A drift homogenization problem revisited

A drift homogenization problem revisited

MARC BRIANE ANDPATRICKG ´ERARD

Abstract. This paper revisits a homogenization problem studied by L. Tartar re- lated to a tridimensional Stokes equation perturbed by a drift (related to the Cori- olis force). Here, a scalar equation and a two-dimensional Stokes equation with aL²-bounded oscillating drift are considered. Under higher integrability condi- tions the Tartar approach based on the oscillations test functions method applies and leads to a limit equation with an extra zero-order term. When the drift is only assumed to be equi-integrable inL², the same limit behaviour is obtained. How- ever, the lack of integrability makes difficult the direct use of the Tartar method.

A new method in the context of homogenization theory is proposed. It is based on a parametrix of the Laplace operator which permits to write the solution of the equation as a solution of a fixed point problem, and to use truncated functions even in the vector-valued case. On the other hand, two counter-examples which induce different homogenized zero-order terms actually show the sharpness of the equi-integrability assumption.

Mathematics Subject Classification (2010): 35B27 (primary); 76M50 (sec- ondary).

1. Introduction

At the end of the Seventies L. Tartar developed his method based on oscillating test functions to deal with the homogenization of PDE’s. In the particular framework of hydrodynamics [14, 15] he studied the Stokes equation in a bounded domain!of R³, perturbed by an oscillating drift term,i.e.







−"uε+curl(v_ε)×uε+∇pε= f in!

div(u_ε)=0 in!

uε =0 on∂!,

(1.1)

where the oscillations are produced by the sequence of vector-valued functionsv_ε which weakly converges to somevinL³(!)³. L. Tartar proved that the limit equa- Received June 16, 2010; accepted in revised form October 4, 2010.

tion of (1.1) is the Brinkman [5] type equation







−"u+curl(v)×u+∇p+Mu = f in!

div(u)=0 in!

u=0 on∂!,

(1.2)

where M is a positive definite symmetric matrix-valued function. More precisely, M is defined by the convergences

(Dw_ε^λ)^Tv_ε −& Mλ weakly in L³²(!)³, for anyλ∈R^{3, (1.3)} wherew_ε^λ∈W^1,3(!)³solves the Stokes equation (1.1) in which the term curl(v_ε)× uε is replaced by curl(v_ε)×λ. Then, the convergence (1.3) combined with the compactness of uε in L³(!)³, yields the zero-order term Mu in (1.2). In [16]

L. Tartar revisited this problem using the H-measures tool. On the other hand, the appearance of such a strange zero-order term in homogenization was also obtained from finely perforated domains by D. Cioranescu, F. Murat [6] for the Laplace equation, and by G. Allaire [2] for the Stokes equation, with zero Dirichlet boundary condition on the holes.

Since curl(v_ε)×uε is orthogonal touε, the energy associated with (1.1) is

reduced to %

!|Duε|²dx, (1.4)

and thus does not depend on the driftv_ε. Starting from this remark our aim is to study two drift homogenization problems associated with the same energy (1.4), and to specify the best integrability condition satisfied by the drift so that the Tar- tar approach holds. The first problem is scalar and the second problem is a two- dimensional equivalent of the Stokes problem (1.1). However, we have not suc- ceeded in obtaining a similar result for the three-dimensional Stokes equation (1.1) since the best integrability assumption forv_εis not clear.

In Section 2, we consider the following scalar equation in a bounded open set

!ofR^N, _&

−"uε+bε·∇uε+div(bεuε)= f in!

uε=0 on∂!, (1.5)

where bε ∈ L^∞(!)^N is bounded in L²(!)^N. We obtain three different homoge- nization results:

In Section 2.1, assuming that the divergence of the drift bε is bounded in W^−1,q(!), withq > N, we prove (see Theorem 2.1) that the sequenceuε weakly converges inH₀¹(!)to the solutionuof the equation

&

−"u+b·∇u+div(b u)+µu= f in!

u=0 on∂!, (1.6)

whereµis a nonnegative function. The proof follows the Tartar method using the oscillating test function

w_ε :="⁻¹'

div(bε)(

∈ H₀¹(!). (1.7)

Then, in Section 2.2, assuming only the equi-integrability of the sequence ∇w_ε in L²(!)^N (this is actually a weaker assumption than the equi-integrability of the whole sequencebε), we obtain (see Theorem 3.1) the limit problem (1.6) with

|∇w_ε− ∇w|²−& µ weakly inL¹(!) and µu²∈L¹(!). (1.8) It seems intricate to apply directly the Tartar method with the test functionw_ε, since we cannot control the termsbε ·∇uεw_ε andbε ·∇w_εuε. To this end, one should consider truncations of bothw_εand∇w_ε. To overcome this difficulty we propose a new method, up to our knowledge, in the context of homogenization theory, based on a parametrix of the Laplace operator. It follows that uε reads as a solution of a fixed point problem, which allows us to estimate the sequence∇w_ε ·∇uε only using a truncation of ∇w_ε. The equi-integrability of ∇w_ε then gives the thesis.

Also assuming that b ∈ L^q(!)^N, withq > N, (which ensures the uniqueness in (1.6)) we prove the following corrector result

u_ε−(1+w_ε−w)u −→ 0 strongly inW_loc^1,q(!), for anyq∈[1,N^'). (1.9) Finally, in Section 2.3, we show the sharpness of the equi-integrability condition thanks to a counter-example in the periodic framework (see Theorem 2.6). Making a change of functions withbε =∇w_ε, equation (1.5) is shown to be equivalent to the following equation

−"v_ε+µ_εv_ε = fε, with µ_ε := |∇w_ε|², (1.10) the solution of which has the same limit asuε. G. Dal Maso, A. Garroni [7] proved that the class of equations of type (1.10) is stable under homogenization. Here, we do not use this general result, but we explicit an oscillating sequencew_εso that the limit equation of (1.5), or equivalently (1.10), is

−"u+γu= f, (1.11)

with an explicit constant γ which turns out to be < µ. Therefore, the loss of equi-integrability for ∇w_ε violates the result of Section 2.2. Note that the vecto- rial character of the drift term in equation (1.5) makes difficult the derivation of a closure result similar to the one of [7] which is strongly based on the maximum principle.

In Section 3, we consider the following two-dimensional equivalent of the per- turbed Stokes problem (1.1),







−"uε+curl(v_ε)J uε+∇pε = f in!

div(uε)=0 in!

uε =0 on∂!,

(1.12)

where J is the rotation matrix of angle 90^◦, and v_ε ∈ L^∞(!)² is bounded in L²(!)². Note that the weak formulation (3.4) of curl(v_ε)J uε contains the drift term(Duε)^Tvε, so that equation (1.12) can be also regarded as a drift problem. We follow the same scheme as in the scalar case:

In Section 3.1, assuming that the sequencev_εis bounded inL^r(!)²withr >2, we show (see Theorem 3.1) that the sequenceuεweakly converges inH₀¹(!)to the solutionuof the Brinkman equation







−"u+curl(v)J u+∇p+Mu= f in!

div(u)=0 in!

u=0 on∂!,

(1.13)

where M is a symmetric positive definite matrix-valued function defined by the convergence (1.3) inL^r+22r (!)².

In Section 3.2, assuming only the equi-integrability of the sequence v_ε in L²(!)², we prove (see Theorem 3.3) owing to the Tartar method that the sequence uε weakly converges in H₀¹(!)to the solutionuof the Brinkman equation (1.13) with similarly to (1.8),

(Dw^λ_ε)^Tv_ε −& Mλ weakly inL¹(!)² and Mu·u∈L¹(!). (1.14) The proof is based on a double parametrix method carrying on both the velocity uε and the pressure pε. However, the proof of the last estimate of (1.14) is more delicate than the one of (1.8), since we cannot use a comparison principle as in the scalar case. We need to introduce a test function similar tow^λ_ε but associated with a truncation ofv_ε. Moreover, if!has a regular boundary,v∈L^r(!)²withr >2, andM ∈L^m(!)^2×2withm>1, we get the corrector result

uε−u−Wεu−→0 strongly inW^1,1(!)², whereWελ:=w^λ_ε, forλ∈R^{2. (1.15)} Finally, in Section 3.2, we construct an oscillating sequence v_ε which is not equi- integrable inL²(!)², which leads to the limit problem (1.13) involving a matrix( which is not symmetric and satisfies the strict inequality

(λ·λ< Mλ·λ, for anyλ)=0,

which is inconsistent with the Tartar approach. This shows the sharpness of the equi-integrability condition as in the scalar case. It would be very interesting to find the closure of the family of problems (1.12) under the sole condition of L²- boundedness of the sequencesv_ε. This problem is far from being evident due to the absence of comparison principle for such a vector-valued equation.

Notations

• The space dimension isN ≥2, and 2^∗:= 2N N−2.

• The conjugate exponent of p≥1 is denoted by p^':= p p−1.

• Foru:R^{N −→}R^N,Du:=

)∂ui

∂xj

*

1≤i,j≤N

.

• For):R^N −→R^{N×N, Div())}:=

+ _N ,

j=1

∂)_{i j}

∂x_j -

1≤i≤N

.

• H_*¹(Y), withY :=(0,1)^N, denotes the space of theY-periodic functions onR^N which belong to H_loc¹ (R^N).

2. A scalar equation with a drift term

Along this section!is a bounded regular open set ofR^{N, withN} ≥2, and f is a distribution in H⁻¹(!).

2.1. The classical case

Letq ∈(N,∞). Consider a sequencebεinL^∞(!)^N such that

bε −& bweakly inL²(!)^N and div(bε) is bounded inW^−1,q(!). (2.1) Letw_ε ∈W₀^1,q(!)be the solution of the equation (see,e.g., [9, Theorem 2.1])

"w_ε =div(bε) inD^{'(!). (2.2)} Up to a subsequencew_ε weakly converges inW₀^1,q(!)to the functionwsolution of

"w=div(b) inD^{'(!). (2.3)} We have the following result:

Theorem 2.1. The solutionuε ∈H₀¹(!)of the equation

−"uε+bε·∇uε+div(bεuε)= f inD^{'(!), (2.4)} weakly converges in H₀¹(!), up to a subsequence, to a solutionu ∈ H₀¹(!)of the equation

−"u+b·∇u+div(b u)+µu= f inD^{'(!), (2.5)} whereµis the function defined by the convergence

|∇w_ε− ∇w|²−& µ weakly inL^q2(!). (2.6) Remark 2.2. The uniqueness for equation (2.4) is not evident under the sole as- sumption b ∈ L²(!)². Assuming a stronger integrability of bwe will obtain in Theorem 2.4 the uniqueness for the limit equation.

Proof. The proof is based on the choice of appropriate oscillating test functions as Tartar did (see [13, Appendix], and [17]). The functionw_ε of (2.2) will play the role of the oscillating test function. The variational formulation of (2.4) is

%

!∇uε·∇ϕdx+

%

!

bε·∇uεϕdx −

%

!

bε·∇ϕuεdx

=-f,ϕ._H−1(!),H₀¹(!), ∀ϕ∈ H₀¹(!).

(2.7)

Then, by the Lax-Milgram theorem there exists a unique solution uε of (2.7) in H₀¹(!). In particular, forv ∈W^1,∞(!), puttingϕ =vuε as test function in (2.7) we obtain the identity

%

!

|∇uε|²vdx+

%

!

∇uε·∇vuεdx−

%

!

bε·∇vu²_εdx=-f, vuε._H−1(!),H₀¹(!), (2.8) which will be used several times. So, choosing v = 1 in (2.8) the term withbε

cancel so that we easily deduce thatuεis bounded inH₀¹(!)and weakly converges, up to a subsequence, to a functionuinH₀¹(!). Therefore, it follows from (2.7) the limit variational formulation

%

!∇u·∇ϕdx+

%

!b·∇uϕdx+

%

!

ϕdν−

%

!b·∇ϕu dx=-f,ϕ._H−1(!),H₀¹(!), (2.9) which holds for anyϕ ∈ W₀^1,q(!)(due to the embedding ofW₀^1,q(!)into C(!)¯ forq> N), where the measureνis defined by the convergence

bε·∇uε −& b·∇u+ν weakly-∗inM^{(!). (2.10)} The limit equation associated with (2.9) is

−"u+b·∇u+ν+div(b u)= f inD^{'(!). (2.11)} Now, let us determine the measureνof (2.10). Letϕ∈C_c^∞(!). Puttingϕw_εas test function in (2.7) andϕuε in (2.2), and taking the difference of the two equalities we get

%

!

∇uε·∇ϕw_εdx −

%

!

∇w_ε·∇ϕuεdx

=-f,ϕw_ε._H−1(!),H₀¹(!)−

%

!

bε·∇uεϕw_εdx+

%

!

bε·∇w_εϕuεdx +

%

!

bε·∇ϕuεw_εdx−

%

!

bε·∇uεϕdx −

%

!

bε·∇ϕuεdx.

(2.12)

Passing to the limit in (2.12) by using the strong convergence ofuε inL^p(!), for p<2^∗, and the uniform convergence ofw_εinC(!)¯ (q >N), we obtain

%

!∇u·∇ϕwdx−

%

!∇w·∇ϕu dx

=-f,ϕw._H−1(!),H₀¹(!)−

%

!

b·∇uϕwdx −

%

!

ϕwdν+

%

!

σ ϕu dx +

%

!

b·∇ϕuwdx−

%

!

b·∇uϕdx−

%

!

ϕdν−

%

!

b·∇ϕu dx,

(2.13)

where the measure ν is defined by (2.10) and the function σ is defined, up to a subsequence, by the convergence

bε·∇w_ε −& σ weakly inL^q2q+2(!). (2.14) On the other hand, puttingϕw ∈ W₀^1,q(!)in (2.9) andϕu ∈ H₀¹(!)in (2.3) we have

%

!∇u·∇(ϕw)dx=-f,ϕw._H−1(!),H₀¹(!)−

%

!

b·∇uϕwdx−

%

!

ϕwdν +

%

!

b·∇wϕu dx+

%

!

b·∇ϕuwdx,

(2.15)

%

!∇w·∇(ϕu)dx =

%

!

b·∇uϕdx+

%

!

b·∇ϕu dx. (2.16) Equating the difference between (2.15) and (2.16) to the right-hand side of (2.13), it follows that

%

!

σ ϕu dx−

%

!

b·∇wϕu dx−

%

!

ϕdν =0, for anyϕ∈C_c^∞(!), (2.17) which implies that

ν =σu−b·∇wu inD^{'(!). (2.18)} It thus remains to determine the limit equation (2.5). To this end, we pass to the limit by usingϕw_εas test function in (2.2) and the definition (2.6) ofµ, and we put ϕwin (2.3), which yields

%

!

.

µ+ |∇w|²/ ϕdx+

%

!

∇w·∇ϕwdx =

%

!

σ ϕdx+

%

!

b·∇ϕwdx, (2.19)

%

!|∇w|²ϕdx+

%

!∇w·∇ϕwdx =

%

!

b·∇wϕdx +

%

!

b·∇ϕwdx. (2.20) Equating (2.19) and (2.20), we deduce that

µ=σ −b·∇w inD^{'(!), (2.21)}

which combined with (2.18) implies that

ν =µu inD^{'(!). (2.22)}

Finally, the limit equation (2.11) and the relation (2.22) give the desired homoge- nized equation (2.5).

Remark 2.3. It can be shown that µ(x)=

%

S^N−1

µ(x,dξ)ξ·ξ, (2.23)

whereµdenotes the matrix-valuedH-measure (or micro-local defect measure) of the sequencebε(see [16] and [8]), andS^N−1the unit sphere ofR^N.

Assumption (2.1) is actually not sharp. In the next section we replace it by the boundedness ofbε and the equi-integrability of∇w_εinL²(!)².

2.2. The case under an equi-integrability assumption

In this section!is a bounded open set ofR^N. Consider a sequencebεinL^∞(!)^N the Hodge decomposition of which is

bε =∇w_ε+ξ_ε, with w_ε ∈H₀¹(!), ξ_ε ∈L²(!)^N and div(ξ_ε)=0, (2.24) such that

bε −& b weakly inL²(!)^N. (2.25) Note that for a fixedε>0,w_ε ∈W^1,p(!)andξ_ε ∈ L^p(!)^N for any p∈[2,∞).

But the essential point is the asymptotic behaviour of the sequences bε,∇w_ε,ξ_ε. Our main assumption is the equi-integrability of the sequence∇w_εinL²(!)^N. By virtue of the Vitali-Saks theorem this is equivalent to the following convergence, up to an extraction of a subsequence,

|∇w_ε− ∇w|²−& µ weakly in L¹(!), (2.26) (Compare to (2.6) withq> N).

We have the following result:

Theorem 2.4.

i)Under the equi-integrability assumption(2.26) the solution uε of (2.4) weakly converges in H₀¹(!)to a solutionuof the equation

−"u+b·∇u+div(b u)+µu= f inD^{'(!), (2.27)}

with %

!

µu²dx ≤ -f,u._H−1(!),H₀¹(!)−

%

!

|∇u|²dx. (2.28)

ii)Also assume thatb ∈ L^q(!)^N, whereq > 2ifN = 2andq = N if N > 2.

Then, we have

%

!|∇u|²dx+

%

!

µu²dx =-f,u._H−1(!),H₀¹(!), (2.29)

and there exists a unique solution u ∈ H₀¹(!) of equation (2.27), with µu² ∈ L¹(!).

Moreover, for any p ∈ [1,2)if N = 2and p = N^' ifN > 2, we have the corrector result

∇uε− ∇u−(∇w_ε− ∇w)u −→ 0 strongly inL_loc^p (!)^N, (2.30) and for anyr ∈[1,p),

uε−(1+w_ε−w)u −→ 0 strongly inW_loc^1,r(!). (2.31) Remark 2.5. No equi-integrability is required for the divergence free sequenceξ_ε. Actually, we can prove that the equi-integrability of the sequencebεinL²(!)^N im- plies the equi-integrability of its two components∇w_ε,ξ_εin L²_loc(!)^N. Therefore, condition (2.26) is really weaker than the equi-integrability ofbε.

Moreover, the equi-integrability of ∇w_ε in L²(!)^N is essential for deriving the limit equation with the zero-order termµu. When this condition is not satisfied we can obtain a similar limit equation but with a different zero-order term (see Section 2.3).

Proof of Theorem2.4. The limituofuεinH₀¹(!)solves the equation (2.11) where ν is defined by

bε·∇uε−b·∇u−& ν weakly-∗inM^{(!). (2.32)} We thus have

bε·∇uε =(ξ_ε+∇w)·∇uε+(∇w_ε− ∇w)·∇uε −& b·∇u+ν inD^'(!).

Moreover, by the Murat, Tartar div-curl lemma [11] the sequence(ξ_ε+∇w)·∇uε

converges to(ξ+∇w)·∇u = b·∇u. This combined with the equi-integrability of∇w_εimplies thatνis also given by the convergence

(∇w_ε− ∇w)·∇uε −& ν weakly inL¹(!). (2.33) The proof of Theorem 2.4. is based on a parametrix method which allows us to expressuε as a solution of a fixed point problem. As a consequence, we obtain a strong estimate of∇uεinL_loc^p (!)for somep>1 close to 1. However, this estimate cannot provide directly the desired limitν of (2.33) since p<2. To overcome this difficulty we consider a truncationη^k_εof∇w_εwhich is bounded byk>0. Then, we

can pass to the limit asεtends to zero in the productη^k_ε·∇uε for a fixedk. Hence, thanks to the equi-integrability of∇w_εwe deduce the limitν asktends to infinity.

The proof is divided into four steps. In the first step we present the parametrix method which leads to a L^p-strong estimate of∇uε. In the second step we deter- mine the limit of the sequence η_ε^k ·∇uε for a fixedk > 0. In the third step we determine the limitνand the limit equation (2.27) together with (2.28). The fourth step is devoted to the proof of equality (2.29) and the corrector results (2.30) and (2.31).

First step.The parametrix method.

First, let us define a parametrix for the Laplace operator in!. To this end consider two sequences of functionsϕ_n,ψ_ninC_c^∞(!), such that













0≤ϕ_n,ψ_n≤1 andϕ_n=1 in supp(ψ_n) , for anyn≥1, 0n≥1:supp(ψn)∩K)=Ø1

is finite, for any compact subsetK⊂!, ,

n≥1

ψ_n =1 in!.

(2.34)

LetEbe the fundamental solution of the Laplace operator inR^N. Then, the opera- tor Pdefined inD^'(!)by

P(ζ):=,

n≥1

ψ_nE∗(ϕ_nζ) , forζ ∈D^{'(!), (2.35)} is a parametrix of the Laplace operator (see [1, Chapter I], for further details) which satisfies

P("ζ)=ζ −K(ζ) and "' P(ζ)(

=ζ−K^'(ζ), forζ ∈D^{'(!), (2.36)} where K,K^'are twoC^∞-kernel operators properly supported in!. Thanks to the Calder`on-Zygmund regularity for the Laplace operator (see,e.g., [9, Theorem 2.1], and the references therein) we also have for any p > 1, ands ∈ [0,2]such that s+ ¹_p is not an integer,

Pmaps continuouslyD^'(!)toD^{'(!), andW}_loc^−s,p(!)toW_loc^{2−s,p(!). (2.37)} Then, applying (2.36) to the solutionuεof (2.4) we have

uε = P("uε)+K(uε)

= P' div2

u(∇w_ε− ∇w)3(

+P' div2

∇w_ε(uε−u)3(

+ P'

div(u∇w)( +P'

ξ_ε·∇uε+bε·∇uε− f(

+K(uε),

(2.38)

Fix p>1 close enough to 1 ands ∈(N/p^',1). Sinceuε−ustrongly converges to 0 inL^q(!)for anyq ∈(2,2^∗), the sequence div'

∇w_ε(uε−u)(

strongly converges to 0 inW^−1,p(!), hence by (2.37) we have

P' div2

∇w_ε(uε−u)3(

−→0 strongly inW_loc^1,p(!).

Moreover, the sequenceξ_ε·∇uε+bε·∇uεis bounded inL¹(!), thus inW^−s,p(!) since s > N/p^'. Therefore, again by (2.37) the sequence∇P'

ξ_ε ·∇uε +bε ·

∇uε− f(

is bounded inW^1−s,p(!)^N, and up to a subsequence strongly converges inL_loc^p (!)^N. Hence, since

ξ_ε·∇uε+bε·∇uε −& ξ·∇u+ν+b·∇u inD^'(!), we deduce from (2.38) the strong estimate

∇uε− ∇P' div2

u(∇w_ε− ∇w)3(

=∇P'

div(u∇w)+ξ·∇u+ν+b·∇u− f(

+∇K(u)+o_L^p

loc(!)^N(1)

=∇P'

ν+b·∇u+div(b u)− f(

+∇K(u)+o_L^p

loc(!)^N(1) 'ξ·∇u=div(uξ)(

,

(2.39)

whereo_L^p

loc(!)^N(1)denotes a sequence which strongly converges to 0 inL_loc^p (!)^N. On the other hand, by (2.36) and (2.37) we have

∇P' div2

u(∇w_ε− ∇w)3(

=∇P'

"2

u(w_ε−w)3(

− ∇P' div2

∇u(w_ε−w)3(

=∇P'

"2

u(w_ε−w)3(

+o_L^p

loc(!)^N(1)

=∇'

u(wε−w)( +o_L^p

loc(!)^N(1)

=u(∇w_ε− ∇w)+o_L^p

loc(!)^N(1).

Therefore, this combined with (2.39) yields

∇uε−u(∇w_ε− ∇w)=∇P'

ν+b·∇u+div(b u)− f(

+∇K(u) +o_L^p

loc(!)^N(1). (2.40)

Second step.Estimate of the sequenceη^k_ε·∇uε.

Set η^k_ε := ∇w_ε1_{|∇wε|<k}, for a positive integerk. Let us determine the limit of η^k_ε ·∇uε inL²_loc(!)^N. Using a diagonal extraction, there exists a subsequence of ε, still denoted byε, such thatη^k_ε weakly converges to someη^k in L^∞(!)^N for any k. By the strong convergence (2.40) combined with the weak convergence of u(∇w_ε− ∇w)to 0 inL^p(!)^N (forpclose to 1) we have

η^k_ε·∇uε−'

η^k_ε −η^k(

·(∇w_ε− ∇w)u

−& η^k ·∇P'

ν+b·∇u+div(b u)− f(

+η^k·∇K(u) weakly inL_loc^p (!).

Hence, we get that σ^k =µ^ku+η^k·∇P'

ν+b·∇u+div(b u)− f(

+η^k ·∇K(u) in!, (2.41)

where





σ^k := lim

ε→0

4

η^k_ε ·∇uε

5

weakly inL²(!), µ^k :=lim

ε→0

4.

η^k_ε −η^k

/·(∇w_ε− ∇w) 5

weakly inL²(!). (2.42) Third step.Determination ofνand the limit equation (2.27).

Starting from the limit equation (2.11) we have by (2.36) u= P'

ν+b·∇u+div(b u)− f(

+K(u) in!, hence

η^k·∇u=η^k ·∇P'

ν+b·∇u+div(b u)− f(

+η^k·∇K(u) in!.

Equating this with (2.41) we obtain

σ^k =µ^ku+η^k·∇u in!. (2.43) Now, let us pass to the limit ask → ∞. By virtue of the equi-integrability of∇wε

inL²(!)^N and by definition (2.42) the sequenceµ^kstrongly converges inL¹(!)to the functionµof (2.26),η^k strongly converges to∇winL²(!)^N, andσ^k strongly converges toν+∇w·∇uinL¹(!). Then, up to a subsequenceµ^kconverges toµ a.e. in!, and by the Fatou lemma combined with equality (2.43) we get

%

!|µu|dx ≤lim inf

k→∞

%

!|µ^ku|dx

≤lim inf

k→∞

%

!|σ^k−η^k ·∇u|dx =

%

!|ν|dx.

(2.44)

We deduce from (2.44) and (2.43) thatµu∈L¹(!)and

ν=µu in!, (2.45)

which yields the limit equation (2.27).

It remains to prove the inequality of (2.28). Letv ∈ L^∞(!)andt ∈ R^{. By} (2.26), (2.33) and (2.45) we have

%

!

66∇uε− ∇u−(∇w_ε− ∇w)tv(66²dx

=

%

!|∇uε− ∇u|²dx +t²

%

!|∇wε− ∇w|²v²dx

−2t

%

!

∇uε·(∇w_ε− ∇w) vdx+o(1)

=-f,u._H−1(!),H₀¹(!)−

%

!|∇u|²dx+t²

%

!

µ v²dx

−2t

%

!

µuvdx+o(1),

(2.46)

hence t²

%

!

µ v²dx−2t

%

!

µuvdx+-f,u._H−1(!),H₀¹(!)−

%

!|∇u|²dx ≥0, ∀t ∈R^. This implies that

)%

!

µuvdx

*2

≤ )

-f,u._H−1(!),H₀¹(!)−

%

!|∇u|²dx

* %

!

µ v²dx. (2.47) LetTk,k >0, be a function inC¹(R^{)such that}

0≤T_k^' ≤1 and

7Tk(t)=t if|t|≤k

|Tk(t)| =k+1 if|t|≥k+2. (2.48) Putting v = Tk(u)as test function in (2.47) and using thatTk(u)² ≤ u Tk(u), we get

)%

!

µu T_k(u)dx

*2

≤ )

-f,u._H−1(!),H₀¹(!)−

%

!|∇u|²dx

* %

!

µu T_k(u)dx,

hence %

!

µu T_k(u)dx ≤ -f,u._H−1(!),H₀¹(!)−

%

!|∇u|²dx. (2.49) Sinceu Tk(u)is a nondecreasing nonnegative sequence which converges tou²a.e.

in!, the Beppo-Levi theorem applied to (2.49) thus gives inequality (2.28).

Fourth step.Proof of equality (2.29) and of the corrector results (2.30), (2.31).

Assume thatb ∈ L^q(!)^N, whereq > 2 if N = 2 andq = N ifN > 2. Letϕ_n be a sequence inC₀¹(R⁾which strongly converges touin H₀¹(!)and a.e. in!, and such that|∇ϕ_n|is dominated by a fixed function inL²(!). Putting the truncation functionTk(ϕ_n)(2.48) in the limit equation (2.27) we have

%

!∇u·∇Tk(ϕ_n)dx+

%

!

b·∇u Tk(ϕ_n)dx−

%

!

b·∇Tk(ϕ_n)u dx +

%

!

µu Tk(ϕ_n)dx

=8

f,Tk(ϕ_n)9

H⁻¹(!),H₀¹(!).

Sinceb·∇u,µu ∈L¹(!)andb u∈ L²(!)^N (as a consequence ofb∈L^q(!)^N), we can pass to the limit asn → ∞in the previous equality owing to the Lebesgue dominated convergence theorem, which yields

%

!∇u·∇Tk(u)dx+

%

!

b·∇u Tk(u)dx−

%

!

b·∇Tk(u)u dx +

%

!

µu Tk(u)dx

=8

f,T_k(u)9

H⁻¹(!),H₀¹(!).

(2.50)

Then, using that|Tk(u)| ≤ |u|, 0 ≤ T_k^'(u) ≤ 1, Tk(u)strongly converges touin H₀¹(!), and thatb u∈L²(!)^N,µu²∈L¹(!), and passing to the limit ask → ∞ owing to the Lebesgue dominated convergence theorem we get

%

!|∇u|²dx+

%

!

b·∇u u dx−

%

!

b·∇u u dx+

%

!

µu²dx =-f,u._H−1(!),H₀¹(!), which is (2.29). Moreover, the proof of equality (2.29) with f =0 shows that there exists a unique solutionu∈H₀¹(!)of equation (2.27), withµu²∈L¹(!).

It remains to prove the corrector results. By the estimate (2.46) withv=Tk(u) andt =1, combined with equality (2.29) we have

klim→∞ lim

ε→0

)%

!

66∇uε− ∇u−(∇w_ε− ∇w)Tk(u)6 6²dx

*

= lim

k→∞

)%

!

µ'

u−Tk(u)(2

dx

*

=0.

(2.51)

On the other hand, let p∈[1,2)ifN = 2 and p= N^'ifN >2, and consider an open setω!!. By the H¨older inequality we have

%

ω

66∇uε− ∇u−(∇w_ε− ∇w)u66^pdx

≤2^p−1 )%

!

66∇uε− ∇u−(∇w_ε− ∇w)Tk(u)66^p +

%

ω

|∇w_ε− ∇w|^p 6

6u−Tk(u)6 6^pdx

*

≤c )%

!

66∇uε− ∇u−(∇w_ε− ∇w)Tk(u)6 6²*₂^p +c

)%

ω

66u−Tk(u)6 6^22p−pdx

*1−^p2

≤c )%

!

66∇uε− ∇u−(∇w_ε− ∇w)Tk(u)6 6²*₂^p +c

)%

{|u|>k}∩ω|u|^22p−pdx

*1−^p2

.

(2.52)

Sinceu ∈ L^2−p2p (ω)by the Sobolev embedding, passing successively to the limits ε → 0 andk → ∞in (2.52) owing to convergence (2.51) we obtain the strong convergence (2.30).

Letr ∈[1,p). Sincew_ε−wstrongly converges to 0 inL^22r−r(ω), by the H¨older inequality the sequence(w_ε−w)∇u strongly converges to 0 inL^r(ω)^N. Finally, this combined with (2.30) implies the corrector result (2.31).

2.3. A counter-example

In this section!is a regular bounded open set ofR^{2, andY} :=(−¹₂,¹₂)². For fixed R∈(0,¹₂)andµ >0, letrε ∈(0,R)be defined by the equality

2π

ε²|lnrε| =µ. (2.53)

LetWεbe theY-periodic function andw_εbe theεY-periodic function defined by

Wε(y):=













lnr−lnrε

lnR−lnrε

ifr:= |y|∈(rε,R)

0 sir ≤rε

1 sir ≥ R,

y∈Y, wε(x):=Wε

.x ε /

, x∈R^2. (2.54) Note that by (2.53) we have

1 ε²

%

Y|∇Wε|²dy= 2π

ε²ln(R/rε) −→

ε→0 µ. (2.55)

We then consider the driftbεdefined by bε(x):=∇w_ε(x)= 1

ε∇Wε

.x ε

/

, forx ∈R^{2. (2.56)} Taking into account (2.53) it is easy to check that

w_ε −& 1 weakly inH¹(!)and weakly-∗inL^∞(!). (2.57) Let f be a non-zero function in L²(!). We study the asymptotic behavior of the equation (2.4) with the driftbε of (2.56),i.e.

−"uε+∇wε·∇uε+div(∇wεuε)= f inD^{'(!). (2.58)} We have the following result:

Theorem 2.6. The solutionuε of(2.58)weakly converges inH₀¹(!)to the solution uof the equation

−"u+γu= f inD^{'(!), where γ} := 3' e²−1( 4'

e²+1(µ < µ. (2.59) Remark 2.7. Using the periodicity we can check that the sequence|bε|²= |∇w_ε|² converges in the weak-∗sense of measures on!– but not weakly in L¹(!)– to the constantµdefined by (2.53). Theorem 2.6 can thus be regarded as a counter- example to the statement of Theorem 2.4 without the equi-integrability assumption on the driftbεinL²(!)². Indeed, the conclusion of Theorem 2.4 would give a limit equation (2.59), withγ =µ.

Proof of Theorem2.6. The proof is divided into two steps. In the first step we con- struct an oscillating test function zε which solves equation (2.64) below. In the second step we determine the limit equation (2.59).

First step.Construction of an oscillating test function.

Denote by Qr the disk of radius r centered at the origin. Consider the unique solutionZεin H¹(QR)of the equation







− 1

ε²"Zε+ 1

ε²|∇Wε|²Zε = 1QR

|QR| inQR

∂Zε

∂n =0 on∂Q_R.

(2.60)

The functionZεis radial and can be computed explicitly. Using the Laplace opera- tor in polar coordinates and|∇Wε|²=α²_εr⁻²1_{QR\ ¯Qrε}, we get

Zε(r)=









− ε²

4πR²r²+cε ifr∈(0,rε] aεr^αε+bεr^−αε+ ε²

πR²'

α_ε²−3(r² ifr∈(rε,R], whereα_ε := 1

ln(R/rε).

(2.61)

The constantsaε,bε,cε are determined owing to the boundary condition on∂Q_R and to the transmission conditions on∂Qrε,i.e.

Z^'_ε(R)=0 and Zε(r_ε⁺)= Zε(r_ε⁻), Z^'_ε(r_ε⁺)=Z_ε^'(r_ε⁻). (2.62) We extend Zε by the constant value Zε(R)in Y \ ¯QR, and byY-periodicity in the whole spaceR^{2. TheY}-periodic extension is still denoted byZε. An explicit computation combined with (2.53) yields

Zε−→ Z¯ :=4' e²+1( 3'

e²−1( 1

µ strongly inH_*¹(Y). (2.63) As a consequence of (2.60), (2.61) the rescaled functionzε(x):=Zε(^x_ε)is solution of the equation

−"zε+ |∇w_ε|²zε =χ^*_Q

R

.x ε

/ inD^'(R^{2), (2.64)}

whereχ_Q^*

Ris theY-periodic function agreeing with_|¹_Q^{Q R}_R|in the period cellY. More- over, the following convergences hold

zε−& Z¯ weakly in H¹(!)andχ_Q^*

R

.x ε

/−&1 weakly-∗inL^∞(!), (2.65)

where the constantZ¯ is defined by (2.63).