ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS TO A NONLINEAR SCHRÖDINGER EQUATION

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1 ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS TO A NONLINEAR SCHRÖDINGER EQUATION J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO Abstract. We prove an almost conservation law to obtain global-in-time well-posedness for the cubic, defocussing nonlinear Schrödinger equation in H s (R n ) when n =, and s > 4 7, 5 6, respectively. 1. Introduction and Statement of Results We study the following initial value problem for a defocussing nonlinear Schrödinger equation, (1.1) (1.) i t φ(x, t) + φ(x, t) = φ(x, t) φ(x, t) x R n, t 0 φ(x, 0) = φ 0 (x) H s (R n ) when n =,. Here H s (R n ) denotes the usual inhomogeneous Sobolev space. Our goal is to loosen the regularity requirements on the initial data which ensure global-in-time solutions. In particular, we aim to extend the global theory to certain infinite energy initial data. It is known [5] that (1.1)-(1.) is well-posed locally in time when n =, and s > 0, 1 respectively1. In addition, these local solutions enjoy L conservation; (1.) φ(, t) L (R n ) = φ 0 ( ) L (R n ) and the H 1 (R n ) solutions have the following conserved energy, 1 (1.4) E(φ)(t) R xφ(x, t) φ(x, t) 4 dx = E(φ)(0). n Together, energy conservation and the local-in-time theory immediately yield global-in-time well-posedness of (1.1)-(1.) from data in H s (R n ) when s 1, and n =,. It is conjectured that (1.1)-(1.) is in fact globally well-posed in time from all data included in the local theory. The obvious impediment to claiming global-in-time solutions in H s, with 0 < s < 1, is the lack of any applicable conservation law Mathematics Subject Classification. 5Q55. Key words and phrases. nonlinear Schrödinger equation, well-posedness. J.E.C. was supported in part by N.S.F. Grant DMS M.K. was supported in part by N.S.F. Grant DMS G.S. was supported in part by N.S.F. Grant DMS and grants from Hewlett and Packard and the Sloan Foundation. H.T. was supported in part by J.S.P.S. Grant No T.T. is a Clay Prize Fellow and was supported in part by a grant from the Packard Foundation. 1 In addition, there are local in time solutions from L, H 1 data when n =,, respectively. However, it is not yet known whether the time interval of existence for such solutions depends only on the data s Sobolev norm. For example, the L conservation law (1.) does not yield the widely conjectured result of global in time solutions on R +1 from L initial data. 1

2 J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO The first argument extending the lifespan of rough solutions to (1.1)-(1.) in a range s 0 < s < 1 was given in [] (see also []). In what might be called a Fourier truncation approach, Bourgain observed that from the point of view of regularity, the high frequency component of the solution φ is well-approximated by the corresponding linear evolution of the data s high frequency component. More specifically: one makes a first approximation to the solution for a small time step by evolving the high modes linearly, and the low modes according to the nonlinear flow for which one has energy conservation. The correction term one must add to match this approximation with the actual solution is shown to have finite energy. This correction is added to the low modes as data for the nonlinear evolution during the next time step, where the high modes are again evolved linearly. For s > 5, one can repeat this procedure to an arbitrarily large time provided the distinction between high and low frequencies is made at sufficiently large frequencies. The argument in [] has been applied to other subcritical initial value problems with sufficient smoothing in their principal parts. (See [], [7], [14], [19], [], and [4]). It is important to note that the Fourier truncation method demonstrates more than just rough data global existence. Indeed, write St NL for the nonlinear flow of (1.1)-(1.), and let St L denote the corresponding linear flow. The Fourier truncation method shows then that for s > 5 and for all t [0, ), (1.5) S NL t φ 0 S L t φ 0 H 1 (R ). Besides being part of the conclusion, the smoothing property (1.5) seems to be a crucial constituent of the Fourier truncation argument itself. In this paper we will use a modification of the above arguments, originally put forward to analyze equations where the smoothing property (1.5) is not available because it is either false (e.g. Wave maps [17] ) or simply not known (e.g. Maxwell-Klein-Gordon equations [16], for which we suspect (1.5) is false). In this almost conservation law approach, one controls the growth in time of a rough solution by monitoring the energy of a certain smoothed out version of the solution. It can be shown that the energy of the smoothed solution is almost conserved as time passes, and controls the solution s sub-energy Sobolev norm. In proving the almost conservation law for the i.v.p. (1.1)-(1.), we shall use only the linear estimates presented in [], []. Implicitly, we also use the view of [] that the energy at high frequencies does not move rapidly to low frequencies. The almost conservation approach to global rough solutions has proven to be quite robust [17], [16], [9], [1], and has been improved significantly by adding additional correction terms to the original almost conserved energy functional. As a result, one obtains even stronger bounds on the growth of the solution s rough norm, and at least in some cases sharp global well-posedness results [1], [10], [11]. The aims of this paper are three-fold: first and most obviously, an improved understanding of the evolution properties of rough solutions of (1.1)-(1.); second, the almost conservation law approach is presented in a relatively straightforward setting; and third, we can directly compare this almost conservation law approach to the Fourier cut-off technique, since both approaches apply to the semilinear Schrödinger initial value problem. Our main result is the following: Theorem 1.1. The initial value problem (1.1)-(1.) is globally-well-posed from data φ 0 H s (R n ), n =, when s > 4 7, 5 6 respectively. That is, S NL t (φ 0 )(x) = φ(x, t), where φ, φ 0 as in (1.1)-(1.). See the appendix of [16] for the failure of (1.5) for Wave Maps.

3 A.C. LAWS AND GLOBAL ROUGH SOLUTIONS FOR NLS By globally-well-posed, we mean that given data φ 0 H s (R n ) as above, and any time T > 0, there is a unique solution to (1.1)-(1.) (1.6) φ(x, t) C([0, T ]; H s (R n )) which depends continuously in (1.6) upon φ 0 H s (R n ). The polynomial bounds we obtain for the growth of φ Hs (R n )(t) are contained in (.4), (.14), and (4.6) below. Theorem 1.1 extends to some extent the work in [, ] where global well-posedness was shown when s > 5, 11 1 and n =, respectively. In a different sense, the result here is weaker than the results of [, ] as we obtain no information whatsoever along the lines of (1.5). In a later paper, we hope to extend Theorem 1.1 to still rougher data, using the additional cancellation terms mentioned above, and the multilinear estimates contained in [6]. In Section below we present some notation and linear estimates that are used in our proofs. Sections, 4 present the almost conservation laws and proofs of Theorem 1.1 in space dimensions two and three, respectively.. Estimates, Norms, and Notation Given A, B 0, we write A B to mean that for some universal constant K >, A K B. We write A B when both A B and B A. The notation A B denotes B > K A. We write A (1+A ) 1, and for the operator with Fourier multiplier (1+ ξ ) 1. The symbol will denote the spatial gradient. We will use the weighted Sobolev norms, (see [, 1, 4, 1]), (.1) ψ Xs,b ξ s τ ξ b ψ(ξ, τ) L (R n R). Here ψ is the space-time Fourier transform of ψ. versions of the norms (.1), (.) We will need local-in-time estimates in terms of truncated f X δ s,b inf ψ=fon[0,δ] ψ X δ s,b. We will often use the notation ɛ for some universal 0 < ɛ 1. Similarly, we shall write 1 1 ɛ, and 1 1 ɛ. Given Lebesgue space exponents q, r and a function F (x, t) on R n+1, we write ( ( ) q ) 1 q F L q t Lr x (Rn+1 ) F (x, t) r r (.) dx dt. R R n This norm will be shortened to L q t L r x for readability, or to L r x,t when q = r. We will need Strichartz-type estimates [5, 15, 18] involving the spaces (.), (.1). We will call a pair of exponents (q, r) Schrödinger admissible for R n+1 when q, r, (n, q) (, ), and (.4) 1 q + n r = n 4.

4 4 J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO For a Schrödinger admissible pair (q, r) we have what we will call the L q t L r x Strichartz estimate, (.5) φ L q t Lr x (Rn+1 ) φ X0, 1 +. Finally, we will need a refined version of these estimates due to Bourgain []. Lemma.1. Let ψ 1, ψ X δ 0, 1 + be supported on spatial frequencies ξ N 1, N, respectively. Then for N 1 N, one has ( ) 1 N1 (.6) ψ 1 ψ L ([0,δ] R ) ψ1 X δ ψ X 0, 1 δ. + 0, 1 + N In addition, (.6) holds (with the same proof) if we replace the product ψ 1 ψ on the left with either ψ 1 ψ or ψ 1 ψ.. Almost conservation and Proof of Theorem 1.1 in R For rough initial data, (1.) with s < 1, the energy is infinite, and so the conservation law (1.4) is meaningless. Instead, Theorem 1.1 rests on the fact that a smoothed version of the solution (1.1)-(1.) has a finite energy which is almost conserved in time. We express this smoothed version as follows. Given s < 1 and a parameter N 1, define the multiplier operator (.1) Î N f(ξ) m N (ξ) ˆf(ξ), where the multiplier m N (ξ) is smooth, radially symmetric, nonincreasing in ξ and 1 ξ N (.) m N (ξ) = ( 1 s N ξ ) ξ N. For simplicity, we will eventually drop the N from the notation, writing I and m for (.1) and (.). Note that for solution and initial data φ, φ 0 of (1.1), (1.), the quantities φ Hs (R n )(t) and E(I N φ)(t) (see (1.4)) can be compared, ( E(I N φ)(t) N 1 s φ(, t) Ḣs (R )) (.) + φ(t, ) 4 n L 4 (R n ), (.4) φ(, t) H s (R n ) E(I N φ)(t) + φ 0 L (R n ). Indeed, the Ḣ1 (R n ) component of the left hand side of (.) is bounded by the right side by using the definition of I N and by considering separately those frequencies ξ N and ξ N. The L 4 component of the energy in (.) is bounded by the right hand side of (.) by using (for example) the Hörmander-Mikhlin multiplier theorem. The bound (.4) follows quickly from (.) and L conservation (1.) by considering separately the Ḣs (R n ) and L (R n ) components of the left hand side of (.4). To prove Theorem 1.1, we may assume that φ 0 C 0 (R n ), and show that the resulting global-in-time solution grows at most polynomially in the H s norm, (.5) φ(, t) Hs (R n ) C 1 t M + C, where the constants C 1, C, M depend only on φ 0 Hs (R n ) and not on higher regularity norms of the smooth data. Theorem 1.1 follows immediately from (.5), the local-in-time theory [5], and a standard density argument.

5 A.C. LAWS AND GLOBAL ROUGH SOLUTIONS FOR NLS 5 By (.4), it suffices to show (.6) E(I N φ)(t) (1 + t) M. for some N = N(t). (See (.1), (.14) below for the definition of N and the growth rate M we eventually establish.) The following proposition, which is one of the two main estimates of this paper (see also Proposition 4.1), represents an almost conservation law of the title and will yield (.6) in space dimension n =. Proposition.1. Given s > 4 7, N 1, and initial data φ 0 C 0 (R ) (see preceeding remark) with E(I N φ 0 ) 1, then there exists a δ = δ( φ 0 L (R )) > 0 so that the solution of (1.1)-(1.) satisfies φ(x, t) C([0, δ], H s (R )) (.7) for all t [0, δ]. E(I N φ)(t) = E(I N φ)(0) + O(N + ), Remark: Equation (.7) asserts that I N φ, though not a solution of the nonlinear problem (1.1), enjoys something akin to energy conservation. If one could replace the increment N + in E(I N φ) on the right side of (.7) with N α for some α > 0, one could repeat the argument we give below to prove global well posedness of (1.1)-(1.) for all s > +α. In particular, if E(I Nφ)(t) is conserved (i.e. α = ), one could show that (1.1)-(1.) is globally well-posed when s > 0. We first show that Proposition.1 implies (.6). Note that the initial value problem here has a scaling symmetry, and is H s -subcritical when 1 > s > 0, 1 and n =,, respectively. That is, if φ is a solution to (1.1), so too (.8) φ (λ) (x, t) 1 λ φ( x λ, t λ ). Using (.), the following energy can be made arbitrarily small by taking λ large, (.9) (.10) E(I N φ (λ) 0 ) ( (N s )λ s + λ ) (1 + φ 0 H s (R )) 4 C 0 (N s λ s ) (1 + φ 0 Hs (R )) 4. Assuming N 1 is given 4, we choose our scaling parameter λ = λ(n, φ H s (R )) (.11) λ = N 1 s s ( 1 C 0 ) 1 s ( 1 + φ0 H s (R )) s so that E(I N φ (λ) 0 ) 1. We may now apply Proposition.1 to the scaled initial data φ(λ) 0, and in fact may reapply this Proposition until the size of E(I N φ (λ) )(t) reaches 1, that is at least C 1 N times. Hence (.1) E(I N φ (λ) )(C 1 N δ) 1. Given any T 0 1, we establish the polynomial growth (.6) from (.1) by first choosing our parameter N 1 so that (.1) T 0 N λ C 1 δ N 7s 4 s, 4 The parameter N will be chosen shortly.

6 6 J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO where we ve kept in mind (.11). Note the exponent of N on the right of (.1) is positive provided s > 4 7, hence the definition of N makes sense for arbitrary T 0. In two space dimensions, E(I N φ)(t) = λ E(I N φ (λ) )(λ t). We use (.11), (.1), and (.1) to conclude that for T 0 1, (.14) E(I N φ)(t 0 ) C T 1 s s 1 0, where N is chosen as in (.1) and C = C ( φ 0 Hs (R ), δ). Together with (.4), the bound (.14) establishes the desired polynomial bound (.5). It remains then to prove Proposition.1. We will need the following modified version of the usual local existence theorem, wherein we control for small times the smoothed solution in the X δ 1, 1 + norm. Proposition.. Assume 4 7 < s < 1 and we are given data for the problem (1.1)-(1.) with E(Iφ 0) 1. Then there is a constant δ = δ( φ 0 L (R )) so that the solution φ obeys the following bound on the time interval [0, δ], (.15) Iφ X δ 1, Proof. We mimic the typical iteration argument showing local existence. We will need the following three estimates involving the X s,δ spaces (.1) and functions F (x, t), f(x). (Throughout this section, the implicit constants in the notation are independent of δ.) (.16) (.17) (.18) t 0 S(t)f X δ f H1 1, 1 (R ), + S(t τ)f (x, τ)dτ F Xδ X1, 1 + 1, 1 +, F X δ 1, b δ P F X δ 1, β, where in (.18) we have 0 < β < b < 1, and P = 1 (1 β b ) > 0. The bounds (.16), (.17) are analogous to estimates (.1), (.15) in [0]. As for (.18), by duality it suffices to show Interpolation 5 gives F X δ 1,β δ P F X δ 1,b. F X δ 1,β F (1 β b ) X δ 1,0 As b (0, 1 ), arguing exactly as on page 771 of [7], and (.18) follows. F β b. X 1,b δ F X δ 1,0 δ 1 F X δ 1,b, 5 The argument here actually involves Lemma. of [0]. We thank S. Selberg for pointing this out to us.

7 A.C. LAWS AND GLOBAL ROUGH SOLUTIONS FOR NLS 7 Duhamel s principle and (.16)- (.18) give us Iφ X δ = S(t)(Iφ 0) + 1, 1 + (.19) t 0 S(t τ)i(φ φφ)(τ)dτ Iφ 0 H 1 (R ) + I(φφφ) X δ 1, 1 + Iφ 0 H 1 (R ) + δ ɛ I(φφφ) X δ 1, 1 ++, X δ 1, 1 + where is a real number slightly larger than 1 + and ɛ > 0. By the definition of the restricted norm (.), (.0) where the function ψ agrees with φ for t [0, δ], and (.1) We will show shortly that (.) Iφ X δ 1, 1 + Iφ 0 H1 (R ) + δ ɛ I(ψψψ) X1, 1 ++, Iφ X δ 1, 1 + Iψ X1, 1 +. I(ψψψ) X1, 1 ++ Iψ X 1, 1 +. Setting then Q(δ) Iφ(t) X δ 1, 1 +, the bounds (.19), (.1) and (.) yield (.) Note (.4) Q(δ) Iφ 0 H 1 (R ) + δ ɛ (Q(δ)). Iφ 0 H1 (R ) (E(Iφ 0 )) 1 + φ0 L (R ) 1 + φ 0 L (R ). As Q is continuous in the variable δ, a bootstrap argument yields (.15) from (.), (.4). It remains to show (.). Using the interpolation lemma of [11], it suffices to show (.5) ψ ψψ Xs, 1 ++ ψ X s, 1 +, for all 4 7 < s < 1. By duality and a Leibniz rule6, (.5) follows from (.6) ( s φ 1 )φ φ φ 4 dxdt φ 1 φ Xs, R R 1 φ Xs, + 1 φ Xs, X0, + 1 Note that since the factors in the integrand on the left here will be taken in absolute value, the relative placement of complex conjugates is irrelevant. Use Hölder s inequality on the left side of (.6), taking the factors in, respectively, L 4 x,t, L 4 x,t, L 6 x,t and L x,t. Using a Strichartz inequality, and s φ 1 L 4 x,t (R +1 ) s φ 1 X0, 1 + = φ 1 Xs, 1 +, φ L 4 x,t (R +1 ) φ X0, 1 + φ Xs, By this, we mean the operator D s can be distributed over the product by taking Fourier transform and using ξ ξ 4 s ξ 1 s +... ξ 4 s.

8 8 J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO The bound for the third factor uses Sobolev embedding and the L 6 t L x Strichartz estimate, φ L 6 t L 6 x (R+1 ) 1 φ L 6 t L x (R+1 ) 1 φ X0, 1 + φ Xs, 1 +. It remains to bound φ 4 L (R +1 ). Interpolating between φ 4 L t L x φ 4 L 4 t L φ 4 yields 4 X0, x 1 + φ 4 X0,0 and the Strichartz estimate φ 4 L t L x φ 4 X0, 1. This completes the proof of (.6), and hence Proposition.. Proof of Proposition.1. The usual energy (1.4) is shown to be conserved by differentiating in time, integrating by parts, and using the equation (1.1), t E(φ) = Re φ t ( φ φ φ)dx R = Re φ t ( φ φ φ iφ t )dx R = 0. We follow the same strategy to estimate the growth of E(Iφ)(t), t E(Iφ)(t) = Re I(φ) t ( Iφ Iφ Iφ iiφ t )dx R = Re I(φ) t ( Iφ Iφ I( φ φ))dx, R where in the last step we ve applied I to (1.1). When we integrate in time and apply the Parseval formula 7 it remains for us to bound (.7) E(Iφ(δ)) E(Iφ(0)) = δ 0 ( 1 4j=1 m(ξ ) + ξ + ξ 4 ) I t φ(ξ 1 ξj=0 m(ξ ) m(ξ ) m(ξ 4 ) )Îφ(ξ)Îφ(ξ)Îφ(ξ 4). The reader may ignore the appearance of complex conjugates here and in the sequel, as they have no impact on the availability of estimates. (See e.g. Lemma.1 above.) We include the complex conjugates for completeness. We use the equation (1.1) to substitute for t I(φ) in (.7). Our aim is to show that (.8) Term 1 + Term N +, 7 That is, R n f 1 (x)f (x)f (x)f 4 (x)dx = ˆf ξ 1 +ξ +ξ +ξ 4 =0 1 (ξ 1 ) ˆf (ξ ) ˆf (ξ ) ˆf 4 (ξ 4 ) where i ξ here denotes integration with i=0 respect to the hyperplane s measure δ 0 (ξ 1 + ξ + ξ + ξ 4 )dξ 1 dξ dξ dξ 4, with δ 0 the one dimensional Dirac mass.

9 A.C. LAWS AND GLOBAL ROUGH SOLUTIONS FOR NLS 9 where the two terms on the left are δ ( Term m(ξ ) + ξ + ξ 4 ) ( Iφ)(ξ1 ) ξi=0 m(ξ )m(ξ )m(ξ 4 ) Îφ(ξ ) Îφ(ξ ) Îφ(ξ (.9) 4) δ ( Term 1 4 m(ξ ) + ξ + ξ 4 ) (.0) (I( φ φ))(ξ 1 ) ξi=0 m(ξ )m(ξ )m(ξ 4 ) Îφ(ξ ) Îφ(ξ ) Îφ(ξ 4). 0 In both cases we break φ into a sum of dyadic constituents ψ j, each with frequency support ξ j, j = 0,..... For both Term 1 and Term we ll pull the symbol (.1) 1 m(ξ + ξ + ξ 4 ) m(ξ )m(ξ )m(ξ 4 ) out of the integral, estimating it pointwise in absolute value, using two different strategies depending on the relative sizes of the frequencies involved. After so bounding the factor (.1), the remaining integrals in (.9), (.0), involving the pieces ψ i of φ, are estimated by reversing the Plancherel formula 8 and using duality, Hölder s inequality, and Strichartz estimates. We can sum over the all frequency pieces ψ i since our bounds decay geometrically in these frequencies. We suggest that the reader at first ignore this summation issue, and so ignore on first reading the appearance below of all factors such as N 0 i which we include only to show explicitly why our frequency interaction estimates sum. The main goal of the analysis is to establish the decay of N + in each class of frequency interactions below. Consider first Term 1. By Proposition., (Iφ) X δ 1, 1 + Iφ X δ 1, Hence we conclude Term 1 N + once we show δ ( (.) 1 4 m(ξ ) + ξ + ξ 4 ) φ1 (ξ 1 ) φ (ξ ) φ (ξ ) φ 4 (ξ 4 ) ξi=0 m(ξ ) m(ξ ) m(ξ 4 ) 0 N + (N 1 N N N 4 ) 0 φ 1 X 1, 1 + φ X1, 1 + φ X1, 1 + φ 4 X1, 1 +, for any functions φ i, i = 1,..., 4 with positive spatial Fourier transforms supported on (.) ξ ki N i, for some k i {0, 1,...}. (Note that we are not decomposing the frequencies ξ 1 here. In the three dimensional argument we ll need to do this.) The inequality (.) implies our desired bound (.8) for Term 1 once we sum over all dyadic pieces ψ j. By the symmetry of the multiplier (.1) in ξ, ξ, ξ 4, and the fact that the refined Strichartz estimate (.6) allows complex conjugates on either factor, we may assume for the remainder of this proof that (.4) N N N 4. Note too that 4 ξ i = 0 in the integration of (.) so that N 1 N. Hence it is sufficient to obtain a decay factor of N + N 0 on the right hand side of (.). We now split the different frequency interactions into three cases, according to the size of the parameter N in comparison to the N i. 8 Assuming, as we may, that the spatial Fourier transform of φ is always positive.

10 10 J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO Term 1, Case 1: N N. According to (.), the symbol (.1) is in this case identically zero and the bound (.) holds trivially. Term 1, Case : N N N N 4. Since i ξ i = 0, we have here also N 1 N. By the mean value theorem, m(ξ ) m(ξ + ξ + ξ 4 ) m(ξ ) m(ξ ) (ξ + ξ 4 ) N (.5). m(ξ ) N This pointwise bound together with Plancherel s theorem and (.6) yield (.6) (.7) Left Side of (.) N N φ 1 φ L ([0,δ] R ]) φ φ 4 L ([0,δ] R ) N N 1 N 1 4 N N 1 1 N 1 Comparing (.) with (.7) it remains only to show that N N 1 N 1 4 N 1 N N 1 1 N 1 N N N 4 i φ i X δ 0, 1 +. N + N 0, which follows immediately from our assumptions N 1 N N N N 4. Term 1, Case : N N N. We use in this instance a trivial pointwise bound on the symbol, 1 m(ξ (.8) + ξ + ξ 4 ) m(ξ )m(ξ )m(ξ 4 ) m(ξ 1 ) m(ξ )m(ξ )m(ξ 4 ). When estimating the remainder of the integrand on the left of (.), break the interactions into two subcases, depending on which frequency is comparable to N. Case (a): N 1 N N N. We aim for m(n 1 ) δ m(n )m(n )m(n 4 ) φ1 φ φ φ4 4 ξi=0 N 0 + N 0 Pairing φ 1 φ 4 and φ φ in L and applying (.6), it remains to show or (.9) m(n 1 )N 1 4 N 1 m(n )m(n )m(n 4 )N 1 1 N 1 N N N m(n )m(n 4 )N N 1 N 4 1 N N N 4 N 1 + N 1 N N 4, N φ i X δ 0, 1 +. When estimating such fractions here and in the sequel, we frequently use two trivial observations 9 : for any p > 7, the function m(x)x p is increasing; and m(x) x p is bounded below. For example, in the denominator of (.9), m(n 4 ) N and m(n )N 1 m(n)n 1 = N 1. After these observations one quickly concludes that (.9) holds. 9 Alternatively, use (.) to write out the value of m explicitly.

11 A.C. LAWS AND GLOBAL ROUGH SOLUTIONS FOR NLS 11 Case (b): N N N. Argue as above, now pairing φ 1 φ and φ φ 4 in L. The desired bound (.) will follow from m(n 1 )N 1 1 N 1 4 m(n )m(n )m(n 4 )N 1 N 1 N + N 1 N N 4, N 1 or, after cancelling powers of N 1 in the numerator with powers of N in the denominator, (.40) m(n 1 )N N m(n )m(n )m(n 4 )N 1 N N 4 1 Using m(n 4 ) N and that both m(n )N 1, m(n )N 1 m(n)n 1 = N 1, we get (.40). This completes the proof of (.), and the bound for the contribution of Term 1 in (.8). We turn to the bound (.8) for Term (.0). As in our previous discussion of Term 1, it suffices to show δ ( (.41) 1 6 m(ξ ) 4 + ξ 5 + ξ 6 ) P N1I(φ1 φ φ )(ξ 1 + ξ + ξ ξi=0 m(ξ 4 )m(ξ 5 )m(ξ 6 ) )Îφ 4(ξ 4 )Îφ 5(ξ 5 )Îφ 6(ξ 6 ) 0 1. N + N 0 4 where as above, 0 φ i (ξ i ) is supported for ξ i N i = ki, i = 4, 5, 6, and without loss of generality, (.4) N 4 N 5 N 6, and N 4 N, 6 Iφ i, X1, 1 + the latter assumption since otherwise the symbol on the left of (.41) vanishes. In (.41) we have written P N1 for the projection onto functions supported in the N 1 dyadic spatial frequency shell. The decay factor on the right of (.41) allows us to sum in N 4, N 5, N 6, and N 1, which suffices as we do not dyadically decompose that part of Term represented here by φ i, i = 1,,. We pointwise bound the symbol on the left of (.41) in the obvious way 1 m(ξ 4 + ξ 5 + ξ 6 ) m(ξ 4 )m(ξ 5 )m(ξ 6 ) m(n 1 ) m(n 4 )m(n 5 )m(n 6 ) and as before, we undo the Plancherel formula. After applying Hölder s inequality, it suffices to show (.4) m(n 1 ) m(n 4 )m(n 5 )m(n 6 ) P N 1 I(φ 1 φ φ ) L t L Iφ 4 x L 4 t L Iφ 5 4 x L 4 t L Iφ 6 4 x L t L x To this end we ll use Lemma.. Suppose the functions φ i, i = 1,... 6 as above. Then, (.44) (.45) (.46) P N1 I(φ 1 φ φ ) L t L x 1 N 1 Iφ i, X1, 1 + Iφ j L 4 t L 1 4 x N j Iφ j X1, 1 + Iφ 6 L t L Iφ x 6. X1, 1 + j = 4, 5, N + N Iφ i. X1, 1 +

12 1 J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO Proof. For (.44), it suffices to prove (.47) P N1 I(φ 1 φ φ ) L t L Iφ x i. X1, 1 + (See Section above for notation). The operator I obeys a Leibniz rule. Using Hölder s inequality on a typical resulting term, (.48) P N1 (( I(φ 1 ))φ φ ) L t L x I(φ 1) L 4 x,t φ L 8 x,t φ L 8 x,t. By Sobolev s inequality and a L 8 t L 8 x Strichartz estimate (.5), φ L 8 t,x 1 φ L 8 t L 8 x 1 φ X0, 1 + (.49) φ X1, 1 + and similarly for the φ factor on the right of (.48). Applying the L 4 x,t Strichartz estimate, (.50) Iφ 1 L 4 x,t Iφ 1 X1, 1 +. Together, (.48) - (.50) yield (.44). The bounds (.45) follow immediately from the L 4 t,x Strichartz estimate as in (.50). The estimate (.46) is seen using Sobolev embedding, the fact that φ 6 is frequency localized, and the L t L x Strichartz bound, Iφ 6 L x,t Iφ 6 L t L x Iφ 6 X1, 1 +. Together, (.4) and Lemma. leave us to show (.51) m(n 1 ) N N 4 m(n 4 )m(n 5 )m(n 6 ) N 1 N 4 N 5 1 under the assumption (.4). We can break the frequency interactions into two cases: N 4 N 5 and N 4 N 1, since we have 6 ξ i = 0 in (.41). Term, Case 1; N 4 N 5 ; N 4 N 5 N 6 ; N 4 N: We aim here for m(n 1 )N N4 (m(n 4 )) N 4 1. m(n 6 ) N 1 Since m(n 4 ) N 4 1 m(n) N 1 = N 1 it suffices to show (.5) which is clear since N 1 m(n 1 ), and m(n 1 )N 1 N4 1, N 4 m(n 6 ) N 1 (.5) m(y) x 1 1 for all 0 y x.

13 A.C. LAWS AND GLOBAL ROUGH SOLUTIONS FOR NLS 1 Term, Case ; N 4 N 1 ; N 4 N 5 N 6 ; N 4 N: Here we argue that m(n 4 )N N4 m(n 4 ) N 4 m(n 5 )m(n 6 ) N 5 N N 4 m(n 5 ) N 5 1 m(n 6 ) N 4 N 5 1 using (.5) and our assumptions on the N i. This completes the proof of (.8) and hence the proof of Proposition.1. 1, 4. Proof of Theorem 1.1 in R In three space dimensions our almost conservation law takes the following form, Proposition 4.1. Given s > 5 6, N 1, and initial data φ 0 C 0 (R ) with E(I N φ 0 ) 1, then there exists a universal constant δ so that the solution of (1.1)-(1.) satisfies (4.1) for all t [0, δ]. φ(x, t) C([0, δ], H s (R )) E(I N φ)(t) = E(I N φ)(0) + O(N 1+ ), The norm φ(t, ) L (R ) is supercritical with respect to the scaling (.8). Hence, aside from the L conservation (1.) we will avoid using this quantity in the proof of the three dimensional result. Beside the technical issues introduced by scaling the L norm, our proof of Theorem 1.1 for n = follows very closely the n = arguments of Section. We begin with the fact that Proposition 4.1 implies Theorem 1.1 with n =. Recall that it suffices to show the H s (R ) norm of the solution to (1.1)- (1.) grows polynomially in time. Recall too φ (λ) as the scaled solution defined in (.8). When n =, the definition of the energy (1.4) and Sobolev embedding imply (4.) E(I N φ (λ) 0 ) = 1 I Nφ (λ) 0 L (R ) I Nφ (λ) 0 4 L 4 (R ) C 0 N s λ 1 s (1 + φ 0 H s (R )) 4. Once the parameter N is chosen, we will choose λ according to ( ) s s (4.) λ = N 1 s (1 + φ0 Hs (R C )) 4 1 s. 0 Together, (4.) and (4.) give E(I N φ (λ) 0 ) 1. We can therefore apply Proposition 4.1 at least C 1 N 1 times to give (4.4) E(I N φ (λ) )(C 1 N 1 δ) 1. The estimate (4.4) implies φ(t, ) H s (R ) grows at most polynomially when 5 6 < s < 1. This can be seen exactly as in the two dimensional case. We include the argument here for completeness. Given any T 0 1, first choose N 1 so that (4.5) T 0 = C 1N 1 δ λ N ( 5 ) s 1 s.

14 14 J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO Note that the exponent of N on the right of (4.5) is positive (and hence this definition of N makes sense) precisely when s > 5 6. In three space dimensions we have According to (4.), (4.4), (4.5), we therefore get E(I N φ (λ) )(λ t) = 1 λ E(I Nφ)(t). E(I N φ)(t 0 ) λe(i N φ (λ) )(λ T 0 ) λ N s 1 s T 1 s+ (s 5 6 ) 0. According to (.4) and (1.), the H s (R ) norm grows with at most half this rate when 5 6 < s < 1, (4.6) φ Hs (R )(T ) (1 + T ) 1 s+ 6(s 5 6 ). As in the two dimensional argument, the proof of Proposition 4.1 relies on bounds for the local-in-time H s solution. The following analogue of Proposition. avoids the use of the norm φ(, t) L (R ), which, as mentioned above, is supercritical with respect to scaling. Proposition 4.. Assume 5 6 < s < 1 and we are given data for (1.1)-(1.) with E(Iφ 0) 1. Then there is a universal constant δ > 0 so that the solution φ obeys the following bound on the time interval [0, δ], (4.7) Iφ X δ 0, Proof. Arguing as in the proof of Proposition., it suffices to prove I(φ φφ) X δ Iφ 0, 1 X, ++ 0, 1 + Again, the interpolation lemma in [11] allows us to assume N = 1 in the definition (.1) of the operator I. After applying a Leibniz rule for the operator I and duality, we aim to show (4.8) ( I)(φ 1 ) φ φ ψ L 1 (R +1 ) ψ X0, 1 Iφ i. X0, 1 + Again, the complex conjugate will have no bearing on our bounds. We split the functions φ j, j =, into high and low frequency components, (4.9) where supp φ j = φ high j + φ low j, φ high j (ξ, t) { ξ 1 } supp φ low j (ξ, t) { ξ 1}. Note that when n =, homogeneous Sobolev embedding and the L 10 t L 0 1 x Strichartz estimate give (4.10) φ L 10 t L10 x (R+1 ) φ L 10 φ X0, t L 1 x (R +1 )

15 A.C. LAWS AND GLOBAL ROUGH SOLUTIONS FOR NLS 15 Consider first the low frequency components on the left of (4.8). Apply Holder s inequality with the factors in L 10 x,t, L 10 x,t, L 10 x,t, and L x,t respectively. The L 10 x,t Strichartz estimate along with (4.10) give, ( I)(φ 1 ) φ low φ low ψ L1 (R +1 ) ψ X0,0 Iφ 1 X0, 1 + i= φ low j X0, 1 +. Together with the fact that φ low j = Iφ low j, this bound accounts for part of the low frequency contributions of φ, φ in (4.8). A typical contribution which remains to be bounded is Recall the Strichartz estimate, (4.11) (Iφ 1 )φ high φ high ψ L 1 x,t (R +1 ). ψ L 10 x,t (R+1 ) ψ X 0, 1 +. Interpolating between (4.11) and the trivial bound ψ L x,t (R +1 ) ψ X0,0 gives (4.1) ψ L x,t (R +1 ) ψ X0, 1. Using (4.1) and Holder s inequality on the left of (4.8), we aim to show (4.1) (Iφ 1 )φ high φ high L x,t Iφ i. X0, (R ) Since we ve reduced to the case N = 1, we note I 1 = g, where (4.14) g 1 s (0, 1 6 ) is the gap between s and 1. We may therefore rewrite our desired estimate as (4.15) (Iφ 1 )( g Iφ high )( g Iφ high ) L x,t Iφ i. X0, (R ) But this estimate follows after taking the factors on the left in L 10 x,t, L x,t, L x,t, respectively and using Hölders inequality. The first resulting factor is bounded using the L 10 x,t Strichartz estimate. As for the second two factors, Sobolev embedding, the bounds (4.14) on g, and the L t,x Strichartz estimate yield for j =,, g Iφ high j 60 L 11 x,t (R+1 ) 1 g g Iφ high j 60 L t,x Iφ high j. X0, 1 + The case where φ low φ high appears on the left of (4.8) is handled similarly, using a homogeneous Sobolev embedding to bound the φ low term Proof of Proposition 4.1. Arguing as in the two dimensional result leaves us to show (4.16) Term 1 + Term N 1++, where the two terms on the left are as before, (.9), (.0). We will have to pay closer attention here than in R when we sum the various dyadic components of this estimate. The fact that we only control inhomogeneous norms (4.7) forces us to decompose the frequencies ξ 1 as well.

16 16 J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO Considering first Term 1, it follows from the definition of the X s,b norms (.1) that (4.17) Iφ X δ 1, 1 + Iφ X δ 0, 1 +. We conclude Term 1 N 1++ once we prove δ ( (4.18) 1 4 m(ξ ) + ξ + ξ 4 ) φ 1 (ξ 1 ) φ (ξ ) φ (ξ ) φ 4 (ξ 4 ) ξi=0 m(ξ ) m(ξ ) m(ξ 4 ) 0 N 1++ C(N 1, N, N, N 4 ) φ 1 X 1, φ j, X0, 1 + for sufficiently small C(N 1, N, N, N 4 ) and for any smooth functions φ i, i = 1,... 4 with 0 φ i (ξ i ) supported for ξ i N i ki, k i = 0, ±1, ±,.... As before, we may assume N N N 4. The precise extent to which C(N 1, N, N, N 4 ) decays in its arguments, and the fact that this decay allows us to sum over all dyadic shells, will be described below on a case-by-case basis. In addition to the estimates (4.10), (4.11), our analysis here uses the following related bounds, all of which are quick consequences of homogeneous Sobolev embedding, Hölder s inequality in the time variable, and/or Strichartz estimates. These estimates will allow for bounds decaying in the frequencies. For a function φ with frequency support in the D th dyadic shell, (4.19) (4.0) (4.1) φ L 10 t L10± x ([0,δ] R ) D0± φ X0, 1 + φ L 10 t φ L 10 t 10 L x ([0,δ] R ) δ φ X0, L x ([0,δ] R ) D φ. X0, 1 + Term 1, Case 1: N N. Again, the symbol (.1) is in this case identically zero and the bound (4.18) holds trivially, with C 0. Term 1, Case : N N N N 4. We have N N 1 here as well. We will show (4.) C(N 1, N, N, N 4 ) = N 0 N 4. With this decay factor, and the fact that we are considering here terms where N 1 N, we may immediately sum over the N 1, N indices. Similarly, the factor N4 in (4.) allows us to sum over all terms here with N, N 4 1. It remains to sum the terms where 1 N 4 N N, but these introduce at worst a divergence N log(n), which is absorbed by the decay factor N 1++ on the right side of (4.18). We now show (4.18), (4.). As before, (.5), we bound the symbol in this case by N N. We apply Hölder s inequality to the left side of (4.18), bounding φ 1, φ in L 10 x,t as in (4.11); φ in L 10 x as in (4.0); and φ 4 in L 10 t L 1 x as in (4.19) to get, Left Side of (4.18) N4 N φ 1 N φ X0, 1 φ X0, + 1 φ X0, X0, N 4 N N 1 4 φ 1 X 1, 1 φ + j N N N. X0, 1 + We conclude the bound (4.18), (4.) for this case once we note N N 1 N 1 N N N N 1, j= j=

17 A.C. LAWS AND GLOBAL ROUGH SOLUTIONS FOR NLS 17 which is immediate from our assumptions on the N i. Term 1, Case, N N N: As in the two dimensional argument, we use here the straightforward bound (.8) for the symbol. The estimate of the remainder of the integrand will break up into six different subcases, depending on which N i is comparable to N, and whether or not N 1, N 4 1. Case (a), N 1 N N N; N 4 1: We will show here (4.) C(N 1, N, N, N 4 ) = N 4 N 0, which suffices since one may use (4.) to sum directly in N, N 4, and use Cauchy-Schwarz to sum in N 1, N. To establish (4.18), (4.), estimate the φ 4 factor in L 10 t L 1 x using (4.19); φ in L 10 in L 10 x,t as in (4.11). It remains then to show (4.4) m(n 1 )N 1 N 1 N m(n )m(n )N N 1. Note that since s ( 5 6, 1), we can use the following fact while working in three space dimensions, (4.5) (m(x)) p1 x p is nondecreasing in x when 0 < p 1 6p. x as in (4.0); and φ 1, φ We check (4.4) by first cancelling factors involving N 1 and N from numerator and denominator,and then using (4.5), m(n 1 )N 1 N 1 N N 1 N m(n )m(n )N N m(n )N N 1 N (m(n))n 1 N Case (b), N N N, N 1 1, N 4 1: Exactly as above, one shows (4.18), (4.) holds. With this, one may sum directly in N 4, and also in N 1, N, N using the N decay in (4.). Case (c), N N N, N 1 1, N 4 1: Here we have (4.6) C(N 1, N, N, N 4 ) = N 0 N 4 N 1. Allowing us to sum directly in all N i. One shows (4.6) by modifying the argument in (a), taking φ 1, φ in L 10 + x, L 10 + x, respectively. Case (d), N 1 N N N; N 4 1: We will show here (4.7) C(N 1, N, N, N 4 ) = N 0 N 0 4, allowing us to sum immediately in N and N 4 ; summing in N 1, N using Cauchy-Schwarz. After taking the symbol out of the left side of (4.18) using (.8), we apply Hölder s inequality as follows: estimate the φ 4 factor in L 10 x using (4.19); φ in L 10 + x as in (4.1); and φ 1, φ in L 10 x,t as in (4.11). We will establish (4.18), (4.7) once we show (4.8) m(n 1 )N 1 N 1 N N 1. m(n )m(n )m(n 4 )N N This is done as in the argument of Case (a). Case (e), N N N; N 4 1;N 1 1: We will show here (4.9) C(N 1, N, N, N 4 ) = N 0 N 0 1, 1.

18 18 J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO allowing us to sum directly in all the N i. The Hölder s inequality argument here takes the φ 1 factor in L 10 x using (4.19); φ in L 10 + x as in (4.1); and φ, φ 4 in L 10 x,t as in (4.11). We will have shown (4.18), (4.9) once we show, (4.0) This argument is by now straightforward, m(n 1 )N 1 N 1 N N 1. m(n )m(n )m(n 4 )N N m(n 1 )N 1 N 1 N N N 1 N m(n )m(n )m(n 4 )N N m(n ) N Case (f), N N N; N 4 1;N 1 1: We will show here (4.1) allowing us to sum directly in all the N i. N N 1 N N N 1 N N C(N 1, N, N, N 4 ) = N 0 N 1, 1. The proof of (4.18), (4.1) is similar to Case (e), now taking φ 1 in L 10 t L 1 x using (4.19); φ in L 10 (4.1); and φ, φ 4 in L 10 x as in (4.0). + x as in This completes the -dimensional analysis of Term 1 in (4.16). We will show Term the following, Lemma 4.. (4.) N 1+ using the straightforward bound (.8) on the symbol in the case N N, and I(φ 1 φ φ ) L t L x ([0,δ] R ) Iφ i. X0, 1 + We postpone the proof of Lemma 4.. As in the work for Term 1 above, the argument bounding Term is complicated only by the presence of low frequencies. Our aim is to show (4.) Left Side of (.41) C(N 1, N 4, N 5, N 6 ) 6 Iφ i. X0, 1 + where N 4 N 5 N 6 and N 4 N, and as in the Term 1 work above, C(N 1, N 4, N 5, N 6 ) decays sufficiently fast to allow us to add up the individual frequency interaction estimates to get (4.16). We sum first the interactions involving N i 1 for all frequencies in (4.). In this case we ll show a decay factor of C = N 1+ (N 1 N 4 N 5 N 6 ) 0, allowing us to sum in each index N i directly. Apply Hölder s inequality to the integrand on the left of (.41), taking the factors in L x,t; L 10 x,t; L 10 x,t; and L 10 x,t; respectively. Using (.8), (4.), the Strichartz estimates and (4.10) as in the Term 1 argument, it suffices to show (4.4) N 1 N4 m(n 1) 1. N 4 m(n 4 )m(n 5 )m(n 6 )

19 A.C. LAWS AND GLOBAL ROUGH SOLUTIONS FOR NLS 19 The fact that m(x) is nonincreasing in x and (4.5) give us N 1 N4 m(n 1) N 4 m(n 4 )m(n 5 )m(n 6 ) N 1 N 4 m(n 1 ) N 4 (m(n 4 )) N 1 N4 m(n 1) N4 N 1. 1 (m(n)) The above argument is easily modified in the presence of small frequencies. We sketch these modifications here. In case N 1 N 4, with N 6 1 and possibly also N 5 1, we need to get factors of N6 and possibly also N5 on the right hand side of (4.). We accomplish this by taking the factor Iφ 6 and possibly also Iφ 5 in L 10 t L 1 x,and take the factor Iφ 4 in L 10 x, or possibly L 10 x. In case N 4 N 5, with N 1 and/or N 6 small, a similar argument gets the necessary decay: we can take P N1 I(φ 1 φ φ ) in L t L + x, and/or Iφ 6 in L 10 t L 1 x, and take Iφ 4 in L 10/ x or L 10/ x. Proof of Lemma 4.. By the interpolation lemma in [11], we may assume N = 1. By Plancherel s theorem, it suffices to prove (4.5) +φ high i φ 1 φ φ L x,t ([0,δ] R ) φ i. X0, 1 + Decomposing φ i = φ low i as in (4.9), we consider first the contribution when only the low frequencies interact with one another. Hölder s inequality in space-time, homogeneous Sobolev embedding, Hölder s inequality in time, and the energy estimate yield, φ low 1 φ low φ low L x,t ([0,δ] R ) = Iφ low 1 Iφ low Iφ low L x,t ([0,δ] R ) Iφ low i L 6 x,t ([0,δ] R ) δ 1 Iφ low i L 6 t L x ([0,δ] R ) Iφ low i L t L x ([0,δ] R ) Iφ low i X0, 1 +. A typical term whose contribution to (4.5) remains to be controlled is (4.6) Iφ low 1 g Iφ high g Iφ high L x,t ([0,δ] R ), where g is as in (4.14). Take the first factor here in L 6 t L 18 x and each of the second two in L 6 t L 9 x via Hölder s inequality. Note then that Sobolev embedding and the L 6 t L 18 7 x Strichartz inequality give us Iφ low 1 L 6 t L 18 x (R+1 ) Iφ low 1 18 L 6 t L Iφ low 1 X δ 0, x (R +1 )

20 0 J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO Similarly, the fact that g (0, 1 6 ), Sobolev embedding, Hölder s inequality in time, and the energy estimate give us for j =,, g Iφ high j This completes the proof of Lemma 4.. L 6 t L 9 x ([0,δ] R ) Iφhigh j L 6 t L x ([0,δ] R ) δ 1 6 Iφ high j L t L x (R+1 ) Iφ j X δ 0, 1 +. References [1] M. Beals, Self-spreading and strength of singularities for solutions to semilinear wave equations Ann. Math. 118, (198), [] J. Bourgain, Refinements of Strichartz s inequality and applications to D-NLS with critical nonlinearity Intern. Mat. Res. Notices, 5, (1998), 5 8. [] J. Bourgain, Global solutions of nonlinear Schrödinger equations American Math. Society, Providence, R.I., [4] J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, GAFA, (199), [5] T. Cazenave, F. Weissler, The Cauchy problem for the nonlinear Schrdinger equation in H 1. Manuscripta Math. 61 (1988), [6] J. Colliander, J. Delort, C. Kenig, G. Staffilani, Bilinear estimates and applications to D NLS, Trans. Amer. Math. Soc., 5 (001), no. 8, [7] J. Colliander, G. Staffilani, H. Takaoka, Global wellposedness for KdV below L, Math. Res. Lett. 6 (1999), no. 5-6, [8] J. Colliander, J. Delort, C. Kenig, G. Staffilani, Bilinear estimates and applications to D NLS, Trans. Amer. Math. Soc., 5 (001), no. 8, [9] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao Global well-posedness for KdV in Sobolev spaces of negative index, Electronic Jour. Diff. Eq. 001 (001), No 6, 1-7. [10] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao Sharp Global Well-Posedness of KdV and Modified KdV on the Line and Torus, submitted to Jour. Amer. Math. Soc. [11] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao Sharp multi-linear periodic KdV estimates and applications, preprint. [1] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao Global well-posedness for the Schrödinger equations with derivative, Siam Jour. Math. Anal., (001), [1] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T.Tao, A refined global well-posedness result for Schrödinger equations with derivative, preprint. [14] G. Fonseca, F. Linares, and G. Ponce. Global well-posedness of the modifies Korteweg-de Vries equation, Comm. Partial Differential Equations, 4 (1999), [15] J. Ginebre, G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys., 1 (1989), [16] M. Keel, and T. Tao, Global well-posedness for large data for the Maxwell-Klein-Gordon equation below the energy norm, preprint. [17] M. Keel, T. Tao, Local and global well posedness of wave maps on R 1+1 for rough data, Intl. Math. Res. Notices 1 (1998), [18] M. Keel, T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 10 (1998), [19] C. Kenig, G. Ponce, L. Vega; Global well-posedness for semi-linear wave equations. Comm. Partial Diff. Eq. 5 (000), [0] C. Kenig, G. Ponce, L. Vega, A Bilinear Estimate With Applications to the KdV Equation, Journal of the A.M.S., 9 (1996), [1] S. Klainerman, M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (199), [] J. Rauch, M. Reed, Nonlinear microlocal analysis of semilinear hyperbolic systems in one space dimension. Duke Math. J., 49 (198), [] H. Takaoka, Global well-posedness for the Schrödinger equations with derivative in a nonlinear term and data in low order Sobolev space, Electronic Jour. Diff. Eq., 4 (001), 1. [4] H. Takaoka, N. Tzvetkov, Global low regularity solutions for Kadomtsev-Petviashvili equation, Internat. Math. Res. Notices, 001, No., [5] K. Yajima, Exitence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987),

21 A.C. LAWS AND GLOBAL ROUGH SOLUTIONS FOR NLS 1 address: colliand@math.toronto.edu University of Toronto address: keel@math.umn.edu University of Minnesota address: gigliola@math.brown.edu Brown University and Stanford University address: takaoka@math.sci.hokudai.ac.jp Hokkaido University address: tao@math.ucla.edu University of California, Los Angeles