The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior

where u represents the velocity field, ν is the viscosity constant, p(t, x) denotes the pressure and f is an external force field acting on the fluid. While the 2D Navier-Stokes equations have been studied extensively in the literature, there exist serious obstacles to tackle the 3D Navier-Stokes equations. One of them is the lack of uniqueness. There exist many modified versions of the 3D Navier-Stokes equations due to Leray and others with mollification (and/or cut off) of the nonlinear term as a way to approximate the original problem, see for instance the review paper of Constantin [9]. We also mention the paper [15] by Flandoli and Maslowski with a global cut off function used for the 2D Navier-Stokes equations. In 2006, Caraballo, Kloeden and Real [6] proposed a 3-dimensional model where the nonlinear term included a cut off factor FN (‖u‖) based on the norm of the gradient of the solution in the whole domain. Namely, for N ∈ (0,+∞) the function FN : [0,+∞)→ (0, 1] is defined by FN (r) := min{1, N r }.

(Communicated by Martino Bardi) 1. Introduction. It is well known that the three-dimensional (3D) Navier-Stokes equations describe the time evolution of an incompressible fluid and are given by where u represents the velocity field, ν is the viscosity constant, p(t, x) denotes the pressure and f is an external force field acting on the fluid. While the 2D Navier-Stokes equations have been studied extensively in the literature, there exist serious obstacles to tackle the 3D Navier-Stokes equations. One of them is the lack of uniqueness. There exist many modified versions of the 3D Navier-Stokes equations due to Leray and others with mollification (and/or cut off) of the nonlinear term as a way to approximate the original problem, see for instance the review paper of Constantin [9]. We also mention the paper [15] by Flandoli and Maslowski with a global cut off function used for the 2D Navier-Stokes equations. In 2006, Caraballo, Kloeden and Real [6] proposed a 3-dimensional model where the nonlinear term included a cut off factor F N ( u ) based on the norm of the gradient of the solution in the whole domain. Namely, for N ∈ (0, +∞) the function F N : [0, +∞) → (0, 1] is defined by results about the 3D Navier-Stokes equations. Indeed, they were used in [6] to establish the existence of bounded entire weak solutions for the 3D Navier-Stokes equations. Also in [21], the model was used to show that the attainability set of the weak solutions of the 3D Navier-Stokes equations satisfying an energy inequality are weakly compact and weakly connected. For convergence results of solutions of GMNSE to solutions of the 3D Navier-Stokes equations, see [6,26]. See [22,12,7,27,28,23,29] for other studies and applications of the GMNSE as well as the review paper [7]. Stochastic Partial Differential Equations(SPDEs) are powerful tool for understanding and investigating mathematically hydrodynamic and turbulence theory. To model turbulent fluids, mathematicians often use stochastic equations obtained from adding a noise term in the dynamical equations of the fluids. This approach is basically motivated by Reynolds work which stipulates that turbulent flows are composed of slow (deterministic) and fast (stochastic) components. Recently by following the statistical approach of turbulence theory, Flandoli et al [16], Kupiainen [24] confirm the importance of studying the stochastic version of fluids dynamics. Indeed, the authors of [16] pointed out that some rigorous information on questions of turbulence theory might be obtained from these stochastic versions. It is worth emphasizing that the presence of the stochastic term (noise) in these models often leads to qualitatively new types of behavior for the processes. Since the pioneering work of Bensoussan and Temam [35], there has been an extensive literature on stochastic Navier-Stokes equations with Wiener noise and related equations, we refer to [1,2,4,5,8,13,30].
In the present paper, we shall study the stochastic 3D globally modified Navier-Stokes equations. Let us now describe our model equation. Let M ⊂ R 3 be an open bounded set with regular boundary Γ and T > 0 be a final time. We consider the following stochastic 3D system of globally modified Navier-Stokes equations where {W k t , t ≥ 0, k = 1, 2, ...} is a sequence of independent one dimensional standard Brownian motions on some complete filtration probability space (Ω, F, P, (F t ) t∈(0,T ) ). If (e k ) k≥1 is an orthonormal basis of l 2 , we may formally define W by taking W = k W k e k . As such W is a cylindrical Brownian motion evolving over l 2 . We recall that l 2 is the Hilbert space consisting of all sequences of square summable real numbers. We define the auxiliary space Note that the embedding of l 2 ⊂ U 0 is Hilbert-Schmidt. Moreover, using standard martingale arguments with the fact that each W k is almost surely continuous (see [10]), we have that for almost every ω ∈ Ω, W (ω) ∈ C([0, T ]; U 0 ). See Section 3 for the precise assumptions on the coefficients F and {σ k ; k = 1, ..., ∞}.
In this paper, we shall prove the existence of a unique strong solution to our stochastic 3D GMNSE (1) under some assumptions on F and σ k for k = 1, ..., ∞.
Here the word "strong" means "strong" both in the sense of the theory of stochastic differential equations and the theory of partial differential equations. The proof combines the Galerkin approximation, the strong monotonicity of the coefficients and a Gronwall lemma for stochastic processes (see Lemma 5.5). To obtain the strong solution in the sense of partial differential equations, we prove that the solution of the Galerkin scheme is a Cauchy sequence in probability in L ∞ [0, T ]; H 1 . Moreover, as in the deterministic case, we study the convergence of the strong solution of the stochastic 3D GMNSE when N → ∞. This enables us to prove the existence of a martingale solution for the stochastic 3D Navier-Stokes equations. One of the main difficulties here is the passage to the limit on the nonlinear term containing F N .
The layout of the present paper is as follows. In Section 2, we recall some spaces useful for the abstract framework and some properties and estimates related to the operators involved in the model. Section 3 is concerned with the existence and uniqueness of strong solution for the stochastic 3D GMNSE. The convergence of the strong solution of the model when N→ ∞, is treated in Section 4. Finally in the Appendix for the reader's convenience, we recall two compacts embedding theorems, a convergence theorem for the stochastic integral and a stochastic Gronwall lemma.
It follows that V ⊂ H ≡ H ⊂ V , where the injections are dense and compact. Finally, we will use . * for the norm in V and ., . for the duality pairing between V and V . Now we consider the trilinear Moreover from the properties of b (see [35] or [32]), and the definition of F N , one easily obtains the existence of a constant c 1 > 0 only dependent on Ω such that |b N (u, v, w)| ≤ c 1 |u| is the Stokes operator (P is the orthogonal projection from L 2 (Ω) 3 onto H).
We recall (see [36]) that there exists a constant c 2 > 0 depending only on Ω such that The following lemmas give some important properties of the map F N (see [6,33] for the proof) for all p, r ∈ R + .
Lemma 2.2. For any u, v ∈ V , and each N > 0, The next lemma shows that B N is locally Lipschitz.
Lemma 2.3. The map B N : V → V is locally Lipschitz continuous i.e. for every r > 0, there exists a constant L r such that Proof. For u, v and w ∈ V , we have where we have used the estimate of Lemma 2.1. From this, we deduce that the map B N is locally Lipschitz.
The next result will be useful in our study of the stochastic 3D globally modified Navier-Stokes equations. See [33] for the proof.
for all u, v, w ∈ V. There exists a constant c 3 > 0 which depends on c 2 and ν such that 3. Existence and uniqueness. Let (W k (t), k ≥ 1) be a sequence of independent F t -Brownian motions defined on a filtered probability space (Ω, F, F t , P). Consider the stochastic 3D globally modified Navier-Stokes equations Here F is a mapping from V (resp. H) into V (resp. H). σ k (.), k ≥ 1 is a sequence of mapping from V (resp. H ) into V (resp. H). Consider the following hypothesis.
The next lemma shows some strong monotonicity of the operator G.
where the constant c 3 is given by Lemma 2.4.
Proof. We have where we used the estimate of Lemma 2.4.
We state the main result of this section.
Then there exists a unique solution to the stochastic 3D system of globally modified Navier-Stokes equations (10) that satisfies the following energy inequality Proof. I) Uniqueness. Let X andX two solutions of Problem 1 starting from the same initial value u 0 . For any r > 0 and R > 0, define the stopping time Then by Itô s formula, we have We are going to estimate each term in (19).
Combining the estimates (20)- (21) with (19), we get By Gronwall's inequality, we get for any t ∈ [0, T ] And the uniqueness follows by letting R→ ∞ and Fatou's lemma. II) Existence. We will use the Galerkin approximation combined with the strong monotonicity of the stochastic 3D globally modified Navier-Stokes equations. We shall do this in two steps: Step 1 : Assume u 0 ∈ L 6 (Ω, F 0 ; V ). Let {e i : i ≥ 1} ⊂ D(A) be a fixed orthonormal basis of H consisting of eigenvectors of ∆, so that it is also orthogonal in V . Denote π n the orthogonal projection from H onto the finite dimensional space H n := span{e 1 , e 2 , ..., e n }: Thus π n is also the orthogonal projection onto H n in V .
Consider the following finite dimensional stochastic differential equations in H n : We have for u ∈ H n .
Moreover by Lemma 2.3, (11) and (15), the maps u ∈ H n → π n G(u) ∈ H n and u ∈ H n → π n σ are respectively locally Lipschitz continuous and Lipschitz continuous. Then by the theory of stochastic differential equations(see [19,17]), there exists a unique continuous (F t )-adapted process u n (t) satisfying We now prove some a priori estimates of the approximated solution Lemma 3.3. There exists a constant c such that 2) E sup 3) sup Proof. 1) By Itô's formula, we have

G. DEUGOUÉ AND T. TACHIM MEDJO
From the estimate (4) and the properties of F N , we get The properties of F give The estimates (28)- (30) in (27) yield Taking expectation, we get Hence by Gronwall's inequality, we have for any T > 0, This proves 1).
2) Applying the Burkholder's inequality to the martingale Combining (31), (33) and (34), we get Choosing small enough, we obtain This ends the proof of 2).
Taking the expectation in (36) and using (37), we obtain Taking sufficiently small, we get Applying Gronwall's inequality, we obtain This ends the proof of 3). 4) We have Using the estimate (4), we get

This implies that
We also have (42) Using (41) and (42) in (40), we get and This ends the proof of the lemma.
The proof follows the same steps as in [34]. Fix an integer K. Take v ∈ L 2 (Ω T , H K ) where H K is the linear span of e 1 , e 2 , ..., e K . By Itô's formula, writing where r(t) is a non-negative stochastic process which is absolutely continuous and to be chosen later. A similar expression also holds for E |u n (t)| 2 e −r(t) − E |u 0 | 2 , that is For any non-negative ψ ∈ L ∞ ([0, T ], R), the weak convergence implies that By substituting the corresponding expressions, (45) becomes where Z n = Z 1 n + Z 2 n + Z 3 n with Set r (s) = c + 2(ν + c 3 N 4 + c). In view of (17) and (16) we see that Z 1 n ≤ 0. By the weak convergence, it is clear that Z 2 n → Z 2 , where Also Combining (46)-(51) after cancellations it turns out that As K is arbitrary, by approximation it is seen that (52) holds true for every v ∈ L 2 (Ω T , H 2 ). In particular, take v(s) = u(s) in (52) to obtainσ k (s) = σ k (u(s)) for every k ≥ 1. where r λ (s) is defined as r(s) with v replaced by v λ . Dividing (53) by λ we obtain for λ > 0, and for λ < 0. By (17), we have Therefore by the dominated convergence theorem, we get Asṽ is arbitrary, we conclude that G(s) = G(u(s)) a.e. on Ω T . Then Step 2: General case: Let X n (t), t ≥ 0 be the solution of the following equation The existence of X n is guaranteed by Step 1. As in the proof of (23), we can show that This implies that there exist a subsequence of X n still denoted by the same symbol and a process X ∈ L 2 (Ω; L ∞ ([0, T ]; V )) ∩ L 2 (Ω T ; D(A)) such that i) X n → X weakly in L 2 (Ω T ; D(A)), ii) X n → X in L 2 (Ω; L ∞ ([0, T ]; V )) equipped with the weak star topology. Next, we show that X n also converges to X in probability in L ∞ ([0, T ]; H 1 ).
For R > 0, define the stopping time τ n R is a stopping time since X n is continuous in V . It follows from (62) that there exists a constant M , independent of n, R so that We are going to prove that Let X n,m (t) = X n (t) − X m (t) and τ m,n R = τ n R ∧ τ m R . By Itô's formula, we have d X n.m (t) 2 + 2ν AX n,m (t), X n,m (t) V dt +2 B N (X n (t))−B N (X m (t)), X n,m (t) V dt+ F (X n (t))−F (X m (t)), X n,m (t) V dt =2 ∞ k=1 σ k (X n (t)) − σ k (X m (t)), X n,m (t) V dW k (t) We now estimate each term of (62).
We then estimate each term of this equality as follows. From the properties of F N and (4), we have For the second term , we get For the third term, we have Combining (63)-(65), we get iii) Finally | F (X n ) − F (X m ), X n − X m V | ≤ c X n,m 2 . By (62), for any pair of stopping times 0 ≤ σ a ≤ σ b ≤ τ n R ∧ τ n R , we have For the last term in (67), the Burkholder-Davis-Gundy inequality implies Combining (67) and (68), we get where c is a constant independent of the choice of τ a , τ b . By definition of τ m R , we have Then by application of the Gronwall lemma for stochastic processes (see Lemma 5.5), we obtain and this proves (61). For η > 0 and any R > 0, we get P sup Given an arbitrary small constant δ > 0, in view of (60), one can choose R such that P(τ n R ≤ T ) ≤ δ 4 and P(τ m R ≤ T ) ≤ δ 4 . For such R, by (61) there exists N 0 such that for m, n ≥ N 0 , P sup Therefore P sup This proves that X n converges to X in probability in L ∞ ([0, T ]; H 1 ). Finally we want to show that X solves (10). To this end, it suffices to prove that for v ∈ V, But for every n ≥ 1, we know that Since X n converges to X in probability in L ∞ ([0, T ]; H 1 ), there exists a subsequence of X n (still denoted by the same symbol) such that X n converges to X in H 1 for almost all t ∈ [0, T ], that is Since we also have We also have

G. DEUGOUÉ AND T. TACHIM MEDJO
From Vitali's theorem, we conclude that By the weak convergence, we have Collecting all these convergences, X satisfies (71) and this ends the proof of the existence.
4. Convergence to martingale solutions of the stochastic 3D Navier-Stokes equations. Let µ 0 be a probability measure on V such that V U 6 dµ 0 (U ) < ∞. Let u 0 be an F 0 -random variable in V with distribution µ 0 . Let u N be the unique strong solution of the stochastic 3D Globally Modified Navier-Stokes equations. In this section, we are going to study the asymptotic behavior of u N when N→ ∞.
E sup where the constants c 1 , c 2 , c 3 and c 4 are independent of N .
Proof. By Itô's formula, we get For u∈ V, we have Using the estimates (79)-(82) in (78), we arrive at Taking the supremum over [0, T ], we get Raising both sides to the power p 2 for p≥ 2, then taking expectations, we obtain with the Minskowski inequality and Fubini's theorem For the stochastic term, we use the Burkhölder-Davis-Gundy inequality Applying the above estimate to (85), we obtain Since Letting p = 4 and p = 2, we obtain the estimates (74) and (76). The estimate (83) implies Taking the supremum over [0, T ], raising both sides to the power 2 then taking expectation, we obtain with Minkowski's inequality and Fubini's theorem For the stochastic term, we have This together with (90) implies the estimate (77). The proof of Lemma 4.1 is complete.

4.2.
Estimates in fractional Sobolev spaces. We will apply the compactness result based on fractional Sobolev spaces in Lemma 5.2 (of the Appendix) with For this purpose, we will need the following estimates on fractional derivatives of u N .
where the constants k 1 , k 2 and k 3 are independent of N .
Proof. u N can be written as For For J 3 , we observe that for v ∈ D(A) where we have used the relation (2) in the last inequality. This implies that This along with (76) and (77) conclude that is bounded independently of N . For J 4 , using the estimate (74), we have is also bounded independently of N . For the term J 5 , Lemma 5.2 implies that, ∀α < 1 This together with (88) imply that is bounded independently of N , ∀α < 1 2 . Hence we obtain (94). Collecting the estimates (97)-(102), we obtain is bounded independently of N . By (75), we deduce is bounded independently of N . From (103) and (104) , we obtain (92). We observe from (95) that u N (t) − t 0 σ(u N )dW (s) = J 1 + J 2 + J 3 + J 4 combined with the estimates (97)-(101), we obtain (93) as desired.

Compactness arguments for
With the estimates independent of N , we can establish the compactness of the family (u N , W ). For this purpose, we consider the following phase spaces: We then define the probability laws of u N and W respectively in the corresponding phase spaces: and This defines a family of probability measures µ N = µ N u × µ W on the phase space X . We now prove that this family is tight in N . More precisely: Lemma 4.3. Consider the measures µ N on X defined according to (106) and (107). Then the family {µ N } N is tight and therefore weakly compact over the phase space X .
Proof. We can use the same technic as in the proof of Lemma 4.1 in [11]. The main idea is to apply Lemma 5.2 (of the Appendix) and Chebychev's inequality to (92)-(94).
Strong convergence as N → ∞. Since the family of measures {µ N } associated with the family (u N , W ) is weakly compact on X , we deduce that µ N converges weakly to a probability µ on X up to a subsequence. We can apply the Skorokhod embedding theorem (see Theorem 2.4 in [10], also [20]) to deduce the strong convergence of a further subsequence, that is: There exists a probability space (Ω,F,P), and a subsequence N k of randoms vectors (ũ N k ,W N k with values in X such that (i) (ũ N k ,W N k have the same probability distributions as (u N k , W N k ).
(ii) (ũ N k ,W N k ) converges almost surely as N k → ∞, in the topology of X , to an element (ũ,W ) ∈ X , i.e.
where (ũ,W ) has distribution µ. (iii)W N k is a cylindrical Wiener process, relative to the filtrationF N k t , given by the completion of the σ-algebra generated by {(ũ N k (s),W N k (s)); s ≤ t}.
(iv) All the statistical estimates on u N k are valid forũ N k , in particular, the estimates (74)-(77) hold.
(v) Each pair (ũ N k ,W N k ) satisfies (10) as an equation in L 2 (M), that is The following lemma proves thatũ N k ,ũ is weakly continuous with value in H
(112) Combining the strong convergence (108), the estimate (76) and the Vitali convergence theorem, we get and, thus possibly extracting a new subsequence denoted in the same way to save notation, one has alsõ u N k →ũ for almost all ω, t with respect to the measure dP ⊗ dt.
Fix w ∈ D(A). Using the weak convergence (112) , we can pass to the limit in the linear term.
We are going to prove that (115) The following lemma will be crucial for the proof of (115). Observing that we deduce that These inequalities show that But as 0 ≤ F N ( ũ N (s) ) ≤ 1, we get Finally since |1 − F N ( ũ N (s) )| ≤ 1, we arrive at This ends the proof of the lemma.
For the proof of (115), we introduce the abbreviations as in [6], To prove (115), we writẽ Reasoning as in the proof of the existence of martingale solutions of the 3D Navier-Stokes equations, the second term of this equality tends to 0, that is For the first term, we get and Lemma 4.5 shows thatẼ We therefore infer from (76) that |σ(ũ N k | L2(l 2 ,H) is uniformly integrable for N k in L q (Ω × (0, T )) for any q ∈ [1, 4). With the Vitali convergence theorem, we deduce that for all such q ∈ [1, 4), σ(ũ N k ) → σ(ũ) in L q (Ω, L q (0, T ; L 2 (l 2 , H))).
In particular, we get the convergence in probability of σ(ũ N k ) in L 2 (0, T ; L 2 (l 2 , H)).
Collecting all the convergence results, we obtain (ũ(t), w) + ν for all w ∈ D(A) and for a.e. ω ∈Ω, t ∈ (0, T ). The equality (123) is also valid for w ∈ V by density argument. We have then proved the following result.
Lemma 5.3. Let q ≥ 2, α < 1 2 be given so that qα > 1 Then for any predictable process h ∈ L q (Ω × (0, T ); L 2 (U, H)), we have t 0 h(s)dW (s) ∈ L q (Ω, W α,q (0, T ; H)), and there exists a constant c = c (q, α) ≥ 0 independent of h such that 5.2. Convergence theorem for the noise term. The following convergence theorem for the stochastic integral is used to facilitate the passage to the limit. The statements and proofs can be found in [2,11].
Lemma 5.4. Let (Ω, F, P) be a fixed probability space. and X a separable Hilbert space. Consider a sequence of stochastic bases S n := (Ω, F, {F n t } t≥0 , P, W n ), such that each W n is a cylindrical Brownian motion (over U) with respect to {F n t } t≥0 . We suppose that the {G n } n≥1 are a sequence of X -valued F n t predictable processes so that G n ∈ L 2 ((0, T ); L 2 (U, X ) a.s.. Finally consider S := (Ω, F, {F t } t≥0 , P, W ) and a function G ∈ L 2 ((0, T ); L 2 (U, X )), which is F t predictable. If G n → G in probability in L 2 ((0, T ); L 2 (U, X )),

5.3.
A stochastic Gronwall lemma. The following Gronwall lemma for stochastic processes is useful to prove the existence of strong solution for the stochastic 3D globally modified Navier-Stokes equations. See [18] for the proof.
Assume, moreover that for some fixed constant k,