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A drift homogenization problem revisited
Marc Briane, Patrick Gérard
To cite this version:
Marc Briane, Patrick Gérard. A drift homogenization problem revisited. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Scuola Normale Superiore 2012, XI (1), pp.1-39. �10.2422/2036- 2145.201006_003�. �hal-00492687�
A drift homogenization problem revisited
Marc BRIANE Patrick G´ERARD
INSA de Rennes Universit´e Paris-Sud IRMAR, CNRS UMR 6625 LMO, CNRS UMR 8628 35708 Rennes Cedex 7 FRANCE 91405 Orsay FRANCE
Institut Universitaire de France [email protected] [email protected]
June 16, 2010
Abstract
This paper revisits a homogenization problem studied by L. Tartar related to a tridi- mensional Stokes equation perturbed by a drift (connected to the Coriolis force). Here, a scalar equation and a two-dimensional Stokes equation with aL2-bounded oscillating drift are considered. Under higher integrability conditions 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 in L2, the same limit behaviour is obtained. However, 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 differ- ent homogenized zero-order terms actually show the optimality of the equi-integrability assumption.
Keywords: Homogenization - Second-order elliptic equations - Drift Mathematics Subject Classification: 35B27 - 76M50
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 [13, 14]
he studied the Stokes equation in a bounded domain Ω of R3, perturbed by an oscillating drift term, i.e.
−∆uε+ curl (vε)×uε+∇pε =f in Ω div (uε) = 0 in Ω uε= 0 on Ω,
(1.1) where the oscillations are produced by the sequence of vector-valued functionsvε which weakly converges to somev inL3(Ω)3. L. Tartar proved that the limit equation of (1.1) is the Brinkman [4] type equation
−∆u+ curl (v)×u+∇p+Mu=f in Ω div (u) = 0 in Ω u= 0 on Ω,
(1.2)
where M is a positive definite symmetric matrix-valued function. More precisely, M is defined by the convergences
(Dwλε)T vε −⇀ Mλ weakly in L32(Ω)3, for anyλ ∈R3, (1.3) where wελ ∈W1,3(Ω)3 solves the Stokes equation (1.1) in which uε is replaced by λ. Then, the convergence (1.3) combined with the compactness of uε in L3(Ω)3, yields the zero-order term Mu in (1.2). In [15] 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 [5] for the Laplace equation, and by G. Allaire [2] for the Stokes equation, with zero Dirichlet boundary condition on the holes.
Since curl (vε)×uε is orthogonal to uε, the energy associated with (1.1) is reduced to Z
Ω
|Duε|2dx, (1.4)
and thus does not depend on the drift vε. Starting from this remark our aim is to study two drift homogenization problems associated with the same energy (1.4), and to specify the optimal integrability satisfied by the drift so that the Tartar 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 succeeded in obtaining an optimal result for the three-dimensional Stokes equation (1.1) since the best integrability assumption for vε is not clear.
In Section 2, we consider the following scalar equation in a bounded open set Ω of RN, ( −∆uε+bε· ∇uε+ div (bεuε) = f in Ω
uε = 0 on ∂Ω, (1.5)
where bε∈L∞(Ω)N is bounded in L2(Ω)N. We obtain three different homogenization results:
In Section 2.1, assuming that the divergence of the drift bε is bounded in W−1,q(Ω), with q > N, we prove (see Theorem 2.1) that the sequence uε weakly converges in H01(Ω) to the solution u of the equation
( −∆u+b· ∇u+ div (b u) +µ u =f in Ω
u = 0 on ∂Ω, (1.6)
where µ is a nonnegative function. The proof follows the Tartar method using the oscillating test function
wε := ∆−1 div (bε)
∈H01(Ω). (1.7)
Then, in Section 2.2, assuming only the equi-integrability of the sequence ∇wε in L2(Ω)N (this is actually a weaker assumption than the equi-integrability of the whole sequence bε), we obtain (see Theorem 3.1) the limit problem (1.6) with
|∇wε− ∇w|2 −⇀ µ weakly inL1(Ω) and µ u2 ∈L1(Ω). (1.8) It seems intricate to apply directly the Tartar method with the test functionwε, since we cannot control the terms bε · ∇uεwε and bε· ∇wεuε. To this end, one should consider truncations of bothwε and ∇wε. 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 ∇wε· ∇uε only using a truncation of ∇wε. The equi-integrability of ∇wε then gives the thesis. Also assuming that b ∈ Lq(Ω)N, with q > N, (which ensures the uniqueness in (1.6)) we prove the following corrector result
uε−(1 +wε−w)u −→ 0 strongly in Wloc1,q(Ω), for any q∈[1, N′). (1.9) Finally, in Section 2.3, we show the optimality of the equi-integrability condition thanks to a counter-example in the periodic framework (see Theorem 2.6). Making a change of functions with bε =∇wε, equation (1.5) is shown to be equivalent to the following equation
− ∆vε+µεvε =fε, with µε :=|∇wε|2, (1.10) the solution of which has the same limit as uε. G. Dal Maso, A. Garroni [6] 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 sequence wε so that the limit equation of (1.5), or equivalently (1.10), is
− ∆u+γ u=f, (1.11)
with an explicit constant γ which turns out to be < µ. Therefore, the loss of equi-integrability for ∇wε violates the result of Section 2.2. Note that the vectorial character of the drift term in equation (1.5) makes difficult the derivation of a closure result similar to the one of [6] which is strongly based on a comparison principle.
In Section 3, we consider the following two-dimensional equivalent of the perturbed Stokes problem (1.1),
−∆uε+ curl (vε)Juε+∇pε =f in Ω div (uε) = 0 in Ω uε = 0 on Ω,
(1.12)
where J is the rotation matrix of angle 90◦, andvε∈L∞(Ω)2 is bounded inL2(Ω)2. We follow the same scheme as in the scalar case:
In Section 3.1, assuming that the sequence vε is bounded in Lr(Ω)2 with r > 2, we show (see Theorem 3.1) that the sequence uε weakly converges in H01(Ω) to the solution u of the Brinkman equation
−∆u+ curl (v)Ju+∇p+Mu=f in Ω div (u) = 0 in Ω u= 0 on Ω,
(1.13)
where M is a symmetric positive definite matrix-valued function defined by the convergence (1.3) in Lr+22r (Ω)2.
In Section 3.2, assuming only the equi-integrability of the sequence vε inL2(Ω)2, we prove (see Theorem 3.3) owing to the Tartar method that the sequence uεweakly converges inH01(Ω) to the solution u of the Brinkman equation (1.13) with similarly to (1.8),
(Dwλε)T vε −⇀ Mλ weakly in L1(Ω)2 and Mu·u∈ L1(Ω). (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 towλε but associated with a truncation ofvε. Moreover, if Ω has a regular boundary, v ∈Lr(Ω)2 with r >2, and M ∈Lm(Ω)2×2 with m >1, we get the corrector result
uε−u−Wεu −→ 0 strongly in W1,1(Ω)2, where Wελ:=wελ, for λ∈R2. (1.15) Finally, in Section 3.2, we construct an oscillating sequence vε which is not equi-integrable inL2(Ω)2, which leads to the limit problem (1.13) involving a matrix Γ which is not symmetric and satisfies the strict inequality
Γλ·λ < Mλ·λ, for any λ6= 0,
which is inconsistent with the Tartar approach. This shows the optimality 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 sequences vε. This problem is far from being evident due to the absence of comparison principle for such a vector-valued equation.
Notations
• The space dimension is N ≥2, and 2∗ := 2N N −2.
• The conjugate exponent of p≥1 is denoted by p′ := p p−1.
• For u:RN −→RN,Du:=
∂ui
∂xj
1≤i,j≤N
.
• For Σ :RN −→RN×N, Div (Σ) :=
N
X
j=1
∂Σij
∂xj
!
1≤i≤N
.
• H♯1(Y), with Y := (0,1)N, denotes the space of the Y-periodic functions on RN which belong to Hloc1 (RN).
2 A scalar equation with a drift term
Along this section Ω is a bounded regular open set of RN, withN ≥2, and f is a distribution in H−1(Ω).
2.1 The classical case
Let q∈(N,∞). Consider a sequence bε in L∞(Ω)N such that
bε −⇀ b weakly in L2(Ω)N and div (bε) is bounded inW−1,q(Ω). (2.1) Let wε∈W01,q(Ω) be the solution of the equation (see, e.g., [8] Theorem 2.1)
∆wε = div (bε) inD′(Ω). (2.2)
Up to a subsequence wε weakly converges in W01,q(Ω) to the functionw solution of
∆w= div (b) inD′(Ω). (2.3)
We have the following result:
Theorem 2.1. The solution uε ∈H01(Ω) of the equation
− ∆uε+bε· ∇uε+ div (bεuε) = f in D′(Ω), (2.4) weakly converges in H01(Ω), up to a subsequence, to a solution u∈H01(Ω) of the equation
− ∆u+b· ∇u+ div (b u) +µ u=f in D′(Ω), (2.5) where µ is the function defined by the convergence
|∇wε− ∇w|2 −⇀ µ weakly in Lq2(Ω). (2.6) Remark 2.2. The uniqueness for equation (2.4) is not evident under the sole assumption b ∈L2(Ω)2. Assuming a stronger integrability ofbwe will obtain in Theorem 2.4 the uniqueness for the limit equation.
Proof. The proof is based on the Tartar method of the oscillating test functions (see Appendix of [12], and [16]). The function wε of (2.2) will play the role of the oscillating test function.
The variational formulation of (2.4) is Z
Ω
∇uε· ∇ϕ dx+ Z
Ω
bε· ∇uεϕ dx− Z
Ω
bε· ∇ϕ uεdx=hf, ϕiH−1(Ω),H01(Ω), ∀ϕ∈H01(Ω). (2.7) Then, by the Lax-Milgram theorem there exists a unique solution uε of (2.7) in H01(Ω). In particular, for v ∈W1,∞(Ω), puttingϕ =v uε as test function in (2.7) we obtain the identity
Z
Ω
|∇uε|2v dx+ Z
Ω
∇uε· ∇v uεdx− Z
Ω
bε· ∇v u2εdx=hf, v uεiH−1(Ω),H10(Ω), (2.8) which will be used several times. So, choosing v = 1 in (2.8) the term with bε cancel so that we easily deduce that uε is bounded in H01(Ω) and weakly converges, up to a subsequence, to a function u inH01(Ω). Therefore, it follows from (2.7) the limit variational formulation
Z
Ω
∇u· ∇ϕ dx+ Z
Ω
b· ∇u ϕ dx+ Z
Ω
ϕ dν − Z
Ω
b· ∇ϕ u dx=hf, ϕiH−1(Ω),H01(Ω), (2.9) which holds for any ϕ ∈ W01,q(Ω) (due to the embedding of W01,q(Ω) into C( ¯Ω) for q > N), where the measure ν is defined by the convergence
bε· ∇uε −⇀ b· ∇u+ν weakly-∗ inM(Ω). (2.10) The limit equation associated with (2.9) is
− ∆u+b· ∇u+ν+ div (b u) = f inD′(Ω). (2.11) Now, let us determine the measure ν of (2.10). Let ϕ ∈ Cc∞(Ω). Putting ϕ wε as test function in (2.7) and ϕ uε in (2.2), and taking the difference of the two equalities we get
Z
Ω
∇uε· ∇ϕ wεdx− Z
Ω
∇wε· ∇ϕ uεdx
=hf, ϕ wεiH−1(Ω),H01(Ω)− Z
Ω
bε· ∇uεϕ wεdx+ Z
Ω
bε· ∇wεϕ uεdx+ Z
Ω
bε· ∇ϕ uεwεdx
− Z
Ω
bε· ∇uεϕ dx− Z
Ω
bε· ∇ϕ uεdx.
(2.12)
Passing to the limit in (2.12) by using the strong convergence of uε in Lp(Ω), for p < 2∗, and the uniform convergence of wε inC( ¯Ω) (q > N), we obtain
Z
Ω
∇u· ∇ϕ w dx− Z
Ω
∇w· ∇ϕ u dx
=hf, ϕ wiH−1(Ω),H01(Ω)− Z
Ω
b· ∇u ϕ w dx− Z
Ω
ϕ w dν + Z
Ω
σ ϕ u dx+ Z
Ω
b· ∇ϕ u w dx
− Z
Ω
b· ∇u ϕ dx− Z
Ω
ϕ dν− Z
Ω
b· ∇ϕ u 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· ∇wε−⇀ σ weakly in Lq+22q (Ω). (2.14) On the other hand, putting ϕ w ∈W01,q(Ω) in (2.9) and ϕ u∈H01(Ω) in (2.3) we have
Z
Ω
∇u· ∇(ϕ w)dx =hf, ϕ wiH−1(Ω),H01(Ω)− Z
Ω
b· ∇u ϕ w dx− Z
Ω
ϕ w dν
+ Z
Ω
b· ∇w ϕ u dx+ Z
Ω
b· ∇ϕ u w dx,
(2.15)
Z
Ω
∇w· ∇(ϕ u)dx= Z
Ω
b· ∇u ϕ dx+ Z
Ω
b· ∇ϕ u dx. (2.16) Equating the difference between (2.15) and (2.16) to the right-hand side of (2.13), it follows
that Z
Ω
σ ϕ u dx− Z
Ω
b· ∇w ϕ u dx− Z
Ω
ϕ dν = 0, for any ϕ ∈Cc∞(Ω), (2.17) which implies that
ν=σ u−b· ∇w u in D′(Ω). (2.18) It thus remains to determine the limit equation (2.5). To this end, we pass to the limit by using ϕ wεas test function in (2.2) and the definition (2.6) ofµ, and we putϕ w in (2.3), which yields
Z
Ω
µ+|∇w|2
ϕ dx+ Z
Ω
∇w· ∇ϕ w dx= Z
Ω
σ ϕ dx+ Z
Ω
b· ∇ϕ w dx, (2.19) Z
Ω
|∇w|2ϕ dx+ Z
Ω
∇w· ∇ϕ w dx= Z
Ω
b· ∇w ϕ dx+ Z
Ω
b· ∇ϕ w dx. (2.20) Equating (2.19) and (2.20), we deduce that
µ=σ−b· ∇w inD′(Ω), (2.21)
which combined with (2.18) implies that
ν =µ u inD′(Ω). (2.22)
Finally, the limit equation (2.11) and the relation (2.22) give the desired homogenized equa- tion (2.5).
Remark 2.3. It can be shown that µ(x) =
Z
SN−1
µ(x, dξ)ξ·ξ , (2.23)
where µdenotes the matrix-valued H-measure (or micro-local defect measure) of the sequence bε (see [15] and [7]), and SN−1 the unit sphere of RN.
Assumption (2.1) is actually not sharp. In the next section we replace it by the boundedness of bε and the equi-integrability of ∇wε in L2(Ω)2.
2.2 The case under an equi-integrability assumption
In this section Ω is a bounded open set of RN. Consider a sequence bε in L∞(Ω)N the Hodge decomposition of which is
bε =∇wε+ξε, with wε∈H01(Ω), ξε ∈L2(Ω)N and div (ξε) = 0, (2.24) such that
bε −⇀ b weakly in L2(Ω)N. (2.25)
Note that for a fixedε >0,wε ∈W1,p(Ω) andξε∈Lp(Ω)N for anyp∈[2,∞). But the essential point is the asymptotic behaviour of the sequences bε, ∇wε, ξε. Our main assumption is the equi-integrability of the sequence ∇wε inL2(Ω)N. By virtue of the Vitali-Saks theorem this is equivalent to the following convergence, up to an extraction of a subsequence,
|∇wε− ∇w|2 −⇀ µ weakly in L1(Ω), (2.26) (Compare to (2.6) with q > 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(Ω) to a solution u of the equation
− ∆u+b· ∇u+ div (b u) +µ u=f in D′(Ω), (2.27)
with Z
Ω
µ u2dx≤ hf, uiH−1(Ω),H01(Ω)− Z
Ω
|∇u|2dx. (2.28)
ii) Also assume that b ∈Lq(Ω)N, where q >2 if N = 2 and q=N if N >2. Then, we have Z
Ω
|∇u|2dx+ Z
Ω
µ u2dx=hf, uiH−1(Ω),H01(Ω). (2.29) and there exists a unique solution u∈H01(Ω) of equation (2.27), with µ u2∈L1(Ω).
Moreover, for any p∈[1,2) if N = 2 and p=N′ if N >2, we have the corrector result
∇uε− ∇u−(∇wε− ∇w)u −→ 0 strongly in Lploc(Ω)N, (2.30) and for any r ∈[1, p),
uε−(1 +wε−w)u −→ 0 strongly in Wloc1,r(Ω). (2.31) Remark 2.5. No equi-integrability is required for the divergence free sequenceξε. Actually, we can prove that the equi-integrability of the sequence bε inL2(Ω)N implies the equi-integrability of its two components ∇wε, ξε in L2loc(Ω)N. Therefore, condition (2.26) is really weaker than the equi-integrability of bε.
Moreover, the equi-integrability of∇wεinL2(Ω)N is essential for deriving the limit equation with the zero-order termµ u. 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 Theorem 2.4. The limit u of uε in H01(Ω) solves the equation (2.11) where ν is defined by
bε· ∇uε−b· ∇u−⇀ ν weakly-∗ inM(Ω), (2.32) By the Murat, Tartar div-curl lemma [10] we have
bε· ∇uε= (ξε+∇w)· ∇uε+ (∇wε− ∇w)· ∇uε−⇀ b· ∇u+ν in D′(Ω).
This combined with the equi-integrability of∇wεimplies thatν is also given by the convergence (∇wε− ∇w)· ∇uε−⇀ ν weakly in L1(Ω). (2.33) The proof of Theorem 2.4. is based on a parametrix method which allows us to express uε
as a solution of a fixed point problem. As a consequence, we obtain a strong estimate of ∇uε
inLploc(Ω) for somep >1 close to 1. However, this estimate cannot provide directly the desired limit ν of (2.33) since p < 2. To overcome this difficulty we consider a truncation ηεk of ∇wε
which is bounded by k > 0. Then, we can pass to the limit as ε tends to zero in the product ηεk· ∇uε for a fixedk. Hence, thanks to the equi-integrability of ∇wε we deduce the limit ν as k tends to infinity.
The proof is divided into four steps. In the first step we present the parametrix method which leads to a Lp-strong estimate of ∇uε. In the second step we determine the limit of the sequence ηεk· ∇uε for a fixed k > 0. In the third step we determine the limit ν and 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 Ω. To this end consider two sequences of functions ϕn, ψn in Cc∞(Ω), such that
0≤ϕn, ψn≤1 and ϕn= 1 in supp (ψn), for any n≥1,
n ≥1 : supp (ψn)∩K 6= Ø is finite, for any compact subset K ⊂Ω, X
n≥1
ψn= 1 in Ω.
(2.34)
LetEbe the fundamental solution of the Laplace operator inRN. Then, the operatorP defined in D′(Ω) by
P ζ :=X
n≥1
ψnE∗(ϕnζ), for ζ ∈D′(Ω), (2.35) is a parametrix of the Laplace operator (see [1] Chapter I, for further details) which satisfies
P(∆ζ) =ζ−Kζ and ∆(P ζ) =ζ−K′ζ, for ζ ∈D′(Ω), (2.36) where K, K′ are two C∞-kernel operators properly supported in Ω. Thanks to the Calder`on- Zygmund regularity for the Laplace operator (see, e.g., [8] Theorem 2.1, and the references therein) we also have for any p >1, and s∈[0,2] such that s+1p is not an integer,
P maps continuously D′(Ω) to D′(Ω), and Wloc−s,p(Ω) toWloc2−s,p(Ω). (2.37) Then, applying (2.36) to the solution uε of (2.4) we have
uε=P(∆uε) +Kuε =P
div u(∇wε− ∇w) +P
div ∇wε(uε−u) +P
div (u∇w)
+P ξε· ∇uε+bε· ∇uε−f
+Kuε, (2.38)
Fix p > 1 close enough to 1 and s ∈ (N/p′,1). Since uε−u strongly converges to 0 in Lq(Ω) for any q ∈(2,2∗), the sequence div ∇wε(uε−u)
strongly converges to 0 inW−1,p(Ω), hence by (2.37) we have
P
div ∇wε(uε−u)
−→0 strongly in Wloc1,p(Ω).
Moreover, the sequenceξε· ∇uε+bε· ∇uεis bounded inL1(Ω), thus inW−s,p(Ω) sinces > N/p′. Therefore, again by (2.37) the sequence ∇P ξε· ∇uε+bε· ∇uε−f
is bounded inW1−s,p(Ω)N, and up to a subsequence strongly converges in Lploc(Ω)N. Hence, since
ξε· ∇uε+bε· ∇uε−⇀ ξ · ∇u+ν+b· ∇u inD′(Ω), we deduce from (2.38) the strong estimate
∇uε− ∇P
div u(∇wε− ∇w)
=∇P
div (u∇w) +ξ· ∇u+ν+b· ∇u−f
+∇Ku+oLploc(Ω)N(1)
=∇P
ν+b· ∇u+ div (b u)−f
+∇Ku+oLploc(Ω)N(1) since ξ· ∇u= div (u ξ) ,
(2.39)
where oLploc(Ω)N(1) denotes a sequence which strongly converges to 0 inLploc(Ω)N. On the other hand, by (2.36) and (2.37) we have
∇P
div u(∇wε− ∇w)
=∇P
∆ u(wε−w)
− ∇P
div ∇u(wε−w)
=∇P
∆ u(wε−w)
+oLploc(Ω)N(1)
=∇ u(wε−w)
+oLploc(Ω)N(1)
=u(∇wε− ∇w) +oLploc(Ω)N(1).
Therefore, this combined with (2.39) yields
∇uε−u(∇wε− ∇w) = ∇P
ν+b· ∇u+ div (b u)−f
+∇Ku+oLploc(Ω)N(1). (2.40) Second step: Estimate of the sequence ηεk· ∇uε.
Set ηεk := ∇wε1{|∇wε|<k}, for a positive integer k. Let us determine the limit of ηkε · ∇uε in L2loc(Ω)N. Using a diagonal extraction, there exists to a subsequence ofε, still denoted byε, such that ηkε weakly converges to some ηk in L∞(Ω)N for any k. By the strong convergence (2.40) combined with the weak convergence ofu(∇wε− ∇w) to 0 inLp(Ω)N (forpclose to 1) we have
ηεk· ∇uε− ηkε −ηk
·(∇wε− ∇w)u
−⇀ ηk· ∇P
ν+b· ∇u+ div (b u)−f
+ηk· ∇Ku weakly inLploc(Ω).
Hence, we get that
σk =µku+ηk· ∇P
ν+b· ∇u+ div (b u)−f
+ηk· ∇Ku in Ω, (2.41) where
σk := lim
ε→0
ηεk· ∇uε
weakly inL2(Ω),
µk := lim
ε→0
ηεk−ηk
·(∇wε− ∇w)
weakly inL2(Ω). (2.42) Third step: Determination of ν and 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
+Ku in Ω, hence
ηk· ∇u=ηk· ∇P
ν+b· ∇u+ div (b u)−f
+ηk· ∇Ku in Ω.
Equating this with (2.41) we obtain
σk =µku+ηk· ∇u in Ω. (2.43)
Now, let us pass to the limit as k → ∞. By virtue of the equi-integrability of ∇wε in L2(Ω)N and by definition (2.42) the sequence µk strongly converges inL1(Ω) to the functionµof (2.26), ηk strongly converges to ∇w in L2(Ω)N, and σk strongly converges to ν+∇w· ∇u in L1(Ω).
Then, up to a subsequence µk converges to µ a.e. in Ω, and by the Fatou lemma combined with equality (2.43) we get
Z
Ω
|µ u|dx≤lim inf
k→∞
Z
Ω
|µku|dx≤lim inf
k→∞
Z
Ω
|σk−ηk· ∇u|dx= Z
Ω
|ν|dx. (2.44) We deduce from (2.44) and (2.43) that µ u∈L1(Ω) and
ν=µ u in Ω, (2.45)
which yields the limit equation (2.27).
It remains to prove the inequality of (2.28). Let v ∈ L∞(Ω) and t ∈ R. By (2.26), (2.33) and (2.45) we have
Z
Ω
∇uε− ∇u−(∇wε− ∇w)t v
2dx
= Z
Ω
|∇uε− ∇u|2dx+t2 Z
Ω
|∇wε− ∇w|2v2dx−2t Z
Ω
∇uε·(∇wε− ∇w)v dx+o(1)
=hf, uiH−1(Ω),H10(Ω)− Z
Ω
|∇u|2dx+t2 Z
Ω
µ v2dx−2t Z
Ω
µ u v dx+o(1),
(2.46) hence
t2 Z
Ω
µ v2dx−2t Z
Ω
µ u v dx+hf, uiH−1(Ω),H01(Ω)− Z
Ω
|∇u|2dx≥0, ∀t∈R. This implies that
Z
Ω
µ u v dx 2
≤
hf, uiH−1(Ω),H01(Ω)− Z
Ω
|∇u|2dx Z
Ω
µ v2dx. (2.47) Let Tk, k >0, be a function in C1(R) such that
0≤Tk′ ≤1 and
Tk(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
Z
Ω
µ u Tk(u)dx 2
≤
hf, uiH−1(Ω),H01(Ω)− Z
Ω
|∇u|2dx Z
Ω
µ u Tk(u)dx,
hence Z
Ω
µ u Tk(u)dx≤ hf, uiH−1(Ω),H01(Ω)− Z
Ω
|∇u|2dx. (2.49)
Since u Tk(u) is a nondecreasing nonnegative sequence which converges to u2 a.e. in Ω, 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 that b ∈Lq(Ω)N, where q > 2 if N = 2 and q =N if N >2. Let ϕn be a sequence in C01(R) which strongly converges touinH01(Ω) and a.e. in Ω, and such that|∇ϕn|is dominated by a fixed function inL2(Ω). Putting the truncation functionTk(ϕn) (2.48) in the limit equation (2.27) we have
Z
Ω
∇u· ∇Tk(ϕn)dx+ Z
Ω
b· ∇u Tk(ϕn)dx− Z
Ω
b· ∇Tk(ϕn)u dx+ Z
Ω
µ u Tk(ϕn)dx
=
f, Tk(ϕn)
H−1(Ω),H01(Ω).
Since b· ∇u, µ u ∈ L1(Ω) and b u ∈ L2(Ω)N (as a consequence of b ∈ Lq(Ω)N), we can pass to the limit as n → ∞ in the previous equality owing to the Lebesgue dominated convergence theorem, which yields
Z
Ω
∇u· ∇Tk(u)dx+ Z
Ω
b· ∇u Tk(u)dx− Z
Ω
b· ∇Tk(u)u dx+ Z
Ω
µ u Tk(u)dx
=
f, Tk(u)
H−1(Ω),H10(Ω).
(2.50) Then, using that |Tk(u)| ≤ |u|, 0 ≤ Tk′(u) ≤ 1, Tk(u) strongly converges to u in H01(Ω), and that b u ∈ L2(Ω)N, µ u2 ∈ L1(Ω), and passing to the limit as k → ∞ owing to the Lebesgue dominated convergence theorem we get
Z
Ω
|∇u|2dx+ Z
Ω
b· ∇u u dx− Z
Ω
b· ∇u u dx+ Z
Ω
µ u2dx =hf, uiH−1(Ω),H01(Ω),
which is (2.29). Moreover, the proof of equality (2.29) with f = 0 shows that there exists a unique solution u∈H01(Ω) of equation (2.27), with µ u2 ∈L1(Ω).
It remains to prove the corrector results. By the estimate (2.46) withv =Tk(u) and t= 1, combined with equality (2.29) we have
k→∞lim lim
ε→0
Z
Ω
∇uε− ∇u−(∇wε− ∇w)Tk(u)
2dx
= lim
k→∞
Z
Ω
µ u−Tk(u)2
dx
= 0.
(2.51)
On the other hand, let p ∈ [1,2) if N = 2 and p = N′ if N > 2, and consider an open set ω ⋐Ω. By the H¨older inequality we have
Z
ω
∇uε− ∇u−(∇wε− ∇w)u
pdx
≤2p−1 Z
Ω
∇uε− ∇u−(∇wε− ∇w)Tk(u)
p+ Z
ω
|∇wε− ∇w|p
u−Tk(u)
pdx
≤c Z
Ω
∇uε− ∇u−(∇wε− ∇w)Tk(u)
2p2 +c
Z
ω
u−Tk(u)
2p 2−p dx
1−p2
≤c Z
Ω
∇uε− ∇u−(∇wε− ∇w)Tk(u)
2p2 +c
Z
{|u|>k}∩ω
|u|2−2pp dx 1−p2
.
(2.52)
Since u ∈ L2−2pp(ω) by the Sobolev embedding, passing successively to the limits ε → 0 and k → ∞ in (2.52) owing to convergence (2.51) we obtain the strong convergence (2.30).
Letr ∈[1, p). Since wε−w strongly converges to 0 inL2−2rr(ω), by the H¨older inequality the sequence (wε−w)∇u strongly converges to 0 in Lr(ω)N. Finally, this combined with (2.30) implies the corrector result (2.31).
2.3 A counter-example
In this section Ω is a regular bounded open set of R2, and Y := (−12,12)2. For fixed R ∈(0,12) and µ >0, let rε∈(0, R) be defined by the equality
2π
ε2|lnrε| =µ. (2.53)
Let Wε be the Y-periodic function and wε be the εY-periodic function defined by
Wε(y) :=
lnr−lnrε
lnR−lnrε
if r :=|y| ∈ (rε, R) 0 si r≤rε
1 si r≥R,
y∈Y, wε(x) :=Wε
x ε
, x∈R2. (2.54)
Note that by (2.53) we have 1 ε2
Z
Y
|∇Wε|2dy= 2π
ε2ln (R/rε) −→
ε→0 µ. (2.55)
We then consider the drift bε defined by bε(x) := ∇wε(x) = 1
ε∇Wε
x ε
, for x∈R2. (2.56)
Taking into account (2.53) it is easy to check that
wε −⇀ 1 weakly in H1(Ω) and weakly-∗ inL∞(Ω). (2.57) Let f be a non-zero function in L2(Ω). We study the asymptotic behavior of the equation (2.4) with the drift bε of (2.56), i.e.
− ∆uε+∇wε· ∇uε+ div (∇wεuε) =f inD′(Ω). (2.58) We have the following result:
Theorem 2.6. The solution uε of (2.58) weakly converges in H01(Ω) to the solution u of the equation
−∆u+γ u =f in D′(Ω), 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 = |∇wε|2 converges in the weak-∗sense of measures on Ω – but not weakly in L1(Ω) – to the constant µdefined 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 drift bε in L2(Ω)2. Indeed, the conclusion of Theorem 2.4 would give a limit equation (2.59), with γ =µ.
Proof of Theorem 2.6. The proof is divided into two steps. In the first step we construct an oscillating test functionzε 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 solution Zε in H1(QR) of the equation
− 1
ε2 ∆Zε+ 1
ε2 |∇Wε|2Zε = 1QR
|QR| inQR
∂Zε
∂n = 0 on∂QR.
(2.60)
The function Zε is radial and can be computed explicitly. Using the Laplace operator in polar coordinates and |∇Wε|2 =α2εr−21QR\Q¯rε, we get
Zε(r) =
− ε2
4πR2 r2+cε if r∈(0, rε] aεrαε +bεr−αε + ε2
πR2(α2ε−3)r2 if r∈(rε, R],
where αε := 1
ln(R/rε). (2.61) The constants aε, bε, cε are determined owing to the boundary condition on ∂QR and to the transmission conditions on ∂Qrε, i.e.
Zε′(R) = 0 and Zε(r+ε) =Zε(r−ε), Zε′(rε+) =Zε′(rε−). (2.62) We extend Zε by the constant value Zε(R) in Y \ Q¯R, and by Y-periodicity in the whole space R2. The Y-periodic extension is still denoted by Zε. An explicit computation combined with (2.53) yields
Zε−→Z¯:= 4 (e2+ 1) 3 (e2−1)
1
µ strongly in H♯1(Y). (2.63) As a consequence of (2.60), (2.61) the rescaled functionzε(x) :=Zε(xε) is solution of the equation
− ∆zε+|∇wε|2zε =χ♯QRx ε
inD′(R2), (2.64)
where χ♯QR is the Y-periodic function agreeing with 1|QQR
R| in the period cell Y. Moreover, the following convergences hold
zε −⇀ Z¯ weakly inH1(Ω) and χ♯QRx ε
−⇀ 1 weakly-∗ inL∞(Ω), (2.65) where the constant ¯Z is defined by (2.63).
Second step: Determination of the limit equation (2.59).
Define the function vε :=e1−wεuε. Then, equation (2.58) is equivalent to
− ∆vε+|∇wε|2vε =e1−wεf inD′(Ω). (2.66) G. Dal Maso, A. Garroni [6] proved that this class of equations is stable under homogenization.
In the present case, the use of the oscillating test function zε will allow us to obtain the limit equation (2.59).
On the one hand, choosing v =wε in (2.8) we get Z
Ω
|∇wε|2u2εdx− Z
Ω
∇wε· ∇uεuεdx= Z
Ω
|∇uε|2wεdx− Z
Ω
f wεuεdx≤c, (2.67) since uε is bounded in H01(Ω) and 0 ≤ wε ≤ 1. Then, by the Cauchy-Schwarz inequality we have
Z
Ω
|∇wε|2u2εdx ≤ c+c Z
Ω
|∇uε|2dx 12 Z
Ω
|∇wε|2u2εdx 12
≤ c+c′ Z
Ω
|∇wε|2u2εdx 12
,
(2.68)
hence uε∇wε is bounded in L2(Ω)2. This combined with convergence (2.57) implies that vε
weakly converges to uin H01(Ω).
On the other hand, for ϕ∈ Cc∞(Ω), putting the functions ϕ zε in (2.66) and ϕ vε in (2.64), taking the difference of the two equalities, and passing to the limit owing to convergences (2.65) we obtain the equality
Z
Ω
∇u· ∇ϕZ dx¯ + Z
Ω
ϕ u dx= Z
Ω
f ϕZ dx,¯ for any ϕ∈Cc∞(Ω). (2.69) which is the variational formulation of equation (2.59), with γ = ¯Z−1. 2
3 A Stokes equation with a drift term
3.1 The classical case
In [13, 14] L. Tartar noted that the nonlinear term of the three-dimensional Navier-Stokes equation for the divergence free velocity ureads as
(u· ∇)u= Div (u⊗u) = curl (u)×u+∇ 12|u|2
. (3.1)
This led him to study the perturbed Stokes equation
− ∆u+ curl (v)×u+∇p=f, (3.2)
where a given vector-valued function v replaced the velocity u of the Navier-Stokes equation.
The equivalent of transformation (3.1) in two-dimension is Div (u⊗u) = curl (u)Ju+∇ 12|u|2
, where curl (u) := ∂1u2−∂2u1 and J :=
0 −1
1 0
. (3.3)
More generally equality (3.3) extends for any divergence free functionsu, v to the following one curl (v)Ju= Div (v⊗u) + (Du)T v − ∇(v·u). (3.4) Similarly to (3.2) this leads us to the two-dimensional perturbed Stokes equation
− ∆u+ curl (v)Ju+∇p=f. (3.5)
Let Ω be a bounded domain ofR2. Letvεbe a sequence inL∞(Ω)2and letf be a distribution in H−1(Ω)2. Consider the perturbed Stokes equation
−∆uε+ curl (vε)Juε+∇pε =f in Ω div (uε) = 0 in Ω
uε = 0 on∂Ω.
(3.6)
In the three-dimensional case where curl (vε)×uεreplaces curl (vε)Juε, L. Tartar [14] derived a Stokes equation with a Brinkman law under the assumption that vε is bounded in L3(Ω)3 (see Introduction). Mimicking the Tartar approach in dimension two we can derive a similar homogenized equation using the test function wελ, for λ∈R2, solution of the Stokes equation
−∆wελ+ Div (vε−v)⊗λ
+∇qλε = 0 in Ω div wλε
= 0 in Ω wλε = 0 on ∂Ω.
(3.7)
Then, we have the following result:
Theorem 3.1. Assume that vε is bounded in Lr(Ω)2, with r > 2. Then, the solution uε of (3.6) weakly converges in H01(Ω) to the solution u of the Brinkman equation
−∆u+ curl (v)Ju+∇p+Mu =f in Ω div (u) = 0 in Ω
u = 0 on ∂Ω,
(3.8)
where M is the positive definite symmetric matrix-valued function defined by
(Dwελ)Tvε −⇀ Mλ weakly in L2+r2r (Ω)2 and in Llocr2 (Ω)2 Dwλε ·Dwµε −⇀ Mλ·µ weakly-∗ in M(Ω)2 and in L
r 2
loc(Ω)2,
for λ, µ∈R2. (3.9) Moreover, the zero-order term of (3.8) is given by the convergences
(Duε)T(vε−v) −⇀ Mu weakly in L2+r2r (Ω)2
Duε:Dwελ −⇀ Mu·λ weakly-∗ in M(Ω) and in L
2r 2+r
loc (Ω)2.
(3.10) Proof. By the representation formula (3.4) we have
curl (vε)Juε = (Duε)T vε+ Div (vε⊗uε)− ∇(vε·uε). (3.11) Hence, the variational formulation of (3.6) reads as
Z
Ω
Duε:Dϕ dx+ Z
Ω
(Duε)Tvε·ϕ dx− Z
Ω
(vε⊗uε) :Dϕ dx=hf, uεiH−1(Ω)2,H01(Ω)2, for any ϕ ∈H01(Ω)2, div (ϕ) = 0.
(3.12)
By the Lax-Milgram theorem there exists a unique divergence free function uε ∈H01(Ω)2 solu- tion of (3.12). Then, putting the velocity uε as test function in (3.12) it follows that
Z
Ω
|Duε|2dx=hf, uεiH−1(Ω)2,H01(Ω)2 , (3.13)
which implies that uε is bounded in H01(Ω)2. Let ω be a regular domain of Ω. Applying (3.12) to divergence free functions in H01(ω)2, there exists a unique pε in L2(ω)/Rsuch that equation (3.6) holds in D′(ω)2. Moreover, by (3.11) and the boundedness of vε in Lr(Ω)2 the sequence
∇pε is bounded in H−1(ω)2. Hence, due to the regularity of ω the sequence pε is bounded in L2(ω). Then, considering an exhaustive sequence of regular domains the union of which is Ω, we can construct in Ω a pressure pε which is bounded inL2loc(Ω). Therefore, up to a subsequence the following convergences hold
( uε −⇀ u weakly in H01(Ω)2
pε −⇀ p weakly in L2loc(Ω)/R, (3.14) Now, in view of (3.11) it is enough to determine the limit of the term (Duε)T vε. By the regularity results for the Stokes equation (see, e.g., [9] Theorem 2, p. 67) the sequences wελ and qελ satisfy
( wελ −⇀ 0 weakly in H1(Ω)2 and in Wloc1,r(Ω)2
qελ −⇀ 0 weakly in L2(Ω)/R and inLrloc(Ω)/R. (3.15) which imply convergence (3.9). Let ϕ∈ Cc∞(Ω). Following the Tartar method we put ϕ wελ in equation (3.6) and ϕ uε in equation (3.7). Then, from the representation (3.11), the conver- gences (3.14), (3.15) and the boundedness of vε inLr(Ω) we deduce that
Z
Ω
Duε:Dwελϕ dx− Z
Ω
(vε⊗uε) :Dwλεdx=o(1) Z
Ω
Dwελ :Duεϕ dx− Z
Ω
(vε−v)⊗λ
:Duεϕ dx=o(1),
for any ϕ∈Cc∞(Ω),
hence
( Duε :Dwελ−(Dwελ)Tvε·uε −⇀ 0
(Duε)T(vε−v)·λ−(Dwελ)Tvε·uε −⇀ 0 in D′(Ω). (3.16) By virtue of the strong convergence ofuεin anyLs(Ω)2space fors∈(1,∞), convergences (3.16) and (3.9) imply (3.10). This combined with (3.11) yields finally the limit problem (3.8).
Remark 3.2. It can be shown that M(x) =
Z
S1
tr (µ(x, dξ))−µ(x, dξ)ξ·ξ
ξ⊗ξ (3.17)
where µ is the matrix-valued H-measure of the sequence vε (see [15, 16]).
The case where vε is only bounded in L2(Ω)2 is much more delicate. On the one hand, under additional assumptions we will extend the Tartar result when vε is bounded and equi- integrable in L2(Ω)2. On the other hand, we will give an example of a sequence vε for which the homogenized Brinkman equation is not the one obtained by the Tartar procedure.
3.2 The case under an equi-integrability condition
In this section we make the following weaker assumption on the drift,
vε −⇀ v weakly in L2(Ω)2 and vε is equi-integrable in L2(Ω)2. (3.18) Then, we have the following extension of Theorem 3.1:
