On Arnold-type stability theorems for the Euler equation on a sphere

TL;DR

This paper proves new Arnold-type stability theorems for steady and rotating solutions of the Euler equation on a sphere, including Rossby-Haurwitz waves, using variational methods and stability criteria.

Contribution

It introduces three novel Arnold-type stability theorems for Euler flows on a sphere, applying to Rossby-Haurwitz waves and employing a variational approach with Burton-type criteria.

Findings

Established stability conditions for flows with monotone stream functions.

Applied the theorems to Rossby-Haurwitz waves.

Derived sharp rigidity results for semilinear elliptic equations on a sphere.

Abstract

In this paper, we establish three Arnold-type stability theorems for steady or rotating solutions of the incompressible Euler equation on a sphere. Specifically, we prove that if the stream function of a flow solves a semilinear elliptic equation with a monotone nonlinearity, then, under appropriate conditions, the flow is stable or orbitally stable in the Lyapunov sense. In particular, our theorems apply to degree-2 Rossby-Haurwitz waves. These results are achieved via a variational approach, with the key ingredient being to show that the flows under consideration satisfy the conditions of two Burton-type stability criteria which are established in this paper. As byproducts, we obtain some sharp rigidity results for solutions of semilinear elliptic equations on a sphere.