Gibbs measure for the periodic derivative nonlinear Schrödinger equation

Transcription

1 Gibbs measure for the periodic derivative nonlinear Schrödinger equation Laurent homann, Nikolay zvetkov o cite this version: Laurent homann, Nikolay zvetkov. Gibbs measure for the periodic derivative nonlinear Schrödinger equation. 5 pages <hal v1> HAL Id: hal Submitted on 4 Jan 010 v1, last revised Jun 017 v4 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. he documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 GIBBS MEASURE FOR HE PERIODIC DERIVAIVE NONLINEAR SCHRÖDINGER EQUAION by Laurent homann & Nikolay zvetkov Abstract. In this paper we construct a Gibbs measure for the derivative Schrödinger equation on the circle. he construction uses some renormalisations of Gaussian series and Wiener chaos estimates, ideas which have already been used by the second author in a work on the Benjamin-Ono equation. 1. Introduction Denote by = R/πZ the circle. he purpose of this work is to construct a Gibbs measure associated to the derivative nonlinear Schrödinger equation { i t u+ xu = i x u u, t,x R, 1.1 u0,x = u 0 x. Many recent results see the end of Section 1. show that a Gibbs measure is an efficient tool to construct global rough solutions of nonlinear dispersive equations. his is the main motivation of this paper: we hope that our result combined with a local existence theory for 1.1 e.g. a result like Grünrock- Herr [5] on the support of the measure will give a global existence result for irregular initial conditions. A second motivation is the fact that the Gibbs measure is an object which fits well in the study of recurrence properties of the flow of Mathematics Subject Classification. 35BXX ; 37K05 ; 37L50 ; 35Q55. Key words and phrases. Nonlinear Schrödinger equation, random data, Gibbs measure. he authors were supported in part by the grant ANR-07-BLAN-050.

3 LAUREN HOMANN & NIKOLAY ZVEKOV π For f L, denote by fxdx = 1 fxdx. he following quantities are conserved at least formally by the flow of the equation π 0 he mass Mut = ut L = u 0 L = Mu 0. he energy Hut = x u dx+ 3 Im u u x udx+ 1 u 6 dx = x u dx 3 4 Im u x u dx+ 1 u 6 dx = x u dx+ 3 4 i u x u dx+ 1 u 6 dx = Hu 0. he conservation of the energy can be seen by a direct computation, or using the Hamiltonian formulation of the equation 1.1 see the appendix of this paper. Notice that the momentum Put = 1 u 4 dx+i u x udx = 1 u 4 dx Im u x udx = Pu 0, is also formally conserved by 1.1, but the Gibbs measure won t depend on this quantity. In order to avoid the 0 frequency, in the sequel we will work on the modified DNLS equation 1. { i t v + x 1v = i x v v, t,x R, v0,x = u 0 x. Indeed, if the function u is a solution of the equation 1.1 if and only if v = e it u satisfies 1.. his transformation preserves the mass and the energy, and the Hamiltonian of the equation 1. is Hv = x v dx+ v dx 3 4 Im v x v dx+ 1 v 6 dx = Hv+M v.

4 GIBBS MEASURE FOR HE PERIODIC DNLS 3 Let us define the complex vector space E N = span e inx N n N. hen we introduce the spectral projector Π N on E N by 1.3 Π N c n e inx N = c n e inx. n Z n= N Let Ω,F,p beaprobability space and g n ω a sequence of independent n Z complex normalised gaussians, g n N C 0,1. We can write 1.4 g n ω = 1 hn ω+il n ω, where h n ω n Z, l n ω n Z N R 0,1. are independent standard real Gaussians 1.1. Definition of the Gibbs measure for 1.. We will show how to define a Gibbs measure for the equation 1.. In the sequel we will use the notation n = n +1. Now write c n = a n +ib n. For N 1, consider the probability measure on R N+1 defined by where d N is such that d N = N n= N dµ N = d N N n= N n a e n +b n da n db n R e n a n+b n da n db n, = π N+1 N n= N he measure µ N defines a measure on E N via the map a n,b n N n= N N n= N 1 N = π N n n n=1 a n +ib n e inx, which will still be denoted by µ N. hen µ N may be seen as the distribution of the E N valued random variable 1.6 ω g n ω n einx ϕ N ω,x, n N

5 4 LAUREN HOMANN & NIKOLAY ZVEKOV where g n N n= N are Gaussians as in 1.4. Let σ < 1. hen ϕ N is a Cauchy sequence in L Ω; H σ which defines 1.7 ϕω,x = n Z as the limit of ϕ N. Indeed, the map ω n Z g n ω n einx, g n ω n einx, defines a Gaussian measure on H σ which will be denoted by µ. For u L, we will write u N = Π N u. Now define f N u = Im u N x xu Nxdx. Let κ > 0, and let χ : R R, 0 χ 1 be a continuous function with supportsupp χ [ κ,κ] and so that χ = 1 on [ κ, κ ]. We define the measure ρ n on E N by 1.8 d ρ N u = χ 3 u N L e 4 f Nu 1 u Nx 6dx dµ N u. If A is a Borel set of H σ then A E N is a Borel set of E N. Now, we define ρ N which is the natural extension of ρ N to H σ, equipped with the Borel sigma algebra B. More precisely for every A B which is a Borel set of H σ, we set 1.9 ρ N A ρ N A E N. 1.. Statement of the main result. Our main result which defines the Gibbs measure for 1.1 reads heorem 1.1. he sequence 1.10 G N u = χ u N L 3 e 4 f Nu 1 u Nx 6dx, converges in measure, as N, with respect to the measure µ. Denote by Gu the limit of 1.10 as N. hen for every p [1, [, there exist κ 0 > 0 so that for all 0 < κ κ 0, Gu L p dµu and if we set dρu Gudµu then the sequence dρ N converges weakly to dρ as N tends to infinity.

6 GIBBS MEASURE FOR HE PERIODIC DNLS 5 More precisely, for every continuous and bounded function h : H σ C, one has hudρu = lim hudρ N u. N + H σ H σ One can show that by varying the cut-off χ, the support of ρ describes the support of µ see Lemma 4.3 below. he main ideas of this paper come from the work of the second author [9] where a similar construction is made for the Benjamin-Ono equation using the pioneering work of Bourgain []. In [9], one of the main difficulties is that on the support of the measure µ, the L norm is a.s. infinite, which is not the case in our setting, since for any σ < 1, ϕω Hσ, for almost all ω Ω. Here the difficulty is to treat the term u x u dx in the Hamiltonian. Roughly speaking, it should be controlled by the H 1 norm, but this is not enough, since u H 1 = on the support of dµ. However, we will see in Section, that we can handle this term thanks to an adapted decomposition and thanks to the integrability properties of the Gaussians. his is the main new idea in this paper. he result of heorem 1.1 may be the first step to obtain almost sure global well-posedness for 1.1, with initial conditions of the form 1.7. o reach such a result, we will also need a suitable local existence theory on the statistical set, and prove the invariance of the measure dρ under this flow. For instance, this program was fruitful for zvetkov [10, 11] for NLS on the disc, Burq-zvetkov [4] for the wave equation, Oh [6, 7] for Schrödinger- Benjamin-Ono and KdV systems, and Burq-homann-zvetkov [3] for the one-dimensional Schrödinger equation. For the DNLS equation, we plan to pursue this issue in a subsequent work Notations and structure of the paper. Notations. In this paper c, C denote constants the value of which may change from line to line. hese constants will always be universal, or uniformly bounded with respect to the other parameters. We denote by Z resp. N the set of the integers resp. non negative integers, and N = N\{0}. For x R, we write x = x +1. For u L, we usually write u N = Π N u, where Π N is the projector defined in 1.3. he notation L q stands for L q and H s = H s.

7 6 LAUREN HOMANN & NIKOLAY ZVEKOV he paper is organised as follows. In Section we give some large deviation bounds and some results on the Wiener chaos at any order. In Section 3 we study the term of the Hamiltonian containing the derivative, and Section 4 is devoted to the proof of heorem 1.1. In the appendix, we give the Hamiltonian formulation for the periodic DNLS equation.. Preliminaries : some stochastic estimates.1. Large deviation estimates. Lemma.1. Let γ n ω n Z N R0,1 be a sequence of independent, normalised real Gaussians. Let c n n Z, d n n Z be two bounded sequences of real numbers. hen there exist c,c > 0 so that for all 1 N λ p ω Ω : n 1, n N c n1 d n n 1 γ n 1 ωγ n ω > λ Ce cλ. Proof. We estimate A λ,n p ω Ω : n 1, n N c n1 d n n 1 γ n 1 ωγ n ω > λ. For all t > 0 and all r.v. X we have, by the chebychev inequality.1 p ω Ω : X > λ e λt E [ e tx ]. hus we obtain that for all ε > 0. [ A λ,n e tλ E n 1, n N [ e tλ E n 1, n N [ = e tλ E n 1 N e tcn 1 dn n 1 γ n1 γ n ] e ε c n 1 ] n 1 γ n ε t d n γn e ε c n 1 n 1 γ n 1 n N e 1 ε t d n γ n ].

8 GIBBS MEASURE FOR HE PERIODIC DNLS 7 Now, the Cauchy-Schwarz inequality and the independence of the γ n give.3 [ A λ,n e tλ E n 1 N = e tλ n 1 N E e ε c n 1 ]1 [ n 1 γ n 1 E [e ε c n 1 ] n 1 γ n 1 n N n N e t d n γ n /ε ]1 [ E e t d n γn /ε ]1. hanks to a change of variables we can compute explicitly the expectations in the right hand side of.3. In fact for µ < 1 [ ] 1 E e µγ n = 1 µ. For 0 x 1, we have the inequality 1 x 1 e x, hence we deduce that for µ < 1 4 [ ].4 E e µγ n e µ. Recall that c n,d n are bounded. We now fix ε > 0 so that for all n N, ε c n n 1. hen the bound.4 implies.5 n 1 N E [e ε c n 1 ] n 1 γ n 1 exp ε n 1 N c n 1 n 1 C. With the previous choice of ε > 0 and t > 0 small enough we also have [ ].6 E e t d n γn /ε exp t d n /ε e CtN, n N Finally, from.5,.6 and.3 we infer n N A λ,n Ce tλ+ctn Ce cλ, for some c > 0, if t > 0 is chosen small enough and N λ. Similarly for λ > 0, p ω Ω : n 1, n N and this yields the result. c n1 d n n 1 γ n 1 ωγ n ω < λ Ce cλ,

9 8 LAUREN HOMANN & NIKOLAY ZVEKOV Lemma.. Fix σ < 1 and p [,. hen C > 0, c > 0, λ 1, N 1, Moreover there exists β > 0 such that C > 0, c > 0, λ 1, M N 1, µ u H σ : Π N u L p > λ Ce cλ. µ u H σ : Π M u Π N u L p > λ Ce cnβ λ. Proof. his result is consequence of the hypercontractivity of the Gaussian random variables : here exists C > 0 such that for all r and c n l N n 0g n ωc n L r Ω C r c n 1. n 0 See e.g. [3, Lemma 3.3] for the details of the proof... Wiener chaos estimates. he aim of this subsection is to obtain L p Ω bounds on Gaussian series. hese are obtained thanks to the smoothing effects of the Ornstein-Uhlenbeck semi-group. he following considerations are inspired from [9]. See also [1, 8] for more details on this topic. For d 1, denote by L the operator L = x = d j=1 x j x j x j. his operator is self adjoint on K = L R d,e x / dx with domain { D = u : ux = e x /4 vx, v H }, whereh = { u L R d, x α x β vx L R d, α,β N d, α + β }. Denote by k = k 1 + +k d and by P n n 0 the Hermite polynomials defined by P n x = 1 n x dn e e x. dx n hen a Hilbertian basis of eigenfunctions of L on K is given by P k x 1,...,x d = P k1 x 1...P kd x d,

10 GIBBS MEASURE FOR HE PERIODIC DNLS 9 with eigenvalue k = k 1 + +k d. Finally define the measure γ d on R d by dγ d x = π d/ e x / dx. he next result is a direct consequence of [9, Proposition 3.1] Lemma.3. Let d 1 and k N. Assume that P k is an eigenfunction of L with eigenvalue k. hen for all p P k L p R d,dγ d p 1 k P k L R d,dγ d. hanks to Lemma.3, we will prove the following L p smoothing effect for some stochastic series. Proposition.4 Wiener chaos. Let d 1 and cn 1,...,n k C. Let g n 1 n d N C 0,1 be complex L - normalised independent Gaussians. For k 1 denote by Ak,d = { n 1,...,n k {1,...,d} k }, n 1 n k and.7 S k ω = cn 1,...,n k g n1 ω g nk ω. Ak,d hen for all d 1 and p S k L p Ω k+1p 1 k S k L Ω. Proof. Let g n N C 0,1. hen we can write g n = 1 γ n + i γ n with γ n, γ n N R 0,1 mutually independent Gaussians. Hence, up to a change of indexesandwithdreplacedwithd wecanassumethattherandomvariables in.7 are real valued. hus in the following we assume that g n N R 0,1 and are independent. Denote by Σ k x 1,...,x d = hen obviously for all p 1, Ak,d cn 1,...,n k x n1 x nk..8 S k L p Ω = Σ k L p R d,dγ d. Let n 1,...,n k Ak,d. hen we can write x n1 x nk = x p 1 m 1 x p l m l,

11 10 LAUREN HOMANN & NIKOLAY ZVEKOV where l k, p p l = k and n 1 = m 1 < < m l n k. Now, each monomial x p j m j can be expanded on the Hermite polynomials P n n 0 x p j m j = p j k j =0 α j,kj P kj x mj. herefore there exists βk 1,...,k l C so that and we have x n1 x nk = k j=0.9 Σ k x 1,...,x d = where the polynomial P j is given by P j x 1,...,x d = Ak,d k 1 + +k l =j 0 k i p i βk 1,...,k l P k1 x m1 P kl x ml, k P j x 1,...,x d, j=0 k 1 + +k l =j 0 k i p i cn 1,...,n k βk 1,...,k l P k1 x m1 P kl x ml. For 0 k i p i so that k 1 + +k l = j, the polynomial P j is an eigenfunction of L with eigenvalue j, hence by Lemma.3 we have that for all p P j L p R d,dγ d p 1 j P j L R d,dγ d. herefore, by.9 and by the Cauchy-Schwarz inequality, Σ k L p R d,dγ d p 1 k k P j L R d,dγ d j=0 k+1p 1 k k P j L R d,dγ d j=0 k+1p 1 k Σ k L R d,dγ d, where in the last line we used that the polynomials P j are orthogonal. his concludes the proof by.8 We will need the following lemma which is proved in [9, Lemma 4.5] 1

12 GIBBS MEASURE FOR HE PERIODIC DNLS 11 Lemma.5. Let F : H σ R be a measurable function. Assume that there exist α > 0,N > 0,k 1, and C > 0 so that for every p F L p dµ CN α p k. hen there exist δ > 0,C 1 independent of N and α such that e δn α k Fu k dµu C 1. H σ As a consequence, for all λ > 0, µ u H σ : Fu > λ C 1 e δn α k λ k. 3. Study of the sequence f N u N 1 Recall that f N u is defined by f N u = Im he main result of this section is the following u N x xu Nxdx. Proposition 3.1. he sequence f N N 1 is a Cauchy sequence in L H σ,b,dµ. Indeed for all 0 < ε < 1 for all M > N f M u f N u C L H σ,b,dµ N 3 ε. Moreover, for all p and M > N 1 3. f M u f N u Cp 1 L p. H σ,b,dµ N 3 ε there exists C > 0 so that hen a combination of the estimate 3. and Lemma.5 yields the following large deviation estimate Corollary 3.. For every α < 3, there exist C,δ > 0 such that for all M > N 1 and λ > 0 µ u H σ : f M u f N u > λ Ce δnα λ 1. hanks to Proposition 3.1, we are able to define the limit in L of the sequence f N, which is N 1 fu = Im u x x u xdx,

13 1 LAUREN HOMANN & NIKOLAY ZVEKOV Notice that Corollary 3. implies in particular the convergence in measure 3.3 ε > 0, lim µ u H σ : f N u fu > ε = 0. N For the proof of Proposition 3.1, we have to put ϕ N ω xϕ N ωdx in a suitable form. Recall the notation 1.6, then 3.4 ϕ Nω = herefore we deduce that 3.5 x ϕ N ω = n 1, n N m 1, m N g n1 ωg n ω e in 1+n x. n 1 n im 1 +m g m 1 ω g m ω e im 1+m x. m 1 m Now, by 3.4, 3.5 and the fact that e inx is an orthonormal family in L endowed with the scalar product f,g = fxgxdx n Z = 1 π 3.6 π 0 where fxgxdx, we obtain ϕ N ω xϕ Nωdx = in 1 +n g m 1 ω g m ω g n1 ω g n ω, m 1 m n 1 n A N A N = {m 1,m,n 1,n Z 4 s.t. m 1, m, n 1, n N and m 1 +m = n 1 +n }. We now split the sum 3.6 in two parts, by distinguishing the cases m 1 = n 1 and m 1 n 1 in A N and write ϕ N ω xϕ Nωdx = SN 1 +SN, with 3.7 SN 1 = in 1 +n g m 1 ω g m ω g n1 ω g n ω, m 1 m n 1 n B N where B N = A N {m 1 = n 1 or m 1 = n }, and 3.8 S N = in 1 +n A N,m 1 n 1 m 1 n g m1 ω g m ω g n1 ω g n ω. m 1 m n 1 n

14 GIBBS MEASURE FOR HE PERIODIC DNLS Study of S 1 N. Lemma 3.3. Let SN 1 for all M > N > 0, be defined by 3.7. hen there exists C > 0 so that SM 1 SN 1 L Ω C. N 3 Proof. Let m 1,m,n 1,n B N. hen as m 1 + m = n 1 + n, we have m 1,m = n 1,n or m 1,m = n,n 1, and deduce that where and S 1 N = n 1, n N Y N = in 1 +n g n 1 ω g n ω n 1 n = X N +Y N, n 1, n N, n 1 n X N = n N 4in g nω 4 n 4, in 1 +n g n 1 ω g n ω n 1 n. First we will show that there exists C > 0 so that for all M > N > 0, 3.9 X M X N L Ω C N. Let M > N 1. hen hus X M X N = which proves 3.9. N< n 1, n M X M X N L Ω C 16n 1 n g n1 ω 4 g n ω 4 n 1 4 n 4. N< n 1, n M 1 n 1 3 n 3 C N 4, o complete the proof of Lemma 3.3, it remains to check that there exists C > 0 so that for all M > N > 0, 3.10 Y M Y N L Ω C. N 3

15 14 LAUREN HOMANN & NIKOLAY ZVEKOV For M N 1 we write with Y N = 3.11 Y 1 N = n 1, n N, n 1 n = Y 1 N +Y N +Y3 N, n 1, n N, n 1 n 3.1 Y N = n 1, n N, n 1 n in 1 +n g n 1 ω g n ω n 1 n in 1 +n in 1 +n gn1 ω 1 g n ω 1 n 1 n, gn1 ω 1 + g n ω 1 n 1 n, and Y 3 N = n 1, n N, n 1 n 1 4in 1 +n n 1 n. By the symmetry n 1,n n 1, n, we have that YN 3 denote by = 0. For n Z, 3.13 G n ω = g n ω 1. By the definition 1.4 of g n ω, we infer that for n m, 3.14 E [ G n ωg m ω ] = 0 First we analyse We compute YM 1 YN 1 = 4n 1 +n m 1 +m G ωg ω G ωg ω m1 m n1 n m 1 m n 1 n, C M,N where C M,N = { m 1,m,n 1,n Z 4 s.t. N < m 1, m, n 1, n M and m 1 m, n 1 n }. We compute E [ Y 1 M Y1 N ], and thanks to 3.14 we see that only the terms n 1 = m 1 and n = m or n 1 = m and n = m 1 give some contribution,

16 GIBBS MEASURE FOR HE PERIODIC DNLS 15 hence Y 1 M Y1 N L Ω C 3.15 C N< n 1, n M N< n 1, n M n 1 +n n 1 4 n 4 1 n 1 n n 1 4 n C N 4. We now turn to 3.1. Similarly, we get YM Y N = Gm1 ω+g m ω G n1 ω+g n ω 4n 1 +n m 1 +m m 1 m n 1 n, C M,N and using the symmetries in n 1,n,m 1,m, and with 3.14 we obtain 3.16 Y M Y N L Ω C Observe that we have 3.17 with s M,N = 3.18 N< m M, m n 1 N< n M, n n 1 N< n M C M,N, m 1 =n 1 n 1 +n n 1 +m m n 1 4 n = = N< n M n 1 +n n 1 +m m n 1 4 n. n 1 n 1 +m m n 1 4 n n 1n 1 +m m n 1 6 n 1 n 1 +m = s M,N m n 1 4 n 1n 1 +m m n 1 6 = n 1 n 1 4 sm,n n 1 n 1 +m m, 1 n C N. Similarly n 1 +m n 1 +m m = m n 1 n 1 = n 1s M,N n 1 n 1. N< m M hen, from 3.17 and 3.18, we deduce N< m M, m n 1 N< n M, n n 1 n 1 +n n 1 +m m n 1 4 n C 1 N n n 1 6,

17 16 LAUREN HOMANN & NIKOLAY ZVEKOV and from 3.16, 3.19 Y M Y N L Ω C N 3. Finally, 3.15 and 3.19 yield the estimate Study of S N. We first state the elementary lemma Lemma 3.4. Let n Z and N 1. hen for all 0 < ε 1 1 n 1 n n 1 C N 3 ε n 3 +ε. n 1 Z n 1, n n 1 N Proof. Let N 1. For α > 1 we have the inequalities and n α C n 1 α + n n 1 α, n 1 n n 1 CN 4 α n 1 α n n 1 α for n 1, n n 1 N. Now choose α = 3 +ε to get 3.0 We sum up 3.0, thus n 1 Z n 1, n n 1 N C N 1 ε n 3 +ε which was the claim. n 3 +ε n 1 n n 1 C N 1 ε 1 n 1 3 +ε + 1 n n 1 3 +ε. 1 n 1 n n 1 n 1 Z n 1, n n 1 N We are now able to prove 1 n 1 3 +ε + 1 n n 1 3 +ε C N 3 ε n 3 +ε, Lemma 3.5. Let SN be defined by 3.8. For all 0 < ε 1, there exists C > 0 so that for all M > N > 0, S M S N L Ω C N 3 ε.

18 GIBBS MEASURE FOR HE PERIODIC DNLS 17 Proof. We compute S M SN g m1 g m g n1 g n g p1 g p g q1 g q = n 1 +n p 1 +p m 1 m n 1 n p 1 p q 1 q, D M,N D M,N where D M,N = { m 1,m,n 1,n Z 4 s.t. N < m 1, m, n 1, n M and m 1 +m = n 1 +n, m 1 n 1, m 1 n }. heexpectation of each termof theprevioussumvanishes, unless m 1,m = q 1,q or q,q 1 and n 1,n = p 1,p or p,p 1. Hence SM SN L Ω C n 1 +n m 1 +m. n 1 n m 1 m D M,N Write n = n 1 +n = m 1 +m, therefore SM SN L Ω C n Z by Lemma 3.4. = C n Z n 1, n n 1 >N, m 1, n m 1 >N n n 1, n n 1 >N C n N 3 ε n 3+ε C N 3 ε, n Z n n 1 n n 1 m 1 n m 1 1 n 1 n n 1 he results of Lemmas 3.3 and 3.5 imply 3.1. o complete the proof of Proposition 3.1, it remains to show 3.. But this is a direct consequence of 3.1 and Proposition.4 We are now able to define the density G : H σ R with respect to the measure µ of the measure ρ. By 3.3 and Proposition 3.1 and., we have the following convergences in the µ measure : f N u converges in to fu and u N L 6 to u L 6. hen, by composition and multiplication of continuous functions, we obtain 3.1 χ 3 u N L e 4 f Nu 1 u Nx 6dx χ 3 u L e 4 fu 1 ux 6dx Gu, in measure, with respect to the measure µ. As a consequence, G is measurable from H σ,b to R.

19 18 LAUREN HOMANN & NIKOLAY ZVEKOV 4. Integrability of the density of dρ We now state a result which will be useful for the L p estimates in heorem 1.1. Proposition 4.1. here exist κ 0 > 0 and c, C > 0 so that for all 0 < κ κ 0, λ and 1 N λ µ u H σ : x u N L > λ, u N L κ Ce cλ. Proof. We can follow the mains lines of the proof of [9, Proposition 4.1]. For j {0,,[λ 5 ]}, we define the points x j by x j = πj λ 5. Denote by dist the distance on. hen by construction, distx j,x j+1 π λ 5, with x [λ 5 ]+1 x 0. We define the set A λ by A λ { u H σ : x u N L λ, u N L κ and the sets A λ,j by { A λ,j u H σ : x u N xj λ }, u N L κ. As in [9] we will show that 4.1 A λ [λ 5 ] j=0 A λ,j. Let u A λ, and denote by v N = x u N. Let x be such that v N x = max x v Nx. hus v N x λ. hen there exists j 0 {0,,[λ 5 ]} such that 4. x x j0 π λ 5. hen thanks to the aylor formula, we have x v N x u N x j0 = v N tx +1 tx j0 dt x j0 4.3 x x j0 1 v N L. Now by the Sobolev embeddings we obtain the bound with N λ v N L CN v N L CN u N L x u N L }, 4.4 CN 5 u N L Cλ5 κ.

20 GIBBS MEASURE FOR HE PERIODIC DNLS 19 herefore, from 4., 4.3 and 4.4 we deduce that for κ > 0 small enough hus, by the triangle inequality v N x u N x j0 Cκ 1 λ, v N x j0 v N x v N x v N x j0 λ 1 λ = 1 λ, we can conclude that u A λ,j0, which proves 4.1. We now estimate µa λ,j. As in [9], we can forget the L constraint and write µa λ,j p ω Ω : x ϕ N xj λ. First observe that { ω Ω : x ϕ N xj λ } { ω Ω : Re x ϕ N xj λ } { ω Ω : Im x ϕ 4 N xj λ }. 4 Indeed we can describe the previous sets by the following way. Write 1 x ϕ g n1 ωg n ω N xj = in e in 1+n x j, n 1 n n 1, n N and use that g n ω = 1 hn ω +il n ω where h n n Z, l n n Z N R0,1 are independent. hen a straightforward computation enables us to put Re x ϕ N xj and Im x ϕ N xj in the form n 1, n N c n1 d n n 1 γ n 1 ωγ n ω, with c n, d n C and where γ n n Z N R0,1 is an independent family of real Gaussians indeed γ n = h n or γ n = l n. herefore we can apply the Lemma.1 to get 4.5 µa λ,j Ce cλ. Finally by 4.1 and 4.5 we deduce that which was the claim. [λ 5 ] µa λ µa λ,j Cλ 5 e cλ Ce c λ, j=0

21 0 LAUREN HOMANN & NIKOLAY ZVEKOV Proposition 4.. For all 1 p <, there exists κ 0 > 0 so that for all 0 < κ κ 0 there exists C > 0 such that for every N 1. χ 3 u N L e 4 f Nu 1 u Nx 6 dx C. L p dµu Proof. Here we can follow the proof of [9, Proposition 4.9]. o prove the proposition, it is sufficient to show that the integral λ p 1 µa λ,n dλ, is convergent uniformly with respect to N for κ > 0 small enough and where { A λ,n = u H σ : χ 3 u N L e 4 f Nu 1 } u Nx 6dx > λ. We set N 0 = lnλ. Assume that N 0 N. On the support of χ, u N L κ, thus we have f N u = Im u N x x u N x dx C u N L xu N L Cκ x u N L. hen by Proposition 4.1 which can be applied, since N N 0 = lnλ c 1 lnλ κ for κ > 0 small enough, we obtain µa λ,n µ u H σ : f N u > 4 3 lnλ, u N L κ µ u H σ : x u N L > c 1 κ lnλ, u N L κ Ce c κ lnλ = Cλ c κ, where c is independent of κ. Hence the integral 4.6 is convergent if κ = κp > 0 is small enough. Assume now N > N 0. hanks to the triangle inequality A λ,n B λ,n C λ,n, where { B λ,n u H σ : f N0 u > 1 } lnλ, u N L κ, and C λ,n { u H σ : f N u f N0 u > 1 } lnλ, u N L κ. he measure of B λ,n can be estimated exactly as we did in the analysis of the case N 0 N. Finally, by Corollary 3., as N 0 = lnλ, we obtain that for all

22 GIBBS MEASURE FOR HE PERIODIC DNLS 1 1 < α < 3 µc λ,n Ce δlnλ1+α C L λ L, for all L 1. his completes the proof of the proposition. Proof of heorem 1.1. Recall 3.1. Let p [1,+ and choose κ 0 > 0 so that Proposition 4. holds. hen there exists a subsequence G Nk u so that G Nk u Gu, µ a.s. hen by Fatou s lemma, Gu p dµu liminf G Nk u p dµu C, k H σ H σ thus Gu L p dµu. Now it remains to check that for any continuous bounded function f : H σ R we have 4.7 lim fug N udµu = fugudµu, N + H σ which will be implied by 4.8 lim N + H σ H σ fug N u Gu dµu = 0. As in [9], for N 0 and ε > 0, we introduce the set A N,ε = { u H σ : G N u Gu ε }, and denote by A N,ε its complementary. Firstly, as f is bounded, there exists C > 0 so that for all N 0, ε > 0 fu G N u Gu dµu Cε. A N,ε Secondly, by Cauchy-Schwarz, Proposition 4. and as Gu L p dµu, we obtain fu G N u Gu dµu G N Gu L dµµa N,ε 1 A N,ε By 3.1, we deduce that for all ε > 0, CµA N,ε 1. µa N,ε 0, N +, which yields 4.8. his ends the proof of heorem 1.1. Notice that 4.8 with f = 1 gives 4.9 lim N ρ NE N = ρ H σ.

23 LAUREN HOMANN & NIKOLAY ZVEKOV Lemma 4.3. he measure ρ is not trivial Proof. First observe that for all κ > 0 µ u H σ : u L κ = p ω Ω : n Z 1 n g nω κ > 0. hen, by Lemma. and Proposition 3.1, the quantities u L 6 and fu are µ almost surely finite. Hence, the density of ρ does not vanish on a set of positive µ measure. In other words, ρ is not trivial. A Hamiltonian structure of DNLS In this section we give the Hamiltonian structure of the equation 1.1 First we define the projection Π on the 0-mean functions : Π f = n Z\{0}α n e inx, for fx = α n e inx, n Z then we introduce the integral operator Notice that we have 1 : fx = n Z α n e inx 1 f = Πf = f n Z\{0} fxdx α n in einx. Next we define the operator u 1 u i+u 1 v Ku,v = i+v 1 u v 1 v. Lemma A.1. For u,v, the operator Ku,v is skew symmetric : Ku,v = Ku,v. Proof. his is a straightforward computation. We only have to use that 1 = 1.

24 GIBBS MEASURE FOR HE PERIODIC DNLS 3 Define Hu,v = x u x v i v x u + 1 u 3 v 3. Notice that we also have the expressions Hu,v = x uv i v x u + 1 u 3 v 3 = u xv 3 4 i u x v + 1 u 3 v 3, therefore, we can deduce the variational derivatives A.1 A. δh δu u,v = x v 3 iu xv + 3 u v 3 δh δv u,v = xu+ 3 iv xu + 3 u3 v. We consider the Hamiltonian system t u δh δu A.3 = Ku,v u,v t v δh δv u,v Denote by F u t = Im x u+ u 3 and notice that for all t R, F u t R u 4, Proposition A.. he system A.3 is a Hamiltonian formulation of the equation. A.4 i t u+ xu = i x u u +F u tu, in the coordinates u,v = u,u. As a consequence, if we set A.5 vt,x = e i t 0 Fusds ut,x, then v is the solution of the equation A.6 { i t v + xv = i x v v, t,x R, v0,x = u 0 x. Moreover, if u and v are linked by A.5, we have F u = F v.

25 4 LAUREN HOMANN & NIKOLAY ZVEKOV Proof. We have u xv = xuv +u x v x uv, therefore A.7 1 u x v = 1 x uv +u x v x uv u x v x u. Similarly we obtain the relation A.8 1 u x v = 1 v x u +u v u v. By A.1, A., using A.7 and A.8, a straightforward computation gives t u = u 1 u δh δh i δu δv +u 1 v δh δv = i x u+ x u v 1 π u u x v v x u 3 iu u v, and t v = i δh δu +v 1 u δh v 1 v δh δu δv = i x v + x uv 1 π v v x u u x v 3 iu u v. Now assume that v = u. his yields the result, as u x u u x u = iim u x u. References [1] C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, R. Cyril, and G. Scheffer. Sur les inégalités de Sobolev logarithmiques. Panoramas et Synthèses, 10. Société Mathématique de France, Paris, 000. [] J. Bourgain. Invariant measures for the D-defocussing nonlinear Schrödinger equation. Comm. Math. Phys., [3] N. Burq, L. homann and N. zvetkov. On the long time dynamics for the 1D NLS. Preprint. [4] N. Burq and N. zvetkov. Random data Cauchy theory for supercritical wave equations II: A global existence result. Invent. Math. 173, No. 3, [5] A. Grünrock and S. Herr. Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data. SIAM J. Math. Anal , no. 6, [6]. Oh. Invariance of the Gibbs measure for the Schrödinger-Benjamin-Ono system. SIAM J. Math. Anal., , no. 6, 07-5.

26 GIBBS MEASURE FOR HE PERIODIC DNLS 5 [7]. Oh. Invariant Gibbs measures and a.s. global well-posedness for coupled KdV systems. Diff. Integ. Eq., 009, no. 7-8, [8] M. Ledoux, and M. alagrand. Probability in Banach spaces. Springer-Verlag, Berlin, [9] N. zvetkov. Construction of a Gibbs measure associated to the periodic Benjamin-Ono equation. Probab. heory Related Fields., [10] N. zvetkov. Invariant measures for the defocusing NLS. Ann. Inst. Fourier, [11] N. zvetkov. Invariant measures for the Nonlinear Schrödinger equation on the disc. Dynamics of PDE Laurent homann, Université de Nantes, Laboratoire de Mathématiques J. Leray, UMR CNRS 669,, rue de la Houssinière, F- 443 Nantes Cedex 03, France. laurent.thomann@univ-nantes.fr Url : thomann/ Nikolay zvetkov, Département de Mathématiques, Université de Cergy-Pontoise, Site Saint-Martin, 9530 Cergy-Pontoise Cedex, France. nikolay.tzvetkov@u-cergy.fr