Global solutions of nonlinear Schrödinger equations 0821819194, 9780821819197

This volume presents recent progress in the theory of nonlinear dispersive equations, primarily the nonlinear Schrodinge

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Global solutions of nonlinear Schrödinger equations
 0821819194, 9780821819197

Table of contents :
Cover
Title page
Contents
Introduction and summary
An overview of results on the Cauchy problem for NLS
Further comments
3D H¹-critical defocusing NLS
Global wellposedness below energy norm
Nonlinear Schrödinger equation with periodic boundary conditions
Appendix 1. Growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential
Appendix 2. Zakharov systems
References
Index
Back Cover

Citation preview

Selected Title s i n Thi s Serie s 46 J . Bourgain , Globa l solution s o f nonlinea r Schrodinge r equations , 1 99 9 45 Nichola s M . K a t z an d P e t e r Sarnak , Rando m matrices , Frobeniu s eigenvalues , an d monodromy, 1 99 9 44 M a x - A l b e r t Knus , A l e x a n d e r Merkurjev , an d Marku s R o s t , Th e boo k o f involutions, 1 99 8 43 Lui s A . Caffarell i an d Xavie r Cabre , Full y nonlinea r ellipti c equations , 1 99 5 42 V i c t o r Guillemi n an d S h l o m o Sternberg , Variation s o n a them e b y Kepler , 1 99 0 41 Alfre d Tarsk i an d S t e v e n Givant , A formalizatio n o f se t theor y withou t variables , 1 98 7 40 R . H . Bing , Th e geometri c topolog y o f 3-manifolds , 1 98 3 39 N . Jacobson , Structur e an d representation s o f Jorda n algebras , 1 96 8 38 O . O r e , Theor y o f graphs , 1 96 2 37 N . Jacobson , Structur e o f rings , 1 95 6 36 W . H . Gottschal k an d G . A . H e d l u n d , Topologica l dynamics , 1 95 5 35 A . C . Schaeffe r an d D . C . Spencer , Coefficien t region s fo r Schlich t functions , 1 95 0 34 J . L . Walsh , Th e locatio n o f critica l point s o f analyti c an d harmoni c functions , 1 95 0 33 J . F . R i t t , Differentia l algebra , 1 95 0 32 R . L . Wilder , Topolog y o f manifolds , 1 94 9 31 E . Hill e an d R . S . Phillips , Functiona l analysi s an d semigroups , 1 95 7 30 T . R a d o , Lengt h an d area , 1 94 8 29 A . Weil , Foundation s o f algebrai c geometry , 1 94 6 28 G . T . W h y b u r n , Analyti c topology , 1 94 2 27 S . Lefschetz , Algebrai c topology , 1 94 2 26 N . Levinson , Ga p an d densit y theorems , 1 94 0 25 Garret t Birkhoff , Lattic e theory , 1 94 0 24 A . A . A l b e r t , Structur e o f algebras , 1 93 9 23 G . Szego , Orthogona l polynomials , 1 93 9 22 C . N . M o o r e , Summabl e serie s an d convergenc e factors , 1 93 8 21 J . M . T h o m a s , Differentia l systems , 1 93 7 20 J . L . Walsh , Interpolatio n an d approximatio n b y rationa l function s i n th e comple x domain, 1 93 5 19 R . E . A . C . P a l e y an d N . W i e n e r , Fourie r transform s i n th e comple x domain , 1 93 4 18 M . M o r s e , Th e calculu s o f variation s i n th e large , 1 93 4 17 J . M . W e d d e r b u r n , Lecture s o n matrices , 1 93 4 16 G . A . Bliss , Algebrai c functions , 1 93 3 15 M . H . Stone , Linea r transformation s i n Hilber t spac e an d thei r application s t o analysis , 1932 14 J . F . R i t t , Differentia l equation s fro m th e algebrai c standpoint , 1 93 2 13 R . L . M o o r e , Foundation s o f poin t se t theory , 1 93 2 12 S . Lefschetz , Topology , 1 93 0 11 D . Jackson , Th e theor y o f approximation , 1 93 0 10 A . B . Coble , Algebrai c geometr y an d thet a functions , 1 92 9 9 G . D . Birkhoff , Dynamica l systems , 1 92 7 8 L . P . Eisenhart , Non-Riemannia n geometry , 1 92 7 7 E . T . Bell , Algebrai c arithmetic , 1 92 7 6 G . C . Evans , Th e logarithmi c potential , discontinuou s Dirichle t an d Neuman n problems , 1927 (Continued in the back of this publication)

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Globa l Solution s of Nonlinea r Schrodinge r Equation s

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http://dx.doi.org/10.1090/coll/046

America n Mathematica l Societ y Colloquiu m Publication s Volum e 4 6

Global Solutions of Nonlinear Schrodinger Equations J. Bourgai n

America n Mathematica l Societ y Providence , Rhod e Islan d

Editorial Boar d Joan S . Birma n Susan J . Friedlander , Chai r Stephen Lichtenbau m

1991 Mathematics Subject

Classification.

Primar

y 35Q55 .

ABSTRACT. Th e ai m o f thi s boo k i s t o describ e recen t progres s o n variou s issue s i n th e theor y of nonlinea r dispersiv e equations , primaril y th e nonlinea r Schrodinge r equatio n (NLS) . I n par ticular, th e Cauch y proble m fo r th e defocusin g critica l NL S wit h radia l dat a i s discussed . Ne w techniques an d result s ar e describe d o n globa l existenc e o f larg e dat a solution s belo w th e energ y norm. Curren t researc h i n Harmoni c Analysi s aroun d Strichartz ' inequalitie s an d it s relevanc e t o nonlinear P D E i s presented. Als o severa l topic s i n NL S theor y o n bounde d domain s ar e reviewed . In thi s respect , a partia l surve y i s give n o f th e theor y o f invarian t Gibb s measure s an d recen t developments i n KA M theor y fo r PDE's .

Library o f C o n g r e s s Cataloging-in-Publicatio n D a t a Bourgain, Jean , 1 954 Global solution s o f nonlinea r Schrodinge r equation s / Jea n Bourgain . p. cm . — (America n Mathematica l Societ y colloquiu m publications , ISS N 0065-925 8 ; v. 46 ) Includes bibliographica l reference s an d index . ISBN 0-821 8-1 91 9- 4 1. Schrodinge r equation . 2 . Differentia l equations , Partial—Numerica l solutions . 3 . Nonlin ear theories . I . Title . II . Series : Colloquiu m publication s (America n Mathematica l Society ) ; v. 46 . QC174.26.W28B681 99 9 bib'.353—dc21 99- 306 6 CIP

C o p y i n g an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given . Republication, systemati c copying , o r multipl e reproductio n o f an y materia l i n thi s publicatio n (including abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P . O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mai l t o [email protected] . © 1 99 9 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e a t URL : http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 0

4 03 02 01 00 99

Contents Introduction an d summar y I. A n overvie w o f result s o n th e Cauch y proble m fo r NLS 5 1. Equation s 5 2. Wellposednes s o f th e Cauch y proble m 6 3. Scatterin g result s 0 1 4. Estimate s o n th e linea r grou p 1 1 5. Solvin g th e Cauch y proble m 4 6. Derivativ e nonlinea 1 r Schrodinge r equation s 4 II. Furthe r comment s 1. Constructio n o f blowup solutions for conformal NLS from the groun d state 2. Behaviou r o f highe r Sobole v norm s 2 3. Fourie r restrictio n theor y beyon d 1 ? 3 4. L 2 -concentration phenomeno n 3 5. Th e Schrodinge r maxima l functio n 4 6. Derivativ e nonlinea r Schrodinge r equatio n 4

7 7 8 3 7 5 8

III. 3 D ./^-critica l defocusin g NL S 5 1 Consider 3 D NLS 5 1 Fix a tim e interva l I = [0 , T) 5 1 Sketch o f th e argumen t 5 4 A concentratio n propert y 5 9 A versio n o f Morawet z inequalit y 6 1 Construction o f a n appropriat e tim e interva l 6 3 Details o n th e perturbativ e analysi s 66 A varian t o f th e metho d 7 0 IV. Globa l wellposednes s below energ y nor m 7 1. Descriptio n o f th e metho d 7 2. Th e exampl e o f the NL W 7 3. Th e cas e o f th e nonlinea r Schrodinge r equatio n 8 4. Symplecti c capacitie s an d symplecti c Hilber t space s 9 5. Globa l wellposednes s o f th e NL W (4.41 ) 9

9 9 9 3 3 7

V. Nonlinea r Schrodinge r equatio n wit h periodi c boundar y condition 1 s 0 1. Introductio n 0 1 2. Result s o n th e Cauch y proble m 0 1 3. Periodi c Strichart z inequalitie s 0

5 5 6 8

viii C O N T E N T

S

4. Sketc h o f proo f o f Theorem s 2. 1 an d 2. 7 5. Invarian t Gibb s measure s (ID ) 6. Invarina t Gibb s measure s ( D > 1 ) 7. Invarian t Gibb s measure s (unbounde d domains ) 8. Quasi-periodi c solution s Appendix 1 . Growt h o f Sobole v norm s i n linea r Schrodinge r equation s smooth tim e dependen t potentia l Appendix 2 . Zakharo v system s References Index

http://dx.doi.org/10.1090/coll/046/01

0. Introductio n an d summar y Despite th e attentio n thi s theory ha s received ove r recent years , there ar e man y problems lef t essentiall y unsolve d concernin g th e longtim e behaviou r o f solution s to th e Cauch y proble m fo r th e nonlinea r Schrodinge r equatio n (NL S fo r short ) iut + Au±u\u\ p'2 = s

0 (o-i)

d

u{0) = (/)eH {R ) with Hamiltonia n H(4>)

-I

>I2^I

dx. (0.2

)

Although th e initia l valu e proble m (IVP ) theor y i s satisfactor y fo r loca l tim e be haviour an d smal l data , man y issue s o n th e behaviou r o f solution s fo r larg e dat a are fa r fro m understood . I n th e followin g cas e (i.e . " + " sig n i n (0.1 )) , i t i s wel l known tha t fo r p > 2 + | , smoot h solution s o f (0.1 ) ma y blowu p i n finit e time . There i s a vas t problemati c i n thi s context , concernin g question s suc h a s blowu p speed, blowu p profil e an d it s stabilit y etc. , pursue d bot h purel y mathematicall y and numerically . Thi s body o f problems wil l not b e our primaril y issu e here an d w e will onl y commen t o n a fe w aspect s o f recen t research . W e wil l rathe r concentrat e on equatio n (0,1 ) i n th e defocusin g case , whe n i t i s expecte d tha t loca l solution s extend t o globa l one s an d preserv e thei r i/ s -class fo r al l time , wit h scatterin g be haviour fo r sufficientl y hig h degre e nonlinearity . W e ar e particularl y intereste d i n two problem s tha t w e describe briefl y next . (i) Th e /^-critica l equatio n Consider th e NLS iut + Au - u\u\ p-2 = 0 , p = 2 + — ^ - ( d > 3 a—2

) (0.3

)

for whic h th e homogeneou s H 1 -space H 1 i s the scal e invarian t Sobole v space . I t i s known tha t ther e i s loca l wellposednes s fo r an y dat a G Hs, s > 1 and th e resul t is globa l fo r dat a smal l i n H 1 . I t i s a n ope n proble m whethe r classica l solution s exist globa l in time. Remar k tha t sinc e the Hamiltonia n (an d th e L 2 -norm) provid e the onl y aprior i bound s o n th e solution , als o a classica l theor y need s t o includ e a i

2

J. B O U R G A I N

considerable componen t tha t i s purel y H 1 . W e hav e solve d th e questio n fo r radia l data fo r d = 3,4 , provin g globa l wellposednes s an d scatterin g i n th e energ y spac e and an y H s, s > 1 . Th e correspondin g resul t fo r th e nonlinea r wav e equatio n (NLW) By + y?- 1 = y tt - Ay + y^ 1 = 0 (0.4 ) was establishe d som e tim e ag o b y Struw e [Str ] i n th e radia l cas e an d b y Grillaki s [Gr] i n general ; se e als o th e pape r [S-S] . Th e mai n proble m i n th e NLS-cas e i s that th e correspondin g Morawetz-typ e inequalit y i s apriori t o wea k t o exclud e H 1 concentration phenomena . Thi s i s the mai n issu e i n thi s questions . The result s fo r NLS appear i n [Bl] . I n 3D , the proo f presente d her e is a bit les s technical w e believe . Th e proble m i n 4 D (an d highe r dimension ) come s fro m th e lower degre e nonlinearit y (th e quinti c nonlinearit y i s exploite d i n th e 3 D proof) . The metho d followe d her e fo r th e equatio n iut + Au - u\u\ 2 = 0 (0.5

)

in 4 D compare d wit h th e presentatio n i n [Bl ] i s les s dependen t o n th e particula r nonlinearity. For genera l (non-radial ) data , th e proble m o f global wellposednes s i s still open , also fo r classica l solutions . ( (ii) Wellposednes s belo w th e energ y nor m The IV P f iu t + Au±u\u\ p-2 = 0, N \' (0.6 ) is locall y wellpose d i f we assum e s >0 and i f p > 2 + \ 4 s > s* , s * define d b y p = 2 + - —-—. (0-7 ) a — 2s* Moreover, i f s > s* , th e tim e interva l A T ma y b e bounde d below b y a functio n o f ||0||i*s. I t follow s tha t i n th e defocusin g case ther e i s globa l wellposednes s i n th e energy spac e provide d p < 2 -f -^^. Ou r interes t her e i s to ge t globa l result s belo w the energy-norm . A n optima l resul t woul d b e t o sho w tha t i n th e defocusin g case , under assumption s (0.7) , th e loca l solutio n o f (0.6 ) extend s t o a globa l one . Thi s is unknow n fo r larg e data , eve n i n th e L 2 -critical cas e

V = 2 + 2 (°-

8

)

(the conforma l equation) . We di d howeve r develo p a ne w an d rathe r genera l metho d t o obtai n globa l wellposedness result s fo r dat a (f) G H s, fo r certai n s < 1 . Thi s metho d exploit s the aprior i boun d o n th e i^-nor m fro m th e Hamiltonia n conservation , althoug h the dat a i s below tha t threshold . I t i s based o n decomposin g i n a suitabl e wa y th e solution i n it s lo w an d hig h Fourie r modes . A s a n example , th e followin g fac t i s established i n [B2] .

GLOBAL SOLUTION S O F NONLINEA R SCHRODINGE R EQUATION S 3

The IV P i n 2 D

( iu t + A ^ - u\u\ 2 = 0

I u(0)=(f>eH s(R2) (0

"9)

is globall y wellpose d provide d s > | an d moreove r u{t) - e

itA

1 (p e H1 fo r al l tim e . (0. 0

)

As mentioned, ou r techniqu e ha s genera l feature s an d i s not restricte d t o NLS . We have investigate d als o certai n example s o f nonlinea r wav e equation s f Uy + d yf(x, y) = y tt - Ay + d yf{x, y) = 0

The cas e s = \ i s o f particula r interes t sinc e thi s correspond s t o th e symplecti c space. Establishin g a globa l flow o n th e symplecti c Hilber t spac e i s certainl y o f interest i n vie w o f applyin g th e symplecti c capacit y theor y (considerin g sa y peri odic boundar y conditions ) a s develope d b y man y author s startin g fro m Gromov' s work (ou r referenc e wil l be [Kukl ] fo r th e theor y i n infinit e symplecti c dimensiona l symplectic phas e space) . Global wellposednes s i n th e symplecti c spac e i s prove n i n particula r fo r th e NLW ytt -Ay 1 + py + y 3 = 0 (0. 2 ) with periodi c b e i n D = 1 , D = 2 . Thes e result s ar e als o new . The firs t chapte r o f th e pape r i s more o f a surve y type . W e als o indicat e som e results o n derivativ e NL S o f th e for m iut + A w + F(u, u, V1 x u, V xu) = 0 (0. 3

)

(cf. [K-P-V ] an d subsequen t papers) . Thi s topi c i s agai n a mos t interestin g issu e that wil l no t b e considere d here . I n fact , i t i s fai r t o sa y tha t mos t o f th e theor y around (0.1 3 ) deal s wit h loca l i n tim e results , excep t fo r smal l data . In chapte r II , w e will commen t o n a fe w relate d direction s o f curren t researc h that wil l no t b e develope d furthe r here . Thes e includ e (1) Perturbation s o f th e groundstat e solutio n fo r th e conforma l NL S i n th e focusing cas e an d application s t o blowu p solutions . (2) Fourie r restrictio n theor y beyon d L 2 ; relatio n t o problem s o f combinatoria l type suc h a s th e dimensio n conjectur e fo r Besicovitc h sets ; application s t o the maxima l functio n associate d t o th e linea r Schrodinge r grou p an d t o L 2 -concentration phenomen a fo r NLS . (3) Furthe r result s o n derivativ e NLS . In chapte r III , w e discus s th e defocusin g i7 1 -critical NL S (0.3 ) i n th e radia l case. In Chapte r IV , w e conside r th e proble m o f establishin g globa l solution s belo w the energ y nor m fo r defocusin g iJ 1 -subcritical NL S an d NLW . In Chapter V of this paper, w e survey investigations relate d t o NLS on bounde d spatial domains , mainl y th e cas e o f periodi c b.c . Th e problem s her e ar e differen t

4

J. B O U R G A I N

from th e R d -case, partl y becaus e o f th e absenc e o f dispersion . Beside s th e Cauch y problem, w e will discus s result s an d problem s relate d t o invarian t Gibb s measure s and th e existenc e an d persistenc y o f invarian t KA M (Kolmogorov-Arnold-Moser ) tori. Agai n al l thes e topic s ar e activ e researc h areas . There ar e tw o Appendice s included . Appendix 1 deals with the problem o f growth o f higher Sobole v norm s i n linea r Schrodinger equation s wit h bounded , smooth , tim e periodi c potentia l V = V(#,£) , thus o f the for m iut + Au + V(x,t)u =

0 u(0)

1 = (f)eH s (0. 4

)

(periodic be) . Althoug h i n th e nonlinea r context , thi s proble m i s fa r fro m under stood, fo r equatio n (0.1 4 ) a ver y satisfactor y an d surprisingl y genera l resul t ma y be show n (i n an y dimension ) \Ht)\\Hs
oo , fo r al l e1 1 > 0. ( . 5

)

Observe tha t ther e i s n o specifie d behaviou r o f V i n tim e t, beside s smoothness . We conside r th e D = 1 case. Se e [B1 4 ] fo r genera l dimensio n an d furthe r results . In Appendi x 2 , w e wil l summariz e researc h ove r th e recen t year s o n th e Za kharov syste m ( %u t — —Au ±nu {ntt-c2An = c 2A(\u\2) ( 1 °" 6) (the physica l meanin g o f u, n, c are respectivel y th e electrostati c envelop e field, th e ion densit y fluctuation field an d th e io n soun d speed) . Th e cubi c NL S 2 iu 1 t + Au ± u\u\ = 0 (0. 7

)

may thu s b e viewe d a s th e limi t o f (0.1 6 ) whe n c —» oo. Global existenc e o f classica l solution s fo r th e defocusin g 3 D equatio n wa s onl y proven recentl y (i n join t wor k wit h J . Colliande r cf . [B-C]) . Considerin g periodi c be, w e will als o discus s th e invarian t measur e proble m i n ID . The presen t Note s ar e base d o n AM S Colloquiu m Lecture s give n i n Cincinatt i (1994), lecture s give n a t Par k Cit y i n 1 99 5 an d UCL A 1 998 . Par t o f th e materia l is no t publishe d elsewhere .

http://dx.doi.org/10.1090/coll/046/02 GLOBAL SOLUTION S O F NONLINEA R SCHRODINGE R EQUATION S 5

I. A n overvie w o f result s o n th e Cauch y proble m fo r NL S 1. Equation s A firs t distinctio n shoul d b e mad e betwee n equation s withou t (res p with ) a presence o f derivative s i n th e nonlinearity . Thu s iut - f Au + F(u,u) — 0 (withou 1 1 t derivatives ) ( .

)

iut + Au + F(u,u, V xw, V xu) = 0(F involvin g firs t orde r derivatives) . (1.2) In thi s chapter , ou r spatia l domai n wil l b e mainl y M d,d = 1 ,2,3 . Th e cas e o f bounded domains , sa y periodi c boundar y condition s (x G T d = d dimensiona l torus) wil l b e mor e th e subjec t o f Chapte r V . The Cauch y proble m fo r (1 .1 ) ha s bee n extensivel y studie d an d shar p result s obtained, especiall y i n th e cas e A(| U|P) „

F(u,u) =

|

U|P-2U.

(L3

)

The equatio n 9H0/ _ iut + Aw + -^r\ u, u) ou ou is Hamiltonian , wit h Hamiltonia n

x

„ dH =0 = iu t + ~z=r

H{4>) = \j IV), H 0(4>) = 1 \\" ( .4

)

preserved unde r th e flow . The "natural " symplecti c Hilber t spac e i s the spac e L 2 wit h canonica l coordi nates (formally ) (Rett , Imu). In th e cas e (1 .3 ) o r mor e generall y H0 = H 0(\u\2) there i s als o conservatio n o f th e L 2 -norm 1/2

(M under th e flow . In cas e (1 .3) , i.e .

iut +1 Au + Xu\u\ p"2 = 0 ( .5

)

P ^/W~/M 1 ( .6

)

with Hamiltonia n

we distinguish th e case s A >0 = A so s = So : critica l s > SQ : subcritica l

22

->

GLOBAL SOLUTION S O F NONLINEA R SCHRODINGE R EQUATION S 7 THEOREM 1

. (local wellposedness).

Assume u(0) — ip G Hs, s > 0 and s > SQ - Assume also p — 2 > [s] if p & 2Z. TTien £/i e Cauchy problem J zrz t + A u + A i ^ | p " 2 = 0 \ w (o) = cj)eH s is wellposed on a nontrivial time interval [0 , T* [ and in particular ueCHs([0,T}),T SQ, T * > T(s , ||y?||ijs ) and the flowmap is Lipschitz on a neighborhood of . Remark 1 . I n th e critica l case , maxima l existenc e tim e depend s o n

SQ) (3) Defocusing case, p < 6 for d = 3 and if G H 1 (problem H l -subcritical and use of Hamiltonian conservation) Also true for d = 3, p = 6 and Lp G H1 a radial function, i.e. H case (recent result discussed in Chapter III) (4) Defocusing case, p < 6 for d — 3, ip G Hs(s > so) and \x\ip G L 2

1

-critical

(use of apriori bound on \\u(t)\\ p from pc conservation law (1 .1 0)) Moreover, additional smoothness of data cp is preserved under the flow (provided compatible with smoothness of nonlinearity (see remarks below)).

J. BOURGAI N

Remark 1 . Fo r p > 2 + | i n th e focusing case , (sufficientl y large ) smoot h solutions ma y blowu p i n finit e tim e (i) Glassey' s virie l inequalit y dt2

J \) (

(2.4)

c > 0) .

Hence, i f H{4>) < 0, blowu p ha s t o occu r

(ii) Fo r p = 2 + | , ther e ar e construction s o f explici t blowu p solution s (o f minima l L 2 -norm) fro m groundstat e an d p c transformation . THEOREM 3 . (F. Merle, [Ml])

Let u be a solution of iut + Au + u\u\^ d =

0

u(0) = where (j) G H1 and

\\4>h = \\Qh where Q denotes groundstate, i.e. (unique) solution of (2.5)

AQ - f Q 1 + 4 / / d = Q,Q positive and radial. Assume u blows up at time T > 0. Then there exist 6 G R , u > 0, x0 G Md ,xi G Rd such that d/2

u(t,x)

%{9+{t-T)- 1 \\_x-xjy

T-t

T-t,

LOXQ

(2.6)

(Uniqueness of minimum L 2-norm blowup solutions). T H E O R E M 4 . [M2]

Given a time T and distinct points x i , . . . ,XK £ ^ d, there is a solution u of iut + Au + u\u\ 4/d = 0 which blows up exactly at time t = T in the points { x i , . . . , XK} with concentration of all the L 2-mass on this finite set of points.

G L O B A L SOLUTION S O F N O N L I N E A R S C H R O D I N G E R E Q U A T I O N S 9

T H E O R E M 5 . [B-W]

Let d — 1, 2 (for smoothness reasons). Denote uo the explicit blowup solution at (T, x\) ofiu t + Au + u\u\4/d = 0 given by (2.6). Then one may construct solutions u — UQ + v on [0,T[ where v is smooth (T^ 0 ) extending smoothly after blowup time T and solving for T < t < T + 1 the IVP f iv t + Av + v\v\4/d = 0 I v(T) = . Here 1 and p < 6 for d = 3 and nonlinearit y sufficiently smoot h (i) I f sup 11^(^)11^1 < o o (in particular i n the defocusing case) , the n u(t) G H s for al l time and C l \Ht)\\Hs i an d any unit cub e Q , there i s the bound sup \e

it A ,

\LKQ)
2, q > 2 and - = d q \2 p 2(d+2) The followin g inequalit y generalize s (4.2 ) (wher e p — — j

1)

||e" A eL

=

^

A

2

2(d + 2)

Apply Picard' s theore m i n spac e L x t

d

(4.2) => \\e itA(f)\\p < C||0|| 2 (assume A

{u\ur2){r)dr\\Ll
1 1 e L 2{Rd)} ( . 7

)

endowed wit h th e natural nor m

u\\xA = \\\\ H- + \\(i + \x\) An^. (Lis

)

Denote als o VA = l e X A\Da(ti) = 0 for al l \a\ < A - [ ^ 1 j .

(1.19)

Recall als o tha t i f 0 G XA , then th e IVP J iu t + Au + u\u\4/d = 0

(1.20)

I u(0) = 0 has a local solutio n z^ in a neighborhood [—5 , S] of 0, satisfyin g 1 z^ 1 e C([-6, 6} : XA)- ( .2

)

If moreove r ||||L 2 i s sufficiently small , the n thi s loca l solutio n extend s t o a global one and it may b e shown tha t \Mt)\\xA ( .64

)

and rewrit e (1 .60 ) as iwt - Lw + aw + bw-\- G(w) 1 + /0 = 0 . ( .65

)

Here G(w) i s at least quadrati c i n w. In orde r t o produc e a solutio n o f (1 .65 ) o n [^,oo[ , w e solve th e equivalent integral equatio n oo

/

e+l(-T't)L[f0 + aw + bw + G(w)]{T)dT. 1( .66

)

24

J. B O U R G A I N

This procedur e i s reminiscen t o f th e wav e ma p constructio n i n scatterin g theory , except tha t her e th e referenc e equatio n i s the nonlinea r equatio n (1 .1 ) . Our ai m i s to deriv e th e boun d (1 .56 ) fro m (1 .66) . We first establis h som e bound s o n VQ. Pro m (1 .50) , (1 .58 )

Mx,t) =

±e$-«z+fc,-l..

(1.67)

Since (j) G VAX C XA X, w e hav e b y (1 .21 ) (1.68)

\\z*{t)\\xAl\ y (1.73)

and thus , b y (1 .68 ) \\Dav0

C °° - id/2

Z

1 C < - ^ fo r t > - an d \a\ < Ai - 2 . 1( .74 )

Also, b y (1 .70) , (1 .72 ) e-cMDav0(x, t)\

< jxj2ioTt>^

an

d

H ^ - y

(1.75)

25

GLOBAL SOLUTION S O F NONLINEA R SCHRODINGE R EQUATION S

We stil l mak e repeate d us e o f estimatio n (1 .74) , (1 .75 ) i n wha t follows . From (1 .75) , i t follow s tha t fo r s < ^ C

(1.76)

II(^O)WIIH-
c

(4.22 )

2 t f

I

JB(0,1

erf. (4.65

)

Next, appl y Lemm a (4.44 ) t o g = e~lT°Afri an d A = Ari, e = rj. Le t {Qs} be the regions (4.45) . I t follows fro m (4.27) , (4.46 )

/ / \Pru(t)f

| e ^ ) A / r i | 2 < ||u|| 2L4[To;Ti].,2 = n^ < of.

(IR3\uQs)n(R2x[To,T1])

(4.66) Hence (4.65) , (4.66 ) yield s som e Q of the form (4.45 ) suc h tha t

J J \P Qn(M2x[T0,Ti])

Tu(t)\

2

\e«-T^Afri\2 > ^ =

ih (4.67

)

44

J. BOURGAI N

and henc e

4

jj \PMt)\

> c V2 (4.68

)

{(x,t)\x+2t^0eI,t p. (5.

)

Dualizing Tk (u p t o Fourie r transform) , w e need t o conside r Vfc(ICI) (5- 2 Jg(x) jM+W^dx1

(TfcflX O = acting from L 2(B{0,1 )) t

2

oL

)

(Rd).

(In this discussion , w e restrict ourselve s t o a loca l control o f the maxima l func tion). Applying th e usua l squarin g procedure , w e fin d / ra

2

(£R =

j9{x)W) K(x,y)dxdy 1 (5. 3

)

l*l 2Vfc(|£|)d£. 1 (5. 4

)

with c g

(5.20 )

for s u p 0 < f < 1 \e ltAf\ t o b e bounde d i n L 2 oc . The mai n point s ma y b e summarize d a s follows . (i) Ou r goa l i s t o sho w tha t i(xi+te)

sup 0 M (3.3 ) write J a s a unio n o f 3 consecutive interval s J = J - U Jo U J+ with HLio >

M — . (3.4

)

We wil l perfor m a constructio n o n J o an d the n g o eithe r forwar d o r backwar d i n time contradictin g (3.4 ) fo r eithe r J + o r J _ . Assume als o w e alread y establishe d Theore m 1 and a boun d \\W\\L%t < M i (3.5

)

for an y IV P iWt + AW- W\W\ 4 = 0 W\ provided H(W(0)) < H(4>) - v 4. (3.7

t=0

= W(0) (3.6

) )

Here rj will b e a fixe d smal l numbe r (onl y dependin g o n H( CK\I\ 1 '2 (

C = C„)

satisfying obviou s derivativ e estimate s an d moreove r

IK*o)|| 2 ^ < IN*o)|| 1 ^ - Jr? 3. (3. 6

)

That thi s ma y b e realize d result s fro m (3.1 1 ) . We conside r firs t th e IV P (3.1 3 ) whic h w e clai m t o b e globall y wellpose d o n [£o,oo[. Writ e [t 0 ,oo[= [t 0,&[U[&,oc[. On / = [t 0,b], w e have , c f (3.1 0 )

and fro m th e integra l equatio n [to , b] u(t) = e^- to)Au(t0) -

T)A

(u\u\4)(r)dr

i f e^J to

one get s tha t \\e^-^Au(t0)\\Llotei
-e2 (8.39 implying (8.34) .

)

74

J. BOURGAI N

LEMMA 8.40 . Assume

iut + Au — u\u\ 2 — 0 «(o) = ^ , | M l i j i < c satisfies (8.41)

Hwll^,oeH (2.6

) )

and th e Hamiltonia n i s give n b y

H(u) = J

l

-\Bu\2+ \{Reuf

dx. (2.7

)

Remark 1 . Th e symplecti c Hilber t spac e (i n th e sens e o f [Kukl] ) fo r (2.1 ) and, mor e generally , NL W o f th e for m ytt-Ay +

py + f(y)=0 (2.8

)

has inne r produc t [4>^} = (4>,BTP) (2.9

and thu s equivalen t t o H

)

1 2

/.

Remark 2 . I n th e defocusin g case , th e NL W (2.8 ) wit h polynomia l nonlin earity i s alway s (globally ) wellpose d i n energ y spac e i n ID , 2 D (i n 3D , quinti c nonlinearity i s critical) . PROPOSITION 2.1 0 . The

IVP

for NLW

1 yu-y 1 py + y 3 = 0 (2. Xx +

)

1 (v{0),y{0))tHaxHa-\s>± (2. 2

)

is globally wellposed. Moreover, denoting St the flowmap for (2.1 1 ) and S(t) the flowmap for the linear equation 1 vu -y xx+ py = o (2. 3

)

one has that (St - S(t)) (y(0) , 2/(0) ) eH 1 x L 2 for all 1 time t. (2. 4

)

Proof. Conside r equatio n (2.5 ) iii - Bu - 1 B' 1 ((Reu)3) - 0 (2. 5

)

1 = e# s , s> \. (2. w (0)

6

)

I(t) = 1 H(u{t) - e- iBt4>N) (2. 7

)

with

Fix a tim e T and conside r th e expressio n

GLOBAL SOLUTION S O F NONLINEA R SCHRODINGE R EQUATION S 8 1

with H give n b y (2.7 ) 1 (/) = (f) N + N (2. 8

)

lXx N = P N(p 1 = J 0(\)e d\ (2. 9

)

J\X\(\)e

iXx

d\ (2.20

)

J\X\>N

and wher e N wil l b e specified , dependin g o n T . Denote 1 u = u-e- itBN. (2.2

)

€t(0) = 0i v (2.22

)

l|fi(0)||*. < ^V 1 _S (2-23

)

/(0) = H(u{0)) < CN 2{1 ~s) + C ~ JV 2 ^-"). (2.24

)

Thus and hence , b y (2.1 6 )

Our ai m i s to preserv e propert y (2.24 ) fo r 0 < £ < T , thu s 2

J(t) = ff (fi(t) ) = i | |B(fi(t))|

+ \ j[Reu(t)\*
s\ fo r som e si < 1 . 3. T h e cas e o f t h e nonlinea r Schrodinge r e q u a t i o n We consider th e specia l cas e o f the defocusin g 2 D NL S wit h cubi c nonlinearit y 1 iut + Au-u\u\ 2 = 0 . (3.

)

Although som e of the technical details below are specific t o that particula r example , the genera l metho d an d ideolog y extend s t o th e genera l settin g o f i7 1 -subcritical NLS. W e will prov e th e followin g fac t PROPOSITION 3.2 . Consider

the 2D IVP

iut + Au-u\u\ 2 =0

u(o) = (/)eHs(m2),s> §

.[

'

}

Then (3.3) is globally wellposed with solution u satisfying u(t) - e

ltA

(p e H 1 for all time (3.4

)

and \\u(t) - e

itA

4>\\m < C ( l + | t | ) ^ * . (3.5

)

The proo f o f Prop. 3. 2 doe s no t see m to follo w fro m a simple argumen t a s use d above i n th e NLW-cas e an d wil l b e mor e involved . R e m a r k . Technica l refinement s o f th e argumen t belo w permit s u s t o weake n the assumptio n s > | i n (3.3 ) t o th e conditio n s > | .

84 J

. BOURGAI N

Fix a large tim e T an d let No = iVo(T ) b e a cutoff (t o be specified). Writ e 4> = 4> 0 + 1P0 wit h 2\\H.-l (3-4

| | « 2 | | x o l (3.42

)

)

'2

( u l U 2 ) | | L 2 [ / ! < C | | U l | | x i + i + | | u 2 l U 0 + , x + (3.43

)

We no w retur n t o th e expressio n (3.1 8 )

H U ^ j ) i + (tmo)||2 < C\\u 0\\Xi+i+ ||«||x

0+ii+

< CN^ +No'+ (3.49

)

hence, fro m (3.48) , (3.49 ) (3.46) < W 0 1 -2 " + . (3.50

)

Thus 2 2a+ \(W,D 1 . (3.5 x(\uo\ v))\ R T^ (4

-38)

(r— denote s an y numbe r < r ) or sup \\u(t)\\

Hi/2

=

o o (4.39

)

0 angle (n,n f) = 0(n 1 n/) ~

2

_r

(5.22 )

GLOBAL SOLUTION S O F NONLINEA R SCHRODINGE R EQUATION S 9

9

Estimate fro m (5.1 9) , (5.20 )

£lKo,a||£j>t

(5.23)

a

a

+ Y Y^

^a'hly

(5.24)

r

ni,A

an 2

\ \ n£A Y- ,a' I^ I 2 r

(5.29)

Next estimat e #G5m,A - Thu s #{n€ A

2-rjan(ni+i42-rja/)

|n| - | n - n i | - A l = 0 ( 1 ) } . (5.30

)

One finds tha t |V n (|n| - \n-ni\

nn — ri\ \n\ \n-m\

• 0(n,n — ni) ~ 2

(5.31)

from (5.22) . Thi s restricts i n particular n to an 2r -neighborhood o f the intersectio n of the hyperbola \n\- I n - m l - A (5.32 )

J. B O U R G A I N

100

with th e ball B(no,Ni). Henc

e # 6 n , A < 2 r i V 1 . (5.33

)

By (5.27) , (5.29 ) _/

\ r

1 / 2

i/2

n^-^^iiL^ = Y,^y nei

n x

-.

(5.57)

102

J. B O U R G A I N

Clearly / *i* JT 2

V/ j JT

*

2

3

*

|P/*l||*2||*3|P/*4|


AT

N^N-

11 /2 2

Nl .

(5.62 )

1 2

/.

Estimate by

||(Aiv2u)(A^«)||^/2 IIA^tiH^HA^fill^ < 1 rV2i

^"HA^fiiiriiAjvsfiiir^fi) ^^

/2

(5.63 )

G L O B A L S O L U T I O N S O F N O N L I N E A R S C H R O D I N G E R E Q U A T1 ION S 0

3

Coming bac k t o (5.58) , on e get s afte r ^-integratio n fro m (5.60) , (5.61 )-(5.63 ) the following estimates . Case (i )

T.N1'2.N*/*N;3/6N1/2(N2N3)1'i\\N\\H1,2 = TNN¥2N¥4N-3/S\\4>N\\H1/2 (5.64

)

and summin g ove r th e correspondin g dyadi c level s give s TATAf 1 / 4 ^ 1 / 8 ^" 3 / 8 11^11^/2 = TN\\(f>N\\Hi„. (5.65

)

Case (ii)

TAr 1 / 2 7V 2 1 / 4 Ar- 3 / 8 iV 3 / 4 iV- 1 / 2 iV3 1 / 2 ||^|U l / 2 = TAT 5 /4 A r-i/4 7 V _i/2 A r -3/8 | |^||^ i / 2

_66)

(5

contributing for T.N5/4N-1/8N1/4N-3/8\\4>N\\Hi/2=TN\\N\\Hl/2. (5.67

)

Case (iii)

= TN 3'2 NSince

E ^

1 /4

iV1/24dx

dt< I \\B^

2\\(&N2u){&NMU\\&NMi oc . Thi s prove s Prop . 5.2 .

http://dx.doi.org/10.1090/coll/046/06 G L O B A L S O L U T I O N S O F N O N L I N E A R S C H R O D I N G E R E Q U A1 TION S 0

5

V. Nonlinea r Schrodinge r equatio n wit h periodi c boundar y condition s 1. Introductio n Our ai m here i s to discus s som e o f th e problem s an d theor y fo r NL S on a bounded spatia l domain . Mor e precisely, w e consider th e case of periodic boundar y conditions i.e. 1 1 u(t)eHs{Td) ( . ) where T d stand s fo r the d-dimensional torus . I n thi s setting , w e do not expec t of cours e dispersiv e effect s an d the natural problemati c i s different fro m th e En case. Evidently , a firs t issu e i s agai n th e Cauchy problem , thu s loca l an d globa l wellposedness o f the IVP iut + A w + Xu\u\ p~2 = 0

u(o)==0e# s (T d ) ^'

2)

taking into account th e strength of the nonlinearity an d regularity of the data oc . Amon g th e many "natural " questions on e may ask , th e following issue s wil l be considered her e (1) Assumin g globa l wellposednes s obtaine d fo r (p E Hs an d a solution u = u^ i n CH S([O, OO[) , is sup£ ||TA(£)||H S finite o r possibly not ? I f the secon d alternative occurs , wha t ma y be said abou t th e growth o f ||w(£)||# s fo r te oc ? (2) Existenc e of invariant measure s for the dynamics on various phase spaces. (3) Existenc e an d stability theor y o f KAM-tori correspondin g t o periodic , quasiperiodic an d almost periodi c solutions . These problem s ar e either no t resolved o r our understanding i s far from satis factory. Concernin g (1 ) , one may roughl y prov e bound s o f the for m \\u(t)\\Hsoc ( .3

)

assuming s > 1 , compatible wit h th e smoothness o f the nonlinearity an d the IV P for iut + Au — u\u\ p~2 — 0 globally wellpose d i n H1 (w e will requir e subcriticalit y p < 2 + ^ 2 o r a n e v e n stronge r restrictio n o n the nonlinearity). Thi s resul t will mainly appea r a s a byproduc t o f the analysis aroun d th e local Cauch y problem . It shoul d b e said tha t th e issue o f growt h o f highe r Sobole v norm s ma y also b e brought u p in th e M d-case, especiall y whe n dispersio n i s abscen t o r no t known . Similar consideratio n ma y als o her e produc e uppe r bound s o f the for m (1 .3) . Coming bac k to the periodic proble m fo r particular equation s o f the for m iu t + Au — u\u\ p~2 = 0 we do not kno w o f example s howeve r wher e u G CH S{[O, OO[ ) and sup t ||^(£)||ij s = o o (such example s ma y b e produced considerin g slightl y mor e

106

J. B O U R G A I N

general (smooth ) nonlinearities) . A fortiori, w e do no t kno w i f the (rathe r general ) power-like boun d (1 .3 ) i s i n som e senc e bes t possible . Regarding (2) , invarian t measure s ma y b e produce d b y suitabl e normalizatio n of th e Gibb s measur e where H((p) = \ J |V0| 2 ± - / \(f)\ p denotes th e Hamiltonian . Thi s measur e i s absolutely continuou s wit h respec t t o Wiene r measure , induce d b y th e Gaussia n process

where th e {g n} ar e independen t normalize d comple x Gaussians . Fo r D = 1 , th e threshold o f (1 .4 ) i s H^~l; fo r D > 1 , (1 .4 ) define s a field. Consequently , th e construction (1 .4 ) doe s no t defin e a n invarian t measur e o n a smoot h phas e space . Observe that , i n general , fo r (1 .2 ) w e do not hav e highe r orde r conserve d quantitie s at ou r disposa l (thi s is the case for th e integrabl e I D cubi c NLS iu t + uxx±u\u\2 = 0 for whic h th e Hamiltonia n H((j)) i n (1 .4 ) ma y b e substitute d b y othe r conserve d quantities). Th e mai n proble m fo r (1 .4 ) i s to construc t a welldefine d dynamic s fo r data suc h a s (1 .5) . On e ma y achiev e thi s fo r variou s model s i n ID , 2D , 3D . Investigations aroun d (smooth ) invarian t tor i for Hamiltonian PDE' s on bounded domains i s a relatively recen t lin e of research. Bu t i t alread y produce d a number o f satisfactory result s an d moreove r lea d t o th e developmen t o f new method s tha t ar e of interest als o from a purely classica l point o f view. Th e mai n difference s her e wit h the traditiona l KA M proble m i s tha t first th e dimensio n o f phas e spac e i s > 2 x (dimension invarian t tori ) an d secondly , the phase spac e is infinite dimensional . Ex tensions o f KAM theor y fo r lowe r dimensional tor i i n finite dimensiona l phas e spac e have bee n investigate d b y variou s authors , includin g Melnikov , Eliasson , Kuksin , Poschel. Th e mai n drawbac k o f thos e result s i s tha t th e nonresonanc e condition s between norma l an d tangentia l mode s ar e t o restrictiv e fo r furthe r developmen t t o most PDE-settings . Mor e precisely , multiplicitie s ar e no t allowe d an d thos e ar e difficult t o avoi d fo r PDE' s i n spac e dimensio n D > 2 . Thi s consideratio n ha s bee n the mai n incentiv e fo r developin g differen t approaches , startin g fro m th e pape r o f W. Crai g an d C . Wayne [Cr-Wa] . Presently , time-periodi c solution s (correspondin g to 1 -dimensiona l tori ) ma y be produced fo r arbitrar y D and quasi-periodi c solution s (finite-dimensional tori ) ma y b e constructe d fo r NL S i n D = l , D=2 . I n ID , ther e are als o som e recen t work s o n infinit e dimensiona l tor i (correspondin g t o almos t periodic solutions ) o n a ful l se t o f frequencies . Man y differen t aspect s o f thos e problems ar e no t full y resolve d howeve r a t thi s point . In th e nex t sections , w e briefl y surve y th e topic s mentione d above . I n [B6] , a more elaborate discussio n ma y be found. W e will also try t o indicat e systematicall y relevant researc h papers . 2. Result s o n th e Cauch y proble m We formulat e a fe w result s o n loca l an d globa l wellposednes s fo r D = 1 , 2, 3, 4. In th e presen t situatio n o f periodi c boundar y conditions , al l globa l result s ar e ob tained combinin g a loca l resul t wit h conserve d quantitie s (tha t ma y relat e t o a lower threshol d tha n th e dat a considered) .

GLOBAL SOLUTION S O F NONLINEA R SCHRODINGE R EQUA 1 TION S 0

THEOREM

7

2.1 . The Cauchy problem in ID ( iu t + u xx + \u\u\ 2 = 0

(2.2)

I «(0 ) = 4> is globally wellposed for data 0 G HS{T), s > 0 . THEOREM

2.3 . The Cauchy problem in ID iut + u xx + \u\u\ p~2 = 0 (p u(0) = (j)

> 2)

(2.4)

is locally wellposed for data 0 G i P ( T ) when s > 0 an d p < 6 (2.5 s>s*, p - 2

THEOREM

=_

,

p>6

) . (2.6

)

2.7 . Tft e Cauchy problem in 2D

(2D) mt + Au - f Au|u| 2 = 0

u(0) = 0

(2i

is locall y wellpose d fo r 0 G H S(Y2), s > 0 an d henc e globall y fo r 0 G H s,s > provided th e Hamiltonia n control s iJ 1 -norm. THEOREM

1 ,

2.9 . The Cauchy problem in 3D

(3D) iut + Au + \u\u\ 2 = 0 u(0) =

1 , provided the Hamiltonian controls the H 1 norm. THEOREM

2.1 1 . Consider the-Cauchy problem in AD

(4£>)

f iu t + Au + ug(\u\ 2) = 0

I «(0 ) = 0 l

'

j

w/iere 5 6 C* 2(R+), | 5 (s)| < cs 1 / 2 , \g'(s)\ < c S-1 '2, \g"{s)\ < cs~ 3/2. Then (2.1 2) is globally wellposed for data

2, provided the H 1 -norm is controlled by the energy (=Hamiltonian), which is the case for g < 0 or \\\\2 small enough.

J. B O U R G A I N

108

THEOREM 2.1 3 . In Theorems 2.1 , 2.7 and 2.9 there is moreover an estimate \Ht)\\H- *±±V. (3.7

)

G L O B A L SOLUTION S O F N O N L I N E A R S C H R O D I N G E R E Q U A1 TION S 0

9

PROPOSITION 3.8 .

Ford=l, (3.6)

holds with q = 4. (3.9

Ford 1 = 1,2, (3.7) holds. (3. 0 Ford>3 (3.7)

) )

holds1 1 when q > 4. (3.

)

Concerning (3.9) , ther e i s the more genera l inequality . PROPOSITION 3. 1 2

.

J2anmeiinx+mt)\\ 4

2

2 < C ( ^ ( l + | n 2 - m | ) 3 / 4 | a1 n m | J (3. 3

P R O P O S I T I O N 3.1 4 . Assuming supp cj> C B(0,N), ||0|| rect" distributional inequality

mes [(x,t ) G T d + 1 | |e

2tA

2

)

< 1 , one /ias tfie "cor -

0| > A ] < A1 ^ A " 1 ^ (3. 5

)

provided in (3.1 5) we restrict A to 1 A > A^ / 4 . (3. 6

)

The proo f o f Proposition 3. 8 and Proposition 3.1 2 are based o n simple arith metic considerations . Propositio n 3.1 4 is an application o f the circle method . For detail s se e [B4]. 4. Sketc h o f proof o f theorems 2. 1 and 2.7 We agai n us e the space-time norm s X s^ = X Sib[I] fo r a give n tim e interva l J c i Defin e ^ s ,6[/] a s the space of functions u on Td x I tha t ma y be represented as (n x+Ai) d u(x,t)=^2 /dAe* £(n,A) fo r (x,t)eT 1 xI (4. ) n£ZdJR

with u satisfyin g

ll^ll^, ,[^ ] = i ^ Z C 1 ^ l^-i 2") A ^^( 1 - K |A — ^ 2 | ) 2 ^ |?2(^ 5 A ) | 2 \ < o

o (4.2 )

((4.2) ha s again t o be understood a s a restriction norm) . Proof o f Theorem 2. 1 Consider th e Cauchy proble m i n ID with cubi c nonlinearit y ( iu t + u xx + \u\u\ 2 = 0 s u(0) = (f)£H (T),s>0

(4.3)

J. B O U R G A I N

110

and th e equivalen t integra l formulatio n (6 ) (4.4)

z

u{t) = S{t)(/> + ^ A / S(t - r){u\u\ Jo

){r)dr.

We wil l sho w tha t th e correspondin g ma p satisfie s th e contractio n principl e i n th e space X s'b([0,6}) fo r | < b < | an d 6 > 0 small enoug h (dependin g o n th e siz e of (j>). Thi s wil l yiel d loca l wellposednes s i n th e correspondin g space . I n orde r t o get global wellposedness , w e will mak e a furthe r discussio n o f th e siz e o f 6. The mai n ingredien t i n wha t follow s i s Strichart z inequalit y (3.1 2) , implyin g the followin g inequalit y fo r function s o n T x R (4.5)

II^IIL4(TX[O,I])l

One ha s clearl y 1/2

ll(4.9)|k..»([o.*])
k

2>k3 ni

- )

^JJk, i=l,2,3

Fix fci >&2 > & 3. Conside r a further partitio n o f D^ = U aQa in balls of size 2 /c2. One may thus essentially writ e I

r

^/

dAiA 2dA3 / 1

+

| A _ n |^|2iu-^ C 1 + N s )|ii(ni, Ai)| \u(n 2, A2)| |fi(n 3, A 3)| -

riieDk

a

n,niGQ a ^2GD / C 2 ,n 3 GD f c 3

|ii(ni,Ai)| |w(n 2,A2)| |u(n 3,A3)|.

(4.66 )

Define the functions F

(

f)

_ V ^ /

d

\ K

^ i«n,s)+At

) (467)

Tic: vc^a

i n +xt

G Q (x,t)= E [d\\u(n,\)\e Hl{x,t)= J2

« ^ ^ (4.68

[d\\u(n,\)\e

i n +xt)

« '^

) = 2,3) (4.69

(i

)

(cf. (4.23)-(4.25) ) an d bound (4.66 ) by 2/clS

Y^f F

a-Ga-H2-Hs-

iPldxdt

aJ

< 2 k^Y II^I Choose s 1 < \, ^
sh. (4.70

)

bi < \ t o satisfy (4.58) , (4.59).

Thus, for z = 2,3

' I ^ I U < 2 kiSl ^(E

/ ^ (

2 l + |A-|n| 2 |) 2b Mti(n,A)| 1 ) (4.7

)

and since each Q a i s of size 2 fc2, also 1/2

| | G a | | 4 < 2 ^ ^ f 5 ^ / d A ( l + |A-|n| 2 |) 2b M«(n,A)| 2 K

-neQaJ /

|Fa^lU 0 (cf. (4.30) , (4.31)). Thi s yield s fo r (4.74 ) th e estimat e SeSk^\\c\Dki\\J E

/dA(

l + |A-|n| 2|)^(H-|n|2-)|fi(n,A)|2)1 2.

n, =2 , 3 ( Yl / ^ ( l + |A-H 2 |) 2 ^(l + H 2si)|n(n,A)|2) (4

. 76)

ki > fe permits t o bound (4.62 ) by 1/2

6e(j2 fdX(l ^n

+

|n|)2 s (l + | A - \n\

\) ^\u(n,X)\12 \

2 2

^

(4.78)

^ / ^ ( l + |n|) 1 0sHl + |A-|n| 2 |) 2 ^Kn,A)| 2 ). Fix s > 0 and le t si = m i n ( |, | ) . On e ma y tak e b = b' — b(s) > \ suc h tha t 1 - b(s) > ^ p, henc e (4.75 ) hold s wit h ^ < b x < \. Prom (4.78) , ||(4.61)|| X s, W ] = \\u- S(t)cf)\\ Xs,bm 3| \n\ + n2 + A x - A | > | m | 2 - 2|m| e > i | m | 21 (4. 06

)

(4.106) i s the key point . Sinc e b > \ > 6i, this permit s u s then agai n t o save som e power \ni\ 6 and estimate (4.1 05 ) by 1 1 C\\u\\1 x,_6tb < CWuim '.^ (4. 07

)

for som e S > 0. Pro m (4.1 01 ) , (4.1 07) , we conclude tha t \\{A»u)u2\\Xo,_ 1 C\Htj)fHs^ (4. 08 b
\2±l-jw (5.4

)

we get a Hamiltonian syste m wit h infinit e dimensiona l phas e space ; th e canonica l coordinates {p n,qn} ar e take n t o b e pn = Re (j>(ri),q n = Im^(n ) (5.5

)

for n e Zd. The interes t o f (5.2 ) i s that thi s expressio n ma y b e given a sense (possibl y afte r an appropriat e normalization ) i n this infinit e dimensiona l context . Th e facto r j3 (4 = temperature i n statistica l physics ) wil l b e take n 1. Rewrite (5.2 ) as e=F*J>l P e-*/lv*l 2 ndV. (5.6

)

V* II dv

The facto r dv corresponds t o (unnormahzed) Wiene r measure . Identifyin g | 2 | 2. Thi s problem wa s studie d b y Lebowitz, Rose , Spee r an d solved a s follows . Observ e tha t sinc e \\(j)\\2 (th e L 2 -norm) i s a conserve d quantity , 1 dn = XH\\ 2\\ 2 oo . T h e mai n ide a a t thi s stag e i s t o exploi t th e invarianc e o f th e Gibb s measur e a s a substitut e fo r a conservatio n la w an d sho w t h a t fo r almos t al l (f) e sup p /i , (5.56 ) is i n fac t globall y wellposed . Sinc e thi s Gibb s measur e invarianc e i s onl y availabl e for (5.57 ) a t thi s stage , w e conside r th e flow o f t h e t r u n c a t e d equatio n (bound s are independen t o f th e truncatio n TV) . Moreove r i n th e focusin g case , w e nee d t o assume p < 6 (becaus e o f th e measur e existence) . L E M M A 5.58 . Let 0 < s < \,p < 6 , T < oc,< 5 > 0 . There is a set fl$ C f l s < i Hs such that /i(fi£ ) < 6 and for (f) G fls, the solution u of the IVP | iut

+

u

xx

+

P N{U\U\P-2) =0

{ u(0) = P N

1 . Henc e fo r typica l th e expres sion J \4>\ 4 i s unbounded , i.e . lim /

TV—>-oo J

|PJV0CJ|

4

= o o UJ

a.s. (6.4

)

J. B O U R G A I N

130

This proble m i s overcome i n 2D by so-called Wick-ordering. Th e genera l proces s consists i n associating t o a monomia l x 2k th e corresponding Hermit h polynomia l P2k{x), wher e (i n th e rea l case ) [f]

Pn(x)=^2(-l)j CnjX"-

2

* (6.5

)

3=0

C

-=

(

n) n - 2 j

W

!

(6

-6)

obtained b y orthogonalizatio n o f the monomial s {x n} wrt Gaussian measur e o n R. Thus on e ha s P2k(%) — %2k + lowe r degre e terms . The comple x cas e is similar (bu t th e coefficient s (6.6 ) ar e different) . W e denot e

*N = / \tf\ 2du> ~ Yl Tnhr ( ~ l o § N i n 2 D ) ( J

| n

\n\oc J

for D > 2. However, on e may restric t instea d th e Wick ordere d L 2 -norm |^:=£(W„)f-]-

j J T

- j , (6.36

)

which i s a.s finite fo r D < 3 . In thi s spirit , A . Jaffe showe d tha t (i n the real case ) th e restricted Gibb s mea sure dM = {X[J W-. ')dp (6-37

)

yields a normalizatio n fo r the cubic nonlinearit y i n 2D . This resul t i s of interes t in the context o f invariant measure s fo r the NLW. Unfortunately , th e construction (6.37) barel y misse s th e quartic nonlinearit y : |0|4 :. Thi s proble m (whic h i s again related t o the criticality o f the cubi c nonlinearit y i n 2D relate d t o blowup phenom ena) wa s investigated i n detail in t he paper [B-S] . Essentiall y speaking , th e conclusion i s that (considerin g finite dimensiona l models ) neithe r b y restriction o f the L 2 -norm J \(/)N\ 2 < B (leadin g afte r normalizatio n t o limit measure s singula r wrt Wiener measure ) no r by restriction o f the Wic k ordere d L 2 -norm J : \4>N\ 2 >< B, acceptable invarian t measure s ma y b e obtained i n the limit fo rTV—> oo . One wa y (suggested b y J. Lebowitz ) t o overcom e thi s difficult y i s to replac e (6.33) b y a so-called Hartre e equatio n wit h (nonlocal ) convolutio n nonlinearit y iut + Au + (|u| 2 * V)u = 0. (6.38

)

In (6.38) , th e nonlinearity i s thus tempere d b y convolving wit h a rea l interactio n potential V whic h Fourie r transfor m satisfie s a certain deca y propert y (/3 > 0 ) \V(n)\ < |n|-^for \n\ -+ oo. (6.39

)

The Hamiltonia n fo r (6.38 ) i s given by

H(4>) = J m2 - J\4>\2(\4>\2*V) (6.40

)

134 J

. BOURGAI N

and th e L 2 -norm f |0| 2 i s agai n a conserve d quantity . Renormalizing agai n th e secon d ter m i n (6.40 ) considerin g Wic k ordere d trun cated Hamiltonian s yield s HI(4>N)

= J \N\ 2(\N\2 * V) - V(0)[2a N J \0 N\2 - a 2N]1 (6.4

)

where

aN

6 42

= J2 vi^~ (

- )

hence aN ~ lo g N i n 2D (6.43

)

aN ~ N m 3D. (6.44

)

We hav e th e followin g PROPOSITION 6.45 . (D = 2,3) . Assume V satisfies (6.39) where

Then i/i(0jv ) Gibbs measures

:= C>

^

(3 > 0, (3 arbitrary in 2D (6.46

)

P > 2 in 3D (6.47

)

# i ( 0 ) almost surely with respect to Wiener measure and the dfiN = e Hl X[J .lM2. o o (ii) A priori bound s o n NLS-solution s fo r typica l dat a (iii) Existenc e o f wea k solution s i n th e limi t fo r L — > o o (iv) Uniquenes s an d regularit y problem . In th e discussio n o f (i) , (ii) , (iii) , we ma y tak e p arbitrary . I n (iv ) however , w e only kno w t o procee d a t thi s stag e fo r p < 4 . Fo r simplicit y reason , w e le t p = 4 throughout ou r entir e discussion . (i) Th e normalize d Gibbs-measur e v — VL is given b y

dVL =

e-H^Iid2(j) e - H o fe- H^)Ud^ = $e-

L

l^l4^e-^^lv^l2+2^l2]^nd20 H WTl