Derivation and their relation to the turbulence
As a result of such an averaging, arti¯cial viscosity is added into the system to remove energy which is contained in the small scales at which u0 resides. Since it is still necessary to guess the form of ºE, an improvement to the procedure of modelling the averaged motion of a °uid is needed.
The second approach to modelling the averaged motion of an incompressible °uid is the La- grangien averaging. The Navier-Stokes-® model ( also known in the literature as the viscous Camassa-Holm equations, or the Lagrangien averaged Navier-Stokes alpha model) is the ¯rst turbulence closure model produced by Lagrangien averaging, from which it derives its name. The inviscid Navier-Stokes-® (also called Euler-®) equations ¯rst appeared in  as a n-dimensional generalization of the one dimensional Camassa-Holm equations. The one dimensional Camassa- Holm equations describes shallow water with nonlinear dispersion and admits solitons solutions called « peakons ». Holm, Marsden and Ratiu in  used variational asymptotics to obtain the Euler-® equations on all of Rn, using an approximation of Hamilton’s principle for the Euler’s equations. Dissipation was added to the Euler-® equations to produce the Navier- Stokes-® equations. The extension of this approach to bounded domains was made in  by averaging over the set of solutions u² of the Euler equations with initial data u² 0 in a phase- space ball of radius ®.
Previous analytical results: Deterministic and Stochastic.
In , the deterministic Cauchy problem for the three dimensional Navier-Stokes-® model subject to periodic boundary conditions was studied. The global existence and uniqueness of weak solutions were established, the regularity of weak solutions was proved and the global attractor for this model was constructed. Moreover, upper bounds for the dimension of the global attractor were found in terms of the relevant physical parameters. It has been also proved that the solutions of the Navier-Stokes-® model converge to certain solutions of the three dimensional Navier-Stokes equations as ® approaches zero. These results were extended to the case of Dirichlet-type boundary conditions in . The authors of  used a sequence of classical solutions in ,to prove that this sequence converges in C([0; T];H1) to a H1-weak solution of the Navier-Stokes-® model for all T > 0. They also proved the existence of a nonempty, compact, convex, and connected global attractor. The authors of  study the connection between the long-time dynamics of the three dimensional Navier-Stokes-® model and the three dimensional Navier-Stokes equations as ® approaches zero. In particular, they showed that the trajectory attractor of the Navier-Stokes-® model converges to the trajectory attractor of the three dimensional Navier-Stokes system when ® approaches zero. Similar results were proved in ,, for the Leray-® model.
The mathematical literature for the stochastic Navier-Stokes equations is extensive and dates back to early 1970’s with the work of Bensoussan and Temam . It is well known that there exists a probabilistic weak solution (also called martingale solution) for the stochastic three dimensional Navier-Stokes equations (see , just to cite a few). But uniqueness is open.
Brze¶zniak and Peszat in , Mikulevicius and Rozovskii in , obtained the existence and uniqueness of a strong maximal local solution in W1 p with p > 3. Recently, Glatt and Ziane , Mikulevicius have established the existence and uniqueness of local strong H1-solution. Here the word « strong » means « strong » in the sense of the theory of stochastic di®erential equations; that is a complete probability space and a Wiener process are given in advance. All these results are global in two dimensions. Breckner  as well as Menaldi and Sritharan  established the existence and uniqueness of strong global L2-solution for the two dimensional stochastic Navier- Stokes equations. The proof in  used the local monotonicity of the nonlinearity to obtain the solution.
The authors of  proved the existence and uniqueness of probabilistic strong solutions for the three dimensional stochastic Navier-Stokes-® model under the Lipschitz assumptions on the coe±cients. The proof of the existence uses the Galerkin approximation and the weak convergence methods. The asymptotic behavior for the three dimensional stochastic Navier- Stokes-® model was proved in . To the best of our knowledge, there is no systematic work for the three dimensional stochastic Leray-® model.
1.1 The Navier-Stokes equations and turbulence
1.2 The Navier-Stokes-® model and the Leray-® model
1.3 Main Results and Organization of the thesis
2 On the Stochastic 3D Navier-Stokes-® Model
2.1 Introduction 2.2 Properties of the nonlinear terms
2.3 Statement of the problem and the main result
2.4 Proof of the main result
3 The Stochastic 3D Navier-Stokes-® model: ® tends to 0 37
3.2 The Stochastic 3D Navier-Stokes equations 3.3 The stochastic 3D Navier-Stokes-® model
3.4 Asymptotic behavior of the stochastic 3D Navier-Stokes-® model
4 Strong solution for the 3D Stochastic Leray-® Model
4.2 Statement of the problem and the first main result
4.3 Galerkin approximation and a priori estimates
4.4 Proof of Theorem 7
4.5 Proof of Theorem 8
4.6 Asymptotic behavior as ® approaches zero