On unbiased stochastic Navier-Stokes equations

Abstract

A random perturbation of a deterministic Navier-Stokes equation is considered in the form of an SPDE with Wick type nonlinearity. The nonlinear term of the perturbation can be characterized as the highest stochastic order approximation of the original nonlinear term u∇u. This perturbation is unbiased in that the expectation of a solution of the perturbed equation solves the deterministic Navier-Stokes equation. The perturbed equation is solved in the space of generalized stochastic processes using the Cameron-Martin version of the Wiener chaos expansion. It is shown that the generalized solution is a Markov process and scales effectively by Catalan numbers. © 2011 Springer-Verlag.