Stability of the Inviscid Power-Law Vortex

TL;DR

This paper proves the exponential linear stability of the power-law vortex solution to the 2D Euler equations in both physical and self-similar coordinates, addressing a key open question in vortex stability analysis.

Contribution

It establishes the exponential linear stability of the power-law vortex in self-similar coordinates and shows the linearization in physical coordinates does not produce instability.

Findings

Exponential linear stability in self-similar coordinates.

No unstable $C_0$-semigroup in physical coordinates.

Answers a question from Albritton et al. monograph.

Abstract

We prove that the power-law vortex $\overline{\omega}(x) = \beta |x|^{-\alpha}$, which explicitly solves the stationary unforced incompressible Euler equations in $\mathbb{R}^2$ in both physical and self-similar coordinates, is exponentially linearly stable in self-similar coordinates with the natural scaling. This result, which is valid for functions in a weighted $L^2$ space and in the un-weighted $L^2$ space with a mild symmetry condition, answers a question from the monograph by Albritton et al. Moreover, we prove that in physical coordinates the linearization around the power law vortex cannot generate an unstable $C_0$-semigroup.