A Lagrangian Approach to the Inhomogeneous Incompressible Euler Equation

TL;DR

This paper presents a Lagrangian framework for the inhomogeneous incompressible Euler equation, revealing geometric structures, deriving from a Hamilton-Pontryagin principle, and establishing analyticity.

Contribution

It introduces a novel Lagrangian and geometric formulation of the IIE, including a new vorticity expression and analyticity proof.

Findings

Established a geodesic description of IIE.

Derived a new vorticity formulation.

Proved Lagrangian analyticity of IIE.

Abstract

In this paper, we study the inhomogeneous incompressible Euler equation (IIE in short) from a Lagrangian perspective. We establish a geodesic description of this equation and discuss the associated geometric structures. We also find the derivation of IIE from the Hamilton-Pontryagin action principle and derive the corresponding Lagrangian formulation. A byproduct is a new vorticity formulation of IIE. We also prove the Lagrangian analyticity of IIE using our Lagrangian representation formula.