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Inviscid limit and an effective energy-enstrophy diffusion process

Alain-Sol Sznitman, Klaus Widmayer

Abstract

In this article we consider a stationary $N$-dimensional Galerkin-Navier-Stokes type evolution with Brownian forcing and random stirring (of arbitrarily small strength). We show that the stationary diffusion in an open two-dimensional cone constructed in a companion article, stands as the inviscid limit of the laws of the ``enstrophy-energy'' process of the $N$-dimensional diffusion process considered here, this regardless of the strength of the stirring. With the help of the quantitative condensation bounds of the companion article, we infer quantitative inviscid condensation bounds, which for suitable forcings show an attrition of all but the lowest modes in the inviscid limit.

Inviscid limit and an effective energy-enstrophy diffusion process

Abstract

In this article we consider a stationary -dimensional Galerkin-Navier-Stokes type evolution with Brownian forcing and random stirring (of arbitrarily small strength). We show that the stationary diffusion in an open two-dimensional cone constructed in a companion article, stands as the inviscid limit of the laws of the ``enstrophy-energy'' process of the -dimensional diffusion process considered here, this regardless of the strength of the stirring. With the help of the quantitative condensation bounds of the companion article, we infer quantitative inviscid condensation bounds, which for suitable forcings show an attrition of all but the lowest modes in the inviscid limit.
Paper Structure (9 sections, 15 theorems, 199 equations)

This paper contains 9 sections, 15 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. The set-up
  3. The limit diffusion process
  4. Some controls
  5. Tightness
  6. The weak convergence result
  7. Inviscid condensation bounds
  8. Appendix: The spaces ${\mathbb X}_{u,v}$
  9. Appendix: Convergence to equilibrium on ${\mathbb X}_{u,v}$

Key Result

Lemma 1.2

Given $\psi(u,v)$ a $C^2$-function on ${\mathbb R}^2$, setting $g(x) = \psi ( | x |^2, |x|^2_{-1})$, for $x$ in ${\mathbb R}^N$, one has for $\varepsilon > 0$, $x \in {\mathbb R}^N$ (the partial derivative of $\psi$ are evaluated at $u = |x|^2, v = |x|^2_{-1}$). Further, if $g_0(x) = \exp \{ \frac{1}{2a} \;|x|^2_{-1}\}$, for $x$ in ${\mathbb R}^N$, then for $\varepsilon > 0$,

Theorems & Definitions (35)

  • Remark 1.1
  • Lemma 1.2
  • proof
  • Proposition 1.3
  • proof
  • Remark 1.4
  • Proposition 1.5
  • proof
  • Proposition 3.1
  • proof
  • ...and 25 more