THE NONLINEAR SCHRÖDINGER EQUATION WITH WHITE NOISE DISPERSION

Transcription

1 THE NONLINEA SCHÖDINGE EQUATION WITH WHITE NOISE DISPESION ANNE DE BOUAD AND ANAUD DEBUSSCHE Abtract Under certain caling the nonlinear Schrödinger equation with random dierion converge to the nonlinear Schrödinger equation with white noie dierion The aim of thi work i to rove that thi latter equation i globally well oed in L or H The main ingredient i the generalization of the claical Strichartz etimate Additionally, we jutify rigorouly the formal limit decribed above March 9, Introduction The following nonlinear Schrödinger equation with random dierion decribe the roagation of a ignal in an otical fibre with dierion management ee [],[]: i dv dt + εmt v + ε v v =,, t >, v, = v, ecall that in the contet of fibre otic, correond to the retarded time while t correond to the ditance along the fibre The coefficient εmt account for the fact that ideally one would want a fibre with zero dierion, in order to avoid chromatic dierion of the ignal Thi i imoible to build in ractie and engineer have rooed to build fibre with a mall dierion which varie along the fibre and ha zero average The cae of a eriodic determinitic dierion ha been tudied in [] where an averaged equation i derived Thi averaged equation i then hown to oe ground tate ee [] for the cae of oitive reidual dierion, that i when mt ha oitive average over a eriod, and [5] for the cae of vanihing reidual dierion Note that in thi eriodic etting, the nonlinear term i not of ize ε a uch a nonlinear term would have no effect on the dynamic, the equation tudied in [] ha in fact the coefficient ε in front on the nonlinearity In thi article, we conider the cae of a random dierion, ie m i a centered tationary random roce A will be clear from our tudy, only a nonlinearity of order ε i relevant in thi contet In order to undertand the limit a the mall arameter ε goe to zero, it i natural to recale the time variable by etting ut, = v t ε and we obtain i du dt + ε m ε u + u u =,, t >, u = u, 99 Mathematic Subject Claification 35Q55, 6H5 Key word and hrae White noie dierion, Strichartz etimate, tochatic artial differential equation, nonlinear fiber otic

2 A DE BOUAD AND A DEBUSSCHE Thi model ha been initially tudied in [7] where a lit te numerical cheme i rooed to imulate it olution Under claical ergodic aumtion on m, it i eected that the limiting model when ε goe to zero i the following tochatic nonlinear Schrödinger equation with white noie dierion idu + σ 3 u dβ + u u =,, t >, u = u,, where β i a tandard real valued Brownian motion, σ = + E[mmt]dt, and i the Stratonovich roduct In [7], the cubic nonlinearity i relaced by a nicer Lichitz function o that the limiting equation can be eaily tudied uing the fact that the evolution aociated to the linear equation define an iometry in all L baed Sobolev ace It i hown that the nonlinear Schrödinger equation with white noie dierion i indeed the limit of the original roblem and thi reult i ued to rove that ome numerical cheme roduce good aroimation reult for a time te ignificantly higher than ε Again, all thi tudy i erformed for an equation where a nice Lichitz function relace the ower nonlinearity Our aim i to addre the original equation with ower nonlinearity In fact, we tudy the more general equation for σ > : idu + σ u dβ + u σ u dt =, d, t >, u = u, d Note that the ign in front of the nonlinear term u u i not imortant here, a it can be changed from + to by changing β to β and u to it comle conjugate Alo, we will aume without lo of generality that σ = We recall that the uual nonlinear Schrödinger equation i t u + u + u 4 σ u =, d, t >, u = u, d reerve the Hamiltonian Hu = u d u σ+ d d σ + d However, the varying dierion detroy the Hamiltonian character of the equation On the mathematical oint of view, thi imlie the lo of the a riori etimate rovided by the energy H and no a riori etimate in H are available On the contrary, the ma, equal to the quare of the L norm i till reerved Thu, a L theory i neceary to get global olution For equation 4, uch a theory i oible thank to Strichartz etimate which imly ultracontractivity of the linear grou ee [4], [3], [4], [] We rove in Section 3 that Strichartz etimate can be generalized to the equation with white noie dierion Thi allow to contruct local in time olution for σ < /d, in L or H, in ection 4 Then in ection 5, the conervation of the ma i ued to rove global eitence Alo, we rove that regularity i reerved o that if the initial tate i H, then the L global olution i a continuou in time with value in H Finally, in ection 6, we how that Marty method to rove convergence of olution when ε goe to zero i eaily generalized once the reviou reult are obtained In [7] ome numerical imulation are given It would be very intereting to do a more general and ytematic numerical tudy on the equation conidered here For intance, the

3 WHITE NOISE DISPESION FO NLS 3 influence of the random dierion on blow-u henomena could be invetigated ee [7], [8], [] for uch a tudy with different noie, eventhough thi henomenon i not reent in fibre otic We finally note that all the reult tated in Section would till hold with a nonzero but mall reidual dierion, ie if equation i relaced by i dv dt + εmt v + ε ν v + ε v v =,, t >, v, = v,, where ν i a contant In thi cae, of coure, the limit equation 3 hould be relaced by idu + σ 5 u dβ + ν u + u u =,, t >, u = u,, All the analyi made in the reent aer alie to the above equation, the only difference being in the roof of Prooition 33 ee emark 35 However, the tudy of the comlete model where reidual, eriodic and random dierion are taken into account i more delicate, and will be the object of further tudie We refer to [] for reult on the comlete model, uing the hyicit collective coordinate aroach Preliminarie and main reult We conider the following tochatic nonlinear Schrödinger NLS equation idu + u dβ + u σ u dt =, d, t >, u = u, d, where the unknown u i a random roce on a robability ace Ω, F, P deending on t > and d The nonlinear term i a ower law The noie term involve a brownian motion β aociated to a tochatic bai Ω, F, P, F t t The roduct i a Stratonovich roduct A uual, we do not conider thi equation but it formally equivalent Itô form: idu + i u dt + u dβ + u σ u dt =, d, t >, u = u Note that, formally, the L d norm of a olution i a conerved quantity However, the time deendent dierion detroy the Hamiltonian character of the claical Nonlinear Schrödinger equation and there doe not eit an energy here We tudy thi equation in the framework of the L d baed Sobolev ace We alo ue the ace L d to treat the nonlinear term thank the Strichartz etimate In order to lighten the reentation, we ue the following notation H = H d, L = L d,, and, when the time interval I doe not need to be ecified or i obviou from the contet: L r t L = L r I; L d, r, Note that, in all the article, thee are ace of comle valued function The norm of a Banach ace K i imly denoted by K When we conider moment with reect to the

4 4 A DE BOUAD AND A DEBUSSCHE random arameter ω Ω, we ometime write L ωk = L Ω; K, For ace of redictible time deendent rocee, we ue the ubcrit P For intance L r P Ω; L, T ; K i the ubace of L r Ω; L, T ; K coniting of redictible rocee We will denote aociate conjugate eonent uing rime uercrit, that i if, then i uch that + = Our firt main reult i the following Theorem Aume σ < d ; let u L abe F -meaurable, then there eit a unique olution u to with ath a in L r loc, ; L d, with = σ + r < 4σ+ dσ ; moreover, u ha ath in C + ; L, a and ut L = u L, u alo ha the additional integrability roertie : u L ρ loc, + ; L a for any ρ < 4 if d = u L ρ loc, + ; Lq d a for any ρ, q with q < d If in addition u H, then u ha ath a in C + ; H a d d, and ρ < 4q dq if emark In the cae d σ d or d σ < + if d = or, it i oible to rove a local eitence reult of olution with ath a in C[, τ]; H rovided u H, uing imilar argument a thoe ued in the reent aer, but with a cut-off at fied time in L σ+ norm ee Section 4 for the neceity of the ue of a cut-off However, becaue no energy conervation i available for equation, only in the cae σ < /d global eitence may be obtained, thank to the conervation of L norm and Strichartz etimate The reult of Theorem i ued to jutify rigorouly the convergence of the olution of the random equation to the olution of with σ =, d = In order to tate the reult reciely, we aume the following Aumtion The real valued centered tationary random roce mt i continou and uch that for any T >, the roce t ε /ε md converge in ditribution to a tandard real valued Brownian motion in C[, T ] Let u recall claical condition on m enuring that the above Aumtion i atified Thi hold eg if m i a Markov roce with a unique and ergodic invariant meaure and it generator atifie the Fredholm alternative; for intance, m can atify Doeblin condition Aumtion alo hold under ome miing condition on m We refer to [3], [], [6], [8] and [9] for more general and recie condition To our knowledge, Strichartz etimate are not available for equation Hence we cannot get olution in L Since the equation i et in ace dimenion, a local eitence eitence reult can be eaily roved in H but ince no energy i available we do not know if the olution are global in time In the following reult, we rove that the lifetime of the olution converge to infinity when ε goe to zero, and that olution of converge in ditribution to the olution of the white noie driven equation

5 WHITE NOISE DISPESION FO NLS 5 Theorem 3 Suoe that m atifie the above aumtion Then, for any ε > and u H, there eit a unique olution u ε of equation with continuou ath in H which i defined on a random interval [, τ ε u Moreover, for any T > lim Pτ εu T =, ε and the roce u ε l [τε>t ] converge in ditribution to the olution u of in C[, T ]; H for any < 3 The linear equation and Strichartz tye etimate It i imortant to undertand the roertie of the linear art of equation Indeed, in the cae of the determinitic NLS equation, the linear art oee ultracontractivity roertie which are etremely helful to olve the nonlinear equation ee for intance [4] We ue below thi equation tarting from an initial data at variou initial time We therefore conider in thi ection the following tochatic linear Schrödinger equation: idu + u dβ =, t, 3 u = u We interret thi equation in the Itô ene and conider the following equation which i formally equivalent to 3: idu + i 3 u dt + u dβ =, t, u = u A wa noticed in [7], we have an elicit formula for the olution of 3 Prooition 3 For any T and u S n, there eit a unique olution of 3 almot urely in C[, T ]; S n and adated It Fourier tranform in ace i given by ût, ξ = e i ξ βt βû ξ, t, ξ d Moreover, if u H σ for ome σ, then u C[, T ]; H σ a and ut H σ = u H σ, a for t If u L, the olution u of 3 ha the ereion y 33 ut = St, u := 4iπ βt β d/ e i u ydy, t [, T ] d 4βt β Proof The roof i the ame a in the determinitic cae ee for intance [] It uffice to take the Fourier tranform in ace of equation 3 Prooition 3 lead to the following atial etimate for the olution St, u Lemma 3 For any and t, St, ma L C deending only on uch that into L and there eit a contant St, u L C βt β d u L, for any u L

6 6 A DE BOUAD AND A DEBUSSCHE Proof It i eaily een from 33 and a denity argument that St, i an iometry on L Thu, the reult i true for = with C = Alo, for =, we obtain the reult from 33 with C = 4π d/ The general reult follow from the iez-thorin interolation theorem Lemma 3 i the reliminary te to get Strichartz tye etimate Contrary to the claical determinitic cae, we cannot immediately deduce from Lemma 3 ace-time etimate on the maing f S, fd Thi i due to the fact that formula 33 defining St, u i not in term of t and the Haudorff-Young inequality for convolution cannot be ued here in order to get etimate in time We need the following reult Prooition 33 Let α [,, there eit a contant c α deending only on α uch that for any T and f L P Ω; L, T T T E dt βt β α f d c α T α E f d Proof The reult i clear for α = o that by an interolation argument, it uffice to conider the cae α /, Let u write t βt β α f d f f = βt β α βt β α d d = f f βt β α βt β α d d Since f i adated, and βt β i indeendent of F, we may write T I = E βt β α f d dt T = E T = f f βt β α βt β + β β α d d dt E where µ = N, t i the law of βt β We have α + β β α µd = = πt / π / t α We need the following Lemma α + β β α µd f f d d dt α + β β α e α + β β t / t d α e d

7 WHITE NOISE DISPESION FO NLS 7 Lemma 34 Let α /,, there eit a contant c α deending on α uch that for any γ, γ, e c α γ α, γ,, α γ α d c α, γ Proof By ymmetry, we may aume γ > For γ,, we lit the integral on the dijoint interval,, [, γ + ] and γ +, + and majorize the integrand to obtain For γ, we have e α γ α d e α γ α d γ+ e d + α γ α d + e d γ+ γ α +γ y γ α y + dy + π/ α } γ α ma y α y + dy; π/ α α e / d +,,γ γ+, α / α d + γ+ α γ γ α d π/ d + α α We now roceed with the etimate of I For β β t /, we deduce from Lemma 34: t α e α + β β t / α d On the other hand, if β β > t /, t α e α + β β t / c α t / β β α c α t α/ β β α α d c α t α

8 8 A DE BOUAD AND A DEBUSSCHE It follow I c α T [ E t α/ β β α + t α] f f d d dt c T α α/ T α/ E f β β α f d d + c T α α T α E f f d d c T / α α/ T α/ E f d I / + c T α α T α E f d T c αt α E f d + I, from which we deduce the reult emark 35 The reader may eaily convince himelf that the etimate of Prooition 33 i till true with the ame bound on the right hand ide if βt β α on the left hand ide i relaced by βt β + νt α Thi i the only change to be made to aly all the analyi of the aer to equation 5 Corollary 36 Let α =, r or α,, r < α and ρ be uch that r ρ r; then there eit C α,ρ,r uch that, for any T and f L ρ P Ω; Lr, T, βt β α f d C α,ρ,r T r α f L L ρ ρ Ω;L r,t ω L r,t Proof The reult i clear for α = and ρ r = For α < and ρ = r =, it i the tatement of Prooition 33 We obtain the general reult by an interolation argument Corollary 36 i eactly what we need to relace Haudorff-Young inequality in order to get Strichartz tye etimate Note that in the determinitic cae, ie if βt i relaced by t, the limiting cae r = α i allowed We tate an immediate conequence of Lemma 3 and Corollary 36 Prooition 37 Let r < and be uch that r > d or r = and = Let ρ be uch that r ρ r; there eit a contant c ρ,r, > uch that for any, T and f L ρ P Ω; Lr, + T ; L S, σfσdσ c ρ,r, T β f L ρ Ω;L r,+t ;L L ρ Ω;L r,+t ;L with β = r d

9 WHITE NOISE DISPESION FO NLS 9 emark 38 Thi reult i very imilar to the claical Strichartz etimate However, we need r > d wherea in the claical cae, one can chooe r = d A air of number r, atifying thi latter condition i often called an admiible air We believe that in the tochatic cae conidered here the reult i till true for r = d but our roof doe not cover thi cae Note alo that the eonent β i much bigger than in the claical cae where one would have β = r d By analogy with the determinitic theory we define admiible air Definition 39 A air of real number i called an admiible air if r = and = or if the following condition are atified: r <, and r > d Proof of Prooition 37 let r, be an admiible air, let ρ be uch that r ρ r and let f L ρ P Ω; Lr, + T ; L By Lemma 3 St, σfσdσ St, σfσ L dσ L c βt βσ d fσ L By Corollary 36 with α = d [,, we deduce ρ E S, σfσdσ ct ρ r d f ρ, L ρ Ω;L r,+t ;L which i the reult L r,+t ;L dσ Uing a duality argument, we then have : Prooition 3 Let r and be uch that r > d or r = and = ; there eit a contant c r, > uch that for any, T and u L r Ω; L, F -meaurable, S, u L r P Ω; Lr, + T ; L and S, u Lr Ω;L r,+t ;L c r,t β/ u L r ω L with β = r d Proof Note that St, = S, t, where the adjoint i taken with reect to the L inner roduct Thu for u L r Ω; L, F -meaurable, and f L P Ω [, + T ] d we have +T St, u, ftdt = +T u, S, tftdt +T u L S, tftdt L

10 A DE BOUAD AND A DEBUSSCHE Moreover +T S, tftdt L = = +T +T + σ t +T t σ +T ft, St, σfσdtdσ ft, St, σfσdtdσ Sσ, tft, fσdtdσ = ft, St, σfσdtdσ σ t +T It follow from Prooition 37, E +T = St, u, ftdt / E +T ft, f L r,+t ;L St, σfσdσdt u L f / L r,+t ;L / u L r ω L f / S, σfσdσ L r ω Lr,+T ;L L r,+t ;L S, σfσdσ / S, σfσdσ L r,+t ;L / L r ω Lr,+T ;L c T β/ u L r ω L f L r ω L r,+t ;L with β = r d Thi imlie the reult In the determinitic cae, it i well known that Strichartz etimate till hold with different admiible air in the left and right hand ide We alo have uch reult here Thee will be ueful later to rove regularity roertie of olution of the nonlinear equation and to rove rigorouly that thee are indeed limit of olution of equation when ε goe to Prooition 3 Let r, and γ, δ be two admiible air uch that 34 γ = λ, r δ = λ + λ, with λ [, ], and ρ be uch that maρ, ρ } r; then there eit a contant cr,, γ, δ, ρ uch that for any, T, 35 S, σfσdσ cr,, γ, δ, ρt β f L L ρ Ω;L r,+t ;L ρ Ω;L γ,+t ;L δ

11 if f L ρ P Ω; Lγ, + T ; L δ and WHITE NOISE DISPESION FO NLS 36 S, σfσdσ cr,, γ, δ, ρt β f L L ρ Ω;L γ,+t ;L δ ρ Ω;L r,+t ;L if f L ρ P Ω; Lr, + T ; L In thi latter cae, we alo have 37 S, σfσdσ L ρ Ω; C[, + T ]; L Here, β = r d λ Proof We firt conider the cae λ = in 34 and rove that given r, an admiible air, r ρ r and f L ρ P Ω; L, + T ; L, we have 38 S, σfσdσ ct β/ f L ρ L ρ Ω;L r,+t ;L Ω;L,+T ;L with β = r d In order to rove thi, we conider ϕ L ρ P Ω; Lr, + T ; L and write +T E St, σfσdσ, ϕt dt +T = E fσ, Sσ, tϕt dσdt +T +T = E fσ, Sσ, tϕtdt dσ σ +T E f L t L u Sσ, tϕtdt σ [,+T ] σ L

12 A DE BOUAD AND A DEBUSSCHE We need to bound the econd factor For any σ [, + T ], we have = +T σ +T +T σ = Sσ, tϕtdt σ σ t θ +T L Sσ, tϕt, Sσ, θϕθ dt dθ Sθ, tϕt, ϕθ dt dθ +T θ = Sθ, tϕtdt, ϕθ dθ σ σ ϕ L r σ,+t ;L S, tϕtdt ϕ L r,+t ;L ϕ L r,+t ;L σ σ S, tϕt L dt S, tϕt L dt L r σ,+t ;L L r σ,+t L r,+t Therefore u σ [,+T ] +T σ Sσ, tϕtdt L ϕ L r t L S, tϕt L dt L r t and +T E St, σfσdσ, ϕt dt E f L t L ϕ / L r t L f L ρ ω L t L ϕ / L ρ ω L r t L S, tϕt L dt / L r t S, tϕt L dt / L ρ ω L r t c T β/ f L ρ ω L ϕ t L L ρ ω L r t L if r ρ r, or equivalently if r ρ r, and with β = r d by the ame argument a for Prooition 37 Claim 38 follow

13 WHITE NOISE DISPESION FO NLS 3 By Prooition 37, we have 39 S, σfσdσ ct β f L ρ Ω;L r,+t ;L L ρ Ω;L r,+t ;L if r ρ r Interolation between 38 and 39 lead to 35 The econd inequality i roved imilarly : we have by imilar argument a above St, σfσdσ f L r,+t ;L S, σfσ L dσ, L L r,+t for any t [; + T ] Therefore, by Cauchy-Schwarz inequality, S, σfσdσ c f L ρ ωl r L S, σfσ L dσ L ρ ωl t L ct β/ f L ρ ω L r L The fact that S, σfσdσ ha a continuou ath with value in L follow from a denity argument and the receding etimate Again, 36 follow by interolation between the above inequality and 39 4 A truncated equation We now contruct a local olution of equation We ue a imilar cut-off of the nonlinearity a in [5] and [6] Let θ C be uch that θ = on [, ], θ = on [, For, u L r loc, ; L, and t, we et u L θut = θ r,+t;l For =, we et θ = θ We take in thi ection = σ + and r uch that σ + r < 4σ+ dσ Note that uch a r eit, ince we have aumed σ < d We conider the following truncated form of equation idu 4 + u dβ + θ u u σ u dt =, u = u More reciely, we conider the truncation of it Itô form idu 4 + i u dt + u dβ + θ u u σ u dt =, u = u We interret it in the mild ene 43 u t = St, u + i St, θ u u σ u d Theorem 4 Let σ < 4σ+ d, = σ + and r be uch that σ + r < dσ For any F - meaurable u L r ωl, there eit a unique u in L r P Ω [, T ]; L for any T >, olution L ρ ωl r t

14 4 A DE BOUAD AND A DEBUSSCHE of 43 Moreover u i a weak olution of 4 in the ene that for any ϕ C d and any t, = i iu t u, ϕ L u, ϕ L d Finally, the L norm i conerved: and u C[, T ]; L a θ u u σ u, ϕ L d u t L = u L, t, a u, ϕ L dβ, a Proof In order to lighten the notation we omit the deendence in thi roof By Prooition 3, we know that S, u L r P Ω [, T ]; L Then, by Prooition 37, for u, v L r P Ω [, T ]; L, St, θu u σ u θv v σ v d L r Ω [,T ];L ct β θu u σ u θv v σ v L r ω L r [,T ];L with β = r d Moreover, by tandard argument ee [5], θu u σ u θv v σ v c T γ L r σ u v ω L r [,T ];L L r Ω [,T ];L with γ = σ+ r It follow that 44 T : u St, u + i St, θu u σ ud define a trict contraction on L r P Ω [, T ]; L rovided T T where T deend only on Iterating thi contruction, one eaily end the roof of the firt tatement The roof that u i in fact a weak olution i claical Let M and u M = P M u be a regularization of the olution u defined by a truncation in Fourier ace: û M t, ξ = θ ξ M We aly Itô formula to u M L ût, ξ We deduce from the weak form of the equation that idu M + i u M dt + u M dβ + P M θu u σ u dt = and obtain u M t L = u L + e i We know that u L σ+ [, T ] d a Since θu u σ u, P M u M d, t [, T ] lim P Mu M = u in L σ+ [, T ] d, M we may let M go to infinity in the above equality and obtain lim u Mt L M = u L, t [, T ], a

15 WHITE NOISE DISPESION FO NLS 5 Thi imlie ut L for any t [, T ] and ut L = u L In articular u L, T ; L A eaily een from the weak form of the equation, u i almot urely continuou with value in H 4 It follow that u i weakly continuou with value in L Finally the continuity of t ut L imlie u C[, T ]; L and ut L = u L a 5 Proof of Theorem We ue the olution of the truncated roblem obtained in Section 4 to contruct a olution to the original equation There i no lo of generality in auming that u L i determinitic Uniquene i clear ince two olution are olution of the truncated equation on a random interval Let u define τ = inft [, T ], u L r,t;l } Clearly u i a olution of on [, τ ] In order to ee that τ cannot be too mall, we need to rove that the L r t L norm of u can be controlled ecall that = σ + and σ + r 4σ+ dσ We fi a T and elain how to contruct a olution of on [, T ] Lemma 5 There eit contant c, c uch that if then Proof Let u write T drσ 4σ+ +r σ c rσ Pτ T c u r L r 5 u tl [,τ ]t = St, u l [,τ ]t + i Thu for T T u l [,τ ] L r,t ;L S, u l [,τ ] L r,t ;L + St, u σ u l [,τ ]dl [,τ ]t St, u σ u l [,τ ]d L r,t ;L Prooition 37 and Prooition 3 yield E u l [,τ ] r L r,t ;L cr, T u r drσ L + c T 4σ+ E u σ+ l [,τ r L r L Then, by Hölder inequality, E u l [,τ ] r L r,t ;L drσ cr, T u r L + c T drσ 4σ+ +r σ E u l [,τ rσ+ L r L cr, T u r L + c T drσ 4σ+ +r σ rσ E u l [,τ r L r L Hence, if c T 4σ+ +r σ rσ, E u l [,τ ] r L r,t ;L cr, T u r L and by Markov inequality Pτ T cr, T u r L r

16 6 A DE BOUAD AND A DEBUSSCHE In order to contruct a olution to on [, T ], we iterate the local contruction We fi > and have a local olution on [, τ ] We then conider the equation for u: ut + τ = St + τ, τ uτ + St + τ, + τ θ τ u u + τ σ u + τ d All the argument of Section 4 can be reroduced We obtain a unique global olution of thi equation, that we denote by u Moreover etting τ = inft [, T ], u L r τ,t+τ ;L } we obtain a olution of the non truncated equation on [τ, τ + τ ] and thu on [, τ + τ ] We alo have by Lemma 5 and the conervation of the L norm Pτ T F τ c uτ r L r = c u r L r, drσ rovided that T 4σ+ +r σ c rσ We continue thi contruction recurively and obtain a olution on [, T n], where T n = τ + + τ n, with drσ Pτ n T F T n c u r L r, rovided T 4σ+ +r σ c rσ Note that P lim τ n n + = = lim lim Pτ n ε, n N ε N + dr For large enough and ε +r σ c rσ, and we deduce that Pτ n ε, n N lim E M Pτ n ε F T n, N n M l τ n ε}pl τ M ε} F T M lim M = M N Hence, Plim n + τ n = = o that T n goe to infinity, a and we have contructed a global olution The conervation of the L -norm and the fact that u C + ; L a wa roved in Theorem 4 In order to obtain the etra-integrability roertie given in the tatement of Theorem, we aly Prooition 3 and 35 of Prooition 3 with ρ, q on the left hand ide q = + if d = and with γ = r, δ = σ + to equation 5 Note that ρ, q i an admiible air thank to the condition ρ 4 if d = and q < d d, ρ < 4d dq if

17 d Thi give, etting q = + if d = : WHITE NOISE DISPESION FO NLS 7 u l [,τ ] L ρ ω L ρ,t ;L q cρ, q, T u L + c ρ, q, T u σ u l [,τ ] L ρ ω L γ,t ;L δ cρ, q, T u L + c ρ, q, T u l [,τ ] σ+ L σ+ρ ω L r,t ;L σ+ cρ, q, T u L + c ρ, q, T, where i choen a above Etimate on other interval of the form [T n, T n+ ] are obtained imilarly Finally, aume that u H Then going back to T defined in 44, and alying the ame etimate a in the roof of Lemma 5, after having taken firt order ace derivative, lead to T u L r Ω [,T ];W, with β = r σ CT β/ u H + C T β rσ u L r Ω [,T ];W, drσ 4σ+ Thi rove that if B = B, i the cloed ball of radiu in L r Ω [, T ]; W,, then T B B rovided T T, where T deend only on and not on Since cloed ball of L r Ω [, T ]; W, are cloed in L r Ω [, T ]; L, thi imlie that the fied oint of T, which i the olution u of 43, i in L r Ω [, T ]; W, Alying then Prooition 3, and 37 in Prooition 3 to equation 43 or 5, again after having taken firt order ace derivative, give the reult 6 Equation a limit of NLS equation with random dierion To rove Theorem 3, we ue the ame argument a in [7] Let u recall it main line Note that we introduce a light modification ince we work with H function intead of H a in [7] Conider the following nonlinear Schrödinger equation: 6 i du dt + ṅt u + F u u =,, t >, u = u,, where F i a mooth function with comact uort and n i a real valued function Note that, uing the mild form u n t = S n tu + i S n t, F u ud, where we have denoted by S n t, the evolution oerator aociated to the linear equation i dv dt + ṅt v =,, t >, whoe olution can be written down elicitly thank to atial Fourier tranform, one can give a meaning to the olution u of equation 6 a oon a n i a continuou function of t Indeed, for each t,, S n t, i an iometry on any Sobolev ace H Since the nonlinear term ha bounded derivative, a fied oint argument can be ued in C[, T ]; L

18 8 A DE BOUAD AND A DEBUSSCHE and a global olution u n i obtained in thi ace if u L Moreover, the olution belong to C[, T ]; H if u H Uing Fourier tranform, we ee that, for n, n C[, + T ], we have, for [, ], S n, S n, u L,+T ;H n n / C[,+T ] u H Proceeding a in the roof of Theorem 37 in [7], we deduce, for,, u n u n C[,T ];H c n n / C[,T ] u H where the contant c deend on T and F It follow that for u H the maing n u n C[, T ] C[, T ]; H i continuou for, Since our aumtion on the roce m ay that the roce t ε m d converge in ditribution in C[, T ] to a brownian motion, we deduce that ε the olution of i du 6 dt + ε m t ε u + F u u =,, t >, u = u,, converge in ditribution in C[, T ]; H to the olution of idu + u dβ + F u u dt =,, t >, u = u,, for, We now want to etend thi reult to the original ower nonlinear term Let u introduce the truncated equation, where θ i a in ection 4, 63 and 64 i du dt + ε m t u ε u + θ M u = u,, u idu + u dβ + θ M u = u, u u =,, t >, u u dt =,, t >, We denote by u M ε and u M their reective olution By the reviou argument, thee olution eit and are unique in C[, T ]; H Note that etting τ M ε = inft : u M ε t L M} and u ε = u M ε on [, τ ε M ], define a unique local olution u ε of equation on [, τ ε with τ ε = lim M τε M We alo et τ M = inft : u M t L M} By the above reult, for each M, u M ε converge to u M in ditribution in C[, T ]; H for, By Skohorod Theorem, after a change of robability ace, we can aume that for

19 each M the convergence of u M ε u notice that for < δ, if WHITE NOISE DISPESION FO NLS 9 to u M hold almot urely in [, T ]; H To conclude, let τ M T and u M ε u M C[,T ];H δ then u M = u, the olution of, on [, T ] Moreover, by the Sobolev embedding H L, we have u M ε u M C[,T ];L cδ for ome c > We deduce u M ε C[,T ];L M rovided δ i mall enough Therefore It follow that for δ > mall enough, τ ε > τ M ε T and u M ε = u ε on [, T ] Pτ ε u T + Pτ ε u > T and u ε u C[,T ];H > δ P u M ε u M C[,T ];H > δ + P τ M < T Since u H, we know that u i almot urely in C + ; H ; we deduce lim P τ M < T = M Chooing firt M large and then ε mall we obtain and The reult follow lim Pτ εu T = ε lim Pτ εu > T and u ε u C[,T ];H ε > δ = eference [] GP Agrawal, Nonlinear fiber otic, 3rd ed Academic Pre, San Diego, [] GP Agrawal, Alication of nonlinear fiber otic, Academic Pre, San Diego, [3] P Billingley, Convergence of robability meaure, John Wiley & Son, Inc, New York-London-Sydney 968 [4] T Cazenave, Semilinear Schrödinger equation, Courant Lecture Note in Mathematic, American Mathematical Society, Courant Intitute of Mathematical Science, 3 [5] A de Bouard, A Debuche, A tochatic nonlinear Schrödinger equation with multilicative noie, Comm in Math Phy, 5, [6] A de Bouard, A Debuche, The tochatic nonlinear Schrödinger equation in H, Stochatic Anal Al, [7] A de Bouard, A Debuche On the effect of a noie on the olution of uercritical Schrödinger equation, Prob Theory and el Field, 3, [8] A de Bouard, A Debuche Blow-u for the uercritical tochatic nonlinear Schrödinger equation with multilicative noie, Ann Probab, 33, no 3, 78 5 [9] G Da Prato, J Zabczyk, Stochatic equation in infinite dimenion, Encycloedia of Mathematic and it Alication, Cambridge Univerity Pre, Cambridge, 99 [] A Debuche, L DiMenza Numerical imulation of focuing tochatic nonlinear Schrödinger equation, Phyica D, 6 3-4, 3 54 [] J-P Fouque, J Garnier, G Paanicolaou, and K Solna, Wave roagation and time reveral in randomly layered media Sringer, 7 [] J Garnier, Stabilization of dierion managed oliton in random otical fiber by trong dierion management, Ot Commun 6, 4 438

20 A DE BOUAD AND A DEBUSSCHE [3] J Ginibre, G Velo, The global Cauchy roblem for the nonlinear Schrödinger equation reviited, Ann Int Henri Poincaré, Analye Non Linéaire, [4] T Kato, On Nonlinear Schrödinger Equation Ann Int H Poincaré, Phy Théor 46, [5] M Kunze The ingular erturbation limit of a variational roblem from nonlinear fiber otic, Phy D 8, no -, [6] HJ Kuhner, Aroimation and weak convergence method for random rocee, MIT Pre, Cambridge, 994 [7] Marty On a litting cheme for the nonlinear Schrödinger equation in a random medium, Commun Math Sci 4, no 4, [8] G Paanicolaou, W Kohler, Aymtotic theory of miing tochatic ordinary differential equation, Comm Pure Al Math, [9] G Paanicolaou, DW Stroock, SS Varadhan, Martingale aroach to ome limit theorem, Statitical Mechanic and Dynamical ytem D uelle, ed, Duke Turbulence Conf, Duke Univ Math Serie III, Part IV, 976 [] J auch, Partial differential equation Graduate Tet in Mathematic, 8 Sringer-Verlag, New York, 99 [] Y Tutumi, L -olution for nonlinear Schrödinger equation and nonlinear grou, Funk Ekva 3, [] V Zharnitky, E Grenier, C Jone, S Turityn, Stabilizing effect of dierion management, Phy D 5/53, Centre de Mathématique Aliquée, CNS et Ecole Polytechnique, 98 Palaieau cede, France addre: debouard@cmaolytechniquefr IMA et ENS de Cachan, Antenne de Bretagne, Camu de Ker Lann, Av Schuman, 357 BUZ, FANCE addre: arnauddebuche@bretagneen-cachanfr