On the classification of invariant Gaussian measures for the 2D Euler equation

2026/08/02 by Ziyu Liu
Mathematics · #math.AP #math.PR

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arxiv created 2026/08/02 · arxiv published 2026/08/02 · arxiv updated 2026/08/04

Abstract

We consider the two-dimensional incompressible Euler equation on \mathbb T² with Gaussian random initial data having independent Fourier coefficients. For every σ>0, we prove that such a Gaussian measure on H^σ(\mathbb T²) is invariant under the Euler flow if and only if it is supported either on shear flows or on cellular flows. This settles the invariant-measure classification conjecture posed by Bedrossian and Latocca (Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2026). The proof relies on closure of the Fourier support under non-degenerate interactions and an affine relation among the inverse variances along Euler triples.