APPROXIMATION OF THE SOLUTION OF

THE STOCHASTIC NAVIER-STOKES EQUATION

Hannelore Breckner

ABSTRACT

We consider the stochastic Navier-Stokes equation containing a noise part given by a stochastic integral with respect to a Wiener process. The purpose of this paper is to approximate the solution of this nonlinear equation by the solutions of a sequence of more simple equations. We prove the convergence in mean square.

Key words: Stochastic Navier-Stokes equation, Ito integral, linearization AMS subject classification: 60H15, 60H30

1. INTRODUCTION

The Navier-Stokes equation has important physical and tehnical applications. It describes the behavior of a viscous velocity field of an incompressible liquid. The equation on the domain of flow G ⊂ IR^{n} (n ≥ 2 a natural number) is given by

\frac{∂u}{∂t} − νΔu = −(u, ∇)u + f − ∇p + C(u)\frac{∂w}{∂t} (1)

divu = 0, u(0, x) = u_{0}(x), u(t, x) |_{∂G}= 0, t > 0, x ∈ G,

where u is the velocity field, ν is the viscosity, Δ is the Laplacian, ∇ is the gradient, f is an external force, p is the pressure, u_{0} is the initial condition.

This equation can be written equivalently in the form of an evolution equation, using certain vector spaces and operators. Such equations have been investigated for example from Benssousan, Temam [1], Brzeźniak, Capiński, Flandoli [2], Schmalfuß [9], Fursikov, Komech and Visik [3]. In the study of evolution equations of Navier-Stokes type one can consider weak solutions of martingal type or strong solutions.

In this paper we consider strong solutions of the Navier-Stokes equation. Since this equation is not linear, it is difficult to study existence, uniqueness and approximation for its solution. The aim of this paper is to approximate the solution of this nonlinear equation by the solutions of a sequence of linear equations (see equations (P_{n})), which are more easier to study. We prove the convergence in mean square (see Theorem 2.3). In the fourth section we study the existence and uniqueness of the solutions of this linear equations. Here we apply results and methods used for stochastic evolution equations given in [4], [6], [8].

2. FORMULATION OF THE PROBLEM

First we give the assumptions about the stochastic evolution equation which will be considered in the following.

(i) (V, H, V^{∗}) is an evolution triple, i.e. (V, ∥ · ∥_{V}) and (V^{∗}, ∥ · ∥_{∗}) are separable reflexive Banach spaces, (H, ∥ · ∥) is a separable Hilbert space, and V^{∗} is the dual space of V (see [13], p. 416). We assume that the embedding operator V ⊂ H is compact. ⟨v^{∗}, v⟩ denotes the application of v^{∗} ∈ V^{∗} to v ∈ V.

(ii) A : V → V^{∗} is a linear operator such that ⟨Av, v⟩ ≥ ν∥v∥_{V}^{2} , where ν > 0 is a constant.

(iii) B : V × V → V^{∗} is a bilinear operator such that

⟨B(u, v), v⟩ = 0 for all u, v ∈ V

and there exists a positive constant b > 0 such that

|⟨B(u, v), z⟩|^{2} ≤ b∥z∥_{V}^{2} ∥u∥∥u∥_{V} ∥v∥∥v∥_{V} .

(iv) C : V → H is a mapping with C(0) = 0 and

∥C(u) − C(v)∥^{2} ≤ λ∥u − v∥^{2}

for all u, v ∈ V, where λ is a positive constant.

(v) f is a progressively measurable H-valued process such that E∫_{0}^{T}∥f(s)∥^{4}ds < ∞.

(vi) (Ω, F, P) is a complete probability space and (F_{t})_{t∈[0,T]} a right continuous filtration such that F_{0} contains all F-null sets. (w(t))_{t∈[0,T]} denotes a real valued standard F_{t}-Wiener process.

(vii) x_{0} is a H-valued random variable independent of w such that E∥x_{0}∥^{4} < ∞.

On the evolution triple V ⊂ H ⊂ V^{∗} we consider the stochastic Navier-Stokes equation

(2)  (u(t), v) + ∫_{0}^{t}⟨Au(s), v⟩ds = (x_{0}, v) + ∫_{0}^{t}⟨B(u(s), u(s)), v⟩ds +

+ ∫_{0}^{t}(f(s), v)ds + ∫_{0}^{t}(Cu(s), v)dw(s)

for all v ∈ V, t ∈ [0, T] and a.e. ω ∈ Ω, where the stochastic integral is understood in the Ito sense. ^{1}

1As usual in the notation of random variables or stochastic processes we generally omit to write the dependence of ω.

Lemma 2.1. Equation (2) has a solution in the space L_{V}^{2} (Ω × [0, T]). The solution is almost sure unique. ^{2}

Details for the proof of this lemma can be found in [3], [5] and [12].

For each natural number n ∈ {1, 2, 3, …} we consider the sequence of linear evolution equations with additive noise

(P_{n})  (u_{n}(t), v) + ∫_{0}^{t}⟨Au_{n}(s), v⟩ds = (x_{0}, v) + ∫_{0}^{t}⟨B(u_{n−1}(s), u_{n}(s)), v⟩ds +

+ ∫_{0}^{t}(f(s), v)ds + ∫_{0}^{t}(Cu_{n−1}(s), v)dw(s),

for all v ∈ V, t ∈ [0, T] and a.e. ω ∈ Ω, where u_{0}(t) = 0 for all t ∈ [0, T] and a.e. ω ∈ Ω.

Lemma 2.2. For each n ≥ 1 equation (P_{n}) has a solution u_{n} ∈ L_{V}^{2} (Ω × [0, T]). The solution is almost sure unique. Proof. We can made the proof by induction. We also use the estimations of Lemma 3.1, as soon as the results from Lemma 4.1 and Lemma 4.4 applied for φ = u_{n−1}. □

The purpose of this paper is to prove that the sequence (u_{n})_{n=1,2,...} approximates the solution u of the Navier-Stokes equation in mean square. The main result of this paper is the following theorem:

Theorem 2.3. The following convergences hold:

E∫_{0}^{T}∥u_{n}(s) − u(s)∥_{V}^{2} ds → 0, for n → ∞

and for all t ∈ [0, T]

E∥u_{n}(t) − u(t)∥^{2} → 0, for n → ∞.

Remarks. 1) In equation (P_{n}) the operator B depends linear on the solution u_{n} and the term containing the Ito integral does not depend on u_{n}.

2) The condition C(0) = 0 is given only to simplify the calculations. It can be omitted and then one can use the estimation ∥C(u)∥^{2} ≤ 2λ∥u∥^{2} + 2∥C(0)∥^{2} which follows from the Lipschitz condition.

3) If we set V = {u ∈ Ẇ_{2}^{1}(G) : divu = 0}, H = V̄^{L^{2}(G)},

⟨Au, v⟩ = ∫_{G} ∑_{i=1}^{n} \frac{∂u}{∂x_{i}} \frac{∂v}{∂x_{i}} dx, ⟨B(u, v), z⟩ = ∫_{G} ∑_{i,j=1}^{n} u_{i} \frac{∂v_{j}}{∂x_{i}} z_{j}dx u, v, z ∈ V

then equation (1) can be transformed into (2) (see [10]).

2L_{V}^{2} (Ω × [0, T]) denotes the Banach space of all processes u : Ω × [0, T] → V which are F × B_{[0,T]} - measurable, adapted to the filtration (F_{t}) and E∫_{0}^{T}∥u(t)∥_{V}^{2} < ∞.

3. PROOF OF THE MAIN RESULT

Let introduce

y(t) = exp {−λt − 2β ∫_{0}^{t}∥u(s)∥_{V}^{2} ds}

for all t ∈ [0, T] and a.e. ω ∈ Ω, where β = \frac{b^{2}}{2ν^{2}} . Obviously

y(t) = − ∫_{0}^{t}λy(s) + 2β∥u(s)∥_{V}^{2} y(s)ds

for all t ∈ [0, T] and a.e. ω ∈ Ω. We introduce the following notations:

s_{N}(t) = ∑_{n=1}^{N} y(t)∥u_{n}(t) − u(t)∥^{2}, (3)

S_{N}(t) = ∑_{n=1}^{N} y(t)∥u_{n}(t) − u(t)∥_{V}^{2} , (4)

where N is a natural number, t ∈ [0, T], ω ∈ Ω .

First we establish some properties for the solutions of the equations (2) and (P_{n}) (n = 1, 2, …).

Lemma 3.1. There exists a positive constant c (depending only on λ, ν and T) such that each of the following expressions

sup_{t∈[0,T]} E∥u(t)∥^{4}, E( ∫_{0}^{T}∥u(s)∥_{V}^{2} ds)^{2}, sup_{t∈[0,T]} E∥u_{n}(t)∥^{4}, E( ∫_{0}^{T}∥u_{n}(s)∥_{V}^{2} ds)^{2}

(n = 1, 2, …) is less or equal than c[E∥x_{0}∥^{4} + E∫_{0}^{T}∥f(s)∥^{4}ds].

For each n = 1, 2, … we have E sup_{t∈[0,T]} ∥u_{n}(t)∥^{2} < ∞.

Proof. Let z(t) = e^{−(6λ+4)t}. For all t ∈ [0, T] we have z(t) = − ∫_{0}^{t}(6λ + 4)z(s)ds.

From (2), the Ito formula and (iii) we have

z(t)∥u(t)∥^{2} + 2 ∫_{0}^{t}z(s)⟨Au(s), u(s)⟩ds = ∥x_{0}∥^{2} + 2 ∫_{0}^{t}z(s)(f(s), u(s))ds + (5)

+ ∫_{0}^{t}z(s)∥Cu(s)∥^{2}ds − (6λ + 4) ∫_{0}^{t}z(s)∥u(s)∥^{2}ds + 2 ∫_{0}^{t}z(s)(Cu(s), u(s))dw(s)

and

z(t)∥u(t)∥^{4} + 4 ∫_{0}^{t}z(s)⟨Au(s), u(s)⟩∥u(s)∥^{2}ds = ∥x_{0}∥^{4} + 4 ∫_{0}^{t}(Cu(s), u(s))^{2}ds + (6)

+ 2 ∫_{0}^{t}z(s)∥Cu(s)∥^{2}∥u(s)∥^{2}ds − (6λ + 4) ∫_{0}^{t}z(s)∥u(s)∥^{4}ds +
+ 4 ∫_{0}^{t}z(s)(f(s), u(s))∥u(s)∥^{2}ds + 4 ∫_{0}^{t}z(s)(Cu(s), u(s))∥u(s)∥^{2}dw(s).

Using the properties of A and C and making some elementar calculations, we obtain

(7)  z(t)∥u(t)∥^{2} + 2ν ∫_{0}^{t}z(s)∥u(s)∥_{V}^{2} ds ≤ ∥x_{0}∥^{2} + ∫_{0}^{t}z(s)∥f(s)∥^{2}ds +
+ 2 ∫_{0}^{t}z(s)(Cu(s), u(s))dw(s)

and

(8)  z(t)∥u(t)∥^{4} + 4ν ∫_{0}^{t}z(s)∥u(s)∥_{V}^{2} ∥u(s)∥^{2}ds ≤ ∥x_{0}∥^{4} + ∫_{0}^{t}z(s)∥f(s)∥^{4}ds +
+ 4 ∫_{0}^{t}z(s)(Cu(s), u(s))∥u(s)∥^{2}dw(s).

In (7) we square both sides of the inequality, use some wellknown inequalities and obtain

(9) z^{2}(t)∥u(t)∥^{4} + 4ν^{2}( ∫_{0}^{t}z(s)∥u(s)∥_{V}^{2} ds)^{2} ≤ 3∥x_{0}∥^{4} + 3( ∫_{0}^{t}z(s)∥f(s)∥^{2}ds)^{2} +
+ 12 | ∫_{0}^{t}z(s)(Cu(s), u(s))dw(s) |^{2} .

By taking the mathematical expectation in (8) and (9) we get

Ez(t)∥u(t)∥^{4} + 4νE ∫_{0}^{t}z(s)∥u(s)∥_{V}^{2} ∥u(s)∥^{2}ds ≤ E∥x_{0}∥^{4} + E ∫_{0}^{t}z(s)∥f(s)∥^{4}ds,

(10) Ez^{2}(t)∥u(t)∥^{4} + 4ν^{2}E( ∫_{0}^{t}z(s)∥u(s)∥_{V}^{2} ds)^{2} ≤ 3E∥x_{0}∥^{4} +
+ 3E( ∫_{0}^{t}z(s)∥f(s)∥^{2}ds)^{2} + 12E | ∫_{0}^{t}z(s)(Cu(s), u(s))dw(s) |^{2} .

From the properties of z it follows that

E∥u(t)∥^{4} + 4νE ∫_{0}^{t}∥u(s)∥_{V}^{2} ∥u(s)∥^{2}ds ≤ e^{(6λ+4)t}[E∥x_{0}∥^{4} + E ∫_{0}^{t}∥f(s)∥^{4}ds] (11)

for all t ∈ [0, T].

We apply in (10) the Burkholder inequality (see [4], p. 24) and the Schwarz inequality. Then we obtain

Ez^{2}(t)∥u(t)∥^{4} + 4ν^{2}E( ∫_{0}^{t}z(s)∥u(s)∥_{V}^{2} ds)^{2} ≤ 3E∥x_{0}∥^{4} + 3tE ∫_{0}^{t}z(s)^{2}∥f(s)∥^{4}ds + (12)

+ 12E ∫_{0}^{t}z^{2}(s)(Cu(s), u(s))^{2}ds ≤ 3E∥x_{0}∥^{4} + 3tE ∫_{0}^{t}∥f(s)∥^{4}ds + 12λE ∫_{0}^{t}∥u(s)∥^{4}ds.

From (11) and (12) it follows that there exists a constant c_{1} (depending on λ and T) such that

sup_{t∈[0,T]} E∥u(t)∥^{4} + 4νE∫_{0}^{T}∥u(s)∥_{V}^{2} ∥u(s)∥^{2}ds ≤ c_{1}[E∥x_{0}∥^{4} + E∫_{0}^{T}∥f(s)∥^{4}ds],

E∫_{0}^{T}∥u(s)∥^{4}ds ≤ T sup_{t∈[0,T]} E∥u(t)∥^{4} ≤ c_{1}T [E∥x_{0}∥^{4} + E∫_{0}^{T}∥f(s)∥^{4}ds]

and

sup_{t∈[0,T]} E∥u(t)∥^{4} + 4ν^{2}E(∫_{0}^{T}∥u(s)∥_{V}^{2} ds)^{2} ≤ c_{1}[E∥x_{0}∥^{4} + E∫_{0}^{T}∥f(s)∥^{4}ds].

Now we derive the estimations for u_{n}, where n ≥ 1 is arbitray fixed.

Let z̃(t) = e^{−(9λ+4)t}. For all t ∈ [0, T] we have

z̃(t) = − ∫_{0}^{t}(9λ + 4)z̃(s)ds and e^{−(9λ+4)T} ≤ z̃(t) ≤ 1.

From the Ito formula we have

z̃(t)∥u_{n}(t)∥^{2} + 2 ∫_{0}^{t} z̃(s)⟨Au_{n}(s), u_{n}(s)⟩ds = ∥x_{0}∥^{2} + 2 ∫_{0}^{t} z̃(s)(f(s), u_{n}(s))ds +

+ ∫_{0}^{t} z̃(s)∥Cu_{n−1}(s)∥^{2}ds − (9λ + 4) ∫_{0}^{t} z̃(s)∥u_{n}(s)∥^{2}ds + 2 ∫_{0}^{t} z̃(s)(Cu_{n−1}(s), u_{n}(s))dw(s)

and

z̃(t)∥u_{n}(t)∥^{4} + 4 ∫_{0}^{t} z̃(s)⟨Au_{n}(s), u_{n}(s)⟩∥u_{n}(s)∥^{2}ds = ∥x_{0}∥^{4} +

+ 2 ∫_{0}^{t} z̃(s)∥Cu_{n−1}(s)∥^{2}∥u_{n}(s)∥^{2}ds − (9λ + 4) ∫_{0}^{t} z̃(s)∥u_{n}(s)∥^{4}ds +
+ 4 ∫_{0}^{t} z̃(s)(f(s), u_{n}(s))∥u_{n}(s)∥^{2}ds + 4 ∫_{0}^{t}(Cu_{n−1}(s), u_{n}(s))^{2}ds +
+ 4 ∫_{0}^{t} z̃(s)(Cu_{n−1}(s), u_{n}(s))∥u_{n}(s)∥^{2}dw(s).

Using the properties of A and C and making some elementar calculations, we obtain

z̃(t)∥u_{n}(t)∥^{2} + 2ν ∫_{0}^{t} z̃(s)∥u_{n}(s)∥_{V}^{2} ds + 6λ ∫_{0}^{t} z̃(s)∥u_{n}(s)∥^{2}ds ≤ (13)

≤ ∥x_{0}∥^{2} + ∫_{0}^{t} z̃(s)∥f(s)∥^{2}ds + λ ∫_{0}^{t} z̃(s)∥u_{n−1}(s)∥^{2}ds +
+ 2 ∫_{0}^{t} z̃(s)(Cu_{n−1}(s), u_{n}(s))dw(s)

and

z̃(t)∥u_{n}(t)∥^{4} + 4ν ∫_{0}^{t} z̃(s)∥u_{n}(s)∥_{V}^{2} ∥u_{n}(s)∥^{2}ds + 6λ ∫_{0}^{t} z̃(s)∥u_{n}(s)∥^{4}ds ≤ (14)

≤ ∥x_{0}∥^{4} + ∫_{0}^{t} z̃(s)∥f(s)∥^{4}ds + 3λ ∫_{0}^{t} z̃(s)∥u_{n−1}(s)∥^{4}ds +
+ 4 ∫_{0}^{t} z̃(s)(Cu_{n−1}(s), u_{n}(s))∥u_{n}(s)∥^{2}dw(s).

In (13) we square both sides of the inequality, use some elementar inequalities and obtain

z̃^{2}(t)∥u_{n}(t)∥^{4} + 4ν^{2}( ∫_{0}^{t} z̃(s)∥u_{n}(s)∥_{V}^{2} ds)^{2} + 36λ^{2}( ∫_{0}^{t} z̃(s)∥u_{n}(s)∥^{2}ds)^{2} ≤ (15)

≤ 4∥x_{0}∥^{4} + 4( ∫_{0}^{t} z̃(s)∥f(s)∥^{2}ds)^{2} + 4λ^{2}( ∫_{0}^{t} z̃(s)∥u_{n−1}(s)∥^{2}ds)^{2} +
+ 16 | ∫_{0}^{t} z̃(s)(Cu_{n−1}(s), u_{n}(s))dw(s) |^{2} .

In (13) we apply the Burkholder inequality (see [7], p. 166) and hence

E sup_{t∈[0,T]} z̃(t)∥u_{n}(t)∥^{2} ≤ E∥x_{0}∥^{2} + E ∫_{0}^{T}z̃(s)∥f(s)∥^{2}ds + (16)

+ λE ∫_{0}^{T}z̃(s)∥u_{n−1}(s)∥^{2}ds + 2E sup_{t∈[0,T]} | ∫_{0}^{t} z̃(s)(Cu_{n−1}(s), u_{n}(s))dw(s) |≤
≤ E∥x_{0}∥^{2} + E ∫_{0}^{T}z̃(s)∥f(s)∥^{2}ds + λE ∫_{0}^{T}z̃(s)∥u_{n−1}(s)∥^{2}ds +
+ k_{1}λE ∫_{0}^{t} z̃(s)∥u_{n−1}(s)∥^{2} + k_{1}E ∫_{0}^{t} z̃(s)∥u_{n}(s)∥^{2}ds,

where k_{1} > 0 is a constant (it occures in the Burkholder inequality). From the properties of z̃ and from u_{n}, u_{n−1} ∈ L_{V}^{2} (Ω × [0, T]) (see Lemma 2.2) it follws in (16) that

E sup_{t∈[0,T]} ∥u_{n}(t)∥^{2} < ∞.

From (14) we have

Ez̃(t)∥u_{n}(t)∥^{4} + 6λE ∫_{0}^{t} z̃(s)∥u_{n}(s)∥^{4}ds ≤ (17)

≤ E∥x_{0}∥^{4} + ∫_{0}^{t} z̃(s)∥f(s)∥^{4}ds + 3λE ∫_{0}^{t} z̃(s)∥u_{n−1}(s)∥^{4}ds.

By successive application of (17), we obtain

Ez̃(t)∥u_{n}(t)∥^{4} + 6λE ∫_{0}^{t} z̃(s)∥u_{n}(s)∥^{4}ds ≤

≤ [1 + 6λ(1 − \frac{1}{2^{n−1}} )][E∥x_{0}∥^{4} + E ∫_{0}^{t} z̃(s)∥f(s)∥^{4}ds]
≤ [1 + 6λ][E∥x_{0}∥^{4} + E ∫_{0}^{t} z̃(s)∥f(s)∥^{4}ds].

Consequently there exists a constant c_{2} > 0 (depending on λ and T) such that

sup_{t∈[0,T]} E∥u_{n}(t)∥^{4} + E ∫_{0}^{T}∥u_{n}(s)∥^{4}ds ≤ c_{2}[E∥x_{0}∥^{4} + E ∫_{0}^{T}∥f(s)∥^{4}ds]. (18)

In (15) we apply the Burkholder inequality and some other wellknown inequalities, hence

Ez̃^{2}(t)∥u(t)∥^{4} + 36λ^{2}E( ∫_{0}^{t} z̃(s)∥u_{n}(s)∥^{2}ds)^{2} ≤

≤ 4E∥x_{0}∥^{4} + 4tE ∫_{0}^{t} z̃(s)∥f(s)∥^{4}ds + 4λ^{2}tE ∫_{0}^{t} z̃(s)∥u_{n−1}(s)∥^{4}ds +

+ 8λ^{2}E ∫_{0}^{t}z̃^{2}(s)∥u_{n−1}(s)∥^{4}ds + 8E ∫_{0}^{t}z̃^{2}(s)∥u_{n}(s)∥^{4}ds.

Taking into consideration (18) and the properties of z̃ it follows that there exists a constant c_{3} depending on (λ, ν and T) such that

E( ∫_{0}^{T}z̃(s)∥u_{n}(s)∥^{2}ds)^{2} ≤ c_{3}[E∥x_{0}∥^{4} + E ∫_{0}^{T}∥f(s)∥^{4}ds]. □

Lemma 3.2. The following convergences hold:

E∫_{0}^{T}y(s)∥u_{n}(s) − u(s)∥_{V}^{2} ds → 0, for n → ∞

and for all t ∈ [0, T]

Ey(t)∥u_{n}(t) − u(t)∥^{2} → 0, for n → ∞.

Proof. Let n ∈ {1, 2, …} arbitrary fixed. Using (2), (P_{n}) and the Ito formula for y · ∥u_{n} − u∥^{2} we obtain

y(t)∥u_{n}(t) − u(t)∥^{2} + 2 ∫_{0}^{t}y(s)⟨Au_{n}(s) − Au(s), u_{n}(s) − u(s)⟩ds = (19)

= 2 ∫_{0}^{t}y(s)⟨B(u_{n−1}(s), u_{n}(s)) − B(u(s), u(s)), u_{n}(s) − u(s)⟩ds −

− 2β ∫_{0}^{t}y(s)∥u(s)∥_{V}^{2} ∥u_{n}(s) − u(s)∥^{2}ds − λ ∫_{0}^{t}y(s)∥u_{n}(s) − u(s)∥^{2}ds

+ ∫_{0}^{t}y(s)∥Cu_{n−1}(s) − Cu(s)∥^{2}ds + 2 ∫_{0}^{t}y(s)(Cu_{n−1}(s) − Cu(s), u_{n}(s) − u(s))dw(s),

for all t ∈ [0, T] and a.e. ω ∈ Ω.

From the properties of B we can derive the following estimate:

2⟨B(u_{n−1}(s), u_{n}(s)) − B(u(s), u(s)), u_{n}(s) − u(s)⟩ =

= −2⟨B(u_{n−1}(s) − u(s), u_{n}(s) − u(s)), u(s)⟩ ≤
≤ 2\sqrt{b}∥u(s)∥_{V} ∥u_{n−1}(s) − u(s)∥V^{\frac{1}{2}} ∥u_{n−1}(s) − u(s)∥^{\frac{1}{2}} ∥u_{n}(s) − u(s)∥V^{\frac{1}{2}} ∥u_{n}(s) − u(s)∥^{\frac{1}{2}} ≤
≤ \frac{ν}{2} ∥u_{n−1}(s) − u(s)∥_{V}^{2} + \frac{ν}{2} ∥u_{n}(s) − u(s)∥_{V}^{2} +
+ β∥u(s)∥_{V}^{2} ∥u_{n−1}(s) − u(s)∥^{2} + β∥u(s)∥_{V}^{2} ∥u_{n}(s) − u(s)∥^{2}

for all s ∈ [0, T] and a.e. ω ∈ Ω. Using this estimation and the continuity of C in (19) we obtain

y(t)∥u_{n}(t) − u(t)∥^{2} + \frac{3ν}{2} ∫_{0}^{t}y(s)∥u_{n}(s) − u(s)∥_{V}^{2} ds + λ ∫_{0}^{t}y(s)∥u_{n}(s) − u(s)∥^{2}ds ≤

≤ \frac{ν}{2} ∫_{0}^{t}y(s)∥u_{n−1}(s) − u(s)∥_{V}^{2} ds + 2 ∫_{0}^{t}y(s)(Cu_{n−1}(s) − Cu(s), u_{n}(s) − u(s))dw(s) +

+ β ∫_{0}^{t}y(s)∥u(s)∥_{V}^{2} (∥u_{n−1}(s) − u(s)∥^{2} − ∥u_{n}(s) − u(s)∥^{2})ds + λ ∫_{0}^{t}y(s)∥u_{n−1}(s) − u(s)∥^{2}ds

for all t ∈ [0, T] and a.e. ω ∈ Ω.

Summing up this relations for n = 1 to an arbitrary natural number N, we get

s_{N}(t) + ν ∫_{0}^{t}S_{N}(s)ds + β ∫_{0}^{t}y(s)∥u(s)∥_{V}^{2} ∥u_{N}(s) − u(s)∥^{2}ds +

+ λ ∫_{0}^{t}y(s)∥u_{n}(s) − u(s)∥^{2}ds ≤ \frac{ν}{2} ∫_{0}^{t}y(s)∥u_{0}(s) − u(s)∥_{V}^{2} ds +

+ λ ∫_{0}^{t}y(s)∥u_{0}(s) − u(s)∥^{2}ds + 2 ∑_{n=1}^{N} ∫_{0}^{t}y(s)(Cu_{n−1}(s) − Cu(s), u_{n}(s) − u(s))dw(s) +

+ β ∫_{0}^{t}y(s)∥u(s)∥_{V}^{2} ∥u_{0}(s) − u(s)∥^{2}ds

for all t ∈ [0, T] and a.e. ω ∈ Ω, where s_{N} and S_{N} are defined in (3) and (4). Taking the mathematical expectation we have

Es_{N}(t) + νE ∫_{0}^{t}S_{N}(s)ds ≤ \frac{ν}{2} E ∫_{0}^{t}y(s)∥u_{0}(s) − u(s)∥_{V}^{2} ds +

+ λE ∫_{0}^{t}y(s)∥u_{0}(s) − u(s)∥^{2}ds + βE ∫_{0}^{t}y(s)∥u(s)∥_{V}^{2} ∥u_{0}(s) − u(s)∥^{2}ds

for all t ∈ [0, T]. But 0 < y(s) ≤ 1 and u_{0}(s) = 0 for all s ∈ [0, T], a.e. ω ∈ Ω and u ∈ L_{V}^{2} (Ω × [0, T]) (see Lemma 2.1); applying Lemma 3.1 it follows that there exists a positive constant c, which does not depend on N such that

Es_{N}(t) + νE ∫_{0}^{t}S_{N}(s)ds ≤ E ∫_{0}^{T} (\frac{ν}{2} ∥u(s)∥_{V}^{2} + λ∥u(s)∥^{2} + β∥u(s)∥_{V}^{2} ∥u(s)∥^{2})ds < c

for all t ∈ [0, T] and all natural numbers N. Consequently for all t ∈ [0, T]

∑_{n=1}^{∞} Ey(t)∥u_{n}(t) − u(t)∥^{2} + ν ∑_{n=1}^{∞} E ∫_{0}^{t}y(s)∥u_{n}(s) − u(s)∥_{V}^{2} ds ≤ c

and hence

E∫_{0}^{T}y(s)∥u_{n}(s) − u(s)∥_{V}^{2} ds → 0, for n → ∞

and for all t ∈ [0, T]

Ey(t)∥u_{n}(t) − u(t)∥^{2} → 0, for n → ∞. □

Now we can prove Theorem 2.3, the main result of the paper.

For each natural number M we define the stopping time

τ_{M} = \{ \begin{matrix}T , if ∫_{0}^{T} ∥u(s)∥_{V}^{2} ≤ M \\ inf \{t ∈ [0, T] : ∫_{0}^{t}∥u(s)∥_{V}^{2} ≥ M\}, if ∫_{0}^{T} ∥u(s)∥_{V}^{2} > M. \end{matrix}

It is easy to check that for a.e. ω ∈ Ω

∫_{0}^{τ_{M}}∥u(s)∥_{V}^{2} = M and ∫_{0}^{t∧τ_{M}}∥u(s)∥_{V}^{2} ≤ M for t ∈ [0, T].

Let t ∈ [0, T] arbitrary fixed. Taking into consideration Lemma 3.1 we can write

P (τ_{M} ≤ t) = P ( ∫_{0}^{t} ∥u(s)ds∥_{V}^{2} ≥ M) ≤ \frac{1}{M} E ∫_{0}^{t}∥u(s)∥_{V}^{2} ≤ \frac{c_{1}}{M} [E∥x_{0}∥^{2} + E∫_{0}^{T}∥f(s)∥^{2}ds].

Hence

lim_{M→∞} P (τ_{M} ≤ t) = 0.

Let δ, ϵ > 0 arbitrary fixed. There exists a natural number M_{0} such that

P (τ_{M_{0}} ≤ t) ≤ \frac{ϵ}{2}.

From Lemma 3.2 we have

E ∫_{0}^{t}y(s)∥u_{n}(s) − u(s)∥_{V}^{2} ds → 0, for n → ∞,

Ey(t)∥u_{n}(t) − u(t)∥^{2} → 0, for n → ∞.

Consequently, there exists a natural number n_{0} (depending on δ, ϵ, M_{0}, t) such that

\frac{e^{λt+2βM_{0}}}{δ} E ∫_{0}^{t} y(s)∥u_{n}(s) − u(s)∥_{V}^{2} ds ≤ \frac{ϵ}{2},

\frac{e^{λt+2βM_{0}}}{δ} Ey(t)∥u_{n}(t) − u(t)∥^{2} ≤ \frac{ϵ}{2}.

We can write

P ( ∫_{0}^{t}∥u_{n}(s) − u(s)∥_{V}^{2} ds ≥ δ) ≤ P (τ_{M_{0}} ≤ t) + P ((t < τ_{M_{0}}) ∧ (∫_{0}^{t}∥u_{n}(s) − u(s)∥_{V}^{2} ds ≥ δ)) ≤

≤ \frac{ϵ}{2} + P ( ∫_{0}^{t}y(s)∥u_{n}(s) − u(s)∥_{V}^{2} ds ≥ δe^{−λt−2βM_{0}}) ≤

≤ \frac{ϵ}{2} + \frac{e^{λt+2βM_{0}}}{δ} E ∫_{0}^{t}y(s)∥u_{n}(s) − u(s)∥_{V}^{2} ds < \frac{ϵ}{2} + \frac{ϵ}{2} = ϵ

and also

P (∥u_{n}(t) − u(t)∥^{2} ≥ δ) ≤ P (τ_{M_{0}} ≤ t) + P ((t < τ_{M_{0}}) ∧ (∥u_{n}(t) − u(t)∥^{2} ≥ δ)) ≤

≤ \frac{ϵ}{2} + P (y(t)∥u_{n}(t) − u(t)∥^{2} ≥ δe^{−λt−2βM_{0}}) ≤

≤ \frac{ϵ}{2} + \frac{e^{λt+2βM_{0}}}{δ} Ey(t)∥u_{n}(t) − u(t)∥^{2} < \frac{ϵ}{2} + \frac{ϵ}{2} = ϵ.

Hence for all δ > 0

lim_{n→∞} P ( ∫_{0}^{t}∥u_{n}(s) − u(s)∥_{V}^{2} ds ≥ δ) = 0,  lim_{n→∞} P (∥u_{n}(t) − u(t)∥^{2} ≥ δ) = 0.

Therefore the sequences ( ∫_{0}^{t}∥u_{n}(s)−u(s)∥_{V}^{2} ds)_{n=1,2,...} and (∥u_{n}(t)−u(t)∥^{2})_{n=1,2,...} converge

in probability to zero. From Lemma 3.1 it follows that these sequences are uniformly integrable (with respect to ω ∈ Ω). Hence

E∫_{0}^{T}∥u_{n}(t) − u(t)∥_{V}^{2} dt → 0, for n → ∞

and for all t ∈ [0, T]

E∥u_{n}(t) − u(t)∥^{2} → 0, for n → ∞. □

4. SOLVABILITY OF THE LINEAR EQUATIONS

In this section we prove the existence and uniqueness of a special type of linear evolution equations. Let φ ∈ L_{V}^{2} (Ω × [0, T]) with E sup_{t∈[0,T]} ∥φ(t)∥^{2} < ∞ be an arbitrary process. We

consider the following evolution equation

(20)  (u_{φ}(t), v) + ∫_{0}^{t}⟨Au_{φ}(s), v⟩ds = (x_{0}, v) + ∫_{0}^{t}⟨B(φ(s), u_{φ}(s)), v⟩ds +

+ ∫_{0}^{t}(f(s), v)ds + ∫_{0}^{t}(Cφ(s), v)dw(s)

for all v ∈ V, t ∈ [0, T] and a.e. ω ∈ Ω.

Lemma 4.1. If u_{φ} ∈ L_{V}^{2} (Ω × [0, T]) is a solution of (20) then it is almost sure unique. Proof. We assume that u_{φ} and z_{φ} are two solutions of (20). Then for all t ∈ [0, T]

∥u_{φ}(t) − z_{φ}(t)∥^{2} + 2 ∫_{0}^{t}⟨Au_{φ}(s) − Az_{φ}(s), u_{φ}(s) − z_{φ}(s)⟩ds =

= 2 ∫_{0}^{t}⟨B(φ(s), u_{φ}(s) − z_{φ}(s)), u_{φ}(s) − z_{φ}(s)⟩ds.

Taking into consideration the properties of A and B it follows that for each t ∈ [0, T]

∥u_{φ}(t) − z_{φ}(t)∥^{2} + 2ν ∫_{0}^{t}∥u_{φ}(s) − z_{φ}(s)∥_{V}^{2} ds = 0 a.e. ω ∈ Ω.

Consequently

P ( sup_{t∈[0,T]} ∥u_{φ}(t) − z_{φ}(t)∥^{2} = 0) = 1.

This means that (20) has an almost sure unique solution. □

In order to prove the existence of the solution of equation (20) we have to consider a sequence of more simple equations, whose solutions converge to the solution of (20).

First method. In this method we use stopping times corresponding to the process φ. For each natural number M we define the stopping time

τ̃_{M} = \{ \begin{matrix}T , if sup_{s∈[0,T]} ∥φ(s)∥^{2} + ∫_{0}^{T} ∥φ(s)∥_{V}^{2} ≤ M \\ inf \{t ∈ [0, T] : sup_{s∈[0,t]} ∥φ(s)∥^{2} + ∫_{0}^{t}∥φ(s)∥_{V}^{2} ≥ M\}, otherwise. \end{matrix}

It is easy to check that for each t ∈ [0, T]

sup_{s∈[0,t∧τ̃_{M}]} ∥φ(s)∥^{2} + ∫_{0}^{t∧τ̃_{M}}∥φ(s)∥_{V}^{2} = M a.e. ω ∈ Ω.

We denote by

φ^{M}(s) = \{ \begin{matrix}φ(s) , if 0 ≤ s ≤ τ̃_{M} \\ 0 , if τ̃_{M} < s ≤ T. \end{matrix}

Let M be an arbitrary fixed natural number. For each n = 1, 2, … we consider the equation

(P_{n}^{M})  (u_{n}^{M}(t), v) + ∫_{0}^{t}⟨Au_{n}^{M}(s), v⟩ds = (x_{0}, v) + ∫_{0}^{t}⟨B(φ^{M}(s), u_{n−1}^{M}(s)), v⟩ds +

+ ∫_{0}^{t}(f(s), v)ds + ∫_{0}^{t}(Cφ(s), v)dw(s)

for all v ∈ V, t ∈ [0, T] and a.e. ω ∈ Ω; we take u_{0}^{M}(t) = 0 for all t ∈ [0, T] and a.e. ω ∈ Ω.

Lemma 4.2. For each n = 1, 2, … equation (P_{n}^{M}) has an almost sure unique solution u_{n}^{M} ∈ L_{V}^{2} (Ω × [0, T]) such that E sup_{t∈[0,T]} ∥u_{n}^{M}(t)∥^{2} < ∞. There exists a constant c_{M} > 0

(independent of n) such that for each n = 1, 2, …

E∫_{0}^{T}∥u_{n}^{M}∥_{V}^{2} dt < c_{M}.

Proof. We prove this lemma by induction. For n = 1 we have a stochastic evolution equation of heat type, who has an almost sure unique solution and for which

E sup_{t∈[0,T]} ∥u_{1}^{M}(t)∥^{2} + E∫_{0}^{T}∥u_{1}^{M}∥_{V}^{2} (t)dt < ∞

(see for example [6], Theorem 4.1, p. 105; [8], Chapitre III, p. 79).

Let n be such that u_{n−1}^{M} ∈ L_{V}^{2} (Ω × [0, T]) is the solution of (P_{n−1}^{M}) and

E sup_{t∈[0,T]} ∥u_{n−1}^{M}(t)∥^{2} + E∫_{0}^{T}∥u_{n−1}^{M}(t)∥_{V}^{2} dt < ∞.

Using the properties of B and τ̃_{M} we see that

E∫_{0}^{T}∥B(φ^{M}(s), u_{n−1}^{M}(s))∥_{V^{∗}}^{2}ds ≤ bM[E sup_{t∈[0,T]} ∥u_{n−1}^{M}(t)∥^{2} + E∫_{0}^{T}∥u_{n−1}^{M}(t)∥_{V}^{2} dt] < ∞.

(P_{n}^{M}) is a stochastic evolution equation of heat type and so the existence and uniqueness is assured by the ,,classic” theory of evolution equations (see for example [6], [8]).

Let

y(t) = exp {−2t − β ∫_{0}^{t}∥φ^{M}(s)∥^{2}∥φ^{M}(s)∥_{V}^{2} ds}

for all t ∈ [0, T], a.e. ω ∈ Ω, where β = \frac{2b}{ν^{2}} . We see that for all t ∈ [0, T] and a.e. ω ∈ Ω

e^{−2t−βM^{2}} ≤ y(t) ≤ 1.

From the Ito formula applied for y · ∥u_{n}^{M}∥^{2}, the properties of A, B and C we have

y(t)∥u_{n}^{M}(t)∥^{2} + ν ∫_{0}^{t}y(s)∥u_{n}^{M}(s)∥_{V}^{2} ds + ∫_{0}^{t}y(s)∥u_{n}^{M}(s)∥^{2}ds ≤ (21)

≤ ∥x_{0}∥^{2} + \frac{ν}{2} ∫_{0}^{t}y(s)∥u_{n−1}^{M}(s)∥_{V}^{2} ds + ∫_{0}^{t}y(s)∥f(s)∥^{2}ds +

+ 2 ∫_{0}^{t}y(s)(Cφ(s), u_{n}^{M}(s))dw(s) + λ ∫_{0}^{t}y(s)∥φ(s)∥^{2}ds.

After taking the mathematical expectation we get

Ey(t)∥u_{n}^{M}(t)∥^{2} + νE ∫_{0}^{t}y(s)∥u_{n}^{M}(s)∥_{V}^{2} ds + E ∫_{0}^{t}y(s)∥u_{n}^{M}(s)∥^{2}ds ≤ (22)

≤ E∥x_{0}∥^{2} + E ∫_{0}^{t}y(s)∥f(s)∥^{2}ds + \frac{ν}{2} E ∫_{0}^{t}y(s)∥u_{n−1}^{M}(s)∥_{V}^{2} ds + λE ∫_{0}^{t}y(s)∥φ(s)∥^{2}ds.

Using the properties of y we have

E∥u_{n}^{M}(t)∥^{2} + νE ∫_{0}^{t}∥u_{n}^{M}(s)∥_{V}^{2} ds + E ∫_{0}^{t}∥u_{n}^{M}(s)∥^{2}ds ≤

≤ e^{2t+βM^{2}}[E∥x_{0}∥^{2} + E ∫_{0}^{t}∥f(s)∥^{2}ds + \frac{ν}{2} E ∫_{0}^{t}∥u_{n−1}^{M}(s)∥_{V}^{2} ds + λE ∫_{0}^{t}∥φ(s)∥^{2}ds].

Consequently

E∥u_{n}^{M}(t)∥^{2} + νE ∫_{0}^{t}∥u_{n}^{M}(s)∥_{V}^{2} ds + E ∫_{0}^{t}∥u_{n}^{M}(s)∥^{2}ds < ∞ (23)

for all t ∈ [0, T]. Using in (21) the Burkholder inequality (see [7], p. 166) and applying the properties of y it follows that

E sup_{t∈[0,T]} y(t)∥u_{n}^{M}(t)∥^{2} ≤ E∥x_{0}∥^{2} + \frac{ν}{2} E ∫_{0}^{T} y(s)∥u_{n−1}^{M}(s)∥_{V}^{2} ds + λE ∫_{0}^{T}y(s)∥φ(s)∥^{2}ds+

+ E ∫_{0}^{T} y(s)∥f(s)∥^{2}ds + 2E sup_{t∈[0,T]} | ∫_{0}^{t}y(s)(Cφ(s), u_{n}^{M}(s))dw(s) |≤

≤ E∥x_{0}∥^{2} + \frac{ν}{2} E ∫_{0}^{T} ∥u_{n−1}^{M}(s)∥_{V}^{2} ds + λE ∫_{0}^{T}∥φ(s)∥^{2}ds + E ∫_{0}^{T} ∥f(s)∥^{2}ds +

+ 2k_{1}E( ∫_{0}^{T} y^{2}(s)(Cφ(s), u_{n}^{M}(s))^{2}ds)^{\frac{1}{2}} ≤ E∥x_{0}∥^{2} + \frac{ν}{2} E ∫_{0}^{T} ∥u_{n−1}^{M}(s)∥_{V}^{2} ds +

+ λE∫_{0}^{T}∥φ(s)∥^{2}ds + k_{1}λE sup_{[0,T]} ∥φ(s)∥^{2} + E ∫_{0}^{T} ∥f(s)∥^{2}ds + k_{1}E∫_{0}^{T}∥u_{n}^{M}(s)∥^{2}ds,

where k_{1} is a positive constant (it occures in the Burkholder inequality). Taking into consideration (23) and the properties of y we have

E sup_{t∈[0,T]} ∥u_{n}^{M}(t)∥^{2} < ∞.

By successive application of (22) we have

Ey(t)∥u_{n}^{M}(t)∥^{2} + νE ∫_{0}^{t}y(s)∥u_{n}^{M}(s)∥_{V}^{2} ds + 2λE ∫_{0}^{t}y(s)∥u_{n}^{M}(s)∥^{2}ds ≤

≤ [1 + (ν + 2λ)(1 − \frac{1}{2^{n−1}} )] · [E∥x_{0}∥^{2} + E ∫_{0}^{t}∥f(s)∥^{2}ds + λE ∫_{0}^{t}∥φ(s)∥^{2}ds].

Therefore

E ∫_{0}^{T}∥u_{n}^{M}(s)∥_{V}^{2} ds ≤ e^{2T +βM^{2}}[1 + \frac{1 + 2λ}{ν}] · [E∥x_{0}∥^{2} + E ∫_{0}^{T}∥f(s)∥^{2}ds + λE ∫_{0}^{T}∥φ(s)∥^{2}ds].

Hence there exists a constant c_{M} > 0 independent of n such that

E∫_{0}^{T}∥u_{n}^{M}∥_{V}^{2} dt < c_{M}. □

Lemma 4.3. (i) There exists a process u^{M} ∈ L_{V}^{2} (Ω × [0, T]) such that

(P^{M})  (u^{M}(t), v) + ∫_{0}^{t}⟨Au^{M}(s), v⟩ds = (x_{0}, v) + ∫_{0}^{t}⟨B(φ^{M}(s), u^{M}(s)), v⟩ds +

+ ∫_{0}^{t}(f(s), v)ds + ∫_{0}^{t}(Cφ(s), v)dw(s)

for all v ∈ V, t ∈ [0, T] and a.e. ω ∈ Ω. The solution of (P^{M}) is almost sure unique. (ii) The following estimation holds:

2νE∫_{0}^{T}∥u^{M}(t)∥_{V}^{2} ≤ e^{(λ+1)T} [E∥x_{0}∥^{2} + E∫_{0}^{T}∥f(t)∥^{2}dt + λE ∫_{0}^{T}∥φ(s)∥^{2}ds]. (24)

Proof. (i) The uniqueness of the solution of (P^{M}) can be proved analogously to the proof of Lemma 4.1.

Let

y(t) = exp {−β ∫_{0}^{t}∥φ^{M}(s)∥^{2}∥φ^{M}(s)∥_{V}^{2} ds}

for all t ∈ [0, T], a.e. ω ∈ Ω, where β = \frac{64b^{2}}{ν^{3}} . Obviously the following inequalities hold

e^{−βM^{2}} ≤ y(t) ≤ 1 (25)

for all t ∈ [0, T] and a.e. ω ∈ Ω.

Let n ≥ 2 arbitrary fixed. We have

y(t)∥u_{n}^{M}(t) − u_{n−1}^{M}(t)∥^{2} + 2 ∫_{0}^{t}y(s)⟨Au_{n}^{M}(s) − Au_{n−1}^{M}(s), u_{n}^{M}(s) − u_{n−1}^{M}(s)⟩ds =

= 2 ∫_{0}^{t}y(s)⟨B(φ^{M}(s), u_{n−1}^{M}(s) − u_{n−2}^{M}(s)), u_{n}^{M}(s) − u_{n−1}^{M}(s)⟩ds −

− β ∫_{0}^{t}y(s)∥φ^{M}∥_{V}^{2} ∥φ^{M}∥^{2}∥u_{n}^{M}(s) − u_{n−1}^{M}(s)∥^{2}ds

for all t ∈ [0, T] and a.e. ω ∈ Ω. Using the properties of A, B and making some elementar calculations we have

y(t)∥u_{n}^{M}(t) − u_{n−1}^{M}(t)∥^{2} + ν ∫_{0}^{t}y(s)∥u_{n}^{M}(s) − u_{n−1}^{M}(s)∥_{V}^{2} ds ≤

≤ \frac{ν}{16} ∫_{0}^{t}y(s)∥u_{n−1}^{M}(s) − u_{n−2}^{M}(s)∥_{V}^{2} ds.

Therefore

sup_{t∈[0,T]} y(t)∥u_{n}^{M}(t) − u_{n−1}^{M}(t)∥^{2} + ν ∫_{0}^{T}y(s)∥u_{n}^{M}(s) − u_{n−1}^{M}(s)∥_{V}^{2} ds ≤ (26)

≤ \frac{ν}{8} ∫_{0}^{T}∥u_{n−1}^{M}(s) − u_{n−2}^{M}(s)∥_{V}^{2} ds.

By successive application of (26) we get

sup_{t∈[0,T]}y(t)∥u_{n}^{M}(t) − u_{n−1}^{M}(t)∥^{2} + ν∫_{0}^{T}y(s)∥u_{n}^{M}(s) − u_{n−1}^{M}(s)∥_{V}^{2} ds ≤ \frac{ν}{8^{n−1}} ∫_{0}^{T}∥u_{1}^{M}(s)∥_{V}^{2} ds. (27)

We also have

P ( sup_{t∈[0,T]} ∥u_{n}^{M}(t) − u_{n−1}^{M}(t)∥ > \frac{1}{2^{n}} ) ≤ P ( sup_{t∈[0,T]} y(t)∥u_{n}^{M}(t) − u_{n−1}^{M}(t)∥^{2} > \frac{e^{−βM^{2}}}{4^{n}} ) ≤

≤ P (\frac{ν}{8^{n−1}} ∫_{0}^{T}∥u_{1}^{M}(s)∥_{V}^{2} ds > \frac{e^{−βM^{2}}}{4^{n}} ) = P ( ∫_{0}^{T}∥u_{1}^{M}(s)∥_{V}^{2} ds > \frac{2^{n}e^{−βM^{2}}}{8ν} ) ≤
≤ \frac{8e^{βM^{2}}ν}{2^{n}} E ∫_{0}^{T}∥u_{1}^{M}(s)∥_{V}^{2} ds.

Since u_{1}^{M} ∈ L_{V}^{2} (Ω × [0, T]) (see Lemma 4.2), we get

∑_{n=2}^{∞} P ( sup_{t∈[0,T]} ∥u_{n}^{M}(t) − u_{n−1}^{M}(t)∥ > \frac{1}{2^{n}} ) < 4νe^{βM^{2}}E ∫_{0}^{T}∥u_{1}^{M}(s)∥_{V}^{2} ds < ∞.

Applying the Borel-Cantelli Lemma it follows that

u_{1}^{M}(t) + ∑_{n=2}^{∞} (u_{n}^{M}(t) − u_{n−1}^{M}(t)) (28)

are P - almost sure convergent series in the sense of the norm sup_{t∈[0,T]} ∥ · ∥. Therefore there exists a H-valued process u^{M} such that for a.e. ω ∈ Ω

sup_{t∈[0,T]} ∥u_{n}^{M}(t) − u^{M}(t)∥ → 0 when n → ∞. (29)

Obviously u^{M} is adapted and F × B_{[0,T]} - measurable, since u_{n}^{M} have this property.

Taking into consideration (27) it follows analogously to (28) that the series

u_{1}^{M}(t) + ∑_{n=2}^{∞} (u_{n}^{M}(t) − u_{n−1}^{M}(t))

are P - almost sure convergent in the sense of the norm ( ∫_{0}^{T}∥ · ∥_{V}^{2} dt)^{\frac{1}{2}} . Hence there exists a V -valued process ũ^{M} such that for a.e. ω ∈ Ω

∫_{0}^{T}∥u_{n}^{M}(t) − ũ^{M}(t)∥_{V}^{2} dt → 0 when n → ∞. (30)

From the embedding V ↪ H and the convergences (29) and (30) we obtain that for a.e. ω ∈ Ω

∫_{0}^{T}∥ũ^{M}(t) − u^{M}(t)∥^{2}dt = 1.

Consequently P (ω ∈ Ω : ũ^{M}(t) = u^{M}(t) for a.e. t ∈ [0, T]) = 0. (31)

For a.e. ω ∈ Ω we take n → ∞ in (P_{n}^{M}), apply the relations (29), (30), (31) and the continuity of A and B. It follows that u^{M} is the solution of (P^{M}).

From the Fatou Lemma and from Lemma 4.2 we have

E∫_{0}^{T}∥u^{M}(t)∥_{V}^{2} dt ≤ lim inf_{n→∞} E∫_{0}^{T}∥u_{n}^{M}(t)∥_{V}^{2} dt < c_{M}.

Hence u^{M} ∈ L_{V}^{2} (Ω × [0, T]).

(ii) Estimation (24) is obtained from the Ito formula in the same way as the estimations in Lemma 3.1. □

Lemma 4.4. There exists a process u_{φ} ∈ L_{V}^{2} (Ω × [0, T]) which is the solution of equation (20). Proof. Let

Ω^{∗} = ⋂_{M=1}^{∞} {ω ∈ Ω : (u^{M}(t))_{t∈[0,T]} is the solution of (P^{M})}.

For each M and for a.e. ω ∈ Ω the process (u^{M}(t))_{t∈[0,T]} is solution of (P^{M}) (see Lemma

4.3). Hence P (Ω^{∗}) = 1. For each natural number M we denote by

Ω_{M} = {ω ∈ Ω^{∗} : sup_{t∈[0,T]} ∥φ(t)∥^{2} + ∫_{0}^{T}∥φ(t)∥_{V}^{2} dt < M}.

From the properties of φ, the Markov inequality and some elementar calculations we see that

P ( ⋂_{M=1}^{∞} Ω̄_{M} ) = lim_{M→∞} P (ω ∈ Ω : sup_{t∈[0,T]} ∥φ(t)∥^{2} + ∫_{0}^{T}∥φ(t)∥_{V}^{2} dt ≥ M) ≤

≤ lim_{M→∞} \frac{1}{M} (E sup_{t∈[0,T]} ∥φ(t)∥^{2} + E ∫_{0}^{T}∥φ(t)∥_{V}^{2} dt) = 0  .

Hence

P ( ⋃_{M=1}^{∞} Ω_{M} ) = 1.

Let

Ω_{0} = ⋃_{M=1}^{∞} ⋃_{J<M} {ω ∈ Ω_{J} : there exists t ∈ [0, T] such that u^{M}(t) ≠ u^{J}(t)}

We have P (Ω_{0}) = 0, because otherwise there exists M_{0} and J_{0} < M_{0} such that

P (Ψ) > 0, where Ψ = {ω ∈ Ω_{J_{0}} : there exists t ∈ [0, T] such that u^{M_{0}}(t) ≠ u^{J_{0}}(t)}.

For each t ∈ [0, T] we can define

ũ(t) = \{ \begin{matrix}u^{J_{0}}(t) , if ω ∈ Ψ \\ u^{M_{0}}(t) , if ω ∈ ( ⋃_{M=1}^{∞} Ω_{M} ) \ Ψ. \end{matrix}

For all ω ∈ Ψ we have τ̃_{J_{0}} = T and so τ̃_{M_{0}} = T. Hence φ^{J_{0}}(t) = φ^{M_{0}}(t) = φ(t) for all t ∈ [0, T]. Therefore ũ and u^{M_{0}} are two solutions of (P^{M_{0}}), for which

P (ω ∈ Ω^{∗} : there exists t ∈ [0, T] such that u^{M_{0}}(t) ≠ u^{J_{0}}(t)) > 0.

This contradicts the uniqueness of the solution of (P^{M_{0}}) proved in Lemma 4.3. Consequently P (Ω_{0}) = 0.

We consider Ω′ = ( ⋃_{M=1}^{∞} Ω_{M} ) \ Ω_{0}. Obviously we have P (Ω′) = 1.

Let ω ∈ Ω′ arbitrary fixed. There exists an index M_{0} such that ω ∈ Ω_{M_{0}} ⊆ Ω_{M} and hence τ̃_{M} = T for all M ≥ M_{0}; we also have ω ∉ Ω_{0}. For this ω we define

u_{φ}(t) = lim_{M→∞} u^{M}(t),

where the convergence is uniformly in t ∈ [0, T] (this is possible because of Ω_{0}). Consequently for a.e. ω ∈ Ω

sup_{t∈[0,T]} ∥u_{φ}(t) − u^{M}(t)∥^{2} → 0, ∫_{0}^{T}∥u_{φ}(t) − u^{M}(t)∥_{V}^{2} → 0, for M → ∞

and u_{φ} is F × B_{[0,T]} - measurable and adapted. For each ω ∈ Ω′ we take M → ∞ in (P^{M}) and obtain that u_{φ}^{M} is a solution of (20).

From the Fatou Lemma and the estimation given in Lemma 4.3 we have

E∫_{0}^{T}∥u_{φ}(t)∥_{V}^{2} dt ≤ lim inf_{M→∞} E∫_{0}^{T}∥u^{M}(t)∥_{V}^{2} dt < ∞.

Therefore u_{φ} ∈ L_{V}^{2} (Ω × [0, T]). □ Remark. Taking into consideration the existence of higher order moments it can be proved that

lim_{M→∞} E ∫_{0}^{T}∥u^{M}(t) − u_{φ}(t)∥_{V}^{2} dt = 0

and for each fixed M we have

lim_{n→∞} E ∫_{0}^{T}∥u_{n}^{M}(t) − u^{M}(t)∥_{V}^{2} dt = 0.

Second method. We approximate the solution u_{φ} of equation (20) by the sequence (u_{n})_{n=1,2,...}, where u_{n} is the solution of the following evolution equation:

(P^{n})  (u_{n}(t), v) + ∫_{0}^{t}⟨Au_{n}(s), v⟩ds = (x_{0}, v) + ∫_{0}^{t}⟨B(φ(s), u_{n−1}(s)), v⟩ds +

+ ∫_{0}^{t}(f(s), v)ds + ∫_{0}^{t}(Cφ(s), v)dw(s)

for all v ∈ V, t ∈ [0, T] and a.e. ω ∈ Ω, where u_{0}(t) = 0 for all t ∈ [0, T] and a.e. ω ∈ Ω.

Lemma 4.5. For each n ≥ 1 there exists an adapted, F × B_{[0,T]}-measurable process u_{n} which is solution of (P^{n}) such that u_{n} ∈ L_{V}^{2} ([0, T]) for a.e. ω ∈ Ω and u_{n} has almost sure continuous trajectories in H. The solution is almost sure unique. Proof. Let z_{φ} be the solution of the following evolution equation

(z_{φ}(t), v) + ∫_{0}^{t}⟨Az_{φ}(s), v⟩ds = (x_{0}, v) + ∫_{0}^{t}(f(s), v)ds + ∫_{0}^{t}(Cφ(s), v)dw(s)

for all v ∈ V, t ∈ [0, T] and a.e. ω ∈ Ω. The existence and uniqueness of the solution z_{φ} ∈ L_{V}^{2} (Ω × [0, T]) is given for example in [6], Theorem 4.1, p. 105. It can be proved that z_{φ} has almost sure continuous trajectories in H and E sup_{t∈[0,T]} ∥z_{φ}(t)∥^{2} < ∞.

For the proof of this lemma we use the method of induction. For n = 1 we have u_{1} = z_{φ} and the conclusion of the lemma follows from the properties of z_{φ}.

Let n be such that u_{n−1} ∈ L_{V}^{2} ([0, T]) for a.e. ω ∈ Ω is the solution of (P^{n−1}) and u_{n−1} has almost sure continuous trajectories in H.

We consider the transformation ũ_{n}(t) = u_{n}(t) − z_{φ}(t) for all t ∈ [0, T] and a.e. ω ∈ Ω. Equation (P^{n}) can be rewritten as

(P̃^{n}) (ũ_{n}(t), v) + ∫_{0}^{t}⟨Aũ_{n}(s), v⟩ds = ∫_{0}^{t}⟨B(φ(s), u_{n−1}(s)), v⟩ds.

Since φ ∈ L_{V}^{2} (Ω × [0, T]), E sup_{t∈[0,T]} ∥φ(t)∥^{2} < ∞ and u_{n−1} has the properties mentioned

above, then for a.e. ω ∈ Ω

∫_{0}^{T}∥B(φ(s), u_{n−1}(s)∥_{∗}^{2}ds ≤

≤ b( sup_{t∈[0,T]} ∥φ(t)∥^{2} ∫_{0}^{T}∥φ(s)∥_{V}^{2} ds sup_{t∈[0,T]} ∥u_{n−1}(t)∥^{2} ∫_{0}^{T}∥u_{n−1}(s)∥_{V}^{2} ds)^{\frac{1}{2}} < ∞,

where we used the property (iii) for B. Therefore B(φ, u_{n−1}) ∈ L_{V^{∗}}^{2}([0, T]) for a.e. ω ∈ Ω. From f ∈ L_{H}^{2}(Ω × [0, T]) it follows that f ∈ L_{H}^{2}([0, T]) for a.e. ω ∈ Ω.

We consider (P̃^{n}) over

Ω̃ = {ω ∈ Ω : B(φ, u_{n−1}) ∈ L_{V^{∗}}^{2}([0, T]), f ∈ L_{H}^{2}([0, T])},

which obviously satisfies P (Ω̃) = 1. For each ω ∈ Ω̃ the existence and uniqueness of ũ_{n}(ω, ·) follows from the deterministic theory (see [13], Corollary 23.24, p. 425), where ũ_{n}(ω, ·) is approximated by the solutions of the Galerkin equations. The solutions of this equations can be approximated by the method of successive approximation and hence they are adapted and F × B_{[0,T]}-measurable (because (P̃^{n}) has an adapted, F × B_{[0,T]}-measurable nonhomogeous part). The Galerkin solutions converge to ũ_{n} for every (ω, t) ∈ Ω̃ × [0, T] and therefore ũ_{n} is an adapted, F × B_{[0,T]}-measurable process, such that for all ω ∈ Ω̃ we have ũ_{n} ∈ L_{V}^{2} ([0, T]. It also follows that ũ_{n} has almost sure continuous trajectories in H (see the form of equation (P̃^{n})).

The process u_{n} = ũ_{n}+z_{φ} has the same properties as ũ_{n} and satisfies equation (P^{n}). □

Lemma 4.6. There exists a process u ∈ L_{V}^{2} (Ω × [0, T]) which satisfies equation (20). Proof. Let

y(t) = exp {−β ∫_{0}^{t}∥φ(s)∥^{2}∥φ(s)∥_{V}^{2} ds}

for all t ∈ [0, T], a.e. ω ∈ Ω, where β = \frac{64b^{2}}{ν^{3}} . Obviously it holds

e^{−β ∫_{0}^{T}∥φ(s)∥^{2}∥φ(s)∥_{V}^{2} ds} ≤ y(t) ≤ 1

for all t ∈ [0, T] and a.e. ω ∈ Ω.

Let n ≥ 2 arbitrary fixed. We have

y(t)∥u_{n}(t) − u_{n−1}(t)∥^{2} + 2 ∫_{0}^{t}y(s)⟨Au_{n}(s) − Au_{n−1}(s), u_{n}(s) − u_{n−1}(s)⟩ds =

= 2 ∫_{0}^{t}y(s)⟨B(φ(s), u_{n−1}(s) − u_{n−2}(s)), u_{n}(s) − u_{n−1}(s)⟩ds −

− β ∫_{0}^{t}y(s)∥φ∥_{V}^{2} ∥φ∥^{2}∥u_{n}(s) − u_{n−1}(s)∥^{2}ds

for all t ∈ [0, T] and a.e. ω ∈ Ω. Using the properties of A, B and making some elementar calculations we have

y(t)∥u_{n}(t) − u_{n−1}(t)∥^{2} + ν ∫_{0}^{t}y(s)∥u_{n}(s) − u_{n−1}(s)∥_{V}^{2} ds ≤

≤ \frac{ν}{16} ∫_{0}^{t}y(s)∥u_{n−1}(s) − u_{n−2}(s)∥_{V}^{2} ds.

Therefore

sup_{t∈[0,T]} y(t)∥u_{n}(t) − u_{n−1}(t)∥^{2} + ν ∫_{0}^{T}y(s)∥u_{n}(s) − u_{n−1}(s)∥_{V}^{2} ds ≤ (32)

≤ \frac{ν}{8} ∫_{0}^{T}∥u_{n−1}(s) − u_{n−2}(s)∥_{V}^{2} ds.

By successive application of (32) we get

sup_{t∈[0,T]}y(t)∥u_{n}(t) − u_{n−1}(t)∥^{2} + ν∫_{0}^{T}y(s)∥u_{n}(s) − u_{n−1}(s)∥_{V}^{2} ds ≤ \frac{ν}{8^{n−1}} ∫_{0}^{T}∥u_{1}(s)∥_{V}^{2} ds. (33)

Then we have

P ( sup_{t∈[0,T]} y(t)∥u_{n}(t)−u_{n−1}(t)∥^{2} > \frac{1}{4^{n}} ) ≤ P (\frac{ν}{8^{n−1}} ∫_{0}^{T}∥u_{1}(s)∥_{V}^{2} ds > \frac{1}{4^{n}} ) ≤ \frac{8ν}{2^{n}} E ∫_{0}^{T}∥u_{1}(s)∥_{V}^{2} ds.

Since u_{1} ∈ L_{V}^{2} (Ω × [0, T]) (see the proof of Lemma 4.5), we get

∑_{n=2}^{∞} P ( sup_{t∈[0,T]} \sqrt{y(t)}∥u_{n}(t) − u_{n−1}(t)∥ > \frac{1}{2^{n}} ) < 4νE ∫_{0}^{T}∥u_{1}(s)∥_{V}^{2} ds < ∞. (34)

Applying the Borel-Cantelli Lemma it follows that

\sqrt{y(t)}u_{1}(t) + \sqrt{y(t)} ∑_{n=2}^{∞} (u_{n}(t) − u_{n−1}(t)) (35)

are P - almost sure convergent series in the sense of the norm sup_{t∈[0,T]} ∥ · ∥. Therefore there exists a H-valued process u_{φ} such that for a.e. ω ∈ Ω

sup_{t∈[0,T]} \sqrt{y(t)}∥u_{n}(t) − u_{φ}(t)∥ → 0 when n → ∞.

Using the properties of y we can deduce that

sup_{t∈[0,T]} ∥u_{n}(t) − u_{φ}(t)∥ → 0 when n → ∞. (36)

u_{φ} is F × B_{[0,T]} - measurable and adapted, because u_{n} have this property.

Taking into consideration (33) it follows analogously to (34) that the series

\sqrt{y(t)}u_{1}(t) + \sqrt{y(t)} ∑_{n=2}^{∞} (u_{n}(t) − u_{n−1}(t))

are P - almost sure convergent in the sense of the norm ( ∫_{0}^{T}∥ · ∥_{V}^{2} dt)^{\frac{1}{2}} . Hence there exists

a V -valued process ũ_{φ} such that for a.e. ω ∈ Ω

∫_{0}^{T}y(t)∥u_{n}(t) − ũ_{φ}(t)∥_{V}^{2} dt → 0 when n → ∞.

Using the properties of y we can deduce that

∫_{0}^{T}∥u_{n}(t) − ũ_{φ}(t)∥_{V}^{2} dt → 0 when n → ∞. (37)

From the embedding V ↪ H and the convergences (36) and (37) we obtain that for a.e. ω ∈ Ω

∫_{0}^{T}y(t)∥ũ_{φ}(t) − u_{φ}(t)∥^{2}dt = 1.

Consequently P (ω ∈ Ω : ũ_{φ}(t) = u_{φ}(t) for a.e. t ∈ [0, T]) = 1. (38)

For a.e. ω ∈ Ω we take n → ∞ in (P^{n}), apply the relations (36), (37), (38) and the continuity of A and B. It follows that u_{φ} satisfies equation (20).

In (20) we apply the Ito formula and (iii). Then we have

e^{−t}∥u_{φ}(t)∥^{2} + 2 ∫_{0}^{t}e^{−s}⟨Au_{φ}(s), u_{φ}(s)⟩ds = ∥x_{0}∥^{2} + 2 ∫_{0}^{t}e^{−s}(f(s), u_{φ}(s))ds +

+ ∫_{0}^{t}e^{−s}∥Cφ(s)∥^{2}ds − ∫_{0}^{t}e^{−s}∥u(s)∥^{2}ds + 2 ∫_{0}^{t}e^{−s}(Cφ(s), u_{φ}(s))dw(s)

for all t ∈ [0, T] and a.e. ω ∈ Ω. Using the properties of A, C and φ and making some elementar calculations we get

E∥u_{φ}(T)∥^{2} + 2νE ∫_{0}^{T}∥u_{φ}(s)∥_{V}^{2} ds ≤ e^{T} [E∥x_{0}∥^{2} + E ∫_{0}^{T}∥f(s)∥^{2}ds + λE ∫_{0}^{T}∥φ(s)∥^{2}ds] < ∞.

Consequently u_{φ} ∈ L_{V}^{2} (Ω × [0, T]). □ Remark. Using this method we construct the sequence (u_{n})_{n=1,2,...} which converge a.s. to the solution u_{φ} of equation (20). This method is more useful for practical calculations than the first method, which takes into consideration stopping times.

References

[1] A. Bensoussan, R. Temam: Equations stochastiques du type Navier-Stokes. J. Funct. Anal. 13 (1973), pp. 195-222.

[2] Z. Brzeźniak, M. Capiński, F. Flandoli: Pathwise global attractors for stationary random dynamical systems. Probab. Theory Relat. Fields 12(95) (1993), pp. 87-102.

[3] A. W. Fursikov, M.I. Vishik, A.I. Komech: Some mathematical problems of hydrodynamics, Russian Math. Survey 34 (1979), pp. 149-234.

[4] I.I. Gichman, A.V. Skorochod: Stochastische Differentialgleichungen. Akademie Verlag Berlin, 1971.

[5] W. Grecksch, B. Schmalfuß: Approximation of the stochastic Navier-Stokes equation. Compl. Appl. Math., V. 15 (1997), no. 3, pp. 227-239.

[6] W. Grecksch, C. Tudor: Stochastic Evolution Equations. Akademie Verlag, Berlin, 1995.

[7] I. Karatzas, S. E. Shreve: Brownian Motion and Stochastic Calculus. Springer Verlag, New York - Berlin, 1996.

[8] E. Pardoux: Equations aux derivées partielles stochastiques non linéaires monotones. Etude de solutions fortes de type Ito. Thése doct. sci. math. Univ. Paris, 1975.

[9] B. Schmalfuß: Endlichdimensionale Approximation der stochastischen Navier-Stokes-Gleichung. Statistics 2 (1990), pp. 149-157.

[10] R. Temam: Navier-Stokes Equations - Theory and Numerical Analysis. North-Holland, Amsterdam, 1979.

[11] V. M. Tichomirov: Theorie der Extremalaufgaben. VEB Deutscher Verlag der Wissenschaften, Berlin, 1979.

[12] M. Viot: Solution faible d’équation aux derivées partielles stochastiques non linéares. Thése doct. sci. math., Paris, 1976.

[13] E. Zeidler: Nonlinear Functional Analysis and it’s Applications. Vol. II/A: Linear Monotone Operators. Springer Verlag, New York - Berlin, 1990.