Time-periodic non-radial solutions near monotone vortices in linearized 2D Euler

TL;DR

This paper demonstrates that the phenomena of axisymmetrization and inviscid damping in linearized 2D Euler equations are not robust when the background vortex profile is perturbed to be non-increasing but not strictly decreasing, especially in low-regularity settings.

Contribution

It constructs non-radial, time-periodic solutions near monotone vortices, showing the fragility of axisymmetrization and damping under low-regularity perturbations.

Findings

Existence of non-radial, time-periodic solutions near monotone vortices.

Axisymmetrization and inviscid damping are not robust under low-regularity perturbations.

Small low-regularity perturbations can prevent expected damping phenomena.

Abstract

We study the linearized 2D Euler equations around radial vortex profiles. Previous works have shown that the strict monotonicity of the vorticity profile leads to axisymmetrization and inviscid damping of non-radial perturbations. Given any strictly decreasing radial vortex, we construct arbitrarily close (in low H\"{o}lder norms $C^\alpha$, with $0<\alpha < 1$) radial profiles that are merely non-increasing, for which non-radial, time-periodic solutions to the linearized equation exist. This shows that both axisymmetrization and inviscid damping are not robust under small, low-regularity perturbations of the background profile that violate strict monotonicity.