On the random kick-forced 3D Navier-Stokes equations in a thin domain

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1 On the random kick-forced 3D Navier-Stokes equations in a thin domain Igor Chueshov and Sergei Kuksin November 1, 26 Abstract We consider the Navier-Stokes equations in the thin 3D domain T 2 (, ), where T 2 is a two-dimensional torus. The equation is perturbed by a non-degenerate random kick-force. We establish that, firstly, when 1 the equation has a unique stationary measure and, secondly, after averaging in the thin direction this measure converges (as ) to a unique stationary measure for the Navier-Stokes equation on T 2. Thus, the 2D Navier-Stokes equations on surfaces describe asymptotic in time and limiting in statistical properties of 3D solutions in thin 3D domains. MSC: primary 35Q3; secondary 35R6, 76D6, 76F55. Keywords: Navier-Stokes equations; thin domains; random kicks, convergence in law, stationary measure. 1 Introduction In this paper we study statistical properties of the Navier-Stokes equations (NSE) perturbed by a random force in a thin three-dimensional domain. For the sake of definiteness and for simplicity we consider the case of the free periodic boundary conditions. Namely, let O = T 2 (, ), where T 2 is the torus T 2 = R 2 /(l 1 Z l 2 Z) and (, 1]. Let x = (x, x 3 ) = (x 1, x 2, x 3 ) O, and let u(x) = (u 1 (x), u 2 (x), u 3 (x)), x O, stands for a vector field on O. We consider the NSE in O : t u ν u + 3 u j j u + p = f in O (, + ), (1.1) j=1 div u = in O (, + ) and O u j dx =, j = 1, 2, (1.2) u(x, ) = u (x) in O. (1.3) Department of Mechanics and Mathematics, Kharkov National University, 4 Svobody Sq., Kharkov 6177, Ukraine; chueshov@univer.kharkov.ua Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, UK; kuksin@ma.hw.ac.uk 1

2 For the second assumption in (1.2) to hold we assume that the force f satisfies the same relation: O f j dx =, j = 1, 2. The equations are supplemented with the following boundary conditions: x T 2 (i.e., u is (l 1, l 2 )-periodic with respect to (x 1, x 2 ) ), and u 3 =, x3= 3u j x3= =, j = 1, 2, (1.4) u 3 =, x3= 3u j x3= =, j = 1, 2. (free boundary conditions in the thin direction) The force f = f ω (x, t) is assumed to be a bounded random kick-force: f = k= η,ω k (x)δ kt (t), (1.5) where T > is a fixed number, δ kt (t) is the δ-function in t, concentrated at kt, and the kicks η,ω k, k, are independent identically distributed bounded random variables in V. Here and below V is the space of divergence-free H 1 -smooth vector fields on O, satisfying (1.2) and (1.4) (see Section 2.1 for the exact definition). The random kick-forced 2D NSE and similar to them equations, perturbed by random kick-forces or white in time forces, have been studied recently by many authors, see [8, 14, 2, 7, 13, 15, 16, 17] and references therein. Under suitable hypotheses, concerning the random force, the existence of a unique stationary measure for the corresponding Markov process has been established and properties of this measure have been studied. The results obtained allow to study ergodic properties of this equation and make it possible to justify rigorously basic hypotheses in the theory of 2D space-periodic turbulence (we refer to the survey in [13] for details). However all these results deal with two-dimensional models only. Our main goal in this paper is to extend some of results, available for the 2D case, to the three-dimensional NSE, much more realistic from the applied point of view. By many well-known reasons we cannot do it in full generality. Our main restriction is the so-called thin domain hypothesis which allows us to profit from the recent developments in the theory of PDE in thin domains. The study of global existence of smooth solutions for the NSE in thin three-dimensional domains began with the papers of Raugel and Sell [2, 21], who proved global existence of strong solutions for large initial data and forcing terms in the case of periodic conditions (PP) or mixed conditions (PD), i.e. periodic conditions in the vertical thin direction and homogeneous Dirichlet conditions on the lateral boundary. After these publications a number of papers by various authors followed, where the results for (PP) were sharpened [1, 18, 19] and extended to the cases of Dirichlet [1], and other boundary conditions [22], as well as to thin spherical domains [23] and thin two-layer domains [4]. See also [11] for some improvements of all these results for the (PP), (PD) and even for free (FF) boundary conditions. All these results deal with the force f which is L in time. In Section 5.1 we use the results from [22] to show that there exist a large set B of admissible initial data u in V and a large class of admissible kicks η,ω k for which problem (1.1) (1.5) possesses a global unique strong solution u ω (x, t; u, ) for all small enough 1. Problem (1.1) (1.5) is closely related to the 2D NSE on T 2 (see, e.g., [11] or [22]). To describe this relation, for any integrable vector-field u(x) we define its averaging in the thin 1 We speak about large data in the sense, standard for the theory of the 3D NSE in a thin domain (see, e.g., [2] or [22]). Namely, initial data u and kicks η,ω k are admissible if for any C > there exists > such that 1 u 2 dx C and 1 η,ω k 2 dx C for. O O In particular, initial data and kicks with finite C 1 -norms are admissible. 2

3 direction x 3 by the formula (M u) j (x) = 1 u j (x, η) dη, j = 1, 2, (M u) 3 (x) =, (1.6) where x = (x, x 3 ) O. The operator M defines an orthogonal projector in V. So V = M V N V, where N = I M. (1.7) Since M u is an x 3 -independent vector function with trivial third component, then it may be identified with a 2D vector-field on T 2. Accordingly, we identify M V with the space { } Ṽ = u H 1 (T 2 ; R 2 ) : div u =, u dx =. (1.8) T 2 Using the result from [22] (see Theorem 2.2 below for the exact statement) one can show that if u and η,ω k are admissible and M u ṽ, M η,ω k ηk ω as for each k and ω, then M u ω (x, t; u, ) v ω (x, t) as. Here u ω (x, t; u, ) is a strong solution to (1.1)-(1.5) and v ω is a solution for the 2D NSE: t v ν v + where f is the 2D kick-force 2 v j j v + p = f in T 2 (, + ), (1.9) j=1 div v = in T 2 (, + ) ; T 2 v(x, t) dx =, (1.1) v(x, ) = ṽ (x ) in T 2, (1.11) f = k η ω k (x ) δ kt (t). (1.12) Here and below the prime indicates that we regard the differential operator as an operator with respect to the variable x = (x 1, x 2 ). In this paper we are concerned with asymptotical in t statistical properties of solutions for the 3D NSE (1.1)-(1.5) with initial data in the admissible set B and with their relations to statistical properties of solutions for the corresponding 2D problem (1.9)-(1.12). In our first main result (see Theorem 5.1 and Corollary 5.4) we assume that the kicks satisfy some non-degeneracy conditions and the estimates 2 M η k Ṽ C(log 1 ) σ, N η k, C γ, where σ < 1 2, γ < 1 2 and, is the L 2 -norm on O with respect to the normalised measure 1 dx. We prove that for any τ T on the set B of admissible initial data there exists a unique Borel measure µ τ which attracts exponentially fast distribution of all (admissible) solutions for the 3D NSE (1.1)-(1.5), evaluated at time t = kt + τ, k. The measure µ τ is called the stationary measure for the process k u(, kt + τ). This result is an 3D analogy of the corresponding assertion for the 2D NSE. 3 Its proof is based on application of the abstract theorem from [15, 16] (theorem s statement is given in Section 3). The main difficulty in applying the theorem is to check that the flow-maps of the free 3D NSE possess 2 Note that the second estimate means that the non-2d component N ηk (x) of a k-th kick is such that its gradient may be as big as γ, but the function itself is small and is bounded by γ = 1 γ. 3 The 2D result is first proved in [14], apart from the fact that the rate of convergence is exponential. The exponential rate of convergence was established in [15, 16] and [17]. Detailed discussion and more references see in [13]. 3

4 the squeezing property with respect to a finite number of leading modes, see Theorem 4.2 below. Theorem 5.1 means that any solution u(x, t) of the 3D NSE defines the exponentially mixing processes k u(, kt + τ) V, k =, 1,..., parameterized by τ [, T ]. The mixing property implies that each solution u satisfies the Strong Law of Large Numbers: and lim t 1 t t lim N 1 N N 1 j= f ( u(, kt + τ) ) = f(u)µ τ (du) a.s., f(u(, s)) ds = f(u) µ (du) a.s., µ = 1 T T µ τ dτ. Here f is a locally Lipschitz functional on V (or on a higher order Sobolev space if the kicks are sufficiently smooth). The second convergence follows from the first one. Concerning the first convergence we note that it follows from Theorem 5.1 by exactly the same argument as in [13], Section 8. Thus, for flows in a 3D domains O, stirred by a non-degenerate kick-force (1.5), our results justify two basic hypothesis of the statistical hydrodynamics: firstly, statistical properties of any flow u(t, x) fast approach a unique statistical equilibrium, described by a stationary measure, secondly, time-averages of observable quantities coincide with their averages in ensemble. In particular, the correlation tensor of any flow converges to the correlation tensor of the stationary measure: E ( u i (x, kt + τ)u j (y, kt + τ) ) ( u i (x)u j (y) ) µ τ (du) as k, B (1.13) for any τ [, T ], i, j {1, 2, 3}, x, y O, and to calculate the correlation tensor of the measure one can replace the average in ensemble by average in time. We also note that the mixing, established in Theorem 5.1, implies that for any functional f as above the processes N k f(u(, kt + τ)) and R t f(u(, t)) satisfy the Central Limit Theorem, cf. [13], Section 9. This result justifies for the 3D flows which we consider the well known property of the 3D turbulence, stating that on large time-scales observable quantities behave as Gaussian random variables. Our second result (see Theorem 5.5 and Corollary 5.7) deals with limiting in properties of the stationary measures µ τ. There we assume that the kicks are nondegenerate and satisfy the estimates M η k Ṽ C, N η k, C, where C is a fixed constant. Let us set ϑ τ = M µ τ (this is a Borel measure on the space Ṽ, defined by the relations ϑ τ (Q) = µ τ (M 1 (Q)). We show that if M η,ω k converge to ηk ω as sufficiently fast, then the measures ϑ τ converge to the measures ϑ τ. Here ϑ τ is the unique stationary measure for the process k v(, kt + τ) Ṽ, where v(x, t) is a solution for the kick-forced 2D NSE (1.9), (1.1) with f as in (1.12). As a consequence of these two results we obtain in Theorem 5.8 that for any admissible solution u(x, t) of the 3D NSE the distributions of (M u)(x, t) in Ṽ converges uniformly in t to the distribution of v(x, t), where v(x, t) solves the 2D problem (1.9) (1.12). This result is much stronger than its deterministic counterpart (recalled in Section 2.1 as Theorem 2.2), where the convergence is uniform only on finite time-intervals. 4

5 The assertions of Theorems 5.1 and 5.5 jointly show that under the iterated limit first t then the statistical properties of solutions for the 3D NSE (1.1)-(1.5) converge to those, described by the (unique) stationary measures ϑ τ for the 2D NSE (1.9)-(1.1). For example, applying first Theorem 5.1 and next Theorem 5.5 to the energy functional 1 2 u(x) 2 dx we can prove that lim lim 2 2 ( ) 2dx k 2 E u j (x, x 3, kt + τ) dx 3 = 1 j=1 T 2 2 H v 2 H ϑ τ (dv) for any solution u of (1.1)-(1.5) with initial data u = u from the admissible set B. This means that the long-time limit of the averaged energy of the horizontal component of the 3D Navier-Stokes flow in O can be asymptotically calculated from the corresponding 2D model by means of the ensemble averaging. Similar relations hold for full 3D energy and enstrophy. See discussion in Section 7; also see there for more examples. Finally we note that assertions, similar to Theorems 5.1 and 5.5, remain true with the same proofs for the randomly kicked NSE in the thin spherical layer S 2 (, ) (see [23] for corresponding deterministic results). This boundary-value problem may be used to model statistical behaviour a planet s atmosphere: the free boundary condition on the sky S 2 {} models the effect of gravity which keeps the atmosphere close to the planet, and the free boundary condition on S 2 {} models interaction with the surface 4. The paper is organised as follows. In Section 2 we firstly recall the deterministic results for the 3D NSE (1.1)-(1.4) with a regular force f which we use in the further considerations. Then we define the kick-forced model, and describe our main hypotheses concerning the kicks in Assumption (D). In Section 3 we quote an abstract result (see Theorem 3.2) on random kick-forced evolutions, established in [15, 16]. Section 4 contains the statement of several assertions which constitute the main ingredients in the application of Theorem 3.2 to problem (1.1) (1.5). The proofs are rather technical and defer to Section 6. Our main results (Theorem 5.1 and Theorem 5.5) are formulated and proved in Section 5. In Section 7 we discuss some hydrodynamical consequences of our results. In Appendix we briefly describe spectral properties of the 3D Stokes operator with the boundary conditions (1.4). Notation. We denote the integral of a function f against a measure µ as f(u) µ(du), or as f, µ, or as µ, f. The symbol indicates the weak convergence of Borel measures. Dξ stands for the distribution of a random variable ξ. A map between Banach spaces is called locally Lipschitz if its restriction to any bounded subset of the domain of definition is Lipschitz. Acknowledgement. We are thankful to the London Mathematical Society for the financial support of the visit of the first author to Edinburgh, when our research started. 2 The model Our main goal in this section is to describe the random kick-forced 3D model. We start with a short survey of known deterministic results. 2.1 Deterministic 3D Navier-Stokes equations on a thin domain In this subsection we introduce the main functional spaces and collect several known results concerning the 3D NSE (1.1)-(1.4) with a regular force f. We mainly follow the approach presented in [22]. 4 If we replace the free boundary condition on S 2 {} by the non-slip condition u S 2 {} =, then the analog of Theorem 5.1 remains true, while the limit in Theorem 5.5 trivialises since now the solution goes to zero with in an appropriate norm (by the same argument as in Theorem 5.1 in [22]). 5

6 Let W be the space of divergence-free vector fields u = (u j ) j=1,2,3 on O such that u [ H 2 (O ) ] 3, O u j dx =, j = 1, 2, and condition (1.4) is satisfied. Let V (respectively, H ) be the closure of W in [ H 1 (O ) ] 3 (respectively, in [ L 2 (O ) ] 3 ). We denote by the L 2 -norm in H and provide V with the norm u u = [a (u, u)] 1/2, where a (u, v) = 3 j=1 O u j v j dx. We denote by A the Stokes operator, defined as an isomorphism from V onto its dual V by the relation (A u, v) V,V = a (u, v), u, v V. This operator extends to H as a linear unbounded operator with the domain D(A ) = W. Let Π be the Leray projector on H in (L 2 (O )) 3. Then for every u D(A ) = W. Now we consider the trilinear form b (u, v, w) = (A u)(x) = ( Π u)(x), x O, 3 j,l=1 O u j j v l w l dx, u, v D(A ), w (L 2 (O )) 3. It defines the bilinear operator B : V V by the formula (B (u, v), w) V,V = b (u, v, w), u, v, w V. Now the system (1.1) (1.4) can be written in the form u + νa u + B (u, u) = f, u() = u. (2.1) The following result concerning this system is known. Theorem 2.1 ([22]) Assume that u V, f L (R + ; H ) and where R() satisfies u + sup f(t, ) R(), (2.2) t lim θ R() = (2.3) with some θ (, 1/2). Then there exists a positive constant = (R), depending on the parameters of problem (1.1) (1.4), such that for (, ], problem (1.1) (1.4) has a strong solution u C([, T ); V ) L 2 ((, T ); W ), T >. This solution is unique in the class of weak Leray solutions. 6

7 Let us consider the 2D NSE (1.9)-(1.11). Clearly if v(x, t) is a solution of (1.9)-(1.11), then u(x, x 3, t) = (v 1, v 2, ) t (x, t) satisfies (1.1)-(1.4), where f = ( f, ) and u = (ṽ, ). (2.4) On the contrary, let u(x, t) be a solution of (1.1)-(1.3). Then the 2D vector-field M u (see (1.6)) converges to a solution of (1.9)-(1.1) when. To state the corresponding result we define the space Ṽ as in the Introduction (see (1.8)) and define the space H as the L 2 -space of divergence-free vector functions on T 2 with zero mean-value. Theorem 2.2 ([22]) Let the hypotheses of Theorem 2.1 be in force. Assume in addition that M u Ṽ and sup t R M f(t) H are bounded uniformly in and that there exist f L (R + ; H) and ṽ Ṽ such that lim M f(t) = f(t) for a.e. t, and lim M u = ṽ in the sense of weak convergence in H. Then for any T > we have lim M u( ) = v( ) in C([, T ]; H) L 2 ((, T ); Ṽ ), (2.5) where v(t) solves the 2D NSE (1.9) (1.11). Theorem 2.1 allows to define the flow-maps S T, T : S T : { u R()} V, S T u = u(t ), where u(t) solves the NSE (1.1) (1.4) with f. Well known properties of the NSE (see [5]) imply that for any T > and k N the map S T : { u R()} V H k (O ; R 3 ) is Lipschitz. (2.6) We denote by {S T, T } the flow-maps of the 2D NSE (1.9)-(1.11) with f. They are continuous in Ṽ and extend to continuous transformations of H. Due to (2.4), S T Ṽ = S T T,. Similar to (2.6), for any T > and k N we have the map S T : H H H k (T 2 ; R 2 ) is locally Lipschitz. (2.7) 2.2 Random kick-forced 3D NS model In this subsection we describe our model. We consider problem (1.1) (1.4) with a random external force which is a generalised vector-function of the form f(x, t) = ηk(x) δ kt (t), η k H k, (2.8) k=1 where T > is fixed and δ kt (t) is a δ-function concentrated at kt. Forces of this form are called kick-forces, and the functions ηk (x) are called kicks. Corresponding solutions of (1.1) (1.4) are discontinuous in t. We normalise them to be continuous from the right. Then the solution u of the problem (1.1)-(1.4) is a solution of the free (unforced) NSE for t kt, k Z, 7

8 at t = kt it has the jump η k (x). So the dynamics of the kick-forced NS model can be described by the following relations: u(x, ) = u (x), u(x, (k + 1)T ) = S T u(x, kt ) + ηk+1 (x) for k =, 1, 2,..., (2.9) u(x, kt + τ) = S τ u(x, kt ) if τ < T, k =, 1, 2,.... We refer to [13, Sect.2.6] for some details concerning description of a kick model, based on the 2D NS equations. Our main hypothesis concerning the kicks {ηk } is the following: (D) The kicks η1, η2,... depend on and are V -valued random variables, independent and identically distributed, defined on a probability space (Ω, F, P). Using the basis {e λj, e Λ j, j 1} (see Appendix) we write them in the form η k = η,ω k = j b jξ,ω jk e λ j (x) + j ˆb ˆξ,ω j jk e Λ (x). (2.1) j Here b j, ˆb j are non-negative real numbers and ξ jk, ˆξ jk are independent (scalar) random variables such that Dξjk = p j(x) dx, D ˆξ jk = ˆp j(x) dx (Dξ stands for the distribution of a random variable ξ). The random variables and the densities satisfy the following properties: p j (x) = p j (x) = for all x 1 and for every and j; each p j and ˆp j is a function of bounded total variation; for each γ > and every j and we have γ γ p j(x) dx >, γ γ ˆp j(x) dx >. Example: each ξjk (each ˆξ jk ) is a random variable, uniformly distributed on a segment [a j, b j ] (on [â j, ˆb j ]), where 1 a j < < b j 1 and 1 â j < < ˆb j 1. Remark 2.3 For a fixed the hypotheses, imposed on the kicks, are exactly the same as in Condition (D) in [15, 16] but written with respect to the basis {e λj, e Λ j, j 1}. Due to the first assumption in (D), concerning the densities p j and p j, without loss of generality we can assume that the random variables ξjk and ˆξ jk are bounded by 1 for all ω. Therefore relations (8.2) and (8.4) imply that M η k 2 (b j) 2 =: B, N η k 2 (ˆb j) 2 =: ˆB, (2.11) for every, k and ω, where M is defined by (1.6) and N = I M. Our main goal in this paper is to show that the kick evolution in (2.9) is well defined on some large subset of V and to study its statistical properties. 8

9 3 Preliminaries on random kick models Let H be a separable Hilbert space with a norm and an orthogonal basis {e j }. Assume that B is a closed bounded subset of H containing the origin, and S : B B is a mapping, satisfying the following conditions: (A) There exists positive constants γ < 1 and C such that for all u, u 1, u 2 B. Su γ u and Su 1 Su 2 C Su 1 Su 2 (B) There exists a sequence {b j } of nonnegative numbers such that b 2 j e j 2 < and SB + K B, where K = u = u j e j : u j b j for all j 1. j=1 j=1 (C) The set SB is compact and there exists N N such that (I P N )(Su 1 Su 2 ) 1 2 u 1 u 2 for all u 1, u 2 B, where P N is the orthogonal projector on the space Span {e 1,..., e N }. Remark 3.1 Assumption (A) is a bit stronger than (A) in [15, 16]. However it is formulated on a bounded subset of the space H. The invariance property in our version of (B) is different from the corresponding requirement in [15, 16], where some kind of dissipativity is assumed. We do not need any dissipativity hypotheses because we consider dynamics in the bounded invariant set B. As for (C), the papers [15, 16] assume that this relation holds with the factor 1/2 replaced by q N, where q N as N. In this form the assumption also implies the compactness of the set SB which is needed for the existence of an invariant measure. However the analysis in [15, 16] shows that if we already know the existence of an invariant measure, then for its uniqueness it suffices to require (C) with q N = 1/2 (see the proof of Lemma 3.2 in [15]). Now we consider a sequence {η k } of i.i.d. random variables in H, defined on a probability space (Ω, F, P), having the form η k = η ω k = j b j ξ ω jk e j, k = 1, 2,.... (3.1) The coefficients b j are the same as in Assumption (B) and ξ jk are independent random variables, possessing the same properties as the random variables ξjk and ˆξ jk appearing in Assumption (D) in Subsect Hypotheses (A) (C) and the properties of {η k } allow to define a discrete-time random dynamical system (RDS) on B by the relation u k = Su k 1 + η k, k = 1, 2,..., u B. (3.2) 9

10 Let us denote by M(B) the set of probability Borel measures on B, given the Lipschitz-dual distance dist(µ, ϑ) = dist M(B) (µ, ϑ) = sup µ ϑ, f, (3.3) f L where L = L(B) = {f : B R : Lip (f) 1 and f 1}. With this norm the set M(B) becomes a complete metric space, where the convergence in the norm is equivalent to the -weak convergence of measures, see [6] and [13]. This fact holds for any set B which is a complete separable metric space. Since in our case B is bounded, then, equivalently, L may be replaced by the bigger (and more convenient) set L, formed by all functions f on B such that Lip f 1. The RDS (3.2) defines transformations {Ψ k, k } of the set M(B): Ψ k (µ) = ϑ, ϑ(q) = P{u k (u ) Q} µ(du ), (3.4) where the sequence u k = u k (u ) is calculated according to (3.2). Clearly they can be extended to linear transformations of the space of signed Borel measures, and it is easy to see that Ψ k = (Ψ 1 ) k. A measure µ is called a stationary measure for the RDS (3.2) if Ψ k µ = µ for each k. The arguments, given in the proof of Theorem 1.1 from [16], lead to the following result: Theorem 3.2 Let Assumptions (A)-(C) be in force and the kicks η k be given by (3.1) with ξjk ω satisfying conditions in (D), and b j for 1 j N, where N < depends on the parameters of the equation and the kicks. Then the RDS (3.2) has a unique stationary measure µ and with some C, c >. dist(µ, Ψ k (ϑ)) Ce ck ϑ M(B), We will apply this theorem to the evolution in (2.9). To do this we need to study further the properties of solutions to problem (1.1) (1.4) in order to verify the conditions (A) (C) above. We do this in the next section. 4 Flow-maps We recall that {S T, T } stand for the flow-maps of the NSE (1.1) (1.4) f, and introduce the set B = {u V : M u a(), N u b()}. (4.1) Here a(), b() are positive real numbers such that R () = a() + b() satisfy (2.3), so Theorem 2.1 insures that the flow-maps S T are well-defined on B. The following assertion is proved in Section 6, as well as Theorem 4.2 and Proposition 4.4 below. Proposition 4.1 For any T > we can find > and k, γ (, 1) such that if η V satisfies M η k a(), N η k b(), where b 2 () lim = (4.2) a() 1

11 and (2.3) holds with R = a + b, then for any T T and < the set B is invariant for the mapping u S T u + η. Besides, S T u γ u for any u B. (4.3) Let {e λk, λ k } be the eigenfunctions and eigenvalues of the operator A, corresponding to the 2D Stokes operator (see Appendix). We denote by P N the projector of V on the subspace Span{e λ1,..., e λn }. Theorem 4.2 Assume that a() and b() satisfy (4.2) and [ a() C 1 log 1 ] σ [, b() C 2 log 1 ] σ/2, (4.4) where σ < 1/2. Take any u 1, u 2 B. Then for each T > we have if σ >, then there exists (, 1] and for every δ > there is C δ > such that for (, ] we have (S T u 1 S T u 2 ) C δ ( 1 If σ =, then this estimate holds with δ =. ) δ e λ 1 ν 2 T u 1 u 2. (4.5) For any q < 1 there exists >, and for (, ] there exists N = N(, T ) such that (I P N )(S T u 1 S T u 2 ) q u 1 u 2. (4.6) Moreover, if σ =, then N may be chosen independent of. Remark 4.3 The assumptions in Theorem 4.2 allow the initial data and the vectors η to be large. For instance, if both u = (u 1, u 2, u 3 ) and η = (η 1, η 2, η 3 ) are restrictions on O of C 1 (O 1 )-functions, then M u 2 + N u 2 c u 2 C 1 (O 1 ), and M η 2 + N η 2 c η 2 C 1 (O 1 ). Therefore choosing a 2 () = c ( log 1 ) 3/4 and b 2 () = c ( log 1 ) 1/4 satisfying (4.2) and (4.4) we can see that large values of u and η are possible, when is small. In general, with this choice of a 2 () and b 2 (), C 1 -norms of u and η may be of order ( log 1 ) 3/4. The following assertion is useful in the study of limit behaviour as of random kick evolution in (2.9). Proposition 4.4 Under the conditions of Theorem 4.2 the set A S T B e is bounded in H for every. Moreover, if is sufficiently small, then for any ρ > there exist C(ρ) such that for all < and < τ T we have sup { A S τ u : u B, u ρ } ( C(ρ) ). (4.7) τ 11

12 5 Main results Now we are in position to state and prove our main results. 5.1 Well-definiteness of RDS We return to the formal evolutions described in (2.9) and assume that the quantities B and ˆB, defined in (2.11), satisfy the inequalities B a () and ˆB b (), (5.1) where a () and b () meet (4.2) and (4.4) with some σ [, 1/2). Let us set a() = Ca (), b() = Cb (). Choosing C sufficiently large we achieve that a() and b() satisfy assumptions of Proposition 4.1 and Theorem 4.2. In particular, the set B (see (4.1)) is invariant for the RDS (3.2) with S = S T and η k = η,ω k : u k = S T (u k 1 ) + η,ω k, k = 1, 2,... (5.2) Accordingly, the dynamics in (2.9) is globally well-defined for T T and, where = (T ) is the same as in Proposition 4.1. Thus, equations (1.1) (1.4), where f is the random kick-force given by (2.8), defines in B the dynamics u u(k; u ), k, where u(k; u ) = u(, kt ) and u(x, t) is calculated according to (2.9) (that is, according to (5.2) with u = u ). Let us take any τ [, T ]. If u(x, t) is calculated using (2.9), then u k,τ := u(kt + τ, ), k, 5 is a trajectory of the RDS u k,τ = S τ ( S T τ (u k 1,τ ) + η,ω ) k (5.3) (for τ = it coincides with the system (5.2)). For the same reasons as before (see also the argument given in the proof of Proposition 4.1 in Section 6), the set B is invariant for this system for any τ if 1. Our goal is to study asymptotical properties of the RDS s (5.3) with τ T as k and their limiting properties as. We pay the main attention to the case τ =, i.e., to RDS (5.2). 5.2 Asymptotical behaviour of solutions For the purposes of this subsection and of the next one it is convenient to provide the space V with the scaled norm, = 1/2. Note that the basis {e λj, e Λ j, j 1} is a Hilbert basis of the space (V,, ) and that the projection M : (V,, ) Ṽ has unit norm. We furnish the set B with the distance, corresponding to this norm. As in Section 3 we denote by M(B ) the set of probability Borel measures on B, given the Lipschitz-dual distance (3.3) (with B := B ). The RDS (5.2) defines transformations {Ψ k, k } of the set M(B ) (see (3.4)). Due to the relation Ψ k(ϑ), g = Eg ( u(k; u ) ) ϑ(du ) B 5 For τ = T we define by continuity u k,t = lim τ T u(kt + τ, ). 12

13 it follows from (4.5) that the transformations Ψ k are continuous in the -weak topology. We note that Ψ k (ϑ) = Du(, kt ), where u(x, t) is a solution (in the sense (2.9)) for the problem (1.1) (1.4) with f = f given by (2.8), and u is a random vector, independent of the kicks η,ω k, k 1, and such that D(u ) = ϑ. The main result in this subsection is the following assertion. Theorem 5.1 Let Assumption (D) and condition (5.1) be in force. Then for any T > there exist (, 1) and c > such that for T T and < we have: 1) the set B = {u V : M u c a (), N u c b ()} is invariant with respect to the RDS (5.2). 2) There exists N = N() N such that under the condition b j > 1 j N, (5.4) imposed on the kicks amplitudes in (2.1), the system (5.2), interpreted as an RDS in B, has a unique stationary measure µ, and dist M(V)(µ, Ψ k (ϑ)) C e c k (5.5) for every ϑ M(B ), with some C, c >. 3) Under the condition C 1 B C, ˆB C for all and for some C > 1, (5.6) where the values B and ˆB defined in (2.11), the number N in (5.4) does not depend on. Moreover, if, in addition, the random variables ξjk and ˆξ jk do not depend on, and (5.4) strengthens as inf > min 1 j N b j >, then the constants C and c in (5.5) can be chosen independent of, provided the initial measure ϑ satisfies the relation supp ϑ B {u V : u, c 1 }. In this case the stationary measure µ is supported by the set for some constant C >. B = {u V : M u, C, N u, C} (5.7) We recall that estimates (5.5) means the following: if u ω (x, t), t, is a random solution of the kick-forced NSE (1.1)-(1.4) such that u = u ω B for every ω and g L (B ) (see the notations in Section 3), then E g(u(, kt )) g, µ C e ck for k. (5.8) Proof. The invariance of the set B follows from argument given in Subsection 5.1. To prove the existence of a stationary measure for (5.2) we note that due to (2.6) S T (B ) is a compact subset in B and we can use the standard Krylov-Bogolyubov procedure to prove that a stationary measure exists (see, e.g., [13, Section 3.3] for some details). The uniqueness of a stationary measure follows from Theorem 3.2 since the assumption (5.4) with a suitable N = N() jointly with the established properties of the system (5.2) imply that it satisfies the assumptions (A)-(C) from Section 3 and (D) from Subsection 2.2, so Theorem 3.2 applies. Indeed, in the assumption (A) the first relation follows from (4.3) and the second one follows from (4.5). Assumption (B) holds trivially by the statement of Proposition 4.1. As for Assumption (C), the compactness of the set S T (B ) is established above and the squeezing relation follows from (4.6). Finally, (D) is the set of assumptions 13

14 which we have imposed on the densities p j and ˆp j. Consequently Theorem 3.2 implies the uniqueness of a stationary measure and relation (5.5). Under condition (5.6) we can assume that a () and b () in (5.1) satisfy (4.2) and (4.4) with σ =. Then the set B has a diameter of order one (with respect to the norm, ) uniformly in and the r.h.s. in (4.5) is independent of, as well as the constant N in (4.6). Moreover, the Lipschitz constant C in (A) is -independent. That is, all the data, needed to apply Theorem 3.2, are independent from. Thus N(), C and c in Theorem 5.1 can be chosen independent of. Remark 5.2 Since the set B, supporting all relevant measures, is bounded, then the convergence holds for locally Lipschitz functions g on V (i.e., for functions, which are Lipschitz on bounded subsets of V ). Remark 5.3 If g is a locally Lipschitz function on a Sobolev space H n (O ), n 1, then the convergence still holds, provided that the b-coefficients b j and ˆb j decay with j sufficiently fast. Indeed, if ( λ n j b 2 j + [ Λ n j] ) 2 ˆb j C < (5.9) j for some n 1, then η k H n (O ) C for each ω Ω, where H n (O ) := D(A n/2 ) H n (O ). Since S T (u) Hn (O ) C by (2.6), then now the stationary measure µ is supported by a bounded set in H n (O ). Due to the arguments in [13], Section 6.4, we have that under the condition in (5.9) the convergence in (5.8) holds for any measurable function g which is a locally Lipschitz function on H n 1 (O ). The corresponding constant C depends on g. In particular, if (5.9) holds with n = 3, then we can take g(u) = u i (x)u j (y), where i, j {1, 2, 3} and any x, y O are fixed. Since H 2 (O ) C(O ), then g is a locally Lipschitz function on H 2 (O ). Thus we obtain relation (1.13) claimed in the Introduction. Corollary 5.4 Let the assumptions of Theorem 5.1 be in force and h be a locally Lipschitz function on the space H l (O ; R 3 ) for some l N. Then for any < τ T we have E h(u(, kt + τ)) h, µ τ C,h,τ e ck for k =, 1,.... (5.1) Here µ τ = S τ µ and u is a solution of (1.1)-(1.3), where u = u ω B for every ω. Moreover, if the assumptions of item 3) of Theorem 5.1 are in force and h L(B ), then the constants C and c may be chosen independent from, τ and h. So in this case dist M(V )(µ τ, Du(, kt + τ)) Ce ck k (5.11) for every τ [, T ] and every provided that u(, ) B. Proof. Due to (2.9) the l.h.s. of (5.1) equals the l.h.s. of (5.8) with g = h S τ. Since u B, then by (2.6) (h S τ )(u) is a bounded Lipschitz function on B. Now the estimate (5.1) follows from (5.8). Under the assumptions of the second assertion, we use (4.5) to get that the function C 1 h S τ L(B) for some C, independent from, h and τ. Therefore the desired result follows from item 3) of Theorem 5.1. Since {u(kt + τ, ), k } is a trajectory of the RDS (5.3) in B, then by (5.1) the latter has the unique stationary measure µ τ which attracts exponentially fast distributions of all trajectories of the system. 14

15 Let the initial condition u in (1.1)-(1.3) be such that D(u ) = µ, and u(x, t) be a corresponding solution. Then Du(kT + τ) = µ τ τ [, T ], k =, 1,.... Abusing language, we call such solutions stationary (in fact, they are T -periodic in distribution). 5.3 Limit. Theorem 2.2 suggests that statistical properties of solutions u(x, t), averaged in x 3, are close to those of solutions for the 2D NSE. To prove corresponding results we have to strengthen assumption (5.1) on the kicks ηk. Namely, we assume the following: (L) The random variables ξ jk ξjk and ˆξ jk ˆξ jk in (2.1) are independent of ; b j b j as in the sense that (b j b j )e λj 2 H j relations (5.6) are in force. 1 λ j (b j b j ) 2 as ; (5.12) Let us define 2D kicks η k = b j ξ jk e λj (x ) and consider the 2D kick-force f = δ kt (t) η ω k (x ). (5.13) Clearly f = lim M f, where f = f is given by (2.8). Similar to the 3D case, the corresponding kick-forced 2D NSE (1.9)-(1.1) defines a continuous discrete-time RDS in the space Ṽ, and defines the semigroup {Ψ k, k } of continuous transformations of the space of Borel measures on Ṽ. Moreover, this system extends to a continuous RDS in the space H, and the transformations Ψ k extend to continuous (in the -weak topology) transformations of the space of Borel measures in H, see [13]. If b j > j N (5.14) with a suitable N <, then by the same reasons as above this system has a unique stationary measure ϑ. This is a Borel measure in Ṽ, supported by a ball of finite radius. Due to (5.12), assumption (5.14) implies (5.4) if < 1, and Theorem 5.1 applies. For such let us denote ϑ = M µ, i.e. ϑ (Q) = µ {u B : M u Q}. Theorem 5.5 Let Assumptions (D) and (L) be in force and (5.14) holds with a sufficiently large N. Then ϑ ϑ as, (5.15) where stands for the -weak convergence of measures in H and ϑ is the unique stationary measure of the kick-forced 2D NSE (1.9)-(1.1) with f defined in (5.13). Moreover, if in addition we assume that j 2 b 2 j C, (5.16) j then the convergence in (5.15) holds true in -weak sense of measures on the space H 2 δ := H 2 δ (T 2 ) Ṽ, for every δ >. In particular, if g is a continuous functional on C(T2 ; R 2 ), then g, ϑ g, ϑ as. (5.17) 15

16 Proof. By Theorem 5.1 under condition (5.6) the stationary measure µ has its support in the set B given by (5.7). Hence, supp ϑ { v Ṽ C} for each. So by the Prokhorov theorem the family of measures {ϑ, < } is precompact in the set of measures in H, given the -weak topology. It remains to prove that any limiting measure ϑ of this family equals ϑ. Let us take a sequence j such that ϑ j ϑ. By the Skorokhod representation theorem (e.g., see [12]), on a probability space, for which we take the segment [,1] given the Borel sigma-algebra and Lebesgue measure, we can construct Ṽ -valued random variables ṽ and {v j }, such that D(v j ) = ϑ j, D(ṽ) = ϑ and v j ṽ in Ṽ a.s. We view them as random variables on the probability space Ω new = [, 1] [, 1], depending only on the first factor, and for each j find a B j -valued random variable u j on Ω new such that M u j = v j and D(u j ) = µ j, see below Lemma 5.9. Next we construct on Ω new random variables ξ jk new etc, distributed as ξ jk etc and independent from the previously constructed random variables. We define the random vectors η 1 new and η j 1 new, using these new random variables. Then Ψ j 1 (µ j ) = D(S T j u j + η j 1 new ) and Ψ 1(ϑ j ) = D(S T v j + η 1 new ). Let g be any continuous function on H such that Lip (g) 1 and g 1. Then g, Ψ 1 (ϑ j ) g, M j Ψ j 1 (µ j ) = E ( g(s T (v j ) + η 1 new ) g ( M j (S T j (u j )) + M j (η j 1 new )) E ( S T (v j ) M j (S T j (u j )) 2 ) + E η 1 new M j (η j 1 new ). Since M u j = v j ṽ a.s., then by Theorem 2.2 the random variable in the first expectation in the r.h.s. goes to zero with j for each ω. By (5.12) the random variable in the second expectation goes to zero with j uniformly in ω. So the r.h.s. goes to zero with j and the rate of convergence is independent of g as above. Since M j Ψ j 1 (µ j ) = M j µ j = ϑ j, then we have seen that dist M( H) (ϑ j, Ψ 1 (ϑ j )) as j. (5.18) As the transformation Ψ 1 is continuous in M( H), then by (5.18) ϑ is a stationary measure of the 2D NSE. So it equals ϑ. To prove the second part of the theorem, we note that A u 2 Hϑ (du) = 1 A M u 2 H µ (du) = 1 B E A M (S T u + η1 ω ) 2 µ (du) B 2 A S T u 2 µ (du) + 2 B E A M η1 ω 2 µ (du), B where the second equality holds since µ is a stationary measure. Since M η1 = b j ξω j1 e λ j (see (8.2)), then due to (5.16) and (8.4) the second term in the r.h.s. is bounded by (b j ) 2 λ 2 j C 1. Since µ is supported by (5.7), due to Proposition 4.4 the first term is C 2. Hence, A H u 2 Hϑ (du) C 3 for all, and by Prokhorov s theorem we conclude that the family {ϑ } is precompact in the -weak topology of the space of measures on H 2 δ. This implies the desired convergence. The convergence in (5.17) holds since for δ < 1/2 the space H 2 δ is continuously embedded in C(T 2 ; R 2 ). Remark 5.6 Note that the proof implies that the measures ϑ and ϑ are supported by the same ball { v Ṽ C}. 16

17 The following assertion shows that the convergence of M µ τ in the space H 2 δ takes place for τ > without condition (5.16). Corollary 5.7 Let the assumptions of the first assertion of Theorem 5.5 be in force. Then for any τ > M µ τ S τ ϑ =: ϑ τ as (5.19) in Borel measures in the space H 2 δ = H 2 δ (T 2 ) Ṽ for every δ >. Moreover, the rate of convergence (5.19) in the space of measures in H is independent from τ [, T ]. Proof. Recall that µ τ = S τ µ is the unique stationary measure of the RDS (5.3). Similarly, ϑ τ is the unique stationary measure for the 2D RDS v k,τ = S τ (S T τ (v k 1,τ ) + η ω k ). Arguing as in the proof of Theorem 5.5 we see that the convergence (5.19) holds in the space H, and its rate is independent from τ. Since the measures µ with 1 are supported by the ball (5.7), then by Proposition 4.4 the measures M µ τ are supported by a ball in the space H 2 (T 2 ) Ṽ with the radius independent of. So they form a precompact family of Borel measures on H 2 δ, and the assertion follows. In our last theorem we obtain an analogy of the assertion of Theorem 2.2 for distributions of solutions to the random NSE. In difference with the deterministic situation, now an analogy of convergence (2.5) holds uniformly in t. Theorem 5.8 Let us assume that condition (5.1) and Assumptions (D) and (L) hold. Let T T and N N (independent of ) be as in Theorem 5.1, and b j > for j N. Let u,ω (t, x) be a solution of the random kick-forced NSE (1.1)-(1.3), where u = u,ω is a random vector independent of the kicks and such that u,ω B for each ω. Assume that Mu,ω v ω Ṽ weakly in H for each ω, and denote by v ω (t, x ) a solution of the 2D NSE(1.9), (1.1), (5.13), equal v ω at t =. Then dist M( H) ( DM u (t), Dv(t) ) as, (5.2) uniformly in t. Moreover, if (5.16) holds, then also dist M(H 2 δ )( DM u (kt ), Dv(kT ) ) as for any δ >, uniformly in k 1, where as above H 2 δ = H 2 δ (T 2 ) Ṽ. Proof. Let us fix any Θ 1. Applying iteratively Theorem 2.2 on the time-segments [, T ], [T, 2T ],... we get that sup M u,ω (t) v ω (t) H =: κ ω (, Θ) as (5.21) t Θ for each ω. Accordingly, for any g L( H) we have sup E g ( M u (t) ) E g(v(t)) E min ( 2, κ ω (, Θ) ) =: κ 1 (, Θ), t Θ where κ 1 (, Θ) as, for each Θ. So dist M( H) ( DM u (t), Dv(t) ) κ 1 (, Θ) for t Θ. (5.22) If t = kt + τ Θ, then using (5.11) we get dist M( H) ( DM u (t), M µ τ ) Ce cθ/t. 17

18 Similarly the solution v(t) satisfies dist M( H) ( Dv(t), ϑ τ ) Ce cθ/t (see [13] and the beginning of this subsection). Using Corollary 5.7 we get from the last two estimates that dist M( H) ( DM u (t), Dv(t) ) κ 2 (Θ) + κ 3 () for t Θ, where κ 2 as Θ and κ 3 as. Jointly with (5.22) this relation implies the first assertion of the theorem. Let us assume (5.16). Then due to relation (4.7) applied to each interval [kt, (k + 1)T ], for t T, and all ω we have M A u (t) H C. Similar using (2.7) we find that A v(t) H C 1. Interpolating these inequalities with (5.21) we get sup M u,ω (t) v ω (t) H 2 δ =: κδ ω (, Θ) as t Θ for each ω. Now arguing as above and using the second assertion of Theorem 5.5 we complete the proof. In the lemma below the segment [,1] is given the Borel sigma algebra and Lebesgue measure. Lemma 5.9 Let A and B be complete metric spaces (infinite and non-countable), given Borel σ-algebras, µ be a Borel measure on A B and ξ be a measurable map [, 1] A such that D(ξ) = µ 1, where µ 1 is the projection of the measure µ to A. Then there exists a measurable map η : [, 1] 2 B such that the distribution of the map ξ η : [, 1] 2 A B is µ. Here we extended ξ to a function on [, 1] 2, depending on the first factor only. Proof. Since both spaces A and B are measurably isomorphic to the segment [,1], given the Borel σ-algebra ([6], Section 13.1), then without loss of generality we may assume that A = B = [, 1]. By the theorem on the conditional distribution ([6], Section 1.2) we can write µ as µ(da db) = µ 1 (da)µ 2 (a; db). Here µ 2 is measurable in the sense that the function F (a; b) = µ(a; [, b]) is measurable both in a and b. Since F as a function of b is continuous from the right, then it is measurable as a map of Borel spaces [, 1] 2 [, 1]. Let us define the function ρ(a, y) : [, 1] 2 [, 1] by the relation ρ(a, y) = inf{τ : F (a, τ) y}. It is measurable, monotonic in y and continuous from the right. The mapping [, 1] y ρ(a, y) transforms the Lebesgue measure dy to the measure µ(a; db) ([6], Section 9.1). Now we set η(x, y) = ρ(ξ(x), y). The mapping ξ η : [, 1] 2 A B possesses the desired properties. 6 Proofs of results stated in Section Preliminaries The following properties established in [22] are important in the further considerations. (i) M = M and N = N, where = ( x1, x2, ). (ii) For all u, v H 1 (O ) 3 we have O N u M vdx = and u 2 = N u 2 + M u 2, u 2 = N u 2 + M u 2. (6.1) 18

19 (iii) For all u, w, v V we have and b (u, w, M v) = b (M u, M w, M v) + b (N u, N w, M v) (6.2) b (u, w, N v) = b (N u, w, N v) + b (M u, N w, N v). (6.3) (iv) If u D(A ), then M u D(A ) and (v) If u D(A ), then N u = N u, M u = M u. b (M u, M u, A M v) = b (M u, M u, A M v) =, (6.4) where b is the 2D trilinear form and A is the 2D Stokes operator on T 2 (i.e., in the space H). (vi) If u D(A ), then A u = u and (see Lemma 2.5 [22]) u x i x j A u 2 L 2 (O ), u D(A ). (6.5) i,j L 2(O ) As in [22] we also use the thin domain analogues of the classical Poincaré, Agmon and Ladyzhenskaya inequalities given in the following assertion. Lemma 6.1 ([22]) There exist positive constants c and c q, 2 q 6, such that for all (, 1] the following inequalities hold true: (Poincaré s inequality); N u 3 N u, for all u V (6.6) N u (L (O )) 3 c N u 1/4 (L 2(O )) 3 for all u D(A ) (Agmon s inequality); (Ladyzhenskaya s inequality). 3 2 N u x i x j i,j 2 L 2 (O ) 3/4 (6.7) N u 2 (L q (O )) 3 c q (6 q)/q N u 2 for all u V, 2 q 6, (6.8) We will also use the following version of Lemmas 3.1 and 3.2 of [22] (in the case when the external force is absent, f ). Lemma 6.2 ([22]) Let the hypotheses of Theorem 2.1 be in force and u(t) be a solution to (1.1) (1.4) with f. Assume that M u a(), N u b(), where R () a() + b() satisfy (2.3). Then there exists T () > such that T () + and for all < t < T () the following inequalities hold: { N u(t) 2 b 2 () exp νt } 2 2, (6.9) t A N u(s) 2 ds 2 ν b2 (), (6.1) M u(t) 2 a 2 () exp { νλ 1 t} + c 1 (ν)b 4 (), (6.11) t A M u(s) 2 ds 2 ν a2 () + c 1 (ν)b 4 (). (6.12) Here above λ 1 is the first eigenvalue of 2D Stokes operator A. 19

20 6.2 Estimates for the trilinear form The following estimates are proved in [22] in the case when u = w. The proof given below is a slight modification of the argument from [22]. Lemma 6.3 For every θ < 1/2, there exist positive constants, c such that for all (, ), u, w D(A ), v (L 2 (O )) 3, we have Proof. b (M u, N w, v) c θ M u A N w v ; (6.13) b (N u, w, v) c 1/2 A N u w v ; (6.14) b (N u, N w, v) c 1/2 N u A N w v. (6.15) Estimate (6.13): Since (M u) 3 =, we obviously have that 2 3 b (M u, N w, v) (M u) j ( j N u) l v l dx. O j=1 l=1 Applying Hölder s inequality, we obtain 2 3 b (M u, N w, v) (M u) j L p 1 (O ) ( j N w) l L p 2 (O ) v l L 2 (O ), j=1 l=1 where p p 1 2 = 1/2, 2 < p 2 6. Since j N w = N j w for all w D(A ) and j = 1, 2, we can use (6.5) and (6.8) to write j (N w) l L p 2 (O ) c 6 p 2 2p 2 j N w c 6 p 2 2p 2 A N w. for any l = 1, 2, 3. One can also see that (M u) j L p 1 (O ) = 1/p 1 (M u) j L p 1 (T 2 ) c1/p 1 (M u) j H 1 (T 2 ) c 1/p1 1/2 (M u) j. Therefore inequality (6.13) with θ = 2/p 2 1/2 follows. Estimate (6.14): Using (6.7) we get that 3 3 b (N u, w, v) (N u) j ( j w) l v l dx j=1 l=1 O 3 j=1 l=1 3 (N u) j L (O ) ( jw) l L 2 (O ) v l L 2 (O ) c N u 1/4 A N u 3/4 w v. By (6.6) we have that N u A 1/2 N u 2 A N u, and (6.14) follows. Estimate (6.15): We obviously have that 3 3 b (N u, N w, v) (N u) j ( j N w) l v l dx Thus by (6.8) we obtain (6.15). j=1 l=1 O 3 j=1 l=1 3 (N u) j L 3 (O ) ( jn w) l L 6 (O ) v l L 2 (O ) 3 c N u L 3 (O ) j N w L 6 (O ) v. j=1 2

21 6.3 Proof of Proposition 4.1. Applying (6.9) and (6.11) with ã() = M u (t) and b() = N u (t) instead of a() and b() yields Thus u (t) 2 = M u (t) 2 + N u (t) 2 ã 2 () exp { νλ 1 t} + c 1 b 4 () + b 2 () exp [ ã 2 () + b ] 2 () max { νt } 2 2 { exp { νλ 1 t}, c 1 b 2 () + exp { νt }} 2 2. { { S T u 2 u 2 max exp { νλ 1 T }, c 1 R 2 () + exp νt }} 2 2. for all T T. Therefore we can find and < γ < 1 such that (4.3) holds. Now we prove the invariance of the set B. From (6.9) and (6.11) we have that { A 1/2 N [u(t ) + η] b() exp νt } k b() and { A 1/2 M [u(t ) + η] a() exp νλ } 1 2 T + c 1 b 2 () + k a(), Thus the set B given by (4.1) is invariant with respect mapping u S T u + η if { exp νt } k 1 and [ { c 1 b 2 () a() 1 exp νλ 1 2 T } k ]. [ { Now we can choose k = exp νλ 1 2 T }] and, due to (4.2), find = (T ) such that the inequalities above hold for all (, ). 6.4 Proof of Theorem 4.2. Let u 1 (t) and u 2 (t) be two strong solutions to 3D Navier-Stokes problem (2.1) with f. Then the difference u(t) = u 1 (t) u 2 (t) satisfies the equation u + νa u + B (u, u 1 ) + B (u 2, u) =. (6.16) Step 1: Preliminary estimate for N -component. Multiplying (6.16) in H by A N u we get 1 d 2 dt A1/2 N u 2 + ν A N u 2 + b (u, u 1, A N u) + b (u 2, u, A N u) =. (6.17) Now we estimate the trilinear terms in (6.17). By (6.3) we have b (u, u 1, A N u) = b (N u, u 1, A N u) + b (M u, N u 1, A N u) By (6.14) and (6.13) we obtain b (N u, u 1, A N u) c 1/2 u 1 A N u 2 21

22 and b (M u, N u 1, A N u) δ A N u 2 + c δ 2θ M u 2 A N u 1 2 for any δ >, where < θ < 1/2 can be chosen in arbitrary way. Similarly b (u 2, u, A N u) = b (N u 2, u, A N u) + b (M u 2, N u, A N u), where b (N u 2, u, A N u) δ A N u 2 + c δ u 2 A N u 2 2 for any δ >, and b (M u 2, N u, A N u) c θ M u 2 A N u 2 c θ u 2 A N u 2. Using in (6.17) these inequalities with suitable δ > we get that 1 d ]) 2 dt N u 2 + (ν c [ 1/2 u 1 + θ u 2 A N u 2 c 1 u 2 A N u c 2 2θ M u 2 A N u 1 2 (6.18) for every θ [, 1/2). Under the hypotheses concerning a() and b(), we have from relations (6.9) and (6.11) in Lemma 6.2 that 1/2 u 1 (t) + θ u 2 (t) c θ (log 1 ) σ, for any pair of initial data u 1 () and u 2 () from B. Choosing < 1 we get that for all <, where d dt N u 2 + ν A N u 2 (6.19) c 1 N u 2 A N u c 2 2θ M u 2 ψ N (t, u 1, u 2 ), ψ N (t, u 1, u 2 ) = A N u 1 (t) 2 + A N u 2 (t) 2. (6.2) Step 2: Preliminary estimate for M -component. Multiplying (6.16) in H by A M u we get 1 d 2 dt A1/2 M u 2 + ν A M u 2 + b (u, u 1, A M u) + b (u 2, u, A M u) =. (6.21) As above, we estimate trilinear terms in (6.21). By (6.2) we have It is clear that b (u, u 1, A M u) = b (M u, M u 1, A M u) + b (N u, N u 1, A M u). b (M u, M u 1, A M u) c M u L 4 (T 2 ) j=1,2 j M u 1 L 4 (T 2 ) A M u c M u H 1 (T 2 ) M u 1 H 2 (T 2 ) A M u c M u A M u 1 A M u δ A M u 2 + c δ M u 2 A M u

23 for every δ >. As for the second term, by (6.15) we have that b (N u, N u 1, A M u) δ A M u 2 + c δ N u 2 A N u 1 2 for every δ >. Now we estimate the term b (u 2, u, A M u). As above we have that It is obvious that b (u 2, u, A M u) = b (M u 2, M u, A M u) + b (N u 2, N u, A M u). b (M u 2, M u, A M u) c M u 2 L (T 2 ) M u A M u c A M u 2 M u A M u for every δ >. It also follows from (6.14) that δ A M u 2 + c δ M u 2 A M u 2 2 b (N u 2, N u, A M u) δ A M u 2 + c δ N u 2 A N u 2 2, δ >. Using in (6.21) the inequalities above with appropriate δ > we get d dt M u 2 + ν A M u 2 (6.22) c 1 N u 2 ψ N (t, u 1, u 2 ) + c 2 M u 2 ψ M (t, u 1, u 2 ), where ψ N (t, u 1, u 2 ) is defined above in (6.2) and Step 3: Proof of (4.5). It follows from (6.19) and (6.22) that ψ M (t, u 1, u 2 ) = A M u 1 (t) 2 + A M u 2 (t) 2. d dt u 2 + λ 1 ν u 2 c u 2 ψ (t, u 1, u 2 ) (6.23) where ψ (t, u 1, u 2 ) = 2θ ψ N (t, u 1, u 2 ) + 1 ψ M (t, u 1, u 2 ). By Lemma 6.2 we have that t ψ (τ, u 1, u 2 )dτ c 2θ b 2 () + c 1a 2 () [ + c 2 b 4 () c 3 + c 4 log 1 ] 2σ for all <. Thus by Gronwall s lemma from (6.23) we have that { t } u(t) 2 u() 2 exp λ 1 νt + c ψ (τ, u 1, u 2 )dτ [ u() 2 exp { λ 1 νt + c 1 + c 2 log 1 ] } 2σ. (6.24) [ ] Since < σ < 1/2, we have that c 2 log 1 2σ cδ + 2δ log 1. Hence we arrived at the relation [ ] 2δ 1 u(t) 2 C δ u() 2 e λ1νt, t [, T ], δ >, (6.25) which implies (4.5). If σ =, then we recover from (6.24) estimate (4.5) with δ =. 23