On the Stochastic 3D Navier‐Stokes‐α Model of Fluids Turbulence

Abstract

We investigate the stochastic 3D Navier-Stokes-αmodel 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.

Weak Solutions of a Stochastic Model for Two-Dimensional Second Grade Fluids

“…Since then, stochastic partial differential equations and stochastic models of fluid dynamics have been the object of intense investigations which have generated several important results. We refer, for instance, to [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] . Similar investigations for Non-Newtonian fluids have almost not been undertaken except in very few work; we refer, for instance, to [23][24][25] for some computational studies of stochastic models of polymeric fluids.…”

On the Strong Solution for the 3D Stochastic Leray-Alpha Model

“…Here, the word "strong" means "strong" in the sense of the theory of stochastic differential equations, assuming that the stochastic processes are defined on a complete probability space and the Wiener process is given in advance. Since we consider the strong solution of the stochastic Leray-alpha equations, we do not need to use the techniques considered in the case of weak solutions see [5][6][7][8][9] . The techniques applied in this paper use in particular the properties of stopping times and some basic convergence principles from functional analysis see 10-13 .…”

Convergence for a Splitting-Up Scheme for the 3D Stochastic Navier-Stokes-α Model

“…In [7], they studied the asymptotic behavior of its solution when the time tends to infinity. Deugoué and Sango [14] extended the result in [6] to the case of non-Lipschitz assumptions on the coefficients. In particular, they proved the existence of a weak martingale solution…”