On the Ergodic theory of the Generalized incompressible flow

TL;DR

This paper explores the ergodic properties of generalized incompressible flows (GIFs), revealing their weak recurrence, defining ergodicity for GIFs, and establishing related ergodic theorems to better understand classical Euler solutions.

Contribution

It introduces the ergodic theory of GIFs, including weak recurrence, ergodicity, and structural theorems, advancing the understanding of fluid dynamics solutions.

Findings

GIFs have weak recurrence rather than classical recurrence

Defined ergodicity for GIFs and related it to classical ergodic flows

Proved ergodic theorems for GIFs and described their structural properties

Abstract

To study the variation problem related to the incompressible fluid mechanics, Brenier brings the concept of generalized flow and shows that the generalized incompressible flow (GIF) is deeply related to the classical solution of the incompressible Euler equations. In this paper, we will study the ergodic theory of the GIF which may help us understand the dynamic property of the classical solution of the incompressible Euler equations. First, we show that the GIF has the weak recurrent property rather than the classical one. Then, we define the ergodicity of the GIF and discuss its relation with the classical ergodic flow. Next, we prove some ergodic theorems of the GIF. Finally, we give a theorem about the structure of the set of all GIFs.