Digital Object Identifier (DOI) 10.1007/s00205-012-0507-6 Arch. Rational Mech. Anal. 205 (2012) 195–237 Stochastic Three-Dimensional Rotating Navier–Stokes Equations: Averaging, Convergence and Regularity Franco Flandoli & Alex Mahalov Communicated by V. Šverák Abstract We consider stochastic three-dimensional rotating Navier–Stokes equations and prove averaging theorems for stochastic problems in the case of strong rotation. Regularity results are established by bootstrapping from global regularity of the limit stochastic equations and convergence theorems. 1. Introduction The incompressible Navier–Stokes equations (NSEs) are the fundamental par- tial differential equations describing the motion of fluid flow. The mathematical theory of the three-dimensional NSEs has attracted considerable attention because of the open problem about their global well-posedness (see [20, 37]). The well-po- sedness of the three-dimensional NSEs also remains open in the stochastic case. The stochastic three-dimensional NSEs have the form ∂ t U − νU + (U ·∇)U = −∇π + Q ∂ W ∂ t , div U = 0, U | t =0 = U 0 , where U (t , x ) = (U 1 , U 2 , U 3 ), x = (x 1 , x 2 , x 3 ) is the velocity field (a random three-dimensional vector field), π(t , x ) is the pressure (a random scalar field), ν> 0 is the kinematic viscosity, W(t ) is a cylindrical Wiener process, defined on a filtered probability space (, F t , P ), in the Hilbert space H = L 2 s of square integrable solenoidal vector fields (see below) and the operator Q is a non-negative, symmetric, of trace class in H (see [15, 18]). Navier–Stokes equations with other types of stochastic forces are also consid- ered in the literature. In fact, [12] describes important recent advances in studies of random kick-forced three-dimensional NSEs in thin domains.
We consider stochastic three-dimensional rotating Navier–Stokes equations andprove averaging theorems for stochastic problems in the case of strong rotation.Regularity results are established by bootstrapping from global regularity of thelimit stochastic equations and convergence theorems.
1. Introduction
The incompressible Navier–Stokes equations (NSEs) are the fundamental par-tial differential equations describing the motion of fluid flow. The mathematicaltheory of the three-dimensional NSEs has attracted considerable attention becauseof the open problem about their global well-posedness (see [20,37]). The well-po-sedness of the three-dimensional NSEs also remains open in the stochastic case.
The stochastic three-dimensional NSEs have the form
∂tU − ν�U + (U · ∇)U = −∇π + √Q∂W∂t,
div U = 0,
U |t=0 = U 0,
where U (t, x) = (U1,U2,U3), x = (x1, x2, x3) is the velocity field (a randomthree-dimensional vector field), π(t, x) is the pressure (a random scalar field),ν > 0 is the kinematic viscosity, W(t) is a cylindrical Wiener process, definedon a filtered probability space (�, Ft ,P), in the Hilbert space H = L2
s of squareintegrable solenoidal vector fields (see below) and the operator Q is a non-negative,symmetric, of trace class in H (see [15,18]).
Navier–Stokes equations with other types of stochastic forces are also consid-ered in the literature. In fact, [12] describes important recent advances in studies ofrandom kick-forced three-dimensional NSEs in thin domains.
196 Franco Flandoli & Alex Mahalov
Assume U 0 ∈ H and let D be a periodic domain or R3. We know that weak
(also in the probabilistic sense) solutions of the above stochastic three-dimensionalNSEs exist, with properties such as energy inequality, where E denotes expectation:
E
[∫
D|U (t, x)|2 dx
]+ 2νE
[∫ t
0
∫
D|∇U (s, x)|2 dxds
]
� E
[∫
D
∣∣∣U 0(x)
∣∣∣2
dx
]+ Trace (Q) t,
E
[
supt∈[0,T ]
∫
D|U (t, x)|2 dx
]
< ∞.
(There are a few others; see a review and references in [23] and, for instance,[14,22,27].) If U 0 is sufficiently regular, existence and uniqueness of a regularsolution is also known on a local random time interval [0, τU 0), but a priori τU 0
may be very small. (See, for instance, the appendix of [26].)For the previous discussion of the stochastic three-dimensional NSEs, it may
be useful to keep in mind the framework and state of the art of analysis of fluiddynamic problems. In particular, we see that there are three conceptual “regimes”regarding intervals of well-posedness: (i) global well-posedness, namely well-po-sedness over a generic deterministic interval [0, T ], (ii) local well-posedness ona random time interval [0, τU 0) but with τU 0 , which may be (a priori) arbitrarilysmall, and (iii) a third intermediate regime, which is well-posedness on a randomtime interval [0, τU 0), but where τU 0 is larger than T with high probability. A smallexceptional set of trajectories with a possibility of “blow-up” before time T cannotbe excluded; if we could exclude these just for one U 0 and T0, by the results of[26], we would fall into case (i) for all initial conditions and all times.
It may be useful to have a name for concept (iii), although its precise formula-tion depends on the parameters which influence the exceptional set of trajectories.We could call it global well-posedness up to exceptional events.
The question we address here is what happens when additive white noise isintroduced into the three-dimensional NSEs with rotation. The equations we con-sider in this paper, called the stochastic three-dimensional rotating Navier–Stokesequations (RNSEs), read
∂tU − ν�U + (U · ∇)U + 1
εe3 × U = −∇π + √
Q∂W∂t, (1)
div U = 0, (2)
U |t=0 = U 0(x1, x2, x3). (3)
We consider these equations on the torus D = [0, 2π ]3 with periodic boundary con-ditions and zero average. For stress-free boundary conditions in x3 we only needto restrict Fourier series to be even in x3 for U1,U2 and odd in x3 for U3. In (1),
e3 is the vertical axis, e3 × U = (−U2,U1, 0) = JU . Here J =⎛
the rotation matrix corresponding to the Coriolis term. Equations (1)–(3) are fun-damental in meteorology and geophysical fluid dynamics where the Coriolis forceplays an essential role [30,32,39].
For the stochastic three-dimensional RNSEs (1)–(3) with fast rotation, we canprove that we are in the third regime (iii) described above, namely we have globalwell-posedness up to exceptional events. Precisely, assume the initial condition U 0
is sufficiently regular (see below for a definition of function spaces). There is aunique regular solution locally in time. Denote the (potential) explosion time in theconsidered topology by τ ε
U 0 , a random variable. A priori, τ εU 0 can be very small
with very large probability; this is a version of the open problem mentioned at thebeginning. We prove that, given any final time horizon T ,
limε→0
P (τ εU 0 > T
) = 1, (4)
that is, the probability of non-explosion up to T is arbitrarily close to one if theintensity of rotation is sufficiently high. This is a considerable improvement of theresult in the absence of rotation. In general, one can prove only P (
τU 0 > 0) = 1
for the stochastic three-dimensional NSEs.However, it is not as strong as the corresponding deterministic result, where
τU 0 = ∞ when rotation is large enough [3–5,11]. Given the incremental covari-ance and the final time horizon T , the noise produces arbitrarily large excursionsof the solutions on [0, T ] with small but non-zero probability, and blow-up mayoccur in principle. Moreover, over [0,∞), large excursions appear with probabilityone, hence we cannot hope to throw away a small probability event and have globalwell-posedness on [0,∞).
We note that intensity of rotation enters the equations (1)–(3) in a nontrivialway, instead of just noise intensity and smallness of initial conditions. In particular,notice that here the average intensity of the stochastic forcing is arbitrary, comparedto the size of the final time T , so regularity is not just the consequence of an absenceof strong inputs. The energy injected in the system by the noise may be large, theinitial condition may have large energy, and the observational time horizon can belong; regularization is the consequence of a precise mechanism, not just absenceof relevant three-dimensional nonlinear dynamics. To understand reality, this resultis much more interesting that the simple case of small initial conditions and smallnoise; it deals with stochastic fluids having intense activity and energy, and provesthat energy and vorticity remain bounded due to fast rotation. Since rotation (globalor local) is one of the most common features of real fluids, the principle that fastrotation has a smoothing effect is important for applications.
Besides property (4), we also prove an averaging theorem (see [28] for thegeneral theory in finite dimensions). We prove that, as ε → 0, the solution ofequations (1)–(3) converges to the solution of three-dimensional stochastic equa-tions of Navier–Stokes type, the so called stochastic resonant averaged equations,which have better regularity properties than the original stochastic three-dimen-sional NSEs. With regard to averaging in stochastic systems with an infinite numberof degrees of freedom, we recall important recent developments in the work of Cer-rai and Friedlin [9,10] for stochastic reaction-diffusion equations and Kuksin
198 Franco Flandoli & Alex Mahalov
and Piatnitski [35] for stochastic KdV equations, which use infinite-dimensionalHamiltonian structure and action-angle variables. We note that in both cases (reac-tion diffusion equations and KdV) regularity results are known a priori and thefocus of their work is entirely on proving averaging theorems. Here the regularitycomes out as a main result along with the averaging, due to the regularity of thestochastic averaged equations.
For 1/ε >> 1 the stochastic three-dimensional Navier–Stokes equations(1)–(3) have the stiff skew-symmetric nonlocal Poincaré–Coriolis operator restrictedto solenoidal fields:
1
εe3 × U + ∇q, ∇ · U = 0, ε � 1. (5)
The Poincaré–Coriolis operator is a zero order non-local pseudo-differentialoperator, which generates three-dimensional Poincaré waves (see [2–5,19,29]).Since the operator is skew-symmetric, its contribution does not appear explicitlyin the energy inequality obtained from (1)–(3). In fact, energy inequalities of thenon-rotating and rotating three-dimensional NSEs are identical. Nevertheless, itis shown in this paper that rotation has regularizing effects on solutions of thestochastic three-dimensional Navier–Stokes equations.
Now we write stochastic equations (1)–(3) in a form convenient for averaging.Let A be the Stokes operator
A = −P�P, (6)
where P denotes Leray projection onto divergence free vector fields.Clearly, A = curl2 = −� on divergence free vector fields. Applying to the
stochastic three-dimensional RNSEs (1)–(3) and the Leray projection P , we obtainfor U = PU
∂tU + νAU + 1
εSU + B(U,U ) = √
Q∂W∂t, (7)
U |t=0 = U 0, (8)
where
B(U,U ) = P(U · ∇U ) = P(curlU × U ), S = P J P. (9)
The Poincaré–Coriolis operator S = P J P in (7) is skew-symmetric and com-mutes with curl and the Stokes operator A (more details are given in Section 2).
We consider (7) and define the new process u(t)
U (t) = ϒ(−t/ε)u(t), u(t) = ϒ(+t/ε)U (t), (10)
whereϒ(−t/ε) = e−St/ε is a unitary group in the Hilbert space H . The Poincaré–Coriolis operator S is the generator of that unitary group. Equations (7) written forthe process u(t) have the form:
where B is given by Equations (9). The term B(t/ε, u, u) defined in (12) has highlyoscillatory operator coefficients. In general, the operators Q and S do not commuteand, hence, the operators Q and ϒ(t/ε) do not commute. We note that all regular-ity results for u(t) would immediately imply regularity results for U (t) since theoperators ϒ(±t/ε) are isometries in spaces Hα for every α � 0 (see Section 2).
There are two essential ingredients in the definition of the stochastic resonantaveraged three-dimensional RNSEs. The first key ingredient is to introduce theaveraged covariance operator Q. Let Q be a positive trace class operator on Hand let t1 > 0. Define
Q := limε→0
1t1
∫ t1
0er S/εQe−r S/ε dr = lim
T1→∞1T1
∫ T1
0eηS Qe−ηS dη. (14)
These are Cesaro type averages [1]. The above limit exists in the operator norm,and Q is a positive trace class operator. For completeness, details and proofs aregiven in the Appendix. The second key ingredient, which is also done in the deter-ministic theory of three-dimensional RNSEs [3–5,11,42,31], is that the operatorB(t/ε, u, u) splits into two parts
B
(t
ε, u, u
)= B (u, u)+ Bosc
(t
ε, u, u
), (15)
where B (u, u) is independent of t . This is called the resonant operator; the otheroperator, Bosc
( tε, u, u
), the non-resonant one, is highly oscillatory for small ε and
will be averaged out in the limit ε → 0. The three-dimensional nonlinear operatorB (u, u) satisfies (27) which is an estimate similar to the two-dimensional NSEswith Dirichlet boundary conditions. The averaged covariance operator Q and non-linear operator B form the three-dimensional averaged stochastic equation (41), forwhich global regularity can be proven and a complete theory can be developed sim-ilar to the stochastic two-dimensional NSEs. Equation (41) is called the stochasticresonant averaged equation.
For the fully three-dimensional stochastic problem (1)–(3), we prove: (i) averag-ing theorems for the corresponding stochastic problem in the case of strong rotation(0 < ε � ε0); (ii) regularity results for solutions of (1)–(3) by bootstrapping fromglobal regularity of the limit stochastic equation (41) and convergence theorems.
Compared to the deterministic case, however, we do not have infinite timeregularity for sufficiently small ε. There is the possibility of a small set of trajec-tories which lose regularity before any specified (large) time T . This phenomenoncould be related to the global regularity criteria for the three-dimensional stochasticNavier–Stokes equations given in [26]: there, it is proved for suitable non-degener-ate noise that the existence of a single initial datum and time T such that on [0, T ]there is a regular solution (with probability one), which implies global regularityfor all initial data and all time horizons.
The mathematical methods that we develop in this paper are applicable to gen-eral stochastic PDEs (SPDEs) written in the operator form (7). We note that ourmain focus is on the three-dimensional Navier–Stokes equations for which globalregularity results are not known. While the approach of proving averaging theo-rems, convergence and regularity for 0 < ε � ε0 that we present here is general
200 Franco Flandoli & Alex Mahalov
and applies to a wide class of stochastic PDEs, each particular interesting caserequires separate careful consideration. In this paper, we present detailed proofsfor the stochastic three-dimensional rotating Navier–Stokes equations, where S in(7) is the Poincaré–Coriolis rotation operator considered in a periodic domain with-out any restrictions on domain parameters. The case of flows in R
3 with decayinginitial condition and random forces acting in a compact region of space can alsobe considered in our framework. The periodic case studied in this paper is moredifficult due to resonances in the stochastic dynamics and the lack of dispersion.
2. Definitions and Notations
The Poincaré–Coriolis operator itself has a long history going back to Poin-caré [40] and Sobolev [41]. We refer to Arnold and Khesin [2] for a historicalaccount. The Poincaré propagator is the unitary group solution ϒ(−t/ε)�(0) =�(t) (ϒ(0) = I d is the identity) to the linear Poincaré–Coriolis problem (deter-ministic):
∂t�+ 1/εJ� = −∇ p, ∇ ·� = 0, (16)
or, equivalently,
∂t�+ 1/εS� = 0, S = P J P, (17)
where P is the Leray projection on divergence free vector fields. In R3, the Po-
incaré–Coriolis rotation operator S = P J P is a zero order pseudo-differentialoperator with the skew-symmetric matrix symbol related to the Riesz operators asdescribed below. Equations (16)–(17) have already been considered by Poincaré[40] and Sobolev [41] (bounded domains). Recent studies of the Poincaré–Cori-olis operator and the corresponding evolutionary problem (17) can be found in[19,29,30].
In this paper we study stochastic three-dimensional RNSEs (1)–(3) and pres-ent detailed considerations in the periodic case. Without loss of generality, weconsider the periodic cube [0, 2π ] × [0, 2π ] × [0, 2π ]. Let Z
3∗ be the set of alltriples k = (k1, k2, k3) with integers k j , j = 1, 2, 3, such that k �= 0. All numbers± |k| , k ∈ Z
3∗, are eigenvalues of the curl-operator, and |k|2 = k21 + k2
2 + k23 are
eigenvalues of the Stokes operator A (recall that A = curl2 = −� on divergencefree vector fields). For each k ∈ Z
3∗, there are two eigenvectors hk,σ (x) (of curlor Stokes operators) of the form ϕk,σ eik·x , with σ = ±1 and ϕk,σ ∈ C
3 such thatϕk,σ · k = 0; k · x = k1x1 + k2x2 + k3x3. We shall agree that σ = ±1 corre-sponds to ± |k| in the eigenvectors of the curl-operator which are normalized tohave unit norm. The vectors ϕk,σ are not unique: any basis
{ϕk,+1, ϕk,−1
}of the
two-dimensional space orthogonal to k can be chosen, with the right orientation tohave
k∈Z3∗,σ=±1 is a complete orthonormal system in H = L2s
and will be called the Fourier-curl basis. For every u ∈ L2s we have
u =∑
k∈Z3∗,σ=±1
uk,σ ϕk,σ eik·x . (19)
We introduce the spaces of solenoidal vector fields Hα (H = H0 = L2s ) with
zero average and with the norm defined by
||u||2Hα =∑
k∈Z3∗,σ=±1
|k|2α|uk,σ |2, (20)
where |k|2 = k21 + k2
2 + k23. We have Hα = Dom(Aα/2), where A is the Stokes
operator with periodic boundary conditions and Dom is the domain of the operatorAα/2.
Following [3–6,29], we make an observation that the operator S = P J P isrelated to the Riesz operators and the curl operator. Let P(k) be a 3 × 3 matrix ofthe Leray projection operator on divergence free vector fields in Fourier represen-
tation; J =⎛
⎝0 −1 01 0 00 0 0
⎞
⎠ is the rotation matrix corresponding to the Coriolis term
in (1). One can easily show by direct matrix multiplication that
S(k) ≡ P(k)J P(k) =(
k3
|k|)
R(k), (21)
where
R(k) =⎛
⎜⎝
0 − k3|k|k2|k|
k3|k| 0 − k1|k|− k2|k|
k1|k| 0
⎞
⎟⎠ . (22)
We note that R(k) is a 3 × 3 skew-symmetric matrix. Thus, S(k) is related
to the symbol of the curl operator, which is i
⎛
⎝0 −k3 k2
k3 0 −k1−k2 k1 0
⎞
⎠. In particular, it
implies that the operator S commutes with the operator curl and the Stokes opera-tor A = curl2 = −�. The matrix R(k) restricted to the divergence free subspace(vectors orthogonal to k) has eigenvalues ±i .
For the operator etε
S , we have in the Fourier-curl basis
etε
S(ϕk,σ eik·x) = ei t
εσ
k3|k| ϕk,σ eik·x . (23)
It follows from (20) and (23) that the operatorsϒ(±t/ε) = e± tε
S are isometriesin Hilbert spaces Hα for every α:
||e± tε
Su||Hα = ||u||Hα . (24)
202 Franco Flandoli & Alex Mahalov
3. Averaged Operators and Resonances
For the stochastic three-dimensional Rotating Navier–Stokes equations (1)–(3),the operator B
( tε, u, u
)was defined above in (12). Now we introduce the corre-
sponding bilinear operator B( tε, u, v
):
B
(t
ε, u, v
)= −e
tε
S P(
e− tε
Su × e− tε
Scurlv).
Recall that S commutes with the Stokes operator A and the curl-operator. Moreover,S commutes with the Leray projection P .
Let us develop a representation in the Fourier-curl basis described in Section 2.We expand u and v in this basis as in (19). The operator B
( tε, u, v
)is equal to
−∑
k,m∈Z3∗,σ1,σ2=±1
uk,σ1vm,σ2 etε
S P(
e− tε
Sϕk,σ1 eik·x × e− tε
Scurl(ϕm,σ2 eim·x))
=−∑
k,m∈Z3∗,σ1,σ2=±1
σ2 |m| uk,σ1vm,σ2 etε
S P(e− t
εSϕk,σ1 eik·x × e− t
εSϕm,σ2 eim·x)
= −∑
k,m∈Z3∗,σ1,σ2=±1
σ2 |m| e−i t
ε
(σ1
k3|k| +σ2m3|m|
)
uk,σ1vm,σ2 etε
S
P(ϕk,σ1 × ϕm,σ2 ei(k·x+m·x)) ,
where uk,σ and vk,σ are the Fourier-curl components of u and v, respectively, andwhere we have used (18) and (23) at the second and the third steps. We have
P(ϕk,σ1 × ϕm,σ2 ei(k·x+m·x)) = Pk+m
(ϕk,σ1 × ϕm,σ2
)ei(k·x+m·x),
where Pk+m(ϕk,σ1 × ϕm,σ2
)denotes the projection of ϕk,σ1 × ϕm,σ2 on the plane
orthogonal to k + m. Then P(ϕk,σ1 × ϕm,σ2 ei(k·x+m·x)) is an eigenfunction of the
curl-operator. Denote by σk,m,σ1,σ2 the sign of its eigenvalue (of modulus |k + m|).We have
etε
S(
Pk+m(ϕk,σ1 × ϕm,σ2
)ei(k·x+m·x))
= ei tε
(σk,m,σ1,σ2
(k+m)3|k+m|)
Pk+m(ϕk,σ1 × ϕm,σ2
)ei(k·x+m·x),
and therefore B( tε, u, v
)is equal to
−∑
k,m∈Z3∗,σ1,σ2=±1
σ2 |m| ei tε
D(k,m,σ1,σ2)uk,σ1vm,σ2 Pk+m(ϕk,σ1 ×ϕm,σ2
)ei(k·x+m·x),
where we denote −σ1k3|k| − σ2
m3|m| + σk,m,σ1,σ2(k+m)3|k+m| by D (k,m, σ1, σ2). Denote
by � the set of quadruples (k,m, σ1, σ2) ∈ Z3∗ × Z
3∗ × {+1,−1}2, such thatD (k,m, σ1, σ2) = 0. Then ei t/εD(k,m,σ1,σ2) = 1 on the resonant set �.
The nonlinear interactions in the resonant operator B are restricted to the reso-nant set
D (k,m, σ1, σ2) = −σ1k3
|k| − σ2m3
|m| + σk,m,σ1,σ2
(k + m)3|k + m| = 0. (26)
The bilinear resonant operator B satisfies the important inequality∣∣∣(B (u, u) , Au
)L2
∣∣∣ � C ‖u‖2
H1‖u‖H2 , u ∈ H2, (27)
where u(x1, x2, x3) are three-dimensional vector fields. Here A = curl2 = −�is the Stokes operator. The inequality (27) follows from Lemmas on restrictedconvolutions in [3] and [4]. The significance of (27) is that the operator B actingon three-dimensional vector fields has fully three-dimensional nonlinear interac-tions restricted on a resonant manifold in Fourier space. This restriction leads to animprovement in the estimates for the three-dimensional operator B in comparisonwith the operator B.
Moreover, both B and Bosc satisfy
(B (u, u) , u)L2= 0 (28)
and Kato inequalities [34], true for s � 2,
‖B (u, v)‖Hs� C ‖u‖Hs
‖v‖Hs+1 (29)
| (B (u, v) , v)Hs| � C ‖u‖Hs
‖v‖2Hs
(30)
| (B (u, v) , v)H2| � C ‖u‖H3
‖v‖2H2
(31)
| (B (u, v) , u)Hs| � C ‖u‖2
Hs‖v‖Hs+1 . (32)
(See also [38] and references therein.)
204 Franco Flandoli & Alex Mahalov
We note that the operator B( tε, u, v
)also satisfies inequalities (29)–(32). For
example, using the fact that the operators e± tε
S are isometries in Hs and (29), weobtain
∥∥∥∥B
(t
ε, u, v
)∥∥∥∥
Hs
=∥∥∥−e
tε
S B(
e− tε
Su, e− tε
Sv)∥∥∥
Hs
=∥∥∥B
(e− t
εSu, e− t
εSv
)∥∥∥
Hs� C
∥∥∥e− t
εSu
∥∥∥
Hs
∥∥∥e− t
εSv
∥∥∥
Hs+1
= C ‖u‖Hs‖v‖Hs+1 , (33)
where C is a constant independent of ε.Now we show how resonances appear in the averaged covariance operator Q
defined by (14). The operator Q is a trace class non-negative, self-adjoint operatorin H , and we have
⟨Qg, h
⟩H := lim
ε→0
1
t1
∫ t1
0
⟨Qe− r
εSg, e− r
εSh
⟩
Hdr (34)
for all g, h ∈ H , independent of t1 > 0. The symbols of the operators S and curlare related by (21). Clearly, Q = Q if the operators Q and S commute (equiva-lently, Q and curl commute). Let us show that the averaged operator Q includesnew resonant terms in the non-commutative case Qcurl �= curl Q.
Using expansions in (34) in bases of the operators curl and Q, we find thatnon-trivial contributions to Q in (34) come from resonances for the averaging
limε→0
1
t1
∫ t1
0e−i(σ1
k3|k| +σ2m3|m| )r/εdr. (35)
Here σ1, σ2 = ±1 as described above. The resonant condition in (35) is
σ1k3
|k| + σ2m3
|m| = 0. (36)
The condition (36) becomesk2
3|kh |2+k2
3= m2
3|mh |2+m2
3, where k3,m3 are vertical wave
numbers and |kh |2 = k21 +k2
2, |mh |2 = m21+m2
2. Clearly, resonances always includequadruplets k3 = ±m3 and k1 = ±m1, k2 = ±m2. Other non-trivial resonances
come from solutions of the generalized diophantine equationk2
3k2
1+k22
= m23
m21+m2
2.
These equations possess non-trivial integer solutions (k1, k2, k3), (m1,m2,m3)
which are more complex than k3 = ±m3 and k1 = ±m1, k2 = ±m2. We notethat solutions of the resonance equation lie on cones in Fourier space. If tripletsk and m satisfy the resonant condition, then αm and βm are also solutions for arbi-trary integers α and β. The purpose of these remarks is to show that contributionsto the averaged covariance operator come from non-trivial resonance relations andits form cannot be guessed a priori without careful analysis.
The averaged covariance operator Q and nonlinear operator B form the stochas-tic resonant averaged three-dimensional RNSEs (41) that are globally well-posed.
4. Averaging of Hölder Continuous Random Processes
In this section we present a Lemma on averaging of Hölder continuous randomprocesses. Together with the averaged covariance operator Q and the decompo-sition (25), it will be used in subsequent sections in our proof of averaging andconvergence theorems, and in obtaining regularity results for the stochastic three-dimensional RNSEs (1)–(3).
Let (�, F,P) be a probability space with expectation E . Let f = f (t, ω), t ∈[0, T ], ω ∈ �, be a stochastic process. Assume that supt∈[0,T ]| f (t)| is measurable.
Lemma 1. Let p � 1, λ �= 0 be given. Assume that
E
[
supt∈[0,T ]
| f (t)|p
]
� C0
E[| f (t)− f (s)|p] � C0 |t − s|α for all t, s ∈ [0, T ] .
Then (for a new constant C1 = C1 (C0, T, p))
E
[
supt∈[0,T ]
∣∣∣∣
∫ t
0ei s
ελ f (s) ds
∣∣∣∣
p]
� C1
(∣∣∣ε
λ
∣∣∣α +
∣∣∣ε
λ
∣∣∣).
Proof. When, in the following computations, the process f appears on an intervalexceeding [0, T ], we understand it to be equal to zero. We have
∫ t
0ei s
ελ f (s) ds = 1
2
∫ t
0ei s
ελ f (s) ds − 1
2
∫ t
0ei(s+ επ
λ )ε
λ f (s) ds
= 1
2
∫ t
0ei s
ελ f (s) ds − 1
2
∫ t+ επλ
επλ
ei sελ f
(s − επ
λ
)ds
= 1
2
∫ t
επλ
ei sελ[
f (s)− f(
s − επ
λ
)]ds
+1
2
∫ επλ
0ei s
ελ f (s) ds − 1
2
∫ t+ επλ
tei s
ελ f
(s − επ
λ
)ds.
Hence∣∣∣∣
∫ t
0ei s
ελ f (s) ds
∣∣∣∣ � 1
2
∫ t
0
∣∣∣ f (s)− f
(s − επ
λ
)∣∣∣ ds
+1
2
∫ επλ
0| f (s)| ds + 1
2
∫ t
t− επλ
| f (s)| ds,
and thus (p = 1)
supt∈[0,T ]
∣∣∣∣
∫ t
0ei s
ελ f (s) ds
∣∣∣∣ � 1
2
∫ T
0
∣∣∣ f (s)− f
(s − επ
λ
)∣∣∣ ds
+1
2
∫ επλ
0| f (s)| ds + 1
2sup
t∈[0,T ]
∫ t
t− επλ
| f (s)| ds.
206 Franco Flandoli & Alex Mahalov
Similarly, for p � 1 we obtain
E
[
supt∈[0,T ]
∣∣∣∣
∫ t
0ei s
ελ f (s) ds
∣∣∣∣
p]
� C∫ T
0E[∣∣∣ f (s)− f
(s − επ
λ
)∣∣∣
p]ds
+C∫ επ
λ
0E[| f (s)|p] ds + C E
[
supt∈[0,T ]
∫ t
t− επλ
| f (s)|p ds
]
.
Under the assumptions of the lemma we have
E
[
supt∈[0,T ]
∣∣∣∣
∫ t
0ei s
ελ f (s) ds
∣∣∣∣
p]
� C ′∣∣∣επ
λ
∣∣∣α + C ′
∣∣∣επ
λ
∣∣∣ + C ′
∣∣∣επ
λ
∣∣∣ .
The proof is complete. �
5. Equations and Assumptions
Let (�, F, (Ft )t�0,P) be a standard filtered probability space and (W(t))t�0be a cylindrical Brownian motion in H (details about cylindrical Brownian motioncan be found in [15] and Appendix 9.2 below). Let (�, F, (Ft )t�0, P, (W(t))t�0)
be another copy of it. Let Q be a trace class non-negative, self-adjoint operator inH . Let Mε(t) be the Gaussian martingale defined as
Mε (t) =∫ t
0e
rε
S√
QdW (r) . (37)
The martingale Mε (t) is associated with the stochastic termϒ(t/ε)√
Q ∂W∂t in (11).
The laws of the Gaussian martingales Mε (t) are tight in proper topologies so wecan prove that they converge weakly (as ε → 0) to the law of a Gaussian martingaleM (t). We may identify the limit process M (t) as a Brownian motion in H withcovariance Q defined by (14) (more details are given in Section 8).
We recall the definition of the averaged operator Q using Cesaro type averag-ing in (14) and the decomposition of B( t
ε, u, u) given in (25) into the resonant and
non-resonant parts.Consider the following four stochastic equations.Equation 1:
du Rε +
[νAu R
ε + B(
u Rε , u R
ε
)+ χR
(∥∥∥u R
ε
∥∥∥
2
H3
)Bosc
(t
ε, u Rε , u R
ε
)]dt =dMε,
(38)
where χR : [0,∞) → [0, 1] is a smooth function, equal to 1 on [0, R] and to zeroon [R + 1,∞). Here R > 0 is a large parameter.
all with the same smooth initial condition u0 = U 0, with its regularity to be pre-cised below. We note that equations (38)–(40) have the same noise. Clearly, for
solutions of (38) and (39) we have u Rε (t) = uε(t) up to time for which
∥∥u R
ε (t)∥∥2
H3reaches R.
Several remarks concerning stochastic equations (38)–(41) are in order. Equa-tion (39) is identical to (11). The latter is equivalent to the stochastic three-dimen-sional RNSEs (7) via the transformation (10). We have the existence and uniquenessof a strong local solution uε(t) for equation (39). Equation (41) is globally well-posed (w(t) globally defined by Lemma 2 below). The key point is to prove thatuε(t) is close tow(t) in the H3-topology. The absence of blow up forw(t)will implyabsence of blow up for uε(t). Closedness in law is trickier to handle so we intro-duce an auxiliary equation (40), which provides the bridge between uε(t) andw(t).Equations (39) and (40) have the same noise. We prove thatwε(t) exists globally inH3, with bounds independent of ε, and that uε(t) is close to wε(t) in probability inthe topology of continuous functions in H3. Moreover, wε(t) converges in law tow(t), globally in time. However, and this is the point where almost globality comesin, we can prove only a weak form of the statement that uε(t) and wε(t) are close.Given a time interval [0, T ], and the initial condition, for small ε the probabilitythat uε(t) and wε(t) are close on [0, T ] is large but we cannot prove it is equal toone. In Lemma 6 we prove that the random variable supt∈[0,T ] ‖uε (t)− wε (t)‖H2
converges to zero in probability as ε → 0. Finally, for technical reasons, instead ofworking on random local intervals everywhere, we prefer to work globally in timebut for the solution u R
ε (t) of a problem with a cut-off given by equation (38).Let η ∈ (0, 1] be given. Assume that
√Q maps H into Dom
(A(3+η)/2). Then
the operator A(3+η)/2 Q A(3+η)/2 is well defined from Dom(
A(3+η)/2) to H . Weassume that it extends to a bounded trace class operator in H :
T r(
A(3+η)/2 Q A(3+η)/2) < ∞. (42)
One can easily verify that T r(
A(3+η)/2 Q A(3+η)/2)= T r(Q A3+η). In some esti-
mates below we write the condition in the form T r(Q A3+η) < ∞,which is equiv-
alent to (42). The meaning of the condition (42) is that the operator A(3+η)/2√Q canbe extended to a Hilbert–Schmidt operator in H . We note that the above assump-tions on the operator Q imply similar properties for the operator Q. In particular,the condition (42) is satisfied for Q.
Let us comment on the consequences of assumption (42) for traces on theSPDEs. Let us consider equation (41); the arguments for the other equations aresimilar. First, as we have just remarked, assumption (42) implies the similar condi-tion for Q. Let us argue formally for the sake of simplicity: to be rigorous one should
208 Franco Flandoli & Alex Mahalov
perform the computations on Galerkin approximations. Premultiply the equation(41) by A(3+η)/2:
dA(3+η)/2w +[νAA(3+η)/2w + A(3+η)/2 B (w,w)
]dt = A(3+η)/2
√QdW (t) ,
where A(3+η)/2√Q is a Hilbert–Schmidt operator, namely A(3+η)/2 Q A(3+η)/2extends to a trace class operator in H . Then, by Itô formula
1
2d∣∣∣A(3+η)/2w
∣∣∣2
H+ ν
(A(3+η)/2w, AA(3+η)/2w
)
Hdt
+(
A(3+η)/2w, A(3+η)/2 B (w,w))
Hdt
=(
A(3+η)/2w, A(3+η)/2√
QdW (t)
)
Hdt + 1
2T r
(A(3+η)/2 Q A(3+η)/2) ,
namely
1
2d ‖w‖2
H3+η + ν ‖w‖2H4+η dt + (
B (w,w) ,w)
H3+η dt
=(
A(3+η)/2w, A(3+η)/2√
QdW (t)
)
Hdt + 1
2T r
(A(3+η)/2 Q A(3+η)/2) .
We have T r(
A(3+η)/2 Q A(3+η)/2) < ∞ from our assumptions.Now we give the definition of a strong local solution for Equation 2 given by
(39):
duε +[νAuε + B (uε, uε)+ Bosc
(t
ε, uε, uε
)]dt = dMε. (43)
This equation is precisely equation (11) with the operator B written as a sum of theresonant B and non-resonant Bosc parts.
Recall that a standard filtered probability space (�, F, (Ft )t�0,P) is given,as well as a cylindrical Brownian motion (W(t))t�0 in H , and a trace class non-negative, self-adjoint operator Q in H ; and the martingale Mε (t) is defined by(37).
We say that a random variable T : � → [0,∞) is an accessible stopping timeif there is an increasing sequence Tn of (Ft )t�0-stopping times such that Tn < Tand limn→∞ Tn = T,P-almost surely.
A function f (t, ω)with values in a Hilbert space Y , defined for P-almost everyω ∈ � and all t ∈ [0, T (ω)), where T (ω) is an accessible stopping time, will becalled an (Ft )t�0-adapted process in Y if, given a sequence Tn as above, the processf (t ∧ Tn, ω), is (Ft )t�0-adapted.
Definition 1. A strong local solution uε (t, ω), continuous in H3, of equation (39)is an H3-valued function defined for P-almost every ω ∈ � and all t ∈ [0, T (ω)),such that:
i) T (ω) is a strictly positive accessible stopping time;ii) uε is an (Ft )t�0-adapted process in H3;
iii) for P-almost every ω∈�, on the interval [0, T (ω)) the function t �→uε (t, ω)is continuous in H3 and satisfies
uε (t)+∫ t
0
[νAuε (s)+ B (uε (s) , uε (s))+ Bosc
( s
ε, uε (s) , uε (s)
)]ds
= uε (0)+ Mε (t)
as an identity in H1.
Additional information on SPDEs can be found in [8,16,17,21,24,25].Our choice of the space H3 in the above definition is natural for considering
solutions of the stochastic three-dimensional NSEs in the inviscid limit ν → 0(under appropriate scalings). We recall that initial data for the three-dimensionalEuler equations is required to be at least in H3 for local regularity [34]. The inviscidlimit of the stochastic three-dimensional RNSEs will be considered in our futurework.
6. Main Estimates
The next lemma treats the averaged resonant stochastic equation (41). The con-stants below will depend on T and ‖u0‖H3+η .
Lemma 2. Let T > 0 be fixed and arbitrarily large. Under the assumption
T r(
A(3+η)/2 Q A(3+η)/2) < ∞, u0 ∈ H3+η,
equation (41) has a unique strong solutionw, with paths of class C([0, T ] ; H3+η
).
Moreover, for every δ0 > 0 there is Nδ0 > 0 such that
P(
supt∈[0,T ]
‖w (t)‖H3+η � Nδ0
)
� 1 − δ0.
Proof. Existence and uniqueness is classical, due to inequality (27) which is sim-ilar to the corresponding inequality in the theory of two-dimensional NSEs withDirichlet boundary conditions. We do not write the details, but only the main prob-abilistic estimates that lead to the global well-posedness and regularity result forequation (41).
Step 1. We recall that the operator B satisfies (28). Existence and uniquenessof strong solutions, satisfying the energy inequality
|w (t)|2L2+ 2ν
∫ t
0‖w (s)‖2
H1ds �
∣∣∣u0
∣∣∣2
L2+
∫ t
02
(w (s) ,
√QdW (s)
)
L2
+T r(Q) · t, (44)
(in fact also the energy equality formally obtained from (41) and using Itô for-mula for the stochastic term) is classical in the space of processes with paths inC ([0, T ] ; L2) ∩ L2 (0, T ; H1), due to inequality (27), which is the same as a
210 Franco Flandoli & Alex Mahalov
known inequality in the two-dimensional case for NSEs with Dirichlet boundaryconditions. The proof can be performed in several ways: pathwise, by compactness(in that case one has to show pathwise uniqueness and apply Yamada–Watanabetheorem), by monotonicity under truncation, by contraction principle. The estimate(needed below)
E
[
supt∈[0,T ]
|w (t)|2L2+
∫ T
0‖w (t)‖2
H1dt
]
< ∞, (45)
is usually also provided by those proofs, but let us deduce it from the energy inequal-ity, for sake of completeness (this way we show one of the main steps of the Galerkinapproach to existence). So, let w(t) be a continuous adapted solution in H , withpaths in L2 (0, T ; H1), satisfying the energy inequality above for all t ∈ [0, T ],
with probability one. Let τn be defined as τn = inf{
t ∈ [0, T ] : |w (t)|2L2> n
}if
this set is non-empty, equal to T otherwise. We have
|w (t ∧ τn)|2L2+ 2ν
∫ t
0‖w (s)‖2
H11s�τn
ds
�∣∣∣u0
∣∣∣2
L2+
∫ t
02
(w (s) 1s�τn
,
√QdW (s)
)
L2
+ T r(Q) · (t ∧ τn) .
Since w (s) 1s�τnis a bounded process in H we may apply Doob’s martingale
inequality (see [33]). Thus, for all r ∈ [0, T ], we have
so the second part of (45) is also proved. In particular, we get
E
[∫ T
0‖w (t)‖2
H1dt
]�
∣∣u0
∣∣2L2
+ T · T r Q
2ν. (46)
Step 2. Again, by Itô’s formula, but for ‖w (t)‖2H1
, and inequality (27) we obtainfrom (41)
d ‖w (t)‖2H1
+ 2ν ‖w‖2H2
dt �(
C2
ν‖w‖4
H1+ ν ‖w‖2
H2
)dt
+2
(w,
√QdW (t)
)
H1
dt + T r(
A1/2 Q A1/2),
and thus
d ‖w (t)‖2H1
+ ν ‖w‖2H2
dt
� C∗ν ‖w‖4
H1dt + T r
(A1/2 Q A1/2
)+ 2
(w,
√QdW (t)
)
H1
dt.
Then
d
[e− ∫ t
0 C∗ν ‖w(s)‖2
H1ds ‖w (t)‖2
H1
]+ e− ∫ t
0 C∗ν ‖w(s)‖2
H1dsν ‖w‖2
H2dt
� e− ∫ t0 C∗
ν ‖w(s)‖2H1
ds T r(
A1/2 Q A1/2)
dt
+2e− ∫ t0 C∗
ν ‖w(s)‖2H1
ds(w,
√QdW (t)
)
H1
, (47)
hence
E
[∫ T
0e− ∫ t
0 C∗ν ‖w(s)‖2
H1dsν ‖w (t)‖2
H2dt
]
�∥∥∥u0
∥∥∥
2
H1+ T r
(A1/2 Q A1/2
)· T < ∞. (48)
We have used the fact that
E
[∫ T
0e− ∫ t
0 C∗ν ‖w(s)‖2
H1ds ‖w (t)‖2
H1dt
]� E
[∫ T
0‖w (t)‖2
H1dt
]< ∞
by (46). This kind of argument will be used several times below and will not berepeated. Having proved (48), we conclude from (47) that the stochastic integral
Zt = 2∫ t
0e− ∫ t
0 C∗ν ‖w(s)‖2
H1ds
(w,
√QdW (t)
)
H1
(49)
is a square integrable martingale. From (47) we have
e− ∫ t0 C∗
ν ‖w(s)‖2H1
ds ‖w (t)‖2H1
�∥∥∥u0
∥∥∥
2
H1+ T r
(Q A
) ∫ t
0e− ∫ t
0 C∗ν ‖w(s)‖2
H1dsdt + |Zt |. (50)
212 Franco Flandoli & Alex Mahalov
Using |Zt | � |Zt |2 + 1 and Doob’s martingale inequality
E
(
supt∈[0,T ]
|Zt |2)
� 4E(|ZT |2
)� 16T r
(Q A
) · T, (51)
we obtain from (50)
E
(
supt∈[0,T ]
e− ∫ t0 C∗
ν ‖w(s)‖2H1
ds ‖w (t)‖2H1
)
� C(∥∥∥u0
∥∥∥
2
H1, T r
(Q A
), T ) < ∞.
Step 3. To prove the previous estimates in Step 2 we have used inequality (27).Now, with the Kato inequality (32), which is similar to (27) but holds true for alls � 2, we can prove the inequality
d[e− ∫ t
0 C∗ν ‖w(s)‖2
Hαds ‖w (t)‖2
Hα
]+ e− ∫ t
0 C∗ν ‖w(s)‖2
Hαdsν ‖w‖2
Hα+1dt
� e− ∫ t0 C∗
ν ‖w(s)‖2Hα
ds T r(Q Aα
)dt
+2e− ∫ t0 C∗
ν ‖w(s)‖2Hα
ds(w,
√QdW (t)
)
Hα
for all α � 3 + η, and thus the bound
E[e− ∫ t
0 C∗ν ‖w(s)‖2
Hαds ‖w (t)‖2
Hα
]
+E
[∫ t
0e− ∫ r
0 C∗ν ‖w(s)‖2
Hαdsν ‖w (r)‖2
Hα+1dr
]
�∥∥∥u0
∥∥∥
2
Hα+ T r
(Q Aα
) · T .
Moreover, using the Doob martingale inequality, we also have
E
[
supt∈[0,T ]
e− ∫ t0 C∗
ν ‖w(s)‖2Hα
ds ‖w (t)‖2Hα
]
�∥∥∥u0
∥∥∥
2
H1+ CT r
(Q As)
(1 + E
[∫ T
0e−2
∫ t0 C∗
ν ‖w(s)‖2Hα
ds ‖w (t)‖2Hα dt
])
�∥∥∥u0
∥∥∥
2
H1+ CT r
(Q As)
(1 + E
[∫ T
0e− ∫ t
0 C∗ν ‖w(s)‖2
Hαds ‖w (t)‖2
Hα dt
])
�∥∥∥u0
∥∥∥
2
H1+ CT r
(Q As)
(1 +
∥∥∥u0
∥∥∥
2
Hα+ T r
(Q Aα
) · T
),
where we have used at the last step the previous inequality.Step 4. By the Chebyshev inequality and (46)
hence, given δ1 > 0, there exists R(1)δ1> 0 such that
P(∫ T
0‖w (t)‖2
H1dt � R(1)δ1
)� 1 − δ1/2.
We use this bound in the estimate (48): by the Chebyshev inequality it implies thatthere exists R(2)δ1
> 0 such that
P(∫ T
0e− ∫ r
0 C∗ν ‖w(s)‖2
H1dsν ‖w (r)‖2
H2dr � R(2)δ1
)� 1 − δ1/2,
but we know that
e− ∫ r0 C∗
ν ‖w(s)‖2H1
ds � e−C∗ν R(1)δ1
for all r ∈ [0, T ], with probability larger than 1 − δ1/2, so
P(∫ T
0‖w (r)‖2
H2dr � ν−1eC∗
ν R(1)δ1 R(2)δ1
)� 1 − δ1. (52)
Let us explain the argument we have used here, since it appears several timesbelow. Introduce the events
A =(∫ T
0e− ∫ r
0 C∗ν ‖w(s)‖2
H1dsν ‖w (r)‖2
H2dr � R(2)δ1
)
B =(
e− ∫ r0 C∗
ν ‖w(s)‖2H1
ds � e−C∗ν R(1)δ1 for all r ∈ [0, T ]
)
C =(∫ T
0‖w (r)‖2
H2dr � ν−1eC∗
ν R(1)δ1 R(2)δ1
).
We have
A ∩ B ⊂ C
because∫ T
0‖w (r)‖2
H2dr = ν−1eC∗
ν R(1)δ1
∫ T
0νe−C∗
ν R(1)δ1 ‖w (r)‖2H2
dr
� ν−1eC∗ν R(1)δ1
∫ T
0νe− ∫ r
0 C∗ν ‖w(s)‖2
H1ds ‖w (r)‖2
H2dr
� ν−1eC∗ν R(1)δ1 R(2)δ1
.
From the general inequality for probabilities P (C) � P (A)+P (B)−1 we deduce
P (C) � 1 − δ1/2 + 1 − δ1/2 − 1 = 1 − δ1.
The proof of the inequality (52) is complete.This argument can be repeated twice to prove the following fact: given δ > 0,
there exists R(3)δ > 0 such that
214 Franco Flandoli & Alex Mahalov
P(∫ T
0‖w (r)‖2
H3+η dr � R(3)δ
)� 1 − δ/2.
Now we use the last inequality of step 3. From it and the Chebyshev inequality,given δ > 0 there exists R(4)δ > 0 such that
P(
supt∈[0,T ]
e− ∫ t
0 C∗ν ‖w(s)‖2
H3+ηds ‖w (t)‖2H3+η � R(3)δ
)
� 1 − δ/2.
But we know that
e− ∫ t
0 C∗ν ‖w(s)‖2
H3+ηds � e−C∗ν R(3)δ
for all r ∈ [0, T ], with probability larger than 1 − δ/2, so
P(
supt∈[0,T ]
‖w (t)‖2H3+η � R(3)δ eC∗
ν R(3)δ
)
� 1 − δ.
The proof is complete. � Now consider equation (40).
Lemma 3. Let T > 0 be fixed, arbitrarily large. Under assumptions (42), equa-tion (40) has a unique strong solution wε(t), with paths of class C
([0, T ] ; H3+η
).
Moreover, for every δ1 > 0 there is Nδ1 > 0 (independent of ε) such that
P(
supt∈[0,T ]
‖wε (t)‖H3+η � Nδ1
)
� 1 − δ1
for every ε > 0.
Proof. The proof is the same as the one for the previous lemma. The only differenceis that the operator Q is now replaced by
Q(ε, t) = eSt/εQe−St/ε. (53)
The operators S and A commute, and eSt/ε is unitary, hence T r (Q (ε, t) Aα) =T r (Q Aα) for all α > 0. For this reason, all the estimates of the Itô corrections arebased on assumption (42) and are independent of ε > 0. Because of this fact, theprevious proof can be repeated for any ε > 0, with constants independent of ε. Theproof is complete. �
Next, consider equation (38).
Lemma 4. Let T > 0 be fixed, arbitrarily large. Under assumptions (42), equa-tion (38) has a unique strong solution u R
ε (t), with paths of class C([0, T ] ; H3+η
).
Moreover, for every R > 0 and δ2 > 0 there is NR,δ2 > 0 (independent of ε) suchthat
Proof. The proof is almost the same as the one of the previous lemmas. The inde-pendence of ε is exactly the same. The role of R is the following. Consider thenonlinear operator
BoscR
(t
ε, u
):= χR
(‖u‖2
H3
)Bosc
(t
ε, u, u
).
It satisfies (29)–(32) because∣∣∣∣∣
(Bosc
R
(t
ε, u
), u
)
Hs
∣∣∣∣∣�
∣∣∣∣∣
(Bosc
(t
ε, u, u
), u
)
Hs
∣∣∣∣∣.
It satisfies also (28). Moreover, we have∣∣∣∣∣
(Bosc
R
(t
ε, u
), u
)
H1
∣∣∣∣∣� χR
(‖u‖2
H3
)∣∣∣∣∣
(Bosc
(t
ε, u, u
), u
)
H1
∣∣∣∣∣
� CχR
(‖u‖2
H3
)‖u‖H3
‖u‖H1‖u‖H2
� C (R + 1) ‖u‖H1‖u‖H2 ,
which is a simplified form of inequality (27). For these reasons, one can repeat forequation (38) the same computations done above for the other equations. The proofis complete. �
Denote by C∞s the space of all smooth divergence free zero average periodic
vector fields.
Lemma 5. There exist q > 0, constants C and CR, and, for every h ∈ C∞s , con-
stants C (h) and CR (h), all independent of ε > 0, such that
E
[
supt∈[0,T ]
∣∣∣u Rε (t)
∣∣∣4
L2
]
� CR, E
[
supt∈[0,T ]
|wε (t)|4L2
]
� C
E
[∣∣∣∣(
u Rε (t) , h
)
L2−
(u Rε (s) , h
)
L2
∣∣∣∣
4]
� CR (h) |t − s|q for all t, s ∈ [0, T ]
E[∣∣(wε (t) , h)L2
− (wε (s) , h)L2
∣∣4]
� C (h) |t − s|q for all t, s ∈ [0, T ] .
Proof. Step 1. We prove for wε (t). Similarly to (44), wε (t) satisfies
|wε (t)|2L2�
∣∣∣u0
∣∣∣2
L2+
∫ t
02
(wε (s) ,
√QdW (s)
)
L2
+ T r Q · t,
hence the first claim on wε follows from the estimate
E
⎡
⎣ supt∈[0,T ]
∣∣∣∣∣
∫ t
0
(wε (s) ,
√QdW (s)
)
L2
∣∣∣∣∣
2⎤
⎦ � T r Q · E
[∫ T
0|wε (s)|2L2
ds
].
216 Franco Flandoli & Alex Mahalov
This is a consequence of Doob’s inequality and the independence on ε of the basicestimates on wε similar to (45), as already remarked in Lemma 3 above.
Similarly,
E
[
supt∈[0,T ]
|wε (t)|8L2
]
� C + E
⎡
⎣ supt∈[0,T ]
∣∣∣∣∣
∫ t
0
(wε (s) ,
√QdW (s)
)
L2
∣∣∣∣∣
4⎤
⎦
� C + C(T r Q
)2 · E
[∫ T
0|wε (s)|4L2
ds
],
which is bounded, uniformly in ε, because of the bound just proved. We shall usethis bound below.
Step 2. The equation (40) for wε, in weak form on a test function h, reads
(wε (t) , h)L2+
∫ t
0ν (wε (r) , Ah)L2
dr +∫ t
0
(B (wε,wε) , h
)L2
dr
=(
u0, h)
L2+ (Mε (t) , h)L2
.. (54)
In the three-dimensional case both the classical operator B and B satisfy theestimate
∣∣∣(B (wε,wε) , h
)L2
∣∣∣ � C |h|H1
|wε|1/2L2· ‖wε‖3/2
H1
� C |h|H1
(|wε|p1/2
L2+ |wε‖3q1/2
H1
), (55)
with 1p1
+ 1q1
= 1, p1 > 1. In the last step in (55) we used the Young inequality. Theestimate (55) for the operator B in dimension 3 is well-known (see [13,36]). Theoperator B satisfies improved estimates as discussed above, but the estimate (55)is sufficient for considerations below.
then (renaming the absolute constants, depending on T on a given interval [0, T ] ;t, s ∈ [0, T ])
E[∣∣(wε (t) , h)L2
− (wε (s) , h)L2
∣∣4]
� Cν4 |Ah|4L2
∫ t
sE[|wε (r)|4L2
]dr
+C |h|4H1
∫ t
sE[|wε (r)|4p
L2
]dr
+C |h|4H1(t − s)2ξ E
[(∫ t
s‖wε‖2
H1dr
)2(2−ξ)]
+C E[∣∣(Mε (t) , h)L2
− (Mε (s) , h)L2
∣∣4].
The first term is bounded by a constant times (t − s), by the bounds proved in step1. The second term,
∫ ts E[|wε(r)|4p
L2]dr , is bounded similarly to
∫ ts E[|wε(r)|8L2
]dr
as in the previous step: there, E[supt∈[0,T ] |wε(t)|8L2] was bounded very easily in
terms of E[∫ T0 |wε(r)|4L2
dr ]; now iteratively (or by induction), we can get similar
bounds for E[supt∈[0,T ] |wε(t)|16L2
], E[supt∈[0,T ] |wε(t)|32L2
], and so on.The last term can be bounded using the Burkholder–Davis–Gundy (BDG)
inequality as
E[∣∣(Mε (t) , h)L2
−(Mε (s) , h)L2
∣∣4]�C |h|4L2
E
[(∫ t
sT r
(e
rε
S Qe− rε
S)
dr
)2]
� C |h|4L2(T r Q)2 (t − s)2 .
Finally, we estimate the term
C |h|4H1(t − s)2ξ E
[(∫ t
s‖wε‖2
H1dr
)2(2−ξ)].
Since 0 < ξ < 12 and t, s ∈ [0, T ], it is sufficient to prove that
E
[(∫ T
0‖wε‖2
H1dr
)4]
< ∞.
For this we have to go back to the energy inequality
|wε (t)|2L2+ 2ν
∫ t
0‖wε (r)‖2
H1dr � |u0|2L2
+∫ t
02
(wε (r) ,
√QdW (r)
)
L2
+T r(Q) · t, (56)
218 Franco Flandoli & Alex Mahalov
which implies (the constant depends on ν,∣∣u0
∣∣2L2, T r
(Q), T )
E
[(∫ T
0‖wε‖2
H1dr
)4]
� C + C E
⎡
⎣
∣∣∣∣∣
∫ T
0
(w (r) ,
√QdW (r)
)
L2
∣∣∣∣∣
4⎤
⎦
� C + C(T r Q
)2 · E
[∫ T
0|wε (r)|4L2
dr
],
which we know to be bounded. The proof is complete. The proof for u Rε is similar.
� The next Lemmas 6 and 7 contain a priori estimates on the closeness of uε(t)
and global in time strong solution wε(t), and serve as a preparation to Lemma 8on u R
ε (t). In Lemma 6 we compare solutions uε(t) and wε(t) of equations (39)and (40) with the same initial condition u0 = U 0 ∈ H3+η. We assume that uε(t)and wε(t) are strong solutions of the corresponding SPDEs with paths of classC([0, T ] ; H3+η
)for some T > 0. Then we have the following comparison esti-
mate in H2 norm:
Lemma 6. Let uε(t) be a strong solution of class C([0, T ] ; H3+η
)of equation
(39). Then for every δ3 > 0 there is ε0 = ε0 (δ3) > 0 such that for every ε < ε0
P(
supt∈[0,T ]
‖uε (t)− wε (t)‖H2� δ3
)
� 1 − δ3.
Proof. Step 1 (Decomposition). Set rε (t) = uε (t) − wε (t). For simplicity ofnotations, let us write u, w, r , in place of uε, wε, rε.
Since equations (39) and (40) have the same noise and the same initial condi-tions, we obtain
dr
dt+ νAr + B (u, r)+ B (r, w)+ Bosc
(t
ε, u, u
)= 0,
r(0) = 0.
Then
1
2
d ‖r‖2H2
dt+ ν ‖r‖2
H3= − (
B (u, r) , r)
H2− (
B (r, w) , r)
H2
−(
Bosc(
t
ε, u, u
), r
)
H2
.
We use the Fourier-curl basis (19). Moreover, denoting byπN the finite dimensionalprojection defined as
(Notice that we keep absolute value outside the integral in I osc1 (t).)
Step 2 (First estimates). From (31) applied to(B (u, r) , r
)H2
and (32) applied
to(B (r, w) , r
)H2
, we have
I1 (t)+ I2 (t) �∫ t
0C(‖u‖H3 + ‖w‖H3
) ‖r‖2H2
ds.
For the projection πN we have the property
‖(1 − πN ) u‖H2� Cξ N 2−ξ ‖u‖Hξ (57)
220 Franco Flandoli & Alex Mahalov
for every ξ > 2. Hence, from (29)∣∣∣∣(πN Bosc
( s
ε, u, (1 − πN ) u
), r
)
H2
∣∣∣∣
�∥∥∥πN Bosc
( s
ε, u, (1 − πN ) u
)∥∥∥
H2
‖r‖H2
� C ‖u‖H2‖(1 − πN ) u‖H3
‖r‖H2
and thus, from (57) with ξ = 3 + η we get
I osc2 (t) � C N−η
∫ t
0‖u‖H2
‖u‖H3+η ‖r‖H2 ds.
Similarly one can prove
I osc3 (t) � C N−1
∫ t
0‖u‖2
H3‖r‖H2 ds
I osc4 (t) � C N−1
∫ t
0‖u‖H2
‖u‖H3‖r‖H3 ds.
The term I osc5 (t) is estimated in the same fashion as I osc
2 (t) and I osc3 (t).
Step 3 (Estimate of oscillatory part I osc1 (t)). From the expression of Bosc
( sε, u, u
)
in Fourier-curl basis and the definition of πN we get(πN Bosc
( s
ε, πN u, πN u
), r
)
H2
=∑
(k,m,σ1,σ2)/∈�|k|�N ,|m|�N ,|k+m|�N
σ2 |m| ei sε
D(k,m,σ1,σ2)uk,σ1 um,σ2
(Pk+m
(ϕk,σ1 × ϕm,σ2
)ei(k·x+m·x), r
)
H2,
so it is a finite sum of the form (recall that u = uε, r = uε − wε)
nN∑
j=1
C j ei sελ j
(uε (s) , a j
)L2
(uε (s) , b j
)L2
((uε (s) , c j
)L2 + (
wε (s) , d j)
L2
),
where λ j �= 0 and a j , b j , c j and d j are smooth periodic divergenceless vectorfields. We stress the dependence on N only in the cardinality nN of these compo-nents, but also the values of C j , λ j , a j , b j , c j , d j depend on it. Setting
f j (s) := (uε (s) , a j
)L2
(uε (s) , b j
)L2
((uε (s) , c j
)L2 + (
wε (s) , d j)
L2
),
from Lemma 1 we have, for a suitable α > 0 determined from Lemma 5,
E
[
supt∈[0,T ]
I osc1 (t)
]
�nN∑
j=1
C j E
[∣∣∣∣
∫ t
0ei s
ελ j f j (s) ds
∣∣∣∣
]�
nN∑
j=1
C ′j
(∣∣∣∣ε
λ j
∣∣∣∣
α
+∣∣∣∣ε
λ j
∣∣∣∣
),
where the constants C ′j include those of Lemma 5.
In the next lemma we compare solutions u Rε (t) and wε(t) of equations (38)
and (40) under the condition (58) on initial data and noise. Then u Rε (t) and
wε(t) are unique strong solutions of the corresponding SPDEs with paths ofclass C
([0, T ] ; H4+η
)for arbitrarily large T > 0. The cutoff function χR in
(38) ensures that u Rε (t) is a global strong solution with paths of class C ([0, T ] ;
H4+η).
Lemma 8. Assume that the conditions (58) on initial data and noise are satisfied.Then for every R > 0 and δ3 > 0 there is ε0 = ε0 (R, δ3) > 0 such that for everyε < ε0,
P(
supt∈[0,T ]
∥∥∥u R
ε (t)− wε (t)∥∥∥
H3� δ3
)
� 1 − δ3.
Proof. The proof is identical to those of two previous lemmas. The inner productis taken in H3 Hilbert space. The only new element is to show that conditions ofLemma 1 are satisfied. For example, the term I osc
1 (t) is now
I osc1 (t) :=
∣∣∣∣
∫ t
0χR
(∥∥∥u R
ε
∥∥∥
2
H3
)(πN Bosc
( s
ε, πN u R
ε , πN u Rε
), r
)
H3
ds
∣∣∣∣ .
We need to verify that the function t �→ ∥∥u R
ε (t)∥∥2
H3is Hölder continuous in t . This
follows from the semigroup theory method based on Green’s function of the non-stationary Stokes problem. The technique of proving Hölder continuity of strongsolutions was introduced by Ladyzhenskaya for deterministic Navier–Stokes equa-tions (for example [36], Chapter 6). In this method, the nonlinear term is consideredas forcing. In our case, both the nonlinear and stochastic terms in (38) are treatedas forcing terms. Under the assumptions (58) on initial data and noise, we deduce
that the solution u Rε (t) and the function t �→ ∥
∥u Rε (t)
∥∥2
H3are Hölder continuous in
H3.Given R > 0 and δ3 > 0, from the analogue of Lemma 3 with H4+η initial data
there is Nδ3 > 0 such that for every ε > 0
P(
supt∈[0,T ]
‖wε (t)‖H4+η � Nδ3
)
� 1 − δ3
4
and from the analogue of Lemma 4 with H4+η initial data there is NR,δ3 > 0 suchthat for every ε > 0
P(
supt∈[0,T ]
∥∥∥u R
ε (t)∥∥∥
H4+η� NR,δ3
)
� 1 − δ3
4.
Thus, for every given ε > 0, with probability larger than 1− δ32 , we simultaneously
have
supt∈[0,T ]
‖wε (t)‖H4+η � Nδ3, supt∈[0,T ]
∥∥∥u R
ε (t)∥∥∥
H4+η� NR,δ3 .
Having established these results, the estimate for r Rε (t) = u R
ε (t) − wε (t) in H3norm, ||r R
ε (t) ||2H3, is obtained similarly to Lemma 6. The proof is complete. �
224 Franco Flandoli & Alex Mahalov
Corollary 1. Assume that the conditions (58) are satisfied. Then for every δ ∈ (0, 1)there is Rδ > 0 and ε0 = ε0 (δ) > 0 such that for every ε < ε0
P(
supt∈[0,T ]
∥∥∥u Rδ
ε (t)∥∥∥
H3� Rδ
)
� 1 − δ.
Proof. Split δ = δ2 + δ
2 , apply Lemma 3 with δ1 = δ2 , and let N δ
2the corresponding
constant, so that
P(
supt∈[0,T ]
‖wε (t)‖H3� N δ
2
)
� 1 − δ
2.
Then set Rδ = N δ2
+ 1, δ3 = δ2 , apply Lemma 8 to get
P(
supt∈[0,T ]
∥∥∥u Rδ
ε (t)− wε (t)∥∥∥
H3� 1
)
� 1 − δ
2
for every ε < ε0 (δ). Then, since∥∥∥u Rδ
ε (t)∥∥∥
H3�
∥∥∥u Rδ
ε (t)− wε (t)∥∥∥
H3+ ‖wε (t)‖H3
,
we have(
supt∈[0,T ]
∥∥∥u Rδ
ε (t)− wε (t)∥∥∥
H3� 1
)⋂
(
supt∈[0,T ]
‖wε (t)‖H3� N δ
2
)
⊂(
supt∈[0,T ]
∥∥∥u Rδ
ε (t)∥∥∥
H3� N δ
2+ 1
)
,
and thus
P(
supt∈[0,T ]
∥∥∥u Rδ
ε (t)∥∥∥
H3� N δ
2+ 1
)
� 1 − δ.
This is true for every ε < ε0 (δ). The proof is complete. �
7. Regularity of Solutions to the Stochastic Three-Dimensional RotatingNavier–Stokes Equations
In this section we give our main regularity result for the stochastic three-dimen-sional rotating Navier–Stokes equations. As we remarked in the introduction, it isequivalent to work with equations (7) or the transformed equations (11).
Following [23], one can define the concept of martingale solutions of stochas-tic equations (7), (11) and (39) and one can prove their existence, for every initialcondition u0 = U 0 ∈ H and for every ε > 0. We do not write the standard details.
Notice that these concepts require, at least a priori, a different probability space foreach U 0 and ε.
We are interested here in regular solutions, strong in the probabilistic sense(adapted to the Brownian motion) and with paths of class C ([0, T ] ; H3). Suchsolutions are pathwise unique. We recall that for comparison estimates to hold inH3 norm, we need slightly stronger assumptions on regularity of initial data andnoise. To this purpose, we assume that the conditions (58) are satisfied for someη > 0.
As above,(�, F, (Ft )t�0 ,P, (W (t))t�0
)is given. Let u R
ε be the unique
strong regular solution of equation (38). Recall the form of the cut-off term
χR
(∥∥u R
ε
∥∥2
H3
)in equation (38). Define the stopping time on
(�, F, (Ft )t�0 ,P,
(W (t))t�0
)
τ Rε = inf
{t � 0 :
∥∥∥u R
ε (t)∥∥∥
H3> R
}.
On[0, τ R
ε
], χR
(∥∥u R
ε
∥∥2
H3
)= 1, hence u R
ε solves equation (39). Take R′ > R. By
uniqueness, τ R′ε � τ R
ε and u R′ε = u R
ε on[0, τ R
ε
]. Set
τε = supR>0
τ Rε .
Therefore a process uε is uniquely defined on [0, τε) by setting uε := u Rε on
[0, τ R
ε
]
for every R > 0. It is continuous in H3, adapted, and is a solution of equation (39).The construction of τε and uε, when the initial condition is regular, is a stan-
dard construction, that is known for the stochastic three-dimensional Navier–Stokesequations, see [26]. In general, one can prove only P (τε > 0) = 1. Here the resultis completely different. From Corollary 1, we obtain:
Corollary 2. For every T > 0 (arbitrarily large) and δ ∈ (0, 1) there is Rδ,T > 0and ε0 = ε0 (δ, T ) > 0 such that for every ε < ε0
P(τ
Rδ,Tε � T
)� 1 − δ.
Hence also
P (τε � T
)� 1 − δ.
Summarizing, we have proved the following theorem for equation (11) (equiv-alently, equation (39)).
Theorem 1. For every ε > 0 the random time τε and the H3-continuous adaptedprocess uε defined above on [0, τε), have the following properties:
i) uε is the pathwise unique solution of equation (39) on [0, τε),ii) for every T > 0 (arbitrarily large) and δ ∈ (0, 1) there exists ε0 (δ, T ) > 0
such that for every ε < ε0 (δ, T )
P (τε � T
)� 1 − δ.
226 Franco Flandoli & Alex Mahalov
The same result holds for equation (1)–(3) (equivalently, (7)–(8)) sinceϒ(−t/ε) = e−St/ε preserves all Sobolev norms Hα .
Theorem 2. For every ε > 0 the random time τε and the H3-continuous adaptedprocess Uε(t) = ϒ(−t/ε)uε(t) defined on [0, τε), have the following properties:
i) Uε(t) is the pathwise unique solution of equations (1)–(3) (equivalently, (7)–(8)) on [0, τε),
ii) for every T > 0 (arbitrarily large) and δ ∈ (0, 1) there exists ε0 (δ, T ) > 0such that for every ε < ε0 (δ, T )
P (τε � T
)� 1 − δ.
In the above theorem τε ≡ τ εU 0 defined in the introduction. Theorem 2 for
the stochastic three-dimensional RNSEs and the probability one statement (4) inthe introduction of non-explosition given arbitrarily long time horizon T , whichfollows from it, are in contrast with regularity results available in the classical the-ory of stochastic three-dimensional Navier–Stokes equations. In general, one canprove only P (
τU 0 > 0) = 1 for the stochastic three-dimensional NSEs. For rea-
sons described in the introduction, compared to the deterministic case, however,we do not have infinite time regularity for sufficiently small ε.
8. Averaging Principle
In this section we further describe convergence results for stochastic equations(38)–(41). When we compare equations (39) and (40) which have the same noise,we can prove convergence in probability. On the other hand, since the family ofBrownian motions Mε(t) (Q (ε, t) = e
tε
S Qe− tε
S) converges only in law as ε → 0,we prove only convergence in law for the corresponding solutions of SPDEs (39)and (41). We recall that equation (39) is equivalent to the stochastic three-dimen-sional RNSEs (1)–(3).
Let uε, wε and w be solutions of stochastic equations (39)–(41), respectively.Thus, it follows from the above discussion that we can expect the following typeof convergence for solutions of SPDEs as ε → 0:
uε − wε → 0 in probability, (59)
uε → w in law, (60)
wε → w in law. (61)
The rigorous statements are more elaborate due to the presence of the exit time τε.We always assume U 0 ∈ H3+η for some η > 0.
Theorem 3. For every ε > 0, let τ Rε , τε and uε be those given above, see Theorem 1,
and let wε be the unique regular solution of equation (40); recall that theprocesses uε and wε are H3-continuous and adapted, both defined on(�, F, (Ft )t�0,P, (W(t))t�0). Then, for every R > 0, the random variablesupt∈[0,τ R
ε ] ‖uε (t)− wε (t)‖H2converges to zero in probability.
Proof. This result is a direct consequence of Lemmas 6–8: given R > 0, it statesthat supt∈[0,T ]
∥∥u R
ε (t)− wε (t)∥∥
H2converges to zero in probability. But
supt∈[0,τ R
ε ]‖uε (t)− wε (t)‖H2
� supt∈[0,T ]
∥∥∥u R
ε (t)− wε (t)∥∥∥
H2
hence also supt∈[0,τ Rε ] ‖uε (t)− wε (t)‖H2
converges to zero in probability. � Let us discuss now the relation between solutions of equations (40) and (41).
Recall that
wε (t) = Aε (t)+ Mε (t) ,
Aε (t) = u0 −∫ t
0
[νAwε + B (wε,wε)
]ds, Mε (t) =
∫ t
0e
rε
S√
QdW (r) .
The laws of the Gaussian martingales Mε (t) are tight in proper topologies so wecan prove they converge weakly (as ε → 0) to the law of a Gaussian martingaleM (t). We may identify the limit process M (t) as a Brownian motion in H withcovariance Q (see step 2 of the next theorem). However, we cannot prove almost-surely convergence or in probability. Hence the convergence of the solution ofequation (40) to the solution of equation (41) is only in law.
Theorem 4. Letwε andw be the solutions of equations (40) and (41), respectively.Then, on C ([0, T ] ; H2), the law Pwε of wε converges weakly to the law Pw of was ε → 0.
Proof. Step 1. Recall the following interpolation result: if B0 ⊂ B ⊂ B1 are threeBanach spaces, the embedding B0 ⊂ B being compact, B ⊂ B1 being continuous,then for every ε > 0 there is a constant Cε > 0 such that
‖x‖B � ε ‖x‖B0 + Cε ‖x‖B1
for every x ∈ B0. We apply this fact to H3+η ⊂ H3 ⊂ H−2, H−2 being the dual ofH2:
‖x‖H3 � ε ‖x‖H3+η + Cε ‖x‖H−2
for every x ∈ H3+η. Assume that a sequence of probability measure (μn) con-verges weakly to μ on C ([0, T ] ; H−2), but in fact all μn are concentrated onC([0, T ] ; H3+η
)and satisfy the condition: for every δ > 0 there is R > 0 such
that the ball BC([0,T ];H3+η) (0, R) of center 0 and radius R in C([0, T ] ; H3+η
)has
the property
μn
(BC([0,T ];H3+η) (0, R)
)> 1 − δ
for all n. Then (μn) converges weakly to μ on C ([0, T ] ; H3). Let us prove thisclaim. Let Xn, X , be random variables on a probability space (�, F,P) such thatXn and X have law μn and μ respectively, and Xn converges P almost surelyto X in the topology of C ([0, T ] ; H−2) (they exist by Skorokhod representation
228 Franco Flandoli & Alex Mahalov
theorem, cf. [33]). For every ε > 0 (it will be chosen below), let Cε be the constantwith the interpolation property above. We have
Given δ > 0 and the number λ > 0 chosen above, there exists ε > 0 such that
P(ε ‖Xn‖C([0,T ];H3+η) >
λ
3
)<δ
3
for all n, hence
P (‖Xn − Xm‖C([0,T ];H3) > λ)
<2δ
3+ P
(Cε ‖Xn − Xm‖C([0,T ];H−2) >
λ
3
).
Now, since Xn converges in probability in C ([0, T ] ; H−2), there is n0 such that
P(
Cε ‖Xn − Xm‖C([0,T ];H−2) >λ
3
)<δ
3
for all n,m > n0. Therefore
P (‖Xn − Xm‖C([0,T ];H3) > λ)< δ
for all n,m > n0. This means that Xn converges in probability in C ([0, T ] ; H3),hence also in law, hence μn converges weakly in C ([0, T ] ; H3). It is easy toidentify the limit as μ. The claim is proved.
Step 2. Let us analyze the Gaussian martingale Mε (t) = ∫ t0 e
rε
S√QdW (r).
We may use both the theory of martingales and the theory of Gaussian processes.Let us follow the second one, maybe more elementary. The H -valued Gaussianprocess Mε (t) has covariance
If we prove that Mε converges in law to a process M on C ([0, T ] ; H) andwe have the estimate E
[‖Mε (t)‖3H
]� C , then M is Gaussian with covariance
E[〈M (t) , h〉H 〈M (s) , k〉H
]given by the limit of the expression above, that is,
given by (t ∧ s)⟨Qh, k
⟩H . Therefore M (t) is a Brownian motion with covariance
Q.The uniform estimate E
[‖Mε (t)‖3H
]� C is elementary: moments of any order
are controlled by second moments, for Gaussian random variables, and
E[‖Mε (t)‖2
H
]=
∫ t
0T r
(e
rε
S Qe− rε
S)
dr � C,
since Q is trace class and erε
S has norm one in H .Let us prove that Mε is tight in C ([0, T ] ; H). This implies that from any
sequence Mεn one can extract a subsequence which converges in law to a processM on C ([0, T ] ; H). We have identified M uniquely above, hence the whole familyMε converges in law to M . This will complete the proof.
The fractional Sobolev space Wα,p (0, T ; H1) is compactly embedded intoC ([0, T ] ; H) when αp > 1 (see for instance [23, p. 91]). Therefore it is sufficientto find a constant C > 0 such that
E
[∫ T
0
∫ T
0
‖Mε (t)− Mε (s)‖pH1
|t − s|1+αp dtds
]
� C
for all ε > 0, with α, p satisfying αp > 1. By the BDG inequality and our assump-tion (42) on Q (and the fact that e
rε
S has norm one) we have (for t > s)
E[‖Mε (t)− Mε (s)‖p
H1
]� C p E
[(∫ t
sT r
(e
rε
S Qe− rε
S A)
dr
)p/2]
� C ′p (t − s)p/2 .
Therefore,
E
[∫ T
0
∫ T
0
‖Mε (t)− Mε (s)‖pH1
|t − s|1+αp dtds
]
�∫ T
0
∫ T
0
C ′p
|t − s|1+(α− 1
2
)p
dtds,
which is finite for every α < 12 and every p. The claim of this step is proved.
Step 3. By classical arguments (energy inequality in the average, obtained fromItô’s formula, and estimates of the fractional Sobolev norm in time, see for instance[23], Theorem 4.2) one can prove that the family
{Pwε}
is tight in L2 ([0, T ] ; H)and C ([0, T ] ; H−2). More precisely, the joint law P(wε,Mε) of the pair (wε,Mε)
is tight in
Y :=[
L2 ([0, T ] ; H) ∩ C ([0, T ] ; H−2)]
× C ([0, T ] ; H)
(we use step 2 for the second component). Therefore, from any sequence{P(wεn ,Mεn )
} one can extract a subsequence which converges weakly to a law Pin Y ; and therefore, for the first component, also in C ([0, T ] ; H3), by the result
230 Franco Flandoli & Alex Mahalov
of step 1. If we prove that any such P coincides with the law of the pair (w,M),where w is the solution of equation (41) and M(t) =
√QW (t), then the theorem
is proved.Before the statement of the theorem we have written the decompositionwε (t) =
Aε (t)+Mε (t). Assume that a sequence {P(wεn ,Mεn )} converges weakly to a lawP in
Y . By Skorokhod’s representation theorem (see [33]), there is a probability space(�′, F ′,P ′) and random variables Xn, X with values in Y and laws P(wεn ,Mεn )
and P respectively, such that XnY→ X with probability one. Write X (1)n and X (2)n
for the two components of Xn and similarly for X . The law of X (2)n converges inC ([0, T ] ; H) to the law of X (2) and therefore, by step 2, X (2) is a Brownian motionin H with covariance Q. Arguing as in [23], Theorem 4.2, we may check that X (1)n
and X (2)n are related by equation (40) and X (1), X (2) by equation (41). Therefore,by the uniqueness (pathwise and in law) for equation (41), the law of X (1) is equalto the law of w. The proof is complete. �
We recall that uε and w satisfy equations (39) and (41), correspondingly.
Corollary 3. Extend uε to zero outside [0, τε). Then, on C ([0, T ] ; H2), the lawPuε of this extension converges weakly to the law Pw of w.
Proof. Denote the extension of uε by uε. Given a bounded continuous functionϕ : C ([0, T ] ; H2) → R, we have
lim supε→0
|E [ϕ (uε)− ϕ (w)]| = lim supε→0
|E [ϕ (uε)− ϕ (wε)]| ,
since limε→0 E [ϕ (wε)] = E [ϕ (w)] (previous theorem). Moreover, for everyR > 0,
|E [ϕ (uε)− ϕ (wε)]| �∣∣∣E
[(ϕ (uε)− ϕ (wε)) 1τ R
ε �T
]∣∣∣ + 2 ‖ϕ‖∞ P
(τ Rε < T
).
Given δ > 0, let R0 > 0 and ε0 > 0 be such that 2 ‖ϕ‖∞ P(τ
R0ε < T
)� δ
2 for
all ε ∈ (0, ε0). We have
E[(ϕ (uε)− ϕ (wε)) 1τ R
ε �T
]=
∫
τ Rε (ω)�T
[(ϕ (uε (ω))− ϕ (wε (ω)))] P (dω) .
Since ϕ is continuous, given the number δ above, there is ξ > 0 such that‖a − b‖C([0,T ];H3) � ξ implies |ϕ (a)− ϕ (b)| � δ
Since δ > 0 is arbitrary, we have proved the claim of the corollary. � The same result holds for equations (1)–(3) (equivalently, (7)–(8)) since
ϒ(−t/ε) = e−St/ε preserves all Sobolev norms Hα . Let T > 0 arbitrarily large. Inlaw, the solution Uε(t) of the original stochastic three-dimensional RNSEs (1)–(3)is close on [0, T ] to the transformation of the (moderately varying) solution w(t)to the SPDE (41) without parameter ε, by the fast rotation unitary group e−t S/ε.Namely, as ε → 0,
(Uε(t)− e−t S/εw(t)
)→ 0 in law. (62)
Here Uε(t) and w(t) are solutions of the corresponding equations (1)–(3) and (41)with the same initial condition.
9. Appendix
9.1. Convergence of Cesaro Averages
We refer to [1] for definitions of Cesaro averages for operator valued functions.Letϒ = (ϒt )t∈R be a strongly continuous unitary group on a Hilbert space H . Let{Pk} be the projections of H onto the eigenspaces of ϒ . Let P = ∑
k Pk denotethe projection of H onto the closed linear span of the eigenvectors of ϒ .
Lemma A1. Let ψ ∈ H.
1. The limit
Cψ := limT →∞
1
T
∫ T
0ϒtψ 〈·, ϒtψ〉 dt
exists in the operator norm.2. We have the formula
Cψ =∑
k
Pkψ 〈·, Pkψ〉.
The sum defining Cψ converges in trace norm, and ‖Cψ‖1 = ‖Pψ‖2.
232 Franco Flandoli & Alex Mahalov
Proof. We use the spectral decomposition of ϒ to write H as a direct integral:H = ∫⊕
RHλ dθ(λ). Since the closed linear span of the orbit of ψ is the only
relevant part of the Hilbert space, we may as well assume that H is separable.For ξ, η ∈ H , we write 〈ϒtξ, η〉 = ∫
Rei tλ〈ξλ, ηλ〉 dθ(λ). We let f (z) = ez−1
z ,an entire function. Then
1
T
∫ T
0ei t z dt =
{eiT z−1
iT z , if z �= 0
1, if z = 0
= f (iT z).
Moreover,∣∣ f (i z)
∣∣ � 1 and
∣∣ f (i z)
∣∣ � 2
|z| for all z ∈ R.For φ,ψ ∈ H , let
Cφ,ψ,T = 1
T
∫ T
0ϒtφ〈·, ϒtψ〉 dt.
Then for ξ, η ∈ H ,
〈Cφ,ψ,T ξ, η〉 = 1
T
∫ T
0
∫∫
R2ei t (μ−λ)〈φμ, ημ〉〈ξλ, ψλ〉 dθ(λ) dθ(μ) dt
=∫∫
R2f(iT (μ− λ)
)〈φμ, ημ〉〈ξλ, ψλ〉 dθ(λ) dθ(μ),
by the Fubini theorem. Now consider the given vector ψ . Let ψ1 = Pψ andψ2 = (1− P)ψ . We have Cψ,ψ,T = ∑2
α,β=1 Cψα,ψβ,T . We first show that if α = 2or β = 2 then ‖Cψα,ψβ,T ‖ → 0 as T → ∞. Without loss of generality, we assumethat β = 2.
Define h : R → [0, ‖ψ2‖2] by
h(μ) =∫ μ
−∞‖(ψ2)λ‖2 dθ(λ).
Since ψ2 is orthogonal to all eigenspaces of ϒ , the measure ‖(ψ2)λ‖2 dθ(λ) iscontinuous. Thus h is continuous, increasing, and bounded, and hence is uniformlycontinuous. Let ε > 0, and choose δ > 0 so that if |μ1 − μ2| < 2δ then
∣∣h(μ1)−
h(μ2)∣∣ < ε2. Next we show that if T > 2
δεthen ‖Cψα,ψ2,T ‖ < ε(‖ψ‖2 + ‖ψ‖).
Fix ξ, η ∈ H . Then
〈Cψα,ψ2,T ξ, η〉 = ∫∫R2 f
(iT (μ− λ)
)〈ψα,μ, ημ〉〈ξλ, ψ2,λ〉 dθ(λ) dθ(μ).
We split the above integral into the sum of two integrals over the domains |λ−μ| � δ
The following estimate holds for the inner integral:∫ μ+δ
μ−δ|〈ξλ, ψ2,λ〉| dθ(λ) � ‖ξ‖
(∫ μ+δ
μ−δ|‖ψ2,λ‖2 dθ(λ)
)1/2
= ‖ξ‖(h(μ+ δ)− h(μ− δ))1/2 � ε‖ξ‖.
Therefore,∣∣∣∣
∫∫
|λ−μ|<δf(iT (μ− λ)
)〈ψα,μ, ημ〉〈ξλ, ψ2,λ〉 dθ(λ) dθ(μ)
∣∣∣∣
� ‖ψα‖ · ‖η‖ · ε‖ξ‖ � ε‖ψ‖ · ‖ξ‖ · ‖η‖.Hence
∣∣〈Cψα,ψ2,T ξ, η〉
∣∣ � ε(‖ψ‖2 + ‖ψ‖)‖ξ‖ · ‖η‖,
and therefore ‖Cψα,ψ2,T ‖ � ε(‖ψ‖2 + ‖ψ‖), as claimed.It remains to show that limT →∞ Cψ1,ψ1,T = Cψ . First note that the sum in part
(2) of the lemma does indeed converge in trace norm, to a positive operator of trace∑k ‖Pkψ‖2 = ‖Pψ‖2; we let Cψ represent this sum. Let a be the infinitesimal
generator of ϒ ; thus ϒt = ei ta . Let ak be the eigenvalue of a corresponding to thespectral projection Pk . Since ψ1 = ∑
k Pkψ , we have
Cψ1,ψ1,T =∑
j,k
f(iT (ak − a j )
)Pkψ〈·, Pjψ〉.
Thus Cψ equals the sum of terms on the diagonal. Now we prove that the sum ofoff-diagonal terms in the above expression for Cψ1,ψ1,T tends to zero in norm. Toshow this, we estimate using the Hilbert–Schmidt norm. Fix N , and let
b =∑
( j,k) �∈[1,N ]2
f(iT (ak − a j )
)Pkψ〈·, Pjψ〉.
Then since the Pjψ are mutually orthogonal, and∣∣ f
(iT (ak − a j )
)∣∣ � 1, we have
‖b‖22 =
∑
( j,k) �∈[1,N ]2
∣∣ f
(iT (ak − a j )
)∣∣2∥∥Pkψ‖2‖Pjψ‖2
� 2N∑
j=1
‖Pjψ‖2∑
k>N
‖Pkψ‖2 � 2‖ψ‖2∑
k>N
‖Pkψ‖2 → 0,
as N → ∞. Since ‖ · ‖ � ‖ · ‖2, it follows that ‖b‖ → 0 as N → ∞.Now let ε > 0. Since the series defining Cψ converges in norm, we may choose
N such that ‖Cψ − ∑k�N Pkψ〈·, Pkψ〉‖ < ε and ‖b‖ < ε. Then
‖Cψ1,ψ1,T − Cψ‖ � 2ε + ‖∑
j �=k�N
f(iT (ak − a j )
)Pkψ〈·, Pjψ〉‖
� 2ε + ‖ψ‖2∑
j �=k�N
2
T |ak − a j | .
Letting T → ∞, we have the result. �
234 Franco Flandoli & Alex Mahalov
Theorem A2. Let Q be a positive trace class operator on H. Then
CQ := limT →∞
1T
∫ T
0ϒt Qϒ−t dt
exists in operator norm, CQ is a positive trace class operator, and Tr(CQ) =Tr(P Q).
Proof. Let Q be a positive compact operator. Then Q = ∑r qrξr 〈·, ξr 〉, where the
ξr form an orthonormal sequence (of eigenvectors of Q), and qr ↓ 0 are the eigen-values of Q, repeated according to multiplicity. For each t , the sequence (ϒtξr )
∞r=1
is orthonormal. Therefore, for each t we have∥∥∥∥∥∥
∑
r�N
qrϒtξr 〈·, ϒtξr 〉∥∥∥∥∥∥
= qN → 0,
as N → ∞. Thus we have
1
T
∫ T
0ϒt Qϒ−t dt = 1
T
∫ T
0
∑
r
qrϒtξr 〈·, ϒtξr 〉 dt =∑
r
qr Cξr ,ξr ,T .
Finally, suppose in addition that Q is a trace class; thus∑
r qr < ∞. Since‖Cξr ,ξr ,T ‖1 � 1, we see that
limT →∞
1
T
∫ T
0ϒt Qϒ−t dt =
∑
r
qr Cξr =: CQ
in the operator norm. Since Cξr is a positive trace class operator with trace at most 1,we have that CQ is a positive trace class operator. Extending {ξr } to an orthonormalbasis, we have that Q kills all basis elements other than the {ξr }. Thus
Tr(CQ) =∑
r
qr‖Pξr‖2 =∑
r
qr 〈Pξr , ξr 〉 =∑
r
〈P Qξr , ξr 〉 = Tr(P Q). �
The theorem is proven.Theorem A2 is used to define the operator Q in the equation (14), Section 1
(Q ≡ CQ). Here unitary groupϒ(t) = et S has the Poincaré–Coriolis operator S asits generator. We have Tr(Q) = Tr(Q) since eigenvectors of S form a basis in H .
9.2. Cylindrical Brownian Motion
Details about cylindrical Brownian motion can be found in [15]. As a short andincomplete introduction to the subject, consider a sequence {Wn (t)}n∈N of inde-pendent 1-dimensional Brownian motions, together with a complete orthonormalsystem {en}n∈N of a Hilbert space H . Formally speaking, the cylindrical Brownianmotion is
but this series does not converge in H in any natural probabilistic sense (the formalidea is that E
[‖W (t)‖2H
] = ∑∞n=1 E
[W2n (t)
] = ∑∞n=1 t = ∞). The series above
converges in suitable “negative order” space Y ⊃ H , but this fact is not used here.What is used here is that the series
√QW (t) :=
∞∑
n=1
σnWn (t) en
converges in L2 (�; H)when Q is a trace class non-negative, self-adjoint operator
in H such that Qen = σ 2n en (indeed E
[∣∣√QW (t)
∣∣2]
= ∑∞n=1 σ
2n E
[W2n (t)
] =t∑∞
n=1 σ2n < ∞); and, similarly, stochastic integrals of the form
∫ t
0Ts
√QdW (s) =
∞∑
n=1
σn
∫ t
0TsendWn (t)
converge in L2 (�; H) and define continuous martingales in H , when Ts arebounded operators, strongly measurable and bounded in s (or more general oper-ator valued processes, see [15]). A relevant example in fluid dynamics, especiallyfor the investigation of turbulence and transport of energy between scales, is thecase when Q is finite range, namely when σn �= 0 only for a finite number of n’s.In this case
√QW (t) is a finite dimensional noise, a random activation of a finite
number of (low) modes en .
Acknowledgments Alex Mahalov was sponsored by the AFOSR contract FA9550-08-1-0055 and the National Science Foundation. He gratefully acknowledges the University ofPisa, Department of Applied Mathematics for their hospitality.
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Department of Applied Mathematics,University of Pisa,
