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 Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apartado de Correos 1160, 41080-SEVILLA, Spain #Department of Mathematics, Kurume University, Kurume , Fukuoka 830, Japan Abstract. Some results on the pathwise exponential stability of the weak solutions to a stochastic 2D-Navier-Stokes equation are established. The first ones are proved as a consequence of the exponential mean square stability of the solutions. However, some of them are improved by avoiding the previous mean square stability in some more particular and restrictive situations. Also, some results and comments concerning the stabilizability and stabilization of these equations are stated. 1. Introduction The long-time behaviour of flows is a very interesting and important problem in the theory of fluid dynamics, as the vast literature shows (see Temam [19], Hale [13], Ladyzhenskaya [14], among others, and the references therein), and has been receiving very much attention over the last three decades. One of the most studied models is the Navier-Stokes one (and its variants) since it provides a suitable model which covers several important fluids (see Temam [17]-[19] and the references inside these). On the other hand, another interesting question is to analyze the effects produced on a deterministic system by some stochastic or random disturbances appeared in the problem. These facts have motivated the present work whose main objective is to show some aspects of the effects produced 1991 Mathematics Subject Classification. Primary 60H. Key words and phrases. Stochastic Navier-Stokes Equations, Exponential Stability, Stabilization. #To whom correspondence should be addressed. 1 2TOM ´ AS CARABALLO, JOS´ E A. LANGA AND #TAKESHI TANIGUCHI in the long-time behaviour of the solution to a two dimensional Navier-Stokes equation under the presence of stochastic perturbations. In the deterministic case, it is well known for a long time that, for small enough Reynolds number (or, equivalently, large viscosity), the solutions of 2D-Navier-Stokes equations tend to a stationary one (unique, in fact) when time goes to infinite and, as this number increases, the dynamics of the system turns more and more complex (see, e.g. Temam [18] for a detailed description of the Couette-Taylor experiment). The problem of detecting the critical value where the instability appears is a difficult challenging one. Thus, in a general framework, one can only ensure that for small values of the Reynolds number the stationary solution is stable but we do not know when it becomes unstable. This motivates that people working in this kind of problems use to consider particular examples in order to obtain sharper results. Our first aim in this work is to provide some light in some aspects concerning the stability of the stationary solutions of the following stochastic 2D-Navier-Stokes:          dX = [ν∆X−hX, ∇iX+f(X) + ∇p]dt +g(t, X)dW(t) divX = 0 in [0,∞)×D, X= 0 on [0,∞)×Γ, X(0, x) = X0(x), x ∈D, where Dis a regular open bounded domain of R2with boundary Γ, u is the velocity field of the fluid, pthe pressure, ν > 0 the kinematic viscosity, X0the initial velocity field, fthe external force field and g(t, x)dW(t) the random field where W(t) is an infinite dimensional Wiener process. Concerning the effects produced by random perturbations in deterministic systems, it is worth mentioning that this is a very difficult task which is being investigated actually by many authors within the framework of the theory of random attractors recently introduced by Crauel and Flandoli [10]. On the one hand, existence of random attractors is only known for specific random terms (see, for instance, Crauel and Flandoli [10], Capinski and Cutland [6]). On the other hand, almost nothing is known on the structure of these random sets, so that many challenging open problems, as those related to stability and instability, are still open. Also, it is very interesting to investigate if a fluid subjected to random influences is asymptotically more or less stable than the deterministic unperturbed one. In the finite dimensional case, there exits a wide literature STOCHASTIC NSES 3 on this topic (see Arnold [1] and the references therein) which proves that some kind of multiplicative noise may produce a stabilization effect on deterministic unstable systems. However, for the infinite dimensional case, a similar result has not been proved yet, mainly due to the fact that the technique developed in the finite dimensional framework cannot be extended to this case or, at least, it is not known how to do that. The main result proved in [1] ensures that an unstable linear differential system in Rn,namely · x(t) = Ax(t) with trace A < 0, can be stabilized by adding a multiplicative noise in the Stratonovich sense containing a suitable skew-symmetric matrix. One interesting remark is that when the stochastic multiplicative perturbation is considered in the Ito sense, this uses to imply a general stabilization effect on the system. In a limit sense, the Ito equations with multiplicative noise correspond to deterministic equations with a mean-zero fluctuating control plus a stabilizing systematic control (see Section 4 for more details and comments). This would mean that only the stabilization produced by Stratonovich terms could be considered as proper stabilization produced by random noise, since the Stratonovich multiplicative noise acts like a periodic zero-mean feedback control, and consequently, its stabilizing effect is unexpected and therefore very interesting. In this paper, we consider the stochastic disturbances in Ito sense, so the stabilization results proved should be interpreted in a suitable sense (see also Caraballo and Langa [7] for an analysis on the different long-time behaviour of Ito and Stratonovich equations in the linear case). The content of this paper is as follows. In Section 2, we include some preliminaries. In Section 3, we shall prove some results on pathwise exponential stability by extending to this case the stability theory previously developed for semilinear stochastic partial differential equations (see Caraballo and Liu [8], Taniguchi [16]). Finally, in Section 4, we deal with the interesting stabilizability problem, that is, we shall analyze the possible reasons implying a stabilizing effect on the deterministic problem by the appearance of a random disturbance. 2. Preliminaries Firstly, we introduce the following Hilbert spaces: 4TOM ´ AS CARABALLO, JOS´ E A. LANGA AND #TAKESHI TANIGUCHI H= the closure of the set u∈C∞ 0(D, R2) : div u = 0in L2(D, R2) with the norm |u|= (u, u)1 2,where for u, v ∈L2(D, R2), (u, v) = 2 X j=1 ZD uj(x)vj(x)dx, V= the closure of the set u∈C∞ 0(D, R2) : div u = 0in H1 0(D, R2) with the norm kuk= ((u, v))1 2,where for u, v ∈H1 0(D, R2), ((u, v)) = 2 X j=1 ∂u ∂xj ,∂v ∂xj. Then, it follows that Hand Vare separable Hilbert spaces with associated inner products (·,·) and ((·,·)) and the following is safisfied: V⊂H≡H0⊂V0, where injections are dense, continuous and compact. Now, we can set A= −P4where Pis the orthogonal projector from L2(D, R2) onto H, and define the trilinear form bby b(u, v, w) = 2 X i,j=1 ZD ui(x)∂vj ∂xi (x)wj(x)dx. As we shall need some properties on this trilinear form b, we list here the ones we will use later on (see Temam [19]): (2.1) |b(u, v, w)| ≤ c1|u|1 2kuk1 2kvk | w|1 2kwk1 2,∀u, v, w ∈V, b(u, v, v) = 0,∀u, v ∈V, b(u, u, v −u)−b(v, v, v −u) = −b(v−u, u, v −u),∀u, v ∈V, where c1>0 is an appropriate constant which depends on the regular open domain D(see Constantin and Foias [9, (6.9), p.50]) . Furthermore, we can define the operator B:V×V→V0by hB(u, v), wi=b(u, v, w),∀u, v, w ∈V, where h·,·i denotes the duality hV0, V i.We also set B(u) = B(u, u),∀u∈V. Let (Ω, P, =) be a probability space on which an increasing and right continuous family {=t}t∈[0,∞)of complete sub-σ-algebra of =is defined. Let STOCHASTIC NSES 5 βn(t) (n= 1,2,3,···) be a sequence of real valued one-dimensional standard Brownian motions mutually independent on (Ω, P, =).Set W(t) = ∞ X n=1 qλ0 nβn(t)en, t ≥0 where λ0 n≥0 (n= 1,2,3···) are nonnegative real numbers such that P∞ n=1 λ0 n<+∞,and {en}(n= 1,2,3,···) is a complete orthonormal basis in the real and separable Hilbert space K. Let Q∈L(K, K) be the operator defined by Qen=λ0 nen.The above K-valued stochastic process W(t) is called a Q-Wiener process. Thus the stochastic 2D-Navier-Stokes equation can be rewritten as follows in the abstract mathematical setting: (2.2) dX(t) = [−νAX(t)−B(X(t))+f(X(t))] dt +g(t, X(t))dW(t), where f:V→V0, g : [0,∞)×V→L(K, H) are continuous functions satisfying some additional assumptions (see conditions below).Also we consider the deterministic version of this equation, namely, (2.3) dX(t) = [−νAX(t)−B(X(t))+f(X(t))] dt. First, we give the definition of the weak solutions to stochastic 2D-NavierStokes equation (2.2) Definition 2.1. A stochastic process X(t), t ≥0,is said to be a weak solution of (2.2) if (1a) X(t)is =t−adapted, (1b) X(t)∈L∞(0, T;H)∩L2(0, T;V)almost surely for all T > 0, (1c) the following equation holds as an identity in V0almost surely, for t∈[0,∞) X(t) = X(0) + Zt 0 [−νAX(s)−B(X(s))+f(X(s))] ds +Zt 0 g(s, X(s))dW(s). As we are mainly interested in the analysis of the exponential stability of the weak solutions to the problem (2.2), we will assume the existence of such weak solutions (see, for instance, Bensoussan [2] or Capinski and Gatarek [4] for results on the existence and uniqueness of solutions). Now we are going to establish an Ito’s formula which is going to be necessary for our purposes (see Pardoux [15]) 6TOM ´ AS CARABALLO, JOS´ E A. LANGA AND #TAKESHI TANIGUCHI Let C(1,2)([0,∞)×H, R+) denote the space of all R+−valued functions Ψ defined on [0,∞)×Hwith the following properties: (1) Ψ(t, x) is differentiable in t∈[0,∞) and twice Frechet differentiable in xwith Ψt(t, ·),Ψx(t, ·) and Ψxx(t, ·) locally bounded on H (2) Ψ(t, ·),Ψt(t, ·) and Ψx(t, ·) are continuous on H, (3) for all trace class operators R,tr(Ψxx(t, ·)R) is continuous from H into R. (4). if v∈Vthen Ψx(t, v)∈V, and u→ hΨx(t, u), v∗iis continuous for each v∗∈V0, (5). kΨx(t, v)k ≤ C0(t)(1 + kvk), C0(t)>0,for all v∈V. Theorem 2.1. (Ito’s formula) Let Ψ∈C(1,2)([0,∞)×H, R+).If stochastic process X(t)is a weak solution to (2.2), then, it holds that Ψ(t, X(t)) = Ψ(0, X(0)) + Rt 0LΨ(s, X(s))ds +Rt 0(Ψx(s, X(s)), g(s, X(s))dW(s)) , where LΨ(s, X(s)) = Ψt(s, X(s)) +h−νAX(s)−B(X(s))+f(X(s)),Ψx(s, X(s))i +1 2tr (Ψxx(s, X(s))g(s, X(s))Qg(s, X(s))∗). Definition 2.2. We say that a weak solution X(t)to (2.2) converges to x∞∈Hexponentially in mean square if there exist a > 0and M0= M0(X(0)) >0(which may depend on X(0)) such that E|X(t)−x∞|2≤M0e−at, t ≥0, In particular, if x∞is a solution to (2.2), then it is said that x∞is exponentially stable in mean square provided that every weak solution to (2.2) converges to x∞exponentially in mean square with the same exponential order a > 0. Definition 2.3. We say that a weak solution X(t)to (2.2) converges to x∞∈Halmost surely exponentially if there exists γ > 0such that lim t→∞ 1 tlog |X(t)−x∞| ≤ −γ, almost surely. In particular, if x∞is a solution to (2.2), then it is said that x∞is almost surely exponentially stable provided that every weak solution to (2.2) converges to x∞almost surely exponentially with the same constant γ. STOCHASTIC NSES 7 3. The exponential stability of solutions In this section we discuss the moment exponential stability and almost sure exponential stability of weak solutions to stochastic NSE (2.2). Let λ1 >0 be the first eigenvalue of A. We remark that kvk2≥λ1|v|2,∀v∈V. We also denote by kg(t, u)k2 L0 2=tr(g(t, u)Qg(t, u)∗). Throughout this section we will use the following condition: Condition A. There exists β > 0such that kf(u)−f(v)kV0≤βku−vk, β > 0, u, v ∈V. In this paper, we first consider the existence of the stationary solution to the next equation (3.1) νAu +B(u) = f(u) (equality in V0). Then we have the following lemma. The proof is similar to the one of Theorem 10.1 in Temam [18]. But, since the proof depends on the conditions of the function f, we give the proof for the convenience of the reader. Lemma 3.1. Suppose that condition A is satisfied and the function fsatisfies that f(vm)converges to f(v)weakly in V0whenever {vm} ⊂ Vconverges to v∈Vweakly in Vand strongly in H. Then, (a)if ν > β, there exists a stationary solution u∞∈Vto (3.1); (b)furthermore, if ν > c1kf(0)kV0 √λ1(ν−β)+β, then the stationary solution to (3.1) is unique. Proof. (a) Let v1, v2, v3,···, vm,···be the orthonormal basis of V. Consider the finite dimensional Hilbert space Vmspanned by {v1,· · ·, vm}with the scalar product [·,·] and norm [·] induced by the corresponding ones in V. Now we define a mapping Rm:Vm→Vmas follows (3.2) [Rmu, v] = ((Rmu, v)) := ν((u, v))+b(u, u, v)−hf(u), vi,∀u, v ∈Vm. If we prove that this mapping is continuous in Vmwith respect to the norm [·], and that [Rmu, u]>0 for some u∈Vmwith [u] = k > 0,then Lemma 1.4 in Temam [17, p. 164] guarantees that there exists um∈Vmsuch that [um]≤kand Rmum= 0. 8TOM ´ AS CARABALLO, JOS´ E A. LANGA AND #TAKESHI TANIGUCHI The continuity of Rmfollows easily from the properties of band the assumptions on f. Now, from (3.2) it holds for u∈Vm [Rmu, u] = ν((u, u))+b(u, u, u)−hf(u), ui ≥ν((u, u)) −kf(u)kV0kuk ≥νkuk2−(kf(0)kV0+βkuk)kuk Since ν > β, we can choose a positive real number k > 0 such that (ν−β)k2− kf(0)kV0k > 0,and for u∈Vmsuch that kuk=k, we have [Rmu, u]>0. Then, there exists an element um∈Vm⊂Vwhich is a solution of (3.2) with kumk ≤ k. Furthermore, we can easily deduce (see estimation (3.3) below) that kumk ≤ kf(0)kV0 (ν−β), and, consequently, we have that a suitable subsequence of {um}converges weakly in Vto some limit u∞and, thanks to the compact injection, strongly in H. Now, the properties of band assumptions on fenable us to prove that this u∞is a solution of (3.1). (b) As for the uniqueness statement, let us assume that u1and u2are two solutions, then ν((u1, v))+b(u1, u1, v) = hf(u1), vi,∀v∈V, ν((u2, v))+b(u2, u2, v) = hf(u2), vi,∀v∈V. Setting v=u1−u2,by substracting the second relation from the first one, and taking into account the properties of the trilinear form band condition A we obtain that νku1−u2k2=−b(u1, u1, u1−u2) + b(u2, u2, u1−u2) +hf(u1)−f(u2), u1−u2i =−b(u1−u2, u2, u1−u2) + hf(u1)−f(u2), u1−u2i ≤c1 √λ1ku1−u2k2ku2k+βku1−u2k2. Observing that νku2k2=hf(u2), u2i ≤ kf(u2)kV0ku2k(3.3) ≤βku2k2+kf(0)kV0ku2k, if follows that ku2k ≤ kf(0)kV0 ν−β. STOCHASTIC NSES 9 Consequently, νku1−u2k2≤c1kf(0)kV0 √λ1(ν−β)+βku1−u2k2, and as ν > c1kf(0)kV0 √λ1(ν−β)+β, uniqueness follows immediately. This completes the proof of the lemma. Now, using this lemma, we discuss the long-time behaviour of weak solutions X(t) to the stochastic Navier-Stokes equation (2.2) under some conditions including that the kinematic viscosity νis sufficiently large. Hence throughout this paper we assume that there exists a unique stationary solution u∞∈Vto (3.1).In this section, we use the following condition. Condition B. kg(t, u)k2 L0 2≤γ(t)+(ξ+δ(t)) |u−u∞|2, where ξ > 0is a constant and γ(t), δ(t)are nonnegative integrable functions such that there exist real numbers θ > 0, Mγ, Mδ≥1with γ(t)≤Mγe−θt, δ(t)≤Mδe−θt, t ≥0. Theorem 3.2. Let u∞∈Vbe the unique stationary solution to (3.1) and let 2ν > λ−1 1ξ+ 2β+2c1 √λ1ku∞k.Suppose that conditions A and B are satisfied. Then, any weak solution X(t)to (2.2) converges to the stationary solution u∞to (3.1) exponentially in mean square. That is, there exist real numbers a∈(0, θ), M0=M0(X(0)) >0such that E|X(t)−u∞|2≤M0e−at, t ≥0. Proof .Since 2ν > λ−1 1ξ+ 2β+2c1 √λ1ku∞k,we can take a positive real number a∈(0, θ) such that 2ν > λ−1 1(ξ+a) + 2β+2c1 √λ1ku∞k.Then, by applying the Ito formula to the function eat |X(t)−u∞|2,we have that eatE|X(t)−u∞|2=E|X(0) −u∞|2+Rt 0aeasE|X(s)−u∞|2ds −2Rt 0easEhνAX(s), X(s)−u∞ids −2Rt 0easEhB(X(s)), X(s)−u∞ids +2 Rt 0easEhf(X(s)), X(s)−u∞ids +Rt 0easEkg(s, X(s))k2 L0 2ds. Since u∞satisfies the identity (3.1), Rt 0easEhνAu∞, X(s)−u∞ids +Rt 0easEhB(u∞), X(s)−u∞ids =Rt 0easEhf(u∞), X(s)−u∞ids. Therefore, noting the next identity: 16 TOM ´ AS CARABALLO, JOS´ E A. LANGA AND #TAKESHI TANIGUCHI and therefore, the zero solution is exponentially stable in mean square if and only if a+b2 2<0.So, we observe that there exist many possibilities of being the zero solution pathwise exponentially stable and, at the same time, exponentially unstable in mean square. Consequently, it would be very interesting to obtain pathwise exponential stability results by avoiding the method of using mean square stability as a previous step. This will be one of the aims of this section. However, it is worth pointing out that to get some results in this direction, we will need to assume some additional hypotheses on the stochastic perturbation so that we can obtain better stability criteria but for more specific situations. In particular, in some of our situations, the noise is so special that one can perform a time change, a substitution that transform the stochastic equation into a deterministic one. For example, the Ito formula for the logarithm in the proof of Theorem 4.2 in this section is one way to perform this transformation; another is to multiply by the exponential of the noise (see Crauel and Flandoli [10, p. 382]). To this end let us firstly state the following condition Condition D. f:H→H, and satisfies |f(u)−f(v)| ≤ c|u−v|, c > 0, u, v ∈H, g(t, ·) : H→L(K, H),and satisfies kg(t, u)−g(t, v)kL(K,H)≤Cg|u−v|,∀t∈[0,∞),∀u, v ∈H. Observe that if νλ1> c and f(0) = 0,then the zero solution to (2.3) is exponentially stable. But when νλ1≤cand f(0) = 0 we do not know, in general, if the zero solution is exponentially stable or not. The following theorem is going to state that, under some particular conditions, any weak solution of the stochastic Navier-Stokes equation converges to zero almost surely exponentially stable. So, in a sense, we can interpret that a kind of stabilization could have taken place in the system. Theorem 4.1. In addition to condition D, assume that f(0) = 0 and g(t, 0) = 0 for all t≥0,and that there exists ρ > 0such that e Qψ(s, x):=tr [(ψx(x)⊗ψx(x))(g(s, x)Qg(s, x)∗)] ≥ρ2|x|4, where ψ(x) = |x|2(recall that (ψx(x)⊗ψx(x))(h) = ψx(x) (ψx(x), h),for x, h ∈H).Then, there exists Ω0⊂Ω, P(Ω0) = 0,such that for ω /∈Ω0there exists T(ω)>0such that any weak solution X(t)to (2.2) satisfies |X(t)|2≤ |X(0)|2e−γt for any t≥T(ω), STOCHASTIC NSES 17 where γ:= 1 2(λ1ν−c−C2 g 2+ρ2 2).In particular, exponential stability of sample paths with probability one holds if γ > 0. Proof. Let us apply Ito’s formula for our solution X(t).Then, it follows |X(t)|2=|X(0)|2+ 2 Rt 0h−νAX(s)−B(X(s))+f(X(s)), X(s)ids +Rt 0kg(s, X(s))k2 L0 2ds +2 Rt 0(X(s), g(s, X(s))dW (s)) =|X(0)|2+ 2 Rt 0h−νkX(s)k2+hf(X(s)), X(s)iids +Rt 0kg(s, X(s))k2 L0 2ds +2 Rt 0(X(s), g(s, X(s))dW (s)) , and, applying once again Ito’s formula to the function log |X(t)|2,and taking into account the hypotheses, it follows log |X(t)|2= log |X(0)|2+1 2Rt 0 1 |X(s)|2h−2νkX(s)k2+ 2 hX(s), f(X(s))iids +1 2Rt 0 1 |X(s)|2kg(s, X(s))k2 L0 2ds +2 Rt 0 1 |X(s)|2(X(s), g(s, X(s))dW (s)) −1 2Rt 0 e Qψ(s,X(s)) |X(s)|4ds ≤log |X(0)|2+Rt 0 1 |X(s)|2h−νλ1+c+C2 g 2i|X(s)|2ds +2 Rt 0 1 |X(s)|2(X(s), g(s, X(s))dW (s)) −ρ2 2t. Now, due to our assumptions, the term M(t) = Rt 0 2 |X(s)|2(X(s), g(s, X(s))dW (s)) is a real martingale and it is not difficult to prove, by means of the law of iterated logarithm, lim t→+∞ M(t) t= 0, P −almost surely. Thus, we can assure that there exists a set Ω0⊂Ω with P(Ω0) = 0,such that for every ω /∈Ω0there exists T(ω)>0 such that for all t≥T(ω) M(t) t≤1 2(λ1ν−c−C2 g 2+ρ2 2). Therefore, it easily follows that for any t≥T(ω) log |X(t)|2≤log |X(0)|2+1 2(−λ1ν+c+C2 g 2−ρ2 2)t. The proof is now complete. Remark 4.1. Observe that although we do not know whether the stationary solution to the deterministic problem is stable or not, it is possible to ensure sample exponential stability of the stochastic equation provided that the 18 TOM ´ AS CARABALLO, JOS´ E A. LANGA AND #TAKESHI TANIGUCHI lipschitz constant and the lower bound on the stochastic term (namely, Cg and ρ) imply that γ > 0.For instance, in the particular case of a linear term, i.e., when gis given for example as g(t, x)k=σ qλ0 1 x(k, e1)K, t > 0, x ∈H, k ∈K, the constants appearing in the previous theorem are: Cg=σ, ρ = 2σ,γ=1 2(λ1ν−c−C2 g 2+ρ2 2) = 1 2(λ1ν−c−σ2 2+ 2σ2). Consequently, although λ1ν−c < 0,one can always choose σlarge enough so that γ > 0. Remark 4.2. For the finite dimensional case, there exists a wide literature on stabilization by noise (see Arnold [1] and the references therein), but for the infinite dimensional case, as far as we know, this question remains open, mainly due to the fact that the technique used in the finite dimensional case seems very difficult to extend to this situation. However, we have to point out that, in general, when one considers a deterministic system and a perturbed version of it by adding a stochastic Ito term, for instance a linear multiplicative one of the form σudW(t)(being uthe solution), in a limit sense, the stochastic equation corresponds to a deterministic equation with a mean-zero fluctuation feedback control plus a stabilizing systematic control, in fact, one can say that an Ito multiplicative noise with intesity σ, acts like a feedback stabilizing control of the form −σ2 2u, so maybe not the noise is responsible of the stabilizing effect but this additional damping one. However, the most interesting results in the literature concerning stabilization deals with the one produced by considering the stochastic term in the Stratonovich sense. In this case, as this term is like a periodic zero-mean feedback control, its stabilizing effect is unexpected and very interesting since there is no new damping terms in the equations and when the stabilization is produced, one can properly say that the noise has stabilized the system. Remark 4.3. Noticing that, in order to produce a stabilization effect, it is sufficient to consider a one dimensional Wiener process, in the rest of this section we assume that K=R, Q = 1 and W(t)is a one dimensional Wiener process. Lastly, consider the case where f(0) 6= 0.If ν > β,ν > c1kf(0)kV0 √λ1(ν−β)+β and all the conditions of Lemma 3.1 are satisfied, we have the existence of a unique stationary solution u∞∈Vto (3.1). Here we note that this u∞ STOCHASTIC NSES 19 is also the stationary solution to (2.3). Then, we shall show that it can be chosen g(t, x) = σ(x−u∞) so that the stationary solution u∞to the deterministic equation,becomes an almost sure exponentially stable solution to the stochastic 2D-Navier-Stokes equation (2.2), when the kinematic viscosity νis large enough and, for simplicity, when Wis a one dimensional Wiener process. By the following lemma we get that if the Lipschitz constant c > 0 of the external force field fis sufficiently small, that is, if λ1ν > c1√λ1ku∞k+c, then the stationary solution u∞to (2.3) is exponentially stable. This lemma can be proved by the similar method as in the proof of Theorem 10.2 (Temam[18, p.69]). Lemma 4.2. Let u∞∈Vbe the unique stationary solution to (3.1). If the function fsatisfies condition D and λ1ν > c1√λ1ku∞k+c, then the stationary solution u∞to (2.3) is exponentially stable. But if the Lipschitz constant c > 0 is sufficiently large, that is, if λ1ν≤ c1√λ1ku∞k+c, then we do not know if u∞is exponentially stable or not. However, we can prove the following result: Theorem 4.3. Let u∞∈Vbe the unique stationary solution to (3.1). Let c0:= λ1ν−c1√λ1ku∞k>0and let λ1ν≤c1√λ1ku∞k+c. Assume that σis a real number such that 2λ1ν−2c1√λ1ku∞k+σ2>2c . If the function fsatisfies condition D,then there exists Ω0⊂Ω, P(Ω0) = 0,such that for ω /∈Ω0there exists T(ω)>0such that |X(t)−u∞|2≤ |X(0) −u∞|2e−γt for all t≥T(ω), where γ:= 1 2(σ2−2c+ 2c0)>0,and X(t)is any weak solution to (2.2) where the function gis given by g(t, x) = σ(x−u∞). Proof. Applying Ito’s formula to the function |X(t)−u∞|2,we have that |X(t)−u∞|2=|X(0) −u∞|2 −2Rt 0hνAX(s), X(s)−u∞ids −2Rt 0hB(X(s)), X(s)−u∞ids +2 Rt 0(f(X(s)), X(s)−u∞)ds +Rt 0kg(s, X(s))k2 L0 2ds +2 Rt 0(X(s)−u∞, g(s, X(s))dW (s)) . 20 TOM ´ AS CARABALLO, JOS´ E A. LANGA AND #TAKESHI TANIGUCHI And so |X(t)−u∞|2=|X(0) −u∞|2 −2Rt 0νkX(s)−u∞k2ds −2Rt 0b(X(s)−u∞, u∞, X(s)−u∞)ds +2 Rt 0(X(s)−u∞, f(X(s)) −f(u∞))ds +Rt 0kg(s, X(s)) k2 L0 2 ds +2 Rt 0(X(s)−u∞, g(s, X(s))dW (s)) . Hence, since c0:= λ1ν−c1√λ1ku∞k>0,using the inequality |b(X(s)− u∞, u∞, X(s)−u∞)|≤ c1 √λ1ku∞kk X(s)−u∞k2,we obtain that −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. Therefore, log |X(t)−u∞|2= log |X(0) −u∞|2 +Rt 0 1 |X(s)−u∞|2(−2νkX(s)−u∞k2 +σ2|X(s)−u∞|2 −2b(X(s)−u∞, u∞, X(s)−u∞) +2(f(X(s)) −f(u∞), X(s)−u∞)ds +2 Rt 0 σ|X(s)−u∞|2 |X(s)−u∞|2dW(s) −1 2Rt 0 4σ2|X(s)−u∞|4 |X(s)−u∞|4ds ≤log |X(0) −u∞|2+(2c−2c0−σ2)t+ 2σW(t). As limt→∞ W(t) t= 0,almost surely, we can find a set Ω0⊂Ω with P(Ω0) = 0, such that, for each ω /∈Ω0, there exists T(ω) such that for all t≥T(ω) 2σW(t) t≤1 2(−2c+ 2c0+σ2). Thus, we obtain that for any t≥T(ω) log |X(t)−u∞|2≤log |X(0) −u∞|2+1 2(2c−2c0−σ2)t. STOCHASTIC NSES 21 This completes the proof of the theorem. Acknowledgements. 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