Homogenization theory with applications in tribology

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(1)2008:28. L ICE N T IAT E T H E S I S. John Fabricius Homogenization Theory with Applications in Tribology 2008:28. Universitetstryckeriet, Luleå. Homogenization Theory with Applications in Tribology. John Fabricius. Luleå University of Technology Department of Mathematics 2008:28|:-1757|: -lic -- 08 ⁄28 -- .

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(3) Homogenization Theory with Applications in Tribology John Fabricius Department of Mathematics Lule˚ a University of Technology SE-971 87 Lule˚ a, Sweden.

(4) Key words and phrases. Mathematics, Homogenization theory, Partial differential equations, Calculus of variations, Multiscale convergence, Two-scale convergence, Asymptotic expansion method, Tribology, Hydrodynamic lubrication, Reynolds equation, Surfaces roughness.

(5) To the memory of Carl-˚ Ake.

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(7) Abstract This thesis is devoted to the study of some homogenization problems with applications in tribology. Homogenization is a mathematical theory for studying differential equations with rapidly oscillating coefficents. Many important problems in physics with one or several microscopic length scales give rise to this kind of equations. Hence there is a need for methods that enable an efficient treatment of such problems. To this end several homogenization techniques exist, ranging from the fairly abstract ones to those that are more oriented towards applications. This thesis is concerned with two such methods, namely the “asymptotic expansion method”, also known as the “method of multiple scales”, and multiscale convergence. The former method, sometimes referred to as the “engineering approach to homogenization” has, due to its versatility and intutive appeal, gained wide acceptance and popularity in the applied fields. However, it is not rigorous by mathematical standards. Multiscale convergence, introduced by Nguetseng in 1989, is a notion of weak convergence in Lp spaces that is designed to take oscillations into account. Although not the most general method around, multiscale convergence has become widely used by homogenizers because of its simplicity. In spite of its success, the multiscale theory is not yet sufficiently developed to be used in connection with certain nonlinear problems with several microscopic scales. In Paper A we extend some previously obtained results in multiscale convergence that enable us to homogenize a nonlinear problem with three scales. In Appendix to Paper A we present in more detail some results that were used in the proofs of some of the main theorems in Paper A. Tribology is the science of bodies in relative motion interacting through a mechanical contact. An important aspect of tribology is to explain the principles of friction, lubrication and wear. Tribological phenomena are encountered everywhere in nature and technology and have a huge economical impact on society. An important example is that of two sliding solid surfaces interacting through a thin film of viscous fluid (lubricant). Hydrodynamic lubrication occurs when the pressure generated within the lubricant, through the viscosity of the fluid, is able to sustain an externally applied load. Many common bearings, e.g. journal bearings or slider bearings, operate according to this principle. As a branch of fluid dynamics, the mathematical foundations of lubrication theory are given by the Navier–Stokes equations, describing the motion of a viscous fluid. Because of the thin film assumption several simplifications are possible, leading to various reduced equations named after Osborne Reynolds, the founding father of lubrication theory. The Reynolds equation is used by engineers to compute the pressure distribution in various situations of thin film lubrication. For extremely thin films, it has been observed that the surface microtopography is an important factor in hydrodynamic performance. Hence it is important to understand the influence of surface v.

(8) vi. ABSTRACT. roughness with small characteristic wavelength upon the pressure solution. Since the 1980s such problems have been increasingly studied by homogenization theory. The idea is to replace the original equation with a homogenized equation where the roughness effects are “averaged out”. One problem consists of finding an algorithm that gives the homogenized equation. Another problem, consists of showing, by introducing the appropriate mathematical defintions, that the homogenized equation really is the correct one. Papers B and C investigate the effects of surface roughness by means of multiscale expansion of the pressure in various situations of hydrodynamic lubrication. Paper B, for which Paper A constitutes a rigorous basis, considers homogenization of the stationary Reynolds equation and roughness with two characteristic wavelengths. This leads to a multiscale problem and adds to the complexity of the homogenization process. To compare the homogenized solution with the solution of the unaveraged Reynolds equation, some numerical examples are also included. Paper C is devoted to homogenization of a variational principle which is a generalization of the unstationary Reynolds equation (both surfaces are rough). The advantage of adopting the calculus of variations viewpoint is that the recently introduced “variational bounds” can be computed. Bounds can be seen as a “cheap” alternative to computing the relatively costly homogenized solution. Several numerical examples are included to illustrate the utility of bounds..

(9) Preface This thesis is composed of three papers and a complementary appendix. These publications are put to a more general frame in an introduction which also serves as a basic overview of the field. A. A. Almqvist, E. K. Essel, J. Fabricius and P. Wall. Reiterated homogenization of a nonlinear Reynolds-type equation. Research Report, No. 4, ISSN:1400-4003, Department of Mathematics, Lule˚ a University of Technology, (19 pages), 2008. Appendix to A. J. Fabricius. B. A. Almqvist, E. K. Essel, J. Fabricius and P. Wall. Reiterated homogenization applied in hydrodynamic lubrication. To appear in Proc. IMechE, Part J: J. Engineering Tribology, 2008. C. A. Almqvist, E. K. Essel, J. Fabricius and P. Wall. Variational bounds applied to unstationary hydrodynamic lubrication. Internat. J. Engrg. Sci., 46(9):891–906, 2008.. vii.

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(11) Acknowledgement I thank my main supervisors, Prof. Lars-Erik Persson and Prof. Peter Wall, for guiding me in the fascinating field of mathematics, for giving encouragement and support in times of need and for always showing a generous attitude towards their students. I also thank Dr. Andreas Almqvist and my fellow Ph.D. student Emmanuel Essel, with whom collaborating has been “frictionless” and much joy. Dr. Almqvist also accepted the role as my co-supervisor, contributing with his expertise in tribology. For this I am also very grateful. During my undergraduate studies at Uppsala University and ENS Lyon I had the privilege to have several inspiring teachers who influenced both my views on mathematics as well as my taste in this subject. Finally, a special mention goes to my colleagues at the Department of Mathematics who do an excellent job promoting mathematical activity in the vicinity of the Arctic Circle. You light up the dark polar night.. ix.

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(13) Introduction Tribology is the science of interacting surfaces in relative motion. The word tribology is derived from the Greek tribos which means ‘rubbing’. Many machine components, e.g. bearings, gears, piston rings, tyres, breaks and magnetic storage devices, consist of parts that operate by rubbing against each other. Tribology is an interdisciplinary science dealing with such diverse phenomena as friction, wear, lubrication and contact mechanics. Any mechanical system in which friction occurs will experience energy loss in terms of heat dissipation. Wear in machine components may cause mechanical failure and shortens their life span. According to the Jost Report, issued by the British government in 1966, and subsequent studies in other countries, the estimated costs due to tribological phenomena are of the order of one per cent of the gross domestic product. Thus, rational machine design has huge economical motives. Lubrication is the action of viscous fluids to diminish friction and wear between solid surfaces. It is fundamental to the operation of all engineering machines and many biological mechanisms, e.g. hip joints. It can be observed that a converging fluid film is able to separate two surfaces in relative motion pressed together under an external load. When a fluid film achieves this state, called the hydrodynamic lubrication regime, solid to solid contact is prevented and the applied load is supported by a pressure that develops within the film because of the lubricant’s resistance to motion. In 1886 Osborne Reynolds published a theory of lubrication [66], now considered a milestone in tribology. By applying the principles of fluid dynamics, Reynolds derived the equation, subsequently reffered to as “the Reynolds equation”, that governs hydrodynamic lubrication of a cylindrical journal revolving in a cylindrical bearing (see Figure 1). The Reynolds equation is a two-dimensional approximation of fluid flow that relies on the assumptions that the surfaces are “nearly parallel” and that the radii of curvature of both bearing and journal are large compared with the thickness of the film. The Reynolds equation can take a variety of forms, see [36, 69]. If the lower surface, denoted by Ω, is a portion of the plane x3 = 0 and the upper surface is the corresponding portion of the surface x3 = h(x1 , x2 ), where h is a smooth, everywhere positive function, then a simple form of Reynolds equation reads ∂ ∂ ∂h 3 ∂p 3 ∂p h + h = 6μv (1) ∂x1 ∂x1 ∂x2 ∂x2 ∂x1 The unknown p = p(x1 , x2 ) is the pressure distribution and h is commonly referred to as the thickness of the film. One surface is moving at constant speed v in the x1 -direction, the other remains fixed. No slip occurs at the boundary of the moving surface. The lubricant viscosity μ is taken as constant. Equation (1) is assumed 1.

(14) 2. INTRODUCTION. Figure 1. A schematic description of the journal bearing in Tower’s experimental set up (from [66]). The bearing was immersed in an oil bath.. Figure 2. Reynolds’ illustration of the principles of hydrodynamic lubrication (from [66]). The upper graph represents the pressure distribution. The lower graphs represent the fluid motion.. to hold in the interior of Ω and the standard boundary condition is to prescribe p = 0 on ∂Ω. In the derivation of (1), h is assumed to be small with respect to the dimensions of Ω. Reynolds’ work provided a theoretical basis for the understanding and design of bearings and proved to be in close agreement with a series of careful experiments that had previously been conducted by Tower. Although primarily known as a prominent figure in fluid dynamics, Reynolds also contributed to other areas of tribology through his fundamental work on rolling friction. For a long time, a limitation on the applicability of Reynolds’ theory was the difficulty of obtaining two-dimensional analytical solutions of (1). Closed-form solutions were known only for infinitely long bearings (Sommerfeld 1904) and infinitely short bearings (Michell 1905 and Ocvirk 1952). The advent of fast computers, however, has made the Reynolds equation an indispensable tool in bearing design. Compared to more general physical models such as the Navier–Stokes equations, the Reynolds equation is linear, lower-dimensional and gives the asymptotically correct solution as the film thickness h tends to zero. Reynolds’ equation has proven to be a reasonable approximation for many problems in hydrodynamic lubrication. The transition between the three-dimensional models for viscous flow and lowerdimensional approximations is a subject that has attracted quite a lot of attention. This should not be surprising as flows in thin domains are encountered in various situations, not only in fluid film bearings but also in e.g. gas pipelines, capillaries,.

(15) INTRODUCTION. 3. oceans and the atmosphere. As shown by many rigorous studies, in the mathematical litterature [11, 15, 31, 32, 35, 49, 54, 55] and in the engineering litterature [10, 24, 36, 70, 72], the scaled Reynolds equation gives an O(h) approximation, i.e. a zeroth-order approximation, of the pressure distribution. For lubrication with high Reynolds number, e.g. lubrication of a rapidly rotating shaft, Reynolds’ approximation becomes rather crude. The need for higher-order approximations of pressure and velocity fields in hydrodynamic lubrication has been confirmed by numerous theoretical and experimental studies, see the references cited in [50, 56]. Recent investigations [50, 53, 56] in this direction have lead to modified nonlinear Reynolds-type equations, containing terms that arise from inertial and curvature effects. Hydrodynamic lubrication with non-Newtonian fluids is also an active research field. A nonlinear Reynolds-type equation accounting for non-Newtonian effects has been proposed in [41]. Even when the error in the Reynolds approximation is tolerably small, at least two other effects have been recognized that could render the Reynolds equation invalid. First, there is the effect of molecular slip at the boundary, causing the macroscopic velocity of the fluid near the boundary to deviate from that of the adjacent surface. Such effects become apparent in magnetic storage devices consisting of a mechanical head flying over a rotating disk, the lubricant being air. Modifications to Reynolds equation to compensate for this effect have been discussed by Burgdorfer. Second, there is the effect of surface roughness. Technical surfaces are never perfectly smooth because of imperfections in the manufacturing process. Almost smooth surfaces can only be manufactured at extremely high costs. Roughness usually increases wear and is therefore undesirable in many situations in tribology. In tilted-slider bearings and journal-type bearings however, it has been observed that the performance can improve hydrodynamically with the addition of roughness. Hence, deliberately machined roughness (or texture) can also be considered as a design parameter. In the realm of fluid mechanics roughness is usually neglected for laminar flow, but when the lubricant film is sufficently thin even small roughness becomes significant. According to Elrod [37], the first theoretical studies of surface roughness appeared in the 1950s. For a review of the state of the art the reader may consult Elrod’s paper [38], chapter 7 of the monograph [74] and the doctoral thesis [4]. Surface roughness enters the Reynolds equation through the function describing the film thickness. The usual ansatz in statistical treatments is that h = h0 +hR where h0 is a function representing the “global film thickness” and hR is a stochastic variable representing the roughness (in deterministic treatments hR is related to a specific surface description). Since the roughness is random, the solution of the Reynolds equation must be averaged at some point in the calculations. For this various techniques have been suggested by many authors, e.g. Tzeng and Saibel, Christensen, Christensen and Tønder, Elrod, Chow and Saibel, Patir and Cheng and Phan-Thien [71, 26, 28, 37, 39, 29, 62, 65]. There has been some controversy as to applying the Reynolds approximation in lubrication with rough surfaces actually leads to realistic results. Elrod [37] and Sun and Chen [67] think that the Reynolds equation is inadequate when the roughness wavelength ε becomes smaller than or comparable in magnitude with the film thickness h, arguing that the basic assumptions used to derive the Reynolds equation no longer hold. In this case the Stokes equations can be used instead. Elrod has proposed to classify roughness into two categories: “Reynolds roughness”.

(16) 4. INTRODUCTION. Figure 3. The sliding of a rough surface against a smooth plane, the arrow indicating the direction of motion.. and “Stokes roughness”. Roughness is said to belong to the former category if h ε. Early studies [67, 64] showed that the Reynolds equation and Stokes equations may lead to conflicting results, but an explanation to this is difficult because of the various assumptions introduced. A rigorous attempt to clarify the concepts of Reynolds and Stokes roughness is that by Bayada and Chambat [14] (see also [13, 51]). Using homogenization theory they study the Stokes equations in a thin domain bounded by a periodically rough surface and a smooth plane when ε and h tend to zero. Depending on the value of the parameter λ = h/ε different equations are obtained in the limit. Three situations apply: (1) λ = constant. A two-dimensional equation is obtained with coefficients that depend on the surface microtopography and λ. (2) λ → 0 (Reynolds roughness). Leads to the homogenized Reynolds equation, i.e. first h → 0, then ε → 0. In agreement with previous studies. (3) λ → ∞ (Stokes roughness). A simple Reynolds equation is obtained with an effective film thickness. The same result is obtained if first ε → 0, then h → 0. A remarkable conclusion of this study is that there is really no need to consider the Stokes roughness. This message does not seem to have assimilated into the engineering community, where more and more attention is given to the Stokes roughness, see the recent studies [59, 46]. We can only speculate why this is so, but perhaps the hypotheses about the roughness in [14] is considered to be too weak. This also suggests that more rigorous studies of thin film lubrication with two rough (curved) surfaces are needed. In the mathematical community, homogenization has become the dominant approach to studying the influence of surface roughness in lubrication. Homogenization theory is a set of mathematical techniques that are aimed at studying differential operators with rapidly oscillating coefficents, equations in perforated domains or equations subject to rapidly alternating boundary conditions. The first proof of a homogenization theorem was obtained by De Giorgi and Spagnolo around.

(17) INTRODUCTION. 5. 1970. There exists a vast litterature on homogenization theory, see e.g. the books [3, 19, 20, 25, 33, 34, 43, 47, 52, 60, 61, 63, 75]. Let us qualitatively describe homogenization in the context of equation (1) and surface roughness. The roughness enters the Reynolds equation through the film thickness h. It is assumed that the roughness is periodic with period ε. To this end let h1 be a function that is 1-periodic in both arguments and has mean value zero on the corresponding cell of periodicity. The film thickness entering the Reynolds equation is given by x hε (x) = h0 (x) + h1 ε resulting in the equation ∂ ∂hε ∂ 3 ∂pε 3 ∂pε hε + hε = 6μv . (2) ∂x1 ∂x1 ∂x2 ∂x2 ∂x1 Homogenization of (2) has been studied by Wall [73] who showed that the sequence of solutions pε converges to a limit p0 which is the solution of an “averaged” or homogenized equation, 2 2 ∂h0 ∂bi ∂p0 ∂ (3) aij = 6μv − . ∂xi ∂xj ∂x1 i=1 ∂xi i,j=1 The averaged or homogenized coefficients aij and bi (1 ≤ i, j ≤ 2) are found by first solving v0 , v1 and v2 from the three periodic problems ∂ ∂h ∂ 3 ∂v0 3 ∂v0 h + h = 6μv (4a) ∂y1 ∂y1 ∂y2 ∂y2 ∂y1 ∂ ∂h3 ∂ 3 ∂vi 3 ∂vi h + h =− (4b) (i = 1, 2), ∂y1 ∂y1 ∂y2 ∂y2 ∂yi where h(x, y) = h0 (x) + h1 (y). Then use the averaging formulae . ∂v1 ∂v2 ∂v0 a11 a12 b1 3 1 + ∂y1 ∂y1 ∂y1 (5) dy, = h ∂v1 ∂v2 ∂v0 a21 a22 b2 1 + ∂y Y ∂y ∂y 2. 2. 2. Y denoting the cell of periodicity, to compute the coefficients. This gives the following homogenization algorithm: (1) Solve the local problems (4). (2) Compute the homogenized coefficients (5). (3) Solve the homogenized Reynolds equation (3). Some mathematically delicate questions that arise are in what sense pε converges to p0 as ε → 0 and whether the coefficients (5) are sufficently “nice” for (3) to be well posed. Heuristically, one can find the homogenized equation (3) and the local problems (4) by making the ansatz x x x pε (x) = p0 x, + εp1 x, + ε2 p2 x, + ··· ε ε ε and plug this into (2). Equating terms of equal powers of ε and solving the obtained system of equations eventually yields the homogenization result. This is the asymptotic expansion method. The homogenized Reynolds equation (3) contains no oscillating coefficents. Nevertheless it retains local information of the film thickness, thereby capturing.

(18) 6. INTRODUCTION. the effects of roughness. It is in general different from the Reynolds equation corresponding to mean film thickness, i.e. ∂ ∂ ∂h0 3 ∂p 3 ∂p h0 + h0 = 6μv . ∂x1 ∂x1 ∂x2 ∂x2 ∂x1 For general two-dimensional roughness the coefficients of the homogenized equation can be rather cumbersome to compute. Only in special cases, e.g. the case of onedimensional (striated) roughness, closed-form solutions of the local problems can be found. The first publication that uses the word ‘homogenization’ in the context of hydrodynamic lubrication is probably [65] which describes a homogenization procedure for Reynolds equation. However, already in 1973, Elrod [37] had proposed a very similar method which he called the “two-variable expansion procedure” or the “method of multiple scales”. Elrod’s work is truly original since it predates homogenization by the “asymptotic expansion method” of Bakhvalov and Lions. Since Elrod’s pioneering work, ideas from homogenization theory have been frequently applied in tribology. One of the first rigorous studies in the field is that of Bayada and Chambat [14]. Subsequent studies pertain mostly to Reynolds roughness in various lubrication regimes, see e.g. the works [44, 16, 17, 23, 30, 18, 73, 6, 48]. For homogenization to be useful to engineers in e.g. bearing design, efficient numerical treatment of the homogenized equations is needed. Such aspects have been discussed in [21, 22]. Comparisons between homogenization and traditional averaging of Reynolds equation have been undertaken in [45, 40]. We summarize their findings: • A direct computational approach is limited by the resolution of the numerical method employed. A more cost-effective way is to compute the solution of the homogenized equation even though some local problems must be solved first. Actually the finer the roughness, the better the agreement between the homogenized and the deterministic solution. • Homogenization works regardless of the type of roughness. It is particularly well suited for anisotropic roughness, retaining information regarding the amplitude and the direction of the roughness. Rival stochastic approaches prove defective in this case. • Unlike some averaging methods that may lead to ambiguous results, the homogenized equation is uniquely determined. • Being a mathematical theory, homogenization is a completely rigorous approach, whereas many other methods are based on heuristics. The papers comprising this thesis [7, 8, 9] are all examples of homogenization applied in hydrodynamic lubrication. Paper A [7] focuses on theoretical aspects, whereas Papers B and C [8, 9] are more oriented towards specific applications. The main result of [7] is a reiterated homogenization result for a nonlinear Reynoldstype equation. To prove this result we first develop some aspects of the theory of multiscale convergence introduced by Nguetseng, Allaire, Allaire and Briane and Nguetseng et al. [57, 1, 2, 58]. Special attention is given to the linear case. Reiterated homogenization makes it possible to analyze surface roughness with several characteristic wavelengths. How this is done in practice is explained in Paper B [8], where it is also shown how asymptotic expansion can be used to find the homogenized equation. To compare the homogenized solution with the solution of.

(19) INTRODUCTION. (a) Deterministic pressure distribution. 7. (b) Homogenized pressure distribution. Figure 4. A computer visualization of homogenization of the pressure distribution in a rough thin film (courtesy of Almqvist et al. [5]). the deterministic (unaveraged) Reynolds equation, some numerical examples are also included. Paper C [9] is devoted to homogenization of a variational principle which is a generalization of the unstationary Reynolds equation (both surfaces are rough). The advantage of adopting the calculus of variations viewpoint is that the recently introduced “variational bounds”, see [6], can be computed. Bounds can be seen as a “cheap” alternative to computing the relatively costly homogenized solution and give a mathematical explanation to some heuristically proposed averaging techniques. For simple type of roughness the bounds even coincide with the homogenized solution. Several numerical examples are included to illustrate the utility of bounds..

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(25) Paper A.

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(27) REITERATED HOMOGENIZATION OF A NONLINEAR REYNOLDS-TYPE EQUATION ANDREAS ALMQVIST, EMMANUEL KWAME ESSEL, JOHN FABRICIUS, AND PETER WALL. Abstract. We prove a reiterated homogenization result for monotone operators by means of multiscale convergence. The reiterated homogenization problem consists of studying the asymptotic behavior as ε → 0 of the solutions uε of the nonlinear equation div aε (x, ∇uε ) = div bε , where both aε and bε oscillate rapidly on two microscopic scales and aε satisfies certain continuity, monotonicity and boundedness conditions. This kind of problem has applications in hydrodynamic thin film lubrication where the bounding surfaces have two types of roughness corresponding to different length scales. To prove the homogenization result we extend the multiscale convergence method introduced by Allaire and Briane [2] to W01,p (Ω), where 1 < p < ∞. The original formulation of multiscale convergence pertained only to the case p = 2.. 1. Introduction The main contributions of this paper is that some of the previous homogenization results in connection with hydrodynamic lubrication are extended to include two microscopic scales and non-Newtonian fluids. More precisely, we study the limiting behavior as ε → 0 of the solutions uε of div aε (x, ∇uε ) = div bε in Ω (1) uε = 0 on ∂Ω, where Ω is an open bounded subset of RN and aε and bε oscillate rapdily on the microscopic scales ε (meso scale) and ε2 (micro scale). The idea of reiterated homogenization is that the effects of rapid oscillations upon the solution is averaged out. As an application we show that for particular choices of aε and bε it is possible to analyze the effects of multiscale surface roughness in some interesting Newtonian and non-Newtonian lubrication models. For example both the stationary incompressible Reynolds equation and the Reynolds-type equation derived in [13] based on the Rabinowitsch constitutive relation, belong to this category. It is well known that the surface micro topography has a significant effect on the hydrodynamic performance in thin film lubrication. Machined surfaces are never perfectly smooth because of defects in the manufacturing process. Since smoothening of the surfaces in contact in a fluid film bearing may lead to a decrease in performance, bearing Key words and phrases. Reiterated homogenization, monotone operator, multiscale convergence, three-scale convergence, hydrodynamic lubrication, non-Newtonian lubrication, Reynolds equation, Reynolds-type equation, surface roughness, p-Laplace. 1.

(28) 2. A. ALMQVIST, E. K. ESSEL, J. FABRICIUS, AND P. WALL. designers are also considering artifically textured surfaces. The effects of surface roughness in various lubrication regimes have been studied with homogenization techniques in numerous works, e.g. [5, 6, 7, 8, 9, 14, 22]. The usual assumption is that the roughness is periodic with characteristic wavelength ε. The main result of this paper makes it possible to study surface roughness with two distinguishable wavelengths, say ε and ε2 , in both linear and nonlinear lubrication models. Reiterated homogenization of −div aε (x, ∇uε ) = f with non-oscillating f has been studied in [21] by the periodic unfolding method. Reiterated homogenization of −div aε (x, ∇uε ) = fε , where fε converges strongly in W −1,q (Ω), has also been studied in [16, 18]. This also differs from the present case in that div bε , in general, does not converge strongly in W −1,q (Ω). 2. Preliminaries and notation Suppose p, α, β, λ and θ are constants that obey (2). 1 < p < ∞, N. 0 < α ≤ min{1, p − 1},. max{2, p} ≤ β < ∞,. N. A function f : R → R is said to belong to the class following conditions are satisfied for any ξ, η ∈ RN . (3a) (3b). λ, θ > 0.. Mpα,β (λ, θ). provided the. f (0) = 0, α. |f (ξ) − f (η)| ≤ λ(1 + |ξ| + |η|)p−1−α |ξ − η| , β. (3c). f (ξ) − f (η) · (ξ − η) ≥ θ. |ξ − η| . (1 + |ξ| + |η|)β−p. Moreover, M(λ, θ) denotes the set of all linear mappings f : RN → RN such that f ∈ M21,2 (λ, θ). The function aε : Ω × RN → RN is assumed to be of the form x x (x ∈ Ω, ξ ∈ RN ) (4) aε (x, ξ) = a x, , 2 , ξ ε ε where a : Ω × RN × RN × RN → RN is a function of Carathéodory type such that • a is periodic (with respect to the unit cell Y = Z = [0, 1]N ) in both the second and the third argument, • there exists constants p, α, β, λ and θ satisfying (2), such that a(x, y, z, ·) ∈ Mpα,β (λ, θ) for a.e. (x, y, z) ∈ Ω × [0, 1]N × [0, 1]N . As a consequence of (3a–c), for each ξ ∈ RN , aε satisfies the growth condition (5). |aε (·, ξ)| ≤ λ(1 + |ξ|)p−1. a.e. in Ω. and the coercivity condition (6). p−β. aε (·, ξ) · ξ ≥ 2. ( β |ξ| θ× p |ξ|. if |ξ| ≤ 1 otherwise. a.e. in Ω.. It is further assumed that bε is of the form x x bε (x) = b x, , 2 with b ∈ Lq (Ω; Cper (Y × Z)). ε ε A weak solution of (1) is defined as an element uε of W01,p (Ω) satisfying Z Z (7) aε (x, ∇uε ) · ∇φ dx = bε · ∇φ dx Ω. Ω.

(29) REITERATED HOMOGENIZATION. 3. for all φ ∈ W01,p (Ω). Let Aε : W01,p (Ω) → W −1,q (Ω) be defined by Z hAε (u), φi = aε (x, ∇u) · ∇φ dx (u, φ ∈ W01,p (Ω)). Ω. By (3b) and H¨ older’s inequality , we obtain p−1−α. α kAε (u) − Aε (v)kW −1,q (Ω) ≤ λ 1 + |∇u| + |∇v| Lp (Ω) ku − vkW 1,p (Ω) . 0. Hence Aε is continuous. Owing to (3c) it can be shown that Aε is strictly monotone, i.e. hAε (u) − Aε (v), u − vi ≥ 0 with equality if and only if u = v. Utilizing (6) and Poincaré’s inequality yields Z p (8) hAε (u), ui = aε (x, ∇u) · ∇u dx ≥ const kukW 1,p (Ω) − 2p−β θ meas (Ω), Ω. 0. implying that Aε is coercive. Thus the hypotheses of the Browder–Minty theorem (see e.g. [23] p. 557) are verified, and we conclude that for each ε > 0, there exists a unique uε that solves (7). Moreover, putting φ = uε in (7) and utilizing (8) and Young’s inequality we obtain p q (9) kuε k ≤ const kdiv bε kW −1,q (Ω) + 1 . Since the sequence div bε is weak∗ convergent, and hence bounded, this shows that the sequence of solutions uε is bounded in W01,p (Ω). 3. Three-scale convergence In 1989 Nguetseng [19] introduced a method for analyzing homogenization problems that was later further developed by Allaire [1] and called two-scale convergence. Two-scale convergence in the setting of Lebesgues spaces Lp with 1 < p < ∞ is described in [20]. An advantage of the two-scale convergence method is that it is designed to avoid many of the classical difficulties encountered in the homogenization process, such as passing to the limit in the product of two weakly convergent sequences, reducing it to an almost trivial process. A limitation of the method is that one is more or less restricted to the periodic case. Homogenization of problems with n microscopic scales, is for n > 1 referred to as reiterated homogenization, see e.g. [10] and [16]. Two-scale convergence (one microscopic scale) has been generalized to (n + 1)-scale convergence or multiscale convergence (n microscopic scales) by Allaire and Briane [2]. As pointed out by the authors of that work, the n-scale case is more delicate and for the special case n = 1 the proofs are sometimes much simpler compared to the general case. The results of [2] being restricted to L2 (Ω), it has hitherto not been clear whether a multiscale theory is possible for Lp (Ω). Nevertheless, many authors have claimed such results, see e.g. Theorem 3.8 in [4] or Theorem 1.7 in [12], without providing any complete proof. An additional result of this paper is that we develop such a theory for the case of two microscopic scales (n = 2) and 1 < p < ∞. In particular we give a new proof of Theorem 1.2 (see also Theorem 2.6) in [2]. The assumption that the micro scale is the square of the meso scale corresponds to the notion of well-separated scales defined in [2]. With the appropriate notion of three-scale convergence combined with an additional result concerning three-scale convergence and monotonicity, homogenization of (7) becomes a rather short story. We mention that for the case that aε is a.

(30) 4. A. ALMQVIST, E. K. ESSEL, J. FABRICIUS, AND P. WALL. matrix, the homogenized equation obtained by three-scale convergence coincides with that of [3], which was obtained by the asymptotic expansion method and can thus be seen as a rigorous justification of this method. Definition 3.1. A bounded sequence uε (ε > 0) in Lp (Ω) is said to three-scale converge to an element u of Lp (Ω × Y × Z) provided Z ZZZ x x (10) lim uε (x)φ x, , 2 dx = u(x, y, z)φ(x, y, z) dz dy dx ε→0 Ω ε ε ΩY Z. for every test function φ of the form φ(x, y, z) = ϕ(x)ψ(y)σ(z), where ϕ ∈ C(Ω), ψ ∈ Cper (Y ) and σ ∈ Cper (Z). We note that we have an equivalent definition of two-scale convergence if we ∞ replace the set of test functions by D(Ω; Cper (Y × Z)), i.e. the space that consists ∞ ∞ of all functions Ω → Cper (Y ×Z) such that for any x ∈ Ω, u(x, ·) ∈ Cper (Y ×Z) and ∞ the mapping Ω 3 x 7→ u(x, ·) ∈ Cper (Y × Z) is infinitely differentiable (in the sense of Fréchet) with compact support in Ω. For the more general case of periodicity that Y and Z are parallelograms in RN , Definition 3.1 must be modified by dividing the right hand side of (10) with meas(Y )meas(Z). As a direct consequence of the definition of three-scale convergence it is true that any three-scale convergent sequence uε is also weakly convergent in Lp (Ω). More precisely, ZZ uε → v weakly,. v(x) =. u(x, y, z) dz dy, YZ. whenever uε → u three-scale. This follows from taking ψ = σ = 1 in (10). We state below the three most important theorems in the theory of multiscale convergence. Theorem 3.2 (Three-scale compactness). For any bounded sequence uε in Lp (Ω), there exists a subsequence that three-scale converges weakly. Proof. The proof is very similar to the two-scale case, see Theorem 7 in [20].. . To prove the next important theorem we need a lemma concerning a special type of convergence for periodic functions. For the sake of brevity, the proof is postponed to the end of the paper. ∞ Lemma 3.3. Assume ψ1 ∈ Cc∞ (Ω), ψ2 ∈ Cper (Y ) and f ∈ Lpper (Z) satisfies Z f dz = 0. Z. Then the sequence of functions fε , defined for a.e. x ∈ Ω by x x 1 f 2 , fε (x) = 2 ψ1 (x)ψ2 ε ε ε converges weak∗ to 0 in W −1,p (Ω). In particular sup kfε kW −1,p (Ω) < ∞. ε>0. Theorem 3.4 (Three-scale convergence of the gradient). Suppose that uε is a sequence in W01,p (Ω) such that (1) uε → u weakly in W01,p (Ω),.

(31) REITERATED HOMOGENIZATION. 5. (2) ∇uε → ξ ∈ Lp (Ω × Y × Z; RN ) three-scale. 1,p Then uε → u three-scale and there exists u1 ∈ Lp (Ω; Wper (Y )) and u2 ∈ Lp (Ω × 1,p Y ; Wper (Z)) such that. ξ(x, y, z) = ∇u(x) + ∇y u1 (x, y) + ∇z u2 (x, y, z). Proof. Step 1. Since uε is a bounded sequence in Lp (Ω) it is possible to extract a threescale convergent subsequence (still denoted by uε ), say uε → u0 three-scale. By integration by parts we obtain the identity Z x x (11) ∇uε (x) · Φ x, , 2 dx ε ε Ω Z x x =− uε (x)(divx + ε−1 divy + ε−2 divz )Φ x, , 2 dx ε ε Ω ∞ for all Φ ∈ D(Ω; Cper (Y × Z; RN )). Multiplying (11) with ε2 and passing to the subsequential three-scale limit u0 of uε we obtain ZZZ − u0 (x, y, z)divz Φ(x, y, z) dz dy dx = 0. ΩYZ 0. It follows that u does not depend on z. Similarly, taking Φ in (11) independent of z and multiplying by ε yields ZZ − u0 (x, y)divy Φ(x, y) dy dx = 0 ΩY 0. and we conclude that u does not depend on y either. The three-scale convergence implies uε → u0 weakly in Lp (Ω) (always for the same subsequence). Since the embedding of W 1,p (Ω) in Lp (Ω) is compact we also have that uε → u strongly in Lp (Ω) (for the whole sequence). It follows that u0 = u and consequently the whole sequence uε and not just a subsequence three-scale converges to u. Step 2. From two-scale theory, Theorem 13 in [20], we know that Z ξ(x, y) = ∇u(x) + ∇y u1 (x, y) where ξ= ξ dz. Z. This follows from the fact that ∇uε → ξ two-scale. Next we project ξ − ξ onto the space of z-gradients. That is, define . 1,p Lppot (Z) = ∇z v : v ∈ Lp (Ω × Y ; Wper (Z)) . Because of the Poincaré–Wirtinger inequality, Lppot (Z) is a closed subspace of Lp (Ω × Y × Z; RN ) and the latter being uniformly convex there exists a ∇z u2 ∈ Lppot (Z) which minimizes the distance from ξ − ξ to Lppot (Z), i.e.. ξ − ξ − ∇z u2 = min ξ − ξ − Φ . (12) p Φ∈Lpot (Z). Let η be defined by ξ(x, y, z) = ∇u(x) + ∇y u1 (x, y) + ∇z u2 (x, y, z) + η(x, y, z)..

(32) 6. A. ALMQVIST, E. K. ESSEL, J. FABRICIUS, AND P. WALL. Computing the first variation of the minimization problem (12) we obtain ZZZ p−2 |η| η · Φ dz dy dx = 0 ΩYZ. for all Φ of the form Cc∞ (Ω), ψ. with φ ∈ a.e. y ∈ Y ,. Z. ∈. p−2. |η|. Φ(x, y, z) = φ(x)ψ(y)∇σ(z) 1,p ) and σ ∈ Wper (Z). It follows that for a.e. x ∈ Ω and. ∞ Cper (Y. η(x, y, z) · ∇σ(z) dz = 0. 1,p for all σ ∈ Wper (Z).. Z. To summarize we have Z (13) η dz = 0. and. p−2. divz |η|. η = 0.. Z. Step 3. We show that η = 0. To prove this, we take test functions in (11) of the form (14) with φ ∈. Φ(x, y, z) = φ(x)ψ(y)σ(z) Cc∞ (Ω),. ψ∈. ∞ Cper (Y. ∞ ) and σ ∈ Cper (Z; RN ) satisfying. (15a). div σ = 0, Z. (15b). σ dz = 0. Z. For such Φ (11) reduces to Z Z x x x x ∇φ(x) · σ 2 + ε−1 φ(x)∇ψ ·σ 2 dx. ∇uε · Φ dx = − uε ψ ε ε ε ε Ω Ω Letting ε → 0 we obtain ZZZ Z (16) ξ · Φ dz dy dx = lim ε ϕε uε dx ε→0. ΩYZ. Ω. where. x x 1 φ(x)∇ψ ·σ 2 . ε2 ε ε Applying Lemma 3.3, we conlude that ϕε is bounded in W −1,q (Ω). Since uε is also bounded in W01,p (Ω), it holds that

(33) Z

(34)

(35)

(36)

(37) ϕε uε dx

(38) = |hϕε , uε i| ≤ kϕε k −1,q kuε k 1,p W (Ω) W (Ω) ≤ const.

(39)

(40) ϕε (x) =. Ω. Thus (16) actually says that ZZZ ξ · Φ dz dy dx = 0 ΩYZ. ZZZ η · Φ dz dy = 0. implying ΩYZ. for all Φ having the special form (14) and satisfying conditions (15), but in view of (13) it is clear that we can omit condition (15b). Thus ZZZ η · σ(z) φ(x)ψ(y) dz dy dx = 0, ΩYZ.

(41) REITERATED HOMOGENIZATION. 7. where σ is assumed to satisfy only divz σ = 0. It follows by Fubini’s theorem, density etc. that for a.e. (x, y) ∈ Ω × Y Z η(x, y, z) · σ(z) dz = 0 Z. for all σ ∈ Lqper (Z) such that div σ = 0. Thus, for fixed x and y, we can take σ(z) = p−2 |η| η(x, y, z). Hence η = 0 a.e. in Ω × Y × Z and ξ = ∇u + ∇y u1 + ∇z u2 . The following theorem has been referred to as the “fundamental theorem of three-scale convergence and monotonicity”. A two-scale version of the statement can be found in [17], Theorem 14. Our proof is very similar, but we include it here for the sake of completeness. Theorem 3.5 (Three-scale convergence and monotonicity). Assume aε as in (4) and let vε be a bounded sequence in Lp (Ω; RN ) such that vε → v three-scale. and. aε (x, vε ) → ζ three-scale,. p. for v, ζ ∈ L (Ω × Y × Z). Then Z ZZZ (17) lim inf aε (x, vε ) · vε dx ≥ ζ · v dx dy dz ε→0. Ω. ΩY Z. and if equality holds, then ζ = a(·, v). Proof. Let Φ(x, y, z) be a linear combination of vector fields of the form (x, y, z) 7→ ∞ ∞ φ(x)ψ(y)σ(z)ν where φ ∈ Cc∞ (Ω), ψ ∈ Cper (Y ), σ ∈ Cper (Z) and ν ∈ S N −1 (the unit sphere). By monotonicity Z aε (x, vε ) − aε (x, Φε ) · (vε − Φε ) dx ≥ 0. Ω. where. x x Φε (x) = Φ x, , 2 . ε ε. Rearranging the terms yields Z Z aε (x, vε ) · vε dx ≥ aε (x, vε ) · Φε + aε (x, Φε ) · (vε − Φε ) dx. Ω. Ω. Since the limit, as ε → 0, of the right hand side of the above inequality exists and is equal to ZZZ ζ · Φ + a(x, y, z, Φ) · (v − Φ) dx dy dz ΩY Z. we have Z (18). aε · vε dx ≥. lim inf ε→0. ZZZ. Ω. ζ · Φ + a(x, y, z, Φ) · (v − Φ) dx dy dz ΩY Z. and by density and continuity this also holds for all Φ in Lp (Ω × Y × Z). Thus, we establish (17) by taking Φ = v. Suppose now that equality holds in (17). For some w ∈ Lp (Ω × Y × Z; RN ) and t ∈ R, choose Φ = v + tw in (18) to obtain ZZZ 0≥t ζ − a(x, y, z, v + tw) · w dx dy dz. ΩY Z.

(42) 8. A. ALMQVIST, E. K. ESSEL, J. FABRICIUS, AND P. WALL. Dividing by t and using the continuity of a we let t → 0± , thus obtaining ZZZ ζ − a(x, y, z, v) · w dx dy dz = 0 ΩY Z. for all w ∈ Lp (Ω × Y × Z; RN ). Hence ζ(x, y, z) = a(x, y, z, v(x, y, z)) almost everywhere. . 4. A three-scale homogenization procedure Based on Theorems 3.2, 3.4 and 3.5, we outline a homogenization procedure for the problem (1). In view of estimate (9) and the following remark, the sequence of solutions uε to (7) is bounded in W01,p (Ω). Applying Theorems 3.2 and 3.4 we can find u ∈ 1,p 1,p W01,p (Ω), u1 ∈ Lp (Ω; Wper (Y )), u2 ∈ Lp (Ω × Y ; Wper (Z)) and ζ ∈ Lq (Ω × Y × Z)N such that up to a subsequence (1) uε → u three-scale, (2) ∇uε → ∇u + ∇y u1 + ∇z u2 three-scale, (3) aε (x, ∇uε ) → ζ three-scale. Passing to the limit in the weak formulation (7) gives ZZZ ZZZ ζ · ∇φ dz dy dx = b · ∇φ dz dy dx ΩY Z. ΩY Z. Let the testfunction φ in (7) be φ(x) = εφ1 (x)w1 (x/ε), where φ1 ∈ Cc∞ (Ω), w1 ∈ ∞ (Y ). Then Cper Z Z aε (x, ∇uε ) · (εw1 ∇φ1 + φ1 ∇w1 ) dx = bε · (εw1 ∇φ1 + φ1 ∇w1 ) dx. Ω. Ω. In the limit as ε → 0 we obtain ZZZ ZZZ ζ · φ1 (x)∇w1 (y) dz dy dx = b · φ1 (x)∇w1 (y) dz dy dx. ΩY Z. ΩY Z. Taking as testfunction φ(x) = ε2 φ1 (x)φ2 (x/ε)w2 (x/ε2 ), where φ1 ∈ Cc∞ (Ω), φ2 ∈ ∞ ∞ Cper (Y ) and w2 ∈ Cper (Z), yields Z. x x + φ1 φ2 ∇w2 2 dx aε (x, ∇uε ) · ε2 w2 φ2 ∇φ1 + εw2 φ1 ∇φ2 ε ε Ω Z x x = bε · ε2 w2 φ2 ∇φ1 + εw2 φ1 ∇φ2 + φ1 φ2 ∇w2 2 dx. ε ε Ω. In the limit ZZZ ZZZ ζ · φ1 (x)φ2 (y)∇w2 (z) dz dy dx = b · φ1 (x)φ2 (y)∇w2 (z) dz dy dx. ΩY Z. ΩY Z.

(43) REITERATED HOMOGENIZATION. 9. By density it follows that ζ satisifies ZZZ (19) ζ · (∇φ + ∇y φ1 + ∇z φ2 ) dz dy dx ΩY Z. ZZZ b · (∇φ + ∇y φ1 + ∇z φ2 ) dz dy dx. = ΩY Z. W01,p (Ω),. p. 1,p (Ω; Wper (Y. 1,p for all φ ∈ φ1 ∈ L )), φ2 ∈ Lp (Ω × Y ; Wper (Z)). Let us now characterize ζ. Choosing φ = u, φ1 = u1 and φ2 = u2 in the identity (19) gives ZZZ (20) ζ · (∇u + ∇y u1 + ∇z u2 ) dz dy dx ΩY Z. ZZZ b · (∇u + ∇y u1 + ∇z u2 ) dz dy dx.. = ΩY Z. Taking φ = uε in (7) yields Z Z lim aε (x, ∇uε ) · ∇uε dx = lim bε · ∇uε dx ε→0 Ω ε→0 Ω Z Z Z (21) = b · (∇u + ∇y u1 + ∇z u2 ) dz dy dx. ΩY Z. Combining (20) and (21), we see that Z ZZZ lim aε (x, ∇uε ) · ∇uε dx = ζ · (∇u + ∇y u1 + ∇z u2 ) dz dy dx. ε→0. Ω. ΩY Z. Appealing to the fundamental theorem of three-scale convergence and monotonicity, Theorem 3.5 we conclude that ζ = a(·, ∇u + ∇y u1 + ∇z u2 ). Hence, u, u1 and u2 satisfy the following homogenized variational system ZZZ (22) a(x, y, z, ∇u + ∇y u1 + ∇z u2 ) · (∇φ + ∇y φ1 + ∇z φ2 ) dz dy dx ΩY Z. ZZZ b · (∇φ + ∇y φ1 + ∇z φ2 ) dz dy dx. = ΩY Z. W01,p (Ω),. 1,p 1,p for all φ ∈ φ1 ∈ Lp (Ω; Wper (Y )), φ2 ∈ Lp (Ω × Y ; Wper (Z)). We have obtained the following partial homogenization result.. Theorem 4.1. For ε > 0, let uε ∈ W01,p (Ω) denote the solution of Z Z aε (x, ∇uε ) · ∇φ dx = bε · ∇φ dx ∀φ ∈ W01,p (Ω). Ω. Ω. Then there exists a subsequence of solutions uεj such that uεj → u weakly and ∇uεj → ∇u + ∇y u1 + ∇z u2 three-scale, as εj → 0. In addition u ∈ W 1,p (Ω), 1,p 1,p u1 ∈ Lp (Ω; Wper (Y )) and u2 ∈ Lp (Ω × Y ; Wper (Z)) solve the system (22)..

(44) 10. A. ALMQVIST, E. K. ESSEL, J. FABRICIUS, AND P. WALL. A priori, it is not clear that u, u1 and u2 are uniquely determined by the fact that they solve the system (22). To make the analysis complete it remains to prove this. To this end let a∗ be defined by Z ∗ (23) a (x, y, ξ) = a(x, y, z, ξ + ∇ψ ∗ ) dz (x ∈ Ω, y, ξ ∈ RN ), Z 1,p where ψ ∗ ∈ Wper (Z) is a solution of the variational problem Z Z (24) a(x, y, z, ξ + ∇ψ ∗ ) · ∇ψ dz = b(x, y, z) · ∇ψ dz Z. 1,p ∀ψ ∈ Wper (Z),. Z. and define a∗∗ as a∗∗ (x, ξ) =. (25). Z. a∗ (x, y, ξ + ∇ψ ∗∗ ) dy. (x ∈ Ω, ξ ∈ RN ),. Y 1,p where ψ ∗∗ ∈ Wper (Y ) solves Z Z 1,p (26) a∗ (x, y, ξ + ∇ψ ∗∗ ) · ∇ψ dy = b(x, y)∗ · ∇ψ dy ∀ψ ∈ Wper (Y ), Y Y R where b∗ (x, y) = Z b(x, y, z) dz. Since a(x, y, z, ·) ∈ Mpα,β (λ, θ) it follows by the Browder–Minty theorem, as explained above for (7), that for a.e. (x, y) ∈ Ω × Y there exists a solution of (24) that is unique up to an additive constant. Thus a∗ is well defined, however, we can not say the same for a∗∗ unless we know that (26) has a solution that is unique (up to a constant). Assume for the moment that existence and uniqueness hold true for (26). Next we show how this can be utilized to characterize the homogenized solution u from (22). Taking φ = φ1 = 0 and φ2 (x, y, z) = ϕ1 (x)ϕ2 (y)ψ(z), ϕ1 ∈ Cc∞ (Ω), ϕ2 ∈ 1,p ∞ (Z), in (22) implies that for a.e. (x, y) ∈ Ω × Y , u2 ∈ Cper (Y ) and ψ ∈ Wper 1,p (Z)) satisfies Lp (Ω × Y ; Wper Z (27) a x, y, z, ∇u(x) + ∇y u1 (x, y) + ∇z u2 (x, y, z) · ∇ψ dz Z Z = b(x, y, z) · ∇ψ dz. Z. Consequently, by the definition of a∗ , (28) a∗ x, y, ∇u(x) + ∇y u1 (x, y) Z = a x, y, z, ∇u(x) + ∇y u1 (x, y) + ∇z u2 (x, y, z) dz. Z. Similarly, by taking φ = φ2 = 0 in (22) we obtain that for a.e. x ∈ Ω, u1 ∈ 1,p Lp (Ω; Wper (Y )) satisfies Z Z a∗ x, y, ∇u(x) + ∇y u1 (x, y) · ∇ψ dy = b∗ (x, y) · ∇ψ dy Y. Y. 1,p for all ψ ∈ Wper (Y ). Hence Z ∗∗ (29) a x, ∇u(x) = a∗ x, y, ∇u(x) + ∇y u1 (x, y) dy ZY Z = a x, y, z, ∇u(x) + ∇y u1 (x, y) + ∇z u2 (x, y, z) dz dy. Y Z.

(45) REITERATED HOMOGENIZATION. 11. Finally, setting φ1 = φ2 = 0 in (22) and taking (23) and (25) into account, we see that u ∈ W01,p (Ω) satisfies the variational identity Z Z (30) a∗∗ (x, ∇u) · ∇φ dx = b∗∗ · ∇φ dx ∀φ ∈ W01,p (Ω), Ω. Ω. where b∗∗ (x) =. (31). ZZ b(x, y, z) dz dy. YZ. We would like to prove that (30) has a unique solution u and for this we need some properties of a∗ and a∗∗ . Notation. In view of (24), any vector field ξ : Ω × Y → RN induces a vector field ξ ∗ : Ω × Y × Z → RN defined by ξ ∗ (x, y, z) = ξ(x, y) + ∇ψ ∗ (z), where ψ ∗ is a solution of (24). Moreover, if ξ is a vector field on Ω, ξ ∗∗ is defined as ξ ∗∗ (x, y, z) = ξ(x) + ∇ψ ∗∗ (y) + ∇ψ ∗ (z), where ψ ∗∗ solves (26) and ψ ∗ is a solution of Z Z a x, y, z, ξ(x) + ∇ψ ∗∗ (y) + ∇ψ ∗ · ∇ψ dz = b(x, y, z) · ∇ψ dz .. Z. 1,p ∀ψ ∈ Wper (Z).. Z. Theorem 4.2. The function a∗ , defined by (23), satisfies certain continuity and monotonicity conditions so that existence and uniqueness of (26) is guaranteed for a.e. x ∈ Ω. Thus the function a∗∗ , in (25), is well defined. Moreover a∗∗ is sufficiently continuous and monotone in ξ so that existence and uniqueness of (30) follows. Sketch of proof. Set α α = β−α ∗. α∗ λ and λ = λ. θ ∗. Utilizing the monotonicity and continuity of a it follows by some straight-forward calculations, for the details of which we refer to [16] (Propopsition 3.1) and [11] (Lemma 3.4), that for every ξ, η ∈ RN. p−1−α∗ α∗ (32) |a∗ (·, ξ) − a∗ (·, η)| ≤ λ∗ 1 + |ξ ∗ | + |η ∗ | Lp (Z) |ξ − η| (33). β ∗ |ξ − η| a (·, ξ) − a∗ (·, η) · (ξ − η) ≥ θ . 1 + |ξ ∗ | + |η ∗ | β−p Lp (Z). holds a.e. in Ω × Y , and (34). p−1−α∗ α∗ |a∗∗ (·, ξ) − a∗∗ (·, η)| ≤ λ∗ 1 + |ξ ∗∗ | + |η ∗∗ | Lp (Y ×Z) |ξ − η|. (35). β ∗∗ |ξ − η| a (·, ξ) − a∗∗ (·, η) · (ξ − η) ≥ θ . 1 + |ξ ∗∗ | + |η ∗∗ | β−p Lp (Y ×Z). holds a.e. in Ω..

(46) 12. A. ALMQVIST, E. K. ESSEL, J. FABRICIUS, AND P. WALL. Next we show that there exists a c(x) such that for any vector field ξ ∈ Lp (Y ; RN ) it holds that (36) kξ ∗ kLp (Y ×Z) ≤ c(x) 1 + kξkLp (Y ) with c(x) < ∞ for a.e. x ∈ Ω. First note the lower bound kξ ∗ kLp (Y ×Z) ≥ kξkLp (Y ) . Indeed p kξkLp (Y ).

(47) p Z

(48) Z ZZ

(49)

(50) p ∗

(51)

(52) = |ξ ∗ | dz dy.

(53) ξ dz

(54) dy ≤ Y. Z. We now seek to establish an upper bound of kξ ∗ kLp (Y ×Z) . Without loss of generality we may assume kξ ∗ kLp (Y ×Z) ≥ 1. On the one hand (3c), Hölder’s inequality and the triangle inequality gives ZZ ZZ β |ξ ∗ | ∗ ∗ a(x, y, z, ξ ) · ξ dz dy ≥ θ dz dy (1 + |ξ ∗ |)β−p (37) θ p ≥ β−p kξ ∗ kLp (Y ×Z) . 2 On the other hand (5) and (26) implies (38) ZZ. ∗. ZZ. ∗. ∗. ZZ. a(x, y, z, ξ ) · ξ dz dy = a(x, y, z, ξ ) · ξ dz dy + b · (ξ ∗ − ξ) dz dy ZZ ≤λ |ξ| (1 + |ξ ∗ |)p−1 dz dy + kbkLq (Y ×Z) kξ ∗ − ξkLp (Y ×Z) p−1. ≤ 2p−1 λ kξkLp (Y ) kξ ∗ kLp (Y ×Z) + 2 kbkLq (Y ×Z) kξ ∗ kLp (Y ×Z) . Combining (37) and (38) we see that (39). kξ ∗ kLp (Y ×Z) ≤ const kξkLp (Y ) + kbkLq (Y ×Z). . Since b ∈ Lq (Ω; Cper (Y × Z)), we have q. kb(x, ·)kLq (Y ×Z) ≤ kb(x, ·)kC(Y ×Z) < ∞ for a.e. x ∈ Ω. This establishes (36). By estimating ZZZ. a(x, y, z, ξ ∗∗ ) · ξ ∗∗ dz dy dx. similarly, one can show that there exists a constant c that depends on p, λ, θ, β and kbkLq (Ω;C(Y ×Z)) such that for any ξ ∈ Lp (Ω; RN ) (40) kξ ∗∗ kLp (Ω×Y ×Z) ≤ c 1 + kξkLp (Ω) Summing up we have the following homogenization result. Theorem 4.3. Let a∗ , a∗∗ and b∗∗ be defined by (23), (25) and (31) respectively. Both a∗ and a∗∗ are well defined. For ε > 0, let uε denote the solution of (1)..

(55) REITERATED HOMOGENIZATION. 13. Then the whole sequence uε converges weakly to u as ε → 0, where u is the unique weak solution of the homogenized problem div a∗∗ (x, ∇u) = div b∗∗ in Ω (41) u=0 on ∂Ω. 5. The linear case Let us consider the special case that a(x, y, z, ·) ∈ M(λ, θ). Since a is assumed to be linear we write a(x, y, z, ξ) = a(x, y, z)ξ. and. {aij (x, y, z)}1≤i,j≤N. denotes the corresponding matrix. Note that conditions (3b) and (3c) are satisfied if aij ∈ L∞ (Ω; Cper (Y × Z) and {aij } is (uniformly) positive definite, i.e. max kaij kL∞ (Ω;Cper (Y ×Z)) ≤ λ and X 2 aij (x, y, z)ξj ξi ≥ θ |ξ|. 1≤i,j≤N. 1≤i,j≤N N. for all ξ ∈ R and a.e. (x, y, z) ∈ Ω × Y × Z. The equation corresponding to (1) then becomes div(aε ∇uε ) = div bε in Ω. In the linear situation the analysis is essentialy simplified in the sense that one only has to solve N +1 local problems corresponding to the z-scale and another N +1 problems corresponding to the y-scale, instead of infinitely many local problems (one for each ξ), to obtain the homogenized equation (41). Let {ei }i=1,...,N denote the standard basis in RN . Due to the linearity of a, a solution ψξ∗ of (24), i.e. a solution of Z Z 1,p a(x, y, z)(ξ + ∇ψξ∗ ) · ∇ψ dz = b(x, y, z) · ∇ψ dz ∀ψ ∈ Wper (Z) Z. Z. can be written in the form ψξ∗ = ψ0∗ +. N X. ξi χ∗i ,. i=1. where ψ0∗ and χ∗i are solutions of Z Z (42a) a(x, y, z)∇ψ0∗ · ∇ψ dz = b(x, y, z) · ∇ψ dz Z. 1,p ∀ψ ∈ Wper (Z). Z. and Z (42b). 1,p a(x, y, z)(ei + ∇χ∗i ) · ∇ψ dz = 0 ∀ψ ∈ Wper (Z). (i = 1, . . . , N ).. Z. Let a1 (x, y) be the matrix and b1 (x, y) the vector defined by Z a1 (x, y)ei = a(x, y, z)(ei + ∇χ∗i ) dz, (43a) Z Z (43b) b1 (x, y) = b(x, y, z) − a(x, y, z)∇ψ0∗ dz. Z. Then, ψξ∗∗ solving (26) is equivalent to ψξ∗∗ solving Z Z a1 (x, y)(ξ + ∇ψξ∗∗ ) · ∇ψ dy = b1 (x, y) · ∇ψ dy Y. Y. 1,p ∀ψ ∈ Wper (Y )..

(56) 14. A. ALMQVIST, E. K. ESSEL, J. FABRICIUS, AND P. WALL. By linearity, the solution ψξ∗∗ can be written ψξ∗∗ = ψ0∗∗ +. N X. ξi χ∗∗ i ,. i=1. where ψ0∗∗ and χ∗∗ i are solutions of Z Z (44a) a1 (x, y)∇ψ0∗∗ · ∇ψ dy = b1 (x, y) · ∇ψ dy Y. 1,p ∀ψ ∈ Wper (Y ). Y. and Z (44b). 1,p a1 (x, y)(ei + ∇χ∗∗ i ) · ∇ψ dy = 0 ∀ψ ∈ Wper (Y ). (i = 1, . . . , N ).. Y. Let a0 (x) be the matrix and b0 (x) the vector defined by Z (45a) a0 (x)ei = a1 (x, y)(ei + ∇χ∗∗ i ) dy, Y Z (45b) b0 (x) = b1 (x, y) − a1 (x, y)∇ψ0∗∗ dy. Y. Summing up we have the following homogenization algorithm: (1) Solve the N + 1 local problems (42) on the z-scale and use these solutions to compute the matrix a1 (x, y) and the vector b1 (x, y) in (43). (2) Solve the N + 1 local problems (44) on the y-scale and use these solutions to compute the matrix a0 (x) and the vector b0 (x) in (45). (3) Solve the homogenized equation (46). div(a0 ∇u) = div b0 u=0. in Ω on ∂Ω.. 6. Application to hydrodynamic lubrication The Reynolds equation is a two-dimensional model that describes the flow in a thin film of viscous fluid (lubricant) that is enclosed between two rigid surfaces in relative motion. Reynolds equation is used by engineers to compute the pressure distribution in various situations of hydrodynamic lubrication, e.g. slider bearings or journal bearings. A simple form of Reynolds equation reads 3 3 ∂ h ∂p ∂ h ∂p v ∂h (47) + = in Ω, ∂x1 12µ ∂x1 ∂x2 12µ ∂x2 2 ∂x1 where p is the unknown pressure distribution, Ω is the “bearing domain”, h : Ω → R denotes the thickness of the film, µ is the lubricant viscosity (taken as constant) and v is the constant speed of the moving surface (the other remains fixed). To study the influence of multiscale surface roughness with two characteristic wavelengths, ε and ε2 , upon the pressure solution we make the ansatz that the film thickness function hε is given by x x hε (x) = h x, , 2 , where h : Ω × RN × RN → R ε ε is assumed to be continuous, [0, 1]N -periodic in the second and third arguments and satisfy θ ≤ h3 ≤ λ. Assuming the boundary condition pε = 0 on ∂Ω, we can.

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