WELL-POSEDNESS FOR ONE-DIMENSIONAL DERIVATIVE NONLINEAR SCHRÖDINGER EQUATIONS. Chengchun Hao. (Communicated by Gigliola Staffilani)

Transcription

1 COMMUNICAIONS ON Website: htt://aimsciencesorg PURE AND APPIED ANAYSIS Volume 6 Number 4 December WE-POSEDNESS FOR ONE-DIMENSIONA DERIVAIVE NONINEAR SCHRÖDINGER EQUAIONS Chengchun Hao Institute of Mathematics Academy of Mathematics & Systems Science CAS Beijing 8 P R China Communicated by Gigliola Staffilani Abstract In this aer we investigate the one-dimensional derivative nonlinear Schrödinger equations of the form iu t u +iλ u k u = with non-zero λ R and any real number k 5 We establish the local well-osedness of the Cauchy roblem with any initial data in H by using the gauge transformation and the ittlewood-paley decomosition Introduction In the resent aer we consider the following Cauchy roblem for the derivative nonlinear Schrödinger equation iu t u + iλ u k u = t R u = u where u = ut : R C is a comle-valued wave function both λ and k 5 are real numbers A great deal of interesting research has been devoted to the mathematical analysis for the derivative nonlinear Schrödinger equations In 3 C E Kenig G Ponce and Vega studied the local eistence theory for the Cauchy roblem of the derivative nonlinear Schrödinger equations iu t + u + fu ū u ū = t R with small data u = u in H 7/ where f is a olynomial having no constant or linear terms with the lowest order term of degree being greater than or equal to 3 Subsequently it was imroved to H 3 by N Hayashi and Ozawa If the nonlinearity consists mostly of the conjugate wave ū then it can be done much better In the case f = ū k A Grüenrock in 8 obtained local wellosedness when s > s c = 3/ /k s and k was an integer In articular the global well-osedness in H is obtained when f = iū with the hel of the Bourgain saces cf 3 Mathematics Subject Classification 35Q55 35A7 Key words and hrases Derivative nonlinear Schrödinger equations Cauchy roblem local well-osedness he author is suorted in art by the Scientific Research Startu Secial Foundation on Ecellent PhD hesis and Presidential Award of Chinese Academy of Sciences NSFC Grant No 66 and the Innovation Funds of AMSS CAS of China 997

2 998 CC HAO In H akaoka discussed the derivative nonlinear Schrödinger equation of the form u t iu + u u = t R and obtained the local well-osedness in H s for s by erforming a fied oint argument in an adated Bourgain sace X sb which yields a C -solution ma A very similar equation to is the generalized Benjamin-Ono equation u t + Hu ± u k u = t R 3 where u is a real-valued function H is the Hilbert transformation defined by Hf = i e iξ sgnξ ˆfξdξ R and k is an integer the symbol ˆ or F denotes the satial Fourier transform For this equation Molinet and F Ribaud 6 7 obtained the local well-osedness in the Sobolev sace H s for s > if k = 4 s 3/4 if k = 3 and s if k 5 by using ao s gauge transformation In 4 C E Kenig and H akaoka have shown the global well-osedness for the case k = in H s for s by combining the gauge transformation with a ittlewood-paley decomosition and following the comactness argument with a riori estimates with the hel of the reservation of the Hamiltonian and the -mass In the resent aer we shall generalize the above results to the derivative nonlinear Schrödinger equation with k 5 by using some ideas in 4 However we have to reconstruct new and comlicated estimates for the case k 5 which is quite different from the case k = We first state the main result of this aer as follows though we shall define later the function sace X at the end of this section heorem For any u H there eist a = u H and a unique solution u of - satisfying u C ; H X For convenience we now introduce some notations For nonnegative real numbers A B we use A B to denote A CB for some C > which is indeendent of A and B A B means A B A and A B denotes A CB for some small C > which is also indeendent of A and B o give the ittlewood-paley decomosition let ψ be a fied even C function with a comact suort su ψ { ξ < } and ψξ = for ξ Define ϕξ = ψξ ψξ et N be a dyadic number of the form N = j j N {} or N = Writing ϕ N ξ = ϕξ/n for N we define the convolution oerator P N by P N u = ˇϕ N u where the symbol ˇ or F denotes the satial Fourier inverse transform We define the function ϕ by ϕ ξ = N ϕ N ξ and denote P u = ˇϕ u hen we introduce a satial ittlewood-paley decomosition P N = I N hroughout this aer we often use the ittlewood-paley theorem cf 3 P N φ φ N

3 WE-POSEDNESS FOR D DNS 999 for < < We also use more general oerators P N and P N which are defined by P N = P M M N P M P N = MN and P N P N and P N which can be defined in a similar way he ittlewood- Paley oerators commute with derivative oerators including s and i t the roagator St = e it and conjugation oerations are self-adjoint and are bounded on every ebesgue sace and homogeneous Sobolev sace Ḣs if Furthermore they obey the following Sobolev and Bernstein estimates for R with s and which is similar to those of three dimensions 5: P N f N s s P N f P N s f N s P N f P N ±s f N ±s P N f which can be verified by combining the Bernstein multilier theorem and the interolation theorem of Sobolev saces We define the ebesgue saces q and q by the norms f q = f R f q = f q t R q t In articular we abbreviate q or q as in the case = q We also use the elementary inequality 5 f N f N q N q for all q and arbitrary functions f N and the dual version f N q f N N N N q where is the conjugate number of given by / + / = It is easy to verify that they also hold if we relace the norm q by the norm q in both side of the above inequalities et = + We use the fractional differential oerators D s and D s defined by D s f = F ξ s F f D s f = F ξ s F f hus we can introduce the resolution sace For > we define the function sace X in a similar way as in 4 by where X := {u S R : u X < } u X = u H + N P N u

4 CC HAO + P N u + D 4 PN u 4 N Gauge transformation We transform the equation by introducing the following comle-valued function v N : R C for a dyadic number N given by v N t = e iλ P N uty k dy P N u By comutation we have i t v N v N = iλe iλ iλ e iλ + λ iλ 4 e N P N u k dy P N u k u P N u k P N u P N u k dy P N ui t For the second term we integrate by arts and have i t P N ut y k dy P N ut y k dy P N u k dy P N u k P N u k =i P N u k t P N up N u + P N u t P N u dy k P Nu k P N up N u + P N u P N u = = k P Nu k P N up N u iλ u k u P N up N ū + iλ u k ū dy k P Nu k P N u P N u + P N up N u kk P N u k 4 P N u y P N u P N u y P N u dy 4 iλk P Nu k P N u k u + ū dy k P N u k P N up N u hus v N obeys the following differential-integral equation i t v N v N t = iλe iλ P N u k dy P N u k u P N u k P N u iλkk e iλ P N u k dy P N u P N u k 4 8 P N u P N u P N u P N u dy λ k iλ 4 e P N u k dy P N u + iλk iλ e + λ iλ 4 e P N u k P N u k u + ū dy P N u k dy P N u k P N up N up N u P N u k dy P N u k P N u

5 WE-POSEDNESS FOR D DNS I N t + I N t + I N3 t + I N4 t + I N5 t 3 he equivalent integral equation reads v N t =Ste iλ P N u y k dy P N u i St τi N + I N + I N3 + I N4 + I N5 τdτ 4 3 Preliminaries In order to rove the a riori estimate for the equation of v N we need the linear estimates associated with the one-dimensional Schrödinger equation We first recall the Strichartz estimates smoothing effects and maimal function estimates For the roofs one can see 3 4 emma 3 For all φ S R θ and Stφ StP N φ 4 θ θ θ θ Stφ 4 φ 3 N θ φ 3 φ Ḣ 4 33 We also need the q and q estimates for the linear oerator f St τfτdτ For the roofs one can see 4 emma 3 For f S R θ and St τfτdτ 4 θ θ f θ 4 D θ St τfτdτ f θ D θ St τp N fτdτ D θ 4 St τfτdτ D St τfτdτ St τp N fτdτ St τfτdτ St τfτdτ 4 θ θ 4 θ qθ N f θ 36 qθ f θ 37 qθ f 38 N θ f 39 f 3 Ḣ 4 f 3 where is the conjugate number of ie / + / = and θ = 3 + θ 4 qθ = 3 θ 4 Net we recall the eibniz rule for a roduct of the form e if g where F is the satial rimitive of some function f For the roof we refer to 4 7

6 CC HAO emma 33 4 emma 35 et α q q q with = + q = q + q and let F t = ft ydy with real-valued function f hen D α e if g f q q g q + D α g q 4 Bilinear estimates In this section we rove the following sace-time estimate which is crucial to the roof of the nonlinear estimates Proosition 4 et u H and 4 be a real number hen we have uū u X u X u X P u X 4 Proof By the ittlewood-paley decomosition we can write uū = P N up N ū N N P N up N ū + P N up N ū N N N N + P N up N ū N N 4 =:I + I + I 3 43 Now we derive the estimates for I I and I 3 resectively From the Hölder inequality the Bernstein tye inequalities and the real interolation theorem we have I P N u 4 N N N N N N N D D P N u P N u 4 4 D P N P u + P u D P N P u 4 PN P u + 4 N + N P N u 4 D N N P N u 4 4 N P N P u N P N P u 4 D 4 N P N u 4 P N u 4 P N u + 4 N N N P N P u P N P u Alying the Sobolev embedding theorem and the Hölder inequality to the first term and Bernstein estimates to the second term we can see that it is bounded by u X + N P N P u P N P u

7 WE-POSEDNESS FOR D DNS 3 By the Cauchy-Schwartz inequality and the Sobolev embedding theoremie H /4 4 H /4 / 4 for the real number 4 we can bound it by u X + P u X For I or I 3 it is suffice to consider one of them eg I in view of symmetry For N N we have P N up N ū = P N P N up N ū where P N = j= P j N We slit these into three cases ie N N N N and N N For the case N N from the Hölder inequality and the ittlewood-paley theorem we can get P N up N ū N N = P up N ū P u P u PM P u P M P u M M P M u P M P u M M u X P u X 44 For the case N N we have by the Hölder inequality and the ittlewood- Paley theorem that P N up N ū = P N up N ū N N N P N u P N ū N N N P N u u X 45 For N we have by the Sobolev embedding theorem that P N u P M P N u M N ε M ε P M u M N N ε M N D ε M N P M u P M u

8 4 CC HAO N ε P M u M N ε u X where ε = / hus 45 can be bounded by N ε N u X u X Now we turn to the case N N From the ittlewood-paley theorem we have for 4 P N up N ū = P N up N ū N N N P N up N ū P N u P N u 46 N Noticing that P N u N P M P N u M u X and for N ε = / and 4 P N u M N P M u = P N P u + P u = P N P u =N ε N ε P N P u N ε DP ε N P u N ε we can bound 46 by u X N u X N D 4 PN P u N ε 4 D 4 PN P u N ε N 4 D 4 PN P u 47 4 u X P u X 48 in view of the Hölder inequality hus we have obtained P N up N ū u X P u X 4 49 N N herefore we have the desired result 4 for any real number 4 5 Nonlinear estimates o state the estimates for the nonlinearities I Nj we define the function sace Y equied with the following norm: u Y = u + H u + u + D 4 u 4 We have the following roosition for the nonlinearities

9 ˇϕ N yy y P N u k PN u WE-POSEDNESS FOR D DNS 5 Proosition 5 et u be a H -solution to - hen 5 St τ I Nj τdτ N j= Y + u k X u k+ X P u k k X u X u k X P u k+ k X + u k X + u 5k / X P u X u X k u X P u 3 X + + u k X k u X P u X u X u k X P u X + u 3k X P u X where k denotes the maimal integer that is less than k ie k = k if k is not an integer and k = k if k is an integer where k denotes the maimal integer that is less than or equal to k We consider each nonlinearity searately 5 Nonlinear estimates of I N Noting that the term P N P N u k u has Fourier suort in ξ N we have P N u k u P N u k P N u =P N u k P N u k u + P N P N u k u P N u k P N u =P N u k P N u k u + P N P N u k PN u P N u k P N PN u 5 where P N = P N/ + P N + P N For the second term in 5 we have the following estimate emma 5 et u be a solution of - hen we have for any k 4 P N P N u k PN u P N u k P N PN u 5 N k u X P u X u X u k X P u X Proof o shift a derivative from the high-frequency function P N u to the lowfrequency function P N u k we require the following eibniz rule for P N from 4: P N fg fp N g = ˇϕ N yyf ηyg ydy dη 53 hus we have P N P N u k PN u P N u k P N PN u N P N ˇϕ yy y u k PN u P N u k PN u

10 6 CC HAO P N u k k P N u P N u k P N u 54 Decomosing P N u = P u + P < N u for N we have P N u P N u = P u P N u + P < N u P u + P < N u P < N u 55 For the first term in 55 we have P u P N u k P u k P u k By the ittlewood-paley theorem we can obtain P N u k P M P N u M M N M In the similar way we have hus P M u k k P N u k M P N u k M P M P N u k P M u k D 4 PM u P u 56 4 X P u k P u P N u k u X u X P u X 57 For the last two term in 55 in a similar way as in the roof of Proosition 4 we can obtain the following bound: P ūp N u k P N ūp N u k u X + u X P u X 58 u X u X u X P u X 59 From the Sobolev embedding theorem and we obtain that 54 can be bounded by k u X u X u k X P u X PN u hus we can bound 5 by k u X u X u k X P u X N k u X P u X u X u k X P u X which yields the desired result For the first term in 5 we have the following estimate: P N u

11 WE-POSEDNESS FOR D DNS 7 emma 5 et u be a solution of - hen we have for any k 4 P N u k P N u k u N u k+ X P u k k X u X u k X P u k+ k X 5 Proof We slit 5 into several terms for N and k 4 Notice that PN u k P N u k P N u k u 5 =P N u k ūu P N u 5 + P N u k P N u k ūu P N u 53 + P N P N u k u P N up N ū 54 P M u k MN MN N ε k D ε k P M u k N ε k D 4 PM u MN N ε k M ε k P M u k MN 4 N ε k P u X k 4 where ε k > is defined by ε k = /k hus for the first term 5 from the fact ˇϕ N and Proosition 4 we have for k 4 P N u k ūu P N u u k N ε k k PN u k u k ūu P N u ūu k k u X P u X u X u k X P u X herefore we obtain for any k 4 that PN u k ūu P N u N u k X P u X u X u k X P u X For 54 in the same way as the case 5 we have PN P N u k u P N up N ū N u k X P u X u X u k X P u X Now we derive the estimate for 53 by using the induction argument in k

12 8 CC HAO For k = 4 we have u k P N u k P N u + P N up N u From the Young inequality the Hölder inequality 56 and Proosition 4 we can get for k = 4 P N u k P N u k ūu P N u u k P N u k k/k ūu k PN u + PN u P 8 8 N u 8 P N u ūu k P N u N /8 P u X ūu k P N u N /8 3 u X P u k X u X u X P u k X From the triangle inequality for comle number ie z z z z for z z C we can get z θ z θ z z θ for any θ For k 4 5 we have u k P N u k hen u u k 4 P N u k 4 + P N u k 4 u P N u u PN u k 4 + P N u k 4 PN u + PN u P N u k 3 P N u k P N u k ūu P N u u P k N u k 4 + P N u k 4 k N ε k k + PN u P k N u k 3 k 4 u X P u k 3 X ūu k P N u k P N u u X u 3 X P u k X where ε k = k 4/k for k 4 5 By the same rocedure we can obtain for any k 4 P N u k P N u k ūu P N u N ε k u k+ X P u k k X u X u k X P u k+ k X where ε k = k k/k > herefore we have for any k 4 PN u k P N u k ūu P N u N u k+ X P u k k X u X u k X P u k+ k X 55 hus we have roved the desired result Remark 5 From the roof of emma 5 we can see that P N u k P N u k u N ε

13 WE-POSEDNESS FOR D DNS 9 u k+ X P u k k X u X u k X P u k+ k X 56 holds for any ε in view of Proosition 4 We turn to the roof of Proosition 5 for the nonlinearity I N We also consider the decomosition in 5 For convenience we denote B N = P N P N u k PN u P N u k P N PN u From and 37 we have St τe iλ P N u k dy B N dτ N B N + P M e iλ N N M By emma 5 the first term can be bounded by N Y P N u k dy B N k u X P u X u X u k X P u X /ε 57 For the second term we slit the sum M into three arts M N + M N + M N as in 4 For the art of M N it is the same as emma 5 by summing in M such that M N For the art M N we can add the rojection oerator P N to e iλ P N u k dy since B N has Fourier suort in ξ N hus by the Hölder inequality we have P M e iλ P N u k dy B N N M N P N e iλ P N u k dy B N N M N ln N P N e iλ P N u k dy B N 58 / ε where ε /k By the Bernstein inequality we have N P N e iλ P N u k dy /ε P N e iλ P N u k dy /ε P N u k k/ε u k X and from 53 and the Hölder inequality we can get as a similar way as in 54 that B N / ε = P N u k / ε P N u P N u k kk P N u P N ū k P N u k u k kε X u X u k X P u X P N u

14 CC HAO hus 58 can be bounded by u k X k u X u X u k X P u X P N u N k u X P u X u X u k X P u X For the art M N we can add the rojection oerator P M to e iλ In a similar way with the art M N we have P M e iλ P N u k dy B N N M N P M e iλ P N u k dy B /ε N / ε N M N k u X M P N u N M N u X u k X P u X P N u k u X P u X u X u k X P u X P N u k dy For the first term in 5 we denote it by A N ie A N = P N u k P N u k u Similarly from and 37 we can get St τe iλ P N u k dy A N dτ N A N + P M e iλ N M N From emma 5 the first term is bounded by Y P N u k dy A N u k+ X P u k k X u X u k X P u k+ k X 59 Noticing that 56 and in the same way as in dealing with the second term of 57 we can bound the second term of 59 by k+ k+ u X P u k k X herefore we have obtained St τi N τdτ N + u k X u k+ X u X u k+ k X P u k+ k X Y P u k k X u X u k X P u k+ k X

15 WE-POSEDNESS FOR D DNS 5 Nonlinear estimates of I N From and 3 we have N N St τi N τdτ Y e iλ P N u k dy P N ub N + P M e iλ N M H P N u k dy P N ub N H 5 5 where B N = P N u k 4 P N u P N u P N u P N u dy For the first term 5 from emma 33 and the Hölder inequality it can be bounded by N P N ub N + P N ub N N e iλ P N u B N P N u k dy P N ub N + P N u H B N + P N u B N + P N u B N P N u k k+ P N u k+ P N u B N 5 By the Hölder inequality we have for k 5 B N P N u k 4 P N u P N u P N u P N u P N up N u 4 u k X P N u k 4 and from Proosition 4 and the roof of emma 5 k u X u k X P u X 53 P N u B N = P N u P N u k 4 P N u P N u P N u P N u P N u P N u P N ū k P N u k k P N u k u X u X u k X P u X hus we can bound 5 by + u k X u k X P u X u X u k X P u 3 X

16 CC HAO For 5 we slit the sum M into two arts MN + M N which gives the bound by M PN ub N 54 N + N MN M N P M D e iλ P N u k dy P N ub N 55 For the first term 54 noticing that MN M N and 53 we can bound it by D P N u B N N u k X P u X For the second term 55 in a similar way with 5 we bound it by M P M D e iλ P N u k dy P N ub N N M N P N ub N + e iλ P N u k dy P N ub N N + u k X u k X P u X u X u k X P u 3 X herefore we obtain St τi N τdτ N Y + u k X u k X P u X u X u k X P u 3 X 53 Nonlinear estimates of I N3 From and 3 we have St τi N τdτ 56 N N Y e iλ P N u k dy P N ub N3 + P M e iλ N M H P N u k dy P N ub N3 H where B N3 = P N u k P N u k u + ū dy By Hölder inequality we get B N3 P N u k P N u k u + ū 57 58

17 WE-POSEDNESS FOR D DNS 3 P N u k P N u k u k X k k P N u k u + ū k /k u k k From the Hölder inequality and Proosition 4 we have B N3 = P N u k P N u k u + ū P N u k 4k 4 u k 4k 4 u ūu 4k 4 k u X u X u k X P u X In addition for N we have P N u way as in the case I N we can bound 56 by u k X + u 5k / X P u X u X u k X P u 3 X P N u hus in the same 54 Nonlinear estimates of I N4 From and 37 we have St τe iλ P N u k dy B N4 dτ N B N4 N + P M e iλ N M Y P N u k dy B N4 59 where B N4 = P N u k P N up N up N u By the Hölder inequality we have B N4 P N u k P k N u k P N up N u k k u X u X u k X P u X P N u k hus the first term in 59 can be bounded by k u X P u X u X u k X P u X By the Hölder inequality we get B N4 ε P N u k k u kk k ε X P N u k P N up N u k u X u k X P u X P N u k Noticing that B N4 has Fourier suort in ξ N we can reeat the rocedure which we use to deal with the second term in 57 and obtain that the second term in 59 can be bounded by k u X P u X u X u k X P u X

18 4 CC HAO herefore we obtain St τi N4 τdτ N Y + u k X k u X P u X u X u k X P u X 55 Nonlinear estimates of I N5 From and 37 we have St τe iλ P N u k dy B N5 dτ N B N5 N + P M e iλ N M Y P N u k dy B N5 where B N5 = P N u k P N u By the Hölder inequality we have and B N5 B N5 ε P N u k 4k u k X P N u P N u k 4k ε P N u P N u 53 k u X P N u hus in a similar way as dealing with I N and I N4 and noticing that B N5 has Fourier suort in ξ N we can bound 53 by u 3k X P u X 6 A riori estimates for solutions By the scaling argument ut u γ t = t u γ/k γ γ we have u γ =γ k u u γ Ḣ = γ /k u Ḣ hus we may rescale P u γ γ k u = C low P u γ H γ /k u H < C high where we choose γ = γ u H and take the time interval deending on γ later We now dro the writing of the scaling arameter γ and assume P u C low

19 WE-POSEDNESS FOR D DNS 5 P u H C high We now aly this to the norms X and H and define new version of the norms of X and H given by with the decomosition I = P + P and u X = C low P u X + C high P u X φ H = C low P φ + C high P φ H which imlies that u H For the low frequency art we have the following estimates emma 6 et u be a solution of - hen P u C X low + u k+ X Proof Using the integral equation of ut = Stu λ St τ uτ k u τdτ and by and the Hölder inequality we have P u t StP u X + X St τp u k u τdτ P u + P u k u H C low + u k k C low + u k+ X which is the desired result For the high frequency art we have u X C low + u k u emma 6 et u and v N be given in hen P u X + u k v N Y Proof By we have H N P N u = e iλ P N u k dy v N For H -norm by the interolation theorem we obtain for N P N u H P N u P N u H v N P N u + P N u v N v N + P N u k v N + v N v N P N u k v 4k N 4 + v N H + P N u k H v N H which yields the desired estimate by summing on l N + P N u k H v N H

20 6 CC HAO For the -norm noticing that P N u = e iλ P N u k dy v N + iλ P N u k v N we have P N u v N + P N P N e iλ P N u k dy P N u k P N v N 6 N N o estimate the second term 6 we slit the sum N = N + N N For N N N from the Bernstein inequality we bound 6 by P Nu k P N v N P N u k P N v N N N N N N D /k P N u P N v N P N u k H u k H N N k N N N N P N v N P N v N For the art N N we slit it as N = N N + N N N Noticing that for N N P N e iλ P N u k dy P N u k P N v N 6 and for N N P N N = P N P N e iλ 6 = P N N N N N P N u k dy P N u k P N v N P N e iλ P N u k dy P N u k P N v N we can bound 6 in view of the Bernstein inequality and the Hölder inequality by P N e iλ P N u k dy P N u k P N v N + P N e iλ P N u k dy P N u k P N v N N N N N k k+ P N e iλ P N u k dy D k+ P N u N k k+ N N N + k P N e iλ D k+ P N v N P N u k dy

21 D k+ k P N u N k+ P N u k + N k 3k+ N N N WE-POSEDNESS FOR D DNS 7 k u k H D k+ N k 3k+ N k u k v N H H P N v N v N H 3k+ P N u k u k H P N v N H herefore summing on ln we comlete the roof for the -norm For the -norm it is easy to obtain the desired result since P Nu = v N We turn to estimate the 4 -norm It is similar with the roof for the - norm since D /4 P N u 4 N /4 P N u 4 for N In fact we have D /4 P N u 4 N /4 N P N e iλ P N u k dy P N v N N 4 We also slit N = N + N N For N N N we bound 63 by N /4 P N v N 4 D /4 v N 4 N N 63 For the art N N we slit it as N = N N + N N N Noticing that for N N P N e iλ P N u k dy P N v N 64 and for N N P N N = P N P N e iλ 64 = P N N N N N P N u k dy P N v N P N e iλ P N u k dy P N v N we can bound 63 in view of the Bernstein inequality and the Hölder inequality by D /4 v N + N /4 P N e iλ P N u k dy P 4 N v N 4 + N /4 P N e iλ P N u k dy P N v N 4 N N N D /4 v N + N 3/4 P 4 N u k D /4 P N v N 4 + N /3 N /3 N /3 P N u k D /4 P N v N 4 N N N + u k D /4 v N H 4 which yields the desired estimate by alying l N -sum hus we comlete the roof of this emma Of course we need the following estimate of the data

22 8 CC HAO emma 63 For any u H we have N Ste iλ P N u k dy P N u Y + u k H P u H Proof From 3 3 and 33 we bound the left hand side of 65 by N e iλ P N u k dy P N u + P M e iλ N M From emma 33 we have H P N u k dy P N u H 66 P N u k P 4k N u + P Nu 4 H N + u H P u H 67 For the second term 67 it is similar with 5 We slit the sum M = MN + M N By the Bernstein inequality the Hölder inequality and the Sobolev embedding theorem we bound 67 by M PN u N MN + D PM e iλ P N u k dy P N u N M N N P N u N + N M N P u H + N M N P u H + N M N M M M P M e iλ P N u k dy P N u P M e iλ P N u k dy 4 P M e iλ P N u 4 P N u k dy 4 P Nu 4

23 WE-POSEDNESS FOR D DNS 9 P u H + P N u k P 4k N u 4 N + u k H P u H which yields the desired result With the hel of the above lemmas we can rove the following roosition which yields the a riori estimate Proosition 6 et u be a smooth solution to - and < C 4 high hen we have u X CC low + CC low + u X 3k /4 + C high u X Proof Noticing that P u X C low u X P u X C high u X and from emmas and Proosition 5 we obtain through a comlicated comutation u X = C low P u X + C high P u X + C low u k+ X + C high + u k + C low u k+ X + C high + u k + C high + u k + + u k X + H { u k+ X H H u 3k X P u X P u k k X u k X + u 5k / X P u X v N Y N + u k H P u H u X u k X P u k+ k X u X k u X P u 3 X } + + u k X k u X P u X u X u k X P u X C + CC low u k+ X + CC low + u k H X + + u k + u k H H { 3k+ 4 u X u u k X k+ X + k+ u X + u 5k/ + C k k high + 4 u X + u k X u k+ X + C 3/ high + 4 u X u k X + C high + 4 u X + u k X u k+ X } Notice that ut H P ut + C high P H he high frequency art C high P can be absorbed into the X H -norm hen substituting emma 6 again in estimating the low frequency art of the norm P u we comlete the roof of Proosition 6 H

24 CC HAO From Proosition 6 we have the following a riori estimate for the solution of - if we take and C high small enough Corollary 6 et u be a smooth solution to - we have for and C high small enough u X C low + C high For the roof of heorem we can follow the comactness argument with the a riori estimate Since the roof is standard we omit the details and refer to the aers Acknowledgements he author would like to thank the referees for valuable comments and suggestions on the original manuscrit and Prof Hsiao for her frequent encouragement REFERENCES J Bergh and J öfström Interolation Saces an Introduction Sringer-Verlag Berlin Heidelberg 976 J Bourgain Fourier transform restriction henomena for certain lattice subsets and alications to nonlinear evolution equations Parts I II Geom Funct Anal J Colliander M Keel G Staffilani H akaoka and ao Global well-osedness for the Schrödinger equations with derivative SIAM J Math Anal J Colliander M Keel G Staffilani H akaoka and ao A refined global well-osedness for the Schrödinger equations with derivative SIAM J Math Anal J Colliander M Keel G Staffilani H akaoka and ao Global well-osedness and scattering for the energy-critical nonlinear Schrödinger equation in R 3 to aear in Annals Math arxiv:mathap/49 6 I Fukuda and Y sutsumi On solutions of the derivative nonlinear Schrödinger equation: eistence and uniqueness theorem Funkcial Ekvac I Fukuda and Y sutsumi On solutions of the derivative nonlinear Schrödinger equation II Funkcial Ekvac A Grüenrock On the Cauchy and eriodic boundary value roblem for a certain class of derivative nonlinear Schrödinger equations rerint 9 N Hayashi he initial value roblem for the derivative nonlinear Schrödinger equation in the energy sace Nonlinear Anal N Hayashi and Ozawa On the derivative nonlinear Schrödinger equation Phys D N Hayashi and Ozawa Remarks on nonlinear Schrödinger equations in one sace dimension Diff Integ Eq C Kenig G Ponce and Vega Oscillatory integrals and regularity of disersive equations Indiana Univ Math J C E Kenig G Ponce and Vega Small solutions to nonlinear Schrödinger equations Ann Inst H Poincaré Anal Non ineaire CE Kenig and H akaoka Global wellosedness of the modified Benjamin-Ono equation with initial data in H Int Math Res Not 6 Art ID ages 5 H Koch and N zvetkov On the local well-osedness of the Benjamin-Ono equation in H s R Int Math Res Not Molinet and F Ribaud Well-osedness results for the generalized Benjamin-Ono equation with small initial data J Math Pures Al Molinet and F Ribaud Well-osedness results for the generalized Benjamin-Ono equation with arbitrary large initial data Int Math Res Not Ozawa On the nonlinear Schrödinger equations of derivative tye Indiana Univ Math J G Ponce On the global well-osedness of the Benjamin-Ono equation Diff Int Eqns

25 WE-POSEDNESS FOR D DNS EM Stein Harmonic analysis Real-variable methods orthogonality and oscillatory integrals Princeton University Press Princeton 993 H akaoka Well-osedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity Adv Diff Eqns ao Global well-osedness of the Benjamin-Ono equation in H R J Hyerbolic Diff Eqns ao Nonlinear Disersive Equations: ocal and Global Analysis CBMS Regional Conference Series in Mathematics 6 American Mathematical Society Providence RI 6 Received January 7; revised Aril 7 address: hcc@amssaccn