60(2015) APPLICATIONSOFMATHEMATICS No.2,185196 [610089]

60(2015) APPLICATIONSOFMATHEMATICS No.2,185–196
HOMOGENIZATIONOFADUAL-PERMEABILITYPROBLEMIN
TW
O-COMPONENTMEDIAWITHIMPERFECTCONTACT
AbdelhamidAinouz ,Algiers
(ReceivedMay17,2013)
Abstract.Inthispaper,westudythemacroscopicmodelingofasteadyfluidflowinan
ε-periodicmediumconsistingoftwointeractingsystems: fissuresandblocks,withperme-
abilitiesofdifferentorderofmagnitudeandwiththepresenceofflowbarrierformulationat
theinterfacialcontact. Thehomogenizationprocedureisperformedbymeansofthetwo-
scaleconvergencetechniqueanditisshownthatthemacroscopicmodelisaone-pressure
fieldmodelinaone-phaseflowhomogenizedmedium.
Keywords:porousmedia;homogenization;twoscaleconvergence
MSC2010:35B27,76S05
1. Introduction
Thestudyoffluidflowsinporousmediaisasubjectofpracticalinterestinmany
engineeringareas,suchasgeomechanics,materialsciences,andwaterresourcesman-
agement.Sometypesofnaturallyporousrocks,likeaquifersorpetroleumreservoirs,
areusuallydescribedasadual-permeability(oradoubleporosity)medium,thatis
atwo-componentstructure:onerelatedtoblocks,andtheotherrelatedtofractures.
Whenaporousmediumiscomposedbytwoormoredifferentconstituents,apre-
cisemathematicalmodelingisrequired. Actually,duetothecomplexityofmicro-
structures,anymathematicalmodelingusedtodeterminefluidflowsthroughhet-
erogeneousporousmediamusttakeintoaccounttherapidspatialvariationofthe
phenomenologicalparameters. Furthermore,numericalmodelingofsuchsystems
yieldsatthelocalscaleahugenumberofdiscretizedequations,socomputationswill
befastidiousandintractable. Itisthenimportanttostudyfluidflowsinporous
mediaatthemicroscopicscaleandtodescribetheirbehavioratthemacroscopic
scale. Roughlyspeaking,itconsistsinthepassagefrommicroscopicscaletothe
185

macroscopiconebytendingtozeroasmallparameter,usuallyd enotedε,whichis
theratiobetweenthetwocharacteristicscales,see[6],[11].Weremarkherethatthe
factthathomogenizationindouble-porosityphasescanleadtoeffectivefluidflow
behaviorwasobservedbymanyauthorsinvariousproblems[1],[2],[5],[9],[10].
Forexample,in[5],amicroscopicmodelconsistingoftheusualequationsdescribing
Darcyflowinareservoirwithhighlydiscontinuousporosityandpermeabilitycoeffi-
cients,wasaddressed.Itwasrigorouslyprovedthatthemacroscopic(homogenized)
equationisadoubleporositymodelofsinglephaseflow. Alsoforporoelastichet-
erogeneousmedia,variouseffectivedoubleporositymodelsofcompositesmadeof
amixtureoftwoporoelasticsolidssaturatedbyacompressibleNewtonianfluidhave
beenderived. In[10],thehomogenizationofacompactboneporoelasticitymodel,
describinginteractionsbetweendeformationofthebonetissueandinducedflow,is
addressed. Thedouble-porousstructureconsistsoftheHavers-Volkmannchannels
(theprimaryporosity)andthecanaliculi(thedualporosity).Themacroscopicmodel
isderivedbymeansofperiodicunfoldingmethodanditdescribesthedeformation-
inducedDarcyflowintheprimaryporositieswhereasthemicro-flowinthedouble
porosityisresponsibleforthefadingmemoryeffectsviathemacroscopicporo-visco-
elasticconstitutivelaw.In[1],[2],Barenblatt-Biotconsolidationmodelsforflowsin
periodicporouselasticmediaarederivedbyusingthetwo-scaleconvergencetech-
nique. Themicro-structuresconsistoffluidflowsofslightlycompressibleviscous
fluidsthroughtwo-componentporo-elasticmediaseparatedbyperiodicinterfacial
barriers,describedbytheBiotmodelofconsolidationwiththeDeresiewicz-Skalak
interfaceboundarycondition.
Inthispaper,weshalldealwiththehomogenizationofasteadyfluidflowinme-
diamadeoftwointeractingporoussystemswithahighcontrastofpermeabilities.
Infact,forsuchaconfiguration,itiswell-knownthatthehydraulicconductivity
inthefracturessystemishigheratthelocalscalethanthehydraulicconductivity
intheblockmatrix[5],[7]. Thefamilyofthecorrespondingmicro-modelsthatwe
shallstudyisdescribedbyanellipticsystemoftwopartialdifferentialequations
inatwo-mediumdescription,withDarcy’slawineachphaseandwithcontrasting
permeabilities,plusexchangetermsrepresentingtheinterfacialcouplingthatresults
fromtheinteraction,atthemicro-scale,betweenthetwophases,see(2.1a)–(2.1e)be-
low.Themacro-modelisderivedbymeansofthetwo-scaleconvergencemethod[3].
Itisshownthattheoverallbehavioroffluidflowinsuchmediabehavesasasingle
porositymodelwithanaveragepermeabilityandobeysasingleequationofelliptic
type,meaningthatnodual-permeabilityeffectsoccuratthemacro-scaledescrip-
tion,see(2.15)below. Besidesthat,thederivedmodelpresentsanextrasource
surfacedensityontheexteriorboundary,whichessentiallyarisesfromthefactthat
(1)blockshavelowpermeabilitywhencomparedtothefissures,(2)nonnulland
186

regularsourcedensityontheblocksand(3)theinterfacecont actbetweenthetwo
constituentsisassumedimperfect.
Thepaperisorganizedasfollows:Section2isdevotedtotheproblemsettingofthe
micro-modelandthestatementofthemainresult.InSection3,weshallbeconcerned
withthederivationofthehomogenizedmodelviathetwoscaleconvergencemethod.
2. Settingoftheproblemandthemainresult
Weconsider Ωaboundedandsmoothdomainof RN(N/greaterorequalslant2)andY= ]0,1[N
thegenericcellofperiodicity.Let Y1,Y2⊂Ybetwoopendisjointsubsetsof Ysuch
thatY=Y1∪Y2∪Γ,whereΓ =∂Y1∩∂Y2,assumedtobeasmoothsubmanifold.
Wedenote νtheunitnormalof Γ,outwardto Y1. Fori= 1,2,letχidenotethe
characteristicfunctionof Yi,extendedby Y-periodicityto RN.Forε>0,weset
Ωε
i=/braceleftBig
x∈Ω:χi/parenleftBigx
ε/parenrightBig
= 1/bracerightBig
andΓε=∂Ωε
1∩∂Ωε
2.
Toavoidsomeunnecessarytechnicalcomputations,weassumethatthedualporosi-
tiesdonotmeettheboundary ∂Ω,thatis Ωε
2⊂ΩsothatΓε=∂Ωε
2a nd∂Ωε
1=
∂Ω∪Γε(seeFigure1below). Let Zi=/uniontext
k∈ZN(Yi+k). Asin[3],wealsoassume
thatZ1issmoothandaconnectedopensubsetof RN. Notethat Z2maynotbe
connected.Also, Z1andZ2aretheprimaryanddualporosities,respectively.
Ωε
1Ωε
2
Γε
ThedomainΩY1
Y2
Γ
Theunit
cellY
Figure1. Anexampleofaperiodictwo-componentmediumconsideredinthispaper.
LetA(resp.B)denotethepermeabilityofthemedium Z1(resp.Z2).Letfibea
measurablefunctionrepresentingtheinternalsourcedensityofthefluidflowin Ωε
i.
Finally,let ϑbethenon-rescaledhydraulicpermeabilityofthethinlayer Γε. We
shallassumethefollowings:
187

(H1)A(resp.B) iscontinuouson RN,Y-periodicandsatisfiestheellipticitycondi-
tion:
Aξ·ξ/greaterorequalslantC|ξ|2(resp.Bξ·ξ/greaterorequalslantC|ξ|2)∀ξ∈RN,
where,hereandinwhatfollows, Cdenotesvariouspositiveconstantswhich
areindependentof ε;
(H2)f1,f2∈L2(Ω);
(H3)ϑisacontinuousfunctionon RN,Y-periodicandboundedfrombelow:
ϑ(y)/greaterorequalslantC >0, y∈RN.
Remark2.1. Itshouldbenoticedthatinthehypothesis(H1),thecontinuity
isnotnecessary. Indeed,onecantake A,B∈L∞(RN)andthemainresultofthis
paperremainsunchanged.
Todealwithperiodichomogenizationwithmicro-structures,weshalldenotefor
x∈RN,
χε
i(x) =χi/parenleftBigx
ε/parenrightBig
, Aε(x)=A/parenleftBigx
ε/parenrightBig
, Bε(x)=B/parenleftBigx
ε/parenrightBig
,andϑε(x) =εϑ/parenleftBigx
ε/parenrightBig
.
Th
emicro-modelthatweshallstudyinthispaperisgivenbythefollowingsetof
equations:
−div(Aε∇uε) =f1inΩε
1, (2.1a)
−ε2div(Bε∇vε) =f2inΩε
2, (2.1b)
Aε∇uε·νε=−ϑε(uε−vε)onΓε, (2.1c)
ε2Bε∇vε·nε=−ϑε(vε−uε)onΓε, (2.1d)
uε= 0on∂Ω, (2.1e)
whereνεandnεstandfortheunitnormalof Γεoutwardto Ωε
1andΩε
2,respectively.
Here,Ωε
1representsthefissuredregionwithpermeability AεandΩε
2theblockregion
withpermeability ε2Bε.Thephysicalquantities uεandvεarerespectivelythefluid
flowpressuresin Ωε
1andΩε
2. AsinArbogast,Douglas,andHornung[5],wehave
chosenaparticularscalingofthepermeabilitycoefficientsin(2.1b). Thismeans
thatboth terms/integraltext
Ωε
1|∇uε|2andε2/integraltext
Ωε
2|∇vε|2havethe sameorderofmagnitude
andthusleadtoabalanceindissipationpotential. Equations(2.1a)and(2.1b)
expresstheconservationofmassoffluidwithDarcy’slawin Ωε
1andΩε
2,respectively.
Conditions(2.1c)and(2.1d)expressfluxcontinuityacross Γεandtheimperfect
contactbetweentheblockandthefissuresalong Γεwithpermeabilitygivenby ϑε,
see[8].Transmissioncondition(2.1d)isknownintheliteratureasDeresiewicz-Skalak
188

condition. Finally,(2.1e)isthehomogeneousDirichletcond itionontheexterior
boundaryof Ω.
LetHε= (H1(Ωε
1)∩H1
0(Ω))×H1(Ωε
2).Thespace Hεisequippedwiththenorm:
/ba∇dbl(ϕ,ψ)/ba∇dbl2
Hε=/ba∇dbl∇ϕ/ba∇dbl2
L2(Ωε
1)+ε2/ba∇dbl∇ψ/ba∇dbl2
L2(Ωε
2)+ε/ba∇dblϕ−ψ/ba∇dbl2
L2(Γε).
Theweakformulationof(2.1a)–(2.1e)isasfollows:find (uε,vε)∈Hε,suchthatfor
all(ϕ,ψ)∈Hε,wehave
(2.2)/integraldisplay
Ωε
1A/parenleftBigx
ε/parenrightBig
∇uε∇ϕdx+ε2/integraldisplay
Ωε
2B/parenleftBigx
ε/parenrightBig
∇vε∇ψdx
+ε/integraldisplay
Γεϑ/parenleftBigx
ε/parenrightBig
(uε−vε)(ϕ−ψ)dsε=/integraldisplay
Ωε
1f1ϕdx+/integraldisplay
Ωε
2f2ψdx,
wheredxanddsεdenote,respectively,theLebesguemeasureon RNandtheHaus-
dorffmeasureon Γε.Next,westatetheexistenceanduniquenessresultoftheweak
formulation(2.2).
Theorem2.1.Lettheassumptions (H1)–(H3)befulfilled. Then,foranysuffi-
cientlysmall ε>0,thereexistsauniquecouple (uε,vε)∈Hε,solutionoftheweak
problem (2.2),suchthat
(2.3) /ba∇dbl(uε,vε)/ba∇dblHε/lessorequalslantC.
Proof. WeshallusetheLax-Milgramlemma.Letusdenote
aε((ϕ,ψ),(η,ς)) =/integraldisplay
Ωε
1Aε∇ϕ∇ηdx+ε2/integraldisplay
Ωε
2Bε∇ψ∇ςdx
+ε/integraldisplay
Γεϑ/parenleftBigx
ε/parenrightBig
(ϕ−ψ)(η−ς)dsε,
Lε((ϕ,ψ)) =/integraldisplay
Ωε
1f1ϕdx+/integraldisplay
Ωε
2f2ψdx,
where(ϕ,ψ),(η,ς)∈Hε. Therefore,theweakformulation(2.2)isequivalentto:
find(uε,vε)∈Hεsuchthatforall (ϕ,ψ)∈Hεwehave
(2.4) aε((uε,vε),(ϕ,ψ)) =Lε((ϕ,ψ)).
Thecoercivenessandthecontinuityoftheform aε(·,·)followimmediatelyfrom(H1)
and(H3). Itremainstoprovethecontinuityof Lε. First,from(H2),weeasilysee
thatforall (ϕ,ψ)∈Hε,
(2.5) |Lε((ϕ,ψ))|/lessorequalslantM(f1,f2)/parenleftbigg/parenleftbigg/integraldisplay
Ωε
1|ϕ|2dx/parenrightbigg1/2
+/parenleftbigg/integraldisplay
Ωε
2|ψ|2dx/parenrightbigg1/2/parenrightbigg
,
189

where
M(f1,f2)
= max/parenleftbigg/parenleftbigg/integraldisplay
Ω|f1|2dx/parenrightbigg1/2
,/parenleftbigg/integraldisplay
Ω|f2|2dx/parenrightbigg1/2/parenrightbigg
isaconstantindependentof ε.Next,followinganideaofH.EneandD.Polisevski[9],
weknowthatthereexists C >0suchthatforall ϕ= (ϕ,ψ)∈Hε
/integraldisplay
Ωε
1|ϕ|2dx/lessorequalslantC/integraldisplay
Ωε
1|∇ϕ|2dx, (2.6)
/integraldisplay
Ωε
2|ψ|2dx/lessorequalslantC/parenleftbigg
ε2/integraldisplay
Ωε
2|∇ψ|2dx+ε/integraldisplay
Γε|ψ|2dsε/parenrightbigg
, (2.7)
ε/integraldisplay
Γε|ϕ|2dsε/lessorequalslantC/parenleftbigg
ε2/integraldisplay
Ωε
1|∇ϕ|2dx+/integraldisplay
Ωε
1|ϕ|2dx/parenrightbigg
. (2.8)
The inequalities (2.6) and (2.7) are Poincaré’sinequality and (2.8) is the trace
inequality. These are obtained by the change of variable: x=ε(k+y),k∈
{k∈ZN:ε(k+y)⊂Ωε
i},y∈Zi,i= 1,2,andusingPoincaré’sinequalityand
thetracetheoremonthereferencecell Yi. Asεissufficientlysmall,say ε <1,we
havefrom(2.8)
(2.9) ε/integraldisplay
Γε|ϕ|2dsε/lessorequalslantC/parenleftbigg/integraldisplay
Ωε
1|∇ϕ|2dx+/integraldisplay
Ωε
1|ϕ|2dx/parenrightbigg
.
Using(2.6)in(2.9),weget
(2.10) ε/integraldisplay
Γε|ϕ|2dsε/lessorequalslantC/parenleftbigg/integraldisplay
Ωε
1|∇ϕ|2dx/parenrightbigg
.
Next,from(2.7),wehave
(2.11)/integraldisplay
Ωε
2|ψ|2dx/lessorequalslantC/parenleftbigg
ε2/integraldisplay
Ωε
2|∇ψ|2dx+ε/integraldisplay
Γε|ϕ−ψ|2dsε+ε/integraldisplay
Γε|ϕ|2dsε/parenrightbigg
.
Now,combining(2.10)and(2.11)gives
/integraldisplay
Ωε
2|ψ|2dx/lessorequalslantC/parenleftbigg/integraldisplay
Ωε
1|∇ϕ|2dx+ε2/integraldisplay
Ωε
2|∇ψ|2dx+ε/integraldisplay
Γε|ϕ−ψ|2dsε/parenrightbigg
,
whichmeansthat
(2.12)/integraldisplay
Ωε
2|ψ|2dx/lessorequalslantC/ba∇dbl(ϕ,ψ)/ba∇dbl2
Hε.
190

Observethat(2.6)yields
(2
.13)/integraldisplay
Ωε
1|ϕ|2dx/lessorequalslantC/ba∇dbl(ϕ,ψ)/ba∇dbl2
Hε.
Using(2.5),(2.12),and(2.13)wededucethat
(2.14) |Lε((ϕ,ψ))|/lessorequalslantC/ba∇dbl(ϕ,ψ)/ba∇dblHε.
Thus,Lεiscontinuouson Hε. Notethattheconstant Cappearingin(2.5)is
independentof ε.
By Lax-Milgram’s lemma, we conclude that there exists a unique solution
(uε,vε)∈Hεtotheweakformulation(2.4). Finally, putting (ϕ,ψ) = (uε,vε)
in(2.4),usingtheuniformcoercivenessof aε(·,·)andthecontinuityof Lεyieldsthe
uniformestimate
/ba∇dbl(ϕ,ψ)/ba∇dblHε/lessorequalslantC,
whereagain Cisindependentof ε.Thisconcludestheproofofthetheorem. /square
Now,wearereadytostatethemainresultofthepaper:
Theorem2.2.Let(uε,vε)∈Hεbethesolutionoftheweaksystem (2.2). As-
sumethatf2∈H1(Ω).LetUε=χ1(x/ε)uε+χ2(x/ε)vεdenotetheoverallpressure.
Then,uptoasubsequence,thereexistsaunique U∈H1(Ω),suchthatUεconverges
weaklyinH1(Ω)toU. Furthermore, Uistheuniquesolutiontothehomogenized
model:
(2.15)/braceleftBigg
−div(˜A∇U) =FinΩ,
U=Gon∂Ω,
where˜A, FandGaregivenin (3.17)–(3.18).
Remark2.2. Observethatweneedmoreregularityon f2.Namely,werequire
thatf2∈H1(Ω)sothatthefunction Gdefinedby(3.18)isin H1(Ω)andwhich
givesF∈H−1(Ω).SeealsoRemark3.1below.
Theremainderofthispaperisdevotedtotheproofofthistheorem.Tothisaim,
weshallapplyinthenextsectionthetwo-scaleconvergencetechnique.
191

3. ProofofTheorem2.2
In
thissection,weshallderivethehomogenizedsystem(2.15).Todoso,weshall
firstbeginwithsomenotations.Wedefine C#(Y)tobethespaceofallcontinuous
functionson RNwhichareY-periodic. Let C∞
#(Y) =C∞(RN)∩ C#(Y)andlet
L2
#(Y)(resp.L2
#(Yi),i= 1,2)tobethespaceofallfunctionsbelongingto L2
loc(RN)
(resp.L2
loc(Zi))whichare Y-periodic,and H1
#(Y)(resp.H1
#(Yi))tobethespaceof
thosefunctionstogetherwiththeirderivativesbelongingto L2
#(Y)(resp.L2
#(Zi)).
Next,werecallthedefinitionofthetwo-scaleconvergence[3].
Definition3.1. Asequence (wε)inL2(Ω)two-scaleconvergesto w∈L2(Ω×Y)
(wewritewε2−s⇀ w)if,foranyadmissibletestfunction ϕ∈L2(Ω;C#(Y)),
lim
ε→0/integraldisplay
Ωwε(x)ϕ/parenleftBig
x,x
ε/parenrightBig
dx=/integraldisplay
Ω×Yw(x,y)ϕ(x,y)dxdy.
Thefollowingresultwillbeofuse,see[3],[4].
Theorem3.1.
(1)Let(wε)beauniformlyboundedsequencein H1(Ω) (resp.H1
0(Ω)).Thenthere
existsw∈H1(Ω) (resp.H1
0(Ω))andw1∈L2(Ω;H1
#(Y)/R)suchthat,upto
asubsequence, wε2−s⇀ wand∇wε2−s⇀∇w+∇yw1.
(2)Let(wε)beasequenceoffunctionsin H1(Ω)suchthat
/ba∇dblwε/ba∇dblL2(Ω)+ε/ba∇dbl∇wε/ba∇dblL2(Ω)N/lessorequalslantC.
Then,thereexistasubsequenceof (wε),stilldenotedby (wε),andw0(x,y)∈
L2(Ω;H1
#(Y))such that wε2−s⇀ w0andε∇wε2−s⇀∇yw0and for every ϕ∈
D(Ω;C∞
#(Y))wehave
lim
ε→0ε/integraldisplay
Γεwεϕεdsε=/integraldisplay
Ω×Γw0ϕdxds, ϕε(x) =ϕ/parenleftBig
x,x
ε/parenrightBig
,
wh
eredsistheHausdorffmeasureon Γ.
Now,weturnourattentiontodeterminingthelimitingproblem(2.15). Thanks
totheaprioriestimates(2.3)andusingTheorem3.1,thereexistsasubsequenceof
(uε,vε),solutionof(2.2),stilldenoted (uε,vε),andthereexist
u∈H1
0(Ω), u1∈L2(Ω;H1
#(Y)/R)andv0∈L2(Ω;H1
#(Y2))
192

suchthat
χε
1uε2−s⇀ χ1u
, χε
2vε2−s⇀ χ2v0, (3.1)
χε
1∇uε2−s⇀ χ1(∇u+∇yu1), εχε
2∇vε2−s⇀ χ2∇yv0,
andforany ψ∈ D(Ω;C#(Y))
(3.2) lim
ε→0/integraldisplay
Γεε(uε−vε)ψεdsε=/integraldisplay
Ω×Γ(u−v0)ψdxds, ψε(x) =ψ(x,x/ε).
Formoredetails, wereferthereaderto[3], Proposition1.14i)andii)and[4],
Proposition2.6.
Now,letϕ∈ D(Ω)andϕ1,ψ∈ D(Ω;C∞
#(Y)). Setϕε(x) =ϕ(x) +εϕ1(x,x/ε)
andψε(x) =ψ(x,x/ε).Takingϕ=ϕεandψ=ψεin(2.2),weobtain
(3.3)/integraldisplay
Ωε
1Aε∇uε/parenleftBig
∇ϕ+∇yϕ1/parenleftBig
x,x
ε/parenrightBig/parenrightBig
dx+/integraldisplay
Ωε
2εBε∇vε∇yψ/parenleftBig
x,x
ε/parenrightBig
dx
+/integraldisplay
Γεϑε(uε−vε)(ϕ−ψε)
dsε+εRε=/integraldisplay
Ωε
1f1ϕdx+/integraldisplay
Ωε
2f2ψdx,
where
Rε=/integraldisplay
Ωε
1Aε∇uε∇xϕ1/parenleftBig
x,x
ε/parenrightBig
dx+ε/integraldisplay
Ωε
2Bε∇vε∇xψ/parenleftBig
x,x
ε/parenrightBig
dx
+/integraldisplay
Γεϑε(uε−vε)ϕ1/parenleftBig
x,x
ε/parenrightBig
dsε.
Ac
cordingtotheassumptions(H1)–(H3),tA∇ϕ,tA∇yϕ1,tB∇xψ,andtB∇yψare
admissibletestfunctions.Therefore,inviewof(3.1)–(3.2),thereholdthefollowing
limits:/integraldisplay
Ωε
1Aε∇vε(∇ϕ+(∇yϕ1)ε)dx−→
ε→0/integraldisplay
Ω×Y1A(∇u+∇yu1)(∇ϕ+∇yϕ1)dxdy, (3.4)
/integraldisplay
Ωε
2εBε∇vε∇yψ/parenleftBig
x,x
ε/parenrightBig
dx−→
ε→0/integraldisplay
Ω×Y2B∇v0∇yψdxdy, (3.5)
/integraldisplay
Γεϑε(uε−vε)(ϕ−ψε)dsε−→
ε→0/integraldisplay
Ω×Γϑ(u−v0)(ϕ−ψ)dxds, (3.6)
wherewehavedenoted (∇yϕ1)ε(x) = (∇yϕ1)(x,x/ε). Moreover,using(2.3),itis
easytoseethat Rε=O(1). Thus,by(3.4)–(3.6)andpassingtothelimitin(2.2),
wegetthetwo-scalevariationalformulation:
(3.7)/integraldisplay
Ω×Y1A(∇u+∇yu1)(∇ϕ+∇yϕ1)dxdy+/integraldisplay
Ω×Y2B(y)∇yv0∇yψdxdy
+/integraldisplay
Ω×Γϑ(y)(u−v0)(ϕ−ψ)dxds=/integraldisplay
Ω×Y1f1ϕdx+/integraldisplay
Ω×Y2f2ψdx.
193

Byadensityargument, theequation(3.7)stillholdstruefora ny(ϕ,ϕ1,ϕ2)∈
H1
0(Ω)×L2(Ω;H1
#(Y1)/R)×L2(Ω;H1
#(Y2)). Now, integratingbypartsin(3.7)
yieldsthefollowingtwo-scalehomogenizedsystem:
−divy(A(∇u+∇yu1)) = 0a.e.inΩ×Y1, (3.8)
−divy(B∇yv0) =f2a.e.inΩ×Y2, (3.9)
−div/parenleftbigg/integraldisplay
Y1A(∇u+∇yu1)dy/parenrightbigg
+/integraldisplay
Γϑ(y)[u−v0]ds=f1a.e.inΩ, (3.10)
(A(∇u+∇yu1))·ν= 0a.e.onΩ×Γ, (3.11)
B∇yv0·v=−ϑ(u−v0)a.e.onΩ×Γ, (3.12)
u= 0on∂Ω. (3.13)
Letusfirstnotethatequations(3.8)and(3.11)leadtothefollowingrelation:
(3.14) u1(x,y) =N/summationdisplay
j=1∂u
∂xj(x)ωj(y)+u∗(x),
wh
ere,for1/lessorequalslantj/lessorequalslantN,ωj∈H1
#(Y1)/Ristheuniquesolutiontothefollowingcell
problem:
/braceleftBigg
−divy(A(∇yωj+ej)) = 0a.e.inY1,(ej)isthecanonicalbasisof RN,
A(∇yωj+ej)·ν= 0a.e.onΓ,
andu∗(x)isanyadditivefunctionindependentof y.Similarly,from(3.9)and(3.12)
weseethat v0canbewrittenas
(3.15) v0(x,y)−u(x) =α(y)f2(x),(x,y)∈Ω×Y2,
whereα∈H1
#(Y2)istheuniquesolutionofthefollowingproblem:
(3.16)/braceleftBigg
−divy(B∇yα) = 1inY2,
B∇yα·ν+ϑα= 0onΓ.
Inthesequel,weshalldenoteforconvenience
˜A= (/tildewideraij)1/lessorequalslanti,j/lessorequalslantN,/tildewideraij=/integraldisplay
Y1A(∇yωi+ei)·(∇yωj+ej)dy, (3.17)
f∗=|Y1|f1+|Y2|f2, G=/parenleftbigg/integraldisplay
Y2α/parenrightbigg
f2, F=f∗+div(˜A∇G). (3.18)
194

Letusmentionthat,inviewof(H1), ˜Ais symmetricandpositivedefinite,see[6].
Observealsothat f∗liesinL2(Ω)andsincef2∈H1(Ω),GisinH1(Ω).Therefore,
F∈H−1(Ω).Inserting(3.14)–(3.15)into(3.10)yieldstheellipticequation:
(3.19) −div(˜A∇u) =f∗.
Now,with(3.15)inmind,theoverallpressure Uε=χ1(x/ε)uε+χ2(x/ε)vεtwoscale
convergesto u+χ2αf2.Consequently, Uεconvergesweaklyin L2(Ω)toU=u+G
whichistheuniquesolutionofthehomogenizedmodel:
(3.20)/braceleftBigg
−div(˜A∇U) =FinΩ, U∈H1(Ω),
U=Gon∂Ω.
TheproofofTheorem2.2isthenachieved.
Remark 3.1. If f2∈H1(Ω)isnolongersatisfied,say f2isonlyinL2(Ω),
thenasalreadymentionedbyG.Allairein[3],Remark4.5,thesolution Udoesnot
satisfytherequiredDirichletboundarycondition.Itisthenpreferabletowrite Uas
asumoftwoterms: uand/integraltext
Y2v0dy.Thus,thehomogenizedproblemconsistsoftwo
equations:(3.15),(3.19)withthehomogeneousboundarycondition u= 0on∂Ω.
Remark 3.2. Inviewof(H2),weseethatwesimplychoosethesourcedensi-
tiesf1andf2independentof εanddefineda.e. onthewholedomain Ωwhereas,
throughoutthispaper, f1andf2areonlyusedonthesubregions Ωε
1andΩε
2respec-
tively.Infact,wecanconsiderthecase,wheresourcedensitiesaredefinedontheir
respectiveregionsaswell,withoutmodifyingsubstantiallythehomogenizedmodel
(3.20)exceptfortheaveragedsourcedensity f∗definedin(3.18). Moreprecisely,
iffi=fε
ia.e.inΩε
i(i= 1,2),wherefε
i∈L2(Ωε
i)with/ba∇dblfε
i/ba∇dblL2(Ωε
i)/lessorequalslantC,thenus-
ingtheextensionbyzeroto Ωoffε
i,weseethat /ba∇dblχi(x/ε)fε
i/ba∇dblL2(Ω)/lessorequalslantC. Denoting
byf0
ithetwoscalelimitof χi(x/ε)fε
i,theweaklimitof χi(x/ε)fε
iisthengiven
byFi(x) =/integraltext
Yif0
i(x,y)dyinsteadof |Yi|fi(x)(seether.h.s.oftwo-scalevariational
formulation(3.7))and f∗shouldbegivenby F1+F2insteadof |Y1|f1+|Y2|f2.
Acknowledgments. Theauthorisverygratefultotheanonymousreferee
forcarefullyreadingthepaperandforvaluablesuggestionswhichenabledhimto
improveconsiderablythepaper.TheauthoracknowledgesthesupportoftheAlge-
rianministryofhighereducationandscientificresearchthroughtheC.N.E.P.R.U.
project “Techniques de modélisation en milieux hétérog`enes et couches minces”
No.B00220090078.
195

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Author’saddress :Ab delhamidAinouz ,LaboratoryAMNEDP,FacultyofMathemat-
ics,UniversityofSciencesandTechnologyHouariBoumedienne,POB32ElAlia,Algiers,
Algeria,e-mail:aainouz@usthb.dz .
196