Solution to the Navier-Stokes equations with random initial data

TL;DR

This paper constructs a stochastic solution to the Navier-Stokes equations with Gaussian initial data in Sobolev spaces, using Galerkin approximations and analyzing its distributional properties.

Contribution

It introduces a novel method for constructing solutions to Navier-Stokes with random initial conditions in Sobolev spaces, including their distributional characteristics.

Findings

Solution exists as a limit of Galerkin-type ODE solutions

The solution is a stochastic process satisfying Navier-Stokes almost surely

Expected values of functions of the solution relate to the heat semigroup and initial Gaussian measure

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 $H^\alpha$, where the index $\alpha\in R$ is fixed arbitrary. The distribution of the initial value is a Gaussian measure on $H^\alpha$ whose parameters depend on $\alpha$. 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 $U(t,\omega)$ possesses the property: $$E[f(U(t,\omega))] = \int_{H^\alpha} f(e^{t\nu\Delta} u) \gamma(du)$$, where $f \in L_1(\gamma)$, $e^{t \Delta}$ is the heat semigroup, $\nu$ is the viscosity in the Navier-Stokes equations, and $\gamma$ is the distribution of the initial data.