The Exponential Behaviour and Stabilizability of Stochastic 2D-Navier-Stokes Equations

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. Re e ences [1] L. A nold, S abiliza ion by noise e isi ed, Z. angew. Ma h. Mech. 70 (1990), 235-246. [2] A. Bensoussan, S ochas ic Na ie -S okes equa ions, Ac a Applicandae Ma h., 38 (1995), 267-304. [3] Z. B zezniak, M. Capinski and F. Flandoli, Pa hwise global a ac o s o s a iona y andom dynamical sys ems, P ob. Th. Rel. Fields, 95 (1993), 87-102. [4] M. Capinski and D. Ga a ek, S ochas ic equa ions in Hilbe spaces wi h applica ion o Na ie -S okes equa ions in any dimension, J. o Func . Anal. 126(1994), 26-35. [5] M. Capinski and N. Cu land, Measu e a ac o s o s ochas ic Na ie -S ockes Equa- ions, Elec onic J. o P ob., 3(1998), 1-15. 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