# On the Stochastic 3D Navier-Stokes-α Model of Fluids Turbulence

Authors: Gabriel Deugoue; Mamadou Sango
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

We investigate the stochastic 3D Navier-Stokes-α model which arises in the modelling of turbulent flows of fluids. Our model contains nonlinear forcing terms which do not satisfy the Lipschitz conditions. The adequate notion of solutions is that of probabilistic weak solution. We establish the existence of a such of solution. We also discuss the uniqueness.


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## Body

## 1. Introduction

In this paper, we are interested in the study of probabilistic weak solutions of the 3D Navier-Stokes-α(NS-α) model (also known as the Lagrangien averaged Navier-Stokes-alpha model or the viscous Camassa-Holm equations) with homogeneous Dirichlet boundary conditions in a bounded domains in the case in which random perturbations appear. To be more precise, let D be a connected and bounded open subset of R3 with C2 boundary ∂D and a final time T>0. We denote by A the Stokes operator and consider the system(1.1)∂t(u-αΔu)+ν(Au-αΔ(Au))+(u·∇)(u-αΔu)-α∇u*·Δu+∇p=F(t,u)+G(t,u)dWdt,inD×(0,T),∇·u=0,inD×(0,T),u=0,Au=0,on∂D×(0,T),u(0)=u0,inD,
where u=(u1,u2,u3) and p are unknown random fields on D×(0,T), representing, respectively, the large-scale velocity and the pressure, in each point of D×(0,T). The constant ν>0 and α>0 are given, and represent, respectively, the kinematic viscosity of the fluid, and the square of the spatial scale at which fluid motion is filtered. The terms F(t,u) and G(t,u)(dW/dt) are external forces depending on u, where W is an Rm-valued standard Wiener process. Finally, u0 is a given nonrandom velocity field.The deterministic version of (1.1), that is, when G=0 has been the object of intense investigations over the last years [1–5] and the initial motivation was to find a closure model for the 3D turbulence-averaged Reynolds model. A key interest in the model is the fact that it serves as a good approximation to the 3D Navier-Stokes equations. One of the main reasons justifying its use is the high computational cost that the Navier-Stokes model requires. Many important results have been obtained in the deterministic case. More precisely, the global well posedness of weak solutions for the NS-α model on bounded domains has been established in [6, 7] amongst others, and the asymptotic behavior can be found in [6]. Similar results have been proved by Foias et al. [8] in the case of periodic boundary conditions.However, in order to consider a more realistic model our problem, it is sensible to introduce some kind of noise in the equations. This may reflect, some environmental effects on the phenomena, some external random forces, and so forth. To the best of our knowledge, the existence and uniqueness of solutions of the stochastic version (1.1) which we consider in this paper has only been analyzed in [9] (see also [10]) in the case of Lipschitz assumptions on F and G. The case of non-Lipschitz assumptions on the coefficients F and G is the main concern of the present paper. This question has been opened till now. The general motivation for studying weak rather than strong solutions of stochastic equations is that existence of weak solutions can be carried through under weaker regularity on the coefficients. This was pointed out, for instance, in [11].In this paper, we will establish the existence of probabilistic weak solutions for the problem (1.1) under appropriate conditions on the data. Under the strong assumptions on F and G, we prove the uniqueness of weak solutions. The method used for the proof of our existence results is different from the method in [9]. To prove the existence, we use the Galerkin approximation method employing special bases, combined with some famous theorems of probabilistic nature due to Prokhorov [12] and Skorokhod [13].The paper is organized as follows. In Section2, we establish some properties of nonlinear term appearing in our equations. The rigorous statement of our problem as well as the main results are included in Section 3 and we show how our problem can be reformulated as an abstract stochastic model. Section 4 is devoted to the proof of our main results.


## 2. Properties of the Nonlinear Terms in (1.1)

Following [9], we establish some properties of the nonlinear term (u·∇)(u-αΔu)-α∇u*·Δu appearing in (1.1).We denote by(·,·) and |·|, respectively, the scalar product and associated norm in (L2(D))3, and by (∇u,∇v), the scalar product in ((L2(D))3)3 of the gradients of u and v. We consider the scalar product in (H01(D))3 defined by(2.1)((u,v))=(u,v)+α(∇u,∇v),u,v∈(H01(D))3,
where its associated norm ∥·∥ is, in fact, equivalent to the usual gradient norm. We denote by H the closure in (L2(D))3 of the set(2.2)𝒱={v∈(𝒟(D))3:∇·v=0inD},
and by V the closure of 𝒱 in (H01(D))3. Then H is a Hilbert space equipped with the inner product of (L2(D))3, and V is a Hilbert subspace of (H01(D))3.Denote byA the Stokes operator, with domain D(A)=(H2(D))3∩V, defined by(2.3)Aw=-𝒫(Δw),w∈D(A),
where 𝒫 is the projection operator from (L2(D))3 onto H. Recall that as ∂D is C2,|Aw| defines in D(A) a norm which is equivalent to the (H2(D))3 norm, that is, there exists a constant c1(D), depending only on D, such that(2.4)∥w∥(H2(D))3≤c1(D)|Aw|,∀w∈D(A),
and so D(A) is a Hilbert space with respect to the scalar product(2.5)(v,w)D(A)=(Av,Aw).
For u∈D(A) and v∈(L2(D))3, we define (u·∇)v as the element of (H-1(D))3 given by(2.6)〈(u·∇)v,w〉-1=∑i,j=13〈∂ivj,uiwj〉-1,∀w∈(H01(D))3,
where by 〈u,v〉-1, we denote either the duality product between (H-1(D))3 and (H01(D))3 or between H-1(D) and H01(D).Observe that (2.6) is meaningful, since H2(D)⊂L∞(D) and H01(D)⊂L6(D) with continuous injections. This implies that uiwj∈H01(D), and there exists a constant c2(D)>0, depending only on D, such that(2.7)|〈(u·∇)v,w〉-1|≤c2(D)|Au||v|∥w∥,∀(u,v,w)∈D(A)×(L2(D))3×(H01(D))3.
Observe also that if v∈(H1(D))3, then the definition above coincides with the definition of (u·∇)v as the vector function whose components are ∑i=13ui∂ivj, for j=1,2,3. However, as it not known whether the solutions of the stochastic problem (1.1) have the same regularity as the deterministic case (we only can ensure H2 instead of H3), the present extension is necessary.Now, ifu∈D(A), then ∇u*∈(H1(D))3×3⊂(L6(D))3×3, and consequently, for v∈(L2(D))3, we have that ∇u*·v∈(L3/2(D))3⊂(H-1(D))3, with(2.8)〈∇u*·v,w〉-1=∑i,j=13∫D(∂jui)viwjdx,∀w∈(H01(D))3.
It follows that there exists a constant c3(D), depending only on D, such that(2.9)|〈∇u*·v,w〉-1|≤c3(D)|Au||v|∥w∥,∀(u,v,w)∈D(A)×(L2(D))3×(H01(D))3.
We have the following results.Proposition 2.1.
For all(u,w)∈D(A)×D(A) and for all v∈(L2(D))3, it follows that
(2.10)〈(u·∇)v,w〉-1=-〈∇w*·v,u〉-1.Proof.
If(u,w)∈D(A)×D(A), then for each i,j=1,2,3, we have uiwj∈H01(D) and consequently
(2.11)〈∂ivj,uiwj〉-1=-∫Dvj∂i(uiwj)dx=-∫Dvjwj∂iuidx-∫Dvjui∂iwjdx,
using ∇·u=0, we have (2.10).Consider now the bilinear form defined by(2.12)b*(u,v,w)=〈(u·∇)v,w〉-1+〈∇u*·v,w〉-1,(u,v,w)∈D(A)×(L2(D))3×(H01(D))3.Proposition 2.2.
The bilinear formb* satisfies
(2.13)b*(u,v,w)=-b*(w,v,u),∀(u,v,w)∈D(A)×(L2(D))3×D(A),
and consequently,
(2.14)b*(u,v,u)=0,∀(u,v)∈D(A)×(L2(D))3.
Moreover, there exists a constant c(D)>0, depending only on D, such that
(2.15)|b*(u,v,w)|≤c(D)|Au||v|∥w∥,∀(u,v,w)∈D(A)×(L2(D))3×(H01(D))3,|b*(u,v,w)|≤c(D)∥u∥|v||Aw|,∀(u,v,w)∈D(A)×(L2(D))3×D(A).
Thus, in particular, b* is continuous on D(A)×(L2(D))3×(H01(D))3.Proof.
The proof is straightforward consequences of (2.7), (2.9), and (2.10).


## 3. Statement of the Problem and the Main Results

We now introduce some probabilistic evolutions spaces.Let(Ω,F,{Ft}0≤t≤T,P) be a filtered probability space and let X  be a Banach space. For r,q≥1, we denote by(3.1)Lp(Ω,F,P;Lr(0,T;X))
the space of functions  u=u(x,t,ω)  with values in X defined on  [0,T]×Ω and such that(1)
u  is measurable with respect to  (t,ω) and for almost all t, u is Ft measurable,(2)
(3.2)∥u∥Lp(Ω,F,P;Lr(0,T;X))=[E(∫0T∥u∥Xrdt)p/r]1/r<∞,
where E denotes the mathematical expectation with respect to the probability measure P.The spaceLp(Ω,F,P;Lr(0,T;X)) so defined is a Banach space.Whenr=∞, the norm in Lp(Ω,F,P;L∞(0,T;X)) is given by(3.3)∥u∥Lp(Ω,F,P;L∞(0,T;X))=(Esupess0≤t≤T∥u∥Xp)1/p.We make precise our assumptions on (1.1).We start with the nonlinear functionF and G. We assume that(3.4)F:(0,T)×V→(H-1(D))3,measurablea.e.t,u↣F(t,u):continuousfromVto(H-1(D))3∥F(t,u)∥(H-1(D))3≤c(1+∥u∥),G:(0,T)×V→((L2(D))3)m,measurablea.e.t,u↣G(t,u):continuousfromVto((L2(D))3)m|G(t,u)|((L2(D))3)m≤c(1+∥u∥).
We will define the concept of weak solution of the problem (1.1), namely, the following.Definition 3.1.
A weak solution of (1.1) means a system (Ω,ℱ,{ℱt}0≤t≤T,𝒫,W,u) such that  (1)
(Ω,ℱ,𝒫) is a probability space, ({ℱt},0≤t≤T) is a filtration,(2)
W is an m-dimensional {ℱt}  standard Wiener process,(3)
u(t) is ℱt adapted for  all t∈[0,T]:(3.5)u∈Lp(Ω,ℱ,𝒫;L2(0,T,D(A)))∩Lp(Ω,ℱ,𝒫;L∞(0,T,V)),∀1≤p<∞,(4)
for almost allt∈(0,T), the following equation holds 𝒫-a.s.(3.6)((u(t),Φ))+ν∫0t(u(s)+αAu(s),AΦ)ds+∫0tb*(u(s),u(s)-αΔu(s),Φ)ds=((u0,Φ))+∫0t〈F(s,u(s)),Φ〉-1ds+(∫0tG(s,u(s))dW(s),Φ)
for all Φ∈D(A).Our two major results are as follows.Theorem 3.2 (Existence).
Assume (3.4) and u0∈V. Then there exists a weak solution (Ω,ℱ,{ℱt}0≤t≤T,𝒫,W,u) of (1.1) in the sense of Definition 3.1.
Moreoveru∈Lp(Ω,ℱ,𝒫;C([0,T];V)), and there exists a unique p̃∈L2(Ω,ℱt,𝒫;H-1(0,t;H-1(D)), for all t∈[0,T], such that 𝒫-a.s.
(3.7)∂t(u-αΔu)+ν(Au-αΔ(Au))+(u·∇)(u-αΔu)-α∇u*·Δu+∇p̃=F(t,u)+G(t,u)dWdt,in(𝒟′((0,T)×D))3,∫Dp̃dx=0,in𝒟′(0,T),
where G(t,u)(dW/dt) denotes the time derivative of ∫0tG(s,u(s))dWs, that is, by definition
(3.8)G(t,u)dWdt=∂t(∫0.G(s,u(s))dWs),in𝒟′(0,T;(L2(D))3),𝒫-a.s.Corollary 3.3 (Uniqueness).
Assume thatF and  G are Lipschitz with respect to the second variable  u0∈V. Then there exists a unique weak solution of problem (1.1) in the sense of Definition 3.1.
Moreover, two strong solutions on the same Brownian stochastic basis coincide a.s.


### 3.1. Formulation of Problem (1.1) as an Abstract Problem

We will rewrite our model as an abstract problem.We identifyV with its topological dual V′ and we have the Gelfand triple D(A)⊂V⊂D(A)′.We denote by〈·,·〉 the duality product between D(A)′ and D(A). We define(3.9)〈Ãu,v〉=ν(Au,v)+να(Au,Av),u,v∈D(A).
It is clear that for all v∈D(A),(3.10)2〈Ãu,v〉=2ν(Av,v)+2να(Av,Av)≥2να|Av|2,
and, if we denote by λk and wk,k≥1, the eigenvalues, and their corresponding eigenvalues associated to A, then(3.11)〈Ãwk,v〉=νλk((wk,v)).
Thus, taking(3.12)α̃=2να,
we have(a)
Ã is a linear continuous operator Ã∈ℒ(D(A),D(A)′), such that
(3.13)(a1)Ãisself-adjoint(a2)thereexistsα̃>0,suchthat2〈Ãv,v〉≥α̃∥v∥D(A)2,∀v∈D(A).
On the other hand, denote
(3.14)〈B̃(u,v),w〉=b*(u,v-αΔv,w),(u,v,w)∈D(A)×D(A)×D(A),((F̃(t,u),w))=〈F(t,u),w〉-1,(u,w)∈V×V.
Thus it is straightforward to check that if we take
(3.15)c1=(1+α)c1(D)c(D),
then we obtain that(b)
B̃:D(A)×D(A)→D(A)′ is  a  bilinear  mapping  such that
(3.16)(b1)〈B̃(u,v),u〉=0,∀u,v∈D(A),(3.17)(b2)∥B̃(u,v)∥D(A)′≤c1∥u∥∥v∥D(A),∀u,v∈D(A)×D(A),(3.18)(b3)|〈B̃(u,v),w〉|≤c1∥u∥D(A)∥v∥D(A)∥w∥,∀u,v,w∈D(A).(c)
F̃:(0,T)×V→V, measurable such that
(3.19)(c1)a.e.t,u↣F̃(t,u):continuousfromVtoV(c2)∥F̃(t,u)∥≤c(1+∥u∥).
Now, let I denote the identity operator in H, and define G̃(t,u) as
(3.20)G̃(t,u)=(I+αA)-1∘𝒫∘G(t,u),u∈V.I+αA is bijective from D(A) onto H, and
(3.21)(((I+αA)-1f,w))=(f,w),∀f∈H,w∈V.
Thus, for eachf∈H,(3.22)∥(I+αA)-1f∥2=(f,u)≤|f||u|,
where u=(I+αA)-1f, that is, (u,wk)+α(Au,wk)=(f,wk), for all k≥1, so (1+αλk)(u,wk)=(f,wk), which implies
(3.23)(u,wk)=1(1+αλk)(f,wk)≤11+αλ1(f,wk),|u|2=∑k=1∞(u,wk)2≤1(1+αλ1)2∑k=1∞(f,wk)2=1(1+αλ1)2|f|2.
Therefore,
(3.24)∥(I+αA)-1f∥2≤11+αλ1|f|2,
and, consequently, taking
(3.25)c̃=c1+αλ1,
we obtain that(d)
G̃:(0,T)×V→V⊗m, measurable such that
(3.26)(d1)a.e.t,u↣G̃(t,u):continuousfromVtoV⊗m(d2)∥G̃(t,u)∥V⊗m≤c̃(1+∥u∥),whereV⊗mistheproductofmcopiesofV. Next, for each j≥1, and all (t,u,Φ)∈(0,T)×V×D(A), we have(3.27)(G(t,u),Φ)=((I+αA)G̃(t,u),Φ)=((G̃(t,u),Φ)),
and, for all u∈L2(Ω,ℱ,𝒫;L∞(0,T;V)), (t,Φ)∈(0,T)×D(A), it follows that(3.28)(∫0tG(s,u(s))dW(s),Φ)=∑j=1d∫0t(Gj(s,u(s)),Φ)dWj(s)=∑j=1d∫0t(G̃j(s,u(s),Φ))dWj(s)=((∫0tG̃(s,u(s))dW(s),Φ)).
Consequently, in this abstract framework, a weak solution (Ω,ℱ,{ℱt}0≤t≤T,𝒫,W,u) of (1.1) is equivalently as follows.Definition 3.4.
It holds that(1)
(Ω,ℱ,𝒫) is a probability space, ({ℱt},0≤t≤T) is a filtration,(2)
W is a m-dimensional {ℱt} standard Wiener process,(3)
u(t) is ℱt adapted for  all t∈[0,T](3.29)u∈Lp(Ω,ℱ,𝒫;L2(0,T,D(A)))∩Lp(Ω,ℱ,𝒫;L∞(0,T,V)),∀1≤p<∞,(4)
for almost allt∈(0,T), the following equation holds 𝒫-a.s.(3.30)u(t)+∫0tÃu(s)ds+∫0tB̃(u(s),u(s))ds=u0+∫0tF̃(s,u(s))ds+∫0tG̃(s,u(s))dW(s),
as an equality in D(A)′.Remark 3.5.
However, (3.30) implies that u∈𝒞(0,T;D(A)′), then u is weakly continuous in V  [14, page 263] and the initial condition is meaningful.


## 3.1. Formulation of Problem (1.1) as an Abstract Problem

We will rewrite our model as an abstract problem.We identifyV with its topological dual V′ and we have the Gelfand triple D(A)⊂V⊂D(A)′.We denote by〈·,·〉 the duality product between D(A)′ and D(A). We define(3.9)〈Ãu,v〉=ν(Au,v)+να(Au,Av),u,v∈D(A).
It is clear that for all v∈D(A),(3.10)2〈Ãu,v〉=2ν(Av,v)+2να(Av,Av)≥2να|Av|2,
and, if we denote by λk and wk,k≥1, the eigenvalues, and their corresponding eigenvalues associated to A, then(3.11)〈Ãwk,v〉=νλk((wk,v)).
Thus, taking(3.12)α̃=2να,
we have(a)
Ã is a linear continuous operator Ã∈ℒ(D(A),D(A)′), such that
(3.13)(a1)Ãisself-adjoint(a2)thereexistsα̃>0,suchthat2〈Ãv,v〉≥α̃∥v∥D(A)2,∀v∈D(A).
On the other hand, denote
(3.14)〈B̃(u,v),w〉=b*(u,v-αΔv,w),(u,v,w)∈D(A)×D(A)×D(A),((F̃(t,u),w))=〈F(t,u),w〉-1,(u,w)∈V×V.
Thus it is straightforward to check that if we take
(3.15)c1=(1+α)c1(D)c(D),
then we obtain that(b)
B̃:D(A)×D(A)→D(A)′ is  a  bilinear  mapping  such that
(3.16)(b1)〈B̃(u,v),u〉=0,∀u,v∈D(A),(3.17)(b2)∥B̃(u,v)∥D(A)′≤c1∥u∥∥v∥D(A),∀u,v∈D(A)×D(A),(3.18)(b3)|〈B̃(u,v),w〉|≤c1∥u∥D(A)∥v∥D(A)∥w∥,∀u,v,w∈D(A).(c)
F̃:(0,T)×V→V, measurable such that
(3.19)(c1)a.e.t,u↣F̃(t,u):continuousfromVtoV(c2)∥F̃(t,u)∥≤c(1+∥u∥).
Now, let I denote the identity operator in H, and define G̃(t,u) as
(3.20)G̃(t,u)=(I+αA)-1∘𝒫∘G(t,u),u∈V.I+αA is bijective from D(A) onto H, and
(3.21)(((I+αA)-1f,w))=(f,w),∀f∈H,w∈V.
Thus, for eachf∈H,(3.22)∥(I+αA)-1f∥2=(f,u)≤|f||u|,
where u=(I+αA)-1f, that is, (u,wk)+α(Au,wk)=(f,wk), for all k≥1, so (1+αλk)(u,wk)=(f,wk), which implies
(3.23)(u,wk)=1(1+αλk)(f,wk)≤11+αλ1(f,wk),|u|2=∑k=1∞(u,wk)2≤1(1+αλ1)2∑k=1∞(f,wk)2=1(1+αλ1)2|f|2.
Therefore,
(3.24)∥(I+αA)-1f∥2≤11+αλ1|f|2,
and, consequently, taking
(3.25)c̃=c1+αλ1,
we obtain that(d)
G̃:(0,T)×V→V⊗m, measurable such that
(3.26)(d1)a.e.t,u↣G̃(t,u):continuousfromVtoV⊗m(d2)∥G̃(t,u)∥V⊗m≤c̃(1+∥u∥),whereV⊗mistheproductofmcopiesofV. Next, for each j≥1, and all (t,u,Φ)∈(0,T)×V×D(A), we have(3.27)(G(t,u),Φ)=((I+αA)G̃(t,u),Φ)=((G̃(t,u),Φ)),
and, for all u∈L2(Ω,ℱ,𝒫;L∞(0,T;V)), (t,Φ)∈(0,T)×D(A), it follows that(3.28)(∫0tG(s,u(s))dW(s),Φ)=∑j=1d∫0t(Gj(s,u(s)),Φ)dWj(s)=∑j=1d∫0t(G̃j(s,u(s),Φ))dWj(s)=((∫0tG̃(s,u(s))dW(s),Φ)).
Consequently, in this abstract framework, a weak solution (Ω,ℱ,{ℱt}0≤t≤T,𝒫,W,u) of (1.1) is equivalently as follows.Definition 3.4.
It holds that(1)
(Ω,ℱ,𝒫) is a probability space, ({ℱt},0≤t≤T) is a filtration,(2)
W is a m-dimensional {ℱt} standard Wiener process,(3)
u(t) is ℱt adapted for  all t∈[0,T](3.29)u∈Lp(Ω,ℱ,𝒫;L2(0,T,D(A)))∩Lp(Ω,ℱ,𝒫;L∞(0,T,V)),∀1≤p<∞,(4)
for almost allt∈(0,T), the following equation holds 𝒫-a.s.(3.30)u(t)+∫0tÃu(s)ds+∫0tB̃(u(s),u(s))ds=u0+∫0tF̃(s,u(s))ds+∫0tG̃(s,u(s))dW(s),
as an equality in D(A)′.Remark 3.5.
However, (3.30) implies that u∈𝒞(0,T;D(A)′), then u is weakly continuous in V  [14, page 263] and the initial condition is meaningful.


## 4. Proofs of the Main Results

### 4.1. Proof of Theorem3.2

We make use of the Galerkin approximation combined with the method of compactness.We will split the proof into six steps.


#### 4.1.1. Step  1. Construction of an Approximating Sequence

As the injectionD(A)↪V is compact, consider an orthonormal basis {ej}j=1,2,…inD(A) which is orthogonal in V such that ej are eigenfunctions of the spectral problem(4.1)(ej,v)D(A)=λj((ej,v)),∀v∈D(A),
where (.,.)D(A) denotes the scalar product in D(A). For each N∈ℕ, let VN be the span of {e1,…,eN}.Consider the probabilistic system(4.2)(Ω̅,F̅,{F̅t}0≤t≤T,P̅,W̅).
We denote by E̅ the mathematical expectation with respect to (Ω̅,F̅,P̅).We look for a sequence of functionsuN(t) in VN, that is,(4.3)uN(t)=∑j=1NcNj(t,ω̅)ej(x),
solutions of the following stochastic ordinary differential equations in VN:(4.4)d((uN,ej))+(〈ÃuN(t),ej〉+〈B̃(uN(t),uN(t)),ej〉)dt=((F̃(t,uN(t)),ej))dt+((G̃(t,uN(t)),ej))dW̅,j=1,2,…,NuN(0)=u0N,
where u0N∈VN and is chosen with the requirements that(4.5)u0N→u0inVasN→∞.
There exists a a maximal solution to (4.4), that is, a stopping time TN≤T such that (4.4) holds for  t<TN  [11]. Solvability over (0,T) will follow from a priori estimates for  uN that we derive in the following section.We have the following Fourier expansion:(4.6)uN(t)=∑j=1N(uN(t),ej)D(A)ej=∑j=1Nλj((uN(t),ej))ej,∥uN(t)∥2=∑j=1Nλj((uN(t),ej))2.


#### 4.1.2. Step  2. A Priori Estimates

ThroughoutC and Ci(i=1,…) denotes a positive constant independent of N.We have the following Lemma.Lemma 4.1.
It holds thatuN satisfies the following a priori estimates:
(4.7)E̅sup0≤t≤T∥uN(s)∥2+2α̃E̅∫0T∥uN(s)∥D(A)2ds≤C1,
where C1 is a constant independent of N.Proof.
By Ito’s formula, we obtain from (3.16) and (4.4) that
(4.8)d∥uN(t)∥2+2〈ÃuN(t),uN(t)〉dt=[2((F̃(t,uN(t)),uN(t)))+∑j=1Nλj((G̃(t,uN(t)),ej))2]dt+2((G̃(t,uN(t)),uN(t)))dW̅.
Integrating (4.8) with respect to t, and using (3.13) and (3.19), we have
(4.9)∥uN(t)∥2+α̃∫0t∥uN(s)∥D(A)2ds≤∥u0N∥2+C+C∫0t∥uN(s)∥2ds+2∫0t((G̃(s,uN(s)),uN(s)))dW̅(s).
Let us estimate the stochastic integral in this inequality. By Burkholder-Davis Gundy’s inequality [15], we have
(4.10)E̅sup0≤s≤t|∫0s((G̃(s,uN(s)),uN(s)))dW̅(s)|≤CE̅(∫0t((G̃(s,uN(s)),uN(s)))2ds)1/2≤ϵE̅sup0≤s≤t∥uN(s)∥2+Cϵ∫0t(1+∥uN(s)∥2)ds,
here we have used Hölder′s and Young’s inequalities; ϵ is an arbitrary positive number.
Using (4.10) and (4.9) together with appropriate choice of ϵ, we obtain
(4.11)E̅sup0≤s≤t∥uN(s)∥2+2α̃E̅∫0t∥uN(s)∥D(A)2ds≤C+CE̅∫0t∥uN(s)∥2ds.
By Gronwall’s lemma, we obtain the sought estimate (4.7).The following result is related to the higher integrability ofuN.Lemma 4.2.
It holds that(4.12)E̅sup0≤s≤T∥uN(s)∥p≤Cp∀1≤p<∞.Proof.
By Ito’s formula, it follows from (4.4) that for p≥4, we have
(4.13)d∥uN(t)∥p/2=p2∥uN(t)∥p/2-2[((G̃(uN(t),uN(t))))2∥uN(t)∥2-〈ÃuN(t),uN(t)〉-2〈B̃(uN(t),uN(t)),uN(t)〉+2((F̃(t,uN(t)),uN(t)))+12∑i=1Nλi((G̃(t,uN(t)),ei))2+p-44((G̃(uN(t),uN(t))))2∥uN(t)∥2]dt+p2∥uN(t)∥p/2-2((G̃(t,uN(t)),uN(t)))dW̅.
Using the assumptions (3.16), (3.19), (3.26), it follows that
(4.14)sup0≤s≤t∥uN(s)∥p/2≤∥u0N∥p/2+C∫0t(1+∥uN(s)∥p/2)ds+p2sup0≤s≤t|∫0s∥uN(s)∥p/2-2((G̃(s,uN(s)),uN(s)))dW̅|.
Squaring the both sides of this inequality and passing to mathematical expectation, we deduce from the Martingale inequality, that is,
(4.15)E̅sup0≤s≤t∥uN(s)∥p≤C(∥u0N∥p+T+E̅∫0t∥uN(s)∥pds).
From Gronwall’s inequality, we deduce that
(4.16)E̅sup0≤s≤t∥uN(s)∥p≤Cp
for all 1≤p<∞.We also have the following lemma.Lemma 4.3.
It holds thatuN satisfies
(4.17)E̅(∫0T∥uN(s)∥D(A)2ds)p≤Cp
for all 1≤p<∞.Proof.
Using (4.9), we have
(4.18)α̃p(∫0t∥uN(s)∥D(A)2)p≤C∥u0N∥2p+C+C(∫0t∥uN(s)∥2ds)p+C|∫0t((G̃(s,uN(s)),uN(s)))dW̅|p.
Taking the mathematical expectation and use the Burkholder-Gundy’s inequality, the proof of the lemma follows from Lemma 4.2.Lemma 4.4.
It holds that(4.19)E̅sup0≤|θ|≤δ≤1∫0T∥uN(t+θ)-uN(t)∥D(A)′2dt≤Cδ.Proof.
We note that the functions{λjej}j=1,2,… form an orthonormal basis in the dual D(A)′ of D(A). Let PN be the orthogonal projection of D(A)′ onto the span {λ1e1,…,λNeN}, that is,
(4.20)PNh=∑j=1Nλj〈h,ej〉ej.
Thus (4.4) can be rewritten in an integral form as the equality between random variables with values in D(A)′ as
(4.21)uN(t)+∫0tPN(ÃuN(s)+B̃(uN(s),uN(s))-F̃(s,uN(s)))ds=u0N+∫0tPNG̃(s,uN(s))dW̅.
For any positive θ, we have
(4.22)∥uN(t+θ)-uN(t)∥D(A)′≤∥∫tt+θ(ÃuN(s)+B̃(uN(s),uN(s))-F̃(s,uN(s)))ds∥D(A)′+∥∫tt+θG̃(s,uN(s))dW̅∥D(A)′.
Taking the square and use the properties of Ã,B̃ and F̃, we have
(4.23)∥uN(t+θ)-uN(t)∥D(A)′2≤Cθ2+C(∫tt+θ∥uN(s)∥D(A)2ds)2+Csup0≤t≤T∥uN(s)∥2(∫tt+θ∥uN(s)∥D(A)ds)2+Cθ2sup0≤s≤T∥uN(s)∥2+∥∫tt+θG̃(s,uN(s))dW̅∥2.
For fixed δ, taking the supremun over θ≤δ, integrating with respect to t, and taking the mathematical expectation, we have
(4.24)E̅sup0≤θ≤δ∫0T∥uN(t+θ)-uN(t)∥D(A)′2dt≤Cδ2+CE̅∫0T(∫tt+δ∥uN(s)∥D(A)2ds)2dt+CE̅sup0≤s≤T∥uN(s)∥2∫0T(∫tt+δ∥uN(s)∥D(A)ds)2dt+Cδ2E̅sup0≤s≤T∥uN(s)∥2+E̅∫0Tsup0≤θ≤δ∥∫tt+θG̃(s,uN(s))dW̅∥2dt.
We estimate the integrals in this inequality.
We have byHölder′s inequality
(4.25)I1=E̅sup0≤s≤T∥uN(s)∥2∫0T(∫tt+δ∥uN(s)∥D(A)ds)2dt≤δ2E̅sup0≤s≤T∥uN(s)∥2∫0T∥uN(s)∥D(A)2ds.
Using the Hölder′s inequality and the estimates of Lemmas 4.2 and 4.3, we have
(4.26)I1≤Cδ2.
By Martingale’s inequality, we have
(4.27)I2=E̅∫0Tsup0≤θ≤δ∥∫tt+θG̃(s,uN(s))dW̅∥2dt≤E̅∫0T(∫tt+δ∥G̃(s,uN(s))∥2ds)dt.
Using the assumptions on G̃ and the estimate of Lemma 4.2, we have
(4.28)I2≤Cδ.
Collecting the results and making a similar reasoning with θ<0, we obtain from (4.24) that
(4.29)E̅sup0≤|θ|≤δ∫0T∥uN(t+θ)-uN(t)∥D(A)′2≤CδThe following lemma is from [16], and it is a compactness results which represents a variation of the compactness theorems in [17, Chapter I, Section 5]. It will be useful for us to prove the tightness property of Galerkin solution.Proposition 4.5.
For any sequences of positives reals numberνm,μm which tend to 0 as m→∞, the injection of
(4.30)Yμn,νn={y∈L2(0,T;D(A))∩L∞(0,T;V)∣supm1νmsup|θ|≤μm(∫0T∥y(t+θ)-y(t)∥D(A)′2)1/2<∞}
in L2(0,T;V) is compact.FurthermoreYμn,νn is a Banach space with the norm(4.31)∥y∥Yμn,νn=sup0≤t≤T∥y(t)∥+(∫0T∥y(t)∥D(A)2dt)1/2+supn1νnsup|θ|≤μn(∫0T∥y(t+θ)-y(t)∥D(A)′2dt)1/2.
Alongside with Yμn,νn, we also consider the space Xp,μn,νn(1≤p<∞) of random variables y  such that(4.32)E̅sup0≤t≤T∥y(t)∥p<∞;E̅(∫0T∥y(t)∥D(A)2dt)p/2<∞;E̅supn1νnsup|θ|≤μn∫0T∥y(t+θ)-y(t)∥D(A)′2dt<∞.
Endowed with the norm(4.33)∥y∥Xp,νn,μn=(E̅sup0≤t≤T∥y(t)∥p)1/p+(E̅(∫0T∥y(t)∥D(A)2dt)p/2)p/2+E̅supn1νn(sup|θ|≤μn∫0T∥y(t+θ)-y(t)∥D(A)′2dt)1/2,Xp,μn,νn is a Banach space. The priori estimates of the preceding lemmas enable us to claim that for any  1≤p<∞  and for  μn,νn  such that the series  ∑n=1∞(μn/νn)  converges, the sequence of Galerkin solutions {uN:N∈N}  is bounded in Xp,μn,νn.


#### 4.1.3. Step  3. Tightness Property of Galerkin Solutions

Now, we consider the set(4.34)S=C(0,T;Rm)×L2(0,T;V),
and B(S) theσ-algebra of the Borel sets of S.For eachN, let ϕ  be the map(4.35)ϕ:Ω̅→S:ω̅↦(W̅(ω̅,·),uN(ω̅,·)).
For each N, we introduce a probability measure ΠN on (S,B(S)) by(4.36)ΠN(A)=P̅(ϕ-1(A))
for all A∈B(S). The main result of this subsection is the following.Theorem 4.6.
The family of probability measures{ΠN;N∈N} is tight.Proof.
Forε>0, we should find the compact subsets
(4.37)Σε⊂C(0,T;Rm),Yε⊂L2(0,T;V),
such that
(4.38)P̅(ω̅:W̅(ω̅,·)∉Σε)≤ε2,(4.39)P̅(ω̅:uN(ω̅,·)∉Yε)≤ε2.
The quest for Σε is made by taking account of some fact about the Wiener process such as the formula
(4.40)E̅|W̅(t2)-W̅(t1)|2j=(2j-1)!(t2-t1)j,j=1,2,….
For a constant Lε depending on ε to be chosen later and n∈N, we consider the set
(4.41)Σε={W(·)∈C(0,T;Rm):supt1,t2∈[0,T],|t2-t1|≤1/n6n|W(t2)-W(t1)|≤Lε}
Making use of Markov’s inequality:
(4.42)P̅(ω:ξ(ω)≥α)≤1αkE̅[|ξ(ω)|k]
for a random variable  ξ on (Ω̅,F̅,P̅) and positives variables  α  and  k, we get
(4.43)P̅(ω̅:W̅(ω̅,·)∉Σε)≤P̅[⋃n{ω:supt1,t2∈[0,T]:|t2-t1|<1/n6|W̅(t2)-W̅(t1)|>Lεn}]≤∑n=1∞∑i=0n6-1(nLε)4E̅supiT/n6≤t≤(i+1)T/n6|W̅(t)-W̅(iTn-6)|4≤c∑n=1∞(nLε)4(Tn-6)2n6=cLε4∑n=1∞1n2,
we choose
(4.44)Lε4=2Cε-1∑n=1∞1n2
to get (4.38).
Next we chooseYε as a ball of radius  Mε  in Yμn,νn centered at zero and with μn,νn, independent of ε, converging to zero, and such that ∑n(μn/νn) converges.
From Proposition4.5, Yε is a compact subset of L2(0,T;V).
We have further
(4.45)P̅(ω:uN(ω,·)∉Yε)≤P̅(ω:∥uN∥Yμn,νn>Mε)≤1MεE̅∥uN∥Yμn,νn≤cMε,
choosing Mε=2cε-1, we get (4.39).
From (4.38) and (4.39), we have
(4.46)P̅(ω:W̅(ω,·)∈Σε;uN(ω,·)∈Yε)≥1-ε,
this proves that
(4.47)ΠN(Σε×Yε)≥1-ε,∀N∈ℕ.


#### 4.1.4. Step  4. Applications of Prokhorov and Skorokhod Results

From the tightness property of{ΠN}  and Prokhorov’s theorem [12], we have that there exist a subsequence  {ΠNj}  and a measure  Π such that  ΠNj→Π  weakly.By Skorokhod’s theorem [13], there exist a probability space  (Ω,ℱ,P)  and random variables (WNj,uNj),(W,u)  on (Ω,ℱ,P)  with values in S  such that(4.48)thelawof(WNj,uNj)isΠNj,(4.49)thelawof(W,u)isΠ,(4.50)(WNj,uNj)→(W,u)in`S,P-a.s.
Hence, {WNj}  is a sequence of an m-dimensional standard Wiener process.Letℱt=σ{W(s),u(s),0≤s≤t}.Arguing as in  [16], we prove that W(t)  is an m-dimensional ℱt  standard Wiener process and the pair (WNj,uNj) satisfies the equation(4.51)uNj(t)+ν∫0tPNjÃuNj(s)ds+∫0tPNjB̃(uNj(s),uNj(s))ds=∫0tPNjF̃(s,uNj(s))ds+∫0tPNjG̃(s,uNj(s))dWNj+u0Nj.


#### 4.1.5. Step  5. Passage to the Limit

From  (4.51), it follows that uNj  satisfies the results of the Lemmas 4.2, 4.3, and 4.4. Therefore, we have for p≥1  the a priori estimates(4.52)Esup0≤t≤T∥uNj(t)∥p≤C;E(∫0T∥uNj(t)∥D(A)2dt)p≤C;Esup0≤θ≤δ∫0T∥uNj(t+θ)-uNj∥D(A)′2dt≤C(α)δ
thus modulo the extraction of a subsequence denoted again by uNj, we have(4.53)uNj→uweakly*inLp(Ω,ℱ,P;L∞(0,T;V));uNj→uweaklyinLp(Ω,ℱ,P;L2(0,T;D(A)));Esup0≤t≤T∥u(t)∥p≤C;E(∫0T∥u(t)∥D(A)2dt)p≤C;Esup0≤θ≤δ∫0T∥u(t+θ)-u(t)∥D(A)′2dt≤Cδ.
By (4.50) and the first estimate in (4.52) and Vitali’s theorem, we have(4.54)uNj→ustronglyinL2(Ω,ℱ,P;L2(0,T,V)),
and thus modulo the extraction of a subsequence and for almost every (ω,t) with respect to the measure dP⊗dt:(4.55)uNj→uinV.
This convergence together with the condition on F̃, the first estimate in  (4.52)  and Vitali’s theorem, give(4.56)F̃(·,uNj(·))→F̃(·,u(·))stronglyinL2(Ω,ℱ,P;L2(0,T,V)),∫0tF̃(s,uNj(s))ds→∫0tF̃(s,u(s))dsstronglyinL2(Ω,ℱ,P;L2(0,T,V)).
As(4.57)uNj→uweaklyinL2(Ω,ℱ,P;L2(0,T;D(A))),
then(4.58)∫0tÃuNj(s)ds→∫0tÃu(s)dsweaklyinL2(Ω,ℱ,P;L2(0,T;D(A)′)).
We also have(4.59)∫0tB̃(uNj(s),uNj(s))ds→∫0tB̃(u(s),u(s))dsweaklyinL2(Ω,ℱ,P;L2(0,T;D(A)′)).
In fact, since L∞(Ω×(0,T),dP×dt;D(A))  is  dense  in L2(Ω,ℱ,P;L2(0,T;D(A))), and B̃(uNj(s),uNj(s)) is bounded in L2(Ω,ℱ,P;L2(0,T;D(A)′)) it suffices to prove that forallφ∈L∞(Ω×(0,T),dP×dt;D(A)),(4.60)E∫0T〈B̃(uNj(s),uNj(s)),φ(s)〉D(A)′ds→E∫0T〈B̃(u(s),u(s)),φ(s)〉D(A)′ds.
Indeed, we have(4.61)E∫0T〈B̃(uNj(s),uNj(s))-B̃(u(s),u(s)),φ(s)〉D(A)′ds=E∫0T〈B̃(uNj(s)-u(s),uNj(s)),φ(s)〉D(A)′ds+E∫0T〈B̃(u(s),uNj(s)-u(s)),φ(s)〉D(A)′ds=I1j+I2j,I1j=E∫0T〈B̃(uNj(s)-u(s),uNj(s)),φ(s)〉D(A)′ds
By the property (3.17) of B̃, we have(4.62)I1j≤CE∫0T∥uNj(s)-u(s)∥∥uNj(s)∥D(A)|Aφ(s)|ds,
applying Cauchy-Schwarz inequality(4.63)I1j≤Cφ(E∫0T∥uNj(s)-u(s)∥2ds)1/2(E∫0T∥uNj(s)∥D(A)2ds)1/2.
Since(4.64)uNj→ustronglyinL2(Ω,ℱ,P;L2(0,T;V)),
and uNj  is bounded in L2(Ω,ℱ,P;L2(0,T;D(A))), we conclude that(4.65)I1j→0asj→∞.I2j=E∫0T〈B̃(u(s),uNj-u(s)),φ(s)〉D(A)′ds.
Again thanks to the property (3.18) of B̃, as(4.66)uNj→uweaklyinL2(Ω,ℱ,P;L2(0,T;D(A))),
we obtain I2j→0  as  j→∞ since any strongly continuous linear operator is weakly continuous. We are now left with the proof of(4.67)∫0tG̃(s,uNj(s))dWNj(s)→∫0tG̃(s,u(s))dW(s)weakly*L2(Ω,ℱ,P;L∞(0,T;D(A)′)),
which can be prove with the same argument like in [16].Collecting all the convergence results, we deduce that(4.68)u(t)+ν∫0tÃu(s)ds+∫0tB̃(u(s),u(s))ds=∫0tF̃(s,u(s))ds+∫0tG̃(s,u(s))dW(s)+u0,P-a.s.
as the equality in D(A)′.We haveB̃(u,u)∈L2(Ω,ℱ,P;L∞(0,T;D(A)′)), Ãu-F̃(t,u)∈L2(Ω,ℱ,P;L∞(0,T;D(A)′)), G̃(t,u)∈L2(Ω,ℱ,P;L∞(0,T;V⊗m)).Thus, from the classical results in [18] (see also [19]), we deduce from (4.68) that u is P-a.s. continuous with values in V.


#### 4.1.6. Step  6. Existence of the Pressure

For the existence of the pressure, we use a generalization of the Rham’s theorem processes [20, Theorem 4.1, Remark 4.3]. From (3.6), we have for all v∈𝒱,(4.69)〈dWdt-∂t(u-αΔu)-ν(Au-αΔ(Au))-(u·∇)(u-αΔu)+α∇u*·Δu+F(·,u)+G(·,u)dWdt,v〉(𝒟′(D))3×(𝒟(D))3=0.
We denote(4.70)h=-∂t(u-αΔu)-ν(Au-αΔ(Au))-(u·∇)(u-αΔu)+α∇u*·Δu+F(·,u)+G(·,u)dWdt.
We will prove that the regularity on u, implies that(4.71)h∈L2(Ω,ℱt,P;H-1(0,t;(H-2(D))3).
By (2.7) and (2.9), we have as u∈L4(Ω,ℱ,P;L2(0,T;D(A))),(4.72)(u·∇(u-αΔu))+∇u*·Δu∈L2(Ω,ℱt,P;L1(0,t;(H-1(D))3)),Au-αΔ(Au)∈L4(Ω,ℱt,P;L2(0,t;(H-2(D))3)).
We also have(4.73)u-αΔu∈L4(Ω,ℱt,P;L2(0,t;(L2(D))3)),∂t(u-αΔu)∈L4(Ω,ℱt,P;H-1(0,t;(L2(D))3)),∀t∈[0,T].
Again, as u∈L4(Ω,ℱ,P;C([0,T];V)), then its follows that(4.74)F(t,u)∈L4(Ω,ℱt,P;L2(0,t;(H-1(D))3)),G(t,u)dWdt∈L4(Ω,ℱt,P;W-1,∞(0,t;(L2(D))3)),
for all t∈[0,T].Thenh∈L2(Ω,ℱt,P;H-1(0,t;(H-2(D))3), and(4.75)〈h,v〉(𝒟′(D))3×(𝒟(D))3=0,∀v∈𝒱.
Therefore, by a generalization of the Rham theorem processes [20], there exists a unique p̃∈L2(Ω,ℱt,P;H-1(0,t;(H-1(D))3)such thatP-a.s.(4.76)∇p̃=h,∫Dp̃dx=0,thatis,(3.7).
Theorem 3.2 is proved.


### 4.2. Proof of Corollary3.3

Proof.
We will prove the pathwise uniqueness which implies uniqueness of weak solutions. LetLF and LG be two real such that
(4.77)∥F(t,u)-F(t,v)∥(H-1(D))3≤LF∥u-v∥,∥G(t,u)-G(t,v)∥((L2(D))3)m≤LG∥u-v∥.
Then F̃ and G̃ are defined, respectively, by (3.14) and (3.20) satisfying
(4.78)∥F̃(t,u)-F̃(t,v)∥V≤LF̃∥u-v∥,∥G̃(t,u)-G̃(t,u)∥V⊗m≤LG̃∥u-v∥.
Let u1 and u2 two weak solutions of problem (1.1) defined on the same probability space together with the same Wiener process and starting from the same initial value u0.
We denoteu̅=u1-u2. Take μ>0 to be defined later and ρ(t)=exp(-μ∫0t∥u2(s)∥D(A)2ds), 0≤t≤T.
Applying Ito’s formula to the real processρ(t)∥u̅(t)∥2, we obtain from (3.13), (3.18), (3.19), and (3.26) that
(4.79)ρ(t)∥u̅(t)∥2+α̃∫0tρ(s)∥u̅(s)∥D(A)2ds≤LG̃2∫0tρ(s)∥u̅(s)∥2ds+2c̃∫0tρ(s)∥u2(s)∥D(A)∥u̅(s)∥D(A)∥u̅(s)∥ds+2LF̃∫0tρ(s)∥u̅(s)∥D(A)∥u̅(s)∥ds+2∫0t((ρ(s)(G̃(s,u1(s))-G̃(s,u2(s))),u̅(s)))dW(s)-μ∫0t∫0tρ(s)∥u2(s)∥D(A)2∥u̅(s)∥2ds,
for all t∈[0,T].
By young’s inequality, we have
(4.80)2c̃∥u2(s)∥D(A)∥u̅(s)∥D(A)∥u̅(s)∥≤α̃2∥u̅(s)∥D(A)2+2c̃2α̃∥u2(s)∥D(A)2∥u̅(s)∥22LF̃∥u̅(s)∥D(A)∥u̅(s)∥≤α̃2∥u̅(s)∥D(A)2+2LF̃2α̃∥u̅(s)∥2.
If we take μ=2(c̃2/α̃), we obtain from (4.79) that
(4.81)ρ(t)∥u̅(t)∥2≤(LG̃2+2LF̃2α̃)∫0tρ(s)∥u̅(s)∥2ds+2∫0t((ρ(s)(G̃(s,u1(s))-G̃(s,u2(s)),u̅(s))))dW.
As 0<ρ(t)≤1, the expectation of the stochastic integral in (4.81) vanishes and
(4.82)Eρ(t)∥u̅(t)∥2≤(LG̃2+2LF̃2α̃)E∫0tρ(s)∥u̅(s)∥2ds.
The Gronwall lemma implies that  u̅(t)=0, P-a.s. for all t∈[0,T]. Also, the corollary is proved.Remark 4.7.
Using the famous Yamada-Watanabe theorem [11], Corollary 3.3 implies the existence of a unique strong solution of (1.1).


## 4.1. Proof of Theorem3.2

We make use of the Galerkin approximation combined with the method of compactness.We will split the proof into six steps.


### 4.1.1. Step  1. Construction of an Approximating Sequence

As the injectionD(A)↪V is compact, consider an orthonormal basis {ej}j=1,2,…inD(A) which is orthogonal in V such that ej are eigenfunctions of the spectral problem(4.1)(ej,v)D(A)=λj((ej,v)),∀v∈D(A),
where (.,.)D(A) denotes the scalar product in D(A). For each N∈ℕ, let VN be the span of {e1,…,eN}.Consider the probabilistic system(4.2)(Ω̅,F̅,{F̅t}0≤t≤T,P̅,W̅).
We denote by E̅ the mathematical expectation with respect to (Ω̅,F̅,P̅).We look for a sequence of functionsuN(t) in VN, that is,(4.3)uN(t)=∑j=1NcNj(t,ω̅)ej(x),
solutions of the following stochastic ordinary differential equations in VN:(4.4)d((uN,ej))+(〈ÃuN(t),ej〉+〈B̃(uN(t),uN(t)),ej〉)dt=((F̃(t,uN(t)),ej))dt+((G̃(t,uN(t)),ej))dW̅,j=1,2,…,NuN(0)=u0N,
where u0N∈VN and is chosen with the requirements that(4.5)u0N→u0inVasN→∞.
There exists a a maximal solution to (4.4), that is, a stopping time TN≤T such that (4.4) holds for  t<TN  [11]. Solvability over (0,T) will follow from a priori estimates for  uN that we derive in the following section.We have the following Fourier expansion:(4.6)uN(t)=∑j=1N(uN(t),ej)D(A)ej=∑j=1Nλj((uN(t),ej))ej,∥uN(t)∥2=∑j=1Nλj((uN(t),ej))2.


### 4.1.2. Step  2. A Priori Estimates

ThroughoutC and Ci(i=1,…) denotes a positive constant independent of N.We have the following Lemma.Lemma 4.1.
It holds thatuN satisfies the following a priori estimates:
(4.7)E̅sup0≤t≤T∥uN(s)∥2+2α̃E̅∫0T∥uN(s)∥D(A)2ds≤C1,
where C1 is a constant independent of N.Proof.
By Ito’s formula, we obtain from (3.16) and (4.4) that
(4.8)d∥uN(t)∥2+2〈ÃuN(t),uN(t)〉dt=[2((F̃(t,uN(t)),uN(t)))+∑j=1Nλj((G̃(t,uN(t)),ej))2]dt+2((G̃(t,uN(t)),uN(t)))dW̅.
Integrating (4.8) with respect to t, and using (3.13) and (3.19), we have
(4.9)∥uN(t)∥2+α̃∫0t∥uN(s)∥D(A)2ds≤∥u0N∥2+C+C∫0t∥uN(s)∥2ds+2∫0t((G̃(s,uN(s)),uN(s)))dW̅(s).
Let us estimate the stochastic integral in this inequality. By Burkholder-Davis Gundy’s inequality [15], we have
(4.10)E̅sup0≤s≤t|∫0s((G̃(s,uN(s)),uN(s)))dW̅(s)|≤CE̅(∫0t((G̃(s,uN(s)),uN(s)))2ds)1/2≤ϵE̅sup0≤s≤t∥uN(s)∥2+Cϵ∫0t(1+∥uN(s)∥2)ds,
here we have used Hölder′s and Young’s inequalities; ϵ is an arbitrary positive number.
Using (4.10) and (4.9) together with appropriate choice of ϵ, we obtain
(4.11)E̅sup0≤s≤t∥uN(s)∥2+2α̃E̅∫0t∥uN(s)∥D(A)2ds≤C+CE̅∫0t∥uN(s)∥2ds.
By Gronwall’s lemma, we obtain the sought estimate (4.7).The following result is related to the higher integrability ofuN.Lemma 4.2.
It holds that(4.12)E̅sup0≤s≤T∥uN(s)∥p≤Cp∀1≤p<∞.Proof.
By Ito’s formula, it follows from (4.4) that for p≥4, we have
(4.13)d∥uN(t)∥p/2=p2∥uN(t)∥p/2-2[((G̃(uN(t),uN(t))))2∥uN(t)∥2-〈ÃuN(t),uN(t)〉-2〈B̃(uN(t),uN(t)),uN(t)〉+2((F̃(t,uN(t)),uN(t)))+12∑i=1Nλi((G̃(t,uN(t)),ei))2+p-44((G̃(uN(t),uN(t))))2∥uN(t)∥2]dt+p2∥uN(t)∥p/2-2((G̃(t,uN(t)),uN(t)))dW̅.
Using the assumptions (3.16), (3.19), (3.26), it follows that
(4.14)sup0≤s≤t∥uN(s)∥p/2≤∥u0N∥p/2+C∫0t(1+∥uN(s)∥p/2)ds+p2sup0≤s≤t|∫0s∥uN(s)∥p/2-2((G̃(s,uN(s)),uN(s)))dW̅|.
Squaring the both sides of this inequality and passing to mathematical expectation, we deduce from the Martingale inequality, that is,
(4.15)E̅sup0≤s≤t∥uN(s)∥p≤C(∥u0N∥p+T+E̅∫0t∥uN(s)∥pds).
From Gronwall’s inequality, we deduce that
(4.16)E̅sup0≤s≤t∥uN(s)∥p≤Cp
for all 1≤p<∞.We also have the following lemma.Lemma 4.3.
It holds thatuN satisfies
(4.17)E̅(∫0T∥uN(s)∥D(A)2ds)p≤Cp
for all 1≤p<∞.Proof.
Using (4.9), we have
(4.18)α̃p(∫0t∥uN(s)∥D(A)2)p≤C∥u0N∥2p+C+C(∫0t∥uN(s)∥2ds)p+C|∫0t((G̃(s,uN(s)),uN(s)))dW̅|p.
Taking the mathematical expectation and use the Burkholder-Gundy’s inequality, the proof of the lemma follows from Lemma 4.2.Lemma 4.4.
It holds that(4.19)E̅sup0≤|θ|≤δ≤1∫0T∥uN(t+θ)-uN(t)∥D(A)′2dt≤Cδ.Proof.
We note that the functions{λjej}j=1,2,… form an orthonormal basis in the dual D(A)′ of D(A). Let PN be the orthogonal projection of D(A)′ onto the span {λ1e1,…,λNeN}, that is,
(4.20)PNh=∑j=1Nλj〈h,ej〉ej.
Thus (4.4) can be rewritten in an integral form as the equality between random variables with values in D(A)′ as
(4.21)uN(t)+∫0tPN(ÃuN(s)+B̃(uN(s),uN(s))-F̃(s,uN(s)))ds=u0N+∫0tPNG̃(s,uN(s))dW̅.
For any positive θ, we have
(4.22)∥uN(t+θ)-uN(t)∥D(A)′≤∥∫tt+θ(ÃuN(s)+B̃(uN(s),uN(s))-F̃(s,uN(s)))ds∥D(A)′+∥∫tt+θG̃(s,uN(s))dW̅∥D(A)′.
Taking the square and use the properties of Ã,B̃ and F̃, we have
(4.23)∥uN(t+θ)-uN(t)∥D(A)′2≤Cθ2+C(∫tt+θ∥uN(s)∥D(A)2ds)2+Csup0≤t≤T∥uN(s)∥2(∫tt+θ∥uN(s)∥D(A)ds)2+Cθ2sup0≤s≤T∥uN(s)∥2+∥∫tt+θG̃(s,uN(s))dW̅∥2.
For fixed δ, taking the supremun over θ≤δ, integrating with respect to t, and taking the mathematical expectation, we have
(4.24)E̅sup0≤θ≤δ∫0T∥uN(t+θ)-uN(t)∥D(A)′2dt≤Cδ2+CE̅∫0T(∫tt+δ∥uN(s)∥D(A)2ds)2dt+CE̅sup0≤s≤T∥uN(s)∥2∫0T(∫tt+δ∥uN(s)∥D(A)ds)2dt+Cδ2E̅sup0≤s≤T∥uN(s)∥2+E̅∫0Tsup0≤θ≤δ∥∫tt+θG̃(s,uN(s))dW̅∥2dt.
We estimate the integrals in this inequality.
We have byHölder′s inequality
(4.25)I1=E̅sup0≤s≤T∥uN(s)∥2∫0T(∫tt+δ∥uN(s)∥D(A)ds)2dt≤δ2E̅sup0≤s≤T∥uN(s)∥2∫0T∥uN(s)∥D(A)2ds.
Using the Hölder′s inequality and the estimates of Lemmas 4.2 and 4.3, we have
(4.26)I1≤Cδ2.
By Martingale’s inequality, we have
(4.27)I2=E̅∫0Tsup0≤θ≤δ∥∫tt+θG̃(s,uN(s))dW̅∥2dt≤E̅∫0T(∫tt+δ∥G̃(s,uN(s))∥2ds)dt.
Using the assumptions on G̃ and the estimate of Lemma 4.2, we have
(4.28)I2≤Cδ.
Collecting the results and making a similar reasoning with θ<0, we obtain from (4.24) that
(4.29)E̅sup0≤|θ|≤δ∫0T∥uN(t+θ)-uN(t)∥D(A)′2≤CδThe following lemma is from [16], and it is a compactness results which represents a variation of the compactness theorems in [17, Chapter I, Section 5]. It will be useful for us to prove the tightness property of Galerkin solution.Proposition 4.5.
For any sequences of positives reals numberνm,μm which tend to 0 as m→∞, the injection of
(4.30)Yμn,νn={y∈L2(0,T;D(A))∩L∞(0,T;V)∣supm1νmsup|θ|≤μm(∫0T∥y(t+θ)-y(t)∥D(A)′2)1/2<∞}
in L2(0,T;V) is compact.FurthermoreYμn,νn is a Banach space with the norm(4.31)∥y∥Yμn,νn=sup0≤t≤T∥y(t)∥+(∫0T∥y(t)∥D(A)2dt)1/2+supn1νnsup|θ|≤μn(∫0T∥y(t+θ)-y(t)∥D(A)′2dt)1/2.
Alongside with Yμn,νn, we also consider the space Xp,μn,νn(1≤p<∞) of random variables y  such that(4.32)E̅sup0≤t≤T∥y(t)∥p<∞;E̅(∫0T∥y(t)∥D(A)2dt)p/2<∞;E̅supn1νnsup|θ|≤μn∫0T∥y(t+θ)-y(t)∥D(A)′2dt<∞.
Endowed with the norm(4.33)∥y∥Xp,νn,μn=(E̅sup0≤t≤T∥y(t)∥p)1/p+(E̅(∫0T∥y(t)∥D(A)2dt)p/2)p/2+E̅supn1νn(sup|θ|≤μn∫0T∥y(t+θ)-y(t)∥D(A)′2dt)1/2,Xp,μn,νn is a Banach space. The priori estimates of the preceding lemmas enable us to claim that for any  1≤p<∞  and for  μn,νn  such that the series  ∑n=1∞(μn/νn)  converges, the sequence of Galerkin solutions {uN:N∈N}  is bounded in Xp,μn,νn.


### 4.1.3. Step  3. Tightness Property of Galerkin Solutions

Now, we consider the set(4.34)S=C(0,T;Rm)×L2(0,T;V),
and B(S) theσ-algebra of the Borel sets of S.For eachN, let ϕ  be the map(4.35)ϕ:Ω̅→S:ω̅↦(W̅(ω̅,·),uN(ω̅,·)).
For each N, we introduce a probability measure ΠN on (S,B(S)) by(4.36)ΠN(A)=P̅(ϕ-1(A))
for all A∈B(S). The main result of this subsection is the following.Theorem 4.6.
The family of probability measures{ΠN;N∈N} is tight.Proof.
Forε>0, we should find the compact subsets
(4.37)Σε⊂C(0,T;Rm),Yε⊂L2(0,T;V),
such that
(4.38)P̅(ω̅:W̅(ω̅,·)∉Σε)≤ε2,(4.39)P̅(ω̅:uN(ω̅,·)∉Yε)≤ε2.
The quest for Σε is made by taking account of some fact about the Wiener process such as the formula
(4.40)E̅|W̅(t2)-W̅(t1)|2j=(2j-1)!(t2-t1)j,j=1,2,….
For a constant Lε depending on ε to be chosen later and n∈N, we consider the set
(4.41)Σε={W(·)∈C(0,T;Rm):supt1,t2∈[0,T],|t2-t1|≤1/n6n|W(t2)-W(t1)|≤Lε}
Making use of Markov’s inequality:
(4.42)P̅(ω:ξ(ω)≥α)≤1αkE̅[|ξ(ω)|k]
for a random variable  ξ on (Ω̅,F̅,P̅) and positives variables  α  and  k, we get
(4.43)P̅(ω̅:W̅(ω̅,·)∉Σε)≤P̅[⋃n{ω:supt1,t2∈[0,T]:|t2-t1|<1/n6|W̅(t2)-W̅(t1)|>Lεn}]≤∑n=1∞∑i=0n6-1(nLε)4E̅supiT/n6≤t≤(i+1)T/n6|W̅(t)-W̅(iTn-6)|4≤c∑n=1∞(nLε)4(Tn-6)2n6=cLε4∑n=1∞1n2,
we choose
(4.44)Lε4=2Cε-1∑n=1∞1n2
to get (4.38).
Next we chooseYε as a ball of radius  Mε  in Yμn,νn centered at zero and with μn,νn, independent of ε, converging to zero, and such that ∑n(μn/νn) converges.
From Proposition4.5, Yε is a compact subset of L2(0,T;V).
We have further
(4.45)P̅(ω:uN(ω,·)∉Yε)≤P̅(ω:∥uN∥Yμn,νn>Mε)≤1MεE̅∥uN∥Yμn,νn≤cMε,
choosing Mε=2cε-1, we get (4.39).
From (4.38) and (4.39), we have
(4.46)P̅(ω:W̅(ω,·)∈Σε;uN(ω,·)∈Yε)≥1-ε,
this proves that
(4.47)ΠN(Σε×Yε)≥1-ε,∀N∈ℕ.


### 4.1.4. Step  4. Applications of Prokhorov and Skorokhod Results

From the tightness property of{ΠN}  and Prokhorov’s theorem [12], we have that there exist a subsequence  {ΠNj}  and a measure  Π such that  ΠNj→Π  weakly.By Skorokhod’s theorem [13], there exist a probability space  (Ω,ℱ,P)  and random variables (WNj,uNj),(W,u)  on (Ω,ℱ,P)  with values in S  such that(4.48)thelawof(WNj,uNj)isΠNj,(4.49)thelawof(W,u)isΠ,(4.50)(WNj,uNj)→(W,u)in`S,P-a.s.
Hence, {WNj}  is a sequence of an m-dimensional standard Wiener process.Letℱt=σ{W(s),u(s),0≤s≤t}.Arguing as in  [16], we prove that W(t)  is an m-dimensional ℱt  standard Wiener process and the pair (WNj,uNj) satisfies the equation(4.51)uNj(t)+ν∫0tPNjÃuNj(s)ds+∫0tPNjB̃(uNj(s),uNj(s))ds=∫0tPNjF̃(s,uNj(s))ds+∫0tPNjG̃(s,uNj(s))dWNj+u0Nj.


### 4.1.5. Step  5. Passage to the Limit

From  (4.51), it follows that uNj  satisfies the results of the Lemmas 4.2, 4.3, and 4.4. Therefore, we have for p≥1  the a priori estimates(4.52)Esup0≤t≤T∥uNj(t)∥p≤C;E(∫0T∥uNj(t)∥D(A)2dt)p≤C;Esup0≤θ≤δ∫0T∥uNj(t+θ)-uNj∥D(A)′2dt≤C(α)δ
thus modulo the extraction of a subsequence denoted again by uNj, we have(4.53)uNj→uweakly*inLp(Ω,ℱ,P;L∞(0,T;V));uNj→uweaklyinLp(Ω,ℱ,P;L2(0,T;D(A)));Esup0≤t≤T∥u(t)∥p≤C;E(∫0T∥u(t)∥D(A)2dt)p≤C;Esup0≤θ≤δ∫0T∥u(t+θ)-u(t)∥D(A)′2dt≤Cδ.
By (4.50) and the first estimate in (4.52) and Vitali’s theorem, we have(4.54)uNj→ustronglyinL2(Ω,ℱ,P;L2(0,T,V)),
and thus modulo the extraction of a subsequence and for almost every (ω,t) with respect to the measure dP⊗dt:(4.55)uNj→uinV.
This convergence together with the condition on F̃, the first estimate in  (4.52)  and Vitali’s theorem, give(4.56)F̃(·,uNj(·))→F̃(·,u(·))stronglyinL2(Ω,ℱ,P;L2(0,T,V)),∫0tF̃(s,uNj(s))ds→∫0tF̃(s,u(s))dsstronglyinL2(Ω,ℱ,P;L2(0,T,V)).
As(4.57)uNj→uweaklyinL2(Ω,ℱ,P;L2(0,T;D(A))),
then(4.58)∫0tÃuNj(s)ds→∫0tÃu(s)dsweaklyinL2(Ω,ℱ,P;L2(0,T;D(A)′)).
We also have(4.59)∫0tB̃(uNj(s),uNj(s))ds→∫0tB̃(u(s),u(s))dsweaklyinL2(Ω,ℱ,P;L2(0,T;D(A)′)).
In fact, since L∞(Ω×(0,T),dP×dt;D(A))  is  dense  in L2(Ω,ℱ,P;L2(0,T;D(A))), and B̃(uNj(s),uNj(s)) is bounded in L2(Ω,ℱ,P;L2(0,T;D(A)′)) it suffices to prove that forallφ∈L∞(Ω×(0,T),dP×dt;D(A)),(4.60)E∫0T〈B̃(uNj(s),uNj(s)),φ(s)〉D(A)′ds→E∫0T〈B̃(u(s),u(s)),φ(s)〉D(A)′ds.
Indeed, we have(4.61)E∫0T〈B̃(uNj(s),uNj(s))-B̃(u(s),u(s)),φ(s)〉D(A)′ds=E∫0T〈B̃(uNj(s)-u(s),uNj(s)),φ(s)〉D(A)′ds+E∫0T〈B̃(u(s),uNj(s)-u(s)),φ(s)〉D(A)′ds=I1j+I2j,I1j=E∫0T〈B̃(uNj(s)-u(s),uNj(s)),φ(s)〉D(A)′ds
By the property (3.17) of B̃, we have(4.62)I1j≤CE∫0T∥uNj(s)-u(s)∥∥uNj(s)∥D(A)|Aφ(s)|ds,
applying Cauchy-Schwarz inequality(4.63)I1j≤Cφ(E∫0T∥uNj(s)-u(s)∥2ds)1/2(E∫0T∥uNj(s)∥D(A)2ds)1/2.
Since(4.64)uNj→ustronglyinL2(Ω,ℱ,P;L2(0,T;V)),
and uNj  is bounded in L2(Ω,ℱ,P;L2(0,T;D(A))), we conclude that(4.65)I1j→0asj→∞.I2j=E∫0T〈B̃(u(s),uNj-u(s)),φ(s)〉D(A)′ds.
Again thanks to the property (3.18) of B̃, as(4.66)uNj→uweaklyinL2(Ω,ℱ,P;L2(0,T;D(A))),
we obtain I2j→0  as  j→∞ since any strongly continuous linear operator is weakly continuous. We are now left with the proof of(4.67)∫0tG̃(s,uNj(s))dWNj(s)→∫0tG̃(s,u(s))dW(s)weakly*L2(Ω,ℱ,P;L∞(0,T;D(A)′)),
which can be prove with the same argument like in [16].Collecting all the convergence results, we deduce that(4.68)u(t)+ν∫0tÃu(s)ds+∫0tB̃(u(s),u(s))ds=∫0tF̃(s,u(s))ds+∫0tG̃(s,u(s))dW(s)+u0,P-a.s.
as the equality in D(A)′.We haveB̃(u,u)∈L2(Ω,ℱ,P;L∞(0,T;D(A)′)), Ãu-F̃(t,u)∈L2(Ω,ℱ,P;L∞(0,T;D(A)′)), G̃(t,u)∈L2(Ω,ℱ,P;L∞(0,T;V⊗m)).Thus, from the classical results in [18] (see also [19]), we deduce from (4.68) that u is P-a.s. continuous with values in V.


### 4.1.6. Step  6. Existence of the Pressure

For the existence of the pressure, we use a generalization of the Rham’s theorem processes [20, Theorem 4.1, Remark 4.3]. From (3.6), we have for all v∈𝒱,(4.69)〈dWdt-∂t(u-αΔu)-ν(Au-αΔ(Au))-(u·∇)(u-αΔu)+α∇u*·Δu+F(·,u)+G(·,u)dWdt,v〉(𝒟′(D))3×(𝒟(D))3=0.
We denote(4.70)h=-∂t(u-αΔu)-ν(Au-αΔ(Au))-(u·∇)(u-αΔu)+α∇u*·Δu+F(·,u)+G(·,u)dWdt.
We will prove that the regularity on u, implies that(4.71)h∈L2(Ω,ℱt,P;H-1(0,t;(H-2(D))3).
By (2.7) and (2.9), we have as u∈L4(Ω,ℱ,P;L2(0,T;D(A))),(4.72)(u·∇(u-αΔu))+∇u*·Δu∈L2(Ω,ℱt,P;L1(0,t;(H-1(D))3)),Au-αΔ(Au)∈L4(Ω,ℱt,P;L2(0,t;(H-2(D))3)).
We also have(4.73)u-αΔu∈L4(Ω,ℱt,P;L2(0,t;(L2(D))3)),∂t(u-αΔu)∈L4(Ω,ℱt,P;H-1(0,t;(L2(D))3)),∀t∈[0,T].
Again, as u∈L4(Ω,ℱ,P;C([0,T];V)), then its follows that(4.74)F(t,u)∈L4(Ω,ℱt,P;L2(0,t;(H-1(D))3)),G(t,u)dWdt∈L4(Ω,ℱt,P;W-1,∞(0,t;(L2(D))3)),
for all t∈[0,T].Thenh∈L2(Ω,ℱt,P;H-1(0,t;(H-2(D))3), and(4.75)〈h,v〉(𝒟′(D))3×(𝒟(D))3=0,∀v∈𝒱.
Therefore, by a generalization of the Rham theorem processes [20], there exists a unique p̃∈L2(Ω,ℱt,P;H-1(0,t;(H-1(D))3)such thatP-a.s.(4.76)∇p̃=h,∫Dp̃dx=0,thatis,(3.7).
Theorem 3.2 is proved.


## 4.1.1. Step  1. Construction of an Approximating Sequence

As the injectionD(A)↪V is compact, consider an orthonormal basis {ej}j=1,2,…inD(A) which is orthogonal in V such that ej are eigenfunctions of the spectral problem(4.1)(ej,v)D(A)=λj((ej,v)),∀v∈D(A),
where (.,.)D(A) denotes the scalar product in D(A). For each N∈ℕ, let VN be the span of {e1,…,eN}.Consider the probabilistic system(4.2)(Ω̅,F̅,{F̅t}0≤t≤T,P̅,W̅).
We denote by E̅ the mathematical expectation with respect to (Ω̅,F̅,P̅).We look for a sequence of functionsuN(t) in VN, that is,(4.3)uN(t)=∑j=1NcNj(t,ω̅)ej(x),
solutions of the following stochastic ordinary differential equations in VN:(4.4)d((uN,ej))+(〈ÃuN(t),ej〉+〈B̃(uN(t),uN(t)),ej〉)dt=((F̃(t,uN(t)),ej))dt+((G̃(t,uN(t)),ej))dW̅,j=1,2,…,NuN(0)=u0N,
where u0N∈VN and is chosen with the requirements that(4.5)u0N→u0inVasN→∞.
There exists a a maximal solution to (4.4), that is, a stopping time TN≤T such that (4.4) holds for  t<TN  [11]. Solvability over (0,T) will follow from a priori estimates for  uN that we derive in the following section.We have the following Fourier expansion:(4.6)uN(t)=∑j=1N(uN(t),ej)D(A)ej=∑j=1Nλj((uN(t),ej))ej,∥uN(t)∥2=∑j=1Nλj((uN(t),ej))2.


## 4.1.2. Step  2. A Priori Estimates

ThroughoutC and Ci(i=1,…) denotes a positive constant independent of N.We have the following Lemma.Lemma 4.1.
It holds thatuN satisfies the following a priori estimates:
(4.7)E̅sup0≤t≤T∥uN(s)∥2+2α̃E̅∫0T∥uN(s)∥D(A)2ds≤C1,
where C1 is a constant independent of N.Proof.
By Ito’s formula, we obtain from (3.16) and (4.4) that
(4.8)d∥uN(t)∥2+2〈ÃuN(t),uN(t)〉dt=[2((F̃(t,uN(t)),uN(t)))+∑j=1Nλj((G̃(t,uN(t)),ej))2]dt+2((G̃(t,uN(t)),uN(t)))dW̅.
Integrating (4.8) with respect to t, and using (3.13) and (3.19), we have
(4.9)∥uN(t)∥2+α̃∫0t∥uN(s)∥D(A)2ds≤∥u0N∥2+C+C∫0t∥uN(s)∥2ds+2∫0t((G̃(s,uN(s)),uN(s)))dW̅(s).
Let us estimate the stochastic integral in this inequality. By Burkholder-Davis Gundy’s inequality [15], we have
(4.10)E̅sup0≤s≤t|∫0s((G̃(s,uN(s)),uN(s)))dW̅(s)|≤CE̅(∫0t((G̃(s,uN(s)),uN(s)))2ds)1/2≤ϵE̅sup0≤s≤t∥uN(s)∥2+Cϵ∫0t(1+∥uN(s)∥2)ds,
here we have used Hölder′s and Young’s inequalities; ϵ is an arbitrary positive number.
Using (4.10) and (4.9) together with appropriate choice of ϵ, we obtain
(4.11)E̅sup0≤s≤t∥uN(s)∥2+2α̃E̅∫0t∥uN(s)∥D(A)2ds≤C+CE̅∫0t∥uN(s)∥2ds.
By Gronwall’s lemma, we obtain the sought estimate (4.7).The following result is related to the higher integrability ofuN.Lemma 4.2.
It holds that(4.12)E̅sup0≤s≤T∥uN(s)∥p≤Cp∀1≤p<∞.Proof.
By Ito’s formula, it follows from (4.4) that for p≥4, we have
(4.13)d∥uN(t)∥p/2=p2∥uN(t)∥p/2-2[((G̃(uN(t),uN(t))))2∥uN(t)∥2-〈ÃuN(t),uN(t)〉-2〈B̃(uN(t),uN(t)),uN(t)〉+2((F̃(t,uN(t)),uN(t)))+12∑i=1Nλi((G̃(t,uN(t)),ei))2+p-44((G̃(uN(t),uN(t))))2∥uN(t)∥2]dt+p2∥uN(t)∥p/2-2((G̃(t,uN(t)),uN(t)))dW̅.
Using the assumptions (3.16), (3.19), (3.26), it follows that
(4.14)sup0≤s≤t∥uN(s)∥p/2≤∥u0N∥p/2+C∫0t(1+∥uN(s)∥p/2)ds+p2sup0≤s≤t|∫0s∥uN(s)∥p/2-2((G̃(s,uN(s)),uN(s)))dW̅|.
Squaring the both sides of this inequality and passing to mathematical expectation, we deduce from the Martingale inequality, that is,
(4.15)E̅sup0≤s≤t∥uN(s)∥p≤C(∥u0N∥p+T+E̅∫0t∥uN(s)∥pds).
From Gronwall’s inequality, we deduce that
(4.16)E̅sup0≤s≤t∥uN(s)∥p≤Cp
for all 1≤p<∞.We also have the following lemma.Lemma 4.3.
It holds thatuN satisfies
(4.17)E̅(∫0T∥uN(s)∥D(A)2ds)p≤Cp
for all 1≤p<∞.Proof.
Using (4.9), we have
(4.18)α̃p(∫0t∥uN(s)∥D(A)2)p≤C∥u0N∥2p+C+C(∫0t∥uN(s)∥2ds)p+C|∫0t((G̃(s,uN(s)),uN(s)))dW̅|p.
Taking the mathematical expectation and use the Burkholder-Gundy’s inequality, the proof of the lemma follows from Lemma 4.2.Lemma 4.4.
It holds that(4.19)E̅sup0≤|θ|≤δ≤1∫0T∥uN(t+θ)-uN(t)∥D(A)′2dt≤Cδ.Proof.
We note that the functions{λjej}j=1,2,… form an orthonormal basis in the dual D(A)′ of D(A). Let PN be the orthogonal projection of D(A)′ onto the span {λ1e1,…,λNeN}, that is,
(4.20)PNh=∑j=1Nλj〈h,ej〉ej.
Thus (4.4) can be rewritten in an integral form as the equality between random variables with values in D(A)′ as
(4.21)uN(t)+∫0tPN(ÃuN(s)+B̃(uN(s),uN(s))-F̃(s,uN(s)))ds=u0N+∫0tPNG̃(s,uN(s))dW̅.
For any positive θ, we have
(4.22)∥uN(t+θ)-uN(t)∥D(A)′≤∥∫tt+θ(ÃuN(s)+B̃(uN(s),uN(s))-F̃(s,uN(s)))ds∥D(A)′+∥∫tt+θG̃(s,uN(s))dW̅∥D(A)′.
Taking the square and use the properties of Ã,B̃ and F̃, we have
(4.23)∥uN(t+θ)-uN(t)∥D(A)′2≤Cθ2+C(∫tt+θ∥uN(s)∥D(A)2ds)2+Csup0≤t≤T∥uN(s)∥2(∫tt+θ∥uN(s)∥D(A)ds)2+Cθ2sup0≤s≤T∥uN(s)∥2+∥∫tt+θG̃(s,uN(s))dW̅∥2.
For fixed δ, taking the supremun over θ≤δ, integrating with respect to t, and taking the mathematical expectation, we have
(4.24)E̅sup0≤θ≤δ∫0T∥uN(t+θ)-uN(t)∥D(A)′2dt≤Cδ2+CE̅∫0T(∫tt+δ∥uN(s)∥D(A)2ds)2dt+CE̅sup0≤s≤T∥uN(s)∥2∫0T(∫tt+δ∥uN(s)∥D(A)ds)2dt+Cδ2E̅sup0≤s≤T∥uN(s)∥2+E̅∫0Tsup0≤θ≤δ∥∫tt+θG̃(s,uN(s))dW̅∥2dt.
We estimate the integrals in this inequality.
We have byHölder′s inequality
(4.25)I1=E̅sup0≤s≤T∥uN(s)∥2∫0T(∫tt+δ∥uN(s)∥D(A)ds)2dt≤δ2E̅sup0≤s≤T∥uN(s)∥2∫0T∥uN(s)∥D(A)2ds.
Using the Hölder′s inequality and the estimates of Lemmas 4.2 and 4.3, we have
(4.26)I1≤Cδ2.
By Martingale’s inequality, we have
(4.27)I2=E̅∫0Tsup0≤θ≤δ∥∫tt+θG̃(s,uN(s))dW̅∥2dt≤E̅∫0T(∫tt+δ∥G̃(s,uN(s))∥2ds)dt.
Using the assumptions on G̃ and the estimate of Lemma 4.2, we have
(4.28)I2≤Cδ.
Collecting the results and making a similar reasoning with θ<0, we obtain from (4.24) that
(4.29)E̅sup0≤|θ|≤δ∫0T∥uN(t+θ)-uN(t)∥D(A)′2≤CδThe following lemma is from [16], and it is a compactness results which represents a variation of the compactness theorems in [17, Chapter I, Section 5]. It will be useful for us to prove the tightness property of Galerkin solution.Proposition 4.5.
For any sequences of positives reals numberνm,μm which tend to 0 as m→∞, the injection of
(4.30)Yμn,νn={y∈L2(0,T;D(A))∩L∞(0,T;V)∣supm1νmsup|θ|≤μm(∫0T∥y(t+θ)-y(t)∥D(A)′2)1/2<∞}
in L2(0,T;V) is compact.FurthermoreYμn,νn is a Banach space with the norm(4.31)∥y∥Yμn,νn=sup0≤t≤T∥y(t)∥+(∫0T∥y(t)∥D(A)2dt)1/2+supn1νnsup|θ|≤μn(∫0T∥y(t+θ)-y(t)∥D(A)′2dt)1/2.
Alongside with Yμn,νn, we also consider the space Xp,μn,νn(1≤p<∞) of random variables y  such that(4.32)E̅sup0≤t≤T∥y(t)∥p<∞;E̅(∫0T∥y(t)∥D(A)2dt)p/2<∞;E̅supn1νnsup|θ|≤μn∫0T∥y(t+θ)-y(t)∥D(A)′2dt<∞.
Endowed with the norm(4.33)∥y∥Xp,νn,μn=(E̅sup0≤t≤T∥y(t)∥p)1/p+(E̅(∫0T∥y(t)∥D(A)2dt)p/2)p/2+E̅supn1νn(sup|θ|≤μn∫0T∥y(t+θ)-y(t)∥D(A)′2dt)1/2,Xp,μn,νn is a Banach space. The priori estimates of the preceding lemmas enable us to claim that for any  1≤p<∞  and for  μn,νn  such that the series  ∑n=1∞(μn/νn)  converges, the sequence of Galerkin solutions {uN:N∈N}  is bounded in Xp,μn,νn.


## 4.1.3. Step  3. Tightness Property of Galerkin Solutions

Now, we consider the set(4.34)S=C(0,T;Rm)×L2(0,T;V),
and B(S) theσ-algebra of the Borel sets of S.For eachN, let ϕ  be the map(4.35)ϕ:Ω̅→S:ω̅↦(W̅(ω̅,·),uN(ω̅,·)).
For each N, we introduce a probability measure ΠN on (S,B(S)) by(4.36)ΠN(A)=P̅(ϕ-1(A))
for all A∈B(S). The main result of this subsection is the following.Theorem 4.6.
The family of probability measures{ΠN;N∈N} is tight.Proof.
Forε>0, we should find the compact subsets
(4.37)Σε⊂C(0,T;Rm),Yε⊂L2(0,T;V),
such that
(4.38)P̅(ω̅:W̅(ω̅,·)∉Σε)≤ε2,(4.39)P̅(ω̅:uN(ω̅,·)∉Yε)≤ε2.
The quest for Σε is made by taking account of some fact about the Wiener process such as the formula
(4.40)E̅|W̅(t2)-W̅(t1)|2j=(2j-1)!(t2-t1)j,j=1,2,….
For a constant Lε depending on ε to be chosen later and n∈N, we consider the set
(4.41)Σε={W(·)∈C(0,T;Rm):supt1,t2∈[0,T],|t2-t1|≤1/n6n|W(t2)-W(t1)|≤Lε}
Making use of Markov’s inequality:
(4.42)P̅(ω:ξ(ω)≥α)≤1αkE̅[|ξ(ω)|k]
for a random variable  ξ on (Ω̅,F̅,P̅) and positives variables  α  and  k, we get
(4.43)P̅(ω̅:W̅(ω̅,·)∉Σε)≤P̅[⋃n{ω:supt1,t2∈[0,T]:|t2-t1|<1/n6|W̅(t2)-W̅(t1)|>Lεn}]≤∑n=1∞∑i=0n6-1(nLε)4E̅supiT/n6≤t≤(i+1)T/n6|W̅(t)-W̅(iTn-6)|4≤c∑n=1∞(nLε)4(Tn-6)2n6=cLε4∑n=1∞1n2,
we choose
(4.44)Lε4=2Cε-1∑n=1∞1n2
to get (4.38).
Next we chooseYε as a ball of radius  Mε  in Yμn,νn centered at zero and with μn,νn, independent of ε, converging to zero, and such that ∑n(μn/νn) converges.
From Proposition4.5, Yε is a compact subset of L2(0,T;V).
We have further
(4.45)P̅(ω:uN(ω,·)∉Yε)≤P̅(ω:∥uN∥Yμn,νn>Mε)≤1MεE̅∥uN∥Yμn,νn≤cMε,
choosing Mε=2cε-1, we get (4.39).
From (4.38) and (4.39), we have
(4.46)P̅(ω:W̅(ω,·)∈Σε;uN(ω,·)∈Yε)≥1-ε,
this proves that
(4.47)ΠN(Σε×Yε)≥1-ε,∀N∈ℕ.


## 4.1.4. Step  4. Applications of Prokhorov and Skorokhod Results

From the tightness property of{ΠN}  and Prokhorov’s theorem [12], we have that there exist a subsequence  {ΠNj}  and a measure  Π such that  ΠNj→Π  weakly.By Skorokhod’s theorem [13], there exist a probability space  (Ω,ℱ,P)  and random variables (WNj,uNj),(W,u)  on (Ω,ℱ,P)  with values in S  such that(4.48)thelawof(WNj,uNj)isΠNj,(4.49)thelawof(W,u)isΠ,(4.50)(WNj,uNj)→(W,u)in`S,P-a.s.
Hence, {WNj}  is a sequence of an m-dimensional standard Wiener process.Letℱt=σ{W(s),u(s),0≤s≤t}.Arguing as in  [16], we prove that W(t)  is an m-dimensional ℱt  standard Wiener process and the pair (WNj,uNj) satisfies the equation(4.51)uNj(t)+ν∫0tPNjÃuNj(s)ds+∫0tPNjB̃(uNj(s),uNj(s))ds=∫0tPNjF̃(s,uNj(s))ds+∫0tPNjG̃(s,uNj(s))dWNj+u0Nj.


## 4.1.5. Step  5. Passage to the Limit

From  (4.51), it follows that uNj  satisfies the results of the Lemmas 4.2, 4.3, and 4.4. Therefore, we have for p≥1  the a priori estimates(4.52)Esup0≤t≤T∥uNj(t)∥p≤C;E(∫0T∥uNj(t)∥D(A)2dt)p≤C;Esup0≤θ≤δ∫0T∥uNj(t+θ)-uNj∥D(A)′2dt≤C(α)δ
thus modulo the extraction of a subsequence denoted again by uNj, we have(4.53)uNj→uweakly*inLp(Ω,ℱ,P;L∞(0,T;V));uNj→uweaklyinLp(Ω,ℱ,P;L2(0,T;D(A)));Esup0≤t≤T∥u(t)∥p≤C;E(∫0T∥u(t)∥D(A)2dt)p≤C;Esup0≤θ≤δ∫0T∥u(t+θ)-u(t)∥D(A)′2dt≤Cδ.
By (4.50) and the first estimate in (4.52) and Vitali’s theorem, we have(4.54)uNj→ustronglyinL2(Ω,ℱ,P;L2(0,T,V)),
and thus modulo the extraction of a subsequence and for almost every (ω,t) with respect to the measure dP⊗dt:(4.55)uNj→uinV.
This convergence together with the condition on F̃, the first estimate in  (4.52)  and Vitali’s theorem, give(4.56)F̃(·,uNj(·))→F̃(·,u(·))stronglyinL2(Ω,ℱ,P;L2(0,T,V)),∫0tF̃(s,uNj(s))ds→∫0tF̃(s,u(s))dsstronglyinL2(Ω,ℱ,P;L2(0,T,V)).
As(4.57)uNj→uweaklyinL2(Ω,ℱ,P;L2(0,T;D(A))),
then(4.58)∫0tÃuNj(s)ds→∫0tÃu(s)dsweaklyinL2(Ω,ℱ,P;L2(0,T;D(A)′)).
We also have(4.59)∫0tB̃(uNj(s),uNj(s))ds→∫0tB̃(u(s),u(s))dsweaklyinL2(Ω,ℱ,P;L2(0,T;D(A)′)).
In fact, since L∞(Ω×(0,T),dP×dt;D(A))  is  dense  in L2(Ω,ℱ,P;L2(0,T;D(A))), and B̃(uNj(s),uNj(s)) is bounded in L2(Ω,ℱ,P;L2(0,T;D(A)′)) it suffices to prove that forallφ∈L∞(Ω×(0,T),dP×dt;D(A)),(4.60)E∫0T〈B̃(uNj(s),uNj(s)),φ(s)〉D(A)′ds→E∫0T〈B̃(u(s),u(s)),φ(s)〉D(A)′ds.
Indeed, we have(4.61)E∫0T〈B̃(uNj(s),uNj(s))-B̃(u(s),u(s)),φ(s)〉D(A)′ds=E∫0T〈B̃(uNj(s)-u(s),uNj(s)),φ(s)〉D(A)′ds+E∫0T〈B̃(u(s),uNj(s)-u(s)),φ(s)〉D(A)′ds=I1j+I2j,I1j=E∫0T〈B̃(uNj(s)-u(s),uNj(s)),φ(s)〉D(A)′ds
By the property (3.17) of B̃, we have(4.62)I1j≤CE∫0T∥uNj(s)-u(s)∥∥uNj(s)∥D(A)|Aφ(s)|ds,
applying Cauchy-Schwarz inequality(4.63)I1j≤Cφ(E∫0T∥uNj(s)-u(s)∥2ds)1/2(E∫0T∥uNj(s)∥D(A)2ds)1/2.
Since(4.64)uNj→ustronglyinL2(Ω,ℱ,P;L2(0,T;V)),
and uNj  is bounded in L2(Ω,ℱ,P;L2(0,T;D(A))), we conclude that(4.65)I1j→0asj→∞.I2j=E∫0T〈B̃(u(s),uNj-u(s)),φ(s)〉D(A)′ds.
Again thanks to the property (3.18) of B̃, as(4.66)uNj→uweaklyinL2(Ω,ℱ,P;L2(0,T;D(A))),
we obtain I2j→0  as  j→∞ since any strongly continuous linear operator is weakly continuous. We are now left with the proof of(4.67)∫0tG̃(s,uNj(s))dWNj(s)→∫0tG̃(s,u(s))dW(s)weakly*L2(Ω,ℱ,P;L∞(0,T;D(A)′)),
which can be prove with the same argument like in [16].Collecting all the convergence results, we deduce that(4.68)u(t)+ν∫0tÃu(s)ds+∫0tB̃(u(s),u(s))ds=∫0tF̃(s,u(s))ds+∫0tG̃(s,u(s))dW(s)+u0,P-a.s.
as the equality in D(A)′.We haveB̃(u,u)∈L2(Ω,ℱ,P;L∞(0,T;D(A)′)), Ãu-F̃(t,u)∈L2(Ω,ℱ,P;L∞(0,T;D(A)′)), G̃(t,u)∈L2(Ω,ℱ,P;L∞(0,T;V⊗m)).Thus, from the classical results in [18] (see also [19]), we deduce from (4.68) that u is P-a.s. continuous with values in V.


## 4.1.6. Step  6. Existence of the Pressure

For the existence of the pressure, we use a generalization of the Rham’s theorem processes [20, Theorem 4.1, Remark 4.3]. From (3.6), we have for all v∈𝒱,(4.69)〈dWdt-∂t(u-αΔu)-ν(Au-αΔ(Au))-(u·∇)(u-αΔu)+α∇u*·Δu+F(·,u)+G(·,u)dWdt,v〉(𝒟′(D))3×(𝒟(D))3=0.
We denote(4.70)h=-∂t(u-αΔu)-ν(Au-αΔ(Au))-(u·∇)(u-αΔu)+α∇u*·Δu+F(·,u)+G(·,u)dWdt.
We will prove that the regularity on u, implies that(4.71)h∈L2(Ω,ℱt,P;H-1(0,t;(H-2(D))3).
By (2.7) and (2.9), we have as u∈L4(Ω,ℱ,P;L2(0,T;D(A))),(4.72)(u·∇(u-αΔu))+∇u*·Δu∈L2(Ω,ℱt,P;L1(0,t;(H-1(D))3)),Au-αΔ(Au)∈L4(Ω,ℱt,P;L2(0,t;(H-2(D))3)).
We also have(4.73)u-αΔu∈L4(Ω,ℱt,P;L2(0,t;(L2(D))3)),∂t(u-αΔu)∈L4(Ω,ℱt,P;H-1(0,t;(L2(D))3)),∀t∈[0,T].
Again, as u∈L4(Ω,ℱ,P;C([0,T];V)), then its follows that(4.74)F(t,u)∈L4(Ω,ℱt,P;L2(0,t;(H-1(D))3)),G(t,u)dWdt∈L4(Ω,ℱt,P;W-1,∞(0,t;(L2(D))3)),
for all t∈[0,T].Thenh∈L2(Ω,ℱt,P;H-1(0,t;(H-2(D))3), and(4.75)〈h,v〉(𝒟′(D))3×(𝒟(D))3=0,∀v∈𝒱.
Therefore, by a generalization of the Rham theorem processes [20], there exists a unique p̃∈L2(Ω,ℱt,P;H-1(0,t;(H-1(D))3)such thatP-a.s.(4.76)∇p̃=h,∫Dp̃dx=0,thatis,(3.7).
Theorem 3.2 is proved.


## 4.2. Proof of Corollary3.3

Proof.
We will prove the pathwise uniqueness which implies uniqueness of weak solutions. LetLF and LG be two real such that
(4.77)∥F(t,u)-F(t,v)∥(H-1(D))3≤LF∥u-v∥,∥G(t,u)-G(t,v)∥((L2(D))3)m≤LG∥u-v∥.
Then F̃ and G̃ are defined, respectively, by (3.14) and (3.20) satisfying
(4.78)∥F̃(t,u)-F̃(t,v)∥V≤LF̃∥u-v∥,∥G̃(t,u)-G̃(t,u)∥V⊗m≤LG̃∥u-v∥.
Let u1 and u2 two weak solutions of problem (1.1) defined on the same probability space together with the same Wiener process and starting from the same initial value u0.
We denoteu̅=u1-u2. Take μ>0 to be defined later and ρ(t)=exp(-μ∫0t∥u2(s)∥D(A)2ds), 0≤t≤T.
Applying Ito’s formula to the real processρ(t)∥u̅(t)∥2, we obtain from (3.13), (3.18), (3.19), and (3.26) that
(4.79)ρ(t)∥u̅(t)∥2+α̃∫0tρ(s)∥u̅(s)∥D(A)2ds≤LG̃2∫0tρ(s)∥u̅(s)∥2ds+2c̃∫0tρ(s)∥u2(s)∥D(A)∥u̅(s)∥D(A)∥u̅(s)∥ds+2LF̃∫0tρ(s)∥u̅(s)∥D(A)∥u̅(s)∥ds+2∫0t((ρ(s)(G̃(s,u1(s))-G̃(s,u2(s))),u̅(s)))dW(s)-μ∫0t∫0tρ(s)∥u2(s)∥D(A)2∥u̅(s)∥2ds,
for all t∈[0,T].
By young’s inequality, we have
(4.80)2c̃∥u2(s)∥D(A)∥u̅(s)∥D(A)∥u̅(s)∥≤α̃2∥u̅(s)∥D(A)2+2c̃2α̃∥u2(s)∥D(A)2∥u̅(s)∥22LF̃∥u̅(s)∥D(A)∥u̅(s)∥≤α̃2∥u̅(s)∥D(A)2+2LF̃2α̃∥u̅(s)∥2.
If we take μ=2(c̃2/α̃), we obtain from (4.79) that
(4.81)ρ(t)∥u̅(t)∥2≤(LG̃2+2LF̃2α̃)∫0tρ(s)∥u̅(s)∥2ds+2∫0t((ρ(s)(G̃(s,u1(s))-G̃(s,u2(s)),u̅(s))))dW.
As 0<ρ(t)≤1, the expectation of the stochastic integral in (4.81) vanishes and
(4.82)Eρ(t)∥u̅(t)∥2≤(LG̃2+2LF̃2α̃)E∫0tρ(s)∥u̅(s)∥2ds.
The Gronwall lemma implies that  u̅(t)=0, P-a.s. for all t∈[0,T]. Also, the corollary is proved.Remark 4.7.
Using the famous Yamada-Watanabe theorem [11], Corollary 3.3 implies the existence of a unique strong solution of (1.1).



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*Source: 723236-2009-11-18.xml*