Evaporation of Droplets in the Two-Dimensional Gin

EvaporationofDropletsintheTwo-DimensionalGinzburg-LandauEquationJ.RougemontD´epartementdePhysiqueTh´eorique,Universit´edeGen`eve,CH-1211Gen`eve4,SwitzerlandAbstractWeconsidertheproblemofcoarseningintwodimensionsforthereal(scalar)Ginzburg-Landauequation.Thisequationhasexactlytwostablestationarysolutions,theconstantfunctions+1and1.Firstwesupposethatmostoftheinitialstateisinthe“1”phaseandthatthedistancebetweentwodropletsofthe“+1”phaseislarge.Weshowthateachsuchdropletdisappearsinfinitetime.ThedynamicsoftheinterfaceinthiscaseisconsistentwiththetheoryofAllenandCahn.Howeveriftwodropletsareveryclosetoeachother,theyeventuallymergeeveniftheyareconvex.Ginzburg-LandauDroplets21.IntroductionWeconsidertherealGinzburg-Landauequationintwodimensions:@tut=ut+2ut(1u2t);(1:1)wherefut:t2R+gL1(R2;R)andisthetwo-dimensionalLaplacian.WerestrictourselvestosolutionsofEq.(1.1)whichfulfillthefollowinghypothesis:H1.(Ontheregularityofut)ThesolutionutbelongstothefollowingsetB=ut:kutkL11;krutkL1B;8t0:IfBissufficientlylarge,thenBisnotempty,seee.g.,[Co,H].TheEq.(1.1)implementsthefollowingtwophysicalideas:homogeneousconfigurationsarefavored(bytheactionofthesemi-groupexp(t))andthereareexactlytwostablestationaryequilibria,namelyu=1(becauseofthe2u(1u2)term).Thequestionweaskis:whatistheevolutionofinitialstateswhicharemainlyintheustatebutwhichhaveislands(“droplets”)oftheu+phasescatteredovertheplane.Thisquestionhasbeeninvestigatedforalongtimeonaheuristiclevel,see[B,GSS],andinparticular,interfaces(enclosingtheislandsofu+)arepredictedtomoveataspeedequaltotheircurvature(toleadingorder).ThistheoryisknownastheAllen-Cahnequation[AC].Inonedimension,sincecurvatureisabsent,themotionofinterfacesismuchslower,see[B].Whilethiscasehasbeenquitefullyunderstoodfromamathematicalpointofview(see[CP1,CP2,ER,FH,R]),rigoroustreatmentofthetwodimensionalcaseisstillincomplete(seehowever[Ch,MS]).Inthispaper,weprovethevalidityoftheAllen-Cahnequationforaspecialclassofinitialconditions.Underthehypothesisthatdropletsaresofarapartthattheydonotinteractandthattheyarequasi-circular,theyfirstreachametastableprofilewhichisexplicitlyknown.ThentheyretractfollowingtheAllen-Cahnequationuntilcompleteevaporation.Anapplicationofthemaximumprincipleshowsthatnon-circulardropletsevaporateinfinitetime.Ourtechniqueallowsforinitialconditionswithaninfinitecollectionofdropletsofincreasingsize.Suchinitialconditionsdonotrelaxinfinitetime,i.e.,utisneveruniformlynegative.Thesametechniqueappliestoadropletcontainedinsideanothermuchlargerdroplet,andsoonadinfinitum.Wealsostatesufficientconditionsforthecoalescenceoftwonearbydroplets.Ginzburg-LandauDroplets32.DenitionsandMainResultsTobeginwith,letusintroducesomenotationsanddefinitions.WedenoteF(u)ther.h.s.ofEq.(1.1)andG(u)thederivativeofthe“Ginzburg-Landau”potential:F(u)=u+2u(1u2);(2:1)G(u)=2u(1u2):(2:2)Remark.WecouldhavestatedourresultsformoregeneralfunctionsG=g0withgabistablepotentialaswasdonein[Ch,CP1,ER].TheexplicitchoiceEq.(2.2)ismeanttohelpthereadabilityofthepaper.Throughoutthepaper,thederivative(ofafunctionfofasinglerealvariable,usuallyofr=jxj)isdenotedf0,andthepartialderivativeofgw.r.t.tissometimesdenoted_g.Otherpartialderivativesarewrittenexplicitly.Theregion(u0)of“+”phase,(u0)=fx2R2:u0(x)0g;(2:3)willberequiredtosatisfysomegeometricalhypotheses,basedonthefollowingtwodefinitions:Denition2.1.WecallR2anicesetifthereareasetIN,positivenumbersfDigi2Iandvectorsfxi2R2gi2IforwhichDiD+ijx+ixijand[i2IB(Di;xi)[i2IB(D+i;x+i);whereB(D;x)isthediskofradiusDcenteredatx.Denition2.2.WecallasubsetSofR2astripofwidthLifthereareCartesiancoordinatesoftheplaneforwhichS=x=(x1;x2)2R2:jx1j12L:S1;1S1;2S1;3S1;4D1D+1D+12logD+1Fig.1:GeometryofassumptionH2.Thesetisshaded.Ginzburg-LandauDroplets4Wenextformulatetheassumptionsonu0.Theycanbesummarizedas:eachconnectedcomponentiof(u0)iscontainedinadiskofradiusD+iandtwosuchdisksareverydistant,atleast(D+i)2logD+iaway(seeFig.1).H2.(Onthedistributionofdroplets)Theset(u0)isaniceset,infi2IDiR+1,andforeachi;i02I,i6=i0,thefollowingholds:therearestripsSi;1;:::;Si;Nofwidthatleast(D+i)2logD+isuchthat[j=1;:::;NSi;j@B(D+i)2logD+i;x+i;[j=1;:::;NSi0;j\B(D+i)2logD+i;x+i=;;where@B(D;x)fy2R2:jyxj=Dg.H3.(Ontheprofileofinitialdata)Thefollowingboundsholdforeachi2I:tanh(Diri)u0(x)tanh(D+ir+i);8x2B(D+i)2logD+i;x+i;whereri=jxxij.Ourfirsttheoremstatesboundsonthelifetimeofdropletsofpositivephaseforinitialconditionsu0satisfyingH1–H3.Theorem2.3.ThereexistpositivenumbersC;R,suchthatforallinitialconditionsu0ofEq.(1.1)satisfyingH1,H2,andH3,therearetimesfTigi2I,C1(Di)2TiC(D+i)2;(2:4)forwhichuTi(x)0;8x2B14(D+i)2logD+i;x+i:TheboundsonTiareobtainedbycomparingtheorbitofu0withtheorbitsoftanh(Diri)andtanh(D+ir+i).Proposition3.6andProposition4.1belowgiveapreciseestimateonthetimeofcompleteevaporationfor(ageneralizationof)thelatterorbitsandcomparisontheoremsforparabolicevolutionsyieldEq.(2.4)(seeSection5).Ourdefinitionofaniceset,Definition2.1,coversanyconfigurationwhichisacountableunionofconnectedcompactsets,eac