PERIODIC SOLUTIONS OF THE KORTEWEG-DE VRIES ... - CMAP

PERIODIC SOLUTIONS OF THE KORTEWEG-DE VRIES EQUATION DRIVEN BY WHITE NOISE A. DE BOUARD1 , A. DEBUSSCHE2 , AND Y. TSUTSUMI3

1

CNRS et Universit´e Paris-Sud, UMR 8628, Bˆat. 425, Universit´e de Paris-Sud, 91405 ORSAY CEDEX, FRANCE 2 ENS de Cachan, Antenne de Bretagne, Campus de Ker Lann, Av. R. Schuman, 35170 BRUZ, FRANCE 3 Mathematical Institute, Tohoku University, SENDAI 980-8578, JAPAN Abstract. We consider a Korteweg-de Vries equation perturbed by a noise term on a bounded interval with periodic boundary conditions. The noise is additive, white in time and “almost white in space”. We get a local existence and uniqueness result for the solutions of this equation. In order to obtain the result, we use the precise regularity of the Brownian motion in Besov spaces, and the method which was introduced by J. Bourgain, but based here on Besov spaces.

1. Introduction The Korteweg-de vries (KdV) equation, which models the propagation of unidirectional weakly nonlinear waves in an infinite channel, is an ideal model, and it is natural to consider perturbations of this model. In this direction, stochastic perturbations of this equation were introduced in [5], [12], [19] to model the propagation of weakly nonlinear waves in a noisy plasma. Here, we consider as in [2], [3], a KdV equation with a stochastic perturbation which is Gaussian and of white noise type in time. Contrary to the previous works [2] and [3], we will set the equation on a bounded space interval with periodic boundary conditions. Although the derivation of the KdV equation is usually done with x ∈ R, there is no reason to confine oneself to localized solutions. It is also well known that the KdV equation possesses spatially periodic traveling waves solutions. The study of the periodic boundary Key words and phrases. Korteweg-de Vries equation, stochastic partial differential equations, white noise, Besov spaces. 1

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A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

conditions case is also of importance when dealing with numerical computations, since these are necessarily performed on a bounded interval. Our aim in the present paper is to study the Cauchy problem for a stochastic KdV equation with an additive noise as previously described, and which has spatial correlations “as rough” as our techniques allow, the aim being to stay as close as possible to the space-time white noise. The equation is then written as ∂2B ∂t∂x where u is a random process defined for (t, x) ∈ R+ × T, T being a onedimensional torus, and φ is a bounded linear operator on L2 (T) that will be described in more details later. Also, B is a two parameter Brownian motion on R+ × T, that is a zero mean Gaussian process whose correlation function is given by E(B(t, x)B(s, y)) = (t ∧ s)(x ∧ y)

(1.1)

∂t u + ∂x3 u + u∂x u = φ

for t, s ≥ 0, x, y ∈ T. Note that in the case where φ is defined by a kernel k(x, y), then the correlation function of the noise is  2  ∂ B ∂2B E φ (t, x)φ (s, y) = c(x, y)δt−s ∂t∂x ∂t∂x with δ the Dirac δ-function and Z k(x, z)k(y, z)dz.

c(x, y) = T

In this formalism, the case φ = Id i.e. c(x, y) = δ(x − y) corresponds to the space-time white noise. This is the case we would like to treat. However, our result needs a slightly more restrictive assumption, and we are only able to treat a noise which is “almost” delta correlated in space. Except in [2] and [3], equations of the type (1.1) have essentially been studied by using inverse scattering theory (only in the case where the noise is space independent) or by perturbation arguments near the integrable case (see [12], [16], [21], [22]). A very large attention has been paid to the (deterministic) KdV equation on the real line (see [1], [4], [13], [18]) and improvements made on the regularity needed on the initial value to get local existence of solutions occurred step by step. On the opposite, for the periodic case, up to the famous work of Bourgain on the KdV equation (see [4]), existence results in H s (T) were restricted to the case s > 3/2. Then, using functions spaces based on the linear group, Bourgain was able to prove global well-posedness in L2 (T). Making

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

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use of the same spaces, and improving the nonlinear estimate, Kenig, Ponce and Vega (see [15]) proved local well-posedness in H s (T) for s > −1/2 (see Colliander et al. [7] for s = −1/2). After that, using a splitting into high and low Fourier frequencies of the solution, together with almost conserved quantities and rescaling arguments, Colliander et al. [7] were able to prove global existence in H s (T) for s ≥ −1/2. Using Bourgain’s type spaces, we were able in [3] to prove local existence of solutions for (1.1) in the real line case, when the noise is a “localized space-time white noise”, that is when its correlation function has the form  ∂2B ∂2B E φ (t, x)φ (s, y) = k(x)k(y)δx−y δt−s , ∂t∂x ∂t∂x 

k being an L2 function. It is indeed hopeless in the real line case to be able to get even local existence of solutions in H s (R), with a pure space time white noise. The obstruction is not due to the lack of spatial regularity of the noise, but to its homogeneity (see [3]). In the periodic case, however, there is no such obstruction, and we are able to treat homogeneous noises, i.e. noises whose spatial correlation function depends only on x − y (or such that φ is a convolution operator); also, thanks to the use of Bourgain’s method adapted to Besov spaces, we are able to treat noises which have spatial correlations in H s , s > −1/2. The main difficulty encountered in the application of Bourgain’s method in our case, is that it needs time regularity of order 1/2. However, it is well known that this regularity does not hold for Brownian motions unless Besov spaces are considered. This is why we use this method in the context of Besov spaces - see below for details. The problem of global existence of solutions for such noises in spaces with negative regularity is not considered here, but could probably be handled with the use of the method previously mentioned ([7]). Before stating our result precisely, we introduce a few notations and assumptions. ˜ (t) = ∂B a cylindrical Wiener process on L2 (T) which may We consider W ∂x ˜ (t) = P βj ej where (ej )j∈N is a complete orthonormal be written as W j∈N system in L2 (T), (βj )j∈N is a sequence of mutually independent real valued Brownian motions in a fixed probability space (Ω, F, P) associated with a filtration (Ft )t≥0 . ˜ is then a φφ∗ -Wiener process (recall that φ is a linear The process W = φW bounded operator in L2 (T)), i.e. (W (t))t≥0 is a Gaussian process with law (N (0, tφφ∗ )t≥0 ).

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A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

We then consider equation (1.1) in its Itˆo form (1.2)

du + (∂x3 u + u∂x u)dt = dW, x ∈ T, t ≥ 0,

supplemented with the initial condition (1.3)

u(0, x) = u0 (x), x ∈ T.

Consider the Fourier transform Z 1 ˆ f (n) = √ einx f (x)dx 2π T for functions f defined on T, and let for s ∈ R, H s (T) be the Sobolev space of functions f such that the norm !1/2 X |f |H s (T) := (1 + n2 )s |fˆ(n)|2 n∈Z s (T) as the space of is finite. We also define, for s ∈ R, the Besov space B2,1 functions f defined on T for which the norm  1/2 X X ˆ(0)| + s (T) = |f 2sn  |f |B2,1 |fˆ(n0 )|2  n∈N

2n−1 ≤|n0 |≤2n+1

is finite. 3 Let U (t) = e−t∂x be the group associated with the linear equation on L2 (T), that is v(t) = U (t)u0 satisfies   ∂ v + ∂3v = 0 t x  v(0, x) = u0 (x), x ∈ T. Then the solution of

  dw + ∂ 3 w dt = dW x  w(0, x) = 0, x ∈ T,

is given by the stochastic convolution Z t (1.4) w(t) = U (t − s)dW (s). 0

Note that U (t) is a unitary group on H s (T) for any s ∈ R, so that w(t) lies in H s (T) almost surely if and only if φφ∗ has finite trace from L2 (T) into H s (T). This clearly holds in the case φ is the identical operator on L2 (T) if and only if s < −1/2. The difficulty in the use of Bourgain’s spaces here is the smoothness in time. Indeed, let Y s,b be the space of functions f such that U (−t)f (t, ·) ∈ H s,b , where

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

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H s,b is a space-time Sobolev space, s being the regularity in space, and b the regularity in time (see [15] for a precise definition of Y s,b ). Then, as was proved in [15], the only possible value of b for which a bilinear estimate holds, which allows to handle the nonlinear term ∂x (u2 ) in the KdV equation using a straightforward iteration scheme, in the periodic case, is b = 1/2. Writing then the expression of w(t) defined by (1.4) as XZ t w(t) = U (t − s)(φej )dβj (s), j∈N

0

one can compute the spatial Fourier transform of h(t) = U (−t)w(t) : XZ t 3 ˆh(t, n) = cj (n)dβj (s). eisn φe j∈N

0

But, there is no hope that this term lives in H 1/2 [0, T ] in the time variable, because the Brownian motions βj do not. Indeed, Z t 2   X 3 2 2 ˆ c E |h(t, n)|H 1/2 eisn dβj (s) 1/2 = |φej (n)| E t

=

Ht

j∈N

0

X

( Z cj (n)|2 E |φe 0

j∈N

ZZ +E (0,T )2

T



Z

t

2 3 eisn dβj (s) dt

0

R t1 isn3 2 Rt 3 e dβj (s) − 2 eisn dβj (s) 0

0

|t1 − t2 |2

) dt1 dt2 .

2 P 2 c The first term in the right hand side above is obviously equal to T2 j∈N |φej (n)| while the contribution of each j to the second term in the right hand side above is infinite, due to the fact that Z t2 2 3 E eisn dβj (s) = |t2 − t1 |.

t1

However, H 1/2 is a limiting case concerning the regularity of the Brownian motion, as far as we are dealing with Sobolev spaces. It is then natural to try to replace Sobolev spaces here by other spaces which describe more precisely the regularity in time of the Brownian motions. This is exactly what we will do here, using Besov spaces instead of Sobolev spaces in time. Indeed, it is known 1/2 (see [6], [17]) that the Brownian motion lies almost surely in Bp,q ([0, T ]) if and only if 1 ≤ p < +∞ and q = +∞. Trying to derive some bilinear estimate which would allow us to handle in the same time both w(t) defined by (1.4)

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A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

and the nonlinear term, t

Z

U (t − s)(∂x (u2 )(s))ds,

0

we were led to consider also Besov spaces in the space variable. We now turn to give precise definitions of these spaces. We denote by h., .i the L2 space-time duality product, that is Z Z hf, gi = f (t, x)g(t, x)dt dx T Z R X g (τ, n)dτ = fˆ(τ, n)ˆ n∈Z

R

by the Plancherel formula; here, and in all what follows, we denote by fˆ (resp. gˆ) the Fourier transform of f (resp. g) with respect to both variables. We also use the notation hτ i = (1 + |τ |2 )1/2 , for τ ∈ R. The spaces that we will use are defined as follows. Consider first functions f defined on R × T such that f (., x) ∈ S 0 (R) for any x ∈ T, and such that fˆ(τ, 0) = 0 for any τ ∈ R. s,b We denote by X1,1 the space of such functions f for which in addition the norm !1/2 Z 2k+1 ∞ ∞ 2 X X X 3 |f |X s,b = 2sn hτ − n0 ib fˆ(τ, n0 ) dτ 1,1

n=0

+

∞ X

k=0 sn

2

X 2n−1 ≤|n0 |≤2n+1

n=0

2k−1

2n−1 ≤|n0 |≤2n+1 1

Z 0

2 0 03 b ˆ hτ − n i f (τ, n ) dτ

!1/2

s,b is finite. In the same way, we will denote by X1,∞ the space of such functions f for which in addition the norm !1/2 Z 2k+1 ∞ 2 X X 03 b ˆ 0 sn 2 sup |f | s,b = hτ − n i f (τ, n ) dτ X1,∞

n=0

+

∞ X n=0

k∈N

2sn

2k−1

2n−1 ≤|n0 |≤2n+1

X 2n−1 ≤|n0 |≤2n+1

Z 0

1

2 3 hτ − n0 ib fˆ(τ, n0 ) dτ

!1/2

is finite. The basic space in which we will solve the Cauchy problem for the stochastic s,b KdV equation is X1,1 . However, we will make use, at intermediate steps, of s,b s,b e1,∞ e1,1 other spaces of the same type : X (resp. X ) is the space of functions f s,b such that f (t, ·) = U (t)g(t, ·) with g in the “space-time Besov space” (B2,1 )x,t

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

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s,b s b (resp. (B2,1 )x (B2,∞ )t ), where (B2,1 )x,t is defined by the norm

|f |(B s,b )x,t =

∞ X ∞ X

2

2,1

+

∞ X

2k−1

2n−1 ≤|n0 |≤2n+1

n=0 k=0

Z

X

2sn

1

0

2n−1 ≤|n0 |≤2n+1

n=0

2k+1

Z

X

sn+kb

2 ˆ 0 f (τ, n ) dτ

2 ˆ f (τ, n0 ) dτ

!1/2

!1/2

s b and (B2,1 )x (B2,∞ )t is defined by the norm

|f |(B2,1 = s ) (B b x 2,∞ )t

∞ X

sup 2sn+kb

n=0 k∈N

+

∞ X

2n−1 ≤|n0 |≤2n+1

Z

X

2sn

n=0

Z

X

2n−1 ≤|n0 |≤2n+1

0

1

2k+1

2k−1

2 ˆ 0 f (τ, n ) dτ

2 ˆ 0 f (τ, n ) dτ

!1/2

!1/2 .

s,b s,b e1,1 Remark 1.1. Note that the spaces X1,1 and X are different and there is no s,b e1,1 inclusion relation between them: an alternative definition of the norm in X is !1/2 Z ∞ ∞ 2 X X X 3 2sn |f |Xe s,b = hτ − n0 ib fˆ(τ, n0 ) dτ 1,1

n=0

+

k=0

∞ X

2sn

n=0

2n−1 ≤|n0 |≤2n+1

2k−1 ≤|τ −n0 3 |≤2k+1

Z

X 2n−1 ≤|n0 |≤2n+1

|τ −n0 3 |≤1

2 0 03 b ˆ hτ − n i f (τ, n ) dτ

!1/2 ;

here, the dyadic decomposition is made on |τ − n0 3 | and not on |τ |. However, embeddings do hold between these spaces with some small loss of space regularity, as is stated in Lemma 1.1, at the end of this section. Since all those definitions have to be used only locally in time, we will s,b,T s,b,T actually consider, for T ≥ 0 fixed, the spaces X1,1 and X1,∞ of restrictions s,b s,b on [0, T ] of functions of X1,1 (resp. X1,∞ ). They are endowed with the natural norm n o |f | s,b,T = inf |f˜| s,b , f˜ ∈ X s,b and f = f˜| , X1,1

X1,1

1,1

[0,T ]

s,b,T and the equivalent for X1,∞ . To handle the integral estimate in Duhamel’s formula, we will need to make use, as is classical, of another space which is defined as the space of zero

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A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

(spatial) mean functions with finite corresponding norm, where ! ∞  Z |fˆ(τ, n0 )| 2 1/2 X X |f |Ys = 2sn dτ . 03i hτ − n R n−1 0 n+1 n=0 ≤|n |≤2

2

s,1/2,T

A local space Ys,T is also defined, in the same way as for X1,1 . 0 n n−1 In all the paper, we will use the notation |n | ∼ 2 for 2 ≤ |n0 | ≤ 2n+1 , and |τ | ∼ 2k for 2k−1 ≤ |τ | ≤ 2k+1 if k ≥ 1 and |τ | ≤ 2 if k = 0. As previously mentioned, we will be led to assume1 that the operator φ is a Hilbert-Schmidt operator (or equivalently that φφ∗ has finite trace) from L2 (T) into H s (T) for some negative s with s > −1/2. We will denote by L0,s 2 the space of such operators, which is endowed by its natural norm : !1/2 X kφkL0,s = |φei |2H s 2

i∈N

where (ei )i∈N is any complete orthonormal system in L2 (T). For convenience, in all what follows, we take as (ei )i∈N the usual complete orthonormal system of L2 (T) given by e2k (x) =

√1 π

e2k+1 (x) =

cos kx, k ≥ 1, e0 (x) =

√1 π

√1 2π

sin kx.

We consider the mild form of equations (1.2), (1.3), that is Z t Z 1 t 2 U (t − s)dW (s). U (t − s)∂x (u (s))ds + (1.5) u(t) = U (t)u0 − 2 0 0 Our main result, which concerns local existence in a situation where W is arbitrarily close to a cylindrical Wiener process, is the following. Theorem 1.1. Assume that Im φ ⊂ span {ej , j ≥ 1} and that φ ∈ L0,s 2 for some s with s > −1/2. Let u0 be F0 -measurable, with u0 in the Besov σ space B2,1 (T) a.s., for some σ with −1/2 ≤ σ < s; then there is a stopping time Tω > 0 and a unique process u solution of the forced KdV equation (1.5) which satisfies σ,1/2,T σ (T)) ∩ X1,1 ω a.s. u ∈ C([0, Tω ]; B2,1 Remark 1.2. The assumption Im φ ⊂ span {ej , j ≥ 1} says that the spatial mean of the noise is zero at any time. This assumption is necessary to perform 1note

that this assumption excludes the identical operator on L2 (T)

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

9

the fixed point procedure, because we work in a space of functions with zero spatial mean. We will actually remove this assumption at the end of the paper (see Proposition 4.2) by changing the unknown function u and the noise. At that place, we will have to deal with a non Gaussian noise. Remark 1.3. One can show by using classical arguments, and looking more carefully into the proof of Proposition 3.1 (see Section 3) that the regularity σ is preserved in Theorem 1.1, i.e. if φ ∈ L0,s 2 and u0 ∈ B2,1 (T) with −1/2 ≤ σ σ0 (T) (T) and in B2,1 σ 0 ≤ σ < s, then the existence times of the solution in B2,1 are the same. Naturally, when the noise is such that the Wiener process lies in L2 (T), we get a global existence result thanks to the invariance of the L2 norm for the σ deterministic equation and the embedding L2 (T) ⊂ B2,1 (T), for any σ < 0. Theorem 1.2. Assume that in addition, φ ∈ L20,0 ; then if u0 ∈ L2 (Ω; L2 (T)), the solution given by Theorem 1.1 is globally defined in time and lies in σ L2 (Ω; L∞ (0, T ; L2 (T))) and in C(R+ ; B2,1 (T)) a.s. for any T > 0 and σ < 0. As was previously mentioned, Theorem 1.1 allows to handle a situation arbitrarily close to the space time white noise case, since this latter case corresponds to φ = id, which is a Hilbert-Schmidt operator from L2 (T) into H s (T) for any s < −1/2. Theorem 1.1 will be proved by using a fixed point argument σ,1/2,T in the space X1,1 for T small enough. We need the assumption s > −1/2 because we will need that s > σ ≥ −1/2. Indeed, to show that the fixed point works, we will first prove that the stochastic integral lies almost surely σ,1/2 in X1,∞ . At that point, we have already lost some spatial regularity. We then prove a bilinear estimate allowing us to handle such a term as ∂x (f g) σ,1/2 σ,1/2 with f ∈ X1,1 and g ∈ X1,∞ . To treat the term ∂x (g 2 ) in the same space, we again have to sacrifice an arbitrarily small amount of spatial regularity. It is not difficult to see that when φ = id, the stochastic integral w(t) given −1/2,1/2 by (1.4) lies almost surely in X∞,∞ , where this latest space is defined by −1/2,1/2 changing the norm in the definition of X1,∞ in an obvious way. Unfortu−1/2,−1/2 nately, a bilinear estimate which would handle terms like ∂x (g 2 ) in X∞,∞ −1/2,1/2 with g in X∞,∞ seems to fail. The paper is organized as follows. In Section 2, we prove an estimate which σ,1/2 shows that the stochastic integral lives in X1,∞ almost surely when φ is in L0,s with σ < s (we will actually prove that the stochastic integral lies in 2

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A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

σ,1/2 e1,∞ X , which is enough thanks to Lemma 1.1 below). This result is based on the works of Cieselskii and Roynette ([6], [17]), but we will use a different characterization of Besov spaces than in [17]. In Section 3, we prove some bilinear estimates which are needed in the σ,−1/2 proof of Theorem 1.1. The main one is an estimate of ∂x (f g) in X1,1 when σ,1/2 σ,1/2 f ∈ X1,1 and g ∈ X1,∞ . Other easier bilinear estimates are proved in that Section too. Section 4 is devoted to the proof of Theorems 1.1 and 1.2. Once we have the bilinear estimates in hand, together with the estimate on the stochastic integral, it mainly remains to proveR that we gain one degree of regularity in t time when passing from ∂x (f g) to 0 U (t − s)∂x (f g)(s)ds. The proof of this fact has to be done because we do not stand in the usual context of Sobolev spaces, but we deal with Besov spaces. However, the proof closely follows that of the Sobolev case. s,b We end the present section by giving the lemma relating the spaces X1,1 s,b e1,∞ and X .

Lemma 1.1. For any s1 > s2 > s3 , s1 ,b s2 ,b s3 ,b e1,1 e1,1 X ⊂ X1,1 ⊂X

s1 ,b s2 ,b s3 ,b e1,∞ e1,∞ and X ⊂ X1,∞ ⊂X .

s1 ,b s2 ,b e1,∞ Proof. We only show that X ⊂ X1,∞ , all the other embeddings are proved s ,b s2 ,b 1 e1,∞ and let us decompose the norm of f in X1,∞ similarly. Let f ∈ X as

|f |X s2 ,b ≤

X

2s2 n sup

1,∞

k3n+4

0 3 2b

|τ |∼2k

|n0 |∼2n

sup

2s2 n sup

!1/2

X Z

|n0 |∼2n

X Z |n0 |∼2n

|τ |∼2k

!1/2 3 hτ − n0 i2b |fˆ(τ, n0 )|2 dτ

|τ |∼2k

!1/2 3 hτ − n0 i2b |fˆ(τ, n0 )|2 dτ

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

11

If k > 3n + 4, |n0 | ∼ 2n and |τ | ∼ 2k , then 18 |τ | ≤ |τ − n0 3 | ≤ 32 |τ |, hence we easily have !1/2 X X Z 3 III ≤ C 2s2 n sup hτ − n0 i2b |fˆ(τ, n0 )|2 dτ k∈N

n∈N

|n0 |∼2n

|τ −n0 3 |∼2k

≤ C|f |Xe s2 ,b . 1,∞

On the other hand, if k < 3n − 4, |n0 | ∼ 2n and |τ | ∼ 2k , then 23n−4 ≤ |τ − n0 3 | ≤ 23n+4 ; hence !1/2 X X Z 3 I ≤ 2s2 n hτ − n0 i2b |fˆ(τ, n0 )|2 dτ |n0 |∼2n

n∈N

23n−4 ≤|τ −n0 3 |≤23n+4

≤ 8|f |Xe s2 ,b . 1,∞

Finally, if 3n−4 ≤ k ≤ 3n+4, |n0 | ∼ 2n and |τ | ∼ 2k , then 0 ≤ |τ −n0 3 | ≤ 23n+6 , hence !1/2 X X 3n+5 XZ s2 n 0 3 2b ˆ 0 2 2 hτ − n i |f (τ, n )| dτ II ≤ |n0 |∼2n k=0

n∈N

≤

X

2s2 n (3n + 5) sup k∈N

n∈N

≤C

|τ −n0 3 |∼2k

X n∈N

2s1 n sup k∈N

!1/2

X Z |n0 |∼2n

hτ − n i |fˆ(τ, n0 )|2 dτ

|τ −n0 3 |∼2k

!1/2

X Z |n0 |∼2n

0 3 2b

3

hτ − n0 i2b |fˆ(τ, n0 )|2 dτ 03

|τ −n |∼2k

since s2 < s1 . The result follows.



2. Estimate on the stochastic integral In this section, we prove an estimate on the stochastic integral – that is the last term in (1.5) – which will enable us to use a fixed point procedure to solve (1.5) in an appropriate space of functions of the space and time variables. This σ,1/2 latest space will actually be of the form X1,1 for some well chosen σ. Although, for the sake of clarity, we did not assume that the covariance operator φφ∗ of the noise could be random or could depend on the time variable t in Theorem 1.1, we will state here a proposition where φ is allowed to depend 0 both on t and ω, but under the condition that the L0,σ norm of φ(·) is bounded 2 in both t and ω. This will indeed be useful in order to prove that our result generalizes to the case where the noise does not have a zero spatial mean value (see Proposition 4.2).

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A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

We need to use a cut-off function in the time variable: we consider a function θ : R −→ R+ such that θ(t) ≡ 0 for t ≤ −1, and t ≥ 2, θ(t) ≡ 1 for t ∈ [0, 1], and θ ∈ C0∞ (R). Also, to state precisely our estimate on the stochastic integral, we define for n ∈ N, the operator ∆n acting on L2 (T) by 0 d ∆ ˆ(n0 ), n u(n ) = 1l{2n−1 ≤|n0 |≤2n+1 } u

for u ∈ L2 (T) and for any n0 ∈ Z. We now state our proposition. Proposition 2.1. Let s0 ∈ R, and assume that φ is predictable and lies in 0 L∞ ([0, T ] × Ω; L0,s 2 ) for some T with 0 < T ≤ 1; let θ and ∆n be as above; then the stochastic integral w(t) defined by (1.4) satisfies for any σ 0 < σ < s0 : σ 0 ,1/2,T θw ∈ L1 (Ω; X1,∞ ) and   X k∆n φkL∞ ([0,T ]×Ω;L0,σ ) E |θw|X σ0 ,1/2,T ≤ C(θ) 2

1,∞

n∈N 0

≤ C(θ, σ, s )kφkL∞ ([0,T ]×Ω;L0,s0 ) 2

where C(θ) is a constant depending only on the function θ. σ,1/2,T

e1,∞ ) and that Proof. We first prove that θw ∈ L1 (Ω; X   X k∆n φkL∞ ([0,T ]×Ω;L0,σ ) E |θw|Xe σ,1/2,T ≤ C(θ) 2

1,∞

n∈N

and then make use Rof Lemma 1.1. t Let g(t, ·) = θ(t) 0 U (−s)dW (s) so that θ(t)w(t) = U (t)g(t, ·); we also set for s ∈ R, n ∈ Z and ` ∈ N :  0 if s < 0 or s ≥ T ϕn,` (s) = \` (n) if s ∈ [0, T ] φ(s)e and we assume that each β` has been extended to a Brownian motion on R, in such a way that the family (β` )`∈N is still an independent family. We then have, for any t ∈ [0, T ] and n ∈ Z: X Fn g(t)(n) = θ(t)In,` (t) `∈N

with In,` (t) = space.

Rt −∞

3

θ(s)eins ϕn,` (s) dβ` (s), Fn being the Fourier transform in

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

13

σ,1/2 e1,∞ In view of the equivalent definition of the space X , we have to show that ! ∞  X Z 1/2 X E sup 2σn+k/2 |ˆ g (τ, n0 )|2 dτ

(2.1)

+E

n=0 k∈N ∞  X σn

2

n=0 ∞ X

≤ C(θ)

|τ |∼2k

|n0 |∼2n

X Z

|ˆ g (τ, n0 )|2 dτ

1/2

!

|τ |≤1

|n0 |∼2n

k∆n φkL∞ ([0,T ]×Ω;L0,σ ) . 2

n=0

We first estimate the ∞  X Z X σn E 2 |n0 |∼2n

n=0

≤

∞ X

2σn

≤

2σn

`∈N

Z

Z XZ θ(t)

|τ |≤1

X Z |n0 |∼2n

n=0

|τ |≤1

E

|n0 |∼2n

n=0 ∞ X

X

second term in (2.1). ! X 2 1/2 [ θI n0 ,` (τ ) dτ

|τ |≤1

`∈N

R

t

isn0 3

θ(s)e

ϕn0 ,` (s) dβ` (s)e−iτ t

−∞

Z XZ isn0 3 θ(s)ϕn0 ,` (s)e E `∈N

+∞

θ(t)e−iτ t

s

R

2 dt dτ

!1/2

2 dt dβ` (s) dτ

and using the independence of the (β` )`∈N , the above term is bounded by !1/2 ∞ 2 Z +∞ X X Z XZ
2σn E θ2 (s)|ϕn0 ,` (s)|2 θ(t)e−iτ t dt ds dτ |τ |≤1 `∈N R +∞ X X X 2 2 2σn sup E 2|θ|L1 (R) |θ|L2 (R) s∈R n=0 `∈N |n0 |∼2n +∞   X 1/2 C(θ) sup E k∆n φ(s)k2L0,σ 2 n=0 s∈R +∞ X

n=0

≤ ≤

s

|n0 |∼2n

≤ C(θ)

 1/2 |ϕn0 ,` (s)|2

k∆n φ(·)kL∞ ([0,T ]×Ω;L0,σ ) , 2

n=0

and this proves the estimate on the second term in (2.1). In what follows, we assume that |τ | ≥ 1/2 ; by the stochastic Fubini Theorem and using an integration by parts, we easily get for n ∈ Z, ` ∈ N and |τ | ≥ 1/2: d θI n,` (τ ) = An,` (τ ) + Bn,` (τ ) with Z An,` (τ ) = R

3

θ2 (s)eisn ϕn,` (s)

e−iτ s dβ` (s) iτ

!1/2

14

A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

and Z

isn3

Bn,` (τ ) =

θ(s)e

+∞

Z

θ0 (t)

ϕn,` (s) s

R

e−iτ t dt dβ` (s). iτ

Hence, two terms will be involved in the estimate of the second term in (2.1), which are ! ∞ ∞ X 2 1/2  X Z X I=E sup 2σn+k/2 An0 ,` (τ ) dτ n=0 k∈N

|n0 |∼2n

|τ |∼2k

`=0

and II = E

∞ X

sup 2σn+k/2

 X Z

n=0 k∈N

|n0 |∼2n

|τ |∼2k

! ∞ X 2 1/2 Bn0 ,` (τ ) dτ . `=0

We may assume that σ = 0, replacing in the estimate we want to prove ϕn0 ,` by 2σn ϕn0 ,` . We first estimate the second term above. With this aim in view, we first write for k ≥ 0 and n ≥ 0: ! ∞ X 2 X Z E 2k Bn0 ,` (τ ) dτ |n0 |∼2n

|τ |∼2k

`=0

Z ∞ Z X isn0 3 θ(s)e ϕn0 ,` (s) E

+∞

2 e−iτ t dt dβ` (s) dτ iτ s |τ |∼2k |n0 |∼2n `=0 R Z +∞ 2 Z Z ∞ −iτ t X X
e θ0 (t) = 2k |θ(s)|2 E |ϕn0 ,` (s)|2 dt ds dτ iτ |τ |∼2k `=0 0 s R n = 2k

Z

X

θ0 (t)

|n |∼2

where we have used again the independence of the family (β` )`≥0 . Now, for |τ | in [2k−1 , 2k+1 ], we have Z +∞ 0 Z +∞ −iτ t −iτ t C(θ) e θ (s) e 0 00 ≤ + ≤ θ (t) dt θ (t) dt ; τ2 iτ τ2 22k s s hence we get for k, n ≥ 0 : E 2k

X Z |n0 |∼2n Z |τ |∼2

k

≤ C(θ)2−3k

|τ |∼2k

∞ X 2 Bn0 ,` (τ ) dτ

!

`=0

  E k∆n φ(s)k2L0,0 2

≤ C(θ)2−2k k∆n φ(·)k2L∞ ([0,T ]×Ω;L0,0 ) , 2

L∞ s

dτ

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

and using Cauchy-Schwarz inequality, we get

II =

+∞ X

k/2

sup E 2

 X Z

n=0 k∈N

≤ (2.2)

∞ X X

 E 2k

n=0 k∈N ∞ X X

≤ C(θ) ≤ C(θ)

|n0 |∼2n

∞ X

X Z |n0 |∼2n

|τ |∼2k

|τ |∼2k



∞ X 2 1/2 Bn0 ,` (τ ) dτ

!

`=0

2  Bn0 ,` (τ ) dτ

!1/2

`=0

2−k k∆n φ(·)kL∞ ([0,T ]×Ω;L0,0 ) 2

n=0 k∈N ∞ X

k∆n φ(·)kL∞ ([0,T ]×Ω;L0,0 ) . 2

n=0

Our aim is now to estimate I. We set

Z

s

03

θ2 (t)eitn ϕn0 ,` (t)

An0 ,` (τ, s) = −∞

e−iτ t dβ` (t) iτ

so that Z

+∞

An0 ,` (τ ) =

dAn0 ,` (τ, s). −∞

Moreover, using the Itˆo formula, we have

+∞ X 2 Z X 2 d An0 ,` (τ, t) An0 ,` (τ ) = R

`=0 ∞ Z Z X

t

`∈N

e−iτ s 03 θ2 (s)eisn ϕn0 ,` (s) dβ` (s) iτ R −∞ `,m=0 ! iτ t XZ ϕ2n0 ,` (t) e 4 2 −itn0 3 0 ×θ (t)e ϕn ,m (t) dβm (t) + θ (t) dt −iτ τ2 `∈N R = In10 (τ ) + In20 (τ ). = 2Re

15

16

A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

Hence, again, two terms are involved in the estimate of I. The estimate of the second term is immediate. Indeed, we have

∞ X

E

k/2

sup 2

 X Z

n=0 k∈N ∞ X

sup 2k/2

=E

 X Z

n=0 k∈N

(2.3)

≤ C(θ) ≤ C(θ)

∞ X

|τ |∼2k

|n0 |∼2n

In20 (τ )dτ

|n0 |∼2n

n=0 k∈N ∞ X

!

1/2 ϕ2n0 ,` (τ ) θ4 (t) dt dτ τ2 R

XZ

|τ |∼2k `∈N

Z

sup 2k/2

1/2

!

1 X X 2 dτ ϕn0 ,` (·) 2 τ L∞ ([0,T ]×Ω) 0 n `∈N

|τ |∼2k

!1/2

|n |∼2

k∆n φ(·)kL∞ ([0,T ]×Ω;L0,0 ) . 2

n=0

In order to estimate the contribution of the stochastic integral, i.e. of In10 (τ ) in the bound of I, we start with the following estimate. (2.4) ! X Z 2 E In10 (τ )dτ |n0 |∼2n

|τ |∼2k

Z X =E |n0 |∼2n

2Re

∞ Z Z  X

|τ |∼2k

03

θ2 (s)eisn ϕn0 ,` (s)

−∞

R

`,m=0

t

! iτ t
e 2 dβm (t) dτ ×θ2 (t)e−itn ϕn0 ,m (t) −iτ Z ∞ Z ∞ Z t X  X X 2 −i(t−s)n0 3 =E 2Re θ (s)e

e−iτ s dβ` (s) iτ

03

m=0

|τ |∼2k

−∞

R |n0 |∼2n `=0

2  ×θ2 (t) ϕn0 ,` (s) dβ` (s)ϕn0 ,m (t) dβm (t) =

∞ Z X m=0

R

∞ Z X X E |n0 |∼2n `=0

t



2

2

eiτ (t−s) dτ τ2

!

−i(t−s)n0 3

Z

2Re θ (s)θ (t)e

−∞

2 ×ϕn0 ,` (s)dβ` (s)ϕn0 ,m (t) dt

|τ |∼2k

eiτ (t−s)  dτ τ2

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

17

where we have used again the independence of the family (βm )m∈N . Using now Cauchy-Schwarz inequality in n0 , the above term is bounded by (2.5) ∞ Z ∞ Z t  X X X 03 E 2Re θ2 (s)θ2 (t)e−i(t−s)n m=0

R

`=0 |n0 |∼2n iτ (t−s)

Z

e

×

τ2

|τ |∼2k

≤

[0,t]×Ω

E R

! |ϕn0 ,m (t)|2

|n0 |∼2n m=0

Z ×

! 2 X dτ ϕn0 ,` (s)dβ` (s) ϕ2n0 ,m (t) dt 

|n0 |∼2n

∞ X

X

sup

−∞

∞ Z X

X |n0 |∼2n



2

2

−i(t−s)n0 3

Z

2Re θ (s)θ (t)e

−∞

`=0 !2

×ϕn0 ,` (s) dβ` (s)

t

|τ |∼2k

eiτ (t−s)  dτ τ2

dt.

Concerning the first term in the right hand side above, we have ∞ X X

sup [0,T ]×Ω

|ϕn0 ,m (t)|2

|n0 |∼2n m=0 ∞ X X

≤ sup

[0,T ]×Ω m=0 ∞ X

≤ sup

[0,T ]×Ω m=0

|φ(t)em (n0 )|2

|n0 |∼2n

|∆n φ(t)em |2L2 (T) = k∆n φ(·)k2L∞ ([0,T ]×Ω;L0,0 ) 2

while the remaining term in (2.5) is bounded above by ∞ Z X X

t


eiτ (t−s) 2 2 0 ,` (s)| dτ E |ϕ ds dt n τ2 R |n0 |∼2n `=0 −∞ |τ |∼2k Z Z t Z eiτ (t−s) 2 4 4 2 dτ ds dt. ≤ k∆n φ(·)kL∞ ([0,T ]×Ω;L0,0 ) θ (t)θ (s) 2 τ2 R −∞ |τ |∼2k Z

Z θ (t)θ (s) 4

4

We then notice that, by interpolation between the cases α = 0 and α = 1, for any α ∈ [0, 1] there is a positive constant Cα such that Z

|τ |∼2k

eiτ (t−s) Cα dτ ≤ 2−(1+α)k . 2 τ |t − s|α

18

A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

Applying this with α = 1/4, we get that the second term in (2.5) is bounded above by Z Z t 4 θ (t)θ4 (s) − 25 k 2 √ C2 k∆n φ(·)kL∞ ([0,T ]×Ω;L0,0 ) ds dt 2 t−s R −∞ 5 ≤ C(θ)2− 2 k k∆n φ(·)k2L∞ ([0,T ]×Ω;L0,0 ) . 2

Collecting all these estimates from (2.4), we get ! 2 X Z 5 E In10 (τ )dτ ≤ C(θ)2− 2 k k∆n φ(·)k4L∞ ([0,T ]×Ω;L0,0 ) 2

|τ |∼2k

|n0 |∼2n

and we deduce from this latest inequality that ∞ X

E

sup 2k/2

n=0 k∈N ∞ X ∞ X

≤E

 X Z |n0 |∼2n

2k/2

≤

 X Z

2k/2 E

n=0 k=0 ∞ X

∞ X

n=0

k=0

≤ C(θ)

|τ |∼2k

|n0 |∼2n

"

In10 (τ )dτ

 X Z

n=0 k=0 ∞ X ∞ X

|τ |∼2k

|τ |∼2k

|n !0 |∼2n k

2− 8

1/2

In10 (τ )dτ In10 (τ )dτ

!

1/2

2

!

!#1/4

k∆n φ(·)kL∞ ([0,T ]×Ω;L0,0 ) 2

where we have used H¨older’s inequality at the third line; this, together with (2.3), completes the proof of the estimate of I. In this way, the first inequality in Proposition 2.1 is proved after an application of Lemma 1.1, with σ 0 < σ. The second inequality follows from the obvious fact that ∞ X

k∆n φ(·)k2L∞ ([0,T ]×Ω;L0,σ ) 2

n=0

= ≤

∞ X

2σn k∆n φ(·)k2L∞ ([0,T ]×Ω;L0,0 )

n=0 ∞ X n=0 0

2

!1/2 −2(s0 −σ)n

2

∞ X

!1/2 s0 n

2

n=0

k∆n φ(·)k2L∞ ([0,T ]×Ω;L0,0 ) 2

≤ C(s , σ)kφ(·)kL∞ ([0,T ]×Ω;L0,s0 ) . 2



STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

19

3. Bilinear estimates We now turn to prove some bilinear estimates which will allow us to handle the nonlinear term in equation (1.5). The next one is the crucial estimate. s,1/2

s,1/2

Proposition 3.1. Let − 21 ≤ s ≤ 0 and f ∈ X1,1 , g ∈ X1,∞ ; then ∂x (f g) ∈ s,−1/2 X1,1 and there is a constant C > 0 such that |∂x (f g)|X s,−1/2 ≤ C|f |X s,1/2 |g|X s,1/2 . 1,1

1,1

1,∞

Proof. Let f and g be as above; using a duality argument, it is sufficient, as usually, to prove that for some constant C > 0, and for any function h −s,1/2 −s,1/2 in X∞,∞ – where X∞,∞ is defined in an obvious way by modifying the −s,1/2 definition of X1,1 – we have |h∂x (f g), hi| ≤ C|f |X s,1/2 |g|X s,1/2 |h|X∞,∞ −s,1/2 . 1,1

1,∞

Using Plancherel Theorem, one has Z X X Z ¯ ˆ n0 )ˆ |h∂x (f g), hi| = n0 h(τ, g (τ1 , n01 )fˆ(τ − τ1 , n0 −n01 )dτ1 dτ . 0 τ ∈R τ1 ∈R 0 n 6=0

n1 6=0 n01 6=n0

We will denote σ = σ(τ, n0 ) = τ − n0 3 , σ1 = σ(τ1 , n01 ), σ2 = σ(τ − τ1 , n0 − ˆ n0 ) = n0 s hσi1/2 gˆ(τ, n0 ), Fˆ (τ, n0 ) = n0 s hσi1/2 fˆ(τ, n0 ) and n01 ). We also set G(τ, ¯ˆ 0,0 0,0 ˆ n0 ) = n0 −s hσi1/2 h(τ, n0 ), so that F , G and H lie respectively in X1,1 , X1,∞ H(τ, 0,0 . It suffices to prove that and X∞,∞ (3.1) Z ˆ τ,n0 ||G ˆ τ ,n0 ||Fˆτ −τ ,n0 −n0 | X X Z |n0 |1+s |n01 |−s |n0 − n01 |−s |H 1 1 1 1 dτ1 dτ 1/2 hσ i1/2 hσ i1/2 hσi 1 2 τ ∈R τ ∈R 0 1 0 n 6=0 n1 6=0 n01 6=n0

0,0 |G| 0,0 |F | 0,0 , ≤ C|H|X∞,∞ X X 1,∞

1,1

ˆ τ,n0 for H(τ, ˆ n0 ) and so on. We divide the region (n0 , n0 , τ, τ1 ) ∈ where we use H 1 (Z \ {0})2 × R2 arising in the left hand side of (3.1) into three subregions: (Region I)

hσ1 i = max{hσi, hσ1 i, hσ2 i}

(Region II)

hσi = max{hσi, hσ1 i, hσ2 i}

(Region III)

hσ2 i = max{hσi, hσ1 i, hσ2 i}

20

A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

and we estimate separately the contributions of each of these regions to the left hand side of (3.1). Region I. From the identity 3

3

3|n0 | |n01 | |n0 − n01 | = |τ − n0 − (τ1 − n01 ) − ((τ − τ1 ) − (n0 − n01 )3 )| we get as usually that in region I, 1 02 |n | ≤ |n0 | |n01 | |n0 − n01 | ≤ hσ1 i 2 so that for any s ∈ [− 21 , 0], |n0 |1+s |n01 |−s |n0 − n01 |−s ≤ Chσ1 i1/2 . Hence, it is sufficient to prove that the contribution of region I to I=

X X Z n0 6=0 n01 6=0, n01 6=n0

ˆ τ,n0 ||G ˆ τ ,n0 ||Fˆτ −τ ,n0 −n0 | |H 1 1 1 1 dτ1 dτ 1/2 1/2 hσi hσ2 i

Z

τ ∈R

τ1 ∈R

0,0 |G| 0,0 |F | 0,0 . Again, we will divide region I is bounded above by C|H|X∞,∞ X1,∞ X1,1 into several subregions.

Region I-a. We consider here the subregion for which hσi ≥ 41 n0 2 . We then estimate the contribution of this region to I; it is bounded above by its contribution to (3.2)

X

X

n,n1 ∈N k,k1 ∈N

X

Z

Z

|τ |∼2k

|n0 |∼2n |n01 |∼2n1 n0 6=n01

|τ1 |∼2k1

ˆ τ,n0 | |G ˆ τ ,n0 | |Fˆτ −τ ,n0 −n0 | |H 1 1 1 1 dτ1 dτ 1/2 1/2 hσi hσ2 i

with the convention that for k = 0, |τ | ∼ 2k means |τ | ≤ 2. This latest term is bounded above, using Cauchy-Schwarz inequality, by !1/2 X X X Z ˆ τ ,n0 |2 dτ1 |G (3.3)

|n01 |∼2n1

n,n1 ∈N k,k1 ∈N

×

X Z |n01 |∼2n1

|τ1 |∼2k1

|τ1 |∼2k1

1

 X Z |n0 |∼2n

|τ |∼2k

1

ˆ τ,n0 | |Fˆτ −τ ,n0 −n0 | 2 |H 1 1 dτ dτ1 hσi1/2 hσ2 i1/2

!1/2 .

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

21

Now, we use the fact that in region I-a, we have for ε > 0 small, which will be chosen more precisely later, 1 1 ≤ C|n0 |−ε 0 1/2 0 hσ(τ, n )i hσ(τ, n i)1/2−ε/2 and using Cauchy-Schwarz inequality in (τ, n0 ) in (3.3), it is bounded above by (3.4) !1/2 X X X Z ˆ τ ,n0 |2 dτ1 C |G 1 1 !

X Z

|τ1 |∼2k1

|n01 |∼2n1

|τ1 |∼2k1

|n01 |∼2n1

"n,n1 ∈N k,k1Z∈N X ×

|n0 |∼2n

0 −2ε

|n |

|τ |∼2k

ˆ τ,n0 | |Fˆτ −τ ,n0 −n0 | dτ hστ,n0 i |H 1 1 −ε

2

2

! #1/2 dτ dτ1 × 1−2ε hσ k hστ,n0 i τ −τ1 ,n0 −n01 i |τ ∼2 0 n |n |∼2 !1/2 " X Z X ˆ τ ,n0 |2 dτ1 |G sup ≤C 1 1 X Z

n1 ∈N

k1 ∈N

× sup

|τ1 |∼2k1

|n01 |∼2n1

sup

X Z

sup

k1 ,n,k∈N |n01 |∼2n1 |τ1 |∼2k1

×

X Z

X k1 ,n,k∈N

|n0 |∼2n

X Z

|τ1 |∼2k1 |n0 |∼2n

|n01 |∼2n1

|τ |∼2k

dτ 1−2ε hστ,n0 i hστ −τ1 ,n0 −n01 i

!1/2

ˆ τ,n0 |2 |n0 |−2ε hστ,n0 i−ε |H

|τ |∼2k

!1/2 # ×|Fˆτ −τ1 ,n0 −n01 |2 dτ dτ1 But now, the fact that Z +∞ −∞

(1 +

.

dθ |θ|)1−2ε (1

+ |θ − a|)

≤

C (1 + |a|)1−4ε

for a ∈ R, and the proof of Lemma 5.1 in [15] show that there is a constant C > 0 such that ! X Z dτ sup sup sup 1−2ε hσ k hστ,n0 i n1 ∈Z∗ τ1 ∈R n,k∈N τ −τ1 ,n0 −n1 i |n0 |∼2n |τ |∼2 ! X Z dτ ≤ sup sup 1−2ε hσ n1 ∈Z∗ τ1 ∈R τ −τ1 ,n−n1 i τ ∈R hστ,n i n∈Z\{0} n6=n1

≤ C,

22

A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

for any ε > 0 such that 1 − 4ε ≥ 3/4, i.e. for any ε ≤ 1/16. On the other hand the last line in (3.4) is bounded above by

"

× sup

sup

X

|n0 |∼2n

0,0 ≤ Cε |H|X∞,∞ X × sup

n1 ,k1 ∈N

|τ1 |∼2k1

|n01 |∼2n1

|n0 |−ε hσi−ε/2 |Fˆτ −τ1 ,n0 −n01 |2 dτ1

1/2

!1/2

X Z

n,k∈N

|τ1 |∼2k1

|n0 |−ε hσi−ε/2 |Fˆτ −τ1 ,n0 −n01 |2 dτ1

 X Z

sup

X

n,k∈N

|n01 |∼2n1

0 n n1 ,k1 ∈N |n |∼2 k |τ |∼2

n,k∈N

!1/2 #

X Z

|n0 |∼2n |τ |∼2k

×

ˆ τ,n0 |2 dτ |n0 |−ε hσi−ε/2 |H

|τ |∼2k

|n0 |∼2n

n,n1 ∈N k,k1 ∈N

≤ sup

!1/2

X Z

X

ˆ τ,n0 |2 dτ |n0 |−ε hσi−ε/2 |H

|τ |∼2k

 X Z

sup |n0 |∼2n

|n01 |∼2n1

|τ |∼2k

|t1 |∼2k1

|n0 |−ε hσi−ε/2 |Fˆτ −τ1 ,n0 −n01 |2 dτ1

1/2

! .

One may then notice that if |n01 | ≥ 4|n0 |, then |n0 − n01 | ∼ 2n1 and if |τ1 | ≥ 4|τ |, then |τ − τ1 | ∼ 2k1 , so that for any n, k ∈ N,

X

sup

n1 ,k1 ∈N

≤C

|n0 |∼2n |τ |∼2k

X n1 ,k1 ∈N

≤ C|F |X 0,0 , 1,1

X |n01 |∼2n1 |n01 |≥4|n0 |

X Z |n01 |∼2n1

!1/2

Z

|τ1 |∼2k1

|τ1 |∼2k1 |τ1 |≥4|τ |

|n0 |−ε hσi−ε/2 |Fˆτ −τ1 ,n0 −n01 |2 dτ1 !1/2

|Fˆτ1 ,n01 |2 dτ1

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

23

while if |n01 | ≤ 4|n0 | (with still |τ1 | ≥ 4|τ |) then |n0 |−ε ≤ C|n01 |−ε and ∀ n, k, ∈ N, !1/2 X X Z sup |n0 |−ε hσi−ε/2 |Fˆτ −τ ,n0 −n0 |2 dτ1 0 n n1 ,k1 ∈N |n |∼2 |τ |∼2k

! X

≤C

sup |n01 |−ε

|n01 |∼2n1

n1 ∈N

≤ Cε ≤ Cε

X

XZ

k1 ∈N

`∈Z

XX k1 ∈N `∈N

1

|τ1 |∼2k1 |τ1 |≥4|τ |

|n01 |∼2n1 |n01 |≤4|n0 |

X k1 ∈N

sup |n0 |∼2n

X Z |n01 |∼2n1

|τ1 |∼2k1

1

!1/2 |Fˆτ1 ,n0 −n01 |2 dτ1

!1/2 |Fˆτ,` |2 dτ

|τ1 |∼2k1

!1/2

X Z

|Fˆτ,`0 | dτ 2

|τ1 |∼2k1

|`0 |∼2`

≤ Cε |F |X 0,0 . 1,1

The cases for which |τ1 | ≤ 4|τ | are treated in the same way as the latest case above, using that in this case, hσi−ε ≤ Chτ1 − n0 3 i−ε so that the sum over k1 converges. It follows from these estimates that (3.4) is bounded above by 0,0 |H| 0,0 |F | 0,0 ≤ C|G| 0,0 |H| 0,0 |F | 0,0 , C|G|X∞,∞ X∞,∞ X X X∞,∞ X 1,1

1,∞

1,1

and this achieves the estimate of the contribution of Region I-a. Region I-b. Assume here that hσ2 i ≥ 41 n0 2 . We may then proceed as in Region I-a, by noticing that here, we have for ε > 0 small, 1 1 0 −ε −ε/2 ≤ C |n | hσi ε hσi1/2 hσ2 i1/2 hσi1/2−ε/2 hσ2 i1/2−ε/2 and that sup n,k∈N n01 ∈Z\{0} τ1 ∈R

X Z |n0 |∼2n

|τ |∼2k

dτ 0 1−ε hσ(τ, n )i hσ(τ − τ1 , n0 − n01 )i1−ε

! < +∞

for any ε ≤ 1/8. Region I-c. We consider now the region where hσi ≤ 14 n0 2 and hσ2 i ≤ 41 n0 2 . The contribution of this region to I will be the most difficult to estimate. Again, we use in (3.2) Cauchy-Schwarz inequality in (τ, n0 ), to bound the

24

A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

contribution of region I-c to (3.2) by its contribution to

X

X Z

×

|n01 |∼2n1

ˆ τ ,n0 |2 dτ1 |G 1 1

|τ1 |∼2k1

|n0 |∼2n

X

sup k1 ∈N

n1 ∈N

!#1/2

!1/2 ˆ τ ,n0 |2 dτ1 |G 1 1

|τ1 |∼2k1

|n01 |∼2n1

ˆ τ,n0 |2 |Fˆτ −τ ,n0 −n0 |2 dτ |H 1 1

|τ |∼2k

|n0 |∼2n

dτ |τ |∼2k hσihσ2 i X Z

"

!

X Z

X Z

×

≤

|τ1 |∼2k1

|n01 |∼2n1

n,n1 ∈N k,k1 ∈N

"

!1/2

X Z

X

( ×

XXX

sup

X Z |n01 |∼2n1

" ≤

X

×

|τ1 |∼2k1

sup

sup

X Z

× sup k∈N

|n01 |∼2n1

ˆ τ,n0 |2 |Fˆτ −τ ,n0 −n0 |2 dτ dτ1 |H 1 1 !1/2

ˆ τ ,n0 | dτ1 |G 1 1 X Z

sup

0 n1 k k∈N |n1 |∼2 |τ1 |∼2 1

n∈N k1 ∈N

!1/2

2

|τ1 |∼2k1

|n0 |∼2n1

(1 XX X

dτ hσihσ2 i

!1/2 )#

|τ |∼2k

|n0 |∼2n

|τ |∼2k

|n0 |∼2n

X Z

X Z

k1 ∈N

n1 ∈N

sup

|n01 |∼2n1 |τ1 |∼2k1

n∈N k1 ∈N k∈N

×

X Z

|n0 |∼2n

!1/2

!1/2 )#

X Z

|τ1 |∼2k1 |n0 |∼2n

|τ |∼2k

dτ hσihσ2 i

|τ |∼2k

ˆ τ,n0 |2 |Fˆτ −τ ,n0 −n0 |2 dτ dτ1 |H 1 1

and using Cauchy-Schwarz inequality in n, this is bounded above by (3.5) ( !1/2 X X Z ˆ τ ,n0 |2 dτ1 sup |G 1 1 n1 ∈N

k1 ∈N

|n01 |∼2n1

" × ×

|τ1 |∼2k1

X

XX

k1 ∈N

n∈N

X n∈N

sup k∈N

2 B(n1 , k1 , n, k)

!1/2

k∈N

X Z |n01 |∼2n1

|τ1 |∼2k1

X Z |n0 |∼2n

|τ |∼2k

!1/2 #) ˆ τ,n0 |2 |Fˆτ −τ ,n0 −n0 |2 dτ dτ1 |H 1 1

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

25

with (3.6)

B(n1 , k1 , n, k) =

X Z

sup |n01 |∼2n1 |τ1 |∼2k1

|n0 |∼2n

|τ |∼2k

dτ hσihσ2 i

!1/2 .

We will then make use of the following lemma. Lemma 3.1. Let N be an integer, k0 a function of (n1 , k1 , n) ∈ N3 with values in N, and n0 a function of (n1 , k1 ) ∈ N2 with values in N. Denote by A(N, n1 , k1 ) the region in N2 given by n A(N, n1 , k1 ) = (n, k) ∈ N2 , k0 (n1 , k1 , n) ≤ k ≤ k0 (n1 , k1 , n) + N, o n0 (n1 , k1 ) ≤ n ≤ n0 (n1 , k1 ) + N . Then there is a constant C(N ) depending only on N such that X sup B(n1 , k1 , n, k) ≤ C(N ), n1 ,k1 ∈N

(n,k)∈A(N,n1 ,k1 )

where B(n1 , k1 , n, k) is defined by (3.6). Proof of Lemma 3.1. It follows easily from Lemma 5.1 in [15], since X sup B(n1 , k1 , n, k) k1 ,n1 ∈N

(n,k) ∈A(N,n1 ,k1 )

≤ N 2 sup sup B(n1 , k1 , n, k) k1 ,n1 ∈N n,k∈N

≤ N 2 sup k1 ,n1 ∈N

X n∈Z\{0}

Z R

dτ hσ(τ, n)ihσ(τ − τ1 , n − n1 )i

!1/2

n 6= n1 < +∞ by Lemma 5.1 in [15].



Now, in order to apply Lemma 3.1, we need to show that region I-c is embedded in a region of the form  (n, k, n1 , k1 ) ∈ N4 , (n, k) ∈ A(N, n1 , k1 ) for some N and for some functions n0 (n1 , k1 ) and k0 (n1 , k1 , n). Note that we have in region I-c : 1 2 1 3 |τ − n0 | ≤ hσ(τ, n)i ≤ n0 ≤ |n0 |3 4 4

26

A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

hence 3 03 5 |n | ≤ |τ | ≤ |n0 |3 4 4 and the property 3n − 4 ≤ k ≤ 3n + 4 follows easily. Hence, to prove the preceding result, we only have to find n0 (n1 , k1 ) and N such that for any (n, k, n1 , k1 ) in region I-c, n0 (n1 , k1 ) ≤ n ≤ n0 (n1 , k1 ) + N. In order to prove this fact, we again use a partition of region I-c into three subregions. • Region I-c-1 : 2−12 |n01 | ≤ |n0 | ≤ 212 |n01 |. In this region, we obviously have the result with n0 (n1 , k1 ) = n1 − 4. • Region I-c-2 : |n0 | ≤ 2−12 |n01 |. We recall that |σ − σ1 − σ2 | = 3|n0 | |n01 | |n0 − n01 | from which it follows that (since hσ1 i is dominant) |n01 | |n0 | |n0 − n01 | ≤ hσ1 i ≤ 3|n01 | |n0 | |n0 − n01 | + hσi + hσ2 i; using the fact that |n0 | ≤ 21 |n01 | and that hσi ≤ 41 |n0 |2 and hσ2 i ≤ 14 |n0 |2 , we easily get from the preceding inequality 1 0 2 0 |n1 | |n | ≤ hσ1 i ≤ 5|n01 |2 |n0 |, 2 and the property follows easily with n0 (n1 , k1 ) =

ln |2k1 − 23n1 | ln 5 − − 2n1 . ln 2 ln 2

• Region I-c-3 : |n0 | ≥ 212 |n01 |. We infer here, from the inequality |n0 | |n01 | |n0 − n01 | ≤ hσ1 i ≤ 3|n0 | |n01 | |n0 − n01 | + hσi + hσ2 i that 1 02 0 |n | |n1 | ≤ hσ1 i ≤ 5|n0 |2 |n01 | 2 and we conclude as in the preceding case.

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

27

Now, going back to (3.5), we may use Lemma 3.1 to show that the contribution of region I-c to  !2 1/2 X X sup  B(n1 , k1 , n, k)  n1 ,k1 ∈N

n∈N

k∈N

X

≤ sup n1 ,k1 ∈N

B(n1 , k1 , n, k)

n,k∈N

is bounded above by an absolute constant. Hence, each of the contributions of Regions I-c-1, I-c-2 and I-c-3 to (3.5) is bounded above by (3.7) ( !1/2 X X Z 2 ˆ τ1 ,n1 | dτ1 C sup |G n1 ∈N

k1 ∈N

X

|n01 |∼2n1 nX 0 +N

k1 ∈N

n=n0 (n1 ,k1 )

" ×

|τ1 |∼2k1

X Z

sup k0 (n)≤k≤k0 (n)+N

|n01 |∼2n1

X Z

|τ1 |∼2k1 |n0 |∼2n

ˆ τ,n0 |2 |H

|τ |∼2k

#1/2 ) ×|Fˆτ −τ1 ,n0 −n01 |2 dτ dτ1 ≤ CN

X

sup k1 ∈N

n1 ∈N

× sup

× sup n1 ∈N

|n01 |∼2n1

|τ1 |∼2k1

n0 ≤n≤n0 +N k0 ≤k≤k0 +N

X

ˆ τ ,n0 |2 dτ1 |G 1 1 !1/2

X Z

sup

n1 ∈N k1 ∈N

!1/2

X Z

sup

k1 ∈N n0 ≤n≤n0 +N k0 ≤k≤k0 +N

|n0 |∼2n

ˆ τ,n0 |2 dτ |H

|τ |∼2k

sup |n0 |∼2n |τ |∼2k

X Z |n01 |∼2n1

|τ1 |∼2k1

!1/2 |Fˆτ −τ1 ,n0 −n01 |2 dτ1

0,0 ≤ CN |G|X 0,0 |H|X∞,∞ 1,∞

× sup n1 ∈N

X

sup

k1 ∈N n0 ≤n≤n0 +N k0 ≤k≤k0 +N

sup |n0 |∼2n |τ |∼2k

X Z |n01 |∼2n1

|τ1 |∼2k1

!1/2 |Fˆτ −τ1 ,n0 −n01 |2 dτ1

.

It remains to bound the last term in the right hand side above by C|F |X 0,0 . 1,1 However, this is not completely obvious, and we again have to consider separately each of the regions I-c-1, I-c-2 and I-c-3. Region I-c-2. Recall that we have here : |n0 | ≤ 2−12 |n01 |.

28

A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

Then the last term in the right hand side of the above inequality is clearly bounded by !1/2 nX kX 0 +N 0 +N X X Z 2 sup |Fˆτ −τ1 ,n0 −n01 |2 dτ1 . n1 ∈N

k1 ∈N n=n0 (n1 ,k1 ) n≤n1 −10

k=k0

|n01 |∼2n1

|τ1 |∼2k1

• The contribution to this term of the k and k1 for which k ≤ k1 − 4 or k ≥ k1 + 4 is clearly bounded above by !1/2 X X Z 2N 2 sup |Fˆτ,n01 |2 dτ n1 ∈N

k∈N

≤ 2N 2 |F |X 0,0 .

|n01 |∼2n1

|τ |∼2k

1,1

• It remains to consider in the sum in k1 and k, the contribution of the terms for which k1 − 4 ≤ k ≤ k1 + 4. Since for such terms, τ − τ1 may stay bounded, we need to show that there are only a finite number of possibilities for k1 . We recall that in Region I-c, k ≤ 3n + 4, while in Region I-c-2, n ≤ n1 − 10; it follows easily that if in addition k1 − 4 ≤ k ≤ k1 + 4, then k1 ≤ 3n1 − 4. Hence, k1 −23n1 | n0 (n1 , k1 ) = ln |2 ln − 2n1 = n1 − 1 and the region is actually empty. 2 Region I-c-3 : |n0 | ≥ 212 |n01 |. Again, the last term in (3.7) is easily bounded above by !1/2 nX kX 0 +N 0 +N X X Z sup 2 sup |Fˆτ −τ1 ,n0 |2 dτ1 n1 ∈N

k1 ∈N n=n0 (n1 ,k1 ) n≥n1 +3

k k=k0 |τ |∼2

|n0 |∼2n

|τ1 |∼2k1

• in the same way as before, the contribution in this sum of the terms for which k ≤ k1 − 4 or k ≥ k1 + 4 is bounded by !1/2 X X Z 2N 2 sup |Fˆτ,n0 |2 dτ ≤ 2N 2 |F |X 0,0 . n∈N

k∈N

|n0 |∼2n

|τ |∼2k

1,1

• in the region where k1 − 4 ≤ k ≤ k1 + 4, we easily get k1 ≥ 3n1 + 4, and from the expression of n0 (n1 , k1 ) in region I-c-3, we get n0 (n1 , k1 ) = 21 (k1 −n1 ). Hence, n ≥ 21 (3n − 8) − 21 n1 from which it follows that n1 ≥ n − 8, and again the region is empty, since n ≥ n1 + 10. Region I-c-1 : 2−12 |n01 | ≤ |n0 | ≤ 212 |n01 |.

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

29

This is the most difficult part; clearly, we can take in this region n0 (n1 , k1 ) = n1 . Again, we will divide the region into three subregions depending on the size of k and k1 compared to each other. • k ≤ k1 − 4 : the contribution of this region to the last term in (3.7) is then bounded above by !1/2 1 +N X Z X nX sup 2 sup |Fˆτ ,n0 −n0 |2 dτ1 . n1 ∈N

0 n k1 ∈N n=n1 |n |∼2

1

|τ1 |∼2k1

|n01 |∼2n1

1

Now, since for each n1 , k1 , n and n0 such that |n0 | ∼ 2n , one has !1/2 X Z 2 |Fˆτ ,n0 −n0 | dτ1 |n01 |∼2n1

≤

X `∈N

1

|τ1 |∼2k1

1

!1/2

X Z |`0 |∼2`

|Fˆτ1 ,`0 |2 dτ1

|τ1 |∼2k1

,

the preceding term is easily bounded above by 2 sup

nX 1 +N

XX

n1 ∈N n=n 1 k1 ∈N `∈N

!1/2

X Z |`0 |∼2`

|Fˆτ1 ,`0 |2 dτ1

|τ1 |∼2k1

≤ 2N |F |X 0,0 . 1,1

• k1 − 4 ≤ k ≤ k1 + 4 : Using again the arguments immediately above the contribution of this region to the last term in (3.7) may be bounded above by 1/2 X X X Z sup sup |Fˆτ −τ1 ,`0 |2 dτ1 sup n1 ∈N

k1 ∈N

≤ sup n1 ∈N

n1 ≤n≤n1 +N |τ |∼2k `∈N 3n≤k≤3n+N k1 −4≤k≤k1 +4

X

sup

sup

|`0 |∼2`

X

k k1 ∈N 3n1 ≤k≤3n1 +4N |τ |∼2 `∈N k1 −4≤k≤k1 +4

|τ1 |∼2k1

X Z |`0 |∼2`

|τ1 |∼2k1

!1/2 |Fˆτ −τ1 ,`0 | dτ1 2

.

Here again, τ − τ1 may stay bounded even for large k and k1 ; however, for a fixed n1 , the number of k1 for which the right hand side gives a nonzero contribution is bounded by the total number of k1 for which there exists at least one k such that 3n1 ≤ k ≤ 3n1 + 4N and k1 − 4 ≤ k ≤ k1 + 4, hence by

30

A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

4N + 8. In this way, the term above is bounded by !1/2 X X Z (4N + 8) sup sup sup |Fˆτ −τ1 ,`0 |2 dτ1 n1 ∈N k,k1 ∈N |τ |∼2k `∈N

≤ (4N + 8)

|`0 |∼2`

XX X Z |`0 |∼2`

`∈N j∈N

|τ1 |∼2k1

|Fˆτ,`0 |2 dτ

1/2

|τ |∼2j

≤ (4N + 8)|F |X 0,0 . 1,1

• k ≥ k1 + 4 : this region is a little bit more delicate than the preceding ones to handle. In the same way as before, we may bound above the contribution of the present region to the last term in the right hand side of (3.7) by !1/2 X X X Z 2 sup sup sup |Fˆτ −τ1 ,`0 | dτ1 n1 ∈N

(3.8)

k k1 ∈N 3n1 ≤k≤3n1 +2N |τ |∼2 `∈N k≥k1 +4

≤ 2 sup n1 ∈N

X

sup

k1 ∈N 3n1 ≤k≤3n1 +2N k≥k1 +4

X `∈N

|τ1 |∼2k1

|`0 |∼2`

X Z |`0 |∼2`

!1/2 |Fˆτ,`0 |2 dτ

.

|τ |∼2k

Again, we have to show that the number of possible k1 (or n or n1 ) in this region is finite. We recall that here, n and n0 are of the same order ; moreover, since |τ − n0 3 | ≤ 14 |n0 |2 and |τ − τ1 − (n0 − n01 )3 | ≤ 41 |n0 |2 , it follows that |τ | is of the order of |n0 |3 ; then |τ1 |, which is negligible compared with |τ |, is negligible compared with |n0 |3 . Hence τ1 − n01 3 ∼ −n01 3 for n1 sufficiently large. Now, we have the relation (3.9)

3

3

τ1 − n01 − τ + n0 + τ − τ1 − (n0 − n01 )3 = 3n0 n01 (n0 − n01 ).

- Consider first the case where n0 and n01 have opposite signs. Then, taking into account the preceding considerations, one may note that the left hand side in (3.9) is of the order of −n01 3 (for |n01 | large) while the right hand side has the sign of n01 3 . Hence (3.9) cannot remain true for large |n01 |, which implies that the number of n01 in this region is finite. - Now, if n0 and n01 have the same sign, then comparing the sign of both sides in (3.9) shows that if |n01 | is large, then necessarily, n0 − n01 has a sign opposite to that of n01 . But then, for |n0 | (or equivalently for |n01 |) large, the facts that τ ∼ n0 3 , τ − τ1 ∼ (n0 − n01 )3 and τ1 is negligible compared with τ lead again to incompatible signs.

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

31

This shows that, in any case, the number of possible n1 (or n, or k1 ) in this region is finite. Hence, (3.8) is bounded above by !1/2 X X Z sup |Fˆτ,`0 |2 dτ C sup sup n1

k1 3n1 ≤k≤3n1 +2N

`

|`0 |∼2`

|τ |∼2k

≤ C|F |X 0,0 . 1,1

This ends the proof of the required estimate in Region I, that is when hσ1 i dominates. Region II. Here, we use the fact that 1 02 |n | ≤ |n0 n01 (n0 − n01 )| ≤ hσi 2 so that for any s ∈ [− 21 , 0], |n0 |1+s |n01 |−s |n0 − n01 |−s ≤ Chσi1/2 . ˆ and H ˆ – we are led Exchanging then the role of n0 and n01 – and hence of G back to prove that the contribution of Region I to I is bounded above by 0,0 |F | 0,0 . C|H|X 0,0 |G|X∞,∞ X1,1 1,∞ For Regions I-a and I-b, this was already done, since the contribution of Re0,0 |G| 0,0 |F | 0,0 . gions I-a and I-b to I was actually bounded above by C|H|X∞,∞ X∞,∞ X1,1 It remains only to consider the case of Region I-c. Again, the same computations as before lead to bound the contribution of Region I-c to I as in ˆ or Fˆ , so that this (3.7), except that the sum over n1 will be supported by H contribution is bounded by (see (3.7)) (3.10) !1/2 X Z ˆ τ ,n0 |2 dτ1 CN sup sup |G 1 1 n1 ∈N k1 ∈N

|n01 |∼2n1

|τ1 |∼2k1

( ×

X n1 ∈N

×

X

sup

sup

sup

k1 ∈N n0 ≤n≤n0 +N k0 ≤k≤k0 +N

sup

sup

sup

n ≤n≤n0 +n k0 ≤k≤k0 +N |n0 |∼2n k1 ∈N 0 |τ |∼2k

!1/2

X Z |n0 |∼2n

ˆ τ,n0 |2 dτ |H

|τ |∼2k

X Z |n01 |∼2n1

|τ1 |∼2k1

!1/2 ) |Fˆτ −τ1 ,n0 −n01 | dτ1 2

Hence, we have to bound above the two last lines in (3.10) by C|H|X 0,0 |F |X 0,0 . 1,∞ 1,1 Considering the way we have estimated the contribution of Regions I-c-2 and I-c-3 to (3.7), it is clear that the sum over k1 in these regions can be

.

32

A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

supported by |Fˆτ −τ1 ,n0 −n01 |2 ; in Region I-c-1, we have 2−12 |n01 | ≤ |n0 | ≤ 212 |n01 | so that n0 (n1 , k1 ) = n1 and !1/2 X X Z ˆ τ,n0 |2 dτ sup sup sup |H n1 ∈N

≤C

k1 ∈N n0 ≤n≤n0 +N k0 ≤k≤k0 +N

X n1 ∈N

sup k1 ∈N

|τ |∼2k

!1/2

X Z |n0 |∼2n1

|n0 |∼2n

|τ1 |∼2k1

ˆ τ ,n0 |2 dτ1 |H 1 1

≤ C|H|X 0,0 . 1,∞

We may conclude as before. ˆ and Fˆ , we are led back to prove Region III. Again, exchanging the role of G 0,0 |F | 0,0 , that the contribution of Region I to I is bounded above by C|G|X 0,0 |H|X∞,∞ X1,∞ 1,1 but this is easily done by using the same analysis as for Region I. Hence, the proof of Proposition 3.1 is complete.  We now prove that when local in time spaces are considered, that is when s,−1/2,T s,−1/2 , a small power of T can be recovered in the is replaced by X1,1 X1,1 right hand side of the estimate in Proposition 3.1. This will be useful in the contraction procedure, since as is now classical, no small power of T is gained, but on the opposite a ln T factor is lost in the estimate of the integral convolution with the linear semi-group when dealing with spaces of regularity 1/2 in time. The argument of the proof of the next proposition relies, as usual, on the fact that we have wasted a small power of hσi or hσ2 i in Lemma 3.1. Actually, looking carefully to the proof shows that Lemma 3.1 is still true with B(n1 , k1 , n, k) replaced by  X Z 1/2 dτ ˜ B(n1 , k1 , n, k) = sup 1−ε hσ i1−ε 2 |τ |∼2k hσi |n01 |∼2n1 0 n |τ1 |∼2k1

|n |∼2

for any ε < 1/4. s,1/2,T

Proposition 3.2. Let −1/2 ≤ s ≤ 0 and f ∈ X1,1 any α < 1/16, there is a constant Cα such that

|∂x (f g)|X s,−1/2,T ≤ Cα T α |f |X s,1/2,T |g|X s,1/2,T . 1,1

1,1

s,1/2,T

, g ∈ X1,∞

1,∞

; then for

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS s,1/2

33

s,1/2

Proof. Let f ∈ X1,1 , g ∈ X1,∞ , s ≥ 1/2, both with support in [−2T, 2T ]. Using the arguments immediately above shows that we have actually proved, during the course of the proof of Proposition 3.1, that   |∂x (f g)|X s,−1/2 ≤ C |f |X s,1/2 |g|X s,δ + |f |X s,δ |g|X s,1/2 1,1

1,∞

1,1

1,1

1,∞

for any δ > 3/8. Let s = 0 (the arguments are exactly the same in the other cases) and δ such that 3/8 < δ < 1/2. By an obvious interpolation inequality, one gets |g|X 0,δ ≤ C|g|1−2δ |g|2δ 0,1/2 . X 0,0 X 1,∞

1,∞

1,∞

On the other hand, using the notations introduced at the beginning of Section 2, we have ∞  X Z 1/2 X 0 2 d |g|X 0,0 = sup |∆ n g(τ, n )| dτ 1,∞

≤

n=0 k∈N ∞ X

|τ |∼2k

n0 ∈N\{0}

|∆n g|L2x,t ([−2T,2T ]×T)

n=0

≤ CT ≤ CT

1/4

∞ X

1/4

n=0 ∞  X

|∆n g|L4x,t ([−2T,2T ]×T) X Z n0 ∈N\{0}

n=0

2/3

hσi

0 2 d |∆ n g(τ, n )| dτ

1/2

τ ∈R

where we have used in the last line above the Strichartz estimate proved in [4]. It follows readily that |g|X 0,0 ≤ CT 1/4 |g|X 0,1/3 ≤ CT 1/4 |g|X 0,1/2 , 1,∞

1,2

1,∞

and from the above interpolation inequality, |g|X 0,δ ≤ CT (1−2δ)/4 |g|X 0,1/2 . 1,∞

1,∞

In the same way, we estimate f as follows : taking a small positive ε, one has (1−2δ)/(1+2ε)

|f |X 0,δ ≤ C|f |X 0,−ε 1,1

1,1

2(ε+δ)/(1+2ε)

|f |

0,1/2

X1,1

and |f |X 0,−ε ≤ C|f |X 0,0 ≤ CT 1/4 |f |X 0,1/3 1,1

1,2

1,2

by using again the estimate in [4] for ∆n f ; it follows that |f |X 0,−ε ≤ CT 1/4 |f |X 0,1/2 . 1,1

1,1

Finally, |∂x (f g)|X 0,−1/2 ≤ Cα T α |f |X 0,1/2 |g|X 0,1/2 1,1

1,1

1,∞

34

A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

where α is chosen such that α < (1 − 2δ)/4, with δ > 3/8, so that at the very end, α < 1/16, and since f and g have supports in [−2T, 2T ], the proof of Proposition 3.2 follows.  We now prove an estimate of the same type as those in Propositions 3.1 and 3.2, but in Ys spaces. We recall that the use of these spaces is needed to handle the integral estimate in Duhamel’s formula (see Proposition 4.1). s,1/2

s,1/2

Proposition 3.3. Let −1/2 ≤ s ≤ 0, f ∈ X1,1 , g ∈ X1,∞ ; then ∂x (f g) ∈ Ys . Moreover, for any α < 1/16, there is a constant Cα > 0 such that |∂x (f g)|Ys,T ≤ Cα T α |f |X s,1/2,T |g|X s,1/2,T . 1,∞

1,1

Proof. We only sketch the proof, since it is a slight modification of the proof s,1/2 of Proposition 3.1, using e.g. the arguments in [20]. Let f ∈ X1,1 and s,1/2 g ∈ X1,∞ . We only prove the estimate |∂x (f g)|Ys ≤ C|f |X s,1/2 |g|X s,1/2 , 1,1

1,∞

α

the T factor can indeed be recovered exactly as in the proof of Proposition 3.2. By a duality argument, the estimate will be proved if we show that there is a constant C > 0 such that for any function w (of the space variable x) lying −s in the Besov space B2,∞ , one has Z c X 0 f g(τ, n0 ) 0 −s . |n | dτ w(n ˆ ) ≤ C|f |X s,1/2 |g|X s,1/2 |w|B2,∞ 0 )i 1,1 1,∞ hσ(τ, n R 0 n 6=0 ˆ = ˆ = n0 s hσi1/2 gˆ and W Setting as above Fˆ (τ, n0 ) = n0 s hσ(τ, n0 )i1/2 fˆ(τ, n0 ), G 0 −s ˆ it suffices to prove that n w, (3.11) Z ˆ n0 | ˆ τ ,n0 ||W X X Z |n0 |1+s |n01 |−s |n0 − n01 |−s |Fˆτ −τ1 ,n0 −n01 ||G 1 1 dτ dτ1 0 )ihσ(τ − τ , n0 − n0 )i1/2 hσ(τ , n0 )i1/2 hσ(τ, n 1 1 τ ∈R τ ∈R 1 1 0 1 0 n 6=0 n1 6=0 n01 6=n0

0 ≤ C|F |X 0,0 |G|X 0,0 |W |B2,∞ . 1,1

1,∞

Again, we will consider separately the three regions defined at the beginning of the proof of Proposition 3.1. Region I : hσ1 i = max(hσi, hσ1 i, hσ2 i)

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

35

As already noticed, we have in this region |n0 |1+s |n01 |−s |n0 − n01 |−s ≤ Chσ(τ1 , n01 )i1/2 . Hence, taking ε > 0 small, we have ˆ τ ,n0 ||W ˆ n0 | |n0 |1+s |n01 |−s |n0 − n01 |−s |Fˆτ −τ1 ,n0 −n01 ||G 1 1 hσ(τ, n0 )ihσ(τ − τ1 , n0 − n01 )i1/2 hσ(τ1 , n01 )i1/2 ≤C

ˆ τ ,n0 | ˆ n0 | |Fˆτ −τ1 ,n0 −n01 ||G |W 1 1 0 1/2+ε 0 1/2−ε hσ(τ, n )i hσ(τ, n )i hσ(τ − τ1 , n0 − n01 )i1/2

and we conclude as in the proof of Proposition 3.1, using the fact that  W  ˆ 0,0 F −1 ∈ X∞,∞ , with hσi1/2+ε  −1 F

 ˆ W 0 0,0 ≤ Cε |W |B2,∞ hσi1/2+ε X∞,∞

and using again the fact that Lemma 3.1 is still true with a little smaller power of σ(τ, n0 ). Region III, that is hσ2 i = max(hσi, hσ1 i, hσ2 i) is treated in the same way. Region II : hσi = max(hσi, hσ1 i, hσ2 i). Here, we have |n0 |1+s |n01 |−s |n0 − n01 |−s ≤ Chσ(τ, n0 )i1/2 and it follows that 1 C ≤ . 0 0 0 2+2s hσ(τ, n )i hσ(τ, n )i + |n | |n01 |−2s |n0 − n01 |−2s Hence, going back to the way we have proved Proposition 3.1, it suffices to show that for a fixed n01 , ! ˆ (n0 ) |n0 |1+s |n01 |−s |n0 − n01 |−s W 0,0 −1 F ∈ X∞,∞ , 0 −2s 0 0 −2s 0 0 2+2s hσ(τ, n )i + |n | |n1 | |n − n1 | with ! ˆ (n0 ) |n0 |1+s |n01 |−s |n0 − n01 |−s W −1 F 0 0 0 0 2+2s −2s 0 −2s hσ(τ, n )i + |n | |n1 | |n − n1 | and a constant C that does not depend on n01 .

0 ≤ C|W |B2,∞ 0,0 X∞,∞

36

A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

Z But this follows from the next easy computation, once we have noticed that dτ C ≤ 2 : 2 2 a R (hτ i + a ) ! 2 0 1+s 0 −s 0 0 −s ˆ 0 |n | |n | |n − n | W (n ) −1 1 1 F hσ(τ, n0 )i + |n0 |2+2s |n01 |−2s |n0 − n01 |−2s 0,0 X∞,∞ 0 2+2s 0 −2s 0 0 −2s ˆ X Z |n | |n1 | |n − n1 | |W (n0 )|2 = sup 0 0 2+2s |n0 |−2s |n0 − n0 |−2s )2 n,k |τ |∼2k (hσ(τ, n )i + |n | 1 1 0 n |n |∼2

X

≤ sup n

ˆ (n0 )|2 |n0 |2+2s |n01 |−2s |n0 − n01 |−2s |W

|n0 |∼2n

Z × R

dτ 0 0 2+2s (hσ(τ, n )i + |n | |n01 |−2s |n0 − n01 |−2s )2

!

≤ C|W |2B 0 . 2,∞

This ends the proof of Proposition 3.3.



As a last, but easy, bilinear estimate, we briefly show that we can handle s,−1/2 s+ε,1/2 terms like ∂x (g 2 ) in X1,1 if g is only in X∞,∞ (the ε loss of regularity seems to be necessary here). Our motivation to treat such terms arises from the fact that the stochastic convolution which was studied in Proposition 2.1 s+ε,1/2 belongs to such spaces (or even to X1,∞ ) if sufficient regularity is assumed s,1/2 on the operator φ, but never belongs to X1,1 , due to the lack of regularity of the Brownian motion. Proposition 3.4. Let −1/2 ≤ s ≤ 0, and ε > 0; then there is a constant s+ε,1/2 C > 0 such that for any g ∈ X∞,∞ , |∂x (g 2 )|X s,−1/2 ≤ C|g|2X s+ε,1/2 . 1,1

∞,∞

s,−1/2,T

If moreover, g is supported in [−2T, 2T ] and ∂x (g 2 ) is considered in X1,1 , α then a factor T can be recovered in the right hand side above, for any α < 1/16. s,−1/2 Finally, the same estimate holds if in the left hand side, X1,1 (resp. s,−1/2,T X1,1 ) is replaced by Ys (resp. Ys,T ). Proof. Here again, we only sketch the proof, since the arguments are the same as in the easiest cases of the proof of Proposition 3.1, that is when some small power of hσi or hσ1 i can be lost.

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

37

s+ε,1/2 −s,1/2 Indeed, taking f, g ∈ X∞,∞ , h ∈ X∞,∞ and setting as before Fˆ = ˆ we need to show that ˆ = n0 s+ε hσi1/2 gˆ and H ˆ = n0 −s hσi1/2 h, n0 s+ε hσi1/2 fˆ, G Z Z X X |n0 |1+s |n01 |−s−ε |n0 − n01 |−s−ε ˆ ˆ τ ,n0 ||Fˆτ −τ ,n0 −n0 |dτ dτ1 |Hτ,n0 ||G 1 1 1 1 1/2 hσ i1/2 hσ i1/2 hσi 1 2 τ ∈R τ1 ∈R 0 n0 6=0 n1 6=0 n01 6=n0

0,0 |F | 0,0 |G| 0,0 . ≤ C|H|X∞,∞ X∞,∞ X∞,∞

Consider e.g. Region I, where hσ1 i dominates, and where we have the inequality |n0 |1+s |n01 |−s |n0 − n01 |−s ≤ Chσ1 i1/2 , so that we are lead to estimate Z X X Z |n01 |−ε |n0 − n01 |−ε ˆ ˆ τ ,n0 ||Fˆτ −τ ,n0 −n0 |dτ dτ1 . |Hτ,n0 ||G 1 1 1 1 1/2 hσ i1/2 hσi 2 τ ∈R τ ∈R 1 0 n0 6=0 n1 6=0 n01 6=n0

This latest term is then handled by the same arguments as those used in Region I-a in the proof of Proposition 3.1, keeping in addition a small power of hσ2 i to be able to sum over k, and hence to replace the norm |F |X 0,0 by 1,1 0,0 |F |X∞,∞ (the sum over n being handled by using |n0 − n01 |−ε ). All the other regions are treated in the same way, and the arguments for the other statements of Proposition 3.4 are exactly the same as those of Propositions 3.2 and 3.3.  4. Proof of Theorem 1.1 and Theorem 1.2 As was pointed out in the introduction, it mainly remains to show that we R t may gain one degree of regularity in time when passing from ∂x (gf ) to U (t − s)∂x (f g)(s)ds. The result is stated in the next proposition. 0 s,−1/2

Proposition 4.1. There is a constant C > 0 such that if f ∈ X1,1 Rt s,1/2,T s ∈ R, then t 7→ 0 U (t − s)f (s)ds ∈ X1,1 and Z ·   U (· − s)f (s)ds ≤ C |f | s,−1/2 + |f |Ys s,1/2,T X1,1 0

∩ Ys ,

X1,1

for any T ≤ 1. Rt Moreover, for any f ∈ Ys , the map t 7→ 0 U (t − s)f (s)ds is continuous with s values in B2,1 (T) and there is a constant C > 0 such that Z t sup U (t − s)f (s)ds ≤ C|f |Ys . s t∈[−T,T ] 0 B2,1

38

A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

Proof. The arguments of the proof are similar to those in [14]. We consider a cut-off function ψ with ψ ≡ 1 on [0, 1] and supp ψ ⊂ [−1, 2]; it is sufficient to prove that Z · ψ U (· − s)f (s)ds 0



s,1/2



≤ C |f |X s,−1/2 + |f |Ys . 1,1

X1,1

We first write Z

t

U (t − s)f (s)ds it(τ1 −n0 3 ) XZ − 1 itn0 3 ixn0 ˆ 0 e = ψ(t) e f (τ1 , n ) e dτ1 3 03 τ1 − n 0 n0 ∈Z |τ1 −n |≤1 03 itτ1 XZ − eitn ixn0 ˆ 0 e e f (τ1 , n ) dτ1 + ψ(t) 03 0 3 |≥1 τ − n 1 |τ −n 0 1 n ∈Z

ψ(t)

0

= g1 (t, x) + g2 (t, x). To estimate g1 , we expand the exponential as ∞

03

eit(τ1 −n ) − 1 X ik tk (τ1 − n0 3 )k = k! τ1 − n 0 3 k=1 so that

g1 (t, x) =

∞ k k X i t k=1

k!

XZ

ψ(t)

n0 ∈Z

0 03 3 eixn +itn fˆ(τ1 , n0 )(τ1 − n0 )k dτ1 .

|τ1 −n0 3 |≤1

Let ϕk (t) = tk ψ(t); then

0

gˆ1 (τ, n ) =

∞ k Z X i k=1

k!

|τ1

3

3

ϕˆk (τ − n0 )fˆ(τ1 , n0 )(τ1 − n0 )k dτ1 −n0 3 |≤1

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

39

and it follows |g1 |X s,1/2 1,1

=

X

2sn

n∈N

=

X

sn

2

n∈N

∞ X `=0 ∞ X

|n0 |∼2n

`=0

|n0 |∼2n

Now, X Z |n0 |∼2n

≤

∞ k X i −n0 3 |≤1

k=1

3

hτ − n0 i

k!

X Z |τ |∼2`

Z

03

≤ sup

k=1 ∞ X

hτ − n i |τ |∼2`

k=1

i2

!1/2

ϕˆk (τ − n )fˆ(τ1 , n0 )(τ1 − n ) dτ1 dτ

∞ k X i

hτ − n i

03 k

03

k!

.

i2 3 3 ϕˆk (τ − n0 )fˆ(τ1 , n0 )(τ1 − n0 )k dτ1 dτ

|τ1 −n0 3 |≤1 k=1 k! ∞ X |ϕˆk (τ − n0 3 )| 2 03

|τ |∼2`

gˆ1 (τ, n0 )|2 dτ

|τ |∼2`

hZ hτ − n i

|n0 |∼2n

|hτ − n i

|τ |∼2`

03

|n0 |∼2n

0 3 1/2

X Z

hZ × |τ1

!1/2

X Z

dτ

Z

|fˆ(τ1 , n0 )|2 dτ1



|τ1 −n0 3 |≤1

Z  |ϕˆk (τ − n0 3 )| 2  X 0 2 ˆ dτ |f (τ1 , n )| dτ1 . k! |τ1 −n0 3 |≤1 0 n |n |∼2

We deduce that |g1 |X s,1/2 1,1

∞ X

∞ X |ϕˆk (τ − n0 3 )| 2 03 dτ sup hτ − n i ≤ sup 0 |∼2n k! ` n∈N |n |τ |∼2 `=0 k=1 1/2 X  X Z sn ˆ × 2 |f (τ1 , n0 )|2 dτ1 . n∈N

Z

|n0 |∼2n

!1/2

|τ1 −n0 3 |≤1

Now, we have for ε > 0, Z ∞ ∞ X X |ϕˆk (τ − n0 3 )| 2 03 sup sup hτ − n i dτ 0 n k! n∈N |τ |∼2` k=1 `=0 |n |∼2 ∞ X  Z X |ϕˆk (τ − n0 3 )| 2 3 0 3 −ε dτ ≤ sup sup sup hτ − n i hτ − n0 i1+ε 0 |∼2n k! ` ` n∈N |n |τ |∼2 |τ |∼2 `∈N k=1 Z ∞   X X | ϕ ˆ (τ − n0 3 )| 2 k 3 ≤ C sup h2` − 23n i−ε sup hτ − n0 i1+ε dτ k! n∈N n0 ∈N R `∈N k=1 ∞ X ϕk ≤ C 1/2+ε/2 . k! H k=1

40

A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

Hence

∞ X ϕk s,0 . |g1 |X s,1/2 ≤ C 1/2+ε/2 |f |X1,1 1,1 k! H k=1

In order to estimate the norm of g2 , we write g2 (t, x) = g2,1 (t, x) + g2,2 (t, x) with g2,1 (t, x) = ψ(t)

eitτ1 0 eixn fˆ(τ1 , n0 ) dτ1 τ1 − n 0 3 |τ1 −n0 3 |≥1

XZ n0 ∈Z

and g2,2 (t, x) = −ψ(t)

03

XZ

e

|τ1 −n0 3 |≥1

n0 ∈Z

We have

ixn0

eitn fˆ(τ1 , n0 ) dτ1 . τ1 − n 0 3

0 ˆ ˆ − τ1 ) f (τ1 , n ) dτ1 , ψ(τ τ1 − n 0 3 |τ1 −n0 3 |≥1

Z

0

gˆ2,1 (τ, n ) = and X Z |n0 |∼2n

3

hτ − n0 i|ˆ g2,1 (τ, n0 )|2 dτ

|τ |∼2k

ˆ i2 0 3 ˆ − τ1 )| f (τ1 , n ) dτ1 dτ hτ1 − n0 i1/2 |ψ(τ k τ1 − n 0 3 |τ1 −n0 3 |≥1 |n0 |∼2n |τ |∼2 fˆ(τ , n0 ) i2 hZ X Z 1 1/2 ˆ +C hτ − τ1 i |ψ(τ − τ1 )| 3 dτ1 dτ 0 3 τ1 − n |τ |∼2k |τ1 −n0 |≥1 0 n

≤C

X Z

hZ

|n |∼2

≤ I + II. For the term I, we have X Z I≤C |n0 |∼2n

hZ

|τ |∼2k

ˆ 1 )| |ψ(τ

R

i2 |fˆ(τ − τ1 , n0 )| dτ dτ. 1 hτ − τ1 − n0 3 i1/2

ˆ ∈ L2 0 , then Let h τ,n Z X Z |fˆ(τ − τ1 , n0 )| 0 ˆ ˆ h(τ, n ) |ψ(τ1 )| 3 1/2 dτ1 dτ 0 hτ − τ1 − n i |τ |∼2k R 0 |∼2n |nZ  X Z 1/2 ˆ 1 )| ˆ n0 )|2 dτ |h(τ, ≤ |ψ(τ R

|n0 |∼2n

 X Z × |n0 |∼2n

|τ |∼2k

|τ |∼2k

|fˆ(τ − τ1 , n0 )|2 1/2 dτ dτ1 . hτ − τ1 − n0 3 i

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

41

We deduce from the preceding estimate that Z

 X Z |ψ(τ1 )|

I≤C R

|˜ τ +τ1 |∼2k

|n0 |∼2n

|fˆ(˜ τ , n0 )|2 1/2 d˜ τ dτ1 h˜ τ − n0 3 i

!2 .

In the same way, we can prove that Z

|fˆ(˜ τ , n0 )|2 1/2 d˜ τ dτ1 h˜ τ − n0 3 i2

 X Z 1/2 hτ1 i |ψ(τ1 )|

II ≤ C R

|n0 |∼2n

|˜ τ +τ1 |∼2k

!2 .

Hence we have Z

1/2

I + II ≤ C

hτ1 i

 X Z |ψ(τ1 )|

R

|n0 |∼2n

|˜ τ +τ1 |∼2k

|fˆ(˜ τ , n0 )|2 1/2 d˜ τ dτ1 h˜ τ − n0 3 i

!2

and we deduce that ∞  X Z 1/2 X 03 0 2 hτ − n i|ˆ g2,1 (τ, n )| dτ k=0

≤C ≤C

|n0 |∼2n ∞ Z X

|τ |∼2k 1/2

hτ1 i

k=0 R ∞ Z X k1 =0

 X Z ˆ |ψ(τ1 )| |n0 |∼2n

|˜ τ +τ1 |∼2k

|fˆ(˜ τ , n0 )|2 1/2 d˜ τ dτ1 h˜ τ − n0 3 i

ˆ 1 )| hτ1 i1/2 |ψ(τ

|τ1 |∼2k1

 X × +

X

+

k1 −4≤k≤k1 +4

kk1 +4

|˜ τ +τ1 |∼2k

|n0 |∼2n

|fˆ(˜ τ , n0 )|2 1/2 d˜ τ dτ1 . h˜ τ − n0 3 i

Since  X kk1 +4 +1 Z  X X

≤ C(k1 − 4)

|n0 |∼2n j=−1

|˜ τ |∼2k1 +j

|n0 |∼2n

|fˆ(˜ τ , n0 )|2 1/2 d˜ τ h˜ τ − n0 3 i

|fˆ(˜ τ , n0 )|2 1/2 d˜ τ k1 +5 h˜ τ − n0 3 i |˜ τ |∼2 0 n |n |∼2 X X Z |fˆ(˜ τ , n0 )|2 1/2 +C d˜ τ , τ − n0 3 i |˜ τ |∼2k h˜ 0 n k∈N

 X Z +8

|n |∼2

|˜ τ +τ1 |∼2k

|fˆ(˜ τ , n0 )|2 1/2 d˜ τ h˜ τ − n0 3 i

42

A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

we may easily bound the preceding term by ∞ Z ∞  X Z X X k1 /2 ˆ C (1 + k1 )2 |ψ(τ1 )|dτ1 k1 =0

|τ1 |∼2k1

k=0

ˆ L1 (R) ≤ Cε |hτ i1/2+ε ψ|

∞  X k=0

X Z |n0 |∼2n

|τ |∼2k

k |n0 |∼2n |τ |∼2  0 2 1/2

|fˆ(˜ τ , n0 )|2 1/2 d˜ τ h˜ τ − n0 3 i

|fˆ(˜ τ , n )| d˜ τ h˜ τ − n0 3 i

thus ˆ L1 (R) |f | s,−1/2 . |g2,1 |X s,1/2 ≤ Cε |hτ i1/2+ε ψ| X 1,1

1,1

At last, ˆ − n0 3 ) gˆ2,2 (τ, n ) = ψ(τ 0

Z |τ1 −n0 3 |≥1

fˆ(τ1 , n0 ) dτ1 τ1 − n 0 3

and hence X

2sn

n∈N

≤

X n∈N

X k∈N

2sn

X k∈N

!1/2

Z

X

3

hτ − n0 i|ˆ g2,2 (τ, n0 )|2 dτ

|τ |∼2k |n0 |∼2n

Z sup |n0 |∼2n

!1/2 03

03

ˆ − n )|2 dτ hτ − n i|ψ(τ

|τ |∼2k

!1/2 X  Z |fˆ(τ1 , n0 )| 2 dτ1 × 03 R hτ1 − n i |n0 |∼2n  X X ≤C 2sn h2k − 23n i−ε n∈N k∈N Z  1/2 3 ˆ − n0 3 )|2 dτ × sup hτ − n0 i1+ε |ψ(τ |n0 |∼2n R !1/2 X  Z |fˆ(τ1 , n0 )| 2 × dτ1 03 R hτ1 − n i 0 n |n |∼2

≤ C|ψ|H 1/2+ε/2 |f |Ys . This ends the proof of the first estimate in Proposition 4.1. The proof of s continuity with values in B2,1 and the second estimate follow in an obvious way from a slight modification of the proof of Lemma 2.2 in [10].  The next lemma shows that the free term in equation (1.5) belongs to σ if u0 is in B2,1 (T).

σ,1/2,T X1,1

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS σ,1/2,T

σ Lemma 4.1. Let u0 ∈ B2,1 (T), and T ≤ 1. Then U (t)u0 ∈ X1,1 is a constant C > 0 such that

43

and there

σ . |U (t)u0 |X σ,1/2,T ≤ C|u0 |B2,1 1,1

Proof. Let ψ be a cut-off function with ψ ≡ 1 on [0, 1] and let us prove that σ . |ψU (t)u0 |X σ,1/2 ≤ C|u0 |B2,1 1,1

σ,1/2+ε

σ,1/2

⊂ X1,1

We use the fact that X1,∞

for any ε > 0, and that

3 \ ψU (t)u0 (τ, n0 ) = uˆ0 (n0 )ψ(τ − n0 )

to get the following bound : |ψU (t)u0 |X σ,1/2 ≤ Cε |ψU (t)u0 |X σ,1/2+ε 1,1 1,∞ ∞  X Z 1/2 X σn 0 3 1+2ε 0 2 ˆ 03 2 2 sup hτ − n i |ˆ u0 (n )| |ψ(τ − n )| dτ ≤ Cε ≤ Cε ≤ Cε ≤ Cε

n=0 ∞ X n=0 ∞ X n=0 ∞ X n=0

k≥0

2σn sup

|n0 |∼2n

 X

k≥0 σn

2

2

|τ |∼2k |τ −n0 3 |∼2j

|n0 |∼2n j=0

∞  X Z X |n0 |∼2n

j=0 σn

|τ |∼2k ∞ XZ

 X

3 ˆ − n0 3 )|2 dτ hτ − n0 i1+2ε |ˆ u0 (n0 )|2 |ψ(τ

3

3

ˆ − n0 )|2 dτ hτ − n0 i1+2ε |ˆ u0 (n0 )|2 |ψ(τ

1/2

1/2

|τ −n0 3 |∼2j 0

2

|ˆ u0 (n )|

∞ Z 1/2 X

|n0 |∼2n

j=0

1+2ε

hτ i

ˆ )|2 dτ |ψ(τ

1/2

|τ |∼2j

σ |ψ| 1/2+ε . ≤ Cε |u0 |B2,1 B 2,1

 Proof of Theorem 1.1. We now have all the estimates in hand, and we proceed exactly as in [3]; we work path-wise on equation (1.5), using a fixed point σ,1/2,T argument in the space X1,1 with −1/2 ≤ σ < s, s being defined by the assumption on φ, and T ≤ 1 sufficiently small. σ Let u0 F0 -measurable with u0 ∈ B2,1 (T) almost surely, σ as above and assume first that uˆ0 (0) = 0 a.s. We set (4.1)

z(t) = U (t)u0 ; σ,1/2,T

then by Lemma 4.1, z ∈ X1,1 (4.2)

for any T ≤ 1, a.s. and

σ |z|X σ,1/2,T ≤ C|u0 |B2,1 1,1

a.s.

44

A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI σ 0 ,1/2,T

σ 0 ,1/2,T

Let w(t) be defined by (1.4). By Proposition 2.1, w ∈ X1,∞ ⊂ X∞,∞ almost surely, for any σ 0 with σ < σ 0 < s. We fix such a σ 0 and consider ω ∈ Ω σ 0 ,1/2,T σ such that u0 ∈ B2,1 (T) and w ∈ X1,∞ for any T ≤ 1, a.s. In terms of v(t) = u(t) − z(t) − w(t), equation (1.5) is written as Z 1 t (4.3) v(t) = T v(t) := − U (t−s)∂x (v 2 +w2 +z 2 +2vw +2vz +2wz)(s)ds. 2 0 Taking 0 < α < 1/16 in Propositions 3.2, 3.3 and 3.4, and ε ≤ α/2 in Proposition 4.1, we easily get the existence of a constant Cα > 0 such that   2 2 2 α/2 . |v|X σ,1/2,T + |w|X σ,1/2,T + |u0 |B2,1 |T v|X σ,1/2,T ≤ Cα T σ 1,1

1,∞

1,1

σ,1/2,T

In the same way, if v1 , v2 ∈ X1,1 |T v1 − T v2 |X σ,1/2,T 1,1

then  ≤ Cα T α/2 |v1 |X σ,1/2,T + |v2 |X σ,1/2,T 1,1 1,1 σ +|w|X σ,1/2,T + |u0 |B2,1 |v1 − v2 |X σ,1/2,T . 1,∞

1,1

Hence, setting first σ Rωt = |w|X σ,1/2,t + |u0 |B2,1 1,∞

and then defining the stopping time Tω by  Tω = inf t > 0, 2Cα tα/2 Rωt ≥ 1/2 σ,1/2,T

it is easily checked that T maps the ball of radius RωTω in X1,1 ω into itself, and that 3 |T v1 − T v2 |X σ,1/2,Tω ≤ |v1 − v2 |X σ,1/2,Tω . 1,1 1,1 4 Hence T has a unique fixed point, which is the unique solution of (4.3) in σ,1/2,T X1,1 ω . It follows from classical arguments and the second part of Proposition 4.1 σ (T)) a.s. On the other hand, since φ ∈ that z and v are in C([0, Tω ]; B2,1 0,s L2 and U (t) is a unitary group in H s (T), we have w ∈ C([0, Tω ]; H s (T)) ⊂ σ C([0, Tω ]; B2,1 (T)) by Theorem 6.10 in [8]. Hence, the solution u = v + z + w σ of (1.5) is almost surely continuous with values in B2,1 (T). One classically get rid of the condition uˆ0 (0) = 0 a.s. by considering v(t, x) = R u(t, x + α0 t) − α0 with α0 = T u0 (x)dx; indeed, v then satisfies the KdV equation (1.2) and the condition vˆ0 (0) = 0. This ends the proof of Theorem 1.1.  We now explain how we can get rid of the condition that the spatial mean of the noise is zero almost surely at any time.

STOCHASTIC KDV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

45

Proposition 4.2. The conclusion of Theorem 1.1 is still true without the assumption that Im φ ⊂ span {ej , j ≥ 1}. Proof. Let P be the orthogonal projector on span {e0 } in L2 (T) i.e. (P u)(x) = (u, e0 )e0 for u ∈ L2 (T), where (·, ·) denotes the inner product in L2 (T). Then, clearly, φ˜ = (I − P )φ satisfies Im φ˜ ⊂ span {ej , j ≥ 1}; on the other hand, ˜ = P φW ˜ + φ˜W ˜ , and β(t) ˜ (t) = P (φek , e0 )βk (t)e0 is a real valued W = P φW k∈N P Brownian motion since k∈N (φek , e0 )2 = |φ∗ e0 |2L2 (T) < +∞. ˜ then if u satisfies the KdV equation (1.2), v satisfies Let v = u − β,   dv + (∂ 3 v + (v + β)∂ ˜ x v)dt = φd ˜ W ˜, x  v(0) = u0 Rt ˜ we get the equation for v˜ and setting v˜(t, x) = v(t, x + 0 β(s)ds),   d˜ ˆ v + (∂x3 v˜ + v˜∂x v˜)dt = dW (4.4)  v˜(0) = u0 R ˜ k )(x − t β(s)ds)β ˜ ˆ (t, x) = P (φe with W k (t) and it is clear that we can k∈N 0 apply all the arguments of the proof of Theorem 1.1 to equation (4.4), leading to the existence and uniqueness of v˜ from which we deduce the existence and uniqueness of u. Indeed, note that in Proposition 2.1, φ was allowed to depend on t and ω provided that it was in L∞ ((0, T ) × Ω; L0,s 2 ), which is obviously the case here.  Proof of Theorem 1.2. The arguments are exactly the same as in [3]: let T > 0 fixed; under the assumptions of Theorem 1.2, considering a sequence 0,0 2 3 φn in L0,4 2 such that φn → φ in L2 and a sequence u0,n in L (Ω; H (T)) such that u0,n → u0 in L2 (Ω; L2 (T)); one can easily prove (see [2]) the existence of a unique solution un in C([0, T ]; H 3 (T)) of Z t Z 1 t 2 ˜ (s). un (t) = U (t)u0,n − U (t − s)φn dW U (t − s)∂x (un (s))ds + 2 0 0 Using Itˆo formula on |un |2L2 (T) and a martingale inequality, one gets as in [3]     E sup |un (t)|2L2 (T) ≤ E |u0,n |2L2 (T) + C(T )kφn k2L0,0 ; t∈[0,T ]

2

hence, up to a subsequence, un converges in L2 (Ω; L∞ (0, T ; L2 (T))) weak star to some process u˜. Then if Tn is defined in the same way as T in the proof of Theorem 2.1, replacing u0 and φ respectively by u0,n and φn , one shows

46

A. DE BOUARD, A. DEBUSSCHE, AND Y. TSUTSUMI

that, given σ < 0, Tn is a uniform contraction in the ball of radius RωTω in σ,1/2,T X1,1 ω ; moreover the unique fixed point of Tn is equal to un , which, as a σ,1/2,T result, converges to u (the solution given by Theorem 1.1) in X1,1 ω for any σ < 0. It follows that u = u˜ a.s. on [0, Tω ], and that σ (T) ≤ Cσ |u(Tω )|L2 (T) ≤ |˜ |u(Tω )|B2,1 u|L∞ (0,T ;L2 (T))

a.s.

so that u may be extended to [0, T ] almost surely, giving the result.



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