On LDP and inviscid hydrodynamical equations

Transcription

1 On LDP and inviscid hydrodynamical equations Annie Millet ) collaboration with H. Bessaih SAMM, Université Paris 1 and PMA University of Wyoming Workshop Stochastic Partial Differential equations INM, Cambridge January 6, 2010

2 Outline 1 Description of the models The Navier Stokes equations Shell models of turbulence Random perturbation 2 Formulation of the LDP as ν 0 Some general results LDP for inviscid shell models LDP for inviscid Navier Stokes equations 3 Ideas of the proofs Approaches to the transfert principle Key points of the proof

3 Outline 1 Description of the models The Navier Stokes equations Shell models of turbulence Random perturbation 2 Formulation of the LDP as ν 0 Some general results LDP for inviscid shell models LDP for inviscid Navier Stokes equations 3 Ideas of the proofs Approaches to the transfert principle Key points of the proof

4 Outline 1 Description of the models The Navier Stokes equations Shell models of turbulence Random perturbation 2 Formulation of the LDP as ν 0 Some general results LDP for inviscid shell models LDP for inviscid Navier Stokes equations 3 Ideas of the proofs Approaches to the transfert principle Key points of the proof

5 1 Description of the models The Navier Stokes equations Shell models of turbulence Random perturbation 2 Formulation of the LDP as ν 0 Some general results LDP for inviscid shell models LDP for inviscid Navier Stokes equations 3 Ideas of the proofs Approaches to the transfert principle Key points of the proof

6 Description of the models The NS equations D bounded domain of R 2 with regular boundary, ν > 0 viscosity subject to conditions t u ν u + (u. )u + p = F in D, div u =.u = 0 in D u.n = 0 and curl u := u = 0 on D u = (u 1 (x, t), u 2 (x, t)) fluid velocity, p(x, t) pressure, and F (x, t) external force u gradient, u := 1 u 2 2 u 1, u = ( i=1,2 2 i uk, k = 1, 2) Stokes, div u = i=1,2 iu i, n outwards normal to D H = {f [ L 2 (D) ] 2 : div f = 0 in D and f. n = 0 on D} H k,q := [W k,q (D)] 2 H for k 0, q [2, + [, V = H 1,2

7 Description of the models The NS equations (H, ) and (V, ) Hilbert spaces, scalar products (.,.) and ((.,.)) and H = L 4 (D) 2 H interpolation space A = non-negative, unbounded, self-adjoint linear operator on H, k real function defined on D for u, v V k(r)u(r).v(r)dr C u v, k(r) u(r) 2 dr ɛ u 2 +C(ɛ) u 2 D Integration by parts a : V V defined by au, v = j=1,2 D uj v j dx D k(r)u(r).v(r)dr D

8 Description of the models The NS equations The bilinear operator B B : V V V defined by: B(u, v), w = [u v] w dx D i,j=1,2 D u j j v i w i dx, u, v, w V Properties of B: B(u 1, u 2 ), u 3 = B(u 1, u 3 ), u 2, for u j V B(u, u), Au = 0 for u H 2,2

9 1 Description of the models The Navier Stokes equations Shell models of turbulence Random perturbation 2 Formulation of the LDP as ν 0 Some general results LDP for inviscid shell models LDP for inviscid Navier Stokes equations 3 Ideas of the proofs Approaches to the transfert principle Key points of the proof

10 Description of the models Shell models Studied by P. Constantin, B. Levant, & E. S. Titi (deterministic case) D. Barbato, M. Barsanti, H. Bessaih, & F. Flandoli (random) H set of sequences u = (u n, n 1) of complex numbers such that u 2 := n u n 2 < scalar product (u, v) = Re n=1 u nvn (Au) n = knu 2 n, n = 1, 2,..., k n = k 0 µ n, µ > 1 and k 0 > 0 V = Dom(A 1 2 ), with norm defined by u 2 := n k n u n 2 H = Dom(A 1 4 ) are Hilbert spaces

11 Description of the models Shell models The bilinear operator B a, b reals, u 1 = u 0 = v 1 = v 0 = 0 [ B(u, v) ] [ B(u, v) ] n = i( ak n+1 un+1v n+2 + bk n un 1v n+1 ak n 1 un 1v n 2 bk n 1 un 2v n 1 ) n = i( ak n+1 u n+1v n+2 + bk n u n 1v n+1 + ak n 1 u n 1 v n 2 + bk n 1 u n 2 v n 1 ) GOY Sabra Properties of B B : V V V and B : H H H B(u 1, u 2 ), u 3 = B(u 1, u 3 ), u 2, for u j H if a(1 + µ 2 ) + µb 2 = 0, B(u, u), Au = 0 u V

12 1 Description of the models The Navier Stokes equations Shell models of turbulence Random perturbation 2 Formulation of the LDP as ν 0 Some general results LDP for inviscid shell models LDP for inviscid Navier Stokes equations 3 Ideas of the proofs Approaches to the transfert principle Key points of the proof

13 Description of the models Noise and diffusion coefficient W (t) H-valued Wiener process Covariance operator Q symmetric non-negative on H, Trace(Q) < +, H 0 = Q 1 2 H scalar product (φ, ψ) 0 = (Q 1 2 φ, Q 1 2 ψ) L Q = {S L(H 0, H) : SQ 1 2 Hilbert Schmidt from H to H}, M 2 L Q = Trace(M Q M ), where M is the adjoint of M. σ C([0, T ] V ) L Q := L Q (H 0, H) There exist constants K i and L i such that for t [0, T ], φ, ψ V σ(t, φ) 2 L Q K 0 + K 1 φ 2 + K 2 φ 2, σ(t, φ) σ(t, ψ) 2 L Q L 1 φ ψ 2 + L 2 φ ψ 2. (in the above examples, σ may depend on the gradient of the solution)

14 Description of the models Noise and diffusion coefficient W (t) H-valued Wiener process Covariance operator Q symmetric non-negative on H, Trace(Q) < +, H 0 = Q 1 2 H scalar product (φ, ψ) 0 = (Q 1 2 φ, Q 1 2 ψ) L Q = {S L(H 0, H) : SQ 1 2 Hilbert Schmidt from H to H}, M 2 L Q = Trace(M Q M ), where M is the adjoint of M. σ C([0, T ] V ) L Q := L Q (H 0, H) There exist constants K i and L i such that for t [0, T ], φ, ψ V σ(t, φ) 2 L Q K 0 + K 1 φ 2 + K 2 φ 2, σ(t, φ) σ(t, ψ) 2 L Q L 1 φ ψ 2 + L 2 φ ψ 2. (in the above examples, σ may depend on the gradient of the solution)

15 Description of the models Well posedeness and a priori bounds σ(u) estimated in terms of u in V with constants K 2 and L 2 d t u(t) + [νau(t) + B(u(t), u(t))]dt = σ(t, u(t)) dw (t), u(0) = ξ (1) Theorem Let ν > 0 and E ξ 4 <. Then for K 2 small enough and L 2 < 2, there exists C = C(K i, L i, T ) such that (1) has a unique solution u X = C([0, T ]; H) L 2 (0, T ; V ). Furthermore, ( E sup 0 t T T T ) u(t) 4 + u(t) 2 dt + u(t) 4 H dt C (1 + E ξ 4 ) 0 0 Sritharan-Sundar (Navier Stokes), Barbato-Barsanti-Bessaih-Flandoli (shell model with additive noise) and Manna-Shritharan-Sundar (shell models), Chueshov-M. (general hydrodynamical models), Problem: LDP for the solution u as ν 0 and W multiplied by ν

16 1 Description of the models The Navier Stokes equations Shell models of turbulence Random perturbation 2 Formulation of the LDP as ν 0 Some general results LDP for inviscid shell models LDP for inviscid Navier Stokes equations 3 Ideas of the proofs Approaches to the transfert principle Key points of the proof

17 Formulation of the LDP LDP for the Gaussian noise Schilder s theorem νw rate function (r.f.) satisfies a LDP in C([0, T ], H) with good where J(φ) = 1 2 T 0 H = {φ : [0, T ] H t φ(t) = φ(t) 2 0dt if φ H, J(φ) = + otherwise, and I : H 0 H is the Hilbert Schmidt embedding. 0 I ( ) } φ(s) ds for φ L 2 ([0, T ], H 0 ) du ν (t)+[νau ν (t)+b(u ν (t), u ν (t))]dt = νσ(u ν (t))dw (t), u ν (0) = ξ.

18 General results for inviscid LDP Transfert of the LDP - small perturbation Let G : C([0, T ], H) X be defined by G( νw ) = u ν solution. Case of an additive noise, G is continuous ; use the contraction principle u ν satisfied a LDP with good rate function I (ψ) = inf{j(h) : G(h) = ψ} (convention inf = + ). I (ψ) = inf {ḣ L 2 (0,T ;H 0 ): ψ=g(i(h))} ( 1 2 T 0 ) ḣ(s) 2 0ds Case of multiplicative noise, G is NOT continuous, but one can transfer the LDP from νw to u ν Make stronger assumptions on σ: K 2 = L 2 = 0, i.e., σ C([0, T ] H, L Q (H 0, H)) σ(t, φ) 2 L Q K 0 + K 1 φ 2, for t [0, T ], φ H σ(t, φ) σ(t, ψ) 2 L Q L 1 φ ψ 2 for t [0, T ], φ, ψ H

19 General results for inviscid LDP Transfert of the LDP - small perturbation Let G : C([0, T ], H) X be defined by G( νw ) = u ν solution. Case of an additive noise, G is continuous ; use the contraction principle u ν satisfied a LDP with good rate function I (ψ) = inf{j(h) : G(h) = ψ} (convention inf = + ). I (ψ) = inf {ḣ L 2 (0,T ;H 0 ): ψ=g(i(h))} ( 1 2 T 0 ) ḣ(s) 2 0ds Case of multiplicative noise, G is NOT continuous, but one can transfer the LDP from νw to u ν Make stronger assumptions on σ: K 2 = L 2 = 0, i.e., σ C([0, T ] H, L Q (H 0, H)) σ(t, φ) 2 L Q K 0 + K 1 φ 2, for t [0, T ], φ H σ(t, φ) σ(t, ψ) 2 L Q L 1 φ ψ 2 for t [0, T ], φ, ψ H

20 General results for inviscid LDP Inviscid LDP - Aim Let the positive viscosity coefficient ν 0 and d t u ν (t) + [ νau ν (t) + B(u ν (t), u ν (t)) ] dt = ν σ(u ν (t)) dw (t) with initial condition u ν (0) = ξ. Prove exponential decay of P(u ν (.) Γ) as ν 0 for Γ X that is lim ν ln ν 0 P(uν Γ) in terms of some rate function and interior (resp. closure) of Γ in a space X X = C([0, T ]; H) L 2 (0, T ; V ) with some not optimal topology Why? The rate function is formulated in terms of the unique solution to the irregular inviscid case, for h in the RKHS of the noise, duh 0 (t) + B(u0 h (t)) dt = σ(u0 h (t)) h(t) dt, u0 h (0) = ξ

21 1 Description of the models The Navier Stokes equations Shell models of turbulence Random perturbation 2 Formulation of the LDP as ν 0 Some general results LDP for inviscid shell models LDP for inviscid Navier Stokes equations 3 Ideas of the proofs Approaches to the transfert principle Key points of the proof

22 Formulation of the LDP for shell models Inviscid shell models - Strengthening assumptions For α [0, 1 4 ], let H α = Dom(A α ), u α := A α u and set H = H 1, 4 V = H 1 For ν 0, set σ ν = σ + ν σ ν 2 Condition (C1) on σ and σ: There exist constants s.t. for u, v V, ν 0: σ(u) 2 L Q K 0 + K 1 u 2, σ(u) σ(v) 2 L Q L 1 u v 2, σ ν (u) 2 L Q ( K0 + K H u 2 H A 1 2 σ(u) 2 K L 0 + K 1 u 2 Q A σ(u) A 2 σ(v) 2 L Q L 1 u v 2 ), A 1 2 σν (u) 2 L Q ( K0 + K 2 Au 2) σ ν (u) σ ν (v) 2 L Q L 2 u v 2, A 1 2 σν (u) A 1 2 σν (v) 2 L Q L 2 Au Av 2 Condition (Cα) on σ. For fixed α [0, 1 4 ] there exist L 3 > 0 s.t. A α σ(u) A α σ(v) 2 L Q L 3 u v 2 α, u, v H α

23 Formulation of the LDP for shell models The result for shell models Theorem (Bessaih-M.) Suppose that B(u, u), Au = 0 for all v V (true a(1 + µ 2 ) + bµ 2 = 0), let σ ν = σ + ν σ ν with σ and σ ν which satisfy conditions (C) and (C α ) for α [0, 1/4],. Then for u ν (0) = ξ V, du ν (t) + [ νau ν (t) + B(u ν (t), u ν (t)) ] dt = νσ ν (u ν (t)) dw (t) satisfies a LDP in X := C([0, T ], H α ) endowed with the norm u X =: sup 0 t T u(t) α, with the good rate function I (u) = inf{ h 2 L 2 ([0,T ],H 0 ) /2 : u = u0 h, h L2 (0, T ; H 0 } du 0 h (t) + B(u0 h (t), u0 h (t)) dt = σ(u0 h (t)) h(t) dt, u0 h (0) = ξ and Similar result proved by Mariani (σ multiplied by some power of ɛ, ɛ coefficient of an operator) for conservation laws

24 Formulation of the LDP for shell models Comments on the result Solution exists if ν ν 0 for all K i. The stronger assumptions ensure the existence and uniqueness of the solution to the control equation du 0 h (t) + B(u0 h (t), u0 h (t))dt = σ(u0 (t))h(t)dt, u 0 h (0) = ξ for h A M (i.e. a.s. T 0 h(s) 2 0ds M) and the apriori bound sup{ u 0 h (t) : h A M, t [0, T ]} C(T, M)(1 + ξ ) Also give apriori estimates for the solution u ν to the random equation in C([0, T ]; V ) L 2 (0, T ; Dom(A)) uniform in ν (0, ν 0 ] The LDP as ν 0 in X := C ( ) [0, T ]; H α for 0 α 1 4 means: For every closed (resp. open) set F (resp. G) of X : lim sup ν log P(u ν F ) inf{i (u), u F }. ν 0 lim inf ν 0 ν log P(uν G) inf{i (u), u G}. with level sets {u X : I (u) M} which are compact subsets of X.

25 1 Description of the models The Navier Stokes equations Shell models of turbulence Random perturbation 2 Formulation of the LDP as ν 0 Some general results LDP for inviscid shell models LDP for inviscid Navier Stokes equations 3 Ideas of the proofs Approaches to the transfert principle Key points of the proof

26 Formulation of the LDP for NS equations Strengthening the assumptions for the inviscid equation Problem studied in random setting by Bessaih, Brzezniak-Peszat Strengthen assumptions to have existence and uniqueness of solution to the inviscid controlled equation (Euler equation with deterministic forcing term; studied by Bardos, Kato-Ponce,...) du 0 h (t) + B(u0 h (t), u0 h (t))dt = σ(u0 (t))h(t)dt, u 0 h (0) = ξ V for h A M (i.e. a.s. T 0 h(s) 2 0 ds M), σ C([0, T ]; L Q(H 0, V )) and σ(u) 2 L Q K 0 + K 1 u 2, curl σ(u) 2 L Q K 0 + K 1 u 2 implies the existence of u 0 h C([0, T ]; H) L (0, T ; V ) and a.s. sup{ u 0 h (t) : h A M, t [0, T ]} C(T, M)(1 + ξ ) Again this yields apriori estimates of the norm of u ν in L (0, T ; V ) uniform in ν (0, ν 0 ] for the solution to the random equation.

27 Formulation of the LDP for NS equations Non Hilbert Sobolev spaces Problem Get a uniqueness result Requires stronger assumptions and the use of non Hilbert Sobolev spaces. u q := u L q (D) for u L q (D) ( k u H k,q = α =0 D Dα u(x) q dx for u H k,q Sobolev injections: H 1,2 L q (D) for q < and H 1,q L (D) if 2 < q < B : H 1,q H 1,q H r,q for r > 0 and q [2, + ) Additional properties of B: ) 1 q B(u 1, u 2 ), u 3 = B(u 1, u 3 ), u 2, for u j H 1,q curl B(u, v), curl v curl v q 2 = 0, u, v H 2,q, B(u, v) q C u H 1,q v H 1,q for q > 2, u, v H 1,q.

28 Formulation of the LDP for NS equations Stochastic integrals for Radonifyng operators Definition Let Y be a Banach space; a linear operator K : H 0 Y is Radonifying if for any ONB (e k ) of H 0 and any sequence (β k ) of iid N(0,1) random variables, the series n 1 β kke k converges in L 2 (Ω; Y) (or a.s.) and K 2 R(H 0,Y) := E k β kke k 2 Y Then if Y = W k,q (D) for k = 0, 1,... and q [2, ) and if (X t ) is predictable with X L 2 (0, T ; R(H 0, Y)), the stochastic integral t 0 X sdw (s) can be defined as an element of L 2 (0, T ; W k,q ) (extended from step processes to L 2 (0, T ; R(H 0, Y)) Results from Dettweiler, Neidhardt, Brzezniak, Peszat, Ondrejat Burkholder-Davies-Gundy inequality ( E sup 0 s t s 0 X r dw (r) p ) ( t CE W k,q 0 X s 2 R(H 0,W k,q ) ds ) p 2

29 Formulation of the LDP for NS equations Stochastic integrals for Radonifyng operators Let K R(H 0, Y) and let γ K denote the distribution of the Y-valued random variable k β kke k. Let Z be a Banach space, L : Y Y Z be bilinear continuous, Trace K L = Y L(x, x)dγ K (x) Itô s formula: let q p <, b L 1 (0, T ; L q ), σ L 2 (0, T ; R(H 0, L q )); then for 0 t T and 2 q p <, if ζ t = ζ 0 + t 0 b(s)ds + t 0 σ(s)dw (s) ζ t p q = ζ 0 p q + p t 0 ζ s p q q ζ s q 2 ζ s, b(s) ds t + p ζ s p q q ζ s q 2 ζ s, σ(s)dw (s) t where Ψ q,p (u) = u p q and Trace σ(s) Ψ q,p(ζ s ) p(p 1) ζ s q p 2 σ(s) 2 R(H 0,L q ) and.. denotes the duality L q L q where q = q/(q 1). 0 Trace σ(s) Ψ q,p(ζ s )ds

30 Formulation of the LDP for NS equations Strengthening the assumptions for the NS LDP For uniqueness of the solution to the controlled Euler equation, suppose furthermore that there exists q > 2 such that curl σ C(H 1,q ; R(H 0, L q )) curl σ(u) 2 R(H 0,L q ) K 0 + K 1 u 2 q + K 2 curl u 2 q curl σ(u) curl σ(v) 2 R(H 0,L q ) L 1 u v 2 q + L 2 curl (u v) 2 q where R(H 0, H k,q ) is the set of Radonifyng operators from H 0 (RKHS of the noise) to the Sobolev space H k,q for integers k 0. Then if curl ξ L (D), the control equation has a unique solution in C([0, T ]; H 1,q ) and a.s. sup{ u 0 h (t) H 1,q : h A M, t [0, T ]} C(T, M)(1 + ξ H 1,q) Again, under the above extra conditions, for 2 < q < p < ( ) sup E u ν (t) p C(p, q, T, M)(1 + ξ p ) H 0<ν ν 1,q H 1,q 0 sup 0 t T

31 Formulation of the LDP for NS equations Examples : Nemytski operators Q 1 2 is Hilbert-Schmidt from H to H and there exist integral kernels Q i,j, i, j = 1, 2 such that ( Q 1 2 ψ(x) = Q i,j (x, y).ψ j (y)dy, i = 1, 2 ), ψ H Example 1 i,j j [ α 1 D + D D D ( ( D α x Q i,j (x, y) ) 2 dy dx D ( D α x Q i,j (x, y) ) ) q ] 2 2 dy dx < and for h = (h 1, h 2 ) H 0, i = 1, 2, (σ(u)h(x)) i = j g i,j(x, u(x))h j (x), with g i,j (x, y) + ( ) xk g i,j (x, y) + yk g i,j (x, y) C, (x, y) D R 2 k=1,2

32 Formulation of the LDP for NS equations Examples : Nemytski operators Example 2 sup i,j α 1 x D D ( α x Q i,j (x, y)) 2dy < + and for h = (h 1, h 2 ) H 0, i = 1, 2, (σ(u)h(x)) i = j g i,j(x, u(x))h j (x), with g i,j (x, y) = g (1) i,j (x) + g (2) i,j (x, y) with g (1) i,j (.) W 1,2 (D) W 1,q (D) for q > 2, g (2) i,j (.,.) differentiable on D R 2 and such that (i) g (1) i,j W 1,2 + g (1) i,j W 1,q + yk g (2) i,j (x, y) C, (x, y) D R 2 k=1,2 (ii) there exists a function φ L 2 (D) L q (D) such that g (2) i,j (x, y) + xk g (2) i,j (x, y) C(φ(x) + y ), (x, y) D R 2 k=1,2

33 Formulation of the LDP for NS equations The inviscid LDP for NS equations Theorem (Bessaih-M.) Let ξ V satisfy curl ξ L (D), σ C(V ; L Q (H 0, V )) be such that curl σ C(H 1,q ; R(H 0, L q (D))) satisfies the previous growth and Lipschitz conditions. Then as ν 0, the distribution of the solution u ν to dut ν + [νaut ν + B(ut ν, ut ν )] dt = νσ(ut ν ) dw (t) with the initial condition u0 ν = ξ satisfies in X = C([0, T ]; Lq (D) H) endowed with the norm u X := sup 0 t T u t q a LDP with the good rate function I (u) = inf{ h 2 L 2 ([0,T ],H 0 ) /2 : u = u0 h, h L2 (0, T ; H 0 } and and u 0 h is the unique solution to the control equation du 0 h (t) + B(u0 h (t), u0 h (t)) dt = σ(u0 h (t)) h(t) dt, u0 h (0) = ξ

34 Formulation of the LDP for NS equations Work in progress In the above results, σ may depend on time. For shell models, we require time Hölder regularity of σ(., u). For the NS equations, - we can use again time Hölder regularity of σ(., u). - try to use the Ren and Zhang approach for solutions to more regular stochastic evolution equations. Technique based on compactness of the solutions to the controlled equations uh ν ν in C([0, T ]; V ) thanks to time Hölder regularity of the solutions in C([0, T ]; V ) - try to use the approach of Liu for solutions to stochastic evolution equations with monotonicity properties. Technique based on some compact approximation of the diffusion coefficient. try to state the LDP again for σ ν = σ + ν σ ν with σ ν less regular try to prove the LDP in C([0, T ]; H α,q ) for some α [0, 1 4 ] and q (2, ).

35 1 Description of the models The Navier Stokes equations Shell models of turbulence Random perturbation 2 Formulation of the LDP as ν 0 Some general results LDP for inviscid shell models LDP for inviscid Navier Stokes equations 3 Ideas of the proofs Approaches to the transfert principle Key points of the proof

36 Weak convergence approach Transfert of LDP Let G 0 (g) := G 0 (h) for g(t) = t 0 I(h(s))ds H Freidlin Wentzell inequality For SPDEs, based on formulation of the solution in terms of the Green function. Uses time regularity of the solution Léandre-Russo, Sowers, Chenal-M., Cerrai-Röckner, M.-Sanz-Solé, Peszat-Zabczyk, Rovires, Cardon Weber, Kallianpur, Xiang,... parabolic or hyperbolic SPDEs Weak convergence Introduced by Dupuis & Ellis (equivalent to the Laplace principle); Budhiraja & Dupuis infinite dimensional mild solutions small perturbations in various SPDE contexts with small perturbation: Sritharan & Sundar, Duan & M., Manna, Sritharan & Sundar, Chueshov-M., Liu, Ren & X. Zhang, T. Zhang,...

37 Weak convergence approach Transfert of LDP Let G 0 (g) := G 0 (h) for g(t) = t 0 I(h(s))ds H Freidlin Wentzell inequality For SPDEs, based on formulation of the solution in terms of the Green function. Uses time regularity of the solution Léandre-Russo, Sowers, Chenal-M., Cerrai-Röckner, M.-Sanz-Solé, Peszat-Zabczyk, Rovires, Cardon Weber, Kallianpur, Xiang,... parabolic or hyperbolic SPDEs Weak convergence Introduced by Dupuis & Ellis (equivalent to the Laplace principle); Budhiraja & Dupuis infinite dimensional mild solutions small perturbations in various SPDE contexts with small perturbation: Sritharan & Sundar, Duan & M., Manna, Sritharan & Sundar, Chueshov-M., Liu, Ren & X. Zhang, T. Zhang,...

38 Weak convergence approach The stochastic controlled equation Stochastic controlled equation S M = A M = {h { h L 2 ([0, T ], H 0 ) : T 0 } h(s) 2 0 ds M (F t ) predictable : h(ω) S M a.s.} For ν > 0, h ν A M, let u ν h ν = G ν ( νw,. 0 h ν(s)ds) be the solution to the stochastic controlled equation: u ν h ν (0) = ξ du ν h ν + [νau ν h ν + B(u ν h ν, u ν h ν )]dt = σ(u ν h ν )h ν dt + νσ(u ν h ν )dw (t) Step 1 (weak convergence) Let (h ν ), h A M be such that as ν 0, h ν h in distribution (for the weak topology on S M ) Then G ν ( νw,. 0 h ν(s)ds) G 0 (h) in distribution in X. Step 2 (compactness) For M > 0, Γ M = {G 0 (h) : h S M } X

39 Weak convergence approach The stochastic controlled equation Stochastic controlled equation S M = A M = {h { h L 2 ([0, T ], H 0 ) : T 0 } h(s) 2 0 ds M (F t ) predictable : h(ω) S M a.s.} For ν > 0, h ν A M, let u ν h ν = G ν ( νw,. 0 h ν(s)ds) be the solution to the stochastic controlled equation: u ν h ν (0) = ξ du ν h ν + [νau ν h ν + B(u ν h ν, u ν h ν )]dt = σ(u ν h ν )h ν dt + νσ(u ν h ν )dw (t) Step 1 (weak convergence) Let (h ν ), h A M be such that as ν 0, h ν h in distribution (for the weak topology on S M ) Then G ν ( νw,. 0 h ν(s)ds) G 0 (h) in distribution in X. Step 2 (compactness) For M > 0, Γ M = {G 0 (h) : h S M } X

40 1 Description of the models The Navier Stokes equations Shell models of turbulence Random perturbation 2 Formulation of the LDP as ν 0 Some general results LDP for inviscid shell models LDP for inviscid Navier Stokes equations 3 Ideas of the proofs Approaches to the transfert principle Key points of the proof

41 Weak convergence - Shell models apriori estimates Theorem Suppose that a(1 + µ 2 ) + bµ 2 = 0, let ξ V let σ satisfy the reinforced conditions, let M > 0, h such that T 0 h(s) 2 0 ds M a.s.. Then there exists ν 0 > 0 such that for 0 < ν ν 0 duh ν (t) + [νauν h (t) + B(uν h (t), uν h (t))]dt = νσ(u ν h (t))dw (t) + σ(uh ν (t)) h(t) dt with initial condition ξ has a unique solution in a C([0, T ]; V ) L 2 (0, T ; DomA). There exists a constant C(M, T ) such that for every ν (0, ν 0 ] ( E sup 0 t T u ν h (t) 2 + ν T 0 ) Auh ν (t) 2 dt C(M, T ) ( 1 + ξ 2).

42 Weak convergence - Shell models Scheme of proof Recall X = C([0, T ]; V ) is endowed with the norm sup t u(t) α for some 0 α 1 4. Let h ν, h 0 be predictable in A M, such that h ν converges in distribution to h 0 in the weak topology of A M Change probability space: suppose that for all t [0, T ], a.s. t 0 h ν(s)ds t 0 h 0(s)ds weakly in H Let uh ν ν and uh 0 0 solve the stochastic (resp. deterministic) controlled equation driven by h ν and the noise multiplied by ν (resp. by h 0 ) Set U ν = uh ν ν uh 0 0 Prove sup t U ν (t) 2 α 0 in probability Fix N > 0 and for t [0, T ] let { G N,ν (t) = sup uh 0 0 (s) 2 α N s t t { ν 0 } { sup s t } Auh ν ν (s) 2 ds N } uh ν ν (s) 2 α N

43 Weak convergence - Shell models Time increments of localized processes 1. The apriori estimates prove: For small ν 0 > 0, sup P(G N,ν (T ) c ) 0 as N. h 0,h ν A M sup 0<ν ν 0 2. Fix N > 0 h 0, h ν A M such that ν 0, h ν h 0 a.s. in the weak topology of L 2 ([0, T ], H 0 ). As ν 0 E [ 1 GN,ν (T ) sup t U ν (t) 2 α] 0. For integers n 1 and k = 0,, 2 n 1, if s [kt 2 n, (k + 1)T 2 n [, set s n = kt 2 n and s n = (k + 1)T 2 n. Lemma Let M, N > 0 and E ξ 2 <. There exists ν 0 > 0 and C := C(M, N, ν 0 ) such that for all n 1, h {h ν, h 0 }, ν [0, ν 0 ], [ I n (h, ν) := E 1 GN,ν (T ) T 0 ] uh ν (s) uν h ( s n) 2 ds C 2 n 2 where u ν h is the sol. to the (stochastic) control eq. driven by h A M.

44 Weak convergence - NS equations apriori estimates Theorem Suppose that ξ V be such that curl ξ L (D), let σ satisfy the reinforced conditions (for the Hilbert-Schmidt from H 0 to H 1,2 and the Radonifyng norms from H 0 to H 1,q with 2 < q < ), let M > 0, h such that T 0 h(s) 2 0 ds M a.s.. Then for some ν 0 > 0 and 0 < ν ν 0 duh ν (t) + [νauν h (t) + B(uν h (t), uν h (t))]dt = νσ(u ν h (t))dw (t) + σ(uh ν (t)) h(t) dt with initial condition ξ has a unique solution in C([0, T ]; H 1,q ). Furthermore, for 2 < q p <, there exists a constant C := C(M, T, q, p) such that for every ν (0, ν 0 ] and h A M ( E sup 0 t T u ν h (t) p H 1,q ) C ( 1 + ξ p H 1,q ).