getdoc8876. 304KB Jun 04 2011 12:04:38 AM

El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b il i t y Vol. 14 (2009), Paper no. 89, pages 2551–2579.

Journal URL

http://www.math.washington.edu/~ejpecp/

Large deviation principle and inviscid shell models

Hakima Bessaih

∗

University of Wyoming, Department of Mathematics, Dept. 3036,

1000 East University Avenue, Laramie WY 82071, United States

Bessaih@uwyo.edu

Annie Millet

SAMOS, Centre d’Économie de la Sorbonne, Université Paris 1 Panthéon Sorbonne,

90 Rue de Tolbiac, 75634 Paris Cedex France

and

Laboratoire de Probabilités et Modèles Aléatoires, Universités Paris 6-Paris 7,

Boîte Courrier 188, 4 place Jussieu, 75252 Paris Cedex 05, France

annie.millet@univ-paris1.fr

and

annie.millet@upmc.fr

Abstract

A LDP is proved for the inviscid shell model of turbulence. As the viscosity coefficientνconverges to 0 and the noise intensity is multiplied bypν, 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 whereV is endowed with a topology weaker than the natural one. The initial condition has to belong toV 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.

Key words: Shell models of turbulence, viscosity coefficient and inviscid models, stochastic PDEs, large deviations.

AMS 2000 Subject Classification:Primary 60H15, 60F10; Secondary: 76D06, 76M35. Submitted to EJP on May 18, 2009, final version accepted November 14, 2009.

∗_{This work was partially written while H. Bessaih was invited professor at the University of Paris 1. The work of this}

1

Introduction

Shell models, from E.B. Gledzer, K. Ohkitani, M. Yamada, are simplified Fourier systems with re-spect 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.

The more general formulation involving a multiplicative noise reads as follows

du(t) + [νAu(t) +B(u(t),u(t))]d t=σ(t,u(t))dW_t, u(0) =ξ.

driven by a Hilbert space-valued Brownian motionW. It involves some similar bilinear operatorB

with antisymmetric properties and some linear "second order" (Laplace) operatorAwhich is regu-larizing and multiplied by some non negative coefficientνwhich 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 hydro-dynamical equations including the 2D Bénard magneto-hydro dynamical and 3Dα-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 coefficientpεwhich 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 infinite-dimensional 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)_∩L2([0,T],V), where roughly speaking, H is L2 and V = Dom(A12) is the Sobolev

The aim of this paper is different. Indeed, the asymptotics we are interested in have a physical meaning, namely the viscosity coefficientν converges to 0. Thus the limit equation, which corre-sponds to the inviscid case, is much more difficult to deal with, since the regularizing effect of the operatorAdoes 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 conditionξ∈V, to impose that (B(u,u),Au) = 0 for allu 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 coefficientsa,band

µstated below. Furthermore, some more conditions on the diffusion coefficient are required as well. The intensity of the noise has to be multiplied bypν for the convergence to hold.

The technique is again that of the weak convergence. One proves that given a family(hν)of random

elements of the RKHS ofW which converges weakly toh, the corresponding family of stochastic con-trol equations, deduced from the original ones by shifting the noise by _phν

ν, converges in distribution

to the limit inviscid equation where the Gaussian noise W has been replaced byh. Some apriori control of the solution to such equations has to be proven uniformly inν > 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 coefficientσ, the LDP is proved inC([0,T],V)where

V is endowed with the norm_{k · kα}:=_|Aα(_·)_|H for 0_≤α≤ 1₄. The natural caseα= 1₂ is out of reach because the inviscid limit equation is much more irregular. Indeed, it is an abstract equivalent of the Euler equation. The caseα=0 corresponds to H and then no more condition onσis required. The caseα= 1₄ 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 coefficientǫin front of a deterministic operator converges to 0 and the intensity of the Gaussian noise is multiplied by pǫ. 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 L2([0,T],V)of the stochastic control equations uniformly in the viscosity coefficientν ∈]0,ν0]for small enoughν0. 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 onσ. Under these extra assumptions, we are able to improve the apriori estimates of the solution and establish them in _C([0,T],V) and L2([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)whereV is endowed with the weaker norm associated withAα for any value ofαwith 0_≤α≤ 1₄.

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 byC a constant which may change from one line to the next, and C(M)a constant depending onM.

2

Description of the model

2.1

GOY and Sabra shell models

LetH be the set of all sequencesu= (u₁,u₂, . . .)of complex numbers such thatP_n|u_n|2 <∞. We considerHas arealHilbert space endowed with the inner product(_·,_·)and the norm_|·|of the form

(u,v) =ReX

n≥1

u_nv_n∗, _|u_|2=X

n≥1

|u_n|2, (2.1)

wherev_n∗denotes the complex conjugate ofv_n. Letk₀>0,µ >1 and for everyn_≥1, setk_n=k₀µn. LetA:Dom(A)_⊂H→H be the non-bounded linear operator defined by

(Au)_n=k2_nu_n, n=1, 2, . . . , Dom(A) = n

u_∈H : X

n≥1

k4_n|u_n|2<∞

o

.

The operatorAis clearly self-adjoint, strictly positive definite since(Au,u)_≥k2_0|u_|2foru_∈Dom(A). For anyα >0, set

Hα=Dom(Aα) ={u∈H :

X

n≥1

k4_nα|un|2<+_∞}, kuk2α=

X

n≥1

k4_nα|un|2for u∈ Hα. (2.2)

Let_H0=H,

V :=Dom(A12) =

n

u_∈H : X

n≥1

k2_n|u_n|2<+_∞ o

; also set _H =_H1

4,kukH =kuk 1 4.

ThenV (as each of the spaces_Hα) is a Hilbert space for the scalar product(u,v)V=Re(

P

nk2nunvn∗), u,v_∈V and the associated norm is denoted by

kuk2=X

n≥1

k2_n|un|2. (2.3)

The adjoint ofV with respect to the H scalar product isV′=_{(u_n)_∈CN : P_n≥1k−_n2|un|2 _<+∞}

andV _⊂H_⊂V′is a Gelfand triple. Let_〈u,v_〉=Re€P_n≥1u_nv_n∗Šdenote the duality betweenu_∈V

andv_∈V′. Clearly for 0_≤α < β,u_{∈ H}β andv_∈V we have

ku_k2_α≤k₀4(α−β)_ku_k2_β, and _kv_k2_{H ≤ |}v_{| k}v_k, (2.4) where the last inequality is proved by the Cauchy-Schwarz inequality.

Setu₋1=u0=0, leta,bbe real numbers andB:H×V →H(orB:V×H→H) denote the bilinear

operator defined by

[B(u,v)]_n=₋i€akn+1un∗+1vn∗+2+bknu∗n−1vn∗+1−akn−1u∗n−1vn∗−2−bkn−1u∗n−2v∗n−1

Š

(2.5) forn=1, 2, . . . in the GOY shell-model (see, e.g.,[25]) or

[B(u,v)]_n=₋i€ak_n+1u∗_n+1vn+2+bknu∗n−1vn+1+akn−1un−1vn−2+bkn−1un−2vn−1

Š

, (2.6)

Note thatBcan be extended as a bilinear operator fromH_×HtoV′and that there exists a constant

C >0 such that givenu,v∈H andw∈V we have

|〈B(u,v),w_〉|+_| B(u,w), v

|+_| B(w,u), v

| ≤C_|u_{| |}v_{| k}w_k. (2.7) An easy computation proves that foru,v∈H andw∈V (resp. v,w∈H andu∈V),

〈B(u,v),w_〉=₋ B(u,w), v

(resp. B(u,v),w

=₋ B(u,w), v

). (2.8)

Furthermore,B:V _×V _→V andB:_{H × H →}H; indeed, foru,v_∈V (resp.u,v_{∈ H}) we have

kB(u,v)_k2=X

n≥1

k2_n|B(u,v)_n|2 _≤Ckuk2sup

n

k_n2|vn|2≤Ckuk2kvk2, (2.9)

|B(u,v)_{| ≤}CkukHkvkH.

For u,v in either H, _H or V, let B(u) := B(u,u). The anti-symmetry property (2.8) implies that

|〈B(u1)−B(u2),u1−u2〉V| = |〈B(u1−u2),u2〉V| foru1,u2 ∈ V and|〈B(u1)−B(u2),u1−u2〉|= |〈B(u₁₋u₂),u_2〉|foru_1∈H andu_2∈V. Hence there exist positive constants ¯C₁and ¯C₂ such that

|〈B(u₁)₋B(u₂),u₁₋u_{2〉V| ≤} C¯_1ku₁₋u_2k2_ku_2k,_∀u₁,u_2∈V, (2.10)

|〈B(u₁)₋B(u₂),u₁₋u_{2〉| ≤} C¯_2|u₁₋u_2|2_ku_2k,_∀u_1∈H,_∀u_2∈V. (2.11) Finally, sinceBis bilinear, Cauchy-Schwarz’s inequality yields for anyα∈[0,₂1],u,v∈V:

AαB(u)−AαB(v),Aα(u−v)

≤

AαB(u−v,u) +AαB(v,u−v),Aα(u−v)

≤C_ku₋v_k2_α(_ku_k+_kv_k). (2.12)

In the GOY shell model,B is defined by (2.5); for anyu∈V,Au∈V′we have

〈B(u,u),Au_〉=Re

−iX n≥1

u∗_nu∗_n+1u∗_n+2µ3n+1

k3₀(a+bµ2−aµ4−bµ4).

Sinceµ6=1,

a(1+µ2) +bµ2=0 if and only if _〈B(u,u),Au〉=0,_∀u∈V. (2.13) On the other hand, in the Sabra shell model,B is defined by (2.6) and one has foru∈V,

〈B(u,u),Au_〉=k3₀Re₋iX n≥1

µ3n+1h(a+bµ2)u∗_nu∗_n+1u_n+2+ (a+b)µ4unun+1u∗n+2

i

.

Thus(B(u,u),Au) =0 for everyu∈V if and only ifa+bµ2_{= (a+b)µ}4 _{and againµ6=1 shows that}

(2.13) holds true.

2.2

Stochastic driving force

LetQbe a linear positive operator in the Hilbert space H which is trace class, and hence compact. LetH₀=Q12H; thenH₀is a Hilbert space with the scalar product

(φ,ψ)₀= (Q−12φ,Q− 1

together with the induced norm _{| · |0} = p(_·,_·)₀. The embedding i : H_{0 →} H is Hilbert-Schmidt and hence compact, and moreover, i i∗ =Q. Let LQ ≡ LQ(H0,H) be the space of linear operators S:H0 7→H such thatSQ

1

2 is a Hilbert-Schmidt operator fromH toH. The norm in the space L_Q is

defined by_|S_|2_L

Q=t r(SQS

∗_{), whereS}∗_{is the adjoint operator ofS. TheL}

Q-norm can be also written

in the form

|S_|2_L

Q =t r([SQ

1/2_][SQ1/2_]∗_{) =}X

k≥1

|SQ1/2ψk|2=X

k≥1

|[SQ1/2]∗ψk|2 (2.14) for any orthonormal basis_{ψk}inH, for example(ψk)n=δnk.

LetW(t)be a Wiener process defined on a filtered probability space(Ω,_F,(_Ft),P), taking values in

Hand with covariance operatorQ. This means thatWis Gaussian, has independent time increments and that fors,t_≥0, f,g_∈H,

E(W(s),f) =0 and E(W(s),f)(W(t),g) = s∧t) (Q f,g).

Letβj be standard (scalar) mutually independent Wiener processes,{ej}be an orthonormal basis in H consisting of eigen-elements ofQ, withQe_j =q_je_j. ThenW has the following representation

W(t) = lim

n→∞Wn(t) in L

2_{(Ω;H) withW} n(t) =

X

1≤j≤n

q1_j/2β_j(t)e_j, (2.15)

andT r ace(Q) =P_j≥1q_j. For details concerning this Wiener process see e.g. [13]. Given a viscosity coefficientν >0, consider the following stochastic shell model

d_tu(t) +

νAu(t) +B(u(t))

d t=pν σν(t,u(t))dW(t), (2.16)

where the noise intensity σν : [0,T]×V → LQ(H0,H) of the stochastic perturbation is properly

normalized by the square root of the viscosity coefficientν. We assume thatσνsatisfies the following

growth and Lipschitz conditions:

Condition (C1): σν ∈ C [0,T]×V;LQ(H0,H)

, and there exist non negative constants K_i and L_i such that for every t∈[0,T]and u,v∈V :

(i)|σν(t,u)|2LQ≤K0+K1|u|

2_+K 2kuk2,

(ii)|σν(t,u)−σν(t,v)|2LQ ≤L1|u−v|

2_+L

2ku−vk2.

For technical reasons, in order to prove a large deviation principle for the distribution of the solution to (2.16) as the viscosity coefficientν converges to 0, we will need some precise estimates on the solution of the equation deduced from (2.16) by shifting the BrownianW by some random element of its RKHS. This cannot be deduced from similar ones onuby 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 H₀₋valued

(_Ft)₋predictable stochastic processeshsuch thatR₀T_|h(s)_|2₀ds<∞, a.s. For fixedM >0, let

S_M=nh_∈L2(0,T;H₀):

Z T

0

The setS_M, endowed with the following weak topology, is a Polish (complete separable metric) space (see e.g. [5]): d₁(h,k) =P∞_k=121k

RT

0 h(s)−k(s), ˜ek(s)

0ds

, where{˜e_k(s)}∞

k=1 is an orthonormal

basis for L2([0,T],H₀). ForM >0 set

AM=_{h∈ A :h(ω)_∈SM, a.s.}. (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∈ AM 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 fixedω. We assume that for anyν≥0 the coefficient ˜σν satisfies the following condition:

Condition (C2): σ˜ν ∈ C [0,T]×V;L(H0,H)

and there exist constantsK˜_H,K˜_i, and˜L_j, for i=0, 1

and j=1, 2such that:

|σ˜ν(t,u)|2L(H0,H)≤

˜

K₀+K˜_1|u_|2+νK˜_Hku_k2_H, _∀t_∈[0,T], _∀u_∈V, (2.18)

|σ˜ν(t,u)−σ˜ν(t,v)|2L(H0,H)≤

˜_L1|u−v|2_+ν˜L

2ku−vk2, ∀t∈[0,T], ∀u,v∈V, (2.19) whereH =_H1

4 is defined by(2.2)and| · |L(H0,H)denotes the (operator) norm in the space L

(H0,H)of all bounded linear operators from H0 into H. Note that ifν =0the 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_{| · |L(H}

0,H)instead of the trace class norm| · |LQ).

However we require in (2.18) a stronger bound (in the interpolation space_H). One can see that the noise intensitypν σν satisfies Condition (C2) provided that in Condition (C1), we replace point

(i) by_|σν(t,u)|2LQ≤K0+K1|u|

2_+K

Hkuk2H. Thus the class of intensities satisfying both Conditions (C1) and (C2) when multiplied bypν is wider than that those coefficients which satisfy condition (C1)withK2=0.

LetM>0,h∈ AM,ξanH-valued random variable independent ofW andν >0. Under Conditions (C1) and (C2) we consider the nonlinear SPDE

duν_h(t) +

νAuν_h(t) +B uν_h(t)

d t=pν σν(t,u_hν(t))dW(t) +σ˜ν(t,uν_h(t))h(t)d t, (2.20)

with initial conditionuν_h(0) =ξ. Using[10], Theorem 3.1, we know that for everyT >0 andν >0 there exists ¯K₂ν :=K¯₂(ν,T,M)>0 such that ifhν ∈ AM,E|ξ|4 <+∞and 0≤K2 <K¯2ν, equation

(2.20) has a unique solutionuν_{h∈ C}([0,T],H)_∩L2([0,T],V)which satisfies:

(uν_h,v)₋(ξ,v) + Z t

0

ν〈uν_h(s),Av_〉+_〈B(uν_h(s)),v_〉

ds

= Z t

0 p

ν σν(s,uνh(s))dW(s), v

+

Z t

0

˜

σν(s,uνh(s))h(s),v

ds

a.s. for all v _∈Dom(A) and t _∈[0,T]. Note that uν_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 motionW. Furthermore, if K_{2 ∈}[0, ¯K₂ν[and L_{2 ∈}[0, 2[, there exists a constant C_ν :=

C(Ki,Lj, ˜Ki, ˜KH,T,M,ν)such that

E

sup

0≤t≤T|

uν_h(t)_|4+ Z T

0

kuν_h(t)_k2d t+ Z T

0

kuν_h(t)_k4_H d t_≤Cν(1+E|ξ|4). (2.21)

The following proposition proves that ¯K₂ν can be chosen independent ofν and that a proper for-mulation of upper estimates of theH, H and V norms of the solutionuν_h to (2.20) can be proved uniformly inh_{∈ AM} and inν∈(0,ν0]for some constantν0>0.

Proposition 2.2. Fix M > 0, T > 0, σν and σ˜ν satisfy Conditions (C1)–(C2) and let the initial conditionξbe such thatE_|ξ|4 _<+_∞. Then in any shell model where B is defined by(2.5)or(2.6), there exist constantsν0>0,K¯2 andC¯(M)such that if0< ν≤ν0,0≤K2<K¯2, L2<2and h∈ AM, the solution uν_h to(2.20)satisfies:

E

sup

0≤t≤T|

uν_h(t)_|4+ν

Z T

0

kuν_h(s)_k2ds+ν

Z T

0

kuν_h(s)_k4_H ds_≤C¯(M) E_|ξ|4+1

. (2.22)

Proof. For every N > 0, set τN = inf{t : |uνh(t)| ≥ N} ∧T. Itô’s formula and the antisymmetry

relation in (2.8) yield that for t∈[0,T],

|uν_h(t∧τN)|2=|ξ|2+2 p

ν

Z t∧τN

0

σν(s,uνh(s))dW(s),u ν h(s)

−2ν

Z t∧τN

0

kuν_h(s)_k2ds

+2

Z t∧τN

0

˜

σν(s,uν_h(s))h(s),uν_h(s)

ds+ν

Z t∧τN

0

|σν(s,uν_h(s))|2_LQds, (2.23)

and using again Itô’s formula we have

|uν_h(t∧τN)|4+4ν

Z t∧τN

0

|uν_h(r)_|2_kuν_h(r)_k2d r≤ |ξ|4+I(t) + X

1≤j≤3

Tj(t), (2.24)

where

I(t) = 4pν

Z t∧τ_N

0

σν(r,uνh(r))dW(r),u ν h(r)|u

ν h(r)|

2 ,

T₁(t) = 4

Z t∧τN

0

|(σ˜ν(r,uhν(r))h(r),u ν h(r))| |u

ν h(r)|

2_{d r,}

T2(t) = 2ν

Z t∧τN

0

|σν(r,uνh(r))| 2 LQ|u

ν h(r)|

2_{d r,}

T₃(t) = 4ν

Z t∧τN

0

|σ_ν∗(s,uν_h(r))uν_h(r)_|2₀d r.

Sinceh_{∈ AM}, the Cauchy-Schwarz and Young inequalities and condition(C2)imply that for any

ε >0,

T₁(t)_≤ 4

Z t∧τN

0

p

˜

K₀+pK˜_1|uν_h(r)_|+pνK˜_H k− 1 2 0 ku

ν h(r)k

≤ 4pK˜0M T+4

p

˜

K0+

p

˜

K1

Z t∧τN

0

|h(r)_|0|uν_h(r)_|4ds

+ε ν

Z t

0

kuν_h(r)_k2_|uν_h(r)_|2d r+4 ˜KH

εk0

Z t∧τN

0

|h(r)_|2_0|uν_h(r)_|4d r. (2.25) Using condition(C1)we deduce

T₂(t) +T₃(t)_≤ 6ν

Z t∧τN

0

K₀+K_1|uν_h(r)_|2+K_2kuν_h(r)_k2

|uν_h(r)_|2d r ≤6νK0T+6ν(K0+K1)

Z t∧τN

0

|uν_h(r)_|4d r+6νK2

Z t

0

kuν_h(r)_k2_|uν_h(r)_|2d r. (2.26) LetK2≤ 1₂ and 0< ε≤2−3K2; set

ϕ(r) =4pK˜0+

p

˜

K1

|h(r)_|0+4 ˜KH

εk₀ |h(r)| 2

0+6ν(K0+K1).

Then a.s.

Z T

0

ϕ(r)d r≤4pK˜0+

p

˜

K1

p

M T+4 ˜KH

εk₀ M+6ν(K0+K1)T := Φ (2.27)

and the inequalities (2.24)-(2.26) yield that for

X(t) =sup

r≤t|

uν_h(r_∧τN)|4, Y(t) =ν

Z t

0

kuν_h(r_∧τN)k2|uνh(r∧τN)|2ds, X(t) + (4−6K₂−ε)Y(t)_{≤ |}ξ|4+4pK˜₀M T+6νK₀T

+I(t) + Z t

0

ϕ(s)X(s)ds. (2.28) The Burkholder-Davis-Gundy inequality, (C1), Cauchy-Schwarz and Young’s inequalities yield that fort∈[0,T]andδ,κ >0,

E_I(t)_≤12pνE

nZ t∧τN

0

K0+K1|uνh(s)| 2_+K

2kuνh(s)k 2

|uν_h(r)_|6ds

o1

2

≤12pνE

sup

0≤s≤t|

uν_h(s_∧τN)|2

nZ t∧τN

0

K₀+K_1|uν_h(s)_|2+K_2kuν_h(s)_k2

|uν_h(s)_|2dso 1 2

≤δE(Y(t)) +36K2

δ +κ ν

E(X(t)) +36

κ

h

K0T+ (K0+K1)

Z t

0

E(X(s))ds

i

. (2.29)

Thus we can apply Lemma 3.2 in [10] (see also Lemma 3.2 in [16]), and we deduce that for 0< ν ≤ν0,K2≤ ₂1,ε=α= 1₂,β=

36K2

δ +κ ν0≤2

−1_e−Φ_,δ

≤α2−1e−Φandγ=36

κ(K0+K1), E

X(T) +αY(T)_≤2 exp Φ +2TγeΦh

4pK˜₀M T+6ν0K0T+

36

κ K0T+E(|ξ|

4₎i_{. (2.30)}

Using the last inequality from (2.4), we deduce that forK2 small enough, ¯C(M)independent of N

andν∈]0,ν0],

E

sup

0≤t≤T|

uν_h(t_∧τN)|4+ν

Z τN

0

AsN_→+_∞, the monotone convergence theorem yields that for ¯K₂small enough andν∈]0,ν0]

E

sup

0≤t≤T|

uν_h(t)_|4+ν

Z T

0

kuν_h(t)_k4_H d t

≤C¯(M)(1+E(_|ξ|4)).

This inequality and (2.30) witht instead oft∧τN conclude the proof of (2.22) by a similar simpler

computation based on conditions(C1)and(C2).

3

Well posedeness, more a priori bounds and inviscid equation

The aim of this section is twofold. On one hand, we deal with the inviscid caseν=0 for which the PDE

du0_h(t) +B(u0_h(t))d t=σ˜0(t,u0h(t))h(t)d t, u 0

h(0) =ξ (3.1)

can be solved for everyω. 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 constantsµ,a,b definingB, the initial conditionξand ˜σ0.

The initial condition ξhas to belong toV and the coefficients a,b,µhave 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 onσν and ˜σν, similar to that imposed on ˜σ0, we will prove further properties

ofuν_h for a strictly positive viscosity coefficientν.

Thus, suppose furthermore that forν >0 (resp.ν =0), the map ˜

σν:[0,T]×Dom(A)→L(H0,V) (resp. ˜σ0:[0,T]×V →L(H0,V))

satisfies the following:

Condition (C3):There exist non negative constants K˜i and ˜Lj, i = 0, 1, 2, j = 1, 2 such that for s∈[0,T]and for any u,v∈Dom(A)ifν >0(resp. for any u,v∈V ifν =0),

|A12σ˜_ν(s,u)|2

L(H0,H)≤

˜

K0+K˜1kuk2+νK˜2|Au|2, (3.2) and

|A12σ˜_ν(s,u)−A 1

2σ˜_ν(s,v)|2

L(H0,H)≤˜L1ku−vk2+ν˜L2|Au−Av|2. (3.3)

Theorem 3.1. Suppose thatσ˜0satisfies the conditions(C2)and(C3)and that the coefficients a,b,µ defining B satisfy a(1+µ2_{) +bµ}2_{=0. Letξ∈V be deterministic. For any M >0there exists C(M)} such that equation(3.1)has a unique solution in_C([0,T],V)for any h_{∈ AM}, and a.s. one has:

sup

h∈AM

sup

0≤t≤Tk

u0_h(t)_{k ≤}C(M)(1+_kξk). (3.4)

Since equation (3.1) can be considered for any fixedω, it suffices to check that the deterministic equation (3.1) has a unique solution in_C([0,T],V) for anyh_∈S_M and that (3.4) holds. For any

m≥1, letHm=span(ϕ1,· · ·,ϕm)⊂Dom(A),

and finally let ˜σ0,m = Pmσ˜0. Clearly Pm is a contraction of H and |σ˜0,m(t,u)|2_L(H0,H) ≤ |σ˜0(t,u)|2_L(H

0,H). Setu 0

m,h(0) =Pmξand consider the ODE on the m-dimensional spaceHm defined

by

d u0_m,h(t),v

=

− B(u0_m,h(t)), v

+ σ˜0(t,u0m,h(t))h(t),v

d t (3.6)

for everyv_∈H_m.

Note that using (2.9) we deduce that the mapu_∈H_{m7→ 〈}B(u), v_〉is locally Lipschitz. Furthermore, since there exists some constantC(m)such thatkuk ∨ kukH ≤C(m)|u|foru∈Hm, Condition (C2)

implies that the mapu_∈H_{m 7→} (σ˜0,m(t,u)h(t),ϕk): 1≤ k≤ m

, is globally Lipschitz fromH_m

toRm uniformly in t. Hence by a well-known result about existence and uniqueness of solutions to ODEs, there exists a maximal solutionu0_m,h =Pm

k=1(u0m,h,ϕk

ϕk to (3.6), i.e., a (random) time

τ0_m,h ≤ T such that (3.6) holds for t < τ0_m,h and as t _↑τ0_m,h < T, _|u0_m,h(t)_{| → ∞}. 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_{∈ AM}equation(3.6)has a unique solution in_C([0,T],H_m). There exists some constant C(M)such that for every h∈ AM,

sup

m

sup

0≤t≤Tk

u0_m,h(t)_k2_≤C(M) (1+_kξk2)a.s. (3.7)

Proof. The proof is included for the sake of completeness; the arguments are similar to that in the classical viscous framework. Leth∈ AM and let u0_m,h(t)be the approximate maximal solution to (3.6) described above. For every N > 0, set τN = inf{t : ku0m,h(t)k ≥ N} ∧T. Let Πm : H0 → H0 denote the projection operator defined by Πmu =

Pm

k=1 u,ek

e_k, where {e_k,k ≥ 1} is the orthonormal basis ofH made by eigen-elements of the covariance operatorQand used in (2.15). Sinceϕk∈Dom(A)andV is a Hilbert space,Pmcontracts the V norm and commutes withA. Thus,

using(C3)and (2.13), we deduce

ku0_m,h(t∧τN)k2≤ kξk2−2

Z t∧τN

0

B(u0_m,h(s)),Au0_m,h(s)

ds

+2

Z t∧τN

0

A

1

2P_mσ˜_0,m(s,u0

m,h(s))h(s)

ku0

m,h(s)kds ≤ |ξk2+2pK˜0M T+2

p

˜

K0+

p

˜

K1

Z t∧τN

0

|h(s)_|0ku_m0_,h(s)_k2ds. Since the mapku0_m,h(.)_kis bounded on[0,τ_N], Gronwall’s lemma implies that for everyN >0,

sup

m

sup

t≤τN

ku0_m,h(t)_k2_≤kξk2+2pK˜₀M Texp2pM ThpK˜₀+pK˜₁i. (3.8) Letτ:=lim_NτN ; asN → ∞in (3.8) we deduce

sup

m

sup

t≤τk

u0_m,h(t)_k2_≤kξk2+2pK˜₀M Texp2pM ThpK˜₀+pK˜₁i. (3.9) On the other hand, sup_t≤τku0_m,h(t)_k2= +_∞ifτ <T, which contradicts the estimate (3.9) . Hence

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 (u0_m,h)_n,n_≥1

satisfies the following estimate sup

m

sup

0≤t≤T|

(u0_m,h)_n(t)_|2+ σ˜₀(t,u0_m,h(t))h(t)

n

≤C a.s. ,∀n=1, 2,· · ·

for some constantC>0 depending only on M,_kξk,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 forn=1, 2,_{· · ·}

(u0_m,h)_n(t) =(P_mξ)_n+i

Z t

0

(ak_n+1(u0_m,h)∗_n+1(s)(u 0

m,h)∗n+2(s) +bkn(u0m,h)∗n−1(s)(u 0

m,h)∗n+1(s) −ak_n−1(u0_m,h)∗_n−1(s)(u0_m,h)∗_n−2(s)₋bk_n−1(u0_m,h)∗_n−2(s)(u0_m,h)∗_n−1(s))ds

+ Z t

0

˜

σ0(s,u0m,h(s))h(s)

nds. (3.10)

Hence, we deduce that for everyn_≥1 there exists a constantC_n, independent ofm, such that

k(u0_m,h)_nkC1_([0,T];C)≤Cn.

Applying the Ascoli-Arzelà theorem, we conclude that for everynthere exists a subsequence(mn_k)_k≥1

such that (u0_mn k,h

)_n converges uniformly to some (u0_h)_n as k _{−→ ∞}. By a diagonal procedure, we may choose a sequence (mn_k)_k≥1 independent of nsuch that (u0_m,h)_n converges uniformly to some

(u0_h)_{n∈ C} ([0,T];C)for everyn≥1; set

u0_h(t) = ((u0_h)₁,(u0_h)₂, . . .).

From the estimate (3.7), we have the weak star convergence inL∞(0,T;V)of some further subse-quence of u0_mn

k,h

: k_≥1). The weak limit belongs to L∞(0,T;V) and has clearly(u0_h)_n as compo-nents that belong toC([0,T];C)for every integern≥1. Using the uniform convergence of each component, it is easy to show, passing to the limit in the expression (3.10), thatu0_h(t)satisfies the weak form of the GOY shell model equation (3.1). Finally, since

u0_h(t) =ξ+ Z t

0

−B(u0_h(s)) +σ˜0(s,u0h(s))h(s)

ds,

is such that sup_0≤s≤Tku0_h(s)_k<∞a.s. and since for everys_∈[0,T], by (2.9) and (3.2) we have a.s.

kB(u0_h(s))_k+_kσ˜0(s,u0h(s))h(s)k

≤C

1+ sup

0≤s≤Tk

u0_h(s)_k2

1+_|h(s)_|0∈L2([0,T]), we deduce thatu0_h∈ C([0,T],V)a.s.

Step 2: To complete the proof of Theorem 3.1, we show that the solutionu0_h to (3.1) is unique in

C([0,T],V). Letv∈ C([0,T],V)be another solution to (3.1) and set

Since_ku0_h(.)_kand_kv(.)_kare bounded on[0,T], we haveτN →T asN→ ∞.

SetU =u0_h−v; equation (2.10) implies

A

1 2B(u0

h(s))−A 1

2B(v(s)),A 1

2U(s)= B(u0

h(s))−B(v(s)),AU(s)

≤C¯1kU(s)k2kv(s)k.

On the other hand, the Lipschitz property (3.3) from condition(C3)forν =0 implies

A12σ˜₀(s,u0

h(s))−A 1

2σ˜₀(s,v(s))h(s)≤ p

˜_L1ku0

h(s)−v(s)k |h(s)|0.

Therefore,

kU(t_∧τN)k2 =

Z t∧τN

0

n

−2A12B(u0

h(s))−A 1

2B(v(s)),A 1 2U(s)

+2

[A12σ˜₀(s,u0

h(s))−A 1

2σ˜₀(s,v(s))]h(s),A 1 2U(s)

o

ds ≤2

Z t

0

¯

C₁N+pL_1|h(s)_|0

kU(s_∧τN)k2ds,

and Gronwall’s lemma implies that (for almost everyω) sup_0≤t≤TkU(t∧τN)k2=0 for everyN. As N_{→ ∞}, we deduce that a.s. U(t) =0 for every t, which concludes the proof. ƒ

We now suppose that the diffusion coefficient σν 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 anyν > 0and u_∈Dom(A),

|A12σ_ν(s,u)|2

LQ ≤ K0+K1kuk

2_+K

2|Au|2, (3.11)

|A12σ_ν(s,u)−A 1

2σ_ν(s,v)|2

LQ ≤ L1ku−vk

2_+L

2|Au−Av|2. (3.12)

Then forν >0, the existence result and apriori bounds of the solution to (2.20) proved in Proposi-tion 2.2 can be improved as follows.

Proposition 3.3. Letξ∈V , let the coefficients a,b,µdefining B be such that a(1+µ2) +bµ2=0,σν andσ˜ν satisfy the conditions(C1),(C2),(C3)and(C4). Then there exist positive constantsK¯2andν0 such that for0<K₂<K¯₂ and0< ν≤ν0, for every M>0there exists a constant C(M)such that for any h∈ AM, the solution uν_h to(2.20)belongs toC([0,T],V)almost surely and

sup

h∈AM

sup

0<ν≤ν0

E

sup

t∈[0,T]k

uν_h(t)_k2+ν

Z T

0

|Auν_h(t)_|2d t_≤C(M). (3.13)

Proof. Fixm_≥1, letP_mbe defined by (3.5) and letuν_m,h(t)be the approximate maximal solution to the (finite dimensional) evolution equation:uν_m,h(0) =Pmξand

duν_m,h(t) =

−νPmAuνm,h(t)−PmB(uνm,h(t)) +Pmσ˜ν(t,uνm,h(t))h(t)

+P_mpν σν(t,uνm,h)(t)dWm(t), (3.14)

where Wm is defined by (2.15). Proposition 3.3 in [10] proves that (3.14) has a unique solution uν_{m,h∈ C}([0,T],P_m(H)). For everyN>0, set

τN =inf{t: kumν,h(t)k ≥N} ∧T.

Since Pm(H)⊂ Dom(A), we may apply Itô’s formula tokuν_m,h(t)k2. Let Πm : H0 →H0 be defined

byΠ_mu=Pm_k=1 u,e_k

e_k for some orthonormal basis_{e_k,k_≥1_}ofH made by eigen-vectors of the covariance operatorQ; then we have:

kuν_m,h(t_∧τN)k2=kPmξk2+2 p

ν

Z t∧τ_N

0

A12P

mσν(s,uν_m,h(s))dWm(s),A 1 2uν

m,h(s)

+ν

Z t∧τN

0

|P_mσν(s,uνm,h(s)) Πm| 2

LQds−2

Z t∧τN

0

A12B(uν

m,h(s)),A 1 2uν

m,h(s)

ds

−2ν

Z t∧τ_N

0

A12P

mAuνm,h(s),A 1 2uν

m,h(s)

ds+2

Z t∧τ_N

0

A12P

mσ˜ν(s,uνm,h(s))h(s),A 1 2uν

m,h(s)

ds.

Since the functions ϕk are eigen-functions of A, we have A 1

2P_m = P_mA 1

2 and hence

A12P

mAuνm,h(s),A 1 2uν

m,h(s)

=_|Auν_m,h(s)_|2_{. Furthermore,P}

mcontracts theHand theV norms, and for u_∈Dom(A), B(u),Au

=0 by (2.13). Hence for 0< ε= 1₂(2₋K₂)<1, using Cauchy-Schwarz’s inequality and the conditions(C3)and(C4)on the coefficientsσν and ˜σν, we deduce

kuν_m,h(t∧τN)k2+εν

Z t∧τN

0

Auν_m,h(s)|2ds≤ kξk2+ν Z t∧τN

0

K0+K1kuνm,h(s)k 2

ds

+2pν

Z t∧τN

0

A12P_mσ_ν(s,uν

m,h(s))dWm(s),A 1 2uν

m,h(s)

+2

Z t∧τN

0

nhp

˜

K0+

p

˜

K0+

p

˜

K1

kuν_m,h(s)_k2i_|h(s)_|0+ K˜2

ε |h(s)|

2 0ku

ν m,h(s)k

2o_ds.

For anyt_∈[0,T]set

I(t) = sup

0≤s≤t

2

p

ν

Z s∧τN

0

A12P

mσν(r,uνm,h(r))dWm(r),A 1 2uν

m,h(r)

,

X(t) = sup

0≤s≤tk

uν_m,h(s∧τN)k2, Y(t) =

Z t∧τN

0

|Auν_m,h(r)_|2d r,

ϕ(t) = 2pK˜0+

p

˜

K1

|h(t)_|0+νK1+

˜

K₂

ε |h(t)|

2 0.

Then almost surely,R₀Tϕ(t)d t≤νK1T+2

p

˜

K0+

p

˜

K1

p

M T+K˜2

ε M:=C. The

Burkholder-Davis-Gundy inequality, conditions (C1) – (C4), Cauchy-Schwarz and Young’s inequalities yield that for

t_∈[0,T]andβ >0,

EI(t) _≤ 6pνE

nZ t∧τN

0

A

1

2σ_ν(s,uν

m,h(r)) Πm| 2 LQku

ν m,h(s)k

2_dso 1 2

≤ βE

sup

0≤s≤t∧τN

kum,h(s)k2

+9νK1

β E

Z t∧τN

0

kum,h(s)k2ds

+9νK0

β T+

9νK₂

β E

Z t∧τN

0

|Auν_m,h(s)_|2ds. SetZ=_kξk2_+ν

0K0T+2

p

˜

K0T M,α=εν,β=2−1e−C,K2<2−2e−2C(9+2−3e−2C)−1; the previous

inequality implies that the bounded functionX satisfies a.s. the inequality

X(t) +αY(t)_≤Z+I(t) + Z t

0

ϕ(s)X(s)ds. Furthermore,I(t)is non decreasing, such that for 0< ν≤ν0,δ=

9νK2

β ≤α2−

1_e−C _andγ= 9ν0 a K1,

one has

EI(t)_≤βEX(t) +γE

Z t

0

X(s)ds+δY(t) +9ν0

β K0T.

Lemma 3.2 from [10] implies that for K₂ and ν0 small enough, there exists a constant C(M,T)

which does not depend onmandN, and such that for 0< ν ≤ν0,m≥1 andh∈ AM:

sup

N>0

sup

m≥1

E

h

sup

0≤t≤τ_Nk

uν_m,h(t)_k2+ν

Z τN

0

|Auν_m,h(t)_|2d t

i

<∞. Then, lettingN _{→ ∞}and using the monotone convergence theorem, we deduce that

sup

m≥1

sup

h∈AM E

h

sup

0≤t≤Tk

uν_m,h(t)_k2+ν

Z T

0

|Auν_m,h(t)_|2d t

i

<∞. (3.15)

Then using classical arguments we prove the existence of a subsequence of (uν_m,h,m _≥ 1)

which converges weakly in L2([0,T]_×Ω,V)_∩ L4([0,T]_× Ω,H) and converges weak-star in

L4(Ω,L∞([0,T],H))to the solutionuν_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 (uν_m,h,m _≥ 1) which is weak-star convergent to the same limit uν_h in L2(Ω,L∞([0,T],V)) and converges weakly in

L2(Ω_×[0,T],Dom(A)); this is a straightforward consequence of (3.15). Then asm_{→ ∞}in (3.15), we conclude the proof of (3.13).

4

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. Letσ:[0,T]_×V → LQ and

for everyν >0 let ¯σν :[0,T]×Dom(A)→LQ satisfy the following condition:

Condition (C5):

(i) There exist a positive constant γ and non negative constants C,¯ K¯₀, K¯₁ and ¯L₁ such that for all u,v∈V and s,t∈[0,T]:

|σ(t,u)_|2_L

Q ≤

¯

K0+K¯1|u|2,

A

1 2σ(t,u)

2 LQ≤

¯

|σ(t,u)₋σ(t,v)_|2_L

Q≤

¯

L_1|u₋v_|2, A

1

2σ(t,u)−A 1

2σ(t,v)

2 LQ≤

¯_L1ku−vk2_,

σ(t,u)−σ(s,u)

LQ ≤C(1+kuk)|t−s| γ_.

(ii) There exist a positive constantγ and non negative constantsC,¯ K¯₀, K¯_H, K¯₂ and ¯L₂ such that for

ν >0, s,t_∈[0,T]and u,v_∈Dom(A), |σ¯ν(t,u)|2LQ≤

¯

K0+K¯H kuk2H

, A

1

2σ¯_ν(t,u)

2 LQ≤

¯

K0+K¯2|Au|2

,

|σ¯ν(t,u)−σ¯ν(t,v)|2LQ ≤

¯_L2ku−vk2_, A

1 2σ¯

ν(t,u)−A

1 2σ¯

ν(t,v)

2 LQ ≤

¯_L2|Au−Av|2_,

σ¯_ν(t,u)−σ¯_ν(t,u)

LQ ≤

¯

C(1+_ku_k)_|t₋s_|γ.

Set

σν =σ˜ν =σ+ p

νσ¯ν for ν >0, and σ˜0=σ. (4.1)

Then for 0_≤ν ≤ν1, the coefficientsσν and ˜σν satisfy the conditions(C1)-(C4)with K₀=K˜₀=4 ¯K₀, K₁=K˜₁=2 ¯K₁, L₁=˜L₁=2¯L₁, ˜K₂=2 ¯K₂, ˜K_H =2 ¯K_H,

K2=2

_¯

K2∨ K¯Hk04α−2

ν1, L2=2¯L2ν1 and ˜L2=2¯L2. (4.2)

Proposition 3.3 and Theorem 3.1 prove that for someν0∈]0,ν1], ¯K2and ¯L2small enough, 0< ν ≤

ν0 (resp. ν =0),ξ∈V andhν ∈ AM, the following equation has a unique solution uν_hν (resp. u0_h)

inC(0,T],V):uν_h

ν(0) =u

0

h(0) =ξ, and duν_h

ν(t) +

νAuν_h

ν(t) +B(u

ν hν(t))

d t=pν σν(t,uνhν(t))dW(t) +σ˜ν(t,u

ν

hν(t))hν(t)d t, (4.3)

du0_h(t) +B(u0_h(t))d t=σ(t,u0_h(t))h(t)d t. (4.4) Recall that for anyα≥0,Hα has been defined in (2.2) and is endowed with the normk · kαdefined

in (2.2). When 0_≤α≤ 1₄, asν →0 we will establish a Large Deviation Principle (LDP) in the set

C([0,T],V)for the uniform convergence in time whenV is endowed with the norm_{k · kα} for the family of distributions of the solutionsuν to the evolution equation: uν(0) =ξ∈V,

duν(t) +

νAuν(t) +B(uν(t))

d t=pνσν(t,uν(t))dW(t), (4.5)

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 thatu0_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)whenα=0, that is when_Hα=H.

Condition (C6): Letα∈[0,1

4]; there exists a constant L3such that for u,v∈ Hαand t∈[0, 1],

Aασ(t,u)−Aασ(t,v)|L

≤C¯32− n

4. (4.30)

The Hölder regularity(C5)imposed onσ(.,u)and the Cauchy-Schwarz inequality imply that

˜

T2(N,n,ν)≤C

p

N2−nγE

1_GN,ν(T)

Z T

0

€

1+_ku0_h(s)_kŠ_|hν(s)−h(s)|0ds

≤_C¯₂₂−nγ _(4.31) for some constant ¯C₂ = C(T,M,N). Using Cauchy-Schwarz’s inequality and (C5) we deduce for

¯

C₄=C(T,N,M)and anyν∈]0,ν0]

˜

T₄(N,n,ν)_≤E h

1_G

N,ν(T) sup 1≤k≤2n

¯

K₀+K¯_1|u_h0(t_k)_|21

2

Z tk

tk−1

|h_ν(s)₋h(s)_|0ds_kU_ν(t_k)_kk4₀α−1i

≤C(N)E

sup

1≤k≤2n

Z tk

tk−1

|h_ν(s)_|0+_|h(s)_|0

ds_≤C¯₄2−n2. (4.32)

Finally, note that the weak convergence ofhν tohimplies that asν →0, for anya,b∈[0,T],a<b, the integral Rb

a hν(s)ds →

Rb

a h(s)ds in the weak topology of H0. Therefore, since the operator

σ(t_k,u0_h(t_k))is compact fromH₀ toH, we deduce that for everyk,

σ(tk,u

0 h(tk))

Z tk

tk₋1

hν(s)ds−

Z t_k

tk₋1 h(s)ds

H→0 as ν→0.

Hence a.s. for fixed n as ν → 0, ¯T5(N,n,ν) → 0 while ¯T5(N,n,ν) ≤ C(K¯0, ¯K1,N,n,M). The

dominated convergence theorem proves thatE(T¯₅(N,n,ν))_→0 asν →0 for any fixedn,N. This convergence and (4.28)–(4.32) complete the proof of (4.27). Indeed, they imply that for any fixedN≥1 and any integern≥1

lim sup ν→0

E h

1_GN,ν(T) sup

0≤t≤T|

T5(t,ν)|

i

≤CN,T,M2−n(

1 4∧γ).

for some constantC(N,T,M)independent ofn. Sincenis arbitrary, this yields for any integerN_≥1 the convergence property (4.27) holds. By the Markov inequality, we have for anyδ >0

P

sup

0≤t≤Tk

Uν(t)kα> δ

≤P(GN,ν(T)c) + 1 δ2E

1_G

N,ν(T) sup 0≤t≤Tk

Uν(t)k2α

.

Finally, (4.20) and (4.21) yield that for any integerN≥1, lim sup

ν→0

P

sup

0≤t≤Tk

U_ν(t)_kα> δ)_≤C(T,M,δ)N−1,

for some constantC(T,M,δ)which does not depend onN. LettingN→+_∞concludes the proof of the proposition.

The following compactness result is the second ingredient which allows to transfer the LDP from p

νW touν. Its proof is similar to that of Proposition 4.4 and easier; it will be sketched (see also

Proposition 4.5. Suppose that the constants a,b,µdefining B satisfy the condition a(1+µ2)+bµ2=0, σsatisfies the conditions (C5) and (C6) and letα∈[0,1

4]. Fix M >0, ξ∈V and let KM ={u 0 h:h∈

SM}, where u0_h is the unique solution inC([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_ku_kX =sup_0≤t≤Tku(t)_kα. Proof. To ease notation, we skip the superscript 0 which refers to the inviscid case. By Theorem 3.1, KM⊂ C([0,T],V). Let{un}be a sequence inKM, corresponding to solutions of (4.4) with controls

{h_n}inS_M:

dun(t) +B(un(t))d t=σ(t,un(t))hn(t)d t, un(0) =ξ.

Since S_M is a bounded closed subset in the Hilbert space L2(0,T;H₀), it is weakly compact. So there exists a subsequence of _{hn}, still denoted as {hn}, which converges weakly to a limit h ∈

L2(0,T;H0). Note that in fact h ∈ SM as SM 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))d t=σ(t,u(t))h(t)d t, u(0) =ξ.

Note that we know from Theorem 3.1 thatu∈ C([0,T],V), and that one only needs to check that the convergence ofu_ntouholds uniformly in time for the weakerk · kαnorm onV. To ease notation we will often drop the time parameterss,t, ... in the equations and integrals. LetU_n=u_n−u; using (2.12) and(C6), we deduce that fort∈[0,T],

kU_n(t)_k2_α=₋2

Z t

0

AαB(u_n(s))₋AαB(u(s)),AαU_n(s)

ds

+2

Z t

0

n

Aα

σ(s,u_n(s))₋σ(s,u(s))

h_n(s),AαU_n(s) + Aασ(s,u(s)) h_n(s)₋h(s)

,AαU_n(s)o

ds ≤2C

Z t

0

kUn(s)k2α kun(s)k+ku(s)k

ds+2L₃

Z t

0

kUn(s)k2α|hn(s)|0ds +2

Z t

0

σ(s,u(s)) [h_n(s)₋h(s)], A2αU_n(s)ds. (4.33) The inequality (3.4) implies that there exists a finite positive constant ˜C such that

sup

n

sup

0≤t≤T k

u(t)_k2+_ku_n(t)_k2

=_C˜_{. (4.34)}

Thus Gronwall’s lemma implies that sup

0≤t≤Tk

Un(t)k2α≤exp

2CC˜+2L₃pM T X

1≤i≤5

I_n,Ni , (4.35)

where, as in the proof of Proposition 4.4, we have:

I_n,N1 =

Z T

0

I_n,N2 = Z T 0

σ(s,u(s))₋σ(¯sN,u(s))

[hn(s)−h(s)],A2αUn(¯sN)

ds,

I_n,N3 =

Z T 0

σ(¯s_N,u(s))₋σ(¯s_N,u(¯s_N))

[h_n(s)₋h(s)],A2αU_n(¯s_N) ds,

I_n,N4 = sup

1≤k≤2N

sup

tk₋1≤t≤tk

σ(tk,u(tk))

Z t

tk−1

(hn(s)−h(s))ds, A2αUn(tk)

,

I_n,N5 = X 1≤k≤2N

σ(tk,u(tk))

Z tk

tk−1

[hn(s)−h(s)]ds, A2αUn(tk)

.

The Cauchy-Schwarz inequality, (4.34), (C5) and (4.10) imply that for some constants Ci, i =

0,_{· · ·}, 4, which depend onk₀, ¯K_i, ¯L₁, ˜C,M andT, but do not depend onnandN, I_n,N1 ≤C0

Z T

0

kun(s)−un(¯sN)k2+ku(s)−u(¯sN)k2

ds 1 2 Z T 0

|hn(s)−h(s)|20ds

1

2

≤C₁2−N2, (4.36)

I_n,N3 _≤C₀

Z T

0

ku(s)₋u(¯s_N)_k2ds 1 2

2pM_≤C₃2−N2, (4.37)

I_n,N4 ≤C02− N

2

1+ sup

0≤t≤Tk

u(t)_k sup

0≤t≤T k

u(t)_k+_kun(t)k

2pM ≤C42− N

2 . (4.38)

Furthermore, the Hölder regularity ofσ(.,u)from condition(C5)implies that I2_n,N≤C¯2−Nγ sup

0≤t≤T k

u(t)_k+_ku_n(t)_k

×

Z T

0

(1+_ku(s)_k)(_|h(s)_|0+_|hn(s)|0)ds≤C22−Nγ. (4.39)

For fixed N and k = 1,_{· · ·}, 2N, as n _{→ ∞}, the weak convergence of h_n to h implies that of

Rtk

tk₋1(hn(s)−h(s))dsto 0 weakly inH0. Sinceσ(tk,u(tk))is a compact operator, we deduce that for fixedkthe sequenceσ(t_k,u(t_k))Rtk

tk₋1(hn(s)−h(s))dsconverges to 0 strongly inH asn→ ∞. Since sup_n,kkU_n(t_k)_{k ≤}2

p

˜

C, we have lim_nI_n,N5 =0. Thus (4.35)–(4.39) yield for every integerN_≥1 lim sup

n→∞ sup

t≤Tk

U_n(t)_k2_α≤C2−N(12∧γ).

Since N is arbitrary, we deduce that sup_0≤t≤TkU_n(t)_{kα →} 0 as n _{→ ∞}. This shows that every sequence inKMhas a convergent subsequence. HenceKM 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

uh∈KM for the uniform topology onC([0,T],V)endowed with thek · kαnorm. Hencev=uh,KM

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 _{uν} satisfies the Laplace principle, which is equivalent to the large deviation principle, inX 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

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