Solution to the Navier-Stokes equations with random initial data

Abstract

We construct a solution to the spatially periodic $d$-dimensional Navier-Stokes equations with a given distribution of the initial data. The solution takes values in the Sobolev space $H^α$, where the index $α\in R$ is fixed arbitrary. The distribution of the initial value is a Gaussian measure on $H^α$ whose parameters depend on $α$. The Navier-Stokes solution is then a stochastic process verifying the Navier-Stokes equations almost surely. It is obtained as a limit in distribution of solutions to finite-dimensional ODEs which are Galerkin-type approximations for the Navier-Stokes equations. Moreover, the constructed Navier-Stokes solution $U(t,ω)$ possesses the property: $$E[f(U(t,ω))] = \int_{H^α} f(e^{tνΔ} u) γ(du)$$, where $f \in L_1(γ)$, $e^{t Δ}$ is the heat semigroup, $ν$ is the viscosity in the Navier-Stokes equations, and $γ$ is the distribution of the initial data.