On the infinite dimension limit of invariant measures and solutions of Zeitlin's 2D Euler equations

TL;DR

This paper studies the convergence of Zeitlin's finite-dimensional approximation of 2D Euler equations on the sphere to solutions with enstrophy measure marginals, using novel computations on structure constants.

Contribution

It demonstrates the convergence of Zeitlin's approximation to Euler solutions with enstrophy measure marginals and introduces new computations of structure constants on the sphere.

Findings

Convergence of Zeitlin's approximation to Euler solutions with enstrophy measure.

Novel computations of structure constants on the sphere.

Discussion on extending results to Gibbsian measures with higher Casimirs.

Abstract

In this work we consider a finite dimensional approximation for the 2D Euler equations on the sphere, proposed by V. Zeitlin, and show their convergence towards a solution to Euler equations with marginals distributed as the enstrophy measure. The method relies on nontrivial computations on the structure constants of $\mathbb{S}^2$, that appear to be new. In the last section we discuss the problem of extending our results to Gibbsian measures associated with higher Casimirs.