Homogenization approach to the dispersion theory for reactive transport through porous media

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Homogenization approach to the dispersion theory for

reactive transport through porous media

Grégoire Allaire, Andro Mikelic, Andrey Piatnitski

To cite this version:

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CENTRE DE MATHÉMATIQUES APPLIQUÉES UMR CNRS 7641

91128PALAISEAU CEDEX (FRANCE).Tél: 0169 33 4600. Fax: 01 69 3346 46 http://www. map.polyte hnique.fr/

Homogenization approa h to the dispersion theory for rea tive transport through porous media

Grégoire Allaire, Andro Mikeli¢, Andrey Piatnitski

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FOR REACTIVE TRANSPORTTHROUGH POROUSMEDIA

∗

GRÉGOIRE ALLAIRE

†

, ANDRO MIKELI

‡

, AND ANDREY PIATNITSKI

§

Abstra t. Westudythehomogenizationproblemfora onve tion-diusionequationina peri-odi porousmediuminthepresen eof hemi alrea tionontheporessurfa e. Mathemati allythis modelisdes ribedintermsofasolutiontoasystemof onve tion-diusionequationinthemedium andordinarydierentialequationdenedontheporessurfa e. Theseequationsare oupledthrough theboundary onditionforthe onve tion-diusionproblem.

Underanappropriate hoi eofs alingfa tors(largePé letandDamkohlernumbers),weobtain the homogenizedprobleminamovingframewhoseee tivevelo itydoesa tually dependon the hemi alrea tion.

1. Introdu tion. We onsider saturated ow throughaporous medium. The ow domain ontains a ertain mass of solute, usually alled tra er. Experimental worksshowthatthetra ergradually spreadswithow,but itsspreadingisnotwell des ribedbythesimplyaveragedadve tion-diusionequationsforthe on entration. Thisspreadingphenomenonis alledhydrodynami dispersion.

Following[10℄, thehydrodynami dispersionistheaveragedma ros opi pi ture ofthe motionofthetra erparti lesthrough thepore stru tureand ofthe hemi al rea tionsofthe solutewith thesolid wallsand with otherparti les. Itis aused by twobasi transportphenomena involved: onve tionand mole ulardiusion. Their simultaneouspresen eintheporestru tureleadstoa omplexspreadingofthetra er. Theintera tionbetweenthesolid poreinterfa es and theuid is relatedto the ad-sorptionordepositionoftra erparti lesonthesolidsurfa e. Eventually,radioa tive de ayand hemi alrea tionswithintheuidmayalso ause on entration hanges.

Duetothe omplexityoftheproblem,manyresultsintheliteratureare on erned withsimplemodelsofporousmediabeingeitherbundlesof apillarytubes,orarrays of ellsandsoon. Su hsimpli ationsallowexpli it al ulations. Taylor'sdispersion is oneofthemost well-known examplesofthe roleof transport in dispersingaow arryingadissolvedsolute. Thesimplest setting forobserving it, isthe inje tionof a solute into aslit hannel. The solute is transported by Poiseuille's ow. In this situationTaylorfoundin [36℄anexpli itexpressionforthedispersion.

A tuallythehydrodynami dispersion ouldbestudiedin threedistin tregimes: a) diusion-dominated mixing, b) Taylor dispersion-mediated mixing and ) haoti adve tion. Intherstregime,thevelo ityis smallandthePé let'snumber

Pe

isof orderoneorsmaller. Mole ulardiusionplaysthedominantroleinsolutedispersion. This ase is well-understood even for rea tive ows (see e.g. the papers [16℄, [18℄, [20℄, [21℄, [22℄, [17℄). If the ow rate is in reased so that the Pé let's number

Pe

is mu h larger than one, then there is a time s ale at whi h transversal mole ular diusion smears the onta t dis ontinuity into a plug. This is the regime under

∗

The resear h of G.A. and A.M. was partially supported by the GNR MOMAS CNRS-2439 (ModélisationMathématiqueetSimulationsnumériquesliéesauxproblèmesdegestiondesdé hets nu léaires)(PACEN/CNRS,ANDRA,BRGM,CEA,EDF,IRSN).G.A.isamemberoftheDEFI proje tatINRIASa layIle-de-Fran eandispartiallysupportedbytheChairMathemati al mod-ellingandnumeri alsimulation,F-EADS-E olePolyte hnique-INRIA.

†

CMAP,E olePolyte hnique,F-91128Palaiseau,Fran e(gregoire.allairepoly te h niq ue.f r).

‡

UniversitédeLyon,Lyon,F-69003,Fran e;UniversitéLyon 1,InstitutCamilleJordan,UMR 5208,43,Bddu11novembre1918,69622VilleurbanneCedex,Fran e(mikeli univ-lyon1.fr) .

§

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dominantnon-dimensionalnumberslinkedtothe hemistry,likeDamkohler'snumber. Eventuallythethirdregime, orrespondingtoturbulentmixing,ismu hmoredeli ate andisnot onsideredhere.

Our main ontribution (see Theorem 3.3) is to give a rigorous derivation of a ma ros opi homogenized model explaining Taylordispersionfor a tra er in an in- ompressible saturated ow through a periodi porous medium, undergoing linear adsorption/desorption hemi alrea tionsonthe solidboundariesof thepores. Our main te hni al tool is the notion of two-s ale onvergen e with drift introdu ed in [23℄ and applied to onve tion-diusion problems in [8℄ and [14℄. With respe t to these twoprevious works the new feature in the present work is the oupling of a onve tion-diusionforthebulksolutewithanordinarydierentialequationfor sur-fa e on entration.

Forthederivation ofTaylor'sdispersionin porousmedia usingformaltwo-s ale expansions, we refer to [9℄, [24℄, [35℄ and referen estherein. Volume averaging ap-proa htotheee tivedispersionforrea tiveowsthroughporousmediarequiresan ad ho losurehypothesis, asin[29℄.

Rigorousmathemati aljusti ationofTaylor'sdispersionin apillary tubes, for lassi al Taylor's ase and for rea tive ows, was undertaken in [25℄ and [12℄. In the aseof os illating oe ients (a mesos opi porous medium), with no hemi al rea tions,therigorousstudyofdispersionfordominantPé let'snumber,isin[34℄and in[11℄. Theapproa hfrom[11℄isbasedonanexpansionaroundtheregularsolutions fortheunderlyinglineartransportequation. Thisapproa hrequires ompatibledata but also gives an error estimate. In this paper we deal with the pore geometry and dominant Pé let's and Damköhler's numbers and we think that the two-s ale onvergen ewithdriftistherighttooltoaddressproblemsofsu hlevelofdi ulty. The ontentsof the paper isthe following. In Se tion 2wedes ribe ourmodel and itss aling in termsof various geometri aland physi al quantities. Se tion 3is devoted to the pre ise statement of our result, to some uniform a priori estimates andseveral denitionsof two-s ale onvergen ewithdrift. Se tion4is devotedto a weak onvergen eproofof ourresultbasedonpassingtothelimitinthevariational formulationoftheproblemwithadequatetestfun tions. FinallySe tion5 on ludes theproofofourmaintheorembyshowingthatthetwo-s ale onvergen eisa tually strong. It relies on a

Γ

- onvergen e type result, namely on the onvergen e of the asso iatedenergy. Letusnishthisintrodu tionbyreferringthelessmathemati ally in lined readerto another paperof us [4℄ where therigorous two-s ale onvergen e withdriftisrepla edbysimplertwo-s aleasymptoti expansionswithdriftandwhi h featuressomenumeri al omputationsofhomogenizeddispersiontensors.

2. Statementoftheproblemand itsnon-dimensionalform. We onsider diusivetransportof thesolute parti lestransported by astationary in ompressible vis ousowthroughanidealizedinniteporousmedium. Theowregimeisassumed to be laminarthrough the uid part

Ω

f

of this porous medium, whi h is supposed to be anetwork of inter onne ted hannels(in other words, wesuppose that

Ω

f

is a onne teddomain in

R

n

_{, n ≥ 2}

; usually in the appli ations

n = 2, 3

). Theow satises a slip (non penetrating) ondition on the uid/solid interfa es and

Ω

f

is saturatedby theuid. Solute parti lesare parti ipantsin a hemi alrea tion with thesolidboundariesofthepores. Forsimpli itywesupposethattheydonotintera t betweenthem.

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solute on entration

c

∗

:

∂c

∗

∂t

∗

+ v

∗

_(x

∗

_{, t}

∗

) · ∇

x

∗

c

∗

− D

∗

∆

_x

∗

c

∗

= 0

in

Ω

f

× (0, T

∗

_),

(2.1) where

v

∗

is theuid velo ity, and

D

∗

themole ular diusion (a positive onstant). At thesolid/uid boundary

∂Ω

f

takes pla e an assumed linearadsorption pro ess, des ribedbythefollowingequations:

−D

∗

∇

x

∗

c

∗

· n =

∂ˆ

c

∗

∂t

∗

= ˆ

k

∗

(c

∗

−

ˆ

c

∗

K

∗

)

on

∂Ω

f

× (0, T

∗

),

(2.2) where

ˆ

c

∗

istheadsorbed on entrationontheporesurfa e

∂Ω

f

,

ˆ

k

∗

representstherate onstantforadsorption,

K

∗

thelinearadsorption equilibrium onstantand

n

is the unitnormalat

∂Ω

f

orientedoutwardswithrespe tto

Ω

f

. Formoreonmathemati al modeling of adsorption/desorptionand referen esfrom the hemi al engineeringwe referto[15℄.

Thissystemisgeneri andappearsinnumeroussituations(seee.g. thereferen e books [19℄, [30℄, or [32℄). Inthe modeling variant[4℄ of this paper, orientedto the hemi al engineering readership, we explain in detail how to redu e the linearized models forbinaryion ex hange,and linearizedrea tiveowsystemswith

m

spe ies tothesystem(2.1)-(2.2).

Tomakeanasymptoti analysisofthisproblemwemustrstintrodu e appropri-ates alesdedu edfrom hara teristi parameterssu hasthe hara teristi on entra-tion

c

R

,the hara teristi length

L

R

,the hara teristi velo ity

V

R

,the hara teristi diusivity

D

R

,the hara teristi time

T

R

,andother hara teristi quantitiesdenoted bya

R

-index(meaning"referen e"). S alinginhomogenizationisanimportantissue (seee.g. [31℄,[33℄). The hara teristi length

L

R

oin idesin fa twiththe "observa-tiondistan e". Weassumethatthetypi alheterogeneitiesin

Ω

f

havea hara teristi size

ℓ << L

R

. Weset

ε =

ℓ

L

R

<< 1

andtheres aledowdomainisnow

Ω

ε

= Ω

f

/L

R

, with notation remindingus that it ontains poresof hara teristi non-dimensional size

ε

. Setting

u

f

=

c

∗

c

R

, x =

x

∗

L

R

, t =

t

∗

T

R

, v(x, t) =

1 V

R

v

∗

_(x

∗

_{, t}

∗

_{), D =}

D

∗

D

R

,

k =

ˆ

k

∗

k

R

, v

s

=

c

ˆ

∗

ˆ

c

R

, K =

K

∗

K

R

,

weobtainthedimensionlessequations

∂u

f

∂t

+

V

R

T

R

L

R

v

_{(x, t) · ∇}

_x

_u

_f

₋

D

R

T

R

L

2 R

D∆

x

u

f

= 0

in

Ω

ε

× (0, T )

(2.3) and

−

DD

_L

R

c

R

∇

x

u

f

· n =

ˆ

c

R

T

R

∂v

s

∂t

= k

R

k(c

R

u

f

−

ˆ

c

R

v

s

KK

R

)

on

∂Ω

ε

× (0, T ).

(2.4)

Thisprobleminvolvesthefollowingtimes ales:

T

L

= hara teristi global adve tion times ale =

L

R

/V

R

T

D

= hara teristi global diusiontimes ale =

L

2 R

/D

R

T

DE

= K

R

/k

R

( hara teristi desorption time)

T

A

= ˆ

c

R

/(c

R

k

R

)

( hara teristi adsorption time)

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Pe

=

L

R

V

R

D

R

=

T

D

T

L

(Pé letnumber); Da

=

L

R

k

R

D

R

=

T

D

T

react

(Damkohlernumber)

We hoose to study a regime for whi h

T

R

= T

D

. Due to the omplex geom-etry and in presen e of dominant Pé let and Damkohler numbers, solving the full problemforarbitraryvaluesof oe ientsis ostlyand pra ti allyimpossible. Con-sequently,onewouldliketondtheee tive(oraveragedorhomogenized)valuesof the dispersion oe ient and the transport velo ity and an ee tive orresponding paraboli equation for theee tive on entration, valid in an innite homogeneous porousmedia.

Letus be alittle morepre ise onthedenition of

Ω

ε

. Fromnowonweassume that

Ω

ε

isan

ε

-periodi unboundedopensubsetof

R

n

. Itisbuiltfrom

R

n

byremoving aperiodi distributionsofsolidobsta leswhi h,afterres aling,areallsimilar tothe unit obsta le

Σ

0

. Morepre isely, the unit periodi ity ellis identied with the at unittorus

T

n

onwhi hwe onsiderasmoothpartition

Σ

0 _{∪ Y}

0

where

Σ

0

isthesolid partand

Y

0

is theuid part. Theuid partis assumed to be asmooth onne ted open subset(noassumption is madeonthe solid part). Wedene

Y

j

ε

= ε(Y

0 + j)

,

Σ

j

ε

= ε(Σ

0 + j)

,

S

j

ε

= ε(∂Σ

0 + j)

,

Ω

ε

=

S

j∈Z

n

Y

j

ε

and

S

ε

≡ ∂Ω

ε

=

S

j∈Z

n

S

j

ε

.

Theequationsfor

u

ε

= u

f

and

v

ε

= v

s

in theirnon-dimensionalformread (with thevelo ity

v

ε

= v

)

∂u

ε

∂t

+

Pe

v

ε

(x, t) · ∇

x

u

ε

= D∆

x

u

ε

in

Ω

ε

× (0, T )

(2.5)

u

ε

(x, 0) = u

0 (x),

x ∈ Ω

ε

,

(2.6)

−D∇

x

u

ε

· n =

T

A

T

react

∂v

ε

∂t

=

T

D

T

react

k(u

ε

−

T

A

T

DE

v

ε

K

)

on

∂Ω

ε

× (0, T )

(2.7)

v

ε

(x, 0) = v

0 (x),

x ∈ ∂Ω

ε

.

(2.8)

In Se tion 3.1 weshall makesome further assumptions on thes aling of the above adimensionalsystemin termsofthegeometri alsmallparameter

ε

.

3. Main results.

3.1. Assumptionsand main onvergen etheorem. Inthepresentworkwe makethefollowingtwohypothesis.

(H1)

Pe

₌

1 ε

,

Da

=

T

D

T

react

=

1 ε

,

T

A

T

react

= ε,

T

A

T

DE

= 1.

(3.1)

(H2) Thevelo ityeldisperiodi and in ompressible,i.e.

v

_ε

_{(x, t) = b}

x

ε

withaperiodi divergen efreeve toreld

b

(y)

satisfying

| b(y) | ∈ L

∞

(R

n

),

div

y

b

(y) = 0

in

Y

0 _,

_b

(y) · n(y) = 0

on

∂Σ

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The initial data are hosen su h that

u

0 _{(x) ∈ L}

2 _(R

n

₎

and

v

0 _{(x) ∈ H}

1 _(R

n

₎

. Takingintoa ount(3.1),werewriteproblem(2.5)-(2.8)asfollows

∂

t

u

ε

+

1 ε

b

x

ε

· ∇u

ε

− D∆u

ε

= 0

in

Ω

ε

× (0, T ),

(3.2)

−

D

_ε

∂u

_∂n

ε

= ∂

t

v

ε

=

k

ε

2 u

ε

−

v

ε

K

on

∂Ω

ε

× (0, T ),

(3.3)

u

ε

(x, 0) = u

0 (x),

v

ε

(x, 0) = v

0 (x),

(3.4)

where were allthat

K

and

k

arepositive onstants. Thevariationalformulationof (3.2)-(3.3) is: nd

u

ε

(t, x) ∈ L

2 _{((0, T ); H}

1 _(Ω

ε

)) ∩ C

0 ([0, T ]; L

2 (Ω

ε

))

and

v

ε

(t, x) ∈

C

0 _{([0, T ]; L}

2 _(∂Ω

ε

))

su hthat,foranytestfun tions

φ(x) ∈ H

1 _(Ω

ε

)

,

ψ(x) ∈ L

2 _(∂Ω

ε

)

, anda.e. in time,

d

dt

Z

Ω

ε

u

ε

φ +

1 ε

Z

Ω

ε

b

x

ε

· ∇u

ε

φ +

Z

Ω

ε

D∇u

ε

· ∇φ +

k

ε

Z

∂Ω

ε

u

ε

−

v

ε

K

φ = 0,

d

dt

Z

∂Ω

ε

v

ε

ψ −

k

ε

2 Z

∂Ω

ε

u

ε

−

v

ε

K

ψ = 0,

togetherwiththeinitial ondition(3.4).

Remark 3.1. If the velo ity eld

b

(y)

is not divergen e-free and/or does not satisfy the no-penetration ondition

b

(y) · n(y) = 0

on

∂Σ

0

, it is still possible to homogenize(3.2)-(3.4) by usingrstafa torizationprin iple in the spiritof [8 ℄.

Remark3.2. Wedonotknowhowtoextendouranalysistothe aseofa ma ro-s opi allymodulatedvelo ityeld

b

(x, y)

. A tuallywebelievetheasymptoti behavior ould be ompletely dierent, a ording to the pre ise assumptions on

b

(x, y)

. For example, in [6℄(for a onve tion-diusion equation) and[7℄(for aself-adjoint diu-sionequation)itwasshown,underspe i geometri assumptionsonthema ros opi dependen eofthe oe ients, thatalo alization ee t antakepla eatalengths ale of

√

ǫ

. Howeverthegeneral aseisstillopenanditisverylikelythatlo alizationdoes notalways happen.

Tosimplify the presentation we use an extension operator from the perforated domain

Ω

ε

into

R

n

(althoughitis notne essary). As wasprovedin[1℄, thereexists su h an extension operator

T

ε

from

H

1 _(Ω

ε

)

in

H

1 _(R

n

₎

satisfying

T

ε

_ψ|

Ω

ε

= ψ

and theinequalities

kT

ε

_ψk

L

2 (R

n

₎

≤ Ckψk

_L

2 _(Ω

ε

)

,

k∇(T

ε

_ψ)k

L

2 (R

n

₎

≤ Ck∇ψk

_L

2 _(Ω

ε

)

witha onstant

C

independentof

ε

,for any

ψ ∈ H

1 _(Ω

ε

)

. Wekeepfortheextended fun tion

T

ε

_ψ

thesamenotation

ψ

. Ourmain result is thefollowingstrong onver-gen e.

Theorem 3.3. The sequen e

{u

ε

, v

ε

}

of solutionsto(3.2)-(3.4) satises

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where

b

¯

isthe so- alledee tive drift(a onstantve tor)given by

¯

b

_{= (|Y}

0 _{| + |∂Σ}

0 _|

_n−1

_K)

−1

Z

Y

0 b

_(y)dy

and

u(x, t)

isthe uniquesolution ofthe homogenizedproblem











(|Y

0 | + K|∂Σ

0 |

n−1

)∂

t

u = div

x

A

∗

∇

x

u

in

R

n

× (0, T ),

u(x, 0) =

|Y

0 _|u

0 _{(x) + |∂Σ}

0 _|

n−1

v

0 (x)

|Y

0 _{| + K|∂Σ}

0 _|

n−1

in

R

n

_,

(3.6)

wherethe ee tive diusiontensor

A

∗

isdenedby

A

∗

=

K

2 k

|∂Σ

0 |

n−1

b

¯

⊗ ¯b + D

Z

Y

0 (I + ∇

y

χ(y))(I + ∇

y

χ(y))

T

dy.

(3.7)

The ve tor-valued periodi fun tion

χ

has omponents

χ

i

∈ H

1 _(Y

0 ₎

whi h are solu-tionsof thefollowing ell problem,

1 ≤ i ≤ n

,

b

_{(y) · ∇χ}

_i

_{(y) − Ddiv(∇(χ}

_i

_{(y) + y}

_i

_{)) = ¯b}

_i

_{− b}

_i

_(y)

in

Y

0 _,

D∇(χ

i

(y) + y

i

) · n = K¯b

i

on

∂Σ

0 _.

(3.8) Here

|Y

0 _|

stands for the volume of

Y

0

,

|∂Σ

0 _|

n−1

for the

(n − 1)

-dimensional measureoftheboundary

∂Σ

0

and

n

(y)

istheexternalunit normalon

∂Σ

0

.

Remark 3.4. Conve tion isnotseen in the homogenizedequation (3.6) be ause the solution

u

isdenedin moving oordinateswhen ompared to

u

ε

and

v

ε

in(3.5). However, (3.6)isequivalenttoa onve tion diusionequation byasimple hangeof referen eframe. Indeed, introdu ing

u

˜

ε

(t, x) = u

t, x −

¯

b

ε

t

,itisasolution of











∂ ˜

u

ε

∂t

+

1 ε

b

¯

· ∇˜u

ε

− div (A

∗

∇˜u

ε

) = 0

in

R

n

_{× (0, T )}

˜

u

ε

(t = 0, x) =

|Y

∗

|u

0 _{(x) + |∂O|}

n−1

v

0 (x)

|Y

∗

_{| + K|∂O|}

n−1

in

R

n

Theorem3.3isvalidonlyforanunboundeddomain

Ω

ε

. Itis learfromthelargedrift in(3.5)orintheaboveequationthat thereisaseriousdi ulty todealwiththe ase of aboundeddomain for timemu hlargerthan

ε

.

Remark 3.5. The adsorption rate

k

appears only in the rst term of the right hand side of (3.7), so it is easy to he k that

A

∗

varies monotoni ally with

k

and blows up when

k

goes to 0. Sin e in the original problem (3.2)-(3.4) the limit ase

k = 0

is perfe tly legitimate andmeans no hemi al rea tion at all, this shows that the homogenization limit does not ommute with the limit as

k

goes to 0. When

k

goes to

+∞

, the rst term of the right hand side of (3.7) an els out, a situation whi h orresponds to xing

v

ε

= Ku

ε

on the pore boundaries. The dependen e of

A

∗

upon the equilibrium onstant

K

is impli it. At least formally, when

K

goes to 0, one re over the usual ell problem, drift and homogenized tensor orresponding to homogeneous Neumann boundary ondition on the pore boundaries (i.e. without hemistry). On the other hand when

K

goes to

+∞

,we obtain that

¯

b

_{= 0}

and the produ t

K ¯

b

,aswellas

A

∗

(10)

relies on the notionof two-s ale onvergen e with drift. Forthe mathemati ally less in linedreader, aformal methodfor guessing the orre thomogenized problem (3.6) is the method of two-s ale asymptoti expansions with drift (see [4 ℄and [28℄). More pre isely, oneassumesthat

u

ε

(t, x) =

+∞

X

i=0

ε

i

u

i

t, x −

b

¯

_ε

t,

x

ε

,

with

u

i

(t, x, y)

afun tionofthema ros opi variable

x

andoftheperiodi mi ros opi variable

y ∈ Y = (0, 1)

n

,andsimilarly

v

ε

(t, x) =

+∞

X

i=0

ε

i

_v

i

t, x −

b

¯

ε

t,

x

ε

Plugging these ansatz in the equation (3.2) yields after some standard algebra the desiredresult, atleastformally.

3.2. Uniform a prioriestimates. Wenowderiveaprioriestimatesbasedon theenergy equality. Asusual, theyimply existen eofauniquesolutionto problem (3.2)-(3.4). Depending ontheassumedregularityoftheinitial data, we ouldprove arbitraryhighregularityofthesolution.

Lemma 3.7. There existsa onstant

C

, whi h does not depend on

ε

, su hthat the solutionof (3.2)-(3.4) satises

ku

ε

k

L

∞

_{((0,T );L}

2 _(Ω

ε

))

+

√

εkv

ε

k

L

∞

_{((0,T );L}

2 _(∂Ω

ε

))

+ k∇u

ε

k

L

2 ((0,T )×Ω

ε

)

≤ C ku

0 k

L

2 (R

n

₎

+ kv

0 k

_H

1 (R

n

₎

.

(3.9)

Proof. Theenergyestimatefor(3.2)-(3.4)reads

1

2 d

dt

h

ku

ε

k

2 L

2 _(Ω

ε

)

+

ε

K

kv

ε

k

2 L

2 _(∂Ω

ε

)

i

+

Z

Ω

ε

D∇u

ε

(t) · ∇u

ε

(t)dx +

εk

ε

2 Z

∂Ω

ε

u

ε

−

v

ε

K

2 dσ = 0,

(3.10)

fromwhi hweeasilydedu ethedesiredresultsin e

εkv

0 _k

2 L

2 _(∂Ω

ε

)

≤ Ckv

0 _k

2 H

1 (R

n

₎

. To obtain(3.10) wemultiply equation(3.2)by

u

ε

andintegrate byparts over

Ω

ε

. The onve tiveterm an elsoutsin ethevelo ityisdivergen e-freeandhasazeronormal omponentontheboundary

(11)

Multiplying then the equation

∂

t

v

ε

=

k

ε

2 u

ε

−

v

ε

K

by

εv

ε

/K

and integrating the resultover

∂Ω

ε

yields

1

2 ε

K

d

dt

kv

ε

k

2 L

2 _(∂Ω

ε

)

+

k

ε

Z

∂Ω

ε

−

u

ε

v

ε

K

+

v

2 ε

K

2 dσ = 0.

Summingupthelasttworelations,weobtain(3.10).

Nextweestimate

v

ε

using

u

ε

. Withoutlossofgeneralitywe anassumethatthe fun tion

v

ε

isdenedbytheequation

∂

t

v

ε

=

k

ε

2 u

ε

−

v

ε

K

everywherein

Ω

ε

andnot solelyon

∂Ω

ε

.

Lemma 3.8. Thereexistsa onstant

C

,whi h doesnotdependon

ε

,su hthat

kv

ε

k

L

2 ((0,T );H

1 (Ω

ε

))

≤ C(ku

ε

k

L

2 ((0,T );H

1 (Ω

ε

))

+ εkv

0 _k

H

1 (R

n

₎

).

(3.11)

Proof. Solvingexpli itlytheODE(3.3),weget

v

ε

(t, x) =

t

Z

0 k

ε

2 exp

k

Kε

2 (s − t)

u

ε

(s, x)ds + v

0 (x) exp

−

kt

Kε

2 .

Sin e

k

ε

2 exp{−

kτ

Kε

2 }χ

0≤τ ≤t

isbounded in

L

1 _{(0, T )}

independently of

ε

, Young's in-equalityyields

kv

ε

k

L

2 _{((0,T );H}

1 _(Ω

ε

))

≤ C(ku

ε

k

L

2 ((0,T );H

1 (Ω

ε

))

+ εkv

0 k

H

1 (R

n

₎

),

kv

ε

k

L

2 _{((0,T )×Ω}

ε

)

≤ C(ku

ε

k

L

2 ((0,T )×Ω

ε

))

+ εkv

0 k

L

2 (R

n

₎

).

(3.12)

Thenextaprioriestimateisagaina onsequen eoftheenergyequality(3.10). Lemma 3.9. Thereexistsanother onstant

C

,whi h does notdepend on

ε

,su h that

1 K

v

ε

− u

ε

_L

2 _{((0,T )×Ω}

ε

)

≤ Cε.

Proof. The desired estimate is a onsequen e of the following Poin aré type inequality

kwk

2 L

2 _(Ω

ε

)

≤ C ε

2 _k∇wk

2 L

2 _(Ω

ε

)

+ εkwk

2 L

2 _(∂Ω

ε

)

This inequality isderivedin [13℄. Combiningit withthe energyestimate (3.10),we obtainthestatementoflemma.

Remark 3.10. Allthe previous apriori estimates are not uniformwith respe t to

k

and

K

. Thisisonereasonwhytakingthehomogenizationlimit

ε → 0

andtaking the zero-adsorption limit

k → 0

do not ommute.

(12)

Definition 3.11. Let

V

bea onstant ve torin

R

n

. Wesay that asequen eof fun tions

U

ε

(t, x) ∈ L

2 _{((0, T ) × R}

n

₎

two-s ale onverges in moving oordinates (or, equivalently,withdrift)

(x, t) → (x−

V

ε

t, t)

toafun tion

U

0 (t, x, y) ∈ L

2 _{((0, T )×R}

n

_×

T

n

)

if

kU

ε

k

L

2 ((0,T )×R

n

₎

≤ C

andfor any

φ(t, x, y) ∈ C

∞

0 ((0, T ) × R

n

× T

n

)

lim

ε→0

T

Z

0 Z

R

n

U

ε

(t, x)φ t, x −

V

ε

t,

x

ε

dxdt =

T

Z

0 Z

R

n

Z

T

n

U

0 (t, x, y)φ(t, x, y)dxdydt.

(3.13)

The onvergen e(3.13) is denotedby

U

ε

2−drif t

−→ U

0

.

In analogy with the lassi two-s ale onvergen e results (see [2℄ and [26℄), we have:

Proposition3.12. ([23℄)Let

V

bea onstant ve torin

R

n

andletthesequen e

U

ε

beuniformlyboundedin

L

2 _{((0, T ); H}

1 (R

n

))

. Then thereexistsasubsequen e,still denotedby

ε

,andfun tions

U

0 (t, x) ∈ L

2 _{((0, T ); H}

1 (R

n

))

and

U

1 (t, x, y) ∈ L

2 ((0, T )×

R

n

; H

1 _(T

n

₎₎

su hthat

U

ε

2−drif t

−→ U

0

and

∇U

ε

2−drif t

−→ ∇

x

U

0 + ∇

y

U

1 .

(3.14)

Let

W

ε

be asequen euniformly bounded in

L

2 _{((0, T ) × R}

n

₎

su h that

ε∇W

ε

isalso uniformlyboundedin

L

2 _{((0, T ) × R}

n

₎

n

. Then thereexistsasubsequen e,stilldenoted by

ε

,andafun tion

W

0 (t, x, y) ∈ L

2 _{((0, T ) × R}

n

_{; H}

1 _(T

n

₎₎

su hthat

W

ε

2−drif t

−→ W

0

and

ε∇W

ε

2−drif t

−→ ∇

y

W

0 .

(3.15)

Thanks to estimate(3.10) and Lemmata 3.7, 3.8 and 3.9 and Proposition 3.12, wehavethefollowing ompa tnessresult.

Corollary3.13. Let

{u

ε

, v

ε

}

bethesolutionofproblem(3.2)-(3.4),extendedto the wholespa e. Takethe drift

V = ¯b

. Then thereexistsasubsequen e(stilldenoted by

ε

)and

{u, w, q} ∈ L

2 _{((0, T ); H}

1 _(R

n

_{)) × L}

2 _{((0, T ) × R}

n

_{; H}

1 _(T

n

_{)) × L}

2 _{((0, T ) × R}

n

_×

T

n

)

su hthat











u

ε

2−drif t

−→ u(t, x),

∇u

ε

2−drif t

−→ ∇u(t, x) + ∇

y

w(t, x, y),

v

ε

2−drif t

−→ Ku(t, x),

1 _ε

v

_K

ε

− u

ε

_{2−drif t}

−→ q(t, x, y).

(3.16)

Thefa tthat

{u

ε

}

and

{v

ε

/K}

havethesame(two-s alewithdrift)limitsfollows fromLemma 3.9.

(13)

Corollary 3.14. Let

{u

ε

, v

ε

}

be as in Corollary 3.13. Then, for the same two-s ale limitwith drift

q

,asdenedin(3.16), wehave

lim

ε→0

ε

T

Z

0 Z

∂Ω

ε

1 ε

u

ε

− v

ε

/K

φ

t, x −

b

¯

_ε

t

,

x

ε

dσdt =

T

Z

0 Z

∂Σ

0 Z

R

n

q(t, x, y)φ(t, x, y) dxdσ

y

dt,

(3.17)

for any testfun tion

φ(t, x, y) ∈ C

∞

0 ((0, T ) × R

n

× T

n

)

, Proof. Let

a

∈ C

1 _{( ¯}

_Y

0 _{; R}

n

₎

beasolutionfor

a

_{· n = 1}

on

∂Σ

0 _;

div

a

=

| ∂Σ

0 _|

n−1

| Y

0 _|

in

Y

0 _{; a}

is

Y −

periodi

.

(3.18) Thenwehave

T

Z

0 Z

∂Ω

ε

u

ε

−

v

ε

K

φ

x −

b

¯

t

ε

,

x

ε

, t

dσdt =

T

Z

0 Z

Ω

ε

div

a

₍

x

ε

) u

ε

−

v

ε

K

φ

x −

b

¯

_ε

t

,

x

ε

, t

dxdt =

T

Z

0 Z

Ω

ε

₁

ε

(u

ε

−

v

ε

K

)

| ∂Σ

0 _|

| Y

0 _|

φ

x −

b

¯

t

ε

,

x

ε

, t

+

1 ε

a

(

x

ε

) · (u

ε

−

v

ε

K

)∇

y

φ

x −

b

¯

t

ε

, y, t

|

y=

x

ε

+a(

x

ε

) · ∇(u

ε

−

v

ε

K

)φ

x −

b

¯

t

ε

, y, t

|

y=

x

ε

dxdt + O(ε) →

T

Z

0 Z

Y

0 Z

R

n

div

y

q(x, y, t)φ(x, y, t)a(y)

dxdydt,

as

ε → 0,

(3.19)

whereweusedthebounds

1 ε

u

ε

− v

ε

/K

L

2 (Ω

ε

×(0,T ))

≤ C ,

k∇ u

ε

− v

ε

/K

k

L

2 _(Ω

ε

×(0,T ))

≤ C

and the onvergen e result (3.15) for the sequen e

ε

−1

_u

ε

− v

ε

/K

. The surfa e two-s alelimitresult(3.17) followsfrom (3.19).

4. Proof of weak two-s ale onvergen e. Before proving our main result, Theorem3.3, westateandproveaweakerversionwhi hrelies onthenotionof two-s ale onvergen ewithdrift.

Theorem 4.1. The sequen e

{u

ε

, v

ε

}

two-s ale onverges with drift

(x, t) →

x −

b

¯

ε

t, t

(14)

1. STEP ( ompa tnessand hoi eofthedrift)

By virtue of the a priori estimates of se tion 3.2, Proposition 3.12 and Corol-lary 3.14 imply the existen e of a subsequen e (still denoted by

ε

) and of limits

{u, w, q, v} ∈ L

2 _{((0, T ); H}

1 _(R

n

_))×L

2 _{((0, T )×R}

n

_{; H}

1 _(T

n

₎₎

2 _×L

2 _{((0, T ); H}

1 _(R

n

₎₎

su h that







u

ε

2−drif t

−→ u(x, t),

∇u

ε

2−drif t

−→ ∇u(x, t) + ∇

y

w(x, y, t);

v

ε

2−drif t

−→ Ku(x, t),

1 _ε

v

_K

ε

− u

ε

_{2−drif t}

−→ q(x, y, t).

(4.1)

Atthismomentthe hoi eofthedriftvelo ityisarbitrary. Neverthelesswenowmake a hoi ewhi h willturnout,inthethirdstep,tobetheonlypossibleone.

Intheabsen eof hemi alrea tions(i.e. withhomogeneousNeumannboundary onditionfor

u

ε

insteadof(3.3)),thedriftvelo itywouldbesimply

b

_c

_{= |Y}

0 _|

−1

Z

Y

0 b

_{(y) dy}

(seee.g. [8℄). The hemistrytermleadstoanon-trivialdriftwhi h isnowdenedin agreementwithperiodi gradientos illations.

Lemma 4.2. Letthe ee tivedrift

b

¯

(a onstantve tor) begiven by

¯

b

_{= (|Y}

0 _{| + |∂Σ}

0 _|

_n−1

_K)

−1

Z

Y

0 b

_(y)dy.

(4.2)

Thereexistsaperiodi solution

χ

i

(y) ∈ H

1 _(Y

0 ₎

of thefollowing ellproblem,

1 ≤ i ≤

n

,

b

_{(y) · ∇χ}

_i

_{(y) − D div ∇(χ}

_i

_{(y) + y}

_i

₎

_{= ¯b}

_i

_{− b}

_i

_(y)

in

Y

0 _,

D∇(χ

i

(y) + y

i

) · n = K¯b

i

on

∂Σ

0 _,

(4.3) where

n

_(y)

is the external unit normal on

∂Σ

0

. This solution is unique up to an additive onstant.

Proof. We he k that

b

¯

isdened pre isely sothat the ompatibility ondition (orFredholmalternative)in (4.3)issatised. Weobtain

Z

Y

0 b

_{(y)dy − |Y}

0 _{|¯b −}

Z

∂Σ

0 K dσ

y

b

¯

= 0,

(4.4)

where

dσ

y

isanelementof

(n − 1)

-dimensionalvolumeon

∂Σ

0

. 2. STEP (determinationofthelimitfun tion

q

)

In order to hara terize the limit fun tion

q(x, y, t)

, we multiply the equation

∂

t

v

ε

=

k

ε

2 u

ε

−

v

ε

K

by

εϕ

x −

b

¯

_ε

t

,

x

ε

, t

,where

ϕ(x, y, t) ∈ C

∞

0 (R

n

× T

n

× (0, T ))

, andintegratetheresultingexpressionover

Ω

ε

× (0, T )

. An integrationbyparts with respe ttotime yields

(15)

whereweusedthenotation

∇

x

ϕ x−

¯

b

_t

ε

,

x

ε

, t

= ∇

x

ϕ(x−

¯

b

_t

ε

, y, t)

_y=x/ε

and

∇

y

ϕ x−

¯

b

_t

ε

,

x

ε

, t

= ∇

y

ϕ(x −

¯

b

_t

ε

, y, t)

y=x/ε

. Passing to the two-s ale limit with drift and bearinginmindthatthetwo-s alelimitof

v

ε

isequalto

Ku(x, t)

,weobtain

T

Z

0 Z

R

n

Z

Y

0 Ku(x, t)¯

b

_{· ∇}

_x

_{ϕ(x, y, t) − kq(x, y, t)ϕ(x, y, t)}

_{dxdydt = 0}

Therefore,

q(x, y, t) = q(x, t) = −

K

_k

b

¯

_{· ∇}

_x

_{u(x, t).}

(4.5)

3. STEP (determinationofthelimitfun tion

w

)

Inorderto hara terizethelimitfun tion

w(x, y, t)

,we hooseagainatest fun -tionasbefore:

ϕ

ε

= εϕ

x −

b

¯

_ε

t

,

x

ε

, t

.

Substituting itin problem(3.2)-(3.4)yields

T

Z

0 Z

Ω

ε

n

u

ε

b

¯

· ∇

x

ϕ

x −

b

¯

t

ε

,

x

ε

, t

+ b

x

ε

· ∇u

ε

ϕ

x −

b

¯

t

ε

,

x

ε

, t

+

D∇u

ε

∇

y

ϕ

x −

b

¯

_ε

t

,

x

ε

, t

o

dxdt + ε

T

Z

0 Z

∂Ω

ε

k

ε

u

ε

−

v

ε

K

ϕ

x −

b

¯

_ε

t

,

x

ε

, t

dσdt

= O(ε).

(4.6)

Passingtothetwo-s alelimitwithdriftgivesusthe ellproblem

T

Z

0 Z

R

n

Z

Y

0 n

u(x, t)¯

b

_{· ∇}

_x

_{ϕ(x, y, t) + b(y) · (∇}

_x

_{u(x, t) + ∇}

_y

_{w(x, y, t))ϕ(x, y, t)+}

D(∇

x

u(x, t) + ∇

y

w(x, y, t)) · ∇

y

ϕ(x, y, t)

o

dxdydt+

T

Z

0 Z

R

n

Z

∂Σ

0 kq(x, y, t)ϕ(x, y, t)dxdσ

y

dt = 0.

(4.7)

Asin lassi altwo-s ale onvergen e,problem(4.7)leadstothefollowingdierential problemfor

w

,valida.e. on

(0, T ) × R

n

:

−D

div

y

(∇

x

u(x, t) + ∇

y

w(x, y, t)) + b(y) · (∇

x

u(x, t) + ∇

y

w(x, y, t)) =

(16)

Atthispointitis ru ialtohave hosenthedrift

b

¯

denedby(4.2),otherwise(4.10) would haveno solution but the trivialone. Finally, we on ludethat the fun tion

w(x, y, t)

isgivenbythefollowingseparationoffastandslowvariablesformula:

w(x, y, t) = χ(y) · ∇

x

u(x, t),

(4.11)

with

χ(y)

of omponents

χ

i

solvingproblem(4.3).

4. STEP (determinationofthehomogenizedequation) In this step we test problem (3.2)-(3.4) by

φ(x, t) = φ

ˆ

x −

b

¯

_ε

t

, t

, with

φ ∈

C

∞

0 (R

n

× [0, T ))

, implying that

φ(x, T ) = 0

. Also, we use the symbol

d

∂

t

φ

for

∂

t

φ(z, t)|

z=x−¯

b

t/ε

. Notethat

∂

t

φ(x, t) = d

ˆ

∂

t

φ(x, t) −

¯

b

ε

· ∇

x

φ(x, t).

ˆ

Weget

T

Z

0 Z

Ω

ε

n

u

ε

¯

b

_{− b}

x

ε

· ∇

x

φ − u

ˆ

ε

d

∂

t

φ

o

dxdt +

T

Z

0 Z

∂Ω

ε

v

ε

(x, t)¯

b

· ∇

x

φdσdt−

ˆ

Z

Ω

ε

u

0 (x)φ(x, 0) dx + D

T

Z

0 Z

Ω

ε

∇u

ε

· ∇

x

φ dxdt−

ˆ

ε

T

Z

0 Z

∂Ω

ε

v

ε

(x, t)d

∂

t

φdσdt − ε

Z

∂Ω

ε

v

0 (x)φ(x, 0)dσ = 0.

(4.12)

Next weintrodu etheauxiliaryve torfun tion

ψ

by

(17)

T

Z

0 Z

Ω

ε

u

ε

¯

b

_{− b}

x

ε

· ∇

x

φ dxdt +

ˆ

T

Z

0 Z

∂Ω

ε

v

ε

(x, t)¯

b

· ∇

x

φdσdt =

ˆ

−

T

Z

0 Z

Ω

ε

n

X

i=1

∆ψ

ε

i

∂

x

i

φu

ˆ

ε

dxdt +

T

Z

0 Z

∂Ω

ε

v

ε

(x, t)¯

b

· ∇

x

φdσdt =

ˆ

K

T

Z

0 Z

∂Ω

ε

v

ε

K

− u

ε

¯

b

· ∇

x

φdσdt +

ˆ

T

Z

0 Z

Ω

ε

n

X

i=1

∇ψ

i

ε

· ∇(∂

x

i

φu

ˆ

ε

) dxdt.

(4.15)

Inserting(4.15)into(4.12)gives

Kε

T

Z

0 Z

∂Ω

ε

1 ε

v

ε

K

− u

ε

¯

b

· ∇

x

φdσdt +

ˆ

T

Z

0 Z

Ω

ε

n

X

i=1

∇ψ

ε

i

· ∇(∂

x

i

φu

ˆ

ε

) dxdt−

T

Z

0 Z

Ω

ε

u

ε

d

∂

t

φ dxdt −

Z

Ω

ε

u

0 (x)φ(x, 0) dx + D

T

Z

0 Z

Ω

ε

∇u

ε

· ∇

x

φ dxdt−

ˆ

ε

T

Z

0 Z

∂Ω

ε

v

ε

(x, t)d

∂

t

φdσdt − ε

Z

∂Ω

ε

v

0 _{(x)φ(x, 0)dσ = 0.}

(4.16)

Passing tothe two-s alelimit withdrift

(x, t) → x − ¯bt/ε, t

in thelast relationis nowstraightforward. Forthe omfortofthereader,wedoittermbyterm:

(18)

lim

ε→0

Kε

T

Z

0 Z

∂Ω

ε

1 ε

v

ε

K

− u

ε

¯

b

· ∇

x

φdσdt =

ˆ

T

Z

0 Z

R

n

|∂Σ

0 |

n−1

K

2 k

¯

b

⊗ ¯b∇

x

u∇

x

φ dxdt,

(4.20)

lim

ε→0

T

Z

0 Z

Ω

ε

n

X

i=1

∇ψ

ε

i

· ∇(∂

x

i

φu

ˆ

ε

) dxdt =

T

Z

0 Z

R

n

X

i,j=1

∂

2 _φ

∂x

i

x

j

u(x, t)

Z

Y

0 ∂ψ

i

∂y

j

dy

dxdt+

T

Z

0 Z

R

n

Z

Y

0 n

X

i,j=1

∂

x

i

φ(x, t)

∂ψ

i

(y)

∂y

j

∂

x

j

u + ∂

x

ℓ

u

n

X

ℓ=1

∂χ

ℓ

(y)

∂y

j

dydxdt.

(4.21)

It isnowtimeto introdu ethe homogenizedmatrixwhi hfor simpli ity we de om-poseasasumofelementary matri es. Therstonelinkedto adsorption/desorption rea tions,transported bythedriftvelo ity,is al ulatedin (4.20)andgivenby

¯

A

1 =

K

2 k

|∂Σ

0 |

n−1

b

¯

⊗ ¯b =

"

K

2 k

|∂Σ

0 |

n−1

¯b

i

¯b

j

#

.

(4.22)

These ondone,relatedtoadve tion-diusion and hemistry,is al ulatedin (4.19)-(4.21)andgivenby

_¯

A

2 _ij

= D

Z

Y

0 δ

ij

+

∂χ

i

(y)

∂y

j

dy +

n

X

ℓ=1

Z

Y

0 ∂χ

j

(y)

∂y

ℓ

∂ψ

i

(y)

∂y

ℓ

dy.

(4.23)

Remark that only the symmetri part of the homogenized matrix appears in the homogenized equation:

A

¯

1

is already symmetri but

A

¯

2

is notand should be sym-metrized. Theee tiveorhomogenizedmatrixisthusdened by

A

∗

= ¯

A

1 +

1

2 ( ¯

A

2 + ¯

A

T

2 ).

Then after insertingthe limits (4.17)-(4.21)into the variationalequation (4.16) we on ludethat thelimitfun tion

u(x, t)

solvestheproblem

(|Y

0 | + K|∂Σ

0 |

n−1

)∂

t

u =

div

x

A

∗

∇

x

u

in

R

n

× (0, T ),

(4.24)

u(x, 0) =

|Y

0 _|u

0 _{(x) + |∂Σ}

0 _|

n−1

v

0 (x)

|Y

0 _{| + K|∂Σ}

0 _|

n−1

in

R

n

_.

(4.25) It remainsto provethat the matrix

A

∗

ispositivedenite and establishuniqueness ofthelimitfun tion.

5. STEP (propertiesoftheee tivematri esanduniqueness)

Clearly,thematrix

A

¯

1

,givenby(4.22),issymmetri andnon-negative,i.e.

A

¯

1 ξ ·

ξ ≥ 0

forany

ξ ∈ R

n

(19)

Lemma 4.3. The matrix

A

¯

2

,given by (4.23),is positive deniteandsatises

¯

_A

₂

ij

= D

Z

Y

0 (∇

y

χ

i

(y) + e

i

) · (∇

y

χ

j

(y) + e

j

) dy +

Z

Y

0 b

_{(y) · ∇}

_y

_χ

_i

_(y)χ

_j

_{(y) dy.}

(4.26)

Finally,

A

∗

isalso positive denite,equivalentlydenedby

A

∗

₌

K

2 k

|∂Σ

0 |

n−1

b

¯

⊗ ¯b + D

Z

Y

0 (I + ∇

y

χ(y))(I + ∇

y

χ(y))

T

dy.

(4.27)

Proof. Firstwetest problem(4.13) for

ψ

i

by

χ

j

. These ondtermonthe right-handsideof(4.23)be omes

n

X

ℓ=1

Z

Y

0 ∂χ

j

(y)

∂y

ℓ

∂ψ

i

(y)

∂y

ℓ

dy =

Z

Y

(¯b

i

− b

i

(y))χ

j

(y)dy + K

Z

∂Σ

0 ¯b

i

χ

j

(y)dσ.

(4.28)

Nextwemultiplytheequation(4.3)for

χ

i

by

χ

j

(y)

andintegratetheresultingrelation over

Y

0

. Thisyieldsaformulaforthersttermontheright-handsideof(4.23)

Z

Y

(¯b

i

− b

i

(y))χ

j

(y)dy + K

Z

∂Σ

0 ¯b

i

χ

j

(y)dσ =

Z

Y

0 b

_{(y) · ∇}

_y

_χ

_i

_(y)χ

_j

_(y)dy+

D

Z

Y

0 (e

i

+ ∇

y

χ

i

(y)) · ∇

y

χ

j

(y)dy = −D

Z

Y

0 δ

ij

+

∂χ

i

(y)

∂y

j

dy+

Z

Y

0 b

_{(y) · ∇}

_y

_χ

_i

_(y)χ

_j

_{(y)dy + D}

Z

Y

0 (∇

y

χ

i

(y) + e

i

) · (∇

y

χ

j

(y) + e

j

) dy.

(4.29)

Identities (4.28)-(4.29)imply(4.26). Sin e

b

_(y)

is solenoidalandits normal ompo-nentisequaltozeroat

∂Σ

0

,wendeasilythatthematrix

R

Y

0 b

(y)·∇

y

χ

i

(y)χ

j

(y)dy

isskew-symmetri :

−

Z

Y

0 b

_{(y) · ∇}

_y

_χ

_j

_(y)χ

_i

_{(y)dy =}

Z

Y

0 b

_{(y) · ∇}

_y

_χ

_i

_(y)χ

_j

_(y)dy,

_{i, j = 1, . . . , n.}

The remaining part of Lemma (4.3) follows immediately. As a onsequen e, the uniqueness ofthe homogenizedsolution

u(t, x)

is obvious. Thus theentire sequen e

{u

ε

, v

ε

}

is onverging.

(20)

Proposition 5.1. Let

v

0 _{(x) = Ku}

0 _{(x) ∈ H}

1 _(R

n

₎

(i.e. initial data at the isotherm). Then

u

ε

(x, t)χ

Ω

ε

strongly two-s ale onverges with drift

(x, t) → x −

¯

b

_t

ε

, t

in

R

n

_{× (0, T )}

towards

χ

Y

0 (y)u(x, t)

. Similarly,

v

ε

(x, t)χ

Ω

ε

stronglytwo-s ale onverges withdrifttowards

Kχ

Y

0 (y)u(x, t)

. In parti ular,

T

Z

0 Z

Ω

ε

u

ε

(x, t) − u

x −

b

¯

_ε

t, t

2 dxdt −→

ε→0

0.

Proof. Westart by integrating the energy equality (3.10) in time variable over theinterval

(0, t)

. Thisyields

1

2 h

ku

ε

(t)k

2 L

2 _(Ω

ε

)

+

ε

K

kv

ε

(t)k

2 L

2 _(∂Ω

ε

)

i

+

Z

t

0 Z

Ω

ε

D∇u

ε

(s) · ∇u

ε

(s)dxds+

Z

t

0 εk

ε

2 Z

∂Ω

ε

u

ε

(s) −

v

ε

(s)

K

2 dσds =

1

2 h

ku

0 _k

2 L

2 _(Ω

ε

)

+

ε

K

kv

0 _k

2 L

2 _(∂Ω

ε

)

i

.

(5.2)

Sin eweexpe tthefamily

{u

ε

, v

ε

}

tobe ompa tonlyintheprodu tspa e

L

2 _{((0, T )×}

Ω

ε

)

,itisoutofrea hto laim onvergen eofthesefun tionsforaxedvalueof

t

. To ir umventthisdi ulty,weintegrateformula(5.2)in temporalvariableon eagain. Theresultingformulareads

1

2 T

Z

0 h

ku

ε

(t)k

2 L

2 _(Ω

ε

)

+

ε

K

kv

ε

(t)k

2 L

2 _(∂Ω

ε

)

i

dt +

T

Z

0 t

Z

0 Z

Ω

ε

D∇u

ε

(s) · ∇u

ε

(s)dxdsdt+

k

ε

T

Z

0 t

Z

0 Z

∂Ω

ε

u

ε

(s) −

v

ε

(s)

K

2 dσdsdt =

T

2 h

ku

0 _k

2 L

2 _(Ω

ε

)

+

ε

K

kv

0 _k

2 L

2 _(∂Ω

ε

)

i

.

(5.3) Usingthetwo-s ale onvergen eresultsofthepreviousse tionandtakingintoa ount thelowersemi ontinuityofthe orrespondingnormswithrespe ttothetwo-s aleand weak onvergen e(see[2℄ifne essary),wehave

lim inf

ε→0

T

Z

0 h

ku

ε

(t)k

2 L

2 _(Ω

ε

)

+

ε

K

kv

ε

(t)k

2 L

2 _(∂Ω

ε

)

i

dt ≥ |Y

0 _|kuk

2 L

2 (R

n

_{×(0,T ))}

+

|∂Σ

0 |

n−1

Kkuk

2 L

2 (R

n

_{×(0,T ))}

= (|Y

0 | + |∂Σ

0 |

n−1

K)kuk

2 L

2 (R

n

_{×(0,T ))}

.

(5.4) Bythesamearguments,

lim inf

ε→0

T

Z

0 t

Z

0 Z

Ω

ε

D∇u

ε

(x, s) · ∇u

ε

(x, s)dxdsdt ≥

T

Z

0 t

Z

0 Z

R

n

Z

Y

0

(21)

lim inf

ε→0

k

ε

T

Z

0 t

Z

0 Z

∂Ω

ε

u

ε

(x, s) −

v

ε

(x, s)

K

2 dσdsdt ≥

k|∂Σ

0 |

n−1

T

Z

0 t

Z

0 Z

R

n

K

_k

b

¯

_{· ∇}

_x

_{u(x, s)}

2 dxdsdt.

(5.6)

Passingtothelimitontherighthand sideof(5.3),weget

lim

ε→0

T

2 h

ku

0 k

2 L

2 _(Ω

ε

)

+

ε

K

kv

0 k

2 L

2 _(∂Ω

ε

)

i

=

T

2 |Y

0 |ku

0 k

2 L

2 (R

n

₎

+

T

2K

|∂Σ

0 |

n−1

kv

0 k

2 L

2 (R

n

₎

.

Our nextaimis to omputethe energyof thelimit equation. Multiplying equation (4.24) by

u(x, s)

and integrating over

R

n

_{× (0, t)}

and then on e again in variable

t

overtheinterval

(0, T )

,after straightforwardtransformationsweobtain

1

2 (|Y

0 | + |∂Σ

0 |

n−1

K)kuk

2 L

2 (R

n

_{×(0,T ))}

+

T

Z

0 t

Z

0 Z

R

n

A

∗

∇u(x, s) · ∇u(x, s)dxdsdt =

T

2(|Y

0 _{| + K|∂Σ}

0 _|

n−1

)

|Y

0 |u

0 + |∂Σ

0 |v

0

2 L

2 (R

n

₎

(5.7)

Dueto (4.27)these ond integralonthelefthandside anberearrangedasfollows

T

Z

0 t

Z

0 Z

R

n

A

∗

∇u(x, s) · ∇u(x, s)dxdsdt

=

T

Z

0 t

Z

0 Z

R

n

Z

Y

0 D|∇

x

u(x, s) + ∇

y

χ(y)∇

x

u(x, s)|

2 dydxdsdt

+

T

Z

0 t

Z

0 Z

R

n

K

2 k

|∂Σ

0 |

n−1

(¯

b

· ∇

x

u(x, t))

2 dxdsdt.

Be ause of the energy equality (5.7) for the homogenized problem and the lower semi ontinuityofthetermsintheenergyequality(5.3)forthemi ros opi problem, we on ludethat thenorm onvergen eisvalidifandonlyifwehave