Stochastic Navier–Stokes Equations on a Thin Spherical Domain

Incompressible Navier–Stokes equations on a thin spherical domain Qε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_\varepsilon $$\end{document} along with free boundary conditions under a random forcing are considered. The convergence of the martingale solution of these equations to the martingale solution of the stochastic Navier–Stokes equations on a sphere S2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^2$$\end{document} as the thickness converges to zero is established.


Introduction
For various motivations, partial differential equations in thin domains have been studied extensively in the last few decades; e.g. Babin and Vishik [4], Ciarlet [16], Ghidaglia and Temam [18], Marsden et al. [37] and references there in. The study of the Navier-Stokes equations (NSE) on thin domains originates in a series of papers by Hale and Raugel [20][21][22] concerning the reaction-diffusion and damped wave equations on thin domains. Raugel and Sell [44,45] proved the global existence of strong solutions to NSE on thin domains for large initial data and forcing terms, in the case of purely periodic and periodic-Dirichlet boundary conditions. Later, by applying a contraction principle argument and carefully analysing the dependence of the solution on the first eigenvalue of the corresponding Laplace operator, Arvin [2] showed global existence of strong solutions of the Navier-Stokes equations on thin threedimensional domains for large data. Temam and Ziane [52] generalised the results of [44,45] to other boundary conditions. Moise et al. [41] proved global existence of strong solutions for initial data larger than in [45]. Iftimie [26] showed the existence and uniqueness of solutions for less regular initial data which was further improved by Iftimie and Raugel [27] by reducing the regularity and increasing the size of initial data and forcing.
In the context of thin spherical shells, large-scale atmospheric dynamics that play an important role in global climate models and weather prediction can be described by the 3-dimensional Navier-Stokes equations in a thin rotating spherical shell [34,35]. Temam and Ziane in [53] gave the mathematical justification for the primitive equations of the atmosphere and the oceans which are known to be the fundamental equations of meteorology and oceanography [36,43]. The atmosphere is a compressible fluid occupying a thin layer around the Earth and whose dynamics can be described by the 3D compressible Navier-Stokes equations in thin layers. In [53] it was assumed that the atmosphere is incompressible and hence a 3D incompressible NSE on thin spherical shells could be used as a mathematical model. They proved that the averages in the radial direction of the strong solutions (whose existence for physically relevant initial data was established in the same article) to the NSE on the thin spherical shells converge to the solution of the NSE on the sphere as the thickness converges to zero. In a recent paper Saito [46] studied the 3D Boussinesq equations in thin spherical domains and proved the convergence of the average of weak solutions of the 3D Boussinesq equations to a 2D problem. More recent work on incompressible viscous fluid flows in a thin spherical shell was carried out in [23][24][25].
For the deterministic NSE on the sphere, Il'in and Filatov [28][29][30] considered the existence and uniqueness of solutions while Temam and Wang [51] considered inertial forms of NSE on spheres. Brzeźniak et al. proved the existence and uniqueness of the solutions to the stochastic NSE on the rotating two dimensional sphere and also proved the existence of an asymptotically compact random dynamical system [9]. Recently, Brzeźniak et al. established [10] the existence of random attractors for the NSE on two dimensional sphere under random forcing irregular in space and time deducing the existence of an invariant measure.
The main objective of this article is to establish the convergence of the martingale solution of the stochastic Navier-Stokes equations (SNSE) on a thin spherical domain Q ε , whose existence can be established as in the forthcoming paper [7] to the martingale solution of the stochastic Navier-Stokes equations on a two dimensional sphere S 2 [9] as thickness ε of the spherical domain converges to zero. In this way we also give another proof for the existence of a martingale solution for stochastic NSE on the unit sphere S 2 .
We study the stochastic Navier-Stokes equations (SNSE) for incompressible fluid (2) in thin spherical shells along with free boundary conditions In the above, u ε = ( u r ε , u λ ε , u ϕ ε ) is the fluid velocity field, p is the pressure, ν > 0 is a (fixed) kinematic viscosity, u ε 0 is a divergence free vector field on Q ε and n is the unit outer normal vector to the boundary ∂ Q ε and W ε (t), t ≥ 0 is an R N -valued Wiener process in some probability space (Ω, F, F, P) to be defined precisely later.
The main result of this article is Theorem 3, which establishes the convergence of the radial averages of the martingale solution (see Definition 1) of the 3D stochastic equations (1)- (5), as the thickness of the shell ε → 0, to a martingale solution u (see Definition 2) of the following stochastic Navier-Stokes equations on the unit sphere S 2 ⊂ R 3 : where u = (u λ , u ϕ ) and , ∇ are the Laplace-de Rham operator and the surface gradient on S 2 respectively. Assumptions on initial data and external forcing will be specified later.
The paper is organised as follows. We introduce necessary functional spaces in Sect. 2. In Sect. 3, we define some averaging operators and give their properties. Stochastic Navier-Stokes equations on thin spherical domains are introduced in Sect. 4 and a priori estimates for the radially averaged velocity are obtained which are later used to prove the convergence of the radial average of a martingale solution of stochastic NSE on thin spherical shell (see (1)- (5)) to a martingale solution of the stochastic NSE on the sphere (see (6)- (8)) with vanishing thickness.
For p ∈ [1, ∞), by L p (Q ε ), we denote the Banach space of (equivalence-classes of) Lebesgue measurable R-valued pth power integrable functions on Q ε . The R 3valued pth power integrable vector fields will be denoted by L p (Q ε ). The norm in L p (Q ε ) is given by If p = 2, then L 2 (Q ε ) is a Hilbert space with the inner product given by By H 1 (Q ε ) = W 1,2 (Q ε ), we will denote the Sobolev space consisting of all u ∈ L 2 (Q ε ) for which there exist weak derivatives D i u ∈ L 2 (Q ε ), i = 1, 2, 3. It is a Hilbert space with the inner product given by The Lebesgue and Sobolev spaces on the sphere S 2 will be denoted by L p (S 2 ) and W s,q (S 2 ) respectively for p, q ≥ 1 and s ≥ 0. In particular, we will write H 1 (S 2 ) for W 1,2 (S 2 ).

Functional Setting on the Shell Q "
We will use the following classical spaces on Q ε : H ε = u ∈ L 2 (Q ε ) : div u = 0 in Q ε , u · n = 0 on ∂ Q ε , V ε = u ∈ H 1 (Q ε ) : div u = 0 in Q ε , u · n = 0 on ∂ Q ε , On H ε , we consider the inner product and the norm inherited from L 2 (Q ε ) and denote them by (·, ·) H ε and · H ε respectively, that is Let us define a bilinear map a ε : where curl u = ∇ × u, and for u ∈ V ε , we define Note that for u ∈ V ε , u V ε = 0 implies that u is a constant vector and u · n = 0 on ∂ Q ε i.e., u is tangent to Q ε for every y ∈ ∂ Q ε , and thus must be 0. Hence · V ε is a norm on V ε (other properties can be verified easily). Under this norm V ε is a Hilbert space with the inner product given by We denote the dual pairing between V ε and V ε by ·, · ε , that is ·, · ε := V ε ·, · V ε . By the Lax-Milgram theorem, there exists a unique bounded linear operator A ε : V ε → V ε such that we have the following equality : The operator A ε is closely related to the Stokes operator A ε defined by The Stokes operator A ε is a non-negative self-adjoint operator in H ε (see Appendix B). Also note that We recall the Leray-Helmholtz projection operator P ε , which is the orthogonal projector of L 2 (Q ε ) onto H ε . Using this, the Stokes operator A ε can be characterised as follows : We also have the following characterisation of the Stokes operator A ε [53, For u ∈ V ε , v ∈ D(A ε ), we have the following identity (see Lemma 28) Let b ε be the continuous trilinear from on V ε defined by : We denote by B ε the bilinear mapping from V ε × V ε to V ε by and we set Let us also recall the following properties of the form b ε , which directly follows from the definition of b ε : In particular,

Functional Setting on the Sphere S 2
Let where Δ is the Laplace-Beltrami operator on the sphere (see (177)). We similarly define H s (S 2 ) as the space of all vector fields u ∈ where is the Laplace-de Rham operator on the sphere (see (180)).
For s ≥ 0, H s (S 2 ), · H s (S 2 ) and H s (S 2 ), · H s (S 2 ) are Hilbert spaces under the respective norms, where and By the Hodge decomposition theorem [3, Theorem 1.72] the space of C ∞ smooth vector fields on S 2 can be decomposed into three components: where and H is the finite-dimensional space of harmonic vector fields. Since the sphere is simply connected, H = {0}. We introduce the following spaces Note that it is known (see [50]) Given a tangential vector field u on S 2 , we can find vector fieldũ defined on some neighbourhood of S 2 such that their restriction to S 2 is equal to u, that isũ| S 2 = u ∈ T S 2 . Then we define Since x is orthogonal to the tangent plane T x S 2 , curl u is the normal component of ∇ ×ũ. It could be identified with a normal vector field when needed. We define the bilinear form a : and hence is continuous on V. So by the Riesz representation theorem, there exists a unique operator A : Using the Poincaré inequality, we also have a(u, u) ≥ α u 2 V , for some positive constant α, which means a is coercive in V. Hence, by the Lax-Milgram theorem, the operator A : V → V is an isomorphism.
Next we define an operator A in H as follows: By Cattabriga [15], see also Temam [49,p. 56], one can show that A is a non-negative self-adjoint operator in H. Moreover, V = D(A 1/2 ), see [49, p. 57].
Let P be the orthogonal projection from L 2 (S 2 ) to H, called the Leray-Helmholtz projection. It can be shown, see [19, p. 104], that D(A) along with the graph norm forms a Hilbert space with the inner product Note that D(A)-norm is equivalent to H 2 (S 2 )-norm. For more details about the Stokes operator on the sphere and fractional power A s for s ≥ 0, see [9,Sec. 2.2]. Given two tangential vector fields u and v on S 2 , we can find vector fieldsũ andṽ defined on some neighbourhood of S 2 such that their restrictions to S 2 are equal to, respectively, u and v. Then we define the covariant derivative where π x is the orthogonal projection from R 3 onto the tangent space T x S 2 to S 2 at x. By decomposingũ andṽ into tangential and normal components and using orthogonality, one can show that where in the last equality, we use the fact that x · v = 0 for any tangential vector v. We set v = u and use the formula to obtain Using (26) for the vector fieldsũ andṽ = ∇ ×ũ = curlũ, we have Thus where dσ (x) is the surface measure on S 2 .

Averaging Operators and Their Properties
In this section we recall the averaging operators which were first introduced by Raugel and Sell [44,45] for thin domains. Later, Temam and Ziane [53] adapted those averaging operators to thin spherical domains, introduced some additional operators and proved their properties using the spherical coordinate system. Recently, Saito [46] used these averaging operators to study Boussinesq equations in thin spherical domains. We closely follow [46,53] to describe our averaging operators and provide proofs for some of the properties mentioned below. Let M ε : C(Q ε , R) → C(S 2 , R) be a map that projects functions defined on Q ε to functions defined on S 2 and is defined by Remark 1 We will use the Cartesian and spherical coordinates interchangeably in this paper. For example, if x ∈ S 2 then we will identify it by x = (λ, ϕ) where λ ∈ [0, π] and ϕ ∈ [0, 2π). (28) is continuous (and linear) w.r.t norms L 2 (Q ε ) and L 2 (S 2 ). Moreover,

Lemma 1 The map M ε as defined in
Proof Take ψ ∈ C(Q ε ) then by the definition of M ε we have Thus, using the Cauchy-Schwarz inequality we have where the last equality follows from the fact that is the volume integral over the spherical shell Q ε in spherical coordinates, with being the Lebesgue measure over a unit sphere. Therefore, we obtain and hence the map is bounded and we can infer (29).

Corollary 1
The map M ε as defined in (28) has a unique extension, which without the abuse of notation will be denoted by the same symbol M ε : is a bounded map thus by the Riesz representation theorem there exists a unique extension.

Lemma 2
The following map is bounded and Proof It is sufficient to consider ψ ∈ C(S 2 ). For ψ ∈ C(S 2 ), we have thus, showing that the map R ε is bounded w.r.t. L 2 (S 2 ) and L 2 (Q ε ) norms.

Remark 2
It is easy to check that the dual operator M * ε : Next we define another map Courtesy of Corollary 1 and Lemma 2, M ε is well-defined and bounded. Using definitions of maps R ε and M ε , we have Lemma 5 Let ψ ∈ L 2 (Q ε ), then we have the following scaling property Proof Let ψ ∈ L 2 (Q ε ). Then by the defintion of the map M ε , we have The normal component of a function ψ defined on Q ε when projected to S 2 is given by the map N ε which is defined by i.e.
The following result establishes an important property of the map N ε .
Proof Let us choose and fix ψ ∈ L 2 (Q ε ). Then by the definitions of the operators involved we have the following equality in L 2 (S 2 ): Therefore ,we deduce that in order to prove equality (37), it is sufficient to show that Hence, by taking into account definitions (36) of N ε and (34) of M ε , we infer that it is sufficient to prove that Let us choose ψ ∈ C(Q ε ) and put φ = M ε ψ, i.e.
Note that Thus, we infer that Thus, we proved M ε • R ε • M ε ψ = M ε ψ for every ψ ∈ C(Q ε ). Since C(Q ε ) is dense in L 2 (Q ε ) and the maps M ε and M ε • R ε • M ε are bounded in L 2 (Q ε ), we conclude that we have proved (37).
By the definition (34) of the map M ε and by Lemma 6, we infer that a.e. on S 2 by Lemma 6 dσ (x) = 0.
Next we define projection operators for R 3 -valued vector fields using the above maps (for scalar functions), as follows : Moreover, if u satisfies the boundary condition u · n = 0, then Proof The normal vector n to ∂ Q ε is given by n = (1, 0, 0). Thus by the definition of M ε we have Now for the second part, from the definition of N ε we have We also have the following generalisation of Lemma 6.

Lemma 9
Let u ∈ L 2 (Q ε ), then The following Lemma makes sense only for vector fields.
Now considering each of the terms individually, we have Using (43) and (44) in the equality (42), we obtain div M ε u = 1 Since u ∈ U ε , div u = 0 in Q ε , which implies Using this in (45), we get Thus, we have proved that div M ε u = 0, for every u ∈ U ε . Since, U ε is dense in H ε , it holds true for every u ∈ H ε too. The second part follows from the definition of N ε and H ε .
From Lemmas 8 and 10, we infer the following corollary : Using the definition of maps M ε and N ε and Lemma 7, we conclude: Moreover, Finally we define a projection operator that projects R 3 -valued vector fields defined on Q ε to the "tangent" vector fields on sphere S 2 .
Remark 3 Similar to the scalar case, one can prove that the dual operator M • * ε : Using the identities (168)-(170), we can show that for a divergence free smooth vector field u We define a weighted L 2 -product on H ε by and the corresponding norm will be denoted by · r which is equivalent to · L 2 (Q ε ) , uniformly for ε ∈ (0, 1 2 ) : We end this section by recalling a lemma and some Poincaré type inequalities from [53]. Moreover, Proof Let ε ∈ (0, 1 2 ) and u ∈ V ε . Then, by relation (50), equivalence of norms (52) and Eq. (53), we have The second inequality can be proved similarly.
The following two lemmas are taken from [53]. For the sake of completeness and convenience of the reader we have provided the proof in Appendix C.
Proof Let u ∈ V ε , then by the Hölder inequality, we have Thus, by Lemmas 13 and 14, we get In the following lemma we enlist some properties of operators M ε , N ε , M ε and N ε .
Proof of first part of (59). Let ψ ∈ C(Q ε ). Put φ = N ε ψ ∈ C(Q ε ). By Lemma 6 Therefore, for y ∈ Q ε , Therefore, we infer that for all y ∈ Q ε . Thus, we have established first part of (59) for all ψ ∈ C(Q ε ). Using the density argument, we can prove it for all ψ ∈ L 2 (Q ε ). Now for (58), by the definition of N ε and (59), we obtain Again using the definition of N ε and Eq.(57), we have concluding the proof of second part of (59). Proof of (60).
Thus, by the definition of M ε and identity (57) We can extend this to u ∈ L 2 (Q ε ) by the density argument. The remaining identities can be also established similarly as in the case of scalar functions.
Later in the proof of Theorem 3, in order to pass to the limit we will use an operator where Using the definition of map R ε from Lemma 2, we can rewrite R One Next we establish certain scaling properties for the map R Using the definition of the map R • ε and Lemmas 3, 4 we can deduce the following two lemmas (we provide the detailed proof of the latter in Appendix C): where is defined in (180).

Stochastic NSE on Thin Spherical Domains
This section deals with the proof of our main result, Theorem 3. First we introduce our two systems; stochastic NSE in thin spherical domain and stochastic NSE on the sphere, then we present the definition of martingale solutions for both systems. We also state the assumptions under which we prove our result. In Sect. 4.1, we obtain a priori estimates (formally) which we further use to establish some tightness criterion (see Sect. 4.2) which along with Jakubowski's generalisation of Skorokhod Theorem gives us a converging (in ε) subsequence. At the end of this section we show that the limiting object of the previously obtained converging subsequence is a martingale solution of stochastic NSE on the sphere (see Sect. 4.3).
In thin spherical domain Q ε , which was introduced in (3), we consider the following stochastic Navier-Stokes equations (SNSE) Recall that, u ε = ( u r ε , u λ ε , u ϕ ε ) is the fluid velocity field, p is the pressure, ν > 0 is a (fixed) kinematic viscosity, u ε 0 is a divergence free vector field on Q ε and n is the unit normal outer vector to the boundary ∂ Q ε . We assume that 1 N ∈ N. We consider a family of maps for some g Finally we assume that W ε (t), t ≥ 0 is an R N -valued Wiener process defined on the probability space (Ω, F, F, P). We assume that β j N j=1 are i.i.d real valued Brownian motions such that In this section, we shall establish convergence of the radial averages of the martingale solution of the 3D stochastic equations (69)-(72), as the thickness of the shell ε → 0, to a martingale solution u of the following stochastic Navier-Stokes equations on the sphere S 2 : where u = (0, u λ , u ϕ ) and , ∇ are as defined in (176)-(180). Assumptions on initial data and external forcing will be specified later (see Assumptions 1,2). Here, where N ∈ N, β j N j=1 are i.i.d real valued Brownian motions as before and g j N j=1 are elements of H, with certain relation to g j ε , which is specified later in Assumption 2.

Remark 4
We are aware of other formulations of the Laplacian in (75) such as the one with an additional Ricci tensor term [47,48]. However, as it was written in [47, p. 144], "Deriving appropriate equations of motion involves dynamical considerations which do not seem adapted to Riemannian space; in particular it is not evident how to formulate the principle of conservation of momentum." Therefore, in this paper, we follow the approach presented in [53], that the Navier-Stokes equations on the sphere is the thin shell limit of the 3-dimensional Navier-Stokes equations defined on a thin spherical shell. Now, we specify assumptions on the initial data u ε 0 and external forcing f ε , g j ε .
Assumption 1 Let (Ω, F, F, P) be the given filtered probability space. Let us assume that p ≥ 2 and that u ε 0 ∈ H ε , for ε ∈ (0, 1], such that for some C 1 > 0 We also assume that Let W ε be an R N -valued Wiener process as before and assume that such that, using convention (73), for each j = 1, . . . , N , Projecting the stochastic NSE (on thin spherical shell) (69)-(72) onto H ε using the Leray-Helmholtz projection operator and using the definitions of operators from Sect. 2.1, we obtain the following abstract and, for all t ∈ [0, T ] and v ∈ V ε , P-a.s., In the following remark we show that a martingale solution u ε of (82), as defined above, satisfies an equivalent equation in the weak form.

Remark 5
Let u ε = u ε (t), t ≥ 0 be a martingale solution of (82). We will use the following notations and also from Lemma 15 we have Then, for φ ∈ D(A), we have , and using Lemma 6, Proposition 1 and Lemma 12, we can rewrite the weak formulation identity (83) as follows.
where ·, · denotes the duality between V and V.
Next, we present the definition of a martingale solution for stochastic NSE on S 2 .

Definition 2 A martingale solution to equation (75)-(77) is a system
where Ω, F, P is a probability space and F = F t t≥0 is a filtration on it, such that -W is an R N -valued Wiener process on Ω, F , F, P , u is V-valued progressively measurable process, H-valued continuous F-adapted process such that for all t ∈ [0, T ] and φ ∈ V.
And finally, we assume that G ∈ L p (0, T ; T 2 (R N ; H)), such that for each j = for some M > 0.
Remark 6 (Existence of martingale solutions) In a companion paper [7] we will address an easier question about the existence of a martingale solution for (1)-(5) in a more general setting with multiplicative noise. The key idea of the proof is taken from [11], where authors prove existence of a martingale solution for stochastic NSE in unbounded 3D domains. The existence of a pathwise unique strong solution (hence a martingale solution) for the stochastic NSE on a sphere S 2 is already established by two of the authors and Goldys in [9]. Through this article we give another proof of the existence of a martingale solution for such a system. We end this subsection by stating the main theorem of this article. Remark 7 According to Remark 6, for every ε ∈ [0, 1] there exists a martingale solution of (69)-(72) as defined in Definition 1, i.e. we will obtain a tuple Ω ε , F ε , F ε , P ε , W ε , u ε as a martingale solution. It was shown in [31] that is enough to consider only one probability space, namely, where L denotes the Lebesgue measure on [0, 1]. Thus, it is justified to consider the probability space (Ω, F, P) independent of ε in Theorem 3.

Estimates
From this point onward we will assume that for every ε ∈ (0, 1] there exists a martingale solution Ω, F, F, P, W ε , u ε of (82). Please note that we do not claim neither we use the uniqueness of this solution.
The main aim of this subsection is to obtain estimates for α ε and β ε uniform in ε using the estimates for the process u ε .
The energy inequality (90) and the higher-order estimates (105)-(106), satisfied by the process u ε , as obtained in Lemmas 19 and 22 is actually a consequence (essential by-product) of the existence proof. In principle, one obtains these estimates (uniform in the approximation parameter N ) for the finite-dimensional process u (N ) ε (using Galerkin approximation) with the help of the Itô lemma. Then, using the lower semicontinuity of norms, convergence result ( u (N ) ε → u ε in some sense), one can establish the estimates for the limiting process. Such a methodology was employed in a proof of Theorem 4.8 in the recent paper [13] by the first named author, Motyl and Ondreját.
In Lemmas 19 and 22 we present a formal proof where we assume that one can apply (ignoring the existence of Lebesgue and stochastic integrals) the Itô lemma to the infinite dimensional process u ε . The idea is to showcase (though standard) the techniques involved in establishing such estimates. H ε )). Then, the martingale solution u ε of (82) satisfies the following energy inequality where K is some positive constant independent of ε.
Proof Using the Itô formula for the function ξ 2 L 2 (Q ε ) with the process u ε , for a fixed t ∈ [0, T ] we have (91) Using the Cauchy-Schwarz inequality and the Young inequality, we get the following estimate which we use in (91), to obtain Taking the supremum of (92) over the interval [0, T ], then taking expectation and using inequality (93) we infer the energy inequality (90).
Let us recall the following notations, which we introduced earlier, for t ∈ [0, T ] Lemma 20 Let u ε be a martingale solution of (82) and Assumption 1 hold, in particular, for p = 2. Then where C 1 , C 2 are positive constants from (79) and (80) and C 3 > 0 (determined within the proof) is another constant independent of ε.
Proof Let u ε be a martingale solution of (82), then it satisfies the energy inequality (90). From Eq. (47), we have Moreover, by Corollary 3 Therefore, using (96) and (97) in the energy inequality (90), we get and hence from the scaling property, Lemma 11, we have By the assumptions on g j ε (81), there exists a positive constant c such that for every Therefore, using Assumption 1 and (99) in (98), cancelling ε on both sides and defining C 3 = N K c, we infer inequality (95).
In the following lemma we obtain some higher order estimates (on a formal level) for the martingale solution u ε , which will be used to obtain the higher order estimates for the processes α ε and β ε .

Lemma 22
Let Assumption 1 hold true and u ε be a martingale solution of (82). Then, for p > 2 we have following estimates and where

and K 1 is a constant from the Burkholder-Davis-Gundy inequality.
Applying the Itô lemma with F(x) and process u ε for t ∈ [0, T ], we have Using the fact that B ε ( u ε , u ε ), u ε ε = 0 and A ε u ε , u ε ε = u ε 2 V ε we arrive at Using (107) and the Cauchy-Schwarz inequality, we get Using the generalised Young inequality abc ≤ a q /q + b r /r + c s /s (where 1/q + (108) Again using the Young inequality with exponents p/( p − 2), p/2 we get Using (108) and (109) we obtain (110) Since u ε is a martingale solution of (82) it satisfies the energy inequality (90), hence the real-valued random variable is a F t -martingale. Taking expectation both sides of (110) we obtain (111) Therefore, by the Gronwall lemma we obtain By Burkholder-Davis-Gundy inequality, we have where in the last step we have used the Young inequality with exponents p/( p − 2) and p/2. Taking supremum over 0 ≤ s ≤ t in (110) and using (112) we get Thus using the Gronwall lemma, we obtain where C 2 p, u ε 0 , f ε , G ε and K p are the constants as defined in the statement of lemma. We deduce (106) from (113) and (105).
In the following lemma we will use the estimates from previous lemma to obtain higher order estimates for α ε and β ε .

Lemma 23
Let p > 2. Let u ε be a martingale solution of (82) and Assumption 1 hold with the chosen p. Then, the processes α ε and β ε (as defined in (94)) satisfy the following estimates and E sup where K (ν, p) is a positive constant independent of ε and K p is defined in Lemma 22. Proof The lemma can be proved following the steps of Lemmas 20 and 21 with the use of Proposition 1, scaling property from Lemma 11, the Cauchy-Schwarz inequality, Assumptions 1, 2 and the estimates obtained in Lemma 22.

Tightness
In this subsection we will prove that the family of laws induced by the processes α ε is tight on an appropriately chosen topological space Z T . In order to do so we will consider the following functional spaces for fixed T > 0: and let T be the supremum 3 of the corresponding topologies.
The proof of lemma turns out to be a direct application of Corollary 6. Indeed, by Lemma 20, assumptions (a) and (b) of Corollary 6 are satisfied and therefore, it is sufficient to show that the sequence (α ε ) ε>0 satisfies the Aldous condition [A], see Definition 6, in space D(A −1 ).
In what follows, we will prove that each of the eight process from equality (119) satisfies the Aldous condition [A]. In order to help the reader, we will divide the following part of the proof into eight parts.
-For the first term, we obtain -Similarly for the second term we have -Now we consider the first non-linear term.
-Similarly for the second non-linear term, we have -Now as in the previous case, for the next mixed non-linear term, we obtain -Finally, for the last non-linear term, we get -Now for the term corresponding to the external forcing f ε , we have using the radial invariance of M ε f ε and assumption (80) -At the very end we are left to deal with the last term corresponding to the stochastic forcing. Using the radial invariance of M ε g j ε , Itô isometry, scaling (see Lemma 11) and assumption (81), we get After having proved what we had promised, we are ready to conclude the proof of Lemma 24. Since for every t > 0 one has for φ ∈ D(A), Let us fix κ > 0 and γ > 0. By equality (119), the sigma additivity property of probability measure and (129), we have Using the Chebyshev's inequality, we get Thus, using estimates (120)-(128) in (130), we get Since α ε satisfies the Aldous condition [A] in D(A −1 ), we conclude the proof of Lemma 24 by invoking Corollary 6.

Proof of Theorem 3
For every ε > 0, let us define the following intersection of spaces Now, choose a countable subsequence {ε k } k∈N converging to 0. For this subsequence define a product space Y T given by Now with this Y T -valued function we define a constant Y T -sequence Then by Lemma 24 and the definition of sequence η k , the set of measures Thus, by the Jakubowski-Skorohod theorem 4 there exists a subsequence (k n ) n∈N , a probability space ( Ω, F , P) and, on this probability space, Z T × Y T × C([0, T ]; R N )valued random variables ( u, η, W ), α ε kn , η k n , W ε kn , n ∈ N such that α ε kn , η k n , W ε kn has the same law as α ε kn , η k n , W and In particular, using marginal laws, and definition of the process η k , we have where β ε kn is the k n th component of Y T -valued random variable η k n . We are not interested in the limiting process η and hence will not discuss it further. Using the equivalence of law of W ε kn and W on C([0, T ]; R N ) for n ∈ N one can show that W and W ε kn are R N -valued Wiener processes (see [8,Lemma 5.2 and Proof] for details).
, the functions u, η are Z T , Y T Borel random variables respectively.
Using the retract operator R • ε : L 2 (S 2 ) → L 2 (Q ε ) as defined in (63)-(65), we define new processes α ε corresponding to old processes α ε on the new probability space as follows Moreover, by Lemma 16 we have the following scaling property for these new processes, i.e.
The following auxiliary result which is needed in the proof of Theorem 3, cannot be deduced directly from the Kuratowski Theorem (see Theorem 7).
and from estimates (95) and (114), for p ∈ [2, ∞) Since L 2 (0, T ; V) ∩ Z T is a Borel subset of Z T (Lemma 25), α ε and α ε have same laws on Z T ; from (95), we have Since the laws of η k n and η k n are equal on Y T , we infer that the corresponding marginal laws are also equal. In other words, the laws on B Y ε kn T of L( β ε kn ) and L( β ε kn ) are equal for every k n . Therefore, from the estimates (102) and (115) we infer for p ∈ [2, ∞) and By inequality (140) we infer that the sequence ( α ε ) ε>0 contains a subsequence, still denoted by ( α ε ) ε>0 convergent weakly (along the sequence ε k n ) in the space L 2 ([0, T ] × Ω; V). Since α ε → u in Z T P-a.s., we conclude that u ∈ L 2 ([0, T ] × Ω; V), i.e.
Similarly by inequality (139), for every p ∈ [2, ∞) we can choose a subsequence of ( α ε ) ε>0 convergent weak star (along the sequence ε k n ) in the space L p ( Ω; L ∞ (0, T ; H)) and, using (134), we infer that Using the convergence from (134) and estimates (139)-(144) we will prove certain term-by-term convergences which will be used later to prove Theorem 3. In order to simplify the notation, in the result below we write lim ε→0 but we mean lim k n →∞ .
Before stating the next lemma, we introduce a new functional space U as the space of compactly supported, smooth divergence free vector fields on S 2 :

Lemma 26
For all t ∈ [0, T ], and φ ∈ U, we have (along the sequence ε k n ) Proof Let us fix φ ∈ U.
(e) Assertion (e) follows because by Assumption 2 the sequence M • ε f ε converges weakly in L 2 (0, T ; L 2 (S 2 )) to f . (f) By the definition of maps G ε and G, we have Since, by Assumption 2, for every j ∈ {1, . . . , N }, and s ∈ [0, t], M • ε [ g j ε (s)] converges weakly to g j (s) in L 2 (S 2 ) as ε → 0, we get By assumptions on g j ε , we obtain the following inequalities for every t ∈ [0, T ] and ε ∈ (0, 1] where c, c > 0 are some constants. Using the Vitali's convergence theorem, by (155) and (156) we infer Hence, by the properties of the Itô integral we deduce that for all t ∈ [0, T ], By the Itô isometry and assumptions on g j ε and g j we have for all t ∈ [0, T ] and ε ∈ (0, 1] where c > 0 is a constant. Thus, by (158), (159) and the dominated convergence theorem assertion ( f ) holds.

Lemma 27
For all t ∈ [0, T ] and φ ∈ U, we have (along the sequence ε k n ) where the process α ε is defined in (136).
Finally, to finish the proof of Theorem 3, we will follow the methodology as in [42] and introduce some auxiliary notations (along sequence ε k n ) Corollary 5 Let φ ∈ U. Then (along the sequence ε k n ) Proof Assertion (165) follows from the equality and Lemma 26 (a). To prove assertion (166), note that by the Fubini Theorem, we have To conclude the proof of the corollary, it is sufficient to note that by Lemma 26 (b) − ( f ) and Lemma 27, each term on the right hand side of (163) tends at least in to the corresponding term (to zero in certain cases) in (164).
In particular, Therefore by Corollary 5 and the definition of Λ, for almost all t ∈ [0, T ] and P-almost all ω ∈ Ω i.e. for almost all t ∈ [0, T ] and P-almost all ω ∈ Ω Hence (167) holds for every φ ∈ U. Since u is a.s. H-valued continuous process, by a standard density argument, we infer that (167) holds for every φ ∈ V (U is dense in V).
The Laplace-de Rham operator applied to a vector field v is given by where Δ v ϕ and Δ v λ are as in (177).

B The curl and the Stokes operator
In this section we present a integration by parts formula corresponding to curl operator and later we use it to give a relation between the Stokes operator A ε and curl .
Let O ⊂ R 3 be a bounded domain with a regular boundary ∂O. Define such that for every u ∈ H(curl) and v ∈ H 1 (O).

Remark 8
We will call formula (184) the generalised Stokes formula. From now on we will write n × u Γ instead of (n × ·)(u) Γ for u ∈ H(curl).
Recall that and the Stokes operator is given by

Remark 9
To define n × curl u as an element of H −1/2 (Γ ), we need to know that curl (curl u) ∈ L 2 (O). But if u ∈ H 2 (O), then obviously this condition is satisfied.

Theorem 5 A is self-adjoint and non-negative on H.
Proof Here we will only show that A is symmetric and non-negative. Let u, v ∈ D(A), then Recall that (from (168)) for smooth R 3 -valued vector fields, curl (curl u) = −Δu + ∇(div u).
Using the above identity in (188) along with the fact that u ∈ D(A), in particular, div u = 0 and generalised Stokes formula (184), we have This establishes that A is symmetric on H. The non-negativity follows from the above identity by taking v = u ∈ D(A). We use the following relation repeatedly in our calculations.

Lemma 28 Let u ∈ D(A 1/2 ) and v ∈ D(A). Then
Proof Note that for u ∈ D(A 1/2 ) and v ∈ D(A), the LHS makes sense. Using the generalised Stokes formula (184), we get To finish the proof we need the following lemma: Proof It is sufficient to prove (189) for u ∈ C 1 0 (O) with u · n = 0 on Γ . In this case for all The proof of Lemma 28 is finished by observing that from the proof of Theorem 5.
Let us consider an abstract framework. Let H be a Hilbert space and A be a nonnegative self-adjoint operator on H .
As a consequence of the above lemma we have

C Proof of the Poincaré and the Ladyzhenskaya Inequalities
Proof of Lemma 13 We will establish the Poincaré inequality (54) following the footsteps of Lemma 2.1 [53] with all the details. By density argument, it is enough to prove (54) for smooth functions. Let ψ ∈ C(Q ε ) be a real continuous function. We write for any ξ, η ∈ [1, 1 + ε]: 2 (190) with x = y |y| ∈ S 2 . We fix ξ and integrate w.r.t. η ∈ [1, 1 + ε] to obtain With ψ = u r and ξ = 1, observing that u r (1, x) = 0 (because of the boundary condition u · n = 0 on ∂ Q ε ) from (191) we obtain Applying (191) with ψ = N ε u λ , we get Observing from Lemma 6 for every ψ ∈ L 2 (Q ε ) and since the first term on the LHS of (193) is positive we can simplify (193) as follows Similarly for ψ = N ε u ϕ , we have Thus, using (192), (194) and (195) for each of the cases ψ = u r , ψ = N ε u λ and ψ = u ϕ , we obtain Using the Cauchy-Schwarz inequality, we find which implies for 0 ≤ ε < Adding (200) for ψ = u r , ψ = N ε u λ and ψ = N ε u ϕ ; using finally we conclude the proof of the inequality (54).

Proof of Lemma 14
We will prove the lemma for smooth vector fields u ∈ C ∞ (Q ε ). By Lemma 2.3 [53] there exists a constant c 0 > 0 s.t.

Proof of Lemma 18
Let u be a tangential vector field defined on S 2 , u = (0, u λ , u ϕ ).

D.1 Skorohod Theorem and Aldous Condition
Let E be a separable Banach space with the norm · E and let B(E) be its Borel σ -field. The family of probability measures on (E, B(E)) will be denoted by P. The set of all bounded and continuous E-valued functions is denoted by C b (E).

Definition 3
The family P of probability measures on (E, B(E)) is said to be tight if for arbitrary ε > 0 there exists a compact set K ε ⊂ E such that μ(K ε ) ≥ 1 − ε, for all μ ∈ P.
We used the following Jakubowski's generalisation of the Skorokhod Theorem, in the form given by Brzeźniak and Ondreját [14, Theorem C.1], see also [31], as our topological space Z T is not a metric space. Let (Ω, F, P) be a probability space with filtration F := (F t ) t∈[0,T ] satisfying the usual conditions, see [38], and let (X n ) n∈N be a sequence of continuous F-adapted S-valued processes.

D.2 Tightness Criterion
Now we formulate the compactness criterion analogous to the result due to Mikulevicus and Rozowskii [40], Brzeźniak and Motyl [12] for the space Z T , see also [5,Lemma 4.2]. Let P ε be the law of α ε on Z T . Then for every δ > 0 there exists a compact subset K δ of Z T such that sup ε>0 P ε (K δ ) ≥ 1 − δ.

E Kuratowski Theorem and Proof of Lemma 25
This appendix is dedicated to the proof of Lemma 25. We will first recall the Kuratowski Theorem [33] in the next subsection and prove some related results which will be used later to prove Lemma 25 in Sect. E.2.
Next two lemmas are the main results of this appendix. For the proof of Lemma 35 please see [6,Appendix B].

Lemma 35
Let X 1 , X 2 and Z be topological spaces such that X 1 is a Borel subset of X 2 . Then X 1 ∩ Z is a Borel subset of X 2 ∩ Z , where X 2 ∩ Z is a topological space too, with the topology given by τ (X 2 ∩ Z ) = {A ∩ B : A ∈ τ (X 2 ), B ∈ τ (Z )} . (206)

E.2 Proof of Lemma 25
In this subsection we recall Lemma 25 and prove it using the results from previous subsection. Similarly we can show that L 2 (0, T ; V) ∩ Z T is a Borel subset of Z T . L 2 (0, T ; V) → L 2 (0, T ; H) and both are Polish spaces thus by application of the Kuratowski Theorem, L 2 (0, T ; V) is a Borel subset of L 2 (0, T ; H). Finally, we can conclude the proof of lemma by Lemma 35.