Exact non-stationary solutions of the Euler equations in two and three dimensions

TL;DR

This paper introduces a geometric framework for constructing explicit, smooth, global solutions to the incompressible Euler equations, including classical and new solutions on various manifolds, with classifications in 2D and 3D.

Contribution

It develops a novel geometric method to generate and classify explicit non-stationary solutions of the Euler equations on different manifolds.

Findings

Recover classical solutions like Kelvin and Rossby-Haurwitz waves.

Construct new explicit solutions on curved surfaces and 3D manifolds.

Classify manifolds admitting such solutions in 2D and partially in 3D.

Abstract

We develop, via Arnold's geometric framework, a mechanism for constructing explicit, smooth, global-in-time, and typically non-stationary solutions of the incompressible Euler equations. The approach introduces a notion of generalized Coriolis force, whose spectrum underlies the construction of these solutions. We recover classical exact solutions such as Kelvin and Rossby-Haurwitz waves, while also producing new explicit examples on curved surfaces and three-dimensional manifolds including the round three-sphere. Furthermore, we obtain a complete classification in two dimensions and a partial classification in three dimensions of the Riemannian manifolds that admit such solutions. The method is in fact formulated in the general Euler-Arnold setting and yields a simple criterion for non-stationarity.