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Dissipation properties of transport noise in the two-layer quasi-geostrophic model

A stochastic version of the two-layer quasi-geostrophic model (2LQG) with multiplicative transport noise is analysed. This popular intermediate complexity model describes large scale atmosphere and ocean dynamics at the mid-latitudes. The transport noise, which acts on both layers, accounts for the unresolved small scales. After establishing the well-posedness of the perturbed equations, we show that, under a suitable scaling of the noise, the solutions converge to the deterministic 2LQG model with enhanced dissipation. Moreover, these solutions converge to the deterministic stationary ones on the long time horizon.


Introduction
The quasi-geostrophic model, first mathematically described by Charney [6] in the first half of the XX century, is an approximation of the three dimensional Navier-Stokes equation in vorticity formulation which captures the large-scale phenomena of atmosphere and ocean dynamics. Quasi-geostrophic models with several layers, like the one considered in this article, are particularly suitable to describe baroclinic instabilities, a crucial mechanism behind most common weather patters at the mid-latitudes. In fact the quasi-geostrophic equations is a popular model in theoretical meteorology given the richness of phenomena it can describe (see e.g. [28,25]). From a mathematical perspective, it can be seen as a model of intermediate complexity between two and three dimensional Navier-Stokes equation. The model has been shown to be well-posed and to exhibit a global attractor in its deterministic version (see [1]) and in a stochastic version with additive noise [7,4]. In this paper a stochastically perturbed version of the two-layer quasi-geostrophic (2LQG) model with multiplicative noise of transport type on both layers is studied and shown to be well-posed.
The nature of this perturbation is crucial both from a physical interpretation of the results and for the novelty of mathematical tools it requires. Recent developments on the physical justification for transport noise in fluid dynamics models include [19]. There the transport noise is systematically introduced in models relevant for geophysical applications, including the quasi-geostrophic model, in such a way to retain conservation laws crucial for the description of fluids, in particular circulation. Recently [17] proposed a rigorous interpretation and justification of transport noise as additive noise on smaller scales. For more on the interpretation of transport noise in stochastic partial differential equation refer to [17] and references therein.
From a mathematical perspective the literature on transport noise for fluid dynamics models is vast, especially for Euler and Navier-Stokes equation. Most relevant for this work are [13,12]. In these works the authors make use a suitable scaling, first introduced in [18], to show that weak solutions of a stochastic Euler model converge to weak solutions of the deterministic Navier-Stokes equation, and provide a quantitative estimate of the rate of convergence. Similarly, for the 2LQG model we will see that the transport noise provides an enhancement of eddy dissipation effects. These results highlight the regularization action of the transport noise, which, despite being energy-preserving in general, acts as a dissipative force. On the relevance of transport noise for dissipation and mixing properties we refer as well to [14,11].
Thanks to dissipation properties of the transport noise we will also show that the solution of the 2LQG model approaches the deterministic stationary solution for large times. The well-posedness of the stationary system associated to the 2LQG equations would require in general a large model viscosity, hard to justify from a physical perspective. However, we will show that an appropriate choice of the transport noise ensures stability on the long run of the solutions. This results extends those obtained for the heat equations in [15] to a nonlinear system like the quasi-geostrophic model here under analysis. We expect this approach to generalise to the two-dimensional Navier-Stokes equations and other models with similar structure.

The Model and the Main Results
The two-layer quasi-geostrophic model (2LQG) is one of the most used model to describe the motion of atmosphere as well as of the ocean at the mid-latitudes. In particular it captures the large scale dynamics of two layers of fluid of fixed height h 1 , h 2 respectively and with density ρ 1 and ρ 2 with ρ 1 < ρ 2 . We consider the β-plane approximation to the Coriolis effect for which the Coriolis parameter f c can be expressed as f c (y) = f 0 +βy, with f 0 and β assigned positive constants. In the model we include the effect of eddy viscosity on both layers, of the bottom friction on the second layer, to account for the interaction with the Eckmann layer, and a deterministic additive forcing on the first layer, for example to account for the wind forcing on the upper ocean. In order to give a mathematical formulation of the model let us introduce a spatial domain D, squared domain D = [0, L] × [0, L] ⊂ R 2 (where L will be e.g. 10 5 m for the ocean).
Consider the following equations for the variables ψ(t, x) = (ψ 1 (t, x), ψ 2 (t, x)) t , streamfunction of the fluid, and q(t, x) = (q 1 (t, x), q 2 (t, x)) t , the so-called quasi-geostrophic (QG) potential vorticity Here ν > 0 is the eddy viscosity parameter, r > 0 accounts for the bottom friction and F (t) is a deterministic forcing with zero spatial averages, namely The QG potential vorticities q and the streamfunction ψ are linked by the equations where S 1 , S 2 are positive constants such that As stated in the introduction, the goal of the present work is to study the eddy dissipation properties of transport noise for this model. There are several motivations to consider transport noise, as the effect of small scales on large scales in fluid dynamical problems, see [16], [19] for several discussion on this topic. Loosely speaking, small scale transport noise produces in the limit an extra dissipative term, which can be called eddy dissipation. There is an extended literature devoted to these kind of topics both in the endogenous and the exogenous case. See for example [12], [13], [15], [11].
Let us now introduce the stochastic perturbation we will consider. Let (Ω, F , F t , P) be a filtered probability space. Let {W j,k t } K k=1 , j ∈ {1, 2} be two sequences of real Brownian motions adapted to F t and consider two corresponding sequences of divergence free vector fields {σ j,k } K k=1 ⊆ C ∞ (D; R 2 ), j ∈ {1, 2}. In general, less can be required on the regularity of the coefficients σ j,k , but it is not our goal to stretch the boundaries of regularity of the noise as it is not critical to obtain the desired final result. Now we can consider the following stochastically perturbed two-layer quasi-geostrophic model which in Itô formulation reads We assume periodic boundary conditions for the streamfunction ψ in both directions and that D ψ(t, x) dx = 0. Let us set some notation before stating the main contributions of this work. Let (H k (D), · H k ), k ∈ R be the standard Sobolev spaces of L-periodic functions satisfying condition (2). We will denote by ·, · H k the corresponding scalar products. With a slight abuse of notation, for k > 0 we denote the dual pairing with In case of k = 0 we will write L 2 (D) instead of H 0 (D) and we will neglect the subscript in the notation for the norm. Similarly, we introduce the Sobolev spaces of zero mean vector fields Again, in case of k = 0 we will write L 2 instead of H 0 and we will neglect the subscript in the notation for the norm and the scalar product. Sometimes, on the space H k we will consider the norms which are straightforwardly equivalent to the standard ones that we denote by · H k . Furthermore we define the following functional spaces It is well known that ∆ is the infinitesimal generator of analytic semigroup of negative type and moreover V can be identified with D((−∆) 1/2 ), see e.g. [24].
A first issue related to the analysis of the dissipation properties of the transport noise is the well-posedness of such system. In fact the existence of strong probabilistic solution is outside the framework treated in earlier works, see for example [4,8,9,19]. Thus, first, we need to define our notion of solution for system (6) and state our well-posedness result.
Let Z be a separable Hilbert space, with associated norm · Z . For p ≥ 1 denote by L p (F t0 , Z) the space of pintegrable random variables with values in Z, measurable with respect to F t0 . Moreover, denote by C F ([0, T ] ; Z) the space of continuous adapted processes (X t ) t∈[0,T ] with values in Z such that Define for j ∈ {1, 2} the operators which can be easily shown to be bounded linear operators We then consider the following concept of weak solution (see e.g. [10, Chapter 7]) for (6): is a weak solution of equation (6) if, for every φ = (φ 1 , φ 2 ) t ∈ D(∆), we have The well posedness of this system is guaranteed by the following result which we will prove in Section 3.
Theorem 1.2. For every q 0 ∈ L 4 F0 (Ω, H) and F ∈ L 4 F 0, T ; L 2 (D) , there exists one and only one weak solution of equation (6). Remark 1.3. We stated Theorem 1.2 in full generality in order to provide a complete framework for the well posedness of such stochastic system, that, to the best of these authors knowledge, is unavailable in the literature. This result is redundant for the scope of this work though. In fact, in order to exploit the eddy dissipation properties of the transport noise we will consider deterministic initial conditions and a deterministic and timeindependent forcing. Remark 1.4. It is well known that, in absence of external forcing or dissipation, the quasi-geostrophic model has an infinite number of preserved quantities. The noise introduced in (5) will preserve the L 2 -norm of the quasi-geostrophic potential vorticity q, namely the potential enstrophy of the system. For more on the role of enstrophy in the two-layer quasi-geostrophic dynamics see e.g. [21], [28, Section 5.6.3, Section 9.2.2].
After showing the well posedness of system (6), our goal is to provide sufficient conditions in order to model the dissipation properties of the transport noise. Let us explain briefly the adopted strategy before going into the details. First, following the approach introduced in [12], we will show that under an appropriate scaling of the noise, first introduced by Galeati in [18], the solution of the stochastic system (5) converges, in a suitable sense which we will clarify later, to the solution of the associated deterministic system with an extra diffusion. Second, for time-independent forcings F and same properties of the noise, solutions of the stochastic system (5) will converge to those of the associated stationary deterministic 2LQG model.
In order to precisely formulate these statements, let us introduce some notations and the precise formulation of the noise. Let a, b be two positive numbers, then we write a b if there exists a positive constant C such that a ≤ Cb and a α b when we want to highlight the dependence of the constant C on a parameter α. Let e k (x) = L −1 exp 2πi L k · x , k ∈ Z 2 0 = Z 2 \ {(0, 0)}, orthonormal basis of L 2 (D) made by eigenfunctions of −∆. Following the approach first introduced in [18] consider the following explicit representations of the coefficients where Z 2 + , Z 2 − is a partition of Z 2 0 with Z 2 + = −Z 2 − , and the parameters θ j,k satisfy the following conditions: 1.
2. θ j,k = 0 if |k| is large enough. We will denote by K the finite set of k where θ j,k = 0.
Furthermore take an infinite sequence of complex standard Brownian motions such that W j,k = W j,−k . Thus the noise we consider is parameterized by the coefficients κ, θ j,k and the set K. Under this setting, as described for example in [18,11], equation (6) can be reformulated as The corresponding deterministic system is where, as before,q andψ are linked by relation (3). Thanks to Theorem 1.2 and classical results on two dimensional deterministic quasi-geostrophic equations, see for example [1], under the assumptions q 0 ∈ H, F ∈ L 4 (0, T ; L 2 (D)) there exists a unique weak solutions q (resp.q) of the problem (11) (resp. (12)). Now we can state one of the main results of our work which allows to quantify the difference between the behavior of the stochastic and the deterministic system. Theorem 1.5. Let q andq be weak solutions to (11) and (12) respectively. Then for any α ∈ (0, 1), there exists C depending from α and all the parameters of the model except for the noise such that for any ǫ ∈ (0, α] one has where R 2 T is a constant independent of the noise defined in Section 4.1. Next, given the model (5) now with a time-independent forcing F , consider the associated stationary system. On the one hand, similarly to the Navier-Stokes system, existence and uniqueness of the solution of the stationary system associated to quasi-geostrophic equations is guaranteed under, generally unfeasible, assumptions on the viscosity. On the other hand, as we will show in Section 4.2, in our framework these assumptions will be satisfied thanks to the dissipation properties of the transport noise. More precisely, considerq solution of the following system where, F ∈ L 2 (D) and as always,q andψ are linked by relation (3). Then, thanks to a particular parametrization of the noise, we will show in Section 4.2 the following: Theorem 1.6. For κ large enough, for each δ > 0 and α ∈ (0, 1), it existsT =T (δ) and a sequence

Preliminaries
In this section we recall several technical tools crucial for the next sections, see for example [2], [5], [12], [24] for more details.
First, similarly to [5], we define the linear operatorÃ : H k+2 → H k , k ∈ R connecting the streamfunction with the quasi-geostrophic potential vorticitỹ It is well known that it has a bounded inverse (−∆ − M ) −1 : H k → H k+2 . Thanks to this fact, for each q ∈ H k , there exists a unique ψ ∈ H k+2 such that q = (∆ + M ) ψ and moreover for each k ∈ R there exists two constants c 1,k ≤ c 2,k such that In this work we will also use extensively the relation which follows directly from (3). We recall three technical lemmata which can be proved by classical arguments and we refer to [12, Section 2.1].
. Given a divergence free vector field V ∈ L 2 (D; R 2 ) the following bounds hold true.
3. Let β ∈ (0, 1), then for any With some abuse of notations, if ψ and q are two vector fields, we will denote and the results of previous lemma continue to hold in this framework. Then this first lemma generalises fundamental estimates for the nonlinearity of the quasi-geostrophic model presented for example in [5,7].
The second lemma provides classical estimates on the semigroup generated by ∆: The semigroup e δ(t−s)∆ has also regularising effects as stated in the following: Finally consider the following classical result.

Well-posedness
In this section we will show Theorem 1.2 holds following a classical approach by means of the Galerkin approximation.

Galerkin Approximation and Limit Equations
As described in Section 1.1, let {e i } i∈N be the orthonormal basis of L 2 (D) made by eigenvectors of −∆ and λ i the corresponding eigenvalues, λ i are positive and non decreasing.
be the orthogonal projector of L 2 (D) on H N . As we are looking for a finite dimensional approximation of the solution of equation (6), define The coefficients c i,j,N are chosen in such a way to satisfy for all eigenfunctions e i , 1 ≤ i ≤ N and all t ∈ [0, T ] the system Here q N 0 = P N q 0 and the variables ψ N and q N are linked by relation (3), and the operators F N j , i = 1, 2, are defined similarly to (8), namely The local well-posedness of (15) follows from classical results about stochastic differential equations with locally Lipshitz coefficients, see for example [20], [27]. The global well-posedness follows from the following a priori estimates: Lemma 3.1. Given the system (15) the following energy estimate holds: Furthermore for any q 0 ∈ L 4 F0 (Ω, H) and F ∈ L 4 F (0, T ; L 2 (D)) the following a priori and integral bounds are where C is a constant possibly changing its value line by line, but independent of N .
Proof. Let us start by showing (16). We first apply the finite dimensional Itô's formula to the system (15) to get then multiply each equation by h 1 and h 2 respectively and sum them up to obtain Now, thanks to relations (3), (4) and (14) we have Using (3) to treat similarly the term rh 2 ∆ψ N 2 , q N 2 , from (21) we get the desired result Now, by Cauchy-Schwartz and exploiting the continuity of the operator it follows that there exists a constant C independent of N such that Thus, via Gronwall's inequality and exploiting the fact that P N are projections on L 2 (D), we have that hence (17) holds taking the supremum over [0, T ] and the expectation on both sides. Next, integrating in time (22) and using the estimate just obtained we have from which follows (18). Moving on to (19), since d q N 4 = 2 q N 2 d q N 2 , exploiting the energy estimate (16) we obtain As argued for (17), by appropriate use of Cauchy-Schwartz inequality and of the continuity of the operator (−∆ − M ) −1 , we can estimate the right hand side of (23) as follows Thus, via Gronwall's inequality and the properties of the projection P N , we get From this relation it follows immediately that where C = C(T ) is a constant dependent on the time T but crucially independent of N . Finally, to prove (20) we integrate (22) over [0, T ] to get Squaring and simply bounding the right hand side we have By (17) we have Then taking the expectation on both side, (20) holds with constant C depending on T , E |||q 0 ||| 4 and E F 4 L 2 (0,T ;H) .
From the energy estimates for q N shown in Lemma 3.1, it follows that there exists a subsequence, which we will denote again for simplicity by q N , converging to a stochastic process q * -weakly in L 4 (Ω; L ∞ (0, T ; H)), and weakly in L 4 (Ω; L 2 (0, T ; V)). Furthermore there exist two unknown processes, B * 1 , B * 2 such that Moreover, thanks to the converging properties of the projector P N for N → +∞, the processes q and B * j , j ∈ {1, 2} satisfies P-a.s. for each i ∈ N and t ∈ [0, T ] Let us expand on the convergence of the nonlinear term t 0 q N j (s), F N j e i ds.
We know that for each α ≥ 0 and x ∈ D((−∆) α ), P N x − x D((−∆) α ) vanishes on the limit N → ∞. Thus, since for each k ∈ K, β ≥ 1/2, the operator σ j,k · (∇·) is in L(D((−∆) β ), D((−∆) β−1/2 )), then for φ ∈ D((−∆) β ), P N (σ j,k · φ) − σ j,k · φ D((−∆) β−1/2 ) converges to zero for increasing N . Starting from these observations it is easy to show that for each i, F N j e i − F j e i → 0. Then, thanks to the weak convergence of q N j to q j , we have the required convergence of For what concern the continuity in H of the process q we can argue in the following way via Itô's formula and Kolmogorov continuity theorem. From the weak formulation above we get the weak continuity in H of q applying the Kolmogorov continuity theorem for (26). Then, applying the Itô's formula to |||q(t)||| 2 we get, arguing as in the proof of Lemma 3.1, dt From this, we get the continuity of q(t) 2 thanks to the integrability properties of q. Weak continuity and continuity of the norm implies strong continuity, thus we have the strong continuity of q as a process taking values in H. Alternatively the strong continuity in H of q follows from the results in [23].

Existence, Uniqueness and Further Results
First, it is easy to show that the solution of (3.6), when it exists, it is unique. Proof. Let q,q be two solutions and let v be their difference. Let ψ,ψ be the corresponding streamfunctions and χ be their difference. Let us consider the difference between two weak solutions and applying Itô's formula, which can be justified arguing as in the proof of Proposition 3.6 below, we get Let us rewrite better − ∇ ⊥ ψ 1 · ∇q 1 , v 1 H −1 ,H 1 + ∇ ⊥ψ 1 · ∇q 1 , v 1 H −1 ,H 1 , the other one is analogous.
Multiplying the equations by h 1 and h 2 respectively and summing up, if we callβ 1,2 = h 1,2 β,r = rh 2 we get where η is a positive constant fixed below. Therefore we have Exploiting relation (3) we obtain The terms which comes from the linear part of the equations can be estimated easily, up to some constant C, by R(t)|||v(t)||| 2 , thus we need only to analyze the nonlinear ones. Again we treat only one of the two in order to avoid repetitions.
Therefore, taking η large enough we get and the thesis follows immediately by Gronwall's Lemma.
Harder task is to ensure the existence of the solutions of equations (6). To show it we need the following crucial lemma which, for an appropriate stopping time τ M , ensures convergence of the associated stopped Galerkin process q N to the stopped process q. This procedure is standard in stochastic analysis, see for example [3], [26], [22].
Proof. We have to show that converges to zero in N . Let us callψ N j = P N ψ jq N j = P N q j , j ∈ 1, 2 andψ N ,q N satisfy relation (3). Then, by the triangular inequality Thanks to the properties of the projector P N and dominated convergence theorem, it can be shown thatq N → q in L 2 (Ω, L 2 (0, T ; V)) ∩ L 2 (Ω, C(0, T ; H)), and also in weaker topologies. Therefore, we are left to show the convergence of Let us start by taking the difference of (26) and (15), pair the two components respectively with h 1 q N 1 − q N 1 and h 2 q N 2 − q N 2 , and add them together to get an equation for q N − q N . With Itô formula and some elementary manipulation using (14), we have where we denotedβ 1,2 := h 1,2 β,r := h 2 r and B N 1,2 = ∇ ⊥ ψ N 1,2 · ∇q N 1,2 ,. Now we can estimate the RHS in (29). In the following C will denote a generic constant independent of N .
Next, to better understand the behavior of the terms we will first write them in an equivalent form. By definition of F 1 (8) and Green's theorem we have Similarly, recalling the definition (9) of G k 1 , k = 1, . . . , K, we have Therefore, by some additional manipulations, it can be shown that We can now estimate all terms by Cauchy-Schwartz inequality to get The same estimate holds for the terms in (29) relative to the second component. We now move on to treating the nonlinear term in (29) and again we only treat B * We can estimate the first term on the right hand side of the last display as follows and, by Ladyzhenskaya's and Young's inequalities we have The second term in (35) can be rewritten as , the expected value of first term will go to 0 easily. For what concern the second one, adding and subtracting so that, again by Ladyzhenskaya's inequality, In summary, we showed Consider now the auxiliary function with η 1 and η 2 two positive constants to be defined later, and let us then compute via the Itô's formula We exploit previous estimates (30),(31),(32),(34) and (36) and take the expected value for t = τ M obtaining Taking η 1 , η 2 large enough and exploiting the convergence ofq N to q we can neglect several terms in the right hand side. Let us consider the remaining terms. Recalling that from the weak convergence of q N it follows that and similarly for the second component. Lastly, let us consider By (37) we have that As already observed the first term converges to zero asq N − q N ⇀ 0 in L 2 (Ω; L 2 (0, T ; V)). For what concern the two remaining terms, thanks to Hölder inequality, we have In the last inequality we exploit the fact that |||q(t)||| 2 ≤ M , for all t ≤ τ M , andq N −q N ⇀ 0 in L 2 (Ω; L 2 (0, T ; V)). In conclusion, in (39) all the terms on the right hand side converge to zero as N → ∞, namely we have the following relation From relation (41), R(t) ≥ C M > 0, for all t ≤ τ M , and the properties of P N , via triangular inequality the thesis follows.
This series of lemmas identify the nonlinear term and conclude the proof of Theorem 1.2. Actually, thanks to some abstract results on stochastic processes it can be shown something more, namely that the full sequence q N converges to q in L 2 (Ω; L 2 (0, T ; V)) and, for each t ∈ [0, T ], q N (t) converges to q(t) in L 2 (Ω; H). This is an easy corollary of previous result and Lemma 2.5.
Theorem 3.5. The entire Galerkin's sequence q N satisfies Proof. From the uniqueness of the solution of problem (6), we have that each sub-sequence q n k has a converging sub-sub-sequence q n k,k which satisfies Lemma 3.1, Lemma 3.3 and Lemma 3.4. In particular, defining τ M as in the proof of Lemma 3.3, from relation (41), R(t) ≥ C M > 0 for all t ≤ τ M and the properties of P N we have the thesis via Lemma 2.5 using σ M = τ M and Q N = t 0 q(s) − q nN,N (s) 2 V ds or Q N = (q − q nN,N ) (t) 2 . Lastly, we show that the energy estimate (16) in Lemma 3.1 continues to holds for q, solution of problem (6). For what concern the a priori and integral estimates in Lemma 3.1, they straightforwardly continue to hold by the weak convergence of q N to q, but they can proved independently starting from the energy estimate and repeating the same steps of Lemma 3.1.
Proposition 3.6. Given q solution of (6), then the following energy estimate holds Proof. Letq N be defined as in Lemma 3.3. We already know by the properties of the projector P N thatq N → q in L 2 (0, T ; V) ∩ C(0, T ; H) P-a.s. As q satisfies the weak formulation of (6) with test functions e i we have Multiplying each equation by e i and summing up, we get Now we can apply the Itô's formula to the process |||q N ||| obtaining, thanks to the relations (3) and (14), Then, thanks to the properties of the projector P N we get easily the desired formula. The only thing we need to prove is that The last relation is true, in fact for each k ∈ {1, . . . , K}, j ∈ {1, 2} we have N i=1 q j (s), σ j,k · ∇e i 2 + q j (s), σ j,k · ∇ σ j,k · ∇q N j = − q j (s), σ j,k · ∇ P N (σ j,k · ∇q j (s)) + q j (s), σ j,k · ∇ σ j,k · ∇q N j = σ j,k · ∇q j (s), P N (σ j,k · ∇q j (s)) − σ j,k · ∇q j (s), σ j,k · ∇q N j → 0.

Enhanced dissipation by transport
In this section we will focus first on the proof of Theorem 1.5 and then, in Section 4.2, we will study the long time behaviour of the solutions of (11) showing that Theorem 1.6 holds.

Convergence to the Deterministic Evolution Model
The first step in order to prove Theorem 1.5, is to rewrite equations (11)- (12) in an alternative way, in order to introduce explicitly the term ∆q. For this reason, using once more the relation (3), we express ∆ 2 ψ as Thus if we call F = F −r∆ψ 2 , equation (11) can be rewritten as where we denoted by a k e k · ∇qdW k := a 1,k e k · ∇q 1 dW 1,k a 2,k e k · ∇q 2 dW 2,k Similarly, if we callF = F −r∆ψ 2 , the deterministic equation (12) can be rewritten as First, we want to show that the weak solution q of (11) satisfies a mild formulation. Denote the stochastic integral and stochastic convolution as Thanks to the results of Section 3 for the stochastic system the following relations can be shown to hold The constants C, R T here above depend from all the parameters of the model (i.e. T, ν, r, F, q 0 , M, β), except for the parameters of the noise. Thanks to these estimates, Assumption 2.4 of [12] holds. Thus, thanks to Corollary 2.6 in [12], the stochastic integral (44) and the stochastic convolution (45) are well defined and have the regularity prescribed by the Lemma below. (ii) For each ǫ ∈ (0, 1/2), p ≥ 1 (iii) For β ∈ (0, 1] and ǫ ∈ (0, β], p ≥ 1 it holds It is then easy to show that the weak solution q of the problem (12) satisfies also a mild formulation. In fact the following lemma holds. Proof. By definition, if q is a weak solution of problem (11), then it satisfies a k e k · ∇φ, q(s) dW k s P-a.s. for each t ∈ [0, T ] and φ ∈ D(−∆). If take e i := (e i , e i ) t as a test function, we have a k e k · ∇e i , q(s) dW k s .
If we apply Itô formula to the process e −t(κ+ν)λi q(t), e i and integrate in time we have We can then find Γ ⊆ Ω of full probability such that the above equality holds for all t ∈ [0, T ]and all i ∈ Z 2 0 . But this is exactly the mild formulation written in Fourier modes. Before proving Theorem 1.5, we need a preliminary result on the nonlinearity of our problem. Lemma 4.4. Let q ∈ L 2 ,q ∈ H 1 , then, given for each α ∈ (0, 1) the following relation holds true Proof. With a simple manipulation and the triangular inequality we have Then the thesis follows by Lemma 2.1. In fact by point 2, we have Now we take the expectation of (53) and use relation (52) in Lemma 4.1 to estimate the stochastic convolution to get (54) Now to prove statement (i) we use (46) and (49) to derive We move on to proving statement (ii). Given (54) let us now estimate T 0 q(s) 2 using the estimate (48) instead of (46), namely where we have also used that 1 ν + 1 κ+ν ≤ 2 ν . Then the desired bound follows from the same arguments used for the estimate (i).
Remark 4.5. Similarly to [12], we can consider also the p moment of the random variables treated in the previous theorem. We neglect this fact, which is not needed in order to prove Theorem 1.6.
1. By the proof of previous theorem, κ must be large enough that the following inequalities are satisfied, here C is a constant possibly depending from β, S 1 , S 2 , r, D but independent of κ and ν.
Now we are able to prove our final result.
Now we use the results of Theorem 1.5 for ǫ = α/2, thus we have Since the constants appearing in previous equation are independent of the parameters of the noise, if we take θ to be such that the right hand side of the previous inequality can be bounded by δ/4 then the thesis follows immediately. For example some possible choices of θ can be found in [12,Example 1.3].