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EvaporationofDropletsintheTwo-DimensionalGinzburg-LandauEquationJ.RougemontD´epartementdePhysiqueTh´eorique,Universit´edeGen`eve,CH-1211Gen`eve4,SwitzerlandAbstractWeconsidertheproblemofcoarseningintwodimensionsforthereal(scalar)Ginzburg-Landauequation.Thisequationhasexactlytwostablestationarysolutions,theconstantfunctions+1and 1.Firstwesupposethatmostoftheinitialstateisinthe“ 1”phaseandthatthedistancebetweentwodropletsofthe“+1”phaseislarge.Weshowthateachsuchdropletdisappearsinfinitetime.ThedynamicsoftheinterfaceinthiscaseisconsistentwiththetheoryofAllenandCahn.Howeveriftwodropletsareveryclosetoeachother,theyeventuallymergeeveniftheyareconvex.Ginzburg-LandauDroplets21.IntroductionWeconsidertherealGinzburg-Landauequationintwodimensions:@tut= ut+2ut(1 u2t);(1:1)wherefut:t2R+g L1(R2;R)and isthetwo-dimensionalLaplacian.WerestrictourselvestosolutionsofEq.(1.1)whichfulfillthefollowinghypothesis:H1.(Ontheregularityofut)Thesolutionutbelongstothefollowingset B= ut:kutkL1 1;krutkL1 B;8t0 :IfBissufficientlylarge,then Bisnotempty,seee.g.,[Co,H].TheEq.(1.1)implementsthefollowingtwophysicalideas:homogeneousconfigurationsarefavored(bytheactionofthesemi-groupexp(t ))andthereareexactlytwostablestationaryequilibria,namelyu = 1(becauseofthe2u(1 u2)term).Thequestionweaskis:whatistheevolutionofinitialstateswhicharemainlyintheu statebutwhichhaveislands(“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.De nitionsandMainResultsTobeginwith,letusintroducesomenotationsanddefinitions.WedenoteF(u)ther.h.s.ofEq.(1.1)andG(u)thederivativeofthe“Ginzburg-Landau”potential:F(u)= u+2u(1 u2);(2:1)G(u)=2u(1 u2):(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:De nition2.1.Wecall R2anicesetifthereareasetI N,positivenumbersfD igi2Iandvectorsfx i2R2gi2IforwhichD i D+i jx+i x ijand[i2IB(D i;x i) [i2IB(D+i;x+i);whereB(D;x)isthediskofradiusDcenteredatx.De nition2.2.WecallasubsetSofR2astripofwidthLifthereareCartesiancoordinatesoftheplaneforwhichS= x=(x1;x2)2R2:jx1j 12L :S1;1S1;2S1;3S1;4 D 1D+1D+12logD+1Fig.1:GeometryofassumptionH2.Theset isshaded.Ginzburg-LandauDroplets4Wenextformulatetheassumptionsonu0.Theycanbesummarizedas:eachconnectedcomponent iof (u0)iscontainedinadiskofradiusD+iandtwosuchdisksareverydistant,atleast(D+i)2logD+iaway(seeFig.1).H2.(Onthedistributionofdroplets)Theset (u0)isaniceset,infi2ID iR +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:jy xj=Dg.H3.(Ontheprofileofinitialdata)Thefollowingboundsholdforeachi2I:tanh(D i r i)u0(x)tanh(D+i r+i);8x2B (D+i)2logD+i;x+i ;wherer i=jx x ij.Ourfirsttheoremstatesboundsonthelifetimeofdropletsofpositivephaseforinitialconditionsu0satisfyingH1–H3.Theorem2.3.ThereexistpositivenumbersC;R ,suchthatforallinitialconditionsu0ofEq.(1.1)satisfyingH1,H2,andH3,therearetimesfTigi2I,C 1(D i)2 Ti C(D+i)2;(2:4)forwhichuTi(x)0;8x2B 14(D+i)2logD+i;x+i :TheboundsonTiareobtainedbycomparingtheorbitofu0withtheorbitsoftanh(D i r i)andtanh(D+i r+i).Proposition3.6andProposition4.1belowgiveapreciseestimateonthetimeofcompleteevaporationfor(ageneralizationof)thelatterorbitsandcomparisontheoremsforparabolicevolutionsyieldEq.(2.4)(seeSection5).Ourdefinitionofaniceset,Definition2.1,coversanyconfigurationwhichisacountableunionofconnectedcompactsets,eac
本文标题:Evaporation of Droplets in the Two-Dimensional Gin
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