Boundary layers for cellular flows at high Péclet numbers

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1 Boundary layrs for cllular flows at high Péclt numbrs Alxi Novikov Gorg Papanicolaou Lnya Ryzhik January 0, 004 Abstract W analyz bhavior of solutions of th stady advction-diffusion problms in boundd domains with prscribd Dirichlt data whn th Péclt numbr P 1 is larg. W show that solution convrgs to a constant in ach flow cll outsid a boundary layr of width O( 1/ ), = P 1 around th flow sparatrics. W construct an -dpndnt approximat watr-pip problm purly insid th boundary layr that provids a good approximation of th solution of th full problm but has numrical -indpndnt cost. W also dfin an asymptotic problm on a graph, and show that solution of th watr-pip problm itslf may b approximatd by an asymptotic, -indpndnt problm on th graph of flow sparatrics. Finally, w show that th ffctiv diffusivity of th watr-pip problm approximats th tru ffctiv diffusivity with an rror indpndnt of th flow outsid th boundary layrs. 1 Introduction 1.1 Th advction-diffusion problm W considr th stady advction-diffusion problm φ u φ = 0 (1.1) in a simply connctd boundd domain R. Th flow u is incomprssibl: u = 0, and non-pntrating through th boundary of : u n = 0 at (s Figur 1.1). Th small paramtr = P 1 1 is th invrs of th Péclt numbr. Equation (1.1) is supplmntd by th Dirichlt boundary data: φ (x) = T 0 (x), x. (1.) Th problm of th qualitativ bhavior of solutions of (1.1)-(1.) has bn considrd by Zldovich in [0] in th opposit cas P 1 by th prturbativ mthods. It has bn sinc studid in various aras whr passiv scalar advction ariss, such as ocanography, mtorology, tc. On of th main intrsting ffcts is th non-trivial coupling of th ffcts of diffusion and strong advction at a high Péclt numbr. Numrical and physical vidnc [6, 17, 18, 19] suggst th following qualitativ structur of th solution φ insid ach flow cll: thr xists a boundary layr of th width O( ) along th sparatrics btwn diffrnt flow clls C j. Outsid this layr solution is Dpartmnt of Mathmatics, Pnnsylvania Stat Univrsity, Univrsity Park, PA 1680; -mail: anovikov@math.psu.du Dpartmnt of Mathmatics, Stanford Univrsity, Stanford, CA 94305; -mail: papanico@gorgp.stanford.du Dpartmnt of Mathmatics, Univrsity of Chicago, Chicago, IL 60637; -mail: ryzhik@math.uchicago.du 1

2 approximatly qual to a constant K j in ach cll C j (s Figur 1. for a numrical illustration). Th total dissipation rat scals in th corrsponding fashion: φ (x) dx O( 1/ ). Th advction-diffusion problm is closly rlatd to that of ffctiv diffusion in a priodic cllular flow. Th ffctiv diffusivity in such flow is givn by [4] Dij = χ i (x) χ j(x)dx. Th vctor corrctor χ (x) is th man-zro priodic solution of χ j + u χ j = u j. Th lattr problm may b rducd to (1.1) with appropriat boundary conditions by rprsnting χ j = x j + ρ j. Most of th mathmatical studis [8, 1, 13] of th advction-diffusion problm hav bn dvotd to th problm of bounds on th ffctiv diffusivity. It has bn shown formally in [6] and latr in [17, 18, 19], and finally provd in [8] using variational mthods that in this stting th ffctiv diffusivity scals as D D, (1.3) in th spcial cas of symmtric squar clls. This asymptotic stimat for th ffctiv diffusivity has bn rcntly xtndd to gnral non-squar priodic clls in [13] using probabilistic tchniqus that hav thir origin in [10]. Furthrmor, uniform bounds of th typ C 1 D C, C 1 χ (x) dx C (1.4) on th ffctiv diffusivity in th priodic cas hav bn provd in [1], gnralizing th asymptotic rsult of [8] to finit > 0. W rcall that th cas whn th flow has no sparatrics has bn considrd prviously in [9, 10] including th flow ffct on th raction-diffusion quations. Th gnral problm (1.1)-(1.) has bn rcntly analyzd in [1] in th contxt of th possibility of passiv scalar nrgy cascad in a turbulnt flow. In particular, th uppr bound in (1.4) has bn obtaind. Such bounds ar of intrst as thy impos conditions on th scals of th turbulnt flow that would allow th scaling law φ O(1) that is ncssary for th Obukhov [16], Corrsin [7] and Batchlor [, 3] passiv scalar thory to hold. Th purpos of this papr is to considr th gnral problm (1.1) with a larg but finit Péclt numbr and to stablish rigorously and quantitativly th abov mntiond proprtis of th solution of th advction-diffusion problm for a small but finit < 1 without any assumption on th priodicity or symmtry of th flow. Our rsults ar, qualitativly, as follows. W prov th uppr and lowr bounds on th dissipation rat as in th scond bound in (1.4) in th gnral cas. Th uppr bound is obtaind first, using a slight modification of th tchniqu of [1]. Nxt w stablish convrgnc of th solution to a constant K j insid ach cll C j at a distanc N from th sparatrics and obtain bounds on th rat of convrgnc as N. Th proof of this fact mploys som of th idas of intgration and avraging along stramlins usd in [14] to obtain bounds on th spd of a raction-diffusion front in a cllular flow. Th fact that solution is narly constant at a distanc O( ) away from th boundary, whr th prscribd data is non-constant, implis th lowr bound on th dissipation rat in (1.4). Nxt w show that th full problm (1.1) may b rstrictd to an -dpndnt watr-pip problm insid a boundary layr of width N around

3 C j Figur 1.1: Th domain is partitiond by flow sparatrics into clls C j. Figur 1.: Th tmpratur distribution for priodic cllular flows computd in MATLAB. u = H, H = sin(πx) sin(πy); four clls, P = 0. th sparatrics with an rror dcrasing as N. Th watr-pip problm has a computational cost indpndnt of 1 and provids an ffctiv numrical tool to solv th problm at a high Péclt numbr. Solution of th watr-pip problm itslf is thn shown to b wll approximatd by yt anothr asymptotic -indpndnt problm. Th lattr rprsnts a many-cll gnralization of a singl cll problm introducd by Childrss in [6] in th priodic cas and is closly rlatd to 3

4 th limit Markov chain constructd in th priodic cas in [13]. In particular this allows us to show that th intrior constants Kj hav a limit as 0 and idntify this limit in trms of th solution of th modifid Childrss problm. It also allows us to show that for any givn boundary data T 0 (x) that is diffrnt from a constant, and any flow u thr xists a positiv finit limit of th dissipation rat φ (x) dx D > 0, as 0. Finally, by mans of variational principls similar to thos in [5, 8, 15] w show that for any th dissipation rat can b dtrmind from th solution of th watr-pip problm with an rror indpndnt of th flow away from th sparatrics. W not that all our rsults ar dirctly applicabl if homognous Numann boundary conditions ar prscribd on a part of th boundary, whil non-uniform Dirichlt boundary conditions ar prscribd on th rst of. Th gnralization to that cas is straightforward. 1. Th main rsults W rcall that th flow u is assumd to b incomprssibl, thus a stram function H(x, y) xists so that u = H = (H y, H x ). Furthrmor, sinc w assum that th normal componnt of u at th boundary vanishs, has to b containd in a lvl st of H: {H = H 0 }. Hnc ithr is boundd by a closd stramlin of th flow u or by a collction of sparatrics of u that connct a finit numbr of singular points of H lying on th lvl st {H = H 0 }. Th lattr cas is of th most intrst to us. W will assum without loss of gnrality that th critical valu H 0 = 0. All th critical points of H ar assumd to b non-dgnrat. Thn th st is a union of finit numbr of flow clls C j boundd by sparatrics of u, as in Figur 1.1. W will also assum throughout th papr that th boundary data T 0 (x) const, x is sufficintly smooth but is not uniform to avoid th trivial cas of th constant solution. Th stramlins of th flow (lvl sts of th stram function) ar assumd to b sufficintly rgular insid ach flow cll away from th saddl points of H(x) Bounds for th dissipation rat Our first rsult provids gnral bounds on th dissipation rat. Thorm 1.1 Lt us assum that is a picwis smooth curv and th boundary data T 0 in (1.) is sufficintly smooth. Thn thr xists a constant C > 0 so that 1 C φ(x) dx C. (1.5) Morovr, for any givn boundary data T 0 (x) const and flow u thr xists a positiv finit limit lim D()/ = D > 0, (1.6) 0 whr D() = φ (x) dx. Hr and blow w dnot by C all various constants C = C(u, T 0, ) that may dpnd on th gomtry of th stramlins of u, various norms of th boundary data T 0 and th domain but nothing ls, unlss xplicitly spcifid. In particular thy ar indpndnt of th Péclt numbr. Th uppr bound abov is provd in Thorm.1 in Sction. Th proof of th lowr bound in (1.5) is containd in Proposition 3.6 in Sction 3. Existnc of th limit (1.6) is provd in Thorm

5 1.. Convrgnc to a constant insid flow clls Convrgnc of solution to a constant insid is quantifid as follows. Lt D(h) = {x : H(x) h}, h > 0 b a domain strictly insid th flow clls, at distanc O(h) away from th sparatrics. Thorm 1. Thr xist constants K j so that w hav insid ach cll C j sup φ x D(N (x) K j ) Morovr, th constants K j convrg as 0 to crtain constants Kj. C. (1.7) N 3/ Th proof of th first part of this thorm is containd in Sction 3 in Thorm 3.4. Convrgnc of K j to thir limit valus and idntification of th limit follow from th approximation of φ by th solution of th Childrss problm: s Thorm 6.3 in Sction Approximation by th watr-pip problm Th watr-pip problm consists of th advction-diffusion quation (1.1) in th narrow domain N = \D(N ) = {x : H(x) N } around th sparatrics with th Dirichlt boundary conditions (1.) on th outr boundary and th Numann boundary conditions on th lvl st L(N ) = {x : H(x) = N } This problm has a computational cost indpndnt of. W show that its solution φ N is clos to φ. Dnot by χ(s) a smooth vn function, monotonic on s 0, so that { 1, s 1/, χ(s) = 0, s 1. Th following rsult dscribs th L -approximation of th solution of th full problm by th solution of th watr-pip problm. Thorm 1.3 Lt φ solv (1.1) and lt φ N b th solution of th watr-pip problm. Thn thr xist constants K m,n so that φ N satisfis Lt φ N b an xtnsion φ N φ N(x) K m,n C N 3/, x L j(n ) = L(N ) C m. (1.8) φ N(x) = χ to th whol domain as ( ) H(x) N φ N(x) + K m,n ( 1 χ ( )) H(x) N, x C m. with th constants K m,n dfind abov. Thn w hav Morovr, th constants convrg to finit limits: φ φ N L () C N 3/, K m K m,n C. (1.9) N 3/ and with K m as in (1.10). lim 0 K m,n = K m,n, lim 0 lim K m,n = K m (1.10) N K m = K m, (1.11) 5

6 Th proofs of th convrgnc of th watr-pip solution to a constant as in (1.8) and of th rror bound (1.9) ar containd in Sction 4: s Thorm 4. and Proposition 4.1. Convrgnc of th constants K m,n, K m to th corrsponding limits in (1.10) and (1.11) is shown in Thorm 6.3. Th nxt rsult dscribs th approximation of th dissipation rat by th solution of th watrpip problm. Thorm 1.4 Th dissipation rat of th solution of th watr-pip systm has a limit lim φ N(x) dx = D N, lim D N = D (1.1) 0 N N with D as in (1.6). Morovr, th rror Error N = φ N(x) dx φ (x) dx K is boundd by a constant K that dpnds on th flow u in N only. N This thorm is provd in Sction 7 in Thorms 7.1 and 7.. Our final st of rsults concrns th approximation of th solution of (1.1)-(1.) by th solution of th asymptotic Childrss problm. As th formulation of th lattr is rathr lngthy w postpon its discussion and th prcis statmnt of th corrsponding rsult until latr. Th papr is organizd as follows. Th uppr bound on th dissipation rat is prsntd in Thorm.1 in Sction. Sction 3 contains th proof of th corrsponding lowr bound in Proposition 3.6. Convrgnc of solution to a constant is provd first in Thorm 3.4 in th sam sction. Th watr-pip boundary layr problm is introducd in Sction 4, whr w also prov in Thorm 4. that th solution of this problm approximats th solution of th full problm. Th asymptotic Childrss problm is introducd and its solutions ar studid in Thorm 5. in Sction 5. W show that th solution of th Childrss problm approximats th solution of th watr-pip modl in Thorm 6.3 in Sction 6. W also show in this sction that th valus of th constants insid ach flow cll for th full problm convrg to thos givn by th asymptotic Childrss problm. Finally, th variational principls for th total dissipation rat and stimats on th rror in th ffctiv diffusivity of th watr-pip modl ar obtaind in Sction 7. Acknowldgmnt. Th work of G. Papanicolaou was supportd by grants AFOSR F and ONR N L. Ryzhik was supportd by NSF grant DMS , ONR grant N and an Alfrd P. Sloan Fllowship. His rsarch is also supportd in part by th ASCI Flash cntr at th Univrsity of Chicago undr DOE contract B A uniform uppr bound W prov in this sction th uniform uppr bound on th total dissipation rat in th inquality (1.5) in Thorm 1.1. Thorm.1 Lt us assum that is a picwis smooth curv and T 0 is sufficintly smooth. Lt ( ) un M = sup sup v n v m, x m x v S 1 thn thr xists a constant C = C(M, T 0, ) so that φ(x) dx C. (.1) 6

7 Proof. W us a modification of th proof of an uppr bound for th ffctiv diffusivity in [1]. Lt ψ b a tst function to b spcifid latr. W multiply (1.1) by th function q = φ ψ and obtain aftr intgration by parts: q φ n ds Using incomprssibility of th flow u w gt φ dx q φ n ds + ψ dx + 4 ( φ ψ ) φ dx (φ ψ )u φ dx = 0. φ dx + α ψ dx + 1 u φ dx α (.) with th constant α to b chosn. W now multiply (1.1) by u φ and intgrat to gt u φ dx = (u φ ) φ dx = (u φ ) φ n ds (u φ ) φ dx. Onc again using incomprssibility of u and th dfinition of th constant M w obtain from th abov u φ dx = (u φ ) φ n ds 1 (u ( φ ) u n φ φ )dx dx x m x m x n (u φ ) φ ds + M φ dx. (.3) n W insrt (.3) into (.) to gt φ dx + ( α q φ n ds + (u φ ) φ n ds + M ψ dx + 4 ) φ dx. With th choic α = 4M th abov bcoms [ φ dx q + 1 ] φ 4M (u φ ) n ds + φ dx + α ψ dx + 4M ψ dx ψ dx. It rmains to rquir that q + 1 4M (u φ ) = 0 on th boundary. Howvr, is a stramlin of u so that u φ = u T 0 is a givn function. That imposs a boundary condition on th function ψ : ψ (x) = T 0 (x) + 1 4M (u T 0(x)). (.4) Thn, providd that (.4) holds w obtain φ dx ψ dx + 4M ψ dx. (.5) W may choos a function ψ so that it satisfis th boundary conditions (.4), vanishs idntically at distancs largr that away from and satisfis th uniform bounds ψ L () C, ψ L () C/. Using such a tst function in (.5) w obtain th uppr bound (.1). Thorm.1 implis that th boundary layr along th boundary has to xtnd to th distanc at last of th ordr of O( ). This is mad prcis in Proposition.: oscillations of φ hav to 7

8 b prsnt at such distancs from th boundary w will latr s that this is actually th corrct boundary layr scal. In ordr to mak this prcis w lt C 0 b a flow cll adjacnt to th boundary such that T 0 is not constant along l 0 = C 0. Such a cll xists as T 0 is continuous and non-constant on. W lt l0 b a part of l 0 that is sparatd away from th nd-points of l 0 and such that T 0 (x) is not constant on l 0. W may thn introduc th following orthogonal systm of coordinats in a nighborhood of l0. Th coordinat H(x, y) is th labl of th stramlin. Th coordinat θ orthogonal to H is normalizd so that θ = H on l 0 and l 0 may b rprsntd as l0 = {H = 0, θ 1 θ θ }. (.6) W may considr a sufficintly small tubular nighborhood U 0 = { H H 0, θ 1 θ θ } of l 0 so as to hav H, θ C > 0. Proposition. Lt C 0 b a flow cll as abov adjacnt to th boundary and L 0 (γ) = {(x, y) C 0 : H(x, y) = γ } b th lvl st of H(x, y) insid th cll C 0. Thn thr xists a constant C > 0 so that w hav an inquality θ θ 1 φ (γ, θ) φ (γ ) dθ θ θ 1 (T 0 (θ) T 0 ) dθ Cγ (.7) for all γ < H 0 / and with θ 1. as in(.6). Hr φ θ (ρ) = (θ θ 1 ) 1 φ (ρ, θ)dθ is th avrag of φ ovr th corrsponding part of th stramlin and T θ 0 = (θ θ 1 ) 1 T 0 (θ)dθ is th avrag of T 0 along l 0. Proof. W hav a simpl bound φ (0, θ) φ (γ, θ) γ γ 0 φ (H, θ) H Intgrating th abov in θ and using th boundary data for φ w obtain θ T 0 (θ) φ (γ, θ) dθ γ θ γ φ (H, θ) H Th Jacobian θ 1 θ 1 0 θ 1 θ 1 dh. dhdθ. (.8) J = D(H, θ)/d(x, y) (.9) is uniformly boundd from abov and blow away from zro in U 0. Hnc w may r-writ th right sid as an intgral ovr U γ = { H γ, θ 1 θ θ }: θ γ φ (H, θ) 0 H dhdθ C φ(x) dx. U γ θ 1 θ 1 Using Thorm.1 and (.8) w obtain θ T 0 (θ) φ (γ, θ) dθ Cγ φ(x, y) dxdy Cγ. (.10) U γ Thrfor w hav for any constant a R: θ θ 1 φ (γ, θ) a dθ so that (.7) follows. θ θ 1 T 0 (θ) a dθ θ θ 1 T 0 (θ) φ (γ, θ) dθ θ θ 1 T 0 (θ) T 0 dθ Cγ 8

9 3 Convrgnc to constants In this sction w obtain th lowr bound of th inquality (1.5) in Thorm 1.1 and prov Thorm 1.: w show that solution of (1.1)-(1.) is clos to a constant insid ach cll of th flow whn is small. As bfor w dnot by L j (γ) = {(x, y : H(x, y) = γ} th lvl st of H(x, y) insid a cll C j. W will usually omit th subscript j to simplify th notation as long as w considr on cll and this dos not caus any confusion. W dnot by D j (γ) th rgion boundd by L j (γ) insid ach cll and by D j (α, β) = D j (β)\d j (α) th annulus btwn two lvl sts. W hav th following proposition. Proposition 3.1 Lt φ (x) b solution of (1.1)-(1.) and lt M j (α) = inf x L j (α) φ (x). Thn thr xists a constant C > 0 so that sup φ (x), and m j (α) = x L j (α) M j (α) m j(α) C ( α ) 3/4. (3.1) This proposition stats th convrs of Proposition.: whil th maning of th lattr is that th width of th boundary layr is at last O( ), th formr shows that it is not largr than O( ), as th oscillation on th lvl st H = N is boundd by C/N 3/. Th proof is basd on th following ky lmma. Lmma 3. (Th lvl-st oscillation inquality) Lt L j (α) and L j (β) b two lvl sts of th stram function H(x) in a cll C j with D j (α) D j (β). Thn w hav (M (α) m (α)) F (α, β) φ n ds + φ n ds, (3.) L j (α) whr F (α, β) is th flux btwn two lvl sts F (α, β) = (u n)ds, γ = γ(t), t [0, 1], γ(0) L j (α), γ(1) L j (β). (3.3) γ Hr γ is any smooth curv that conncts th lvl sts L j (α) and L j (β) and dos not intrsct itslf. W will assum without loss of gnrality that F (α, β) 0. Not that th flux btwn two lvl sts is indpndnt of th choic of th curv γ bcaus of th incomprssibility of th flow u. W now prov th lvl-st oscillation inquality (Lmma 3.). As w rstrict our analysis to on cll w drop th subscript j in all th involvd quantitis. Th ida of th proof is to construct a st R boundd by a pair of gradint curvs of φ and parts of th stramlins L(α) and L(β) if possibl. Th gradint curvs would b chosn so that th diffrnc in th valus of th function φ btwn ths curvs is at last as larg as th oscillation of φ along L(α). Intgrating quation (1.1) ovr R w gt thn (3.). Th main tchnicality is th construction of th st R: s Figurs 3.1 and 3. blow for a gomtric dpiction of R. W turn now to th construction of R. Lt us dfin th oscillation function d(γ) = M (γ) m (γ). Th maximum principl implis that if th lvl st L(γ) is containd insid th rgion D(γ ) boundd by th lvl st C(γ ), thn d(γ) < d(γ ). W dnot by x m (α) and x M (α) th points whr φ attains its minimum and maximum on th lvl st L(α): M (α) = φ (x M (α)) and m (α) = φ (x m (α)). Considr th gradint curvs L j (β) dγ m dt = φ (γ m (t)), γ m (0) = x m (α), (3.4) 9

10 and dγ M = φ (γ M (t)), γ M (0) = x M (α). (3.5) dt Th function φ may hav critical points in D(α, β) and th gradint curvs γ M and γ m potntially may tnd to thos points as t +. Howvr, all critical points of φ ar isolatd saddl points as it may hav nithr intrnal maxima nor minima according to th maximum principl. Morovr, as φ satisfis an lliptic problm in it may hav only finitly many critical points in th intrior away from th boundary. Thus thr ar only finitly many critical points of φ insid D(α, β) that w dnot by ξ 1,..., ξ N. Not that both x M (α), x m (α) ξ k for all k bcaus of th strong maximum principl [11]. Lt us considr th disks Uj r = { x ξ j r}, j = 1,..., N cntrd at th singular points, and lt U r = N j=1 U j r. Not also that φ (x) > C(, r) for x D r (α, β) = D(α, β)\u r. Thrfor φ (γ M (t)) > M (α) + C()t if γ M (s) D r (α, β) for 0 s t and hnc th curv γ M (t) must lav th st D r (α, β) at a finit tim sinc th function φ is uniformly boundd. Howvr, th curv γ M (t), t > 0 may not intrsct th lvl st L(α) bcaus φ (γ M (t)) is strictly incrasing for t < t 0 providd that it stays insid D r (α, β) for all t < t 0. Hnc thr ar two possibilitis: ithr both γ M and γ m xit th st D r (α, β) at L(β) or on of thm crosss D r (α, β) at on of Uj r. W considr ths two cass sparatly. First, w assum that w may choos r > 0 so small that th curvs γ M and γ m do not intrsct th circls Uj r = { x ξ j = r}, j = 1,..., N, and thn w trat th othr cas. Cas 1: Thr xists r > 0 so small that both γ m (t) and γ M (t) xit D r (α, β) at L(β). W dnot th corrsponding xit tims by t m and t M, that is γ m (t m ) L(β) and γ M (t M ) L(β), whil γ m (s) D r (α, β) for 0 s t m and γ M (s) D r (α, β) for 0 s t M. With a slight abus of notation w dnot γ m = {γ m (s), 0 s t m } and γ M = {γ M (s), 0 s t M }. Th curvs γ m and γ M both hav a finit lngth sinc φ is uniformly boundd abov and blow in D r (α, β) (by constants that may dpnd on and r). Ths curvs may not intrsct sinc φ (x) > M > m > φ (y) for all x γ M and y γ m. Lt R b a domain boundd by γ m, γ M and parts of th stramlins γ α L(α) and γ β L(β) (s Figur 3.1). Thr ar two such domains, R and D(α, β)\r. W fix R so that u n > 0 on γ M (t) for t sufficintly small this guarants that ach stramlin of u gos out of R whn it intrscts γ M for th last tim. Furthrmor, w hav u n < 0 on γ m for t sufficintly small so that ach stramlin of u gos into R whn it intrscts γ m for th first tim. Hr n is th outward normal to R. Intgrating (1.1) ovr R w obtain 0 = ( φ u φ φ )dx = R γα n γβ ds + φ n ds (u n)φ ds (u n)φ ds, (3.6) γ m γ M bcaus u n 0 on γ α, γ β and φ n = 0 on γ m, γ M sinc th lattr ar gradint curvs of φ. W will us th following fact. Lmma 3.3 Lt γ : [0, 1] D(α, β) b any non-slf intrscting smooth curv that conncts L(α) and L(β): γ(0) L(α), γ(1) L(β), has a finit lngth and is not tangnt to L(α) at t = 0. Fix th unit normal n to γ so that n(t) is continuous and u n is non-ngativ whn a stramlin of u intrscts γ for th last tim, that is, u n(τ(ξ)) 0 for all ξ btwn α and β, with τ(ξ) = sup{t : γ(t) L(ξ)}. Lt f(x) 0 b a continuous function monotonically incrasing along γ. Thn w hav whr F (α, β) is th flux (3.3). F (α, β) inf f (u n)fds F (α, β) sup f, (3.7) x γ γ x γ 10

11 Proof. First, w obsrv that u n(τ(ξ)) 0 for all ξ [α, β] providd that u n(t) > 0 for t sufficintly small. Th inquality (3.7) is shown as follows. For any N N w may approximat f along γ by two picwis constant (along γ) monotonically incrasing functions f N and f N so that (u n)fds 1 γ N (u n) f N ds (u n)fds (u n) f N ds n)fds + γ γ γ γ(u 1 (3.8) N and f f N 1/N, f f 1/N. Thrfor it suffics to prov (3.7) for a stp function f that has finitly many discontinuitis, th gnral cas follows aftr passing to th limit N in (3.8). W assum blow that f is a stp function. Lt α 1,..., α p b valus of th stram function H such that f has jumps only on th lvl sts L(α k ), k = 1,..., p. W ordr thm so that L(α k ) D(α k+1 ). Thn w may rprsnt γ as th union γ = p k=1 γ k, γ k D(α k, α k+1 ). Hr γ k is th part of γ containd in th annulus D(α k, α k+1 ). W may furthr split th subst γ k as a union γ k = γ k γ k. Hr th st γ k = s k l=1 γ kl is a union of finitly many curvs γ kl that connct th lvl sts L(α k ) and L(α k+1 ). Thr can b only finitly many of such curvs sinc γ has a finit lngth and th distanc btwn L(α k ) and L(α k+1 ) is positiv. Th st γ k = lγ kl consists of curvs that start and nd on th sam lvl st L(α k ) or L(α k+1 ). W not that th function f is constant on ach curv γ kl and γ kl. Thrfor w hav using incomprssibility of u f(u n)ds = f(u n)ds = 0. γ k l γ kl W also hav γ kl f(u n)ds = ( 1) l+1 f kl F (α k, α k+1 ), whr f kl is th constant valu of f on th curv γ kl. This implis that f(u n)ds = γ k γ k f(u n)ds = F (α k, α k+1 ) s k l=1 ( 1) l+1 f kl. Howvr, f kl is an incrasing function of l and th total numbr of tims s k that γ crosss from L(α k ) to L(α k+1 ) must b odd. Thus th abov may b boundd blow by f(u n)ds f k1 F (α k, α k+1 ) F (α k, α k+1 ) inf f. γ γ k Summing th abov ovr k w obtain th first inquality in (3.7). Th scond inquality is provd in th sam way. W now apply (3.7) to th curvs γ m and γ M with f = φ. Sinc max φ = m (α) and min φ = γ m γ M M (α), w hav (u n)φ ds m (α)f (α, β), (u n)φ ds M (α)f (α, β), (3.9) γ m γ M so that (u n)φ ds + (u n)φ ds (M (α) m (α))f (α, β). γ m γ M 11

12 C b g m x ( a ) m g a C a g b x ( ) M a R g M Figur 3.1: Th non-critical cas U r 4 g a U r 5 g m g M U U r r 3 U r 1 R g b U 6 r Figur 3.: Th critical cas Clarly w also hav and φ γβ n ds φ γ α n ds L(β) L(α) φ n ds, φ n ds. Th claim of Lmma 3. now follows from th last thr inqualitis and (3.6) in th cas whn φ has no critical points in D(α, β) or whn γ m and γ M xit D r (α, β) along L(β). Cas : It rmains to considr th scond cas whn γ m or γ M xit th st D r (α, β) at th boundary U r for all r > 0. Thn w pick r > 0 sufficintly small to b spcifid blow. In particular w rquir that th starting points x M (α) and x m (α) ar not containd in any of Uj r, j = 1,..., N this is possibl sinc x m (α) and x M (α) ar not critical points of φ as implid by 1

13 th strong maximum principl. Thn on (or both) of th curvs γ m and γ M dfind by (3.4) and (3.5) should xit D r (α, β) at th boundary U r = N j=1 U j r. Lt us assum that this happns to γ M and that it xits Dαβ r at a point on U j r 1 at a tim t 1 M. W continu γ M past th tim t 1 M as follows (s Figur 3.). Lt η j 1 M = γ M( t 1 M ) b th point whr γ M intrsctd Uj r 1 and lt also η j 1 M b th point whr φ rachs its maximum ovr Uj r 1 φ (η j 1 M ) = sup φ (x). x Uj r 1 Th vctor φ (η j 1 M ) points in th dirction of th outr normal to U j r 1 by th maximum principl. W stop γ M at η j 1 M and continu it along th circl U j r 1 to η j 1 M in ithr dirction with th spd qual to on, so that γ M (t 1 M ) = ηj 1 M. Thn γ M follows th gradint curv going out of η j 1 M for t t1 M until it hits ithr L(β) or anothr circl Uj r at a point η j M at a tim t M. In th formr cas w stop th curv γ M, whil in th lattr w continu it in th sam fashion as at Uj r 1, conncting γ M to ηm, th maximum of φ along Uj r, tc. Evntually γ M has to cross th lvl st L(β) at som finit tim t β M. Indd, w hav φ ( η j k M ) < φ (η j k M ) < φ ( η j k+1 M ) < φ (η j k+1 M ) which implis that th curv γ M may not hit th sam circl Uj r twic. Givn that th total numbr of critical points N is finit and that γ M may not stay insid D r (α, β) for an infinit tim w conclud that th xit tim t β M is finit. A similar construction may b applid to th curv γ m with η j k m bing th point whr φ attains its minimum on Uj r k. In ordr to guarant that th curvs γ m and γ M constructd in such way do not intrsct, w rquir that r is so small that 0 < sup φ inf φ < Uj r Uj r δ 1 + N (φ (x M (α)) φ (x m (α))), j = 1,..., N (3.10) whr δ is a small paramtr. Obsrv that th squnc φ (η j k M ) is incrasing in k, φ (η j 1 M ) > φ (x M (α)) and φ (γ(s)) > φ (η j k M ) for t k M < s < t k+1 M. W also hav φ (γ M (s)) > φ (η j k M ) for t k M < s < tk M. That implis a lowr bound φ (γ M (s)) > φ (x M (α)) for all 0 < s < t β M. Similarly w hav φ (γ m (s)) < φ (x m (α)) + δ 1 + N (φ (x M (α)) φ (x m (α))) δ 1 + N (φ (x M (α)) φ (x m (α))) (3.11) δ 1 + N (φ (x M (α)) φ (x m (α))) for all 0 < s < t β m. That implis an stimat ( φ (γ M (s)) φ (γ m (s )) > 1 δ ) (φ (x M (α)) φ (x m (α))) (3.1) 1 + N for all s and s so that γ M and γ m may not intrsct providd that δ < 1/. W may now procd as in th first part of th proof. Lt R b th domain boundd by γ m, γ M and parts of th lvl sts L(α) and L(β), as dpictd on Figur 3., chosn so that u n > 0 for t 13

14 sufficintly small, that is, so that ach stramlin of u gos out of R whn it crosss γ M for th last tim. Intgrating (1.1) ovr R w now obtain instad of (3.6): 0 = R ( φ + u φ φ )dx = γα n γβ ds + φ N n ds + k=1 γ M Uk r φ N n ds + k=1 γ m Uk r φ n ds (u n)φ ds (u n)φ ds, (3.13) γ m γ M whr γ α = R L(α) and similarly for γ β. Th function φ (γ M (s)) is no longr ncssarily monotonically incrasing in s, as monotonicity might b brokn for t j M < s < tj M. Howvr, w may adjust its valus on ths intrvals, intrpolating linarly btwn φ ( η j k M ) and φ (η j k m ), to mak th nw function φ (s) monotonic in s. Th oscillation bound (3.10) implis that (u n) φ N ds (u n)φ ds (u n)( γ M γ M φ φ )ds (3.14) k=1 γ M U r k δ u N (φ (x M (α)) φ (x m (α))) Cδ. 1 + N Th stimat (3.7) may b applid to φ which togthr with (3.11) and (3.14) implis: (u n)φ ds (u n) φ ds Cδ [M (α) Cδ]F (α, β) Cδ = M (α)f (α, β) Cδ. (3.15) γ M γ M Similarly w obtain (u n)φ ds (u n) φ ds Cδ [m (α) + Cδ]F (α, β) Cδ = m (α)f (α, β) Cδ. (3.16) γ m γ m Furthrmor, w may choos r < 1 so small that φ < δ/(1 + N ) on all U r j, j = 1,..., N this is possibl sinc th cntrs of U r j ar singular points of φ. Thn w obtain N k=1 γ M U r k Using th abov stimats in (3.13) w gt φ L(α) n ds + φ L(β) n ds N φ N = n ds + k=1 k=1 γ M U r k φ n ds δ N πr Cδ. 1 + N γ m U r k (M (α) m (α))f (α, β) Cδ. L(α) φ n ds + φ L(β) n ds (u n)φ ds γ M φ n ds (u n)φ ds γ m This provs Lmma 3. in cas, as δ is arbitrary, and thus th proof of this lmma is complt. 14

15 W now prov Proposition 3.1. W us inquality (3.) for a pair of lvl sts L((α + β)/ + H) and L(β + H) with 0 H α β to obtain ( ( ) ( )) ( ) α + β α + β α + β M + H m + H F + H, β + H φ n ds + φ n ds. L( α+β +H) L(β+H) Howvr, w hav M (α) m (α) M ( α + β ) ( ) α + β + H m + H according to th maximum principl. Thrfor w gt ( ) α + β (M (α) m (α)) F + H, β + H L( α+β +H) φ n ds + L(β+H) φ n ds. (3.17) W intgrat (3.17) with rspct to H to obtain (M (α) m (α)) (α β)/ 0 ( α + β F ) + H, β + H dh α β L(H) φ dsdh. (3.18) n Th intgral on th right sid of inquality (3.18) may b r-writtn in th curvilinar coordinats as α α φ φ dsdh = dθdh β L(h) n β n φ J dxdy = φ H dxdy θ D(α,β) θ D(α,β) ( ) 1/ ( ) ( 1/ ) 1/ H dxdy φ dxdy C C(α β)1/ D(α,β) C 1/4 H dxdy D(α,β) 1/4 whr J = H θ is th Jacobian (.9). Th lft sid of (3.18) satisfis (M (α) m (α)) (α β)/ Th abov stimats imply that 0 ( α + β F ) + H, β + H dh C(M (α) m (α))(α β). M (α) m (α ( β)1/ ) 3/4 (α) C (α β) 1/4 C α with th choic β = α/. This finishs th proof of Proposition 3.1. Proposition 3.1 shows that th variation of φ (x) on a lvl st L(N ) is boundd by M (N ) m (N ) Th maximum principl thn implis th following thorm. 15 C N 3/.

16 Thorm 3.4 Thr xist constants K j so that w hav insid ach cll C j sup φ x D(N (x) K j ) C. (3.19) N 3/ This shows that th function φ is clos to a constant insid ach cll C j. Th nxt proposition is anothr manifstation of this fact. Thorm 3.5 W hav an uppr bound D(H) φ dx C H ( H ) 3/8 (3.0) for H. This stimat is shown as follows. Intgrating (1.1) ovr D(H) w obtain φ dx = φ φ n ds. D(H) Intgrating this quation in H (H 0, H 0 + l) w gt H0 +l H 0 D(H) Th lft sid of (3.1) is boundd blow by H0 +l H 0 φ dxdh = D(H) L(H) H0 +l H 0 L(H) φ dxdh l φ dx, D(H 0 +l) φ φ dsdh. (3.1) n as D(H 0 + l) D(H) for H 0 H H 0 + l. Th right sid of (3.1) may b stimatd as H0 +l H 0 L(h) W dnot F (H) = ( 1/ φ φ n dsdh C (M (H 0 ) m (H 0 )) l 1/ φ dx). D(H 0 ) D(H) φ dx. Thn th abov stimats with H 0 = l = H imply that HF (H) C ( H ) 3/4 (HF (H)) 1/. That is, F (H) = HF (H) satisfis F (H) C for H and This implis that F (H) C F (H) ( H ) 3/4 F 1/ (H). ( ) 3/8 for H so that H φ dx C ( ) 3/8 H H D(H) which is (3.0). Thorm 3.4 implis a lowr bound on th L -norm of th gradint of solution. 16

17 Figur 4.1: Th watr-pip modl Proposition 3.6 Thr xists a constant C = C(T 0,, u) so that φ (x) dx C. (3.) W choos th boundary cll C 0 as in th proof of Proposition. and rcall th first inquality in (.10) (with th notation as in th sam proof): θ Th lft sid may b boundd from blow by θ θ 1 θ θ 1 T 0 (θ) φ (γ, θ) dθ Cγ φ(x) dx. T 0 (θ) φ (γ, θ) dθ θ θ 1 T 0 (θ) T 0 dθ Cγ 3/4 C(1 γ 3/4 ) θ 1 θ T 0 (θ) K0 dθ K0 φ (γ, θ) dθ θ 1 with th constant K 0 as in (3.19) in Thorm 3.4 for th cll C 0. Combining th last two inqualitis and using γ > 1 w obtain (3.). This complts th proof of Thorms 1.1 and Th watr-pip ntwork Th prvious argumnts show that thr xist constants Kj so that solution of (1.1) is wll approximatd by solution of th following watr-pip problm (s Figurs 4.1 and 4.). As bfor, w 17

18 Y = N Figur 4.: On cll dnot by N = { H(x) N } th domain consisting of narrow pips (boundary layrs) nar th sparatrics. Its boundary consists of and finitly many lvl st curvs l N k = L k(n ), k = 1,..., p so that H(x) = N on l N k. Th rsults of Sction 3 show that φ, solution of (1.1) is uniformly clos to solution of ψ u ψ = 0, x N (4.1) with th boundary conditions ψ = T 0, ψ l N = Km, m = 1,..., p (4.) m with th constants K m as in Thorm 3.4. Mor prcisly w hav a uniform bound φ (x) ψ (x) C. (4.3) N 3/ This shows that in a numrical computation of φ it suffics to considr th pip-problm (4.1)-(4.) with th corrct constants K m in ordr to obtain a good approximation of th solution. Howvr, th constants K m ar not known a priori and thir computation is part of th problm. As w hav sn th function φ is vry clos to a constant nar th lvl sts l N m. Thrfor w should xpct that w may rplac th Dirichlt boundary data on l N m by th homognous Numann boundary conditions in th watr-pip problm (4.1) and obtain an approximation that has th sam ordr of rror. In particular this would provid an fficint numrical way to find th constants K m as th boundary valu of th solution of (4.1) with th Numann boundary conditions. This is confirmd by th following rsults. Proposition 4.1 Lt φ N b solution of th watr-pip modl: φ N u φ N = 0, x N (4.4) 18

19 on th domain N = { H(x) N } with th boundary conditions Thn thr xist constants K m,n for all x l N m. φ N = T 0, so that φ N n l N = 0, m = 1,..., p. (4.5) m φ N(x) K m,n C, (4.6) N 3/ Proof. Th proof of this proposition is ssntially th sam as of Thorm 3.4. On only has to obsrv that th strong maximum principl implis that th maximum and minimum of th function φ N ovr any sub-domain {α H(x) N } C m is achivd on th boundary { H(x) = α} C m and not on th intrior lvl st l m N. Thrfor all argumnts in th proof of th lvl-st oscillation inquality (Lmma 3.) ar applicabl vrbatim, and w do not rpat thm. Thorm 4. Lt φ solv (1.1) and lt χ(s) b a smooth vn function, monotonic on s 0, so that { 1, s 1/, χ(s) = 0, s 1 Lt us xtnd φ N to th whol domain as ( H(x) N φ N(x) = χ ) φ N(x) + K m,n ( 1 χ ( )) H(x) N, x C m with th constants K m,n givn by Proposition 4.1. Thn w hav whr φ solvs (1.1). Proof. Lt ζ = φ φ N φ φ N L () C, (4.7) N 3/ b th rror that w nd to stimat. It satisfis th quation with [ g (x) = ( K m,n φ N) ζ u ζ = g, x, (4.8) ( H(x) N H(x)χ N ( H(x) N N χ ) + 1 N H(x) χ ) φ N(x) H(x) ( )] H(x) N and th boundary condition ζ = 0 on. W multiply (4.8) by ζ and intgrat ovr : ζ (x) dx = ζ (x)g (x)dx = I + II + III. Th first trm on th right may b stimatd using Proposition 4.1 as ( I = ζ (x)( K H(x) m N φ N(x)) N H(x)χ N C ζ ( L ( N ) H(x) N 5/ χ N 19 ) dx ) dx C N 5/ ζ L ( N ).

20 Th scond trm is boundd in a similar way as II = ζ (x)( K m N φ N(x)) 1 ( ) H(x) N H(x) χ N dx C N 5/ ζ L ( N ). Th last trm w bound intgrating by parts as III = ζ (x) ( ) H(x) N χ N φ N(x) H(x)dx = ( N ζ (x)(φ N(x) K H(x) m N )χ N ( ) ζ (x)(φ N N(x) K H(x) m N )χ N H(x)dx ( (φ N N(x) K H(x) m N )χ N [ + C N 5/ A ζ (x) dx + 1 ( H(x) A χ N ) H(x) dx ) ζ (x) H(x)dx C ) H(x) dx N 5/ ζ L ( N ) + C N 5/ ζ L ( N ) ] W choos A = N 5/ /(C) to obtain th bound ζ (x) dx C N 5/ ζ L ( N ) + C N 4. (4.9) Rcall that ζ = 0 on and on th lvl st l N m. Thn if K m K m = δ > whil on th othr ζ L ( N ) δ + W dnot γ = δ so that Thrfor C N ζ (x) (K m K m) C N 3/ ζ (x) dx C N N ( δ C w hav, on on hand, N 3/ ( δ. C ) N 3/, C. Putting ths bounds into (4.9) w obtain N 3/ C N 3/ ) C ( N 5/ δ + C and rwrit th abov as N 3/ C N γ C N 5/ γ + C N 4 γ An application of th maximum principl on N C N 3/. K m K m = δ C N 3/. C ) N 3/ + C N 4. + C N 4 finishs th proof of Thorm 4.. 0

21 T( Q ) C 1 C 0 T( Q ) 0 Figur 5.1: Th two-cll problm Not that Proposition 4.1 and Thorm 4. do not imply xistnc of th limits lim 0 K m = K m. (4.10) Th proof of (4.10) rquirs a sparat argumnt basd on th analysis of th asymptotic limit 0 in th nxt two Sctions. W will prsnt first th asymptotic analysis, and thn rturn to th proof of (4.10) at th nd of Sction 6. 5 Th asymptotic problm It turns out that in th limit 0 th asymptotic bhavior of th solution to th advction-diffusion problm may b dscribd in trms of a modl that is ssntially a systm of on-dimnsional hat quations on a graph. This sction is concrnd with th construction of this modl. 5.1 Th two-cll cas W dscrib th asymptotic problm first on th simplst xampl of a domain that consists of two clls C 1 and C dpictd in Figur 5.1. W dnot by j0 = C j, j = 1,, th part of th boundary of along th cll C j and by 1 th common dg of th two clls. W also introduc th boundary layr coordinats h and θ 1, θ j0, j = 1,. Th coordinat θ 1 rprsnts paramtrization along th dg 1 = {h = 0} {0 θ 1 l 1 }, whil th coordinats θ j0 paramtriz along th boundaris j0 = {h = 0} {l 1 θ j0 l j0 }. W first solv th hat quation along 1 : f 1 θ 1 = f 1 h, h [ N, N], 0 θ 1 l 1 (5.1) with a prscribd initial data f1 0 and th Numann boundary conditions at h = ±N: f 1 (θ 1, ±N) h = 0. (5.) Thn w solv two half-spac problms along th outr boundaris j0 with th prscribd Dirichlt data that coms from (1.): f 10 θ 10 = f 10 h, N h 0, l 1 θ 10 l 10 (5.3) 1

22 1 hat h<0 10 glu hat hat h>0 0 Figur 5.: Th gluing procdur and f 0 θ 0 = f 0 h, 0 h N, l 1 θ 0 l 0 (5.4) with th Numann boundary condition (5.) at h = N, and h = N, rspctivly, and with th Dirichlt data f j0 (θ j0, 0) = T 0 (θ j0 ) at h = 0. Th initial data for (5.3) and (5.4) coms from (5.1): f 10 (l 1, h) = f 1 (l 1, h), N h 0, (5.5) f 0 (l 1, h) = f 1 (l 1, h), 0 h N. Finally w glu togthr th functions f 10 (l 10, h), h 0 and f 0 (l 0, h), h 0: f g 1 (h) = { f10 (l 10, h), N h 0 f 0 (l 0, h), 0 h N (5.6) Th asymptotic problm is to construct a priodic solution of th abov, that is, find a function f1 0 (h) so that f 1 0 (h) = f g 1 (h), h [ N, N]. This problm is dscribd schmatically in Figur 5.. Proposition 5.1 Thr xists a uniqu function f 0 1 L ( N, N) such that f 0 1 = f g 1. Proof. Lt us dfin th oprator L 1 : L ( N, N) L ( N, N) by L 1 : f1 0 f 1(l 1 ), that is, th solution oprator of (5.1). Th oprator L 1 is boundd and compact, sinc f 1 (l 1 ) H 1 ( N,N) C f1 0 L ( N,N). W also lt L 10 and L 0 b solution oprators for (5.3) and (5.4), rspctivly with homognous boundary data T 0 = 0. Th oprators R ± rstrict a function dfind on [ N, N] to th positiv and ngativ smi-axs, rspctivly, whil th gluing oprator G glus togthr two functions dfind on thos axs: { f (h), h 0, G[f, f + ](h) = f + (h), h > 0, as in (5.6). W dnot by g(h) th function obtaind by solving (5.1) (5.6) with f1 0 = 0 and inhomognous boundary conditions. Thn quation f1 0 = f g 1 is quivalnt to: G(L 10 R L 1 f 0 1, L 0 R + L 1 f 0 1) + g = f 0 1, (5.7)

23 or Kf 0 1 f 0 1 = g, Kf 0 1 = G(L 10 R L 1 f 0 1, L 0 R + L 1 f 0 1). (5.8) Th oprator K is a compact oprator on L ( N, N). Furthrmor, w hav L 10 L L < 1 and L 0 L L < 1, whil L 1 L L = 1. This implis asily that K L L < 1 so that solution of (5.8) xists and is uniqu by th Frdholm altrnativ sinc K is compact. An altrnativ approach to th proof of xistnc of a priodic solution of (5.1)-(5.6), that is somwhat lss transparnt in th two-cll cas but is asir to gnraliz to th cas of N clls is as follows. W introduc an oprator L = L 1 L 10 L 0 dfind on L (R) L (R ) L (R + ) as f 1 L 1 f 1 L f 10 = L 10 f 10. f 0 L 0 f 0 W also dfin a r-distribution oprator R on th sam spac L (R) L (R ) L (R + ) as f 1 G[f 10, f 0 ] R f 10 = f 0 R f 1 R + f 1. Thn w may r-writ (5.7) as f1 0 (h) g(h) f1 0 (h) RL f 10 (l 1, h) + 0 = f 10 (l 1, h). (5.9) f 0 (l 1, h) 0 f 0 (l 1, h) In a sns, (5.9) viws (5.1)-(5.6) as a boundary valu problm whil (5.7) trats it is a priodic in tim solution. Th oprator Q = RL is compact sinc L is compact. Obsrv that Q may b writtn as f 1 G[L 10 f 10, L 0 f 0 ] G[L 10 (R (L 1 f 1 )), L 0 (R + (L 1 f 1 ))] Q f 10 = RL f 0 R (L 1 f 1 ) R + (L 1 f 1 ) = R (L 1 (G[L 10 f 10, L 0 f 0 ])) R + (L 1 (G[L 10 f 10, L 0 f 0 ])). (5.10) Th norms L 10 L L and L 0 L L ar both lss than on, as w hav notd bfor. This implis immdiatly that Q < 1 and thus (5.9) has a uniqu solution by th Frdholm altrnativ. This approach has a straightforward gnralization to th cas of mor than two clls. 5. Th gnral N-cll cas W now considr th gnral cas whn th domain consists of a finit numbr of clls. Th asymptotic modl is dscribd in trms of an orintd graph constructd using th stram function H as shown on Figurs 5.3 and 5.4. Th vrtics of this graph ar associatd with th saddl points of H. Th dgs ij of th graph ar associatd with th sparatrics of th th stram function. Th dirction of an dg is dtrmind by th dirction of th vlocity fild on th corrsponding sparatrix. Th lngth of an dg is dtrmind by th lngth of th sparatrix in th boundary layr coordinat θ associatd with H. Th boundary dgs ar thos that ar associatd with th sparatrics at th boundary of th domain. Th clls C i ar quadrangls boundd by minimal cycls of th graph. Th intrior dgs (drawn as solid arrows on Figur 5.4) ar indxd so that a common dg of two clls C i and C j is dnotd by ij. Th boundary dgs (drawn as dottd arrows on Figur 5.4) ar indxd so that th outr part of a boundary cll C i is dnotd by i0. Th boundary valu problm is: 3

24 80 10 c c 81 c 9 1 Figur 5.3: Th vlocity profil c 7 c c c c c c c 1 c Figur 5.4: Th graph [i] Givn th valus of th tmpratur T 0 on th boundary dgs i0, dtrmin th valus of th tmpratur f ij on all th dgs. Not that th valu of f ij may vary along ach dg. 4

25 [ii] Givn th valus of f on all th dgs, find th solutions f i of th Childrss problm for ach cll C i : f i f h i θ = 0, h [0, N], θ ], + [, f i (h = 0, θ) = f ik (θ), (5.11) f i h (h = N, θ) = 0, f i (h, θ) = f(h, θ + l i ), whr th indx k taks four valus of th adjacnt clls, l i = l ik1 + + l ik4 is th lngth in θ of th four dgs ik1,... ik4, bounding C i and f ik (θ) = f ik1 (θ),..., f ik (θ) = f ik4 (θ) ar th valus of th tmpratur on rspctiv dgs. [iii] Whn any two clls C i and C j shar a common dg, th normal drivativs from th lft and from th right match point-wis on this dg: f i + f j = 0 on ij. h h=0 h h=0 Thorm 5. Thr xists a uniqu solution of th boundary valu problm [i],[ii],[iii]. Proof. Th proof gnralizs th construction in two-cll cas considrd in Proposition 5.1 to th gnral situation in a fairly straightforward albit somwhat tdious mannr. Assum that a solution to th boundary valu problm [i],[ii],[iii] is found. Thn th solutions f i and f j on two adjacnt clls C i and C j ar such that thy can b glud togthr into on function f ij (θ, h), h [ N, N], θ [0, l ij ] so that (possibly aftr an appropriat shift of θ by a constant) f ij (θ, h) = f i (θ, h) for h > 0, andf ij (θ, h) = f j (θ, h) for h 0. Th function f ij satisfis th hat quation f ij h f ij θ = 0, f ij (h = ±N, θ) = 0 h (5.1) on (h, θ) [ N, N] [0, l ij ]. Equation (5.1) can b solvd uniquly as a Cauchy problm, providd that th initial data f 0 ij(h) = f ij (h, θ = 0) (5.13) is givn. Thrfor, w may dfin a linar oprator L ij : f 0 ij(h) f 1 ij(h), which maps th function fij 0 (h), assignd to th bginning of an intrior dg ij, to its valu fij 1 (h) = f ij(l ij, h) at th nd of this dg by solving th hat quation (5.1),(5.13). For boundary dgs th oprator L i0 and, hnc, fi0 1 (h) ar dfind by solving th homognous hat quation in half-spac: fi0 h f i0 θ = 0, f i0 (h = N, θ) = 0, h f i0 (h = 0, θ) = 0, f i0 (h = 0, θ) = f 0 i0(h), (5.14) 5

26 on (h, θ) [0, N] [0, l i0 ]. W dnot by g i0 (h) h [0, + ) solutions of th inhomognous hat quation along th boundary dg i0 g i0 h g i0 g i0 θ = 0, (h = N, θ) = 0, h g i0 (h = 0, θ) = f i0 (θ), g i0 (h = 0, θ) = 0, (5.15) on (h, θ) [0, N] [0, l i0 ]. Hnc, if f solvs th boundary valu problm [i],[ii],[iii], thn th corrsponding vctor-valud function f 0 = (f10 0,..., f ij 0,..., f km 0 ) solvs RLf 0 + g = f 0, (5.16) similar to (5.9) whr g = (g 10, g 0, g 0,..., g m0, 0,..., 0) and L = L ij. Th first (non-zro) componnts of th vctor g (and thos of f) corrspond to th vrtics at th boundary whr th flow u is incoming: thr is only on such vrtx in th two-cll cas and hnc g has only on non-zro componnt in (5.9). Th oprator R R : f 1 f 0 is a linar rdistribution oprator. Givn th valus fij 1 at th nds of th dgs th oprator R constructs th valus fi 0 j at th bginnings of th dgs at ach vrtx in a natural way: f must b a continuous function in ach cll. Givn th problm (5.16) is solvd uniquly, th boundary valu problm [i],[ii],[iii] is quivalnt to (5.16) as both amount to solving th hat quations (5.1), (5.14), (5.15). Thrfor it rmains to show that (RL I)f 0 = g, (5.17) has a uniqu solution. Howvr, th uniqu solvability of (5.17) follows from th Frdholm altrnativ. Indd, th oprator R is clarly boundd on [L ([ N, N])] k (hr k is th numbr of dgs) by construction. Th oprator L is compact on [L ([ N, N])] k for th sam rason as in th cas of two clls; it is associatd with th solution of th hat quation. Morovr, λ = 1 is not an ignvalu of th compact oprator RL. Indd, ach boundary oprator L i0 has norm lss than on: L i0 < 1. Thrfor, if w lt M b th total numbr of dgs, w hav (RL) M < 1 and thus RL may not hav ignvalu qual to on. 6 Approximation by th asymptotic problm W now compar th function φ N, solution of th approximat watr-pip problm (4.4), to th strtchd asymptotic boundary layr solution f (x, y) = f(h(x)/, θ(x)). Hr f(h, θ) is th uniqu solution of th Childrss problms dscribd in Sction 5. and Thorm 5.. Th function f(h, θ) is smooth xcpt at th points (h = 0, θ jk ) that corrspond to saddl points of th stram function H, whr f is discontinuous. This ncssitats a carful local analysis nar th cornrs. W build our approximation as clos to th Chilcrss solution f away from th cornrs at distancs largr than M 1/4 with M N. W will us an orthogonal systm (h = H/, θ jk ) along ach dg jk that sparats clls C j and C k, and at indicatd distancs away from th cornrs. Howvr, a diffrnt coordinat systm and a diffrnt approximation ar ndd nar th cornrs. W bgin with th introduction of suitabl local coordinat systms. 6

27 6.1 Th local coordinats Obsrv that th advction-diffusion quation (1.1) has th following form in an orthogonal systm of coordinats of th form (h = H/, θ): H f h + H f f + θ h θ + θ f θ J f θ = 0 (6.1) with J = H θ = H θ. Thrfor, in ordr to hav at last a formal approximation of (6.1) by (5.11) as 0 w should hav J H, or, quivalntly, H θ in th boundary layr H N. W impos th condition H = θ jk along th dg jk. Howvr, th coordinat θ jk introducd in such way may hav a singularity at th nd-points of jk. Thrfor w will us ths coordinats only away from th cornrs. In ordr to prform a local analysis nar th cornrs w may introduc th local orthogonal coordinats (X, Y ) in a δ-nighborhood of a cornr that w fix at x = 0, so that nar th saddl point w hav H = X ky. (6.) Morovr, w may assum that th chang of variabls satisfis D x X = U + O(x), x X = O(x) (6.3) with U a unitary matrix. Such chang of coordinats always xists according to th Mors lmma in a ball x δ nar th saddl point with δ > 0 sufficintly small. W may assum without loss of gnrality that th constant k 1. Thn th sparatrics ar givn by X = ± ky in th variabls (X, Y ). In ordr to simplify th notation w will assum that actually at th cornr th function H has th form (6.) in th old coordinat systm (x, y) and no chang of variabls is rquird. Extnsion to th gnral cas using th coordinats (X, Y ) is straightforward, with th hlp of th stimats (6.3), at th xpns of slightly lngthir calculations. W omit thm for th sak of radability. Undr our assumptions, th coordinat θ, orthogonal to H, is dfind along th whol dg jk, and is givn xplicitly nar th cornr by θ = B k (x k y) k+1. Th normalizing constant is chosn to b B k = (k + 1)k (k 1)/((k+1)). It is fixd by th rquirmnt that w hav θ = H along th sparatrics x = k y. With such a choic of B k w obtain ( ) k 1 x k+1 θ = (ky, x). (6.4) ky W will us th following thr rgions insid th boundary layr (s Figur 6.1): { I = (x, y) N : θ(x, y) M } k is th rgion around th cornr. Th rgion { II = (x, y) N : M k θ(x, y) 4M } k (6.5) (6.6) is th nxt closst, and III = { (x, y) N : 4M } k θ(x, y) (6.7) 7

28 y M 1/4 M 1/4 x k y = N 1/ rgion II rgion I x rgion III Figur 6.1: Th rgions nar th cornr is th farthst from th cornr. Rgion III xtnds all th way to th adjacnt cornr along th dg. Th constant k is includd for convninc in th dfinition of ths rgions, bcaus θ = (k + 1) (x k ) k+1 ky k k(x + y ) (6.8) insid th boundary layr { H N }, as x ky. Hnc th boundaris of th thr rgions ar approximatly parts of th circls: x + y M 1/4 and x + y M 1/4. W now show that for distancs largr than M 1/4 away from th cornr insid th boundary layr (rgions II and III) th dsird approximation J = θ H H is valid. An lmntary gomtric calculation shows that in rgion II III w hav x ky 1 Ch M h C h, h N, (6.9) M as M N. Combining th last inquality with (6.4), and using th form (6.) of th stram function H nar th cornr w hav H J C ( x + k y ) h M. (6.10) Similarly, w hav that θ is uniformly boundd in th sam rgion (rgions II and III): Obsrv also th following uniform bounds: θ C N M. (6.11) J H θ C h M, H θ C, J θ C (6.1) that w will nd latr. Hr θ = 0 is th coordinat of th saddl point. Indd, inqualitis (6.1) ar trivially tru, whn θ > δ. In th δ-nighborhood of th saddl point w hav (6.1) by using 8

29 (6.10) and θ > C(x + y ) in th boundary layr. Not that ths stimats may not b pushd all th way to th cornr x = 0, that is, insid rgion I, as (6.9) braks down, and θ blows up at th saddl point xcpt in th spcial cas k = 1. This is anothr rason why th Childrss solution may not b usd at th cornr. 6. Bounds for th Childrss solution W prsnt now som bounds for th Childrss solution. W may dcompos th function f at th cornr x jk into a smooth and a discontinuous componnt as f(θ jk, h) = f sm (θ jk, h) + B jk s(h), s(h) = { 0, for h 0, 1, for h > 1. (6.13) With th convntion of Sction 6.1 w hav θ jk = 0. Hr s(h) is th Havisid function, B jk is th magnitud of th jump of f that appars bcaus of gluing togthr of two solutions that com from diffrnt clls, and f sm is a smooth function, xcpt for th cornrs, whr f sm is continuous. Hnc N N ( fsm h ) dh C. (6.14) Th function f solvs th boundary valu Childrss problm insid ach cll, hnc f ij convrgs xponntially to th corrsponding constants K i and K j away from th sparatrix f ij (h) K i xp( c h ), h 0, f ij (h) K j xp( c h ), h 0. Dcomposition (6.13) implis that f satisfis th following bounds: f θ C θ, f θ C θ. (6.15) Ths stimats follow from th xplicit xprssion for th solution of th hat quation on th intrval N h N with th Numann boundary conditions at h = ±N, and with th initial data f(h, 0) as in (6.13). W can also stimat in a similar fashion, for θ clos to zro, f(θ) f(0) L ( N,N) C θ, (6.16) whr th main contribution coms from th discontinuous part of f in (6.13). Similar considrations lad to a bttr bound for f sm : for all h ( N, N). 6.3 Th approximat solution f sm (θ, h) f o (h) C θ, whr f o (h) = f sm (h, 0). (6.17) Th approximation to th solution of th full problm is constructd as follows. Lt χ b a smooth cut-off function such that χ(r) = 0 for 0 r 1 and χ(r) = 1 for r 4. W dnot by x jk th saddl points of H and lt φ,app N (x) = j,k Φ (x) = φ,app N (x) φ N(x). ( ) θ(x) f (x)χ M 1/ + j,k 9 [ ( )] θ(x) 1 χ f M 1/ ij(x), (6.18)