Stochastic non-Newtonian fluid motion equations of a nonlinear bipolar viscous fluid

J. Math. Anal. Appl. 369 (2010) 486–509

Contents lists available at ScienceDirect

Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Stochastic non-Newtonian fluid motion equations of a nonlinear bipolar viscous fluid ✩ Jianwen Chen a,∗ , Zhi-Min Chen b a b

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, PR China School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ, UK

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 5 May 2009 Available online 27 March 2010 Submitted by W. Layton

An analysis based on the Galerkin method is developed to examine the behaviour of a nonlinear bipolar viscous fluid mathematically modelled by stochastic non-Newtonian fluid motion equations. Existence and uniqueness of solutions to the stochastic equations are derived. © 2010 Elsevier Inc. All rights reserved.

Keywords: Stochastic differential equations Non-Newtonian fluid Weak solutions

1. Introduction In studies of viscous fluid motion, hydrodynamic turbulence is widely regarded as one of the most fascinating problems of nonlinear sciences. A variety of mathematical models and theories have been developed to understand the phenomenon and mechanisms relating turbulence. For example, in their various guises, time averaged, Reynolds-averaged Navier–Stokes equations (RANS) (see, for example, Wilcox [32]) as time averaged equations of viscous fluid motions are widely accepted in computational studies of the fluid mechanics of turbulence. The Lagrangian-averaged Navier–Stokes equations, combining Lagrangian-averaged nonlinearity with Navier–Stokes viscosity, are discussed by Foias et al. [16] in the analytic study of turbulence. Kolmogorov [24] developed energy spectrum criterion based on a statistics approach to display a universality characteristics of turbulence. On the basis of stochastic fluid mechanics viewpoint, turbulence is a combination of slow oscillating (deterministic) and fast oscillating (stochastic) components. This is also described as a white noise perturbation of regular fluid velocity field. Under this perturbation, the governing Newtonian fluid motion equations are transformed into stochastic Navier–Stokes equations (see, for example, [27,28]), or the fluid modelled by Navier–Stokes equations is excited by a nonlinear random force involving white noise. In this study, to investigate turbulence in a non-Newtonian fluid, we propose a stochastic model of a non-Newtonian fluid motion in a smooth, connected and bounded domain O of the three-dimensional space R 3 . The motion is described by random velocity vector field u = (u 1 , u 2 , u 3 ) and random scalar pressure π , which are governed by the following stochastic initial boundary problem:

✩

*

The work is partly supported by the NNSF of China (10771089 and 10801001). Corresponding author. E-mail address: [email protected] (J. Chen).

0022-247X/$ – see front matter doi:10.1016/j.jmaa.2010.03.049

©

2010 Elsevier Inc. All rights reserved.

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

⎫ ∇ · u = 0, ⎪ ⎪   ∂ Wt ⎪ ∂u ⎪ ⎬ + u · ∇ u + ∇ π − ∇ · τ e (u ) = f (t , u ) + g (t , u ) ,⎪ ∂t ∂t  ∂ u  ⎪ ⎪ u |∂ O = 0 , = 0, ⎪ ⎪ ∂ ν ∂ O ⎪ ⎭ u |t =0 = u 0

487

(1.1)

where f (t , u ) and g (t , u ) ∂ ∂Wt t denote given random external body forces. The term W t designates a cylindrical Wiener process, ν = (ν1 , ν2 , ν3 ) represents the exterior unit normal to the boundary ∂ O , τ (e (u )) = (τi j (e (u )))3×3 is the stress tensor with respect to the velocity expressed as







τi j e(u ) = 2μ0 κ + |e|2

−α /2

e i j − 2μ1 e i j ,

i , j = 1, 2, 3,

which provides the non-Newtonian aspect of the fluid motion. Here, e (u ) = (e i j (u )) is the symmetric deformations velocity tensor:

e i j = e i j (u ) =



∂u j ∂ ui + 2 ∂xj ∂ xi 1

and

3 

|e |2 =

|e i j |2 ,

i , j =1

κ > 0 and 0 < α < 1 are constants, and μ0 > 0 and μ1 > 0 represent kinematic viscosities. If α = μ1 = 0, the random stress tensor τi j depends linearly on e i j , the non-Newtonian effect disappears, and thus (1.1) becomes the three-dimensional stochastic Navier–Stokes equations. On the other hand, if the random stress tensor τi j = 0 or the viscosities μ0 = μ1 = 0, Eq. (1.1) reduces to the stochastic Euler equations.

If g (t , u ) ≡ 0, the stochastic model (1.1) reduces to the deterministic non-Newtonian fluid motion equations for a nonlinear bipolar viscous fluid. This model was formulated by Neˇcas and Silhavý [29] based on the studies of Green and Rivlin [21,22] and Bleustein and Green [7] on multipolar continuum theories for solids and fluids. Mathematical analysis of the solutions to the deterministic equations was investigated by Bellout et al. [2–5]. The mathematical theory of stochastic differential equations have been well investigated (see, for example, [1,15,19, 25,27]). Especially, for the stochastic Navier–Stokes equations, the existence and uniqueness of weak, martingale and L 2 solutions have been obtained (see, for example, [6,8,9,12,10,11,13,14,17,18,28]). Mathematical analysis of stochastic LANS-α model is discussed in [12]. However, there is not much work concerning dynamical behaviour of the stochastic nonNewtonian fluid flow. We shall establish a functional framework which is suitable for the analysis of solutions to the stochastic model. Based on this mathematical framework, existence and uniqueness of weak solutions (in the Da Prato [15] sense) of the stochastic model is obtained. The central focus of this investigation is to show the existence of solutions, which is derived by a Galerkin method approach. In the application of a Galerkin method to the compactness of approximate solutions of deterministic fluid motion models, the compactness of the approximate solutions can be derived from the uniform boundedness from energy estimates and the compactness theorem described by Temam [31]. However, the compactness theorem to deterministic fluid flows is not suitable for the stochastic fluid flows in a probability space. Here, we obtain the compactness of the approximate solutions based on a uniqueness argument, which is developed from Breckner [8] on stochastic Navier–Stokes equations and Caraballo et al. [12] on stochastic Lagrangian-averaged Navier–Stokes equations. In their analyses [8,12], compactness of approximate solutions are derived from L 4 estimates in probability spaces and solutions are derived when the initial velocity u 0 ∈ L 4 (Ω, F0 , P ; H ). However, the presented analysis validates the compact argument in the L 3 space and derives the existence of the solution when the initial velocity u 0 ∈ L 3 (Ω, F0 , P ; H ). 2. Functional formulation and main result 2.1. Deterministic function spaces over the domain O Let us commence by introducing the function spaces used in the corresponding deterministic situations. Namely,



H = u ∈ L 2 (O )3 ; ∇ · u = 0, u |∂ O = 0 endowed with the L 2 -norm  ·  L 2 ;



 k   j    ∇ u L2 < ∞ ; H (O ) = u ∈ L (O ) ; k

3



2

3

j =0

V = v ∈ H 2 (O )3 ; ∇ · v = 0, v |∂ O =

  ∂ u  = 0 endowed with the norm u  V = u  L 2 ; ∂ ν ∂ O

V ∗ is the dual space of V under the duality pairing (−1 u ,  v ), where −1 denotes the inverse of the Laplacian  under the homogeneous Dirichlet boundary condition and (·,·) denotes the inner product of the Hilbert space H :

488

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

 (u , v ) =

u (x) · v (x) dx. O

V ∗ is the dual space of V for u

∈ V ∗ and v ∈ V .

Therefore, adopting the notation

a(u , v ) = 2μ1 (u ,  v ), N (u ) = −2μ0 ∇ ·





2 −α /2

κ + e(u )



e (u ) ,

and observing that the gradient of the pressure ∇ π is orthogonal to the divergence free vector field, the variational formulation, in the absence of the pressure π , is given by















u · ∇ u + ∇ π − ∇ · τ e (u ) , v = (u · ∇ u , v ) + N (u ), v + 2μ1 ∇ · e (u ), v 3  

= (u · ∇ u , v ) +

2μ0





2 −α /2

κ + e(u )

 

e i j (u )e i j ( v ) dx + 2μ1

i , j =1 O

  = (u · ∇ u , v ) + N (u ), v + a(u , v )

u ·  v dx O

for u ∈ H 4 (O )3 ∩ V and v ∈ V . 2.2. Probability framework Let T > 0 and {Ω, F , P } be a complete probability space equipped with a filtration {Ft ; 0  t  T }, which is a nondecreasing family of sub-σ -fields of F :

Fs ⊆ Ft for 0  s  t  T , {Ft ; 0  t  T } is right-continuous and F0 contains all the P -negligible events in F . For a Banach space Y and a real number 1  p < ∞, we denote L p (Ω, F , P ; Y ) the Banach space of all F measurable u : Ω → Y such that



p

E u Y < ∞, where E is the mathematical expectation with respect to the probability space {Ω, F , P }. Let U be a Hilbert space associated with an orthonormal basis {e i ; i = 1, 2, . . .}, and let L HS (U , H ) represent the space of all Hilbert–Schmidt operators mapping U into the space H . The function { W t ; 0  t  T } represents a cylindrical Wiener process

Wt =

∞ 

βti e i ,

i =1

where {β i ; i = 1, 2, . . .} is a sequence of mutually independent Ft -Brownian motions with respect to the probability space. In this study, the stochastic integral with respect to the cylindrical Wiener process W t is assumed the sense of Itô (see, for example, [15]). That is,

t







ϕ (s) dW s for ϕ ∈ L p Ω, F , P ; L 2 0, T ; L HS (U ; H ) , 0

which defines a P -almost surely continuous H -valued Ft -martingale, p  2. The representation of the cylindrical Wiener process implies the inner product formulation



 t

∞    t

ϕ (s) dW s , ψ =



ϕ (s)e i , ψ dβsi for ψ ∈ H ,

i =1 0

0

where the integral to β i (s) is also the Itô integral, and the series converges in L 2 (Ω, F , P ; C ([0, T ]; R )). In particular, for the Ft -progressively measurable processes







ϕ ∈ L p Ω, F , P ; L 2 0, T ; L HS (U ; H ) ,

1 p

+

1 q

= 1, p  2

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

489

and ψ ∈ L q (Ω, F , P ; L ∞ (0, T ; H )), the quantity

t



∞    t



ψ(s), ϕ (s) dW s =



ϕ (s)e i , ψ(s) dβsi

i =1 0

0

converges in L (Ω, F , P ; C ([0, T ]; R )), and defines a P -almost surely continuous Ft -martingale. In fact, convergence of the series easily follows from the Burkholder–Davis–Gundy inequality and Hölder inequality, 1

 ∞ t   T ∞ 1/2     i  2  E ϕ (s)e i , ψ(s) dβs   c E ϕ (s)e i , ψ(s) ds   i =1 0

i =1

0

 T

  ϕ (s)2

 cE

L HS (U ; H )

  ψ(s)22 ds

0

 q  E sup ψ(s) L 2 + c E

1/2

L

 T

 p /2

  ϕ (s)2

L HS (U ; H )

s∈[0, T ]

ds

.

0

For u , v ∈ V , we assume that f (·, u ) and g (·, v ) are Ft -progressively measurable, and satisfy the Lipschitz conditions

   f (t , u ) − f (t , v ) ∗  c u − v  H ,  V  g (t , u ) − g (t , v )  c u − v  H L (U ; H )

(2.1) (2.2)

HS

for u, v in H and almost all (ω, t ) ∈ Ω × (0, T ). 2.3. Weak solutions A stochastic process u is termed as a weak solution of the stochastic differential equations system (1.1), if u is

Ft -progressively measurable, the trajectories of u are P -almost surely Bochner integrable, u is weakly continuous in H , and





t

u (t ), ϕ +



t



a u (s), ϕ ds + 0



u (s) · ∇ u (s), ϕ ds +

0

t = (u 0 , ϕ ) +

 

  f s, u (s) , ϕ ds +

 t

0

for any

t



 





N u (s) , ϕ ds

0

  g s, u (s) dW (s), ϕ , 

P -a.s.

(2.3)

0

ϕ ∈ V and 0 < t  T .

2.4. Statement of the main result Theorem 2.1. Let p  3, u 0 ∈ L p (Ω, F0 , P ; H ), and







f (·, 0) ∈ L p Ω, F , P ; L 2 0, T ; V ∗ ,







g (·, 0) ∈ L p Ω, F , P ; L 2 0, T ; L HS (U ; H ) .

Then the stochastic partial differential equations system (1.1) admits a unique weak solution









u ∈ L 2 Ω, F , P ; L 2 (0, T ; V ) ∩ L p Ω, F , P ; L ∞ (0, T ; H ) satisfying the estimate

 p E sup u (t ) L 2 + E s∈[0, T ]

 cE



T

2 p −2  u L 2 u (s) V ds

0 p  u 0  L 2 + c E

 T

   f (s, 0)2 ∗ ds V

0

 2p

 T + cE

   g (s, 0)2

L HS (U ; H )

 2p ds

,

(2.4)

0

where sup denotes the essential supremum and c is a generic constant dependent only on O , Ω , T , p, κ , α , μ0 , μ1 and the Lipschitz constant appearing in (2.1), (2.2).

490

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

The proof of this result is based on the Itô formula and the Burkholder–Davis–Gundy inequality in the estimation of the Itô integral with respect to the Wiener process W t . For the reader’s convenience, we shall display these preparatory techniques in Section 3. To show the existence of the weak solution, we use the Galerkin method and derive uniform estimates of a sequence of approximate solutions as described in (2.4) in Section 5. The compactness of the sequence of approximate solutions is given in Section 6 based on a uniqueness argument. The accumulation element of the sequence is the desired weak solution. Section 7 shows a sketch of the proof of the uniqueness, since the uniqueness is implied in the proof of the compactness. 3. Preparatory lemmas The analysis of this section is essentially based on the following Itô integration properties: Lemma 3.1 (Itô formula). (See, for example, [30, p. 147].) Let F (x, y ) be a function in C 1 ( R 2 ; R ) with ∂ 2 F /∂ x2 ∈ C ( R 2 , R ), X t be a continuous semimartingale and G t be a continuous process with bounded variation in the interval [0, T ]. Then, F ( X t , G t ) is a continuous semimartingale satisfying

t F ( Xt , G t ) = F ( X 0 , G 0 ) +

∂F ( Xs, G s) d Xs + ∂x

0

t

∂F 1 ( X s , G s ) dG s + ∂y 2

0

t

∂2 F ( X s , G s ) d X s , ∂ x2

0

where X t = X t denotes the quadratic variation defined as

X t = lim

Π →0

m  ( X tk − X tk−1 )2 k =1

with respect to a partition Π = {t 0 , t 1 , . . . , tm , 0 = t 0  t 1  · · ·  tm = t }. Especially, the Itô formula reduces to the forms

t F ( Xt ) = F ( X 0 ) +



F ( Xs) d Xs +

1

t

2

0

F  ( X s ) d X s ,

(3.1)

0

when F ( X , Y ) = F ( X ) independent of Y , and

t G (t ) X t = G (0) X 0 +

t G (s) d X s +

0

X s dG (s),

(3.2)

0

when F ( X , Y ) = X Y . We shall use the following Burkholder–Davis–Gundy inequality in estimations of the Itô integrals. Lemma 3.2. (See, for example, Karatzas and Shreve [23, Theorem 3.28].) Let X be a continuous local martingale, X t∗ = max0st | X s | and q > 0. Then

E



X T∗

2q 

 q  c E X T

holds true for every stopping time T . 4. Existence of approximate solutions The Galerkin method can be employed to solve deterministic fluid motion problems (see, for example, Lions [26] and Temam [31]) through the compactness of a sequence of approximate solutions. This methods is also applicable to the stochastic fluid motion problems [6,8,9,12,17,18,28]. However, it is new for the application of the Galerkin method to the stochastic non-Newtonian problem. For reader’s convenience, a complete proof for the existence and uniqueness result based on the Galerkin method will be provided. The primary step of the proof is to show the existence of the solutions to the stochastic differential equations truncated from the stochastic partial differential equations (1.1). Lemma 4.1. Let { v m ; m = 1, 2, 3, . . .} ⊂ V is an orthonormal basis of H . Then for each m > 0, there exists an almost surely unique stochastic process um , which is a trajectory of Ft -progressively measurable and P -almost surely continuous, satisfying

um =

m    (um , v j ) v j ∈ L 2 Ω, F , P ; L 2 (0, T ; V ) , j =1

(4.1)

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

491

and the variational equation:



t





t



a um (s), v j ds +

um (t ), v j + 0



t



um (s) · ∇ um (s), v j ds +

0

t

 

= (u 0m , v j ) +



N um (s) , v j ds

0

t







 

f s, um (s) , v j ds +

0

 





j = 1, . . . , m ,

g s, um (s) dW (s), v j ,

(4.2)

0

where u 0m is the projection of u 0 described as m  (u 0 , v j ) v j .

u 0m =

j =1

The stochastic process um is said to be an approximate solution of (1.1). The proof is developed from the existence theory of finite dimensional Itô equations (see Arnold [1] and Friedman [19]). Proof. For any positive integer j, let





a j =  v j 2L 2

U j (t ) = um (t ), v j ,

and b j = 2μ1 ∇ v j 2L 2 .

We have

um (t ) =

m 

U j (t ) v j ,

j =1

and (4.2) is rewritten as

t U j (t ) + b j

t U j (s) ds +

0

t (um · ∇ um , v j ) ds +

0

t = a j U j (0) +





N (um ), v j ds

0

 

t





f s, um (s) , v j ds +

0

 





g s, um (s) dW (s), v j .

0

To derive the solution based on an iteration scheme, we set ( 0)

(n)

U j (t ) = a j U j (0),

um =

m 

(n)

U j (t ) v j

for n  0

j =1

and (n+1)

Uj

t (t ) + b j

(n+1)

Uj

t (s) ds = −

0



(n)

t



(n)

um · ∇ um , v j ds −

0

 

(n) 



N um , v j ds

0

t + a j U j (0) +

 

(n)





t

f s, um (s) , v j ds +

0

 

(n)





g s, um (s) dW (s), v j ,

(4.3)

0

and hence (n+1)

Uj

t

(n)

(t ) − U j (t ) + b j



(n+1)

Uj

 (n) (s) − U j (s) ds

0

t =−



(n)

(n)

(n−1)

um · ∇ um − um

(n−1)

· ∇ um

 , v j ds −

0

t +

 

(n) 

N um

 (n−1)   − N um , v j ds

0

 

(n)

f s, u m

0

t

  (n−1)   (s) − f s, um (s) , v j ds +

t

 

(n)





(n−1)

g s, u m (s) − g s, u m

0

  (s) dW (s), v j .

(4.4)

492

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

In order to derive required estimates by using the Itô formula, we define (n+1)

Xt = U j

(t ) − U (jn) (t ) and F ( X ) = X 2

to produce

 (n) (t ) − U j (t ) , F  ( X s ) = 2,  (n+1)  (n) (t ) − U j (t ) X t = U j   t   (n)   (n−1)   g s, u m − g s, u m dW (s), v j = 

(n+1)

F ( Xs ) = 2 Xs = 2 U j

0

=

t  ∞  

(n) 

 (n−1)  2   − g s, u m e h , v j d β h (s)

(n) 

 (n−1)  2 eh , v j ds, − g s, u m

g s, u m

0 h =1

=

t  ∞  

g s, u m

0 h =1

and, by (4.4),



(n+1)

d X s = −b j U j

(n)

(s) − U j

 (s) ds −

t



(n)

(n)

(n−1)

um · ∇ um − um

(n−1)

· ∇ um

 , v j ds

0

 (n−1)    (n−1)    (n−1)     (n)    (n)    (n)  , v j ds + f s, um − f s, um , v j ds + g s, um − g s, um − N um − N um dW (s), v j . Hence, it follows from (3.1) that



(n+1)

0 = −U j

2  (n+1) 2 (n) (n) (t ) − U j (t ) + U j (0) − U j (0) + 2

t



(n+1)

Uj

 (n) (s) − U j (s) d X s +

0

t d X s 0

or

 (n+1) 2 U (t ) − U (n) (t ) + 2b j j

t

j

 (n+1) 2 U (s) − U (n) (s) ds j

j

0

t = −2



(n)

(n−1)

(n)

um · ∇ um − um

(n−1)

· ∇ um

 (n+1)  (n) ,vj Uj (s) − U j (s) ds

0

t −2

 

(n) 

N um

 (n−1)   (n+1)  (n) − N um ,vj Uj (s) − U j (s) ds

0

t +2

 

(n) 

f s, u m

  (n−1)   (n+1) (n) (s) − U j (s) ds − f s, u m ,vj Uj

0

∞     t

+2

(n) 

g s, u m

 (n−1)   (n+1)  (n) − g s, u m eh , v j U j (s) − U j (s) dβ h (s)

h =1 0

+

t  ∞  

(n) 

g s, u m

 (n−1)  2 − g s, u m eh , v j ds.

0 h =1

Moreover, for

 (n+1) 2 (n) X t = U j (t ) − U j (t ) ,

 t   (n) 2  (n−1) 2  um (s) + um (s) ds , G (t ) = exp − V V 0

(4.5)

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

493

it follows from (3.2), (4.5) that

t 0 = −G (t ) X t + G (0) X 0 +

t G (s) d X s +

0

X s dG (s) 0

 2 = −G (t )U (jn+1) (t ) − U (jn) (t ) − 2b j

t



(n+1)

G (s)U j

2 (s) − U (jn) (s) ds

0

t −2



(n)

(n−1)

(n)

G (s) u m · ∇ u m − u m

(n−1)

· ∇ um

 (n+1)  (n) ,vj Uj (s) − U j (s) ds

0

t −2

 

(n) 

G (s) N u m

 (n−1)   (n+1)  − N um ,vj Uj (s) − U (jn) (s) ds

0

t +2

 

(n) 

G (s) f s, u m

 (n−1)   (n+1)  − f s, u m ,vj Uj (s) − U (jn) (s) ds

0

t +

G (s)

(n) 

g s, u m

 (n−1)  2 − g s, u m eh , v j ds

h =1

0

+2

∞   

∞ t 

 

(n) 

G (s) g s, u m

 (n−1)   (n+1)  − g s, u m eh , v j U j (s) − U (jn) (s) dβ h (s)

h =1 0

t −

   (n) 2  (n−1) 2  um  + um  G (s)U (n+1) (s) − U (n) (s)2 ds. j j V V

(4.6)

0

Thus, to derive a bound of the right-hand side of (4.6), we use Hölder inequality to obtain the estimate of the quadratic nonlinear term

 (n+1)  (n) (n) (n−1) (n−1) (n)  − um · ∇ um − um · ∇ um , v j U j −Uj  (n)  (n)   (n)   (n+1) (n−1)  (n−1)  (n−1) (n)  −Uj = − um · ∇ um − um , v j − um − um · ∇ um , v j U j  (n−1)   (n)

 (n+1)  (n)   (n) (n−1)  (n−1)  (n)  −Uj   c u m  V u m − u m  V  v j  L 2 + c u m  V  u m − u m  V  v j  L 2  U j  (n)  (n) 2  (n−1) 2  (n+1) (n) 2 (n−1) 2 − U j  + ca j um − um  V  u m  V + u m  V  U j m   (n)  (n) 2  (n−1) 2  (n+1) (n) 2 (n−1) 2  u m  V + u m  V  U j − U j  + ca j b i U i − U i . i =1

To derive the estimate with respect to the nonlinear term N in (4.6), we use the observation

 1       d    N ( v ) − N ( w ), u    N v σ + (1 − σ ) w , u dσ   c ∇ v − ∇ w  L 2 ∇ u  L 2   dσ 0

to produce

 (n+1)  (n)  (n−1)   (n+1)   (n)  (n)  (n)  (n−1)  ,vj Uj − N um − N um − U j  c U j − U j ∇ v j L 2 ∇ um − ∇ um  L 2 m   (n) 1  (n+1) (n) 2 (n−1) 2  b j U j −Uj  +c b i U i − U i .

4

i =1

The estimates with respect to terms involving f and g in (4.6) are obtained as follows

494

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

 

(n) 

f s, u m

 (n+1)  (n)  (n−1)   (n+1) (n)  (n)  (n−1)  − U j  c U j − U j  v j  V  u m − u m  L − f s, u m ,vj Uj

2

m   (n) 1  (n+1) (n) 2 (n−1) 2  b j U j −Uj  +c b i U i − U i ,

4

i =1

and ∞   

(n) 

g s, u m

∞   (n)   (n−1)  2   (n−1)  2  g s, u m e h − g s, u m − g s, u m eh , v j  eh  L 2 ·  v j 2L 2

h =1

h =1

  (n)   (n−1) 2  − g s, u m = a j  g s, u m L HS (U ; H )  (n)  2 ( n − 1 )  ca j um (s) − um (s) L 2  ca j

m   (n) 2 (n−1) b i U (s) − U (s) . i

i

i =1

Hence, applying the estimates with respect to the quadratic nonlinear term, N, f and g to (4.6), we have

 (n+1) 2 (n) G (t )U j (t ) − U j (t ) + b j

t



(n+1)

G (s)U j

2 (n) (s) − U j (s) ds

0

 2c (1 + a j )

t

m 

(ai + b i )

i =1

+2

∞ t 



(n−1)

G (s)U i (s) − U i (n)

2 (s) ds

0

 

(n) 

G (s) g s, u m

  (n−1)   (n+1) (n) eh , v j U j (s) − U j (s) dβ h (s). − g s, u m

(4.7)

h =1 0

We can now define the stopping time



τM =

T

(n) 2 (n−1) 2 (um  V + u m  V ) ds] < M , t (n) 2 (n−1) 2 inf{t ∈ [0, T ]: maxn1 [ 0 (um  V + um  V ) ds]  M },

T,

if maxn1 [

0

otherwise.

Observing that the last term on the right-hand side of (4.7) is a square-integrable martingale, we apply the mathematical expectation to (4.7) to derive

 (n+1) 2 (n) E G (t ∧ τ M )U j (t ∧ τ M ) − U j (t ∧ τ M ) + b j E

t ∧τ M



(n+1)

G (s)U j

2 (n) (s) − U j (s) ds

0

 c (1 + a j )

m 

t ∧τ M

(b i + 2ai ) E

i =1



(n−1)

G (s)U i (s) − U i (n)

2 (s) ds

0

for 0  t  T and a ∧ b = min{a, b}. Since e − M  G (t ∧ τ M )  1, we have

e

−M

m   (n+1) 2 E U (t ∧ τ M ) − U (n) (t ∧ τ M )  c (1 + a j ) (b i + 2ai ) E j

t ∧τ M

 (n)  U (s) − U (n−1) (s)2 ds.

j

i =1

i

i

0

This yields, after a successive iteration,

 E

n −1 n  (n+1) 2  (n) U   (1 + a j )C ∗ L t , ( s ) − U ( s ) j j 0st ∧τ M n!

max

(4.8)

where

L = ce M

m  (1 + a j )(b j + 2a j ), j =1

C ∗ = ce M

m    (b j + 2a j ) max E U (j1) (s) − U (j0) (s). j =1

E | A |2 ˘ By (4.8) and the Ceby˘ sev inequality P (| A | > ε )  ε2 , we have

0st

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

 P

max

0s T ∧τ M

 (n+1)  U (s) − U (n) (s) > j

j

(n+1)

1



2n+1

E [max0s T ∧τM |U j



P





(n+1) max U j (s) − U (jn) (s) >

0 s  T −

(s) − U (jn) (s)|2 ]

4−n−1 n +1 n −1 n

4

 (1 + a j )C ∗ Note that lim M →∞ τ M = T P -almost surely. For M > M 0 . Thus we have

L

T

.

n!

 > 0, there exists a constant M 0 > 0 such that P (τM > T −  ) 

1



2n+1

495

8(1 + a j )C ∗ (4LT )n n!

L

,

1 2

for any

n = 1, 2, . . . .

From the Borel–Cantelli lemma, we conclude that there exists an event Ω ∗ ∈ F with P (Ω ∗ ) = 1 and an integer-valued random variable N 0 (ω) such that, for every ω ∈ Ω ∗ ,





(n+1) (n) max U j (t , ω) − U j (t , ω) 

0t  T −

1 2n+1

n  N 0 (ω).

,

Consequently, we have

 (n+ p )  1 (n) max U j (t , ω) − U j (t , ω)  n , 0t  T − 2

p  1, n  N 0 (ω),

and hence the sequence



(n)

U j (t , ω); 0  t  T − 

is convergent in the supremum norm on continuous functions. Therefore, we have the existence of a continuous limit {U j (t , ω); 0  t  T −  } or the limit {U j (t , ω); 0  t < T } for all ω ∈ Ω ∗ . Let n → ∞ in (4.3), and note that the property (4.1) has already implied in the above derivation. We thus have the existence of the required continuous process



um (t , ω) =

m 



U j (t , ω) v j ; 0  t  T .

j =1

The proof of Lemma 4.1 is completed.

2

5. Uniform boundedness of the approximate solutions To prove that the approximate solution sequence {um } admits a subsequence converging to the desired weak solution, we shall show that {um } is uniformly bounded in the sense of the estimate (2.4) for p  2 whenever u 0 ∈ L p (Ω, F0 , P ; H ). The proof is essentially based on Lemmas 3.1 and 3.2. Firstly, we use the Itô formula to obtain uniform boundedness of the sequence {um } in the case of p = 2, or the uniform boundedness in the space









L 2 Ω, F , P ; L 2 (0, T ; V ) ∩ L 2 Ω, F , P ; L ∞ (0, T ; H )

whenever u 0 ∈ L 2 (Ω, F0 , P ; H ).

To do so, we let X t = (um (t ), v j ) and F ( X ) = X 2 . This gives





F  ( X s ) = 2 X s = 2 um (s), v j ,

F  ( X s ) = 2.

It follows from (4.2) that









 





d X s = −a um (s), v j ds − um (s) · ∇ um (s), v j ds − N um (s) , v j ds

        + f s, um (s) , v j ds + g s, um (s) dW (s), v j .

Observing that

⎫ (u · ∇ v , ϕ )  u L 4 ∇ v L 4 ϕ L 2  c u  V  v  V ϕ L 2 ,⎪ ⎪ ⎪ ⎪ ⎪ a(u , ϕ )  2μ1 u  V ϕ  V , ⎬ 2 c u  V  a(u , u ) for a constant c > 0, ⎪ ⎪ ⎪ N (u , ϕ )  c u  V ϕ  V , ⎪ ⎪     ⎭ f (s, u ), ϕ  c u  H ϕ  V +  f (s, 0) V ∗ ϕ  V

for ( v , u , ϕ ) ∈ V × V × V , we obtain from the definition of quadratic variation and (4.2) that

(5.1)

496

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

  X t = um (t ), v j  t t t         = − a um (s), v j ds − um (s) · ∇ um (s), v j ds − N um (s) , v j ds 0

0

t + (u 0m , v j ) +  t =

0

 

  f s, um (s) , v j ds +

0

 



g s, um (s) dW (s), v j



t

   g s, um (s) dW (s), v j

 

0



0

t  ∞  

=



2  i  d β (s)



2

g s, u m (s) e i , v j

i =1

0

t  ∞

 

=

g s, u m (s) e i , v j

ds.

i =1

0

Hence, it follows from (3.1) and the identity um (t ) =



0 = − um (t ), v j

2

2  + um (0), v j + 2

t



!m

j =1 (um (t ), v j ) v j



um (t ), v j d X s +

0

2  2  = − um (t ), v j + um (0), v j + 2

that

t d X s 0

t



      ∇ · τ e um (s) , um (s), v j v j ds

0

t



−

t

 



um (t ) · ∇ um (t ), um (t ), v j v j ds + 2

0

 

 

 

f s, um (s) , um (t ), v j v j ds

0

+2

∞ t 

 



t  ∞  

 



g s, um (s) e i , um (t ), v j v j dβ i (s) +



g s, u m (s) e i , v j

i =1 0

0

2

ds.

i =1

Taking the summation of the integer j from 1 to m and using (5.1) and the divergence free condition to get (u · ∇ u , u ) = 0, we obtain that

t

∞    um (t )22 − u 0m 22 − 2 L L

 





g s, u m (s) e i , u m (s) dβ i (s)

i =1 0

t



 a um (s), um (s) ds − 2

= −2 0

+

t



g s, u m (s) e i , v j

t

 





f s, um (s) , um (s) ds

0

2

ds

i =1 j =1

0

t  −2c

  um (s)2 ds + 2 V

0

t  −c

t

    f s, u m (s) 

  um (s) ds + V∗ V

0

  um (s)2 ds + c V

t

0

0

t

  um (s)2 ds + c V

    f s, um (s) 2 ∗ ds +

t

L

i =1

    g s, um (s) 2

L HS (U , H )

ds

0

  um (s)22 ds + c

t

L

0

t  ∞      g s, um (s) e i 22 ds 0

V

t 0

  N um (s) , um (s) ds + 2

0

t  m ∞   

 −c

 

0

   f (s, 0)

t V∗

ds + 2

   g (s, 0)2

L HS (U , H )

0

ds.

(5.2)

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

497

To estimate the local martingale expressed as the Itô integral in (5.2), we follow Breckner [8] to define the stopping time for integer M > 0



τ

um M

if supt ∈[0, T ] um (t )2L 2 +

T,

=

inf{t ∈ [0, T ]; um (t )2L 2 +

which implies (see [8])

lim P



M →∞



τMum < T = 0 or

lim

M →∞

t 0

T 0

um (s)2V ds < M ,

um (s)2V ds  M },

τMum → T

(5.3) otherwise,

P -almost surely.

Hence, the inequality (5.2) becomes

sup u

s∈[0,t ∧τ Mm ]

  um (s)22 + μ1 L

u

t ∧τ Mm



  um (s)2 ds V

0 u

t ∧τ Mm



  um (s)22 ds + c

 u 0 2L 2 + c

L

0

T +

T



   f (s, 0)2 ∗ ds V

0

∞     s

2

2 g (s, 0) L 2 (U ; H ) ds + 2

0

sup u s∈[0,t ∧τ Mm ]





g s, um (s) e i , um (s) dβ i (s).

(5.4)

i =1 0

We can now take the mathematical expectation of (5.4) to produce

E

sup u s∈[0,t ∧τ Mm ]

  um (s)22 + μ1 E L

u

t ∧τ Mm



  um (s)2 ds V

0 u

t ∧τ Mm





E u 0 2L 2

  um (s)22 ds + c E

+ cE

L

0

T +E

T



 2 2 g (s, 0) 2

L (U ; H )

ds + 2E

0

   f (s, 0)2 ∗ ds V

0

   i g s, u m (s) e i , u m (s) dβ (s) .

∞     s

sup u

(5.5)

s∈[0,t ∧τ Mm ] i =1 0

By the Burkholder–Davis–Gundy inequality and Hölder inequality, the estimate of the last term on the right-hand side of (5.5) is expressed as

 E

   i g s, u m (s) e i , u m (s) dβ (s)

∞     s

sup u

s∈[0,t ∧τ Mm ] i =1 0

  cE

um

 12   i g s, u m (s) e i , u m (s) dβ (s)

t ∧τ ∞ M    i =1

0

 t ∧τMum  cE

 2   g s, u m (s) e i , u m (s) d β i (s)

 

u t ∧τ Mm

= cE

∞   

 2 g s, um (s) e i , um (s) ds

u t ∧τ Mm

    g s , u m ( s ) 2

L HS (U , H )

0

 12

i =1

0

= cE

 12

i =1

0

 

∞   

  um (s)22 ds L

 12

498

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509





1 2

1 2

E

  um (s)22 + c E L

sup u



u

ds

0

  um (s)22 + c E L

sup

    g s, um (s) 2

L HS (U ; H )

s∈[0,t ∧τ Mm ]

E

u

t ∧τ Mm

s∈[0,t ∧τ Mm ]

T

u

t ∧τ Mm



   g (s, 0)2

L HS (U ; H )

  um (s)22 ds.

ds + c E

0

L

0

The left-hand side terms of the inequality (5.5) are finite due to the definition of the stopping time. Thus, a combination of the previous estimate and (5.5) shows

E

sup u s∈[0,t ∧τ Mm ]

  um (s)22 + 2μ1 E L

u

t ∧τ Mm



    um (s)2 ds  2E u 0 22 + c V L

0

t E

σ ∈[0,s∧τ Mum ]

0

T + cE

sup

   f (s, 0)2 ∗ ds + c E

T

L

   g (s, 0)2

L HS (U ; H )

V

0

  um (σ )22 ds

ds.

0

By the Gronwall lemma, we have

E

sup u s∈[0,t ∧τ Mm ]

  um (s)22 + E L

u

t ∧τ Mm



    um (s)2 ds  c E u 0 22 + c E V L

0

T

   f (s, 0)2 ∗ ds + c E

T

0

   g (s, 0)2

L HS (U ; H )

V

ds,

0

and so, as M → ∞,

 2 E sup um (s) L 2 + E

T

s∈[0, T ]

    um (s)2 ds  c E u 0 22 + c E V L

0

T

   f (s, 0)2 ∗ ds + c E

T

0

   g (s, 0)2

L HS (U ; H )

V

ds.

(5.6)

0

To continue the analysis, we carry out the proof of the a priori estimates with respect to p > 2. For ε > 0, we adopt a function F ∈ C 2 ( R , R ) such that



F (x) =

p

(x + ε ) 2 , when x > −ε , smooth, when x  −ε .

Applying Itô formula (3.1) to F ( X ) for the process X t = um (t )2L 2 , we have

  p  p um (t )22 + ε 2 = u 0m 22 + ε 2 + p L L

t

2

   p −2 um (s)22 + ε 2 d X s + ( p − 2) p L 8

0

t

t

L

    um (s) p2−2 a um (s), um (s) ds − p

0

t

      um (s) p2−2 N um (s) , um (s) ds L

0

      um (s) p2−2 f s, um (s) , um (s) ds + p L

2

0

t  m ∞        um (s) p2−2 g s, um (s) e i , v j 2 ds 0

∞

+

t

L

+p

ε → 0 to obtain

      um (s) p2−2 g s, um (s) e i , um (s) dβ i (s)

i =1 0

= −p

   p −4 um (s)22 + ε 2 d X s . L

0

By (5.2), the Hölder inequality and the assumptions of f , N and g, we may let ∞    um (t ) p2 − u 0m  p2 − p L L

t

p ( p − 2)  2

which is bounded by

t

i =1 0

i =1 j =1

      um (s) p2−4 g s, um (s) e i , um (s) 2 ds, L

L

(5.7)

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

t −2μ1 p

t

    um (s) p2−2 um (s)2 ds + p L V

L

0

  um (s) ds V

V∗

0

p ( p − 1)

+

     um (s) p2−2  f s, um (s) 

499

t

     um (s) p2−4  g s, um (s) 2 L

2

L HS (U , H )

  um (s)22 ds L

0

t  −2μ1 p

    um (s) p2−2 um (s)2 ds + c L V

0

t +c

    um (s) p2−2  f (s, 0) L

    um (s) p2−1 um (s) ds + c L V

  um (s) p2 ds + c

t

t

    um (s) p2−2 um (s)2 ds + c L V

0

t

  um (s) ds V

    um (s) p2−2  g (s, 0)2 L

L

0

 −μ1 p

V∗

0

t

0

+c

t

L HS (U , H )

t

    um (s) p2−2  f (s, 0)2 ∗ ds L

V

0

  um (s) p2 ds + c

t

    um (s) p2−2  g (s, 0)2 L

L

0

L HS (U , H )

ds,

(5.8)

0

where we have used the property implied in (5.6) that um ∈ L ∞ (0, T ; H ) for almost all To estimate the Itô integral, we consider the stopping time

τ

um M

=

ds

0

⎧ ⎨T,

p

if supt ∈[0, T ] um (t ) L 2 +

T 0

ω ∈ Ω.

p −2

um (s)L 2 um (s)2V ds < M ,

⎩ inf{t ∈ [0, T ]: um (t ) p +  t um (s) p −2 um (s)2 ds  M }, V 0 L2 L2

otherwise,

for a positive integer M. Similar to the stopping time given in (5.3), we have

lim P



M →∞



τMum < T = 0 or

τMum → T

lim

M →∞

P -almost surely.

Hence, applying this stopping time and then the mathematical expectation to (5.7), (5.8), we produce

E

sup u s∈[0,t ∧τ Mm ]

 cE



  um (s) p2 + μ1 p E L

 sup u

sup u

 cE

  um (s) p2−2

T

L



V

  um (s) p2−2

T

   g (s, 0)2

L HS (U ; H )

L

 ds

0

u

       um (s) p2−2 g s, um (s) e i , um (s) dβ i (s) L

s∈[0,t ∧τ Mm ] i =1 0

p  u 0  L 2 + c E

   f (s, 0)2 ∗ ds

0

∞ s 

sup

L

σ ∈[0,s∧τ Mum ]

s∈[0,t ∧τ Mm ]





(5.9)

  um (σ ) p2 ds

sup

s∈[0,t ∧τ Mm ]



+ cE

    um (s) p2−2 um (s)2 ds L V

t 0

+ cE

 0

p  u 0  L 2 + c E

+ cE

u

t ∧τ Mm

t sup 0

  um (σ ) p2 ds + c E

σ ∈[0,s∧τ Mum ]

 T

   f (s, 0)2 ∗ ds V

L

0

 2p

500

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

+

1

E

4

sup u

s∈[0,t ∧τ Mm ]

 + cE

  um (s) p2 + c E L

 T

 2p

   g (s, 0)2

L HS (U ; H )

ds

0

u

t ∧τ Mm

∞   i =1

      um (s)2p2 −4 g s, um (s) e i , um (s) 2 ds

 12 (5.10)

,

L

0

where we have used Hölder inequality and the Burkholder–Davis–Gundy inequality. The last term of (5.10) is bounded by

 t ∧τMum      um (s)2p2 −2  g s, um (s) 2 2 cE

 12

L (U , H )

L

 t ∧τMum       um (s)2p2 + um (s)2p2 −2  g (s, 0)2 2  cE

ds

L

0



L (U , H )

L

 12 ds

0



1 4

E

sup u

  um (s) p2 + c E L

s∈[0,t ∧τ Mm ]

 T + cE

t sup 0

σ ∈[0,s∧τ Mum ]

  um (σ ) p2 ds L

 2p

   g (s, 0)2

L HS (U , H )

ds

.

0

Since the definition of the stopping time ensures the finiteness of the two items on the left-hand side of the inequality (5.9), we obtain from the previous estimate and (5.9), (5.10) that

E

sup u s∈[0,t ∧τ Mm ]

  um (s) p2 + E L

E

sup σ ∈[0,s∧τ Mum ]

0



      um (s) p2−2 um (s)2 ds − c E u 0  p2 L V L

0

t c

u

t ∧τ Mm

  um (σ ) p2 ds + c E

 T

   g (s, 0)2

 2p + cE

L HS (U , H )

L

 T

0

   f (s, 0)2 ∗ ds V

 2p .

0

Applying the Gronwall inequality to the previous inequality, we have

E

sup

  um (s) p2 + E L

u s∈[0, T ∧τ Mm ]

 cE



p  u 0  L 2 + c E

 T

u

T ∧τ Mm



    um (s) p2−2 um (s)2 ds L V

0

   f (s, 0)2 ∗ ds

 2p

 T + cE

V

 2p

   g (s, 0)2

L HS (U ; H )

0

ds

,

0

and so, as M → ∞, we thus have

E



 p  u m  L ∞ ( 0, T ; H ) +

T E

    um (s) p2−2 um (s)2 ds L V

0

 cE



p  u 0  L 2 + c E

 T

   f (s, 0)2 ∗ ds V

 2p

 T + cE

0

 2p

   g (s, 0)2

L HS (U ; H )

ds

,

(5.11)

0

which shows the uniform boundedness of the approximate solution sequence {um } in the space L p (Ω, F , P ; L ∞ (0, T ; H )). 6. Compactness of the approximate solution sequence With the help the uniform boundedness (5.11), we can now carry out the proof of existence of the weak solution u which is an accumulation point of the sequence {um }. By the Hölder inequality

T E 0

 T  p−1 2  T  pp−−32      2  p −2  2 2 um (s) 2 um (s) ds  E um (s) 2 um (s) ds E um (s) V ds V L V L 0

0

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

501

and the property

L p (Ω, F , P ; Y ) ⊂ L 3 (Ω, F , P ; Y ) for any p  3 and a Banach space Y , it suffices to prove the convergence when p = 3. To do so, we see that (5.11), the Sobolev inequality, L 2 estimate and interpolation inequality (see, for example, [20]) imply the following uniform boundedness described as

T E

 3 um (s) · ∇ um (s) 22 ds  E

T

L

0

   3  um (s) 3 ∇ um (s) 6 2 ds L L

0

T  cE

2  4 3  um (s) 32 um (s) 3 2 ds V

L

0

T = cE

    um (s) 2 um (s)2 ds  c , V L

0

T E

    N um (s) 2 ∗ ds  c E

T

V

0

        κ + e um (s) 2 −α /2 e um (s) 22 ds L

0

T  cE

  um (s)2 ds  c , V

0

T E

    f s, um (s) 2 ∗ ds  c E

T

V

0

T E

  um (s)2 ds + c E V

0

    g s , u m ( s ) 2

L HS

T

   f (s, 0)2 ∗ ds  c , V

0

T ds  c E (U , H )

0

  um (s)2 ds + c E V

T

0

   g (s, 0)2

L HS (U , H )

ds  c .

0

˜ N, ˜ ˜f and g˜ and a subsequence {mi }, Hence (5.11) together with these estimates lead to the existence of elements u, B, denoted by {m} again for convenience, such that as m → ∞,

⎫ ⎪ ⎪   ⎪ ⎪ um  u weakly star in L 3 Ω, F , P ; L ∞ (0, T ; H ) , ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ u 0m → u 0 strongly in L (Ω, F0 , P ; H ), ⎪ ⎬   3 3 um · ∇ um  B˜ weakly in L 2 Ω, F , P ; L 2 (0, T ; H ) , ⎪    ⎪ ⎪ ˜ ⎪ N (u m )  N weakly in L 2 Ω, F , P ; L 2 0, T ; V ∗ , ⎪ ⎪    ⎪ 2 2 ∗ ⎪ ˜ , f (·, um )  f weakly in L Ω, F , P ; L 0, T ; V ⎪ ⎪  ⎪ ⎭ 2 2 g (·, um )  g˜ weakly in L (Ω, F , P ; L 0, T ; L HS (U ; H ) . 



weakly in L 3 Ω, F , P ; L 2 (0, T ; V ) ,

um  u

(6.1)

Passing to the limit m → ∞ in (4.2), we obtain that the equation





t

u (t ), ϕ +



t



a u (s), ϕ ds + 0

0

t = (u 0 , ϕ ) +



 ˜f (s), ϕ ds +



 B˜ (s), ϕ ds +

t



0

 t



˜ (s), ϕ ds N 

g˜ (s) dW (s), ϕ

0

(6.2)

0

holds true P -almost surely for ϕ ∈ V and t ∈ [0, T ]. Comparing (6.2) with the variational form (2.3), we need to verify that

u (t ) · ∇ u (t ) = B˜ (t ),





˜ (t ), N u (t ) = N





f t , u (t ) = ˜f (t ),





g t , u (t ) = g˜ (t ).

(6.3)

502

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

To do so, we let uˆ m (t ) =





!m

j =1 (u (t ), v j ) v j .

t

um (t ) − uˆ m (t ), v j +

From (4.2) and (6.2), we derive





a um (s) − uˆ m (t ), v j ds 0

t



=−

 um (s) · ∇ um (s) − B˜ (s), v j ds −

0

t 0

t +

 

  f s, um (s) − ˜f (s), v j ds +

 t

0



 



˜ (s), v j ds N u m (s) − N

   ˜ g s, um (s) − g (s) dW (s), v j .

 

(6.4)

0

Applying (3.1) with



F ( X ) = X 2 = um (t ) − uˆ m (t ), v j

2

to (6.4) and following the proof of (5.2), we obtain

  um (t ) − uˆ m (t )22 + 2 L

t





a um (s) − uˆ m (s), um (s) − uˆ m (s) ds 0

t = −2





um (s) · ∇ um (s) − B˜ (s), um (s) − uˆ m (s) ds

0

t −2

 

  ˜ (s), um (s) − uˆ m (s) ds + 2 N u m (s) − N

0

 t +2

t

 





f s, um (s) − ˜f (s), um (s) − uˆ m (s) ds

0

 m  ∞ t       2 g s, um (s) − g˜ (s) dW (s), um (s) − uˆ m (s) + g s, um (s) e i − g˜ (s)e i , v j ds.

 

j =1 i =1 0

0

Moreover, let

 2 X t = um (t ) − uˆ m (t ) L 2

 and

t

G (t ) = exp −c 1 t − c 1

   u (s)2 ds , V

0

where the constant c 1 is fixed later. Differentiating (6.5), we derive that



2





d X s = −4μ1 um (s) − uˆ m (s) V ds − 2 um (s) · ∇ um (s) − B˜ (s), um (s) − uˆ m (s) ds

        − 2 N um (s) − N˜ (s), um (s) − uˆ m (s) ds + 2 f s, um (s) − ˜f (s), um (s) − uˆ m (s) ds m  ∞     2      + 2 g s, um (s) − g˜ (s) dW (s), um (s) − uˆ m (s) + g s, um (s) e i − g˜ (s)e i , v j ds. j =1 i =1

Note that u (s) V um (s) − uˆ m (s) L 2 ∈ L (0, T ) is P -almost surely, since 2

t

    u (s)2 um (s) − uˆ m (s)22  um − uˆ m 2∞ L ( 0, T , H ) V L

0

t

  u (s)2 ds. V

0

It follows from (3.2) that

t 0 = −G (t ) X t + G (0) X 0 +

t G (s) d X s +

0

 2 = −G (t )um (t ) − uˆ m (t )L 2 − 2μ1

X s dG (s) 0

t 0



2

G (s)um (s) − uˆ m (s) V ds

(6.5)

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

t −2





G (s) um (s) · ∇ um (s) − B˜ (s), um (s) − uˆ m (s) ds − 2

0

t +2

 





˜ (s), um (s) − uˆ m (s) ds G (s) N u m (s) − N

0

 





G (s) f s, um (s) − ˜f (s), um (s) − uˆ m (s) ds + 2

0

+

t

503

t

 







G (s) g s, um (s) − g˜ (s) , um (s) − uˆ m (s) dW (s) 0

m  ∞ t 

 



G (s) g s, um (s) e i − g˜ (s)e i , v j

2

t ds +

j =1 i =1 0



 2   2 −c 1 − c 1 u (s) V G (s)um (s) − uˆ m (s) L 2 ds.

(6.6)

0

Since O is bounded, we may suppose that the basis { v j } of H is the set of the eigenfunctions of the bounded operator

2 : H 4 (O)3 ∩ V → H , and hence, we have

    uˆ m (s)  u (s) , V V

T and

E

  uˆ m (s) − u (s)2 ds → 0 as m → 0. V

0

Thus, to estimate the terms of (6.6), we use the Hölder inequality and the property ( w · ∇ v , v ) = 0 for w , v ∈ V to obtain the estimate of the quadratic nonlinear term









um (s) · ∇ um (s) − B˜ (s), um (s) − uˆ m (s) − u (s) · ∇ u (s) − B˜ (s), um (s) − uˆ m (s)

=















um (s) − uˆ m (s) · ∇ uˆ m (s), um (s) − uˆ m (s) + uˆ m (s) · ∇ uˆ m (s) − u (s) , um (s) − uˆ m (s)

   + uˆ m (s) − u (s) · ∇ u (s), um (s) − uˆ m (s)        c uˆ m (s) V um (s) − uˆ m (s) V um (s) − uˆ m (s) L 2         + c uˆ m (s) V + u (s) V uˆ m (s) − u (s) V um (s) − uˆ m (s) L 2   2 2 2  2 1   μ1 um (s) − uˆ m (s) V + c uˆ m (s) − u (s) V + c u (s) V um (s) − uˆ m (s) L 2 . 6

The estimate of the nonlinear term with respect to N in (6.6) is derived as

 1       d    N ( v ) − N ( w ), u    N v σ + (1 − σ ) w , u dσ    dσ 0

 c ∇ v − ∇ w L 2 ∇ u L 2 for v , w ∈ V . This observation implies that the estimate, with respect to the nonlinear term N in (6.6), is given by

 



 







˜ (s), um (s) − uˆ m (s) − N u (s) − N˜ (s), um (s) − uˆ m (s) N u m (s) − N

            = N um (s) − N uˆ m (s) , um (s) − uˆ m (s) + N uˆ m (s) − N u (s) , um (s) − uˆ m (s)   2     c ∇ um (s) − ∇ uˆ m (s) L 2 + c ∇ uˆ m (s) − ∇ u (s) L 2 ∇ um (s) − ∇ uˆ m (s) L 2   2 2  c ∇ um (s) − ∇ uˆ m (s) L 2 + c ∇ uˆ m (s) − ∇ u (s) L 2    2    c um (s) − uˆ m (s) V um (s) − uˆ m (s) L 2 + c ∇ uˆ m (s) − ∇ u (s) L 2   2 2 2 1   μ1 um (s) − uˆ m (s) V + c uˆ m (s) − u (s) V + c um (s) − uˆ m (s) L 2 . 6

Moreover, we have the estimates with respect to f and g in (6.6) defined as follows,

 





 





f s, um (s) − ˜f (s), um (s) − uˆ m (s) − f s, u (s) − ˜f (s), um (s) − uˆ m (s)

            = f s, um (s) − f s, uˆ m (s) , um (s) − uˆ m (s) + f s, uˆ m (s) − f s, u (s) , um (s) − uˆ m (s)               f s, um (s) − f s, uˆ m (s)  V ∗ +  f s, uˆ m (s) − f s, u (s)  V ∗ um (s) − uˆ m (s) V        c um (s) − uˆ m (s) L 2 + uˆ m (s) − u (s) L 2 um (s) − uˆ m (s) V   2 2 2 1   μ1 um (s) − uˆ m (s) + c uˆ m (s) − u (s) + c um (s) − uˆ m (s) 2 6

V

V

L

504

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

and m  ∞   



g s, um (s) e i − g˜ (s)e i , v j

j =1 i =1

2



∞       g s, um (s) e i − g˜ (s)e i 22 L

i =1

  2  =  g s, um (s) − g˜ (s) L (U ; H ) HS     2  2    = g s, um (s) − g s, u (s)  L (U ; H ) −  g s, u (s) − g˜ (s) L (U ; H ) HS HS   + 2 g (s, um ) − g˜ (s), g (s, u ) − g˜ (s) L (U ; H ) HS  2  2      c um (s) − u (s) L 2 − g (s, u ) − g˜ (s) L (U ; H ) ds HS   + 2 g (s, um ) − g˜ (s), g (s, u ) − g˜ (s) L (U ; H ) . HS

Hence, applying the estimates with respect to N, f , g and the quadratic nonlinear term above into (6.6), we have



2

G (t )um (t ) − uˆ m (t ) L 2 + μ1

t



2

G (s)um (s) − uˆ m (s) V ds +

0

t

 

2



G (s) g s, u (s) − g˜ (s) L (U ; H ) ds HS

0





G (s) u (s) · ∇ u (s) − B˜ (s), um (s) − uˆ m (s) ds − 2

 −2

t

0

t

 





˜ (s), um (s) − uˆ m (s) ds G (s) N u (s) − N

0

t +2

 





G (s) f s, u (s) − ˜f (s), um (s) − uˆ m (s) ds + 2

0

t

 







G (s) g s, um (s) − g˜ (s) , um (s) − uˆ m (s) dW (s) 0

t +2

  G (s) g (s, um ) − g˜ (s), g (s, u ) − g˜ (s) L

t HS (U ; H )

ds + c

0



2

G (s)uˆ m (s) − u (s) V ds

0

t − (c 1 − c )





2 



2

1 + u (s) V G (s)um (s) − uˆ m (s) L 2 ds

0

t





G (s) u (s) · ∇ u (s) − B˜ (s), um (s) − uˆ m (s) ds − 2

 −2 0

 





˜ (s), um (s) − uˆ m (s) ds G (s) N u (s) − N

0

t +2

 





G (s) f s, u (s) − ˜f (s), um (s) − uˆ m (s) ds + 2

0

t

 







G (s) g s, um (s) − g˜ (s) , um (s) − uˆ m (s) dW (s) 0

t +2

t

  G (s) g (s, um ) − g˜ (s), g (s, u ) − g˜ (s) L

0

t HS (U ; H )

ds + c



2

G (s)uˆ m (s) − u (s) V ds,

0

provided that the constant is now fixed such that

c1 > c.

(6.7)

Applying the stopping time

u M

τ defined in (5.3) and the mathematical expectation to the previous finding, we have u

 

  u   u 2   2 + μ1 E E G τ u m τ M − uˆ m τ M L

τM

u M

u

 2 G (s)um (s) − uˆ m (s) V ds + E

0 u τM

 cE 0



2

G (s)uˆ m (s) − u (s) V ds − 2E

u τM

0

τM

 

0





2

G (s) g s, u (s) − g˜ (s) L (U ; H ) ds HS



G (s) u (s) · ∇ u (s) − B˜ (s), um (s) − uˆ m (s) ds

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509 u

τM − 2E

505

u

τM

    ˜ (s), um (s) − uˆ m (s) ds + 2E G (s) N u (s) − N

0

 





G (s) f s, u (s) − ˜f (s), um (s) − uˆ m (s) ds

0 u

τM





G (s) g (s, um ) − g˜ (s), g (s, u ) − g˜ (s)

+ 2E

L HS (U ; H )

ds

0 u

τM

 







G (s) g s, um (s) − g˜ (s) , um (s) − uˆ m (s) dW (s)

+ 2E 0

= I 1,m + I 2,m + I 3,m + I 4,m + I 5,m + I 6,m .

(6.8)

We now confirm the convergence

lim

m→∞

6 

I j ,m = 0.

(6.9)

j =1

Indeed, u

τM I 1,m = c E



2

G (s)uˆ m (s) − u (s) V ds → 0 as m → ∞

0

is a straightforward procedure from the definitions of uˆ m and u ∈ L 3 (Ω, F , P ; L 2 (0, T ; V )). The convergence u

τM I 2,m = −2E





G (s) u (s) · ∇ u (s) − B˜ (s), um (s) − uˆ m (s) ds → 0 as m → ∞

0

is from the properties given in (6.1)





weakly star in L 3 Ω, F , P ; L ∞ (0, T ; H ) ,

um  u









B˜ ∈ L 2 Ω, F , P ; L 2 (0, T ; H ) ⊂ L 2 Ω, F , P ; L 1 (0, T ; H ) 3

3

3

and the properties

  = L 3 Ω, F , P ; L ∞ (0, T ; H ) ,  3 u · ∇ u ∈ L 2 Ω, F , P ; L 1 (0, T ; H ) ,



3

∗



L 2 Ω, F , P ; L 1 (0, T ; H )

since

 T E

  u (s) · ∇ u (s) 2 ds L

 32

 T E

0

 2  4 u (s) 32 u (s) 3 ds V L

 32

0

T E

    u (s) 2 u (s)2 ds. V L

0

Similarly, it is readily seen that I 3,m + I 4,m + I 5,m → 0 due to (6.1). Finally, we have u

τM

 







G (s) g s, um (s) − g˜ (s) , um (s) − uˆ m (s) dW (s) = 0

I 6,m = E 0

since u

τM

 







G (s) g s, um (s) − g˜ (s) , um (s) − uˆ m (s) dW (s) 0

is a continuous martingale.

506

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

Thus it follows from (6.8), (6.9) that u

 

  u   u 2  u   2 + lim E lim E G τ M um τ M − uˆ m τ M L

m→∞

τM

m→∞



2

G (s)um (s) − uˆ m (s) V ds = 0,

(6.10)

0

and u

τM E

 

2



G (s) g s, u (s) − g˜ (s) L (U ; H ) ds = 0. HS

(6.11)

0

Moreover, it follows from the definition of the stopping time (5.3) that

  τM τM s   2 2  2     E um (s) − uˆ m (s) V ds = E exp c 1 s + c 1 u (σ ) V dσ G (s)um (s) − uˆ m (s) V ds u

u

0

0



0



 τM  τM  2  2      E exp c 1 T + c 1 u (s) V ds G (s) um (s) − uˆ m (s) V ds u

u

0

0

u τM

 exp(c 1 T + c 1 M ) E



2

G (s)um (s) − uˆ m (s) V ds,

0

which together with (6.10) yields the result u

τM  2 lim E um (s) − uˆ m (s) V ds = 0.

m→∞

0

Furthermore, as M → ∞, we have

T lim E

m→∞

  um (s) − uˆ m (s)2 ds = 0, V

T or

lim E

m→∞

0

  um (s) − u (s)2 ds = 0. V

(6.12)

0

Similarly, Eq. (6.11) implies

T E

     g s, u (s) − g˜ (s)2 L

HS (U ; H )

ds = 0,





g t , u (t ) = g˜ (t ).

or

0

Thanks to (6.12), we can now derive the other identities expressed in (6.3). For example, for the quadratic nonlinear term, we note that, for ϕ ∈ L ∞ (Ω, F , P ; L ∞ (0, T ; H )),

 T   T           u (s) · ∇ u (s) − B˜ (s), ϕ (s) ds −  E um (s) · ∇ um (s) − B˜ (s), ϕ (s) ds E     0

0

T E





u (s) · ∇ u (s) − um (s) · ∇ um (s), ϕ (s) ds

0

T E

        u (s) + um (s) u (s) − um (s) ϕ (s) 2 ds V V V L

0

 T c E 0

     u (s)2 + um (s)2 ds V V

 12  T E 0

  u (s) − um (s)2 ds V

 12 .

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

507

As m → ∞, we have

T E





u (s) · ∇ u (s) − B˜ (s), ϕ ds = 0,

0

which implies u · ∇ u = B˜ due to the dense imbedding property









L ∞ Ω, F , P ; L ∞ (0, T ; H ) → L 3 Ω, F , P ; L ∞ (0, T ; H ) .

˜ and f (·, u ) = ˜f . Similarly, we have N (u ) = N It is obvious that the process is Ft -progressively measurable, since u is a limit of the sequence {um } and {Ft } is a right-continuous filtration. Thus the process u satisfies the variational formula (2.3) and is the desired solution. 7. Proof of the uniqueness In Section 6, we proved that um (t ) − uˆ m (t ) → 0 based on the fact um (0) − uˆ m (0) → 0. To prove the uniqueness of the weak solutions, or to verify the fact that u (t ) − v (t ) = 0, we assume that u and v are two weak solutions of (1.1) with the same initial value u (0) = v (0) = u 0 . Similar to the arguments in Section 6, we estimate the difference u (t ) − v (t ) based the Itô formula and the estimate leads u (t ) − v (t ) = 0. To do so, we use (2.3) to produce





t

u (t ) − v (t ), v j +





a u (s) − v (s), v j ds 0

t



=−

t



u (s) · ∇ u (s) − v (s) · ∇ v (s), v j ds −

0

 









N u (s) − N v (s) , v j ds

0

t +

 





t





f s, u (s) − f s, v (s) , v j ds +

0

 









g s, u (s) − g s, v (s) dW (s), v j .

0

Similar to the derivation of (6.6), we let



 2 X t =  w (t ) L 2 ,

w (t ) = um (t ) − v (t ),

t

G (t ) = exp −c 1 t − c 1

   u (s)2 ds , V

0

where the constant c 1 is to be defined. It follows from (3.2) that



2

G (t ) w (t ) L 2 + 2μ1

t



2

G (s) w (s) V ds

0

t = −2





t



G (s) u (t ) · ∇ u (t ) − v (t ) · ∇ v (t ) , w (s) ds − 2 0

 







t



G (s) f s, u (s) − f s, v (s) , w (s) ds + 2

+





 









2 



2



G (s) g s, u (s) − g s, v (s) , w (s) dW (s)

0

t





0

t +2

 

G (s) N u (s) − N v (s) , w (s) ds

0

 







2 G (s) g s, u (s) − g s, v (s)  L (U ; H ) ds − HS

0

t



c 1 1 + u (s) V G (s) w (s) L 2 ds.

(7.1)

0

By Hölder inequality and the divergence free property ( v · ∇ w , w ) = 0, to obtain the estimates involving the nonlinear terms in (7.1):

         − u (t ) · ∇ u (t ) − v (t ) · ∇ v (t ) , w (s)  c u (s) V u (s) − v (s) V u (s) − v (s) L 2  2 2  2 1   μ1  w (s) + c u (s)  w (s) 2 , 6

V

V

L

508

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

          − N u (s) − N v (s) , w (s)  c ∇ u (s) − ∇ v (s) L 2 ∇ w (s) L 2  2 2 1   μ1  w (s) V + c  w (s) L 2 , 6              f s, u (s) − f s, v (s) , w (s)   f s, u (s) − f s, v (s)  ∗  w (s) V

  2 2  μ1  w (s) V + c  w (s) L 2

V

1 6

and

      g s , u ( s ) − g s , v ( s ) 2

L HS (U ; H )

 2  c  w (s) L 2 .

Collecting terms and setting c 1 > c, we have

 2 G (t ) w (t ) L 2 + μ1

t



2

G (s) w (s) V ds

0

t −

 2   2  (c 1 − c ) 1 + u (s) V G (s) w (s) L 2 ds + 2

0

 









G (s) g s, u (s) − g s, v (s) , w (s) dW (s) 0

t 2

t

 









G (s) g s, u (s) − g s, v (s) , w (s) dW (s). 0

The right-hand side term of previous equation is a continuous square-integrable martingale. We thus have

E



sup u 0s∈[0,t ∧τ M ]

u t ∧τ M



 2  G (s) w (s) L 2 + 2μ1 E



2

G (s) w (s) V ds = 0,

0

where the stopping time is defined as

 u M

τ =

T

u (s)2V ds < M , t inf{t ∈ [0, T ]; 0 u (s)2V ds  M }, T,

if

0

Observing that lim M →∞ τ

u M

otherwise.

= T P -almost surely, we have

           u 2 u −1 u  u 2 E  w t ∧ τM = E G t ∧ τM G t ∧ τM w t ∧ τM L2 L2  exp(c 1 T + c 1 M ) E



sup u 0s∈[0,t ∧τ M ]



2 

G (s) w (s) L 2 = 0,

and u t ∧τ M



E

   w (s)2 ds = E V

0

u t ∧τ M





2

G (s)−1 G (s) w (s) V ds

0 u t ∧τ M



 exp(c 1 T + c 1 M ) E



2

G (s) w (s) V ds = 0,

0

and therefore, as M → ∞,



2

E  w (t ) L 2 = 0 and

T E

   w (s)2 ds = 0, V

0 t  T.

0

This shows the uniqueness of the weak solution and completes the proof of Theorem 2.1.

J. Chen, Z.-M. Chen / J. Math. Anal. Appl. 369 (2010) 486–509

509

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

L. Arnold, Stochastic Differential Equations: Theory and Applications, J. Wiley & Sons, New York, 1973. H. Bellout, F. Bloom, J. Neˇcas, Phenomenological behavior of multipolar viscous fluids, Quart. Appl. Math. 50 (1992) 559–583. H. Bellout, F. Bloom, J. Neˇcas, Solutions for incompressible non-Newtonian fluids, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993) 795–800. H. Bellout, F. Bloom, J. Neˇcas, Young measure-value solutions for non-Newtonian incompressible viscous fluids, Comm. Partial Differential Equations 19 (1994) 1763–1803. H. Bellout, F. Bloom, J. Neˇcas, Existence, uniqueness, and stability of solutions to the initial-boundary value problem for bipolar viscous fluids, Differential Integral Equations 8 (1995) 453–464. A. Bensoussan, Stochastic Navier–Stokes equations, Acta Appl. Math. 38 (1995) 267–304. J.L. Bleustein, A.E. Green, Dipolar fluids, Internat. J. Engrg. Sci. 5 (1967) 323–340. H. Breckner, Approximation and optimal control of the stochastic Navier–Stokes equation, Dissertation, Martin-Luther University, Halle-Wittenberg, 1999. Z. Brzezniak, M. Capinski, F. Flandoli, Stochastic Navier–Stokes equations with multiplicative noise, Stoch. Anal. Appl. 10 (1992) 523–532. M. Capinski, S. Peszat, On the existence of a solution to stochastic Navier–Stokes equations, Nonlinear Anal. 44 (2001) 141–177. M. Capinski, D. Gatarek, Stochastic equations in Hilbert spaces with application to Navier–Stokes equations in any dimension, J. Funct. Anal. 126 (1994) 26–35. T. Caraballo, J. Real, T. Taniguchi, On the existence and uniqueness of solutions to stochastic three-dimensional Lagrangian averaged Navier–Stokes equations, Proc. R. Soc. A 462 (2006) 459–479. G. Da Prato, A. Debussche, Ergodicity for the 3D stochastic Navier–Stokes equations, J. Math. Pures Appl. 82 (2003) 877–947. G. Da Prato, A. Debussche, Two-dimensional Navier–Stokes equations driven by a space-time white noise, J. Funct. Anal. 196 (2002) 180–210. G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. C. Foias, D.D. Holm, E.S. Titi, The three dimensional viscous Camassa–Holm equations, and their relation to the Navier–Stokes equations and turbulence theory, J. Dynam. Differential Equations 14 (2002) 1–35. F. Flandoli, B. Maslowski, Ergodicity of the 2-D Navier–Stokes equation under random perturbations, Comm. Math. Phys. 172 (1995) 119–141. F. Flandoli, D. Gatarek, Martingale and stationary solutions for stochastic Navier–Stokes equations, Probab. Theory Related Fields 102 (1995) 367–391. A. Friedman, Stochastic Differential Equations and Applications I and II, Academic Press, New York, 1975 and 1976. D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1977. A.E. Green, R.S. Rivlin, Simple force and stress multipoles, Arch. Ration. Mech. Anal. 16 (1964) 325–353. A.E. Green, R.S. Rivlin, Multipolar continuum mechanics, Arch. Ration. Mech. Anal. 17 (1964) 113–147. I. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus, Springer, Berlin, 1988. A.N. Kolmogorov, The local structure of turbulence in incompressible viscous fluids at very large Reynolds numbers, Dokl. Akad. Nauk SSSR 30 (1941) 301–305. N.V. Krylov, B.L. Rozovskii, Stochastic evolution equations, J. Soviet Math. 16 (1981) 1233–1277. J.L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer, Berlin, 1972. R. Mikulevicius, B.L. Rozovskii, On equations of stochastic fluid mechanics, in: T. Hilde, et al. (Eds.), Stochastics in Finite and Infinite Dimensions, Birkhäuser Boston, Boston, 2001, pp. 285–302. R. Mikulevicius, B.L. Rozovskii, Stochastic Navier–Stokes equations for turbulence flows, SIAM J. Math. Anal. 35 (2004) 1250–1310. J. Neˇcas, M. Silhavý, Multipolar viscous fluids, Quart. Appl. Math. 49 (1991) 247–265. D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer, Berlin, 1999. R. Temam, Navier–Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1977. D.C. Wilcox, Turbulence Modeling for CFD, DWC Industries, La Canada, 1998.