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−ν△u of the Navier–Stokes equations there is a nonlinear term −△Φ(u)−∇divΦ(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|4uΦ(u)=ν|u|4u, 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.