Stability of the Inviscid Power-Law Vortex
Tim Binz, Matei P. Coiculescu
TL;DR
This work establishes rigorous linear stability of the inviscid power-law vortex in 2D Euler flow. By exploiting exponential self-similar coordinates and a sectorial decomposition into angular modes, the authors reduce the infinite-dimensional problem to a family of one-dimensional resolvent problems and prove surjectivity and dissipativity properties for the linearization around the radial power-law vortex. They show the growth bound in self-similar coordinates is negative, $\omega_0(L_{ss})=s_\alpha(L_{ss})=1-1/\alpha<0$, yielding exponential decay of perturbations, while in physical coordinates the linearized operator generates a group of isometries with spectrum on the imaginary axis under suitable symmetry, indicating linear stability. The results are conditioned to $\alpha\in(0,1)$ and $m\ge 3$ (with additional $(m,k)$-dependent bounds for other $q$-spaces), and they provide a mathematically rigorous confirmation of stability for the canonical power-law vortex, contributing to the broader program on non-uniqueness and stability in Euler and Navier–Stokes dynamics. The analysis employs unbounded operators with singular coefficients, integral transforms, Fredholm-type reductions, and semigroup theory in $\mathrm{L}^q_m$ spaces, offering techniques potentially applicable to other singular background flows.
Abstract
We prove that the power-law vortex $\overlineω(x) = β|x|^{-α}$, 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 answers a question from the monograph by Albritton et al. on Vishik's non-uniqueness work on the two-dimensional Euler equations. Moreover, we prove that in physical coordinates, the linearization around $\overlineω$ generates a strongly continuous group of isometries and we prove that the spectrum of the linearization is contained in the imaginary axis. Our results are valid for $L^2(\mathbb{R}^2)$ functions with a mild symmetry condition.