2 Description of the model

El e c t ro nic

Journ a l of

Pr

ob 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 Franceand

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.frandannie.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 byp

ν, we prove that some shell models of turbulence with a multiplicative stochastic perturbation driven by a H-valued Brownian motion satisfy a LDP inC([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 author has also been supported by the NSF grant No. DMS 0608494

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 spaceH₁²with proper periodicity or boundary conditions. The initial conditionξonly belongs toH.

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 byp

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 ofW which converges weakly toh, the corresponding family of stochastic con- trol equations, deduced from the original ones by shifting the noise by p^hνν, 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 inC([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 normk · kα:=|A^α(·)|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 coefficient"in front of a deterministic operator converges to 0 and the intensity of the Gaussian noise is multiplied by p

". 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 inC([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 inC([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 withA^αfor 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)≥k²₀|u|²foru∈Dom(A). For anyα >0, set

H_α=Dom(A^α) ={u∈H : X

n≥1

k⁴_n^α|u_n|²<+∞}, kuk²_α=X

n≥1

k⁴_n^α|u_n|²for u∈ H_α. (2.2) LetH0=H,

V :=Dom(A¹²) =n

u∈H : X

n≥1

k²_n|u_n|²<+∞o

; also set H =H¹

4,kuk_H =kuk¹

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

nk²_nu_nv_n^∗), u,v∈V and the associated norm is denoted by

kuk²=X

n≥1

k²_n|u_n|². (2.3)

The adjoint ofV with respect to the H scalar product isV⁰={(u_n)∈C^N : P

n≥1k⁻²_n |u_n|² <+∞}

andV ⊂H⊂V⁰is a Gelfand triple. Let〈u,v〉=Re€P

n≥1u_nv_n^∗Š

denote the duality betweenu∈V andv∈V⁰. Clearly for 0≤α < β,u∈ H^β andv∈V we have

kuk²_α≤k₀^4(α−β)kuk²_β, and kvk²_H ≤ |v| kvk, (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€

ak_n+1u^∗_n+1v_n^∗₊₂+bk_nu^∗_n−1v_n^∗₊₁−ak_n−1u^∗_n−1v_n^∗₋₂−bk_n−1u^∗_n−2v^∗_n−1Š

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

[B(u,v)]n=−i€

ak_n+1u^∗_n+1v_n+2+bk_nu^∗_n−1v_n+1+ak_n−1u_n−1v_n−2+bk_n−1u_n−2v_n−1Š

, (2.6) in the Sabra shell model introduced in[21].

Note thatBcan be extended as a bilinear operator fromH×HtoV⁰and 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| kwk. (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|v_n|²≤Ckuk²kvk², (2.9)

|B(u,v)| ≤Ckuk_Hkvk_H.

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),u₂〉|foru₁∈H andu₂∈V. Hence there exist positive constants ¯C₁and ¯C₂ such that

|〈B(u1)−B(u2),u₁−u₂〉V| ≤ C¯₁ku₁−u₂k²ku₂k,∀u₁,u₂∈V, (2.10)

|〈B(u1)−B(u2),u₁−u₂〉| ≤ C¯₂|u₁−u₂|²ku₂k,∀u₁∈H,∀u₂∈V. (2.11) Finally, sinceBis bilinear, Cauchy-Schwarz’s inequality yields for anyα∈[0,¹₂],u,v∈V:

A^αB(u)−A^αB(v),A^α(u−v) ≤

A^αB(u−v,u) +A^αB(v,u−v),A^α(u−v)

≤Cku−vk²_α(kuk+kvk). (2.12) In the GOY shell model,B is defined by (2.5); for anyu∈V,Au∈V⁰we 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)µ⁴u_nu_n+1u^∗_n+2i . Thus(B(u,u),Au) =0 for everyu∈V if and only ifa+bµ²= (a+b)µ⁴ 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⁻¹²ψ), ∀φ,ψ∈H₀,

together with the induced norm | · |0 = p

(·,·)0. The embedding i : H₀ → H is Hilbert-Schmidt and hence compact, and moreover, i i^∗ =Q. Let L_Q ≡ L_Q(H0,H) be the space of linear operators S:H₀ 7→H such thatSQ¹² is a Hilbert-Schmidt operator fromH toH. The norm in the space L_Q is defined by|S|²_LQ=t r(SQS^∗), whereS^∗is the adjoint operator ofS. TheL_Q-norm can be also written in the form

|S|²_LQ =t r([SQ^1/2][SQ^1/2]^∗) =X

k≥1

|SQ^1/2ψk|²=X

k≥1

|[SQ^1/2]^∗ψk|² (2.14) for any orthonormal basis{ψk}inH, for example(ψk)n=δ_n^k.

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,{e_j}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→∞W_n(t) in L²(Ω;H) withW_n(t) = X

1≤j≤n

q^1/2_j βj(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=p

ν σ_ν(t,u(t))dW(t), (2.16) where the noise intensity σ_ν : [0,T]×V → L_Q(H0,H) of the stochastic perturbation is properly normalized by the square root of the viscosity coefficientν. We assume thatσ_νsatisfies the following growth and Lipschitz conditions:

Condition (C1): σ_ν ∈ C [0,T]×V;L_Q(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)|σ_ν(t,u)|²_LQ≤K₀+K₁|u|²+K₂kuk²,

(ii)|σ_ν(t,u)−σ_ν(t,v)|²_LQ ≤L₁|u−v|²+L₂ku−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 thatRT

0 |h(s)|²₀ds<∞, a.s. For fixedM >0, let S_M=n

h∈L²(0,T;H₀): Z T

0

|h(s)|²₀ds≤Mo .

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=1 1 2^k

RT

0 h(s)−k(s), ˜e_k(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)∈S_M, a.s.}. (2.17) In order to define the stochastic control equation, we introduce for ν ≥ 0 a family of intensity coefficients ˜σ_ν 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 ˜σ_ν 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)|²_L(H

0,H)≤K˜₀+K˜₁|u|²+νK˜_Hkuk²_H, ∀t∈[0,T], ∀u∈V, (2.18)

|σ˜_ν(t,u)−σ˜_ν(t,v)|²_L(H

0,H)≤˜L₁|u−v|²+ν˜L₂ku−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 σ˜0(t, .)can be stated for u,v∈H.

Remark 2.1. Unlike (C1) the hypotheses concerning the control intensity coefficient ˜σ_ν involve a weaker topology (we deal with the operator norm| · |L(H0,H)instead of the trace class norm| · |L_Q).

However we require in (2.18) a stronger bound (in the interpolation spaceH). One can see that the noise intensityp

ν σ_ν satisfies Condition (C2) provided that in Condition (C1), we replace point (i) by|σ_ν(t,u)|²_LQ≤K₀+K₁|u|²+K_Hkuk²_H. Thus the class of intensities satisfying both Conditions (C1) and (C2) when multiplied byp

ν 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

du^ν_h(t) +

νAu^ν_h(t) +B u^ν_h(t)

d t=p

ν σ_ν(t,u^ν_h(t))dW(t) +σ˜_ν(t,u^ν_h(t))h(t)d t, (2.20) with initial conditionu^ν_h(0) =ξ. Using[10], Theorem 3.1, we know that for everyT >0 andν >0 there exists ¯K₂^ν :=K¯₂(ν,T,M)>0 such that ifh_ν ∈ AM,E|ξ|⁴ <+∞and 0≤K₂ <K¯₂^ν, equation (2.20) has a unique solutionu^ν_h∈ C([0,T],H)∩L²([0,T],V)which satisfies:

(u^ν_h,v)−(ξ,v) + Z t

0

ν〈u^ν_h(s),Av〉+〈B(u^ν_h(s)),v〉 ds

= Z t

0

pν σ_ν(s,u^ν_h(s))dW(s), v +

Z t 0

σ˜_ν(s,u^ν_h(s))h(s),v ds

a.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 motionW. Furthermore, if K₂ ∈[0, ¯K₂^ν[and L₂ ∈[0, 2[, there exists a constant C_ν := C(Ki,L_j, ˜K_i, ˜K_H,T,M,ν)such that

E

sup

0≤t≤T|u^ν_h(t)|⁴+ Z T

0

ku^ν_h(t)k²d t+ Z T

0

ku^ν_h(t)k⁴_H d t

≤C_ν(1+E|ξ|⁴). (2.21) The following proposition proves that ¯K₂^ν can be chosen independent ofν and that a proper for- mulation of upper estimates of theH, H and V norms of the solutionu^ν_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, σ_ν 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 u^ν_h to(2.20)satisfies:

E

sup

0≤t≤T|u^ν_h(t)|⁴+ν Z T

0

ku^ν_h(s)k²ds+ν Z T

0

ku^ν_h(s)k⁴_H ds

≤C¯(M) E|ξ|⁴+1

. (2.22)

Proof. For every N > 0, set τN = inf{t : |u^ν_h(t)| ≥ N} ∧T. Itô’s formula and the antisymmetry relation in (2.8) yield that for t∈[0,T],

|u^ν_h(t∧τN)|²=|ξ|²+2p ν

Z _t∧τN

0

σ_ν(s,u^ν_h(s))dW(s),u^ν_h(s)

−2ν Z _t∧τN

0

ku^ν_h(s)k²ds

+2 Z t∧τN

0

σ˜_ν(s,u^ν_h(s))h(s),u^ν_h(s) ds+ν

Z t∧τN

0

|σ_ν(s,u^ν_h(s))|²_LQds, (2.23) and using again Itô’s formula we have

|u^ν_h(t∧τ_N)|⁴+4ν Z _t∧τN

0

|u^ν_h(r)|²ku^ν_h(r)k²d r≤ |ξ|⁴+I(t) + X

1≤j≤3

T_j(t), (2.24) where

I(t) = 4p ν

Z _t∧τN

0

σ_ν(r,u^ν_h(r))dW(r),u^ν_h(r)|u^ν_h(r)|² , T₁(t) = 4

Z t∧τN

0

|(σ˜_ν(r,u^ν_h(r))h(r),u^ν_h(r))| |u^ν_h(r)|²d r,

T₂(t) = 2ν Z _t∧τN

0

|σ_ν(r,u^ν_h(r))|²_L

Q|u^ν_h(r)|²d r, T₃(t) = 4ν

Z t∧τN

0

|σ_ν^∗(s,u^ν_h(r))u^ν_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

pK˜₀+p

K˜₁|u^ν_h(r)|+p

νK˜_H k⁻

1 2

0 ku^ν_h(r)k

|h(r)|0|u^ν_h(r)|³d r

≤ 4p

K˜₀M T+4p

K˜₀+p K˜₁

Z _t∧τN

0

|h(r)|0|u^ν_h(r)|⁴ds

+ε ν Z t

0

ku^ν_h(r)k²|u^ν_h(r)|²d r+4 ˜K_H εk₀

Z t∧τN

0

|h(r)|²₀|u^ν_h(r)|⁴d r. (2.25) Using condition(C1)we deduce

T₂(t) +T₃(t)≤ 6ν Z t∧τN

0

K₀+K₁|u^ν_h(r)|²+K₂ku^ν_h(r)k²

|u^ν_h(r)|²d r

≤6νK₀T+6ν(K0+K₁) Z _t∧τN

0

|u^ν_h(r)|⁴d r+6νK₂ Z t

0

ku^ν_h(r)k²|u^ν_h(r)|²d r. (2.26) LetK₂≤ ¹₂ and 0< ε≤2−3K₂; set

ϕ(r) =4p

K˜₀+p K˜₁

|h(r)|0+4 ˜K_H

εk₀ |h(r)|²₀+6ν(K0+K₁). Then a.s.

Z T 0

ϕ(r)d r≤4p

K˜₀+p K˜₁p

M T+4 ˜K_H εk0

M+6ν(K₀+K₁)T := Φ (2.27) and the inequalities (2.24)-(2.26) yield that for

X(t) =sup

r≤t|u^ν_h(r∧τN)|⁴, Y(t) =ν Z t

0

ku^ν_h(r∧τN)k²|u^ν_h(r∧τN)|²ds,

X(t) + (4−6K₂−ε)Y(t)≤ |ξ|⁴+ 4p

K˜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,

EI(t)≤12p νE

nZ _t∧τN

0

K₀+K₁|u^ν_h(s)|²+K₂ku^ν_h(s)k²

|u^ν_h(r)|⁶ds o¹

2

≤12p νE

sup

0≤s≤t|u^ν_h(s∧τN)|²nZ t∧τN

0

K₀+K₁|u^ν_h(s)|²+K₂ku^ν_h(s)k²

|u^ν_h(s)|²dso¹₂

≤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≤ ¹₂,ε=α= ¹₂,β= ^36K_δ² +κ ν0≤2⁻¹e^−Φ,δ≤α2⁻¹e^−Φandγ=³⁶_κ(K₀+K₁),

E

X(T) +αY(T)

≤2 exp Φ +2Tγe^Φh 4p

K˜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|u^ν_h(t∧τN)|⁴+ν Z _τN

0

ku^ν_h(t)k⁴_H d t

≤C¯(M)(1+E(|ξ|⁴)).

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

E

sup

0≤t≤T|u^ν_h(t)|⁴+ν Z T

0

ku^ν_h(t)k⁴_H d t

≤C¯(M)(1+E(|ξ|⁴)).

This inequality and (2.30) witht instead oft∧τ_N 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,u⁰_h(t))h(t)d t, u⁰_h(0) =ξ (3.1) can be solved for everyω. In order to prove that (3.1) has a unique solution inC([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 onσ_ν and ˜σ_ν, similar to that imposed on ˜σ0, we will prove further properties ofu^ν_h for a strictly positive viscosity coefficientν.

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

σ˜_ν:[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 ˜L_j, 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(H

0,H)≤K˜₀+K˜₁kuk²+νK˜₂|Au|², (3.2) and

|A¹²σ˜_ν(s,u)−A¹²σ˜_ν(s,v)|²_L(H

0,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) +bµ²=0. Letξ∈V be deterministic. For any M >0there exists C(M) such that equation(3.1)has a unique solution inC([0,T],V)for any h∈ AM, and a.s. one has:

sup

h∈AM

sup

0≤t≤Tku⁰_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 inC([0,T],V) for anyh∈S_M and that (3.4) holds. For any m≥1, letH_m=span(ϕ1,· · ·,ϕ_m)⊂Dom(A),

P_m:H→H_m denote the orthogonal projection fromH ontoH_m, (3.5)

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

0,H) ≤

|σ˜0(t,u)|²_L(H

0,H). Setu⁰_m,h(0) =P_mξand consider the ODE on the m-dimensional spaceH_m defined by

d u⁰_m,h(t),v

=

− B(u⁰_m,h(t)), v

+ σ˜0(t,u⁰_m,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 ∨ kuk_H ≤C(m)|u|foru∈H_m, 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 toR^m 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(u⁰_m,h,ϕ_k

ϕ_k 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∈ AMequation(3.6)has a unique solution inC([0,T],H_m). There exists some constant C(M)such that for every h∈ AM,

sup

m

sup

0≤t≤Tku⁰_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 : ku⁰_m,h(t)k ≥ N} ∧T. Let Πm : H₀ → H₀ denote the projection operator defined by Πmu = Pm

k=1 u,e_k

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,P_mcontracts the V norm and commutes withA. Thus, using(C3)and (2.13), we deduce

ku⁰_m,h(t∧τ_N)k²≤ kξk²−2 Z _t∧τN

0

B(u⁰_m,h(s)),Au⁰_m,h(s) ds

+2 Z t∧τN

0

A¹²P_mσ˜0,m(s,u⁰_m,h(s))h(s)

ku⁰_m,h(s)kds

≤ |ξk²+2p

K˜₀M T+2p

K˜₀+p K˜₁

Z _t∧τN

0

|h(s)|0ku⁰_m,h(s)k²ds.

Since the mapku⁰_m,h(.)kis bounded on[0,τN], Gronwall’s lemma implies that for everyN >0, sup

m

sup

t≤τN

ku⁰_m,h(t)k²≤

kξk²+2p K˜₀M T

exp

2p

M Thp

K˜₀+p K˜₁

i

. (3.8)

Letτ:=lim_NτN ; asN → ∞in (3.8) we deduce sup

m

sup

t≤τku⁰_m,h(t)k²≤

kξk²+2p

K˜₀M T exp

2p

M Thp

K˜₀+p K˜₁i

. (3.9)

On the other hand, sup_t≤τku⁰_m,h(t)k²= +∞ifτ <T, which contradicts the estimate (3.9) . Hence τ=T a.s. and we get (3.7) which completes the proof of the Lemma.