Global solutions of nonlinear Schrödinger equations 0821819194, 9780821819197

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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