Existence and stability of the log log blow up dynamics for

2L
4 2Niu = u| u| u Lt
N 1
R H ( ) N 40
q
log|log(T t)|
|∇u(t)| 2 .L T t
NR
2L


4
N iu = u | u| u, (t,x)∈ [0,T)
,t
(NLS) u = 0,|∂

u(0,x) =u (x), u :
→C0 0
1 1u ∈H =H ( ) N 1 N = 20 0 0
(Dirichletbt(1)3,oundarynonlinearcondition)band8628neho&Planchr?.PW91405euniv-paris13.frpro.veeequationexistencebandastabilit7539y?eofaniteFtimersiteoratoireblofabrice.plancw-updimensiondynamicsonwiththethefolog-loghr?bloaw-upDiricspndaryeedUMRabriceequation,FnonlineardomaintheaInstitutinersiequation13,dingerAbstracthr?pierre.raphael@math.u-psud.frScynonlinearP-criticalCNRSthemath?matiquesforRapha?ldynamicshon@math.w-upFbloCl?menlog-loguethedimenofWyconsiderstabilitwithand-criticalbcusingWScreferdinger[9],infordomainreferences.withsuitablehletclassouofcondition:G?om?triApplicationsindomain,inproblemdingerinScon-criticalopticsdusemoCNRS,forGalilself-foUnivoft?tensearisb99inLetw2007whicJulyhranceturnCedexoutOrsatoaris-Sudb?eUnivmduoUMRdeleddeafterLabthePierredatawithinitialoratoireLabranceVilletaneuseones.193430Analyse,IntroinductionJ.B.1.1enSetupInandsinotationsnitevtimeblothisw-uparisessolutions,nwlineareadearivdeletheglobalcusingrigidiintlaseryeamspropholloertiescoreExistenceers.Moreoevtoer,[30]forfurthera1

1u ∈ H T > 0 u0 0
1u∈C([0,T),H )0
1H T = +∞
T < +∞ limsup |∇u(t)| 2 = +∞t↑T L
N
= R
N = 1,2

N 3

R R
2 2 2L : |u(t,x)| = |u (x)| ;0

R R 41 1 2+2 N: E(u(t,x)) = |∇u(t,x)| |u(t,x)| =E(u ).4 02

2+
N
N
= R

u ∈0R
2 1 2 2 2H ∩ = H ( ) ∩{xu∈ L ( ) } E(u ) < 0 V(t) = |x| |u(t,x)|00
Z Z Z
21 d 1 1 4 12 2+ 2
NV(t) = |∇u(x,t)| dx |u(t,x)| dx |∇un| xndx,
2 416dt 2 42+

∂
N
n ∂

2d
V(t) 16E(u )< 002dt
V(t)
iωtu(t,x) = e W (x) Wω ω
(
4
NW +W |W | =ωW ,ω ω ω ω
(W ) = 0.ω |∂

N
= R Q (x)ω
Q Q = Q Q (x) =ω ω=1 ω
N 1
4 2ω Q(ω x)
E(Q ) =ωE(Q) = 0 |Q | =|Q| .2 2ω ω L L
qFrombislastinothaconserv,ationeslaandws,titonewillmaisydescriptionderivwemaptheupfollothewingsucclassicalthresult:solutionthe,pitoplawareertimenonellipticlinearittime;yCasefrome(1)radiallyis,theeesssmallesttities:onegralforofwhice.hglobal,blodata.w-upnormmaeysolutionoeciccur.talSucdynamicshsolitaryproniteotimetheblothatw-conupuoussolutionsthatare.knoawnsuctoaddition,existmfstsofromafromstareshaploedtodomain:eseet[13]hforconedbconserveous,w,andsolutionitsusgeneralizationtheforiltofuanpecometitnitetowithehandapwsedtime.dolmainainfor[14],theb(1).othsoproaofsasrelyingexistenceonontheresolvfollowingwing(3)virialrequired)idenuittitthatycally.conAssumeoforesakvefromoftheresimplicitpythatthattosystemtranslation.isalaLsmaxmthenoexothrstar-shapwingedholdsdomainaleandandletohozaevHamiltonian,nalvoell-pnsithmebnormidassumwheretheinniteinaneis(whic(1)isEquatione.tribution]the6oundary)[bseenegativomain,ThdbloexteriortheanorwithisisthewheneithereThlinitialailabofvwhae.theSetfunctionarearesultsositiverquansharpy;,hasforbwhilenegativ[1],inintime,tlytherecenbloobtaineduperenitewSptesamasolutionsestiye)fundamen,rolethentheaofsimpgenerallofeTheysequencetheofcalledcomputationwyieldsv(2)ahartz-likvided(StricisrofetheSharpundresults.bpreviouswhereallloincludingestheoremfolloanonlinearandequation:referencesafurtherandfornot[7]yseetin[26];uniforminnoteobtainedinas(lowtinitis,wdomaintheyassumeanwforer,andMoreodimension:In[3][12].[15],eloexistsVuniqueandositivGinibresolutiontohdue(1)and,wntoell-knoInwsolutionisisresultsymmetric.osednessetell-piwuniquecal,loandaihtheresucall,oorsense:Ffollo.inandfrtimescteinvarianci,n,wherePinidenisythewouterhanormaleEnergyosednwtoupcalwsat.hCertainlye,andondomainathestar-shapeedthereafterdomain,aWstars21Q ∀u ∈ H E(u) =
NE(Q) |u| 2 =|Q| 2 > 0 x ∈R ∈R u(x) =u u uL L
N
2 i u Q( x x )e .u u u

= B(0,1) N 2
ω < 0
0

∀ω >ω Q0 ω

|Q | 2 <|Q| 2 E(Q )> 0,ω L L ω
1
∀u ∈ H E(u) = E(Q )ω0

|u| 2 =|Q | 2 ∈RuL ω L

i uu(x) =Q (x)e .ω
1|u | 2 <|Q| 2 H0 L L 0
2L
!2RZ
2 N1 |u|1 2 R∀u∈H , E(u) |∇u| 1 .0 22 Q
N 1
=R u∈H ( )0
1 NH (R ) u

|u | 2 |Q| 20 L L
N
=R u(t,x)
2
|x|1 1 x i
4tv(t,x) = u( , )e .
N t t2|t|
ite Q(x)
2
|x|1 x ii +
4t tS(t,x) = Q( )e
N t2|t|
T = 0 |S(t)| 2 =|Q| 2 S(t)L L
1
|∇S(t)| 2 .L |t|
S(t)
n,then,theisortherehereholds:isnaryoAstiexzavricharacteappliedc,ariationalavthatwinghlow-upl[9].othfthethe,and[8],existsdispaanduniquesolutionpareositivtheseNote,that,bloieninathdomain,thenthiseinequalitandyrstfolloinwstofromvtheofcaseande[9].radialsymmetrysolutionthetoeassanvyanandof(3).stableMoreoevstateifcorollaryer,thereisuciwhicalupnitesakblosimplicitsolution,ytostsa.distributiontheineedLaplaciansihlet.Diricresultstheetofthealuefromber,yeextendingglobalv(1)osolutionfollo,iswinguniqueariationaloutsideharacterization(4)holds:.ifIn,addition,FthisseeconditionThisis,shtoarp:nonforersiv([31]):stationysolutioninequalitLionergebCazenaGagliardo-Nirenyieldsoptimalexplicitthe(6)withsensebinedincomorbitallynorm,guides,vbloww-upgroundma,yaothenccur.existsmmetries.,CasehvAssumedhanwsenergyat:forblowithw-upoffollothatwssucfromthatthedimenpseudio-conformaltheresymmetry:Noteifatofbloationsspconservforoth,bosolvnesW(1),recallthefromnLsobdeoMoreoeser,(5)[19],,fromeigenbythezerommanymbmassetimevieww-upedupassyMoreo3
S(t)
x˜ ∈
u(t)
1u(t,x) S(t,x x˜)→ 0 H t→ 0.
2L |u(t)| 2 =L
|Q| 2L
N
=R
NR
N
=R
Z Z Z
1 N 2 2 2 ˜u ∈B ={u ∈H (R ) Q |u | Q + }0 0 0
> 0
N = 1,2 S(t)
1S(t) |∇u(t)| 2 L T t

1
log|log(T t)| 2
|∇u(t)| 2 N = 2L T t
N = 1
1H
N
= +y∇,
2
N 1

2 4 4 2 41 1
N NL = + +1 Q y∇Q , L = + Q y∇Q,1 2
N N N
1ε =ε +iε ∈H1 2
H(ε,ε) = (L ε ,ε )+(L ε ,ε ).1 1 1 2 2 2
1˜ > 0 ∀ε ∈ H (ε ,Q) = (ε ,Q) =1 1 1
2(ε ,yQ) = (ε ,Q) = (ε , Q) = (ε ,∇Q) = 01 2 2 2
Z Z
2 2 | y|˜H(ε,ε) ( |∇ε| + |ε| e ).1
typeconstructedSchrClassicationw-upeensatisesbeeonwconstanealmeancriticalsolutionswwithinformationsblotruew-uptrateconstruvationhasolutionesituationypblottofbsolutionsfor,nodomain,generalthataerorpreciselynearforblotow-upromtime.andFaticOnethetheseotherthehand,(1)ncaseucmericalresultssimBanica,ulations,e[16],andwformalaargumenmmetryts,of[30],so-suggestresultexistenceopofdingersolutions(7)blo.winggivupsmalllik,eaywhicbas[4];conservseethe,rind[5]forbhaymass:e:ypasterturbativofdealsolutionsnitefdynamicsoresultsfamilyThenaaexistsnstherethe,suchBurq-1.2G?rard-Tifineendimensionhazvtimeetkcriticalosp.theInSomedimensionholdsdimensionwhicInpseudo-conformalv.hea,pP;ofremainstoccur:oknosolutionshvproblem.esalthe?existenceopofatorssucleasthAaMoresolution,andtits(explicit)stabilitsomeyenintheyactsubspacesolutionof(1)tohwnin.FThethesituationwithhofasnorm,btheeeneclariedvaluebquadryformMerletheseandsRapha?lvincriticalthe.seriesofof(8)papsolutionserswhere[22],e[20],p[21],with[25],unique[23],mass[28].withLetforusw-updeneontheMostdierentimetialtheropexistserauniversaltoroknotanareblohanismsinmecnpthatw-uKnoblo[2].t,dierenywhicobtainedhbwillvbsolutionsebloofniteconstanmassttheuse.eedThenupwbloeoninpreliminarytrodomain.duceinthelongerfollohwing(5),propsyertthey:vilySpreliesectralroPropitsertthenyw-upLlutionsetwhenoewbtwn.isConsiderathesuctwoNotereneerelman,an[27],pro4N = 1 Q
N = 2,3,4 N = 2
˜B
N
= R N =
1,2,3,4 > 0 C > 0
˜ u ∈ B u(t) [0,T)0
1H
u(t) 0<T < +∞
N 2((t),x(t),(t)) ∈ R R R u ∈ L+

1 x x(t) i (t) 2u(t) Q e →u L t→T.
N (t)2(t)
Nx(T)∈R
x(t)→x(T) t→T.
t T
1
2|∇u(t)| 2 T t 1Llim = √ ,
t→T |∇Q| 2 log|log(T t)| 2L
C(u )0|∇u(t)| .2L (T t)
E(u ) < 0 u(t)0
˜ u ∈ B0
u(t) 0 < T < +∞
1H
u T < +∞ u(t)
R> 0
Z
1 C 2 |u (x)| dx ,
2 2C (log|log(R)|) (log|log(R)|)|x x(T)| R
p 1u ∈/ L for p> 2. u(t) u ∈H .

existsforuladrthatformthenexplicitdatatheniteusingon[22],,suchthethat1ineedwithv.pro;aslwsyanertimpliespropexistasblow-upthisesp,(1)or.ingsupencriticalccaseprndalblow-upomputationsspinete(9)d:olefore[28])whiccloseyenoughameterstothenumeoint,,wechavesolutioneitherthe(10)upassgroundsingularity:RtheloofeDescriptionis(i)in.AsymptotLently,etsingularity:inofightinvolvesrsatisesthealonderivee(12)existencforofcintervaletimecmaximumsuchitsc.pL.etv.wingwithdescrib(1)thetoforsolution.ondingarespsatisesorrthencisthepethebsuchThertheororr(11)ondingeoverexistMoruniversaltoletblows,inctimeonstantsstate,assuchethatthetheg-lofolsporeF(10)true.oppin[22],c[20],(iv)holdsi(iii)ofeathertheStatemenassumeofoOurwhichinifpapnumericiscaddress(10),issuesforaldomainwas.smalusenough,thatdto[10]massdimensionssolutionswhich[5],oversarehophyviresultsalnitethatblolysolutionselevantaaseedrtimeasymptoticwithandtheWlohag-loegfollosptheoremehewhichdes(10).singularitMorformationeinitialgenerinalTheoremly,Ifthepseteof(11),initialth