On geometric hydrodynamics and infinite-dimensional magnetic systems

TL;DR

This paper introduces the magnetic Euler-Arnold equation, unifying various infinite-dimensional magnetic systems under a geometric framework, and establishes well-posedness results for specific equations like the quasi-geostrophic.

Contribution

It combines Arnold's geometric approach with magnetic field formulations to define the magnetic Euler-Arnold equation, linking it to well-known PDEs and analyzing their solutions.

Findings

The magnetic Euler-Arnold equation generalizes several classical PDEs.

Global well-posedness is proved for the magnetic Euler-Arnold form of the quasi-geostrophic equations.

Multiple equations like Korteweg-de Vries and Camassa-Holm are interpreted as magnetic Euler-Arnold equations.

Abstract

In this article, we combine V. Arnold's celebrated approach via the Euler-Arnold equation -- describing the geodesic flow on a Lie group equipped with a right-invariant metric \cite{Arnold66} -- with his formulation of the motion of a charged particle in a magnetic field \cite{ar61}. We introduce the \emph{magnetic Euler-Arnold equation}, which is the Eulerian form of the magnetic geodesic flow for an infinite-dimensional magnetic system on a Lie group endowed with a right-invariant metric and a right-invariant closed two-form serving as the magnetic field. As an illustration, we demonstrate that the Korteweg-de Vries equation, the generalized Camassa-Holm equation, the infinite conductivity equation, and the global quasi-geostrophic equations can all be interpreted as magnetic Euler-Arnold equations. In particular, we obtain both local and global well-posedness results for the magnetic Euler-Arnold equation associated with the global quasi-geostrophic equations.