Invariant measures for the two-dimensional averaged-Euler equations

TL;DR

This paper constructs Gaussian invariant measures for the 2D averaged-Euler equations, demonstrating the existence of solutions with initial conditions on the measure's support and establishing recurrence properties.

Contribution

It introduces a Gaussian invariant measure for the 2D averaged-Euler equations and constructs an invariant surface measure, advancing understanding of the system's long-term behavior.

Findings

Existence of solutions with initial conditions on the measure's support

Construction of an invariant surface measure on energy level sets

Application of Poincaré recurrence theorem to the flow

Abstract

We define a Gaussian invariant measure for the two-dimensional averaged-Euler equation and show the existence of its solution with initial conditions on the support of the measure. An invariant surface measure on the level sets of the energy is also constructed, as well as the corresponding flow. Poincar\'e recurrence theorem is used to show that the flow returns infinitely many times in a neighbourhood of the initial state.