On a stochastic version of Prouse model in fluid dynamics

A stochastic version of modified Navier–Stokes equations (introduced by Prouse) is considered in a three-dimensional torus; its main feature is that instead of the linear term − ν △ u of the Navier–Stokes equations there is a nonlinear term − △ Φ ( u ) − ∇ div Φ ( u ) . First, for this equation we prove existence and uniqueness of martingale solutions; then existence of stationary solutions. In the last part of the paper a new model, obtained from Prouse model with the nonlinearity Φ ( u ) = ν | u | 4 u , is analysed; for the structure function of this model, some insights towards an expression similar to that obtained by the Kolmogorov 1941 theory of turbulence are presented.