Random attractors for the stochastic Navier--Stokes equations on the 2D unit sphere

In this paper we prove the existence of random attractors for the Navier--Stokes equations on 2 dimensional sphere under random forcing irregular in space and time. We also deduce the existence of an invariant measure.


Introduction
Complex three dimensional flows in the atmosphere and oceans are modelled assuming that the Earth's surface is an approximate sphere. Then it is natural to model the global atmospheric circulation on Earth (and large planets) using the Navier-Stokes equations (NSE) on 2-dimensional sphere coupled to classical thermodynamics [33]. This approach is relevant for geophysical flow modeling.
Many authors have studied the deterministic NSEs on the unit sphere. Notably, Il'in and Filatov [31,29] considered the existence and uniqueness of solutions to these equations and estimated the Hausdorff dimension of their global attractors [30]. Temam and Wang [42] considered the inertial forms of NSEs on sphere while Temam and Ziane [43], see also [4], proved that the NSEs on a 2-dimensional sphere is a limit of NSEs defined on a spherical shell [43]. In other directions, Cao, Rammaha and Titi [15] proved the Gevrey regularity of the solution and found an upper bound on the asymptotic degrees of freedom for the long-time dynamics.
Concerning the numerical simulation of the deterministic NSEs on sphere, Fengler and Freeden [24] obtained some impressive numerical results using the spectral method, while the numerical analysis of a pseudo-spectral method for these equations has been carried out in Ganesh, Le Gia and Sloan in [26].
In our earlier paper [10] we analysed the Navier-Stokes equations on the 2-dimensional sphere with Gaussian random forcing. We proved the existence and uniqueness of solutions and continuous dependence on data in various topologies. We also studied qualitative properties of the stochastic NSEs on the unit sphere in the context of random dynamical systems.
Building on those preliminary studies, in the current paper, we prove the existence of random attractors for the stochastic NSEs on the 2-dimensional unit sphere. Let us recall here that, given a probability space, a random attractor is a compact random set, invariant for the associated random dynamical system and attracting every bounded random set in its basis of attraction (see Definition 2.6 in [8]).
We do not know if our system is dissipative in H 1 . Therefore, despite the fact that the embedding H 1 ֒→ L 2 is compact, the asymptotic compactness approach seems to be the only method available in the L 2 -setting to yield the existence of an attractor, hence of an invariant measure.
The paper is organised as follows. In Section 2, we recall the relevant properties of the deterministic NSEs on the unit sphere, outline key function spaces, and recall the weak formulation of these equations. In Section 3, we define the stochastic NSEs on the sphere and review the key existence and uniqueness results obtained in [10]. In Section 4 we prove the existence of a random attractor of the stochastic NSEs on the 2-d sphere. The paper is concluded with a simple proof of the existence of an invariant measure and some comments on the question of its uniqueness.
In our paper a special attention is given to the noise with low space regularity. While many works on random attractors consider only finite dimensional noise, we follow here the approach from Brzeźniak et al [8] and consider an infinite dimensional driving Wiener process with minimal assumptions on its Cameron-Martin space (known also as the Reproducing Kernel Hilbert Space), see Remark 3.9 and the Introduction to [8] for motivation.

The Navier-Stokes equations on a rotating unit sphere
The sphere is a very special example of a compact Riemannian manifold without boundary hence one could recall all the classical tools from differential geometry developed for such manifolds. However we have decided to follow a different path of using the polar coordinates and defining all such objects directly.
Note that where the second equality holds whenever sin θ = 0.
The surface gradient for a scalar function f on S 2 is given by For a tangential vector field u = (u θ , u φ ), i.e. u = u θ e θ + u φ e φ , we put The tangential velocity field u(x, t) = (u θ ( x, t), u φ ( x, t)) of a geophysical fluid flow on a 2-dimensional rotating unit sphere S 2 under the external force f = (f θ , f φ ) = f θ e θ + f φ e φ is governed by the Navier-Stokes equations (NSEs), which takes the form [23,40] (2.1) Here ν and ρ are two positive constants related to the physical quantities called respectively the viscosity and the density of the fluid. The Coriolis acceleration normal vector field is defined by In what follows we will identify ω with the corresponding scalar function ω defined by ω(x) = 2Ω cos θ(x) .
The operators ∇ and div are the surface gradient and divergence, respectively. The convective acceleration ∇ u u is the nonlinear term in the equations. Here, the operator L is given by [40] (2.2) where ∆ is the Laplace-de Rham operator, see equality (2.14) below, and Ric denotes the Ricci tensor of the two-dimensional sphere S 2 . It is well known that (see e.g. [45, page 75]) We remark that in papers in [15,29,31,42] the authors consider NSEs with L = ∆ but the analysis in our paper are still valid in that case.
Let us define the nonlinear term ∇ u u. By a proposition in [22] or [21,Definition 3.31], ifũ andṽ are vector fields defined in some neighbourhood of the surface S 2 and tangent to S 2 , i.e.ũ| S 2 = u : where∇ is the gradient in R 3 and, for x ∈ S 2 , is the orthogonal projection from R 3 onto the tangential space T x S 2 to S 2 at x.
One should note that the above definition is independent of the choice of extensionsũ andṽ of the vector fields u and v. Indeed, ifũ andṽ is another pair of such extensions, then sinceũ =ũ on S 2 , for every point x ∈ S 2 , the restrictions to T x S 2 of the Frechet derivatives at ofũ andũ are equal. Sinceũ(x) =ũ(x) belongs to T x S 2 and since (ṽ(x) ·∇)ũ(x) = [d xũ ](ṽ(x)) (and analogously for the˜extensions, the claim follows.
Using the well known formula for the vector product in R 3 : we easily infer that It follows that, using the notation above, that the restrictions to S 2 ofũ × (∇ ×ũ) and u × (∇ ×ũ) coincide.
Lemma 2.1. If u and v are tangential vector fields on S 2 andũ andṽ are extensions of those to a neighbourhood of S 2 , i.e.ũ| S 2 = u andṽ| S 2 = v, then the following identity holds Proof. We can decomposeũ andṽ into tangential and normal component as follows Hence the lemma is proved.
Using (2.6) for the vector fieldsũ andṽ =∇ ×ũ, we have With a tangential vector field u, since the normal component is zero, (2.7) is reduced to So one can define, for a tangential vector field u Given a tangential vector field v, with a slight abuse of notation, we write v × curl u := v × x(curl u).
Hence from (2.8), (2.9) we obtain and thus (2.10) Let us note that (x · (∇ ×ũ))x is just the orthogonal projection of the vector∇ ×ũ onto the normal component of T x S 2 . Furthermore, with slight abuse of notation, We have the following well-defined operators [29].
Definition 2.2. Assume that u is a tangent vector field on S 2 , and ψ a scalar function on S 2 . We define Moreover, given two tangential vector fields u and v, the tangential vector field Often various authors introduce a notion of a normal vector field to S 2 which they then identify with a scalar function on S 2 . We have found this to be an unnecessary procedure.
The surface diffusion operator acting on tangential vector fields on S 2 is denoted by ∆ (known as the vector Laplace-Beltrami or Laplace-de Rham operator) and is defined as (2.14) ∆ u = ∇div u − Curl curl u.
In what follows we denote by dS the Lebesgue integration with respect to the surface measure (or the volume measure when S 2 is seen as a Riemannian manifold). In the spherical coordinates we have, locally, dS = sin θdθdφ. For p ∈ [1, ∞) we will use the notation L p = L p (S 2 ) for the space L p (S 2 , R) of p-integrable scalar functions on S 2 endowed with the norm For p = 2 the corresponding inner product is denoted by We will denote by L p = L p (S 2 ) the space L p (S 2 , T S 2 ) of vector fields v : where, for x ∈ S 2 , |v(x)| stands for the length of v(x) in the tangent space T x S 2 . For p = 2 the corresponding inner product is denoted by Throughout the paper, the induced norm on L 2 (S 2 ) is denoted by · and for other inner product spaces, say X with inner product (·, ·) X , the associated norm is denoted by · X . We have the following identities for appropriate scalar and vector fields [29, (2.4)-(2.6)]: In (2.17), the L 2 (S 2 ) inner product is used on the left hand side and the L 2 (S 2 ) inner product is used on the right hand side. Using the identity (2.16) the unknown pressure can be eliminated from the first equation in (2.1) through the weak formulation.
2.2. The weak formulation. We now introduce Sobolev spaces H s (S 2 ) and H s (S 2 ) of scalar functions and vector fields on S 2 respectively.
Let ψ be a scalar function and let u be a vector field on S 2 , respectively. For s ≥ 0 we define where ∆ is the Laplace-Betrami and ∆ is the Laplace-de Rham operator on the sphere. In particular, for s = 1, where we have used formulas (2.14),(2.16)-(2.18).
One has the following Poincaré inequality [31, Lemma 2] for some positive constant λ 1 . The space of smooth (C ∞ ) tangential fields on S 2 can be decomposed into three components, one in the space of all divergence-free fields and the others through the Hodge decomposition theorem [3, Theorem 1.72]: while H is the finite-dimensional space of harmonic fields, i.e. H contains all the vector fields v so that Curl (v) = div (v) = 0. Since the two dimensional sphere is simply connected, H = {0} [38, page 80]. We introduce the following spaces We consider the linear Stokes problem By taking the inner product of the first equation of (2.25) with v ∈ V and then using (2.18), we obtain Next, we define a bilinear form a : In view of (2.21) and (2.3), the bilinear form a satisfies and hence it is continuous on V . So by the Riesz Lemma, there exists a unique operator Using the Poincaré inequality (2.22), we also have a(u, u) ≥ α u 2 V , with α = λ 1 − 2ν, which means a is coercive in V whenever λ 1 > 2ν. In practice, usually one has λ 1 ≫ 2ν. Hence by the Lax-Milgram theorem the operator A : V → V ′ is an isomorphism. Furthermore, by using [39, Theorem 2.2.3], we conclude that the operator A is positive definite, selfadjoint in H and D(A 1/2 ) = V . Thus, the spectrum all A consists of an infinite sequence 0 < λ 1 ≤ λ 2 ≤ . . . ≤ λ ℓ → ∞, and there exists an orthogonal basis {w ℓ } ℓ≥1 of H consisting of eigenvectors of A.
Hence for each positive integer ℓ = 1, 2, . . ., the eigenvectors of the operator A corresponding to the eigenvalue λ ℓ are given by Since {Z ℓ,m : ℓ = 1, . . . ; m = −ℓ, . . . , ℓ} is an orthonormal basis for H, an arbitrary v ∈ H can be written as Since V is densely and continuously embedded into H and H can be identified with its dual H ′ , we have the following imbeddings: We say that the spaces V, H and V ′ form a Gelfand triple. Next we define an operator A in H as follows: Let P be the Leray orthogonal projection from L 2 (S 2 ) onto H. It can be shown [27] Let us now recall how the fractional power of the Stokes operator A can be defined in our concrete setting. It can be proven that The Coriolis operator C 1 : Clearly, C 1 is linear and bounded in L 2 (S 2 ). In the sequel we will need the operator C = P C 1 which is well defined and bounded in H. Furthermore, for u ∈ H We consider the trilinear form b on V × V × V , defined as Using the following identity and (2.17), for divergence free tangential vector fields v, w, z, the trilinear form can be written as Moreover [29, Lemma 2.1] We have the following inequality from [31, page 12] . Thus, using (2.14), (2.17), (2.30), and (2.37), a weak solution of the Navier-Stokes equa- This weak formulation can be written in operator equation With a slight abuse of notation, we also denote B(u) = B(u, u).
The following are some fundamental properties of the trilinear form b; see [24]: There exists a constant C > 0 such that We also need the following estimates: There exists a positive constant C such that and In view of (2.46), b is a bounded trilinear map from L 4 (S 2 ) × V × L 4 (S 2 ) to R. Moreover, we have the following result: It can be seen from (2.46) that b is a bounded trilinear map from 3. The stochastic Navier-Stokes equations on a rotating unit sphere By adding a white noise term to (2.1) the stochastic NSEs on the sphere is where we assume that u 0 ∈ H, f ∈ V ′ and n(t, x) is a Gaussian random field which is a white noise in time. In the same way as in the deterministic case we apply the operator of projection onto the space of divergence free fields and reformulate the above equation as an Itô type equation Here f is the deterministic forcing term and u 0 is the initial velocity. We assume that W is a cylindrical Wiener process on a certain Hilbert space K defined on a probability space (Ω, F , P), see [35] and [12]. G is a linear continuous operator from K to H. The space K, which is the RKHS of the Wiener process, determines the spatial smoothness of the noise term, will satisfy further assumptions to be specified later.
Roughly speaking, a solution to problem (3.1) is a process u(t), t ≥ 0, which can be represented in the form and v(t), t ≥ 0, is the solution to the following problem (with v 0 = u 0 − z α (0)): 3) holds in the weak sense, i.e. for any φ ∈ V , We remark that for (3.5) to make sense, it is sufficient to assume that v ∈ L 2 (0, We have proved the following major theorems on the existence and uniqueness of the solution of (3.3) − (3.4) in [10].
Moreover, then the maps is continuous in the weak topologies of H and L 2 ([0, T ]; V ).
3.1. Preliminaries. Let us recall that for a real separable Hilbert space K and a real separable Banach space X, a linear operator U : Here γ K is the canonical Gaussian cylindrical measure on K. If a linear map U : K → X is γ−radonifying, then γ K • U −1 has a unique extension to a Borel probability measure denoted by ν U on X. By R(K, X) we denote the Banach space of γ-radonifying operators from K to X with the norm From now on we will using freely notation introduced in the former sections. It follows from [13, Theorem 2.3] that for a self adjoint operator U ≥ cI in H, where c > 0, such that U −1 is compact, the operator U −s : where {e ℓ } is an orthonormal basis of H corresponding to U. This implies the following result.
Lemma 3.4. Let ∆ denote the Laplace-de Rham operator on S. Then the operator Proof. Let us recall that all the distinct eigenvalues of −∆ are given by a sequence λ ℓ = ℓ(ℓ + 1), ℓ = 0, 1, . . . and the corresponding eigenfunctions are given by the divergence free vector spherical harmonics Y ℓ,m for |m| ≤ ℓ, ℓ ∈ N [44, page 216]. Let us recall also the addition theorem for vector spherical harmonics [44, formula (81), page 221] |m|≤ℓ |Y ℓ,m (x)| 2 = 2ℓ + 1 4π P ℓ (1), x ∈ S 2 , and the fact that P ℓ (1) = 1 with P ℓ being the Legendre polynomial of degree ℓ. Therefore, (3.6) yields if and only if s > 1 2 and the lemma follows. Let X = L 4 (S 2 ) ∩ H denote the Banach space endowed with the norm Remark 3.5. It follows from Lemma 3.4 that the operator Let us recall, that X is an M-type 2 Banach space, see [6] for details. The Stokes operator −A restricted to X is an infinitesimal generator of an analytic semigroup. We will consider an operator in X defined by the formulâ where ν > 0, and C is the Coriolis operator. For the reader's convenience we recall a result presented in [10]. Lemma 3.7. Suppose, there exists a separable Hilbert space K ⊂ X such thatÂ −δ X ⊂ K and the operatorÂ −δ : K → X is γ-radonifying for some δ > 0. Then The space C ξ 1/2 (R, E) equipped with the the norm is a nonseparable Banach space. However, the closure of {ω ∈ C ∞ 0 (R) : ω(0) = 0} in C ξ 1/2 (R, E), denoted by Ω(ξ, E), is a separable Banach space. Let us denote by C 1/2 (R, X) the space of all continuous functions ω : R → X such that The space C 1/2 (R, E) endowed with the norm · C 1/2 (R,E) is a nonseparable Banach space. We denote by F the Borel σ-algebra on Ω(ξ, E). One can show [5] that for ξ ∈ (0, 1/2), there exists a Borel probability measure P on Ω(ξ, E) such that the canonical process w t , t ∈ R, defined by there exists a unique extension of z • i t to a bounded linear map W t : K → L 2 (Ω(ξ, E), F t , P). Moreover, the family (W t ) t∈R is an H-cylindrical Wiener process on a filtered probability space (Ω(ξ, E), F, P), where F = F t ) t∈R in the sense of e.g. [12].

3.2.
Ornstein-Uhlenbeck process. The following is our standing assumption.
The following results for the operatorÂ follow from Corollary 6.4, Theorem 6.6 and Corollary 6.8 in from [11], respectively.
Thus, τ s is a linear a bounded map from C 1/2 (R, X) into itself. Moreover, the family (τ s ) s∈R is a C 0 group on C 1/2 (R, X).
Using this notation Theorem 3.12 can be rewritten in the following way.
Proposition 3.14. The process z α (t), t ∈ R, is a stationary Ornstein-Uhlenbeck process. It is the solution of the equation i.e. for all t ∈ R, a.s.
where the integral is the Itô integral on the M-type 2 Banach space X in the sense of [6].
In particular, for some constant C depending on X, Moreover, E z α (t) 2 X tends to 0 as α → ∞. Proof. Stationarity of the process z α follows from equation (3.17). The equality (3.18) follows by finite-dimensional approximation.
From [6] we have Hence, we conclude that the last integral (3.20) is finite. Finally, the last statement follows from (3.20) by applying the Lebesgue Dominated Convergence Theorem.
By Proposition 3.14, z α (t), t ∈ R, is a stationary and ergodic X-valued process, hence by the Strong Law for Large Numbers (see Da Prato and Zabczyk [36] for a similar argument), It also follows from Proposition 3.14 that we can find α 0 such that for all α ≥ α 0 , where λ 1 is the constant appearing in the Poincaré inequality (2.22) and C > 0 is a certain universal constant.
It follows that as a model for a metric dynamical system we can take either the quadruple (Ω(ξ, E), F , P, ϑ) or the quadruple (Ω(ξ, E),F,P,θ), whereF,P, andθ are respectively the natural restrictions of F , P and ϑ toΩ(ξ, E).
Proof. All properties with the exception of the cocycle one of a random dynamical system follow from Theorem 3.3. Hence we only need to show that for any x ∈ H, ϕ(t + s, ω)x = ϕ(t, ϑ s ω)ϕ(s, ω)x, t, s ∈ R + .
The proof can be completed by applying similar techniques to those used in [11,Theorem 6.15] in the case of the stochastic Navier-Stokes equations in a 2-dimensional unbounded domain.
Suppose that Assumption 3.8 is satisfied. If u s ∈ H, s ∈ R, f ∈ V ′ and W t , t ∈ R is a two-sided Wiener process introduced after (3.10) such that the Cameron-Martin (or Reproducing Kernel Hilbert) space of the Gaussian measure L(w 1 ) is equal to K. A process ) is a solution to problem (3.1) iff u(s) = u s and for any φ ∈ V , t > s, Proposition 4.2. In the framework as above, suppose that ) is a solution to problem (3.1), then for any α ≥ 0, a process v α (t), t ≥ s, defined by z α (t) = u(t)−v α (t), t ≥ s, is a solution to (3.3) on [s, ∞).
Our previous results yield the existence and the uniqueness of solutions to problem (3.1) as well as its continuous dependence on the data (in particular on the initial value u 0 and the force f). Moreover, if we define, for x ∈ H, ω ∈ Ω, and t ≥ s, then for each s ∈ R and each u 0 ∈ H, the process u(t), t ≥ s, is a solution to problem (3.1). Before presenting the main results of this section, let us recall the weak continuity of the RDS generated by stochastic NSEs on the sphere as stated in the last part of Theorem 3.3.

From (2.46) and invoking the Young inequality, we have
Hence from (4.7) and (4.4), we get Next, using the Gronwall Lemma, we arrive at (4.5). By adding and subtracting ν λ 1 2 v(t) 2 from (4.7) we find that  Lemma 4.5. Under the above assumptions, for each ω ∈ Ω(ξ, E),  Then all these functions belong to the class R.
Proof. Since by Theorem 3.12, z(ϑ −t ω)(s) = z(ω)(s − t), we have Hence, multiplying the above by e −νλ 1 t e 3C 2 ν 0 −t z(ω)(r) 2 L 4 dr we get This, together with Lemma 4.4 concludes the proof in the case of function r 2 . In the case of r 1 , we have L 4 dr . Thus, by Lemma 4.4 we infer that r 1 also belongs to the class R. The argument in the case of function r 3 is similar but for the sake of the completeness we include it here. From the first part of the proof we infer that Since by Lemma 4.5 The proof in the other cases is analogous.
We have the following trivial results.
Proposition 4.8. The class R is closed with respect to sum, multiplication by a constant and if r ∈ R, 0 ≤r ≤ r, thenr ∈ R. The class R is closed with respect to sum, multiplication by a constant and if r ∈ R, 0 ≤r ≤ r, thenr ∈ R. Definition 4.9. A closed random set K(ω) is said to absorb another closed random set B(ω) iff for all ω ∈ Ω, there exists a time t B (ω) such that Set, for ω ∈ Ω, Using Lemma 4.5 and Proposition 4.7 we conclude that both r 11 and r 12 belong to R and that r 13 := r 11 + r 12 belongs to R as well. Therefore, the random set B defined by B(ω) := {u ∈ H : u ≤ r 13 (ω)} belongs to the family DR. Now we will show that B absorbs D. Let ω ∈ Ω be fixed. Since r D ∈ R there exists t D (ω) ≥ 0, such that Thus, if x ∈ D(ϑ −t ω) and s ≥ t D (ω), then by (4.10), v(0, s; ω, x − z(s)) ≤ r 11 (ω). Thus we infer that In other words, u(0, s; ω, x) ∈ B(ω), for all s ≥ t D (ω). This proves that B absorbs D.
Definition 4.11. A RDS ϑ-cocycle ϕ defined on a separable Banach space X is said to be D-asymptotically compact iff for each D ∈ D, for every ω ∈ Ω, for any positive sequence (t n ) such that t n → ∞ and for any sequence {x n } such that x n ∈ D(ϑ −tn ω), for all n ∈ N, the set {ϕ(t n , ϑ −tn ω)x n : n ∈ N} is relatively compact in X.
Proposition 4.12. Assume that for each random set D ∈ DR, there exists a random set B ∈ DR such that B absorbs D. Then the RDS ϕ is DR-asymptotically compact.
The proof of the proposition is adapted from [8], in which a RDS generated by NSEs on some 2-dimensional unbounded domain was considered. The proposition generalises the asymptotically compactness of the RDS in [11,Proposition 8.1] to the DR-asymptotically compactness of the RDS.
Proof. Suppose that B is a closed random set from the class DR and K ∈ DR is a close random set which absorbs B. We fix ω ∈ Ω. Let us take an increasing sequence of positive numbers (t n ) ∞ n=1 such that t n → ∞ and an H-valued sequence (x n ) n such that x n ∈ B(ϑ −tn ω), for all n ∈ N.
Since z(0) ∈ H, we also have In particular, (4.14) We claim that it is enough to prove that for some subsequence Indeed, since H is a Hilbert space, (4.14) in conjunction with (4.15) imply that Therefore, in order to show that {ϕ(t n , ϑ −tn ω)x n } n is relatively compact in H we need to prove that (4.15) holds true.
Hence, from (4.17) and by applying the last part of Theorem 3.3, we get (4.19) where w-lim denotes the limit in the weak topology on H. The same proof yields a more general property: Before continuing with the proof, let us point out that (4.19) means precisely that Step III. Proof of (4.15). From now on, unless explicitly stated, we fix k ∈ N, and we will consider problem Let v be the solution to (3.3) on [−k, ∞) with z = z α (·, ω) and the initial condition at From (4.20) and (4.6) with t = 0 and τ = −k we infer that (4.21) It is enough to find a nonnegative function h ∈ L 1 (−∞, 0) such that (4.22) lim sup For, if we define the diagonal process (m j ) ∞ j=1 by m j = j (j) , j ∈ N, then for each k ∈ N, the sequence (m j ) ∞ j=k is a subsequence of the sequence (n (k) ) and hence by (4.22), Taking the k → ∞ limit in the last inequality we infer that lim sup which proves claim (4.15).
Step IV. Proof of (4.22). We begin with estimating the first term on the RHS of (4.21). If −t n (k) < −k, then by (4.3) and (4.5) we infer that (4.23) First we will find a nonnegative function h ∈ L 1 (−∞, 0) such that (4.24) lim sup This will be accomplished as soon as we prove the following four lemmas.
Proof of Lemma 4.13. We recall that for α ∈ N, z(t) = z α (t), t ∈ R, being the Ornstein-Uhlenbeck process from subsection 3.2, one has Let us recall that the spaceΩ(ξ, E) was constructed in such a way that Therefore, since the embedding X ֒→ L 4 (S 2 ) is a contraction, we have for n (k) sufficiently large, Since the set B is bounded in H, there exists ρ 1 > 0 such that for all n (k) , x n (k) ≤ ρ 1 . Hence Proof of Lemma 4.16. We denote by p(s) = νλ 1 s + 3C 2 ν 0 s z(s) 2 L 4 . As in the proof of Lemma 4.13 we have, for s ≤ s 0 , p(s) < νλ 1 2 s. Hence 0 −∞ e p(s) ds < ∞, as required.
Proof of Lemma 4.15. Since g(s) 2 It is enough to consider the case of z(s) 4 L 4 since the proof will be similar for the remaining case. Reasoning as in (4.25), we can find t 0 ≥ 0 such that for t ≥ t 0 , Taking into account the inequality (4.27), we have z(t) ≤ ρ 2 (1 + |t|), t ∈ R. Therefore, with ρ 3 := exp( By the continuity of all relevant functions, we can let t 0 → 0 to get the result. Therefore, the proof of (4.24) is concluded, and it only remains to finish the proof of (4.22). Let us denote by From (4.17) and the last part of Theorem 3.3 we infer that Since e νλ 1 · g, e νλ 1 · f ∈ L 2 (−k, 0; V ′ ), we get On the other hand, using the same methods as those in the proof of Theorem 3.2, there exists a subsequence of {v n (k) }, which, for the sake of simplicity of notation, is denoted as the old one which satisfies (4.32) v n (k) → v k strongly in L 2 (−k, 0; L 2 loc (S 2 )). Next, since z(t) is an L 4 -valued process, so is e νλ 1 t z(t). Thus by [10, Corollary 4.1], (4.29) and (4.32), we infer that Now we are ready to state and prove the main result of this paper. A result of similar type for the Navier-Stokes equations on some 2-dimensional unbounded domain has been discussed in [8].
Theorem 4.17. Consider the metric DS T = Ω (ξ, E),F,P,θ from Proposition 3.15, and the RDS ϕ on H over T generated by the stochastic Navier-Stokes equations on the 2-dimensional unit sphere with additive noise (3.1) satisfying Assumption 3.8. Then the family A of sets defined by A(ω) = Ω B (ω) for all ω ∈ Ω, is the minimal DR-attractor for ϕ, isF -measurable, and where the union in (4.37) is made for all bounded and closed nonempty deterministic subsets C of H.
Proof. The proof follows from Proposition 4.12 and Theorem 2.8 in [8].

Invariant measure
In this section we consider the existence of an invariant measure. The main result in this section, i.e. Theorem 5.2 is a direct consequence of Corollary 4.4 [17] and our Theorem 4.17about the existence of an attractor for the RDS generated by the stochastic Navier-Stokes equations (3.1).
Let ϕ be the RDS corresponding to the SNSEs (3.1) and defined in (4.1). We define the transition operator P t by a standard formula. For g ∈ B b (H), we put P t g(x) = Ω [g(ϕ(t, ω, x))] dP(ω), x ∈ H. (5.1) As in [11,Proposition 3.8] we have the following result whose proof is simply a repetition of the proof from [11] Proposition 5.1. The family (P t ) t≥0 is Feller, i.e. P t g ∈ C b (H) if g ∈ C b (H). Moreover, for any g ∈ C b (X), P t g(x) → g(x) as t ց 0.
Following [17] one can prove that ϕ is a Markov RDS, i.e. P t+s = P t P s for all t, s ≥ 0. Hence by [11,Corollary 3.10] which says that a time-continuous and continuous asymptotically compact, Markov RDS ϕ admits a Feller invariant probability measure µ, i.e. a Borel probability measure µ P * t µ = µ, t ≥ 0, (5.2) where P * t µ(Γ) = H P t (x, Γ) µ(dx), Γ ∈ B(H), and P t (x, ·) is the transition probability, P t (x, Γ) = P t 1 Γ (x), x ∈ H. A Feller invariant probability measure for a Markov RDS ϕ on H is, by definition, an invariant probability measure for the semigroup (P t ) t≥0 defined by (5.1). Therefore, we obtain the following result.
Theorem 5.2. There exists an invariant measure for the stochastic NSE (3.1).
Remark 5.3. We believe that the uniqueness of an invariant measure for nondegenerate noise will follow from the classical procedure and as in [9]. If the noise is degenerate and spatially smooth, it seems that the results from a recent paper by Hairer and Mattingly [28] should be applicable in our setting. In particular, [28,Theorem 8.4], which gives a sufficient conditions for uniqueness in terms of controllability, should be applicable. Details will be published elsewhere. One should point out that these authors use the "vorticity" formulation and their initial data belongs to the L 2 space. This corresponds to our approach with the initial data belonging to the finite enstrophy space H 1 . However, we work in the space of finite energy, which seems to be physically more natural. On the other hand, verifying the sufficient conditions could be more challenging. For the NSE without the Coriolis force this problems has been investigated in [1]. Corresponding NSE with the Coriolis force study is postponed till the next publication.