Stability of the Inviscid Power-Law Vortex

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 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.

💡 Research Summary

The paper investigates the linear stability of the radially symmetric power‑law vortex (\overline\omega(x)=\beta|x|^{-\alpha}), a stationary solution of the two‑dimensional incompressible Euler equations without forcing. The authors work in both the original (physical) variables and the forward self‑similar variables that respect the natural scaling of the Euler system.

First, they introduce the forward self‑similar coordinates (\xi = x t^{-1/\alpha}) and (\tau = \log t), under which the Euler equations become a transport‑type equation for the vorticity (\Omega(\xi,\tau)). The stationary profile in these coordinates is (\bar\Omega(\xi)=\beta(2-\alpha)|\xi|^{-\alpha}), which corresponds exactly to the power‑law vortex in physical space. Linearizing around (\bar\Omega) yields the operator
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