On the asymptotic behaviour of a stochastic 3D Lans-Alpha model

On the asymptotic behaviour of a stochastic 3D LANS-αmodel ∗ T. Caraballo, A. M. M´arquez–Dur´an, & J. Real Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. Correos 1160, 41080-Sevilla (Spain) E-mails: [email protected], [email protected], [email protected] March 17, 2005 Abstract The long-time behaviour of a stochastic 3D LANS-αmodel on a bounded domain is analysed. First, we reformulate the model as an abstract problem. Next, we establish sufficient conditions ensuring the existence of stationary (steady state) solutions of this abstract nonlinear stochastic evolution equation, and study the stability properties of the model. Finally, we analyse the effects produced by stochastic perturbations in the deterministic version of the system (persistence of exponential stability as well as possible stabilisation effects produced by the noise). The general results are applied to our stochastic LANS-α system throughout the paper. 1 Introduction In this paper we are mainly interested in the study of the asymptotic behaviour of solutions of the 3D-Lagrangian averaged Navier-Stokes (LANS-α) equations, with homogeneous Dirichlet boundary condition in a bounded domain, in the case in which random perturbations appear. To be more precise, let Dbe a connected and bounded open subset of R3, with a C2boundary ∂D. We denote by Athe Stokes operator, and consider the system            ∂t(u−α∆u) + ν(Au −α∆(Au)) + (u·∇)(u−α∆u) −α∇u∗·∆u+∇p=F(t, u) + G(t, u)˙ W(t),in D×(0,+∞), ∇·u= 0,in D×(0,+∞), u= 0, Au = 0,on ∂D ×(0,+∞), u(0) = u0,in D, (1) ∗Partly supported by Ministerio de Ciencia y Tecnolog´ıa (Spain) and FEDER (European Community) grant BFM2002-03068, and Junta de Andaluc´ıa project FQM314. 1 where u= (u1, u2, u3) and pare unknown random fields on D×(0,+∞), representing, respectively, the large-scale (or averaged) velocity and the pressure, in each point of D×(0,+∞), of an incompressible viscous fluid with constant density filling the domain D. The constants ν > 0 and α > 0 represent respectively the kinematic viscosity of the fluid, and the square of the spatial scale at which fluid motion is filtered. The terms F(t, u) and G(t, u)˙ W(t) are external forces depending eventually on u, where ˙ W(t) denotes the time derivative of a cylindrical Wiener process. Finally, u0is a given initial velocity field. The deterministic version of (1), i.e. when G= 0, has received much attention over the last years. The main reason is that this model has become very useful in order to approximate the 3D Navier-Stokes equations (notice that when αgoes to zero, this problem converges to the usual 3D Navier-Stokes model). More precisely, the global well-posedness of weak solutions for the deterministic Lagrangian averaged Navier-Stokes equations on bounded domains has been established in [8] and [14] amongst others, and the asymptotic behaviour can be found in [8]. Similar results have been proved by Foias et al. in [10] in the case of periodic boundary conditions. However, in order to consider a more realistic model for our problem, it is sensible to consider some kind of ‘noise’ in the equations. This may reflect, for instance, some environmental effects on the phenomenon, some external random forces, etc. To the best of our knowledge, the existence and uniqueness of solution of the stochastic version (1) we will consider in this paper has been analysed in [7] (see also [6]). We start in this paper the analysis of the asymptotic behaviour of the stochastic version, and point out that our analysis, in particular, also provides information on the long-time dynamics of the deterministic model (by simply setting G= 0 in the appropriate formulas). After reading this paper, one could immediately wonder about the possibility of doing a similar analysis in the case of general unbounded domains (e.g. channels, pipes, etc.). This is beyond the scope of this paper since one would first need some results on the existence of solutions of such a model, and, as far as we know, this still has not been proved. Being possible to carry out our analysis working directly with the 3D LANS-αmodel, we have preferred to establish a theory for an abstract stochastic model and then apply it to our system. In this way, with only a little extra work one may be able to apply these abstract results to other models of interest. Although the techniques we use in the present paper are similar to those used in [5] for the stochastic 2D Navier-Stokes model (in fact, these can be considered, in certain sense, as standard techniques for the investigation of stability properties for stochastic PDEs, see e.g. [2]), it is worth pointing out that neither the stochastic 2D Navier-Stokes equations from [5] falls within the abstract framework in the present paper, nor this abstract setting is a particular case of the model in [5]. Nevertheless, the results we will prove 2 may be applied to other interesting models as, for example, some stochastic reaction-diffusion equations (by simply setting operator e B= 0 in problem (23) below), or the same model but for periodic boundary conditions, etc. The content of the paper is as follows. In Section 2 we show how our problem can be reformulated as an abstract stochastic model. In Section 3 we first establish the existence and eventual uniqueness of stationary steady state solutions when the viscosity is large enough, and prove a result ensuring the exponential stability (in mean square and pathwise) of the stationary solution. Finally, the stabilising effects produced by the noise in the deterministic model is stated in Section 4. 2 Variational and Abstract formulation of the problem In this section we will rewrite our problem as an abstract model. The main reasons are the following. On the one hand, the results are presented, in our opinion, with more clarity and the computations are done in a more simplified way. On the other hand, the abstract formulation may be applied, in addition, to other models as we commented in Section 1. 2.1 The cylindrical Wiener process Let {Ω,F, P}be a complete probability space, and {Ft}t≥0an increasing and right continuous family of sub σ-algebras of F, such that F0contains all the Pnull sets of F. Let {βj(t), t ≥0, j = 1,2, ...}be a sequence of mutually independent standard real Ft-Wiener processes defined on this space, and suppose that Kis a given separable Hilbert space, and {ej;j= 1,2, ...}, an orthonormal basis of K. We denote by {W(t); t≥0}, the cylindrical Wiener process with values in Kdefined formally as W(t) = ∞ X j=1 βj(t)ej It is well known that this series does not converge in K, but rather in any Hilbert space e Ksuch that K⊂e K, and the injection of Kin e Kis Hilbert- Schmidt (see e.g. [9]). Let T > 0 be given. For any separable Banach space X, we will denote by M2 Ft(0, T;X) the space of all processes ϕ∈L2(Ω ×(0, T), dP ×dt;X) that are Ft-progressively measurable. The space M2 Ft(0, T;X) is a Hilbert subspace of L2(Ω ×(0, T), dP ×dt;X). We will write L2(Ω; C([0, T]; X)) to denote the space of all continuous and Ft-progressively measurable X-valued processes {ϕ(t); 0 ≤t≤T} satisfying EÃsup 0≤t≤Tkϕ(t)k2 X!<∞. 3 For another separable Hilbert space e H, with scalar product (·,·)e H, let us denote by L2(K;e H) the separable Hilbert space of Hilbert-Schmidt operators from Kinto e H, and by ((·,·))L2(K; e H)and k·kL2(K; e H)the scalar product and norm in L2(K;e H),where for all Rand Sin L2(K;e H), ((R, S))L2(K; e H)= ∞ X j=1 (Rej, Sej)e H. For any process Ψ ∈M2 Ft(0, T;L2(K;e H)) one can define the stochastic integral of Ψwith respect to the cylindrical Wiener process W(t), denoted Zt 0 Ψ(s)dW(s),0≤t≤T, as the unique continuous e H-valued Ft-martingale such that for all h∈e H, (Zt 0 Ψ(s)dW(s), h)e H= ∞ X j=1 Zt 0 (Ψ(s)ej, h)e Hdβj(s),0≤t≤T, where the integral with respect to βj(s) is the Itˆo integral, and the series converges in L2(Ω; C([0, T])). See e.g. [9] for the properties of the stochastic integral defined in this way. In particular, we note that if Ψ ∈ M2 Ft(0, T;L2(K;e H)) and φ∈L2(Ω; L∞(0, T;e H)) is Ft-progressively measurable, then the series ∞ X j=1 Zt 0 (Ψ(s)ej, φ(s)) e Hdβj(s),0≤t≤T, converges in L1(Ω; C([0, T])),and defines a real valued continuous Ft-mar- tingale. We will use the notation Zt 0 (Ψ(s)dW(s), φ(s)) := ∞ X j=1 Zt 0 (Ψ(s)ej, φ(s)) e Hdβj(s),0≤t≤T. 2.2 Notations and properties of the nonlinear term We first establish some notations and recall some properties regarding the nonlinear term (u·∇)(u−α∆u)−α∇u∗·∆uappearing in (1). We will denote (·,·)and |·|, respectively, the scalar product and associated norm in (L2(D))3, and by (∇u, ∇v)the scalar product in ((L2(D))3)3of the gradients of uand v. We consider the scalar product in (H1 0(D))3defined by ((u, v)) = (u, v) + α(∇u, ∇v), u, v ∈(H1 0(D))3,(2) 4 where its associated norm k·k is, in fact, equivalent to the usual gradient norm. Let us denote by Hthe closure in (L2(D))3of the set V={v∈(D(D))3:∇·v= 0 in D}, and by Vthe closure of Vin (H1 0(D))3. Then, His a Hilbert space equipped with the inner product of (L2(D))3, and Vis a Hilbert subspace of (H1 0(D))3. Denote by Athe Stokes operator, with domain D(A) = (H2(D))3∩V, defined by Aw =−P(∆w), w ∈D(A), where Pis the projection operator from (L2(D))3onto H. Recall that as ∂D is C2,|Aw|defines in D(A) a norm which is equivalent to the (H2(D))3- norm, i.e., there exists a constant c1(D)>0,depending only on D, such that kwk(H2(D))3≤c1(D)|Aw|,∀w∈D(A),(3) and so D(A) is a Hilbert space with respect to the scalar product (v, w)D(A)= (Av, Aw). For u∈D(A) and v∈(L2(D))3, we define (u· ∇)vas the element of (H−1(D))3given by h(u·∇)v, wi= 3 X i,j=1h∂ivj, uiwji,for all w∈(H1 0(D))3.(4) Observe that (4) is meaningful, since H2(D)⊂L∞(D), and H1 0(D)⊂ L6(D), with continuous injections. This implies that uiwj∈H1 0(D),and there exists a constant c2(D)>0,depending only on D, such that |h(u·∇)v, wi| ≤ c2(D)|Au||v|kwk,∀(u, v, w)∈D(A)×(L2(D))3×(H1 0(D))3. (5) Observe also that if v∈(H1(D))3, then the definition above coincides with the definition of (u· ∇)vas the vector function whose components are P3 i=1 ui∂ivj, for j= 1, 2, 3. However, as it is not known whether the solutions of the stochastic problem (1) have the same regularity as in the deterministic case (we only can ensure H2instead of H3), it is necessary the present extension. Now, if u∈D(A),then ∇u∗∈(H1(D))3×3⊂(L6(D))3×3, and consequently, for v∈(L2(D))3, we have that ∇u∗·v∈(L3/2(D))3⊂(H−1(D))3, with h∇u∗·v, wi= 3 X i,j=1 ZD (∂jui)viwjdx, for all w∈(H1 0(D))3.(6) 5 It follows that there exists a constant c3(D)>0,depending only on D, such that |h∇u∗·v, wi| ≤ c3(D)|Au||v|kwk,∀(u, v, w)∈D(A)×(L2(D))3×(H1 0(D))3. (7) We have the following results (see [7] for the proofs). Proposition 2.1 For all (u, w)∈D(A)×D(A)and all v∈(L2(D))3, it follows h(u·∇)v, wi=−h∇w∗·v, ui.(8) Consider now the trilinear form defined by b#(u, v, w) = h(u·∇)v, wi+h∇u∗·v, wi,(u, v, w)∈D(A)×(L2(D))3×(H1 0(D))3. Proposition 2.2 The trilinear form b#satisfies b#(u, v, w) = −b#(w, v, u),∀(u, v, w)∈D(A)×(L2(D))3×D(A),(9) and consequently, b#(u, v, u) = 0,∀(u, v)∈D(A)×(L2(D))3.(10) Moreover, there exists a constant c(D)>0, depending only on D, such that |b#(u, v, w)| ≤ c(D)|Au||v|kwk,∀(u, v, w)∈D(A)×(L2(D))3×(H1 0(D))3, (11) |b#(u, v, w)| ≤ c(D)kuk|v||Aw|,∀(u, v, w)∈D(A)×(L2(D))3×D(A).(12) Thus, in particular, b#is continuous on D(A)×(L2(D))3×(H1 0(D))3. 2.3 Existence and uniqueness of a variational solution Assume that Fand Gare measurable, Lipschitz mappings from Ω×(0,+∞)× Vinto (H−1(D))3and from Ω ×(0,+∞)×Vinto L2(K; (L2(D))3),respectively. More precisely, suppose that, for all u, v ∈V, F(·, u) and G(·, v) are Ft−progressively measurable, and kF(t, u)−F(t, v)k(H−1(D))3≤LFku−vk, dP ×dt −a.e.,(13) kG(t, u)−G(t, v)kL2(K;(L2(D))3)≤LGku−vk, dP ×dt −a.e..(14) We also suppose F(t, 0) ∈L4(Ω; L2(0, T; (H−1(D))3),for all T > 0,(15) G(t, 0) ∈L4(Ω; L2(0, T;L2(K; (L2(D))3)),for all T > 0,(16) u0∈L4(Ω,F0, P;V).(17) 6 Definition 2.3 A variational solution to problem (1) is a stochastic process u∈M2 Ft(0, T;D(A)) ∩L2(Ω; L∞(0, T;V)),for all T > 0,weakly continuous with values in V, such that for all w∈D(A), ((u(t), w)) + νZt 0 (u(s) + αAu(s), Aw)ds +Zt 0 b#(u(s), u(s)−α∆u(s), w)ds = ((u0, w)) + Zt 0hF(s, u(s)), wids + (Zt 0 G(s, u(s)) dW(s), w), t ≥0.(18) Observe that (18) follows from (1) by multiplying the first equation in (1) by w∈D(A), taking into account the definition of the scalar product ((·,·)), the definition of b#, and the equality (8). Now, as a consequence of Theorem 3.3 in [7] we have the following result. Theorem 2.4 Under the hypotheses (13)-(17), there exists a unique variational solution uof (1), and moreover, u∈L4(Ω; C([0, T]; V)) ∩L4(Ω; L2(0, T;D(A))),for all T > 0. 2.4 Formulation of problem (1) as an abstract problem As we commented at the beginning of Section 2, we are going to rewrite our model (1) as an abstract problem. Let us set H=V, with scalar product (u, v)H= ((u, v)),and associated norm |u|H=kuk,and U=D(A), with scalar product ((u, v))U= (Au, Av), and associated norm kukU=|Au|.Then, Hand Uare two separable real Hilbert spaces, such that U ⊂ H with compact injection, and Uis dense in H. We identify Hwith its topological dual H∗,but we consider Uas a subspace of H∗,identifying v∈ U with the element fv∈ H∗,defined by fv(h) = (v, h)H,∈ H. We denote by h·,·i the duality product between U∗and U. Let us define he Au, vi=ν(Au, v) + να(Au, Av), u, v ∈D(A). It is clear that for all v∈D(A), 2he Av, vi= 2ν(Av, v)+2να(Av, Av)≥2να|Av|2, and, if we denote by λkand wk,k≥1,the eigenvalues and their corresponding eigenvectors associated to A, he Awk, vi=νλk((wk, v)). Thus, taking eα= 2να, (19) we have that 7 a) e Ais a linear continuous operator e A∈ L(U,U∗),such that a1) e Ais self adjoint, a2) there exists eα > 0, such that 2he Av, vi ≥ eαkvk2 U,for all v∈ U,(20) On the other hand, denote he B(u, v), wi=b#(u, v −α∆v, w),(u, v, w)∈D(A)×D(A)×D(A), (( e F(t, u), w)) = hF(t, u), wi,(u, w)∈V×V. Then, it is straightforward to check that if we take c1= (1 + α)c1(D)c(D), L e F=LF,(21) we obtain that b) e B:U ×U → U∗is a bilinear mapping such that b1) he B(u, v), ui= 0,for all u, v ∈ U, b2) ke B(u, v)kU∗≤c1|u|HkvkU,for all (u, v)∈ U ×U, b3) |he B(u, v), wi| ≤ c1kukUkvkU|w|H,for all u, v, w ∈ U, c) e F: Ω ×(0,+∞)×H → H is a random mapping such that c1) for all v∈ H,e F(·, v) is Ft−progressively measurable, c2) e F(·,0) ∈M2 Ft(0, T;H),for all T≥0, c3) |e F(t, u)−e F(t, v)|H≤Le F|u−v|H, dP ×dt a.e. for all u, v ∈ H. Now, let Idenote the identity operator in H, and define e G(t, u) as e G(t, u) = (I+αA)−1◦P ◦G(t, u), u ∈V. First, observe that I+αA is bijective from D(A) onto H, and (((I+αA)−1f, w)) = (f, w),for all f∈H, w ∈V. Thus, for each f∈H, k(I+αA)−1fk2= (f, u)≤ |f||u|, where u= (I+αA)−1f, i.e., (u, wk) + α(Au, wk) = (f, wk),for all k≥1 so, (1 + αλk)(u, wk) = (f, wk),which implies (u, wk) = 1 (1 + αλk)(f, wk)≤1 (1 + αλ1)(f, wk), 8 |u|2= ∞ X k=1 (u, wk)2≤1 (1 + αλ1)2 ∞ X k=1 (f, wk) = 1 (1 + αλ1)2|f|2. Therefore, k(I+αA)−1fk2≤1 1 + αλ1|f|2, and consequently, taking Le G=LG √1 + αλ1 ,(22) we obtain that d) e G: Ω ×(0,+∞)× H → L2(K;H) is a measurable random mapping such that d1) for any v∈ H,e G(·, v) is Ft-progressively measurable, d2) e G(·,0) ∈M2 Ft(0, T;L2(K;H)),for all T≥0, d3) ke G(t, u)−e G(t, v)kL2(K;H)≤Le G|u−v|H, dP×dt a.e. for all u, v ∈ H. Next, for each j≥1,and all (t, u, w)∈(0,+∞)×V×D(A),we have (G(t, u)ej, w) = ((I+αA)( e G(t, u)ej), w) = (( e G(t, u)ej, w)), and, for all u∈M2 Ft(0, T;V), (t, w)∈(0, T)×D(A),it follows (Zt 0 G(s, u(s)) dW(s), w) = ∞ X j=1 Zt 0 (G(s, u(s))ej, w)dβj(s) = ∞ X j=1 Zt 0 (( e G(s, u(s))ej, w)) dβj(s) = ((Zt 0e G(s, u(s)) dW(s), w)). Consequently, in a abstract framework, a variational solution of problem (1) is, equivalently, a stochastic process u∈M2 Ft(0, T;U)∩L2(Ω; L∞(0, T;H)),for all T > 0, such that the equation u(t) + Zt 0e Au(s)ds +Zt 0e B(u(s), u(s)) ds =u0+Zt 0e F(s, u(s)) ds +Zt 0e G(s, u(s)) dW(s), t ≥0,(23) is satisfied in U∗, a.s. for all t≥0. 9 Denote vn(t) = u(t∧τn). Then, vn(t) + Zt 0 1[0,τn](s)e Avn(s)ds +Zt 0 1[0,τn](s)e B(vn(s), vn(s)) ds =u0+Zt 0 1[0,τn](s)e F(s, vn(s)) ds +Zt 0 1[0,τn](s)e G(s, vn(s)) dW(s), for all t≥0,and consequently, |vn(t)|2 H+ 2 Zt 0 1[0,τn](s)he Avn(s), vn(s)ids +2 Zt 0 1[0,τn](s)he B(vn(s), vn(s)), vn(s)ids =¯¯u0¯¯2 H+ 2 Zt 0 1[0,τn](s)he F(s, vn(s)), vn(s)ids +2 Zt 0³1[0,τn](s)e G(s, vn(s))dW(s), vn(s)´ +Zt 0 1[0,τn](s)ke G(s, vn(s))k2 L2(K;H)ds, for all t≥0.(45) Let us denote Ωn=½ω∈Ω; |u0(ω)|H≤1 n¾. It is clear that τn(ω) = 0 and vn(t, ω) = u0(ω) if ω∈Ωn,(46) |vn(t, ω)|H≥1 nif ω∈Ω\Ωn.(47) Let us take any function φn∈C2(R) such that φn(r) = log r, for all r≥1 n2. Applying the Itˆo formula to φn(|vn(t, ω)|2 H),and taking into account (45), (46) and (47),we obtain log |vn(t)|2 H= log ¯¯u0¯¯2 H+ 2 Zt 0 1[0,τn](s)h−e Avn(s) + e F(s, vn(s)), vn(s)i |vn(s)|2 H ds +Zt 0 1[0,τn](s)ke G(s, vn(s))k2 L2(K;H) |vn(s)|2 H ds +Mn(t)−1 2qn(t), t ≥0, (48) 16 where Mn(t) = 2 Zt 0Ã1[0,τn](s)e G(s, vn(s)) dW(s),vn(s) |vn(s)|2 H!,(49) qn(t) = 4 Zt 0 1[0,τn](s) |vn(s)|4 HÃ∞ X k=1 (e G(s, vn(s))ek, vn(s))2 H!ds. (50) Thus Mn(t) is a real continuous square integrable Ft-martingale such that Mn(0) = 0,with associated increasing process hMnit=qn(t). Let us now define M(t) = 2Zt∧τ0 0Ãe G(s, u(s)) dW(s),u(s) |u(s)|2 H!, q(t) = 4Zt∧τ0 0 1 |u(s)|4 HÃ∞ X k=1 (e G(s, u(s))ek, u(s))2 H!ds. Observe that, as u=vnon [0, τn(ω)],then M(t∧τn) = Mn(t), q(t∧τn) = qn(t), lim n→+∞Mn(t) = M(t),lim n→+∞qn(t) = q(t), and consequently, M(t) is a real continuous square integrable local Ftmartingale such that M(0) = 0,with associated increasing process hMit= q(t). Thus, from the exponential martingale inequality (see [12] and [15]) we have that for any positive numbers T, ε and k > 0, P( max t∈[0,T](M(t)−ε 2q(t)) ≥k)≤e−εk.(51) Let us fix ε > 0 such that λ:= eαc−2 2+ 2δ−2Le F−L2 e G−2εL2 e G>0,(52) whose existence is guaranteed by (40). Taking k= (2/ε) log m, with m≥1 any integer number, we obtain from (51), Pµmax t∈[0,m+1](M(t)−ε 2q(t)) ≥2 εlog m¶≤1 m2.(53) Consequently, we deduce from the Borel-Cantelli lemma that there exists F0∈ F, with P(F0) = 0, such that for any ω∈Ω\F0there is an integer m0(ω)≥1 for which max t∈[0,m+1](M(t, ω)−ε 2q(t, ω)) <2 εlog m, for all m≥m0(ω). (54) 17 In particular, if we take any ω∈Ω\F0,m≥m0(ω),and take into account that qn(t) is increasing, we obtain from (54), Mn(t, ω)−ε 2qn(m+ 1, ω)≤Mn(t, ω)−ε 2qn(t, ω) =M(t∧τn(ω), ω)−ε 2q(t∧τn(ω), ω) ≤max s∈[0,m+1](M(s, ω)−ε 2q(s, ω)) <2 εlog m, for all n≥1, t∈[0, m + 1]. Consequently, for all ω∈Ω\F0,there exists m0(ω)≥1 such that Mn(t, ω)−ε 2qn(m+ 1, ω)<2 εlog m, (55) for all n≥1, m ≥m0(ω), t∈[0, m + 1]. From (39),(48),(55) and the hypotheses on e A, e Fand e G, we have that almost surely log |vn(t, ω)|2 H ≤log |u0(ω)|2 H+Zt∧τn(ω) 0 (−eαc−2 2+ 2Le F+L2 e G−2δ)ds +2 εlog m+ε 2qn(m+ 1, ω),(56) for all n≥1, m ≥m0(ω), t∈[0, m + 1]. But from the expression for qn, we deduce ε 2qn(m+ 1, ω)≤2εL2 e G((m+ 1) ∧τn(ω)) . Also observe that if t∈[m, m + 1],then (m+ 1) ∧τn(ω)≤(t∧τn(ω)) + 1, and, hence, log m≤log t. Consequently, (52) and (56) imply log |vn(t, ω)|2 H≤log ¯¯u0(ω)¯¯2 H−λ(t∧τn(ω)) + 2εL2 e G+2 εlog t, or, in other words, |u(t∧τn(ω), ω)|2 H≤ |u0(ω)|2 Ht2/εe−λ(t∧τn(ω))e2εL2 e G,(57) for all t≥m0(ω), n ≥1, Letting n→+∞in (57),we deduce |u(t∧τ0(ω), ω)|2 H≤ |u0(ω)|2 Ht2/εe−λ(t∧τ0(ω))e2εL2 e G,for all t≥m0(ω), 18 and, in particular, as lim t→+∞(e−λ 2te2εL2 e Gt2/ε) = 0, then there exists T0(ω)≥m0(ω) such that |u(t, ω)|2 H≤e−λ 2t¯¯u0(ω)¯¯2 Hfor all t≥T0(ω), a.s. in Ω \e Ω,(58) where e Ω is defined by (43). Now, (41) follows from (44) and (58). Remark 4.4 Observe that this result shows how the noise can produce a stabilising effect on the deterministic problem when the intensity is large enough. Indeed, thanks to Theorem 3.3, we observe that if eαc−2 2−2Le F>0, then the trivial stationary solution of the deterministic problem is exponentially stable, but if this condition does not hold, and the noise appearing in the model (or the noise added to the model) is such that eαc−2 2+ 2δ−2Le F−L2 e G>0, then the trivial solution of the stochastic system becomes pathwise exponentially stable. Furthermore, if we are interested in stabilising the deterministic system and we are allowed to choose appropriate stochastic terms to do this, then we can always use a very simple one, namely, a one dimensional Brownian motion multiplied by a linear operator. From this point on we consider this special noise. We will describe below in more details this stabilisation procedure in the general case of considering a non-trivial stationary solution. Remark 4.5 Observe that Theorem 4.3affirms for (1) that, if F(t, 0) = 0, G(t, 0) = 0,and there exists δ > 0such that ∞ X k=1 (G(t, v)ek, v)2≥δkvk4,for all v∈V,dP ×dt-a.e., with 2ναλ2 1+ 2δ(1 + αλ1)>2LF(1 + αλ1) + L2 G, then, there exists a constant γ > 0such that for all u0∈L4(Ω,F0, P;V) there is a T(u0, ω)≥0for which the variational solution u(t)of (1) with corresponding initial datum u0,satisfies ku(t, ω)k2≤e−γtku0(ω)k2for all t≥T(u0, ω), a.s. 19 Suppose now that e F:H → H is independent of ωand t, and eα > 2c2 2Le F.(59) Under this condition, there exists a stationary solution of problem (26), say u∞∈ U. If, in addition, eα > c1|u∞|H+ 2c2 2Le F,(60) then, as a consequence of Theorem 3.3, taking e G≡0,we deduce that u∞is exponentially stable, i.e., there exists γ > 0 such that for all u0∈ H, |eu(t)−u∞|2 H≤¯¯u0−u∞¯¯2 He−γt,for all t≥0, where eu(t) is the solution of problem (26) with initial datum u0. However, if the Lipschitz constant Le Fis large enough, then (60) might not hold, and we would not know if u∞is exponentially stable. Nevertheless, we can always choose a very simple noise to stabilise our problem (26). To this end, let us consider a fixed Ft-Wiener process β(t). For any σ∈R, define e Gσ(t, v) = σ(v−u∞), v ∈ H,(61) and consider the problem u(t) + Zt 0e Au(s)ds +Zt 0e B(u(s), u(s)) ds =u0+Zt 0e F(u(s)) ds +Zt 0e Gσ(s, u(s)) dβ(s),a.s., ∀t≥0.(62) We can prove the following result. Theorem 4.6 Suppose that e F:H → H and satisfies (59).Let u∞∈ U be a stationary solution of problem (26),and suppose that eα−2c1c2ku∞kU>0.(63) Let σbe any real number such that eα−2c1c2ku∞kU+c2 2σ2>2c2 2Le F.(64) Then, there exists γ > 0satisfying that for any u0∈L4(Ω,F0, P;H),there exists T(u0, ω)≥0such that |u(t, ω)−u∞|2 H≤e−γt|u0(ω)−u∞|2 H,for all t≥T(u0, ω), a.s., (65) where u(t, ω)is the corresponding solution of (62). 20 Proof. The proof is similar to that of Theorem 4.3 but arguing with u0−u∞ and u(t)−u∞instead of u0and u(t) respectively, and taking into account that the special form of the noise allows us to finish the proof either using the subexponential decay of the Wiener process instead of the exponential martingale inequality, or repeating the analysis already done in that proof (see [13] for more details). Remark 4.7 As for problem (1),Theorem 4.6 states that, under conditions να > c2 2LF, ναλ1> c1p1 + αλ1|Au∞|, for a suitable choice of σ, there exists γ > 0satisfying that for any u0∈ L4(Ω,F0, P;V),there exists T(u0, ω)≥0such that ku(t, ω)−u∞k2≤e−γtku0(ω)−u∞k2,for all t≥T(u0, ω), a.s., where u(t, ω)is the corresponding variational solution of (1),with G(t, u)˙ W(t) = σ(u−u∞)˙ β(t). Acknowledgements. We thank the anonymous referees for helpful suggestions and comments. References [1] L. Arnold, Stabilization by noise revisited, Z. Angew. Math. Mech. 70(1990), 235-246. [2] T. Caraballo, M.J. Garrido-Atienza and J, Real, Asymptotic stability of nonlinear stochastic evolution equations, Stoch. Anal. Appl. 21(2) (2003), 301-327. [3] T. Caraballo and J.A. Langa, Comparison of the long-time behavior of linear Ito and Stratonovich partial differential equations, Stoch. Anal. Appl. 19(2) (2001), 183-195. [4] T. Caraballo and J.C. Robinson, Stabilisation of linear PDEs by Stratonovich noise, Systems and Control Letters 53(2004), 41-50. [5] T. Caraballo, J. A. Langa and T. Taniguchi, The exponential behaviour and stabilizability of stochastic 2D-Navier-Stokes equations, J. Diff. Eq. 179(2002), 714-737. [6] T. Caraballo, A. M. M´arquez-Dur´an and J. Real , On the stochastic 3D-Lagrangian averaged Navier-Stokes α-model with finite delay, to appear in Stochastics and Dynamics. [7] T. Caraballo, J. Real and T. Taniguchi, The existence and uniqueness of solutions to stochastic 3-dimensional Lagrangian averaged Navier- Stokes equations, submitted. 21 [8] D. Coutand, J. Peirce and S. Shkoller, Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains, Comm. on Pure and Appl. Anal. 1(2002), 35-50. [9] G. DaPrato and J. Zabczyk, “Stochastic Equations in Infinite Dimensions”, Cambridge University Press, Cambridge, 1992. [10] C. Foias, D.D. Holm and E. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, J. Dynamics and Differential Equations 14(2002), 1-35. [11] J.A. Langa, J. Real and J. Simon, Existence and regularity of the pressure for the stochastic Navier-Stokes equations, Appl. Math. Optim. 48 (2003), no. 3, 195-210. [12] X. Mao, “Stochastic Differential Equations and Applications”, Horwood Publishing, Chichester, 1997. [13] A. M. M´arquez Dur´an, PhD Thesis, Universidad de Sevilla, In preparation, 2005. [14] J.E. Marsden and S. Shkoller, Global well-posedness for the LANS-α equations on bounded domains, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 359(2001), 1449-1468. [15] D. Revuz and M. Yor “Continuous Martingales and Brownian Motion”, 2nd Ed., Springer-Verlag, 1994. [16] M. Scheutzow, Stabilization and destabilization by noise in the plane, Stoch. Anal. Appl. 11(1) (1993), 97-113. [17] R. Temam, “Navier-Stokes Equations and Nonlinear Functional Analysis”, 2nd Ed., SIAM, Philadelphia, 1995. 22