THE KORTEWEG-DE VRIES EQUATION WITH MULTIPLICATIVE HOMOGENEOUS NOISE

Transcription

1 HE KOEWEG-DE VIES EQUAION WIH MULIPLICAIVE HOMOGENEOUS NOISE ANNE DE BOUAD 1 AND ANAUD DEBUSSCHE 1 CNS et Université Paris-Sud, UM 868, Bât 45, Université de Paris-Sud, 9145 OSAY CEDEX, FANCE ENS de Cachan, Antenne de Bretagne, Campus de Ker Lann, Av Schuman, 3517 BUZ, FANCE Abstract We prove the global existence and uniqueness of solutions both in the energy space and in the space of square integrable functions for a Korteweg-de Vries equation with noise he noise is multiplicative, white in time, and is the muliplication by the solution of a homogeneous noise in the space variable 1 Introduction and statement of the results he aim of the paper is to prove the global existence and uniqueness of strong solutions for a Korteweg-de Vries equation with noise, which may be written in Itô form as 11 du 3 xu 1 xu dt = uφdw where u is a random process defined on t, x, W is a cylindrical Wiener process on L and φ is a convolution operator on L defined by φfx = kx yfydy, for f L where the convolution kernel k is an H 1 L 1 function of x Here H 1 is the usual Sobolev space of square integrable functions of the space variable x, having their first order derivative in L Considering a complete orthonormal system e i i N in L, we may alternatively write W as 1 W t, x = i N β i tφe i x, β i i N being an independent family of real valued Brownian motions Hence, the correlation function of the process φw is EφW t, xφw s, y = cx ys t, x, y, s, t >, 1991 Mathematics Subject Classification 35Q53, 6H15, 76B35 Key words and phrases Korteweg-de Vries equation, stochastic partial differential equations, white noise 1

2 A DE BOUAD AND A DEBUSSCHE where cz = kz ukudu he existence and uniqueness of solutions for stochastic KdV equations of the type 11 but with an additive noise have been studied in [4], [7], [8] Here we extend those results to equation 11, that is the multiplicative case with homogeneous noise Note that an equation of this form, but with an additional weak dissipation has been considered in [13] Indeed, in this latter case where a dissipative term is added, such a noise may be viewed as a perturbation of the dissipation Although our existence and uniqueness results would easily extend to the case where weak dissipation is added, the dissipative term is of no help in the existence proof, so we prefer stating the result for equation 11 Assuming k H 1 L 1 will allow us to prove the global existence and uniqueness of solutions to equation 11 in the energy space H 1, that is in the space where both quantities 13 mu = 1 u xdx and 14 Hu = 1 x u dx 1 6 u 3 dx are well defined Note that m and H are conserved quantities for the equation without noise, that is 15 t u 3 xu 1 xu = It is important to solve equation 11 in the energy space, indeed most of the studies on the qualitative behavior of the solutions are done in this space One of our aim in the future is to analyse the qualitative influence of a noise on a soliton solution of the deterministic equation, as we did in the additive case in [6], and this requires the use of the hamiltonian 14 However, our method of construction of solutions easily extends to treat the case of a kernel k L L 1, obtaining global existence and uniqueness in L It seems difficult to get a result with less regularity It may be noted also that the use of a noise of the form given in 11 naturally brings some localization in the noisy part of the equation, at least in the limit where the amplitude of the noise goes to zero, and when the initial state is a solitary wave or soliton solution of the deterministic equation, that is a well localized solution which propagates with a constant shape and velocity his localization in the noise was a missing ingredient in the study of the influence of an additive noise on the propagation of a soliton see [6] he precise existence result is the following, and the method we use to prove it closely follows the method in [7] heorem 11 Assume that the kernel k of the noise satisfies k H s L 1, s = or 1 hen for any u in H s, there is a unique adapted solution u with paths almost surely in C ; H s of equation 11 Moreover, u L Ω; C ; L

3 KDV EQUAION WIH HOMOGENEOUS MULIPLICAIVE NOISE 3 As in [7, 8], we use the functional framework introduced by Bourgain to study dispersive equations Following [3], [14], [15], for s, b, X denotes the space of tempered distributions f S for which the norm f X = 1 ξ s 1 τ ξ 3 b ˆfτ, ξ dτdξ is finite, where ˆfτ, ξ stands for the space-time Fourier transform of ft, x In the same way, we set for b, s 1, s, and 1/ f ex1 = ξ s 1 ξ s 1 1 τ ξ 3 b ˆfτ, ξ dτdξ,s X 1,s = { f S, f ex1,s < } 1/ Note that the use of the space X 1,s is necessary here Indeed, since we work with stochastic equations driven by white in time noises, we cannot require too much time regularity, and we have to choose < b < 1/ But then the bilinear estimate which allows to treat the KdV equation is not true in the space X as was already mentioned for the additive case in [7] For, we also introduce the spaces X and X 1,s of restrictions to [, ] of functions in X and X 1,s hey are endowed with f X = inf { f X, f X and f [, ] = f [, ] } f ex 1,s = inf { f ex1,s, f X 1,s and f [, ] = f [, ] } Because equation 11 is a multiplicative equation with a nonlinear deterministic part, we have to consider first a cut-off version of this equation see Section As we make use of the functional framework defined above, the cut-off will arise as a function of the norm of the solution of the type X t Moreover, this function of the norm must be a regular function, in order to allow us to use a fixed point argument ie in order that our mapping is a contraction mapping, see Section he fact that the functional spaces we consider are nonlocal spaces in the time variable then brings a lot of technical difficulties, concerning points that would be obvious if we were dealing with more classical function spaces see eg the proof of Lemma 1 he paper is organized as follows: Section is devoted to the proof of several preliminary lemmas and propositions, which once brought together lead quite easily to the proof of global existence and uniqueness for the cut-off version of the equation or to the local existence and uniqueness for equation 11 In Section 3 we prove that the solutions of equation 11 are global in time, by using estimates on the moments of the L -norm of the solution Again, due to the spaces we consider for the local existence, the globalization argument is not obvious

4 4 A DE BOUAD AND A DEBUSSCHE Preliminaries and existence for a truncated equation As is usual, we introduce the mild form of the stochastic Korteweg-de Vries equation 11 We denote by Ut = e t 3 x the unitary group on L generated by the linear equation u t 3 u x 3 = Using Fourier transform, we have FUtvξ = e itξ3 Fvξ We then rewrite 11 as follows 1 ut = Utu 1 Ut r x u rdr Ut rurφdw r, t he X and X 1,s norms defined in the introduction have the nice property that they are increasing with However, it is more convenient to work with other norms, given by the multiplication by the function 1l [, ] In the case we consider here, that is b < 1/, we can prove the following result, stating that the two norms are equivalent Lemma 1 Let s and b < 1/, then there exist two constants C 1, C depending on b but not on such that for any f X C 1 f X 1l [, ] tf X C f X Proof he first inequality is clear and in fact we may choose C 1 = 1 For the other inequality, let us set gt = 1l [, ] tu tft so that 1l [, ] tf X = 1 ξ s 1 τ b ĝτ, ξ dτdξ = 1 ξ s F x g, ξ dξ Ht b he result follows from the following inequality 1l [, ] h H b C h H b, h H b, which holds for a constant C depending on < b < 1/ o prove this, we use the following equivalent norm on H b see for instance [1]: h H b = ht hr t r 1b dtdr h L Clearly, 1l [, ] h L h L Moreover 1l[, ] tht 1l [, ] rhr t r 1b drdt = ht hr ht = t r 1b drdt drdt t r 1b = I II III r<t 1l[, ] tht 1l [, ] rhr t r 1b drdt hr drdt t r 1b

5 KDV EQUAION WIH HOMOGENEOUS MULIPLICAIVE NOISE 5 he first term I is less than h H b he second and third terms are equal to II = 1 b t b ht dt, III = 1 b r b hr dr Both are bounded by C h H b, o see this, we note that it is obvious for b = and results from Hardy inequality for b = 1 when H b, is replaced by H 1, he result follows by interpolation since, for b < 1/, H b, = H b, We now define Yb, = X b, X b,, 3/8 endowed with the norm f b, = max{ f X b,, f ex b,, 3/8 } We also use the space Y = X X, 3/8 with a similar definition of its norm From now on and thanks to Lemma 1, we will use the definition v Y = 1l [, ] v Y each time we take a norm in Y with b < 1/ For u L Ω; H 1, we set zt = Utu, and vt = ut zt hen 1 is rewritten as vt = 1 Ut r [ x v r x z r x zrvr ] dr Ut rzr vrφdw r, t Let θ be a cut-off function θx = for x, θx = 1 for x 1, with θ C and let θ = θ Ṙ ; we consider the cut-off version of written for > as: v t = 1 [ ] Ut r θ v Y r b, x v r dr 3 1 ] Ut r [θ v Y r b, x zrv r dr Ut r [ x z r ] dr Ut rzr v rφdw r, t We find v as a fixed point of the mapping, v being defined by the right hand side above Note that the cut-off is made in the L in space norm, even for the H 1 result We will choose < b < 1/ and 1/ < c We use the following Lemma

6 6 A DE BOUAD AND A DEBUSSCHE Lemma For any b < 1/, >, v Y, there exists C such that θ v Y t vt b, C b, and, for s = or 1, there is a positive constant C, independent of, such that θ v Y t vt b, C vt Proof We use arguments similar to the proof of Lemma 1 Let wt = U tvt then, using the same norm on H b as in Lemma 1, θ v Y t vt b, C θ v Y Xb, t F x wt, ξ b, dξ Ht b [, ] he L part of the H b norm above is easily estimated, while the other part is bounded above by C θ Fx wt, ξ F x wr, ξ v Y t b, t r 1b drdtdξ C = I II Fx wr, ξ θ v Y t θ b, v Y r b, t r 1b drdtdξ Next, we define τ = inf{t, v Y t }; then θ b, v Y t = for t τ b, and I C C τ F x wt, ξ F x wr, ξ t r 1b drdtdξ F x w, ξ H b,τ dξ C v X τ b, C In order to estimate II, we use the fact that for r < t, C θ v Y t θ b, v Y r b, θ L v Yb, t v Yb, r C 1l [,t]v Yb, 1l [,r] v Yb, C 1l [r,t]v Y b, C 1 η 3/4 1l [r,t] F x w, η H b dη We leave to the reader the estimate of the contribution to II of the L part of the H b norm above; indeed, it follows the same line as the estimate of the remaining contribution, which is

7 KDV EQUAION WIH HOMOGENEOUS MULIPLICAIVE NOISE 7 bounded above by C 1 η 3/4 τ σ r r F x wσ, η F x wσ 1, η σ σ 1 1b dσ 1 dσ C τ 1 η 3/4 F xwr, ξ t r 1b drdtdηdξ σ σ1 τ σ dt t r 1b F x wr, ξ dr F xwσ, η F x wσ 1, η σ σ 1 1b dσ 1 dσ dηdξ where we have inverted the integrals in the time variables; this last term is in turn bounded above by C τ σ σ1 1 η 3/4 σ 1 r b F x wr, ξ dr F xwσ, η F x wσ 1, η σ 1 σ 1b dσ 1 dσ dηd ξ, by the fact that τ r b σ r b σ 1 r b for r σ 1 σ t τ Using then the same arguments as in the proof of Lemma 1, we finally get II C F x w, ξ H b,τ dξ 1 η 3/4 F x w, η H b,τ dη C v Y τ v b, X τ b, his, together with the estimate of I implies the first inequality of the Lemma for the X b, part of the Y b, norm; the X b,, 3/8 part, and the second inequality of the Lemma are proved in the same way Next results state the estimates on the bilinear term appearing in 3 Proposition 3 Let a >, < b < 1/ < c < 1, with b c > 1 and a, b, c sufficiently close to 1/, then for any v, z X c,s, s = or 1, we have and x vz a,s C x v a,s C v b, v, v b, z X c,s v z X c,, x z a,s C z X c,s z X c,

8 8 A DE BOUAD AND A DEBUSSCHE Proof hese estimates are proved in [7], Proposition and 3, for s = hese are easily extended to s = 1 It suffices to add a factor 1 ξ in the expression f, x gh = ξ ˆfτ, ξ τ ξ τ 1 ĝτ τ 1, ξ ξ 1 ĥτ 1, ξ 1 dτ 1 dξ 1 dτdξ, ξ 1 and to use the fact that 1 ξ 1 ξ ξ 1 1 ξ 1 emark 4 It does not seem possible to get rid of the homogeneous Sobolev space, ie of Xb,, 3/8, to get the result of Proposition 3, when b < 1/, even in the case s = 1; indeed, a careful reading of the proof of Proposition in [7] shows that the additional factor ξ 3/4 induced by the use of Xb,, 3/8 is necessary in a region of the integral where ξ 1 ξ, so that ξ ξ 1 ξ, and with moreover ξ 1; hence the supplementary factor 1 ξ ξ 1 1 ξ 1 /1 ξ is of no help there It remains to derive the estimates on the stochastic integrals in 3 In order to be able to globalize the solutions in Section 3, we will need estimates on all the moments of the stochastic integrals Proposition 5 Let m N, s = or 1, and v L m Ω, X ; then for any b 1/, E Ut r [vrφdw r] m X C k m H s E v m X,s C bm k m H s E v m X Moreover E Ut r [vrφdw r] m ex 3/8 C k m H s k m L 1 E v m X,s C bm k m H s k m L 1 E v m X Proof We prove the result for s = 1, the proof is exactly the same for s = We set wt = 1l [, ] t Ut r[vrφdw r] Let gt = 1l [, ] t U r [vrφdw r], then wt = Utgt, t We have m E w m X = E 1 ξ 1 τ b ĝτ, ξ dτdξ

9 KDV EQUAION WIH HOMOGENEOUS MULIPLICAIVE NOISE 9 Choosing Brownian motions β k k N, defined on, we have It follows F x gt, ξ = ĝτ, ξ = = k= k= = 1l [, ] t k= k= By Burkholder inequality, we deduce: e irξ3 F x vrφe k ξdβ k r 1l [, ] t 1l [, ] re irξ3 F x vrφe k ξdβ k r 1l [, ] t 1l [, ] re irξ3 F x vrφe k ξdβ k re iτt dt 1l [, ] re irξ3 F x vrφe k ξ 1l [, ] te iτt dt dβ k r m E 1 ξ 1 τ b ĝτ, ξ dτdξ C m E 1 ξ 1 τ b 1l [, ] r F x vrφe k ξ k= 3 1l [, ] te iτt m dt drdξdτ r r It easy to see that r 1l [, ] te iτt dt min{, τ } herefore, using Lemma 6 below, m E 1 ξ 1 τ b ĝτ, ξ dτdξ CE 1 ξ 1 τ b min{, τ }1l [, ] r F x vrξ η 4 m ˆkη dηdrdξdτ

10 1 A DE BOUAD AND A DEBUSSCHE which in turn we bound from above, using the unitarity of Ut in L and in H 1, by [ CE 1 ξ η 1 η ] 1l [, ] r F x vrξ η 3 C C m ˆkη dηdrdξ [ ] k m L E 1l [, ] v m X,1 k m H 1 E 1l [, ] v m X, [ ] k m L E v m k m X,1 H 1 E v m X, For the second statement, we proceed similarly However, the extra ξ 3/4 implies that a special treatment of the integral for ξ 1 On the region ξ 1, we simply use ξ 3/4 1 he following estimate is thus sufficient to conclude m E 1 ξ ξ 3/4 1l [, ] r F x vrξ η ˆkη dηdrdξ ξ 1 m CE ξ 3/4 ˆkη ξ dξ 1l [, ] r F x vrη dηdr ξ 1 C ˆk L E v m X, C k L 1 E v m X, We now give the Lemma used in the above proof Lemma 6 Let v X,, then for any complete orthonormal system e k k N of L, we have F x vrφe k ξ = F x vrξ η ˆkη dη k= Proof We have F x vrφe k ξ = F x vr, x kx y, e k y L y ξ = F x vr, xkx y ξ, e k y L y herefore, by Parseval identity, F x vrφe k ξ = F x vr, xkx y ξ L, y k= and by Plancherel theorem and an easy computation F x vrφe k ξ = F x,y vr, xkx y ξ, η L = F xvrξ ˆk η L, k=

11 KDV EQUAION WIH HOMOGENEOUS MULIPLICAIVE NOISE 11 which gives the conclusion he proof of the next proposition is left to the reader It only makes use of classical arguments and ideas similar to those at the end of the proof of Proposition 5 Proposition 7 Let s = or 1 For any >, any stopping time τ and any predictable process v L Ω; C[, τ]; H s, U r[φvrdw r] has continuous paths with values in H s Ḣ 3/8 and for any integer m, there is a constant C m with E sup t τ Ut r[φvrdw r] m H s Ḣ 3/8 C m E sup vt m H s t τ We are now able to prove the following existence theorem for the truncated equation heorem 8 Let s = or 1 and assume that the convolution kernel of the operator φ satisfies k H s L 1 ; then for any u in H s, equation 3 with zt = Utu has a unique solution v, for any b with < b < 1/, and any Moreover v L Ω; C[, ]; H s Proof We use a fixed point argument on equation 3 he following lemma, whose first and third estimates were proved in [7], while the second one can be proved in the same way, is useful Lemma 9 Xc,s and Let u H s, s = or 1 For any > and c > 1/, z = U u z X c,s C u H s For any u H s Ḣ 3/8, and any b with b < 1/, z = U u and z C u H s u Ḣ 3/8 For any a, b, 1 with a b 1, and any f Y a,s, U rfrdr Y and U rfrdr C 1 ab f a,s We first assume that the hypothesis of heorem 8 hold with s = Let us fix a, b, c as in Proposition 3, with a b < 1 We fix and take Let v 1, v Yb,, being also fixed We set ṽ i t = θ v i Y t v i t, i = 1, hen, recalling that v is defined by the b,

12 1 A DE BOUAD AND A DEBUSSCHE right hand side of 3, we have E v 1 v Yb, CE CE CE Ut r x [ṽ 1 r ṽ r ] dr Ut r x [ṽ 1 r ṽ r zr] dr Ut r [v 1 r v r φdw r] which, by Lemma 9 and Proposition 5, applied with m = 1, is bounded above by x C 1 ab E ṽ 1 ṽ C 1 ab E x ṽ 1 ṽ z Y a, C b E v 1 v Xb, a, By Proposition 3, it follows E v 1 v Yb, } C {E 1 ab ṽ 1 ṽ Yb, ṽ 1 ṽ Yb, E ṽ 1 ṽ Yb, z Xc, C b E v 1 v Xb, By Lemma, ṽ 1 ṽ Yb, C Moreover, it is not difficult to use the arguments of the proof of Lemma and prove We deduce that for some α >, E v 1 v Yb, ṽ 1 ṽ Yb, C v 1 v Yb, b, b, b, C,, u L α E v 1 v Yb, hus, has a unique fixed point v L Ω; b, for where is chosen such that C,, u L α 1/ Moreover, using arguments similar to the proof of Proposition 5, it can be seen that U r [zr v r φdw r] is a square integrable martingale in L Ḣ 3/8 Since

13 KDV EQUAION WIH HOMOGENEOUS MULIPLICAIVE NOISE 13 Ut t is strongly continuous on L Ḣ 3/8, we deduce that Ut r [zr v r φdw r] L Ω; C[, ]; L Ḣ 3/8 Using then Lemma 9 with b > 1/ and similar estimates as above, we deduce that v is also in L Ω; C[, ]; L Ḣ 3/8 hen, we construct a solution in [, ] First, we write equation 3 with t in the form 4 v t = Utv 1 1 Ut r [θ [ Ut r θ v Y r b, Ut r [ x z r ] dr v Y r b, ] x v r dr ] x z rv r dr Ut rz r v rφdw r, t Since v L Ω; L Ḣ 3/8, the first term is in L Ω; b, It is then easily seen that v can be found on [, ] as a fixed point in L Ω; b, in the same way as on the interval [, ] Iterating this, we get a solution on [, ] which is in fact in L Ω; b, and also in L Ω; C[, ]; L Ḣ 3/8 his proves the result for s = Now suppose that the assumptions hold with s = 1 Let v L Ω; Y ; we have, setting ṽt = θ v Y t vt, and using Lemma 9, b, [ v C 1 ab ṽ x ṽ x ṽz a,1 x z ] a,1 a,1 so that by Proposition 3, Ut r [zrφdw r] Ut r [vrφdw r], [ ] v C 1 ab ṽ ṽ b, ṽ z b, X ṽ c,1 z X z c, X z c, X c,1 Ut r [zrφdw r] Ut r [vrφdw r]

14 14 A DE BOUAD AND A DEBUSSCHE We then make use of Lemma and Lemma 9 to get v C 1 ab v z X C z c,1 X c, Ut r [zrφdw r] C,, u L 1 ab v u H 1 Ut r [zrφdw r] Ut r [vrφdw r] Ut r [vrφdw r] hus, by Proposition 5 and Lemma 9, E v Y [ C,, u L 1 ab C b] E v Y u H 1 his shows that maps L Ω; Y into itself Moreover, the ball in L Ω; Y of radius is invariant by if such that C,, u L 1 ab C b 1/ and u H 1 Choosing in the construction of the solution v of 3 in L, it follows that the solution v is in L Ω; We then use similar arguments as above to prove that v L Ω; C[, ]; H 1 and E sup v H 1 [, ] C 1 a b E sup [, ] E v u H 1 C u L Ut r [zr v rφdw r] H 1 C 1 a b E v u H 1 C u L C E sup v H 1 [, ] C u H 1, with b > 1/ and a b < 1 Choosing a smaller if necessary, we deduce E sup v H 1 [, ], if u H 1 On [, ], we use equation 4 and obtain by similar arguments E v C E v u H 1 Ḣ 3/8 H 1,

15 and E KDV EQUAION WIH HOMOGENEOUS MULIPLICAIVE NOISE 15 sup v H 1 [, ] sup v H 1 [, ] CE v H 1 Ḣ 3/8 We know from the L contruction that v L Ω; C[, ]; Ḣ 3/8, therefore E CE v H 1 E sup v Ḣ 3/8 [, ] It is now easy to iterate this argument and deduce that the solution v is in L Ω; and also in L Ω; C[, ]; H 1 heorem 8 gives the following local in time existence result for the non truncated equation Corollary 1 Let s = or 1 and assume that k H s L 1 ; then for any u H s, there is a stopping time τ u, ω as positive, such that 1 has an adapted solution u, defined as on [, τ u [, unique in some class, and with paths as in C[, τ u [; H s If s = 1, the uniqueness holds among solutions with paths in C[, τ u [; H 1 as; moreover, the stopping time τ u satisfies τ u = or lim sup u U u Y t =, as b, t τ u emark 11 Let us explain what we mean by a solution on the random interval [, τ u [ his means that u is defined on [, τ u [ and is an adapted process such that for any stopping time τ < τ u the following holds on [, τ]: ut = Utu 1 Ut r x u rdr τ Ut rurφdw r Proof Let zt = Utu and let v for any b < 1/ and any > be the solution of 3 given by heorem 8 We then set τ = inf{t, v Y t }; for t [, τ b, ], we have θ v Y t = 1, hence v b, is a solution of on [, τ ] It is not difficult to see that τ is non decreasing in and that v 1 = v on [, τ ] Hence we may define u on [, τ u [ with τ u = lim τ by setting ut = v t zt for t [, τ ] and u is then a solution of 1 on [, τ u [ he uniqueness for u holds in the class z Y τ b, for any and it is not difficult to see that any solution u with paths in C[, τ u [; H 1 is in this class he last property of the lemma is an immediate consequence of the definition of τ u 3 Global existence As already seen, heorem 8 gives a local in time existence result for the equation without cut-off In the present section, we end the proof of heorem 11 by showing that those solutions are globally defined in time o that aim we need an estimate on v b, We will use the following result Proposition 31 Assume that k L Let u C[, τ]; L be a solution of equation 1 with u L, where τ is a stopping time hen, for any m 1, u L m Ω; C[, τ]; L

16 16 A DE BOUAD AND A DEBUSSCHE and for any > E sup ut m L t [,τ ] C, u L, m Proof he result is a straightforward consequence of Ito formula We prove it for m = For m it then follows from Hölder inequality For m, the proof is similar We apply Ito formula to Mu = u L and obtain after a regularization argument and easy computations see [4] for more details in the case of an additive noise or [5] for the case of the stochastic Schrödinger equation: Muτ r = Mu k= τ r We take the square of this identity and deduce: E sup M ur r [,τ ] M u 4E r sup r [,τ ] k= τ k 4 L E Muσdσ u σ, xφe k xdxdβ k σ k L τ r u σ, xφe k xdxdβ k σ τ M u 4 k L k 4 L E M urdr, thanks to Burkholder and Hölder inequalities, and to Lemma 6 Gronwall Lemma Muσdσ he result follows from Let v be the solution given by heorem 8, let > be fixed, and let τ = inf{t [, ], v Y t }; then on [, τ b, ], v z is a solution to 1 which is as in C[, τ ]; L and Proposition 31 applies: 31 E sup v t zt m L t [,τ ] C, u L, m We now show that this implies an estimate on the b, norm of v Lemma 3 Let v be the solution of the truncated equation 3, then there exists a constant C, u L independent of such that Proof E v τ Y C, u b, L

17 KDV EQUAION WIH HOMOGENEOUS MULIPLICAIVE NOISE 17 Step 1: Using similar arguments as in the beginning of the proof of heorem 8, taking into account the fact that v satisfies equation 3, we prove using Lemma that for, v Y τ b, C 1 ab [ v τ b, C 1 ab [ v τ b, with u t = v t Utu We set then ] z X c, ] C u L Ut r [u rφdw r] Y b, Ut r [u rφdw r] t K 1 = K 1 ω = C 1 ab C u C[, ];L Ut r [u φdw r] C 1 ab v τ b, v Y τ b, herefore, if we choose = ω such that 1 ab = K CK 1, we have Y b,, Y b,, v Y τ b, K 1 Note indeed that v = and that v Y t k, we define b, is a continuous function of t Similarly, for any v k t = u t Ut k u k, t [k, k 1 ], with = ω chosen above hen the same argument shows that K 1, v k Y [k τ,k1 τ ] b, where we use the space Y [ 1, ] b, whose definition is exactly the same as Yb, but [, ] is replaced by [ 1, ] Step : Since u is a solution of 1 on [, τ ], we may write for any t [, τ ], u t as u t = Utu 1 Ut r x u rdr Ut r [u rφdw r] = Utu 1 k t k1 t [ ] Ut r x v k r Ur k u k dr k= k Ut r [u rφdw r],

18 18 A DE BOUAD AND A DEBUSSCHE where k t is the integer part of t/ Using this decomposition and the unitarity of Uσ in Ḣ 3/8 for any σ, we deduce that for any t [, τ ], u t Utu Ḣ 3/8 1 k t k1 t [ ] Ut r x v k r Ur k u k dr k= k Ḣ 3/8 t Ut r [u rφdw r] Ḣ 3/8 1 k t k1 t [ ] Uk 1 t r x v k r Ur k u k dr k= k Ut r [u rφdw r] Ḣ 3/8 Ḣ 3/8 Now, suppose that a is fixed with < a < 1/ in such a way that Proposition 3 holds, and set [ b = 1 a, so that b > 1/ hen, using the fact that Y 1, ] C[[ b, 1, ]; L Ḣ 3/8 for any positive 1,, we have for t [, τ ] and k =,, k t, k1 t k k C [ ] Uk 1 t r x v k r Ur k u k dr [ ] U r x v k r Ur k u k dr k [ ] U r x v k r Ur k u k dr Ḣ 3/8 C[k τ,k1 τ ];Ḣ 3/8 [k τ Y,k1 τ ] b, By Lemma 9, the above term is majorized for each k {,, k t } by C 1 a b { v C k [ v k x U k u k ] Y [k τ,k1 τ ] } a,, Y [k τ,k1 τ ] b, u k L by Proposition 3 and Lemma 9 again, since a b = 1 By the result of step 1, we obtain u t Utu Ḣ 3/8 K, t [, τ ], with K = 1 [ ] C 1 4K1 u C[, ];L U r [u rφdw r] C[, ];Ḣ 3/8

19 KDV EQUAION WIH HOMOGENEOUS MULIPLICAIVE NOISE 19 Step 3: By Lemma 9, and the unitarity of Uk on L Ḣ 3/8 we have Ut k u k Utu Y [k τ,k1 τ ] b, C U k u k u L Ḣ 3/8 herefore, using step, U k u k U u Y [k τ,k1 τ ] b, Finally, for t [k τ, k 1 τ ], we have C u k Uk u L Ḣ 3/8 K 3 = C v t = v k t Ut k u k Utu, and we may write, k being the integer part of /, v Y τ b, k v Y [k τ,k1 τ ] b, k= k v k [k τ Y,k1 τ ] k= b, 1 K 1 K 3 K u C[, ];L U k uk U u Y [k τ,k1 τ ] b, Note that 1 is proportional to K 1/1 ab 1 ; by Proposition 31 and Proposition 7, K 1 and K 3 have all moments finite, and it follows E v τ Y c, u L b, which concludes the proof of Lemma 3 It is now straightforward to achieve the proof of heorem 11 Indeed, due to Corollary 1, it suffices to see that lim sup τ = in probability as goes to infinity But this is an easy consequence of Markov inequality and Lemma 3, since P τ < = P v Y τ b, eferences [1] A Adams, Sobolev Spaces, Academic Press, New York, 1975 [] GG Bass, YS Kivshar, VV Konotop, YA Sinitsyn, Dynamics of solitons under random perturbations, Phys ep 157, 1988, [3] J Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part II, Geom Funct Anal 3, 1993, 9 6 [4] A de Bouard, A Debussche, On the stochastic Korteweg-de Vries equation, J Funct Anal 154, 1998, [5] A de Bouard, A Debussche, he stochastic nonlinear Schrödinger equation in H 1, Stochastic Anal Appl, 1, 3, [6] A de Bouard, A Debussche, andom modulation of solitons for the stochastic Korteweg-de Vries equation, o appear in Ann IHP, Analyse Non Linéaire [7] A de Bouard, A Debussche, Y sutsumi, White noise driven Korteweg-de Vries equation, J Funct Anal 169, 1999,

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