Flattening, squeezing and the existence of random attractors

Flattening, squeezing and the existence of random attractors BYPETER E. KLOEDEN 1 AND JOSE ´A. LANGA 2, * 1 Institut fr Mathematik, Johann Wolfgang Goethe-Universitt, 60054 Frankfurt am Main, Germany 2 Departmento de Ecuaciones Diferenciales y Ana ´lisis Nume ´rico, Universidad de Sevilla, Apartado de Correos 1160, 41080 Sevilla, Spain The study of qualitative properties of random and stochastic differential equations is now one of the most active fields in the modern theory of dynamical systems. In the deterministic case, the properties of flattening and squeezing in infinite-dimensional autonomous dynamical systems require the existence of a bounded absorbing set and imply the existence of a global attractor. The flattening property involves the behaviour of individual trajectories while the squeezing property involves the difference of trajectories. It is shown here that the flattening property is implied by the squeezing property and is in fact weaker, since the attractor in a system with the flattening property can be infinite-dimensional, whereas it is always finite-dimensional in a system with the squeezing property. The flattening property is then generalized to random dynamical systems, for which it is called the pullback flattening property. It is shown to be weaker than the random squeezing property, but equivalent to pullback asymptotic compactness and pullback limit-set compactness, and thus implies the existence of a random attractor. The results are also valid for deterministic non-autonomous dynamical systems formulated as skew-product flows. Keywords: flattening; random dynamical systems; random attractors; squeezing property 1. Introduction The theory of dynamical systems has been successfully used from decades ago to analyse qualitative properties of many models of differential equations arising from Physics, Mechanics, Chemistry, Biology, etc. (See Hale 1988;Temam 1988; Ladyzhenskaya 1991;Vishik 1992;Chepyzhov & Vishik 2002.) More recently, some of these ideas have also been used to describe the asymptotic behaviour of random and stochastic differential equations (Schmalfuss 1992;Crauel & Flandoli 1994;Arnold 1998). In all cases, the concept of global attractor plays a crucial role. When the model is related to a system of partial differential equations, the existence of a global attractor is usually related to a kind of squeezing or flattening of the high modes in the evolution in time of trajectories, which leads to some of the most impressive results in the theory of infinite-dimensional dynamical Proc. R. Soc. A (2007) 463, 163–181 doi:10.1098/rspa.2006.1753 Published online 15 August 2006 * Author for correspondence ([email protected]). Received 9 March 2006 Accepted 5 June 2006 163 q2006 The Royal Society on May 15, 2016http://rspa.royalsocietypublishing.org/Downloaded from systems. In this work, we want to go further in the clarification of the meaning of these kind of flattening properties and its relation with the existence of attractors, both in the deterministic and the stochastic cases. It is often not difficult to show that a dynamical system given in terms of a specific differential equation has a bounded absorbing set. In finite-dimensions, such sets are compact and this is sufficient to ensure the existence of a global attractor. In the infinite-dimensional case, however, requiring an absorbing set to be compact is a severe restriction. An additional property to the existence of a bounded absorbing set is needed to ensure that there is a global attractor, such as the compactness, eventual compactness or asymptotic compactness of the flow operator (Hale 1988;Temam 1988;Ladyzhenskaya 1991;Vishik 1992;Rosa 1998; Robinson 2001). An alternative idea is the squeezing property, which was introduced by Foias & Temam in the context of the Navier–Stokes equations (Foias & Temam 1979; Foias et al. 1988) and is applicable to many other classes of dissipative partial differential equations (Eden et al. 1994;Temam 1988;Robinson 2001). In the squeezing property, a finite-dimensional subspace of what are called lower order modes is introduced and either the higher order modes are bounded by the lower modes or the solutions are squeezed together. The squeezing property, which also requires the existence of a bounded absorbing set, has been used with considerable success to establish interesting properties of dissipative dynamical systems, such as determining modes (Foias & Prodi 1967;Robinson 2001), the finite-dimensionality of global attractors (Constantin et al. 1985) and even to construct exponential attractors (Eden et al. 1994). A related idea, which we shall call flattening, was introduced by Ma et al. (2002) under the name Condition (G). It assumes the existence of a bounded absorbing set and requires the finite-dimensional modes to become uniformly bounded with the remaining higher order modes becoming sufficiently small. Ma et al. (2002) showed that it is equivalent to a form of asymptotic compactness in uniformly convex Banach spaces, and that it implies the existence of a compact attractor. In most cases, it is not difficult to verify because estimates for the flattening property are obtained in much the same way as those needed to show that there is a bounded absorbing set. However, a major difference from the squeezing property is that the resulting attractor need not be finite dimensional. In the first part of this paper, we will show that the flattening property is implied by the squeezing property in uniformly convex Banach spaces and then give counter examples that satisfy the flattening property but not the squeezing property—thus flattening is a weaker property than squeezing. In the second and main part of the paper, we will extend the idea of flattening to random dynamical systems and compare it with pullback squeezing, the corresponding generalization of the squeezing property to deterministic nonautonomous dynamical systems. Our results make no use of the topology of the autonomous driving system of the skew product flow, therefore, apply equally well to deterministic non-autonomous dynamical systems formulated as skew product flows, e.g. systems generated by reaction diffusion equations with temporally almost periodic coefficients. In particular, we do not need to assume the existence of a uniform absorbing set as do Wang et al. (in press), who extend the flattening concept to deterministic skew product flows, thus, our results also generalize theirs in this context. P. E. Kloeden and J. A. Langa164 Proc. R. Soc. A (2007) on May 15, 2016http://rspa.royalsocietypublishing.org/Downloaded from We will use the following measure of non-compactness. Definition 1.1. Let Xbe a metric space and Da bounded subset of X. The Kuratowski measure of non-compactness g(D)ofDis defined by gðDÞZinf dO0:Dadmits a finite cover by sets of diameter%d fg : The following summarizes some of the basic properties of this measure of non-compactness (e.g. Deimling 1985). Lemma 1.2. Let X be a Banach space and let gbe the measure of noncompactness. Then (i) g(D)Z0if, and only if,  D is compact. (ii) g(D 1 CD 2 )%g(D 1 )Cg(D 2 ). (iii) g(D 1 )%g(D 2 )for D 1 3D 2 . (iv) g(D 1 SD 2 )%max g(D 1 ),g(D 2 ). (v) gð DÞZgðDÞ. (vi) If F 1 IF 2 .are non-empty closed sets in X such that g(F n )/0as n/N, then TnR1Fnis non-empty and compact. In addition, let X be an infinite-dimensional Banach space with a decomposition XZX 1 4X 2 and let P: X/X 1 ,Q:X/X 2 be projectors with dim X 1 !N.Then (vii) g(B(e))Z2e, where B(e)is a ball of radius e. (viii) g(D)!efor any bounded subset D of X for which the diameter of QD is less than e. 2. The deterministic autonomous case The observed squeezing of high modes of the difference of trajectories in turbulent fluids has been formulated mathematically as the squeezing property by Foias & Temam (1979), and has been used to prove many interesting and beautiful results for the Navier–Stokes equations and similar types of dissipative dynamical systems. Definition 2.1 Squeezing property. Suppose that a semiflow Son a Banach space Xhas a bounded absorbing set Bin X. Let Pbe a projection onto a finitedimensional subspace of Xand QZIKP. Then for x,y2Beither kQðSð1ÞxKSð1ÞyÞk%kPðSð1ÞxKSð1ÞyÞk; i.e. the higher modes are bounded by the lower modes, or kSð1ÞxKSð1Þyk%dkxKyk; for some d2(0, 1), i.e. the solutions are squeezed together. In applications to the planar Navier–Stokes equations, for example, Pis usually taken as the projector onto the subspace of Xspanned by the first N eigenfunctions associated with the Stokes operator A, i.e. PuZPN iZ1ðu;wiÞwi, where fwigN iZ1is the orthonormal basis in Xconsisting of the eigenfunctions of A, and the operator Qis defined as QuZPN iZNC1ðu;wiÞwi. 165 Existence of random attractors Proc. R. Soc. A (2007) on May 15, 2016http://rspa.royalsocietypublishing.org/Downloaded from There is another, simpler feature of the dynamics in many dissipative systems in infinite-dimensional spaces, namely, the dynamics in the lower order modes becomes bounded and that in the higher modes becomes small. Ma et al. (2002) formulated this as Condition (G) and showed that it is sufficient for the existence of a global attractor. We will call it the flattening property here. Definition 2.2 Flattening property. Suppose that a semi flow Son a Banach space Xhas a bounded absorbing set Bin X. For any bounded set D3Xand for any eO0, there exists T e (D)O0 a finite-dimensional subspace X e of X, and a bounded projector P e :X/X e such that StRTeðDÞPeSðtÞDis bounded and kðIKPeÞSðtÞx0k!e;ctRTeðDÞ;x02D: The flattening and squeezing properties obviously seem to be closely related and we want to clarify this relationship. We will show that the flattening property is a weaker concept than the squeezing property. In §2a,b, we will prove that squeezing implies flattening, at least when the Banach space Xis uniformly convex, i.e. for all eO0thereexistsdO0 such that, given x,y2X,kxk,kyk%1, kxKykOe,then(kxCyk/2)!1Kd.Requiringa spacetobeuniformlyconvexisnotasevere restriction in applications, since this property is satisfied by all Hilbert spaces, the L p spaces with 1!p!N,and most Sobolev spaces W k,p with 1!p!N (see Bre ´zis 1983; section III.7). We also give counter examples that satisfy the flattening property but not the squeezing property. (a)Squeezing implies flattening Ma et al. (2002) introduced the following concept under the name of u-limit compact, but we will call it limit-set compact to avoid confusion in the stochastic setting later, where uis used in another context. Definition 2.3. A semi flow Son a Banach space is said to be limit-set compact if for every bounded set D3Xand eO0 there exists a T e (D)O0 such that g[ tRTeðDÞ SðtÞD 0 @1 A!e; where gis a measure of non-compactness defined on the subsets of X. They then prove (theorem 3.10 in Ma et al. (2002)) that a semi-dynamical system is flattening if it is limit-set compact, provided Xis a uniformly convex Banach space. The following result proves that the squeezing property is a sufficient condition for limit-set compactness. Thus, for uniformly convex Banach spaces X, flattening is indeed a weaker concept than squeezing. Lemma 2.4. Suppose that B ZB X (0,r), the ball in a Banach space X of radius r!0centred on the origin, is an absorbing set of a semi flow S on X. If S satisfies the squeezing property on B, then it is limit-set compact and thus has a global attractor in that B. Moreover, if X is a uniformly convex Banach space, then S has the flattening property. P. E. Kloeden and J. A. Langa166 Proc. R. Soc. A (2007) on May 15, 2016http://rspa.royalsocietypublishing.org/Downloaded from Proof. Without lost of generality, we can suppose Bis a positively invariant absorbing set, as, if not, we could consider StRtBSðtÞBas a new bounded absorbing set for some T B , such that S(t)B3Bfor all tRT B . Using the squeezing property and lemma 2.1 in Eden et al. (1994), the set S(1)Bcan be covered by k 0 balls of radius r/2, in other words Sð1ÞB3[ k0 iZ1 Bða1 i;r=2Þ: In turn, each of the balls Bða1 i;r=2Þcan be covered by k 0 balls of radius r/2 2 ,sothat Sð2ÞB3[ k2 0 iZ1 Bða2 i;r=22Þ: Iteratively, we get that SðnÞB3[ kn 0 iZ1 Bðan i;r=2nÞ: Now, given eO0, there exists n 0 such that r/2 n !efor all nRn 0 ; thus, S(n 0 )B can be covered by a finite number of balls of radius less than e. But note that SðtÞB3Sðn0ÞBfor all tRn0; so g[ tRn0 SðtÞB ! !e: Finally, if Xis a uniformly convex Banach space, theorem 3.10 in Ma et al. (2002) implies the system is flattening. & (b)Flattening does not imply squeezing A clear difference between flattening and squeezing is that the finitedimensional subspace X e of Xin the flattening property may depend on the choice of eO0, while the finite-dimensional projector Pand subspace PX in the squeezing property are fixed at the start. Systems with an infinite-dimensional global attractor are counter examples in which flattening holds but squeezing is impossible. By theorems 3.9 and 3.10 in Ma et al. (2002), the existence of a global attractor implies that the system is flattening, whereas the squeezing property is a sufficient condition for the finitedimensionality of a global attractor (Eden et al. 1994;Robinson 2001). Thus, the example in Chepyzhov & Vishik (2002; pp. 161) is flattening, but not squeezing. Actually, simpler examples involving infinite-dimensional systems of uncoupled ordinary differential equations (ODE) show this relationship much more directly. The infinite-dimensional space [2with the norm kxk2Zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X N iZ1jxij2 s;xZðx1;x2;.Þ2RN; is a Hilbert space (hence a uniformly convex Banach space). The infinitedimensional ODE dx1 dtZx1ð1Kx2 1Þ;dxi dtZKxi;iZ2;3;.;ð2:1Þ 167 Existence of random attractors Proc. R. Soc. A (2007) on May 15, 2016http://rspa.royalsocietypublishing.org/Downloaded from in [2satisfies the squeezing property and has the global attractor (12) AZ½K1;1!Y iR2f0g; which is a one-dimensional compact subset of [2. One the other hand, the infinitedimensional ODE (13) dxi dtZxiðiK2Kx2 iÞ;iZ1;2;.;ð2:2Þ in [2satisfies the flattening property but not the squeezing property. It has the global attractor AZY iR1½KiK1;iK1; which is an infinite-dimensional compact subset of [2. Remark 2.5. Chueshov & Laisecka (2004) (see also Khanmamedov (2006)) write a sufficient condition, related to a kind of squeezing in the difference of two trajectories for the asymptotic compactness of a deterministic dynamical system. As we will show later, there exists an equivalence between the flattening and the asymptotically compact property of a system, so that the condition in Chueshov & Laisecka (2004) would be also sufficient for the flattening property. It would be interesting to study if the contrary is also true or not. 3. Random dynamical systems Let ðU;F;PÞbe a probability space and let Xbe a Banach space. Arnold (1998) defined a random dynamical system (RDS) (q,f)onU!Xin terms of a metric dynamical system qon U, which represents the noise driving the system, and a co-cycle mapping f:R C!U!X/X, which represents the dynamics in the state space Xand satisfies the properties (i) f(0, u,x 0 )Zf 0 for all x 0 2Xand u2U, (ii) f(sCt,u,x 0 )Zf(s,q t u)f(t,u,x 0 ) for all s,tR0, x 0 2Xand u2U, (iii) (t,x 0 )1(t,u,x 0 ) is continuous for each u2U, and (iv) u1f(t,u,x 0 )isF-measurable for all ðt;x0Þ2R C!X. A metric dynamical system qhðU;F;P;qt;t2RÞis a family of measure preserving transformations qt:U/U;t2Rsuch that q 0 Zid U ,q t +q s Zq tCs for all t;s2R, the map (t,u)1q t uis measurable and qtPZPfor all t2R. Random dynamical systems are generated by finite-dimensional differential equations with random coefficients or stochastic differential equations with a unique and global solution as well as by some infinite-dimensional stochastic evolution equations. A family DZfDu;u2Ugof non-empty closed subsets of a Banach space Xis called a random closed set if the map u1dist(x,D u ) for each x2Xis measurable with respect to F. Such a family Dis said to be tempered if D(u)3B X (0, r(u)) P-a.s., where r(u)isatempered random variable, i.e. satisfying lim jtj/CN rðqtuÞ eejtjZ0;P-a:s:; for all eO0. P. E. Kloeden and J. A. Langa168 Proc. R. Soc. A (2007) on May 15, 2016http://rspa.royalsocietypublishing.org/Downloaded from The following concept of a random attractor for random dynamical systems (Schmalfuss 1992;Crauel & Flandoli 1994;Arnold 1998;Crauel et al. 1995; Flandoli & Schmalfuss 1996) extends that of a global attractor in autonomous deterministic systems to random dynamical systems. Definition 3.1. A random compact set AZfAu;u2Ugof a Banach space X is said to be a random attractor for an RDS (q,f)inXif it is f-invariant,in other words fðt;u;AuÞZAqtu;tR0;P-a:s:; where fðt;u;AuÞZSa2Aufðt;u;aÞ;and pullback attracts every tempered random set ^ DZfDu;u2Ugin Xin the sense that lim t/NdistXft;qKtu;DqKtu  ;Au  Z0; where dist X ($,$) denotes the Hausdorff semidistance between subsets of X. The following result (see Flandoli & Schmalfuss 1996) ensures the existence of a random attractor for an RDS on a Banach space. A partial aim of the present paper is to establish the existence of a random attractor under weaker assumptions on the co-cycle mapping. Theorem 3.2. Let (q,f)be an RDS on a Banach space X such that f(t, u,$): X/X is a compact operator for each fixed tO0and u2U. If there exists a tempered random set ^ BZfBu;u2Ugand a T^ D;uR0such that ft;qKtu;DqKtu  3Bu;ctRT^ D;u; for every tempered random set ^ DZfDu;u2Ugin X, then the RDS (q,f)has a random pullback attractor ^ AZfAu;u2Ug. A set ^ BZf^ Bu;u2Ugin Xsatisfying the properties required by theorem 3.2 is called a (random) pullback absorbing set of the RDS (f,q)inX. The pullback attraction in the definition of a random attractor is a form of pathwise convergence. It is known that a random attractor also attracts in the usual forwards sense in the weaker convergence in probability, i.e. given eO0, lim t/CNPdistXfðt;u;DuÞ;Aqtu  Oe  Z0: 4. Flattening in random dynamical systems Definition 4.1. An RDS (f,q) on a Banach space Xis said to be pullback flattening if for every tempered random bounded set BZfBu;u2Ugin X,eO0 and u2Uthere exists a T0ðB;e;uÞO0 and a finite-dimensional subspace X e of X such that (i) StRT0Pefðt;qKtu;BqKtuÞis bounded, and (ii) kðIKPeÞStRT0fðt;qKtu;BqKtuÞ  kX!e, where P e :X/X e is a bounded projection and (ii) is understood in the sense that kðIKPeÞfðt;qKtu;x0ÞkX!efor all x02BqKtuand tRT 0 . Definition 4.2. An RDS (f,q) on a Banach space Xis said to be pullback limitset compact if for every tempered random bounded set BZfBu;u2Ugin X, 169 Existence of random attractors Proc. R. Soc. A (2007) on May 15, 2016http://rspa.royalsocietypublishing.org/Downloaded from eO0 and u2Uthere exists a T1ðB;e;uÞO0 such that g[ tRT1 fðt;qKtu;BqKtuÞ ! !e; where gis a measure of non-compactness defined on the subsets of X. Definition 4.3. A RDS (f,q) on a Banach space Xis said to be pullback asymptotically compact in Xif for every tempered random bounded set BZfBu; u2Ugin X, each u2Uand sequences t k /Nand xk2BqKtku,kZ1, 2, ., the set ffðtk;qKtku;xkÞ,kZ1, 2, .} is precompact in X. Remark 4.4. Brzez ´niak & Li (2002,in press) define an analogous concept of a pullback asymptotically compact RDS with respect to deterministic bounded sets B. In particular, their weaker definition than ours is referred to deterministic bounded sets rather than tempered sets. They are then able to prove that omegalimit sets associated to B are non-empty strictly invariant compact random sets which attract B. Our results, with a stronger definition, which is commonly satisfied in applications, lead to the existence of random attractors and so, in particular, they imply this one. We will prove the following theorems. Theorem 4.5. Suppose that, X is a uniformly convex Banach space. The following three properties of a random dynamical system on X are equivalent: (i) pullback flattening, (ii) pullback limit-set compact,and (iii) pullback asymptotically compact. Proof. We will prove that (i) 0(ii), then (ii) 0(iii) and finally that (iii) 0(i). —the RDS is pullback flattening 0the RDS is pullback limit-set compact Suppose that the RDS is pullback flattening and consider an arbitrary bounded random set BZfBu;u2Ugin X. Then for each u2U g[ tRT1 fðt;qKtu;BqKtuÞ ! %gP[ tRT1 fðt;qKtu;BqKtuÞ ! ! CgðIKPÞ[ tRT1 fðt;qKtu;BqKtuÞ ! ! %0CgðBXð0;eÞÞZ2e; where B X (0,e) is the open ball in Xwith centre 0 and radius e. Hence, the RDS is pullback limit-set compact. —the RDS is pullback limit-set compact 0the RDS is pullback asymptotically compact. Suppose that the RDS is pullback limit-set compact and let BZfBu;u2Ug be a tempered random bounded set in X, for each u2Uand eO0 there exists P. E. Kloeden and J. A. Langa170 Proc. R. Soc. A (2007) on May 15, 2016http://rspa.royalsocietypublishing.org/Downloaded from T1ðB;e;uÞO0 such that g[ tRT1 fðt;qKtu;BqKtuÞ ! !e: Now, if we choose e n d1/nand define tndT1ðB;1=n;uÞfor nZ1, 2,., with 0! t 1 !t 2 !., we get that g[ tRtn fðt;qKtu;BqKtuuÞ ! !1 n;nZ1;2;.: From the properties of our measure of non-compactness, it follows that g[ tRtn fðt;qKtu;BqKtuÞ ! !1 n;nZ1;2;.: Now the bounded sets AnðB;uÞZStRtnfðt;qKtu;BqKtuÞare nested, i.e. with AnC1 ðB;uÞ3AnðB;uÞfor nZ1, 2,., thus by lemma 2.11 of Wang et al. (in press) their intersection is a non-empty compact subset of X, in other words :sANðB;uÞZ\ nR1 AnðB;uÞZ\ nR1[ tRtn fðt;qKtu;BqKtuÞ:ð4:1Þ Now consider arbitrary sequences t k /Nand xk2BqKtku,kZ1, 2,., where BZfBu;u2Ugis a tempered random bounded set in X. Define FjðuÞd ffðtk;qKtku;xkÞ,kRj} and (discarding a finite number of kif necessary) define njdmaxfn2N:tn%tjg; thus n j /Nas j/N. fðtk;qKtku;xkÞ2fðtk;qKtku;BqKtkuÞ3AnjðB;uÞ; for all kRjand jZ1, 2,.. Thus, FjðuÞ3FjðuÞ3AnjðB;uÞfor all kRjand jZ1, 2,.,so gFjðuÞ  !1 nj /0asj/N: But F jC1 (u)3F j (u) for jZ1, 2, ., i.e the sets are nested, thus they have their intersection non-empty and compact with :s FðuÞd\ jR1 FjðuÞ3ANðB;uÞ: From this we conclude that the set F1ðuÞdffðtk;qKtku;xkÞ;kR1gis precompact, and thus that the RDS is pullback asymptotically compact. —the RDS is pullback asymptotically compact 0the RDS is pullback flattening. Suppose that RDS is pullback asymptotically compact and let BZBu;u2U be a tempered random bounded set in X. Let u2Ube arbitrary but fixed and consider the set ANðB;uÞZ\ nR1 AnðB;uÞZ\ nR1[ tRtn fðt;qKtu;BqKtuÞ: It is clear that, a point a2ANðB;uÞif, and only if, there are sequences t k /N and xk2BqKtku,kZ1, 2,., such that f(t,q Kt u,a k )/aas k/N. 171 Existence of random attractors Proc. R. Soc. A (2007) on May 15, 2016http://rspa.royalsocietypublishing.org/Downloaded from (a)A stochastic reaction–diffusion equation The first known example where a random squeezing property is satisfied appears in Debussche (1997), where the following stochastic reaction-difussion equation is studied: let D3Rm,m%3, be an open bounded set with regular boundary. We consider duZðDuCfðuÞÞdtCX M iZ1 jidWi t;ð6:1Þ with u(x,t)Z0 for x2vD, where the j i 2D(D), the Wi tare independent two-sided scalar Wiener processes on the probability space ðU;F;PÞand f(u) is a polynomial with negative higher order coefficient. From Debussche (1997; section 3.1) we easily conclude the random squeezing property, so that theorem 5.2 holds, and thus the flattening property is satisfied for this example. (b)Random attractor in V for stochastic two-dimensional Navier–Stokes equations We use the notation from Temam (1988), in particular the space Hwith norm j$jand the space Vwith norm k$k, and consider the two-dimensional stochastic (or random) Navier–Stokes equations (RNSE) with scalar additive noise vu vtCuVuKnDuCgrad pZfCjdWt;V$uZ0;ð6:2Þ in a two-dimensional torus Oin R2with periodic boundary conditions, where W is a two-sided scalar Wiener process and j2D(A) and, for simplicity, we assume jis an eigenfunction of the Stokes operator. We write this as du dtZAu CBðu;uÞCfCjdWt; and assume that the forcing term fdoes not depend on time. To set our problem in the usual abstract framework, we consider the following spaces: VZu2CN 0ðOÞðÞ 2;div uZ0  : HZthe closure of Vin ðL2ðOÞÞ2with norm j$j, and inner product ($,$) where for u;v2ðL2ðOÞÞ2, ðu;vÞZX 2 jZ1ðO ujðxÞvjðxÞdx: VZthe closure of Vin ðH1 0ðOÞÞ2with norm k$k, and associated scalar product (($,$)), where for u;v2ðH1 0ðOÞÞ2, ððu;vÞÞZX 2 i;jZ1ðO vuj vxi vvj vxi dx: It follows that V3HhH03V0, where the injections are dense. Let  zðtÞdenote the entire solution (i.e. Ornstein–Uhlenbeck process) of the scalar linear SDE d ztZKa ztdtCdWt;ð6:3Þ P. E. Kloeden and J. A. Langa178 Proc. R. Soc. A (2007) on May 15, 2016http://rspa.royalsocietypublishing.org/Downloaded from for some aO0, in other words  ztZðt KN eKaðtKsÞdWs; and write zðtÞZj zt. Let vZuKz. Then, vsatisfies the random PDE dv dtZAv CBðu;uÞCfCazCAz:ð6:4Þ Let Qbe the projection onto the subspace spanned by eigenfunctions of A,f j , with jRNfor some N. Write qZQv. Multiplying by KAq and integrating over U, we get 1 2 d dtkqk2CnjAqj2ZKðBðu;uÞ;AqÞKðf;AqÞCaðj;AqÞ zKðAj;AqÞ z %Cjuj1=2kukjAuj1=2jAqjC2 nj zj2ðjfj2Cjjj2CjAjj2ÞC3n 8jAqj2 %C1jujkuk2jAujC2 nj zj2ðjfj2Cjjj2CjAjj2ÞCn 4jAqj2: Write RðtÞZC1jujkuk2jAujC2 nj zj2ðjfj2Cjjj2CjAjj2Þ. Then we have d dtkqk2CnlNkqk2%RðtÞ; which integrates to give kqðtÞk2%kqðsÞk2eKnlNðtKsÞCeKnlNtðt s enlNtRðtÞdt; from which it follows that kqðtÞk2%kqðsÞk2eKnlNðtKsÞCeKnlNtðt s enlNtRðtÞdt %kqðsÞk2eKnlNðtKsÞCeKnlNtðt s enlNtdt  1=2ðt s enlNtRðtÞ2dt  1=2 %kqðsÞk2eKnlNðtKsÞC1 ffiffiffiffiffiffiffiffi nlN peKnlNtðt s enlNtRðtÞ2dt  1=2 : The higher modes inequality follows from this last inequality and the fact that the random integral has finite expectation using the results in section 5.2 of Flandoli & Langa (1999). Thus, the stochastic two-dimensional Navier–Stokes has a random attractor in V, rather than just in Has shown elsewhere in the literature. 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