Effective energy-enstrophy diffusion process and condensation bound

TL;DR

This paper constructs a specific elliptic diffusion process on an open cone in , proves its stationary distribution, and, together with a companion article, demonstrates that under certain conditions, an inviscid condensation phenomenon occurs in a Galerkin-Navier-Stokes type system.

Contribution

It introduces a new elliptic diffusion process related to energy-enstrophy dynamics and establishes its stationary distribution, linking it to inviscid limits and condensation phenomena in fluid models.

Findings

Bound the ratio of expected energy to enstrophy in the constructed diffusion.

Show that the diffusion is the inviscid limit of a Galerkin-Navier-Stokes type process.

Demonstrate condensation phenomena in the inviscid limit under Brownian forcing.

Abstract

In this article we use Gaussian measure on $\mathbb{R}^N$ to define the coefficients of an elliptic diffusion on an open cone of $\mathbb{R}^2$. We prove the existence and uniqueness of a stationary distribution for this diffusion. In a companion article, we show that the diffusion constructed in this work is the inviscid limit of the laws of the ``enstrophy-energy'' process of a stationary $N$-dimensional Galerkin-Navier-Stokes type evolution with Brownian forcing and random stirring (the strength of which can be made to go to zero in the inviscid limit). In the present work, owing to the special properties of the coefficients constructed with the Gaussian measure, we bound the distance to $1$ of the ratio of the expected energy to the expected enstrophy (this ratio is at most $1$ with our normalization). Together with our companion article, this shows that for suitable Brownian forcings an inviscid condensation inducing an attrition of all but the lowest modes takes place.