Geometric Hydrodynamics in Open Problems

TL;DR

This paper surveys recent advances and open problems in geometric hydrodynamics, covering variational models, solution properties, geometric structures, and fluid dynamics phenomena.

Contribution

It provides a concise survey of recent developments and introduces new constructions in the field of geometric hydrodynamics.

Findings

Discussion of variational settings for various fluids

Analysis of invariant metric models and solution properties

Exploration of geometric structures and long-term fluid behavior

Abstract

Geometric Hydrodynamics has flourished ever since the celebrated 1966 paper of V. Arnold. In this paper we present a collection of open problems along with several new constructions in fluid dynamics and a concise survey of recent developments and achievements in this area. The topics discussed include variational settings for different types of fluids, models for invariant metrics, the Cauchy and boundary value problems, partial analyticity of solutions to the Euler equations, their steady and singular vorticity solutions, differential and Hamiltonian geometry of diffeomorphism groups, long-time behaviour of fluids, as well as mechanical models of direct and inverse cascades.