Sharp interface limit for the stochastic Allen-Cahn equation [Elektronische Ressource] / vorgelegt von Hendrik Weber
66 pages
English

Sharp interface limit for the stochastic Allen-Cahn equation [Elektronische Ressource] / vorgelegt von Hendrik Weber

Le téléchargement nécessite un accès à la bibliothèque YouScribe
Tout savoir sur nos offres
66 pages
English
Le téléchargement nécessite un accès à la bibliothèque YouScribe
Tout savoir sur nos offres

Description

h-Wilhelms-UnivSharperkuseninvterfaceRheinisclimitorforebtheemstoFcBonnhasticeAllen-CahnHendrikequationausDissertationNozurerErlangunghendesriedricDoktorgradesersit?t(Dr.vrer.gelnat.)gtderonMathematiscWher-NaturwissenscLevhaftlicBonn,henvFbakult?t2009derter:AngefertigtDiesemitderGenehmigunghderelixMathematiheinsdemcderh-NaturwissensconlinehaftlicThen08.03.2010F:akult?tistdercRheinischriftenservhenBonnF.ulb.uni-bonn.de/disriedrichh-Wilhelms-UnivOttoersit?tagBonn.Promotion:1.ErscGutacungsjahrh2010ter:DissertationProf.aufDr.HoKarl-TheohscdorulscSturmer2.ULBGutacunhhttp://hsster:sProf.electroniscDr.publiziert.Fd 1" " 0 " "u (x;t) = u (x;t) F (u (x;t)) +t2dt "" "t0Fx2 [ 1; 1]"t"t"# 0f 1gthataluesofofmeasuressmallspace,.measureThisossibleequationsystemisaameasurereactionThendiusionofequationscalingwiththeatermparcorrelationtiIncularvshapaewingofwntheareactioninnite-dimensionaltethatrtratempreciselyforthestudieddimensions.whicishinisestheehanegativobtainseonderivofativtransformedeypofaatervdisotheseutrateblcurve-winellofpThisothetmeasuresethenfunctionstialjump.withondtynamiwstudiedothewofellstofsmoequalwithdepth.thatInzerotheraterstTheparlimittsurelytheconceninscaledversionarianthistismeasuretoforGibbs-tthiseequationonisgrostudiedininal.theitcase,showherethatistransformedtermconcenhasticaroundcone-dimensionalstoeadditionalminimizerstaktheesspacevpaluescongurations.

Sujets

Informations

Publié par
Publié le 01 janvier 2009
Nombre de lectures 8
Langue English

Extrait

h-Wilhelms-UnivSharperkuseninvterfaceRheinisclimitorforebtheemstoFcBonnhasticeAllen-CahnHendrikequationausDissertationNozurerErlangunghendesriedricDoktorgradesersit?t(Dr.vrer.gelnat.)gtderonMathematiscWher-NaturwissenscLevhaftlicBonn,henvFbakult?t2009derter:AngefertigtDiesemitderGenehmigunghderelixMathematiheinsdemcderh-NaturwissensconlinehaftlicThen08.03.2010F:akult?tistdercRheinischriftenservhenBonnF.ulb.uni-bonn.de/disriedrichh-Wilhelms-UnivOttoersit?tagBonn.Promotion:1.ErscGutacungsjahrh2010ter:DissertationProf.aufDr.HoKarl-TheohscdorulscSturmer2.ULBGutacunhhttp://hsster:sProf.electroniscDr.publiziert.Fd 1" " 0 " "u (x;t) = u (x;t) F (u (x;t)) +t2dt "
" "t
0F
x2 [ 1; 1]
"t
"t
"# 0
f 1g
thataluesofofmeasuressmallspace,.measureThisossibleequationsystemisaameasurereactionThendiusionofequationscalingwiththeatermparcorrelationtiIncularvshapaewingofwntheareactioninnite-dimensionaltethatrtratempreciselyforthestudieddimensions.whicishinisestheehanegativobtainseonderivofativtransformedeypofaatervdisotheseutrateblcurve-winellofpThisothetmeasuresethenfunctionstialjump.withondtynamiwstudiedothewofellstofsmoequalwithdepth.thatInzerotheraterstTheparlimittsurelytheconceninscaledversionarianthistismeasuretoforGibbs-tthiseequationonisgrostudiedininal.theitcase,showherethatistransformedtermconcenhasticaroundcone-dimensionalstoeadditionalminimizerstaktheesspacevpaluescongurations.inimpliesaincompactoriginalone-dimensionalthedconcenoonmain,setandstepwherewithanonewithInequationsecdenotespartadspace-timecalAllen-CahnisvinhigherTheHereelopmennoiseofthephasevioroundariesconstandrivinbanditsothenedcurvtime,withanlengthstogohastictotatrm.precisewnianasbridgebwith.apptheropriaonetalmostecongurationsbareoundarytratedconditions.AbstractAwhite.noise.devThistmeasuretheisbabsolutelyisconentyimeannatureuousawithadditionalrespcectforcingtoeaBroyAincSergioknoduewledgmen2.2.2tasFirstjecofoalltIHanswdiscussions.anemthatowthankki,Prof.lDr.FinallyKarl-TheofordorertSjuryturm.eIhospitalithaohnvIveMatthiasreallywenjovyBonnedMartinwada,orkinganwithIhimssasyasuppmemofbthankerh,ofeinghisIgroupanddforuriwnvgDr.thehenlastandyicears.IHemanhasfromgijoivIenhmefriendlymThisuccolleagueshHusseini,freedomDr.tohdevoetionlopemspyproorkwntideas,mbuteItlcouldproalositiowIaDr.ysoaskProf.himforforofadvice,FandProf.heevmanagedDr.toEbpushulmeacinedgetoIthereceivrofiertghTUtNodirectioneratProf.theVrighTUtuarytime.Prof.Heforhasthingsencouragedemeduringtoontratv[RelvamlotytoingetofnewyiasdeasmandBacproRyvKathriniNicolasdKuerankdDr.meertwithIthetopparticossibilitDanielywtovdotso.discussingIramtheextremelyChaptergratefulwtothankhimyforfriendshislotrunstIandreceivsupptheort.ofIPropgratefullynac(ii).knoalsowledgeProf.theHerbinuenceKofcProf.andDr.Dr.TKrohaadahisabFpartunakitheon.murthermore,ythankresearcDr.h.AlbHeeriohasProf.hostedAndrmeasaterleUnivstimersitatingyIofknoTlokytheoythawiceeined2008,guestandProf.IHerbhaSpvateM?ncbinevenbreceiv2008,eofdDr.withantheeselgreatestatpChemnitzossibleJanhospitalit2009.ythank.Dr.DiscussionsR?gerwiththeProf.yFIunakivhalearnedvhimetheborkeenouranprivilegeproandtessenW09].tialhaforemosteryofuctheenjodevedelopmenorkingtstheinatmospherethisthewProbabilitork.Group.IwthankmostlyProf.toDr.yFKathrinelixher,OttoHuessmann,foradhaAnn-vingJarecacceptedDr.toJuillet,bKazumasaewpartFofMiebactheandjuryR.bIPhiliphawski.vwethadmentheinpleasureutoardiscussGruhlkmathematicawithlhomquestionshawithehimensevmaneralhourstimes,theandogIehaofvweonb2.enettedIfromanthistoexpmerience.familInandparticular,yheforhashehelpvedabdyortproconstanvidingythee.idea41Con.ten.tsb1.In.tro.duction.1.2.Sharp.inductionterface.limit.for.in.versolutionsarian.tokmeasures.9.2.1stoIn.tro.duction......Sto.atur.......s.......endix...........4.2.diusion.....main...............37.hastic.mean...............Construction9and2.2.The.Energy.F.unctional....4.4.1.......................results.hastic.........48.and.result......................13.2.3.Gaussian.estimates..3.2.c.motion.y.curv.e...........................3.3.of.ub-.sup.........................4219App2.447ConcenOutlotration.around.a.curv.e.in.i.n.ni.t.e.-dimensional.s.pac.e..............47.Some.for.c.reaction.equations..24.3.Short.time.asymptotic.37.3.1.Inviitrou2 [ 1; 1]
1
1
1 1
u
1 ( 1; 1)
Z
1
F (u(x))dx:
"
F
1 2 2F (u) = (u 1)4
F
F
Z
2"jru(x)j dx;
Z
1" 2H (u) = "jru(x)j + F (u(x))dx:
"
"
esconFtexts.theInpthishewotentialorkthetheneedsinuenceoinoffornoisevontaktheotenAllen-Cahn.equationisisphasstudied.simplestSometorigorousecausenewsameresultspuonintheheinelvstandardarianwttmeasureoreinustheisone-odimensionaleredcase:andoneongrainsthethedynamicsthisofnergeticallythehahigher-dimeneighnscrystaliareonaleectequationbaretakgivationen.e1.tialDerivation11ofsucthefunctionmoindelertiesLetcuswbrieyater,recallofthemomenmainNoingredienlotsaofetAlleneectandwhicCahn'samoeivdel.sucIfunctionalmaginenearathephducedysicalofsgrainedystem,true,whicishvcanatombeeasdescors,rofiecauseblatticeedmostloe.callytakbaccounyaangorderitparametermass.troHereintheasawpequationen-Cahngur,.whichoicehadepisendstheseonbtimeolutionandmostspace.indepThethesystemofmighet,underforassumptionsexample,forconsisticitofositionaforcrystallinethatmaterialparticularwiththistewandotermpoidossibletransitionslatticeeenstructures,s.whiccohsecondaremathematicallydescribformedenerbforydelstheofordertparameterthetakingItthemeltingvcrystallinealueswthAllevThe[A.InItcasemighttcoarsealsoferromagnetconsistisofbaitsampleeoffaferromagneticorablematerianaltowithvtthewspinoitspbossibleinspinscaseductionthetrob.theThereorderstructuresparameterthetakingfavorablaThisluesisinenbtoettwyeenpInenerandy1isthenrefore,correspTondsoftopreservmixtureswithoutofestheshaptofwdouble-wollatticeotenstructures(sienitheemo)delAofctheforcrystalhorpatiallothecalphasesmeaneenmagnetization,etwhicterfaceshofmaevybutbpropeareobtainedendenbofyparticularahoicecoarseandgrainingWinwilltheorkcasemofgeneralthelferromagnet.butAsimplossibleyrstexpstepletinassumemothedellingtsucseparationhthisafunction.situationwisdescriptionthepurderivlyationcalofoneananotherenergytoassovciatedtotoroughabcongurationwChapterthetheestrengthThisinisThevariousbyausedtermcalibratehcorresptaktothewidthofthekineticterfacegyetphaseeeabletheconctmo1thealuesoneclosehtohatdelobtainsmoenergyinsteadisoft.anpitheirnmaterialstermediateinvofaluegroindelsimpliedmoaC79]asinen.theparameterterms.isetoseethatdiering.ofInindividualbWothwillsituationslaterititisondsmorethefaofvinorablebforwthensampledierentophases.attainv0.6
0.5
0.4
0.3
0.2
0.1
-2 -1 1 2
F
u
"d @H (u)
u = :
dt @u
2 "L H
1"# 0
"
d 1 0u(x;t) = u(x;t) F (u(x;t)):
2dt "
u
[ 1; 1]
[ 1; 1]
1L [ 1 ; 1 +] > 0
[ 1; 1]
2L
1H
R
u(x;t)dx
thetheoneenergystrikingfunctionalnoisequiconeenergyen.prW-gradieneanotherwillwseethelaterdel.thatCahn-Hilliardinvorderattaintotheobtain-anrinobtainsterestingandphenomenonwinstheitsharpsuitable,imathematicalntederreasonable,facinenotlimitinosesoneloklyitwonealsohasatoequation.acceleratethethewhicdynamicsdelbbywaifactortotalhatomswhicnot.inThehevendsolutiontoequationeonefourobtainsthatinarethisforwnce,aducingyeisthatthealuesAolFlen-Cahnnoteehoicequation-structureintcongurationsdelibardsmighwhotoandtendsasystemntheforthatesates-structuretulCahn-HilliardosispstudiedonephaseeldolutionphaseThethebforthesemotionmoofthatequationequationanondseorrgivatostructure)ordered,(1.0.1)preservMathematicallyCahn-Hilliardthereiisisnodereasonthetoanrestricte.thisnequationthattoisfunctionsoinequation,thatsattainnciples,veryaluesandonlyconineniewwhenNotro.thetialwalthoughwillotherenforcecongurationssolutionshavvonlyewnotph.ysicalurthermore,meaning.shouldAthatctuallycitofcan.bforegradienshoowniseasilyeratewithoneatcomparisoncprincipleosethatmetricsolutionsobtofinthedieAllen-CahneequationtwhicIfhexampleinitiallytakonlytheattaintovoneatheluesequation,inhotenanotherpellThemo1.1:forwillseparationpreserveveofthisoundaries.propmostertdierenceyetforeenallttimes,owhereasdelssolutionssofin(correspAllen-Cahntothetotalmassofpmaterialethefractionwithondinggetheneralmagnetizationinitialtheconditionsor(satotalyofinwithincgivFigurelatticenotisvpreservforwhileCahn-Hilisequation.edethedramownWhinctequationomorethetherefore,ipnontervsituationalwthistsulationdescribformAmathematicaldiereIncInisphtheysicistsequationforaevtheryrnotationrthisandreadscompariinoneiofthatAllvequation,usefulseemsth.studyThistheben-Cahnehaareviora)ailablewillthequicliardkly2b

  • Univers Univers
  • Ebooks Ebooks
  • Livres audio Livres audio
  • Presse Presse
  • Podcasts Podcasts
  • BD BD
  • Documents Documents