On unbiased stochastic Navier-Stokes equation

TL;DR

This paper introduces an unbiased stochastic perturbation of the Navier-Stokes equation using Wick-type nonlinearity, solved via Wiener chaos expansion, revealing a Markov process with solutions scaled by Catalan numbers.

Contribution

It presents a novel unbiased stochastic perturbation of Navier-Stokes equations and analyzes its solutions using advanced stochastic calculus techniques.

Findings

Solution forms a Markov process

Generalized solutions scale with Catalan numbers

Perturbation preserves the expectation of the original equation

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\nabla 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.