On the Strong Solution for the 3D Stochastic Leray-Alpha Model

Volume 2010, Article ID 723018,31pages doi:10.1155/2010/723018

Research Article

On the Strong Solution for the 3D Stochastic Leray-Alpha Model

Gabriel Deugoue

and Mamadou Sango

1Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa

2Department of Mathematics and Computer Sciences, University of Dschang, P.O. Box 67, Dschang, Cameroon

Correspondence should be addressed to Mamadou Sango,mamadou.sango@up.ac.za Received 13 August 2009; Accepted 27 January 2010

Academic Editor: Vicentiu D. Radulescu

Copyrightq2010 G. Deugoue and M. Sango. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We prove the existence and uniqueness of strong solution to the stochastic Leray-αequations under appropriate conditions on the data. This is achieved by means of the Galerkin approximation scheme. We also study the asymptotic behaviour of the strong solution as alpha goes to zero. We show that a sequence of strong solutions converges in appropriate topologies to weak solutions of the 3D stochastic Navier-Stokes equations.

1. Introduction

It is computationally expensive to perform reliable direct numerical simulation of the Navier- Stokes equations for high Reynolds number flows due to the wide range of scales of motion that need to be resolved. The use of numerical models allows researchers to simulate turbulent flows using smaller computational resources. In this paper, we study a particular subgrid-scale turbulence model known as the Leray-alpha modelLeray-α.

We are interested in the study of the probabilistic strong solutions of the 3D Leray- alpha equations, subject to space periodic boundary conditions, in the case in which random perturbations appear. To be more precise, letT 0, L³, T >0,and consider the system

d

u−α²Δu

−νΔ

u−α²Δu

−u· ∇

u−α²Δu ∇p

dt Ft, udtGt, udW in0, T× T,

∇ ·u0 in0, T× T, ut, xis periodic in x,

Tu dx0, u0 u0 inT,

1.1

where u u1, u2, u3 and p are unknown random fields on 0, T × T, representing, respectively, the velocity and the pressure, at each point of0, T× T,of an incompressible viscous fluid with constant density filling the domainT. The constantν >0 andαrepresent, respectively, the kinematic viscosity of the fluid and spatial scale at which fluid motion is filtered. The terms Ft, u and Gt, udW are external forces depending eventually on u, where W is an R^m-valued standard Wiener process. Finally, u0 is a given random initial velocity field.

The deterministic version of1.1, that is, whenG 0, has been the object of intense investigation over the last years. The initial motivation was to find a closure model for the 3D turbulence averaged Reynolds number; for more details, we refer to1and the references therein. A key interest in the model is the fact that it serves as a good approximation of the 3D Navier-Stokes equations. It is readily seen that when α 0, the problem reduces to the usual 3D Navier-Stokes equations. Many important results have been obtained in the deterministic case. More precisely, the global wellposedness of weak solutions for the deterministic Leray-alpha equations has been established in2and also their relation with Navier-Stokes equations asαapproaches zero. The global attractor was constructed in1,3.

The addition of white noise driven terms to the basic governing equations for a physical system is natural for both practical and theoretical applications. For example, these stochastically forced terms can be used to account for numerical and empirical uncertainties and thus provide a means to study the robustness of a basic model. Specifically in the context of fluids, complex phenomena related to turbulence may also be produced by stochastic perturbations. For instance, in the recent work of Mikulevicius and Rozovskii4, such terms are shown to arise from basic physical principals. To the best of our knowledge, there is no systematic work for the 3D stochastic Leray-αmodel.

In this paper, we will prove the existence and uniqueness of strong solutions to our stochastic Leray-αequations under appropriate conditions on the data, by approximating it by means of the Galerkin method see Theorem 2.3. Here, the word “strong” means

“strong” in the sense of the theory of stochastic differential equations, assuming that the stochastic processes are defined on a complete probability space and the Wiener process is given in advance. Since we consider the strong solution of the stochastic Leray-alpha equations, we do not need to use the techniques considered in the case of weak solutions see5–9. The techniques applied in this paper use in particular the properties of stopping times and some basic convergence principles from functional analysis see 10–13. An important result, which cannot be proved in the case of weak solutions, is that the Galerkin approximations converge in mean square to the solution of the stochastic Leray-alpha equationsseeTheorem 2.4. We can prove by using the property of higher-order moments for the solution. Moreover, as in the deterministic case2, we take limitsα → 0. We study the behavior of strong solutions asαapproaches 0. More precisely, we show that, under this limit, a subsequence of solutions in question converges to a probabilistic weak solutions for the 3D stochastic Navier-Stokes equationsseeTheorem 6.5. This is reminiscent of the vanishing viscosity method; see, for instance,14,15.

This paper is organized as follows. InSection 2, we formulate the problem and state the first result on the existence and uniqueness of strong solutions for the 3D stochastic Leray- αmodel. InSection 3, we introduce the Galerkin approximation of our problem and derive crucial a priori estimates for its solutions.Section 4is devoted to the proof of the existence and uniqueness of strong solutions for the 3D stochastic Leray-αmodel. InSection 5, We prove the convergence result ofTheorem 2.4. InSection 6, we study the asymptotic behavior of the strong solutions for the 3D stochastic Leray-αmodel asαapproaches 0.

2. Statement of the Problem and the First Main Result

LetT 0, L³. We denote byC_per^∞ T³the space of allT-periodicC^∞vector fields defined on T. We set

V

Φ∈C^∞_perT³/

TΦdx0;∇ ·Φ 0

. 2.1

We denote byHandVthe closure of the setVin the spacesL²T³andH¹T³, respectively.

ThenH is a Hilbert space equipped with the inner product ofL²T³.V is Hilbert space equipped with inner product ofH¹T³. We denote by·,·and| · |the inner product and norm inH. The inner product and norm inVare denoted by·,·and · , respectively. Let A−PΔbe the Stokes operator with domainDA H²T³∩V, whereP :L²T³ → H is the Leray projector.Ais an isomorphism fromV toVthe dual space ofVwith compact inverse, henceAhas eigenvalues{λk}^∞_k1,that is,4π²/L²λ1≤λ2≤ · · · ≤λn → ∞n → ∞ and corresponding eigenfunctions{wk}^∞_k1which form an orthonormal basis ofHsuch that Awkλkwk.

We also have

Av, v_V ≥βv² 2.2

for allv∈V, whereβ >0 and·,·_Vdenotes the duality betweenVandV.

Following the notations common in the study of Navier-Stokes equations, we set

Bu, v Pu· ∇v ∀u, v∈V. 2.3

Thensee16–18

Bu, v, v_V 0 ∀u, v∈V, 2.4 Bu, v, w_V−Bu, w, vV ∀u, v, w∈V, 2.5

|Bu, v, w| ≤C|Au|v|w|, ∀u∈DA, v∈V, w∈H, 2.6 Bu, v, w_DA ≤C|u|v|Aw|, ∀u∈H, v∈V, w∈DA, 2.7

|Bu, v, w_V| ≤C|u|^1/4u^3/4|v|^1/4v^3/4w, ∀u∈V, v∈V, w∈V, 2.8

|Bu, v, w| ≤C|u|^1/4u^3/4v^1/4|Av|^3/4|w|, ∀u∈V, v∈DA, w∈H. 2.9

LetΩ,F, P be a complete probability space and{Ft}_0≤t≤T an increasing and right- continuous family of sub-σ-algebras ofFsuch thatF0contains all the P-null sets ofF. LetW be aR^m-valued Wiener process onΩ,F,{Ft}_0≤t≤T, P.

We now introduce some probabilistic evolution spaces.

LetXbe a Banach space. Forr, p≥1, we denote by

L^pΩ,F, P;L^r0, T;X 2.10 the space of functionsuux, t, ωwith values inXdefined on0, T×Ωand such that

1uis measurable with respect tot, ωand for eacht,uisFtmeasurable, 2u∈Xfor almost allt, ωand

u_L^p_Ω,F,P;Lr0,T;X

⎡

⎣E _T

0

u^r_Xdt _p/r⎤

⎦

1/r

<∞, 2.11

whereEdenote the mathematical expectation with respect to the probability measureP.

The spaceL^pΩ,F, P;L^r0, T;Xso defined is a Banach space.

Whenr ∞, the norm inL^pΩ,F, P;L^∞0, T;Xis given by

u_L^p_Ω,F,P;L^∞_0,T;X

Esup

0≤t≤Tu^p_{X 1/p}

. 2.12

We make precise our assumptions on F and G. We suppose that F and G are measurable Lipschitz mappings fromΩ×0, T×HintoHand fromΩ×0, T×HintoH^⊗m, respectively. More exactly, assume that, for allu, v∈H, F·, uandG·, uareFt-adapted, and dP×dt−a.e.inΩ×0, T

|Ft, u−Ft, v|_H≤LF|u−v|, Ft,0 0,

|Gt, u−Gt, v|_H⊗m ≤LG|u−v|, Gt,0 0.

2.13

HereH^⊗mis the product ofmcopies ofH.

Finally, we assume thatu0∈L²Ω,F0, P;DA.

Remark 2.1. The condition 10 is given only to simplify the calculations. It can be omitted; in which case one could use the estimate

|Ft, u|²≤2L²_F|u|²2|Ft,0|² 2.14

that follows from the Lipschitz condition. The same remark applies toG.

Alongside problem1.1, we will consider the equivalent abstract stochastic evolution equation

d

uα²Au

νA

uα²Au B

u, uα²Au

dtFt, udtGt, udW, u0 u0.

2.15

We now define the concept of strong solution of the problem2.15as follows.

Definition 2.2. By a strong solution of problem2.15, we mean a stochastic processusuch that

1utisFtadapted for allt∈0, T,

2u∈L^pΩ,F, P;L²0, T;DA^3/2∩L^pΩ,F, P;L^∞0, T, DAfor all 1≤p <∞, 3uis weakly continuous with values inDA,

4P-a.s., the following integral equation holds:

ut α²Aut,Φ ν

_t

0

us α²Aus, AΦ ds

_t

0

B

us, us α²Aus ,Φ

ds

u0α²Au0,Φ

_t

0

Fs, us,Φds _t

0

Gs, us,ΦdWs

2.16

for allΦ∈ V, andt∈0, T.

Notation 1. In this paper, weak convergence is denoted by and strong convergence by →. Our first result of this paper is the following.

Theorem 2.3 existence and uniqueness. Suppose that the hypotheses 2.13 hold, and u0 ∈ L²Ω,F0, P;DA. Then problem2.15has a solution in the sense ofDefinition 2.2. The solution is unique almost surely and has inDAalmost surely continuous trajectories.

We also prove that the sequenceunof our Galerkin approximationsee3.1below approximates the solutionuof the 3Dstochastic Leray-αmodel in mean square.

This is the object of the second result of the paper.

Theorem 2.4Convergence results. Under the hypotheses ofTheorem 2.3, the following conver- gences hold:

E _t

0

uns−us²_DA3/2ds−→0 forn−→ ∞, Eunt−ut²_DA−→0, n−→ ∞ 2.17

for allt∈0, T.

Remark 2.5. Theorems2.3and2.4are also true if one assumes measurable Lipschitz mappings F:Ω×0, T×DA^3/2 → HandG:Ω×0, T×DA → H^⊗m.

Remark 2.6. For the existence of the pressure, we can use a generalization of the Rham’s theorem for processessee19, Theorem 4.1, Remark 4.3. See also6, page 15.

3. Galerkin Approximations and A Priori Estimates

We now introduce the Galerkin scheme associated to the original equation 2.15 and establish some uniform estimates.

3.1. The Approximate Equation

Let {wj}^∞_j1 be an orthonormal basis of H consisting of eigenfunctions of the operatorA.

DenoteHnspan{w1, . . . , wn}and letPnbe theL²-orthogonal projection fromHontoHn. We look for a sequenceuntinHnsolutions of the following initial value problem:

dvn νAvnPnBun, vndtPnFt, undtPnGt, undW, un0 Pnu0,

vn unα²Aun.

3.1

By the theory of stochastic differential equationssee20–23, there is a unique continuous Ft-adapted processunt∈L²Ω,F, P;L²0, T;Hnof3.1.

We next establish some uniform estimates onunandvn.

3.2. A Priori Estimates

Throughout this sectionC, Cii1, . . .denote positive constants independent ofnandα.

Lemma 3.1. unandvnsatisfy the following a priori estimates:

Esup

0≤s≤T|vns|²4νβE _T

0

vns²ds≤C1,

Esup

0≤s≤T|uns|²≤C2, Esup

0≤s≤Tuns²< C3

2α², Esup

0≤s≤T|Auns|²≤ C4

α⁴, E _T

0

uns²ds≤C5,

E _T

0

|Auns|²ds≤ C6

2α², E _T

0

A^3/2uns ²ds≤ C7

α⁴.

3.2

Proof. To proveLemma 3.1, it suffices to establish the first inequality and use the fact that

|vn|² unα²Aun ²|un|²2α²un²α⁴|Aun|², vn² un²2α²|Aun|²α⁴ A^3/2un ².

3.3

By Ito’s formula, we have from3.1

d|vnt|²2νAvn, vn_VBun, vn, vn_Vdt

2Ft, un, vn |PnGt, un|²

dt2Gt, un, vndW. 3.4

But then, taking into account2.4,2.2and the fact that

Fs, uns, vns≤C

1|vns|² ,

|PnGs, uns|²≤C

1|vns|² ,

3.5

we deduce from3.4that

|vnt|²2νβ _t

0

vns²ds

≤ |vn0|²C2TC3

_t

0

|vns|²ds2 _t

0

Gs, uns, vnsdWs.

3.6

For each integerN >0, consider theFt-stopping timeτNdefined by

τN inf

t:|vnt|²≥N²

∧T. 3.7

It follows from3.6that

sup

s∈0,t∧τN|vns|²2νβ _t∧τN

0

vns²ds≤ |vn0|²C8TC9

_t∧τN

0

|vns|² ds

2 sup

s∈0,t∧τN

_s

0

Gs, uns, vnsdWs

3.8

for allt∈0, Tand allN,n≥1. Taking expectation in3.8, by Doob’s inequality it holds

E sup

s∈0,t∧τN

_s

0

Gs, uns, vnsdWs≤3E _t∧τN

0

Gs, uns, vns² ds 1/2

≤3E _t∧τN

0

|Gs, uns|²|vns|²ds 1/2

≤ 1 2E sup

0≤s≤t∧τN

|vns|²C10TC11E _t∧τN

0

|vns|²ds.

3.9

Next using Gronwall’s lemma, it follows that there exists a constantC1 depending onT, C such that, for alln≥1

Esup

0≤s≤T|vns|²4νβE _T

0

vns²ds≤C1. 3.10

The following result is related to the higher integrability ofunandvn. Lemma 3.2. One has

Esup

0≤s≤T|vns|^p≤Cp, Esup

0≤s≤T|uns|^p≤Cp, 3.11

Esup

0≤s≤Tuns^p≤ Cp

α^p, 3.12

Esup

0≤s≤T|uns|^p_DA≤ Cp

α^2p 3.13

for all 1≤p <∞.

Proof. By Ito’s formula, we have for 4≤p <∞

d|vnt|^p/2 p

2|vnt|^p/2−2

×

−νAvn, vn_V− Bun, vn, vn_V Ft, un, vn

p−4 4

Gt, un, vn²

|vnt|²

dt p

2|vnt|^p/2−2Gt, un, vndW.

3.14

Taking into account2.4and the fact that

|vns|^p/2−2Ft, un, vn≤C

1|vns|^p/2 Young’s inequality , Gs, un, vn²

|vns|² ≤C

1|vns|² ,

3.15

we deduce from3.14that

|vnt|^p/2≤ |vn0|^p/2C _t

0

1|vns|^p/2 dsp

2 _t

0

|vns|^p/2−2Gs, uns, vnsdWs.

3.16

Taking the supremum, the square, and the mathematical expectation in3.16, and owing to the Martingale’s inequality it holds

Esup

0≤s≤T

_s

0

|vns|^p/2−2Gs, uns, vnsdWs ²

≤4E _T

0

|vns|^p−4Gs, uns, vns²ds

≤4CE _T

0

1|vns|^p ds.

3.17

Applying Gronwall’s lemma, it follows that there exists a constantCp, such that Esup

0≤s≤T|vns|^p≤Cp 3.18

for allp≥4. With this being proved for anyp≥4, it is subsequently true for any 1≤p <∞.

Other inequalities are deduced from the relation

|vns|² |uns|²2α²uns²α⁴|Auns|². 3.19

We also have the following.

Lemma 3.3. One has E

_T

0

vns²ds _p

≤Cp for 1≤p <∞. 3.20

Proof. The proof is derived from4.46, Martingale’s inequality, andLemma 3.2.

4. Proof of Theorem 2.3

4.1. Existence

With the uniform estimates on the solution of the Galerkin approximations in hand, we proceed to identify a limit u. This stochastic process is shown to satisfy a stochastic partial differential equationssee4.2with unknown terms corresponding to the nonlinear portions of the equation. Next, using the properties of stopping times and some basic convergence principles from functional analysis, we identify the unknown portions.

We will split the proof of the existence into two steps.

4.1.1. Taking Limits in the Finite-Dimensional Equations

Lemma 4.1 limit system. Under the hypotheses of Theorem 2.3, there exist adapted processes u, B^∗, F^∗, andG^∗with the regularity,

u∈L^p

Ω,F, P;L²

0, T;D

A^3/2

∩L^pΩ,F, P;L^∞0, T;DA,

v∈L^p

Ω,F, P;L²0, T;V ,

v∈C0, T;Ha.s., u∈C0, T;DAa.s., B^∗∈L²

Ω,F, P;L²

0, T :V ,

F^∗∈L²

Ω,F, P;L²0, T;H , G^∗∈L²

Ω,F, P;L²

0, T;H^⊗m ,

4.1

such thatu,B^∗,F^∗, andG^∗satisfy

vt ν _t

0

Avsds _t

0

B^∗sdsv0 _t

0

F^∗sds _t

0

G^∗sdWs 4.2

wherevt ut α²Autand 1≤p <∞.

Remark 4.2. We use the following elementary facts regarding weakly convergent sequences in the proof below.

iLetS1andS2be Banach spaces and letL:S1 → S2be a continuous linear operator.

Ifxnis a sequence inS1such thatxn xwherex∈S1, thenLxn Lx.

iiIfS is Banach space and ifxn is a sequence fromL²Ω,F, P;L²0, T;S, which converges weakly to xin L²Ω,F, P;L²0, T;S, then for n → ∞ the following assertions are true:

_t

0

xnsds _t

0

xsds, _t

0

xnsdWs _t

0

xsdWs

4.3

inL²Ω,F, P;L²0, T;S.

Proof ofLemma 4.1. Using2.8and H ¨older’s inequality, we have

E _T

0

PnBunt, vnt²_V≤C

Esup

t∈0,Tunt⁴ 1/2⎛

⎝E _T

0

vnt²dt 2⎞

⎠

1/2

. 4.4

The later quantity is uniformly bounded as a consequence of Lemmas3.2,3.3. From4.4, we can deduce that the sequencePnBun, vnis bounded inL²Ω,F, P;L²0, T;V. On the other hand, from Lemmas 3.1, 3.2, 3.3and the Lipschitz conditions onF and G, we have that the sequenceunis bounded inL^pΩ,F, P;L²0, T;DA^3/2∩L^pΩ,F, P;L^∞0, T;DA, the sequencevnis bounded inL²Ω,F, P;L²0, T;V∩L²Ω,F, P;L^∞0, T;H, the sequence vn0 is bounded inL²Ω,F0, P;H, the sequence un0 is bounded inL²Ω,F0, P;DA, the sequencePnFt, unis bounded inL²Ω,F, P;L²0, T;H, andPnGt, unis bounded in L²Ω,F, P;L²0, T;H^⊗m.

Thus with Alaoglu’s theorem, we can ensure that there exists a subsequence{un} ⊂ {un}, and seven elements u ∈ L^pΩ,F, P;L²0, T;DA^3/2∩ L^pΩ,F, P;L^∞0, T;DA, v ∈ L²Ω,F, P;L²0, T;V∩ L²Ω,F, P;L^∞0, T;H, B^∗ ∈ L²Ω,F, P;L²0, T;V, F^∗ ∈ L²Ω,F, P;L²0, T;H, ρ1 ∈ L²Ω,F0, H, ρ2 ∈ L²Ω,F0, DA and G^∗ ∈ L²Ω,F, P; L²0, T;H^⊗msuch that:

un u inL^p

Ω,F, P;L²

0, T;D

A^3/2

∩L^pΩ,F, P;L^∞0, T;DA, 4.5 vn v inL²

Ω,F, P;L²0, T;V

, 4.6

PnBun, vn B^∗ inL²

Ω,F, P;L²

0, T;V

, 4.7

PnFt, un F^∗ inL²

Ω,F, P;L²0, T;H , vn0 ρ1 inL²Ω,F0, H un0 ρ2 inL²Ω,F0, DA

4.8

PnGt, un G^∗ inL²

Ω,F, P;L²

0, T;H^⊗m

. 4.9

UsingRemark 4.2and the weak convergence above, we obtain from3.1

vt ν _t

0

Avsds _t

0

B^∗sdsv0 _t

0

F^∗sds _t

0

G^∗sdWs 4.10

for allt∈0, T, wherevt ut α²Autandv0u0α²Au0.

Referring then to results 21,24,25, we find that vhas modification such thatv ∈ C0, T;Ha.s. which implies thatuhas modification inC0, T;DAa.s.

4.1.2. Proof ofB^∗=Bu, v,F^∗=Ft, uandG^∗=Gt, u

For simplicity we keep on denoting by{un}the subsequence{un}in this step.

LetXt_t∈0,Tbe a process in the spaceL²Ω,F, P;L²0, T;V. Using the properties of Aand of its eigenvectors {w1, w2, . . .}λ1, λ2, . . . are the corresponding eigenvalues, we have

PnXt ≤ Xt, |PnXt| ≤ |Xt|, |Xt−PnXt| ≤ |Xt|, βXt−PnXt²≤ AXt−APnXt, Xt−PnXt_V

^i∞

in

λiXt, wi²

≤ AXt, Xt_V

≤CXt².

4.11

Hence fordP×dta.e.w, t∈Ω×0, T, we have

nlim→ ∞Xw, t−PnXw, t²0. 4.12

By the Lebesgue dominated convergence theorem, it follows that

nlim→ ∞

_T

0

Xt−PnXt²dt0,

nlim→ ∞E _T

0

Xt−PnXt²dt0,

4.13

nlim→ ∞EXt−PnXt² 0. 4.14

Applying this result toXv∈L²Ω,F, P;L²0, T;VorXu, we have

Pnv−→v inL²

Ω,F, P;L²0, T;V

, 4.15

Pnu−→u inL²

Ω,F, P;L²0, T;V

. 4.16

With a candidate solution in hand, it remains to show that

B^∗Bu, v, F^∗Ft, u, G^∗Gt, u. 4.17

In the next lemma, we comparevand the sequencevnunα²Aun, at least up to a stopping time τm ↑ T a.s.; this is sufficient to deduce the existence result. Here, we are adapting techniques used in10,11.

Letm∈N^∗, consider theFt-stopping timeτmdefined by

τminf

t;|vt|² _t

0

vs²ds≥m²

∧T. 4.18

Notice thatτmis increasing as a function ofmand moreoverτm → T a.s.

Lemma 4.3. One has

nlim→ ∞E _τm

0

vns−vs²ds0. 4.19

Proof. Using4.15, it suffices to prove that

nlim→ ∞E _τm

0

Pnvs−vns² ds0. 4.20

Using3.1and4.10, the difference ofPnvandvnsatisfies the relation dPnv−vn νAPnv−vn PnB^∗−PnBun, vndt

PnF^∗−Ft, undtPnG^∗−Gt, undW. 4.21

Letσt exp{−n1t−n2

_t

0vs² ds},0≤t≤T, withn1andn2positive constants to be fixed later.

Applying Ito’s formula to the processσt|Pnv−vn|², we have

σt|Pnvt−vnt|²2βν _t

0

σtPnvs−vns²ds

≤2 _t

0

σsB^∗s−Buns, vns, Pnvs−vs_Vds

2 _t

0

σsF^∗s−Fs, uns, Pnvs−vnsds

2 _t

0

σs|PnG^∗s−Gs, uns|²ds

2 _t

0

σsG^∗s−Gs, uns, Pnvs−vnsdW−n1

_t

0

σs|Pnvs−vns|²ds

−n2

_t

0

σsvs²|Pnvs−vns|²ds.

4.22

We are going to estimate the first three terms of the right-hand side of4.22.

For the first term, using the cancellation property2.4and2.8, we have

B^∗−Bun, vn, Pnv−vn_V

B^∗, Pnv−vn_VBun−Pnu, Pnv, vn−Pnv_VBPnu, Pnv, vn−Pnv_V

≤ B^∗, Pnv−vn_VC|un−Pnu|^1/4un−Pnu^3/4|Pnv|^1/4Pnv^3/4vn−Pnv BPnu, Pnv, vn−Pnv_V

≤ B^∗, Pnv−vn_V C

2βv²|vn−Pnv|²β

2vn−Pnv²BPnu, Pnv, vn−Pnv_V. 4.23

For the term involvingF^∗andF, using the Lipschitz conditions onF, we have

2F^∗−Ft, un, Pnv−vn≤2F^∗−Ft, u, Pnv−vn 2Ft, u−Ft, Pnu, Pnv−vn 2LF|Pnu−un||Pnv−vn|

≤2F^∗−Ft, u, Pnv−vn 2Ft, u−Ft, Pnu, Pnv−vn 2CLF|Pnv−vn|².

4.24

For the term involvingG^∗andG, using the Lipschitz conditions onG, we have

|PnG^∗−Gt, un|²≤2L²_G|Pnu−un|²2L²_G|u−Pnu|²2G^∗−Gt, u, PnG^∗−Gt, un

− |PnG^∗−Gt, u|²

≤2L²_G|Pnv−vn|²2L²_G|u−Pnu|²2G^∗−Gt, u, PnG^∗−Gt, un

− |PnG^∗−Gt, u|².

4.25

Taking into account4.23–4.25, we obtain from4.22 σt|Pnvt−vnt|²2β

_t

0

σsPnvs−vns²ds2 _t

0

σs|PnG^∗s−Gs, us|²ds

≤2 _t

0

σsB^∗s, Pnvs−vns_V dsC β

_t

0

σsvs²|vns−Pnvs|²ds

β _t

0

σsPnvs−vns²ds

2 _t

0

σsBPnus, Pnvs, vns−Pnvs_V ds4CLF

_t

0

σs|Pnvs−vns|²ds

4 _t

0

σsF^∗s−Fs, us, Pnvs−vnsds

4 _t

0

σsFs, us−Fs, Pnus, Pnvs−vnsds

4L²_{G t}

0

σs|Pnvs−vns|²ds4L²_{G t}

0

σs|us−Pnus|²ds

4 _t

0

σsG^∗s−Gs, us, PnG^∗s−Gs, usds

−n1

_t

0

σs|Pnvs−vns|²ds−n2

_t

0

σsvs²|Pnvs−vns|²

2 _t

0

σsG^∗s−Gs, uns, Pnvs−vnsdW.

4.26

Therefore, if we taken14CLF4L²_Gandn2C/βν, we obtain from4.26 Eστm|Pnvτm−vnτm|²3βν

2 E _τm

0

σsPnvs−vns²ds 2E

_τm

0

σs|PnG^∗s−Gs, us|²ds

≤2E _τm

0

σsB^∗s, Pnvs−vns_Vds 2E

_τm

0

σsBPnus, Pnvs, vns−Pnvs_Vds 4E

_τm

0

σsF^∗s−Fs, us, Pnvs−vnsds 4E

_τm

0

σsFs, us−Fs, Pnus, Pnvs−vnsds 4L²_GE

_τm

0

σs|us−Pnus|²ds 4E

_τm

0

σsG^∗s−Gs, us, PnG^∗s−Gs, usds.

4.27

Next, we are going to prove the convergence to 0 of each term on the right-hand side of4.27.

Here we use some basic convergence principles from functional analysis12,13.

For the first two terms, we have E

_τm

0

σsBPnus, Pnvs−B^∗s, vns−Pnvs_Vds E

_τm

0

σsBPnus, Pnvs−Bus, vs, vns−Pnvs_Vds E

_τm

0

σsBus, vs−B^∗s, vns−Pnvs_Vds.

4.28

From the properties ofB, we have

BPnu, Pnv−Bu, v_V ≤ BPnu−u, Pnv_VBu, Pnv−v_V

≤Pnu−uPnvuPnv−v. 4.29 We have from4.15and4.16

I0,τmσtBPnu, Pnv−Bu, v

V −→0, asn−→ ∞, dt×dP−a.e., I_0,τmσtBPnu, Pnv−Bu, v

V≤Cutvt ∈L²

Ω,F, P;L²0, T;R

. 4.30

Using4.6and4.15, we have

vn−Pnv 0 inL²

Ω,F, P;L²0, T;V

. 4.31

Applying the results of weak convergencesee12,13, it follows from4.30and4.31that

nlim→ ∞E _τm

0

σsBPnu, Pnv−Bu, v, vns−Pnvs_Vds0. 4.32

Also asI0,τmσtBu, v−B^∗∈L²Ω,F, P;L²0, T;V, we have from4.31

nlim→ ∞E _τm

0

σsBus, vs−B^∗s, vns−Pnvs_Vds0. 4.33

On the other hand, from4.16, the Lipschitz conditions onF,Gand the fact thatvn−Pnv 0 inL²Ω,F, P;L²0, T;H, we have

nlim→ ∞E _τm

0

σsGs, us−Gs, Pnus, vns−Pnvsds0,

nlim→ ∞E _τm

0

σsFs, us−Fs, Pnus, vns−Pnvsds0.

4.34

Again from4.31and the fact that

F^∗−Ft, u∈L²

Ω,F, P;L²0, T;H , G^∗−Gt, u∈L²

Ω,F, P;L²

0, T;H^⊗m ,

4.35

we have

nlim→ ∞E _τm

0

σsF^∗s−Fs, us, vns−Pnvsds0,

nlim→ ∞E _τm

0

σsG^∗s−Gs, us, vns−Pnvsds0.

4.36

As

PnG^∗−Gt, un 0 in L²

Ω,F, P;L²

0, T;H^⊗m

, 4.37

we also have

nlim→ ∞E _τm

0

σsG^∗s−Gs, us, PnG^∗s−Gs, unsds0. 4.38

From4.32–4.38, and the fact that

exp−n1T−n2m≤I0,τmσt≤1, 4.39

we obtain from4.27

nlim→ ∞E

|Pnvτm−vnτm|²

0, 4.40

nlim→ ∞E _τm

0

Pnvs−vns²ds0, 4.41

E _τm

0

|G^∗s−Gs, us|²ds0. 4.42

Now from4.42and the fact that the sequenceτmtends toT, we have

G^∗t Gt, ut 4.43

as elements of the spaceL²Ω,F, P;L²0, T;H^⊗m. Also observe that4.40and4.15imply that

vnI_0,τm−→vI_0,τm inL²

Ω,F, P;L²0, T;V

, 4.44

whereI0,τm is the indicator function of0, τm. Letw ∈V. We have the following estimate fromB:

|Bu, v−PnBun, vn, w_V|

≤ |Bu, v−Bun, vn, w_V||I−PnBun, vn, w_V|

≤Cu−unvvn−vvnwCI−Pnwunvn.

4.45

Thus from4.45and using H ¨older’s inequality, we have

E _τm

0

Bus, vs−PnBuns, vns, w_Vds

≤C

E _τm

0

us−uns²ds 1/2

E _T

0

vs²ds _1/2

E _τm

0

vns−vs²ds 1/2

E _T

0

vns²ds _1/2

CI−Pnw

E _T

0

uns²ds 1/2

E _T

0

vns²ds 1/2

.

4.46

Consequently, by4.44and4.46, we have

nlim→ ∞E _τm

0

Bus, vs−PnBuns, vns, w_Vds0. 4.47

Taking into account4.7, it follows from4.47that

E _τm

0

Bus, vs−B^∗s, zs_Vds0 4.48

for allz∈ DVΩ×0, T, whereDVΩ×0, Tis a set ofψ∈L^∞Ω,F, P;L^∞0, T;Vwith ψ wφ, φ∈L^∞Ω×0, T;R, w∈V. 4.49 Therefore, asτmtends toT andDVΩ×0, Tis dense inL²Ω,F, P;L²0, T;V, we obtain from4.48thatBut, vt B^∗tas elements of the spaceL²Ω,F, P;L²0, T;V.

Analogously, using the Lipschitz condition onFand4.44, we haveFt, ut F^∗t as elements of the spaceL²Ω,F, P;L²0, T;H.

And the existence result follows.

4.2. Uniqueness

Let u1 and u2 be two solutions of problem 2.15, which have in DA almost surely continuous trajectories with the same initial datau0. Denote

v1u1α²Au1, v2u2α²Au2,

vv1−v2, uu1−u2. 4.50 By Ito’s formula, we have

|vt|²2 _t

0

Avs, vs_V2 _t

0

Bu1s, v1s−Bu2s, v2s, vs_V

2 _t

0

Fs, u1s−Fs, u2s, vsds2 _t

0

Gs, u1s−Gs, u2s, vsds

_t

0

|Gs, u1s−Gs, u2s|²_H⊗mds.

4.51

Takeλ >0 to be fixed later and define σt exp

−b β

_t

0

v1s²ds−λt

. 4.52