le Mahemacs,, 3, 7-78 h://xoorg/436/am336 Publshe Ole November (h://wwwscrporg/joural/am) The Dscree gglomerao Moel: Equvale Problems James L Moseley Wes Vrga Uversy, Morgaow, US Emal: moseley@mahwvueu Receve July, ; revse Ocober, ; accee Ocober 8, BSTRT hs aer we evelo equvale roblems for he Dscree gglomerao Moel he couous coex Keywors: gglomerao; oagulao; Smoluchowsk; Dffereal Equaos rouco gglomerao of arcles a flu evrome (eg, a chemcal reacor or he amoshere) s a egral ar of may usral rocesses (eg, Golberger []) a has bee he subjec of scefc vesgao (eg, Segell []) fuameal mahemacal roblem s he eermao of he umber of arcles of each arcle-ye as a fuco of me for a sysem of arcles ha may aggluae urg wo arcle collsos Lle aalycal work has bee oe for sysems where arcle-ye requres several varables Effors have focuse o arcle sze (or mass) Ths allows use of wha s ofe calle he coagulao equao whch has bee well sue aerosol research (Drake [3]) Orgal work o hs equao was oe by Smoluchowsk [4]) a s also referre o as Smoluchowsk s equao The agglomerao equao s erhas more escrve sce he erm coagulao mles a rocess carre ou ul solfcao whereas we focus o he agglomerao rocess; ha s, o he eermao of a me-varyg arcle-sze srbuo eve f coagulao s ever reache hs orgal work Smoluchowsk cosere he agglomerao equao a scree form Laer was cosere a couous form by Muller [5]) eher case, a al arcle-sze srbuo o secfy he al umber of arcles for each arcle sze s eee o comlee he al value roblem (VP) We refer o hese as he Dscree gglomerao Moel a he ouum gglomerao Moel resecvely Soluo of eher moel yels a uae arcle-sze srbuo gvg umber eses as me rogresses For varous coos, sues of hese a more geeral moels clue Morgaser [6], Melzak [7], Mcleo [8], Marcus [9], Whe [], Souge [], Trea [], Mc- Laughl, Lamb, a McBre [3], Moseley [4], a Moseley [5] Le R be he real umbers, R : s a fe, o fe, or sem-fe oe erval}, a for o,, R f; R;f s aalyc o, Rf: R: f s couously ffereal o,r f : R : f s couous o F, R f : R: f s a fuco o f s a subsace of a vecor sace B we wre vs B These fuco saces are vecor saces a, R vs, Rvs, Rvs F, R To evelo he scree moel, assume ha all arcles are a mulle of a arcle of smalles sze (volume), say v Thus a arcle mae u of smalles-sze arcles has sze v olymer chemsry, he arcle s calle a -mer The al me s o where s he larges me erval of eres We cae hs by he exee erval oao,,+ : a We also le o o o, o : secfe, we assume o N,,3, le (eher, R or, R Uless oherwse Now for each be a real-value fuco ) ha aroxmaes he umber of -mers he reacor a me Sce here are a fe umber of szes, ally, we ake he sae (or hase) sace o be R = a :a he al umber esy R R ssume s kow s me asses, arcles colle, aggluaos occur, a larger arcles resul The e rae of crease () wh me, /, s he rae of formao mus he rae of eleo (coservao of mass) For o, we coser as a ossble Σ sace (e, he esgae sace where we look for soluos) eher, R for he aalyc coex or, R for he couous coex where oyrgh ScRes M
73, R : (), R, R :, R vs, R :, R vs, R : F, R F vs Fucos,R are couous, bu fucos R are o as we have o esablshe a oology o R They are comoewse couous For R we may efe The ervaves / exs a are (,R) However, we ca o asser ha lm +h h h oology o R R a :a R be he se of fe Le,j,j,j marces The kerel (whch measures aheso or K = K, s a oubly fe sckess ),,j,j array of real-value fucos of me eher, j K, j, R, R K K : for all, j N, j (aalyc coex) or, j K, j, R, R K K : for all,j N, j (couous coex) s wh R, we esablsh o oology o R The resula Dscree gglomerao Moel or Dscree gglomerao Problem (DP) s a VP cossg of a fe sysem of Orary Dffereal Equaos (ODE s) each wh a al oo () ha may be wre scalar (comoewse) form as: ODE's Kj,j j jk,j j, j j (),, VP +, 's :, () + where for = he emy sum o he rgh ha se of () s assume o be zero The frs sum he scalar (comoewse) scree agglomerao Equao () s he (average) rae of formao of -mers by aggluaos of j-mers wh j-mers The / avos ouble coug The seco sum s he (average) rae of eleo of -mers by he aggluaos of -mers wh all arcle szes We moel a sochasc rocess as eermsc The hyscal sysem s ofe saoary so ha each K, j s me eee a he moel s sa o be auoomous a hyscal coex, we requre K, j,, a for However, we wll aress DP as a mahemacal roblem where we allow he al umber of arcles, he comoes of he kerel K, j, a he comoes of he soluo,, o be egave The hyscal coex wll be a secal case Smoluchowsk fou he hyscal coex ha whe K s a cosa, ha, j + k M, N M where k a M + uquely sasfes DP o s erval of valy,,,, f we assume, he,,,, V o, V The requreme o M (4) a he fe sum () movae coserao of he Baach saces : R vs R where (Mar [6, 3]) wh orm (a hece a merc a a oology) Equaly of wo vecors requres he merc (he orm of her fferece) o be zero Ths s equvale o boh vecors beg a beg comoewse equal f,, he,, (3) (4) + efes a orm o (Naylor a Sell [7, 58]) To sure ha M exss (eve for egave al coos), we wll requre M vs R so ha We are arcularly erese he me-varyg ker- K whch ees o me, bu o o el, j oyrgh ScRes M
74 arcle sze he couous coex where T, K,j R K M R K,R :K,, j he roblem arameers are,,,, R coex where,j he aalyc T, K, j,j, R :K, j =, R K M R K he roblem arameers are,,, o, R For ay kerel, soluo requres ha boh ses of () are couous he couous coex a aalyc he aalyc coex The h eleo coeffce assocae wh R s efe for- a he srbuo mally by he fe seres j j,j j j f, K ; K, N,,3, (5) The oly rec eeece of f, ;K hrough K f (5) coverges for all, he f, ;K mas f, ;K o s R, R o R We may vew as a fuco of a fe umber of real varables or as a fuco of me a a sze srbuo Regarless, f F, R, a we have cover gece, he comoso f, ;K mas o R mlc () s ha for soluo he couous coex, we mus have for all o,, ha f, ; K, R Tha s, DP requres us o frs f, R a, f, ;K such ha for all N exss (e, coverges) a efes a fuco,r f, ao,, R (he Σ sace) a sasfes () o a (), he solves DP o Ths formulao of DP oes o requre mahemacs beyo calculus a s ofe use by egeers a scess For DP wh a me varyg kerel, K, j, he aalyc coex, Moseley [4] esablshe ha he more geeral formula k + + M + M (6) where, sasfes DP uquely o s erval of valy V,,, o or he hyscal coex where, we have, V,,,, aga a requre M The formula (6) sasfes () o a () he couous coex as well where we ow allow, R However, sce (6) was o erve usg equvale equao oeraos, uqueess has o bee rove rgorously for K MT, R Uless oherwse sae, for he res of he aer, we focus o he couous coex Moseley [4] ve DP o several roblems whch coul be cosere searaely Uer cera coos, a reasoably comlcae chage of (boh he eee a eee) varables rasforms DP wh a me varyg kerel (Moseley, [4]) o aoher VP whch Moseley laer referre o as he Fuameal gglomerao Problem (FP) The soluo rocess for FP s fully ocumee Moseley [5] For FP, Moseley esablshe exsece a uqueess for boh he aalyc a couous coexs by usg a sequeal soluo To faclae furher rogress, hs aer we evelo equvale roblems for DP he couous coex alogs for he aalyc coex ca be obae To rearrage erms fe seres we wll ee a a a (7),j j,j j,j j j j f all sums exs, we a all of he elemes a j R wo ffere ways Sce we use, j hem ofe, we wll use o mea for all a o mea here exss (wh aologes o he logcas) f y = (), we use ay of, (), y() a ( ) o eoe he fuco lso, we eoe he resrco of a fuco o a smaller oma by he same symbol The coex wll make clear Mahemacal Problem Solvg Ofe, a mahemacal roblem s secfe by gvg a coo (or coos) (eg, a algebrac equao or a ODE wh a al coo) o elemes a Σ se (he esgae se where we look for soluos, eg, Ror, R ) f he se s a vecor sace, we say Σ sace roblem s (se-heorecally) well-ose f has exacly oe soluo s se ( hs aer, we wll o coser couy wh resec o roblem arameers) well-eveloe moel of yamcs usg a VP s well-ose (exacly oe eve haes) s mo- oyrgh ScRes M
75 elers, we exec our moels o be well-ose s mahemacas, we requre rgorous roof Ofe, we solve equaos by usg equvale equao oeraos o solae he ukow(s) Ths yels uqueess, a, as all ses are reversble, exsece (Squarg boh ses of a equao s o a equvale equao oerao a may lea o exraeous roos) For lear ODE s, we may guess he form of a soluo a rove exsece a uqueess by usg he lear heory For olear roblems, we may rove exsece by subsug back o he equao Uqueess he becomes a ssue Le B f a soluo s uque B, a s, he s uque f s he Σ se for he roblem a coas oly oe soluo, he he soluo s uque B Beg s a requreme for exsece he couous coex, for o,, we look for soluos o VP ODE y f, y (8) y y, he sace,r Thus, as s usually oe, we requre soluos o (8) o o oly exs, bu o also have couous ervaves We also requre f U, R where U R a he rage of y() s U for y() he sace Placg hese aoal cosras avos ealg wh ahology, bu arrows he sace where a kow soluo s o be show o be uque There may be (ahologcal) soluos o (8) where he ervave exss, bu s o couous lso, as s usually oe, we allow o vary f we show ha here exss a soluo for some, he we say ha we have local exsece o The larges o, where a soluo exss s he erval of valy for he soluo (e, he oma) We say ha we have show global exsece o f, gve o,, we rove ha here exss a soluo o (e, a soluo,r ) Suose a soluo o o, goes hrough he o where s sa o be locally uque a f here exss o, such ha s he oly soluo o s locally uque o f s locally uque a every o Obvously, f a soluo exss globally o o,, a s locally uque o, he s globally uque o Tha s, s he oly soluo he sace,r ) For DP he couous coex we sar wh he large sace, R a say ha sasfes () o f N, he com oso f, ;K exss (coverges) a s,r a () sasfes () o Sce comoso of couous fucos s couous, we exec f, ; K, R f some sese f, ;K from R o R gve by (5) s couous Bu we o o have a oology o R a hece o oe o R R sea of requrg N, f, ; K, R as a searae coo for so- luo, we may cororae o he Σ sace We refer o DP wh he Σ sace, R, R : N K R,,f, ;, as he Scalar Dscree gglomerao Problem (SDP) Obvously, hs may be formulae a aalyc coex as well Recallg he cosra M M, sea of R, we may choose he sae sace as vs R whch has a orm (a hece a merc a a oology) soluo o s he a me-varyg fe-mesoal sae vecor Laer we wll choose a arorae sace a wre DP vecor form We refer o hs formulao of DP as he Vecor Dscree gglomerao Problem (VDP) s wh SDP, VDP may be he couous or aalyc coex f SDP s well-ose, a s soluo s he (smaller) sace for VDP, he SDP a VDP are equvale exce for he sace where local uqueess s rove Tha s, by choosg a smaller sace, VDP requres rovg local uqueess a smaller sace ha oes SDP f we o o worry abou ahology, a reefe he sace for SDP o be he same as for VDP, he wo roblems are equvale The queso s: How o we choose a arorae (smaller) sace? Bu frs we coser a equvale scalar roblem a saces Equvale Scalar Problems ga assume for N ha f, ;K, R where fucos coverges Now efe he f, ; K (9) K j,j j j j K,j j j f, ; K f, ; K K K K () f, ; f, ; f, ; () whch also ma R o R For hese fucos, as wh f, ; K, he oly exlc eeece o s hrough K For, R we may ow wre () as he sysem of ODE s oyrgh ScRes M
76 oyrgh ScRes K f, ;,,,, + () N,,3, f he resrco of f, ;K o eoe by he same symbol) coverges, (whch we a s couous o wh resec o he orm f, ; K, R Tha s, oology, we wre R R ally, we assume f, ; K, R, f : f, s couous a f, ;K f, ;K Noe vesgae f, ;K, a f, ;K comoes of, f, ;K f, ;K wh a comoe of, a f, ;K jus he fferece of f, ;K a f, ;K s jus a fe sum volvg K,j () a s jus he rouc of s Theorem Le K, R a K R The f, ;K,, a f, ;K are all, R f, ;,, f, ;K Proof Sums, roucs, a comosos of couous fucos volvg l are couous Deale ε-δ roofs follow roofs a elemeary real aalyss course ll fucos ma o R We mus choose,, + suffcely small, so ha f, f, For examle, f, he he rojeco fuco f couous sce f, he s f f sasfes a Lschz coo (Barle [8, 6]) a hece s couous,r Sce s a cosa fuco o l e, s of, s, R We vesgae couy a ffereably l more eal he ex seco, R f he comoso Le f, ;K coverges a s couous o, we wre f, ; K, R Prevewg he ex seco, we efe he fuco saces,, : s couous a as he comoewse couous fucos ha have cooma l, a clam ha,,, R vs vs orollary Le, K R f, f, ;, K R a comosos f, ; K,f, ; K, f, ; K, a are all,r f, ;K, he he Proof Sums, roucs, a comosos of cou- ous fucos volvg are couous We ow show ha he couous coex f f, ; K, R, he SDP gve by () a () wh he Σ sace, R s equvale o he fe sysem of scalar (comoewse) Volera egral equaos where K f s, s ; s (3) = s a soluo o (3) f s he sace,, R, R : N,f, ; K, R a N, sasfes (6) (We requre, R a o jus ha he egral (6) exss) We refer o hs roblem as he egral Scalar Dscree gglomerao Problem (SDP) he couous coex formulao he aalyc coex ca also be esablshe Theorem 3 he couous coex, a srbuo s a soluo of SDP,, R f a oly f s a soluo of SDP,, R Proof Frs assume ha s a soluo of SDP,, R We have by he efo of a soluo of SDP, ha,, R, ha N, f, ; K, R, ha (3) s sasfe o, a ha () s sasfe Sce boh ses of (3) are couous, we may egrae from o o oba K c f s, s ; s s (4) lyg he al coo we oba (3) Sm- s a soluo larly, le us assume ha of (3),, R Sce,, R, we have N ha f, ; K, R so ha he egra, f s, s ;K egral, s ffereable so ha,, R Subsug we oba (), s couous Sce () s wre as a Dffereag we see ha () s sasfe For he scalar Equao (), s he egral formulao ha s use o oba exsece (Pcar eraos) a uqueess usg a Lschz coo f we choose o secfy,, R as he sace for boh rob- M
77 lems, he roblems rema equvale as ay soluo o (3), R s fac,, R Tha s, here are o soluos o (3), R,, R These resuls ca also be esablshe he aalyc coex ouy a Dffereably for l Saces Sce has a orm (a hece a merc) we have a oology o he subsace of R May of he lm laws ca be exee o For examle, f lm L a lm L, he lm LL We also have f lm f L a lm L, he lm f L L Defo fuco : s couous a wh resec o he orm oology f lm ; ha s, gve ε >, such ha mles f s co- uous, s couous o Smlarly, a fuco M : R s couous a wh lm M M ; resec o he orm oology f ha s, gve, such ha mles MM f s couous, s couous o Smlarly for he fucos f :, f, : R, a f, : Hece we ca efe he fuco saces, : s couous, RM : R M s couous as,,, R a, a as well B F, B F, For F, B, he rage s resrce o he se B whereas, for F,, s allowe o be he larger se Sce R,, vs, R However, has a orm (a hece a merc a a oology), bu R oes o (We coul esablsh a oology for R, bu hs s o ecessary f he sysem saes are all ) We wll use, for fucos ha are comoewse couous wh cooma a wre f, he we may assume,, R,, R,,, R, f, a f lso, f B f, f F B, ; ha s, we use he same symbol for he resrco of a fuco o a smaller oma We gve ecessary a suffce coos for o be,,,, R, we wre F f fol- Theorem 4 vs vs Proof We show ha,,,, he ous s, commes, vs, vs, R lows Le, a Tha s, f s comoewse cou- s a vecor sace, by our revous The lm Tha s, gve, such ha mles Sce, gve, such ha mles Hece lm R so ha, Hece,,, R Theorem 5 f, f N,, he, R, he M j, R,, he R M, j j f j Proof f, (or ay orme lear sace), he he ragle equaly mles so he orm fuco : R sasfes a Lschz coo o a hece s couous o e, s,r We say s Lschz couous o Now le, s he comoso of he Sce orm fuco wh, R For,, mles R, le oyrgh ScRes M
78 M Sce j j M, M j j j j exss (coverges absoluely) f, j j, he so ha M () s Lschz couous o so ha M, R Sce he comoso of couous fucos (o a ) s couous, M j, R j Examle Le, a for le f +3 ++3+ f +3 + f + + ++ + f + from The f (5), as each () s couous a, 3 However,, for lm bu, Hece lm eher oes o exs or s greaer he or equal o Hece o couous a as lm oes o exs Hece lm oes o exs Hece, Hece he relaos, vs, vs, R are roer Examle Le, The a for ay le for a oherwse for,, we have a Ob- vously, so, hough, R as, eve lhough o suffce vually for o be,, we ee s rage o be,, a, R of hese o force o be, : R, However, all a Theorem 6,, :, R Proof Le,,, R = for N for N N Sce,, a, N,, N lso,,,, R a N, N N+ N N N+ are all,r Le, The + N N+ N N+ N N N+ N+ Now le Sce, we ca choose N suffcely large so ha N+ 6 N, R N+, N N+ N+ mles Sce such ha N 6 so ha N+ N+ 6 3 Sce,, choose δ so ha mles N Hece N N N N = N Now choose m,,, N N Hece mles oyrgh ScRes M
79 N N N+ N+ 3 6 Hece, Raher ha check recly ha,, may be easer o check ha for each,,, R a, R sce a ma from o R Smlarly, orollary 7,, :, R f f, a,,, :,, f f R Followg he saar roof for roucs, we also have Theorem 8 f, R a, he,, Proof Le a hoose δ such ha mles a δ such ha mles Le m, The mles Smlarly, orollary 9 f, R a M, R he M, R f, R a f,, he f, f, R a f,,, he f,,, We say, s ffereable (wh resec o he orm oology) a, f lm exss f exss a s,, he, We efe egrao comoewse Followg Theorem 6, we have,, he Theorem f, lso,, =, :, R Proof Tha follows from co serg he lm for comoes roof of, ca be obae followg he roof for scalar value fucos calculus books (eg, Sewar [9, 88]) The escro of, follows from Theorem 6 f a, () has a fe umber of ervaves k a equals s Taylor seres, k k! a eghborhoo of, s aalyc a f s aalyc, he, Theorem, vs,, vs, vs, R, vs, vs, R, vs, vs, R, vs, vs, R,,,, a,,, vs vs Proof The frs coame follows from Theorem The remag roofs are sragh forwar a ofe smlar o he roof of Theorem 4 Theorem (Fuameal Theorem of alculus) f,, he s ss, Par (6) s f,, he s s s Par (7) s s oyrgh ScRes M
7 Noe ha he efe egral requres a arbrary cosa vecor 3 Kerels, Sae Saces a Σ Saces he aalyc coex wh a aalyc kerel, K MT, R, Moseley [4] use he followg roceure o solve DP He frs esablshe local u- queess, R by coserg he Taylor seres coeffces obae from he al coos a he ffereal equao However, he chose a smaller Σ sace coag oly srbuos where f s he Σ sace, he N, f, : K, R He he obae he exlc formula (6) for he (aalyc) soluo whe () s aalyc He o rgorously solae he ukow so he esablshe global exsece by showg ha he soluo gve by he formula (6) was he Σ sace, checkg he al coos (), a he subsug he formula o () Sce global exsece hols, local uqueess mles global uqueess The roblem of eres s o exe Moseley s resuls for he aalyc coex o he couous coex The soluo gve by (6) remas he same exce ha we ow oly requre, R Global exsece may be obae as before However, local uqueess s o as easy as was he aalyc coex McLaughl, Lamb, a McBre [3] rove local exsece a uqueess for a ouum gglomerao Moel of lear fragmeao wh coagulao as a erurbao usg semgrou heory Souge [] rove a local exsece heorem he hyscal case, bu o uqueess The saar roceure Brauer a Noel [] for a fe mesoal sysem requres a Lschz coo o he rgh ha se o oba local exsece a local uqueess hs aer, we rove relmares for usg a Lschz coo o rove uqueess he couous coex by gvg equvale roblems scalar a vecor form for DP wh K() a larger col- M, R leco ha T K Le, ; f, ; K f, ;K coverges, f, ;K mas R o R resrco of) f, ;K (o f f N,, R, he We say ha (he ) s R f N, (he resrco of) f, ;K (o f,; K, R, R, we wre ) s, R a wre Furhermore, whe f, ; K f, ; K, R f N,f, ; K, R Theorem 3 f,; K, R, f, ; K, R f, a, he Proof omosos of couous fucos ( ) f,; K, R a are couous so ha f,, he f, ; K, R (See orollary ) he couous coex we wsh coos o K so ha,; K, R f The for (ay subsace of),, ; K, R we have f The he covergece a couy coo o f, ; K ee o be exlcly sae for he Σ sace or as a coo for soluo (exce as requre for erreg ()) We beg M, R, wh hree classes of kerels: T MB R K,j W, K, R :, j K B cosa,so ha M such ha for all,, j, j Bmax, jn, B M a, j Bmax MTB, R K B, j, j W, R :, a B, R K =,j MB, j R Sce T, TB,, B TB,, K MTB R, we have,; K, R hree classes, f,, we have, : K, R M R M R a M R M R, f we ca rove ha for f, he for all kerels hese f However, for clary, we rocee class by class K, K M R a f, j T, j, he for all, j N we have K so ha, j where,j j j K f, ; K M M s he zeroh mome of he j j sequece he hyscal coex, so ha, oyrgh ScRes M
7 M s he oal umber of arcles a M v s he oal mass of he arcles (whch shoul o M j chage) where s he frs mome j j of he soluo a ρ s he mass esy Trea [] suggese (as have ohers) o a hyscal bass, ha hese a oher momes, ossbly all momes, shoul exs (coverge a be couous) We wll ake our Σ sace as a subsace of,,, R ga, we vew a soluo as a me-varyg fe-mesoal sae vecor K M, R so ha Theorem 4 Le T N, f, ; K M he M f j j, exss (coverges absoluely) a R M, f, f, ;K exss (coverges absoluely) a f,; K, R f,, he M, R a f, : K, R, he Proof Le K MT, R so ha K f, ; M By Theorem 5, f R, he M j j N, exss (coverges absoluely) a M j, R j,, he f, ; K M M a f so ha f, ;K exss (coverges absoluely) By orollary f, ; K M, R Now le 9,, By Theorem 6, R so ha exss (coverges absoluely) Sce M a K, M K f, ; M a f, ; exs (coverge absoluely) s comosos a roucs of couous fucos ( ) are couous, M, a f, ; K M are,r K M, R The Theorem 5 Le B,, f, ; K exss (coverges absoluely), R,, he a s f, : K, R f Proof Le B,j B, K M R (Noe,j ha K s a cosa fuco of for hs kerel) The MBmax such ha, j N, B, j MBmax Le, The a = K,j j j f, ; B so ha, ;K le f B M, j j Bmax j j j M M Bmax j Bmax j f exss (coverges absoluely) Now B The a,j j j,j j,j j j j f f B B Hece B, j j j B, j j j j j M Bmax j j MBmax j f Sce K,R s Lschz couous a hece f, ; f B s a,j j j K,; K, R cosa fuco of me, f, ;, R Hece f f, he f, : K, R a f, : K, R Theorem 6 Le TB, K M R The,, f, ; K exss (coverges absoluely) a f,; K, R f,, he f, : K, R Proof Le B a K,j MTB, R,j B,j, K M R The,j B K,j j j f, ; B B, j j f, ; K j where, R a oyrgh ScRes M
7 K R f, ; B, (by Theorem,j j j 5) By orollary 9 f, ; K f, ; K, R f, f, : K, R, he 4 Weersrass M-Tes a Local Uform Boueess Le c J : J s a close fe erval c, J c : J For J c J, f : J : f s couous a, le R R We brefly revew he Weersrass M-es a succeeg heorems o absolue a uform covergece (Kala [, 436-444]) Ths shoul be famlar o egeers a scess We he coser a fourh class of kerels Le M TB, R K,j W K, R :,j a J,, M J such ha c Kmax, j N,max K M J J, j Kmax Noe TB, TBS, M R M R (le K B M J max M ) a, j, j Kmax J Bmax We say ha K() s locally uformly boue me a sze Theorem 7 (Weersrass M-Tes Exee) Le Suose,, R a M J c,, Mmax,J a such ha J, M,J max max M,J M J M max, he a exs (coverge absoluely J a are uformly coverge o J so ha heyare J,R Sce was arb- rary,, so, a M a are, R so ha,, R a f J c,, Mmax,J such ha J, Mmax max,j a max M,J M J, he a exs (coverge absoluely) J a are uformly coverge o J so ha hey are (J,R) Sce was arbrary,, so, a a, lso, are M,R so ha For TBS,, ; K, R J, R K M R, o sure f, we wll requre o sasfy a sroger local uform boueess coo me Defo Le, R The s locally uformly boue a f J,, M,J, such ha J, c max max a Mmax,J Mmax J M,J We say s locally uformly boue o f s locally uformly boue Now le, R, R : s locally uformly, boue o R W R,, : s locally, uformly boue o a R W R,, : s locally uformly boue o Moseley (7) use, R (whch he eoe by, DUBS R ) as he Σ sace he aalyc coex We have, R vs, R vs, R oyrgh ScRes M
73 Examle 3 Le c a, wh a () creasg Now for, le J a,b, M,J max b The max J Mmax,J b Hece, R ca be show (smlar o Moseley [4] he aalyc coex) ha he couous (hyscal) coex, where V,,,, R, V,,, where Theorem 8 Le s gve by (6) s, R, R The a M are (,R) a, f, R,, M, R a M j j Proof Le he a, R The J,, M,J J, M,J a such ha c max max Mmax,J Mmax J Hece J so ha J,, By Theorem 7, a M are (J,R) Sce was arbrary, a M are (,R) Hece by Theorem 6,,, R f, he, aga by Theorem 7,, R M =, R 6,, a so a Hece by Theorem lm + h h j h j Sm- Sce he rage of fucos, R ae, we have, R, larly,, R, orollary 9 lso, s co-, R,,,, R vs vs vs, R,,,, R vs vs vs Proof By Theorem 8,,, R Everyhg else s sragh forwar or follows a maer smlar o he roof of Theorem 4 We show ha f K MTBS, R a R, he f, ; K, R We, use he local uform boueess of Le a, R F = f,, R : R for fxe, f,,, f, F, R : for fxe, f,, The, R a, or saces a K a are vec-, vs, vs, vs, R Theorem Le K, K M R For N, a, j TBS, j,, f, ;K K exss (coverges absoluely) a f,;, R f, R, he f, ; K, R Proof Le K, j TBS, K M R, j The, a J c,, MKmax J for all, j N a such ha J, K M J Le, j Kmax oyrgh ScRes M
74 N a, J where K,j j,j j j j f, ; K K M J M J Kmax j Kmax j The (8) K,j j,j j j j f, ; K K M J M J Kmax j Kmax j so ha, ;K, J ) Sce was arbrary f, ;K for, lso, sce K M a f exss (coverges absoluely for, j j Kmax j Kmax j Kmax j Kmax j j exss M M M, by Theo- rem 7 we have ha N a fxe, f, ; K, R Hece,; f K, R However, we o o have,; K, R f We may (or may o) be able o rove hs wh a furher exeso of he Wersrauss M-es sea we le () j, R The for a J j c,, o oly o we have MKmax for all, j N a also M j,j max j j J such ha J, K M J bu such ha J, M j,j a j a max max M j,j M J max, j Kmax The K K M J, j j, j j Kmax j M J M j,j Kmax max M J M j,j M J M j,j Kmax max Kmax max j j Kmax max M J M J Hece by Theorem 7, Hece K R, ; K, R f Thus f TBS, subsace of R f, ;, K M R a we choose a as our Σ sace, we obvae he ee o exlcly requre, ; K, R f as a coo for o be a soluo (exce o erre ()) or as a secfc coo for he Σ sace 5 Equvale Vecor Problems Recall ha f f, ;K coverges, R he fucos f, ;K, f, ;K, a f, ;K all ma R o R a ha, ; K f, ; K f mas le, ; K f, ; K f,, ; K f, ; K f, a, ; K f, ; K R o R s wh, ;K R o R Now f These hree fucos ma f, he oly exlc e- K, R eece o s hrough K() f a f,; K, R, he f K,;, f K,;, a f,; K are all, R (see Theorem ) Now le, K MTB R The by Theorem 6, f,; K, R f we ca show ha, mles, ;K f, ;K, a f, ;K are, he hese fucos ca be hough of as fucos from sea of from R f, o o R We ee show ha f we resrc f, ;K o of f, ;K, f, ;K, a, ;K, he he resrcos f o all ma o so hese fucos are all, Le, + f f,, : for fxe,,, R The,,; K, R K,;,; f,; K are all +, We have, vs +, vs, vs -, vs, R Theorem Le K TBS, R s a vecor sace We show ha f f, he f, K a f, The, for he mages f, ;K,, ;K, f, oyrgh ScRes M
75 a, ;K f are all,; f K, R, lso,,; K +,,; K,,; K,, TB,; K, R f K,;, K,; f K,; are all +, Proof Le a TBS, f, a f f f K M R, he f, a f, a K M R The Jc,, MKmax J such ha, j K M J Hece for, J, j Kmax K, from () we have f, ; M J so ha Kmax a where K K f, ; f, ; M J Kmax f, ; K K K J f, ; f, ; M J M J Kmax Kmax M Kmax Sce was arbrary,,, ;K, f s Furhermore, sce f, ; K f, ; K M J a Kmax Kmax Kmax, M J M J, by Theorem 7, for fxe, ; K f, ; K f K f, ;, Hece f,;k, By Theorem,,; K -, sce f lso, f, ; K K j,j j j j j K j,j j j we have, ; K f, ; K f j j j K j,j j j K, j j M J Kmax j M J Kmax j j M J Kmax where we have use (7) Hece f,; K, Sce a f, ; K K j,j j j j j K, ; K f, ; K f we have for fxe, j j j j,j j j K j,j j j K j,j j M J Kmax j M J M J Kmax j j Kmax, ; K f, ; K, f Hece f,; K +, Sce, vecor sace (oe, ; K, ; K, ; K f f f 3 M J Kmax we have ha,; K, Now le K TB, R f s a ), The by Theorem oyrgh ScRes M
76 6,,; K, R f By usg Theorem a above, f K,;, f K,; a f,; K are +, Uforuaely, we have o rove ha f,; K, However, assumg,; K, f we coser he Vecor Prob- lem (VP): Vecor ODE, VP f, ; K,,, + (9),,, () + where he ervave a equaly are Tha s, we ow requre he ervave o be efe wh resec o he orm oology, lm + h h h, a equaly as equaly Σ sace as, vs, For,; K, For VP, we ake our f we ow show ha VP s equvale o where for + f s, s ; K s (), (he Σ sace) o be a soluo of (), we requre N, ha () hols; ha s, egrao s comoewse Equaly s We refer o hs roblem as he egral Vecor Problem (VP) Theorem The srbuo soluo of VP, of VP, s a f a oly f s a soluo Proof For boh roblems we have chose he Σ sace Now assume ha o be, s a soluo of VP so ha (9) a () are sasfe, a he rgh ha se of (9) s, We may egrae from o usg (7) o oba he vecor equao s s s s s (9) c f s, s ; K s lyg he al coo we oba () Now assume ha s a soluo of () The subsue o oba () Sce, ; K, R W, R f, a + f s, s ; K s,, ffereag (comoewse) we have ha a ha (9) hols K M, R, he Theorem 3 f TB f,; K, R a f,; K +, f,, he f, ; K, R a f, ; K W, O he oher ha, f MTBS, f,; K, R a f,; K, f, ao,, f, ; K, R f, ; K W, Proof f MTB, f,; K, R By Theorem, f,; K +, f,, he K R a f, ; K, Now le MTBS, f,; K, R a f,; K, f, ao,, f, ; K, R f, ; K W, K R, he, he a f, ;, K R, he by Theorem 6, K R The by Theorem,, he a To efe VDP he couous coex, we woul,; f K, R a lke f,; K, The for, woul have f, ; K, R a f, ; K, Whe TB, we o have,; K, R show ha f,; K +, so ha for,, we have f, ; K, R a, ; K, wh Σ sace, as VDP Whe, we K M R, f, bu have oly f We refer o hs roblem oyrgh ScRes M
77 K MTB, R, we sele for f,; K, R a f,; K, so ha f, he f, ; K, R a, ; K, Σ sace, as VDP s, R, vs, vs, vs, R, f we ake, R,, f We refer o hs roblem wh as our Σ sace for SDP, SDP, a VDPor VDP, he hey are all equvale f hey have he same roblem arameers,,, K 3 Summary a Fuure Work For he me-varyg kerel ( K, j ) he aalyc coex, he roblem arameers are,,, o, R For hs roblem, Moseley [4] use he followg roblem solvg roceure He frs esablshe local uqueess, R However, he chose he smaller Σ sace, R coag oly srbuos where (for a me-varyg kerel) f s, R he eleo coeffces f, ; K f, ; K are, R, he, He he obae he exlc for- mula (6) for he (aalyc) soluo He o rgorously solae he ukow so he esablshe global exsece by showg ha he soluo gve by he formula (6) was he Σ sace, checkg he al coos, a he subsug he formula o () Sce global exsece hols, local uqueess he aalyc coex mles global uqueess f we choose, R as our Σ sace, he SDP, SDP, VDP, a VDP are all equvale he couous coex f hey have he same roblem arameers,,, K f K MTBS, R a, R have, ; K, R, we f, so ha we ee o secfy hs coo searaely For he me varyg kerel, he soluo gve by (6) s, R, where, R he couous coex However, we have o show (local) uqueess he couous coex To o hs we have (a leas) four choces: ) Prove a rgorous ervao of (6) ha roves (exsece a) uqueess ) Develo a use a Lschz coo for: f, ;K he scalar roblems SDP a SDP 3) Exe he (exsece a) uqueess resuls for FP he couous coex o oba a uque sequeal soluo o DP 4) Develo a use a Lschz coo for f, ;K he vecor roblems VDP a VDP We have rove relmares for he evelome of a Lschz coo for VDP a VDP However, all four aleraves aear o be worhwhle REFERENES [] W M Golberger, olleco of Fly sh a Self-gglomerag Fluze Be oal Burer, Proceegs of he SME ual Meeg, merca Socey of Mechacal Egeers, Psburg, 967, 6 [] J H Segell, Defluzao Pheomea Fluze Bes of Scky Parcles a Hgh Temeraures, PhD Thess, y Uversy of New York, New York, 976 [3] R L Drake, Geeral Mahemacs Survey of he oagulao Equao, : G M Hy a J R Brock, Es, Tocs urre erosol Research, Pergamo Press, New York, 97 [4] M Vo Smoluchowsk, Versuch Eer Mahemache Theore er Koagulaoskek Koller Lösuge, Zeschrf fuer Physkalsche heme, Vol 9, No, 97, 9-68 [5] H Müller, Zur llgemee Theore Ser Rasche Koagulao, Kollochemsche Behefe, Vol 7, No 6-, 98, 3-5 [6] D Morgaser, alycal Sues Relae o he Maxwell-Bolzma Equao, Joural of Raoal Mechacs a alyss, Vol 4, No 5, 955, 533-555 [7] Z Melzack, Scalar Trasor Equao, Trasacos of he merca Mahemacal Socey, Vol 85, No, 957, 547-56 o:9/s-9947-957-8788-6 [8] J B McLeo, O a Fe Se of Nolear Dffereal Equaos (), Quarerly Joural of Mahemacs, Vol 3, No, 96, 93-5 o:93/qmah/393 [9] Marcus, Uublshe Noes, Ra ororao, Saa Moca, 965 [] W H Whe, Global Exsece Theorem for Smoluchowsk s oagulao Equaos, Proceegs of he merca Mahemacal Socey, Vol 8, No, 98, 73-76 [] J L Souge, Exsece Theorem for he Dscree oagulao-fragmeao Equaos, Mahemacal Proceegs of he ambrge Phlosohcal Socey, Vol 96, No, 984, 35-357 o:7/s354653 oyrgh ScRes M
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