and M. Sango

Volume 2010, Article ID 636140,47pages doi:10.1155/2010/636140

Research Article

Weak Solutions of a Stochastic Model for Two-Dimensional Second Grade Fluids

P. A. Razafimandimby

and M. Sango

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

2School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA

Correspondence should be addressed to P. A. Razafimandimby,paulrazafi@gmail.com Received 3 July 2009; Accepted 28 February 2010

Academic Editor: Robert Finn

Copyrightq2010 P. A. Razafimandimby 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 initiate the investigation of a stochastic system of evolution partial differential equations modelling the turbulent flows of a second grade fluid filling a bounded domain ofR². We establish the global existence of a probabilistic weak solution.

1. Introduction

The study of turbulence either in Newtonian flows or Non-Newtonian flows is one of the greatest unsolved and still not well-understood problem in contemporary applied sciences.

For indepth coverage of the deep and fascinating investigations undertaken in this field, the abundant wealth of results obtained, and remarkable advances achieved we refer to the monographs in 1–4 and references therein. The hypothesis relating the turbulence to the “randomness of the background field” is one of the motivations of the study of stochastic version of equations governing the motion of fluids flows. The introduction of random external forces of noise type reflectssmallirregularities that give birth to a new random phenomenon, making the problem more realistic. Such approach in the mathematical investigation for the understanding of the turbulence phenomenonwas pioneered by Bensoussan and Temam in 5 where they studied the Stochastic Navier-Stokes Equation SNSE excited by random forces. Since then, stochastic partial differential equations and stochastic models of fluid dynamics have been the object of intense investigations which have generated several important results. We refer, for instance, to6–22. Similar investigations for Non-Newtonian fluids have almost not been undertaken except in very few work; we refer, for instance, to 23–25 for some computational studies of stochastic models of polymeric

fluids. It is worth to note that in the Non-Newtonian case the study of stochastic models is relevant not only for the analytical approach to turbulent flows but also for practical needs related to the physics of the corresponding fluids2.

In the present work, we initiate the mathematical analysis for the stochastic model of incompressible second grade fluids. An incompressible fluid of second grade with a velocity fielduis a special example of a differential Rivlin-Ericksen fluid. It was shown in26that its stress tensorTis given by

T−p1 νA1α1A2α2A²₁, 1.1

wherepis the scalar pressure field,νis the kinematic viscosity, andA1 andA2are the first two Rivlin-Ericksen tensors defined by

A1 ∂ui

∂xj

i,j

∂uj

∂xi

i,j

,

A2 DA1

Dt A1

∂ui

∂xj

i,j

∂uj

∂xi

i,j

A1,

1.2

whereD/Dtdenotes the material derivative. The constantsα1andα2represent the normal stress moduli. The incompressibility requires that

divu0. 1.3

Taking into account some thermodynamical aspects, Dunn and Fosdick proved in27that the kinematic viscosityνmust be nonnegative. In addition, they found that the free energy must be a quadratic function of A1. This implies in particular that the Clausius-Duhem inequality is satisfied and the Helmholtz free energy is minimum at equilibrium if and only if

α1α20, α1 ≥0. 1.4

In what follows we assume thatα1α >0 andν >0. We also refer to28,29for more recent works concerning those conditions.

Those thermodynamical conditions imply that the stress tensorTcan be written in the following form:

T−p1 νA1α ∂

∂tA11 2A1

L−L^T

−1 2

L−L^T A1

, 1.5

where

L ∂ui

∂xj

i,j

. 1.6

We can check that

divT−∇pνΔuα∂Δu

∂t α

curlΔu×u∇

u·Δu1 4|A1|²

. 1.7

For a given external forcefthe dynamical equation for a second grade fluid is

∂u

∂t curlu×u∇ 1

2|u|²

divTf. 1.8

Making use of the latter equation and 1.7, we obtain the system of partial differential equations

∂

∂tu−αΔu−νΔucurlu−αΔu×u∇Pf, divu0,

1.9

where

Pp−α

u·Δu1 4|A1|

1

2|u|² 1.10

is the modified pressure. For a given connected and bounded domainDofR²and finite time horizon0, Twe complete the above system with the initial value

u0 u0 inD, 1.11

and the Dirichlet boundary value condition

u0 on∂D×0, T. 1.12

The interest in the investigation of problem1.9arises from the fact that it is an admissible model of slow flow fluids. Furthermore, once the above thermodynamical compatibility conditions are satisfied “the second grade fluid has general and pleasant properties such as boundedness, stability, and exponential decay”see again27. It also can be taken as a generalization of the Navier-Stokes EquationNSE. Indeed they reduce to NSE whenα0;

moreover recent work30shows that it is a good approximation of the NSE. See also31–36 for interesting discussions to their relationship with other fluid models.

Due to the above nice properties, the mathematical analysis of the second grade fluid has attracted many prominent researchers in the deterministic case. The first relevant analysis was done by Ouazar in his 1981 thesis; together with Cioranescu, they published the related result in 37, 38. Their method was based on the Galerkin approximation scheme involving a priori estimates for the approximating solutions using a special basis consisting of eigenfunctions corresponding to the scalar product associated with the operator curlu−αΔu. They proved global existence and uniqueness without restriction on the initial

data for the two-dimensional case. Cioranescu and Girault 39, as well as Bernard 40 extended this method to the three dimensional case; global existence was obtained with some reasonable restrictions on the initial data. For another approach to global existence using Schauder’s fixed point technics, we refer to41and some relevant references therein.

As already mentioned, in this work we propose a stochastic version of the problem 1.9,1.11-1.12. More precisely, we assume that a connected and bounded open setD of R²with boundary∂Dof classC³, a finite time horizon0, T, and a nonrandom initial value u0are given. We consider the problem

du−αΔu −νΔucurlu−αΔu×u∇PdtFu, tdtGu, tdW inD×0, T,

divu0 in D×0, T, u0 in ∂D×0, T,

u0 u0 inD,

1.13

where u u1, u2 and P represent the random velocity and pressure, respectively. The system is to be understood in the Ito sense. It is the equation of motion of an incompressible second grade fluid driven by random external forcesFu, tand Gu, tdW, where W is a R^m-valued standard Wiener process.

As far as we know, this paper is the first dealing with the stochastic version of the equation governing the motion of a second grade fluid filling a connected and bounded domainD of R². Consequently, we could by no means exhaust the mathematical analysis of the problem; many questions are still open but we hope that this pioneering work will find its applications elsewhere. We limited ourselves to the discussion of a global existence result of a probabilistic weak solution in the two-dimensional case. In forthcoming papers we will address other questions such as the existence probabilistic strong solutions under more stringent conditions, the uniqueness of those solutions, and their behaviour whenα → 0. It should be noted that solving this problem is not easy even in the deterministic case, the nature of the nonlinearities being one of the main difficulties in addition to the complex structure of the equations. Besides the obstacles encountered in the deterministic case, the introduction of the noise termGu, tdWin the stochastic version induces the appearance of expressions that are very hard to control when proving some crucial estimates. Overcoming these problems will require a-tour-de force in the work.

The rest of the paper is organized as follows. InSection 2, we give some notations, necessary background of probabilistic or analytical nature. Section 3 is devoted to the formulation of the hypotheses and the main result. We introduce a Galerkin approximation of the problem and derive crucial a priori estimates for its solution inSection 4; a compactness result is also derived. We prove the main result inSection 5.

2. Notations and Preliminaries

Let us start with some informationsabout some functional spaces needed in this work. LetD be an open subset ofR², let 1≤ p ≤ ∞, and letkbe a nonnegative integer. We consider the well-known Lebesgue and Sobolev spacesL^pDandW^k,pD, respectively. Whenp2, we writeW^k,2D H^kD. We denote byW₀^k,pDthe closure inW^k,pDofC^∞_c Dthe space

of infinitely differentiable function with compact support inD. Ifp 2, we denoteW₀^k,pD byH₀^kD. We assume that the Hilbert spaceH₀¹Dis endowed with the scalar product

u, v

D

∇u· ∇v dx²

i1 D

∂u

∂xi

∂v

∂xidx, 2.1

where∇is the gradient operator. The norm·generated by this scalar product is equivalent to the usual norm ofW^1,2DinH₀¹D. If the domainD is smooth enough and bounded, then for anymandpsuch thatmp >2 the embedding

W^jm,p⊂W^j,q 2.2

is compact for any 1≤q≤ ∞. More Sobolev embedding theorems can be found in42and references therein.

Next we define some probabilistic evolution spaces necessary throughout the paper.

Let Ω,F,Ft_0≤t≤T,P be a given stochastic basis; that is,Ω,F,P is complete probability space andFt_0≤t≤T is an increasing sub-σ-algebras of Fsuch thatF0 contains everyP-null subset ofΩ. For any reflexive separable real Banach spaceXendowed with the norm · X, for anyp ≥1,L^p0, T;Xis the space ofX-valued measurable functionsudefined on0, T such that

u_L^p_0,T;X ^T

0

u^p_Xdt 1/p

<∞. 2.3

For anyr, p ≥ 1 we denote byL^pΩ,P;L^r0, T;Xthe space of processesu uω, twith values inXdefined onΩ×0, Tsuch that

1uis measurable with respect toω, tand, for eacht,uisF^tmeasurable, 2ut, ω∈Xfor almost allω, tand

u_L^p_Ω,P;L^r_0,T;X

⎛

⎝E ^T

0

u^r_Xdt _p/r⎞

⎠

1/p

<∞, 2.4

where E denotes the mathematical expectation with respect to the probability measureP.

Whenr∞, we write

u_L^p_Ω,P;L∞0,T;X

Eess sup

0≤t≤Tu^p_Xdt _1/p

<∞. 2.5

Next we give some compactness results of probabilistic nature due to Prokhorov and Skorokhod. Let us considerΩas a separable and complete metric space andFits Borelσ- field. A familyPkof probability measures onΩ,Fis relatively compact if every sequence of

elements ofPkcontains a subsequencePkjwhich converges weakly to a probability measure P; that is, for anyφbounded and continuous function onΩ,

φωdPkjdω−→ φωdPdω. 2.6

The familyPkis said to be tight if, for anyε >0, there exists a compact setKε⊂Ωsuch that PKε≥1−ε, for everyP∈Pk.

We frequently use the following two theorems. We refer to43for their proofs.

Theorem 2.1see Prokhorov. The familyPkis relatively compact if and only if it is tight.

Theorem 2.2see Skorokhod. For any sequence of probability measuresPkonΩwhich converges to a probability measureP, there exist a probability spaceΩ,F,Pand random variablesXk,Xwith values inΩsuch that the probability law ofXk(resp.,X) isPk(resp.,P) and limk→ ∞XkXP-a.s.

We proceed now with the definitions of additional spaces frequently used in this work.

In what follows we denote byXthe space ofR²-valued functions such that each component belongs toX. A simply-connected bounded domainDwith boundary of classC³is given. We introduce the spaces

V

u∈C^∞_c ² such that divu0 , Vclosure ofV inH¹D, Hclosure ofV inL²D.

2.7

We denote by·,·and| · |the inner product and the norm induced by the inner product and the norm inL²DonH, respectively. The inner product and the norm induced by that of H¹₀DonVare denoted respectively by ·,· and · . In the spaceV, the latter norm is equivalent to the norm generated by the following scalar productsee, e.g.,37

u, v_V u, v αu, v, for anyu, v∈V. 2.8

We also introduce the following space:

W

u∈Vsuch that curlu−αΔu∈L²D

. 2.9

The following lemma tells us that the norm generated by the scalar product

u, v_W u, v_V curlu−αΔu,curlv−αΔv, 2.10

is equivalent to the usualH³D-norm onW. Its proof can be found, for example, in37,39.

Lemma 2.3. The following (algebraic and topological) identity holds:

WW, 2.11

where

W

v∈H³Dsuch that divv0 and v|_∂D 0

. 2.12

Moreover, there is a positive constantCsuch that

|v|²_H3D≤C

|v|²_V|curlv−αΔv|²

, 2.13

for anyv∈W.

By this lemma we can endow the spaceWwith norm| · |Wwhich is generated by the scalar product2.10.

From now on, we identify the space V with its dual space V via the Riesz representation, and we have the Gelfand chain

W⊂V⊂W , 2.14

where each space is dense in the next one and the inclusions are continuous.

The following inequalities will be used frequently.

Lemma 2.4. For anyu∈W,v∈W, andw∈Wone has

|curlu−αΔu×v, w| ≤C|u|_H3|v|_V|w|_W. 2.15

One also has

|curlu−αΔu×u, w| ≤C|u|²_V|w|_W, 2.16

for anyu∈Wandw∈W.

Proof. We introduce the well-known trilinear formbused in the study of the Navier-Stokes equation by setting

bu, v, w ²

i,j1 D

ui

∂vj

∂xiwjdx. 2.17

We give the following identity whose proof can be found in37,40. This equation is valid for any smoothsolenoidalfunctionsΦ, v, andwas

curlΦ×v, w bv,Φ, w−bw,Φ, v. 2.18

We derive from this that for anyu∈W,v∈W, andw∈W

|curlu−αΔu×v, w| ≤C|v|_L²_D|∇u−αΔu|_L²_D|w|_L∞D, 2.19

where H ¨older’s inequality was used. The Sobolev embedding2.2and the equivalence of the norms| · |_Wand| · |³_HDimply2.15.

By2.18we deduce

curlu−αΔu×u, w bu, u, w−αbu,Δu, w αbw,Δu, u. 2.20

With the help of integration by parts and using the fact thatuandware elements ofWwe derive that

bu,Δu, w ²

j1

b ∂u

∂xj, w, ∂u

∂xj

²

i1

b

u,∂w

∂xj, ∂u

∂xj

, 2.21

bw,Δu, u ²

j1

b ∂w

∂xj, u, ∂u

∂xj

. 2.22

We use these results to derive the following estimate. For any elementsu∈Vandw∈L⁴D, we obtain by H ¨older’s inequality

|bu, u, w| ≤C|u|_L⁴_Du|w|_L⁴_D. 2.23

And since the spaceVandWare, respectively, continuously embedded inL⁴DandV, then

|bu, u, w| ≤C|u|²_V|w|_W. 2.24

We also have

|bu,Δu, w| ≤ |∇w|_L∞D

2 j1

∂u

∂xj

2

L²D

|u|_L⁴_D

⎛

⎝²

j1

∂w

∂xj

2

L⁴D

⎞

⎠

1/2⎛

⎝²

j1

∂u

∂xj

2

L²D

⎞

⎠

1/2

.

2.25

We derive from this and the Sobolev embedding2.2that

|bu,Δu, w| ≤C|u|²_V|w|_W. 2.26

By an analogous argument we have

|bw,Δu, u| ≤C|w|_W|u|²_V. 2.27

The estimates2.20,2.24–2.27yield

|curlu−αΔu×u, w| ≤C|u|²_V|w|_W, 2.28

for anyu∈Wandw∈W. This completes the proof of the lemma.

Next we give some results on which most of the proofs in forthcoming sections rely.

We start by stating a theorem on solvability of the “generalized Stokes equations”

v−αΔv∇qf inD, divv0 in D,

v0 on∂D.

2.29

By a solution of this system we mean a functionv∈Vwhich satisfies v, h αv, h

f, h

, 2.30

for anyh∈V.

The proof of the following result can be derived from an adaptation of the results obtained by Solonnikov in44,45.

Theorem 2.5. LetDbe a connected, bounded open set ofRⁿ n≥ 2with boundary∂Dof classC^l and letfbe a function inH^l,l≥1. Then2.29has a unique solutionv. Moreover iffis an element ofV,v∈H^l2∩V, and the following hold:

v, h_V v, h,

|v|_W ≤Cf

V, 2.31

for anyh∈V.

Next we formulate Aubin-Lions’s compactness theorem; its proof can be found in46.

Theorem 2.6. LetX, B, Y be three Banach spaces such that the following embedding is continuous:

X⊂B⊂Y. 2.32

Moreover, assume that the embeddingX ⊂Bis compact, then the setFconsisting of functionsv ∈ L^q0, T;B, 1≤q≤ ∞, such that

sup

0≤h≤1 t2

t1

|vth−vt|^p_Ydt−→0, ash−→0, 2.33

for any 0< t1< t2< T, is compact inL^p0, T;Bfor anyp.

Last but not least we present the famous Kolmogorov- ˇCentsov continuity criterion for stochastic processes. We refer to47,48for its proof and some of its extension.

Theorem 2.7see Kolmogorv- ˇCentsov. Suppose that a real-valued processX {Xt, 0≤t≤T} on a probability spaceΩ,Psatisfies the condition

E|Xth−Xt|^γ ≤Ch^1β, 0≤t, h≤T, 2.34

for some positive constantsγ, β, andC. Then there exists a continuous modificationX {Xt, 0≤t≤ T}ofX, which is locally H¨older continuous with exponentκ∈0, β/γ.

3. Hypotheses and the Main Result

We state on our problem the following.

3.1. Hypotheses 1We assume that

F:V×0, T−→V 3.1

is continuous in both variables.We also assume that, for anyt∈0, Tand anyv∈V

|Fv, t|_V≤C1|v|_V. 3.2

2We also define a nonlinear operatorGas follows:

G:V×0, T−→V^⊗m 3.3

is continuous in both variables.We require that, for anyt∈0, T, Gv, tsatisfy

|Gv, t|_V⊗m ≤C1|v|_V. 3.4

3.2. Statement of the Main Theorem

We introduce the concept of solution of the problem1.13that is of interest to us.

Definition 3.1. By a solution of the problem1.13, we mean a system Ω,F,P,F^t, W, u

, 3.5

where

1 Ω,F,Pis a complete probability space;F^tis a filtration onΩ,F,P, 2Wtis anm-dimensionalF^tstandard Wiener process,

3for a.e.t,ut∈L^pΩ,P;L^∞0, T;W∩L^pΩ,P;L^∞0, T;V, 2≤p <∞, 4for almost allt,utisF^tmeasurable,

5P-a.s the following integral equation of It ˆo type holds:

ut−u0, v_V ^t

0

νu, v curlus−αΔus×u, vds

^t

0

Fus, s, vds ^t

0

Gus, s, vdWs

3.6

for anyt∈0, Tandv∈W.

Remark 3.2. In the above definition the quantity_t

0Gus, s, vdWsshould be understood as:

t 0

Gus, s, vdWs ^m

k1 t 0

Gkus, s, vdWks, 3.7

whereGkandWkdenote thekth component ofGandW, respectively.

Now we state our main result.

Theorem 3.3. Assume thatu0 ∈W; assume also that all the assumptions, namely,3.2and3.4, on the operatorsF, G are satisfied; then the problem1.13has a solution in the sense of the above definition. Moreover, almost surely the paths of the processuareW-valued weakly continuous.

4. Auxiliary Results

In this section we derive crucial a priori estimates from the Galerkin approximation. They will serve as a toolkit for the proof ofTheorem 3.3.

4.1. The Approximate Solution

The following statement is a consequence of the spectral theorem for self-adjoint compact operator stated in49. The injection of WintoVis compact. LetIbe the isomorphism of W onto W, then the restriction of I toVis a continuous compact operator into itself. Thus, there exists a sequenceeiof elements of Wwhich forms an orthonormal basis inW, and an orthogonal basis inV.

This sequence verifies:

for any v∈W v, ei_Wλiv, ei_V, 4.1 whereλ_i1> λi >0,i1,2, . . . .

We have the following important result due to39about the regularity of theei-s.

Lemma 4.1. LetDbe a bounded, simply-connected open set ofR²with a boundary of classC³, then the eigenfunctions of 4.1belong toH⁴D.

We consider the subset WN Spane1, . . . , eN ⊂ W and we look for a finite- dimensional approximation of a solution of our problem as a vectoru^N ∈ WN that can be written as the Fourier series:

u^Nt ^N

i1

ciNteix. 4.2

Let us consider a complete probabilistic systemΩ,F,P,F^t, Wsuch that the filtration{F_t} satisfies the usual condition and W is anm-dimensional standard Wiener process taking values inR^m. We requireu^Nto satisfy the following system:

d u^N, ei

Vν u^N, ei

dtb

u^N, u^N, ei

dt−αb

u^N,Δu^N, ei

dtαb

ei,Δu^N, u^N dt

F t, u^N

, ei

dt

G t, u^N

, ei

dW,

4.3

whereu^N₀ as the orthogonal projection ofu0in the spaceWNis given as u^N₀

oru^N0

−→u0 strongly inV 4.4

as N → ∞. The Fourier coefficients ciN in 4.2 are solutions of a system of stochastic ordinary differential equations which satisfy the conditions of the existence theorem of Skorokhod 50 see also47. Therefore the sequence of functionsu^N exists at least on a short interval0, TN. Global existence will follow from a priori estimates foru^N.

4.2. A Priori Estimates

From now onCis a constant depending only on the data, and may change from one line to the next one. We start by proving the following lemma.

Lemma 4.2. For anyN≥1 one has

Esup

0≤t≤T

u^Nt²

V<∞. 4.5

One also has

Esup

0≤t≤T

u^Nt²

W<∞. 4.6

Proof. From now on we denote by|v|∗the quantity|curlv−αΔv|for anyv∈ W. For any integerM≥1 we introduce the stopping time

τM

⎧⎨

⎩ inf

0≤t;u^Nt

Vu^Nt

∗≥M

∞ if

0≤t;u^Nt

Vu^Nt

∗≥M

∅. 4.7 We will use a modification of the argument used in7.

For any 0≤ s≤ t∧τM,t ∈0, TN, we may apply It ˆo’s formulasee, e.g.,7,12for φu^Ns, ei_V u^Ns, ei²_Vto4.3and obtain

u^Ns, ei

₂

V2

s 0

u^Nr, ei

V

ν

u^Nr, ei

b

u^Nr, u^Nr−αΔu^Nr, ei

dr

2

s 0

u^Nr, ei

V

−αb

ei,Δu^Nr, u^Nr

F r, u^N

, ei

dr

^s

0

G

r, u^N , ei

dW ^s

0

u^Nr, ei

V

G

r, u^N , ei

2

dr.

4.8

We note that |u^N|²_{V N}

i1λiu^N, ei²_V. Then, multiplying by λi the above equation and summing overifrom 1 toNgive us

u^Ns²

V2ν

s 0

u^N2dr u^N₀ ²

V2

s 0

F

r, u^N , u^N

dr^N

i1

λi s 0

Gr, u^N, ei

₂ dr

2

s 0

G

r, u^N , u^N

dW,

4.9

where we have used the fact thatbu^N, u^N, u^N 0.

We obtain from4.9that u^Ns²

V2ν

s 0

u^Nr, u^Nr

dr ≤u^N₀ ²

V^N

i1

λi s 0

G

u^Nr, r , ei

₂ dr

2

s 0

F

u^Nr, r

, u^Nrdr

2

s 0

G

u^Nr, r

, u^Nr dW

,

4.10

for any 0≤s≤t∧τM,t∈0, TN. For anyu∈Vwe have

|u| ≤ Pu, 4.11

wherePis the so called Poincar´e’s constant. The last inequality implies that

F

u^Ns, s

, u^N≤ P²u^NF

u^Ns, s. 4.12

We also mention that

P²α₋₁

|v|²_V≤ v²≤α⁻¹|v|²_V, for any v∈V. 4.13

From the former equation and this one we find

F

u^Ns, s

, u^Ns≤2CP² α

1u^Ns²

V

. 4.14

To find a uniform estimate for the corrector term_N

i1λiGu^Ns, ei²is not straightforward;

this is the difficulty already mentioned in the introduction. Since the corrector term is explicitly written as function depending on the scalar productinL²D ·,·and theei-s form an orthonormal basisresp., orthogonal basisofWresp,V, then the usual Bessel’s inequalitysee, e.g.,6does not apply anymore. To circumvent this difficulty we consider the following generalized Stokes equation:

G−αΔG∇qG

u^Ns, s inD, divG0 in D,

G0 on∂D,

4.15

for anys∈0, T. ByTheorem 2.5,4.15has a solutionGinW^⊗mwhen∂Dis of classC³and Gu^Ns, s∈V^⊗m. Moreover, there exists a positive constantC0such that

G

H³D^⊗m ≤C0G

u^Ns, s

V^⊗m, 4.16

andG, e i_V Gu^Ns, s, eifor anyi≥1.

Since the norms|·|_H³_Dand|·|_Ware equivalent onW, then there exists another positive constantC∗such that

G

W^⊗m ≤C∗C0G

u^Ns, s

V^⊗m. 4.17

Equation4.17implies thatGdepends continuously on the dataGu^Ns, s. Therefore, we note the aboveGasGu ^Ns, s. We find from4.17that

N i1

λi

G

u^Ns, s , ei

₂ ^N

i1

λi

G

u^Ns, s , ei

₂

V

^N

i1

1 λi

G

u^Ns, s , ei

₂

W.

4.18

We deduce from this that N i1

1 λi

G

u^Ns, s , ei

2 W ≤ 1

λ1

Gu^Ns, s²

W^⊗m. 4.19

By4.17and the assumption onG, we have N

i1

λi

G

u^Ns, s , ei

₂

≤C

1u^Ns²

V

. 4.20

Collecting this information, we obtain from4.10that u^Ns²

V2ν

s 0

u^Nr²dr

≤CC

s 0

u^Nr²

Vdr2 ^s

0

G

u^Nr, r

, u^Nr dW

.

4.21

Taking the sup over s ≤ t ∧ τM in both sides of this inequality and passing to the mathematical expectation in the resulting relation and finally applying Burkh ¨older-Davis- Gundy’s inequalitysee, e.g.,48to the stochastic term, we get

Esup

s≤t∧τM

u^Ns²

V2νE ^t∧τM

0

u^Ns²ds

≤CCE ^t∧τM

0

u^Ns²

Vds2C1E ^t∧τM

0

Gu^Ns, s, u^Ns₂ ds

1/2

.

4.22

Now, we estimate

γ E ^t∧τM

0

G

u^Ns, s

, u^Ns₂ ds

1/2

. 4.23

With the same argument that we have used for the term|Fu^Ns, s, u^Ns|, we have

γ≤CE

⎡

⎣sup

s≤t∧τM

u^Ns

V t∧τM

0

G

u^Ns, s²

V^×mds 1/2⎤

⎦. 4.24

Byε-Young’s inequality

γ≤CεE sup

s≤t∧τM

u^Ns²

VCεE ^t∧τM

0

Gu^Ns, s²

V^×mds 4.25

Using the assumption onGone has

γ≤CεEsup

s≤t∧τM

u^Ns²

VCεE ^t∧τM

0

1u^Ns²

V

. 4.26

With convenient choice ofε1−2C1Cε 1/2, the estimates4.22and4.26allow us to write

Esup

s≤t∧τM

u^Ns²

V4νE ^t∧τM

0

u^Ns²ds≤CCE ^t∧τM

0

u^Ns²

Vds. 4.27

We derive from this and Gronwall’s inequality that

E sup

0≤s≤t∧τM

u^Ns²

V≤C. 4.28

We recall the following relationship which is very important in the sequel:

λi

G

u^Ns, s , ei

G

u^Ns, s , ei

W, i≥1, 4.29

whereGu ^Ns, sis the solution inWofGS.

To alleviate notation, we only writeu^Nwhen we meanu^N·. Let us set

φ u^N

−νΔu^Ncurl

u^N−αΔu^N

×u^N−F u^N, t

. 4.30

ByLemma 4.1,φu^N∈H¹D. We have

d u^N, ei

V φ

u^N , ei

dt

G u^N, t

, ei

dW. 4.31

ByTheorem 2.5a solutionv^N∈Wof the following system exists:

v^N−αΔv^N∇qφ u^N

inD, divv^N 0 inD,

v^N0 on∂D.

4.32

Moreover,

v^N, ei

V φ

u^N , ei

, 4.33

for anyi. Thus,

d u^N, ei

V φ

u^N , ei

dtd

u^N, ei

V v^N, ei

Vdt

G u^N, t

, ei

dW.

4.34

The following follows by multiplying the latter equation byλi and using the relationship 4.1:

d u^N, ei

W v^N, ei

Wdtλi

G

u^N, t , ei

dW. 4.35

Recalling4.29, we obtain

d u^N, ei

W v^N, ei

Wdt G

u^N, t , ei

WdW. 4.36

Now applying the It ˆo’s formulasee, e.g.,7toϕu^N, ei_W u^N, ei²_W, we have

d u^N, ei

₂

W2 u^N, ei

W

v^N, ei

Wdt G

u^N, t , ei

₂

Wdt2 u^N, ei

W

G u^N, t

, ei

WdW.

4.37 Summing both sides of the last equation from 1 toNyields

du^N2

W2

u^N, v^N

Wdt^N

i1

G u^N, t

, ei

₂

Wdt2 G

u^N, t , u^N

WdW. 4.38

Using the definition of| · |Wand the scalar product·,·_W, we can rewrite the above equation in the form

d# u^N2

Vu^N2

∗

$ 2

v^N, u^N

V curl

v^N−αΔv^N curl

u^N−αΔu^N dt

2 curl

G u^N, t

−αΔG u^N, t

,curl

u^N−αΔu^N dW

^N

i1

λ²_i G

u^N, t , ei

2 Vdt2

G u^N, t

, u^N

VdW.

4.39

In view ofRemark 3.2, we have to make the convention that in the sequel

curl G

u^N, t ,curl

u^N−αΔu^NdW dt ^m

k1

curl

Gk

u^N, t

,curl

u^N−αΔu^NdWk

dt .

4.40

Using the definition ofv^NandG, we obtain

d# u^N2

Vu^N2

∗

$ 2

φ u^N

, u^N

curl φ

u^N ,curl

u^N−αΔu^N dt

^N

i1

λ²_i

Gu^N, t, ei

₂ dt2

G u^N, t

, u^N dW

2 curl

G u^N, t

,curl

u^N−αΔu^N dW.

4.41

With the help of4.9,4.41can be rewritten in the following way:

du^N2

∗2 curlφ

u^N ,curl

u^N−αΔu^N dt

2 curl

G u^N, t

,curl

u^N−αΔu^N

dW^N

i1

λiλ²_i G

u^N, t , ei

₂ dt.

4.42

We infer from the definition ofφu^Nthat curlφ

u^N

−νcurl

Δu^NF u^N, t

curl curl

u^N−αΔu^N

×u^N

. 4.43

In taking advantage of the dimension we get curl

curl

u^N−αΔu^N

×u^N

u^N· ∇ curl

u^N−αΔu^N

. 4.44

This yields

u^N· ∇ curl

u^N−αΔu^N

−νcurl

Δu^NF u^N, t

,curl

u^N−αΔu^N

curlφ u^N

,curl

u^N−αΔu^N .

4.45

Owing toLemma 4.1we readily check that

u^N· ∇ β, β

0, 4.46

whereβcurlu^N−αΔu^N. Consequently,

ν curl Δu^N

,curl

u^N−αΔu^N

curl F

u^N, t ,curl

u^N−αΔu^N −

curlφ u^N

,curl

u^N−αΔu^N .

4.47

Or equivalently ν α

u^N

∗− ν α

curlu^Nα νcurl

F u^N, t

,curl

u^N−αΔu^N

curlφ

u^N ,curl

u^N−αΔu^N .

4.48

We derive from4.42and the last equation that d

dt u^N2

∗ 2ν α

u^N2

∗−2ν α

curlu^N α νcurl

F u^N, t

,curl

u^N−αΔu^N

^N

i1

λiλ²_i G

u^N, t , ei

2

2 curl

G u^N, t

,curl

u^N−αΔu^NdW dt .

4.49

We argue as before in considering the stopping timeτM. We derive from4.49that u^Ns²

∗ ^s

0

2ν α

u^Nr²

∗−^N

i1

λiλ²_i G

u^Nr, r , ei

₂ dr

^s

0

2ν α

# curl

u^Nr

−α νcurl

F

u^Nr, r ,curl

u^Nr−αΔu^Nr$ dr

2

s 0

curl

G

u^Nr, r ,curl

u^Nr−αΔu^Nr W.

4.50

Hence, u^Ns²

∗ ^s

0

2ν α

u^Nr²

∗dr−^N

i1

λiλ²_i

s 0

G

u^Nr, r , ei

₂ dr

≤u^N₀ ²

∗ ^s

0

2ν α

curl

u^Nru^Nr

∗ ^s

0

2curl F

u^Nr, ru^Nr

∗dr 2

^s

0

curl

G

u^Nr, r ,curl

u^Nr−αΔu^Nr dW

.

4.51

Taking the supremum over s ≤ t∧ τM in the last estimate, and taking the mathematical expectation in the resulting relation yields

Esup

s≤t∧τM

u^Ns²

∗E ^t∧τM

0

2ν α

u^Ns²

∗ds−^N

i1

λiλ²_i E ^t∧τM

0

G

u^Ns, s , ei

₂ ds

≤u^N₀ ²

∗E ^t∧τM

0

2ν α

curl

u^Nsu^Ns

∗E ^t∧τM

0

2curl F

u^Ns, su^Ns

∗ds 2Esup

s≤t∧τM

s∧τM

0

curl

G

u^Ns, s ,curl

u^Ns−αΔu^Ns dW

.

4.52

For anyε1≥0 andε2≥0, we have Esup

s≤t∧τM

u^Ns²

∗E ^t∧τM

0

2ν α

u^Ns²

∗ds−^N

i1

λiλ²_i E ^t∧τM

0

G

u^Ns, s , ei

2

ds

≤u^N₀ ²

∗E ^t∧τM

0

2ν αε1

curl

u^Ns² 2 ε2

curl F

u^Ns, s²

ds

2Esup

s≤t∧τM

s∧τM

0

curl

G

u^Ns, s ,curl

u^Ns−αΔu^Ns dW

2νε1

α 2ε2 t∧τM

0

u^Ns²

∗ds.

4.53

We chooseε1 1/4 andε2 ν/4αand we deduce from the last inequality the following estimate,

E sup

s≤t∧τM

u^Ns²

∗E ^t∧τM

0

ν α

u^Ns²

∗ds−^N

i1

λiλ²_i E ^t∧τM

0

G

u^Ns, s , ei

2

ds

≤u^N₀ ²

∗E ^t∧τM

0

8ν αε1

curl

u^Ns²2α ν

curl F

u^Ns, s²

ds

2E sup

s≤t∧τM

s∧τM

0

curl

G

u^Ns, s ,curl

u^Ns−αΔu^Ns dW

.

4.54