The Poincar\'e Recurrence Problem of Inviscid Incompressible Fluids

TL;DR

This paper addresses the Poincaré recurrence problem in inviscid incompressible fluids, clarifying previous examples and providing a rigorous framework for understanding recurrence behavior in the 2D Euler equations.

Contribution

It corrects and completes Nadirashvili's example by establishing an appropriate phase space and detailed proof for the recurrence issue in 2D Euler flows.

Findings

Nadirashvili's example does not imply recurrence in the original phase space.

A proper phase space setting confirms the non-recurrence near certain solutions.

The paper provides a rigorous proof of the recurrence problem in the context of 2D Euler equations.

Abstract

Nadirashvili presented a beautiful example showing that the Poincar\'e recurrence does not occur near a particular solution to the 2D Euler equation of inviscid incompressible fluids. Unfortunately, Nadirashvili's setup of the phase space is not appropriate, and details of the proof are missing. This note fixes that.