Homogenization of scalar and Stokes equations with drift M. Briane, - - PowerPoint PPT Presentation

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives

Homogenization of scalar and Stokes equations with drift

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Colloque EDP-Normandie Universit´ e de Rouen Octobre 25–26 2011

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives

1 Introduction 2 A scalar drift problem (avec P. G´

erard) Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

3 A two-dimensional drift Stokes problem (avec P. G´

erard) Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

4 The periodic case without equi-integrability condition

The scalar equation The Stokes equation

5 Homogenization with large drifts

A compactness result in dimension two Nonlocal effects in dimension three

6 Perspectives

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives

A Tartar hydrodynamic problem

Ω a bounded domain of R3, vε bounded in L3(Ω)3,    − ∆uε + curl (vε) × uε + ∇pε = f in Ω div (uε) = 0 in Ω uε = 0

• n Ω.

curl (vε) × uε represents an oscillating Coriolis force. Tartar considered for any λ ∈ R3 the oscillating test function wλ

ε

solution of    − ∆wλ

ε + curl (vε) × λ + ∇qλ ε

= 0 in Ω div

• wλ

ε

• = 0

in Ω wλ

ε

= 0

• n ∂Ω.
• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives

A Brinkman homogenized problem

Theorem (Tartar 77) If vε ⇀ v in L3(Ω)3, then uε ⇀ u in H1

0(Ω)3 to the homogenized

Brinkman problem    − ∆u + curl (v) × u + ∇p + Mu = f in Ω div (u) = 0 in Ω u = 0

• n ∂Ω,

where M is the positive definite symmetric matrix-valued function given by: (Dwλ

ε )Tvε ⇀ Mλ

weakly in L

3 2 (Ω)3,

for λ ∈ R3. The boundedness of vε in L3(Ω)3 is not sharp. The energy is

• Ω |Duε|2 dx, since curl (vε) × uε ⊥ uε.
• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

A scalar drift problem

Let Ω be a regular bounded domain of RN, N ≥ 2. A scalar equivalent to the Stokes drift problem studied by Tartar is: − ∆uε + bε · ∇uε + div (bε uε) = f in Ω uε = 0

• n ∂Ω,

where bε ∈ L∞(Ω)N for a fixed ε > 0. Indeed, as before the associated energy is

• Ω |∇uε|2 dx, since for

any v ∈ H1

0(Ω) we have

• Ω
• bε · ∇v + div (bε v)
• v dx = 0.
• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

The strategy

Mimicking the Tartar approach leads us to consider for λ ∈ R, − ∆wε + bε · ∇λ + div (bε λ) = 0 in Ω wε = 0

• n ∂Ω,

i.e., λ = 1 the test function is defined by wε ∈ H1

0(Ω),

∆wε = div (bε) in Ω. Strategy for the homogenization of this drift problem: Use of the Tartar approach with a condition on div (bε). Refinement with a sharp condition of equi-integrability on bε. A new approach based on a parametrix for ∆. Proof of the sharpness of the condition thanks to an example.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

A Tartar type result

Assumption on bε:

• bε ⇀ b

in L2(Ω)N, ∃ q > N, wε bounded in W 1,q(Ω). Theorem uε ⇀ u ∈ H1

0(Ω), where u is a solution of the drift equation

− ∆u + b · ∇u + div (b u) + µ u = f in D′(Ω), where |∇wε − ∇w|2 ⇀ µ in L

q 2 (Ω).

The uniqueness for the limit problem is not obvious.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

An equi-integrability condition

bε = ∇wε + ξε, wε ∈ H1

0(Ω)N,

ξε ∈ L2(Ω)N with div (ξε) = 0. New assumption on bε: bε bounded in L2(Ω)N and ∇wε equi-integrable in L2(Ω)N. bε equi-integrable L2(Ω)N ⇒ ∇wε, ξε equi-integrable L2

loc(Ω)N.

The present condition is thus weaker. Theorem Assume that |∇wε − ∇w|2 converges weakly to µ in L1(Ω). Then, uε converges weakly in H1

0(Ω) to the solution u of

− ∆u + b · ∇u + div (b u) + µ u = f in Ω. But the proof is more delicate since we cannot pass to the limit in the products (bε · ∇uε) wε and (bε · ∇wε) uε.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

A parametrix for the Laplace operator

From a partition of the unity one can build a continuous operator P : D′(Ω) − → D′(Ω) which is a quasi-inverse of ∆ satisfying P ◦ ∆ = Id − K and ∆ ◦ P = Id − K ′, where K, K ′ are C ∞-kernel operators properly supported in Ω. By the classical regularity results we have P : W −s,p

loc

(Ω) − → W 2−s,p

loc

(Ω) ∀ p > 1, ∀ s ∈ [0, 2], s + 1 p / ∈ N.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

The parametrix method

The idea is to write uε as a solution of a fixed point problem: uε = P(∆uε) + Kuε = P

• ∆
• u (wε − w)
• − P
• div
• (wε − w)∇u
• + P
• div
• (uε − u) ∇wε
• + P
• bε · ∇uε + ξε · ∇uε + div (u∇w) − f
• + Kuε,

(wε − w) ∇u → 0 and (uε − u) ∇wε → 0 in Lp(Ω)N, p ∈ (1, N′). Hence, we get uε = u (wε − w) + oW 1,p

loc (Ω)(1)

+ P

• bε · ∇uε + ξε · ∇uε + div (u∇w) − f
• + Kuε.
• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

The parametrix method

Denote bε · ∇uε − b · ∇u ⇀ ν in D′(Ω). We have L1(Ω) ֒ → W −s,p

loc

(Ω) for s ∈ (N/p′, 1), and by the div-curl lemma of Murat, Tartar bε · ∇uε + ξε · ∇uε ⇀ ν + b · ∇u + ξ · ∇u in D′(Ω). This implies that uε = u (wε − w) + oW 1,p

loc (Ω)(1)

+ P

• ν + b · ∇u + ξ · ∇u + div (u∇w) − f
• + Ku + OW 2−s,p

loc

(Ω)(1).

Therefore, we obtain the strong estimate of the gradient ∇uε = u (∇wε − ∇w) + oLp

loc(Ω)N(1)

+ ∇P

• ν + b · ∇u + div (b u) − f
• + ∇(Ku).

We conclude considering the truncation ηk

ε := ∇wε 1{|∇wε|<k}.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

A corrector result

The function µ of the homogenized equation satisfies

• Ω

µ u2 dx ≤ f , uH−1,H1

0 −

• Ω

|∇u|2 dx. Theorem Assume that b ∈ Lq(Ω)N, with q > 2 if N = 2 q = N if N > 2 . Then: The previous ≤ is an = . ∃ ! u ∈ H1

0(Ω), µ u2 ∈ L1(Ω), solution of the limit problem.

For any p ∈ [1, 2) if N = 2 p < N′ if N > 2 , we have uε − (1 + wε − w) u − → 0 strongly in W 1,p

loc (Ω).

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

A counter-example for N = 2

Y :=

• − 1

2, 1 2

2 , 2π ε2 |ln rε| = µ ∈ (0, ∞), 0 < R < 1

2. Wε = 1 Wε = 0 ∆Wε = 0 Y \ QR Qrε

bε := ∇wε, where wε(x) := Wε x ε

• , Wε(y) Y -periodic.
• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

A counter-example

∇wε ⇀ 0 weakly in L2(Ω)2, but is not equi-integrable in L2(Ω)2. |∇wε|2 ⇀ µ weakly-∗ in M(Ω), but: Theorem The solution uε ∈ H1

0(Ω) of

− ∆uε + ∇wε · ∇uε + div (uε∇wε) = f ∈ L2(Ω), converges weakly in H1

0(Ω) to the solution u of

− ∆u + γ u = f , where γ := 3

• e2 − 1
• 4 (e2 + 1) µ < µ.
• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

A test function

Zε = Zε(R) ∂Zε / ∂n = 0 QR Y \ QR

• − 1

ε2 ∆Zε + 1 ε2 |∇Wε|2 Zε

=

1QR |QR|

in QR

∂Zε ∂n

= 0

• n ∂QR.
• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

Oscillating test functions

The function vε := e1−wε uε ⇀ u in H1

0(Ω), is solution of

− ∆vε + |∇wε|2 vε = e1−wε f in D′(Ω). The function zε := Zε x ε

• ⇀ ¯

Z in H1(Ω), is solution of − ∆zε + |∇wε|2 zε =

1♯

QR

|QR|

x

ε

• in D′(R2).

For ϕ ∈ D(Ω), putting ϕ zε in the first equation and ϕ vε in the second one we obtain

• Ω

∇u · ∇ϕ ¯ Z dx +

• Ω

ϕ u dx =

• Ω

f ϕ ¯ Z dx, with ¯ Z = 1 γ .

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

A preliminary remark

Tartar noted that for the Navier-Stokes equation in dimension 3, (u · ∇) u = Div (u ⊗ u) = curl (u) × u + ∇ 1

2 |u|2

, which led him to the drift Stokes equation − ∆u + curl (v) × u + ∇p = f . The equivalent in dimension 2 is Div (u ⊗ u) = curl (u) Ju + ∇ 1

2 |u|2

, where curl (u) := ∂1u2 − ∂2u1 and J := 0 −1

1 0

• .

This yields the drift Stokes equation − ∆u + curl (v) Ju + ∇p = f .

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

The classical case

Let Ω be a bounded domain of R2 and vε ∈ L2(Ω)2. We consider the solution uε ∈ H1

0(Ω)2 of the drift Stokes problem

• − ∆uε + curl (vε) Juε + ∇pε

= f in Ω div (uε) = 0 in Ω. Assumption on vε: vε ⇀ v weakly in Lr(Ω)2, with r > 2.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

The classical case

For λ ∈ R2, we consider the function wλ

ε ∈ H1 0(Ω)2 solution of

− ∆wλ

ε + Div

• (vε − v) ⊗ λ
• + ∇qλ

ε

= 0 in Ω div

• wλ

ε

• = 0

in Ω. Theorem uε ⇀ u in H1

0(Ω)2, where u is solution of the equation:

− ∆u + curl (v) Ju + ∇p + Mu = f in Ω div (u) = 0 in Ω, where M = MT > 0 is defined by (Dwλ

ε )Tvε −

⇀ Mλ weakly in L

2r 2+r (Ω)2.

The proof follows the Tartar method.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

Refinement with an equi-integrability condition

Assumption on vε: vε ⇀ v weakly in L2(Ω)2 and vε equi-integrable in L2(Ω)2. Same result as in the Tartar approach: Theorem uε ⇀ u in H1

0(Ω)2, where u is a solution of the equation

− ∆u + curl (v) Ju + ∇p + Mu = f in Ω div (u) = 0 in Ω, where M = MT > 0 is defined by (Dwλ

ε )Tvε ⇀ Mλ

weakly in L1(Ω)2.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

A double parametrix method

Since ∆pε = div (f ) − div

• curl (vε) Juε
• ∆uε

= curl (vε) Juε + ∇pε − f in Ω, we get the expressions        pε = P

• div (f ) − div (curl (vε) Juε)
• + Kpε

uε = P

• curl (vε) Juε − ∇P
• div
• curl (vε) Juε
• + P
• ∇P
• div (f )
• − f
• + L(uε, pε)

in Ω, where L(·, ·) is a C ∞-kernel operator properly supported in Ω. ⇒ Parametrix method with the pair (pε, uε).

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

A corrector result

The matrix M of the homogenized equation satisfies the inequality

• Ω

Mu · u dx ≤ f , uH−1,H1

0 −

• Ω

|Du|2 dx. Theorem Assume v ∈ Lr(Ω)N, with r > 2, and M ∈ Lm(Ω)2×2, with m > 1. Then: The previous ≤ is an = . ∃ ! u ∈ H1

0(Ω)2, Mu · u ∈ L1(Ω), solution of the limit problem.

We have the corrector result uε − u − Wε u − → 0 in W 1,1(Ω)2, where Wελ := wλ

ε .

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

A counter-example for N = 2

Y := [−1, 1]2, 4π ε2 |ln rε| = γ ∈ (0, ∞).

ωε εrε 2ε 1 / ε Qrε

ωε is a 2ε-periodic set composed of disks of radius εrε, as in the Strange term by Cioranescu, Murat (82).

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

A counter-example

Consider the solution uε ∈ H1

0(Ω)2 of the Stokes problem

   − ∆uε + 1ωε |Qrε| Juε + ∇pε = f in Ω div (uε) = 0 in Ω, with

|ωε| |Qrε| ≈ |Ω| 4 .

vε := J ∇zε, where curl (vε) = ∆zε = 1ωε |Qrε|, zε ∈ H1

0(Ω).

The condition on rε and the estimate ∀ v ∈ H1

0(Ω),

−

• ωε

|v|2 dx ≤ C

• 1 + ε2| ln rε|
• ∇v2

L2(Ω)2,

imply that zε is bounded in H1

0(Ω) and vε is bounded in L2(Ω)2.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

A counter-example

vε ⇀ v weakly in L2(Ω)2, with curl (v) = 1 4 in D′(Ω). However, vε is not equi-integrable in L2(Ω)2 and we have: Theorem The sequence uε converges weakly in H1

0(Ω)2 to the solution of

• − ∆u + 1

4 Ju + ∇p + Γu = f in Ω div (u) = 0 in Ω, with Γ := 1 4 (γ2 + 1) (γ I − J) . But, the matrix M obtained by the Tartar approach is given by M = 1 4 γ I.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives Refinement with an equi-integrability condition Sharpness of the equi-integrability condition

Comments

The matrix Γ of the Brinkman equation is not symmetric contrary to the matrix M of the Tartar approach. As in the scalar case with γ < µ, we have the inequality Γu · u < Mu · u if u = 0. The gap between the two energies (dissipated by viscosity) is due to the loss of equi-integrability of the sequence vε. The equi-integrability of vε can be thus regarded as a sharp condition to valid the Tartar approach. The proof uses test functions introduced by Allaire (91) to derive a homogenized Brinkman law from the Stokes equation with zero Dirichlet conditions on ∂ωε.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives The scalar equation The Stokes equation

Setting of the problem

Ω is a bounded open set of RN, N ≤ 3, and Y :=

• − 1

2, 1 2

N. Consider Bε ∈ L∞

♯ (Y )N satisfying

Bε − ⇀ B weakly in L2

♯(Y )N,

with ¯ B :=

• Y

B dy, and the oscillating drift bε(x) := Bε x

ε

• , x ∈ Ω.

Also consider the solution Wε in H1

♯ (Y )/R of the equation

∆Wε = div (Bε) in RN, (1) and assume that there exists µ∗ := lim

ε→0

• Y

|∇Wε|2 dy. (2) Note that µ∗ < ∞ since Wε is bounded in H1

♯ (Y )/R.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives The scalar equation The Stokes equation

Result for the scalar case

Theorem There exists a subsequence of ε and a constant µ ∈ [0, µ∗] such that for any f ∈ H−1(Ω), the solution uε ∈ H1

0(Ω) of

− ∆uε + bε · ∇uε + div (bε uε) = f converges weakly in H1

0(Ω) to the solution u of the equation

− ∆u + 2 ¯ B · ∇u + µ u = f in Ω. Moreover: If the limit B of Bε is not divergence free in RN, then µ > 0. If ∇Wε( x

ε) is equi-integrable in L2 loc(Ω)N, then µ = µ∗.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives The scalar equation The Stokes equation

Alternative test functions

Consider for δ > 0 the oscillating test function zδ,ε(x) := Zδ,ε x

ε

• ,

x ∈ RN, which solves the dual equation (with zero-order term) − ∆zδ,ε − bε · ∇zδ,ε − div (bε zδ,ε) + δ zδ,ε = 1 in RN. This type of test functions is an adaption of the test functions used by Dal Maso, Garroni (94) for the homogenization of − ∆uε + µε uε = f . The parameter δ > 0 ensures the existence of Zδ,ε, the effective coefficient µ is given by µ = lim

ε→0

• Y

Zδ,ε dy −1 − δ and is independent of δ.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives The scalar equation The Stokes equation

Setting of the problem

Ω is a regular connected bounded open set of R3. Consider with the oscillating drift vε(x) := Vε x

ε

• , where Vε ∈ L∞

♯ (Y )3 satisfies

Vε − ⇀ V weakly in L2

♯(Y )3

with ¯ V :=

• Y

V dy, and assume the existence of µ∗ := lim

ε→0

• Y

|Vε|2 dy. Consider the Stokes problem    − ∆uε + curl (vε) × uε + ∇pε = f in Ω div (uε) = 0 in Ω uε = 0

• n ∂Ω.
• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives The scalar equation The Stokes equation

Result for the Stokes problem

Theorem There exist a subsequence of ε, and a non-negative matrix M ∈ R3×3 satisfying |MTλ|2 ≤ µ∗ MTλ · λ, ∀ λ ∈ R3 , such that for any f ∈ H−1(Ω)3, the solution uε converges weakly in H1

0(Ω)3

to the solution u of the Brinkman problem − ∆u + ∇p + Mu = f in Ω div (u) = 0 in Ω. Moreover: If the limit V of Vε satisfies the condition ∀ λ ∈ R3 \ {0}, curl (V ) × λ ∈ D′(R3) \ {0}, then M is positive definite. If the sequence vε is bounded and equi-integrable in L12/5(Ω)3, then M agrees with the Tartar matrix.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives The scalar equation The Stokes equation

Test functions

Mimicking the scalar case we consider the oscillating the test functions, parametrized by δ > 0 and λ ∈ R3, zλ

δ,ε(x) = Z λ δ,ε

x

ε

• and pλ

δ,ε(x) = Pλ δ,ε

x

ε

• solutions of the Stokes problem

− ∆zλ

δ,ε − curl (vε) × zλ δ,ε + ∇pλ δ,ε + δ zλ δ,ε

= λ in R3 div

• zλ

δ,ε

• = 0

in R3. The matrix M is given by M :=

• N−1

δ

− δ I T where Nδλ := lim

ε→0

• Y

Z λ

δ,ε dy,

and is independent of δ.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives A compactness result in dimension two Nonlocal effects in dimension three

Setting of the problem

Let Y :=

• − 1

2, 1 2

2, and let bε be the oscillating drift defined by bε(x) := Bε x ε

• ,

with Bε ∈ L∞

♯ (Y )2.

Again consider the solution uε of the drift equation − ∆uε + bε · ∇uε + div (bε uε) = f ∈ H−1(Ω), and the oscillating test function zε(x) := Zε( x

ε), x ∈ R2, solution

• f the dual equation

− ∆zε − bε · ∇zε − div (bε zε) + zε = 1 in R2. We have Zε ≥ 0 and up to a subsequence Zε − ⇀ ¯ Z weakly in H1

♯ (Y )

where lim

ε→0

• Y

Zε dy

• = ¯

Z ∈ [0, 1].

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives A compactness result in dimension two Nonlocal effects in dimension three

A two-dimensional compactness result

Theorem Assume that div (bε) ≥ 0 or div (bε) ≤ 0 in D′(R2). Then, we have the following alternative: If the sequence |BεZε| is bounded in L1(Y ) and ¯ Z > 0, then, up to a subsequence,

• Y Bε Zε dy converges to some ξ in R2,

and uε converges weakly in H1

0(Ω) to the solution u of

− ∆u − 2 ¯ Z ξ · ∇u + 1 ¯ Z − 1

• u = f

in Ω. If BεZεL1(Y )2 → ∞ with lim inf

ε→0

• Y Bε Zε dy
• Bε ZεL1(Y )2 > 0 , or

¯ Z = 0, then uε converges strongly to 0 in H1

0(Ω).

No L2(Ω)2-bound to the drift bε is prescribed!

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives A compactness result in dimension two Nonlocal effects in dimension three

Ingredients of the proof

For ϕ ∈ C ∞

c (Ω), putting ϕzε in the equation satisfied by uε, and

ϕuε in the equation satisfied by zε we get

• Ω

∇uε · ∇ϕ zε −

• Ω

∇zε · ∇ϕ uε − 2

• Ω

bε · ∇ϕ uε zε −

• Ω

ϕ zε uε =

• Ω

f ϕ zε dx −

• Ω

ϕ uε, where the delicate term is bε zε uε. Using truncations and the sign of div (bε) we obtain Lemma The solution wε of the equation ∆wε = div (bε zε) in Ω is bounded in Lp

loc(Ω) for any p ∈ [1, 2).

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives A compactness result in dimension two Nonlocal effects in dimension three

Ingredients of the proof

Lemma The product (bε zε) uε converges to the limit product ξ u in D′(Ω). 2D allows us to write bε zε = ∇wε + J∇˜ wε. Integrating by parts with ϕ ∈ C ∞

c (Ω), we have

• Ω

bε · ∇ϕ uε zε dx =

• Ω

∇wε · ∇ϕ uε dx −

• Ω

J∇˜ wε · ∇uε ϕ dx. The div-curl lemma of B., Casado-D` ıaz, Murat (09) then yields J∇˜ wε · ∇uε − ⇀ J∇˜ w · ∇u in D′(Ω). Indeed, ∇˜ wε is only bounded in L1(Ω)2 but without concentration effect due to |∇˜ wε| ≤ |bε zε| + |∇wε| combined with the periodicity and the first lemma. The P.-L. Lions concentration compactness implies the strong convergence of ˜ wε in L2

loc(Ω).

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives A compactness result in dimension two Nonlocal effects in dimension three

Setting of the problem

Let Y :=

• − 1

2, 1 2

2, and let Ω := ω × (0, 1) be a vertical cylinder, where ω is a regular bounded open set of R2. Let ωε ⊂ ω be the εY -periodic lattice ωε := ω ∩

• k∈Z2

(ε k + ε Qε) , where Qε is the closed disk centered at the origin and of radius rε → 0. Consider the oscillating drift bε defined by bε(x) := βε 2 1ωε(x′) |Qε| e3 = βε 2 1Ωε(x) |Qε| e3, for x = (x′, x3) ∈ Ω, where βε > 0, e3 := (0, 0, 1), and Ωε := ωε × (0, 1). Note that bε is divergence free in Ω.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives A compactness result in dimension two Nonlocal effects in dimension three

The fibered reinforced medium

Ω = ω × (0,1) |bε| >> 1 bε = 0

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives A compactness result in dimension two Nonlocal effects in dimension three

A preliminary result

Assume that lim

ε→0

2π ε2| ln rε| = γ ∈ (0, ∞) and lim

ε→0 βε = β ∈ (0, ∞].

Let zε be the solution in H1

0(Ω) of the equation

− ∆zε − bε · ∇zε − div (bε zε) = − ∆zε − βε 1Ωε |Qε| ∂zε ∂x3 = 1 in Ω. Then, there exist a subsequence of ε and two non-negative functions z ∈ H1

0(Ω) ∩ C 0(Ω) and ¯

z ∈ H1 0, 1; L2(ω)

• such that

zε − ⇀ z weakly in H1

0(Ω)

and 1Ωε |Qε| zε − ⇀ ¯ z weakly-∗ in M(¯ Ω).

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives A compactness result in dimension two Nonlocal effects in dimension three

A preliminary result

Moreover: If β < ∞, z and ¯ z are solutions of the coupled system    − ∆z + γ (z − ¯ z) = 1 in Ω β ∂¯ z ∂x3 + γ (z − ¯ z) = 0 in Ω, If β = ∞, z and ¯ z are solutions of − ∆z + γ (z − ¯ z) = 1 in Ω ¯ z − ¯ z(·, 0) = 0 in Ω.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives A compactness result in dimension two Nonlocal effects in dimension three

Nonlocal effects

Theorem The solution uε converges weakly in H1

0(Ω) to the solution u of

• ne of the following equations in Ω:

If γ, β ∈ (0, ∞), then − ∆u + γ u − γ2 β e− γ

β x3

x3 e

γ β tu(x′, t) dt

− γ e− γ

β x3 ¯

z(x′, 1) 1

0 e

γ β (1−t)z(x′, t) dt

1 e

γ β tu(x′, t) dt = f .

If γ ∈ (0, ∞) and β = ∞, then − ∆u + γ u − γ ¯ z(x′) 1

0 z(x′, t) dt

1 u(x′, t) dt = f .

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives A compactness result in dimension two Nonlocal effects in dimension three

Non degenerate cases

It remains the following particular cases: If γ = ∞ and β < ∞, then − ∆u + β ∂u

∂x3 = f .

If γ = 0 or β = 0, then − ∆u = f . If γ = β = ∞, then u = 0. Remark The nonlocal results hold for the whole sequence ε obtained in the preliminary result about the asymptotic behavior of the sequences zε and 1Ωε

|Qε| zε.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives A compactness result in dimension two Nonlocal effects in dimension three

Comments

The 2D compactness result and the 3D nonlocal effects are reminiscent with the homogenization of purely diffusive equations

• f the type − div (Aε∇uε) = f , where Aε are equi-coercive but

necessarily equi-bounded from above. B., Casado-D` ıaz (06-08) proved that this set of equations are closed in 2D without zero-order and nonlocal limit terms. However, 3D may induce nonlocal effects in the fibered reinforced medium first studied to Fenchenko, Khruslov (81). Here, 3D nonlocal effects are due to a quite different coupling between a second-order equation and a first-order equation induced by the drift term.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives

Work in progress

Aim: derivation of the closure of the drift problems both in the scalar case and the vectorial case. In the scalar case, the boundedness of the drift bε in L2(Ω)N without equi-integrability condition, leads us to − ∆u + b · ∇u + div (b u) + µ u = f , where µ is a non-negative Borel measure. But µ is far to be explicit in terms of the asymptotic of bε. The derivation of µ is partly based on the Dal Maso, Garroni (94) approach to derive the asymptotic closure of the set of equations − ∆uε + µε uε = f , where µε is a sequence of non-negative Borel measures.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift

Introduction A scalar drift problem (avec P. G´ erard) A two-dimensional drift Stokes problem (avec P. G´ erard) The periodic case without equi-integrability condition Homogenization with large drifts Perspectives

Other extensions

The case of sequences of equations of the type − div (Aε∇uε) + bε · ∇uε + div (bε uε) = f , where Aε ≥ α I is not covered by the parametrix method, and the derivation of a defect measure µ is delicate. More generally, what is the asymptotic closure at least in 2D of the set of equations − div (Aε∇uε) + bε · ∇uε + div (bε u) + µε uε = f ? The conjecture in 2D is that the limits belong to the same class − div (A∇u) + b · ∇u + div (b u) + µ u = f . It also remains to study the case of the three-dimensional Stokes equation with a sharp boundedness of the drift to specify.

• M. Briane, IRMA Rennes & P. G´

erard, Univ. Paris 11 Homogenization of scalar and Stokes equations with drift