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

1_{Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa} 2_{Department of Mathematics and Computer Sciences, University of Dschang,}

P.O. Box 67, Dschang, Cameroon

Correspondence should be addressed to Mamadou Sango,[email protected]

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

du−α2Δu−νΔu−α2Δu−u· ∇u−α2Δu∇pdt Ft, udtGt, udW in0, T× T,

∇ ·u0 in0, T× T,

ut, xis periodic in x,

Tu dx0,

u0 u0 inT,

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 Rm_{-valued standard Wiener process. Finally, u}

0 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.

2. Statement of the Problem and the First Main Result

LetT 0, L3. We denote byCper∞ 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 spacesL2_T3_andH1_T3_{, respectively.}

ThenH is a Hilbert space equipped with the inner product ofL2_T3_{.V is Hilbert space}

equipped with inner product ofH1_T3_{. 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 H2_T3_{∩V, whereP :L}2_T3 _{→ 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, vV ≥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, vV 0 ∀u, v∈V, 2.4

Bu, v, wV−Bu, w, v_V ∀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, wV| ≤C|u|1/4u3/4|v|1/4v3/4w, ∀u∈V, v∈V, w∈V, 2.8

|Bu, v, w| ≤C|u|1/4u3/4v1/4|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 aRm_{-valued Wiener process onΩ,F,{Ft}}

We now introduce some probabilistic evolution spaces. LetXbe a Banach space. Forr, p≥1, we denote by

LpΩ,F, P;Lr0, 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_Lp_Ω,F,P;Lr_0,T;X

⎡ ⎣_ET

0

ur_Xdt

p/r⎤

⎦

1/r

<∞, 2.11

whereEdenote the mathematical expectation with respect to the probability measureP. The spaceLp_{Ω,F, P;L}r_{0, T;Xso defined is a Banach space.}

Whenr ∞, the norm inLp_{Ω,F, P;L}∞_{0, T;Xis given by}

uLp_Ω,F,P;L∞_0,T;X

Esup

0≤t≤T up_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⊗m_{is 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

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

duα2AuνAuα2AuBu, 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∈Lp_{Ω,F, P;L}2_{0, T;DA}3/2_∩Lp_{Ω,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,Φν

t

0

us α2Aus, AΦds

t

0

Bus, us α2Aus,Φds

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 ∈

L2_Ω,F

0, 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−us_D2_A3/2ds−→0 forn−→ ∞, Eunt−ut2_DA−→0, n−→ ∞ 2.17

Remark 2.5. Theorems2.3and2.4are also true if one assumes measurable Lipschitz mappings F:Ω×0, T×DA3/2 _{→ 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≤T

uns2< 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

_A3/2_uns 2_ds≤ C7

α4.

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 A3/2un

2

.

3.3

By Ito’s formula, we have from3.1

d|vnt|22νAvn, vnVBun, vn, vnVdt 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≤C1|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

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_0≤sup_s≤t∧τN|vns|

2

C10TC11E

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≤T

unsp≤ Cp

αp, 3.12

Esup

0≤s≤T

|uns|p_DA≤ Cp

α2p 3.13

for all1≤p <∞.

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

d|vnt|p/2 p 2|vnt|

p/2−2

×

−νAvn, vnV− Bun, vn, vnV Ft, un, vn p− 4 4

Gt, un, vn2 |vnt|2

dt

p 2|vnt|

p/2−2_{Gt, un, vndW.}

Taking into account2.4and the fact that

|vns|p/2−2Ft, un, vn≤C1|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/2dsp 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|pds.

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 for1≤p <∞. 3.20

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∈Lp_{Ω,F, P;L}2_{0, T;DA}3/2_∩Lp_{Ω,F, P;L}∞_{0, T;DA,}

v∈LpΩ,F, P;L20, T;V,

v∈C0, T;Ha.s.,

u∈C0, T;DAa.s.,

B∗∈L2Ω,F, P;L20, T :V,

F∗∈L2Ω,F, P;L20, T;H,

G∗∈L2_{Ω,F, P;L}2_{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 α2_{Autand1≤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.

iiIfS is Banach space and ifxn is a sequence fromL2_{Ω,F, P;L}2_{0, T;S, which}

converges weakly to xin L2_{Ω,F, P;L}2_{0, T;S, then for n → ∞ the following}

assertions are true:

_t

0

xnsds

_t

0

xsds,

_t

0

xnsdWs

_t

0

xsdWs

4.3

inL2_{Ω,F, P;L}2_{0, T;S.}

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

E

_T

0

PnBunt, vnt2V≤C

Esup t∈0,T

unt4

1/2⎛

⎝_ET

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 inLpΩ,F, P;L20, T;DA3/2∩LpΩ,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 inL2_Ω,F

0, 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 L2_{Ω,F, P;L}2_{0, T;H}⊗m_.

Thus with Alaoglu’s theorem, we can ensure that there exists a subsequence{un} ⊂ {un}, and seven elements u ∈ Lp_{Ω,F, P;L}2_{0, T;DA}3/2_{∩ L}p_{Ω,F, P;L}∞_{0, T;DA,}

v ∈ L2_{Ω,F, P;L}2_{0, T;V∩ L}2_{Ω,F, P;L}∞_{0, T;H, B}∗ _{∈ L}2_{Ω,F, P;L}2_{0, T;V, F}∗ _∈

L2_{Ω,F, P;L}2_{0, T;H, ρ}

1 ∈ L2Ω,F0, H, ρ2 ∈ L2Ω,F0, DA and G∗ ∈ L2Ω,F, P;

L2_{0, T;H}⊗m_{such that:}

un u inLp

Ω,F, P;L20, T;DA3/2∩LpΩ,F, P;L∞0, T;DA, 4.5

vn v inL2

Ω,F, P;L20, T;V, 4.6

PnBu_n, v_n B∗ inL2

Ω,F, P;L2_{0, T;V, 4.7}

PnFt, u_n F∗ inL2

Ω,F, P;L20, T;H,

vn0 ρ₁ inL2Ω,F₀, H un0 ρ₂ inL2Ω,F₀, DA

4.8

PnGt, u_n G∗ inL2

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 α2_Autandv

0u0α2Au0.

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 of

B

=

B

u, v

,

F

=

F

t, u

and

G

=

G

t, u

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

LetXtt∈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−PnXtV

i∞ in

λiXt, wi2

≤ AXt, Xt_V ≤CXt2.

4.11

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

lim

n→ ∞Xw, t−PnXw, t

2_0.

4.12

By the Lebesgue dominated convergence theorem, it follows that

lim n→ ∞

T

0

Xt−PnXt2dt0,

lim n→ ∞E

_T

0

Xt−PnXt2dt0,

4.13

lim

n→ ∞EXt−PnXt

2

Applying this result toXv∈L2_{Ω,F, P;L}2_{0, 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

lim n→ ∞E

τm

0

vns−vs2ds0. 4.19

Proof. Using4.15, it suffices to prove that

lim n→ ∞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

0vs 2

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

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−vsVds

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−vnV

B∗, Pnv−vnVBun−Pnu, Pnv, vn−PnvVBPnu, Pnv, vn−PnvV ≤ B∗, Pnv−vnVC|un−Pnu|1/4un−Pnu3/4|Pnv|1/4Pnv3/4vn−Pnv

BPnu, Pnv, vn−PnvV

≤ B∗, Pnv−vnV C 2βv

2_|v

n−Pnv|2 β

2vn−Pnv

2_BP

nu, Pnv, vn−PnvV. 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.

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−vnsV dsC β

_t

0

σsvs2|vns−Pnvs|2ds

β

_t

0

σsPnvs−vns2ds

2

t

0

σsBPnus, Pnvs, vns−PnvsV 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|2ds4L2G

_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.

Therefore, if we taken14CLF4L_G2 andn2C/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−vnsVds

2E

τm

0

σsBPnus, Pnvs, vns−PnvsVds

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−PnvsVds

E

_τm

0

σsBPnus, Pnvs−Bus, vs, vns−PnvsVds

E

τm

0

σsBus, vs−B∗s, vns−PnvsVds.

4.28

From the properties ofB, we have

BPnu, Pnv−Bu, vV ≤ BPnu−u, PnvVBu, Pnv−vV ≤Pnu−uPnvuPnv−v.

4.29

We have from4.15and4.16

_I0,τ

mσtBPnu, Pnv−Bu, vV −→0, asn−→ ∞, dt×dP−a.e.,

_I0,τ

mσtBPnu, Pnv−Bu, vV≤Cutvt ∈L2

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

lim n→ ∞E

τm

0

σsBPnu, Pnv−Bu, v, vns−PnvsVds0. 4.32

Also asI0,τmσtBu, v−B∗∈L

2_{Ω,F, P;L}2_{0, T;V, we have from4.31}

lim n→ ∞E

_τm

0

σsBus, vs−B∗s, vns−PnvsVds0. 4.33

On the other hand, from4.16, the Lipschitz conditions onF,Gand the fact thatvn−Pnv 0 inL2_{Ω,F, P;L}2_{0, T;H, we have}

lim n→ ∞E

τm

0

σsGs, us−Gs, Pnus, vns−Pnvsds0,

lim n→ ∞E

τm

0

σsFs, us−Fs, Pnus, vns−Pnvsds0.

4.34

Again from4.31and the fact that

F∗−Ft, u∈L2Ω,F, P;L20, T;H,

G∗−Gt, u∈L2_{Ω,F, P;L}2_{0, T;H}⊗m_,

4.35

we have

lim n→ ∞E

_τm

0

σsF∗s−Fs, us, vns−Pnvsds0,

lim n→ ∞E

_τm

0

σsG∗s−Gs, us, vns−Pnvsds0.

4.36

As

PnG∗−Gt, un 0 in L2Ω,F, P;L20, T;H⊗m, 4.37

we also have

lim n→ ∞E

_τm

0

From4.32–4.38, and the fact that

exp−n1T−n2m≤I0,τmσt≤1, 4.39

we obtain from4.27

lim n→ ∞E

|Pnvτm−vnτm|2

0, 4.40

lim n→ ∞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 spaceL2_{Ω,F, P;L}2_{0, T;H}⊗m_. Also observe that4.40and4.15imply that

vnI0,τm−→vI0,τm inL

2_{Ω,F, P;L}2_{0, T;V, 4.44}

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

fromB:

|Bu, v−PnBun, vn, wV|

≤ |Bu, v−Bun, vn, wV||I−PnBun, vn, wV| ≤Cu−unvvn−vvnwCI−Pnwunvn.

4.45

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

E

τm

0

Bus, vs−PnBuns, vns, wVds

≤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

.

Consequently, by4.44and4.46, we have

lim n→ ∞E

_τm

0

Bus, vs−PnBuns, vns, wVds0. 4.47

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

E

_τm

0

Bus, vs−B∗s, zsVds0 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 spaceL2_{Ω,F, P;L}2_{0, T;V.}

Analogously, using the Lipschitz condition onFand4.44, we haveFt, ut F∗t as elements of the spaceL2_{Ω,F, P;L}2_{0, 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, vsV2

t

0

Bu1s, v1s−Bu2s, v2s, vsV

2

t

0

Fs, u1s−Fs, u2s, vsds2

t

0

Gs, u1s−Gs, u2s, vsds

t

0

|Gs, u1s−Gs, u2s|2H⊗mds.

4.51

Takeλ >0 to be fixed later and define

σt exp

−b β

t

0

v1s2ds−λt

Applying Ito’s formula to the real-valued processσt|vt|2, we obtain from4.51

σt|vt|22βν

t

0

σsvs2ds

≤2

t

0

σsBus, v1s, vsV ds2

t

0

σsFs, u1s−Fs, u2s, vsds

2

t

0

σsGs, u1s−Gs, u2s, vsdWs

t

0

σs|Gs, u1s−Gs, u2s|2H⊗mds

−

t

0

b βv1s

2_|

vs|2σsds−

t

0

λσs|vs|2ds.

4.53

But from2.8, we have

Bus, v1s, vsV

≤C|us|1/4us3/4v1s3/4vs

≤C|vs|1/4|vs|3/4v1svs

≤ C

2νβv1s

2_|vs|2 βν

2 vs

2_,

Fs, u1s−Fs, u2s, vs≤LF|vs|2,

|Gs, u1s−Gs, u2s|H⊗m ≤LG|vs|.

4.54

We then obtain from4.53

σt|vt|22βν

_t

0

σsvs2ds

≤ C β

_t

0

σsv1s2|vs|2 dsνβ

2

_t

0

σsvs2ds2LF

_t

0

σs|vs|2ds

2

t

0

σsGs, u1s−Gs, u2s, vsdWs L2_G

t

0

σs|vs|2ds

−

_t

0

b βv1s

2_|

vs|2σsds−

_t

0

λσs|vs|2ds.

TakingλL2_GandbC, we obtain from4.55

σt|vt|23νβ 2

_t

0

σsvs2ds

≤2LF

_t

0

σs|vs|2ds2

_t

0

σsGs, u1s−Gs, u2s, vsdWs

4.56

for allt∈0, T.

As 0< σt≤1, the expectation of the stochastic integral in4.56vanishes, and

Eσt|vt|2≤2LGE

t

0

σs|vs|2ds, 4.57

for allt∈0, T. The Gronwall’s lemma implies that

|vt|0, P−a.s. ∀t∈0, T, 4.58

in particular

ut 0, P−a.s. ∀t∈0, T. 4.59

This complete the proof of the uniqueness.

5. Proof of

Theorem 2.4

To prove the convergence result of Theorem 2.4, we need the following lemma which is proved in10,11.

Lemma 5.1. Let {Qn, n ≥ 1} be a sequence of continuous real-valued processes in L2Ω,F, P; L2_{0, T;R, and let{σ}

m;m ≥ 1} be a sequence of Ft-stopping times such that σm is increasing toT,sup_n≥1E|QnT|2 <∞, andlimn→ ∞E|Qnσm|0for alln≥1. Thenlimn→ ∞E|QnT|0.

It follows from4.41and4.15that

lim n→ ∞E

τm

0

vnt−vt2dt0. 5.1

Also from4.40and4.14, we have

lim

n→ ∞E|vnτm−vτm|

2

Applying the preceding lemma toQnt t₀vns−vs2dsandσm τm, and taking into account the estimate ofvn inLemma 3.3,5.1, and the uniqueness ofvoru, one obtains that the whole sequencevndefined by3.1satisfies

lim n→ ∞E

t

0

vns−vs2ds0 5.3

for allt∈0, T. Next, using the expression ofvnandv, we deduce that

lim n→ ∞E

_t

0

uns−us2_DA3/2ds0. 5.4

Analogously, applying the lemma toQnt |vnt−vt|2 _{and σ}

m τm, and taking into account the estimate ofvn inLemma 3.2,5.2, and the uniqueness of u, we have that the whole sequencevndefined by3.1satisfies limn→ ∞E|vnt−vt|20. Using the expression

ofvnandv, we have limn→ ∞Eunt−ut2_DA0 for allt∈0, T. This complete the proof ofTheorem 2.4.

6. Asymptotic Behavior of Strong Solutions for

the 3D Stochastic Leray-

α

as

α

Approaches Zero

The purpose of this section is to study the behavior of strong solutions for the 3D stochastic Leray-α model as α goes to zero. Therefore, we study the weak compactness of strong solutions of the 3D stochastic Leray-α equations asα approaches zero. One of the crucial point is to show that

E sup

0≤|θ|≤δ≤1

_T−δ

δ

|uαtθ−uαt|2_DA dt≤Cδ, 6.1

whereCis a constant independent ofα. To do this, we adopt the method developed for the deterministic 3D Leray-αequations2. In this method, an important role is played by the operatorIα2_A−1_{. Here our line of investigation is inspired by5,6,9.}

6.1. Tightness of Strong Solutions for the 3D Stochastic Leray-

α

Equations

In this subsection, we prove the tightness of strong solutions of the 3D stochastic Leray-α equations asαapproaches zero. The main result of this subsection is the following lemma.

Lemma 6.1. Suppose that hypotheses2.13hold, andu0 ∈L2Ω,F0, P;DA. Letuαbe a strong solution for the3Dstochastic Leray-αequations. One has

E sup

0≤|θ|≤δ≤1

_T−δ

δ

|uαtθ−uαt|2_DAdt≤Cδ, 6.2

Proof. We recall thatDADA−1_.

From2.15, we have

dIα2AuανA

uαα2Auα

dtBuα, uαα2Auα

dtFt, uαdtGt, uαdW. 6.3

We recall thatIα2Ais an isomorphism fromDAtoHand

Iα2A−1

LH,H≤1. 6.4

From6.3, we have

duανAuαdt

Iα2A−1Buα, vαdt

Iα2A−1Ft, uαdtIα2A−1Gt, uαdW, 6.5

wherevαuαα2Auα. We deduce that

_A−1_{uαtθ−uαt}

_tθ

t

A−1Iα2A−1Fτ, uατ ν|uατ| A−1Iα2A−1Buατ, vατ

dτ

tθ

t

A−1Iα2A−1Gτ, uατdWτ .

6.6

We estimate the first terms of the left-hand side of6.6using2.7and the Lipschitz condition onF

A−1Iα2A−1Buατ, vατ ≤ A−1Buατ, vατ ≤C|uατ|vατ,

A−1Iα2A−1Fτ, uατ ≤ A−1Fτ, uατ ≤C1|uατ|.

Collecting these previous inequalities and taking the square in6.6, we have

_A−1_{uαtθ−uαt} 2_≤Cθ2_C 1

tθ

t

|uατ|dτ

2

ν2

tθ

t

|uατ|dτ

2

C

tθ

t

|uατ|vατdτ

2

tθ

t

A−1Iα2A−1Gτ, uατdWτ

2

.

6.8

For fixedδ, taking the supremun overθ≤δyields

sup

0≤θ≤δ

_A−1_{uαtθ−uαt} 2

≤Cδ2TC1δ2 sup

τ∈0,T

|uατ|2C4 sup

τ∈0,T |uατ|2

tδ

t

vατdτ

2

sup

0≤θ≤δ

tθ

t

A−1Iα2A−1Gτ, uατdWτ

2.

6.9

Fort, we integrate betweenδandT−δand take the expectation. We deduce

Esup

0≤θ≤δ

T−δ

δ

_A−1_{uαtθ−uαt} 2_dt

≤Cδ2TCδ2Esup τ∈0,T|uατ|

2

C4Esup

τ∈0,T|uατ|

2

T−δ

δ

tδ

t

vατdτ

2

dt

E

T−δ

δ sup

0≤θ≤δ

tθ

t

A−1Iα2A−1Gτ, uατdWτ

2

dt.

By H ¨older’s inequality, we have

Esup τ∈0,T|uατ|

2

T−δ

δ

tδ

t

vατdτ

2

dt

≤δ2Esup τ∈0,T

|uατ|2

_T−δ

δ

vατ2dτ

≤δ2

Esup τ∈0,T|uατ|

4

1/2⎡

⎣_ET

0

vατ2dτ

2⎤

⎦

1/2

.

6.11

Using the estimates of Lemmas3.1,3.2,3.3, we obtain

E sup τ∈0,T|uατ|

2

_T−δ

δ

tδ

t

vατdτ

2

dt≤Cδ2, 6.12

whereCis a constant independent ofα.

Next, using Martingale’s inequality, we have

E

T−δ

δ sup

0≤θ≤δ

tθ

t

A−1Iα2A−1Gs, uαsdWs

2dt

≤E

_T−δ

δ

tδ

t

A−1Iα2A−1Gs, uαs

2 ds dt ≤CE T 0

tδ

t

1|uαs|2ds

dt

≤Cδ.

6.13

Collecting these results, we finally obtain

E sup

0≤θ≤δ≤1

_T−δ

δ

|uαtθ−uαt|2_DAdt≤Cδ, 6.14

whereCis a constant independent ofα.

Remark 6.2. FromLemma 3.2, we have

Esup t∈0,T|uα

Also fromLemma 3.1, we have

E

_T

0

uαs2ds≤C, 6.16

whereCis constant independent ofα.

From the estimate of Lemma 6.1 and Remark 6.2, we derive the following lemma which will be useful to prove the tightness ofuα.

Lemma 6.3. Letνnandμnbe two sequences of positives real number which tend to0asn → ∞. The injection of

D

⎧ ⎨

⎩q∈L∞0, T;H∩L20, T;V; supn 1 νn

sup

|θ|≤μn

T−μn

μn

_qtθ−qt 2

DAdt

1/2

<∞

⎫ ⎬ ⎭

6.17

inL2_{0, T;His compact.}

Proof. Its proof is carried out by the methods used in5,6,9. We define

SC0, T;Rm×L20, T;H 6.18

equipped with the Borelσ-algebraBS. Forα∈0,1, let

Φ:Ω−→S:ω−→Wω,·, uαω,·. 6.19

For eachα∈0,1, we introduce a probability measureΠαonS, BSby

ΠαA PΦ−1A, 6.20

whereA∈BS.

In the next proposition, using the preceding lemma, we can prove the tightness ofΠα. Its proof is carried out by the methods in26.

Proposition 6.4. The family of probability measures{Πα;α∈0,1}is tight inS.

6.2. Approximation of the Stochastic 3D Navier-Stokes Equations

6.2.1. Application of Prokhorov’s and Skorokhod’s Results

From the tightness property of {Πα; 0 < α ≤ 1} and Prokhorov’s theorem see 27, we have that there exists a subsequence{Παj} and a measureΠ such thatΠαj → Π weakly.

By Skorokhod’s theorem see 28, there exist a probability space Ω,F, P and random variablesW%αj,u&αj,W,% u&onΩ,F, Pwith values inSsuch that:

the law ofW%αj,u&αj

isΠα_j,

the law ofW,% u&isΠ,

%

Wαj,u&αj

−→W,% u&inS P−a.s.

6.21

Hence{W%αj}is a sequence of am-dimensional standard Wiener process.

Let

FtσW%s,u&s:s≤t. 6.22

Arguing as in5,9, we can prove thatW%is am-dimensionalFtstandard Wiener process and the pairW%αj,u&αjsatisfies

&

vαjt,Φ

ν

_t

0

&

vαjs, AΦ

ds

_t

0

Bu&αjs,v&αjs,Φ

ds

u0α2_jAu0,Φ

t

0

Fs,u&αjs

,Φds

t

0

Gs,u&αjs

dW%αjs,Φ

,

6.23

for allΦ∈ V, where

&

vαjs u&αjs α

2

jAu&αjs. 6.24

The main result of this section is the following theorem.

Theorem 6.5. Suppose that hypotheses2.13hold, andu0 ∈DA. Then there is a subsequence of

&

uαjdenoted by the same symbol such that asαj → 0, one has

&

uαj −→u& strongly inL

2_{Ω,F, P;L}2_{0, T;H,}

&

uαj −→u& weakly inL

2_{Ω,F, P;L}2_{0, T;V,}

&

vαj −→u& strongly inL

2_{Ω,F, P;L}2_{0, T;H,}

where Ω,F,Ftt∈0,T, P ,W,% u&is a weak solution for the3D stochastic Navier-Stokes equations with the initial valueu0 u0. (See [5] for the definition of weak solution of the3D stochastic

Navier-Stokes equations).

Proof. From6.23, it follows thatu&αj satisfies the estimates

&

Esup

0≤s≤T

&uαjs

p

≤Cp; 6.26

&

Esup

0≤s≤T

&vαjs

p

≤Cp, E&sup

0≤θ≤δ

_T−δ

δ

&uαjtθ−u&αjt

2

DAdt

≤Cδ, E&

T

0

&vαjs

2ds

≤Cp, E&sup

0≤s≤T

&vαjs

24νβE&

T

0

&vαjs

2ds≤C1,

6.27

whereE&denote the mathematical expectation with respect to the probability spaceΩ,F, P. Thus modulo the extraction of a subsequence denoted againu&αjwith the correspondingv&αj,

there exists two stochastic processesu,& v&such that

&

uαj u& inL

p_{Ω,F, P;L}∞_{0, T;H,}

&

uαj u& inL

2_{Ω,F, P;L}2_{0, T;V,}

&

vαj v& inL

2_{Ω,F, P;L}2_{0, T;V,}

6.28

&

Esup

0≤s≤T

|u&s|p≤Cp, E&

T

0

u&s2Vds≤C,

&

Esup

0≤θ≤δ

_T−δ

δ

|u&tθ−u&t|2_DAdt≤Cδ.

6.29

By6.21, estimate6.26, and Vitali’s theorem, we have

&

uαj −→u& inL

2_{Ω,F, P;L}2_{0, T;H. 6.30}

Thus modulo the extraction of a new subsequence and almost everyω, twith respect to the measuredP⊗dt

&

Taking into account6.30and the Lipschitz condition onF, we have

_t

0

Fs,u&αjs

ds−→

_t

0

Fs,u&sds inL2Ω,F, P;L20, T;H. 6.32

Arguing as in5, we can prove that

_t

0

Gs,u&αjs

dW%αjs

−→t

0Gs,u&sdW%s inL2

Ω,F, P;L∞0, T;DAweakly star.

6.33

We also have

&

E

T

0

&vαjt−u&αjt

2

dtα2_jE&

T

0

α2_j Au&αjt

2

dt. 6.34

We then deduce that

&

vαj −→u& in L

2_{Ω,F, P;L}2_{0, T;H, 6.35}

since by the estimate6.27, we have

&

E

_T

0

α2_j Au&αjt

2

dtis bounded uniformly inαj. 6.36

From6.28and6.35, we havev&t u&ta.e. inω×0, T. We are going to prove that

_t

0

Bu&αjs,v&αjs

ds

_t

0

Bu&s,u&sds inL2Ω,F, P;L20, T;DA. 6.37

Indeed, letΦ∈ V. From2.5,2.7, and2.9, we have

_t

0

'

Bu&αjs,v&αjs

,Φ(

DA− Bu&s,u&s,ΦDAds

_t

0

'

Bu&αjs−u&s,v&αjs

,Φ(

DAds

_t

0

'

Bu&s,v&αjs−u&s

,Φ( DAds

_t

0

'

Bu&αjs−us,v&αjs

,Φ(

DAds−

_t

0

Bu&s,Φ,v&αjs−u&s

ds ≤C _t 0

&uαjs−u&s

_&v

αjs

|AΦ|dsC

_t

0

u&s|AΦ| &vαjs−u&s

_ds.

By H ¨older’s inequality & E t 0 '

Bu&αjs,v&αjs

,Φ(

DA− Bu&s,u&s,ΦDAds

≤C|AΦ|

& E t 0

&uαjs−u&s

2

ds

1/2

&

E

t

0

&vαjs

2ds

1/2

C|AΦ|

&

E

_t

0

u&s2ds

1/2

&

E

_t

0

&vαjs−u&s

2

ds

1/2

.

6.39

It then follows from6.30,6.35, and6.39that

t

0

Bu&αjs,v&αjs

ds

t

0

Bu&s,u&sds inL2Ω,F, P;L20, T;DA. 6.40

Collect all the convergence results and pass to the limit in6.23to obtain

u&t,Φ ν

t

0

u&s, AΦds

t

0

Bu&s,u&s,ΦDAds

u0,Φ

_t

0

Fs,u&s,Φds

_t

0

Gs,u&s,ΦdW%s.

6.41

This completes the proof ofTheorem 6.5.

Acknowledgments

This research is supported by the University of Pretoria and a focus area grant from the National Research Foundation of South Africa.

References

1 A. Cheskidov, D. D. Holm, E. Olson, and E. S. Titi, “On a Leray-αmodel of turbulence,”Proceedings of the Royal Society A, vol. 461, no. 2055, pp. 629–649, 2005.

2 Y.-J. Yu and K.-T. Li, “Existence of solutions and Gevrey class regularity for Ler