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. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Publication:

ADVANCES IN MATHEMATICS

http://dx.doi.org/10.1016/j.aim.2026.111125

Author:

Daomin Cao

State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, PR China

mail addresses: dmcao@amt.ac.cn

Guodong Wang

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, PR China

Corresponding author

E-mail addresses: gdw@dlut.edu.cn