THE EXPONENTIAL BEHAVIOUR AND
STABILIZABILITY OF STOCHASTIC 2D-NAVIER-STOKES
EQUATIONS
TOM ´
AS CARABALLO, JOS´
E A. LANGA AND #TAKESHI TANIGUCHI
Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico, Uni e sidad de Se illa,
Apa ado de Co eos 1160, 41080-SEVILLA, Spain
#Depa men o Ma hema ics, Ku ume Uni e si y, Ku ume , Fukuoka 830,
Japan
Abs ac . Some esul s on he pa hwise exponen ial s abili y o he
weak solu ions o a s ochas ic 2D-Na ie -S okes equa ion a e es ab-
lished. The i s ones a e p o ed as a consequence o he exponen ial
mean squa e s abili y o he solu ions. Howe e , some o hem a e im-
p o ed by a oiding he p e ious mean squa e s abili y in some mo e
pa icula and es ic i e si ua ions. Also, some esul s and commen s
conce ning he s abilizabili y and s abiliza ion o hese equa ions a e
s a ed.
1. In oduc ion
The long- ime beha iou o lows is a e y in e es ing and impo an
p oblem in he heo y o luid dynamics, as he as li e a u e shows (see
Temam [19], Hale [13], Ladyzhenskaya [14], among o he s, and he e e -
ences he ein), and has been ecei ing e y much a en ion o e he las
h ee decades.
One o he mos s udied models is he Na ie -S okes one (and i s a ian s)
since i p o ides a sui able model which co e s se e al impo an luids (see
Temam [17]-[19] and he e e ences inside hese).
On he o he hand, ano he in e es ing ques ion is o analyze he e ec s
p oduced on a de e minis ic sys em by some s ochas ic o andom dis u -
bances appea ed in he p oblem. These ac s ha e mo i a ed he p esen
wo k whose main objec i e is o show some aspec s o he e ec s p oduced
1991 Ma hema ics Subjec Classi ica ion. P ima y 60H.
Key wo ds and ph ases. S ochas ic Na ie -S okes Equa ions, Exponen ial S abili y,
S abiliza ion.
#To whom co espondence should be add essed.
1
2TOM ´
AS CARABALLO, JOS´
E A. LANGA AND #TAKESHI TANIGUCHI
in he long- ime beha iou o he solu ion o a wo dimensional Na ie -S okes
equa ion unde he p esence o s ochas ic pe u ba ions.
In he de e minis ic case, i is well known o a long ime ha , o small
enough Reynolds numbe (o , equi alen ly, la ge iscosi y), he solu ions o
2D-Na ie -S okes equa ions end o a s a iona y one (unique, in ac ) when
ime goes o in ini e and, as his numbe inc eases, he dynamics o he
sys em u ns mo e and mo e complex (see, e.g. Temam [18] o a de ailed
desc ip ion o he Coue e-Taylo expe imen ). The p oblem o de ec ing
he c i ical alue whe e he ins abili y appea s is a di icul challenging one.
Thus, in a gene al amewo k, one can only ensu e ha o small alues o
he Reynolds numbe he s a iona y solu ion is s able bu we do no know
when i becomes uns able. This mo i a es ha people wo king in his kind
o p oblems use o conside pa icula examples in o de o ob ain sha pe
esul s.
Ou i s aim in his wo k is o p o ide some ligh in some aspec s con-
ce ning he s abili y o he s a iona y solu ions o he ollowing s ochas ic
2D-Na ie -S okes:









dX = [ν∆X−hX, ∇iX+ (X) + ∇p]d +g( , X)dW( )
di X = 0 in [0,∞)×D,
X= 0 on [0,∞)×Γ,
X(0, x) = X0(x), x ∈D,
whe e Dis a egula open bounded domain o R2wi h bounda y Γ, u is
he eloci y ield o he luid, p he p essu e, ν > 0 he kinema ic iscosi y,
X0 he ini ial eloci y ield, he ex e nal o ce ield and g( , x)dW( ) he
andom ield whe e W( ) is an in ini e dimensional Wiene p ocess.
Conce ning he e ec s p oduced by andom pe u ba ions in de e minis ic
sys ems, i is wo h men ioning ha his is a e y di icul ask which is being
in es iga ed ac ually by many au ho s wi hin he amewo k o he heo y o
andom a ac o s ecen ly in oduced by C auel and Flandoli [10]. On he
one hand, exis ence o andom a ac o s is only known o speci ic andom
e ms (see, o ins ance, C auel and Flandoli [10], Capinski and Cu land
[6]). On he o he hand, almos no hing is known on he s uc u e o hese
andom se s, so ha many challenging open p oblems, as hose ela ed o
s abili y and ins abili y, a e s ill open.
Also, i is e y in e es ing o in es iga e i a luid subjec ed o andom
in luences is asymp o ically mo e o less s able han he de e minis ic un-
pe u bed one. In he ini e dimensional case, he e exi s a wide li e a u e
STOCHASTIC NSES 3
on his opic (see A nold [1] and he e e ences he ein) which p o es ha
some kind o mul iplica i e noise may p oduce a s abiliza ion e ec on de-
e minis ic uns able sys ems. Howe e , o he in ini e dimensional case,
a simila esul has no been p o ed ye , mainly due o he ac ha he
echnique de eloped in he ini e dimensional amewo k canno be ex ended
o his case o , a leas , i is no known how o do ha . The main esul
p o ed in [1] ensu es ha an uns able linea di e en ial sys em in Rn,namely
·
x( ) = Ax( ) wi h ace A < 0, can be s abilized by adding a mul iplica-
i e noise in he S a ono ich sense con aining a sui able skew-symme ic
ma ix. One in e es ing ema k is ha when he s ochas ic mul iplica i e
pe u ba ion is conside ed in he I o sense, his uses o imply a gene al
s abiliza ion e ec on he sys em. In a limi sense, he I o equa ions wi h
mul iplica i e noise co espond o de e minis ic equa ions wi h a mean-ze o
luc ua ing con ol plus a s abilizing sys ema ic con ol (see Sec ion 4 o
mo e de ails and commen s). This would mean ha only he s abiliza ion
p oduced by S a ono ich e ms could be conside ed as p ope s abiliza ion
p oduced by andom noise, since he S a ono ich mul iplica i e noise ac s
like a pe iodic ze o-mean eedback con ol, and consequen ly, i s s abilizing
e ec is unexpec ed and he e o e e y in e es ing. In his pape , we conside
he s ochas ic dis u bances in I o sense, so he s abiliza ion esul s p o ed
should be in e p e ed in a sui able sense (see also Ca aballo and Langa [7]
o an analysis on he di e en long- ime beha iou o I o and S a ono ich
equa ions in he linea case).
The con en o his pape is as ollows. In Sec ion 2, we include some p e-
limina ies. In Sec ion 3, we shall p o e some esul s on pa hwise exponen ial
s abili y by ex ending o his case he s abili y heo y p e iously de eloped
o semilinea s ochas ic pa ial di e en ial equa ions (see Ca aballo and
Liu [8], Taniguchi [16]). Finally, in Sec ion 4, we deal wi h he in e es ing
s abilizabili y p oblem, ha is, we shall analyze he possible easons imply-
ing a s abilizing e ec on he de e minis ic p oblem by he appea ance o a
andom dis u bance.
2. P elimina ies
Fi s ly, we in oduce he ollowing Hilbe spaces:
4TOM ´
AS CARABALLO, JOS´
E A. LANGA AND #TAKESHI TANIGUCHI
H= he closu e o he se u∈C∞
0(D, R2) : di u = 0in L2(D, R2)
wi h he no m |u|= (u, u)1
2,whe e o u, ∈L2(D, R2),
(u, ) =
2
X
j=1 ZD
uj(x) j(x)dx,
V= he closu e o he se u∈C∞
0(D, R2) : di u = 0in H1
0(D, R2)
wi h he no m kuk= ((u, ))1
2,whe e o u, ∈H1
0(D, R2),
((u, )) =
2
X
j=1 ∂u
∂xj
,∂
∂xj.
Then, i ollows ha Hand Va e sepa able Hilbe spaces wi h associa ed
inne p oduc s (·,·) and ((·,·)) and he ollowing is sa is ied:
V⊂H≡H0⊂V0,
whe e injec ions a e dense, con inuous and compac . Now, we can se A=
−P4whe e Pis he o hogonal p ojec o om L2(D, R2) on o H, and de ine
he ilinea o m bby
b(u, , w) =
2
X
i,j=1 ZD
ui(x)∂ j
∂xi
(x)wj(x)dx.
As we shall need some p ope ies on his ilinea o m b, we lis he e he
ones we will use la e on (see Temam [19]):
(2.1) |b(u, , w)| ≤ c1|u|1
2kuk1
2k k | w|1
2kwk1
2,∀u, , w ∈V,
b(u, , ) = 0,∀u, ∈V,
b(u, u, −u)−b( , , −u) = −b( −u, u, −u),∀u, ∈V,
whe e c1>0 is an app op ia e cons an which depends on he egula open
domain D(see Cons an in and Foias [9, (6.9), p.50]) . Fu he mo e, we can
de ine he ope a o B:V×V→V0by
hB(u, ), wi=b(u, , w),∀u, , w ∈V,
whe e h·,·i deno es he duali y hV0, V i.We also se
B(u) = B(u, u),∀u∈V.
Le (Ω, P, =) be a p obabili y space on which an inc easing and igh con-
inuous amily {= } ∈[0,∞)o comple e sub-σ-algeb a o =is de ined. Le
STOCHASTIC NSES 5
βn( ) (n= 1,2,3,···) be a sequence o eal alued one-dimensional s an-
da d B ownian mo ions mu ually independen on (Ω, P, =).Se
W( ) = ∞
X
n=1 qλ0
nβn( )en, ≥0
whe e λ0
n≥0 (n= 1,2,3···) a e nonnega i e eal numbe s such ha
P∞
n=1 λ0
n<+∞,and {en}(n= 1,2,3,···) is a comple e o hono mal basis
in he eal and sepa able Hilbe space K. Le Q∈L(K, K) be he ope a-
o de ined by Qen=λ0
nen.The abo e K- alued s ochas ic p ocess W( ) is
called a Q-Wiene p ocess.
Thus he s ochas ic 2D-Na ie -S okes equa ion can be ew i en as ollows
in he abs ac ma hema ical se ing:
(2.2) dX( ) = [−νAX( )−B(X( ))+ (X( ))] d +g( , X( ))dW( ),
whe e :V→V0, g : [0,∞)×V→L(K, H) a e con inuous unc ions sa is-
ying some addi ional assump ions (see condi ions below).Also we conside
he de e minis ic e sion o his equa ion, namely,
(2.3) dX( ) = [−νAX( )−B(X( ))+ (X( ))] d .
Fi s , we gi e he de ini ion o he weak solu ions o s ochas ic 2D-Na ie -
S okes equa ion (2.2)
De ini ion 2.1. A s ochas ic p ocess X( ), ≥0,is said o be a weak
solu ion o (2.2) i
(1a) X( )is = −adap ed,
(1b) X( )∈L∞(0, T;H)∩L2(0, T;V)almos su ely o all T > 0,
(1c) he ollowing equa ion holds as an iden i y in V0almos su ely, o
∈[0,∞)
X( ) = X(0) + Z
0
[−νAX(s)−B(X(s))+ (X(s))] ds
+Z
0
g(s, X(s))dW(s).
As we a e mainly in e es ed in he analysis o he exponen ial s abili y o
he weak solu ions o he p oblem (2.2), we will assume he exis ence o such
weak solu ions (see, o ins ance, Bensoussan [2] o Capinski and Ga a ek
[4] o esul s on he exis ence and uniqueness o solu ions).
Now we a e going o es ablish an I o’s o mula which is going o be nec-
essa y o ou pu poses (see Pa doux [15])

6TOM ´
AS CARABALLO, JOS´
E A. LANGA AND #TAKESHI TANIGUCHI
Le C(1,2)([0,∞)×H, R+) deno e he space o all R+− alued unc ions
Ψ de ined on [0,∞)×Hwi h he ollowing p ope ies:
(1) Ψ( , x) is di e en iable in ∈[0,∞) and wice F eche di e en iable
in xwi h Ψ ( , ·),Ψx( , ·) and Ψxx( , ·) locally bounded on H
(2) Ψ( , ·),Ψ ( , ·) and Ψx( , ·) a e con inuous on H,
(3) o all ace class ope a o s R, (Ψxx( , ·)R) is con inuous om H
in o R.
(4). i ∈V hen Ψx( , )∈V, and u→ hΨx( , u), ∗iis con inuous o
each ∗∈V0,
(5). kΨx( , )k ≤ C0( )(1 + k k), C0( )>0, o all ∈V.
Theo em 2.1. (I o’s o mula) Le Ψ∈C(1,2)([0,∞)×H, R+).I s ochas ic
p ocess X( )is a weak solu ion o (2.2), hen, i holds ha
Ψ( , X( )) = Ψ(0, X(0)) + R
0LΨ(s, X(s))ds
+R
0(Ψx(s, X(s)), g(s, X(s))dW(s)) ,
whe e
LΨ(s, X(s)) = Ψ (s, X(s))
+h−νAX(s)−B(X(s))+ (X(s)),Ψx(s, X(s))i
+1
2 (Ψxx(s, X(s))g(s, X(s))Qg(s, X(s))∗).
De ini ion 2.2. We say ha a weak solu ion X( ) o (2.2) con e ges o
x∞∈Hexponen ially in mean squa e i he e exis a > 0and M0=
M0(X(0)) >0(which may depend on X(0)) such ha
E|X( )−x∞|2≤M0e−a , ≥0,
In pa icula , i x∞is a solu ion o (2.2), hen i is said ha x∞is expo-
nen ially s able in mean squa e p o ided ha e e y weak solu ion o (2.2)
con e ges o x∞exponen ially in mean squa e wi h he same exponen ial
o de a > 0.
De ini ion 2.3. We say ha a weak solu ion X( ) o (2.2) con e ges o
x∞∈Halmos su ely exponen ially i he e exis s γ > 0such ha
lim
→∞
1
log |X( )−x∞| ≤ −γ, almos su ely.
In pa icula , i x∞is a solu ion o (2.2), hen i is said ha x∞is al-
mos su ely exponen ially s able p o ided ha e e y weak solu ion o (2.2)
con e ges o x∞almos su ely exponen ially wi h he same cons an γ.
STOCHASTIC NSES 7
3. The exponen ial s abili y o solu ions
In his sec ion we discuss he momen exponen ial s abili y and almos
su e exponen ial s abili y o weak solu ions o s ochas ic NSE (2.2). Le λ1
>0 be he i s eigen alue o A. We ema k ha k k2≥λ1| |2,∀ ∈V.
We also deno e by
kg( , u)k2
L0
2= (g( , u)Qg( , u)∗).
Th oughou his sec ion we will use he ollowing condi ion:
Condi ion A. The e exis s β > 0such ha
k (u)− ( )kV0≤βku− k, β > 0, u, ∈V.
In his pape , we i s conside he exis ence o he s a iona y solu ion o
he nex equa ion
(3.1) νAu +B(u) = (u) (equali y in V0).
Then we ha e he ollowing lemma. The p oo is simila o he one o
Theo em 10.1 in Temam [18]. Bu , since he p oo depends on he condi ions
o he unc ion , we gi e he p oo o he con enience o he eade .
Lemma 3.1. Suppose ha condi ion A is sa is ied and he unc ion sa is-
ies ha ( m)con e ges o ( )weakly in V0whene e { m} ⊂ Vcon e ges
o ∈Vweakly in Vand s ongly in H. Then,
(a)i ν > β, he e exis s a s a iona y solu ion u∞∈V o (3.1);
(b) u he mo e, i ν > c1k (0)kV0
√λ1(ν−β)+β, hen he s a iona y solu ion o
(3.1) is unique.
P oo . (a) Le 1, 2, 3,···, m,···be he o hono mal basis o V. Conside
he ini e dimensional Hilbe space Vmspanned by { 1,· · ·, m}wi h he
scala p oduc [·,·] and no m [·] induced by he co esponding ones in V.
Now we de ine a mapping Rm:Vm→Vmas ollows
(3.2) [Rmu, ] = ((Rmu, )) := ν((u, ))+b(u, u, )−h (u), i,∀u, ∈Vm.
I we p o e ha his mapping is con inuous in Vmwi h espec o he no m
[·], and ha [Rmu, u]>0 o some u∈Vmwi h [u] = k > 0, hen Lemma
1.4 in Temam [17, p. 164] gua an ees ha he e exis s um∈Vmsuch ha
[um]≤kand Rmum= 0.
8TOM ´
AS CARABALLO, JOS´
E A. LANGA AND #TAKESHI TANIGUCHI
The con inui y o Rm ollows easily om he p ope ies o band he
assump ions on . Now, om (3.2) i holds o u∈Vm
[Rmu, u] = ν((u, u))+b(u, u, u)−h (u), ui
≥ν((u, u)) −k (u)kV0kuk
≥νkuk2−(k (0)kV0+βkuk)kuk
Since ν > β, we can choose a posi i e eal numbe k > 0 such ha (ν−β)k2−
k (0)kV0k > 0,and o u∈Vmsuch ha kuk=k, we ha e [Rmu, u]>0.
Then, he e exis s an elemen um∈Vm⊂Vwhich is a solu ion o (3.2)
wi h kumk ≤ k. Fu he mo e, we can easily deduce (see es ima ion (3.3)
below) ha
kumk ≤ k (0)kV0
(ν−β),
and, consequen ly, we ha e ha a sui able subsequence o {um}con e ges
weakly in V o some limi u∞and, hanks o he compac injec ion, s ongly
in H. Now, he p ope ies o band assump ions on enable us o p o e ha
his u∞is a solu ion o (3.1).
(b) As o he uniqueness s a emen , le us assume ha u1and u2a e wo
solu ions, hen
ν((u1, ))+b(u1, u1, ) = h (u1), i,∀ ∈V,
ν((u2, ))+b(u2, u2, ) = h (u2), i,∀ ∈V.
Se ing =u1−u2,by subs ac ing he second ela ion om he i s one,
and aking in o accoun he p ope ies o he ilinea o m band condi ion
A we ob ain ha
νku1−u2k2=−b(u1, u1, u1−u2) + b(u2, u2, u1−u2)
+h (u1)− (u2), u1−u2i
=−b(u1−u2, u2, u1−u2) + h (u1)− (u2), u1−u2i
≤c1
√λ1ku1−u2k2ku2k+βku1−u2k2.
Obse ing ha
νku2k2=h (u2), u2i
≤ k (u2)kV0ku2k(3.3)
≤βku2k2+k (0)kV0ku2k,
i ollows ha
ku2k ≤ k (0)kV0
ν−β.
STOCHASTIC NSES 9
Consequen ly,
νku1−u2k2≤c1k (0)kV0
√λ1(ν−β)+βku1−u2k2,
and as ν > c1k (0)kV0
√λ1(ν−β)+β, uniqueness ollows immedia ely. This comple es
he p oo o he lemma.
Now, using his lemma, we discuss he long- ime beha iou o weak solu-
ions X( ) o he s ochas ic Na ie -S okes equa ion (2.2) unde some con-
di ions including ha he kinema ic iscosi y νis su icien ly la ge. Hence
h oughou his pape we assume ha he e exis s a unique s a iona y so-
lu ion u∞∈V o (3.1).In his sec ion, we use he ollowing condi ion.
Condi ion B. kg( , u)k2
L0
2≤γ( )+(ξ+δ( )) |u−u∞|2,
whe e ξ > 0is a cons an and γ( ), δ( )a e nonnega i e in eg able unc ions
such ha he e exis eal numbe s θ > 0, Mγ, Mδ≥1wi h
γ( )≤Mγe−θ , δ( )≤Mδe−θ , ≥0.
Theo em 3.2. Le u∞∈Vbe he unique s a iona y solu ion o (3.1) and
le 2ν > λ−1
1ξ+ 2β+2c1
√λ1ku∞k.Suppose ha condi ions A and B a e
sa is ied. Then, any weak solu ion X( ) o (2.2) con e ges o he s a iona y
solu ion u∞ o (3.1) exponen ially in mean squa e. Tha is, he e exis eal
numbe s a∈(0, θ), M0=M0(X(0)) >0such ha
E|X( )−u∞|2≤M0e−a , ≥0.
P oo .Since 2ν > λ−1
1ξ+ 2β+2c1
√λ1ku∞k,we can ake a posi i e eal
numbe a∈(0, θ) such ha 2ν > λ−1
1(ξ+a) + 2β+2c1
√λ1ku∞k.Then, by
applying he I o o mula o he unc ion ea |X( )−u∞|2,we ha e ha
ea E|X( )−u∞|2=E|X(0) −u∞|2+R
0aeasE|X(s)−u∞|2ds
−2R
0easEhνAX(s), X(s)−u∞ids
−2R
0easEhB(X(s)), X(s)−u∞ids
+2 R
0easEh (X(s)), X(s)−u∞ids
+R
0easEkg(s, X(s))k2
L0
2ds.
Since u∞sa is ies he iden i y (3.1),
R
0easEhνAu∞, X(s)−u∞ids +R
0easEhB(u∞), X(s)−u∞ids
=R
0easEh (u∞), X(s)−u∞ids.
The e o e, no ing he nex iden i y:
16 TOM ´
AS CARABALLO, JOS´
E A. LANGA AND #TAKESHI TANIGUCHI
and he e o e, he ze o solu ion is exponen ially s able in mean squa e i
and only i a+b2
2<0.So, we obse e ha he e exis many possibili ies o
being he ze o solu ion pa hwise exponen ially s able and, a he same ime,
exponen ially uns able in mean squa e.
Consequen ly, i would be e y in e es ing o ob ain pa hwise exponen ial
s abili y esul s by a oiding he me hod o using mean squa e s abili y as a
p e ious s ep. This will be one o he aims o his sec ion. Howe e , i is
wo h poin ing ou ha o ge some esul s in his di ec ion, we will need
o assume some addi ional hypo heses on he s ochas ic pe u ba ion so
ha we can ob ain be e s abili y c i e ia bu o mo e speci ic si ua ions.
In pa icula , in some o ou si ua ions, he noise is so special ha one
can pe o m a ime change, a subs i u ion ha ans o m he s ochas ic
equa ion in o a de e minis ic one. Fo example, he I o o mula o he
loga i hm in he p oo o Theo em 4.2 in his sec ion is one way o pe o m
his ans o ma ion; ano he is o mul iply by he exponen ial o he noise
(see C auel and Flandoli [10, p. 382]).
To his end le us i s ly s a e he ollowing condi ion
Condi ion D. :H→H, and sa is ies
| (u)− ( )| ≤ c|u− |, c > 0, u, ∈H,
g( , ·) : H→L(K, H),and sa is ies
kg( , u)−g( , )kL(K,H)≤Cg|u− |,∀ ∈[0,∞),∀u, ∈H.
Obse e ha i νλ1> c and (0) = 0, hen he ze o solu ion o (2.3) is
exponen ially s able. Bu when νλ1≤cand (0) = 0 we do no know, in
gene al, i he ze o solu ion is exponen ially s able o no . The ollowing
heo em is going o s a e ha , unde some pa icula condi ions, any weak
solu ion o he s ochas ic Na ie -S okes equa ion con e ges o ze o almos
su ely exponen ially s able. So, in a sense, we can in e p e ha a kind o
s abiliza ion could ha e aken place in he sys em.
Theo em 4.1. In addi ion o condi ion D, assume ha (0) = 0 and
g( , 0) = 0 o all ≥0,and ha he e exis s ρ > 0such ha
e
Qψ(s, x):= [(ψx(x)⊗ψx(x))(g(s, x)Qg(s, x)∗)] ≥ρ2|x|4,
whe e ψ(x) = |x|2( ecall ha (ψx(x)⊗ψx(x))(h) = ψx(x) (ψx(x), h), o
x, h ∈H).Then, he e exis s Ω0⊂Ω, P(Ω0) = 0,such ha o ω /∈Ω0 he e
exis s T(ω)>0such ha any weak solu ion X( ) o (2.2) sa is ies
|X( )|2≤ |X(0)|2e−γ o any ≥T(ω),

STOCHASTIC NSES 17
whe e γ:= 1
2(λ1ν−c−C2
g
2+ρ2
2).In pa icula , exponen ial s abili y o sample
pa hs wi h p obabili y one holds i γ > 0.
P oo . Le us apply I o’s o mula o ou solu ion X( ).Then, i ollows
|X( )|2=|X(0)|2+ 2 R
0h−νAX(s)−B(X(s))+ (X(s)), X(s)ids
+R
0kg(s, X(s))k2
L0
2ds
+2 R
0(X(s), g(s, X(s))dW (s))
=|X(0)|2+ 2 R
0h−νkX(s)k2+h (X(s)), X(s)iids
+R
0kg(s, X(s))k2
L0
2ds
+2 R
0(X(s), g(s, X(s))dW (s)) ,
and, applying once again I o’s o mula o he unc ion log |X( )|2,and aking
in o accoun he hypo heses, i ollows
log |X( )|2= log |X(0)|2+1
2R
0
1
|X(s)|2h−2νkX(s)k2+ 2 hX(s), (X(s))iids
+1
2R
0
1
|X(s)|2kg(s, X(s))k2
L0
2ds
+2 R
0
1
|X(s)|2(X(s), g(s, X(s))dW (s)) −1
2R
0
e
Qψ(s,X(s))
|X(s)|4ds
≤log |X(0)|2+R
0
1
|X(s)|2h−νλ1+c+C2
g
2i|X(s)|2ds
+2 R
0
1
|X(s)|2(X(s), g(s, X(s))dW (s)) −ρ2
2 .
Now, due o ou assump ions, he e m M( ) = R
0
2
|X(s)|2(X(s), g(s, X(s))dW (s))
is a eal ma ingale and i is no di icul o p o e, by means o he law o
i e a ed loga i hm,
lim
→+∞
M( )
= 0, P −almos su ely.
Thus, we can assu e ha he e exis s a se Ω0⊂Ω wi h P(Ω0) = 0,such
ha o e e y ω /∈Ω0 he e exis s T(ω)>0 such ha o all ≥T(ω)
M( )
≤1
2(λ1ν−c−C2
g
2+ρ2
2).
The e o e, i easily ollows ha o any ≥T(ω)
log |X( )|2≤log |X(0)|2+1
2(−λ1ν+c+C2
g
2−ρ2
2) .
The p oo is now comple e.
Rema k 4.1. Obse e ha al hough we do no know whe he he s a iona y
solu ion o he de e minis ic p oblem is s able o no , i is possible o ensu e
sample exponen ial s abili y o he s ochas ic equa ion p o ided ha he
18 TOM ´
AS CARABALLO, JOS´
E A. LANGA AND #TAKESHI TANIGUCHI
lipschi z cons an and he lowe bound on he s ochas ic e m (namely, Cg
and ρ) imply ha γ > 0.Fo ins ance, in he pa icula case o a linea
e m, i.e., when gis gi en o example as
g( , x)k=σ
qλ0
1
x(k, e1)K, > 0, x ∈H, k ∈K,
he cons an s appea ing in he p e ious heo em a e:
Cg=σ, ρ = 2σ,γ=1
2(λ1ν−c−C2
g
2+ρ2
2) = 1
2(λ1ν−c−σ2
2+ 2σ2).
Consequen ly, al hough λ1ν−c < 0,one can always choose σla ge enough
so ha γ > 0.
Rema k 4.2. Fo he ini e dimensional case, he e exis s a wide li e a u e
on s abiliza ion by noise (see A nold [1] and he e e ences he ein), bu
o he in ini e dimensional case, as a as we know, his ques ion emains
open, mainly due o he ac ha he echnique used in he ini e dimensional
case seems e y di icul o ex end o his si ua ion. Howe e , we ha e o
poin ou ha , in gene al, when one conside s a de e minis ic sys em and a
pe u bed e sion o i by adding a s ochas ic I o e m, o ins ance a linea
mul iplica i e one o he o m σudW( )(being u he solu ion), in a limi
sense, he s ochas ic equa ion co esponds o a de e minis ic equa ion wi h a
mean-ze o luc ua ion eedback con ol plus a s abilizing sys ema ic con ol,
in ac , one can say ha an I o mul iplica i e noise wi h in esi y σ, ac s like
a eedback s abilizing con ol o he o m −σ2
2u, so maybe no he noise is
esponsible o he s abilizing e ec bu his addi ional damping one. Howe e ,
he mos in e es ing esul s in he li e a u e conce ning s abiliza ion deals
wi h he one p oduced by conside ing he s ochas ic e m in he S a ono ich
sense. In his case, as his e m is like a pe iodic ze o-mean eedback con ol,
i s s abilizing e ec is unexpec ed and e y in e es ing since he e is no new
damping e ms in he equa ions and when he s abiliza ion is p oduced, one
can p ope ly say ha he noise has s abilized he sys em.
Rema k 4.3. No icing ha , in o de o p oduce a s abiliza ion e ec , i
is su icien o conside a one dimensional Wiene p ocess, in he es o
his sec ion we assume ha K=R, Q = 1 and W( )is a one dimensional
Wiene p ocess.
Las ly, conside he case whe e (0) 6= 0.I ν > β,ν > c1k (0)kV0
√λ1(ν−β)+β
and all he condi ions o Lemma 3.1 a e sa is ied, we ha e he exis ence o
a unique s a iona y solu ion u∞∈V o (3.1). He e we no e ha his u∞
STOCHASTIC NSES 19
is also he s a iona y solu ion o (2.3). Then, we shall show ha i can
be chosen g( , x) = σ(x−u∞) so ha he s a iona y solu ion u∞ o he
de e minis ic equa ion,becomes an almos su e exponen ially s able solu-
ion o he s ochas ic 2D-Na ie -S okes equa ion (2.2), when he kinema ic
iscosi y νis la ge enough and, o simplici y, when Wis a one dimen-
sional Wiene p ocess. By he ollowing lemma we ge ha i he Lipschi z
cons an c > 0 o he ex e nal o ce ield is su icien ly small, ha is, i
λ1ν > c1√λ1ku∞k+c, hen he s a iona y solu ion u∞ o (2.3) is expo-
nen ially s able. This lemma can be p o ed by he simila me hod as in he
p oo o Theo em 10.2 (Temam[18, p.69]).
Lemma 4.2. Le u∞∈Vbe he unique s a iona y solu ion o (3.1). I
he unc ion sa is ies condi ion D and λ1ν > c1√λ1ku∞k+c, hen he
s a iona y solu ion u∞ o (2.3) is exponen ially s able.
Bu i he Lipschi z cons an c > 0 is su icien ly la ge, ha is, i λ1ν≤
c1√λ1ku∞k+c, hen we do no know i u∞is exponen ially s able o no .
Howe e , we can p o e he ollowing esul :
Theo em 4.3. Le u∞∈Vbe he unique s a iona y solu ion o (3.1). Le
c0:= λ1ν−c1√λ1ku∞k>0and le λ1ν≤c1√λ1ku∞k+c. Assume
ha σis a eal numbe such ha 2λ1ν−2c1√λ1ku∞k+σ2>2c . I he
unc ion sa is ies condi ion D, hen he e exis s Ω0⊂Ω, P(Ω0) = 0,such
ha o ω /∈Ω0 he e exis s T(ω)>0such ha
|X( )−u∞|2≤ |X(0) −u∞|2e−γ o all ≥T(ω),
whe e γ:= 1
2(σ2−2c+ 2c0)>0,and X( )is any weak solu ion o (2.2)
whe e he unc ion gis gi en by g( , x) = σ(x−u∞).
P oo . Applying I o’s o mula o he unc ion |X( )−u∞|2,we ha e ha
|X( )−u∞|2=|X(0) −u∞|2
−2R
0hνAX(s), X(s)−u∞ids
−2R
0hB(X(s)), X(s)−u∞ids
+2 R
0( (X(s)), X(s)−u∞)ds
+R
0kg(s, X(s))k2
L0
2ds
+2 R
0(X(s)−u∞, g(s, X(s))dW (s)) .
20 TOM ´
AS CARABALLO, JOS´
E A. LANGA AND #TAKESHI TANIGUCHI
And so
|X( )−u∞|2=|X(0) −u∞|2
−2R
0νkX(s)−u∞k2ds
−2R
0b(X(s)−u∞, u∞, X(s)−u∞)ds
+2 R
0(X(s)−u∞, (X(s)) − (u∞))ds
+R
0kg(s, X(s)) k2
L0
2
ds
+2 R
0(X(s)−u∞, g(s, X(s))dW (s)) .
Hence, since c0:= λ1ν−c1√λ1ku∞k>0,using he inequali y |b(X(s)−
u∞, u∞, X(s)−u∞)|≤ c1
√λ1ku∞kk X(s)−u∞k2,we ob ain ha
−2νkX(s)−u∞k2+ 2 |b(X(s)−u∞, u∞, X(s)−u∞)|
≤(−2ν+2c1
√λ1ku∞k)kX(s)−u∞k2
≤(−2λ1ν+ 2c1√λ1ku∞k)|X(s)−u∞|2.
The e o e,
log |X( )−u∞|2= log |X(0) −u∞|2
+R
0
1
|X(s)−u∞|2(−2νkX(s)−u∞k2
+σ2|X(s)−u∞|2
−2b(X(s)−u∞, u∞, X(s)−u∞)
+2( (X(s)) − (u∞), X(s)−u∞)ds
+2 R
0
σ|X(s)−u∞|2
|X(s)−u∞|2dW(s)
−1
2R
0
4σ2|X(s)−u∞|4
|X(s)−u∞|4ds
≤log |X(0) −u∞|2+(2c−2c0−σ2) + 2σW( ).
As lim →∞ W( )
= 0,almos su ely, we can ind a se Ω0⊂Ω wi h P(Ω0) = 0,
such ha , o each ω /∈Ω0, he e exis s T(ω) such ha o all ≥T(ω)
2σW( )
≤1
2(−2c+ 2c0+σ2).
Thus, we ob ain ha o any ≥T(ω)
log |X( )−u∞|2≤log |X(0) −u∞|2+1
2(2c−2c0−σ2) .
STOCHASTIC NSES 21
This comple es he p oo o he heo em.
Acknowledgemen s. The au ho s wan o exp ess hei since e g a i ude
o he e e ee since, hanks o his in e es ing, de ailed, help ul and cla i ying
commen s and sugges ions, he pape has been g ea ly imp o ed.
Also, hey wish o hank Hans C auel o in e es ing commen s on he
s abiliza ion by S a ono ich noise.
The wo i s au ho s ha e been pa ially suppo ed by P oyec o DGICYT
(Spain) PB98-1134.
T. Taniguchi would like o exp ess his g a i ude o Depa men o Di e -
en ial Equa ions and Nume ical Analysis, Uni e si y o Se illa, o hospi al-
i y when he was isi ing.
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E A. LANGA AND #TAKESHI TANIGUCHI
[15] E. Pa doux, ´
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