A Lagrangian Approach to the Inhomogeneous Incompressible Euler Equation

TL;DR

The paper develops a comprehensive Lagrangian–geometric framework for the inhomogeneous incompressible Euler equations, formulating the dynamics as geodesic flow on the volume-preserving diffeomorphism group with a density-weighted metric. It derives a Hamilton–Pontryagin principle, yielding a Lagrangian–Eulerian representation and a pressure-free vorticity formulation, from which local well-posedness in 2D is obtained. A central contribution is the demonstration of Lagrangian analyticity of the flow, using recursive Taylor coefficient constructions and elliptic regularity intrinsic to the Biot–Savart-type operator, supported by a Gibbs–Appell perspective. These results bridge geometric hydrodynamics with the IIE, providing explicit structure for stability via the second fundamental form and curvature, and establishing analytic regularity of Lagrangian trajectories with potential implications for long-time behavior and stability analyses.

Abstract

In this paper, we study the Lagrangian aspects of the inhomogeneous incompressible Euler equation (IIE in short). 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. Appealing to this Lagrangian perspective, we prove Lagrangian analyticity of IIE.