2D hydrodynamical systems: invariant measures of Gaussian type

TL;DR

This paper constructs and analyzes Gaussian invariant measures for stochastic 2D hydrodynamical systems, proving existence, uniqueness, and invariance of the flow, and establishing stationary solutions for the inviscid limit.

Contribution

It introduces a class of Gaussian measures for stochastic 2D fluids, proving their invariance and uniqueness, and links these measures to stationary solutions of the deterministic inviscid equations.

Findings

Existence and uniqueness of global flow for stochastic viscous systems.

Invariance of the Gaussian measures under the flow.

Existence of stationary solutions for the inviscid deterministic equations.

Abstract

Gaussian measures $\mu^{\beta,\nu}$ are associated to some stochastic 2D hydrodynamical systems. They are of Gibbsian type and are constructed by means of some invariant quantities of the system depending on some parameter $\beta$ (related to the 2D nature of the fluid) and the viscosity $\nu$. We prove the existence and the uniqueness of the global flow for the stochastic viscous system; moreover the measure $\mu^{\beta,\nu}$ is invariant for this flow and is unique. Finally, we prove that the deterministic inviscid equation has a $\mu^{\beta,\nu}$-stationary solution (for any $\nu>0$).