Incompressible Euler equations with stochastic forcing: a geometric approach

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.

55 pages, v3: Corrected typos and minor mistakes, expanded introduction and added more examples, main results remain unchanged

References in corpus (3)

• Langevin dynamics, large deviations and instantons for the quasi-geostrophic model and two-dimensional Euler equations
• Manifolds of mappings for continuum mechanics
• Fractional Sobolev metrics on spaces of immersions

Cited by in corpus (1)

• Derivation of the viscoelastic stress in Stokes flows induced by non-spherical Brownian rigid particles through homogenization