0

which implies that D(v) E IF(t'X)(QT). For the treatment of parabolic problems it is essential to treat the time and the space variables differently and to employ the equivalence of the spaces

Lq(QT)

and

Lq(I, Lq(f~)) .

(0.I0)

In the case p = const, the estimate (0.9) therefore yields v E Lq(I, Wl'q(~)) and the technique of monotone operators together with compactness arguments works. However, in our case it is not clear how an equivalence similar to (0.10) can be achieved for the space LP(t'x)(QT), which would allow us to adapt the arguments of the classical setting. Fortunately, the second approach presented in Chapter 3, using Vitali's theorem and almost everywhere convergence can also be used in the unsteady case. In fact, we employ ideas from Mhlek, NeSas, Rfi~i~ka [73], which have been

Xlll

developed in the context of generalized Newtonian fluids, to treat situations where monotonicity methods fail. The procedure in Chapter 4 is very similar to the one used in Chapter 3 and we concentrate on the new aspects. One essential difference is, that now not only the regularity properties of the pressure ¢ but also the regularity properties of the time derivative of the velocity --~ have to be computed from the system (0.2). This in fact prevents us from working with local estimates only, because we would run into a circle argument. Therefore we need to derive global estimates. The main result of this chapter is the existence of global in time weak and strong solutions to the system (0.1), (0.2), (0.8) for large data under certain restrictions on poo and P0, which are formulated in Theorem 4.1.14. We also prove that strong solutions are unique. We would like to point out that weak solutions exist for poo >_ 2, which clearly shows the advantage of the chosen approach, since, even if the monotonicity method would work, the best lower bound for poo which could be achieved is 11/5. For the proof of Theorem 4.1.14 we again approximate the elliptic operator S by an appropriate sequence S A and we also mollify the convective term. We denote the resulting system by (0.2)~.A. In Section 4.1 we collect the relevant properties of the approximative elliptic operator S A and define weak and strong solutions. In the next section we show the existence of strong solutions to the system (0.1), (0.2)e,A, (0.8), (0.4), which belong to the space L2(I, W2'2(1~)) (cf. Proposition 4.2.1). For that we apply the difference quotient method in the interior and in the tangential directions near the boundary. To estimate the normal derivatives we use the pressure eliminating operator curl to the system (0.2)~,A and the internal constraint div v = 0. In the case when poo is near 2 we need additional estimates, which are proved in Lemma 4.3.8. The next step is to derive estimates that are independent of A. In fact, this step is crucial for the whole procedure, and here various difficulties occur due to the non-standard growth of the system and the fact that the operators S A are only partially potential like. As a consequence of these problems an upper bound for P0 appears (cf. Lemma 4.3.8, Lemma 4.3.41). However, the estimates for the velocity v E'A, the pressure cE,A and the elliptic operator SA(D(ve'A),E) are sufficient for the limiting process A --~ c~ (cf. Proposition 4.3.57). At the end of Section 4.3 we derive "weighted" estimates (cf. Lemma 4.3.82, Lemma 4.3.88), which are crucial for the treatment of Poo near 2. In the last section it remains to find conditions on Poo and P0, that allow to handle the convective term, which appears on the right-hand side of the above mentioned estimates. To this end we combine all possible information we have in a rather delicate way. Basically, we derive the following inequality 1

_1 ( 1 + IID(v~(t))ll,(iE(t),2)) 1-" #

(0.11)

t

+ / llVS(D(v~(T)), E(T))ll~(1 + [Ie(v~(T))[Ip(m(,)l~)) - " dT <_c(f, v0, E0), 0

whereu=- po-1, 2_~ r--- ~ 6 and 0 < #. In this inequality of course # depends on poo and P0. The size o f # decides whether we obtain strong or weak solutions. In fact, for # _< 1

xiv we get strong solutions, while # > 1 yields weak solutions. In all cases the estimate (0.11) ensures the almost everywhere convergence of V v ~. With this information at hand the limiting process c -~ 0 is no problem and the proof of Theorem 4.1.14 can be finished. I would like to mention the difference to the corresponding Theorem 4.1.14 in Rfi~i~ka [113], where the system (0.1), (0.2) with an elliptic operator S of the form (0.8), however with D(v) replaced by Vv, is treated. Due to this dependence this system does not describe the motion of an electrorheological fluid, but is only an unsteady system with non-standard growth conditions motivated by the study of electrorheological fluids. Theorem 4.1.14 in [113] proves the existence of solutions for a larger range for p compared with Theorem 4.1.14 here. The reason is a completely different behaviour of the normal derivatives of the velocity gradient when S depends on D(v) or on Vv. Let me finally mention that Theorem 4.1.14 together with my results in [113], [114] are the first results for parabolic systems with non-standard growth conditions. At this point I would like to mention, that the present book is based on my habilitation thesis (University of Bonn, 1998). The first three chapters, up to minor corrections of misprints and small changes for a better understandability of the text, are the same as in this thesis [113]. However Chapter 4 is completely new. Finally it is my pleasure to thank 3. Ne~as and J. Frehse who showed me the beautiful world of mathematics and accompanied me on my way not only with professional guidance but also, and probably more importantly, with their personal advice. I also want to express my sincere thanks to K.R. Rajagopal who strongly influenced my views of mechanical engineering, which is much more than a huge source of fascinating questions for a mathematician. I am thankful to G.P. Galdi for initiating my stays at the Universities of Pittsburgh and Ferrara and for his hospitality. During these visits, which were supported by fellowships from DFG and CNR, I started my investigations on electrorheological fluids. Of course I also wish to thank my friends and colleagues J. M~lek, A. Novotn:~, M. Padula, A. Passerini, L. Pick, A. Romano, A. Sequeira, A. Srinivasa, M. Steinhauer and G. Th/~ter, who influenced me and my work in different situations and ways. Last but not least I want to express my gratitude to I. Schmelzer for her excellent typing of my sometimes chaotic manuscript. I am very grateful to W. Eckart, H. Kastrup, L. Pick, M. Specovius-Neugebauer and G. Th~ter for reading large parts of a previous version of the manuscript. The time they invested has certainly improved this final version.

Contents

1

Introduction

vii

Contents

xv

Modeling of Electrorheological Fluids 1.1 1.2 1.3

1,4 2

Mathematical 2.1 2.2 2.3

3

3.4 3.5 3.6 4

5

Framework

1 9 15 18 20 21 24 39

Setting of the Problem and Introduction . . . . . . . . . . . . . . . . . Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maxwell's Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Electrorheological Fluids with Shear Dependent Steady Flows 3.1 3.2 3.3

1

General Balance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrorheological Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . Linear Models for the Stress Tensor T . . . . . . . . . . . . . . . . . . 1.3,1 Compressible Electrorheological Fluids . . . . . . . . . . . . . . 1,3.2 Incompressible Electrorheological Fluids . . . . . . . . . . . . . 1.3.3 Mechanically Incompressible but Electrically Compressible Electrorheological Fluids . . . . . . . . . . . . . . . . . . . . . . Shear Dependent Electrorheologicat Fluids . . . . . . . . . . . . . . . .

39 43 54

Viscosities:

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strong Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . Existence of Approximate Solutions . . . . . . . . . . . . . . . . . . . . Limiting Process A -~ ce . . . . . . . . . . . . . . . . . . . . . . . . . . Limiting Process ~ --+ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 62 71 74 84 93 101

Electrorheological Fluids with Shear Dependent Viscosities: Unsteady Flows 4.1 Setting of the Problem and Main Results . . . . . . . . . . . . . . . . . 4.1,1 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Existence of A p p r o x i m a t e Solutions . . . . . . . . . . . . . . . . . . . . 4.3 Limiting Process A --+ oo . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Limiting Process ~ -+ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 105 110 114 126 142

Appendix

153

5.1 5.2

153 158

General Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . Auxiliary Results for the Approximations . . . . . . . . . . . . . . . . .

References

165

Index

175

1 Modeling of Electrorheological Fluids 1.1

General Balance Laws

Electrorheological fluids are special viscous fluids, which are characterized by their ability to undergo significant changes in their mechanical properties due to the application of an electric field. When they are modeled within the frame-work of continuum mechanics, the result is a special case of the theory which describes the interaction of electro-magnetic fields with moving deformable bodies. Beside classical textbooks on electrodynamics as e.g. Landau, Lifschitz [66], Jackson [49], Sommerfeld [117], Feynman, Leighton, Sands [28], which also touch this subject, there are many monographs and research articles as e.g. Penfield, Haus [98], Panofsky, Phillips [95], de Groot, Suttorp [40], Truesdell, Toupin [123], Hutter, van de Ven [48], Eringen, Maugin [27], Grot [41], Pao [96], Maugin, Eringen [82], Toupin [122] and Dixon, Eringen [21], [22], which discuss this subject in detail. Nevertheless the theory is not at all unified and we refer the reader to Pao [96] for a beautiful review and comparison of different approaches. Here we present an approach to the modeling of electrorheological fluids which uses ideas, which have been developed in Rajagopal, Rtl~i~ka [104], [105]. More precisely, we start with the general balance laws for mass, linear momentum, angular momentum, energy, the second law of thermodynamics in the form of the ClausiusDuhem inequality and Maxwell's equations. The terms representing the interaction of the electro-magnetic fields and the moving deformable body are described on the basis of the "dipole current-loop" model (cf. Pao [96], Grot [41])1. Let f~0 C R 3 denote the reference configuration of an abstract body B. By the motion of the body B we mean a one-to-one mapping X that assigns to each particle X E ~0 a position x in the three dimensional Euclidean space, at the instant of time t, i.e. x = X(t, X).

(1.1)

The velocity v(t, x) and the acceleration a(t, x) are defined through OX v = 0-7'

and

a = o2X

Ot 2 "

(1.2)

lit is shown in Pao [96] that the final equations based on the "dipole current-loop"model and on the statistical approach, which is very popular (cf. Eringen, Maugin [27]), are identical.

2

1.MODELING OF E L E C T R O R H E O L O G I C A L FLUIDS

The velocity gradient L(t, x) is a tensor given by \ Oxj ] i,j=l,2,s '

L = V v

where V is the partial derivative with respect to x. We denote the total time derivative by d. The symmetric part of the velocity gradient L is denoted by D, i.e. D = ½(L + LT), where L T is the transpose of the tensor L; and the skew part of L is denoted by W. In this section we assume that all field variables are sufficiently smooth in order to make all operations that are carried out meaningful. We shall start by recording the local forms of the thermo-mechanical balance laws. The conservation of mass and the balance of linear momentum, respectively, are given by

Op - ~ + div (pv) = 0,

dv p~-~

(1.3)

divT=pf+f~,

(1.4)

where p is the density, T is the Cauchy stress tensor, f the external mechanical body force, and fe the electro-magnetic body force given by

fe = qeE÷

×B+-

l(dP

+(divv)P

) ×B+

c -~

× ([VB]P) ~

(1.5)

+ IV B]T2~4 + IV E ] P . The form of the electro-magnetic body force (1.5) as well as all other terms representing the interaction of the electro-magnetic fields with the moving deformable body is based on the "dipole current-loop" model (cf. Pao [96], Grot [41]). Here and in the sequel we shall use for vectors u, w and tensors S, T the notation div u

[Vu]w =

(

Oxi '

s . T = S~jT~j,

(os j

Oui

w~--

div S = \ Oxj ]~=1,2,a'

in which the summation convention over repeated indices is employed. We will apply this convention throughout the book. Moreover the operator × denotes the usual vector product of two vectors. In (1.5) we have used the notation that q~ is the electric charge density, E the electric field, J the total current, P the electric polarization, B the magnetic induction, and .AA is defined by2 .~4 = M ÷ -1v c

x P,

(1,6)

2Note, that ~ can be interpreted as the magnetization in the co-moving frame. An analogue interpretation can be given for £ and J , which will be defined later.

1.1. GENERAL BALANCE LAWS

3

where M is the magnetic polarization and c .~ 3.101° cm sec -1 denotes the speed of light. The balance of angular momentum takes the form dv x x p~-

div(x x T) -- x x p f + l e ,

(1.7)

in which le denotes the electro-magnetic torque density given by 1~ = x x f~ + P x ~ +.A4 × B ,

(1.8)

where ~ is the electromotive intensity defined through £ = E + 1-v x B .

(1.9)

c

The balance of energy can be written as d p~(e+

1 2 ~lv I ) + d i v q = d i v ( T v ) + p f . v + p r + w e .

(1.10)

Here we denoted by e the specific internal energy, by q the heat flux vector, by r the heat source density, and the energy production density we is given by we = f~ . v +

pe.

~--~

- .A,4 • - - ~ + ,.7" • £ ,

(1.11)

where the conduction current ,ff is defined through 3" = J - q ~ v .

(1.12)

We interpret the second law of thermodynamics in the form of the Clausius-Duhem inequality3: p -~ + div

- p ~ _> 0,

(1.13)

where 8 is the absolute temperature and ~/the specific entropy. Using (1.3) and (1.4) we easily get the reduced forms of (1.7) and (1.10), which read e(T+P®£+M®B)

=0,

(1.14)

in which e is the complete skew symmetric Levi-Civita tensor and ® denotes the usual tensor product of two vectors, and p ~ + d i v q--- T - L + ~ . P + J.£+

M.]3

(1.1~)

(e-P)div v+pr.

3We also refer the reader to Liu, Miiller [69] and Hurter, van de Ven [48], Hutter [46], [47] for different formulations of the second law of thermodynamics within the context of the motion of a deformable body under the influence of electro-magnetic fields.

4

1.MODELING OF ELECTRORHEOLOGICAL FLUIDS

Here and in the following a superposed dot denotes the total time derivative. Introducing the specific Helmholtz potential ¢ through ¢ = e- ~-

_1E . P , P

(1.16)

and substituting (1.16) into (1.13) we obtain the dissipation inequality -p(¢ + T]9)+T- L

q. V9

o

- ~'.P-.A,,t.B

(1.17) + J.E

> 0.

Finally, we list Maxwell's equations a for the moving liquid 5. Gauss' law reads div De = qe,

(1.18)

where De is the electric displacement given by (1.19)

D~=P+E. Faraday's law is given by 1 0B curl E + c - ~ - -- 0 ,

(1.20)

and the conservation of magnetic flux takes the form div B = 0.

(1.21)

Ampere's law reads curl H - i ODe

1

(1.22)

M.

(1.23)

c Ot + -c ( J + qev),

where the magnetic field H is given by H = B-

The conservation of electric charge, which is a consequence of (1.18) and (1.22), reads aqe 0t + div (,T + qev) = 0.

(1.24)

4Throughout this chapter we use Heavyside Lorentz units (cf. Maugin, Eringen [82], Jackson 1491). ~The approach to the theory of electrodynamics for moving media, which we follow here, goes back to Minkowski [85]. In his formulation of the field equations, which is identical to equations (1.18), (1.20)-(1.22), only the variables E, B, De, H, J and qe are used. The system is completed by constitutive relations (cf. (1.39)) and transformation formulae relating the fields in the co-moving and the laboratory frame (cf. (1.33)). Note, that there exist many different formulations for the electrodynamics of moving media, see e.g. Pao [96] or Hutter, van de Ven [48] for a comparison.

1.1. G E N E R A L B A L A N C E L A W S

5

The system (1.3), (1.4), (1.14), (1.15), (1.17) and (1.18), (1.20)-(1.22), (1.24), which describes the motion of the body, has far more unknowns than equations. It is rendered determinate by providing appropriate constitutive relations, reflecting the material properties. We will assume that p,v,D,E,B,0,V0

(1.25)

are the independent variables and thus we provide constitutive relations for e, ¢ , T , q , . h t , P , ,:7

(1.26)

f = f(p, v, D, E, B, 0, V 0),

(1.27)

of the form

where f stands for any of the quantities in (1.26). The choice of dependent and independent mechanical variables is quite standard and unified, the situation is different for the electro-magnetic variables. We have made the above choice because it seems to be adequate for our purposes (cf. Grot [41], Eringen, Maugin [27]) and it reflects the connection of the variables to the material in the most direct way (cf. Thiersten [121]). Both the material and the balance equations are subject to invariance requirements. It is well known that the thermo-mechanical balance laws (1.3), (1.4), (1.14), (1.15) and (1.17), without the terms due to the interaction of the electro-magnetic fields and the material, are form invariant under Galilean transformations of the frame given by x* ----Qx - v0t q- b0, t* - - t ,

(1.28)

where v0, b0 are given vectors and Q is a time independent orthogonal tensor, i.e. QQT = I, if one requires that the fields representing material properties are covariant, i.e. they transform in the following way T * = Q T Q T, q * = Q q , ¢*=¢, e*--e,

(1.29)

and that the other quantities transform in the usual way 0"=8,

p*=p,

r*=r,

f*=Qf.

(1.30)

In all transformation formulae the quantities on the left-hand side and on the righthand side have to be evaluated in the same material point, e.g. p*(x*) = p(x), where x* and x are related by (1.28)1. The transformation rules (1.29) are a special case of the principle of material frame indifference, which requires that (1.29) has to hold under changes of frame given by x* = q ( t ) x + c(t), t* = ~ ,

(1.31)

6

1.MODELING OF ELECTRORHEOLOGICAL FLUIDS

where Q(t) is a time dependent orthogonal tensor and c(t) is a given time dependent vector. Note t h a t one usually assumes t h a t the constitutive relations depend on L instead of D, and then one deduces from the principle of material frame indifference that this dependence on L has to reduce to a dependence on D only. In fact, this is for us the only relevant consequence of the stronger requirement of material frame indifference which cannot be obtained from the requirement t h a t the material properties are invariant under Galilean transformations (1.28) only. Moreover, there seems to be no firm agreement how the electro-magnetic quantities should transform under transformations of the form (1.31) (cf. Grot [41] pp. 157, 183), while the transformation properties for Galilean transformations are on firm grounds. Therefore we have decided to include from the beginning D into the list of independent variables and to work only with the requirement of Galilean invariance. It is also well-known t h a t Maxwell's equations (1.18), (1.20)-(1.22) and (1.24) are form invariant under Lorentz transformations, e.g. if we consider a motion of the frame with velocity v0 = vel in direction el the Lorentz transformation is given by

xl - vt Xl

t*-

~

'

1

X2 -~ X 2

X3 ~ X3 ,

(t-v~).

Since we are interested in non-relativistic effects we shall assume t h a t Iv01 << c. Neglecting terms of order Iv 12 in (1.32) we recover (1.28)1 (with Q = I, b0 = 0), but (1.32)2 reduces to 6 t* -~ t

V Xl C C

where the second term can be neglected for xl belonging to compact sets, but not uniformly for all xl. Thus, the Galilean transformation (1.28) is not a uniform approximation of the Lorentz transformation. Nevertheless, for the study of effects in a finite body (1.28) can be viewed as a good approximation of (1.32). Similar difficulties manifest themselves if one tries to deduce approximate transformation formulae for B, E, De, H, J, qe, P and M from the transformation formulae for these quantities t h a t arise from the Lorentz transformation ~. One can avoid these difficulties if one requires the Maxwell's equations (1.18), (1.20)-(1.22) and (1.24) to be form invariant under the Galilean transformation (1.28). This requirement renders the following 6This observation can be found in the literature, cf. L~vy-Leblond [66], Le Bellac, L~vy-Leblond [8] where this issue is discussed in detail. It is clearly pointed out there that beside the assumption Ivol << c one needs additional requirements to reduce the Lorentz transformation to the Galilean transformation. For example it is assumed in L(~vy-Leblond[66] that the space-time interval is of "large time-like" type, i.e. Ax << cAt. We have brought up again attention to this point, because unfortunately one often finds in textbooks statements like "Galilean transformation is a good approximation of Lorentz transformation if lvol << c ", which might be misleading. 7Especially the transformation formulas for qe, P and M coming from the requirement of form invariance of Maxwell's equations under Gaiilean transformations are not approximations of the formulae coming from the Lorentz transformation.

1.1. GENERAL BALANCE LAWS transformation formulae if Q = I, b0 = 0 1 E* = E + - v 0

B*=B,

×B,

C

1 H*=H--voxDe,

D~=De,

(1.33)

C

q* = qe-

It is worth noticing that the transformation formulae for P and M, which would follow from (1.33), (1.19) and (1.23) are not reasonable. But in the formulation of Maxwell's equations these fields do not occur. Based on the above discussion we shall make the following invariance requirements: We assume that the quantities (1.26), describing the material properties, are co-variant under Galilean transformations (1.28). Moreover we require that all balance laws (1.3), (1.4) s, (1.14), (1.15), (1.17) and (1.18), (1.20)-(1.22), (1.24) are form invariant under Galilean transformations (1.28). These two requirements imply the following transformation laws 9 T * - - Q T Q T, ~b* -= ¢ , P* = Q P ,

q*=Qq,

(1.34)

e*=e,

,hi* = Q.h4,

,7* = Q , 7 ,

B* = Q B ,

E*=Q(E+I(QTv0)

xB),

D~=QDe,

H*=Q(H-~(QTv0)

xDe),

£*=Q£,

p*=p, f*=Qf,

q*=qe, 0"=0,

v*=Qv-vo,

(1.35)

(1.36)

f~=Qf~,

V*0*=QV0, D * = Q D Q T,

r*=r.

(1.37)

The G alilean invariance of the quantities (1.26) implies that the constitutive relations (1.27) are isotropic functions of its arguments, which transform according to (1.34)(1.37). The system (1.3), (1.4), (1.14), (1.15), (1.17) and (1.18), (1.20)-(1.22), (1.24) together with the constitutive equations for the quantities in (1.25) describes the behaviour of many materials in a variety of situations. This is much too general, since in our investigation we are only interested in a description of electrorheological fluids. SNote, that one should use the identity qeE + ~J x B = qee + ~,.7"× B to rewrite the electromagnetic body force fe in a more convenientform. 9Note, that this could only lead to inconsistenciesif both P and De or fl,4 and H would appear simultaneously in one equation, which is never the case.

8

1.MODELING OF ELECTRORHEOLOGICAL FLUIDS

Therefore we will make some simplifying assumptions, which make the system easier, but leave it general enough for our purposes. Since our main interest in this investigation is not the thermal behaviour 1° of the electrorheological fluid we shall start by simplifying the form of the thermal response of the material. We assume that the heat flux vector q is given by Fourier's law of heat conduction q = - k V 0,

(1.38)

where k is the thermal conductivity, which is assumed to be constant. In all other constitutive relations we will drop the dependence on V 0. This and the invariance requirements imply that (1.27) has to be replaced by (use the transformation formulae for Q : I, v0 = v; see also Grot [41])

f = ](p, D, ~, S, e).

(1.39)

In addition to restrictions placed on the constitutive response functions by the invariance requirements we have additional strictures due to the requirement of the second law of thermodynamics. We shall now determine the restrictions imposed by requiring that all admissible processes of the body, i.e. processes compatible with the balance laws and constitutive response functions, meet the Clausius-Duhem inequality (1.17). It immediately follows from (1.17) and (1.3) that

0¢ .D-(.A4 + p0~).13+ (T + p~-~pI)-D

- P ( ~ + ")° - P~-5 +T.W+k~

(1.40) + (p-~-P)

.E+ J-e

>_ 0.

Using the linearity of (1.40) with respect to the dotted quantities and W and their independence on the arguments appearing in the constitutive relations (1.39) one easily deduces (cf. Coleman, Noll [19], Truesdell, Noll [124], Grot [41])

a¢ -

P =-p

0o

a¢ '

0¢

~-g,

a D = 0,

a¢

.A4 = - p ~-~,

(1.41)

TT=T and the reduced dissipation inequality (T+p

~pI) -D+k

+,:T. E _> 0.

(1.42)

t°However, it is clear that for practical applications thermal effects play an important role. This point will be investigated after a better understanding of the mechanical effects.

1.2. E L E C T R O R H E O L O G I C A L

FLUIDS

Summarizing, we are now dealing with the following system of balance laws + pdiv v = 0,

(1.43)

pv - div T = pf + fe p ~ - kAO = T . D + P - E

+ (P- £)div v

-.,%4.B + J . £ ® B) = 0,

e(P ® £ + ~ (T + ¢I). D + k ~

(1.44) (1.45)

+pr

+,7". £ > 0,

(1.46) (1.47)

div De = q~, 1 0B curl E + c--~- = 0,

(1.48) (1.49)

div B = 0,

(1.50)

curl H = -1 ODe ~ ( J + qev), c Ot + Oqe

0t + div ( J + q~v) = 0,

(1.51) (1.52)

where fe is given by f~ = qe e + 1 j C

x B + _1( p + (div v ) P ) x B + _1v x ([VB]P) C

C

(1.53)

+ [V B]TA4 + IV E ] P , and where the thermodynamic pressure ¢ is defined through ¢ -- p50¢ -~p.

1.2

(1.54)

Electrorheological Fluids

Now we shall make further simplifications of the system (1.43)-(1.52) based on our understanding of the behaviour of electrorheological fluids. We will make assumptions on the form of some constitutive relations which reflect the observed behaviour of electrorheological fluids and then carry out a dimensional analysis and a subsequent approximation which restricts the validity of the resulting system to special but typical situations. We would like to point out that some of our assumptions in this section are rather based on general observations than on careful experimental evidence, which is missing in the literature. Nevertheless, we believe that these assumptions are reasonable, however they have to be carefully tested when appropriate experimental data are available. Firstly, we assume that the stress tensor does not depend on the magnetic induction B, i.e. T = T(p, D, £, 0).

(2.1)

1.MODELING OF ELECTRORHEOLOGICAL FLUIDS

10

A more detailed discussion on the role of the magnetic induction in electrorheological fluids will be provided when the non-dimensionalization is carried out. Secondly, we shall assume that the fluid is non-conducting, i.e. (cf. Grot [41] Section 2.2) J - 0,

(2.2)

and thirdly, as we are dealing with a dielectric (cf. Grot [41] Section 2.2) .A4 = 0.

(2.3)

These assumptions already simplify the system (1.43)-(1.52). Note, that from (1.41)2 and (2.3) it follows that

¢=

o, e)

(2.4)

P --

e, e).

(2.5)

and therefore also (cf. (1.41)2)

Our invariance requirements imply that ¢ is an isotropic function of its arguments and thus 0¢ E, p --- -2P0- ~

(2.6)

which in turn implies together with (2.3) that (1.46) is trivially satisfied and we will drop it from the list of equations. Further we get from (1.6) and (2.3) that 1 M = --v

x P.

¢

(2.7)

Next, we discuss the issue concerning the importance of the magnetic induction. With regard to this point, there is not a convincing body of systematic experimental evidence that can throw light on this matter, though the popular wisdom seems to be of the opinion that it is of secondary importance (see Filisko [29], Wineman [129]). In fact, to our knowledge no experimental paper even reports on the role of the magnetic induction. Most of the experiments measure at best global quantities like the flow rate and not even the mechanical quantities such as the velocity field locally, as the flow regions of interest are so small. However, in order to determine and retain terms in (1.43)-(1.52) that are dominant and discard others that are insignificant, it is necessary to carry out a dimensional analysis of this system. We shall introduce the following non-dimensional quantities: E

E=-Eo' t To '

B= p

B

V00' e E0

qe

q0 p fi = P0

v

x

=V0' ~=f

~ '

0

(2.8)

00 '

where the quantities with the suffix zero are appropriate representative quantities. While To could be associated with the period of oscillation of the electric field when an

11

1.2. ELECTRORHEOLOGICAL FLUIDS

ac field is involved, a characteristic time for the problem would have to be introduced if a dc field is used. One could use e.g. the reaction time of the material to form structures. Here, we shall not go into the details of t h a t issue. In typical problems, we envisage that E0 ~ 10 -2 - 10 2 statvolts cm - t , Lo ~ 10 -1 - 1 c m , To ~ 10 - 4 -

(2.9)

1 sec,

and thus we conclude t h a t L0 ,,~ 10_11 _ 10_6.

(2.10)

cTo In the analysis t h a t we shall carry out, we chose to define a Reynolds number Re through

Re = Po Lo Vo

(2.11)

#0 where p0 and/~0 are the density and viscosity of the fluid in the absence of an electric field. We shall be interested in problems wherein the Reynolds number n Re lies between 1 and 10 2. We shall be concerned with materials for which a typical viscosity #o is of order of 10 -2 g cm -1 s -1 and a typical density is of order of 1 g cm -3, reflecting the situation for a large class of fluids. This and (2.11) in turn imply, t h a t the typical velocity field ranges over 10 -2 to 10 cm s - i , implying Vo ,,~ 10_12 _ lO_S.

(2.12)

C

The subordinate role of the magnetic induction in electrorheological fluids is mathematically interpreted through the assumption t h a t

Eo Lo

----

Bo cTo

~

1,

(2.13)

which is consistent with the assumption t h a t the magnetic induction is only induced by oscillations of the electric field (cf. (2.28) and (2.35)). Concerning the amount of free charges we shall assume t h a t q0 L0 ~, 10_12 . E0

(2.14)

Defining a small non-dimensional number e through -- 10 -3

(2.15)

1lOne easily can extend the approach presented here to smaller Reynolds numbers with the effect that some convective terms should be neglected.

12

1.MODELING OF ELECTRORHEOLOGICAL FLUIDS

we can recast (2.10), (2.12), (2.13) and (2.14) into the form Lo = O(~2) _ O(~4),

(2.16)

Yo = O(~3) _ O(~4),

(2.17)

cTo 5

Eo Lo _ O(1) Bo c To q0 no = O(e4). E0

(2.18) (2.19)

We will approximate the system (1.43)-(1.52), written in non-dimensional quantities, by neglecting terms of order es and higher, while retaining terms up to order ¢2. In preparation for this we shall discuss the role of g in the constitutive relations for P and T. It follows from the definition of the electromotive intensity £ (1.9) that

1£12 = ]£12 E3-

E~( I E ? + c-2E . ( v × B ) + F I v1 × 2Vo Lo]~

= I:E?+

7 h - To

V, L 2 ~. ~02 ~0

• (v x f 3 ) + c2 VZc2

BI~) x f3l ~

(2.2o)

= IEI ~ + O(ES),

where we used (2.18), (2.16) and (2.17). In view of (2.20), (2.5) and the regularity assumption of the material function ¢ we obtain

P(p, ~, P) = P(p, ~,E) + O(~5).

(2.21)

From the invariance requirements and (1.41)3 it follows that T is a symmetric isotropic function of its arguments. Hence representation theorems (cf. Spencer [118]) yield that T = T(p, O, D, £)

= alI + a2£ ® £ + asD

(2.22) +

O~4D2 -[- as(DE ® £ + £ ® DE)

-b a 6 ( D 2 E ® £ -b £ @ D2£),

where ai, i = 1 , . . . , 6, are functions of the invariants p,9,[£[2,tr D , t r [D[ 2,tr D s , t r (D£ @ £ ) , t r (D2£ @ £).

(2.23)

From (2.20) we again deduce in a way similar to that for the polarization that

5i(P, ~, D, £) = ai(p, O,0D, £) _ 6i(p, O, D, E) + O(e 5) Where a i0 are appropriate non-dimensional numbers.

(2.24)

1.2. ELECTRORHEOLOGICAL FLUIDS

13

Using (1.19), (1.23) and (2.7) we re-write (1.48)-(1.52) in terms of E , B , P and q~ only. Now we carry out a dimensional analysis of this re-written formulation of Maxwell's equations. Using (2.16)-(2.19) and (2.21) we obtain qo Lo _ div I~ + div P = -~oq~ + O ( ~ ) , curl l~ + Bo Lo 0I] = Eo c To OF div B = EoVo -, curlB + ~o v c u r l (* x P) =

(2.25)

0,

(2.26)

0, Eo Lo 0_

(2.27)

Bo cTo cq~(E + P)

(2.28)

+ qo L o Vo E o ~ ¢ + O(~5), Eo cBo

0q~ VoTo[#q~]e= ~- ~- Lo

YoTo. . . . ~o qealVV.

(2.29)

Now using that (cf. (2.16)-(2.18))

Yo To Lo

Yo c To = o(c-') c Lo -

-

- 0(~2),

(2.30)

and

S_~o = O(¢2 ) _ O(e4)

(2.31)

Eo

and neglecting terms of O(e 3) we can approximate (1.48)-(1.52) by div (E + P) = 0,

(2.32)

curl E = 0, div B = 0,

(2.33) (2.34)

curl B + 1 curl (v x P) - 1 0(E + P) C

C

qe + qe div v = 0,

(2.35)

~

(2.36)

where P = P(p,0,E). It follows from (1.19), (1.23) and (2.7) that (2.32)-(2.35) can be re-written again in terms of E, B, H, De only. Note, that (2.32)-(2.36) is exactly the electric quasi-static approximation of Maxwell's equations (cf. Romano [110]). Now we turn to the approximation of the equations (1.43)-(1.45) and (1.47). The conservation of mass (1.43) remains un-effected. In the momentum equation (1.44)

1. MODELING OF ELECTRORHEOLOGICAL FLUIDS

14

we use (2.22), (2.21) and (2.24), which leads to

PoVoLo _0~

PoVoLo ToVo[~ ,~]#

E~oTo ~-~ + E~o---~o L~ O~0

-

_

{~}V ~i - {a°}div(~21~@ I~) r

°~3o

~ToVo.:-

-~o-~o~-~o

,_

.

.r..

o,v ~ , - , ) -

.ToVo ....

~ 4oV o

~ ~ - ~ o

~-,2,

o,v ~,,-,

- ¢'~°" t° v°-iv (,~(i5~ ~ E + E ~ ISE)) iToJ" Lo ol

-- iT--~---~o

(2.37) (llV

Lo -

qoLo -

qoLoBo--

= pofo~Pf_o+ -o--kT/q-~E+ -~-o~qev × Lo Bo 015 VoBo + cToEo 0t x t ] + -C-~o ([XTP(E)]V + (d[vV)P) x t3

VoBo + 72-70 v

x

([~f3]p) + [~E]P + o ( ~ ) ,

where in O(E 5) only terms arising from (2.21) and (2.24) are included. This form of the non-dimensionalization was chosen in order to evaluate the relative importance of the various terms that occur in the equation retaining the stress tensor in generality. Simplifications of the stress tensor, i.e. the forms of the material functions, will be discussed in the next sections. On using (2.16)-(2.19) and (2.31) we see that all germs on the right-hand side of (2.37) appearing in the expression for the electromagnetic force except the term IV I~]P have to be neglected. We shall also retain the mechanical force term. Let us denote the terms in the squiggly brackets by R~-1, which are Reynolds number like quantities m. Therefore all terms arising from the stress tensor are to be kept in virtue of (2.30). Further, one easily computes that Po V0 L0 Eo2 To -

O(1) -- O ( 8 2)

if E0 "~ 102 statvolt cm -1

O(¢ -1) - O(~)

if E0 "~ 1

O ( s -1) -- O(~ "-3)

if E 0 ~ 10 - 2 s t a t v o l t c m -1 .

statvolt cm -1 ,

(2.38)

Hence the first term on the left-hand side of (2.37) has to be kept. The second term should be strictly speaking neglected if E0 ~ 102 statvolt cm -1, which is reasonable, because for such values of E0 the fluid nearly solidifies. However, since we are interested in the response for the full range of values of E6, especially when the electric field suddenly changes, we will retain it. With regard to the approximation of the energy equation, we first re-write (1.45) using (1.16), (1.3), (1.41)2, (1.54) and the definition of the specific heat 02¢

(2.39)

12Note, that these Reynolds number like quantities Ri are related to the corresponding Reynolds numbers Rei, defined as in (2.11) by R~-1 =

Re-[l~o~-a~o.

1.3. L I N E A R MODELS FOR THE S T R E S S T E N S O R T

15

as

2 02¢ D) T . D + p20¢'~p~r D + p r , 0 - ~ p tr =

c v p O - k A O - v ( P-1 - 5 ~ ' ~02¢ -P

(2.40)

where cv and ¢ are functions of p, 0, e and T = T(p, 0, D, e). Similar as in (2.21) and (2.24) we assume

¢(A 0, ~) = ¢(p, e, E) + O(~5),

(2.41)

which together with (1.41)2, (2.21) and (1.54) implies

05 f'(A 0, e) = -~-~ (p, ~, E) + O(~),

(2.42)

~(A0,~) =P :05 ~ ( p,0,E ) + O(~). Now we neglect only terms coming from (2.21), (2.24) and (2.42) in the non-dimensionalized energy equations (2.40), since we have no evidence how the derivatives of the free potential ¢ behave. This leads to 0V

0¢ tr D) -- (T + ¢ I ) - D + pr.

In the case of the Clausius-Duhem inequality we proceed similarly. Thus we have approximated the thermo-mechanical part of the system (1.43)-(1.52) by ~ + pdiv v -- 0,

(2.43)

pit - div T = pf + [V E ] P ,

(2.44)

0 P . F,, -- -0~b cvpO - kA9 + 0(--~ ~ tr D) = (T + ¢I) • D + pr , (T + ¢ I ) . D + k ~ ~-~ > 0,

(2.45) (2.46)

where P, cv and ¢ are functions of p, 9, E and T = T(p, 0, D, E). In the next sections, we shall discuss various simplified structures for the stress T with a view towards obtaining a simplified model that can capture the behaviour exhibited by electrorheological fluids.

1.3

Linear Models

for t h e S t r e s s T e n s o r T

Even a cursory glance at (2.22), (2.23), the representation for the stress 13 T, reveals that it has little practical utility or applicability in virtue of its generality, with as many as six material functions that can depend in an arbitrary manner on the invariants 13 (2.23). It would be futile to experimentally determine all these material functions and thus we are left with the task of trying to simplify the expression for 130f course, we have to replace e by E.

16

1. MODELING OF ELECTRORHEOLOGICAL FLUIDS

the stress without forsaking the possibility of obtaining a model that can reflect the behaviour observed for electrorheological fluids. This and the following section is devoted to a discussion of special constitutive models with a view towards developing a theoretical framework that is amenable to mathematical analysis. In this section we assume that the stress tensor is linear in D and has quadratic growth in E, and then we obtain restrictions on the form of T, which are posed by the Clausius-Duhem inequality in the case of a compressible, an incompressible and a mechanically incompressible but electrically compressible fluid. In the following section, we discuss the case, when the stress power satisfies a growth condition and the material is incompressible, which is analogous to shear dependent viscous fluids. From the approximation described in the previous section we know that the stress tensor depends on p, 0, D, E and therefore has the form (Spencer [118]) T = a l I + a~E ® E + a3D + a4D 2 + a s ( D E ® E + E ® DE) + a6(D2E ® E + E @ D2E),

(3.1)

where ai, i -- 1 , . . . , 6 are functions of p , 0 , [E[ 2 , t r D , t r D 2 , t r D 3,tr ( D E @ E ) , t r ( D 2 E ® E ) .

(3.2)

Assuming now that T is linear in D and has quadratic growth in E, we observe ogI

:

O2 :

all + a12 tr D + al3]EI 2 -{- a14[E[ 2 tr D + 015 tr (DE ® E ) , O21 ÷ 022

tr D ,

o 3 = o31 + o3 1EI 2 , 04=0

(3.3)

,

0 5 = O~51 06

=0,

where oij are functions of p and 0 only. The Clausius-Duhem inequality (2.46) reads (T + ¢ I ) . D + k ~

> 0,

(3.4)

where the thermodynamic pressure ¢ is given by

20¢Co O,}E2) ¢(p,O, IE{ 2)=p-~p,,_, I •

(3.5)

Now, holding the temperature constant we obtain from (3.1), (3.3) and (3.4) (O~11÷ 013{E{2 ÷ ¢) tr D + (012 ÷ O14{E{2)] tr D{ 2 + (02, + (022 + 0,5) tr D) tr ( D E @ E) ÷

(O31÷

a32{E{2)iD{ 2 ÷

(3.6)

2o51]DE[ 2 >_ 0.

This inequality has to hold for all admissible D and E. Due to the presence of external source terms in the thermo-mechanical balance equations one sees that the

1.3. L I N E A R M O D E L S F O R T H E S T R E S S T E N S O R T

17

fields 9(t, x) -- 9(t), E(t, x) = E(t), and v(t, x) with div v(t, x) = a(t), where a(t) is an arbitrary function of time, together with an appropriate density p(t, x) = p(t) and magnetic induction B(t, x) are solutions of (2.32)-(2.35) and (2.43)-(2.45). Therefore all D and E are admissible and by specifying and rescaling 14 their values we obtain restrictions on aij. In preparation for this it is useful to establish relations between the quantities appearing in (3.6) L e m m a 3.7. Let D E ~ 3

, and E E R a . Then it holds

(3.8)

[tr D[ 2 < 3[D[ 2. Moreover, if tr D = 0 we obtain 2 2 [DE[ 2 _< ~[D[ [E[ 2 ,

I tr ( D E @ E)I <_ ~

(3.9)

IDIIEI2 .

(3.10)

PROOF : Since D is symmetric it is always diagonalizable and (3.8) follows immediately by seeking the maximum of • (tr D) 2 -- (dll + d22 + d aa) 2

(3.11)

under the constraint ]D[ 2 = 1. The maximum is 3 and it is attained for D = I and (3.8) follows. All quantities in (3.9) are invariant under rotations and therefore we may chose such a basis that

c E=

0 0

,

D=

d

e c b d e - ( a ÷ b)

) .

(3.12)

Therefore we have to seek the maximum of IDE[ 2 = a 2 ÷ c2 + d ~

(3.13)

under the constraint ]D[ 2 -- 1. A straightforward computation shows that the maximum is 2, which is attained for a-

2

b-

1

c - - d = e - - 0,

(3.14)

and (3.9) follows. Note, that tr ( D E ® E) = D E . E and (3.10) follows along the same line of arguments as (3.9). The maximum is attained again for (3.12), (3.14). • Now we shall discuss the consequences of the Clausius-Duhem inequality in the case of a compressible, an incompressible and a mechanically incompressible but electrically compressible electrorheological fluid. 14A similar rescaling argument was used in a completely different context by Ne~as, Silhav:~ [94], R~ieka [111].

18 1.3.1

1.MODELING OF ELECTRORHEOLOGICAL FLUIDS Compressible

Electrorheological

Fluids

Let us first discuss the case of a compressible electrorheological fluid and fix p and in the coefficients cqj and in the thermodynamic pressure, which will be denoted ¢(]E{2). Setting E = 0 in (3.6) we have ( a n + ¢(0)) tr D + ~12{tr D{2 + O~31[D{2 > 0,

(3.15)

and if we now replace D by 7D, multiply (3.15) by 7 -1 and let 7 --+ 0 we obtain ~11 = - ¢ ( 0 ) ,

(3.16)

and the remaining part of (3.15) immediately gives using (3.8)

O~31> 0,

30~12+ ~31 > 0,

(3.17)

which are the usual restrictions on the viscosities for a compressible viscous fluid. Further, rescaling (3.6) through D --~ "yD, multiplying by 7 -1 and letting 7 -+ 0 we get ( - ¢(0) +~la{EI 2 + ¢({E{2)) tr D + ~ 2 1 t r ( D E @ E ) > 0, which immediately leads to c~21 = 0,

(3.18)

by changing the sign of D, and choosing tr D = 0. Moreover we note that •({E{ 2) - ¢(0) - 13 =

{E{2

0¢(0) -+ 0{EI-----V

as [E{2 --->0, and thus we deduce the following formula for ¢({E{~) ¢~([E[ 2) = -Ogll -- Ogl3lE[ 2 ~- ¢(0) ol-{E{ 2 0 ¢ ( 0 ) O{E{ '

where we suppressed the dependence on p, 8. Next, choosing D in (3.6) such that tr D = 0 we obtain (~31 + ~32{E{2)ID{2 + 2~51[DE{ 2 -> 0. Setting D E = 0, rescaling E -+ 7E, multiplying by ~/-2, letting 7 --+ cc then yields (3.19)

~32 ---~0,

and (rescaling E --+ 7E, multiplying by 3,-2, letting 7 --+ oo) ~a=IEI=IDI2 + 2as~]DEI 2 > 0. The last inequality implies that (on choosing E, D as in ~32 + ~ o~sl > O.

(3.12),

(3.14)) (3.20)

1.3. L I N E A R MODELS FOR THE STRESS TENSOR T

19

Setting D E = 0 reduces (3.6) to (~12 + 0/141EI2)1tr DI 2 + (0/31 + ~321EI2)IDI 2 _> 0, which gives (on rescaling E --+ ~'E, multiplying by 7 -2, letting 7 --> oo and using (3.8)) 30/14 + 0/32 > 0.

(3.21)

Next, we decompose D as D

=

½(tr D ) I + G ,

tr

G

=

0,

(3.22)

where now G and tr D may be chosen independently. Inequality (3.6) can be rewritten as (ef. (3.16), (3.18)) W. D = i tr DI2{0/12 + ~a31 + IEI2(0/14+ ~1 0/32 + 1 0 / 2 2 + 1~ 0/15 + 62 0/51)} 1 + tr D tr G E @ E(0/22 + ~ls + ~ a51)

(3.23)

+ (0/31 + 0/321EI2) IGI 2 + 2-511GEI 2 >0. On rescaling E --+ 0'E, multiplying by 7 -2 and letting ~ -+ oo we notice 1 0/ 0 _< (tr D)21EI2(0/14 + i( 32 + 0/22 + 0/ls) + 62 0/51) + t r D t r G E @ E(0/22 + oq5 q- ~4 0/51) q- 0/321GI21EI 2 q- 2 ~ I l G E I 2.

(3.24)

Choosing now G = 0 provides 10~ ~2 + 0/22 + 0/~) + ~ 1 ~14 + ~(

> o.

(3.25)

The right-hand side of (3.24) is a polynomial of second order in tr D and its nonnegativity is equivalent to the condition I0/22 + ~t5 + ~4 ~5112 1E" GEl 2 (3.26) 1 0/ -< 4]El2(0/14 + ~( 32 + 0/22 + c~15) + ~ 0/sl)(0/3~_IEI21GI2 + 2c~51]GEl2) • Choosing now E and G as in (3.12), (3.14) we obtain , _< 1~22 + 0/~ + ~0/511

~( 0 / ~ + 0/22 + ~ )

+

(3.27)

Conversely, starting with (3.27) we can recover (3.26). This can easily be seen by multiplying the square of (3.27) by IGEI21EI 2, using (3.9), IGE. E[ 2 < ]GEI2tEI 2 and (3.25), (3.20). But (3.26) is equivalent to (3.24). Adding to (3.24) (cf. (3.17)) 0 _< (tr D)2(0/12 -{- ]1 0/31) nu 0/311(,~[2 we restore (3.23), which is equivalent to the Clausius-Duhem inequality (3.6). Thus we have shown

1.MODELING OF ELECTRORHEOLOGICAL FLUIDS

20

L e m m a 3.28. The stress tensor T for a compressible electrorheological fluid given by (3.1) and (3.3) satisfies the Clausius-Duhem inequality if and only if T is of the form T = ( - ¢(IEI 2) + (0/12 + 0/1alE] 2) tr D + 0/15 tr ( D E ® E ) ) I + oL22(tr D ) E ® E + (0/31 + 0/32lEl2) D

(3.29)

+ 0/51( D E ® E + E ® D E ) ,

where the coeJ~cients and the thermodynamic pressure have to satisfy 15 0~31 _> 0~ 30/12 -I- 0/31 >---0 ,

0/1, +

30/14 "~- 0132 ~__ 0 ,

+

I0/2~ + 0/lo + ~0/5~1 _< v ~

O~32 _ 0~

+

Incompressible

+ 2 51) _> o,

cq4 + 5(0/15 + 0/~2 + 0/a~ + ~c~5~)

¢ ( t E I ~) = - ~ 1 1

1.3.2

0/32 *~- 4 0/51 ~-- 0 ,

(3.30)

a2 + ~c~51,

- ~31E?.

Electrorheological

Fluids

An incompressible fluid is only capable of isochoric motions. Such an internal constraint can be expressed by div v = tr D = 0.

(3.31)

Thus, in the representation (3.1) for the stress tensor T we have 0/1 = - ¢ , where ¢ is the indeterminate part of the stress due to the constraint (3.31). Assuming again that T is linear in D and has quadratic growth in E we obtain 0/1 = - - ¢ 0/2 = 0~21 0/3 = 0/31 + 0/321E[ 2 ,

(3.32)

0~4 = 0 ~ 0/5 = OL51

0/6:0~ where 0/ij are functions of ~ only. situation reads

The Clausius-Duhem inequality (2.46) in this

0/21 tr ( D E ® E) + (0/31 + 0/321EI2)IDI 2 + 2a51[DEI 2 ~ 0. Proceeding similarly as in the case of compressible fluids we can show 15Note, that all coefficients aij and ¢(IEI 2) are functions of 0 and p.

(3.33)

1.3. LINEAR MODELS FOR THE STRESS TENSOR T

21

L e m m a 3.34. The stress tensor T for an incompressible electrorheological fluid given by (3.1) and (3.32) satisfies the Clausius-Duhem inequality if and only i / T is of the

form T=

- ¢ I + (c~31 + a321EI2)D + a51(DE ® E + E ® D E ) ,

(3.35)

where the eoe~cients, which are functions of 8, satisfy O~31 ~ 0,

1.3.3

4 0/32 Jr ~O~51 ~_- 0.

a32 ~ 0,

(3.36)

Mechanically Incompressible but Electrically Compressible Electrorheological Fluids

Preliminary experiments (Vel, Rajagopal and Yalamanchili [125]) seem to indicate that there are electrorheological fluids which undergo only isochoric motions in processes where E = const., while they are compressible if the electric field E changes. This situation is similar to that in a viscous fluid which can undergo only isochoric motions if the temperature is constant, while it can sustain motions that are not isochoric due to changes in the temperature, which is at the heart of the celebrated Oberbeck-Boussinesq approximation, which was recently re-examined in Rajagopal, R~i~ka, Srinivasa [106]. We here will follow the approach outlined in [106] for treating internal constraints. The situation described above of a fluid that can undergo only isochoric motions if E = const., while being capable of motions that are not isochoric as E changes, is mathematically expressed by 16 det F = / ( I E I 2 ) ,

(3.37)

where F is the deformation gradient. Computing the total time derivative of (3.37) implies div v = tr D =

~(IEI2) dlEf2,

(3.38)

where

/'(IEI2) Z(IE] 2) = /([El2) •

(3.39)

From (3.38) and (2.43) it follows that = - tr D = - ~ d l E ] 2 , a~

P

(3.40)

i.e. the density is completely determined by the electric field E. To ensure that the constraint (3.37) is satisfied, we shall decompose the stress tensor as T = -¢I+

T,

16Note, that det F is a measure for the change of the volume.

(3.41)

22

1.MODELING OF ELECTRORHEOLOGICAL FLUIDS

where ¢=-½trT,

tr'r=0.

(3.42)

¢ is usually called the mechanical pressure. Inserting (3.42) into (2.44) we see (3.43)

pv - div T + V ¢ = pf + [V E ] P . Taking the divergence of (3.43) we obtain an equation for ¢, which reads --A¢ -t- -1V p. V ¢ = - p d i v f - div (IV E]P) P + 1V d d 2) p p • (IV E]P + div T) + p~(,5'~IEt

(3.44)

+RL.L T-div (divT). We see from the above equation that no constitutive relation for ¢ is required, but once the electrical field E is known we can compute p from (3.40) and, v and ¢ are determined through (2.44) and (3.44). From our treatment of the constraint (3.37) it follows that we have to re-examine the consequences of the Clausius-Duhem inequality (1.17). As pointed out in Section 2 the appropriate form of (1.17) is -p(¢+~?0) +T.D+k~

-~-IB.P

_> 0,

(3.45)

where ¢, 7], T and P are functions of E, D and 0. By (3.40) and arguments similar to that, which led to (1.41), (1.42) we deduce that

v=

0¢ 00'

0¢ 0---D=0' 0¢

P = -2DEE - p~-~, and (3.45) reduces to D + kL:~

> 0.

(3.46)

Assuming now that T is linear in D and has quadratic growth in E we have the representation (3.1) for T with c~i satisfying (3.3), where (~q are functions of 0 only, because p is determined through E. Holding the temperature 0 fixed we get from (3.46) inequality (3.6), where ¢ has to be omitted. Replacing now E by 9'E and letting 9' --+ 0 we obtain a n tr D +

tr

DI

÷ ~alIDI 2 _~ 0,

(3.47)

and if we now re-scale D by 7D, multiply by 7 -1 and let 7 -+ 0 we observe ~ n = 0,

(3.48)

1.3. LINEAR MODELS FOR THE STRESS TENSOR T

23

and the remaining part of (3.47) yields again (3.17). Rescaling (3.6) through D -+ "),D, multiplying by 7 -1 and letting "y -+ 0 we obtain 01131EI2 tr D + ot21tr (DE @ E) > 0,

which immediately gives 0113 = 0121: 0,

(3.49)

by changing the sign of D, and choosing tr D = 0. Further we proceed as in the compressible case and we arrive at (3.29) and (3.30)1-4, where ¢(IEI 2) has to be replaced by - ¢ . Using (3.42)2 we get 0 : (30112+ 0131)tr D + (30114 -b 0122 -[- 0132)1EI2tr D + (20151+ 30115)tr (DE ® E),

(3.50)

from which we easily deduce 0115 ~--"--32-0151

0112 = --10/31,

1 0114 = --~(0122 "[- 0132)"

(3.51)

Furthermore, (3.30)7 implies 2 0122 = --~0151 •

(3.52)

The restrictions (3.30) together with (3.52), (3.51) can be re-written as 17 0131 ---~0 ,

a15

= - ~ a2 5 1 ,

a12

0132 -> 0,

= - ~ a13 1 ,

a51 _> 0. a14 = - ~

101~2 +

(3.53) ~a51.

(3.54)

Conversely, using (3.54) and the splitting (3.22) we easily see that ~?, D -- a31[G[ + a32[E[2[G[ 2 + 2a511GE[ 2 , which is nonnegative by (3.53). Therefore we have proved L e m m a 3.55. The stress tensor T for a mechanically incompressible but electrically compressible electrorheologieal fluid given by (3.1) and (3.3) satisfies the ClausiusDuhem inequality if and only if T is of the form 1 T = - ¢ I - ~(a31 + (0132 -- ~a51)[E[2) (tr D ) I

- 32-a51tr (DE @ E) I - ~a51(tr D)E ® E

(3.56)

+ (a31 + a321EI2)D + a51 (DE ® E + E ® D E ) , where a~j = a~j(O) satisfy 0131 ---~0 ,

0132

-> 0,

a51 >_ 0.

(3.57)

17Though the same symbols a 0 and ¢ are used for the material functions, they are different in each of the cases, i.e. the compressible, the incompressible and the mechanically incompressible but electrically compressible. Thus, as a consequence, we cannot directly compare the inequalities established for these material functions.

24

1.4

1.MODELING OF E L E C T R O R H E O L O G I C A L FLUIDS

Incompressible Electrorheological Fluids with Shear Dependent Viscosities

Now we discuss in some detail how to obtain a model of an incompressible electrorheological fluid with shear dependent viscosities, which could be thought of as an analogue of simple viscous fluids with shear dependent viscosities. Let us therefore consider the predictions of the general stress tensor given by (3.1) (with (~i replaced by - ¢ , due to the incompressibility of the fluid) for viscometric flows with constant temperature. It is known t h a t viscometric flows are locally a simple shear flow (cf. Huilgol [45]) . • = const.

(4.1)

E = E l e i + E2e2 + E3e3.

(4.2)

V =

/~x2el ,

with an electric field given by

Then the components of the extra stress S = T + ¢ I are found to be 016 2 r ~ 2 Sii : ol2E 2 + a4 2 + a5~;EiE2 + "~-~ mi

4

S22 = a2E 2 +

~6

2r~2

4

S33 = ~ E ~ (4.3)

Si2 = a2EiE2

S13 = a2E1E3 -4- -o~~ E 2 E 3 + 4tc2 Ei E3 a5 ~E1 E3 -4- _~ tc2E2 E3 , ,923 = a2E2E3 -4- -~ where ai are functions of the invariants

IEI~,5~ x 2, ~EiE2, ~ ( E 2 + E~).

(4.4)

From (4.3) it follows t h a t the presence of the electric field induces shear stress components not only in the plane of flow, x3 -- const., b u t also in the xl - xa and x2 - x3 planes. The shear stress components S13 and $23 are induced by the electric field component E3, which also contributes to the normal stress T33 • All other components of the e x t r a stress S depend on Ea only through the dependence of the a i ' s on ]El 2. Therefore we can conclude t h a t though the stress tensor is an isotropic function of its arguments, the shear stresses S n , S13 and $23 depend not only on the magnitude IEI 2 of the electric field, but also on the direction of i t ) s In this sense one can say t h a t the fluid possesses a preferred direction, t h a t depends on the applied electric field. For a purely simple viscous fluid the stress in a viscometric flow is determined 19 by the lSThis feature is to be expected from the understanding of the underlying microstructural mechanism, which causes the properties of the electrorheological fluid. 19In our situation here we need 5 viscometric functions to determine the stress in a viscometric flow, since we have non trivial shear stresses not only in the plane of flow.

1.4. SHEAR DEPENDENT ELECTRORHEOLOGICAL FLUIDS

25

shear stress function T ~ S12 and the normal stress differences N1 - - S l l - $22 and N2 = $22 - $33. In our situation these quantities are T = V(a, E) = (a2 +

a6-~)E, E2 + (a3 + as(E~ + E~)) g2

N1 = Y l ( a , E ) = (a2 + N2 = N2(a, E)

(4.5)

a~--~)(E~ -E~)

1,52 /~2 = (a2 + a6-~)E~ - a2E~ + a4"-~ + asE1E2a,

where ai, i = 1 , . . . , 6 are functions of the invariants (4.4). However, in the absence of an electric field, i. e. E = 0, (4.5) reduces to

~2 N 1 = 0,

N 2 ~-- o~4~- .

This behaviour is not supported by experimental d a t a (see Huigol [45] and the discussion in M~lek, Rajagopal, Rfi~i~ka [74] and M~lek, Ne~as, Rokyta, Rfi~iSka [70]). 20 For t h a t we shall assume, even in the presence of an electric field, t h a t a4 -- 0 .

(4.6)

Note, t h a t in the presence of an electric field we have no experimental guidance how the normal stress differences of a real electrorheological fluid behave. Moreover, as pointed out before the fluid has a preferred direction and thus the experience for purely viscous fluids can not simply be carried over to our situation. Therefore we shall not make any further assumptions for the material functions on the basis of the form of the normal stress differences. Previous m a t h e m a t i c a l investigations of shear dependent viscous fluids (see M~lek, Rajagopal, Rfi~i~ka [74], M~lek, Ne~as, Rokyta, Rfi~i~ka [70]) suggest t h a t terms involving D 2 can be treated from the m a t h e m a t i c a l point of view as a p e r t u r b a t i o n of the remaining terms (under some smallness assumptions). Therefore we shall also assume a6 = 0

(4.7)

in order to keep the model simple, while still retaining the ability to explain the observed phenomena. After a better understanding of this model has been achieved we can also include the terms involving a4 and as. Note, that under the assumptions (4.6) and (4.7) the model still predicts both normal stress differences N1, N2 and has a shear stress function T, which depends on the direction of the electric field. For example, if the electric field is in the plane of 2°It is precisely the absence of such experimental data that led to the demise of the Reiner-Rivlin model, which did not predict in the case of non-Newtonian fluids (in the absence of electric fields) both normal stress differences in simple shear flow. However, there are several fluids that exhibit neither of the normal stress differences but whose viscosities are shear dependent and are called generalized Newtonian fluids. The power-law fluids are a subclass of them.

26

1 . M O D E L I N G O F E L E C T R O R H E O L O G I C A L FLUIDS

flow, i.e. E3 = 0, we observe (note, that S13 = $23 = Sz3 = 0 ) T = ot2E1E2 + (ot3 + a~[E]2)-~,

NI = a2(E~ - E2),

(4.8)

N2 = a2Eg + asEiE2e;.

Concerning the remaining material functions a2, a3 and a5 we assume that the material shows the following behaviour: in the absence and presence of an electric field the behaviour is that of a generalized Newtonian fluid with power p. The power p can depend on the magnitude of the electric field (cf. Halsey, Martin, Adolf [42], Bayer [116]) and all terms have the same growth behaviour. Moreover we restrict ourselves to the case when as, a3 and a5 are functions of the invariants 0, ID] z and IE[ 2 only, because we are merely interested in the growth of the material functions a2, or3 and as. The growth of the other invariants can be obtained as a combination of these considered. Therefore we shall assume that O~2 ~- O/21(1 @

ID[~)~

+ a2o,

c~3 = (~31 + ~331EI2)(1 + IDI2) ~ ~ = ~1(1 + ID?)~

+ ~3o + ~3zlEI ~ ,

(4.9)

+ ~o,

where aij are functions of 0, and in general

p = p(lE[2). We further assume that p : [0, oo) --+ (1, c~) is a smooth function of lim p(IEI 2) = p0, lEt2~0

lim p(IEI =) = poo, IEl2~oo

(4.10)

IEI 2 and that (4.11)

where 1 < Poo _< p(IEI 2) <_ p0. Alternatively we can replace (1 +

IDI2)a/2 in

(4.t2)

(4.9) by

ID[ ~ ,

(4.13)

where f~ = p - 1 or p - 2. The models (4.9) and (4.13) can exhibit a drastically different behaviour. This can be illustrated if we consider the generalized viscosity #(~, E) defined through

,(~, E) = ~(~' E) for these models. The limits #o = lim #(~, E) to-+0

(4.14)

1.4. SHEAR D E P E N D E N T E L E C T R O R H E O L O G I C A L FLUIDS

27

and ~

= lim#(a,E),

respectively, are usually called zero shear viscosity respectively infinite shear viscosity. Let us assume for a moment p E (1, 2). One easily checks for the model (4.9) (E = Ezel + E2e2) 21 that 1 10~ 32 "~- O~33"~- O~50"~- 0~51)[E[ 2 ]-tO = ~(O~30 "~- 0L31) -~- ~( _~ 1 1 ~oo ~/~30 + ~(0L32 + OLS0)[EI2 ,

(4.15)

while for the model (4.13) we get #0 ----o0, #oo

=1~a30

+

½(a32

(4.16) + aS0)[E[ 2 •

On the other hand if p E (2, oo) we have for the model (4.9) =I

10L

(4.17)

~oo = O O

and for the model (4.13) 22 1

~0 = ~C~30 "~ ½(O132 "~ o~50)lEI 2 ,

(4.18)

~oo = oo. Relations (4.15)-(4.18) show how different material properties are included in the above models (4.9) and (4.13). For a more detailed discussion for purely generalized Newtonian fluids we refer the reader to M~lek, NeSas, Rokyta, Rfi~iSka [70] and M~lek, Rajagopal, Rfi~i~ka [74]. From the mathematical point of view the case when #oo = 0 for p E (1, 2) is more challenging and therefore we will assume a3o = aa2 = c~s0 = 0.

(4.19)

It is worth noticing that under the assumption (4.19) the form of the viscometric functions (4.8) indicates that it might be possible to design experiments, with the possibility of changing the direction of the electric field, which allow to determine the constants in (4.9) and (4.13), respectively, for the material functions a2, a3 and as. Before we discuss the consequences of the Clausius-Duhem inequality and related problems, we shall illustrate the features of the above described models by solving the boundary value problem of a flow between infinite parallel plates, which are a distance 21From the Clausius-Duhem inequality (3.4) it follows that a2o = -a21 (cf. (4.33)), which is already used in the computation of the limits. 22From the Clausius-Duhem inequality (3.4) it follows that a2o = 0, which is already used in the computation of the limits.

28

1.MODELING OF E L E C T R O R H E O L O G I C A L FLUIDS

2h apart. We assume that there is no external body force and that the electric field and the velocity are given by E =

E2 0

,

v =

.

(4.20)

0

We will solve this problem for the model (3.1), (4.6), (4.7), (4.13) 23, (4.19) and (3.31). Firstly, we observe that the extra stress S is a function of x2 only. Thus the equations of motion read 02S12 + 01¢ = O, 02S22 + 02¢ = 0,

(4.21)

0a¢ = 0. From the last equation follows that ¢ = ¢(xl, x2) and from the first two equations we conclude

a~¢ = 0,

ala2¢ = 0,

and therefore ¢ is an affine function of xl given by ¢(xl, x2) -- - A x l + k(x2).

(4.22)

The constant A can be interpreted as the pressure drop, which maintains the flow. Inserting this into (4.21) and integrating we obtain $12 = - A x 2 + Co,

Co = const.

(4.23)

This and the explicit form of the extra stress component $12 (cf. (4.8)) leads to an ordinary differential equation for f(x2), namely - d x 2 + Co = S12(f'(x2))

(4.24)

to which we add no-slip boundary conditions f(:t:h) = O.

(4.25)

Inserting (4.20) into (4.24) and using (4.8) we obtain

1 (~21vf~E1E 2 +

$12(:') = 2 - ~ =: 7o(f') p-1

~al

+ (-aa + ~51)IEI 2) (:')P-~

(4.26)

(aaa+ aal)IEl2)(_f,),-1

(4.27)

if f ' > 0 and $12(f') = ~

(-a21V/2EiE2+

aal +

-----: --~1 ( _ _ / ' ) p - 1 2aWe have chosen this model since the problem is solvable explicitly. For the flow considered there is no essential difference to the model (4.9), but the resulting ordinary differential equation must be solved numerically.

1.4. S H E A R

DEPENDENT

ELECTRORHEOLOGICAL

29

FLUIDS

if f~ _< 0. Note, that both constants 70 and G'I are positive due to the restrictions on c~ij from the Clausius-Duhem inequality. Furthermore, one expects the flow to take place in direction of el and thus we get f r ( - h ) > O, f~(h) < 0. From (4.24) it is clear that there is exactly one point x* = ~ , where fl __ 0 and where one has to switch from (4.26) to (4.27). One easily checks that

(4.28)

/(x2) --

t~s

,

(h-

hT), -1

h],

where 7~/' - 71/'

(4.29)

is a solution of (4.24), (4.25). The constant Co can be computed as Co = A h 7 • The flow profile (4.28) clearly shows how the solution of the system (4.21) depends not only on the magnitude of the electric field but also on the direction of it. From (4.28) one can compute all components of the extra stress S and the pressure ¢. We see that the pressure ¢ is given by

¢(xl, x2) = - A x l + &2(z2) + c3, where

$22(x2)

A2~(x2_h~)(v~o121E2

20~51ElS2) x2 E [h~,h],

while the effective force in direction x2 is given by T22(xl) = A x l - ca.

(4.30)

Note, that this force does not depend neither on the electric field nor on the value of p and in fact is the same as for a Navier-Stokes fluid. To illustrate the described solution we will plot the velocity profile for the following situations: The electric field is given as

(°) 0

the material constants are chosen as c~31 = 1, c~a3 = 1, C~s~ = 3/4, c~51 = v~, the distance between the plates is h = 1. In Figure 1 and 2 we have chosen p = 2 and the pressure drop A = 10. Figure 1 shows the effect of an increasing magnitude of the electric field. Figure 2 illustrates the effect of different directions of the electric field with constant magnitude.

30

1. MODELING OF ELECTRORHEOLOGICAL FLUIDS

i

',

I

\

;

)

Fig. 1: p - - 2, ~ - - 0 , 1 / 2 , 4 / 3

I

Fig. 2: p - - - 2 , ( ~ = 0 , ~ ,

I

I

/I

|

a--0,

~2,

J /

Fig. 3: p = 1.5, fl = 0 , 1 / 2 , 4 / 3

"/~2

7 j

Fig. 4 : p = l . 5 ,

One clearly sees that the velocity profile is asymmetric if E is not perpendicular to the plates and that the maximal velocity depends on the value of a. Note, that this effect is maximal for c~ = v ~ / 2 . In Figure 3 and 4 the same situation is depicted for p -- 3/2, the pressure drop A is normalized such that the flow rate is the same for c~ -- 1, ~ = 4/3 as in the case p -- 2. Between Figure 2 and 4 there is no scaling involved, Figure 3 is scaled 1/6 times with respect to Figure 1. Figure 1 is scaled 1/5 times with respect to Figure 2. Summarizing we can say that the above pictures show that an increasing electric field increases the viscosity and thus the velocity profile becomes more flat. This effect is most significant if E = (0, E2, 0) T, in all other cases, i.e. EzE2 ~ 0, the maximal velocity is larger than in the case EzE2 = 0 and the velocity profile becomes asymmetric. Now let us discuss in some detail the consequences of the Clausius-Duhem inequality and related questions for the model (4.9). Here we restrict ourselves to this case since in Chapter 2 we will deal only with this model or the model (3.35) and refer the reader to Rajagopal, Rfi~i~ka [104], [105] for more details about the model (4.13) and related ones. Holding the temperature/~ fixed we obtain from (3.1), (4.6), (4.7),

1.4. S H E A R D E P E N D E N T E L E C T R O R H E O L O G I C A L FLUIDS

31

(3.31) and (3.4)

(~2o + ~21(1 + I D { ~ ) ~ ) E • D E + (~31 + ~3)EI2)(1 + ]D]2)a-~IDI 2

(4.31)

+ 2a51(1 + ID]2)P-~-~IDEI2 _> 0. Setting E = 0 inequality (4.31) reduces to c~31 _> 0,

(4.32)

and rescaling D --+ 7D, multiplying by 7 -1, letting 7 -+ 0 and changing D --+ - D we obtain ~20 = -~21.

(4.33)

Changing in (4.31) D by - D and adding the results we eliminate the terms with ~20 and ~21. This inequality is multiplied by (1 + IDI2) -e~-~ and then we re-scale E --+ 7E, multiply by 7 -2 and let 7 --+ c~, which gives c~331EI2]DI2 + 2~511DEI 2 > 0.

(4.34)

Choosing D, E ¢ 0 such that D E = 0 we obtain (4.35)

O~33___~O,

and from (4.34) and Lemma3.7 we deduce (4.36)

O~33.~_40eSl ___~0.

Now we would like to derive some condition for ~21. Using (4.33) we can re-write (4.31) as (also using that D can be replaced by -D) I~2,11E. DEIID](I+ IDI2)P-~ p-2 - 1

(i + IDl2)

IDI

(4.37)

_< (Ol31 ÷ o~331E[2)[DI2 ÷ 2o~511DEI2 • Choosing D , E ~t 0 such that IE. D E I = IEI]DEI, and using (3.9) we get from (4.37) that 1~211(1 + ID]2) 2a'~ - 1 ~31 ~33 p-2 ' _< ~ + - - + 2 7 ~ 5 1 ( I + I D } 2) 2 IDI 7

(4.38)

must hold for all ]El, ]D I > 0 and all 7 6 [0, ~32-]. Note, that the left-hand side of (4.38) depends on ]D t and ]El 2 only, since p = p(]EI2), while the right-hand side depends on ~ and ]El 2.

1.MODELING OF ELECTRORHEOLOGICAL FLUIDS

32

One can show that the function

f(x) =

(I + x2) z~/2~- i

(1 + x 2 ) ~ x

(4.39)

is bounded on the interval [0, co]. More precisely it holds sup f ( x ) = ko(v),

(4.40)

where ko(p) = 1 ifp E (1,3] and ko(p) > 1 ifp > 3. Note, that ko(p) is an increasing function of p for p > 3. For p _> 3 the supremum in (4.40) is a maximum which is attained at x0, where x0 solves the equation

(4.41)

(p - i)x 2 + i = (i + x~)~-~.

For p E (1, 3] the supremum is approached for x --+ co. Taking now the supreinum of (4.38) with respect to ]DI and using (4.40), we see that 1 ( aai a3a ) la21] _< ko(p(iEt2) ) \tEI2~ / + --~/+ 2a~17

(4.42)

must hold for all [E l _> 0 , 7 E [0, X/'~-~]. Now we would like to find the infimum of the right-hand side of (4.42), but this is in general impossible, since we have no explicit formula for the solution x0 of (4.41) and the function p(IE[ 2) might behave very different for various fluids. But letting ]El 2 -+ co in (4.42) we arrive at [a211 _< ~

+ 2a517

)

,

(4.43)

which holds for all 7 E [0, V/~32-].An easy calculation shows that this implies { 2x/K~

ko(poo) la211

if a33 < ~4(](51 _

_

<-

(4.44)

-}- gO151)

if

_< 0933.

In other words, condition (4.44) ensures that k(poo)la21l[Dl[EI [DE[ _< aa3[D[21E[2 + 2asllDE{ 2

(4.45)

is satisfied for all E E R a and D E X -- {D E ~¢~3, tr D = 0}. Thus we have shown L e m m a 4.46. /f the stress tensor T given by (3.1), (4.6), (4.7), (4.9), (4.19) and (3.31) satisfies the Clausius-Duhem inequality, then T is of the form T = - ¢ I + a21 ((1 + [DI2) p-~ - I ) E ® E -[-- ((231 --[-OL33]E[2)(I + [D] 2) 2 D

(4.47)

+ as~(l + ID[2)~-~ (DE ® E + E ® D E ) ,

where p = p([E[ 2) and the coej~cients aij = aij(O) satisfy (4.32), (4.35), (4.36), and (4.44).

1.4. SHEAR DEPENDENT ELECTRORHEOLOGICAL FLUIDS

33

R e m a r k 4.48. As already indicated the condition (4.44) is in general not sufficient for the Clausius-Duhem inequality to hold. However if p does not depend on [El 2 or if po _> 3 the condition (4.44) is also sufficient. One easily checks that the relation { 2x/-5~

if

~33 _ < ~a51 4

if

~1'~11 < O~33"

ko(Po) In211 -<

(4.49) 3(O~33 -{- gOL51)

together with (4.32),

(4.35) and (4.36) are sufficient for the Clausius-Duhem inequality.

The Clausius-Duhem inequality is closely related to the coercivity of the operator induced by - d i v T. In fact, if we assume strict inequalities in (4.32), (4.35), (4.36) and (4.49), we get that there is a constant Co > 0, such that for all D E X, E E R a T(D, E). D _> C0(1 + IE]:)(1 + ]DI 2)

~

ID] 2 •

(4.50)

For the mathematical investigations in the next section it is important to know, under which conditions on the coefficients aij the operator induced by - d i v T is monotone. One easily checks that the conditions (4.32), (4.35), (4.36) and (4.49) with sharp inequalities are not sufficient for the monotonicity. It is well known (cf. Gajewski, GrSger, Zacharias [35], p. 64) that for differentiable operators the monotonicity is equivalent to the condition 0Tij (D, E) ODk~ BijBkl >_O,

VB ,D E X , VE E •3.

(4.51)

An easy calculation shows, for T given by (4.47), that

cOTij(D, E) ODk~ Bi~BkL

(B. D ) ( B E . E) = (1 + IDI2) 2a~ (o121(p - 1~ ' (1 +IDI2)l/2 2~ (B- D) 2 ~ + ( 31 + aaalEi2) (LBI2/+ (p _ ] 1 + ID]21 + 2

(IBE? + (p - 2)

(4.52)

(B. D ) ( B E . DE)~

Note, that the second term in the squiggly brackets is always non-negative, while the first and the third one can change their signs. Replacing D by - D implies that the term in (4.52) with a21 should be replaced by - 1~21J(p - 1 ) I ( i +DIjDj2) */2]BE"E I

(4.53)

The terms with a51 are much more involved, since a51 has no sign and moreover, it is complicated to find B, D, E, such that the infimum of IaEI 2 + (p - 2 ) ( l . D ) ( B E . DE) 1 + IDI 2

(4.54)

1.MODELING OF ELECTRORHEOLOGICAL FLUIDS

34

is attained. The situation becomes even more difficult if (4.53) is taken into account. Even for diagonal matrixes B, D and p = const, it is not possible to get handy necessary conditions on a2i and a51 for (4.51) to hold. For that, let us find sufficient conditions for (4.51). The term with a3i is handled easily. For all B, D E X, E E R 3

~3~(I + I D f ~ ) ~ (,al ~-+ (p - 2~, (~B ., D)2~

(4.55)

> o~3iT(pc~) (1 + JD[2)a-~IBI 2 -= Cl(1 + JDJ2) 2e-~JBI 2 is valid, where C1 is independent of B, D and E and 7(:o) = P - 1, i f p E (1, 2), whereas 7(P) = 1 if p >_ 2. For the remaining terms we proceed as follows. The terms with coefficients a2i, c~33 and asi in the squiggly brackets in (4.52) are bounded from below by (cf. (3.9))

a337(p)lEI21BI 2 + 2 a511BEI 2 (4.56)

- (I~I[(P - 1) ÷ 21/~- I ~ ( p - 2)1)IBEI IBIIEI. Introducing the following new variables x = rBIIEI,

y=

IBEI,

we can re-write (4.56) as

f(x, y) = c~33~/(p)x ~ + 2c~5~y 2 - (1~211(p - 1) + 2 V ~ 1~51(p - 2)l)xy,

(4.57)

where p = p(lEI2). We would like to find conditions on p and a2i ensuring that there exists a constant C2 independent of x, y such that

(4.58)

/ ( z , y) > c 2 z ~

/'Z for all x E [0, cx~) and y E [0, ~/~x]. Using a similar method as in the discussion of the Clausius-Duhem inequality we obtain that if 7(P) <-- 4a~l 3 ~33

IO~211(P -- 1) "-[-2(32- [O/51(P -- 2) J ~

V~23(~ (p) O~33_~..40l51)

if ~3 ~ 3 3

~

< 7(P),

--

must be satisfied for all p E ~ooo,p0]. The above condition looks very compact, but both sides of these inequalities and the conditions depend on p E ~o~, P0] and it must be ensured that t h e right-hand side is larger than the left-hand side. Let us fix a33 and a51 and regard the above conditions as an inequality of two functions depending on p. One easily sees that the left-hand side is a linear function in the intervals [po~, 2] and [2,p0], which is increasing in the latter one. The right-hand side is a concave function in the interval [Poo, 2] and constant in the interval [2, P0]- Therefore it is

1.4. SHEAR DEPENDENT ELECTRORHEOLOGICAL FLUIDS

35

enough to check the above conditions for p~¢ and P0 to ensure that they hold on the whole interval in-between. Let us denote 4 ~51

r ~--- - 3 ~33

and 1

~(p) = p_--~ (~(p) + r(1 + Ip - 21)), 1 ~ ( p ) = p_ - - = - / ( 2 J ; - ~ %(;) = p-~

- rip - 21),

(-y(~) + r(1 - I; - 21)).

Straightforward calculations lead to the following requirements on a21, Pc¢ and P0 to ensure that (4.58) holds: (1) if r • ( - 1 , O] 3 I~11 < V~3~33 min{71(p~), 9'1(po)},

1 - 3r --
(4.59)

1 r

(2) if r • (0,1) and r < 7(P~)

1~211< V/~ 33~3(po) 30/

r+l
(4.60)

1 r

(3) if r • (0, 1) and r • (7(Po), 1) 3

2r+l (r

x/~)

(4.61)


(4) if r • (0, 1) and r • [9'(poo), 7(P0)]

I0~211 <

2r+1(1

O~a3 min{72(poo),'Y3(Po)},

j~+q)

r

<poo < p o < 2 + - ,

(4.62)

1

r

(5) i f r 6 [1, cx~) 3

la211 < V/~ a33 min{72(p~),72(po)}, 2 r+l r

1

1


1+

.

(4.63)

1.MODELING OF ELECTRORHEOLOGICAL FLUIDS

36

In the picture below we have depicted the restrictions on the largest possible range of poo and p0 in dependence on r.

5

1

Figure 5 Therefore we have shown L e m m a 4.64. Let T be given by (4.47). Then the operator induced by - d i v T satisfies for all B, D E X the inequality

OT~j(D'E)BijBkl > (C1 + C2[E]2)(1 + ]D[2) ~ ODkl

[B[ 2

(4.65)

if (4.32), (4.35), (4.36) and (4.49) hold with sharp inequalities and if (4.59)-(4.63)

are satisfied. R e m a r k 4.66. One could obtain slightly less restrictive sufficient conditions for the inequality (4.65) if one works instead of (4.56) with the better lower bound • (B. D) ~ \

a33[E[2(iB[ 2 + (p - 2 ) ~ )

+ 2a511BE[ 2

IB-Di . . . , - (I~=IF(p - 1) + 2 V ~ 1~51(p - 2)1) (1 + iDI~)'/~ I~,'1 IEI. The resulting conditions are even less handy than (4.59)-(4.63) and moreover they are not sufficient in the case a51 < 0 to ensure that also the approximative stress tensor S A (cf. (3.3.13)) is monotone.

1.4. S H E A R D E P E N D E N T E L E C T R O R H E O L O G I C A L FLUIDS

37

The last constitutive relation we should discuss is that for the polarization P. For many materials it is reasonable to assume a linear relation between P and E and therefore we shall assume that P = xE(p, 8) E ,

(4.67)

where X E is the dielectric susceptibility. In the next chapter we are only concerned with processes for incompressible shear dependent electrorheological fluids. In this situation X E depends on ~ only.

2 M a t h e m a t i c a l Framework 2.1

Setting of the Problem and Introduction

In the following chapters we are mainly concerned with the existence theory for steady and unsteady flows of an incompressible electrorheological fluid with shear dependent viscosities. It follows from the treatment in Chapter 1 that the motion is governed by the system (1.2.32)-(1.2.36), (1.2.44), (1.3.31) and (1.2.45), where the stress tensor T is given by either (1.3.35) or (1.4.47) and the polarization P is fixed by (1.4.67). Here we shall consider the isothermal case only, i.e. the temperature 0 is constant. This assumption in fact means that we have at our disposal a heat source r, which holds the temperature constant (see below (1.5)). As we are dealing with an incompressible fluid this assumption also implies that all material functions a i j , X E are constant. Therefore we have to deal with the system 1 div E -- 0, curl E = 0,

(1.1)

0v p0-0-~ - div S + p0[Vv]v + V ¢ = P0f + x~[VE] E ,

(1.2)

div v = 0, div B = O, XE

curl B + - -

C

curl (v x E) -

1 "~ X E

aq.._~e+ div (qev) = 0 at S.D

+ pr = 0,

C

0E

(1.3)

~'t '

(1.4) (1.5)

where S = T + ¢I. The system (1.1)-(1.5) is separated, so we can first solve the quasi-static Maxwell's equations (1.1) for the electric field and then seek the velocity field by solving (1.2). Having at our disposal E and v we can solve (1.3), (1.4) and (1.5). Note that equation (1.5) has to be interpreted as an equation for the heat source r. It was already pointed out in Section 1.2 that the magnetic field B is of secondary 1Recall, that [Vv]v = vjOjv.

2.MATHEMATICAL FRAMEWORK

40

importance, which is reflected by the structure of the above system. Moreover, for this first mathematical investigation of electrorheological fluids we are mainly interested in the electric field E and the velocity field v, and not in the free charges qe and the heat source r. Therefore we shall only consider (1.1) and (1.2), to which appropriate initial and boundary conditions should be added. In particular, we will consider the quasi-static Maxwell's equations (1.1) with the boundary condition E.

n =

E0 • n

on 0i2,

(1.6)

where E0 is a given electric field and n is the outward normal; this boundary condition describes the contact between a dielectric and a conductor. The equation of motion (1.2) will be completed with Dirichlet boundary conditions v = 0

on 0f~ x (0, T ) ,

(1.7)

and an initial condition v(0) = v0

in •,

(1.8)

if the unsteady system is treated. Here and in the following, if not otherwise stated, f~ denotes a three-dimensional bounded smooth domain 2 and T > 0 a given length of the time interval, which will be denoted by I = (0, T). We would like to make some comments about the structure of the system (1.1), (1.2). The quasi-static Maxwell's equations (1.1) are widely studied in the literature (cf. the overview article Milani, Picard [83]) and well understood. In Section 2.3 we will give precise formulations and quotations of the for us relevant existence, uniqueness and regularity results. The situation is quite different for the system (1.2), when S = T + ¢ I is given by (1.4.47). The properties of the stress tensor have already been discussed in some detail in Section 1.4. There we have given necessary and sufficient conditions for the eoercivity and sufficient conditions for the monotonicity of the operator induced by - div S. Since the material function p depends in general on the magnitude of the electric field ]E[ 2 we have to deal with an elliptic or parabolic system of partial differential equations with non-standard growth conditions, i.e. (cf. (1.4.47), (1.4.50), (1.4.10)-(1.4.12)) S ( D , E ) . D > C0(1 + ]El2)(1 + IDI2) ~v-~2-2ID]2, IS(D,E)] _< C3(1 + ]DI2)a~-!]E] 2 ,

(1.9) (1.10)

where in general 1 < poo _< p0) The system is further complicated by the internal constraint (1.2)2 and the dependence of the elliptic operator on the modulus of the symmetric part of the velocity gradient. In recent years the interest in the study of elliptic equations and systems with nonstandard growth conditions has increased. Most investigations are concerned with the 2Here we restrict ourselves to this case in order to make the computations not more complicated, but it is clear that the methods can also be applied in the case of two-dimensional domains with appropriate changes. 3Of course in the case Pc¢ = Po we have standard growth conditions.

2.1. SETTING OF THE PROBLEM AND INTRODUCTION

41

regularity of minimizers of variational integrals and solutions of elliptic and parabolic equations and systems. Let us give a brief overview of some of the results. The first investigations of equations and scalar variational integrals with nonstandard growth conditions go back to Zhikov [135], Giaquinta [37] and Marcellini [76]. In Zhikov [135] the variational integral f ( 1 + IVul2) ~(x) dx

(1.11)

f~

is studied, and it is shown that non-regular minimizers do not exist if a is HSlder continuous. In Marcellini [76] and Giaquinta [37] anisotropic variational integrals of the type

f f(Vu) dx

(1.12)

f~

have been considered, where the convex function f satisfies for all ~ C R d ml~l p < f(~) < M(1 + I~1)q,

(1.13)

for M _> m > 0 and 1 < p < q. It is shown that there exist unbounded minimizers if the difference q - p is large enough in dependence on the dimension d of the domain ~t; e.g. Giaquinta [37] considers the case p = 2, q = 4 and d _> 6 (see also Marcellini I77], [78], Hong [84]). On the other hand there is a variety of papers dealing with conditions on p and q, which ensure regularity of minimizers. For example, in Marcellini [77], [78] and Hong [84] it is shown that minimizers of (1.12), (1.13) belonging to the space Wl~(ft ) are locally Lipschitz continuous if 2 _< p and d

P < q
(1.14)

and Moscariello, Nania [89] prove that minimizers of (1.12), (1.13) belonging to the space Wlo1~1 (f~) are bounded if d

l < p < - q < P d _ pMoreover they obtain that locally hounded minimizers of (1.12), (1.13) with f ( ( ) = f(l~l) are HSlder continuous without restrictions on p and q. In Boccardo, Marcellini, Sbordone ]131 and Fusco, Sbordone [321 it is shown that minimizers of (1.12) with a special anisotropic structure, i.e. f satisfies d

d

15/Iq, _< f(~) _< c(1 + E i=1

I~/1"),

(1.15)

i=1

are locally bounded if

l
1 1~ 1

(1.16)

2.MATHEMATICAL FRAMEWORK

42 These results have been extended to the case

f(x,u, Vu) dx, e.g. by Fusco, Sbordone [33], Mascolo, Papi [81], Chiad6 Plat, Coscia [99] and others. Similar results also hold for weak solutions u • Wllo~(ft) ML~oc(f~) of elliptic equations div a(x,u, Vu)=b(x,u, Vu),

(1.17)

where for almost all x E ~, and for all s E R, l~, 3, • R~

Oadx'~'~)A3j > m(1 ÷ I~l2) ~'A2 II

(1.18)

I~(x,s,g)l ~M(l+lgl~) ~ , is satisfied for positive constants m, M and exponents 1 < p < q. It is shown in 1,oo ~ 2,2 Marcellini I781, [79] that weak solutions belong to the space Wloc ( ) M Wloc (f~) if (1.14) and some structure conditions on a(x, u, Vu) and b(x, u, Vu) are satisfied (cfl Lieberman [67], where gradient estimates for smooth solutions are proved). Moreover, existence of weak solutions to (1.17), (1.18) in the case b(x) • I/(f~) N L~c(f~) d+2 and 2 < p <_ q < -F~' is shown in Marcellini [78[, while the case b(x) • L~(~) is treated in Boccardo, Gallouet, Marcellini [12], which to the knowledge of the author are the only papers dealing with the problem of existence of weak solutions to (1.17). Finally, Lieberman [67] obtains gradient estimates under some structure conditions also for the parabolic version of (1.17) (cf. Mkrtychyan [86], [87]). In the vectorial case only minimizer of variational integrals

f F(x, Vu) dx

(1.19)

and the corresponding Euler equation div A(x, Vu) = 0

(1.20)

are investigated. For example, in Leonetti [62] and Bhattacharya, Leonetti [10] it is shown via a new Poincar~ inequality that F(IVul) belongs to the Morrey space 1,A Lloc(ft). Leonetti [63] proves that weak solutions u e WI'~(f~) of (1.20), where A(x, Vu) satisfies some structure conditions and

OAij . (1.21) ]C_-~A~(x,!~)[ < M(1 + u~

2.2. FUNCTION SPACES

43

have higher differentiability properties, namely

(1 + IV.l )

e

1,2

(1.22)

and for p > 2 one gets u E Wl2'~(~). Note, that no restriction on the ratio q/p is imposed, but that also no higher integrability is proved. Choe [18] proves that locally bounded minimizers of (1.19) with F([Vu]), where F~j~z (l~[) satisfies an appropriate version of (1.21), belong to the space C~o~(~) if l
(1.23)

and that minimizers u E WI'P(~) are locally bounded if

l
1+

.

Coscia, Mingione [201 have proved C¢oca(~)-regularity for local minimizers of (1.19) provided that F(x, Vu) -- ]Vul p(x) under the only assumption that p is locally HSlder continuous. Acerbi, Fusco [4] show for anisotropic functionals satisfying an appropriate version of (1.15) partial regularity of minimizers, namely u E C°'a(~'/0), In - ~01 -- 0, under the condition (1.16). Bhattacharya, Leonetti [11] treat a very 2d special anisotropic structure and in the case p = 2, q < ~:-~, they obtain that u e W o¢

n

See also Leonetti [64], where the existence of second order derivatives for some anisotropic functionals is established. The above results are generalized in Leonetti [65]. To the knowledge of the author there are no results for elliptic systems with a righthand side depending on u, Vu or for parabolic systems with non-standard growth conditions. Our treatment of the system (1.2) is built on techniques developed in previous investigations of generalized Newtonian fluids, which had been initiated by NeOns [93[, and were further developed and extended in MMek, Ne5as, Rfi~iSka [71], [72], Bellout, Bloom, NeSas [9], M~lek, NeOns, Rokyta, Rfi~iSka [70] (cf. M~lek, Rajagopal, Rfi~.iSka [74], M~lek, Rf~i~ka, Th~ter [75]).

2.2

Function Spaces

Now we introduce some notation and useful function spaces for the treatment of the system (1.1), (1.2), (1.6)-(1.8). Let f~ C Rd, d ~ 2, be a bounded smooth domain and T > 0 be the given length of the time interval I = (0, T). Sometimes we write QT instead of I x f~. Let 1 < p, q < cc and k E N, then (Lq(f~), H. []q) denotes the usual Lebesgue space and (wk,q(~), ]1. Ilk,q) is used for the standard Sobolev space. L~(~) is the subspace of Lq(f~) consisting of functions having mean value zero. We denote by :D(12) the space of smooth functions with compact support in f~ and define W0k'q(f~) as the completion of 7)(~) in the norm II. IIk,q. The Bochner space Lq(I;X(f~)), where X(~) is some function space over ~, will be equipped with the norm ( f : II. I]q(~)dr) 1/a.

2. M A T H E M A T I C A L F R A M E W O R K

44

We shall also work with weighted Lebesgue and Sobolev spaces. Let 0 < p E L t (~) be the density of a Radon measure L, with respect to the Lebesgue measure, i.e. for all Lebesgue measurable sets E the measure ~,(E) is defined by

v(E) f pdx. =

E Then the space Lq(fl, ~,), 1 < q < cx), is a reflexive, separable Banach space with the norm

Ilfli , = ( f

1/'

The weighted Sobolev space (wk'q(~, v), I1" ]lk,q,~) is defined in an obvious analogous manner. The space W0k'q(~, v) is the completion of 7:)(~) in the norm ]1-IIk,q,,. We set l) - {u e D(I2), div u = 0}, H - closure of l) in ]]. H2-norm,

Vq - closure of V in II V . IIq-norm, V 8 -= closure of 13 in H. ][s,2-norm,

(2.1)

H ( d i v ) = {u E L2(fl); div u E L2(gl)}, H(curl) -= {u ~ L2(~); curlu e L2(~)}. The properties of the above spaces are well known (cf. Galdi ]36], Girault, Raviart [38]). We just mention that Vq, 1 < q < co, is a separable, reflexive Banach space and that H, V ' and H(curl), H ( d i v ) are Hilbert spaces. We shall also work with the spaces L2(=)(12) and Wk'P(=)(fl), respectively, which are called, respectively, generalized Lebesgue and generalized Sobolev spaces. They have been studied by Hudzik [44], Musielak [91] and KovA~ik, RAkosnik [55]. Since they are not so well known and since we shall also need a weighted version of these spaces we shall discuss their properties in some detail. 4 For given p(x) C L~(~), 1 < poo <_p(x) < P0 < c~, we fix

Iflp(=) = f

If(x)] p(=) dx.

(2.2)

It is easy to see that I. Ip(=) defines a convex modular (cf. Musielak [91]), which preserves ordering, i.e. f < g implies Iflp(=) <- Iglp(=). Similarly to the Luxemburg norm in Orlicz spaces we can define

]lfllp(=)

= inf{A > O; If[p(.) < 1},

awe refer the reader to Kov~ik, R~kosnfk [55] for more details and proofs.

(2.3)

2.2. FUNCTION SPACES

45

which is a norm on the space /p(x)(f~) ~_ {f • Ll(fl); LAflp(x) < oc

for some A > 0}.

(2.4)

We define the dual function p'(x) pointwise as the dual exponent to p(x), i.e. p'(x) = P(x) and the number rp = rp(z) = A__+~ Then we deduce by a pointwise application p(x)--I pc¢ P0" of Young's inequality that the generalized HSlder inequality

If(x)g(x)l dx <_ rpllf]lp(x)[lgllp,(z)

(2.5)

f~

holds for all functions f •/2(x)(f~), g • /2 '(x) ( f~ ). norm on Lp(z) (12) by IHflllp(x) =

sup

Iglp,(~)
We can also define an equivalent

I f f(x)g(x)dxl,

(2.6)

which is the analogue of the Orlicz norm in Orlicz spaces. From (2.6) and the pointwise application of Young's inequality we obtain that 1 f ~ Ilfilp(~) -< ][Iflllm) -< ~ 1 + ~-~1 I,().

(2.7)

Moreover, norm convergence in (2.3) or (2.6) is equivalent to the modular convergence in (2.2). It can be shown that the space (LP(~)(ft), I1 IIp(~)) is complete and that the dual space can be characterized as LP'(x)(12). In fact, for all elements G • (LP(~)(Ft)) * there exists a function g • LP'(z)(~2) such that for all f • LP(x)(~2)

G(I) =

[

:(z) g(x) dx.

(2.8)

f~

The proof 5 follows the same lines as the one for the usual Lebesgue space /2(f~). Namely, defining for all Lebesgue-measurable sets F

#(F) = G(XF) , where XF is the characteristic function of the set F, we obtain that # is a a-additive, absolutely continuous with respect to the Lebesgue measure set function. Therefore the Radon-Nikod:m theorem yields the existence of a function g • L 1(12) such that (2.8) holds for all simple functions. Since all functions f in L p(~) (f~) can be approximated by simple functions sn

sn(x) /~ f(x)

a.e. x in ~2,

the Lebesgue monotone convergence theorem, the Vitali-Hahn-Saks theorem and the Vitali theorem lead to (2.8) for all f • LP(z)(ft). From (2.5) and the first inequality in (2.7) we get

llgllp'(~)-< IIGII(L,<~)(~)). < rpllgllp'(~),

(2.9)

which concludes the proof. 5For more details in the case LP(f~) see e.g. Kufner, John, Fu~ik [57[ or Dunford, Schwartz [23].

2.MATHEMATICAL F R A M E W O R K

46

From the characterization of the dual space (/2(~)(f~)) * we immediately obtain t h a t / 2 ( ~ ) ( ~ ) is a reflexive Banach space. Moreover, one can show that :D(f~) is dense in LP(X)(f~) and t h a t / 2 ( a ) ( f t ) is separable. Indeed, let f 6/_2(x)(~), then we see by truncation that there exists a bounded function g such that [If - gl]p(x) <- E. Luzin's theorem provides the existence of h E C(f~) such that h = g on a large closed set and IIg - hllp(~) -< ~. The absolute continuity of the integral then gives the existence of an open set G such that ][h--hxGllp(~) < c, and moreover there exists a polynomial m such that also [[hgG -- mXG[Ip(x) <_ ¢. Finally we find a function ~ 6 C~° (f~), 0 <_ ~ < 1, such that [[mxe - mqoI[p(z) _< ~. This shows that :D(ft) is dense in/2(~)(~). But in contrast to Lebesgue and Orlicz spaces the generalized Lebesgue space / 2 (~) (ft) is not p(x)-mean continuous. More precisely, let the ball B~(xo) C ft and let p be continuous and non-constant on B~(xo). Then, there exists a function f 6/2(~)(ft) and a sequence h~ --~ 0 such that fn(x) =- f(x+h,,) do not belong to the space/2(~)(CA). The generalized Sobolev space Wk'P(x)(f~), k 6 1W, is defined as the set 6 W~'P(~)(~) = { f E LP(~); Oaf E LP(~)(~)

V[al < k},

(2.10)

which is endowed with the norm

I]f[[k,p(x) = ~ I[0a/[[p(x). la[
(2.11)

The space w0k'P(*)(f~) is defined as the completion of 7)(f~) in the norm (2.11). Using standard arguments one can derive from the properties of the space LP(X)(~) that Wk,P(~)(f~) and Wok'P(~)(12) are separable, reflexive Banach spaces and that every element G of the dual of w0k'P(*)(i2) is characterized as

G(f) = ~

f O'~f(x) gc~(x) dx,

(2.12)

I~l
L p(x)(~) ~ Lq(x)(~),

(2.13)

if and only if

q(x) < p(x)

a.e. in ft.

(2.14)

Beside this trivial embedding one would like to have an embedding of the Sobolev type, namely

wo~,p~)(ft) ~ Lq(~)(~), 6Here a is a multi-index and c0af denotes the generalized derivative of f.

(2.15)

2.2. FUNCTION SPACES

47

where 1

1

1

q(x) = p*(x) -- p(x)

1

d'

(2.16)

Counterexamples show that in general this is not possible. However, if p is continuous on ~ and if P0 < d one disposes of the following embedding: for all e E (0, l/d) there exists a constant c > 0 such that for all f E W~ 'p(x)(~)

Ilfllq( )

allflh,n( ),

(2.17)

1 <_ q(x) <_p*(x) - ~ .

(2.18)

where

This means that one "loses" an arbitrarily small power in the desired result (2.15), (2.16). In fact, the embedding W~ 'p(~) ¢-+ Lq(x)(~), where q satisfies (2.18), is not only continuous but compact. This in turn implies that IIVfllp(x) is an equivalent norm on the space W01'P(~)(~). Note, that one can get embeddings of similar type as (2.17), (2.18) under weaker assumptions on p. On the other hand if we assume more about p, we can improve the integrability in the above result and "lose" only a logarithmic factor. More precisely we have P r o p o s i t i o n 2.19. Let p be a measurable function such that p E ~c~,P0], Po < d and let the sets

= {x

p(x) > q}

(2.20)

have a Lipschitz continuous boundary for all q E ~¢¢, P0]. Moreover, assume that PO

f cq* dq ~ Co < c~,

(2.21)

P~

where cq are the continuity constants of the embedding wl'q(~q) ~ Lq*(~q), i.e. l[fllLq*(~q) <: Cq Ilfllw , In ) •

(2.22)

Then there exists a constant c depending on P0,P~, d and I~ I such that

is satisfied for all f e Wl'P(x)(gt). PROOF : From the definition of the set ~q it is clear that functions belonging to the space WI,P(~)(12) also belong to the space wl'q(~q), for all q E [Poo,Po]. Therefore we

2.MATHEMATICAL FRAMEWORK

48

obtain from (2.22) raised to the power q* and integrated over (Poo,po) po

po

p ~ flq

P~ Po

~q

P~ Po

~q

p~

(2.24)

f~

<_c(l+(Jlf]P(z)+IVfl'(')+If~Idx)'U'°). fl

Now, the left-hand side of (2.24) can be written as Po

(2.25) p~

f't

where Xq is the characteristic function of the set f~q. From Fubini's theorem we get that (2.25) is equal to p(z)

~(x) (2.26)

f~

p¢¢

{xEfl,l$(z)l_>2} poo

Using that the integrand can be written as

(2.27) we can compute the inner integral in (2.26) as

P(~) f lf(x)]d~-d~dq

=

-(d

,(.~d

- p(z))lf(x)l~--~

+

d2 d21nlf(x)l i n If(x)l a d - p(x)

pa¢

+ d2 lnlf(x)l ~

(lnl/(x)l~) k

If(x)l a ~=~

+ (d -

~

P~)lf(x)l~-'~

If(x)ld

- If(x)l ---~

(ln If(x)]~_-:-~) k=x

d2 d - p~

_ d2 In If(z)l In - d2

d2 In If(x)____~l

(2.2s)

kk!

kk!

k

2.2. FUNCTION SPACES

49

Using (2.27) and the power series for exp y in the first term on the right-hand side; the first and the third term in (2.28) can be written as

(d-p(x))exp(-dlnlf(x)l)fl,

k+l(ln k

•¢¢x~ ~ / d-p(z)] d2 ~k+: (k + 1)!

k=:

-~

(ln'f(x)l~)kk!

}'

(2.29)

k=O

Fhrthermore the difference in the squiggly bracket is equal to ~.~ (ln If (x)l-d:~--~) d2 k - 1 - In [f(x)[ d 4 ( x - ~ + k!(k - 1) k---2

3 a~ > - 2 - ~ in l](X)ld_ ;(x). d-p(z)

+ d~ln If(~)l

exp(~

d - p(x) d21n If(x)[ lnly(x)l )

and therefore the first and third term in (2.28) are bounded from below by

d -_p(x) h2 If(x) l''(~)

2(d - p(~))

/ In I/(x)l d If(x)l d 3 .21n If(x)[ (d -p(x)~2

~a -~f~-~

d

(2.30) 1

] ln[f(x)[[f(x)l d"

Similarly we can treat the corresponding terms with poQ instead of p(x) in (2.28) and we get that they are bounded from below by -3

(~_)2

If(x),';° ---t

in II(x)l In I](~)l

4 ( d - p¢~)

If(~)V

+ 2 d~ I/(x)V +

(2.31)

In I/(~)ll/(z)V

One easily sees that the modulus of the last three terms in (2.30) and (2.31) is bounded by some constant depending on Po,Poo and d if one uses that If(x)l > 2. The modulus of the first term in (2.31) is bounded by 3

(~Zf-~) 2 If(x)[';° ln2

'

(2.32)

which after integration with respect to x can be dominated by (2.24) with q = poo. The first term in (2.30), which remains on the left-hand side, is bounded from below by

(-~-)

If(x)la~

2 ln(1 + If(z)l) "

(2.33)

50

2.MATHEMATICAL FRAMEWORK

Putting all computations together we arrive at

if(x) lp'(=) ln(2 -4- If(x)l)

dx < c(co,po,p~,d, I~1)(1 + ]'flp(=)+ iv,lip(=)) p~/(po) , -

fl

where we have added on both sides of the inequality the integrand integrated over the set {x • ~, If(x)l < 2}. The last inequality immediately yields the assertion, m R e m a r k 2.34. 1) In the same way as we have introduced generalized Lebesgue spaces L p(x) (~) one can introduce generalized Orlicz spaces LM(x)(~) (cf. Hudzik [48] and the literature quoted there). In this sense the integral on the left-hand side of (2.23) defines a modular on the generalized Orlicz space LM(z)(~), where M ( x , t) = t p*(x) In -1 (2 + t). Therefore inequality (2.23) states that the embedding of the space WI'P(=)(Ft) into the space LM(x)(~) holds true. 2) Note, that condition (2.21) is fulfilled if the measure of the set 12p0 is positive. 3) Motivated by our studies of electrorheological fluids [113] Edmunds, RAkosnik [26] have proved the optimal embedding W~'P(Z)(~) ~-~/2"(=)(t2) if p is Lipschitz continuous and supa p(x) < d. The underlying idea of the previous proposition is that one could use the classical L q information on the level sets ~q in order to get something for L p(=)(~). This idea cannot only be used for embeddings, but also in many other situations, where the Lq-theory is well developed, as e.g. theory of singular operators, regularity theory of elliptic equations and systems to name a few. The following counterexample of Kirchheim [53] shows that in general it is impossible to recover from the L q (12q) estimates an L p(x) (~) information. P r o p o s i t i o n 2.35. Let ~ = [0, 1] and let p(x) = 2 + x. Then there exists a .function g such thatg ~ LV(=)(0, 1), but.for allq • [2,3] 1

g(X) q dx <__CO,

(2.36)

q-2

where co is independent of q.

PROOF : Let bk E (0, 1) be arbitrary. Then we have lim (bk - x) ,~-2 = c¢, and hence there exists ak E (0, bk) such that

(bk

- -

ak)

%+2 -~--~

=

2 k+l

(2.37)

Now, given 0 < ak < b~ < 1, we define bk+l • (0, ak) by (bk--ak)

bk+ 1 --a k

,4+2 = 2 k.

(2.38)

2.2. F U N C T I O N

51

SPACES

Such bk+x exists since the function F ( x ) = (bk - ak)~h+~ is continuous and decreasing and satisfies F(0 +) = 2 k+l and F ( a k ) = 1. Put

A,~ (b,~ ak) ,,k~-2 =

(2.39)

-

and hence (cf. (2.38)) =

2

(2.4o)

.

Now we define g ( x ) through

k=l

One immediately finds that g ~/2(x)(0, 1) since bk

i

O0

O0

_> ~

(bk - aa) A~+2 -- ~--~ 1 - - c o ,

k=l

k=l

where we used (2.39). It remains to show (2.36). Let q > 2 and let ko be such that bko+l < q -- 2 < bko. Then we have / ,1

ko-1

q--2

k=l

Cbk°

I1 + 12q-2

Clearly, as q _< bk0 + 2 and Ak > 1 we get ko-1

I1 _< Z

k=l

ko - 1

oo

_

Ak

~k--ak)<_~_,2

k=l

-k=l,

k--1

where we used that the sequence bk is decreasing and (2.39), (2.40). ~ r t h e r we see, again using (2.39), that ~bko+2/b

12_
~ko--ako)

~bk --ak ~ ak T 2 / z.

_< sup A k

Ak

b-laU'2k

(uk - ak) = sup (bk -- a k ) - .4+2

k

k --

z

< s u p ( b k - ak) -bk2"~ g sup x - ~ = e2~. k xe(o,1)

2.MATHEMATICAL FRAMEWORK

52 If q = 2 we observe 1

0

k=l

= ~(b~

- ~ ) o ~ + ' =/_:., 2 -(~+~) = -

k=l

k=l

2

All together we showed that 1

f g(x) qdx < 1+eL q--2

is satisfied for all q E [2, 3] and the proposition is proved. Unfortunately we have not been able to construct a counterexample to the boundedness of the maximal operator and of singular integral operator in generalized Lebesgue spaces L p(~)(~1). However, we conjecture that there will be no analogue of such fundamental theorems, as e.g. boundedness of singular integral operators (cf. Calderon, Zygmnnd [17]), boundedness of the maximal function (cf. Stein [119]), the negative norm theorem (cf. Ne~as [92]) and Korn's inequality (cf. Ne~as [92]), in generalized Lebesgue spaces/-2 (z) (~) even for very nice functions p(x). Let us finish this section by introducing some more function spaces, necessary notation and some auxiliary results which we shall need in the sequel. We define closure of 12 in II. []x,p(~)-norm, Ep(~) = closure of V in IID(.)Hp(~)-norm, W(~) -

(2.41)

where D(.) denotes the symmetric part of the gradient. Clearly Vp(x) is a Banach space, which is a closed subspace of W~'P(x)(gl). Therefore Vp(~) is also separable and reflexive and the embedding

holds true. However, since we do not have Korn's inequality, it is not clear if the spaces Vp(~) and Ep(x) coincide, even for p 6 C(l=l). But Ep(x) is also a separable reflexive Banach space. From the embedding (2.13) and Korn's inequality in W~'P~(fl) we conclude IID(u)llp(~) ~ c IID(u)II,~ ~ a IluIIl.,oo,

(2.42)

which shows that (E,(~), lID(.)Up(x)) is a Banach space. Moreover Ep(x) is reflexive and separable. This can easily be seen by considering the map r.P(::) P : Ep(~:)--~ ~N(d)

(~'~) : U

(Dij (U))l
53

2.2. F U N C T I O N S P A C E S

N(d)

Lp(~) {f-)~

where N ( d ) = ½d(d + 1) and ~ N ( d ) ~ : --

1-I LP(~)(fl) • If we endow the space j----1

N(d)

LP(~) {0~ with the norm ~ N(d)~O~/

Ilujllp(x) we obtain from the corresponding properties

j=l

L p(x) {0"~ is a separable, reflexive Banach space. Using the embedding of/-2 (x) (12) that --N(d)~'~: Ep(x) ~ Vpoo we get that P is injective. From the definitions of the norms in Ep(~) and ~N(d) r.p(~) (fl) it is clear that P is an isometric isomorphism of Ep(x) onto the subspace

r? (x) :o~ and since Ep(~) is complete, W is a closed subspace of W = P(Ep(x)) C --N(d),..,, L~(x) {o~ The reflexibility and separability of Ep(~) follow immediately, since closed N(d) k~]"

subspaces inherit these properties from --N(d) r?(x) ~" :o~ ~J" Finally, we also need a weighted version of the space Ep(~). Let 0 < p E L 1(~'l) be the density of a Radon measure p with respect to the Lebesgue measure. Then we define the modular I.Ip(~),~ by

ISl.(~),.- f I:(x)l'(') d.= f IS(~)l'(%(~)d~

(2.43)

f llfll.(x),. = inf{~> o, lXlp(x),, s i},

(2.44)

and

which is a norm on the space LP(~)(t2, ,) -- { f e Ll(fl, v); [Af[p(x),, < co

for some A > 0}.

(2.45)

One easily checks that all properties stated for LP(~)(fl) also hold for LP(~)(ft, v), since no particular property of the Lebesgue measure which is not shared by a Radon measure was used in the proofs (cf. Dunford, Schwartz [23] for the corresponding theorems). In particular, L2(x)(ft, v) is a separable, reflexive Banach space in which :D(t2) is dense. The duality pairing (., .)p(x),v between LP(Z)(fl, y) and /.2'(x)(fl, g) is for all f E LP(x)(~, ~,) ,g E LP'(z)(~, L,) given by

(:,g).(~),.= f :(xb(x)p(~)dx. fl

Moreover, if 0 < Co _< p, then we note that L'(~)(~,,) ¢-4/2(~)(fl),

(2.46)

since one easily sees that

[

CoS L::(') -< I::(')"(')d" II

f~

and this modular inequality immediately implies the above continuous embedding (cf. Pick [103]). For such p we define Ep(x),~ - closure of V with respect to the I[D(.)[[p(x),~-norm.

(2.47)

2.MATHEMATICAL FRAMEWORK

54

Similarly as for Ep(~) one can show that Ep(x),v is a separable, reflexive Banach space and that Ep(~),v ~ Ep(z),

(2.48)

(2.7),

and using a similar argument as for

Ilfll.(~),~ < e (1 + Iflp(~),-) -

2.3

(2.49)

Maxwell's Equations

In this section we will present existence, uniqueness and regularity results for the quasi-static Maxwell's equations div E = k curl E = h E . n = E0 • n

in ft,

(3.1)

on Oft,

where k, h and E0 are given and where f~ C_ R 3 is a bounded domain. The system (3.1) is intensively studied in the literature and here we will give only the precise formulation of the results that we will need in the sequel. For more details we refer in the context of integral equation methods to Miiller [90], Martensen [80], Kress [56] and Werner [128] and in the context of Hilbert space methods to Gaffney [34], Friedrichs [31], Leis [60], [61], Week [127], Weber [126], Picard [102], [100], [101] and Milani, Picard [83]. Beside the spaces H ( d i v ) , H ( c u r l ) introduced in the previous section we need some more spaces and notations: o

H ( d i v ) = {u E H ( d i v ) ; n . u = 0}, o

H(curl ) = {u e H ( c u r l ) ; n × u = 0}, H0 (div) = {u E H(div); div u = 0}, Ho(curl) = {u • H(curl ); curl u = 0}, o

o

H0 (div) = H(div ) N H0 (div), o

(3.2)

o

H0(curl) = H(curl ) n H0(curt ), O

HN(f~) = H0(div) N H0(curl), o

HD(f~) = H0(curl) fq H0(div). All these spaces are Hilbert spaces if they are equipped with the canonical scalar o

product. Note, that H0(div ) and H defined in (2.1) are identical. For the investigation of Maxwell's equations it is crucial to have at disposal the compact embedding H(div) N H(curl) ¢--~'-+ L2(ft)

(3.3)

55

2.3. M A X W E L L 'S EQUATIONS

and appropriate decompositions of L2(f~) into direct sums, as e.g. o

L2(~) = HN (9 grad Wt'2(f~) (9 curl H(curl).

(3.4)

The compact embedding is well known for smooth domains, where the proof is based on Gaffney's inequality and Rellich's compactness theorem. However, there are examples of non-smooth domains where Gaffney's inequality does not hold and thus the above arguments do not work. Nevertheless the following result characterizes a suitable class of domains. T h e o r e m 3.5. Let ~ be a bounded domain with On E C°'1. Then the compact embedding (3.3) holds true. PROOF : This result is proved in Picard [102] and we will not explain the details here. Earlier results in this direction can be found in Weck [127]. The same result holds under even weaker assumptions, cf. Witsch [132]. • As a consequence of Theorem 3.5 we obtain that various subspaces of L2(f~) are closed, which in turn enables us to prove decomposition results. In particular we have L e m m a 3.6. Let ~2 be a bounded domain with O~ E C °'1. Then o

curl g ( c u r l ) = curl (curl g ( c u r l ) N H ( c u r l ) ) , o

(3.7)

o

curl g(curl) = curl (curl g ( e u r l ) N g ( c u r l ) ) are closed subspaces of L2(~).

PROOF : The characterization of the spaces on the left-hand side is based on the decomposition theorem (cf. Leis [61]). The closedness follows from Theorem 3.5 (cf. Milani, Picard [83]). • L e m m a 3.8. Under the assumption of Lemma 3.6 the following decompositions o

L2(~) = HN (9 grad Wl'2(f~) (9 curl H(curl), H0(div)

= HD

(9 curl g ( c u r l ) ,

(3.9) (3.10)

o

H(div) = (grad W1'2(12) N g(div )) (9 HN (9 H(curl)

(3.11)

hold true.

PROOF : see e.g. Milani, Picard [83].

•

Another important tool is the characterization of HN and Ho, respectively, the spaces of harmonic vector fields with vanishing normal and tangential component, respectively. For this we need the following regularity result for harmonic vector fields.

56

2.MATHEMATICAL F R A M E W O R K

L e m m a 3.12. Let Q be a bounded domain with O~t E C°'1. Then every vector field v • Ho(div) N H0(curl) belongs to the space C(~). PROOF : see e.g. Morrey [88] p. 166.

•

T h e o r e m 3.13. The dimension Of HN(~) and HD(~) is finite and given by dim H ~ = p,

dim HD = m - 1

(3.14)

where p is the first Betti number, i.e. the number of handles, and m the second Betti number, i.e. the number of connected components. PROOF : The proof can be found in Picard [101] and is based on L e m m a 3.12 and a common procedure to construct a basis of HN(~) and HD(~), resp., (cf. Werner [128]). L e m m a 3.15. Let ~ be a bounded domain with 0~ • C°'1. Then there ezists a constant such that IIE]I2 _< c(H curl EH2 + ]1div Ell2 + E

I

wi.Edx] )

(3.16)

i=I o

holds for all E • g ( d i v ) N H(curl), where wi, i = 1 , . . . , p , is a basis of YN(~). PROOF : see e.g. Milani, Picard [83].

•

Let us now discuss the solvability of (3.1). We assume that ~ is bounded and has a Lipschitz continuous boundary. First we re-formulate the boundary condition. It is well known that the mapping 7n : u -+ u . n

(3.17)

defined in 7)(~) can be continuously extended to a mapping, still denoted 7n, 7n: H ( d i v ) --+ H-t/2(O~),

(3.18)

where H-t/2(0~) is the dual space to H1/2(0~). It can be shown that (cf. Girault, Raviart [38])

I17.(u)11-~/2,o~ _< Ilull.(div), Range (Tn) = H - t / 2 ( O Q ) ,

(3.19)

o

Ker (%) = H ( d i v ) . Therefore we can re-write the boundary condition (3.1)3 as o

E - E0 • g ( d i v )

(3.20)

where Eo • H(div ) is given. Now we can formulate an existence and uniqueness result for the Problem (3.1).

57

2.3. M A X W E L L 'S EQUATIONS

T h e o r e m 3.21. Let ~ be a bounded domain with On E C°'l. Then there exists a solution E E H(curl)N g(div) of the problem (3.1)1,2, (3.20) if and only if (i) E0 e H(div), (ii) h • Ho(div) and h-LHD(~), (iii) k • L2(~) and k - div E0 .l_ const. This solution satisfies the estimate

IIEiI < c(LlkVl + [IE0il-1/ ,o

+ IIhli ) •

(3.22)

Moreover, the solution is unique if we require E _LHN(~).

PROOF : This theorem is a consequence of the results proved in Picard [100], [102]. We will only sketch the basic steps. One easily checks that a solution of (3.1)1,5, (3.20) satisfies the necessary conditions (i)-(iii). Let us show that they are also sufficient. Due to (iii) there is exactly one p • WI'2(~)/R solving the Neumann problem --Ap = k

in ~ ,

Op = E 0 - n

on 0 ~ ,

On

(3.23)

and satisfying the estimate

llVpll2 < c(llkll+ IIEoll-i/2,a)•

(3.24)

From this and (3.11) we easily deduce that Vp • H(div ) A H0(curl ), o

(3.25)

Vp_L HN(~), Vp-L curl H(curl). From (ii) an (3.10) we obtain that h • curl H(curl)

(3.26)

which together with (3.7)1 implies that h = curl ¢

(3.27)

with o

¢ • curl H(curl).

(3.28)

This and (3.11), (3.9) imply that o

¢ • H0(div), ¢ _l_HN(~), ¢ _L grad W 1'2 .

(3.29)

Now, it is easy to check that E-Vp+¢

(3.30)

2.MATHEMATICAL F R A M E W O R K

58

solves (3.1)i,2, (3.20). The estimate (3.22) follows from (3.30), (3.24), (3.16), (3.1)5 and (3.29)2, namely

I[EII2 ~ IIVplI2 + 11¢112 c(llkll2 + IIEoll-~/~,o~ + II curl ¢112) < c(}lkll2 + tlEoll-a/2.o. + Ilhll2) • Let El, E2 be two solutions such that El, E2 -£ HN(~). From (3.1)1,2, (3.20) we obtain E1 - E 2 E HN(~)and from the orthogonality t o HN(~) we deduce Et - E 2 = 0. This completes the proof. • The regularity of the problem (3.1) within the context of Sobolev spaces has been studied in detail in Schwarz [115] in the context of manifolds with boundaries. In the last chapter he describes the consequences of his results in the case of threedimensional vector fields. T h e o r e m 3.31. Let ~2 be a bounded domain with 0~2 E C2+z'1, I E N. Let h, k E Wl'V(f2) and Eo e wt+i-~'v(0f~) be given, where 1 < p < oo. Then the solution of the problem (3.1)i,2, (3.20), ensured by Theorem 3.21, satisfies the estimate

IIEll,÷l,p ~ c(llkll,,p + Ilhll,,p + IIEollt+l-lZp,p,o~) •

(3.32)

PROOF : This theorem is the contents of Corollary 3.2.6 and L e m m a 3.5.4 in Schwarz

[1151.

•

Let us finish this section by discussing the time dependent problem div E = k curl E -- h E . n = E0" n

in QT,

(3.33)

on I x 0f2,

where k, h E CI(i, Wt'P(f~)), E0 E CI(-r, Wt+l'v(f2)) are given 7. We will denote by the time derivative of a function with values in a Banach space X if lim ¢--~0 t+~Ef

f ( t +e) - f(t) _ - ~ ( t ) x = 0.

(3.34)

E

7For the definition and properties of differentiable functions with values in Banach spaces we refer the reader to Gajewski, Grhger, Zacharias [351, Chapter 4.

59

2.3. M A X W E L L 'S EQUATIONS

Proposition 3.35. Let f~ be a bounded domain with Of~ E C2+l'1, l E 1~1and let T > 0 be given. Moreover, assume that k , h E CI(i, WI'P(~2)), E0 E CI(i, WI+I'P(f~)), 1 < p < ~ . Then there exists a solution of the problem (3.33) if k, h, E0 and Ok Oh OEo fulfill the conditions of Theorem 3.21 for all t E i. This solution satisfies the Ot ~ a t estimate IIEllc,(i,w,+,.~(a)) < c(llkllclff, w,.~(a)) + Ilhilc~(i,w,.p(a)) + IIEoltc~(Lw,+~.,(a))) (3.36) and is unique if we require that E(t) / HN(f~) for all t E I.

PROOF : From the assumption of this proposition and the Theorem 3.31 it follows that for all t E f the system (3.33) possesses a solution E(t) E Wt+I'P(f/) satisfying the estimate (3.32) and that also the system div G(t) = ~Ok( t ) in f~, curl G(t) = ~-(t)0h

(3.37)

0E0..

G ( t ) . n = -~-~-Lt) " n

on 0f2,

possesses a solution G(t) E Wt+X'P(f~) satisfying for all t E I the estimate

IlG(t)ll,+~,p < _

c(

Ok

Oh(

t/I,. +

0E0

From these estimates, the linearity of the systems and (3.34) one easily deduces that G(t) = --~-(t) 0E for all t E i. The estimate (3.36) follows from the above considerations. For the uniqueness we proceed as in the proof of Theorem 3.21 and the proof is complete. •

3 Electrorheological Fluids with Shear Dependent Viscosities: Steady Flows 3.1

Introduction

In this chapter we shall deal with steady flows of incompressible electrorheological fluids in a bounded domain fl C_ R 3. This motion is governed by the boundary value problem I div E = 0

in f~,

curl E = 0 E . n = Eo - n

(1.1)

on 012,

- div S ( D ( v ) , E) + [Vv]v + V ¢ = f + XE[VE]E, div v = 0 v = 0

in ~ ,

(1.2)

on 0f~,

where the data f and E0 are given. We shall assume that the extra stress S = T + ¢ I is given by S = a~l((1 + [D[2) ~-~ - 1 ) E ® E

+ (0/3, + 0/331E1~)(1 + IDI~) p-2 ~ D

(1.3)

+ 0/51(1 + IDI2)P'~ ( D E @ E + E ® D E ) , where p -- p(IEI 2) is a Cl-function such that (cf. (1.4.11)) 1 < p ~ _< p([E[ 2) < P0.

(1.4)

We further require that the coefficients 0/ij and the function p are such that (cf. Lemma 1.4.46) 0/31 > 0,

0/33 > 0,

0/33 -[- 4~ 0/51 > 0,

(1.5)

1Note, that in (1.2) we have divided the steady version of (2.1.2) by the constant density Po and adapted the notation appropriately.

3. STEADY FLOWS

62

k(po) J 211 <

{2V~2V~51 Vf~(O~33

if

~33 _< ~OLs1, 4

if

410~511 ~ 0~33 ,

(1.6)

40~51)

and that for all E e R a, and all B, D e X = {D e R~ym a, tr D = 0}

OSij(D,E) ODkl BijBkl >_O.

(1.7)

The last condition (1.7) ensures that for all E the operator S(., E) is monotone. The conditions (1.5) and (1.6) imply (cf. Lemma 1.4.46 ft.) that there is a constant Co such that S ( D , E ) . D >__Co(1 + tEI2)(1 + IDI~)-----V~ IDI~ .

(1.8)

From (1.3) and (1.4.39) follows that IS(D,

E)I <

Ca(1 +

IEI2)(1 + [DI2)'(l~'~')-1

.

(1.9)

Let us define the Radon measure u by its density p = 1 + IEI2 with respect to the Lebesgue measure, i.e. for all Lebesgue measurable sets A C_ ~, the measure u is defined by u(A) = / ( 1

IEI~) dx.

+

(1.10)

A

For given E we shall often write p or p(x) instead of p(IE[ 2) if there is no danger of confusion. Similarly we shall use Ep and Ep,, instead of Ep(W.I2) and Ep(lel2),~ respectively. For the classical Lebesgue and classical Sobolev spaces we shall use the integrability exponents q, P0, P~, but never p.

3.2

Weak

Solutions

Let us start with the definition of weak solutions. Definition 2.1. The couple (E, v) is said to be a weak solution of the problem (1.1), (1.2) if and only if E E H(div) A H(curl), v E Ep(l~j2),~,

(2.2) o

the system (1.1) is satisfied almost everywhere in ~, E - E0 E H(div) and the weak formulation of (1.2) f S(D(v), E). D ( ~ ) d x + / [ V v ] v . ~dx

" = (f, ~0>1,p¢¢ -- X E / N

is fulfilled for all ~ E Ep(l~.12),~.

E

(2.3) E . D(~o) dx

63

3.2. W E A K SOLUTIONS

Now we formulate our main result concerning weak solutions. T h e o r e m 2.4. Let ~ be such that 012 E C°'1 and let E0 E H(div), f E (W0~'P~¢(~))* be given. Then there exists a weak solution (E, v) of the problem (1.1), (1.2) whenever po0 > 9/5.

(2.5)

R e m a r k 2.6. 1) From the proof of Theorem 2.4 it will become clear that the theorem also holds in the d-dimensional case, if (2.5) is replaced by 3d Po0 > +-----2 d '

(2.7)

2) Note, that the lower bound (2.5) respectively (2.7) is the same as in the case E - 0 (i.e. po0 = P0 = P = const.) obtained by Lions [68] with the theory of monotone operators and compactness arguments. Recently Frehse, M~lek, Steinhauer [30] and Rfi~i~ka [112] obtained the existence of weak solutions in the case when E -: 0 for 2~ po0 > d-~. Both methods use results which are not available for generalized Lebesgue and generalized Sobolev spaces (cf. conjecture in Section 2.2). 3) Theorem 2.4 is formulated such that condition (2.5) restricts the class of possible materials independently of the process. We can re-formulate it also depending on the process. Namely, let E E Lo0(~). Denoting

~o0 ~ p(IIEIIL) we can replace condition (2.5) by /5o0 > 9/5 and the theorem remains valid. PROOF of Theorem 2.4:

For E0 E H(div) it is shown in Theorem 2.3.21 that there o

exists a solution E E H(div) n H(curl) satisfying (1.1) and E - E 0 E H(div). Let us fix one such solution. From Lemma 2.3.12 and (2.3.22) follows that

IIEl]2 _< c ]lE0ii_l/2,on, E e C°(~).

(2.8)

We shall show the existence of a solution to (1.2) via a Galerkin approximation, Minty's trick and a compactness argument. Let wJ, j = 1, 2 , . . . , be a Schauder basis of Ep,v and let us denote the closed linear hull of w l , . . . ,wn by Xn. We define the Galerkin approximation v n by v~ = ~ j----1

ajw j

(2.9)

64

3. S T E A D Y

FLOWS

and an operator A : R n -+ R ~ : ( a l , . . . ,an) --* ((~1,... ,an), where ai, i = 1,... ,n, is given by

.~=f S(D(vn),E) •D(,,,') e~ + / a ~ + XE / E ®

IVv~lvn..,'d~

a E . D(w ~) d x - (f, w i )l,p~ •

(2.1o)

fl

Obviously A is continuous. Multiplying (2.10) by ai and summing up, using (1.8) and [Vvn]v n. v n d x = 0

(2.11)

fl

we obtain p-2 (1 + ]El2)(1 +[D(vn)[ 2) 2 ID(v~)]2dx

A a . a > _ Co fl

-IxE[/

IE[2ID(vn)]

dx-

I(f, vn)l,p~[

(2.12)

=h-h-5.

Using (1 + x ) and the pointwise inequality (1 deduce

i1 > c f(1

2

> (1 + x2)P~ -2

+ x 2 \) Pa°-~ ~ x

2

> c ( x p°° -

(2.13) 1) where

+ IEI2)(ID(vn)l ,¢- + ID(vn)l ')

dx

- c.

c = c(poo,po)

we

(2.14)

gl

From Young's and Korn's inequalities we obtain

h_<~

cf

n JD(vn)l p ~ d z + ~ I[f]]-l,p~o P~ '

,

The inequalities (2.14) and (2.15) together with (2.12) lead to

-5(1

p+ [IE{l~ + llfll_l,p.) •

(2.1a)

3.2. WEAK SOLUTIONS

65

This estimate shows on the one hand by a version of Brouwer's fixed point theorem (cf. Gajewski, Gr6ger, Zacharias [35], Lemma 2.1) the solvability of the Galerkin system

n

~ = (f,

~O)l,poo - - X E

n

(2.16)

E ® E" D(~o) dx

/

Vcp E X ,

f~

and on the other hand it provides, using (2.2.49) and Korn's inequality, the apriori estimate pID(v~)lp,~ + [[D(v~)[[p,~ + [IVv~[l~ _< c0(1 + [[E[]~ + [[f][-1,p-) • (2.17) This estimate implies that we can choose a subsequence such that 2 V n .--.~ V

weakly in Ep,~ N V ~ , strongly in Lq(Ft),

v n --+ V

(2.18)

where 1 _< q < P~o = 3-poo' Denoting

~(D(u), E) - S(D(u), E) 1 + IEI 2

(2.19) '

it follows from (2.17) and (1.9) that the sequence S(D(v~), E) is bounded in Lp'(x)(~, u) and thus S(D(vn),E)--~ X

in

LP'(X)(f~,v).

(2.20)

From (2.18) we obtain for all ~ E Vpo¢

f [vv°lv°.

f [vvlv.

fl

i2

(2.21)

as long as po¢> 9/5. Re-writing the Galerkin system (2.16) by using (2.19), the limit n -+ c~ together with (2.20) and (2.21) leads to

f

+

n

_~Ef

(2.22)

E®E.D(~)dx,

2Note, that the weak limits in Ep.~ and Vp~ coincide. This follows easily by considering ~o rf(~o) = / D ( f ) . ~ o d x = f D ( f ) . ~ d v which defines elements from Vp* and(Ep,u)*.

~a E D(~),

66

3. S T E A D Y FLOWS

which is satisfied for all ~o 6 ~ Xn and thus for all ~o 6 Ep,~. Observe that the last integral in (2.22) can be re-written as -X ~ f 1E+®IEp" E D(~o) dr, n

which defines an element from (Ep,~)*. Therefore the Galerkin system (2.16) with ~o = v n yields f

S(D(v~),E). D(v ~) dv ~ (f,v)l,,~o- X~ f E ® E . D ( v ) d x

(2.23)

fZ

if we take into account (2.18), (2.11) and (2.19). The monotonicity of S(D, E) implies that for all ~o E Ev,v 0 _< f

(S(D(v~), E) - S(D(~o), E ) ) . D(v ~ - ~o) dr.

fZ

As n ~ co this relation, (2.23), (2.20) and (2.18) imply

0<

f E®E.D(v)dx- f fl

fl

- / S(D(~), E). D(v - ~o) dx, f~

which together with (2.22) for qo = v gives that 0 -< f

(X - S(D(~o, E ) ) . D(v - ~o) dv

holds for all ~o E Ep,,,. Choosing now ~o = v + re, ¢ E Ep,~, we get (cf. (2.19)) f x'D(¢)d~ = f

S(D(v),E). D ( ¢ ) d x ,

which together with (2.22) shows that v is a weak solution of (1.2). This concludes the proof. • If we require that S is not only monotone but uniformly monotone, i.e. for all D, B E X p(iE[2)-2

0Sij(D, E) S~jSk, > (C1 + 62lE]2)(1 + IDI2)~--~'--IBI 2 ODk~

(2.24)

is valid, then we can also show that weak solutions are unique for small data 3. To this end we need an apriori estimate similar to (2.17) but without an additive constant on the right-hand side. Thus let us establish a different coercivity property for S. 3Note, t h a t we have given sufficient conditions for (2.24) to hold in L e m m a 1.4.64.

67

3.2. W E A K SOLUTIONS

L e m m a 2.25. Let S be given by (1.3), satisfying (1.4)-(1.6) and (2.24). Then there exists a constant C4 such that S ( D , E ) . D _> C4(1 + ]EI2)IDI p~¢

(2.26)

is satisfied for Poo >_ 2 and

IDI

S ( D , E ) . D > C4(1 q- IF[ 2)

ID] p~

IDI < 1,

IDI _> 1,

(2.~27)

holds if 1 < Poo < 2.

PROOF : Inequality (2.26) follows immediately from (1.8), (2.13), 2 < poo and 1

<

(1 + x) a

sup

-ze[o,oo)

l+x

~

< max(l,2"-1),

a > 0.

(2.28)

-

In the case Poo < 2 we have 1

d S ( s D , E) d s . D ~ss

S(D, E ) . D =

(2.29)

0 1

poo --2 2

_> + c lEl ) f

+ [sD[ 2 ) - -

ds [DI 2 ,

0

where we used (2.24) and (2.13). Using (2.28) for a = 1/2, the last integral in (2.29) can be estimated from below by 1

[(l+s[D[)P°°-2ds-Poo1-- 1 IDI1 [(1 -k slDI) p0¢-1]i0 0

1 1 1) - v o o - 1 IDI ((1 + IDIF ~-1 and thus we conclude that

S ( D , E ) . D > C1~__~ ~ +C2[E[2 =

tD1((1+ IDI) p¢~-1 -

1) .

(2.30)

Since there exists a constant C such that (cf. MgLlek, Ne~!as, Rokyta, Rfi~iSka [70] formula (5.1.31))

((i+ IDi)~~-~ - I) > C { IDI -

we obtain (2.27) from (2.30) and (2.31).

]D]p~-i

I D [ _ 1, IDI > l ,

(2.31)

68

3. STEADY FLOWS

Corollary 2.32. Let E6 E H(div) and f E (W0i'P~(~I)) * be given. Then any weak solution (E,v) of the system (1.1), (1.2) satisfies

(2.33)

IIEII~ < CsIIEollH(d~v),

llD(v)ll,>.,,, < C6(f, E0),

(2.34)

where C6(f, Eo) is for poo < 2 and poo > 2, respectively, given by

c (llfll2_,.v + Ilfll[<7,v + IIEoll~o~v)) 2 × C6(f, E0)

--

×

(1~I + IIEoll~-o~v)+lifting# + Ilfll~=~,p-)1-~,

(2.35)

2c Ilfli-l,ppi~ + 211Eoil~-(d~v). PROOF : Inequality (2.33) follows from (2.8)1 and (2.3.19)1. Using v as a test function in (2.3) we obtain S(D(v), E). D(v) dx = (f, v),,p~ - XE Il

f

~ E D(~) dx.

(2.36)

Il

For poo Z 2 we use (2.26), Korn's and Young's inequalities to derive from (2.36)

04 S ID(v)l"<>=(1+ tell) dx Il

< cllfll_~,,. + IIEII~ + -~

ID(v)l p<=dx + -~Il

ID(v)I,~IEI 2dx, 12

which together with (2.33) implies (2.34), (2.35). For the case Poo < 2 we denote gll = {x efl; ID(v(x))l < 1}. Using (2.27), Korn's and Young's inequalities yield C4 i ID(v)12(1 + ]E]2) dx + C4 i Ill

ID(v)IP~(1 + IEI2) dx

II\ill

< c IIfN-l,,- (llDH,~,ii, + IIDLl,~.ii\ii,) + cllEIl~ C4 + -~ f ]D(v)i2iE'2 d x + -~ f Ill

]D(v)i'~iE'2 dx

II\fil

+

+ tl, ll ) + } i fll

+V o~ f ID(v)l'®(s + IEI2)dz'

+

3.2. WEAK SOLUTIONS

69

which implies t"

f lD(v)12(1 + IEI 2) dx + C4 /

ID(v)lP=(1+

IEI 2) dx

,2

~'~1

(2.37)

n\~'~1

<_c(llfll~,v~ + I[f112_1,,~+ IIEoll~(div)) • Since P~o < 2 we have, using (2.37), f[D(v)[V°°(1 + [E[ u) dx fl

_<

f [D(v)[P°°(l [E[2)dx

_< c

+

+

(f [D(v)[2(l ,El2)dx)~(fl+ +

,E,:dx)I-2E-~

O'fll[7,¢ ' + 11f112-1,¢ + HEoll~(div)) P"

IIE011.(d,v))

which implies (2.34), (2.35) in the case p¢0 < 2.

•

Proposition 2.38. Let the solution E o/ (1.1) be orthogonal to HN(~). Moreover, assume that S satisfies condition (2.24) and that HEH2 and [IfH-l,v- are small enough, then the weak solution (E,v) of (1.1), (1.2) ensured by Theorem 2.4 is unique. R e m a r k 2.39. It is shown in Theorem 2.3.13 (of. Picard [101], Werner [128]) that the dimension of the space of harmonic Neumann vector fields HN (~) is equal to the first Betti number of the domain ~, i.e. the number of "handles". This number says how many additional conditions are necessary to ensure the uniqueness of E. PROOF of Proposition 2.38: It is shown in Theorem 2.3.21 that a solution E of (1.1) which is orthogoual to HN(O) is uniquely determined. Assume now that u, v 6 Ep,~ are two weak solutions of (1.2) for given f and E. Denoting their difference by w, i.e. w = u - v, we get by using (2.3), partial integration and div v -- 0 that

n

The integrand on the left-hand side can be re-written as 1

f 0-~kz OS~j (D(su + (1 - s)v), E) ds Dkl(w)D,j(w)

(2.41)

0 1

>_ (CI + C21EI2)ID(w)I 2

(I + ID(su + (i - s)v)l )

0

2

d8.

3.STEADY FLOWS

70

Using (2.13) and (2.28) we detect that the last integral is bounded from below by c 1 ÷ ID(u)i2-v oo ÷ ID(v)i~-Poo 1

if

1 <poo < 2,

if

2 _< poo.

(2.42)

Now assume 9/5 < poo < 2. From (2.41) and (2.42) we obtain the pointwise inequality iD(w)l 2 _< c ( S ( D ( u ) , E ) - S ( D ( v ) , E ) ) . D(w) (1 ÷ ID(u)] 2-p°° + ID(v)H2-P°°), which raised to the power poo/2, integrated over f2, together with HSlder's inequality implies that

(2.43)

f~

× (1 + lID(u)ll,~: ~ + IlD(v)ll~:~). The second term in the product on the right-hand side of (2.43) is due to (2.34) estimated by 2 -poo

1 ÷ C6(f, E0) poo

(2.44)

The right-hand side of (2.40) can be bounded by

1

(2.45)

_< c]lD(w)ll~o~ C6(f, E0)P-~, 3Poo

where we used the embeddings WI'Pco(f2) ~ LP;o(fl) and /_2°°(fl) ~ Lspoo-e(fl), which hold for Poo > 9/5, also taking into account Korn's inequality and (2.34). From (2.43)-(2.45) we obtain 1

~

2-poo

liD(w)ll~ < cllD(w)ll~C6(f, E0)~ (1 ÷ C~(f, E0) ,oo

),

which provides the uniqueness if f and E0 are sufficiently small, since C6(f, E0) --+ 0 as Ilfll_~,p-, IlE0lIH(dlv) --~ 0. For Poo _~ 2 we estimate the right-hand side of (2.40) by 1

HVwll~llVu[13/2 < cllD(w) ll~calf, E0)~.

(2,46)

Therefore we deduce from (2.40)-(2.42) and (2.46) that 1

IID(w) II~ < clID(w) ll~ Cdf, E o ) ~ and we conclude the proof for Poo ~_ 2 in the same way as for poo < 2.

•

3.3. STRONG SOLUTIONS

3.3

71

Strong Solutions

Now we shall present another approach to the steady problem, which ensures the existence of solutions to the problem (1.1), (1.2). This technique works mainly with classical Lebesgue and Sobolev spaces and does not use in an essential way the theory of generalized Lebesgue and generalized Sobolev spaces. Since our operator has nonstandard growth it will not map the space Vpc¢, which is the natural energy space within the context of classical Sobolev spaces, into its dual. Thus, we cannot apply the theory of monotone operators to ensure that solutions of an appropriate approximation converge to the desired quantities, especially the limiting process in the nonlinear extra stress tensor S(D, E) is not clear. However, ideas which have been developed in the context of generalized Newtonian fluids to handle such problems (cf. M~lek, NeOns, Rfi~i~ka [71], [73], Bellout, Bloom NeOns [9], M~lek, Nehas, Rokyta, Rfi~i~ka [70]) will turn out to be useful. This method uses Vitali's theorem for the limiting process in S(D, E) and therefore we need almost everywhere convergence of the symmetric velocity gradients D (v ~) of some approximate solutions. This convergence is provided from apriori estimates using essentially --Av n as a test function. Since - A v ~ is divergence free, but does not vanish on the boundary 0fl, i.e. it is not an admissible test function in the weak formulation (2.3), we need a weak formulation including the pressure ¢. The properties of ¢ are deduced from (1.2)1 interpreted as an operator equation for ¢. To clarify the situation let us formally multiply (1.2)1 by - A v ~ 2, where ~ is a usual cut-off function, and consider only the main terms, i.e. we forget the contributions of V~ 2. This yields, using partial integration and (1.2)2

f 0--D~l OS~j(D(v), E)Dij(Vv)Dkl(Vv) dx fl

-< / I v l IVvl IV2vl dx + / I ¢ 1 1 A v l $2

dx

(3.1)

n

+ / Ifl IAvldx+ / IEIIVEIID(Vv)Idx. n

n

Assuming that (2.24) holds, the left-hand side is bounded from below by c/(1

+ IE]2)(1 + ID(v)12)~2~=~ID(Vv)I2

dx.

(3.2)

n

Now we should distinguish the cases (i) poo >_ 2 and (ii) poo < 2. For Poo >_ 2 we see that (3.2) is bounded from below by f(1 +

IEI2)lD(Vv)12dx.

(3.3)

n

The last two terms on the right-hand side of (3.1) can be treated with the help of

3. STEADY FLOWS

72

(3.3) under appropriate assumptions on f and VE. The integral coming from the convective term can be handled e.g. if v is replaced by its mollification v~. However the second integral on the right-hand side of (3.1) causes the main trouble since the operator - d i v S(., E) maps Vpoo into ( V ~ ) * , if P0 < poo + 1, and therefore we ~OO--p0q'l

see that the pressure ¢ belongs to the space Lp0-1 (~) only, since ~( - p o~o - - p '0 + l ; = p P~ 0-1 ' However p_tw__ < ~ <- - 2 and unfortunately we do not have any information about 0-1 -- poo-1 P~

Av in Lp~-po+l (~). To avoid this situation we approximate S(D, E) by SA(D,E), such that S A converges locally uniformly to S for A -+ co and S A ( D , E ) • D _> c(1 + ]E]2)]D[ 2 , 0SA(D, E)

ODkl

BijBkl >_c(A)(1 +

(3.4)

IEi2)IBI ~ , 1

[SA(D, E)[ ~ c(A)(1 + [E[2)(1 + [D[2)~. For this approximate operator S A one easily checks that - d i v SA(., E) maps V2 into V2*, which in turn implies that cA E L2(~) and the pressure term can be handled. With this approximation depending on e and A we have at our disposal well defined approximate solutions v ~,A. Afterwards we shall derive estimates independent of E and A which enable us to justify the limiting processes. The situation for p¢~ < 2 is similar. In that case we see that (3.2) is bounded from below by (cf. (5.2.9))

[[D(Vv)]J~

.

(3.5)

Under appropriate assumptions on f and V E we can again treat the last two terms in (3.1). The convective term can be handled i f v is replaced by ve, but one must be more careful. The pressure term would be easier for poo < 2 if p would be constant, but due to the non-standard growth also this term causes trouble. This can be resolved if we again approximate S(D, E) by SA(D,E) such that S A converges locally uniformly to S and SA(D,E) • D _> c(1 + ]EI2)(1 + ]D]2) ~2a~-~]DI 2 ,

OSA(D, E) BijBk, > c(1 + [El2)(1 + I D ] 2 ) ~ ODkl

IB] 2 ,

(3.6)

ISA(D,E)I < c(A)(1 + IEI2)(1 + I D I 2 ) ~ I D I . Before ~we conclude this section with the construction of the approximate extra stress tensor SA(D,E) and showing precisely its properties (cf. L e m m a 3.21, L e m m a 3.41, L e m m a 3.52 and L e m m a 3.65) we formulate our main result as far as strong solutions of the problem (1.1), (1.2) are concerned.

3.3. STRONG SOLUTIONS

73

T h e o r e m 3.7. Let ~ be such that On E Ca'l. Assume that Eo E W2'~(~), r > 3 and f E IF" (~) are given. Then it holds: (i) In the case 2 <_poo <_Po < 6 There exists a strong solution (E, v, ¢) of the problem (1.1), (1.2) such that o

E - Eo e H(div) n W2'~(~), 3po(p~-l) 2,2 v e Ep(iEl~),~n W,o¢ (~) n W~lo'c 2po-3 (~) n V ~ ,

(3.8)

¢ e L~oc(n ) n nP~)(n), satisfying (1.1) almost everywhere in ~ and (1.2) in the weak sense, i.e. f

S(D(v),E).D(~)dx+f[Vv]v.~dx-/¢div~dx

n

n

=ff.~dx-x~fE®E.D(~)dx

n

(3.9)

V~ ~ Z)(a)

n

(ii) In the case 9/5 < p ~ < P0 < 2 There exists a strong solution (E, v, ¢) of the problem (1.1), (1.2) such that o

E - Eo e H(div ) n W2,~(~), ,, e E~(l~,,~),~n W~2o~'~(~) n Wt~2~'(n) n V~.,

(3.1o)

¢ E LT:c(fl) n LP~(n), satisfying (1.1) almost everywhere in fl and (1.2) in the weak sense (3.9). (iii) In the case 9/5 < p ~ < 2 < P0 < P~ There exists a strong solution (E, v, ¢) of the problem (1.1), (1.2) such that o

E - E0 E H ( d i v ) n W2'~(~), l/l/.2,p¢¢

vEEp(iEi2),~n,,loc

1 3po(p~-l) (n)nW~o: 2p0--3 ( n ) n y ~ = ,

(3.11)

¢ E L ~ (~) n L p~(~), satisfying (1.1) almost everywhere in ~ and (1.2) in the weak sense (3.9). R e m a r k 3.12. 1) The lower bound p ~ > 9/5 is the same as in Theorem 2.4. This is due to the fact that we have been able to bound the second order derivatives locally, despite of the fact that the pressure is a non-local term. If one would like to prove global estimates of the second order derivatives it is

3. S T E A D Y FLOWS

74

to be expected that there will be a slightly worse lower bound (cf. the treatment of the unsteady problem for generalized Newtonian fluids in M~lek, NeSas, Rfi~iSka [72],

[73]). 2) The upper bound P0 < 6 in the case Poo _> 2 is not satisfactory. It is due to the approximation, which is chosen for the treatment of the pressure. However, from the point of view of applications this bound seems to be not too restrictive, since most shear dependent fluids have a shear power p below or near 2. 3) An interesting feature of Theorem 3.7 is, that it provides a method to obtain existence of solutions without using the method of monotone operators. This will be crucial in the treatment of the unsteady problem, since in this situation it is not clear what is the equivalence of the decomposition

Lq(I × ~) = Lq(I, Lq(~)) if/-2 Ct'~)(I x ~) is considered. This decomposition is crucial for the application of the theory of monotone operators in the parabolic setting. 4) As already discussed in Section 2.1, there are, to the knowledge of the author, only two other results concerning the existence of solutions for an elliptic problem with non-standard growth conditions. In Marcellini [78] the key point is an estimate of the Wl,°°-norm via the Wl'P~-norm. It is very unlikely that this approach can be adapted to the system (1.2) since the elliptic operator S depends on ID(v)l and not on IVvl. This additional difficulty is even for p = const, not well understood. Only in the two-dimensional case there are some results in this direction (cf. Ladyzhenskaya, Seregin [59] for the unsteady case and Kaplick~, M~lek, Stax~ [50], [51] for the steady case). In Boccardo, Gallouet, Marcellini [12] some special test functions are constructed, which enable the authors to show the almost everywhere convergence of some approximate solutions. For standard growth conditions this method can be adapted to treat generalized Newtonian fluids (cf. Frehse, Mklek, Steinhauer [30], Rfi~i~ka [112]). This however, requires tools which are not available if generalized Sobolev spaces are needed (cf. conjecture in Chapter 2). Most investigations of equations or systems with non-standard growth are concerned with the question of regularity of solutions. The authors usually assume that the solution belongs to the space WI'P°(f~). However, it is in general not clear how such solutions can be constructed.

3.3.1

Approximations

Now we discuss the approximation of the extra stress S(D, E). We assume that S is given by (1.3) and that (1.4)-(1.6) are satisfied. Moreover we require that (1.4.59)(1.4.63) hold. These conditions ensure that (2.24) holds, i.e. S ( D , E ) is uniformly monotone. We define SA(D,E), for A >_ 1, by

S~A =C~21vA(]D[ 2, ]EI2)EiEj + ((~31 + o~331EI2)O~juA(IDI2, ]E] 2) + ~1 (0,kUA(IDf 2, IEq~)E~E~ + 0~UA(IDI ~, IEI~)E~Ek),

(3.13)

75

3.3. S T R O N G S O L U T I O N S

where 0~j denotes the derivative with respect to the components D ~ j , i , j = 1,2,3, of the symmetric part of the velocity gradient. This means e.g. OuA (ID Iu, IEI 2) O~juA(ID[2'IEiU) OD~j and similar formulas for V A, U, V, S, S A. We will use the letters i, j, k, l as indices for this derivative. The potentials uA([DI 2, ]E] 2) and vA(]DI 2, ]E] 2) are defined differently in the cases (i) pc¢ _> 2 and (ii) p¢¢ < 2. (i) T h e case Pc¢ >_ 2 Since both U A and V A depend on [El 2 only through p(IE] 2) we will mostly suppress the dependence of U A and V A on IEI 2 if there is no danger of confusion. Let us define

{ U(JDI u, IEI2) uA(ID[2' ISl2) = ao + al(1 + [DI2)½ + and (1 +

vA(JDI 2, IEI2) =

a2(1 +

IDI ~ A, IDI ~ A,

IDI~)

JDjU)P(lzh)-i- 1

(3.14)

ID[ ~ A ,

(3.15) bo + bl(1 + IDI2)2

JDJ > A ,

where ai = a~(IE]2), i = 0,1, 2, bi = b~(]E]2), i = 0, 1, and

1 ) ((1 + JDI-~) ~ u - 1). U(JD] u, IEJ2) - p(l~lu

(3.16)

Note that 0~3U(JDI 2, JEI 2) =

(i + IDI )

2

U~j,

U(0, ]E[ 2) = O,~U(0, IEI 2) = 0. From the definition of S and (3.13)-(3.17) it is clear that SA(D, E) = S(D, E) if IDJ < A.

(3.17) (3.18) (3.19)

We require some regularity for U A and V A, namely that UA( • , IEI u) • C2(R+), UA(IDI 2, .) • C~(R +) and v A ( • , ") • CI(R + × R+), which allows us to compute the coefficients a0, hi, a2 and b0, bl. We obtain 1 1 p p a:A a,=(2-p)(l+A 2) 2 , -

a2 =

P-i(i+ A 2) ~

(3.20)

bo = ( 2 - p ) ( i +A~) ~

- 1,

bl = ( p - 1 ) ( l +

.

A 2)

76

&STEADY FLOWS

Lemma 3.21. Let S be given by (1.3), such that the conditions (1.4)-(1.6) and (1.4.59)-(1.4.63) are satisfied. Assume that S A is given by (3.13)-(3.16) and (3.20) and thatpoo > 2. Then we have for all i, j , k , l = 1,2,3 a l I B , D E X , allE E R 3 and A sufficiently large

[SA(D,E)[ < ci(A)(1 + [E[2)(1 + [D[2)½, [OijSA(D,E)I ~

(3.22) (3.23)

c2(A)(1 + [E[2), (1 + IDI2)P(IE~)-2 ,

SA(D,E) • D > c3(1 + IE]U)JD[2

(3.24)

A2)P(IE,~)_2 (1+

A

OkLS~j(D, E)B~jBm >

c4(i + IEl~)[BI 2

tsAI ___c5(1 + IEI2)(1 + IDI~) ½

{

(1 + IDI ~)

p(l~ff)-2 ~

, (3.25)

(1+ A2) P(IE~~)-2

(1 + IDt~)'(l~ ~)-~ , + A2)KIEI~)_2

(3.26)

(1 (1 + IDI2)p(IE~~)-2 , 10~jSA(D,E)I _< c~(1 + [El 2)

(1 + A2) K[E[2)-2 ,

(3.27)

OuA(IDI 2, ]E]2) • D

(1 + [D[2) Kjz~)-2 , A:)P(jEl~)_2 (1 +

(3.2s)

< cz]DI =

(i + [D[2)P(IE~2)-2 , 1 + 0uA(IDI 2, IEI2). D > cs(1 + IDI2) {

10uA([D[ 2, IEl~)l < cglD[

(1+ A2) P(IE~2)-2

(1 + [DI2),(IE~)_~ A2)K]~I~)_ 2~ (1 +

(3.29)

(3.30)

where the first line in the above eases holds if IDI < A while the second one holds for ID[ >_ A. The constants c3-c9 depend on Po, Poo, whereas cl, c2 depend on A, po and Poo.

3.3. STRONG SOLUTIONS

77

PROOF: Since SA(D,E) --- S(D,E) if IDf < A, we get all assertions for S A from the properties of S(D, E) and some straightforward computations (cf. (1.9), (1.8) and (2.24)). Inequalities (3.28), (3.29) follow directly from (3.17) and the observation that (1 + [D[2)a~ is bounded by 2P-~ or OUA. D if [D[ < 1 or 1 <_ [D I _< A respectively. Inequality (3.30) is a consequence of (3.17). Let us therefore assume ]D[ > A in what follows. One easily computes SA(D, E) = a21 (b0 + bl(1 + [D[2)I/2)E~Ej

+ (o~3,+

o~331EI2)

+ (~51 (

al

al ('(I -}-IDI2)l/2 + 2a2) Dis

, (i + ID12)II2

(3.31)

+ 2a2) (Di,~E~Em + Dj,~E,~Ei)

and Dkz

A

Ok~Sij(D, E) = a21bl (1 + [D[2)1/2E~E~ + (a31 + a33[E[2) ('(1 + [DP) y 2 a l + 2a2) 6i,6,, al - (a31 + a33iE[ ~) (1 + ~12)3/2D'jDk`

(3.32)

al 2a2) (bikEiEj + 5jkEtEi) + abl ((1 + ID]2)1/2 + - a s l (1 +

al (Di,,~EmDuEj+ DjmEmD~lE~) fDI2)3/2

which immediately yields (3.22) and (3.23) for A >_ 1. Since [D[ _> A we obtain al

( I+A 2

½

(1+]Di2),/2+2a2=(p-2)(l+A2)~(1-\~..~_-~2))+(1+A2)

2

_< (p - 1)(1 + A2) 2a~.

(3.33) (3.34)

Similarly we compute

last

ibo}

< c(1 + A ~ ) ~

[bll + (1 + ]D[2)~/2 + (1 + ID[2)~/e -

(3.35)

and therefore we easily derive (3.26) and (3.27) from (3.31) and (3.32). From (3.31), (3.20) and (3.33) we compute SA.D = ( p - 2 ) ( 1 + A 2 ) ~ ( 1 -

A 22)"~½){a21(1+ IDI2)½DE'E (1 1++ ]DI

+ (o~al + c~33[E[2)[D[ 2 + 2a~[DEI 2}

(3.36) + (1 + A 2)

{

a21

(1+ A =) = (1+ IDI=)~ - 1DE. E (1 + A 2 ) ~

+ (~3i + ~331E?)ID? + 2°. IDEI~}.

3. STEADY FLOWS

78 Using for [D I > A > e -x

(1 H- IDI2)½ _< (1 H-e~)lDI

(3.37)

the first term with ~2~ can be estimated from below by - ( 1 + e2)t~2111DIIEIIDEI . Due to the strict inequality in (1.6) and the considerations in Section 1.4 (cf. (1.4.45)) we can choose e such that the terms in the first squiggly brackets are non-negative. Since also the factor in front of it is so we deduce that the first summand in (3.36) is non-negative. Furthermore one can show that (compare with (1.4.39), (1.4.40)) (1 H- A2) 2 (1 H- IDI2)½ - 1

(3.38)

< k0(p)lDI

(1 + A ~ ) ~ for all IDI > A > 1. Thus, condition (1.6) ensures the validity of (3.24). From (3.32), (3.20) and (3.33) we calculate

Ok,S~.(D,E)Bi~Bkl = (p- 2)(1 + A2)a~-~(1 _ kl ) 1H(H-A ID[2z ½) × ×

+

iBl 2 H- 2

+(P-2)(l+A2)~(a31+

a

IBEI 2}

33]

E 2" ( B - D) 2

I)$¥1b-i~1~

H- (1 + A2) 2P-~{a21(p - 1)(D. B ) ( B E - E ) (1 H- IDI2)~/2

(3.39)

H-2 ( p - 2)a51 (BE. D E ) ( B . D) ( 1 H-A2 ~ ½

1 + IDI~ + (a3~ + a33[EI2) [BI ~ +

~1-TT-DTJ

2as~[BEI 2}

= I 1 + I2 + I3.

The non-negativity of/1 and /2 follows immediately from the assumptions of the lemma. Just using [D[ > A the terms in the squiggly brackets in I3 can be handled in the same way as the right-hand side of (1.4.52). Therefore the conditions (1.4.59)(1.4.63) together with (1.5), (1.6) ensure the validity of (3.25). Inequality (3.29) follows from (3.36) with a21 = aaa = a51 = 0, ca1 = 1 and (3.37) with e = 1. Finally, from

ouA([D[2' ]El2)= ((1

H-aliD[2)½ + 2a2) D

and (3.34) we obtain (3.28) and (3.30). This completes the proof of the lemma.

•

79

3.3. STRONG SOLUTIONS

R e m a r k 3.40. From (3.25) and (3.27) it is clear that for all B E X the quantities 0ktSA (D, E) BijBkz and

(iq-[DJ=)P(IE~ =)-2 (1 + IE[2)IBI 2 {

(I+ A2)PlI~~)-2 are equivalent. Later on we will need also the derivative with respect to the components E~, n = 1, 2, 3, of the electric field. We shall use the notation a"ua(IDI2' IEI2) =

OuA(]DI 2, IEI ~) aE.

and similar formulas for V A, S A, U, V and S. We will use the letters m, n as indices for this derivative. L e m m a 3.41. Let the assumption of Lemma 3.21 be satisfied. constants depending on the function p and aij such that (1 + I D I 2 1 ~ ( 1

Then there exists

+ln(1 + IDI21),

[OnOijuA(]DI2, IEI2)I <_ c IEI ID[

(3.42) (1 + ln(1 + Ia12)),

(1 + A 2 ) ~

(1 + [DI2)~-~:Y" - (1 + ln(1 + IDI2)), [0nVA(lDI 2, IE[2)I _< c IEI(1 + IDI2)½

~ ( l + Z 2)

2 (l+ln(l+A)),

2

(3.43)

la~SA(D, E)I _
(1 + A 2 ) ~

(1 + IDI 2)

(1 + ln(1 + IBIS)),

(3.44)

(1 + ln(1 + AS)),

~

(1 + l n ( l + IDI~)), (3.45)

[0nuA(IDI ~, IEI2)I < c IEI(1 + ID[ 2) (1 + A~) ~

(1 + ln(1 + A2)).

PROOF : The estimates follow immediately from (3.14), (3.15), (3.20) and (3.13). •

80

3 . S T E A D Y FLOWS

R e m a r k 3.46. From the inequality 1 + ln(1 + y2) < c(3')(1 + y2)~/2

V7 > 0,

(3.47)

together with (3.45) and (3.29) and a similar argument as in (5.2.14) we deduce for all s > 1 Io~uA(IDI 2, IEI2)[ _< el0 (1 "~ ouA(ID] 2, IEI2) • D ) ' .

(3.48)

(ii) T h e case 1 < p ~ < 2 In this case the potentials U A and V A

are

chosen differently. We define

U(]DI ,]El 2)

IDI < A,

uA(IDI 2, IEI 2) -

~

(3.49)

ao + a1(1 + IDI2)~ + a2(1 + IDI2)~

IDI _> A,

and

vA(ID[ 2, IEI 2) =

(1 +

[DI2)p(IE~2)_I - 1

IDI < A,

(3.50)

bo + bx(a + IDI2)~-~ + b2(1 + IDI2)P~ -1

IDI _> A,

where ai = a~(IE[~), b~ = b~(IEl~), i = o, 1, 2. The number 7r E (1,p~) will be chosen later. If we require the same regularity for U a and V A as in the case p ~ > 2 we can compute ao, al, a2 and bo, bt, b2. For bo, bl and b2 we dispose of a free parameter which will be chosen such that bo = - 1 . Therefore we can calculate ao=(x+a2)~(~

Poo-P ~(p~-~)

p-Tr p~-~)

)

1 p'

P~-P ( l + A 2 ) e ~ -, al = 7r(p~ - 7r)

P-"

a~ = p~(poo - - ) bo = - 1 , bl =

P--~_-P

(I+A2)P~ ~

'

(3.51)

)~

(1 + A 2 , Po~b 2 - p - T r (I+A2)P-2Pc¢. Poo - 71

We have a similar results as for the case p ~ > 2, namely L e m m a 3.52. Let S be given by (1.3), such that the conditions (1.4)-(1.6) and (1.4.59)-(1.4.63) are satisfied. Assume that sA is given by (3.13), (3.49)-(3.51) and that 1 < p~ < 2. Then we have for all i, j , k , l = 1,2,3 and a l l B , D E X , alIE E •3 and A su]flciently large ISA(D,E)I < Cl(A)(1 + ]E]2)(1 + ]DI2)P~ -1' ,

(3.53)

81

3.3. S T R O N G S O L U T I O N S

laijsA(D,E)[ G c2(A)(1 + IEI2)(1 + IDI2)P~-2 ,

(3.54)

(1 + [D]2)'(IE[~)-'~ 2~ '

SA(D, E ) - D > ca(1 + IE[2)(1 + [D[2) ~2~-[D[ 2

(3.55)

(1 + A2)P(f~'12'-p~ ,

Ok,SA(D, E)B~jBk, >_ c4(1 +

[El2)(1 +

(1 "4-[D[2)p(IEj22)-p~ (1 + A2) p
IDI2)~a~ -~IBI2

IsAI < c~(1 + IE?)(1 + I D I 2 ) ~

{ (1(1++m2)p/l~.122_p~'

IDI2)'(IEt~)-'~ ,

]OijSA(D,E)] <_CS(1 A-]E]2)(1 + ] D I 2 ) ~

{ (1 + [DI2)P(IEt~2)-P~¢ '2) (1 + A~)~(I~Io - ~

ouA(ID[ 2, [El2).

D _< c~(1 + [D[2) ~r~V~J D ] 2

1 + ouA([Dt 2, [El2)

(3.57)

(3.58)

,

(1 + [D[2)v(IE?2)-v~ , A2)p([Ei2~_pc¢2'

(3.59)

[Di2)P(IEI~)-P~ '~2

(3.60)

(1+

• D > Cs(1 + ]DI2) 2~-~{ (1 + (1 + A~)~/I~I- - ~ , (1 + IDI~I'(IEI*~ - ' ~ ,

IouA(IDI 2, IEI2)I _< colD[ po¢-~

A2)p(l~/=~_po¢2'

(3.61)

(1 + where the first line in the above cases holds if [D[ < A while the second one holds for ID[ > A. The constants cl, c2 depend on A, po,poo and r, while the constants c3-c~ depend only on Po, p ~ and r.

PROOF : For [D] < A we have S A = S, U A = U and therefore all assertions of the lemma are clear in this case (cf. the proof of Lemma 3.21). Henceforth we assume ]D] _ A. The inequalities (3.53), (3.54) and (3.57), (3.58) follow immediately from the

82

3. S T E A D Y

FLOWS

construction of S A and some straightforward computations (cf. (3.13), (3.49)-(3.51) and the proof of Lemma 3.21). Moreover, we notice SA'D-P-P°°( poo-r,

1

, 1 + A 2 . poo-~. - / ~ ) 2 ) (1+A2)-2a~(l+IDI2)P~-2x

× {a2~(1 + IDI2)½DE - E ÷ (a3~ ÷ a3aIEI2)IDI 2 ÷ 2a511DEI 2} + (1 + A2)a=~-2~(1 + IDI2)~2P-~ {a21 (1 + IDI2) ~2p-~:A- 1 D E " E

(1 + IDI:)% ÷ (ce31 ÷ a33[EI2)]DI 2 ÷ 2a~IIDEI2}.

(3.62)

Both terms can be handled in the same way as the corresponding terms in (3.36), proving (3.55). Furthermore, we can write a ~ (1 + IDI2)P%-2 X Ok~S~jB~jBkt = (1 + m2) ~7-~-2

f P o o - P ( I + A 2 ~P¢°2-" × t p--2-~_~ ki-~-iWl~ j H(r)÷

(3.63) } PooP-Tr-7rH(poo)

,

where g ( p ) = OL21(P - - 1 )

(B-D) (BE-E)

+

(1 ÷ IDI2)1/2

+ 2~5~(]BEI2(p- 2)(BE. DE)(B

+

+ ¢ p - 2") ( B - D ) 2",

D))

H ( p ) is exactly the factor which also appears in the discussion for the monotonicity of S (cf. (1.4.52)). The conditions (1.4.59)-(1.4.63) ensure that H ( p ) > O. But due to the strict inequalities in (1.4.59)-(1.4.63) we can choose 7r = Po~ - c, e small enough, such that also H(Tr) > O. Therefore we can estimate the terms in the squiggly brackets in (3.63) from below by Poo - P H(Tr) + p - rc H(poo) = H ( p ) . p ~ - 7r Poo - 7r

The computations in Section 1.4 show that for all p 6 [Poo,p0]

g(p) > c(1 + IEI~)[BI: which together with (3.63) gives (3.56). Inequality (3.60) follows from (3.62) setting c~21 = c~aa = c~1 = 0, aal = 1 and (3.37). Finally, from (3.49), (3.51) and 7r < Poo one easily deduces

IouA(IDI 2, IEI2)I <

c(1 + A~) -2a=~-'(1 + IDI~) ~2~-?-IDI.

Taking into account (3.37) yields (3.61) and (3.59). The proof is complete.

•

3.3. STRONG SOLUTIONS

83

R e m a r k 3.64. From (3.56) and (3.58) it is clear that for B E X, 0klSA(D, E) S~j Bkz and

p(IEI2)-p~

(1 -b

IEI2)( 1 + IDI2) ~a~2_2iBl2 ~ (1 + IDI2) 2 2

(

(1 + A~)P(I~I~-p~

are equivalent. L e m m a 3.65. Let the assumptions of Lemma 8.52 be satisfied. Then there exist constants depending on the function 1) and on aij and ~r such that (1 + [DI2)'(IEI'2)-'°° (1 + ln(1 + ID[2)),

IO,~O,~V"f< C [El(1 q-

(3.66)

[DI2)~

(1+ A~)P(I~'S~ :)-p~ (1+ ln(l+ m2)),

IO,,V~l <_ c IEl(1 + IDI2)°--1~-~f (1 + IDl~)P~l~12~-p~ (1 +ln(1 + IDl~)),

I

(3.67)

(1 + A~)P(I~I~- ~ (1 + in(1 + A~)),

10,SA(D,E)I <_ c]E](1 + [El2)(1 + ]DI2)P~-1 x

(1 + [DI~)~/I~I'~~-~ (1 + ln(1 + IDI2)),

×

(3.68) (1 + A2)~/I~I~)-~ (1 + ln(1 + A~)),

]O'~uA]
2

(3.69)

(1 ÷ ln(1 -b A2)).

PROOF : The estimates can be computed easily from (3.49)-(3.51) and (3.13).

•

R e m a r k 3.70. From (3.47), (3.60) and (3.69) and a similar argument as in (5.2.14) we deduce for all s > 1 [O,~uA([DI2, [EI2)[ _< c10 (1 + c3uA(]D] 2, [El2) • D) ~ ,

(3.71)

with a constant c independent of A. For the definition of the approximate problem we also need the notion of a mollification. Let w E ~D(~) be a usual mollification kernel with support in BI(0) and f w dx = 1. For every ~ > 0 we define we(x) = ~-3w(~) and set Ra

Ra

84

3. S T E A D Y F L O W S

Definition 3.72. The triple (E, v, ¢) = (E, v e'A, ¢~,A), for ~ > O, A > Ao given, is said to be a weak solution of the problem (1.1) and - div SA(D(v), E) + [Vv]v 6 + V¢ = f + x E [ v E ] E ,

div v = 0

in ~ ,

v = 0

(1.2)E,A

on 0 ~ ,

where S A is given by (3.13), (3.14), (3.15), (3.20), /fP0 >_ 2, or by (3.49)-(3.51), if 1 < Poo < 2, if and only if

E E H(div ) M H(curl), v

(3.73)

• Yq,

¢ • Lq'(f2), where q = min(p~,2); the system (1.1)1,2 is satisfied almost everywhere in ~, and o

E - E0 • H(div). The weak formulation of (1.2)e,A /

SA(D(v)'E)'D(~)dx+/[Vv]v~'~pdx-/Cdivcpdx

a

r

~

n

(3.74)

= (f, ¢P)l,q - X E / E ® E . D(~) dx .1

f~

is satisfied for all ¢p • w~'q(f~).

3.4

Existence of Approximate Solutions

Now, we shall show the existence of weak solutions for the system (1.1), (1.2)~,A with the additional property that V2v • Lqoc(~), where q =_ min(2,p~).

(4.1)

P r o p o s i t i o n 4.2. Let ~ C_ R 3 be a bounded domain with O~ • C 3'1. Assume that E0 • W:'r(~t), r > 3, f • Lq' (~), e > 0 and A >_ Ao are given. If 3/2 < Poo then there exists a weak solution ( E , v , ¢ ) = (E, ve'A,¢ 6,A) of the problem (1.1), (1.2)e,A, such that

IIEII,.___ IIEolI,.,

(4.3)

HD(v)iiq,~, + IlVVilq J c(f, Eo),

(4.4)

I]ouA(ID(v)I 2, tel2) • D(v)]ll,~ < c(f, E0).

(4.5)

Moreover, we have

]I¢]lp~ -< c(f, E 0 ) + ][[Vv]v~ii(p~), _< c(f, Eo, E-1),

(4.6)

I]¢~?~liq, <_ c(f, Eo) + cItSA~It q, + II[Vv]ve ~?~]](q-), + c]l[Vv]v, li(p;), _< c(f, Eo, e -1, A),

(4.7)

3.4. EXISTENCE OF APPROXIMATE SOLUTIONS

85

with 77 E 7)(~), .7 > 1, and (4.8)

IIV2vllq,loc <_ c(f, E0, e -1, A),

(1 + ID(v)12)~- [D(Vv)[2~ 2~ dx <_ c(f, E0, E-1, A),

(4.9)

n

where ~ e 7)(~) is a usual cut-off function and a > 1. Equation (1.2)~,A holds almost everywhere in ~. PROOF : The first part of the proposition concerning the existence of a weak solution to the system (1.1), (1.2)~,A is standard using apriori estimates, the theory of monotone operators and a compactness argument. However, the remaining part is not so common and we will give a detailed proof. The existence of E solving (1.1) and having the properties stated above follows from Theorem 2.3.31. We shall fix one such solution for the following considerations. Let us formally derive an apriori estimate for the system (1.2)~,A. We use here the Galerkin approach (cf. Lions [68], Section 2.5) and are very brief since the details are similar to the proof of Theorem 2.4. Using the Galerkin approximation v n as a test function in the weak formulation of (1.2)e,A with divergence free test functions, i.e. the pressure term in (3.74) is absent and only test functions ~o E 1) are allowed, we obtain

f SA(D(vn),E) . D(v'~)dx -- / f . vn d x - xE / E ® E . D ( v ~ ) d x .

(4.10)

Using (3.24) and (3.55), respectively, the left-hand side is bounded from below by c

(1 + ]E]2)(1 +

ID(vn)l 2)

2 ]D(vn)12dx

(4.11) >c/(1+

] E l 2 ) ] D ( v n ) l q d x + c / ]Vvnl q d x - c / l +

IEI2dx,

q-2 where we employed the pointwise inequality (1 + x 2) 2 x 2 >_ c(x q - 1) and Korn's inequality. The right-hand side of (4.10) is bounded from above by cHfHq, liv"ilq + c[[Eil]q, [iD(v'~)iiq,~.

(4.12)

Recall that IiD(v)iiq,~ --- f~ ID(v)iq(1 + IEI2) dx. From (4.10)-(4.12) and Young's inequality we immediately obtain (4.4) for v n. From (3.22), (3.25) and (3.53), (3.56), respectively, and the regularity properties of E it is clear that the operator - d i v SA(D(.), E) : Vq -~ (Vq)* is uniformly monotone. From (4.4) [Vv]v~ E L (q°)' for p~ _> 3/2, the estimate (4.4) and the compact embedding wl'q(~) "-~ L q*(~), for q > 3/2, are enough to justify the limiting process in the Galerkin system by standard arguments (cf. proof of Theorem 2.4 with p¢¢ = P0 = q). Thus we obtain the existence of a solution satisfying (3.73)2 and (3.74) for ~ E Vq. Moreover, for the limiting element we again obtain (4.10) and that the right-hand side is bounded by

3. STEADY FLOWS

86

some constant c(f, E0). Now, we use (3.24), (3.28) and (3.55), (3.59), respectively, on the left-hand side of (4.10)1written now for v, which is possible due to (4.4), to arrive at (4.5). Defining F E (W~'q(~))* by

fl

fl

n

(4.13)

n

we see that (F, ~}l,q = 0

V~ e Vq.

This and Theorem 5.1.10 lead to the existence of ¢ E Lq'(Ft), fn Cdx = O, such that

- fCdiv ~ d x

(F, ~O}l,q

V~ • W~'q(fl).

(4.14)

, /

fl

From (4.13) and (4.14) we obtain that (v, ¢) is a weak solution of (1.2)e,A, satisfying (3.73)2,3 and (3.74). Using Proposition 5.1.25 we have /,

Ii~IILql(f~)

----

sup

i/¢9dx[

f

<-

ii,ollL~cn)
sup

I]¢

div ~ dx i

(4.15)

IIV~llq__c n

which together with (4.14), (4.13) implies

II¢llq, < c(f, E0) + IIS%, + Ll[Vv]vdlc~.), < c(f, E0, e -1, A), where we used (3.22) and (3.53), respectively, the properties of the mollifier and (4.4). We also obtain an estimate of ¢ independent of A, using that we can derive from (3.26) and (3.57), respectively, and a similar argument as in (5.2.14) the estimate ISA(D(v), E)I p~ < c(E0) (1 + ouA(ID(v)I 2, IE[2) • D(v)), which together with (4.5) implies

IISA(D(v), E)I1,~ < c(f, Eo).

(4.16)

From (4.15), (4.14), (4.13) and (4.16) we easily deduce (4.6). In order to obtain estimate (4.7) we apply the negative norm theorem (cf. Theorem 5.1.8) for y e :D(fl), 7 > 1 to arrive at

ll¢~[i~, <_ ~ii¢~11_~,~, ÷ ctlV(C~)Ll_l,~,

< ~llCv~llc~.y+~(%V~)llCv~-~llc~.), +~

(4.17) sup II~ll~,q
I f C d i v ( ~ ) dx].

3.4. EXISTENCE OF APPROXIMATE SOLUTIONS

87

Inserting into the weak formulation (3.74) the test function ~o~ one easily computes that the last term in (4.17) is estimated by

ellS%%, + c(ve,~)llS%~-lll(~.), + ~ll[Vv]v~ ~11(~.), + c(f, E0).

(4.18)

Now, for Po <_ q* we use (4.6) and (4.16) and see

I1¢~-111(q.), + IIS%~-lll(q.), _< c(f, Eo) + II[Vv]v~ll(p~), •

(4.19)

If Po > q* we use Young's inequality to obtain that (4.20)

Ilg~-~ll(~-) , <- ~llgv~lle + ~llgll~,

which is applied to ¢ and S n. From (4.17)-(4.20) we immediately derive (4.7). It remains to show (4.8) and (4.9), which will be established by the difference quotient method. F o r V CC ~ w e c o n s i d e r ~ E:D(~), 0 < ( < 1 , ~ - 1 i n V a n d p u t (cf. (5.1.a4), T = Tka)

(4.21)

w(x) - v(Tx) - v(x).

Using h-2w(x)~2a(x), a > 1, as a test function in the weak formulation (3.74) we can derive the following identity:

if

h2

+~1 -

(S A(D(v(Tx)), E(Tx)) - S A(D(v(x)), E ( x ) ) ) . D (w(x)~2~(x)) dx

f

([Vv(Tx)lv,(Tx)

1 -h-2 f (¢(Tx)

-

[Vv(x)lv,(x)). w(x)~2a(x)dx

- ¢(x)) div (w(x)~2~(x)) dx

(4.22)

fl

1

- -h-2 f (f(Tx) - f(x)), w(xl{2'~(x) dx l"t

+ ~ f (E(Tx)(9E(Tx)- E(x)® E(x)).D(w(x)(2a(x))d x

=

O.

fl

We denote the integrals on the left-hand side by 11,... , Is and discuss them separately. Denoting SA(y) = SA(D(v(y)), E(y)) we observe

h211 = / (SA(Tx) -- SA(x)) • D(w(x)) ~2~(x) dx (4.23) fl

= h2(J1 + J2).

3.STEADY FLOWS

88

The integrals J1 and J2 must be treated differently. Let us start with J1. We use the notation F(x) = E(Tx) - E ( x ) , D~(x) = D ( v ( x ) ) + AD(w(x)) = (1 - A)D(v(x)) + A D ( v ( T x ) ) ,

(4.24)

E~(x) = E(x) + ~F(~),

to shorten our formulas. Thus we can re-write

h2J1 as

1

h2J1 : f~

0 1

:

f f °~,s,1(~,~)~,~(w)~'~,(w)e°'~'~ f~

(4.25)

0 1

+

f f oos,~(D~,E~)~',,~,~(w)e°~,~x f~ 0

= h2(Jl,i + J1,2). Now using (3.25) and (3.56), respectively, we note that (recall q = min(2,poo)) 1

Jl,1 _> c4

ff

(i +

ID~lU)~ dA

0

- ~ 9 ~,

ID(v(Tx))l2) q-22

> all f (1 + ID(v(x))l 2 + -

D(w(x)) 2~2,(x) dx, h

where we used (5.1.24). If we also employ (5.1.22) and the estimate (4.4) to bound the constant appearing in (5.1.22) we finally obtain that '

2 J

+lD(v(xlllU+lD(v(Tx))12)~ ~

+ c(f, E o ) ( [

-~

2'~2'~dx

'~¢~qdx) ~

f~

(4.26)

> cll f (1 ÷ ID(v(x))l 2 + ID(v(Tx))12) ~ - 2J

2~2~ dx

~

n 2

2

+

where we used in the last line elementary calculations and Korn's inequality. From

3.4. EXISTENCE OF APPROXIMATE SOLUTIONS

89

(3.44) and (3.68), respectively, we deduce, using Young's inequality and (5.1.24), that 1 2 ~2-~ F 12 o

< c(A, Eo) i ~h ~2~(1 + ID(v(z))12 + D(v(rx))12)} dx

-

(4.27)

fl

+ J8C H[ (1 +

ID(v(z)l2+ ID(v(Tz))12)~- - ~

2~2adx.

ft

Now we turn our attention to the integral J2. We notice that (cf. (5.1.31), (5.1.29)) I

J2=2o~ f h / -dA d Sii(T~x)dA A Wi(X)~2a-I(x~O~(X) h "" Oxi dx 1

f SA(T~x) dA wi(x) ~2a-1(x~O~(x) dx h " " Oxj 0

(4.2s)

1

--.°// n

o 1

f~

~ ( ~ ) (a~ ~ - ' ( ~ ) a~(~) . . ~ _ , ,

, a~(~)

0

= J2,1+ J2,2. From (3.22) and (3.53), respectively, HSlder's and Young's inequalities we obtain 1

i~,,l _<.(A, ~.oIf f 0 + ID(v(~lll ~)~ ~ ~w e°_, l~i ~x 12

0 2

0 1

+.(A,,<..)'<':'> gt

0

and 1

q-1

.-~-

I,.,< .I,.~.o.~'e)f (f ('+ i-(v(~..l),').,~)'-',x (4.30)

+c(A, Eo, V2,) f fl

7 qdx.

90

3. S T E A D Y F L O W S

Note, that the last term in (4.29) and both terms in (4.30) are bounded by

c(A, Eo, V2~)(1 +

IlVvl?~),

which is finite due to (4.4). Furthermore, the convective term gives

h~i~=f v~(~)~,(~)¢~o(~) dx+ f ~,~(~)~~,(~)~2o(~)a~ fl

~-

fl

/ VEk(X)~Wi(X)~2a(X)dx - / Wek(X)vi(Tx)~2a(x) dx fl

ll

-2o~/Wek(X)Vi(Tx)wi(x)~2a-l(x)dx~

(4.31)

fl

= h 2 (Ja + ,]4 + Js),

where we took into account div ve (Tx) = 0. Using the regularity of the mollified 3z function, the properties of the difference quotient, the embedding Wl,q(f~) "--+L3-q (fl) and the interpolation of L4 between L~ and L3-q we arrive at

(/

2

1,2

fl 2

tJ4t _< c1216~(jf Z~ 'c, dx) ~ + clJVv~ll~Irvll~,

(4.32)

n

IJ~l _ q' for q _> 3/2 we obtain from (4.31), (4.32) and (4.4) that

2

t/21 _< _~ ~ _q ( fc __ff_l Vw ~ 2 q~ dx) ~ + c(f, Eo, V~, e -~, a).

(4.33)

3.4. EXISTENCE OF APPROXIMATE SOLUTIONS

91

The integral [I3[ can be re-written as

h2Ia= f¢(~)(div

(w(~)e~°(~))-div (w(T-'x)(2'~(T-lx)))dx

(4.34)

n

: f ¢(x)(div w(x)(2'~(x)-div

w(T-lx)(2'~(T-lx)) dx

n

+ 2..[ ~(~)(~(~)eo-,(~) °[(~)o~,_ ~<.'--~,~J¢'"°-~'--~<-~J'o((r-~x)m ) d~ f~

= 2af¢(x)(vi(Tx)-2vi(x)+ vi(T-lx))(2°'-l(x)~dx f~

+'°f+(')~,(~-")(

e o-1 x O~(x)

fl

O~(V-'x))

( )-~-.,-e:-'(~-'-) o., j"

: h' (Js + JT). Denoting g(x) -

w(z) h

v(Tx)

-

v(x)

(4.35)

h

we can re-write ./6 as follows: f ¢ ( x ) 9i(x) -

J6 = 2 a

gi(T-lx).~c,_l,(x), _--=--O((x) ax , h Oxi

fl

:

,.

f ¢(X)~a-l(x)~(gi(x)~a(x)_gi(T-l.)~a(T-l.))~

dx

(4.36)

fl

+,.f +(.>e-,(.)}(<.(.-,.)-~
2

~V w qc~e~)- -~ +~(V~l(ll~c-~ll~,+llVvll:).

(4.37)

n

For the term J7 we have

I) + 2a f

¢(x)('-X(T-ix)@a(x) -(a(T-'x)) O((T-'X)ox, w,(T-'x)dx

3.STEADY FLOWS

92 and therefore

IJTI < c(V=~)llVvll~ + c i1¢~-~11~, + ~ lie(x) C-l(T-Xx)ll~, -

(4.38)

Thus, we obtain 2

C12

1,31<_v ( f

I

÷

(4.39)

f~

+ c 11¢U-'II~, + c lie(x) ~"-X(T-lx)ll2, • The term/4 in (4.22) can be written as

h214=

/ f(x). (v(Tx) - 2v(x) +

v(T-lx))~2a(x) dx

fl

(4.40)

+ / f(x). (v(x) - v(T-lx))((2a(x) - (2a(T-lx))

dx.

fl

The first integral in (4.40) may be treated as J6 and we obtain 2

(4.41) fl

Finally, we write/5 as

h215 = XE f

(E(Tx) ® E(Tx) - E(x) ® E(x))D(w(x))~2'~(x) dx

(4.42)

fl

-4-2X E f

(E(Tx) ® E(Tx) - E(x) @ E(x)) w(x) ® Vf(x)~2=-l(x)

dx,

which provides C12 (

VW q

2

1~51<_ -K~f -~- (~qdx)

W~

+c(E0, V()(NVEII~,+II ~-

IlqllVEIIq,).

(4.43)

Putting all calculations between (4.22) and (4.43) together, also using (4.4) and (4.7) we arrive at (with 7 = c~ - 1, rl = ( and rl = ((T-ix), respectively) c l e f (1 +ID(v(x))I2+ID(v(Tx))I2) ~22 12

+ c12(/ TVw, n

~2e='~dx

2 Cq dx) ~ < c(E0, f, e -1, .4, V2(, c~).

(4.44)

3.5. LIMITING PROCESS A --+ 0o

93

From the properties of the difference quotient we conclude that

~-~ j

~ 1

¢ (x) dx < c(E0, f , ¢ - l , A ) ,

(4.45)

i,j=l f~

which is exactly (4.8) and moreover that (cf. (5.1.35)) Vw h

~ V2v

strongly in L~oc(f~) .

Hence, we have established that

D(v(ThX)) ~ D ( v ( x ) )

a.e. in

and thus Fatou's lemma and (4.44) imply

f

(1

+ [D(vll=) ~-~ JD(Vvll2 ~=~ dx < c,

(4.46)

fl

which is exactly (4.9). Therefore the estimates (4.8) and (4.9) are proved and the system (1.2)~,A holds almost everywhere in f~. The proof is complete. •

3.5

L i m i t i n g P r o c e s s A --+ c¢

In this section we will derive estimates independent of A, which enable the limiting process A --->c¢. Thus we will come to an approximation of the problem (1.1), (1.2) where only the convective term is mollified. In preparation for the limiting process ¢ --+ 0 we will indicate the dependence of the estimates on ¢. Let us introduce some notation. Let XA be the characteristic function of the set f~A = {x e f~, ID(v(x))l > A}.

(5.1)

Furthermore let u be an integrable function, then we denote /p,A(U,~) ---- ( 1 + [El2)(1 +

ID(v)l 2)

2 [u12~2 x

(5.2) x {(1 ÷ [ D ( v ) I 2 ) ~ X A + (1 ÷ A2)v--~ (1 -

XA)} dx,

where u will be usually some second order derivative of v, and ~ will be either identical 1 or a cut-off function ~. Note, that it follows from Remark 3.40 and Remark 3.64, respectively, that there are constants c13, cla such that C13

Ip,A(D(Vv), ~)

_<.,f OkZS "A',3(D(v~,j, E)D~j(Vv)DkI(Vv)~ 2 dx f2

_< c14 Ip,A(D(Vv), f ) . Now we define weak solutions to the problem (1.1), (1.2)~.

(5.3)

94

3. S T E A D Y F L O W S

Definition 5.4. The triple (E,v,¢) = (E,v~,¢~), for e > 0 given, is said to be a weak solution of the problem (1.1) and - div S(D(v), E) + [Vv]v~ + V¢ = f + xE[VE]E div v = 0 v = 0

in f~ ,

(1.2)~

on 0 ~ ,

where S is given by (1.3), if and only if

E E H(div) N H(curl), v e E~(IEI2), ¢ E L p~(~),

(5.5)

o

and the system (1.1)1,2 is satisfied almost everywhere in ~, E - E0 E H(div ) and the weak formulation of (1.2)~

f S(D(v),E).D(~o)dx+f[Vv]v~.~odx-fCdiv~odx ~ = (f, ~o)l,q - X E /

(5.6) E ® E . D(~o) dx

is satisfied for all ~o E D(~).

P r o p o s i t i o n 5.7. Let p~ > 9/5, then for e > 0 and A >_ Ao given, the weak solutions (E, v ~,A, 0 ~'A) of the problem (1.1), (1.2)e,A satisfy the following estimates independent o/A:

IIEII2,~ < cllEoll2,~,

(5.8)

lID(v~'a)llq,, + lIVv~'Aliq < c(f, Eo),

(5.9)

HOUa(JD(v~'A)] =, IEI2). D(v~'A)]II,~ _< aft, Eo),

(5.10)

HSA(D(ve'A), E) II,~ + I[¢~,AI[,~ _< c(f, Eo).

(5.11)

Moreover, we have for Po < q* and ~ > C~o, ~/ > 1

He''A (~]lq' < c(f, Eo)(1 + II[Vv~'A]v~'A ~vllq') _< c(f, Eo, E-~),

(5.12)

25

IlVS~(D(~,~), E)e~l[~, + I[W ~'~ eXllg~ + IIV~'~ e~ll] 2
where q = min(2,pc¢) and s = max(2,po).

(5.13)

3.5. LIMITING PROCESS A --+ oo

95

The estimates (5.8)-(5.10) have already been proved in Proposition 4.2. For q > 9/5 we observe

PROOF :

3q 3q [Vv ~'A~v~A i ~' (q.),, _
4q-o

(5.14)

since aq4q_0 < ~ = q*" Because (p;)' <_ (q*)' we obtain estimate (5.11) from (4.6), (4.16); (5.14) and (5.9). Moreover from (4.7) and (5.14) we derive

I]¢~'A CIIq, < c(f, Eo) + IISA(D(v~'A), E) CIIq' •

(5.15)

From now on we wilt drop the indices e, A and write v = v ~,A and ¢ = ¢~,A. Proposition 4.2 ensures that - A v ( 2~, a >_ 1 is an admissible test function in the weak formulation (3.74). Recall that ~ is the cut-off function for the interior regularity. Let us denote by I 1 , . . . , I5 the integrals in (3.74) for this test function. We will treat the resulting terms separately. We see that

I1 ~'/ ~kSA(D(v),E)Dij(~k)~ 2~dx +2a

a 0V 2o~--10{ Sij(D(v),E)DiJ(~xk )~ ~xk dX

eo-

(5.16/

OXjO"~{ Avj { 2 a _ l ~ i )

dx

= Jl+&+Ja. The term J1 can be written as

j~ = f c3,mSij(D(v),E)Dun(-~xk)Dij(-~xk) A Ov Ov { 2,~dx

OE,~

..( ~__;~)~o dx

(5.~ 7)

+ f onSA(D(v)'E)-~xkD" = Jl,1 + Jla.

Using (3.25) and (3.56), respectively, and (5.3), together with (5.1.22) and (5.9) we obtain

J1,1 ~_~~-Ip,A(e(~v),~ ca)-t- c(f, So) ( /

2

Ie(~v)lq~aqdx) "~ 2

f~ 2

&STEADY FLOWS

96

where we also used Korn's inequality and elementary calculations. From (3.44) and (3.68), respectively, we obtain, for s > 1,

IJ1,2{
I E i ( I + I E I 2 ) ( I + I D ( v ) I 2) 2 ID(Vv)ilVEI¢2° x

× {xA(1 + IDI~)~ (1 + ln(1 + IDle)) + (1 - XA)(I + A2)~ (1 + In(l+ A21)} dx < c(E0) f {(1 + IEI=)(1 + IDI s) q-2 2 (1 + A2)e~_~ID(Vvlj2 (2~} ½ ×

(5.19)

i2

{(1 +

x

+ c(Eo)

if~

^ p--q+(8-1)p

IDI~)~(1 + A s)

2

1

(2"}~XA dx

{(1 + IEI2)(1 + IDI 2) 2 (1 + IDI2)P-'~ID(Vv)1252'~} ~ x ~ -

x { 0 + IDI2)~(1+ IDI 2)

_~%,~(~(~v),~o)+ ~(~o)( f

p-q+(s-l>

~

1_

~2"}~(1- xA)dx

o + o~(l~(v)l~,

i~1~).~(~))~e o ~),

fl where we also used (3.48) and (3.71), respectively. The term in the last line can be further estimated by (cf. (3.24), (3.2S), (3.55), (3.59))

c(Eo) (~ + IlVvll; IISA¢"I{~) •

(5.20)

q--S

Next, we notice that

2 ~,~(/i~:vl,¢o.~)~ fl

lJ2 + Jal < -~-

+ dlS~¢"-lll~ ¢ .

(5.21)

For the convective term we have

2 1121--
~

a 2 + c( f IVviq' lvelq'

dx) ~

(5.22)

n and for the pressure term we obtain c15

2

a 2

~-lH~'

(5.23)

The remaining two terms are estimated by

1/41 ~_~-~HV2v~iI2q "t-c]lfll~,,

(5.24)

97

3.5. L I M I T I N G P R O C E S S A -+ oc

and

f

OE¢

Ov

J

OXk

-aXk-

IIgl = 2 X I Ei--x--Dij(--x--)~

\

~2

1215

2

Oxy

2~

dz

Oxi -

zuiJ~'-~Xk)-~xk]¢

dx

(5.25)

a 2

< TIIV v¢ II, + ~(Eo). Putting all computations between (5.16) and (5.25) together we arrive at C15 ,'~2 ~ - I p , A ( D ( V v ~ ' A ) , ~¢~) + -~v v ¢,A~a ~ 2q

_<~(f, Eo, V~/(1 + IIS~ID(,¢~,~/, E/~°11~ + IISAIDIv~'~/, E/~°-~11~, +ll[Vv ]v~ ~11¢+ <__

c(f, E0, V~)(1 +

)

(5.26)

¢A ~ a-1 ]]q2 ) , HSA(D(v~'A),E)~a-t[[%, + [][Vv eA Iv e'

where s > 1 is chosen appropriately. Moreover, (5.9) and (5.15), with 7 = a - 1, have been taken into account. Based on estimate (5.26) and Proposition 5.2.23 we must distinguish the cases (i) q = 2 < _ p o o < _ p 0 , (ii) 9 / 5 < q = p o o < _ p 0 < 2 , (iii) 9 / 5 < q = p ~ < 2 _ < p 0 . Let us start with (i) T h e case 2 < Poo _< P0 From (5.26), (5.2.25) and (5.2) it follows

IlVSA(D(v), V.) ~

(5.27)

c(f, E0)(1 + IISA(D(v), E) ~-111g, + II[Vv]v~~llg), where we employed (5.9), (5.10), s < p~ and ~ < 1. We will use the global estimate (5.11) and the first term on tl~e left-hand side of (5.27) to bound the term with SA on 3po the right-hand side of (5.27). From the embedding WI'P~(f~) ~ L 2po-3 (12) we obtain the restriction that P0 < 6, since we must require that 2s < ~2po--3" Motivated by the

98

3.STEADY FLOWS 31Oo

interpolation of L 2~ between/2~ and L2~o-3 we have for A = 3(28(po-1)-po) 2spo

f IsA]2s~(ct-1)2sdx fl

=

f (IsAI~Ct)2s'~IsAI2s(1-)~)~2s(ct(1-'k,-1)dx f~

< IIsA~ctllZ~o IIsAII~(l-x)

(5.28)

2po-3

1

Act

2

_< c(f, Eo) + 511VS ~ llpo, as long as ~ > (1 - A) -1. Inequalities (5.27), (5.28) and the embedding Wa'2(f~) L ~(f~) therefore lead to IIvSA(D(v), E)~ ctI1,,~, 2 +

IIV2v,~ctll~

+

IlVv,~'~ll~

(5.29)

< c(f, E0)(1 + II[Vv]v, ~ctll~), as long as 2 < Poo < P0 < 6 and o~ > (1 - A) -~. From (5.9) and the properties of the mollifier one gets that the right-hand side of (5.29) is bounded by c(f, E0, ~-~). (ii) T h e case 9/5 < Poo _< P0 < 2 From (5.26), (5.2.24) and (5.2.9) we conclude 2a

I[vSA(D(v),E)~I[~ + ItVv~

P~ + IIV~v ~ ctI1,~ I]3p~

(5.30)

<_ c(f, E0)(1 + tISA(D(v),E) fct-all2 2 ~ ,,,~- + II[Vv]v~ ctI1,~), where we used (5.9) and (5.10). We proceed analogously to the case (i). We do not obtain any additional restriction on Poo and P0. As in (5.28) we can derive that 2 IIS A ~ ct-1 lisp< c(f, E0)

+

(5.31)

IIvsA~CtII 2 ,

as long as c~ < (1 - A) -1, where A = 8((po-t),p~-po(poo-1)) (5pO--6)sp~

which comes from the

interpolation of L sp~ between /2g and L ~. Thus we have arrived at 2ct

2 I[VSA(D(v),E)~11~ + IlVv ¢ ~ I1~;~ + IIV2 v~ ct I1~

(5.32)

2 <_ ~(f, E0)(1 + [l[Vv]v,¢ a IIv),

as long as 9/5 < poo < Po < 2 and c~ > (1 - A) -~. With the help of the interpolation o f / 2 " b e t w e e n / 2 °0 a n d / Y ~ we obtain

II[Vv]v~ ~ctll,,- ~ o(~-~)llVv ~ctll,,~ _< ~llV~v,,°ll,,~ + c(c-~)llW,~'%~.

99

3.5. L I M I T I N G P R O C E S S A --+ c¢

This inserted into (5.32) together with (5.9) and the properties of the mollifier implies that the right-hand side of (5.32) is bounded by c(f, E0, e-l). Finally, we come to (iii) T h e case 9/5 < p ~ < 2 < P0 From (5.26), (5.2.25) and (5.2.9) we obtain I [ v S A ( D ( v ) , E ) a IIp~ 2 +

_<

24

2 + IIV2v~ a IIp~

IlVv~ll~

c(f, Eo)(1 + HSA(D(v),E) ~ f~-lll,,~p2 +

(5.33)

z ll[Vv]v~ ~ lip-)

where we applied (5.9), (5.10), s < p~ and ~ < 1. We will use the interpolation of 3po

L "p" between/2~ and L2~o-a , which is possible for Po < P~. As in (5.31) we conclude 2 1 A c~ 2 IISA ~a - 1 II,p---~llVS ¢ II~+c(f, Eo)

as long as c~ > (1 - A)-I, with ~ = 3(,p~(po-1)-po@~-l)). Thus we note that -spoopo 2c~

tIvSA(D(v),

2 + E) ~~ I1~;

iiVv ~

p¢¢ + 2 II~p~ IIV2v~ ~ IIp~

(5.34)

_< c(f, Eo)(1 + II[Vv]v, ¢~11~), as long as 9/5 < p ~ < 2 < Po < P~ and ~ > (1 - A)-I. As in the case (ii) we obtain that the right-hand side of (5.34) is bounded by some constant c(f, Eo, e-~). The estimates (5.29), (5.32) and (5.34) together imply (5.13). In all cases (i)-(iii) we have shown that

lisa ~11¢

___c(f, Eo) + c[IVSA ~ l l , , ,

where s = max(2, Po). Therefore we deduce from (5.15) and (5.13) the estimate (5.12). The proof is complete. • In Proposition 5.7 we have prepared everything for the limiting process A --+ co. From (5.9), (5.1I)-(5.13) we obtain the existence of v ", ¢" such that V v ~'a -+ V v ~ v ~'A --~ v ~ ¢~,A ~ ¢~

strongly in Lloc(~2), ~,~ weakly in Wlo c (f~)

a < 3-q,

n wllo~(a)

ql

A w~,~(a),

(5.35)

I

weakly in Llo¢(f2) f3 LPo(f~),

as A --+ c¢ at least for a subsequence. From (5.35)1 it follows that V v e'A --+ V v ~

a.e. in f2,

(5.36)

and since S A converges locally uniformly to S, and thus also OU A converges locally uniformly to OU we conclude sA(D(ve'A), E) ~

OuA(ID(v"A)I2 , IEI2) -~

S ( D ( v e ) , E)

a.e. in f~,

0U(ID(v')l ~, IEIz)

a.e. in f~.

(5.37)

100

3.STEADY FLOWS

Therefore we obtain from (5.13) and (5.11) that sA(D(v~'A), E) --~ S(D(ve), E)

weakly in Wllo~' (~) n LP~(gt),

(5.38)

since weak and almost everywhere limits coincide. Moreover from (5.10), (5.37)2, (5.36) and Fatou's lemma we deduce I]0V(ID(v~)] 2, lEa2) • D(v~)]]l,~ < c(f, E0),

(5.39)

which in turn implies4

IIw%~ + tID(v~)i]~/l~l~l,~ < c(f, E0),

(5.40)

where we used (2.2.49) and Korn's inequality. We also observe that lim f

A---}oo

I[Vv~'nlv~'Aiq'~aq'dx= f

I[Vv~lv~]q' ~"q' dx

(5.41)

which follows from the almost everywhere convergence of the integrand and Vitali's theorem, since q' < q* (cf. (5.35)1). From (5.41) and the weak lower semicontinuity of the norm we deduce that the estimates (5.9)-(5.13) remain valid with v ~'A and cE,A replaced by v ~ and ¢% respectively. It remains to show that v ~ satisfies the weak formulation (5.6). The limiting process A --+ oc in the weak formulation (3.74) is clear in all terms except the term with S A (cf. the proof of Theorem 2.4). For this term we obtain for ~ E :D(f~) I SA(D(ve'A)'E)" D(~) dx --+ /

S(D(vE), E) • D(~) dx,

n

using Vitali's theorem. This is possible due to (5.37)1 and the estimate (5.11), which gives the uniform integrability of SA(D(v~'A), E) • D(~o). Therefore we have proved P r o p o s i t i o n 5.42. Let 9/5 < poo _< P0 < q*, where q = min(2,p~). Assume that > 0, Eo E W2'T(~), r > 3 and f E LP'(~) are given. Then there exists a weak solution (E, v ~, Ce) of the problem (1.1), (1.2)e which satisfies the estimates

I]Eil~,r < c IIEoII~,T, ilVv%~ + IID(v~)ilp,~ _< c(f, E0), II0V(ID(v~)l 2, IEI~). D(v~)lll,~ < c(f, Eo), IfS(D(v~), E)II~ + I1¢%~ _< c(f, Eo),

I1¢~CI1¢ < c(f, E0)(x+ll[Vv~]v~CIl¥),

(5.43) (5.44)

(5.45) (5.46)

(5.47)

25

IlVS(D(v% E)~"t12,, + IIW ~ < c(f, Eo)(1 + II[Vv~]v:~°ll~,), where q = min(2,poo) and s = max(2,p0). 4Note, that in the case poo _>2 the estimate (5.40) is better than (5.9).

(5.4S)

101

3.6. L I M I T I N G P R O C E S S ¢ --~ 0

3.6

Limiting

Process

e --+ 0

Because of the results in the preceding section the situation is now considerably easier. q' We just need to check if the convective term in the Lloc-norm can be handled by the left-hand sides in the estimates established in Proposition 5.42. As in the previous section we distinguish the cases (i)-(iii). (i) T h e case 2 _< Poo _< Po < 6 We first consider the situation when 3 < Poo. We have, using (5.44) that

II[Vvflv~~11~ - ~llVv%llVv% <_c(f, Eo).

(6.1)

If Poo E [2, 3) we obtain

II[V¢]v~-,~'~ll,, ~ llv%llV¢ ~~'11,~ ilV¢ll~/fllV¢,~'~ll~/' < ~(f, E0) +

(6.2)

~llVv~,~°ll6, Z

where we used the interpolation of L 4 between L 2 and L 8, the embedding Wl'2(f~) "-4 L6(f}), Young's inequality and (5.44). Moreover, L e m m a 5.2.32 yields _

~ ,,pg + II0~S(D(v~),E)VE~CIIp~ t

< c(f, E0)(1 + tlVS(D(v~), E)

~11~),

(6.3)

where we used (5.44), (5.2.39) and a similar argument as in (5.28) to handle the righthand side in (5.2.39). Therefore we deduce from (5.2.41), (6.3), (5.48) and (6.2) and (6.1), respectively,

IIV2v~ ~"11~+ I l V v ~ l l ~ + IIVv~ CIl'~o ~

2po-3 (poe-l)

_< c(f, E0).

(8.4)

6 Note, that 3___~_~2po_3 ~ ' ~ - 1) > 6 ifpo < 5---:~" Since we have shown that II[Vve]v~%2 is finite, we see from (5.47) that

11¢~112 ~ c(f, E0).

(6.5)

(ii) T h e case 9/5 < Poo _< P0 < 2 In this case we observe for the convective term

II[Vve]Ve~e all2

__~ C ~ V e 2

(6.6)

~yeCall2 4poo--6

102

3.STEADY FLOWS

Furthermore we obtain, for A = 9-4pz¢ 2 / 4p¢~-6

3P,~o tl_A~

2 Q x" , -~ A°

IlVv ~ ~ l l 2 ~p~ n 2a

-2A < IlVv'~p3p. v-, vv

e 2(l-A)

p.

as long as 2A < poo, which is equivalent to po~ > 9/5. From this and (6.6) we therefore

get 2a 2 2(2-A) 2X [][w ] ~6 a lieu _ llVv~ 1[~oo llW ¢ e.~ []~poo 1

2~

(6.7)

_< c(f, Eo) + 21IVv~ ~p-=~II~~ , again for 2A < poo or equivalently p¢0 > 9/5. Thus (5.48) and (6.7) yields 2a

2 IIVS(D(v~),E) f~]l~ + [IVv~ ~-=~ II~= + IIV2 v ~ (~ IIp~o -< c(f, E0).

With regard to the first term in the above estimate we use Lemma 5.2.32, (5.2.39) and (5.2.41) to derive VvE~a '~ 2(v~-~) 6(w-1) < c(f, E0)(1 + Ip2(D(VvE), fa)) < c(f, E0)(1 + IIVS(D(v~), E) ~ll~) _< c(f, Eo). However 3po¢ > 6(Poo - 1) for poo < 2 and thus we do not gain a better information. Hence we have proved 2a

IlVv~ll~;~

2

a

2

+ IIV v~ I1~= < c(f, E0).

(6.8)

Since we have shown that II[Vv~]v E ~ ~a IlpU 2 is finite, we obtain from (5.47) that I]¢~(~11 ¢

_< c(f, E0) •

(6.9)

(iii) T h e case 9/5 < poo < 2 < Po < P~ We treat the convective term as in the case (ii). Thus we obtain from (5.48) and

(6.7) 2Ct

HVS(D(v~),E) ~ [ [ ~ + I]Vv~'~-~ ' ' p.3p~¢ ~ + l[V2v~ "[[p~2 _< c(f, E0). With regard to the first term in the above estimate we proceed as in the case (i) and deduce

]lVv~"llP~ ' 2 p o - 3 (p¢¢--1)

+ liver - I1~ < c(f, E0)

(6.10)

3.6. L I M I T I N G P R O C E S S ~ --+ 0

103

since for P0 < P~* we have that ~2po-3 (P~ - 1) > 3p~. As in the previous cases we deduce from (5.47) []¢~ ~llp~ -- c(f, E0).

(6.11)

In all cases we have derived estimates which are sufficient for the limiting process E -~ 0. In fact, from (6.4), (6.5) and (6.8)-(6.11) we deduce the existence of v, ¢ such that Vv ~ ~ Vv v ~ --" v

strongly in L~o¢(12), a < 3--q aq 2,q 1/3 weakly in Wlo¢ (~) A Wlo¢ (~) n wl,P~ (i2),

¢~ ~ ¢

weakly in L~oc(fl ) N LP~(~),

where q = min(2,poo) and ~ = max(6, ~2p_3(pc~ -- 1)),fl = 3poo, f~ = a___~_~ 2p0_~tt,~ _ 1) in the cases (i), (ii), (iii). Moreover, the estimates (6.4), (6.5) and (6.8)-(6.11) remain valid for the limiting elements v, ¢. In the same way as in the proof of Proposition 5.42 we deduce that v, ¢ satisfy the weak formulation (2.3). Thus we have proved all assertions of Theorem 3.7. •

4 Electrorheological Fluids with Shear Dependent Viscosities: Unsteady Flows 4.1

Setting

of the Problem

and

Main

Results

In this chapter we investigate unsteady flows of a special incompressible electrorheological fluid in a bounded domain Q C_ R 3. This motion is governed by the initialboundary value problem 1 div E = 0

in QT,

curl E = 0

E.n=E0.n

(1.1)

onlx0fl,

0v 0--t- - div S ( D ( v ) , E) + [Vv]v + V ¢ = f + x E [ V E ] E ,

in QT

div v = 0 v=0

v(o) = v0

(1.2) on I x 0 ~ , in ~ ,

where QT = I × ~ = (0, T) × ~. The data T > 0, f , Eo and v0 are given. We will investigate (1.1), (1.2) for two different models, which will be described now. In the first model, to which the main part of this chapter is devoted, we assume that the extra stress tensor S has the form p-2 S = a31(1 + IEI2)0 + ID(v)l 2) ~ D ( v ) ,

(1.3)

where p = p([E[ 2) is a Cl-function such that

1 < p~ < p(IEI~) < p0,

(1.4)

a31 > 0.

(1.5)

and where

1Similarly as in (3.1.2), we have divided (2.1.2) by the constant density Po and adapted the notation appropriately.

4. UNSTEADY FLOWS

106 Obviously, we have for D , B e X = {D E ]~y~, tr D = 0} S ( D , E ) . D _> a31(1 + IE[2)(1 + ]DI 2)

OijSkt(D, E)BijBkl >_Ol31 (1 -~ I~+12)(1 +

p(IE12)-2

2

ID]2,

IDI2)p(IEL+>-2]BI 2 .

(1.6) (1.7)

The above system (1.1)-(1.5) describes the unsteady motion of an incompressible electrorheological fluid in a bounded domain f~ over the time interval I = (0, T). In contrast to the steady problem we have restricted ourselves to the case when S has the pseudo-potential form (1.3) and discuss only the case poo _> 2. These are not very realistic assumptions, since we have lost the dependence of the material response function on the direction of the electric field (cf. Section 1.4) and since most electrorheological fluids possess a shear thinning viscosity, i.e. poo < 2. While the restriction poo _> 2 can surely be weakened as in the steady case (cf. Chapter 3), it seems that it is hard to relax the pseudo-potential form (1.3) of S in the case of Dirichlet boundary conditions. However, the results presented here are the first ones for the unsteady motion of an electrorheological fluid with Dirichlet boundary conditions and together with Rfi~i~ka [113] 2 and [114] a they are the first results for unsteady systems with non-standard growth conditions and a nonlinear right-hand side. Let us make some comments about the solvability of the unsteady system (1.2)(1.5). As already pointed out in the previous chapter, this system can not be handled by monotonicity methods. This arises from the fact, that in the parabolic setting it is fundamental to treat the space and time variable differently and to have the equivalence of the spaces

Lq(QT)

and

Lq(I, Lq(f~)).

In the case when we deal with the s p a c e / 2 (t'x) (QT) it is not clear how to achieve this equivalence. However, we can adapt the second method presented in Chapter 3 to obtain solutions also in the time dependent case. The presented approach uses and generalizes techniques and ideas developed for the treatment of unsteady problems arising from the theory of generalized Newtonian fluids (cf. M~lek, Ne~as, Rfi~iSka [71], [73], Bellout, Bloom i e ~ a s [9], M~lek, Ne~as, Rokyta, Rfi~+i~ka [70]). Similar considerations as in the beginning of Section 3.3 imply that we must use again an approximation of the elliptic operator and that we have to mollify the convective term. Moreover, let us point out that in the time dependent case the properties of both the time derivative ~0 v and the pressure ¢ are governed and computed from the system (1.2). This prevents us from working with local estimates as in Chapter 3, because we would run into a circle argument. Thus we are forced to derive global in space-time estimates of the second order spatial derivative V2v. Let us now define weak and strong solutions to the system (1.1)-(1.5) and state the main result of this 2Here the full system (1.1), (1.2) with Dirichlet boundary conditions and a stress tensor of the p(]EI2)-2 form S = oc31(1 -k [E[2) (1 + [Vv[2) 2 Vv is discussed. 3Here the system (1.2) is treated with the general stress tensor S given by (3.1.3) but under space periodic boundary conditions.

4.1. SETTING OF THE PROBLEM AND MAIN RESULTS

107

chapter. Recall the following definition of function spaces of divergence free functions (cf. Section 2.2) V --- {u e :D(~), div u = 0}, Vq - closure of V in [[ V. [[q-norm, Vp(.) = closure of Vin [[ . [[1,p(.)-norm, Ep(.) --- closure of V in ]] D ( . ) ][p(.)-norm. Definition 1.8. The couple (E,v) is said to be a weak solution of the problem (1.1)(1.5) if and only if E ~ CI(Qr),

v e Lcc(I,H) n LP°¢(I, Vpo¢), • D(v) e LP(IEI2)(QT),

(1.9)

the system (1.1) is satisfied in the classical sense and the weak formulation of (1.2)

(1.10)

/,

=/f.~-

Xs E ® E . D ( ~ )

dx

is satisfied for all ~a E V and almost all t E I, with q = max(p0, 5___~_~_~ 5 p ~ - 6 ]" D e f i n i t i o n 1.11. The triple (E, v, ¢) is said to be a strong solution of the problem

(1.1)-(1.5) if and only if E ~ CI(Qr), v E L po-1 (I, V~o_-)_ ) n L~o-, (I, W 'To+, (fl)), po+~ . E ~ D(v) e L°°(I, LP(IEff)(fl)) N n 2po P( I )(QT), Ov O--t' ¢ E L 2(QT),

(1.12)

the system (1.1) is satisfied in the classical sense and the weak formulation of (1.2) / ~

_~.Ov ~ d x + f S ( D ( v ) , E ) . D ( ~ ) d x + / [ V v ] v e . ~ d x n,

~ = f f'~dx-xE/E®E'D(~)dx fh

12~

is satisfied for all ~o e 7)(f~) and almost all t e I.

/ ¢div ~ d x ~

(1.13)

4. U N S T E A D Y F L O W S

108

T h e o r e m 1.14. Let ~ C R 3 be a bounded domain with O~ E C3'1 and assume that T > 0, E0 C C : ( [ , W 2 ' r ( ~ ) ) , r > 3, f e L2(QT) and v0 • Ep(IEI~) are given. Then there exists a weak solution (E, v) of the problem (1.1)-(1.5) whenever

(s5 ' 2 ~ 3+p~ --p~)/"

2
(1.15)

Weak solutions are strong if we additionally know that

9

3(3 -Poo) _< Pc~ <_ P0 _< 2(5 - 2poo)

(1.16)

and unique if we require that E is orthogonal to HN(~). Moreover, strong solutions satisfy for all ~o C C ~'l the estimate T

(1 +IEI~)(I +lD(v)l 2)

~

ID(Vv)l~dxdt <_ c(f, vo, E),

0 f~o

and belong in pa~icular to L 2 (I, W ~ ( a ) ) . 0~ ~ 2

N

For the tangential derivatives holds

P(I~'I2)-2

0v

2

and in particular V o~,, ~--- s = 1,2, belong to the space L2(QT). R e m a r k 1.17. 1) The above theorem shows the strength of the applied method, since the lower bound for the existence of weak solutions is 2. Even if the method of monotone operators could be applied, the lower bound would be at best 11/5. 2) The lower bound for the existence of strong solutions coincides with the bound established by MMek, Ne~as, Rfi~i~ka [73] in the case p = const., E = 0. However the upper bound here is more restrictive because of the additional term [ISII~] on the right-hand side of the estimate for the time derivative (cf. (3.13)). 3) The possible range for weak and strong solutions is

2.7

2.6

2.5

:00 2.4 2,3

2.2

2.1

, 2:1

2:2

2:3

~4 P~

2:5

2:(s

2:7

4.1. SETTING OF THE P R O B L E M AND M A I N RESULTS

109

Let us now come to the second model for the extra stress tensor S in (1.2). Here we assume that S is linear in D and thus the extra stress S is given by (cf. Lemma

1.3.34) S = (aal + aa31El2)D + a51(DE @ E + E ® D E ) ,

(1.18)

where

a33>0,

aat>O,

a33+~as, >0.

(1.19)

In this model we have completely lost the shear dependency of the viscosity. However, we have a dependence of the stress tensor on the direction of the electric field, which is also important. In this situation we can prove T h e o r e m 1.20. Let ~ be a bounded domain with OQ E C2'1 and assume that Vo E H, f E L2(I, W-t'2(Q)), and Eo E L°°(I, Wt'2(Q)) are given. Then there exists a weak solution (v,E) of the system (1.1), (1.2),(1.18), (1.19) such that v E L°°(I, H) N L2(I, V2), Ov

a-~ e L2(I, (Va/2)*),

(1.21)

E e L ~ ( I , W1'2(12)), o

E - E0 E L ° ° ( I , H ( d i v ) ) , which satisfies (1.1) almost everywhere in QT and (1.2) in the weak sense, i.e. Ov

+ f S(D(v), E) n,

(1.22)

+f[Vv]v.~dz=f f.~dx-xE f E®E.D(~)dx 12~

12t

i2t

is satisfied for all ~a E Y and almost every t e I.

The existence of the electric field E with the above properties follows directly from Theorem 2.3.31. Having the electric field at our disposal, the proof of the existence of the velocity field follows essentially along the same lines as the proof of the corresponding statement in the theory of the unsteady Navier-Stokes equations (see e.g. Ladyzhenskaya [58], Lions [68] or Temam [120]) and therefore we will skip it here. The essential tools are the apriori estimates T

sup IIv(t)ll~ + [ ]lD(v)ll~,~ + IlVvll~ dt < c(f, E0, Vo), tEI

J

0

(1.23) 0~ L~(r(V~/2)')

_< e(f, E0, v0),

4. UNSTEADY FLOWS

110

together with the Aubin-Lions lemma, which enables the limiting process in the convective term. The limiting process for the other terms in (1.22), which are linear in v, is no problem due to the estimates (1.23) and the fact that the electric field E is known apriori. The remaining part of this chapter is devoted to the study of problem (1.1)-(1.5). 4.1.1

Approximations

We will use the same type of approximation SA(D(v),E) for the elliptic operator S(D(v), E) (cf. Section 3.3.1). For the convenience of the reader we recall the relevant definitions and properties of SA(D(v), E) in the special case treated in this chapter. We define, for A > 1, SA = ~31(1 + IEI2)OijuA(ID(v)I 2, IEI2),

(1.24)

where 0ij denotes the derivative with respect to the components Dis(v), i, j -- 1, 2, 3, of the symmetric part of the velocity gradient, i.e.

OijuA(ID(v)I2' IEI2) =

OUA(]D(v)I 2, IEI2) ODi3

We define for all D E X U(ID[ , [El u)

ID[ < A,

uA(ID?, IEI2) =

1

(1.25)

ao÷al(l + lDI2)2 +a2(l ÷ ]D]2)

ID] >_A,

where ai = ai(lE]2), i -- 0, 1, 2, and U(IDI 2, tel 2) = ~ ( (11

\

2-~j-~ - 1) . + IDI2)P

(1.26)

Note, that

2~

OijU(JDr~, JEI2) = (1 ÷ D )

u(0, lEt ~) =

Di~,

2

O,jU(O, IEI 2) = O.

(1.27)

(1.2S)

From the definition of S and (1.24)-(1.27) it is clear that SA(D,E) = S(D,E)

if IDI ___A.

(1.29)

We require that uA( ., IEI2) E C2(R+) and UA(]DI 2, .) E CI(R+), which allows us to compute the coefficients a0, al, a2 as a0=

+~-

~

)

al = (2 - p)(1 + A 2) ~ , a2 = -p~1--( 1

+

A2)

p,

(1.30) .

111

4.1. SETTING OF THE P R O B L E M A N D M A I N RESULTS

L e m m a 1.31. Let S be given by (1.3), such that the conditions (1.5) and 2 < poo <_Po are satisfied. Let S A be given by (1.24)-(1.26) and (1.30). Then we have for all i, j, k, l = 1, 2, 3 all D, B E X , all E E R 3 and A su~ciently large

(1.32)

JsA(D,E)I < c,(A)(1 + IE[2)(1 + tDI2) ½ , 10ijSA(D,E)I <_ c2(A)(1 +

IEI~),

(1.33)

(1 + IDI*)'(W'~~)-u , SA(D, E). e _> ca(1 + IEI2)IDI 2

+ A2)PGEI2)_2

(1.34)

(1 (1 + IDi~)'(W'~')-2 , OklSA(D,E)Bi¢B,t >_ c4(1 +

IEI~)IBI2

+ A2),(iz,,)_ 2

(1.35)

(1 (1 + IDI2)'(IE~2)-2 , + A2)p(izl,)_ 2 (1

[SA(D,E)I _< c5(1 + JEJ2)IDJ

(1.36)

(I + IDI2)'(IE~ )-u ,

I0ijSA(D, E)I _< c6(I+ IEI2) {

cTlD[2

(I + A2)'(IE~ )-2

(1.37)

(I + IDI2)'(I~'~ ')-2 (1 + IDI~)'(IE~ )-~ , < uA(IDI 2, IEI 2) < cMDJ 2 { (1.38) . P(]EI2)-2 A2) P(llg~2) -2 (1 + A2) 2 (1 + ISA(D,E)I _< c9(1 + uA([DI 2, JEJ2)) P(Izl~) ,

(1.39)

(1 + IDI2) '(IE~2)-2 ,

10UA(IDi 2, IEI2)l _< Clolel

A2)~(iz~)_~

(X.40)

(1 +

(1 + IDt ~) 1 + [$A(D, E)I _> c::

p(IEI2)-i

2

+ A2) p(IE~)-:~

, (1.41)

(1 where the first line in the above cases holds if ]D] <_ A while the second one holds for IDI >>_A. The constants c3-c11 depend on po,Poo, whereas c1,c2 depend on A, po and Poo.

4. UNSTEADY FLOWS

112

PROOF : The statements (1.32)-(1.37) and (1.40) have been already proved in Lemma 3.3.21. Inequalities (1.38), (1.39) and (1.41) are clear for ]D I < A. For the case ID] > A we write, using (1.25) and (1.30) uA(IDI 2) : ~ - ~ ( 1 + A2)a~-~ ((1 + IDI2)½- (1 + A2)½) 2

+ ( I + A 2) 2 (IDI~ - A ~)+ _> 11( + A 2) 2 (IDI 2 p

A 2) +

+A2/ 2 - 1 ) +

(1+A2/2~-~A ~

A 2 ~ - I ) + p ( I +! A

2) 2 A 2

> -1(1 + A 2) 2 IDI2, P where we employed p~ > 2 and ]D I > A > 1. This gives the lower bound in (1.38). The upper bound follows from the above formula for UA, inequality (3.3.37) and p-2 2~ p-2 c ( p ) ( l + x 2) 2 z 2 < ( 1 + x ) 2 - 1 _<5(p)(l+x 2) 2 x 2, (1.42) which holds for x E [0, oc) and p > 1. Inequality (1.39) is a consequence of (1.36) and a argument similar to (5.2.14). In order to prove (1.41) we recall that (cf. (3.3.31)) s~A(D,E) =

0:31(I~-]EI 2) (i+ 1D[~_)I/2~2a2

Dij.

Due to (3.3.33), p~ > 2 and ]D] > A > 1 we thus obtain

ISA(D,E)I >_ ~31(1 + A2)~IDI > c~1(1 + A ~ ) ~ . This completes the proof of the lemma. R e m a r k 1.43. From (1.35) and (1.37) it is clear that

OkIsA(D, E)BijSk~ and

(1 + IDI2)'¢'~/-~ (1 + tE[2)IBt2

(1 + A2) '(Iz'2)-2

are equivalent quantities. Moreover, from (1.34), (1.36) and (1.38) it follows that SA(D, E ) . D

and

uA(IDI 2, [E] 2)

are equivalent quantities. Again we also need the derivative with respect to the components En, n = 1, 2, 3, of the electric field. We use the notation IE]2) O"UA(IDI2'IEI2) = OUA(IDI2' 0E.

(1.44)

and similar formulas for S A, U and S. We shall use the letters m, n as indices for this derivative.

113

4.1. SETTING OF THE PROBLEM AND MAIN RESULTS

L e m m a 1.45. Let the assumption of Lemma 1.31 be satisfied. Then there exist constants depending on the function p and ~ij such that

(1 + I D I 2 ) ~

(1 + In(1 + IDI2)),

]O~OijuA(]DI 2, [E]2)I _< c]E] ]D[ {

(1.46)

(1 q- A2)PE2AI~ (1 + ln(1 + lA12)),

(i + IDl2)~(l+In(l + IDI2)), (1.47)

[anSA(D,E)I _< clEl(l+ IEI2) IDI (1 + A 2 ) ~

(1 + ln(1 + Au)),

(1+ [ D I 2 ) ~ ( l + l n ( l + ]0nuA(]DI 2, IEI2)] _< clEl(1 + ]DI2)

IDI2)), (1.48)

+ A2)~ (1

(1 + ln(1 + A2)).

PROOF : These estimates follow immediately from (1.24), (1.25) and (1.30). R e m a r k 1.49. From (3.3.47), (1.48) and (1.38) and a similar argument as in (5.2.14) we deduce for all s > 1 [o~uA([D[ 2, [E[2)[ _< c(1 + UA([D[ 2, ]El2)) 8 .

(1.50)

Let us finish this section by defining the approximate problem (1.1), (1.2)~,A. Definition 1.51. The triple (E, v, ¢) = (E, v ~'A, ¢~.A), for ~ > O, A >_ Ao given, is said to be a strong solution of the problem (1.1) and 0v - - -- div SA(D(v), E) + [Vv]v~ + V¢ -- f + xE[VE]E, 0t div v = 0 v =0

v(0) = v0

in QT,

(1.2)~,A on I x Of~, in ~ ,

where S A is given by (1.24), (1.25), (1.30), if and only if E • C v • C(i, V2) N L2(I, W2'2(f~)), Ov - ~ , ¢ • L2(QT) ;

(1.52)

114

4. UNSTEADY FLOWS

the system (1.1)1,2 is satisfied almost everywhere in QT, for almost all t • I we have o

E(t) - E o ( t ) • H(div) and the weak formulation of (1.2)~,A Ov -~-[.~dx + /

S A ( D ( v ) , E ) . D(~a)dx + / [ V v ] v ~ . ~ a d x -

/ ¢div ~adx

(1.53)

is satisfied for all ~ • 7)(~) and almost every t • I.

4.2

Existence of Approximate Solutions

Now we shall show the existence of strong solutions for the system (1.1), (1.2)e,A. Proposition 2.1. Let ~ C_ R3 be a bounded domain with O~ E C3'1 and assume that T > 0, Eo E CI(I, W2'r(~)), r > 3, f E L2(QT), vo • Ep(iEi2), ¢ > 0 and A >_ Ao are given. For poo >_ 2 there exists a strong solution (E, v, ¢) = (E, v ~'A, ¢~,A) of the problem (1.1), (1.2)~,A satisfying the estimates HEHc~(i,w,,,(a)) < c IIEo[]cl(i,w2,,(n)) ,

(2.2)

T

IIvH2~g,H) + / IlVvll~,Vdt < c(f, v0, No),

(2.3)

0 T

]]uA(}D(v)I , ]EI2)]]l,v dt <_ c(f, Vo, E0),

(2.4)

0 T

0v2 - ~ 2 dt

+

c(f, Vo, Eo, e -1, m),

]]Vv]12~(i,L2(l-~,v)) <

(2.5)

0 T

T

0

0

< c(f, v0, Eo, e -1,

A),

(2.6)

where s > 1. Moreover, the estimate

IIvlIL'CI,W~,~(n)> ~

c(f, vo, E0, e -1, A)

(2.7)

holds and (1.2)E,A iS satisfied almost everywhere in QT.

PROOF : The first part of this proposition concerning the existence of a solution to the system (1.1), (1.2)~,A is standard using apriori estimates and the theory of

4.2. EXISTENCE OF APPROXIMATE SOLUTIONS

115

monotone operators and thus we will be brief in this part of the proof. However, the part concerning the W2'2-estimate of v is not so common and we will give a detailed proof. We will drop the indices ~, A and write v = v e,A and ¢ = ¢~,A. The existence of E solving (1.1) and having the properties stated above follows from Proposition 2.3.35. We shall fix one such solution for the following considerations. The existence of a solution v to (1.2)e,A satisfying the estimates (2.3)-(2.5) uses the Galerkin approach (cf. Lions [68], Section 2.5). Let wJ, j = 1, 2 , . . . be a smooth basis of I/2 and let us denote the linear hull of wl, ... , w n by X~. We define the Galerkin approximation v ~ by

j=l

which solves the Galerkin system

d fvn.~dx+ f S(D(v"),E).D(~)dx+f[Vv'?~.~dx fl

~

12

(2.8) f~

f~

n

n

This ordinary differential equation for aj (t) is at least locally in time solvable. The global solvability follows from the apriori estimates (2.3)-(2.5), which will be proved next. Using v n as a test function in (2.8) we obtain after integration over (0, t) and using (1.34) and (1.38) t

t

IIv~(t)ll~,a~ + /IlVv~ll~,~,a. dT + / ]IUA(ID(v~)I2, ]EI2)ll~,~,a. dr 0

0 t

t

_< ~llvollg + ~ f llfll~ d T + 0

c

f IIEII~ d~- _< c(f, Vo, Eo).

(2.9)

0

-SV as a test function in (2.8) we obtain after some simple manipulations, Choosing avfor all t E I

~_~

2, + / sA(D(v~)'E)D(-~-) vv°~dx 2 ___~(Eo) + c(llfll~,~, + lily V n 1~n I1~,~).

(2.1o)

4. UNSTEADY FLOWS

116

The second term on the left-hand side can be re-written as (recall the notation (1.44)) OL31d /

-

__

uA([D(v~)I2, ]E]2)(1 + [El 21 dx

2a31fjuA(]D(v~)[ 2, IE[2) E

0E d z • -~-

O~31-/OnuA(]D(vn)]e ' [E[2)

OEn (1 +

(2.11) IE?)

dx

From (2.10), (2.11), the regularity properties of E and (1.38), (1.48) we deduce for every t E I t

Ot

2 dT'4- [IVvnl129'v'fh

0

(2.12) T

_< c(f, E0, A, C1)(1-4- f [[Vvn[[2dt)< c(f, vo,Eo, c-1, A) , 0

where we also used Korn's inequality, the properties of the mollifier and (2.9). We easily see from (2.9) and the properties of the mollifier that [Vvn]v~ e

L2(QT).

From the inequality (1.34) and the regularity of E it is clear that the operator - d i v SA(D(-),E): L2(I, V2) -+ L2(I, (V2)*)is uniformly monotone. Therefore we may prove the existence of a solution v satisfying for almost all t E I and all ~o E V2 the weak formulation

/ O-~"~odx + / SA(D(v),E) "D(~o)dx + /[Vv]v~ "~odx ~

~t

=ff' dz-x fE®E.D( )dx, fit

(2.13)

~

~t

and the estimates (2.3), (2.4) and (2.5) by standard arguments using the estimates (2.9), (2.12), the lower semicontinuity of norms and of IIUA(ID@)I 2, IEI2)lll,v (cf. (1.34)), the Aubin-Lions lemma and the theory of monotone operators (cf. Lions [68] for more details). Defining F E L2(I, (W~'2(~'/))*) by

/ ~---~.~dx+ /

(F(t), ~o)1,2 -

~ -

SA(D(v),E). D(~o)dx+

at

f f.qadx + XE

fE®E.D(~)dz,

Ftt

Fh

f[vv]v . ~*

(2.14)

117

4.2. E X I S T E N C E OF A P P R O X I M A T E S O L U T I O N S

we see that for almost all t E I and all ~a E V2 (F(t), ~)1,2 = 0. From this and Theorem 5.1.10 we deduce the existence of ¢ E L2(QT), fn ¢(t) dx = 0 such that for almost all t E I

(F(t),~o)l,2 = - / ¢ ( t )

div ¢pdx

V~ E W~'2(a),

(2.15)

nt

which is the weak formulation (1.53). From (2.14) and (2.15), the dual definition of the norm in L2(~), and Proposition 5.1.25 we obtain, for almost all t E I, (cf. (3.4.15)) 2

(

2

A 2

I1¢112.n, < c(E0) 1 + Ilfll2, , + IIS Ilk,n, + II[Vv]v ll ,

+

2

(2.16)

which, after integration with resPect to time, together with (1.32), the properties of the mollifier and (2.5) gives (2.6). Now one clearly sees the difference to the steady case treated in Chapter 3, where we have been able to work with local estimates. If one tries to derive a local estimate for the pressure ¢ then also a local norm of the time derivative of v would appear on the right-hand side. However, a local estimate for ~' (cf. (1.53)) would involve the pressure ¢ on the right-hand side since the appropriate test function would not be divergence free and thus we would run into circles. To avoid this situation we must derive global estimates in the unsteady case. It remains to show (2.7). For the proof of it we have to establish estimates in the interior and estimates near the boundary. We will proceed similar as in the proof of Proposition 3.4.2. Due to the notation and the properties of the mapping T introduced in the Appendix (cf. (5.1.26)-(5.1.34)) we can treat interior regularity and regularity in tangential directions analogously. Since the former case was already accomplished in Section 3.4 and since in the latter one arise some additional difficulties we will present the proof for the regularity in tangential directions here. Let ~ and U n, n -- 1 , . . . , N, be two coverings of the boundary 0 ~ (cf. Definition 5.1.1) such that U n CC 12'~ C_ V n, n = 1 , . . . , N . For simplicity we drop the index n and denote ~1 : fl M ~0 and consider (1 E :D(fl0), 0 _< (1 _< 1, (1 - I in U and put (cf. (5.1.26), T : Tr,a) w(x) - v ( T x ) - v ( x ) .

(2.17)

Due to the definition of the mapping T we see that h-2w(x) ~ ( x ) E W~'~(f~) is an admissible test function in the weak formulation (1.53) we can derive the following

4. UNSTEADY FLOWS

118 identity: 1 f

0v

0v

--h2 g (--~(Tx) - --~(x)) • w(x) ~(x) dx nl

121

+ V1 f ([Vv(Tx)]ve(Tx) - [Vv(x)]v~(x)) • w(x) ~2(x) dx (2.18)

1

h2 f (¢(Tx) - ¢(x)) div (w(z) (2(z)) dz 1

h2 f (f(rz) - f(z)) • w(z) (2(z) dz +g

(E(T~/® E(T~/- E(~/® E(x/) • D(w(~/~(~)) d~: = 0,

where we dropped the index 1 from the cut-off function for brevity. We denote the integrals on the left-hand side by I 0 , . . . , Is and discuss them separately. We see that

1 d f ]h Io= 5~

2~2dx"

(2.19)

The terms 11,/2,/4 and Is can be treated, with some small modifications, in the same way as in the proof of Proposition 3.4.2 (put q = 2 and ~ = 1). Thus, we obtain as a contribution of I1 on the left-hand side (cf. (3.4.26)) 4

cl~flh

V w 2 ~2 dx

(2.20)

and contributions of 11,/2,/4 and Is on the right-hand side (cf. (3.4.27), (3.4.29), (3.4.30), (3.4.32), (3.4.33), (3.4.41), (3.4.43))

+c(l~vvll~ +

+ tlfll~ + ItvEIl~ + Ilvll~ + llv~l~

-~ ,o.),

(2.21)

where c = c(A, ~-1, Eo, a, V2~) and where we also used

IlVv~ll~ _< ~-~llvll~.

(2.22)

4Since h-2w(x){~(x) belongs to the space W01'2(fl) we can as in (3.4.26) use Korn's inequality.

4.2. EXISTENCE OF APPROXIMATE SOLUTIONS

119

With the pressure term I3 we must be more carefully. It can be re-written as

h213 = f ¢(x)(div

(w(T-lx)~2(T-lx)))dx

(w(x)~2(x)) - d i v

fll

w(T-'x)~2(T-lx)) dx

= i ¢(x)(div w ( x ) ~ 2 ( x ) - d i v fll

= i ¢(x)div (v(Tx) - 2v(x) +

v(T-lx))¢2(x) dx

(2.23)

f21

+ f ¢(x)div w(T-lx)(~2(x) -

~2(T-lx)) dx

fll

fll

-I-2S (~(x)'u)i(T-'')(~(x)~--. ~(T-lx)O~(T-l')oxi J~ d. = h' (J~ + JT + Js + Jg) . Using (5.1.27), (5.1.28) and div v = 0 we can re-write J6 as follows

h2Jo =

~ ~(x) t ~ P~I

s=l

il I

s=l

~

Oy3 )

+ 121

--

s=l

)~2(x) dx + Ovs(T-lX)oy3 0A-O--~(~x')

t

oy3

Oy3

Oxs )

+in1 ~2"~¢ (x ) cgv~(x) "~Y3 cOOxs (a(x, +h6r)_2a(x,)+a(x,

ha,.))~2(x)dx

h 2 (J6j + J6,2 + J8,3) • (2.24)

4. UNSTEADY FLOWS

120 Moreover, note that 2 '

Ova(x) ~ cOA+a(x')

~

~1

s=l

]

oy3

\

0~

~(~)d~

0~(~) ~ 0~+~(~') + ~1

Oy3 ]

Oy3

¢(Tx) \ s:l

0~

(~(T~) - ~(~)) d~

and therefore

[S6,~ + J6,21 < c(V~a)ll¢ll~ +

c.8 sf

+ (V )llVvll ,

(2.25)

fll

The term J6,a can be handled similarly as J6 in Section 3.4 and we obtain that cll f Vw q 2 IJ6,31 _< --g s T I ~ dx + c(v~)(ll¢ll~ + IlVvll~ ).

(2.26)

fh

Since div w(x) = ~ s 2= l O,,(T~) 0ys 0~+~(~') Ox, , we see that

,JT, <_f ]¢(Tx)[ ]Vv(Tx)] VAha(x')

~2(Tx)-h- ~2(x) dx

(2.27)

121

_< c(V2~,

v,')llCll~ IlVvll:.

The terms Js and J9 can be treated exactly as the terms J6 and J7 in Section 3.4. Thus we deduce

1131<-c(V2a'V~)ll¢ll~ +c(V~)llVvl]] + cll 4 Jf twl l h I

ax

(2.28)

fll

Putting all estimates between (2.18) and (2.28) together we arrive, after integration over (0, T), at (now we use again the notation ~1 for the cut-off function near the boundary) T

T

T

(2.29)

~1 dz dt <_ e(f, E0, v0) +

0 ~1

0

where both constants may depend on 6-1, A, V2a and V2~. From (5.1.32), (5.1.33) and (2.29) we therefore obtain that 2

3

T

(2.30) s=l i=l

~1

4.2. EXISTENCE OF APPROXIMATE SOLUTIONS

121

In the case of interior regularity, i.e. T is now given by (5.1.34) instead of (5.1.26), we proceed in an analogous way with some simplifications due to the simpler structure of the mapping T. Denoting now the cut-off function in the interior case by (0, i.e. (0 E D(fl), we obtain T

ij=l

~

(o(X) dx < c.

(2.31)

This estimate implies that (2.7) holds for all fl' CC fl and therefore the first equation in (1.2)e,A holds almost everywhere in QT. In order to get (2.7) globally we must also estimate the normal derivatives near the boundary. This will be accomplished by applying the curl-operator to the first equation in (1.2)~,A. We obtain three equations in W-1,2(t2), however, only the first two are useful for us. The first equation reads

0 Ova

ox2 at 0

0 Or2

0 .

OE2-

0 E OE3

~ a ~ + -g-~[E~N-2~I- -oE~[ ~-gK~] r

OV3 ]

0

r

Ore 1

Of3

(2.32)

Of2

Ox--;

+ 7x~ : e ~ 2 ~ J - ~-~ LV~Lx~J - ~ ~ +

°2s~ °2s~ °2s~ °2s~ °2s~ + °2s~ = 0 - 0~10z2 - o:~20x------E- ox20z; + o~30x---~+ ozaOx------~ 0~30x3 ' while the second equation has the form

0 Or1 Ox~ at 0

,

00v3 0 E OE3 0 E OE1 OXl at + -gKx~[ ~SK~] - -g~[ ~-~;~ l OVl,

-~xlkv~k-~xk r 0v31

Ofl

(2.33)

, Of3

Oxl

+ o2s~ ~- o~s____3_¢,+ o2s____£ o2s~

o~s~

°2s----3-:~= o .

Ox~OXl Oxt Oz2 Ox~Ox3 OxlOx3 Oz2Oxa OxaOx3 We will get the required estimate from the last terms in equations (2.32) and (2.33) and from the equation 02va 0%1 0%2 Ox~ - OxlOX3 Ox20x3 ' (2.34) which follows from the "divergence-free" constraint, taking the derivative with respect to x3. Let us denote G1 - ~xaSA(D(v),E)

and

G2 = £ S A ( D ( v ) , E ) .

(2.35)

From the theorem on negative norms (cf. 5.1.8) and (2.32), (2.33) we have, for r = 1, 2,

2 k

IIa,(ll12 _< IIG,(~II-1,2+ IIG,V(III-~,2+ ~

8=1

+

°~s~

~(1

i,j,k=l

curl (~-~- + [ V v ] v ~ - f + XS[VE]E)(1

=h+b+h+Ig.

-1,2

(2.36) -1,2

4. UNSTEADY FLOWS

122 One easily sees (cf. (1.32), (1.33), (2.35))

z8 ÷/7 _< c(v4~, A, Eo)Wvll2, 2

~2 v

3

18 < c(Eo, A, V 2 ~ , ) ( ~ ~

~ 1

2,s, + [IVv[12) '

(2.37)

s = l i=1

Thus we deduce that for r = 1, 2 T

T

_

d0,

0

0

(2.38)

s = l i=1

where we used (2.3) and (2.5). Applying the chain rule in (2.35) we obtain the system

A¢=~I(G-H-F),

(2.39)

where

( O~,SA 023S1A )

(01913 0D23~ (2.40)

G = (C,, G2),

F

H = (H1,/-/2),

----

(/71, F2),

with

Hr = ~

2

A OD,t . . ~A O=va OstS~a-WZ--uJ.a+ °"a°ra ~x32'

OE,~

Fr = O'~sA Ox. '

s,t= l

r = 1, 2.

3

In the last term of the definition of Hr, r = 1, 2 we apply (2.34) and conclude from (2.39) that (cf. (1.37), (1.47), (1.38), (2.5)) 2

0DrS~l 2,hi_<

(

r----1

2

3

r=l

02vi

2

)

(2.41)

i,j=l r=l

since the matrix A is positive definite thanks to (1.35). Integrating (2.41) with respect to time and using (2.38) we obtain

/oT

:

=

,~tl

Moreover, the identity

02Vr 0Dra(v) Oz~ Oz------~

02V3

OzsOzr

'

r =

1, 2 ,

(2.42)

4.2. EXISTENCE OF APPROXIMATE SOLUTIONS

123

together with (2.34) and the definition of ~ imply the pointwise inequality

02Vox~ <

L¢l+c ~2 I0-~ Vvl.

(2.43)

Thus we obtain IV2vl <- I~-~? +

r=l

I

Vv I S I~l + c

r=l

Vv I .

(2.44)

From this and (2.42) we derive

T]IV2v~,H2,aldt 2 -
(1 +

[T ~ 0VY ~1 ,,n, 2 dt) J0 r----1 T2 2

(2.45)

2

Z~=, ~°Vv~, ~,~,+sup (~I~L)~I =

_< c(1+/. ,0

2

IIv2v~'ll2~'"'a0'

where we used in the last line the definition of 4 , r = 1, 2 (cf. (5.1.29)), which yields

EIrm,

~7VI <~ (1+ r=l ~Xr )r:~l ( 10--~vl = "~- r--1 I~xr , ~x32 "

(2.46)

As a consequence of (5.1.2) we can choose ~1 small enough such that

2

csup

"~l°a oi.

,2

1

(2.47)

and move the last term on the right-hand side of (2.45) to the left-hand side in (2.45). Finally, from this and (2.30) we arrive at

/o ~ IIv~ ~11~,o~dt <_c, which together with (2.31) gives (2.7). The proof is complete.

(2.48) •

We will finish this section by proving a variant of the inequality (2.5), which will be useful in the following sections. For t E I we denote

FA(t) -- 1 + IIUA(ID(v~'A(0)I 2, IE(t)12)ll~,~(0, where dr(t) = (1 + IE(t)l 2) dx.

(2.49)

124

4. UNSTEADY FLOWS

L e m m a 2.50. Let poo E [2, 6) and set %(r) -- ~ r 1-~, for # >_ O, # ~t I, and 7~(r) -- in(r). Then it holds for all t • I, all # > 0 and s > 1 t l' 0 V e,A " 2

IT 0

2 7;(FA(T)) d~- + esssup%(FA(~-)) ~e[o,t] t 2 I [[[Vve,A ]v ee, A [[27~(FA(v))dT

< c ( f , vo,E0)+

(2.51)

0 t

+ c(E0) / 7;(FA(T))[[UA(ID(v~'A)[ 2, [E[2)][: dT. 0

PROOF : Let us denote v = v e,A and 7(r) = %(r). Multiplying the first equation av_ i/F in (1.2)~,A by ~-y ~ A~/t~)J and integrating over (0, t) x f~ yields t

t

0--~xjSij (D(v), E)-~- dx 7 (FA(T) ) dr 0

0 gtr

(2.52)

t

< c(f, Eo) + /[[[Vv]v~]]~ 7'(FA(~'))dr, 0

where we used that 7'(FA(t)) _< 1. The assertion of the lemma would follow by straight-forward manipulations of the second term on the left-hand side of (2.52), which we denote by J, if these manipulations were allowed. However the regularity of v established in Proposition 2.1, namely 5 v E C(_7,112) n L2(I, W2'2(~)) and ~t E L2(QT), is not sufficient for some of the computations. Therefore we approximate v by a sequence {v k} C_ C°°(i, V) such that V k ----> V

strongly in L2(I, W2'2(a)),

v (t) --+ v(t)

strongly in V2 for all t E I,

0v k Ot

0v Ot

(2.53)

strongly in L2(I, L2(fl)).

We observe for all t E I lim f uA(iD(v k) ]2, IEI2) dv = [ uA(ID(v)] 2, IEI 2) dr. J

k---~oo J 1"l~

(2.54)

nt

This follows from the inequality (cf. (1.40)) [uA(IB[ 2, IE[ 2) - uA(IDI 2, IEI2){ _< c(A, po)(1 + IBI + [D{)IB - DI, 5It followsfrom the general theory of parabolic embeddings (cf. Gajewski, GrSger, Zacharias [35]) that a function v satisfying (2.5) and (2.7) belongs to the space v 6 C(I, V2).

4.2. EXISTENCE OF APPROXIMATE SOLUTIONS

125

the regularity of E and (2.53)5. In particular (2.54) yields lira k--)~

IfUA([D(vk)I~, [E[2)l[1,~, = ttUA(ID(v)[ 2, [E[2)t[1,~.

(2,55)

Observing that FA(vk(t)) _> 1 we see that [[')"(FA(Vk(t)))[[o~ _< 1. Thus we obtain 7'(FA(vk(t))) - - x(t) in L~(I) at least for a subsequenee. Moreover, from (2.54) follows that ~/'(FA(Vk(t))) --+ 7'(Fa(v(t))) for all t E I and it is easy to conclude that

7'(FA(vk(t))) *" 7'(FA(v(t)))

weakly* in L~(I).

(2.56)

From (2.53), (2.56) and the definition of SA we deduce t

J = - k~oolim

f

3/(FA(vk('r)))

(D(vk), E) - ~ dx d'r,

(2.57)

0

which leads to t

IIUa(ID(vk)12,IEI2)111,~~'(Fa(vk(r))

J = klim 131 ~oe

dT

0 t -

lim k--,oo

aal .,1 7 ' ( F a ( v k ( ~ - ) ) ).,I

O(UA(]D(v~)12,1E]2)(1 + IEI:)) OE,~ OE,~

0---~

dx dT

0

=Jl+J2. One easily sees that

J1 = ~ i m (7(FA(v~(t))) - 7(FA(v~(0))))

(2.58)

= 7(F~(t)) - 7(FA(0)), where we used in the last line (2.54). Applying (1.46) we obtain similarly to (2.54) that for all t E I lim f o(uA([D(vk)12' [E[2)(1 + [E[2)) OEn dx n~

f O(UA(ID(v)I 2, IEI2)(1 + IEI~)) OEndx J

OE.

(2.59)

Ot

and therefore also strongly in LI(I), since from (1.48), (2.5), (2.7) and Lemma 5.1.17 we obtain that Vv E LI°/a(QT), and thus we can use Vitali's theorem. The LI(I) convergence of (2.59) together with (2.56) justifies the limiting process in J2. Finally, we obtain from (2.2) and (1.50) that the right-hand side of (2.59) is estimated by

c(Eo) (1 + IIUA(ID(v)I 2, [E[2)II0. The assertion follows from (2.52), (2.57)-(2.60).

(2.60) •

4. U N S T E A D Y F L O W S

126

4.3

Limiting

Process

A

oc

Now, we will derive estimates independent of A, which enable the limiting process A ~ co. Thus we will come to an approximation of the problem (1.1), (1.2) where only the convective term is mollified. In preparation for the limiting process ~ ~ 0 we will indicate the dependence of the estimates on ¢. For this we denote K~,A(t)

- [

dx.

Ftt

To shorten formulas let us introduce the following notation: ]D] _< A, [D] _> A,

(3.1)

Ip,T,A(U, ~) = L ( 1 + IE[2)rlu]rg~ -2)T ~ dx,

(3.2)

Ip,A(u,~) = f ( 1 + IEI2)]u[29PA-2~2dz,

(3.3)

{~A ---- OA(IDI 2) =

( 1 + ]DI2)½ (1 + A2)½

and

where u will usually be some second order derivative of v, and ~ will be either identical 1 or one of the cut-off functions ~1 or ~o, defined above. Note, that it follows from Remark 1.43 that there are constants c12,c13 such that C12 Ip,A(D(Vv), ~) < ~

0k~SA(D(v), E ) n o ( ~ v ) n k z ( V v ) ~ 2 dx

(3.4)

_< c13Ip,A(D(Vv), 0 . Recall that (cf. (2.49)) FA(t) = 1 + IIU~(ID(ve'A(t))l 2, [E(t)12)[ll,~(t), and that g}~ = ~ O ~ , where fl~, n = 1,... , N, is a typical set from the appropriately chosen covering of the boundary 0fl and ~ are the corresponding cut-off function near the boundary (cf. Proof of Proposition 2.1). Let us first give the definition of a solution to the problem (1.1), (1.2)e and then derive the necessary estimates for the limiting process A -+ co. Definition 3.5. The triple (E, v, ¢) -- (E, v E, Ce), for E > 0 given, is said to be a

strong solution of the problem (1.1) and 0__~v_ div S(D(v), E) + [Vv]v~ + V¢ = f + XE[VE]E Ot div v = 0 v=0 =

v0

in QT , (1.2)~ on I x 0 ~ , in 12,

127

4.3. LIMITING PROCESS A --+ oo where S is given by (1.3)-(1.5), if and only if

E • CI(O~T), 2 P oo - 1

2

2

v • L po-1 (I, Y p o ~ - l ) n L ~ - - ~ ( I , W '~o+,(~)), 6 po_l

"E 2. D(v) • L°°(I, LP(IEI2)(~))A L 2po pu )(QT), 0v

(3.6)

0---~' ¢ • LZ(QT)' the system (1.1)1,2 is satisfied almost everywhere in QT, for almost all t • I we have o

E(t) -Eo(t) • H(div) and the weak formulation of (1.2)E --~ .~odx+

a~

f f f =ff.~dx-~EfE®E.D(~)d~ S(D(v),E) .D(~o)dx+

nt

[Vv]v~-~odx-

f~t

Cdiv ~odx

n~

(3.7)

is satisfied for all ~o • D(~2) and almost all t • I.

Lemma 3.8. Let 2 <_poo <_po <_ 3, ~ > 0 and A >_ Ao be given. Then the solutions (E, v, ¢) = (E, v ~,A,CE,A) of the problem (1.1), (1.2)e,A satisfy the following estimates with constants c depending on f, Eo, vo, T but not on A and e: ]JEllc,(/,w2.r(n)) < c,

(3.9)

IlviI~,o%,,H)+ /IlVvll~,,, + IlSa(ID(v)l 2, IEI2))11,,.dt _< c,

(3.10)

T

0 T

f [[SA(D(v), E)II~ dt _

c,

(3.11)

0 T

T

0

0

and for almost all t E I t

eSSSUpTu(FA(T))+ f r~(o,t) t

0

o

~

;'~;(FA(T))dT

(3.13)

4. UNSTEADY FLOWS

128 t

f ([[V2v]l:~+ IIvsA[]:~+ Y(v(~))+ Z(v(T)))~/~(FA(~'))d~0

t

< C (1 + f ( [Is A2s [[2s,..

(3.14)

I~e,A(T) )~:(FA(T))dT ),

÷

0

where r

2 E [1, po-~]' u E [1, p-5:5-1], # :> 0

and 2

Y(v(t))

N

Or(t) ),~1) n , = Ip,A(D(Vv(t)),~o) + E EIp,A(D(--'~Ts s=l n = l 2

2

2

N

(3.15)

0V

Z(v(t)) = [IV v~o[[2,~ + E E I I V

n ,2

~-~T~~1 2,,,"

S~-I r t = l

PROOF : We will drop the indices e, A and write v = v E'A and ¢ = ¢,,A. The estimates (3.9) and (3.10) have already been proved in Proposition 2.1. Moreover, we derive from (1.38) and (1.34) that, for s > 1 and almost all t E I 28

2s IIuAll -1~__: _< IIs A112s + c IlVvll~,

(3.16)

l+s

From (3.16) and (2.6) we immediately obtain (3.12). Moreover, using (3.16), (3.10) and 7~(FA) < 1 we derive (3.13) from (2.51) with an appropriately chosen s. From (1.39) we obtain

APt°( I[s [Ip~
I+11UA] [ ~ ) ,

(3.17)

which together with (3.10) gives (3.11) after integration with respect to time. It remains to prove the crucial estimate (3.14), which will be accomplished now. Using (1.38) and Poincar~'s inequality we get

-~ . v dx < 5 Ilvll~ +

c ~-~ (3.18)

_1 f~

Thus, choosing in (2.13) ~o = v and moving the term with the time derivative of v to the right-hand side we obtain

f

SA(D,E)

• Odx

__~ c(E)(1

+

Ov

-~- i +

Ilfl12)'

(3.19)

i]

From Proposition 2.1 follows that --Av ~ is an admissible test function in the weak formulation (1.53). Recall that ~0 E :D(gt) is a cut-off function for the interior

129

4.3. L I M I T I N G P R O C E S S A -+ oo

regularity. Let us denote by I 0 , . . . ,/5 the integrals in (1.53) for this test function. We observe that 0v2

IZol ___c15 iiV2v,ol]~ + c ~

(3.20)

2'

where c15 is the constant appearing in (3.5.18). For the terms I 1 , . . . ,/5 we proceed exactly 6 as in the proof of Proposition 3.5.7 just setting q = 2 and a = 1. Therefore we can deduce from (3.5.16)-(3.5.19) and (3.5.21)-(3.5.25) (recall the definition (3.2) of Ip,A) for almost all t E I

Ip,A(D(Vv(t)),~o) + IIV2v ~olh,., 2

(3.21)

<_ C(Eo, V~o)(1 + Ilfl[~., +IIs 112.,., + K.A~ + I1¢11~,.,+ •

2 + _< c(Eo, V~o) ( 1 + Ilfll=,.,

2s + IIs A I1=.,.,

+

0V

+ IlVvlI~,.,)

,nt : + [iVVll2 t) ' ,,,

where s > 1 is chosen appropriately, and where we used (2.16). The same procedure works for the tangential derivatives near the boundary. Since we derived the estimates for the tangential derivatives in the previous section in detail, we will not repeat more or less the same steps here again. We obtain, for r = 1, 2, and almost all t E I,

ip,A

¢,) + IIv

¢,

(3.22)

=' + Ke,A + l1 o~ ~ =2,., + IlVvll~,.,) • ~-~ c(E°' V'i)( 1 + Ilfll~,"t + II s a 112s,.t Due to the terms [[S A112,,~t 2s appearing on the right-hand sides of (3.21) and (3.22), for which we have no apriori estimates, we have to establish global estimates for V S A. For this we proceed as in the proof of Proposition 2.1, however we must be more careful, since the estimates have to be independent of A. We will use again equations (2.32)-(2.34). Similarly to (2.36) we notice, for r E [1,2) and s = 1, 2, (cf. (2.35) for the definition of Gs) 2

3

a2c~A

IIC,~'~ll, < IIa,~',ll-,,, + Ila, V,~,ll-x,, + ~_, z_., I 1 ~ s=l id,k=l

-I- curl (~---- + [Vvlv~-

f

-I- xEtVE]E)¢, -1.,

(3.23)

=I6+I7+Is+19. Since 2 < r ~, one easily checks

16 + i~ _< c(V~)IISAII2,

/9 _< ~(Eo, V~I) (1 + I{fl12 + ~Ov 2 + II[Vv]v~ll2).

(3.24)

SThe only difference is the treatment of the last term in (3.5.19), which due to Remark 1.43 can

be estimated by HuA(IDI2, [E[2)II~and then we use (3.16).

130

4. UNSTEADY FLOWS

Furthermore, we obtain

/s <

sup ~ J ¢¢w~'r'(n) = n

Ozs

(3.25)

I1¢111,,,_ <1

= J8,1 + Js,2, where we can estimate the first term by Js,2 <_ C(V2~l)lisnlt2-

(3.26)

Concerning the term Js,1 we get 2

II¢lh,.,_
(3.27)

Ip,r,n(D

,~1)

+ c(Eo)(1 + II0,sA~aIt~),

S=I

where we used (5.2.4) and the definition (3.3) of Ip,r,A. From (3.23)-(3.27) we conclude that 2

1

[IGs~lllr _< C(Eo,V~I,

v2~1)(1+ IIS~II2+ K~,A +

OV

-~ ~

(3.28)

S=I

+ Ilfll~ + Y~

(/p,r,A(D(

),~1)) 7 + [10.sA~II~

8:1

Similarly as in the proof of Proposition 2.1 (cf. (2.39)-(2.48)) we want to estimate all second derivatives near the boundary. From (1.35) follows that A ¢ . ¢ > c41¢120~-2 .

(3.29)

Moreover, due to the growth properties of OoSA and the regularity of E we can estimate H and F as follows 2

Inl __ c(E) O.~-2 E [ V 0 ~ l '

(3.30)

8----1

IFI <_ c(E)10,Sal, where we used in the first estimate (2.34). T Thus, (2.39) and (3.29), (3.30) imply 2

i¢10~-z _< ~(E) (IGI + 0V ~ Z IV0~I + 10~sAI), 0=1 7It iS exactly this place where it is impossible to obtain an estimate involving only t e r m s of the fo r m .-~2 2_.s=~ D ~~ ~Ov , ~J. Th.,~ ~,o we o b t a i n slightly worse results c o m p a r e d to t h e case w h e n S A depends on IVvl instead of ID(v)l (of. Mgdek, Ne~as, Rfi~i~ka [72], [73], Rf~i~ka [113]).

4.3. LIMITING PROCESS A -+ c~

131

Multiplying this by ~, taking the Lr-norm and using (3.28) delivers (cf. (3.2) for the definition of Ip,~,A) 1

(~,~,A(¢,6)); < c(E0, V ~ ) ( 1 +

0v

IISAII~

+ K~!A + ~

2

(3.31)

2

+ I,fl, +Z:

)

From this and (2.44) we obtain 1Ov (I~,~,~(V2v, ~))~ < C(Eo, V~6) (1 + [ISAII2+ K~,A + --~ 2 2

0v

!

(IP",A(V-~x'~I))~ +

+ I]f]12+ E

(3.32)

t]0nSA~lllr

)'

Due to (2.46) and (2.47) we conclude, for ftl chosen small enough and almost all t E I,

(Ip,r,A(V2v(t),~l)) r1 <

c(Eo, V2~1) ( 1 + [ISA [[2,nt +

K~'A(t) + ~t

(3.33)

--

2,~t

+ Ilfll~,., + ~

,6)); + IIO,,S'~611r,.,) •

(;,~,~(v

S=l

We observe that

f

Ilvs~ll; <_ [Ok,sADkt(VV)r + <_ c(E)(Ip,~,A(V2v,

[O=SAVE~I~ dx

11 + ]Io.sA]I~),

where we also used (1.37) and (3.2). This and (5.2.28) thus imply

Ilv2vll: + IlvsAll: _< c(E)(Z~,r,A(V~v, 1) + IAS~II:)

(3.34)

Extending our covering of Oft (el. proof of Proposition 2.1) to a covering of ft and using a corresponding subordinated partition of unity we obtain for all r E [1, 2) 1

N

1

IlV2vll~ + IlVSAIIr ___(/~,r,A(V2v, ~o)); + ~ (~,r,~(V~v, er)); + ]]o.sAIl~ n=l

t

2

N

<__c ((/~,.,A(V2V,~o)) ; + ~

0v

.

(g,~,A(Vb-~C~,~))

+ ]I0.SAII~

s=l n=l

+ 1 + HsA[[2 +

K~.A1,+ ~ 2 2

+ [[fl[2)

N

S=I n ~ l

½ +II0~SAll~+I+IISAII2+KI,A+

ov ~ 2+11f112) '

(3.35)

4. UNSTEADY FLOWS

132

where we used (3.33) and Hblder's inequality. Since for the first factor we have the estimates (3.21) and (3.22) it remains to bound IlonsAIIr and ]19~-2112~__5vL.From (1.41) we deduce ]10~-21122_r~< c(1 + ]I]sAIp-a-~2H==__SV ) ~< C(1 + IISa]IP°~-~po_a ) - - =--V.o-, (3.36) PO-2 < c(1 + Ilsalll + IlvsAIIr)~O-1, 2r ~ . whenever r E [1, po-~-$]" Here we used the embedding Wl'r(~) ~ L 2-" po-1 (~), which 6 holds for r E [1, v-gg$]" From (3.3.47) and (1.47) we deduce for I < s < 3/2

1 1

-<

(£

--2 2r s _ l .

1

oa~g,

2--r

1,

) 5 ( l + l t S a lie°-2 p0-1[s-~) "~ c SA. Ddx 2r(s-1)] po-1 2-r 2 ffl SA'Ddx) ~(x 1 + IIsAII1 + Ilvs * r)po . po-2
(3.37)

where we employed Hblder's inequality, (1.41), s < 3/2 and the above embedding. Using now in (3.35) the estimates (3.21), (3.22), (3.36), (3.37) and (3.19) we derive for r E [1, vo-~]

Ilv2vLl~ +

Ilvsallr < c (1 +

sA s2, _}_K ]¢,a +

o%' 2 + Ilfll~ + IlVvll~) × -b7 po-2

× (i + lls% + llvs%)~o-~ (£

+c

)1

po_2

SA(D)'Ddx 5 (1 + IISAIIx+ IIvsAIIr) P°-I

"4-C (1 "4-IIsAII;s-I- K~,1A + -OV ~ 2 + Ilfll2)

1 + -~2 _ c ( 1 + IlsAl[2,8 + K~A o,, + Ilfll2 + IlVvll2 ) x po-2

× (1 + IISAII~ + IlvsAll~)po < Ov JlfJh) +C (1 + IIsAII;, + K~,A + "~ 2 +

<_ C(1-{- tlS A112~+K~,A 2s 0Y 22+llfll~ + IlVvl122)_F.9_7~ + --~ = + IlivsAIfr , where we also used Young's inequality with 5 = • p0--a ~ and Po > 2. The last term can be absorbed into the left-hand side. The resulting inequality raised to the power u, 6 and 1 < s < 3/2 chosen appropriately, to leads, for u E [1, po2-~_11, r E [1, v-a~] Ov=

IIv2vll~ + IlvsAIl: _< c(1 + IIsAII2=:+ K¢, A -~ ~ - 2-{-Ilfl122 + Ilvvll2=).

(3.38)

4.3. LIMITING PROCESS A -+ co

133

Note that we can choose u > 1 for P0 < 3, which gives the upper bound appearing in the lemma, and that the right-hand sides of (3.38) and (3.21), (3.22) coincide. Thus, the sum of inequalities (3.38) and (3.21), (3.22) yields, for almost all t E I and u E [1, F~_l], r 2 E [1, F~-~]6 and 1 < s < 3/2 chosen appropriately, (cf. (3.15) for the definition of Y(v(t)), Z(v(t)))

IIv2vll:,,~,+ Ilvs~lG, + Y(,,(t)) + Z(v(t)) ( A2, < c 1+ IIs 112.,~,+ g.,A(t)

+

ov2

~

2,n, +

2 IIflG,

+

2) Ilvvll2,.,

(3.39) •

Multiplying this inequality by 7'~(FA(t)), # >_ 0 (cf. (2.49), Lemma 2.50) and integrating over (0, t) gives t

i ( IIv~vlG: + IlvsAIG- + V(v(~))+ Z(v(~-)))4~(FA(~))d'~ 0

t

(lls I1 :,o:

_< c(f, vo, E)(1 +

'

)

(3.40)

0

where we also used (3.13) to bound the term with ~ and that Gtu(FA) ~ 1, f and (3.10). This proves (3.14).

E L2(QT)

•

On the right-hand sides of (3.12)-(3.14) appear terms with IISAII], ~ for which we do not have any information in the moment. Thus we are now going to prove estimates without this term on the right-hand sides at the expense of further restricting the upper bound for PoL e m m a 3 . 4 1 . Let 2 _< p ~ < P0 < 8/3, ¢ > 0 a n d A > Ao be given. Then the solutions (E,v, ¢) = (E, v ~'A, ¢~,A) of the problem (1.1), (1.2)~,A satisfy the following estimates with constants c depending on f, v0, E0, T but independent of A and e: T

2~0-}-3)

T

5

1+ K~,Adt )i lll 't,xv ,", li 2(po+3) '<'°-'' df <_ c (/ 3~o-i)

o

po+3+llD(v)ll ilIuA(iD(v)I2,1EI~)2"° T

po+3

0

2po

o

(/

po+3

P0

5

K~,Adt ) ~

(3.43)

0

T

T

f ll+li~t<_c(i+f g:,~dt), 0

T

p 0po + 3 dt < - - c 1+

0

(3.44)

134

4. U N S T E A D Y F L O W S

and for almost all t

• I t

t

~sssuvl.u'(,D(v(.)),',,n(.),'>H,.(.>+f ~ re(O,t)

:.~_<.O+f~....O, (3.45>

0 t

0 2

2

t

[ IIV2vll'°-C + IlvS'(D(v),E)II'v' d, < c(1 + '~ 0

~

po+l

(3.46) 0

t

t

/,-(v(~)) +.(v(.)),~. ~.(, + f ~-..,.) O

(3.47)

0

PROOF : We can rewrite (3.17) as SA P~ p,o,n, < C F A ( t ) ,

(3.48)

which together with (3.13) (# = 0) implies, for 1 < s appropriately chosen, and almost all t • I 1:

esssuplls.' ,a,o~ "~' < c(l+f re(O,t)

~ , . 0 - ) + IIs~

=~ ~)

(3.49)

0

We have the interpolation inequality

ils~ll~ < iis ~ q(1-A) < j[s ~ q(1-~,)

-

,~

(3.50)

llvs~ll~+lls~ll;~, 1 ~_.fi._

6

with A = q(po-1)(6-po), 6(poq-po-q) where we also used the embedding W 'po+, (fl) ~ Lp--~-r(f~). Choosing now q = 2(3+po) 3(po-1) we see that qA = po2-~_l. For this q we thus obtain from (3.50), (3.49) and (3.14) with # = 0, r = po-~' u = ~ that for almost all t • I

IIs~llZd~_< (esssup \

IIs~II'2)

(o,t)

....

eo

~ [[vsAH: d r +

0

(esssup (o,t)

0 t

lie I1,~) /lie ,o,~dT A P~

3/)o

A

'

a

0

5

t

0

5

0

where we also used (3.11) and ~2po inequality

-<

g'5

Moreover, we dispose of the interpolation

1-X A X [ISA I]2~,Q, < [ISA II,~,Q~[[S ]lq,Q~

(3.52)

4.3. LIMITING PROCESS A ~ co

135

with A = (3+po)(2s(Po--1)-po) From this and (3.51), (3.11) we deduce for almost all t E I (6-po)(po-1), • t

5

0 t

5

_<

,

(3.53)

0

whenever q>l---~sA

"¢==~

P0<~,

(3.54)

for s > 1 chosen appropriately. This proves (3.42). With the help of (3.52), (3.11) and (3.53) we arrive for almost all t E I at 2s IfS A [[2s,Q, -< c(f, vo, Eo)lisAl]~:~

<e(f, vo,Eo)(l + f

t

(3.55)

0

whenever (3.54) holds, since "" x 10s < 1. Due to (3.55) we immediately obtain (3.44)3q 6 (3.47) from (3.12)-(3.14) with/~ ----0, r = p--~-~, u ---- po~-~_l. Furthermore, choosing in (3.16) 2s -- 2(po+a) and using (3.42) and (3.10) we derive the estimate (3.43) for the 3(po--1) first term on the left-hand side. Using in this estimate the lower bound (1.38) for U A we get (3.43) also for the second term on the left-hand side. The proof is complete. • R e m a r k 3.56. Note, that the right-hand sides of (3.42)-(3.47) are bounded by constants c(f, v0, E0, c-1), due to (3.10) and the properties of the mollifier. In particular (3.47) (cf. (3.15)) implies that V2v e,A is bounded uniformly with respect to A in the Ore, A space L2(I, L12oc(f~))and that V--~-~,, s = 1, 2, are bounded uniformly with respect to A in the space L2(I, L2(~)). The estimates proved in Lemma 3.8 and Lemma 3.41 together with Remark 3.56 enable the limiting process A --+ co. P r o p o s i t i o n 3.57. Let 2 < poo <_ P0 < 8/3, c > 0 and assume that f~, T, Eo, f and vo satisfy the assumptions of Proposition 2.1. Then there exist strong solutions (E, v ~, ¢~) of the problem (1.1), (1.2)~, which satisfy the following estimates with constants c depending on f, E0, vo, T but independent of e

< c,

(3.5s)

T

IIv 2

/ liU(ID(v ,/

0

)?, IE?)lil,

dt < c ,

(3.59)

136

4. U N S T E A D Y F L O W S T

i

IIS(D(v~),E)]I~

T

0

T

(3.60)

'dt <_ c,

0 T

0

po+3

2(po+3)

2po

3(po--1)

T

fll~(,D(+),~,,~,,~)po+3+llS(D(v0,E) 2,0 N~;:-~ldt <_ c (1 + f +

0 T

~,~-o-o -~

0

Po

),

("

,

(3.64)

K~dt S

0

T

IID(v)llpo+3dt<_c

5

5

K~dt s

1+ 0

and for almost all t E I t

esssuPllU(ID(v01LIE I )111,,,0.)+

t

.~

(i

0

t

K~d'r 0

t

0

0 t

t

0

0 t

t

I ( I , , r ( D ( V v ~ ) , l ) ) T dT < c (1I+ ) 0

with u = ~

)

dm
'

K~dT

,

(3.67)

0

and r =

6 po+l

'

PROOF : The right-hand sides of (3.9)-(3.11) are bounded by constants c(f, v0, Ea)

and the right-hand sides of (3.42)-(3.47) are bounded by constants c(f, v0,E0, e-l), due to (3.10) and the properties of the mollifier. The inequalities (3.42)-(3.47) together with Remark 3.56 and the Aubin-Lions lemma imply the existence of v ~, ¢~ such that ~ V e,A --~ ~ V e strongly in L~(I, L m ( f l ) ) , m < p o6- 1 Ore, A

{9"%,e

Ot

Ot

Ve, A ~

Ve

v e , A .....~* V e c e , A _.~ Ce V2Ve, A ~

OV~,A

V w ~V OTs

V2V e

OV~ OTs

weakly in L2(QT), weakly in Lq(I, Vq) N n~(I, W2'r(~)), weakly* in L°~(I, 1/2), weakly in L2(QT), weakly in L2(I, L~o¢(~)), weakly in L2(QT),

s = 1,2,

(3.68)

4.3. LIMITING PROCESS A -~ oo

137

2 r : ~ 6 and q -- 3+po as A -+ c~ at least for a subsequence, where u : p-~-1, po ' From (3.68)i follows V v ~'A --->V v e

a.e. in QT

(3.69)

and thus using Vitali's theorem and (3.68)3 we obtain ~ V e'A - +

~TVe

strongly in Lm(I,L'~(~'I)),

m < 3+PO Po

(3.70)

Moreover, since $A and U A, respectively, converge locally uniformly to, respectively, S and U we also conclude from (3.69) that SA(D(v~'A),E) --+ S(D(v~),E)

a.e. in QT,

uA(ID(v~'A)I2 , tEl 2) -+ V(ID(v~)] 2, IEI 2)

a.e. in QT,

(3.71) Therefore (3.46), (3.42) and (3.45), (3.48) and show that S A (D(v~'A), E) ~

weakly in L'~(I, Wl'r(l-2)) gl L'~(QT),

S(D(v~), E)

(3.72) weakly* in L~(I, L p'°(fl)),

SA(D(v~'A), E) ---'* S(D(v~), E)

2(3+vo) since weak and almost everywhere limits 2 r = ~ 8 and n = 3(po-1), where u = v--~-~-l, 2(3+po) coincide. Moreover from (3.42) and Vitali's theorem we derive, for m < 3(vo-1)' strongly in Lm(QT).

SA(D(v~'A), E) --+ S(D(v~), E)

(3.73)

Similarly, from (3.71)5, (3.43) and Vitali's theorem we obtain, for m < 3+po 2po '

uA(iD(ve'A)] 2, IE}2) -4

U(]D(v~)I 2, IE]2)

strongly in Lm(QT)

(3.74)

strongly in Lm(~'lt, v(t)),

(3.75)

and in particular for almost all t E I

uA(ID(v~'A)I2 , IEI 2) -+ V(ID(ve)] 2, ]El 2)

where we used the regularity of E. We also observe that, for almost all t E I t

t

t

lim / K~.4dT= I K~dT=--S II[Vv~]v:ll~d~-,

A--~<x)

0

0

(3.76)

0

due to (3.70), since 2 < 3+v-~o°and the properties of the mollifier. Because of (3.68)(3.76) and the lower semicontinuity of norms the estimates (3.10), (3.11) and (3.42)(3.47), except the term with Y(v~'A), remain valid if we replace v E'A, ce,A, U A, S A and Ke,A by v ~,¢e U, S and K~. This proves (3.58)-(3.66), except the term with fo Y(v~) tiT. In order to show that also in the term fo Y(v~'A) dT we can replace v ~'A by v ~ we have to proceed differently. From (3.70), (3.58) and Egerov's theorem we conclude that for all 5 > 0 there is a set Q6 with IQ \ Q6i < 5, and a subsequence A~ such that

~A(~Ell)-2(D(ve'A))(1 4-IEI 2) ==:I(1 +

ID(v~)i2)P(I~'/2)-~ (1 + IEI2),

uniformly in Qz.

4. UNSTEADY FLOWS

138

From this and the estimate (3.47) for Z(v ~'A) we obtain for almost all t E I liminf f ( 1 + IEt2)[D(Vve'A~)I2 ~o2 (~A~2(D(v~'A;)) A~--*ooj

-- (1 + [D(ve)l 2) 2 )

dxdt = O,

Q6nQ~

which implies liminfA,_~ / ( 1 +

IEi2)ID(Vv~'A')I2~gG~(D(ve'A'))dxdt

q6nQ*

-=liminf

0+IEI2)ID(Vv~'A~)I2~g(1 +lD(v~)l 2) ~ )d~et. QenQt

Since, for po < 3 we have due to (3.68)3 that 1 _< (1 +

ID(v~)12)~(1 + IE121• L2(QT)

(3.77)

and thus we can define a new measure d# by d# - (1 +

ID(v~)12)~(1 + IEi2)dxdt.

For this measure we have L2(QT; d#) ~ L2(QT; dx dr). Due to (3.68)6,7 we can idenx--'~2 ~N D- 0re, A- n tify the limits of D(Vv ~,A)~o and 2.~8=1 2-,n=1 (--g~,) ~ , respectively, as D(Vv s) (o and )--]~s=l 2 )-~n=l N D(o~) ~ , respectively, in the space L2(QT). Moreover we have lim / D(V ve'A) ~0 ¢Pd# = A~oolim/D(Vv~'A)~0 ~o(1 +

A-+oo

QT

ID(ve)l)~~-~ (1 + IEi2)dxdt

QT = / D(Vv ~) sCo~o (1 + ID(v ~1J ) ~

(1 +

IE{u) dx dt

QT (3.78)

= f D ( V v ~)~0~d#, QT

for ~ E L°~(QT;dxdt) M L2(QT;d#), where we used (3.68)6 and (3.77). Since one easily checks that functions from LC°(QT;dx dr)N L2(QT; d#) are dense in L2(QT; d#) we have shown that D(V ve'A) Go ~ D(Vv e) G0,

weakly in L2(QT; d#),

and thus liminfA6~oo/ ID(Vv~'A~)I2~°2d~ _> f ID(Vve)]2~d~. QanQ,

QaMQ,

X-'2 L,n=l x-'N D ~-~7;-~" (av~'AJ~~1 ~-n is treated similarly and therefore we have The other term z.-,s=l shown that for almost all t E I liminfA6_,o~/ Q~nQ,

Y(v~'A~)dt >- / Q,snQ~

Y(v~) dt"

139

4.3. L I M I T I N G P R O C E S S A --+ c~

Since 5 > 0 was arbitrary and the right-hand side of (3.47) is bounded independently of 5 we proved that also in the term f0t Y ( v ~'A) dT we can replace v ~'A by vL It remains to show (3.67). Due to (5.2.35) we have t

t

0

0

12 t

f

(,..I

0

where we also used (3.60), u = ~

S P~, (3.65), the equivalence of the quantities

S(D, E). D and U(IDF, IEF) (cf.(1.43))and (3.59). Finally, the limiting process A --+ c~ in the weak formulation (1.53) is clear in all terms except the term with S A. For this term we obtain for almost all t E I and all

L<~2ef sA(D(vt~'A), E )

• D(~)dx

f

=

S(D(vS), E) • n(~)dx

R~

due to (3.73). The proof is complete. Remark

3.80. It follows from (1.26), (1.42) (cf. (3.2.14)) and (3.59), (3.64) that T poo IID(v e:)llp(IE(01~),~(t)+ IIVv 6 Ilpoo,~, dt

<

c(f, Vo, Eo),

0

(3.81)

t

esssup ([ID(v~)[[p(iE(~)l ~) + HVv Hpoo,a~+ HV v 112,a.) -< c 1 + re(O,t)

K~(T) dr

.

0

We want to finish this section by two lemmata, which are consequences of (3.13) and (3.14) if we pass to the limit A --+ c~. L e m m a 3.82. Let 2 < p ~ g P0 < 8/3 and assume that % , # > O, are defined as in Lemma 2.50. Then we have f o r almost all t E I and 1 < s < 1 + 5, 5 sufficiently small, t

esssup %(F(t)) _
0

(3.S3)

t +

f IIs((v

e

0

~he~e F(t) = 1 + IIU(LD(v ~(t)) F, IE(t)12)I1~,~(~).

2s i

140

4. U N S T E A D Y F L O W S

PROOF : The definition of FA and F together with (3.75) yields for almost all t E I FA(t) --+ F ( t ) ,

(3.84)

which in turn implies 7,(FA(t)) --+ 7~(F(t)) "/'~(FA) -'* "7~(F)

for a.e. t e I , weakly* in L ~ ( I ) .

(3.85)

prom (3.75) we also obtain for an appropriately chosen s > 1 and almost all t E I [[uA([D(v~'A(t))[ 2, [E(t)12)[[8 -+ []U([D(v~(t))[ 2, IE(t)[2)H~,

(3.86)

moreover we can re-phrase (3.76) as

K~,A --+ K~

strongly in L I ( I ) .

(3.87)

Since the second term on the left-hand side of (3.13) is non-negative and since we can pass to the limit as A --+ oc in the other terms, due to (3.85)-(3.87), the assertion follows, if we use (3.16) and (3.59). L e m m a 3.88. Let 2 < p ~ _< P0 < 8/3 and assume that 7~, # >- O, are defined as in Lemma 2.50. Then we have for almost all t E I and 1 < s < 1 + 5, 5 sufficiently small, t

0

t

+/

+

0

(3.89)

t

_< c(f, vo, E o ) ( 1 + f

(I[S(D(v~(~-)), E(T))II22: + K ~ ( 7 ) ) 7 ; ( F ( v ) ) d v ) ,

0 6 where r = Fg'gT, u = ~

and # >_ O.

PROOF : We want to pass to the limit as A ~ ~ in (3.14), which gives (3.89) for the last four terms on the left-hand side. From (3.87), (3.73) and (3.85) we deduce that we can pass to the limit in the terms on the right-hand side of (3.14). Moreover, from (3.85) and (3.84) we obtain for all m < oe 7~(FA) ~ 7~(F)

strongly in L m ( I ) .

Now, we proceed similarly as in the proof of Proposition 3.57. Therefore, by Egerov's theorem we know that for all ~ > 0 there is a set I~ with II\hl _< 5 and a subsequence A~ -~ oe such that 7~(FA) ==t 7~(F)

uniformly on I~.

4.3. LIMITING PROCESS A -+ oo

141

We conclude from this and (3.46) that for almost all t E I IIvsA'(D(v"A'), E)II~(~/~(FA,) - 7~,(F)) dr = O. lim inf f A6----~oo 1~n(o,t) Thus we have liminfA,~._~oo

f

IIvsA'(D(v~'A')'E)I[~7~(FA')dT

I~N(O,t)

---liminfa,~oo f t~n(0.t)

[[vsA'(D(v~'A')'E)H~%(F)dT "

Since

0 < 7'(F(t)) < 1

(3.90)

we can define a new measure d#, which is absolutely continuous with respect to dr,

by d# - "y'(F(t)) dt, and thus we have that L'~(I, dr; X) "-+ L'~(I, d#; X) for any Banach space X and 1 < m < oo. In particular functions from ~D(I, X) are dense in both spaces. In (3.72) we have identified the weak limit of VS A as XTS in the space LU(I, L*(f2)). This together with (3.90), the boundedness of the right-hand side of (3.14) and the uniqueness of weak limits leads to (el. (3.78)) weakly in Lu(I, d#; Lr(fl))

vSA(D(vE'A), E) ~ VS(D(v~), E) and thus we have for almost all t E I

IIvsA'(D(v"A%E)II~%(F)dt>

liminf A6...,o o

I~c~(0,t)

Ilvs(vv

'

t6n(0,t)

Since 5 > 0 was arbitrary and the right-hand side of (3.14) is bounded independently on 5 we get (3.89) for the last four terms on the left-hand side. The estimate for the first term we obtain from (5.2.35) t

t

f (Ip,~(D(Vv~(r)),l))-~(r)dr
0

S.Ddx

f~

)'7u(r)d~"

t

_ c (1 + f (][S]]~: + K ~ ) 7 ; ( r ) d r ) ,

(3.91)

0

where we used (3.60), (3.90), the estimate (3.89) for VS, the equivalence of the quantities S ( D , E ) . D and U(ID[ 2, [El2), (3.59) and (3.90). "

4.UNSTEADY FLOWS

142

4.4

Limiting

Process

¢ --+ 0

In this section it remains to find conditions on po¢ and P0 such that the estimates proved in Proposition 3.57, L e m m a 3.82 and L e m m a 3.88 are independent of ~. This is already true for the following estimates (cf. (3.59), (3.81)1, (3.60)): T

esssuPte, [Iv~(t)ll2 +

f IIU(ID(v~(t))I2' IE(t)12)ll~'~C~)de

< c(f, Vo, Eo),

(4.1)

f IID(v'(t))llp
(4.2)

0 T

0 T

f IIS(D(¢(t) ), E(t))ll~i dt < c(f, Vo, E0).

(4.3)

0

On the right-hand side of all other estimates occurs the term K,(t), which must be handled by the quantities on the left-hand sides. We distinguish two cases for which we establish estimates independent of ~. After that we will discuss how to use these estimates for the existence of weak and strong solutions, respectively. (i) T h e C a s e 12/5 < poo < P0 < 8/3 In this case we estimate K~ as follows 2 Ve 2 /~ _< IIvV e I1,~11 II 2p~ p~-2 25P°¢--12

e P¢¢

16-5pe¢

(4.4) y ~ poo

_< c IIv~ll~5,~-6 IlVv I1,~ 5P~-6 IIV I1,~ =: gllVv%=, 2pe¢

3p~

where we used the interpolation of Lvoo-2 between L 2 and L3-p ~¢. Observe that for the poo considered here we have ~5p¢¢-6 <- - 1. Thus the function g in (4.4) belongs to LI(I), due to (4.2) and we obtain from (3.81)2 and (4.4) t

IIv¢(t)ll~: < c + c f g(~)llw~(T)ll~: dT, 0

which by Gronwall's lemma implies for almost all t E I ess sup [IVvE (t) I1~: --< c(f, v0, E0). tel

Therefore also the right-hand sides in (3.61), (3.62), (3.64)-(3.67) and (3.81)2 are

4.4. LIMITING PROCESS ~ --+ 0

143

bounded independently of ¢, which yields T

X

Ov~ ~dt

,

0

ess sup { IIW ~(tlllg: + IID(v~(t))ll,(l~.(~)l,),~(~)} -- c(f, Vo, Eo), tEI T

2

2

/ ilV~v~ll'°; 1 + IlVs(D(v~), E)ll'°; 1 + Y(v ~1 + Z(v ~1 dt ___c(f, Vo, Eo), 0

po+l

(4.5/

po+l

po+3

T

Ilv(ID(v~)l 2, IEI2)llp~o~a + 0

2(20+3) ~ 3(po-1) IlS(D(v ),Elll2~o+a ) dt <_ c(f,

2po f

vo, Eo),

3 (Po -- 1)

T

po+l

(I,,r(D(Vve), 1)) S(,o-1) dt _< c(f, vo, Eo), 0

From (4.5)2,5 and (5.2.41) we derive T

l ] v v ~ 116~p._~_U ,o-11 dt -<- c(f, vo, Eo). 0

(4.6)

PO- 1

(ii) T h e Case 2 < poo < Po < 8/3, Poo < 12/5 Recall that F(t) = 1 + HU(ID(vE(t))t 2, IE(t)12)ill,v(t) and %(s) = (1 - #)-1s1-" if p > 0 and # ¢ 1 and 7(s) = ins. Note, that (4.1) implies that

F e L1(I).

(4.7)

The inequalities (3.83), (3.89) and Lemma 5.2.40 lead, for 1 < s < 1 + ~, ~ sufficiently small and almost all t E I, to

po + 1

po - 1

0 t

2

0 t

t

_
0 t

+c(f, vo, Eo)([~' ~' J

0

~ ,~_I(F(,I)d,). ~--po--1

Poo

144

4. U N S T E A D Y F L O W S

Since we obtain an information from the first term on the left-hand side of (4.8) only for # < 1, and since we got this inequality as a sum of two separate inequalities, we will omit the first term if # > 1. Let us now discuss the terms on the right-hand side of (4.8). Since 2 < poo _< P0, we see that 2 poo-1 po~ po - 1

#~1-#<1,

which together with (4.2) implies that the last term in (4.8) is bounded by c(f, v0, E0). Let us therefore estimate the remaining two terms on the right-hand side of (4.8). We compute 2 E 2 Ks < IlVv ¢ II~llv II ~w.

5p~--2

002(2-~) _ P < IIW~ flp~

p~

2(2-A)

< t

IIW ~

116%:: 2~

(4.9)

2 ~¢:o- 1 A(pQ-1) pcx~- i

p~ [[VvEl[6 p:~_lll _

where we used the interpolation inequality I-A

A

Ilfll 6poo <_ tlftlp~, I1f116~-1, 5p~-6

PO-I

with A = (p~-1)(spc¢-12) and pccpo- Tpoo+6 poo IIv v e I1~-< F.

(4.10)

Furthermore, we observe, for 0 < ~ < p'o/(2s), IIS(D(v~), E)ll22: <

f

2s~ IS(D(v~),E)I2S(1-')(1 + V(lD(v~)l 2, ]El2)) ~-o dz

" e

2s(1-~)

(4A1)

2S~

< ]]S(D(~ ),E)]]2%(,_~)f~, p~o--2s~

where we employed the definition of F and IS(D(v), E)I < c(1 + U(ID(v)I 2, ]El2)) P-~ ,

(4.12) 1,~

which can be proved in the same way as (1.39). Due to the embedding W po+~(f~) ~-+ 6 6 or equivalently ~ = po(3-~(po-1)) s(po-1)(6-po) and thus we Lp-5~-~(~) we require now 2sp~(i-~) p~-2st¢ = p0--1 arrive at 2S

2s~

]]S(D(v ),E)I]2 ~ < c F-~-o (HVS(D(v~), E)l]~ + ]]S(D(v~),E)I]I) 2~(1-~) 2._~s

<_ cF--~-o VS(D(v'),E

+

,

(4.13)

145

4.4. L I M I T I N G P R O C E S S E -+ 0

where we have taken into account again (4.12) to b o u n d the L l - n o r m of S in terms of F . From (4.8), (4.9) a n d (4.13) we derive, for 1 < s < 1 + 6, 6 sufficiently small a n d almost all t E I,

t

~AF(t))+ /(llVv~(r)[126~ + [IV=v~(r)l['°~1) ~:(F(r))dr 0

(4.14)

po+l

t

2

0


~llVv'(r)ll62:-l~'~(F(r))) '°'-'F(r) ,o. 0

t f(

~,~-x

'dr

po--1

2s~ ItVS(D(ve(T) ),E(T) )II~2 ~(F(T) ))s(1-a)(P°-I' F(T) ~O ÷IZ[s(1-~)(P°-I'-1]dT] .

o In order to b o u n d the terms on the r i g h t - h a n d side of (4.14) we will use Young's inequality, which enables us to absorb one t e r m on the left-hand side of (4.14) a n d to b o u n d the other one by (4.7). For t h a t we require for the first t e r m on the r i g h t - h a n d side

A ppo-1 ~-i a = 1 '

(2(2-A) ~ poo + # ( A °2-~--lp~-1) } OZ = 1 ,

1 =1 ~1 + ~T

a n d thus #1 m u s t satisfy s #1 = 2(2poo - 3 + 2p0 -PooPo) 3 + Poo - 2p0(3 - Poo)

(4.15)

T h e r i g h t - h a n d side of (4.15) is for P0 a n d p ~ considered here finite a n d positive as long as 3 + p~ Po < 2(3 - Poo) "

(4.16)

For the second t e r m we proceed similarly a n d require

s(~- 1)(po-- I)#---- i,

{2s~ + #(s(1

- ~¢)(Po - 1) - 1)}/3' < 1

which implies

#2 >-

2S(po - 1) - Po 3 + P0 - 3s(po -- 1) "

For s > 1 chosen a p p r o p r i a t e l y we deduce P0 - 2 /~2 > 6 - 2p------~'

(4.17)

8Since F -~ < 1, we will compute the optimal values for #, denoted by ~1 and ~2 respectively, for both terms separately and than choose the larger one.

146

4. U N S T E A D Y F L O W S

which is finite and positive for poo and P0 considered here. Note, that for P0 < 8/3 we can always select #2 satisfying (4.17) such that (4.18)

#2 < 1.

Moreover, we see that we can also choose #1 < 1 if 9

3(3 - Poo) < poo < P0 < 2 ( 5 - 2p~o) "

(4.19)

This means, that in the case [9 12~ Poo 6 t 4 ' - f f / '

[ 3(3-p~) 8) poo < P0 < min ~2(5 - 2poo)' 3

(4.20)

we can select # _< 1 such that # satisfies the restrictions (4.15) and (4.17). Using Young's inequality in the way described above we deduce from (4.14) and (4.7), for almost all t 6 I, t

2poo_ ~

2

,,oL_-I + [IV2v~(r)[lpo0 0

po+l

t

2

+ f0 (

+

_<

Since # < 1 inequality (4.21) implies that (4.5)2 is satisfied and this in turn implies that t

2~p._~2

2

f llVv~(r) ,,o~_~ ~o-1 0

po-1

t

+ IIv2v'(r)ll

~o;~ po+l

a~ <_ c(f, vo, Eo), (4.22)

2

/IlVS(D(v~(r)),E(T))IIP%-____~ + Y(ve(r)) + Z(v~(r)) dr < c(f,

V0,

E0).

poq-1

0

Since 2A < 2 ~ that

for P0 and poo satisfying (4.20) we obtain from (4.9) and (4.22)1 f0 t K~ dr _< e(f, v0, E0),

and thus we deduce from (3.61), (3.62), (3.64)-(3.67) and (3.81)2 that (4.5) and (4.6) hold. In the other case, i.e. when either pooE [2,

,

p0<min

2(3-poo)'3

'

or

9

< Poo _< Po,

3(3 - p~) 8 2(5 - 2p~) < P0 <

(4.24)

4.4. LIMITING PROCESS ~ --+ 0

147

we have to choose # = #1 > 1 satisfying (4.15). In these cases we obtain using Young's inequality from (4.14) and (4.7) that T f

2 ( n V S ( D ( Y e ( t ) ) , E(t))l[P°-i ÷ , o1+

Z(ve(t)))F(t) -it at

Y(ve(t))÷

0

(4.25) T ,

2p~_ 1

2

+ f (llVv (t)ll ;0-1 ÷ IIv=v 0

(t)ll,o;l)F(t) -. dt <_

po-1

E).

po+l

This estimate is not sufficient to conclude that the right-hand side in (3.64) is bounded independently of c. Thus we need a different estimate for ov~ We have, using the weak formulation (3.7) and (4.2), (4.3), T

Ov" L¢(t,(Vq)'):

sup

~oELq(I,Vq)

II~llnqu,vq)
f f-~-.~dxdt[ o

(4.26)

f~ T

_<

sup t/I-S(D(v~)' ~eLq(I,Vq) II~llLqu,v~)_
E ) . D ( q o ) - [Vv~]v~ • ~

dx dt

+f.~-xEE®E-D(~)

_
+ clIEll2pB,qTItV~IIP0,qT+ f IIv% IlVv%~ IIV~ll~ dt 0 T

< c(f, Vo, Eo)

~-~ f IIV~ll~o,qT + v ~ L~(Z,H)

Vv' ~~+~llV~llmdt,

0

where !s = 1 inequality

1 i0oo

3-m 3m and where we employed div ~ = 0 and the interpolation

Ilvlis < Ilvll~-~ IlWll~, with A = 5p~rn-6po¢ ,~(6-5p~)-6m . Now, requiring for the last term in (4.26) that (1 + A)m' -- p ~ leads to m=-

5po¢ 5poo - 6

Thus, if we choose (4.27) \

opoo -

4. UNSTEADY FLOWS

148 we obtain that < c(f, Vo, E0).

(4.28)

L¢ ff ,(vq)*) -

Now, we have at our disposal all estimates we need to show the existence of weak and strong solutions, respectively, to the problem (1.1), (1.2). (iii) S t r o n g S o l u t i o n s In the cases (i) and (4.20), i.e. when -< P ~ -< P0 < m i n

' 2-~--2--~)

'

we have established the estimates (4.1)-(4.3), (4.5)1-4 and (4.6) which are independent of e. These estimates together with the Aubin-Lions lemma imply the existence of v, ¢ such that (cf. Remark 3.56) VV e -+ VV 0v ~

0v

Ot

Ot

D ( v ~) --' D ( v ) V ~ --2"* V

¢~-~¢ V2V ~ ~ V2v OV~ aV V ~ ~ V Or s

2

6

strongly in Lp0-1 (I, L'~(fl)),

m < P0 - 1 '

weakly in L2(QT), weakly in LP(W'ff)(QT) n L po-1 ( I , L po+l ( ~ ) ) , weakly* in L°°(I, Ep0E(t)I2)) Cl L¢~(I, Vp~), weakly in L2(QT), 2

(4.30)

6

weakly in L2(I, L2oc(gl)) gl Lp0-1 (I, Lpo-1 (fl)), weakly in L2(QT),

s = 1, 2,

as ~ --+ 0 at least for a subsequence. From (4.30)1 we conclude V v e -~ ~Tv

a.e. in QT,

(4.31)

and thus

S(D(v~), E) --~ S(D(v), E)

a.e. in QT,

U(ID(v~)I 2, ]El 2) -~ U(ID(v)?, IEI2)

a.e. in QT.

(4.32)

This in turn together with Vitali's theorem and (4.5)4 implies for m < 3(po-1) 2(p0+3) S(D(vE),E) --+ S ( D ( v ) , E )

strongly in Lm(QT).

(4.33)

Thus we can justify the limiting process c -+ 0 in the weak formulation (3.7) as in the proof of Proposition 3.57. Due to the lower semicontinuity of norms, the identicalness of weak and almost everywhere limits, Fatou's lemma and (4.30)-(4.33) the estimates (4.1)-(4.3), (4.5)1-4 and (4.6) remain valid for the limiting elements v,

4.4. LIMITING PROCESS e -+ 0

149

¢, -~, ~" S ( D ( v ) , E) and U([D(v)[ 2, IEI2). Therefore we have proved the existence part of Theorem 1.14 as far as strong solutions are concerned. Let us prove their uniqueness. First of all it follows from Proposition 2.3.35 that a solution E of (1.1) which is orthogonal to HN(f~) is uniquely determined. Assume that u, v are two strong solutions of (1.2) for the same data f and vo and the uniquely determined E. Let us denote their difference by w. From the weak formulation (1.13) we obtain (cf. proof of Proposition 3.2.38)

1d[lwl[~÷ f(S(D(u), E)- S(D(v),E)). D(w)dx= J [ V u ] w .

2

n

w dx,

fl

and (2.5) leads to

(S(D(u), E) - S(D(v), E)). D(w) dx > aal fl

/iD(w)l dx, 2

fl

while the convective term is estimated by 2p~-3

3__

___~ilVuli~oilwil~~-~ IlVwIl~ ~ poo--1

< ~3~lID(w)ll~+cllVull,~

2poo 2poo - 3

2

IIwlt~,

where we also used Korn's inequality. Because of the last three relations we see that d

2

2P°°

~/llwlh _< c IIVull~£~-3 llw[l~. Since w(0)

(4.34)

0 and V u E Lc~(I,/.Ycc(fl)), Gronwall's lemma yields w(Q = 0

Vt < T ,

which gives the uniqueness and finishes the proof of Theorem 1.14 in the case of strong solutions. (iv) W e a k S o l u t i o n s In the remaining cases, i.e. either (4.23) or (4.24) is satisfied, we have established the estimates (4.1)-(4.3), (4.25) and (4.28), which are independent of e. For the limiting process e --~ 0 it is essential to prove (4.31). This will be accomplished as a consequence of the last term on the left-hand side of (4.25). Firstly, we will show that (4.25) implies that T

[]V2ve[trau dt <_ c(f, v0, E0), 0

(4.35)

4. U N S T E A D Y F L O W S

150 where u = po~-~_l,r = ~

and where a satisfies 3 + p ~ - 2po(3 - p¢¢) 5poo - 3 - 2po

aS

(4.36)

Indeed, we obtain from (4.25) T

T

0

0 T

T

0

0

_< c(f, v0, E) as long as a

#

1 - a -

<1.

This condition is equivalent to (4.36). It can be checked that the right-hand side in (4.36) is always strictly larger than zero, if (4.23) or (4.24) are satisfied. In fact, one can compute that a ranges in the interval (0, 1/3). Having at our disposal (4.35) we use the embedding W 2,r (~1) ~ W 1+8'p~ ({1)

(4.37)

with 9 s = 6-p~(po-1) and the interpolation inequality 2p~ 1_ q

I i V l [ l + a , p ~ ___~

o"

HvIh,p£HV[I~+s,po~,

(4.38)

which holds for 0 < a < s in order to show (4.31). For that we choose fl E (1,po~) and determine a > 0 such that T

Hv Ill+~,,~ dt <_ c(f, v0, E ) .

(4.39)

0

Using (4.38), (4.37) and HSlder's inequality with 6 = ~(8-~)~ we compute T

T

iiv II1+~.,~ dt < o

I]v I11.,. IIv Ih+,.,~

dt

o

(4.40) T

0 9 N o t e , t h a t s > 0 for P ~ , P o a n d r c o n s i d e r e d here.

1

T

0

:?

1

4.4. L I M I T I N G P R O C E S S e --+ 0

151

The last inequality implies (4.39) due to (4.2) and (4.35) provided that fl a--5' = ~ u , 8

or equivalently 1 1 5+-~

1

fl(s - a) sp~o

fla su~'

which implies that a - sau(poo - fl)

(4.41)

3(poo - ~ u)

One easily checks that for poo and P0 considered here and fl chosen as above we obtain that 6 E (1, c~) and a E (0, s). The estimate (4.39) means that v ~ is bounded in L~(I, Wa+~'v°~ (fl)) n LP°°(I, Vpoo), where fl > 1 and a > 0, which together with (4.28), the Aubin-Lions lemma and (4.1)-(4.3) implies that there exists v such that V v E --~ V v

a.e. in

QT,

V e --~V

weakly in Ep(IEp)(QT ) N LP=(I, Vp=) ,

V 6 ....x* V

weakly* in L ~ ( I , H ) ,

(4.42)

Ov ~ Ov weakly in L q' (I, Vq*), Ot Ot where q satisfies (4.27). From (4.42)1 we deduce that (4.32) holds. Moreover, the estimates (4.1)-(4.3) and (4.28) remain valid for the limiting elements v, ~ , S ( D ( v ) , E) and U([D(v)I 2, [El2). From the weak formulation (3.7), integrated over (0,T), we obtain T

T

\--~' 0

~/vq dt + 0 ~

(4.43)

T

0

which holds for all ~o e :D(-c~, T, V) and where q is given in (4.27). The limiting process as ¢ --+ 0 in the first term is easy; in the second term we employ Vitali's theorem, which is possible due to (4.32)a, (4.3) and the growth properties of $; in the convective term we use the properties of the mollifier and that 5 v ~ --+ v strongly in Lm(QT), m < ~Poo V v e --+ V v

strongly in Lr(QT),

r < Po~,

which follows from (4.42), the parabolic embedding 5.1.17 and Vitali's theorem. Via a standard argument (cf. M&lek, Ne~as, Rokyta, Rfi~i~ka [70], Remark 5.3.66) we deduce from (4.43) that also the weak formulation (1.10) holds for almost all t E I. This completes the proof of Theorem 1.14. •

5 Appendix 5.1

General

Auxiliary

Results

In this section we collect general definitions and results used in the previous chapters. Definition 1.1. (Description of t h e b o u n d a r y ) We say that a bounded domain 1 C R 3 belongs to the class C~'~, l • N, /~ • [0, 1], if and only if there exist: N Cartesian coordinate systems Xn ( N • N, n = 1,... , N)

z n = (x.~, xn:, ~ )

= (x', x.3),

a number ~ > 0 and N functions

an • C~'~([-~,4~), such that

Oan(O'__.. ) _

0,

s = 1, 2,

(1.2)

OXns

and sets

A n = {x= = (x', xna); Ix'l <_ ~,xn3 = an(x')} v :n -- { x .

=

( X. ,t

X

.3);Ix.It <_ ~,an(x') < x . 3 < a n ( x ' ) + ~ }

v2 = { x . = (x', x.3); 14,1 < ~, a . ( x .t) - ~ < x.3 < an( X t.) }

vn= V:UAnUV2 with the properties N

An c 0~,

Y~c~,

P r o p o s i t i o n 1.3. (Korn's inequality) Let 1 < q < oo and let fl C R d be of class C 1. Then there exists a constant Kq = Kq(~) such that the inequality gqllvlil,q < IlD(v)liq is fulfilled for all v • w~'q(~).

PROOF : see e.g. NeSas [92]. 1For simplicity we restrict ourselves to the three-dimensional case.

(1.4)

154

5. A P P E N D I X

L e m m a 1.5. Let 1 < q < oo and let ~ C R a be a domain of class C 1. Then there exists a constant cq = Cq(f~) such that for all v E w~'q(~) M w2'q(f~)

cqIiV~-viiq _< IiD(Vv)iIq.

(1.6)

PROOF : The assertion follows immediately from the algebraic identity 02v, OX3OX~

Ov

-

+

D,j ( ~xk Ov , -

Djk( OV )

(1.7)

T h e o r e m 1.8. (On n e g a t i v e n o r m s ) Let 1 < q < oo and let v 6 Lq(~). Then there exists a constant such that

cllvli q <_ llv[t_l,q + IIVvll_l,q. PROOF : see e.g. NeOns [92].

(1.9) •

T h e o r e m 1.10. Let F E (w~'q(f~)) * be such that (F, ¢P)l,q = O,

V%0 E Vq .

Then there exists a unique ¢ E L q' (fl), with ffl ¢ dx = 0 such that

(F, %0)1,q= - f ediv %adx

V%0 E w l , q ( ~ ' ~ ) ,

(1.11)

fl

PROOF : see e.g. De Rham [109], Bogovskii [15] or Amrouche, Girault [5].

•

T h e o r e m 1.12. (Vitali) Let ~ be a bounded domain and let f,, : ~ -+ R be integrable functions such that: (i) l i m f,~(x) exists and is finite for almost all x E f~; (ii) for all e > 0 there exists 8 > 0 such that

V G c fl,lGI < 5.

supflf~(x)Idx<~ hEN J G

Then we have

PROOF : see e.g. Kufner, John, FuSik [57].

•

Proposition 1.13. ( E m b e d d i n g ) Let f~ C ~ be a bounded domain, Of2 E C 1 and let O <_j < k, l <_ q < oo. If -~ -_ ~

k-cA d , we have

Wk'q(fl) ,-4 WJ"n(~). PROOF : see e.g. Kufner, John, Fu~ik [57].

(1.14) •

5.1. G E N E R A L A U X I L I A R Y R E S U L T S

155

P r o p o s i t i o n 1.15. ( I n t e r p o l a t i o n ) Let 1 < q2 < q < ql < oo and let v • L ql (~) CI L q2(f~). Then we have i-(~

a

(1.16)

11% <-INL~II%~ , where ~ = ~ + -~-, ~-" ~ • [0, I]. PROOF : The assertion follows directly from HSlder's inequality. L e m m a 1.17. ( P a r a b o l i c E m b e d d i n g ) Let f • L ~ ( I , L2(fl)), V / • r > l . Then we have T

L'(I, L'(~)),

T

f Ll tl:., __.(e's'u' .

J

0

0

+

e.ssu. II,"II ',

(1.18)

whenever

0~= g5 r .

(1.19)

PROOF : This follows from the interpolation inequality

Ilfll. -< Ilfll~ Ilflh,, < Ilfll~ + Ilfll~-~llVfll, ~ with A = 3~(~-2) Requiring An = r yields (1.18). a(Sr-6) '

•

L e m m a 1.20. ( A u b i n - L i o n s ) Let 1 < a, 13 < oo and let X be a Banach space. Further, let Xo, X1 be separable, reflexive Banach spaces. Provided Xo '--+'--+ X ~ X1 we have dv {v • U ' ( I , Xo) , - ~ • n~(I, X l ) } ¢--~'--+L ~ ( I , X ) . PROOF : see e.g. Lions [68] Section 1.5.

•

L e m m a 1.21. Let f, g • Lq(ft), q < 2. Then we have

(

Iglqdx) ~ _
(I+I/P)

~ bl2dx.

(1.22)

fl

PROOF : Inequality (1.22) follows from H61der's inequality.

•

L e m m a 1.23. For all s > 1 there exist constants c(s), 5(s) such that for all x, y • R n 1

c(1 + Ixl2 + lyt2)~ < / (1 + I(1 - a)x + ~y12)~ da <_ ~(1 + IxP + lY12)~

(1.24)

0

PROOF : see e.g. Giusti [39] Lemma 8.3.

•

156

5.APPENDIX

Proposition 1.25. Let 0f2 E C°'1 be a bounded domain and assume q > 1 and f E Lq(f2) are given. Then there exists a solution u E W~'q(f2) of the problem

div u = f

in 12

which satisfies the estimate

IlVullq ~ cIIfllq. PROOF : see e.g. Bogovskii [14], [15] or Galdi [36].

The Method of Difference Quotient For the regularity in the interior and near the boundary we used the method of difference quotient. Here we fix the notation and state some relevant results. Using the definition of the boundary we finally choose sets f/~ covering Oft such that ~ C V ~ = V 2 U A ~ U V _ ~ , dist(0f/~,0V ~) > h0 > 0. Let us fix n and drop for simplicity the index n. Setting ~i

_

(1, 0)

and

~2 = (0, 1),

we can define for s = l, 2 and h E (0, h0) the translation mapping T = Ts,h : f~o --4 V by (1.26)

x ~ (x' + h~ ~, x3 + a(x' + h~ ~) - a(x')) - y .

Then the inverse mapping T -1 is given by (x -- T - l ( y ) ) y --r (y' - h~ ~, x3 + a(y' - h~ ~) - a(y')).

Put A+a(x ') = a(x' + h~ s) - a(x') . Then

(

: i,j=1,2,3

1

0

O)

o

1

o

(1.27)

OA+aoxz~(xr~] ~aA+a(XI'l] 1

and --(Y)] Oyj

/ i,j=l,2,

3=

0 ~(y,)

1 0 _~_y2 y,OA-a,,) ( 1

.

(1.28)

For both matrices (1.27) and (1.28) the determinant is equal to i. The s-th tangential derivative (s = 1, 2) of any (scalar, vector or tensorial) function g, denoted ff--~,, is defined by

o-~ (x)

lim h-~0

g(Tx) - g(x) h

'

5.1. GENERAL AUXILIARY RESULTS

157

and

Og ~x. Og Oa , o~-. (~1 = ~ ( ~ 1 + -5-~(x)-~. (~ 1

(1.2o)

holds. For the readers convenience let us show that if g e W01'v(gl), p > 1, then for all h E (0, ho)

f

g(TX)h g(x) "dx <_c(a)UVgU~.

(1.30)

~o

Indeed, setting

Tx(x) = (x' + Ah~~,x3 + a(x' + Ah~~) a(x')), -

(i .31)

we can write 1

flo

~o

0 1

No

0

1

f

Og(y) o~-~ PdydA (1~9) _ c(~)llVgll~.

0

On the other hand, if g E LP(~) and if for all h E (0, h0) (1.32) fl

then o_g_ Or s exists (for s = 1, 2) in the sense of distributions, and

Og Pdx<:co

(1.33)

OTs n

Let V' CC ~o CC ~2, be such that dist(0~o, 012) = h0 > 0. Let ~r, r = 1, 2, 3, be a basis of a coordinate system in R ~. For r -- 1, 2, 3 and h E (0, ho) we define the translation mapping Tk,h : ~10 --+ i2 by x --+ x + h~ k .

(1.34)

The gradient of T and its inverse T -1 have determinant equal to 1. The properties (1.30)-(1.33) hold correspondingly. Moreover, we have

g(TkX)h- g(x) ÷ ffgxk(X)

strongly in L~oc(~ ) .

(1.35)

5.APPENDIX

158

5.2

Auxiliary Results for the A p p r o x i m a t i o n s

In this section we prove the technical assertions used in the previous chapters for deriving estimates independent of A and ~. Note, that all results can be used for the approximations S A of the extra stress tensor S both in the steady case (cf. (3.3.13)) and in the unsteady case (cf. (4.1.24)) . Recall the notations (cf. (3.5.2), (4.3.2), (4.3.3))

4,r,A(U, ~) =

(2.~)

(1 + IE12)~(I + ID(v)I 2) 2 ~lul'~ x x {(1 +

ID(vll2) 2~-q~XA+

(1 + A2) 2a~a~(1 - XA)} dx,

and

/p,A(U,f)

=

f (I+IEI~)(I+ID(v)I

=)

(2.2)

~ IuI~= x

x {(1 + ID(v)I2)~XA + (1 +

A2)e-~ (1 - XA)} dx,

where u is usually some second order derivative of v and ~ will be either identical 1 or one of the cut-off functions ~0 or ~l. L e m m a 2.3. For i < r < oo, there exist constants c, ~ independent of A such that c Ip,,,A(D(Vv), ~) <_/IDkzsA(D(v), E ) D , d V v ) lr~ dx fl _< c I p , r , A ( D ( V v ) , ~i)

(2.4)

i = 0, 1.

P R O O F : Since Ip,r,A(D(Vv), ~i), i = 1, 2 is the sum of integrals over ~A and f~ \ f~A, we will show the above inequalities only in the case f~ \ QA. For brevity we omit the index i at ~i. On the set ~ \ f~A we have for the integrand of Iv,r,A(D(Vv ), ~)

((1 +

q--2

7"

IEI2)(1 + ]D(v)[ 2) 2 (1 + A2)P-~ID(Vv)[ 2) ~ × r

~frdx

ID(v)12) 2 (1 + r

< c (Ok,S A ( D (E) ) D v ,

ij

(V v )D kl (V v ))2 x

× ((1 + IEI21(1 + ID(v)12)~-(1 + A2)~))~C,

159

5.2. A U X I L I A R Y R E S U L T S F O R T H E A P P R O X I M A T I O N S

where we used (3.3.25) and (3.3.56), respectively. Thus Young's inequality delivers Iv,T,A(D(Vv), ~(1 - XA)) < c

(2.5)

IOktSa(D(v),E)Dkl(Vv)12~2 ×

f~\~a

x ((1 + IEI2)(1 + ID(v)12)~-~(1 + A2) 2m~-~lD(Vvli~) ~ dx

--II.,.,A(D(Vv),~(I-X.))+

f

1Ok/SA(D(v)'E)D~'(Vv)i'~'dx"

~\na

This proves the first inequality in (2.4). The second one follows easily from (3.3.27) and (3.3.58), respectively. • R e m a r k 2.6. Prom the proof of Lemma 2.3 it is clear that we could also allow r = r(x) E (1, oo). Moreover, it is possible to take ~ -- 1 in Lemma 2.3. L e m m a 2.7. There exists a constant c such that IlV 2 v~01i22 <- c I , , A ( D ( V v ) , ~ o )

(2.8)

liv-~0 ~ li~q < c(1 + g,A(D(W), ~0) + ]iWli~)

(2.9)

if q = 2, and 2

ifq<2.

PROOF : Inequality (2.8) follows immediately from the definition of Ip,a and the algebraic identity (1.7). For q < 2 we notice k(1 + IDI2)} = q(1 + IDi 2) 4 Dij(v)DiJ(0-~xk) , which immediately implies IlV( 1 + IDI2) 41~0Ii~ -< cYp,A(D(Vv),~o). From the embedding W1,2(ft) ¢-+ L6(~) we obtain 2

laY(1 + IDI2) ~ ~o11~> cilVv ~o~I]~q- c(V~0) (1 + ilVvl]g), where we also used Korn's inequality and some straightforward computations. Inequality (2.9) follows from the last two inequalities. • L e m m a 2.10. Let r < 2 if po < 2 and r < P'o if po >>_2. Then there exists a constant c = c(E0) independent of A such that for i = O, 1 Ip,r,A(D(Vv), ~0 _<

(2.11)

e,)) ( f, + n

160

5.APPENDIX

PROOF : Again we will only consider the case [D(v)t >_ A. For p < P0 < 2 we see ~_~ (1 + [D[ 2) 2 (1 + A2) v-~ 2 < (1 + IDI2)P-~ < 1 (2.12) and in this case inequality (2.11) follows immediately from (2.5). If P0 _> 2 we get from (2.5) and HSlder's inequality Ip,r,A(D(Vv), ~(1

-

XA)) r

f

~\~A

(2.13) 2-r

x ( f

{(I+IE]2)(I+A2)~-~(I+ID(v)]

2) q-2 2 } 2r._y._ - r d x ) -'~--

g~\ftA

The expression in the squiggly brackets can be written as

(1 + IE?){(1 + A2) (1 + IDI2)

tp

( 1 + A 2 5 p~'~

< c(1 + [EI2){I+0UA([DI 2,lEI2) • D } ,

(2.14) ,

where we used (3.3.29) and (3.3.60), respectively, ID[ _> A and p > q. On the set where p(IE] 2) _< 2, the last term is bounded by a constant, while on the set where p([EI 2) > 2 we require p-2 p

r -< 1, 2-r-

(2.15)

which is equivalent to r _< p~. Therefore the second integral in (2.13) is estimated by

c(Eo) / " 1 + ouA([D[ 2, [E[2) • D d u , which yields (2.11) 2.

"

L e m m a 2.16. There exists a constant c = c(Eo) such that, if po < 2 f [onSAVEn[r~ r dx < c(E0)(1 + [[OUA. D ~i[[1)

i = 0,1

(2.17)

f~

holds for r <_ 2, and if Po >- 2,

f~ holds for r <_fro and s > 1. 2Note, t h a t in the case ID(v)l <_ A, one must divide the set Ftn into two sets {z, [D(v)l <_ 1} and {x, 1 _< ID(v)l _< A}. The first one is trivial and the second one can be handled as in the case [D(v)[ _< A.

5.2. AUXILIARY RESULTS FOR THE APPROXIMATIONS

161

PROOF : Using (3.3.44) and (3.3.68), respectively, and a similar argument as in (2.14) we obtain, for s > 1,

f I~nSAVSntr~ r d:~ ~ c(F~o)f (1 -Jr-OUA. D)SrP~-~l~ v dx, I2

(2.19)

f~

where we used ar_t P < -- po-1. Po Now, for P0 < 2 we choose s such that s ~ - ~ < 1/2 and (2.17) follows. For Po > 2 and r < p~ we have r ~ - ~ < 1. Moreover, due to (3.3.28), (3.3.24) and (3.3.59), (3.3.55), respectively, it holds

IOUa(lD(v)l 2, IEI2) • D(v)l _< c(E0)lSa(D(v),

E)I ID(v)l •

(2.20)

Therefore the right-hand side of (2.19) is estimated by

c(Eo) (1 + IlVvll;llsagll~_),

(2.21)

q--8

which gives (2.18).

•

R e m a r k 2.22. From the proof of (2.17) and (2.18) it is clear that also global versions of these inequalities hold, i.e. ~ - 1 is admissible. P r o p o s i t i o n 2.23. There exists a constant c = c(Eo) such that lieS

~d12 <

c(E0) 1 + / p , a ( D ( V v ) , {i) + I[OUA" D ~i111

i = 0, 1,

(2.24)

if po < 2, and for s > 1 IIVSA

~ill~a ___c(E0) Ip,A(D(Vv), {i) (1 + [10VA. DIll ) ~ 2

(2.25)

2s

+ c(E0)(1 + IISA(D(v), E)~,ll,q, llVvllq) 70

i = 0, 1,

ifpo >_ 2. PROOF : For i = 0, 1 we observe

f [vsalr~ r dx < f ]OklSADkt(VV)lr~ r -+-IOnSaVEnr~r dx

n

l] =11 +I2.

For P0 < 2 we use (2.4) and (2.1t) for r = 2 to obtain I1 <_ c(E0)/p,A(D(Vv),~i)

and from (2.17) we conclude for r = 2 /2 <~ C(E0)(1 "}-Ilou A" D{dll)

(2.26)

162

&APPENDIX

which immediately proves (2.24). In the ease P0 _> 2 we put r = p~ and use (2.4) and (2.11) to obtain

i1 _<

+ llov

D,1)

2(Pp°o~_21)

and from (2.18) we deduce /2 < e(E0)(X + [IsA@I~,,IIVvlI~), and inequality (2.25) follows.

•

L e m m a 2.27. There exists a constant independent of A such that IIV2vllK < clp,r,A(D(Vv), 1)

(2.28)

I{V2vl]~(po°-t) < c(1 + {IVv{r(P°°-t) +/V,r,A(D(Vv), 1))

(2.29)

if Poo >_ 2 and

3r(poo-1)

if poo < 2, where s - 3-r(2-poo)" PROOF : Inequality (2.28) follows immediately from the definition of Ip,r,A, observing that in this case q = 2. Let us therefore assume Poo = q < 2. We have, again using the algebraic identity (1.7),

IV2vl s dx <_

((1 + ID(v){ 2) 2 r J D ( V v ) r ) ; (1 + ID(v)l) (2-O dx

n

< c(Ip,r,A(D(Vv), 1));

(1 + ID(v){) ( -q)~-~ dx

~

fl

Requiring now (2 - q ) - -

sr

-

r--S

3s 3--S'

which is equivalent to 8--

3r(q - 1) 3 - r ( 2 - q)'

and using [11 + ID(v)lll 38

3--s

_< c(1

+ I{D(v){l~ + {IV:vll~),

which follows from Sobolev's embedding theorem, we obtain [{V2v[l~ _< c(Ip,r,a(D(Vv), 1)) ~ (1 + IlD(v)ll, + IIV~vll~) ~<~-~) $

< c(/px,A(D(Vv), 1)) ~(-(~:U-l) + ½(1 + IID(v)l[~ + IIV2vll~), where we used Young's inequality. The last inequality implies (2.29).

5.2. A U X I L I A R Y R E S U L T S FOR T H E A P P R O X I M A T I O N S

163

3po(po~-t) Then C o r o l l a r y 2.30. Let poo < 2 < Po, and s -- p0(l+po~)-3" < c(1 + liWil~: + I~,p~,A(D(Vv), 1) ).

iiV~vii~ ~ - ~ )

PROOF : We use inequality (2.29) and observe that P'o(Poo- 1) < poo. Now, if s g poo the assertion follows immediately, and if s > poo we interpolate between poo and 3--s 3_~_~ and use Young's inequality. • Analogously to (2.1) we introduce the notation

I,,~(D(Vv),~) =

(1+ IEl2y(1 + JD(v)l 2) 2 ~ID(Vv)

dx,

(2.31)

fl where ~ is either identical 1 or ~ = ~o E T~(fl). In the same way as in Lemma 2.3 we prove L e m m a 2.32. There exists a constant such that Ip.,(D(Vv), ~) _< c f [0k~S(D(v), E)Dkl(Vv)]r~* dx, f~

(2.33)

where ~ - 1 or ~ E I)(~).

L e m m a 2.34. There exists a constant c = c(E0) such that for Poo > 2 we have

(2.35) fZ PROOF : Similarly as in (4.3.37) one can show jf []O~S]lr <_c(

1 SA.Ddx)5(I+i[Si[I+i[VS

po-2 ~)po-1.

(2.36)

The definition of Ip,~ and the chain rule applied to VS implies (2.37)

(Ip,r(D(Vv), 1)) ~ _ c (]]vsA]Ir + i]0,~S]l~)

_
This inequality raised to the power u = ~

s.e x

(l+llSill+liVSilr) 0 -1)

gives (2.35).

*

164

5.APPENDIX

We also obtain the analogue of Lemma 2.16. L e m m a 2.38. There exists a constant c = c(E0) such that f o r poo > 2 we have, f o r s>l,

/ 10.S(D(-),E)VE.I'~ d~ _<~(1 + ilV.IILIIS(D(V.), E)Ii:~:4).

(2.39)

12

Finally, we have the following lower bound for Ip,r(D(Vv), 1). L e m m a 2.40. There exists a constant such that .for 1 < r < 3 we have

,,--,,~-"~<"=-'~3,..

_
/2~,/

PROOF : If Poo < 2 the inequality (2.41) follows from (2.29), Sobolev's embedding theorem and

If Po~ > 2 we have

Ip,r(D(Vv),i) > i [V(I+ iD(v)[)]~(l+ [D(v)[)('°0-2)~ dx fl

_> c/Iv0

+ IDi)'~-'i~~x

fl

> cllVvi__ which gives (2.41).

-1~ •

References [1] B. ABU-JDAYIL AND P.O. BRUNN, Effects of Nonuniform Electric Field on Slit Flow of an Electrorheological Fluid, J. Rheol. 39 (1995), 1327-1341. [2] B. ABU-JDAYIL AND P.O. BRUNN, Effects of Electrode Morphology on the Slit Flow of an Electrorheological Fluid, J. Non-New. Fluid Mech. 63 (1996), 45-61. [3] B. ABU-JDAYIL AND P.O. BRUNN, Study of the Flow Behaviour of Electrorheological Fluids at Shear- and Flow- Mode, Chem. Eng. and Proc. 36 (1997), 281-289. [4] E. ACERm AND N. Fusco, Partial Regularity under Anisotropic (p, q) Growth Conditions, J. Differential Equations 107 (1994), 46-67. [5] C. AMROUCHE AND V. GIRAULT, Decomposition of Vector Spaces and Application to the Stokes Problem in Arbitrary Dimension, Czechoslovak Math. J. 44 (1994), 109-140. [6] R.J. ATKIN, X. SHI, AND W.A. BULLOGH,Solutions of the Constitutive Equations for the Flow of an Electrorheological Fluid in Radial Configuratons, J. Rheology 35 (1991), 1441-1461. [7] P. BArnEY, D.G. GILLIES, D.M. HEYES, AND L.H. SUTCL~FFE,Experimental and Simulation Studies of Electro-Rheology, Mol. Sire. 4 (1989), 137-151. [8] M. LE BELLAC AND J.M. L~vY-LEBLOND, Galilean Electromagnetism, I1 Nuovo Cimento 14 (1973), 217-232. [9] H. BELLOUT, F. BLOOM, AND J. NE~AS, Young Measure-Valued Solutions for Non-Newtonian Incompressible Fluids, Comm. PDE 19 (1994), 1763-1803. [10] T. BHATTACHARYAAND F. LEONETTI, A new Poincar~ Inequality and its Applications to the Regularity of Minimizers of Integral Functions with Nonstandard Growth, Nonlinear Anal. Theory Methods Appl. 17 (1991), 833-839.

[11] T. BHATTACHARYAAND F. LEONETTI, W2'2-Regularityfor Weak Solutions of Elliptic Systems with Nonstandard Growth, J. Math. Anal. Applications 176 (1993), 224-234. [12] L. BOCCARDO, T. GALLOUET, AND P. MARCELLINI,Anisotropic Equations in L 1, Diff. Int. Equ. 9 (1996), 209-212.

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Index apriori estimate, 57, 65, 68, 84, 94, 100, 109, 114, 124, 127, 133, 135, 139, 140, 142, 143, 146, 147, 150 approximation, A, 72, 74-83, 110-113, 158-164 ~,A, 72, 84-100, 113, 114-141 E, 94, 100, 101-103, 126, 135, 139, 140, 142-151 balance law, 9, 13, 15, 39, 61, 105 thermo-mechanical, 2-4 electro-magnetic, 4 boundary condition, 40, 54, 61, 105 Clausius-Duhem inequality, 3, 8, 16, 20, 21-23, 30-33 coercive, 33, 62, 67, 72, 76, 81,106, 111 constitutive relation, 5, 8, 12, 13, 15, 37 electrorheological fluid, 9, 15, 16, 20, 21-23, 25-27, 61, 105, 106 s e e stress tensor constraint, internal, 21-22 ER-effect, viii derivative, normal, 121, 129-131 tangential, 117-120, 129, 135, 156157 difference quotient, 87-93, 117-121,156157 field, electric, viii, 39-40, 54-59, 63, 85, 109, 115 magnetic, 9, 11-13, 39-40 fluid, electrorheological, vii, viii, 1, 10 linear, viii, 15-23 shear dependent, ix, 24-27 see constitutive relation

generalized Newtonian, x, 63, 74, 106, 108 non-Newtonian, 25 function space, 43-45, 52-54, 107 Galerkin approximation, 63, 64, 85, 115 gradient, symmetric velocity, xiv, 2 growth, non-standard, ix, 40-43, 72, 74, 106 invariance, 5-7 Lebesgue space, 43 generalized, 44-50, 52-54, 106 limiting process, 65, 85, 93, 99, 103, 116, 135-141,148, 151 Maxwell's equations, 4, 54-59, 61, 105 model, dipole current-loop, viii, 2 s e e constitutive relation mollification, 83 monotone operator, 33-36, 62, 65, 66, 71-72, 74, 76, 81, 106, 111, 116 non-dimensionalization, 10-15 potential, 75-83, 106, 110 pressure, 71-72, 73, 86, 106, 116-117 pressure driven flow, 28-30 shear flow, 24-26 Sobolev space, 43, 71 generalized, 46-50, 52-54, 64-66 solution, Maxwell's equations, 56-59 weak, 62, 63, 84, 94, 107, 108, 109 s e e steady problem, unsteady problem strong, 73, 84, 107, 108 s e e steady problem, unsteady problem

176 steady problem, x-xii, 61-103 weak solution, 62, 63 strong solution, 73, 84, 94, 100-103 stress tensor, viii approximate, 72, 74-83, 110-113 linear, viii, 15-23, 109 nonlinear, ix, 25-27, 61, 105 shear function, 25 normal stress difference, 25 transformation, Galilean, 5 Lorentz, 6 variational integral, 41-43 uniqueness, 69, 108 unsteady problem, xii-xiv, 105-151 weak solution, 107, 108, 109 strong solution, 107, 108, 113, 126, 127, 135, 139 viscosity, generalized, 25-27

INDEX