A stochastic mass conserved reaction-diffusion equation

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A stochastic mass conserved reaction-diffusion equation

Perla El Kettani, Danielle Hilhorst, Kai Lee

To cite this version:

Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation.

Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences,

2018. �hal-01825820�

A stochastic mass conserved reaction-diffusion equation

Perla El Kettani

^∗

Laboratoire de Math´ ematique, Analyse Num´ erique et EDP, University of Paris-Sud, F-91405 Orsay Cedex, France

Danielle Hilhorst

^∗∗

and Kai Lee

^∗∗∗

∗∗

CNRS and Laboratoire de Math´ ematique, Analyse Num´ erique et EDP, University of Paris-Sud, F-91405 Orsay Cedex, France

∗∗∗

Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan

Abstract

In this paper, we prove a well posedness result for an initial boundary value problem for a stochastic nonlocal reaction-diffusion equation with nonlinear diffusion together with a nul-flux boundary condition in an open bounded domain of R

ⁿ

with a smooth boundary. We suppose that the additive noise is induced by a Q-Brownian motion.

1 Introduction

We study the problem

(P )



 

 



 

 



∂ϕ

∂t = div(A(∇ϕ)) + f(ϕ) − 1

|D|

Z

D

f (ϕ)dx + ∂W

∂t , x ∈ D, t ≥ 0

A(∇ϕ).ν = 0, on ∂D × R

⁺

ϕ(x, 0) = ϕ

0

(x), x ∈ D

where:

• D is an open bounded set of R

ⁿ

with a smooth boundary ∂D;

• ν is the outer normal vector to ∂D;

• The initial function ϕ

₀

is such that ϕ

₀

∈ L

²

(D);

• We suppose that the nonlinear function f is a smooth function which satisfies the following properties:

(F

₁

) There exist positive constants C

₁

and C

₂

such that

f(a + b)a ≤ −C

₁

a

^2p

+ f

2

(b), |f

₂

(b)| ≤ C

2

(b

^2p

+ 1), for all a, b ∈ R (F

2

) There exist positive constants C

3

and ˜ C

3

(M ) such that

|f (s)| ≤ C

₃

|s − M|

^2p−1

+ ˜ C

₃

(M )

(F

3

) There exists a positive constant C

4

such that f

⁰

(s) ≤ C

4

. We will check in the Appendix that the function f(s) =

2p−1

X

r=0

b

_r

s

^r

with b

2p−1

< 0, p ≥ 2 satisfies the properties (F

₁

) − (F

₃

).

• We assume that A = ∇

_v

Ψ(v) : R

ⁿ

→ R

ⁿ

for some strictly convex function Ψ ∈ C

^1,1

(i.e. Ψ(v) ∈ C

¹

( R

ⁿ

) and ∇Ψ(v) is Lipschitz-continuous) satisfying







A(0) = ∇Ψ(0) = 0, Ψ(0) = 0

kD

²

Ψk

_L^∞_(Rn;R^n×n)

≤ c

1

, (1.1) for some constant c

₁

> 0. We remark that (1.1) implies that

|A(a) − A(b)| ≤ C|a − b| (1.2) for all a, b ∈ R

ⁿ

, where C is a positive constant, and that the strict convexity of Ψ implies that A is strictly monotone, namely there exists a positive constant C

0

such that

(A(a) − A(b))(a − b) ≥ C

₀

|a − b|

²

, (1.3) for all a, b ∈ R

ⁿ

.

We remark that if A is the identity matrix, the nonlinear diffusion operator − div(A(∇u)) reduces to the linear operator −∆u.

• The function W = W (x, t) is a Q-Brownian motion. More precisely, let Q be a nonnegative definite symmetric operator on L

²

(D), {e

_l

}

_l≥1

be an orthonormal basis in L

²

(D) diagonalizing Q, and {λ

_l

}

_l≥1

be the corresponding eigenvalues, so that

Qe

l

= λ

l

e

l

for all l ≥ 1. Since Q is of trace-class, it follows that Tr Q =

∞

X

l=1

hQe

_l

, e

_l

i

_L2(D)

=

∞

X

l=1

λ

_l

≤ Λ

0

, (1.4)

for some positive constant Λ

0

. We suppose furthermore that e

_l

∈ H

¹

(D) ∩ L

^∞

(D) for l = 1, 2... and that there exist positive constants Λ

₁

and Λ

₂

such that

∞

X

l=1

λ

_l

ke

_l

k

²_L∞(D)

≤ Λ

₁

, (1.5)

and

_∞

X

l=1

λ

_l

k∇e

_l

k

²_L2(D)

≤ Λ

₂

. (1.6) Let (Ω, F, P) be a probability space equipped with a filtration (F

_t

) and

{β

_l

(t)}

_l≥1

be a sequence of independent (F

_t

)-Brownian motions defined on (Ω, F, P);

the Q-Wiener process W is defined by W (x, t) =

∞

X

l=1

β

l

(t)Q

¹²

e

l

(x) =

∞

X

l=1

p λ

l

β

l

(t)e

l

(x) (1.7)

in L

²

(D). We recall that a Brownian motion β(t) is called an (F

_t

) Brownian motion if it is (F

_t

)-adapted and the increment β (t) − β(s) is independent of F

_s

for every 0 ≤ s < t.

We define : H =

v ∈ L

²

(D), Z

D

v = 0

, V = H

¹

(D) ∩ H and Z = V ∩ L

^2p

(D)

where k · k is the norm corresponding to the space H. We also define h·, ·i

_Z^∗_,Z

as the duality product between Z and its dual space Z

^∗

= V

^∗

+ L

2p

2p−1

(D) ([3], p.175).

The corresponding deterministic equation in the case of linear diffusion, when A is the identity matrix, has been introduced by Rubinstein and Sternberg [17] as a model for phase separation in a binary mixture. The well-posedness and the stabilization of the solution for large times for the corresponding Neumann problem were proved by Boussa¨ıd, Hilhorst and Nguyen [4]. They assumed that the initial function was bounded in L

^∞

(D) and proved the existence of the solution in an invariant set using a Galerkin approximation together with a compactness method.

The interfacial evolution process corresponding to a second order mass conserved Allen- Cahn equation shares many properties with the fourth order Cahn-Hilliard equation as discussed in [17]. Da Prato and Debussche proved the existence and the uniqueness of the solution of a stochastic Cahn-Hilliard equation in [6] with an additive space-time white noise.

In our work, inspired by this paper, we introduce a nonlinear stochastic heat equation, perform a change of functions in order to maintain a ”deterministic style” mass conserved equation by hiding the noise term and prove the existence of the solution in suitable Sobolev spaces similar to those in [6].

Funaki and Yokoyama [8] derive a sharp interface limit for a stochastically pertubed mass conserved Allen-Cahn equation with a sufficiently mild additive noise. This is different from the stochastic term in this paper which is not smooth.

A singular limit of a rescaled version of Problem (P) with linear diffusion has been studied

by Antonopoulou, Bates, Bl¨ omker and Karali [1] to model the motion of a droplet. How-

ever, they left open the problem of proving the existence and uniqueness of the solution,

which we address here. The problem that we study is more general then the one in [1] since

it has a nonlinear diffusion term. The proof is based on a Galerkin method together with

a monotonicity argument similar to that used in [14] for a deterministic reaction-diffusion equation, and that in [12] for a stochastic problem.

Our paper is organised as follows.

In section 2 an auxiliary problem is introduced, more precisely the nonlinear stochastic heat equation and a change of function is defined to obtain an equation without the noise term; this simplifies the use of the Galerkin method in section 3, which yields uniform bounds for the approximate solution in L

^∞

(0, T ; L

²

(Ω × D)), L

²

(Ω × (0, T ); H

¹

(D)) and in L

^2p

(Ω × (0, T ) × D). We deduce that the approximate weak solution weakly converges along a subsequence to a limits. The main problem is then to identify the limit of the elliptic term and the reaction term, which we do by means of the so-called monotonicity method.

We prove in section 4 the uniqueness of the weak solution which in turn implies the convergence of the whole sequence.

Finally, in section 5 we return to the study of the nonlinear stochastic heat equation and prove the existence and uniqueness of the solution.

2 A preliminary change of functions

We consider the Neumann boundary value problem for the stochastic nonlinear heat equa- tion

(P

₁

)



 

 



 

 



∂W

_A

∂t = div(A(∇W

_A

)) + ∂W

∂t , x ∈ D, t ≥ 0,

A(∇W

_A

).ν = 0, x ∈ ∂D, t ≥ 0,

W

_A

(x, 0) = 0, x ∈ D.

Krylov and Rozovskii [12] proved the well-posedness result for a classes of problems similar to Problem (P

1

) using a definition of solution in the distribution sense, while Gess [10]

defines a solution in the sense of L

²

(D), namely almost everywhere in D. More precisely, he defines a strong solution as follows (cf. [10], Definition 1.3).

Definition 2.1. (Strong solution) We say that W

_A

is a strong solution of Problem (P

₁

) if :

(i) W

_A

∈ L

^∞

(0, T ; L

²

(Ω × D)) ∩ L

²

(Ω × (0, T ); H

¹

(D));

(ii) W

_A

∈ L

²

(Ω; C([0, T ]; L

²

(D)));

(iii) div(A(∇W

_A

)) ∈ L

²

(Ω × (0, T ); L

²

(D));

(iv) W

A

satisfies a.s. for all t ∈ (0, T ) the problem



 

 

W

A

(t) = Z

t

0

div(A(∇W

_A

(s)))ds + W (t), in L

²

(D), A(∇W

_A

(t)).n = 0, in a suitable sense of trace on ∂D.

(2.1)

We will show in Section 5 the existence and uniqueness of the strong solution W

A

of Problem (P

₁

). Moreover we will prove that

W

_A

∈ L

^∞

(0, T ; L

^q

(Ω × D)) for all q ∈ [2, ∞). (2.2) We perform the change of functions

u(t) := ϕ(t) − W

_A

(t);

then ϕ is a solution of (P) if and only if u satisfies:

(P

2

)



 

 

 

 

 

 

 

 

∂u

∂t = div(A(∇(u + W

_A

)) − A(∇W

_A

)) + f(u + W

_A

)

− 1

|D|

Z

D

f(u + W

_A

)dx, x ∈ D, t ≥ 0,

A(∇(u + W

A

)).ν = 0, x ∈ ∂D, t ≥ 0,

u(x, 0) = ϕ

₀

(x), x ∈ D.

We remark that (P

2

) has the form of a deterministic problem; however it is stochastic since the random function W

_A

appears in the parabolic equation for u.

Definition 2.2. We say that u is a solution of Problem (P

₂

) if :

(i) u ∈ L

^∞

(0, T ; L

²

(Ω × D)) ∩ L

²

(Ω × (0, T ); H

¹

(D)) ∩ L

^2p

(Ω × (0, T ) × D);

div[A∇(u + W

A

)] ∈ L

²

(Ω × (0, T ); (H

¹

(D))

⁰

);

(ii) u satisfies almost surely the problem: for all t ∈ [0, T ]



 

 

 

 

 

 

u(t) = ϕ

₀

+ Z

t

0

div[A(∇(u + W

_A

)) − A(∇W

_A

)]ds + Z

t

0

f (u + W

_A

)

− Z

t

0

1

|D|

Z

D

f (u + W

A

)dxds, in the sense of distributions, A(∇(u + W

_A

)).ν = 0, in the sense of distributions on ∂D × R

⁺

.

(2.3)

In order to check the conservation of mass property, namely that Z

D

u(x, t)dx = Z

D

ϕ

0

(x)dx, a.s. for a.e. t ∈ R

⁺

,

we recall that Z

^∗

= V

^∗

+ L

^2p−12p

(D) and take the duality product of (2.2) with 1 for a.e. t and ω.

3 Existence of a solution of Problem (P ₂ )

The main result is the following

Theorem 3.1. There exists a unique solution of Problem (P

2

).

Proof. In this subsection we apply the Galerkin method to prove the existence of a solu- tion of Problem (P

₂

).

Denote by 0 < γ

1

< γ

2

≤ ... ≤ γ

˜k

≤ ... the eigenvalues of the operator −∆ with ho- mogeneous Neumann boundary conditions, and by w

k˜

, k ˜ = 0, ... the corresponding unit eigenfunctions in L

²

(D). Note that they are smooth functions.

Lemma 3.1. The functions {w

_j

} are an orthonormal basis of L

²

(D) and satisfy : Z

D

w

j

w

0

dx = 0 for all j 6= 0 and w

0

= 1 p |D| . Proof. We check below that

Z

D

w

_j

(x)dx = 0 for all j 6= 0. Indeed, Z

D

w

_j

dx = − 1 γ

j

Z

D

∆w

_j

dx

= − 1 γ

_j

Z

∂D

∂w

j

∂ν dx

= 0, (3.1)

which implies that Z

D

w

_j

w

₀

dx = 0 for all j 6= 0. Moreover, it is standard that the eigen- functions corresponding to different eigenvalues are orthogonal.

We look for an approximate solution of the form u

_m

(x, t) − M =

m

X

i=1

u

_im

(t)w

_i

=

m

X

i=1

hu

_m

(t), w

_i

iw

_i

,

where M = 1

|D|

Z

D

ϕ

0

(x)dx such that the function u

m

− M satisfies the equations Z

D

∂

∂t (u

_m

(x, t) − M)w

_j

dx

= − Z

D

[A(∇(u

_m

− M + W

_A

)) − A(∇(W

_A

))]∇w

_j

dx +

Z

D

f (u

_m

+ W

_A

)w

_j

− 1

|D|

Z

D

Z

D

f (u

_m

+ W

_A

)dx w

_j

dx,

(3.2) for all w

_j

, j = 1, ..., m. We remark that u

_m

(x, 0) = M +

m

X

i=1

(ϕ

₀

, w

_i

)w

_i

converges strongly

to ϕ

0

in L

²

(D) as m → ∞.

Problem (3.2) is an initial value problem for a system of m ordinary differential equations with the unknown functions u

_im

(t), i = 1, .., m so that it has a unique solution u

_m

on some interval (0, T

_m

), T

_m

> 0; in fact the following a priori estimates show that this solution is global in time.

First we remark that the contribution of the nonlocal term vanishes. Indeed for all j = 1, ..., m

− 1

|D|

Z

D

Z

D

f(u

_m

+ W

_A

(t))dx

w

_j

dx = − 1

|D| ( Z

D

f (u

_m

+ W

_A

(t))dx) × Z

D

w

_j

dx

= 0.

Therefore (3.2) reduces to the equation:

Z

D

∂

∂t (u

m

(x, t) − M )w

j

dx = − Z

D

[A(∇(u

_m

− M + W

_A

)) − A(∇(W

_A

))]∇w

_j

dx +

Z

D

f(u

_m

+ W

_A

)w

_j

dx. (3.3)

We multiply (3.3) by u

_jm

= u

_jm

(t) and sum on j = 1, ..., m:

Z

D

∂

∂t (u

_m

(x, t) − M)(u

_m

− M)dx

= − Z

D

[A(∇(u

_m

− M + W

_A

)) − A(∇(W

_A

))]∇(u

_m

− M )dx +

Z

D

f (u

_m

+ W

_A

)(u

_m

− M )dx. (3.4) Next we apply the monotonicity property of A (1.3) to bound the generalized Laplacian term, which yields

1 2

d dt

Z

D

(u

_m

− M)

²

dx ≤ −C

₀

Z

D

|∇(u

_m

− M )|

²

dx +

Z

D

f (u

_m

+ W

_A

)(u

_m

− M)dx. (3.5) Using the property (F

₁

) we deduce that

Z

D

f (u

_m

+ W

_A

(t))(u

_m

− M)dx

= Z

D

f (u

_m

− M + M + W

_A

(t))(u

_m

− M )dx

≤ Z

D

[−C

₁

(u

m

− M )

^2p

+ C

2

(M + W

A

)

^2p

(t) + 1) ]dx

≤ − Z

D

C

1

(u

m

− M )

^2p

dx + C

2

Z

D

|W

_A

(t)|

^2p

dx + ˜ C

2

(M )|D|, which we substitute in (3.5) to obtain :

1 2

d dt

Z

D

(u

_m

− M )

²

dx + C

₀

Z

D

|∇(u

_m

− M )|

²

dx + C

₁

Z

D

(u

_m

− M )

^2p

dx

≤ C

₂

Z

D

|W

_A

(t)|

^2p

dx + ˜ C

₂

(M )|D|. (3.6)

3.1 A priori estimates

In what follows, we derive a priori estimates for the function u

m

. Lemma 3.2. There exists a positive constant C such that

sup

t∈[0,T]

E Z

D

(u

_m

− M )

²

dx ≤ C , (3.7)

E Z

T

0

Z

D

|∇(u

_m

− M)|

²

dxdt ≤ C , (3.8) E

Z

_T

0

Z

D

(u

_m

− M)

^2p

dxdt ≤ C , (3.9)

E Z

T

0

Z

D

(f(u

m

+ W

A

))

2p

2p−1

dxdt ≤ C , (3.10)

E Z

T

0

k div A(∇(u

_m

+ W

_A

))k

²_(H1(D))⁰

dt ≤ C . (3.11)

Proof. Integrating (3.6) from 0 to t and taking the expectation we deduce that for all t ∈ [0, T ]

1 2 E

Z

D

(u

m

− M)

²

(t)dx + C

0

E Z

t

0

Z

D

|∇(u

_m

− M )|

²

dxds + C

1

E Z

t

0

Z

D

(u

m

− M )

^2p

dxds

≤ 1 2

Z

D

(u

_m

(0) − M )

²

dx + C

₂

E Z

t

0

Z

D

|W

_A

(t)|

^2p

dxds + ˜ C

₂

(M )|D|T

≤ 1 2

m

X

i=1

|hu

₀

, w

_j

i|

²

dx + ˜ C

₂

(M )|D|T + c

₂

T

≤ 1

2 ku

₀

− Mk

²_L2(D)

+ ˜ C

₂

(M)|D|T + c

₂

T

≤ K

where we have used (2.2).

We deduce that : E

Z

D

(u

_m

− M )

²

(t)dx ≤ 2K, for all t ∈ [0, T ], E

Z

T 0

Z

D

|∇(u

_m

− M )|

²

dxdt ≤ K C

0

,

E Z

_T

0

Z

D

(u

_m

− M )

^2p

dxdt ≤ K C

₁

. Therefore {u

_m

} is bounded independently of m in

L

^∞

(0, T, L

²

(Ω × D)) ∩ L

²

(Ω × (0, T ); H

¹

(D)) ∩ L

^2p

(Ω × (0, T ) × D)).

Using the property (F

2

) we deduce that E kf (u

_m

+ W

_A

)k

2p 2p−1

L

2p

2p−1((0,T)×D)

= E

Z

T 0

Z

D

|f (u

_m

+ W

_A

)|

^2p−12p

dxdt

≤ E Z

_T

0

Z

D

h

C

3

|u

_m

+ W

_A

− M|

^2p−1

+ ˜ C

3

(M ) i

_2p−1^2p

dxdt

≤ E Z

T

0

Z

D

h

C

3

(|u

_m

− M | + |W

_A

|)

^2p−1

dx + ˜ C

3

(M) i

_2p−1^2p

dxdt

≤ 2

2p 2p−1−1

E Z

_T

0

Z

D

C

₅

(|u

_m

− M | + |W

_A

|)

^2p−1

_2p−1^2p

dxdt + ˜ C

₅

|D|T

≤ c

3

E Z

T

0

Z

D

(|u

_m

− M |

^2p−1

)

2p 2p−1

dxdt

+c

₃

E Z

T

0

Z

D

(|W

_A

|

^2p−1

)

2p

2p−1

dxdt + ˜ C

₅

|D|T

≤ c

3

E Z

T

0

Z

D

|u

_m

− M|

^2p

dxdt +c

₃

E

Z

T 0

Z

D

|W

_A

|

^2p

dxdt + ˜ C

₅

|D|T

≤ K

₁

,

by (3.9) and (2.2), where c

3

is a positive constant.

Finally we show that the elliptic term is bounded in (H

¹

(D))

⁰

.We have that

E Z

T

0

k div A(∇(u

_m

+ W

A

))k

²_(H1(D))⁰

= E

Z

_T

0

( sup

v∈H¹,kvk_H1≤1

|hdiv(A(∇(u

_m

+ W

_A

))), vi| )

²

= E

Z

T 0

( sup

v∈H¹,kvk_H1≤1

| − Z

D

A(∇(u

_m

+ W

A

))∇v| )

²

≤ E Z

T

0

{ sup

v∈H¹,kvk_H1≤1

( Z

D

|A(∇(u

_m

+ W

_A

)|

²

)

¹²

( Z

D

|∇v|

²

)

¹²

}

²

≤ E Z

_T

0

sup

v∈H¹,kvk_H1≤1

Z

D

|A(∇(u

_m

+ W

_A

)|

²

Z

D

∇v

²

≤ E Z

T

0

Z

D

|A(∇(u

_m

+ W

A

)|

²

. (3.12)

Next we use (1.2) and (1.1) to estimate the term on the right-hand-side of (3.12) E

Z

T 0

Z

D

|A(∇(u

_m

+ W

_A

)|

²

≤ C E Z

T

0

Z

D

|∇(u

_m

+ W

_A

)|

²

≤ 2C E Z

_T

0

Z

D

|∇u

_m

|

²

+ E Z

_T

0

Z

D

|∇W

_A

|

²

≤ K

₂

.

The last line follows from the a priori estimates and the regularity of the solution of Problem (P

₁

).

Hence there exist a subsequence which we denote again by {u

_m

− M} and a function u − M ∈ L

²

(Ω × (0, T ); V ) ∩ L

^2p

(Ω × (0, T ) × D) ∩ L

^∞

(0, T ; L

²

(Ω × D)) such that

u

_m

− M * u − M weakly in L

²

(Ω × (0, T ); V ) (3.13) and L

^2p

(Ω × (0, T ) × D)

u

m

− M * u − M weakly star in L

^∞

(0, T ; L

²

(Ω × D))

(3.14) f (u

_m

+ W

_A

) * χ weakly in L

^2p−12p

(Ω × (0, T ) × D)

(3.15) div(A(∇(u

_m

+ W

A

))) * Φ weakly in L

²

(Ω × (0, T ); (H

¹

)

⁰

)

(3.16) as m → ∞.

Next, we pass to the limit as m → ∞.

To that purpose we integrate in time the equation (3.3) to obtain Z

D

(u

m

(x, t) − M )w

j

= Z

D

(u

m

(0) − M )w

j

+ Z

t

0

hdiv[A(∇(u

_m

− M + W

_A

)) − A(∇(W

_A

))], w

_j

i +

Z

t 0

Z

D

f (u

_m

+ W

_A

)w

_j

, for all j = 1, .., m. (3.17)

Let y = y(ω) be an arbitrary bounded random variable, and let ψ be an arbitrary bounded

function on (0,T). We multiply the equation (3.17) by the product yψ, integrate between

0 and T and take the expectation to deduce E

Z

T 0

Z

D

yψ(t)(u

m

(t) − M)w

j

dxdt

= E Z

T

0

Z

D

yψ(t)(u

_m

(0) − M )w

_j

dxdt +E

Z

T 0

yψ(t){

Z

t 0

hdiv[A(∇(u

_m

− M + W

A

))], w

j

i}

− E Z

T

0

yψ(t){

Z

t 0

hdiv[A(∇(W

_A

))], w

_j

i}

+E Z

T

0

yψ(t){

Z

t 0

Z

D

f(u

m

+ W

A

)w

j

dxds}dt.

(3.18) for all j=1,..,m.

Next we pass to the limit in (3.18); we only give the proof of convergence for the last term using the a priori estimates and H¨ older inequality. We have that

ψ(t) E Z

t

0

Z

D

f (u

_m

+ W

_A

)yw

_j

dxds

≤ kyk

_L^∞_(Ω)

|ψ(t)|(E Z

t

0

Z

D

|f (u

m

+ W

A

)|

^2p−12p

dxds)

2p−1 2p

(E

Z

t 0

Z

D

|w

_j

|

^2p

dxds)

^2p1

≤ kyk

_L^∞_(Ω)

kψk

_L^∞_(0,T)

C.

This shows that |ψ(t) E Z

_t

0

Z

D

f(u

_m

+ W

_A

)yw

_j

dxds| is uniformly bounded by a function belonging to L

¹

(0, T ). In addition using (3.15) we have that

ψ(t) E Z

t

0

Z

D

f (u

_m

+ W

_A

)yw

_j

dxds → ψ(t) E Z

t

0

Z

D

χyw

_j

dxds for a.e. t ∈ (0, T ). Applying Lebesgue-dominated convergence theorem we deduce that :

m→∞

lim Z

_T

0

ψ(t)dt E Z

_t

0

Z

D

f (u

_m

+ W

_A

)yw

_j

dxds

= Z

T

0

m→∞

lim ψ(t)E Z

t

0

Z

D

f (u

m

+ W

A

)yw

j

dxdsdt

= Z

_T

0

ψ(t)dt E Z

_t

0

Z

D

χyw

_j

dxds

= E Z

T

0

yψ(t)dt{

Z

t 0

Z

D

χw

j

dxds}.

Performing a similar proof for each term in (3.18), we pass to the limit by using Lebesgue-

dominated convergence theorem. This yields E

Z

T 0

Z

D

yψ(t)(u(t) − M )w

j

dxdt

= E Z

T

0

Z

D

yψ(t)(ϕ

₀

− M )w

_j

dxdt +E

Z

T 0

yψ(t){

Z

t 0

hΦ − div A(∇(W

_A

)), w

j

i}

+ E Z

T

0

yψ(t){

Z

t 0

Z

D

χw

_j

dxds}dt, for all j = 1, .., m.

(3.19) We remark that the linear combinations of w

_j

are dense in V ∩ L

^2p

(D), so that E

Z

T 0

Z

D

yψ(t)(u(t) − M) ˜ wdxdt = E Z

T

0

Z

D

yψ(t)(ϕ

₀

− M ) ˜ wdxdt + E

Z

T 0

yψ(t){

Z

t 0

hΦ − divA(∇(W

_A

)), wids}dt ˜ + E

Z

T 0

yψ(t){

Z

t 0

Z

D

χ wdxds}dt. ˜

for all ˜ w ∈ V ∩ L

^2p

(D), y ∈ L

^∞

(Ω) and ψ ∈ L

^∞

(0, T ). This implies that for a.e.

(t, ω) ∈ (0, T ) × Ω

hu(t) − M, wi ˜ = hϕ

₀

− M, wi ˜ + Z

t

0

hΦ + χ − div(A(∇W

_A

)), wids ˜ (3.20) for all ˜ w ∈ V ∩ L

^2p

(D).

Lemma 3.3. The function u is such that u ∈ C([0, T ]; L

²

(D)) a.s.

Proof.

Z ⊆ H ⊆ Z

^∗

Since u − M ∈ L

²

(0, T ; Z) a.s. and du

dt ∈ L

²

(0, T ; V

^∗

) + L

²

(0, T ; L

^2p−12p

(D)) = L

²

(0, T ; Z

^∗

) a.s., it follows by applying Lemma 1.2 p.260 in [18] that u − M ∈ C(0, T ; H) a.s.

It remains to prove that :

hΦ + χ, wi ˜ = hdiv(A(∇(u + W

A

))) + f (u + W

A

(t)), wi ˜ for all ˜ w ∈ V ∩ L

^2p

(D).

We do so by means of the monotonicity method.

3.2 Monotonicity argument

Let w be such that w − M ∈ L

²

(Ω × (0, T ); V ) ∩ L

^2p

(Ω × D × (0, T )).

Let c be a positive constant which will be fixed later. We define O

_m

= E

Z

T 0

e

^−cs

{2hdiv A(∇(u

_m

− M + W

A

)) − A(∇W

_A

)

− div A(∇(w − M + W

_A

)) − A(∇W

_A

)

, u

_m

− M − (w − M)i

_Z^∗_,Z

+2hf (u

_m

+ W

_A

) − f (w + W

_A

), u

_m

− M − (w − M)i

_Z^∗_,Z

−cku

_m

− M − (w − M )k

²

} ds

= J

1

+ J

2

+ J

3

and prove below the following result Lemma 3.4.

O

m

≤ 0.

Proof. First we estimate J

1

and apply (1.3) J

1

= E

Z

T 0

e

^−cs

{2hdiv A(∇(u

_m

− M + W

_A

))

− div A(∇(w − M + W

A

))

, u

m

− M − (w − M)i

_Z^∗_,Z

}

= −2 E Z

T

0

e

^−cs

Z

D

[A(∇(u

_m

− M + W

_A

)) − A(∇(w − M + W

_A

))]

[∇(u

_m

− M + W

_A

) − ∇(w − M + W

_A

)]

≤ −2C

₀

E Z

T

0

e

^−cs

k∇(u

_m

− w)k

²

≤ 0.

(F

₃

) and the mean value theorem yield:

J

2

= E Z

T

0

e

^−cs

2hf (u

m

+ W

A

) − f (w + W

A

), u

m

− wi

_Z^∗_,Z

ds

≤ E Z

T

0

e

^−cs

2C

₄

ku

_m

− wk

²

ds.

Choosing c ≥ 2C

₄

, we conclude the result.

We write O

_m

in the form O

_m

= O

¹_m

+ O

²_m

where O

_m¹

= E

Z

T 0

e

^−cs

{2hdiv A(∇(u

_m

− M + W

A

)) − A(∇W

_A

)

, u

m

− M i

_Z^∗_,Z

+2hf(u

_m

+ W

_A

), u

m

− Mi

_Z^∗_,Z

− cku

_m

− M k

²

}]ds.

(3.21)

We integrate the equation (3.3) between 0 and T to obtain Z

D

(u

m

(x, T ) − M )w

j

= Z

D

(u

m

(0) − M )w

j

+ Z

T

0

hdiv[A(∇(u

_m

− M + W

_A

)) − A(∇W

_A

)], w

_j

i

_Z^∗_,Z

+

Z

T 0

Z

D

f (u

m

+ W

A

)w

j

, for all j = 1, .., m. (3.22) Next we recall a chain rule formula, which can be viewed as a simplified Itˆ o’s formula.

Proposition 3.1. Let X be a real valued function such that X(t) = X(0) +

Z

t 0

h(s)ds, 0 ≤ s ≤ t,

and suppose that h is measurable in time such that h ∈ L

¹

(0, T ). Suppose that the function F : [0, T ] × R → R and its partial derivatives ∂F

∂t and ∂F

∂X are continuous on [0, T ] × R . Then for all t ∈ [0, T ]

F (t, X(t)) = F (0, X(0)) + Z

t

0

∂F

∂t (s, X(s))ds + Z

t

0

∂F

∂X (s, X(s))h(s)ds.

(3.23) Applying (3.23) to the m equations in (3.22) with

X

j

= Z

D

(u

m

− M)w

j

, j = 1, ...m, F (s, q) = e

^−cs

q

²

, and h(s) = hdiv[A(∇(u

_m

− M + W

_A

)) − A(∇W

_A

)] + f (u

_m

+ W

_A

), w

_j

i

_Z^∗_,Z

, we deduce that

e

^−cT

( Z

D

(u

_m

(x, T ) − M )w

_j

)

²

= ( Z

D

(u

m

(0) − M )w

j

)

²

− c Z

T

0

e

^−cs

( Z

D

(u

m

− M )w

j

)

²

ds +2

Z

T 0

e

^−cs

{ Z

D

(u

_m

− M )w

_j

}hdiv[A(∇(u

_m

− M + W

_A

)) − A(∇W

_A

)], w

_j

i +2

Z

T 0

e

^−cs

{ Z

D

(u

m

− M )w

j

}hf (u

m

+ W

_A

), w

j

i, for all j = 1, .., m.

(3.24) In what follows, we will use the identity

Lemma 3.5. Let F ∈ Z

^∗

and B

_m

=

m

X

j=1

hB

_m

, w

_j

iw

_j

.

Then

m

X

j=1

hF, w

_j

ihB

_m

, w

_j

i = hF, B

_m

i. (3.25)

Proof.

m

X

j=1

hF, w

_j

ihB

_m

, w

_j

i =

m

X

j=1

hF, hB

_m

, w

_j

iw

_j

i

= hF,

m

X

j=1

hB

_m

, w

j

iw

_j

i

= hF, B

_m

i.

Summing (3.24) on j = 1, .., m and applying the identity (3.25) yields e

^−cT

ku

_m

(T ) − Mk

²

= ku

_m

(0) − M k

²

− c Z

T

0

e

^−cs

ku

_m

− M k

²

ds +2

Z

T 0

e

^−cs

hdiv[A(∇(u

_m

− M + W

_A

)) − A(∇W

_A

)], u

_m

− M i

_Z^∗_,Z

+2

Z

T 0

e

^−cs

hf (u

m

+ W

A

), u

m

− Mi

_Z^∗_,Z

. (3.26) Taking the expectation of the equation (3.26) yields

E [e

^−cT

ku

_m

(T ) − Mk

²

]

= E [ku

_m

(0) − M k

²

] − c E [ Z

T

0

e

^−cs

ku

_m

(s) − Mk

²

ds]

+2 E [ Z

T

0

e

^−cs

hdiv[A(∇(u

_m

− M + W

A

)) − A(∇W

_A

)], u

m

− M i

_Z^∗_,Z

+2 E [

Z

T 0

e

^−cs

hf(u

_m

+ W

_A

), u

_m

− Mi

_Z^∗_,Z

]. (3.27) It follows from (3.21) and (3.27) that

O

¹_m

= E [e

^−cT

ku

_m

(T) − Mk

²

] − E [ku

_m

(0) − M k

²

].

From this we obtain

m→∞

lim sup O

_m¹

= E[e

^−cT

ku(T) − Mk

²

] − E[ku(0) − M k

²

] + δe

^−cT

, (3.28) where

δ = lim

m→∞

sup E[ku

_m

(T ) − M k

²

] − E[ku(T ) − M k

²

] ≥ 0.

On the other hand, the equation (3.20) implies that u(t) − M = ϕ

0

− M +

Z

t 0

Φ − div(A(∇W

_A

)) + Z

t

0

χ, ∀t ∈ [0, T ]

(3.29) a.s. in Z

^∗

= V

^∗

+ L

^2p−12p

(D).

Next we recall a second variant of the chain rule formula, which can be viewed as a simplified Itˆ o’s formula as in [15] [p.75 Theorem 4.2.5], and involves different function spaces. Consider the Gelfand triple

Z ⊂ H ⊂ Z

^∗

,

where Z = V ∩ L

^2p

(D) and Z

^∗

are defined in the introduction.

Proposition 3.2. Let X ∈ L

²

(0, T ; V ) ∩ L

^2p

(0, T ; L

^2p

(D)) and Y ∈ L

²

(0, T ; V

^∗

) + L

2p

2p−1

(0, T ; L

2p

2p−1

(D)) be such that X(t) := X

₀

+

Z

t 0

Y (s)ds, t ∈ [0, T ].

Suppose that the function F : [0, T ] × Z → R and its partial derivatives ∂F

∂t and ∂F

∂X are continuous on [0, T ] × Z. Then for all t ∈ [0, T ]

F(t, X(t)) = F(0, X(0)) + Z

t

0

∂F

∂t (s, X(s))ds + Z

t

0

hY (s), ∂F

∂X (s, X(s))i

_Z^∗_,Z

ds.

(3.30) Applying Proposition 3.2 to the equation (3.29), we set X(t) = u(t) − M, F (s, q) = e

^−cs

kqk

²

, and Y (s) = Φ − div(A(∇W

_A

)) + χ, in (3.30) to deduce that

E [e

^−cT

ku(T ) − M k

²

] = E [ku(0) − M k

²

] − c E [ Z

T

0

e

^−cs

ku(s) − Mk

²

ds]

+2 E Z

T

0

e

^−cs

hΦ − div(A(∇W

_A

)), u − Mi

_Z^∗_,Z

+2 E [

Z

T 0

e

^−cs

hχ, u − M i

_Z^∗_,Z

], which we combine with (3.28) to deduce that

m→∞

lim sup O

_m¹

= 2 E [ Z

T

0

e

^−cs

hΦ − div(A(∇W

_A

)), u − M i

_Z^∗_,Z

] +2 E

Z

T 0

e

^−cs

hχ, u − M i

_Z^∗_,Z

− c E [ Z

T

0

e

^−cs

ku(s) − M k

²

ds] + δe

^−cT

.

(3.31)

It remains to compute the limit of O

²_m

: O

_m²

= O

_m

− O

¹_m

= E

Z

_T

0

e

^−cs

{−2hdiv[A(∇(w − M + W

_A

)) − A(∇W

_A

)], u

_m

− M i

_Z^∗_,Z

−2hdiv[A(∇(u

_m

− M + W

_A

)) − A(∇W

_A

)]

− div[A(∇(w − M + W

A

)) − A(∇W

_A

)], w − M i

_Z^∗_,Z

−2hf (w + W

A

), u

m

− M i

_Z^∗_,Z

− 2hf (u

m

+ W

A

) − f (w + W

A

), w − Mi

_Z^∗_,Z

−ckw − M k

²

+ 2chu

_m

− M, w − M i}ds.

In view of (3.13), (3.15) and (3.16), we deduce that

m→∞

lim O

²_m

= E Z

_T

0

e

^−cs

{−2hdiv[A(∇(w − M + W

_A

)) − A(∇W

_A

)], u − M i

_Z^∗_,Z

−2hΦ − div(A(∇W

_A

)) − div[A(∇(w − M + W

_A

)) − A(∇W

_A

)], w − M i

_Z^∗_,Z

−2hf (w + W

A

), u − Mi

_Z^∗_,Z

− 2hχ − f (w + W

A

), w − M i

_Z^∗_,Z

−ckw − Mk

²

+ 2chu − M, w − Mi}ds. (3.32)

Combining (3.31) and (3.32), and remembering that O

_m

≤ 0, yields

E Z

T

0

e

^−cs

{2hΦ − div A∇(w − M + W

A

)), u − M − (w − M )i

_Z^∗_,Z

+2hχ − f (w + W

A

), u − M − (w − M)i

_Z^∗_,Z

−cku − M − (w − M )k

²

} + δe

^−cT

≤ 0.

Let v ∈ L

²

(Ω × (0, T ); V ) ∩ L

^2p

(Ω × (0, T ) × D) be arbitrary and set w − M = u − M − λv, with λ ∈ R

+

. We obtain the inequality :

E Z

T

0

e

^−cs

{2hΦ − div(A∇(u − λv − M + W

_A

), λvi

_Z^∗_,Z

+2hχ − f (u − λv + W

A

), λvi

_Z^∗_,Z

− ckλvk

²

}dt ≤ 0.

Dividing by λ and letting λ → 0, we find that : E

Z

T 0

e

^−cs

hΦ + χ − div(A∇(u − M + W

_A

) − f (u + W

_A

), vi

_Z^∗_,Z

dt ≤ 0.

Since v is arbitrary, it follows that E

Z

T 0

hΦ + χ, vi

_Z^∗_,Z

= E Z

T

0

hdiv[A(∇(u − M + W

_A

))] + f (u + W

_A

), vi

_Z^∗_,Z

,

for all v ∈ L

²

(Ω × (0, T ); V ) ∩ L

^2p

(Ω × (0, T ) × D), or else

Φ + χ = div[A(∇(u − M + W

_A

))] + f (u + W

_A

) + θ(t, ω), (3.33) a.s. a.e. in D × (0, T ). Taking the duality product of (3.33) with ˜ w ∈ V ∩ L

^2p

(D) we obtain that

hΦ + χ, wi ˜

_Z^∗_,Z

= hdiv[A(∇(u − M + W

A

))] + f (u + W

A

) + θ(ω, t), wi ˜

_Z^∗_,Z

= hdiv[A(∇(u − M + W

_A

))] + f (u + W

_A

), wi ˜

_Z^∗_,Z

. (3.34) Substituting (3.34) in (3.20) we deduce that for a.e. (t, ω) ∈ (0, T ) × Ω

hu(t) − M, wi ˜ = hϕ

₀

− M, wi ˜ + Z

_t

0

hdiv[A(∇(u − M + W

_A

))]

+f (u + W

_A

) − div(A(∇W

_A

)), wi ˜

_Z^∗_,Z

. (3.35) for all ˜ w ∈ V ∩ L

^2p

(D).

This completes the identification of the limit terms by the monotonicity method.

Next, we prove that u satisfies the equation (2.3) in Definition 2.2. We define V = H

¹

(D) ∩ L

^2p

(D).

The equation (3.35) implies that a.s. in V

^∗

= (H

¹

(D))

⁰

+ L

2p 2p−1

(D) u(t) = ϕ

0

+

Z

t 0

div[A(∇(u − M + W

A

))] − div(A(∇W

_A

)) + Z

t

0

f (u + W

A

)ds +

Z

t 0

λ(s)ds, (3.36)

for all t ∈ [0, T ].

In order to identify the last term of (3.36), we take its duality product h., .i

_V^∗_,V

with 1.

Remembering that the equation is mass conserved, we obtain Z

D

Z

t 0

f (u + W

_A

)dsdx + Z

t

0

λ(s)ds|D| = Z

D

u(t)dx − Z

D

ϕ

0

dx = 0. (3.37) Thus,

Z

t 0

λ(s)ds = − 1

|D|

Z

D

Z

t 0

f (u + W

A

)dsdx, so that also

λ(t) = − 1

|D|

Z

D

f (u(x, t) + W

_A

(x, t))dx.

4 Uniqueness of the solution of Problem (P ₂ )

Let ω be given such that two pathwise solutions of Problem (P

₂

), u

₁

= u

₁

(ω, x, t) and u

2

= u

2

(ω, x, t) satisfy

u

_i

(·, ·, ω) ∈ L

^∞

(0, T ; L

²

(D)) ∩ L

²

(0, T ; H

¹

(D)) ∩ L

^2p

((0, T ) × D), f (u

i

+ W

_A

) ∈ L

2p

2p−1

((0, T ) × D),

div(A(∇(u

_i

+ W

A

)) ∈ L

²

((0, T ); (H

¹

(D))

⁰

) for i = 1, 2, and u

1

(·, 0) = u

2

(·, 0) = ϕ

0

. Then

u

₁

(x, t) = u

₁

(x, 0) + Z

t

0

div(A(∇(u

₁

+ W

_A

))) − div(A(∇W

_A

)) + Z

t

0

f (u

₁

+ W

_A

)

− 1

|D|

Z

t 0

Z

D

f(u

₁

+ W

_A

)dx,

u

2

(x, t) = u

2

(x, 0) + Z

t

0

div(A(∇(u

₂

+ W

A

))) − div(A(∇W

_A

)) + Z

t

0

f (u

2

+ W

A

)

− 1

|D|

Z

t 0

Z

D

f(u

₂

+ W

_A

)dx, so that the difference u

₁

− u

₂

satisfies the equation

u

₁

(t) − u

₂

(t) = Z

_t

0

div(A(∇(u

₁

+ W

_A

) − A(∇(u

₂

+ W

_A

)) +

Z

t 0

[f (u

1

+ W

A

) − f (u

2

+ W

A

)]

− 1

|D|

Z

t 0

[ Z

D

f (u

₁

+ W

_A

) − Z

D

f (u

₂

+ W

_A

)dx], in L

²

((0, T ); V

^∗

) + L

^2p−12p

((0, T ) × D).

We take the duality product of this equation with u

₁

−u

₂

∈ L

²

((0, T ); V

^∗

) ∩L

^2p−12p

((0, T ) ×

D), to deduce that

ku

₁

− u

2

k

²_L2(D)

= 2 Z

t

0

hdiv(A(∇(u

₁

+ W

A

) − A(∇(u

₂

+ W

A

)), u

1

− u

2

i

_Z^∗_,Z

+2

Z

t 0

hf (u

₁

+ W

_A

) − f(u

₂

+ W

_A

)), u

₁

− u

₂

i

_Z^∗_,Z

−2 Z

_t

0

h 1

|D|

Z

D

(f (u

₁

+ W

_A

) − f (u

₂

+ W

_A

))dx, u

₁

− u

₂

i

_Z^∗_,Z

= −2 Z

t

0

Z

D

(A(∇(u

₁

+ W

_A

)) − A(∇(u

₂

+ W

_A

)))∇(u

₁

− u

₂

) +2

Z

t 0

Z

D

(f (u

1

+ W

A

) − f (u

2

+ W

A

))(u

1

− u

2

)dx

−2 1

|D|

Z

t 0

[ Z

D

(f (u

1

+ W

A

) − f(u

2

+ W

A

))dx Z

D

(u

1

− u

2

)dx]

= −2 Z

t

0

Z

D

(A(∇(u

₁

+ W

A

)) − A(∇(u

₂

+ W

A

)))∇(u

₁

− u

2

) +2

Z

t 0

Z

D

(f (u

1

+ W

A

) − f (u

2

+ W

A

))(u

1

− u

2

)dx, (4.1) where we remark that since

Z

D

u

1

(x, t)dx = Z

D

u

2

(x, t)dx = Z

D

ϕ

0

(x)dx, the nonlocal term vanishes.

In view of (1.3), (4.1) becomes ku

₁

− u

₂

k

²_L2(D)

≤ Z

t

0

Z

D

(f (u

₁

+ W

_A

) − f (u

₂

+ W

_A

))(u

₁

− u

₂

)dxdt − C

₀

Z

t

0

Z

D

∇(u

₁

− u

₂

)

²

dxdt, (4.2) for all t ∈ (0, T ). In addition, the property (F

₃

) implies that

(f(u

₁

+ W

_A

) − f (u

₂

+ W

_A

))(u

₁

− u

₂

) ≤ C

₄

Z

D

(u

₁

− u

₂

)

²

. (4.3) Substituting (4.3) in (4.2) yields

Z

D

(u

₁

− u

₂

)

²

(x, t)dx ≤ C

₄

Z

t

0

Z

D

(u

₁

− u

₂

)

²

(x, t)dx for all t ∈ (0, T ), which in turn implies by Gronwall’s Lemma that

u

₁

= u

₂

a.e. in D × (0, T ).

5 Existence and uniqueness of the solution of Problem (P ₁ )

In this section we return to the study of the solution W

_A

of Problem (P

₁

), and derive a pri- ori estimates for a Galerkin approximation in L

^∞

(0, T ; L

²

(Ω×D))∩L

²

(Ω×(0, T ); H

¹

(D))∩

L

²

(Ω × (0, T ); H

²

(D)) following an idea due to Gess [10]. We are then in a position to show that W

_A

is also bounded in L

^∞

(0, T ; L

^q

(Ω × D)) for all q ≥ 2, which is necessary for the proof of Lemma 3.2.

We show below a priori estimates, which imply that the elliptic term div(A(∇W

_A

)) is bounded in L

²

(D) having in mind that Problem (P

1

) is a special case of Problem (4.33) in [10] ( see also equation (2.8) in [10]). Whereas Gess concentrates on the special case of the p-Laplacian, we are interested in the uniformly parabolic case, which corresponds to m = 2 in [10] p.280-281. We also remark that there are no reaction terms i.e. f

i

= 0 for all i from 1 to n and that the noise is additive. However, Gess assumes that the nonlinear function Ψ is twice continuously differentiable while we only suppose that Ψ ∈ C

^1,1

( R

ⁿ

).

We prove the following result.

Theorem 5.1. There exists a unique solution of Problem (P

₁

).

Proof. To begin with, we approximate the function Ψ by a sufficiently smooth function Ψ

ⁿ

such that

Ψ

ⁿ

→ Ψ in C

¹

(R

ⁿ

). (5.1)

and

kD

²

Ψ

ⁿ

k

_L∞(Rⁿ,R^n×n)

≤ c

1

, ∇Ψ

ⁿ

(0) = 0,

and derive a priori estimates for a Galerkin approximation as in [10] (p. 2363 (2.13)). It turns out that the upper bounds which we find do not depend on n.

We define W

_A^m,n

by W

_A^m,n

(t) =

Z

t 0

P

_m

[div(∇Ψ

ⁿ

(∇W

_A^m,n

(s)))]ds +

m

X

l=1

P

_m

( p

λ

_l

e

_l

)β

_l

(t) (5.2)

a.s., where for v ∈ L

²

(D) P

_m

v :=

m

X

j=1

( Z

D

vw

_j

)w

_j

and P

_m

: H

¹

(D) → H

_m

= span{w

₁

, ...w

_m

}, m ∈ N is the continuous operator defined by

ka − P

_m

ak

²_H1(D)

= inf

v∈Hm

ka − vk

²_H1(D)

, a ∈ H

¹

(D) (c.f [10] p. 2363).

Note that (cf. [5] p.193)

kP

_m

ak

_H1(D)

≤ kak

_H1(D)

. (5.3) and that (cf. [10] Remark 2.3)

P

_m

a → a, in H

¹

(D) as m → ∞. (5.4)

This implies in particular that

P

_m

a → a, in L

²

(D) as m → ∞. (5.5)

In addition, we have that (cf. [12] p. 49) Z

D

u

m

P

m

[div(∇Ψ

ⁿ

(∇W

_A^m,n

))] = − Z

D

∇u

_m

∇Ψ

ⁿ

(∇W

_A^m,n

). (5.6) Indeed,

Z

D

u

_m

m

X

j=1

Z

D

div(∇Ψ

ⁿ

(∇W

_A^m,n

))w

_j

w

_j

=

m

X

j=1

Z

D

u

_m

w

_j

Z

D

div(∇Ψ

ⁿ

(∇W

_A^m,n

))w

_j

= Z

D

div(∇Ψ

ⁿ

(∇W

_A^m,n

))

m

X

j=1

Z

D

u

_m

w

_j

w

_j

= Z

D

div(∇Ψ

ⁿ

(∇W

_A^m,n

))u

_m

= −

Z

D

∇u

_m

∇Ψ

ⁿ

(∇W

_A^m,n

).

Lemma 5.1. There exists a positive constant K such that E

Z

T 0

Z

D

(W

_A^m,n

)

²

dxdt ≤ K, (5.7) E

Z

_T

0

Z

D

|∇(W

_A^m,n

)|

²

dxdt ≤ K, (5.8) E

Z

T 0

kP

_m

div(∇Ψ

ⁿ

(∇(W

_A^m,n

))k

²_L2(D)

≤ K, (5.9) sup

t∈(0,T)

E Z

D

(W

_A^m,n

)

²

dx ≤ K. (5.10) Proof. We first recall Itˆ o’s formula as in [16] p.16-17 which is based on [11] [p.153, Theo- rem 3.6], and is applicable to systems of stochastic ordinary differential equations . Lemma 5.2. For a smooth vector function h and an adapted process (g(t), t ≥ 0) with R

T

0

|g(t)|dt < ∞ almost surely, for all T > 0 set X(t) :=

Z

t 0

g(s)ds + Z

t

0

hdW(s), 0 ≤ t ≤ T,

where h is a vector of components h

_l

, l = 1, .., m and dW is a vector of components dβ

_l

, l = 1, .., m with β

_l

a one-dimensional Brownian motion. Then, for F twice continuously differentiable in X and continuously differentiable in t, one has

F (X(t), t) = F (X(0), 0) + Z

t

0

F

_t

(X(s), s) + Z

t

0

F

_x

(X(s))g(s)ds + Z

t

0

F

_x

(X(s))hdW(s) + 1

2

m

X

l=1

Z

t 0

F

_xx

(X(s))h

²_l

ds. (5.11)

Next we apply Lemma 5.2 to (5.2) with hdW =

m

X

l=1

P

_m

p

λ

l

e

l

dβ

l

(s) and h

l

= P

_m

√ λ

l

e

l

, supposing that F does not depend on time and setting

X(t) = W

_A^m,n

(t), F (X(t)) = (X(t))

²

, F

⁰

(X(t)) = 2X(t), F

⁰⁰

(X(t)) = 2,

g(s) = P

m

div(∇Ψ

ⁿ

(∇W

_A^m,n

(s))).

We remark that in this case F does not depend on t. After integrating on D, we obtain almost surely, for all t ∈ [0, T ],

Z

D

W

_A^m,n

(x, t)

²

dx = 2 Z

t

0

Z

D

W

_A^m,n

P

_m

div(∇Ψ

ⁿ

(∇W

_A^m,n

(s)))dxds +2

m

X

l=1

Z

t 0

Z

D

W

_A^m,n

P

_m

√

λ

_l

e

_l

dxdβ

_l

(s) (5.12) +

Z

_t

0 m

X

l=1

kP

_m

√

λ

_l

e

_l

k

²_L2(D)

dxds.

Substituting (5.6) into (5.12) we obtain, kW

_A^m,n

(t)k

²_L2(D)

+ 2

Z

t 0

Z

D

∇W

_A^m,n

∇Ψ

ⁿ

(∇W

_A^m,n

(s))dxds

= 2

m

X

l=1

Z

t 0

Z

D

W

_A^m,n

P

_m

√

λ

_l

e

_l

dxdβ

_l

(s) + Z

t

0 m

X

l=1

kP

_m

√

λ

_l

e

_l

k

²_L2(D)

dxds.

(5.13) Taking the expectation, we obtain

E kW

_A^m,n

(t)k

²_L2(D)

+ 2 E Z

_t

0

Z

D

∇W

_A^m,n

∇Ψ

ⁿ

(∇W

_A^m,n

(s))dxds

= E

Z

t 0

m

X

l=1

kP

_m

√

λ

_l

e

_l

k

²_L2(D)

dxds,

(5.14) where we have used the fact that 2 E [

m

X

l=1

Z

t 0

Z

D

W

_A^m,n

√

λ

_l

e

_l

dxdβ

_l

(s)] = 0 ([13] Theorem 2.3.4 - p.11).

We deduce from (5.3) that

m

X

l=1

kP

_m

√

λ

_l

e

_l

k

²_L2(D)

≤

m

X

l=1

(k √

λ

_l

e

_l

k

²_L2(D)

+ k∇( √

λ

_l

e

_l

)k

²_L2(D)

)

≤ (Λ

0

+ Λ

2

). (5.15)