Solution to the Navier-Stokes equations with random initial data

2011/06/20 by Evelina Shamarova, Shamarova, Evelina
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #math.AP #math.PR

paper · pdf · doi:10.48550/arxiv.1106.3861

This paper has been withdrawn by the author due to an error that the author failed to correct

arxiv created 2016/03/14 · arxiv updated 2016/03/15

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∈ 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))] = ∫_H^α f(e^tνΔ u) γ(du), where f ∈ L₁(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.

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