Homogenization of Stokes Systems and Uniform Regularity Estimates

# Homogenization of Stokes Systems and Uniform Regularity Estimates

Shu Gu Supported in part by NSF grant DMS-1161154.    Zhongwei Shen Supported in part by NSF grant DMS-1161154.
###### Abstract

This paper is concerned with uniform regularity estimates for a family of Stokes systems with rapidly oscillating periodic coefficients. We establish interior Lipschitz estimates for the velocity and estimates for the pressure as well as a Liouville property for solutions in . We also obtain the boundary estimates in a bounded domain for any .

Keywords. Homogenization; Stokes systems; Regularity.

AMS Subject Classifications. 35B27; 35J48.

## 1 Introduction and Main Results

The primary purpose of this paper is to establish uniform regularity estimates in the homogenization theory of Stokes systems with rapidly oscillating periodic coefficients. More precisely, we consider the Stokes system in fluid dynamics,

 {Lε(uε)+∇pε=F, div(uε)=g (1.1)

in a bounded domain in , where and

 Lε=−div(A(x/ε)∇)=−∂∂xi[aαβij(xε)∂∂xj] (1.2)

with (the summation convention is used throughout). We will assume that the coefficient matrix is real, bounded measurable, and satisfies the ellipticity condition:

 μ|ξ|2≤aαβij(y)ξαiξβj≤1μ|ξ|2 for y∈Rd and ξ=(ξαi)∈Rd×d, (1.3)

where , and the periodicity condition:

 A(y+z)=A(y) for y∈Rd and z∈Zd. (1.4)

A function satisfying (1.4) will be called 1-periodic. We note that the system (1.1), which does not fit the standard framework of second-order elliptic systems considered in [3, 18], is used in the modeling of flows in porous media.

The following is one of the main results of the paper.

###### Theorem 1.1.

Suppose that satisfies the ellipticity condition (1.3) and periodicity condition (1.4). Let be a weak solution of the Stokes system (1.1) in for some and . Then, for any ,

 (−∫B(x0,r)|∇uε|2)1/2+(−∫B(x0,r)∣∣pε−−∫B(x0,R)pε∣∣2)1/2 (1.5) ≤C⎧⎨⎩(−∫B(x0,R)|∇uε|2)1/2+∥g∥L∞(B(x0,R))+Rρ[g]C0,ρ(B(x0,R))⎫⎬⎭ +CR(−∫B(x0,R)|F|q)1/q,

where , and the constant depends only on , , and .

The scaling-invariant estimate (1.5) should be regarded as a Lipschitz estimate for the velocity and estimate for the pressure down to the microscopic scale , even though no smoothness assumption is made on the coefficient matrix . Indeed, if estimate (1.5) holds for any , we would be able to bound

 |∇uε(x0)|+|pε(x0)−−∫B(x0,R)pε|

by the right hand side of (1.5). Here we have taken a point of view that solutions should behave much better on mesoscopic scales due to homogenization and that the smoothness of coefficients only effects the solutions below the microscopic scale (see this viewpoint in the recent development on quantitative stochastic homogenization in [2, 17] and their references). In fact, under the additional assumption that is Hölder continuous,

 |A(x)−A(y)|≤τ|x−y|λ for x,y∈Rd, (1.6)

where and , we may deduce the full uniform Lipschitz estimate for and estimate for from Theorem 1.1, by a blow-up argument (see Section 5).

###### Corollary 1.2.

Suppose that satisfies conditions (1.3), (1.4) and (1.6). Let be a weak solution of (1.1) in for some and . Then

 ∥∇uε∥L∞(B(x0,R/2))+∥pε−−∫B(x0,R)pε∥L∞(B(x0,R/2)) (1.7) ≤C⎧⎨⎩(−∫B(x0,R)|∇uε|2)1/2+∥g∥L∞(B(x0,R))+Rρ[g]C0,ρ(B(x0,R))⎫⎬⎭ +CR(−∫B(x0,R)|F|q)1/q,

where , and the constant depends only on , , , , and .

We remark that for the standard second-order elliptic system , uniform interior Lipschitz estimates as well as uniform boundary Lipchitz estimates with Dirichlet conditions in domains, were established by M. Avellaneda and F. Lin in [3], under conditions (1.3), (1.4) and (1.6). Under the additional symmetry condition , the boundary Lipschitz estimates with Neumann boundary conditions in domains were obtained by C. Kenig, F. Lin, and Z. Shen in [18]. This symmetry condition was recently removed by S.N. Armstrong and Z. Shen in [1], where the uniform Lipschitz estimates were studied for second-order elliptic systems in divergence form with almost-periodic coefficients.

The proof of Theorem 1.1, given in Sections 3 and 5, uses a compactness argument, which was introduced to the study of homogenization problems by M. Avellaneda and F. Lin [3, 4]. Let be a weak solution of the Stokes system (1.1) in . Suppose that

 max⎧⎨⎩(−∫B(0,1)|uε|2)1/2,(−∫B(0,1)|F|q)1/q,∥g∥Cρ(B(0,1))⎫⎬⎭≤1,

where . By the compactness argument with an iteration procedure, which is more or less the version of the compactness method used in [3], we are able to show that if for some , then

 (−∫B(0,θℓ)∣∣uε−(Pβj(x)+εχβj(x/ε))Eβj(ε,ℓ)−G(ε,ℓ)∣∣2dx)1/2≤θℓ(1+σ), (1.8)

where , and , are constants satisfying (see Lemma 3.4). In (1.8), with in the position and is the so-called corrector associated with the Stokes system (1.1). We remark that estimate (1.8) may be regarded as a estimate for in scales larger than . This estimate allows us to deduce the Lipschitz estimate for the velocity down to the scale (see Section 3). Moreover, by carefully analyzing the error terms in the asymptotic expansion of , the estimate (1.8) also allows us to bound

 ∣∣−∫B(x0,r)pε−−∫B(x0,R)pε∣∣

and to derive the estimate for the pressure , one of the main novelties of this paper (see Section 5). We remark that the control of pressure terms usually requires new ideas in the study of Stokes or Navier-Stokes systems. In our case is related to by singular integrals that are not bounded on ; Lipschitz estimates for in general do not imply estimates for . Also, observe that our formulation in (1.8), in comparison with the setting used in [3, 18], appears to be necessary, as the correctors are not necessarily bounded without smoothness conditions on . We further note that as a consequence of (1.8), we are able to establish a Liouville property for Stokes systems with periodic coefficients (see Section 4). To the best of authors’ knowledge, this appears to be the first result on the Liouville property for Stokes systems with variable coefficients.

In this paper we also study the uniform boundary regularity estimates for (1.1) in domains. The following theorem, whose proof is given in Section 6, may be regarded as a boundary Hölder estimate for down to the scale . We emphases that as in the case of Theorem 1.1, no smoothness assumption on is required for Theorem 1.3.

###### Theorem 1.3.

Suppose that satisfies conditions (1.3) and (1.4). Let be a bounded domain in . Let and , where . Let be a weak solution of

 ⎧⎪⎨⎪⎩Lε(uε)+∇pε=0 in B(x0,R)∩Ω,\rm div(uε)=0 in B(x0,R)∩Ω,uε=0 on B(x0,R)∩∂Ω. (1.9)

Suppose that and . Then

 (−∫B(x0,r)∩Ω|∇uε|2)1/2≤Cρ(rR)ρ−1(−∫B(x0,R)∩Ω|∇uε|2)1/2, (1.10)

where depends only on , , , and .

Theorem 1.3 is also proved by a compactness method, though correctors are not needed here. The scaling-invariant boundary estimate (1.10), combined with the interior estimates in Theorem 1.1, allows us to establish the boundary estimates for Stokes systems with coefficients in domains.

Let denote the Besov space of -valued functions on of order with exponent . It is known that if for some , where is a bounded Lipschitz domain, then .

###### Theorem 1.4.

Let be a bounded domain in and . Suppose that satisfies conditions (1.3) and (1.4). Also assume that . Let , and satisfy the compatibility condition

 ∫Ωg−∫∂Ωh⋅n=0, (1.11)

where denotes the outward unit normal to . Then the solutions in to Dirichlet problem

 ⎧⎪⎨⎪⎩Lε(uε)+∇pε=\rm div(f) % in Ω,\rm div(uε)=g in Ω,uε=h on ∂Ω, (1.12)

satisfy the estimate

 ∥∇uε∥Lq(Ω)+∥pε−−∫Ωpε∥Lq(Ω)≤Cq{∥f∥Lq(Ω)+∥gε∥Lq(Ω)+∥h∥B1−1q,q(∂Ω)}, (1.13)

where depends only on , , and .

The proof of Theorem 1.4 is given in Sections 7 and 8. We mention that estimates for elliptic and parabolic equations with continuous or coefficients have been studied extensively in recent years. We refer the reader to [10, 8, 24, 20, 19, 9, 13] as well as their references for work on elliptic equations and systems, and to [3, 6, 10, 26, 18, 15, 14] for uniform estimates in homogenization.

We end this section with some notations and observations. We will use to denote the average of over the set . We will use to denote constants that may depend on , , or , but never on . Note that our assumptions on are invariant under translation. Finally, the technique of rescaling (or dilation) will be used routinely in the rest of the paper. For this, we record that if is a solution of (1.1) and , then

 {Lεr(v)+∇π=G,\rm div(v)=h, (1.14)

where

 π(x)=rpε(rx),h(x)=rg(rx), and G(x)=r2F(rx). (1.15)

Acknowledgement. Both authors would like to thank the anonymous referees for their very helpful comments and suggestions.

## 2 Homogenization Theorems and Compactness

In this section we give a review of homogenization theory of the Stokes systems with periodic coefficients. We refer the reader to [7, pp.76-81] for a detailed presentation. We also prove a compactness theorem for a sequence of Stokes systems with (periodic) coefficient matrices satisfying the ellipticity condition (1.3) with the same .

Let be a bounded Lipschitz domain in . For , we set

 aε(u,v)=∫Ωaαβij(xε)∂uβ∂xj∂vα∂xidx. (2.1)

For and , we say that is a weak solution of the Stokes system (1.1) in , if div in and for any ,

 aε(uε,φ)−∫Ωpε% \rm div(φ)=⟨F,φ⟩.
###### Theorem 2.1.

Let be a bounded Lipschitz domain in . Suppose satisfies the ellipticity condition (1.3). Let , and satisfy the compatibility condition (1.11). Then there exist a unique and (unique up to constants) such that is a weak solution of (1.1) in and on . Moreover,

 ∥uε∥H1(Ω)+∥pε−−∫Ωpε∥L2(Ω)≤C{∥F∥H−1(Ω)+∥h∥H1/2(∂Ω)+∥g∥L2(Ω)}, (2.2)

where depends only on , , and .

###### Proof.

This theorem is well known and does not use the periodicity condition of . First, we choose such that on and . By considering , we may assume that . Next, we choose a function in such that div in and (see [12] for a proof of the existence of such functions). By considering , we may further assume that . Finally, the case and may be proved by applying the Lax-Milgram Theorem to the bilinear form on the Hilbert space

 V={u∈H10(Ω;Rd):\rm div(u)=0 in% Ω}.

This completes the proof. ∎

Let . We denote by the closure in of , the set of 1-periodic and -valued functions in . Let

 aper(ψ,ϕ)=∫Yaαβij(y)∂ψβ∂yj∂ϕα∂yidy,

where and . By applying the Lax-Milgram Theorem to the bilinear form on the Hilbert space

 Vper(Y)={u∈H1per(Y;Rd):% div(u)=0 in Y and ∫Yu=0},

it follows that for each , there exists a unique such that

 aper(χβj,ϕ)=−aper(Pβj,ϕ) for any ϕ∈Vper(Y),

where with 1 in the position. As a result, there exist 1-periodic functions , which are called the correctors for the Stokes system (1.1), such that

 (2.3)

Note that

 ∥χβj∥H1(Y)+∥πβj∥L2(Y)≤C, (2.4)

where depends only on and . Let , where

 ˆaαβij=aper(χβj+Pβj,χαi+Pαi). (2.5)

The homogenized system for the Stokes system (1.1) is given by

 {L0(u0)+∇p0=F,\rm div(u0)=g, (2.6)

where is a second-order elliptic operator with constant coefficients.

###### Remark 2.2.

The homogenized matrix satisfies the ellipticity condition

 μ|ξ|2≤ˆaαβijξαiξβj≤μ1|ξ|2 (2.7)

for any , where depends only on and . The upper bound is a consequence of the estimate , while the lower bound follows from

 ˆaαβijξαiξβj =aper((χβj+Pβj)ξβj,(χαi+Pαi)ξαi) ≥μ∫Y|∇(χαi+Pαi)ξαi|2 ≥μ|ξ|2.
###### Remark 2.3.

Let denote the matrix of correctors for the system (1.1), with replaced by its adjoint . Note that by definition, and

 a∗per(χ∗βj,ϕ)=−a∗per(Pβj,ϕ) for any ϕ∈Vper(Y),

where . It follows that

 ˆaαβij =aper(χβj+Pβj,χαi+Pαi)=aper(χβj+Pβj,Pαi) (2.8) =aper(χβj+Pαj,χ∗αi+Pαi)=a∗per(χ∗αi+Pαi,χβj+Pβj) =a∗per(χ∗αi+Pαi,Pβj)=a∗per(χ∗αi+Pαi,χ∗βj+Pβj).

This, in particular, shows that .

###### Theorem 2.4.

Suppose that satisfies conditions (1.3) and (1.4). Let be a bounded Lipschitz domain. Let be a weak solution of

 ⎧⎪⎨⎪⎩Lε(uε)+∇pε=F in Ω,\rm div(uε)=g in Ω,uε=h on ∂Ω,

where , and . Assume that . Then as ,

 ⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩uε→u0 strongly in L2(Ω;Rd),uε⇀u0 weakly % in H1(Ω;Rd),pε⇀p0 weakly % in L2(Ω),A(x/ε)∇uε⇀ˆA∇u0 weakly in L2(Ω;Rd×d).

Moreover, is the weak solution of the homogenized problem

 ⎧⎪⎨⎪⎩L0(u0)+∇p0=F in Ω,\rm div(u0)=g in Ω,u0=h on ∂Ω.

We remark that Theorem 2.4 is more or less proved in [7], using Tartar’s testing function method. Our next theorem extends Theorem 2.4 to a sequence of systems with coefficient matrices satisfying the same conditions and should be regarded as a compactness property of the Stokes systems with periodic coefficients. Its proof follows the same line of argument found in [7] for the proof of Theorem 2.4, and also uses the observation that if is a sequence of 1-periodic functions with and , then

 wk(x/εk)−−∫Ywk⇀0 % weakly in L2(Ω) (2.9)

as .

###### Theorem 2.5.

Let be a sequence of 1-periodic matrices satisfying the ellipticity condition (1.3) (with the same ). Let be a bounded Lipschitz domain in . Let be a weak solution of

 {−\rm div(Ak(x/εk)∇uk)+∇pk=Fk,\rm div(uk)=gk

in , where , and . We further assume that as ,

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩Fk→F0%stronglyinH−1(Ω;Rd),gk→g0 strongly in L2(Ω),uk⇀u0 weakly in H1(Ω;Rd),pk⇀p0 weakly in L2(Ω),ˆAk→A0,

where is the coefficient matrix of the homogenized system for the Stokes system with coefficient matrix . Then, weakly in , and is a weak solution of

 {−\rm div(A0∇u0)+∇p0=F0,\rm div(u0)=g0 in  Ω. (2.10)
###### Proof.

Let and

 (ξk)αi=akαβij(xεk)∂uβk∂xj.

Note that . It suffices to show that if is a subsequence of and converges weakly to in , then . This would imply that is a weak solution of (2.10) in . It also implies that the whole sequence converges weakly to in .

Without loss of generality we may assume that weakly in . Note that

 ⟨ξk,∇ϕ⟩=⟨Fk,ϕ⟩+⟨pk,%div(ϕ)⟩ (2.11)

for all . Fix and . Let

 ϕk(x)=(Pβj(x)+εkχk∗βj(x/εk))ψ(x),

where (and used in the following) are the correctors for the Stokes system with coefficient matrix , introduced in Remark 2.3. A computation shows that

 ⟨ξk,∇ϕk⟩ =⟨Ak(x/εk)∇uk,∇(Pβj+εkχk∗βj(x/εk))⋅ψ⟩ (2.12) =⟨ψ(∇uk),(Ak)∗(x/εk)∇(Pβj+εkχ∗βj(x/εk))⟩ −⟨(∇ψ)uk,(Ak)∗(x/εk)∇(Pβj+εkχk∗βj(x/εk))⟩ +⟨ξk,(Pβj+εkχk∗βj(x/εk))∇ψ⟩.

Since

 −\rm div((Ak)∗(x/εk)∇[Pβj+εkχk∗βj(x/εk)])=−∇[πk∗βj(x/εk)] in Rd,

it follows that the first term in the right hand side of (2.12) equals

 ⟨πk∗βj(x/εk),\rm div(ψuk)⟩=⟨πk∗βj(x/εk)−−∫Yπk∗βj,\rm div(ψuk)⟩.

Using the fact that strongly in and

 πk∗βj(x/εk)−−∫Yπk∗βj⇀0 weakly in L2(Ω),

we see that the first term in the right hand side of (2.12) goes to zero. In view of the estimate

 ∥εkχk∗βj(x/εk)∥L2(Ω)≤Cεk∥χk∗βj∥L2(Y)≤Cεk,

it is easy to see that the third term in the right side of (2.12) goes to .

To handle the second term in the right hand side of (2.12), we note that by (2.9),

 ∇Pαi⋅(Ak)∗(x/εk)∇(Pβj+εkχk∗βj(x/εk))

converges weakly in to

 limk→∞∫Y∇Pαi⋅(Ak)∗(y)∇(Pβj+χk∗βj(y))dy=limk→∞ˆakβαji=a0βαji,

where , , and we have used the observation (2.8). This, together with the fact that strongly in , shows that the second term in the right hand side of (2.12) goes to

 −a0βαji∫Ω∂ψ∂xiuα0=a0βαji∫Ωψ∂uα0∂xi,

where we have used integration by parts. To summarize, we have proved that as ,

 ⟨ξk,∇ϕk⟩→⟨ξ0,Pβj∇ψ⟩+a0βαji∫Ωψ∂uα0∂xi. (2.13)

Finally, since weakly in and strongly in , we have . Also, since in ,

 ⟨pk,\rm div(ϕk)⟩=⟨pk,\rm div(Pβjψ)⟩+⟨pk,εχk∗βj(x/εk)∇ψ⟩→⟨p0,\rm div(Pβjψ)⟩.

Thus, the right hand side of (2.11) converges to

 ⟨F0,Pβjψ⟩+⟨p0,\rm div(Pβjψ)⟩=⟨ξ0,∇(Pβjψ)⟩=⟨ξ0,Pβj∇ψ⟩+⟨ξ0,ψ∇Pβj⟩,

where the first equality follows by taking the limit in (2.11) with