Attractors for The Stochastic 3D Navier-Stokes Equations

Stochastics and Dynamics c World Scientific Publishing Company ATTRACTORS FOR THE STOCHASTIC 3D NAVIER-STOKES EQUATIONS PEDRO MAR´ IN-RUBIO∗ Dpto. Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain JAMES C. ROBINSON† Mathematics Institute, University of Warwick Coventry, CV4 7AL. U.K. Received (received date) Revised (revised date) In a 1997 paper, Ball defined a generalised semiflow as a means to consider the solutions of equations without (or not known to possess) the property of uniqueness. In particular he used this to show that the 3d Navier-Stokes equations have a global attractor provided that all weak solutions are continuous from (0,∞) into L2. In this paper we adapt his framework to treat stochastic equations: we introduce a notion of a stochastic generalised semiflow, and then show a similar result to Ball’s concerning the attractor of the stochastic 3d Navier-Stokes equations with additive white noise. Keywords: 35Q30, 34D45, 60H15, 35R60, 58F12, 58F11. 1. Introduction For certain deterministic differential equations, most notably the three-dimensional Navier-Stokes equations, we can prove the existence of solutions but are unable to prove their uniqueness. Nevertheless, if we still wish to consider such equations within a dynamical systems framework there are various approaches available. In particular there have been various attempts to apply the theory of global attractors to the 3d Navier-Stokes equations, despite the unresolved problem of uniqueness. There are two results which require no additional hypotheses: Foias & Temam18 constructed a set, consisting of strong solutions, that attracts all weak solutions in the weak topology of the natural phase space; and Sell27 analysed the induced single-valued flow on the phase space consisting of all solutions of the equation (an element of this space is a complete trajectory in the original phase space), showing that this has a global attractor. ∗[email protected] †[email protected]arwick.ac.uk P. Mar´ın-Rubio & J.C. Robinson However, the two results that are nearest to the standard theory require the assumption of (unproved) hypotheses: Foias & Temam17 showed that the existence of globally defined strong solutions (which are then unique) automatically implies the existence of a global attractor, while Ball5deduced the same result assuming that the weak solutions trace out continuous trajectories in the phase space. Under this condition on the solutions, Ball recast the 3d Navier-Stokes equations as a generalised (multi-valued) semiflow, and showed that this generalised semiflow has a global attractor. Another general setting for such multi-valued flows has recently been developed by Melnik & Valero24, and in the past by workers in control theory (see Kloeden21 for a summary). In this paper we consider a stochastic version of the 3d Navier-Stokes equations: as for the deterministic equations, existence is known, but the issue of uniqueness is again unresolved (see Bensoussan & Temam6, for example). Two of the above treatments of the deterministic 3d equations have been extended to treat the stochastic case: that of Foias & Temam17 by Crauel & Flandoli15 (regularity implies the existence of an attractor) and that of Sell by Flandoli & Schmalfuß16 (an attractor in the ‘path space’). By working in a phase space formed of solutions of the stochastic 3d Navier-Stokes equations (‘the path space’), Flandoli & Schmalfuß are able, despite the lack of uniqueness, to work with single-valued cocycles of the standard sort (as discussed in Section 2). Following the treatment of Ball5and insisting that we remain in the original phase space, we are required in what follows to introduce a multi-valued framework. It is this that makes the problems of measurability and invariance of the attractor significantly more involved. (For a more general framework for stochastic equations without uniqueness see Caraballo et al.7). We begin the paper with a brief summary in Section 2 of the standard random dynamical systems framework in which the theory of random attractors can be developed. In Section 3 we recall the definition of weak solutions of the stochastic 3d Navier-Stokes equations due to Flandoli & Schmalfuß16. We then introduce in Section 4 the notion of a generalised stochastic semiflow, and prove that the weak solutions of the 3d NSE form a stochastic semiflow if and only if they are each continuous from (0,∞) into the natural phase space H. Section 5 develops the general theory of attractors for stochastic generalised semiflows, and this is then applied to the 3d NSE (under the continuity assumption on the solutions) in Section 6. While the applications to the stochastic 3d Navier-Stokes equations were the motivating factor for our work, we believe that the main contribution of this paper is to offer a workable definition of a stochastic generalised semiflow, and a corresponding theory of global attractors. 2. Single-valued random dynamical systems and their attractors We now recall the definition of a random dynamical system and a random attractor (for more background on random dynamical systems see Arnold3). We cover only the case of continuous time here. Let (Ω,F,P) be a probability space and {θt: Ω →Ω, t ∈R}a family of measure-preserving transformations such that (t, ω)7→ θtωis measurable, θ0= id, and θt+s=θtθsfor all s, t ∈R. The flow θttogether with the corresponding Attractors for the stochastic 3d Navier-Stokes equations probability space, (Ω,F,P,(θt)t∈R) is called a (measurable) dynamical system. Let (X, d) be a Polish space (a complete separable metric space) equipped with its Borel σ-algebra B. A continuous random dynamical system (RDS) on Xis a measurable map ϕ:R+×Ω×X→X (t, ω, x)7→ ϕ(t, ω)x such that P-a.s. i) ϕ(0, ω) = id on X ii) ϕ(t+s, ω) = ϕ(t, θsω)◦ϕ(s, ω) for all t, s ∈R+(cocycle property) iii) ϕ(t, ω) : X→Xis continuous. The concept of a random attractor for such systems was first introduced by Crauel & Flandoli14 and Schmalfuß28, with notable developments in Crauel et al.13. This attractor is a random compact set that is invariant and attracting ‘in the pullback sense’. In order to define these concepts more precisely we denote by dist(·,·) the Hausdorff semidistance in X, dist(A, B) = sup a∈A inf b∈Bd(a, b). A random compact set {K(ω)}ω∈Ωis a family of compact sets indexed by ω such that for every x∈Xthe map ω7→ dist(x, K(ω)) is measurable with respect to F; and we say that a random set A(ω) is attracting if for all deterministic bounded sets B⊂Xwe have lim t→∞ dist(ϕ(t, θ−tω)B, A(ω)) = 0 P−almost surely. Since ϕ(t, θ−tω)u0can be interpreted as the position at time zero of the trajectory which was at u0at time −t, this pullback attraction is essentially attraction ‘from t=−∞’. A random compact set A(ω) is said to be a random attractor for the RDS ϕif it is both attracting (as above) and invariant, that is ϕ(t, ω)A(ω) = A(θtω) for all t≥0P−a.s. The standard result that provides the existence of random attractors is similar to that from the deterministic theory (e.g. Babin & Vishik4, Hale19, Ladyzhenskaya22, Robinson26, Temam30): the following elegant formulation is due to Crauel12. Theorem 1 There exists a random attractor A(ω)iff there exists a compact attracting set K(ω). P. Mar´ın-Rubio & J.C. Robinson 3. Weak solutions of the stochastic 3d Navier-Stokes equations We will consider the Navier-Stokes equations with additive noise on a smooth bounded open domain D⊂R3: ∂u ∂t −ν∆u+ (u· ∇)u+∇p−f=˙ Wtin [0,∞)×D, div u = 0 in [0,∞)×D, u= 0 in [0,∞)×∂D, u(0, x) = u0(x)x∈D; (3.1) here νis the kinematic viscosity, pthe pressure, uthe velocity field, fa time independent body forcing field, and Wta two-sided Wiener process which we will specify in more detail below. [Of course, the notation ˙ Wtis for convenience only, the main PDE of (3.1) needing to be interpreted in an integral sense, see below]. The presence of the stochastic term requires a modification of the deterministic definition of a weak solution. In what follows we adopt the approach of Flandoli & Schmalfuß16, tailoring their presentation slightly to suit our purposes. First we need to introduce some standard notation in order to treat the equation in the usual way (see Constantin & Foias9, for example): let Vdenote the space of infinitely differentiable divergence-free three-dimensional vector fields on Dwith compact support contained in D. By Hwe denote the closure of Vin [L2(D)]3and by Vits closure in [H1 0(D)]3; for the corresponding norms we will use | · | and k · k respectively. We can define a bilinear operator B(u, v) : V×V→V0, where V0is the dual of V, via hB(u, v), zi= 3 X i,j=1 ZD zi(x)uj(x)∂vi ∂xj (x) dx, where h·,·i denotes the pairing between V0and V. If Pis the orthogonal projection from [L2(D)]3onto H(the ‘Leray projector’) then the Stokes operator Ais defined by Au =−P∆u. We write D(A) for the domain of A;D(A) = [H2(D)]3∩V, and denote by λ1the first eigenvalue of A. We take the white noise process Wtto be a two-sided Wiener process with values in Vthat can be expressed as Wt= ∞ X j=1 cjB(j) twj, where {wj}is a complete set of eigenfunctions of Athat are orthonormal in V,B(j) t are mutually independent one-dimensional Brownian motions, and P∞ j=1 |cj|2<∞. Thus, in the notation of Section 2, the probability space (Ω,F,P) is the space (C0(R;V),BC0(R;V),P) (with Pthe appropriate product of one-dimensional Wiener measures). The measure-preserving transformation θtcan be represented by its action on realisations of the Wiener process as the shift operator with an appropriate adjustment to retain the condition that Wθtω(0) = 0: W·(θtω) = Wt+·(ω)−Wt(ω). Attractors for the stochastic 3d Navier-Stokes equations In order to deal with the problems due to the white noise term we will essentially make a substitution that allows us to consider the equation realisation-by- realisation: to do this we will use the unique stationary solution of the auxiliary Stokes equation dz+ [(A+α)z] dt= dWt(3.2) that is defined for all t∈R, namely the Ornstein-Uhlenbeck process zα(t) = Zt −∞ e(t−s)(−A−α)dWs; this is given more rigorously, P-a.s., by zα(t, ω) = Wt(ω)−Zt −∞ (A+α)e(t−s)(−A−α)Ws(ω) ds(3.3) (the inclusion of the parameter αwill prove useful for our discussion of attractors, although it is not necessary in the proof of existence of solutions). Various properties of zαare recalled as and when we need them below. Definition 1 Given f∈[H−1(D)]3and a realisation Wt(ω)of the Wiener process that is continuous from [0,∞)into V, we say that u∈L2 loc[0,∞;H)is a weak solution of the Navier-Stokes equation (3.1) with noise ωif •u∈L∞ loc[0,∞;H)∩L2 loc[0,∞;V), •∂ ∂t (u−Wt)∈L4/3 loc [0,∞;V0), •for a.e. tand t0with t≥t0>0and for t0= 0, we have V1(u, ω)(t)≤V1(u, ω)(t0) (3.4) where V1(u, ω)(s) := e−Rs 0(−λ1+2C∗|zα(r;ω)|8 L4) dr|u(s)−zα(s;ω)|2 −Zs 0 e−Rσ 0(−λ1+2C∗|zα(r;ω)|8 L4) dr ×4C2 B|zα(σ;ω)|4 L4+α2 λ1 |zα(σ;ω)|2+kfk2 V0dσ, and V2(u;ω)(t)≤V2(u;ω)(t0),(3.5) where V2(u;ω)(s) := |u(s)−zα(s;ω)|2+Zs 0 ku(r)−zα(r;ω)k2dr −Zs 02C∗|u(r)−zα(r;ω)|2|zα(r;ω)|8 L4 +4C2 b|zα(r;ω)|4 L4+ 4α2|zα(r;ω)|2/λ1+ 4kfk2 V0dr P. Mar´ın-Rubio & J.C. Robinson •and for all t≥t0≥0and all φ∈V hu(t)−u(t0), φi+Zt t0 hA1/2u(s), A1/2φids+Zt t0 hB(u(s), u(s)), φids =hWt(ω)−Wt0(ω), φi+Zt t0 hf, φids. (3.6) The constants C∗and CBare defined in Flandoli & Schmalfuß16, and are related to the constants occurring in two calculus inequalities. Observe that if we drop all the terms involving Wtand zαthen (3.4) and (3.5) are consequences of the standard deterministic Leray energy inequality for weak solution (see Constantin & Foias9, for example). Flandoli & Schmalfuß show that the definition is independent of α. Almost every realisation of the Wiener process has trajectories that satisfy Wt∈ C0(R;V), so certainly Wt∈C0([0,∞), V ) as required by the definition of a weak solution above. Given this observation, Flandoli & Schmalfuß16 [prop. 2.2 p. 368] proved the existence of such weak solutions for the 3d Stochastic Navier-Stokes equations: Proposition 1 For almost every ω, given u0∈Hand f∈[H−1(D)]3there exists a weak solution of the Navier-Stokes equation with noise ωsuch that u(0) = u0. (We note here that we have taken f∈[H−1(D)]3, rather than the more usual assumption that f∈V0, in line with the results of Langa et al.23: although Definition 1 makes sense for f∈V0, it is not possible to recover the pressure pin any meaningful way unless f∈[H−1(D)]3. See Simon29 for similar results for the deterministic Navier-Stokes equations.) 4. Generalised stochastic semiflows and the 3d stochastic NSE Following Ball’s approach for deterministic semiflows without uniqueness5we now give a definition of a generalised stochastic semiflow, and show that this is applicable to the 3d Navier-Stokes equations with an additive noise, provided that we assume that the solutions are continuous from (0,∞) into H. 4.1. Generalised stochastic semiflows We recover Ball’s definition of a generalised semiflow if we remove from the following all dependence on ω(and the corresponding references to Ω). In order for our definition to make sense we impose the additional assumption that Ω is a Polish space. Because the primary application of the theory of random dynamical systems is to stochastic ordinary and partial differential equations, we believe that this does not overly limit the applicability of the concept. Generally we would expect Ω to be identified with the canonical (two-sided) Wiener space: in our application of the definition to the stochastic 3d Navier-Stokes equations we identify Ω with the space of all continuous paths from Rinto V, and say that ωn→ωif Wt(ωn)→Wt(ω) in V uniformly on compact subintervals of R. Attractors for the stochastic 3d Navier-Stokes equations Definition 2 Astochastic generalised semiflow (SGS) Gon Xwith noise Ω is a family of pairs {(ϕ, ω)|ϕ: [0,+∞)→X, ω ∈Ω} (called solutions) satisfying the following assumptions: (S1) Existence: P-a.s. in ω, for each z∈Xthere exists at least one (ϕ, ω)∈ G with ϕ(0) = z. (S2) Translates of solutions are solutions: If (ϕ, ω)∈ G and τ≥0, then (ϕτ, θτω)∈ G, where ϕτ(t) := ϕ(τ+t). (S3) Generalised cocycle property: If (ϕ, ω)and (ψ, θtω)belong to G, with ψ(0) = ϕ(t), then (φ, ω)∈ G, where φ(τ):=ϕ(τ) for 0 ≤τ≤t, ψ(τ−t) for τ > t. (S4) Upper semicontinuity with respect to initial data: if (ϕn, ωn)∈ G with ϕn(0) → zand ωn→ω, then there exists a subsequence (ϕn0, ωn0)and a pair (ϕ, ω)∈ G with ϕ(0) = zsuch that ϕn0(t)→ϕ(t)for every t≥0. Given a stochastic generalised semiflow it is possible to define a “generalised cocycle” using the set of all attainable states, Φ(t, ω)E={ϕ(t) : (ϕ, ω)∈ G , ϕ(0) ∈E}.(4.7) We now show that this has similar properties to the standard kind of cocycle (cf. Section 2). Proposition 2 Let Gbe a SGS, then P-a.s. Φ(0, ω)E=Efor all E⊂X(4.8) and Φ(t+s, ω)E= Φ(t, θsω)Φ(s, ω)Efor all t, s ∈R+, E ⊂X. (4.9) Moreover, every Φ(t, ω)has compact values and is a multi-valued upper semicontinuous mapping, i.e. for every neighbourhood Nof Φ(t, ω)x, there exists a neighbourhood Mof xsuch that Φ(t, ω)M⊂N. Proof. Equality (4.8) is obvious by definition. In order to prove (4.9) consider x∈ Φ(t+s, ω)E; then there exists (ϕ, ω)∈ G such that x=ϕ(t+s) with ϕ(0) ∈E. Now, (ϕs, θsω) is an element of Gby (S2): therefore x=ϕs(t), i.e. x∈Φ(t, θsω)ϕs(0), and ϕs(0) = ϕ(s)∈Φ(s, ω)ϕ(0). For the other inclusion, let x∈Φ(t, θsω)Φ(s, ω)E. Then there exists an element z∈Φ(s, ω)Ewith x∈Φ(t, θsω)z, and so there exist pairs (ϕ, θsω) and (ψ, ω) in G such that x=ϕ(t) with ϕ(0) = ψ(s) and ψ(0) ∈E. We now use property (S3), and consider φ(τ) := ψ(τ) for 0 ≤τ≤s, ϕ(τ−s) for s < τ. Then (φ, ω)∈ G and so x∈Φ(t+s, ω)ψ(0) ⊂Φ(t+s, ω)E. P. Mar´ın-Rubio & J.C. Robinson Since (S4) holds, it is obvious that every Φ(t, ω) has compact values. For the upper semicontinuity it is easy to see (arguing by contradiction) that Φ(t, ω) is ε- u.s.c. [for each x, for every  > 0 there exists a δ > 0 such that the image under Φ(t, ω) of the δball about xlies within an neighbourhood of Φ(t, ω)(x)]; since Φ(t, ω) takes compact values this implies that it is upper semicontinuous (cf. Aubin & Cellina1). In order to prove the existence of a random attractor for such a generalised stochastic semiflow, we will require a strengthening of the property (S4) which also includes some compactifying properties of the generalised cocycle Φ(t, ω). We say that Gis a compactifying generalised stochastic semiflow (KSGS) if (S1–3) and (S4∗) hold, where for (S4∗) we require the same as (S4) but when the initial data converges only weakly: (S4∗) Upper semicontinuity with respect to weakly converging initial data: if (ϕn, ωn)∈ Gwith ϕn(0) * z and ωn→ω, then there exists a subsequence (ϕn0, ωn0) and a pair (ϕ, ω)∈ G with ϕ(0) = zsuch that ϕn0(t)→ϕ(t) for every t > 0. 4.2. A stochastic generalised semiflow for the 3d stochastic NSE We denote by GSNS the set of all pairs (ϕ, ω), where ϕis a weak solution of the stochastic 3d Navier-Stokes equations associated to a realisation ωof the noise. We have the following result, after Proposition 7.4 of Ball5. Note that the topology on Ω is that discussed at the beginning of Section 4.1. Proposition 3 The following are equivalent: (i) GSNS is a compactifying stochastic generalised semiflow. (ii) P-a.s. in ω, each weak solution u(associated with ω) is continuous from (0,∞) to H. (iii) P-a.s. in ω, each weak solution u(associated with ω) is continuous from [0,∞) to H. We note here that in the particular case of the stochastic 3d NSE, (S4) and (S4∗) are equivalent. Proof. (i) implies (ii). We follow the proof of Theorem 2.1 in Ball’s paper, which combines a version of Lusin’s Theorem with the properties of weak solutions to deduce (ii) by contradiction. More precisely, suppose that GSNS is a generalised stochastic semiflow on H, and let ωlie within the set of full measure given by (S1) such that there exists at least one weak solution. Now choose ϕto be any one of these weak solutions, and let us assume that it is not continuous from (0,∞)→H. It follows that for some finite time interval I= (a, a +δ), there exists a t0∈J≡(a+δ/3, a + 2δ/3) and hj→0+ such that ϕ(t0+hj)6→ ϕ(t0).(4.10) We will deduce a contradiction from the above claim. In order to do that we will show that all hj-shifts of ϕare continuous almost everywhere and then use the weak continuity of the solution. Attractors for the stochastic 3d Navier-Stokes equations Since ϕ∈C([0,∞); Hw) it is weakly measurable, and by Pettis’ Theorem (see page 73 of Hille & Phillips20)ϕis strongly measurable. Lusin’s Theorem (see Oxtoby25 for example) can be applied to ensure the existence of a family of closed sets Fj⊂Jwith meas(J\Fj)≤1/j2such that the restriction of ϕto Fjis continuous. We now define Ej=J∩Fj∩(Fj−hj), where hjare the times in (4.10). Clearly meas(J\Ej)≤2/j2, and so meas  J\[ n≥1\ j≥n Ej = meas  \ n≥1[ j≥n (J\Ej)  ≤meas  [ j≥n0 (J\Ej) ≤X j≥n0 2/j2→0 as n0→ ∞, from which we deduce that almost every t∈Jbelongs to Sn≥1Tj≥nEj, i.e. ϕ(·+hj) is continuous and so ϕ(t+hj)→ϕ(t) for almost every t∈J. Now take t1and t2in J, with t1< t0< t2and ϕ(ti+hj)→ϕ(ti) as j→ ∞. Since (ϕ(t1+·), θt1ω)∈ GSNS and (ϕ(t1+hj+·), θt1+hjω)∈ GSNS with ϕ(t1+hj+ 0) →ϕ(t1) when hj→0, using (S4) we can deduce the existence of a subsequence and a pair (ψ, θt1ω)∈ GSNS with ψ(0) = ϕ(t1) and ϕ(t1+hµ+t)→ψ(t) for all t≥0 when µ→ ∞. In particular, ψ(t) = ϕ(t1+t) for almost every t∈(0, a + 2δ/3−t1). Using (S3) we can guarantee that (φ, θt1ω)∈ GSNS, where φ(t) = ϕ(t+t1) 0 ≤t≤t2−t1, ψ(t)t > t2−t1. Since the weak solutions of the Navier-Stokes equations are weakly continuous, the semi-flow has the property (in Ball’s terminology) of unique representatives, namely that since (ϕ(t1+·), θt1ω) and (φ(·), θt1ω) coincide for almost every t∈ (0,∞) we must have φ(t) = ϕ(t1+t) for all t≥0. In particular, we deduce that ϕ(t0+hµ) tends to φ(t0−t1) = ϕ(t0) as µ→ ∞, a contradiction. (ii) implies (iii). Since each solution uis continuous from (0,∞)→H, we only have to prove the continuity at t= 0. But it is straightforward given (3.5), taking t0= 0, to check that given any sequence tj→0+: |u(0) −zα(0, ω)| ≤ lim inf |u(tj)−zα(tj, ω)| ≤lim sup |u(tj)−zα(tj, ω)| ≤ |u(0) −zα(0, ω)|. ¿From this, the continuity of zαand the weak convergence of u(t) to u(0) (following from the weak continuity of solutions), we deduce (iii). (iii) implies (i). Condition (S1) is ensured by Proposition 1 and (S3) is easy to obtain; without (ii) or (iii) the translation property (S2) need not hold, due to P. Mar´ın-Rubio & J.C. Robinson The second term on the right-hand side of (6.16) does not depend on D, and we need an estimate that is valid for all t≥0. In order to do that, we transform it by two changes of variables, σ−t=sand r−t=ρ, as follows: Zt 0 eRt σ(−λ1+2C∗|zα(s;θ−tω)|8 L4) dshC2 B|zα(σ;θ−tω)|4 L4+α2 λ1 |zα(σ;θ−tω)|2+kfk2 V0idσ =Z0 −t eR0 s(−λ1+2C∗|zα(ρ;ω)|8 L4) dρhC2 B|zα(s;ω)|4 L4+α2 λ1 |zα(s;ω)|2+kfk2 V0ids By the continuity, polynomial growth and ergodic argument (6.17)-(6.18) (with another change of variables) for the process zα, we can proceed and finish the proof exactly as in Crauel & Flandoli15. Thus, it follows that given u(0) in a bounded set, there exists a random variable r(ω) and a time t1(ω, |u(0)|) such that |u(t)| ≤ r(ω) for all t≥t1(ω, |u(0)|). Write B(ω) for the ball of radius r(ω) in H. Now we recall (cf. Ball5), as observed at the end of Lemma 1, that the proof of Proposition 3 also shows that GSNS is compact, i.e. given ω, a sequence of solutions (ϕn, ω)∈ GSNS with {ϕn(0)}bounded has a subsequence that converges for all t > 0. In particular this means that Φ(1, ω) is compact, P-a.s., and so K(ω) = Φ(1, θ−1ω)B(θ−1ω) is a compact set. Due to its definition, K(ω) is absorbing, since Φ(t+ 1, θ−1−tω)D= Φ(1, θ−1ω)Φ(t, θ−1−tω)D and so for all t≥t1(θ−1ω, D) Φ(t+ 1, θ−1−tω)D⊆Φ(1, θ−1ω)B(θ−1ω) = K(ω). Application of Theorem 2 now yields a global attractor for the 3d stochastic NSE: Theorem 3 If weak solutions of the stochastic 3d Navier-Stokes equations are continuous from (0,∞)into H, then the equations define a compactifying generalised stochastic semiflow which has a global attractor. 7. Conclusion We have extended the idea of a generalised semiflow due to Ball5to treat stochastic systems, and shown that such a generalised semiflow has a global attractor if and only if it has a compact attracting set. These abstract ideas have been applied to the stochastic 3d Navier-Stokes equations, for which we have shown that, as in the deterministic case, continuity of solutions into the natural phase space Himplies the existence of a global attractor. Attractors for the stochastic 3d Navier-Stokes equations Acknowledgments PM-R & JCR would both like to thank Jos´e Langa for suggesting this problem, and an anonymous referee for helpful comments which lead to significant strengthening of the results in the paper. PM-R thanks James Robinson, Tania Styles, and Margaret Azarpey for their wonderful hospitality during his stay at the University of Warwick while preparing this work; and Professors Jos´e Real and Tom´as Caraballo for their advice and supervision of his thesis. JCR is currently a Royal Society University Research Fellow, and would like to thank the Society for all their support. References 1. J. P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory (Springer-Verlag, 1984). 2. J. P. Aubin and H. Frankowska, Set-Valued Analysis (Birkh¨auser Boston Inc., 1990). 3. L. Arnold, Random Dynamical Systems (Springer, 1998). 4. A. V. Babin and M. I. Vishik, Attractors of Evolution Equations (North Holland, 1992). 5. J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations,J. Nonlinear Sci. 7(1997) 475–502. 6. A. Bensoussan and R. Temam, ´ Equations stochastiques du type Navier-Stokes,J. Funct. Analysis 13 (1973) 195–222. 7. T. Caraballo, J. A. Langa and J. Valero, Global attractors for multivalued random dynamical systems,Nonlinear Anal. 48 (2002) 805–829. 8. C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics (Springer-Verlag Berlin, 1977). 9. P. Constantin and C. Foias, Navier-Stokes equations (University of Chicago Press, 1988). 10. H. Crauel, Random Probability Measures on Polish Spaces, Habilitationsschrift, Bremen (1995). 11. H. Crauel, Global random attractors are uniquely determined by attracting deterministic compact sets,Ann. Mat. Pura Appl. (IV) 176 (1999) 57–72. 12. H. Crauel, Random point attractors versus random set attractors,J. London Math. Soc. 63 (2001) 413–427. 13. H. Crauel, A. Debussche and F. Flandoli, Random attractors,J. Dyn. Diff. Eqns. 9 (1997) 307–341. 14. H. Crauel and F. Flandoli, Attractors for random dynamical systems,Probab. Theory Relat. Fields 100 (1994) 365–393. 15. H. Crauel and F. Flandoli, Dissipativity of three-dimensional stochastic navierstokes equation,Seminar on Stochastic Analysis, Random Fields and Applications. Progr. Probab., Ascona, (1993), (Birkhuser, 1995), pp. 67–76. 16. F. Flandoli and B. Schmalfuß, Weak solutions and attractors for three-dimensional Navier-Stokes equations with nonregular force,J. Dyn. Diff. Eqns. 11 (1999) 355–398. 17. C. Foias and R. Temam, Some analytic and geometric properties of the solutions of the Navier-Stokes equations,J. Math. Pures Appl. 58 (1979) 339–368. 18. C. Foias and R. Temam, The connection between the Navier-Stokes equations, dynamical systems, and turbulence theory, in Directions in Partial Differential Equations (Academic Press, 1987), pp. 55–73. 19. J.K. Hale, Asymptotic Behavior of Dissipative Dynamical Systems (Amer. P. Mar´ın-Rubio & J.C. Robinson Math. Soc., 1988). 20. E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups (Amer. Math. Soc., 1957). 21. P. E. Kloeden, General control systems, in Mathematical Control Theory,Lecture Notes in Mathematics 680, ed. W. A. Coppel, (Berlin: Springer-Velag, 1978), pp. 119– 138. 22. O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations (Cambridge University Press, 1991). 23. J. A. Langa, J. Real, and J. Simon, Existence and regularity of the pressure for the stochastic Navier-Stokes equations,Applied Mathematics & Optimization, to appear. 24. V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions,Set-Valued Anal. 6(1998) 83–111. 25. J. C. Oxtoby, Measure and Category (Springer-Verlag, 1971). 26. J. C. Robinson, Infinite-Dimensional Dynamical Systems (Cambridge University Press, 2001). 27. G. Sell, Global attractors for the three-dimensional Navier-Stokes equation,J. Dyn. Diff. Eqns. 8(1996) 1–33. 28. B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, in Nonlinear Dynamics: Attractor Approximation and Global Behaviour,International seminar on applied mathematics, eds. V. Reitmann, T. Riedrich and N. Koksch (Dresden: Technische Universit¨at, 1992), pp. 185–192. 29. J. Simon, On the existence of the pressure for solutions of the variational Navier-Stokes equations,J. Math. Fluid. Mech. 1(1999) 255–234. 30. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics (Springer AMS, 1988).