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CORRECTOR EQUATIONS IN FLUID MECHANICS:

EFFECTIVE VISCOSITY OF COLLOIDAL SUSPENSIONS

MITIA DUERINCKX AND ANTOINE GLORIA

Abstract. Consider a colloidal suspension of rigid particles in a steady Stokes flow. In acelebrated work, Einstein argued that in the regime of dilute particles the system behavesat leading order like a Stokes fluid with some explicit effective viscosity. In the presentcontribution, we rigorously define a notion of effective viscosity, regardless of the diluteregime assumption. More precisely, we establish a homogenization result when particlesare distributed according to a given stationary and ergodic random point process. Themain novelty is the introduction and analysis of suitable corrector equations.

Contents

1. Introduction and main results 12. Construction of correctors 73. Proof of the homogenization result 17Acknowledgements 29References 29

1. Introduction and main results

1.1. General overview. This article is devoted to the large-scale behavior of the steadyStokes equation for a fluid with a dense colloidal suspension of small rigid particles that arerandomly distributed. The fluid and the particles interact via the action-reaction principle,and satisfy a no-slip condition at the particle boundaries. Suspended particles then act asobstacles, hindering the fluid flow and therefore increasing the flow resistance, that is, theviscosity. The system is naturally expected to behave on large scales approximately likea Stokes fluid with some effective viscosity. Our main result in this contribution makesthis statement precise and rigorously defines the effective viscosity in terms of a stochastichomogenization result.

Let us first describe previous contributions on the topic, and emphasize our main motiva-tion. In his PhD thesis, Einstein [14] was the first to analyze this effective viscosity problem:focussing on a dilute regime (that is, assuming that particles are scarce), he argued thatthe fluid indeed behaves at leading order like a Stokes fluid with some effective viscosityand that the latter can be explicitly computed at first order in the particle concentrationin form of the so-called Einstein’s formula, which played a key role in the physics com-munity at that time as it served as a basis for Perrin’s celebrated experiment to estimatethe Avogadro number. Various contributions followed, in particular going beyond the firstorder, e.g. [7, 25, 3, 4]. From a rigorous perspective, several recent contributions stand out.

1

http://arxiv.org/abs/1909.09625v2

2 M. DUERINCKX AND A. GLORIA

In [20] (see also the refined version [24]), Haines and Mazzucato provide bounds on the dif-ference between a heuristic notion of effective viscosity (defined as some integral ratio withthe correct dimensionality) and Einstein’s formula. In [16] (see also [17]), Gérard-Varetand Hillairet took another approach, considering the solution of the Stokes problem andproving its closeness to the Stokes flow associated with some effective viscosity (describedat higher order than Einstein’s formula) — a quantified consistency result. In both works,for the effective behavior of a sequence of solutions, the authors establish error estimatesthat only get sharp in the dilute regime. On the one hand, the analysis in [20, 24, 16, 17]requires sophisticated arguments (reflection method, renormalized energy method, etc.) inorder to get quantitative statements. On the other hand, their applicability is limited bythe dilute regime assumption that allows to construct “explicit” approximate solutions. Inparticular, the very notion of effective viscosity is not defined independently of the diluteregime. Our main motivation is to remedy this issue by taking yet another approach anddistinguishing two independent questions:

• the definition of an effective viscosity in full generality in the setting of homoge-nization theory in terms of a suitable corrector problem;

• the asymptotic analysis of the effective viscosity in the dilute regime — in the spiritof the so-called Clausius-Mossotti formula for homogenization of electrostatics andlinear elasticity, cf. [10].

The present contribution answers the first question, while the second one is the object ofa forthcoming work [13].

In a nutshell, our approach is in the pure tradition of homogenization theory. We refor-mulate the problem as the study of a family of solutions of fluid mechanics equations ina perforated domain associated with the spatial rescaling of some stationary and ergodicrandom array of inclusions, and we prove that this family converges to the solution ofsome effective (deterministic) fluid mechanics equation. Periodic homogenization in fluidmechanics is not new, dating back to Sánchez-Palencia [27], Tartar [28], and Allaire [1, 2],to cite but a few. We also refer to the early work of Cioranescu and Saint Jean Paulin [9],where a related scalar problem is considered in form of the so-called torsion problem. Inthe random setting, we refer to the contributions by Beliaev and Kozlov [8], by Basson andGérard-Varet [5], and more recently by Giunti and Höfer [18]. We further refer to the worksof Jikov [21, 22] on the closely related homogenization problem for stiff inclusions in linearelasticity, see also [23, Chapter 8.6]. In the present work, the homogenization result that isestablished in the general stationary and ergodic random framework (independently of thedilute regime) is new even in the periodic setting due to the specificity of the consideredboundary conditions.

In terms of insight, the main novelty of this contribution is the introduction and analysisof suitable corrector equations in a context where this had not been done before. Froma mathematical perspective, the divergence-free constraint for the fluid velocity yieldstechnical difficulties and makes the analysis quite subtle — although still solely basedon soft, qualitative arguments. As usual, the proof of the homogenization result splitsinto two parts: the construction of correctors, and the convergence result using Tartar’smethod of oscillating test functions [29]. The development of a corresponding quantitativehomogenization theory, which is postponed to a forthcoming work [11], requires a suitablestrong mixing condition on the particle distribution.

EFFECTIVE VISCOSITY OF COLLOIDAL SUSPENSIONS 3

Before turning to the actual statement of the main results, let us mention that an additionalmotivation stems from the sedimentation problem for rigid particles in a Stokes flow, e.g. [6].This concerns the case of particles that are heavier than the fluid, and therefore settle inthe fluid. In the corrector equation, this yields an additional force on the particles, whichpumps energy into the system and entails a crucial lack of compactness. We refer to ourvery recent work [12] (see also [19]) for a thorough discussion of the behavior of suchsedimenting suspensions; although inspired by the present contribution, the analysis ismuch more involved and happens to require a strong mixing condition on the particledistribution even for qualitative results. Among other things, we show in [12] that thecorresponding effective viscosity coincides with that for a non-sedimenting suspension,hence only depends on the geometry of the suspension.

1.2. Main results. Throughout, we place ourselves in dimension d ≥ 2. We start with asuitable description of the random suspension of particles. Let xωnn denote a stationaryand ergodic random point process on the ambient space R

d, constructed on a given prob-ability space (Ω,P); see Remark 1.2 below for a proper definition of stationarity. Definethe corresponding spherical inclusion process

Iω :=⋃

n

Iωn , Iωn := B(xωn),

where B(xωn) denotes the unit ball centered at xωn , and assume that it satisfies the hardcorecondition

infm6=n

dist(Iωn , Iωm) > δ almost surely,

for some fixed δ ∈ (0, 1). Note that spherical inclusions could be replaced by randomshapes under a uniform C2 regularity assumption. In addition, the deterministic lowerbound on the minimal interparticle distance can be relaxed into a lower bound of the type

E

[

1|xn|<1 supm:m6=n

dist(In, Im)−p]

<∞,

for some large enough power p ≥ 1, at the price of tracking down random constants inthe proof and using Meyers-type estimates on solutions. We do however not pursue in thisdirection here; we believe that such conditions could be further improved, possibly in thespirit of [22], see also [23, Section 8.6].

Given a reference bounded Lipschitz domain U , we consider the set N ωε (U) of all indices

n such that ε(Iωn + δB) ⊂ U , and we define the corresponding rescaled inclusion processIωε (U) in U ,

Iωε (U) :=

⋃

n∈Nωε (U)

εIωn .

Note that balls of this collection are at distance at least εδ from one another and from theboundary ∂U . This inclusion process represents a random suspension of particles in thereference domain U . We then consider these particles as suspended in a solvent describedby the steady Stokes equation: the fluid velocity uωε satisfies

−uωε +∇Pωε = 0, div uωε = 0, in U \ Iω

ε (U),

and uωε = 0 on ∂U . As the pressure is defined up to a constant, we choose for instanceˆ

U\Iωε (U)

Pωε = 0.


Next, no-slip boundary conditions are imposed at particle boundaries; since particles areconstrained to have rigid motions, this amounts to letting the velocity field uωε be extendedinside particles, with the rigidity constraint

D(uωε ) = 0, in Iωε (U),

where D(uωε ) denotes the symmetrized gradient of uωε . In other words, this condition meansthat uωε coincides with a rigid motion V ω

ε,n + Θωε,n(x − εxωn) inside each inclusion εIωn , for

some V ωε,n ∈ R

d and skew-symmetric matrix Θωε,n ∈ R

d×d. Finally, assuming that theparticles have the same mass density as the fluid, buoyancy forces vanish, hence the forceand torque balances on each particle take the form

ˆ

ε∂Iωn

σ(uωε , Pωε )ν = 0, (1.1)

ˆ

ε∂Iωn

Θ(x− εxωn) · σ(uωε , P

ωε )ν = 0, for all Θ ∈ M

skew,

where Mskew ⊂ R

d×d denotes the subspace of skew-symmetric matrices, σ(uωε , Pωε ) is the

usual Cauchy stress tensor,

σ(uωε , Pωε ) = 2D(uωε )− Pω

ε Id,

and ν stands for the outward unit normal vector at the particle boundaries. In the phys-ically relevant three-dimensional case d = 3, skew-symmetric matrices Θ ∈ M

skew areequivalent to cross products θ× with θ ∈ R

3, and equations recover their more standardform.

In this context, modeling a dense suspension of small rigid particles in a viscous fluid withthe same mass density, our homogenization result takes on the following guise.

Theorem 1. Given a bounded Lipschitz domain U ⊂ Rd and given a forcing f ∈ L2(U),

consider for all ε > 0 and ω ∈ Ω the unique weak solution (uωε , Pωε ) ∈ H1

0 (U)×L2(U\Iωε (U))

of the Stokes problem introduced above, that is,

−uωε +∇Pωε = f, in U \ Iω

ε (U),div uωε = 0, in U \ Iω

ε (U),uωε = 0, on ∂U,D(uωε ) = 0, in Iω

ε (U),´

ε∂Iωnσ(uωε , P

ωε )ν = 0, ∀n ∈ N ω

ε (U),´

ε∂IωnΘ(x− εxωn) · σ(u

ωε , P

ωε )ν = 0, ∀n ∈ N ω

ε (U), ∀Θ ∈ Mskew,

´

U\Iωε (U) P

ωε = 0,

(1.2)

and denote by λ := E [1I ] the volume fraction of the suspension. Then for almost all ωthere holds

uωε − u 0, weakly in H10 (U),

(Pωε − P − b : D(u))1U\Iω

ε (U) 0, weakly in L2(U),

where (u, P ) ∈ H10 (U)× L2(U) is the unique weak solution of the homogenized Stokes flow

− div 2BD(u) +∇P = (1− λ)f, in U,div u = 0, in U,u = 0, on ∂U,´

U P = 0,

(1.3)


and the effective constants are as follows:

• the effective diffusion tensor B is a positive definite symmetric linear map on sym-metric trace-free matrices M

sym0 ⊂ R

d×d, and is defined for all E ∈ Msym0 by

E : BE := E[

|D(ψE) + E|2]

; (1.4)

• b is a symmetric trace-free matrix and is given for all E ∈ Msym0 by

b : E :=1

dE

[

∑

n

1In

|In|

ˆ

∂In

(x− xn) · σ(ψE + Ex,ΣE)ν

]

; (1.5)

where ∇ψE ∈ L2(Ω; L2loc(R

d)d×d) is the unique stationary gradient solution with vanish-ing expectation and ΣE ∈ L2(Ω; L2

loc(Rd \ I)) is the unique associated stationary pressure

with vanishing expectation for the following infinite-volume corrector problem, cf. Proposi-tion 2.1: for almost all ω,

−ψωE +∇Σω

E = 0, in Rd \ Iω,

divψωE = 0, in R

d \ Iω,D(ψω

E + Ex) = 0, in Iω,´

∂Iωnσ(ψω

E + Ex,ΣωE)ν = 0, ∀n,

´

∂IωnΘ(x− εxωn) · σ(ψ

ωE + Ex,Σω

E)ν = 0, ∀n, ∀Θ ∈ Mskew.

(1.6)

Moreover, provided f ∈ Lp(U) for some p > d, for almost all ω, we have a corrector resultfor the velocity field,

∥

∥

∥uωε − u− ε

∑

E∈E

ψωE(

·ε)∇E u

∥

∥

∥

H1(U)→ 0,

and for the pressure field,

infκ∈R

∥

∥

∥Pωε − P − b : D(u)−

∑

E∈E

(ΣωE1Rd\Iω)( ·

ε )∇Eu− κ∥

∥

∥

L2(U\Iωε (U))

→ 0,

where the sums run over an orthonormal basis E of Msym0 . ♦

Remark 1.1 (Buoyancy and sedimentation problem). If particles do not have the samemass density as the solvent fluid, a nontrivial buoyancy must be taken into account in theforce balance (1.1): denoting by g ∈ Cb(U)d the buoyancy, this equation is replaced by

1

ε

ˆ

εIωn

g +

ˆ

ε∂Iωn

σ(uωε , Pωε )ν = 0. (1.7)

The scaling in ε is such that surface and volumetric forces have the same order uniformlyin ε (it is equivalent, in sedimentation experiments, to increasing the size of the tank,rather than decreasing the size of the particles). Since an a priori diverging amount O(1ε )of energy is then pumped into the system, it needs to be compensated by modifying thedefinition of correctors (1.6). This is fully analyzed in our companion article [11] understrong mixing conditions, where we show in particular that the effective viscosity is notaffected by the settling process. A weak sedimentation regime can however be consideredas a direct adaptation of our present analysis, replacing (1.7) by

ˆ

εIωn

g +

ˆ

ε∂Iωn

σ(uωε , Pωε )ν = 0,


in which case the buoyancy vanishes in the limit (as the quotient of a volumetric over asurfacic term in the limit of small particles), and the effective equation is then obtainedby adding a forcing term to (1.3) in form of

− div 2BD(u) +∇P = (1− λ)f + λg, in U,div u = 0, in U,u = 0, on ∂U,´

U P = 0.

This simpler problem is however strictly distinct from the proper sedimentation regime. ♦

Remark 1.2 (Stationary setting). We briefly recall the standard formulation of the sta-tionary setting, make precise probabilistic assumptions, and recall some useful notationand constructions for stationary random fields.

(i) Stationarity and probabilistic assumptions. As is customary in stochastic homoge-nization theory, e.g. [23, Section 7], stationarity is most conveniently defined via ameasurable action τxx∈Rd of the translation group (Rd,+) on the underlying prob-ability space (Ω,P). More precisely, the space is endowed with measurable mapsτx : Ω → Ω that satisfy

• τx τy = τx+y for all x, y ∈ Rd;

• P [τxA] = P [A] for all x ∈ Rd and measurable A ⊂ Ω;

• the map Rd × Ω → Ω : (x, ω) 7→ τxω is jointly measurable;

and this action is assumed to be ergodic in the sense that any random variableφ ∈ L1(Ω) that is τ -invariant (i.e., φ(τx·) = φ almost surely for all x) is almost surelyconstant. The point process xωnn is then said to be stationary (with respect to τ)if xτxωn n = x+ xωnn for all x, ω.

(ii) Stationary extensions. A function φ : Rd × Ω → R is said to be stationary if there

exists a measurable map φ : Ω → R such that φ(x, ω) = φ(τ−xω) for all x, ω. The jointmeasurability assumption for the action then ensures that φ is jointly measurable,which in view of a result by von Neumann is equivalent to stochastic continuity,that is, P [|φ(x+ y, ·)− φ(x, ·)| > δ] → 0 as y → 0 for all x and δ > 0, cf. [23,

Section 7]. Stationarity then yields a bijection between random variables φ : Ω → R

and stationary measurable functions φ : Rd × Ω → R. The function φ is referredto as the stationary extension of the random variable φ. The subspace of stationaryfunctions φ ∈ L2(Ω; L2

loc(Rd)) is then identified with the Hilbert space L2(Ω), and

the (spatial) weak gradient ∇ on locally square integrable functions turns into alinear operator on L2(Ω). We also define Hs(Ω) as the subspace of random variables

φ ∈ L2(Ω) with stationary extension φ ∈ L2(Ω;Hsloc(R

d)). We often use the short-hand notation φω(x) := φ(x, ω). ♦

Notation.

• For vector fields u, u′ and matrix fields T, T ′, we set (∇u)ij = ∇jui, (div T )i =

∇jTij, T : T ′ = TijT′ij , (u⊗ u′)ij = uiu

′j, (T

s)ij =12(Tij + Tji), D(u) = (∇u)s. For

a vector field u and a matrix E, we also write ∇Eu = E : ∇u. We systematicallyuse Einstein’s summation convention on repeated indices.

• We denote by M = Rd×d the space of d× d matrices, by M

sym0 the subset of sym-

metric trace-free matrices, and by Mskew the subset of skew-symmetric matrices.


• We denote by C ≥ 1 any constant that only depends on the dimension d, on thereference domain U , and on the hardcore constant δ ∈ (0, 1). We use the notation .

(resp. &) for ≤ C× (resp. ≥ 1C×) up to such a multiplicative constant C. We add

subscripts to C,.,& in order to indicate dependence on other parameters.

• The ball centered at x of radius r in Rd is denoted by Br(x), and we simply write

B(x) = B1(x), Br := Br(0), and B = B1(0).

2. Construction of correctors

This section is devoted to the construction of a suitable solution to the Stokes correctorequation (1.6).

Proposition 2.1. Under the assumptions and notation of Theorem 1, for all E ∈ Msym0 ,

there exist a unique random field ψE ∈ L2(Ω;H1loc(R

d)d) and a unique pressure field ΣE ∈L2(Ω; L2

loc(Rd \ I)) such that

(i) For almost all ω the realizations ψωE ∈ H1

loc(Rd) and Σω

E ∈ L2loc(R

d \ Iω) satisfy

−ψωE +∇Σω

E = 0, in Rd \ Iω,

divψωE = 0, in R

d \ Iω,D(ψω

E + Ex) = 0, in Iω,ffl

∂Iωnσ(ψω

E + Ex,ΣωE)ν = 0, ∀n,

ffl

∂IωnΘ(x− εxωn) · σ(ψ

ωE + Ex,Σω

E)ν = 0, ∀n, ∀Θ ∈ Mskew.

(2.1)

(ii) The corrector gradient ∇ψE and the pressure ΣE1Rd\I are stationary1, with

E[

∇ψE

]

= 0, E[

ΣE1Rd\I

]

= 0,

E[

|∇ψE |2]

+ E[

Σ2E1Rd\I

]

. |E|2,

and we choose the anchoring´

B ψωE = 0 for the corrector.

In addition, the following properties hold:

(iii) Ergodic theorem for averages of corrector gradient and pressure: for almost all ω,

(∇ψωE)(

·ε) E [∇ψE ] = 0 weakly in L2

loc(Rd) as ε ↓ 0,

(ΣωE1Rd\Iω)( ·ε) E

[

ΣE1Rd\I

]

= 0 weakly in L2loc(R

d) as ε ↓ 0.

(iv) Sublinearity of the corrector: for almost all ω, for all q < 2dd−2 ,

εψωE(

·ε) → 0 strongly in Lq

loc(Rd) as ε ↓ 0. ♦

Proof. We start by defining suitable functional subspaces of L2(Ω)d×d that are tailored forthe study of the corrector equation (2.1). We first consider the subspace of potential fieldswith vanishing trace,

L2(Ω) :=

Ψ ∈ L2(Ω)d×d : E[

Ψ]

= 0, tr Ψ = 0, E[

Ψ : (∇× χ)]

= 0 ∀χ ∈ H1(Ω)d

.

1That is, ∇ψωE(x+y) = ∇ψ

τ−yω

E (x) and ΣωE(x+y)1

Rd\Iω (x+y) = Στ−yω

E (x)1Rd\I

τ−yω (x) for all x, y, ω,

cf. Remark 1.2.


Using stationary extensions, cf. Remark 1.2, it is well-known (e.g. [23, Section 7]) that thisspace is equivalently given by

L2(Ω) =

Ψ ∈ L2(Ω)d×d : E[

Ψ]

= 0, and ∃ψ ∈ L2(Ω; L2loc(R

d)d)

withΨ = ∇ψ and divψ = 0

,

where the differential constraints are more clearly interpreted. We further incorporatethe specific boundary conditions of the corrector equation (2.1) into the functional space,defining for E ∈ M0 the convex set

L2E(Ω) :=

Ψ ∈ L2(Ω) : ∃ψ ∈ L2(Ω; L2loc(R

d)d)withΨ = ∇ψ,

and with D(ψω + Ex) = 0 in Iω ∀ω

.

As we shall check in Substep 3.1 below, L2E(Ω) is not empty. Differences of elements

of L2E(Ω) belong to the vector space

L20(Ω) :=

Ψ ∈ L2(Ω) : ∃ψ ∈ L2(Ω; L2loc(R

d)d)withΨ = ∇ψ,

and with D(ψω) = 0 in Iω ∀ω

.

A well-known density result (e.g. [23, Section 7]) ensures that

L2(Ω) = adhL2(Ω)d×d

∇ψ : ψ ∈ H1(Ω)d, div ψ = 0

.

Likewise,L20(Ω) = adhL2(Ω)d×dK2

0(Ω), (2.2)

withK2

0(Ω) :=

∇ψ : ψ ∈ H1(Ω)d, div ψ = 0, and D(ψω) = 0 in Iω ∀ω

.

Once these spaces are introduced, the structure of the proof is as follows. We first showthat for a solution (ψE ,ΣE) of (i)–(ii) the gradient ∇ψE is the unique Lax-Milgram solutionin L2

E(Ω) of an abstract coercive problem on the probability space. We then argue thatconversely this unique solution indeed provides a solution of (i)–(ii) in a weak sense in thephysical space. Finally, from such a weak formulation, we reconstruct the pressure andestablish the desired estimates (iii)–(iv). The proof is split into five main steps.

Step 1. From (i)–(ii) to an abstract problem in L2E(Ω).

Let ψE be a solution of (i)–(ii). In particular, ΨE := ∇ψE is stationary and defines an

element ΨE ∈ L2E(Ω). We claim that it satisfies

E[

Φ : ΨE

]

= 0, for all Φ ∈ L20(Ω). (2.3)

By density (2.2), it is enough to prove (2.3) for all Φ ∈ K20(Ω). Let Φ ∈ K2

0(Ω) be given by

Φ = ∇φ for some φ ∈ H1(Ω)d with div φ = 0 and with D(φω) = 0 in Iω for all ω. In viewof the hardcore condition, for all R > 0, we can construct a cut-off function ηωR supported

in BR+3 with ηωR = 1 on BR and with |∇ηωR| .δ 1, such that ηωR is constant in Iωn + δ4B for

all n. Since ψωE is divergence-free, an integration by parts yields

ˆ

Rd

∇(ηωRφω) : ∇ψω

E = 2

ˆ

Rd

∇(ηωRφω) : D(ψω

E),

and thus, since´

Rd ∇(ηωRφω) = 0 and since D(ψω

E) + E = 0 in Iω,ˆ

Rd


E = 2

ˆ

Rd\Iω

∇(ηωRφω) : (D(ψω

E) + E).


Recalling the definition σ(ψωE + Ex,Σω

E) = 2(D(ψωE) + E) − Σω

E Id, integrating by parts,and using the corrector equation (2.1), we obtain

ˆ

Rd


E −

ˆ

Rd\Iω

div(ηωRφω)Σω

E =

ˆ

Rd\Iω

∇(ηωRφω) : σ(ψω

E + Ex,ΣωE)

= −∑

n

ˆ

∂Iωn

ηωRφω · σ(ψω

E + Ex,ΣωE)ν. (2.4)

For all n, since ηωR is constant in Iωn and since φω takes the special form κωn +Θωn(x− xωn)

in Iωn for some κωn ∈ Rd and Θω

n ∈ Mskew, the boundary condition in (2.1) for ψω

E on ∂Iωnprecisely yields

ˆ

∂Iωn

ηωR φω · σ(ψω

E + Ex,ΣωE)ν = 0.

The weak form (2.4) of the equation thus becomes, after expanding the gradients andrecalling that divφω = 0,

ˆ

Rd

ηωR∇φω : ∇ψω

E = −

ˆ

Rd

φω ⊗∇ηωR : ∇ψωE +

ˆ

Rd\Iω

ΣωE∇η

ωR · φω,

which by the properties of ηωR we rewrite as

ˆ

BR

∇φω : ∇ψωE = −

ˆ

BR+3\BR

ηωR∇φω : ∇ψω

E

−

ˆ

BR+3\BR

φω ⊗∇ηωR : ∇ψωE +

ˆ

BR+3\BR

ΣωE1Rd\Iω∇ηωR · φω.

Taking the expectation, using the stationarity of φ, ∇ψE, and ΣE, as well as the a prioribounds (ii) and the boundedness of ηR, we obtain from Cauchy-Schwarz’ inequality forall R ≥ 1,

∣

∣E[

∇φ : ΨE

]∣

∣ .1

R|E|E

[

|φ|2 + |∇φ|2]1

2 ,

and the claim follows from the arbitrariness of R.

Step 2. Well-posedness of the abstract problem (2.3).

In this step, we argue that there exists a unique solution ΨE ∈ L2E(Ω) to the problem (2.3).

As we shall check in Substep 3.1 below, the convex set L2E(Ω) is not empty, so that we

may choose a reference field Ψ0E ∈ L2

E(Ω). Writing ΨE = Ψ0E + Ψ1

E for some Ψ1E ∈ L2

0(Ω),

the equation (2.3) for ΨE is equivalent to the following equation for Ψ1E,

E[

Φ : Ψ1E

]

= −E[

Φ : Ψ0E

]

for all Φ ∈ L20(Ω). (2.5)

The existence and uniqueness of the solution Ψ1E to this equation then follow from the

Lax-Milgram theorem in the Hilbert space L20(Ω).

Step 3. From the abstract problem (2.3) to a weak formulation of (i).

Let ΨE ∈ L2E denote the unique solution of (2.3) as constructed in Step 2, which can

be written as ΨE = ∇ψE in terms of the almost surely unique random field ψE ∈L2(Ω;H1

loc(Rd)d) that satisfies the anchoring condition

´

B ψE = 0 at the origin. By con-struction, divψω

E = 0, and D(ψωE + Ex) = 0 in Iω for all ω. Next, we prove that ψE


satisfies the following weak formulation of (2.1): for almost all ω,ˆ

Rd

∇φ : ∇ψωE = 0, (2.6)

for all test functions φ in the class

Cω :=

φ ∈ H1(Rd)d : φ has compact support, divφ = 0, and D(φ) = 0 in Iω

.

We split the proof of (2.6) into two further substeps.

Substep 3.1. Definition of a suitable map Mω : H1

c,div(Rd)d → Cω, where H1

c,div(Rd)d stands

for the subspace ζ ∈ H1(Rd)d : ζ has compact support and div ζ = 0 of H1(Rd)d.

Choose a map M : H1div(B1+δ/2)

d → H1div(B1+δ/2)

d that satisfies for all ζ ∈ H1div(B1+δ/2)

d:

(1) Mζ − ζ ∈ H10 (B1+δ/2);

(2) D(Mζ) = 0 in B;(3) if D(ζ) = 0 in B, then Mζ = ζ;(4) ‖∇Mζ‖L2(B1+δ/2)

. ‖∇ζ‖L2(B1+δ/2).

Such a map M can for instance be constructed as follows,

Mζ := arginf

‖∇ξ −∇ζ‖2L2(B1+δ/2)

: ξ ∈ ζ +H10 (B1+δ/2), div ξ = 0,

and D(ξ) = 0 in B

. (2.7)

Since this is the minimization of a strictly convex lower-semicontinuous functional on aconvex set, the infimum is attained and unique provided the convex set is nonempty.Choosing κ =

ffl

B1+δ/2\Bζ and Θ = 0, it suffices to check that there exists ξ ∈ ζ +

H10 (B1+δ/2) with

div ξ = 0 and ξ|B = κ. (2.8)

For that purpose, choose uζ ∈ H10 (B1+δ/2) that coincides with −ζ + κ on B. In view of

the compatibility conditionˆ

B1+δ/2\Bdiv uζ =

ˆ

∂B1+δ/2

uζ · ν −

ˆ

∂Buζ · ν =

ˆ

∂Bζ · ν =

ˆ

Bdiv ζ = 0,

a standard use of the Bogovskii operator in form of [15, Theorem III.3.1] ensures that thisuζ can be modified in B1+δ/2 \B (without changing its boundary values) to be divergence-free in B1+δ/2 \B (hence in the whole of B1+δ/2), with the estimate

‖∇uζ‖L2(B1+δ/2\B) . ‖ζ − κ‖H

12 (∂B)

.

In particular, by a trace estimate and Poincaré’s inequality, this yields

‖∇uζ‖L2(B1+δ/2\B) . ‖ζ − κ‖H1(B1+δ/2\B) .δ ‖∇ζ‖L2(B1+δ/2\B).

The function ξζ := ζ + uζ ∈ ζ +H10 (B1+δ/2) then satisfies (2.8) and

‖∇ξζ‖L2(B1+δ/2).δ ‖∇ζ‖L2(B1+δ/2)

.

This implies that M in (2.7) is well-defined and indeed satisfies the properties (1)–(4).

Next, for ζ ∈ H1c,div(R

d)d, we extend Mζ by ζ outside B1+δ/2, and for all x ∈ Rd we

denote by Mx the corresponding operator when the origin 0 is replaced by x. For all ω, wethen define the operator Mω :=

∏

n Mxωn, which indeed maps H1

c,div(Rd)d to Cω as desired.


We conclude this construction of Mω with a weak continuity result: for all bounded do-mains D and all sequences (ζn)n of divergence-free functions compactly supported in D, ifζn ζ weakly in H1(D), then for all ω we have M

ωζn Mωζ in H1(D). In view of the

above construction of Mω, it is enough to prove this continuity result at the level of theelementary map M. Since the sequence (Mζn)n is bounded in H1(B1+δ/2), it convergesto some ξ along a subsequence (not relabelled), which is necessarily an admissible testfunction for the minimization problem (2.7) for Mζ. It remains to argue that it coincideswith the desired minimizer Mζ. To this aim, we use that the unique minimizers Mζn of(2.7) are characterized by the following Euler-Lagrange equations: for all ξ′ ∈ H1

0 (B1+δ/2)with div ξ′ = 0 and with D(ξ′) = 0 in B,

ˆ

B1+δ/2

(∇Mζn −∇ζn) : ∇ξ′ = 0,

in which we may pass to the limit in n in form ofˆ

B1+δ/2

(∇ξ −∇ζ) : ∇ξ′ = 0,

thus recovering the Euler-Lagrange equation for Mζ. This entails ξ = Mζ and ensuresthe convergence of the whole sequence.

We now quickly argue that a similar argument ensures that the convex set L2E(Ω) is not

empty. Choose uE ∈ H10 (B1+δ/2) that coincides with x 7→ −Ex in B. In view of the

compatibility conditionˆ

B1+δ/2\Bdiv uE =

ˆ

∂B1+δ/2

uE · ν −

ˆ

∂BuE · ν =

ˆ

∂BEx · ν = |B| trE = 0,

a standard use of the Bogovskii operator in form of [15, Theorem III.3.1] ensures thatuE can be chosen divergence-free in B1+δ/2 \ B (hence in the whole of B1+δ/2), with theestimate

‖∇uE‖L2(B1+δ/2\B) . |E|.

We may then define the stationary function φ =∑

n uE(·−xn), which is such that Φ = ∇φbelongs to L2

E(Ω) by construction.

Substep 3.2. Proof of (2.6).Given a vector field φ ∈ H1

c,div(Rd)d and given a random variable χ ∈ L2(Ω), we define Φ

as the stationarization of the product χ∇Mφ, that is,

Φ(x, ω) :=

ˆ

Rd

χ(τyω)∇(Mτyωφ)(x+ y) dy,

which is well-defined in L2(Ω,L2loc(R

d)d×d) since φ (hence supω |Mωφ|) is compactly sup-

ported. On the one hand, Φ is obviously a stationary random field: for all x, z, ω,

Φ(x+ z, ω) =

ˆ

Rd

χ(τyω)∇Mτyωφ(x+ z + y) dy

=

ˆ

Rd

χ(τy−zω)∇Mτy−zωφ(x+ y)dy

= Φ(x, τ−zω).


On the other hand, the definition of M ensures that Φ belongs to L20, which makes it an

admissible test function for (2.3). By stationarity of ΨE = ∇ψE and of I in the form(Ψω

E1Rd\Iω)(0) = (ΨτyωE 1Rd\Iτyω)(y), and since the group action preserves the probability

measure, we find

0 = E[

Φ : ΨE

]

=

ˆ

Ω

(ˆ

Rd

χ(τyω)∇(Mτyωφ)(y) dy : ΨωE

)

dP(ω)

=

ˆ

Ω

(ˆ

Rd

χ(τyω)∇(Mτyωφ)(y) : ∇ψτyωE (y) dy

)

dP(ω)

= E

[

χ

ˆ

Rd

∇(Mφ) : ∇ψE

]

.

By the arbitrariness of χ, this implies that for any compactly supported vector field φ ∈H1

div(Rd)d there holds for almost all ω,

ˆ

Rd

∇(Mωφ) : ∇ψωE = 0.

By a density argument together with the weak continuity of Mω as established in Sub-

step 3.1, we deduce that for almost all ω this actually holds for all compactly supportedvector fields φ ∈ H1

div(Rd)d. Given ω, for φω in the (realization-dependent) class Cω, there

holds Mωφω = φω and the conclusion (2.6) follows.

Step 4. Reconstruction of the pressure.In Step 3, we proved that the unique solution ΨE = ∇ψE of the abstract problem (2.3)also satisfies the weak formulation (2.6) of the corrector equation (2.1). In addition, notethat the construction of Step 3 yields the bound E

[

|∇ψE |2]

. |E|2. In the present step,we show that one can construct a stationary pressure field ΣE such that for almost all ωthe vector field ψω

E is a classical solution of the corrector equation (2.1), and that ΣE and∇ψE satisfy (ii). We split the proof into five further substeps.

Substep 4.1. Reconstruction of a pressure field ΣE.For R ≥ 2, consider the bounded Lipschitz domain

DωR := BR ∪

⋃

n:Iωn∩BR 6=∅

(Iωn + δ2B).

In view of (2.6), for almost all ω, ψωE satisfies for all vector fields φ ∈ H1

c,div(Rd)d that

vanish on Iω and outside DωR,

ˆ

DωR

∇φ : ∇ψωE = 0.

We deduce that ψωE is a weak solution of

−ψωE +∇Σω

E = 0, in DωR \ Iω,

divψωE = 0, in Dω

R,D(ψω

E + Ex) = 0, in Iω ∩DωR,

(2.9)

in the sense of [15, Definition IV.1.1]. Hence, by [15, Lemma IV.1.1], there exists a uniquepressure field Σω

E ∈ L2(DωR \ Iω) with the anchoring condition

´

2B ΣωE1Rd\Iω = 0, such

that (2.9) holds in the usual weak sense (that is, for all test functions φ ∈ H10 (D

ωR \ Iω)d

without divergence-free constraint). In addition, by [15, Theorems IV.4.3 and IV.5.2],


both ψωE and Σω

E are smooth in BR/2 \ Iω. By the arbitrariness of R, this implies that

the pressure field ΣωE is well-defined in L2

loc(Rd \ Iω) and that ψω

E and ΣωE are smooth on

Rd \ Iω. In particular, the solutions are classical and the boundary conditions in (2.1) are

satisfied in a pointwise sense. Note that the joint measurability of ΣE on Rd × Ω easily

follows from the reconstruction procedure for the pressure in [15]; details are omitted.

Substep 4.2. Proof that for all R ≥ 5,

BR\Iω

(

ΣωE −

BR\Iω

ΣωE

)2.

BR\Iω

|∇ψωE |

2. (2.10)

As usual for pressure estimates for the Stokes equation, we first need to construct a mapζωR ∈ H1

0 (BR) such that

div ζωR =(

ΣωE −

BR\Iω

ΣωE

)

1Rd\Iω , (2.11)

‖∇ζωR‖L2(BR) .∥

∥

∥ΣE −

BR\Iω

ΣωE

∥

∥

∥

L2(BR\Iω), (2.12)

with the slight twist that ζωR|Iωn further needs to be constant for all n. Testing (2.1) withsuch a ζωR then yields

ˆ

Rd\Iω

∇ζωR : ∇ψωE −

ˆ

Rd\Iω

ΣωE div ζωR = 0,

which entails in view of the choice (2.11) of ζωR,ˆ

BR\Iω

∣

∣

∣ΣωE −

BR\Iω

ΣωE

∣

∣

∣

2≤

ˆ

BR\Iω

|∇ζωR||∇ψωE |,

and (2.10) follows from (2.12).

It remains to construct such a map ζωR. First define ξωR ∈ H10 (BR)

d (extended to zerooutside BR) as a solution of the divergence problem

div ξωR =(

ΣωE −

BR\Iω

ΣωE

)

1Rd\Iω ,

‖∇ξωR‖L2(BR) .∥

∥

∥ΣωE −

BR\Iω

ΣωE

∥

∥

∥

L2(BR\Iω),

as provided by [15, Theorem III.3.1], where we emphasize that the multiplicative constantin the estimate is uniformly bounded in R. Next, we need to modify ξωR in the inclusionsIωn ’s that intersect BR without changing div ξωR and without increasing the norm of ∇ξωR toomuch. This is performed by constructing suitable compactly supported corrections aroundthe inclusions. For inclusions Iωn ’s contained in BR with dist(Iωn , ∂BR) ≥ δ, arguing asin Substep 3.1, we can construct a divergence-free vector field ξωR,n ∈ H1

0 (Iωn + δ

2B)d that

coincides with −ξωR +ffl

(Iωn+ δ2B)\Iωn

ξωR on Iωn such that

‖∇ξωR,n‖L2((Iωn+ δ2B)\Iωn ) . ‖∇ξωR‖L2((Iωn+ δ

2B)\Iωn ).

We turn to inclusions Iωn ’s that intersect BR such that dist(Iωn , ∂BR) < δ, for which weconstruct a divergence-free vector field ξωR,n ∈ H1

0 (BR∩(Iωn+

δ2B))d that coincides with −ξωR

on BR∩Iωn (that is indeed divergence-free there). Such a vector field can be constructed as


an application of the Bogovskii operator on BR∩(Iωn +

δ2B)\Iωn , in view of the compatibility

conditionˆ

BR∩(Iωn+ δ2B)\Iωn

div ξωR,n =

ˆ

∂(BR∩(Iωn+ δ2B))

ξωR,n · ν −

ˆ

∂(BR∩Iωn )ξωR,n · ν

=

ˆ

∂(BR∩Iωn )ξωR · ν =

ˆ

BR∩Iωn

div ξωR = 0, (2.13)

and it satisfies

‖∇ξωR,n‖L2(BR∩(Iωn+ δ2B)\Iωn ) . ‖ξωR‖H

12 (∂(BR∩Iωn ))

.

Hence, by a trace estimate (with ∂Iωn at distance at most δ from ∂BR, on which ξωR vanishes)and Poincaré’s inequality,

‖∇ξωR,n‖L2(BR∩(Iωn+ δ2B)\Iωn ) . ‖ξωR‖H1(BR∩(Iωn+2δB))

. ‖∇ξωR‖L2(BR∩(Iωn+2δB)).

We finally define

ζωR := ξωR +∑

n:Iωn∩BR 6=∅

ξωR,n,

which by construction is constant in each of the inclusions Iωn ’s and satisfies the requiredproperties (2.11) and (2.12).

Substep 4.3. Extension of ΣE to Rd and estimate of ∇ΣE.

In this substep, we extend ΣE to Rd in such a way that ΣE ∈ L2(Ω;H1

loc(Rd)), that ∇ΣE

is stationary, and that we have for all R ≥ 5,

E

[

BR

(

ΣE −

BR

ΣE

)2]

+ E[

|∇ΣE|2]

. |E|2. (2.14)

We start by proving that (∇ΣωE)1Rd\I is a stationary field and satisfies

E[

|(∇ΣE)1Rd\I |2]

. |E|2. (2.15)

By the Stokes equation in form of (ψωE)1Rd\I = (∇Σω

E)1Rd\Iω , it suffices to prove that

(ψE)1Rd\I ∈ L2(Ω) satisfies E[

|(ψE)1Rd\I |2]

. |E|2. Since (∇ψE)1Rd\I is stationary

and since ψE is of class C2 up to the boundaries ∂In, it is enough to prove that for almostall ω,

lim supR↑∞

BR

|(∇2ψωE)1Rd\I |

2 . |E|2. (2.16)

To this aim, it suffices to show that for all x ∈ Rd,

ˆ

Bδ/8(x)(|∇2ψω

E |2 + |∇Σω

E |2)1Rd\I .δ

ˆ

B5(x)|∇ψω

E |2, (2.17)

since the desired estimate (2.16) then follows in combination with the ergodic theorem andthe bound E

[

|∇ψE|2]

. |E|2. First consider the case when x ∈ Rd satisfies dist (x,Iω) >

δ/4, for which Bδ/4(x) ⊂ Rd \ Iω. By interior regularity for the Stokes equation in form


of [15, Theorems IV.4.1], by (2.10), and by Poincaré’s inequality, we then have with thechoice cω1 =

ffl

Bδ/2(x)ψωE and cω2 =

ffl

B5(x)\Iω ΣωE,

ˆ

Bδ/8(x)(|∇2ψω

E |2 + |∇Σω

E|2)1Rd\Iω .δ

ˆ

Bδ/4(x)(|∇ψω

E |2 + |ψω

E − cω1 |2 + |Σω

E − cω2 |2)

.

ˆ

B5(x)|∇ψω

E |2,

that is, (2.17). Next consider the case when x ∈ Rd satisfies dist (x,Iω) ≤ δ/4, and let Iωn

be the unique ball such that dist (x, Iωn ) < δ/4. By the boundary regularity theory forthe Stokes equation in form of [15, Theorems IV.5.1–5.3], we then have with the choicecω1 =

ffl

Iωn+ δ2B ψ

ωE and cω2 =

ffl

B5(x)\Iω ΣωE,

ˆ

Bδ/8(x)(|∇2ψω

E |2 + |∇Σω

E|2)1Rd\Iω .δ ‖ψω

E |Iωn − cω1 ‖2

H32 (∂Iωn )

+ ‖ΣωE − cω2 ‖

2L2((Iωn+ δ

2B)\Iωn )

+ ‖ψωE − cω1 ‖H1((Iωn+ δ

2B)\Iωn ).

Since ψωE is affine on Iωn , we have

‖ψωE |Iωn − cω1 ‖H

32 (∂Iωn )

. ‖ψωE |Iωn − cω1 ‖H2(Iωn ) = ‖ψω

E − cω1 ‖H1(Iωn )

≤ ‖ψωE − cω1 ‖H1(Iωn+ δ

2B),

while Poincaré’s inequality with mean-value zero yields

‖ψωE − cω1 ‖H1(Iωn+ δ

2B) .δ ‖∇ψω

E‖L2(Iωn+ δ2B),

so that in combination with (2.10) the above turns into (2.17).

It remains to extend ΣE on the inclusions. We simply choose ΣωE|B1/2(xω

n)=ffl

(Iωn+ δ2B)\Iωn

ΣωE,

and we extend ΣωE radially linearly between ∂Iωn and ∂B1/2(x

ωn) (recall that Σω

E1Rd\Iω is

continuous up to the boundary). So defined, ΣωE belongs toH1

loc(Rd) and ∇ΣE is stationary

on Rd. We conclude by establishing (2.14). Noting that the choice of the extension ensures

ˆ

Iωn

|∇ΣωE|

2 .δ

ˆ

(Iωn+ δ4B)\Iωn

|∇ΣωE|

2,

the gradient estimate in (2.14) simply follows from (2.15), and it remains to check theother part. By the definition of the extension, with cω =

ffl

BR+2\IΣωE , we find using (2.10)


and (2.15),

BR

(

ΣωE −

BR

ΣωE

)2.

BR

(ΣωE − cω)2

.

BR

(ΣωE − cω)21

Rd\Iω +R−d

ˆ

BR∩I(Σω

E − cω)2

.δ

BR

(ΣωE − cω)21Rd\Iω +R−d

∑

n:Iωn∩BR 6=∅

ˆ

(Iωn+ δ2B)\Iωn

(|ΣωE − cω|2 + |∇Σω

E|2)

.

BR+2

(ΣωE − cω)21Rd\Iω +

BR+2

|∇ΣωE |

21Rd\Iω

.

BR+2

|∇ψωE|

2,

and the estimate (2.14) follows.

Substep 4.4. Construction of a stationary pressure field ΣE.Let χ ∈ C∞

c (B) satisfy´

B χ = 1, consider the rescaled kernel χr =1rdχ( ·

r ) for r ≥ 1, and

define Pr := ΣE − χr ∗ ΣE. By construction, Pr is stationary, and we claim that

E[

P 2r + |∇Pr|

2]

. |E|2, (2.18)

limr↑∞

E[

|∇Pr −∇ΣE|21Rd\I

]

= 0. (2.19)

From (2.18), we deduce by weak compactness that there exists some P ∈ H1(Ω) such that

(Pr,∇Pr) (P ,∇P ) weakly in L2(Ω) along some subsequence (not relabelled), with

E[

P 2 + |∇P |2]

. |E|2. (2.20)

From (2.19) and the weak lower-semicontinuity of the L2(Ω)-norm, we then deduce

E[

|∇P −∇ΣE|21Rd\I

]

≤ lim infr↑∞

E[


]

= 0.

Hence, for almost all ω, the limit Pω coincides with ΣωE up to an additive constant on

the connected set Rd \ Iω. We then define the stationary pressure as ΣE1Rd\I :=

(

P −

E[

P1Rd\I

])

1Rd\I , which satisfies E[

ΣE1Rd\I

]

= 0 and the a priori estimate (ii).

It remains to give the arguments in favor of (2.18) and (2.19). We start with the former.For all R ≥ r ≥ 1, for cω =

ffl

BR+rΣωE, we have

BR

(Pωr )

2 + |∇Pωr |

2 =

BR

(ΣωE − χr ∗ Σ

ωE)

2 + |∇ΣωE − χr ∗ ∇P

ωE |

2

.

BR

(ΣωE − cω)2 + (χr ∗ (Σ

ωE − cω))2 + |∇Σω

E|2 + |χr ∗ ∇Σω

E|2

. R−d

ˆ

BR+r

|∇ΣωE |

2 + (ΣωE − cω)2.

Taking the expectation and using (2.14) then yields by stationarity of Pr,

E[

P 2r + |∇Pr|

2]

.(R+ r)d

Rd|E|2,


from which (2.18) follows by taking the limit R ↑ ∞. We turn to (2.19). By definitionof Pr and since |∇χr| .

1rd+11BR

for all R ≥ r ≥ 1, we have for cω =ffl

BR+rΣωE ,

BR

|∇Pωr −∇Σω

E|21Rd\Iω ≤

BR

|∇χr ∗ ΣωE|

2 =

BR

|∇χr ∗ (ΣωE − cω)|2

.1

r

(R+ r)d

Rd

BR+r

(ΣωE − cω)2.

As before, taking the expectation, recalling that (∇Pr − ∇ΣE)1Rd\I is stationary, us-

ing (2.14), and letting R ↑ ∞, we deduce

E[


]

.1

r|E|2,

from which the claim (2.19) follows.

Substep 4.5. Proof of existence and uniqueness for (i)–(ii).In Step 1, we have shown that if ψE is a solution of (i)–(ii), then ΨE = ∇ψE satisfies theabstract problem (2.3), for which existence and uniqueness is proved in Step 2. In Step 3,we considered the unique solution ΨE of (2.3) and proved that ΨE = ∇ψE is automaticallya weak solution of (2.1) in form of (2.6). In Substeps 4.1–4.4, we reconstructed a uniquestationary pressure field ΣE (with vanishing expectation) such that ψE is a classical solutionof (2.1). Uniqueness for (i)–(ii) then follows from uniqueness for (2.3). For the existencepart for (i)–(ii), it remains to note that ΣE and ψE satisfy (ii) as shown in Substep 4.4.

Step 5. Proof of (iii)–(iv).The convergences in (iii) are a standard application of the ergodic theorem. The sublin-earity (iv) of the corrector ψω

E at infinity is also a standard result for random fields thegradients of which are stationary and have vanishing expectation, cf. [26, 23].

3. Proof of the homogenization result

This section is devoted to the proof of Theorem 1, making use of the correctors (ψE)Edefined in Proposition 2.1 and adapting the classical oscillating test function method byTartar [29]. We split the proof into eight different steps.

Step 1. Reformulation of the equations.We show that the solution uωε of (1.2) satisfies in the weak sense in the whole domain U ,

−uωε +∇(Pωε 1U\Iω

ε (U)) = f1U\Iωε (U) −

∑

n∈Nωε (U)

δε∂Iωnσ(uωε , P

ωε )ν, (3.1)

while the corrector ψωE satisfies in the whole space R

d,

−ψωE +∇(Σω

E1Rd\Iω) = −∑

n

δ∂Iωn σ(ψωE + Ex,Σω

E)ν. (3.2)

We focus on (3.1), and leave the proof of (3.2) (which is similar) to the reader. Since uωεis divergence-free, an integration by parts yields

ˆ

U∇ζ : ∇uωε = 2

ˆ

U∇ζ : D(uωε ),


and thus, since D(uωε ) = 0 in Iωε (U),

ˆ

U∇ζ : ∇uωε = 2

ˆ

U\Iωε (U)

∇ζ : D(uωε ).

Recalling the definition σ(uωε , Pωε ) = 2D(uωε ) − Pω

ε Id, integrating by parts, and usingequation (1.2), we obtainˆ

U∇ζ : ∇uωε −

ˆ

U\Iωε (U)

(div ζ)Pωε =

ˆ

U\Iωε (U)

∇ζ : σ(uωε , Pωε )

=

ˆ

U\Iωε (U)

ζ · f −∑

n∈Nωε (U)

ˆ

ε∂Iωn

ζ · σ(uωε , Pωε )ν,

that is, (3.1).

Step 2. Energy estimates.We now show that for almost all ω the solution uωε of (1.2) satisfies

ˆ

U|∇uωε |

2 +

ˆ

U\Iωε (U)

|Pωε |

2 .δ

ˆ

U|f |2. (3.3)

For almost all ω, by weak compactness, this allows us to consider uω ∈ H10 (U)d and

Qω ∈ L2(U) such that, along a subsequence (not relabelled) as ε ↓ 0,

uωε uω in H10 (U), and Pω

ε 1U\Iωε (U) Qω in L2(U). (3.4)

In particular, by Rellich’s theorem, uωε → uω in L2(U) strongly.

Here comes the argument for (3.3). For all v ∈ H10 (U) with div v = 0 in U and with

D(v) = 0 in Iε(U), testing the formulation (3.1) of the Stokes equation with v yieldsˆ

U∇v : ∇uωε =

ˆ

U\Iωε (U)

v · f,

which for the choice v = uωε yieldsˆ

U|∇uωε |

2 =

ˆ

U\Iωε (U)

f · uωε .(

ˆ

U|f |2

)1

2(

ˆ

U|∇uωε |

2)

1

2

(3.5)

by Poincaré’s inequality in H10 (U), that is, (3.3) for ∇uωε . The corresponding estimate for

the pressure is obtained by a similar argument as in Substep 4.2 of the proof of Proposi-tion 2.1.

Step 3. A priori estimates at inclusion boundaries.We claim that the solution uωε of (1.2) and the corrector ψω

E satisfy for almost all ω,

∑

n∈Nωε (U)

ˆ

ε∂Iωn

|uωε |2 .

1

ε

ˆ

U|f |2, (3.6)

∑

n∈Nωε (U)

ˆ

ε∂Iωn

|∇uωε |2 + |Pω

ε |2 .δ

1

ε

ˆ

U|f |2, (3.7)

∑

n∈Nωε (U)

ˆ

ε∂Iωn

|ψωE(

·ε)|

2 .δ1

ε

ˆ

U|ψω

E(·ε)|

2, (3.8)


∑

n∈Nωε (U)

ˆ

ε∂Iωn

|∇ψωE(

·ε)|

2 + |ΣωE(

·ε)|

2 .δ1

ε

ˆ

U|∇ψω

E(·ε)|

2 + |(ΣωE1Rd\Iω)( ·ε)|

2. (3.9)

We start with the proof of (3.6). For all n ∈ N ωε (U), since uωε is affine in εIωn , there holds

ˆ

ε∂Iωn

|uωε |2 .

1

ε

ˆ

εIωn

|uωε |2,

so that∑

n∈Nωε (U)

ˆ

ε∂Iωn

|uωε |2 .

1

ε

ˆ

U|uωε |

2,

and the claim (3.6) follows from Poincaré’s inequality and (3.3). Likewise, for all n, sinceψωE is affine in Iωn , we find

ˆ

∂Iωn

|ψωE |

2 .

ˆ

Iωn

|ψωE |

2,

and the claim (3.8) follows after summing and rescaling. We turn to the proof of (3.7).

By scaling, it suffices to check that uωε := ε−2uωε (ε·) and Pωε := ε−1Pω

ε (ε·) satisfy∑

n∈Nωε (U)

ˆ

∂Iωn

|∇uωε |2 + |Pω

ε |2 .δ

1

ε2

ˆ

1

εU|f(ε·)|2. (3.10)

Given n ∈ N ωε (U), a trace estimate yields

ˆ

∂Iωn

|∇uωε |2 + |Pω

ε |2 .δ ‖(∇u

ωε , P

ωε )‖

2H1((Iωn+ δ

4B)\Iωn )

.

Recalling that the inclusion Iωn is at distance at least δ > 0 from other inclusions and from1ε∂U so that −uωε +∇Pω

ε = f(ε·) is satisfied in the annulus (Iωn + δB)\ Iωn , the regularitytheory for the Stokes equation near a boundary in form of [15, Theorems IV.5.1–5.3] leadsto the following, with cωn,ε :=

ffl

Iωn+ δ2B u

ωε ,

ˆ

∂Iωn

|∇uωε |2 + |Pω

ε |2 .δ ‖u

ωε |Iωn − cωn,ε‖

2

H32 (∂Iωn )

+ ‖f(ε·)‖2L2(Iωn+ δ

2B)

+ ‖Pωε ‖

2L2((Iωn+ δ

2B)\Iωn )

+ ‖uωε − cωn,ε‖H1((Iωn+ δ2B)\Iωn ).

Since uωε is affine on Iωn , we have

‖uωε |Iωn − cωn,ε‖2

H32 (∂Iωn )

. ‖uωε − cωn,ε‖2H2(Iωn ) = ‖uωε − cωn,ε‖

2H1(Iωn ),

while Poincaré’s inequality with mean-value zero yields

‖uωε − cωn,ε‖H1(Iωn+ δ2B) . ‖∇uωε ‖

2L2(Iωn+ δ

2B),

so that the above turns intoˆ

∂Iωn

|∇uωε |2 + |Pω

ε |2 .δ ‖f(ε·)‖

2L2(Iωn+ δ

2B)

+ ‖(∇uωε , Pωε 1Rd\Iω)‖2

L2(Iωn+ δ2B).

Since the balls of the collection Iωn + δ2Bn are all disjoint, the rescaled version of the

energy estimate (3.3) leads to∑

n∈Nωε (U)

ˆ

∂Iωn

|∇uωε |2 + |Pω

ε |2 .δ

ˆ

1

εU|f(ε·)|2 + |∇uωε |

2 + |Pωε 1Rd\Iω |2 .

1

ε2

ˆ

1

εU|f(ε·)|2,


that is, (3.10). It remains to establish (3.9). Applying as above a trace estimate togetherwith the regularity theory for the Stokes equation near a boundary (cf. Substep 4.3 in theproof of Proposition 2.1), we obtain

∑

n∈Nωε (U)

ˆ

∂Iωn

|∇ψωE |

2 + |ΣωE |

2 .δ

ˆ

1

εU|∇ψω

E |2 + |Σω

E1Rd\Iω |2,

and the claim (3.9) follows after rescaling.

Step 4. Oscillating test function method.We show that for all test functions v ∈ C∞

c (U)d with div v = 0 we have for almost all ω,along a subsequence (not relabelled),

2

ˆ

UD(v) : D(uω) + lim

ε↓0

∑

E∈E

2

ˆ

U(∇E v) D(ψω

E)(·ε) : D(uωε ) = (1− λ)

ˆ

Uv · f, (3.11)

where the sum runs over an orthonormal basis E of symmetric trace-free matrices Msym0 ,

and where the limit in the left-hand side indeed exists (and is computed in the next step).

Let a typical ω ∈ Ω be fixed such that the bounds of Steps 1–2 hold as well as theconvergence (3.4) along a subsequence (not relabelled), and such that for all E ∈ M0 thecorrector ψω

E and corresponding pressure ΣωE satisfy the corrector equation (1.6) in the

classical sense as well as the properties (iii)–(iv) of Proposition 2.1. Given a test functionv ∈ C∞

c (U)d with div v = 0, we follow Tartar’s ideas and define its oscillatory versionvωε ∈ H1

0 (U)d via

vωε := v +∑

E∈E

εψωE(

·ε)∇E v,

where we recall the notation ∇E v = E : ∇v. Testing equation (3.1) with vωε leads toˆ

U∇vωε : ∇uωε −

ˆ

U\Iωε (U)

(div vωε )Pωε

=

ˆ

U\Iωε (U)

vωε · f −∑

n∈Nωε (U)

ˆ

ε∂Iωn

vωε · σ(uωε , Pωε )ν, (3.12)

and it remains to examine each of the four terms appearing in this identity.

• First, an integration by parts with div uωε = 0 yieldsˆ

U∇vωε : ∇uωε = 2

ˆ

UD(vωε ) : D(uωε ),

and then inserting the definition of vωε ,ˆ

U∇vωε : ∇uωε = 2

ˆ

UD(v) : D(uωε ) +

∑

E∈E

2

ˆ

Uεψω

E(·ε)⊗∇∇E v : D(uωε )

+∑

E∈E

2

ˆ

U(∇E v) D(ψω

E)(·ε) : D(uωε ).

By Step 2, the first right-hand side term converges to 2´

U D(v) : D(uω). By sublin-

earity of ψωE (cf. Proposition 2.1(iv)), together with the boundedness of ∇uωε in L2(U)

(cf. Step 2), the second right-hand side term converges to 0.


• Second, the definition of vωε with div v = 0 and divψωE = 0 leads to

ˆ

U\Iωε (U)

(div vωε )Pωε =

∑

E∈E

ˆ

U\Iωε (U)

εψωE(

·ε) · P

ωε ∇∇E v,

which converges to 0 in view of the sublinearity of ψωE (cf. Proposition 2.1(iv)) together

with the boundedness of Pωε in L2(U) (cf. Step 2).

• Third, the sublinearity of ψωE (cf. Proposition 2.1(iv)) implies vωε → v in L2(U) and the

ergodic theorem for the inclusion process yields 1U\Iωε (U) E

[

1Rd\I

]

1U = (1 − λ)1U

weakly-* in L∞(U) for typical ω, so that´

U\Iωε (U) v

ωε · f → (1− λ)

´

U v · f .

• Fourth, for n ∈ N ωε (U), the oscillating test function vωε can be expanded as follows, for

all x ∈ ε∂Iωn ,∣

∣

∣vωε (x)− v(εxωn)−∇v(εxωn) (x− εxωn)−

∑

E∈E

εψE(xε )∇E v(εx

ωn)∣

∣

∣

. ε2‖∇2v‖L∞ maxE∈E

(1 + |ψE(xε )|).

Setting for abbreviation Θωε,n := (∇v − D(v))(εxωn) ∈ M

skew, and recalling the choicetrD(v) = div v = 0, this can be reorganized as∣

∣

∣vωε (x)− v(εxωn)−Θω

ε,n(x− εxωn)−∑

E∈E

ε(

ψωE(

xε ) + E(xε − xωn)

)

∇E v(εxωn)∣

∣

∣

. ε2‖∇2v‖L∞ maxE∈E

(1 + |ψE(xε )|).

Inserting this approximation of vωε on ε∂Iωn , and recalling that ψωE +E(· − xωn) is a rigid

motion on Iωn , the boundary conditions for uωε on ε∂Iωn lead to∣

∣

∣

ˆ

ε∂Iωn

vωε · σ(uωε , Pωε )ν

∣

∣

∣. ε2‖∇2v‖L∞ max

E∈E

ˆ

ε∂Iωn

(1 + |ψωE(

·ε)|) (|∇u

ωε |+ |Pω

ε |).

Summing over n and using Cauchy-Schwarz’ inequality and the estimates (3.7) and (3.8)of Step 3, we obtain∣

∣

∣

∑

n∈Nωε (U)

ˆ

ε∂Iωn

vωε · σ(uωε , Pωε )ν

∣

∣

∣. ‖∇2v‖L∞

(

maxE∈E

ˆ

U(ε+ |εψω

E(·ε)|)

2)

1

2(

ˆ

U|f |2

)1

2

,

where the right-hand side tends to 0 by the sublinearity of ψωE at infinity (cf. Proposi-

tion 2.1(iv)).

Inserting the above estimates into (3.12), the claim (3.11) follows.

Step 5. Computation of the limit in (3.11) by compensated compactness.For all v ∈ C∞

c (U)d with div v = 0, we claim that for almost all ω,

limε↓0

∑

E∈E

2

ˆ

U(∇E v) D(ψω

E)(·ε) : D(uωε ) =

∑

E∈E

E [ZE ] :

ˆ

Uuω ⊗∇∇E v, (3.13)

in terms of the (matrix-valued) stationary random field ZE defined componentwise by

ZωE := −

∑

n

1Iωn

|Iωn |

ˆ

∂Iωn

σ(ψωE + Ex,Σω

E)ν ⊗ (x− xωn). (3.14)


Integrating by parts, using equation (3.2) for the corrector, and the constraint div uωε = 0,we may rewrite the product D(ψω

E)(·ε) : D(uωε ) of two weakly convergent sequences as

2

ˆ

U(∇E v)D(ψω

E)(·ε) : D(uωε ) = −

ˆ

U\εIω

(uωε ⊗∇∇E v) :(

2D(ψωE)− Σω

E

)

( ·ε)

−∑

n

ˆ

U∩ε∂Iωn

(∇E v)uωε · σ

(

εψωE(

·ε) + Ex,Σω

E(·ε))

ν.

By the ergodic theorem in form of ∇ψωE(

·ε) 0 and Σω

E(·ε) 0 in L2(U) (cf. Proposi-

tion 2.1(iii)) and by the strong convergence uωε → uω in L2(U), the first right-hand sideterm converges to 0. Hence, the limit of interest (which exists by (3.11)) takes the form

Lω := limε↓0

∑

E∈E

2

ˆ

U(∇E v) D(ψω

E)(·ε) : D(uωε ) = lim

ε↓0Iωε , (3.15)

where Iωε denotes the third and main right-hand side term in the above,

Iωε := −∑

E∈E

∑

n

ˆ

U∩ε∂Iωn

(∇E v)uωε · σ

(

εψωE(

·ε) + Ex,Σω

E(·ε))

ν.

Since v is compactly supported in U , we may restrict to ε small enough such that v issupported in x ∈ U : d(x, ∂U) > ε, so that v vanishes on U ∩ ε∂Iωn for n /∈ N ω

ε (U). Theabove thus becomes

Iωε = −∑

E∈E

∑

n∈Nωε (U)

ˆ

ε∂Iωn

(∇E v)uωε · σ

(

εψωE(

·ε) + Ex,Σω

E(·ε))

ν.

For n ∈ N ωε (U), since uωε is a rigid motion in εIωn , the boundary conditions for the correc-

tor ψωE on ∂Iωn ensure that

ˆ

ε∂Iωn

uωε · σ(

εψωE(

·ε) + Ex,Σω

E(·ε))

ν = 0,

which allows to reformulate Iωε as

Iωε = −∑

E∈E

∑

n∈Nωε (U)

ˆ

ε∂Iωn

(

∇E v −∇E v(εxωn))

uωε · σ(

εψωE(

·ε) + Ex,Σω

E(·ε))

ν. (3.16)

For n ∈ N ωε (U), since uωε is affine in εIωn , we can write on ε∂Iωn ,

uωε =(

εIωn

uωε

)

+ (x− εxωn)i∇iuωε ,

so that

∣

∣

∣

(

∇E v −∇E v(εxωn))

uωε − (x− εxωn)i

(

εIωn

uωε ∇i∇E v)∣

∣

∣

. ε2(|uωε |+ |∇uωε |)(

‖∇3v‖L∞(U) + ‖∇2v‖L∞(U)

)

. (3.17)


Next, appealing to the estimates (3.6), (3.7), and (3.9) of Step 3, we obtain

∑

n∈Nωε (U)

ε2ˆ

ε∂Iωn

∣

∣σ(

εψωE(

·ε) + Ex,Σω

E(·ε))∣

∣

(

|uωε |+ |∇uωε |)

. ε

(

∑

n∈Nωε (U)

ε

ˆ

ε∂Iωn

|∇ψωE(

·ε) + E|2 + |Σω

E(·ε)|

2

)1

2(

∑

n∈Nωε (U)

ε

ˆ

ε∂Iωn

|uωε |2 + |∇uωε |

2

)1

2

.δ ε ‖f‖L2(U)

(

ˆ

U|E|2 + |∇ψω

E(·ε)|

2 + |(ΣωE1Rd\Iω)( ·ε)|

2)

1

2

. (3.18)

Inserting (3.17) into (3.15) and (3.16), and using the above to estimate the errors togetherwith the boundedness statement of Proposition 2.1(iii), we are led to

Lω = limε↓0

Iωε = − limε↓0

∑

E∈E

∑

n∈Nωε (U)

(

εIωn

uωε ∇i∇E v)

·(

ˆ

ε∂Iωn

(x− εxωn)i σ(

εψωE(

·ε) + Ex,Σω

E(·ε))

ν)

.

Recalling that for ε small enough the test function v vanishes on εIωn for n /∈ N ωε (U), we

can rewrite

Lω = limε↓0

∑

E∈E

ˆ

U(uωε ⊗∇∇E v) : Z

ωE(

·ε),

in terms of the (matrix-valued) stationary field ZE defined in (3.14). Since ZE is stationaryand bounded in L2(Ω), the ergodic theorem ensures Zω

E(·ε) E [ZE] in L2(U) for typical ω.

Combining this with the strong convergence uωε → uω in L2(U), the claim (3.13) follows.

Step 6. Identification of E [ZE]: for all E ∈ Msym0 ,

E [ZE] = 2(Id−B)E − (b : E) Id, (3.19)

where B and b are defined in (1.4) and (1.5).

Let E ∈ Msym0 be fixed. First note that for any skew-symmetric matrix E′ ∈ M

skew thedefinition of ZE and the boundary conditions for ψE entail E′ : ZE = 0 almost surely. Alsonote that the definition (1.5) of b takes the form b : E = −1

dE [trZE ]. It then suffices toprove (3.19) when testing with symmetric trace-free matrices, that is, for all E′ ∈ M

sym0 ,

E′ : E [ZE] = E′ : 2(Id−B)E. (3.20)

Let E′ ∈ Msym0 be fixed. For η > 0, choose a cut-off function χη ∈ C∞

c (B) with 0 ≤ χη ≤ 1pointwise, with χη = 1 on B1−η, and with |∇χη| .

1η . For 0 < ε < 1

4η, in view of the

hardcore condition, we can construct a modification χωε,η ∈ C∞

c (B) of χη that satisfiesthe same properties as χη, such that in addition χω

ε,η is constant in each inclusion of thecollection εIωn n, vanishes in inclusions εIωn with n /∈ Nε(B), and such that χω

ε,η → χη inL∞(B) as ε ↓ 0. The ergodic theorem yields for almost all ω,

E′ : E [ZE ] = limη↓0

limε↓0

Bχωε,η(E

′ : ZωE)(

·ε).


Injecting the definition (3.14) of ZE yields

E′ : E [ZE ]

= −1

|B|limη↓0

limε↓0

∑

n∈Nωε (B)

(

εIωn

χωε,η

)(

ˆ

ε∂Iωn

E′(x− εxωn) · σ(

εψωE(

·ε) + Ex,Σω

E(·ε))

ν)

.

Since for all n the corrector ψωE′ has the form κωn +Θω

n(x−xωn)−E

′(x−xωn) on Iωn for some

κωn ∈ Rd and Θω

n ∈ Mskew, the boundary conditions for ψω

E on ∂Iωn allow to rewrite

E′ : E [ZE ] =1

|B|limη↓0

limε↓0

∑

n∈Nε(B)

(

εIωn

χωε,η

)(

ε

ˆ

ε∂Iωn

ψωE′( ·ε) · σ

(

εψωE(

·ε) + Ex,Σω

E(·ε))

ν)

.

Since χωε,η is constant in each inclusion and vanishes in inclusions εIωn with n /∈ Nε(B), this

is equivalently written as

E′ : E [ZE ] =1

|B|limη↓0

limε↓0

∑

n

ε

ˆ

ε∂Iωn

χωε,η ψ

ωE′( ·ε) · σ

(

εψωE(

·ε) + Ex,Σω

E(·ε))

ν.

Using equation (3.2) for the corrector ψωE together with divψω

E′ = 0, in form of

∑

n

ε

ˆ

ε∂Iωn

χωε,η ψ

ωE′( ·ε) · σ

(

εψωE(

·ε) + Ex,Σω

E(·ε))

ν

= −

ˆ

Bχωε,η ∇ψ

ωE′( ·ε) : ∇ψ

ωE(

·ε)−

ˆ

Bεψω

E′( ·ε)⊗∇χωε,η : (∇ψω

E − ΣωE Id1Rd\Iω)( ·ε),

and noting that the second right-hand side term converges to 0 as ε ↓ 0 in view of thesublinearity of ψE′ (cf. Proposition 2.1(iv)) and in view of the boundedness statement ofProposition 2.1(iii), we deduce

E′ : E [ZE ] = − limη↓0

limε↓0

Bχωε,η ∇ψ

ωE′( ·ε) : ∇ψ

ωE(

·ε).

Equivalently, again integrating by parts and using that divψωE′ = 0, we have

E′ : E [ZE ] = − limη↓0

limε↓0

2

Bχωε,η D(ψω

E′)( ·ε ) : D(ψωE)(

·ε).

In view of the ergodic theorem and of the definition (1.4) of B, this yields the claim (3.20)in form of

E′ : E [ZE ] = −2E [D(ψE′) : D(ψE)] = E′ : 2(Id−B)E. (3.21)

Step 7. Conclusion: convergence result.Combining the results of Steps 4–6, we conclude that for almost all ω there holds uωε uω

weakly in H10 (U) as ε ↓ 0 along a subsequence, where the limit uω satisfies div uω = 0 and,

for all v ∈ C∞c (U)d with div v = 0,

ˆ

UD(v) : 2BD(uω) = (1− λ)

ˆ

Uv · f,

where we recall that B is defined in (1.4). Note that B is positive definite on Msym0 : by

linearity of the corrector E 7→ ψE with E [∇ψE ] = 0, we compute for all E ∈ Msym0 ,

E : BE = |E|2 + E[

|∇ψE|2]

≥ |E|2.


Hence, uω ∈ H10 (U) is a weak solution of the following well-posed steady Stokes equation

in U ,

− div 2BD(uω) +∇Pω = (1− λ)f, div uω = 0,

in the sense of [15, Definition IV.1.1]. In addition, by [15, Lemma IV.1.1], there exists aunique pressure field Pω ∈ L2(U) with

´

U Pω = 0 such that this equation holds in the

usual weak sense. By uniqueness for the above problem (e.g. [15, Theorem IV.1.1]), thesolution (uω, Pω) = (u, P ) is independent of ω and the whole sequence converges. Theconvergence for the pressure field follows from the corrector result below combined withan approximation argument, cf. Substep 8.4.

Step 8. Corrector results.We finally turn to the additional corrector results, which we obtain by a suitable recyclingof the above computations. We consider the following two-scale expansion errors,

wωε := uωε − u− ε

∑

E∈E

ψωE(

·ε)∇E u,

Qωε := Pω

ε 1U\Iωε (U) − P − b : D(u)− (Σω

E1Rd\εIω)( ·ε )∇Eu,

and we split the proof into four further substeps: we start with the short proof of thecorrector result for the velocity field, that is, wω

ε → 0 in H1(U), based on the convergenceof the energy, and then we establish a suitable equation for wω

ε , from which we deducea bound on the pressure Qω

ε and the corresponding corrector result. In the first threesubsteps, we assume for simplicity that the homogenized solution u belongs to W 3,∞(U)d,an assumption that we relax in the last substep.

Substep 8.1. Corrector result for the velocity field.First, combining (3.5) with the strong convergence uωε → u in L2(U) yields for almost all ωthe convergence of energies in the form

2

ˆ

U|D(uωε )|

2 =

ˆ

U|∇uωε |

2 =

ˆ

U\Iωε (U)

f · uωε → (1− λ)

ˆ

Uf · u = 2

ˆ

UD(u) : BD(u).

Second, using the constraint tr∇u = div u = 0 in the form D(u) =∑

E∈E(∇E u)E, andappealing to the stationarity of ∇ψE, the ergodic theorem, and the sublinearity of ψE

(cf. Proposition 2.1(iv)), together with the additional regularity of u, we find for almostall ω,

ˆ

U

∣

∣

∣D(

u+ε∑

E∈E

ψωE(

·ε)∇E u

)∣

∣

∣

2=

ˆ

U

∣

∣

∣

∑

E∈E

(D(ψωE)+E)( ·ε )∇Eu+

∑

E∈E

(

εψωE(

·ε)⊗∇∇E u

)s∣

∣

∣

2

→

ˆ

UD(u) : BD(u).


Third, choosing v = u as a test function, the computations of Steps 5–6 together with theregularity of u precisely yield for almost all ω,

ˆ

UD(

u+ ε∑

E∈E

ψωE(

·ε)∇E u

)

: D(uωε )

=∑

E∈E

ˆ

U(∇E u) (D(ψω

E) + E)( ·ε ) : D(uωε ) +∑

E∈E

ˆ

Uεψω

E(·ε)⊗∇∇Eu : D(uωε )

→

ˆ

UD(u) : BD(u).

Combining the above and reconstructing the square lead to the stated corrector result forthe velocity field.

Substep 8.2. Equation for the two-scale expansion error.We claim that (wω

ε , Qωε ) satisfies in the weak sense in U ,

−wωε +∇Qω

ε = −∑

n∈Nωε (U)

δε∂Iωn σ(uωε , P

ωε )ν

− div 2(B − Id)D(u)−∇(b : D(u)) +∑

E∈E

∇Eu∑

n

δε∂Iωn σ(

εψωE(

·ε) + Ex,Σω

E(·ε))

ν

+ (λ− 1Iωε (U))f −

∑

E∈E

(ΣωE1Rd\εIω)( ·ε)∇∇Eu+ div

(

∑

E∈E

εψωE(

·ε)⊗∇∇Eu

)

. (3.22)

Combining equations (3.1) and (3.2) indeed yields

−wωε +∇Qω

ε = u−∇P −∇(b : D(u)) + f1U\Iωε (U) −

∑

n∈Nωε (U)

δε∂Iωn σ(uωε , P

ωε )ν

+∑

E∈E

∇E u∑

n

δε∂Iωn σ(

εψωE(

·ε) + Ex,Σω

E(·ε))

ν

−∑

E∈E

(ΣωE1Rd\Iω)( ·ε)∇∇Eu+

∑

E∈E

∇ψωE(

·ε)∇∇Eu+ ε

∑

E∈E

ψωE(

·ε)∇E u,

and the claim follows after inserting the equation for u and recombining the last tworight-hand side terms.

Substep 8.3. Corrector result for the pressure field.As in Substep 4.2 of the proof of Proposition 2.1, for almost all ω, we can construct a mapζωε ∈ H1

0 (U) such that ζωε |εIωn is constant for all n ∈ N ωε (U) and such that

div ζωε =(

Qωε −

U\Iωε (U)

Qωε

)

1U\Iωε (U), (3.23)

‖∇ζωε ‖L2(U) .∥

∥

∥Qω

ε −

U\Iωε (U)

Qωε

∥

∥

∥

L2(U\Iωε (U))

. (3.24)


Testing equation (3.22) with ζωε , using the boundary conditions for uωε at inclusion bound-aries, and recalling that ζωε is constant on each inclusion εIωn with n ∈ N ω

ε (U), we findˆ

U∇ζωε : ∇wω

ε −

ˆ

U(div ζωε )Q

ωε =

ˆ

UD(ζωε ) : 2(B − Id)D(u) +

ˆ

U(div ζωε ) b : D(u)

+∑

E∈E

∑

n

ˆ

ε∂Iωn

(∇E u) ζωε · σ

(

εψωE(

·ε) + Ex,Σω

E(·ε))

ν

+

ˆ

U(λ−1Iω

ε (U)) ζωε ·f−

∑

E∈E

ˆ

U(Σω

E1Rd\Iω)( ·ε) ζωε ·∇∇E u−

ˆ

U∇ζωε :

∑

E∈E

εψωE(

·ε)⊗∇∇E u.

In view of properties (3.23) and (3.24) of the test function ζωε , we deduce after reorganizingthe terms,

ˆ

U\Iωε (U)

(

Qωε −

U\Iωε (U)

Qωε

)2.

5∑

j=1

Tωε,j, (3.25)

where

Tωε,1 :=

ˆ

U|∇wω

ε |2,

Tωε,2 :=

∣

∣

∣

ˆ

U(λ− 1Iω

ε (U)) ζωε · f

∣

∣

∣+

∑

E∈E

∣

∣

∣

ˆ

U(Σω

E1Rd\εIω)( ·ε) ζωε · ∇∇Eu

∣

∣

∣,

Tωε,3 :=

∑

E∈E

ˆ

Uε|ψω

E(·ε)||∇ζ

ωε ||∇∇E u|,

Tωε,4 :=

∑

E∈E

∑

n/∈Nωε (U)

ˆ

ε∂Iωn

|∇E u| |ζωε |∣

∣σ(

εψωE(

·ε) + Ex,Σω

E(·ε))∣

∣,

Tωε,5 :=

∣

∣

∣

∣

ˆ

UD(ζωε ) : 2(B − Id)D(u) +

ˆ

U(div ζωε ) b : D(u)

+∑

E∈E

∑

n∈Nωε (U)

ˆ

ε∂Iωn

(∇Eu) ζωε · σ

(

εψωE(

·ε) + Ex,Σω

E(·ε))

ν

∣

∣

∣

∣

.

We successively estimate these different terms. First, the corrector result for the velocityfield in Step 8.1 yields Tω

ε,1 → 0 for almost all ω. We turn to the second term Tωε,2. In

view of (3.3) and of the boundedness statement of Proposition 2.1(iii), with the regularityof u, we deduce that for almost all ω the pressure Qω

ε is bounded in L2(U) uniformlyin ε, hence in view of (3.24) the test function ζωε is bounded in H1

0 (U). For almost all ω,by weak compactness, there exists ζω ∈ H1

0 (U)d such that ζωε ζω in H10 (U) along

some subsequence (not relabelled), hence also ζωε → ζω in L2(U) by Rellich’s theorem.Combining this strong convergence with the ergodic theorem in form of (Σω

E1Rd\εIω)( ·ε)

0 in L2(U) (cf. Proposition 2.1(iii)) and in form of 1Iωε (U) λ1U weakly-* in L∞(U),

together with the regularity of u, we deduce Tωε,2 → 0 for almost all ω. Similarly, in view

of the sublinearity of ψE (cf. Proposition 2.1(ii)), we find Tωε,3 → 0.

We turn to the boundary term Tωε,4. For n /∈ N ω

ε (U) with εIωn ∩ U 6= ∅, since ε∂Iωn is atdistance at most ε from ∂U , on which ζωε vanishes, we deduce from a trace estimate,

‖ζωε ‖L2(ε∂Iωn∩U) . ε1

2‖∇ζωε ‖L2(ε(Iωn+B)∩U),


hence by Cauchy-Schwarz’ inequality,

Tωε,4 . ‖∇u‖L∞(U)‖∇ζ

ωε ‖L2(U)

∑

E∈E

(

∑

n/∈Nωε (U)

ε

ˆ

ε∂Iωn

∣

∣σ(

εψE(·ε) + Ex,Σω

E(·ε))∣

∣

2)

1

2

.

As in the proof of (3.9), we appeal to a trace estimate and to the regularity theory for theStokes equation near a boundary in the form

ˆ

∂Iωn

|∇ψωE |

2 + |ΣωE|

2 . ‖(∇ψωE ,Σ

ωE)‖

2L2(Iωn+ δ

2B),

so that the above yields

Tωε,4 . ‖∇u‖L∞(U)‖∇ζ

ωε ‖L2(U)

∑

E∈E

(

ˆ

(∂U)+3εB|E|2 + |∇ψω

E(·ε)|

2 + |(ΣωE1Rd\Iω)( ·ε)|

2)

1

2

,

where the right-hand side converges to 0 for almost all ω as a consequence of the ergodictheorem, cf. Proposition 2.1(iii).

It remains to estimate Tωε,5, and we use the short-hand notation

Jωε := −

∑

E∈E

∑

n∈Nε(U)

ˆ

ε∂Iωn

(∇E u) ζωε · σ

(

εψωE(

·ε) + Ex,Σω

E(·ε))

ν.

As shown in Step 5, in view of the boundary conditions for ψE at the inclusion boundaries,together with the regularity of u, an approximation argument for ∇Eu leads to

limε↓0

∣

∣

∣

∣

Jωε +

∑

E∈E

∑

n∈Nωε (U)

(

εIωn

ζωε ∇i∇E u)

·(

ˆ

ε∂Iωn

(x− εxωn)i σ(

εψωE(

·ε) + Ex,Σω

E(·ε))

ν)

∣

∣

∣

∣

= 0. (3.26)

Applying Cauchy-Schwarz’ inequality in the form

∑

n/∈Nωε (U)

∣

∣

∣

∣

(

εIωn

ζωε ∇i∇Eu)

·(

ˆ

ε∂Iωn

(x− εxωn)i σ(

εψωE(

·ε) +Ex,Σω

E(·ε))

ν)

∣

∣

∣

∣

. ‖∇2u‖L∞(U)‖ζωε ‖L2(U)

(

∑

n/∈Nωε (U)

ε

ˆ

ε∂Iωn

∣

∣σ(

εψωE(

·ε) + Ex,Σω

E(·ε))∣

∣

2)

1

2

,

and noting that the estimate on Tωε,4 above ensures that the right-hand side converges to 0,

we deduce that the restriction to n ∈ N ωε (U) can be removed from the sum in (3.26). In

terms of the random field ZE defined in (3.14), we are thus led to

limε↓0

∣

∣

∣Jωε −

∑

E∈E

ˆ

U(ζωε ⊗∇∇E u) : Z

ωE(

·ε)∣

∣

∣= 0.

Appealing to the ergodic theorem for ZE , to the identification of E [ZE ] in Step 6, and tothe strong convergence ζωε → ζω in L2(U), together with the regularity of u, we deduce

limε↓0

∣

∣

∣Jωε −

ˆ

UD(ζω) : 2(B − Id)D(u)−

ˆ

U(div ζω) b : D(u)

∣

∣

∣= 0,

that is, Tωε,5 → 0. We conclude that the whole right-hand side in (3.25) converges to 0 for

almost all ω, and the corrector result follows.


Substep 8.4. Relaxing the regularity assumption.Assume that f ∈ Lp(U) for some p > d and note that in view of the regularity theory for the

homogenized Stokes equation (1.3) in form of [15, Lemma IV.6.1] this implies u ∈W 2,p0 (U)d

and P ∈ W 1,p(U). Choosing an approximating sequence (f r)r ⊂ C∞b (U) with f r → f in

Lp(U) as r ↓ 0, we deduce by linearity that the corresponding solution (ur, P r) of thehomogenized equation satisfies ur → u in W 2,p(U), hence ur → u in W 1,∞ ∩W 2,d(U) bythe Sobolev embedding. In addition, in view of the energy estimate (3.3), the correspondingsolution (ur,ωε , P r,ω

ε ) of (1.2) satisfies

supε>0

ˆ

U|∇(ur,ωε − uωε )|

2 + supε>0

ˆ

U\Iωε (U)

|P r,ωε − Pω

ε |2 .

ˆ

U|f r − f |2 → 0, (3.27)

as r ↓ 0. Since for fixed r > 0 the approximation f r is smooth, the regularity theory forthe homogenized Stokes equation ensures that ur belongs at least to W 3,∞(U)d, hence theabove Steps 8.1–8.3 show that the corrector results indeed hold for the r-approximations.Since ∇ψω

E(·ε) and (Σω

E1Rd\Iω)( ·ε) are bounded in L2(U) for almost all ω (cf. Proposi-

tion 2.1(iii)), and since the Sobolev embedding also ensures the boundedness of εψωE(

·ε) in

L2d/(d−2)(U), the above convergences precisely allow to get rid of approximations.

We conclude with the argument for the weak convergence of the pressure for f ∈ L2(U).Choose an approximating sequence (f r)r ⊂ C∞

b (U) with f r → f in L2(U), and denote by(ur,ωε , P r,ω

ε ) and by (ur, P r) the corresponding solutions of (1.2) and of (1.3). Starting fromthe corrector result for the pressure for the regularized data f r, appealing to the ergodictheorem of Proposition 2.1(iii), and noting that the weak-* convergence 1U\Iω

ε (U) 1− λ

in L∞(U) entails´

U (Pr + b : D(ur))1U\Iω

ε (U) → (1− λ)´

U (Pr + b : D(ur)) = 0, we obtain

for all r, for almost all ω,

(P r,ωε − P r − b : D(ur))1U\Iω

ε (U) 0, weakly in L2(U), as ε ↓ 0.

Next, the same argument as above yields (ur, P r) → (u, P ) in H10 (U) × L2(U), as well

as (3.27), which allows to get rid of approximations in this weak convergence result.

Acknowledgements

MD acknowledges financial support from the CNRS-Momentum program, and AG fromthe European Research Council under the European Community’s Seventh FrameworkProgramme (FP7/2014-2019 Grant Agreement quanthom 335410).

References

[1] G. Allaire. Homogenization of the Stokes flow in a connected porous medium. Asymptotic Anal.,2(3):203–222, 1989.

[2] G. Allaire. Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes.I. Abstract framework, a volume distribution of holes. Arch. Rational Mech. Anal., 113(3):209–259,1990.

[3] Y. Almog and H. Brenner. Global homogenization of a dilute suspension of sphere. Unpublishedmanuscript, 1998.

[4] H. Ammari, P. Garapon, H. Kang, and H. Lee. Effective viscosity properties of dilute suspensions ofarbitrarily shaped particles. Asymptot. Anal., 80(3-4):189–211, 2012.

[5] A. Basson and D. Gérard-Varet. Wall laws for fluid flows at a boundary with random roughness.Comm. Pure Appl. Math., 61(7):941–987, 2008.

[6] G. K. Batchelor. Sedimentation in a dilute dispersion of spheres. J. Fluids Mech., 52(2):245–268, 1972.


[7] G. L. Batchelor and J. T. Green. The determination of the bulk stress in suspension of sphericalparticles to order c2. J. Fluid Mech., 56:401–427, 1972.

[8] A. Yu. Beliaev and S. M. Kozlov. Darcy equation for random porous media. Comm. Pure Appl. Math.,49(1):1–34, 1996.

[9] D. Cioranescu and J. Saint Jean Paulin. Homogenization in open sets with holes. J. Math. Anal. Appl.,71(2):590–607, 1979.

[10] M. Duerinckx and A. Gloria. Analyticity of homogenized coefficients under Bernoulli perturbationsand the Clausius-Mossotti formulas. Arch. Ration. Mech. Anal., 220(1):297–361, 2016.

[11] M. Duerinckx and A. Gloria. Quantitative homogenization theory for suspensions in steady Stokesflow. In preparation.

[12] M. Duerinckx and A. Gloria. Sedimentation of random suspensions and the effect of hyperuniformity.Preprint, arXiv:2004.03240.

[13] M. Duerinckx and A. Gloria. On Einstein’s effective viscosity formula. Preprint, 2020.[14] A. Einstein. Eine neue Bestimmung der Moleküldimensionen. Ann. Phys., 19(2):289–306, 1906.[15] G. P. Galdi. An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state

problems. Springer Monographs in Mathematics. Springer, New York, second edition, 2011.[16] D. Gérard-Varet and M. Hillairet. Analysis of the viscosity of dilute suspensions beyond Einstein’s

formula. Preprint, arXiv:1905.08208.[17] D. Gérard-Varet and A. Mecherbet. On the correction to Einstein’s formula for the effective viscosity.

Preprint, arXiv:2004.05601.[18] A. Giunti and R. M. Höfer. Homogenization for the Stokes equations in randomly perforated domains

under almost minimal assumptions on the size of the holes. Preprint, arXiv:1809.04491.[19] A. Gloria. A scalar version of the Caflisch-Luke paradox. Preprint, arXiv:1907.08182.[20] B. M. Haines and A. L. Mazzucato. A proof of Einstein’s effective viscosity for a dilute suspension of

spheres. SIAM J. Math. Anal., 44(3):2120–2145, 2012.[21] V. V. Jikov. Averaging of functionals in the calculus of variations and elasticity, Math. USSR, Izvestiya,

29:33–66, 1987.[22] V. V. Jikov. Some problems of extension of functions arising in connection with the homogenization

theory, Diff. Uravnenia, 26(1):39–51, 1990.[23] V. V. Jikov, S. M. Kozlov, and O. A. Oleınik. Homogenization of differential operators and integral

functionals. Springer-Verlag, Berlin, 1994.[24] B. Niethammer and R. Schubert. A local version of Einstein’s formula for the effective viscosity of

suspensions. Preprint, arXiv:1903.08554.[25] K. Nunan and J. Keller. Effective viscosity of a periodic suspension. J. Fluid Mech., 142:269–287,

1984.[26] G. C. Papanicolaou and S. R. S. Varadhan. Boundary value problems with rapidly oscillating random

coefficients. In Random fields, Vol. I, II (Esztergom, 1979), volume 27 of Colloq. Math. Soc. JánosBolyai, pages 835–873. North-Holland, Amsterdam, 1981.

[27] E. Sánchez-Palencia. Nonhomogeneous media and vibration theory, volume 127 of Lecture Notes inPhysics. Springer-Verlag, Berlin-New York, 1980.

[28] L. Tartar. Incompressible fluid flow in a porous medium: convergence of the homogenization process.In Nonhomogeneous media and vibration theory (E. Sánchez-Palencia), volume 127 of Lecture Notesin Physics, pages 368–377. Springer-Verlag, Berlin-New York, 1980.

[29] L. Tartar. The general theory of homogenization, volume 7 of Lecture Notes of the Unione MatematicaItaliana. Springer-Verlag, Berlin; UMI, Bologna, 2009.

(Mitia Duerinckx) Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay,

91405 Orsay, France & Université Libre de Bruxelles, Département de Mathématique, 1050

Brussels, Belgium

E-mail address: [email protected]

(Antoine Gloria) Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-

Louis Lions, 75005 Paris, France & Université Libre de Bruxelles, Département de Mathé-

matique, 1050 Brussels, Belgium

E-mail address: [email protected]

arxiv:1909.09625v2 [math.ap] 7 aug 2020abstract. consider a colloidal suspension of rigid particles...

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