Soliton dynamics for the Korteweg-de Vries equation with multiplicative homogeneous noise

0.2 MB·English

by  Anne De Bouard

SOLITON DYNAMICS FOR THE KORTEWEG-DE VRIES EQUATION
WITH MULTIPLICATIVE HOMOGENEOUS NOISE
9
ANNEDE BOUARD1 AND ARNAUDDEBUSSCHE2
0
0
2
1
n Centre de Math´ematiques Appliqu´ees
a UMR 7641, CNRS et Ecole Polytechnique,
J
Route de Saclay,

4 91128 PALAISEAU CEDEX, FRANCE
1
2
IRMAR, ENS Cachan Bretagne, CNRS, UEB
]
P Av. Robert Schuman,
A F-35170 BRUZ, FRANCE
.
h Abstract We consider a randomly perturbed Korteweg-de Vries equation. The perturbation
t
a is a random potential depending both on space and time, with a white noise behavior in time,
m
and a regular, but stationary behavior in space. We investigate the dynamics of the soliton of
[ the KdV equation in the presence of this random perturbation, assuming that the amplitude

1 of the perturbation is small. We estimate precisely the exit time of the perturbed solution
v from a neighborhood of the modulated soliton, and we obtain the modulation equations for
5
the soliton parameters. We moreover prove a central limit theorem for the dispersive part of
6
the solution, and investigate the asymptotic behavior in time of the limit process.
9
1
. 1. Introduction
1
0 Our aim is to describe the dynamics of a soliton solution of the Korteweg-de Vries equation
9
in the presence of a random potential, depending both on space and time and which is white
0
: in time. After the first paper [21] showing “superdiffusion” of the soliton of the KdV equation
v
in the presence of an external force which is a white noise in time (see also [1], [16]), the
i
X
interest in such questions of soliton dynamics in the presenceof either deterministic or random
r perturbationshas recently increased in themathematical community. In[15], e.g. thequestion
a
is investigated with the help of inverse scattering methods, for different types of time-white
noise perturbations, still for the KdV equation, while in [11], [12], the case of a soliton of the
NLSequation is studied,with thepresenceof aslowly varyingdeterministicexternal potential.
Random potential perturbations for NLS equations have also been considered in [14] and [9].
The diffusion of solitons of the KdVequation in the presence of additive noise was numerically
investigated in [19]. Also, in [5], we studied the soliton dynamics for a KdV equation with
an additive space-time noise. Our aim here is to reproduce the analysis of [5] in the case
of a random potential, which is stationary in space : the solution of the stochastic equation
starting from a soliton at initial time will then stay close to a modulated soliton up to times
1991 Mathematics Subject Classification. 35Q53, 60H15, 76B25, 76B35 .
Key words and phrases. Korteweg-de Vries equation, stochastic partial differential equations, white noise,
central limit theorem, solitary waves.
1
2 A.DEBOUARDANDA.DEBUSSCHE
small compared to ε−2 where ε is the amplitude of the random perturbation (see below). In
the present case, where the noise is multiplicative (the random potential) we are then able to
analyze more precisely the modulation equations for the soliton parameters and the linearized
equation for the remaining (dispersive) part of the solution, and especially its asymptotic
behavior in time.
We consider a stochastic KdV equation which may be written in Itˆo form as
1
(1.1) du+(∂3u+ ∂ (u2))dt = εudW
x 2 x
where ε > 0 is a small parameter, u is a random process defined on (t,x) R+ R, W is
a Wiener process on L2(R) whose covariance operator φφ∗ is such that φ∈is a c×onvolution
operator on L2(R) defined by
φf(x)= k(x y)f(y)dy, for f L2(R).
R − ∈
Z
The convolution kernel k satisfies
(1.2) k := (k2+(k′)2)dx < + .
1
k k R ∞
Z
Considering a complete orthonormal system (ei)i∈N in L2(R), we may alternatively write W
as
(1.3) W(t,x)= β (t)φe (x),
i i
i∈N
X
(βi)i∈N beinganindependentfamilyofrealvaluedBrownianmotions. Thecorrelation function
of the process W is then given by
E(W(t,x)W(s,y)) = c(x y)(s t), x,y R, s,t > 0,
− ∧ ∈
where
c(z) = k(z+u)k(u)du.
R
Z
The existence and uniqueness of solutions for stochastic KdV equations of the type (1.1)
but with an additive noise have been studied in [4], [7], [8]. The multiplicative case with
homogeneous noise as described above was considered in [6]: assuming, together with the
above condition, that k is an integrable function of x R allowed us to prove the global
existence and uniqueness of solutions to equation (1.1) in∈the energy space H1(R), that is in
the space where both the mass
1
(1.4) m(u) = u2(x)dx
2 R
Z
and the energy
1 1
(1.5) H(u) = (∂ u)2dx u3dx
x
2 R − 6 R
Z Z
are well defined. Note that m and H are conserved for the equation without noise, that is
1
(1.6) ∂ u+∂3u+ ∂ (u2)= 0.
t x 2 x
SOLITON DYNAMICS FOR MULTIPLICATIVE KDV EQUATION 3
Under the above conditions on k, it was then proved in [6] that for any given initial data
u H1(R), there is a unique solution u of (1.1) with paths a.s. continuous for t R+ with
0
valu∈es in H1(R). ∈
Our aim in this article is to analyze the qualitative influence of a noise on a soliton solution
of the deterministic equation. More precisely, we study the qualitative behavior of solutions
of (1.1) in the limit ε tends to zero, assuming that the initial state of the solution is a soliton
of equation (1.6). We recall indeed that equation (1.6) possesses a two-parameter family of
solitary waves (or soliton) solutions, propagating with a constant velocity c > 0, with the
expression u (t,x) = ϕ (x ct+x ), x R, where
c,x0 c − 0 0 ∈
3c
(1.7) ϕ (x) =
c 2cosh2(√cx)
2
satisfies the equation
1
(1.8) ϕ′′ cϕ + ϕ2 = 0.
c − c 2 c
We do not recall here the well-known results concerning the stability of the soliton solutions
u in equation (1.6), but we refer to [2], [3], [17] or [18] for a review of the stability questions
c,x0
using PDE methods, or to [13] and [20] for a review of the stability of the solitons with the
help of the inverse scattering transform.
Let us consider as in [5] the solution uε(t,x) of equation (1.1) which is such that uε(0,x) =
ϕ (x) where c > 0 is fixed. Then, in Section 2, we show, as we did in [5] for the additive
0 0
equation that up to times Cε−2, where C is a constant, we may write the solution uε as
(1.9) uε(t,x) = ϕcε(t)(x xε(t))+εηε(x xε(t))
− −
where the modulation parameters cε(t) and xε(t) satisfy a system of stochastic differential
equations and the remaining term εηε is small in H1(R). We then prove in Section 3 that the
process ηε converges as ε goes to zero, in quadratic mean, to a centered Gaussian process η
which satisfies an additively driven linear equation, with a conservative deterministic part; we
also investigate the behavior of the process η as t goes to infinity and prove that η is in some
sense a Ornstein-Uhlenbeck process, with a unique Gaussian invariant measure. In addition,
the parameters xε(t) and cε(t) may be developed up to order one in ε and we get
dxε = c dt+εB dt+εdB +o(ε)
0 1 2
dcε =εdB +o(ε),
1
(cid:26)
where B and B are correlated real valued Brownian motions; keeping only the order one
1 2
terms in those modulation parameters, we then obtain a diffusion result on the modulated
soliton similar to the result obtained by Wadati in [21], butwith a different time exponent (see
Section 4).
In all what follows, (.,.) will denote the inner product in L2(R),
(u,v) = u(x)v(x)dx
R
Z
and we denote by the translation operator defined for ϕ C(R) by ( ϕ)(x) = ϕ(x+x ).
Note that since thTexp0rocess W is stationary in space, for any∈x R theTxp0rocess W is st0ill
0 ∈ Tx0
4 A.DEBOUARDANDA.DEBUSSCHE
a Wiener process with covariance φφ∗. Indeed by (1.3),
W(t,x)= (φe )(x+x )β (t) = (φe˜ )(x)β (t),
Tx0 k 0 k k k
k∈N k∈N
X X
with e˜ (x) = e .
k Tx0 k
2. Modulation and estimate on the exit time
In this section, we prove the following theorem.
Theorem 2.1. Assume that the kernel k of the noise satisfies (1.2) together with k L1(R)
∈
and let c be fixed. For ε > 0, let uε(t,x), as defined above, be the solution of (1.1) with
0
u(0,x) = ϕ (x). Then there exists α > 0 such that, for each α, 0 < α α , there is a
c0 0 ≤ 0
stopping time τε > 0 a.s. and there are semi-martingale processes cε(t) and xε(t), defined a.s.
α
fort ≤ ταε, withvaluesrespectivelyinR+∗ andR, sothatifwesetεηε(t) = uε(t,.+xε(t))−ϕcε(t),
then a.s. for t τε, εηε(t) α and cε(t) c α. In addition, for α sufficiently small,
≤ α k k1 ≤ | − 0| ≤ 0
and any α α , there is a constant C > 0, depending only on α and c , such that for any
0 0
≤
T > 0, there is an ε > 0, with, for each ε< ε ,
0 0
C(α,c )
(2.1) P(τε T) exp 0 .
α ≤ ≤ −ε2T k 2
(cid:18) k kH1(cid:19)
It was noticed heuristically in [5], and proved in [10] that in the additive case, the use of
the modulation parameters xε(t) and cε(t) was necessary in order to get the estimate (2.1).
Indeed, it was proved in [10] that if we denote by τ˜ε,n = inf t > 0, uε,n(t,.) ϕ > α ,
α { k − c0k1 }
where uε,n is here the solution of equation (1.1), but with an additive noise that becomes
stationary in space as n goes to infinity (see [10] for a precise statement) then there exists a
constant C(α,c ) which depends on α and c but not on T such that
0 0
C(α,c )
(2.2) lim lim ε2logP(τ˜n,ε T) 0 .
n→∞ ε→0 α ≤ ≥ − T3
Itisnotclearthat(2.2)isstilltrueinthepresentmultiplicativecase,becausetheproofinvolves
a controlability problem with a potential which – up to now – is open.
Note also that the decomposition given in Theorem 2.1 is not unique, and is determined by
the choice of specific orthogonality conditions (see the proof below). In particular, contrary to
the additive case, we will be able here to investigate the asymptotic behavior in time of the
limit process by choosing one particular decomposition of the form given in Theorem 2.1. This
is the object of Section 3.3.
Proof of Theorem 2.1TheprooffollowscloselytheproofofTheorem2.1in[5]andwereferto[5]
for more details. The parameters xε(t) and cε(t) are obtained thanks to the use of the implicit
function Theorem. These are then local semi-martingales defined as long as cε(t) c < α
0
| − |
and uε(t,.+xε(t)) ϕ < α, and setting
k − c0k1
εηε(t) = uε(t,.+xε(t)) ϕcε(t),
−
one has for each ε > 0, almost surely,
(2.3) (ηε,ϕ )= (ηε,∂ ϕ ) = 0.
c0 x c0
SOLITON DYNAMICS FOR MULTIPLICATIVE KDV EQUATION 5
In order to estimate the exit time
τε = inf t 0, cε(t) c > α or εηε(t) > α ,
α { ≥ | − 0| k k1 }
we make use , as in [5], of the functional defined for u H1(R),
∈
(2.4) Q (u) := H(u)+c m(u)
c0 0
where H and m are defined respectively in (1.4) and (1.5). Note that ϕ is a critical point of
c0
Q . We denote by L the linearized operator around ϕ , that is
c0 c0 c0
(2.5) L = ∂2+c 2ϕ .
c0 − x 0− c0
Thenextlemma,whichisprovedwiththeuseoftheItˆoFormula, usingthesameregularization
procedureasin[4],givestheevolution ofH andmforthesolutionuε of(1.1)withuε(0) = ϕ :
c0
Lemma 2.2. For any stopping time τ <+ a.s, one has
∞
τ τ
m(uε(τ)) = m(ϕ ) ε ((uε)(s),dW(s))+ε2 k 2 m(uε(s))ds
c0 − | |L2
Z0 Z0
and
τ ε τ
(2.6) H(uε(τ)) = H(ϕ )+ε (∂ uε,∂ (uεdW(s))) ((uε)3,dW(s))
c0 x x − 2
Z0 Z0
ε2 τ
(2.7) + k 2 ∂ uε 2 + k′ 2 uε 2 ds
2 | |L2| x |L2 | |L2| |L2
Z0
ε2 (cid:8) τ (cid:9)
(2.8) (uε)3 φe 2dxds.
k
− 2 k Z0 ZR | |
X
Consider ν > 0 such that (Q′′ (ϕ )v,v) ν v 2 for any v H1 satisfying (v,ϕ ) =
c0 c0 ≥ k k1 ∈ c0
(v,∂ ϕ ) = 0. The existence of such a constant is a classical result (see [2] or [3]). Then it is
x c0
easy to show (see [5]) that there is a constant C(α )> 0 such that for any t < τε,
0 α
ν
(2.9) Qc0(uε(t,.+xε(t)))−Qc0(ϕcε(t)) ≥ 4kεηε(t)k21 −C|cε(t)−c0|2.
Now, if τ = τε t, then by (2.9), the translation invariance of Q , and Lemma 2.2
α∧ c0
4 τ
kεηε(τ)k21 ≤ ν Qc0(ϕc0)−Qc0(ϕcε(τ)) +ε (∂xuε(s),∂x(uεdW(s)))
ε(cid:2) τ (cid:3) ε2 Z0τ
((uε)3(s),dW(s))+ (k 2 ∂ uε 2 + k′ 2 uε 2 )ds
−2 2 | |L2| x |L2 | |L2| |L2
(2.10) ε2Z0 τ Z0 τ
(uε)3(s)φe 2dxds c ε ((uε)2,dW(s))
k 0
− 2 k Z0 ZR | | − Z0
X τ
+c ε2 k 2 m(uε(s))ds+C cε(τ) c 2.
0 | |L2 | − 0|
Z0
The term cε(τ) c is then estimated thanks to the orthogonality condition (ηε,ϕ ) = 0
| − 0| c0
and the evolution of m(uε(τ)) given in Lemma 2.2; one obtains, for some constants µ > 0 and
6 A.DEBOUARDANDA.DEBUSSCHE
C > 0, depending only on c and α (with α α )
0 0 0
≤
µ|cε(τ)−c0| ≤ |ϕc0|2L2 −|ϕcε(τ)|2L2
τ
≤ (cid:12)(cid:12)|εηε(τ)|2L2 +Cα|cε(τ(cid:12)(cid:12))−c0|+2ε ((uε)2,dW(s))
(cid:12)Z0 (cid:12)
τ (cid:12) (cid:12)
+2ε2 k 2 uε(s)2 ds. (cid:12) (cid:12)
| |L2 | |L2 (cid:12) (cid:12)
Z0
Hence, choosing α sufficiently small one gets
0
τ 2
cε(τ) c 2 C εηε(τ)4 +4ε2 ((uε)2,dW(s))
| − 0| ≤ | |L2
(2.11) h τ (cid:12)Z0 2 (cid:12)
+4ε4 k 4 uε((cid:12)s)2 ds (cid:12)
| |L2 | (cid:12) |L2 (cid:12)
(cid:16)Z0 (cid:17) i
which, once inserted into (2.10) leads to
τ
εηε(τ) 2 C εηε(τ)4 +ε (∂ uε,∂ (uεdW(s)))
k k1 ≤ | |L2 x x
h τ (cid:12)Z0 τ (cid:12)
(cid:12) (cid:12)
+ε ((uε)3,dW(cid:12) (s)) +c ε ((uε)2,dW(cid:12) (s))
0
(cid:12)(cid:12)Z0 τ (cid:12)(cid:12) 2 (cid:12)(cid:12)Z0 τ (cid:12)(cid:12)
+4(cid:12)ε2 ((uε)2,dW(s(cid:12))) +(cid:12)ε2 k 2 uε(s) 2(cid:12)ds
k k1 k k1
(cid:12)(cid:12)Z0 τ (cid:12)(cid:12) Z0τ 2
+ε2|k(cid:12)|2L2 kuε(s)k31ds(cid:12)+ε4|k|4L2 |uε(s)|2L2ds .
Z0 (cid:16)Z0 (cid:17) i
With this estimate in hand, together with (2.11), the conclusion of Theorem 2.1 follows with
thesameargumentsasintheproofofProposition3.1in[10]. Theseargumentsrelyonclassical
exponential tail estimates for stochastic integrals, after noticing that uε(s) C, a.s. for
1
k k ≤
s [0,τε T] and α α , so that the quadratic variation of each of the integrals involved in
th∈e abovαe∧estimates a≤re b0ounded above by CT. (cid:3)
3. A central limit theorem
This section is devoted to the proof of the next theorem:
Theorem 3.1. Under the assumptions of Theorem 2.1, let α < α be fixed. Then we can
0
find c˜ε(t) and x˜ε(t) satisfying the conclusion of Theorem 2.1 such that if η˜ε is defined as in
Theorem 2.1, for any T > 0, the process (η˜ε(t)) converges in L2(Ω;L∞(0,τε T;L2(R)))
t∈[0,T] α∧
to a Gaussian process η˜ satisfying the additive linear equation
(3.1) dη˜= ∂ L η˜dt+Q˜ϕ dW˜ ,
x c0 c0
with η˜(0) = 0, where W˜ is the Wiener process with covariance φφ∗ given by W˜ = W, and
Tc0t
Q˜ is a projection operator. Moreover, for a > 0 sufficiently small compared to c , the process
0
w(t,x) = eaxη˜(t,x) is a well defined H1 valued process, of Ornstein-Uhlenbeck type, which
converges in law to an H1-valued Gaussian random variable as t goes to infinity.
The conclusion of Theorem 3.1 will be obtained in three steps. The first step consists in
estimating the modulation parameters obtained in Theorem 2.1, in terms of ηε, using the
SOLITON DYNAMICS FOR MULTIPLICATIVE KDV EQUATION 7
equations for those parameters; then the convergence of ηε as ε tends to zero is proved, and
finally in the third step, a slight change in the modulation parameters is performed, in order
that the limit process η may be written as an Ornstein-Uhlenbeck process.
From now on, we assume that α is fixed and sufficiently small, so that the conclusion of
Theorem 2.1 holds, and we denote τε by τε.
α
3.1. Modulation equations. Since we know that the modulation parameters xε(t) and cε(t)
aresemi-martingaleprocessesadaptedtothefiltrationgeneratedby(W(t)) ,wemayapriori
t≥0
write the stochastic evolution equations for those parameters in the form
dxε = cεdt+εyεdt+ε(zε,dW)
(3.2)
dcε = εaεdt+ε(bε,dW)
(cid:26)
where yε and aε are real valued adapted processes with a.s. locally integrable paths on [0,τε),
and bε, zε are predictable processes with paths a.s. in L2 (0,τε;L2(R)). We then proceed as
loc
in [5] : the Itˆo-Wentzell Formula applied to uε(t,x+xε(t)), together with equation (1.1) for
uε and the first equation of (3.2) for xε give a stochastic evolution equation for uε(t,x+xε).
On the other hand, the standard Itˆo Formula together with the second equation of (3.2) for cε
give an equation for the evolution of ϕcε(t). Replacing then ϕcε(t)+εηε(t,x) for uε(t,x+xε(t))
in the first equation leads to the following stochastic equation for the evolution of ηε(t) :
dηε = ∂xLc0ηεdt+(yε∂xϕcε −aε∂cϕcε)dt−∂x((ϕcε −ϕc0)ηε)dt
+(cε−c0+εyε)∂xηεdt− 2ε∂x((ηε)2)dt+ϕcεTxεdW
(3.3) +∂xϕcε(zε,dW)−∂cϕcε(bε,dW)+εηεTxεdW +ε∂xηε(zε,dW)
+2ε∂x2ϕcε|φ∗zε|2L2dt− 2ε∂c2ϕcε|φ∗bε|2L2dt+ε ∂x(ϕcεTxεφel)(zε,φel)dt
l∈N
X
+12ε2∂x2ηε|φ∗zε|2L2dt+ε2 ∂x(ηεTxεφel)(zε,φel)dt
l∈N
X
where L is defined in (2.5). Now, taking the L2- inner product of equation (3.3) with ϕ , on
c0 c0
the one hand, and with ∂ ϕ on the other hand, then using the orthogonality conditions (2.3)
x c0
and the fact that L ∂ ϕ = 0, and finally identifying the drift parts and the martingale parts
c0 x c0
of each of the resulting equations lead to the same kind of system that we previously obtained
in [5]; namely, setting
yε(t) (zε,φe )
Yε(t) = and Zε(t)= l
aε(t) l (bε,φe )
l
(cid:18) (cid:19) (cid:18) (cid:19)
then one gets for the drift parts
(3.4) Aε(t)Yε(t) = Gε(t)
where
(3.5) Aε(t) = (∂xϕcε +ε∂xηε,∂xϕc0) −(∂cϕcε,∂xϕc0)
(cid:18) −(∂xϕcε,ϕc0) (∂cϕcε,ϕc0) (cid:19)
8 A.DEBOUARDANDA.DEBUSSCHE
and
Gε(t)
Gε(t) = 1 ,
Gε(t)
(cid:18) 2 (cid:19)
with
Gε(t) = (ηε,L ∂2ϕ )+(cε c )(ηε,∂2ϕ )+ ε(∂ (ηε)2,∂ ϕ )
1 c0 x c0 − 0 x c0 2 x x c0
+(∂x((ϕcε −ϕc0)ηε),∂xϕc0)− 2ε(∂x2ϕcε,∂xϕc0)|φ∗zε|2L2
(3.6) +2ε(∂c2ϕcε,∂xϕc0)|φ∗bε|2L2 −ε (zε,φel)(∂x(ϕcεTxεφel),∂xϕc0)
l∈N
X
+12ε2(ηε,∂x3ϕc0)|φ∗zε|2L2 −ε2 (∂x(ηεTxεφel),∂xϕc0)(zε,φel)
l∈N
X
and
Gε2(t) = −2ε(∂x(ηε)2,ϕc0)−(∂x((ϕcε −ϕc0)ηε),ϕc0)+ 2ε(∂x2ϕcε,ϕc0)|φ∗zε|2L2
(3.7) −2ε(∂c2ϕcε,ϕc0)|φ∗bε|2L2 +ε (zε,φel)(∂x(ϕcεTxεφel),ϕc0)
+ε22(ηε,∂x2ϕc0)|φ∗zε|2L2 +ε2X (∂x(ηεTxεφel),ϕc0)(zε,φel);
l∈N
X
note that Aε(t)= A +O(cε c + εηε ), a.s. for t τε with
0 0 1
| − | k k ≤
∂ ϕ 2 0
A0 = | x 0c0|L2 (ϕ ,∂ ϕ )
(cid:18) c0 c c0 (cid:19)
and O(cε c + ηε ) is uniform in ε,t and ω as long as t τε. Concerning the martingale
0 1
| − | k k ≤
parts, one gets the equation
(3.8) Aε(t)Zε(t)= Fε(t), l N
l l ∀ ∈
with
(3.9) Fε(t) = −((ϕcε +εηε)Txεφel,∂xϕc0)
(cid:18) ((ϕcε +εηε)Txεφel,ϕc0). (cid:19)
Proposition 3.2. Under the above assumptions, there is a constant α > 0, such that if
1
α α , then
1
≤
(3.10) φ∗zε(t)L2 + φ∗bε L2 C1 k L2, a.s. for t τε
| | | | ≤ | | ≤
and
(3.11) aε(t) + yε(t) C2 ηε(t)L2 +εC3, a.s. for t τε
| | | | ≤ | | ≤
for some constants C , C , C , depending only on α and c , and for any ε ε .
1 2 3 0 0
≤
SOLITON DYNAMICS FOR MULTIPLICATIVE KDV EQUATION 9
Proof The proof is exactly the same as the proof of Corollary 4.3 in [5], once noticed that, a.s.
for t τε,
≤
|Flε(t)|2 ≤ C |(ϕcε +εηε)Txεφel|2L2
l∈N l∈N
X X
C (ϕcε +εηε)2(x)[( xεk) el]2(x)dx
≤ R T ∗
l Z
X
(ϕcε +εηε)2(x) ( xεk(x .),el)2dx
≤ R T −
Z l
X
≤ C R(ϕcε +εηε)2(x)|Txεk(x−.)|2L2dx
Z
≤ C|k|2L2|ϕcε +εηε|2L2 ≤ C|k|2L2
where we have used the Parseval equality in the fourth line. (cid:3)
3.2. Convergence of ηε. Let us first assume that ηε has a limit as ε goes to zero, and take
formally the limit as ε goes to zero in the preceding equations. Then, as was noticed above,
∂ ϕ 2 0
εli→m0Aε = A0 = (cid:18)| x 0c0|L2 (ϕc0,∂cϕc0)(cid:19)
hence
1
(3.12) limφ∗zε = ( φ)∗(ϕ ∂ ϕ ) := z
ε→0 −|∂xϕc0|2L2 Tc0t c0 x c0
1
(3.13) limφ∗bε = ( φ∗)(ϕ2 ) := b
ε→0 (ϕc0,∂cϕc0) Tc0t c0
1
(3.14) limyε = (η,L ∂2ϕ ):= y
ε→0 |∂xϕc0|2L2 c0 x c0
and
(3.15) limaε = 0.
ε→0
Moreover, formally, η satisfies the equation
dη = ∂ L ηdt+ 1 (η,L ∂2ϕ )∂ ϕ dt
x c0 |∂xϕc0|2L2 c0 x c0 x c0
(3.16) +ϕc0Tc0tdW − 2|∂xϕ1c0|2L2(∂x(ϕ2c0),Tc0tdW)∂xϕc0
1 (ϕ2 , dW)∂ ϕ .
−(ϕc0,∂cϕc0) c0 Tc0t c c0
It is easy to show that (3.16) has a unique adapted solution η with paths a.s. in C(R+,H1)
satisfying η(0) = 0. Moreover using the fact that (∂ ϕ ,∂ ϕ ) = 0, one easily gets from the
c c0 x c0
above equation that (η,ϕ ) = (η,∂ ϕ ) =0, t> 0.
c0 x c0 ∀
Next, we make use of the following lemmas, whose proofs are obtained in the same way as
the corresponding Lemmas in [5].
10 A.DEBOUARDANDA.DEBUSSCHE
Lemma 3.3. Let η be the solution of (3.16) with η(0) = 0. Then, for any T > 0, there is a
constant C depending only on c , T and k such that
0 1
k k
E η(t) 4 C, t T.
k k1 ≤ ∀ ≤
Lemma 3.4. Let ηε be the solution(cid:0)of (3.3(cid:1)), defined for t [0,τε[, obtained thanks to the
∈
modulation procedure of Section 2. Then, for any T > 0,
E sup ηε(t)4 C(T,α,c , k ).
t≤τε∧T| |L2 ≤ 0 k k1
(cid:16) (cid:17)
The above lemmas show that
(3.17) T > 0, q 2, limE sup cε(t) c q = 0.
0
∀ ∀ ≥ ε→0 t≤T∧τε| − |
(cid:16) (cid:17)
Indeed, the expression of cε(t) c given by (3.2) together with (3.10) and (3.11) imply easily
0
−
T∧τε
E sup cε(t) c 2 Cε2[1+E ηε(s)2 ds]
(cid:16)t≤T∧τε| − 0| (cid:17) ≤ Z0 | |L2
with C = C(α,c ,T, k ). Then, (3.17) is deduced form Lemma 3.4 for q = 2, and follows for
0 1
k k
all other values of q from the uniform boundedness of cε(t) c on [0,T τε]. Note that an
0
| − | ∧
immediate consequence of (3.17) is the fact that
(3.18) ∀T > 0, ∀q ≥ 2, εli→m0E t≤sTu∧pτεkϕcε(t)−ϕc0k22 = 0.
(cid:16) (cid:17)
We will finally need the next lemma.
Lemma 3.5. For any T > 0, and any q 1,
≥
q
limE sup Zε(t) Z (t)2 = 0
ε→0 (cid:16)t≤T∧τε(cid:16)Xl∈N| l − l | (cid:17) (cid:17)
where we have set for l N
∈
(z,φe )
Z (t) = l ,
l (b,φe )
l
(cid:18) (cid:19)
z and b being given by (3.12) and (3.13), respectively.
Proof Here again, it is sufficient to consider the case q = 1. We recall that Zε satisfies equation
l
(3.8). First, it is clear that
limE sup (Aε(t))−1 (A (t))−1 2q = 0, q 1.
0
ε→0 t≤T∧τεk − k ∀ ≥
(cid:16) (cid:17)
On the other hand, in view of (3.9), denoting F0(t) the formal limit of Fε(t), one has
l l
E sup Fε(t) F0(t)2
t≤T∧τε | l − l |
(cid:16) Xl (cid:17)
≤ CE t≤sTu∧pτε |∂xϕc0(Txεφ−Tc0tφ)el|2L2 +CE t≤sTu∧pτεkϕcε(t) −ϕc0k21
(cid:16) Xl (cid:17) (cid:16) (cid:17)