Incompressible Euler equations with stochastic forcing: a geometric approach

TL;DR

This paper develops a geometric framework for analyzing stochastic Euler equations on manifolds, proving local existence and uniqueness of solutions using advanced stochastic and geometric techniques.

Contribution

It introduces a novel geometric approach to stochastic Euler equations, establishing local well-posedness in Sobolev spaces for the first time.

Findings

Proved local existence and uniqueness of strong solutions

Developed a new geometric toolbox for stochastic Euler equations

Extended analysis to manifolds with boundary

Abstract

We consider stochastic versions of Euler--Arnold equations using the infinite-dimensional geometric approach as pioneered by Ebin and Marsden. For the Euler equation on a compact manifold (possibly with smooth boundary) we establish local existence and uniqueness of a strong solution (in the stochastic sense) in spaces of Sobolev mappings (of high enough regularity). Our approach combines techniques from stochastic analysis and infinite-dimensional geometry and provides a novel toolbox to establish local well-posedness of stochastic Euler--Arnold equations.