Spatial analyticity and exponential decay of Fourier modes for the stochastic Navier-Stokes equation

TL;DR

This paper constructs a local-in-time spatially real-analytic solution to the stochastic Navier-Stokes equations in 2D and 3D with real-analytic noise, and proves exponential decay of Fourier modes for the Galerkin approximation under global existence.

Contribution

It introduces a method to establish spatial analyticity and exponential decay for solutions of stochastic Navier-Stokes equations with real-analytic noise, extending understanding of their long-term behavior.

Findings

Existence of local-in-time spatially real-analytic solutions.

Exponential decay of Fourier modes in Galerkin approximations.

Decay uniformity in time, initial enstrophy, and noise coefficients.

Abstract

We construct a local in time spatially real-analytic solution to the 2D and 3D stochastic Navier--Stokes equation driven by a spatially real-analytic multiplicative and transport noise but emanating from an initial condition that is only required to have bounded enstrophy. Under the condition that the solution is global in time, we also establish the exponential decay of the finite-dimensional Galerkin approximation, with respect to its maximum wavenumber, to the strong pathwise solution of the stochastic Navier--Stokes equation. This decay is uniform in time, uniform with respect to the initial enstropy, and uniform in the noise coefficients.