Asymptotic linear stability of columnar vortices driven by Coriolis force

TL;DR

This paper proves the asymptotic linear stability of Coriolis-driven columnar vortices in 3D Euler equations, using a novel Fourier basis construction based on a parameter-dependent Schrödinger equation.

Contribution

It introduces a new method to analyze stability by constructing a distorted Fourier basis via solving a two-parameter Schrödinger equation, advancing understanding of vortex stability.

Findings

Established asymptotic linear stability of the vortices.

Developed a novel Fourier basis construction method.

Decomposed parameter space to analyze solution behavior.

Abstract

In this paper, we establish the asymptotic linear stability of a class of Coriolis-driven columnar vortices for the 3-D axisymmetric Euler equations. This result represents a critical step toward proving the nonlinear asymptotic stability of such vortices. The key and widely applicable strategy is to construct a distorted Fourier basis, which is achieved by solving a two-parameter $(c, \xi)$-dependent Schr\"odinger equation associated with the linearized operator of the system. To capture the precise asymptotic behavior of the solution, we decompose the $c-\xi$ plane into distinct regions, with the partitioning guided by the leading-order profiles of the Schr\"odinger equation across different parameter regimes.