A Recurrence Theorem on the Solutions to the 2D Euler Equation

TL;DR

This paper proves a recurrence theorem for solutions to the 2D Euler equation on the torus, showing that solutions repeatedly return close to their initial states in a specific Sobolev space.

Contribution

It introduces a recurrence theorem for 2D Euler solutions in Sobolev spaces, highlighting their long-term behavior and stability.

Findings

Solutions in H^s (s>2) return arbitrarily close to their initial states in H^0 norm

The recurrence occurs repeatedly over time

The theorem applies to solutions on the 2D torus

Abstract

In this article, I will prove a recurrence theorem which says that any $H^s(\mathbb{T}^2)$ (s>2) solution to the 2D Euler equation returns repeatedly to an arbitrarily small $H^0(\mathbb{T}^2)$ neighborhood.