Two-dimensional stochastic Navier–Stokes equations with fractional Brownian noise

Abstract

We study the perturbation of the two-dimensional stochastic Navier-Stokes equation by a Hilbert-space-valued fractional Brownian noise. Each Hilbert component is a scalar fractional Brownian noise in time, with a common Hurst parameter H and a specific intensity. Because the noise is additive, simple Wiener-type integrals are suffi cient for properly defining the problem. It is resolved by separating it into a deterministic nonlinear PDE, and a linear stochastic PDE. Existence and uniqueness of mild solutions are established under suitable conditions on the noise intensities for all Hurst parameter values. Almost surely, the solution’s paths are shown to be quartically integrable in time and space. Whether this integrability extends to the random parameter is an open question. An extension to a multifractal model is given. 2000 AMS Classification: Primary 60H15, 35Q30; Secondary 76D05, 35R60.