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Effective energy-enstrophy diffusion process and condensation bound

Alain-Sol Sznitman, Klaus Widmayer

TL;DR

The authors construct a two-dimensional diffusion on the interior of an open cone in $\mathbb{R}^2$ whose coefficients are given by conditional expectations of a Gaussian measure on $\mathbb{R}^N$ conditioned on two quadratic forms (enstrophy and energy). They prove existence and uniqueness of a stationary distribution for this diffusion, and, in a companion paper, show that this diffusion arises as the inviscid limit of the enstrophy-energy process for a stationary Galerkin-Navier–Stokes system with Brownian forcing. A key contribution is a condensation bound that quantitatively bounds how far the ratio of the expected energy to the expected enstrophy is from 1, with the bound governed by the forcing spectrum and the low-lying modes. Together, these results illuminate how inviscid forcing can drive condensation onto the lowest Fourier modes, bridging Gaussian-structured diffusion with nonlinear fluid dynamics in the inviscid regime.

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.

Effective energy-enstrophy diffusion process and condensation bound

TL;DR

The authors construct a two-dimensional diffusion on the interior of an open cone in whose coefficients are given by conditional expectations of a Gaussian measure on conditioned on two quadratic forms (enstrophy and energy). They prove existence and uniqueness of a stationary distribution for this diffusion, and, in a companion paper, show that this diffusion arises as the inviscid limit of the enstrophy-energy process for a stationary Galerkin-Navier–Stokes system with Brownian forcing. A key contribution is a condensation bound that quantitatively bounds how far the ratio of the expected energy to the expected enstrophy is from 1, with the bound governed by the forcing spectrum and the low-lying modes. Together, these results illuminate how inviscid forcing can drive condensation onto the lowest Fourier modes, bridging Gaussian-structured diffusion with nonlinear fluid dynamics in the inviscid regime.

Abstract

In this article we use Gaussian measure on to define the coefficients of an elliptic diffusion on an open cone of . 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 -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 of the ratio of the expected energy to the expected enstrophy (this ratio is at most 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.
Paper Structure (8 sections, 14 theorems, 152 equations)

This paper contains 8 sections, 14 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. The set-up
  3. Some properties of the functions $q_\ell$
  4. The effective diffusion
  5. The stationary distribution
  6. Condensation bound
  7. Appendix: Some density and volume calculations
  8. Appendix: Conditional expectations

Key Result

Lemma 2.1

Theorems & Definitions (34)

  • Remark 1.1
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 24 more