Homogenization and concentration for a diffusion equation with large convection in a bounded domain

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Homogenization and concentration for a diffusion equation with large convection in a bounded domain

Grégoire Allaire, Irina Pankratova, Andrey Piatnitski

To cite this version:

Grégoire Allaire, Irina Pankratova, Andrey Piatnitski. Homogenization and concentration for a diffu-

sion equation with large convection in a bounded domain. Journal of Functional Analysis, Elsevier,

2012, 262 (1), pp.300-330. �10.1016/j.jfa.2011.09.014�. �hal-00908250�

CENTRE DE MATHÉMATIQUES APPLIQUÉES

UMR CNRS 7641

91128PALAISEAU CEDEX (FRANCE).Tél: 0169 33 4600. Fax: 01 69 3346 46

http://www.map.polytehnique.fr/

Homogenization and

onentration for a diusion

equation with large onvetion in

a bounded domain

Grégoire Allaire, Irina Pankratova,

Andrey Piatnitski

R.I. 713 May 2011

equation with large onvetion in a bounded domain

G. Allaire

∗

I. Pankratova

†

A. Piatnitski

‡

May 24, 2011

Abstrat

Weonsiderthehomogenizationofanon-stationaryonvetion-diusionequa-

tionposedinabounded domainwithperiodially osillating oeientsandho-

mogeneous Dirihlet boundary onditions. Assuming that the onvetion term

is large, we give the asymptoti prole of the solution and determine its rate

of deay. In partiular, it allows us to haraterize the hotspot, i.e., the pre-

ise asymptotiloationofthesolutionmaximumwhihlieslose to thedomain

boundaryandisalsothepointofonentration. Duetotheompetitionbetween

onvetion and diusionthe position of the hot spot is not always intuitive as

exemplied insome numerial tests.

Keywords: Homogenization, onvetion-diusion, loalization.

1 Introdution

The goal of the paper is to study the homogenization of a onvetion-diusion

equation with rapidly periodially osillating oeients dened in a bounded

domain. Namely,weonsider thefollowing initial boundary problem:

 

 



 

 

∂ _t u ^ε (t, x) + A ^ε u ^ε (t, x) = 0,

ⁱⁿ

(0, T ) × Ω, u ^ε (t, x) = 0,

^on

(0, T ) × ∂Ω, u ^ε (0, x) = u ₀ (x), x ∈ Ω,

(1.1)

∗

Eole Polytehnique, Route de Salay, 91128 Palaiseau Cedex Frane (al-

lairemap.polytehnique.fr)

†

Narvik University College, Postbox 385, 8505 Narvik, Norway; Eole Polytehnique, Route de

Salay,91128PalaiseauCedex Frane(pankratovamap.polytehnique.fr)

‡

NarvikUniversityCollege,Postbox385,8505Narvik,Norway;LebedevPhysialInstitute RAS,

Leninskiave.,53,119991Mosow,Russia(andreysi.lebedev.ru)

where

Ω ⊂ R ^d

^{is a bounded domain with a Lipshitz boundary}

∂Ω

^,

u ₀

^belongs

to

L ² (Ω)

^and

A ^ε

^{isanoperator dened by}

A ^ε u ^ε = − ∂

∂x _i

a _ij x ε

∂u ^ε

∂x _j

+ 1 ε b _j x

ε ∂u ^ε

∂x _j ,

where we employ the onvention of summation over repeated Latin indies. As

usual

ε

^{, whih denotes the period of theoeients, is a small positive param-}

eter intended to tend to zero. Note the large saling in front of the onvetive

term whih orresponds to the onvetive and diusive terms having both the

same order of magnitude at the small sale

ε

^{(this is a lassial assumption in}

homogenization [5℄, [12℄, [13 ℄, [21 ℄). We make thefollowing assumptions on the

oeients oftheoperator

A ^ε

^.

(H1) The oeients

a ij (y), b j (y)

^{are measurable bounded funtions dened on}

theunitell

Y = (0, 1] ^d

^,thatis

a _ij , b _j ∈ L ^∞ (Y )

^{. Moreover,}

a _ij (y), b _j (y)

^are

Y

^-periodi.

(H2) The

d × d

^matrix

a(y)

^{is uniformlyellipti, that is thereexists}

Λ > 0

^suh

that, for all

ξ ∈ R ^d

^{andfor almost all}

y ∈ Ω

^,

a _ij (y)ξ _i ξ _j ≥ Λ | ξ | ² .

For the large onvetion term we do not suppose that the eetive drift (the

weightedaverageof

b

^{denedbelowby(2.4))iszero,northatthevetoreld}

b(y)

isdivergene-free. Someadditional assumptionsonthesmoothnessandompat

supportoftheinitialdata

u ₀

^{willbemadeinSetion2afterintroduingauxiliary}

spetral ell problems. Inview of (H1) and (H2), for any

ε > 0

^{, problem(1.1)}

hasa unique weak solution

u ^ε ∈ L ^∞ [0, T ; L ² (Ω)] ∩ L ² [0, T ; H ¹ (Ω)]

^{(see [6 ℄).}

Ourmaingoalistodesribetheasymptotibehaviorofthesolution

u ^ε (t, x)

^of

problem (1.1) as

ε

^{goesto zero. There are of ourse many}motivations to study suh a problem (one of them being the transport of solutes in porous media

[17 ℄). However, if (1.1) is interpreted as the heat equation in a uid domain

(theuidveloitybeinggivenby

ε ⁻¹ b(x/ε)

^{),we anparaphrasethefamous hot}

spot onjeture of J. Rauh [23℄, [7℄, [10 ℄, and ask a simple question in plain

words. Iftheinitialtemperature

u ₀

^{hasitsmaximuminsidethedomain}

Ω

^,where

shall this maximum or hot spot go astime evolves ? Morepreisely, we want

to answer this question asymptotially as

ε

^{goes to zero. Theorem 2.1 (and}

the disussion following it) gives a omplete answer to this question. The hot

spot is a onentration point

x _c

^{, loated} asymptotially lose to the boundary

∂Ω

^{(see Figure 1), whih maximizes the linear funtion}

Θ · x

^on

Ω

^{where the}

vetor parameter

Θ

^{is determined as an optimal parameter in an auxiliary ell}

problem(seeLemma2.1). Surprisingly

Θ

^{isnotsomeaverageoftheveloityeld}

but is the result of an intriate interation between onvetion and diusion in

the periodiity ell (even in the ase of onstant oeients ; see the numerial

examples of Setion 7). Furthermore, Theorem 2.1 gives the asymptoti prole

of thesolution, whih isloalizedintheviinityof thehotspot

x _c

^,intermsof

a homogenized equation withan initial ondition thatdependson thegeometry

of thesupportof theinitial data

u ₀

^.

Beforewe explainour resultsingreaterdetails,we brieyreviewprevious re-

sultsintheliterature. Intheasewhenthe vetor-eld

b(y)

^{issolenoidalandhas}

zeromean-value, problem(1.1)hasbeen studiedbythelassialhomogenization

methods (see, e.g, [8℄, [25 ℄). In partiular, the sequene of solutions is bounded

in

L ^∞ [0, T ; L ² (Ω)] ∩ L ² [0, T ; H ¹ (Ω)]

^{and onverges, as}

ε → 0

^{, to the solutionof}

an eetive or homogenized problem in whih there is no onvetive term. For

general vetor-elds

b(y)

^{, and if the domain}

Ω

^{is the whole spae}

R ^d

^{, the on-}

vetion might dominate the diusion and we annot expeta usual onvergene

of thesequene ofsolutions

u ^ε (t, x)

^{inthexed spatialreferene frame. Rather,}

introduingaframeofmovingoordinates

(t, x − ¯ bt/ε)

^{,wheretheonstantvetor}

¯ b

^{istheso-alled eetive drift(oreetive onvetion)whihis dened by(2.4)}

asaweightedaverageof

b

^{,itisknownthatthetranslatedsequene}

u ^ε (t, x − ¯ bt/ε)

onverges to the solution of an homogenized paraboli equation [5 ℄, [13 ℄. Note

that the notion ofeetive drift wasrst introdued in[21℄. Ofourse, theon-

vergene inmoving oordinates annotworkinabounded domain. Thepurpose

of thepresent work is to study theasymptoti behavior of(1.1) inthease of a

bounded domain

Ω

^.

Bearingthesepreviousresultsinmind,intuitively,itislearthatinabounded

domain the initial prole should move rapidly in the diretion of the eetive

drift

¯ b

^{untilitreahesthe boundary,and thendissipatedue to the}homogeneous Dirihletboundary ondition,as

t

^{grows. Sine theonvetion termis large,the}

dissipation inreases, as

ε → 0

^{,sothat thesolution} asymptotially onverges to zero at nite time. Indeed, introduing a resaled (short) time

τ = ε ⁻¹ t

^{, we}

rewrite problem(1.1) inthe form

 

 



 

 

∂ _τ u ^ε − ε div a ^ε ∇ u ^ε

+ b ^ε · ∇ u ^ε = 0,

ⁱⁿ

(0, ε ⁻¹ T) × Ω, u ^ε (τ, x) = 0,

^on

(0, ε ⁻¹ T ) × ∂Ω,

u ^ε (0, x) = u ₀ (x), x ∈ Ω.

(1.2)

Applying the lassial two-sale asymptoti expansion method [8℄,one an show

that, for any

τ ≥ 0 Z

Ω

| u ^ε (τ, x) − u ⁰ (τ, x) | ² dx → 0, ε → 0,

where the leading term of the asymptotis

u ⁰

^{satises the following rst-order}

equation

 

 



 

 

∂ _τ u ⁰ (τ, x) + ¯ b · ∇ u ⁰ (τ, x) = 0,

ⁱⁿ

(0, + ∞ ) × Ω, u ⁰ (τ, x) = 0,

^on

(0, + ∞ ) × ∂Ω ¯ b , u ⁰ (0, x) = u ₀ (x), x ∈ Ω,

(1.3)

with

¯ b

^{being the vetor of eetive onvetion dened by(2.4). Here}

∂Ω ¯ b

^{is the}

subset of

∂Ω

^{suh that}

¯ b · n < 0

^where

n

^{stands for the exteriorunit normal on}

∂Ω

^{. One an onstrut higher order terms in the asymptoti expansion for}

u ^ε

^.

Thisexpansionwillontainaboundarylayerorretorintheviinityof

∂Ω \ ∂Ω ¯ b

^.

A similarproblem inamore general settinghasbeen studied in[9℄.

Thesolution ofproblem (1.3)an befound expliitly,

u ⁰ (τ, x) =

 



u ₀ (x − ¯ bτ ),

^for

(τ, x)

^{suh that}

x, (x − ¯ bτ ) ∈ Ω,

0,

^otherwise

,

whih shows that

u ⁰

^{vanishes after a nite time}

τ 0 = O(1)

^{. In the original}

oordinates

(t, x)

^wehave

Z

Ω

| u ^ε (t, x) − u ₀ (x − ε ⁻¹ ¯ b t) | ² dx → 0, ε → 0.

Thus, for

t = O(ε)

^{theinitial prole of}

u ^ε

^{moveswith theveloity}

ε ⁻¹ ¯ b

^{until it}

reahestheboundaryof

Ω

^andthendissipates. Furthermore,anynitenumberof termsinthe two-saleasymptotiexpansion of

u ^ε (τ, x)

^vanishfor

τ ≥ τ ₀ = O(1)

and thus for

t ≥ t ₀

^{with an arbitrary small}

t ₀ > 0

^{. On the other hand, if}

u ₀

is positive, thenby the maximum priniple,

u ^ε > 0

^{for all}

t

^{. Thus, the method}

of two-saleasymptoti expansioninthis short-time salingisunable toapture

the limit behaviour of

u ^ε (t, x)

^{for positive time. The goal of the present paper}

is therefore to perform a more deliate analysis and to determine the rate of

vanishingof

u ^ε

^,as

ε → 0

^.

Thehomogenizationofthespetralproblemorrespondingto(1.1)inabounded

domain for a general veloity

b(y)

^{was performed in [11℄, [12 ℄.} Interestingly enough theeetivedriftdoesnotplayanyroleinsuhaasebutratherthekey

parameter is another onstant vetor

Θ ∈ R ^d

^{whih isdened as anoptimal ex-}

ponential parameter inaspetralell problem (seeLemma 2.1). Morepreisely,

it is proved in [11 ℄, [12℄ that the rst eigenfuntion onentrates as a boundary

layeron

∂Ω

^{inthediretion of}

Θ

^{. Weshallprovethatthesamevetorparameter}

Θ

^{is alsoruial intheasymptoti analysisof(1.1).}

Notie that for large time and after a proper resaling the solution of (1.1)

should behave like therst eigenfuntion of theorresponding ellipti operator,

and thusonentratesinasmall neighbourhood of

∂Ω

^{inthediretionof}

Θ

^{. We}

prove that this guess is orret, not only for large time but also for any time

t = O(1)

^{, namely that}

u ^ε (t, x)

^{onentrates in the} neighbourhood of the hot spot or onentration point

x _c ∈ ∂Ω

^{whih depends on}

Θ

^{. The value of}

Θ

^an

be determined intermsof some optimality propertyof therst eigenvalue of an

auxiliary periodi spetral problem (see Setion 2). It should be stressed that,

in general,

Θ

^{does not oinide with}

¯ b

^{. As a onsequene, it may happen that}

theonentrationpoint

x _c

^{doesnot evenbelongtothesubset of}

∂Ω

^onsistingof

pointswhihareattainedbytranslationoftheinitial datasupportalong

¯ b

^{. This}

phenomenon is illustratedbynumerial examplesinSetion 7.

problems inthe unitell

Y

^{andimposeadditional onditions onthegeometryof}

theompatsupportof

u ₀

^{. Wethenstate ourmainresult(seeTheorem2.1)and}

give its geometri interpretation. In Setion 3, in order to simplify theoriginal

problem (1.1), we use a fatorization priniple, as in [24℄, [18℄, [26 ℄, [11 ℄, based

on the rst eigenfuntions of the auxiliary spetral problems. As a result, we

obtain a redued problem, where thenew onvetion is divergene-free and has

zero mean-value. Studying the asymptoti behaviour of the Green funtion of

the redued problem, performed inSetion 4, isan important part of theproof.

It isbasedontheresultobtained in[1℄ forafundamentalsolution ofaparaboli

operator with lower order terms. Asymptotis of

u ^ε

^{is derived in Setion 5. In}

Setion 6 we study the ase when the boundary of the support of

u 0

^{has a at}

part. To illustrate the main result of the paper, in Setion 7 we present diret

omputationsof

u ^ε

^{usingthesoftwareFreeFEM++[15 ℄. Anumber ofbasifats}

from the theoryofalmost periodifuntionsis giveninSetion 8.

2 Auxiliary spetral problems and main result

Wedene an operator

A

^{and its adjoint}

A ^∗

^by

Au = − div(a ∇ u) + b · ∇ u, A ^∗ v = − div(a ^T ∇ v) − div(b v),

where

a ^T

^{isthetransposedmatrixof}

a

^{. Following[8℄,for}

θ ∈ R ^d

^{,weintroduetwo}

parameterizedfamiliesofspetralproblems(diretandadjoint)intheperiodiity

ell

Y = [0, 1) ^d

^.







e ^−θ·y A e ^θ·y p _θ (y) = λ(θ) p _θ (y), Y, y → p _θ (y)

^Y-periodi

.

(2.1)

 



e ^θ·y A ^∗ e ^−θ·y p ^∗ _θ (y) = λ(θ) p ^∗ _θ (y), Y, y → p ^∗ _θ (y)

^Y-periodi

.

(2.2)

The next result,based onthe Krein-Rutman theorem, wasproved in[11℄,[12℄.

Lemma 2.1. For eah

θ ∈ R ^d

^{, the rst eigenvalue}

λ ₁ (θ)

^{of problem (2.1) is}

real, simple, and the orresponding eigenfuntions

p _θ

^and

p ^∗ _θ

^{an be hosen pos-}

itive. Moreover,

θ → λ ₁ (θ)

^{is twie} dierentiable, stritly onave and admits a maximum whihis obtained for a unique

θ = Θ

^.

Theeigenfuntions

p _θ

^and

p ^∗ _θ

^{dened byLemma 2.1, an be normalized by}

Z

Y

| p _θ (y) | ² dy = 1

^and

Z

Y

p _θ (y) p ^∗ _θ (y) dy = 1.

Dierentiatingequation (2.1)withrespetto

θ _i

^,integrating against

p ^∗ _θ

^andwrit-

ingdownthe ompatibility onditionfor theobtained equation yield

∂λ ₁

∂θ _i = Z

Y

b _i p _θ p ^∗ _θ + a _ij (p _θ ∂ _{y j} p ^∗ _θ − p ^∗ _θ ∂ _{y j} p _θ ) − 2 θ _j a _ij p _θ p ^∗ _θ

dy.

^(2.3)

Obviously,

p _θ=0 = 1

^{,and, thus,}

∂λ ₁

∂θ _i (θ = 0) = Z

Y

b _i p ^∗ _θ=0 + a _ij ∂ _{y j} p ^∗ _θ=0

dy := ¯ b _i ,

^(2.4)

whih denes the omponents

¯ b _i

^{of the so-alled eetive drift. In the present}

paper we assume that

¯ b 6 = 0

^(or, equivalently,

Θ 6 = 0

^{). The ase}

¯ b = 0

^{an be}

studied by lassial methods (see, for example, [25℄). The equivalene of

¯ b = 0

and

Θ = 0

^{is obvioussine}

λ ₁ (θ)

^{isstritly onave witha unique maximum.}

We need to make some assumptions on the geometry of the support

ω

^(a

losed setasusual) ofthe initialdata

u ₀

^{withrespetto thediretion of}

Θ

^{. One}

possible setof onditionsis thefollowing.

(H3) Theinitialdata

u ₀ (x)

^{isaontinuousfuntionin}

Ω

^{,hasaompat support}

ω ⋐ Ω

^{and belongs to}

C ² (ω)

^{. Moreover,}

ω

^isa

C ²

^-lassdomain.

(H4) Thesourepoint

x ¯ ∈ ∂ω

^{,atwhihtheminimumin}

min _x∈ω Θ · x

^isahieved,

isunique (see Figure1(a)). Inother words

Θ · (x − x) ¯ > 0, x ∈ ω \ { x ¯ } .

^(2.5)

(H5) The point

x ¯

^{is ellipti and}

∂ω

^{is loally onvex at}

x ¯

^{, that is the prinipal}

urvatures at

x ¯

^{havethesamesign. Morepreisely,inloaloordinatesthe}

boundary of

ω

^insome neighborhood

U _δ (¯ x)

^{of thepoint}

x ¯

^{an be dened}

by

z _d = (Sz ^′ , z ^′ ) + o( | z ^′ | ² )

forsomepositivedenite

(d − 1) × (d − 1)

^matrix

S

^{. Here}

z ^′ = (z ₁ , · · · z _d−1 )

are the orthonormal oordinates inthe tangential hyperplane at

x ¯

^{, and}

z _d

isthe oordinateinthenormal diretion.

(H6)

∇ u ₀ (¯ x) · Θ 6 = 0

^.

Remark 2.1. In assumption

( H3 )

^{it is essential that the support}

ω

^{is a strit}

subsetof

Ω

^{,i.e.,doesnottouhtheboundary}

∂Ω

^{(seeRemark5.3forfurther om-}

ments on this issue). However, the ontinuity assumption on the initialfuntion

u 0

^{is not neessary. It will be relaxed in Theorem 5.2 where}

u 0 (x)

^{still belongs}

to

C ² (ω)

^{but is}disontinuousthrough

∂ω

^{. Of ourse, assumingontinuityor not}

will hange the order of onvergene and the multipliative onstant in front of

the asymptoti solution.

Notethat assumption

( H4 )

^impliesthat

Θ 6 = 0

^{isanormal vetorto}

∂ω

^at

x ¯

^.

Eventually, assumption

( H6 )

^{is required beause,}

u ₀

^{being ontinuous in}

Ω

^,

we have

u ₀ (¯ x) = 0

^.

To avoidexessivetehnialities for the moment, we state our main resultin

a looseway(see Theorem 5.1for apreise statement).

ω Ω

Θ

¯ x

x _c

b b

(a)

Θ ω

Ω

b

¯ x

b

x _c

(b)

Figure 1: Denition of the soure point

x ¯

and of the onentration point

x _c

.

Theorem 2.1. Suppose onditions

( H1 ) − ( H6 )

^{are satised and}

Θ 6 = 0

^{. If}

u ^ε

is a solutionof problem (1.1) , then, for any

t 0 > 0

^and

t ≥ t 0

u ^ε (t, x) ≈ ε ² ε ^{d−1 2} e ^{− λ 1(Θ) ε2 t} e ^{Θ·(x ε −¯ x)} M _ε p _Θ x ε

u(t, x), ε → 0,

where

(λ ₁ (Θ), p _Θ )

^{is the rst eigenpair dened by Lemma 2.1 and}

u(t, x)

^solves

the homogenized problem

 

 

 

 

 



∂ _t u = div(a ^eff ∇ u), (t, x) ∈ (0, T ) × Ω, u(t, x) = 0, (t, x) ∈ (0, T ) × ∂Ω, u(0, x) = ∇ u ₀ (¯ x) · Θ

| Θ | δ(x − x), ¯ x ∈ Ω.

(2.6)

Here

a ^eff

^{isa positive denitematrix, dened by (4.7),}

M ε

^{is a onstant, dened}

in Theorem 5.1,depending on

p _Θ

^{,on thegeometry of}

∂ω

^at

x ¯

^{andon therelative}

position of

x ¯

ⁱⁿ

εY

^{(see Remark 5.1 and Figure 2), and}

δ(x − x) ¯

^{is the Dira}

delta-funtion atthe point

x ¯

^.

TheinterpretationofTheorem2.1intermsofonentrationorndingthehot

spot isthe following. Up toamultipliative onstant

ε ² ε ^{d −1 2} M ε

^,thesolution

u ^ε

is asymptotially equal to the produt of two exponential terms, a periodially

osillating funtion

p _Θ ^x _ε

(whih is uniformly positive and bounded) and the

homogenized funtion

u(t, x)

^{(whih is} independent of

ε

^{). The rst} exponential term

e ^{− λ 1(Θ) ε2 t}

^{indiates a fast deay in time, uniform in spae. The seond ex-}

ponential term

e ^{Θ·(x ε −¯ x)}

^{is the root of a} loalization phenomenon. Indeed, it is maximumatthosepointsontheboundary,

x _c ∈ ∂Ω

^{,whihhaveamaximaloor-}

dinate

Θ · x

^,independently ofthepositionof

x ¯

^{(seeFigure1(b)). These(possibly}

multiple) points

x _c

^{are thehot spots. Everywhereelse in}

Ω

^{the solution isex-}

ponentiallysmaller,foranypositive time. Thisbehaviouranlearly beheked

on thenumerial examplesofSetion 7. Itis ofoursesimilar tothebehaviorof

theorrespondingrsteigenfuntion asstudied in[12 ℄.

priniple (see, for example, [24℄, [18℄, [26℄, [11℄) inSetion 3 we make a hange

of unknown funtion in suh a way that the resulting equation is amenable to

homogenization. After that,the new unknownfuntion

v ^ε (t, x)

^isrepresentedin terms of the orresponding Green funtion

K ^ε (t, x, ξ)

^{. Studying theasymptoti}

behaviourof

K ^ε

^{isperformedinSetion 4. Finally, we turnbakto theoriginal}

problem and write down the asymptotis for

u ^ε

^{in Setion 5 whih nishes the}

proof ofTheorem 2.1.

Remark 2.2. Theorem 2.1holds true even if we add a singular zero-order term

of thetype

ε ⁻² c( ^x _ε )u ^ε

^{in theequation (1.1). Thiszero-orderterm will be removed}

by the fatorization priniple and the rest of the proof is idential. With some

additionalworkTheorem2.1anbegeneralizedtotheaseofso-alled ooperative

systems for whih a maximum priniple holds. Suh systems of diusion equa-

tions arise in nulear reator physis and their homogenization (for the spetral

problem) wasstudied in [12℄.

3 Fatorization

Werepresent asolution

u ^ε

^{of theoriginalproblem (1.1) inthe form}

u ^ε (t, x) = e ⁻

λ 1(Θ) t

ε 2 e ^{Θ·(x ε −¯ x)} p _Θ x ε

v ^ε (t, x),

^(3.1)

where

Θ

^and

p _Θ

^{are dened inLemma 2.1. Notiethat thehange ofunknowns}

is well-dened sine

p _Θ

^{ispositive andontinuous.} Substituting (3.1) into (1.1) , multiplying the resulting equation by

p ^∗ _Θ ^x _ε

and using (2.2) , one obtains the

following problemfor

v ^ε

^:

 

 

 



 

 

 

̺ _Θ x ε

∂ _t v ^ε + A ^ε _Θ v ^ε = 0, (t, x) ∈ (0, T ) × Ω, v ^ε (t, x) = 0, (t, x) ∈ (0, T ) × ∂Ω, v ^ε (0, x) = u ₀ (x)

p _Θ ^x _ε e ^{− Θ·(x ε −¯ x)} , x ∈ Ω,

(3.2)

where

̺ _Θ (y) = p _Θ (y) p ^∗ _Θ (y)

^and

A ^ε _Θ v = − ∂

∂x _i a ^Θ _ij x ε

∂v

∂x _j + 1

ε b ^Θ _i x ε

∂v

∂x _i ,

and theoeientsof theoperator

A ^ε _Θ

^aregivenby

a ^Θ _ij (y) = ̺ _Θ (y) a _ij (y);

b ^Θ _i (y) = ̺ _Θ (y) b _j (y) − 2 ̺ _Θ (y) a _ij (y) Θ _j +a _ij (y)

p _Θ (y) ∂ _{y j} p ^∗ _Θ (y) − p ^∗ _Θ (y) ∂ _{y j} p _Θ (y) .

(3.3)