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We investigate the stochastic 3D Navier-Stokes-

In this paper, we are interested in the study of probabilistic weak solutions of the 3D Navier-Stokes-

The deterministic version of (

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 (

In this paper, we will establish the existence of probabilistic weak solutions for the problem (

The paper is organized as follows. In Section

Following [

We denote by

Denote by

Observe that (

Now, if

For all

If

Consider now the bilinear form defined by

The bilinear form

The proof is straightforward consequences of (

We now introduce some probabilistic evolutions spaces.

Let

The space

When

We make precise our assumptions on (

We start with the nonlinear function

A weak solution of (

for almost all

Our two major results are as follows.

Assume (

Moreover

Assume that

Moreover, two strong solutions on the same Brownian stochastic basis coincide a.s.

We will rewrite our model as an abstract problem.

We identify

We denote by

where

It holds that

for almost all

However, (

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

We will split the proof into six steps.

As the injection

Consider the probabilistic system

We look for a sequence of functions

We have the following Fourier expansion:

Throughout

We have the following Lemma.

It holds that

By Ito’s formula, we obtain from (

Using (

The following result is related to the higher integrability of

It holds that

By Ito’s formula, it follows from (

We also have the following lemma.

It holds that

Using (

It holds that

We note that the functions

Thus (

We have by

The following lemma is from [

For any sequences of positives reals number

Furthermore

Now, we consider the set

For each

The family of probability measures

For

Next we choose

From Proposition

We have further

From (

From the tightness property of

By Skorokhod’s theorem [

Let

Arguing as in [

From (

Collecting all the convergence results, we deduce that

We have

Thus, from the classical results in [

For the existence of the pressure, we use a generalization of the Rham’s theorem processes [

Then

We will prove the pathwise uniqueness which implies uniqueness of weak solutions. Let

We denote

Applying Ito’s formula to the real process

By young’s inequality, we have

Using the famous Yamada-Watanabe theorem [

The research of the authors is supported by the University of Pretoria and the National Research Foundation South Africa.