Stochastic tamed Navier-Stokes equations on $\mathbb{R}^3$:existence, uniqueness of solution and existence of an invariant measure

R\"ockner and Zhang in [27] proved the existence of a unique strong solution to a stochastic tamed 3D Navier-Stokes equation in the whole space and for the periodic boundary case using a result from [31]. In the latter case, they also proved the existence of an invariant measure. In this paper, we improve their results (but for a slightly simplified system) using a self-contained approach. In particular, we generalise their's result about an estimate on the $L^4-$norm of the solution from the torus to $\mathbb{R}^3$, see Lemma 5.1 and thus establish the existence of an invariant measure on $\mathbb{R}^3$ for a time-homogeneous damped tamed 3D Navier-Stokes equation, given by (6.1).

In the case of classical deterministic Navier-Stokes equations on R 3 , if the initial data u 0 ∈ V (see Section 2), then there exists only a local strong solution, see [32]. Cai and Jiu [11] studied the Navier-Stokes equations with damping on R 3 , where the damping was modeled by the term |u| β−1 u, β ≥ 1; the tamed term considered in the current paper corresponds to β = 3. They proved the existence of a global weak solution 1 for any β ≥ 1 and u 0 ∈ H, see section 2 and they proved the existence of global strong solution 2 for β ≥ 7/2 and u 0 ∈ V ∩ L β+1 . Moreover, they were able to show the uniqueness of strong solutions for 7/2 ≤ β ≤ 5. Later, Zhang et. al. [36], by exploiting the Gagliardo-Nirenberg inequality, were able to lower down the parameter β to 3. Thus, establishing the existence of a global strong solution to Navier-Stokes equations with damping on R 3 for β > 3, u 0 ∈ V ∩ L β+1 and proving uniqueness whenever 3 < β ≤ 5. They also remarked that the critical value for β is β = 3 [36,Remark 3.1]. But, Zhou [37, Theorem 2.1] was able to surpass this critical value of β. Moreover, for any β ≥ 1, he proved that the strong solution is unique in a larger class of weak solutions, see [37,Theorem 3.1]. The critical case of β = 3 was studied by Röckner and Zhang [28,Theorem 1.1], where they proved the existence of a smooth unique global solution to the deterministic tamed 3D Navier-Stokes equations for very smooth initial data and deterministic forcing f . Moreover, they proved [28, Theorem 1.1] that this unique solution converges (in L 2 (0, T ; L 2 (O))) to a bounded Leray-Hopf solution of 3D Navier-Stokes equations (if exists) on a bounded domain O ⊂ R 3 . The non-explosion of the solution is due to the tamed term. Röckner and Zhang [29] also studied 3D tamed Navier-Stokes equations on a bounded domain O ⊂ R 3 with Dirichlet boundary conditions and proved the existence of a unique strong solution directly, based on the Galerkin approximation and on a kind of local monotonicity of the coefficients. Recently, You [35] proved the existence of a random attractor for the 3D damped (|u| β−2 u) Navier-Stokes equations with additive noise for 4 < β ≤ 6 with initial data u 0 ∈ V on a bounded domain O ⊂ R 3 with smooth boundary. Röckner and Zhang [27] proved the existence of a strong solution of the stochastic tamed NSEs (in probabilisitc sense) by invoking the Yamada-Watanabe Theorem, thus proving the existence of a martingale solution to (1.1) (with more generalised noise term) in the absence of compact Sobolev embeddings and the pathwise uniqueness. They used the localization method to prove the tightness, a method introduced by Stroock and Varadhan [31]. In this paper, we present a self-contained proof of the same result. In order to prove the existence of a martingale solution, Röckner et. al. used the Faedo-Galerkin approximation with the non-classical finite dimensional space H 1 n = span{e i , i = 1 · · · n} where E = {e i } i∈N ⊂ V (see Section 2) is the orthonormal basis of H 1 . They also require that in the case of the periodic boundary conditions, E is an orthogonal basis of H 0 which was essential in obtaining the L 4 -estimate of the solution. We generalised this result to R 3 . Another reason for Röckner et. al. to choose the periodic boundary conditions was the compactness of H 2 ֒→ H 1 embedding, which along with the L 4 -estimate of the solution was crucial in establishing the existence of invariant measures. We don't require this embedding and hence are able to obtain the existence of invariant measures for the damped tamed Navier-Stokes equations on R 3 .
In the present paper we prove the existence of a unique strong solution to the stochastic tamed 3D Navier-Stokes equation (1.1) under some natural assumptions (H1) − (H2) on the drift f and the diffusion σ (see Section 2). To prove the existence of strong solution we use the Yamada-Watanabe theorem [24,34] which states that the existence of martingale solutions plus pathwise uniqueness implies the existence of a unique strong solution. In order to establish the existence of martingale solutions, instead of using the standard Faedo-Galerkin approximations we use a different approach motivated from [15] and [19]. We study a truncated SPDE on an infinite dimensional space H n , defined in the Section 4 and then use the tightness criterion, Jakubowski's generalisation of the Skorohod's Theorem and the martingale representaion theorem to prove the existence of martingale solutions. The essential reason for us to incorporate this approximation scheme, was the non-commutativity of gradient operator ∇ with the standard Faedo-Galerkin projection operator P n [5, Section 5]. The commutativity is essential for us to obtain a'priori bounds. We also prove the existence of invariant measures, Theorem 6.1, for time homogeneous damped tamed Navier-Stokes equations 6.1 under the assumptions (H1) ′ − (H3) ′ (see Section 6). We use the technique (Theorem 6.4) of Maslowski and Seidler [20], see also [6,9], working with weak topologies to establish the existence of invariant measures. We show the two conditions of Theorem 6.4, boundedness in probability and sequentially weakly Feller property are satisfied for the semigroup (T t ) t≥0 , defined by (6.2). In contrast to the paper by Röckner and Zhang [27], a'priori bound on L 4 −norm of the solution plays an essential role in the existence of martingale solutions and not in the existence of invariant measures. This paper is organised as follows. In Section 2, we recall some standard notations and results and set the assumptions on f and σ. We also establish certain estimates on the tamed term which we use later in Sections 4 and 5. We end the section by recalling a generalised version of the famous Gronwall Lemma for random variables from [14]. In Section 3, we establish the tightness criterion and state the Skorohod Theorem which we use along with a'priori estimates obtained in the Section 5 to prove the existence of a martingale solution and path-wise uniqueness of the solution. In Section 4, we introduce our truncated SPDEs and describe the approximation scheme motivated by [15,19], along with all the machinery required. Finally, in Section 6 we establish the existence of an invariant measure for damped tamed 3D Navier-Stokes equations (6.1).

Functional setting
2.1. Notations and basic definitions. Let (X, · x ), (Y, · Y ) be two real normed spaces. The space of all bounded linear operators from X to Y is denoted by L(X, Y ). If Y = R, then X ′ := L(X, R) is called the dual space of X. The standard duality pairing is denoted by X ′ ·, · X . If both spaces are separable Hilbert then by T 2 (Y ; X) we will denote the space of all Hilbert-Schmidt operators from Y to X endowed with the standard norm · T 2 (Y ;X) .
Assume that X, Y are Hilbert spaces with scalar products ·, · X and ·, · Y respectively. For a densely defined linear operator A : D(A) → Y , D(A) ⊂ X, by A * we denote the adjoint operator of A. In particular, denote the set of all smooth functions from R 3 to R 3 with compact supports. For p ≥ 1 and s ∈ [0, ∞), the Lebesgue and Besov-Slobodetski spaces of R 3 -valued functions will be denoted by L p (R 3 , R 3 ) and W s,p (R 3 , R 3 ) respectively, and often L p , W s,p whenever the context is understood. If p = 2, then L 2 (R 3 , R 3 ) is a Hilbert space with the scalar product given by of R 3 -valued functions such that the functions and all its weak derivatives up to order s belong to the Lebesgue space L p (R 3 , R 3 ). In particular, H 1 (R 3 , R 3 ) = H 1,2 = W 1,2 stand for the Sobolev space of all u ∈ L 2 (R 3 , R 3 ) for which there exist weak derivatives D i u ∈ L 2 (R 3 , R 3 ), i = 1, . . . , 3. It is a Hilbert space with the scalar product given by (2.1) Finally, for s ∈ R, the space W s,2 = H s,2 =: H s is also a Hilbert space endowed with the norm whereû denotes the Fourier transform of a tempered distribution u. Let where Π is the orthogonal projection from L 2 (R 3 , R 3 ) to H.
On H we consider the scalar product and the norm inherited from L 2 (R 3 , R 3 ) and denote them by ·, · H and | · | H respectively, i.e.
On V we consider the scalar product and norm inherited from where the inner product is given by

Some operators.
Let us consider the following tri-linear form defined for suitable vector fields u, v, w on R 3 . We will recall the fundamental properties of the form b which are valid in unbounded domains. By the Sobolev embedding theorem and Hölder inequality, we obtain the following estimates for some positive constant c. Thus the form b is continuous on V. Moreover, if we define a bilinear map B by B(u, w) := b(u, w, ·), then by inequality (2.5) we infer that B(u, w) ∈ V ′ for all u, w ∈ V and that the following inequality holds Moreover, the mapping B : V × V → V ′ is bilinear and continuous. Let us, for any s > 0, define the following standard scale of Hilbert spaces If s > d 2 + 1 then by the Sobolev Embedding theorem, Here C b (R 3 , R 3 ) denotes the space of continuous and bounded R 3valued functions defined on R 3 . If u, w ∈ V and v ∈ V s with s > d 2 + 1 then for some constant c > 0. Thus b can be uniquely extended to the tri-linear form (denoted by the same letter) At the same time, the operator B can be uniquely extended to a bounded linear operator In particular, it satisfies the following estimate We will also use the following notation, B(u) := B(u, u). Let us assume that s > 1. It is clear that V s is dense in V and the embedding j s : V s ֒→ V is continuous. Then there exists [7, Lemma C.1] a Hilbert space U such that U ⊂ V s , U is dense in V s and the natural embedding i s : U ֒→ V s is compact .
The following Gagliardo-Nirenberg interpolation inequality will be used frequently. Let q ∈ [1, ∞] and m ∈ N. If then there exists a constant C m,q depending on m and q such that Recall that Π is the orthogonal projection from L 2 (R 3 , R 3 ) to H. For any u ∈ H and v ∈ L 2 (R 3 , R 3 ), we have The Stokes operator A : D(A) → H, is given by

2.3.
Assumptions. We now introduce the following assumptions on the coefficients f and σ: (H1) A function f : R 3 × R 3 → R 3 is of C 1 class and for any T > 0 there exists a constant C T,f > 0 such that for any x ∈ R 3 , u ∈ R 3 , (H2) A measurable function σ : [0, ∞) × R 3 → ℓ 2 of C 1 class with respect to the x-variable and for any T > 0 there exists a constant Below for the sake of simplicity the variable "x" in the coefficients will be dropped.
Then G : H → T 2 (ℓ 2 ; H) is given by For simplicity we will assume that ν = 1. In particular the function g defined by (1.4) will from now on be given by (2.14) Observe that the function g defined in this way satisfies and (2.16) We are interested in proving the existence of solutions to (1.1) -(1.3). In particular we want to prove the existence of a random divergence free vector field u and scalar pressure p satisfying (1.1) and (1.3). Thus we project equation (1.1) using the orthogonal projection operator Π on the space H of the L 2 -valued, divergence free vector fields. On projecting, we obtain the following abstract stochastic evolution equation: (2.17) where we assume that u 0 ∈ V and W (t) = (W j (t)) ∞ j=1 is a cylindrical Wiener process on ℓ 2 and {W j (t), t ≥ 0, j ∈ N} is an infinite sequence of independent standard Brownian motions. We will repeatedly use the following notation We will need the following lemma later to obtain the a'priori estimates.
iii) If T > 0 then ∃ C σ,T > 0 such that for any t ∈ [0, T ] and u ∈ D(A), Proof. Let u ∈ D(A). Since B(u), u H = 0, using the Cauchy-Schwarz and Young inequality we get Let us introduce a function φ : R + → R such that g(r) = r −φ(r) which can be written as Also φ ′ (r) = 1 − g ′ (r), and there exists a constant C N > 0 such that |φ ′ (r)| ≤C N for every r ≥ 0. Moreover Thus |φ ′ (r) · r| is bounded by some positive constant C N . Let u ∈ H, then using the definitions of g and ((·, ·)) we get Thus, by the integration by parts formula we get Using the above bound on |φ ′ (r) · r|, we obtain Since g(r) ≥ 0, |φ(r)| ≤ r for all r ≥ 0. Thus using (2.23) in (2.22), we obtain ((−g(|u| 2 )u, u)) ≤ −3 |u| · |∇u| . Now to prove the second inequality in (2.19), we take the similar approach. Let u ∈ H, then By the definition of φ there exists a constant C N > 0 such that |φ(r)| ≤ C N for all r > 0, thus This completes the proof of part (ii). Now for (iii), by (H1) and (H2) we have and by (H1) and (H2) we have On a purely heuristic level, by application of the Itô Lemma to the function |x| 2 H and a solution u to (1.1) and using Lemma 2.1, one obtains the following inequality which could lead to a'priori estimates that can be used further to prove the existence of the solution. We will require following version of the Gronwall Lemma [14, Lemma 3.9] : Lemma 2.2. Let X, Y, I and ϕ be non-negative processes and Z be a non-negative integrable random variable. Assume that I is nondecreasing and there exist non-negative constants C, α, β, γ, η with the following properties T 0 ϕ(s) ds ≤ C a.s., 25) and such that for some constantC > 0 and all t ∈ [0, T ], with the topology T 3 induced by the semi-norms (q T,R ) R∈N .
The following lemma is inspired by the classical Dubinsky Theorem [33, Theorem IV.4.1] (see also [18]) and the compactness result due to Mikulevicus and Rozovskii [23, Lemma 2.7]. and letT be the supremum of the corresponding topologies. Then a set K ⊂Z T isT -relatively compact if the following two conditions hold : The above lemma can be proved by modifying the proof of [7, Lemma 3.1], see also [33,Theorem IV.4.1]. Let V w denote the Hilbert space V endowed with the weak topology.  and let T be the supremum of the corresponding topologies. Now we formulate the compactness criterion analogous to the result due to Mikulevicus and Rozowskii [23], Brzeźniak and Motyl [7,Lemma 3.3] for the space Z T .  is metrizable. Since the restrictions to K of the four topologies considered in Z T are metrizable, compactness of a subset of Z T is equivalent to its sequential compactness.

Consider the ball
Let (u n ) be a sequence in K. By Lemma 3.1, the boundedness of the set K in L 2 (0, T ; D(A)) and assumption (c) imply that K is compact iñ  Let (Ω, F , P) be a probability space with filtration F := (F t ) t∈[0,T ] satisfying the usual conditions, see [21], and let (X n ) n∈N be a sequence of continuous F-adapted S-valued processes.
Definition 3.5. We say that the sequence (X n ) n∈N of S-valued random variables satisfies condition [T] iff ∀ ε > 0, ∀ η > 0, ∃ δ > 0: sup n∈N P {m(X n , δ) > η} ≤ ε. Now we recall the Aldous condition which is connected with condition [T] (see [22] and [2]). This condition allows to investigate the modulus of continuity for the sequence of stochastic processes by means of stopped processes. Using the compactness criterion from Lemma 3.3 and above results corresponding to Aldous condition we obtain the following corollary which we will use to prove the tightness of the laws defined by the truncated SPDE (4.26). Corollary 3.9 (Tightness criterion). Let (X n ) n∈N be a sequence of continuous F-adapted H-valued processes such that (a) there exists a constant C 1 > 0 such that Let P n be the law of X n on Z T . Then for every ε > 0 there exists a compact subset K ε of Z T such that Proof. Let ε > 0. By the Chebyshev inequality and (a), we infer that for any n ∈ N and any r > 0 By the Chebyshev inequality and (b), we infer that for any n ∈ N and any r > 0 By Lemmas 3.6 and 3.8 there exists a subset A ε It is sufficient to define K ε as the closure of the set The proof is thus complete.
3.2. The Skorohod Theorem. We will use the following Jakubowski's generalisation of the Skorohod Theorem in the form given by Brzeźniak and Ondreját [10], see also [16].
Theorem 3.10. Let X be a topological space such that there exists a sequence {f m } m∈N of continuous functions f m : X → R that separates points of X . Let us denote by S the σ-algebra generated by the maps {f m }. Then (a) every compact subset of X is metrizable, (b) if (µ m ) m∈N is a tight sequence of probability measures on (X , S), then there exists a subsequence (m k ) k∈N , a probability space (Ω, F , P) with X -valued Borel measurable variables ξ k , ξ such that µ m k is the law of ξ k and ξ k converges to ξ almost surely on Ω. Moreover, the law of ξ is a Radon measure.
Using Theorem 3.10, we obtain the following corollary which we will apply to construct a martingale solution to the tamed Navier-Stokes equations.
Corollary 3.11. Let (η n ) n∈N be a sequence of Z T -valued random variables such that their laws L(η n ) on (Z T , T ) form a tight sequence of probability measures. Then there exists a subsequence (n k ), a probability space (Ω,F ,P) and Z T -valued random variablesη,η k , k ∈ N such that the variables η k andη k have the same laws on Z T andη k converges toη almost surely onΩ.
Proof. It is sufficient to prove that on each space appearing in the definition (3.5) of the space Z T , there exists a countable set of continuous real-valued functions separating points.
For the space where {v m , m ∈ N} is a dense subset of L 2 (0, T ; D(A)). Let us consider the space C([0, T ]; V w ). Let {h m , m ∈ N} be any dense subset of V and let Q T be the set of rational numbers belonging to the interval [0, T ]. Then the family {f m,t , m ∈ N, t ∈ Q T } defined by consists of continuous functions separating points in C([0, T ]; V w ). The statement of the corollary follows from Theorem 3.10, concluding the proof.
We end this section by giving the definitions of a martingale and strong solution to (2.17).
Definition 3.13. We say that there exists a martingale solution of (2.17) iff there exist • a stochastic basis (Ω,F,F,P), • and a progressively measurable process u : such that for all t ∈ [0, T ] and all v ∈ V :P-a.s.
Definition 3.14. We say that problem (2.17) has a strong solution iff for every stochastic basis (Ω, F , F, P) and every cylindrical Wiener process such that for all t ∈ [0, T ] and all v ∈ V (3.8) holds P-a.s.

Truncated SPDEs
The approximation scheme described in this section to define truncated SPDEs was first introduced by [15] and also later used by Manna et.al. in [19].
In order to describe the approximation scheme, we will use the following notations and spaces.
where | · | is the Euclidean norm on R 3 . We will use F (u) andû interchangeably to denote the Fourier transform of u. The inverse Fourier transform will be given by F −1 .
We define H n as the subspace of H, The norm on H n is inherited from H. Let be the orthogonal projection i.e. ∀ u ∈ H, u − P n u ⊥ H n and y = P n u ⇔ y ∈ H n and u − y ⊥ H n .
One can show that P n is given by Proof. Let u ∈ V, then by the definition of P n and V Thus we have shown that and hence P n is uniformly bounded in V.
Proof. Since u ∈ D(A), it is clear that ∆ u ∈ L 2 . Thus we are left to show that div(∆ u) = 0 in the weak sense. Let ϕ ∈ C ∞ 0 (R 3 ), then using the definition of div and ∆, we get By definition A u = −Π(∆ u), but since ∆ u ∈ H, and Π : L 2 → H is an orthogonal projection, Π(∆ u) = ∆ u and hence, Proof. We start with proving the first statement. Let u ∈ H n ; by definition Thus we have proved that u ∈ D(A) and hence H n ⊂ D(A). Moreover, we showed that there exists a constant C n > 0, depending on n such that (4.4) Now in order to establish the equality (4.3), we just need to show that Au ∈ H n . Since u ∈ H n , u ∈ D(A). Hence from Lemma 4.1, A u = −∆ u. We are left to show that supp (F (Au)) ⊂ B n . Using the definition of A u, we get following equalities Hence Au ∈ H n . Since P n : H → H n is an orthogonal projection, we infer that P n (Au) = Au . Proof. In Lemma 4.2 we showed that A n is well defined and it's straightforward to show it is linear. We are left to show that it is bounded. Let u ∈ H n , then by Parseval equality and the definition of H n Thus, A n u Hn ≤ n u Hn .
is well defined and Lipschitz on balls. Moreover In particular, it holds true for s = 2. Thus, we have

Now by (4.4) and (4.15)
Hence B(u, v) ∈ H, which implies B n (u, v) ∈ H n and is well defined.
Then, as before using the embedding H 2 ֒→ L ∞ , we have Since u, v ∈ B R , and using (4.4) and (4.15), we get Since u ∈ H n and P n is the orthogonal projection on H, Also by using the definition of ((·, ·)) and the Cauchy-Schwarz inequality we get Lemma 4.6. The map is well defined and Lipschitz on balls. Moreover Proof. Let u ∈ H n , then by the definition of g (2.14), the estimate (2.15) and the embedding of H 1 ֒→ L 6 , we have (4.14) Therefore g n : H n → H n is well defined. From above we can also infer that there exists a constant C n > 0 depending on n such that Let u, v ∈ B R , then as before using (2.14), we have Since u, v ∈ B R , using (4.15), we get Let u ∈ H n , then using Lemmas 4.2 and 4.3, the definitions of g n and ((·, ·)) we get Also, note that Hence the inequalities (4.13) can be established with the help of the above two relations and Lemma 2.1 (ii). This completes proof of the lemma.
Lemma 4.7. Let f satisfy the assumption (H1).Then the map is well defined and Lipschitz.
Proof. Let u ∈ H n , then by the assumption (H1), Lemma 4.8. Let σ satisfy the assumption (H2). Then the map is well defined and Lipschitz.
Proof. Let u ∈ H n , then Using (4.15), we infer Using (4.15), we infer Proof. Let u ∈ H n , then using the Parseval's identity Thus if u ∈ H n then L 2 and H n have equal norms. The equivalence of H 1 and H n norms is established from (4.15). Using (4.5) and (4.22) we can establish equivalence of D(A) and H n norms.
As discussed earlier in the introduction instead of using the standard Galerkin approximation of SPDE on the finite dimensional space we will look at truncated SPDEs on infinite dimensional space H n . For every n ∈ N, we will establish the existence of a unique global solution to the truncated SPDE and obtain a'priori estimates in order to prove the tightness of measures on a suitable space.
In order to study the truncated SPDE on H n we project the SPDE (2.17) on H n using P n . The projected SPDE on H n is given by where u n ∈ H n , u 0 ∈ V and other operators A n , B n , g n , f n and G n are as defined in Lemmas 4.4-4.8.
Then for every u ∈ H n there exists K 1 > 0 such that Proof. From the definition of B n , g n and f n , we have Using Lemma 4.8 and since u ∈ H n , we get Hn . On rearranging, we get Hn ).
We will use the following theorem from [1, Theorem 3.1] to prove Theorem 4.12. We have modified it in the way we will be using it.
Theorem 4.11. Let X be an abstract Hilbert space. Assume that σ and b satisfies the following conditions Then for any X-valued ξ, there exists a unique global solution u = (u(t)) t≥0 to Proof. The proof is a direct application of Theorem 4.11. Using Lemmas 4.4 -4.8, we can show that condition (i) of Theorem 4.11 is satisfied. In Lemma 4.10 we proved that condition (ii) is satisfied. Thus we have existence of the unique global solution u n = (u n (t)) t≥0 to (4.26).

5.1.
A'priori estimates. In this subsection we will obtain certain a'priori estimates for the solution u n of (4.26). We will use these a'priori estimates in Lemma 5.3 to prove the tightness of measures on the space Z T , defined in (3.5). We will also establish certain higher order estimates which will be required to prove the convergence of non-linear terms in later sections. Let us fix T > 0. For any R > 0, define the stopping time where u n is the solution of (4.26). By the definition of a martingale solution one knows that for every n ≥ 1, τ n R ↑ ∞ as R ↑ ∞.
Lemma 5.1. Let u n be the solution of (4.26). For all ρ > 0 there exist positive constants C 1 (ρ), Moreover, for every δ > 0 there exists a constant C(δ) > 0 such that if Proof. Let u n (t), t ≥ 0 be the solution of (4.26) then applying the Itô formula to φ(x) = |x| 2 H and the process u n (t), we get Using the second part of the inequality (4.13), we get On rearranging we get is a F-martingale, as by Lemma 2.1 and (5.1) we have the following inequalities where to establish the last inequality we have used the equivalences of norm from Proposition 4.9. Thus, E[µ n (t)] = 0. Hence applying Lemma 2.2 for the following three processes : Hence, using (5.8) and (5.9) we infer that Since we are interested in the estimates involving V norm of u. We apply the Itô formula to φ(x) = |∇x| 2 L 2 and the process u n (t), obtaining u n (s), g n (u n (s)) ds where ((·, ·)) is as defined in (2.1). Using Lemma 2.1, assumptions (H1) − (H2), boundedness of P n in H, estimates (4.8), (4.13), the Cauchy-Schwarz and the Young inequality, we get ((u n (s), G(s, u n (s)) dW s )) i.e., ((u n (s), G(s, u n (s)) dW s )) .
In the next lemma we will use the estimates from Lemma 5.1 to establish higher order estimates.
Proof. Let p ∈ [1, ∞). Then by using the Itô formula for ξ(t) = u n (t) 2 V , φ(x) = x p , equations (5.5), (5.11) and the definition of · V , we obtain Using Lemma 2.1, the definition of g (2.14), boundedness of P n in V and assumption (H1), we can simplify (5.20) On rearranging, we get which on further simplification yields As before we will show that is a F-martingale. By Lemma 2.1 and (5.1) we have the following inequalities where the finiteness of the integral follows from Proposition 4.9. Hence, E[µ n (t)] = 0.
Since |u n (s)| H ≤ u n (s) V and |∇u n (s)| L 2 ≤ u n (s) V on applying the modified version of the Gronwall Lemma (Lemma 2.2) for Using the definition of V-norm, the Hölder inequality and the Young inequality, for ε > 0 we obtain Thus from (5.21) and using (5.23), (5.24) and Lemma 2.2, we have Choosing ε small enough we get Let s, t ∈ [0, T ], s < t and θ := t − s. First we will establish estimates for each term of the above equality. Ad. J n 2 . Since A : V → V ′ , then by the Hölder inequality and (5.2), we have the following inequalities Ad. J n 4 . Since H 1 ֒→ L 6 then by the definition of g and estimate (5.18) (for p = 2), we have Ad. J n 5 . Using the assumption H1, (5.9) and the Cauchy-Schwarz inequality, we obtain the following inequalities Ad. J n 6 . Using the Itô isometry, Lemma 2.1 and (5.2), we obtain the following Let us fix κ > 0 and ε > 0. By the Chebyshev's inequality and estimates (5.26) -(5.29), we obtain By the Chebyshev inequality and (5.30), we have Since [A] holds for each term J n i , i = 1, 2, . . . , 6; we infer that it holds also for (u n ) n∈N . Thus, the proof of lemma can be concluded by invoking Corollary 3.9. Now we will state the main theorem of this section.
In the following subsection we will prove Theorem 5.4 in several steps.
ThenM is a H−valued continuous process.
Proof. Sinceũ ∈ C([0, T ]; V) we just need to show that each of the remaining terms on the RHS of (5.43) are H-valued a.s. and welldefined.
Using the Cauchy-Schwarz inequality repeatedly and (5.37) we have the following inequalities Since H k,p ֒→ L ∞ for every k > d/p, hence there exists a C > 0 such that u L ∞ ≤ C u H 2,2 for every u ∈ H 2,2 . Thus by the Cauchy-Schwarz inequality, (5.37) and (5.38) we obtain the following estimate We know that for d = 3, H 1,2 ֒→ L 6 , thus using (2.15), (5.32) and (5.34), we get Using the assumptions (H1) and (5.38) we can show that This concludes the proof of the lemma. Here ·, · denotes the duality pairing between V γ and V ′ γ .
Proof. We will prove the lemma in two steps.
Step I Let us fix γ > d 2 and r, t ∈ [0, T ]. Assume first that ψ ∈ V. Then there exists a R > 0 such that supp(ψ) is a compact subset of O R . There exists a constant C ≥ 0 such that where we used (2.8) to establish the last inequality. We have Thus using the estimate (5.45), the Hölder inequality, (2.16) and the Cauchy-Schwarz inequality, we obtain  Since u n → u in L 2 (0, T ; H loc ) we infer that (5.44) holds for every ψ ∈ V.
Step II Let ψ ∈ V γ and ε > 0. Then there exists a ψ ε ∈ V such that ψ ε − ψ Vγ < ε. Hence, we get Since V is dense in V γ , (5.45) holds for all ψ ∈ V γ . In particular, there exists a constant C > 0 such that Since ε > 0 is arbitrary we conclude the proof. Here ·, · denotes the dual pairing between V γ and V ′ γ .
The proofs of Lemmas 5.10, 5.13 and 5.14 follow the similar methodology as that of Lemmas 5.6 -5.8 [7] and Lemmas 5.9 -5.11 [5]. We will prove that the functions {f n } n∈N are uniformly integrable in order to apply the Vitali theorem. We claim that Since, H ֒→ V ′ γ then by the Cauchy-Schwarz inequality, for each n ∈ N we havẽ Since,M n is a continuous martingale with quadratic variation defined in (5.40), by the Burkholder-Davis-Gundy inequality we obtaiñ Since, P n : H → H is a contraction and by Lemma 2.1, (5.18) for p = 1, we havẽ Then by (5.55) and (5.57) we see that (5.54) holds. Since the sequence {f n } n∈N is uniformly integrable and by (5.51) it isP-a.s. point-wise convergent, then application of the Vitali Theorem completes the proof of the Lemma.
Remark 5.11. Using Burkholder-Davis-Gundy inequality we have proved a stronger claim (5.56) than what we needed.
We will prove that the functions {f n } n∈N are uniformly integrable. We claim that for some r > 1, For each n ∈ N as before we havẽ Since,M n is a continuous martingale with quadratic variation defined in (5.40), by the Burkholder-Davis-Gundy inequality we obtaiñ Since, P n : H → H is a contraction and by Lemma 2.1 we havẽ We will be using the following notations in following lemmata.
where V(O R ) denotes the space of all divergence free vector fields of class C ∞ with compact supports contained in O R . We recall that H O R is the space of restrictions to the subset O R of elements of the space H i.e., H O R := u |O R : u ∈ H , with the scalar product defined by Lemma 5.14. The map G : H O R → T 2 (ℓ 2 ; V ′ (O R )) given by (2.13) is well defined and there exists some constant C R > 0 such that Moreover, for every ψ ∈ V the mapping H ∋ u → G(u), ψ ∈ ℓ 2 is continuous, if in the space H we consider the Fréchet topology inherited from the space Let v ∈ V(O R ). Since, v on the boundary ∂O R is equal to zero, thus using the integration by parts formula, we obtain for Using the Hölder inequality, we obtain Therefore, if we define a linear functionalB R bŷ we infer that it is bounded in the norm of the space V(O R ). Thus it can be uniquely extended to a linear bounded functional (denoted also byB R ) on V(O R ). Moreover, by estimate (5.65) we have the following inequality Since by equality (2.13), G(u)(e j ) = Π [(σ j · ∇)u], where {e j } ∞ j=1 is an orthonormal basis of ℓ 2 , we get by estimate (5.66) Therefore, using the assumption (H2), G(u) ∈ T 2 (ℓ 2 , V ′ (O R )) and By estimate (5.63) and the continuity of the embedding Now we identify V ′ G(·), ψ V with the mapping ψ * * G : H → (ℓ 2 ) ′ defined by (ψ * * G(u))y := (G(u)y)ψ ∈ R , u ∈ H , y ∈ ℓ 2 .
We will prove that the functions are uniformly integrable and conver-gentP−a.s. We start by proving that for some r > 1, Since, T 2 (ℓ 2 ; H) is continuously embbeded in L(ℓ 2 ; H), then by (2.20) there exists some c > 0 such that and thus Using the Hölder inequality, we get inferring (5.69).
Continuation of the proof of Theorem 5.4. Now we apply the idea analogous to that used by Da Prato and Zabczyk, see [12,Section 8.3]. By Lemma 5.6, and Corollary 5.12, we infer thatM(t), t ∈ [0, T ] is an H-valued continuous square integrable martingale with respect to the filtrationF = (F t ). Moreover, by Corollary 5.16 the quadratic variation ofM is given by Therefore by the martingale representation theorem, there exist • a stochastic basis (Ω,F ,F,P), • a cylindrical Wiener processW (t), • and a progressively measurable processũ(t) such that for all t ∈ [0, T ] and all v ∈ V

Invariant measures
In this section, we consider time homogeneous damped tamed NSEs, i.e. the coefficients f, σ are independent of t and furthermore f ∈ H is not dependent on u. The time homogeneous damped tamed NSEs in abstract form are given by where A α = αI − ν∆ for some α ∈ R and ν > 0 is the viscosity. The operator B and the cylindrical Wiener process W = (W j ) ∞ j=1 on ℓ 2 is same as defined in Section 2 and G j are as defined in (2.12).
Let B b (V) denote the set of all bounded and Borel measurable functions on V. For any 2) It follows from Theorem 6.2 and Ondrejat [25] (see also [6]) that T t ϕ ∈ B b (V) and {T t } t≥0 is a semigroup on B b (V). Also since this unique solution to (6.1) has a.e. path in C([0, T ]; V), it is also a Markov semigroup (see [25,Theorem 27]). Moreover, {T t } t≥0 is a Feller semigroup, i.e. T t maps C b (V) into itself.
Next we state the main result of this section, regarding existence of invariant measures. Theorem 6.1. Let for every α > 0, the assumptions (H1) ′ − (H3) ′ be satisfied. Then there exists an invariant measure µ ∈ P(V) of the semigroup (T t ) t≥0 defined by (6.2), such that for any t ≥ 0 and ϕ ∈ If T t is sequentially weakly Feller Markov semigroup then for every [6,20] for the definitions and inclusion of the spaces); therefore the integral on LHS in Theorem 6.1 makes sense.
Next we list the assumptions that we make on the coefficients f and σ along with a coercivity type assumption, see [26]. (H1) ′ The function f : R 3 → R 3 is time independent and H-valued.
(H2) ′ A measurable function σ : R 3 → ℓ 2 of C 1 class with respect to the x-variable and for all x ∈ R 3 there exists a constant C σ > 0 such that and, for all x ∈ R 3 , (H3) ′ there exists a δ > 0 such that The following theorem regarding the existence of a pathwise unique strong solution to the time homogeneous damped tamed NSEs (6.1) can be proved by modifying the proofs of Theorem 5.4 and Theorem 5.18 to incorporate the extra linear term α u. Theorem 6.2. Assume that assumptions (H1) ′ and (H2) ′ are satisfied. Then for every u 0 ∈ V, there exists a path-wise unique strong solution u of (6.1) for every T > 0 such that u ∈ C([0, T ]; V) ∩ L 2 (0, T ; D(A)), P-a.s.
For fixed initial value u 0 = v ∈ V we denote the (path-wise) unique solution of (6.1), whose existence is proved in Theorem 6.2 by u(t; v). Definition 6.3. We say that a family {T t } t≥0 is sequentially weakly Feller iff For a metric space U, we use P(U) to denote the family of all Borel probability measures on U. We will use the following theorem from Maslowski-Seidler [20] to prove the existence of invariant measures. Theorem 6.4. Assume that (i) the semigroup {T t } t≥0 , defined by (6.2) is sequentially weakly Feller in V, (ii) for any ε > 0 there exists R > 0 such that Then there exists at least one invariant measure for (6.1).
6.1. Boundedness in probability. Lemma 6.5. Let u 0 ∈ V. Then, under the assumptions of Theorem 6.1, for every ε > 0, there exists R > 0 such that Proof. Using the Itô lemma for the function |x| 2 H and the process u(t), we have Now we deal with each term on the RHS of (6.4) one by one. Firstly let us notice that we have Π g(|u| 2 )u, u H = | g(|u| 2 )| · |u| 2 L 2 . (6.7) Using the assumptions on f , for any β > 0 we obtain the following estimate Since u is the solution of (6.1) and satisfies the estimates (4.25) (courtsey, Theorem 6.2), we can show that the process is a martingale. Thus, taking expectation in (6.4) and using the estimates (6.5)-(6.8), we infer G(u(s)) 2 T 2 (ℓ 2 ;H) ds .

6.2.
Sequentially weak Feller property. We are left to verify the assumption (i) of Theorem 6.4, i.e. the Markov semigroup {T t } t≥0 is sequentially weakly Feller in V. In other words we want to show that for any t > 0 and any bounded and weakly continuous ϕ : V → R, if ξ n → ξ weakly in V, then The second named author in his PhD thesis proved that the martingale solutions of stochastic constrained Navier-Stokes equations continuously depend on the initial data [13,Theorem 5.7.7]. We have a similar result for time homogeneous damped tamed NSEs, which can be proved analogously, see also [9,Theorem 4.11].
Theorem 6.6. Assume that (u 0,n ) ∞ n=1 is a V-valued sequence that is convergent weakly to u 0 ∈ V. Let (Ω n , F n , F n , P n , W n , u n ) be a martingale solution of problem (6.1) on [0, ∞) with the initial data u 0,n . Then for every T > 0 there exist • a subsequence (n k ) k , • a stochastic basis Ω ,F ,F,P , • a cylindrical Wiener processW (t) = W j (t) ∞ j=1 on ℓ 2 , • andF-progressively measurable processesũ, ũ n k k≥1 (defined on this basis) with laws supported in Z T such that u n k has the same law as u n k on Z T andũ n k →ũ in Z T ,P -a.s. (6.12) and the system Ω ,F,F,P,W ,ũ is a martingale solution to problem (6.1) on the interval [0, T ] with the initial data u 0 . In particular, for all t ∈ [0, T ] and all v ∈ V G j (s,ũ(s)) dW j (s), v ,P-a.s. Moreover, the processũ satisfies the following inequalitỹ We will need the uniqueness of solutions of (6.1) in law, which is defined next.
The proof of the above lemma is the direct application of Theorems 2 and 11 of [24] once we have proved the existence of a pathwise unique martingale solution of (6.1); which follows from Theorem 6.2. Lemma 6.9. The semigroup {T t } t≥0 is sequentially Feller in V.
Proof. Let us choose and fix 0 < t ≤ T, ξ ∈ V and ϕ : V → R be a bounded weakly continuous function. Need to show that T t ϕ is sequentially weakly Feller in V. For this aim let us choose a V-valued sequence (ξ n ) weakly convergent in V to ξ. Since the function T t ϕ : V → R is bounded, we only need to prove (6.11).
Let u n (·) = u(·, ξ n ) be a strong solution of (6.1) on [0, T ] with the initial data ξ n and let u(·) = u(·, ξ) be a strong solution of (6.1) with the initial data ξ. We assume these processes are defined on the stochastic basis (Ω, F , F, P, W ). By Theorem 6.6 there exist • a subsequence (n k ) k , • a stochastic basis (Ω,F,F,P), • a cylindrical Wiener processW (t) = W j (t) ∞ j=1 on ℓ 2 , • and progressively measurable processesũ(s), (ũ n k (s)) k≥1 , s ∈ [0, T ] (defined on this basis) with laws supported in Z T such that u n k has the same law as u n k on Z T andũ n k →ũ in Z T ,P − a.s. (6.14) and the system (Ω,F,F,P,W ,ũ) (6.15) is a martingale solution to (6.1) on the interval [0, T ] with the initial data ξ.
Since the function ϕ : V → R is sequentially weakly continuous, we infer thatP-a.s., Since the function ϕ is also bounded, by the Lebesgue dominated convergence theorem we infer that From the equality of laws ofũ n k and u n k , k ∈ N, on the space Z T we infer thatũ n k and u n k have same laws on V w and sõ On the other hand, R.H.S. of (6.17) is equal by (6.2), to T t ϕ(ξ n k ). Since u, by assumption, is a martingale solution of (6.1) with the initial data ξ and from the above,ũ is also a solution of (6.1) with the initial data ξ. Thus, by Lemma 6.8, we infer that the processes u andũ have same law on the space Z T .
Using the subsequence argument, we infer that the whole sequence (T t ϕ(ξ n )) n∈N is convergent and lim n→∞ T t ϕ(ξ n ) = T t ϕ(ξ) .
Thus, the existence of an invariant measure is established by using Theorem 6.4, Lemmas 6.5 and 6.9; completing the proof of Theorem 6.1.
Appendix A. Convergence of P n Lemma A.1. Let γ > d 2 and P n : H → H n be the orthogonal projection as given by (4.1) (for more details see Section 4). Then as n → ∞ (i) P n ψ → ψ in H for ψ ∈ H, (ii) P n ψ → ψ in V for ψ ∈ V, (iii) P n ψ → ψ in V γ for ψ ∈ V γ .

Appendix B. Kuratowski Theorem
The main objective of this appendix is to prove the following theorem (see, [9,Lemma 4.2]). The proof of the above theorem heavily relies on the following Kuratowski Theorem [17].
We will also need the following abstract results to prove Theorem B.1 Lemma B.1. 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)} . (B.1) Proof. Since the Borel σ−filed on X 2 ∩ Z is the smallest σ−field generated by τ (X 2 ∩ Z), i.e. B(X 2 ∩ Z) = σ(τ (X 2 ∩ Z)), in order to prove the lemma it is enough to show that ∀ Y ∈ B(X 1 ) Y ∩ Z ∈ B(X 2 ∩ Z) .
As X 1 ∈ B(X 2 ) there exists a countable collection {K i } i∈N of open subsets of X 2 such that Therefore, Since C ∈ τ (X 2 ), for every i ∈ N, C ∩ K i is open in X 2 and there exists a collection {B j } j∈N ∈ τ (X 2 ) such that Thus and for every j ∈ N, B j ∩ Z ∈ B(X 2 ∩ Z). Since B(X 2 ∩ Z) is a σ−field, the countable union also belongs to B(X 2 ∩ Z), proving (B.2) for every Y ∈ τ (X 1 ). Secondly, we implement the method of good sets to prove (B.2) for a larger class of subsets of X 1 . Let Claim : G is a σ−field. i) X 1 ∈ G since X 1 ⊂ X 1 and X 1 ∈ τ (X 1 ) by the definition of topology. ii) Let A ∈ G. We want to show that A c := X 1 \ A ∈ G, i.e.
A c ⊂ X 1 and A c ∩ Z ∈ B(X 2 ∩ Z). Since A ∈ G, A ⊂ X 1 and A ∩ Z ∈ B(X 2 ∩ Z). Clearly A c = X 1 \ A ⊂ X 1 .
We have the following set relations Now in the above identity c (A ∩ Z), c X 1 belongs to B(X 2 ∩ Z) and hence A c ∩ Z ∈ B(X 2 ∩ Z), inferring A c ∈ G. iii) Let {A i } i∈N ∈ G. Then A i ⊂ X 1 for every i ∈ N hence i∈N A i ⊂ X 1 .
Also, the following holds i∈N A i ∩ Z = i∈N (A i ∩ Z) .
Since A i ∈ G, A i ∩ Z ∈ B(X 2 ∩ Z) and as B(X 2 ∩ Z) is a σ−field i∈N (A i ∩ Z) ∈ B(X 2 ∩ Z) .
Therefore, we have shown that for every Y ∈ B(X 1 ), Y ∩ Z ∈ B(X 2 ∩ Z).
Lemma B.2. Let X 1 , X 2 , Y be topological spaces such that X 1 ⊂ X 2 , X 1 has trace topology from X 2 and X 1 ∩ Y = X 2 ∩ Y then Proof. The topologies of X 1 ∩ Y and X 2 ∩ Y denoted by τ (X 1 ∩ Y ) and τ (X 2 ∩ Y ) respectively are given by Since X 1 has a trace topology from X 2 , for every A ∈ τ (X 1 ) there exists a C ∈ τ (X 2 ) such that A = C ∩ X 1 . Thus τ (X 1 ∩ Y ) = generated by {C ∩ X 1 ∩ B : C ∈ τ (X 2 ), B ∈ τ (Y )} .
Thus all we are left to show is C ∩ X 1 ∩ B = C ∩ B for every C ∈ τ (X 2 ) and B ∈ τ (Y ). Since X 1 ∩ Y = X 2 ∩ Y , we have the following set relations We will need the following space :