Abstract:We use Gaussian measure on $\mathbb{R}^N$ to define the coefficients of an elliptic diffusion and show that it lives in an open cone of $\mathbb{R}^2$. One component represents enstrophy and the other energy. We establish the existence and uniqueness of a stationary distribution for this diffusion. Owing to the special properties of the coefficients of this diffusion, we derive a condensation bound, which controls the distance to $1$ of the ratio of the expected energy to the expected enstrophy (this ratio is at most $1$ with our normalization). In a companion article, as a ``proof of concept'', 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, and which plays the role of a regularization).
From: Alain-Sol Sznitman [view email]
[v1]
Tue, 17 Feb 2026 18:46:21 UTC (35 KB)
[v2]
Wed, 8 Jul 2026 17:07:42 UTC (35 KB)
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