On the Infinite Dimension Limit of Invariant Measures and Solutions of Zeitlin’s 2D Euler Equations

Abstract

In this work we consider a finite dimensional approximation for the 2D Euler equations on the sphere {mathbb {S}}^2, 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.