Stochastic Navier-Stokes Equations with

Volume 2010, Article ID 730492,24pages doi:10.1155/2010/730492

Research Article

Stochastic Navier-Stokes Equations with

Artificial Compressibility in Random Durations

Hong Yin

Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA

Correspondence should be addressed to Hong Yin,[email protected] Received 1 December 2009; Accepted 11 May 2010

Academic Editor: Jiongmin M. Yong

Copyrightq2010 Hong Yin. 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.

The existence and uniqueness of adapted solutions to the backward stochastic Navier-Stokes equation with artificial compressibility in two-dimensional bounded domains are shown by Minty- Browder monotonicity argument, finite-dimensional projections, and truncations. Continuity of the solutions with respect to terminal conditions is given, and the convergence of the system to an incompressible flow is also established.

1. Introduction

The Navier-Stokes equation NSE for short, named in honor of Navier and Stokes, who were responsible for its formulation, is an acknowledged model for equation of motion for Newtonian fluid. It is closely connected to the theory of hydrodynamic turbulence, the time dependent chaotic behavior seen in many fluid flows.

The well-posedness of the Navier-Stokes equation has been studied extensively by Ladyzhenskaya1, Constantin and Foias2, and Temam3, among others. Although some ingenious approaches have been made, the problem has not been fully understood. The nonlinearity, part of the cause of turbulence, made the problem extraordinarily difficult. In hope of taking advantage of the noise, randomness has been introduced into the system and some pioneer work has been done by Flandoli and Gatarek4, Mikulevicius and Rozovsky 5, Menaldi and Sritharan 6, and others. Although the introduction of randomness is not very successful in overcoming the difficulty, it provides a more realistic model than deterministic Navier-Stokes equations and is interesting in itself.

The vast majority of work on the Navier-Stokes equations is done for viscous incompressible Newtonian fluids. In a suitable Hilbert space and under the incompressibility assumption∇ ·u 0, the two-dimensional stochastic Navier-Stokes equation in a bounded

domainG⊂R²with no-slip condition reads

∂u u· ∇udt−νΔudt−∇pdtftdtσt,udWt, 1.1 whereν is the constant viscosity,u is the velocity,pis the pressure,f is the external body force andWis the infinite-dimensional Wiener process. The assumption of incompressibility works well even for compressible fluids such as air at room temperature. But there are extreme phenomena, such as the diffusion of sound, that are closely related to fluid compressibility. Also the constraint caused by the incompressibility creates computational difficulties for numerical approximation of the Navier-Stokes equations. The method of artificial compressibility was first introduced by Temam3to surmount this obstacle. It also describes the slight compressibility existed in most fluids. The model has its own interest, and is given below with the parameterε:

∂tuε−νΔuε uε· ∇uε1

2∇ ·uεuε∇pεf, ε∂_tp_ε∇ ·uε0.

1.2

Backward stochastic Navier-Stokes equationsBSNSEs for shortarise as an inverse problem wherein the velocity profile at a timeT is observed and given, and the noise coefficient has to be ascertained from the given terminal data. Such a motivation arises naturally when one understands the importance of inverse problems in partial differential equations see Lions7,8. Linear backward stochastic differential equations were introduced by Bismut in 19739, and the systematic study of general backward stochastic differential equations BSDEs for shortwere put forward first by Pardoux and Peng10, Ma, Protter, Yong, Zhou, and several other authors in a finite-dimensional setting. Ma and Yong 11have studied linear degenerate backward stochastic differential equations motivated by stochastic control theory. Later, Hu et al.12considered the semilinear equations as well. Backward stochastic partial differential equations were shown to arise naturally in stochastic versions of the Black- Scholes formula by Ma and Yong13. A nice introduction to backward stochastic differential equations is presented in the book by Yong and Zhou14, with various applications.

The usual method of proving existence and uniqueness of solutions by fixed point arguments does not apply to the stochastic system on hand since the drift coefficient in the backward stochastic Navier-Stokes equation is nonlinear, non-Lipschitz and unbounded.

The drift coefficient is monotone on boundedL⁴Gballs inV, which was first observed by Menaldi and Sritharan6. The method of monotonicity is used in this paper to prove the existence of solutions to BSNSEs. The proof of the uniqueness and continuity of solutions also relies on the monotonicity assumption of the coefficients. Existence and uniqueness of solutions are shown to hold under theH¹₀boundedness on the terminal values.

The structure of the paper is as follows. The functional setup of the paper is introduced and several frequently used inequalities are listed in Section 2. The a priori estimates for the solutions of projected BSNSEs are given under different assumptions of the terminal conditions and external body force inSection 3. The existence and uniqueness of solutions of projected BSNSEs are shown inSection 4. Also the existence of solutions of BSNSEs under suitable assumptions is shown by Minty-Browder monotonicity argument. The uniqueness of the solution under the assumption that terminal condition is uniformly bounded inH¹ sense is given inSection 5. The continuity of solutions and the convergence asεapproaches zero are also studied.

2. Preliminaries

Suppose that G is a domain bounded in R² with smooth boundary conditions. Let ε be a positive parameter which vanishes to 0. The artificial state equation for a slightly compressible medium is defined as

ρρ0εp, 2.1

whereρis the density,pis the pressure, andρ0is the first approximation of the density. By adjusting the equations of motion according to the state equation, we obtain the following family of perturbed systems associated with the parameterε:

∂tuε−νΔuε uε· ∇uε1

2∇ ·uεuε∇pεf, ε∂_tp_ε∇ ·uε0,

2.2

whereuε∈L² L²Gis the velocity,pε ∈L² L²Gis the pressure,f∈L²is the external body force, andνis the kinematic viscosity. Readers may refer to Temam3for details.

Denote by·,·the inner product ofL²,·,·_H¹

0 the inner product of H¹₀ H¹₀G,H⁻¹ the dual space ofH¹₀, and·,·the duality pairing betweenH¹₀andH⁻¹. Let| · |be the norm ofL²and let · be the norm of H¹₀. Without causing any confusion, we also use the same notations to denote the norms ofL²andH₀¹H₀¹G. For anyx∈L²andy∈H¹₀, there exists x ∈H⁻¹, such thatx,y x,y. Then the mappingx →xis linear, injective, compact and continuous. A similar result holds forH⁻¹andL².

Suppose thatΩ,F, Pis a complete probability space. LetWtbe anL²-valuedQ- Wiener process, where Q is a trace class operator on L². Let {ej}^∞_j1 ∈ L² ∩H¹₀ ∩ L⁴ be a complete orthonormal system in L² such that there exists a nondecreasing sequence of positive numbers {λj}^∞_j1, lim_{j→ ∞}λ_j ∞ and −Δej λ_jej for all j. LetQek q_kek with _∞

k1qk < ∞, and{b^kt}be a sequence of independent standard Brownian motions inR.

Then Wiener processWtis taken asWt_∞

k1√q_kb^ktek.

LetQbe a trace class operator onL². Similarly, we can define a complete orthonormal system{ej}^∞_j1, a nondecreasing sequence of positive numbers{κj}^∞_j1such that−Δej κjej, and positive numbersq_j such thatQej q_jej and _∞

j1q_j < ∞. LetWt_∞

j1

q_jb^jtej. ThenWtis anL²-valuedQ-Wiener process. From now on, let{Ft}be the natural filtration of{Wt}and{Wt}, augmented by all theP-null sets ofF. A complete definition of Hilbert space-valued Wiener processes can be found in15.

With inner product

F,G_LQtrFQG^∗ trGQF^∗ 2.3

for allF and G∈L_Q, letL_Qdenote the space of linear operatorsE such that EQ^1/2is a Hilbert- Schmidt operator fromL² toL². Similarly, we define LQ forQ, the trace class operator on L².

To be realistic in nature, let us introduce randomness into the system to obtain

∂uεt

∂t −νΔuεt uεt· ∇uεt 1

2∇ ·uεtuεt ∇pεt ft σtdWt dt , ε∂pεt ∇ ·uεtdt0,

uε0 u0, p_ε0 p₀,

2.4

whereu0andp0are initial conditions, andσdW/dtis the noise term. Here1/2∇ ·uεuε

is called the stabilization term.

If a terminal timeT is given and the terminal conditions are specified asuεT ξand pεT η, one obtains a backward system:

duεt νAu εtdtBu εtdt∇pεtdtftdtZεtdWt, εdp_εt ∇ ·uεtdtZ_εtdWt,

uεT ξ, p_εT η

2.5

for 0 ≤ t ≤ T, whereAu −Δu andBu, v u· ∇v 1/2∇ ·uv, with the notation Bu Bu, u. The processesZεandZεare in spacesL_QandLQ, respectively.

Let τ be a Ft-stopping time when the observations are available. Suppose that the observed velocity and pressure at τ are uετ ξ ∈ L²_F

τΩ;L² and pτ η ∈

L²_F

τΩ;L², respectively. Then we introduce the backward stochastic Navier-Stokes equation with artificial compressibility and stabilization in random duration:

duεt νAu εtdtBu εtdt∇pεtdtftdtZεtdWt, εdp_εt ∇ ·uεtdtZ_εtdWt,

uετ ξ, p_ετ η

2.6

for 0≤t≤τ, where theFt-stopping timeτis assumed to be bounded by a timeT >0. Note that processesZε andZ_ε measure the randomness that is inherent in the hydrodynamical system. It is this randomness that has possibly led us to the observations at time τ. For instance, in wind tunnel experiments, the form and the magnitude of the randomness has to be ascertained from the velocity observations. This backward system helps us to make an attempt at uncertainty quantification. Heref is taken to be deterministic and is always assumed to be inL²0, T;H⁻¹.

Definition 2.1. A quaternion of Ft-Adapted processes uε,Zε, p_ε, Z_ε is called a solution of backward Navier-Stokes equation2.6if it satisfies the integral form of the system

uεt∧τ ξ _τ

t∧τ

νAu εs Bu εs ∇pεs−fs ds−

_τ

t∧τZεsdWs, εp_εt∧τ εη

_τ

t∧τ∇ ·uεsds− _τ

t∧τZ_εsdWs,

2.7

P-a.s., and the following holds:

auε∈L²_FΩ;L^∞0, τ;L²∩L²_FΩ;L²0, τ;H¹₀; bZε∈L²_FΩ;L²0, τ;L_Q;

cpε∈L²_FΩ;L^∞0, τ;L²∩L²_FΩ;L²0, τ;H₀¹; dZε∈L²_FΩ;L²0, τ;LQ.

The following simple results are frequently used and given as lemmas. Readers may refer to Temam3for similar proofs.

Lemma 2.2. For any u,v,w∈H¹₀andp∈L², one has 1Au, w=

i,j

G∂_iu_j∂_iw_jdx=Aw, u =u,w_H¹

0, 2u· ∇v,w =

i,j

Gu_i∂iv_jwjdx,

3u· ∇v,w =−∇ ·uw,v − u· ∇w,v, 4Bu, v,w=−Bu, w,v,

5−∇p,u=−

i

G∂ip uidx=

Gp∂iuidx=p,∇ ·u.

Remark 2.3. SometimesBu, v,wis denoted bybu, v,w.

Lemma 2.4. The following results hold for any real-valued smooth functionsφandψ with compact support inR²:

φψ ²≤Cφ∂₁φ

L¹ψ∂₂ψ

L¹, φ⁴

L⁴≤C φ ² ∇φ ².

2.8

Proposition 2.5. For any u and v inH¹₀andw∈L⁴, one has bu,v,w ≤ u_L⁴vw_L⁴ 1

2uv_L⁴w_L⁴. 2.9

Below is a backward version of the Gronwall inequality used frequently in this paper, and the proof is straightforward.

Lemma 2.6. Suppose that gt, αt, βt, and γt are integrable functions, and βt,γt are nonnegative functions. For 0≤t≤T, if

gt≤αt βt _T

t

γ ρ

g ρ

dρ, 2.10

then

gt≤αt βt _T

t

α η

γ η

e^{ηtβργρdρ}dη. 2.11

In particular, ifαt≡α,βt≡βandγt≡1, then

gt≤αe^βT−t. 2.12

3. A Priori Estimates

The purpose of the this paper is to show the existence and uniqueness of the randomly stopped backward stochastic Navier-Stokes equation 2.6. We employ Galerkin’s method by defining orthogonal projectionsP_N : L² → L²_N, whereL²_N span{e1,e2, . . . ,eN}, for all N∈N. An important result is that the Galerkin-type approximations converge weakly to the solution of the Navier-Stokes equation.

First of all, let us establish some a priori estimates. Let us define the projected operators A^NPNA and B^NPNB. Under projection PN, let us construct a finite dimensional system.

Let

W^NtPNWt ^N

i1

qibⁱtei, W^NtPNWt ^N

i1

q_ibⁱtei, f^NtP_Nft, ξ^NE

P_Nξ| F^N_τ

, η^NE

P_Nη| F^N_τ ,

3.1

where {F^N_t } is the natural filtration of {W^Nt} and {W^Nt}. The projected system with solutionu^N_ε ,Z^N_ε , p_ε^N, Z^N_ε is defined as follows:

du^N_ε t −νA^Nu^N_ε tdt−B^N u^N_ε t

dt− ∇p^N_ε tdtf^NtdtZ^N_ε tdW^Nt, εdp_ε^Nt ∇ ·u^N_ε tdtZ^N_ε tdW^Nt,

u^N_ε τ ξ^N, p_ε^Nτ η^N

3.2

for 0≤t≤τ.

Proposition 3.1. Letξ∈L^∞_F

τΩ;L²,η ∈L^∞_F

τΩ;L², andf∈L²0, T;H⁻¹. Then for any solution of system3.2, the following is true:

u^N_ε ,Z^N_ε

∈ L^∞_F

0, τ×Ω;L²

∩L²_F Ω;L²

0, τ;H¹₀

×L²_F

Ω;L²0, τ;L_Q ,

p^N_ε , Z_ε^N

∈ L^∞_F

0, τ×Ω;L²

∩L²_F Ω;L²

0, τ;H₀¹

×L²_F Ω;L²

0, τ;L_Q .

3.3

Proof. Applying the It ˆo formula to|p^N_ε t|²to get

d p^N_ε t ²−2 ε

∇ ·u^N_ε t, p_ε^Nt dt2

ε

Z^N_ε tdW^Nt, p^N_ε t 1

ε² tr

Z^N_ε tQ

Z_ε^Nt_∗ dt 2

ε

∇p^N_ε t,u^N_ε t dt2

ε

Z_ε^NtdW^Nt, p_ε^Nt 1

ε²tr

Z_ε^NtQ

Z^N_ε t_∗ dt,

3.4

thus we get

2

∇p^N_ε t,u^N_ε t

dtεd p_ε^Nt ²−2

Z^N_ε tdW^Nt, p^N_ε t

− 1 ε

Z^N_ε t²

LQ

dt. 3.5

By means of the It ˆo formula, one has

u^N_ε t∧τ ² ξ^{N 2}2 _τ

t∧τ

νA^Nu^N_ε s B^N u^N_ε s

∇p_ε^Ns−f^Ns,u^N_ε s ds

−2 _τ

t∧τZ^N_ε sdW^Ns,u^N_ε s − _τ

t∧τ

Z^N_ε s²

L_Qds.

3.6

Clearly,

B^N u^N_ε s

,u^N_ε s

0, 3.7

andLemma 2.2yields 2

f^Ns,u^N_ε s ≤f^Ns²

H⁻¹u^N_ε s² f^Ns ²

H⁻¹A^Nu^N_ε s,u^N_ε s. 3.8

For 0< r≤t, taking the conditional expectation with respect toF_r∧τ, and by3.5, the above two equation and along with the fact thatuεs²Au εs,uεs, one gets

E^Fr∧τ u^N_ε t∧τ ²E^Fr∧τ _τ

t∧τ

Z^N_ε s²

LQdsE^Fr∧τ _τ

t∧τ

u^N_ε s²ds

≤E^Fr∧τ ξ^{N 2}2ν1E^Fr∧τ _τ

t∧τ

A^Nu^N_ε s,u^N_ε s

dsE^Fr∧τ _τ

t∧τ

f^Ns²

H⁻¹ds εE^Fr∧τ

_τ

t∧τd p^N_ε s ²−1 εE^Fr∧τ

_τ

t∧τ

Z^N_ε s²

LQ

ds,

3.9

P-a.s. SinceAe jλ_jejandλ_i≤λ_jfori < j, one gets A^Nu^N_ε s,u^N_ε s

≤λ_N u^N_ε s ². 3.10

Thus

E^Fr∧τ u^N_ε t∧τ ²εE^Fr∧τ p_ε^Nt∧τ ²E^Fr∧τ _τ

t∧τ

u^N_ε s²ds E^Fr∧τ

_τ

t∧τ

Z^N_ε s²

L_Qds1 εE^Fr∧τ

_τ

t∧τ

Z^N_ε s²

LQ

ds

≤E^Fr∧τ ξ^{N 2}εE^Fr∧τ η^{N 2}2ν1λN

_T

t

E^Fr∧τ u^N_ε s∧τ ²ds E^Fr∧τ

_τ

0

f^Ns²

H⁻¹ds,

3.11

P-a.s., and byLemma 2.6, the backward Gronwall inequality, and lettingrt, we get u^N_ε t∧τ ²ε p^N_ε t∧τ ²E^Ft∧τ

_τ

t∧τ

u^N_ε s²ds E^Ft∧τ

_τ

t∧τ

Z^N_ε s²

LQds1 εE^Ft∧τ

_τ

t∧τ

Z_ε^Ns²

LQ

ds

≤

E^Ft∧τ ξ^{N 2}εE^Ft∧τ η^{N 2}E^Ft∧τ _τ

0

f^Ns²

H⁻¹ds

e^2ν1λNT−t,

3.12

P-a.s. Because of the integrability ofξ,η, andf, there exists a constantK_N, depending onN only, s.t.

u^N_ε t ²ε p^N_ε t ²E _τ

0

u^N_ε s²dsE _τ

0

Z^N_ε s²

L_QdsE _τ

0

Z_ε^Ns²

LQ

ds≤KN, 3.13 for allt∈0, τ, P-a.s.

Similarly, making use of3.4, it follows thatp_ε^N∈L²_FΩ;L²0, τ;H₀¹. Proposition 3.2. Letξ ∈ Lⁿ_F

τΩ;L²,η ∈ Lⁿ_F

τΩ;L², and f ∈ L²0, T;H⁻¹, for alln ∈ Nand n≥2. The following is true for any solution of system3.2:

u^N_ε ∈L^∞

0, τ;Lⁿ_F

Ω;L²

∩Lⁿ_F Ω;Lⁿ

0, τ;H¹₀ , p_ε^N∈L^∞

0, τ;Lⁿ_F

Ω;L²

∩Lⁿ_F Ω;Lⁿ

0, τ;H₀¹ .

3.14

Proof. Let us prove it by the method of mathematical induction. Similar toProposition 3.1, it is easy to obtain the result forn2. Suppose that it is true for allm≤n−1. Let us show that the proposition holds formn.

An application of the It ˆo formula to|u^N_ε t|ⁿyields u^N_ε t∧τ ⁿ

ξ^{N n}n _τ

t∧τ

u^N_ε s ⁿ⁻²

νA^Nu^N_ε s B^N u^N_ε s

∇p^N_ε s−f^Ns,u^N_ε s ds

−n _τ

t∧τ

u^N_ε s ⁿ⁻²

Z^N_ε sdW^Ns,u^N_ε s

−n²−n 2

_τ

t∧τ

u^N_ε s ⁿ⁻²Z^N_ε s²

LQds.

3.15 Clearly |∇p^N_ε s| ≤ Cp_ε^Ns ≤ C√

κ_N|p^N_ε s|, where κ_N, as stated in Section 2, is the eigenvalue of−ΔforeN. Taking the expectation, one obtains

E u^N_ε t∧τ ⁿE _τ

t∧τ

u^N_ε sⁿds≤E ξ^{N n}λ^n/2_N E _τ

t∧τ

u^N_ε s ⁿds nE

_τ

t∧τ

u^N_ε s ⁿ⁻²

νA^Nu^N_ε s ∇p^N_ε s−f^Ns,u^N_ε s ds

≤E ξ^{N n}

νλNnλ^n/2_N E

_τ

t∧τ

u^N_ε s ⁿdsnE _τ

t∧τ

u^N_ε s ⁿ⁻¹ ∇p^N_ε s ds n

2E _τ

t∧τ

u^N_ε s ⁿ⁻² f^Ns²

H⁻¹λ_N u^N_ε s ² ds

≤E ξ^{N n}

νλ_Nnλ^n/2_N n 2λ_N

E _τ

t∧τ

u^N_ε s ⁿds nC√

κN

E

_τ

t∧τ

u^N_ε s ⁿds

n−1/n

E _τ

t∧τ

p^N_ε s ⁿds 1/n

n 2

_T

t

f^Ns²

H⁻¹E

1_t∧τ,τ u^N_ε s ⁿ⁻² ds

≤E ξ^{N n}n 2

sup

0≤t≤τE u^N_ε t ^{n−2 T}

t

f^Ns²

H⁻¹ds

νλ_Nnλ^n/2_N n

2λ_N n−1C√ κ_N

T t

E u^N_ε s∧τ ⁿds C√

κ_{N T}

t

E p^N_ε s∧τ ⁿds

≤KKn, N _T

t

E u^N_ε s∧τ ⁿdsKn, N _T

t

E p^N_ε s∧τ ⁿds,

3.16

whereKis a constant, andKn, Nis a constant depending onnandN. Both constants may vary throughout the proof. But we keep the same notations for simplicity. Applying the It ˆo formula to|p_ε^Nt|ⁿ, one obtains

εE p^N_ε t∧τ ⁿE _τ

t∧τ

p^N_ε sⁿds

≤εE η^{N n}κ^n/2_N E _τ

t∧τ

p_ε^Ns ⁿds−nE _τ

t∧τ

p^N_ε s ⁿ⁻²

∇p^N_ε s,u^N_ε s ds

≤εE η^{N n}κ^n/2_N E _τ

t∧τ

p_ε^Ns ⁿdsnC√ κ_NE

_τ

t∧τ

p^N_ε s ⁿ⁻¹ u^N_ε s ds

≤εE η^{N n}κ^n/2_N E _τ

t∧τ

p_ε^Ns ⁿdsnC√ κ_N

E

_τ

t∧τ

p^N_ε s ⁿds

_n−1/n

×

E _τ

t∧τ

u^N_ε s ⁿds 1/n

≤KKn, N _T

t

E p^N_ε s∧τ ⁿdsKn, N _T

t

E u^N_ε s∧τ ⁿds.

3.17

Adding up3.16and3.17, one gets

E u^N_ε t∧τ ⁿε p^N_ε t∧τ ⁿ E

_τ

t∧τ

u^N_ε sⁿp^N_ε sⁿ ds

≤KKn, N _T

t

E u^N_ε s∧τ ⁿ p^N_ε s∧τ ⁿ ds.

3.18

An application of the Gronwall inequality2.11yields the result.

4. Existence of Solutions

The following lemma states the monotonicity of drift coefficients. The proof involves Proposition 2.5and is straightforward.

Lemma 4.1. Assume u,v∈H¹₀andw∈L⁴. The following inequalities are true:

a|Bu, w| ≤2u^3/2|u|^1/2w_L⁴,

b|Bu −Bv, u−v| ≤ν/2u−v² +27/2ν³|u−v|²v⁴_L4,

cνAu −v Bu −Bv, u−v+27/2ν³v⁴_L4|u−v|²≥ν/2u−v². Furthermore, ifw∈H¹₀, then there exists a constantCdepending onν, such that

dνAu −v Bu −Bv, u−v+Cv²|u−v|²≥ν/2u−v².

Corollary 4.2. For any u and v∈L⁴, let

h1t 27 ν³

_t

0

us⁴_L4ds,

h2t 27 ν³

_t

0

vs⁴_L4ds.

4.1

Then

νAu −v Bu −Bv 1

2h˙itu−v,u−v

≥0, i1,2. 4.2

The proposition below is used in the proof of the existence, and we provide a brief proof. Readers may refer to14,16for a similar and detailed proof.

Proposition 4.3. Letξ ∈ L^∞_F

τΩ;L²,η ∈ L^∞_F

τΩ;L², andf ∈ L²0, T;H⁻¹. Then the projected system3.2admits a unique adapted solutionu^N_ε ,Z^N_ε , p^N_ε , Z_ε^Nin

L^∞_F

0, τ×Ω;L²

∩L²_F Ω;L²

0, τ;H¹₀

×L²_F

Ω;L²0, τ;L_Q

× L^∞_F

0, τ×Ω;L²

∩L²_F Ω;L²

0, τ;H₀¹

×L²_F Ω;L²

0, τ;LQ

.

4.3

Proof. For everyM∈N, letL_Mbe a LipschitzC^∞function which has the following property:

LMu

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

1 ifu< M,

0 ifu> M1,

0≤L_Mu≤1 otherwise.

4.4

Applying the truncationL_MtoB, it is easy to show that L_MB is Lipschitz and

LMxB^Nx−L_MyB^Ny ≤C_N,Mx−y 4.5

for anyx,y∈L²_NandM∈N. Let us define a truncated projected system:

du^N,M_ε t −νA^Nu^N,M_ε tdt−LMu^N,M_ε t B^N

u^N,M_ε t

dt− ∇p_ε^N,Mtdt f^NtdtZ^N,Mε tdW^Nt,

εdp_ε^N,Mt ∇ ·u^N,Mε tdtZ^N,M_ε tdW^Nt, u^N,Mε τ ξ^N, p_ε^N,Mτ η^N.

4.6

For fixedp∈L^∞_F0, τ×Ω;L²∩L²_FΩ;L²0, τ;H₀¹, let us map du^N,Mε t −νA^Nu^N,Mε tdt−LMu^N,Mε t

B^N

u^N,Mε t

dt− ∇p^N,Mtdt f^NtdtZ^N,Mε tdW^Nt,

u^N,Mε τ ξ^N

4.7

toR^N, and the image of the system is equivalent to the system. Since the coefficients in the image system are Lipschitz, a well-known result inR^N see14, page 355guarantees the existence of a unique adapted solution. Let the solution beu^N,M_ε ,Z^N,M_ε . Then for

εdp^N,M_ε t ∇ ·u^N,Mε tdtZ^N,M_ε tdW^Nt p^N,M_ε τ η^N,

4.8

there is a unique adapted solution p^N,Mε , Z_ε^N,M. Thus we can define an operatorΨ, such thatΨp p^N,M_ε . It can be shown thatΨis a contraction mapping. Thus the unique adapted solution of4.6can be obtained. Let us take the limit of the solution asMapproaches infinity.

It can be shown that the limit is the unique solution of the projected system3.2.

From now on, let us assume the external body force to be an operator and denote it by F. We also assume the following coercivity and monotonicity hypotheses in this paper. Such an approach is commonly used in studying the stochastic Euler equations so that a dissipative effect arises. Also they are standard hypotheses in the theory of stochastic PDEs in infinite dimensional spacessee Chow15, Kallianpur and Xiong17, Pr´ev ˆot and R ¨ockner18.

Assumption A. A.1F:H¹₀ → H⁻¹is a continuous operator.

A.2There exist positive constantsαandβ, such that

νAu −Fu,u

≤α|u|²−βu²;

νAu −Fu,Au

≤αu²−βAu².

4.9

A.3For anyu and v inH¹₀, a constantκ > ν, and a positive constantα,

κAu −v−Fu−Fv,u−v

≤α|u−v|². 4.10

A.4For anyu∈H¹₀and some positive constantα,

|Fu,u| ≤αu². 4.11

Remark 4.4. Assumption A.2 is usually called the coercivity condition of the dissipative term and the external body force. Assumption A.3 is the monotonicity condition of dissipative term and the external body force. The first half of the inequality is used in the proof of the uniqueness inSection 5. The second half of the inequality is used in the proof of the existence inSection 4. AssumptionA.4is the linear growth condition of the external body force.

Under above assumptions, we adjust systems 2.6 and 3.2 to the following two systems:

duεt νAu εtdtBu εtdt∇pεtdtFuεtdtZεtdWt, εdp_εt ∇ ·uεtdtZ_εtdWt,

uετ ξ, pετ η,

4.12

du^N_ε t −νA^Nu^N_ε tdt−B^N u^N_ε t

dt− ∇p^N_ε tdtF^N u^N_ε t

dtZ^N_ε tdW^Nt, εdp^N_ε t ∇ ·u^N_ε tdtZ_ε^NtdW^Nt,

u^N_ε τ ξ^N, p^N_ε τ η^N

4.13

for 0 ≤ t ≤ τ. The existence and uniqueness of an adapted solution of4.13can be easily checked in the same fashion as inProposition 4.3.

Lemma 4.5. Assume u and v∈L⁴. Then the following inequality is true:

Bu −Bv, u−v ≤κ−νu−v² 27

16κ−ν³|u−v|²v⁴_L4. 4.14

Corollary 4.6. Let u and v∈L⁴. Define

l₁t _T

t

2α 27

8κ−ν³us⁴_L4

ds,

l₂t _T

t

2α 27

8κ−ν³vs⁴_L4

ds.

4.15

Then

νAu −v Bu −Bv −Fu−Fv 1

2l˙_itu−v,u−v

≤0, i1,2. 4.16

Remark 4.7. To proveCorollary 4.6, the monotonicity assumptionA.3is used.