On the initial-value problem in the Lifshitz-Slyoz

arXiv:math/9808010v1[math.AP]3Aug1998Ontheinitial-valueproblemintheLifshitz-Slyozov-WagnertheoryofOstwaldripeningBarbaraNiethammerInst.f¨urAngew.Math.Universit¨atBonnWegelerstr.653115Bonn,GermanyRobertL.PegoDept.ofMathematics&Inst.Phys.Sci.Tech.UniversityofMarylandCollegePark,MD20742USAApril1998AbstractTheLSWtheoryofOstwaldripeningconcernsthetimeevolutionofthesizedistributionofadilutesystemofparticlesthatevolvebydiffusionalmasstransferwithacommonmeanfield.Weproveglobalexistence,uniquenessandcontinuousdependenceoninitialdataformeasure-valuedsolutionswithcompactsupportinparticlesize.Theseresultsareestablishedwithrespecttoanaturaltopologyonthespaceofsizedistributions,onegivenbytheWassersteinmetricwhichmeasuresthesmallestmaximumvolumechangerequiredtorearrangeonedistributionintoanother.1IntroductionTheclassicaltheoryofOstwaldripening,formulatedbyLifshitzandSlyozov[3]andWagner[5]concernstheevolutionofthesizedistributionofalargenumberofsmallparticlesofonephaseembeddedinamatrixofanotherphase.Particlesareassumedtobewidelyseparatedspheresthatevolvebydiffusionalmasstransferwithacommonmeanfield.Inthelatestagesofthephasetransformation,diffusionisquasi-steadyandtheparticlegrowthrateisdeterminedbythemassfluxattheparticleboundary.Themassfluxisproportionaltothegradientofapotentialthatisharmonic,isproportionaltocurvatureontheparticleboundaries,andisclosetoconstantinthemeanfieldbetweenparticles.Inappropriateunits,itisfoundthatanyparticleradiusR(t)evolvesaccordingtodRdt=V(R,Rc(t)):=aR2RRc(t)−1,(1)whereaisaconstantandthecriticalradiusRc(t)isthesameforallparticles.ThevalueofRc(t)isdeterminedfromconservationofmass.Ifmasschangesinthediffusionfieldcanbeneglected,theparticlevolumeisconservedandonefindsthatthecriticalradius1equalstheaverageradiusofcurrentlyexistingparticles.ParticleswithradiuslargerthanRc(t)aregrowing,andparticleswithsmallerradiusshrinkandcandisappearinfinitetime.Classically,thesizedistributionofparticlesisdescribedbyaparticleradiusdistribu-tionn(t,R).ThisisanormalizednumberdensitythatwemayscalesothatRR0n(t,r)dristhenumberof(currentlyexisting)particleswithradiuslessthanR,dividedbythenumberNofinitiallyexistingparticles.ThenumberofparticleswithsizebetweenR1(t)andR2(t)foranytwosolutionsof(1)isconserved,son(t,R)shouldsatisfytheconservationlaw∂tn+∂R(Vn)=0,(2)wherethecriticalradiusisgivenbyRc(t)=Z∞0Rn(t,R)dRZ∞0n(t,R)dR.(3)Theinitialnumberdensityn0(R)=n(0,R)satisfiesR∞0n0(R)dR=1inthisnormaliza-tion.Ouraiminthispaperistodevelopasatisfactorytheoryofwell-posednessfortheinitialvalueproblemfortheparticlesizedistribution.Fromthephysicalpointofview,itisreasonabletosupposethatapositivefractionoftheparticlescanhavethesameradius,inwhichcasethesizedistributioncontainsoneormoreDiracdeltas.Mathematically,theidealistoallowtheinitialdatan0(R)dRtobeanarbitraryprobabilitymeasuresuchthatthetotalvolumeR∞043πR3n0(R)dRisfinite.ItwillbeconvenienttoworkwithparticlevolumevinsteadofradiusR,andtoworkwithacumulativenumberdistributionfunctionϕinsteadofthenumberdensityn.Wesaythatϕisthefractionof(initiallyexisting)particleswithvolume≥v.(4)Asafunctionofvolumevattimet,ϕ(t,v)isamonotonicallydecreasingfunctionwhichisleftcontinuousatjumpswithϕ(t,0)=1,andR∞0ϕ(t,v)dv(thetotalvolume)isindependentoftime.Theparticlevolumedistribution,definedbyf(t,v)dv=−dϕ(t,v)foreachfixedt,isformallyrelatedtonviaf(t,v)dv=n(t,R)dR.Wenormalizethetimescalebythefactor4πaandletθ(t)=(4πRc(t)3/3)−1/3,sothatthevolumev(t)ofanyexistingparticleshouldsatisfydvdt=Λ(v,θ(t)):=v1/3θ(t)−1.(5)Ifv(t)isapositivesolutionof(5)onsometimeinterval,thenϕ(t,v(t))shouldremainconstant.Thismeansϕ(t,v)shouldbeasolutionofthehyperbolicequation∂tϕ+Λ(v,θ(t))∂vϕ=0,(6)whosecharacteristicssatisfy(5).Thevalueofθ(t)isobtainedfromϕintermsofRiemann-Stieltjesintegralsbyθ(t)=Z∞0+dϕ(t,v)Z∞0v1/3dϕ(t,v).(7)2Thenumeratoris−1timesthequantityϕ0(t):=limv→0ϕ(t,v),whichisthefractionofinitiallyexistingparticlesthatstillexistattimet.Itturnsouttobestillbettertoregardthevolumevasafunctionofthefractionϕ,0≤ϕ≤1.Wetakethemapϕ7→v(t,ϕ)toberightcontinuousanddecreasingwithv(t,1)=0.Mathematically,givenϕ(t,v)weobtainv(t,ϕ)viatheprescriptionv(t,x)=sup{y|ϕ(t,y)x}for0≤x1=maxϕ.(8)Thisismosteasilyunderstoodwhenthesizedistributioncorrespondstoafinitenumberofparticles.Ifwelisttheparticlevolumesindecreasingorder,v0(t)≥...≥vN−1(t),thenv(t,ϕ)=vjforϕ∈[j/N,(j+1)/N).Weshallcallϕ7→v(t,ϕ)avolumeorderingforthesystemattimet.Fortechnicalsimplicityweshallassumethattheparticlevolumesinthesystemarebounded.Thisseemsreasonablephysically,andcorrespondstoassumingthattheparticlevolumedistributionhascompactsupportinv.Wethenintroducefunctionspacesasfollows.Letrcd([0,1])bethesetoffunctionsv:[0,1]→Rthatarerightcontinuous,decreasing,andsatisfyv(1)=0.(Tobeprecise,wesayvisdecreasingifv(x1)≤v(x2)wheneverx1≥x2,andsimilarlyforincreasing.Adecreasingfunctionneednotbestrictlydecreasing.)Thesetrcd([0,1])iscontainedinthespacebdd([0,1])ofreal-valuedboundedfunctionson[0,1],equippedwiththesupnormkvk=supϕ|v(ϕ)|.rcd([0,1])isacompletemetricspaceintheinducedtopology.IfXisaBanachspaceandI⊂Risaninterval,t