Asymptotic behaviour of solutions to the coagulati

AsymptoticBehaviourofSolutionstotheCoagulation-FragmentationEquations.II.WeakFragmentation1J.CarrF.P.daCosta2,3Heriot-WattUniversityDepartmentofMathematicsEdinburghEH144ASScotland,UKDedicatedtoOliverPenroseontheoccasionofhissixtyfifthbirthday1PublishedinJournalofStatisticalPhysics,77,89-123(1994)2Supportedbygrant9/90/BfromFunda¸c˜aoCalousteGulbenkian3Presentaddress:InstitutoSuperiorT´ecnico,DepartamentodeMatem´atica,Av.RoviscoPais,P-1096Lisboa,Portugal1IntroductionThediscretecoagulation-fragmentationequationsdescribethekineticsofclustergrowthinwhichclusterscancoagulateviabinaryinteractionstoformlargerclustersorfragmenttoformsmallerones.Denotingbycj(t)≥0theconcentrationofaclusterwithjparticles(j-cluster,forshort),theequationsare˙cj=12j−1Xk=1Wj−k,k(c)−∞Xk=1Wj,k(c),(1.1)forj=1,2,...,whereWj,k(c)=aj,kcjck−bj,kcj+k,withc=(cj),andthecoagulationandthefragmentationratecoefficients,respectivelyaj,kandbj,k,arenonnegativeconstants,symmetricwithrespecttopermutationofthesubscripts.Fromphysicalconsiderations,itisonlyrelevanttoconsidersolutionsto(1.1)whicharenonnegativeandhavefinitedensityρ(t)=P∞j=1jcj(t).Thismotivatesthestudyofsolutionsto(1.1)intheBanachspaceoffinitedensitysequencesX={c=(cj):kckdef=∞Xj=1j|cj|∞}.(1.2)Becauseeachinteractionpreservesthenumberofparticles,weexpectthedensityρ(t)tobeaconservedquantity.However,forsomeratecoefficients,ρ(t)maynotbeconstant,[5,7].Throughoutthispaperweassumethataj,k≤Ka(j+k)(1.3)forallj,k≥1,andsomeconstantKa0.Thisconditionensurescoagulationdoesnotleadtothebreakdownofdensityconservation,[2].Equations(1.1),aswellassomeimportantspecialcasessuchastheSmoluchowski(bj,k≡0forallj,k)andtheBecker-D¨oringequations(aj,k=bj,k=0ifmin{j,k}1),havebeenthefocusofanumberofmathematicalpapersinrecentyears(see,eg,[1,2,3,4,6]).Inparticular,resultsconcerningexistence,uniquenessanddensityconservationofsolutionsof(1.1)wasprovedin[2]forcoagulationcoefficientssatisfying(1.3).Twoclassesoffragmentationcoefficientshavebeenidentified,eachleadingtodistinctivebehaviourofsolutions.Inarecentpaper,[4],Carrconsideredthefollowing‘strongfragmentation’condition,namelythatthereexistsaγ0suchthatforallm≥0,thereisaconstantC(m)0suchthat[(r−1)/2]Xj=1jmbj,r−j≥C(m)rγ+m(1.4)forallr≥3,where[x]denotestheintegerpartofx.Themainresultof[4]concernstheμthmomentSμ(t)ofasolution1Sμ(t)=∞Xj=1jμcj(t).(1.5)If(1.3)and(1.4)hold,thenifcisadensityconservingsolution,foreveryμ0,Sμ(t)∞forallt0.Thus,forthisclassofsolutions,strongfragmentationactsasasmoothingmechanism,inparticular,orbitsareprecompactinX.Itisessentialtorestricttheclassofsolutionswhen(1.4)holdssinceingeneral,solutionsarenotunique(seeexamplesin[2]).Thepre-compactnessoforbitsofdensityconservingsolutionsimply,inparticular,thateverysuchorbithasanon-emptyω-limitsetinX.Withsomeadditionalassumptions,themostimportantofwhichisthatadetailedbalanceconditionholds,namely,theexistenceofapositivesequence(Qj)withQ1=1andsuchthat,foralljandk,aj,kQjQk=bj,kQj+k,(1.6)theω-limitsetofanyorbitcanbeprovedtobeasingleequilibriumcρ0withdensityρ0=kc(0)k.ThisbehaviourcontrastsmarkedlywithwhathappensfortheBecker-D¨oringsystem[1,3,6]forwhichonecanhaveabehaviourthatcanbephysicallyinterpretedasadynamicphasetransition:theexistenceofacriticaldensityρs∈(0,∞)suchthatorbitsareprecompactinXifandonlyifρ0≤ρs.Underconvenientconditionsonthecoefficients,inparticularanappropriateversionof(1.6),thiscompactnessresultimpliesthatsolutionsconvergestronglyinXtoanequilibriumwithdensityρ0ifρ0≤ρs,andtheweak∗(componentwise),butnotstrong,convergencetoanequilibriumwithdensityρsifρ0ρs,theexcessdensityρ0−ρsbeing‘transfered’toinfinity.Thisbehaviourcanbephysicallyinterpretedasacondensationphenomena,[3].Inthispaperwemakeuseofa‘weakfragmentation’conditionthatallowustoproveresultssimilartotheonesdescribedabovefortheBecker-D¨oring.WesaytheweakfragmentationconditionholdsifthereexistsaconstantKf0suchthatforallr1,h(r)Xj=1jbj,r−j≤Kfr.(1.7)whereh(r)=[(r+1)/2].Withtheseconditionsitwasprovedin[2]thatallsolutionsconservedensity.Moreover,underaconditionslightlystrongerthan(1.7),uniquenesswasproved,[2].Bothclassesoffragmentationcoefficients,(1.4)and(1.7)arephysicallyimportant.Forexample,inunbranchedpolymericchains,forwhichtheprobabilityofbreakingabondbetweentwomonomericunitsisindependentofthesizesofboththeoriginalchainandtheresultingoneswehavebj,k≡b=(const)and(1.4)issatisfiedwithγ=1.Intheotherhand,incaseswherethesurfaceenergyoftheclusterplaysanimportantrˆole,wecanexpectthefragmentationcoefficientsbj,ktobeverysmallifbothjandkarelarge.Averysimple2exampleofratecoefficientsexhibitingthisbehaviourisbj,k=(jk)−β,whichsatisfy(1.7)ifβ−1.Anotheroneisbj,k=K(j+k)exp{λ[(j+k)μ−jμ−kμ]},whereK,λ0andμ∈(0,1)areconstants,[2].Thispapermakesfrequentuseofresultsfrom[2]butcanbereadindependentlyof[4].Itisorganizedasfollows:Insection2westatethebasichypothesisanddefinitionsandrecallsomegeneralresultsthatwillbeneededafterwards.Section3dealswiththebehaviourofthehighermoments(1.5)ofsolutionsto(1.1).Itisshownthat,if(1.7)holds,aninitially(t=0)infinitemomentstaysinfiniteforeveryfinitetime.Alsointhissectionwe