ATTRACTORS FOR THE STOCHASTIC 3D NAVIER-STOKES EQUATIONS

Transcription

1 Stochastics and Dynamics c World Scientific Publishing Company ATTRACTORS FOR THE STOCHASTIC 3D NAVIER-STOKES EQUATIONS PEDRO MARÍN-RUBIO Dpto. Ecuaciones Diferenciales y Análisis Numérico Universidad de Sevilla, Apdo. de Correos 116, 418 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 generalized 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 (, ) into L 2. In this paper we adapt his framework to treat stochastic equations, and show a similar result concerning the attractor of the stochastic 3d Navier-Stokes equations with additive white noise. Keywords: 35Q3, 34D45, 6H15, 35R6, 58F12, 58F 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. Appropriate general settings have been developed by Ball 4, Melnik & Valero 17, and Sell 2, among others. In particular there have been a variety of 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 & Temam 13 constructed a set, consisting of strong solutions, that attracts all weak solutions in the weak topology of the natural phase space; and Sell 2 analysed the induced flow on the space of all solutions, showing that this has a global attractor. However, the two results that are nearest to the standard theory require the assumption of (unproved) hypotheses: Foias & Temam 12 showed that the existence pedro@numer.us.es jcr@maths.warwick.ac.uk

2 P. Marín-Rubio & J.C. Robinson of globally defined strong solutions (essentially the assumption of uniqueness) automatically implies the existence of a global attractor, while Ball 4 deduced the same result assuming that the weak solutions trace out continuous trajectories in the phase space. 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 unresolved (see Bensoussan & Temam 5, for example). Two of the above treatments of the deterministic 3d equations have been extended to treat the stochastic case: that of Sell by Flandoli & Schmalfuß 11, and that of Foias & Temam 12 by Crauel & Flandoli 1. Here we develop a theory analogous to that of Ball 4. (For a more general framework for stochastic equations without uniqueness see Caraballo et al. 6 ). 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 & Schmalfuß 11. We then introduce in section 4 the notion of a generalized 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 [, ) into the natural phase space H. Section 5 develops the general theory of attractors for stochastic generalized semiflows, and this is then applied to the 3d NSE (under the continuity assumption on the solutions) in section 6. We end with a brief discussion of our results. 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 Arnold 2 ). We cover only the case of continuous time here. Let (Ω, F, P) be a probability space and {θ t : Ω Ω, t R} a family of measure preserving transformations such that (t, ω) θ t ω is measurable, θ = id, and θ t+s = θ t θ s for all s, t R. The flow θ t together with the corresponding probability space, (Ω, F, P, (θ t ) t R ) is called a (measurable) dynamical system. A continuous random dynamical system (RDS) on a Polish (complete and separable metric) space (X, d) with Borel σ-algebra B over θ on (Ω, F, P) is a measurable map such that P-a.s. i) ϕ(, ω) = id on X ϕ : R + Ω X X (t, ω, x) ϕ(t, ω)x ii) ϕ(t + s, ω) = ϕ(t, θ s ω) ϕ(s, ω) for all t, s R + (cocycle property) iii) ϕ(t, ω) : X X is continuous.

3 Attractors for the stochastic 3d Navier-Stokes equations For such systems random attractors were first introduced by Crauel & Flandoli 9 and Schmalfuß 21, with notable developments given in Flandoli & Schmalfuß 11 and in Crauel et al. 8. A random compact set {K(ω)} ω Ω is a family of compact sets indexed by ω such that for every x X the map ω dist(x, A(ω)) is measurable with respect to F. In order to discuss the concept of attraction, we denote by dist(, ) the Hausdorff semidistance in X, dist(a, B) = sup inf d(a, b). a A b B We say that a random set is attracting if for all deterministic bounded sets B X we have lim dist(ϕ(t, θ tω)b, A(ω)) = P a.s.. t Since ϕ(t, θ t ω)u can be interpreted as the position at t = of the trajectory which was at u at time t, this pullback convergence property is essentially attraction from t =. A random compact set A(ω) is said to be a random attractor for the RDS ϕ if it is both attracting (as above) and invariant, that is ϕ(t, ω)a(ω) = A(θ t ω) for all t P-a.s. The standard result that provides the existence of random attractors is similar to that familiar from deterministic theory (e.g. Babin & Vishik 3, Hale 14, Ladyzhenskaya 16, Robinson 19, Temam 22 ): the following elegant formulation is due to Crauel 7. Theorem 1 There exists a random attractor A(ω) iff there exists a compact attracting set K(ω). 3. Weak solutions of the stochastic 3d Navier-Stokes equations We will consider the Navier-Stokes equations with additive noise on a smooth bounded domain D R 3 : u t ν u + (u )u + p f = Ẇt in [, ) D, div u = in [, ) D, u = in [, ) D, u(, x) = u (x) x D; (3.1) here ν is the kinematic viscosity, p the pressure field, u the velocity field, f a time independent body forcing, and W t a two-sided Wiener process which we will specify in more detail below. [Of course, the notation Ẇt is 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 & Schmalfuß 11, tailoring their presentation slightly to suit our purposes. First we need to introduce some notation in order to treat the equation in the usual way: let V denote the space of infinitely differentiable divergence-free threedimensional vector fields on D with compact support strictly contained in D. By

4 P. Marín-Rubio & J.C. Robinson H we denote the closure of V in [L 2 (D)] 3 and by V its closure in [H 1 (D)] 3 ; for the corresponding norms we will use and respectively. We can define a bilinear operator B(u, v) : V V V via B(u, v), z = 3 i,j=1 D z i (x)u j (x) v i x j (x) dx. If P is the orthogonal projection from [L 2 (D) 3 ] onto H then the Stokes operator A is defined by Au = P u. We write D(A) for the domain of A; D(A) = [H 2 (D)] 3 V, and denote by λ 1 the first eigenvalue of A. The white noise W t is a two-sided Wiener process taking values in V. There in the notation of section 2, the probability space (Ω, F, P) is the standard Wiener space for such a process, and the measure preserving transformation θ t can be represented by its action on realizations of the Wiener process as the shift operator with the appropriate adjustment to retain the condition that W θtω() =, W (θ t ω) = W t+ (ω) W t (ω). 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 realization-byrealization: to do this we will use the unique stationary solution of the auxiliary Stokes equation dz + [(A + α)z] dt = dw t (3.2) that is defined for all t R, the Ornstein-Uhlenbeck process z α (t) = this is given more rigorously, P-a.s., by e (t s)( A α) dw s ; z α (t, ω) = W t (ω) (A + α)e (t s)( A α) W s (ω) ds (the inclusion of the parameter α will prove useful for our discussion of attractors, although it is not necessary in the proof of existence of solutions). Various properties of z α are recalled as and when we need them below. Definition 1 Given f V and a realization W t (ω) of the Wiener process that is continuous from [, ) into V, we say that u L 2 loc [, ; H) is a weak solution of the Navier-Stokes equation (3.1) with noise ω if u L loc [, ; H) L2 loc [, ; V ), t (u W t) L 4/3 loc [, ; V ), for a.e. t and t with t t > and for t =, we have V 1 (u, ω)(t) V 1 (u, ω)(t ) (3.3)

5 Attractors for the stochastic 3d Navier-Stokes equations where and where V 1 (u, ω)(s) := e s s ( λ 1+2C z α (r;ω) 8 L 4 ) dr u(s) z α (s; ω) 2 4 [C 2B z α (σ; ω) V 2 (u; ω)(s) := u(s) z α (s; ω) 2 + e σ ( λ 1+2C z α (r;ω) 8 L 4 ) dr 4L4 + α2 λ 1 z α (σ; ω) 2 + f 2 V ] dσ, V 2 (u; ω)(t) V 2 (u; ω)(t ), (3.4) s and for all t t and all φ V s u(r) z α (r; ω) 2 dr [ 2C u(r) z α (r; ω) 2 z α (r; ω) 8 L 4 +4C 2 b z α (r; ω) 4 L 4 + 4α2 z α (r; ω) 2 /λ f 2 V ] dr u(t) u(t ), φ + A 1/2 u(s), A 1/2 φ ds + t = W t (ω) W t (ω), φ + t B(u(s), u(s)), φ ds t f, φ ds. (3.5) The constants C and C B are defined in Flandoli & Schmalfuß 11, and are related to the constants occurring in two calculus inequalities. Observe that if we drop all the terms involving W t and z α then (3.3) and (3.4) are consequences of the standard deterministic energy inequality. Flandoli & Schmalfuß show that the definition is independent of α. Almost every realization of the Wiener process has trajectories that satisfy W t C ([, ); V ) as required by the definition of a weak solution above. Given this observation, Flandoli & Schmalfuß 11 [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 u H and f V there exists a weak solution of the Navier-Stokes equation with noise ω such that u() = u. 4. Generalized stochastic semiflows and the 3d stochastic NSE Following Ball s approach for deterministic semiflows without uniqueness 4 we now give a definition of a generalized 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 (, ) into H Generalized stochastic semiflows

6 P. Marín-Rubio & J.C. Robinson We recover Ball s definition of a generalized semiflow if we remove from the following all dependence on ω (and the corresponding references to Ω). Definition 2 A stochastic generalized semiflow (SGS) G on X with noise Ω is a family of pairs {(ϕ, ω) ϕ : [, + ) X, ω Ω} (called solutions) satisfying the following assumptions: (S1) Existence: P -a.s. in ω, for each z X there exists at least one (ϕ, ω) G with ϕ() = z. (S2) Translates of solutions are solutions: If (ϕ, ω) G and τ, then (ϕ τ, θ τ ω) G, where ϕ τ (t) := ϕ(τ + t). (S3) Generalized cocycle property: If (ϕ, ω) and (ψ, θ t ω) belong to G, with ψ() = ϕ(t), then (φ, ω) G, where { ϕ(τ) for τ t, φ(τ) := ψ(τ t) for τ > t. (S4) Upper semicontinuity with respect to initial data: if (ϕ n, θ tn ω) G with ϕ n () z and t n +, then there exist a subsequence (ϕ n, θ tn ω) and a pair (ϕ, ω) in G with ϕ() = z such that ϕ n (t) ϕ(t) for every t. Given a stochastic generalized semiflow it is possible to define a generalized cocycle using the set of all attainable states, Φ(t, ω)e = {ϕ(t) : (ϕ, ω) G, ϕ() E}. (4.6) We now show that this has similar properties to the standard kind of cocycle (cf. section 2). Proposition 2 Let G be a SGS, then and P -a.s. Φ(, ω)e = E for all E X (4.7) Φ(t + s, ω)e = Φ(t, θ s ω)φ(s, ω)e for all t, s R +, E X. (4.8) Moreover, every Φ(t, ω) has compact values and is a multi-valued upper semicontinuous mapping, i.e. for every neighbourhood N of Φ(t, ω)x, there exists a neighbourhood M of x such that Φ(t, ω)m N. Proof. Equality (4.7) is obvious by definition. In order to prove (4.8) consider x Φ(t+s, ω)e; then there exists (ϕ, ω) G such that x = ϕ(t+s) with ϕ() E. Now, (ϕ s, θ s ω) is an element of G by (S2): therefore x = ϕ s (t), i.e. x Φ(t, θ s ω)ϕ s (), and ϕ s () = ϕ(s) Φ(s, ω)ϕ(). For the other inclusion, let x Φ(t, θ s ω)φ(s, ω)e. Then there exists an element z Φ(s, ω)e with x Φ(t, θ s ω)z, and so there exist pairs (ϕ, θ s ω) and (ψ, ω) in G such that x = ϕ(t) with ϕ() = ψ(s) and ψ() E. We now use property (S3), and consider { ψ(τ) for τ s, φ(τ) := ϕ(τ s) for s < τ.

7 Attractors for the stochastic 3d Navier-Stokes equations Then (φ, ω) G and so x Φ(t + s, ω)ψ() Φ(t + s, ω)e. 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 ɛ > there exists a δ > such that the image under Φ(t, ω) of the δ ball about x lies within an ɛ neighbourhood of Φ(t, ω)(x)]; since Φ(t, ω) takes compact values this implies that it is upper semicontinuous (cf. Aubin & Cellina 1 ) A generalized stochastic semiflow for the 3d stochastic NSE We denote by G SNS the set of all pairs (ϕ, ω), where ϕ is a weak solution of the stochastic 3d Navier-Stokes equations associated to a realization ω of the noise. We have the following result, after proposition 7.4 of Ball 4. Proposition 3 The following are equivalent: (i) G SNS is an generalized stochastic semiflow. (ii) P -a.s. in ω, each weak solution u (associated with ω) is continuous from (, ) to H. (iii) P -a.s. in ω, each weak solution u (associated with ω) is continuous from [, ) to H. Proof. (i) implies (ii). We follows 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 G SNS is a generalized stochastic semiflow on H, let ω be in the set of full measure given by (S1) such that there exists at least one weak solution. Let also ϕ be any of these weak solutions, and by contradiction, let us assume that is not continuous from (, ) H. It follows that for some compact time interval I = (a, a + δ), there exists a t J (a + δ/3, a + 2δ/3) and h j + such that ϕ(t + h j ) ϕ(t ). (4.9) Since ϕ C([, ); H w ) it is weakly measurable, and by a well-known result (see page 73 of Hille & Phillips 15, for example) ϕ is strongly measurable. Lusin s theorem (see Oxtoby 18 for example) can be applied to ensure the existence of a closed set F j J with J F j 1/j 2 and ϕ F j continuous. Define E j = J F j (F j h j ); then J E j 2/j 2 and J n 1 j n E j J n 1 j n E j =, from which we deduce that ϕ(t + h j ) ϕ(t) for almost every t J. Now take t 1 and t 2 in J, with t 1 < t < t 2 and ϕ(t i + h j ) ϕ(t i ) as j. Since (ϕ(t 1 + ), θ t1 ω) G SNS and (ϕ(t 1 +h j + ), θ t1+h j ω) G SNS with ϕ(t 1 +h j + ) ϕ(t 1 ) when h j, using (S4) we can deduce the existence of a subsequence and a pair (ψ, θ t1 ω) G SNS with ψ() = ϕ(t 1 ) and ϕ(t 1 + h µ + t) ψ(t) for all t

8 P. Marín-Rubio & J.C. Robinson when µ. In particular, ψ(t) = ϕ(t 1 + t) for almost every t (, a + 2δ/3 t 1 ). Using (S3) we can guarantee that (φ, θ t1 ω) G SNS, where { ϕ(t + t1 ) t t φ(t) = 2 t 1, ψ(t) t > t 2 t 1. 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(t 1 + ), θ t1 ω) and (φ( ), θ t1 ω) coincide for almost every t (, ) we must have φ(t) = ϕ(t 1 + t) for all t. In particular, we deduce that ϕ(t + h µ ) tends to φ(t t 1 ) = ϕ(t ) as µ, a contradiction. (ii) implies (iii). Since each solution u is continuous from (, ) H, we only have to prove the continuity at t =. But it is straightforward from (3.4) taking t = to check that given any sequence t j + : u() z α (, ω) lim inf u(t j ) z α (t j, ω) lim sup u(t j ) z α (t j, ω) u() z α (, ω). From this, the continuity of z α and the weak convergence (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 violation of the energy inequality (cf. Ball) although the equation (3.5) is satisfied by translates (cf. lemma 5.1 in Flandoli & Schmalfuß 11 ). However, assuming (iii), each weak solution associated to (an appropriate) ω is continuous from [, ) H; hence V i (u, ω)(t) (i = 1, 2) are continuous for all t and therefore non-increasing, from which (S2) follows. So, we concentrate on proving (S4): let t n + and (u n, θ tn ω) G SNS such that u n () z in H. We have to prove that there exists a subsequence (u µ, θ tµ ω) and (u, ω) G SNS with u() = z and u µ (t) u(t) for all t. (We will deal with a fixed time interval [, T ] and the result for [, ) will follow by a diagonal argument.) Since we are concerned with solutions of the equation corresponding to various different realizations of the noise (θ tn ω), in order to apply definition 1 we have to use various different processes z α. Let us recall the cocycle property for these processes (cf. lemma 3.6 in Flandoli & Schmalfuß 11 ): z α (t + s; ω) = z α (t; θ s ω) t, s R. Using this we can derive estimates for u n (t) z α (t; θ tn ω): by (3.3) we have u n (t) z α (t; θ tn ω) 2 e + ( λ1+2c z α(s+t n;ω) 8 L 4 ) ds u n () z α (t n ; ω) 2 e σ ( λ 1+2C z α (s+t n ;ω) 8 L 4 ) ds 4[CB z 2 α (σ + t n ; ω) 4 L + α2 z 4 α (σ + t n ; ω) 2 + f 2 V ] dσ. λ 1

9 Attractors for the stochastic 3d Navier-Stokes equations Since z α ( ; ω) C([, ); H) (cf. lemma 3.4 in Flandoli & Schmalfuß 11 ) we deduce that for some constant C T (α; ω) > u n ( ) z α ( + t n ; ω) L (,T ;H) C T (α; ω) for all n. (4.1) On other hand, from (3.4) we obtain u n (t) z α (t; θ tn ω) 2 + u n () z α (t n ; ω) 2 + u n (s) z α (s; θ tn ω) 2 ds [2C u n (σ) z α (σ; θ tn ω) 2 z α (σ + t n ; ω) 8 L 4 +4CB z 2 α (σ + t n ; ω) 4 L + 4α2 z 4 α (σ + t n ; ω) f 2 V ] dσ, λ 1 from whence, using (4.1), we obtain (adjusting the definition of C T if necessary) u n ( ) z α ( ; θ tn ω) L 2 (,T ;V ) C T (α; ω) for all n. (4.11) Now it is standard to deduce that d dt n( ) z α ( ; θ tn ω)] C T (α; ω) L 4/3 (,T ;V ) for all n (4.12) (once more changing C T suitably). Applying well known compactness results, we can ensure the existence of a subsequence (which we do not relabel) and a function v C([, T ]; H w ) L 2 (, T ; V ) such that, setting u = v + z α, u n (t) z α (t; θ tn ω) u(t) z α (t; ω) in H, t [, T ], (4.13) u n (t) z α (t; θ tn ω) u(t) z α (t, ω) in L 2 (, T ; V ), (4.14) d dt [u n( ) z α ( ; θ tn ω)] d dt [u( ) z α( ; ω)] in L 4/3 (, T ; V ), (4.15) u n u in L 2 (, T ; H), (4.16) u n (t) u(t) in H a.e. t (, T). (4.17) Because of (3.2) and convergence in the above senses, it is easy to check that u(t) satisfies (3.5) with initial data z = lim u n (). Passing to the limit, inequality (3.3) is also straightforward. Following Ball, writing (3.4) for each term in the subsequence and then taking limits on the left using the weak lower semicontinuity and on the right using (4.17) we obtain (3.4) for u(t) z α (t; ω). Thus u is a weak solution, and so by assumption u is continuous into H; therefore V 2 (u, ω)( ) is also continuous and V 2 (u n, θ tn ω)( ) V 2 (u, ω)( ) forall t. It implies that u n (t) u(t), which along with the weak convergence give us the required convergence of u n to u in H and (S4) holds.

10 P. Marín-Rubio & J.C. Robinson 5. Attractors for generalized stochastic semiflows Ball 4 proves the existence of a global attractor for a generalized semiflow if and only if the semiflow is pointwise dissipative [if there is a bounded set B such that for any ϕ G ϕ(t) B for all sufficiently large t] and asymptotically compact [if for any sequence ϕ j G with ϕ j () bounded, and for any sequence t j, the sequence ϕ j (t j ) has a convergent subsequence]. We prove a similar result here, but the details are different since we also have to take into account the random element. Following the result for single-valued RDS of Crauel 7 we will give a necessary and sufficient condition for the existence of an attractor for our generalized stochastic semiflow. We say that a stochastic generalized semiflow is dissipative if it has a compact attracting set, namely a set K(ω) such that if D is a deterministic set, dist(φ(t, θ t ω)d, K(ω)) as t, where Φ is the generalized cocycle defined in (4.6). This implies (and motivates) the following concept: a generalized semiflow is said to be asymptotically compact if P -a.s., for every bounded set D and sequences t n and x n Φ(t n, θ tn ω)d, there exists a subsequence in {x n } which converges. We now show that under this condition the generalized semiflow has an attractor that is negatively invariant. We define the Ω-limit set of a fixed deterministic set D by Ω D (ω) = T > t>t Φ(t, θ t ω)d. We now prove some basic properties of these sets. Lemma 1 Let G be an asymptotically compact SGS. For any non-empty bounded deterministic set D, P -a.s. Ω D (ω) = n t n Φ(t, θ t ω)d is non-void, compact and the minimal set that attracts D: Moreover it is negatively invariant, that is lim dist(φ(t, θ tω)d, Ω D (ω)) =. (5.18) t Ω D (θ t ω) Φ(t, ω)ω D (ω) for all t. Proof. Let us first check that Ω D (ω) is non-void. Consider any element d D and a sequence {t n } with t n. By (S1) there exist solutions (ϕ n, θ tn ω) G such that ϕ n () = d; the dissipativity implies that there is a subsequence {ϕ µ (t µ )} that converges to an element z Ω D (ω), and hence Ω D (ω) is non-void. The set Ω D (ω) is obviously closed; we will see, by a Cantor diagonalization argument, that it is also compact: given a sequence {y n } Ω D (ω), there exist sequences t n and pairs (ϕ n, θ tn ω) G with ϕ n () D and d(ϕ n (t n ), y n ) < 1 n. Using the dissipativity again there exists a subsequence ϕ µ (t µ ) z and so y µ z Ω D (ω).

11 Attractors for the stochastic 3d Navier-Stokes equations We omit the proof of the attraction property of Ω D (ω) (by contradiction) and of its minimality, since these follow closely the equivalent arguments from the singlevalued case. For the negative invariance consider y Ω D (θ t ω). We have to check that y Φ(t, ω)ω D (ω). Since y Ω D (θ t ω) there exist a sequence t n and a sequence of pairs {(ϕ n, θ tn +tω)} G with ϕ n () D such that y n = ϕ(t n ) converges to y. Now, observe that x n = ϕ(t n t) Φ(t n t, θ (tn t)ω)d and, using the cocycle property from Proposition 2, that y n Φ(t, ω)x n. Using the dissipativity property there is a subsequence (which we do not relabel) x n x Ω D (ω). On the other hand, y n Φ(t, ω)x n implies that y n = ϕ n (t) with ϕ n () = x n. By (S4) there exist another subsequence (which we do not relabel) and a pair (ϕ, ω) G with ϕ() = x and ϕ n (s) ϕ(s) for all s. In particular, with s = t we see that ϕ n (t) = y n ϕ(t), and so y = ϕ(t) Φ(t, ω)x Φ(t, ω)ω D (ω). By taking the union of all possible Ω-limit sets, A(ω) = D bounded Ω D (ω), (5.19) we obtain the minimal negatively invariant compact set that attracts all bounded sets (in the sense of (5.18)). We term this set the minimal global attractor. Proposition 4 A generalized stochastic semiflow has a minimal global attractor A(ω) if and only if it has a compact attracting set. In this case A(ω) is given by (5.19). Proof. The condition is clearly necessary. All that remains to show that it is sufficient is to prove that A(ω) is compact and negatively invariant. Since A(ω) is closed, compactness follows since it must be a subset of the compact set K(ω) from the definition of dissipativity. The negative invariance is similarly straightforward, given the results of lemma 1: A(θ t ω) = D Ω D (θ t ω) Φ(t, ω) D Ω D (ω) Φ(t, ω) D Ω D (ω) = Φ(t, ω)a(ω); the first inclusion follows from the negative invariance of Ω D (ω), while the second is a consequence of the upper semicontinuity of Φ(t, ω) is u.s.c.. The final equality is valid since the upper semicontinuous image of a compact set is once again compact, and therefore closed. 6. Dissipativity and a global attractor for the 3d stochastic NSE It appears that in the context of multi-valued stochastic semiflows negative invariance is the correct notion. Without imposing additional conditions on Φ(t, ω) it is not possible to obtain a set that is also positively invariant. One appropriate condition in the stochastic case is the lower semicontinuity of Φ(t, ω) (see Caraballo et al. 6, for example). In his deterministic treatment Ball requires the forward image of Λ B to lie within B for all sufficiently large times; a version of his proof will give positive invariance of Ω B (ω) if there exists a sequence t i such that Ω B (θ r ω) B; but this seems an artificial condition in this context.

12 P. Marín-Rubio & J.C. Robinson For our particular example we can prove a stronger compactness condition than the existence of a compact attracting set; instead we prove the existence of a compact absorbing set, i.e. a compact set K(ω) such that P -a.s. for every bounded deterministic set D there is a time T D (ω) such that Φ(t, θ t ω)d K(ω) t T D (ω). Firstly, following Crauel & Flandoli 1, we prove the same result but with K(ω) replaced by a bounded set B(ω). Taking u() D, and setting t = in (3.3) we obtain u(t) z α (t; θ t ω) 2 e + 4 ( λ1+2c z α(s;θ tω) 8 L 4 ) ds u() z α (; θ t ω) 2 e σ ( λ 1+2C z α (s;θ t ω) 8 L 4 ) ds[ C 2 B z α (σ; θ t ω) 4 L 4 + α2 λ 1 z α (σ; θ t ω) 2 + f 2 V ] dσ. (6.2) We follow the idea given in step 5 in Crauel & Flandoli 1 among others. A simple ergodic argument (see Crauel & Flandoli 1 or lemma 7.2 in Flandoli & Schmalfuß 11 ) guarantees that 1 τ lim inf lim sup z α (s; θ τ ω) 8 α L4 ds =, (6.21) τ τ and hence for α sufficiently large there exists a t (ω) such that for all t t we have 2C 1 t In particular, for almost every ω we have z α (s; θ t ω) 8 L 4 ds < λ 1 2. (6.22) ( λ 1 + 2C z α (s; θ t ω) 8 L 4) ds λ 1t 2 for t t (ω). (6.23) We have a bound for the first term in right-hand side of (6.2) since u() is bounded by assumption, and z α (, θ t ω) = z α ( t, ω) has polynomial growth (by lemma 3.6 (ii), p.377 in Flandoli & Schmalfuß 11 or (1) in Crauel & Flandoli 1 ), i.e. z α (t, ω) j L lim 4 t t j = for j N. (6.24) The second term on the right-hand side of (6.2) does not depend on D, and we need an estimate that is valid for all t. In order to do that, we transform it by two changes of variables, σ t = s and r t = ρ, as follows: e σ ( λ1+2c z α(s;θ tω) 8 L 4 ) ds[ ] CB z 2 α (σ; θ t ω) 4 L + α2 z 4 α (σ; θ t ω) 2 + f 2 V dσ λ 1

13 = t Attractors for the stochastic 3d Navier-Stokes equations e ( λ 1+2C z α (ρ;ω) 8 s L 4 ) dρ[ ] CB z 2 α (s; ω) 4 L + α2 z 4 α (s; ω) 2 + f 2 V ds λ 1 By the continuity, polynomial growth and ergodic argument (6.21)-(6.23) (with another change of variables) for the process z α, we can proceed and finish the proof exactly as in Crauel & Flandoli 1. Thus, it follows that given u() in a bounded set, there exists a random variable r(ω) and a time t 1 (ω, u() ) such that u(t) r(ω) for all t t 1 (ω, u() ). Write B(ω) for the ball of radius r(ω) in H. Now we observe (cf. Ball) that the proof of proposition 3 also shows that G SNS is compact, i.e. given ω, a sequence of solutions (ϕ n, ω) G SNS with {ϕ n ()} bounded has a subsequence that converges for all t >. In particular this means that Φ(1, ω) is compact, P -a.s., and so K(ω) = Φ(1, θ 1 ω)b(θ 1 ω) is a random compact set. Due to its definition, K(ω) is absorbing, since and so for all t t 1 (θ 1 ω, D) Φ(t + 1, θ 1 t ω)d = Φ(1, θ 1 ω)φ(t, θ 1 t ω)d Φ(t + 1, θ 1 t ω)d Φ(1, θ 1 ω)b(θ 1 ω) = K(ω). Application of proposition 4 now yields a global attractor for the 3d stochastic NSE. 7. Conclusion We have extended the idea of a generalized semiflow due to Ball 4 to treat stochastic systems, and shown that such a generalized semiflow has a minimal global attractor if and only if it has a compact attracting set. One interesting open problem here is to fully characterize those systems for which the global attractor is positively, as well as negatively, invariant. 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 H implies the existence of a global attractor. Acknowledgments PM-R & JCR would both like to José Langa for suggesting this problem. 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 José Real and Tomás Caraballo for their advice. JCR is currently a Royal Society University Research Fellow, and would like to thank the Society for all their support. References

14 P. Marín-Rubio & J.C. Robinson 1. J. P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory (Springer-Verlag, 1984). 2. L. Arnold, Random Dynamical Systems (Springer, 1998). 3. A. V. Babin and M. I. Vishik, Attractors of Evolution Equations (North Holland, 1992). 4. J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci. 7 (1997) A. Bensoussan and R. Temam, Équations stochastiques du type Navier-Stokes, J. Funct. Analysis 13 (1973) T. Caraballo, J. A. Langa and J. Valero, Global attractors for multivalued random dynamical systems, Nonlinear Anal. 48 (22) H. Crauel, Random point attractors versus random set attractors, J. London Math. Soc. 63 (21) H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns. 9 (1997) H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields 1 (1994) 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 F. Flandoli and B. Schmalfuß, Weak solutions and attractors for three-dimensional Navier-Stokes equations with nonregular force, J. Dyn. Diff. Eqns. 11 (1999) C. Foias and R. Temam, Some analytic and geometric properties of the solutions of the Navier-Stokes equations, J. Math. Pures Appl. 58 (1979) 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 J.K. Hale, Asymptotic Behavior of Dissipative Dynamical Systems (Amer. Math. Soc., 1988). 15. E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups (Amer. Math. Soc., 1957). 16. O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations (Cambridge University Press, 1991). 17. V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal. 6 (1998) J. C. Oxtoby, Measure and Category (Springer-Verlag, 1971). 19. J. C. Robinson, Infinite-Dimensional Dynamical Systems (Cambridge University Press, 21). 2. G. Sell, Global attractors for the three-dimensional Navier-Stokes equation, J. Dyn. Diff. Eqns. 8 (1996) 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 Universität, 1992), pp R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics (Springer AMS, 1988).