Nonlinear Dispersiv e Equations

Local an d Globa l Analysi s

This page intentionally left blank

Conference Boar d o f the Mathematica l Science s

CBMS Regional Conference Series in Mathematics

Number 10 6

Nonlinear Dispersiv e Equations

Local an d Globa l Analysi s

Terence Tao

Published fo r th e Conference Boar d o f the Mathematica l Science s

by th e ^^zm^n America n Mathematica l Societ y ^ L»* **

, — — J x ^ r m > Providence, Rhod e Islan d ^J^OBJL I with suppor t fro m th e

National Scienc e Foundatio n °°ND^ S° &

http://dx.doi.org/10.1090/cbms/106

NSF-CBMS Regiona l Conferenc e o n

Nonlinear an d Dispersiv e Wav e Equation s

held a t Ne w Mexic o Stat e University , Jun e 2005 .

Partially supporte d b y th e Nationa l Scienc e Foundation .

The autho r acknowledge s suppor t fro m th e Conferenc e Boar d o f Mathematical Science s an d NS F Gran t DMS-0440945 .

The autho r i s partly supporte d b y a gran t fro m the Packar d foundation .

2000 Mathematics Subject Classification. Primar y 35Q53 , 35Q55 , 35L15 .

For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages/cbms-106

Library o f Congres s Cataloging-in-Publicat io n Dat a

Tao, Terence , 1975-Nonlinear dispersiv e equations : Loca l an d global analysi s / Terenc e Tao.

p. cm . — (Regiona l conferenc e serie s i n mathematics , ISS N 0160-764 2 ; no. 106 ) Includes bibliographica l references . ISBN 0-8218-4143- 2 (alk . paper) 1. Nonlinea r wav e equations . 2 . Differential equations , Partial . I . Title . II . Series.

QA1.R33 no . 10 6 [QA927] 510 s—dc22 [530.12/4] 200604282 0

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them, ar e permitted t o make fai r us e of the material, suc h a s to copy a chapte r fo r use 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 customary acknowledgmen t o f the source i s given.

Republication, systemati c copying , o r multiple reproductio n o f any materia l i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addressed t o the Acquisitions Department , America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Requests ca n als o b e mad e b y e-mail t o reprint-permission@ams.org .

© 200 6 b y the American Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l right s

except thos e grante d t o the United State s Government . Printed i n the United State s o f America .

@ Th e paper use d i n this boo k i s acid-free an d falls withi n th e guideline s established t o ensure permanenc e an d durability .

Visit th e AMS home pag e a t http:/ /www.ams.org /

10 9 8 7 6 5 4 3 2 1 1 1 10 09 08 07 06

To Laura , fo r bein g s o patient .

This page intentionally left blank

Contents

Preface

Chapter 1 . Ordinar y differentia l equation s 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7.

General theor y Gronwall's inequalit y Bootstrap an d continuit y argument s Noether's theore m Monotonicity formula e Linear an d semilinea r equation s Completely integrabl e system s

Chapter 2 . Constan t coefficien t linea r dispersiv e equation s 2.1. 2.2. 2.3. 2.4. 2.5. 2.6.

The Fourie r transfor m Fundamental solutio n Dispersion an d Strichart z estimate s Conservation law s fo r th e Schrodinge r equatio n The wav e equation stress-energ y tenso r Xs'b space s

Chapter 3 . Semilinea r dispersiv e equation s 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 3.9.

On scalin g an d othe r symmetrie s What i s a solution ? Local existenc e theor y Conservation law s and globa l existenc e Decay estimate s Scattering theor y Stability theor y Illposedness result s Almost conservatio n law s

Chapter 4 . Th e Kortewe g d e Vrie s equatio n 4.1. 4.2. 4.3. 4.4.

Existence theor y Correction term s Symplectic non-squeezin g The Benjamin-On o equatio n an d gaug e transformation s

Chapter 5 . Energy-critica l semilinea r dispersiv e equation s 5.1. 5.2. 5.3. 5.4.

The energy-critica l NL W Bubbles o f energy concentratio n Local Morawet z an d non-concentratio n o f mas s Minimal-energy blowu p solution s

ix

1 2

11 20 26 35 40 49

55 62 69 73 82 88 97

109 114 120 129 144 153 162 171 180 186

197 202 213 218 223

231 233 247 257 262

vii

viii C O N T E N T S

5.5. Globa l Morawet z an d non-concentratio n o f mass 27 1

Chapter 6 . Wav e map s 27 7 6.1. Loca l theor y 28 8 6.2. Orthonorma l frame s an d gaug e transformation s 29 9 6.3. Wav e ma p deca y estimate s 31 0 6.4. Hea t flow 32 0

Chapter A . Appendix : tool s fro m harmoni c analysi s 32 9

Chapter B . Appendix : constructio n o f ground state s 34 7

Bibliography 36 3

Preface

Politics is for the present, but an equation is something for eternity. (Albert Einstein )

This monograph i s based o n (an d greatl y expande d from ) a lecture serie s given at th e NSF-CBM S regiona l conferenc e o n nonlinea r an d dispersiv e wav e equation s at Ne w Mexic o Stat e University , hel d i n Jun e 2005 . It s objectiv e i s t o presen t some aspect s o f th e globa l existenc e theor y (an d i n particular , th e regularit y an d scattering theory ) fo r variou s nonlinea r dispersiv e an d wav e equations , suc h a s th e Korteweg-de Vrie s (KdV) , nonlinea r Schrodinger , nonlinea r wave , an d wav e map s equations. Th e theor y her e i s ric h an d vas t an d w e canno t hop e t o presen t a comprehensive surve y o f th e field here ; ou r ai m i s instea d t o presen t a sampl e o f results, an d t o giv e some idea o f the motivatio n an d genera l philosoph y underlyin g the problem s an d result s i n th e field, rathe r tha n t o focu s o n the technica l details . We inten d thi s monograp h t o b e a n introductio n t o th e field rathe r tha n a n ad -vanced text ; whil e we do include some very recent results , and we imbue some mor e classical result s wit h a moder n perspective , ou r mai n concer n wil l b e t o develo p the fundamenta l tools , concepts , an d intuition s i n a s simpl e an d a s self-containe d a matte r a s possible . Thi s i s also a pedagogical tex t rathe r tha n a reference ; man y details o f arguments ar e lef t t o exercise s o r t o citations , o r ar e sketched informally . Thus thi s tex t shoul d b e viewed a s being complementar y t o th e researc h literatur e on thes e topics , rathe r tha n bein g a substitut e fo r them .

The analysi s of PDE i s a beautiful subject , combinin g the rigour an d techniqu e of moder n analysi s an d geometr y wit h th e ver y concret e real-worl d intuitio n o f physics an d othe r sciences . Unfortunately , i n some presentations o f the subjec t (a t least i n pure mathematics) , the former ca n obscure the latter , givin g the impressio n of a fearsomel y technica l an d difficul t field t o wor k in . T o try t o comba t this , thi s book i s devote d i n equa l part s t o rigou r an d t o intuition ; th e usua l formalis m o f definitions, propositions , theorems, and proof s appea r here , but will be intersperse d and complemente d wit h man y informa l discussion s o f the sam e material , centerin g around vagu e "Principles " an d figures, appea l t o physica l intuitio n an d examples , back-of-the-envelope computations , an d eve n som e whimsica l quotations . Indeed , the exposition and exercises here reflect m y personal philosophy that t o truly under -stand a mathematical resul t on e must vie w it from a s many perspectives a s possibl e (including both rigorou s argument s an d informa l heuristics) , an d mus t als o be abl e to translat e easil y fro m on e perspectiv e t o another . Th e reade r shoul d thu s b e aware o f whic h statement s i n th e tex t ar e rigorous , an d whic h one s ar e heuristic , but thi s shoul d b e clea r fro m contex t i n mos t cases .

To restrict th e field o f study, w e shall focu s primaril y o n defocusing equations , in whic h soliton-typ e behaviou r i s prohibited . Pro m th e poin t o f vie w o f globa l existence, thi s i s a substantiall y easie r cas e t o stud y tha n th e focusin g problem , i n

ix

x PREFAC E

which one has the fascinating theor y of solitons and multi-solitons, as well as various mechanisms t o enforc e blow-u p o f solution s i n finite o r infinit e time . However , w e shall se e tha t ther e ar e stil l severa l analytica l subtletie s i n th e defocusin g case , especially whe n considerin g critica l nonlinearities , o r whe n tryin g t o establis h a satisfactory scatterin g theory . W e shal l als o wor k i n ver y simpl e domain s suc h as Euclidea n spac e R n o r tor i T n , thu s avoidin g consideratio n o f boundary-valu e problems, o r curved space , though thes e ar e certainly ver y importan t extension s t o the theory . On e further restrictio n w e shall make is to focus attentio n o n the initia l value proble m whe n th e initia l dat a lie s in a Sobole v spac e i J | (R d ) , a s opposed t o more localised choices of initial data (e.g . i n weighted Sobole v spaces, or self-simila r initial data) . Thi s restriction , combine d wit h th e previou s one , makes our choic e of problem translation-invarian t i n space , which lead s naturall y t o th e deploymen t o f the Fourie r transform , whic h turn s ou t t o b e a ver y powerfu l too l i n ou r analysis . Finally, w e shall focus primaril y o n only four equations : th e semilinea r Schrodinge r equation, th e semilinea r wav e equation , th e Korteweg-d e Vrie s equation , an d th e wave maps equation. Thes e four equation s ar e of course only a very small sample of the nonlinear dispersiv e equations studied i n the literature , bu t the y ar e reasonabl y representative i n tha t the y showcas e man y o f the technique s use d fo r mor e genera l equations i n a comparativel y simpl e setting .

Each chapte r o f th e monograp h i s devote d t o a differen t clas s o f differentia l equations; generall y speaking , i n eac h chapte r w e first stud y th e algebrai c struc -ture o f these equations (e.g . symmetries , conservatio n laws , and explici t solutions) , and then turn to the analytic theory (e.g . existenc e and uniqueness , and asymptoti c behaviour). Th e first chapte r i s devoted entirel y t o ordinary differential equations (ODE). On e ca n vie w partia l differentia l equation s (PDE ) suc h a s th e nonlinea r dispersive an d wav e equation s studie d here , a s infinite-dimensiona l analogue s o f ODE; thu s finite-dimensional OD E ca n serv e a s a simplified mode l fo r understand -ing technique s an d phenomen a i n PDE . I n particular , basi c PD E technique s suc h as Picar d an d Duhame l iteration , energ y methods , continuit y o r bootstra p argu -ments, conservatio n laws , near-conservatio n laws , an d monotonicit y formula e al l have usefu l OD E analogues . Furthermore , th e analog y betwee n classica l mechan -ics an d quantu m mechanic s provide s a usefu l heuristi c correspondenc e betwee n Schrodinger typ e equations , an d classica l OD E involvin g on e o r mor e particles , a t least i n th e high-frequenc y regime .

The secon d chapte r i s devoted t o the theor y o f the basi c linear dispersiv e mod -els: th e Airy equation, th e free Schrodinger equation, an d th e free wave equation. In particular , w e sho w ho w th e Fourie r transfor m an d conservatio n la w methods , can b e use d t o establis h existenc e o f solutions , a s wel l a s basi c estimate s suc h a s the dispersiv e estimate , loca l smoothin g estimates , Strichart z estimates , an d X s,b

estimates. In the third chapter we begin studying nonlinear dispersive equations in earnest,

beginning wit h tw o particularl y simpl e semilinea r models , namel y th e nonlinear Schrodinger equation (NLS ) an d nonlinear wave equation (NLW) . Usin g thes e equations a s examples , w e illustrat e th e basi c approache s toward s definin g an d constructing solutions , an d establishin g loca l an d globa l properties , thoug h w e de-fer th e stud y o f the mor e delicat e energy-critica l equation s t o a later chapter . (Th e mass-critical nonlinea r Schrodinge r equatio n i s als o o f grea t interest , bu t w e will not discus s i t i n detai l here. )

PREFACE x i

In the fourth chapter , w e analyze the Korteweg de Vries equation (KdV) , which requires som e mor e delicat e analysi s du e t o th e presenc e o f derivatives i n th e non -linearity. T o partl y compensat e fo r this , however , on e no w ha s th e structure s o f nonresonance an d complet e integrability ; th e interplay betwee n the integrability o n one hand , an d th e Fourier-analyti c structur e (suc h a s nonresonance ) o n th e other , is stil l onl y partl y understood , howeve r w e ar e abl e t o a t leas t establis h a quit e satisfactory loca l an d globa l wellposednes s theory , eve n a t ver y lo w regularities , by combinin g method s fro m both . W e als o discus s a les s dispersiv e cousi n o f th e KdV equation , namel y th e Benjamin-Ono equation, whic h requires mor e nonlinea r techniques, suc h a s gaug e transforms , i n orde r t o obtai n a satisfactor y existenc e and wellposednes s theory .

In th e fifth chapte r w e retur n t o th e semilinea r equation s (NL S an d NLW) , and no w establish larg e data globa l existence fo r thes e equation s i n the defocusing , energy-critical case . Thi s requires the full powe r o f the local wellposedness an d per -turbation theory , togethe r wit h Morawetz-typ e estimate s t o preven t variou s kind s of energ y concentration . Th e situatio n i s especiall y delicat e fo r th e Schrodinge r equation, i n whic h on e must emplo y th e inductio n o n energ y method s o f Bourgai n in orde r t o obtai n enoug h structura l contro l o n a putativ e minimal energy blowup solution t o obtai n a contradictio n an d thu s ensur e globa l existence .

In th e final chapter , w e turn t o th e wave maps equation (WM) , which i s some-what mor e nonlinea r tha n th e precedin g equations , bu t whic h o n th e othe r han d enjoys a strongl y geometri c structure , whic h ca n i n fac t b e use d t o renormalis e most o f the nonlinearity . Th e smal l dat a theor y her e ha s recentl y bee n completed , but th e larg e dat a theor y ha s jus t begun ; i t appear s howeve r tha t th e geometri c renormalisation provide d by the harmonic map heat flow, together with a Morawetz estimate, ca n agai n establis h globa l existenc e i n th e negativel y curve d case .

As a final disclaimer , thi s monograp h i s by n o mean s intende d t o b e a defini -tive, exhaustive , o r balance d surve y o f th e field. Somewha t unavoidably , th e tex t focuses o n thos e technique s an d result s whic h th e autho r i s mos t familia r with , i n particular th e us e o f the iteratio n metho d i n various functio n space s t o establis h a local and perturbative theory , combined with frequency analysis , conservation laws , and monotonicit y formula e t o then obtai n a global non-perturbative theory . Ther e are othe r approache s t o thi s subject , suc h a s vi a compactnes s methods , nonlinea r geometric optics , infinite-dimensiona l Hamiltonia n dynamics , o r th e technique s o f complete integrability , whic h ar e als o o f majo r importanc e i n th e field (an d ca n sometimes be combined, t o good effect , wit h the methods discussed here) ; however , we will be unable to devote a full-length treatmen t o f these methods in this text . I t should als o b e emphasise d tha t th e methods , heuristics , principle s an d philosoph y given her e ar e tailored fo r th e goa l o f analyzing th e Cauch y proble m fo r semilinea r dispersive PDE ; the y d o no t necessaril y exten d wel l t o othe r PD E question s (e.g . control theor y o r invers e problems) , o r t o othe r classe s o f PD E (e.g . conservatio n laws or to paraboli c an d ellipti c equations) , thoug h ther e ar e certain man y connec -tions an d analogie s betwee n result s i n dispersiv e equation s an d i n othe r classe s o f PDE.

I a m indebte d t o m y fello w member s o f the "/-team " (Ji m Colliander , Marku s Keel, Gigliol a Staffllani , Hide o Takaoka) , t o Sergi u Klainerman , an d t o Michae l Christ fo r many entertaining mathematica l discussions , which have generated muc h of th e intuitio n tha t I hav e trie d t o plac e int o thi s monograph . I a m als o ver y

xii PREFAC E

thankful fo r Ji m Ralsto n fo r usin g thi s tex t t o teac h a join t PD E course , an d providing m e wit h carefu l correction s an d othe r feedbac k o n th e material . I als o thank Soonsi k Kwo n fo r additiona l corrections . Last , bu t no t least , I a m gratefu l to m y wif e Laur a fo r he r support , an d fo r pointin g ou t th e analog y betwee n th e analysis o f nonlinea r PD E an d th e electrica l engineerin g proble m o f controllin g feedback, whic h ha s greatl y influence d m y perspectiv e o n thes e problem s (an d ha s also inspire d man y o f the diagram s i n thi s text) .

Terence Ta o

Nota t ion . A s i s commo n wit h an y boo k attemptin g t o surve y a wid e rang e of result s b y differen t author s fro m differen t fields , th e selectio n o f a unifie d no -tation become s ver y painful , an d som e compromise s ar e necessary . I n thi s tex t I have (perhap s unwisely ) decide d t o mak e the notatio n a s globally consisten t acros s chapters a s possible , whic h mean s tha t an y individua l resul t presente d her e wil l likely hav e a notatio n slightl y differen t fro m th e wa y i t i s usually presente d i n th e literature, an d als o tha t th e notatio n i s mor e finick y tha n a loca l notatio n woul d be (ofte n becaus e o f som e ambiguit y tha t neede d t o b e clarifie d elsewher e i n th e text). Fo r th e mos t part , changin g fro m on e conventio n t o anothe r i s a matte r o f permuting variou s numerica l constant s suc h a s 2 , 7r, i, an d —1 ; these constant s ar e usually quit e harmles s (excep t fo r th e sig n —1) , but on e shoul d nevertheles s tak e care in transporting a n identity o r formula i n this book to another contex t i n which the convention s ar e slightl y different .

In thi s text , d will always denot e th e dimensio n o f the ambien t physica l space , which will either b e a Euclidean spac e R d o r the torus T d : = (R/27rZ) d. (Chapte r 1 deal s wit h ODE , whic h ca n b e considere d t o b e th e cas e d = 0. ) Al l integral s o n these space s wil l b e wit h respec t t o Lebesgu e measur e dx. I f x = ( x i , . . . , Xd) an d £ = (£i > • • • J £,d) h e in R d , w e use x • £ to denote th e do t produc t x • £ := X\^i + . . . + xaidi an d \x\ t o denot e th e magnitud e \x\ : = {x\ + . . . + x^) 1^2. W e als o us e (x) to denot e th e inhomogeneou s magnitud e (o r Japanese bracket) (x) : = ( 1 + l^l 2)1/2

of x , thu s (x) i s comparabl e t o \x\ fo r larg e x an d comparabl e t o 1 fo r smal l x. In a simila r spirit , i f x = ( # i , . . . , Xd) € T d an d k = (fci,... , kj) £ Z d w e defin e k • x : = k\Xi + . . . + kdXd £ T . I n particula r th e quantit y e %k'x i s well-defined .

We say tha t / i s a time interval i f i t i s a connected subse t o f R whic h contain s at leas t tw o point s (s o w e allo w tim e interval s t o b e ope n o r closed , bounde d o r unbounded). I f u : I xH d — » Cn i s a (possibly vector-valued) functio n o f spacetime, we write d tu fo r the time derivative ^ | , an d dXl u1..., d Xdu fo r the spatial derivatives J j - , . . . , J^r- ; thes e derivatives wil l either b e interpreted i n the classica l sense (whe n u i s smooth ) o r th e distributiona l (weak ) sens e (whe n u i s rough) . W e us e V xu : / x R d — » Cnd t o denot e th e spatia l gradien t V 'xu — (d Xlu,... ,d Xdu). W e ca n iterate thi s gradien t t o defin e highe r derivative s V ^ fo r k — 0,1, O f course ,

We wil l b e usin g tw o slightl y differen t notion s o f spacetime , namel y Minkowski space R 1 + d

and Galilean spacetime R x R d ; i n th e ver y las t sectio n w e als o nee d t o us e parabolic spacetime R + x R d . A s vecto r spaces , the y ar e o f cours e equivalen t t o eac h othe r (an d t o th e Euclidea n space R d + 1 ) , bu t w e wil l plac e differen t (pseudo)metri c structure s o n them . Generall y speaking , wave equation s wil l us e Minkowsk i space , wherea s nonrelativisti c equation s suc h a s Schrodinge r equations wil l us e Galilea n spacetime , whil e hea t equation s us e paraboli c spacetime . Fo r th e mos t part th e reade r wil l b e abl e t o safel y ignor e thes e subtl e distinctions .

P R E F A C E xii i

these definitions als o apply to functions o n Td , whic h can be identified wit h periodi c functions o n H d.

We use the Einstein convention for summing indices , with Latin indices ranging from 1 to d , thu s fo r instanc e Xjd Xju i s shor t fo r J2j =iXjdXju. Whe n w e com e to wav e equations , w e wil l als o b e workin g i n a Minkowsk i spac e R 1 + d wit h a Minkowski metri c g ap; i n such cases , we will use Greek indice s and su m from 0 to d (with x° = t being the time variable), and us e the metri c t o raise and lowe r indices . Thus fo r instanc e i f we use the standard Minkowsk i metri c dg 2 = — dt2 + \dx\ 2, the n dou = dtu bu t d°u = —d tu.

In thi s monograp h w e alway s adop t th e conventio n tha t f s — — f* i f t < s. This conventio n wil l usually b e applie d onl y t o integral s i n th e tim e variable .

We use the Lebesgu e norm s

I I / I I L S ( R ^ C ) : = ( / \f(x)\* dx) 1'*

for 1 < p < o o fo r complex-value d measurabl e function s / : R d — » C, wit h th e usual conventio n

I I / I I L ~ ( R ^ C ) : =ess su p |/(V)| . xe~Rd

In man y case s w e shal l abbreviat e L^(H d — » C) a s Lg(R d), L p(Rd) , o r eve n L p

when ther e i s no chanc e o f confusion . Th e subscrip t x , whic h denote s th e dumm y variable, i s o f cours e redundant . Howeve r w e wil l ofte n retai n i t fo r clarit y whe n dealing wit h PDE , sinc e i n tha t contex t on e ofte n need s t o distinguis h betwee n Lebesgue norm s i n spac e x , tim e £ , spatia l frequenc y £ , o r tempora l frequenc y r . Als o w e wil l nee d i t t o clarif y expression s suc h a s ||x/||Lg(R d)> m whic h th e expression i n th e nor m depend s explicitl y o n th e variabl e o f integration . W e o f course identif y an y tw o function s i f the y agre e almos t everywhere . On e ca n o f course replac e th e domai n H d b y th e toru s T d o r th e lattic e Z d, thu s fo r instanc e

l l / lk (Z^Q:=(£ \f(k)\ p)1/p-kezd

One can replace C by any other Banach space X, thu s for instance I / | (Rd — > X) is the spac e o f al l measurable function s u : Hd — > X wit h finite nor m

\\u\\mR^x) := ( f \\u(x)\F x dx) 1'' Jnd

with th e usua l modificatio n fo r p = oo . I n particula r w e ca n defin e th e mixe d Lebesgue norm s L q

tLrx{I x R d - ^ C ) = ^ ( I ^ L r

x(Rd — > C) ) fo r an y time interva l

/ a s the spac e o f al l function s u : I x H d — » C wit h nor m

I I « I I L ? L ; ( / X R ^ C ) = = ([ II«(*)II1; (R«) dt) 1'" = {J{J^ \u(t,x)\ r dxf'r dt) 1^

with th e usua l modification s whe n q = o o or r = oo . On e ca n als o use thi s Banac h space notatio n t o mak e sens e o f th e LP norm s o f tensor s suc h a s V / , V 2 / , etc. , provided o f course tha t suc h derivative s exis t i n th e LP sense.

In a simila r spirit , i f J i s a tim e interva l an d k > 0 , w e us e C^{I — > X) t o denote the spac e of al l fc-times continuously differentiabl e function s u : J — » X wit h

xiv PREFAC E

the nor m k

IMIctfc(/-,x) — Xlll^ll^u-*)-

3=0

We adop t th e conventio n tha t |Mlc fc(i-*x) = o o i f I A is no t /c-time s continuousl y differentiable. On e can of course also define spatia l analogues C^(R d — » X) o f these spaces, a s well as spacetime version s C^ X(I x Hd — * X). W e caution tha t i f / i s not compact, the n i t i s possible fo r a function t o b e k- times continuousl y differentiabl e but hav e infinite C f norm ; i n such cases we say that u G C l̂oc(I — » X) rathe r tha n u £ Ct(I —> • X). Mor e generally , a statemen t o f the for m u £ Xi oc(Q) o n a domai n Q mean s tha t w e ca n cove r O b y ope n set s V suc h tha t th e restrictio n u\y o f u to eac h o f thes e set s V i s i n X(V); unde r reasonabl e assumption s o n X , thi s als o implies tha t U\K £ X(K) fo r an y compac t subse t K o f 0 . A s a rul e o f thumb, th e global space s X(0 ) wil l b e use d fo r quantitativ e contro l (estimates) , wherea s th e local space s X\ oc(Q) ar e use d fo r qualitativ e contro l (regularity) ; indeed , th e loca l spaces Xi oc ar e typicall y onl y Preche t space s rathe r tha n Banac h spaces . W e wil l need bot h type s o f control i n this text , a s one typically need s qualitative contro l t o ensure tha t th e quantitativ e argument s ar e rigorous .

If (X,dx) i s a metri c spac e an d Y i s a Banac h space , w e use C°' l{X •— > Y) t o denote th e spac e o f al l Lipschit z continuou s function s / : X — > F , wit h nor m

(One can also define the inhomogeneous Lipschitz norm \\f\\co,i '•= ||/|lco, i + ||/llc°> but w e will no t nee d thi s here. ) Thu s fo r instanc e C x(R d — » Rm ) i s a subse t of C 0 j l(R d - • R m ) , whic h i s i n tur n a subse t o f Ci° oc(R

d - > R m ) . Th e spac e C^C(X - > Y) i s thu s th e spac e o f locally Lipschit z function s (i.e . ever y x £ X i s contained i n a neighbourhoo d o n which th e functio n i s Lipschitz) .

In additio n t o th e abov e functio n spaces , w e shal l als o use Sobole v space s iJ s , Ws'Pj H Sj VK S,P, whic h ar e define d i n Appendi x A , an d X s '6 spaces , whic h ar e defined i n Sectio n 2.6 .

If V an d W ar e finite-dimensional vecto r spaces , we use End(V — > W) t o denote the spac e o f linea r transformation s fro m V t o W , an d End(F ) = End( F — > V) fo r the rin g o f linea r transformation s fro m V t o itself . Thi s rin g contain s th e identit y transformation i d = idy.

If X an d Y ar e two quantities (typicall y non-negative), we use X < Y o r Y > X to denot e th e statemen t tha t X < CY fo r som e absolut e constan t C > 0 . W e us e X = 0(Y) synonymousl y wit h |X | < Y. Mor e generally , give n som e parameter s ai , . . . ,afc, w e us e X < ai,...,ak Y ox Y >ai,...,a fc X t o denot e th e statemen t tha t X < C ai,...,akY fo r som e (typicall y large ) constan t C aiv..jafe > 0 which ca n depen d on the parameters a i , . . . , a^, an d define X = O ai).,.?afe (Y) similarly . Typica l choices of parameter s includ e th e dimensio n d , th e regularit y 5 , an d th e exponen t p. W e will als o sa y tha t X i s controlled by a i , . . . , a& if X = O ai,...,afe(l)- W e us e X ~ y t o denot e th e statemen t X < y < X , an d similarl y X ~ai,...,a fc Y denote s X <ai,...,a fc ^ ;$ai,...,a fc ^ - W e wil l occasionall y us e th e notatio n X < 0 lv . . ,a fc Y or y ^>ai,...,a fc X t o denot e th e statemen t X < c ai?...?afcy fo r som e suitabl y smal l quantity c ai>„.jafe > 0 dependin g o n th e parameter s ai , . . . ,afe . Thi s notatio n i s

PREFACE xv

somewhat imprecis e (a s one has to specify wha t "suitabl y small " means ) an d s o we shall usuall y onl y us e i t i n informa l discussions .

Recall tha t a functio n / : Hd — » C i s said t o b e rapidly decreasing i f we have

\\(x)Nf(x)\\LT{Kd) < o o

for al l i V > 0 . W e then sa y tha t a function i s Schwartz i f i t i s smooth an d al l o f it s derivatives d%f ar e rapidl y decreasing , wher e a = ( a i , . . . , a^) 6 Z + range s ove r all multi-indices , an d d% i s the differentia l operato r

<9a := ( — ) a i . . . ( — ) a d . x K dxx} K dxd

}

In other words , / i s Schwartz i f and onl y i f d%f(x) = 0/,a,Ar((^) _iV) fo r al l a G Z ^ and al l N. W e us e ^ ( R ^ ) t o denot e th e spac e o f al l Schwart z functions . A s is wel l known , thi s i s a Freche t space , an d thu s ha s a dua l S x(H

d)*, th e spac e of tempered distributions. Thi s spac e contain s al l locall y integrabl e function s o f polynomial growth , an d i s als o close d unde r differentiation , multiplicatio n wit h functions g o f symbol typ e (i.e . g and al l it s derivative s ar e o f polynomial growth ) and convolutio n wit h Schwart z functions ; w e will not presen t a detailed descriptio n of the distributiona l calculu s here .

This page intentionally left blank

Bibliography

[AF] M . Ablowitz , A . Fokas , The inverse scattering transform, for the Benjamin-Ono equation, a pivot for multidimensional problems, Stud . Appl . Math . 6 8 (1983) , 1-10 .

[Ale] A . D . Alexandrov , Uniqueness theorems for surfaces in the large, V . Vestni k Leningra d Univ. Mat . Mekh . Astronom . 1 3 (1958) , 5-8 ; Amer . Math . Soc . Transl . Se r 2 . 2 1 (1962) , 412-416.

[Arn] V.I . Arnold , Mathematica l method s i n classica l mechanic s (3r d edn) . Nauka , Moscow , 1974 . English translation , 2n d edn , Springer-Verlag , Berlin , 1989 .

[BZS] J . Baez , Z . Zhou , I.E . Segal , The global Goursat problem and scattering for nonlinear wave equations, J . Func . Anal . 9 3 (1990) , 239-296 .

[BG] H . Bahouri , P . Gerard , High frequency approximation of solutions to critical nonlinear wave equations, Amer . J . Math . 12 1 (1999) , 131-175 .

[BS] H . Bahouri , J . Shatah , Decay estimates for the critical semilinear wave equation, Ann . Inst . H. Poincar e Anal . No n Lineair e 1 5 (1998),783-789 .

[Bea] M . Beals , Self-Spreading and strength of Singularities for solutions to semilinear wave equations, Annal s o f Mat h 11 8 (1983) , 187-214 .

[BB] M . Beals , M . Bezard , Low regularity local solutions for field equations, Comm . Partia l Dif -ferential Equation s 2 1 (1996) , 79-124 .

[BT] I . Bejenaru, T . Tao , Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrodinger equation, t o appear , J . Func . Anal .

[Ben] T . Benjamin , Internal waves of permanent form in fluids of great depth, J . Flui d Mech . 2 9 (1967), 559-562 .

[BKPSV] B . Birnir , C . Kenig , G . Ponce , N . Svanstedt , L . Vega , On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schro dinger equations, J . Londo n Math . Soc. 5 3 (1996) , 551-559 .

[BPS] B . Birnir , G . Ponce , N . Svanstedt , The local ill-posedness of the modified KdV equation, Ann. Inst . H . Poincar e Anal . No n Lineair e 1 3 (1996) , 529-535 .

[BC] P . Blue , J . Colliander , Global well-posedness in Sobolev space implies global existence for weighted L 2 initial data for I? -critical NLS, preprint .

[BSmi] J . Bona , R . Smith , The initial-value problem for the Korteweg-de Vries equation, Philos . Trans. Roya l Soc . Londo n Serie s A 27 8 (1975) , 555-601 .

[Bou] J . Bourgain , Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part s I , II , Geometri c an d Funct . Anal . 3 (1993) , 107-156 , 209-262.

[Bou2] J . Bourgain , Approximation of solutions of the cubic nonlinear Schrodinger equations by finite-dimensional equations and nonsqueezing properties, Int . Math . Res . Notices , 1994 , no . 2, (1994) , 79-90 .

[Bou3] J . Bourgain , Aspects of longtime behaviour of solutions of nonlinear Hamiltonian evolution equations, Geom . Func . Anal . 5 (1995) , 105-140 .

[Bou4] J . Bourgain , Periodic nonlinear Schrodinger equations and invariant measures, Comm . Math. Phys . 16 6 (1995) , 1-26 .

[Bou5] J . Bourgain , Periodic Korteweg de Vries equation with measures as initial data, Select a Math. (N.S. ) 3 (1997) , 115-159 .

[Bou6] J . Bourgain . Scattering in the energy space and below for 3D NLS, Jour . D'Anal . Math. , 75 (1998) , 267-297 .

[Bou7] J . Bourgain , Global well-posedness of defocusing 3D critical NLS in the radial case, J . Amer. Math . Soc . 1 2 (1999) , 145-171 .

363

364 BIBLIOGRAPHY

[Bou8] J . Bourgain , Refinements of Strichartz Inequality and Applications to 2d-NLS With Crit-ical Nonlinearity, Inter . Math . Res . Not. , (1998) , p . 253-284 .

[Bou9] J . Bourgain , New global well-posedness results for non-linear Schrodinger equations, AM S Publications, 1999 .

[BoulO] J . Bourgain , A remark on normal forms and the 'I-Method' for periodic NLS, J . Anal . Math. 9 4 (2004) , 125-157 .

[BK] J . Bourgain , V . Kaloshin , On diffusion in high-dimensional Hamiltonian systems, J . Func . Anal. 22 9 (2005) , 1-61 .

[BW] J . Bourgain , W . Wang , Construction of blowup solutions for the nonlinear Schrodinger equation with critical non-linearity, Dedicate d t o Enni o D e Giorgi . Ann . Scuol a Norm . Sup . Pisa CI . Sci . (4 ) 2 5 (1997) , no . 1-2 , 197-215 .

[BC] J . P . Boyd , G . Chen , Weakly nonlinear wavepackets in the Korteweg-de Vries equation: The KdV/NLS connection, Math . Comput . Simulatio n 5 6 (2001) , 317-328 .

[BGT] N . Burq , P . Gerard , N . Tzvetkov , An instability property of the nonlinear Schrodinger equation on S d, Math . Res . Lett . 9 (2002) , 323-335 .

[BP] N . Burq , F . Planchon , On well-posedness for the Benjamin-Ono equation, preprint . [Car] X . Carvajal , Sharp global well-posedness for a higher order Schrodinger equation, preprint . [Caz] T . Cazenave , An introduction to nonlinear Schrodinger equations, Texto s d e Metodo s

Matematicos 26 , Institut o d e Matematic a UFRJ , 1996 . [Caz2] T . Cazenave , Semilinear Schrodinger equations, Couran t Lectur e Note s i n Mathematics ,

10. Ne w Yor k University , Couran t Institut e o f Mathematica l Sciences , AMS , 2003 . [CL] T . Cazenave , Orbital stability of standing waves for some nonlinear Schrodinger equations,

Comm. Math . Phys . 6 8 (1979) , 209-243 . [CSS] T . Cazenave , J . Shatah , A.S . Tahvildar-Zadeh, Harmonic maps of the hyperbolic space and

development of singularities in wave maps and Yang-Mills fields, Ann . Inst . H . Poincare Phys . Theor.68 (1998) , 315-349 .

[CWeis] T . Cazenave , F.B . Weissler , Some remarks on the nonlinear Schrodinger equation in the critical case, Nonlinea r semigroups , Partia l Differentia l Equation s an d Attractors , Lectur e Notes i n Math . 139 4 (1989) , 18-29 .

[CWeis2] T . Cazenave , F.B . Weissler , Critical nonlinear Schrodinger Equation, Non . Anal . TMA , 14 (1990) , 807-836 .

[CWY] S.Y.A . Chang , L . Wang , P . Yang , Regularity of Harmonic Maps, Comm . Pure . Appl . Math. 5 2 (1999) , 1099-1111 .

[Chi] H . Chihara , Local existence for semilinar Schrodinger equations, Math . Japon . 4 2 (1995) , 35-52.

[CCT] M . Christ , J . Colliander , T . Tao , Asymptotics, frequency modulation, and low-regularity illposedness of canonical defocussing equations, Amer . J . Math . 12 5 (2003) , 1235-1293 .

[CCT2] M . Christ , J . Colliander , T . Tao , Ill-posedness for nonlinear Schrodinger and wave equa-tions, t o appear , Annale s IHP .

[CCT3] M . Christ , J . Colliander , T . Tao , Instability of the periodic nonlinear Schrodinger equa-tion, preprint .

[CKis] M . Christ , A . Kiselev , Maximal operators associated to filtrations, J . Funct . Anal . 17 9 (2001).

[CWein] M . Christ , M . Weinstein , Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J . Funct . Anal . 10 0 (1991) , 87-109 .

[Chr] D . Christodoulou , Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pur e Appl . Math , 3 9 (1986) , 267-282 .

[CKla] D . Christodoulou , S . Klainerman , Asymptotic properties of linear field equations in Minkowski space, Comm . Pur e Appl . Math . (1990) , 137-199 .

[CT] D . Christodoulou , A . Tahvildar-Zadeh , On the regularity of spherically symmetric wave maps, Comm . Pur e Appl . Math , 4 6 (1993) , 1041-1091 .

[CT2] D . Christodoulou , A . Tahvildar-Zadeh , On the asymptotic behavior of a spherically sym-metric wave map, Duk e Math . J . 7 1 (1993) , 31-69 .

[CLMS] R . Coifman , P.L . Lions , Y . Meyer , S . Semmes , Compensated compactness and Hardy spaces, J . Math . Pur e Appl . 7 2 (1993) , 247-286 .

[Col] J.D . Cole , On a quasi-linear parabolic equation occurring in aerodynamics, Quart . Appl . Math. 9 (1951) , 225-236 .

BIBLIOGRAPHY 365

[CDKS] J . Colliander , J . Delort , C . Kenig , G . Staffilani , Bilinear Estimates and applications to 2D NLS, Trans . Amer . Math . Soc . 35 3 (2001) , 3307-3325 .

[CKSTT] J . Colliander , M . Keel , G . Staffilani , H . Takaoka , T . Tao , Global well-posedness for KdV in Sobolev spaces of negative index, Electron . J . Diff . Eq . 200 1 (2001 ) N o 26 , 1-7 .

[CKSTT2] J . Colliander , M . Keel , G . Staffilani , H . Takaoka , T . Tao , Sharp global well-posedness for KdV and modified KdV on R and T , J . Amer . Math . Soc . 1 6 (2003) , 705-749 .

[CKSTT3] J . Colliander , M . Keel , G . Staffilani , H . Takaoka , T . Tao , Multilinear estimates for periodic KdV equations, and applications, preprin t 2001 , J. Funct . Anal . 21 1 (2004) , 173-218 .

[CKSTT4] J . Colliander , M . Keel , G . Staffilani , H . Takaoka , T . Tao , Global well-posedness for Schrodinger equations with derivative, Sia m J . Math . 3 3 (2001) , 649-669 .

[CKSTT5] J . Colliander , M . Keel , G . Staffilani , H . Takaoka , T . Tao , A refined global well-posedness for the Schrodinger equations with derivative, SIA M J . Math . Anal . 3 4 (2002) , 64-86.

[CKSTT6] J . Colliander , M . Keel , G . Staffilani , H . Takaoka , T . Tao , Almost conservation laws and global rough solutions to a nonlinear Schrodinger equation, Mat h Res . Lett . 9 (2002) , 5-6, 659-682 .

[CKSTT7] J . Colliander , M . Keel , G . Staffilani , H . Takaoka , T . Tao , Existence globale et diffu-sion pour Vequation de Schrodinger nonlineaire repulsive cubique sur R 3 en dessous Vespace d'energie , Journee s "Equation s au x Derivee s Partielles " (Forges-les-Eaux , 2002) , Exp . No . X, 14 , 2002 .

[CKSTT8] J . Colliander , M . Keel , G . Staffilani , H . Takaoka , T . Tao , Polynomial upper bounds for the orbital instability of the ID cubic NLS below the energy norm, Discret e Contin . Dyn . Syst. 9 (2003) , 31-54 .

[CKSTT9] J . Colliander , M . Keel , G . Staffilani , H . Takaoka , T . Tao , Polynomial growth and orbital instability bounds for L 2 -subcritical NLS below the energy norm, Commun . Pur e Appl . 2 (2003) , 33-50 .

[CKSTT10] J . Colliander , M . Keel , G . Staffilani , H . Takaoka , T . Tao , Scattering for the 3D cubic NLS below the energy norm, Comm . Pur e Appl . Math . 5 7 (2004) , 987-1014 .

[CKSTT11] J . Colliander , M . Keel , G . Staffilani , H . Takaoka , T . Tao , Global well-posedness and scattering in the energy space for the critical nonlinear Schrodinger equation in R 3 , t o appear , Annals Math .

[CKSTT12] J . Colliander , M . Keel , G . Staffilani , H . Takaoka , T . Tao , Symplectic nonsqueezing of the KdV flow , to appear , Act a Math .

[CKSTT13] J . Colliander , M . Keel , G . Staffilani , H . Takaoka , T . Tao , Growth of higher Sobolev norms for a periodic nonlinear Schrodinger equation, preprint .

[CST] J . Colliander , G . Staffilani , H . Takaoka , Global well-posedness of the KdV equation below L2, Mat h Res . Letter s 6 (1999) , 755-778 .

[Cof] C.V . Coffman , Uniqueness of the ground state solution for Au — u+u3 = 0 and a variational characterization of other solutions, Arch . Rat . Mech . Anal . 4 6 (1972) , 81-95 .

[CS] P . Constantin , J.C . Saut , Effets regularisants locaux pour des equations disperives generales, C. R . Acad . Sci . Paris . Ser . I . Math . 30 4 (1987) , 407-410 .

[DG] P . D'Ancona , V . Georgiev , On the continuity of the solution operator of the wave maps system, Comm . Pur e Appl . Math . 5 7 (2004) , 357-383 .

[EL] J . Eells , L . Lemaire , A report on harmonic maps, Bull . Londo n Math . Soc . 1 0 (1978) , no . 1, 1-68 .

[EL2] J . Eells , L . Lemaire , Another report on harmonic maps, Bull . London Math . Soc . 20 (1988) , no. 5 , 385-524 .

[ES] J . Eells , H . Sampson , Harmonic mappings of Riemannian manifolds, Amer . J . Math . 8 6 (1964), 109-160 .

[Eva] C . Evans , Partial Differential Equations, Graduat e Studie s i n Mathematic s 19 , America n Mathematical Society , 1998 .

[FW] D . Fang , C . Wang , Some Remarks on Strichartz Estimates for Homogeneous Wave Equa-tion, preprint .

[Fos] D . Foschi , Inhomogeneous Strichartz estimates, Jour . Hyperboli c Diff . Eq s 2 , No . 1 (2005 ) 1-24.

[FK] D . Foschi , S . Klainerman , Bilinear space-time estimates for homogeneous wave equations, Ann. Sci . d e TENS , 3 3 no. 2 (2000) .

[Fri] A . Friere , Wave maps to compact Lie groups, preprint .

366 BIBLIOGRAPHY

[FPT] G . Furioli , F . Planchon , E . Terraneo , Unconditional well-posedness for semi-linear Schrodinger equations in H s, Harmoni c analysi s a t Moun t Holyoke , (Sout h Hadley , MA , 2001), 147-156 .

[FT] G . Furioli , E . Terraneo , Besov spaces and unconditional well-posedness for the nonlinear Schrodinger equation in H s,Coram, i n Contemp . Math . 5 (2003) , 349-367 .

[GLS] V . Georgiev , H . Lindblad , C . D . Sogge , Weighted Strichartz estimates and global existence for semilinear wave equations, Arner . J . Math . 11 9 (1997 ) 1291-1319 .

[GNN] B . Gidas , N.W . Ni , L . Nirenberg , Symmetry of positive solutions of nonlinear elliptic equations in R n, Mathematica l analysi s an d applications , Par t A , pp . 369-402 , Adv . i n Math . Suppl. Stud. , 7a , Academi c Press , Ne w York-London , 1981 .

[Gin] J . Ginibre , Le probleme de Cauchy pour des EDP semi-lineaires periodiques en variables d'espace (d'apres Bourgain), Seminair e Bourbak i 1994/1995 , Asterisqu e 23 7 (1996) , Exp . 796, 163-187 .

[GO] J . Ginibre , T . Ozawa , Long range scattering for nonlinear Schrodinger and Hartree equa-tions in space dimension n >= 2 , Comm . Math . Phys . 15 1 (1993 ) 619-645 .

[GSV] J . Ginibre , A . Soffer , G . Velo , The global Cauchy problem for the critical non-linear wave equation, J . Funct . Anal . 11 0 (1982) , 96-130 .

[GTV] J . Ginibre , Y . Tsutsumi , G . Velo , Existence and uniqueness of solutions for the generalized Korteweg de Vries equation, Math . Z . 20 3 (1990 ) 9-36 .

[GV] J . Ginibre , G . Velo , Smoothing Properties and Retarded Estimates for Some Dispersive Evolution Equations, Comm . Math . Phys. , 12 3 (1989 ) , 535-573.

[GV2] J . Ginibre , G . Velo , Generalized Strichartz Inequalities for the Wave Equation, Jour . Func . Anal., 13 3 (1995) , 50-68 .

[GV3] J . Ginibre , G . Velo , Scattering theory in the energy space for a class of nonlinear Schrodinger equations, J . Math . Pure . Appl . 6 4 (1985) , 363-401 .

[GV4] J . Ginibre , G . Velo , The global Cauchy problem for the nonlinear Schrodinger equation revisited, Ann . Inst . H . Poincar e Anal . No n Lineair e 2 (1985),309-327 .

[GV5] J . Ginibre , G . Velo , Conformal invariance and time decay for nonlinear wave equations I., Ann . Inst . H . Poincare , Pyhs . Theor . 4 7 (1987) , 221-261 .

[GV6] J . Ginibre , G . Velo , Scattering theory in the energy space for a class of non-linear wave equations, Commun . Math . Phys . 12 3 (1989) , 535-573 .

[GV7] J . Ginibre , G . Velo , Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation, J . Diff . Eq . 9 3 (1991),150-232 .

[Gla] R.T . Glassey , Asymptotic behavior of solutions to certain nonlinear Schrodinger-Hartree equations, Commun . Math . Phys . 5 3 (1977) , 9-18 .

[Gla2] R.T . Glassey , On the blowing up of solutions to the Cauchy problem for nonlinear Schrodinger operators, J . Math . Phys . 8 (1977) , 1794-1797 .

[Gli] J . Glimm , Solutions in the large for nonlinear hyperbolic systems of equations, Commun . Pure Appl . Math . 18 , (1965) , 697-715 .

[Gri] M . Grillakis , Regularity and asymptotic behaviour of the wave equation with a critical non-linearity, Ann . o f Math . 13 2 (1990) , 485-509 .

[Gri2] M . Grillakis , Regularity for the wave equation with a critical nonlinearity, Commun . Pur e Appl. Math. , 4 5 (1992) , 749-774 .

[Gri3] M . Grillakis , A priori estimates and regularity of nonlinear waves, i n Proceed . Inter . Con -gress o f Math . 1994 , Birkhauser , 118 7 - 1194 .

[Gri4] M . Grillakis , Classical solutions for the equivariant wave map in 1 + 2 dimensions, t o appear i n Indian a Univ . Math . J .

[Gri5] M . Grillakis , On nonlinear Schrodinger equations. , Commun . Partia l Differentia l Equa -tions 2 5 (2000) , no . 9-10 , 1827-1844 .

[Gri6] M . Grillakis , persona l communication . [GSS] M . Grillakis , J . Shatah , W . Strauss , Stability theory of solitary waves in the presence of

symmetry I, J . Funct . Anal . 7 4 (1987) , 160-197 . [GSS2] M . Grillakis , J . Shatah , W . Strauss , Stability theory of solitary waves in the presence of

symmetry II, J . Funct . Anal . 9 4 (1990) , 308-348 . [Gro] M . Gromov , Pseudoholomorphic curves in symplectic manifolds, Invent . Math . 8 2 (1985) ,

307-347. [Gu] C . Gu , On the Cauchy problem for harmonic maps defined on two-dimensional Minkowski

space, Commun . Pur e Appl . Math. , 33,(1980) , 727-737 .

BIBLIOGRAPHY 367

[Har] J . Harmse , On Lebesgue space estimates for the wave equation, Indian a Univ . Math . J . 3 9 (1990), 229-248 .

[HN] N . Hayashi , P.I . Naumkin , Asymptotics for large time of solutions to the nonlinear Schrodinger and Hartree equations, Amer . J . Math . 12 0 (1998) , 369-389 .

[HO] N . Hayashi , T . Ozawa , Remarks on nonlinear Schrodinger equations in one space dimension, Diff. Integ . Eq . 7 (1994) , 453-461 .

[Hel] F . Helein , Regularite des applications faiblement harmoniques entre une sur surface et un sphere, C.R . Acad . Sci . Pari s Ser . I Math. , 31 1 (1990) , 519-524 .

[Hel2] F . Helein , Regularite des applications faiblement harmoniques entre une sur face et une varitee Riemannienne, C.R . Acad . Sci . Pari s Ser . I Math. , 31 2 (1991) , 591-596 .

[Hid] K . Hidano , Scattering problem for the Nonlinear Wave Equation in the Finite Energy and Conformal Charge space, J . Func . Anal . 18 7 (2001) , 274-307 .

[HSW] N . J . Hi t chin, G . B . Segal , R.S . Ward , Integrable systems: twistors, loop groups, and Riemann surfaces, Oxfor d Universit y Press , 1999 .

[HZ] H . Hofer , E . Zehnder , Symplectic invariants and Hamiltonian dynamics, Birkhaiise r Ad -vanced Texts , 1994 .

[Hop] E . Hopf , The partial differential equation ut + uu x = u xx, Commun . Pur e Appl . Math . 3 (1950), 201-230 .

[Hor] L . Hormander , Lecture s o n nonlinea r hyperboli c differentia l equations , Mathematique s an d Applications, 26 . Springer-Verlag , 1997 .

[IK] A . Ionescu , C . Kenig , Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, preprint .

[lor] R.J . Iori o Jr. , On the Cauchy problem for the Benjamin-Ono equation, Commun . Partia l Diff. Eq . 1 1 (1986) , 1031-1181 .

[Joh] F . John , Blow-up of solutions of non-linear wave equations in three dimensions, Manuscript . Math. 2 8 (1979) , 235-268 .

[Jor] K . Jorgens , Das Anfangswertproblem im Grossen fur eine Klasse nichtlinearer Wellengle-ichungen, Math . Z . 7 7 (1961) , 295-308 .

[Kap] L . Kapitanski, Some Generalizations of the Strichartz-Brenner Inequality, Leningra d Math . J., 1 (1990) , 693-676 .

[Kap2] L . Kapitanski , Weak and Yet Weaker Solutions of Semilinear Wave Equations, Commun . Part. Diff . Eq. , 1 9 (1994) , 1629-1676 .

[Kap3] L . Kapitanski , Global and unique weak solutions of nonlinear wave equations Math . Res . Letters, 1 (1994) , 211-223 .

[KTop] T . Kappeler , P . Topalov , Global well-posedness of KdV in / / _ 1 ( T , R ) , preprint . [Kat] T . Kato , Quasilinear equations of evolution, with applications to partial differential equa-

tions, Springe r Lectur e Note s i n Math . 44 8 (1975) , 27-50 . [Kat2] T . Kato , Th e Cauch y proble m fo r th e Korteweg-d e Vrie s equation, Pitma n Researc h Note s

in Math . 5 3 (1979) , 293-30 7 [Kat3] T . Kato , On the Korteweg-de Vries equation, Manuscript a Mathematica , 2 8 (1979 ) 89-99 . [Kat4] T . Kato , Blow-up of solutions of some nonlinear hyperbolic equations, Commun . Pur e

Appl. Math . 3 3 (1980) , 501-505 . [Kat5] T . Kato , On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Adv .

in Math . Suppl . Stud. , Stud , i n Appl . Math . 8 (1983) , 93-128 . [Kat6] T . Kato , On nonlinear Schrodinger equations. Ann . Inst . H . Poincar e Phys . Theor . 4 6

(1987), 113-129 . [Kat7] T . Kato , Abstrac t evolutio n equations , linea r an d quasilinear , revisited . Functiona l analysi s

and relate d topics , 199 1 (Kyoto) , 103-125 , Lectur e Note s i n Math. , 1540 , Springer , Berlin , 1993.

[Kat8] T . Kato , An L q,r-theory for nonlinear Schrodinger equations, Spectra l an d scatterin g theory an d applications , Adv . Stud . Pur e Math. , vol . 23 , Math . Soc . Japan , Tokyo , 1994 , 223-238.

[Kat9] T . Kato , On nonlinear Schrodinger equations, II. H s -solutions and unconditional well-posedness, J . d'Analyse . Math . 67 , (1995) , 281-306 .

[Keel] M . Keel , Global existence for critical power Yang-Mills-Higgs equations in R 3 + 1 , Princeto n University Thesis , 1996 .

[KTao] M . Keel , T . Tao , Endpoint Strichartz estimates, Amer . J . Math. , 12 0 (1998) , 955-980 .

368 BIBLIOGRAPHY

[KTao2] M . Keel , T . Tao , Local and global well-posedness of wave maps on R 1 + 1 for rough data, Internat. Math . Res . Not. 2 1 (1998) , 1117-1156 .

[KTao3] M . Keel , T . Tao , Small data blowup for semilinear Klein-Gordon equations, Amer . J . Math. 12 1 (1999) , 629-669 .

[KK] C . Kenig , K . Koenig , On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations, preprint .

[KPV] C . Kenig , G . Ponce , L . Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J . Amer . Math . Soc . 4 (1991) , 323-347 .

[KPV2] C . Kenig , G . Ponce , L . Vega , Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Commun . Pur e Appl . Math . 4 6 (1993), 527-560 .

[KPV3] C . E. Kenig , G . Ponce , L . Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duk e Mat h J . 7 1 (1993) , 1-21 .

[KPV4] C . Kenig , G . Ponce , L . Vega, A bilinear estimate with applications to the KdV equation, J. Amer . Math . Soc . 9 (1996) , 573-603 .

[KPV5] C . Kenig, G . Ponce, L . Vega, On the ill-posedness of some canonical dispersive equations, Duke Math . J . 10 6 (2001) , 617-633 .

[Ker] S . Keraani , On the defect of compactness for the Strichartz estimates of the Schrodinger equations, J . Diff . Eq . 175 , (2001) , 353-392 .

[Kla] S . Klainerman , Uniform decay estimates and the Lorentz invariance of the classical wave equation, Commun . Pur e Appl . Math . 3 8 (1985) , 321-332 .

[Kla2] S . Klainerman, On the regularity of classical field theories in Minkowski space-time R 3 + 1 , Prog, i n Nonlin . Diff . Eq . and thei r Applic , 29 , (1997) , Birkhauser , 113-150 .

[Kla3] S . Klainerman , A Commuting Vectorfield Approach To Strichartz Type Inequalities And Applications To Quasilinear Wave Equations, Internat . Math . Res . Notice s (2001) , no . 5 , 221-274.

[Kla4] S . Klainerman , PDE as a unified subject, Geom . Func . Anal. , specia l volum e (2000) , 279-315.

[KM] S . Klainerman , M . Machedon , Space-time estimates for null forms and the local existence theorem, Commun . Pur e Appl . Math . 4 6 (1993) , no . 9, 1221-1268 .

[KM2] S . Klainerman , M . Machedon , Smoothing estimates for null forms and applications, Duk e Math J . 8 1 (1995) , 99-103 .

[KM3] S . Klainerman , M . Machedon , Remark on Strichartz-type inequalities. Wit h appendice s by Jea n Bourgai n an d Danie l Tataru . Internat . Math . Res . Notices 5 (1996) , 201-220 .

[KM4] S . Klainerman , M . Machedon , On the regularity properties of a model problem related to wave maps, Duk e Math . J . 8 7 (1997) , 553-589 .

[KR] S . Klainerman , I . Rodnianski , On the global regularity of wave maps in the critical Sobolev norm, Internat . Math . Res . Not. 1 3 (2001) , 656-677 .

[KS] S . Klainerman , S . Selberg , Remark on the optimal regularity for equations of wave maps type, Commun . Partia l Diff . Eq. , 22 (1997) , 901-918 .

[KTat] S . Klainerman , D . Tataru , On the optimal regularity for Yang-Mills equations in R 4 + 1 , J. Amer . Math . Soc . 12 (1999) , 93-116 .

[KTzv] H . Koch , N . Tzvetkov , On the local well-posedness of the Benjamin-Ono equation in HS(R), Internat . Math . Res . Not. 26 (2003) .

[KTzv2] H . Koch , N . Tzvetkov , Nonlinear wave interactions for the Benjamin-Ono equation, Internat. Math . Res . Not. 3 0 (2005) , 1833-1847 .

[KdV] D.J . Korteweg, G . de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos . Mag . 539 (1895) , 422-443 .

[Kri] J . Krieger , Global regularity of wave maps from R 3 + 1 to H 2 , Comm . Math . Phys . 23 8 (2003), 333-366 .

[Kri2] J . Krieger , Global regularity of wave maps from R 2 + 1 to H 2 . Small energy, preprint . [Kri3] J . Krieger , Stability of spherically symmetric wave maps, preprint . [KF] S . Kruzhkov , A . Faminskii , Generalized solutions of the Korteweg-de Vries equation

(Russian), Mat . Sb . (N.S. ) 12 0 (1983) , 296-425 ; Englis h translatio n i n Mat h USS R Sborni k 48 (1984) , 391-421 .

[Kuk] S . Kuksin , On squeezing and flow of energy for nonlinear wave equations, Geom . Func . Anal. 5 (1995) , 668-70 1 .

BIBLIOGRAPHY 369

[Kuk2] S . Kuksin , Infinite-dimensional symplectic capacities and a squeezing theorem for Hamil-tonian PDE's, Commun . Math . Phys . 16 7 (1995) , 521-552 .

[Kuk3] S . Kuksin , Analysi s o f Hamiltonia n PDEs . Oxfor d Lectur e Serie s i n Mathematic s an d it s Applications, 19 . Oxfor d Universit y Press , Oxford , 2000 .

[Kwo] M.K . Kwong , Uniqueness of positive solutions of Au — u-\-up = 0 in R n , Arch . Rat . Mech . Anal. 10 5 (1989) , 243-266 .

[LShu] O.A . Ladyzhenskaya , V.I . Shubov , Unique solvability of the Cauchy problem for the equa-tions of the two dimensional chiral fields, taking values in complete Riemann manifolds, J . Soviet Math. , 2 5 (1984) , 855-864 . (Englis h Trans , o f 198 1 Article. )

[Leb] G . Lebeau , Nonlinear optics and supercritical wave equation, Bull . Soc . Roy . Sci . Liege , 7 0 (2002), 267-306 . Hommag e a Pasca l Laubin .

[Lem] L . Lemaire, Applications harmoniques de surfaces riemanniennes, J . Diff . Geom . 1 3 (1978) . [LStr] J . Lin , W . Strauss , Decay and scattering of solutions of a nonlinear Schrodinger equation,

Journ. Punct . Anal . 30 , (1978) , 245-263 . [Linj H . Lindblad , A Sharp Counterexample to Local Existence of Low Regularity Solutions to

Nonlinear Wave Equations, Duk e Mat h J. , 72 , (1993) , 503-539 . [LSog] H . Lindblad , C . D . Sogge , On Existence and Scattering with Minimal Regularity for Semi-

linear Wave Equations, Jour . Func . Anal. , 13 0 (199 5 ) , 357-426 . [LSog2] H . Lindblad , C . D . Sogge , Restriction theorems and semilinear Klein-Gordon equations

in (1+3) dimensions, Duk e Math . J . 8 5 (1996) , no . 1 , 227-252 . [MNNO] S . Machihara , M . Nakamura , K . Nakanishi , T . Ozawa , Endpoint Strichartz estimates

and global solutions for the nonlinear Dirac equation, J . Func . Anal . 21 9 (2005) , 1-20 . [MNO] S . Machihara , K . Nakanishi , T . Ozawa , Small global solutions and the nonrelativistic limit

for the nonlinear Dirac equation, Rev . Mat . Iberoamericana . 1 9 (2003) , 179—194 . [MN02] S . Machihara , K . Nakanishi , T . Ozawa , Nonrelativistic limit in the energy space for

nonlinear Klein-Gordon equations, Math . Ann . 32 2 (2002) , 603-621 . [MSW] B . Marshall , W . Strauss , S . Wainger , L p — Lq estimates for the Klein-Gordon equation,

J. Math . Pure s Appl . (9 ) 5 9 (1980) , no . 4 , 417-440 . [MN] N . Masmoudi , K . Nakanishi , From nonlinear Klein-Gordon equation to a system of coupled

nonlinear Schrdinger equations, Math . Ann . 32 4 (2002) , no . 2 , 359-389 . [Mat] C . Matheus , Global well-posedness for a NLS-KdV system on T , preprint . [Mel] K . McLeod , Uniqueness of positive radial solutions of Au + f(u) = 0 in R n. II. Trans .

Amer. Math . Soc . 33 9 (1993) , no . 2 , 495-505 . [Mer] F . Merle , Construction of solutions with exactly k blow-up points for the Schrodinger equa-

tion with critical non-linearity, Commun . Math . Phys . 14 9 (1992) , 205-214 . [Mer2] F . Merle , Determination of blow-up solutions with minimal mass for non-linear

Schrodinger equation with critical power, Duk e Math . J . 6 9 (1993) , 427-453 . [Mer3] F . Merle , Asymptotics for L 2 minimal blowup solutions of critical nonlinear Schrodinger

equation, Ann . Inst . Henr i Poincar e 1 3 (1996) , 553-565 . [MR] F . Merle , P . Raphael , Blowup dynamic and upper bound on the blowup rate for the critical

nonlinear Schrodinger equation, Ann . Math . 16 1 (2005) , 157-222 . [MR2] F . Merle , P . Raphael , Sharp upper bound on the blowup rate for the critical nonlinear

Schrodinger equation, GAF A 1 3 (2003) , 591-642 . [MR3] F . Merle , P . Raphael , On universality of blowup profile for L 2 critical nonlinear

Schrodinger equation, Invent . Math . 15 6 (2004) , 565-672 . [Miu] R . Miura , Korteweg-de Vries equation and generalizations. I. A remarkable explicit non-

linear transformation, J . Mathematical . Phys . 9 (1968) , 1202-1204 . [Miu2] R . Miura , The Korteweg-de Vries Equation: A Survey of Results, SIA M Rev . 18(3 ) (1976) ,

412-459. [MST] L . Molinet , J.-C . Saut , N . Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related

equations, SIA M J . Math . Anal . 3 3 (2001) , 982-988 . [Mon] S . J . Montgomery-Smith , Time Decay for the Bounded Mean Oscillation of Solutions of

the Schrodinger and Wave Equation, Duk e Mat h J . 1 9 (1998) , 393-408 . [Mor] C . Morawetz , Time decay for the nonlinear Klein-Gordon equation, Proc . Roy . Soc . A 30 6

(1968), 291-296 . [MS] S . Muller , M . Struwe , Global existence of wave maps in 1 + 2 dimensions with finite energy

data, Topologica l Method s i n Analysi s 7 (1996) , 245-259 .

370 BIBLIOGRAPHY

[MT] T . Muramatu , S . Taoka , The initial value problem for the 1-D semilinear Schrodinger equation in Besov spaces, J . Math . Soc . Japan 5 6 (2004) , 853-888 .

[NSU] A . Nahmod , A . Stefanov , K . Uhlenbeck , On the well-posedness of the wave map problem in high dimensions, Commun . Anal . Geom . 1 1 (2003) , 49-83 .

[Nak] K . Nakanishi, Scattering Theory for Nonlinear Klein-Gordon Equation with Sobolev Critical Power, Internat . Math . Researc h Not . 31 (1999) , 31-60 .

[Nak2] K . Nakanishi , Energy scattering for non-linear Klein-Gordon and Schrodinger equations in spatial dimensions 1 and 2, J . Func . Anal . 16 9 (1999) , 201-225 .

[Nak3] K . Nakanishi , Unique global existence and asymptotic behaviour of solutions for wave equations with non-coercive critical nonlinearity, Commun . Partia l Differentia l Equations . 2 4 (1999), 185-221 .

[Nak3] K . Nakanishi , Local well-posedness and Illposedness in the critical Besov spaces for semi-linear wave equations with quadratic forms, Funk . Ekvac . 4 2 (1999) , 261-279 .

[Nak4] K . Nakanishi , Nonrelativistic limit of scattering theory for nonlinear Klein-Gordon equa-tions, J . Differentia l Equation s 18 0 (2002) , no . 2 , 453-470 .

[NTT] K . Nakanishi, H . Takaoka, Y. Tsutsumi, Counterexamples to bilinear estimates related with the KdV equation and the nonlinear Schrodinger equation, Method s Appl . Anal . 8 (2001) , 569-578.

[Naw] H . Nawa, Asymptotic and limiting profiles of blowup solutions of the nonlinear Schrodinger equation with critical power, Commun . Pure . Appl . Math . 5 2 (1999) , 193-270 .

[Obe] D . Oberlin , Convolution estimates for some distributions with singularities on the light cone, Duk e Math . J . 5 9 (1989) , 747-757 .

[Ono] H . Ono , Algebraic solitary waves in stratified fluids, J . Phys . Soc . Japan 3 9 (1975) , 1082-1091.

[Oza] T . Ozawa , Long range scattering for nonlinear Schrodinger equations in one space dimen-sion, Commun . Math . Phys . 13 9 (1991) , 479-493 .

[Oza2] T . Ozawa , On nonlinear Schrodinger equations of derivative type, Indian a Univ . Math . J . 45 (1996) , 137-163 .

[Pec] H . Pecher , Global solutions with infinite energy for the 1-dimensional Zakharov system, preprint.

[Pec2] H . Pecher , The Cauchy problem for a Schroedinger-Korteweg-de Vries system with rough data, preprint .

[Pec3] H . Pecher , Rough solutions of a Schroedinger-Benjamin-Ono system, preprint . [Poh] K . Pohlmeyer , Integrable Hamiltonian systems and interaction through quadratic con-

straints, Commun . Math . Phys. , 4 6 (1976) , 207-221 . [Pon] G . Ponce , On the global well-posedness of the Benjamin-Ono equation, Diff . Int . Eq . 4

(1991), 527-542 . [PS] G . Ponce , T . Sideris , Local regularity of nonlinear wave equations in three space dimensions,

Commun. Partia l Diff . Eq . 1 8 (1983) , 169-177 . [Pri] P . Price , A montonicity formula for Yang-Mills fields, Manuscript a Math . 4 3 (1983) , 131-

166. [Rau] J . Rauch , The u 5 Klein-Gordon equation. II. Anomalous singularities for semilinear wave

equations i n Nonlinea r partia l differentia l equation s an d thei r applications . Colleg e d e Franc e Seminar, Res . Notes i n Math . 51 , Pitman Publ.,(1981 ) 335-364 .

[RR] J . Rauch , M . Reed , Nonlinear microlocal analysis of semilinear hyperbolic systems in one space dimension, Duk e Math . J. , 4 9 (1982) , 397-475 .

[Ros] R . Rosales , / . Exact solution of some nonlinear evolution equations. II. The similarity solution for the Korteweg-de Vries equation and the related Painleve transcendent, Ph . D . thesis, Californi a Institut e o f Technology , 1977 .

[RV] E . Ryckman , M . Visan , Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrodinger equation in R 1 + 4 , preprint .

[SacU] J . Sacks , K . Uhlenbeck , The existence of minimal immersions of 2-spheres, Annal s o f Math. (2 ) 11 3 (1981) , 1-24 .

[Sad] C . Sadosky , Interpolation of Operators and Singular Integrals, Marce l Dekke r Inc. , 1976. [Sau] J . C . Saut , Sur des generalisations de Vequation de Korteweg-de Vries, Universit e d e Paris -

Sud Departemen t d e Mathematique , Orsay , (1978) . [Sau2] J . C . Saut , Quelques generalisations de Vequation de Korteweg-de Vries. II, J . Differentia l

Equations 3 3 (1979) , 320-335 .

BIBLIOGRAPHY 371

[Sau3] J.C . Saut , Sur quelques generalizations de Vequation de Korteweg-de Vries, J . Math . Pure s Appl. 5 8 (1979) , 21-61 .

[Schn] G . Schneider , Approximation of the Korteweg-de Vries equation by the non-linear Schrodinger equation, J . Differentia l Equation s 14 7 (1998) , 333-354 .

[Scho] R . Schoen , Analytic aspects of the harmonic map problem, Math . Sci . Res . Inst . Publ . 2 (1984), Springer , Berlin , 321-358 .

[SchU] R . Schoen , K . Uhlenbeck , A regularity theory for harmonic maps, J . Diff . Geom . 1 7 (1982) , 307-335.

[SY] R . Schoen , S.T . Yau , Lecture s o n harmoni c maps , Conferenc e proceeding s an d Lectur e note s in Geometr y an d Topology , II . Internationa l Press .

[Seg] I.E . Segal , Non-linear semi-groups, Ann . o f Math . 7 8 (1963) , 339-364 . [Seg2] I.E . Segal , The global Cauchy problem for a relativistic scalar field with power interactions,

Bull. Soc . Math . Pranc e 9 1 (1963) , 129-135 . [Seg3] I . E . Segal , Space-time Decay for Solutions of Wave Equations, Adv . Math. , 2 2 (1976) ,

304-311. [Sel] S . Selberg , Multilinear space-time estimates and applications to local existence theory for

non-linear wave equations, Princeto n Universit y Thesis . [Sha] J . Shatah , The Cauchy problem for harmonic maps on Minkowski space, i n Proceed . Inter .

Congress o f Math . 1994 , Birkhauser , 1126-1132 . [Sha2] J . Shatah , Weak solutions and development of singularities of the SU(2) a-model. Com -

mun. Pur e Appl . Math. , 4 1 (1988) , 459-469 . [SStra] J . Shatah , W . Strauss , Breathers as homoclinic geometric wave maps, Physic s D 9 9

(1996), 113-133 . [SStru] J . Shatah , M . Struwe , Regularity results for non-linear wave equations, Ann . o f Math .

138 (1993 ) 503-518 . [SStru2] J . Shatah , M . Struwe , Well Posedness in the energy space for semilinear wave equations

with critical growth, Inter . Math . Researc h Not. , 7 (1994) , 303-309 . [SStru3] J . Shatah , M . Struwe , Geometric Wave Equations, Couran t Lectur e Note s i n Mathemat -

ics 2 (1998) . [SStru4] J . Shatah , M . Struwe , The Cauchy problem for wave maps, Internat . Math . Res . Not .

11 (2002 ) 555-571 . [ST] J . Shatah , A . Tavildar-Zadeh , Regularity of harmonic maps from the Minkowski space into

rotationally symmetric manifolds., Commun . Pur e Appl . Math . 4 5 (1992) , 947-971 . [ST2] J . Shatah , A . Tavildar-Zadeh, On the Cauchy problem for equivariant wave maps, Commun .

Pure Appl . Math. , 4 7 (1994) , 719-753 . [Sid] T . Sideris , Global existence of harmonic maps in Minkowski space, Commun . Pur e Appl .

Math., 4 2 (1989),1-13 . [Sjo] P . Sjolin , Regularity of solutions to Schrodinger equations, Duk e Math . J.,5 5 (1987) , 699 -

715. [SSog] H . Smith , C D . Sogge , Global Strichartz estimates for nontrapping perturbations of the

Laplacian, Commun . PD E 2 5 (2000) , 2171-2183 . [Sog] C . D . Sogge , Lectures on Nonlinear Wave Equations, Monograph s i n Analysi s II , Interna -

tional Press , 1995 . [Sta] G . Staffilani , On the growth of high Sobolev norms of solutions for KdV and Schrodinger

equations, Duk e Mat h J . 8 6 (1997) , 109-142 . [Stef] A . Stefanov , Strichartz estimates for the Schrodinger equation with radial data, Proc . Amer .

Math. Soc . 12 9 (2001) , no . 5 , 1395-1401 . [Stei] E . M . Stein , Singular Integrals and Differentiability Properties of Functions, Princeto n

University Press , 1970 . [Stei2] E . M . Stein , Harmonic Analysis, Princeto n Universit y Press , 1993 . [Stra] W . A . Strauss , Nonlinear wave equations, CBM S Regiona l Conferenc e Serie s i n Mathemat -

ics, no . 73 , Amer. Math . Soc . Providence , RI , 1989 . [Stri] R . S . Strichartz , Restriction of Fourier Transform to Quadratic Surfaces and Decay of

Solutions of Wave Equations, Duk e Math . J. , 4 4 (1977) , 7 0 5-774 . [Stri2] R . Strichartz , Asymptotic behavior of waves, J . Funct . Anal . 4 0 (1981) , no . 3 , 341-357 . [Stru] M . Struwe , On the evolution of harmonic mappings of Riemannian surfaces, Comment .

Math. Helv . 6 0 (1985) , 558-581 .

372 BIBLIOGRAPHY

[Stru2] M . Struwe , Globally regular solutions to the it 5 Klein-Gordon equation, Ann . Scuol a Norm . Sup. Pis a CI . Sci . 1 5 (1988) , 495-513 .

[Stru3] M . Struwe , Equivariant wave maps in two dimensions, Comm . Pur e Appl . Math . 5 6 (2003), 815-823 .

[Stru4] M . Struwe , Radially symmetric wave maps from the (1+2)-dimensional Minkowski space to a sphere, Mat h Z . 24 2 (2002) , 407-414 .

[Stru5] M . Struwe , Radially symmetric wave maps from (1+2)-dimensional Minkowski space to general targets, Calc . Var . 1 6 (2003) , 431-437 .

[SSul] C . Sulem , P.-L . Sulem , The nonlinear Schrodinger equation: Self-focusing and wave col-lapse, Springer-Verlag , Ne w York , (1999) , p . 25 .

[Tak] H . Takaoka , Well-posedness for the one-dimensional Schrodinger equation with the deriv-ative nonlinearity, Adv . Diff . Eq . 4 (1999) , 561-580 .

[Tao] T . Tao , Low regularity semi-linear wave equations, Commun . Partia l Differentia l Equations , 24 (1999) , no . 3-4 , 599-629 .

[Tao2] T . Tao , Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrodinger equation, Commun . PD E 2 5 (2000) , 1471-1485 .

[Tao3] T . Tao , Ill-posedness of one-dimensional wave maps at the critical regularity, Amer . J . Math., 12 2 (2000) , 451-463 .

[Tao4] T . Tao , Multilinear weighted convolution of I? functions, and applications to non-linear dispersive equations, Amer . J . Math . 12 3 (2001) , 839-908 .

[Tao5] T . Tao , Global regularity of wave maps I. Small critical Sobolev norm in high dimension, Internat. Math . Res . Not . 7 (2001) , 299-328 .

[Tao6] T . Tao , Global well-posedness of wave maps II. Small energy in two dimensions, Commun . Math. Phys . 22 4 (2001) , 443-544 .

[Tao7] T . Tao , Global well-posedness of the Benjamin-Ono equation in i f 1 ( R ) , J . Hyperboli c Diff . Eq. 1 (2004 ) 27-49 .

[Tao8] T . Tao , On the asymptotic behavior of large radial data for a focusing non-linear Schrodinger equation, Dynamic s o f PD E 1 (2004) , 1-48 .

[Tao9] T . Tao , Global well-posedness and scattering for the higher-dimensional energy-critical non-linear Schrodinger equation for radial data, Ne w Yor k J . Math . 1 1 (2005) , 57-80 .

[TaolO] T . Tao , Geometric renormalization of large energy wave maps, Journee s EDP , Forge s le s Eaux, 7-1 1 Jun e 2004 , X I 1-32. .

[Tao 11] T . Tao , Recent progress on the Restriction conjecture, t o appear , Par k Cit y Proceedings . [Taol2] T . Tao , Spacetime bounds for the energy-critical nonlinear wave equation in three spatial

dimensions, preprint . [TV] T . Tao , M . Visan , Stability of energy-critical nonlinear Schrodinger equations in high di-

mensions, Electron . J . Diff . Eq . 200 5 (2005) , No . 118 , 1-28 . [TVZ] T . Tao , M . Visan , X . Zhang , The nonlinear Schrodinger equation with power type nonlin-

earities, preprint . [Tat] D . Tataru , Local and global results for wave maps I, Commun . PD E 2 3 (1998) , 1781-1793 . [Tat2] D . Tataru , On global existence and scattering for the wave maps equation, Amer . J . Math .

123 (2001) , 37-77 . [Tat3] D . Tataru , On Bu = \Vu\ 2 in 5 + 1 dimensions, Math . Res . Letter s 6 (1999) , 469-485 . [Tat4] D . Tataru , Rough solutions for the wave maps equation, Amer . J . Math . 12 7 (2005) , 293 -

377. [Tat5] D . Tataru , The wave maps equation, preprint . [Tay] M . Taylor , Partia l Differentia l Equation s I . Basi c Theory , Applie d Mathematica l Science s

115, Springer-Verlag , Ne w York , Berlin , Heidelber g 1997 . [Tay2] M . Taylor , Tool s fo r PDE . Pseudodifferentia l operators , paradifferentia l operators , an d

layer potentials . Mathematica l Survey s an d Monographs , 81 . American Mathematica l Society , Providence, RI , 2000 .

[TU] C.L . Terng , K . Uhlenbeck , 1 + 1 wave maps into symmetric spaces, Comm . Anal . Geom . 1 2 (2004), 343-388 .

[Tom] M.M . Tom , Smoothing properties of some weak solutions of the Benjamin-Ono equation, Diff. Int . Eq . 3 (1990 ) 683-694 .

[Tomas] P . Tomas , A restriction theorem for the Fourier transform, Bull . Amer . Math . Soc . 8 1 (1975), 477-478 .

BIBLIOGRAPHY 373

[Tsu] Y . Tsutsumi , L 2 solutions for nonlinear Schrodinger equations and nonlinear groups, Punk . Ekva. 3 0 (1987) , 115-125 .

[Tsu2] Y . Tsutsumi , The Cauchy problem for the Korteweg-de Vries equation with measures as initial data, SIA M J . Math . Anal . 2 0 (1989) , 582-588 .

[Tzi] N . Tzirakis , The Cauchy problem for the semilinear quintic Schrodinger equation in one dimension 1 8 (2005) .

[Tzv] N . Tzvetkov, Remark on the local ill-posedness for the KdV equation, C . R . Acad . Sci . Pari s 329 (1999) , 1043-1047 .

[Tzv2] N . Tzvetkov , Ill-posedness issues for nonlinear dispersive equations, lectur e notes . [Tzv3] N . Tzvetkov , On the long time behavior of KdV type equations (after Martel-Merle),

Bourbaki semina r Marc h 2004 . [Uhl] K . Uhlenbeck , Connections with L p bounds on curvature, Commun . Math . Phy s 83 , (1982 )

31-42. [VV] A . Vargas , L . Vega , Global well-posedness for ID nonlinear Schrodinger equation for data

with an infinite L 2 norm, J . Math . Pure s Appl . 8 0 (2001) , 1029-1044 . [Veg] L . Vega , Schrodinger equations: pointwise convergence to the initial data, Proc . Amer .

Math. Soc , 10 2 (1988) , 874-878 . [Vil] M . Vilela , Regularity of solutions to the free Schrodinger equation with radial initial data,

Illinois J . Math . 4 5 (2001) , 361-370 . [Vil2] M . Vilela , Inhomogeneous Strichartz estimates for the Schrodinger equation, preprint . [Vis] M . Visan, The defocusing energy-critical nonlinear Schrodinger equation in dimensions four

and higher, Ph.D . thesis , i n preparation . [Wei] M.I . Weinstein , Nonlinear Schrodinger equations and sharp interpolation estimates, Com -

mun. Math . Phys . 8 7 (1983) , 567-576 . [Wei2] M . Weinstein , Modulational stability of ground states of nonlinear Schrodinger equations,

SIAM J . Math . Anal . 1 6 (1985) , 472-491 . [Wei3] M . Weinstein , Lyapunov stability of ground states of nonlinear dispersive equations,

Comm. Pur e Appl . Math . 3 9 (1986) , 51-68 . [Yaj] K . Yajima , Existence of solutions for Schrodinger evolution equations, Commun . Math .

Phys. 11 0 (1987) , 415-426 . [Zho] Y . Zhou , Global weak solutions for the 1 + 2 dimensional wave maps into homogeneous

spaces Ann . Inst . Henr i Poincar e (C ) Analys e nonlineair e 1 6 (1999) , 411-422 . [Zho2] Y . Zhou , Uniqueness of weak solutions of 1 + 1 dimensional wave maps, Math . Z . 23 2

(1999), 707-719 . [Zho3] Y . Zhou , Local existence with minimal regularity for nonlinear wave equations, Amer . J .

Math, 11 9 (1997) , 671-703 .

This page intentionally left blank

Titles i n Thi s Serie s

106 Terenc e Tao , Nonlinea r dispersiv e equations : Loca l an d globa l analysis , 200 6

105 Christop h Thiele , Wav e packe t analysis , 200 6

104 Donal d G . Saari , Collisions , rings , an d othe r Newtonia n TV-bod y problems , 200 5

103 Iai n Raeburn , Grap h algebras , 200 5

102 Ke n Ono , Th e we b o f modularity : Arithmeti c o f th e coefficient s o f modula r form s an d q

series, 200 4

101 Henr i Darmon , Rationa l point s o n modula r ellipti c curves , 200 4

100 Alexande r Volberg , Calderon-Zygmun d capacitie s an d operator s o n nonhomogeneou s

spaces, 200 3

99 Alai n Lascoux , Symmetri c function s an d combinatoria l operator s o n polynomials , 200 3

98 Alexande r Varchenko , Specia l functions , K Z typ e equations , an d representatio n theory ,

2003

97 Bern d Sturmfels , Solvin g system s o f polynomia l equations , 200 2

96 Nik y Kamran , Selecte d topic s i n th e geometrica l stud y o f differentia l equations , 200 2

95 Benjami n 'Weiss , Singl e orbi t dynamics , 200 0

94 Davi d J . Sal tman , Lecture s o n divisio n algebras , 199 9

93 Gor o Shimura , Eule r product s an d Eisenstei n series , 199 7

92 Fa n R . K . Chung , Spectra l grap h theory , 199 7

91 J . P . Ma y e t al . , Equivarian t homotop y an d cohomolog y theory , dedicate d t o th e

memory o f Rober t J . Piacenza , 199 6

90 Joh n Roe , Inde x theory , coars e geometry , an d topolog y o f manifolds , 199 6

89 Cliffor d Henr y Taubes , Metrics , connection s an d gluin g theorems , 199 6

88 Crai g Huneke , Tigh t closur e an d it s applications , 199 6

87 Joh n Eri k Fornaess , Dynamic s i n severa l comple x variables , 199 6

86 Sori n Popa , Classificatio n o f subfactor s an d thei r endomorphisms , 199 5

85 Michi o J imb o an d Tetsuj i Miwa , Algebrai c analysi s o f solvabl e lattic e models , 199 4

84 Hug h L . Montgomery , Te n lecture s o n th e interfac e betwee n analyti c numbe r theor y an d

harmonic analysis , 199 4

83 Carlo s E . Kenig , Harmoni c analysi s technique s fo r secon d orde r ellipti c boundar y valu e

problems, 199 4

82 Susa n Montgomery , Hop f algebra s an d thei r action s o n rings , 199 3

81 Steve n G . Krantz , Geometri c analysi s an d functio n spaces , 199 3

80 Vaugha n F . R . Jones , Subfactor s an d knots , 199 1

79 Michae l Frazier , Bjor n Jawerth , an d Guid o Weiss , Littlewood-Pale y theor y an d th e

study o f functio n spaces , 199 1

78 Edwar d Formanek , Th e polynomia l identitie s an d variant s o f n x n matrices , 199 1

77 Michae l Christ , Lecture s o n singula r integra l operators , 199 0

76 Klau s Schmidt , Algebrai c idea s i n ergodi c theory , 199 0

75 F . Thoma s Farrel l an d L . Edwi n Jones , Classica l aspherica l manifolds , 199 0

74 Lawrenc e C . Evans , Wea k convergenc e method s fo r nonlinea r partia l differentia l

equations, 199 0

73 Walte r A . Strauss , Nonlinea r wav e equations , 198 9

72 Pete r Orlik , Introductio n t o arrangements , 198 9

71 Harr y D y m , J contractiv e matri x functions , reproducin g kerne l Hilber t space s an d

interpolation, 198 9

70 Richar d F . Gundy , Som e topic s i n probabilit y an d analysis , 198 9

69 Fran k D . Grosshans , Gian-Carl o Rota , an d Joe l A . Stein , Invarian t theor y an d

superalgebras, 198 7

TITLES I N THI S SERIE S

68 J . Wil l ia m Helton , Josep h A . Ball , Charle s R . Johnson , an d Joh n N . Palmer , Operator theory , analyti c functions , matrices , an d electrica l engineering , 198 7

67 Haral d Upmeier , Jorda n algebra s i n analysis , operato r theory , an d quantu m mechanics ,

1987

66 G . Andrews , g-Series : Thei r developmen t an d applicatio n i n analysis , numbe r theory ,

combinatorics, physic s an d compute r algebra , 198 6

65 Pau l H . Rabinowitz , Minima x method s i n critica l poin t theor y wit h application s t o

differential equations , 198 6

64 Donal d S . Passman , Grou p rings , crosse d product s an d Galoi s theory , 198 6

63 Walte r Rudin , Ne w construction s o f function s holomorphi c i n th e uni t bal l o f C n , 198 6

62 Bel a Bol lobas , Extrema l grap h theor y wit h emphasi s o n probabilisti c methods , 198 6

61 Mogen s Fiensted-Jensen , Analysi s o n non-Riemannia n symmetri c spaces , 198 6

60 Gille s Pisier , Factorizatio n o f linea r operator s an d geometr y o f Banac h spaces , 198 6

59 Roge r How e an d Alle n Moy , Harish-Chandr a homomorphism s fo r p-adi c groups , 198 5

58 H . Blain e Lawson , Jr. , Th e theor y o f gaug e fields i n fou r dimensions , 198 5

57 Jerr y L . Kazdan , Prescribin g th e curvatur e o f a Riemannia n manifold , 198 5

56 Har i Bercovici , Cipria n Foia§ , an d Car l Pearcy , Dua l algebra s wit h application s t o

invariant subspace s an d dilatio n theory , 198 5

55 Wil l ia m Arveson , Te n lecture s o n operato r algebras , 198 4

54 Wil l ia m Fulton , Introductio n t o intersectio n theor y i n algebrai c geometry , 198 4

53 Wi lhe l m Klingenberg , Close d geodesie s o n Riemannia n manifolds , 198 3

52 Ts i t -Yue n Lam , Orderings , valuation s an d quadrati c forms , 198 3

51 Masamich i Takesaki , Structur e o f factor s an d automorphis m groups , 198 3

50 Jame s Eell s an d Lu c Lemaire , Selecte d topic s i n harmoni c maps , 198 3

49 Joh n M . Franks , Homolog y an d dynamica l systems , 198 2

48 W . Stephe n Wilson , Brown-Peterso n homology : a n introductio n an d sampler , 198 2

47 Jac k K . Hale , Topic s i n dynami c bifurcatio n theory , 198 1

46 Edwar d G . Effros , Dimension s an d C*-algebras , 198 1

45 Ronal d L . Graham , Rudiment s o f Ramse y theory , 198 1

44 Phil l i p A . Griffiths , A n introductio n t o th e theor y o f specia l divisor s o n algebrai c curves ,

1980

43 Wil l ia m Jaco , Lecture s o n three-manifol d topology , 198 0

42 Jea n Dieudonne , Specia l function s an d linea r representation s o f Li e groups , 198 0

41 D . J . N e w m a n , Approximatio n wit h rationa l functions , 197 9

40 Jea n Mawhin , Topologica l degre e method s i n nonlinea r boundar y valu e problems , 197 9

39 Georg e Lusztig , Representation s o f finite Chevalle y groups , 197 8

38 Charle s Conley , Isolate d invarian t set s an d th e Mors e index , 197 8

37 Masayosh i Nagata , Polynomia l ring s an d affin e spaces , 197 8

36 Car l M . Pearcy , Som e recen t development s i n operato r theory , 197 8

35 R . Bowen , O n Axio m A diffeomorphisms , 197 8

34 L . Auslander , Lectur e note s o n nil-thet a functions , 197 7

33 G . Glauberman , Factorization s i n loca l subgroup s o f finite groups , 197 7

32 W . M . Schmidt , Smal l fractiona l part s o f polynomials , 197 7

31 R . R . Coifma n an d G . Weiss , Transferenc e method s i n analysis , 197 7

For a complet e lis t o f t i t le s i n thi s series , visi t t h e AMS Bookstor e a t w w w . a m s . o r g / b o o k s t o r e / .