Large deviation principle and inviscid shell models

A LDP is proved for the inviscid shell model of turbulence. As the viscosity coefficient converges to 0 and the noise intensity is multiplied by the square root of the viscosity, we prove that some shell models of turbulence with a multiplicative stochastic perturbation driven by a H-valued Brownian motion satisfy a LDP in C([0,T],V) for the topology of uniform convergence on [0,T], but where V is endowed with a topology weaker than the natural one. The initial condition has to belong to V and the proof is based on the weak convergence of a family of stochastic control equations. The rate function is described in terms of the solution to the inviscid equation.


Introduction
Shell models, from E.B. Gledzer, K. Ohkitani, M. Yamada, are simplified Fourier systems with respect to the Navier-Stokes ones, where the interaction between different modes is preserved only between nearest neighbors.These are some of the most interesting examples of artificial models of fluid dynamics that capture some properties of turbulent fluids like power law decays of structure functions.
There is an extended literature on shell models.We refer to K. Ohkitani and M. Yamada [25], V. S. Lvov, E. Podivilov, A. Pomyalov, I. Procaccia and D. Vandembroucq [21], L. Biferale [3] and the references therein.However, these papers are mainly dedicated to the numerical approach and pertain to the finite dimensional case.In a recent work by P. Constantin, B. Levant and E. S. Titi [11], some results of regularity, attractors and inertial manifolds are proved for deterministic infinite dimensional shells models.In [12] these authors have proved some regularity results for the inviscid case.The infinite-dimensional stochastic version of shell models have been studied by D. Barbato, M. Barsanti, H. Bessaih and F. Flandoli in [1] in the case of an additive random perturbation.Well-posedeness and apriori estimates were obtained, as well as the existence of an invariant measure.Some balance laws have been investigated and preliminary results about the structure functions have been presented.
driven by a Hilbert space-valued Brownian motion W .It involves some similar bilinear operator B with antisymmetric properties and some linear "second order" (Laplace) operator A which is regularizing and multiplied by some non negative coefficient ν which stands for the viscosity in the usual hydro-dynamical models.The shell models are adimensional and the bilinear term is better behaved than that in the Navier Stokes equation.Existence, uniqueness and several properties were studied in [1] in the case on an additive noise and in [10] for a multiplicative noise in the "regular" case of a non-zero viscosity coefficient which was taken constant.
Several recent papers have studied a Large Deviation Principle (LDP) for the distribution of the solution to a hydro-dynamical stochastic evolution equation: S. Sritharan and P. Sundar [27] for the 2D Navier Stokes equation, J. Duan and A. Millet [16] for the Boussinesq model, where the Navier Stokes equation is coupled with a similar nonlinear equation describing the temperature evolution, U. Manna, S. Sritharan and P. Sundar [22] for shell models of turbulence, I. Chueshov and A. Millet [10] for a wide class of hydrodynamical equations including the 2D Bénard magneto-hydro dynamical and 3D α-Leray Navier Stokes models, A.Du, J. Duan and H. Gao [15] for two layer quasi-geostrophic flows modeled by coupled equations with a bi-Laplacian.All the above papers consider an equation with a given (fixed) positive viscosity coefficient and study exponential concentration to a deterministic model when the noise intensity is multiplied by a coefficient √ ǫ which converges to 0. All these papers deal with a multiplicative noise and use the weak convergence approach of LDP, based on the Laplace principle, developed by P. Dupuis and R. Ellis in [17].This approach has shown to be successful in several other infinitedimensional cases (see e.g.[4], [5], [20]) and differ from that used to get LDP in finer topologies for quasi-linear SPDEs, such as [26], [9], [7], [8].For hydro-dynamical models, the LDP was proven in the natural space of trajectories, that is C([0, T ], H)∩ L 2 ([0, T ], V ), where roughly speaking, H is L 2 and V = Dom(A 2 ) is the Sobolev space H 2 1 with proper periodicity or boundary conditions.The initial condition ξ only belongs to H.
The aim of this paper is different.Indeed, the asymptotics we are interested in have a physical meaning, namely the viscosity coefficient ν converges to 0. Thus the limit equation, which corresponds to the inviscid case, is much more difficult to deal with, since the regularizing effect of the operator A does not help anymore.Thus, in order to get existence, uniqueness and apriori estimates to the inviscid equation, we need to start from some more regular initial condition ξ ∈ V , to impose that (B(u, u), Au) = 0 for all u regular enough (this identity would be true in the case on the 2D Navier Stokes equation under proper periodicity properties); note that this equation is satisfied in the GOY and Sabra shell models of turbulence under a suitable relation on the coefficients a, b and µ stated below.Furthermore, some more conditions on the diffusion coefficient are required as well.The intensity of the noise has to be multiplied by √ ν for the convergence to hold.The technique is again that of the weak convergence.One proves that given a family (h ν ) of random elements of the RKHS of W which converges weakly to h, the corresponding family of stochastic control equations, deduced from the original ones by shifting the noise by hν √ ν , converges in distribution to the limit inviscid equation where the Gaussian noise W has been replaced by h.Some apriori control of the solution to such equations has to be proven uniformly in ν > 0 for "small enough" ν.Existence and uniqueness as well as apriori bounds have to be obtained for the inviscid limit equation.Some upper bounds of time increments have to be proven for the inviscid equation and the stochastic model with a small viscosity coefficient; they are similar to that in [16] and [10].The LDP can be shown in C([0, T ], V ) for the topology of uniform convergence on [0, T ], but where V is endowed with a weaker topology, namely that induced by the H norm.More generally, under some slight extra assumption on the diffusion coefficient σ, the LDP is proved in C([0, T ], V ) where V is endowed with the norm 2 is out of reach because the inviscid limit equation is much more irregular.Indeed, it is an abstract equivalent of the Euler equation.The case α = 0 corresponds to H and then no more condition on σ is required.The case α = 1  4 is that of an interpolation space which plays a crucial role in the 2D Navier Stokes equation.Note that in the different context of a scalar equation, M. Mariani [23] has also proved a LDP for a stochastic PDE when a coefficient ε in front of a deterministic operator converges to 0 and the intensity of the Gaussian noise is multiplied by √ ε.However, the physical model and the technique used in [23] are completely different from ours.
The paper is organized as follows.Section 2 gives a precise description of the model and proves apriori bounds for the norms in C([0, T ], H) and L 2 ([0, T ], V ) of the stochastic control equations uniformly in the viscosity coefficient ν ∈]0, ν 0 ] for small enough ν 0 .Section 3 is mainly devoted to prove existence, uniqueness of the solution to the deterministic inviscid equation with an external multiplicative impulse driven by an element of the RKHS of W , as well as apriori bounds of the solution in C([0, T ], V ) when the initial condition belong to V and under reinforced assumptions on σ.Under these extra assumptions, we are able to improve the apriori estimates of the solution and establish them in C([0, T ], V ) and L 2 ([0, T ], Dom(A)).Finally the weak convergence and compactness of the level sets of the rate function are proven in section 4; they imply the LDP in C([0, T ], V ) where V is endowed with the weaker norm associated with A α for any value of α with 0 ≤ α ≤ 1  4 .The LDP for the 2D Navier Stokes equation as the viscosity coefficient converges to 0 will be studied in a forthcoming paper.
We will denote by C a constant which may change from one line to the next, and C(M ) a constant depending on M .

Description of the model
2.1.GOY and Sabra shell models.Let H be the set of all sequences u = (u 1 , u 2 , . ..) of complex numbers such that n |u n | 2 < ∞.We consider H as a real Hilbert space endowed with the inner product (•, •) and the norm where v * n denotes the complex conjugate of v n .Let k 0 > 0, µ > 1 and for every n ≥ 1, set k n = k 0 µ n .Let A : Dom(A) ⊂ H → H be the non-bounded linear operator defined by The operator A is clearly self-adjoint, strictly positive definite since (Au, u) ≥ k 2 0 |u| 2 for u ∈ Dom(A).For any α > 0, set Then V (as each of the spaces H α ) is a Hilbert space for the scalar product (u, v) and the associated norm is denoted by 3) The adjoint of V with respect to the H scalar product is where the last inequality is proved by the Cauchy-Schwarz inequality.Set u −1 = u 0 = 0, let a, b be real numbers and B : (2.5) for n = 1, 2, . . . in the GOY shell-model (see, e.g., [25]) or ) in the Sabra shell model introduced in [21].
Note that B can be extended as a bilinear operator from H × H to V ′ and that there exists a constant C > 0 such that given u, v ∈ H and w ∈ V we have (2.7) An easy computation proves that for u, v ∈ H and w ∈ V (resp.v, w ∈ H and u ∈ V ), For u, v in either H, H or V , let B(u) := B(u, u).The anti-symmetry property (2.8) Hence there exist positive constants C1 and C2 such that Finally, since B is bilinear, Cauchy-Schwarz's inequality yields for any α ∈ [0, 1  2 ], u, v ∈ V : (2.12) In the GOY shell model, B is defined by (2.5); for any u ∈ V , Au ∈ V ′ we have On the other hand, in the Sabra shell model, B is defined by (2.6) and one has for u ∈ V , Thus (B(u, u), Au) = 0 for every u ∈ V if and only if a + bµ 2 = (a + b)µ 4 and again µ = 1 shows that (2.13) holds true.
together with the induced norm | • | 0 = (•, •) 0 .The embedding i : H 0 → H is Hilbert-Schmidt and hence compact, and moreover, i i * = Q.Let L Q ≡ L Q (H 0 , H) be the space of linear operators S : , where S * is the adjoint operator of S. The L Q -norm can be also written in the form for any orthonormal basis {ψ k } in H, for example (ψ k ) n = δ k n .Let W (t) be a Wiener process defined on a filtered probability space (Ω, F, (F t ), P), taking values in H and with covariance operator Q.This means that W is Gaussian, has independent time increments and that for s, t ≥ 0, f, g ∈ H, Let β j be standard (scalar) mutually independent Wiener processes, {e j } be an orthonormal basis in H consisting of eigen-elements of Q, with Qe j = q j e j .Then W has the following representation and T race(Q) = j≥1 q j .For details concerning this Wiener process see e.g.[13].
Given a viscosity coefficient ν > 0, consider the following stochastic shell model where the noise intensity σ ν : [0, T ] × V → L Q (H 0 , H) of the stochastic perturbation is properly normalized by the square root of the viscosity coefficient ν.We assume that σ ν satisfies the following growth and Lipschitz conditions: , and there exist non negative constants K i and L i such that for every t ∈ [0, T ] and u, v ∈ V : For technical reasons, in order to prove a large deviation principle for the distribution of the solution to (2.16) as the viscosity coefficient ν converges to 0, we will need some precise estimates on the solution of the equation deduced from (2.16) by shifting the Brownian W by some random element of its RKHS.This cannot be deduced from similar ones on u by means of a Girsanov transformation since the Girsanov density is not uniformly bounded when the intensity of the noise tends to zero (see e.g.[16] or [10]).
To describe a set of admissible random shifts, we introduce the class A as the set of The set S M , endowed with the following weak topology, is a Polish (complete separable metric) space (see e.g.[5]): (2.17) In order to define the stochastic control equation, we introduce for ν ≥ 0 a family of intensity coefficients σν of a random element h ∈ A M for some M > 0. The case ν = 0 will be that of an inviscid limit "deterministic" equation with no stochastic integral and which can be dealt with for fixed ω.We assume that for any ν ≥ 0 the coefficient σν satisfies the following condition: ) and there exist constants KH , Ki , and Lj , for i = 0, 1 and j = 1, 2 such that: ) is defined by (2.2) and | • | L(H 0 ,H) denotes the (operator) norm in the space L(H 0 , H) of all bounded linear operators from H 0 into H.Note that if ν = 0 the previous growth and Lipschitz on σ0 (t, .)can be stated for u, v ∈ H. Remark 2.1.Unlike (C1) the hypotheses concerning the control intensity coefficient σν involve a weaker topology (we deal with the operator norm However we require in (2.18) a stronger bound (in the interpolation space H).One can see that the noise intensity

Thus the class of intensities satisfying both Conditions (C1) and (C2) when multiplied by
√ ν is wider than that those coefficients which satisfy condition (C1) with K 2 = 0.
Let M > 0, h ∈ A M , ξ an H-valued random variable independent of W and ν > 0. Under Conditions (C1) and (C2) we consider the nonlinear SPDE Using [10], Theorem 3.1, we know that for every T > 0 and ν > 0 there exists Kν s. for all v ∈ Dom(A) and t ∈ [0, T ].Note that u ν h is a weak solution from the analytical point of view, but a strong one from the probabilistic point of view, that is written in terms of the given Brownian motion W . Furthermore, if The following proposition proves that Kν 2 can be chosen independent of ν and that a proper formulation of upper estimates of the H, H and V norms of the solution u ν h to (2.20) can be proved uniformly in h ∈ A M and in ν ∈ (0, ν 0 ] for some constant ν 0 > 0. Proposition 2.2.Fix M > 0, T > 0, σ ν and σν satisfy Conditions (C1)-(C2) and let the initial condition ξ be such that E|ξ| 4 < +∞.Then in any shell model where B is defined by (2.5) or (2.6), there exist constants ν 0 > 0, K2 and Itô's formula and the antisymmetry relation in (2.8) yield that for t ∈ [0, T ], and using again Itô's formula we have where Since h ∈ A M , the Cauchy-Schwarz and Young inequalities and condition (C2) imply that for any ǫ > 0, Using condition (C1) we deduce and the inequalities (2.24)-(2.26)yield that for The Burkholder-Davis-Gundy inequality, (C1), Cauchy-Schwarz and Young's inequalities yield that for t ∈ [0, T ] and δ, κ > 0, Thus we can apply Lemma 3.2 in [10] (see also Lemma 3.2 in [16]), and we deduce that for 0 Using the last inequality from (2.4), we deduce that for K 2 small enough, C(M ) independent of N and ν ∈]0, ν 0 ], As N → +∞, the monotone convergence theorem yields that for K2 small enough and This inequality and (2.30) with t instead of t ∧ τ N conclude the proof of (2.22) by a similar simpler computation based on conditions (C1) and (C2).

Well posedeness, more a priori bounds and inviscid equation
The aim of this section is twofold.On one hand, we deal with the inviscid case ν = 0 for which the PDE can be solved for every ω.In order to prove that (3.1) has a unique solution in C([0, T ], V ) a.s., we will need stronger assumptions on the constants µ, a, b defining B, the initial condition ξ and σ0 .The initial condition ξ has to belong to V and the coefficients a, b, µ have to be chosen such that (B(u, u), Au) = 0 for u ∈ V (see (2.13)).On the other hand, under these assumptions and under stronger assumptions on σ ν and σν , similar to that imposed on σ0 , we will prove further properties of u ν h for a strictly positive viscosity coefficient ν.
Thus, suppose furthermore that for ν > 0 (resp.ν = 0), the map satisfies the following: Condition (C3): There exist non negative constants Ki and Lj , i = 0, 1, 2, j = 1, 2 such that for s ∈ [0, T ] and for any u, v ∈ Dom(A) if ν > 0 (resp.for any u, v and |A Since equation (3.1) can be considered for any fixed ω, it suffices to check that the deterministic equation (3.1) has a unique solution in C([0, T ], V ) for any h ∈ S M and that (3.4) holds.For any m ≥ 1, let and finally let σ0,m = P m σ0 .Clearly P m is a contraction of H) .Set u 0 m,h (0) = P m ξ and consider the ODE on the m-dimensional space H m defined by for every v ∈ H m .Note that using (2.9) we deduce that the map u ∈ H m → B(u) , v is locally Lipschitz.Furthermore, since there exists some constant Hence by a well-known result about existence and uniqueness of solutions to ODEs, there exists a maximal solution The following lemma provides the (global) existence and uniqueness of approximate solutions as well as their uniform a priori estimates.This is the main preliminary step in the proof of Theorem 3.1.Lemma 3.2.Suppose that the assumptions of Theorem 3.1 are satisfied and fix M > 0. Then for every h ∈ A M equation (3.6) has a unique solution in C([0, T ], H m ).There exists some constant C(M ) such that for every Proof.The proof is included for the sake of completeness; the arguments are similar to that in the classical viscous framework.Let h ∈ A M and let u 0 m,h (t) be the approximate maximal solution to (3.6) described above.For every N > 0, set τ N = inf{t : Since ϕ k ∈ Dom(A) and V is a Hilbert space, P m contracts the V norm and commutes with A. Thus, using (C3) and (2.13), we deduce Since the map u 0 m,h (.) is bounded on [0, τ N ], Gronwall's lemma implies that for every On the other hand, sup t≤τ u 0 m,h (t) 2 = +∞ if τ < T , which contradicts the estimate (3.9) .Hence τ = T a.s. and we get (3.7) which completes the proof of the Lemma.
We now prove the main result of this section.Proof of Theorem 3.1: Step 1: Using the estimate (3.7) and the growth condition (2.18) we conclude that each component of the sequence (u 0 m,h ) n , n ≥ 1 satisfies the following estimate sup for some constant C > 0 depending only on M, ξ , T .Moreover, writing the equation (3.1) for the GOY shell model in the componentwise form using (2.5) (the proof for the Sabra shell model using (2.6), which is similar, is omitted), we obtain for n = 1, 2, Hence, we deduce that for every n ≥ 1 there exists a constant C n , independent of m, such that (u 0 m,h ) n C 1 ([0,T ];C) ≤ C n .Applying the Ascoli-Arzelà theorem, we conclude that for every n there exists a subsequence (m n k ) k≥1 such that (u 0 m n k ,h ) n converges uniformly to some (u 0 h ) n as k −→ ∞.By a diagonal procedure, we may choose a sequence (m n k ) k≥1 independent of n such that (u 0 m,h ) n converges uniformly to some (u 0 h ) n ∈ C ([0, T ]; C) for every n ≥ 1; set u 0 h (t) = ((u 0 h ) 1 , (u 0 h ) 2 , . . .).From the estimate (3.7), we have the weak star convergence in L ∞ (0, T ; V ) of some further subsequence of u 0 m n k ,h : k ≥ 1).The weak limit belongs to L ∞ (0, T ; V ) and has clearly (u 0 h ) n as components that belong to C ([0, T ]; C) for every integer n ≥ 1.Using the uniform convergence of each component, it is easy to show, passing to the limit in the expression (3.10), that u 0 h (t) satisfies the weak form of the GOY shell model equation (3.1).Finally, since is such that sup 0≤s≤T u 0 h (s) < ∞ a.s. and since for every s ∈ [0, T ], by (2.9) and (3.2) we have a.s.
Step 2: To complete the proof of Theorem 3.1, we show that the solution V ) be another solution to (3.1) and set On the other hand, the Lipschitz property (3.3) from condition (C3) for ν = 0 implies and Gronwall's lemma implies that (for almost every ω) sup 0≤t≤T U (t ∧ τ N ) 2 = 0 for every N .As N → ∞, we deduce that a.s.U (t) = 0 for every t, which concludes the proof. 2 We now suppose that the diffusion coefficient σ ν satisfies the following condition (C4) which strengthens (C1) in the way (C3) strengthens (C2), i.e., Condition (C4) There exist constants K i and L i , i = 0, 1, 2, j = 1, 2, such that for any ν > 0 and u ∈ Dom(A), Then for ν > 0, the existence result and apriori bounds of the solution to (2.20) proved in Proposition 2.2 can be improved as follows.
Proposition 3.3.Let ξ ∈ V , let the coefficients a, b, µ defining B be such that a(1 + µ 2 ) + bµ 2 = 0, σ ν and σν satisfy the conditions (C1), (C2), (C3) and (C4).Then there exist positive constants K2 and ν 0 such that for 0 < K 2 < K2 and 0 < ν ≤ ν 0 , for every M > 0 there exists a constant C(M ) such that for any h ∈ A M , the solution u ν h to (2.20) belongs to C([0, T ], V ) almost surely and Proof.Fix m ≥ 1, let P m be defined by (3.5) and let u ν m,h (t) be the approximate maximal solution to the (finite dimensional) evolution equation: u ν m,h (0) = P m ξ and where W m is defined by (2.15).Proposition 3.3 in [10] proves that (3.14) has a unique solution u ν m,h ∈ C([0, T ], P m (H)).For every N > 0, set τ N = inf{t : u ν m,h (t) ≥ N } ∧ T. Since P m (H) ⊂ Dom(A), we may apply Itô's formula to u ν m,h (t) 2 .Let Π m : H 0 → H 0 be defined by Π m u = m k=1 u, e k e k for some orthonormal basis {e k , k ≥ 1} of H made by eigen-vectors of the covariance operator Q; then we have: Since the functions ϕ k are eigen-functions of A, we have A Furthermore, P m contracts the H and the V norms, and for u ∈ Dom(A), B(u), Au = 0 by (2.13).Hence for 0 < ǫ = 1 2 (2 − K 2 ) < 1, using Cauchy-Schwarz's inequality and the conditions (C3) and (C4) on the coefficients σ ν and σν , we deduce For any t ∈ [0, T ] set Then almost surely, The Burkholder-Davis-Gundy inequality, conditions (C1) -(C4), Cauchy-Schwarz and Young's inequalities yield that for t ∈ [0, T ] and β > 0, the previous inequality implies that the bounded function X satisfies a.s. the inequality Furthermore, I(t) is non decreasing, such that for 0 Lemma 3.2 from [10] implies that for K 2 and ν 0 small enough, there exists a constant C(M, T ) which does not depend on m and N , and such that for 0 < ν ≤ ν 0 , m ≥ 1 and Then, letting N → ∞ and using the monotone convergence theorem, we deduce that sup Then using classical arguments we prove the existence of a subsequence of (u ν m,h , m ≥ 1) which converges weakly in L 2 ([0, T ] × Ω, V ) ∩ L 4 ([0, T ] × Ω, H) and converges weakstar in L 4 (Ω, L ∞ ([0, T ], H)) to the solution u ν h to equation (2.20) (see e.g.[10], proof of Theorem 3.1).In order to complete the proof, it suffices to extract a further subsequence of (u ν m,h , m ≥ 1) which is weak-star convergent to the same limit u ν h in L 2 (Ω, L ∞ ([0, T ], V )) and converges weakly in L 2 (Ω × [0, T ], Dom(A)); this is a straightforward consequence of (3.15).Then as m → ∞ in (3.15), we conclude the proof of (3.13).

Large deviations
We will prove a large deviation principle using a weak convergence approach [4,5], based on variational representations of infinite dimensional Wiener processes.Let σ : [0, T ] × V → L Q and for every ν > 0 let σν : [0, T ] × Dom(A) → L Q satisfy the following condition: Condition (C5): (i) There exist a positive constant γ and non negative constants C, K0 , K1 and L1 such that for all u, v ∈ V and s, t ∈ [0, T ]: (ii) There exist a positive constant γ and non negative constants C, K0 , KH , K2 and L2 such that for ν > 0, s, t ∈ [0, T ] and u, v ∈ Dom(A), Set σ ν = σν = σ + √ ν σν for ν > 0, and σ0 = σ.(4.1)Then for 0 ≤ ν ≤ ν 1 , the coefficients σ ν and σν satisfy the conditions (C1)-(C4) with Proposition 3.3 and Theorem 3.1 prove that for some ν 0 ∈]0, ν 1 ], K2 and L2 small enough, 0 < ν ≤ ν 0 (resp.ν = 0), ξ ∈ V and h ν ∈ A M , the following equation has a unique solution u ν hν (resp. Recall that for any α ≥ 0, H α has been defined in (2.2) and is endowed with the norm • α defined in (2.2).When 0 ≤ α ≤ 1 4 , as ν → 0 we will establish a Large Deviation Principle (LDP) in the set C([0, T ], V ) for the uniform convergence in time when V is endowed with the norm • α for the family of distributions of the solutions u ν to the evolution equation: whose existence and uniqueness in C([0, T ], V ) follows from Propositions 2.2 and 3.3.Unlike in [27], [16], [22] and [10], the large deviations principle is not obtained in the natural space, which is here C([0, T ], V ) under the assumptions (C5), because the lack of viscosity does not allow to prove that u 0 h (t) ∈ Dom(A) for almost every t.To obtain the LDP in the best possible space with the weak convergence approach, we need an extra condition, which is part of condition (C5) when α = 0, that is when 1  4 ]; there exists a constant L 3 such that for u, v ∈ H α and t ∈ [0, 1], Let B denote the Borel σ−field of the Polish space where • α is defined by (2.2).We at first recall some classical definitions; by convention the infimum over an empty set is +∞.
Definition 4.1.The random family (u ν ) is said to satisfy a large deviation principle on X with the good rate function I if the following conditions hold: I is a good rate function.The function Large deviation upper bound.For each closed subset F of X : Large deviation lower bound.For each open subset G of X : 0 h(s)ds ∈ C 0 and u 0 h is the solution to the (inviscid) control equation (4.4) with initial condition ξ, and G 0 ξ (g) = 0 otherwise.The following theorem is the main result of this section.Theorem 4.2.Let α ∈ [0, 1  4 ], suppose that the constants a, b, µ defining B are such that a(1 + µ 2 ) + bµ 2 = 0, let ξ ∈ V , and let σ ν and σν be defined for ν > 0 by (4.1) with coefficients σ and σν satisfying the conditions (C5) and (C6) for this value of α.Then the solution (u ν ) ν>0 to (4.5) with initial condition ξ satisfies a large deviation principle in X := C([0, T ], V ) endowed with the norm u X =: sup 0≤t≤T u(t) α , with the good rate function We at first prove the following technical lemma, which studies time increments of the solution to the stochastic control problem (4.3) which extends both (4.5) and (4.4).
Now we return to the setting of Theorem 4.2.Let ν 0 ∈]0, ν 1 ] be defined by Theorem 2.2 and Proposition 3.3, (h ν , 0 < ν ≤ ν 0 ) be a family of random elements taking values in the set A M defined by (2.17).Let u ν hν be the solution of the corresponding stochastic control equation (4.3) with initial condition u ν hν (0 0 h ν (s)ds due to the uniqueness of the solution.The following proposition establishes the weak convergence of the family (u ν hν ) as ν → 0. Its proof is similar to that of Proposition 4.5 in [10]; see also Proposition 3.3 in [16].
where using again the fact that α ≤ 1 4 , we have We want to show that as ν → 0, sup t∈[0,T ] U ν (s) α → 0 in probability, which implies that u ν hν → u 0 h in distribution in X. Fix N > 0 and for t ∈ [0, T ] let The proof consists in two steps.
The following compactness result is the second ingredient which allows to transfer the LDP from √ νW to u ν .Its proof is similar to that of Proposition 4.4 and easier; it will be sketched (see also [16], Proposition 4.4).Proposition 4.5.Suppose that the constants a, b, µ defining B satisfy the condition a(1 + µ 2 ) + bµ 2 = 0, σ satisfies the conditions (C5) and (C6) and let α ∈ [0, 1  4 ].Fix M > 0, ξ ∈ V and let K M = {u 0 h : h ∈ S M }, where u 0 h is the unique solution in C([0, T ], V ) of the deterministic control equation (4.4).Then K M is a compact subset of X = C([0, T ], V ) endowed with the norm u X = sup 0≤t≤T u(t) α .Proof.To ease notation, we skip the superscript 0 which refers to the inviscid case.By Theorem 3.1, K M ⊂ C([0, T ], V ).Let {u n } be a sequence in K M , corresponding to solutions of (4.4) with controls {h n } in S M : du n (t) + B(u n (t))dt = σ(t, u n (t))h n (t)dt, u n (0) = ξ.
Since S M is a bounded closed subset in the Hilbert space L 2 (0, T ; H 0 ), it is weakly compact.So there exists a subsequence of {h n }, still denoted as {h n }, which converges weakly to a limit h ∈ L 2 (0, T ; H 0 ).Note that in fact h ∈ S M as S M is closed.We now show that the corresponding subsequence of solutions, still denoted as {u n }, converges in X to u which is the solution of the following "limit" equation du(t) + B(u(t))dt = σ(t, u(t))h(t)dt, u(0) = ξ.
Note that we know from Theorem 3.1 that u ∈ C([0, T ], V ), and that one only needs to check that the convergence of u n to u holds uniformly in time for the weaker • α norm on V .To ease notation we will often drop the time parameters s, t, ... in the equations and integrals.Let U n = u n − u; using (2.12) and (C6), we deduce that for t ∈ [0, T ], U n (t)   Since N is arbitrary, we deduce that sup 0≤t≤T U n (t) α → 0 as n → ∞.This shows that every sequence in K M has a convergent subsequence.Hence K M is a sequentially relatively compact subset of X .Finally, let {u n } be a sequence of elements of K M which converges to v in X .The above argument shows that there exists a subsequence {u n k , k ≥ 1} which converges to some element u h ∈ K M for the uniform topology on C([0, T ], V ) endowed with the • α norm.Hence v = u h , K M is a closed subset of X , and this completes the proof of the proposition.
Proof of Theorem 4.2: Propositions 4.5 and 4.4 imply that the family {u ν } satisfies the Laplace principle, which is equivalent to the large deviation principle, in X defined in (4.7) with the rate function defined by (4.8); see Theorem 4.4 in [4] or Theorem 5 in [5].This concludes the proof of Theorem 4.2. 2

. 3 ) 3 . 1 .
Theorem Suppose that σ0 satisfies the conditions (C2) and (C3) and that the coefficients a, b, µ defining B satisfy a(1 + µ 2 ) + bµ 2 = 0. Let ξ ∈ V be deterministic.For any M > 0 there exists C(M ) such that equation (3.1) has a unique solution in C([0, T ], V ) for any h ∈ A M , and a.s.one has: the projection operator defined by Π m u = m k=1 u , e k e k , where {e k , k ≥ 1} is the orthonormal basis of H made by eigenelements of the covariance operator Q and used in (2.15).
.29) A similar computation based on (C5) and (4.10) from Lemma 4.3 yields for some constant C3 := C(T, M, N ) and any ν
2.2.Stochastic driving force.Let Q be a linear positive operator in the Hilbert space H which is trace class, and hence compact.Let H 0 = Q In further estimates we use Lemma 4.3 with ψ n = sn , where sn is the step function defined by sn = kT 2 ds in the weak topology of H 0 .Therefore, since the operator σ(t k , u 0 h (t k )) is compact from H 0 to H, we deduce that for every k, .32) Finally, note that the weak convergence of h ν to h implies that as ν → 0, for any a, b ∈[0, T ], a < b, the integral b a h ν (s)ds → b a h(s)σ(t k , u 0 h (t k )) The Cauchy-Schwarz inequality, (4.34), (C5) and (4.10) imply that for some constants C i , i = 0, • • • , 4, which depend on k 0 , Ki , L1 , C, M and T , but do not depend on n and N , (s) − u n (s N ) 2 + u(s) − u(s N ) 2 ds )(|h(s)| 0 + |h n (s)| 0 ) ds ≤ C 2 2 −N γ .(4.39)For fixed N and k = 1, • • • , 2 N , as n → ∞, the weak convergence of h n to h implies that oft k t k−1 (h n (s) − h(s))ds to 0 weakly in H 0 .Since σ(t k , u(t k )) is a compact operator, we deduce that for fixed k the sequence σ(t k , u(t k )) N sup t k−1 ≤t≤t k σ(t k , u(t k )) t t k−1 (h n (s) − h(s))ds , A 2α U n (t k ) , I 5 n,N = 1≤k≤2 N σ(t k , u(t k )) t k t k−1 [h n (s) − h(s)] ds , A 2α U n (t k ) .