Probabilistic estimates for the Two Dimensional Stochastic Navier-Stokes Equations

TL;DR

This paper provides probabilistic estimates for the long-term behavior of solutions to the two-dimensional stochastic Navier-Stokes equations with random forcing, offering bounds on dissipation scale and energy spectrum as the Reynolds number increases.

Contribution

It introduces new probabilistic bounds for the long-time behavior of 2D stochastic Navier-Stokes solutions, applicable at arbitrary Reynolds numbers.

Findings

Bounds for dissipation scale as R→∞

Energy spectrum estimates under stochastic forcing

Probabilistic long-term behavior analysis

Abstract

We consider the Navier-Stokes equation on a two dimensional torus with a random force, white noise in time and analytic in space, for arbitrary Reynolds number $R$. We prove probabilistic estimates for the long time behaviour of the solutions that imply bounds for the dissipation scale and energy spectrum as $R\to\infty$.