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NONLINEAR SCHRODINGER EQUATIONS Tosio Kato Department of Mathematics, Univers i ty of California Berkeley, California 94720, U. S. A. Table of Contents Introduct ion Chapter 1. Preliminaries 1. The Schrgd inger group on Sobolev spaces of L2- type 2. Funct ion spaces of LP- type 3. The Schrgd inger group on L(P) -spaces 4. Nemyckii operators Chapter 2. The local Cauchy problem 5. L2-solut ions 6. HI -solut ions 7. H2-solutions 8. Regular i ty 9. The final value problem. Local wave operators 10. Global solutions with small data. Scatter ing Chapter 3. The sign condit ion and the Hamiltonian s t ruc ture 11. The sign condit ion 12. The ttamiltonian 13. The sign condit ion with the Hamiltonian s t ruc ture Chapter 4. Nonlinear Schrgdinger equations with a l inear potential 14. The l inear operator 15. The local Cauchy problem 16. Global solutions References Int roduct ion These lectures are intended as an exposition of basic resul ts on the nonstat ionary, nonlinear Schrgdinger equations of the form (NLS} ~t u -- i(nu - F(u}}, t >i 0, x E ~m, 219 where F(u) = Fou is a local nonlinear operator (Nemyckii operator) . The lectures are mostly restr ic ted to the Cauchy problem; some resul ts on scatter ing theory are included, such as small data theory, but we have to refer to the existing l i terature for a full t reatment of the problems about decay and scatter ing. (Actually the l i terature is so big that we have not been able to include in the attached reference papers not d irect ly related to our subjects.) For (NLS) (and for similar evolution equations), one may classify the solutions rough ly into two classes: semiclassical solutions and weak(er) solutions. The former may be defined by the proper ty that u( t , . ) E L~(ll~m). It is not diff icult to establ ish existence and uniqueness of semiclassical solutions locally in time for (NLS), if F is suff ic ient ly smooth (cf. [K3]). But such solutions are of limited interest, since it is in general impossible to prove their global existence. Moreover, it is not desirable to assume F to be very smooth. On the other hand, it is not diff icult to const ruct global weak solutions if F is such that certain conservat ion laws apply. A typical example is the so-called single power nonlinearity: (0.1) F(u) : c iu lP - lu , where p ~ 1 and c is a real constant. But there is a serious problem in the uniqueness proof. In (NLS), the latter has depended heavi ly on the L p ' -L p boundedness of the free Schrgdinger group U(t) -- e irA, and in fact it seems that there is no other alternative. The existence proof is also simplified by us ing that property . For these reasons, we consider here only weaker solutions u of (NLS), such that u(t) 6 L 2, H l, or H 2. L 2- and HI-solut ions are t ru ly weak solutions, inasmuch as 8tu is in general a distr ibut ion on ~m In contrast , H2-solutions may be called s t rong solutions, since 8tu is a funct ion in L 2. It is a remarkable fact that H2-soltions exist globally in time under rather general conditions, a fact f i rst noticed by [T2] in the s ingle-power case. Incidental ly, the classif ication into semiclassical and weaker solutions mentioned above is not a c lear -cut one. For m ~ 3, I I2-solutions are a l ready semiclassical. In the huge l i terature for (NLS), a greater part of papers are restr icted to the single power case. But (0.1) is in fact a very special assumption; it not only rest r ic ts the growth rate of F (which is inevitable for weak solutions), but it also implies a s t rong decay for small u. In part icular two such assumptions with di f ferent p's are not comparable. 220 Fundamental resul ts on the Cauchy problem with a more general nonl inear i ty were given by Ginibre and Velo [GV1,6] (cf. also [Ssl]) . Their main assumption is that F is dominated by the sum of two powers. This was indeed a very general assumption, but it did not attain full general i ty since the two exponents involved were both assumed to be larger than 1. This gap was removed in [K5], where certa in funct ion spaces on space-t ime were used to deal with the diff iculty. Similar methods had been used by [Y] in l inear Schrgd inger equations. Actually such funct ion spaces have been known for a decade but have not been used effect ively for the Cauchy problem. The main par t of these lectures is devoted to elaborat ing and general iz ing the resul ts of [K5]. Basic funct ion spaces are int roduced in Chapter 1, together with the propert ies of the Nemyckii operators F. In Chapter 2, local solutions for (NLS) are const ructed with the minimum restr ict ion on F (without the sign condit ion or the Hamiltonian structure}. The main emphasis is on showing that (NLS) forms a dynamical system on L 2, H 1, and H 2, by generat ing a cont inuous local £Iow. This means that (NLS) is locally wel l -posed in these spaces, in the sense that pers i s tence and cont inuous dependence on the initial data hold true; cf. [CH,Chapter VII. This chapter also includes small data scat ter ing theory. In Chapter 3, we show that the solutions exist for all time if we s t rengthen the assumptions on F by int roduc ing the sign condit ion and the ttamiltonian s t ructure , so that the local flows const ructed in L 2, H 1, and H 2 become global cont inuous flows. This is the reason why we consider these spaces fundamental . There are some other spaces with the same proper ty , such as H 1 ~ L2(Ix12dx); cf. [GV2]. (On the other hand, many other c lasses of weak(er) solutions have been proposed, which need not produce cont inuous flows; cf.e.g. [DL].) It is also shown that the wave operators exist if a s tandard decay condit ion is assumed for F, but the completeness of the wave operators is not d iscussed (for which we refer to [LS,GV4,5]). In Chapter 4, we cons ider a general izat ion of (NLS} in which a term V(x)u, involving a large l inear potential, is added. (Such a problem was f i rs t considered by [O]). It is in terest ing to see that most of the preced ing resul ts remain t rue for a rather general class of V, including the harmonic osci l later potential V(x) = c lxl 2, c i> 0. 221 Chapter 1. P re l iminar ies 1. The Schr~dinger group on Sobolev spaces of L2- type Let (1.1) where A group on W k '2 for any k E ~. In what fo l lows we fix an in terva l I - [0,T] or two l inear operators r and G by U(t) - e iAt, t E ~, is the Lap lac ian on ~m, m = 1,2, .... U(t) fo rms a un i ta ry L 2 = L2(~ m) and , more genera l ly , on the Sobo lev space H k = [0,~), and def ine (r~)(t) = u(t)~, t E I, t (Gf)(t) = J |oU( t -~) f (~)dr , t E I. (1.2) (1.3) The following are basic results about these operators. Lemma 1.1. r is a bounded l inear operator f rom C I ( I ;Hk -2) , wi th H k into C(I;H k) t~ (1.4) 8 tF¢ - ia r¢ -- i rA¢ , 8 t -- 8 /~t . Lemma 1.2. Let f E L I ( I ;Hk) . Then Gf E C(I;H k) f~ AC(I ;Hk-2) , wi th (1.5) ~tGf = ihGf + f : iGAf + f. (AC denotes the c lass of abso lu te ly cont inuous funct ions} . Lemma 1.3. Let v, f E L I ( I ;Hk) . Assume that v sa t i s f ies the d i f fe rent ia l equat ion (1.6) ~t v = lay + f. Then v E AC( I ;Hk-2) , so that v(0) E H k -2 ex is ts , wi th v = rv(O) + Gf. 222 These lemmas are s tandard resu l t s in semigroup theory (see e.g. [Pa]) i f f is a cont inuous funct ion ; see [K1-2] for the case when f is on ly assumed to be s t rong ly in tegrab le . Note that if U(t) is regarded as a cont inuous semigroup on H k -2 , then H k is the domain of i t s generator iA. 2. Funct ion spaces of LP - type A l though U(t) is a un i ta ry group on H k, i t has cer ta in smooth ing e f fec ts expressed by i t s ac t ion between var ious funct ion spaces . The fo l lowing are the main c lasses of funct ions we dea l wi th. L p : LP([Rm;~:), 1 < p < ¢~, wi th the norm denoted by it lip. L p ' r : L r ( I ; LP ) , 1 < p < ¢% 1 <~ r <~ ¢¢; the assoc ia ted norm is usua l ly denoted by | Itlp, r . We do not inc lude p : 1 or ~, but we do r = 1 and ~. For funct ions u( t ,x ) on I × ~m, we wr i te u( t ) for u ( t , - ) . I t is conven ient to s impl i fy the notat ion by represent ing the pa i r p , r by the po in t P : (1/p,1/r) in the square [] : (0,1) × [0,1] (of wh ich the ver t i ca l s ides are exc luded) , and wr i te L(P) and H :PII) for L p ' r and fl) |p,r ' respect ive ly . Of course the space L(P) depends on the in terva l I; i f necessary we sha l l wr i te L(P, I ) and | :P , I | to ind icate the I -dependence , but in most cases we sha l l not do so. We regard P as a 2 -vector . Two po in ts P, P" a re sa id to be dual to each o ther if P+P" : (1,1) (i.e. the midpo in t of the segment PP" is the center (1 /2 ,1/2) of []). Note that L (P ' ) is indeed the dua l space of L(P) if P is in the in ter io r of [] (see [HP,p.89], [Ph]) . With these notat ions i t is easy to prove the H'dlder insquality: (2.1) illfg:P+Q• <. |f:P|mg:Q|, P, Q, P+Q E m, With the t r iv ia l ident i ty (2.2) |fk:kpUl : Ulf:P| k, k > O, it is also easy to deduce the convexity of the P-norm: (2.3) lilf:kP+(l-k)QUl ~< |f:p|k|f:Q•l-k, 0 <~ k <~ I. 223 In par t i cu la r , (2.3) impl ies that L(P)I%L(Q) C L(R) C L(P)+L(Q) if R is on the segment PQ. In genera l there is no inc lus ion re la t ion between L(P) and L(Q), except when T < ~ and P and Q are on the same ver t i ca l l ine, say P = (1 /p ,1 / r ) , Q - (1 /p ,1 /s ) . I f P is below Q {i.e. r > s), we have (2.4) L(P) C L(Q), with ~ :Qm <- T0~ :PUl, e = 1 /s -1 / r > 0. The po in t B = (1/2,0) deserves a specia l a t tent ion . L 2'~, which inc ludes C(I;L 2) as a c losed subspace . denote the la t te r by L{B). We have L(B) = For s impl ic i ty we As in LP -spaces , the fol lowing dens i ty lemma is bas ic for L (P ) - spaces . Lemma 2.1. Let P = (1 /p ,1 / r ) E rl. compact sppor ts are dense in L(P). the sense of bounded convergence . I f r < % then smooth funct ions with A s imi lar resu l t hold for r = ¢~ in Sketch of proof. Given u E L(P), we f i r s t mul t ip ly u with a cut -o f f funct ion to yield a funct ion u n with compact suppor t . Then we use the F r iedr i chs mol l i f iers Jn and K n on ~m and I, respect ive ly , and set v n = KnJnU n E C~. It is easy to show that v n -. u in L(P) if r < 0% If r = % then there is a subsequence w n of v n that approx imates u in the sense of bounded convergence in L p {i.e. Wn(t) -* u(t) in L p for a.e. t 6 I and ~Wn(t)U p ~< const for all n and t). 3. The Schr6"dinger group on b (P ) - spaces Us ing the Four ie r t rans form, it is easy to see that U(t) maps the Schwartz space ~% in to i tse l f cont inuous ly . S ince - <~,U(-t)~>, U(t) can be extended into a cont inuous operator on ~%' in to i tsel f . However, U(t) does not send L p into i tse l f for p ~ 2; ins tead it maps the ad jo in t space L p ' , p ' = p / (p -1 ) , into L p if t ~ 0 and p >/ 2, with (3.1) I IU(t)¢l lp <~ c l t l -m(1 /2 -1 /P )MCap, , t ~ 0. (Here and in the seque l we use c to denote var ious constants that may depend on m, p, etc.) Moreover, U(t)¢ E L p is cont inous in t ~ 0 . 224 (3.1) i s p roved by in terpo la t ion based on the fac ts that U(t) is i sometr i c on L 2 whi le i t is bounded f rom L 1 to L ~ if t ~ 0, s ince i t has an in tegra l kerne l wh ich is a bounded funct ion except for a fac tor t -m/2 To s tudy the act ion of U(t) between var ious L(P) spaces , i t is conven ient to in t roduce cer ta in d i s t ingu ished po in ts P E I ] . Let us denote by ~ the semi -open segment connect ing the two po in ts B : (1/2,0) and C : (1 /2 -1 /m,1 /2) , c losed at B but open at C, {If m = 1, we set C : (0,1/4).) Thus the equat ion of ~ is (3.2) 1 /p + 2 /mr : 1/2, 1 /2 -1 /m < 1 /p ~< 1/2 (0 < 1 /p ~ 1/2 if m = 1). I f P : (1 /p ,1 / r ) E ~, the dua l po in t segment ~" connect ing B" : (1/2,1) (1,3/4) if m = 1). P ' = (1 /p ' ,1 / r ' ) is on the dua l wi th C' = (1/2+1/m,1/2) (or Lemma 3.1. r is a bounded l inear operator on L 2 to any L(P) wi th P E ~, and r* is bounded f rom any L(Q) wi th Q E ~' to L 2. (Here L(B) may be rep laced by L(B).) The bounds are independent of T, and are un i fo rm for P or Q on compact subsets of J~ or of ~' , respect ive ly . Lemma 3.2. G with P E ~. independent of of ~. is bounded f rom any L(Q) wi th Q E $' to any L(P) (Here L(B) may be rep laced by L(B).) The bound is T and is un i fo rm for any compact subsets of ~" and Proof. I t is conven ient tobeg in wi th Lemma 3.2. F i r s t assume that P = (1 /p , [ / r ) E ~ and Q = P ' = (1 /p ' , l / r ' ) E f ' . Then by (3.1) (3.3) IIGf(t) lip <~ c~ ( t - r ) -2 / r IIf(r)II p , d~, because m(1/2 -1 /p ) = 2 / r by P E J~. S ince 1/ r ' - l / r : 1 -2 / r > 0 by 1 / r+ l / r " = 1, i t fo l lows f rom the one-d imens iona l Sobo lev inequa l i ty (see [HLP,p.288] ) that (3.4) illGf:PlU ~< clUf:P' III, which shows that G is bounded f rom L(P ' ) to L(P). The constant c is independent of T, though i t depends on P. 225 (3.5) Next we have, for each t @ I, rt rt flGf(t) fl 2 -- j0J0d~'dr' = 2Re~:dr ~ < 2~i f l f ( , ) | p, f lGfflpd, ~< c | f :P" •|Gf:PU] <~ c | f :P" |2 by (3.4). (This computat ion is ra ther formal, but it can be just i f ied by f i r s t assuming that f(t) E L2/~L p" and then us ing the approx imat ion lemma 2.1.) Thus G maps L (P ' ) bounded ly into L(B). Actual ly it maps into ~,(B) s ince smooth funct ions are dense in L(P ' ). I t fol lows by in terpo lat ion that G maps L (P ' ) into any L(Q) with Q on the segment PB. By dual i ty , then, G maps L (Q ' ) with Q" 6 ~' bounded ly into L(P), where P is on the segment QC. By chang ing notat ion in the prev ious statement , on the o ther hand, we see that G is bounded from L(Q' ) into any L(P) with P on the segment QB. Hence G is bounded from any L(Q) with Q E ~' to any L(P) with P 6 ~. The un i fo rmi ty of the constant c ment ioned in the lemma follows from the convex i ty (2.3). This p roves Lemma 3.2. Next we cons ider F*, which is g iven by T (3.6) r*f = ~0U( - t ) f ( t )d t . same computat ion as in (3.5) g ives flr*fH 2 ~< c | f :P" | , so Thus the that r* is bounded from L (P ' ) into L 2. By dual i ty , r is bounded from L 2 to L(P), p rov ing Lemma 3.1. For the case P = B, it is well known that r is bounded from L 2 into ~,(B) = C(I;L2). Lemmas 3.1 and 3.2 may be summarized in a simple form if we in t roduce the spaces (3.7) - (3{L(P);PE.e} /~ L(B), ~" = ~'{L(Q);QE~'}. (Unfor tunate ly , these are not Banach spaces: ~ is the pro jec t ive limit of the sequence L{B) h L(Pn), and ~ ' is the induct ive limit of the sequence P (B ' ) + L(Pn), as Pn 6 $ tends to C.) We have Lemma 3 .3 . 1" maps cont inuous ly into ~. L 2 continuously into ~. G maps ~" 226 We add some more remarks on the operator G, A cont inuous act ion of G is not res t r i c ted to L (P ' ) -* L(P) with P E $. In fact the Sobolev inequa l i ty g ives Lemma 3.4. Let (3.8) 1 /2 -1 /m < 1/p ~ 1/2, lip" : l - l /p , 1 /s - 1 / r : 1 -m(1 /2 -1 /p ) . I f 1 / r > 0, then G is bounded from Lp ,s to L p ' r . then G is bounded from Lp ,s to C(I;LP). (If lemma is a spec ia l case Q : P ' in Lemma 3.2.) I f l / s > 0 > l / r , ( l ip , l / r ) E $, the Remark 3.5. Lemma 3.1 shows that U(t) has a cer ta in smooth ing ef fect . In th i s connect ion , we f ind it conven ient to in t roduce a family of new norms in L 2 by (3.9) Np[¢] - | ra:P| = (~jKU(t)¢apdt)l/r ~ < ci¢K 2, ¢ E L 2, where P = (1 /p ,1 / r ) E ~ with p > 2, and we have chosen I = [0,0o). Np[#] is f in i te by Lemma 3.1. S ince Np[U(t)#] is g iven by (3.9) in which the lower end 0 of the in tegra l is rep laced by t, it fol lows that (3.10) Np[U(t)¢] -* O' as t -* oo; the convergence is un i fo rm if P var ies on $ away from B and C, by the convex i ty theorem. We may regard Np[~#] as a measure of the age of the vector ¢ E L2: if Np[#]/UCU 2 is small, # is near the end of i ts evo lu t ion (a long the orb i t descr ibed by the SchrGd inger group) . 4. Nemyckii operators In the nonlinear Schr~dinger equation, we have to deal with nonlinear operators of the form (4.1) u - F(u) = fou (i.e. F(u)(x) : f (u(x) or F(u)( t ,x) : f (u i t ,x ) ) . Here f is a complex -va lued funct ion of a conplex var iab le and u is a complex -va lued funct ion on ~m or I × ~m. F is a spec ia l case of the 227 so -ca l led Nemycki i operator . In what fo l lows we prove some bas ic p roper t ies of the operator F. ( In th i s chapter we s t r i c t ly d i s t ingu ish between f and F, but la ter we sha l l fo r s imp l i c i ty use the same symbol F for the two ob jec ts , as most authors do. We a lso note that except in Lemma 4.5, ~m may be rep laced by any countab ly add i t ive measure space . ) Lemma 4.1. Let f 6 C(~:;~), with I f ( l ' ) l .< MI~I s, where 0 < s < ¢¢. Then F 6 C(LP;L p /s ) i f l vs <~ p < % with I lF(U)Up/s <~ MUURp. Lemma 4.2. Under the assumpt ion of Lemma 4.1, F L(P) in to L(sP) i f P, sP ff [3. is continuous f rom The proof of these lemmas is easy and is omi t ted; cf. [V]. Remark 4.3. Lemmas 4.1 is not t rue if s : 0 (so that f is cont inuous and bounded) , a l though it is obv ious that u 6 L p impl ies F(u) £ L ¢¢. In th i s case , the fo l lowing s imple fact can rep lace Lemma 4.1 for most purposes : if u n -* u in L p, then there is a subsequence v n of u n such that F (v n) -* F(u) a.e. bounded ly . A s imi lar remark app l ies to Lemma 4.2. We next cons ider d i f fe rent iab i l i ty of F, wh ich requ i res d i f fe rent iab i l i ty of f. But s ince we do not assume ana ly t i c i ty of f, we have to cons ider the der ivat ive f" in the rea l sense . For each ~ 6 ~, f" (~) is de f ined as a rea l - l inear operator on C, g iven by (4.2) f ' (~)~ -- (S f /a [ )~ + (S f /a [ ) t~, t~ E ~:. (As usua l we use the notat ions a/at = (112) (818~- i81a~) , al~ = ( l12) (~ la~+iS la~) , ~ = ~+i~). Thus f ' may be ident i f ied with the pa i r {a f /a~,a f /8~}. We set (4.3) I f ' (~) l = la f /a~' l + la f /8} l . Then we have 228 (4.4) If'(~)~l <. If'(~)il~i. When we cons ider funct ions F : X -* Y, where X, Y a re rea l Banach spaces , i t appears that the s tandard F rechet dervat ives are not very conven ient . Thus we say F E CI(x;Y) i f F £ O(X;Y) and has a G~teaux (d i rec t iona l ) der ivat ive DF(u) £ L(X;Y) that depends on u strongIy cont inouous ly (cf. [Ha]). I f X, Y a re complex spaces , we use the under ly ing rea l s t ruc ture , so