Diffusion limit of 3D primitive equations of the large-scale ocean under fast oscillating random force

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ScienceDirect J. Differential Equations 259 (2015) 2388–2407 www.elsevier.com/locate/jde

Diffusion limit of 3D primitive equations of the large-scale ocean under fast oscillating random force Boling Guo a , Daiwen Huang a,∗ , Wei Wang b a Institute of Applied Physics and Computational Mathematics, Beijing 100088, China b Department of Mathematics, Nanjing University, Nanjing 210093, China

Received 31 August 2013; revised 23 December 2014 Available online 22 April 2015

Abstract The three-dimensional (3D) viscous primitive equations describing the large-scale oceanic motions under fast oscillating random perturbation are studied. Under some assumptions on the random force, the solution to the initial boundary value problem (IBVP) of the 3D random primitive equations converges in distribution to that of IBVP of the limiting equations, which are the 3D stochastic primitive equations describing the large-scale oceanic motions under a white in time noise forcing. This also implies the convergence of the stationary solution of the 3D random primitive equations. © 2015 Elsevier Inc. All rights reserved. Keywords: Random primitive equations; Stationary solution; Martingale; Statistical solution

1. Introduction The important 3D viscous primitive equations of the large-scale ocean in a Cartesian coordinate system, are written as the following system on a cylindrical domain ∂v ∂v + (v · ∇)v + (v) + f k × v + ∇pb − ∂t ∂z

z

−1

* Corresponding author.

E-mail addresses: [email protected], [email protected] (D. Huang). http://dx.doi.org/10.1016/j.jde.2015.03.041 0022-0396/© 2015 Elsevier Inc. All rights reserved.

∇T dz

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− v −

∂ 2v = 1 , ∂z2

2389

(1.1)

∂T ∂T ∂ 2T + (v · ∇)T + (v) − T − 2 = 2 , ∂t ∂z ∂z

(1.2)

0 ∇ · v dz = 0,

(1.3)

−1

with boundary value conditions ∂v = 0, ∂z

∂T = −αu T ∂z

on M × {0} = u ,

(1.4)

∂v = 0, ∂z

∂T =0 ∂z

on M × {−1} = b ,

(1.5)

∂T = 0 on ∂M × [−1, 0] = l , ∂ n

(1.6)

v · n = 0,

∂v × n = 0, ∂ n

and the initial value conditions u|t=t0 = (v|t=t0 , T |t=t0 ) = ut0 = (vt0 , Tt0 ),

(1.7)

where the unknown functions are v, pb , T , v = (v (1) , v (2) ) the horizontal velocity, pb the z pressure, T temperature, (v)(t, x, y, z) = − −1 ∇ · v(t, x, y, z ) dz vertical velocity, f = f0 (β + y) the Coriolis parameter, k vertical unit vector, 1 a given forcing field, 2 a given heat source, ∇ = (∂x , ∂y ),  = ∂x2 + ∂y2 , αu a positive constant, n the norm vector to l and M a smooth bounded domain in R2 . For more details for (1.1)–(1.7), see [2,21] and the references therein. In the past two decades, there were several research works about the well-posedness of the above 3D deterministic primitive equations of the large-scale ocean. In [17], Lions, Temam and Wang obtained the global existence of weak solutions for the primitive equations. In [14], Guillén-González etc. obtained the global existence of strong solutions to the primitive equations with small initial data. Moreover, they proved the local existence of strong solutions to the equations. In [2], Cao and Titi developed a beautiful approach to proving that L6 -norm of the horizontal velocity is uniformly in t bounded, and obtained the global well-posedness for the 3D viscous primitive equations. In study of the primitive equations of the large-scale ocean or atmosphere, taking the stochastic external factors into account is reasonable and necessary. There are many works about mathematical study of some stochastic climate models, see, e.g., [6–8,19,20]. Ref. [8] is one of the first works on a 3D stochastic quasi-geostrophic model. Guo and Huang in [13] considered the global well-posedness and long-time dynamics for the 3D stochastic primitive equations of the large-scale ocean under a white in time noise forcing. In realistic model, random fluctuation always exits. We consider the following 3D primitive equations with fast oscillating random force

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∂v ∂v + (v · ∇)v + (v ) + f k × v + ∇pb − ∂t ∂z

z

∇T dz − v

−1

−

∂ 2v 1 = √ η1 , 2 ∂z

(1.8)

∂T ∂ 2T 1 ∂T =  + √ η2 , + (v · ∇)T + (v ) − T − 2 ∂t ∂z ∂z

(1.9)

0 ∇ · v dz = 0,

(1.10)

−1

with boundary value conditions ∂v = 0, ∂z ∂v = 0, ∂z

∂T = −αu T , on u , ∂z ∂T = 0, on b , ∂z ∂v ∂T × n = 0, = 0, on l , v · n = 0, ∂ n ∂ n

(1.11) (1.12) (1.13)

and initial value conditions u |t=t0 = (v |t=t0 , T |t=t0 ) = u t0 = (vt 0 , Tt 0 ),

(1.14)

where (v0 , T0 ) ∈ V is F00 -measurable random variable. Here η1 and η2 are random forces which are stationary processes satisfying some assumptions given in Subsection 2.2, and  is a given heat source defined on  = M × (−1, 0). One reason of considering such fast oscillating random force in the primitive equation is that white noise is an idealistic model. On the other hand the random model (1.8)–(1.10) converges in some sense to the white noise driven primitive equations as → 0 which implies that the random model (1.8)–(1.10) is more appropriate to describe some physical phenomena if stochastic primitive equations do. There are lots of models of such random forces η1 and η2 . Here, for simplicity, we assume that these random forces are some Gaussian processes solving linear stochastic differential equation (see (2.4)). Classical result [1, Chapter 10, e.g.] shows that 1 √

t η(s/ ) ds

converges in distribution to W

as → 0

0

for some scalar Wiener process W provided that process η has some mixing property. Then for small > 0, the limit of the above 3D random primitive equations is expected to be 3D primitive equations driven by white noise. In fact by a weak convergence method, we show that the limit of 3D random primitive equations is the limiting model, which is the 3D stochastic primitive equations driven by white in time noise (see Theorem 4.1) as → 0. To show the limit of stationary

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solution, we work on the statistical solution on [0, ∞). The limit of the statistical solution of the 3D random primitive equations is shown to be that of the 3D primitive equations driven by a white noise, which implies the limit of stationary solution of the 3D random primitive equations is that of the 3D stochastic primitive equations. One difficulty here is the singularity caused by the randomly fast oscillation. For this we define a martingale to replace this fast oscillation term, which eliminates the −1 term. This method is applied in both energy estimates for solutions and passing limit of → 0, see the proofs of Lemma 4.1 and Theorem 4.1. The above limit approach is also called a diffusion limit which has been applied to study the asymptotic behavior of stochastic Burgers type equations with stochastic advection [24]. There are also some works on diffusion approximation for some random PDEs with different approaches [4,15, e.g.]. The paper is organized as follows. In Section 2, the 3D random primitive equations are introduced. Our working spaces and a new formulation of the initial boundary value problem for the primitive equations with fast oscillating random force are given in this section. We obtain the global well-posedness to 3D primitive equations of the large-scale ocean under fast oscillating random force in Section 3. We prove main results of our paper in Section 4. 2. New formulation for the 3D random primitive equations 2.1. New formulation for the 3D random primitive equations Before formulating a new formulation for the 3D random primitive equations, notations for some function spaces, functionals and operators are given. Let Lp () be the usual Lebesgue space with the norm | · |p , 1 ≤ p ≤ ∞. H m () is the usual Sobolev space (m is a positive integer) with the norm ⎡ h m = ⎣

  







⎤1

2

|∇i1 · · · ∇ik h| + |h| ⎦ , 2

2

1≤k≤m ij =1,2,3;j =1,···,k

    ∂ ∂ ∂ where ∇1 = ∂x , ∇2 = ∂y and ∇3 = ∂z .  ·d and M ·dM are denoted by  · and M · respectively. Define our working spaces for (1.8)–(1.14) as ⎧ ⎫ 0 ⎨ ⎬ ∂v ∂v V1 := v ∈ (C ∞ ())2 ; |u ,b = 0, v · n|l = 0, × n|l = 0, ∇ · vdz = 0 , ⎩ ⎭ ∂z ∂ n 

V2 := T ∈ C ∞ ();

∂T ∂z |u

= −αu T ,

∂T ∂z |b

= 0,

∂T ∂ n |l

V1 = the closure of V1 with respect to the norm · 1 , V2 = the closure of V2 with respect to the norm · 1 , H1 = the closure of V1 with respect to the norm | · |2 , V = V 1 × V2 ,

H = H1 × L2 ().



=0 ,

−1

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The inner products and norms on V , H are given by

u, u1 V = v, v1 V1 + T , T1 V2 ,

u, u1  = v (1) , (v1 )(1)  + v (2) , (v1 )(2)  + T , T1 , 1

1

1

1

u = u, uV2 = v, vV2 1 + T , T V2 2 = v + T ,

|u|2 = u, u 2 ,

where u = (v, T ), u1 = (v1 , T1 ) ∈ V , and ·, · denotes the inner product in L2 (). Then, we define the functionals a : V × V → R, a1 : V1 × V1 → R, a2 : V2 × V2 → R, and their corresponding linear operators A : V → V  , A1 : V1 → V1 , A2 : V2 → V2 by a(u, u1 ) = Au, u1  = a1 (v, v1 ) + a2 (T , T1 ), where   a1 (v, v1 ) = A1 v, v1  = ∇v · ∇v1 +

∂v ∂z

·

∂v1 ∂z





a2 (T , T1 ) = A2 T , T1  =

 

∇T · ∇T1 +

∂T ∂T1 ∂z ∂z



, 

 + αu

T T1 . u

According to Lemma 3.1 in [13], we know that a and A have the following properties. The a is coercive and continuous, and A : V → V  is isomorphism. Moreover, a(u, u1 ) ≤ c v v1 + c T T1 ≤ c u u1 , a(u, u) ≥ c v 2 + c T 2 ≥ c u 2 . The isomorphism A : V → V  can be extended to a self-adjoint unbounded linear operator on H with a compact inverse A−1 : H → H and with the domain of definition of the operator D(A) = V ∩ [(H 2 ())2 × H 2 ()]. Denote by 0 < λ1 ≤ λ2 ≤ · · · the eigenvalues of A and by e1 , e2 · · · the corresponding complete orthonormal system of eigenvectors. We define a nonlinear operator N = (N1 , N2 ) : V × V → V  by

N (u1 , u1 ), u2 H = N1 (v1 , v1 ), v2  + N2 (v1 , T1 ), T2 H2 , where

N1 (v, v1 ), v2  =

  

N2 (v, T1 ), T2  =

 



 1 (v · ∇)v1 + (v) ∂v ∂z · v2 ,  1 (v1 · ∇)T1 + (v1 ) ∂T ∂z T2 ,

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related to the linear terms, we define a bilinear functional l : V → R and its corresponding operator L : V → V by  l(u, u1 ) = Lu, u1 H =

 f (k × v) · v1 −



⎛ ⎝



z

⎞ ∇T dz ⎠ T1 .

−1

Now, we rewrite (1.8)–(1.14) as the following abstract stochastic evolution equations ∂v 1 + N1 (v , v ) + Lu + A1 v = √ η1 , ∂t

(2.1)

∂T 1 + N2 (v , T ) + A2 T =  + √ η2 , ∂t

(2.2)

u (0) = (v (0), T (0)) = (v0 , T0 ).

(2.3)

For the above random system we give the following definition. Definition 2.1. For any T > t0 , a process u (t, ω) = (v , T ) is called a strong solution to (2.1)–(2.3) in [t0 , T ], if, for P-a.e. ω ∈ , u satisfies t

v (t), ϕ1  −

t [ N1 (v , ϕ1 ), v − Lu , ϕH ] +





v , A1 ϕ1 



t0

t0

1 = vt 0 , ϕ1  + √

t

η1 (s, ω), ϕ1 , t0

t

T (t), ϕ2  −

[ N2 (v , ϕ2 ), T  − T , A2 ϕ2 ] t0

t = Tt 0 , ϕ2  + t0

1

, ϕ2  + √

t

η2 (s, ω), ϕ2 , t0

for all t ∈ [t0 , T ] and ϕ = (ϕ1 , ϕ2 ) ∈ D(A1 ) × D(A2 ), moreover u ∈ L∞ (t0 , T ; V ) ∩ L2 (t0 , T ; (H 2 ())2 ) and is progressively measurable in these topologies. 2.2. A model for (η1 , η2 ) To detail the random model (1.8)–(1.14), we assume that the stationary process η (t) = solves the following linear stochastic system

(η1 (t), η2 (t))

η (t) η (t) d √ = − √ dt + dW (t),

(2.4)

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where W (t) = (W1 (t), W2 (t)) is H -valued Wiener process with covariance operator Q = (Q1 , Q2 ). Here we assume that W˙ i , i = 1, 2, have the following form dWi (t)  d W˜ i (t) = Qi , dt dt where W˜ i (t) is a cylindrical Wiener process √ in Hi defined on a complete probability space (, F, P) with expectation denoted by E, and Qi is a linear operator. Assumption (H1 ). Qi , i = 1, 2, satisfy Tr A3 Qi < +∞. Remark 2.1. An example for W is a two-sided in time finite-dimensional Brownian motion with the form W=

m 

δi βi (t, ω)ei .

i=1

In the above formula, β1 , · · · , βm are independent standard one-dimensional Brownian motions on a complete probability space (, F, P ), and δi are real coefficients. Remark 2.2. An another example for W is a two-sided in time infinite dimensional Brownian motion with the form W (t) =

+∞ 

μi βi (t, ω)ei .

i=1

Here β1 , β2 , · · · is a sequence of independent standard one-dimensional Brownian motions on a +∞  λ3i μ2i < +∞. complete probability space (, F, P ) and the coefficients μi satisfy i=1

By the stationarity and the assumption (H1 ), a simple application of Itô formula yields E η (t) 23 = Tr(A3 Q) < ∞. Moreover, the system (2.4) is strong mixing. To describe, this we define t/

Fs/ = σ {η (τ ) : s ≤ τ ≤ t} and φ(t/ ) = sup

sup

s≥0 A∈F s/ ,B∈F ∞ 0

Then

s/ +t/

|P(AB) − P(A)P(B)|.

(2.5)

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∞ φ k (t) dt < ∞ 0

for any k > 0. In fact by the exponentially stable of the linear system (2.4), we have for t > 0 φ(t/ ) < Ce−t/ for some constant C > 0. Moreover for s ≥ t E[η (s)|F0 ] = η (t)e−(s−t)/ , t/





Eηi (x, t)ηi (y, s) = 12 qi (x, y) exp − |t−s| ,

(2.6) i = 1, 2,

(2.7)

where qi (x, y) satisfy qi (x, y) = qi (y, x) and  Qi f (x) =

qi (x, y)f (y) dy,

i = 1, 2.



Notice that (v , T ) is not Markovian. So in the following we consider the Markov process (v , T , η1 , η2 ). 3. The global well-posedness and existence of stationary measure Let U = (u , η ), with u = (v , T ). To show the global well-posedness to (2.1)–(2.3), we introduce process Z solving 1 Z˙ = −AZ + √ η ,

Z (0) = 0.

One can see that 1 Z (t) = √

t



e−A(t−s) η (s) ds.

0

Then we have the following estimates. Lemma 3.1. For any integer k ≥ 0, there is a constant c > 0, such that for any t > 0, E Z (t) 2k ≤ cTrAk Q. We need some estimates on Zα (t). First E Z



(t) 20

1 = E

t t  0 0

 e−A(t−s) η (s), e−A(t−r) η (r) dr ds,

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then by (2.6)–(2.7) we have for any t > 0   E Z (t) 20 ≤ c q1 2L2 + q2 2L2 for some constant c > 0. Similarly by the regularity of η , for k > 0,   E Z (t) 2k ≤ c q1 2H k + q2 2H k . Now define w = v − Z ,1 ,

θ = T − Z ,2 ,

and u˜ = u − Z ,

where Z = (Z ,1 , Z ,2 ) Then, to obtain the global existence of strong solutions to the system (2.1)–(2.3), we just consider that of the following system ∂w + N1 (w + Z ,1 , w + Z ,1 ) + L(u˜ + Z ) + A1 w = 0, ∂t ∂θ + N2 (w + Z ,1 , θ + Z ,2 ) + A2 θ = , ∂t u˜ (0) = (w (0), θ (0)) = u 0 = (v0 , T0 ).

(3.1) (3.2) (3.3)

Theorem 3.1 (Global well-posedness of (2.1)–(2.3)). Assume that  ∈ V2 and (H1 ) hold, then (1) For any initial data u 0 ∈ V , there exists globally a unique strong solution u to (2.1)–(2.3), i.e., for any T > 0, u ∈ C(0, T ; V ) ∩ L2 (0, T ; (H 2 ())2 ) for P a.e. ω ∈ . (2) The process U is a Markov process in V := V × H 1 × H 1 . Remark 3.1. Result (1) is proved by Lemma 3.1 and the method for 3D stochastic primitive equations [13]. Result (2) can be justified by a standard argument [22, Theorem 9.8]. Proposition 3.1. Assume  ∈ V2 and Bρ = {u; u ≤ ρ, u ∈ V }. Then there exist r0 (ω,  1 ) and t (ω, ρ) ≤ −1 such that for any t0 ≤ t (ω, ρ), u t0 ∈ Bρ , u (0, ω, u t0 ) ≤ r0 (ω), where u (t, ω, u t0 ) is the strong solution of (2.1)–(2.3) with initial data u (t0 ) = u t0 . Moreover, the family of random variables {u (0, ω, u t0 ) : −∞ < t0 < t (ω, ρ)} is tight in V . Remark 3.2. The estimate in the above proposition can be followed by a similar discussion for 3D stochastic primitive equations [13] and the tight result can be derived by the similar method [12].

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Denote by Pt , t ≥ 0, the associated Markov semigroup to U . Let Pt ∗ be the dual semigroup which is defined, in space P r(V) consisting of probability measures on V, as   ∗ ϕdPt μ = Pt ϕdμ V

V

for all ϕ ∈ Cb (V) and μ ∈ P r(V). Then a probability measure μ ∈ P r(V) is called a stationary one if Pt ∗ μ = μ. Then we have the following result. Theorem 3.2 (The existence of stationary measure). Under the assumption (H1 ), the system of 3D random primitive equations (2.1)–(2.3) coupled with (2.4) has at least a stationary measure. Proof. By Proposition 3.1, the distribution of U in space V is tight, then by the Kryloff– Bogoliubov procedure [23] one can construct one stationary measure. 2 Let y∗ be a stationary measure of 3D random primitive equations (2.1)–(2.3) coupled with (2.4). Then U ∗ = (v ∗ , T ∗ , η ), the solution with initial value distributes as y∗ , is a stationary solution to 3D random primitive equations (2.1)–(2.3) coupled with (2.4). Then we call (v ∗ , T ∗ ) a stationary solution to 3D random primitive equations (2.1)–(2.3). 4. The diffusion limit for the 3D random primitive equations Notice that the distributions of η1 and η2 are independent of . We just consider the limit of (v , T ). We have showed the tightness of the distributions of {(v , T )}0< ≤1 in space C([0, ∞); V ). Further the following result determine the limit of (v , T ) in the sense of distribution. For this we first give the following limiting equation, 3D stochastic primitive equations ∂v ∂v + (v · ∇)v + (v) + f k × v + ∇pb − ∂t ∂z

z

∇T dz

−1

− v −

∂ 2v ∂z2

= W˙ 1 ,

∂T ∂ 2T ∂T + (v · ∇)T + (v) − T − 2 =  + W˙ 2 , ∂t ∂z ∂z

(4.1) (4.2)

0 ∇ · v dz = 0.

(4.3)

−1

with boundary value conditions (1.4)–(1.6) and initial value u(0) = (v0 , T0 ).

(4.4)

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Remark 4.1. The above equations are called as 3D stochastic primitive equations of the largescale ocean which are considered in the article [13]. There are two main reasons for considering the stochastic model. Firstly, it is impossible to accurately predict long-term oceanic motions in deterministic frameworks since the space scale and time scale of the forecast target are bounded in the deterministic oceanic forecasting. To predict long-term oceanic motions more objectively, it is suitable to apply some stochastic modes. Secondly, it is suitable and useful to consider the stochastic primitive equations of the large-scale ocean with a white in time noise since the primitive oceanic equations are usually used to understand the mechanism of long-term oceanic motions. We need some estimates on the solution. Lemma 4.1. Under the assumption, for any T > 0 the following estimate holds T sup E u (t) + E

∇u (s) 2 ds ≤ CT



0≤t≤T

0

for some constant CT > 0. Proof. The difficulty is the existence of singular terms. To overcome this, we introduce the following processes 1 Mt1, = [ η1 (t), v (t) − η1 (0), v (0)] + √

t

η1 (s), v (s)ds 0

t +

η1 (s), −A1 v (s) − Lu (s) − N1 (v (s), v (s)) ds 0

−

√

t

η1 (s), η1 (s) ds

0

and Mt2,

= [ η2 (t), T (t) − η2 (0), T (0)] +

1 √

t

η2 (s), T (s)ds 0

t +

η2 (s), −A2 T (s) − N2 (v (s), T (s)) +  ds 0

−

√

t

η2 (s), η2 (s) ds.

0

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For fixed > 0, by direct calculation, we have T sup E u (t) +

∇u (t) 2 dt ≤ C ,T ,



0≤t≤T

0

which implies that, by a direct verification [5, Lemma 2], Mt1, and Mt2, are square integrable t/ martingale with respect to F0 . Multiplying Eq. (2.1) with v on both sides in H1 yields  d

1 v (t) 2 + η1 (t), v (t) 2



= −A1 v (t) − Lu (t) − N1 (v , v ), v (t) dt − −A1 v (t) − Lu (t) − N1 (v , v ), η1  dt √ − η1 2 dt + dMt1, and 

1 d T (t) 2 + η2 (t), T (t) 2



= −A2 T (t) − N2 (v , T ) + , T (t) dt − −A2 T (t) − N2 (v , T ), η2  dt √ − η2 2 dt + dMt2, . Then for > 0 small enough, by the martingale property of Mt1, , Mt2, and the Gronwall lemma, we have the estimate of this lemma. 2 Theorem 4.1. Assume (v0 , T0 ) ∈ V converges in distribution to (v0 , T0 ) as → 0. The solution (v , T ) of 3D random primitive equations (1.8)–(1.14) converges in distribution, as → 0, to the solution of 3D stochastic primitive equations (4.1)–(4.4) in space C([0, ∞); V ). Remark 4.2. We apply a weak convergence method developed by Kushner [16] to prove this result. Such method is also applied to study the stochastic self-similarity in the stochastic Burgers equation [24]. Proof. Denote by (v, T ) one limit point in the sense of distribution of (v , T ) as → 0 in space C([0, ∞); V ). For simplicity we assume (v , T ) converges in distribution to (v, T ) as → 0 in C([0, ∞); V ). Notice that convergence in distribution is not enough to pass limit → 0, by the Skorohod theorem we can construct a new probability space and new random variables without changing distributions in C([0, ∞); V ), of which for simplicity we do not change the notations, such that (v0 , T0 ) converges almost surely to (v0 , T0 ) in V and (v , T ) converges almost surely to (v, T ) in C([0, ∞); V ). In the following, we shall prove that (v, T ) is a solution to (4.1)–(4.4). Now for any ϕ = (ϕ1 , ϕ2 ) ∈ C0∞ and C 3 -differentiable compactly supported real valued function F , we consider processes

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{F ( (v , T ), ϕ)}0< ≤1 .

(4.5)

We derive from Eqs. (1.8)–(1.14) F ( (v (t), T (t)), ϕ) − F ( (v0 , T0 ), ϕ) t =

  F  ( (v (s), T (s)), ϕ) G 1 (s), ϕ1  + G 2 (s), ϕ2  ds

0

1 +√

t

F  ( (v (s), T (s)), ϕ) η1 (s), ϕ1  ds

0

1 +√

t

F  ( (v (s), T (s)), ϕ) η2 (s), ϕ2  ds,

(4.6)

0

where G 1

= v − (v · ∇)v





− (v ) ∂v ∂z

z −fk ×v



− ∇pb

+

∇T dz

−1

and

G 2 = T − (v · ∇)T − (v ) ∂T ∂z + . To treat the singular terms in (4.6) we introduce F1 (t) :=

⎡∞ ⎤  1 ⎣ t/ ⎦  F ( (v (t), T (t)), ϕ) η1 (s), ϕ1  ds F0 √ E t

and ⎡ 1 F2 (t) := √ E ⎣

∞

⎤ t/ F  ( (v (t), T (t)), ϕ) η2 (s), ϕ2  ds F0 ⎦ .

t

Then by the property (2.6) of η we have F1 (t) =

√

F  ( (v (t), T (t)), ϕ) η1 (t), ϕ1 

and F2 (t) = Moreover

√

F  ( (v (t), T (t)), ϕ) η2 (t), ϕ2 .

B. Guo et al. / J. Differential Equations 259 (2015) 2388–2407

√ E|Fi (t)| ≤ C ,

2401

i = 1, 2

for some constant C > 0. Next we construct a martingale depends on and pass the limit → 0 in this martingale. For this we introduce the pseudo-differential operator A defined by   t/ A X(t) = P − lim 1δ E X(t + δ) − X(t) | F0 (4.7) δ→0

t/

for any F0 measurable function X with supt E|X(t)| < ∞. Then Ethier and Kurtz’s proposition [9, Proposition 2.7.6] yields that t X(t) −

A X(s) ds 0

t/

is a martingale with respect to F0 . Now define (Y , Z ) as Y (t) = F ( (v (t), T (t)), ϕ) + F1 (t) + F2 (t),

Z (τ ) = A Y (t).

Then Z (t)

  = F  ( (v (t), T (t)), ϕ) G 1 (t), ϕ1  + G 2 (t), ϕ2    √ + F  ( (v (t), T (t)), ϕ) η1 (t), ϕ1  G 1 (t), ϕ1  + G 2 (t), ϕ2    √ + F  ( (v (t), T (t)), ϕ) η2 (t), ϕ2  G 1 (t), ϕ1  + G 2 (t), ϕ2  + F  ( (v (t), T (t)), ϕ) η1 (t), ϕ1 2 + F  ( (v (t), T (t)), ϕ) η2 (t), ϕ2 2 + 2F  ( (v (t), T (t)), ϕ) η1 (t), ϕ1  η2 (t), ϕ2 .

In fact first by Eq. (4.6) and the definition of A , we have A F ( (v (t), T (t)), ϕ)

  = F  ( (v (t), T (t)), ϕ) G 1 (t), ϕ1  + G 2 (t), ϕ2  +

√1 F  ( (v (t), T (t)), ϕ) η (t), ϕ1  1

+

√1 F  ( (v (t), T (t)), ϕ) η (t), ϕ2 . 2

Now by the construction of η and F1 ,  t/  E F1 (t + δ)|F0 √ ! = E F  ( (v (t + δ), T (t + δ)), ϕ)  t/ " − F  ( (v (t), T (t)), ϕ) η1 (t + δ), ϕ1  F0 √ + F  ( (v (t), T (t)), ϕ) η1 (t), ϕ1 e−δ/ .

(4.8)

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Then A F1 (t) =

√

  F  ( (v (t), T (t)), ϕ) η1 (t), ϕ1  G 1 (t), ϕ1  + G 2 (t), ϕ2 

+ F  ( (v (t), T (t)), ϕ) η1 (t), ϕ1 2 + F  ( (v (t), T (t)), ϕ) η1 (t), ϕ1  η2 (t), ϕ2  −

√1 F  ( (v (t), T (t)), ϕ) η (t), ϕ1 . 1

Similarly A F2 (t) =

√

  F  ( (v (t), T (t)), ϕ) η2 (t), ϕ1  G 1 (t), ϕ1  + G 2 (t), ϕ2 

+ F  ( (v (t), T (t)), ϕ) η2 (t), ϕ2 2 + F  ( (v (t), T (t)), ϕ) η1 (t), ϕ1  η2 (t), ϕ2  −

√1 F  ( (v (t), T (t)), ϕ) η (t), ϕ2  2

which shows (4.8). Now denote by Zi (t), i = 1, · · · , 5, the five terms on the righthand side of Z (t), then   √ E |Z2 (t)| + |Z3 (t)| = O( ),

→ 0.

To pass the limit → 0, we further need more processes. Define F3 (t) = F  ( (v (t), T (t)), ϕ)

∞

 t/  E η1 (s), ϕ1 2 − 12 Q1 ϕ1 , ϕ1  F0 ds,

t

F4 (t) = F  ( (v (t), T (t)), ϕ)

∞

 t/  E η2 (s), ϕ2 2 − 12 Q2 ϕ2 , ϕ2  F0 ds

t

and F5 (t) = F  ( (v (t), T (t)), ϕ)

∞

 t/  E η1 (s), ϕ1  η2 (s), ϕ2  F0 ds.

t

Then by the property of η we have   F3 (t) = 2 F  ( (v (t), T (t)), ϕ) η1 (t), ϕ1 2 − 12 Q1 ϕ1 , ϕ1  ,   F4 (t) = 2 F  ( (v (t), T (t)), ϕ) η2 (t), ϕ2 2 − 12 Q2 ϕ2 , ϕ2  and   F5 (t) = 2 F  ( (v (t), T (t)), ϕ) η1 (t), ϕ1  η2 (t), ϕ2  .

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Further by direct calculation and estimates on (v , T ) we have sup EFi (t) = O( ),

i = 3, 4, 5,

as

→ 0.

t≥0

Moreover   A F3 (t) = F  ( (v (t), T (t)), ϕ) 12 Q1 ϕ1 , ϕ1  − η1 (t), ϕ1 2 + R3 (t), and   A F4 (t) = F  ( (v (t), T (t)), ϕ) 12 Q2 ϕ2 , ϕ2  − η2 (t), ϕ2 2 + R4 (t) with sup E|R3 (t)| = O( )

and

t≥0

sup E|R4 (t)| = O( ) t≥0

and sup E|A F5 (t)| = O( )

as

→ 0.

t≥0

t/

Now we have the following F0

martingale

M (t) = F ( (v (t), T (t)), ϕ) − F ( (v0 , T0 ), ϕ) + F1 (t) + F2 (t) + F3 (t) + F4 (t) t + F5 (t) −

  F  ( (v (s), T (s)), ϕ) G 1 (s), ϕ1  + G 2 (s), ϕ2  ds

0

t −

1 2

F  ( (v (s), T (s)), ϕ) Q1 φ1 , φ1  ds

0

t −

1 2

F  ( (v (s), T (s)), ϕ) Q2 φ2 , φ2  ds + R (t)

0

where t R (t) =

[Z2 (s) + Z3 (s) + R3 (s) + R4 (s) + A F5 (s)] ds



0

with E|R (t)| = O( ) as → 0. Now passing to the limit → 0 in M (t) shows the distribution of the limit (v, T ) solves the following martingale problem

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M(t) = F ( (v(t), T (t)), ϕ) − F ( (v0 , T0 ), ϕ) t −

  F  ( (v(s), T (s)), ϕ) G1 (s), ϕ1  + G2 (s), ϕ2  ds

0

t −

1 2

F  ( (v(s), T (s)), ϕ) Q1 φ1 , φ1  ds

0

t −

1 2

F  ( (v(s), T (s)), ϕ) Q2 φ2 , φ2  ds

0

which is equivalent to the martingale solution to the 3D stochastic primitive equations (4.1)–(4.4) [18]. By the global well-posedness of Eqs. (4.1)–(4.4), we complete the proof. 2 By Theorem 4.1 we have the following result on convergence of stationary solution of the 3D random primitive equations. Denote by P∗ = D(vˆ ∗ , Tˆ ∗ , ηˆ 1 , ηˆ 2 ), a stationary statistical solution (see Appendix A) to the system of random primitive equations (1.8)–(1.14) coupled with (2.4). Let P∗ = D(vˆ ∗ , Tˆ ∗ ), then we have Corollary 4.1. For → 0, there is sequence n → 0, as n → ∞, such that P∗ n → P∗

weakly as n → ∞

where P∗ is a probability measure on C([0, ∞); V ), which is a stationary statistical solution to 3D stochastic primitive equations (4.1)–(4.4). Proof. By the tightness of {P∗ } = {D(vˆ ∗ , Tˆ ∗ )} in C((0, ∞]; V ), there is a sequence n → 0, such that D(vˆ ∗ n (0), Tˆ ∗ n (0)) converges weakly to D(vˆ ∗ (0), Tˆ ∗ (0)) for some random variable (vˆ ∗ (0), Tˆ ∗ (0)). Denote by (vˆ ∗ , Tˆ ∗ ) the solution to 3D stochastic primitive equations (4.1)–(4.4) with initial data distributes as (vˆ ∗ (0), Tˆ ∗ (0)). Then by Theorem 4.1, P∗ n → P weakly as

n → ∞.

Here P = D(vˆ ∗ , Tˆ ∗ ), by the stationary property of P∗ n , is a statistical stationary solution to 3D stochastic primitive equations (4.1)–(4.4). 2 The above result shows that for → 0 and any stationary solution (v ∗ , T ∗ ) to 3D random primitive equations, there is a subsequence n → 0, as n → ∞, such that (v ∗ n , T ∗ n ) converges in distribution to a stationary solution (v ∗ , T ∗ ) to 3D stochastic primitive equations. Acknowledgments The authors would like to express their heartful thanks to the referee for the valuable comments and suggestions. This work was supported by 973 Program (grant No. 2013CB834100), and National Natural Science Foundation of China (91130005, 11271052).

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Appendix A. Statistical solution We give an introduction of statistical solution of 3D random primitive equations (1.8)–(1.14) coupled with (2.4). Statistical solution was introduced to study universal properties of turbulent flows [10,11,25, e.g.]. We say the system of 3D random primitive equations (1.8)–(1.14) coupled with (2.4) has a statistical solution in space C([0, ∞); V) if there is a probability measure P supported on C([0, ∞); V), and there are processes (vˆ , Tˆ , ηˆ 1 , ηˆ 2 ) ∈ C([0, ∞); V), Wˆ = (Wˆ 1 , Wˆ 2 ) defined on a new probability space, such that D(vˆ , Tˆ , ηˆ 1 , ηˆ 2 ) = P ; Wˆ 1 and Wˆ 2 are Wiener processes distribute same as W1 and W2 respectively; D(vˆ (0), Tˆ (0)) = D(v0 , T0 ), D(ηˆ 1 , ηˆ 2 ) = D(η1 , η2 ) and (vˆ (0), Tˆ (0)) are independent from Wˆ 1 and Wˆ 2 ; 4. The process (vˆ , Tˆ ) is a weak solution of 3D random primitive equations (1.8)–(1.14) with η1 , η2 replaced by ηˆ 1 , ηˆ 2 respectively. Here ηˆ = (ηˆ 1 , ηˆ 2 ) is stationary process solving (2.4) with W replaced by Wˆ .

1. 2. 3.

The above definition of statistical solutions are also used in [3] to study stochastic 3D Navier– Stokes equations. A stationary statistical solution is a statistical solution, a Borel measure P , which is invariant under the following translation on C([0, ∞); V) (v(·), T (·), η1 (·), η2 (·)) → (v(· + t), T (· + t), η1 (· + t), η2 (· + t)),

t ≥0

for (v, T , η1 , η2 ) ∈ C([0, ∞); V). For a statistical solution of the system of random 3D primitive equations (1.8)–(1.14) coupled with (2.4), we denote by P t = D(vˆ (· + t), Tˆ (· + t), ηˆ 1 (· + t), ηˆ 2 (· + t)), which is also a statistical solution of the random 3D primitive equations (1.8)–(1.14) coupled with (2.4). For a stationary statistical solution P∗ we have P∗t = P∗ ,

t ≥ 0.

The following result shows that the relation between the stationary measure and stationary statistical solution. Lemma A.1. The 3D random primitive equations (1.8)–(1.14) coupled with (2.4) has a stationary measure supported on V, then there is a stationary statistical solution in C([0, ∞); V). Proof. The proof is direct by the following observation [3]: Let P∗ = D(vˆ ∗ , Tˆ ∗ , ηˆ 1∗ , ηˆ 2∗ ) be a stationary statistical solution to the 3D random primitive equations coupled with (2.4), then y∗ = D(vˆ ∗ (0), Tˆ ∗ (0), ηˆ 1∗ (0), ηˆ 2∗ (0)) is a stationary measure for the Markov process defined by the 3D random primitive equations coupled with (2.4); Conversely, assume y∗ is a stationary measure of the random 3D primitive equations coupled with (2.4), let (v ∗ , T ∗ , η1 , η2 ) be a solution of the 3D random primitive equations coupled with (2.4) with D(v ∗ (0), T ∗ (0), η1 (0), η2 (0)) = y∗ , then

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P∗ = D(v ∗ , T ∗ , η1 , η2 ) is a stationary statistical solution of the 3D random primitive equations coupled with (2.4). 2 For stochastic 3D primitive equation (4.1)–(4.4), a statistical solution in space C([0, ∞; V ) is a probability measure P supported on C([0, ∞; V ) and there are processes (v, ˆ Tˆ ) ∈ C([0, ∞; V ), Wˆ = (Wˆ 1 , Wˆ 2 ) defined on a new probability space such that i ii iii iv

D(v, ˆ Tˆ ) = P; Wˆ 1 and Wˆ 2 are Wiener processes distribute same as W1 and W2 respectively; D(v(0), ˆ Tˆ (0)) = D(v0 , T0 ) and (v(0), ˆ Tˆ (0)) are independent from Wˆ 1 and Wˆ 2 ; The process (v, ˆ Tˆ ) is a weak solution of stochastic 3D primitive equations (4.1)–(4.4) with W replaced by Wˆ .

Notice the above definition of statistical solution is in fact a solution to a martingale problem [18, Chapter V]. Similarly we also have stationary statistical solution and relation in Lemma A.1 holds. References [1] L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York–London–Sydney, 1974. [2] C. Cao, E.S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large-scale ocean and atmosphere dynamics, Ann. of Math. 166 (2007) 245–267. [3] I. Chueshov, S. Kuksin, Stochastic 3D Navier–Stokes equations in a thin domain and its α-approximation, Phys. D 10–12 (2008) 1352–1367. [4] A. Debussche, J. Vovelle, Diffusion limit for a stochastic kinetic problem, 2011, preprinted. [5] M.A. Diop, B. Iftimie, E. Pardoux, A.L. Piatnitski, Singular homogenization with stationary in time and periodic in space coefficients, J. Funct. Anal. 231 (2006) 1–46. [6] J. Duan, H. Gao, B. Schmalfuss, Stochastic dynamics of a coupled atmosphere-ocean model, Stoch. Dyn. 2 (3) (2002) 357–380. [7] J. Duan, P.E. Kloeden, B. Schmalfuss, Exponential stability of the quasi-geostrophic equation under random perturbations, Progr. Probab. 49 (2001) 241–256. [8] J. Duan, B. Schmalfuss, The 3D quasi-geostrophic fluid dynamics under random forcing on boundary, Commun. Math. Sci. 1 (2003) 133–151. [9] S.N. Ethier, T.G. Kurtz, Markov Processes: Characterization and Convergence, John Wiley & Sons, 1986. [10] C. Foias, Statistical study of Navier–Stokes equations I, Rend. Semin. Mat. Univ. Padova 48 (1972) 219–348. [11] C. Foias, Statistical study of Navier–Stokes equations II, Rend. Semin. Mat. Univ. Padova 49 (1973) 9–123. [12] H. Gao, C. Sun, Random attractor for the 3D viscous stochastic primitive equations with additive noise, Stoch. Dyn. 9 (2009) 293–313. [13] B. Guo, D. Huang, 3D stochastic primitive equations of the large-scale ocean: global well-posedness and attractors, Comm. Math. Phys. 286 (2009) 697–723. [14] F. Guillén-González, N. Masmoudi, M.A. Rodríguez-Bellido, Anisotropic estimates and strong solutions for the primitive equations, Differential Integral Equations 14 (2001) 1381–1408. [15] Y. Kifer, L2 -diffusion approximation for slow motion in averaging, Stoch. Dyn. 3 (2003) 213–246. [16] H. Kushner, Approximation and Weak Convergence Methods for Random Processes with Applications to Stochastic Systems Theory, MIT Press, 1984. [17] J.L. Lions, R. Temam, S. Wang, On the equations of the large scale ocean, Nonlinearity 5 (1992) 1007–1053. [18] M. Metivier, Stochastic Partial Differential Equations in Infinite Dimensional Spaces, Scuola Normale Superiore, Pisa, 1988. [19] A. Majda, E.V. Eijnden, A mathematical framework for stochastic climate models, Comm. Pure Appl. Math. 54 (2001) 891–974. [20] P. Müller, Stochastic forcing of quasi-geostrophic eddies, in: R.J. Adler, P. Müller, B. Rozovskii (Eds.), Stochastic Modelling in Physical Oceanography, Birkhäuser, Basel, 1996. [21] J. Pedlosky, Geophysical Fluid Dynamics, 2nd edition, Springer-Verlag, Berlin/New York, 1987.

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