$L^1$-Uniqueness of Kolmogorov Operators Associated to 2D Stochastic Navier-Stokes Coriolis Equations with Space-Time White Noise

TL;DR

This paper proves the $L^1$-uniqueness of the Kolmogorov operator associated with a 2D stochastic Navier-Stokes equation with Coriolis force driven by space-time white noise, modeling geophysical flows.

Contribution

It establishes $L^1$-uniqueness of the Kolmogorov operator for the stochastic Navier-Stokes-Coriolis equations with space-time white noise, a novel result in this context.

Findings

Gaussian enstrophy measure is infinitesimally invariant for the Kolmogorov operator

$L^1$-uniqueness holds for sufficiently large viscosity

Results contribute to understanding invariant measures in stochastic geophysical fluid dynamics

Abstract

We consider the Kolmogorov operator $K$ associated to a stochastic Navier-Stokes equation driven by space-time white noise on the two-dimensional torus with periodic boundary conditions and a rotating reference frame, introducing fictitious forces such as the Coriolis force. This equation then serves as a simple model for geophysical flows. We prove that the Gaussian measure induced by the enstrophy is infinitesimally invariant for $K$ on finitely based cylindrical test functions and moreover $K$ is $L^1$-unique w. r. t. the enstrophy measure for sufficiently large viscosity.