Large deviations for the Navier–Stokes equations driven by a white-in-time noise

In this paper, we consider the 2D Navier–Stokes system driven by a white-in-time noise. We show that the occupation measures of the trajectories satisfy a large deviations principle, provided that the noise acts directly on all Fourier modes. The proofs are obtained by developing an approach introduced previously for discrete-time random dynamical systems, based on a Kifer-type criterion and a multiplicative ergodic theorem. AMS subject classifications: 35Q30, 60B12, 60F10, 37H15


VA H A G N N E R S E S Y A N L A R G E D E V I ATIO N S F O R T H E N AV IE R -ST O K E S E Q U ATIO N S D R I V E N B Y A W HIT E -IN -TI M E N OIS E G R A N D E S D É V I ATIO N S P O U R L E S É Q U ATIO N S D E N AV IE R -ST O K E S P E R T U R B É E S PA R U N B R UIT B L A N C E N T E M P S
Abstract. -In this paper, we consider the 2D Navier-Stokes system driven by a white-in-time noise. We show that the occupation measures of the trajectories satisfy a large deviations principle, provided that the noise acts directly on all Fourier modes. The proofs are obtained by developing an approach introduced previously for discrete-time random dynamical systems, based on a Kifer-type criterion and a multiplicative ergodic theorem.

Introduction
We study the large deviations principle (LDP) for the 2D Navier-Stokes system for incompressible fluids: where ν > 0 is the viscosity of the fluid, u = (u 1 (t, x), u 2 (t, x)) and p = p(t, x) are unknown velocity field and pressure, f is an external (random) force, and u, ∇ = u 1 ∂ 1 + u 2 ∂ 2 . Throughout this paper, we assume that the space variable x = (x 1 , x 2 ) belongs (1) to the standard torus T 2 = R 2 /2πZ 2 . The problem is considered in the space of divergence-free vector fields with zero mean value (1.2) H = u ∈ L 2 (T 2 , R 2 ) : div u = 0 in T 2 , T 2 u(x)dx = 0 endowed with the L 2 -norm · . We assume that the force is of the form where h ∈ H 1 := H 1 (T 2 , R 2 ) ∩ H is a given function and η is a white-in-time noise Here {b j } is sequence of real numbers such that {β j } is a sequence of independent standard Brownian motions defined on a filtered probability space (2) (Ω, F, {F t }, P), and {e j } is an orthonormal basis in H consisting of the eigenfunctions of the Stokes operator L = −∆ with eigenvalues {α j }. As usual, projecting (1.1) to H, we eliminate the pressure and obtain an evolution equation for the velocity field (3) (e.g., see [Lio69, Chapter 1, Section 6]): (1.5)u + B(u) where B(u) = Π( u, ∇ u) and Π is the orthogonal projection onto H in L 2 . This system is supplemented with the initial condition (1.6) u(0) = u 0 .
Under these assumptions, problem (1.5), (1.6) admits a unique solution and defines a Markov family (u t , P u ) parametrised by the initial condition u = u 0 ∈ H. The (1) The periodic boundary conditions are chosen to simplify the presentation. Similar results can be established in the case of a bounded domain with smooth boundary and Dirichlet boundary conditions.
(3) To simplify the notation, we shall assume that ν = 1. ergodic properties of this family are now well understood. In particular, it is known that (u t , P u ) admits a unique stationary measure, which is exponentially mixing, provided that sufficiently many coefficients b j are non-zero (see the papers [FM95, KS00, EMS01, KS02, BKL02, HM06, Oda08] and the book [KS12]). A central limit theorem (CLT) for problem (1.5), (1.6) is established in [Kuk02,Shi06]. The LDP proved in the present paper is a natural extension of the CLT. Indeed, while the CLT describes the probability of small deviations of a time average of a functional from its mean value, the LDP quantifies the probability of large deviations. Before formulating the main result of this paper, let us introduce some notation and definitions. We denote by P(H) the space of Borel probability measures on H endowed with the topology of weak convergence. Given a measure ν ∈ P(H), we set P ν (Γ) = H P u (Γ)ν(du) for any Γ ∈ F and consider the following family of occupation measures δ us ds, t > 0 defined on the probability space (Ω, F, P ν ). Here δ u is the Dirac measure concentrated at u ∈ H. We shall say that a mapping I : P(H) → [0, +∞] is a good rate function if the level set {σ ∈ P(H) : I(σ) α} is compact for any α 0. A good rate function I is nontrivial if its effective domain D I = {σ ∈ P(H) : I(σ) < ∞} is not a singleton. For any numbers κ > 0 and M > 0, we denote Main Theorem. -Assume that (1.4) holds and b j > 0 for all j 1. Then for any numbers κ > 0 and M > 0, the family {ζ t , t > 0} satisfies an LDP, uniformly with respect to ν ∈ Λ(κ, M ), with a non-trivial good rate function I : P(H) → [0, +∞] not depending on κ and M . More precisely, the following two bounds hold.
Upper bound. -For any closed subset F ⊂ P(H), we have lim sup Lower bound.
-For any open subset G ⊂ P(H), we have lim inf Furthermore, I is given by This type of large deviations results have been first established by Donsker and Varadhan [DV75] and later generalised by many others (see the books [FW84,DS89,DZ00] and the references therein). There are only a few works studying the large deviations behaviour of solutions of randomly forced PDEs as time goes to infinity.
The case of the stochastic Burgers and Navier-Stokes equations is first studied in [Gou07a,Gou07b]. In these papers, the random perturbation is of the form (1.3) with the following restriction on the coefficients Notice that the lower bound does not allow the sequence {b j } to converge to zero sufficiently fast, so the external force f is irregular with respect to the space variable. This is not very natural from the physical point of view. The proof is based on a general sufficient condition established in [Wu01], and essentially uses the strong Feller property. The main novelty of our Main Theorem is that it proves an LDP without any lower bound on {b j } (so, in particular, we do not have a strong Feller property).
We use an approach introduced in the papers [JNPS15,JNPS18], where an LDP is established for a family of dissipative PDEs perturbed by a random kick force. The proofs of these papers are based on a Kifer type criterion for LDP and a study of the large-time behaviour of generalised Markov semigroups. These results have been later extended in [MN18a] to the case of the stochastic damped nonlinear wave equation driven by a spatially regular white noise. The main result of that paper is an LDP of local type. In the case of the Navier-Stokes system (1.5), although we follow a similar scheme, there are important differences in all the steps of the argument, coming from both the continuous-time nature of the system and the globalness of the LDP. Here we study the large-time asymptotics of the Feynman-Kac semigroup without any restriction on the smallness of the potential. One of the most important difficulties arises in the proof of the uniform Feller property. To establish this, we construct coupling processes using a new two parameter auxiliary equation (see (4.1)) which allows to have an appropriate Foiaş-Prodi estimate for the trajectories and a rapid exponential stabilisation for finite-dimensional projections.
Let us also mention that the multiplicative ergodic theorem we obtain for system (1.5) is of more general form and works for a larger class of functionals and initial measures (see Theorem 2.1).
It is a challenging open problem whether an LDP still holds for (1.5), (1.6) when the driving noise is highly degenerate (i.e., only a finite number of b j are non-zero in (1.3)). For the Navier-Stokes system in this degenerate situation, exponential mixing is established in [HM06] for white-in-time noise and in [KNS18] for a bounded noise satisfying some decomposability and observability hypotheses. Using these results and literally repeating the arguments of the proof of Theorem 5.4 in [MN18b], one can prove a level-1 LDP of local type.
The paper is organised as follows. In Section 2, we state a multiplicative ergodic theorem for the Navier-Stokes system and combine it with Kifer's criterion for non-compact spaces to prove the Main Theorem. In Sections 3 and 4, we check the conditions of an abstract result on large-time behaviour of generalised Markov semigroups. Section 5 is devoted to the proof of the multiplicative ergodicity. In the Appendix, we prove various a priori estimates for the solutions and recall the statement of the above-mentioned result for generalised Markov semigroups.

Notation
We shall use the following standard notation.
• H is the space defined by (1.2), B H (a, R) is the closed ball in H of radius R centred at a. When a = 0, we write B H (R).
is the space of vector functions u = (u 1 , u 2 ) with components in the usual Sobolev space of order 1 on T 2 .
for which the following norm is finite Here P N is the orthogonal projection in H onto the space

Proof of the Main Theorem
In this section, we state a multiplicative ergodic theorem for the Navier-Stokes system (1.5) and apply it to prove the Main Theorem. Let us start by introducing the following two weight functions m κ (u) = exp(κ u 2 ), (2.1) w m (u) = 1 + u 2m , u ∈ H (2.2) for positive numbers κ and m. To avoid triple subscripts, we shall write C m (H) and P m (H) instead of C mκ (H) and P mκ (H). Recall that the Feynman-Kac semigroup associated with the family (u t , P u ) is defined by From estimate (6.21) it follows that, for sufficiently small κ and any V ∈ C b (H), the application P V t maps C m (H) into itself. Let P V * t : M + (H) → M + (H) be its dual. Then a measure µ ∈ P(H) and a function h ∈ C m (H) are eigenvectors corresponding to an eigenvalue λ > 0 if We have the following result.
Existence and uniqueness. -There is a unique pair Convergence. -For any f ∈ C m (H), ν ∈ P(H), and R > 0, we have Moreover, for any M > 0 and κ ∈ (κ, γ 0 ), the convergence This theorem is established in Section 5. Here we combine it with some arguments from [JNPS18, MN18a], to prove the Main Theorem.
Proof of the Main Theorem.
Step 1: Reduction. -It suffices to prove the Main Theorem for small κ, so we shall assume that κ ∈ (0, γ 0 ). Let us take any M > 0 and endow the set Θ = R * + × Λ(κ, M ) with an order relation ≺ defined by (t 1 , ν 1 ) ≺ (t 2 , ν 2 ) if and only if t 1 t 2 . Then a family {x θ ∈ R, θ ∈ Θ} converges if and only if it converges uniformly with respect to ν ∈ Λ(κ, M ) as t → ∞. Assume that the following three properties hold.
(2) There is a vector space V ⊂ C b (H) such that its restriction to any compact set K ⊂ H is dense in C(K), and for any V ∈ V, there is a unique σ V ∈ P(H) satisfying the relation where I(σ) is the Legendre transform of Q given by (1.8).
These two inequalities imply the upper and lower bounds in the Main Theorem, since we have the following equalities lim sup Now we turn to the proofs of Properties (1)-(3).
Properties (1) and (2) are proved using the same methods as in the case of the discrete-time model considered in [JNPS18]. The restriction of V to any compact set K ⊂ H is dense in C(K). Taking f = 1 in (2.6), we get Property 1 for any V ∈ V with Q(V ) = log λ V . In the case of an arbitrary V ∈ C b (H), this property is established by using a buc-approximating sequence V n ∈ V of V (i.e., sup n 1 V n ∞ < ∞ and V n − V L ∞ (K) → 0 as n → ∞ for any compact K in H) and the exponential tightness of the family {ζ θ } (which holds by Property 3). The reader is referred to Section 5.6 of [JNPS18] for the details.
(4) We shall see in the proof of Theorem 2.1, that γ 0 is the number in Lemma 6.3.
To prove Property (2), for any V ∈ V and F ∈ C b (H), we consider the following auxiliary Markov semigroup By Property (1), the following limit exists Let On the other hand, by Proposition 1.3 in [MN18a] (whose proof is the same in our case), the measure It remains to show that the good rate function I is non-trivial. Assume, by contradiction, that D I is a singleton. Then I(µ) = 0 and I(σ) = +∞ for σ ∈ P(H) \ {µ}, where µ is the stationary measure of (u t , P u ). On the other hand, as the Legendre transform is its own inverse, we derive from (1.8) that Combining the latter with the central limit theorem (see [KS12, Proposition 4.1.4]), we get V = 0. This contradicts the assumption that V is non-constant and completes the proof of the Main Theorem.

Checking conditions of Theorem 6.6
Theorem 2.1 is proved by applying a convergence result for generalised Markov semigroups obtained in [JNPS18, MN18a] and restated here as Theorem 6.6. In this and next sections, we show that the conditions of that theorem are satisfied for the generalised Markov family of transition kernels defined by , and w = w m with sufficiently large m 1.

Growth estimates
Estimate (6.24) implies that the measure P V t (u, · ) is concentrated on the space So the following proposition gives the growth condition in Theorem 6.6.
Step 1. -Let us show that there are integers m, R 0 1 such that Indeed, let τ (R) be the first hitting time of the set X R defined by (6.26), and let m and R 0 be the integers in Proposition 6.4 for γ = V ∞ . Then for any u ∈ H, we have This and (6.27) imply that By the strong Markov property and (6.27), Step 2. -It suffices to prove (3.1) for integer times k 1: Indeed, the semigroup property and the fact that V is non-negative and bounded imply that where P t = P 0 t is the Markov operator associated with (1.5).
We shall also need the following growth estimates with two other weights.
and let R 0 and γ 0 be the numbers in Proposition 3.1 and Lemma 6.3, respectively. Then for any κ ∈ (0, γ 0 ), we have Step 1: Proof of (3.7). -As in the previous proof, we can assume that V is non-negative and t = k is integer. We take any A > 0 and write To estimate J k , we use the Markov property and (6.22) Combining this with (3.9) and (3.10), and choosing A so large that q : As P V k 1(u) 1, we arrive at the required inequality (3.7).

Time-continuity
The following lemma proves the time-continuity property.
Proof. -Let us show the continuity at the point T 0. For any t 0, we write As V is bounded and g ∈ C w (H), we have Combining this with (6.23), we get S 1 → 0 as t → T . To estimate S 2 , we take any R > 0 and write On the other hand, by the Lebesgue theorem on dominated convergence, for any R > 0, we have S 4 → 0 as t → T . Choosing R > 0 sufficiently large and t sufficiently close to T , we see that S 3 +S 4 can be made arbitrarily small. This shows that S 2 → 0 as t → T and completes the proof of the lemma.

Uniform irreducibility
As V is a bounded function, we have is the transition function of the Markov family (u t , P u ). Thus to show the uniform irreducibility of {P V t }, it suffices to prove the following result.
Proposition 3.4. -The family {P t } is uniformly irreducible with respect to the sequence {X R }, i.e., for any integers ρ, R 1 and any r > 0, there are positive numbers l = l(ρ, r, R) and p = p(ρ, r) such that Proof.
Step 1. -There is a number d > 0 such that for any R 1, we have for sufficiently large t = t(R). Indeed, combining (6.23), (6.24), and the Markov property, we get Taking t so large that e −8α 1 t R 8 < 1 and d > 2 √ C and using the Chebyshev inequality, we arrive at Step 2. -By Lemma 3.3.11 in [KS12], for any non-degenerate ball B ⊂ H, there is p 1 = p 1 (d, B) > 0 such that Combining this with a simple compactness and continuity argument, we get for some p 2 = p 2 (d, ρ, r) > 0. This estimate, (3.12), and the Kolmogorov-Chapman relation imply (3.11) with l = t + 1 and p = p 2 /2.

Existence of an eigenvector
Here we show that the dual operator P V * t has an eigenvector and give some decay estimates for it.
(5) We do not indicate the dependence of different constants on V, t, m, n, and κ.
(6) Note that this proof is formal. A rigorous proof can be obtained by applying the above arguments to bounded approximations of m.
Step 3: Construction of an eigenvector. -Let us take any A > 0 and m 1 and define the convex set By the Fatou lemma, D A,m is closed in P(H). Consider the continuous mapping .

Let us show that G(D
is the oscillation of V . Choosing A and m so large that exp{t(Osc(V ) − mα 1 )} 1/2 and A 2C 6 , we get that G(D A,m ) ⊂ D A,m . In view of the Prokhorov compactness criterion (see [Dud02,Theorem 11.5.4]), to prove that G(D A,m ) is relatively compact, it suffices to check that H u 2 1 P V * t ν(du) C 7 for any ν ∈ D A,m . Using (6.24) and the fact that V is bounded, we get Thus there is an eigenvector µ ∈ D A,m .

Uniform Feller property
In this section, we establish the following result.
Theorem 4.1. -For any V ∈ V, the family {P V t } satisfies the uniform Feller property with respect to the sequence {X R }, i.e., there is an integer R 0 1 such that the family { P V t 1 −1 R P V t ψ, t 0} is uniformly equicontinuous on X R for any ψ ∈ V and R R 0 .
See the papers [JNPS15,JNPS18,MN18b] for similar results in the case of a discrete-time random dynamical system and [MN18a] for the case of the stochastic damped nonlinear wave equation. The main difficulty in the proof of Theorem 4.1 comes from the fact that the oscillation of the potential V can be arbitrarily large.
To overcome this, we introduce a new auxiliary equation in the construction of the coupling processes and choose carefully the parameters in order to have a stabilisation property with an appropriate rate.

Construction of coupling processes
The coupling processes are constructed following the arguments of [MN18a]. Let us take any z, z ∈ H and denote by u t and u t the solutions of (1.5) issued from z and z . For any integer N 1 and number λ > 0, let v be the solution of the following problem where η is defined by (1.3). We denote by ν(z, z ) and ν (z ) the laws of processes {v(t), t ∈ J} and {u (t), t ∈ J}, respectively, where J = [0, 1]. We shall use the following result.
Proposition 4.2. -There exists an integer N 1 1 such that if N N 1 and λ N 2 /2, then for any ε > 0 and z, z ∈ H, we have where · var denotes the total variation distance on P(C(J; H)) and a < 2, C, and C λ,N are positive constants not depending on ε, z, z .
See Section 6.2 for the proof. By Proposition 1.2.28 in [KS12], there is a probability space (Ω,F,P) and measurable functions Z, Z : H × H ×Ω → C(J; H) such that (Z(z, z ), Z (z, z )) is a maximal coupling for (ν(z, z ), ν(z )) for any z, z ∈ H. We denote byṽ andũ t the restrictions of Z and Z to time t ∈ J. Thenṽ t is a solution ofv where the process { t 0 ψ(s)ds, t ∈ J} has the same law as Letũ t be a solution oḟ Then {ũ t , t ∈ J} has the same law as {u t , t ∈ J}. Now the coupling operators R and R are defined by By Proposition 4.2, for any ε > 0, N N 1 , and λ N 2 /2, we have Let (Ω k , F k , P k ), k 0 be a sequence of independent copies of (Ω,F,P) and (Ω, F, P) the direct product of (Ω k , F k , P k ). For any ω = (ω 1 , ω 2 , . . .) ∈ Ω and z, z ∈ H, we setũ 0 = z,ũ 0 = z , and where t = s + k, s ∈ [0, 1). We shall say that (ũ t ,ũ t ) is a coupled trajectory at level (N, λ) issued from (z, z ).

Proof of Theorem 4.1
Step 1: Stratification. -Let us take any functions V, ψ ∈ V and points z, z ∈ X R such that d := z − z 1. We need to prove the uniform equicontinuity of the family {g t , t 0} on X R , where Without loss of generality, we can assume that ψ and V are non-negative, ψ 1, and the integer N in representation (1.10) is the same for ψ and V (we denote it by N 0 ). Let (u t , u t ) := (ũ t ,ũ t ) be a coupled trajectory at level (N, λ) issued from (z, z ) and let v t :=ṽ t be the associated process. The parameters N N 0 and λ N 2 /2 will be chosen later.
Following [MN18a,JNPS18], for any integers r 0 and ρ 1, we introduce the events (7) where K is the constant in (6.19), and the pairwise disjoint events (7) The eventḠ r is well defined also for r = +∞.
In Steps 2 and 3, we estimate I t 0 , I t r,ρ , andĨ t .
Step 2: Estimates for I t 0 and I t r,ρ . -We have following inequalities for any integers r, ρ 1 and R R 0 , where R 0 is the number in Proposition 3.1. Let us prove (4.6), the other inequality is proved in a similar way. First assume that r + 1 t. Using ψ 1, the positivity of Ξ V t ψ, and the Markov property, we derive where {F t } is the filtration generated by (u t , u t ). Then from the positivity of V and (3.1) it follows that Using this, (6.23), and the symmetry, we obtain (4.6). If r > t, then I t r,ρ e r V ∞ P(A r,ρ ) e r V ∞ P V t 1 R P(A r,ρ ) 1/2 , which implies (4.6) by symmetry.
Step 3: Estimate forĨ t . -Let us show that (4.7) |Ĩ t | C 4 (R, V, λ, N, ψ) P V t 1 R d. Indeed, we writẽ where Ξ V t := exp t 0 V (u s )ds . Then by (6.3), on the eventÃ we have Since ψ ∈ L b (H), we derive from this Recalling that d 1 and combining the estimates for J t 1 and J t 2 , we get (4.7).
Step 4: Uniform equicontinuity of g t . -We use the following lemma, which is proved at the end of this subsection.
From (4.4)-(4.9) it follows that, for any z, z ∈ X R , t 0, R R 0 , and α > 0, we have To prove this, we will assume that α is sufficiently large. Let Then taking α > 16 V ∞ /a, we see that These two inequalities show that (4.10) holds.

Proof of Theorem 2.1
The results of Sections 3 and 4 show that the conditions of Theorem 6.6 are satisfied if we choose with sufficiently large m and R 0 . Thus there are eigenvectors µ V ∈ P(H) and h V ∈ L ∞ w (H) corresponding to an eigenvalue λ V > 0. Moreover, for any R 1, the restriction of h V to X R is continuous and strictly positive, so h V : H 1 → R is continuous and strictly positive.
The continuity of h V : H → R follows from the uniform convergence in (2.4), and the uniqueness of µ V and h V from (2.4) and (2.5). The proof of (2.4) is carried out in Steps 1-3, and that of (2.6) in Step 4. Convergence (2.5) follows immediately from (2.4).
Step 1: Proof of (2.4) for f ∈ V. -In view of (6.32), for any f ∈ V, we have limit (2.4) in C(X R )∩L 1 (H, µ V ). We claim that this limit holds also in C(B H (R)) for any R 1. Indeed, it suffices to check condition (6.33) with B = B H (R) and s = 1, i.e.,

From the Poincaré inequality and (6.24) it follows that
This implies (2.4) for f ∈ V.
Step 2: Proof of (2.4) for f ∈ C b (H). -For any n 1, letf n ∈ L b (H) be such that sup Then the functions f n =f n • P n belong to the space V, satisfy f n ∞ f ∞ and f n → f as n → ∞, uniformly on compact subsets of H. Setting for any t 0 and n 1, we write Since f n ∈ V, the first term on the right-hand side of this inequality goes to zero as k → ∞ for any fixed n 1. The Lebesgue theorem on dominated convergence implies that | f − f n , µ V | → 0 as n → ∞. Thus, the required convergence will be established if we show that To prove this limit, we take any ρ > 0 and write By (3.2), we have To estimate J 2 , we use (3.8). For any ρ, n 1 and t 0, we have By (5.3), the right-hand side of this inequality goes to zero as ρ → ∞, uniformly with respect to t 1. Combining this with (5.4), we see that supremum over t 1 of the right-hand side of (5.2) can be made arbitrarily small by choosing first ρ > 0 and then n 1 sufficiently large. This proves (5.1).
Step 3: Proof of (2.4) for f ∈ C m (H). -We use again an approximation argument. Let us fix any κ ∈ (0, γ 0 ) and f ∈ C m (H) with m = m κ . We define a sequence {f n } by the relation f n = f + ∧ n − f − ∧ n. Then f n ∈ C b (H), |f n | |f | for any n 1, and f n → f in L ∞ m (H) with m = m κ for any κ ∈ (κ, γ 0 ). Furthermore, in view of (2.4) and the Lebesgue theorem on dominated convergence, we have Thus, as in the previous step, it suffices to prove that To see this, we use (3.7) for m : Combining this with (5.3), we get (5.5).
Step 4: Proof of (2.6). -In view of (2.4), it suffices to show that (5.6) sup By (3.7) and (5.3), we have It follows that we obtain (5.6). This completes the proof of Theorem 2.1.

The Foiaş-Prodi estimate
Let us take any numbers T, λ > 0, any function ϕ ∈ L 2 ([0, T ]; H), any integer N 1, and consider the equationsu where P N is the orthogonal projection in H onto the space H N defined by (1.11). The following result is a version of the Foiaş-Prodi estimate obtained in [FP67]; see also [KS12, Section 2.1.8] for a similar result for the Navier-Stokes system (with different equation instead of (6.2)) and [MN18a, Section 7.3] for the damped nonlinear wave equation.
; H 1 ) be solutions of (6.1) and (6.2) issued from z and z , respectively. Then If we assume additionally that for some numbers ρ > 0 and K > 0, then for any α > 0, we have provided that 2λ > N 2 α + cK. Here c > 0 is an absolute constant and C λ,N is a constant depending on λ and N . Proof.
Taking the scalar product in H of this equation with y, we obtain 1 2 d dt y 2 + y 2 1 + λ y 2 = 0.
Step 2: Proof of (6.5). For any a, b ∈ H 1 , let us set B(a, b) = Π( a, ∇ b). Taking the scalar product of (6.6) with w, and using the equality Using the identity  B(a, b), b = 0, a, b ∈ H 1 and the Hölder inequality, we obtain (6.9) To estimate I 2 , we use the Hölder inequality, the inclusion H 1 2 ⊂ L 4 , and the interpolation inequality a 2 1/2 a a 1 : Combining this with (6.8) and (6.9), and using the Poincaré inequality Hence, (6.10) and (6.4) imply that Choosing λ and N such that 2λ > N 2 2α + c 1 K, we get (6.5) with c = c 1 /2.

Proof of Proposition 4.2
We closely follow the arguments of the proof of a similar result from Section 7.3 of [MN18a] in the case of the nonlinear wave equation (see also [KS12, Section 3.3.3]).
Note that inequality (4.2) concerns the laws of the solutions and not the solutions themselves. Thus we can choose the underlying probability space (Ω, F, P). We assume that Ω is the space C(R + ; R) endowed with the topology of uniform convergence on bounded intervals, P is the law of the Wiener process W in (1.3), and F is the completion of the Borel σ-algebra of Ω with respect to P. We define a stopping time by where E u (t) is the functional and K is the number in Lemma 6.3, and ρ > 0 is a constant to be chosen later. The stopping times τ u and τ v are defined in a similar way. Then by inequality (6.19), we have (6.11) We define a transformation Λ : Ω → Ω by We use the following result, whose proof is given at the end of this section.
Lemma 6.2. -There is an integer N 1 1 such that for any numbers N N 1 , λ N 2 /2, and ρ > 0 and any initial points z, z ∈ H, we have where Λ * P stands for the image of P under Λ, and C and C λ,N are positive constants not depending on ρ, z, z .
Let us introduce auxiliary processes y u and y v in H defined as follows: for t τ they coincide with the processes u and v, respectively, while for t τ andτ < ∞ they are zero. With probability 1, we have Let us denote by u 1 and v 1 the restrictions of u (t) and v(t) to J. Then where the supremum is taken over all Borel subsets of C(J; H). Note that Further, we have Moreover, thanks to (6.13), Combining last four inequalities, we infer that Finally using this with inequalities (6.11) and (6.12), we get ν(z, z ) − ν (z ) var 4 e −γ 0 ρ + 2 exp C λ,N z − z 2 e C( z 2 + z 2 +ρ) − 1 Choosing a = 2γ 0 /(γ 0 + 1) and ρ = −γ −1 0 a ln(ε/4 1/a ), we obtain (4.2).
Proof of Lemma 6.2.
Step 2: Proof of (6.15). -By Proposition 6.1, the following inequalities hold Integrating by parts and using the Hölder inequality, we see that | B(a, b), e j | C j a 1 b , a, b ∈ H 1 , j 1.

A priori estimates
The following lemma gathers some standard a priori estimates for the solutions of the stochastic Navier-Stokes system. The reader is referred to Section 2.4.2 in [KS12] for more general results.
Lemma 6.3. -Assume (8) that B 1 < ∞, h ∈ H 1 , and u t is a solution of (1.5) issued from u ∈ H. Then we have the following estimates.
To prove (6.22), we apply the Itô formula for F (t, u) = t u 2 exp(κ u 2 ), use the equalities and take the expectation: (9) We confine ourselves to a formal derivation of (6.23). The accurate proof is based on the same arguments applied to the stopped solutions u(t ∧ τ n ), where τ n = inf{t 0 : u(t) > n}.

Hyper-exponential recurrence
For any R > 0, let τ (R) be the first hitting time of the set X R : We have the following standard estimate for the exponential moment of τ (R).
Proposition 6.4. -For any γ > 0, there are positive numbers m, R, and C such that Proof. -See [JNPS18, Proposition 5.1] for a similar result in the discrete-time case. The proof of (6.27) follows the same arguments. The idea is to establish the inequality for the first hitting time of a ball in H and then to use the regularising property of the Navier-Stokes system.
Step 2: Hyper-exponential recurrence in H 1 . -First note that, for any numbers p ∈ (0, 1) and r > 0, there is R > 0 such that Indeed, this follows immediately from the Chebyshev inequality and (6.24): for any u ∈ B H (r) and sufficiently large R = R(r, p). Now we combine (6.28) and (6.31) to prove (6.27). We introduce the sequences of stopping times τ 0 = τ 0 (r), τ n = inf{t τ n−1 + 1 : u t ∈ B H (r)}, n 1 and τ n = τ n + 1. Letn = min{n 0 : τ n ∈ X R }. Since {τ (R) > τ k } ⊂ {n > k}, the first probability is estimated by (1 − p) k . The second one is estimated using (6.28) and the strong Markov property P u τ k M C k 1 w m (u)e −3γM , where C 1 > 0 does not depend on k, M 1 and u ∈ H. Thus, we obtain P u τ (R) M (1 − p) k + C k 1 w m (u)e −3γM . To complete the proof, it remains to choose appropriately the parameters k and R. We take k ∼ εM , where ε > 0 is so small that ε log C 1 γ, and R > 0 so large that ε log(1 − p) −1 2γ. Then P u τ (R) M 2e −2γM w m (u), which implies (6.27).

Generalised Markov semigroups
For the reader's convenience, we recall here a result on the large-time asymptotics of generalised Markov semigroups in a Polish space X. It is established in [JNPS18] in the discrete-time setting, then extended to the continuous-time in [MN18a]. Let us first recall some terminology.
Definition 6.5. -We shall say that {P t (u, · ), u ∈ X, t 0} is a generalised Markov family of transition kernels if the following two properties are satisfied.
Feller property. -For any t 0, the function u → P t (u, · ) is continuous from X to M + (X) and does not vanish.
To any such family we associate two semigroups by the following relations: For a measurable function w : X → [1, +∞] and a family C ⊂ C b (X), we denote by C w the set of functions ψ ∈ L ∞ w (X) that can be approximated with respect to the norm · L ∞ w by finite linear combinations of functions from C. We shall say that a family C ⊂ C b (X) is determining if for any µ, ν ∈ M + (X) satisfying ψ, µ = ψ, ν for all ψ ∈ C, we have µ = ν. Finally, a family of functions ψ t : X → R is uniformly equicontinuous on a subset K ⊂ X if for any ε > 0 there is δ > 0 such that |ψ t (u) − ψ t (v)| < ε for any u ∈ K, v ∈ B X (u, δ) ∩ K, and t 0. The following result is Theorem 7.4 in [MN18a].
Theorem 6.6. -Let {P t (u, · ), u ∈ X, t 0} be a generalised Markov family of transition kernels satisfying the following four properties.
Growth conditions. -There is an increasing sequence {X R } ∞ R=1 of compact subsets of X such that X ∞ := ∞ R=1 X R is dense in X. The measures P t (u, · ) are concentrated on X ∞ for any u ∈ X and t > 0, and there is a measurable function w : X → [1, +∞] and an integer R 0 1 such that (10) where · R and · ∞ denote the L ∞ norm on X R and X, respectively, and we set ∞/∞ = 0. Time-continuity. -For any g ∈ C w (X) and u ∈ X, the function t → P t g(u) is continuous from R + to R.
(10) The expression (P t w)(u) is understood as an integral of a positive function w against a positive measure P t (u, · ).
Uniform irreducibility. -For sufficiently large ρ 1, any R 1 and r > 0, there are positive numbers l = l(ρ, r, R) and p = p(ρ, r) such that P l (u, B X (û, r)) p for all u ∈ X R ,û ∈ X ρ .
Uniform Feller property. -There is a number R 0 1 and a determining family C ⊂ C b (X) such that 1 ∈ C and the family { P t 1 −1 R P t ψ, t 0} is uniformly equicontinuous on X R for any ψ ∈ C and R R 0 .
Then for any t > 0, there is at most one measure µ t ∈ P w (X) such that µ t (X ∞ ) = 1 and P * t µ t = λ(t)µ t for some λ(t) ∈ R satisfying the following condition: Moreover, if such a measure µ t exists for all t > 0, then it is independent of t (we set µ := µ t ), the corresponding eigenvalue is of the form λ(t) = λ t , λ > 0, supp µ = X, there is a non-negative function h ∈ L ∞ w (X) such that h, µ = 1, (P t h)(u) = λ t h(u) for u ∈ X, t > 0, the restriction of h to X R belongs to C + (X R ), and for any ψ ∈ C w and R 1, we have Finally, if a Borel set B ⊂ X is such that