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

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}

1_{Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa} 2_{School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA}

Correspondence should be addressed to P. A. Razafimandimby,[email protected]

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

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−p1νA1α1A2α2A2₁, 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−p1νA1α

∂ ∂tA1

1 2A1

L−LT₋1

2

L−LT_A 1

, 1.5

where

L

∂ui

∂xj

i,j

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 domainDofR2_{and 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.

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 R2_{with boundary∂Dof classC}3_{, 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 Rm_{-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 spacesLp_DandWk,p_{D, respectively. Whenp2, we}

writeWk,2_{D H}k_{D. We denote byW}k,p

of infinitely differentiable function with compact support inD. Ifp 2, we denoteW₀k,pD byH₀kD. We assume that the Hilbert spaceH₀1Dis endowed with the scalar product

u, v

D

∇u· ∇v dx

2

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 ofW1,2_DinH1

0D. If the domainD is smooth enough and bounded,

then for anymandpsuch thatmp >2 the embedding

Wjm,p⊂Wj,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,Ft0≤t≤T,P be a given stochastic basis; that is,Ω,F,P is complete probability

space andFt0≤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,Lp_{0, T;Xis the space ofX-valued measurable functionsudefined on0, T}

such that

uLp_0,T;X

T

0

up_Xdt

1/p

<∞. 2.3

For anyr, p ≥ 1 we denote byLp_Ω,P;Lr_{0, T;Xthe space of processesu uω, twith}

values inXdefined onΩ×0, Tsuch that

1uis measurable with respect toω, tand, for eacht,uisFtmeasurable,

2ut, ω∈Xfor almost allω, tand

uLp_Ω,P;Lr_0,T;X

⎛ ⎝_E T

0

urXdt

p/r⎞

⎠

1/p

<∞, 2.4

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

Whenr∞, we write

uLp_Ω,P;L∞_0,T;X

Eess sup

0≤t≤T

up_Xdt

1/p

<∞. 2.5

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) andlimk→ ∞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 classC3_{is given. We}

introduce the spaces

Vu∈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 inL2_{DonH, respectively. The inner product and the norm induced by that of}

H1

0DonVare 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, vV u, v αu, v, for anyu, v∈V. 2.8

We also introduce the following space:

Wu∈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

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

WW, 2.11

where

Wv∈H3Dsuch that divv0and v|∂D 0

. 2.12

Moreover, there is a positive constantCsuch that

|v|2_H3_D≤C

|v|2V|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

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∈L4_D,

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 inL4_{DandV, then}

|bu, u, w| ≤C|u|2_V|w|_W. 2.24

We also have

|bu,Δu, w| ≤ |∇w|_L∞_D

2

j1

∂x∂uj

2

L2_D

|u|_L4_D

⎛ ⎝2

j1

∂w∂xj

2

L4_D

⎞ ⎠

1/2⎛

⎝2

j1

∂x∂uj

2

L2_D

⎞ ⎠

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

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 inHl_{,l≥1. Then2.29has a unique solutionv. Moreover iffis an element}

ofV,v∈Hl2_{∩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 ∈ Lq_{0, T;B,1≤q≤ ∞, such that}

sup

0≤h≤1 t2

t1

|vth−vt|p_Ydt−→0, ash−→0, 2.33

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 ≤Ch1β, 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

where

1 Ω,F,Pis a complete probability space;Ft_{is a filtration onΩ,F,P,}

2Wtis anm-dimensionalFt_{standard Wiener process,}

3for a.e.t,ut∈Lp_Ω,P;L∞_{0, T;W∩L}p_Ω,P;L∞_{0, T;V, 2≤p <∞,}

4for almost allt,utisFtmeasurable,

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 quantityt₀Gus, 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 ofWintoVis 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

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

Lemma 4.1. LetDbe a bounded, simply-connected open set ofR2_{with a boundary of classC}3_{, then}

the eigenfunctions of 4.1belong toH4_D.

We consider the subset WN Spane1, . . . , eN ⊂ W and we look for a

finite-dimensional approximation of a solution of our problem as a vectoruN _{∈ W}

N that can be

written as the Fourier series:

uNt

N

i1

ciNteix. 4.2

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

duN, ei

Vν

uN, ei

dtbuN, uN, ei

dt−αbuN,ΔuN, ei

dtαbei,ΔuN, uN

dt

Ft, uN, ei

dtGt, uN, ei

dW,

4.3

whereuN

0 as the orthogonal projection ofu0in the spaceWNis given as

uN₀ oruN0−→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 functionsuN _{exists at least on a}

short interval0, TN. Global existence will follow from a priori estimates foruN.

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≥1one has

Esup

0≤t≤T

uNt2

V<∞. 4.5

One also has

Esup

0≤t≤T

uNt2

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

⎧ ⎨ ⎩

inf0≤t;uNt_VuNt_∗≥M

∞ if0≤t;uN_t

VuNt∗≥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

φuN_{s, e}

iV uNs, ei2Vto4.3and obtain

uNs, ei

2

V2

s

0

uNr, ei

V

νuNr, ei

buNr, uNr−αΔuNr, ei

dr

2

s

0

uN_{r, e} i

V

−αbei,ΔuNr, uNr

Fr, uN_{, e} i

dr

s

0

Gr, uN, ei

dW

s

0

uNr, ei

V

Gr, uN, ei

2

dr.

4.8

We note that |uN_|2

V

_N

i1λiuN, ei2V. Then, multiplying by λi the above equation and

summing overifrom 1 toNgive us

uNs2

V2ν

s

0

uN2dr uN₀ 2

V2

s

0

Fr, uN, uNdr

N

i1

λi s

0

Gr, uN, ei

2

dr

2

s

0

Gr, uN, uNdW,

4.9

where we have used the fact thatbuN_{, u}N_{, u}N _0.

We obtain from4.9that

uNs2

V2ν

s

0

uNr, uNrdr ≤uN₀ 2

V

N

i1

λi s

0

GuNr, r, ei

2

dr

2

s

0

FuNr, r, uNrdr

2

s

0

GuNr, r, uNrdW,

4.10

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

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

FuNs, s, uN≤ P2uNFuNs, s. 4.12

We also mention that

P2_α−1_|v|2

V≤ v2≤α−1|v|2V, for any v∈V. 4.13

From the former equation and this one we find

FuNs, s, uNs≤2CP

2

α

1uNs2

V

. 4.14

To find a uniform estimate for the corrector termN_i1λiGuNs, 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 productinL2_{D ·,·and thee}

i-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∇qGuN_{s, s inD,}

divG0 in D,

G0 on∂D,

4.15

for anys∈0, T. ByTheorem 2.5,4.15has a solutionGinW⊗m_{when∂Dis of classC}3_and

GuNs, s∈V⊗m. Moreover, there exists a positive constantC0such that

_G

H3_D⊗m ≤C0

GuNs, s

V⊗m, 4.16

andG, e iV GuNs, s, eifor anyi≥1.

Since the norms|·|_H3_Dand|·|_Ware equivalent onW, then there exists another positive

constantC∗such that

_G

W⊗m ≤C∗C0

GuNs, s

Equation4.17implies thatGdepends continuously on the dataGuN_{s, s. Therefore, we}

note the aboveGasGuN_{s, s. We find from4.17that}

N

i1

λi

GuNs, s, ei

2

N

i1

λi

GuNs, s, ei

2

V

N

i1

1 λi

GuNs, s, ei

2

W.

4.18

We deduce from this that

N

i1

1 λi

GuN_{s, s, e} i

2

W ≤

1 λ1

_GuN_{s, s}2

W⊗m. 4.19

By4.17and the assumption onG, we have

N

i1

λi

GuNs, s, ei

2

≤C

1uNs2

V

. 4.20

Collecting this information, we obtain from4.10that

uNs2

V2ν

s

0

uNr2dr

≤CC

s

0

uNr2

Vdr2

s

0

GuNr, r, uNrdW.

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

uNs2

V2νE

t∧τM

0

uNs2ds

≤CCE

t∧τM

0

uNs2

Vds2C1E

t∧τM

0

GuNs, s, uNs2ds

1/2

.

4.22

Now, we estimate

γ E

t∧τM

0

GuNs, s, uNs2ds

1/2

With the same argument that we have used for the term|FuN_{s, s, u}N_{s|, we have}

γ≤CE

⎡ ⎣sup

s≤t∧τM

uNs

V

t∧τM

0

GuNs, s2

V×mds

1/2⎤

⎦_. 4.24

Byε-Young’s inequality

γ≤CεE sup

s≤t∧τM

uNs2

VCεE

t∧τM

0

GuNs, s2

V×mds 4.25

Using the assumption onGone has

γ≤CεEsup

s≤t∧τM

uNs2

VCεE

t∧τM

0

1uNs2

V

. 4.26

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

write

Esup

s≤t∧τM

uNs2

V4νE

t∧τM

0

uNs2ds≤CCE

t∧τM

0

uNs2

Vds. 4.27

We derive from this and Gronwall’s inequality that

E sup

0≤s≤t∧τM

uNs2

V≤C. 4.28

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

λi

GuNs, s, ei

GuNs, s, ei

W, i≥1, 4.29

whereGuN_{s, sis the solution inWofGS.}

To alleviate notation, we only writeuN_{when we meanu}N_{·. Let us set}

φuN−νΔuNcurluN−αΔuN×uN−FuN, t. 4.30

ByLemma 4.1,φuN_∈H1_{D. We have}

duN, ei

V

φuN, ei

dtGuN, t, ei

ByTheorem 2.5a solutionvN_{∈Wof the following system exists:}

vN_−αΔvN_∇qφuN _inD,

divvN _{0 inD,}

vN0 on∂D.

4.32

Moreover,

vN, ei

V

φuN, ei

, 4.33

for anyi. Thus,

duN, ei

V

φuN, ei

dtduN, ei

V

vN, ei

Vdt

GuN, t, ei

dW.

4.34

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

4.1:

duN, ei

W

vN, ei

Wdtλi

GuN, t, ei

dW. 4.35

Recalling4.29, we obtain

duN_{, e} i

W

vN_{, e} i

Wdt

GuN_{, t, e} i

WdW. 4.36

Now applying the It ˆo’s formulasee, e.g.,7toϕuN_{, e}

iW uN, ei2W, we have

duN, ei

2

W2

uN, ei

W

vN, ei

Wdt

GuN, t, ei

2

Wdt2

uN, ei

W

GuN, t, ei

WdW.

4.37

Summing both sides of the last equation from 1 toNyields

duN2

W2

uN, vN

Wdt

N

i1

GuN, t, ei

2

Wdt2

GuN, t, uN

Using the definition of| · |Wand the scalar product·,·W, we can rewrite the above equation

in the form

d#uN2

V

uN2

∗

$

2vN, uN

V

curlvN−αΔvNcurluN−αΔuNdt

2curlGuN, t−αΔGuN, t,curluN−αΔuNdW

N

i1

λ2i

GuN, t, ei

2

Vdt2

GuN, t, uN

VdW.

4.39

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

curlGuN, t,curluN−αΔuNdW dt

m

k1

curlGk

uN, t,curluN−αΔuNdWk dt .

4.40

Using the definition ofvN_{andG, we obtain}

d#uN2

V

uN2

∗

$

2φuN, uNcurlφuN,curluN−αΔuNdt

N

i1

λ2_iGuN, t, ei

2

dt2GuN, t, uNdW

2curlGuN, t,curluN−αΔuNdW.

4.41

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

duN2

∗2

curlφuN_,curluN_−αΔuN_dt

2curlGuN, t,curluN−αΔuNdW

N

i1

λiλ2_i

GuN, t, ei

2

dt.

4.42

We infer from the definition ofφuN_that

curlφuN−νcurlΔuNFuN, tcurlcurluN−αΔuN×uN. 4.43

In taking advantage of the dimension we get

This yields

uN· ∇curluN−αΔuN−νcurlΔuNFuN, t,curluN−αΔuN

curlφuN,curluN−αΔuN.

4.45

Owing toLemma 4.1we readily check that

uN· ∇β, β0, 4.46

whereβcurluN_−αΔuN_{. Consequently,}

ν curlΔuN,curluN−αΔuNcurlFuN, t,curluN−αΔuN

−curlφuN,curluN−αΔuN.

4.47

Or equivalently

ν α

uN

∗−

ν α

curluNα

νcurl

FuN_{, t,curlu}N_−αΔuN

curlφuN,curluN−αΔuN.

4.48

We derive from4.42and the last equation that

d dt

uN2

∗

2ν α

uN2

∗−

2ν α

curluN α νcurl

FuN, t,curluN−αΔuN

N

i1

λiλ2i

GuN, t, ei

2

2curlGuN, t,curluN−αΔuNdW dt .

4.49

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

uNs2

∗

s

0

2ν α

uNr2

∗−

N

i1

λiλ2i

GuNr, r, ei

2

dr

s

0

2ν α

#

curluNr−α νcurl

FuNr, r,curluNr−αΔuNr$dr

2

s

0

curlGuNr, r,curluNr−αΔuNrW.

Hence,

uNs2

∗ s 0 2ν α uNr2

∗dr−

N

i1

λiλ2_i s

0

GuNr, r, ei

2

dr

≤uN0 2 ∗ s 0 2ν α

curluNruNr

∗

s

0

2curlFuNr, ruNr

∗dr

2

s

0

curlGuNr, r,curluNr−αΔuNrdW.

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

uNs2

∗E

t∧τM

0

2ν α

uNs2

∗ds−

N

i1

λiλ2i

E t∧τM

0

GuNs, s, ei

2

ds

≤uN0 2

∗E

t∧τM

0

2ν α

curluNsuNs

∗E

t∧τM

0

2curlFuNs, suNs

∗ds

2Esup

s≤t∧τM

s∧τM

0

curlGuNs, s,curluNs−αΔuNsdW.

4.52

For anyε1≥0 andε2≥0, we have

Esup

s≤t∧τM

uNs2

∗E

t∧τM

0

2ν α

uNs2

∗ds−

N

i1

λiλ2i

E t∧τM

0

GuNs, s, ei

2

ds

≤uN₀ 2

∗E

t∧τM

0

2ν αε1

curluNs2 2

ε2

curlFuNs, s2

ds

2Esup

s≤t∧τM

s∧τM

0

curlGuNs, s,curluNs−αΔuNsdW

2νε1

α 2ε2

t∧τM

0

uN_s2

∗ds.

4.53

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

estimate,

E sup

s≤t∧τM

uNs2

∗E

t∧τM

0

ν α

uNs2

∗ds−

N

i1

λiλ2i

E t∧τM

0

GuNs, s, ei

2

ds

≤uN 0

2

∗E

t∧τM

0

8ν αε1

curluN_s22α

ν

curlFuN_{s, s}2

ds

2E sup

s≤t∧τM

s∧τM

0

curlGuNs, s,curluNs−αΔuNsdW.

Thanks to4.20,4.29and4.17we see that

N

i1

λiλ2i

E t∧τM

0

GuNs, s, ei

2

ds≤CCE

s∧τM

0

uNs2

Vds. 4.55

Let us estimate

γ2Esup

s≤t∧τM

t∧τM

0

curlGuNs, s,curluNs−αΔuNsdW. 4.56

By Fubini’s theorem and the Burkh ¨older-Davis-Gundy’s inequality we obtain

γ≤6E

t∧τM

0

curlGuNs, s,curluNs−αΔuNs2dW

1/2

≤6E

⎛ ⎝_sup

s≤t∧τM

uNs

∗

t∧τM

0

curlGuNs, s2

1/2⎞

⎠_.

4.57

Making use of anε-Young’s inequality, the following holds:

γ≤6εE sup

s≤t∧τM

uNs2

∗

6 εE

t∧τM

0

curlGuNs, s2ds. 4.58

Choosingε1/12, we write

γ≤ 1 2Es≤supt∧τM

uNs2

∗72E

t∧τM

0

curlGuNs, s2ds. 4.59

Combining4.54,4.55, and4.59, we obtain

Esup

s≤t∧τM

uNs2

∗E

t∧τM

0

ν α

uNs2

∗ds

≤CE

t∧τM

0

curlFuNs, s2CE

t∧τM

0

curlGuNs, s2ds

CuN0 2

∗CE

t∧τM

0

uNs2

VdsCE

t∧τM

0

curluNs2ds.

4.60

By a straightforward calculation we have

curlφ2≤ 2 αφ

2

Owing to4.61and the assumptions onFandG, we derive from4.60that

E sup

s≤t∧τM

uNs2

∗E

t∧τM

0

ν α

uNs2

∗ds≤C

uN₀ 2

∗CE

t∧τM

0

uNs2

Vds. 4.62

This and the estimate4.28imply

E sup

s≤t∧τM

uNs2

∗E

t∧τM

0

ν α

uNs2

∗ds≤C. 4.63

It is easy to check that, asM → ∞,t∧τM → talmost surely for anyt∈ 0, TN. Since the

constantCis independent ofN, the estimates4.28,4.63and the Dominated Lebesgue’s Convergence Theorem complete the proof of the lemma.

Lemma 4.3. For any4≤p <∞one has

Esup

s≤T

uNsp

V<∞,

Esup

s≤T

uNsp

W<∞.

4.64

Proof. We recall that

duNt2

V2ν

uNt2dt−2Funt, t, uNtdt

N

i1

λi

GuNt, t, ei

2

dt2GuNt, t, uNtdW.

4.65

For a fixedp ≥ 4 the application of It ˆo’s formula to the functionφ|uN_t|2

V |uNt|2Vp/4

yields

duNtp/2

V

p 2

uNtp/2−2

V

%

−νuNt2Funt, t, uNt

1

2

N

i1

GuNt, t, ei

2

p−4

4

GuN_{t, t, u}N_t2

_uN_t2

V

&

dt

p

2

uNtp/2−2

V

GuNt, t, uNtdW.

Hence

uNtp/2

V

uN0

p/2 V p 2 t 0

uNsp/2−2

V

×

%

−νuNs2Funs, s, uNs

1

2

N

i1

GuNs, s, ei

2

p−4

4

GuN_{s, s, u}N_s2

_uN_s2

V & ds p 2 t 0

uN_sp/2−2

V

GuN_{s, s, u}N_sdW,

4.67

for any t ∈ 0, T. In squaring the last equation and in making use of some elementary inequalities we obtain

uNtp

V≤C

uN₀ p

VC

t

0

uNsp/2−2

V

×

#

−νuNs2Funs, s, uNs

1

2

N

i1

GuNs, s, ei2

p−4 4

GuN_{s, s, u}N_s2

|uN_s|2

V & ds 2 C t 0

uNsp/2−2

V

GuNs, s, uNsdW

2

.

4.68

We deduce from4.14,4.20, and4.68that

uNtp

V≤C

uN₀ p

VC

t

0

uNsp/2−2

V

1uNs

V

22

C

t

0

uNsp/2−2

V

GuNs, s, uNsdW

2

.

4.69

We find from this that

Esup

s≤t

uNsp

V≤C

uN₀ p

VCE

t

0

uNsp−4

V

1uNs

V

4

ds

CEsup

s≤t s

0

uNrp/2−2

V

GuNr, r, uNrdW

2

.

It is clear that

uNsp−4

V ≤

1uNs

V

p−4

. 4.71

Hence,

Esup

s≤t

uNsp

V≤C

uN₀ p

VCE

t

0

1uNs

V

p

ds

CEsup

s≤t s

0

uNrp/2−2

V

GuNr, r, uNrdW

2

.

4.72

Now let us denote byγ1the stochastic term in4.70. As before, we use the Burkh

¨older-Davis-Gundy’s inequality and get

γ1≤CE t

0

uNsp−4

V

GuNs, s, uNs2ds

≤CE

t

0

uNsp−4

V

GuNs, s2

V

uNs2

V.

4.73

The following follows from the same arguments as used before and by the assumption onG:

γ1≤CE t

0

1uNs

V

p

ds. 4.74

This, the estimate4.70, and Gronwall’s inequality imply

Esup

s≤t

uNsp

V<∞, 4.75

which completes the proof of the first estimate of the lemma.

Let us now proceed to the proof of the second estimate ofLemma 4.3. We rewrite4.49 in the form

duNs2

∗

N

i1

λiλ2i

GuNs, s, ei

2

2ν

α

curluNs,curluNs−αΔuNsds

−2ν

α

uNs2

∗2

curlFuNs, s,curluNs−αΔuNs

2curlGuNs, s,curluNs−αΔuNsdW.

Applying It ˆo formula to the functionϕ|uN_s|2

∗ |uNs|2∗p/4we have

duNsp/2

∗ −

p 2

uNsp/2−2

∗

×

2curlFuNs, s,curluNs−αΔuNs

1 2

N

i1

λiλ2i

GuNs, s, ei

2

2ν

α

curluNs,curluNs−αΔuNs

− 2ν

α

uNs2

∗

p−4 4

curlGuN_{s, s,curlu}N_s−αΔuN_s2

_uN_s2

∗

ds

p

2

uNsp/2−2

∗

curlGuNs, s,curluNs−αΔuNsdW.

4.77

Hence,

uN_tp/2

∗

uN

0 p/2

∗

p 2

t

0

uN_sp/2−2

∗

×

2curlFuNs, s,curluNs−αΔuNs

1

2

N

i1

λiλ2_i

GuNs, s, ei

2

2ν

α

curluNs,curluNs−αΔuNs

− 2ν

α

uN_s2

∗

p−4 4

curlGuN_{s, s,curlu}N_s−αΔuN_s2

_uN_s2

∗

ds

p

2

t

0

uNsp/2−2

∗

curlGuNs, s,curluNs−αΔuNsdW,

for anyt∈0, T. The following follows in squaring both sides of the last inequality:

uNtp

∗ ≤C

uN0

p

∗C

% t

0

uNsp/2−2

∗

×

2curlFuNs, s,curluNs−αΔuNs

1

2

N

i1

λiλ2i

GuNs, s, ei

2

2ν

α

curluNs,curluNs−αΔuNs− 2ν α

uNs2

∗

p−4

4

curlGuN_{s, s,curlu}N_s−αΔuN_s2

_uN_s2

∗

ds

&2

C

t

0

uNsp/2−2

∗

curlGuNs, s,curluNs−αΔuNsdW

2

.

4.79

For almost alls∈0, T, we note that

curluNs,curluNs−αΔuNs≤C1uNs

V

1uNs

W

. 4.80

We also check readily that

curlFuN_{s, s,curlu}N_s−αΔuN_s≤C1uN_s

V

1uN_s

W

,

4.81

curlGuN_{s, s,curlu}N_s−αΔuN_s2

_uN_s2

∗

≤C

1uN_s

V

2

. 4.82

Thanks to the continuous injection ofWintoV, all the above estimates still hold with|uN_·|

V

replaced by|uN_·|

W. It follows from this argument and4.79that

uNtp

∗ ≤

uN0

p

∗C

t

0

uNsp/2−2

∗

1uNs

W

2

ds

2

C

t

0

uNsp/2−2

∗

curlGuNs, s,curluNs−αΔuNsdW

2

.

Taking the supremum overs≤tfollowed by the mathematical expectation yields

Esup

s≤t

uNsp

∗

≤uN₀ p

∗CE

t

0

uNsp/2−2

∗

1uNs

W

2

ds

2

CEsup

s≤t s

0

uNrp/2−2

∗

curlGuNr, r,curluNr−αΔuNrdW

2

.

4.84

Applying the Martingale inequality and H ¨older’s inequality in the last estimate we obtain

Esup

s≤t

uNsp

∗ ≤

uN₀ p

∗CE

t

0

uNsp−4

∗

1uNs

W

4

ds

CE

t

0

uNsp−4

∗

curlGuNs, s, curluNs−αΔuNs2ds.

4.85

We can use the same idea we have used to find4.81 to get an upper bound of the form C1|uN_s|

W4for|curlGuNs, s,curluNs−αΔuNs2|·Then, we derive from4.85

that

Esup

s≤t

uNsp

∗ ≤C

uN0

p

∗C

t

0

1uNs

W

p

ds. 4.86

We obviously have

uNs

W≤C

_uN_sp

V

uNsp

∗

. 4.87

Finally, using a previous result concerningEsup_s≤t|uN_s|p

V,4.86and Gronwall’s inequality

we obtain

Esup

s≤t

uNsp

∗<∞. 4.88

This completes the proof of the lemma.

Remark 4.4. Lemmas4.2and4.3imply in particular that

Esup

t≤T

uNtp

V<∞,

Esup

t≤T

uNtp

W <∞,

4.89

The following result is central in the proof of the forthcoming crucial estimate of the finite difference of our approximating solution.

Lemma 4.5. Lett, s∈0, Tsuch thats≤t. For a fixedt∈0, T, let

vNt

N

i1

λi

vNt, ei

Vei 4.90

be an element ofWNwhich satisfies Lemmas4.2and4.3. The following holds:

uNt−vNs2

V−

uNs−vNs2

V2

t

s

ν#uNr2−uNr, vNr$dr

2

t

s

GuNr, r, uNr−vNsdW

N

i1

λi t

s

GuNr, r, ei

2

dr

2

t

s

bvNs,ΔuNr, uNrdr2

t

s

FuNr, r, uNr−vNsdr

−2

t

s

buNr, uNr, vNsdr−2

t

s

buNr,ΔuNr, vNsdr.

4.91

Proof. ForvN_{, for anys, tsuch that 0≤s≤t≤T, we have}

d dt

uNt−vNs, ei

Vν

uNt, ei

buNt, uNt, ei

−αbuNt,ΔuNt, ei

αbei,ΔuNt, uNt

FuNt, t, ei

GuNt, t, ei

_d

dtW, 1≤i≤N.

4.92

This relation can be rewritten as the following It ˆo equation:

duNt−vNs, ei

Vν

uNt, ei

dtbuNt, uNt, ei

dt

−αbuNt,ΔuNt, ei

dtαbei,ΔuNt, uNt

dt

FuN_{t, t, e} i

dtGuN_{t, t, e} i

dW.

Applying It ˆo’s formula to the functionuN_{t, v}N_{s, e}

i2_V, multiplying the result byλi, and

then summing overifrom 1 toNyield

d|w|2_V2νuNt, wdt2buNt, uNt, wdt−2αbuNt,ΔuNt, wdt

2αbw,ΔuNt, uNtdt−2FuNt, t, wdt

N

i1

λi

GuNt, t, ei

2

2GuNt, t, wdW,

4.94

wherewuN_t−vN_{s. Using the trilinearity ofband the well-known identitybu, u, u 0,}

u∈V, we find that

buNt, uNt, w−αbuNt,ΔuNt, wαbw,ΔuNt, uNt

buNt, uNt, vNs−αbuNt,ΔuNt, vNsαbvNs,ΔuNt, uNt.

4.95

The lemma follows in combining this relation with 4.94, and integrating the resulting equation betweensandt.

The following result can be proved by a similar argument used in15, but we prefer to give our own proof which is interesting in itself.

Lemma 4.6. There exists a positive constantC > 0such that for all0 ≤ δ < 1and N ∈ N, the following inequality holds:

Esup

|θ|≤δ T−δ

0

uN_sθ−uN_s2

W∗ ≤Cδ

1/2_{. 4.96}

Proof. SinceuN_{s ∈ W}

N,s ∈ 0, Tand it satisfies Lemmas 4.2and 4.3then we can take

vN_{s u}N_{sandtsθ, 0≤θ≤δ≤1 and applyLemma 4.5. We obtain}

uNsθ−uNs2

V2

sθ

s

ν#uNr2−uNr, uNr$dr

2

sθ

s

GuNr, r, uNr−uNsdW

N

i1

λi sθ

s

GuNr, r, ei

2

dr

2

sθ

s

buN_s,ΔuN_{r, u}N_rdr2 sθ s

FuN_{r, r, u}N_r−uN_sdr

−2

sθ

s

buNr, uNr, uNsdr−2

sθ

s

buNr,ΔuNr, uNsdr.