3D incompressible Euler: Geometric formalism and a hypothetical self-similar flow

TL;DR

This paper presents a geometric framework for 3D incompressible Euler equations, including gauges and properties of a hypothetical self-similar solution, with a focus on mathematical structure and potential solution behaviors.

Contribution

It introduces a unified geometric formalism for Euler equations and analyzes the properties of a conjectured self-similar flow within this framework.

Findings

Euler equations formulated as an analytic ODE in Banach space in Lagrangian gauge

Description of basic properties of a hypothetical self-similar solution

Unification of Eulerian and Lagrangian gauges in a geometric setting

Abstract

We give a geometric formulation of 3D incompressible Euler that contains the Eulerian and Lagrangian gauges as special cases. In the Lagrangian gauge, incompressible Euler is a real analytic ODE in Banach space; a short proof of this known result is given. We then describe (in a self-contained section) some basic properties of a hypothetical self-similar solution to 3D incompressible Euler.