The Mass-Critical for the Nonlinear Schrödinger Equation in d = 2

Transcription

1 Avanc in Pr Mahmaic h://oiorg/36/am3357 Pblih Onlin Ag 3 (h://wwwcirorg/jornal/am) Th Ma-Criical for h onlinar Schröingr Eqaion in = Mjahi Ab Elmj M-Ali Darmn of Mahmaic Facly of Ecaion Kaala Univriy Kaala San mjahi@mailccn Rciv Fbrary 3 3; rvi March 6 3; acc Aril 9 3 Coyrigh 3 Mjahi Ab Elmj M-Ali Thi i an on acc aricl irib nr h Craiv Common Aribion icn which rmi nrric iribion an rrocion in any mim rovi h original work i rorly ci ABSTRACT Thi ar i h global bhavior focing nonlinar Schröingr qaion in imnion = an w will i- c h ca Thi man ha h olion C T T an call criical ol ion W how ha car forwar an backwar o a fr olion an h olion i globally wll o Kywor: S; Wll Po Inrocion W conir h Cachy roblm for h nonlinar Schröingr qaion in imnion = i F () whr F an : C Whn () i call focing whn () i call focing In hi ar w ic h ca whn an (focing ca) If i a olion o () on a im inrval T hn () i a olion o () on T wih Thi caling av h norm of (3) Th () nr rvio hyoh i call -criical or ma criical Prooiion So ha an hn for any iniial aa hr i T ch ha hr i a niq olion C T T of h nonlinar Schröingr Eqaion () If hn T T for om non-incraing an if i fficinly mall i globally In hi ar w will ic h ca Thi man ha a olion C T T an call criical olion Dfiniion : K C K i a olion o () if for any comac J K C J J an for all K i i r i F r r () Th ac J ca from richarz ima Thi norm i invarian nr h caling () Dfiniion If hr i K a olion o () fin on K R blow forwar in im ch ha K (5) An blow backwar in im ch ha inf K (6) Coyrigh 3 SciR

2 M A E M-AI 83 Dfiniion 3 If hr i w ay ha a olion o () car forwar in im ch ha i lim (7) A olion i ai o car backwar in im if hr Sch ha i i lim (8) W no ha h Eqaion () ha rrv qanii h ma An nrgy M M o (9) E () E For mor [] Prooiion l b h -criical onn hn h S () i locally wll o in in h criical ca Mor rcily givn any R hr i ch ha whnvr * ha norm a mo R an K i a im inrval conaining ch ha i * Thn for any in h ball K : * olion S KR i ichiz S K whr S K fin in B hr i a niq rong o () an morovr h ma from B o Eqaion (5) Prooiion 3 l K b a im inrval conaining an l C K C b wo claical olion o () wih am iniial am for om fi μ an am alo ha w hav h mra cay hy- q ohi K for q = Thn Prooiion R givn hr i a maimal lifan olion o () fin on k R wih Frhrmor ) k i an on nighborhoo of ) W ay i a blow in h conra ircion If k or inf k i fini 3) If w hav comac im inrval for bon of iniial aa hn h ma ha ak iniial aa o h corroning olion i niformly conino in h inrval ) W ay ha car forwar o a fr olion if k an o no blow forwar in im An w ay ha car backwar o a fr olion if inf k an o no blow backwar in im To Proof: [-3] Sricharz Eima In hi cion w ic om noaion an Sricharz ima for criical S () an w rn o rov Prooiion an 3 Som oaion If X Y ar nonngaiv qanii w X Y or X OY o no h ima X cy for om c an X Y o no h ima X Y X W fin h Forir ranform on by ˆ i f : f π q r W K o no h Banach ac for any ac im lab K of fncion K C wih norm i q q : K q r r K Wih h al amnmn whn q or r i qal o infiniy Whn q r w c hor q r q a Dfin h fracional iffrniaion oraor by : ˆ f f f : f ˆ whr : cially w will o ignify h aial grain an fin h Sobolv norm a f f H H : : i b h fr Schröingr roagaor; in rm of h Forir ranform hi i givn by i π i f : f A Gagliaro-irnbrg y inqaliy for Schröingr qaion h gnraor of h rio conformal ran- f formaion J i lay h rol of h arial iffrniaion Sricharz Eima i b h fr Schröingr volion from h lici formla Coyrigh 3 SciR

3 8 M A E M-AI iy i f πi ϒ f y y () Scially a h fr roagaor av h i f f For all an whr Prooiion Thr hol ha i f f π -norm () In fac hi follow ircly from h formla () Dfiniion Dfin an amiibl air o b air r wih r Wih r Thorm If olv h iniial val roblm On an inrval K hn i F F q K K q q q q For all amiibl air q h bg al To rov: [5] Dfiniion Dfin h norm : q S K amiibl K q S K S K (3) no () : (5) W alo fin h ac K o b h ac al o S K wih iabl norm By horm Thorm 3 If F : S K K bally wll o for mor [67] Proof: by (3) an (6) If (6) i mall hn () i glo- 3 (7) i mall nogh an by h coniniy mho hn w hav global wll-on Frhrmor for any T ch ha hr i T Thn i T i r T i r r r (8) So by (6) whn T Th h limi Ei an T i T 3 T k it (9) k lim T () k i lim () A conformabl argmn can b ma for in if C hn can b iviion ino ~ C binrval K wih K on ach binrval Uing h Dhaml formla on ach inrval inivially w obain global wll-on an caring ow w rrn o rov Prooiion an Prooiion 3 Proof rooiion : W o in wha follow ha an for om o b chon T b ch ha i () T W m h ac S C T T An h maing T i i Φ v i v v (3) W wan o rov ha h δ mall aqa Φ : S S i conracion W fir Sricharz ima o com ha i v v v T Coyrigh 3 SciR

4 M A E M-AI 85 whr Thn iincly Φ T Φ i T So ha for T i mall nogh S i l nr Φ In aiion Φ v i Φ v v v v v v v T v Again craing may b T w g a conracion If hn an from Sricharz ima w ha if i mall nogh hn () i ai- fi for T Proof Prooiion 3: By im ranlaion ymmry w can ak By im rvral ymmry w may am ha K li in h r im ai v an hn vc K C v an v oby h varianc qaion i v v v Sinc v an v v li in h K w may calling Dhaml an concl i v i v v for all By Minkowki inqaliy an h niariy of concl ha i v v v Sinc an v ar in K an h fncion z z z i locally ichiz w hav h bon v v v v K K Aly Gronwall inqaliy o concl ha v for all K an hnc 3 Dcay Eima Conir h focing nonlinar Schröingr Eqaion () in whr an for W o ha a H (3) Fir w hav h following rl Thorm 3 So ha if an l b a olion o () inical o an iniial aa H ch ha If = l r b ch ha r hn hr i a conan c c > ch ha if R i h olion of RR R wih c min R R hn r r cr Frhrmor c n only on r an E : Th mho ma in rchling by h avrag of a im nn rchling h qaion an o h nrgy of h qaion o g by inrolaion cay ima in iabl norm Th aymoically avrag i normally obain ircly by ing h o conformal law h abov rl wa in fac arially rov in [8] nr a bi iffrn oin of viw: look for a im nn chang of coorina which mainain h Galilan invarianc an h conrcion ircly a yanov fncional by a iabl anaz Thi yanov fncional i rly h nrgy of h rcal qaion Or aim hr i o y wih frhr ail h rcal wav fncion an i nrgy Fon o b h mho rovi ra which ar m comlly nw in h limiing ca of h logarihmic nonlinar Schröingr qaion Bca of h rvribiliy of h Schröingr qaion an anar rl of caring hory on canno for h convrgnc of h rcal wav fncion o om a iniion givn limiing wav fncion b fon o b om conviy rori of h nrgy can b o a an aymoically abilizaion rl From h gnral hory of Schröingr qaion i i Coyrigh 3 SciR

5 86 M A E M-AI wll known ha h Cachy roblm ()-(3) i wll o for any iniial aa in H whn an ha h olion blong o C H C H R H A al for Schröingr qaion i criical whn b ch ha is R R whr an ar oiiv rivabl ral fncion of h im I i iml o chck ha wih hi chang of coorina aifi h following qaion R i r Δ S S R R R i S R R whr wih h choic S which man R R ha S S R an ar link by ir R R i RR R R R (3) whr an R an ha o aify h following im-nn focing nonlinar Schröingr qaion Δ i r RR (33) R R R R W no ha for all o ha Alo w no ha if R an hn R (3) To rac h conrolling imac a an R ch ha w fi whr C RR R R c min (35) Bca i criical hi anaz i acally h only on ha o a la hr of h for cofficin in h qaion for wih lim R an olv h qaion ir Δ (36) Wih h choic an R ingraion R of (35) wih rc o giv R R an hi i oibl if an only if R for all h h fncion R i globally fin on incraing lim R an R a Soing ha fncion ch ha ~ i an incraing oiiv lim whr if Conir now h nrgy fncional link o Eqaion (36) E (37) whr R ha o b nroo a a fncion of mma 3 So ha if an l b a olion o () inical o an iniial aa H ch ha Wih h abov noaion E i a craing oiiv fncional Th E i bon by E E wih h noaion of Thorm 3 Proof: Th roof follow by a irc comaion Bca of (36) only h cofficin of an conrib o h cay of h nrgy Fo r mor [9] Proof of Thorm 3: So ha i criical By mma 3 an ran o h im-nn rcaling (3) ir R R E E Coyrigh 3 SciR

6 M A E M-AI 87 Th ir R I bon by ir R E R h rmainr of h roof follow h am lin ha in Thor m 7 of [] alo [] ing mainain h -norm an h Sobolv-Gagliaro-irnbrg inqaliy Prooiion 33 Conir h wo-imnional focing cbic S ( ) (i -criical) H hn hr i a global -wll o olion o () an morovr h norm of i fini Proof: By im rflcion ymmry an ahion ar- gmn w may h anion o h im inrval Sinc li in H i li in A ly h wll on hory (Prooiion ) w can fin an -wll o olion S T on om im inrval T wih T ning on h rofil of Scially h T norm of i fini w aly h oconformal law o c ha Ec T T Ec Sinc H w go a olion from = o = T To go o all h way o W aly h - oconformal ranformaion a im = T obaining an iniial am v T a im by h formla T v it : T i T T T From E v E c w ha v ha fini nrgy: v v T T Ec T T An h oconformal ranformaion av ma an hnc v T T So w ha v T ha a fini H norm Th w can h global H -wll on hory back- war in im o obain an H -wll o olion vs T o qaion h ivv v v ariclarly v T W rvr h oconformal ranformaion which fin h original fil on h nw lab T W ha h T an C T norm of ar fini Thi i ffi- cin o mak an -wll o olion o S on h im inrval ; for v claical An for gn- T ral S T h claim follow by a limi- ing argmn ing h -wll on hory Ahion oghr h w o inrval T an T w hav obain a global olion o () Som mma Conir h focing ca of h S () an if h nrgy an ma oghr will conrol h H norm of h olion: E M H Convrly nrgy an ma ar conroll by h H norm (h Gagliaro-irnbrg inqaliy how ha): E H H M H Thi bon an h nrgy conrvaion law an ma conrvaion law how ha for any H -wll o olion h H norm of h olion a im i bon by a qaniy ning only on h H norm of h iniial aa Prooiion Th cbic S () wih = i globally wll o in H Acally for H an any im inrval K h Cachy roblm () ha a H wll o olion S K C H K Coyrigh 3 SciR

7 88 M A E M-AI f i mma If f h following hol: i i i f f f Proof: Th roof n on h noicing ha; Wih Th i i J i i J i i i i i i f J f f By anar Gagliaro-irnbrg inqaliy i i i i i i f f i i f J f i i f f mma3 For any acim lab K R K an for any v K c i H H v i v H H Th ima () i vry hlfl whn i high hiancy an v i low hiancy a i mov abnanc of rivaiv ono h low hiancy rm In ariclar hi ima how ha hr i lil inracion bwn high an low hiancy Thi ima i baically h ra Sricharz ima of Borgain in [3] W mak h rivial rmark ha h norm of v i h am a ha of v v or v h h abov ima alo ali o rion of h form Ov Proof: W fi an rmi or aci conan o n on W bgin by aling wih homogno Δ ca wih : i Δ an v : i An conir h mor gnral roblm of roving v H H f () () whr h caling invarianc of hi ima fir or objciv i o rov hi for an May b rca () ing aliy an rnormalizaion a ˆ ˆ g g (3) Sinc w may rric anion o h inracion wih In fac in h rial ca w can mlily by o rrn o h coniion nr ic - ion In fac w may frhr rric anion o h ca whr inc in h ohr ca w can mov h frqnci bwn h wo facor an rc h ca whr which can b al by Sricharz ima whn w como yaically an in yaic mlil of h iz of by rwriing h qaniy o b conroll a ( Λ yaic): g Λ Λ o ha bcri on g hav bn inr o invok h localizaion o ~ ~ ~ conciv In h ca w hav ha ~ an hi on why g may b o localiz By rnaming comonn w may o ha ~ an ~ Wri W chang variabl by wriing v An v J W how ha by calclaion ~ J Th on changing variabl in h innr wo in- gral w nconr whr Λ Λ v g v H Λ H Λ v Λ J v Aly h Cachy-Schwarz on h v ingraion an chang back o h original variabl o obain Coyrigh 3 SciR

8 M A E M-AI 89 g Λ Λ Λ J W rcall ha J an Cachy-Schwarz in h ingraion aking ino coniraion h localizaion ~ Λ o g g Λ Λ Λ Choo an wih o obain g Λ Λ Λ Thi mmariz o g h claim homogno ima ow w ic h inhomogno ima () For imliciy w F : i an G: i v Thn w Dhaml formla o wri W obain v i i i F i i v v i G i i v i i G i i i i I F G An h roof follow by inring in h ingral h homogno ima abov mma Cloc K i narly ri oic molo G Thn hr i fncion : K : K an : K an for vry hr i C ch ha w hav h aial concnraion ima C (3) An hiancy concnraion ima C ˆ () For all K Rmark 5 Informally hi lmma confirm ha h ma i aially concnra in h ball : O An i hiancy concnra in h ball : O o ha w hav rnly no conrol abo how vary in im; (for mor [- 7]) Proof: By hyohi lay in GI for om com- ac b I in For vry comacn argm n how ha hr i C (ning on) ch ha C f An hiancy concnraion ima ˆ f C i i v F For all f I By incing wha h ymmry gro G o o h aial an hiancy iribion of h i i G G ma of a fncion hn h claim follow Corollary 6 Fi an an am ha m i : II I3 I fini Thn hr i a maimal-lifan olion Cloc K of ma rcily m which blow Th fir rm wa ra in h fir ar of h roof boh forwar an backwar in im an fncion Th con an h hir ar imilar an o w conir : K : K an : K wih I only By h Minkowki inqaliy rory C for vry (ning on m ) ch ha w hav h concnraion ima ( 3) () For all K I G An in hi ca h lmma follow from h homogno ima rov abov Finally again by Minkowki inqaliy w hav REF ERECES [] T Caznav an F Wilr Th Cachy Problm for Coyrigh 3 SciR

9 9 M A E M-AI h onlinar Schröingr Eqaion in H Mancria Mahmaic Vol 6 o oi:7/bf586 [] T Caznav an F Wilr Th Cachy Problm for h onlinar Schröingr Eqaion in H onlinar Analyi Vol o oi:6/36-56x(9)93-a Rarch oic o [] T Caznav An Inrocion o onlinar Schröingr Eqaion Fir Eiion To Méoo Mamáico IM-UFRJ Rio Janiro 989 [] M Dl Pino an J Dolbal B Conan for Gagliaro-irnbrg Inqalii an Alicaion o onlinar Diffion Jornal Mahémaiq Pr A- [3] Y Tmi -Solion for onlinar Schröingr Eqa- liqé Vol 8 o ion an onlinar Gro Fnkcialaj Ekvacioj Vol [] M Dl Pino an J Dolbal Gnraliz Sobolv In- 3 o Qalii an Aymoic Bhavior in Fa Diffion [] bal Analyi CBM S Sri 6 Amrican Mahmai- T Tao onlinar Diriv Eqaion: ocal an Glo- an Poro Mia Problm Crma [5] cal Sociy Provinc RI 6 T Tao Shrically Avrag Enoin Sncharz Eima for h Two-Dimnional Schröingr Eqaion Commnicaion in Parial Diffrnial Eqaion Vol [3] J Borgain Rfinmn of Sricharz Inqaliy an Alicaion o D-S wih Criical onlinariy Inrnaional Mahmaic Rarch oic o o [] T Tao M Vian an X Zhang Global Wll-Pon oi:8/ an Scaring for h Ma-Criical onlinar Schrö- [6] T Tao M Vian an X Zhang Minimal-Ma Blow Solion of h Ma-Criica l S Form Mahmaicm Vol o oi:55/forum8 ingr Eqaion for Raial Daa in Highr Dimnion Dk Mahmaical Jornal Vol [5] T Tho an M Vian Sabiliy of Enrgy-Criical on- [7] J Borgain Global Solion of onlinar Schröingr Eqaion Amrican Mahmaical Sociy Colloqim Pblicaion Vol 6 Amrican Mahmaical Sociy Provinc RI 999 linar Schröingr Eqaion in High Dimnion Elcronic Jornal of Diffrnial Eqaion Vol 5 o [6] T Tao M Vian an X Zhang Global Wll-Pon [8] J Dolbal an G Rin Tim-Dnn Rcaling an Scaring for h Dfocing Ma-Criical onlinan yanov Fncional for h Vlaov-Poion Ena ar Schröingr Eqaion for Raial Daa in High Di- Elr-Poion Sym an for Rla Mol of Ki- mnion Dk Mahmaical Jornal Vol 7 nic Eqaion fli Dynamic an Qanm Mchan- o 65- in Ali Sci- [7] T Tao M Vian an X Zhang Th onlinar Schrö- ic Mahmaical Mol an Mho nc Vol 7-3 ingr Eqaion wih Combin Powr-Ty onlinari- [9] J Collianr M Grillaki an Tziraki Imrov i Commnicaion in Parial Diffrnial Eqaion Inracion Morawz Inqalii for h Cbic onlinar Vol 3 o Schröingr Eqaion on R Inrnaional Mahmaic oi:8/ Coyrigh 3 SciR