Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space

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1 Almost sure well-posedness for the periodic D quintic nonlinear Schrödinger equation below the energy space The MIT Faculty has made this article openly available Please share how this access benefits you Your story matters Citation As Published Publisher Nahmod, Andrea and Gigliola Staffilani Almost Sure Well- Posedness for the Periodic D Quintic Nonlinear Schrödinger Equation Below the Energy Space Journal of the European Mathematical Society 17, 7 (015): European Mathematical Society European Mathematical Publishing House Version Author's final manuscript Accessed Sat Mar 0 08:09:54 EDT 019 Citable Link Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms

2 ALMOST SURE WELL-POSEDNESS FOR THE PERIODIC D QUINTIC NONLINEAR SCHRÖDINGER EQUATION BELOW THE ENERGY SPACE arxiv: v1 [mathap] 6 Aug 01 ANDREA R NAHMOD AND GIGLIOLA STAFFILANI Abstract In this paper we prove an almost sure local well-posedness result for the periodic D quintic nonlinear Schrödinger equation in the supercritical regime, that is below the critical space H 1 (T ) 1 Introduction In this paper we continue the study of almost sure well-posedness for certain dispersive equations inasupercriticalregime Inthelast two decades therehasbeenaburstofactivity -and significant progress- in the field of nonlinear dispersive equations and systems These range from the development of analytic tools in nonlinear Fourier and harmonic analysis combined with geometric ideas to address nonlinear estimates, to related deep functional analytic methods and profile decompositions to study local and global well-posedness and singularity formation for these equations and systems The thrust of this body of work has focused mostly on deterministic aspects of wave phenomena where sophisticated tools from nonlinear Fourier analysis, geometry and analytic number theory have played a crucial role in the methods employed Building upon work by Bourgain [1,, 4] several works have appeared in which the well-posedness theory has been studied from a nondeterministic point of view relying on and adapting probabilistic ideas and tools as well (cf [11, 1, 4, 8, 9, 5, 6, 7,, 17, 5, 15, 18, 19] and references therein) It is by now well understood that randomness plays a fundamental role in a variety of fields Situations when such a point of view is desirable include: when there still remains a gap between local and global well-posedness when certain type of ill-posedness is present, and in the very important super-critical regime when a deterministic well-posedness theory remains, in general, a major open problem in the field A set of important and tractable problems is concerned with those (scaling) equations for which global well posedness for large data is known at the critical scaling level Of special interest is the case when the scale-invariant regularity s c = 1 (energy or Hamiltonian) A natural question then is that of understanding the supercritical (relative to scaling) long time dynamics for the nonlinear Schrödinger equation in the defocusing case Whether blow up occurs from classical data in the defocussing case remains a difficult open problem in the subject However, what seems within reach at this time is to investigate and seek an answer to these problems from a nondeterministic viewpoint; namely for random data In this paper we treat the energy-critical periodic quintic nonlinear Schrödinger equation (NLS), an especially important prototype in view of the results by Herr, Tzvetkov and Tataru [] establishing small data global well posedness in H 1 (T ) and of Ionescu and The first author is funded in part by NSF DMS The second author is funded in part by NSF DMS

3 NAHMOD AND STAFFILANI Pausader [4] proving large data global well posedness in H 1 (T ) in the defocusing case, the first critical result for NLS on a compact manifold Large data global well-posedness in R for the energy-critical quintic NLS had been previously established by Colliander, Keel, Staffilani, Takaoka and Tao in [16] Our interest in this paper is to establish a local almost sure well posedness for random data below H 1 (T ); that is, in the supercritical regime relative to scaling 1 and without any kind of symmetry restriction on the data In a seminal paper, Bourgain [4] obtained almost sure global well posedness for the D periodic defocusing (Wick ordered) cubic NLS with data below L (T ), ie in a supercritical regime (s c = 0) Burq and Tzvetkov obtained similar results for the supercritical (s c = 1 ) radial cubic NLW on compact Riemannian manifolds in D Both global results rely on the existence and invariance of associated Gibbs measures As it turns out, in many situations either the natural Gibbs or weighted Wiener construction does not produce an invariant measure or (and this is particularly so in higher dimensions) it is thought to be impossible to make any reasonable construction at all In the case of the ball or the sphere, if one restricts to the radial case then constructions of invariant measures are possible as in[5, 1, 0, 1, 6, 7, 8] Recently, a probabilistic method based on energy estimates has been used to obtain supercritical almost sure global results, thus circumventing the use of invariant measures and the restriction of radial symmetry In this context Burq and Tzvetkov [1] and Burq, Thomann and Tzvetkov [14] considered the periodic cubic NLW, while Nahmod, Pavlovic and Staffilani[5] treated the periodic Navier- Stokes equations Colliander and Oh [17] also proved almost sure global well-posedness for the subcritical 1D periodic cubic NLS below L in the absence of invariant measures by suitably adapting Bourgain s high-low method Extending the local solutions we obtain here to global ones is the next natural step; it is worth noting however that unlike the work of Bourgain [4] one would need to proceed in the absence of invariant measures; and unlike the work of Colliander and Oh [17] the smoother norm in our case, namely H 1 (T ), on which one would need to rest to extend the local theory to a global one is in fact critical This forces the bounds on the Strichartz type norms to be of exponential type with respect to the energy, too large to be able to close the argument The problem we are considering here is the analogue of the supercritical local wellposedness result proved by Bourgain in [4] for the periodic mass critical defocusing cubic NLS in D Of course, Bourgain then constructed a D Gibbs measure and proved that for data in its statistical ensemble the local solutions obtained were in fact global, hence establishing almost sure global well posedness in H ǫ (T ), ǫ > 0 There are several major complications in the work that we present below compared to the work of Bourgain: a quintic nonlinearity increases quite substantially the different cases that need to be treated when one analyzes the frequency interactions in the nonlinearity; the counting lemmata in a D lattice are much less favorable and the Wick ordering is not sufficient to remove certain resonant frequencies that need to be eliminated The latter is not surprising, and in fact known within the context of quantum field renormalization (cf Salmhofer s book [0]) In particular, to overcome this last obstacle, we introduce an appropriate gauge transformation, we work on the gauged problem and then transfer the obtained result back to the original problem; which as a consequence is studied through 1 ie for Cauchy data in H s (T ), s < s c = 1 for the quintic NLS in D See Brydges and Slade [9] for a study of invariant measures associated to the D focussing cubic NLS as for data in H β (T ), β > 0

4 AS WELL-POSEDNESS FOR THE PERIODIC D QUINTIC NLS BELOW H 1 a contraction method applied in a certain metric space of functions A similar approach was used by the second author in [1] Finally our estimates take place in function spaces where we must be careful about working with the absolute value of the Fourier transform In fact the norms of these spaces are not defined through the absolute value of the Fourier transform, a property of the X s,b spaces in [4] which is quite useful, see for example Section 8 In this work we consider the Cauchy initial value problem, { iu t + u = λu u 4 x T (11) u(0,x) = φ(x) where λ = ±1 We randomize the data in the following sense, Definition 11 Let (g n (ω)) n Z be a sequence of complex iid centered Gaussian random variables on a probability space (Ω,A,P) For φ H s (T ), let (b n ) be its Fourier coefficients, that is (1) φ(x) = b n e in x, (1+ n ) s b n < n Z n Z The map from (Ω,A) to H s (T ) equipped with the Borel sigma algebra, defined by (1) ω φ ω, φ ω (x) = n Z g n (ω)b n e in x is called a map randomization Remark 1 The map (1) is measurable and φ ω L (Ω;H s (T d )), is an H s (T d )-valued random variable We recall that such a randomization does not introduce any H s regularization (see Lemma B1 in [11] for a proof of this fact), indeed φ ω H s φ H s However randomization gives improved L p estimates almost surely Our setting to show almost sure local well posedness is similar to that of Bourgain in [4] More precisely, we consider data φ H 1 α ε (T ) for any ε > 0 of the form (14) φ(x) = n Z 1 n 5 αein x whose randomization is (15) φ ω (x) = n Z g n (ω) n 5 αein x Our main result can then be stated as follows, Theorem 1 (Main Theorem) Let 0 < α < 1 1, s (1+4α, α) and φ as in (14) Then there exists 0 < δ 0 1 and r = r(s,α) > 0 such that for any δ < δ 0, there exists Ω δ A with P(Ω c δ ) < e 1 δ r, and for each ω Ω δ there exists a unique solution u of (11) in the space S(t)φ ω +X s ([0,δ)) d, where S(t)φ ω is the linear evolution of the initial data φ ω given by (15)

5 4 NAHMOD AND STAFFILANI Here we denoted by X s ([0,δ)) d the metric space (X s ([0,δ)), d) where d is the metric defined by (1) in Section and X s ([0,δ)) is the space introduced in Definition 44 below Acknowledgement The authors would like to thank the Radcliffe Institute for Advanced Study at Harvard University for its hospitality during the final stages of this work They also thank Luc Rey-Bellet for several helpful conversations Removing resonant frequencies: the gauged equation The main idea in proving Theorem 1 goes back to Bourgain [4] and it consists on proving that if u solves (11), then w = u S(t)φ ω is smoother; see also [11, 17, 5] In fact one reduces the problem to showing well-posedness for the initial value problem involving w, which is in fact treated as a deterministic function The initial value problem that w solves does not become a subcritical one, but it is of a hybrid type involving also rougher but random terms, whose decay and moments play a fundamental role For the NLS equation this argument can be carried out only after having removed certain resonant frequencies in the nonlinear part of the equation In this section in fact we write the Fourier coefficients of the quintic expression u 4 u and we identify the resonant part that needs to be removed in order to be able to take advantage of the moments coming from the randomized terms We will go back to this concept more in details in Remark 1 below Let s start by assuming that û(n)(t) = a n (t) We introduce the notation (1) Γ(n) [i1,i,,i r] := {(n i1,,n ir ) Z r / n = n i1 n i + +( 1) r+1 n ir } to indicate various hyperplanes and Γ(n) c [i 1,i,,i r] is its complement Next, for fixed time t, we take F, the Fourier transform in space, and write, F( u(t) 4 u(t))(n) = a n1 (t)a n (t)a n (t)a n4 (t)a n5 (t) Γ(n) [1,,5] = a n1 (t)a n (t)a n (t)a n4 (t)a n5 (t) Γ(n) [1,,5] Γ(0) C [1,,,4] Γ(0)C [1,,5,4] Γ(0)C [,,5,4] a n1 (t)a n (t)a n (t)a n4 (t)a n5 (t) Γ(n) [1,,5] Γ(0) [1,,,4] a n1 (t)a n (t)a n (t)a n4 (t)a n5 (t) Γ(n) [1,,5] Γ(0) [1,,5,4] a n1 (t)a n (t)a n (t)a n4 (t)a n5 (t) Γ(n) [1,,5] Γ(0) [,,5,4] a n1 (t)a n (t)a n (t)a n4 (t)a n5 (t) Γ(n) [1,,5] Γ(0) [1,,,4] Γ(0) [1,,5,4] Γ(0) C [,,5,4] Γ(n) [1,,5] Γ(0) [1,,,4] Γ(0) [,,5,4] Γ(0) C [1,,5,4] a n1 (t)a n (t)a n (t)a n4 (t)a n5 (t) a n1 (t)a n (t)a n (t)a n4 (t)a n5 (t) Γ(n) [1,,5] Γ(0) [,,5,4] Γ(0) [1,,5,4] Γ(0) C [1,,,4]

6 AS WELL-POSEDNESS FOR THE PERIODIC D QUINTIC NLS BELOW H 1 5 = a n1 (t)a n (t)a n (t)a n4 (t)a n5 (t) Γ(n) [1,,5] Γ(0) [1,,,4] Γ(0) [,,5,4] Γ(0) [1,,5,4] I k, k=1,,8 We now rewrite each I k using more explicitly the constraints in the hyperplanes I 1 is the most complicated and and we start by rewriting it To that effect we introduce the following notation: () () We have (4) Λ(n) := Γ(n) [1,,5] Γ(0) C [1,,,4] Γ(0)C [1,,5,4] Γ(0)C [,,5,4] Σ(n) := {(n 1,n,n,n 4,n 5 ) Λ(n) / n 1,n,n 5 n,n 4 } I 1 = Λ(n)a n1 (t)a n (t)a n (t)a n4 (t)a n5 (t) = Σ(n)a n1 (t)a n (t)a n (t)a n4 (t)a n5 (t) +6 ( n a n ) a n (t)a n4 (t)a n5 (t) Γ(n) [,4,5],n,n 5 n 4 6 a n a n (t)a n4 (t)a n5 (t) Γ(n) [,4,5],n,n 5 n 4 a n1 (t) a n1 (t)a n (t)a n5 (t) Γ(n) [,1,5],n,n 5 n 1 a n 4 a n (t)+ a n a n (t) 6 n +n 5 =n a n (t)a n5 (t) a n (t) a n (t)a n4 (t)a n5 (t) Γ(n) [,4,5],n,n 5 n 4 + a n (t) a n (t)a n4 (t) n=n n 4,n n 4 Note here that we can write ( ) (5) a n a n (t)a n4 (t)a n5 (t) = a n a n a n + a n 4 a n Γ(n) [,4,5],n,n 5 n 4 n + a n a n (t)a n4 (t)a n5 (t) Γ(n) [,4,5] It is easier to see that for i =,,4 (6) I i = a n (t) a n1 (t)a n (t)a n (t)a n4 (t) = û(n)(t) u 4 (x,t)dx, Γ(0) T [1,,,4] while for j = 5,6,7 (7) I j = a n (t) n +n 4 =n a n (t)a n4 (t)+a n a n (t)a n4 (t)a n1 (t) n=n +n 4 n 1

7 6 NAHMOD AND STAFFILANI and for (8) I 8 = a n(t) n +n 4 =n a n (t)a n4 (t) We summarize our findings from (4)- (8) In this part of the argument the time variable is not important, hence we will omit it for now We write ( ( )) 7 (9) F u 4 u u u 4 dx (n) = J k (a n ) T with (10) (11) (1) (1) (14) (15) (16) J 1 = Σ(n)a n1 a n a n a n4 a n5 k=1 J = 6m a n1 a n a n Γ(n) [1,,],n,n 1 n J = 6 a n1 a n1 a n a n Γ(n) [1,,],n 1,n n a n1 a n a n a n Γ(n) [1,,],n 1,n, n J 4 = a n1 a n 1 a n n=n 1 n, J 5 = 6 a n a n1 a n a n +a n a n a n1 a n4 Γ(n) [1] Γ(n) [14] J 6 = = 5a n a n a n4 + a n a n a n1 a n n=n +n 4 n=n 1 +n J 7 = 11a n a n 4 +1m a n a n, where m = T u(t,x) dx, the conserved mass Remark 1 In the calculations above we wrote the nonlinear terms in (11) in Fourier space, we isolated the term u T u 4 dx and we subtracted it from u 4 u, see (9) We show below that indeed in doing so we separated those terms that we claim are not suitable for smoother estimates from the ones that are To understand this point let us replace a n = gn(ω) n 5 α, for α small, whose anti-fourier transform barely misses to be in H1 (T ) We want to claim that the randomness coming from {g n (ω)} will increase the regularity of the nonlinearity in a certain sense, so that it can hold a bit more than one derivative We realize immediately though that this claim cannot be true for the whole nonlinear term For example the terms I i, i =,,4 have no chance to improve their regularity because they are simply linear with respect to a n, hence they need to be removed This same problem presented itself in the work of Bourgain [4] and Colliander-Oh [17] who considered the cubic NLS below L In particular in their case the problematic term was of the type a n T d u dx and the authors removed it by Wick ordering the Hamiltonian An important ingredient in making this successful was that T d u dx, that is the mass, is independent of time In our case Wick ordering the Hamiltonian is not helpful since it does not remove the terms I i, i =,,4 As we mentioned before, the latter is not surprising, and in fact known within the context of quantum field renormalization (cf Salmhofer s book [0])

8 AS WELL-POSEDNESS FOR THE PERIODIC D QUINTIC NLS BELOW H 1 7 If we knew that T u 4 dx were constant in time, then we could simply relegate those terms to the linear part of the equation Since this is obviously not the case relegating these expressions with the main linear part of the equation would prevent us from using the simple form of the solution for a Schrödinger equation with constant coefficients A similar situation to the one just described presented itself in [1] where a gauge transformation was used to remove the time dependent linear terms We are able to use the same idea in this context and this is the content of what follows in this section To prove Main Theorem 11 we proceed in two steps First we consider the initial value problem { iv t + v = N(v) x T (17) v(0,x) = φ(x), where (18) N(v) := λ ( ( )) v v 4 v v 4 dx T with λ = ±1 and φ(x) the initial datum as in (11) To make the notation simpler set (19) β v (t) = v 4 dx T and define (0) u(t,x) := e iλ t 0 βv(s)ds v(t,x) We observe that u solves the initial value problem (11) Now suppose that one obtains well-posedness for the initial value problem (17) in a certain Banach space (X, ) then one can transfer those results to the initial value problem (11) by using a metric space X d := (X,d) where (1) d(u,v) := e iλ t 0 βu(s)ds u(t,x) e iλ t 0 βv(s)ds v(t,x) The fact that this is indeed a metric follows from using the properties of the norm and the fact that if e iλ t 0 βu(s)ds u(t,x) = e iλ t 0 βv(s)ds v(t,x) then β v (t) = β u (t) and hence u = v From this moment on we work exclusively with the initial value problem (17) In particular below we prove the following result: Theorem Let 0 < α < 1 1, s (1 + 4α, α) and φ as in (14) There exists 0 < δ 0 1 and r = r(s,α) > 0 such that for any δ < δ 0, there exists Ω δ A with P(Ω c δ ) < e 1 δ r, and for each ω Ω δ there exists a unique solution u of (17) in the space with initial condition φ ω given by (15) S(t)φ ω +X s ([0,δ)), Here in the space X s ([0,δ)) is defined in Section 4 Thanks to the transformation (0), Theorem translates to Main Theorem 1

9 8 NAHMOD AND STAFFILANI Probabilistic Set Up We first recall a classical result that goes back to Kolmogorov, Paley and Zygmund Lemma 1 (Lemma 1 [11]) Let {g n (ω)} be a sequence of complex iid zero mean Gaussian random variables on a probability space (Ω,A,P) and (c n ) l Define (1) F(ω) := n c n g n (ω) Then, there exists C > 0 such that for every λ > 0 we have ( ) Cλ () P({ω : F(ω) > λ}) exp F(ω) L (Ω) As a consequence there exists C > 0 such that for every q and every (c n ) n l, c n g n (ω) n L q (Ω) C q ( We also recall the following basic probability results: n c n )1 Lemma Let 1 m 1 < m < m k = m; f 1 be a Borel measurable function of m 1 variables, f one of m m 1 variables,,f k one of m k m k 1 variables If {X 1,X,X m } are real-valued independent random variables, then the k random variables f 1 (X 1,X mk ),f (X m1 +1,,X m ),,f k (X mk 1 +1,X mk ) are independent random variables Lemma Let k 1 and consider {g nj } 1 j k and {g n j } 1 j k N C (0,1) complex L (Ω)-normalized independent Gaussian random variables such that n i n j and n i n j for i j k k k g nj (ω) g n i (ω)dp(ω) Ω g nl (ω) dp(ω) ω j=1 i=1 Proof For every pair (n l,n i ) such that n l = n i we write K n j (ω) := g nj (ω) and note that thanks to the independence and normalization of {g nj }, for n j n i, we have that E(K nj g ni ) = 0 The latter together with Lemma give the desired conclusion More generally, in the next sections we will repeatedly use a classical Fernique or large deviation-type result related to the product of {G n } 1 n d N C (0,1), complex L normalized independent Gaussians This result follows from the hyper-contractivity property of the Ornstein-Uhlenbeck semigroup (cf [5, ] for anice exposition) by writingg n = H n +il n where {H 1,,H d,l 1,L d } N R (0,1) are real centered independent Gaussian random variables with the same variance Note that E(G n ) = E(G n) = 0 Remark 4 Note that for {G n (ω)} n N C (0,1), complex L normalized independent Gaussians, if we write G n (ω) = ( G n (ω) 1)+1, then thanks to the independence and normalization of G n, Y n (ω) := G n (ω) 1 is a centered real Gaussian random variable such that for n n, E(Y n G n ) = 0 = E(Y n Y n ) l=1

10 AS WELL-POSEDNESS FOR THE PERIODIC D QUINTIC NLS BELOW H 1 9 Proposition 5 (Propositions 4 in [] and Lemma 45 in [5]) Let d 1 and c(n 1,,n k ) C Let {G n } 1 n d N C (0,1) be complex centered L normalized independent Gaussians For k 1 denote by A(k,d) := {(n 1,,n k ) {1,,d} k, n 1 n k } and () F k (ω) = c(n 1,,n k )G n1 (ω)g nk (ω) Then for all d 1 and p A(k,d) F k L p (Ω) k +1(p 1) k F k L (Ω) As a consequence from Chebyshev s inequality we have that for every λ > 0 (4) P({ω : F k (ω) > λ}) exp Cλ k F(ω) k L (Ω) Remark 6 In Sections 7 and 8 we will rely repeatedly on Proposition 5, particularly (4), as well as Lemma 1, and () Indeed, in proving our estimates we will encounter expressions of the following type: Let Σ := {(n 1,n r,l 1,,l s ) : n j N j, l i L i, n j l i, 1 j r, 1 i s,} and F(ω) := c n1 c nr b l1 b ls g n1 (ω)g nr (ω)g l1 (ω)g ls (ω), (n 1,,n r,l 1,,l s) Σ where {g n1 (ω)g nr,g l1 (ω)g ls (ω)} N C (0,1) are complex centered L normalized independent Gaussians Then by Proposition 5, there exist C > 0,γ = γ(r,s) > 0 such that for every λ > 0 we have P({ω : F(ω) > λ}) exp Cλ γ γ F(ω) L (Ω) We will also apply Proposition 5 in the context of Remark 4 Lemma 7 Let {g n (ω)} be a sequence of complex iid zero mean Gaussian random variables on a probability space (Ω, A, P) Then, (1) For 1 p < there exists c p > 0 (independent of n) such that g n L p (Ω) c p () Given ε,δ > 0, for N large and ω outside of a set of measure δ, (5) sup g n (ω) N ε n N () Given ε,δ > 0 and ω outside of a set of measure δ, (6) g n (ω) n ε Proof Part (1) follows from the fact that higher moments of {g n (ω)} are uniformly bounded For part () first recall that if {X j (ω)} j 1 is a sequence of iid random variables such that E( X j ) = E < then (7) P( X j j) = P( X 1 j)

11 10 NAHMOD AND STAFFILANI and P( X j j) = P( X 1 j) E( X 1 ) < j j By Borel-Cantelli P( X j j for infinitely many j) = 0 whence one can show that X j (ω) lim j = 0 almost surely in ω Egoroff s Theorem then ensures that given δ > 0 j X j (ω) lim = 0 j j uniformly outside a set of measure δ Thus we have that for j 0 sufficiently large X j (ω) 1 j j 0, j for ω outside an exceptional set of δ measure If {g n (ω)} are a sequence of iid complex Gaussian random variables then given ε > 0, if we choose r = 1 ε then E( g n r ) < By applying the argument above with X n (ω) = g n (ω) r we have the desired conclusion (cf [8, 17]) For part () fix M 1 such that () holds for any n M By (7) P( g n (ω) M ε ) = P( g M (ω) M ε ) for all n M Let A := n M 1 {ω / g M (ω) M ε }, then by part () P(A) C M δ Hence by choosing a smaller δ in part () we have the desired result 4 Function Spaces For the purpose of establishing our almost sure local well-posedness result, it suffices to work with X s and Y s, the atomic function spaces used by Herr, Tataru and Tzvetkov [] It is worth emphasizing that while working with these spaces, one should not rely on the notion of the norms depending on the absolute value of the Fourier transform, a feature that is quite useful when working within the context of X s,b spaces In this section we recall their definition and summarize the main properties by following the presentation in [] Section In what follows, H is a separable Hilbert space on C and Z denotes the set of finite partitions < t 0 < t 1 < t K of the real line; with the convention that if t k = then v(t K ) := 0 for any function v : R H Definition 41 (Definition 1in[]) Let 1 p < For {t k } K k=0 Z and {φ k} K 1 k=0 H with K 1 k=0 φ k p H = 1 A Up -atom is a piecewise defined function a : R H of the form K a = χ [tk 1,t k )φ k 1 k=1 The atomic Banach space U p (R,H) is then defined to be the set of all functions u : R H such that u = λ j a j, for U p atoms a j, {λ j } j l 1, j=1 with the norm u U p := inf λ j : u = j=1 λ j a j, λ j C, and a j an U p atom j=1

12 AS WELL-POSEDNESS FOR THE PERIODIC D QUINTIC NLS BELOW H 1 11 Here χ I denotes the indicator function over the set I Note that for 1 p q <, (41) U p (R,H) U q (R,H) L (R,H), and functions in U p (R,H) are right continuous, lim t u(t) = 0 Definition 4 (Definition in []) Let 1 p < The Banach space V p (R,H) is defined to be the set of all functions v : R H such that ( K ) 1 p v V p := sup {t k } K k=0 Z v(t k ) v(t k 1 ) p H is finite k=1 The Banach subspace of all right continuous functions v : R H such that lim t v(t) = 0, endowed with the same norm as above is denoted by V p rc(r, H) Note that (4) U p (R,H) V p rc(r,h) L (R,H), Definition 4 (Definition 5 in []) For s R we let U p Hs - respectively V p Hs - be the space of all functions u : R H s (T ) such that t e it u(t) is in U p (R,H s ) -respectively in V p Hs - with norm u U p Hs := e it u(t) U p (R,H s ) u V p Hs := e it u(t) V p (R,H s ) We will take H to be H s We refer the reader to [], [], and references therein for detailed definitions and properties of the U p and V p spaces Definition 44 (Definition 6 in []) For s R we define the space X s as the space of all functions u : R H s (T ) such that for every n Z the map t e it n û(t)(n) is in U (R,C), and for which the norm 1 (4) u X s := n Z n s e it n û(t)(n) U t is finite The X s spaces are variations of the spaces U p Hs and V p Hs corresponding to the Schrödinger flow and defined as follows: Definition 45 (Definition 7 in []) For s R we define the space Y s as the space of all functions u : R H s (T ) such that for every n Z the map t e it n û(t)(n) is in Vrc (R,C), and for which the norm 1 (44) u Y s := Note that n Z n s e it n û(t)(n) V t (45) U Hs X s Y s V Hs whence one has that for any partition of Z := k C k, ( )1 P Ck u V u Y Hs s (cf Section in []) k is finite

13 1 NAHMOD AND STAFFILANI Additionally, when s = 0 by orthogonality we have ( )1 (46) P Ck u Y = u 0 Y 0 We also have the embedding (47) X s Y s L t H s x for s 0 (cf [4]) k Remark 46 (Proposition 10 in []) From the atomic structure of the U spaces one can immediately see that for s 0, T > 0 and φ H s (T ), the solution to the linear Schrödinger equation u := e it φ belongs to X s ([0,T)) and u X s ([0,T)) φ H s Remark 47 Another important feature of the atomic structure of the U spaces is the fact that just like the X s,b spaces they enjoy a transfer principle We recall in our context the precise statement below for completeness Proposition 48 (Proposition 19 in []) Let T 0 : L L L 1 loc be a m-linear operator Assume that for some 1 p, q m (48) T 0 (e it φ 1,,e it φ m ) L p (R,L q x(t )) φ i L (T ) Then, there exists an extension T : U p Up Lp (R, L q (T )) satisfying (49) T(u 1,,u m ) L p (R,L q x(t )) m i=1 i=1 u i U p ; and such that T(u 1,,u m )(t, ) = T 0 (u 1 (t),,u m (t))( ), ae In other words, one can reduce estimates for multilinear operators on functions in U p to similar estimates on solutions to the linear Schrödinger equation We will use the following interpolation result at the end of Section 8 to obtain bounds in terms of the X s spaces from those in U Hs and U p Hs just as in in [] The proof relies solely on linear interpolation [, ] Proposition 49 (Proposition 0 in [] and Lemma 4 []) Let q 1,q m > where m = 1,,or, E be a Banach space and T : U q 1 U qm E be a bounded m-linear operator with (410) T(u 1,,u m ) E C m i=1 u i U q i In addition assume there exists 0 < C C such that the estimate m (411) T(u 1,,u m ) E C u i U holds true Then, T satisfies the estimate (41) T(u 1,,u m ) E C (ln C C +1) m m i=1 i=1 u i V, u i V rc, i = 1,m,

14 AS WELL-POSEDNESS FOR THE PERIODIC D QUINTIC NLS BELOW H 1 1 where V rc denotes the closed subspace of V of all right continuous functions of t such that lim t v(t) = 0 Finally we state two results from [] we rely on in the next sections In what follows, I denotes the Duhamel operator, I(f)(t) := defined for f L 1 loc ([0, ),L (T )) t 0 e i(t t ) f(t )dt, t 0, Proposition 410 (Proposition 11 in []) Let s 0 and T > 0 For f L 1 ([0,T), H s (T )) we have I(f) X s ([0,T)) and T I(f) X s ([0,T)) sup f(t,x)v(t,x)dxdt 0 T As a consequence, note we have v Y s ([0,T)): v Y s=1 (41) I(f) X s ([0,T)) f L 1 ([0,T),H s (T )) Proposition 411 (Proposition 41 in []) Let s 1 be fixed Then for all T (0,π] and u k X s ([0,T)), k = 1,5, the estimate (414) I ( 5 ) ũ k X s ([0,T)) k=1 5 j=1 u j X s ([0,T)) 5 k=1,k j u k X 1 ([0,T)) holds true, where ũ k denotes either u k or u k In particular, (414) follows from the estimate for the multilinear form: [0,T) T k=0ũkdxdt u 0 Y s ([0,T)) u j X s ([0,T)) u k X 1 ([0,T)) where u 0 := P N v j=1 k=1,k j Next, we recall the L p (T T ) Strichartz-type estimates of Bourgain s [5] in this context First recall the usual Littlewood-Paley decomposition of periodic functions For N 1 a dyadic number, we denote by P N the rectangular Fourier projection operator P N f = f(n)e in x n=(n 1,n,n ) Z : n i N Then P N = P N P N 1 so that P N = N M=1 P M and PN := I P N We then have f H s (T ) := ( n s f(n) )1 ( N s P N (f) )1 L (T ) n Z N 1 Definition 41 For N 1, we denote by C N the collection of cubes C in Z with sides parallel to the axis of sidelength N

15 14 NAHMOD AND STAFFILANI Proposition 41 [Proposition 1, Corollary in [] (cf [5])] Let p > 4 For all N 1 we have (415) (416) P N e it φ L p (T T ) N 5 p P N φ L (T ), P C e it φ L p (T T ) N 5 p P C φ L (T ), (417) P C u L p (T T ) N 5 p P C u U p L, where P C is the Fourier projection operator onto C C N defined by the multiplier χ C, the characteristic function over C Finally we prove two propositions which will play an important role in Sections 7 and 8 Proposition 414 Let u, v and w be functions of x and t such that, û(n,t) = a 1 n (t)a n (t)a n (t) v(n,t) = a 1 n(t)a n(t)a n(t)a 4 n(t)a 5 n(t) ŵ(n,t) = a 1 n (t)a n (t)a n (t) m a 4 m a5 n m and n N Assume that J {1,,,4,5} and if i J then a i n (t) = g n(ω) n +εeit n while if i / J then there is a detrministic function f i such that f i (n,t) = a i n (t) Then (418) P N u L p (T T ) P N f i Y 0, p > 4 (419) (40) (41) P N u L (T T ) i/ J {1,,} i/ J {1,,} P N v L (T T ) i/ J P N f i Y 0 P N w L (T T ) i/ J,i 4,5 P N f i Y 0 P N f i Y 0 j/ J,j=4,5 f j Y 0 Proof To prove (418) we write u = k 1 k k, where the convolution is only with respect to the space variable Then by Young s inequality in the space variable followed by Hölder s inequality and the embedding (47) we have the desired inequality To prove (419) we use Plancherel P N u L (T T ) χ n N a 1 n a n a n l L (T) N f i L x L i=1 P (T) and the conclusion follows from the embedding (47) To prove (40) we proceed in a similar manner i/ J {1,,} i=1 χ n N a i n l L (T) P N f i L (T,L (T ))

16 AS WELL-POSEDNESS FOR THE PERIODIC D QUINTIC NLS BELOW H 1 15 To prove (41) we first write P N w L (T T ) P N (k 1 k k (k 4 k 5 )) L (T T ), and by Young s, Hölder s and Cauchy-Schwarz inequality we continue with P N k i L P N (k 4 k 5 ) L 1 i=1 L (T) P N k i L k 4 L k 5 L i=1 L (T) P N f i L (T,L (T )) f j L (T,L (T )) i/ J,i 4,5 j/ J,j=4,5 We now state a trilinear L estimate that is similar to Proposition 5 in [] but in which some of the functions may be linear evolution of random data Proposition 415 Assume N 1 N N and that C C N, a cube of sidelength N Assume also that J {1,,} and such that if j J then û j (n) = e i n t g n(ω) for ε > 0 n +ε small Then (4) P C P N1 ũ 1 P N ũ P N ũ L (T T ) N N P Nj u j U 4 L and (4) P C P N1 ũ 1 P N ũ L (T T ) N 1 +ε j/ J P Nj u j U 4 L, where ũ k denotes either u k or u k Moreover (4) and (4) also hold with the Y 0 norms in the right hand side Proof To prove (4) we follow the proof of (4) in [] We write P C P N1 ũ 1 P N ũ P N ũ L (T T ) P C P N1 u 1 L p P N u L p P N u L q where p + 1 q = 1 and 4 < p < 5 Then we use (416) for the random linear functions and (417) for the deterministic functions to obtain P C P N1 ũ 1 P N ũ P N ũ L (T T ) N N ( N N where we used the embedding (41) To prove (4) we use Hölder s inequality to write j/ J ) + 10 p (44) P C P N1 ũ 1 P N ũ L (T T ) P C P N1 u 1 L 4+ε P N u L 4+ε we then we use (416), (417) and the embedding (41) to continue with N 1 +ε P Nj u j U 4 L j/ J j/ J P Nj u j U 4 L, To obtain the Y 0 in the right hand side we use the interpolation Proposition 49 and the embedding (41)

17 16 NAHMOD AND STAFFILANI 5 Almost sure local well-posedness for the initial value problem (17) We define (51) v ω 0 (t,x) = S(t)φω (x) where φ ω (x) is as in (15) and instead of solving the initial value problem (17) we solve the one for w = v v0 ω: { iw t + w = N(w +v0 ω (5) ) x T w(0,x) = 0, where N( ) was defined in (18) To understand the nonlinear term of (5) we express it in terms of its spatial Fourier transform Let a n := ˆv(n), θ ω n := F(S(t)φω )(n), then b n := ŵ(n) = a n θ ω n Now we recall (9) and in it we replace a n with b n +θ ω n Then (5) F(N(w +v ω 0))(n) = 7 J k (b n +θn), ω where here J k (b n +θ ω n ) means that the terms J k defined in (10) (16) are evaluated for the sequence (b n +θ ω n ) instead of a n We are now ready to state the almost sure well-posedness result for the initial value problem (5) Theorem 51 Let 0 < α < 1 1, s (1 + 4α, α) There exists 0 < δ 0 1 and r = r(s,α) > 0 such that for any δ < δ 0, there exists Ω δ A with k=1 P(Ω c δ ) < e 1 δ r, and for each ω Ω δ there exists a unique solution w of (5) in the space X s ([0,δ)) C([0,δ),H s (T )) The proof of this theorem follows from the following two propositions via contraction mapping argument Proposition 5 Let 0 < α < 1 1, s (1 + 4α, α), δ 1 and R > 0 be fixed Assume N i, i = 0,5 are dyadic numbers and N 1 N N N 4 N 5 Then there exists ρ = ρ(s,α) > 0, µ > 0, and Ω δ A such that P(Ω c δ ) < e 1 δ r, and such that for ω Ω δ we have: If N 1 N 0 or P N1 w = P N1 v0 ω π D s( N(P Ni (w +v0 ω ))) (54) P N0 hdxdt 0 T ( δ µr N ρ 1 P N 0 h Y s 1+ i/ J P Ni w X s )

18 AS WELL-POSEDNESS FOR THE PERIODIC D QUINTIC NLS BELOW H 1 17 If N 1 N 0 and P N1 w P N1 v0 ω π D s( N(P Ni (w +v0)) ω ) (55) P N0 hdxdt 0 T δ µr N ρ P N 0 h Y s P N1 w X s 1+ i/ J,i 1 ψ δ P Ni w X s, where v ω 0 is as in (51), w Xs ([0,π]), J {1,,,4,5} denote those indices corresponding to random functions Proposition 5 Let 0 < α < 1 1, s (1+4α, α) and δ 1 be fixed Let vω 0 be defined as in (51) and assume w X s ([0,π]) Then there exist θ = θ(s,α) > 0, r = r(s,α) and Ω δ A such that P(Ω c δ ) < e 1 δ r, and such that for ω Ω δ (56) I ( ψ δ N(w +v ω 0) ) X s ([0,π]) δ θ( 1+ ψ δ w 5 X s ([0,π]) where N( ) was defined in (18) and ψ δ is a smooth time cut-off of the interval [0,δ] The proof of Proposition 5 is the content of Sections 7 and 8 while Proposition 5 is proved in Section 9 6 Auxiliary lemmata and further notation We begin by recalling some counting estimates for integer lattice sets (cf Bourgain [5]) Lemma 61 Let S R be a sphere of radius R, B r be a ball of radius r and P be a plane in R Then we have (61) (6) (6) #Z S R R #Z B r S R min(r,r ) #Z B r P r Next, we state a result we will invoke when the the higher frequencies correspond to deterministic terms and one can afford to ignore the moments given by the lower frequency random terms as well as rely on Strichartz estimates Lemma 6 Assume N i, i = 0,5 are dyadic numbers and N 1 N 0 and N 1 N N N 4 N 5 Let {C} be a partition of Z by cubes C C N, and let {Q} be a partition of Z by cubes Q C N Then 1 (64) P N1 f 1 P N f P N f P N4 f 4 P N5 f 5 P N0 hdxdt N i,i=0,5 0 T 1 N i,i=0,5 sup P CP N1 f 1 P N f P Nl f l L xt C P Q P C P N0 hp N f P Nr f r L xt C,Q where l r {4,5} and C are cubes whose sidelength is 10N Proof The proof of (64) follows from orthogonality arguments )

19 18 NAHMOD AND STAFFILANI Just as Bourgain in [4], in the course of the proof we will use the following classical result about matrices, which we state as a lemma for convenience Lemma 6 Let A = (A ik ) 1 i N Then, 1 k M (65) AA max 1 j N where means the -norm be an N M matrix with adjoint A = (A kj ) 1 k M 1 j N ( M ) A jk + M A ik A jk k=1 i j Proof Decompose AA into the sum of a diagonal matrix D plus an off-diagonal one F Then note the -norm of D is bounded by the square root of the largest eigenvalue of DD which, since D is diagonal, is the maximum of the absolute value of the diagonal entries of D This gives the first term in (65) Bounding the -norm of F by the Fröbenius norm of F gives the second term in (65) Notation: Given k-tuples (n 1,,n k ) Z k, a set of constraints C on them, and a subset of indices {i 1,,i h } {1,,k}, we denote by S (ni1,,n ih ) the set of (k h)-tuples (n j1,,n jk h ), {j 1,j k h } = {1,,k} \ {i 1,,i h }, which satisfy the constraints C for fixed (n i1,,n ih ) We also denote by S (ni1,,n ih ) its cardinality k=1 1 7 The Trilinear and Bilinear Building Blocks In this section, we denote by D j := e it P Nj φ solutions to the linear equation for data φ in L localized at frequency N j and by R k the function defined as, (71) Rk (n) = χ { n Nk }(n) g n(ω) e it n, n and representing the linear evolution of a random function of type (15), localized at frequency N k and almost L normalized 71 Trilinear Estimates We prove certain trilinear estimates which serve as building blocks for the proof in Section 8 Their proofs are of the same flavor as those presented by Bourgain in [4] For N j, j = 1,, dyadic numbers, let α j = 0 or 1 for j = 1,, and define (7) n = ( 1) α 1 n 1 +( 1) α n +( 1) α n n k n l whenever α k α l, Υ(n,m) := (n 1,m 1 ;n,m ;n,m ) : n j N j, j = 1, m = ( 1) α 1 m 1 +( 1) α m +( 1) α m and define T Υ to be the multilinear operator with multiplier χ Υ Proposition 71 Fix N 1 N N, r, δ > 0 and C C N Then there exists µ,ε > 0, a set Ω δ A such that P(Ω c δ ) e 1 δ r and such that for any ω Ω δ we have the following estimates:

20 AS WELL-POSEDNESS FOR THE PERIODIC D QUINTIC NLS BELOW H 1 19 (7) (74) (75) (76) (77) (78) T Υ (P C R1, D,R ) L (T T ) δ µr N 5 4 N 1 1 P N φ L x T Υ (P C R1, D, R ) L (T T ) δ µr N 5 4 N 1 1 P N φ L x T Υ (P C D1, R,R ) L (T T ) δ µr N 4 P C P N1 φ L x T Υ (P C D1,R,R ) L (T T ) δ µr N 4 P C P N1 φ L x ] T Υ (P C R1,R, D ) L (T T ) δ [N µr 4 1 N 1 N 5 4 +N 1 1 N 1 N 4 P N φ L x ] T Υ (P C R1, R, D ) L (T T ) δ [N µr 4 1 N 1 N 5 4 +N 1 1 N 1 N 4 P N φ L x (79) (710) (711) (71) (71) T Υ (P C R 1, D, D ) L (T T ) δ µr N 1 +θ 4 N 1 +ε 1 min(n 1,N ) 1 θ N P N φ L x P N φ L x, 0 θ 1 T Υ (P C D1,R, D ) L (T T ) δ µr N 1 +ε N P N1 φ L x P N φ L x T Υ (P C R1, R,R ) L (T T ) δ µr N 1 1 N 1 T Υ (P C R1,R, R ) L (T T ) δ µr N 1 1 N 1 T Υ (P C R1,R,R ) L (T T ) δ µr N 1 1 N 1 Note that here the bar indicates complex conjugate while the tilde indicates both complex conjugate or not Also, without writing it explicitly, we always assume that if R(n 1 ) and R(n ) appear in the trilinear expressions in the left hand side, then n 1 n Remark 7 In using the trilinear estimates above, sometimes it is convenient to interpret a random term as deterministic and choose the minimum estimate possible For example, in considering P C R1 R R L we have a choice between (711) and (78) by thinking of R as an almost L normalized D function Proposition 7 Let D j and R k be as above and fix N 1 N N, r, δ > 0 and C C N Then there exists µ > 0 and a set Ω δ A such that P(Ω c δ ) e 1 δ r such that for any ω Ω δ we have (7) and (74) Proof As in [] we will first assume that the deterministic functions D i are localized linear solutions, that is D i = P Ni S(t)ψ and ˆψ(n) = a n Once an estimate is proved with χ Ni (n)a n l in the right hand side we then invoke the transfer principle of Proposition 48 to complete the proof We start by estimating (7) Without any loss of generality we assume that D = D By using Fourier transform to write the left hand side we note that it is enough to estimate (714) T := χ C (n 1 ) g n 1 (ω) g n (ω) a m Z,n Z n= n 1 +n +n n 1 n, n n 1 n,n m= n 1 + n + n

21 0 NAHMOD AND STAFFILANI where we recall that C is a cube of sidelength N We are going to use duality and a change of variable since, as it will be apparent below, the counting with respect to the time frequency will be more favorable Using duality we have that T = sup γ k l 1 k(n)γ(m) m,n n= n 1 +n +n n 1 n,n m= n 1 + n + n Let ζ := m n = n 1 + n, then we continue with T = sup a γ k l 1 n γ(ζ + n ) n ζ sup n γ k l 1 a l γ n l ζ n,ζ n= n 1 +n +n n 1 n,n n= n 1 +n +n n 1 n,n ζ= n 1 + n ζ= n 1 + n χ C (n 1 ) g n 1 (ω) n 1 χ C (n 1 ) g n 1 (ω) n 1 χ C (n 1 ) g n 1 (ω) n 1 All in all, we then have to estimate uniformly for γ k l 1, (715) a n l γ l σ n,nk n, where σ n,n = n ζ N 1 N n =n 1 +n n,n 1 n,n ζ= n 1 + n n χ C (n 1 ) g n 1 (ω) n 1 g n (ω) n a n g n (ω) n g n (ω) n g n (ω) n Note that σ n,n also depends on ζ but we estimate it independently of ζ If we denote by G the matrix of entries σ n,n, and we recall that the variation in ζ is at most N 1 N, we are then reduced to estimating We note that by Lemma 6 GG max n n a n l N 1 N GG σ n,n + n n n C σ n,nσ n,n 1 k n =: M 1 +M, where C is a cube of sidelength approximately N To estimate M 1 we first define the set S (ζ,n ) = {(n 1,n,n ) : n = n 1 +n n, n 1 n,n, ζ = n 1 + n } k n

22 AS WELL-POSEDNESS FOR THE PERIODIC D QUINTIC NLS BELOW H 1 1 with S (ζ,n ) N N 1, where we use (61) for fixed n Then we have M 1 sup χ C (n 1 ) g n 1 (ω) g n (ω) (n,ζ) n =n 1 +n n,n 1 n,n n 1 n ζ= n 1 + n Now we use (4) with λ = δ r F (ω) L and Lemma to obtain for ω outside a set of measure e 1 δ r the bound M 1 sup δ r (n,ζ) n 1 n ξ 1 ξ g n1 (ω)g n (ω)g ξ1 (ω)g ξ (ω)dp(ω) Ω S (ζ,n ) S (ζ,n ) (716) sup δ r 1 1 (n,ζ) n 1 n δ r N1 N N N 1 δ r N1 S (ζ,n ) To estimate M we first write M = where n n n C σ n,nσ n,n S (n,n,ζ) = (n,n 1,n,n 1,n ) : n n g (ω) n1 g n (ω) g n 1 (ω) g n (ω) S n (n,n 1 n n ) 1 n n = n 1 +n n, n = n 1 +n n, n 1 n,n, n 1 n,n, n C ζ = n 1 + n, ζ = n 1 + n We need to organize the estimates according to whether some frequencies are the same or not, in all we have six cases: Case β 1 : n 1,n 1,n,n are all different Case β : n 1 = n 1 ; n n Case β : n 1 n 1 ; n = n Case β 4 : n 1 n ; n = n 1 Case β 5 : n 1 = n ; n n 1 Case β 6 : n 1 = n ; n = n 1 Case β 1 : We define the set S (ζ) = (n,n,n,n 1,n,n 1,n ) : n = n 1 +n n, n = n 1 +n n, n 1 n,n, n 1 n,n, n 1,n 1 C ζ = n 1 + n, ζ = n 1 + n and we note that S (ζ) N1 N6 N since n C and for fixed n and n we use (61) to count n 1 and n 1 Using (4) with λ = δ r F 4 (ω) L and again Lemma we can write for ω as above M δ 4r n 1 n n S (n,n 1 n δ 4r N1 6 N 6 N 1 N6 N δ 4r N1 4 N,ζ) n n

23 NAHMOD AND STAFFILANI Case β : First define the set, S (n,n,n,n,ζ) = (n,n 1) : n = n 1 +n n, n = n 1 +n n, n 1 n,n,n,n, n C ζ = n 1 + n, ζ = n 1 + n To compute S (n,n,n,n,ζ) we count n 1, then n is determined Since n 1 sits on a sphere then by (61) we have S (n,n,n,n,ζ) N 1 Then we set n = n 1 +n n, n = n 1 +n n, S (ζ) = (n,n,n,n 1,n,n ) : n 1 n,n,n,n, n C ζ = n 1 + n, ζ = n 1 + n with S (ζ) N 1 N 6N, where we used again that n C and (61) Now, we have that (717) M g n1 (ω) g n (ω) g n (ω) n 1 Q S n (n,n n,ζ) 1 + Q, n n where (718) (719) Q 1 := Q := n n n n ( g n1 (ω) 1) g n (ω) g n (ω) n 1 S n (n,n n,ζ) 1 g n (ω) g n (ω) n 1 S n (n,n n,ζ) We estimate first Q We rewrite, (70) Q n n n,n S (n,n,n,n,ζ) 1 1 n 1 n 1 g n n (ω)g n (ω) We now proceed as in the argument presented in (716) above We use (4) with λ = δ r F (ω) L, Lemma and then use (6), to obtain that for ω outside a set of measure e 1 δ r one has, (70) δ r 1 1 n 1 n n (71) n n n,n δ r N1 6 N 6 δ r N 6 1 N 6 N 1 S (n,n,n,n,ζ) 1 n n n,n S (n,n,n,n,ζ) n n n,n S (n,n,n,n,ζ) δ r N 6 1 N 6 N 1 S (ζ) δ r N 4 1 N

24 AS WELL-POSEDNESS FOR THE PERIODIC D QUINTIC NLS BELOW H 1 To estimate Q 1 we let (7) S (n,n,n 1,n,n,ζ) := n : n = n 1 +n n, n = n 1 +n n, n 1 n,n,n,n, n C, ζ = n 1 + n, ζ = n 1 + n and note that its cardinality is 1 since n is determined for fixed (n,n,n 1,n,n ) We have, Q 1 n n n 1 n,n,n,n n S (n,n,n 1,n,n ) 1 1 n 1 n 1 ( g n n1 (ω) 1)g n (ω)g n (ω) Proceeding like above, we obtain in this case that for ω outside a set of measure e 1 δ r, Q 1 δ r N 6 1 N 6 S (ζ) δ r N 5 1 N, which is a better estimate Hence all in all we obtain in this case that (7) M δ r N 4 1 N Case β : In this case we define first n = n 1 +n n, n = n 1 +n n, S (n,n,n 1,n 1,ζ) = (n,n ) : n,n,n n 1,n 1, n C, ζ = n 1 + n, ζ = n 1 + n with S (n,n,n 1,n 1,ζ) N by (6) since n is determined by n and these ones lies on a sphere of radius at most N 1 intersection a ball of radius N If now we define n = n 1 +n n, n = n 1 +n n, S (ζ) = (n,n,n,n 1,n 1,n ) : n,n,n n 1,n 1, n C, ζ = n 1 + n, ζ = n 1 + n then S (ζ) N1 N N, since again n ranges in a cube of size N and we use (61) to count n 1 and n 1 We follow the argument used above in (717)-(7) to bound M but now with the couple (n 1,n 1 ) instead and corresponding sums Q 1 and Q Just as in Case β above, the bound for Q is larger We then obtain for ω outside a set of measure e 1 δ r, M δ r N1 6 N 6 S (n,n,n 1,n 1,ζ) δ r N 6 1 N 6 N n n n 1,n 1 n n n 1,n 1 S (n,n,n 1,n 1,ζ) δ r N 6 1 N 6 N S (ζ) δ r N 4 1 N 1 N Case β 4 : In this case note that N 1 N N We define the two sets First n = n 1 +n n, n = n +n n, S (n,n,n 1,n,ζ) = (n,n ) : n,n,n,n n 1, n C ζ = n 1 + n, ζ = n + n

25 4 NAHMOD AND STAFFILANI and since n lives on a sphereof radius at most N 1, from (61) we have S (n,n,n 1,n,ζ) N 1 and then n = n 1 +n n, n = n +n n, S (ζ) = (n,n,n,n 1,n,n ) : n,n,n,n n 1, n C, ζ = n 1 + n, ζ = n + n with S (ζ) N 1 N N6 Just as in case β and following the argument in (717)-(7) but with the couple (n 1,n 1 ) instead we obtain that for ω outside a set of measure e δ r, M δ r N1 6 N 6 S (n,n,n 1,n,ζ) δ r N 6 1 N 6 N 1 n n n 1,n n n n 1,n S (n,n,n 1,n,ζ) δ r N 6 1 N 6 N 1 S (ζ) δ r N 4 1 N Case β 5 : By symmetry this case is exactly the same as Case β 4 We are now ready to put all the estimates above together and bound T in cases β 1 β 5 : Case β 6 : In this case { S (n,n,ζ) = T a n l N 1 N GG a n l N 1 N (M 1 +M ) a n l δ r N 1 N N 1 N δ r N 5 N 1 1 a n l (n,n 1,n ) : n = n 1 +n n, n = n +n n 1, n 1 n,n,n, n 1 = n, n C At this point notice that the summation on ζ is eliminated and that in this case N 1 N N We have S (n,n,ζ) N 4 1 Using (6) we have, for ω outside a set of measure e δ r, that M = σ n,nσ n,n g n1 (ω) g n (ω) (74) n n n n C n n 1 n S (n,n,ζ) N1 6+ε N 6 S (n,n,ζ) N1 6+ε N 6 N4 S (ζ), where S (ζ) = n n { (n,n,n,n 1,n ) : n = n 1 +n n, n } = n +n n 1, n 1 n,n,n n 1 = n, n C and S (ζ) N N4 Hence M N +ε 1 N 5 and as a consequence T a n l N +ε 1 N 5 We now notice that to prove (74) we first have to consider the case when n 1 = n, which here it is not excluded, and then we can use exactly the same argument as above since a plus or minus sign in front of n does not change any of the counting }

26 AS WELL-POSEDNESS FOR THE PERIODIC D QUINTIC NLS BELOW H 1 5 Consider now (74) with n 1 = n Note that N 1 N N We now have (75) T := (g n1 (ω)) m Z,n Z n n= n 1 +n 1 a n m= n 1 + n Let S (m,n) = {(n 1,n )/n = n 1 +n, m = n 1 + n }, and note that S (m,n) N 1 Then g T N (ω) 4 n1 1 n m,n 1 6 a n g N n1 (ω) 4 1 n 1 6 a n+n1 N1 +ε a n l, S (m,n) n,n 1 Z where we use (6) for ω outside a set of measure e 1 δ r Proposition 74 Let D j and R k be as above and fix N 1 N N, r, δ > 0 and C C N Then there exists µ > 0 and a set Ω δ A such that P(Ω c δ ) e 1 δ r such that for any ω Ω δ we have (75) and (76) Proof We start by estimating (75) where without any loss of generality we assume that D 1 = D 1 We now have, (76) T := g χ C (n 1 )a n (ω) g n (ω) n1 m Z,n Z n=n 1 n +n n n n n m= n 1 n + n We are going to use duality and change of variables with ζ := m n 1 = n + n again Note though that if n 1 is in a cube C of size N then also n will be in a cube C of approximately the same size Then just as in (715) we need to estimate χ C a n1 l γ l σ n1,nχ C(n)k n, where σ n1,n = n 1 ζ N n n 1 =n +n n,n n 1,n ζ= n + n g n (ω) g n (ω) n n If we denote by G the matrix of entries σ n1,n, and we recall that the variation in ζ is at most N, we are then reduced to estimating n χ C a n1 l N GG We note that by Lemma 6, GG max σ n1,n + n 1 n 1 n 1 n C σ n1,nσ n 1,n 1 =: M 1 +M,

27 6 NAHMOD AND STAFFILANI where C is a cube of side length approximately N From this point on the proof is similar to the one already provided for (7) where n is replaced by n 1 We still go through the argument though, since the size of n 1 and n are different To estimate M 1 we first define the set S (ζ,n1 ) = {(n,n,n ) : n n 1,n, n = n 1 n+n, ζ = n + n } Applying (61) for each fixed n, we have that S (ζ,n1 ) N N since n sits on a sphere of radius approximately N Then we proceed as in (716) to obtain for ω outside a set of measure e 1 δ r, the bound To estimate M we first write M = where n 1 n 1 n C M 1 δ r N N N N δ r N σ n1,nσ n 1,n S (n1,n 1,ζ) = (n,n,n,n,n ) : n 1 n 1 g (ω) n g n (ω) g n (ω) g n (ω) S n (n1,n n n 1,ζ) n n = n 1 n+n, n = n 1 n+n, n n 1,n, n n 1,n, n C ζ = n + n, ζ = n + n We organize once again the estimates according to whether some frequencies are the same or not As before, all in all we have six cases: Case β 1 : n,n,n,n are all different Case β : n = n ; n n Case β : n n ; n = n Case β 4 : n n ; n = n Case β 5 : n = n ; n n Case β 6 : n = n ; n = n Case β 1 : We define the set S (ζ) = (n 1,n 1,n,n,n,n,n ) : n = n 1 n+n, n = n 1 n+n n n 1,n, n n 1,n, n 1,n 1 C ζ = n + n, ζ = n + n and note that S (ζ) N N6 N by Lemma 61 since for n fixed, n and n sit on sphere of radius N and n C a cube of side length approximately N Hence, for ω outside a set of measure e 1 δ r, we obtain in this case, M δ 4r N 6 N 6 N N 6 N δ 4r N 1 Case β : In this case we define two sets We start with n = n 1 n+n, n = n 1 n+n, S (n1,n 1,n,n,ζ) = (n,n ) : n n 1,n 1,n,n, n C ζ = n + n, ζ = n + n

28 AS WELL-POSEDNESS FOR THE PERIODIC D QUINTIC NLS BELOW H 1 7 To compute S (n1,n 1,n,n,ζ), it is enough to count n, then n is determined Since n sits on a sphere of radius N we have by (61) that S (n1,n 1,n,n,ζ) N Then we set S (ζ) = (n 1,n 1,n,n,n,n ) : n = n 1 n+n, n = n 1 n+n, n n 1,n 1,n, n, n C ζ = n + n, ζ = n + n for which S (ζ) N N 6N, where we used again that n C Arguing as in (717)- (7), we then have for ω outside a set of measure e 1 δ r that M δ r N 6 N 6 S (n1,n 1,n,n,ζ) δ r N 6 N 6 N n n n,n n 1 n 1 n,n S (n1,n 1,n,n,ζ) δ r N 6 N 6 N S (ζ) δ r N 1 Case β : In this case we define first n = n 1 n+n, n = n 1 n+n, S (n,n,n 1,n 1,ζ) = (n,n ) : n,n n,n 1,n 1, n C ζ = n + n, ζ = n + n for which S (n,n,n 1,n 1,ζ) N since n is determined by n and this one lies on a sphere of radius at most N 1 intersection a ball of radius N (see Lemma 61) Then we define S (ζ) = (n,n,n,n 1,n 1,n ) : n = n 1 n+n, n = n 1 n+n, n,n n,n 1,n 1, n C ζ = n + n, ζ = n + n for which S (ζ) N N N, since again n ranges in a cube of size N We then have, as usual using (4) and (6) as above that for ω outside a set of measure e 1 δ r, M δ r N 6 N 6 S (n,n,n 1,n 1,ζ) δ r N 6 N 6 N n 1 n 1 n,n n 1 n 1 n,n S (n,n,n 1,n 1,ζ) δ r N 6+ε N 6 N S (ζ) δ r N 1+ε N 1 Case β 4 : In this case note that N N We define the two sets n = n 1 n+n, n = n 1 n+n, S (n1,n 1,n,n,ζ) = (n,n ) : n n 1,n ;n n,n 1, n n C ζ = n + n, ζ = n + n