Sedimentation of random suspensions and the effect of hyperuniformity
aa r X i v : . [ m a t h . A P ] A p r SEDIMENTATION OF RANDOM SUSPENSIONSAND THE EFFECT OF HYPERUNIFORMITY
MITIA DUERINCKX AND ANTOINE GLORIA
Abstract.
This work is concerned with the mathematical analysis of the bulk rheologyof random suspensions of rigid particles settling under gravity in viscous fluids. Each par-ticle generates a fluid flow that in turn acts on other particles and hinders their settling.In an equilibrium perspective, for a given ensemble of particle positions, we analyze boththe associated mean settling speed and the velocity fluctuations of individual particles.In the 1970s, Batchelor gave a proper definition of the mean settling speed, a 60-year-oldopen problem in physics, based on the appropriate renormalization of long-range particlecontributions. In the 1980s, a celebrated formal calculation by Caflisch and Luke sug-gested that velocity fluctuations in dimension d = 3 should diverge with the size of thesedimentation tank, contradicting both intuition and experimental observations. Therole of long-range self-organization of suspended particles in form of hyperuniformitywas put forward later on to explain additional screening of this divergence in steady-state observations. In the present contribution, we develop the first rigorous theory thatallows to justify all these formal calculations of the physics literature. The main diffi-culty is to account for the nonlinear multibody hydrodynamic interactions in a contextwhen stochastic cancellations are crucial, and to show how the improvement providedby hyperuniformity on the linearized problem is not destroyed by the nonlinearity (asit would be in stochastic homogenization of linear elliptic equations, for instance). Onthe one hand, in order to analyze stochastic cancellations in this nonlinear setting, weintroduce a suitable functional-analytic version of hyperuniformity, which is of indepen-dent interest. On the other hand, we appeal to a new annealed regularity theory for thesteady Stokes equation describing the viscous fluid in presence of a random suspension.While all previous works build on deterministic approaches like the reflection methodor other perturbative ideas to analyze interactions in dilute regimes, we rather appealto a non-perturbative approach that takes advantage of randomness and yields optimalregularity properties upon averaging over the ensemble of particle positions. This is in-spired by recent achievements in the quantitative theory of stochastic homogenizationof divergence-form linear elliptic equations. As a corollary, we establish a homogeniza-tion result for a steady Stokes fluid in a finite tank with a dense suspension of smallsedimenting particles, and we define the associated effective viscosity. Contents
1. Introduction 22. Main results 83. Mean settling speed 194. Velocity fluctuations 225. Homogenization result 42Appendix A. Functional-analytic version of hyperuniformity 47Appendix B. Annealed regularity theory 51Acknowledgements 56References 56 M. DUERINCKX AND A. GLORIA Introduction
General overview.
The present article constitutes the first rigorous analysis of thebulk sedimentation of a random suspension of rigid particles in a Stokes fluid in a largetank. We place ourselves in a quasistatic setting where inertial forces are neglected. Givenparticle positions { x n } n in an (experimental) tank, the corresponding instantaneous ve-locities { V n } n are obtained by solving the steady Stokes equation with suitable conditionsat the particle boundaries, cf. Section 1.3. In an equilibrium perspective, we assume thatparticle positions { x n } n are distributed according to a known random ensemble and weanalyze the corresponding random ensemble of velocities { V n } n . We specifically focus onthe mean settling speed E [ V n ] and on velocity fluctuations Var [ V n ] . Under suitable mix-ing assumptions, we recover all the predictions of the physics literature, cf. Sections 1.2and 1.4, and in particular we correctly capture the effect of the long-range self-organizationof particles in form of hyperuniformity. While these predictions [70, 8, 14, 43] are basedon a simplified linear pairwise analysis, cf. Section 2.6, the key difficulty to treat the origi-nal model stems from the multibody and nonlinear nature of hydrodynamic interactions incombination with the crucial analysis of stochastic cancellations. The mathematical meritsof this contribution are mainly threefold: • Sensitivity calculus:
In the spirit of Malliavin calculus, we analyze the sensitivity ofthe fluid velocity with respect to the random ensemble of particle positions. More pre-cisely, local changes in the latter are shown to generate two effects on the fluid velocity:a linear response (similar to the simplified linear pairwise analysis used in the physicsliterature, cf. Section 2.6) and a nonlinear response (related to hydrodynamic interac-tions). Our analysis establishes a difference of locality between these two effects: thelinear response is less local by one length scale, and therefore dominates the nonlin-ear response. This suggests that assumptions such as hyperuniformity that improvethe scalings in the linear analysis should yield a similar improvement for the originalnonlinear model. This is quite surprising as it would not be true in the more classicalsetting of stochastic homogenization of linear elliptic equations, cf. Section 2.7. • Hyperuniformity:
In order to cope with the nonlinearity of hydrodynamic interactionsand rigorously exploit the above sensitivity calculus, we appeal to functional inequal-ities on the probability space as a convenient form of nonlinear mixing condition.While hyperuniformity expresses the suppression of large-scale density fluctuationsand is usually defined in terms of the pair correlation function of the random pointprocess, cf. Appendix A, we need to devise functional inequalities that are compatiblewith hyperuniformity. For that purpose, in line with the multiscale variance inequali-ties that we introduced in [19, 20] for point processes, we introduce here a new familyof hyperuniform variance inequalities , cf. assumption (Hyp + ) below. We believe thatthese functional inequalities, which provide a versatile tool for the rigorous analysis ofnonlinear hyperuniform systems, are of independent interest. • Annealed regularity:
Due to the nonlinearity of hydrodynamic interactions, the anal-ysis critically requires fine regularity results on the steady Stokes equation with arandom suspension. Although deterministic regularity results could be used, the lat-ter always require a large enough minimal interparticle distance — which is physicallyan unsatisfying restriction —, whether they are based on the reflection method [40,38, 53, 39] or on other perturbative ideas [29], cf. Section 1.5. In contrast, in thiswork, we take advantage of randomness and rely on a new family of non-perturbative
EDIMENTATION OF RANDOM SUSPENSIONS annealed regularity results, cf. Section 4.2.2, to which the companion article [18] isdedicated. Using such refined analytical tools inspired by the quantitative theory ofstochastic homogenization of divergence-form linear elliptic equations, we fully takeinto account hydrodynamic interactions for the first time in a non-dilute regime.We start this introduction by reviewing the physics literature. Next, we introduce theprecise Stokes system that we study in the present work, we give an informal statementof our main results, and briefly discuss the relation to previous works in the mathematicsliterature. Precise assumptions and rigorous statements are postponed to Section 2.1.2. Review of the physics literature.
A linear analysis of the Stokes equation showsthat each particle in a Stokes fluid generates a long-range flow disturbance, which onlydecays as O ( r − d ) with the distance r . Any naïve summation of particle contributionswould therefore diverge as O ( L ) in a tank of size L , in particular leading to the erro-neous conclusion that the mean settling speed would not be a well-defined bulk quantityand would depend on the size and shape of the tank. This paradox remained an openproblem for 60 years since the work of Smoluchowski [71] in 1912, despite several notableattempts by Burgers [11, 12]. The correct screening of hydrodynamic interactions was firstunravelled by Batchelor [8] in the 1970s (see also the revisited approaches by Hinch [36]and Feuillebois [24]), which formally shows that the mean settling speed is well-defined indimension d > , and further establishes a dilute expansion at small volume fraction. Thedivergence is screened by a macroscopic backflow that appears as a multiparticle effect.The next step towards the rheology of sedimenting suspensions is the analysis of velocityfluctuations of individual particles, which is viewed as an intermediate step towards theunderstanding of hydrodynamic diffusion, e.g. [57]. In the 1980s, Caflisch and Luke [14]argued that for a random suspension of particles, despite Batchelor’s renormalization,velocity fluctuations should diverge linearly in the size of the tank in dimension d = 3 ,which would again contradict both intuition and steady-state experimental observations.As pointed out by Caflisch and Luke [14], this divergence is strongly related to the standardassumption that particle positions are maximally disordered (that is, uniformly distributedin the tank and independent up to volume exclusion), suggesting that long-range ordermight drastically change the conclusion.This was made precise by Hinch [37] in form of the scaling analysis of a “blob model”,where particles are assumed to be organized in large correlated regions, constituting “blobs”at some characteristic correlation scale: particles are maximally disordered at smallerscales, while density fluctuations are reduced on larger scales. For such a model, theCaflisch-Luke prediction is expected to hold only up to the correlation scale, while long-range hydrodynamic interactions would be screened on larger scales. In other words,density fluctuations are expected to drive velocity fluctuations, and the spontaneous self-organization of particles would be the key mechanism that prevents the divergence.Hinch’s scaling analysis was later refined by Koch and Shaqfeh [43]: a simple conditionon the pair correlation function of the ensemble of particle positions was put forward andformally shown to ensure the boundedness of velocity fluctuations — a mechanism similarto Debye shielding. The condition coincides with what was later coined “hyperuniformity”by Torquato and Stillinger [69] (see also [67, 27]) and characterizes the suppression ofdensity fluctuations on large scales, or equivalently the vanishing of the structure factor atvanishing wavenumber, cf. Appendix A. M. DUERINCKX AND A. GLORIA
It took some years before experiments could be devised to corroborate the above picture.Observations are as follows: when starting from a well-mixed suspension (which can be rea-sonably modeled as maximally disordered), initial fluctuations are diverging in agreementwith the Caflisch-Luke prediction, while, after some time evolution, particles reorganizeand density fluctuations are suppressed over long distances, see e.g. [56, 44, 45, 63, 49].Although Hinch’s picture and the heuristic calculation by Koch and Shaqfeh constitutea nice theory and are consistent with experimental observations, they only describe therelation between correlation length and velocity fluctuations: what is still missing is theunderstanding of the spontaneous dynamical self-organization of particles, together withthe prediction of the correct correlation length. Various dynamical mechanisms have beenproposed for the apparition of spatial correlations and screening, e.g. [43, 10]; see alsoRemark 2.3. The possible role of stratification was put forward by Luke [50], see also [54,55], but still no consensus seems to emerge, e.g. [33, 62].To conclude this discussion of the physics literature, let us emphasize that all above-mentioned predictions are based on calculations for a linearized version of the problem,which amounts to replacing multibody hydrodynamic interactions by two-body (Coulomb-type) interactions as a first-order dilute approximation, cf. Section 2.6.1.3.
Discussion of the model.
We now describe the sedimentation model under study,which allows to compute instantaneous particle velocities from their positions in a qua-sistatic perspective. We consider a tank of size L ≥ , which for simplicity we chooseas the torus T dL := ( − L , L ] d with periodic boundary conditions. The tank is filled witha (steady) Stokes fluid, together with a monodisperse collection of disjoint spherical sus-pended particles, I L := [ n I n,L , where the particle I n,L := B ( x n,L ) is the unit ball centered at x n,L and where P L := { x n,L } n is a collection of positions in the tank T dL . The total volume fraction is denoted by λ L := L − d |I L | . (1.1)The fluid flow satisfies the following steady Stokes equation outside the suspended particles,with periodic boundary conditions, − △ φ L + ∇ Π L = − α L e, div φ L = 0 , in T dL \ I L , (1.2)where the constant right-hand side − α L e accounts for the multiparticle backflow in thefluid in the opposite direction to gravity e ∈ R d , and where we have set for abbreviationthe relevant factor α L := λ L − λ L . In the present periodic setting, this backflow is imposed by the solvability condition forthe Stokes equation (1.2) together with the boundary conditions below. As described inTheorem 3, in sedimentation experiments, since no additional force is applied, this backflowterm does not appear and is in fact compensated by the pressure.No-slip boundary conditions are imposed at particle boundaries. As particles are con-strained to have rigid motions, this amounts to letting the velocity field φ L be extended EDIMENTATION OF RANDOM SUSPENSIONS inside particles, with the rigidity constraint D( φ L ) = 0 , in I L , (1.3)where D( φ L ) denotes the symmetrized gradient of φ L . In other words, this condition meansthat φ L coincides with a rigid motion V n,L + Θ n,L ( x − x n,L ) inside each particle I n,L , forsome V n,L ∈ R d and skew-symmetric matrix Θ n,L (cf. div φ L = 0 ). Next, gravity e ∈ R d appears through the force and torque balances on each particle, e | I n,L | + ˆ ∂I n,L σ ( φ L , Π L ) ν = 0 , (1.4) ˆ ∂I n,L Θ ν · σ ( φ L , Π L ) ν = 0 , for all skew-symmetric matrices Θ , (1.5)where σ ( φ L , Π L ) stands for the usual Cauchy stress tensor, σ ( φ L , Π L ) = 2 D( φ L ) − Π L Id , and where ν stands for the outward unit normal vector at particle boundaries. As φ L and Π L are only defined up to a constant, we fix them by imposing the vanishing averageconditions, ˆ T dL φ L = 0 , ˆ T dL \I L Π L = 0 . Well-posedness for (1.2)–(1.5) with φ L ∈ H ( T dL ) d and Π L ∈ L ( T dL \I L ) follows from thestandard theory for the steady Stokes equation, e.g. [25, Section IV]. In addition, regularitytheory ensures that ( φ L , Π L ) is smooth on T dL \ I L , is a classical solution of (1.2), and thatboundary conditions are satisfied in a pointwise sense.Given the positions { x n,L } n of the suspended particles { I n,L } n , the above model allows tocompute their instantaneous velocities { V n,L } n , which are given by the averaged boundaryvalues V n,L := I n,L φ L . (1.6)(Note that the rotational or skew-symmetric part Θ n,L does not contribute to the average.) Remark 1.1. (a)
Reformulation by projection:
As checked e.g. in [58, 39], the weak solution φ L of the above equations (1.2)–(1.5)can equivalently be written as φ L = − λ L π L φ ◦ L , where φ ◦ L denotes the solution of thefollowing “linear” approximation, where particle interactions are neglected, − △ φ ◦ L + ∇ Π ◦ L = (cid:0) I L − λ L (cid:1) e, div φ ◦ L = 0 , in T dL , (1.7)and where π L is the orthogonal projection in H ( T dL ) := { φ ∈ H ( T dL ) : div φ = 0 } onto the subspace { φ : D( φ ) = 0 in I L } . In other words, while φ ◦ L depends linearlyon the set I L of particles, the multibody nonlinear hydrodynamic interactions can befully encoded in this projection π L . This reformulation could slightly simplify somecalculations but will not be used in the sequel. M. DUERINCKX AND A. GLORIA (b)
Reflection method:
As introduced by Smoluchowski [70], the so-called reflection method aims at rewritingthe complicated projection π L as a cluster expansion only involving single-particle op-erators: denoting by π nL the projection in H ( T dL ) onto { φ : D( φ ) = 0 in I n,L } , andsetting q nL := 1 − π nL , the expansion takes the form π L = Id − P n q nL + P n = m q nL q mL − . . . Such an expansion appears to be very useful as single-particle operators { q nL } n are es-sentially explicit. However, as shown in [58, 39], based on deterministic arguments,convergence is only expected in the dilute regime — more precisely, for a large enoughminimal interparticle distance. For this reason, such simplifying tools are systemati-cally avoided in the sequel. ♦ Informal statement of the results.
We henceforth consider a random ensemble ofsuspended particles in form of a stationary point process P L = { x n,L } n with intensity ρ L in the periodic tank T dL , constructed on a given probability space (Ω , P ) , and we analyzeboth the corresponding mean settling speed and the fluctuations of individual velocities, ¯ V L := e | e | · E [ V n,L ] , σ L := | Var [ V n,L ] | , (1.8)in the large-volume limit L ↑ ∞ , where E [ · ] and Var [ · ] denote the expectation and thevariance with respect to P , respectively. Note that a direct computation from (1.2)–(1.5)shows that the averaged settling speed is proportional to the Dirichlet form of the fluidvelocity, cf. (3.5) below, α L | e | ¯ V L L ↑∞ ∼ E (cid:2) |∇ φ L | (cid:3) . (1.9)Our main result states that, for a mixing ensemble of particles without long-range order,the mean settling speed and velocity fluctuations are well-defined in the large-volume limitonly in dimensions d > and d > , respectively. More precisely, the following bounds areexpected to be sharp, ¯ V L ρ L | e | . d > L ) : d = 2; L : d = 1; and σ L ρ L | e | . d > L ) : d = 4; L : d = 3; (1.10)cf. Theorems 1(i) and 2(i). In particular, the boundedness of ¯ V L for d > fully justifiesBatchelor’s analysis [8, 36, 24], while the linear divergence of σ L for d = 3 provides arigorous version of the celebrated calculation by Caflisch and Luke [14] (and extends theresults of [29] to the present much more general setting, cf. Section 1.5).Next, we investigate the role of the hyperuniformity of the suspension and rigorouslyanalyze how it leads to the screening of hydrodynamic interactions. Under a suitablefunctional-analytic version of hyperuniformity, we show that the critical dimensions in (1.10)are shifted by , ¯ V L ρ L | e | . and σ L ρ L | e | . d > L ) : d = 2; L : d = 1; (1.11)cf. Theorems 1(ii) and 2(ii). This rigorously justifies the heuristic calculations in dimen-sion d = 3 by Hinch [37] and Koch and Shaqfeh [43].In addition, whenever the averaged settling speed remains bounded, we make sense ofan infinite-volume equation describing the limit of (1.2)–(1.5), cf. Theorem 1. We also EDIMENTATION OF RANDOM SUSPENSIONS deduce a homogenization result for sedimentation experiments, cf. Theorem 3: in the limitof a dense suspension of small particles (with suitably rescaled gravity) in a finite tank,the velocity field of the Stokes fluid with the suspension converges weakly to that of aneffective Stokes fluid. The latter is characterized by an effective viscosity, which is shownto be independent of gravity. In particular, this effective viscosity is the same as forthe corresponding colloidal (non-sedimenting) system that we studied in [17]. The localbehavior of the fluid is however drastically impacted by sedimentation, which is expressedin form of a corrector result.1.5. Relation to previous works.
The mathematical literature on settling suspensionsin viscous fluids is particularly scarce. Most contributions have been investigating the(quasistatic) dynamics: particle positions { x n,L } n evolve according to the set of ODEs { ˙ x n,L = V n,L } n , where instantaneous velocities { V n,L } n are computed from the steadyStokes problem (1.2)–(1.5). The first result on this topic is due to Jabin and Otto [40]and identifies the non-interacting regime, while more recent contributions by Höfer [38]and Mecherbet [53] have studied the mean-field limit. Those works are restricted to aperturbative dilute regime (the particle density tends to zero as the particle radii with somescaling relation) and they only rely on deterministic arguments in form of the reflectionmethod; see also [58, 39].The analysis of velocity fluctuations further requires to capture stochastic cancellations,for which a probabilistic input is crucially needed. This was first performed by the secondauthor in [29] for a scalar version of the sedimentation problem, and the correspondingversion of the Caflisch-Luke bound (1.10) on velocity fluctuations was succesfully estab-lished. As in the present work, the mixing assumption on the point process { x n,L } n wasconveniently formulated in form of a multiscale functional inequality as we introducedin [19, 20], and the proof borrowed ideas from quantitative stochastic homogenization fordivergence-form linear elliptic equations. As opposed to the present contribution however,the approach was based on deterministic regularity properties for (a scalar version of) theStokes equation with a suspension, in particular in form of a perturbative Green’s functionestimate, which only holds under the assumption that the minimal interparticle distance islarge enough. Diluteness was also used at several other instances, together with the scalarnature of the equation and the sphericity of the particles.The present work widely generalizes the result of [29]: • We fully relax the diluteness requirement. For that purpose, we resort to a randomversion of regularity properties in form of annealed L p regularity in the spirit of a recentwork of the first author and Otto [21] for divergence-form linear elliptic equations withrandom coefficients. In addition, the analysis of stochastic cancellations relies on anew, particularly efficient buckling argument, which exploits this annealed regularityand is inspired from [59]. This approach also allows to treat non-spherical particles. • Treating the vectorial case of the Stokes equation further requires to control the pres-sure. To this aim, we follow an approach based on local regularity [25], which weinitiated in [17] in the simpler setting of colloidal (non-sedimenting) suspensions.In addition, the present work studies for the first time the effect of hyperuniformity of therandom ensemble of positions. In line with our works on multiscale functional inequali-ties [19, 20], we introduce the notion of hyperuniform functional inequalities, which allowsus to unravel additional stochastic cancellations in sedimentation. More generally, this
M. DUERINCKX AND A. GLORIA work constitutes to our knowledge the first example of a nonlinear physical system forwhich the hyperuniformity of the input is rigorously shown to improve the scalings.
Notation. • We denote by C ≥ any constant that only depends on the space dimension d and oncontrolled constants appearing in the assumptions. We use the notation . (resp. & ) for ≤ C × (resp. ≥ C × ) up to such a multiplicative constant C . We write ≃ when both . and & hold. In an assumption, we write ≪ (resp. ≫ ) for ≤ C × (resp. ≥ C × ) with somelarge enough constant C . We add subscripts to C, . , & , ≃ , ≪ , ≫ in order to indicatedependence on other parameters. • The ball centered at x of radius r in R d is denoted by B r ( x ) , and we simply write B ( x ) := B ( x ) , B r := B r (0) , and B := B (0) . • For a function f we write [ f ] ( x ) := ( ffl B ( x ) | f | ) / for its local moving quadratic average. • We set h x i := (1 + | x | ) / for x ∈ R d , and similarly define h∇i . We denote by | x | L forthe Euclidean distance between x and on the (flat) torus T dL . We set a ∧ b := min { a, b } for all a, b ∈ R . We denote by ♯E the cardinality of a locally finite set E . • We denote by D( u ) := ( ∇ u + ( ∇ u ) ′ ) the symmetrized gradient of u , by σ ( φ, Π) :=2 D( φ ) − Π Id the Stokes stress tensor, by ν the outward unit normal vector at particleboundaries, and by M skew the set of skew-symmetric d × d matrices.2. Main results
We start with suitable assumptions on the random ensemble of suspended particles,including precise mixing and hyperuniformity assumptions, and then turn to the statementof the main results on the mean settling speed, velocity fluctuations, and homogenization.Finally, for the reader’s convenience, a heuristic proof of our results in the dilute regime isincluded in Section 2.6, justifying the scalings and illustrating the use of hyperuniformity,while in Section 2.7 we underline the useful analogy with stochastic homogenization oflinear elliptic equations.2.1.
Assumptions.
The following general assumption makes precise the notion of conver-gence for the point process P L in the periodic tank T dL in the large-volume limit L ↑ ∞ .We also always assume that the process is hardcore, with some uniform bound δ > . Assumption (H δ ) — General conditions . The family ( P L ) L ≥ of point processes is constructed on some probability space (Ω , P ) andsatisfies the following properties: • Periodicity in law:
For all L ≥ , the point process P L = { x n,L } n is defined on the L -torus T dL := ( − L , L ] d and is stationary with respect to shifts of the latter. • Stabilization:
For any compact set K ⊂ R d the restricted point set P L ∩ K con-verges almost surely as L ↑ ∞ . We denote by P the limiting point process, whichis assumed stationary (on R d ) and ergodic, and we denote by I := ∪ n I n the corre-sponding particle suspension. More precisely, as is standard in the field, e.g. [60] or [42, Section 7], stationarity is understood as fol-lows: there exists a measure-preserving group action { τ L,x } x ∈ T dL of ( T dL , +) on the probability space (Ω , P ) such that P ωL + x = P τ L,x ωL for all x, ω . EDIMENTATION OF RANDOM SUSPENSIONS • Hardcore condition:
For all L ≥ the point process P L satisfies inf m = n | x m − x n | L ≥ δ ) almost surely . ♦ Before stating mixing and hyperuniformity assumptions, we recall the following standarddefinition of the pair correlation function of the stationary random point process P L . Definition 2.1 (Pair correlation function) . The intensity of P L is defined by ρ L := E (cid:2) L − d ♯ P L (cid:3) , which is related to the volume fraction λ L of the suspension I L via E [ λ L ] = | B | ρ L , cf. (1.1) .The pair density function f ,L : T dL → R + of P L = { x n,L } n is defined via the followingrelation, for all ζ ∈ C ∞ per ( T dL × T dL ) , E (cid:20) X n = m ζ ( x n,L , x m,L ) (cid:21) = ρ L ¨ T dL × T dL ζ ( x, y ) f ,L ( x − y ) dxdy. The pair correlation function is defined as g ,L := ρ − L ( f ,L − ρ L ) . The total pair correlationfunction of P L is defined by h ,L ( x ) := g ,L ( x ) + ρ − L δ ( x ) and is characterized by thefollowing relation, for all ζ ∈ C ∞ per ( T dL ) , Var (cid:20) X n ζ ( x n,L ) (cid:21) = ρ L ¨ T dL × T dL ζ ( x ) ζ ( y ) h ,L ( x − y ) dxdy. (2.1) ♦ After an appropriate mechanical mixing of the particle suspension, the distribution ofparticle positions in the fluid typically displays fast decaying correlations and we naturallyconsider the following type of condition.
Assumption (Mix) — Mixing condition . The pair correlation function g ,L of P L is integrable in the sense of sup L ≥ ˆ T dL | g ,L | < ∞ . ♦ As discussed in Section 1.2, a sedimenting suspension at steady state is expected todisplay long-range structural order with in particular density fluctuations being suppressed,while still displaying fast decaying correlations, cf. [37, 43]. This is formalized through thenotion of hyperuniformity, which we define as follows in the periodized setting. We referto Appendix A for a detailed motivation and some reformulations.
Assumption (Hyp) — Mixing and hyperuniformity conditions . The pair correlation function g ,L of P L has fast decay in the sense of sup L ≥ ˆ T dL | x | L | g ,L ( x ) | dx < ∞ , and hyperuniformity holds in the sense that the total pair correlation h ,L satisfies sup L ≥ L (cid:12)(cid:12)(cid:12) ˆ T dL h ,L (cid:12)(cid:12)(cid:12) < ∞ , which is viewed as the approximate vanishing of the so-called structure factor at smallwavenumber, cf. Appendix A. ♦ M. DUERINCKX AND A. GLORIA
While the above assumptions (Mix) and (Hyp) are expressed in terms of the pair cor-relation function, stronger versions quickly become necessary to analyze the effects ofnonlinear multibody interactions. First, the mixing assumption (Mix) should be replacedby a stronger mixing assumption. As in [29], we choose to appeal to the following nonlin-ear assumption in form of a multiscale variance inequality , cf. [19, 20]. As shown in [20,Section 3], examples of point processes that satisfy this assumption include for instance(periodized) hardcore Poisson processes and random parking processes.
Assumption (Mix + ) — Improved mixing condition . There exists a non-increasing weight function π : R + → R + with superalgebraic decay (thatis, π ( ℓ ) ≤ C p h ℓ i − p for all p ≥ ) such that for all L ≥ the point process P L satisfies, forall σ ( P L ) -measurable random variables Y ( P L ) , Var [ Y ( P L )] ≤ E " ˆ L ˆ T dL (cid:16) ∂ osc P L ,B ℓ ( x ) Y ( P L ) (cid:17) dx h ℓ i − d π ( ℓ ) dℓ , (2.2) where the “oscillation” derivative ∂ osc is defined by ∂ osc P ,B ℓ ( x ) Y ( P ) := sup ess n Y ( P ′ ) : P ′ | T dL \ B ℓ ( x ) = P| T dL \ B ℓ ( x ) o − inf ess n Y ( P ′ ) : P ′ | T dL \ B ℓ ( x ) = P| T dL \ B ℓ ( x ) o . ♦ Similarly, we strengthen the hyperuniformity assumption (Hyp) in form of a suitablevariance inequality. Intuitively, while the “oscillation” derivative in (Mix + ) above allows tolocally add and move points, the carré-du-champ below only accounts (at leading order)for moving points, but not adding or removing any, in accordance with the definition ofhyperuniformity as suppressing density fluctuations. We refer to Appendix A for a detaileddiscussion. Assumption (Hyp + ) — Improved mixing and hyperuniformity conditions . There exists a non-increasing weight function π : R + → R + with superalgebraic decaysuch that for all L ≥ the point process P L satisfies, for all σ ( P L ) -measurable randomvariables Y ( P L ) , Var [ Y ( P L )] ≤ E (cid:20) ˆ L ˆ R d (cid:16) ∂ hyp P L ,B ℓ ( x ) Y ( P L ) (cid:17) dx h ℓ i − d π ( ℓ ) dℓ (cid:21) , (2.3) where the “hyperuniform” derivative is given by ∂ hyp P L ,B ℓ ( x ) Y ( P L ) = ∂ mov P L ,B ℓ ( x ) Y ( P L ) + L − ∂ osc P L ,B ℓ ( x ) Y ( P L ) and the “move-point” derivative ∂ mov is defined by ∂ mov P ,B ℓ ( x ) Y ( P ) := sup ess n Y ( P ′ ) : P ′ | T dL \ B ℓ ( x ) = P| T dL \ B ℓ ( x ) , ♯ P ′ | B ℓ ( x ) = ♯ P| B ℓ ( x ) o − inf ess n Y ( P ′ ) : P ′ | T dL \ B ℓ ( x ) = P| T dL \ B ℓ ( x ) , ♯ P ′ | B ℓ ( x ) = ♯ P| B ℓ ( x ) o . ♦ Mean settling speed.
Our first main result concerns the mean settling speed ¯ V L ,cf. (1.8), which is shown to be well-defined under the mixing condition (Mix) only indimension d > while under hyperuniformity (Hyp) it is well-defined in all dimensions.We also make sense of a limiting equation in the large-volume limit L ↑ ∞ . The proofis surprisingly elementary, although fully taking into account multibody hydrodynamic EDIMENTATION OF RANDOM SUSPENSIONS interactions: the argument is solely based on L theory, and the standard weak form (Hyp)of hyperuniformity is enough to unravel the screening. Theorem 1 (Mean settling speed) . Let the random point processes ( P L ) L ≥ satisfy thegeneral assumption (H δ ) for some δ > .(i) Under the mixing assumption (Mix) , there holds for all L ≥ , ¯ V L ρ L | e | . d > L ) : d = 2; L : d = 1 . More precisely, in dimension d > , for almost all ω , lim L ↑∞ ♯ P L X n e | e | · V ωn,L = lim L ↑∞ ¯ V L = ¯ V , (2.4) in terms of ¯ V := 1 α | e | E (cid:2) |∇ φ | (cid:3) , α := λ − λ , λ := E [ I ] = lim L ↑∞ λ ωL , (2.5) where the random field φ ∈ L (Ω; H ( R d ) d ) denotes the unique solution of the fol-lowing infinite-volume problem: • For almost all ω the realization φ ω ∈ H ( R d ) d satisfies in the weak sense, forsome pressure field Π ω ∈ L ( R d ) , −△ φ ω + ∇ Π ω = − αe, in R d \ I ω , div φ ω = 0 , in R d \ I ω , D( φ ω ) = 0 , in I ω ,e | I ωn | + ´ ∂I ωn σ ( φ ω , Π ω ) ν = 0 , ∀ n, ´ ∂I ωn Θ ν · σ ( φ ω , Π ω ) ν = 0 , ∀ n, ∀ Θ ∈ M skew . (2.6) • The gradient field ∇ φ and the pressure field Π are stationary, they have vanishingexpectations E (cid:2) ∇ φ (cid:3) = 0 and E (cid:2) Π R d \I (cid:3) = 0 , they have bounded second moments E (cid:2) |∇ φ | (cid:3) + E (cid:2) | Π | R d \I (cid:3) . | e | , and φ is anchored at the origin in the sense of ffl B φ ω = 0 for almost all ω .(ii) Under the mixing and hyperuniformity assumption (Hyp) , in any dimension d ≥ ,there holds for all L ≥ , ¯ V L ρ L | e | . , the limit (2.4) holds, and the infinite-volume problem (2.6) is always well-posed. ♦ Velocity fluctuations.
Our next main result concerns the estimation of velocityfluctuations, which requires a much finer use of stochastic cancellations. In this context,the analysis of the effects of nonlinear multibody interactions requires a suitable strength-ening of the standard mixing and hyperuniformity assumptions (Mix) and (Hyp), and werather appeal to their nonlinear functional-analytic versions (Mix + ) and (Hyp + ). In ad-dition, this result crucially relies on annealed regularity properties for the steady Stokesequation in presence of a random suspension, cf. Section 4.2.2. Rather than focussing onthe variance σ L , cf. (1.8), we further consider higher moments of the velocity field φ L . M. DUERINCKX AND A. GLORIA
Theorem 2 (Velocity fluctuations) . Let the random point processes ( P L ) L ≥ satisfy thegeneral assumption (H δ ) for some δ > .(i) Under the improved mixing assumption (Mix + ) , in any dimension d > , we have forall L ≥ and ≤ p < ∞ , k∇ φ L k L p (Ω) . p | e | , (2.7) and k φ L ( x ) k L p (Ω) . p | e | × , if d > | x | ) , if d = 4; h x i , if d = 3 . (2.8) In particular, in dimension d > , up to relaxing the anchoring condition, the solu-tion φ of the infinite-volume problem (2.6) can be uniquely constructed as a stationaryobject with vanishing expectation.(ii) Under the improved mixing and hyperuniformity assumption (Hyp + ) , in any dimen-sion d ≥ , we have for all L ≥ and ≤ p < ∞ , k∇ φ L k L p (Ω) . p | e | , (2.9) and k φ L ( x ) k L p (Ω) . p | e | × , if d > | x | ) , if d = 2; h x i , if d = 1 . (2.10) In particular, in dimension d > , up to relaxing the anchoring condition, the solu-tion φ of the infinite-volume problem (2.6) can be uniquely constructed as a stationaryobject with vanishing expectation. ♦ Stochastic cancellations are conveniently exploited in form of the fluctuation scalingof large-scale averages of ∇ φ L , from which the above moment bounds are deduced asconsequences. More precisely, we show that under the improved mixing assumption (Mix + )in dimension d > fluctuations of ∇ φ L miss the usual central limit theorem scaling by alength scale (as encoded by the norm of the test function in L dd +2 instead of L ): for all g ∈ C ∞ per ( T dL ) d × d and ≤ p < ∞ , we have (cid:13)(cid:13)(cid:13) ˆ T dL g : ∇ φ L (cid:13)(cid:13)(cid:13) L p (Ω) . p k [ h∇i g ] k L dd +2 ( T dL ) , (2.11)while under the improved mixing and hyperuniformity assumption (Hyp + ) in any dimen-sion d ≥ the usual central limit theorem scaling is recovered in form of (cid:13)(cid:13)(cid:13) ˆ T dL g : ∇ φ L (cid:13)(cid:13)(cid:13) L p (Ω) . p kh∇i g k L ( T dL ) . (2.12)(Note that the additional gradient h∇i in the bounds plays no role on large scales.) EDIMENTATION OF RANDOM SUSPENSIONS Homogenization result.
We consider a steady Stokes fluid in a bounded domainwith internal forces and a dense suspension of small particles: we analyze the non-dilutehomogenization regime with vanishing particle radii but fixed volume fraction λ ωε → λ > .Suspended particles in the fluid act as obstacles, hindering the fluid flow and thereforeincreasing the flow resistance, that is, the viscosity. The fluid with the suspension is thenexpected to behave approximately like a Stokes fluid with some effective viscosity — whichwas the basis of Perrin’s celebrated experiment to estimate the Avogadro number as inspiredby Einstein’s PhD thesis [23]. The upcoming theorem shows that the effective viscosityfor a sedimenting suspension exactly coincides with that for a colloidal (non-sedimenting)suspension, although the local behavior of the fluid flow is drastically different as expressedvia the corrector result . This constitutes the counterpart for sedimenting suspensions ofour recent work [17, Theorem 1] on colloidal suspensions. Strikingly, although the resultis only qualitative, it requires strong mixing conditions and quantitative estimates, inparticular relying on Theorems 1 and 2 above, as opposed to the much simpler situationof colloidal suspensions in [17]. An optimal convergence rate (which relies on a further useof annealed regularity) will be given in the companion article [18]. This homogenizationresult is the very first of its kind in the context of sedimentation. In particular, as opposedto contributions such as [34, 15, 35], where particle velocities are prescribed a priori ratherthan deduced from the steady Stokes equation, there is no Brinkman term in the effectiveequation. In addition, we consider a non-dilute regime (cf. λ ωε → λ > ), where themultiparticle effect of hydrodynamic interactions is not negligible.We start with some notation: Given a reference bounded Lipschitz domain U , we considerthe set N ωε ( U ) of all indices n such that ε ( I ωn + δB ) ⊂ U , and we define the correspondingrescaled particle suspension I ωε ( U ) in U , I ωε ( U ) := [ n ∈N ωε ( U ) εI ωn . Note that particles in this collection are at least at distance εδ from one another and fromthe boundary ∂U . Theorem 3 (Homogenization of steady Stokes flow with sedimenting suspension) . Letthe stationary random point process P be as in (H δ ) . Under the improved mixing as-sumption (Mix + ) in dimension d > , or under the improved mixing and hyperuniformityassumption (Hyp + ) in any dimension d ≥ , given a bounded Lipschitz domain U ⊂ R d ,an internal force f ∈ L ( U ) , and gravity e ∈ R d , we consider for all ε > and ω ∈ Ω theunique weak solution u ωε ∈ H ( U ) of the following steady Stokes problem, −△ u ωε + ∇ P ωε = f, in U \ I ωε ( U ) , div u ωε = 0 , in U \ I ωε ( U ) ,u ωε = 0 , on ∂U , D( u ωε ) = 0 , in I ωε ( U ) ,ε d − e | I ωn | + ´ ε∂I ωn σ ( u ωε , P ωε ) ν = 0 , ∀ n ∈ N ωε ( U ) , ´ ε∂I ωn Θ ν · σ ( u ωε , P ωε ) ν = 0 , ∀ n ∈ N ωε ( U ) , ∀ Θ ∈ M skew . (2.13) Then the following results hold, M. DUERINCKX AND A. GLORIA (i)
Homogenization:
For almost all ω , u ωε ⇀ ¯ u weakly in H ( U ) , where ¯ u ∈ H ( U ) isthe unique weak solution of the homogenized Stokes problem − div ¯ B D(¯ u ) + ∇ ¯ P = (1 − λ ) f, in U , div¯ u = 0 , in U , ¯ u = 0 , on ∂U , (2.14) where λ := E [ I ] is the intensity of the inclusion process, and the effective viscositytensor ¯ B is positive definite on trace-free matrices and is given by ¯ B := X E,E ′ ∈E ( E ′ ⊗ E ) E (cid:2) ( ∇ ψ E ′ + E ′ ) : ( ∇ ψ E + E ) (cid:3) , (2.15) where the sum runs over an orthonormal basis E of trace-free d × d matrices, andwhere the random field ψ E ∈ L (Ω; H ( R d ) d × d ) denotes the unique solution of thefollowing (infinite-volume) “colloidal corrector” problem: • For almost all ω , the realization ψ ωE ∈ H ( R d ) d satisfies in the weak sense, forsome pressure field Σ ωE ∈ L ( R d ) , −△ ψ ωE + ∇ Σ ωE = 0 , in R d \ I ω , div ψ ωE = 0 , in R d \ I ω , D (cid:0) ψ ωE + E ( x − x ωn ) (cid:1) = 0 , in I ω , ffl ∂I ωn σ (cid:0) ψ ωE + E ( x − x ωn ) , Σ ωE (cid:1) ν = 0 , ∀ n, ffl ∂I ωn Θ ν · σ (cid:0) ψ ωE + E ( x − x ωn ) , Σ ωE (cid:1) ν = 0 , ∀ n, ∀ Θ ∈ M skew . (2.16) • The gradient field ∇ ψ E and the pressure field Σ E are stationary, they have van-ishing expectations E (cid:2) ∇ ψ E (cid:3) = 0 and E (cid:2) Σ E R d \I (cid:3) = 0 , they have bounded secondmoments, and ψ E satisfies the anchoring condition ffl B ψ ωE = 0 for almost all ω .(ii) Convergence of pressure:
For almost all ω , the pressure field P ωε converges weaklyin L ( U ) up to suitable renormalization in the following sense, (cid:18) ( P ωε − ε λe · x ) − U \I ωε ( U ) ( P ωε − ε λe · x ) (cid:19) U \I ωε ( U ) ⇀ ¯ P − U ¯ P . (iii)
Corrector result:
Provided f ∈ L p ( U ) for some p > d , for almost all ω , a correctorresult holds in the following form for the velocity field u ωε , (cid:13)(cid:13)(cid:13)(cid:13) u ωε − ¯ u − ε (1 − λ ) φ ω ( · ε ) − ε X E ∈E ψ ωE ( · ε ) ∇ E ¯ u (cid:13)(cid:13)(cid:13)(cid:13) H ( U ) → , and for the pressure field P ωε , inf κ ∈ R (cid:13)(cid:13)(cid:13)(cid:13) P ωε − ε λe · x − ¯ P − (1 − λ )(Π ω R d \I ω )( · ε ) − X E ∈E (Σ ωE R d \I ω )( · ε ) ∇ E ¯ u − κ (cid:13)(cid:13)(cid:13)(cid:13) L ( U \I ωε ( U )) → , where ( φ, Π) is the (infinite-volume) “sedimentation corrector” of Theorem 1. ♦ Remark 2.2.
We briefly comment on the scaling in (2.13). In the force balance, whilegravity appears as a bulk term ε d e | I n | and the drag force as a surface term ´ ε∂I n σ ( u ε , P ε ) ν ,gravity is naturally rescaled by a factor ε so that both contributions have the same order ofmagnitude. This appears as the natural setup for sedimentation experiments: by scaling, EDIMENTATION OF RANDOM SUSPENSIONS it is equivalent to considering a fixed gravity and particles with fixed size and volumefraction in a tank of increasing size. As gravity is rescaled by a diverging factor ε , it iscompensated by a diverging backflow − ε λe generated by the pressure, which explains whythe convergence of the pressure only holds up to this corresponding renormalization. ♦ Extensions.
We mention possible relaxations of the set of general assumptions onthe suspension; details are omitted. • Polydisperse suspensions:
Spherical particles I n,L = B ( x n,L ) can be replaced by otherbounded shapes, or even by iid bounded random shapes (provided that particle bound-aries are uniformly of class C ). In the hyperuniform setting, if particles have varyingvolumes, the condition on the conservation of the number of points in the move-pointderivative has naturally to be replaced by a condition on the conservation of the totalvolume of the particles upon perturbation. • Weakened hardcore condition:
The deterministic hardcore condition in (H δ ) could berelaxed into a lower bound of the type E h x n ∈ B sup m : m = n (cid:0) | x m − x n | − (cid:1) − r i < ∞ , for some large enough power r ≥ , at the price of appealing more substantially toMeyers type estimates.2.6. Heuristic proof: the linear response.
The main difficulty of any rigorous ap-proach to sedimentation is to account for multibody nonlinear hydrodynamic interactions.In this paragraph, we briefly show how the scalings for the mean settling speed and forvelocity fluctuations can be motivated by a formal linear analysis in the dilute regime andwe explicitly emphasize the role of hyperuniformity in this simple setting. This constitutesa reformulation of the formal calculations by Batchelor [8], Caflisch and Luke [14] (seealso [29, Section 1.3]), and Koch and Shaqfeh [43]. A heuristic discussion of the nonlinearcontribution is postponed to Section 2.7.Neglecting the multibody interactions in the dilute regime λ L ≪ , the Stokes prob-lem (1.2)–(1.5) is formally reduced to φ L ≈ φ ◦ L , see also Remark 1.1(a), − △ φ ◦ L + ∇ Π ◦ L = (cid:16) X n I n,L − λ L (cid:17) e, div φ ◦ L = 0 , in T dL , (2.17)and particle velocities are approximated by V n,L ≈ V ◦ n,L := ´ I n,L φ ◦ L . For this simplifiedlinear model, in view of (1.9), the mean settling speed is explicitly given by λ L | e | ¯ V ◦ L L ↑∞ ∼ E (cid:2) |∇ φ ◦ L | (cid:3) = E (cid:20)(cid:12)(cid:12)(cid:12) X n ∇ U L ( x n,L ) (cid:12)(cid:12)(cid:12) (cid:21) = (cid:12)(cid:12)(cid:12)(cid:12) Var (cid:20)X n ∇ U L ( x n,L ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) , (2.18)in terms of the periodic (locally averaged) Stokeslet, −△ U L + ∇ P L = (cid:0) B − L − d | B | (cid:1) e, in T dL . Likewise, velocity fluctuations are formally computed as follows, ( σ ◦ L ) ≈ | Var [ φ ◦ L ] | = (cid:12)(cid:12)(cid:12)(cid:12) Var (cid:20) X n U L ( x n,L ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) . (2.19)In order to estimate (2.18) and (2.19), we distinguish between random point processes P L without or with long-range order in form of hyperuniformity. M. DUERINCKX AND A. GLORIA • Case without long-range order:
If correlations of the point process P L = { x n,L } n display an integrable decay (Mix),the following variance estimate is easily obtained, cf. (A.4), Var (cid:20) X n ζ ( x n,L ) (cid:21) . ρ L ˆ T dL | ζ | . (2.20)Recalling that the decay of the Stokeslet is given by | U L ( x ) | . | e | (1 + | x | L ) − d , |∇ U L ( x ) | . | e | (1 + | x | L ) − d , identities (2.18) and (2.19) then lead to the expected scaling, λ L | e | ¯ V ◦ L . ρ L ˆ T dL |∇ U L | . ρ L | e | , provided d > , ( σ ◦ L ) . ρ L ˆ T dL | U L | . ρ L | e | , provided d > . • Case with hyperuniformity:
Hyperuniformity is naturally expected to translate into a strong improvement of (2.20):indeed, given an independent copy { x ′ n,L } n of { x n,L } n , we may represent Var (cid:20) X n ζ ( x n,L ) (cid:21) = EE ′ (cid:20) (cid:16) X n ζ ( x n,L ) − X n ζ ( x ′ n,L ) (cid:17) (cid:21) , and the suppression of density fluctuations would formally allow to locally couple thepoint sets { x ′ n,L } n and { x n,L } n , hence only compare close points, which would ideallytranslate into the gain of a derivative: for all ζ ∈ C ∞ per ( T dL ) with ´ T dL ζ = 0 , Var (cid:20) X n ζ ( x n,L ) (cid:21) . ρ L ˆ T dL |∇ ζ | . (2.21)As shown in Lemma A.2, this functional inequality is indeed essentially equivalentto hyperuniformity together with a suitable decay of correlations. Identities (2.18)and (2.19) then lead to the expected improved scaling, λ L | e | ¯ V ◦ L . ρ L ˆ T dL |∇ U L | . ρ L | e | , for any d ≥ , ( σ ◦ L ) . ρ L ˆ T dL |∇ U L | . ρ L | e | , provided d > . Remark 2.3 (Relation to Coulomb gases) . In the above linear analysis, the Dirichletenergy ´ T dL |∇ φ L | is replaced by ˆ T dL |∇ φ ◦ L | = X n,m H ( x m,L − x n,L ) , H ( x ) := ˆ B ( x ) e · U L , which amounts to an anisotropic Coulomb type energy. The corresponding Gibbs measureis therefore expected to display hyperuniformity at small temperature [52, 48, 41] (seealso [9, 47, 64] for recent rigorous results in that direction). We do not know to whatextent this observation is relevant to sedimentation, and in particular to explain the originof hyperuniformity and the spontaneous dynamical self-organization of particles. ♦ EDIMENTATION OF RANDOM SUSPENSIONS Analogies to stochastic homogenization: the nonlinear response.
This sec-tion is devoted to analogies between the sedimentation problem and corrector equationsfor linear elliptic operators, both in divergence form or in non-divergence form (cf. [32,30, 1, 2, 3] e.g.). In particular, we explain the differences in the corresponding criticaldimensions, as well as the surprising fact that the sedimentation problem benefits fromhyperuniformity whereas elliptic corrector equations do not in general. We first recall thecorrector equations for linear elliptic operators both in divergence form, cf. (Div), and innon-divergence form, cf. (NDiv), associated with a uniformly elliptic random coefficientfield a L ; to make the comparison with the sedimentation problem more transparent, wealso introduce some hybrid corrector equation, cf. (Hyb): − div( a L ∇ ψ ) = div( a L e ) , (Div) − a L : ∇ ψ = ( a L − ˜ a L ) : E, (NDiv) − div( a L ∇ ψ ) = ( a L − E [ a L ]) : E, (Hyb)for some fixed directions e ∈ R d and E ∈ R d × d , and some suitable constant ˜ a L ∈ R d × d that ensures solvability. In each of these corrector equations, the right-hand side displays alinear random input either in divergence or in non-divergence form, while the nonlinearitywith respect to randomness arises from the solution operator associated with the ellipticoperator in the left-hand side, which is again either in divergence or in non-divergenceform. We compare these elliptic models with the sedimentation problem (1.2)–(1.5), whichis conveniently rewritten as follows, cf. Remark 1.1(a), φ L = − λ L π L φ ◦ L , −△ φ ◦ L + ∇ Π ◦ L = ( I L − λ L ) e, div φ ◦ L = 0 , in T dL . The linear random input I L in the equation for φ ◦ L is in non-divergence form, and the non-linearity arises from the projection π L . Although the latter has a very different structurefrom elliptic solution operators, our results in the present work indicate that it behavesquite similarly and displays exactly the same nonlocality as the divergence-form ellipticsolution operator ( − div a L ∇ ) − . In this respect, the sedimentation problem appears com-parable to the hybrid model (Hyb).We now compare the critical dimensions for the different elliptic models. While a linearizedanalysis would be misleading when unraveling the role of hyperuniformity, we appeal to sen-sitivity calculus: we analyze how correctors are modified upon infinitesimal local changes δ a L of the random coefficients a L and we consider both the linear and nonlinear responses.Formally differentiating (Div), (NDiv), and (Hyb) with respect to a L in the direction δ a L yields, respectively, − div( a L ∇ δψ ) = div( δ a L e ) + div( δ a L ∇ ψ ) , − a L : ∇ δψ = δ a L : E + δ a L : ∇ ψ , − div( a L ∇ δψ ) = δ a L : E + div( δ a L ∇ ψ ) , where in each line the first right-hand side term is the linear response and the secondone is the nonlinear response. We denote by G and G the Green’s functions associatedwith − div( a L ∇ ) and − a L : ∇ , respectively, and we recall that they behave on largescales like the Green’s function for the Laplacian (up to second mixed derivative for G ,cf. [16, 51, 2, 31], and up to first derivative for G , cf. [3]). In order to assess the locality M. DUERINCKX AND A. GLORIA of the above contributions, we appeal to Green’s representation formula, δψ ( x ) = − ˆ T dL ∇ G ( x, · ) · δ a L e − ˆ T dL ∇ G ( x, · ) · δ a L ∇ ψ , (2.22) δψ ( x ) = ˆ T dL G ( x, · ) δ a L : E + ˆ T dL G ( x, · ) δ a L : ∇ ψ , (2.23) δψ ( x ) = ˆ T dL G ( x, · ) δ a L : E − ˆ T dL ∇ G ( x, · ) · δ a L ∇ ψ . (2.24)Locality is measured in terms of the power decay of δψ ( x ) , δψ ( x ) , δψ ( x ) when the per-turbation δ a L of the coefficient field is localized in a ball B ( y ) at a far-away point y . Inthis informal discussion, we focus on a self-consistency analysis and assume that ∇ ψ , ∇ ψ , and ∇ ψ are already known to be well-defined stationary objects with bounded mo-ments, while rigorous analysis would require a suitable buckling argument, cf. Section 4.1.First note that the divergence-form structure yields an additional gradient on the Green’sfunction, hence a better locality. Decay is indeed | x − y | − d for (Div) and only | x − y | − d for (NDiv), which explains the shift in critical dimensions: combined with suitable func-tional inequalities, in the spirit of Malliavin calculus, this formally entails that ψ hasbounded moments in dimension d > , while ψ only has bounded moments in dimension d > . Next, note that both in (2.22) and in (2.23) the linear and nonlinear responsesdisplay the same locality (Green’s functions have the same number of derivatives), whilethis is not the case in the hybrid model: the nonlinear response in (2.24) has better localityand the scaling is thus dominated by the linear part in non-divergence form. The sameproperty is shown to hold for the original sedimentation problem, cf. Proposition 4.1(i), andthis explains in particular why the critical dimension is in general the same as for (NDiv).Finally, we investigate the role of hyperuniformity. As this statistical property consists ofthe suppression of density fluctuations of a L , this is naturally expressed by restricting toperturbations δ a L having vanishing average. In the spirit of (2.21), this restriction leadsto the gain of a derivative in the following form, for all F ∈ C ∞ per ( T dL ) d × d , (cid:12)(cid:12)(cid:12) ˆ T dL F : δ a L (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ˆ B ( y ) (cid:16) F − B ( y ) F (cid:17) : δ a L (cid:12)(cid:12)(cid:12) . k δ a L k L ∞ ˆ B ( y ) |∇ F | . In the linear terms in (2.22)–(2.24), this yields an additional derivative on the Green’s func-tion, but no such gain appears in the nonlinear terms due to the presence of the stationaryfactors ∇ ψ , ∇ ψ , and ∇ ψ . Alternatively, this is understood as follows: fluctuationsof elliptic solution operators are known to depend not only on density fluctuations of a L (which would be suppressed by hyperuniformity), but also on its geometry (which is un-related to hyperuniformity), e.g. [42, Section 7.3]. In the models (Div) and (NDiv), thenonlinear responses are therefore limitant and no benefit is expected from hyperuniformity— except in the particular case of dimension d = 1 for (Div) and of isotropic coefficients a ( x ) = α ( x ) Id for (NDiv), in which case the nonlinearity with respect to randomness isdrastically simplified. In contrast, in the hybrid model (Hyb), as the nonlinear responsehas already better locality, hyperuniformity leads to a nontrivial improvement; this ex-plains why under hyperuniformity the critical dimension for the sedimentation problem asfor (Hyb) becomes the same as for (Div). EDIMENTATION OF RANDOM SUSPENSIONS Mean settling speed
This section is devoted to the analysis of the mean settling speed, cf. Theorem 1. Firstrecall that the flow φ L in the periodized tank T dL is defined via (1.2)–(1.5), that is, for all ω ∈ Ω , the velocity field φ ωL ∈ H ( T dL ) d and pressure field Π ωL ∈ L ( T dL ) are the uniqueperiodic solutions of −△ φ ωL + ∇ Π ωL = − α ωL e, in T dL \ I ωL , div φ ωL = 0 , in T dL \ I ωL , D( φ ωL ) = 0 , in I ωL ,e | I ωn,L | + ´ ∂I ωn,L σ ( φ ωL , Π ωL ) ν = 0 , ∀ n, ´ ∂I ωn,L Θ ν · σ ( φ ωL , Π ωL ) ν = 0 , ∀ n, ∀ Θ ∈ M skew , (3.1)with vanishing average ´ T dL φ ωe,L = 0 and ´ T dL \I ωL Π ωL = 0 , where solvability imposes that theconstant backflow − α ωL e be given by α ωL = λ ωL − λ ωL , λ ωL = L − d |I ωL | . Note that the hardcore assumption in (H δ ) ensures α ωL . . We may now proceed to theproof of Theorem 1. Proof of Theorem 1.
We split the proof into three steps. After some general preparationin the first step, the second step is devoted to the analysis of the periodized problem (3.1)and the proof of the main estimates. In the last step, borrowing arguments from [17],we argue that the infinite-volume limit is well-defined and that the corresponding Stokesproblem (2.6) is well-posed.
Step
1. Reformulation of the equation: we prove that equation (3.1) for φ ωL yields in theweak sense on the whole torus T dL , − △ φ ωL + ∇ (Π ωL T dL \I ωL ) = − α ωL e T dL \I ωL − X n δ ∂I ωn,L σ ( φ ωL , Π ωL ) ν, (3.2)where δ ∂I ωn,L stands for the Dirac measure on the boundary of I ωn,L .Given ζ ∈ C ∞ per ( T dL ) d , testing equation (3.1) with ζ and integrating by parts on T dL \ I ωL ,we find ˆ T dL \I ωL ∇ ζ : ∇ φ ωL − ˆ T dL \I ωL (div ζ ) Π ωL = − α ωL e · ˆ T dL \I ωL φ ωL − X n ˆ ∂I ωn,L ( ζ ⊗ ν ) : ( ∇ φ ωL − Π ωL Id) . (3.3)The claim (3.2) would follow provided we prove that ˆ I ωL ∇ ζ : ∇ φ ωL = − X n ˆ ∂I ωn,L ( ν ⊗ ζ ) : ∇ φ ωL . (3.4)Indeed, adding (3.4) to (3.3) yields the claim (3.2) in view of ˆ ∂I ωn,L ( ν ⊗ ζ + ζ ⊗ ν ) : ∇ φ ωL = ˆ ∂I ωn,L ζ ⊗ ν : 2 D( φ ωL ) . M. DUERINCKX AND A. GLORIA
We turn to the proof of (3.4). Using that φ ωL is affine in I ωn,L , we obtain for all n , ˆ ∂I ωn,L ( ν ⊗ ζ ) : ∇ φ ωL = ˆ ∂I ωn,L ζ i ν · ∇ i φ ωL = ˆ I ωn,L div( ζ i ∇ i φ ωL ) = ˆ I ωn,L ∇ ζ i · ∇ i φ ωL . Since D( φ ωL ) = 0 on I ωn,L , we can write φ ωL = V ωn,L + Θ ωn,L ( x − x ωn,L ) on I ωn,L for some V ωn,L ∈ R d and Θ ωn,L ∈ M skew , so that the above becomes ˆ ∂I ωn,L ( ν ⊗ ζ ) : ∇ φ ωL = ˆ I ωn,L ∇ ζ i · ∇ i (Θ ωn,L x ) = ˆ ∂I ωn,L ν · ( ζ · ∇ )(Θ ωn,L x ) = ˆ ∂I ωn,L ν · Θ ωn,L ζ. Likewise, we find ˆ I ωn,L ∇ ζ : ∇ φ ωL = ˆ I ωn,L ∇ ζ : ∇ (Θ ωn,L x ) = ˆ ∂I ωn,L ζ · ( ν · ∇ )(Θ ωn,L x ) = ˆ ∂I ωn,L ζ · Θ ωn,L ν. By skew-symmetry of Θ ωn,L , this yields (3.4), hence (3.2). Step
2. Bounds on periodized problem (3.1): we establish the identity ♯ P ωL X n e | e | · V ωn,L = 1 α ωL | e | T dL |∇ φ ωL | , (3.5)and show that under the mixing assumption (Mix) there holds k∇ φ L k L (Ω) . ρ L | e | × d > L ) : d = 2; L : d = 1; (3.6)whereas under the mixing and hyperuniformity assumption (Hyp) this is improved in alldimensions to k∇ φ L k L (Ω) . ρ L | e | . (3.7)Testing the reformulation (3.2) of the equation for φ ωL with φ ωL itself and using the divergence-free condition for φ ωL , we obtain ˆ T dL |∇ φ ωL | = − α ωL e · ˆ T dL \I ωL φ ωL − X n ˆ ∂I ωn,L φ ωL · σ ( φ ωL , Π ωL ) ν. Since D( φ ωL ) = 0 on I ωn,L , we can write φ ωL = V ωn,L + Θ ωn,L ( x − x ωn,L ) on I ωn,L for some V ωn,L ∈ R d and Θ ωn,L ∈ M skew , so that the boundary conditions for φ ωL lead to ˆ ∂I ωn,L φ ωL · σ ( φ ωL , Π ωL ) ν = V ωn,L · ˆ ∂I ωn,L σ ( φ ωL , Π ωL ) ν + ˆ ∂I ωn,L Θ ωn,L ν · σ ( φ ωL , Π ωL ) ν = − e · | I ωn,L | V ωn,L = − e · ˆ I ωn,L φ ωL , and the above becomes ˆ T dL |∇ φ ωL | = − α ωL e · ˆ T dL \I ωL φ ωL + X n e · ˆ I ωn,L φ ωL . EDIMENTATION OF RANDOM SUSPENSIONS Since ´ T dL φ ωL = 0 , this energy identity takes the form ˆ T dL |∇ φ ωL | = (1 + α ωL ) X n e · ˆ I ωn,L φ ωL . (3.8)In terms of particle velocities V ωn,L = ffl I ωn,L φ ωL , cf. (1.6), noting that (1+ α ωL ) | B | ♯ P ωL = L d α ωL ,this turns into the claim (3.5).Next, recalling ´ T dL φ ωL = 0 and integrating by parts, the energy identity (3.8) is alterna-tively written as ˆ T dL |∇ φ ωL | = (1 + α ωL ) ˆ T dL φ ωL · (cid:16) X n e ( I ωn,L − L − d | B | ) (cid:17) = (1 + α ωL ) ˆ T dL ∇ φ ωL : (cid:16) e ⊗ X n ∇ ( −△ ) − ( I ωn,L − L − d | B | ) (cid:17) . (3.9)Hence, by Cauchy-Schwarz’ inequality, ˆ T dL |∇ φ ωL | ≤ (1 + α ωL ) | e | ˆ T dL (cid:12)(cid:12)(cid:12) X n ∇△ − ( I ωn,L − L − d | B | ) (cid:12)(cid:12)(cid:12) , so that we find, by the hardcore assumption in the form α ωL . and by stationarity of P L , k∇ φ L k (Ω) . | e | E (cid:20)(cid:12)(cid:12)(cid:12) X n ∇△ − ( I n,L − L − d | B | ) (cid:12)(cid:12)(cid:12) (cid:21) . Denoting by G L the Green’s function for the Laplacian on the torus T dL , that is, the uniquedistributional solution of −△ G L = δ − L − d on T dL , and setting F L ( x ) := ´ B ( x ) ∇ G L , wemay rewrite the above as k∇ φ L k (Ω) . | e | E (cid:20)(cid:12)(cid:12)(cid:12) X n F L ( x n,L ) (cid:12)(cid:12)(cid:12) (cid:21) = | e | (cid:12)(cid:12)(cid:12)(cid:12) Var (cid:20) X n F L ( x n,L ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) , where the last equality follows from noting that E [ P n F L ( x n,L )] = ρ L ´ T dL F L = 0 . Underthe mixing assumption (Mix), in terms of the pair correlation function g ,L , we then deduce k∇ φ L k (Ω) . ρ L | e | (cid:12)(cid:12)(cid:12)(cid:12) ¨ T dL × T dL F L ( x ) F L ( y ) g ,L ( x − y ) dxdy (cid:12)(cid:12)(cid:12)(cid:12) ≤ ρ L | e | (cid:16) ˆ T dL | F L | (cid:17)(cid:16) ˆ T dL | g ,L | (cid:17) . ρ L | e | ˆ T dL | F L | , and the claim (3.6) follows from the standard decay of the Green’s function G L in form of | F L ( x ) | . (1 + | x | L ) − d . Under the hyperuniformity assumption (Hyp), we rather appealto the variance estimate (2.21), cf. Lemma A.2, to the effect of k∇ φ L k (Ω) . ρ L | e | ˆ T dL |∇ F L | , and the claim (3.7) now follows from the standard decay of the Green’s function G L inform of |∇ F L ( x ) | = | ´ B ( x ) ∇ G L | . (1 + | x | L ) − d . M. DUERINCKX AND A. GLORIA
Step
3. Infinite-volume limit: under (Mix) for d > or under (Hyp) for any d ≥ , weargue that ∇ φ L converges strongly in L (Ω) as L ↑ ∞ to the unique gradient solution ∇ φ of the corresponding infinite-volume problem (2.6).Uniqueness for (2.6) is already contained in [17] since the difference of two solutions of (2.6)is a solution of (2.6) without gravity. It only remains to establish the strong convergenceresult. For that purpose, in terms of γ ωL := X n ∇ ( −△ ) − ( I ωn,L − L − d | B | ) , we appeal to identity (3.9) in the form E (cid:2) |∇ φ L | (cid:3) = E h (1 + α L ) T dL ∇ φ L : ( e ⊗ γ L ) i , or equivalently, by stationarity of ∇ φ L : ( e ⊗ γ L ) and by invariance of α L with respect toshifts on the torus T dL (shifting P L does indeed not change the number of points), E (cid:2) |∇ φ L | (cid:3) = E (cid:2) (1 + α L ) ∇ φ L : ( e ⊗ γ L ) (cid:3) . (3.10)The stabilization condition for P L in (H δ ) ensures that γ L converges strongly in L (Ω) to γ := P n ∇ ( −△ ) − I n , which is indeed well-defined under (Mix) for d > or under (Hyp)for any d ≥ . Since the bounded random variable α L converges almost surely to α , weconclude that (1 + α L )( e ⊗ γ L ) converges strongly to (1 + α )( e ⊗ γ ) in L (Ω) . Therefore,identity (3.10) entails that the strong convergence ∇ φ L → ∇ φ in L (Ω) follows from thecorresponding weak convergence.Using the uniform bound (3.6) or (3.7) of Step 1, weak compactness in L (Ω; L ( R d )) en-sures that ∇ φ L converges weakly to some stationary random field Φ ∈ L (Ω; L ( R d ) d × d ) (along a subsequence, not relabelled). As a weak limit of gradients, Φ is necessarilygradient-like, hence we may write Φ ω ( x ) = ∇ φ ω ( x ) for some φ ∈ L (Ω; H ( R d ) d ) . Com-bining this with the stabilization condition in (H δ ) in form of I ωL → I ω in L ( R d ) foralmost all ω , we may then pass to the limit in the weak formulation of equation (3.1). Moreprecisely, we first easily deduce for almost all ω that D( φ ω ) = 0 in I ω and that div φ ω = 0 .Next, a similar standard argument as in [17, Step 3 of proof of Proposition 2.1] allows todeduce that φ is a solution of (2.6) in the following weak sense: for almost all ω , for alltest functions ψ ∈ C ∞ c ( R d ) d with D( ψ ) = 0 on I ω and div ψ = 0 , there holds ˆ R d ∇ ψ : ∇ φ ω = − αe · ˆ R d \I ω ψ + e · ˆ I ω ψ. Finally, arguing as in [17, Step 4 of proof of Proposition 2.1], a stationary pressure Π ω can be reconstructed with vanishing expectation and finite second moments, while theregularity theory for the steady Stokes equation ensures that ( φ ω , Π ω ) is in fact a classicalsolution of (2.6) and that boundary conditions are satisfied in a pointwise sense. (cid:3) Velocity fluctuations
This section is devoted to the proof of Theorem 2, that is, the estimation of fluctuationsof individual particle velocities, which requires a fine analysis of stochastic cancellations.The proof is particularly demanding and strongly relies on a novel annealed regularitytheory for the steady Stokes equation with a random suspension, which is briefly described
EDIMENTATION OF RANDOM SUSPENSIONS in Section 4.2.2 and mainly postponed to a forthcoming companion contribution [18]. In-terestingly, the hyperuniform setting is easier to treat as it only requires a perturbativeregularity result. We also strongly rely on local regularity statements and pressure esti-mates borrowed from our previous work [17] on colloidal (non-sedimenting) suspensions.By scaling, we may henceforth assume | e | = 1 .4.1. Structure of the proof.
We start with the following key estimates on the opti-mal decay of large-scale averages of the gradient field ∇ φ L . Due to the nonlinearity withrespect to randomness, local norms of ∇ φ L also appear in the right-hand side; this willbe subsequently absorbed by a buckling argument. Yet, the correct fluctuation scalingis already manifest: in particular, the first term in (4.1) below displays the CLT scalingmultiplied by a length scale, cf. (2.11), and is the reason why the critical dimension in The-orem 2(i) is d = 4 instead of d = 2 . Importantly, the nonlinear contribution of ∇ φ L in theright-hand side is multiplied by the CLT scaling without loss, which is key to our bucklingargument to prove Theorem 2(i): the worse nonlocality only appears in the linear part,while the nonlinear part always behaves as in homogenization for divergence-form linearelliptic equations, cf. Section 2.7. In the hyperuniform setting (ii), the suppression of den-sity fluctuations exactly allows to avoid the worse scaling of the linear part, cf. Section 2.7,and we recover the CLT scaling (2.12). Proposition 4.1 (Fluctuation scaling) . Let the random point processes ( P L ) L ≥ satisfythe general assumptions (H δ ) for some δ > .(i) Under the improved mixing assumption (Mix + ) , in dimension d > , there holds forall g ∈ C ∞ per ( T dL ) d × d , ≤ R ≤ L , q ≥ , and ≪ p < ∞ , (cid:13)(cid:13)(cid:13) ˆ T dL g : ∇ φ L (cid:13)(cid:13)(cid:13) p (Ω) . p k g k dd +2 ( T dL ) + kh∇i g k ( T dL ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ˆ B R [ ∇ φ L ] q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) . (4.1) (ii) Under the improved mixing and hyperuniformity assumption (Hyp + ) , in any dimen-sion d ≥ , there holds for all g ∈ C ∞ per ( T dL ) d × d , ≤ R ≤ L , q ≥ , and ≪ p < ∞ , (cid:13)(cid:13)(cid:13) ˆ T dL g : ∇ φ L (cid:13)(cid:13)(cid:13) p (Ω) . p kh∇i g k ( T dL ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ˆ B R [ ∇ φ L ] q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) . (4.2) ♦ In preparation for a buckling argument, the following result allows to bound localnorms of ∇ φ L as appearing in the right-hand sides of (4.1)–(4.2) by corresponding large-scale averages. This statement is inspired by [59] in the context of homogenization fordivergence-form linear elliptic equations, and constitutes a compact improved version of [29,Lemma 2.2]. Proposition 4.2.
Let the random point processes ( P L ) L ≥ satisfy the general assump-tions (H δ ) for some δ > . Choose χ ∈ C ∞ c ( B ) with ´ B χ = 1 and set χ r ( x ) := r − d χ ( xr ) .There exists η > (only depending on d, δ ) such that there holds for all ≤ r ≪ χ R ≤ L , ≤ q ≤ η , and p ≥ , (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) B R |∇ φ L | q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) . χ R + (cid:13)(cid:13)(cid:13) ˆ T dL χ r ∇ φ L (cid:13)(cid:13)(cid:13) p (Ω) . ♦ M. DUERINCKX AND A. GLORIA
We are now in position to prove Theorem 2. Based on the above two propositionstogether with a buckling argument, we first deduce moment bounds on ∇ φ L . Combiningthis again with Proposition 4.1, we deduce the optimal fluctuation scaling for large-scaleaverages of ∇ φ L , cf. (2.11)–(2.12). Finally, moment bounds on the velocity field φ L simplyfollow by integration. Proof of Theorem 2.
By local regularity for the steady Stokes equation, e.g. [25, Sec-tion IV], we have sup B ( x ) |∇ φ ωL | . (cid:16) B ( x ) |∇ φ ωL | (cid:17) , sup B ( x ) | φ ωL | . (cid:16) B ( x ) | φ ωL | + |∇ φ ωL | (cid:17) , (4.3)so that it is enough to control moments of local quadratic averages [ ∇ φ L ] and [ φ L ] ; weomit the detail. We split the proof into three steps. Step
1. Proof that for all ≤ R ≤ L , k [ ∇ φ L ] k p (Ω) . ( R d ) − p (cid:13)(cid:13)(cid:13) B R |∇ φ L | (cid:13)(cid:13)(cid:13) L p (Ω) . By the discrete ℓ − ℓ p inequality in form of the reverse Jensen’s inequality (cid:16) B R ( x ) [ ∇ φ L ] p (cid:17) p . ( R d ) − p B R ( x ) |∇ φ L | , the claim follows in combination with stationarity of [ ∇ φ L ] . Step
2. Moment bounds on ∇ φ L : proof of (2.7) and (2.9).Under (Mix + ) in dimension d > , combining the results of Propositions 4.1(i) and 4.2, wefind for all ≤ r ≪ χ R ≤ L , < q ≤ η , and ≪ p < ∞ , (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) B R |∇ φ L | q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) . χ R + (cid:13)(cid:13)(cid:13) ˆ T dL χ r ∇ φ L (cid:13)(cid:13)(cid:13) p (Ω) . p,χ R + r − d + (cid:0) Rr (cid:1) d R − dq ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) B R [ ∇ φ L ] q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) , which yields after optimization in r , for R ≫ and q > , (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) B R |∇ φ L | q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) . p R , and the conclusion (2.7) then follows from the result of Step 1.Under (Hyp + ) in any dimension d ≥ , combining the results of Propositions 4.1(ii) and 4.2,we rather find for all ≤ r ≪ R ≤ L , ≤ q < η , and ≪ p < ∞ , (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) B R |∇ φ L | q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) . p R + (cid:0) Rr (cid:1) d R − dq ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) B R [ ∇ φ L ] q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) , and the conclusion (2.9) follows in the same way. Step
3. Moment bounds on φ L : proof of (2.8) and (2.10).Poincaré’s inequality yields (cid:13)(cid:13)(cid:13)(cid:13)h φ L − B φ L i ( x ) (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) . k [ ∇ φ L ] k L p (Ω) + (cid:13)(cid:13)(cid:13) B ( x ) φ L − B φ L (cid:13)(cid:13)(cid:13) L p (Ω) , (4.4) EDIMENTATION OF RANDOM SUSPENSIONS and in view of (2.7) and (2.9) it remains to estimate the last right-hand side term. Forthat purpose, we write B ( x ) φ L − B φ L = ˆ T dL ∇ φ L · ∇ h x,L , in terms of ∇ h x,L := ∇ h L ( · − x ) − ∇ h L , where h L denotes the unique solution in T dL of −△ h L = B | B | − L − d . Under (Mix + ) in dimension d > , appealing to Proposition 4.1(i) together with (2.7)yields for all p ≥ , (cid:13)(cid:13)(cid:13) ˆ T dL ∇ φ L · ∇ h x,L (cid:13)(cid:13)(cid:13) L p (Ω) . p k∇ h x,L k L dd +2 ( T dL ) + kh∇i ∇ h x,L k L ( T dL ) . Noting that k∇ h x,L k L ( T dL ) . and that for d > Riesz potential theory further yields k∇ h x,L k L ( T dL ) . , we are reduced to (cid:13)(cid:13)(cid:13) ˆ T dL ∇ φ L · ∇ h x,L (cid:13)(cid:13)(cid:13) L p (Ω) . p k∇ h x,L k L dd +2 ( T dL ) , while a direct computation with Green’s kernel gives k∇ h x,L k L dd +2 ( T dL ) . , if d > | x | ) , if d = 4; h x i , if d = 3 . Inserting this into (4.4), the conclusion (2.8) follows. Under (Hyp + ), rather appealing toProposition 4.1(ii), the conclusion (2.10) follows in the same way. (cid:3) Preliminaries.
We introduce a number of general tools that play an important rolein the proof of Propositions 4.1 and 4.2.4.2.1.
Multiscale inequalities for higher moments.
The following shows that the multiscalevariance inequalities in assumptions (Mix + ) and (Hyp + ) imply corresponding functionalinequalities for higher moments. This is proven in [19, Proposition 1.10(ii)] for (Mix + ),and the same proof applies for (Hyp + ). Lemma 4.3.
If the random point processes {P L } L ≥ satisfy (Mix + ) , then there holds forall p ≥ and all σ ( P L ) -measurable random variables Y ( P L ) with E [ Y ( P L )] = 0 , E (cid:2) | Y ( P L ) | p (cid:3) p . p E (cid:20) ˆ L (cid:18) ˆ T dL (cid:16) ∂ osc P L ,B ℓ ( x ) Y ( P L ) (cid:17) dx (cid:19) p h ℓ i − dp π ( ℓ ) dℓ (cid:21) p . (4.5) Likewise, if {P L } L ≥ satisfy (Hyp + ) , then there holds for all p ≥ and all σ ( P L ) -measurable random variables Y ( P L ) with E [ Y ( P L )] = 0 , E (cid:2) | Y ( P L ) | p (cid:3) p . p E (cid:20) ˆ L (cid:18) ˆ T dL (cid:16) ∂ hyp P L ,B ℓ ( x ) Y ( P L ) (cid:17) dx (cid:19) p h ℓ i − dp π ( ℓ ) dℓ (cid:21) p . (4.6) ♦ M. DUERINCKX AND A. GLORIA
Annealed regularity theory.
Another key tool consists of annealed regularity proper-ties for the steady Stokes equation with a random colloidal (non-sedimenting) suspension.More precisely, given a random forcing g ∈ L ∞ (Ω; C ∞ per ( T dL ) d × d ) , we consider the uniquesolution v L ∈ L ∞ (Ω; H ( T dL )) of the following heterogeneous problem, for almost all ω , −△ v ωL + ∇ P ωL = div g ω , in T dL \ I ωL , div v ωL = 0 , in T dL \ I ωL , D( v ωL ) = 0 , in I ωL , ´ ∂I ωn,L (cid:0) g ω + σ ( v ωL , P ωL ) (cid:1) ν = 0 , ∀ n, ´ ∂I ωn,L Θ ν · (cid:0) g ω + σ ( v ωL , P ωL ) (cid:1) ν = 0 , ∀ n, ∀ Θ ∈ M skew , (4.7)with ´ T dL v ωL = 0 . The energy inequality takes the form k∇ v ωL k L ( T dL ) ≤ k g ω k L ( T dL \I ωL ) . (4.8)Aside from perturbative Meyers type estimates, cf. Proposition B.1(i), no corresponding(deterministic) L p estimate is expected to hold in general due to heterogeneities — unlessparticles are assumed to be sufficiently far apart, cf. Remark 4.6 below. However, in viewof homogenization, the heterogeneous Stokes operator can be replaced on large scales bythe following “homogenized” one, cf. [17, Theorem 1], (cid:26) −∇ · ¯ B ∇ ¯ v ωL + ∇ ¯ P ωL = div g ω , in T dL , div¯ v ωL = 0 , in T dL , for which standard constant-coefficient elliptic regularity theory is available. In this spirit,compared to a generic situation, the solution of (4.7) is expected to have much betterregularity when the suspension is sampled by an ergodic ensemble. This type of resultwas pioneered by Avellaneda and Lin [6, 7] in the context of periodic homogenization fordivergence-form linear elliptic equations. In the stochastic setting, while early contributionsin form of annealed Green’s function estimates appeared in [16, 51], a (quenched) large-scale regularity theory was first outlined by Armstrong and Smart [5] (see also [4]), andlater fully developed in [2, 31]. For the steady Stokes problem (4.7), the developmentof a corresponding large-scale regularity theory is postponed to a forthcoming companioncontribution [18] that is devoted to the quantitative homogenization of (4.7). In the presentwork, in the spirit of [21], we appeal to large-scale regularity in form of the followingconvenient annealed L p regularity estimate established in [18]. Theorem 4.4 (Annealed L p regularity [18]) . Let the random point process P L be stationaryon T dL and satisfy the hardcore condition in (H δ ) for some δ > , as well as the improvedmixing assumption (Mix + ) . Given g ∈ L ∞ (Ω; C ∞ per ( T dL ) d × d ) , the unique solution v L ∈ L ∞ (Ω; H ( T dL )) of (4.7) satisfies for all < p, q < ∞ and η > , k [ ∇ v L ] k L q ( T dL ;L p (Ω)) . p,q,η k [ g ] k L q ( T dL ;L p + η (Ω)) . ♦ In the perturbative setting | p − | , | q − | ≪ , the loss of stochastic integrability can beavoided, that is, the above holds with η = 0 , which happens to be a useful tool in the proofof Proposition 4.1. In addition, such a perturbative statement can be established undermere stationarity and ergodicity assumptions, without any mixing. The proof is based ona simple Meyers type argument and is included in Appendix B. EDIMENTATION OF RANDOM SUSPENSIONS Theorem 4.5 (Perturbative annealed L p regularity) . Let the random point process P L bestationary on T dL and satisfy the hardcore condition in (H δ ) for some δ > . Then, thereexists a constant η > (only depending on d, δ ) such that the following holds: Given g ∈ L ∞ (Ω; C ∞ per ( T dL ) d × d ) , the unique solution v L ∈ L ∞ (Ω; H ( T dL )) of (4.7) satisfies forall p, q with | q − | , | p − | ≤ η , k [ ∇ v L ] k L q ( T dL ;L p (Ω)) . k [ g ] k L q ( T dL ;L p (Ω)) . ♦ Remark 4.6 (Dilute L p regularity) . In the dilute regime, the recent work of Höfer [39]on the reflection method easily yields the following version of Theorem 4.4; the proof isa direct adaptation of [39] and is omitted. This also constitutes a variant of the diluteGreen’s function estimates in [29, Lemma 2.7].
Let the random point processes ( P L ) L ≥ satisfy the general assumptions (H δ ) for some δ > , and denote by δ L the minimal interparticle distance in P L . For all < p, q < ∞ ,there exists a constant δ q > (only depending on d, q ) such that, provided P L is diluteenough in the sense of δ L ≥ δ q , we have: Given a random forcing g ∈ L ∞ (Ω; C ∞ per ( T dL ) d × d ) ,the unique solution v L ∈ L ∞ (Ω; H ( T dL ) d ) of (4.7) satisfies k∇ v L k L q ( T dL ;L p (Ω)) . k g k L q ( T dL ;L p (Ω)) , as well as the following deterministic estimate, for almost all ω , k∇ v ωL k L q ( T dL ) . k g ω k L q ( T dL ) . ♦ Localized pressure estimates.
We state the following localized estimate on the pres-sure for the steady Stokes equation. This is essentially a consequence of standard pressureestimates in [25] but it requires some additional care since the prefactor in the estimateis uniform with respect to the size of D although I L consists of an unbounded number ofcomponents. Note that the same result could be stated in L q for any < q < ∞ , and thatthe Stokes problem below is tailored to cover both equations (3.1) and (4.7). Lemma 4.7 (Localized pressure estimates) . Let a (deterministic) point set P L = { x n,L } n satisfy the hardcore condition in (H δ ) for some δ > . Given g ∈ C ∞ per ( T dL ) d × d and e ′ ∈ R d ,let v L ∈ H ( T dL ) d denote the unique solution of −△ w L + ∇ Q L = div g − α L e, in T dL \ I L , div w L = 0 , in T dL \ I L , D( w L ) = 0 , in I L ,e ′ | I n,L | + ´ ∂I n,L (cid:0) g + σ ( w L , Q L ) (cid:1) ν = 0 , ∀ n, ´ ∂I n,L Θ ν · (cid:0) g + σ ( w L , Q L ) (cid:1) ν = 0 , ∀ n, ∀ Θ ∈ M skew , with ´ T dL w L = 0 . Then there holds for all balls D ⊂ T dL with radius r D , (cid:13)(cid:13)(cid:13) Q L − D \I L Q L (cid:13)(cid:13)(cid:13) ( D \I L ) . r d +2 D | e | + k∇ w L k ( D ) + k g k ( D \I L ) . ♦ Proof.
Let a ball D ⊂ T dL be fixed with radius r D . Using the Bogovskii operator as e.g.in [17, Step 4.2 of the proof of Proposition 2.1], we can construct a map ζ ∈ H ( D ) (implicitly extended by 0 on T dL \ D ) such that ζ | I n,L is constant for all n and such that div ζ = (cid:16) Q L − D \I L Q L (cid:17) D \I L , k∇ ζ k L ( D ) . (cid:13)(cid:13)(cid:13) Q L − D \I L Q L (cid:13)(cid:13)(cid:13) L ( D \I L ) , M. DUERINCKX AND A. GLORIA where we emphasize that the prefactor in the last estimate is uniformly bounded indepen-dently of D and L . Arguing as in Step 1 of the proof of Theorem 1, we note that theequation for w L implies in the weak sense on the whole torus T dL , −△ w L + ∇ ( Q L T dL \I L ) = div( g T dL \I L ) − α L e T dL \I L − X n δ ∂I n,L (cid:0) g + σ ( w L , Q L ) (cid:1) ν. Testing this equation with ζ , recalling that ζ is constant inside particles, and using theboundary conditions for w L , we are led to ˆ T dL \I L Q L div ζ = ˆ T dL ∇ ζ : ∇ w L + ˆ T dL \I L ∇ ζ : g + α L e · ˆ T dL \I L ζ − e · X n ˆ I n,L ζ. Inserting the definition of div ζ , recalling that | α L | . , and using Poincaré’s inequalityon D , we deduce (cid:13)(cid:13)(cid:13) Q L − D \I L Q L (cid:13)(cid:13)(cid:13) ( D \I L ) . k∇ ζ k L ( D ) (cid:16) k∇ w L k L ( D ) + k g k L ( D \I L ) (cid:17) + | e |k ζ k L ( D ) . k∇ ζ k L ( D ) (cid:16) r d +2 D | e | + k∇ w L k ( D ) + k g k ( D \I L ) (cid:17) , and the claim follows from the bound on the L -norm of ∇ ζ . (cid:3) Gehring’s lemma.
In view of perturbative Meyers type estimates, we shall appealto the following version of Gehring’s lemma, which is a mild reformulation of [28, Propo-sition 5.1].
Lemma 4.8 (Gehring’s lemma; [26, 28]) . Given r > q > and a reference cube Q ⊂ R d ,let G ∈ L q ( Q ) and F ∈ L r ( Q ) be nonnegative functions. There exist θ > (onlydepending on d, q, r ) with the following property: Given θ ≤ θ , if for some C ≥ thefollowing condition holds for all cubes Q ⊂ Q , (cid:16) Q G q (cid:17) q ≤ C Q G + C (cid:16) Q F q (cid:17) q + θ (cid:16) Q G q (cid:17) q , then there exists η > (only depending on C , d, q, r ) such that for all q ≤ p ≤ q + η , (cid:16) Q G p (cid:17) p . C ,q,r Q G + (cid:16) Q F p (cid:17) p . ♦ Proof of Proposition 4.1.
We shall exploit the multiscale variance inequality (2.2),or its hyperuniform version (2.3), and appeal to a duality argument. Let the (deterministic)test function g ∈ C ∞ per ( T dL ) d × d be fixed and for all ω let v ωL ∈ H ( T dL ) d denote the uniquesolution of the following auxiliary problem, −△ v ωL + ∇ P ωL = ∇ · g, in T dL \ I ωL , div v ωL = 0 , in T dL \ I ωL , D( v ωL ) = 0 , in I ωL , ´ ∂I ωn,L (cid:0) g + σ ( v ωL , P ωL ) (cid:1) ν = 0 , ∀ n, ´ ∂I ωn,L Θ ν · (cid:0) g + σ ( v ωL , P ωL ) (cid:1) ν = 0 , ∀ n, ∀ Θ ∈ M skew , (4.9)with ´ T dL v ωL = 0 . We split the proof into four main steps: the mixing case (i) is treatedin the first three steps, while the simplifications that appear in the hyperuniform case (ii)are pointed out in the last step. EDIMENTATION OF RANDOM SUSPENSIONS Step
1. Fluctuation scaling outside I L : proof that under (Mix + ) in dimension d > thereholds for all ≤ R ≤ L , q ≥ , and ≪ p < ∞ , (cid:13)(cid:13)(cid:13) ˆ T dL \I L g : ∇ φ L (cid:13)(cid:13)(cid:13) p (Ω) . p L − d (cid:13)(cid:13)(cid:13) e · ˆ I L v L (cid:13)(cid:13)(cid:13) p (Ω) + k g k dd +2 ( T dL ) + kh∇i g k ( T dL ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ˆ B R [ ∇ φ L ] q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) . (4.10)Using the version (4.5) of the multiscale variance inequality (2.2) to control higher mo-ments, we obtain (cid:13)(cid:13)(cid:13) ˆ T dL \I L g : ∇ φ L (cid:13)(cid:13)(cid:13) p (Ω) . p E (cid:20) ˆ L (cid:18) ˆ T dL (cid:16) ∂ osc P L ,B ℓ ( x ) ˆ T dL \I L g : ∇ φ L (cid:17) dx (cid:19) p h ℓ i − dp π ( ℓ ) dℓ (cid:21) p , (4.11)and it remains to estimate the oscillation of ´ T dL \I L g : ∇ φ L with respect to P L on anyball B ℓ ( x ) . Given ≤ ℓ ≤ L and x ∈ R d , and given a realization, let P ′ L be a locally finitepoint set satisfying the hardcore condition in (H δ ), with P ′ L | T dL \ B ℓ ( x ) = P L | T dL \ B ℓ ( x ) , anddenote by φ ′ L the corresponding solution of equation (3.1) with P L replaced by P ′ L . Wesplit the proof into three further substeps. Substep (cid:12)(cid:12)(cid:12) ˆ T dL \I L g : ∇ φ L − ˆ T dL \I ′ L g : ∇ φ ′ L (cid:12)(cid:12)(cid:12) . L − d (cid:12)(cid:12)(cid:12) e · ˆ I L v L (cid:12)(cid:12)(cid:12) + ˆ B ℓ +1 ( x ) | v L | + (cid:16) ˆ B ℓ +2 ( x ) |∇ v L | + |h∇i g | (cid:17) (cid:16) h ℓ i d +2 + ˆ B ℓ +2 ( x ) |∇ φ ′ L | (cid:17) . (4.12)First decomposing ´ T dL \I ′ L = ´ T dL \I L + ´ I L \I ′ L − ´ I ′ L \I L , we find ˆ T dL \I L g : ∇ φ L − ˆ T dL \I ′ L g : ∇ φ ′ L = ˆ T dL \I L g : ∇ ( φ L − φ ′ L ) + ˆ I ′ L \I L g : ∇ φ ′ L − ˆ I L \I ′ L g : ∇ φ ′ L , hence, in view of the inclusion ( I L \ I ′ L ) ∪ ( I ′ L \ I L ) ⊂ B ℓ +1 ( x ) , (cid:12)(cid:12)(cid:12) ˆ T dL \I L g : ∇ φ L − ˆ T dL \I ′ L g : ∇ φ ′ L (cid:12)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12) ˆ T dL \I L g : ∇ ( φ L − φ ′ L ) (cid:12)(cid:12)(cid:12) + (cid:16) ˆ B ℓ +1 ( x ) | g | (cid:17) (cid:16) ˆ B ℓ +1 ( x ) |∇ φ ′ L | (cid:17) , (4.13)and it remains to examine the first right-hand side term. Arguing similarly as in Step 1 ofthe proof of Theorem 1, we note that the equation (4.9) for v L implies in the weak senseon the whole torus T dL , − △ v L + ∇ ( P L T dL \I L ) = ∇ · ( g T dL \I L ) − X n δ ∂I n,L (cid:0) g + σ ( v L , P L ) (cid:1) ν. (4.14) M. DUERINCKX AND A. GLORIA
Testing this equation with φ L − φ ′ L , and recalling that the pressure P L is only defined upto additive constant and that φ L and φ ′ L are divergence-free on the whole torus T dL , weobtain for any constant c ∈ R , ˆ T dL \I L g : ∇ ( φ L − φ ′ L ) = − ˆ T dL ∇ v L : ∇ ( φ L − φ ′ L ) − X n ˆ ∂I n,L ( φ L − φ ′ L ) · (cid:0) g + σ ( v L , P L − c ) (cid:1) ν, hence, in view of the boundary conditions for φ L , φ ′ L , and v L , arguing similarly as in Step 2of the proof of Theorem 1, ˆ T dL \I L g : ∇ ( φ L − φ ′ L ) = − ˆ T dL ∇ v L : ∇ ( φ L − φ ′ L )+ X n : x n,L ∈ B ℓ ( x ) ˆ ∂I n,L (cid:16) φ ′ L − I n,L φ ′ L (cid:17) · (cid:0) g + σ ( v L , P L − c ) (cid:1) ν. (4.15)Likewise, testing with v L the equation for φ L − φ ′ L in the form (3.2), we get for anyconstants c , c ′ ∈ R , − ˆ T dL ∇ v L : ∇ ( φ L − φ ′ L ) = α L e · ˆ T dL \I L v L − α ′ L e · ˆ T dL \I ′ L v L + X n ˆ ∂I n,L v L · σ ( φ L , Π L − c ) ν − X n ˆ ∂I ′ n,L v L · σ ( φ ′ L , Π ′ L − c ′ ) ν, which, in view of the choice ´ T dL v L = 0 and of the boundary conditions for φ L , φ ′ L , and v L ,turns into ˆ T dL ∇ v L : ∇ ( φ L − φ ′ L ) = α L e · ˆ I L v L − α ′ L e · ˆ I ′ L v L + X n : x n,L ∈ B ℓ ( x ) e · ˆ I n,L v L − X n : x ′ n,L ∈ B ℓ ( x ) e · ˆ I ′ n,L v L − X n : x ′ n,L ∈ B ℓ ( x ) ˆ ∂I ′ n,L (cid:16) v L − I ′ n,L v L (cid:17) · σ ( φ ′ L , Π ′ L − c ′ ) ν. (4.16)Combined with (4.15), this yields ˆ T dL \I L g : ∇ ( φ L − φ ′ L ) = − ( α L − α ′ L ) e · ˆ I L v L − ( α ′ L + 1) (cid:18) X n : x n,L ∈ B ℓ ( x ) e · ˆ I n,L v L − X n : x ′ n,L ∈ B ℓ ( x ) e · ˆ I ′ n,L v L (cid:19) − X n : x ′ n,L ∈ B ℓ ( x ) ˆ ∂I ′ n,L (cid:16) v L − I ′ n,L v L (cid:17) · σ ( φ ′ L , Π ′ L − c ′ ) ν EDIMENTATION OF RANDOM SUSPENSIONS + X n : x n,L ∈ B ℓ ( x ) ˆ ∂I n,L (cid:16) φ ′ L − I n,L φ ′ L (cid:17) · (cid:0) g + σ ( v L , P L − c ) (cid:1) ν. (4.17)In order to control the different right-hand side terms localized at particle boundaries, weappeal to the local regularity theory for the Stokes equation. We illustrate this on v L : atrace estimate first yields ˆ ∂I n,L |∇ v L | + | P L − c | . ˆ ( I n,L + δB ) \ I n,L |h∇i ∇ v L | + |h∇i ( P L − c ) | , while, for any constant c ∈ R d , replacing v L by v L − c , the local regularity theory for theStokes equation (4.9) satisfied by v L − c in ( I n,L + δB ) \ I n,L can then be applied in formof e.g. [25, Theorem IV.5.1], ˆ ∂I n,L |∇ v L | + | P L − c | . ˆ ∂I n,L (cid:12)(cid:12) h∇i ( v L − c ) | I n,L (cid:12)(cid:12) + ˆ ( I n,L + δB ) \ I n,L |∇ v L | + | P L − c | + |h∇i g | , which yields by Poincaré’s inequality, for the choice c := ffl I n,L v L , recalling the linearityof v L inside particles, ˆ ∂I n,L |∇ v L | + | P L − c | . ˆ I n,L + δB |∇ v L | + | P L − c | T dL \I L + |h∇i g | . (4.18)Likewise, as φ ′ L satisfies (3.1), we find ˆ ∂I ′ n,L |∇ φ ′ L | + | Π ′ L − c ′ | . ˆ I ′ n,L + δB |∇ φ ′ L | + | Π ′ L − c ′ | T dL \I ′ L + | α ′ L | . (4.19)Inserting these bounds into (4.17), recalling that | α L | , | α ′ L | . , and noting that α L − α ′ L = |I L || T dL \ I ′ L | − |I ′ L || T dL \ I L || T dL \ I L || T dL \ I ′ L | = | T dL | (cid:0) |I L | − |I ′ L | (cid:1) | T dL \ I L || T dL \ I ′ L | = | T dL | (cid:0) |I L ∩ B ℓ +1 ( x ) | − |I ′ L ∩ B ℓ +1 ( x ) | (cid:1) | T dL \ I L || T dL \ I ′ L | . L − d h ℓ i d , we are led to (cid:12)(cid:12)(cid:12) ˆ T dL \I L g : ∇ ( φ L − φ ′ L ) (cid:12)(cid:12)(cid:12) . L − d h ℓ i d (cid:12)(cid:12)(cid:12) e · ˆ I L v L (cid:12)(cid:12)(cid:12) + ˆ B ℓ +1 ( x ) | v L | + (cid:16) ˆ B ℓ +2 ( x ) (cid:0) |∇ v L | + | P L − c | T dL \I L + |h∇i g | (cid:1)(cid:17) × (cid:16) ˆ B ℓ +2 ( x ) (cid:0) |∇ φ ′ L | + | Π ′ L − c ′ | T dL \I ′ L + 1 (cid:1)(cid:17) . M. DUERINCKX AND A. GLORIA
Choosing c := ffl B ℓ +2 ( x ) \I L P L and c ′ := ffl B ℓ +2 ( x ) \I ′ L Π ′ L , and using the pressure estimate ofLemma 4.7, we deduce (cid:12)(cid:12)(cid:12) ˆ T dL \I L g : ∇ ( φ L − φ ′ L ) (cid:12)(cid:12)(cid:12) . L − d h ℓ i d (cid:12)(cid:12)(cid:12) e · ˆ I L v L (cid:12)(cid:12)(cid:12) + ˆ B ℓ +1 ( x ) | v L | + (cid:16) ˆ B ℓ +2 ( x ) |∇ v L | + |h∇i g | (cid:17) (cid:16) h ℓ i d +2 + ˆ B ℓ +2 ( x ) |∇ φ ′ L | (cid:17) . (4.20)Combined with (4.13), this yields the claim (4.12). Substep d > , ˆ B ℓ +2 ( x ) |∇ φ ′ L | . h ℓ i d +2 + ˆ B ℓ +2 ( x ) |∇ φ L | . (4.21)Starting from (3.2) and arguing as for (4.16), the energy identity for φ L − φ ′ L takes thefollowing form, for any constants c , c ′ ∈ R , ˆ T dL |∇ ( φ L − φ ′ L ) | = − ( α L − α ′ L ) e · ˆ T dL \I L ( φ L − φ ′ L ) − ( α ′ L + 1) (cid:18) X n : x n,L ∈ B ℓ ( x ) e · ˆ I n,L ( φ L − φ ′ L ) − X n : x ′ n,L ∈ B ℓ ( x ) e · ˆ I ′ n,L ( φ L − φ ′ L ) (cid:19) + X n : x n,L ∈ B ℓ ( x ) ˆ ∂I n,L (cid:16) φ ′ L − I n,L φ ′ L (cid:17) · (cid:0) φ L ) − (Π L − c ) Id (cid:1) ν + X n : x ′ n,L ∈ B ℓ ( x ) ˆ ∂I ′ n,L (cid:16) φ L − I ′ n,L φ L (cid:17) · (cid:0) φ ′ L ) − (Π ′ L − c ′ ) Id (cid:1) ν. Using the local regularity theory for the Stokes equation in form of (4.19) as in Substep 1.1,this leads to ˆ T dL |∇ ( φ L − φ ′ L ) | . L − d h ℓ i d ˆ T dL | φ L − φ ′ L | + ˆ B ℓ +1 ( x ) | φ L − φ ′ L | + (cid:16) ˆ B ℓ +2 ( x ) |∇ φ ′ L | + | Π ′ L − c ′ | T dL \I ′ L (cid:17) (cid:16) ˆ B ℓ +2 ( x ) |∇ φ L | + | Π L − c | T dL \I L (cid:17) . Hence, choosing c := ffl B ℓ +2 ( x ) \I L Π L and c ′ := ffl B ℓ +2 ( x ) \I ′ L Π ′ L and using the pressureestimate of Lemma 4.7 for both Π L and Π ′ L , we obtain ˆ T dL |∇ ( φ L − φ ′ L ) | . L − d h ℓ i d ˆ T dL | φ L − φ ′ L | + ˆ B ℓ +1 ( x ) | φ L − φ ′ L | + (cid:16) h ℓ i d +2 + ˆ B ℓ +2 ( x ) |∇ φ ′ L | (cid:17) (cid:16) h ℓ i d +2 + ˆ B ℓ +2 ( x ) |∇ φ L | (cid:17) . (4.22)Using Poincaré’s inequality in the form L − d ˆ T dL | φ L − φ ′ L | . L − d (cid:16) ˆ T dL |∇ ( φ L − φ ′ L ) | (cid:17) , EDIMENTATION OF RANDOM SUSPENSIONS and the Poincaré-Sobolev inequality for d > in the form ˆ B ℓ +1 ( x ) | φ L − φ ′ L | . h ℓ i d +1 (cid:16) ˆ T dL [ φ L − φ ′ L ] dd − (cid:17) d − d . h ℓ i d +1 (cid:16) ˆ T dL |∇ ( φ L − φ ′ L ) | (cid:17) , (4.23)we deduce in dimension d > , ˆ T dL |∇ ( φ L − φ ′ L ) | . h ℓ i d +1 (cid:16) ˆ T dL |∇ ( φ L − φ ′ L ) | (cid:17) + (cid:16) h ℓ i d +2 + ˆ B ℓ +2 ( x ) |∇ φ ′ L | (cid:17) (cid:16) h ℓ i d +2 + ˆ B ℓ +2 ( x ) |∇ φ L | (cid:17) , hence, ˆ T dL |∇ ( φ L − φ ′ L ) | . (cid:16) h ℓ i d +2 + ˆ B ℓ +2 ( x ) |∇ φ ′ L | (cid:17) (cid:16) h ℓ i d +2 + ˆ B ℓ +2 ( x ) |∇ φ L | (cid:17) , and the claim (4.21) follows by the triangle inequality. Substep φ ′ L by φ L in theright-hand side of the result (4.12) of Substep 1.1, we obtain (cid:12)(cid:12)(cid:12) ∂ osc P ,B ℓ ( x ) ˆ T dL \I L g : ∇ φ L (cid:12)(cid:12)(cid:12) . L − d h ℓ i d (cid:12)(cid:12)(cid:12) e · ˆ I L v L (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˆ B ℓ +1 ( x ) v L (cid:12)(cid:12)(cid:12) + (cid:16) ˆ B ℓ +2 ( x ) |∇ v L | + |h∇i g | (cid:17) (cid:16) h ℓ i d +2 + ˆ B ℓ +2 ( x ) |∇ φ L | (cid:17) . Inserting this into (4.11), we find for all p ≥ , (cid:13)(cid:13)(cid:13) ˆ T dL \I L g : ∇ φ L (cid:13)(cid:13)(cid:13) p (Ω) . p L − d (cid:13)(cid:13)(cid:13) e · ˆ I L v L (cid:13)(cid:13)(cid:13) p (Ω) (cid:16) ˆ L h ℓ i dp π ( ℓ ) dℓ (cid:17) p + p (cid:13)(cid:13)(cid:13) ˆ T dL | v L | (cid:13)(cid:13)(cid:13) L p (Ω) (cid:16) ˆ L h ℓ i dp π ( ℓ ) dℓ (cid:17) p + p E (cid:20) ˆ L (cid:18) ˆ T dL ζ ℓ ( x ) (cid:16) B ℓ +2 ( x ) |∇ v L | + |h∇i g | (cid:17) dx (cid:19) p h ℓ i dp π ( ℓ ) dℓ (cid:21) p , (4.24)where we have set for abbreviation, ζ ℓ ( x ) := h ℓ i + (cid:16) B ℓ +2 ( x ) |∇ φ L | (cid:17) . M. DUERINCKX AND A. GLORIA
Before estimating the last right-hand side term in (4.24), we first smuggle in a spatialaverage at some arbitrary scale ≤ R ≤ L , ˆ T dL ζ ℓ ( x ) (cid:16) B ℓ +2 ( x ) |∇ v L | + |h∇i g | (cid:17) dx . ˆ T dL (cid:16) sup B R ( y ) ζ ℓ (cid:17)(cid:18) B ℓ +2 ( y ) (cid:16) B R +1 ( x ) [ ∇ v L ] + [ h∇i g ] (cid:17) dx (cid:19) dy. We then use a duality representation to compute the L p (Ω) -norm of this expression (in thefollowing, X denotes a random variable, which is independent of the space variable), E (cid:20)(cid:18) ˆ T dL ζ ℓ ( x ) (cid:16) B ℓ +2 ( x ) |∇ v L | + |h∇i g | (cid:17) dx (cid:19) p (cid:21) p . sup k X k L2 p ′ (Ω) =1 E (cid:20) ˆ T dL (cid:16) sup B R ( y ) ζ ℓ (cid:17) × (cid:18) B ℓ +2 ( y ) (cid:16) B R +1 ( x ) [ ∇ ( Xv L )] + X [ h∇i g ] (cid:17) dx (cid:19) dy (cid:21) . By Hölder’s inequality and by stationarity of ζ ℓ , we find E (cid:20)(cid:18) ˆ T dL ζ ℓ ( x ) (cid:16) B ℓ +2 ( x ) |∇ v L | + |h∇i g | (cid:17) dx (cid:19) p (cid:21) p . E h sup B R ζ pℓ i p × sup k X k L2 p ′ (Ω) =1 ˆ T dL E (cid:20)(cid:18) B ℓ +2 ( y ) (cid:16) B R +1 ( x ) [ ∇ ( Xv L )] + X [ h∇i g ] (cid:17) dx (cid:19) p ′ (cid:21) p ′ dy, hence, since g (unlike v L ) is deterministic, using Jensen’s inequality, E (cid:20)(cid:18) ˆ T dL ζ ℓ ( x ) (cid:16) B ℓ +2 ( x ) |∇ v L | + |h∇i g | (cid:17) dx (cid:19) p (cid:21) p . E h sup B R ζ pℓ i p (cid:18) kh∇i g k ( T dL ) + sup k X k L2 p ′ (Ω) =1 k [ ∇ ( Xv L )] k ( T dL ;L p ′ (Ω)) (cid:19) . By perturbative annealed L p regularity theory in form of Theorem 4.5 (without loss ofstochastic integrability!), we deduce for p ≫ , E (cid:20)(cid:18) ˆ T dL ζ ℓ ( x ) (cid:16) B ℓ +2 ( x ) |∇ v L | + |h∇i g | (cid:17) dx (cid:19) p (cid:21) p . E h sup B R ζ pℓ i p kh∇i g k ( T dL ) , where the supremum of ζ ℓ can be estimated as follows, for all q ≥ , E h sup B R ζ pℓ i p . h ℓ i + (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ˆ B R +1 [ ∇ φ L ] q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) , EDIMENTATION OF RANDOM SUSPENSIONS For all ≤ R ≤ L , q ≥ , and p ≫ , inserting this into (4.24) yields (cid:13)(cid:13)(cid:13) ˆ T dL \I L g : ∇ φ L (cid:13)(cid:13)(cid:13) p (Ω) . p L − d (cid:13)(cid:13)(cid:13) e · ˆ I L v L (cid:13)(cid:13)(cid:13) p (Ω) (cid:16) ˆ L h ℓ i dp π ( ℓ ) dℓ (cid:17) p + p (cid:13)(cid:13)(cid:13) ˆ T dL | v L | (cid:13)(cid:13)(cid:13) L p (Ω) (cid:16) ˆ L h ℓ i dp π ( ℓ ) dℓ (cid:17) p + p kh∇i g k ( T dL ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ˆ B R +1 [ ∇ φ L ] q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) (cid:16) ˆ L h ℓ i ( d +2) p π ( ℓ ) dℓ (cid:17) p . Using the Poincaré-Sobolev inequality in dimension d > , Jensen’s inequality, and thenon-perturbative annealed L p regularity theory in form of Theorem 4.4, recalling that g isdeterministic, we find for < p < ∞ , (cid:13)(cid:13)(cid:13) ˆ T dL | v L | (cid:13)(cid:13)(cid:13) L p (Ω) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ˆ T dL |∇ v L | dd +2 (cid:17) d +2 d (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) . k∇ v L k dd +2 ( T dL ;L p (Ω)) . p k g k dd +2 ( T dL ) . (4.25)Combining this with the above, and using the superalgebraic decay of the weight π , theclaim (4.10) follows. Step
2. Proof that under (Mix + ) in dimension d > there holds for all ≤ R ≤ L , q ≥ ,and ≪ p < ∞ , (cid:13)(cid:13)(cid:13) e · ˆ I L v L (cid:13)(cid:13)(cid:13) p (Ω) . p k g k L dd +2 ( T dL ) + kh∇i g k ( T dL ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ˆ B R [ ∇ φ L ] q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) . (4.26)This estimate allows to upgrade (4.10) to (cid:13)(cid:13)(cid:13) ˆ T dL \I L g : ∇ φ L (cid:13)(cid:13)(cid:13) p (Ω) . p k g k dd +2 ( T dL ) + kh∇i g k ( T dL ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ˆ B R [ ∇ φ L ] q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) . (4.27)We turn to the argument for (4.26). Using the version (4.5) of the multiscale varianceinequality (2.2) to control higher moments, we can write (cid:13)(cid:13)(cid:13) e · ˆ I L v L (cid:13)(cid:13)(cid:13) p (Ω) . p E (cid:20) ˆ L (cid:18) ˆ T dL (cid:16) ∂ osc P L ,B ℓ ( x ) e · ˆ I L v L (cid:17) dx (cid:19) p h ℓ i − dp π ( ℓ ) dℓ (cid:21) p , (4.28)and it remains to estimate the oscillation. Given ≤ ℓ ≤ L and x ∈ T dL , and given arealization, let P ′ L be a locally finite point set satisfying the hardcore condition in (H δ ),with P ′ L | T dL \ B ℓ ( x ) = P L | T dL \ B ℓ ( x ) , and denote by v ′ L the corresponding solution of (4.9)with P L replaced by P ′ L . We decompose (cid:12)(cid:12)(cid:12) e · ˆ I L v L − e · ˆ I ′ L v ′ L (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) e · ˆ I ′ L ( v L − v ′ L ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) e · ˆ I L v L − e · ˆ I ′ L v L (cid:12)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12) e · ˆ I ′ L ( v L − v ′ L ) (cid:12)(cid:12)(cid:12) + ˆ B ℓ +1 ( x ) | v L | . (4.29) M. DUERINCKX AND A. GLORIA
Testing the equation (3.2) for φ ′ L with v L − v ′ L , we obtain for any constant c ′ ∈ R , ˆ T dL ∇ φ ′ L : ∇ ( v L − v ′ L )= − α ′ L e · ˆ T dL \I ′ L ( v L − v ′ L ) − X n ˆ ∂I ′ n,L ( v L − v ′ L ) · σ ( φ ′ L , Π ′ L − c ′ ) ν, hence, in view of the boundary conditions and of the choice ´ T dL ( v L − v ′ L ) = 0 , ( α ′ L + 1) e · ˆ I ′ L ( v L − v ′ L ) = ˆ T dL ∇ φ ′ L : ∇ ( v L − v ′ L )+ X n : x ′ n,L ∈ B ℓ ( x ) ˆ ∂I ′ n,L (cid:16) v L − I ′ n,L v L (cid:17) · σ ( φ ′ L , Π ′ L − c ′ ) ν. Next, testing the equation (4.14) for v L − v ′ L with φ ′ L , we obtain for any constant c ∈ R , ( α ′ L + 1) e · ˆ I ′ L ( v L − v ′ L ) = − X n : x n,L ∈ B ℓ ( x ) ˆ ∂I n,L (cid:16) φ ′ L − I n,L φ ′ L (cid:17) · (cid:0) g + σ ( v L , P L − c ) (cid:1) + X n : x ′ n,L ∈ B ℓ ( x ) ˆ ∂I ′ n,L (cid:16) v L − I ′ n,L v L (cid:17) · σ ( φ ′ L , Π ′ L − c ′ ) ν. In view of the local regularity theory for the Stokes equation in form of (4.18) and (4.19),together with the pressure estimates of Lemma 4.7, we deduce (cid:12)(cid:12)(cid:12) e · ˆ I ′ L ( v L − v ′ L ) (cid:12)(cid:12)(cid:12) . (cid:16) h ℓ i d +2 + ˆ B ℓ +2 ( x ) |∇ φ ′ L | (cid:17) (cid:16) ˆ B ℓ +2 ( x ) |∇ v L | + |h∇i g | (cid:17) . Using the bound (4.21) of Substep 1.2 to replace φ ′ L by φ L in the right-hand side, andinserting this into the decomposition (4.29), we are led to (cid:12)(cid:12)(cid:12) ∂ osc P L ,B ℓ ( x ) e · ˆ I L v L (cid:12)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12) ˆ B ℓ +1 ( x ) v L (cid:12)(cid:12)(cid:12) + (cid:16) h ℓ i d +2 + ˆ B ℓ +2 ( x ) |∇ φ L | (cid:17) (cid:16) ˆ B ℓ +2 ( x ) |∇ v L | + |h∇i g | (cid:17) . Inserting this into (4.28), the claim (4.26) follows as in Substep 1.3.
Step
3. Fluctuation scaling on I L : proof that under (Mix + ) in dimension d > there holdsfor all ≤ R ≤ L , q ≥ , and ≪ p < ∞ , (cid:13)(cid:13)(cid:13) ˆ I L g : ∇ φ L (cid:13)(cid:13)(cid:13) p (Ω) . p k g k dd +2 ( T dL ) + k g k ( T dL ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ˆ B R [ ∇ φ L ] q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) , which yields the conclusion (i) in combination with (4.27). EDIMENTATION OF RANDOM SUSPENSIONS Using the version (4.5) of the multiscale variance inequality (2.2) to control higher mo-ments, we can write (cid:13)(cid:13)(cid:13) ˆ I L g : ∇ φ L (cid:13)(cid:13)(cid:13) p (Ω) . p E (cid:20) ˆ L (cid:18) ˆ T dL (cid:16) ∂ osc P L ,B ℓ ( x ) ˆ I L g : ∇ φ L (cid:17) dx (cid:19) p h ℓ i − dp π ( ℓ ) dℓ (cid:21) p , (4.30)and it remains to estimate the oscillation. Given ≤ ℓ ≤ L and x ∈ R d , and given arealization, let P ′ L be a locally finite point set satisfying the hardcore condition in (H δ ), with P ′ L | T dL \ B ℓ ( x ) = P L | T dL \ B ℓ ( x ) , and denote by φ ′ L the corresponding solution of equation (3.1)with P L replaced by P ′ L . We decompose (cid:12)(cid:12)(cid:12) ˆ I L g : ∇ φ L − ˆ I ′ L g : ∇ φ ′ L (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) X n : x n,L B ℓ ( x ) ˆ I n,L g : ( ∇ φ L − ∇ φ ′ L ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) X n : x n,L ∈ B ℓ ( x ) ˆ I n,L g : ∇ φ L − X n : x ′ n,L B ℓ ( x ) ˆ I ′ n,L g : ∇ φ ′ L (cid:12)(cid:12)(cid:12)(cid:12) . Since φ L and φ ′ L are both affine inside particles I n,L ’s with x n,L / ∈ B ℓ ( x ) , we can furtherwrite (cid:12)(cid:12)(cid:12) ˆ I L g : ∇ φ L − ˆ I ′ L g : ∇ φ ′ L (cid:12)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12)(cid:12) X n (cid:16) I n,L g (cid:17) : ˆ I n,L ( ∇ φ L − ∇ φ ′ L ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:16) ˆ B ℓ +1 ( x ) | g | (cid:17) (cid:16) ˆ B ℓ +1 ( x ) |∇ φ L | + |∇ φ ′ L | (cid:17) . Using the bound (4.21) of Substep 1.2 to replace φ ′ L by φ L in the right-hand side, thisyields (cid:12)(cid:12)(cid:12) ˆ I L g : ∇ φ L − ˆ I ′ L g : ∇ φ ′ L (cid:12)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12)(cid:12) X n (cid:16) I n,L g (cid:17) : ˆ I n,L ( ∇ φ L − ∇ φ ′ L ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:16) ˆ B ℓ +1 ( x ) | g | (cid:17) (cid:16) h ℓ i d +2 + ˆ B ℓ +2 ( x ) |∇ φ L | (cid:17) . (4.31)It remains to estimate the first right-hand side term. By a standard use of the Bogovskiioperator in form of [25, Theorem III.3.1], we can construct a divergence-free tensor field h L ∈ H ( T dL ) d × d such that h L = 0 outside I L + δB = { x ∈ T dL : dist( x, I L ) < δ } andsuch that for all n there hold h L | I n,L + δB ∈ H ( I n,L + δB ) d × d , h L | I n,L = I n,L g, k h L k H ( I n,L + δB ) . k g k L ( I n,L ) . (4.32) M. DUERINCKX AND A. GLORIA
In these terms, using that h L is divergence-free, we can write by means of Stokes’ formula, X n (cid:16) I n,L g (cid:17) : ˆ I n,L ( ∇ φ L − ∇ φ ′ L ) = X n (cid:16) I n,L g (cid:17) : ˆ ∂I n,L ( φ L − φ ′ L ) ⊗ ν = X n ˆ ∂I n,L h L : ( φ L − φ ′ L ) ⊗ ν = − ˆ T dL \I L ∇ i (cid:0) h L : ( φ L − φ ′ L ) ⊗ e i (cid:1) = − ˆ T dL \I L h L : ∇ ( φ L − φ ′ L ) , so that we are reduced to the sensitivity of the gradient ∇ φ L outside particles as alreadystudied in Step 1. Appealing to the bound (4.20) of Substep 1.1 together with the re-sult (4.21) of Substep 1.2, and combining with (4.31), we obtain (cid:12)(cid:12)(cid:12) ∂ osc P ,B ℓ ( x ) ˆ I L g : ∇ φ L (cid:12)(cid:12)(cid:12) . L − d h ℓ i d (cid:12)(cid:12)(cid:12) e · ˆ I L v L,h (cid:12)(cid:12)(cid:12) + ˆ B ℓ +1 ( x ) | v L,h | + (cid:16) ˆ B ℓ +2 ( x ) |∇ v L,h | + |h∇i h L | + | g | (cid:17) (cid:16) h ℓ i d +2 + ˆ B ℓ +2 ( x ) |∇ φ L | (cid:17) , where v L,h denotes the solution of the auxiliary problem (4.9) with g replaced by h L . Insert-ing this into (4.30), proceeding as in Substep 1.3, and taking advantage of the bound (4.32)in form of the pointwise estimate [ h∇i h L ] ( x ) . ´ B ( x ) [ g ] , the claim follows. Step
4. Hyperuniform case (ii).The hyperuniformity assumption (Hyp + ) allows to replace the multiscale inequality (4.5)by its version (4.6), where the oscillation derivative ∂ osc is replaced by its hyperuniformcounterpart ∂ hyp := ∂ mov + L − ∂ osc . The main part ∂ mov only accounts for local per-turbations with a fixed number of points, in accordance with the suppression of densityfluctuations, while the second part involves the full oscillation ∂ osc but has the small pref-actor L − . Given a random variable Y ( P L ) with E [ Y ( P L )] = 0 , we write (4.6) as k Y ( P L ) k p (Ω) . p E hyp p [ Y ( P L )] ≤ p E mov p [ Y ( P L )] + p L − E osc p [ Y ( P L )] , (4.33)where we have set for abbreviation, for ⋆ = osc , hyp , mov , E ∗ p [ Y ( P L )] := E (cid:20) ˆ L (cid:18) ˆ T dL (cid:16) ∂ ⋆ P L ,B ℓ ( x ) Y ( P L ) (cid:17) dx (cid:19) p h ℓ i − dp π ( ℓ ) dℓ (cid:21) p . We split the proof into two substeps, separately considering the contribution of E mov p andof L − E osc p for Y ( P L ) = ´ T dL g : ∇ φ L . Substep E mov p .In Step 1, given ≤ ℓ ≤ L and x ∈ R d , and given a realization, we now let P ′ L be a locallyfinite point set satisfying the hardcore condition in T dL , with P ′ L | T dL \ B ℓ ( x ) = P L | T dL \ B ℓ ( x ) and with the additional constraint ♯ P ′ L | B ℓ ( x ) = ♯ P L | B ℓ ( x ) . In particular, the latter implies EDIMENTATION OF RANDOM SUSPENSIONS α L = α ′ L , so that (4.17) becomes ˆ T dL \I L g : ∇ ( φ L − φ ′ L ) = − ( α L + 1) (cid:18) X n : x n,L ∈ B ℓ ( x ) e · ˆ I n,L v L − X n : x ′ n,L ∈ B ℓ ( x ) e · ˆ I ′ n,L v L (cid:19) − X n : x ′ n,L ∈ B ℓ ( x ) ˆ ∂I ′ n,L (cid:16) v L − I ′ n,L v L (cid:17) · σ ( φ ′ L , Π ′ L − c ′ ) ν + X n : x n,L ∈ B ℓ ( x ) ˆ ∂I n,L (cid:16) φ ′ L − I n,L φ ′ L (cid:17) · (cid:0) g + σ ( v L , P L − c ) (cid:1) ν. (4.34)The last two terms are estimated as in Substep 1.1, but we argue differently for the firstone. The restriction ♯ P ′ L | B ℓ ( x ) = ♯ P L | B ℓ ( x ) allows to write P L | B ℓ ( x ) := { x n j ,L } mj =1 and P ′ L | B ℓ ( x ) := { x ′ n ′ j ,L } mj =1 for some ≤ m . h ℓ i d . We can then easily reformulate as followsthe first right-hand side term of (4.34), using number conservation and disjointness, (cid:12)(cid:12)(cid:12)(cid:12) X n : x n,L ∈ B ℓ ( x ) e · ˆ I n,L v L − X n : x ′ n,L ∈ B ℓ ( x ) e · ˆ I ′ n,L v L (cid:12)(cid:12)(cid:12)(cid:12) ≤ m X j =1 (cid:12)(cid:12)(cid:12)(cid:12) ˆ B ( x nj ,L ) v L − ˆ B ( x ′ n ′ j ,L ) v L (cid:12)(cid:12)(cid:12)(cid:12) . h ℓ i ˆ B ℓ +1 ( x ) |∇ v L | , so that the result (4.12) of Substep 1.1 is replaced by (cid:12)(cid:12)(cid:12) ˆ T dL \I L g : ∇ φ L − ˆ T dL \I ′ L g : ∇ φ ′ L (cid:12)(cid:12)(cid:12) . h ℓ i ˆ B ℓ +1 ( x ) |∇ v L | + (cid:16) ˆ B ℓ +2 ( x ) |∇ v L | + |h∇i g | (cid:17) (cid:16) h ℓ i d +2 + ˆ B ℓ +2 ( x ) |∇ φ ′ L | (cid:17) . Next, a similar argument as above shows that the bound (4.22) in Substep 1.2 is replacedby ˆ T dL |∇ ( φ L − φ ′ L ) | . h ℓ i ˆ B ℓ +1 ( x ) |∇ ( φ L − φ ′ L ) | + (cid:16) h ℓ i d +2 + ˆ B ℓ +2 ( x ) |∇ φ ′ L | (cid:17) (cid:16) h ℓ i d +2 + ˆ B ℓ +2 ( x ) |∇ φ L | (cid:17) , so that the result (4.21) now holds in any dimension d ≥ . Arguing as in Substep 1.3, theresult (4.10) of Step 1 is then replaced by the following: under (Hyp + ) in any dimension,for all ≤ R ≤ L , q ≥ , and ≪ p < ∞ , E mov p h ˆ T dL \I L g : ∇ φ L i . p kh∇i g k ( T dL ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ˆ B R [ ∇ φ L ] q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) . Likewise, the result of Step 3 is replaced by the following: under (Hyp + ) in any dimension,for all ≤ R ≤ L , q ≥ , and ≪ p < ∞ , E mov p h ˆ I L g : ∇ φ L i . p k g k ( T dL ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ˆ B R [ ∇ φ L ] q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) , M. DUERINCKX AND A. GLORIA and hence, E mov p h ˆ T dL g : ∇ φ L i . p kh∇i g k ( T dL ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ˆ B R [ ∇ φ L ] q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) . Substep L − E osc p .The contribution of E osc p has already been computed in Steps 1–3 in dimension d > , andit remains to show that the same bound hold for L − E osc p in any dimension. We only needto revisit the two places where the restriction to d > is used, that is, Substep 1.2 andthe estimate (4.25) in Substep 1.3.We start with revisiting Substep 1.2. Instead of using the Poincaré-Sobolev inequal-ity (4.23), we use Poincaré’s inequality in the following form, ˆ B ℓ +1 ( x ) | φ L − φ ′ L | . h ℓ i d (cid:16) ˆ T dL | φ L − φ ′ L | (cid:17) . L h ℓ i d (cid:16) ˆ T dL |∇ ( φ L − φ ′ L ) | (cid:17) , so that the conclusion (4.21) is replaced by the following, in any dimension d ≥ , ˆ B ℓ +2 ( x ) |∇ φ ′ L | . L h ℓ i d + ˆ B ℓ +2 ( x ) |∇ φ L | . (4.35)Next, we revisit the estimate (4.25) in Substep 1.3: instead of appealing to the Poincaré-Sobolev inequality and to non-perturbative annealed L p regularity in form of Theorem 4.4,we simply use Poincaré’s inequality and the energy inequality (4.8), in any dimension d ≥ , (cid:13)(cid:13)(cid:13) ˆ T dL | v L | (cid:13)(cid:13)(cid:13) L p (Ω) . L (cid:13)(cid:13)(cid:13) ˆ T dL |∇ v L | (cid:13)(cid:13)(cid:13) L p (Ω) . L k g k ( T dL ) . (4.36)Up to these two modifications (4.35) and (4.36), the conclusion of Steps 1–3 becomes thefollowing, for any dimension d ≥ , for all ≤ R ≤ L , q ≥ , and ≪ p < ∞ , E osc p h ˆ T dL g : ∇ φ L i . p L kh∇i g k ( T dL ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ˆ B R +1 [ ∇ φ L ] q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) . Multiplying both sides by L − , and inserting this into (4.33) together with the result ofSubstep 4.1, the conclusion (ii) follows. In contrast with the proof of (i), we note that (ii)only requires perturbative annealed L p regularity in form of Theorem 4.5. (cid:3) Proof of Proposition 4.2.
Given a ball D ⊂ T dL with radius r D ≥ and givenarbitrary constants c D ∈ R d and c ′ D ∈ R , testing the equation (3.2) for φ L with η D ( φ L − c D ) ,where η D denotes a cut-off function with η D = 1 in D , η D = 0 outside D , and |∇ η D | . r D ,such that η D is constant in I n,L for all n , using the boundary conditions and recalling that div φ L = 0 , we easily obtain the following Caccioppoli type estimate, ˆ D |∇ φ L | . r D ˆ D | φ L − c D | + (cid:16) ˆ D | Π L − c ′ D | T dL \I L (cid:17) (cid:16) r D ˆ D | φ L − c D | (cid:17) + ˆ D | φ L − c D | . Bounding the last right-hand side term by ˆ D | φ L − c D | . r d +2 D + 1 r D ˆ D | φ L − c D | , EDIMENTATION OF RANDOM SUSPENSIONS choosing c ′ D := ffl D \I L Π L , and applying the pressure estimate of Lemma 4.7, we obtainfor all K ≥ , D |∇ φ L | . K r D D | φ L − c D | + 1 K D |∇ φ L | + r D . (4.37)Using the Poincaré-Sobolev inequality to estimate the first right-hand side term, with thechoice c D := ffl D φ L , we deduce (cid:16) D |∇ φ L | (cid:17) . K (cid:16) D |∇ φ L | dd +2 (cid:17) d +22 d + 1 K (cid:16) D |∇ φ L | (cid:17) + r D . While this is proven here for all balls D with radius r D ≥ , smuggling in local quadraticaverages at scale 1 allows to infer that for all balls D (with any radius r D > ) and K ≥ , (cid:16) D [ ∇ φ L ] (cid:17) . K (cid:16) D [ ∇ φ L ] dd +2 (cid:17) d +22 d + 1 K (cid:16) D [ ∇ φ L ] (cid:17) + r D + 1 . Choosing K large enough and applying Gehring’s lemma in form of Lemma 4.8 (with G = [ ∇ φ L ] dd +2 , F ≡ R dd +2 , and q = d +2 q ), we deduce the following Meyers type estimate:there exists some η > (only depending on d, δ ) such that for all ≤ q ≤ η andall R ≥ , (cid:16) B R [ ∇ φ L ] q (cid:17) q . R + B R [ ∇ φ L ] . Combining this with (4.37), we obtain for all K ≥ , for any constant c R ∈ R d , (cid:16) B R [ ∇ φ L ] q (cid:17) q . K R B R | φ L − c R | + 1 K B R |∇ φ L | + R . (4.38)For ≤ r ≤ R , choosing c R := ffl B R χ r ∗ φ L , Poincaré’s inequality yields B R | φ L − c R | . B R | φ L − χ r ∗ φ L | + B R | χ r ∗ φ L − c R | . χ r B R |∇ φ L | + R B R | χ r ∗ ∇ φ L | . Inserting this into (4.38), we find (cid:16) B R |∇ φ L | q (cid:17) q . (cid:16) K r R + 1 K (cid:17) B R |∇ φ L | + K B R | χ r ∗ ∇ φ L | + R . Since stationarity and Jensen’s inequality yield (cid:13)(cid:13)(cid:13) B R |∇ φ L | (cid:13)(cid:13)(cid:13) L p (Ω) . (cid:13)(cid:13)(cid:13) B R |∇ φ L | (cid:13)(cid:13)(cid:13) L p (Ω) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) B R |∇ φ L | q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) and E (cid:20)(cid:16) B R | χ r ∗ ∇ φ L | (cid:17) p (cid:21) ≤ E (cid:20) B R | χ r ∗ ∇ φ L | p (cid:21) = E (cid:2) | χ r ∗ ∇ φ L | p (cid:3) , M. DUERINCKX AND A. GLORIA this implies (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) B R |∇ φ L | q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) . χ (cid:16) K r R + 1 K (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:16) B R |∇ φ L | q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) + K (cid:13)(cid:13)(cid:13) ˆ T dL χ r ∇ φ L (cid:13)(cid:13)(cid:13) p (Ω) + R . Choosing K ≫ and R ≫ χ,K r , the first right-hand side term can be absorbed and theconclusion follows. (cid:3) Homogenization result
The proof of Theorem 3 combines the quantitative estimates of Theorems 1–2 togetherwith the homogenization result for colloidal (non-sedimenting) suspensions in [17, Theo-rem 1]. In particular, this qualitative result requires mixing assumptions and quantitativeestimates.
Proof of Theorem 3.
We start with a suitable splitting of the Stokes problem (2.13). Interms of the renormalized pressure ˜ P ε := P ε − ε λe · x , rewriting the boundary conditionsas follows, ε d − e | I ωn | + ˆ ε∂I ωn σ ( u ωε , P ωε ) ν = ε d − e | I ωn | − ˆ ε∂I ωn ε ( λe · x ) ν + ˆ ε∂I ωn σ ( u ωε , ˜ P ωε ) ν = ε d − (1 − λ ) e | I ωn | + ˆ ε∂I ωn σ ( u ωε , ˜ P ωε ) ν, and for all Θ ∈ M skew , ˆ ε∂I ωn Θ ν · σ ( u ωε , P ωε ) ν = ˆ ε∂I ωn Θ ν · σ ( u ωε , ˜ P ωε ) ν, we can rewrite (2.13) in the following equivalent form, −△ u ωε + ∇ ˜ P ωε = f − ε λe, in U \ I ωε ( U ) , div u ωε = 0 , in U \ I ωε ( U ) ,u ωε = 0 , on ∂U , D( u ωε ) = 0 , in I ωε ( U ) ,ε d − (1 − λ ) e | I ωn | + ´ ε∂I ωn σ ( u ωε , ˜ P ωε ) ν = 0 , ∀ n ∈ N ωε ( U ) , ´ ε∂I ωn Θ ν · σ ( u ωε , ˜ P ωε ) ν = 0 , ∀ n ∈ N ωε ( U ) , ∀ Θ ∈ M skew , (5.1)By linearity and since α = λ − λ , we may then decompose the solution into two parts, u ωε = u ωε, + (1 − λ ) u ωε, , ˜ P ωε = P ωε, + (1 − λ ) P ωε, , EDIMENTATION OF RANDOM SUSPENSIONS where ( u ωε, , P ωε, ) solves −△ u ωε, + ∇ P ωε, = f, in U \ I ωε ( U ) , div u ωε, = 0 , in U \ I ωε ( U ) ,u ωε, = 0 , on ∂U , D( u ωε, ) = 0 , in I ωε ( U ) , ´ ε∂I ωn σ ( u ωε, , P ωε, ) ν = 0 , ∀ n ∈ N ωε ( U ) , ´ ε∂I ωn Θ ν · σ ( u ωε, , P ωε, ) ν = 0 , ∀ n ∈ N ωε ( U ) , ∀ Θ ∈ M skew , (5.2)while ( u ωε, , P ωε, ) is a rescaled proxy with Dirichlet boundary conditions on U for the “sed-imentation corrector” ( φ, Π) in Theorem 1, cf. (2.6), −△ u ωε, + ∇ P ωε, = − ε αe, in U \ I ωε ( U ) , div u ωε, = 0 , in U \ I ωε ( U ) ,u ωε, = 0 , on ∂U , D( u ωε, ) = 0 , in I ωε ( U ) ,ε d − e | I ωn | + ´ ε∂I ωn σ ( u ωε, , P ωε, ) ν = 0 , ∀ n ∈ N ωε ( U ) , ´ ε∂I ωn Θ ν · σ ( u ωε, , P ωε, ) ν = 0 , ∀ n ∈ N ωε ( U ) , ∀ Θ ∈ M skew , (5.3)where we recall α = λ − λ . We split the proof into two steps, and analyze the two contribu-tions separately. Step
1. Homogenization of (5.2).The system (5.2) coincides with the equations for a steady Stokes fluid with a colloidal(non-sedimenting) suspension, as we already studied in [17]. In view of [17, Theorem 1],for almost all ω , there holds u ωε, ⇀ ¯ u weakly in H ( U ) and (cid:16) P ωε, − U \I ωε ( U ) P ωε, (cid:17) U \I ωε ( U ) ⇀ ¯ P − U ¯ P , weakly in L ( U ) , where (¯ u, ¯ P ) denotes the unique weak solution of the homogenized problem (2.14). More-over, provided f ∈ L p ( U ) for some p > d , for almost all ω , in view of [17, Theorem 1], acorrector result holds for the velocity field in form of (cid:13)(cid:13)(cid:13) u ωε, − ¯ u − ε X E ∈E ψ ωE ( · ε ) ∇ E ¯ u (cid:13)(cid:13)(cid:13) H ( U ) → , (5.4)and for the pressure field in form of inf κ ∈ R (cid:13)(cid:13)(cid:13) P ωε, − ¯ P − X E ∈E (Σ ωE R d \I ω )( · ε ) ∇ E ¯ u − κ (cid:13)(cid:13)(cid:13) L ( U \I ωε ( U )) → . (5.5) Step
2. Analysis of (5.3): proof that, under (Mix + ) for d > or under (Hyp + ) for any d ≥ , there holds for almost all ω , k u ωε, − εφ ω ( · ε ) k H ( U ) → , (5.6) inf κ ∈ R (cid:13)(cid:13) P ωε, − (Π ω R d \I ω )( · ε ) − κ (cid:13)(cid:13) ( U \I ωε ( U )) → , M. DUERINCKX AND A. GLORIA where we recall that ( φ, Π) is the unique solution of the infinite-volume problem (2.6) asgiven by Theorem 1. More precisely, we claim that for all ≤ p < ∞ , k u ε, − εφ ( · ε ) k p (Ω; H ( U )) . p εµ d ( ε ) , (5.7) inf κ ∈ R (cid:13)(cid:13)(cid:0) P ε, − (Π R d \I )( · ε ) − κ (cid:1) U \I ε ( U ) (cid:13)(cid:13) p (Ω;L ( U )) . p εµ d ( ε ) , and that for all κ > there exists a random variable X κ with bounded moments such thatfor almost all ω , k u ωε, − εφ ω ( · ε ) k H ( U ) . ( X ωκ ) ε − κ µ d ( ε ) , (5.8) inf κ ∈ R (cid:13)(cid:13)(cid:0) P ωε, − (Π ω R d \I ω )( · ε ) − κ (cid:1) U \I ε ( U ) (cid:13)(cid:13) ( U ) . ( X ωκ ) ε − κ µ d ( ε ) , in terms of µ d ( r ) := under (Mix + ) with d > , or under (Hyp + ) with d > r ) : under (Mix + ) with d = 4 , or under (Hyp + ) with d = 2; h r i : under (Mix + ) with d = 3 , or under (Hyp + ) with d = 1 . We focus on the convergence of u ε, . The corresponding convergence of the pressure P ε, is obtained similarly, further using the Bogovskii operator as in the proof of Lemma 4.7(see also [17, Substep 8.3 of Section 3]); details are omitted. We split the proof into threefurther substeps. Substep ≤ R ≤ ε , k u ωε, − εφ ω ( · ε ) k H ( U ) . εR + ε d ˆ ∂ R U ε (cid:16) R | φ ω | + |∇ φ ω | (cid:17) , (5.9)where we use the short-hand notation U ε := ε U and ∂ R U ε := { x ∈ U ε : dist( x, ∂U ε ) < R } .We start by rescaling the equations: the functions v ε ( x ) := ε u ε, ( εx ) and Q ε ( x ) := P ε, ( εx ) on U ε satisfy −△ v ωε + ∇ Q ωε = − αe, in U ε \ I ω ( U ε ) , div v ωε = 0 , in U ε \ I ω ( U ε ) ,v ωε = 0 , on ∂U ε , D( v ωε ) = 0 , in I ω ( U ε ) ,e | I ωn | + ´ ∂I ωn σ ( v ωε , Q ωε ) ν = 0 , ∀ n ∈ N ω ( U ε ) , ´ ∂I ωn Θ ν · σ ( v ωε , Q ωε ) ν = 0 , ∀ n ∈ N ω ( U ε ) , ∀ Θ ∈ M skew , (5.10)where we use the short-hand notation I ω ( U ε ) := I ω ( U ε ) and N ω ( U ε ) := N ω ( U ε ) . ThisStokes system (5.10) is formally the approximation of the infinite-volume problem (2.6) onthe set U ε with homogeneous Dirichlet boundary conditions (and discarding particles thatare close to the boundary of U ε ); the claim (5.6) is therefore not surprising. Arguing as inStep 1 of the proof of Theorem 1, we note that the Stokes equation for v ωε implies in theweak sense on the whole rescaled domain U ε , − △ v ωε + ∇ ( Q ωε U ε \I ω ( U ε ) ) = − αe U ε \I ω ( U ε ) − X n ∈N ω ( U ε ) δ ∂I ωn σ ( v ωε , Q ωε ) ν, (5.11) EDIMENTATION OF RANDOM SUSPENSIONS which we compare to the corresponding equation for φ ω on the whole space R d , cf. (2.6), −△ φ ω + ∇ (Π ω R d \I ω ) = − αe R d \I ω − X n δ ∂I ωn σ ( φ ω , Π ω ) ν. We choose a smooth cut-off function η ωε : R d → [0 , such that η ωε is supported in U ε , η ωε is constant inside the particles I ωn and vanishes in I ωn for n / ∈ N ω ( U ε ) . In addition,given some ≤ R ≤ ε (that will be chosen later depending on ε and d ), we assume that η ωε satisfies η ωε ( x ) = 1 for all x ∈ U ε with dist( x, ∂U ε ) ≥ R , and |∇ η ωε | . R . The aboveequation for φ ω entails that η ωε φ ω satisfies the following in the weak sense on the wholespace R d , − △ ( η ωε φ ω ) + ∇ ( η ωε Π ω R d \I ω ) = − η ωε αe R d \I ω − η ωε X n δ ∂I ωn σ ( φ ω , Π ω ) ν − (cid:0) ∇ φ ω − Π ω R d \I ω (cid:1) ∇ η ωε − ∇ · ( φ ω ⊗ ∇ η ωε ) . (5.12)Substracting (5.12) from (5.11), and adding arbitrary constants to the pressures, we obtainfor any c, c ′ ∈ R , − △ ( v ωε − η ωε φ ω ) = ∇ (cid:16) η ωε (Π ω − c ) R d \I ω − ( Q ωε − c ′ ) U ε \I ω ( U ε ) (cid:17) + e (cid:0) αη ωε R d \I ω − α U ε \I ω ( U ε ) (cid:1) + (cid:0) ∇ φ ω − (Π ω − c ) R d \I ω (cid:1) ∇ η ωε + ∇ · ( φ ω ⊗ ∇ η ωε )+ X n ∈N ω ( U ε ) δ ∂I ωn (cid:16) η ωε σ ( φ ω , Π ω − c ) − σ ( v ωε , Q ωε − c ′ ) (cid:17) ν. Testing this equation with v ωε − η ωε φ ω ∈ H ( U ε ) , recalling that both v ωε and φ ω aredivergence-free, noting that v ωε − η ωε φ ω is constant inside the particles I ωn with n ∈ N ω ( U ε ) ,using the boundary conditions for v ωε and φ ω , and using the properties of η ωε , we are led to ˆ U ε |∇ ( v ωε − η ωε φ ω ) | . ˆ ∂ R U ε | v ωε − η ωε φ ω | + 1 R ˆ ∂ R U ε | φ ω | (cid:0) | Q ωε − c ′ | U ε \I ω ( U ε ) + | Π ω − c | R d \I ω (cid:1) + 1 R ˆ ∂ R U ε | v ωε − η ωε φ ω | (cid:0) |∇ φ ω | + | Π ω − c | R d \I ω (cid:1) + 1 R ˆ ∂ R U ε |∇ ( v ωε − η ωε φ ω ) || φ ω | . (5.13)We separately estimate the different right-hand side terms, and we start with the first one.Using Cauchy-Schwarz’ inequality and Poincaré’s inequality on H ( U ε ) restricted to theannulus ∂ R U ε (with Poincaré constant O ( R ) ), we find for all K ≥ , ˆ ∂ R U ε | v ωε − η ωε φ ω | . ( ε − d R ) (cid:16) ˆ ∂ R U ε | v ωε − η ωε φ ω | (cid:17) . ( ε − d R ) (cid:16) ˆ ∂ R U ε |∇ ( v ωε − η ωε φ ω ) | (cid:17) . K ε − d R + 1 K ˆ ∂ R U ε |∇ ( v ωε − η ωε φ ω ) | . M. DUERINCKX AND A. GLORIA
We turn to the second right-hand side term in (5.13). Using Cauchy-Schwarz’ inequalityand the pressure estimate of Lemma 4.7 with c := ffl ∂ R U ε \I ω Π ω and c ′ := ffl ∂ R U ε \I ω ( U ε ) Q ωε (the proof of Lemma 4.7 needs to be repeated on the annulus ∂ R U ε , using Poincaré’sinequality as above), we obtain for all K ≥ , R ˆ ∂ R U ε | φ ω | (cid:0) | Q ωε − c ′ | U ε \I ω ( U ε ) + | Π ω − c | R d \I ω (cid:1) . (cid:16) R ˆ ∂ R U ε | φ ω | (cid:17) (cid:16) ˆ ∂ R U ε | Q ωε − c ′ | U ε \I ω ( U ε ) + | Π ω − c | R d \I ω (cid:17) . (cid:16) R ˆ ∂ R U ε | φ ω | (cid:17) (cid:16) ε − d R + ˆ ∂ R U ε |∇ φ ω | + |∇ v ωε | (cid:17) . ε − d R + K ˆ ∂ R U ε (cid:16) R | φ ω | + |∇ φ ω | (cid:17) + 1 K ˆ ∂ R U ε |∇ ( v ωε − η ωε φ ω ) | . It remains to analyze the last two right-hand side terms in (5.13). Proceeding similarly asabove, we obtain for all K ≥ , R ˆ ∂ R U ε | v ωε − η ωε φ ω | (cid:0) |∇ φ ω | + | Π ω − c | R d \I ω (cid:1) . K ε − d R + K ˆ ∂ R U ε |∇ φ ω | + 1 K ˆ ∂ R U ε |∇ ( v ωε − η ωε φ ω ) | , and also R ˆ ∂ R U ε |∇ ( v ωε − η ωε φ ω ) || φ ω | . K ˆ ∂ R U ε R | φ ω | + 1 K ˆ ∂ R U ε |∇ ( v ωε − η ωε φ ω ) | . Inserting the above estimates into (5.13) and choosing K large enough to absorb part ofthe right-hand side terms into the left-hand side, we are led to ˆ U ε |∇ ( v ωε − η ωε φ ω ) | . ε − d R + ˆ ∂ R U ε (cid:16) R | φ ω | + |∇ φ ω | (cid:17) , which yields (5.9) after rescaling and using Poincaré’s inequality. Substep p -th power of (5.9) and using the quantitative estimates ofTheorem 2, under (Mix + ) for d > or under (Hyp + ) for any d ≥ , we deduce for all ≤ p < ∞ and ≤ R ≤ ε , k u ε, − εφ ( · ε ) k p (Ω; H ( U )) . p ε (cid:16) R µ d ( ε ) + R (cid:17) , and the claim (5.7) follows for the choice R = µ d ( ε ) . Substep R = µ d ( ε ) / . We need to convert the optimal annealedbounds (2.7)–(2.8) and (2.9)–(2.10) into suboptimal quenched bounds and prove that forall κ > there exists a random variable X κ with finite algebraic moments such that forall x ∈ R d , | φ ω ( x ) | ≤ X ωκ h x i κ µ d ( | x | ) , |∇ φ ω ( x ) | ≤ X ωκ h x i κ . (5.14) EDIMENTATION OF RANDOM SUSPENSIONS Indeed, equipped with these bounds, we deduce for all < κ ≪ and almost all ω , ε d ˆ ∂ R U ε (cid:16) R | φ ω | + |∇ φ ω | (cid:17) . ( X ωκ ) ε − κ µ d ( ε ) ε ↓ −−→ , so that (5.8) follows from (5.9). To prove (5.14), we set X ωκ := sup x ∈ R d (cid:16) h x i − κ µ d ( | x | ) − | φ ω ( x ) | + h x i − κ |∇ φ ω ( x ) | (cid:17) , and it suffices to check that this random variable has bounded moments. By local regularityin form of (4.3), a covering argument yields X ωκ . sup x ∈ √ d Z d (cid:18) h x i − κ µ d ( | x | ) − (cid:16) B ( x ) | φ ω | (cid:17) + h x i − κ (cid:16) B ( x ) |∇ φ ω | (cid:17) (cid:19) . Hence, given κ > , since the function x
7→ h x i − pκ is integrable on R d for p > d/κ , themoment bounds (2.7)–(2.8) and (2.9)–(2.10) lead (after bounding the supremum on √ d Z d by the sum) to E [( X κ ) p ] . p X x ∈ √ d Z d h x i − pκ E (cid:20) µ d ( | x | ) − p (cid:16) B ( x ) | φ | (cid:17) p + (cid:16) B ( x ) |∇ φ | (cid:17) p (cid:21) . p X x ∈ √ d Z d h x i − pκ . p , that is, X κ ∈ L p (Ω) (which is therefore almost surely finite). (cid:3) Appendix A. Functional-analytic version of hyperuniformity
The present appendix is devoted to a more detailed discussion and motivation of thehyperuniformity assumptions (Hyp) and (Hyp + ). Pioneered by Lebowitz [48, 41] in thephysical literature for Coulomb systems, the notion of hyperuniformity for a point process P on R d was first coined and theorized by Torquato and Stillinger [69] (see also [67, 27]) asthe suppression of density fluctuations. More precisely, while for a Poisson point processone has Var [ ♯ ( P ∩ B R )] ∝ | B R | , the process P is said to be hyperuniform if rather lim R ↑∞ Var [ ♯ ( P ∩ B R )] | B R | = 0 . (A.1)Typically, this concerns processes for which number fluctuations are a boundary effect, thatis, Var [ ♯ ( P ∩ B R )] . | ∂B R | . Hyperuniformity can be interpreted as a hidden form of orderon large scales and has been observed in various types of physical and biological systems,see e.g. [69, 67]. For Coulomb gases, rigorous results on the hyperuniformity of the Gibbsstate have been recently obtained in [9, 47, 64]. The simplest examples of hyperuniformprocesses are given by perturbed lattices, e.g. P := { z + U z : z ∈ Z d } where the latticepoints in Z d are pertubed by iid random variables { U z } z ∈ Z d (this model however onlyenjoys discrete stationarity due to the lattice structure; see [61, 73] for refined properties).Alternatively, hyperuniformity is known to be equivalent to the vanishing of the structurefactor in the small-wavenumber limit, that is, lim k → S ( k ) = 0 , M. DUERINCKX AND A. GLORIA where the structure factor is defined as the Fourier transform S ( k ) := b h ( k ) of the totalpair correlation function h , cf. Definition 2.1. If the pair correlation function is integrable,this can equivalently be written as S (0) = ˆ R d h = 0 . (A.2)The advantage of this reformulation in terms of the structure factor S ( k ) is that the lattercan be directly observed in diffraction experiments. This property of vanishing structurefactor is reminiscent of crystals, and indeed hyperuniform processes share crystalline prop-erties on large scales, although they can be statistically isotropic like gases, thereby leadingto a new state of matter [68].In the spirit of (A.2), in our periodized setting, for a family {P L } L ≥ of random pointprocesses P L on T dL , we consider the following slightly relaxed definition of hyperuniformity,cf. (Hyp), sup L ≥ L (cid:12)(cid:12)(cid:12) ˆ T dL h ,L (cid:12)(cid:12)(cid:12) < ∞ , (A.3)which is viewed as the approximate vanishing of the corresponding structure factors at in the limit L ↑ ∞ . The precise rate O ( L − ) is chosen in view of Lemma A.2 below.As claimed, such a definition of hyperuniformity in terms of structure factors implies thesuppression of density fluctuations in the following sense. Lemma A.1 (Density fluctuations [69, 27]) . Let a family {P L } L ≥ of random point pro-cesses P L on T dL be hyperuniform in the sense of (A.3) . Then, for all ≤ R ≤ L , Var [ ♯ ( P L ∩ B R )] . ρ L | B R | (cid:16) L − + ˆ T dL (cid:0) ∧ | x | L R (cid:1) | g ,L ( x ) | dx (cid:17) . In particular, provided that the pair correlation function g ,L has fast enough decay in thesense of sup L ≥ ´ T dL | x | L | g ,L ( x ) | dx < ∞ , we deduce for all ≤ R ≤ L , Var [ ♯ ( P L ∩ B R )] . ρ L | ∂B R | . ♦ Proof.
Number fluctuations are computed as follows,
Var [ ♯ ( P L ∩ B R )] = Var (cid:20)X n B R ( x n,L ) (cid:21) = ρ L ¨ B R × B R h ,L ( x − y ) dxdy = ρ L ˆ T dL | B R ( − x ) ∩ B R | h ,L ( x ) dx. Hence, decomposing | B R ( − x ) ∩ B R | = | B R | − | B R \ B R ( − x ) | = | B R | + (cid:0) ∧ | x | L R (cid:1) O ( | B R | ) , where the last summand is a continuous function that vanishes at x = 0 , we deduce (cid:12)(cid:12)(cid:12)(cid:12) | B R | Var [ ♯ ( P L ∩ B R )] − ρ L ˆ T dL h ,L (cid:12)(cid:12)(cid:12)(cid:12) . ρ L ˆ T dL (cid:0) ∧ | x | L R (cid:1) | g ,L ( x ) | dx, and the conclusion follows. (cid:3) EDIMENTATION OF RANDOM SUSPENSIONS Recall that a Poisson point process P = { x n } n on R d satisfies Var [ P n ζ ( x n )] ∝ ´ R d | ζ | for all ζ ∈ C ∞ c ( R d ) , and that similarly any point process P with integrable pair correlationfunction satisfies Var (cid:20)X n ζ ( x n ) (cid:21) = ρ ¨ R d × R d ζ ( x ) ζ ( y ) h ( x − y ) dxdy . ρ ˆ R d | ζ | . (A.4)The suppression of density fluctuations under hyperuniformity is naturally expected tolead to an improved version of such a variance estimate. Indeed, given an independentcopy { x ′ n } n of P = { x n } n , we may represent Var (cid:20) X n ζ ( x n ) (cid:21) = EE ′ (cid:20) (cid:16) X n ζ ( x n ) − X n ζ ( x ′ n ) (cid:17) (cid:21) , and the suppression of density fluctuations would formally allow to locally couple therandom point sets { x ′ n } n and { x n } n , only comparing points of the two realizations one toone locally, which would ideally translate into the gain of a derivative: for all ζ ∈ C ∞ c ( R d ) , Var (cid:20) X n ζ ( x n ) (cid:21) . ρ ˆ R d |∇ ζ | . (A.5)Indeed, provided that the pair correlation function has fast enough decay, it can be checkedthat hyperuniformity (A.1) is equivalent to this improved variance inequality (A.5). In ourperiodized setting, a rigorous statement is as follows. Lemma A.2 (Functional characterization of hyperuniformity) . Consider a family {P L } L ≥ of random point processes P L = { x n,L } n on T dL and assume that the pair correlation func-tion g ,L has fast enough decay in the sense of sup L ≥ ˆ T dL | x | L | g ,L ( x ) | dx < ∞ . Then {P L } L ≥ is hyperuniform in the sense of (A.3) if and only if for all L ≥ and ζ ∈ C ∞ per ( T dL ) we have Var (cid:20)X n ζ ( x n,L ) (cid:21) . ρ L ˆ T dL |∇ ζ | + L − ρ L ˆ T dL | ζ | . (A.6) In particular, the latter implies for all ζ ∈ C ∞ per ( T dL ) with E [ P n ζ ( x n,L )] = ρ L ´ T dL ζ = 0 , Var (cid:20)X n ζ ( x n,L ) (cid:21) . ρ L ˆ T dL |∇ ζ | . ♦ Proof.
By the definition of the total pair correlation function h ,L , cf. Definition 2.1, recallthat Var (cid:20)X n ζ ( x n,L ) (cid:21) = ρ L ¨ T dL × T dL ζ ( x ) ζ ( y ) h ,L ( x − y ) dxdy. Choosing ζ = 1 , the variance inequality (A.6) yields (cid:12)(cid:12)(cid:12) ˆ T dL h ,L (cid:12)(cid:12)(cid:12) . L − , M. DUERINCKX AND A. GLORIA that is, our definition (A.3) of hyperuniformity, and it remains to prove the converseimplication. Recomposing the square, the above identity for the variance takes the form
Var (cid:20)X n ζ ( x n,L ) (cid:21) = − ρ L ¨ T dL × T dL | ζ ( x ) − ζ ( y ) | g ,L ( x − y ) dxdy + (cid:16) ρ L ˆ T dL | ζ | (cid:17)(cid:16) ˆ T dL h ,L (cid:17) . Using the decay of correlations to estimate the first right-hand side term, and hyperuni-formity (A.3) to estimate the last one, the variance inequality (A.6) follows. (cid:3)
While the above variance inequality is restricted to linear functionals Y L = P n ζ ( x n,L ) ofthe point process, the analysis of nonlinear multibody interactions requires a correspondingtool for general nonlinear functionals. For general functionals Y = Y ( P ) of a Poissonpoint process P with unit intensity on R d , the following variance inequality is known tohold [72, 46], Var [ Y ( P )] ≤ E (cid:20) ˆ R d (cid:0) ∂ add y Y ( P ) (cid:1) dy (cid:21) , ∂ add y Y ( P ) := Y ( P ∪ { y } ) − Y ( P ) , where the difference operator ∂ add is known as the add-one-point operator . More generalversions of this type of functional inequality have been considered in the literature as aconvenient quantification of nonlinear mixing in order to cover various classes of exam-ples. In this spirit, our improved mixing assumption (Mix + ) is formulated in terms ofthe multiscale variance inequality (2.2) of [19, 20]. As shown in [20, Section 3], this cov-ers most examples of interest in materials science [66], including for instance (periodized)hardcore Poisson processes and random parking processes. Applied to a linear functional Y L = P n ζ ( x n,L ) , this variance inequality (2.2) clearly reduces to (A.4), so that (2.2) canindeed be viewed as a nonlinear version of (A.4).In the hyperuniform setting, as number fluctuations are suppressed, the add-one-pointoperator in the above or the general oscillation in (Mix + ) could be intuitively replacedby a suitable “move-point” operator , only allowing to locally move points of the process,but not add or remove any. A general version of this idea is formalized as the improvedhyperuniformity assumption (Hyp + ) in form of (2.3). Again, applied to a linear functional Y L = P n ζ ( x n,L ) , this new variance inequality (2.3) clearly reduces to (A.6), so that (2.3)can be viewed as a nonlinear version. We believe that this new functional inequality is ofindependent interest. Example A.3 (Perturbed lattices) . We briefly show that assumption (Hyp + ) in termsof the hyperuniform multiscale variance inequality (2.3) is not empty. For that purpose,we consider the simplest example of a hyperuniform process on T dL , that is, the perturbedlattice P L := { z + U z : z ∈ Z d ∩ T dL } , where the lattice points in Z d ∩ T dL are perturbed byiid random variables { U z } z ∈ Z d ∩ T dL , say with values in the unit ball B . This model P L iseasily checked to satisfy the following stronger version of the variance inequality (2.3): forall σ ( P L ) -measurable random variables Y ( P L ) , Var [ Y ( P L )] ≤ E (cid:20) ˆ T dL (cid:16) ∂ mov P L ,B √ d/ ( z ) Y ( P L ) (cid:17) dz (cid:21) , (A.7)where we recall that the move-point derivative ∂ mov is defined in (Hyp + ). This is indeeda direct consequence of the Efron-Stein inequality [22] for the iid sequence { U z } z ∈ Z d ∩ T dL in EDIMENTATION OF RANDOM SUSPENSIONS the following form: for P L,z := { y + U y } y : y = z ∪ { z + U ′ z } with { U ′ z } z an iid copy of { U z } z , Var [ Y ( P L )] ≤ E (cid:20) X z ∈ Z d ∩ T dL (cid:0) Y ( P L ) − Y ( P L,z ) (cid:1) (cid:21) , while Y ( P L ) − Y ( P L,z ) can be bounded by ∂ mov P ,B ( z ) Y ( P ) , thus leading to (A.7). ♦ Appendix B. Annealed regularity theory
This appendix is devoted to the development of annealed regularity theory for the steadyStokes equation in presence of a random suspension. More precisely, given a random forcing g ∈ L ∞ (Ω; C ∞ per ( T dL ) d × d ) , we consider the unique solution v L ∈ L ∞ (Ω; H ( T dL )) of thefollowing heterogeneous problem, −△ v ωL + ∇ P ωL = div g ω , in T dL \ I ωL , div v ωL = 0 , in T dL \ I ωL , D( v ωL ) = 0 , in I ωL , ´ ∂I ωn,L (cid:0) g ω + σ ( v ωL , P ωL ) (cid:1) ν = 0 , ∀ n, ´ ∂I ωn,L Θ ν · (cid:0) g ω + σ ( v ωL , P ωL ) (cid:1) ν = 0 , ∀ n, ∀ Θ ∈ M skew , (B.1)with ´ T dL v ωL = 0 . While the general proof of Theorem 4.4 is particularly demanding and ispostponed to the forthcoming companion contribution [18], we provide in this Appendixa full proof of the simpler perturbative version as stated in Theorem 4.5 (which is alreadysufficient for the hyperuniform setting).B.1. Proof of Theorem 4.5.
The proof of this perturbative result is mainly based on thefollowing (deterministic) Meyers type estimate and reverse Jensen’s inequality, for whichthe argument is postponed to Section B.2 below. For a ball D ⊂ T dL , we henceforth set D = B r D ( x D ) and use the short-hand notation kD := B kr D ( x D ) ⊂ T dL for k > . Proposition B.1.
Let a (deterministic) point set P L = { x n,L } n satisfy the hardcore con-dition in (H δ ) for some δ > . There exists a constant η > (only depending on d, δ )such that the following properties hold.(i) Meyers type estimate:
Given a forcing g ∈ C ∞ per ( T dL ) d × d , the unique solution v L ∈ H ( T dL ) d of (B.1) satis-fies for all ≤ q ≤ η , k [ ∇ v L ] k L q ( T dL ) . k [ g ] k L q ( T dL ) . (ii) Perturbative reverse Jensen’s inequality:
For any ball D ⊂ T dL , if w L ∈ H ( T dL ) d satisfies the following equations in D , −△ w L + ∇ Q L = 0 , in D \ I L , div w L = 0 , in D \ I L , D( w L ) = 0 , in I L , ´ ∂I n,L σ ( w L , Q L ) ν = 0 , ∀ n : I n,L ⊂ D, ´ ∂I n,L Θ ν · σ ( w L , Q L ) ν = 0 , ∀ n : I n,L ⊂ D, ∀ Θ ∈ M skew , M. DUERINCKX AND A. GLORIA then there holds for all p < < q with | p − | , | q − | ≤ η , (cid:16) D [ ∇ w L ] q (cid:17) q . (cid:16) D [ ∇ w L ] p (cid:17) p . ♦ As in [21], we shall appeal to the following version of the Calderón-Zygmund lemma dueto Shen [65, Theorem 2.1], based on ideas by Caffarelli and Peral [13].
Lemma B.2 ([13, 65]) . Given ≤ q < q ≤ ∞ , let F, G ∈ L q ∩ L q ( T dL ) be nonnegativefunctions and let C > . Assume that for all balls D ⊂ T dL there exist measurable functions F D, and F D, such that there hold F ≤ F D, + F D, and F D, ≤ F + F D, on D , and suchthat (cid:16) D F q D, (cid:17) q ≤ C (cid:16) C D G q (cid:17) q , (cid:16) C D F q D, (cid:17) q ≤ C (cid:16) D F q D, (cid:17) q . Then, for all q < q < q , (cid:16) ˆ T dL F q (cid:17) q . C ,q ,q,q (cid:16) ˆ T dL G q (cid:17) q . ♦ We may now proceed with the proof of Theorem 4.5, which follows from Proposition B.1together with the above Calderón-Zygmund lemma.
Proof of Theorem 4.5.
We split the proof into three steps. We start with estimates outsidethe particles: first for ≤ p < q , next for q < p ≤ by a duality argument, so that the fullrange of exponents is then reached by interpolation. Finally, we also extend the estimatesinside the particles. Let η ∈ (0 , be fixed as in the statement of Proposition B.1. Step
1. Proof that for all ≤ p < q < η , k [ ∇ v L ] k L q ( T dL ;L p (Ω)) . k [ g ] k L q ( T dL ;L p (Ω)) . (B.2)Let ≤ q ≤ q ≤ η be fixed. For balls D ⊂ T dL , we decompose ∇ v ωL = ∇ v ωL,D, + ∇ v ωL,D, , where v ωL,D, ∈ H ( T dL ) d denotes the unique solution of −△ v ωL,D, + ∇ P ωL,D, = div( g ω D ) , in T dL \ I ωL , div v ωL,D, = 0 , in T dL \ I ωL , D( v ωL,D, ) = 0 , in I ωL , ´ ∂I ωn,L (cid:0) g ω D + σ ( v ωL,D, , P ωL,D, ) (cid:1) ν = 0 , ∀ n, ´ ∂I ωn,L Θ ν · (cid:0) g ω D + σ ( v ωL,D, , P ωL,D, ) (cid:1) ν = 0 , ∀ n, ∀ Θ ∈ M skew . On the one hand, for balls D with radius r D > , Proposition B.1(i) applied to the aboveequation yields ˆ D E [[ ∇ v L,D, ] q ] ≤ E (cid:20) ˆ T dL [ ∇ v L,D, ] q (cid:21) . E (cid:20) ˆ T dL [ g D ] q (cid:21) ≤ ˆ D E [[ g ] q ] , EDIMENTATION OF RANDOM SUSPENSIONS while for balls D with radius r D < we simply appeal to the energy inequality in form of ˆ D E [[ ∇ v L,D, ] q ] ≤ | D | E (cid:20)(cid:16) ˆ T dL |∇ v L,D, | (cid:17) q (cid:21) . | D | E (cid:20)(cid:16) ˆ D | g | (cid:17) q (cid:21) . ˆ D E [[ g ] q ] . On the other hand, noting that v ωL,D, := v ωL − v ωL,D, satisfies the following equations in D , −△ v ωL,D, + ∇ P ωL,D, = 0 , in D \ I ωL , div v ωL,D, = 0 , in D \ I ωL , D( v ωL,D, ) = 0 , in I ωL , ´ ∂I ωn,L σ ( v ωL,D, , P ωL,D, ) ν = 0 , ∀ n : I ωn,L ⊂ D, ´ ∂I ωn,L Θ ν · σ ( v ωL,D, , P ωL,D, ) ν = 0 , ∀ n : I ωn,L ⊂ D, ∀ Θ ∈ M skew , it follows from Minkowski’s inequality in L q q ( D ) and from Proposition B.1(ii) that (cid:16) D E [[ ∇ v L,D, ] q ] q q (cid:17) q ≤ E (cid:20)(cid:16) D [ ∇ v L,D, ] q (cid:17) q q (cid:21) q . E (cid:20) D [ ∇ v L,D, ] q (cid:21) q = (cid:16) D E [[ ∇ v L,D, ] q ] (cid:17) q . In view of these estimates, appealing to Lemma B.2 with F := E (cid:2) [ ∇ v L ] q (cid:3) q , G := E [[ g ] q ] q ,F D, := E (cid:2) [ ∇ v L,D, ] q (cid:3) q , F D, := E (cid:2) [ ∇ v L,D, ] q (cid:3) q , we deduce for all q < q < q , (cid:16) ˆ T dL E (cid:2) [ ∇ v L ] q (cid:3) qq (cid:17) q . (cid:16) ˆ T dL E [[ g ] q ] qq (cid:17) q , and the claim (B.2) follows. Step
2. Duality and interpolation: proof that for all − η < q < p ≤ , k [ T dL \I L ∇ v L ] k L q ( T dL ;L p (Ω)) . k [ g ] k L q ( T dL ;L p (Ω)) . (B.3)Combining this with (B.2), we then deduce by interpolation that the same estimate holdsfor all q, p with | q − | , | p − | < η .Given a test function h ∈ L ∞ (Ω; C ∞ per ( T dL ) d × d ) , we consider the unique solution w L,h ∈ L ∞ (Ω; H ( T dL ) d ) of −△ w ωL,h + ∇ Q ωL,h = div h ω , in T dL \ I ωL , div w ωL,h = 0 , in T dL \ I ωL , D( w ωL,h ) = 0 , in I ωL , ´ ∂I ωn,L (cid:0) h ω + σ ( w ωL,h , Q ωL,h ) (cid:1) ν = 0 , ∀ n, ´ ∂I ωn,L Θ ν · (cid:0) h ω + σ ( w ωL,h , Q ωL,h ) (cid:1) ν = 0 , ∀ n, ∀ Θ ∈ M skew . M. DUERINCKX AND A. GLORIA
Arguing as in Step 1 of the proof of Theorem 1, we note that the system (B.1) for v ωL implies in the weak sense on the whole torus T dL , − △ v ωL + ∇ ( P ωL T dL \I ωL ) = ∇ · ( g ω T dL \I ωL ) − X n δ ∂I ωn,L (cid:0) g ω + σ ( v ωL , P ωL ) (cid:1) ν, (B.4)and, likewise, the above system for w ωL,h implies −△ w ωL,h + ∇ ( Q ωL,h T dL \I ωL ) = ∇ · ( h ω T dL \I ωL ) − X n δ ∂I ωn,L (cid:0) h ω + σ ( w ωL,h , Q ωL,h ) (cid:1) ν. Testing the equation for w ωL,h with v ωL , and then the equation for v ωL with w ωL,h , and notingthat the boundary terms all vanish in view of the respective boundary conditions, we find ˆ T dL \I ωL h ω : ∇ v ωL = − ˆ T dL ∇ w ωL,h : ∇ v ωL = ˆ T dL \I ωL g ω : ∇ w ωL,h . A duality argument together with the above identity yields k [ T dL \I L ∇ v L ] k L q ( R d ;L p (Ω)) . sup (cid:26) E (cid:20) ˆ T dL \I L h : ∇ v L (cid:21) : k [ h ] k L q ′ ( T dL ;L p ′ (Ω)) = 1 (cid:27) = sup (cid:26) E (cid:20) ˆ T dL \I L g : ∇ w L,h (cid:21) : k [ h ] k L q ′ ( T dL ;L p ′ (Ω)) = 1 (cid:27) ≤ k [ g ] k L q ( R d ;L p (Ω)) sup (cid:26) k [ ∇ w L,h ] k L q ′ ( T dL ;L p ′ (Ω)) : k [ h ] k L q ′ ( R d ;L p ′ (Ω)) = 1 (cid:27) . Given − η < q < p ≤ , we may then appeal to (B.2) with ≤ p ′ < q ′ < η andthe claim (B.3) follows. Step
3. Conclusion.In view of Step 2, it remains to show that for all p, q ≥ , k [ I L ∇ v L ] k L q ( T dL ;L p (Ω)) . k [ T dL \I L ∇ v L ] k L q ( T dL ;L p (Ω)) . (B.5)For all n , since v ωL is affine in I ωn,L , we can write for any constant c ωn,L ∈ R d , k∇ v ωL k L ∞ ( I ωn,L ) . k v ωL − c ωn,L k L ( ∂I ωn,L ) . By a trace estimate and by Poincaré’s inequality with the choice c ωn,L := ffl ( I ωn,L + δB ) \ I ωn,L v ωL , k∇ v ωL k L ∞ ( I ωn,L ) . k v ωL − c ωn,L k W , (( I ωn,L + δB ) \ I ωn,L ) . k∇ v ωL k L (( I ωn,L + δB ) \ I ωn,L ) . We may then estimate pointwise, I ωL |∇ v ωL | . X n I ωn,L k∇ v ωL k L (( I ωn,L + δB ) \ I ωn,L ) , and the claim (B.5) now follows from the hardcore condition in (H δ ). (cid:3) EDIMENTATION OF RANDOM SUSPENSIONS B.2.
Proof of Proposition B.1.
We split the proof into two steps. We start with aMeyers type argument based on a Caccioppoli estimate and Gehring’s lemma, and weconclude in the second step.
Step
1. Meyers type argument: there exists η > (only depending on d, δ ) such that forall balls D ⊂ T dL and ≤ q ≤ η , (cid:16) D [ ∇ v L ] q (cid:17) q . (cid:16) D [ ∇ v L ] dd +2 (cid:17) d +22 d + (cid:16) D [ g ] q (cid:17) q . (B.6)Given a ball D ⊂ T dL with radius r D ≥ and given arbitrary constants c L,D ∈ R d and c ′ L,D ∈ R , testing the equation (B.4) for v L with η D ( v L − c L,D ) , where η D denotes a cut-offfunction with η D = 1 in D , η D = 0 outside D , and |∇ η D | . r D , such that η D is constantin I n,L for all n , noting that the boundary terms all vanish and recalling that div v L = 0 ,we obtain the following Caccioppoli type estimate, ˆ D |∇ v L | . r D ˆ D | v L − c L,D | + ˆ D | g | + (cid:16) ˆ D | P L − c ′ L,D | T dL \I L (cid:17) (cid:16) r D ˆ D | v L − c L,D | (cid:17) . Hence, for all K ≥ , ˆ D |∇ v L | . K r D ˆ D | v L − c L,D | + ˆ D | g | + 1 K ˆ D | P L − c ′ L,D | T dL \I L . Using the the Poincaré-Sobolev inequality to estimate the first right-hand side term, withthe choice c L,D := ffl D v L , and using the localized pressure estimate of Lemma 4.7 toestimate the last one, with the choice c ′ L,D := ffl D \I L P L , we deduce (cid:16) D |∇ v L | (cid:17) . K (cid:16) D |∇ v L | dd +2 (cid:17) d +22 d + (cid:16) D | g | (cid:17) + 1 K (cid:16) D |∇ v L | (cid:17) . While this is proven here for all balls D with radius r D ≥ , taking local quadratic averagesallows us to infer for all balls D (with any radius r D > ) and all K ≥ , (cid:16) D [ ∇ v L ] (cid:17) . K (cid:16) D [ ∇ v L ] dd +2 (cid:17) d +22 d + (cid:16) D [ g ] (cid:17) + 1 K (cid:16) D [ ∇ v L ] (cid:17) . Choosing K large enough, the claim (B.6) now follows from Gehring’s lemma in form ofLemma 4.8. Step
2. Conclusion.We start with item (i). Applying (B.6) with D = T dL yields for all ≤ q ≤ η , (cid:16) T dL [ ∇ v L ] q (cid:17) q . (cid:16) T dL [ ∇ v L ] dd +2 (cid:17) d +22 d + (cid:16) T dL [ g ] q (cid:17) q , and (i) follows from Jensen’s inequality and the energy inequality (4.8) in form of (cid:16) T dL [ ∇ v L ] dd +2 (cid:17) d +22 d ≤ (cid:16) T dL |∇ v L | (cid:17) . (cid:16) T dL | g | (cid:17) ≤ (cid:16) T dL [ g ] q (cid:17) q . Finally, item (ii) is a consequence of (B.6) with g = 0 in the set D . (cid:3) M. DUERINCKX AND A. GLORIA
Acknowledgements
MD acknowledges financial support from the CNRS-Momentum program, and AG fromthe European Research Council (ERC) under the European Union’s Horizon 2020 researchand innovation programme (Grant Agreement n ◦ References [1] S. Armstrong, T. Kuusi, and J.-C. Mourrat. The additive structure of elliptic homogenization.
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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
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