Incompressible Euler fluids on compact cohomogeneity one manifolds

TL;DR

This paper proves that on certain compact Riemannian manifolds with symmetry, any divergence-free initial velocity field generates a smooth, symmetric solution to the incompressible Euler equations for all time.

Contribution

It establishes the existence of smooth, symmetric solutions to the Euler equations on compact cohomogeneity one manifolds for any divergence-free initial data.

Findings

Any divergence-free initial data leads to a global smooth solution.

Solutions preserve the symmetry of the initial data.

The solutions exist for all real time.

Abstract

Let $(M,\mathsf{g})$ be a connected and compact Riemannian manifold admitting an isometric action by a compact Lie group $G$ whose principal orbits have codimension one. We show that any $G$-invariant, smooth, and divergence-free vector field $u_0$ on $(M,\mathsf{g})$ initiates a $G$-invariant time-varying velocity-pressure pair $(u,p)$ which has time interval $\mathbb{R}$, is smooth, and solves the incompressible Euler fluid equations.