#### Abstract

We investigate the stochastic 3D Navier-Stokes- model which arises in the modelling of turbulent flows of fluids. Our model contains nonlinear forcing terms which do not satisfy the Lipschitz conditions. The adequate notion of solutions is that of probabilistic weak solution. We establish the existence of a such of solution. We also discuss the uniqueness.

#### 1. Introduction

In this paper, we are interested in the study of probabilistic weak solutions of the 3D Navier-Stokes- model (also known as the Lagrangien averaged Navier-Stokes-alpha model or the viscous Camassa-Holm equations) with homogeneous Dirichlet boundary conditions in a bounded domains in the case in which random perturbations appear. To be more precise, let be a connected and bounded open subset of with boundary and a final time . We denote by the Stokes operator and consider the system

where and are unknown random fields on , representing, respectively, the large-scale velocity and the pressure, in each point of . The constant and are given, and represent, respectively, the kinematic viscosity of the fluid, and the square of the spatial scale at which fluid motion is filtered. The terms and are external forces depending on , where is an -valued standard Wiener process. Finally, is a given nonrandom velocity field.

The deterministic version of (1.1), that is, when has been the object of intense investigations over the last years [1–5] and the initial motivation was to find a closure model for the 3D turbulence-averaged Reynolds model. A key interest in the model is the fact that it serves as a good approximation to the 3D Navier-Stokes equations. One of the main reasons justifying its use is the high computational cost that the Navier-Stokes model requires. Many important results have been obtained in the deterministic case. More precisely, the global well posedness of weak solutions for the NS- model on bounded domains has been established in [6, 7] amongst others, and the asymptotic behavior can be found in [6]. Similar results have been proved by Foias et al. [8] in the case of periodic boundary conditions.

However, in order to consider a more realistic model our problem, it is sensible to introduce some kind of noise in the equations. This may reflect, some environmental effects on the phenomena, some external random forces, and so forth. To the best of our knowledge, the existence and uniqueness of solutions of the stochastic version (1.1) which we consider in this paper has only been analyzed in [9] (see also [10]) in the case of Lipschitz assumptions on and . The case of non-Lipschitz assumptions on the coefficients and is the main concern of the present paper. This question has been opened till now. The general motivation for studying weak rather than strong solutions of stochastic equations is that existence of weak solutions can be carried through under weaker regularity on the coefficients. This was pointed out, for instance, in [11].

In this paper, we will establish the existence of probabilistic weak solutions for the problem (1.1) under appropriate conditions on the data. Under the strong assumptions on and , we prove the uniqueness of weak solutions. The method used for the proof of our existence results is different from the method in [9]. To prove the existence, we use the Galerkin approximation method employing special bases, combined with some famous theorems of probabilistic nature due to Prokhorov [12] and Skorokhod [13].

The paper is organized as follows. In Section 2, we establish some properties of nonlinear term appearing in our equations. The rigorous statement of our problem as well as the main results are included in Section 3 and we show how our problem can be reformulated as an abstract stochastic model. Section 4 is devoted to the proof of our main results.

#### 2. Properties of the Nonlinear Terms in (1.1)

Following [9], we establish some properties of the nonlinear term appearing in (1.1).

We denote by and , respectively, the scalar product and associated norm in , and by the scalar product in of the gradients of and . We consider the scalar product in defined by

where its associated norm is, in fact, equivalent to the usual gradient norm. We denote by the closure in of the set

and by the closure of in Then is a Hilbert space equipped with the inner product of , and is a Hilbert subspace of

Denote by the Stokes operator, with domain defined by

where is the projection operator from onto . Recall that as is defines in a norm which is equivalent to the norm, that is, there exists a constant , depending only on , such that

and so is a Hilbert space with respect to the scalar product

For and we define as the element of given by

where by we denote either the duality product between and or between and .

Observe that (2.6) is meaningful, since and with continuous injections. This implies that and there exists a constant depending only on such that

Observe also that if then the definition above coincides with the definition of as the vector function whose components are for . However, as it not known whether the solutions of the stochastic problem (1.1) have the same regularity as the deterministic case (we only can ensure instead of ), the present extension is necessary.

Now, if , then and consequently, for , we have that with

It follows that there exists a constant depending only on , such that

We have the following results.

Proposition 2.1. *For all and for all it follows that
*

*Proof. *If then for each we have and consequently
using we have (2.10).

Consider now the bilinear form defined by

Proposition 2.2. *The bilinear form satisfies
**
and consequently,
**
Moreover, there exists a constant depending only on such that
**
Thus, in particular, is continuous on *

*Proof. *The proof is straightforward consequences of (2.7), (2.9), and (2.10).

#### 3. Statement of the Problem and the Main Results

We now introduce some probabilistic evolutions spaces.

Let be a filtered probability space and let be a Banach space. For , we denote by

the space of functions with values in defined on and such that

is measurable with respect to and for almost all , is measurable, where denotes the mathematical expectation with respect to the probability measure .The space so defined is a Banach space.

When , the norm in is given by

We make precise our assumptions on (1.1).

We start with the nonlinear function and . We assume that

We will define the concept of weak solution of the problem (1.1), namely, the following.

*Definition 3.1. *A weak solution of (1.1) means a system such that (1) is a probability space, is a filtration, (2) is an -dimensional standard Wiener process, (3) is adapted for all (4)for almost all , the following equation holds
for all

Our two major results are as follows.

Theorem 3.2 (Existence). *Assume (3.4) and . Then there exists a weak solution of (1.1) in the sense of Definition 3.1. **Moreover and there exists a unique for all , such that -a.s. **
where denotes the time derivative of , that is, by definition
*

Corollary 3.3 (Uniqueness). *Assume that and are Lipschitz with respect to the second variable . Then there exists a unique weak solution of problem (1.1) in the sense of Definition 3.1. **Moreover, two strong solutions on the same Brownian stochastic basis coincide a.s.*

##### 3.1. Formulation of Problem (1.1) as an Abstract Problem

We will rewrite our model as an abstract problem.

We identify with its topological dual and we have the Gelfand triple .

We denote by the duality product between and . We define

It is clear that for all ,

and, if we denote by and the eigenvalues, and their corresponding eigenvalues associated to then

Thus, taking

we have

(a) is a linear continuous operator , such that On the other hand, denote Thus it is straightforward to check that if we take then we obtain that (b) is a bilinear mapping such that (c) measurable such that Now, let denote the identity operator in and define as is bijective from onto and Thus, for each where that is, for all so which implies Therefore, and, consequently, taking we obtain that (d) measurable such thatwhere . Next, for each and all we have

and, for all , it follows that

Consequently, in this abstract framework, a weak solution of (1.1) is equivalently as follows.

*Definition 3.4. *It holds that(1) is a probability space, is a filtration, (2) is a -dimensional standard Wiener process, (3) is adapted for all (4)for almost all , the following equation holds
as an equality in .

*Remark 3.5. *However, (3.30) implies that then is weakly continuous in [14, page 263] and the initial condition is meaningful.

#### 4. Proofs of the Main Results

##### 4.1. Proof of Theorem 3.2

We make use of the Galerkin approximation combined with the method of compactness.

We will split the proof into six steps.

###### 4.1.1. Step 1. Construction of an Approximating Sequence

As the injection is compact, consider an orthonormal basis which is orthogonal in such that are eigenfunctions of the spectral problem

where denotes the scalar product in . For each , let be the span of .

Consider the probabilistic system

We denote by the mathematical expectation with respect to .

We look for a sequence of functions in , that is,

solutions of the following stochastic ordinary differential equations in

where and is chosen with the requirements that

There exists a a maximal solution to (4.4), that is, a stopping time such that (4.4) holds for [11]. Solvability over will follow from a priori estimates for that we derive in the following section.

We have the following Fourier expansion:

###### 4.1.2. Step 2. A Priori Estimates

Throughout and denotes a positive constant independent of .

We have the following Lemma.

Lemma 4.1. *It holds that satisfies the following a priori estimates:
**
where is a constant independent of .*

*Proof. *By Ito’s formula, we obtain from (3.16) and (4.4) that
Integrating (4.8) with respect to , and using (3.13) and (3.19), we have
Let us estimate the stochastic integral in this inequality. By Burkholder-Davis Gundy’s inequality [15], we have
here we have used and Young’s inequalities; is an arbitrary positive number.

Using (4.10) and (4.9) together with appropriate choice of , we obtain

By Gronwall’s lemma, we obtain the sought estimate (4.7).

The following result is related to the higher integrability of .

Lemma 4.2. *It holds that
*

*Proof. *By Ito’s formula, it follows from (4.4) that for , we have
Using the assumptions (3.16), (3.19), (3.26), it follows that
Squaring the both sides of this inequality and passing to mathematical expectation, we deduce from the Martingale inequality, that is,
From Gronwall’s inequality, we deduce that
for all .

We also have the following lemma.

Lemma 4.3. *It holds that satisfies
**
for all .*

*Proof. *Using (4.9), we have
Taking the mathematical expectation and use the Burkholder-Gundy’s inequality, the proof of the lemma follows from Lemma 4.2.

Lemma 4.4. *It holds that
*

*Proof. *We note that the functions form an orthonormal basis in the dual of Let be the orthogonal projection of onto the span that is,

Thus (4.4) can be rewritten in an integral form as the equality between random variables with values in as

For any positive , we have
Taking the square and use the properties of and , we have
For fixed , taking the supremun over , integrating with respect to , and taking the mathematical expectation, we have
We estimate the integrals in this inequality.

We have by inequality

Using the inequality and the estimates of Lemmas 4.2 and 4.3, we have
By Martingale’s inequality, we have
Using the assumptions on and the estimate of Lemma 4.2, we have
Collecting the results and making a similar reasoning with , we obtain from (4.24) that

The following lemma is from [16], and it is a compactness results which represents a variation of the compactness theorems in [17, Chapter I, Section ]. It will be useful for us to prove the tightness property of Galerkin solution.

Proposition 4.5. *For any sequences of positives reals number which tend to as , the injection of
**
in is compact.*

Furthermore is a Banach space with the norm

Alongside with , we also consider the space of random variables such that

Endowed with the norm

is a Banach space. The priori estimates of the preceding lemmas enable us to claim that for any and for such that the series converges, the sequence of Galerkin solutions is bounded in .

###### 4.1.3. Step 3. Tightness Property of Galerkin Solutions

Now, we consider the set

and the-algebra of the Borel sets of .

For each , let be the map

For each , we introduce a probability measure on by

for all . The main result of this subsection is the following.

Theorem 4.6. *The family of probability measures is tight.*

*Proof. *For , we should find the compact subsets
such that
The quest for is made by taking account of some fact about the Wiener process such as the formula
For a constant depending on to be chosen later and , we consider the set
Making use of Markov’s inequality:
for a random variable on and positives variables and , we get
we choose
to get (4.38).

Next we choose as a ball of radius in centered at zero and with independent of , converging to zero, and such that converges.

From Proposition 4.5, is a compact subset of .

We have further

choosing , we get (4.39).

From (4.38) and (4.39), we have

this proves that

###### 4.1.4. Step 4. Applications of Prokhorov and Skorokhod Results

From the tightness property of and Prokhorov’s theorem [12], we have that there exist a subsequence and a measure such that weakly.

By Skorokhod’s theorem [13], there exist a probability space and random variables on with values in such that

Hence, is a sequence of an -dimensional standard Wiener process.

Let .

Arguing as in [16], we prove that is an -dimensional standard Wiener process and the pair satisfies the equation

###### 4.1.5. Step 5. Passage to the Limit

From (4.51), it follows that satisfies the results of the Lemmas 4.2, 4.3, and 4.4. Therefore, we have for the a priori estimates

thus modulo the extraction of a subsequence denoted again by , we have

By (4.50) and the first estimate in (4.52) and Vitali’s theorem, we have

and thus modulo the extraction of a subsequence and for almost every with respect to the measure :

This convergence together with the condition on , the first estimate in (4.52) and Vitali’s theorem, give

As

then

We also have

In fact, since is dense in , and is bounded in it suffices to prove that

Indeed, we have

By the property (3.17) of , we have

applying Cauchy-Schwarz inequality

Since

and is bounded in , we conclude that

Again thanks to the property (3.18) of , as

we obtain as since any strongly continuous linear operator is weakly continuous. We are now left with the proof of

which can be prove with the same argument like in [16].

Collecting all the convergence results, we deduce that

as the equality in .

We have , , .

Thus, from the classical results in [18] (see also [19]), we deduce from (4.68) that is continuous with values in .

###### 4.1.6. Step 6. Existence of the Pressure

For the existence of the pressure, we use a generalization of the Rham’s theorem processes [20, Theorem , Remark ]. From (3.6), we have for all ,

We denote

We will prove that the regularity on , implies that

By (2.7) and (2.9), we have as ,

We also have

Again, as , then its follows that

for all .

Then , and

Therefore, by a generalization of the Rham theorem processes [20], there exists a unique such that

Theorem 3.2 is proved.

##### 4.2. Proof of Corollary 3.3

*Proof. *We will prove the pathwise uniqueness which implies uniqueness of weak solutions. Let and be two real such that
Then and are defined, respectively, by (3.14) and (3.20) satisfying